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- http://ufdc.ufl.edu/AA00029808/00001
## Material Information- Title:
- Buckling and delamination analyses of stiffened composite panels in axial compression
- Creator:
- Park, Oung, 1953-
- Publication Date:
- 1999
- Language:
- English
- Physical Description:
- vii, 128 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Boundary conditions ( jstor )
Buckling ( jstor ) Delamination ( jstor ) Design analysis ( jstor ) Eggshells ( jstor ) Laminates ( jstor ) Modeling ( jstor ) Pipe flanges ( jstor ) Skin ( jstor ) Stiffness ( jstor ) Aerospace Engineering, Mechanics, and Engineering Science thesis, Ph. D ( lcsh ) Composite construction -- Research ( lcsh ) Composite materials -- Delamination ( lcsh ) Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1999.
- Bibliography:
- Includes bibliographical references (leaves 119-127).
- General Note:
- Printout.
- General Note:
- Vita.
- Statement of Responsibility:
- by Oung Park.
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43757189 ( OCLC )
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BUCKLING AND DELAMINATION ANALYSES OF STIFFENED COMPOSITE PANELS IN AXIAL COMPRESSION By OUNG PARK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999 ACKNOWLEDGEMENTS I wish to express my sincere appreciation to the members of my supervisory committee. Without the experienced academic advice, patience, encouragement of Dr. Bhavani V. Sankar, chairman of the committee, and Dr. Raphael T. Haftka, cochairman, this work would not have been possible. In addition to the their exceptional guidance, they gave me a wonderful opportunity to interact with other eminent scholars in various ways. Furthermore, they have provided the funding necessary to complete my doctoral study. Dr. Ibrahim K. Ebcioglu not only taught me several courses in solid mechanics but also always made me feel comfortable whenever I talked with him. Dr. Peter G. Ifju showed me his research enthusiasm in experimental mechanics. He always helped me whenever I needed his help. Dr. Fernando E. Fegundo, Jr., was willing to serve as a member of my advisory committee, reviewed this dissertation and gave me helpful suggestions and comments. My special appreciation extends to Dr. James H. States, Jr. and Dr. Cheryl Rose of NASA Langley Research Center. Besides co-authoring and reviewing my publications, they helped me learn STAGS. I would like to acknowledge the great help obtained from Dr. David Bushnell, the developer of PANDA2 at the Lockheed Martin Co.; Dr. William H. Greene, a developer of ABAQUS at HKS Co.; Dr. Mark Hillburger, Mr. Allen Waters at NASA Langley Research Center; and Dr. T. Krishnamurthy at Applied Research Associates. I am grateful to my professors and friends in the Department of Aerospace Engineering, Mechanics, and Engineering Science at the University of Florida, who have taught me and have inspired me during my study. I sincerely appreciate the Agency for Defense Development, where I devoted seventeen years of my young life. I extend my special thanks to Dr. Y. S. Lee, Dr. D. S. Kim, Dr. M. J. Shin, and many other supervisors and colleagues. I am deeply indebted to my beautiful wife (Mee), two children (Chan and Kyoung), and our parents for their endurance, support and prayers. TABLE OF CONTENTS pa ACKNOW LEDGM ENTS..................................................................................................ii ABSTRACT ....................................................................................................................... vi CHAPTERS 1 INTRODUCTION ........................................................................................................... 1 1.1 Background ............................................................................................................ 1 1.2 Literature Survey.................................................................................................... 2 1.2.1 Buckling and Postbuckling Analysis of Stiffened Panels.............................2 1.2.2 Buckling and Postbuckling Analysis of Laminated Composite Plate with Delam ination......................................................................................... 12 1.3 Objective and Scope............................................................................................. 16 2 REVIEW OF THE NONLINEAR FINITE ELEMENT METHOD.............................. 19 2.1 Nonlinear Finite Element Formulations Based on Continuum Mechanics.......... 19 2.1.1 Equation of Equilibrium .............................................................................. 20 2.1.2 Eigenvalue Buckling Prediction.................................................................. 21 2.2 Nonlinear Solution M ethods ............................................................................... 24 2.2.1 Newton-Raphson m ethod............................................................................ 25 2.2.2 Arc-length M ethod ...................................................................................... 27 2.3 Solution Strategy ................................................................................................ 29 3 COMPUTATION OF ENERGY RELEASE RATE..................................................... 33 3.1 Introduction ........................................................................................................ 33 3.2 Strain Energy Derivative M ethod........................................................................ 36 3.3 Path Independent J-Integral................................................................................. 37 3.4 Zero-volum e J-integral........................................................................................ 40 3.5 G from Energy Density ..................................................................................... 43 3.6 G in term s of Crack-tip force .............................................................................. 44 3.7 Virtual Crack Closure Technique........................................................................ 48 3.8 Extension to Delam inated Plates......................................................................... 50 3.9 Sum m ary ........................................................................................................... 54 4 ANALYTICAL AND EXPERIMENTAL CORRELATION OF A STIFFENED COMPOSITE PANEL IN AXIAL COMPRESSION................................................56 4.1 Introduction ......................................................................................................... 56 4.2 Stiffened Panel D efinition................................................................................... 57 4.3 Test Specimen and Test Procedures.................................................................... 60 4.4 Linear Buckling Analysis .................................................................................... 60 4.4.1 PANDA2 and STAGS .................................................................................. 60 4.4.2 Finite Elem ent M odel................................................................................... 62 4.5 Results of Linear Buckling Analysis................................................................... 64 4.5.1 Effect of Geometric Imperfections and Shear Load..................................... 64 4.5.2 Effect of Boundary Conditions and Material Properties..............................65 4.5.3 Summary of Differences between Design Model and Test Model ..............71 4.6 N onlinear A nalysis.............................................................................................. 72 4.6.1 Compressive Load versus End-shortening ....................................................73 4.6.2 Compressive Load versus Out-of-plane Deformations.................................76 4.6.3 Contact between the Panel and Loading Platen ............................................77 4.7 C onclusion ........................................................................................................... 83 5 BUCKLING AND POSTBUCKLING ANALYSIS OF A STIFFENED PANEL WITH SKIN-STIFFENER DEBOND....................................................................... 84 5.1 Introduction ......................................................................................................... 84 5.2 Finite Elem ent M odel.......................................................................................... 85 5.3 Buckling Analysis Results of Multiple Delaminated Plate................................. 88 5.4 Buckling and Postbuckling Analysis Results of Plate with Single D elam ination...................................................................................................... 92 5.5 Buckling Analysis Results of Stiffened Panel with Debond...............................91 5.5.1 Buckling Analysis Results of Debonded Stiffened Panels...........................99 5.5.2 Postbuckling Analysis Results ................................................................... 105 5.6 Comparison of Energy Release Rate................................................................. 107 5.7 C onclusion ......................................................................................................... 112 6 CONCLUSIONS AND FUTURE WORK ................................................................. 117 R E FE R E N C E S ................................................................................................................ 119 BIOGRAPHICAL SKETCH........................................................................................ 128 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BUCKLING AND DELAMINATION ANALYSES OF STIFFENED COMPOSITE PANELS IN AXIAL COMPRESSION By OUNG PARK December, 1999 Chairman: Dr. Bhavani V. Sankar Major Department: Department of Aerospace Engineering, Mechanics, and Engineering Science The major objective of this study is to analyze buckling and delamination behavior of composite stiffened panels subjected to axial compression. First, a combined analytical and experimental study of a blade stiffened composite panel subjected to axial compression was conducted. The effects of the differences between a simple model used to design the panel and the actual experimental conditions were examined. It was found that in spite of many simplifying assumptions the design model did reasonably well in that the experimental failure load was only 10% higher than the design load. Several structural analysis programs, including PANDA2, STAGS, and ABAQUS, were used to obtain high fidelity analysis results. The buckling loads from STAGS agreed well with the experimental failure loads. However, substantial differences were found in the out-of-plane displacements of the panel. Efforts were made to identify the source of these differences. Implementing non-uniform load introduction with general contact definition in the STAGS finite element model improved correlation between the measured and predicted out-of-plane deformations. Next, a new method called Crack Tip Force Method (CTFM) is derived for computing point-wise energy release rate along the delamination front in delaminated plates. The CTFM is computationally simple as the G is computed using the forces transmitted at the crack-tip between the top and bottom sub-laminates and the sublaminate properties. Finally, buckling and postbuckling of a blade-stiffened composite panel under axial compression with a partial skin-stiffener debond are investigated. Two different finite element models, where nodes of the panel skin and the stiffener flange are located on the mid-plane or at the interface between skin and flange, are used. Linear buckling analysis is conducted using both STAGS and ABAQUS. Postbuckling analysis is conducted with STAGS. Comparison between the present results and previous buckling analysis results show a good correlation. Buckling analysis results for various stiffener geometries and debond ratios are presented. CHAPTER 1 INTRODUCTION 1.1 Background Stiffened laminated composite panels have been considered for use in weightsensitive structures such as aircraft and missile structural components. The main advantage of the stiffeners is the increased structural efficiency of the structure with a minimum of additional material. Due to the high stiffness of fiber composites, stiffened composite panels are usually thin. Thus, buckling characteristics are critical considerations for the optimum design of composite structures made of laminated composite plates. Buckling depends on a variety of factors, such as the geometry of the members, boundary conditions and material properties. Because of geometric complexity, stiffened laminated composite panels in axial compression have several buckling modes including general instability, local skin buckling and rolling of stiffeners. Furthermore, stiffened laminated composite flat panels usually exhibit stable postbuckling behavior which, in general, leads to significant differences between buckling load and ultimate failure load. The correct estimation of the load carrying capacity of stiffened panels is therefore very complicated. Advanced fiber-reinforced composite materials such as graphite-epoxy have relatively low transverse tensile and interlaminar shear strengths compared to in-plane 2 strength. Therefore, stiffened panels made of laminated composite are very susceptible to delamination. Delamination, also referred to as interface cracking or debonding where adjacent laminae become separated from one another, has been one of the major known weaknesses of laminated composite structures. Delamination, coming from initial manufacturing imperfections or in-service damage such as foreign object impact, can significantly reduce the stiffness and strength of the composite structures. It is also well known that delamination and skin-stiffener separation are common failure modes of a stiffened composite panel in axial compression. When a delamination is present, it is very important to identify not only its global influence on the load carrying capacity of the structure but also its local behavior under the applied load. Energy release rate has been accepted as a measure for predicting delamination propagation. Most available methods of computing energy release rate use 2-D or 3-D solid finite elements. However it is computationally very expensive to use solid elements for modeling entire complicated structures made of laminated composite materials. Thus simplified beam, plate and shell theories are frequently used for structural analysis of those structures. Therefore, it may be a good idea to also use the same theory to calculate energy release rate. 1.2 Literature Survey 1.2.1 Buckling and Postbuckling Analysis of Stiffened Panels Research on the buckling and postbuckling behavior of stiffened panels has been of interest for many years, with many researchers exploring the response of the stiffened 3 panels. However, due to their geometric complexity and the many parameters involved, a complete understanding of all aspects of behavior is not yet fully achieved. Several researchers [1-5] compiled surveys on buckling and postbuckling behavior of composite panels. Leissa [1] compiled extensive results on buckling and postbuckling behavior of laminated composite panels. Analytical effects of various boundary conditions, stacking sequences, and transverse shear deformation on buckling and postbuckling behavior were explored. Noor and Peters [2] reviewed two aspects of the numerical simulation of the buckling and postbuckling responses of composites structures. The first aspect was exploiting non-traditional symmetries exhibited by composite structures, and strategies for reducing the size of the model and the cost of buckling and postbuckling analyses in the presence of symmetry-breaking conditions (e.g., asymmetry of the material, geometry, and loading). The second aspect was the prediction of onset of local delamination in the postbuckling range and accurate determination of transverse shear stresses in the structure. Bushnell [3] divided the literature in the field of buckling of stiffened panel into three categories. One in which structural analysis is emphasized, a second in which optimum design is emphasized, and a third in which test results are emphasized. Recently, Knight and Starnes [4] reviewed some of the historic developments of shell buckling analysis and design and identified key research directions for reliable and robust methods in shell stability analysis and design. Bedair [5] presented an extensive literature review on stability of stiffened panel under uniform compression. He classified the literature into two categories, analysis and design. The objective of the first category is to develop numerical or analytical formulations to predict the global and local buckling load of structures. In doing so, several assumptions 4 are postulated in idealizing the structure in order to facilitate a solution. The objective of the second group of researchers is to develop simplified models to predict the ultimate strength, or collapse load of the structures. Several simplified models have been developed for that purpose. The main objective of this section is to review briefly previous works in the area of buckling and postbuckling behavior of stiffened composite panels. Surveys are divided into two categories, analysis methods and correlation of analysis and experiment. 1.2.1.1 Analysis methods The analysis of stiffened panel can be performed by either smeared or discrete stiffener approach. A smeared stiffener approach converts the stiffened plate into an equivalent plate with constant thickness by smearing out the stiffeners. This method provides accurate analysis results of global buckling load of stiffened panels when stiffeners are identical and closely spaced. However, this approach ignores the discrete nature of the structures and does not consider all potential buckling modes. The discrete stiffener approach does not have limitation of stiffener spacing and uniformity. In early studies of buckling analysis of stiffened panels, Wittrick and William [6] developed a general-purpose computer program, VIPASA, for determining the critical buckling stresses and natural frequencies of thin prismatic structures that consist of a series of flat plate connected rigidly along their longitudinal edges. The response of each plate element making up the stiffened panel is obtained from an exact solution of thinplate theory. This approach is commonly referred to as exact finite strip method. The analysis used in VIPASA is similar to that which Viswanathan et al. [7] incorporated in 5 their computer program BUCLASP2. BUCLASP2 assumes that panel elements are orthotropic and have balanced laminates, material is linear elastic and thin-plate theory. Stroud and Agranoff [8-9] proposed a simplified analysis based on buckling of othotropic plates with simply supported boundary conditions. The global buckling analysis of stiffened panel was conducted as an orthotropic plate with smeared stiffeners, assuming as a wide column. In contrast with the simplified analysis, the VIPASA analysis code provides a high-quality buckling analysis that considers all buckling modes and ensures continuity of the buckling pattern across the intersection of neighboring plate elements. Stroud and Anderson [10] developed a stiffened panel design code PASCO, which uses VIPASA analysis as well as design techniques from the early simplified methods and expanded capabilities such as initial bow-type imperfection, bending moments, and temperature loading. The major known drawback of the VIPASA code is underestimation of the shear buckling modes when a buckling half-wave length is equal to the panel length [10,12]. To overcome inaccuracy involving shear load and anisotropy exhibited by VIPASA and PASCO, Anderson et al. [11] developed the computer program VICON. The analysis of VICON assumes that the deflection of the plate assembly can be expressed as a Fourier series, which can be used to calculate the forces at the longitudinal junctions between the plates by the same stiffness matrices that result from the VIPASA analysis. The total energy of the panel is expressed in terms of VIPASA stiffness matrices plus conventional stiffness of the supporting structure. Then the total energy is minimized subject to the constraints, using Lagrange multipliers, to obtain the governing equations. Bushnell [13-14] developed PANDA, an early version of PANDA2, where 6 buckling loads are computed by the use of simple assumed displacement functions used in conjunction with Donnell-type kinematic relation. Several types of general and local buckling modes were included. VIPASA, PASCO, and PANDA cannot perform nonlinear postbuckling analysis of the stiffened panel. Bushnell [15-17] released PANDA2, which incorporated nonlinear theory for prediction of behavior of locally imperfect panels. Nonlinear strain-displacement relations analogous to those developed in 1946 by Koiter for perfect panels were extended to handle panels with imperfections in the form of critical local bifurcation buckling mode. In PANDA2, local and general buckling loads are calculated with use of either closed-form expression [14] or with use of discretized models of panel cross-sections [15-17]. The discretized model is based on one-dimensional discretization that is commonly referred to as approximate finite strip method. Approximate finite strip method assumes displacement of strip can be expressed with combination of crosswise polynomial function and lengthwise trigonometric function. Dowe and his group [18-22] have used the method in conjunction with a non-linear analysis based on first-order shear deformation theory. The Finite strip method, which lies between the conventional Rayleigh-Ritz method and the finite element method, provide a means of solving prismatic plate-structure problems with an attractive blend of accuracy, economy and ease of modeling [22]. The finite element method is the most powerful method to predict the buckling and postbuckling behavior of structures. The historic developments of shell buckling analysis using finite element up to1997 can be found in Knight and Starnes [5]. With advancing computer power, the finite element method is widely used to solve various engineering problems. Many current commercially available finite element codes such 7 as MSC/NASTRAN [23] and HKS/ABAQUS [24] provide buckling and postbuckling analysis capability of the stiffened composite panels as one of several analysis options. Knight and Starnes [5] pointed out in their review paper that STAGS [25] is perhaps the premier shell analysis code that focused primarily on shell analysis and solution procedures for shell problems. 1.2.1.2 Analytical and experimental correlation Williams and Stein [26] examined J- and blade-stiffened graphite/epoxy panels experimentally as well as analytically using several analysis codes such as VIPASA, BUCLASP-2 and an early version of STAGS, which was based on a two dimensional finite difference technique. In their study, correlation of experimental and analytical results, which included inplane displacement restraints, indicated that the buckling strain of J-stiffened specimens were 75% to 80% of analytical values and that of bladestiffened panels were 84% to 97% of the analytical values. A nonlinear response was exhibited by several of the specimens in which large lateral displacement in the order of one-quarter of the thickness of the panel plate segments were observed. Starnes and coworkers [27-28] investigated the postbuckling behavior of flat and curved stiffened graphite-epoxy panels loaded in compression. Panels with four equally spaced I-shaped stiffeners and quasi-isotropic skin were tested. Failure of all panels initiated in a skin-stiffener interface region. They showed that analytical results obtained from PASCO as well as STAGS correlate well with typical postbuckling test results up to failure. Their results also showed that modeling of the stiffener components with plate elements having appropriate stiffness is required to obtain satisfactory correlation with the postbuckling test results. 8 Romeo [29] conducted several tests on graphite-epoxy hat-and blade-stiffened panels under uniaxial compression and wing-box beams under pure bending to verify the accuracy of the theoretical analysis. Overall buckling, local buckling and torsional buckling were determined separately using a simple engineering formula, and interactions between these modes were not considered. Adequate correlation with experimental results was obtained for axial compression when the Euler or torsional buckling mode was critical; buckling occurred at lower strain values than predicted when the local buckling mode was critical. Furthermore, he showed that simple compression tests could not represent the load conditions of wing-box compression panel properly; in particular, the bending curvature causes a distributed load perpendicular to panels that could reduce the longitudinal load at which buckling occurred. Bushnell et al. [30] conducted optimum design, fabrication, and test of graphiteepoxy curved, locally buckled panels in axial compression. Three nominally identical large panels were tested. Two of three tests gave reasonably good agreement between test and theory, both with regard to loads at which the panels failed and the mode of failure. They also conducted experimental comparison between specimens with stitched skin-flange combination and specimens with adhesive-bonded skin-flange combination. They found that load carrying capacity of stitched specimens were lower than those of adhesively bonded specimens. Wieland et al. [31] investigated the buckling, postbuckling and crippling of AS4/3502 graphite-epoxy Z-section stiffeners as a function of specimen structural parameters. Variables considered were flange and web widths, flange-to-web corner 9 parameters. Variables considered were flange and web widths, flange-to-web corner radius, thickness and stacking sequence. Analytical model was based on a classical model on local buckling and the numerical analyses were conducted with the ABAQUS finite element code. The results showed that the nature of the load redistribution after buckling and its effect on postbuckling stiffness are related to the geometric variables. The agreement between the analytical and experimental buckling loads was generally good. The agreement degrades as the ratio of flange width with respect to web width is reduced and as the section corner radius is increased. Fan et al. [32] performed the pre- and post-buckling analysis for stiffened panels. Both the thick-wall stiffeners with square cross section and the thin-wall blade stiffeners were employed in their study. After linear buckling analysis, they used an incremental analysis along the load path with a special iteration technique, called initial value method to improve the normal Newton-Raphson method. The computational results of both displacement control and load control were presented. The computed local buckling load and buckling mode agreed well with the test results. However, the computed global buckling load with uniform end-shortening displacement was much higher than the test results. But, the computed global buckling load with load control was close to the test results. The reason of this discrepancy was not clearly explained. Three failure criteria of composite stiffened panel, which are maximum strain, debond failure, and combined global-local buckling criterion were proposed. Nagendra et al. [33] studied the optimum design of blade stiffened panels with holes under buckling and strain constraints. They used PASCO for buckling analysis and optimization with continuous thickness design variables and the Engineering Analysis 10 Language (EAL) finite element analysis code [34] for calculating strain and their derivatives with respect to design variables. Later their optimally designed panels with and without centrally located holes were tested and analytical and experimental results were compared focusing prebuckling behavior of the panel [35]. Prebuckling stiffness from test were about 10% lower than analytical values and failure loads from test was also 10% lower than that from design. Since the uncertainties in the geometric and material properties did not account for the discrepancy between analytical and experimental buckling loads, they hypothesized that geometric imperfections and eccentricities may had reduced the buckling load. Chow and Atluri [36] proposed failure criterion of mixed-mode stress intensity factors for the postbuckling strength of stiffened panel. They showed that post-buckling strength of the stiffened panels compare quite favorably with the experimental results of Starnes et al. [27] and the standard deviation of the error was less than 10 %. Young and Hyer [37] presented modeling procedures that predict the postbuckling response of composite panels with skewed stiffeners. Five panel configurations with various combinations of skin and stiffener orientation were tested. A uniform end shortening displacement was applied to the upper end of the panel in the axial direction, and the axial displacement of the lower end was restrained. The upper and lower ends were clamped and the unloaded sides were simply supported. The individual effect of shell elements, potted load introduction, material properties, and initial geometric imperfections was examined. The results showed that inaccurate modeling assumptions and anomalies in the test such as the support fixtures, the loading frame, and the load introduction of the test specimen could cause the predicted response and the measured response to differ substantially. Divila et al. [38] conducted progressive failure analysis for the simulation of damage initiation and growth in stiffened thick-skin stitched graphite-epoxy panels loaded in axial compression. Failure indices approach, proposed by Chang and Chang [39], was adopted to evaluate the failure mode and location corresponding to all of the major composite laminate failure modes except delamination. Superposed layers of shell elements with multiple integration points through the thickness were used to separate the failure modes for each ply orientation and to obtain the correct effect of bending loads on damage progression. The analysis results were compared with experimental results for three stiffened panels with notches oriented at 0, 15, and 30 degrees to the panel width dimension and found to be in excellent correlation with the experimental results. The local reinforcing effect of Kevlar stitches was simulated in the finite element model by multiplying the fiber buckling strength allowable value, independent of the other stress components, by a stitch factor that is determined empirically. A parametric study was performed to investigate the damage growth retardation characteristics of the Kevlar stitch lines in the panels. The debond between the stiffener flange and the skin were not modeled. Hence, the predicted results were found to be less accurate after the damage zone reached the stiffener flange. Singer et al. [40] presented conventional and less conventional experimental methods in buckling of a vast variety of thin-walled structures in considerable detail. The parameters, which may influence the test results, were systematically highlighted: imperfections, boundary conditions, loading conditions, effect of holes and cutouts. Though authors deals primarily with experimental methods and test results, the 12 theoretical concepts of the basic instability phenomena and numerical methods were also briefly reviewed. Sleight [41] analyzed a composite blade-stiffened panel with a discontinuous stiffener loaded in axial compression. A progressive failure analyses using Hashin's criterion [42] was performed on the blade-stiffened panel. The progressive failure analysis and test results showed good correlation up to the load where local failures occurred. The progressive failure analysis predicted failures around the hole region at the stiffener discontinuity. The final failure of the experiment showed that local delaminations and debonds were present near the hole, and edge delaminations were present near the panel midlength. The progressive failure analysis results did not compare well to the test results since delamination failure modes were not included in this progressive failure approach. 1.2.2 Buckling and Postbuckling Analysis of Laminated Composite Plate with Delamination 1.2.2.1 One dimensional single delamination Chai et al. [43] developed a one-dimensional analytical model to predict throughthe-width delamination buckling and growth based on Euler beam theory. Kardomateas [44] investigated effects of transverse shear and end fixity of delaminated composite by improving the model used Chai et al. A first-order shear deformation theory using variational principle was proposed by Chen [45]. Yin and his group [46-48] investigated the effects of bending-extension coupling on postbuckling behavior. They evaluated strain energy release rate by using J-integral over a surface that encloses the delamination boundary. Simitses et al. [47] developed a one-dimensional model similar to one used by Chai et al. [42] to predict critical loads for delaminated homogeneous plates with both simply supported and clamped ends. They showed that the buckling loads could serve in certain cases as a measure of the load carrying capacity of the delaminated configurations. In other cases, the buckling load is very small and delamination growth is a strong possibility, depending on the toughness of the material. Yin et al. [48] found that a delamination length is short and located near midplane of the plate, the buckling load of the delaminated plate is close to the lower bound of the ultimate axial load capacity. When a delamination length is long and locates near surface of plate, the postbuckling axial loads can be considerably greater than the buckling loads, while the failure of plate may or may not be governed by delamination growth. The effects of bending-extension coupling as well as imperfection were investigated by Sheinman and Shouffer [49]. They found that the coupling effect reduces the load carrying capacity, and imperfection sensitivity of global postbuckling deformation is very high. Wang [50] proposed the concept of a continuous analysis for determining interface stresses and strain energy release rate for the delamination at the interface of skin and flange. A shear deformable beam finite element with nodes offset to either the top or bottom side was proposed by Sankar [51]. Kyoung and Kim [52] investigated asymmetric delamination with respect to the center of the beam-plate. In their study, a variational principle based on shear deformation theory was used to calculate buckling 14 load of orthotropic laminated beam-plate with through-the-width delamination. Recently, Gu and Chattopadhyay [53] carried out compression tests on graphite/epoxy composites plates with delaminations to evaluate the critical load and the actual postbuckling load-carrying capacity. They observed that composite laminates can carry higher loads after buckling. For the particular case they studied, the ultimate load is found to be as high as three times the buckling load. 1.2.2.2 One-dimensional multiple delaminations Kutlu and Chang [54-55] investigated the compressive response of composite laminates with multiple delaminations. They found that multiple delaminations can reduce the load-carrying capacity more compared to a single delamination. Suemasu [56-57] developed closed form solution for linear bifurcation buckling load based on energy method. He found that in the case of multiple delaminations, size and location significantly affect the buckling load. Lee et al. [58] proposed a layer-wise approach for computing the buckling loads and corresponding buckling mode shapes. It was found that the anti-symmetric buckling mode is dominant for a composite laminate having short multiple delaminations. They also addressed the effects of initial imperfection and anisotropy on buckling and postbuckling response of delaminated composite plates. An analysis procedure for determining the buckling load of beam-plates having multiple delaminations was also presented by Wang et al. [59]. 1.2.2.3 Two-dimensional single delamination Whitcomb [60] studied delamination growth caused by local buckling in composite laminates with near surface delamination, using geometrically nonlinear finite 15 element analysis. He showed that delamination extension does not occur until buckling is significantly progressed. A plane finite element was developed by Gim [61] based on lamination theory that included the effects of transverse shear deformation. In the modeling of two-dimensional delaminations in laminated plates, the undelaminated regions was modeled by a single layer of plate elements while the delaminated region was modeled by two layers of plate elements with node offset. Sankar and Sonik [62] proposed a simple expression for the point-wise strain energy release rate along the delamination front using Irwin's crack closure technique. They applied this technique in analyzing stitched double cantilever beam specimens as well as elliptic delaminations in composite plates. Klug et al. [63] investigated efficient modeling of postbuckling delamination growth using plate elements and gap elements. Energy release rate was computed using virtual crack closure technique. From this, a procedure to simulate a successive delamination growth was proposed. Kim [64] presented a modeling approach to study the postbuckling behavior of composite laminate with embedded delamination using two-dimensional shell element and rigid beam elements. 1.2.2.4 Two-dimensional multiple delaminations Suemasu et al. [65-66] analyzed the compressive behavior of plates with mutiple delaminations of different sizes. They showed that the effect of variation of the size of delamination on the compressive behavior is significant and postbuckling behavior is different from that of plates with equal sizes of delamination. Zheng and Sun [67] proposed a triple plate finite element model to analyze delamination interaction in laminated composite structures. Energy release rate was obtained by using virtual crack 16 closure technique. The compatibility conditions between interfaces of plates were imposed by multi-point constraint equation. The results for delamination interaction in a composite laminated circular plate under three point bending were obtained. Lee et al. [68] developed a nonlinear finite element code, DELAM3D, with a three-dimensional layered solid element based on an updated Lagrangian formulation. They simulated the compressive response of a laminated composite plate with mutiple delaminations. Contacts of delaminating interfaces, delamination growth, and fibermatrix failure were also considered in their computation. Double cantilever beam (DCB) and end notched flexure (ENF) tests were conducted to verify the energy release rate. Test results with various crack numbers, size, location, and layer orientation compared well with the numerical results. 1.3 Objective and Scope The first objective of this study is to develop stiffened panel models that can be used to predict buckling and postbuckling behavior with and without delamination. The second objective is to investigate the delamination growth of the stiffened panel based on fracture mechanics using several methods of computing strain energy release rate. Among the several configurations commonly used for stiffened panels such as hat-stiffened panel, J-stiffened panel, and I-stiffened panel, blade stiffened panel has a simple geometry compared with other stiffened panels. Therefore, blade stiffened panel was chosen in this study. However, the present analysis method will be also applicable to any kind of stiffener configurations that have a flange attached to the skin. 17 Two finite element modeling approaches for stiffened panels are commonly used in the literature. One approach is to model the skin with plate elements and to model the stiffener with beam elements. The other is to model both skin and stiffener with plate elements. The second approach appears more attractive for modeling the debonded region between skin and flange. In this study, the panel skin and blade stiffeners are modeled with plate elements. The nodal penetration of the delaminated skin-stiffener interface can be prevented either by adjusting spring constants of fastener elements or by gap elements. Furthermore, an energy release rate for calculating delamination extension is computed using several methods based on fracture mechanics. In order to validate the present modeling approach, a plate with a through-thewidth delamination was modeled and linear bifurcation and nonlinear postbuckling analyses were conducted. Results were compared with the available experimental results. In Chapter 2, basic finite element formulation for buckling problem and nonlinear solution algorithms for postbuckling analysis are briefly described. Chapter 3 provides several methods for computing energy release rate in plate-like structures based on fracture mechanics. In Chapter 4, effects of boundary conditions, material properties, and initial geometric imperfections on buckling and nonlinear prebuckling behavior of blade stiffened panel are investigated. Chapter 5 describes modeling of delamination with elastic spring element, linear buckling analysis results of both delaminated composite plates and debonded stiffened panels, and effects of delamination length, stiffener geometries and stacking sequences on buckling load. Strain energy release rate was computed using the strain energy derivative method, the virtual crack closure 18 method, and the crack-tip force method. Finally, Chapter 6 includes the summary of the present study, concluding remarks, and suggestions for future work. CHAPTER 2 REVIEW OF THE NONLINEAR FINITE ELEMENT METHOD 2.1 Nonlinear Finite Element Formulations Based on Continuum Mechanics Stiffened laminated composite panels deform continuously under compressive loads. In prebuckling stage, deformation and rotation can be considered as infinitesimal in general. Thus, prebuckling response of stiffened laminated composite panel is almost linear. Classical buckling analysis is generally used to estimate the critical loads of stiff structures such as the Euler column subjected to axial compression, which carry design loads by axial or membrane strength rather than bending strength. The out-of-the plane deformation before buckling is therefore almost negligible in the stiff structures. After buckling, stiffened laminated composite panels exhibit large deformation and rotation. Thus, nonlinear formulation is required in order to include the effects of large deformation and rotation. Many researchers [69-72] have efficiently implemented general nonlinear finite element formulations based on the principles of continuum mechanics. Two different approaches have been used in incremental non-linear finite element formulation. The first approach is generally called Eulerian or updated formulation where static and kinematic variables are referred to an updated configuration in each load step. The second approach is called the Lagrangian formulation, where all static and kinematic variables are referred to the initial configuration. The updated Lagrangian is more suitable for analysis of the structures 19 20 that involve very large deformations while the total Lagrangian is more convenient for analyzing the structures with moderately large deformations. The primary objective of this section is to review briefly the non-linear finite element formulation based on continuum mechanics. The detailed description can be found in References [69, 70]. 2.1.1 Equation of Equilibrium Using the principle of minimum total potential energy one can derive the finite element equations. Assume that there exists a total potential energy of the form for linear elastic analysis HI = {e}' {}dV-(JV{u {fb}dV+ f{u}T{fs}dS) (2.1) V where {u} is the displacement vector, {fb and {fS} are body and surface force vectors. The relationship between stresses and strain is of the form: {o = [C]{e} (2.2) where [C] is the constitutive matrix. The condition of equilibrium requires that the first variation of the total potential energy vanish: &fl = [&e]T {a}dV-( {(6u}rT{fb}dV + s{u}T{fs}dS)=0 (2.3) V V From Zienkiewicz [69], strain can be expressed in matrix notation as {e} =[B]J{u} =[Bo]+[B,l]{u} (2.4) 21 where [Bo ] is the matrix for the linear infinitesimal strain and matrix [BL ] contains the nonlinear strain components. Using Equation (2.4) we can rewrite Equation (2.3) as: = T {u)(f [B]{c}dV-( {fb}dV+ f {f s }dS)) = 0 (2.5) V V S 2.1.2 Eigenvalue Buckling Prediction The stability criterion can be obtained from the second variation of the total potential energy. If the second variation of the total potential energy has a positive value, then a system is stable. Conversely, If the second variation of the total potential energy has a negative value then a system is unstable. Computing the second variation of total potential energy from Equation (2.5) as S{}(f ] {=[(a}dV + [B]T{r}IT dV) (2.6.1) V 621= {&T (f []T{a}dV + J[B]T[C][B] {u}T dV) (2.6.2) V From Zienkiewicz [69], the first integral of Equation (2.6.2) can generally be written as v 6[B]r {}dV = [K,]{u} (2.7) where [Ko] is geometric stiffness matrix. Substituting Equation (2.4) into the second integral of Equation (2.6.2) and rearranging, the second variation of the total potential energy can be written as: (2 = {&uT[KT]{&j} (2.8) In Equation (2.8), the tangential stiffness matrix [KT ] can be written as [KT]= [K,] + [Ko] + [K,]. [Ko = [Bo]T[C][Bo]dV (2.9) V [KLI f ([Bo]T[C][BL] +[BL,]T [C][Bo] +[B LT[C][BLI)dV (2.10) V where [Ko] is the small displacement stiffness matrix and [KL] is the large displacement stiffness matrix. A critical point is obtained when the tangent stiffness matrix [KT] has at least one zero eigenvalue. The stability of an equilibrium configuration can be determined solving the eigenvalue problem at the current equilibrium state, [K ] {u (r)} = A(r){u(r)} (2.11) where 11r) is the rth eigenvalue and {u(r) } is the corresponding eigenvector. Computation of the critical point must be done in two steps. First the equilibrium configuration associated with a given load level P is computed. Next the stability of tangent stiffness matrix is examined by computing the eigenvalue of the tangent stiffness matrix at given load level P. This method of determining the stability of a conservative system gives accurate results. However, it is computationally expensive because it involves the solution of a quadratic eigenvalue problem for the critical load. Linearized buckling analysis calculates critical buckling loads based on a linear extrapolation of the structural behavior at a small load level. Thus it is computationally inexpensive. From the fact that geometric stiffness matrix [K, I and 23 large deformation stiffness matrix [KL] depend on the load level P, linearized buckling analysis approximates the tangent stiffness matrix at given load level P as [40]: P [KT(P)] = [K0] + ([K, (AP)] + [KL (AP)]) (2.12) AP where both geometric stiffness matrix and large displacement matrix are computed at small load level AP. If we assume that the critical load is equal to A AP, then the condition for a singular point becomes a standard eigenvalue problem. There are two widely used numerical methods for extracting eigenvalues, the Lanczos method and the subspace iteration method. The Lanczos method is generally faster when a large number of eigenmodes is needed for a system with many degrees of freedom. The subspace iteration method is effective for computing a small number of eigenmodes. Based on the assumption that the displacements {u} are infinitesimal for the small load AP, classical buckling problem further simplifies Equation (2.12) as: ([Ko] + .r)[K, (AP)]){ u (r) } = 0 (2.13) where the large displacement stiffness matrix [KL] in Equation (2.12) is ignored. However the applications of Equation (2.13) should be limited in practical engineering problems. In order to avoid the erroneous computation of the stability points in real engineering applications, the stability problem should be investigated using full tangent stiffness matrix in Equation (2.11) [69]. 2.2 Nonlinear Solution Methods The solution of nonlinear finite element problems includes a series of load steps as well as iterations to establish equilibrium at the new load level. In some nonlinear static analyses the equilibrium configurations corresponding to load levels can be calculated without solving for other equilibrium configurations. However, when the analysis includes path-dependent nonlinear conditions, the equilibrium relation needs to be solved throughout the history of interest. The solution may be obtained by using either the Newton-Raphson or the modified Newton-Raphson methods. The NewtonRaphson method requires evaluation of the tangent stiffness matrix at each iteration, which is computationally expensive. On the other hand, the modified Newton-Raphson method evaluates the tangent stiffness matrix at each load step, thus improving the computational efficiency compared to the Newton-Raphson method. However, the Newton type methods fail to provide a solution in the neighborhood of a global bifurcation or limit points when the tangent stiffness matrix becomes singular (see Figure 2.1). The arc-length method, proposed by Riks [73] and Wempner [74], and modified by Criesfield [75,76], is an effective solution procedure to search equilibrium path beyond the limit points. An important aspect in arc-length methods is that the load level is treated as a variable in addition to the unknown displacements at each iteration of a load step. Thus, an additional constraint equation comprising the displacements and loads is required to calculate the load level. L 0 A D DISPLACEMENT Figure 2.1 Nonlinear response from load versus displacement. 2.2.1 Newton-Raphson Method A system of nonlinear equilibrium equation can be written as S(u) = I(u)- f (2.14) where internal forces I(u) is defined as I(u)= f [B ]{cr}dV (2.15) V Unbalance forces Y(u) represents the difference between internal and external forces. The basic problem is to find solutions that satisfy the nonlinear equilibrium equation, T(u )= 0. Since Equation (2.14) cannot be solved directly for the displacement of u, both an incremental equation of equilibrium from Equation (2.14) and iterative 26 procedure are generally used for its solution. The Newton-Raphson method utilizes the first-order approximation of Equation (2.14) and can be written at load step n+l as T (u '+ q ))_T ( + ~ n(u ) f+I ) +) ( u +I = 0 (2.16) au Here i is the number of iteration, and the tangent stiffness matrix is defined as - KT (2.17) au au From Equation (2.16) we have the following iterative correction as K' +u' = -T,'+ (2.18) where K. is the tangent stiffness matrix at the it iteration. Thus the improved solution can be computed as Si+1 U+ = u +' + Ou n (2.19) The convergence of the Newton-Raphson method is generally very fast. However, the cost of computation is usually high due to the calculation and factorization of the tangent stiffness matrix at each iteration. To reduce the burden of computational cost, the modified Newton-Raphson was introduced in which the stiffness matrix is approximated as a constant: K = K,. There are many possible choices of the approximated stiffness matrix K, For instance, K, can be chosen as either the matrix corresponding to the first iteration or some previous load step. Schematics of Newton-Raphson method and modified Newton-Raphson method are shown in Figure 2.2. A i I I /II Aun K' II I II ~I I 711 III III Ill II III Ill II I III Ill -J- 1 2 3 u 1 3 5 Un /1 Un Un+1 Un Un+ln+l n+1 Figure 2.2 Schematics of Newton-Raphson and modified Newton-Raphson methods. 