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PAGE 1 1 COMPARISON OF CONE MODEL AND MEASURED DYNAMIC IMPEDANCE FUNCTIONS OF SHAL LOW FOUNDATIONS By PATRICK WILLIAM DUNN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Patrick William Dunn PAGE 3 3 To Mom and Dad, thank you for your love, support, and inspiration. PAGE 4 4 ACKNOWLEDGMENTS I thank m y advisor for his guidance and patie nce during my studies over the years. I would like to thank my committee for their assistan ce with this project. Financial backing from the U. S. National Science Founda tion is greatly apprec iated. I thank my parents who have always been supportive throughout my life. Appreciation is due to my uncle and aunt for their continued encouragement. My friends and the rest of my family are also due a great deal of gratitude as well for their moral support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4LIST OF TABLES................................................................................................................. ..........7LIST OF FIGURES.........................................................................................................................8ABSTRACT...................................................................................................................................14CHAPTER 1 INTRODUCTION..................................................................................................................151.1 Problem Statement.......................................................................................................... ..151.2 Hypothesis........................................................................................................................171.3 Objectives.........................................................................................................................171.4 Scope.................................................................................................................................171.5 Organization of Dissertation.............................................................................................182 FOUNDATION IMPEDANCE FUNCTIONS...................................................................... 202.1 Definition................................................................................................................. .........202.2 Computation................................................................................................................ .....222.3 Measurement.....................................................................................................................263 CONE MODEL PREDICTIONS........................................................................................... 403.1 Wolf and Deeks (2004)..................................................................................................... 403.2 Wong and Luco (1978)..................................................................................................... 423.3 Mita and Luco (1989)....................................................................................................... 443.4 Wong and Luco (1985)..................................................................................................... 453.5 Apsel and Luco (1987).....................................................................................................483.6 Nii (1987)..........................................................................................................................483.7 Crouse, et al. (1990)...................................................................................................... ....514 SITE CHARACTERIZATION.............................................................................................. 994.1 Site Investigation..............................................................................................................994.1.1 Traditional Investigation........................................................................................ 994.1.2 Seismic Investigation............................................................................................1004.2 Foundation Selection and Construction..........................................................................1014.3 Shakers............................................................................................................................103 PAGE 6 6 5 MEASURED IMPEDANCE FUNCTIONS......................................................................... 1105.1 Footing 5.................................................................................................................. .......1105.1.1 Vertical Loading................................................................................................... 1105.1.2 Horizontal Loading...............................................................................................1115.2 Footing 6.................................................................................................................. .......1155.2.1 Vertical loading....................................................................................................1155.2.2 Horizontal Loading...............................................................................................1176 IMPEDANCE FUNCTION COMPARISON AND ASSESSMENT.................................. 1586.1 Cone Model Development.............................................................................................. 1586.1.1 Soil Profile............................................................................................................1586.1.2 CONAN Model....................................................................................................1596.2 Cone Model and Measured Impedance Function Comparison....................................... 1636.2.1 Footing 5...............................................................................................................1636.2.2 Footing 6...............................................................................................................1666.3 Assessment.....................................................................................................................1706.3.1 Measurement Error............................................................................................... 1706.3.2 Prediction Parameters........................................................................................... 1716.3.3 Model Assumptions.............................................................................................. 1727 CLOSURE............................................................................................................................1897.1 Summary of Findings.....................................................................................................1897.2 Conclusions.....................................................................................................................1907.3 Recommendations...........................................................................................................191LIST OF REFERENCES.............................................................................................................192BIOGRAPHICAL SKETCH.......................................................................................................195 PAGE 7 7 LIST OF TABLES Table page 31 Parameters for Eight Wong and Luco (1985) Comparisons.............................................. 54 32 Soil Profile for Apsel and Luco (1987) Model.................................................................. 55 33 Model Footings of Nii (1987)............................................................................................56 61 Free Field Soil Properties for Footing 5 and Footing 6................................................... 174 62 CONAN Input File for F ooting 5 Vertical Model. .......................................................... 175 63 CONAN Input File for Footing 6 Horizontal and Rotational Model............................... 176 PAGE 8 8 LIST OF FIGURES Figure page 21 Foundation / Soil Diagram.................................................................................................35 22 Foundation Free Body Diagram......................................................................................... 36 23 Cone Diagrams...................................................................................................................37 24 Reflected and Refr acted Wave Diagram ............................................................................ 38 25 Embedded Stacked Disks................................................................................................... 39 31 Vertical Impedance Functions for Model 1....................................................................... 57 32 Horizontal Impedance Functions for Model 1................................................................... 58 33 Rocking Impedance Functions for Model 1....................................................................... 59 34 Torsion Impedance Functions for Model 1........................................................................ 60 35 Vertical Impedance Functions for Model 4 Length/Width=4............................................ 61 36 Torsion Impedance Functions for Model 4 Length/Width=4............................................ 62 37 Horizontal Impedance F unctions in Long Direction fo r Model 4 Length/W idth=4.......... 63 38 Rocking Impedance Functions About Short Axis. for Model 4 Length/W idth=4.............64 39 Horizontal Impedance F unctions in Short Direction for Model 4 Length/W idth=4......... 65 310 Rocking Impedance Functions About Long Axis for Model 4: Length/Width=4............. 66 311 Vertical Impedance Functions for Model 3 Embedded..................................................... 67 312 Horizontal Impedance Func tions for Model 3 Embedded ................................................. 68 313 Rocking Impedance Functions for Model 3 Embedded.................................................... 69 314 Torsional Impedance Functions for Model 3 Embedded................................................... 70 315 Coupled Impedance Functions for Model 3 Embedded.................................................... 71 316 Vertical Impedance Functions for Model 1: Basis Model ................................................. 72 317 Horizontal Impedance Functi ons for Model 1: Basis Model ............................................. 73 318 Rocking Impedance Functions for Model 1: Basis Model................................................ 74 PAGE 9 9 319 Torsional Impedance Functions for Model 1: Basis Model............................................... 75 320 Vertical Impedance Functions for Model 4: Layer Thickness....................................... 76 321 Horizontal Impedance Functions for Model 4: Layer Thickness .................................. 77 322 Rocking Impedance Functions for Model 4: Layer Thickness...................................... 78 323 Torsional Impedance Functions for Model 4: Layer Thickness .................................... 79 324 Vertical Impedance Functions for Model 6: Higher Vs Contrast ......................................80 325 Horizontal Impedance Functions for Model 6: Higher Vs Contrast .................................. 81 326 Rocking Impedance Functions for Model 6: Higher Vs Contrast ..................................... 82 327 Torsional Impedance Functions for Model 6: Higher Vs Contrast.................................... 83 328 Vertical Impedance Functions fo r Model 8: Poissons Ratio=0.45 ...................................84 329 Horizontal Impedance Functions for Model 8: Poissons Ratio=0.45 ............................... 85 330 Rocking Impedance Functions for Model 8: Poissons Ratio=0.45 ..................................86 331 Torsional Impedance Functions for Model 8: Poissons Ratio=0.45 ................................ 87 332 Horizontal Impedance F unctions for Layered Model ........................................................ 88 333 Rocking Impedance Functions for Layered Model...........................................................89 334 Coupled Impedance Functions for Layered Model........................................................... 90 335 Vertical Impedance Functions for Circular........................................................................ 91 336 Vertical Impedance Functions for Circular........................................................................ 92 337 Vertical Impedance Functions for Rectangular................................................................. 93 338 Vertical Impedance Functions for Rectangular................................................................. 94 339 Vertical Impedance Functions for Surface, L/W=2 Model Footing.................................. 95 340 Vertical Impedance Functions for Station 6...................................................................... 96 341 Horizontal Impedance Functions for Station 6 .................................................................. 97 342 Rocking Impedance Functions for Station 6...................................................................... 98 41 Seismic, SPT, and profile................................................................................................. 105 PAGE 10 10 42 Footing 5.................................................................................................................. ........106 43 Footing 6.................................................................................................................. ........107 44 Reinforcement of Footing 6.............................................................................................107 45 ANCO Model MK12......................................................................................................108 46 NSF NEES Thumper mobile shaker............................................................................ 108 47 NSF NEES TRex mobile shaker................................................................................. 109 51 Vertical Experiment for Footing 5 with ANCO shaker. ..................................................121 52 ANCO vertical results for Footing 5................................................................................ 121 53 ANCO vertical results for Footing 5................................................................................ 122 54 Horizontal Experiment for Footing 5 with Model 400 ElectroSeis shaker. ...................123 55 ElectroSeis horizontal loading resu lts for Footing 5 at elevation 1 ft ............................124 56 ElectroSeis horizontal loading resu lts for Footing 5 at elevation 2 ft ............................125 57 ElectroSeis horizontal loading resu lts for Footing 5 at elevation 3 ft ............................126 58 ElectroSeis horizontal loading resu lts for Footing 5 at elevation 4 ft ............................127 59 ElectroSeis horizontal loading results for t he onedegreeoffreedom analyses of Footing 5 at elevation 1 ft................................................................................................ 128 510 ElectroSeis horizontal loading results for t he onedegreeoffreedom analyses of Footing 5 at elevation 2 ft................................................................................................ 129 511 ElectroSeis horizontal loading results for t he onedegreeoffreedom analyses of Footing 5 at elevation 3 ft................................................................................................ 130 512 ElectroSeis horizontal loading results for t he onedegreeoffreedom analyses of Footing 5 at elevation 4 ft................................................................................................ 131 513 ElectroSeis horizontal lo ading results for the none qual coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 2 ft................................................ 132 514 ElectroSeis horizontal lo ading results for the none qual coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 3 ft................................................ 133 515 ElectroSeis horizontal lo ading results for the none qual coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 4 ft................................................ 134 PAGE 11 11 516 ElectroSeis horizontal lo ading results for the none qual coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 3 ft................................................ 135 517 ElectroSeis horizontal lo ading results for the none qual coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft................................................ 136 518 ElectroSeis horizontal lo ading results for the none qual coupling twodegreeoffreedom analyses of Footing 5 at elevations 3 ft and 4 ft................................................ 137 519 ElectroSeis horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 2 ft.............................................................. 138 520 ElectroSeis horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 3 ft.............................................................. 139 521 ElectroSeis horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 4 ft.............................................................. 140 522 ElectroSeis horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 3 ft.............................................................. 141 523 ElectroSeis horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft.............................................................. 142 524 ElectroSeis horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 3 ft and 4 ft.............................................................. 143 525 ElectroSeis horizontal loading result s for t he onedegreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footi ng 5 at elevations 2 ft and 4 ft............................... 144 526 Vertical Experiment with Thumper shaker...................................................................... 145 527 Vertical Experiment with Thumper shaker...................................................................... 145 528 Thumper vertical results for Footing 6............................................................................ 146 529 Thumper vertical results for Footing 6............................................................................ 146 530 Vertical Experiment with TRex shaker.......................................................................... 147 531 Vertical Experiment with TRex shaker.......................................................................... 147 532 TRex vertical results for Footing 6................................................................................. 148 533 TRex vertical results for Footing 6................................................................................. 148 534 Horizontal Experiment for Footing 6 with Model MK12 ANCO shaker m ounted flush..................................................................................................................................149 PAGE 12 12 535 Horizontal Experiment for Footing 6 with Model MK12 ANCO shaker on stand. ....... 