2.2.2 Arc-Length Methods The basic feature behind the standard arc-length method is that load level A is treated as a variable rather than constant during a load increment. Thus the governing equilibrium equation (2.14) can be rewritten as .+I(u,2) I(un1) 2.+1fo = 0 (2.20) with u n+ = Un + Au n n+, = An + A2n (2.21) where A u n is the total incremental displacement vector at the nh iteration and A 2n is the total incremental load factor at iteration n. Since the load level is treated as a variable, we need an extra equation that constrains the iterative displacements to 28 follow a specified path towards a converged solution. The constraint equation proposed by Riks [73] may be expressed as: (Au,)T (Au,) + A2nf0f0 = Al2 (2.22) where A 1 is a user defined 'incremental length' in the space of n+l dimensions. Criesfield [75] has proposed a modified constraint equation that includes displacement components alone as: (Au,)T (Au) = Al2 (2.23) It is possible to add the constraint equation (2.23) to the system of equation directly and the iterative incremental method could be used again, however this may destroy the symmetry and banded structure of the equilibrium equation. Hence Criesfield [75] proposed an indirect approaches to avoid this difficulty. In their approach, the displacements at a given iteration i is written as: u = K I ,, ( A + JA ) n I = -K ( ,,+1 (A ) SA ,, fo ) (2.24) iF,' (A )+ 8 (5 i ), 8 i = K i fo where 5iF,' are the iterative displacements corresponding to the residual forces ,' .1 and the tangent stiffness matrix K i- 1 is formed using modified Newton-Raphson at the beginning of each increment and kept fixed for all iterations within the increment. By substituting equation (2.24) in which 8A2 is still undetermined into the constraint equation (2.23), we have (Au'-' + 6u )(Au '- + 6u) = Al 2 (2.25) Expanding Eq. (2.25) gives a quadratic equation for the unknown iterative load factor 82 The details of this solution procedure are given in Ref. [75]. n " sphere An++I? U U 1I I I I II n1 q+1 t+1 n+1 Au3 I n n I I Aun . Figure 2.3 One-dimensional interpretation of spherical arc-length procedure. 2.3 Solution Strategy The buckling analysis provides information about the load level at which bifurcation occurs. In some cases the structure withstand far above the buckling load without significant damage. In other cases the structure collapses well below this load due to imperfection sensitivity. Stability loss at a bifurcation point occurs only if the corresponding deformation mode is not contained in the deformation mode for 30 arbitrarily small loads. For example, a flat plate with in-plane loading exhibits no lateral displacements in the pre-buckling range. Thus it does not contain the lateral displacement modes of the buckled plate. Likewise, if the structure as well as loading is symmetric about some plane, all deformation modes antisymmetric with respect to that plane are possible bifurcation buckling modes. In such cases, we may choose to perform a buckling analysis with a nonlinear basic stress state. Sometimes when a bifurcation point does not exist at all, the bifurcation buckling approach may still considered as an acceptable measure of the critical load. If the structure is statically indeterminate and thus allows favorable redistribution of the stresses (e.g., shells with cutouts), then the bifurcation approach is too conservative. If the stiffness of the shell deteriorates with increasing load (e.g., long cylinders under bending), the bifurcation approach gives unconservative results. The bifurcation buckling analysis with a linear stress state is probably a good approximation for any case in which the squares of the rotations in the linear solution are small in comparison to the membrane strains at the load level corresponding to bifurcation. Most of commercially available finite element codes provide an option to perform the stress analysis first, save the data for a certain number of load steps on tape and later decide for which of those load steps buckling loads should be obtained [2325] This option may save some computing time. First, it may be easier to decide on the load levels at which eigenvalues are desired after the results of the stress analysis have been conducted. Next, it makes possible to find additional eigenvalues in a subsequent run. When eigenvalues are computed in a later run, the data deck for the nonlinear prestress analysis can be used. In addition, the user has the option to select certain data 31 sets saved on tape for eigenvalue solution. Changes in boundary conditions are also permitted at this time. In most practical applications one range of eigenvalues is particularly important, especially to a sequence of the lowest eigenvalues. If there exists a symmetry plane, in loading as well as in geometry, the size of the problem can be reduced and significant savings in the total computational effort can be achieved. If the structure on one side of the symmetry plane is considered, only the frequencies of symmetric modes are obtained. If the eigenvalue analysis is based on a nonzero prestress analysis with symmetry conditions and an eigenvalue analysis with boundary conditions corresponding to anti-symmetry. The eigenvalue approach for bifurcation buckling analysis with linear stress state is slightly more complicated than the vibration problem because eigenvalues can be negative as well as positive. Often the analyst is only interested in one eigenvalue, the lowest positive one. If the analysis is performed without a shift, it may happen that only negative eigenvalues are found because these are smaller in magnitude. In that case, the analysis has to be repeated with a positive shift. In choosing the shift for a repeated run the user can utilize the fact that the smallest positive eigenvalue is larger in magnitude than the largest of the negative eigenvalues that were found. Sometimes the buckling loads are symmetric with respect to zero. This is the case, for example, if a plate or a cylinder is subjected to uniform a shear load. It may often be advisable to request more than one eigenvalue also in bucking analysis. If the structure shows insufficient strength and only the lowest eigenvalue and corresponding mode are known, reinforcements may be introduced that have little effect on secondary buckling mode with the eigenvalue below the design load. 32 Nonlinear analysis is computationally expensive compared to linear analysis. In order to get a sound analysis results from nonlinear analysis, analyst should have better insight into the behavior of the analysis model. It is usually possible to save time as well as computer cost by preliminary determination of approximate values for the buckling load, under negative as well as positive load, before a large scale analysis is carried out. A linear analysis with a rather coarse grid will give some idea about the stress distribution and verify the nature of the behavior prior to executing nonlinear analysis with refined model. The size of the finite element model should be determined based on the requirement of accuracy, the efficiency, and the time constraint. Prior knowledge of the geometric modeling will increase the efficiency of an analysis. Type of element and size of element should be carefully selected to obtain high accuracy. Further, analyst should identify the type of nonlinearity and localize the nonlinear region for computational efficiency. To identify the type of nonlinearity, it is also helpful to examine the deformed shapes at various stages of loading (pre-buckling, critical buckling, limit load, and postbuckling) [25]. CHAPTER 3 COMPUTATION OF ENERGY RELEASE RATE 3.1 Introduction Fracture mechanics concepts have been successfully applied to predict the loads which initiates the delamination extension, and also for predicting their stability. The energy release rate G has been accepted as a measure for predicting delamination propagation. In the context of fracture mechanics, the delamination extension is assumed to occur when the computed G is greater than the experimentally determined critical energy release rate G,. Most of the available methods of computing G use 2-D or 3-D solid elements. However, It is computationally very expensive to use solid elements for modeling the entire complicated structure of an aircraft or an automobile. For example, consider 2-D plane strain elements for computing stress intensity factor to estimate the strain energy release rate of a double cantilever beam. A fine mesh must be used around crack-tip in order to capture the stress gradient ahead of crack-tip. Figure 3.1 shows the amount of mesh density required to calculate the stresses to compute the stress intensity factor. Thus a simplified beam, plate or shell theory are frequently used for structural analysis of complicated structures. Therefore, it may me a good idea to use the same theory to calculate the energy release rate also. There are three methods commonly used for computing energy release rate using 2-D or 3-D solid elements. These are Strain 34 Energy Derivative Method (SEDM), J-intergral, and Virtual Crack Closure technique (VCCT). SEDM, first proposed by Dixon and Pook [77], evaluates the change of potential energy as a crack progresses. Implementation of SEDM in a finite element analysis is straightforward [78]. However, it gives only an average value of the energy release rate along the delamination front. Further, this method requires two computations of potential energy, before and after crack propagation. A direct evaluation of energy release rate requiring only a single computation was proposed by Rice [79]. This involves the calculation of an integral on an arbitrary path surrounding the crack tip. This integral, known as the J-integral, is path independent. The Virtual Crack Closure Technique (VCCT), proposed by Irwin [80], is a method for computing energy release rate for self-similar crack extension. This method assumes that the strain energy release during the crack extension is equal to the work required to close the opened crack surfaces. Many investigators [51-68] have proposed VCCT for computing energy release rate using beam and plate elements. Based on plate theory, Sankar and Sonik [62] proposed the Point-wise Strain Energy Density Method (PSEDM). PSEDM suggests that the point-wise strain energy release rate along the crack front is the difference between strain energy densities behind and ahead of crack front. In this chapter a new method called Crack Tip Force Method (CTFM) based on plate theory is introduced. The application of the various methods of computing G for laminated composite structures is discussed. Figure 3.1 Stress field (ayy) near crak-tip of double cantilever beam using 2-D plane strain elements. 3.2 Strain Energy Derivative Method The strain energy derivative method utilizes the change in total strain energy, U, with change in crack length from a to a +Aa (see Figure 3.2). The energy release rate can be obtained in a straight forward manner for the case of displacement control as (3.1) dU UsA. -Ua dAG = = st-deflection ~1A A const.-deflection where AA is the increase in crack area due to change Aa in crack length. Load Control Displacement Control ja.a+Aa P/2 Figure 3.2 Strain energy derivative method. * a f. As shown in Equation (3.1), this method needs two analyses to compute energy release rate and Aa should be small enough to obtain accurate results In the case of load control, the expression for G is modified as G dU_ Ua+a -Ua (3.2) dA AA con .-force 3.3 Path Independent J-Integral Consider a homogeneous body of linear or nonlinear elastic material without singularity shown in Fig. 3.3. The strain energy density Uo is defined as r Uo = U (e)= ide (3.3) 0 where T ii is the stress tensor and is the infinitesimal strain tensor. The Jintegral to compute the energy release rate G is defined as [79] J = J(Uonx a tin uix)ds i = 1,2j = 1,2 (3.4) r where u i is the displacement. n i are the direction cosines of the outward normal along the path F The indices, i and j or x and z are used interchangeably for convenience. Further, summation is performed over repeated indices. An application of divergence theorem gives: J = (U o 15 (ijUi,x)njds = J a"(U S Ij i" ,x)dA (3.5) 1Ai Ui x Differentiating the strain energy density, _Uo Uo Oij DiU SX iX "- a X (3.6) x [ E( Ox 2 Ox ax j x axi S aU a aui (xOxi) 3x (i Bx 1 ax ax1 ax The area integral in Equation (3.5) vanishes. Therefore Equation (3.4) is equal to zero for any closed contour F In order to understand the zero volume integral described in subsequent section easily, consider the conservation integral around a region with singularity as shown in Fig. 3.4. The J integral along closed paths F1 through F4 surrounding crack-tip vanishes as J = (U onx 7 it nj u j .)ds = 0 (3.7) r, + r2 + 3 + 4 But t = 0 and n = 0 along path F2 and F4 Thus the integral along F, clockwise and the integral along F3 counterclockwise sum to zero. J = (Uon nU )ds+J (U nx iinx) ds + (U onx iinuix)ds = 0 (3.8) From equation (3.8) we can show that J integral along path and J integral along path F3 have the same value: J = (U on, ijnjui,x)ds = (U onx a'njui.x)ds = (Uomx omij m uix)ds r3 where normal vector mx is in the opposite direction with respect to normal vector nx in path F 3. (3.9) A Figure 3.3 Conservation integral around a region with no singularities. V bX Figure 3.4 Conservation integral around a region with singularities. 3.4 Zero-Volume J-Integral Consider a portion of the delaminated beam as shown in Fig. 3.5. The beam example is used to minimize the complexity of derivation but this method can be extended easily to delaminated plates. It can be assumed that the delamination length or crack length a is much larger that the thickness h of the thicker sublaminate. If the path of the integral ABCDEF is away from crack-tip, then the beam theory stresses along this path are reasonably accurate compared to exact elasticity solutions. Further, the Jintegral will vanish along the two horizontal paths BC and DE. Thus the integral is given as the sum of integrals along the three vertical paths: AB, CD and EF. Next, it will be shown that these vertical paths can be moved near the crack-tip without losing any computational accuracy of G. The vanishing of the J-integral around a closed path in an elastic material under small strain assumptions is a consequence of the two differential equations of equilibrium satisfied by the stress components: o + "Z=0 ax az (3.10) 8 r a or "z + r -=0 ax az The stress field in a laminated beam given by the shear deformable beam theory may not be accurate near the crack tip, however the stress components satisfy the 41 equilibrium equations exactly. This is because that the transverse shear stresses TxZ in the beam are computed actually by substituting for o xx and then integrating the first equilibrium equations. Thus, the first of Equation (3.10) is satisfied. The shear stresses at a cross section are proportional to the shear stress Xz that is constant along each ligament of the delaminated beam as well as the intact beam ahead of the crack tip. Thus the shear stresses z are independent of x in each of the sublaminates and the first term in the second equilibrium equation is zero. Since beam theory assumes that o', are negligible, the second term is also zero. Thus, second equilibrium equation is also identically satisfied. Then the J-integral evaluated using beam theory around the closed path ABCDEF in Figure 3.5 is identically equal to zero. Further if we decompose delaminated beam with three sublaminates as shown in Figure 3.6 and consider Jintegral for Sublaminate 2. Because the integrals along the horizontal paths are zero, it can be shown that JAB =JHG. Similarly it can be shown that Jco = JKL and JEF = JMN Thus it is now possible to move the three vertical paths AB, CD and EF to near the crack tip (HG, KL and MN) without loss of accuracy. The J-integral evaluated around the Paths 2,3, and 1 (HGKLMN) near the crack tip in Figure 3.5 has been called the zero-volume J-integral or zero-area J-integral, and it is given by G = J () + J (2) + j (3) (3.11) where superscript (1), (2), and (3) represents the paths 1, 2, and 3, respectively. Z L K C M. M2 B X Figure 3.5 Force and moment resultant in a delaminated beam. Figure 3.5 Force and moment resultants in a delaminated beam. IV L N 4-- G 2 = AB JBG JGH JHA = 0 J = =0 j' jG BG HA JAB GH JAB = HG Figure 3.6 Three sublaminates of delaminated beam. Elf 3 - I m w 3.5 G from Energy Densities Consider the J-Integral along Path 1 in Figure 3.5. Along this path nx= -1 and nz=O. Hence the Integral can be written as: N j" = (-Uo + Efx x _wx)ds (3.12) jV) oz ~ rw )ds (3.12) M We will add and subtract to the integrand in Equation (3.12): N N J"= f(-Uo +OxUx + T (wx + u,))ds fru zds (3.13) M M The term u,z can be identified as the rotation at the crack tip, if which is common to all the three paths 1, 2 and 3. Further the second and third terms in the first integral in Equation (3.13) equal to 2U 0 Hence Equation (3.13) can be written as: N N j) = Uods -V fz, ds (3.14) M M Equation (3.14) can be further simplified as: j(1) = U L (1) tV,1 (3.15) where U () and V are the strain energy per unit length and shear force resultant at the cross section 1. Similar results can be derived for paths 2 and 3 as follows: J (2) L(2) tV2 (2) =- U L (3.16) j (3) = -U L (3) + V/,V3 It should be noted that in Equation (3.16) switches signs because of change in the sign of nx from -1 to +1 for the Path 3. Adding all the three integrals and nothing that the shear force resultants must satisfy the equilibrium condition V1+V2 = V3, we find that J = j ) + j (2) + j(3) U (1) + U 2) (3 (3.17) LU'+U2 L U1L (3.17) Thus the energy release rate is the difference between the strain energy densities just behind and just ahead of the crack tip. The strain energy density in the context of beam refers to strain energy per unit length of the beam, U L 3.6 G in terms of Crack Tip Force Consider a very small segment of the beam of length 2 Ax surrounding the crack-tip (see Figure 3.3). It will be convenient to shift the xz coordinates such that the xy plane coincides with the plane of delamination. Further, we will divide the laminate into 4 sub-laminates 2 behind and 2 ahead of the crack tip as shown in Figure 3.7. Let 45 the force and moment resultants near the crack tip in any sub-laminate be represented by a column matrix F such that F T = [P, M V ] where P, M and V are the axial force, z 4 IX 3 Figure 3.7 Sub-laminates in a delaminated beam and the coordinate system. bending moment and shear force resultants. An underscore denotes matrix and a superscript T denotes transpose of a matrix. It should be noted that the force and moment resultants are resolved about the x-axis, which lies in the delamination plane. Thus there is an offset between the laminate mid-planes and the xy plane. The force resultants in each sublaminate are denoted by El, F2, E3 and F4. The compliance matrix of the top and bottom sub-laminates will be denoted by Ct and Cb The deformation in a sublaminate is then given by e = C F (3.18) where the deformations are defined by: 46 T e = [exo, rx Jxz ] (3.19) where the components of the deformations are the strain along the x-axis (not the sublaminate mid-plane), rate of change of rotation and the transverse shear strain, respectively. The force resultants are related by the equilibrium conditions F, + F2 = F3 + F4 (3.20) Further, since the sub-laminates 3 and 4 are intact (not delaminated) the deformations in them should be identical, i.e. e3 = e4 and hence Cb F3 = C, F4 (3.21) If Fi1 and F2 are given, then E3 and F4 can be calculated using Equations. (3.20), and (3.21). The strain energy per unit length in any sub-laminate is given by: UL= -1 F T CF (3.22) 2 Substituting Equation (3.22) into Equation (3.17) we obtain G = F CF +- F TCbF2 --FCbF3--F 4CIF4 (3.23) 2 2 2 2 2 2 Using the relations in Equations (3.20) and (3.21) an interesting expression for G can be derived as G 1 T G = (F4 -F T )(C, + Cb)(F4 1 ) (3.24) 2 4 The term (F4 F ) is actually the force transmitted through the crack tip between the top and bottom sub-laminates, and can be called the crack-tip force, F,. If a rigid link 47 is used to connect the top and bottom crack tip nodes in a finite element model, then the forces transmitted by the rigid link will be exactly equal to the above crack-tip forces. It may be noted that the crack tip force vector F have three components, an axial force, a couple and a transverse force, corresponding to each degree of freedom of the crack tip nodes, u, i and w. Another important implication of Equation (3.23) is that although there are 6 independent forces P, VI,M1, P2,V2 and M2that can be applied to the two delaminated beam ligaments (see Figure 3.5), G depends only on three crack tip force components (see Figure 3.8). If the forces F and F such that e, = e2, i.e., C, F = Cb F2 using Equations (3.18) and (3.19) one can show that F = F4, and then G = 0 If the forces on the top and bottom sub-laminates I and 2 are such that they produce conforming deformations (eI = e 2 ), then the same forces act in sublaminates 4 and 3, respectively, producing conforming deformations (e3 = e4 ). Thus there is no need for any interaction between the top and bottom laminates at the crack-tip, and hence G = 0. 4 3vv mm"~ , Crack.Tin 2-D SOLID BEAM MODEL Figure 3.8 Crack-tip forces in beam 3.7 Virtual Crack Closure Technique The virtual crack closure technique has been used for plate and beam fracture problems by many researchers. We will derive the VCCT from the Crack-Tip Force Method. The expression for G in Equation (3.23) can be written as: G = I-F T.. G = -F (C,(F4 F1 ) + Cb(F 2 F3)) (3.25) where Fc is the matrix of crack tip forces, and Equation (3.18) is used in deriving Equation (3.25). Using the compatibility quation (3.19) in (3.24), we obtain G = FT (-C, Fl + Cb F 2) (3.26) 2 - Since C F denote the deformations we can write (3.26) as (1) + (2) -O' 1 F1G = 1 7) (2) G=2 c ,tx o1, x (3.27) (1) (2) + W ) In deriving the last term of the column matrix in Equation (3.27), we have used the fact that the beam rotation at the crack tip is same for both ligaments 1 and 2, i.e. V,/ = i//2. Multiplying and dividing the right hand side of Equation (3.27) by Ax where Ax is a small length used in the virtual crack closure method, we obtain (1) (t) (2) (t) 2Ax-c(w"' +w)-(w2 w- ) (328 The superscript (t) in Equation (3.28) denotes displacements and rotation at the crack tip, and superscript (1) and (2) denote respectively the displacements of the top and bottom ligaments at a distance Ax from the crack tip. In deriving Equation (3.28) we have used the finite difference approximation of the type, (t) (1) (1) 0- UO) Uox Ax= (3.29) Ax 50 Canceling the crack tip displacements in Equation (3.28) we obtain the equations for the virtual crack closure method as: (1) (2) 3.8 Extension to Delaminated Plates In the case of delaminations in a plate the energy release rate G varies along the delamination front. Formulas similar to Equations (3.10) (SEDM) and (3.22) (VCCT) were derived by Sankar and Sonik [62]. In this section we will derive an additional result for G(s) similar to Equation (3.23) derived for beam. We can use the same notation as we used for beams with the understanding that there are eight force and moment resultants and eight deformation components: [FT]=[Nx NY Nxy Mx My Mxy Qx Qy] [eT]=[ExO y0 Y xy0 Kx Ky 1xy x z (3.31) The laminate compliance matrix [C] will be an 8x8 symmetric matrix, and it relates the force resultants and deformations: 51 e = CF (3.32) A B 0 = B D 0 {F} -0 0 Kwhere, the [A], [B] and [D] are the classical 3x3 laminate stiffness matrices and [K] is the 2 by 2 transverse shear stiffness matrix. In the context of plates the strain energy density is defined as strain energy per unit area of the plate and is given by 1FT U A = F CF (3.33) 2 A formula for G(s) similar to that in Equation (3.23) is given by G(s) = (U (s) + U 2(s) 3 () -(S U (4)(s)) (3.34) where the superscripts denote the four sub-laminates behind and ahead of the delamination front, and s denotes the location of the point along the delamination front. Sub-laminates 1 and 4 are above the delamination plane, and 2 and 3 are below the delamination plane. From Equation (3.34) one can derive another expression for G(s) as: 1 T G = -(F4 F )(C, + Cb )(F4 1 ) (3.35) 2 4 - 52 The derivation of Equation (3.35) is very similar to that of Equation (3.23). As before, the term (F4 F1 ) is the matrix of crack-tip forces. They also represent the jump in force and moment resultants that occur across the delamination front. Sankar and Sonik [62] showed that three of the eight force resultants will be continuous across the delamination front. Assume a coordinate system such that the xaxis is normal to the crack front, y-axis is tangential to the crack front and z- is the thickness direction. Then the continuous force resultants are N y, M and Q Thus the jumps in these force resultants are zero, i.e., (4) (1) (2) (3) = 0 Ny- Ny= Ny- Ny 0 S M 0 (3.36) Q(4) Q(1) =.. Q(2) Q(3) = 0 Thus there will be only five components to the crack tip forces (see Figure 3.9): three forces in the x, y and z directions; two couples, about x and y axes, respectively. The three forces will be the jump in Nx, Nxy and Qx across the delamination front, either in the top laminates (1 and 4) or bottom laminates (2 and 3). The two crack tip couples are the jumps in M x, M x Since the jumps in N MY and Q are equal to zero and they do not contribute to the crack-tip forces, we can delete the 2nd, 5th and 7th rows and columns in Ct and Cb; we will denote them by C, and C b,, . Fz Fy Figure 3.9 Crack-tip forces in delaminated plate. Then from Equation (3.33) an expression of point-wise energy release rate can be derived as: G(s) = F (C, + CTb)FC (3.37) where the crack tip forces Fc are given by: [F T] [(N(4) (1)) (4) 1) ( 4) (M(1) -c =- x xy -Nxy ),(Mx -Mx , (4) (1) (Q (4) xy xy x (3.38) Q I))] The compliance matrices C will take the form: Cl C13 Cl4 cl6 Cl8 C13 C33 C34 C36 C38 C C14 C34 C44 C46 C48 (3.39) C16 C36 C46 C66 C68 (3.39) _C18 C38 C48 C68 C88 where the Cij are the coefficents of the full compliance matrix C, or Cb It should be mentioned that the xyz coordinates should be moved along the delamination front while using Equation(3.30) for computing pointwise G(s). We have derived three formulars Equations (3.27), (3.31), and (3.34) for computing the point-wise energy release rate in a delaminated plate. Out of the three, Equations (3.32) and (3.35) are exact, and their accuracy is limited only by the methods used compute the force and moment resultants or the strain energy densities ahead and behind the delamination front. The accuracy of the VCCT given by Equation (3.30) is limited by not only the crack tip forces but also the mesh size which will define the length of virtual crack growth. 3.9 Summary In this chapter a new method called Crack Tip Force Method (CTFM) was introduced and derived. Three methods of computing G for laminated composites structures are also discussed. A Crack Tip Force Method is derived for computing 55 point-wise energy release rate along the delamination front in delaminated plates. Actually the method can be derived from the Virtual Crack Closure Technique or the previously derived Strain Energy Density Method. However the CTFM is computationally simple as the G is computed using the forces transmitted at the cracktip between the top and bottom sub-laminates and the sub-laminate properties. An evaluation of the aforementioned methods, their applicability to general laminates containing delaminations, and debonded stiffened panels will be presented in chapter 5. CHAPTER 4 ANALYTICAL AND EXPERIMENTAL CORRELATION OF A STIFEENED COMPOSITE PANEL IN AXIAL COMPRESSION 4.1 Introduction Buckling and imperfection sensitivity are expensive to calculate with general finite element models. Consequently, the optimization of stiffened panels often employs simplified models that are exact only for idealized geometries and boundary conditions (e.g., PASCO [10], or PANDA2 [15-17]). Nagendra et al. [33] studied the optimum design of blade stiffened panels with holes subjected to buckling and strain constraints. They used the panel analysis and sizing code (PASCO), based on a linked plate model, for the buckling analysis and optimization with continuous thickness design variables, and the Engineering Analysis Language (EAL [34]) finite element analysis code for calculating strains and their derivatives with respect to design variables. Later, the optimally designed panels with and without centrally located holes were tested, and analytical and experimental results were compared [35]. Nagendra et al. [80] continued the optimum design study of blade stiffened panel using PASCO for analysis, and a genetic algorithm (GA) for the optimization of the panel laminate stacking sequences. Several designs obtained with GA were about 8% lower in weight compared to designs previously obtained with a continuous optimization procedure. 57 Recently, three of the panels designed by Nagendra et al. were fabricated and tested by the Structural Mechanics Branch at NASA Langley Research Center. The experimental failure loads differed by up to about 10% from the design load. However, there were significant differences in loading and boundary conditions between the design conditions and the test conditions. The principal objective of this chapter is to understand the effects of differences between the simplified assumptions made in the design model and the actual test conditions. Another objective is to assess the effectiveness of the simplified PASCO model originally used to design the panel, and this is aided by comparing the results obtained from several structural analysis programs including PANDA2, STAGS, and ABAQUS. 4.2 Stiffened Panel Definition The basic configuration of the panel designated as the baseline design corresponds to the design in the 9th row in Table 7 of Reference [80]. This panel, designated as GA2461 (referring to the design weight of 24.61 lb.), is 30-inches long and 32-inches wide with four equally spaced blade stiffeners (see Figure 4.1). The laminates used in Reference [95] for the skin, stiffener blade, and stiffener flange for the baseline design were balanced, symmetric laminates consisting of 00, 450 and 900 plies. The skin has 40 plies with a stacking sequence [+45/90,/+453/90J45]s and the stiffener flange and blade have an identical stacking sequence of [45/(45/04)2 /902/04/(45/02)2/02/45s with a total of 68 plies. Properties of the AS4/3502 graphite epoxy material used in Reference 5 are given in Table 4.1. Table 4.1 Hercules AS4/3502 graphite epoxy lamina material properties. Young's modulus El =18.50 x 106 psi (longitudinal) Young's modulus El =1.64 x 106 psi (transverse) Shear modulus G12 =0.87 x 106 psi Poisson's ratio vl2 = 0.3 Density p =0.057 lb in3 Ply thickness tply = 0.0052 in The baseline design was designed to support an axial load Nx of 20,000 lb./in. In addition, in order to account for off design conditions, imperfections and modeling inaccuracies, a shear load (N,r = 5000 lb./in) and a longitudinal bow type (3% of the panel length) imperfections were added. The baseline design panel was assumed to be simply supported along the four edges, which is the only boundary condition that can be accurately modeled using PASCO. 32" N, V 0L qip Figure 4.1 Blade stiffened panel with four equally spaced stiffeners under compression and shear loads. All dimensions are in inches. 3 --Y N, 4.3 Test Specimen and Test Procedures The test specimens were fabricated from Hercules AS4/3502 unidirectional graphite/epoxy preimpregnated tape material. The skin (32-in. x 32-in.) and the stiffeners were cured separately in an autoclave. The stiffeners were machined to a length of 32 inches, and then bonded to the skin with FM-300 film adhesive. The panel edges perpendicular to the stiffeners were potted with an aluminum filled epoxy resin to prevent end failure. The length of the potted area was 1 in. on each side. Thus the effective gage length of the test specimen was reduced to 30 inches. The test specimen was loaded in compression using a 1,000,000-lb-capacity hydraulic testing machine. The specimen was flat-end tested without lateral edge supports, and no deliberate imperfection was introduced. Electrical resistance strain gages were used to monitor the strains, and direct current differential transformers (DCDTs) were used to monitor longitudinal in-plane and out-of-plane displacements at selected locations as shown in Fig. 4.2. All electrical signals and corresponding applied loads were recorded automatically at regular time intervals during the tests. 4.4 Linear Buckling Analysis 4.4.1 PANDA2 and STAGS The analyses performed in this study include both buckling and nonlinear postbuckling calculations. Linear buckling analyses for the baseline design were conducted by using both PANDA2 and STAGS. Input files for STAGS linear buckling analysis were generated by PANDA2. Next, the effect of the shear load and the 0 DCDTs (1-7) for out-of-plane deformation of skin. DCDTs (8-11) for out-of-plane deformation of stiffener. DCDT (12) for end-shortening displacement 2~ 8 1 I6 9 2 U3 11 7 1( 5 3 U2 L -4 2U3 Figure 4.2 Layout of the displacement measurement instrumentation for the test panel. Reference plane (ABAQUS) N Reference plane (STAGS) Figure 4.3 Reference plane of ABAQUS and STAGS model. 62 imperfection on the buckling loads of the baseline design were investigated using PANDA2, which employs analysis techniques with similar level of fidelity to that of PASCO. In PANDA2, local and general buckling loads are calculated by either closedform expressions or by discretized models of panel cross sections based on an energy method [15]. STAGS is a finite element code for general purpose analysis of shell structures of arbitrary shape and complexity [25]. STAGS has a variety of finite elements suitable for the analysis of stiffened plates and shells. Four node quadrilateral plate elements with cubic lateral displacement variations (called 410- and 411-Elements) are efficient for the prediction of buckling response of thin shells. For thick plates in which transverse shear deformation is important, the assumed natural strain (ANS) nine node element (480-Element) can be selected [16]. The panel investigated here warrants the use of 480-Element, however 411-Element was also used as the panel was designed by PASCO, which does not model shear deformation. STAGS results were post-processed by PATRAN, which is a commercial software for pre- and post-processing of finite element simulations [82]. 4.4.2 Finite Element Model The STAGS finite element model for the panel had a total of 20 branched shell units, and each branched shell unit had 65 x5 nodes (for the 32-in. long panel) or 61 x 5 nodes (for the 30-in. long baseline design panel), respectively. The axial compressive design load (640,000 lb) was applied with a uniform end-shortening constraint along with compatibility conditions for adjacent shell unit interfaces. In the test, the load was 63 introduced through the potted ends. To simulate this boundary condition, the displacement along the z-direction and the rotation along the x-direction were constrained at the nodal points in the potted region. The adhesive film used to bond the stiffener to the skin in the test panel was modeled by adding an isotropic layer to the model to simulate the bondline between the skin and flange with a thickness of 0.0121 inches. The skin middle surface was used as reference surface on which the nodes lie, and the offset distance of the middle surfaces of the skin-flange combination was modeled as an eccentricity. An alternative finite element modeling approach with ABAQUS suggested by Greene of HKS, Inc. was also used. In this method instead of locating the reference plane at the mid-plane of skin, the bottom plane of blade stiffener was taken as the reference plane. In order to handle the offset distance of the mid-plane of skin, as well as skin-flange combination, an additional 00 ply, with negligibly low stiffness was added to both skin and skin-flange laminates, as shown in Figure 4.3. Both the nine-node thin shell element (S9R5) and the four-node general shell element (S4) in ABAQUS were selected for the stiffened panel models. Both shell elements can account for transverse shear deformations and should therefore be compared to the 480-Element in STAGS. Instead of applying a compressive load at the panel end, uniform compressive displacements were applied at the nodal points along the loaded edge. In order to ensure a uniform state of stress along the entire panel length and also to prevent bending during the pre-buckling stage, the incremental boundary condition option available in ABAQUS was chosen. The buckling load factor was computed from the sum of the reaction forces at the boundary node set. 4.5 Results of Linear Buckling Analysis In this section the effects of geometric imperfections, additional in-plane shear loads, boundary conditions and material property variations on the buckling load of the stiffened panel are discussed. The results are intended to shed light on probable reasons for the discrepancies between predicted buckling loads and corresponding experimental results. Furthermore, the effect of the assumed imperfections and the addition of inplane shear loads on the robustness of the design is also noted. 4.5.1 Effect of Geometric Imperfections and Shear Load A summary of the local buckling load factors with and without shear load, and with and without the initial bow type geometric imperfections (3 % of the panel length) obtained from PANDA2 is given in Table 4.2. The first row in Table 4.2 also includes a comparison of PANDA2 results and the STAGS results (both 480- and 411-Elements) for the perfect panel without the shear load. It may be noted that the PANDA2 results, both Koiter type analysis and BOSOR4* analysis, agree well with the STAGS 411Element results. The buckling mode for the perfect panel obtained using STAGS 480Element is shown in Figure 4.4. According to PANDA2 results, the lowest buckling load corresponded to local buckling, which suggests that the differences in boundary conditions between the analysis and the experiment will not have a large effect on the results. The 10% difference between the results for the 480-Element and the 411* BOSOR4 analysis routine in PANDA2 calculates local buckling load for the single panel module from BOSOR4type strip theory.6 65 Element (Table 4.2) is probably due to transverse shear deformation since the thickness of the skin-flange combination is 0.56 inch. Shear deformation was not included in the original panel design, and this difference indicates that that effect is substantial. The panel with negative bow-type imperfection had a concave surface in the middle of the panel. Thus, the blade tip is subjected to less axial compression and skin is under more axial compression than that of the perfect panels in the neighborhood of mid-length in the axial direction. Similarly, the blade tip near the boundary is under more compression and the skin near the boundary is under less compression than that of the perfect panel. The opposite holds for the positive bow-type imperfection. From Table 4.2, it is clear that the effect of the shear load on the buckling load is small, but the effect of the imperfection is very significant. From the last two rows of Table 4.2 one can note that a 3% positive imperfection results in a very low buckling load factor. The buckling load factor reduces from 1.256 to 0.394. A 3% negative imperfection also reduces the buckling load (from 1.256 to 0.856), but the reduction is smaller than that for a positive imperfection. It should be mentioned that a 3% imperfection is very large for a 32-in. stiffened panel, and thus will lead to very conservative designs. 4.5.2 Effects of Boundary Condition and Material Properties There were slight differences in material properties, panel dimensions and the boundary conditions between the baseline deign and the actual test conditions. In order to understand the effects of these differences, analyses were carried out using both sets of input data. The differences in material properties and dimensions are summarized in Table 4.3 and Table 4.4, respectively. While the changes in the material properties can 66 be input directly, the differences in the thickness of the skin or flange are accounted by implementing a proportional change in the ply thickness in the model. A detailed discussion of this procedure can be found in Ref. 4. The results in Table 4.5 indicate that the effects of boundary conditions and material properties on the buckling load factors are not very significant. Comparison of the first two rows of Table 4.5 reveals the effects of changes in the boundary conditions. Similarly, results in the last two rows show the effects of changes in material properties and panel dimensions. The buckling mode shape of the baseline design (simply supported on 4 sides) predicted by STAGS is shown in Figure 4.4. The overall buckling mode shape obtained from ABAQUS (Figure 4.5) agrees well with that of STAGS. The computed lowest buckling load factor is slightly higher than that of STAGS (1.218 for ABAQUS vs. 1.168 for STAGS). This small difference may be due to modeling differences as discussed in the following section. The STAGS prediction of the buckling mode shape of the test panel with potted boundary conditions (other two edges being free) is shown in Figure 4.6. Table 4.2 Summary of the local buckling load factor from PANDA2 and the lowest buckling load factor from STAGS (4 edges simple supported). PANDA2 PANDA2 STAGS STAGS (Koiter analysis) (BOSOR4 analysis) (480 (411 element*) element) Loading combination with/without Panel end Panel Panel Panel element*) element) imperfection mid-length. end mid-length. N,=20,000 lb/in, Nx-0 without 1.256 1.256 1.328 1.328 1.168 1.296 imperfection NI=20,0001b/in, Ny=5000 lb/in without 1.234 1.234 1.048 1.048 imperfection N,=20,0001b/in, Nx=5000 lb/in with 1.234 0.356 1.048 0.346 imperfection (+3%) Nx=20,0001b/in, N.=5000 lb/in with 1.234 0.856 1.048 0.920 imperfection (-3%) Nx=20,0001b/in, Nx,=0 lb/in with 1.256 0.394 1.328 0.398 imperfection (+3%) Nx=20,0001b/in, Ny0 lb/in with 1.256 1.026 1.328 1.083 imperfection (-3%) *includes shear deformation. Table 4.3 Material properties of baseline designs and test specimen. Table 4.4 Geometric parameters of baseline design and test specimen. Panel Design Test specimen Panel length (in) 30 32 Panel width (in) 32 32 Blade height (in) 3.0705 3.0723 Skin ply thickness (in) 0.00520 0.00555 Blade ply thickness (in) 0.00520 0.00503 Flange ply thickness (in) 0.00520 0.00491 Adhesive thickness (in) 0.0 0.0121 E,, (Msi) E22(Msi) G,2 (Msi) V12 V21 Skin Design 18.50 1.64 0.87 0.3 0.0270 Test 17.333 1.64 0.8151 0.3 0.0284 specimen Blade Design 18.50 1.64 0.87 0.3 0.0270 Test 19.125 1.64 0.8994 0.3 0.0257 specimen Flange Design 18.50 1.64 0.87 0.3 0.0270 Test 19.593 1.64 0.9214 0.3 0.0251 specimen Adhes- Test 0.5 0.5 0.192 0.3 0.3 ive specimen Table 4.5 Buckling load factors obtained from STAGS with various boundary conditions and material properties. Boundary conditions shell unit shell unit (480 element*) (411 element) 4 edges simply supported 1.166 1.296 (baseline design) 2 edges free, 2 edges clamped 1.197 1.340 (baseline design) 2 edges free, 2 edges clamped 1.154 1.286 (potted region, test material, 32-inch panel length) Includes shear deformation. Figure 4.4 The STAGS predicted buckling mode shape of the perfect baseline design panel subjected to uniaxial compression (load factor= 1.166). MSWATFA N V.one.2 03Jnb6 20 45 PE OFl rTIO 0 bb d4* ii Leip O Mob; 1, C OCMI 0 C1 1ME-1, bh d481 h.f.i 1. E0, TW abl. i SM, - S1 .e -o .--I 69 TCMKMNU 022WO I 195.2 FMM*e B W 4tnt,En.2 244: Rllb, o ct ie-VMa ABAOUS 800 Figure 4.5 The predicted buckling mode shape of the blade stiffened panel with 9-node shell elements from the ABAQUS (load factor=1.218). !.TPATit4NVsebn921 9Jn@ 0D :10 FR GE Oms: g2481, Lo p 0, Msa 1 Cri Im Lad: 0 1 54E 01, ga2401 gl 01 : Ei~.nea Tambirat. (VECM4G) -4 3EFOtMATI O 248 8. Ismisl0,M ko 1, CrimI LUmd:u543E 1 .2401. g01Eieec Termimi PATtN2 assa 80o7 '3 Figure 4.6 The predicted buckling mode shape of the blade stiffened panel from the STAGS (load factor- .154). An examination of the STAGS model in Figure 4.3 shows that there is doublecounting of material between the blade stiffener and the flange-skin combination. This is because of the way the nodes are located in the blade elements and in the element that represents the flange-skin combination. Both elements have mid-plane nodes leading to an overlap equal to the thickness of the blade with a width equal to half-thickness of the flange-skin combination. This overlap is avoided in the modeling approach described earlier for the ABAQUS model. This additional material due to the overlap in the model is expected to increase the pre-buckling axial stiffness of the panel. In order to estimate this increase the STAGS model was modified similarly to the ABAQUS model, and the linear buckling analysis was performed on the modified model. The results are presented in Table 4.6 and are also compared with the experimental results. The increase in area due to the overlap contributed to about one half of the total stiffness difference between the analysis and test results. Table 4.6 Comparison of prebuckling stiffness and buckling load factors (STAGS results). EA/L (kip/in) EA (kip) Buckling load factor Model with 3931.2 125,800 1.15 overlap Model with 3684.4 117,900 1.23 no overlap Experiment 3453.9 110,500 1.09 4.5.3 Summary of Differences between Design Model and Test Model In summary, the PASCO model used for designing the panel had several modeling simplifications and compensating factors; their effects are listed in Table 4.7. The two major model simplifications were: 1. PASCO does not account for shear deformation, which is significant for thick composite panels, and this reduces the buckling load by 11%. 2. PASCO employs simple support boundary conditions. The difference due to boundary conditions was only about 1% because the buckling mode is local. To obtain a more robust design, the PASCO model was subjected to an additional shear load and 3% imperfection. Both had substantial effects on the design load. However, because the panel was designed only for imperfection of one sign, it became more sensitive to an imperfection of the opposite sign. Finally, because of the substantial thickness of the blade, it was also found that a centerline modeling, which is common in thin walled structures, produces about seven percent increase in the prebuckling stiffness, and a seven percent reduction in buckling load. The opposite effects are explained by the fact that the overlap draws more loads into the blade, which is the critical element. Overall, the more accurate analytical model predicts a buckling load higher by about 18% than the design load; however, this does not take into account any imperfections. The actual buckling load was only about 10% higher than the design load. It can be concluded that for this panel the simplified model used in PASCO, together with the shear and imperfection loading added for robustness, worked reasonably well. The other two designs that were tested also buckled at or slightly above the design load Table 4.7 Differences between baseline design analysis and actual test panel analysis. Baseline design Test panel Effect on buckling load Loading (lbs/inch) Axial compression 20,000 20,000 Inplane shear load 5,000 0 -20% Imperfection -3% of panel length Not measured -30% (with shear (initial bow type) load) -18% (w/o shear load) Boundary Condition 1% (with test Loaded edges Simple support Clamped (potted) material) Unloaded edges Simple support Unsupported Transverse shear no yes 11% deformation 4.6 Nonlinear Analysis Although the linear buckling loads provide a measure of the compressive load carrying capacity of the stiffened panels, the test results indicate that the panels underwent substantial nonlinear transverse deformations prior to failure. Hence it was decided to perform a nonlinear analysis in order to understand the effects of boundary conditions including the eccentricity in load application. The nonlinear analysis was started without applying any initial imperfection, but the differences in the stacking sequence and the differences in material properties between the skin and blades induce bending deformation. The modified Riks path following algorithm in STAGS was used for the nonlinear analysis. The computation time required for the nonlinear analysis was an order of magnitude higher than for the linear analysis, indicating significant nonlinearity (probably near the buckling load). In the following sub-sections we discuss the results of the nonlinear analyses, and make a comparison between experimental and analytical results. 4.6.1 Compressive Load Versus End-shortening The compressive load versus end-shortening deflection curves from the STAGS nonlinear static analyses, the test, and a linear fit (regression analysis) of the measured data are shown in Fig. 4.7. Recall that the test panel designated as GA2461 in Reference [80] is the baseline design for the present paper. In comparison to the baseline panel, two other test panels (GA2414, GA2458) from Reference [80] have slightly different geometries and stacking sequences. As expected, their compressive load versus end-shortening curves from the tests exhibit a similar trend except at the initial stage of loading. The prebuckling stiffness (EA) was calculated from the slopes of the linear portions of the experimental load versus end-shortening curves for the three panels and by multiplying the slopes by the panel length. The prebuckling stiffness from the experiments is compared in Table 4.8 with the prebuckling stiffness predicted using STAGS for test panel GA2461. It is observed that the prebuckling stiffness of the test panels is about 6% lower than the analytical value. Table 4.8 Comparison of the prebuckling stiffness and buckling load results from STAGS and experiments. EA/L (kip/in) EA (kip) Buckling load factor STAGS nonlinear 3931.2 117,900 1.23 (GA2461) Experimental 3453.9 110,500 1.09 Result (GA261) Experimental 3347.8 107,100 0.94 Result (GA2414) Experimental 3333.8 106,700 1.07 Result (GA2458) Load vs. End-shortening Figure 4.7 Compressive load versus end-shortening from analysis and experiments. Load vs. Out-of-Plane displacement (mid panel) -0.2 -0.15 -0.1 -0.05 0 0.05 Out-of-Plane Displacement (Inch) Figure 4.8 Load versus out-of-plane displacements of the skin at the center bay. -0.05 0.00 0.05 0.10 0.15 0.20 E nd-shortening (inch) 0.25 0.30 Load vs. Out-of-Plane displacement (stiffener) -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 Out-of-plane displacement (inch) Figure 4.9 Load versus out-of-plane displacements of the stiffener at selected location in Fig.2. Variation of Out-of-Plane Displacement 0.00 -0.02 4) -0.04 E -0.06 E e-0.08 O w-0 10 -, 02 0 -0.12 -0.14 Panel Length (inch) Figure 4.10 Variation of the out-of-plane displacements along the panel lengthwise direction. 4.6.2 Compressive Load Versus Out-of -Plane Deformations The layout of the DCDTs used to measure displacements in the test panel is shown in Figure 4.2. The out-of-plane displacements, measured from DCDTs at selected locations in Figure 4.2, are shown in Figures. 4.8 and 4.9. The results in Figure 4.8 show that out-of-plane displacements were initiated at an early stage of the loading and increase linearly in proportion to the loading. This observation suggested the possibility of loading eccentricities along the load introduction edge or rigid body rotation of the panel with respect to the clamped edge in addition to the effects of geometric imperfections. Figure 4.9 shows that the out-of-plane displacements of the blade stiffeners were also started at an early stage of the loading. Except DCDT 11, the out-ofplane deformations were an order-of-magnitude lower than those of the skin in Figure 4.8. Furthermore, significant nonlinear response was only exhibited near the failure load. The large nonlinear response of DCDTI 1 throughout the axial loading was probably due to the effect of the unsupported side-edge boundaries. The out-of-plane displacement variations along the length of the panel (DCDTs 1-4 in Figure 4.2) at selected load levels are shown in Figure 4.10. The results in Figure 4.10 indicate that bending occurred in the test panel in addition to the end shortening due to the axial compressive load. The load versus out-of-plane displacements across the panel mid-length can be found in Ref. [83]. In order to explain the substantial prebuckling bending, a combination of different geometric imperfections and loads applied at a small angle to the axial direction were analyzed to determine their influence on the observed out-of-plane displacements. The capability of STAGS to model geometric imperfections in the shape 77 of the buckling modes was used for these analyses. Various combinations of imperfection amplitudes and load angles were considered. Although for some combinations we could reproduce the test results partially [83], obtaining the right imperfection and the load introduction angle seemed elusive. This difficulty suggests that we look elsewhere for the source of the prebuckling bending. 4.6.3 Contact Between the Panel and Loading Platen Hilburger [97] investigated the effects of non-uniform load introduction and boundary condition imperfections on the compression response of composite cylindrical shells with cutouts. He defined the non-uniform load distribution as anything other than uniform axial displacement of cylinder's loading surface and found two sources of non-uniform load introduction. One was due to lack of planarity in the loading surfaces of the specimen and the loading platens. The other source was due to tilt of the loading platen with respect to the specimen before the loading began. He measured the top and bottom loading surface imperfections as well as potting thickness. Then the imperfection data was fit to curves and input into the STAGS models. Furthermore, the test frame loading platen was modeled as rigid flat plates and generalized contact definitions* given in STAGS were used. A similar modeling approach was used in the present study in order to identify the causes of the substantial out-of-plane deformations in addition to nonlinear end shortening during the early stage of the test. The loading platen was modeled as a rigid * Generalized contact definition means that contact points are calculated by STAGS rather than specified by the user. 78 flat plate in the STAGS analysis. Because the loading surface imperfections were not measured before the test, they were not considered in this study. Instead, it was assumed that the rigid loading platen was initially contacted at the tip of the blade. Stiffeners were assumed to have a small tilt angle with respect to the load introduction edge of the test specimen, as shown in Figure 4.11. STAGS uses generalized contact definitions to check for contact and to construct actual contact elements coupling contacting points with contacted shell elements as the analysis progresses. In doing this, STAGS uses penalty functions to enforce a displacement-compatibility constraint between each contact point and each element with which it is in contact. STAGS utilizes analyst-supplied stiffness versus displacement information to compute the forces resulting from the small contact-surface penetration that may occur. A contact element is conceptually a nonlinear spring connecting the contact point to the surface of the contacted element. This nonlinear spring typically has a low stiffness and generates a small force when the contact-surface penetration is small, but it gets progressively stiffer and generates a larger force as the penetration increases [23]. Generalized contact definitions were implemented to the finite elements that simulate the rigid platen and the load introduction edge of the test specimen. The selection of proper stiffnesses of the contact elements for present analyses is rather arbitrary. Thus, several nonlinear analyses were performed to simulate the observed outof-plane deformation response of the test specimen by changing both tilt angles and the stiffnesses of contact elements between the loading platen and load introduction edge. The combinations of the tilt angles and stiffnesses of the contact elements used for the 79 analyses are summarized in Table 4.9, and the corresponding load versus end shortening results are shown in Figure 4.12. The results in Figure 4.12 suggest that the computed end-shortening response strongly depend on the user-supplied input data in Table 4.9. The response of Model 9 was the closest to that of the test panel in Figure 4.12. The computed out-of-plane displacements of the skin and stiffeners of Model 9 at the selected DCDTs location are further examined as shown in Figures. 4.13 and 4.14, respectively. It is observed that the response of Model 9 in Figure 4.13 shows good correlation with those of the skin of the test panel in Figure 4.8 during the early stages of the load. However, Model 9 exhibits considerable nonlinear behavior of the out-ofplane deformations of skin when the compressive load is above 400,000 lb, which was not present in the response of the test panel in Figure 4.8. Table 4.9 Summary of the tilt angles and the stiffness of contact element. Panel length Tilt angle Stiffness of contact element (inch) (degree) (lb/inch) Model 7 30 0.01 Disp. 0.005 0.05 0.1 0.2 1.0 Force 1.0e3 1.0e5 1.0e7 2.0e8 2.0e8 Model 8 30 0.01 Disp. 0.0001 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e7 1.0e8 2.0e8 2.0e8 Model 9 30 0.01 Disp. 0.005 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e5 1.0e8 2.0e8 2.0e8 Model 10 30 0.005 Disp. 0.005 0.03 0.05 0.2 1.0 SForce 1.0e3 1.0e5 1.0e8 2.0e8 2.0e8 The computed load versus out-of-plane deformation of the stiffener at location of the DCDT 11 shows significant nonlinear behavior in Figure 4.14. This significant nonlinear response was also observed from the measured response of the actual test panel in Figure 4.9. In general, the finite element model with the generalized contact definitions improved correlation between the measured and predicted out-of-plane 80 deformation. However, the details of the displacements are considerably different. It is concluded that some combination of tilt angles and the contact stiffnesses can produce the observed pattern, but there may be some other contribution to the out-of-plane displacements. TIL LOADING PLATEN SKIN BLADE z x Figure 4.11 Schematic of blade stiffened panel and loading platen. Load vs. End-shortening 8.000E+5 7.000E+5 6.000E+5 5.000E+5 S4.000E+5 3.000E+5 2.000E+5 1.000E+5 0.000E+0 0.00 0.05 0.10 0.15 0.20 End-shortening (inch) 0.25 0.30 Figure 4.12 Compressive load versus end-shortening from analysis with contact models and experiment. Model numbers refer to Table 9. Figure 4.13 Load versus out-of-plane displacements of the skin at the center bay from Model 9 analysis. Load vs. Out-of-plane displacement -0.3 -0.25 -0.2 -0.15 Out-of-plane displacement (inch) -0.1 -0.05 0 0.05 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Out-of-plane displacement (inch) Figure 4.14 Load versus out-of plane displacements of the stiffeners at selected locations from Model 9 analysis. 4.7 Conclusion Analytical models using several structural analysis models were used to assess the adequacy of the design model and the correlation with experimental results for a stiffened panel designed using the PASCO program. Of the effects neglected by the simple model, shear deformation was the most important, accounting for about 11% difference in buckling load. The effect of simplified (simple support) boundary conditions was small. The addition to the design model of shear loads and imperfections to improve the robustness of the result did help, even though the inclusion of one-sided imperfection apparently induced sensitivity to imperfection of the opposite sign. Overall, the simplified model did produce designs that in the experiments failed slightly above the design load. The most significant difference between the analytical predictions and experimental measurements was the substantial out-of-plane pre-buckling deformations. To explain these differences, imperfections, load eccentricities, and loading platen tilt angles were considered. Of these the loading platen tilt produced similar patterns of deformation, but these had more nonlinear characteristics than the measured deformations. CHAPTER 5 BUCKLING AND POSTBUCKLING ANALYSIS OF A STIFFENED PANEL WITH SKIN-STIFFENER DEBOND 5.1 Introduction In contrast to most of simplified approaches discussed in literature survey in Chapter 1, finite element based approach can be applicable with high fidelity to more general and realistic structures. Therefore, the axial compressive behavior of a stiffened panel with skin-stiffener debond was explored in this chapter by employing finite element model. Two different finite element models, where nodes of the panel skin elements and the stiffener flange elements are located on the mid-plane of each element or interface between the skin elements and flange elements with offset, were used. The nodes corresponding to the top of the skin and bottom of flange are connected with either the elastic spring fastener elements or multi-point constraint equations. In order to verify present finite element modeling approach, laminated composite plates with/without the through-the-width delamination were first modeled. Both single delamination and multiple delaminations were considered. Then, stiffened composite panels with skin-stiffener debond were modeled. Stiffened composite panels with a single stiffener as well as multiple stiffeners were also considered. Buckling and postbuckling analyses were conducted using STAGS. Comparison was made with available buckling analysis results. Next, numerical examples of computing energy 85 release rate in the context of plate as discussed in Chapter 3 are given for predicting debond extension using the strain energy derivative method, the virtual crack closure technique, and the crack-tip force method. 5.2 Finite Element Model Two finite element-modeling approaches for the stiffened panel are commonly used in the literature. One approach is to model the skin with plate elements and to model the stiffener with beam elements. The other is model both skin and stiffener with plate elements. The second modeling approach appears more attractive for modeling debonded region between skin and flange, and it was chosen in this study. Furthermore, two different finite element models, designated as Model I and Model II, respectively, were also considered (see Figure 5.1). Both skin and flange elements of Model I have offset nodes with small gap at the interface region, while nodes in Model II are located on the mid-planes of skin and flange, respectively. In Model I, in order to satisfy compatibility conditions of intact interface nodes located directly above the skin and below the flange, each nodal degree of freedom was constrained with elastic spring fastener elements with very high spring constants. Nodes located on debonded interface between skin and flange were connected with elastic fastener elements, which have only an axial degree of freedom with very high stiffness in compression and zero stiffness in tension. This can prevent physically unrealistic nodal penetration between skin and flange during the postbuckling analysis. Friction against sliding of the debonded surface was not considered. In Model II, Multi-point constraints were imposed at the interface of intact skin and corresponding flange nodes to satisfy the 86 displacement compatibility condition as h h, u, + L =u 2 s 2 V" X 'S 2 VX, f v + Vy = v V 2 2 ws = wf (5.1) where u, v, and w are displacements in x, y, and z direction, EPx and y are rotations, h is thickness, and subscripts s andf denote skin and flange, respectively. In the debonded region, the same modeling approach as Model I was used. One advantage of model I over model II is such that displacement compatibility conditions at interface nodes in Equation (5.1) is not necessary. The other advantage of model I is that computation of displacements behind crack tip is easy. However, Model I cannot handle multiple delaminations located through-the-thickness direction while Model H can. Model I Nodes are located on interface Model II Nodes are located on mid-plane of skin Figure 5.1 Two different modeling approach for blade stiffened panel. 5.3 Buckling Analysis Results of Multiple Delaminated Plate A plane woven fabric glass fiber reinforced composite panel with three through-the-width delaminations located in the middle of the plate from Suemasu [56] (as shown in Figure 5.2) was examined using Model II approach. The material properties are shown in Table 5.1. The STAGS finite element model for the plate has a total of 4 branched shell units with four node shell elements (element 410), and each branched shell unit has 41 by 11 nodes. Total of 297 elastic fastener elements (element 130) was used to prevent penetration of contact surfaces in debonded region of the plate. The compressive stiffness of an elastic fastener element used in this study is 113 MN/m. In the intact region of plate, a total of 3168 constraint equations was used to satisfy the compatibility conditions of shell unit interfaces. Axial compressive load (2500 N) was applied to load introduction edge with uniform end-shortening constraint. In order to ensure a uniform stress condition for the entire panel length, an incremental boundary condition option was chosen to prevent axial bending during the prebuckling stage. The computed buckling load and buckling load from Reference [56] are given in Table 5.2. The first and second buckling mode shapes from Reference [56] and present analysis are shown in Fig. 5.3 and Fig. 5.4, respectively. It is seen that present buckling analysis results agree well with the corresponding buckling analysis results from Reference [56]. However, the buckling load from the experiment is lower than the computed buckling load. Suemasu [56] indicated that insufficient clamped condition during his experiment might be responsible for low buckling load. Table 5.1 Plane woven fabric fiber reinforced laminates material properties [Ref. 56]. Young's modulus (longitudinal) El = 20.2 GPa Young's modulus (transverse) E2 = 21.0 GPa Young's modulus (through-thickness) E3 = 10.0 GPa Shear modulus (inplane) Gl2 = 4.15 GPa Shear modulus (through-thickness) G13= 4.0 GPa Poisson's ratio (inplane) vl2 = 0.16 Poisson's ratio (through-thickness) v3= 0.3 20 mm 40 mm 160mm 40 mm Figure 5.2 Glass fiber reinforced composite specimen [Ref. 56]. Table 5.2 Comparison of buckling load for delaminated plates ( Pcr=2500 N). Buckling load factor Present Suemasu [56] Mode I 0.440 0.432 Mode II 0.801 0.795 Figure 5.3 Buckling modes of the plate with three delamaninations from Ref. [56]. .N U 4, *on 0.45 a. = 0.5 M CPATMAN V n e220 147 22M44 0AINE: C im: .31.1, Ladp: 0. Mcd 1. Otci Lan O 105005+01 ,pa 1 stal EIenlow Tnsti0al (Z:OMP) -PA FOMmATII 30 N M p1twe 0, lI,Qd Lad01301105p0 1 PIIsa 1 g. nT00Ep mm Tl A fl20 .86se 77 W78 4298 (a) cass 1455 *1A N WlonO 2 200047 233250 0535E P9141. Lmdalp 0. .112, OO La 02=070E .1 p i W b=D eo d0 T15d (Z42Mp PA 493 z (a) Figure 5.4 Computed buckling modes using model II approach. (a) First lowest buckling mode. (b) Second1a2 lowest buckling mode.PA ses (b) Figure 5.4 Computed buckling modes using model II approach. (a) First lowest buckling mode. (b) Second lowest buckling mode. 5.4 Buckling and Postbuckling Analysis Results of Plate with Single Delamination In order to see the effects of the delamination location through thickness, a unidirectional graphite/epoxy laminated plate with single through-the-width delamination was considered (see Figure 5.5). The geometry and material properties used for the graphite/epoxy laminates are given in Table 5.3. Axial compressive load (1,250 lb) was applied to load introduction edges. Three different thickness ratios of top and bottom subraminates were considered and buckling load factors are summarized in Table 5.4. The computed buckling modes agree well with deformed shapes obtained from the test in Reference [53] (see Figures 5.6, 5.7). As expected, the delamination located near the surface of the plate has the lowest buckling load, where local buckling is the dominant buckling mode. As the thickness ratio of sublaminates increases, the buckling mode changes from local buckling to mixed buckling mode (see Figure 5.8) and then from mixed to global buckling mode. delamination 0.15 h 14h 1.25 < 5.0 Thickness of top sublaminate (h 1) Thickness of bottom sublaminate (h2) Figure 5.5 Graphite/epoxy laminated plate with single through-the-width delamination. All dimensions are inch. TABLE 5.3 Graphite epoxy lamina material properties Young's modulus (longitudinal) El =19.50 x 106 psi Young's modulus (transverse) E2 =1.48 x 106 psi Shear modulus G12 --0.80 x 106 psi Poisson's ratio Vl2 = 0.3 Plate length x width 5.0 x 1.25 in. Plate total thickness 0.15 in. Delamination length 2.5 in. Table 5.4 Buckling load delaminated plates with different thickness ratios of sublaminates (Pavol.=1,250 lb). |

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43
3.5 G from Energy Densities Consider the J-Integral along Path 1 in Figure 3.5. Along this path n^= -1 and nz=0. Hence the Integral can be written as: N y(I) = J(-Â£/0 +oxux + Tzxwx)ds (3.12) M We will add and subtract to the integrand in Equation (3.12): N N J(1) = J(-t/0 + The term u,z can be identified as the rotation at the crack tip, \j/, which is common to all the three paths 1, 2 and 3. Further the second and third terms in the first integral in Equation (3.13) equal to 2U 0 Hence Equation (3.13) can be written as: N N 7(l) = ju0ds y/t jr^ds (3.14) M M Equation (3.14) can be further simplified as: Jm =ULm (3.15) where U 1 and Vl are the strain energy per unit length and shear force resultant at the cross section 1. Similar results can be derived for paths 2 and 3 as follows: 64 4.5 Results of Linear Buckling Analysis In this section the effects of geometric imperfections, additional in-plane shear loads, boundary conditions and material property variations on the buckling load of the stiffened panel are discussed. The results are intended to shed light on probable reasons for the discrepancies between predicted buckling loads and corresponding experimental results. Furthermore, the effect of the assumed imperfections and the addition of in plane shear loads on the robustness of the design is also noted. 4.5.1 Effect of Geometric Imperfections and Shear Load A summary of the local buckling load factors with and without shear load, and with and without the initial bow type geometric imperfections (3 % of the panel length) obtained from PANDA2 is given in Table 4.2. The first row in Table 4.2 also includes a comparison of PANDA2 results and the STAGS results (both 480- and 411-Elements) for the perfect panel without the shear load. It may be noted that the PANDA2 results, both Koiter type analysis and BOSOR4* analysis, agree well with the STAGS 411- Element results. The buckling mode for the perfect panel obtained using STAGS 480- Element is shown in Figure 4.4. According to PANDA2 results, the lowest buckling load corresponded to local buckling, which suggests that the differences in boundary conditions between the analysis and the experiment will not have a large effect on the results. The 10% difference between the results for the 480-Element and the 411- ' BOSOR4 analysis routine in PANDA2 calculates local buckling load for the single panel module from BOSOR4- type strip theory.6 5 their computer program BUCLASP2. BUCLASP2 assumes that panel elements are orthotropic and have balanced laminates, material is linear elastic and thin-plate theory. Stroud and Agranoff [8-9] proposed a simplified analysis based on buckling of othotropic plates with simply supported boundary conditions. The global buckling analysis of stiffened panel was conducted as an orthotropic plate with smeared stiffeners, assuming as a wide column. In contrast with the simplified analysis, the VIPASA analysis code provides a high-quality buckling analysis that considers all buckling modes and ensures continuity of the buckling pattern across the intersection of neighboring plate elements. Stroud and Anderson [10] developed a stiffened panel design code PASCO, which uses VIPASA analysis as well as design techniques from the early simplified methods and expanded capabilities such as initial bow-type imperfection, bending moments, and temperature loading. The major known drawback of the VIPASA code is underestimation of the shear buckling modes when a buckling half-wave length is equal to the panel length [10,12]. To overcome inaccuracy involving shear load and anisotropy exhibited by VIPASA and PASCO, Anderson et al. [11] developed the computer program VICON. The analysis of VICON assumes that the deflection of the plate assembly can be expressed as a Fourier series, which can be used to calculate the forces at the longitudinal junctions between the plates by the same stiffness matrices that result from the VIPASA analysis. The total energy of the panel is expressed in terms of VIPASA stiffness matrices plus conventional stiffness of the supporting structure. Then the total energy is minimized subject to the constraints, using Lagrange multipliers, to obtain the governing equations. Bushnell [13-14] developed PANDA, an early version of PANDA2, where 96 MSC/PATRAN Veron 8 5 03 53 *5 t: Com: b**e Loadstep 0 Mode: 1. CnHcal Lo*d: 0 54160E*01. base.eKjOl mv Case. bane. Lod*p. 0. Mode- !. Crtficaf Loed O 54180E*01. hese.eig 01. Tianel#6o*iMNON-LAYERÂ£.D) (M 1.0000 [_ {NON-LAYERED) 9.3301 | 867-01 8 00-01 7.33-01 687-01 6.00-01 1 5.33-01 I 467-01 I 4.00-01 3.33-01 I 2.67-01 2 0001 133-01 667-02 1 1.1007 | d.1jH_Frin9# Max 1.00*00 (g-Nd 2542 Mm 0 Nd 1 ltoM&^0<^dS42 Figure 5.8 Mixed buckling mode from analysis (hl/h2=0.8). End-shortening Figure 5.9 Load versus end-shortening with various thickness ratios of sublaminates. 70 An examination of the STAGS model in Figure 4.3 shows that there is double counting of material between the blade stiffener and the flange-skin combination. This is because of the way the nodes are located in the blade elements and in the element that represents the flange-skin combination. Both elements have mid-plane nodes leading to an overlap equal to the thickness of the blade with a width equal to half-thickness of the flange-skin combination. This overlap is avoided in the modeling approach described earlier for the ABAQUS model. This additional material due to the overlap in the model is expected to increase the pre-buckling axial stiffness of the panel. In order to estimate this increase the STAGS model was modified similarly to the ABAQUS model, and the linear buckling analysis was performed on the modified model. The results are presented in Table 4.6 and are also compared with the experimental results. The increase in area due to the overlap contributed to about one half of the total stiffness difference between the analysis and test results. Table 4.6 Comparison of prebuckling stiffness and buckling load factors (STAGS results). EA/L (kip/in) EA (kip) Buckling load factor Model with overlap 3931.2 125,800 1.15 Model with no overlap 3684.4 117,900 1.23 Experiment 3453.9 110,500 1.09 110 G distributions for G/E (0 deg.) DCB specimens Figure 5.22 G distributions for graphite/epoxy (0 degree) DCB specimens. G distributions for G/E (90 degree) DCB specimens Figure 5.23 G distributions for graphite/epoxy (90 degree) DCB specimens. BUCKLING AND DELAMINATION ANALYSES OF STIFFENED COMPOSITE PANELS IN AXIAL COMPRESSION By OUNG PARK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999 ACKNOWLEDGEMENTS I wish to express my sincere appreciation to the members of my supervisory committee. Without the experienced academic advice, patience, encouragement of Dr. Bhavani V. Sankar, chairman of the committee, and Dr. Raphael T. Haftka, cochairman, this work would not have been possible. In addition to the their exceptional guidance, they gave me a wonderful opportunity to interact with other eminent scholars in various ways. Furthermore, they have provided the funding necessary to complete my doctoral study. Dr. Ibrahim K. Ebcioglu not only taught me several courses in solid mechanics but also always made me feel comfortable whenever I talked with him. Dr. Peter G. Ifju showed me his research enthusiasm in experimental mechanics. He always helped me whenever I needed his help. Dr. Fernando E. Fegundo, Jr., was willing to serve as a member of my advisory committee, reviewed this dissertation and gave me helpful suggestions and comments. My special appreciation extends to Dr. James H. Starnes, Jr. and Dr. Cheryl Rose of NASA Langley Research Center. Besides co-authoring and reviewing my publications, they helped me learn STAGS. I would like to acknowledge the great help obtained from Dr. David Bushnell, the developer of PANDA2 at the Lockheed Martin Co.; Dr. William H. Greene, a developer 11 of ABAQUS at HKS Co.; Dr. Mark Hillburger, Mr. Allen Waters at NASA Langley Research Center; and Dr. T. Krishnamurthy at Applied Research Associates. I am grateful to my professors and friends in the Department of Aerospace Engineering, Mechanics, and Engineering Science at the University of Florida, who have taught me and have inspired me during my study. I sincerely appreciate the Agency for Defense Development, where I devoted seventeen years of my young life. I extend my special thanks to Dr. Y. S. Lee, Dr. D. S. Kim, Dr. M. J. Shin, and many other supervisors and colleagues. I am deeply indebted to my beautiful wife (Mee), two children (Chan and Kyoung), and our parents for their endurance, support and prayers. in TABLE OF CONTENTS page ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 1.1 Background 1 1.2 Literature Survey 2 1.2.1 Buckling and Postbuckling Analysis of Stiffened Panels 2 1.2.2 Buckling and Postbuckling Analysis of Laminated Composite Plate with Delamination 12 1.3 Objective and Scope 16 2 REVIEW OF THE NONLINEAR FINITE ELEMENT METHOD 19 2.1 Nonlinear Finite Element Formulations Based on Continuum Mechanics 19 2.1.1 Equation of Equilibrium 20 2.1.2 Eigenvalue Buckling Prediction 21 2.2 Nonlinear Solution Methods 24 2.2.1 Newton-Raphson method 25 2.2.2 Arc-length Method 27 2.3 Solution Strategy 29 3 COMPUTATION OF ENERGY RELEASE RATE 33 3.1 Introduction 33 3.2 Strain Energy Derivative Method 36 3.3 Path Independent J-Integral 37 3.4 Zero-volume J-integral 40 3.5 G from Energy Density 43 3.6 G in terms of Crack-tip force 44 3.7 Virtual Crack Closure Technique 48 3.8 Extension to Delaminated Plates 50 3.9 Summary 54 IV 4 ANALYTICAL AND EXPERIMENTAL CORRELATION OF A STIFFENED COMPOSITE PANEL IN AXIAL COMPRESSION 56 4.1 Introduction 56 4.2 Stiffened Panel Definition 57 4.3 Test Specimen and Test Procedures 60 4.4 Linear Buckling Analysis 60 4.4.1 PANDA2 and STAGS 60 4.4.2 Finite Element Model 62 4.5 Results of Linear Buckling Analysis 64 4.5.1 Effect of Geometric Imperfections and Shear Load 64 4.5.2 Effect of Boundary Conditions and Material Properties 65 4.5.3 Summary of Differences between Design Model and Test Model 71 4.6 Nonlinear Analysis 72 4.6.1 Compressive Load versus End-shortening 73 4.6.2 Compressive Load versus Out-of-plane Deformations 76 4.6.3 Contact between the Panel and Loading Platen 77 4.7 Conclusion 83 5 BUCKLING AND POSTBUCKLING ANALYSIS OF A STIFFENED PANEL WITH SKIN-STIFFENER DEBOND 84 5.1 Introduction 84 5.2 Finite Element Model 85 5.3 Buckling Analysis Results of Multiple Delaminated Plate 88 5.4 Buckling and Postbuckling Analysis Results of Plate with Single Delamination 92 5.5 Buckling Analysis Results of Stiffened Panel with Debond 91 5.5.1 Buckling Analysis Results of Debonded Stiffened Panels 99 5.5.2 Postbuckling Analysis Results 105 5.6 Comparison of Energy Release Rate 107 5.7 Conclusion 112 6 CONCLUSIONS AND FUTURE WORK 117 REFERENCES 119 BIOGRAPHICAL SKETCH 128 v Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BUCKLING AND DELAMINATION ANALYSES OF STIFFENED COMPOSITE PANELS IN AXIAL COMPRESSION By OUNG PARK December, 1999 Chairman: Dr. Bhavani V. Sankar Major Department: Department of Aerospace Engineering, Mechanics, and Engineering Science The major objective of this study is to analyze buckling and delamination behavior of composite stiffened panels subjected to axial compression. First, a combined analytical and experimental study of a blade stiffened composite panel subjected to axial compression was conducted. The effects of the differences between a simple model used to design the panel and the actual experimental conditions were examined. It was found that in spite of many simplifying assumptions the design model did reasonably well in that the experimental failure load was only 10% higher than the design load. Several structural analysis programs, including PANDA2, STAGS, and ABAQUS, were used to obtain high fidelity analysis results. The buckling loads from STAGS agreed well with the experimental failure loads. However, substantial differences were found in the out-of-plane displacements of the panel. Efforts were made to identify the source of these differences. Implementing non-uniform load introduction with general vi contact definition in the STAGS finite element model improved correlation between the measured and predicted out-of-plane deformations. Next, a new method called Crack Tip Force Method (CTFM) is derived for computing point-wise energy release rate along the delamination front in delaminated plates. The CTFM is computationally simple as the G is computed using the forces transmitted at the crack-tip between the top and bottom sub-laminates and the sub laminate properties. Finally, buckling and postbuckling of a blade-stiffened composite panel under axial compression with a partial skin-stiffener debond are investigated. Two different finite element models, where nodes of the panel skin and the stiffener flange are located on the mid-plane or at the interface between skin and flange, are used. Linear buckling analysis is conducted using both STAGS and ABAQUS. Postbuckling analysis is conducted with STAGS. Comparison between the present results and previous buckling analysis results show a good correlation. Buckling analysis results for various stiffener geometries and debond ratios are presented. Vll CHAPTER 1 INTRODUCTION 1.1 Background Stiffened laminated composite panels have been considered for use in weight- sensitive structures such as aircraft and missile structural components. The main advantage of the stiffeners is the increased structural efficiency of the structure with a minimum of additional material. Due to the high stiffness of fiber composites, stiffened composite panels are usually thin. Thus, buckling characteristics are critical considerations for the optimum design of composite structures made of laminated composite plates. Buckling depends on a variety of factors, such as the geometry of the members, boundary conditions and material properties. Because of geometric complexity, stiffened laminated composite panels in axial compression have several buckling modes including general instability, local skin buckling and rolling of stiffeners. Furthermore, stiffened laminated composite flat panels usually exhibit stable postbuckling behavior which, in general, leads to significant differences between buckling load and ultimate failure load. The correct estimation of the load carrying capacity of stiffened panels is therefore very complicated. Advanced fiber-reinforced composite materials such as graphite-epoxy have relatively low transverse tensile and interlaminar shear strengths compared to in-plane 1 2 strength. Therefore, stiffened panels made of laminated composite are very susceptible to delamination. Delamination, also referred to as interface cracking or debonding where adjacent laminae become separated from one another, has been one of the major known weaknesses of laminated composite structures. Delamination, coming from initial manufacturing imperfections or in-service damage such as foreign object impact, can significantly reduce the stiffness and strength of the composite structures. It is also well known that delamination and skin-stiffener separation are common failure modes of a stiffened composite panel in axial compression. When a delamination is present, it is very important to identify not only its global influence on the load carrying capacity of the structure but also its local behavior under the applied load. Energy release rate has been accepted as a measure for predicting delamination propagation. Most available methods of computing energy release rate use 2-D or 3-D solid finite elements. However it is computationally very expensive to use solid elements for modeling entire complicated structures made of laminated composite materials. Thus simplified beam, plate and shell theories are frequently used for structural analysis of those structures. Therefore, it may be a good idea to also use the same theory to calculate energy release rate. 1.2 Literature Survey 1.2.1 Buckling and Postbuckling Analysis of Stiffened Panels Research on the buckling and postbuckling behavior of stiffened panels has been of interest for many years, with many researchers exploring the response of the stiffened 3 panels. However, due to their geometric complexity and the many parameters involved, a complete understanding of all aspects of behavior is not yet fully achieved. Several researchers [1-5] compiled surveys on buckling and postbuckling behavior of composite panels. Leissa [1] compiled extensive results on buckling and postbuckling behavior of laminated composite panels. Analytical effects of various boundary conditions, stacking sequences, and transverse shear deformation on buckling and postbuckling behavior were explored. Noor and Peters [2] reviewed two aspects of the numerical simulation of the buckling and postbuckling responses of composites structures. The first aspect was exploiting non-traditional symmetries exhibited by composite structures, and strategies for reducing the size of the model and the cost of buckling and postbuckling analyses in the presence of symmetry-breaking conditions (e.g., asymmetry of the material, geometry, and loading). The second aspect was the prediction of onset of local delamination in the postbuckling range and accurate determination of transverse shear stresses in the structure. Bushnell [3] divided the literature in the field of buckling of stiffened panel into three categories. One in which structural analysis is emphasized, a second in which optimum design is emphasized, and a third in which test results are emphasized. Recently, Knight and Starnes [4] reviewed some of the historic developments of shell buckling analysis and design and identified key research directions for reliable and robust methods in shell stability analysis and design. Bedair [5] presented an extensive literature review on stability of stiffened panel under uniform compression. He classified the literature into two categories, analysis and design. The objective of the first category is to develop numerical or analytical formulations to predict the global and local buckling load of structures. In doing so, several assumptions 4 are postulated in idealizing the structure in order to facilitate a solution. The objective of the second group of researchers is to develop simplified models to predict the ultimate strength, or collapse load of the structures. Several simplified models have been developed for that purpose. The main objective of this section is to review briefly previous works in the area of buckling and postbuckling behavior of stiffened composite panels. Surveys are divided into two categories, analysis methods and correlation of analysis and experiment. 1.2.1.1 Analysis methods The analysis of stiffened panel can be performed by either smeared or discrete stiffener approach. A smeared stiffener approach converts the stiffened plate into an equivalent plate with constant thickness by smearing out the stiffeners. This method provides accurate analysis results of global buckling load of stiffened panels when stiffeners are identical and closely spaced. However, this approach ignores the discrete nature of the structures and does not consider all potential buckling modes. The discrete stiffener approach does not have limitation of stiffener spacing and uniformity. In early studies of buckling analysis of stiffened panels, Wittrick and William [6] developed a general-purpose computer program, VIPASA, for determining the critical buckling stresses and natural frequencies of thin prismatic structures that consist of a series of flat plate connected rigidly along their longitudinal edges. The response of each plate element making up the stiffened panel is obtained from an exact solution of thin- plate theory. This approach is commonly referred to as exact finite strip method. The analysis used in VIPASA is similar to that which Viswanathan et al. [7] incorporated in 5 their computer program BUCLASP2. BUCLASP2 assumes that panel elements are orthotropic and have balanced laminates, material is linear elastic and thin-plate theory. Stroud and Agranoff [8-9] proposed a simplified analysis based on buckling of othotropic plates with simply supported boundary conditions. The global buckling analysis of stiffened panel was conducted as an orthotropic plate with smeared stiffeners, assuming as a wide column. In contrast with the simplified analysis, the VIPASA analysis code provides a high-quality buckling analysis that considers all buckling modes and ensures continuity of the buckling pattern across the intersection of neighboring plate elements. Stroud and Anderson [10] developed a stiffened panel design code PASCO, which uses VIPASA analysis as well as design techniques from the early simplified methods and expanded capabilities such as initial bow-type imperfection, bending moments, and temperature loading. The major known drawback of the VIPASA code is underestimation of the shear buckling modes when a buckling half-wave length is equal to the panel length [10,12]. To overcome inaccuracy involving shear load and anisotropy exhibited by VIPASA and PASCO, Anderson et al. [11] developed the computer program VICON. The analysis of VICON assumes that the deflection of the plate assembly can be expressed as a Fourier series, which can be used to calculate the forces at the longitudinal junctions between the plates by the same stiffness matrices that result from the VIPASA analysis. The total energy of the panel is expressed in terms of VIPASA stiffness matrices plus conventional stiffness of the supporting structure. Then the total energy is minimized subject to the constraints, using Lagrange multipliers, to obtain the governing equations. Bushnell [13-14] developed PANDA, an early version of PANDA2, where 6 buckling loads are computed by the use of simple assumed displacement functions used in conjunction with Donnell-type kinematic relation. Several types of general and local buckling modes were included. VIPASA, PASCO, and PANDA cannot perform nonlinear postbuckling analysis of the stiffened panel. Bushnell [15-17] released PANDA2, which incorporated nonlinear theory for prediction of behavior of locally imperfect panels. Nonlinear strain-displacement relations analogous to those developed in 1946 by Koiter for perfect panels were extended to handle panels with imperfections in the form of critical local bifurcation buckling mode. In PANDA2, local and general buckling loads are calculated with use of either closed-form expression [14] or with use of discretized models of panel cross-sections [15-17], The discretized model is based on one-dimensional discretization that is commonly referred to as approximate finite strip method. Approximate finite strip method assumes displacement of strip can be expressed with combination of crosswise polynomial function and lengthwise trigonometric function. Dowe and his group [18-22] have used the method in conjunction with a non-linear analysis based on first-order shear deformation theory. The Finite strip method, which lies between the conventional Rayleigh-Ritz method and the finite element method, provide a means of solving prismatic plate-structure problems with an attractive blend of accuracy, economy and ease of modeling [22], The finite element method is the most powerful method to predict the buckling and postbuckling behavior of structures. The historic developments of shell buckling analysis using finite element up to 1997 can be found in Knight and Starnes [5], With advancing computer power, the finite element method is widely used to solve various engineering problems. Many current commercially available finite element codes such 7 as MSC/NASTRAN [23] and HKS/ABAQUS [24] provide buckling and postbuckling analysis capability of the stiffened composite panels as one of several analysis options. Knight and Starnes [5] pointed out in their review paper that STAGS [25] is perhaps the premier shell analysis code that focused primarily on shell analysis and solution procedures for shell problems. 1.2.1.2 Analytical and experimental correlation Williams and Stein [26] examined J- and blade-stiffened graphite/epoxy panels experimentally as well as analytically using several analysis codes such as VIPASA, BUCLASP-2 and an early version of STAGS, which was based on a two dimensional finite difference technique. In their study, correlation of experimental and analytical results, which included inplane displacement restraints, indicated that the buckling strain of J-stiffened specimens were 75% to 80% of analytical values and that of blade- stiffened panels were 84% to 97% of the analytical values. A nonlinear response was exhibited by several of the specimens in which large lateral displacement in the order of one-quarter of the thickness of the panel plate segments were observed. Starnes and coworkers [27-28] investigated the postbuckling behavior of flat and curved stiffened graphite-epoxy panels loaded in compression. Panels with four equally spaced I-shaped stiffeners and quasi-isotropic skin were tested. Failure of all panels initiated in a skin-stiffener interface region. They showed that analytical results obtained from PASCO as well as STAGS correlate well with typical postbuckling test results up to failure. Their results also showed that modeling of the stiffener components with plate elements having appropriate stiffness is required to obtain satisfactory correlation with the postbuckling test results. 8 Romeo [29] conducted several tests on graphite-epoxy hat-and blade-stiffened panels under uniaxial compression and wing-box beams under pure bending to verify the accuracy of the theoretical analysis. Overall buckling, local buckling and torsional buckling were determined separately using a simple engineering formula, and interactions between these modes were not considered. Adequate correlation with experimental results was obtained for axial compression when the Euler or torsional buckling mode was critical; buckling occurred at lower strain values than predicted when the local buckling mode was critical. Furthermore, he showed that simple compression tests could not represent the load conditions of wing-box compression panel properly; in particular, the bending curvature causes a distributed load perpendicular to panels that could reduce the longitudinal load at which buckling occurred. Bushnell et al. [30] conducted optimum design, fabrication, and test of graphite- epoxy curved, locally buckled panels in axial compression. Three nominally identical large panels were tested. Two of three tests gave reasonably good agreement between test and theory, both with regard to loads at which the panels failed and the mode of failure. They also conducted experimental comparison between specimens with stitched skin-flange combination and specimens with adhesive-bonded skin-flange combination. They found that load carrying capacity of stitched specimens were lower than those of adhesively bonded specimens. Wieland et al. [31] investigated the buckling, postbuckling and crippling of AS4/3502 graphite-epoxy Z-section stiffeners as a function of specimen structural parameters. Variables considered were flange and web widths, flange-to-web corner 9 parameters. Variables considered were flange and web widths, flange-to-web comer radius, thickness and stacking sequence. Analytical model was based on a classical model on local buckling and the numerical analyses were conducted with the ABAQUS finite element code. The results showed that the nature of the load redistribution after buckling and its effect on postbuckling stiffness are related to the geometric variables. The agreement between the analytical and experimental buckling loads was generally good. The agreement degrades as the ratio of flange width with respect to web width is reduced and as the section comer radius is increased. Fan et al. [32] performed the pre- and post-buckling analysis for stiffened panels. Both the thick-wall stiffeners with square cross section and the thin-wall blade stiffeners were employed in their study. After linear buckling analysis, they used an incremental analysis along the load path with a special iteration technique, called initial value method to improve the normal Newton-Raphson method. The computational results of both displacement control and load control were presented. The computed local buckling load and buckling mode agreed well with the test results. However, the computed global buckling load with uniform end-shortening displacement was much higher than the test results. But, the computed global buckling load with load control was close to the test results. The reason of this discrepancy was not clearly explained. Three failure criteria of composite stiffened panel, which are maximum strain, debond failure, and combined global-local buckling criterion were proposed. Nagendra et al. [33] studied the optimum design of blade stiffened panels with holes under buckling and strain constraints. They used PASCO for buckling analysis and optimization with continuous thickness design variables and the Engineering Analysis 10 Language (EAL) finite element analysis code [34] for calculating strain and their derivatives with respect to design variables. Later their optimally designed panels with and without centrally located holes were tested and analytical and experimental results were compared focusing prebuckling behavior of the panel [35]. Prebuckling stiffness from test were about 10% lower than analytical values and failure loads from test was also 10% lower than that from design. Since the uncertainties in the geometric and material properties did not account for the discrepancy between analytical and experimental buckling loads, they hypothesized that geometric imperfections and eccentricities may had reduced the buckling load. Chow and Atluri [36] proposed failure criterion of mixed-mode stress intensity factors for the postbuckling strength of stiffened panel. They showed that post-buckling strength of the stiffened panels compare quite favorably with the experimental results of Starnes et al. [27] and the standard deviation of the error was less than 10 %. Young and Hyer [37] presented modeling procedures that predict the postbuckling response of composite panels with skewed stiffeners. Five panel configurations with various combinations of skin and stiffener orientation were tested. A uniform end shortening displacement was applied to the upper end of the panel in the axial direction, and the axial displacement of the lower end was restrained. The upper and lower ends were clamped and the unloaded sides were simply supported. The individual effect of shell elements, potted load introduction, material properties, and initial geometric imperfections was examined. The results showed that inaccurate modeling assumptions and anomalies in the test such as the support fixtures, the loading frame, and the load introduction of the test specimen could cause the predicted response 11 and the measured response to differ substantially. Dvila et al. [38] conducted progressive failure analysis for the simulation of damage initiation and growth in stiffened thick-skin stitched graphite-epoxy panels loaded in axial compression. Failure indices approach, proposed by Chang and Chang [39], was adopted to evaluate the failure mode and location corresponding to all of the major composite laminate failure modes except delamination. Superposed layers of shell elements with multiple integration points through the thickness were used to separate the failure modes for each ply orientation and to obtain the correct effect of bending loads on damage progression. The analysis results were compared with experimental results for three stiffened panels with notches oriented at 0, 15, and 30 degrees to the panel width dimension and found to be in excellent correlation with the experimental results. The local reinforcing effect of Kevlar stitches was simulated in the finite element model by multiplying the fiber buckling strength allowable value, independent of the other stress components, by a stitch factor that is determined empirically. A parametric study was performed to investigate the damage growth retardation characteristics of the Kevlar stitch lines in the panels. The debond between the stiffener flange and the skin were not modeled. Hence, the predicted results were found to be less accurate after the damage zone reached the stiffener flange. Singer et al. [40] presented conventional and less conventional experimental methods in buckling of a vast variety of thin-walled structures in considerable detail. The parameters, which may influence the test results, were systematically highlighted: imperfections, boundary conditions, loading conditions, effect of holes and cutouts. Though authors deals primarily with experimental methods and test results, the 12 theoretical concepts of the basic instability phenomena and numerical methods were also briefly reviewed. Sleight [41] analyzed a composite blade-stiffened panel with a discontinuous stiffener loaded in axial compression. A progressive failure analyses using Hashins criterion [42] was performed on the blade-stiffened panel. The progressive failure analysis and test results showed good correlation up to the load where local failures occurred. The progressive failure analysis predicted failures around the hole region at the stiffener discontinuity. The final failure of the experiment showed that local delaminations and debonds were present near the hole, and edge delaminations were present near the panel midlength. The progressive failure analysis results did not compare well to the test results since delamination failure modes were not included in this progressive failure approach. 