150 536 ANCO horizontal loading result s for Footing 6 flush m ounting..................................... 151 537 ANCO horizontal loading results for Footing 6 elevated m ounting................................ 152 538 ANCO horizontal loading results for the one degreeoffreedom analyses of Footing 6 flush mounting.............................................................................................................. 153 539 ANCO horizontal loading results for the one degreeoffreedom analyses of Footing 6 elevated mounting.........................................................................................................154 540 ANCO horizontal loading results for th e nonequal coupling twodegreeoffreedom analyses of Footing 6....................................................................................................... 155 541 ANCO horizontal loading results for th e equal coupling twodegreeoffreedom analyses of Footing 6....................................................................................................... 156 542 ANCO horizontal loading results for onedeg reeoffreedom at flushed and elevated positions, independent coupling twodegreeoffreedom, and equal coupling twodegreeoffreedom analyses of Footing 6........................................................................ 157 61 ANCO vertical comparison for Footing 5....................................................................... 177 62 ElectroSeis horizontal comparison fo r the onedegreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footi ng 5 at elevations 2 ft and 4 ft............................... 178 63 ElectroSeis horizontal comparison fo r the onedegreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footi ng 5 at elevations 2 ft and 4 ft............................... 179 64 ElectroSeis horizontal comparison fo r the onedegreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footi ng 5 at elevations 2 ft and 4 ft............................... 180 65 Thumper and TRex vertical comparison for Footing 6.................................................. 181 66 ANCO horizontal comparison for the onedeg reeoffreed om flush mount, independent coupling twodegreeoffree dom, the equal coupling twodegreeoffreedom analyses of Footing 6.........................................................................................182 67 ANCO horizontal comparison for the onedeg reeoffreed om flush mount, independent coupling twodegreeoffree dom, the equal coupling twodegreeoffreedom analyses of Footing 6.........................................................................................183 PAGE 13 13 68 ANCO horizontal comparison for the onedeg reeoffreed om flush mount, independent coupling twodegreeoffree dom, the equal coupling twodegreeoffreedom analyses of Footing 6.........................................................................................184 69 Footing 5 Vertical and 1DoF Error Band ....................................................................... 185 610 Footing 5 2DoF Error Band............................................................................................186 611 Footing 6 Vertical and 1DoF Error Band ....................................................................... 187 612 Footing 6 2DoF Error Band............................................................................................188 PAGE 14 14 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPARISON OF CONE MODEL AND MEASURED DYNAMIC IMPEDANCE FUNCTIONS OF SHAL LOW FOUNDATIONS By Patrick William Dunn May 2010 Chair: Dennis R. Hiltunen Major: Civil Engineering Developments in seismic testing methods a nd increased electronic recording capabilities have reached a level that allows computati on and verification of dynamic impedance function predictions for shallow foundations. Prior to thes e developments very little data was available for verification of impedance functions thus field design capabilities with respect to dynamic foundation response remains impractical. This study draws a positive rela tionship between cone models and multiple impedance function prediction t echniques. Cone models are then utilized to compare to the limited data available from previ ous field studies. Next, field data from an experiment designed for impedance function measur ement is discussed. For the experiment two square and embedded shallow foundations have been dynamically loaded at the National Geotechnical Experiment Site (NGES) on the campus of Texa s A&M University (TAMU) via four dynamic loading devices implemented in several configurations. The dynamic load and vibration responses of the foundations were measur ed via appropriate elec tronic instrumentation. Utilizing appropriate equations of motion and sy stem parameters, and the measured load and vibration responses, dynamic impedance functio ns were backcalculated for the vertical, horizontal sliding, and rocking mode s of vibration. Cone model predictions were compared to the backcalculated impedance functions with promising results. PAGE 15 15 CHAPTER 1 INTRODUCTION 1.1 Problem Statement Recent decades of research have pro duced sign ificant advancements in both laboratory and in situ characterization of geotechnical sites, and in methodologies for the prediction of dynamic foundation behavior. Improvements have yet to be fully realized even though these methods are the starting points for virtually all calculations of dynamic founda tion response, from sensitive instrument bases to earthquake shaking analyses. Multiple numerical models for predicting dynamic foundation responses base d upon elastodynamic theories are now available which seem to exhibit agreement with each other. Coupled with greatly enhanced geophysical testing capabilities which yield excelle nt soil characterization, dynami c footing response predictions may offer the available accur acy for response predictions and possibly foundation design. For example, the cone models presented by Wolf and Deeks (2004) recommend a strength of materials approach for predicting shallo w foundation responses to dynamic loading. The method, documented by Wolf and Deeks, is similar to bar and beam theories commonly applied in structural engineering and is based on one dimensional truncat ed cone modeling techniques. Cone modeling assumes excitation of a foundation creates wave propagation in the form of a vertical cone that increases in radius with respect to depth. From this assumption, fundamental principles of wave propagation are accounted for and a single solu tion is produced. Though some accuracy is sacrificed with cone models when compared to three dimensional elastodynamic methods, this loss is offset by reducti ons in theoretical complexity which leads to ease of use. The simplified cone solutions can be applied to a wide range of site conditions and foundation configurations making them more practical for dynamic foundation design. PAGE 16 16 Foundation excitation and soilstruct ure interaction can be estimated for a wide range of layer configurations, frequencies, embedment depths and scales using cone modeling. There are, however, shortcomings in the fi eld of dynamic footing responses. The most critical being there are no substantial test results based on full size m odels or typical design shapes applied in practice and found in field cons truction. It is impossible to determine whether these theories and methods are applicable to act ual construction and design situations without realistic, controlled results being analyzed. Though the few available studies appear to indicate agreement between theory and observed results, the available research is extrem ely limited in scope for various reasons. For example, some small scale laboratory based test s were reported by Nii (1985). However, these foundations were only a few centimeters in size, and tests were run with model foundations on a model half space constructed of va rious atypical materials (e.g. acryl ic resin and silicone rubber). In addition to material concerns, the model ha lf space had finite dimensions which may alter foundation response. Fry (1963) conducted realistic fu llscale tests but th ese studies lack adequate geophysical surveys to characterize the soil conditions. In thes e experiments seismic properties of the site are sparse and the accuracy of the data could be questioned. More thorough geophysical testing with present testing techniques would ensure more accuracy in any model. That accuracy would most likely carry over to foundation response predictions The same improved capabilities applied to measuring time data used for geotechnical surveys can be applied to observing the excitation experienced by a foundation. Thus the accuracy of measurements during testing is increased making any study potentially more effective. Both the model and any full scale experiments benefit from the refined technology reducing re liance on previous, less accurate studies. PAGE 17 17 Foundations of realistic scales properties, embedment depths and shapes need to be constructed and tested to conf irm current theories of dynamic foundation response. Performing tests under well characterized and controll ed conditions could confirm theories and methodologies that, until now, are assumed to be accurate and useful. 1.2 Hypothesis Strength of m aterials based cone models can be used to accurately predict the response of an in situ shallow foundation excited by a dynamic load. 1.3 Objectives The prim ary objective of this research is to assess the ability of the simplified strength of materials based cone models for predicting sh allow footing responses to dynamic loading accurately. Specific objectives of th is research include the following: Determine reliable techniques for dynamically loading in situ sh allow foundations of typical scales, embedments and shapes in ve rtical and coupled horizontalrocking modes. Determine reliable techniques for measuring foundation responses during excitation and back compute the dynamic impedance func tions of the corresponding measurements. Predict dynamic responses of measured in si tu shallow foundation excitation using cone modeling with comprehensive geotechnical ch aracterization of the site and footing specifications. Conduct detailed comparison of cone model predictions versus measured shallow foundation responses via dynamic impedance functions. 1.4 Scope Two shallow foundations were dynam ically loaded at the National Geotechnical Experiment Site (NGES) on the Texas A&M Univ ersity (TAMU) Riverside Campus in Bryan Texas. The primary advantages and subsequent reasoning for testing th is site begin with previous geotechnical investigations already completed. Data from these tests saved a great deal of time as well as funding and provided valuable information which was not only used in the PAGE 18 18 study but also allowed for greater effectiveness during the experimental design stage. More importantly the completed investigation contai ned seismic data from crosshole testing and surface wave analysis. The seismic data gave an especially valuable insight into the site during experimental design. Another advantage was found in footings that were already constructed. While some were destroyed during prior testing, at least one was still capable of be ing tested in the required manner. Footings tested were loaded dynamically by at least two different shak ers. Each shaker had advantages that were exploited to produce reliable displacements in multiple directions and modes. Prior to shaking, footings were instru mented in order to monitor their displacements. Data pertaining to both the force of the shak ers and motion of the footing were collected simultaneously. Collected data was then proc essed to yield impedance function values and compared to predicted values. Measurement te chniques were also note d during experimentation and studied for effectiveness. 1.5 Organization of Dissertation An overview of the following chapters fo llows. Chapter tw o provides the methodology behind dynamic foundation modeling and a discussi on of steps taken to measure foundation response and calculate impedance function values from those re sponses. Coinciding with and supporting the discussion of technique and measurement is the literature review. Chapter three presents the physical characteristics of the site and any foundations subjected to testing. Among the information of interest discussed are traditional soil testing, seismic testing, foundation physical characteristics, and foundation selecti on and construction. Chapter four demonstrates the relationship between cone modeling and other impedance function prediction methods. With this relationshi p established, the cone model is selected to represent predicted impedance function values in a comparison against measured values. These PAGE 19 19 comparisons further support the relationship between predicted and measured values, thus leading to the conclusion that the tests performed at the NGES s ite are pertinent to the study of cone modeling with respect to pred iction of dynamic impedance functions. PAGE 20 20 CHAPTER 2 FOUNDATION IMPEDANCE FUNCTIONS 2.1 Definition Dyna mic foundation displacement predictions ty pically focus on impedance functions, also referred to as impedances. These impedances are used to create a model of the soil by representing the soil on or in which the foundation rests as an e quivalent system of springs and dampers. Impedances are determined for each type of movement in all directions. For example, to predict the reaction of a foundation excited in the vertic al mode, a spring and damping coefficient are calculated which represent the resistance produced by the soil in the vertical direction. Similar calculations can produce impeda nce values in all three primary directions. Impedances can also be calculated that desc ribe the rotational soil reaction on the foundation about all three axis as well. The response of a footing with respect to dyna mic loading can be modeled starting with a soil, foundation system. A foundation rests on or is embedded in a soil halfspace or multi layered soil system as illustrated in the diag ram of Figure 21. The foundation is excited and results in a displacement. An incomplete list of factors that determine impedances include soil properties such as stiffness and layer thickness as well as foundation parameters such as size, shape and rigidity. Perhaps the most importa nt factor accounted for during foundation response investigation is the frequency of the load. Impedance values can vary s ubstantially with regards to the frequency of the excitation force and thus need to be measured or calculated for any frequency of interest. To predict the foundations respon se to an excitation force, a free body diagram such as the one in Figure 22 is developed containing t ypical necessary inform ation regarding the foundation. Dynamic loads are depicted as forces acting at appropriate locations on the footing. PAGE 21 21 The soil is accounted for by assuming that any pr essure between the footing and the soil can be represented by a reaction force acting at a particular point on the footing. That points location is dictated by the pressure dist ribution of the soil on the f oundation. For example an upward vertical force through the footing centroid is assumed to represent pressure on the bottom of the footing during downward vertical dynamic loading on the top of the center of the footing. Once the free body diagram is complete, dynamic equilibrium can be expressed using Newtons second law (Equation 21), as demo nstrated in the Gazetas (1991). (t)F (t)m (t)Pz z z (21) In the case of vertical exc itation through the center of gravity there are two forces. The components Fz(t) and Pz(t) represent the dynamic load and soil pressure force respectfully with the z subscript indicating the ve rtical direction. Th e acceleration of the mass in the vertical direction is accounted for in the equation by the component mz(t). While the dynamic load and the acceleration of mass are fairly simple to comprehend and qualify, the soil force requires furt her investigation. The interact ion between the foundation and soil is estimated to have a linear relationship to small strain foundation displacement presented in Equation 22. (t)Pz= Kz (t)uz (22) The variable, Kz is typically known as the dynamic ver tical impedance and is expressed in units of force per length. Substitution of equation two into equation one allows for solution of the differential equation for the dynamic displaceme nt, provided that an adequate quantification of Kz is available. Dynamic impedance is a complex number, and further analysis allows Kz to be represented by a spring and dashpot model leading to Equation 23 where K z is the dynamic stiffness. Dynamic stiffness accounts for both the so ils stiffness as well as inertia properties. PAGE 22 22 Material and geometric damping are accounted for by Cz known as the dashpot coefficient. It also turns out that for soilfoundation systems, th e dynamic stiffness and dashpot coefficients are dependent on the excitation force frequency. Kz = K z + i Cz (23) Multiple notation conventions are instituted when studying dynamic impedance functions and often depend on the author. Th e convention presented to this point is that of the Gazetas (1991) text. Another example of notation for disc ussing cone modeling is presented in Wolf and Deeks (2004). They present dynamic impedance in a complex form and using dimesionless coefficients in Equation 24 where S( a0) is the frequency dependent dynamic impedance, k ( a0) is dimensionless dynamic stiffness, and c ( a0) is the dimensionless dampi ng coefficient. The term a0 is dimensionless frequency and can be calculated using Equation 25 where r0 is a length of the foundation and cs 1 is the shear wave velocity of the first layer of soil. The K in equation 4 is a coefficient that assigns a dimension of fo rce per length to the dynamic impedance and is typically some value depending on foundation syst em characteristics such as width or the underlying soil shear modulus. It is typically referred to as the static stiffness. S( a0) = K [ k ( a0) + i a0 c ( a0)] (24) 1 0 0 sc r a (25) 2.2 Computation Methods for determ ining impedance functions vary in approach and scope of effectiveness. For example certain methods may be appropria te for calculating impedance functions when dealing with a surface footing on a half space, while other methods would be necessary to produce impedances for a layered embedded syst em. To develop an understanding of cone PAGE 23 23 models, techniques are most conveniently organi zed into analytical and numeric solutions assuming elastodynamic soil behavior and approximate solutions which determine impedances based on simplified models. Initial impedance function methods focused on analytical solutions. Multiple response prediction approaches have been developed be ginning in the thirties when Reissner (1990) proposed predictions for a simple circular surf ace footing on a linearly elastic isotropic half