1.2.2 Buckling and Postbuckling Analysis of Laminated Composite Plate with Delamination 1.2.2.1 One dimensional single delamination Chai et al. [43] developed a one-dimensional analytical model to predict through- the-width delamination buckling and growth based on Euler beam theory. Kardomateas [44] investigated effects of transverse shear and end fixity of delaminated composite by improving the model used Chai et al. A first-order shear deformation theory using variational principle was proposed by Chen [45], Yin and his group [46-48] investigated the effects of bending-extension coupling on postbuckling behavior. They evaluated strain energy release rate by using J-integral over a surface that encloses the 13 delamination boundary. Simitses et al. [47] developed a one-dimensional model similar to one used by Chai et al. [42] to predict critical loads for delaminated homogeneous plates with both simply supported and clamped ends. They showed that the buckling loads could serve in certain cases as a measure of the load carrying capacity of the delaminated configurations. In other cases, the buckling load is very small and delamination growth is a strong possibility, depending on the toughness of the material. Yin et al. [48] found that a delamination length is short and located near mid plane of the plate, the buckling load of the delaminated plate is close to the lower bound of the ultimate axial load capacity. When a delamination length is long and locates near surface of plate, the postbuckling axial loads can be considerably greater than the buckling loads, while the failure of plate may or may not be governed by delamination growth. The effects of bending-extension coupling as well as imperfection were investigated by Sheinman and Shouffer [49]. They found that the coupling effect reduces the load carrying capacity, and imperfection sensitivity of global postbuckling deformation is very high. Wang [50] proposed the concept of a continuous analysis for determining interface stresses and strain energy release rate for the delamination at the interface of skin and flange. A shear deformable beam finite element with nodes offset to either the top or bottom side was proposed by Sankar [51]. Kyoung and Kim [52] investigated asymmetric delamination with respect to the center of the beam-plate. In their study, a variational principle based on shear deformation theory was used to calculate buckling 14 load of orthotropic laminated beam-plate with through-the-width delamination. Recently, Gu and Chattopadhyay [53] carried out compression tests on graphite/epoxy composites plates with delaminations to evaluate the critical load and the actual postbuckling load-carrying capacity. They observed that composite laminates can carry higher loads after buckling. For the particular case they studied, the ultimate load is found to be as high as three times the buckling load. 1.2.2.2 One-dimensional multiple delaminations Kutlu and Chang [54-55] investigated the compressive response of composite laminates with multiple delaminations. They found that multiple delaminations can reduce the load-carrying capacity more compared to a single delamination. Suemasu [56-57] developed closed form solution for linear bifurcation buckling load based on energy method. He found that in the case of multiple delaminations, size and location significantly affect the buckling load. Lee et al. [58] proposed a layer-wise approach for computing the buckling loads and corresponding buckling mode shapes. It was found that the anti-symmetric buckling mode is dominant for a composite laminate having short multiple delaminations. They also addressed the effects of initial imperfection and anisotropy on buckling and postbuckling response of delaminated composite plates. An analysis procedure for determining the buckling load of beam-plates having multiple delaminations was also presented by Wang et al. [59], 1.2.2.3 Two-dimensional single delamination Whitcomb [60] studied delamination growth caused by local buckling in composite laminates with near surface delamination, using geometrically nonlinear finite 15 element analysis. He showed that delamination extension does not occur until buckling is significantly progressed. A plane finite element was developed by Gim [61] based on lamination theory that included the effects of transverse shear deformation. In the modeling of two-dimensional delaminations in laminated plates, the undelaminated regions was modeled by a single layer of plate elements while the delaminated region was modeled by two layers of plate elements with node offset. Sankar and Sonik [62] proposed a simple expression for the point-wise strain energy release rate along the delamination front using Irwins crack closure technique. They applied this technique in analyzing stitched double cantilever beam specimens as well as elliptic delaminations in composite plates. Klug et al. [63] investigated efficient modeling of postbuckling delamination growth using plate elements and gap elements. Energy release rate was computed using virtual crack closure technique. From this, a procedure to simulate a successive delamination growth was proposed. Kim [64] presented a modeling approach to study the postbuckling behavior of composite laminate with embedded delamination using two-dimensional shell element and rigid beam elements. 1.2.2.4 Two-dimensional multiple delaminations Suemasu et al. [65-66] analyzed the compressive behavior of plates with mutiple delaminations of different sizes. They showed that the effect of variation of the size of delamination on the compressive behavior is significant and postbuckling behavior is different from that of plates with equal sizes of delamination. Zheng and Sun [67] proposed a triple plate finite element model to analyze delamination interaction in laminated composite structures. Energy release rate was obtained by using virtual crack 16 closure technique. The compatibility conditions between interfaces of plates were imposed by multi-point constraint equation. The results for delamination interaction in a composite laminated circular plate under three point bending were obtained. Lee et al. [68] developed a nonlinear finite element code, DELAM3D, with a three-dimensional layered solid element based on an updated Lagrangian formulation. They simulated the compressive response of a laminated composite plate with mutiple delaminations. Contacts of delaminating interfaces, delamination growth, and fiber- matrix failure were also considered in their computation. Double cantilever beam (DCB) and end notched flexure (ENF) tests were conducted to verify the energy release rate. Test results with various crack numbers, size, location, and layer orientation compared well with the numerical results. 1.3 Objective and Scope The first objective of this study is to develop stiffened panel models that can be used to predict buckling and postbuckling behavior with and without delamination. The second objective is to investigate the delamination growth of the stiffened panel based on fracture mechanics using several methods of computing strain energy release rate. Among the several configurations commonly used for stiffened panels such as hat-stiffened panel, J-stiffened panel, and I-stiffened panel, blade stiffened panel has a simple geometry compared with other stiffened panels. Therefore, blade stiffened panel was chosen in this study. However, the present analysis method will be also applicable to any kind of stiffener configurations that have a flange attached to the skin. 17 Two finite element modeling approaches for stiffened panels are commonly used in the literature. One approach is to model the skin with plate elements and to model the stiffener with beam elements. The other is to model both skin and stiffener with plate elements. The second approach appears more attractive for modeling the debonded region between skin and flange. In this study, the panel skin and blade stiffeners are modeled with plate elements. The nodal penetration of the delaminated skin-stiffener interface can be prevented either by adjusting spring constants of fastener elements or by gap elements. Furthermore, an energy release rate for calculating delamination extension is computed using several methods based on fracture mechanics. In order to validate the present modeling approach, a plate with a through-the- width delamination was modeled and linear bifurcation and nonlinear postbuckling analyses were conducted. Results were compared with the available experimental results. In Chapter 2, basic finite element formulation for buckling problem and nonlinear solution algorithms for postbuckling analysis are briefly described. Chapter 3 provides several methods for computing energy release rate in plate-like structures based on fracture mechanics. In Chapter 4, effects of boundary conditions, material properties, and initial geometric imperfections on buckling and nonlinear prebuckling behavior of blade stiffened panel are investigated. Chapter 5 describes modeling of delamination with elastic spring element, linear buckling analysis results of both delaminated composite plates and debonded stiffened panels, and effects of delamination length, stiffener geometries and stacking sequences on buckling load. Strain energy release rate was computed using the strain energy derivative method, the virtual crack closure 18 method, and the crack-tip force method. Finally, Chapter 6 includes the summary of the present study, concluding remarks, and suggestions for future work. CHAPTER 2 REVIEW OF THE NONLINEAR FINITE ELEMENT METHOD 2.1 Nonlinear Finite Element Formulations Based on Continuum Mechanics Stiffened laminated composite panels deform continuously under compressive loads. In prebuckling stage, deformation and rotation can be considered as infinitesimal in general. Thus, prebuckling response of stiffened laminated composite panel is almost linear. Classical buckling analysis is generally used to estimate the critical loads of stiff structures such as the Euler column subjected to axial compression, which carry design loads by axial or membrane strength rather than bending strength. The out-of-the plane deformation before buckling is therefore almost negligible in the stiff structures. After buckling, stiffened laminated composite panels exhibit large deformation and rotation. Thus, nonlinear formulation is required in order to include the effects of large deformation and rotation. Many researchers [69-72] have efficiently implemented general nonlinear finite element formulations based on the principles of continuum mechanics. Two different approaches have been used in incremental non-linear finite element formulation. The first approach is generally called Eulerian or updated formulation where static and kinematic variables are referred to an updated configuration in each load step. The second approach is called the Lagrangian formulation, where all static and kinematic variables are referred to the initial configuration. The updated Lagrangian is more suitable for analysis of the structures 19 20 that involve very large deformations while the total Lagrangian is more convenient for analyzing the structures with moderately large deformations. The primary objective of this section is to review briefly the non-linear finite element formulation based on continuum mechanics. The detailed description can be found in References [69, 70]. 2.1.1 Equation of Equilibrium Using the principle of minimum total potential energy one can derive the finite element equations. Assume that there exists a total potential energy of the form for linear elastic analysis J{Â£}r {a)dV (Jv{}r {fh }dV + u}T[fs}dS) (2.1) where {u} is the displacement vector, [fb} and [/s] are body and surface force vectors. The relationship between stresses and strain is of the form: {<7} = [C]{Â£} (2.2) where [C] is the constitutive matrix. The condition of equilibrium requires that the first variation of the total potential energy vanish: S = \[Se]T {(j}dV -(Â¡{Su}T {fb)dV + js{Su}T {fs}dS)=0 (2.3) V V From Zienkiewicz [69], strain can be expressed in matrix notation as {Â£} = [Â£]{ll} = [fl0] + [flJ{M} (2.4) 21 where [B() ] is the matrix for the linear infinitesimal strain and matrix [Bi ] contains the nonlinear strain components. Using Equation (2.4) we can rewrite Equation (2.3) as: XI = {Su}T (J [B]{a}dV {\{fh }dV + J{/s }dS)) = 0 (2.5) V V s 2.1.2 Eigenvalue Buckling Prediction The stability criterion can be obtained from the second variation of the total potential energy. If the second variation of the total potential energy has a positive value, then a system is stable. Conversely, If the second variation of the total potential energy has a negative value then a system is unstable. Computing the second variation of total potential energy from Equation (2.5) as S2n = {8u }T (l 8[B? {a}dv + j [B? 8{aY dV) (2.6.1) V 82n={SuY {[8{B]t {(J}dv + j[B]T[C][B]]5{uYdV) (2.6.2) v From Zienkiewicz [69], the first integral of Equation (2.6.2) can generally be written as \v8[B]T{o}dV = [Ka]{8u) (2.7) where [Ka] is geometric stiffness matrix. Substituting Equation (2.4) into the second integral of Equation (2.6.2) and rearranging, the second variation of the total potential energy can be written as: 82U = {8uY[KT]{8u} (2.8) 22 In Equation (2.8), the tangential stiffness matrix [Kj ] can be written as [KT] = [Ka] + [K0] + [KL]. [K0] = j[B0?[C][B0]dV (2.9) v [KL] = \([B0]T[C][BL] + [BL]T[C][B0] + [BLf[C][BL])dV (2.10) v where [AT0] is the small displacement stiffness matrix and [TJ is the large displacement stiffness matrix. A critical point is obtained when the tangent stiffness matrix [KT] has at least one zero eigenvalue. The stability of an equilibrium configuration can be determined solving the eigenvalue problem at the current equilibrium state, [A:r]{(r)} = A(r){M equilibrium configuration associated with a given load level P is computed. Next the stability of tangent stiffness matrix is examined by computing the eigenvalue of the tangent stiffness matrix at given load level P. This method of determining the stability of a conservative system gives accurate results. However, it is computationally expensive because it involves the solution of a quadratic eigenvalue problem for the critical load. Linearized buckling analysis calculates critical buckling loads based on a linear extrapolation of the structural behavior at a small load level. Thus it is computationally inexpensive. From the fact that geometric stiffness matrix [Ka] and 23 large deformation stiffness matrix [KL\ depend on the load level P, linearized buckling analysis approximates the tangent stiffness matrix at given load level P as [40]: [/sTr(P)] = [^0] + -^-([^(AP)] + [^(AP)]) (2.12) AP where both geometric stiffness matrix and large displacement matrix are computed at small load level AP. If we assume that the critical load is equal to AP, then the condition for a singular point becomes a standard eigenvalue problem. There are two widely used numerical methods for extracting eigenvalues, the Lanczos method and the subspace iteration method. The Lanczos method is generally faster when a large number of eigenmodes is needed for a system with many degrees of freedom. The subspace iteration method is effective for computing a small number of eigenmodes. Based on the assumption that the displacements {u} are infinitesimal for the small load AP classical buckling problem further simplifies Equation (2.12) as: ([A'0] + A(r,[A: However the applications of Equation (2.13) should be limited in practical engineering problems. In order to avoid the erroneous computation of the stability points in real engineering applications, the stability problem should be investigated using full tangent stiffness matrix in Equation (2.11) [69]. 2.2 Nonlinear Solution Methods 24 The solution of nonlinear finite element problems includes a series of load steps as well as iterations to establish equilibrium at the new load level. In some nonlinear static analyses the equilibrium configurations corresponding to load levels can be calculated without solving for other equilibrium configurations. However, when the analysis includes path-dependent nonlinear conditions, the equilibrium relation needs to be solved throughout the history of interest. The solution may be obtained by using either the Newton-Raphson or the modified Newton-Raphson methods. The Newton- Raphson method requires evaluation of the tangent stiffness matrix at each iteration, which is computationally expensive. On the other hand, the modified Newton-Raphson method evaluates the tangent stiffness matrix at each load step, thus improving the computational efficiency compared to the Newton-Raphson method. However, the Newton type methods fail to provide a solution in the neighborhood of a global bifurcation or limit points when the tangent stiffness matrix becomes singular (see Figure 2.1). The arc-length method, proposed by Riks [73] and Wempner [74], and modified by Criesfield [75,76], is an effective solution procedure to search equilibrium path beyond the limit points. An important aspect in arc-length methods is that the load level is treated as a variable in addition to the unknown displacements at each iteration of a load step. Thus, an additional constraint equation comprising the displacements and loads is required to calculate the load level. 25 Figure 2.1 Nonlinear response from load versus displacement. 2.2.1 Newton-Raphson Method A system of nonlinear equilibrium equation can be written as Â¥ (m ) = / (u) f (2.14) where internal forces I(u) is defined as I(u) = j[B]{cj}dV (2.15) v Unbalance forces ^(m) represents the difference between internal and external forces. The basic problem is to find solutions that satisfy the nonlinear equilibrium equation, Viu )= 0. Since Equation (2.14) cannot be solved directly for the displacement of u, both an incremental equation of equilibrium from Equation (2.14) and iterative 26 procedure are generally used for its solution. The Newton-Raphson method utilizes the first-order approximation of Equation (2.14) and can be written at load step n+1 as y (;;',)= 'r(<) + = o <2.i6> d u Here i is the number of iteration, and the tangent stiffness matrix is defined as ay a/ ^ A. 'j' du du (2.17) From Equation (2.16) we have the following iterative correction as (2.18) where Kj is the tangent stiffness matrix at the ith iteration. Thus the improved solution can be computed as u i+i n + l Un + l + Su'n + l (2.19) The convergence of the Newton-Raphson method is generally very fast. However, the cost of computation is usually high due to the calculation and factorization of the tangent stiffness matrix at each iteration. To reduce the burden of computational cost, the modified Newton-Raphson was introduced in which the stiffness matrix is approximated as a constant: A!-). ~ Kt. There are many possible choices of the approximated stiffness matrix Kt. For instance, Kt can be chosen as either the matrix corresponding to the first iteration or some previous load step. Schematics of Newton-Raphson method and modified Newton-Raphson method are shown in Figure 2.2. 27 Am, n Am n ^ Figure 2.2 Schematics of Newton-Raphson and modified Newton-Raphson methods. 2.2.2 Arc-Length Methods The basic feature behind the standard arc-length method is that load level A is treated as a variable rather than constant during a load increment. Thus the governing equilibrium equation (2.14) can be rewritten as (2.20) with (2.21) where A u n is the total incremental displacement vector at the nth iteration and A An is the total incremental load factor at iteration n. Since the load level is treated as a variable, we need an extra equation that constrains the iterative displacements to 28 follow a specified path towards a converged solution. The constraint equation proposed by Riks [73] may be expressed as: (Aun)T(Aun) + A2A/07o = A/2 (2.22) where A / is a user defined incremental length in the space of n+1 dimensions. Criesfield [75] has proposed a modified constraint equation that includes displacement components alone as: (AJr (AO = A/2 (2.23) It is possible to add the constraint equation (2.23) to the system of equation directly and the iterative incremental method could be used again, however this may destroy the symmetry and banded structure of the equilibrium equation. Hence Criesfield [75] proposed an indirect approaches to avoid this difficulty. In their approach, the displacements at a given iteration i is written as: Su ; = K ,,(A X, + sx '. ) = K i'i'V :(4l) SX /) (2.24) = -'(ai;) + sx ; (Sun), s[ f'/ where 8ir'n are the iterative displacements corresponding to the residual forces VF(J+1, and the tangent stiffness matrix K 71 is formed using modified Newton-Raphson at the beginning of each increment and kept fixed for all iterations within the increment. By substituting equation (2.24) in which 8 is still undetermined into the constraint equation (2.23), we have (A + 8un)( Au-1 + 8un)= A l2 (2.25) 29 Expanding Eq. (2.25) gives a quadratic equation for the unknown iterative load factor^/l The details of this solution procedure are given in Ref. [75]. . Figure 2.3 One-dimensional interpretation of spherical arc-length procedure. 2.3 Solution Strategy The buckling analysis provides information about the load level at which bifurcation occurs. In some cases the structure withstand far above the buckling load without significant damage. In other cases the structure collapses well below this load due to imperfection sensitivity. Stability loss at a bifurcation point occurs only if the corresponding deformation mode is not contained in the deformation mode for 30 arbitrarily small loads. For example, a flat plate with in-plane loading exhibits no lateral displacements in the pre-buckling range. Thus it does not contain the lateral displacement modes of the buckled plate. Likewise, if the structure as well as loading is symmetric about some plane, all deformation modes antisymmetric with respect to that plane are possible bifurcation buckling modes. In such cases, we may choose to perform a buckling analysis with a nonlinear basic stress state. Sometimes when a bifurcation point does not exist at all, the bifurcation buckling approach may still considered as an acceptable measure of the critical load. If the structure is statically indeterminate and thus allows favorable redistribution of the stresses (e.g., shells with cutouts), then the bifurcation approach is too conservative. If the stiffness of the shell deteriorates with increasing load (e.g., long cylinders under bending), the bifurcation approach gives unconservative results. The bifurcation buckling analysis with a linear stress state is probably a good approximation for any case in which the squares of the rotations in the linear solution are small in comparison to the membrane strains at the load level corresponding to bifurcation. Most of commercially available finite element codes provide an option to perform the stress analysis first, save the data for a certain number of load steps on tape and later decide for which of those load steps buckling loads should be obtained [23- 25] This option may save some computing time. First, it may be easier to decide on the load levels at which eigenvalues are desired after the results of the stress analysis have been conducted. Next, it makes possible to find additional eigenvalues in a subsequent run. When eigenvalues are computed in a later run, the data deck for the nonlinear prestress analysis can be used. In addition, the user has the option to select certain data 31 sets saved on tape for eigenvalue solution. Changes in boundary conditions are also permitted at this time. In most practical applications one range of eigenvalues is particularly important, especially to a sequence of the lowest eigenvalues. If there exists a symmetry plane, in loading as well as in geometry, the size of the problem can be reduced and significant savings in the total computational effort can be achieved. If the structure on one side of the symmetry plane is considered, only the frequencies of symmetric modes are obtained. If the eigenvalue analysis is based on a nonzero prestress analysis with symmetry conditions and an eigenvalue analysis with boundary conditions corresponding to anti-symmetry. The eigenvalue approach for bifurcation buckling analysis with linear stress state is slightly more complicated than the vibration problem because eigenvalues can be negative as well as positive. Often the analyst is only interested in one eigenvalue, the lowest positive one. If the analysis is performed without a shift, it may happen that only negative eigenvalues are found because these are smaller in magnitude. In that case, the analysis has to be repeated with a positive shift. In choosing the shift for a repeated run the user can utilize the fact that the smallest positive eigenvalue is larger in magnitude than the largest of the negative eigenvalues that were found. Sometimes the buckling loads are symmetric with respect to zero. This is the case, for example, if a plate or a cylinder is subjected to uniform a shear load. It may often be advisable to request more than one eigenvalue also in bucking analysis. If the structure shows insufficient strength and only the lowest eigenvalue and corresponding mode are known, reinforcements may be introduced that have little effect on secondary buckling mode with the eigenvalue below the design load. 32 Nonlinear analysis is computationally expensive compared to linear analysis. In order to get a sound analysis results from nonlinear analysis, analyst should have better insight into the behavior of the analysis model. It is usually possible to save time as well as computer cost by preliminary determination of approximate values for the buckling load, under negative as well as positive load, before a large scale analysis is carried out. A linear analysis with a rather coarse grid will give some idea about the stress distribution and verify the nature of the behavior prior to executing nonlinear analysis with refined model. The size of the finite element model should be determined based on the requirement of accuracy, the efficiency, and the time constraint. Prior knowledge of the geometric modeling will increase the efficiency of an analysis. Type of element and size of element should be carefully selected to obtain high accuracy. Further, analyst should identify the type of nonlinearity and localize the nonlinear region for computational efficiency. To identify the type of nonlinearity, it is also helpful to examine the deformed shapes at various stages of loading (pre-buckling, critical buckling, limit load, and postbuckling) [25], CHAPTER 3 COMPUTATION OF ENERGY RELEASE RATE 3.1 Introduction Fracture mechanics concepts have been successfully applied to predict the loads which initiates the delamination extension, and also for predicting their stability. The energy release rate G has been accepted as a measure for predicting delamination propagation. In the context of fracture mechanics, the delamination extension is assumed to occur when the computed G is greater than the experimentally determined critical energy release rate Gc. Most of the available methods of computing G use 2-D or 3-D solid elements. However, It is computationally very expensive to use solid elements for modeling the entire complicated structure of an aircraft or an automobile. For example, consider 2-D plane strain elements for computing stress intensity factor to estimate the strain energy release rate of a double cantilever beam. A fine mesh must be used around crack-tip in order to capture the stress gradient ahead of crack-tip. Figure 3.1 shows the amount of mesh density required to calculate the stresses to compute the stress intensity factor. Thus a simplified beam, plate or shell theory are frequently used for structural analysis of complicated structures. Therefore, it may me a good idea to use the same theory to calculate the energy release rate also. There are three methods commonly used for computing energy release rate using 2-D or 3-D solid elements. These are Strain 33 34 Energy Derivative Method (SEDM), J-intergral, and Virtual Crack Closure technique (VCCT). SEDM, first proposed by Dixon and Pook [77], evaluates the change of potential energy as a crack progresses. Implementation of SEDM in a finite element analysis is straightforward [78]. However, it gives only an average value of the energy release rate along the delamination front. Further, this method requires two computations of potential energy, before and after crack propagation. A direct evaluation of energy release rate requiring only a single computation was proposed by Rice [79]. This involves the calculation of an integral on an arbitrary path surrounding the crack tip. This integral, known as the J-integral, is path independent. The Virtual Crack Closure Technique (VCCT), proposed by Irwin [80], is a method for computing energy release rate for self-similar crack extension. This method assumes that the strain energy release during the crack extension is equal to the work required to close the opened crack surfaces. Many investigators [51-68] have proposed VCCT for computing energy release rate using beam and plate elements. Based on plate theory, Sankar and Sonik [62] proposed the Point-wise Strain Energy Density Method (PSEDM). PSEDM suggests that the point-wise strain energy release rate along the crack front is the difference between strain energy densities behind and ahead of crack front. In this chapter a new method called Crack Tip Force Method (CTFM) based on plate theory is introduced. The application of the various methods of computing G for laminated composite structures is discussed. 35 Figure 3.1 Stress field (Gyy) near crak-tip of double cantilever beam using 2-D plane strain elements. 36 3.2 Strain Energy Derivative Method The strain energy derivative method utilizes the change in total strain energy, U, with change in crack length from a to a +Aa (see Figure 3.2). The energy release rate can be obtained in a straight forward manner for the case of displacement control as G = - dU dA Ua+Aa~V0 AA const deflection (3.1) where AA is the increase in crack area due to change Aa in crack length. Load Control Figure 3.2 Strain energy derivative method. 37 As shown in Equation (3.1), this method needs two analyses to compute energy release rate and A a should be small enough to obtain accurate results In the case of load control, the expression for G is modified as G dU dA u^-ut AA const force (3.2) 3.3 Path Independent J-Integral Consider a homogeneous body of linear or nonlinear elastic material without singularity shown in Fig. 3.3. The strain energy density Uo is defined as Â£ U0 =U(e) = \crijdeij (3.3) o where <7 y is the stress tensor and Â£y is the infinitesimal strain tensor. The J- integral to compute the energy release rate G is defined as [79] J = \ (u onx <7 ij" ji,x)ds i = 1,2; = 1,2 (3.4) r where w, is the displacement, n, are the direction cosines of the outward normal along the path T The indices, i and j or x and z are used interchangeably for convenience. Further, summation is performed over repeated indices. An application of divergence theorem gives: J = $r(USH ~ .,)>* = A cr,,u:.AdA (3.5) Differentiating the strain energy density, 38 w0 at/0 agg (3.6) 3* dx The area integral in Equation (3.5) vanishes. Therefore Equation (3.4) is equal to zero for any closed contour V In order to understand the zero volume integral described in subsequent section easily, consider the conservation integral around a region with singularity as shown in Fig. 3.4. The J integral along closed paths T, through r4 surrounding crack-tip vanishes as J = (3.7) r, + r2 + r3 + r4 But (J Â¡ji = 0 and nx = 0 along path T2 and T4 Thus the integral along r, clockwise and the integral along T3 counterclockwise sum to zero. r, r From equation (3.8) we can show that J integral along path T, and J integral along path r3 have the same value: 39 J = Â¡(U 0nx o ijnjui x)ds = \ (U 0nx o ijnjui x)ds r, r 3 = j(U mx- a a171 jui,x)ds r3 where normal vector m x is in the opposite direction with respect to normal vector n x in path T3. Figure 3.3 Conservation integral around a region with no singularities. x Figure 3.4 Conservation integral around a region with singularities. 40 3.4 Zero-Volume J-Integral Consider a portion of the delaminated beam as shown in Fig. 3.5. The beam example is used to minimize the complexity of derivation but this method can be extended easily to delaminated plates. It can be assumed that the delamination length or crack length a is much larger that the thickness h of the thicker sublaminate. If the path of the integral ABCDEF is away from crack-tip, then the beam theory stresses along this path are reasonably accurate compared to exact elasticity solutions. Further, the J- integral will vanish along the two horizontal paths BC and DE. Thus the integral is given as the sum of integrals along the three vertical paths: AB, CD and EF. Next, it will be shown that these vertical paths can be moved near the crack-tip without losing any computational accuracy of G. The vanishing of the J-integral around a closed path in an elastic material under small strain assumptions is a consequence of the two differential equations of equilibrium satisfied by the stress components: dx dx - + + iLi dz = 0 (3.10) The stress field in a laminated beam given by the shear deformable beam theory may not be accurate near the crack tip, however the stress components satisfy the 41 equilibrium equations exactly. This is because that the transverse shear stresses T xz in the beam are computed actually by substituting for <7 ^ and then integrating the first equilibrium equations. Thus, the first of Equation (3.10) is satisfied. The shear stresses at a cross section are proportional to the shear stress T xz that is constant along each ligament of the delaminated beam as well as the intact beam ahead of the crack tip. Thus the shear stresses T Iz are independent of x in each of the sublaminates and the first term in the second equilibrium equation is zero. Since beam theory assumes that (7 a are negligible, the second term is also zero. Thus, second equilibrium equation is also identically satisfied. Then the J-integral evaluated using beam theory around the closed path ABCDEF in Figure 3.5 is identically equal to zero. Further if we decompose delaminated beam with three sublaminates as shown in Figure 3.6 and consider J- integral for Sublaminate 2. Because the integrals along the horizontal paths are zero, it can be shown that Jab=^hg- Similarly it can be shown that J CD = J KL and J ef = J mn Thus it is now possible to move the three vertical paths AB, CD and EF to near the crack tip (HG, KL and MN) without loss of accuracy. The J-integral evaluated around the Paths 2,3, and 1 (HGKLMN) near the crack tip in Figure 3.5 has been called the zero-volume J-integral or zero-area J-integral, and it is given by G = y(l) + y(2) + y(3) (3.ii) where superscript (1), (2), and (3) represents the paths 1,2, and 3, respectively. 42 V, Figure 3.5 Force and moment resultants in a delaminated beam. J 2 ~ J AB J BG JGH ^HA ~ ^ JBG JHA ^ J AB = ~ J GH J AB ~ J HG Figure 3.6 Three sublaminates of delaminated beam. 43 3.5 G from Energy Densities Consider the J-Integral along Path 1 in Figure 3.5. Along this path n^= -1 and nz=0. Hence the Integral can be written as: N y(I) = J(-Â£/0 +oxux + Tzxwx)ds (3.12) M We will add and subtract to the integrand in Equation (3.12): N N J(1) = J(-t/0 + The term u,z can be identified as the rotation at the crack tip, \j/, which is common to all the three paths 1, 2 and 3. Further the second and third terms in the first integral in Equation (3.13) equal to 2U 0 Hence Equation (3.13) can be written as: N N 7(l) = ju0ds y/t jr^ds (3.14) M M Equation (3.14) can be further simplified as: Jm =ULm (3.15) where U 1 and Vl are the strain energy per unit length and shear force resultant at the cross section 1. Similar results can be derived for paths 2 and 3 as follows: y(2) = u l(2) y/,V2 J0) = -uL0)+ 44 (3.16) It should be noted that in Equation (3.16) switches signs because of change in the sign of nx from -7 to +/ for the Path 3. Adding all the three integrals and nothing that the shear force resultants must satisfy the equilibrium condition Vj+V2 = Vj, we find that J = y<'> + y(2) + y(3) = U y + U [2) U [3) (3.17) Thus the energy release rate is the difference between the strain energy densities just behind and just ahead of the crack tip. The strain energy density in the context of beam refers to strain energy per unit length of the beam, U L . 3.6 G in terms of Crack Tip Force Consider a very small segment of the beam of length 2 A* surrounding the crack-tip (see Figure 3.3). It will be convenient to shift the xz coordinates such that the xy plane coincides with the plane of delamination. Further, we will divide the laminate into 4 sub-laminates 2 behind and 2 ahead of the crack tip as shown in Figure 3.7. Let 45 the force and moment resultants near the crack tip in any sub-laminate be represented by a column matrix F such that F_ = [ P, M V ] where P, M and V are the axial force, Z Figure 3.7 Sub-laminates in a delaminated beam and the coordinate system. bending moment and shear force resultants. An underscore denotes matrix and a superscript T denotes transpose of a matrix. It should be noted that the force and moment resultants are resolved about the x-axis, which lies in the delamination plane. Thus there is an offset between the laminate mid-planes and the xy plane. The force resultants in each sublaminate are denoted by F], F2, F3 and F4. The compliance matrix of the top and bottom sub-laminates will be denoted by C, and The deformation in a sublaminate is then given by e = C_ F_ (3.18) where the deformations are defined by: 46 * = [Â£^x,ya\ (3.19) where the components of the deformations are the strain along the x-axis (not the sub laminate mid-plane), rate of change of rotation and the transverse shear strain, respectively. The force resultants are related by the equilibrium conditions Â£l + h.= h.+0.20) Further, since the sub-laminates 3 and 4 are intact (not delaminated) the deformations in them should be identical, i.e. e3 = e4 and hence ChF, = C,F4 (3.21) If F] and Â£2 are given, then Fj and F_4 can be calculated using Equations. (3.20), and (3.21). The strain energy per unit length in any sub-laminate is given by: Vi=\l TCF (3.22) Substituting Equation (3.22) into Equation (3.17) we obtain G=^fJC,F* +F_lCl,F2-F_Â¡CbF, 1 T -fIc.f 2 (3.23) Using the relations in Equations (3.20) and (3.21) an interesting expression for G can be derived as G = F_r, )(C, + CXF4 F,) (3.24) The term (F_4 F_,) is actually the force transmitted through the crack tip between the top and bottom sub-laminates, and can be called the crack-tip force, F_c. If a rigid link 47 is used to connect the top and bottom crack tip nodes in a finite element model, then the forces transmitted by the rigid link will be exactly equal to the above crack-tip forces. It may be noted that the crack tip force vector F_c have three components, an axial force, a couple and a transverse force, corresponding to each degree of freedom of the crack tip nodes, u, ^ and w. Another important implication of Equation (3.23) is that although there are 6 independent forces Pi, V,, M ,, P2, V2 and M 2 that can be applied to the two delaminated beam ligaments (see Figure 3.5), G depends only on three crack tip force components (see Figure 3.8). If the forces F_, and F_2 such that e, = e2 > i.e., C_,F_i C_bf^2 using Equations (3.18) and (3.19) one can show that F_, = F_A, and then G 0 If the forces on the top and bottom sub-laminates 1 and 2 are such that they produce conforming deformations (e_, = e_2), then the same forces act in sublaminates 4 and 3, respectively, producing conforming deformations (e3 = e4). Thus there is no need for any interaction between the top and bottom laminates at the crack-tip, and hence G = 0 . 48 2-D SOLID BEAM MODEL Figure 3.8 Crack-tip forces in beam 3.7 Virtual Crack Closure Technique The virtual crack closure technique has been used for plate and beam fracture problems by many researchers. We will derive the VCCT from the Crack-Tip Force Method. The expression for G in Equation (3.23) can be written as: G = J EJc (Q., (Z-4 ZL\) + C_h (F_2 F_3)) (3.25) where F_c is the matrix of crack tip forces, and Equation (3.18) is used in deriving Equation (3.25). Using the compatibility quation (3.19) in (3.24), we obtain g = ^eU-c,f, +CF2) (3.26) 49 Since CFdenote the deformations we can write (3.26) as G = - (3.27) In deriving the last term of the column matrix in Equation (3.27), we have used the fact that the beam rotation at the crack tip is same for both ligaments 1 and 2, i.e. . Multiplying and dividing the right hand side of Equation (3.27) by Ax where Ax is a small length used in the virtual crack closure method, we obtain G = 1 JjT F c 1 2Ax (Mo)-o))-(m2) -4) (y/w -i//(,))-(y/(2) \f/(,)) > (w(1)+/))-(/)-/)) (3.28) The superscript (t) in Equation (3.28) denotes displacements and rotation at the crack tip, and superscript (/) and (2) denote respectively the displacements of the top and bottom ligaments at a distance Ax from the crack tip. In deriving Equation (3.28) we have used the finite difference approximation of the type, u (i) . 0.x U Q U Ax (/) 0 (3.29) 50 Canceling the crack tip displacements in Equation (3.28) we obtain the equations for the virtual crack closure method as: G = '(C-O' (w(1)-w(2)) (3.30) 3.8 Extension to Delaminated Plates In the case of delaminations in a plate the energy release rate G varies along the delamination front. Formulas similar to Equations (3.10) (SEDM) and (3.22) (VCCT) were derived by Sankar and Sonik [62], In this section we will derive an additional result for G(s) similar to Equation (3.23) derived for beam. We can use the same notation as we used for beams with the understanding that there are eight force and moment resultants and eight deformation components: [FT] = [NX V, Nv Mx Uy Qx Qy\ [r ] = [Â£, e, ro /cv yu yv ] (131) The laminate compliance matrix [C] will be an 8x8 symmetric matrix, and it relates the force resultants and deformations: e = C F 51 (3.32) A B B D 0 0 0 0 K {f} where, the [A], [B] and [D] are the classical 3x3 laminate stiffness matrices and [AH is the 2 by 2 transverse shear stiffness matrix. In the context of plates the strain energy density is defined as strain energy per unit area of the plate and is given by 1 r U,=-FtCF 2 (3.33) A formula for G(s) similar to that in Equation (3.23) is given by G(J) = (1/ V>(i) + V 2,(i) -Ufui (3.34) where the superscripts denote the four sub-laminates behind and ahead of the delamination front, and s denotes the location of the point along the delamination front. Sub-laminates 1 and 4 are above the delamination plane, and 2 and 3 are below the delamination plane. From Equation (3.34) one can derive another expression for G(s) as: c = y (Â£l Fj )(C, + C)(Â£4 F,) (3.35) 52 The derivation of Equation (3.35) is very similar to that of Equation (3.23). As before, the term (F_4 F_, ) is the matrix of crack-tip forces. They also represent the jump in force and moment resultants that occur across the delamination front. Sankar and Sonik [62] showed that three of the eight force resultants will be continuous across the delamination front. Assume a coordinate system such that the x- axis is normal to the crack front, y-axis is tangential to the crack front and z- is the thickness direction. Then the continuous force resultants are N y,M v and Q v. Thus the jumps in these force resultants are zero, i.e., N{4) -N = N{y2) -N[3) = 0 M{4) M(l) = M(2) M(3) =0 1V1 y 1V1 y ivi y 1V1 y u (3 36) ~ Qf = Qf ~ Q = 0 Thus there will be only five components to the crack tip forces (see Figure 3.9): three forces in the x, y and z directions; two couples, about x and y axes, respectively. The three forces will be the jump in N x, N xy and Qx across the delamination front, either in the top laminates (1 and 4) or bottom laminates (2 and 3). The two crack tip couples are the jumps in M x, M ^ Since the jumps in N y,M y and Q y are equal to zero and they do not contribute to the crack-tip forces, we can delete the 2n(^, 5^ and 7^ rows and columns in Q and Q,; we will denote them by C, and Cb . 53 Figure 3.9 Crack-tip forces in delaminated plate. Then from Equation (3.33) an expression of point-wise energy release rate can be derived as: G(s) = ~Fl(CJ_ + C)Fc (3.37) where the crack tip forces F_c are given by: [Â£c J = [(AT a'!'1 ). (- v), (m;4) (AC-ACuer-ei1)] (i) (4) (3.38) The compliance matrices will take the form: 54 C r ''13 C, 4 C, 00 ^13 r ''33 C34 ^36 r ^38 C,4 r ^34 C44 C46 r ^48 Q r ''36 q6 r ^68 *-18 r ^38 Qs ^68 c ^88 (3.39) where the Cij are the coefficents of the full compliance matrix C, or Ch It should be mentioned that the xyz coordinates should be moved along the delamination front while using Equation(3.30) for computing pointwise G(s). We have derived three formulars Equations (3.27), (3.31), and (3.34) for computing the point-wise energy release rate in a delaminated plate. Out of the three, Equations (3.32) and (3.35) are exact, and their accuracy is limited only by the methods used compute the force and moment resultants or the strain energy densities ahead and behind the delamination front. The accuracy of the VCCT given by Equation (3.30) is limited by not only the crack tip forces but also the mesh size which will define the length of virtual crack growth. 3.9 Summary In this chapter a new method called Crack Tip Force Method (CTFM) was introduced and derived. Three methods of computing G for laminated composites structures are also discussed. A Crack Tip Force Method is derived for computing 55 point-wise energy release rate along the delamination front in delaminated plates. Actually the method can be derived from the Virtual Crack Closure Technique or the previously derived Strain Energy Density Method. However the CTFM is computationally simple as the G is computed using the forces transmitted at the crack- tip between the top and bottom sub-laminates and the sub-laminate properties. An evaluation of the aforementioned methods, their applicability to general laminates containing delaminations, and debonded stiffened panels will be presented in chapter 5. CHAPTER 4 ANALYTICAL AND EXPERIMENTAL CORRELATION OF A STIFEENED COMPOSITE PANEL IN AXIAL COMPRESSION 4.1 Introduction Buckling and imperfection sensitivity are expensive to calculate with general finite element models. Consequently, the optimization of stiffened panels often employs simplified models that are exact only for idealized geometries and boundary conditions (e.g., PASCO [10], or PANDA2 [15-17]). Nagendra et al. [33] studied the optimum design of blade stiffened panels with holes subjected to buckling and strain constraints. They used the panel analysis and sizing code (PASCO), based on a linked plate model, for the buckling analysis and optimization with continuous thickness design variables, and the Engineering Analysis Language (EAL [34]) finite element analysis code for calculating strains and their derivatives with respect to design variables. Later, the optimally designed panels with and without centrally located holes were tested, and analytical and experimental results were compared [35], Nagendra et al. [80] continued the optimum design study of blade stiffened panel using PASCO for analysis, and a genetic algorithm (GA) for the optimization of the panel laminate stacking sequences. Several designs obtained with GA were about 8% lower in weight compared to designs previously obtained with a continuous optimization procedure. 56 57 Recently, three of the panels designed by Nagendra et al. were fabricated and tested by the Structural Mechanics Branch at NASA Langley Research Center. The experimental failure loads differed by up to about 10% from the design load. However, there were significant differences in loading and boundary conditions between the design conditions and the test conditions. The principal objective of this chapter is to understand the effects of differences between the simplified assumptions made in the design model and the actual test conditions. Another objective is to assess the effectiveness of the simplified PASCO model originally used to design the panel, and this is aided by comparing the results obtained from several structural analysis programs including PANDA2, STAGS, and ABAQUS. 4,2 Stiffened Panel Definition The basic configuration of the panel designated as the baseline design - corresponds to the design in the 9th row in Table 7 of Reference [80], This panel, designated as GA2461 (referring to the design weight of 24.61 lb.), is 30-inches long and 32-inches wide with four equally spaced blade stiffeners (see Figure 4.1). The laminates used in Reference [95] for the skin, stiffener blade, and stiffener flange for the baseline design were balanced, symmetric laminates consisting of 0, 45 and 90 plies. The skin has 40 plies with a stacking sequence [45/90/1453/90/145,]s and the stiffener flange and blade have an identical stacking sequence of [45/(145/04)2 /902/04/(45/02)2/02/45]s with a total of 68 plies. Properties of the AS4/3502 graphite epoxy material used in Reference 5 are given in Table 4.1. 58 Table 4.1 Hercules AS4/3502 graphite epoxy lamina material properties. Youngs modulus (longitudinal) E, =18.50 x 106 psi Youngs modulus (transverse) E, =1.64 x 106 psi Shear modulus G,2 =0.87 x 106 psi Poissons ratio V12 = 0.3 Density p =0.057 lb in3 Ply thickness tpiy = 0.0052 in The baseline design was designed to support an axial load Nx of 20,000 lb./in. In addition, in order to account for off design conditions, imperfections and modeling inaccuracies, a shear load (Nxy = 5000 lb./in) and a longitudinal bow type (3% of the panel length) imperfections were added. The baseline design panel was assumed to be simply supported along the four edges, which is the only boundary condition that can be accurately modeled using PASCO. 59 32 N, N, | f* (J- | 1 l i IF m Fir 30 -* Y N, Figure 4.1 Blade stiffened panel with four equally spaced stiffeners under compression and shear loads. All dimensions are in inches. 4.3 Test Specimen and Test Procedures 60 The test specimens were fabricated from Hercules AS4/3502 unidirectional graphite/epoxy preimpregnated tape material. The skin (32-in. x 32-in.) and the stiffeners were cured separately in an autoclave. The stiffeners were machined to a length of 32 inches, and then bonded to the skin with FM-300 film adhesive. The panel edges perpendicular to the stiffeners were potted with an aluminum filled epoxy resin to prevent end failure. The length of the potted area was 1 in. on each side. Thus the effective gage length of the test specimen was reduced to 30 inches. The test specimen was loaded in compression using a 1,000,000-lb-capacity hydraulic testing machine. The specimen was flat-end tested without lateral edge supports, and no deliberate imperfection was introduced. Electrical resistance strain gages were used to monitor the strains, and direct current differential transformers (DCDTs) were used to monitor longitudinal in-plane and out-of-plane displacements at selected locations as shown in Fig. 4.2. All electrical signals and corresponding applied loads were recorded automatically at regular time intervals during the tests. 4,4 Linear Buckling Analysis 4.4.1 PANDA2 and STAGS The analyses performed in this study include both buckling and nonlinear postbuckling calculations. Linear buckling analyses for the baseline design were conducted by using both PANDA2 and STAGS. Input files for STAGS linear buckling analysis were generated by PANDA2. Next, the effect of the shear load and the DCDTs (1-7) for out-of-plane deformation of skin. DCDTs (8-11) for out-of-plane deformation of stiffener. DCDT (12) for end-shortening displacement 61 Figure 4.2 Layout of the displacement measurement instrumentation for the test panel. Figure 4.3 Reference plane of ABAQUS and STAGS model. 62 imperfection on the buckling loads of the baseline design were investigated using PANDA2, which employs analysis techniques with similar level of fidelity to that of PASCO. In PANDA2, local and general buckling loads are calculated by either closed- form expressions or by discretized models of panel cross sections based on an energy method [15]. STAGS is a finite element code for general purpose analysis of shell structures of arbitrary shape and complexity [25]. STAGS has a variety of finite elements suitable for the analysis of stiffened plates and shells. Four node quadrilateral plate elements with cubic lateral displacement variations (called 410- and 411-Elements) are efficient for the prediction of buckling response of thin shells. For thick plates in which transverse shear deformation is important, the assumed natural strain (ANS) nine node element (480-Element) can be selected [16]. The panel investigated here warrants the use of 480-Element, however 411-Element was also used as the panel was designed by PASCO, which does not model shear deformation. STAGS results were post-processed by PATRAN, which is a commercial software for pre- and post-processing of finite element simulations [82], 4.4.2 Finite Element Model The STAGS finite element model for the panel had a total of 20 branched shell units, and each branched shell unit had 65 x5 nodes (for the 32-in. long panel) or 61 x 5 nodes (for the 30-in. long baseline design panel), respectively. The axial compressive design load (640,000 lb) was applied with a uniform end-shortening constraint along with compatibility conditions for adjacent shell unit interfaces. In the test, the load was 63 introduced through the potted ends. To simulate this boundary condition, the displacement along the z-direction and the rotation along the x-direction were constrained at the nodal points in the potted region. The adhesive film used to bond the stiffener to the skin in the test panel was modeled by adding an isotropic layer to the model to simulate the bondline between the skin and flange with a thickness of 0.0121 inches. The skin middle surface was used as reference surface on which the nodes lie, and the offset distance of the middle surfaces of the skin-flange combination was modeled as an eccentricity. An alternative finite element modeling approach with ABAQUS suggested by Greene of HKS, Inc. was also used. In this method instead of locating the reference plane at the mid-plane of skin, the bottom plane of blade stiffener was taken as the reference plane. In order to handle the offset distance of the mid-plane of skin, as well as skin-flange combination, an additional 0 ply, with negligibly low stiffness was added to both skin and skin-flange laminates, as shown in Figure 4.3. Both the nine-node thin shell element (S9R5) and the four-node general shell element (S4) in ABAQUS were selected for the stiffened panel models. Both shell elements can account for transverse shear deformations and should therefore be compared to the 480-Element in STAGS. Instead of applying a compressive load at the panel end, uniform compressive displacements were applied at the nodal points along the loaded edge. In order to ensure a uniform state of stress along the entire panel length and also to prevent bending during the pre-buckling stage, the incremental boundary condition option available in ABAQUS was chosen. The buckling load factor was computed from the sum of the reaction forces at the boundary node set. 64 4.5 Results of Linear Buckling Analysis In this section the effects of geometric imperfections, additional in-plane shear loads, boundary conditions and material property variations on the buckling load of the stiffened panel are discussed. The results are intended to shed light on probable reasons for the discrepancies between predicted buckling loads and corresponding experimental results. Furthermore, the effect of the assumed imperfections and the addition of in plane shear loads on the robustness of the design is also noted. 4.5.1 Effect of Geometric Imperfections and Shear Load A summary of the local buckling load factors with and without shear load, and with and without the initial bow type geometric imperfections (3 % of the panel length) obtained from PANDA2 is given in Table 4.2. The first row in Table 4.2 also includes a comparison of PANDA2 results and the STAGS results (both 480- and 411-Elements) for the perfect panel without the shear load. It may be noted that the PANDA2 results, both Koiter type analysis and BOSOR4* analysis, agree well with the STAGS 411- Element results. The buckling mode for the perfect panel obtained using STAGS 480- Element is shown in Figure 4.4. According to PANDA2 results, the lowest buckling load corresponded to local buckling, which suggests that the differences in boundary conditions between the analysis and the experiment will not have a large effect on the results. The 10% difference between the results for the 480-Element and the 411- ' BOSOR4 analysis routine in PANDA2 calculates local buckling load for the single panel module from BOSOR4- type strip theory.6 65 Element (Table 4.2) is probably due to transverse shear deformation since the thickness of the skin-flange combination is 0.56 inch. Shear deformation was not included in the original panel design, and this difference indicates that that effect is substantial. The panel with negative bow-type imperfection had a concave surface in the middle of the panel. Thus, the blade tip is subjected to less axial compression and skin is under more axial compression than that of the perfect panels in the neighborhood of mid-length in the axial direction. Similarly, the blade tip near the boundary is under more compression and the skin near the boundary is under less compression than that of the perfect panel. The opposite holds for the positive bow-type imperfection. From Table 4.2, it is clear that the effect of the shear load on the buckling load is small, but the effect of the imperfection is very significant. From the last two rows of Table 4.2 one can note that a 3% positive imperfection results in a very low buckling load factor. The buckling load factor reduces from 1.256 to 0.394. A 3% negative imperfection also reduces the buckling load (from 1.256 to 0.856), but the reduction is smaller than that for a positive imperfection. It should be mentioned that a 3% imperfection is very large for a 32-in. stiffened panel, and thus will lead to very conservative designs. 4.5.2 Effects of Boundary Condition and Material Properties There were slight differences in material properties, panel dimensions and the boundary conditions between the baseline deign and the actual test conditions. In order to understand the effects of these differences, analyses were carried out using both sets of input data. The differences in material properties and dimensions are summarized in Table 4.3 and Table 4.4, respectively. While the changes in the material properties can I 66 be input directly, the differences in the thickness of the skin or flange are accounted by implementing a proportional change in the ply thickness in the model. A detailed discussion of this procedure can be found in Ref. 4. The results in Table 4.5 indicate that the effects of boundary conditions and material properties on the buckling load factors are not very significant. Comparison of the first two rows of Table 4.5 reveals the effects of changes in the boundary conditions. Similarly, results in the last two rows show the effects of changes in material properties and panel dimensions. The buckling mode shape of the baseline design (simply supported on 4 sides) predicted by STAGS is shown in Figure 4.4. The overall buckling mode shape obtained from ABAQUS (Figure 4.5) agrees well with that of STAGS. The computed lowest buckling load factor is slightly higher than that of STAGS (1.218 for ABAQUS vs. 1.168 for STAGS). This small difference may be due to modeling differences as discussed in the following section. The STAGS prediction of the buckling mode shape of the test panel with potted boundary conditions (other two edges being free) is shown in Figure 4.6. Table 4.2 Summary of the local buckling load factor from PANDA2 and the lowest buckling load factor from STAGS (4 edges simple supported). PANDA2 (Koiter analysis) PANDA2 (BOSOR4 analysis) STAGS (480 element*) STAGS (411 element) Loading combination with/without imperfection Panel end Panel mid-length. Panel end Panel mid-length. Nx=20,000 lb/in, Nxy=0 without imperfection 1.256 1.256 1.328 1.328 1.168 1.296 Nx=20,0001b/in, Nxy=5000 lb/in without imperfection 1.234 1.234 1.048 1.048 Nx=20,0001b/in, Nx>=5000 lb/in with imperfection (+3%) 1.234 0.356 1.048 0.346 Nx=20,0001b/in, Nxy=5000 lb/in with imperfection (-3%) 1.234 0.856 1.048 0.920 Nx=20,0001b/in, Nxy=0 lb/in with imperfection (+3%) 1.256 0.394 1.328 0.398 Nx=20,0001b/in, Nxy=0 lb/in with imperfection (-3%) 1.256 1.026 1.328 1.083 includes shear deformation. 67 Table 4.3 Material properties of baseline designs and test specimen. E(Msi) E22 (Msi) G12 (Msi) Vl2 V21 Skin Design 18.50 1.64 0.87 0.3 0.0270 Test specimen 17.333 1.64 0.8151 0.3 0.0284 Blade Design 18.50 1.64 0.87 0.3 0.0270 Test specimen 19.125 1.64 0.8994 0.3 0.0257 Flange Design 18.50 1.64 0.87 0.3 0.0270 Test specimen 19.593 1.64 0.9214 0.3 0.0251 Adhes ive Test specimen 0.5 0.5 0.192 0.3 0.3 Table 4.4 Geometric parameters of baseline design and test specimen. Panel Design Test specimen Panel length (in) 30 32 Panel width (in) 32 32 Blade height (in) 3.0705 3.0723 Skin ply thickness (in) 0.00520 0.00555 Blade ply thickness (in) 0.00520 0.00503 Flange ply thickness (in) 0.00520 0.00491 Adhesive thickness (in) 0.0 0.0121 68 Table 4.5 Buckling load factors obtained from STAGS with various boundary conditions and material properties. Boundary conditions shell unit (480 element*) shell unit (411 element) 4 edges simply supported (baseline design) 1.166 1.296 2 edges free, 2 edges clamped (baseline design) 1.197 1.340 2 edges free, 2 edges clamped (potted region, test material, 32-inch panel length) 1.154 1.286 * Indue es shear deformation. Figure 4.4 The STAGS predicted buckling mode shape of the perfect baseline design panel subjected to uniaxial compression (load factor=1.166). 69 IASC7PA7RANVreton 8 2 30-1^1-98 1952:12 FRINGE: Bucfcto, SIp2,lAx3l .EonValu*.20 <82: Olotmalbn, Diiptao*mn1s (VEC-WAG) ABAQUS 06FORMATION: Btcfcla. Stop2,lbtodI.EienV*kj=20.4a2 CMornafon, Dt^tocnwnl -ABAQUS Figure 4.5 The predicted buckling mode shape of the blade stiffened panel with 9-node shell elements from the ABAQUS (load factor=1.218). Figure 4.6 The predicted buckling mode shape of the blade stiffened panel from the STAGS (load factor=1.154). 70 An examination of the STAGS model in Figure 4.3 shows that there is double counting of material between the blade stiffener and the flange-skin combination. This is because of the way the nodes are located in the blade elements and in the element that represents the flange-skin combination. Both elements have mid-plane nodes leading to an overlap equal to the thickness of the blade with a width equal to half-thickness of the flange-skin combination. This overlap is avoided in the modeling approach described earlier for the ABAQUS model. This additional material due to the overlap in the model is expected to increase the pre-buckling axial stiffness of the panel. In order to estimate this increase the STAGS model was modified similarly to the ABAQUS model, and the linear buckling analysis was performed on the modified model. The results are presented in Table 4.6 and are also compared with the experimental results. The increase in area due to the overlap contributed to about one half of the total stiffness difference between the analysis and test results. Table 4.6 Comparison of prebuckling stiffness and buckling load factors (STAGS results). EA/L (kip/in) EA (kip) Buckling load factor Model with overlap 3931.2 125,800 1.15 Model with no overlap 3684.4 117,900 1.23 Experiment 3453.9 110,500 1.09 4.5.3 Summary of Differences between Design Model and Test Model 71 In summary, the PASCO model used for designing the panel had several modeling simplifications and compensating factors; their effects are listed in Table 4.7. The two major model simplifications were: 1. PASCO does not account for shear deformation, which is significant for thick composite panels, and this reduces the buckling load by 11%. 2. PASCO employs simple support boundary conditions. The difference due to boundary conditions was only about 1% because the buckling mode is local. To obtain a more robust design, the PASCO model was subjected to an additional shear load and 3% imperfection. Both had substantial effects on the design load. However, because the panel was designed only for imperfection of one sign, it became more sensitive to an imperfection of the opposite sign. Finally, because of the substantial thickness of the blade, it was also found that a centerline modeling, which is common in thin walled structures, produces about seven percent increase in the prebuckling stiffness, and a seven percent reduction in buckling load. The opposite effects are explained by the fact that the overlap draws more loads into the blade, which is the critical element. Overall, the more accurate analytical model predicts a buckling load higher by about 18% than the design load; however, this does not take into account any imperfections. The actual buckling load was only about 10% higher than the design load. It can be concluded that for this panel the simplified model used in PASCO, together with the shear and imperfection loading added for robustness, worked reasonably well. The other two designs that were tested also buckled at or slightly above the design load 72 Table 4.7 Differences between baseline design analysis and actual test panel analysis. Baseline design Test panel Effect on buckling load Loading (lbs/inch) Axial compression Inplane shear load 20,000 5,000 20,000 0 -20% Imperfection (initial bow type) -3% of panel length Not measured -30% (with shear load) -18% (w/o shear load) Boundary Condition Loaded edges Unloaded edges Simple support Simple support Clamped (potted) Unsupported 1% (with test material) Transverse shear deformation no yes 11% 4.6 Nonlinear Analysis Although the linear buckling loads provide a measure of the compressive load carrying capacity of the stiffened panels, the test results indicate that the panels underwent substantial nonlinear transverse deformations prior to failure. Hence it was decided to perform a nonlinear analysis in order to understand the effects of boundary conditions including the eccentricity in load application. The nonlinear analysis was started without applying any initial imperfection, but the differences in the stacking sequence and the differences in material properties between the skin and blades induce bending deformation. The modified Riks path following algorithm in STAGS was used for the nonlinear analysis. The computation time required for the nonlinear analysis was an order of magnitude higher than for the linear analysis, indicating significant nonlinearity (probably near the buckling load). In the following sub-sections we discuss the results of the nonlinear analyses, and make a comparison between experimental and analytical results. 73 4.6.1 Compressive Load Versus End-shortening The compressive load versus end-shortening deflection curves from the STAGS nonlinear static analyses, the test, and a linear fit (regression analysis) of the measured data are shown in Fig. 4.7. Recall that the test panel designated as GA2461 in Reference [80] is the baseline design for the present paper. In comparison to the baseline panel, two other test panels (GA2414, GA2458) from Reference [80] have slightly different geometries and stacking sequences. As expected, their compressive load versus end-shortening curves from the tests exhibit a similar trend except at the initial stage of loading. The prebuckling stiffness (EA) was calculated from the slopes of the linear portions of the experimental load versus end-shortening curves for the three panels and by multiplying the slopes by the panel length. The prebuckling stiffness from the experiments is compared in Table 4.8 with the prebuckling stiffness predicted using STAGS for test panel GA2461. It is observed that the prebuckling stiffness of the test panels is about 6% lower than the analytical value. Table 4.8 Comparison of the prebuckling stiffness and buckling load results from STAGS and experiments. EA/L (kip/in) EA (kip) Buckling load factor STAGS nonlinear (GA2461) 3931.2 117,900 1.23 Experimental Result (GA261) 3453.9 110,500 1.09 Experimental Result (GA2414) 3347.8 107,100 0.94 Experimental Result (GA2458) 3333.8 106,700 1.07 74 Load vs. End-shortening -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 E nd-shortening (inch) Figure 4.7 Compressive load versus end-shortening from analysis and experiments. Load vs. Out-of-Plane displacement (mid panel) Out-of-Plane Displacement (inch) Figure 4.8 Load versus out-of-plane displacements of the skin at the center bay. 75 Load vs. Out-of-Plane displacement (stiffener) 8.0615 - \ 6.0E+5I S' 5.OE+5I- o 4.0E+51 DCDT9, in. -i 3.0E+5 t DCDT10, in 2.0E+5 f DCDT 11, in DCDT 8, in. 1.0E+5 l O.OEiO 1 1 1 -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 Out-of-plane displacement (inch) Figure 4.9 Load versus out-of-plane displacements of the stiffener at selected location in Fig-2. Variation of Out-of-Plane Displacement Figure 4.10 Variation of the out-of-plane displacements along the panel lengthwise direction. 4.6.2 Compressive Load Versus Out-of -Plane Deformations 76 The layout of the DCDTs used to measure displacements in the test panel is shown in Figure 4.2. The out-of-plane displacements, measured from DCDTs at selected locations in Figure 4.2, are shown in Figures. 4.8 and 4.9. The results in Figure 4.8 show that out-of-plane displacements were initiated at an early stage of the loading and increase linearly in proportion to the loading. This observation suggested the possibility of loading eccentricities along the load introduction edge or rigid body rotation of the panel with respect to the clamped edge in addition to the effects of geometric imperfections. Figure 4.9 shows that the out-of-plane displacements of the blade stiffeners were also started at an early stage of the loading. Except DCDT 11, the out-of plane deformations were an order-of-magnitude lower than those of the skin in Figure 4.8. Furthermore, significant nonlinear response was only exhibited near the failure load. The large nonlinear response of DCDT11 throughout the axial loading was probably due to the effect of the unsupported side-edge boundaries. The out-of-plane displacement variations along the length of the panel (DCDTs 1-4 in Figure 4.2) at selected load levels are shown in Figure 4.10. The results in Figure 4.10 indicate that bending occurred in the test panel in addition to the end shortening due to the axial compressive load. The load versus out-of-plane displacements across the panel mid-length can be found in Ref. [83], In order to explain the substantial prebuckling bending, a combination of different geometric imperfections and loads applied at a small angle to the axial direction were analyzed to determine their influence on the observed out-of-plane displacements. The capability of STAGS to model geometric imperfections in the shape 77 of the buckling modes was used for these analyses. Various combinations of imperfection amplitudes and load angles were considered. Although for some combinations we could reproduce the test results partially [83], obtaining the right imperfection and the load introduction angle seemed elusive. This difficulty suggests that we look elsewhere for the source of the prebuckling bending. 4.6.3 Contact Between the Panel and Loading Platen Hilburger [97] investigated the effects of non-uniform load introduction and boundary condition imperfections on the compression response of composite cylindrical shells with cutouts. He defined the non-uniform load distribution as anything other than uniform axial displacement of cylinders loading surface and found two sources of non-uniform load introduction. One was due to lack of planarity in the loading surfaces of the specimen and the loading platens. The other source was due to tilt of the loading platen with respect to the specimen before the loading began. He measured the top and bottom loading surface imperfections as well as potting thickness. Then the imperfection data was fit to curves and input into the STAGS models. Furthermore, the test frame loading platen was modeled as rigid flat plates and generalized contact definitions* given in STAGS were used. A similar modeling approach was used in the present study in order to identify the causes of the substantial out-of-plane deformations in addition to nonlinear end shortening during the early stage of the test. The loading platen was modeled as a rigid * Generalized contact definition means that contact points are calculated by STAGS rather than specified by the user. 78 flat plate in the STAGS analysis. Because the loading surface imperfections were not measured before the test, they were not considered in this study. Instead, it was assumed that the rigid loading platen was initially contacted at the tip of the blade. Stiffeners were assumed to have a small tilt angle with respect to the load introduction edge of the test specimen, as shown in Figure 4.11. STAGS uses generalized contact definitions to check for contact and to construct actual contact elements coupling contacting points with contacted shell elements as the analysis progresses. In doing this, STAGS uses penalty functions to enforce a displacement-compatibility constraint between each contact point and each element with which it is in contact. STAGS utilizes analyst-supplied stiffness versus displacement information to compute the forces resulting from the small contact-surface penetration that may occur. A contact element is conceptually a nonlinear spring connecting the contact point to the surface of the contacted element. This nonlinear spring typically has a low stiffness and generates a small force when the contact-surface penetration is small, but it gets progressively stiffer and generates a larger force as the penetration increases [23]. Generalized contact definitions were implemented to the finite elements that simulate the rigid platen and the load introduction edge of the test specimen. The selection of proper stiffnesses of the contact elements for present analyses is rather arbitrary. Thus, several nonlinear analyses were performed to simulate the observed out- of-plane deformation response of the test specimen by changing both tilt angles and the stiffnesses of contact elements between the loading platen and load introduction edge. The combinations of the tilt angles and stiffnesses of the contact elements used for the 79 analyses are summarized in Table 4.9, and the corresponding load versus end shortening results are shown in Figure 4.12. The results in Figure 4.12 suggest that the computed end-shortening response strongly depend on the user-supplied input data in Table 4.9. The response of Model 9 was the closest to that of the test panel in Figure 4.12. The computed out-of-plane displacements of the skin and stiffeners of Model 9 at the selected DCDTs location are further examined as shown in Figures. 4.13 and 4.14, respectively. It is observed that the response of Model 9 in Figure 4.13 shows good correlation with those of the skin of the test panel in Figure 4.8 during the early stages of the load. However, Model 9 exhibits considerable nonlinear behavior of the out-of plane deformations of skin when the compressive load is above 400,000 lb, which was not present in the response of the test panel in Figure 4.8. Table 4.9 Summary of the tilt angles and the stiffness of contact element. Panel length (inch) Tilt angle (degree) Stiffness of contact element (lb/inch) Model 7 30 0.01 Disp. 0.005 0.05 0.1 0.2 1.0 Force 1.0e3 1.0e5 1.0e7 2.0e8 2.0e8 Model 8 30 0.01 Disp. 0.0001 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e7 1.0e8 2.0e8 2.0e8 Model 9 30 0.01 Disp. 0.005 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e5 1.0e8 2.0e8 2.0e8 Model 10 30 0.005 Disp. 0.005 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e5 1.0e8 2.0e8 2.0e8 The computed load versus out-of-plane deformation of the stiffener at location of the DCDT 11 shows significant nonlinear behavior in Figure 4.14. This significant nonlinear response was also observed from the measured response of the actual test panel in Figure 4.9. In general, the finite element model with the generalized contact definitions improved correlation between the measured and predicted out-of-plane 80 deformation. However, the details of the displacements are considerably different. It is concluded that some combination of tilt angles and the contact stiffnesses can produce the observed pattern, but there may be some other contribution to the out-of-plane displacements. i TILT ANGLE Figure 4.11 Schematic of blade stiffened panel and loading platen. 81 Load vs. End-shortening 0.00 0.05 0.10 0.15 0.20 0.25 0.30 End-shortening (inch) Figure 4.12 Compressive load versus end-shortening from analysis with contact models and experiment. Model numbers refer to Table 9. Load vs. Out-of-plane displacement Out-of-plane displacement (inch) Figure 4.13 Load versus out-of-plane displacements of the skin at the center bay from Model 9 analysis. 82 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Out-of-plane displacement (inch) Figure 4.14 Load versus out-of plane displacements of the stiffeners at selected locations from Model 9 analysis. 4.7 Conclusion 83 Analytical models using several structural analysis models were used to assess the adequacy of the design model and the correlation with experimental results for a stiffened panel designed using the PASCO program. Of the effects neglected by the simple model, shear deformation was the most important, accounting for about 11% difference in buckling load. The effect of simplified (simple support) boundary conditions was small. The addition to the design model of shear loads and imperfections to improve the robustness of the result did help, even though the inclusion of one-sided imperfection apparently induced sensitivity to imperfection of the opposite sign. Overall, the simplified model did produce designs that in the experiments failed slightly above the design load. The most significant difference between the analytical predictions and experimental measurements was the substantial out-of-plane pre-buckling deformations. To explain these differences, imperfections, load eccentricities, and loading platen tilt angles were considered. Of these the loading platen tilt produced similar patterns of deformation, but these had more nonlinear characteristics than the measured deformations. CHAPTER 5 BUCKLING AND POSTBUCKLING ANALYSIS OF A STIFFENED PANEL WITH SKIN-STIFFENER DEBOND 5.1 Introduction In contrast to most of simplified approaches discussed in literature survey in Chapter 1, finite element based approach can be applicable with high fidelity to more general and realistic structures. Therefore, the axial compressive behavior of a stiffened panel with skin-stiffener debond was explored in this chapter by employing finite element model. Two different finite element models, where nodes of the panel skin elements and the stiffener flange elements are located on the mid-plane of each element or interface between the skin elements and flange elements with offset, were used. The nodes corresponding to the top of the skin and bottom of flange are connected with either the elastic spring fastener elements or multi-point constraint equations. In order to verify present finite element modeling approach, laminated composite plates with/without the through-the-width delamination were first modeled. Both single delamination and multiple delaminations were considered. Then, stiffened composite panels with skin-stiffener debond were modeled. Stiffened composite panels with a single stiffener as well as multiple stiffeners were also considered. Buckling and postbuckling analyses were conducted using STAGS. Comparison was made with available buckling analysis results. Next, numerical examples of computing energy 84 85 release rate in the context of plate as discussed in Chapter 3 are given for predicting debond extension using the strain energy derivative method, the virtual crack closure technique, and the crack-tip force method. 5.2 Finite Element Model Two finite element-modeling approaches for the stiffened panel are commonly used in the literature. One approach is to model the skin with plate elements and to model the stiffener with beam elements. The other is model both skin and stiffener with plate elements. The second modeling approach appears more attractive for modeling debonded region between skin and flange, and it was chosen in this study. Furthermore, two different finite element models, designated as Model I and Model II, respectively, were also considered (see Figure 5.1). Both skin and flange elements of Model I have offset nodes with small gap at the interface region, while nodes in Model II are located on the mid-planes of skin and flange, respectively. In Model I, in order to satisfy compatibility conditions of intact interface nodes located directly above the skin and below the flange, each nodal degree of freedom was constrained with elastic spring fastener elements with very high spring constants. Nodes located on debonded interface between skin and flange were connected with elastic fastener elements, which have only an axial degree of freedom with very high stiffness in compression and zero stiffness in tension. This can prevent physically unrealistic nodal penetration between skin and flange during the postbuckling analysis. Friction against sliding of the debonded surface was not considered. In Model II, Multi-point constraints were imposed at the interface of intact skin and corresponding flange nodes to satisfy the 86 displacement compatibility condition as h h f us + ~Â£V'*,s = uf v + ^-w = v ^-w f * 2 Y y-s f 2 Y y' ws =wf (5.1) where u, v, and w are displacements in x, y, and z direction, *FX and % are rotations, h is thickness, and subscripts s and/ denote skin and flange, respectively. In the debonded region, the same modeling approach as Model I was used. One advantage of model I over model II is such that displacement compatibility conditions at interface nodes in Equation (5.1) is not necessary. The other advantage of model I is that computation of displacements behind crack tip is easy. However, Model I cannot handle multiple delaminations located through-the-thickness direction while Model II can. Model I Nodes are located on interface I I Model II Nodes are located on mid-plane of skin Figure 5.1 Two different modeling approach for blade stiffened panel. 5.3 Buckling Analysis Results of Multiple Delaminated Plate 88 A plane woven fabric glass fiber reinforced composite panel with three through-the-width delaminations located in the middle of the plate from Suemasu [56] (as shown in Figure 5.2) was examined using Model II approach. The material properties are shown in Table 5.1. The STAGS finite element model for the plate has a total of 4 branched shell units with four node shell elements (element 410), and each branched shell unit has 41 by 11 nodes. Total of 297 elastic fastener elements (element 130) was used to prevent penetration of contact surfaces in debonded region of the plate. The compressive stiffness of an elastic fastener element used in this study is 113 MN/m. In the intact region of plate, a total of 3168 constraint equations was used to satisfy the compatibility conditions of shell unit interfaces. Axial compressive load (2500 N) was applied to load introduction edge with uniform end-shortening constraint. In order to ensure a uniform stress condition for the entire panel length, an incremental boundary condition option was chosen to prevent axial bending during the prebuckling stage. The computed buckling load and buckling load from Reference [56] are given in Table 5.2. The first and second buckling mode shapes from Reference [56] and present analysis are shown in Fig. 5.3 and Fig. 5.4, respectively. It is seen that present buckling analysis results agree well with the corresponding buckling analysis results from Reference [56], However, the buckling load from the experiment is lower than the computed buckling load. Suemasu [56] indicated that insufficient clamped condition during his experiment might be responsible for low buckling load. 89 Table 5.1 Plane woven fabric fiber reinforced laminates material properties [Ref. 56]. Youngs modulus (longitudinal) E, = 20.2 GPa Youngs modulus (transverse) E2 = 21.0 GPa Youngs modulus (through-thickness) E3= 10.0 GPa Shear modulus (inplane) Gi2 = 4.15 GPa Shear modulus (through-thickness) Gi3 = 4.0 GPa Poissons ratio (inplane) V12 = 0.16 Poissons ratio (through-thickness) Vi3 = 0.3 Figure 5.2 Glass fiber reinforced composite specimen [Ref. 56], 90 Table 5.2 Comparison of buckling load for delaminated plates ( Pcr=2500 N). Buckling load factor Present Suemasu [56] Mode I 0.440 0.432 Mode II 0.801 0.795 N = 4,o = 0.25,ore ss 0.5 Ar = 0-432 Figure 5.3 Buckling modes of the plate with three delamaninations from Ref. [56]. (a) (b) Figure 5.4 Computed buckling modes using model II approach, (a) First lowest buckling mode, (b) Second lowest buckling mode. 92 5.4 Buckling and Postbuckling Analysis Results of Plate with Single Delamination In order to see the effects of the delamination location through thickness, a unidirectional graphite/epoxy laminated plate with single through-the-width delamination was considered (see Figure 5.5). The geometry and material properties used for the graphite/epoxy laminates are given in Table 5.3. Axial compressive load (1,250 lb) was applied to load introduction edges. Three different thickness ratios of top and bottom subraminates were considered and buckling load factors are summarized in Table 5.4. The computed buckling modes agree well with deformed shapes obtained from the test in Reference [53] (see Figures 5.6, 5.7). As expected, the delamination located near the surface of the plate has the lowest buckling load, where local buckling is the dominant buckling mode. As the thickness ratio of sublaminates increases, the buckling mode changes from local buckling to mixed buckling mode (see Figure 5.8) and then from mixed to global buckling mode. Thickness of top sublaminate (hi) Thickness of bottom sublaminate (h2) Figure 5.5 Graphite/epoxy laminated plate with single through-the-width delamination. All dimensions are inch. 93 TABLE 5.3 Graphite epoxy lamina material properties Youngs modulus (longitudinal) Ei =19.50 x 106 psi Youngs modulus (transverse) E2 =1.48 x 106 psi Shear modulus G12 =0.80 x 106 psi Poissons ratio V12 = 0.3 Plate length x width 5.0 x 1.25 in. Plate total thickness 0.15 in. Delamination length 2.5 in. Table 5.4 Buckling load delaminated plates with different thickness ratios of sublaminates (Pappl.=l ,250 lb). hl/h2=0.5 hl/h2=0.8 hl/h2=1.0 Buckling load factor 3.44 5.42 5.79 94 [msopai Es kTRAN Version 5 01 -Sep- 00 JM 07 Cm* beaeO. Loodstoc 0 Mode: 1. CrOcel Load 0 57858EMM. i)0 elg 01: Ceee beteO LoedMep: 0. Mode: I. Crtfcsl Load: O.STS9*E01. beee0.elg.0l TraneMMral-(NON-LAYERED) t.0000 ! sMionel (NON-LAYERED) 9.33-01 I \L. d^.*_FHnge M1 00*00 g-N; Mm 0 0*ld1 . Me* rOO^'N8M1 (b) Figure 5.6 Global buckling mode from test and analysis, (a) Buckling mode in mid-plane delamination [53]. (b) Buckling mode from analysis (hl/h2=1.0). 95 (b) Figure 5.7 Local buckling mode from test and analysis, (a) Buckling mode in near-surface delamination [53], (b) Buckling mode from analysis (hl/h2=0.5). 96 MSC/PATRAN Veron 8 5 03 53 *5 t: Com: b**e Loadstep 0 Mode: 1. CnHcal Lo*d: 0 54160E*01. base.eKjOl mv Case. bane. Lod*p. 0. Mode- !. Crtficaf Loed O 54180E*01. hese.eig 01. Tianel#6o*iMNON-LAYERÂ£.D) (M 1.0000 [_ {NON-LAYERED) 9.3301 | 867-01 8 00-01 7.33-01 687-01 6.00-01 1 5.33-01 I 467-01 I 4.00-01 3.33-01 I 2.67-01 2 0001 133-01 667-02 1 1.1007 | d.1jH_Frin9# Max 1.00*00 (g-Nd 2542 Mm 0 Nd 1 ltoM&^0<^dS42 Figure 5.8 Mixed buckling mode from analysis (hl/h2=0.8). End-shortening Figure 5.9 Load versus end-shortening with various thickness ratios of sublaminates. 97 Postbuckling analysis was also conducted to see the load carrying capacity of the delaminated plates. Imperfection was assumed in the form of the first mode shape obtained from linear buckling analysis and the maximum magnitude of imperfection for postbuckling analysis was 1 % of the top sublaminte thickness. The modified Riks algorithm in STAGS was used for this computation. Loads versus end-shortening curves with different thickness ratios of sublaminates are shown in Figure 5.9. From postbuckling analysis results of Figure 5.9, one can find that the ultimate load carrying capacity of the composite plate with a small thickness ratio of sublaminates (e.g., hl/h2=0.5) is significantly higher than the buckling load obtained from linear bifurcation analysis. This trend was also observed from the test results in Reference [53], 5.5 Buckling Analysis Results Stiffened Panel with Debond Two blade stiffened panels, available from References [83] and [86], respectively, were examined in detail. The configurations of blade stiffened panel, material properties, and stacking sequences from References [83] and [86] are summarized in Table 5.5. The computed energy release rate data is available but the buckling load is not available in Reference [86]. Conversely, experimentally determined buckling load data for panel without skin-stiffener debond is available but energy release rate is not available in Reference [83]. Thus, both panels with single blade stiffener were considered. The panel with a single blade stiffener (Figure 5.11) has a total of 544 nine- node shell element (Element 480) with 2341 nodes, two node fastener elements (Element 130) were used for the joining skin and corresponding flange nodes. An axial compressive load of 3.503MN/m (20,0001b/in) for the panel from Reference [83] was 98 TABLE 5.5 Graphite epoxy lamina material properties from Ref. [83] and Ref. [86] Ref. [86] Ref. [83] Youngs modulus (longitudinal) E, =19.50 x 106 psi E, =18.50 x 106 psi Youngs modulus (transverse) E2 =1.48 x 106 psi E2 = 1.48 x 106 psi Shear modulus G|2 =0.80 x 106 psi G12 =0.80 x 106 psi Poissons ratio v12 = 0.3 < to II o u> TABLE 5.6 Geometric parameters of a blade stiffened panel from Ref. [83] and Ref. [86] Panel Ref. [86] Ref. [83] Panel length (in) 21 30 Panel width (in) 5 8 Blade height (in) 1.45 3.0705 Skin thickness (in) 0.083 0.208 Blade thickness (in) 0.105 0.3536 Flange width (in) 2.0 2.4 Flange thickness (in) 0.067 0.3536 99 applied with uniform end-shortening constraint. In load introduction edges, Poisson expansion was allowed. Both symmetric and free boundary conditions of unloaded side edges were examined. The symmetric boundary conditions represent infinitely wide panel but with debonded stiffeners at uniform spacing. Rigid body motion was constrained along one side edge. 5.5.1 Buckling analysis results of debonded stiffened panels Buckling mode of perfect panel with symmetry boundary condition agree well with buckling mode obtained in Reference [83] (see Figures 5.10 and 5.11). Figures 5.12-5.13 show buckling modes of debonded stiffened panels. As the length of the debond increases, the buckling mode changes from global buckling mode, where blade tip buckling is dominant, to mixed buckling mode and local buckling mode, respectively. This mode transition is expected because the axial and bending stiffness of the blade-flange combination is higher than the stiffness of skin. Thus, the buckling mode of the debonded skin is similar to that of one-dimensional thin-film analysis. This suggests that one dimensional beam-plate model may be still useful to predict buckling load of this problem when local buckling is dominant. In order to see the effect of free side edges on the buckling load, the two aforementioned cases of boundary conditions were examined, and buckling load variations with respect to debond ratio (debond length divided by panel length) are shown in Figure 5.14. Based on the results in Figure 5.14, it can be seen that there is a critical debond length which does not reduce noticeably the buckling load, and this 100 critical debond length varies with the boundary conditions. Furthermore, local buckling is relatively insensitive to boundary conditions compared to global buckling. Variations of buckling loads with respect to different stiffener geometry and stacking sequence for the panel configuration from Ref. 86 are shown in Figures 5.15 and 5.16, respectively. As expected, buckling loads increase in proportion to the increase of stiffener thickness. When the debond ratio is greater than about 0.4, contributions of increased stiffener thickness on buckling load are relatively small due to the local buckling. Figure 5.16 shows that the panel with all 0 degree plies shows small buckling loads compared to the panels with cross plies or balanced symmetry stacking sequences when debond ratios are less than 0.2. This suggests that stiffened panel with all 0 degree plies can lose stability with ease in spite of high axial stiffness. Buckling analysis of the corresponding panel without debond was also conducted using PANDA2 [15-17]. The predicted buckling load factors from PANDA2 with and without neglecting the redistribution of axial compressive load due to local buckling are 0.81 and 0.92, respectively. Those buckling load factors are fairly close to finite element analysis results, 0.95, in Figure 5.16. Figure 5.10 Buckling mode of stiffened panel from Reference [83] Figure 5.11 Global buckling mode 102 Figure 5.12 Mixed buckling mode 103 MSOPATRAN Version .2 30-May 8 18-58:45 FRIN3E Ca*sl*cMlE_ol, LoadSlapO, Mad* I .Crlcal Load 0 88501 E-01 ,Elch41s_ottagOl: Egarwedois, Transbliona DEFORMATION: Caw: sllch4ls_o11, LoacKlap 0, Moda: I .Crlfcal Load: 0.88501 E-01 ,s!lcl>4ls_o.eig0l: Eganvadors, TransMonal-Pfl .83331 Figure 5.13 Local buckling mode. Debond ratio (aL) Figure 5.14 Variations of buckling load with respect to debond ratio. 104 Debond ratio (at) baseline flnagewdh (+10%) flange widh (+10%), thickness (-t25/<) -"flange thickness (+100%) Figure 5.15 Effects of the flange geometry on buckling loads Figure 5.16 Effects of the Stacking Sequences on Buckling Loads 105 5.5.2 Postbuckling analysis results As mentioned previously, stiffened laminated composite flat panels usually exhibit stable postbuckling behavior, which may lead to significant differences between buckling load and ultimate failure load. If the structure exhibits considerable nonlinear prebuckling behavior due to initial imperfection or excessive bowing associated with local buckling, then it is necessary to perform nonlinear analysis. A nonlinear analysis was performed for the panel made up of all 0 degree ply laminates from Reference [86]. An axial compressive load, 22,24IN (50001b), was applied with uniform end-shortening constraint. Initial imperfection was assumed as first mode shape obtained from linear buckling analysis. Load versus end-shortening curves with various debond ratios are shown in Figure 5.17. Figures 5.18-19 show loads versus out-of-plane deformations with several magnitudes of imperfection and debond ratios, respectively. The load versus out-of-plane deformation exhibits stable nonlinear postbuckling behavior. This suggests that linear buckling analysis results of these specific problems can provide overly conservative estimation of load carrying capacity. Therefore, postbuckling analysis is essential to predict structural failure. We also need a failure criterion such as critical stress or critical energy release rate to identify the failure load. 106 End shortening (mm) Figure 5.17 Load versus end-shortening with various debond ratios. Figure 5.18 Load versus out-of-plane deformation 107 5.6 Comparison of Energy Release Rate First, in order to verify the three methods of computing G, using SEDM, VCCT, and proposed CFTM, a double cantilever beam modeled plate with offset node elements was considered. The dimension of DCB specimens (see Figure 5.19) were that used by Raju et al [87] and are as follows: total length 101.6mm; delamination length 50.8 mm; width 24.4 mm; total specimen thickness 3.3 mm; sub-laminate thickness 1.65 mm. The material properties used are listed in Table 5.7. In addition to unidirectional graphite/epoxy, a 16-layer angle ply laminate with the lay-up [+45, -45 }g and an isotropic DCB were also analyzed. The transverse force applied to each ligament of DCB is 1 N/m. The normalized energy release rate values computed using the average G values for the various specimens are shown in Figures 5.20 to 5.23, respectively. The average values used for normalization were compared with those of the 3-D analysis results of Raju et al. [87] in Table 5.8. The average G values from CTFM are closer to those from 3-D than the average G values from VCCT. The accuracy of CTFM depends on the refinement of the finite element along the crack front while the accuracy of VCCT depends not only the refinement of the finite element along the crack front but also that of ahead and behind crack-tip. This may be the reason why the average values of G from VCCT deviate from those of 3-D. However, the comparison of normalized G distribution along the crack front is very good for the example considered. 108 Table 5.7 Elastic material properties [87] Resin Aluminum Graphite/Epoxy El GPa 3.4 71.0 134 E2 GPa 3.4 71.0 13.0 Gl2 GPa 1.3 27.3 6.4 V12 0.3 0.3 0.34 Table 5.8 Average strain energy release rates for DCB specimens (104 J/m2) Isotropic (aluminum) Graphite/Epoxy (0 degree) Graphite/Epoxy (+/- 45 degree) Graphite/Epoxy (90 degree) 3-D 1.0 0.572 2.54 N/A CFTM 0.98 0.541 2.71 5.33 VCCT 0.90 0.522 2.01 5.14 Figure 5.19 Deformed shape of DCB specimen. 109 G distributions for Isotropic (aluminum) DCB specimens y/b Figure 5.20 G distributions for isotropic DCB specimens. G distributions for G/E (+/-45) DCB specimens y/b Figure 5.21 G distributions for graphite/epoxy (+/-45 degree) DCB specimens. 110 G distributions for G/E (0 deg.) DCB specimens Figure 5.22 G distributions for graphite/epoxy (0 degree) DCB specimens. G distributions for G/E (90 degree) DCB specimens Figure 5.23 G distributions for graphite/epoxy (90 degree) DCB specimens. Ill Next, the energy release rates of stiffened panel with skin-stiffener debond from Reference [86] was computed under the same loading and boundary conditions in Reference [86] where stretching displacements, 25.4x1 O'5 mm. were prescribed to the model. Figure 5.24 shows the deformed shape of debonded stiffened panel using 3-D solid elements. On the other hand, deformed shape of the debonded stiffened panel using shell elements is shown in Figure 5.25. Energy release rates along the debond front was computed using VCCT with various mesh refinement, and were compared with those of Reference [86] in Figure 5.26. The comparisons show that the computed G distributions from both the shell elements and the 3-D solid elements are similar to each other. However, the computed G from shell elements is about 20 % lower than that from 3-D solid elements. Wang and Raju [85] proposed a procedure to reduce this discrepancy. However It is still n Finally, under the assumption that debond fronts move along the length-wise direction of stiffener, distributions of energy release rate along debond front with debond ratios, a/L=0.35 and 0.5, were obtained from virtual crack closure techniques (as shown in Figures 5.27-5.29). It should be noted that aforementioned assumption is valid when local buckling is dominant buckling mode. As the load increases, the energy release rate underneath the stiffener increases rapidly. This comes from the fact that the bending stiffness of the flange underneath the stiffener is greater than that of the flange away from the stiffener. Thus, the differences of displacements and rotations for a pair of nodes between skin and flange underneath stiffener is much larger than between those away from the stiffener. 