space. Advances continued with research cove ring the effects of damping (Bycroft [1990]) and finite layer depths. Dynamic finite element me thods dealing with embedment in layered systems were first developed for strip a nd circular footings, and requir ed a shallow rigid layer as a boundary (Waas (1972); Kausel (1974); Luco (1976) Lysmer et al. (1975). This was followed by more analytical and semianalytical multilayered solutions which included rectangular footing shapes, but were unable to account for embedment (Luco [1976]; Gazetasand Roesset [1976, 1979]). Next, analytical and numerical methods were combin ed in an effort to exploit the strengths of both (Kausel [1981] ; Lysmer [1981]; Tassoulas [ 1981]). Today, better finite element and boundary element methods are availabl e for 3D solutions of embedded footings in layered soil deposits. Though generally effective, the analytical and nu merical methods require considerable unique preparation to each set of co nditions to generate so lutions. Hence a great deal of time and funding may be consumed in orde r to apply the above theories to an individual project (Gazetas [1991]). Approximate solutions provide relief from the detailed input and execution of analytical and numerical methods and also provide clarity of the problem that rigorous methods do not allow (Roesset [Foreword]). However, some as pects can not be accounted for with approximate methods, such as soil separation from the founda tion, and thus are ideal for simplification by an PAGE 24 24 approximate solution. Though accuracy is sacrificed, in many cases this may be acceptable if the use of simpler models saves time as well as othe r resources and still provides a reliable safe result (Gazetas [1991]). The a pproximate methods incl ude the halfspace an alogs of Richart, Hall, and Woods (1970), the work of Novak and his colleagues that appears in the computer programs PILAY, PILAY2, and DYNA5, the table and chart solutions of Gazetas (1991), and the cone models described by Wolf and Deek s (2004). Cone models are likely the most advanced of the approximate methods for applica tion in engineering practice. Wolf and Deeks (2004) provide a detailed description of the implementation of cone models to foundation vibration problems. A summary of the methodology is provided in the following paragraphs. To analyze the vibrations of a foundation on the surface of or embedde d in a layered halfspace with cone models, an approach using conica l bars and beams, called cones, is developed. The complicated exact formulation of threedi mensional elastodynamics is replaced by the simple onedimensional description of the theory of the strength of materials, postulating the deformation behavior (plane sections remain plane). The halfspace with linear elastic behavior a nd hysteretic material damping can consist of any number of horizontal layers either overlying a halfspace or fixed at its base. Besides cylindrical foundations, axisymm etric configurations of arbitrary embedment shape can be processed, with the wall and base of the embedded foundation assumed to be rigid. The dynamicstiffness coefficients describing the inte raction forcedisplacement relationship and the effective foundation input motion fo r vertically propagating Sand Pwaves in seismic excitation are calculated for all frequencies. Only approximations of the onedimensiona l strengthofmaterials approach based on wave propagation in cones apply. No other assu mptions are made. For each degree of freedom PAGE 25 25 only one type of body wave exists: for the horiz ontal and torsional motions Swaves propagating with the shearwave velocity; and for the verti cal and rocking motions Pwaves propagating with the dilatationalwave velocity. Two building blocks are require d to construct the procedure to analyze the vibrations of a foundation in a layered halfspace. The first addresses the outw ard wave propagation occurring from a disk embedded in a fullspace modeled as a double cone. 23 il lustrates the sectional property of these initial cones increases in the direction of wave propagation, modeling the spreading of the disturbance in the medium. The cones are thus radiating. The opening angl e of the cone is determined by equating the staticstiffness coefficient of the truncated semii nfinite cone to that of a disk on a halfspace determined using the threedimensional theory of elasticity. The openi ng angle depends only on Poissons ratio (and the degree of freedom). In the case of nearlyincompressible and compressible material (Poissons ratio larger than 1/3), the wave velocity is limited to twice the shearwave velocity and a trapped mass and ma ss moment of inertia are introduced for the vertical and rocking degrees of freedom, respectively. The second building block addresses the wa ve mechanism generated at a material discontinuity corresponding to an interface betw een two layers. When the incident wave propagating in the initial cone (d escribed in the first building bl ock) encounters a discontinuity, a reflected wave and a refracted wave, each propaga ting in its own cone, are created. The cone diagram presented in Figure 24 illustrates both reflect ed and refracted waves. Enforcement of compatibility of displacement and equilibrium of the interface permits the reflected and refracted waves to be expressed as a function of the incident wave. The reflection coefficient, defined as the ratio of the reflect ed wave to the incident wave, depends on the PAGE 26 26 frequency and on the properties of the two material s present at the interface, in particular their impedances. The reflected and refracted waves generated at an interface will also encounter material discontinuities as incident waves at a later stage, yielding additi onal reflections and refractions. By tracking the reflections and refractions seque ntially, the superimposed wave pattern can be established for a layered site up to a certain st age. The termination criterion addresses the number of cone segments in which the waves ha ve propagated and the magnitude of the created waves. The embedded foundation is modeled with a stack of disks in that part of the soil which will be excavated as illustrate d in Figure 25. This leads to a primary dynamic system with redundants acting on the embedded disks. As in the force method of stru ctural analysis, the dynamic flexibility of the free fiel d with respect to the displacem ents of the disks caused by the redundants is established addressing the wave pattern in the layered ha lfspace. Inversion of this relationship, enforcement of the rigidbody moti on of the foundation (con sidering the freefield motion of the seismic waves, if present) and excavation of the trapped material yield the dynamicstiffness coefficients of the embedde d foundation and the effective foundation input motion. Wolf and Deeks (2004) provide a complete MATLAB implementation of the method, together with a comprehensive description of th e workings of each function. These can be used along with the builtin facilities of MATLAB, such as the fast Fourier transformation, to perform all necessary analyses in the MATLAB environment. 2.3 Measurement There are a very sm all number of studies documented in the literature in which dynamic impedance functions for shallow foundations have b een determined in situ at sites in which the PAGE 27 27 soil conditions and propert ies are well characterized. These st udies include Crouse, et al. (1990); De Barros and Luco (1995); Luco, Trifunac, and Wong (1988); Luco and Wong (1990); and Wong, Trifunac, and Luco (1988). Crouse, et al. (1990) report experimental impedance functions for two small concrete foundations supported by soil and in real field co nditions. Both foundations were utilized to support earthquake accelerograph stations. One foundation (Cholame 1E) was located in central California, and consisted of a 10cm (4in.) thick, 1.27m (50in.) square slab resting on corner piers and embedded in a moderately stiff alluvial deposit. The ot her foundation (Station 6) was a 15cm (6in.) thick, 1.22m x 1.14m (48in. x 45i n.) nearly square slab, embedded 7.6 cm (3 in.) in a softer deposit in sout hern California. Shear wave velocity profiles were determined for each site via in situ SASW tests, and reported in detail by Crouse, et al. (1990). The soils at both sites were assumed by Crouse, et al. to have a unit weight of 17,300 N/m3 (110 lb/ft3), a Poissons ratio of 1/3, and a material damping ratio of 0.015. To determine impedance functions, forced harm onic vibration tests were performed on the foundations. Harmonic forces were induced with a small (156 N, 35 lb) eccentricmass vibration shaker bolted to the foundation surface. The me asured response was determined via triaxial accelerometers mounted at each corner of the f oundation. The impedance functions were then determined from the induced force and measured response via appropriate equations of motion for the dynamic system. The impedance functions are presented for the ve rtical, horizontal, and rocking modes of vibration and for frequencies from 10 to 60 Hz. De Barros and Luco (1995) describe testi ng a one quarter scale model of a nuclear containment facility constructed in Hualin, Ta iwan. Detailed procedures and appropriate equations of motion for impedance function dete rmination are provided then, predicted and PAGE 28 28 measured impedances from the test are compared. The model was cylindrical and had a height of 16.13 m with a base of diamet er 10.82 m. The base was 3.00 m thick and rested on a layer of concrete 0.15 m thick. The mass of the base and thin layer of concrete was 695,000 kg. The containment shell was reinforced concrete of wall thickness 0.30 m, outer diameter of 10.52 m, and a height of 11.63 m. The estimated ma ss of the shell was 264,000kg. The roof was a cylindrical slab of diameter 13.28 m with th ickness of 1.50 m and a mass of 505,000kg. In the center was a 2.20 by 2.20 m hole. Four beams w ith dimensions 0.60 m by 0.30 m supported the roof. The thin slab underneath the base rest ed on soil 5.15 m below grade. The surrounding soil was banked with the toe 1.50m from the base. De Barros and Luco (1995) pr ovide a table of soil properties relevant to the study of impedance functions. Cross hole testing provided shear wave velocities for the site. The soil directly below th e base is reported as having a unit weight of 23,740 N/m3, Poissons ratio of 0.47, a nd a damping ratio of 0.02. Horizontal dynamic loading was performed by an eccentric mass shaker on the models base and top in both the NorthS outh direction and the EastWest direction. The horizontal force frequency varied from 2 to 20 Hz and the shaker force exerted relied on the frequency. Vertical dynamic loads were imparted on the base with frequencies ranging from 2 to 25 Hz. A discussion of foundation response reco rding methods was not available. DeBarros and Luco (1995) present methodology for calculating impedance functions from measured data. Calculations of appropriate forces and displacements for determining impedances are introduced. Impedances determ ined from measured responses were then normalized to represent a test conducted with a dy namic force of 9806 N at all frequencies. The measured impedances were compared to those submitted by volunteers who made predictions based on reported soil characteristics of the site and design aspects of the containment model. PAGE 29 29 Luco, Trifunac, and Wong (1988) de monstrate that it is possible to separate the effects of a structure during dynamic loading from the resp onse of a foundation during dynamic testing thereby being able to produce impedance functio ns from measure responses. The Millikan Library Building on the California Institute of Technology campus was loaded dynamically and the responses of the structure and the foundation were monitored. The library is a ninestory reinforced concrete building at a height of 43.9 m above grade with a basement slab 4.3 m below grade. The footprint of the building is genera lly 21 m by 23 m and the building structure has a mass of 10.7 million kg. The foundation consis ts of a 9.75 m wide pad 1.22 m deep running EastWest through the middle of the footprint a nd two 3 m wide by 0.61 m deep footings running under the exterior columns parall el to the central footing the full length of the building. The exterior beams are connected to the central pad by stepped beams. The ce ntral pads interface is 7 m below grade. The total mass of the foundation is 0.14 million kg. Soil conditions are reported as a medium to dense sand mixed with gravel from grade to a depth of 275 m. An eccentricmass shaker was attached to the ro of and a velocity transducer monitored the response of the structure at the roof while three velocity transd ucers monitored the foundation. Tests were conducted in both th e NorthSouth direction and Ea stWest direction. The force imparted on the roof was dependent on the freque ncy at which the shaker was performing. Luco, Trifunac, and Wong (1988) then analyze the data from both th e structure and foundation to calculate impedance functions which accurately model the soil. Impedance functions for frequencies from 0.8 to 2.5 Hz were produced in the horizontal and rocking modes. Wong, Trifunac, and Luco (1988) compare measured and predicted impedance functions for the Millikan library. The te sts and data recorded in Luco Trifunac, and Wong (1988) were PAGE 30 30 used to produce impedance functions and were compared to predicted values. Impedance functions comparisons are presented for frequencies ranging from 0.8 to 2.5 Hz. Luco and Wong (1990) report displacements during dynamic loading for a model of a nuclear containment structure in Lotung, Taiwan and compare them to various displacement predictions for a 98,000 N dynamic force. The model consisted of an elastic structure with an external diameter of 10.50 m a nd a wall thickness of 0.30 m. The height was 14.33 m. The top of the structure was a rigid disc of diameter 9.9 m and thickness of 1.35 m. The structures mass was 598,000 kg. It rested on a base that was 1.06 m thick and had a diameter of 10.80 m with a mass of 234,000 kg. The base and structure comb ination was embedded a total of 4.72 m. Soil conditions at the site required for impedance function predictions were cited but not reported. Dynamic forces were applied to the roof of the structure by a shaker covering various frequency ranges and force magnitudes. Th e structure and foundati on were monitored by accelerometers and converted to displacements Displacements were determined from measurements and results normalized to co rrespond to a dynamic force of 98,000 N. A comparison between measured displacements and predicted displacements is presented ranging from 1 to 30 Hz. For the vertical mode of vibration, a singl edegreeoffreedom model of rigid body motion is reasonable, and determination of an impeda nce function is elementa ry with computational tools that easily handle the mathematics of complex numbers, e.g., Mathcad, Matlab. Equation 26 presents the equation for singledegr eeoffreedom vertical impedance, Sv, and is simply defined as the force imparted on the soil by the foundation, Fv, divided by the vertical displacement of the foundation, Uv. v s vU F S (26) PAGE 31 31 The soil force, Fs, is a function of the applied vert ical force and the inertia of the foundation due to vertical movement as presented in Equation 27. FT is the vertical force applied to the foundation, M0 is the mass of the foundation and Uv is the maximum vertical displacement of the foundation, and is the angular frequency of the system. v 0 2 TsUMFF (27) For horizontal or moment loading contained wi thin a single plane, the horizontal sliding and rocking modes are often coupled, and a tw odegreeoffreedom dynamic model of rigid body motion is required. In general, these equatio ns involve four impedances, namely, horizontal, rocking, the coupling of rocking with horizontal slidi ng, and the coupling of horizontal sliding with rocking, to describe the horizontal disp lacement and rotation response of the foundation. Initially, this strategy is problematic since onl y two independent responses can be measured (e.g., horizontal displacement and rotation), yet four unknown impedances are to be determined. As a remedy, it is possible to assume that th e crosscoupling impedance s are negligible, thus allowing the horizontal sliding and rocking im pedances to be computed from the two measurements (similar to vertical mode disc ussed above) as presented in Equation 28 and Equation 29. h s hU F S (28) b s rM S (29) In this case, Uh is the maximum horizontal displacement of the foundation bottom interface with the soil. The horizontal soil force, represented by Fs, is similar to the vertical soil force because of footing inertia. Calculation of Fs becomes more complicated as it is necessary to account for changes in sliding force due to rotation of the founda tion and the translation of any PAGE 32 32 structure attached to the footing. Thus Equa tion 210 represents the sliding force used to calculate horizontal impedance. }U]{M[}1{)bMUM(FFT2 b 20 b 0 2 Ts (210) The horizontal force impart ed on the foundation is FT. The second term accounts for foundation motion where is still the angular frequency of the system, M0 remains foundation mass, bU is the total motion of the instrume ntation measuring horizontal motion, b2 is the distance from the instrume ntation to the center of gravity of the foundation, b is the maximum rotation of the foundation. The third term account s for structural effects on the foundation. To calculate structural effects on th e foundation, the structure is se parated into individual points each with mass properties. This separation is performed because the frame may not be considered rigid. Each point is treated as a poi nt that will move a unique magnitude and thus, impart a force on the foundation. The angular frequency matrix of point masses [M] and their corresponding maximum displacements { U} make up this term. The matrix {1} is an identity matrix to allow the multiplication of [M] and { U}. The rocking impedance of Equation 9 is the measure of the moment produced by the soil per amount of rotation. The rotation of the foundation is represented by b. The Rocking moment, Ms, is defined in Equation 11. }U]{M)[}1{h}h({)bbMI(UbMF)hH(MT 0 T2 b 2100 2 b 10 2 T0Fs (211) The rocking moment is calculated around th e foundation soil interf ace at the horizontal center of the footing. Many variables used to calculate the horizontal sliding force and the rocking moments are shared. Those that are not shared include the height of the force acting on the frame which is represented by HF, thickness of the foundation, h0, the distance from the PAGE 33 33 center of gravity of the foundation to its bottom, b1, the mass moment of inertia about the axis that the foundation is rotating, I0. As with dynamic horizontal force calculations, dynamic moment calculations require the influence of the st ructure. The structure is separated into point masses with corresponding elevations and moments which, along with displacements and angular frequency, are used to calculate the dyna mic effect of the frame on the foundation. The only unmentioned variable to this point is {h} which is a matrix representing the height of the corresponding point masses. Alternatively, the foundation re sponse can be predicted vi a a twodegreeoffreedom system. This requires measurements from tw o independent loading configurations. By combining the results of the two experiment s, the four measured response (i.e., two displacements and two rotations) can be used to determine the four unknown impedances as follows in Equation 212 through Equation 215. 