112 Comparison of energy release rates versus end-shortening using strain energy derivative method and virtual crack closure technique was made in Figure 5.30. In general, energy release rate from strain energy derivative method agrees well with average energy release rate from virtual crack closure technique. The critical energy release rate must be obtained by experiment. If the assumed critical energy release rate is about Gct =200 J/m2, then debond extension will be first started underneath the blade stiffener below the buckling load. From the fact that distributions of energy release rate in the debond front are not uniform, using the energy release rate from virtual crack closure technique may give more accurate estimation of debond extension than that from the strain energy derivative method. 5.7 Conclusion Buckling and postbuckling behavior of a stiffened panel with a partial skin- stiffener debond are investigated using the finite element method. The present model shows good correlation with the results of existing buckling analyses of a delaminated plate. There exists a critical debond ratio such that if the ratio of debond is less than a critical value, then the buckling load is unchanged. Furthermore, this critical debond ratio depends on geometry, boundary conditions and material properties. Three methods for computing the energy release rate along the crack front with plate elements are proposed to predict debond extension during postbuckling. When local buckling mode is dominant, the maximum energy release rate, which occurs below the stiffener blade, can be much greater than the average energy release rate. 113 Figure 5.24 Deformed shape of debonded stiffened panel using3-D solid elements. Figure 5.25 Deformed shape of the debonded stiffened panel using shell elements 114 Figure 5.26 Energy release rates along the debond front. Distribution of Energy Release Rate (a/L^0.34) 1.5 2 2.5 3 3.5 Through-the-width position of stiffener Figure 5.27 Distribution of Energy Release Rate (a/L=0.34) (0/-45/45/90)s 115 Through-the-width position of stiffener *25.4 mm Pcr/P=0.10 Pcr/P=0.176 Pcr/P=0.297 Pcr/P=0.420 Pcr/P=0.528 Pcr/P=0.614 Pcr/P=0.693 Pcr/P=0.794 Pcr/P=0.870 Pcr/P=0.922 Pcr/P=0.995 Pcr/P=1.11 Pcr/P=1.322 Figure 5.28 Distribution of energy release rate in width direction of stiffener (a/L=0.35) 1.5 2.5 3.5 "P/Papp.=1.139 -P/Papp.=1.139, average ' P/Papp.=0.805 P/Papp.=0.805, average -P/Papp.=0.613 -P/Papp.=0.613, average Through-the-width position of Stiffener *25.4 mm Figure 5.29 Distribution of energy release rate in width direction of stiffener (a/L=0.5) 116 Figure 5.30 Comparison of energy release rate with end-shortening using strain energy derivative method and virtual crack closure technique. CHAPTER 6 CONCLUSIONS AND FUTURE WORK The major objective of this study was to analyze buckling and delamination of composite stiffened panels subjected to axial compression. First, analytical models using several structural analysis models were used to assess the adequacy of the design model and the correlation with experimental results for a stiffened panel designed using the PASCO program. Of the effects neglected by the simple model, shear deformation was the most important, accounting for about 11% difference in buckling load. The effect of simplified (simple support) boundary conditions was small. The addition to the design model of shear loads and imperfections to improve the robustness of the result did help, even though the inclusion of one-sided imperfection apparently induced sensitivity to imperfection of the opposite sign. Overall, the simplified model did produce designs that in the experiments failed slightly above the design load. Furthermore, one of important finding from the analytical and experimental correlation study of stiffened panel is that an approximate modeling of skin-stiffener intersection with common nodes can predict prebuckling response of the stiffened panel with non-negligible error. The most significant difference between the analytical predictions and experimental measurements was the substantial out-of-plane pre-buckling deformations. 117 118 To explain these differences, imperfections, load eccentricities, and loading platen tilt angles were considered. Of these the loading platen tilt produced similar patterns of deformation, but these had more nonlinear characteristics than the measured deformations. Next buckling and postbuckling behavior of a stiffened panel with a partial skin- stiffener debond were investigated using the finite element method. The present model shows good correlation with the results of existing buckling analyses of a delaminated plate. There exists a critical debond ratio such that if the ratio of debond is less than a critical value, then the buckling load is almost unchanged. This critical debond ratio depends on geometry, boundary conditions and material properties. The average G values from CTFM are closer to those from 3-D analysis than the average G values from VCCT. The accuracy of CTFM depends on the refinement of the finite element along the crack front while the accuracy of VCCT depends not only the refinement of the finite element along the crack front but also that of ahead and behind crack-tip. This may be the reason why the average values of G from VCCT deviate from those of 3-D models. However, the agreement between the three methods of normalized G distribution along the crack front is very good for the example considered.. When local buckling mode is dominant, the maximum energy release rate, which occurs below the stiffener blade, can be much greater than the average energy release rate. 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T., Efficient modeling of postbuckling delamination growth in composite laminates using plate elements, ALAA J. 34(1) (1996): 178-184. 64. Kim, H., Postbuckling analysis of composite laminates with delamination, Comp. Struct 62(6) (1997):975-983. 65.Suemasu, H., Kumagai, T. and Gozu, K., Compressive behavior of multiply delaminated composite laminates parti: Experimental and analytical development, ALAA J. 36 (1998): 1279-1285. 66.Suemasu, H., Kumagai, T. and Gozu, K., Compressive behavior of multiply delaminated composite laminates part2: finite element analysis, AIAA J. 36 (1998): 1286-1290. 67.Zheng, S. and Sun, C. T., Delamination interaction in laminated structures, Eng. Frac. Mech. 59(2) (1998):225-240. 68.Lee., Y. J., Lee, C. H., and Fu, W. S., Study on the compressive strength of laminated composite with through-the-width delamination, Comp. Struct. 41 (1998):229-241. 69.Zienkiewicz, The Finite Element method, 3rd ed, McGraw-Hill, London, 1977. 70.Bathe, K. J., Finite element formulations for large deformation dynamic analysis, Int. J. Num. Meth. Eng. 19, (1983): 1269-1289. 126 71.Surana, K. S., Geometrically nonlinear formulation for the three dimensional solid- shell transition elements, Comput. Struct 15(5) (1982):549-566. 72. Chao, W. C. and Reddy, J. N., Analysis of laminated composite shells using a degenerated 3-D element, Int. J. Num. Meth. Eng. 20 (1984): 1991-2007. 73. Riks, E., The application of Newtons method to the problem of elastic stability, J. Appl. Mech. 39 (1972): 1060-1066. 74.Wempner, G. A., Discrete approximations related to nonlinear theories of solid, Int. J. Solids Struct. 7 (1971): 1581-1599. 75.Crisfield, M. A., A fast incremental/iterative solution procedure that handle snap through, Comp. Struct., 13 (1981):55-62. 76.Crisfield, M. A., An arc-length method including line searches and accelerations, Int. J. Num. Meth. Eng. 19 (1983): 1269-1289. 77.Dixon, J. R. and Pook, L. P., Stress intensity factors calculated generally by the finite element technique, Nature 224 (1969): 166. 78.Parks, D. F., A stiffness derivative finite element method in linear fracture mechanics, Eng. Fract. Mech., 2 (1970): 173-176. 79.Rice, J. R., A path-independent integral and approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968):379-386. 80.Nagendra, S., Jestin, D., Giirdal, Z., Haftka, R. T. and Watson, L. T., Improved genetic algorithm for the design of stiffened composite panels, Comp. Struct. 58(3) (1996):543-555. 81.Anon., ABAQUS/Standard Users Manual, Hibbit, Karlsson & Sorensen, Inc. (1996). 127 82.Anon., P3/PATRAN User Manual, The MacNeal-Schwendler Co., Los Angeles, CA, (May, 1995) 83.Park, O., Haftka, R. T., Sankar, B. V., and Nagendra, S., Analytical and experimental study of a blade stiffened panel in axial compression, AIAA-98-1993-CP, 1998 84.Hilburger, M. W., Numerical and experimental study of the compression response of composite cylindrical shells with cutouts, Doctoral Dissertation, Aerospace Engineering, Univ. of Michigan, 1998. 85.Wang, J. T. and Raju, I. S., Strain energy release rate formulae for skin-stiffener debond modeled with plate elements, Eng. Frac. Mech. 54(2) (1996):211-228. 86.Raju, I. S., Sistla, R., and Krishinamurthy, T., Fracture mechanics analyses for skin- stiffener debonding, Eng. Frac. Mech. 54(3) (1996):371-385. BIOGRAPHICAL SKETCH Oung Park was born in Milyang, Republic of Korea on August 4, 1953. After graduating from Pusan High School in Pusan, South Korea in February 1973, he attended Pusan National University in Pusan, and received his B.S. in Mechanical Design Engineering in February 1977. He started his career as a researcher of Agency for Defense Development (ADD), unique research and development organization of Ministry of National Defense in Korea, in March 1977. He was promoted to a senior researcher in January 1984. In September 1984, he was appointed as candidate of scientist and engineer exchange program between Korea and US government. For one year he studied advanced military research technology at Air Force Armament Laboratory in US. In March 1987, he entered the Graduate School of ChungNam National University, Taejeon, Korea. He presented a thesis entitled Low Velocity Impact Response of Laminated Composite Plate using a Higher Order Shear Deformation Theory to the Faculty of the Graduate School in August 1989 and was received M.S in mechanical engineering at same time. After finishing his 17-year career at ADD in July 1993 for further graduate study, he then entered the Graduate School of the University of Florida in August 1993. He is married to his beautiful wife (Mee) since April 1977 and has a son (Chan) and daughter (Kyoung). 128 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bhavani V. Sankar, Chair Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Raphael T. Haftka, Cochair Distinguished Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ibrahim ICEbcioglu Professor Emeritus of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy^ Peter J. Ifju Assistant Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Civil Engineerin a This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1999 M. J. Ohanian Dean, College of Engineering Winfred M. Phillips Dean, Graduate School 95 (b) Figure 5.7 Local buckling mode from test and analysis, (a) Buckling mode in near-surface delamination [53], (b) Buckling mode from analysis (hl/h2=0.5). 28 follow a specified path towards a converged solution. The constraint equation proposed by Riks [73] may be expressed as: (Aun)T(Aun) + A2A/07o = A/2 (2.22) where A / is a user defined incremental length in the space of n+1 dimensions. Criesfield [75] has proposed a modified constraint equation that includes displacement components alone as: (AJr (AO = A/2 (2.23) It is possible to add the constraint equation (2.23) to the system of equation directly and the iterative incremental method could be used again, however this may destroy the symmetry and banded structure of the equilibrium equation. Hence Criesfield [75] proposed an indirect approaches to avoid this difficulty. In their approach, the displacements at a given iteration i is written as: Su ; = K ,,(A X, + sx '. ) = K i'i'V :(4l) SX /) (2.24) = -'(ai;) + sx ; (Sun), s[ f'/ where 8ir'n are the iterative displacements corresponding to the residual forces VF(J+1, and the tangent stiffness matrix K 71 is formed using modified Newton-Raphson at the beginning of each increment and kept fixed for all iterations within the increment. By substituting equation (2.24) in which 8 is still undetermined into the constraint equation (2.23), we have (A + 8un)( Au-1 + 8un)= A l2 (2.25) 127 82.Anon., P3/PATRAN User Manual, The MacNeal-Schwendler Co., Los Angeles, CA, (May, 1995) 83.Park, O., Haftka, R. T., Sankar, B. V., and Nagendra, S., Analytical and experimental study of a blade stiffened panel in axial compression, AIAA-98-1993-CP, 1998 84.Hilburger, M. W., Numerical and experimental study of the compression response of composite cylindrical shells with cutouts, Doctoral Dissertation, Aerospace Engineering, Univ. of Michigan, 1998. 85.Wang, J. T. and Raju, I. S., Strain energy release rate formulae for skin-stiffener debond modeled with plate elements, Eng. Frac. Mech. 54(2) (1996):211-228. 86.Raju, I. S., Sistla, R., and Krishinamurthy, T., Fracture mechanics analyses for skin- stiffener debonding, Eng. Frac. Mech. 54(3) (1996):371-385. CHAPTER 5 BUCKLING AND POSTBUCKLING ANALYSIS OF A STIFFENED PANEL WITH SKIN-STIFFENER DEBOND 5.1 Introduction In contrast to most of simplified approaches discussed in literature survey in Chapter 1, finite element based approach can be applicable with high fidelity to more general and realistic structures. Therefore, the axial compressive behavior of a stiffened panel with skin-stiffener debond was explored in this chapter by employing finite element model. Two different finite element models, where nodes of the panel skin elements and the stiffener flange elements are located on the mid-plane of each element or interface between the skin elements and flange elements with offset, were used. The nodes corresponding to the top of the skin and bottom of flange are connected with either the elastic spring fastener elements or multi-point constraint equations. In order to verify present finite element modeling approach, laminated composite plates with/without the through-the-width delamination were first modeled. Both single delamination and multiple delaminations were considered. Then, stiffened composite panels with skin-stiffener debond were modeled. Stiffened composite panels with a single stiffener as well as multiple stiffeners were also considered. Buckling and postbuckling analyses were conducted using STAGS. Comparison was made with available buckling analysis results. Next, numerical examples of computing energy 84 (a) (b) Figure 5.4 Computed buckling modes using model II approach, (a) First lowest buckling mode, (b) Second lowest buckling mode. 55 point-wise energy release rate along the delamination front in delaminated plates. Actually the method can be derived from the Virtual Crack Closure Technique or the previously derived Strain Energy Density Method. However the CTFM is computationally simple as the G is computed using the forces transmitted at the crack- tip between the top and bottom sub-laminates and the sub-laminate properties. An evaluation of the aforementioned methods, their applicability to general laminates containing delaminations, and debonded stiffened panels will be presented in chapter 5. 18 method, and the crack-tip force method. Finally, Chapter 6 includes the summary of the present study, concluding remarks, and suggestions for future work. 86 displacement compatibility condition as h h f us + ~Â£V'*,s = uf v + ^-w = v ^-w f * 2 Y y-s f 2 Y y' ws =wf (5.1) where u, v, and w are displacements in x, y, and z direction, *FX and % are rotations, h is thickness, and subscripts s and/ denote skin and flange, respectively. In the debonded region, the same modeling approach as Model I was used. One advantage of model I over model II is such that displacement compatibility conditions at interface nodes in Equation (5.1) is not necessary. The other advantage of model I is that computation of displacements behind crack tip is easy. However, Model I cannot handle multiple delaminations located through-the-thickness direction while Model II can. Ill Next, the energy release rates of stiffened panel with skin-stiffener debond from Reference [86] was computed under the same loading and boundary conditions in Reference [86] where stretching displacements, 25.4x1 O'5 mm. were prescribed to the model. Figure 5.24 shows the deformed shape of debonded stiffened panel using 3-D solid elements. On the other hand, deformed shape of the debonded stiffened panel using shell elements is shown in Figure 5.25. Energy release rates along the debond front was computed using VCCT with various mesh refinement, and were compared with those of Reference [86] in Figure 5.26. The comparisons show that the computed G distributions from both the shell elements and the 3-D solid elements are similar to each other. However, the computed G from shell elements is about 20 % lower than that from 3-D solid elements. Wang and Raju [85] proposed a procedure to reduce this discrepancy. However It is still n Finally, under the assumption that debond fronts move along the length-wise direction of stiffener, distributions of energy release rate along debond front with debond ratios, a/L=0.35 and 0.5, were obtained from virtual crack closure techniques (as shown in Figures 5.27-5.29). It should be noted that aforementioned assumption is valid when local buckling is dominant buckling mode. As the load increases, the energy release rate underneath the stiffener increases rapidly. This comes from the fact that the bending stiffness of the flange underneath the stiffener is greater than that of the flange away from the stiffener. Thus, the differences of displacements and rotations for a pair of nodes between skin and flange underneath stiffener is much larger than between those away from the stiffener. 34 Energy Derivative Method (SEDM), J-intergral, and Virtual Crack Closure technique (VCCT). SEDM, first proposed by Dixon and Pook [77], evaluates the change of potential energy as a crack progresses. Implementation of SEDM in a finite element analysis is straightforward [78]. However, it gives only an average value of the energy release rate along the delamination front. Further, this method requires two computations of potential energy, before and after crack propagation. A direct evaluation of energy release rate requiring only a single computation was proposed by Rice [79]. This involves the calculation of an integral on an arbitrary path surrounding the crack tip. This integral, known as the J-integral, is path independent. The Virtual Crack Closure Technique (VCCT), proposed by Irwin [80], is a method for computing energy release rate for self-similar crack extension. This method assumes that the strain energy release during the crack extension is equal to the work required to close the opened crack surfaces. Many investigators [51-68] have proposed VCCT for computing energy release rate using beam and plate elements. Based on plate theory, Sankar and Sonik [62] proposed the Point-wise Strain Energy Density Method (PSEDM). PSEDM suggests that the point-wise strain energy release rate along the crack front is the difference between strain energy densities behind and ahead of crack front. In this chapter a new method called Crack Tip Force Method (CTFM) based on plate theory is introduced. The application of the various methods of computing G for laminated composite structures is discussed. CHAPTER 2 REVIEW OF THE NONLINEAR FINITE ELEMENT METHOD 2.1 Nonlinear Finite Element Formulations Based on Continuum Mechanics Stiffened laminated composite panels deform continuously under compressive loads. In prebuckling stage, deformation and rotation can be considered as infinitesimal in general. Thus, prebuckling response of stiffened laminated composite panel is almost linear. Classical buckling analysis is generally used to estimate the critical loads of stiff structures such as the Euler column subjected to axial compression, which carry design loads by axial or membrane strength rather than bending strength. The out-of-the plane deformation before buckling is therefore almost negligible in the stiff structures. After buckling, stiffened laminated composite panels exhibit large deformation and rotation. Thus, nonlinear formulation is required in order to include the effects of large deformation and rotation. Many researchers [69-72] have efficiently implemented general nonlinear finite element formulations based on the principles of continuum mechanics. Two different approaches have been used in incremental non-linear finite element formulation. The first approach is generally called Eulerian or updated formulation where static and kinematic variables are referred to an updated configuration in each load step. The second approach is called the Lagrangian formulation, where all static and kinematic variables are referred to the initial configuration. The updated Lagrangian is more suitable for analysis of the structures 19 68 Table 4.5 Buckling load factors obtained from STAGS with various boundary conditions and material properties. Boundary conditions shell unit (480 element*) shell unit (411 element) 4 edges simply supported (baseline design) 1.166 1.296 2 edges free, 2 edges clamped (baseline design) 1.197 1.340 2 edges free, 2 edges clamped (potted region, test material, 32-inch panel length) 1.154 1.286 * Indue es shear deformation. Figure 4.4 The STAGS predicted buckling mode shape of the perfect baseline design panel subjected to uniaxial compression (load factor=1.166). of ABAQUS at HKS Co.; Dr. Mark Hillburger, Mr. Allen Waters at NASA Langley Research Center; and Dr. T. Krishnamurthy at Applied Research Associates. I am grateful to my professors and friends in the Department of Aerospace Engineering, Mechanics, and Engineering Science at the University of Florida, who have taught me and have inspired me during my study. I sincerely appreciate the Agency for Defense Development, where I devoted seventeen years of my young life. I extend my special thanks to Dr. Y. S. Lee, Dr. D. S. Kim, Dr. M. J. Shin, and many other supervisors and colleagues. I am deeply indebted to my beautiful wife (Mee), two children (Chan and Kyoung), and our parents for their endurance, support and prayers. in 4.3 Test Specimen and Test Procedures 60 The test specimens were fabricated from Hercules AS4/3502 unidirectional graphite/epoxy preimpregnated tape material. The skin (32-in. x 32-in.) and the stiffeners were cured separately in an autoclave. The stiffeners were machined to a length of 32 inches, and then bonded to the skin with FM-300 film adhesive. The panel edges perpendicular to the stiffeners were potted with an aluminum filled epoxy resin to prevent end failure. The length of the potted area was 1 in. on each side. Thus the effective gage length of the test specimen was reduced to 30 inches. The test specimen was loaded in compression using a 1,000,000-lb-capacity hydraulic testing machine. The specimen was flat-end tested without lateral edge supports, and no deliberate imperfection was introduced. Electrical resistance strain gages were used to monitor the strains, and direct current differential transformers (DCDTs) were used to monitor longitudinal in-plane and out-of-plane displacements at selected locations as shown in Fig. 4.2. All electrical signals and corresponding applied loads were recorded automatically at regular time intervals during the tests. 4,4 Linear Buckling Analysis 4.4.1 PANDA2 and STAGS The analyses performed in this study include both buckling and nonlinear postbuckling calculations. Linear buckling analyses for the baseline design were conducted by using both PANDA2 and STAGS. Input files for STAGS linear buckling analysis were generated by PANDA2. Next, the effect of the shear load and the 3 panels. However, due to their geometric complexity and the many parameters involved, a complete understanding of all aspects of behavior is not yet fully achieved. Several researchers [1-5] compiled surveys on buckling and postbuckling behavior of composite panels. Leissa [1] compiled extensive results on buckling and postbuckling behavior of laminated composite panels. Analytical effects of various boundary conditions, stacking sequences, and transverse shear deformation on buckling and postbuckling behavior were explored. Noor and Peters [2] reviewed two aspects of the numerical simulation of the buckling and postbuckling responses of composites structures. The first aspect was exploiting non-traditional symmetries exhibited by composite structures, and strategies for reducing the size of the model and the cost of buckling and postbuckling analyses in the presence of symmetry-breaking conditions (e.g., asymmetry of the material, geometry, and loading). The second aspect was the prediction of onset of local delamination in the postbuckling range and accurate determination of transverse shear stresses in the structure. Bushnell [3] divided the literature in the field of buckling of stiffened panel into three categories. One in which structural analysis is emphasized, a second in which optimum design is emphasized, and a third in which test results are emphasized. Recently, Knight and Starnes [4] reviewed some of the historic developments of shell buckling analysis and design and identified key research directions for reliable and robust methods in shell stability analysis and design. Bedair [5] presented an extensive literature review on stability of stiffened panel under uniform compression. He classified the literature into two categories, analysis and design. The objective of the first category is to develop numerical or analytical formulations to predict the global and local buckling load of structures. In doing so, several assumptions 120 9.Stroud, W. J., Agranoff, N. and Anderson, M. S., Minimum-mass design of filamentary composite panel under combined loads: design procedure based on a rigorous buckling analysis, NASA TN D-8417, Hampton VA. 1977. 10.Stroud, W. J. and Anderson, M. S., PASCO: Structural Panel Analysis and Sizing Code, capability and analytical foundations, NASA TM-80181, NASA, Hampton VA, 1981. 11.Anderson, M. S., Williams, F. W. and Wright, C. J., Buckling and vibration of any prismatic assembly of shear and compression loaded anisotropic plate with an arbitrary supporting structure, Int. J. Mech. Sci. 25(8) (1983):585-596. 12.Stroud, W. J., Greene, W. H. and Anderson, M. S., Buckling loads of stiffened panels subjected to combined longitudinal compression and shear: results obtained with PASCO, EAL, and STAGS computer programs, NASA TP-2215, Hampton VA 1984. 13.Bushnell, D., PANDA-interactive program for minimum weight design of stiffened cylinderical panels and shells, Comp. Struct. 16(1) (1983): 167-185. 14.Bushnell, D., Theoretical basis of the Panda computer program for preliminary design of stiffened panels under combined in-plane loads, Comp. Struct. 27(4) (1987):541-563. 15.Bushnell, D., PANDA2-program for minimum weight design of stiffened, composite, locally buckled plate, Comp. Struct 25(4) (1987):469-605. 16.Bushnell, D., Nonlinear equilibrium of imperfect, locally deformed stringer-stiffened panels under combined in-plane loads, Comp. Struct 27(4) (1987):519-539. 17.Bushnell, D., Optimization of composite, stiffened, imperfect panels under combined load for service in the postbuckling regime, Comp. Meth. Appl. Mech. Eng. 103 (1993): 43-114. 18.Dawe, D. J. and Craig, T. J., Buckling and vibration of shear deformable prismatic structures by a complex finite strip method, Int. J. Mech. Sci. 30(2) (1988):77-99. 118 To explain these differences, imperfections, load eccentricities, and loading platen tilt angles were considered. Of these the loading platen tilt produced similar patterns of deformation, but these had more nonlinear characteristics than the measured deformations. Next buckling and postbuckling behavior of a stiffened panel with a partial skin- stiffener debond were investigated using the finite element method. The present model shows good correlation with the results of existing buckling analyses of a delaminated plate. There exists a critical debond ratio such that if the ratio of debond is less than a critical value, then the buckling load is almost unchanged. This critical debond ratio depends on geometry, boundary conditions and material properties. The average G values from CTFM are closer to those from 3-D analysis than the average G values from VCCT. The accuracy of CTFM depends on the refinement of the finite element along the crack front while the accuracy of VCCT depends not only the refinement of the finite element along the crack front but also that of ahead and behind crack-tip. This may be the reason why the average values of G from VCCT deviate from those of 3-D models. However, the agreement between the three methods of normalized G distribution along the crack front is very good for the example considered.. When local buckling mode is dominant, the maximum energy release rate, which occurs below the stiffener blade, can be much greater than the average energy release rate. Future work in this area could be in developing progressive damage models for stiffened composite panels as well as application of CTFM on delamination propagation. 14 load of orthotropic laminated beam-plate with through-the-width delamination. Recently, Gu and Chattopadhyay [53] carried out compression tests on graphite/epoxy composites plates with delaminations to evaluate the critical load and the actual postbuckling load-carrying capacity. They observed that composite laminates can carry higher loads after buckling. For the particular case they studied, the ultimate load is found to be as high as three times the buckling load. 1.2.2.2 One-dimensional multiple delaminations Kutlu and Chang [54-55] investigated the compressive response of composite laminates with multiple delaminations. They found that multiple delaminations can reduce the load-carrying capacity more compared to a single delamination. Suemasu [56-57] developed closed form solution for linear bifurcation buckling load based on energy method. He found that in the case of multiple delaminations, size and location significantly affect the buckling load. Lee et al. [58] proposed a layer-wise approach for computing the buckling loads and corresponding buckling mode shapes. It was found that the anti-symmetric buckling mode is dominant for a composite laminate having short multiple delaminations. They also addressed the effects of initial imperfection and anisotropy on buckling and postbuckling response of delaminated composite plates. An analysis procedure for determining the buckling load of beam-plates having multiple delaminations was also presented by Wang et al. [59], 1.2.2.3 Two-dimensional single delamination Whitcomb [60] studied delamination growth caused by local buckling in composite laminates with near surface delamination, using geometrically nonlinear finite 75 Load vs. Out-of-Plane displacement (stiffener) 8.0615 - \ 6.0E+5I S' 5.OE+5I- o 4.0E+51 DCDT9, in. -i 3.0E+5 t DCDT10, in 2.0E+5 f DCDT 11, in DCDT 8, in. 1.0E+5 l O.OEiO 1 1 1 -0.040 -0.020 0.000 0.020 0.040 0.060 0.080 Out-of-plane displacement (inch) Figure 4.9 Load versus out-of-plane displacements of the stiffener at selected location in Fig-2. Variation of Out-of-Plane Displacement Figure 4.10 Variation of the out-of-plane displacements along the panel lengthwise direction. 89 Table 5.1 Plane woven fabric fiber reinforced laminates material properties [Ref. 56]. Youngs modulus (longitudinal) E, = 20.2 GPa Youngs modulus (transverse) E2 = 21.0 GPa Youngs modulus (through-thickness) E3= 10.0 GPa Shear modulus (inplane) Gi2 = 4.15 GPa Shear modulus (through-thickness) Gi3 = 4.0 GPa Poissons ratio (inplane) V12 = 0.16 Poissons ratio (through-thickness) Vi3 = 0.3 Figure 5.2 Glass fiber reinforced composite specimen [Ref. 56], 81 Load vs. End-shortening 0.00 0.05 0.10 0.15 0.20 0.25 0.30 End-shortening (inch) Figure 4.12 Compressive load versus end-shortening from analysis with contact models and experiment. Model numbers refer to Table 9. Load vs. Out-of-plane displacement Out-of-plane displacement (inch) Figure 4.13 Load versus out-of-plane displacements of the skin at the center bay from Model 9 analysis. 79 analyses are summarized in Table 4.9, and the corresponding load versus end shortening results are shown in Figure 4.12. The results in Figure 4.12 suggest that the computed end-shortening response strongly depend on the user-supplied input data in Table 4.9. The response of Model 9 was the closest to that of the test panel in Figure 4.12. The computed out-of-plane displacements of the skin and stiffeners of Model 9 at the selected DCDTs location are further examined as shown in Figures. 4.13 and 4.14, respectively. It is observed that the response of Model 9 in Figure 4.13 shows good correlation with those of the skin of the test panel in Figure 4.8 during the early stages of the load. However, Model 9 exhibits considerable nonlinear behavior of the out-of plane deformations of skin when the compressive load is above 400,000 lb, which was not present in the response of the test panel in Figure 4.8. Table 4.9 Summary of the tilt angles and the stiffness of contact element. Panel length (inch) Tilt angle (degree) Stiffness of contact element (lb/inch) Model 7 30 0.01 Disp. 0.005 0.05 0.1 0.2 1.0 Force 1.0e3 1.0e5 1.0e7 2.0e8 2.0e8 Model 8 30 0.01 Disp. 0.0001 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e7 1.0e8 2.0e8 2.0e8 Model 9 30 0.01 Disp. 0.005 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e5 1.0e8 2.0e8 2.0e8 Model 10 30 0.005 Disp. 0.005 0.03 0.05 0.2 1.0 Force 1.0e3 1.0e5 1.0e8 2.0e8 2.0e8 The computed load versus out-of-plane deformation of the stiffener at location of the DCDT 11 shows significant nonlinear behavior in Figure 4.14. This significant nonlinear response was also observed from the measured response of the actual test panel in Figure 4.9. In general, the finite element model with the generalized contact definitions improved correlation between the measured and predicted out-of-plane 114 Figure 5.26 Energy release rates along the debond front. Distribution of Energy Release Rate (a/L^0.34) 1.5 2 2.5 3 3.5 Through-the-width position of stiffener Figure 5.27 Distribution of Energy Release Rate (a/L=0.34) (0/-45/45/90)s 78 flat plate in the STAGS analysis. Because the loading surface imperfections were not measured before the test, they were not considered in this study. Instead, it was assumed that the rigid loading platen was initially contacted at the tip of the blade. Stiffeners were assumed to have a small tilt angle with respect to the load introduction edge of the test specimen, as shown in Figure 4.11. STAGS uses generalized contact definitions to check for contact and to construct actual contact elements coupling contacting points with contacted shell elements as the analysis progresses. In doing this, STAGS uses penalty functions to enforce a displacement-compatibility constraint between each contact point and each element with which it is in contact. STAGS utilizes analyst-supplied stiffness versus displacement information to compute the forces resulting from the small contact-surface penetration that may occur. A contact element is conceptually a nonlinear spring connecting the contact point to the surface of the contacted element. This nonlinear spring typically has a low stiffness and generates a small force when the contact-surface penetration is small, but it gets progressively stiffer and generates a larger force as the penetration increases [23]. Generalized contact definitions were implemented to the finite elements that simulate the rigid platen and the load introduction edge of the test specimen. The selection of proper stiffnesses of the contact elements for present analyses is rather arbitrary. Thus, several nonlinear analyses were performed to simulate the observed out- of-plane deformation response of the test specimen by changing both tilt angles and the stiffnesses of contact elements between the loading platen and load introduction edge. The combinations of the tilt angles and stiffnesses of the contact elements used for the 90 Table 5.2 Comparison of buckling load for delaminated plates ( Pcr=2500 N). Buckling load factor Present Suemasu [56] Mode I 0.440 0.432 Mode II 0.801 0.795 N = 4,o = 0.25,ore ss 0.5 Ar = 0-432 Figure 5.3 Buckling modes of the plate with three delamaninations from Ref. [56]. 115 Through-the-width position of stiffener *25.4 mm Pcr/P=0.10 Pcr/P=0.176 Pcr/P=0.297 Pcr/P=0.420 Pcr/P=0.528 Pcr/P=0.614 Pcr/P=0.693 Pcr/P=0.794 Pcr/P=0.870 Pcr/P=0.922 Pcr/P=0.995 Pcr/P=1.11 Pcr/P=1.322 Figure 5.28 Distribution of energy release rate in width direction of stiffener (a/L=0.35) 1.5 2.5 3.5 "P/Papp.=1.139 -P/Papp.=1.139, average ' P/Papp.=0.805 P/Papp.=0.805, average -P/Papp.=0.613 -P/Papp.=0.613, average Through-the-width position of Stiffener *25.4 mm Figure 5.29 Distribution of energy release rate in width direction of stiffener (a/L=0.5) 12 theoretical concepts of the basic instability phenomena and numerical methods were also briefly reviewed. Sleight [41] analyzed a composite blade-stiffened panel with a discontinuous stiffener loaded in axial compression. A progressive failure analyses using Hashins criterion [42] was performed on the blade-stiffened panel. The progressive failure analysis and test results showed good correlation up to the load where local failures occurred. The progressive failure analysis predicted failures around the hole region at the stiffener discontinuity. The final failure of the experiment showed that local delaminations and debonds were present near the hole, and edge delaminations were present near the panel midlength. The progressive failure analysis results did not compare well to the test results since delamination failure modes were not included in this progressive failure approach. 1.2.2 Buckling and Postbuckling Analysis of Laminated Composite Plate with Delamination 1.2.2.1 One dimensional single delamination Chai et al. [43] developed a one-dimensional analytical model to predict through- the-width delamination buckling and growth based on Euler beam theory. Kardomateas [44] investigated effects of transverse shear and end fixity of delaminated composite by improving the model used Chai et al. A first-order shear deformation theory using variational principle was proposed by Chen [45], Yin and his group [46-48] investigated the effects of bending-extension coupling on postbuckling behavior. They evaluated strain energy release rate by using J-integral over a surface that encloses the 9 parameters. Variables considered were flange and web widths, flange-to-web comer radius, thickness and stacking sequence. Analytical model was based on a classical model on local buckling and the numerical analyses were conducted with the ABAQUS finite element code. The results showed that the nature of the load redistribution after buckling and its effect on postbuckling stiffness are related to the geometric variables. The agreement between the analytical and experimental buckling loads was generally good. The agreement degrades as the ratio of flange width with respect to web width is reduced and as the section comer radius is increased. Fan et al. [32] performed the pre- and post-buckling analysis for stiffened panels. Both the thick-wall stiffeners with square cross section and the thin-wall blade stiffeners were employed in their study. After linear buckling analysis, they used an incremental analysis along the load path with a special iteration technique, called initial value method to improve the normal Newton-Raphson method. The computational results of both displacement control and load control were presented. The computed local buckling load and buckling mode agreed well with the test results. However, the computed global buckling load with uniform end-shortening displacement was much higher than the test results. But, the computed global buckling load with load control was close to the test results. The reason of this discrepancy was not clearly explained. Three failure criteria of composite stiffened panel, which are maximum strain, debond failure, and combined global-local buckling criterion were proposed. Nagendra et al. [33] studied the optimum design of blade stiffened panels with holes under buckling and strain constraints. They used PASCO for buckling analysis and optimization with continuous thickness design variables and the Engineering Analysis 74 Load vs. End-shortening -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 E nd-shortening (inch) Figure 4.7 Compressive load versus end-shortening from analysis and experiments. Load vs. Out-of-Plane displacement (mid panel) Out-of-Plane Displacement (inch) Figure 4.8 Load versus out-of-plane displacements of the skin at the center bay. 59 32 N, N, | f* (J- | 1 l i IF m Fir 30 -* Y N, Figure 4.1 Blade stiffened panel with four equally spaced stiffeners under compression and shear loads. All dimensions are in inches. 125 60.Whitcomb, J. D. and Shivakumar, K. N., Strain-energy release rate analysis of a laminate with a postbuckled delamination, NASA TM 89091, 1987. 61. Gim, C. K., Plate finite element modeling of laminated plates, Comp. Struct. 52(1) (1994): 157-168. 62. Sankar, B. V. and Sonik, V., Pointwise energy release rate in delaminated plates, ALAA J. 33 (1995): 1312-1318. 63. Klug, J., Wu, X. X. and Sun, C. T., Efficient modeling of postbuckling delamination growth in composite laminates using plate elements, ALAA J. 34(1) (1996): 178-184. 64. Kim, H., Postbuckling analysis of composite laminates with delamination, Comp. Struct 62(6) (1997):975-983. 65.Suemasu, H., Kumagai, T. and Gozu, K., Compressive behavior of multiply delaminated composite laminates parti: Experimental and analytical development, ALAA J. 36 (1998): 1279-1285. 66.Suemasu, H., Kumagai, T. and Gozu, K., Compressive behavior of multiply delaminated composite laminates part2: finite element analysis, AIAA J. 36 (1998): 1286-1290. 67.Zheng, S. and Sun, C. T., Delamination interaction in laminated structures, Eng. Frac. Mech. 59(2) (1998):225-240. 68.Lee., Y. J., Lee, C. H., and Fu, W. S., Study on the compressive strength of laminated composite with through-the-width delamination, Comp. Struct. 41 (1998):229-241. 69.Zienkiewicz, The Finite Element method, 3rd ed, McGraw-Hill, London, 1977. 70.Bathe, K. J., Finite element formulations for large deformation dynamic analysis, Int. J. Num. Meth. Eng. 19, (1983): 1269-1289. 32 Nonlinear analysis is computationally expensive compared to linear analysis. In order to get a sound analysis results from nonlinear analysis, analyst should have better insight into the behavior of the analysis model. It is usually possible to save time as well as computer cost by preliminary determination of approximate values for the buckling load, under negative as well as positive load, before a large scale analysis is carried out. A linear analysis with a rather coarse grid will give some idea about the stress distribution and verify the nature of the behavior prior to executing nonlinear analysis with refined model. The size of the finite element model should be determined based on the requirement of accuracy, the efficiency, and the time constraint. Prior knowledge of the geometric modeling will increase the efficiency of an analysis. Type of element and size of element should be carefully selected to obtain high accuracy. Further, analyst should identify the type of nonlinearity and localize the nonlinear region for computational efficiency. To identify the type of nonlinearity, it is also helpful to examine the deformed shapes at various stages of loading (pre-buckling, critical buckling, limit load, and postbuckling) [25], 16 closure technique. The compatibility conditions between interfaces of plates were imposed by multi-point constraint equation. The results for delamination interaction in a composite laminated circular plate under three point bending were obtained. Lee et al. [68] developed a nonlinear finite element code, DELAM3D, with a three-dimensional layered solid element based on an updated Lagrangian formulation. They simulated the compressive response of a laminated composite plate with mutiple delaminations. Contacts of delaminating interfaces, delamination growth, and fiber- matrix failure were also considered in their computation. Double cantilever beam (DCB) and end notched flexure (ENF) tests were conducted to verify the energy release rate. Test results with various crack numbers, size, location, and layer orientation compared well with the numerical results. 1.3 Objective and Scope The first objective of this study is to develop stiffened panel models that can be used to predict buckling and postbuckling behavior with and without delamination. The second objective is to investigate the delamination growth of the stiffened panel based on fracture mechanics using several methods of computing strain energy release rate. Among the several configurations commonly used for stiffened panels such as hat-stiffened panel, J-stiffened panel, and I-stiffened panel, blade stiffened panel has a simple geometry compared with other stiffened panels. Therefore, blade stiffened panel was chosen in this study. However, the present analysis method will be also applicable to any kind of stiffener configurations that have a flange attached to the skin. 93 TABLE 5.3 Graphite epoxy lamina material properties Youngs modulus (longitudinal) Ei =19.50 x 106 psi Youngs modulus (transverse) E2 =1.48 x 106 psi Shear modulus G12 =0.80 x 106 psi Poissons ratio V12 = 0.3 Plate length x width 5.0 x 1.25 in. Plate total thickness 0.15 in. Delamination length 2.5 in. Table 5.4 Buckling load delaminated plates with different thickness ratios of sublaminates (Pappl.=l ,250 lb). hl/h2=0.5 hl/h2=0.8 hl/h2=1.0 Buckling load factor 3.44 5.42 5.79 35 Figure 3.1 Stress field (Gyy) near crak-tip of double cantilever beam using 2-D plane strain elements. 29 Expanding Eq. (2.25) gives a quadratic equation for the unknown iterative load factor^/l The details of this solution procedure are given in Ref. [75]. . Figure 2.3 One-dimensional interpretation of spherical arc-length procedure. 2.3 Solution Strategy The buckling analysis provides information about the load level at which bifurcation occurs. In some cases the structure withstand far above the buckling load without significant damage. In other cases the structure collapses well below this load due to imperfection sensitivity. Stability loss at a bifurcation point occurs only if the corresponding deformation mode is not contained in the deformation mode for 103 MSOPATRAN Version .2 30-May 8 18-58:45 FRIN3E Ca*sl*cMlE_ol, LoadSlapO, Mad* I .Crlcal Load 0 88501 E-01 ,Elch41s_ottagOl: Egarwedois, Transbliona DEFORMATION: Caw: sllch4ls_o11, LoacKlap 0, Moda: I .Crlfcal Load: 0.88501 E-01 ,s!lcl>4ls_o.eig0l: Eganvadors, TransMonal-Pfl .83331 Figure 5.13 Local buckling mode. Debond ratio (aL) Figure 5.14 Variations of buckling load with respect to debond ratio. 63 introduced through the potted ends. To simulate this boundary condition, the displacement along the z-direction and the rotation along the x-direction were constrained at the nodal points in the potted region. The adhesive film used to bond the stiffener to the skin in the test panel was modeled by adding an isotropic layer to the model to simulate the bondline between the skin and flange with a thickness of 0.0121 inches. The skin middle surface was used as reference surface on which the nodes lie, and the offset distance of the middle surfaces of the skin-flange combination was modeled as an eccentricity. An alternative finite element modeling approach with ABAQUS suggested by Greene of HKS, Inc. was also used. In this method instead of locating the reference plane at the mid-plane of skin, the bottom plane of blade stiffener was taken as the reference plane. In order to handle the offset distance of the mid-plane of skin, as well as skin-flange combination, an additional 0 ply, with negligibly low stiffness was added to both skin and skin-flange laminates, as shown in Figure 4.3. Both the nine-node thin shell element (S9R5) and the four-node general shell element (S4) in ABAQUS were selected for the stiffened panel models. Both shell elements can account for transverse shear deformations and should therefore be compared to the 480-Element in STAGS. Instead of applying a compressive load at the panel end, uniform compressive displacements were applied at the nodal points along the loaded edge. In order to ensure a uniform state of stress along the entire panel length and also to prevent bending during the pre-buckling stage, the incremental boundary condition option available in ABAQUS was chosen. The buckling load factor was computed from the sum of the reaction forces at the boundary node set. 36 3.2 Strain Energy Derivative Method The strain energy derivative method utilizes the change in total strain energy, U, with change in crack length from a to a +Aa (see Figure 3.2). The energy release rate can be obtained in a straight forward manner for the case of displacement control as G = - dU dA Ua+Aa~V0 AA const deflection (3.1) where AA is the increase in crack area due to change Aa in crack length. Load Control Figure 3.2 Strain energy derivative method. 2.2 Nonlinear Solution Methods 24 The solution of nonlinear finite element problems includes a series of load steps as well as iterations to establish equilibrium at the new load level. In some nonlinear static analyses the equilibrium configurations corresponding to load levels can be calculated without solving for other equilibrium configurations. However, when the analysis includes path-dependent nonlinear conditions, the equilibrium relation needs to be solved throughout the history of interest. The solution may be obtained by using either the Newton-Raphson or the modified Newton-Raphson methods. The Newton- Raphson method requires evaluation of the tangent stiffness matrix at each iteration, which is computationally expensive. On the other hand, the modified Newton-Raphson method evaluates the tangent stiffness matrix at each load step, thus improving the computational efficiency compared to the Newton-Raphson method. However, the Newton type methods fail to provide a solution in the neighborhood of a global bifurcation or limit points when the tangent stiffness matrix becomes singular (see Figure 2.1). The arc-length method, proposed by Riks [73] and Wempner [74], and modified by Criesfield [75,76], is an effective solution procedure to search equilibrium path beyond the limit points. An important aspect in arc-length methods is that the load level is treated as a variable in addition to the unknown displacements at each iteration of a load step. Thus, an additional constraint equation comprising the displacements and loads is required to calculate the load level. 