1b 2y0 2b 1y0 2s 1b 1s 2b hUU FF S (212) 2b 1y0 1b 2y0 2s1y01s2y0 rUU MUMU S (213) 1b 2y02b1y0 2s 1b 1s 2b rhUU MM S (214) 2b 1y0 1b 2y0 2s1y01s2y0 hrUU FUFU S (215) The impedances are horizontal, Sh, rocking, Sr, and two coupling terms, Srh and Shr. Impedances are calculated using the same input s and responses used to calculate uncoupled impedances. Subscripts 1 and 2 designate from which configuration variables are measured or calculated. Variables include the force imparted on the foundation, Fs, the moment applied to the PAGE 34 34 foundation, Ms, the rotation of the foundation, b, and the displacement of the foundations center of gravity, U0y, calculated in Equation 16. 2 b b y0bUU (216) Of course, this strategy assumes that the impe dances appropriately describe the foundationsoil interaction for each experiment configurat ion. An executable program (CONAN) is also provided, along with complete details of its use. To define the coupling terms via th e method above, it is assumed that Srh and Shr are independent of each other. Another approach assumes that Srh and Shr are equal and overdetermined. Hence, a least s quares solution can be used to calculate the four impedances. PAGE 35 35 Figure 21. Foundation / Soil Diagram. (Gazetas [1991]) PAGE 36 36 Figure 22. Foundation Free Body Diagram. (Gazetas [1991]) PAGE 37 37 Figure 23. Cone Diagrams. (Wolf and Deeks [2004]) PAGE 38 38 Figure 24. Reflected and Refracted Wave Diagram. (Wolf and Deeks [2004]) PAGE 39 39 Figure 25. Embedded Stacked Disks. (Wolf and Deeks [2004]) PAGE 40 40 CHAPTER 3 CONE MODEL PREDICTIONS Many dynam ic impedance function prediction met hods are available. Precise numerical methods tend to be extremely cumbersome often rendering them unpractical for widespread use. Cone models on the other hand are relatively simp le when compared to other methods potentially making them highly effective for typical desi gn scenarios. The following chapter first demonstrates similarities between predicted impedance functions produced via numerical methods and those produced by cone modeling. Cone model solutions for square and rectangular footings on the surface of and embe dded in both homogeneous and layered profiles are compared with theoretical solutions of Wong and Luco (1978, 1985), Mita and Luco (1989), and Apsel and Luco (1987) availabl e in the literature. Next cone model solutions for surface and embedded round, square, and rectangular footings ar e compared with actual measured behavior. While the database is limited, the welldoc umented work of Nii (1987) for smallscale foundations, and Crouse, et al. (1990) for footi ngs of actual size, provide useful measured foundation behavior to compare w ith cone model predictions. These evaluations will further demonstrate that sufficient engineering accuracy can be obtained via the strengthofmaterials based analyses of cone models. 3.1 Wolf and Deeks (2004) To analyze the vibration s of a foundation on the surface of or embedde d in a layered halfspace, Wolf and Deeks (2004) have presented an approach using conical bars and beams, called cones. With these models, the complicated, exact formulation of threedimensional elastodynamics is replaced by simple one dimensional description of the theory of strength of materials, postulating the deforma tion behavior (plane sections remain plane). A halfspace with linear elastic behavior and hystere tic material damping can consis t of any number of horizontal PAGE 41 41 layers either overlying a halfspace or fixed at its base. Besides cylindrical foundations, axisymmetric configurations of arbitrary embedment shape can be processed, with the wall and base of the embedded foundation assumed to be rigid. Dynamicstiffness coefficients describing the interaction forcedisplacement relationship ar e calculated for all frequencies. Only approximations of the onedimensional strengt hofmaterials approach based on wave propagation in cones apply, and no other assumpti ons are made. For each degree of freedom, only one type of body wave exists: for the horiz ontal and torsional motions, Swaves propagating with the shearwave velocity; and for the vert ical and rocking motions, Pwaves propagating with the dilatationalwave veloc ity. The corresponding di splacements can be formulated directly in closed form as a function of the depth of the site, without any spatial Fourier transformation into the wave number domain. Wolf and Deeks (2004) have presented exte nsive evaluation of their methodology. In these evaluations, cone model predictions are compared with result s from rigorous, threedimensional, elastodynamic solutions, usually the thinlayer finite element method. The evaluations are presented for axisymmetric foundations, for a ll degrees of freedom, and for dimensionless frequencies typically up to six. The halfspace materials are assumed to be linear elastic with hysteretic material damping. Em bedded foundations are assumed to be in full contact with surrounding soil. The sp ecific cases examined are as follows: Both surface and embedded cylindrical f oundations on/in a homogeneous halfspace. Both surface and embedded cylindrical founda tions on/in a homogene ous layer fixed at its base. Cylindrical foundations on the surf ace of a layered halfspace. Cylindrical foundations embedded in a layered halfspace. PAGE 42 42 A cylindrical foundation embedded in an incompressible, layered halfspace. A hemiellipsoid embedded in a homogeneous halfspace. A sphere embedded in a homogeneous fullspace. For all of these cases, the deviations in impedance functions between cone model predictions and the rigorous so lutions are within the range %. Wolf and Deeks (2004) conclude that this is suffi cient engineering accuracy. 3.2 Wong and Luco (1978) W hile the extensive evaluation results pr esented by Wolf and Deeks (2004) are convincing, it is also fair to obser ve that most real shallow founda tions are not axisymmetric. In fact, the majority of real shallo w foundations are probably rectangular in shape. Hence, the first evaluation presented herein is to assess the ab ility of cone models to determine impedance functions for nonaxisymmetric founda tions, in this case rectangular. Wong and Luco (1978) have presented a detaile d tabulation of numerical values for the impedance functions for a massless, rigid, rect angular foundation perfectly bonded to the surface of a viscoelastic, homogeneous halfspace. Impe dance functions are provided in normalized form for four aspect ratios, namely length/width=1, 2, 3, and 4, and for dimensionless frequencies up to 10. The computations have been performed for two values of Poissons ratio, =0.33 and 0.45, and for three values of hysteretic damping ratio, =0, 0.02, and 0.05. Five specific rectangular foundation models were constructed to compare cone model predictions with the tabulated results of Wong and Luco (1978) Models 14 consisted of foundations with aspect ratios of 1, 2, 3, a nd 4, respectively, and a Poissons ratio of =0.33. Model 5 was a rectangular foundation with an as pect ratio of 4, and a Poissons ratio of =0.45. PAGE 43 43 All models were constructed using a foundation widt h of 2 m, a shear wave velocity of 125 m/s, a mass density of 1600 kg/m3, and a hysteretic damping ratio, =0.05. For each of the five models, cone model im pedance functions for vertical, horizontal, rocking, and torsional degrees of freedom were calculated using the CONAN executable program provided by Wolf and Deeks (2004). Please note that the cone m odels assume that the coupling impedance of a surface f oundation is zero. Indeed, the coupling impedances tabulated by Wong and Luco (1978) are small in comparison to the other degrees of freedom. Also, since the cone model impedances are calculated for circ ular disk (axisymmetric) foundations, each of the five rectangular foundations were converted to an equivalent disk for CONAN computations. The equivalency was based upon contact area for tr anslational modes (verti cal, horizontal), and moment of inertia for rotati onal modes (rocking, torsion). By way of example, Figure 31 to Figure 310 present impedance function comparisons for Models 1 and 4, respectively. Each figure displa ys impedance functions for a degree of freedom as computed by CONAN. In each figure, impedance functions are expressed by three graphs and are compared with the corres ponding Wong and Luco (1978) data. Real (stiffness) and imaginary (damping) components are represente d in Graphs A and B respectively. Graph C plots the magnitude of the impedance functi on in dimensional form versus frequency. Impedance function magnitude is simply the ve ctor summation of the real and imaginary components. It is observed that for all cases of imaginary components and magnitude presented, the % engineering accuracy documented by Wo lf and Deeks (2004) is maintained for these foundations on the surface of a homogeneous halfspace. Many of the real components also demonstrate agreement within this accuracy whil e those that do not are of comparable magnitude as well as trend. While it may not be surprising that results for a square shape are very similar to PAGE 44 44 an axisymmetric solution (Figure 31 through Figure 34), it is also evident that the cone model solutions are in good agreement for a rectangular f ooting with an aspect ra tio as large as four (Figure 35 through Figure 310). It should also be noted that the comparison results for the remaining Models 2, 3, and 5 not shown herein are of equal or better quality. 3.3 Mita and Luco (1989) To explore the influence of foundation em bedmen t, Mita and Luco (1989) have presented a detailed tabulation of numerical values for the impedance functions of a massless, rigid, square foundation embedded in and perfectly welded to a viscoelastic, homogeneous halfspace. Impedance functions for horizontal, coupling, rock ing, vertical, and torsio nal degrees of freedom are provided in normalized fo rm for embedment to halfwidth ratios up to 1.5, and for dimensionless frequencies up to three. The com putations have been performed for three values of Poissons ratio, =0.25, 0.33 and 0.4. All computations a ssumed hysteretic damping ratios of 0.001 for shear waves, and 0.0005 for compression waves. The numerical results were calculated using a hybrid approach described by Mita and Luco (1987).Five specific square foundation models were construc ted to compare cone model pr edictions with the tabulated results of Mita and Luco (1989). Models 13 co nsisted of foundations with embedment to halfwidth ratios of 0, 0.5, and 1.5, respec tively, and a Poissons ratio of =0.33. Model 4 was a square foundation with an embe dment to halfwidth ratio of 1.5, and a Poissons ratio of =0.25, while Model 4 was a square foundation with an embedment to halfwidth ratio of 1.5, and a Poissons ratio of =0.4. All models were constructed us ing a foundation width of 2 m, a shear wave velocity of 125 m/s, a mass density of 1600 kg/m3, and hysteretic damping ratios as noted above for the Mita and Luco (1989) results. As with the comparisons presented above, cone models for these square footings were based upon equivalent circular disks. PAGE 45 45 By way of example, Figure 311 thr ough Figure 15 present impedance function comparisons for Model 3, a case with maximum embedment. In each figure, impedance functions for each degree of freedom and as computed by CONAN are compared with the corresponding Mita and Luco (1989) data. As in the previous sections, plotted in each figure are the real and imaginary components represented in Graphs A and B respectively and, in Graph C, the magnitude of the impedance function in dimens ional form versus frequency. It is observed that for all cases presented, with the exception of Figure 315 A, the % engineering accuracy documented by Wolf and Deeks (2004) is also maintained for these square foundations embedded in a homogeneous halfspace. Though a segment of the plot in Figure 315 A does not fall within the desired engineering accuracy, the graph demonstrates a magnitude and trend that is comparable between CONA N and Mita and Luco (1989). It should also be noted that the comparison results for the remaining Models 1, 2, 4, and 5 not shown herein are of equal or better quality. 3.4 Wong and Luco (1985) To explore the influence of soil layering, W ong a nd Luco (1985) have presented a detailed tabulation of normalized horizontal, coupling, rocking, vertical, and torsional impedance functions for a rigid, massless, square foundation re sting on two types of la yered viscoelastic soil models. The first soil model consists of a uni form layer over a uniform halfspace, while the second model consists of a layer with linearly va rying properties over a uniform halfspace. The impedance functions have been calculated base d upon an extension to the case of layered media of the approach proposed by Wong and Luco (1 976). The approach computes a numerical solution of an integral equation involving the Greens functions for a layered medium. The rigid, square foundation is assumed to be perfectly bo nded to the surface of the layered halfspace. PAGE 46 46 The impedance function results are presented for two Poissons ratio combinations: 1) 1=0.33 for the layer, and 2=0.33 for the halfspace, and 2) 1=0.45 for the layer, and 2=0.33 for the halfspace. Three values for the contrast in shear wave velocity between layer and halfspace are provided, namely Vs1/Vs2=0.3, 0.6, and 0.8. The density ratio between the half space and the layer is fixed at 2/ 1=1.13, and the material damping cons tants are fixed at 0.03 and 0.05 for the halfspace and layer, respectively. In each of the tables, the real and imaginar y parts of the normalized impedance functions are presented for 21 dimensionless frequencies ra nging from 0.1 to 5.0, and for five values for the ratio of layer thickness (H) to foundation halfwidth (a): H/a=0.5, 1, 2, 3, and 4. Eight specific square foundation models were constructed to compare cone model predictions with the tabulated re sults of Wong and Luco (1985). For simplicity, all eight models consisted of a uniform layer over a halfspace. In addition, all cone models were assigned a fixed density ratio of 1.13, and fixed material damping values of 0.03 and 0.05, as noted above for the Wong and Luco (1985) results. As with the comparisons presented above, cone models for these square footings were based upon equivalent circular disks. A dditional parameters for the eight models are shown in Table 31. Model 1 was chosen as a basis model with the parameters indicated. For Models 28, the parameters shown in Table 31 are those that are different from Model 1, while all other parameters remain the same. Models 2 and 3 br iefly examine the issue of scale, since the Wong and Luco (1985) results are pres ented in dimensionless format. Model 2 has a foundation half the size of Model 1, while Model 3 has increased th e shear wave velocity of the layer by a factor of two. Models 4 and 5 address the influence of layer thickness, Models 6 and 7 the influence of stiffness contrast, and Model 8 the influence of Poissons ratio. While these eight models PAGE 47 47 certainly do not exhaust the full range of possi bilities presented by the Wong and Luco (1985) data, these models explore the ra nges in principal parameters. By way of example, Figure 316 to Figure 331 present impedance function comparisons for Models 1, 4, 6, and 8, respectively. Results fo r the remaining Models 2, 3, 5, and 7 are also discussed below. In each figure, impedance functions for each degree of freedom and as computed by CONAN are compared with the co rresponding Wong and Luco (1985) data. As in the previous sections, plotted in each graph is bo th real and imaginary components as well as the magnitude of the impedance function in dimensional form versus frequency. It is observed that for the vast majority of cases presented, th e % engineering accura cy documented by Wolf and Deeks (2004) is also maintained for thes e square foundations on the surface of a layered halfspace. The only notable exceptions for ma gnitude are possibly the rocking mode in Model 4, and the vertical mode in Model 8. As for the accuracy of the components of the impedance functions, there are exceptions th at tend to exist at higher fre quencies but plots between CONAN and Wong and Luco (1985) are simila r in trends and order of magnit ude. It should also be noted that the remaining Models 2, 3, 5, and 7 that are not presented herein, display equal if not better comparisons than shown in Figure 316 to Figure 331. The results for Models 2 and 3 indicate that scale is not a factor in these comparisons. Indeed, while the impedance function magnitudes change with scale, the comparisons between cone model and Wong and Luco (1985) are identical to those shown in Figure 31 to Figure 34. Also, the results for Models 5 and 7 are better than for Models 4 and 6, respectively. This is to be expected since Model 5 has a much thicker layer than either Models 1 or 4, and Mode l 7 has a lower stiffness contrast than either Models 1 or 6. PAGE 48 48 3.5 Apsel and Luco (1987) To evaluate the com bined influence of layering and foundation embedment, Apsel and Luco (1987) present impedance functions fo r a cylindrical foundation embedded in a multilayered soil profile. The f oundation model was 24.4 m (80 ft) in diameter, embedded 5.5 m (18 ft) below the ground surface, and was assumed to be in full contact with the surrounding soil. The characteristics of the soil model are shown below in Table 32. The impedance functions are presented for horizontal, rocking, and coupled horizontalrocking degrees of freedom for dimensionless frequencies up to 6, and were co mputed via an integral equation approach described in detail by Apsel and Luco (1987) Based upon these parameters, cone model impedance functions were computed via CONA N for comparison with the Apsel and Luco (1987) results, and the comparison is presented in Figure 332 to Figure 334. As in the previous sections, plotted in each figure are three graphs representing real and imaginary components and the magnitude of the impedance function in dimens ional form versus frequency. It is observed that the results are generally in good agreement. In fact, th e comparison appears very good for the horizontal mode. The cone model appears to slightly over predict the impedances in the higher frequency range for rocking. The coupling impedances are unde r predicted by CONAN for all frequencies investigated, though the trends and shape of the impedance versus frequency relationship are in good agreement with the exception of stiffness. Stiffness seems to be somewhat lower for CONAN than for Apsel and Luco (1987) though the shape and trend of the curves seem similar. 3.6 Nii (1987) While all of the im pedance function solutions presented above are based upon theoretical models, Nii (1987) reported expe rimental results for vertical dynamic impedance of smallscale footings attached to or embedded in a model hal fspace. The halfspace was constructed of room PAGE 49 49 temperature vulcanizing silicone rubber, and was cast in a steel tank 1.7 m long, 1.4 m wide, and 0.4 m high. The mechanical properties of the halfspace are reported as follows: Vs = 11.3 m/s Vr = 10.8 m/s Vp = 1000 m/s 0.5 Damping ratio ( ) = 0.031 = 0.98 g/cm3 The model footings were constructed of acrylic resin, with dimensions as shown in Table 33. Nii (1987) reports that ev en though the acrylic footings were smooth, the surface of the halfspace rubber was slightly adhesive, and the footings and halfspace behaved as though they were bonded. The circular and rect angular footings were forced into vertical vibration via a mechanical shaker. The experiments were c onducted at frequencies where the effects of reflected waves from the boundaries of the model ha lfspace were negligible or diminished (Nii [1987]), and results of the expe riments are reported for dime nsionless frequencies up to 8. Vertical impedances were computed with cone models for five of the experiment footings shown in Table 33: two circular footings, one surface and one embedded, and three rectangular footings with aspect ratios of 1, 1.5, and 2. Sinc e the cone model impedances are calculated for circular disk (axisymmetric) f oundations, cone models for the rectangular footings were based upon equivalent circular disks, in this case based upon contact area for the vertical mode of vibration. Cone model vertical impedance solutions are compared with the experimental results of Nii (1987) in Figure 335 to Figur e 339 for each of the five footings considered. Each figure PAGE 50 50 plots real and imaginary components of the impedance function as well as its magnitude in dimensional form versus frequency. Impedanc e function magnitude is simply the vector summation of the real and imaginary components. It is observed that agreement between cone model predictions and the Nii (1987) experimental results is quite good at low frequencies for all five cases. There is a consiste nt underprediction of the magnit ude at higher frequencies for all five cases. Evaluation of the real (stiffness) and imaginary (damping) components of the impedance functions also reveals a consistent pa ttern for all five cases: the real component is overpredicted, while the imaginary component is underpredicted, with a net underprediction of the overall magnitude as noted in the figures. However, the % engineering accuracy for magnitude documented by Wolf and Deeks (2004) appears to be maintained, except for maybe the embedded, circular footing (Figure 336 C). Real and imaginary components of Luco and Mita (1987) deviate from Nii (1987) and CONAN values at hi gher frequencies but still produce nearly matching magnitudes. Also shown in figur es 335, 337 and 339 are cases that impedance functions from the literature that were dete rmined via more rigorous, threedimensional, elastodynamic solution techniques. The solution for the surface case (Fi gure 335) was obtained from Luco and Mita (1987). The two solutions for the rectangular f ootings (Figure 337 and Figure 339) were obtained from Wong and Luco (1978). Solutions for the two remaining cases (circular, fully embedded and rect angular, surface, length/width = 1.5) were not readily available for Poissons ratio near 0.5. First, it is noted that the three imp edance functions (experimental, rigorous, and cone) are in good accord for the three cases presented (circular, square, and rectangular). This is encouraging. On the other hand, as one might expect, there is closer agreement between the rigorous impedance func tion solutions from the literature and the experimental results, than betw een the approximate cone model solutions and the experimental PAGE 51 51 results. However, it is interesting to note that the relative agreement between impedances for all three cases presented is very similar. This mi ght suggest that the error in the cone model impedances is not due to approximating a square or rectangular footing with an equivalent disk. 3.7 Crouse, et al. (1990) While com parison with the experimental re sults of Nii (1987) is encouraging, Niis experiments employed smallscale laboratory te sts using foundation and support materials that were not concrete and soil. It is likely that the bond between foundation and support material in these experiments is more ideal than between co ncrete and soil, and thus more closely matched the conditions assumed in the theoretical models. It is widely recognized that intimate bond is typically not realized in real fullscale geotechnical foundatio n conditions, particularly for embedded foundations along the foundation sidewall/s oil interface. Further, these experiments were conducted using a support material that is likely very homogenous compared to soil, and represents the simplest case to model. Crouse, et al. (1990) report experimental impedance functions for two small concrete foundations supported by soil and in real field co nditions. Both foundations were utilized to support earthquake accelerograph stations. One foundation (Cholame 1E) was located in central California, and consisted of a 10cm (4in.) thick, 1.27m (50in.) square slab resting on corner piers and embedded in a moderately stiff alluvial deposit. The ot her foundation (Station 6) was a 15cm (6in.) thick, 1.22m x 1.14m (48in. x 45i n.) nearly square slab, embedded 7.6 cm (3 in.) in a softer deposit in sout hern California. Shear wave velocity profiles were determined for each site via in situ SASW tests, and reported in detail by Crouse, et al. (1990). The soils at both sites were assumed by Crouse, et al. to have a unit weight of 17,300 N/m3 (110 lb/ft3), a Poissons ratio of 1/3, and a material damping ratio of 0.015. PAGE 52 52 To determine impedance functions, forced harm onic vibration tests were performed on the foundations. Harmonic forces were induced with a small (156 N, 35 lb) eccentricmass vibration shaker bolted to the foundation surface. The me asured response was determined via triaxial accelerometers mounted at each corner of the f oundation. The impedance functions were then determined from the induced force and measured response via appropriate equations of motion for the dynamic system. The impedance functions are presented for the ve rtical, horizontal, and rocking modes of vibration and for frequencies from 10 to 60 Hz. Using the foundation and soil support physical characteristics and properties as input, impedance functions were computed with cone models for the Station 6 foundation. Since the cone model impedances are calculated for circular disk (axisymmetric ) foundations, the actual square foundation was converted to an equivale nt disk for cone model computations. The equivalency was based upon cont act area for translational mode s (vertical, horizontal), and moment of inertia for rotational modes (rocking) Cholame 1E was not investigated for this study since the corner piers supporting the conc rete slab produce a more complex foundation system than is likely appropriate for the cone model approach. Cone model impedance solutions are compared w ith the experimental results of Crouse, et al. (1990) in Figure 340 to Figure 342. Each graph plots real and imaginary components along with the magnitude of the impedance function in dimensional form versus frequency. Impedance function magnitude is simply the vector summati on of the real and imaginary components. The three sets of data for each of the Crouse, et al. impedance functions correspond to the three different shaker eccentricities used during the experiments. The induced load increases with increasing frequency for an eccentricmass shaker, thus the eccentricity was lowered from an PAGE 53 53 initial setting in two increments as frequency increased to prevent significant nonlinear response of the dynamic system. While agreement between cone model predictions and experiment results are not as good as with the Nii (1987) experiments, it is observed that the agreement is credib le and encouraging. The impedances are all of similar magnitude, an d the trends are similar for the vertical and horizontal modes. There is a disparity in shape for the rocking mode. Unlike the simple homogeneous foundation support system in the Nii ( 1987) experiments, the soil profile at Station 6 is nonhomogeneous and multilayered, yet the c one model predictions appear reasonable. Perhaps the most notable disparity for Station 6 is that the cone models generally overpredict the impedance magnitude, which is opposite th e findings presented above for the Nii (1987) experiments. Overprediction for Station 6 coul d be due to many contributing factors. Both ideal bond between foundation and soil, and effici ent radiation of ener gy from the vibrating foundation through the supporting soil would create a larger predicted impedance magnitude. Neither of these can be expected from a real geotechnical foundation system. Despite these encouraging results, it is probably fair to conclude that the 0% engineering accuracy documented by Wolf and Deeks (2004) is not main tained for these comparisons with measured foundation behavior. Obviously, a much larger data base of comparisons is desired before more definitive conclusions can be drawn. PAGE 54 54 Table 31. Parameters for Eight Wong and Luco (1985) Comparisons Model no. Parameters* Comment 1 Vs1/Vs2 = 0.6, 1 = 0.33, H/a = 2 a = 2 m, Vs1 = 125 m/s, 1 = 1600 kg/m3 Basis model 2 a = 1 m Influence of scale: 1/2 footing size 3 Vs1 = 250 m/s Influence of scale: 2x shear wave velocity 4 H/a = 0.5 Influence of layer thickness: 1/4 layer thickness 5 H/a = 4 Influence of layer thickness: 2x layer thickness 6 Vs1/Vs2 = 0.3 Influence of stiffness contrast: higher 7 Vs1/Vs2 = 0.8 Influence of stiffness contrast: lower 8 1 = 0.45 Influence of Poissons ratio: higher Note: For Models 28, the parameters shown are those that are different from the parameters for the Basis Model 1. PAGE 55 55 Table 32. Soil Profile for Apsel and Luco (1987) Model Wave velocity (m/s) Material damping ratio Layer no. Layer thickness (m) Unit weight (kN/m3) S P S P 1 1.8 20.89 192.0 384.0 0.01 0.005 2 3.7 20.89 338.3 676.7 0.01 0.005 3 4.3 20.89 420.6 841.2 0.01 0.005 4 10.4 20.89 487.7 975.4 0.01 0.005 5 82.3 20.89 609.6 1219.2 0.01 0.005 6 16.2 20.89 762.0 1524.0 0.01 0.005 7 20.89 944.9 1889.8 0.01 0.005 PAGE 56 56 Table 33. Model Footings of Nii (1987) Shape Circular Rectangular on surface Location Radius (cm) Thickness (cm) Length/Width Width (cm) Thickness (cm) Surface 1*, 2, 3, & 4 1 1 1*, 2, 3, & 4 1 1.5 1*, 2, & 3 1 Fully embedded 2* 2 2 1* & 2 1 Note: Analyzed via cone model herein. PAGE 57 57 A B C Figure 31. Vertical Impedance Functions for Model 1, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 58 58 A B C Figure 32. Horizontal Impedan ce Functions for Model 1, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 59 59 A B C Figure 33. Rocking Impedance Functions for M odel 1, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 60 60 A B C Figure 34. Torsion Impedance Functions for Mo del 1, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 61 61 A B C Figure 35. Vertical Impedance Functions for Model 4 Length/Width=4, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 62 62 A B C Figure 36. Torsion Impedance Functions for Model 4 Length/Width=4, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 63 63 A B C Figure 37. Horizontal Impedan ce Functions in Long Directi on for Model 4 Length/Width=4, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 64 64 A B C Figure 38. Rocking Impedance Functions Abou t Short Axis. for Model 4 Length/Width=4, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 65 65 A B C Figure 39. Horizontal Impedan ce Functions in Short Directi on for Model 4 Length/Width=4, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 66 66 A B C Figure 310. Rocking Impedance Functions A bout Long Axis for Model 4: Length/Width=4, Wong and Luco (1978). A) Real. B) Imaginary. C) Magnitude. PAGE 67 67 A B C Figure 311. Vertical Impedance Functions fo r Model 3 Embedded, Mita and Luco (1989). A) Real. B) Imaginary. C) Magnitude. PAGE 68 68 A B C Figure 312. Horizontal Impedance Functions for Model 3 Embedded, Mita and Luco (1989). A) Real. B) Imaginary. C) Magnitude. PAGE 69 69 A B C Figure 313. Rocking Impedance Functions for Model 3 Embedded, Mita and Luco (1989). A) Real. B) Imaginary. C) Magnitude. PAGE 70 70 A B C Figure 314. Torsional Impedance Functions fo r Model 3 Embedded, Mita and Luco (1989). A) Real. B) Imaginary. C) Magnitude. PAGE 71 71 A B C Figure 315. Coupled Impedance Functions for Model 3 Embedded, Mita and Luco (1989). A) Real. B) Imaginary. C) Magnitude. PAGE 72 72 A B C Figure 316. Vertical Impedance Functions for Model 1: Basis Model, Wong and Luco (1985). A) Real. B) Imaginary. C) Magnitude. PAGE 73 73 A B C Figure 317. Horizontal Impedance Functions for Model 1: Basis Model, Wong and Luco (1985). A) Real. B) Imaginary. C) Magnitude. PAGE 74 74 A B C Figure 318. Rocking Impedance Functions for Model 1: Basis Model, Wong and Luco (1985). A) Real. B) Imaginary. C) Magnitude. PAGE 75 75 A B C Figure 319. Torsional Impedance Functions for Model 1: Basis Model, Wong and Luco (1985). A) Real. B) Imaginary. C) Magnitude. PAGE 76 76 A B C Figure 320. Vertical Impedance Functions for Model 4: Layer Thickness, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 77 77 A B C Figure 321. Horizontal Impedance Functions for Model 4: Layer Thickness, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 78 78 A B C Figure 322. Rocking Impedance Functions for Model 4: Layer Thickness, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 79 79 A B C Figure 323. Torsional Impedance Functions for Model 4: Layer Thickness, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 80 80 A B C Figure 324. Vertical Impedance Functions for Model 6: Higher Vs Contrast, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 81 81 A B C Figure 325. Horizontal Impedance Functions for Model 6: Higher Vs Contrast, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 82 82 A B C Figure 326. Rocking Impedance Functions for M odel 6: Higher Vs Contrast, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 83 83 A B C Figure 327. Torsional Impedance Functions for Model 6: Higher Vs Contrast, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 84 84 A B C Figure 328. Vertical Impedance Functions fo r Model 8: Poissons Ratio=0.45, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 85 85 A B C Figure 329. Horizontal Impedance Functions for Model 8: Poissons Ratio=0.45, Wong and Luco (1985). A) Real. B) Imaginary. C) Magnitude. PAGE 86 86 A B C Figure 330. Rocking Impedance Functions for Model 8: Poissons Ratio=0.45, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 87 87 A B C Figure 331. Torsional Impedance Functions fo r Model 8: Poissons Ratio=0.45, Wong and Luco (1985). A) Real. B) Im aginary. C) Magnitude. PAGE 88 88 A B C Figure 332. Horizontal Impedance Functions for Layered Model, Apsel and Luco (1987). A) Real. B) Imaginary. C) Magnitude. PAGE 89 89 A B C Figure 333. Rocking Impedance Functions for La yered Model, Apsel and Luco (1987). A) Real. B) Imaginary. C) Magnitude. PAGE 90 90 A B C Figure 334. Coupled Impedance Functions for La yered Model, Apsel and Luco (1987). A) Real. B) Imaginary. C) Magnitude. PAGE 91 91 A B C Figure 335. Vertical Impedance Functions for Circular, Surface Model Footing, Nii (1987). A). Real. B) Imaginary. C) Magnitude. PAGE 92 92 A B C Figure 336. Vertical Impedance Functions for Circular, Fully Embedded Model Footing, Nii (1987). A). Real. B) Im aginary. C) Magnitude. PAGE 93 93 A B C Figure 337. Vertical Impedance Functions for Rectangular, Surface, L/W=1 Model Footing, Nii (1987). A). Real. B) Im aginary. C) Magnitude. PAGE 94 94 A B C Figure 338. Vertical Impedance Functions fo r Rectangular, Surface, L/W=1.5 Model Footing, Nii (1987). A). Real. B) Imaginary. C) Magnitude. PAGE 95 95 A B C Figure 339. Vertical Impedance Functions fo r Surface, L/W=2 Model Footing, Nii (1987). A). Real. B) Imaginary. C) Magnitude. PAGE 96 96 A B C Figure 340. Vertical Impedance Functions for Station 6, Crouse, et al. (1990). A) Real. B) Imaginary. C) Magnitude. PAGE 97 97 A B C Figure 341. Horizontal Impedance Functions for Station 6, Crouse, et al. (1990). A) Real. B) Imaginary. C) Magnitude. PAGE 98 98 A B C Figure 342. Rocking Impedance Functions for St ation 6, Crouse, et al. (1990). A) Real. B) Imaginary. C) Magnitude. PAGE 99 99 CHAPTER 4 SITE CHARACTERIZATION Two shallow foundations, the firs t 1 m x 1m in plan and embedded 0.9 m, and the second 3 m x 3 m in plan and embedded 0.9 m, have been dynamically loaded at the National Geotechnical Experiment Site (NGES) on the campus of Texas A&M University (TAMU). Foundation responses were recorded in order to determine measured values for impedance functions. This chapter presents a description of the site, foundations, and equipment used to perform tests and also supplies necessary info rmation for calculating impedance functions. 4.1 Site Investigation The TAMU NGES site has been ex tensively ch aracterized via both in situ and laboratory soil testing techniques, including shear wave velocity via both borehole and surface wave methods. The site consists of an upper layer of approximately 10 m of medium dense, silty, fine sand followed by hard clay. The water table is approximately 5 m below the ground surface. A more detailed description of site conditions is presented in the following paragraphs. 4.1.1 Traditional Investigation The NGES s ite was investigated by both trad itional and seismic geotechnical methods. Traditional methods are outlined in the proceedings of Briaud and Gibbens (1994). Of particular interest is the soils profile which consists of a layer (Layer 1) of tan medium dense silty fine sand from existing grade to approximately two m. Directly beneath the first layer is a tan medium dense silty sand with clay (Layer 2) to approximately 10.5 m below grade. The water table is encountered in the sec ond layer at an approximate depth of five m. Below the second layer is a dark gray very hard clay (Layer 3) extending to the bottom of the borings all of which terminate 15.2 m below grade. PAGE 100 100 Values for unit weights of each layer were eith er identified in Briaud and Gibbens (1994) or calculated from information available in the proceedings. For layers one and two they give values of unit weight above the water table as 15.28 KN/m2 and 15.65 KN/m2 respectively. Below the water table, values were calculated based on saturation and void ratio of 19.11 KN/m2. Properties of the clay were assumed based on boring log descriptions due to the lack of tests performed on the clay in the proceedings. Prior to the construction of foundations, between approximately 0.5 m and 1.5 m of overburden was removed from the site. Many additional studies were conducted. Field tests such as standa rd penetration tests (SPT) and cone penetration tests (CPT) were perf ormed along with others. Laboratory tests were also conducted such as triaxial tests and those used to calculate the information above. Results for all tests can be found in Briaud and Gibbens (1994). 