45 the force and moment resultants near the crack tip in any sub-laminate be represented by a column matrix F such that F_ = [ P, M V ] where P, M and V are the axial force, Z Figure 3.7 Sub-laminates in a delaminated beam and the coordinate system. bending moment and shear force resultants. An underscore denotes matrix and a superscript T denotes transpose of a matrix. It should be noted that the force and moment resultants are resolved about the x-axis, which lies in the delamination plane. Thus there is an offset between the laminate mid-planes and the xy plane. The force resultants in each sublaminate are denoted by F], F2, F3 and F4. The compliance matrix of the top and bottom sub-laminates will be denoted by C, and The deformation in a sublaminate is then given by e = C_ F_ (3.18) where the deformations are defined by: 4 are postulated in idealizing the structure in order to facilitate a solution. The objective of the second group of researchers is to develop simplified models to predict the ultimate strength, or collapse load of the structures. Several simplified models have been developed for that purpose. The main objective of this section is to review briefly previous works in the area of buckling and postbuckling behavior of stiffened composite panels. Surveys are divided into two categories, analysis methods and correlation of analysis and experiment. 1.2.1.1 Analysis methods The analysis of stiffened panel can be performed by either smeared or discrete stiffener approach. A smeared stiffener approach converts the stiffened plate into an equivalent plate with constant thickness by smearing out the stiffeners. This method provides accurate analysis results of global buckling load of stiffened panels when stiffeners are identical and closely spaced. However, this approach ignores the discrete nature of the structures and does not consider all potential buckling modes. The discrete stiffener approach does not have limitation of stiffener spacing and uniformity. In early studies of buckling analysis of stiffened panels, Wittrick and William [6] developed a general-purpose computer program, VIPASA, for determining the critical buckling stresses and natural frequencies of thin prismatic structures that consist of a series of flat plate connected rigidly along their longitudinal edges. The response of each plate element making up the stiffened panel is obtained from an exact solution of thin- plate theory. This approach is commonly referred to as exact finite strip method. The analysis used in VIPASA is similar to that which Viswanathan et al. [7] incorporated in 67 Table 4.3 Material properties of baseline designs and test specimen. E(Msi) E22 (Msi) G12 (Msi) Vl2 V21 Skin Design 18.50 1.64 0.87 0.3 0.0270 Test specimen 17.333 1.64 0.8151 0.3 0.0284 Blade Design 18.50 1.64 0.87 0.3 0.0270 Test specimen 19.125 1.64 0.8994 0.3 0.0257 Flange Design 18.50 1.64 0.87 0.3 0.0270 Test specimen 19.593 1.64 0.9214 0.3 0.0251 Adhes ive Test specimen 0.5 0.5 0.192 0.3 0.3 Table 4.4 Geometric parameters of baseline design and test specimen. Panel Design Test specimen Panel length (in) 30 32 Panel width (in) 32 32 Blade height (in) 3.0705 3.0723 Skin ply thickness (in) 0.00520 0.00555 Blade ply thickness (in) 0.00520 0.00503 Flange ply thickness (in) 0.00520 0.00491 Adhesive thickness (in) 0.0 0.0121 CHAPTER 6 CONCLUSIONS AND FUTURE WORK The major objective of this study was to analyze buckling and delamination of composite stiffened panels subjected to axial compression. First, analytical models using several structural analysis models were used to assess the adequacy of the design model and the correlation with experimental results for a stiffened panel designed using the PASCO program. Of the effects neglected by the simple model, shear deformation was the most important, accounting for about 11% difference in buckling load. The effect of simplified (simple support) boundary conditions was small. The addition to the design model of shear loads and imperfections to improve the robustness of the result did help, even though the inclusion of one-sided imperfection apparently induced sensitivity to imperfection of the opposite sign. Overall, the simplified model did produce designs that in the experiments failed slightly above the design load. Furthermore, one of important finding from the analytical and experimental correlation study of stiffened panel is that an approximate modeling of skin-stiffener intersection with common nodes can predict prebuckling response of the stiffened panel with non-negligible error. The most significant difference between the analytical predictions and experimental measurements was the substantial out-of-plane pre-buckling deformations. 117 Model I Nodes are located on interface I I Model II Nodes are located on mid-plane of skin Figure 5.1 Two different modeling approach for blade stiffened panel. ACKNOWLEDGEMENTS I wish to express my sincere appreciation to the members of my supervisory committee. Without the experienced academic advice, patience, encouragement of Dr. Bhavani V. Sankar, chairman of the committee, and Dr. Raphael T. Haftka, cochairman, this work would not have been possible. In addition to the their exceptional guidance, they gave me a wonderful opportunity to interact with other eminent scholars in various ways. Furthermore, they have provided the funding necessary to complete my doctoral study. Dr. Ibrahim K. Ebcioglu not only taught me several courses in solid mechanics but also always made me feel comfortable whenever I talked with him. Dr. Peter G. Ifju showed me his research enthusiasm in experimental mechanics. He always helped me whenever I needed his help. Dr. Fernando E. Fegundo, Jr., was willing to serve as a member of my advisory committee, reviewed this dissertation and gave me helpful suggestions and comments. My special appreciation extends to Dr. James H. Starnes, Jr. and Dr. Cheryl Rose of NASA Langley Research Center. Besides co-authoring and reviewing my publications, they helped me learn STAGS. I would like to acknowledge the great help obtained from Dr. David Bushnell, the developer of PANDA2 at the Lockheed Martin Co.; Dr. William H. Greene, a developer 11 126 71.Surana, K. S., Geometrically nonlinear formulation for the three dimensional solid- shell transition elements, Comput. Struct 15(5) (1982):549-566. 72. Chao, W. C. and Reddy, J. N., Analysis of laminated composite shells using a degenerated 3-D element, Int. J. Num. Meth. Eng. 20 (1984): 1991-2007. 73. Riks, E., The application of Newtons method to the problem of elastic stability, J. Appl. Mech. 39 (1972): 1060-1066. 74.Wempner, G. A., Discrete approximations related to nonlinear theories of solid, Int. J. Solids Struct. 7 (1971): 1581-1599. 75.Crisfield, M. A., A fast incremental/iterative solution procedure that handle snap through, Comp. Struct., 13 (1981):55-62. 76.Crisfield, M. A., An arc-length method including line searches and accelerations, Int. J. Num. Meth. Eng. 19 (1983): 1269-1289. 77.Dixon, J. R. and Pook, L. P., Stress intensity factors calculated generally by the finite element technique, Nature 224 (1969): 166. 78.Parks, D. F., A stiffness derivative finite element method in linear fracture mechanics, Eng. Fract. Mech., 2 (1970): 173-176. 79.Rice, J. R., A path-independent integral and approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968):379-386. 80.Nagendra, S., Jestin, D., Giirdal, Z., Haftka, R. T. and Watson, L. T., Improved genetic algorithm for the design of stiffened composite panels, Comp. Struct. 58(3) (1996):543-555. 81.Anon., ABAQUS/Standard Users Manual, Hibbit, Karlsson & Sorensen, Inc. (1996). BIOGRAPHICAL SKETCH Oung Park was born in Milyang, Republic of Korea on August 4, 1953. After graduating from Pusan High School in Pusan, South Korea in February 1973, he attended Pusan National University in Pusan, and received his B.S. in Mechanical Design Engineering in February 1977. He started his career as a researcher of Agency for Defense Development (ADD), unique research and development organization of Ministry of National Defense in Korea, in March 1977. He was promoted to a senior researcher in January 1984. In September 1984, he was appointed as candidate of scientist and engineer exchange program between Korea and US government. For one year he studied advanced military research technology at Air Force Armament Laboratory in US. In March 1987, he entered the Graduate School of ChungNam National University, Taejeon, Korea. He presented a thesis entitled Low Velocity Impact Response of Laminated Composite Plate using a Higher Order Shear Deformation Theory to the Faculty of the Graduate School in August 1989 and was received M.S in mechanical engineering at same time. After finishing his 17-year career at ADD in July 1993 for further graduate study, he then entered the Graduate School of the University of Florida in August 1993. He is married to his beautiful wife (Mee) since April 1977 and has a son (Chan) and daughter (Kyoung). 128 contact definition in the STAGS finite element model improved correlation between the measured and predicted out-of-plane deformations. Next, a new method called Crack Tip Force Method (CTFM) is derived for computing point-wise energy release rate along the delamination front in delaminated plates. The CTFM is computationally simple as the G is computed using the forces transmitted at the crack-tip between the top and bottom sub-laminates and the sub laminate properties. Finally, buckling and postbuckling of a blade-stiffened composite panel under axial compression with a partial skin-stiffener debond are investigated. Two different finite element models, where nodes of the panel skin and the stiffener flange are located on the mid-plane or at the interface between skin and flange, are used. Linear buckling analysis is conducted using both STAGS and ABAQUS. Postbuckling analysis is conducted with STAGS. Comparison between the present results and previous buckling analysis results show a good correlation. Buckling analysis results for various stiffener geometries and debond ratios are presented. Vll 50 Canceling the crack tip displacements in Equation (3.28) we obtain the equations for the virtual crack closure method as: G = '(C-O' (w(1)-w(2)) (3.30) 3.8 Extension to Delaminated Plates In the case of delaminations in a plate the energy release rate G varies along the delamination front. Formulas similar to Equations (3.10) (SEDM) and (3.22) (VCCT) were derived by Sankar and Sonik [62], In this section we will derive an additional result for G(s) similar to Equation (3.23) derived for beam. We can use the same notation as we used for beams with the understanding that there are eight force and moment resultants and eight deformation components: [FT] = [NX V, Nv Mx Uy Qx Qy\ [r ] = [Â£, e, ro /cv yu yv ] (131) The laminate compliance matrix [C] will be an 8x8 symmetric matrix, and it relates the force resultants and deformations: 27 Am, n Am n ^ Figure 2.2 Schematics of Newton-Raphson and modified Newton-Raphson methods. 2.2.2 Arc-Length Methods The basic feature behind the standard arc-length method is that load level A is treated as a variable rather than constant during a load increment. Thus the governing equilibrium equation (2.14) can be rewritten as (2.20) with (2.21) where A u n is the total incremental displacement vector at the nth iteration and A An is the total incremental load factor at iteration n. Since the load level is treated as a variable, we need an extra equation that constrains the iterative displacements to CHAPTER 3 COMPUTATION OF ENERGY RELEASE RATE 3.1 Introduction Fracture mechanics concepts have been successfully applied to predict the loads which initiates the delamination extension, and also for predicting their stability. The energy release rate G has been accepted as a measure for predicting delamination propagation. In the context of fracture mechanics, the delamination extension is assumed to occur when the computed G is greater than the experimentally determined critical energy release rate Gc. Most of the available methods of computing G use 2-D or 3-D solid elements. However, It is computationally very expensive to use solid elements for modeling the entire complicated structure of an aircraft or an automobile. For example, consider 2-D plane strain elements for computing stress intensity factor to estimate the strain energy release rate of a double cantilever beam. A fine mesh must be used around crack-tip in order to capture the stress gradient ahead of crack-tip. Figure 3.1 shows the amount of mesh density required to calculate the stresses to compute the stress intensity factor. Thus a simplified beam, plate or shell theory are frequently used for structural analysis of complicated structures. Therefore, it may me a good idea to use the same theory to calculate the energy release rate also. There are three methods commonly used for computing energy release rate using 2-D or 3-D solid elements. These are Strain 33 21 where [B() ] is the matrix for the linear infinitesimal strain and matrix [Bi ] contains the nonlinear strain components. Using Equation (2.4) we can rewrite Equation (2.3) as: XI = {Su}T (J [B]{a}dV {\{fh }dV + J{/s }dS)) = 0 (2.5) V V s 2.1.2 Eigenvalue Buckling Prediction The stability criterion can be obtained from the second variation of the total potential energy. If the second variation of the total potential energy has a positive value, then a system is stable. Conversely, If the second variation of the total potential energy has a negative value then a system is unstable. Computing the second variation of total potential energy from Equation (2.5) as S2n = {8u }T (l 8[B? {a}dv + j [B? 8{aY dV) (2.6.1) V 82n={SuY {[8{B]t {(J}dv + j[B]T[C][B]]5{uYdV) (2.6.2) v From Zienkiewicz [69], the first integral of Equation (2.6.2) can generally be written as \v8[B]T{o}dV = [Ka]{8u) (2.7) where [Ka] is geometric stiffness matrix. Substituting Equation (2.4) into the second integral of Equation (2.6.2) and rearranging, the second variation of the total potential energy can be written as: 82U = {8uY[KT]{8u} (2.8) 40 3.4 Zero-Volume J-Integral Consider a portion of the delaminated beam as shown in Fig. 3.5. The beam example is used to minimize the complexity of derivation but this method can be extended easily to delaminated plates. It can be assumed that the delamination length or crack length a is much larger that the thickness h of the thicker sublaminate. If the path of the integral ABCDEF is away from crack-tip, then the beam theory stresses along this path are reasonably accurate compared to exact elasticity solutions. Further, the J- integral will vanish along the two horizontal paths BC and DE. Thus the integral is given as the sum of integrals along the three vertical paths: AB, CD and EF. Next, it will be shown that these vertical paths can be moved near the crack-tip without losing any computational accuracy of G. The vanishing of the J-integral around a closed path in an elastic material under small strain assumptions is a consequence of the two differential equations of equilibrium satisfied by the stress components: dx dx - + + iLi dz = 0 (3.10) The stress field in a laminated beam given by the shear deformable beam theory may not be accurate near the crack tip, however the stress components satisfy the 106 End shortening (mm) Figure 5.17 Load versus end-shortening with various debond ratios. Figure 5.18 Load versus out-of-plane deformation TABLE OF CONTENTS page ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 1.1 Background 1 1.2 Literature Survey 2 1.2.1 Buckling and Postbuckling Analysis of Stiffened Panels 2 1.2.2 Buckling and Postbuckling Analysis of Laminated Composite Plate with Delamination 12 1.3 Objective and Scope 16 2 REVIEW OF THE NONLINEAR FINITE ELEMENT METHOD 19 2.1 Nonlinear Finite Element Formulations Based on Continuum Mechanics 19 2.1.1 Equation of Equilibrium 20 2.1.2 Eigenvalue Buckling Prediction 21 2.2 Nonlinear Solution Methods 24 2.2.1 Newton-Raphson method 25 2.2.2 Arc-length Method 27 2.3 Solution Strategy 29 3 COMPUTATION OF ENERGY RELEASE RATE 33 3.1 Introduction 33 3.2 Strain Energy Derivative Method 36 3.3 Path Independent J-Integral 37 3.4 Zero-volume J-integral 40 3.5 G from Energy Density 43 3.6 G in terms of Crack-tip force 44 3.7 Virtual Crack Closure Technique 48 3.8 Extension to Delaminated Plates 50 3.9 Summary 54 IV REFERENCES 1.Leissa, A. W Buckling of laminated composite plates and shell panels, AFWAL-TR- 85-3069, Air Force Wright Aeronautical Lab., Wright-Patterson AFB 1985. 2.Noor, A. K. and Peters, J. M., Buckling and postbuckling analyses of laminated anisotropic structures, Int. J. Num. Meth. Eng. 27 (1989):383-401. 3.Bushnell, D. and Bushnell, W. D., Minimum-weight design of a stiffened panel via PANDA2 and evaluation of the optimized panel via STAGS, Comput. Struct. 50 (1994):569-602. 4.Knight, F. N., Jr. and Starnes, J. H., Jr., Developments in cylindrical shell stability analysis, ALAA-97-1076-CP, 1997. 5.Bedair, O. K., A contribution to the stability of stiffened plates under uniform compression, Comput. Struct. 66(5) (1998):535-570. 6.Wittrick, W. H. and Williams, F. W., Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings, Int. J. Mech. Sci. 16 (1974):209-239. 7.Viswanathan, A. V. and Tamekumi, M., and Tripp, L. L., Elastic buckling stability of biaxially loaded longitudinally stiffened composite structures, AIAA J. 11(11) (1973): 1553-1559. 8.Stroud, W. J. and Agranoff, N., Minimum-mass design of filamentary composite panel under combined loads: Design procedure based on simplified buckling equation, NASA TN D-8257, Langley Research Center, Hampton VA.1976. 119 2 strength. Therefore, stiffened panels made of laminated composite are very susceptible to delamination. Delamination, also referred to as interface cracking or debonding where adjacent laminae become separated from one another, has been one of the major known weaknesses of laminated composite structures. Delamination, coming from initial manufacturing imperfections or in-service damage such as foreign object impact, can significantly reduce the stiffness and strength of the composite structures. It is also well known that delamination and skin-stiffener separation are common failure modes of a stiffened composite panel in axial compression. When a delamination is present, it is very important to identify not only its global influence on the load carrying capacity of the structure but also its local behavior under the applied load. Energy release rate has been accepted as a measure for predicting delamination propagation. Most available methods of computing energy release rate use 2-D or 3-D solid finite elements. However it is computationally very expensive to use solid elements for modeling entire complicated structures made of laminated composite materials. Thus simplified beam, plate and shell theories are frequently used for structural analysis of those structures. Therefore, it may be a good idea to also use the same theory to calculate energy release rate. 1.2 Literature Survey 1.2.1 Buckling and Postbuckling Analysis of Stiffened Panels Research on the buckling and postbuckling behavior of stiffened panels has been of interest for many years, with many researchers exploring the response of the stiffened 48 2-D SOLID BEAM MODEL Figure 3.8 Crack-tip forces in beam 3.7 Virtual Crack Closure Technique The virtual crack closure technique has been used for plate and beam fracture problems by many researchers. We will derive the VCCT from the Crack-Tip Force Method. The expression for G in Equation (3.23) can be written as: G = J EJc (Q., (Z-4 ZL\) + C_h (F_2 F_3)) (3.25) where F_c is the matrix of crack tip forces, and Equation (3.18) is used in deriving Equation (3.25). Using the compatibility quation (3.19) in (3.24), we obtain g = ^eU-c,f, +CF2) (3.26) 82 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Out-of-plane displacement (inch) Figure 4.14 Load versus out-of plane displacements of the stiffeners at selected locations from Model 9 analysis. 97 Postbuckling analysis was also conducted to see the load carrying capacity of the delaminated plates. Imperfection was assumed in the form of the first mode shape obtained from linear buckling analysis and the maximum magnitude of imperfection for postbuckling analysis was 1 % of the top sublaminte thickness. The modified Riks algorithm in STAGS was used for this computation. Loads versus end-shortening curves with different thickness ratios of sublaminates are shown in Figure 5.9. From postbuckling analysis results of Figure 5.9, one can find that the ultimate load carrying capacity of the composite plate with a small thickness ratio of sublaminates (e.g., hl/h2=0.5) is significantly higher than the buckling load obtained from linear bifurcation analysis. This trend was also observed from the test results in Reference [53], 5.5 Buckling Analysis Results Stiffened Panel with Debond Two blade stiffened panels, available from References [83] and [86], respectively, were examined in detail. The configurations of blade stiffened panel, material properties, and stacking sequences from References [83] and [86] are summarized in Table 5.5. The computed energy release rate data is available but the buckling load is not available in Reference [86]. Conversely, experimentally determined buckling load data for panel without skin-stiffener debond is available but energy release rate is not available in Reference [83]. Thus, both panels with single blade stiffener were considered. The panel with a single blade stiffener (Figure 5.11) has a total of 544 nine- node shell element (Element 480) with 2341 nodes, two node fastener elements (Element 130) were used for the joining skin and corresponding flange nodes. An axial compressive load of 3.503MN/m (20,0001b/in) for the panel from Reference [83] was 4.6.2 Compressive Load Versus Out-of -Plane Deformations 76 The layout of the DCDTs used to measure displacements in the test panel is shown in Figure 4.2. The out-of-plane displacements, measured from DCDTs at selected locations in Figure 4.2, are shown in Figures. 4.8 and 4.9. The results in Figure 4.8 show that out-of-plane displacements were initiated at an early stage of the loading and increase linearly in proportion to the loading. This observation suggested the possibility of loading eccentricities along the load introduction edge or rigid body rotation of the panel with respect to the clamped edge in addition to the effects of geometric imperfections. Figure 4.9 shows that the out-of-plane displacements of the blade stiffeners were also started at an early stage of the loading. Except DCDT 11, the out-of plane deformations were an order-of-magnitude lower than those of the skin in Figure 4.8. Furthermore, significant nonlinear response was only exhibited near the failure load. The large nonlinear response of DCDT11 throughout the axial loading was probably due to the effect of the unsupported side-edge boundaries. The out-of-plane displacement variations along the length of the panel (DCDTs 1-4 in Figure 4.2) at selected load levels are shown in Figure 4.10. The results in Figure 4.10 indicate that bending occurred in the test panel in addition to the end shortening due to the axial compressive load. The load versus out-of-plane displacements across the panel mid-length can be found in Ref. [83], In order to explain the substantial prebuckling bending, a combination of different geometric imperfections and loads applied at a small angle to the axial direction were analyzed to determine their influence on the observed out-of-plane displacements. The capability of STAGS to model geometric imperfections in the shape 5.3 Buckling Analysis Results of Multiple Delaminated Plate 88 A plane woven fabric glass fiber reinforced composite panel with three through-the-width delaminations located in the middle of the plate from Suemasu [56] (as shown in Figure 5.2) was examined using Model II approach. The material properties are shown in Table 5.1. The STAGS finite element model for the plate has a total of 4 branched shell units with four node shell elements (element 410), and each branched shell unit has 41 by 11 nodes. Total of 297 elastic fastener elements (element 130) was used to prevent penetration of contact surfaces in debonded region of the plate. The compressive stiffness of an elastic fastener element used in this study is 113 MN/m. In the intact region of plate, a total of 3168 constraint equations was used to satisfy the compatibility conditions of shell unit interfaces. Axial compressive load (2500 N) was applied to load introduction edge with uniform end-shortening constraint. In order to ensure a uniform stress condition for the entire panel length, an incremental boundary condition option was chosen to prevent axial bending during the prebuckling stage. The computed buckling load and buckling load from Reference [56] are given in Table 5.2. The first and second buckling mode shapes from Reference [56] and present analysis are shown in Fig. 5.3 and Fig. 5.4, respectively. It is seen that present buckling analysis results agree well with the corresponding buckling analysis results from Reference [56], However, the buckling load from the experiment is lower than the computed buckling load. Suemasu [56] indicated that insufficient clamped condition during his experiment might be responsible for low buckling load. e = C F 51 (3.32) A B B D 0 0 0 0 K {f} where, the [A], [B] and [D] are the classical 3x3 laminate stiffness matrices and [AH is the 2 by 2 transverse shear stiffness matrix. In the context of plates the strain energy density is defined as strain energy per unit area of the plate and is given by 1 r U,=-FtCF 2 (3.33) A formula for G(s) similar to that in Equation (3.23) is given by G(J) = (1/ V>(i) + V 2,(i) -Ufui (3.34) where the superscripts denote the four sub-laminates behind and ahead of the delamination front, and s denotes the location of the point along the delamination front. Sub-laminates 1 and 4 are above the delamination plane, and 2 and 3 are below the delamination plane. From Equation (3.34) one can derive another expression for G(s) as: c = y (Â£l Fj )(C, + C)(Â£4 F,) (3.35) DCDTs (1-7) for out-of-plane deformation of skin. DCDTs (8-11) for out-of-plane deformation of stiffener. DCDT (12) for end-shortening displacement 61 Figure 4.2 Layout of the displacement measurement instrumentation for the test panel. Figure 4.3 Reference plane of ABAQUS and STAGS model. 4 ANALYTICAL AND EXPERIMENTAL CORRELATION OF A STIFFENED COMPOSITE PANEL IN AXIAL COMPRESSION 56 4.1 Introduction 56 4.2 Stiffened Panel Definition 57 4.3 Test Specimen and Test Procedures 60 4.4 Linear Buckling Analysis 60 4.4.1 PANDA2 and STAGS 60 4.4.2 Finite Element Model 62 4.5 Results of Linear Buckling Analysis 64 4.5.1 Effect of Geometric Imperfections and Shear Load 64 4.5.2 Effect of Boundary Conditions and Material Properties 65 4.5.3 Summary of Differences between Design Model and Test Model 71 4.6 Nonlinear Analysis 72 4.6.1 Compressive Load versus End-shortening 73 4.6.2 Compressive Load versus Out-of-plane Deformations 76 4.6.3 Contact between the Panel and Loading Platen 77 4.7 Conclusion 83 5 BUCKLING AND POSTBUCKLING ANALYSIS OF A STIFFENED PANEL WITH SKIN-STIFFENER DEBOND 84 5.1 Introduction 84 5.2 Finite Element Model 85 5.3 Buckling Analysis Results of Multiple Delaminated Plate 88 5.4 Buckling and Postbuckling Analysis Results of Plate with Single Delamination 92 5.5 Buckling Analysis Results of Stiffened Panel with Debond 91 5.5.1 Buckling Analysis Results of Debonded Stiffened Panels 99 5.5.2 Postbuckling Analysis Results 105 5.6 Comparison of Energy Release Rate 107 5.7 Conclusion 112 6 CONCLUSIONS AND FUTURE WORK 117 REFERENCES 119 BIOGRAPHICAL SKETCH 128 v xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EWOGXVX4I_RBI2EO INGEST_TIME 2015-04-20T20:21:46Z PACKAGE AA00029808_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 10 Language (EAL) finite element analysis code [34] for calculating strain and their derivatives with respect to design variables. Later their optimally designed panels with and without centrally located holes were tested and analytical and experimental results were compared focusing prebuckling behavior of the panel [35]. Prebuckling stiffness from test were about 10% lower than analytical values and failure loads from test was also 10% lower than that from design. Since the uncertainties in the geometric and material properties did not account for the discrepancy between analytical and experimental buckling loads, they hypothesized that geometric imperfections and eccentricities may had reduced the buckling load. Chow and Atluri [36] proposed failure criterion of mixed-mode stress intensity factors for the postbuckling strength of stiffened panel. They showed that post-buckling strength of the stiffened panels compare quite favorably with the experimental results of Starnes et al. [27] and the standard deviation of the error was less than 10 %. Young and Hyer [37] presented modeling procedures that predict the postbuckling response of composite panels with skewed stiffeners. Five panel configurations with various combinations of skin and stiffener orientation were tested. A uniform end shortening displacement was applied to the upper end of the panel in the axial direction, and the axial displacement of the lower end was restrained. The upper and lower ends were clamped and the unloaded sides were simply supported. The individual effect of shell elements, potted load introduction, material properties, and initial geometric imperfections was examined. The results showed that inaccurate modeling assumptions and anomalies in the test such as the support fixtures, the loading frame, and the load introduction of the test specimen could cause the predicted response 66 be input directly, the differences in the thickness of the skin or flange are accounted by implementing a proportional change in the ply thickness in the model. A detailed discussion of this procedure can be found in Ref. 4. The results in Table 4.5 indicate that the effects of boundary conditions and material properties on the buckling load factors are not very significant. Comparison of the first two rows of Table 4.5 reveals the effects of changes in the boundary conditions. Similarly, results in the last two rows show the effects of changes in material properties and panel dimensions. The buckling mode shape of the baseline design (simply supported on 4 sides) predicted by STAGS is shown in Figure 4.4. The overall buckling mode shape obtained from ABAQUS (Figure 4.5) agrees well with that of STAGS. The computed lowest buckling load factor is slightly higher than that of STAGS (1.218 for ABAQUS vs. 1.168 for STAGS). This small difference may be due to modeling differences as discussed in the following section. The STAGS prediction of the buckling mode shape of the test panel with potted boundary conditions (other two edges being free) is shown in Figure 4.6. Table 4.2 Summary of the local buckling load factor from PANDA2 and the lowest buckling load factor from STAGS (4 edges simple supported). PANDA2 (Koiter analysis) PANDA2 (BOSOR4 analysis) STAGS (480 element*) STAGS (411 element) Loading combination with/without imperfection Panel end Panel mid-length. Panel end Panel mid-length. Nx=20,000 lb/in, Nxy=0 without imperfection 1.256 1.256 1.328 1.328 1.168 1.296 Nx=20,0001b/in, Nxy=5000 lb/in without imperfection 1.234 1.234 1.048 1.048 Nx=20,0001b/in, Nx>=5000 lb/in with imperfection (+3%) 1.234 0.356 1.048 0.346 Nx=20,0001b/in, Nxy=5000 lb/in with imperfection (-3%) 1.234 0.856 1.048 0.920 Nx=20,0001b/in, Nxy=0 lb/in with imperfection (+3%) 1.256 0.394 1.328 0.398 Nx=20,0001b/in, Nxy=0 lb/in with imperfection (-3%) 1.256 1.026 1.328 1.083 includes shear deformation. 99 applied with uniform end-shortening constraint. In load introduction edges, Poisson expansion was allowed. Both symmetric and free boundary conditions of unloaded side edges were examined. The symmetric boundary conditions represent infinitely wide panel but with debonded stiffeners at uniform spacing. Rigid body motion was constrained along one side edge. 5.5.1 Buckling analysis results of debonded stiffened panels Buckling mode of perfect panel with symmetry boundary condition agree well with buckling mode obtained in Reference [83] (see Figures 5.10 and 5.11). Figures 5.12-5.13 show buckling modes of debonded stiffened panels. As the length of the debond increases, the buckling mode changes from global buckling mode, where blade tip buckling is dominant, to mixed buckling mode and local buckling mode, respectively. This mode transition is expected because the axial and bending stiffness of the blade-flange combination is higher than the stiffness of skin. Thus, the buckling mode of the debonded skin is similar to that of one-dimensional thin-film analysis. This suggests that one dimensional beam-plate model may be still useful to predict buckling load of this problem when local buckling is dominant. In order to see the effect of free side edges on the buckling load, the two aforementioned cases of boundary conditions were examined, and buckling load variations with respect to debond ratio (debond length divided by panel length) are shown in Figure 5.14. Based on the results in Figure 5.14, it can be seen that there is a critical debond length which does not reduce noticeably the buckling load, and this 105 5.5.2 Postbuckling analysis results As mentioned previously, stiffened laminated composite flat panels usually exhibit stable postbuckling behavior, which may lead to significant differences between buckling load and ultimate failure load. If the structure exhibits considerable nonlinear prebuckling behavior due to initial imperfection or excessive bowing associated with local buckling, then it is necessary to perform nonlinear analysis. A nonlinear analysis was performed for the panel made up of all 0 degree ply laminates from Reference [86]. An axial compressive load, 22,24IN (50001b), was applied with uniform end-shortening constraint. Initial imperfection was assumed as first mode shape obtained from linear buckling analysis. Load versus end-shortening curves with various debond ratios are shown in Figure 5.17. Figures 5.18-19 show loads versus out-of-plane deformations with several magnitudes of imperfection and debond ratios, respectively. The load versus out-of-plane deformation exhibits stable nonlinear postbuckling behavior. This suggests that linear buckling analysis results of these specific problems can provide overly conservative estimation of load carrying capacity. Therefore, postbuckling analysis is essential to predict structural failure. We also need a failure criterion such as critical stress or critical energy release rate to identify the failure load. 116 Figure 5.30 Comparison of energy release rate with end-shortening using strain energy derivative method and virtual crack closure technique. 31 sets saved on tape for eigenvalue solution. Changes in boundary conditions are also permitted at this time. In most practical applications one range of eigenvalues is particularly important, especially to a sequence of the lowest eigenvalues. If there exists a symmetry plane, in loading as well as in geometry, the size of the problem can be reduced and significant savings in the total computational effort can be achieved. If the structure on one side of the symmetry plane is considered, only the frequencies of symmetric modes are obtained. If the eigenvalue analysis is based on a nonzero prestress analysis with symmetry conditions and an eigenvalue analysis with boundary conditions corresponding to anti-symmetry. The eigenvalue approach for bifurcation buckling analysis with linear stress state is slightly more complicated than the vibration problem because eigenvalues can be negative as well as positive. Often the analyst is only interested in one eigenvalue, the lowest positive one. If the analysis is performed without a shift, it may happen that only negative eigenvalues are found because these are smaller in magnitude. In that case, the analysis has to be repeated with a positive shift. In choosing the shift for a repeated run the user can utilize the fact that the smallest positive eigenvalue is larger in magnitude than the largest of the negative eigenvalues that were found. Sometimes the buckling loads are symmetric with respect to zero. This is the case, for example, if a plate or a cylinder is subjected to uniform a shear load. It may often be advisable to request more than one eigenvalue also in bucking analysis. If the structure shows insufficient strength and only the lowest eigenvalue and corresponding mode are known, reinforcements may be introduced that have little effect on secondary buckling mode with the eigenvalue below the design load. 47 is used to connect the top and bottom crack tip nodes in a finite element model, then the forces transmitted by the rigid link will be exactly equal to the above crack-tip forces. It may be noted that the crack tip force vector F_c have three components, an axial force, a couple and a transverse force, corresponding to each degree of freedom of the crack tip nodes, u, ^ and w. Another important implication of Equation (3.23) is that although there are 6 independent forces Pi, V,, M ,, P2, V2 and M 2 that can be applied to the two delaminated beam ligaments (see Figure 3.5), G depends only on three crack tip force components (see Figure 3.8). If the forces F_, and F_2 such that e, = e2 > i.e., C_,F_i C_bf^2 using Equations (3.18) and (3.19) one can show that F_, = F_A, and then G 0 If the forces on the top and bottom sub-laminates 1 and 2 are such that they produce conforming deformations (e_, = e_2), then the same forces act in sublaminates 4 and 3, respectively, producing conforming deformations (e3 = e4). Thus there is no need for any interaction between the top and bottom laminates at the crack-tip, and hence G = 0 . 109 G distributions for Isotropic (aluminum) DCB specimens y/b Figure 5.20 G distributions for isotropic DCB specimens. G distributions for G/E (+/-45) DCB specimens y/b Figure 5.21 G distributions for graphite/epoxy (+/-45 degree) DCB specimens. CHAPTER 4 ANALYTICAL AND EXPERIMENTAL CORRELATION OF A STIFEENED COMPOSITE PANEL IN AXIAL COMPRESSION 4.1 Introduction Buckling and imperfection sensitivity are expensive to calculate with general finite element models. Consequently, the optimization of stiffened panels often employs simplified models that are exact only for idealized geometries and boundary conditions (e.g., PASCO [10], or PANDA2 [15-17]). Nagendra et al. [33] studied the optimum design of blade stiffened panels with holes subjected to buckling and strain constraints. They used the panel analysis and sizing code (PASCO), based on a linked plate model, for the buckling analysis and optimization with continuous thickness design variables, and the Engineering Analysis Language (EAL [34]) finite element analysis code for calculating strains and their derivatives with respect to design variables. Later, the optimally designed panels with and without centrally located holes were tested, and analytical and experimental results were compared [35], Nagendra et al. [80] continued the optimum design study of blade stiffened panel using PASCO for analysis, and a genetic algorithm (GA) for the optimization of the panel laminate stacking sequences. Several designs obtained with GA were about 8% lower in weight compared to designs previously obtained with a continuous optimization procedure. 56 46 * = [Â£^x,ya\ (3.19) where the components of the deformations are the strain along the x-axis (not the sub laminate mid-plane), rate of change of rotation and the transverse shear strain, respectively. The force resultants are related by the equilibrium conditions Â£l + h.= h.+0.20) Further, since the sub-laminates 3 and 4 are intact (not delaminated) the deformations in them should be identical, i.e. e3 = e4 and hence ChF, = C,F4 (3.21) If F] and Â£2 are given, then Fj and F_4 can be calculated using Equations. (3.20), and (3.21). The strain energy per unit length in any sub-laminate is given by: Vi=\l TCF (3.22) Substituting Equation (3.22) into Equation (3.17) we obtain G=^fJC,F* +F_lCl,F2-F_Â¡CbF, 1 T -fIc.f 2 (3.23) Using the relations in Equations (3.20) and (3.21) an interesting expression for G can be derived as G = F_r, )(C, + CXF4 F,) (3.24) The term (F_4 F_,) is actually the force transmitted through the crack tip between the top and bottom sub-laminates, and can be called the crack-tip force, F_c. If a rigid link 4.5.3 Summary of Differences between Design Model and Test Model 71 In summary, the PASCO model used for designing the panel had several modeling simplifications and compensating factors; their effects are listed in Table 4.7. The two major model simplifications were: 1. PASCO does not account for shear deformation, which is significant for thick composite panels, and this reduces the buckling load by 11%. 2. PASCO employs simple support boundary conditions. The difference due to boundary conditions was only about 1% because the buckling mode is local. To obtain a more robust design, the PASCO model was subjected to an additional shear load and 3% imperfection. Both had substantial effects on the design load. However, because the panel was designed only for imperfection of one sign, it became more sensitive to an imperfection of the opposite sign. Finally, because of the substantial thickness of the blade, it was also found that a centerline modeling, which is common in thin walled structures, produces about seven percent increase in the prebuckling stiffness, and a seven percent reduction in buckling load. The opposite effects are explained by the fact that the overlap draws more loads into the blade, which is the critical element. Overall, the more accurate analytical model predicts a buckling load higher by about 18% than the design load; however, this does not take into account any imperfections. The actual buckling load was only about 10% higher than the design load. It can be concluded that for this panel the simplified model used in PASCO, together with the shear and imperfection loading added for robustness, worked reasonably well. The other two designs that were tested also buckled at or slightly above the design load 113 Figure 5.24 Deformed shape of debonded stiffened panel using3-D solid elements. Figure 5.25 Deformed shape of the debonded stiffened panel using shell elements y(2) = u l(2) y/,V2 J0) = -uL0)+ 44 (3.16) It should be noted that in Equation (3.16) switches signs because of change in the sign of nx from -7 to +/ for the Path 3. Adding all the three integrals and nothing that the shear force resultants must satisfy the equilibrium condition Vj+V2 = Vj, we find that J = y<'> + y(2) + y(3) = U y + U [2) U [3) (3.17) Thus the energy release rate is the difference between the strain energy densities just behind and just ahead of the crack tip. The strain energy density in the context of beam refers to strain energy per unit length of the beam, U L . 3.6 G in terms of Crack Tip Force Consider a very small segment of the beam of length 2 A* surrounding the crack-tip (see Figure 3.3). It will be convenient to shift the xz coordinates such that the xy plane coincides with the plane of delamination. Further, we will divide the laminate into 4 sub-laminates 2 behind and 2 ahead of the crack tip as shown in Figure 3.7. Let 22 In Equation (2.8), the tangential stiffness matrix [Kj ] can be written as [KT] = [Ka] + [K0] + [KL]. [K0] = j[B0?[C][B0]dV (2.9) v [KL] = \([B0]T[C][BL] + [BL]T[C][B0] + [BLf[C][BL])dV (2.10) v where [AT0] is the small displacement stiffness matrix and [TJ is the large displacement stiffness matrix. A critical point is obtained when the tangent stiffness matrix [KT] has at least one zero eigenvalue. The stability of an equilibrium configuration can be determined solving the eigenvalue problem at the current equilibrium state, [A:r]{(r)} = A(r){M equilibrium configuration associated with a given load level P is computed. Next the stability of tangent stiffness matrix is examined by computing the eigenvalue of the tangent stiffness matrix at given load level P. This method of determining the stability of a conservative system gives accurate results. However, it is computationally expensive because it involves the solution of a quadratic eigenvalue problem for the critical load. Linearized buckling analysis calculates critical buckling loads based on a linear extrapolation of the structural behavior at a small load level. Thus it is computationally inexpensive. From the fact that geometric stiffness matrix [Ka] and Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BUCKLING AND DELAMINATION ANALYSES OF STIFFENED COMPOSITE PANELS IN AXIAL COMPRESSION By OUNG PARK December, 1999 Chairman: Dr. Bhavani V. Sankar Major Department: Department of Aerospace Engineering, Mechanics, and Engineering Science The major objective of this study is to analyze buckling and delamination behavior of composite stiffened panels subjected to axial compression. First, a combined analytical and experimental study of a blade stiffened composite panel subjected to axial compression was conducted. The effects of the differences between a simple model used to design the panel and the actual experimental conditions were examined. It was found that in spite of many simplifying assumptions the design model did reasonably well in that the experimental failure load was only 10% higher than the design load. Several structural analysis programs, including PANDA2, STAGS, and ABAQUS, were used to obtain high fidelity analysis results. The buckling loads from STAGS agreed well with the experimental failure loads. However, substantial differences were found in the out-of-plane displacements of the panel. Efforts were made to identify the source of these differences. Implementing non-uniform load introduction with general vi 42 V, Figure 3.5 Force and moment resultants in a delaminated beam. J 2 ~ J AB J BG JGH ^HA ~ ^ JBG JHA ^ J AB = ~ J GH J AB ~ J HG Figure 3.6 Three sublaminates of delaminated beam. 6 buckling loads are computed by the use of simple assumed displacement functions used in conjunction with Donnell-type kinematic relation. Several types of general and local buckling modes were included. VIPASA, PASCO, and PANDA cannot perform nonlinear postbuckling analysis of the stiffened panel. Bushnell [15-17] released PANDA2, which incorporated nonlinear theory for prediction of behavior of locally imperfect panels. Nonlinear strain-displacement relations analogous to those developed in 1946 by Koiter for perfect panels were extended to handle panels with imperfections in the form of critical local bifurcation buckling mode. In PANDA2, local and general buckling loads are calculated with use of either closed-form expression [14] or with use of discretized models of panel cross-sections [15-17], The discretized model is based on one-dimensional discretization that is commonly referred to as approximate finite strip method. Approximate finite strip method assumes displacement of strip can be expressed with combination of crosswise polynomial function and lengthwise trigonometric function. Dowe and his group [18-22] have used the method in conjunction with a non-linear analysis based on first-order shear deformation theory. The Finite strip method, which lies between the conventional Rayleigh-Ritz method and the finite element method, provide a means of solving prismatic plate-structure problems with an attractive blend of accuracy, economy and ease of modeling [22], The finite element method is the most powerful method to predict the buckling and postbuckling behavior of structures. The historic developments of shell buckling analysis using finite element up to 1997 can be found in Knight and Starnes [5], With advancing computer power, the finite element method is widely used to solve various engineering problems. Many current commercially available finite element codes such 39 J = Â¡(U 0nx o ijnjui x)ds = \ (U 0nx o ijnjui x)ds r, r 3 = j(U mx- a a171 jui,x)ds r3 where normal vector m x is in the opposite direction with respect to normal vector n x in path T3. Figure 3.3 Conservation integral around a region with no singularities. x Figure 3.4 Conservation integral around a region with singularities. 85 release rate in the context of plate as discussed in Chapter 3 are given for predicting debond extension using the strain energy derivative method, the virtual crack closure technique, and the crack-tip force method. 5.2 Finite Element Model Two finite element-modeling approaches for the stiffened panel are commonly used in the literature. One approach is to model the skin with plate elements and to model the stiffener with beam elements. The other is model both skin and stiffener with plate elements. The second modeling approach appears more attractive for modeling debonded region between skin and flange, and it was chosen in this study. Furthermore, two different finite element models, designated as Model I and Model II, respectively, were also considered (see Figure 5.1). Both skin and flange elements of Model I have offset nodes with small gap at the interface region, while nodes in Model II are located on the mid-planes of skin and flange, respectively. In Model I, in order to satisfy compatibility conditions of intact interface nodes located directly above the skin and below the flange, each nodal degree of freedom was constrained with elastic spring fastener elements with very high spring constants. Nodes located on debonded interface between skin and flange were connected with elastic fastener elements, which have only an axial degree of freedom with very high stiffness in compression and zero stiffness in tension. This can prevent physically unrealistic nodal penetration between skin and flange during the postbuckling analysis. Friction against sliding of the debonded surface was not considered. In Model II, Multi-point constraints were imposed at the interface of intact skin and corresponding flange nodes to satisfy the 102 Figure 5.12 Mixed buckling mode 77 of the buckling modes was used for these analyses. Various combinations of imperfection amplitudes and load angles were considered. Although for some combinations we could reproduce the test results partially [83], obtaining the right imperfection and the load introduction angle seemed elusive. This difficulty suggests that we look elsewhere for the source of the prebuckling bending. 