4.1.2 Seismic Investigation Seism ic tests of the NGES site were perfor med. Briaud and Gibbens (1994) present data for crosshole seismic testing to a depth of ten meters. Testing did not include the clay layer which starts at 10.5 m thus did not allow adequate seismic charac terization of the layer. An accurate model below ten meters would not be possible without further seismic testing to characterize the clay layer. Hori zontal crosshole rays were initia ted at a depth of two meters. Depths for each subsequent ray were increased at two meter intervals which provided a general idea of subsurface conditions but, these interval s were not sufficient to produce the accuracy desired for the seismic modeling of the soil. Bo reholes used for the test were measured by an inclinometer to determine a precise distance between crosshole equipment at each depth. Resonant column tests were also performed on sa mples from the site and are available in Briaud and Gibbens (1994). PAGE 101 101 A second set of crosshole seismic tests were performed in March of 2007 to enhance the available velocity profile and exte nd the velocity profile into the cl ay layer. The results of these additional tests are displayed in Figure 41. Horizontal rays were measured in the same boreholes used for the first tests and another in clinometer survey was performed to accurately calculate the distance between cr osshole equipment at each dept h. The second set of crosshole tests differed from the first in depth and incremen t. The holes were surveyed to a depth of 14.6 m which was 4.6 m deeper than the initial test and 4.1 m into the clay. Tran and Hiltunen (2008) outline three seismic surface wave surveys conducted at the site in October of 2006. SASW, MASW, and passive MASW surveys were able to provide shear wave data for both footings. This was accomplished by laying out a single survey line that all surface wave tests were conducted along. The survey line was placed in close proximity to both footings assuring the most effective representa tion of the subsurface with regards to shear modulus. A detailed account of the methodology of all three techniques along with a comprehensive report of test procedures is given in Tran and Hiltunen (2008). Of most importance is the general agreement of all four tests indicating that th e tests were performed well thereby bolstering confidence in shear wave velocities used in impedance function prediction calculations. Traditional investigation data such as the bor ehole log and SPT values also seem to agree with seismic methods. Similarities noted by Tran and Hiltunen (2008) between two somewhat different approaches to measuri ng soil strength further suggest th at results provided by seismic analysis are reliable and appropriate fo r use in impedance function calculations. 4.2 Foundation Selection and Construction Field testing required certain characteristics for a footing to be acceptable. Acceptable characteristics were those found in typical construction practices such as scale. There is no PAGE 102 102 specific guideline for such a footing so some estimation was required pertaining to what is common. Embedded footings were necessary as a surface footing would have neglected the study of coupling impedances. A dditionally, the footing had to e ssentially behave as a rigid body under dynamic loading conditions. The NGES site was selected for its previ ously conducted site investigations and foundations that were constructe d for previous research. The intention was to minimize time, cost and effort by utilizing available resources. Generally these intentio ns were realized. Soil data was readily available though some additional test s were required to expand it. In the case of existing footings, a visual inspection of the site revealed that many of th e footings described in Briaud and Gibbons (1994) were un suitable due to severe cracking. Of the footings that were considered for testing, only one, previously named Footing 5, remained in tact, Figure 42. That footing was a small unreinforced 0.9 m by 0.9 m by 1.2 m solid concrete footing embedded 0.7 m. Its center was approximately 3 m from the seismic surface wave survey line making its location convenient for impedance function predictions. A larger footing was desired for shaking. Unfortunately prior testing had damaged any other acceptable footings at the site. The decisi on to construct a larger footing was made. An arbitrary design was decided upon that was a pproximately 3 m by 3 m by 1 m and fully embedded, Figure 43. The design is fully reinfo rced with # 6 reinforcing steel every 5 in, Figure 44, concrete block of 4,000 psi concrete which avoids failure of the foundation during testing and assures rigid behavior. A contractor was contacted that was capable of assisting with a design. The footing, deemed Footing 6, was c onstructed with the east edge 6 m from the seismic surface wave survey line. Though not as close as Footing 5, Footing 6 was adequately positioned to make use of the SASW survey line. PAGE 103 103 Both footings were to be te sted in the vertical directi on with shakers resting on their surfaces. After vertical tests were concluded, shakers were to te st in the horizontal direction requiring anchor bolts to be sunk in order to fast en the shakers to the footings. These anchor bolts were also to be used to attach frames for additional shaker configurations. In the case of Footing 6 multiple anchoring positions were to be created. 4.3 Shakers Four dynamic loading devices im plemented in several configurations have been used to excite each footing in vertical, horizontal sliding, and rocking vibration modes. The four devices are listed as follows: 1) an APS Model 400 Elec tro Seis presented in Figure 42 capable of applying a constant maximum load amplitude of 70 lb over a frequency range from 1 to 100 Hz, 2) an ANCO Model MK12 rotating mass vibrator presented in Figure 45 with a maximum applied load of 10,000 lb, and a frequency range fr om 0 to 100 Hz, 3) the NSF NEES Thumper mobile shaker presented in Figure 46 operated by the University of Texas (Stokoe, et al.), and 4) the NSF NEES TRex mobile shaker presented in Figure 47 operated by the University of Texas (Stokoe, et al.). The Thumper and TRex shakers were used to apply vertical loads to the footings, while the APS and ANCO devices were configured to conduct independent experiments in both the vertical and horizontal directions. In addition, the APS and ANCO shakers configured in horizontal mode were moun ted directly on the footings, and in an elevated position via an attached frame in order to accen tuate the rocking mode of vibration, and to provide a second means to excite the coupled hor izontal sliding and rocking modes by at least two independent means as described above. The complete experimental program produced impedance functions for the vertical, horizontal sliding, rocking, and c oupled sliding/rocking vibration modes and for each of the two embedded foundations. For some vibration mode s, independent experiments were conducted PAGE 104 104 using more than one of the available vibrati on shakers. Results for these experiments are presented in the following chapter. PAGE 105 105 Figure 41. Seismic, SPT, and prof ile Tran and Hiltunen [2008] PAGE 106 106 Figure 42. Footing 5 PAGE 107 107 Figure 43. Footing 6 Figure 44. Reinforcement of Footing 6 PAGE 108 108 Figure 45. ANCO Model MK12 Figure 46. NSF NEES Thum per mobile shaker PAGE 109 109 Figure 47. NSF NEES TRex mobile shaker PAGE 110 110 CHAPTER 5 MEASURED IMPEDANCE FUNCTIONS Dyna mic loads were applied to foundations to determine dynamic impedance functions as presented in Chapter 4. Two shallow foundations were chosen for this investigation and four dynamic load devices were available. The speci fic experiments are described in the following paragraphs from which dynamic impedance functions were determined herein in accordance with the calculations of Chapter 2. 5.1 Footing 5 5.1.1 Vertical Loading Figure 51 presents an experim ent for the ve rtical loading of Footing 5 using an ANCO Model MK12 rotating mass shaker mounted flush to the footing surface. The shaker was configured to load in the vertical direction. The magnitude of the load was determined by the speed and eccentricity settings of the rotating masses. The photogra ph also displays the velocity transducers deployed at each of the four corner s of the footing surface for measurement of the footing response to load. Figure 52 presents measurements of foundation load and vertical displacement. Figure 52a displays continuously increasing amplitude of vertical load as frequency increases, which is characteristic behavior for a rotatingmasstype shaker. The resulting vertical displacement response of the foundation is disp layed in Figures 52b and 52c. The complex number (real and imaginary) format implicitly contains both the amplitude of the sinusoidal displacement response, as well as the phase lag of the displ acement from the applied sinusoidal loading. In operation, the loading was conducted in a steadystat e mode wherein several cycles of load were applied at a constant frequency, beginning at the lowest frequenc y. The dynamic response of the foundation was measured simultaneously via the velocity transducers, and recorded via a PAGE 111 111 dynamic signal analyzer capable of characterizing the signals in complex number format. The displacement response was determined by integrati on of the velocity tran sducer measurements, and the loading was repeated at small increments of frequency through th e frequency range noted in Figures 52b and 52c. Fina lly, it is noted that the real and imaginary components of the displacement responses are displayed for each of the four corners of the foundation. The four responses are indeed very similar, which in part indicates that the founda tion is moving vertically in a rigidbodylike fashion. The average f oundation displacement is calculated from the response of all the footing corner s, which in turn is used to calculate the impedance relationship. Figure 53 presents the real and imaginary components of the dynamic impedance function computed from the vertical load, response measurements, and the foundation characteristics as outlined in Equation 26. Vertical force was calculated in accordance with Equation 27. 5.1.2 Horizontal Loading Figure 54 presents an experim ent for the hor izontal loading of Footing 5 using an APS Dynamics Model 400 ElectroSeis shaker suspended from a tripod and configured to load in the horizontal direction. The horizonta l load was applied to a steel frame attached to the footing surface, and the magnitude of the applied load was measured via a load cell attached between the shaker and frame connection. The photograph also displays the velocity transducers deployed at each of the four corners of the footing surface for measurement of the footing response to load. Figures 55, 56, 57, and 58 present measurements of foundation load, horizontal displacement, and vertical displacement for shaker elevations of 1, 2, 3, and 4 ft above the top surface of the footing. The loadi ng and measurement process was very similar to that described above for previous tests, and the results are di splayed in the same complex number (real and imaginary) format. The responses at each of th e foundation corners are presented individually in PAGE 112 112 the figures. It should be noted that the four horizontal displace ments appear very similar for each elevation, indicating that th e foundation is moving horizontally in a rigidbodylike fashion. The vertical displacements ar e due to the rocking motion of the foundation. Here, it is observed that there are two pairs of similar data for each elevation. The similar responses are for the two corners along a foundation edge that is parallel to the axis of rocking. Of course, there are two such edges, and they ar e each on opposite sides of the axis of rocking. These responses should be at similar amplitude, but 180 degrees out of phase if the rocking axis is contained in the same vertical plane as the center of the f oundation. Indeed, the data possess this symmetry: the real and imaginary pairs are both of similar magnitude, but opposite in sign (i.e., 180 degrees out of phase). This behavior also suggests a rigidbodylike rocking moti on, and with this as an assumption, the rotation of the foundation can be calculated from the geometry and the vertical displacement responses. Some other observations regarding these data are appropriate. First, it should be noted that the applied shaker force decreases significantl y with increasing frequency. The APS 400 is ideally suited to apply constant force amplit ude of about 70 pounds (340 KN) over a frequency range from about 1 to 10 Hz. Above 10 Hz, the force output diminishes significantly due to the limitations of the power amplifier. Second, it is ob served that a system resonant response occurs at a frequency between 40 and 45 Hz. At resonance, the real component of the displacement passes through zero, and a local peak occurs in the imaginary component. The resonance effect is also noted in the force response for shaker posit ions at 3 and 4 ft. Un fortunately, the low force output coupled with a system resonance produces erratic behavior in the measured data at frequencies above approximately 35 Hz. The erratic behavior is more significant as the shaker becomes closer to the footing surface. PAGE 113 113 Using the measured load and displacement re sponses, impedance functions were calculated following the procedures described in Chapter 2. First, Figures 59, 510, 511, and 512 present the real and imaginary components of the horizontal and rocking dynamic impedance functions for each of the four shaker positions and computed from the horizontal load and res ponse measurements as well as the foundation characteristics. The impedance functions were calcu lated in accordance with the single degree of freedom method expressed in Equations 28 and 29, and thus assume that the crosscoupling impedances are zero. Because of the erratic behavior of the da ta discussed above, the impedance functions are displayed only to a maximum frequency of 30 Hz. While there is some variability, it is observe d that the impedance functions are consistent in both magnitude and variation with frequency (i.e., shape) for each of the four shaker positions. It is encouraging that the results are similar for each of the shaker positions, and suggests that the assumption of a linear relationship between forc e and displacement appears reasonable for this experimental configuration. Second, impedance functions were calculated fr om the measured load and displacement responses following the twodegreeoffreedom approach presented in Equations 12 through 15 in Chapter 2. These calculations require load and displacement data fr om experiments conducted at two independent shaker positions. Figures 13 through 18 present results for each of the six possible pairs of shaker position measurements (e.g., 1 ft and 2 ft, 1 ft and 3 ft, 1 ft and 4 ft, etc.), and assumes that the crosscoupling impedances are independent. Similarly, Figures 19 through 24 present results assuming the cros s coupling impedances are equal to one another. As for the onedegreeoffreedom results presented above, all impedances are displayed to a maximum frequency of 30 Hz. PAGE 114 114 First, it is observed in both sets of results that many of the impedanc e functions calculated based on experimental data from the 1 ft shaker po sition are suspect. It is recalled that data from this position are most erratic. Several of these suspect impedances displa y erratic variation in magnitude with changing frequency, and in a few cas es negative values for the real (stiffness) or imaginary (damping) components are produced, which are very unlikely to be indicative of real physical behavior. On the other hand, the result s produced from shaker position pairs of 2 ft and 3 ft, 2 ft and 4 ft, and 3 ft and 4 ft appear consistent in both magnitudes and shape, and do not display the negative values previously noted. Finally, as a means of comparison between solution techniques, Figure 25 presents impedance functions from a oned egreeoffreedom solution at pos ition 4 ft (Figure 12), a twodegreeoffreedom solution with independent crosscoupling using shaker positions 2 ft and 4 ft (Figure 17), and a twodegreeoffreedom solu tion assuming equal crosscoupling and using shaker positions 2 ft and 4 ft (Figure 23). A last issue needs to be addressed regardi ng the calculation of impedance functions from the measured load and response data presented he rein, and following the procedures detailed in Chapter 2. In Chapter 2, Equations 10 and 11 en able calculation of the horizontal soil force, Fs, and the rocking moment, Ms, referenced to the foundation soil interface at the horizontal center of the footing. The calculation of Fs and Ms for a given frequency, are based upon several parameters that include the footing mass and geom etric characteristics, th e input shaker force, and the measured displacements of the foundatio n, assuming that the foundation moves as a rigid body. As noted above, the assumption of rigid b ody motion appears reasonable. In addition, the calculation of Fs and Ms also depends on the mass and ge ometric characteristics and the displacements of the attached frame for those cases in which a frame was utilized in the PAGE 115 115 experiment. Two approaches were utilized a nd compared for determination of the frame displacements. First, like th e foundation, the frame was assumed to behave as a rigid body, and thus the frame displacements were as determined from the responses of the geophones attached to the footing. Second, the frame displacements were calculated using a structural dynamics model available in Paz and Leigh (2004) that will account for the flexibility of the frame rather than assume rigid body motion. The model assume d the frame was a cantilever beam subdivided into elements, each with appropriate lumped ma ss and stiffness properties. Utilizing these two sets of frame displacements, two sets of impedance functions were calculated. It was found that the two impedance function sets were essentially identical, which indicates that the inertial effects due to flexibility of the frame were inc onsequential in these experiments. All of the impedance functions presented herein are base d upon frame displacements calculated using the structural dynamics model. 5.2 Footing 6 5.2.1 Vertical loading Figure 526 and 527 present the experim ent for the vertical loading of Footing 6 using the shaker named Thumper provided by the University of Texas. The shaker was configured to load in the vertical directi on. Magnitude and frequency of the lo ad were determined by an Interface Model 1020 load cell. In the phot ograph in Figure 525 three veloci ty transducers can be seen deployed at each of the four corners of the footing surface for measur ement of the footing response to load. While it appe ars the tires are resting on the footing, Thumper is actually supported by four black bladders two of which can be seen compressed in Figure 526. The bladders allow the shaker to dynamically load the footing while isol ating the truck from vibration. PAGE 116 116 Figure 528 presents measurements of foundation load and vertical displacement in similar fashion to Figure 52. Figure 528a displays am plitude of vertical load as a function of frequency. The resulting vertical displacement re sponse of the foundation is displayed in Figures 528b and 528c. The complex number (real a nd imaginary) format containing both the amplitude of the sinusoidal displacement response as well as the phase lag of the displacement from the applied sinusoidal loading is again used. As with the ANCO shaker, loading was conducted in a steadystate mode to allow several cycles of load to be applied at a constant frequency, beginning at the lowest frequency. The dynamic response of the foundation was measured simultaneously via the velocity tr ansducers, and recorded via a dynamic signal analyzer capable of characterizing the signals in complex number format. The displacement response was determined by integration of the velocity transducer measurements, and the loading was repeated at small increments of frequenc y through the frequency range noted in Figures 528b and 528c. Again, it is noted that the displa cement responses are displayed for each of three corners of the foundation. Data for the fourth corner was not av ailable due to a malfunctioning geophone. The three responses are similar, indica ting that the foundation is moving vertically in a rigidbodylike fashion. Figure 529 presents the real and imaginary co mponents of the vertical dynamic impedance function calculated from the Thumper data. Verti cal impedances for the test of Footing 6 using thumper were computed and displayed in the same fashion as those of Section 5.1. Figures 530 and 531 are images of an experi ment for the vertical loading of Footing 6 using the shaker provided by the University of Texas named TRex. The shaker was configured to load in the vertical direction. Magnitude and frequency of the load were determined by an Interface Model 1240 load cell exhibited in Figure 531. Th e photographs in Figure 530 and PAGE 117 117 Figure 531 also exhibit three velocity transducer s deployed at each of the four corners of the footing surface for measurement of the footing response to load. Figure 532 presents measurements of foundati on load and vertical displacement. Figure 532a displays amplitude of vertic al load for each frequency. Th e resulting vertical displacement responses of the foundation are found in Figures 532b and 532c. As with the test configurations discussed above, the complex number (real and imag inary) format contains both the amplitude of the sinusoidal displacement response, as well as the phase lag of the displacement from the applied sinusoidal lo ading. During the experiment, loading was conducted in a steadystate mode wherein several cycles of load were applied at a constant frequency, beginning at the lowest frequency. The dynamic response of the foundation was measured concurrently via the velocity transduc ers, and recorded via a dynamic signal analyzer capable of characterizing the signals in comple x number format. The displacement response was determined by integration of the velocity transd ucer measurements, and the loading was repeated at small increments of freque ncy through the frequency range not ed in Figures 532b and 532c. The displacement responses are presented for eac h of the four corners of the foundation. The four responses are similar, indicating that the foundation is mo ving vertically as a rigidbody. Figure 533 presents the real and imaginary co mponents of the vertical dynamic impedance function. Impedances for the test of Footing 6 using TRex were calculate d and presented in the same fashion as those from Sections 5.1 and 5.3. 5.2.2 Horizontal Loading Figure 534 and Figure 535 present two experiments for the hor izontal loading of Footing 6 using an ANCO Model MK12 rotating m ass shak er configured to load in the horizontal direction. The first horizontal loading configuration was perfor med with the shaker attached directly to the foundation as illustrated by Figure 534. The second loading configuration PAGE 118 118 includes a frame attached to the foundation and the shaker attached to the top of the frame pictured in Figure 535. The magnitude of the load was determined by the speed and eccentricity settings of the rotating masses. The photograp h in Figure 534 also displays the velocity transducers deployed at each of the four corner s of the footing surface for measurement of the footing response to load. Figures 536 and 537 present measurements of foundation load, horiz ontal displacement, and vertical displacement for the flush mounting and the elevated mounting. The loading and measurement process was very similar to those of previously mentioned loading experiments, and the results displayed in Figure 536 and Fi gure 537 are in the same complex number (real and imaginary) format. The responses at each of the foundation corners are displayed individually in the figures. It should be noted that the four ho rizontal displacements are very similar for each shaker arrangement, which in part indicates that th e foundation is moving horizontally in a ri gidbodylike fashion. The vertical displacements are due to the rocking motion of the foundation. As with the horizontal test of Section 5.2, it is observed that there ar e two pairs of simila r data. The similar responses are for the two corners al ong a foundation edge that is para llel to the axis of rocking. There are two such edges, and they are each on opposite sides of the axis of rocking. These responses should be at similar amplitude, but 180 degrees out of phase if the rocking axis is contained in the same vertical plane as the center of the foundation. The displacements possess this symmetry: the real and imaginary pairs are both of similar magnitude, but opposite in sign. This behavior suggests rigidbody rocking motion, a nd with this as an assumption, the rotation of the foundation can be calculated from the geom etry and the vertical displacement responses. PAGE 119 119 Using the measured load and displacement re sponses, impedance functions were calculated following the procedures described in Chapter 2. Figures 538 and 539 present the real and im aginary components of the horizontal and rocking dynamic impedance functions for each of the two shaker positions and computed from the horizontal load and response measurements as well as the foundation characteristics. The impedance functions were calculated in accord ance with the singledegreeoffreedom method expressed in Equations 28 and 29, and thus a ssume that the crosscoupling impedances are zero. While there is some variability, it is observe d that the impedance functions are similar in both magnitude and variation with frequency (i.e ., shape) for the two shaker positions. If anything, the behavior of the hor izontal impedance function for th e elevated position displays more erratic behavior fo r frequency above 25 Hz. Impedance functions were calculated from th e measured load and displacement responses following the twodegreeoffreedom approach pres ented in Equations 12 through 15 in Chapter 2. These calculations require load and displace ment data from experiments conducted at two independent shaker positions. Figure 40 presents results assuming that the crosscoupling impedances are independent, while results shown in Figure 41 assume the cross coupling impedances are equal to one anothe r. While there are some instan ces of erratic behavior, it is observed that the impedance functions are similar in both magnitude and variation with frequency (i.e., shape) for the two solution techniques. As a means of comparison between solution techniques, Figure 42 presents impedance functions from onedegreeoffreedom solutions for both flush and elevated mountings (Figures PAGE 120 120 38 and 39), and the twodegreeoffreedom solution with independent crosscoupling (Figure 40) and the twodegreeoffreedom solution assuming equal crosscoupling (Figure 41). PAGE 121 121 Figure 51. Vertical Experiment for Footing 5 with ANCO shaker. A B C Figure 52. ANCO vertical results for Footing 5. A) Vertical Loa d. B) Vertical Displacement, Real Component. C) Vertical Di splacement, Imaginary Component. PAGE 122 122 A B Figure 53. ANCO vertical results for Footing 5. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imaginary Component. PAGE 123 123 Figure 54. Horizontal Experi ment for Footing 5 with M odel 400 ElectroSeis shaker. PAGE 124 124 A B C D E Figure 55. ElectroSeis horizontal lo ading results for Footing 5 at elevation 1 ft. A) Horizontal Load. B) Horizontal Displacement, Real Component. C) Horizontal Displacement, Imaginary Component. D) Vertical Displ acement, Real Component. E) Vertical Displacement, Imaginary Component. PAGE 125 125 A B C D E Figure 56. ElectroSeis horizontal lo ading results for Footing 5 at elevation 2 ft. A) Horizontal Load. B) Horizontal Displacement, Real Component. C) Horizontal Displacement, Imaginary Component. D) Vertical Displ acement, Real Component. E) Vertical Displacement, Imaginary Component. PAGE 126 126 A B C D E Figure 57. ElectroSeis horizontal lo ading results for Footing 5 at elevation 3 ft. A) Horizontal Load. B) Horizontal Displacement, Real Component. C) Horizontal Displacement, Imaginary Component. D) Vertical Displ acement, Real Component. F) Vertical Displacement, Imaginary Component. PAGE 127 127 A B C D E Figure 58. ElectroSeis horizontal lo ading results for Footing 5 at elevation 4 ft. A) Horizontal Load. B) Horizontal Displacement, Real Component. C) Horizontal Displacement, Imaginary Component. D) Vertical Displ acement, Real Component. E) Vertical Displacement, Imaginary Component. PAGE 128 128 A B C D Figure 59. ElectroSeis horizonta l loading results for the onedegreeoffreedom analyses of Footing 5 at elevation 1 ft. A) Horizontal Impedance, R eal Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. PAGE 129 129 A B C D Figure 510. ElectroSeis horizonta l loading results for the onedegreeoffreedom analyses of Footing 5 at elevation 2 ft.A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. PAGE 130 130 A B C D Figure 511. ElectroSeis horizonta l loading results for the onedegreeoffreedom analyses of Footing 5 at elevation 3 ft.A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. PAGE 131 131 A B C D Figure 512. ElectroSeis horizonta l loading results for the onedegreeoffreedom analyses of Footing 5 at elevation 4 ft.A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. PAGE 132 132 A B C D E F Figure 513. ElectroSeis horizon tal loading results for the no nequal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 2 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 133 133 A B C D E F Figure 514. ElectroSeis horizon tal loading results for the no nequal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 3 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 134 134 A B C D E F Figure 515. ElectroSeis horizon tal loading results for the no nequal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 4 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 135 135 A B C D E F Figure 516. ElectroSeis horizon tal loading results for the no nequal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 3 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 136 136 A B C D E F Figure 517. ElectroSeis horizon tal loading results for the no nequal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 137 137 A B C D E F Figure 518. ElectroSeis horizon tal loading results for the no nequal coupling twodegreeoffreedom analyses of Footing 5 at elevations 3 ft and 4 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 138 138 A B C D E F Figure 519. ElectroSeis horizon tal loading results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 2 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 139 139 A B C D E F Figure 520. ElectroSeis horizon tal loading results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 3 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 140 140 A B C D E F Figure 521. ElectroSeis horizon tal loading results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 1 ft and 4 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 141 141 A B C D E F Figure 522. ElectroSeis horizon tal loading results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 3 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 142 142 A B C D E F Figure 523. ElectroSeis horizon tal loading results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 143 143 A B C D E F Figure 524. ElectroSeis horizon tal loading results for the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 3 ft and 4 ft. A) Ho rizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 144 144 A B C D E F Figure 525. ElectroSeis horizon tal loading results for the on edegreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft. A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 145 145 Figure 526. Vertical Experiment with Thumper shaker. Figure 527. Vertical Experiment with Thumper shaker. PAGE 146 146 A B C Figure 528. Thumper vertical results for Footing 6. A) Vertical Load. B) Vertical Displacement, Real Component. C) Vertical Di splacement, Imaginary Component. A B Figure 529. Thumper vertical results for Footing 6. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imaginary Component. PAGE 147 147 Figure 530. Vertical Experiment with TRex shaker. Figure 531. Vertical Experiment with TRex shaker. PAGE 148 148 A B C Figure 532. TRex vertical results for Footing 6. A) Vertical Load. B) Vertical Displacement, Real Component. C) Vertical Di splacement, Imaginary Component. A B Figure 533. TRex vertical results for Footing 6. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imaginary Component. PAGE 149 149 Figure 534. Horizontal Experi ment for Footing 6 with Model MK12 ANCO shaker mounted flush. PAGE 150 150 Figure 535. Horizontal Experi ment for Footing 6 with Model MK12 ANCO shaker on stand. PAGE 151 151 A B C D E Figure 536. ANCO horizontal loadi ng results for Footing 6 flush mounting. A) Horizontal Load. B) Horizontal Displacement, Real Co mponent. C) Horizontal Displacement, Imaginary Component. D) Vertical Displ acement, Real Component. F) Vertical Displacement, Imaginary Component. PAGE 152 152 A B C D E Figure 537. ANCO horizontal load ing results for Footing 6 elev ated mounting. A) Horizontal Load. B) Horizontal Displacement, Real Component. C) Horizontal Displacement, Imaginary Component. D) Vertical Displ acement, Real Component. E) Vertical Displacement, Imaginary Component. PAGE 153 153 A B C D Figure 538. ANCO horizontal loadi ng results for the onedegreeoffreedom analyses of Footing 6 flush mounting. A) Horizontal Impeda nce, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. E) Rocking Impedance, Imaginary Component. PAGE 154 154 A B C D Figure 539. ANCO horizontal loadi ng results for the onedegreeoffreedom analyses of Footing 6 elevated mounting. A) Hori zontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. PAGE 155 155 A B C D E F Figure 540. ANCO horizontal load ing results for the nonequal coupling twodegreeoffreedom analyses of Footing 6. A) Horizontal Impe dance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 156 156 A B C D E F Figure 541. ANCO horizontal load ing results for the equal coupling twodegreeoffreedom analyses of Footing 6. A) Horizontal Impe dance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 157 157 A B C D E F Figure 542. ANCO horizontal loadi ng results for onedegreeoffree dom at flushed and elevated positions, independent coupling twodegreeoffreedom, and equal coupling twodegreeoffreedom analyses of Footi ng 6. A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imaginary Component. E) Coupling Impedance, Real Component. F) Coupling Impedance, Imaginary Component. PAGE 158 158 CHAPTER 6 IMPEDANCE FUNCTION COMPARISON AND ASSESSMENT Insitu im pedance functions were calculated for two shallow foundations as presented in Chapter 5. In Chapter 3, cone models demonstrated agreement with more rigorous prediction techniques. Predictions for th e Foundations 5 and 6 were pr oduced using the cone model executable program CONAN. Development of cone models and comparison of cone model predictions to insitu results are pr esented in the following paragraphs. 6.1 Cone Model Development 6.1.1 Soil Profile Im pedance function predictions were calculated via cone modeling. With accuracy suitable to represent other pred iction techniques and relativel y simple implementation, cone modeling was selected for comparison to measured insitu values of impedance functions. The cone models required certain physical characterist ics of the soil profile and foundation. Table 61 presents the properties of the soil surrounding Footings 5 and 6 utilized for CONAN predictions. The properties were either pr ovided by, calculated from, or assumed based on results from Briard and Gibbens (1994), Tran an d Hiltunen (2008) or test ing performed primarily for impedance function research. Layer interface depths were determined from a combination of information from Briard and Gibbens (1994) and Tran and Hiltunen (2008). At depths where parameters changed a layer interface was assumed. For example, Briard a nd Gibbens (1994) reported a soil density and friction angle change at a depth of 1.80 m thus, there a layer interf ace was determined to exist. The MASW survey provided by Tran and Hiltunen (2008) indicated at a depth of 8.38 m shear wave velocity changes which required a layer interface. PAGE 159 159 The shear modulus, Gs, was calculated as a function of shear wave velocity, cs, and layer density, to form Equation 61. Shear wave veloc ity for each layer was determined via an MASW survey as described in Chapter 4. Soil density was reported by Briaud and Gibbens (1994) for Layers 1 through 5 of Table 61. Fo r Layers 6 and 7 no measured densities were available thus, soil density was a ssumed based on descriptions avai lable from the boring logs of Briard and Gibbens (1994). The de scription of a very hard clay led to the assumption for density of 1,886 kg/m3. 