4.6.3 Contact Between the Panel and Loading Platen Hilburger [97] investigated the effects of non-uniform load introduction and boundary condition imperfections on the compression response of composite cylindrical shells with cutouts. He defined the non-uniform load distribution as anything other than uniform axial displacement of cylinders loading surface and found two sources of non-uniform load introduction. One was due to lack of planarity in the loading surfaces of the specimen and the loading platens. The other source was due to tilt of the loading platen with respect to the specimen before the loading began. He measured the top and bottom loading surface imperfections as well as potting thickness. Then the imperfection data was fit to curves and input into the STAGS models. Furthermore, the test frame loading platen was modeled as rigid flat plates and generalized contact definitions* given in STAGS were used. A similar modeling approach was used in the present study in order to identify the causes of the substantial out-of-plane deformations in addition to nonlinear end shortening during the early stage of the test. The loading platen was modeled as a rigid * Generalized contact definition means that contact points are calculated by STAGS rather than specified by the user. 37 As shown in Equation (3.1), this method needs two analyses to compute energy release rate and A a should be small enough to obtain accurate results In the case of load control, the expression for G is modified as G dU dA u^-ut AA const force (3.2) 3.3 Path Independent J-Integral Consider a homogeneous body of linear or nonlinear elastic material without singularity shown in Fig. 3.3. The strain energy density Uo is defined as Â£ U0 =U(e) = \crijdeij (3.3) o where <7 y is the stress tensor and Â£y is the infinitesimal strain tensor. The J- integral to compute the energy release rate G is defined as [79] J = \ (u onx <7 ij" ji,x)ds i = 1,2; = 1,2 (3.4) r where w, is the displacement, n, are the direction cosines of the outward normal along the path T The indices, i and j or x and z are used interchangeably for convenience. Further, summation is performed over repeated indices. An application of divergence theorem gives: J = $r(USH ~ .,)>* = A cr,,u:.AdA (3.5) 92 5.4 Buckling and Postbuckling Analysis Results of Plate with Single Delamination In order to see the effects of the delamination location through thickness, a unidirectional graphite/epoxy laminated plate with single through-the-width delamination was considered (see Figure 5.5). The geometry and material properties used for the graphite/epoxy laminates are given in Table 5.3. Axial compressive load (1,250 lb) was applied to load introduction edges. Three different thickness ratios of top and bottom subraminates were considered and buckling load factors are summarized in Table 5.4. The computed buckling modes agree well with deformed shapes obtained from the test in Reference [53] (see Figures 5.6, 5.7). As expected, the delamination located near the surface of the plate has the lowest buckling load, where local buckling is the dominant buckling mode. As the thickness ratio of sublaminates increases, the buckling mode changes from local buckling to mixed buckling mode (see Figure 5.8) and then from mixed to global buckling mode. Thickness of top sublaminate (hi) Thickness of bottom sublaminate (h2) Figure 5.5 Graphite/epoxy laminated plate with single through-the-width delamination. All dimensions are inch. 98 TABLE 5.5 Graphite epoxy lamina material properties from Ref. [83] and Ref. [86] Ref. [86] Ref. [83] Youngs modulus (longitudinal) E, =19.50 x 106 psi E, =18.50 x 106 psi Youngs modulus (transverse) E2 =1.48 x 106 psi E2 = 1.48 x 106 psi Shear modulus G|2 =0.80 x 106 psi G12 =0.80 x 106 psi Poissons ratio v12 = 0.3 < to II o u> TABLE 5.6 Geometric parameters of a blade stiffened panel from Ref. [83] and Ref. [86] Panel Ref. [86] Ref. [83] Panel length (in) 21 30 Panel width (in) 5 8 Blade height (in) 1.45 3.0705 Skin thickness (in) 0.083 0.208 Blade thickness (in) 0.105 0.3536 Flange width (in) 2.0 2.4 Flange thickness (in) 0.067 0.3536 20 that involve very large deformations while the total Lagrangian is more convenient for analyzing the structures with moderately large deformations. The primary objective of this section is to review briefly the non-linear finite element formulation based on continuum mechanics. The detailed description can be found in References [69, 70]. 2.1.1 Equation of Equilibrium Using the principle of minimum total potential energy one can derive the finite element equations. Assume that there exists a total potential energy of the form for linear elastic analysis J{Â£}r {a)dV (Jv{}r {fh }dV + u}T[fs}dS) (2.1) where {u} is the displacement vector, [fb} and [/s] are body and surface force vectors. The relationship between stresses and strain is of the form: {<7} = [C]{Â£} (2.2) where [C] is the constitutive matrix. The condition of equilibrium requires that the first variation of the total potential energy vanish: S = \[Se]T {(j}dV -(Â¡{Su}T {fb)dV + js{Su}T {fs}dS)=0 (2.3) V V From Zienkiewicz [69], strain can be expressed in matrix notation as {Â£} = [Â£]{ll} = [fl0] + [flJ{M} (2.4) 72 Table 4.7 Differences between baseline design analysis and actual test panel analysis. Baseline design Test panel Effect on buckling load Loading (lbs/inch) Axial compression Inplane shear load 20,000 5,000 20,000 0 -20% Imperfection (initial bow type) -3% of panel length Not measured -30% (with shear load) -18% (w/o shear load) Boundary Condition Loaded edges Unloaded edges Simple support Simple support Clamped (potted) Unsupported 1% (with test material) Transverse shear deformation no yes 11% 4.6 Nonlinear Analysis Although the linear buckling loads provide a measure of the compressive load carrying capacity of the stiffened panels, the test results indicate that the panels underwent substantial nonlinear transverse deformations prior to failure. Hence it was decided to perform a nonlinear analysis in order to understand the effects of boundary conditions including the eccentricity in load application. The nonlinear analysis was started without applying any initial imperfection, but the differences in the stacking sequence and the differences in material properties between the skin and blades induce bending deformation. The modified Riks path following algorithm in STAGS was used for the nonlinear analysis. The computation time required for the nonlinear analysis was an order of magnitude higher than for the linear analysis, indicating significant nonlinearity (probably near the buckling load). In the following sub-sections we discuss the results of the nonlinear analyses, and make a comparison between experimental and analytical results. 41 equilibrium equations exactly. This is because that the transverse shear stresses T xz in the beam are computed actually by substituting for <7 ^ and then integrating the first equilibrium equations. Thus, the first of Equation (3.10) is satisfied. The shear stresses at a cross section are proportional to the shear stress T xz that is constant along each ligament of the delaminated beam as well as the intact beam ahead of the crack tip. Thus the shear stresses T Iz are independent of x in each of the sublaminates and the first term in the second equilibrium equation is zero. Since beam theory assumes that (7 a are negligible, the second term is also zero. Thus, second equilibrium equation is also identically satisfied. Then the J-integral evaluated using beam theory around the closed path ABCDEF in Figure 3.5 is identically equal to zero. Further if we decompose delaminated beam with three sublaminates as shown in Figure 3.6 and consider J- integral for Sublaminate 2. Because the integrals along the horizontal paths are zero, it can be shown that Jab=^hg- Similarly it can be shown that J CD = J KL and J ef = J mn Thus it is now possible to move the three vertical paths AB, CD and EF to near the crack tip (HG, KL and MN) without loss of accuracy. The J-integral evaluated around the Paths 2,3, and 1 (HGKLMN) near the crack tip in Figure 3.5 has been called the zero-volume J-integral or zero-area J-integral, and it is given by G = y(l) + y(2) + y(3) (3.ii) where superscript (1), (2), and (3) represents the paths 1,2, and 3, respectively. 7 as MSC/NASTRAN [23] and HKS/ABAQUS [24] provide buckling and postbuckling analysis capability of the stiffened composite panels as one of several analysis options. Knight and Starnes [5] pointed out in their review paper that STAGS [25] is perhaps the premier shell analysis code that focused primarily on shell analysis and solution procedures for shell problems. 1.2.1.2 Analytical and experimental correlation Williams and Stein [26] examined J- and blade-stiffened graphite/epoxy panels experimentally as well as analytically using several analysis codes such as VIPASA, BUCLASP-2 and an early version of STAGS, which was based on a two dimensional finite difference technique. In their study, correlation of experimental and analytical results, which included inplane displacement restraints, indicated that the buckling strain of J-stiffened specimens were 75% to 80% of analytical values and that of blade- stiffened panels were 84% to 97% of the analytical values. A nonlinear response was exhibited by several of the specimens in which large lateral displacement in the order of one-quarter of the thickness of the panel plate segments were observed. Starnes and coworkers [27-28] investigated the postbuckling behavior of flat and curved stiffened graphite-epoxy panels loaded in compression. Panels with four equally spaced I-shaped stiffeners and quasi-isotropic skin were tested. Failure of all panels initiated in a skin-stiffener interface region. They showed that analytical results obtained from PASCO as well as STAGS correlate well with typical postbuckling test results up to failure. Their results also showed that modeling of the stiffener components with plate elements having appropriate stiffness is required to obtain satisfactory correlation with the postbuckling test results. 53 Figure 3.9 Crack-tip forces in delaminated plate. Then from Equation (3.33) an expression of point-wise energy release rate can be derived as: G(s) = ~Fl(CJ_ + C)Fc (3.37) where the crack tip forces F_c are given by: [Â£c J = [(AT a'!'1 ). (- v), (m;4) (AC-ACuer-ei1)] (i) (4) (3.38) The compliance matrices will take the form: 49 Since CFdenote the deformations we can write (3.26) as G = - (3.27) In deriving the last term of the column matrix in Equation (3.27), we have used the fact that the beam rotation at the crack tip is same for both ligaments 1 and 2, i.e. . Multiplying and dividing the right hand side of Equation (3.27) by Ax where Ax is a small length used in the virtual crack closure method, we obtain G = 1 JjT F c 1 2Ax (Mo)-o))-(m2) -4) (y/w -i//(,))-(y/(2) \f/(,)) > (w(1)+/))-(/)-/)) (3.28) The superscript (t) in Equation (3.28) denotes displacements and rotation at the crack tip, and superscript (/) and (2) denote respectively the displacements of the top and bottom ligaments at a distance Ax from the crack tip. In deriving Equation (3.28) we have used the finite difference approximation of the type, u (i) . 0.x U Q U Ax (/) 0 (3.29) 15 element analysis. He showed that delamination extension does not occur until buckling is significantly progressed. A plane finite element was developed by Gim [61] based on lamination theory that included the effects of transverse shear deformation. In the modeling of two-dimensional delaminations in laminated plates, the undelaminated regions was modeled by a single layer of plate elements while the delaminated region was modeled by two layers of plate elements with node offset. Sankar and Sonik [62] proposed a simple expression for the point-wise strain energy release rate along the delamination front using Irwins crack closure technique. They applied this technique in analyzing stitched double cantilever beam specimens as well as elliptic delaminations in composite plates. Klug et al. [63] investigated efficient modeling of postbuckling delamination growth using plate elements and gap elements. Energy release rate was computed using virtual crack closure technique. From this, a procedure to simulate a successive delamination growth was proposed. Kim [64] presented a modeling approach to study the postbuckling behavior of composite laminate with embedded delamination using two-dimensional shell element and rigid beam elements. 1.2.2.4 Two-dimensional multiple delaminations Suemasu et al. [65-66] analyzed the compressive behavior of plates with mutiple delaminations of different sizes. They showed that the effect of variation of the size of delamination on the compressive behavior is significant and postbuckling behavior is different from that of plates with equal sizes of delamination. Zheng and Sun [67] proposed a triple plate finite element model to analyze delamination interaction in laminated composite structures. Energy release rate was obtained by using virtual crack 4.7 Conclusion 83 Analytical models using several structural analysis models were used to assess the adequacy of the design model and the correlation with experimental results for a stiffened panel designed using the PASCO program. Of the effects neglected by the simple model, shear deformation was the most important, accounting for about 11% difference in buckling load. The effect of simplified (simple support) boundary conditions was small. The addition to the design model of shear loads and imperfections to improve the robustness of the result did help, even though the inclusion of one-sided imperfection apparently induced sensitivity to imperfection of the opposite sign. Overall, the simplified model did produce designs that in the experiments failed slightly above the design load. The most significant difference between the analytical predictions and experimental measurements was the substantial out-of-plane pre-buckling deformations. To explain these differences, imperfections, load eccentricities, and loading platen tilt angles were considered. Of these the loading platen tilt produced similar patterns of deformation, but these had more nonlinear characteristics than the measured deformations. 25 Figure 2.1 Nonlinear response from load versus displacement. 2.2.1 Newton-Raphson Method A system of nonlinear equilibrium equation can be written as Â¥ (m ) = / (u) f (2.14) where internal forces I(u) is defined as I(u) = j[B]{cj}dV (2.15) v Unbalance forces ^(m) represents the difference between internal and external forces. The basic problem is to find solutions that satisfy the nonlinear equilibrium equation, Viu )= 0. Since Equation (2.14) cannot be solved directly for the displacement of u, both an incremental equation of equilibrium from Equation (2.14) and iterative 73 4.6.1 Compressive Load Versus End-shortening The compressive load versus end-shortening deflection curves from the STAGS nonlinear static analyses, the test, and a linear fit (regression analysis) of the measured data are shown in Fig. 4.7. Recall that the test panel designated as GA2461 in Reference [80] is the baseline design for the present paper. In comparison to the baseline panel, two other test panels (GA2414, GA2458) from Reference [80] have slightly different geometries and stacking sequences. As expected, their compressive load versus end-shortening curves from the tests exhibit a similar trend except at the initial stage of loading. The prebuckling stiffness (EA) was calculated from the slopes of the linear portions of the experimental load versus end-shortening curves for the three panels and by multiplying the slopes by the panel length. The prebuckling stiffness from the experiments is compared in Table 4.8 with the prebuckling stiffness predicted using STAGS for test panel GA2461. It is observed that the prebuckling stiffness of the test panels is about 6% lower than the analytical value. Table 4.8 Comparison of the prebuckling stiffness and buckling load results from STAGS and experiments. EA/L (kip/in) EA (kip) Buckling load factor STAGS nonlinear (GA2461) 3931.2 117,900 1.23 Experimental Result (GA261) 3453.9 110,500 1.09 Experimental Result (GA2414) 3347.8 107,100 0.94 Experimental Result (GA2458) 3333.8 106,700 1.07 23 large deformation stiffness matrix [KL\ depend on the load level P, linearized buckling analysis approximates the tangent stiffness matrix at given load level P as [40]: [/sTr(P)] = [^0] + -^-([^(AP)] + [^(AP)]) (2.12) AP where both geometric stiffness matrix and large displacement matrix are computed at small load level AP. If we assume that the critical load is equal to AP, then the condition for a singular point becomes a standard eigenvalue problem. There are two widely used numerical methods for extracting eigenvalues, the Lanczos method and the subspace iteration method. The Lanczos method is generally faster when a large number of eigenmodes is needed for a system with many degrees of freedom. The subspace iteration method is effective for computing a small number of eigenmodes. Based on the assumption that the displacements {u} are infinitesimal for the small load AP classical buckling problem further simplifies Equation (2.12) as: ([A'0] + A(r,[A: However the applications of Equation (2.13) should be limited in practical engineering problems. In order to avoid the erroneous computation of the stability points in real engineering applications, the stability problem should be investigated using full tangent stiffness matrix in Equation (2.11) [69]. 65 Element (Table 4.2) is probably due to transverse shear deformation since the thickness of the skin-flange combination is 0.56 inch. Shear deformation was not included in the original panel design, and this difference indicates that that effect is substantial. The panel with negative bow-type imperfection had a concave surface in the middle of the panel. Thus, the blade tip is subjected to less axial compression and skin is under more axial compression than that of the perfect panels in the neighborhood of mid-length in the axial direction. Similarly, the blade tip near the boundary is under more compression and the skin near the boundary is under less compression than that of the perfect panel. The opposite holds for the positive bow-type imperfection. From Table 4.2, it is clear that the effect of the shear load on the buckling load is small, but the effect of the imperfection is very significant. From the last two rows of Table 4.2 one can note that a 3% positive imperfection results in a very low buckling load factor. The buckling load factor reduces from 1.256 to 0.394. A 3% negative imperfection also reduces the buckling load (from 1.256 to 0.856), but the reduction is smaller than that for a positive imperfection. It should be mentioned that a 3% imperfection is very large for a 32-in. stiffened panel, and thus will lead to very conservative designs. 4.5.2 Effects of Boundary Condition and Material Properties There were slight differences in material properties, panel dimensions and the boundary conditions between the baseline deign and the actual test conditions. In order to understand the effects of these differences, analyses were carried out using both sets of input data. The differences in material properties and dimensions are summarized in Table 4.3 and Table 4.4, respectively. While the changes in the material properties can I Differentiating the strain energy density, 38 w0 at/0 agg (3.6) 3* dx The area integral in Equation (3.5) vanishes. Therefore Equation (3.4) is equal to zero for any closed contour V In order to understand the zero volume integral described in subsequent section easily, consider the conservation integral around a region with singularity as shown in Fig. 3.4. The J integral along closed paths T, through r4 surrounding crack-tip vanishes as J = (3.7) r, + r2 + r3 + r4 But (J Â¡ji = 0 and nx = 0 along path T2 and T4 Thus the integral along r, clockwise and the integral along T3 counterclockwise sum to zero. r, r From equation (3.8) we can show that J integral along path T, and J integral along path r3 have the same value: 94 [msopai Es kTRAN Version 5 01 -Sep- 00 JM 07 Cm* beaeO. Loodstoc 0 Mode: 1. CrOcel Load 0 57858EMM. i)0 elg 01: Ceee beteO LoedMep: 0. Mode: I. Crtfcsl Load: O.STS9*E01. beee0.elg.0l TraneMMral-(NON-LAYERED) t.0000 ! sMionel (NON-LAYERED) 9.33-01 I \L. d^.*_FHnge M1 00*00 g-N; Mm 0 0*ld1 . Me* rOO^'N8M1 (b) Figure 5.6 Global buckling mode from test and analysis, (a) Buckling mode in mid-plane delamination [53]. (b) Buckling mode from analysis (hl/h2=1.0). 104 Debond ratio (at) baseline flnagewdh (+10%) flange widh (+10%), thickness (-t25/<) -"flange thickness (+100%) Figure 5.15 Effects of the flange geometry on buckling loads Figure 5.16 Effects of the Stacking Sequences on Buckling Loads 108 Table 5.7 Elastic material properties [87] Resin Aluminum Graphite/Epoxy El GPa 3.4 71.0 134 E2 GPa 3.4 71.0 13.0 Gl2 GPa 1.3 27.3 6.4 V12 0.3 0.3 0.34 Table 5.8 Average strain energy release rates for DCB specimens (104 J/m2) Isotropic (aluminum) Graphite/Epoxy (0 degree) Graphite/Epoxy (+/- 45 degree) Graphite/Epoxy (90 degree) 3-D 1.0 0.572 2.54 N/A CFTM 0.98 0.541 2.71 5.33 VCCT 0.90 0.522 2.01 5.14 Figure 5.19 Deformed shape of DCB specimen. 123 39.Chang, F.-K., and Chang, K. Y., A progressive damage model for laminated composites containing stress concentration, J. Comp. Mat. 21 (1987):834-855. 40.Singer, J., Arbocz, J., and Weller, T., Buckling experiments: experimental methods in buckling of thin-walled structures: basic concepts, columns, beams and plates, John Wiley & Sons, Chichester, 1998 41.Sleight, D.W., Knight, N. F., Jr., and Wang, J. T., Evaluation of a progressive failure analysis methodology for laminated composite structures, AIAA-97-1187-CP, 1997. 42.Hashin, Z., Failure criteria for unidirectional fiber composites, J. Appl. Mech. 47 (1980):329-334. 43.Chai, H., Babcock, C. D. and Knauss, W., One-dimensional modeling of failure in laminated plates by delamination buckling, Int. J. Sol. Struc. 17 (1981): 1069-1083. 44.Kardomateas, G. A., End fixity effects on the buckling and postbuckling of delaminated composites, Comp. Sci. Tech. 34 (1989): 113-128. 45.Chen, H. -P., Shear Deformation Theory for Compressive Delamination Buckling and Growth, ALAA J. 29 (1991 ):813-819. 46.Yin, W. L. and Wang J. T. S., The energy release rate in the growth of a one dimensional delamination, J. Appl. Mech. 51 (1984):939-941. 47.Simitses, G. H., Sallam, S. and Yin, W. L., Effect of delamination of axially loaded homogeneous laminated plates, AIAA J. 23 (1985): 1437-1444. 48.Yin, W. L., Sallam, S. and Simitses, G. H., Ultimate axial loaded capacity of a delaminated beam-plate, AIAA J. 24 (1986): 184-189. 49.Sheinman, I. and Soffer, M., Effect of delamination on the nonlinear behavior of composite laminated beams, J. Engng Mater. Technol. 112 (1987):392-397. CHAPTER 1 INTRODUCTION 1.1 Background Stiffened laminated composite panels have been considered for use in weight- sensitive structures such as aircraft and missile structural components. The main advantage of the stiffeners is the increased structural efficiency of the structure with a minimum of additional material. Due to the high stiffness of fiber composites, stiffened composite panels are usually thin. Thus, buckling characteristics are critical considerations for the optimum design of composite structures made of laminated composite plates. Buckling depends on a variety of factors, such as the geometry of the members, boundary conditions and material properties. Because of geometric complexity, stiffened laminated composite panels in axial compression have several buckling modes including general instability, local skin buckling and rolling of stiffeners. Furthermore, stiffened laminated composite flat panels usually exhibit stable postbuckling behavior which, in general, leads to significant differences between buckling load and ultimate failure load. The correct estimation of the load carrying capacity of stiffened panels is therefore very complicated. Advanced fiber-reinforced composite materials such as graphite-epoxy have relatively low transverse tensile and interlaminar shear strengths compared to in-plane 1 BUCKLING AND DELAMINATION ANALYSES OF STIFFENED COMPOSITE PANELS IN AXIAL COMPRESSION By OUNG PARK A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999 8 Romeo [29] conducted several tests on graphite-epoxy hat-and blade-stiffened panels under uniaxial compression and wing-box beams under pure bending to verify the accuracy of the theoretical analysis. Overall buckling, local buckling and torsional buckling were determined separately using a simple engineering formula, and interactions between these modes were not considered. Adequate correlation with experimental results was obtained for axial compression when the Euler or torsional buckling mode was critical; buckling occurred at lower strain values than predicted when the local buckling mode was critical. Furthermore, he showed that simple compression tests could not represent the load conditions of wing-box compression panel properly; in particular, the bending curvature causes a distributed load perpendicular to panels that could reduce the longitudinal load at which buckling occurred. Bushnell et al. [30] conducted optimum design, fabrication, and test of graphite- epoxy curved, locally buckled panels in axial compression. Three nominally identical large panels were tested. Two of three tests gave reasonably good agreement between test and theory, both with regard to loads at which the panels failed and the mode of failure. They also conducted experimental comparison between specimens with stitched skin-flange combination and specimens with adhesive-bonded skin-flange combination. They found that load carrying capacity of stitched specimens were lower than those of adhesively bonded specimens. Wieland et al. [31] investigated the buckling, postbuckling and crippling of AS4/3502 graphite-epoxy Z-section stiffeners as a function of specimen structural parameters. Variables considered were flange and web widths, flange-to-web corner 30 arbitrarily small loads. For example, a flat plate with in-plane loading exhibits no lateral displacements in the pre-buckling range. Thus it does not contain the lateral displacement modes of the buckled plate. Likewise, if the structure as well as loading is symmetric about some plane, all deformation modes antisymmetric with respect to that plane are possible bifurcation buckling modes. In such cases, we may choose to perform a buckling analysis with a nonlinear basic stress state. Sometimes when a bifurcation point does not exist at all, the bifurcation buckling approach may still considered as an acceptable measure of the critical load. If the structure is statically indeterminate and thus allows favorable redistribution of the stresses (e.g., shells with cutouts), then the bifurcation approach is too conservative. If the stiffness of the shell deteriorates with increasing load (e.g., long cylinders under bending), the bifurcation approach gives unconservative results. The bifurcation buckling analysis with a linear stress state is probably a good approximation for any case in which the squares of the rotations in the linear solution are small in comparison to the membrane strains at the load level corresponding to bifurcation. Most of commercially available finite element codes provide an option to perform the stress analysis first, save the data for a certain number of load steps on tape and later decide for which of those load steps buckling loads should be obtained [23- 25] This option may save some computing time. First, it may be easier to decide on the load levels at which eigenvalues are desired after the results of the stress analysis have been conducted. Next, it makes possible to find additional eigenvalues in a subsequent run. When eigenvalues are computed in a later run, the data deck for the nonlinear prestress analysis can be used. In addition, the user has the option to select certain data 121 19.Dawe, D. J. and Peshkam, V., Buckling and vibration of finite-length composite prismatic plate structures with diaphragm ends, Part I: finite strip formulation, Comp. Meth. Appl. Mech. Eng. 77 (1989): 1-30. 20.Dawe, D. J. and Peshkam, V., Buckling and vibration of long plate structures by complex finite strip methods, Int. J. Mech. Sci. 32(9) (1990):743-766. 21.Dawe, D. J., Lam,S. S. E. and Azizzan, Z. G., Finite strip post-local-buckling analysis of composite prismatic plate structures, Comp. Struct. 48(6) (1993): 1011-1023. 22.Mohd, S. and Dawe, D. J., Finite strip vibration analysis of composite prismatic shell structures with diaphragm ends, Comp. Struct. 49(5) (1993):753-765. 23.http://www.mcsch.com 24.http://www.hks.com 25.Brogan, F. A., Rankin, C. C. and Cabiness, H. D., STAGS users manual, LMSC P032594, Version 2.0, Lockheed Palo Alto Research Lab., Palo Alto, CA, June, 1995. 26.William, J. G. and Stein, M., Buckling behavior and structural efficiency of open- section stiffened composite compression panels, AIAA J. 14(11) (1976): 1618-1626. 27.Starnes, J. H. Jr., Knight, N. F. Jr. and Rouse, M., Postbuckling behavior of selected flat stiffened graphite-epoxy panels loaded in compression, AIAA J. 23 (1985): 1236-1246. 28.Knight, N. F. Jr. and Starnes, J. H. Jr., Postbuckling behavior of selected curved stiffened graphite-epoxy panels loaded in compression, AIAA J. 26 (1988):344-352. 29.Romeo, G., Experimental investigation on advanced composite stiffened structures under uniaxial compression and bending, AIAA J. 24(11) (1986): 1823-1830. 54 C r ''13 C, 4 C, 00 ^13 r ''33 C34 ^36 r ^38 C,4 r ^34 C44 C46 r ^48 Q r ''36 q6 r ^68 *-18 r ^38 Qs ^68 c ^88 (3.39) where the Cij are the coefficents of the full compliance matrix C, or Ch It should be mentioned that the xyz coordinates should be moved along the delamination front while using Equation(3.30) for computing pointwise G(s). We have derived three formulars Equations (3.27), (3.31), and (3.34) for computing the point-wise energy release rate in a delaminated plate. Out of the three, Equations (3.32) and (3.35) are exact, and their accuracy is limited only by the methods used compute the force and moment resultants or the strain energy densities ahead and behind the delamination front. The accuracy of the VCCT given by Equation (3.30) is limited by not only the crack tip forces but also the mesh size which will define the length of virtual crack growth. 3.9 Summary In this chapter a new method called Crack Tip Force Method (CTFM) was introduced and derived. Three methods of computing G for laminated composites structures are also discussed. A Crack Tip Force Method is derived for computing 80 deformation. However, the details of the displacements are considerably different. It is concluded that some combination of tilt angles and the contact stiffnesses can produce the observed pattern, but there may be some other contribution to the out-of-plane displacements. i TILT ANGLE Figure 4.11 Schematic of blade stiffened panel and loading platen. 69 IASC7PA7RANVreton 8 2 30-1^1-98 1952:12 FRINGE: Bucfcto, SIp2,lAx3l .EonValu*.20 <82: Olotmalbn, Diiptao*mn1s (VEC-WAG) ABAQUS 06FORMATION: Btcfcla. Stop2,lbtodI.EienV*kj=20.4a2 CMornafon, Dt^tocnwnl -ABAQUS Figure 4.5 The predicted buckling mode shape of the blade stiffened panel with 9-node shell elements from the ABAQUS (load factor=1.218). Figure 4.6 The predicted buckling mode shape of the blade stiffened panel from the STAGS (load factor=1.154). epoxy material used in Reference 5 are given in Table 4.1. 58 Table 4.1 Hercules AS4/3502 graphite epoxy lamina material properties. Youngs modulus (longitudinal) E, =18.50 x 106 psi Youngs modulus (transverse) E, =1.64 x 106 psi Shear modulus G,2 =0.87 x 106 psi Poissons ratio V12 = 0.3 Density p =0.057 lb in3 Ply thickness tpiy = 0.0052 in The baseline design was designed to support an axial load Nx of 20,000 lb./in. In addition, in order to account for off design conditions, imperfections and modeling inaccuracies, a shear load (Nxy = 5000 lb./in) and a longitudinal bow type (3% of the panel length) imperfections were added. The baseline design panel was assumed to be simply supported along the four edges, which is the only boundary condition that can be accurately modeled using PASCO. 112 Comparison of energy release rates versus end-shortening using strain energy derivative method and virtual crack closure technique was made in Figure 5.30. In general, energy release rate from strain energy derivative method agrees well with average energy release rate from virtual crack closure technique. The critical energy release rate must be obtained by experiment. If the assumed critical energy release rate is about Gct =200 J/m2, then debond extension will be first started underneath the blade stiffener below the buckling load. From the fact that distributions of energy release rate in the debond front are not uniform, using the energy release rate from virtual crack closure technique may give more accurate estimation of debond extension than that from the strain energy derivative method. 5.7 Conclusion Buckling and postbuckling behavior of a stiffened panel with a partial skin- stiffener debond are investigated using the finite element method. The present model shows good correlation with the results of existing buckling analyses of a delaminated plate. There exists a critical debond ratio such that if the ratio of debond is less than a critical value, then the buckling load is unchanged. Furthermore, this critical debond ratio depends on geometry, boundary conditions and material properties. Three methods for computing the energy release rate along the crack front with plate elements are proposed to predict debond extension during postbuckling. When local buckling mode is dominant, the maximum energy release rate, which occurs below the stiffener blade, can be much greater than the average energy release rate. 107 5.6 Comparison of Energy Release Rate First, in order to verify the three methods of computing G, using SEDM, VCCT, and proposed CFTM, a double cantilever beam modeled plate with offset node elements was considered. The dimension of DCB specimens (see Figure 5.19) were that used by Raju et al [87] and are as follows: total length 101.6mm; delamination length 50.8 mm; width 24.4 mm; total specimen thickness 3.3 mm; sub-laminate thickness 1.65 mm. The material properties used are listed in Table 5.7. In addition to unidirectional graphite/epoxy, a 16-layer angle ply laminate with the lay-up [+45, -45 }g and an isotropic DCB were also analyzed. The transverse force applied to each ligament of DCB is 1 N/m. The normalized energy release rate values computed using the average G values for the various specimens are shown in Figures 5.20 to 5.23, respectively. The average values used for normalization were compared with those of the 3-D analysis results of Raju et al. [87] in Table 5.8. The average G values from CTFM are closer to those from 3-D than the average G values from VCCT. The accuracy of CTFM depends on the refinement of the finite element along the crack front while the accuracy of VCCT depends not only the refinement of the finite element along the crack front but also that of ahead and behind crack-tip. This may be the reason why the average values of G from VCCT deviate from those of 3-D. However, the comparison of normalized G distribution along the crack front is very good for the example considered. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bhavani V. Sankar, Chair Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Raphael T. Haftka, Cochair Distinguished Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ibrahim ICEbcioglu Professor Emeritus of Aerospace Engineering, Mechanics, and Engineering Science 11 and the measured response to differ substantially. Dvila et al. [38] conducted progressive failure analysis for the simulation of damage initiation and growth in stiffened thick-skin stitched graphite-epoxy panels loaded in axial compression. Failure indices approach, proposed by Chang and Chang [39], was adopted to evaluate the failure mode and location corresponding to all of the major composite laminate failure modes except delamination. Superposed layers of shell elements with multiple integration points through the thickness were used to separate the failure modes for each ply orientation and to obtain the correct effect of bending loads on damage progression. The analysis results were compared with experimental results for three stiffened panels with notches oriented at 0, 15, and 30 degrees to the panel width dimension and found to be in excellent correlation with the experimental results. The local reinforcing effect of Kevlar stitches was simulated in the finite element model by multiplying the fiber buckling strength allowable value, independent of the other stress components, by a stitch factor that is determined empirically. A parametric study was performed to investigate the damage growth retardation characteristics of the Kevlar stitch lines in the panels. The debond between the stiffener flange and the skin were not modeled. Hence, the predicted results were found to be less accurate after the damage zone reached the stiffener flange. Singer et al. [40] presented conventional and less conventional experimental methods in buckling of a vast variety of thin-walled structures in considerable detail. The parameters, which may influence the test results, were systematically highlighted: imperfections, boundary conditions, loading conditions, effect of holes and cutouts. Though authors deals primarily with experimental methods and test results, the 17 Two finite element modeling approaches for stiffened panels are commonly used in the literature. One approach is to model the skin with plate elements and to model the stiffener with beam elements. The other is to model both skin and stiffener with plate elements. The second approach appears more attractive for modeling the debonded region between skin and flange. In this study, the panel skin and blade stiffeners are modeled with plate elements. The nodal penetration of the delaminated skin-stiffener interface can be prevented either by adjusting spring constants of fastener elements or by gap elements. Furthermore, an energy release rate for calculating delamination extension is computed using several methods based on fracture mechanics. In order to validate the present modeling approach, a plate with a through-the- width delamination was modeled and linear bifurcation and nonlinear postbuckling analyses were conducted. Results were compared with the available experimental results. In Chapter 2, basic finite element formulation for buckling problem and nonlinear solution algorithms for postbuckling analysis are briefly described. Chapter 3 provides several methods for computing energy release rate in plate-like structures based on fracture mechanics. In Chapter 4, effects of boundary conditions, material properties, and initial geometric imperfections on buckling and nonlinear prebuckling behavior of blade stiffened panel are investigated. Chapter 5 describes modeling of delamination with elastic spring element, linear buckling analysis results of both delaminated composite plates and debonded stiffened panels, and effects of delamination length, stiffener geometries and stacking sequences on buckling load. Strain energy release rate was computed using the strain energy derivative method, the virtual crack closure 100 critical debond length varies with the boundary conditions. Furthermore, local buckling is relatively insensitive to boundary conditions compared to global buckling. Variations of buckling loads with respect to different stiffener geometry and stacking sequence for the panel configuration from Ref. 86 are shown in Figures 5.15 and 5.16, respectively. As expected, buckling loads increase in proportion to the increase of stiffener thickness. When the debond ratio is greater than about 0.4, contributions of increased stiffener thickness on buckling load are relatively small due to the local buckling. Figure 5.16 shows that the panel with all 0 degree plies shows small buckling loads compared to the panels with cross plies or balanced symmetry stacking sequences when debond ratios are less than 0.2. This suggests that stiffened panel with all 0 degree plies can lose stability with ease in spite of high axial stiffness. Buckling analysis of the corresponding panel without debond was also conducted using PANDA2 [15-17]. The predicted buckling load factors from PANDA2 with and without neglecting the redistribution of axial compressive load due to local buckling are 0.81 and 0.92, respectively. Those buckling load factors are fairly close to finite element analysis results, 0.95, in Figure 5.16. 62 imperfection on the buckling loads of the baseline design were investigated using PANDA2, which employs analysis techniques with similar level of fidelity to that of PASCO. In PANDA2, local and general buckling loads are calculated by either closed- form expressions or by discretized models of panel cross sections based on an energy method [15]. STAGS is a finite element code for general purpose analysis of shell structures of arbitrary shape and complexity [25]. STAGS has a variety of finite elements suitable for the analysis of stiffened plates and shells. Four node quadrilateral plate elements with cubic lateral displacement variations (called 410- and 411-Elements) are efficient for the prediction of buckling response of thin shells. For thick plates in which transverse shear deformation is important, the assumed natural strain (ANS) nine node element (480-Element) can be selected [16]. The panel investigated here warrants the use of 480-Element, however 411-Element was also used as the panel was designed by PASCO, which does not model shear deformation. STAGS results were post-processed by PATRAN, which is a commercial software for pre- and post-processing of finite element simulations [82], 4.4.2 Finite Element Model The STAGS finite element model for the panel had a total of 20 branched shell units, and each branched shell unit had 65 x5 nodes (for the 32-in. long panel) or 61 x 5 nodes (for the 30-in. long baseline design panel), respectively. The axial compressive design load (640,000 lb) was applied with a uniform end-shortening constraint along with compatibility conditions for adjacent shell unit interfaces. In the test, the load was 13 delamination boundary. Simitses et al. [47] developed a one-dimensional model similar to one used by Chai et al. [42] to predict critical loads for delaminated homogeneous plates with both simply supported and clamped ends. They showed that the buckling loads could serve in certain cases as a measure of the load carrying capacity of the delaminated configurations. In other cases, the buckling load is very small and delamination growth is a strong possibility, depending on the toughness of the material. Yin et al. [48] found that a delamination length is short and located near mid plane of the plate, the buckling load of the delaminated plate is close to the lower bound of the ultimate axial load capacity. When a delamination length is long and locates near surface of plate, the postbuckling axial loads can be considerably greater than the buckling loads, while the failure of plate may or may not be governed by delamination growth. The effects of bending-extension coupling as well as imperfection were investigated by Sheinman and Shouffer [49]. They found that the coupling effect reduces the load carrying capacity, and imperfection sensitivity of global postbuckling deformation is very high. Wang [50] proposed the concept of a continuous analysis for determining interface stresses and strain energy release rate for the delamination at the interface of skin and flange. A shear deformable beam finite element with nodes offset to either the top or bottom side was proposed by Sankar [51]. Kyoung and Kim [52] investigated asymmetric delamination with respect to the center of the beam-plate. In their study, a variational principle based on shear deformation theory was used to calculate buckling 52 The derivation of Equation (3.35) is very similar to that of Equation (3.23). As before, the term (F_4 F_, ) is the matrix of crack-tip forces. They also represent the jump in force and moment resultants that occur across the delamination front. Sankar and Sonik [62] showed that three of the eight force resultants will be continuous across the delamination front. Assume a coordinate system such that the x- axis is normal to the crack front, y-axis is tangential to the crack front and z- is the thickness direction. Then the continuous force resultants are N y,M v and Q v. Thus the jumps in these force resultants are zero, i.e., N{4) -N = N{y2) -N[3) = 0 M{4) M(l) = M(2) M(3) =0 1V1 y 1V1 y ivi y 1V1 y u (3 36) ~ Qf = Qf ~ Q = 0 Thus there will be only five components to the crack tip forces (see Figure 3.9): three forces in the x, y and z directions; two couples, about x and y axes, respectively. The three forces will be the jump in N x, N xy and Qx across the delamination front, either in the top laminates (1 and 4) or bottom laminates (2 and 3). The two crack tip couples are the jumps in M x, M ^ Since the jumps in N y,M y and Q y are equal to zero and they do not contribute to the crack-tip forces, we can delete the 2n(^, 5^ and 7^ rows and columns in Q and Q,; we will denote them by C, and Cb . 124 50.Wang, J. T. S. and Huang, J. T., Skin/stiffener interface delamination using continuous analysis, Comp. Struct. 30 (1995):319-328. 51.Sankar, B. V., A finite element for modeling delamination in composite beam, Comp. Struct. 38(2) (1991): 1414-1426. 52.Kyoung, W. and Kim, C., Delamination buckling and growth of composite laminated plates with transverse shear deformation, J. Comp. Mat. 29 (1995):2047-2068. 53.Gu, H. and Chattopadhyay, A., An experimental investigation of delamination buckling and postbuckling of composite laminates, Comp. Sci. Tech. 59 (1999):903-910. 54.Kutlu, Z. and Chang, F. K., Composite panels containing multiple through-the-width delaminations and subjected to compression. Part Lanalysis, Comp. Struct. 31 (1995):273-296. 55.Kutlu, Z. and Chang, F. K., Composite panels containing multiple through-the-width delaminations and subjected to compression ILexperiments, Comp. Struct. 31 (1995):297-314. 56.Suemasu, H., Effects of multiple delamination on compressive buckling behavior of composite panel, J. Comp. Mat. 27(12) (1993): 1077-1096. 57.Suemasu, H., Postbuckling behavor of composite panels with multiple delaminations, J. Comp. Mat. 27(12) (1993): 1172-1192. 58.Lee, J., Gurdal, Z. and Griffin O. H. Jr., Layer-wise approach for the bifurcation problem in laminated composite with delaminations, ALAA J. 32 (1993):331-338. 59.Wang, J. T., Pu, H. N., and Lin, C. C, Buckling of beam-plates having multiple delaminations, J. Comp. Mat. 31(10) (1997): 1002-1025. 122 30.Bushnell, D., Holmes, A. M. C, Flaggs, D. L. and McCormick, P. J., Optimum design, fabrication and test of graphite-epoxy, curved, stiffened, locally buckling panels loaded in axial compression, Buckling of structures, edited by I. Elisakoff, Elservier Science Publishers B. V., Amsterdam (1988): 61-131. 31.Wieland, T. M., Morton, J. and Starnes, J. H. Jr., Scale effects in buckling, postbucking and crippling of graphite-epoxy z-section stifferers,ALAA-91-0912-CP, 1991. 32.Fan, S., Kroplin, B. and Geier, B., Buckling, postbuckling and failure behavior of composite-stiffened panels under axial compression, ALAA-92-2285-CP, 1992. 33.Nagendra, S., Haftka, R. T., Gurdal, Z. & Starnes, J. H. Jr., Design of stiffened composite panels with a hole, Comp. Struct. 18 (1991): 195-219. 34.Whetstone, J. D., EISI EAL-Engineering language reference manual, EISI-EAL System Level 2091, Engineering Information Systems Inc., Saratoga, NY, (July 1983) 35.Nagendra, S., Haftka, R. T., Gurdal, Z. & Starnes, J. H. Jr., Buckling and failure characteristics of compression-loaded stiffened composite panels with a hole, Comp. Struct. 28 (1994): 1-17. 36.Chow,W. T. and Atluri, S. N., Prediction of post-buckling strength of stiffened laminated composite panels based on the criterion of mixed-mode stress intensity factor, Comp. Mech. 18 (3) (1996):215-224. 37.Young, R. D. and Hyer, M. W., Accurate modeling of the postbuckling response of composite panels with skewed stiffeners, AIAA-97-1306-CP, 1997. 38.Davila, C. G., Ambur, D. R. and McGowan, D. M., Analytical prediction of damage growth in notched composite panels loaded in axial compression, AIAA-99-1435-CP 1999. 26 procedure are generally used for its solution. The Newton-Raphson method utilizes the first-order approximation of Equation (2.14) and can be written at load step n+1 as y (;;',)= 'r(<) + = o <2.i6> d u Here i is the number of iteration, and the tangent stiffness matrix is defined as ay a/ ^ A. 'j' du du (2.17) From Equation (2.16) we have the following iterative correction as (2.18) where Kj is the tangent stiffness matrix at the ith iteration. Thus the improved solution can be computed as u i+i n + l Un + l + Su'n + l (2.19) The convergence of the Newton-Raphson method is generally very fast. However, the cost of computation is usually high due to the calculation and factorization of the tangent stiffness matrix at each iteration. To reduce the burden of computational cost, the modified Newton-Raphson was introduced in which the stiffness matrix is approximated as a constant: A!-). ~ Kt. There are many possible choices of the approximated stiffness matrix Kt. For instance, Kt can be chosen as either the matrix corresponding to the first iteration or some previous load step. Schematics of Newton-Raphson method and modified Newton-Raphson method are shown in Figure 2.2. Figure 5.10 Buckling mode of stiffened panel from Reference [83] Figure 5.11 Global buckling mode 57 Recently, three of the panels designed by Nagendra et al. were fabricated and tested by the Structural Mechanics Branch at NASA Langley Research Center. The experimental failure loads differed by up to about 10% from the design load. However, there were significant differences in loading and boundary conditions between the design conditions and the test conditions. The principal objective of this chapter is to understand the effects of differences between the simplified assumptions made in the design model and the actual test conditions. Another objective is to assess the effectiveness of the simplified PASCO model originally used to design the panel, and this is aided by comparing the results obtained from several structural analysis programs including PANDA2, STAGS, and ABAQUS. 4,2 Stiffened Panel Definition The basic configuration of the panel designated as the baseline design - corresponds to the design in the 9th row in Table 7 of Reference [80], This panel, designated as GA2461 (referring to the design weight of 24.61 lb.), is 30-inches long and 32-inches wide with four equally spaced blade stiffeners (see Figure 4.1). The laminates used in Reference [95] for the skin, stiffener blade, and stiffener flange for the baseline design were balanced, symmetric laminates consisting of 0, 45 and 90 plies. The skin has 40 plies with a stacking sequence [45/90/1453/90/145,]s and the stiffener flange and blade have an identical stacking sequence of [45/(145/04)2 /902/04/(45/02)2/02/45]s with a total of 68 plies. Properties of the AS4/3502 graphite I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy^ Peter J. Ifju Assistant Professor of Aerospace Engineering, Mechanics, and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Professor of Civil Engineerin a This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1999 M. J. Ohanian Dean, College of Engineering Winfred M. Phillips Dean, Graduate School |