2 ssc G (61) Poissons Ratio was not available in Briard and Gibbens (1994), thus, as with density, this property was assumed based on borehole data. Layers 2 through 9 indicated medium dense sand with silt and clay content varying with depth; as a result a ratio of 0.333 was assumed. For Layers 10 and 11 a ratio of 0.45 is typical for hard clay below the water table. Soil Damping was assumed to be 3 percent ba sed on soil descriptions from Briard and Gibbens (1994) and typical values provided by Pr akash and Puri (1988). This term indicates only mechanical energy losses in the soil as geometric damping is accounted for by the cone models. An investigation of the damping term was conducted to assess its effects on CONAN predicted impedance functions. Three additiona l damping values were inputted to CONAN with all other properties remaining constant. The resulting impedance functio ns varied slightly indicating material damping has a limited effect on cone model predictions for Footings 5 and 6. 6.1.2 CONAN Model Upon completion of a soil profile, the information was used to create CONAN input files. Some data was corrected due to differences be tween survey and test conditions. Table 62 PAGE 160 160 presents CONAN input file data for the Footings 5 vertical input file. Th e input data represented here corresponds to testing perf ormed using the Thumper shaker. Seven physical properties were required for each layer of the cone m odels, the first being the layer type which is designated by one of four letters as follows: F indi cates the free surface, L a finite layer, H a halfspace, and R a perfectly rigid halfspace. The bottom layer must be designated by either H or R. For each layer, a radius was defined. Layers containing the foundation were assigned a nonzero radius and layers below th e foundation were assigned a zero radius. It should be noted that though CONAN required a radi us, it was not necessary for the foundation to be circular. The equivalent radius for a rectangular footi ng depended on its geometry and motion. For the radius of vertical and horizontal motion, rvh, the area of the botto m of the foundation, Af, was calculated. As presented in Equation 62, the footing area was then treated as a circular area for calculating the equivalent radius. For ro cking motions the equivalent radius, rr, was calculated much the same as with horizontal and vertical motions, the difference bei ng that area moment of inertia xI, was used instead of area yielding Equation 63. f vhA r (62) 4 x rI4 r (63) The shear modulus for each layer was corrected for the effects of increased effective stress experienced by the soil. The increase in effective stress was du e to two factors of construction and testing. First, soil was removed and replaced with concrete which has a higher density thereby increasing the effective stress under the footings. Second, an increase in stress was experienced from equipment resting on the footi ngs. The corrected vertical effective stress, vc, PAGE 161 161 was calculated by first employing Bou ssinesqs solution of vertical stress at a point caused by a rectangular loaded area and adding it to the free field vertical effective stress, vff. The implementation of Boussinesqs solution for rectangular load areas is cu rrent standard practice though the footing is not rigid and the material is not homogeneous. Next, the at rest earth pressure coefficient, K0, for sand layers and clay layers wa s calculated utilizing Equations 64 and 65 respectively. Horizontal effective stresses, h, for both before and after the stress increase were determined via Equation 66. The mean stress, 0, was then calculated for free field and post loading arrangements as presente d in Equation 67. A corrected shear wave velocity, Vsc, was computed utilizing the MASW free field velocity, Vsff, 0ff, and 0c as in Equation 68. Finally, the corrected shear modulus, Gsc, was calculated via Equation 69. sin1Ksand0 (64) 1 Kclay0 (65) v0 hK (66) 3 2' h v 0 (67) sff 4 ff0 oc scV V (68) 2 scscV G (69) The next three layer properties were determ ined from Table 61. Poissons Ratio, soil density, and damping were inserted directly into the CONAN input file with no corrected values necessary. PAGE 162 162 Finally thickness is required for any finite layer. The nine layer thicknesses required were determined by a combination of boring logs and MASW survey results. Layers not containing the foundation radius were simply calculated by s ubtracting the bottom depth of a layer from that of the layer above it. For la yers containing the foundation, a maximum layer thickness, d, adjacent to the footing was required. Equation 610 determined layer thickness utilizing shear wave velocity, cs, and the maximum angular velocity, max, of the dynamic force to be imparted on the footing. In the case of foundation em bedment depth being greater than the maximum allowable thickness, the soil around the foundation is separated into additional layers until Equation 610 is satisfied. max s5 c d (610) Upon completing the model inputs, CONAN was run three separate times for each footing to calculate impedance functions for vertical, horizontal, and rocking. Additionally, coupling terms were calculated with the rocking mode Depending on the mode of excitation the appropriate radius was used. Impedance functio ns were produced for a specified range of frequency. For Footing 5, impedance functions were calculated from 8 Hz to 50 Hz at a 1 Hz interval. Footing 6 impedance functions were ca lculated for frequencies ranging from 10 Hz to 40 Hz at a 1 Hz interval. Table 63 contains CONAN input data for Footing 6 horizontal testing. The input file corresponds to the test confi guration performed with the ANC O shaker elevated. It was developed in the same manner as the input file of table 62. PAGE 163 163 6.2 Cone Model and Measured Impedance Function Comparison 6.2.1 Footing 5 Cone model vertical impedance solutions are compared to the experimental results for Footing 5. Figure 61 represents the real value in A, the imaginary value in B, and the dashpot coefficient, or viscous damping, in C. The da shpot coefficient was simp ly calculated by dividing the imaginary component by thereby removing the frequency eff ect. Each plot contains values for testing using the ANCO shaker indicated by points and values predicted by CONAN represented by the curve. Predicted and measured impeda nce functions and dashpot coe fficients exhibit fairly good agreement. With the exception of peaks in the vicinity of the resona nce at 25 Hz, magnitudes plot well within an order of magnitude of each ot her. An important characteristic of Figure 61 B is predicted values plotting lower though excep tions exist at high a nd low frequencies. Predicted and measured real values both demonstr ate a decrease with resp ect to frequency. The decrease is noticeably more extreme in measured values. Imaginary values reveal an increase with frequency and predicted and measured trends agree somewhat well, the peak at 25 Hz being the largest difference. Figure 61 C demonstrates much of the same agreement and many of the exceptions noticed in A and B. Values are within a magnitude of each othe r with the peak near 25 Hz being the greatest excepti on. Predicted and measured da shpots tend to be constant regardless of frequency. This is consistent with dashpot plots for square footings found in Gazetas (1991). Figure 62 through Figure 64 graph the re al and imaginary components and dashpot coefficients for Footing 5 horizontal testing using the ElectroSeis shaker positioned at 2 ft and 4 ft. Cone model predictions are presented for hor izontal, rocking, and coupling terms. Measured values include the onedegreeoffreedom solu tion at 4 ft, and two twodegreeoffreedom PAGE 164 164 solutions for horizontal and rocking modes. Independent and equal crosscoupling terms are included for measured values in Figure 64. To a lesser extent than the vertical mode and with the exception of the onedegreeoffreedom solution, measured impedance functions demonstrate some agreement with horizontal cone model predictions via comparable magnitude s in Figure 62. They are separated by much less than a factor of ten. It should be noted, both real and im aginary values are consistently predicted low when compared to the twodegree offreedom model. A positive aspect is that measured value trends possess similar slopes to predicted values until frequencies begin experiencing resonance effects at 28 Hz. Again as expected, negative and positive slopes are observed in the real and imaginar y plots respectively. When looking at damping, an increase in slope of the measured values is noticeable whil e the predicted curve remains generally straight causing a much greater difference in damping values at higher frequencies. Dashpot coefficients seem to possess characteristics of the Gazetas (1 991) graphs. Predicted and measured trends remain constant with a change in frequency unt il approximately 28 Hz which is near resonance. Figure 63 A and B present the real and imaginary values for predicted and measured rocking impedance functions. They compare much the same as the horizontal impedance functions though, in this case th e real twodegreeoffreedom i ndependent solution seems to be the exception instead of the onedegreeoffreedom solution. The magnitudes of values for real and imaginary are within a factor of ten but, as with horizontal, cone models under predict impedance functions. As for curve shapes, meas ured trend slopes and predicted slopes seem to be agreement. The real onedegreeoffreedom and real twodegreeo ffreedom overdetermined measured values demonstrate a negative slope wh ich is expected. The imaginary values behave as expected with positive slopes and generally agree with one another until 28 Hz which is PAGE 165 165 approaching resonance. Plot C presents dashpot coefficients that agree somewhat with Gazetas (1991) plots for rocking dashpot coefficients. Values are noticeably higher at the lowest frequencies for predicted values but eventually become constant. Measured values demonstrate a trend with a gradually diminishing negative sl ope. Though differing from each other, predicted and measured behavior both possess charac teristics of the Gazetas (1991) plots. Figure 64 A, B, C, and D present the real and imaginary values for predicted and measured coupling impedance functions. Figures 64 C and D contain the same data as A and B, but have been altered to focus on the predicte d, onedegreeoffreedom, and independent twodegreeoffreedom. As observed in A and B th e overdetermined twodegreeoffreedom values were much larger than the others, thus did not compare well with predicted, onedegreeoffreedom, and overdetermined twodegreeoffr eedom value. Studying Figure 64 C and D magnitudes for real components are fairly similar and for imaginary compone nts the similarity is more noticeable. Predicted and measured values are within a factor of ten of each other, the exceptions being at high frequencie s for both real and imaginary valu es. Predictions seem to be higher for real values but, when observing imagin ary values lower in th e low frequency range. Real values demonstrate a drop with an increas e in frequency for both predicted and measured values, though the measured values demonstrate a mu ch steeper drop. The severity of the rate of this drop increases with frequency. Imaginary values for predicted and measured impedances possess a similar slope until reaching 28 Hz. Figure 64 E and F are treated much the same as the previous plots by shifting focus from the inde pendent twodegreeoffr eedom dashpot values. Studying F it should be noted that predictions are lower with the ex ception of high frequencies. All three dashpots are well within a magnitude of each other. Their trends with respect to PAGE 166 166 frequency are also similar exhibiting a cons tant value with increased frequency until approximately 28 Hz. 6.2.2 Footing 6 Cone model vertical impedance solutions are compared to the experimental results for Footing 6. Figure 65 represents the real value in A, the imaginary value in B, and the dashpot coefficient, or viscous damping, in C. The dashpot coefficient was calculated by dividing the imaginary component by thereby removing the frequency effect. Each plot contains values for testing using the Thumper and TRex shakers indicated by points a nd values predicted by CONAN represented by the curves. Predicted and measured impeda nce functions and dashpot coe fficients demonstrate fairly good agreement for real values and good agreement for imaginary values. Real and imaginary values plot within a magnitude of each other. The measured real component for Thumper agrees well with its corresponding predictions at low frequency before becoming consistently lower. TRex measured real values are consistently low but remain comparable to predicted values. The imaginary component of both Thumper and TRe x compare very well with their respective predictions though slightly low. When observing trends with respect to frequency, Thumper drops as frequency increases though predicted values tend to dr op until approximately 10 Hz and slightly rise thereafter. TRex demonstrates agreement with predic ted values dropping slightly at low frequencies then beginning to rise in the higher frequenc y ranges. Imaginary components exhibit good agreement between pr edicted and measured values. The slopes of the curve and trend lines are in the same direction and seem to in crease with frequency at the same rate. Figure 65 C demonstrates much of the same agreement and many of the exceptions noticed in B. Values are within a magnitude of each other. Predicted and meas ured dashpots seem to approach PAGE 167 167 zero at low frequencies and become constant at high frequencies. This is consistent with dashpot plots for square footings found in Gazetas (1991). Figure 66 through Figure 68 present the r eal and imaginary components and dashpot coefficients for Footing 6 horizontal testing using the ANCO shaker positioned flush to the footing surface and elevated on a frame. Cone model predictions are presented for horizontal, rocking, and coupling terms. Measured values include the onedegreeoffreedom solution with the shaker flush, and two twodegreeoffreedom solutions for horizontal and rocking modes. Independent and equal crosscoupling terms are included for measured values in Figure 68. Though not as strong as the ve rtical mode, measured impe dance functions demonstrate some agreement with horizontal cone model pred ictions via comparable magnitudes in Figure 66. Values are separated by much less than a factor of ten. It should be noted, the real component is slightly higher at low frequenc y than predicted values but, imaginary values are consistently predicted low when compared to all three solutions The decrease of real values with respect to frequency is also significant. Predicted real values do not decreas e as sharply as those measured. A similar but opposite relationshi p is apparent when observing the imaginary components. Predicted values possess a positive sl ope but the trend of measured values is to rise more sharply with respect to frequency. Figure 66 C reveals dashpots that are somewhat consistent with each other and with Gazetas (1991). The coefficients a ppear to maintain a cons tant value with regard to frequency and plots found in Gazetas (1991). The cone predictions, however, are significantly lower than the measured values. Figure 67 A and B present the real and imaginary values for predicted and measured rocking impedance functions and demonstrat e some agreement between the two. The magnitudes of values for real and imaginary are within a factor of ten. The real predicted curve PAGE 168 168 is observed lying near the median of the oned egreeoffreedom and i ndependent twodegreeoffreedom real components and within a magnitude of the overdetermined twodegreeoffreedom real components. Again, the cone model predic tions are consistently lower than the measured values. Measured trend slopes and predicted slopes seem to be in some agreement. Real components for predicted and measured values d ecrease with an increas e in frequency, though measured values tend decrease at a higher rate. The overdetermined twodegreeoffreedom values begin to rise at 30 Hz. The imaginary co mponent rises at a simila r but slower rate with respect to frequency. Plot C presents dashpot co efficients. Dashpot coe fficients are noticeably lower at the lowest frequencies for predicted valu es but eventually become constant. Measured values demonstrate a trend with a gradually dimi nishing positive slope. Again, the cone model predictions are consistently lower than the measured values. Figure 68 A and B present the real and imaginary values for predicted and measured coupling impedance functions. Bo th real and imaginary component s are predicted high with few exceptions. Many measured real values seem to pl ot erratically with some points falling close to the predicted curve and others fa lling more than a magnitude away Of the three measured real components, the independent twodegreeoffreed om rh values seem to match the predicted curve the best. The measured imaginary componen ts seem to compare only slightly better than the real. While some values fall near the predicted curve many fall close to a magnitude away. Again the independent twodegreeof freedom rh values seem to match the curve best. Predicted real component changes with respect to frequency exhibit a slight similarity to measured values but, seem to not match well. The predicted values plot nearly horizontal while measured values seem to drop more severely with respect to frequency. Predicted imaginary values seem to rise with increased frequency which contrasts to measured values falling. Observing C, the predicted PAGE 169 169 dashpot rises slightly at low fre quencies then begins to become c onstant at higher frequencies. The measured values seem erratic and posse ss very little apparent shape or trend. In summary, it would appear measured impeda nce functions compare fairly well to cone predicted impedance functions in many aspects according to the plots in this section, but there are exceptions. Perhaps the two most important differences between predicted and measured impedances are the consistent under prediction of damping and the sharp decrease of measured real values demonstrated with Footing 6. The under prediction is noticeable in multiple modes, both footings, and varying severities. As for the drop in measured stiffness compared to predicted, this phenomenon is also recognized in multiple modes, both footings, and a variety of severities. Positive characteristics are observed with respect to the magnitude of both predicted and measured values. Nearly all the plots pr esent comparisons that are within the same magnitude. When studying differences between th e two footings, the most noticeable is in stiffness. Footing 5 measured value trends seem to share a somewhat common slope with predicted curves while in most cases significan tly under predicting the magnitudes. Footing 6 stiffness predictions demonstrate the opposite by be ing fairly accurate with respect to magnitude while not displaying the severe drop with respect to frequency. Expected differences between the footings outlined in Gazetas (1 991) were noticed. For example, the stiffness of Footing 6 is larger than that of 5 which is consistent with their difference in size. Another difference is the quality of the coupling data. Gazetas (1991) states that as the depthwi dth ratio increases the coupling term should increase. Footing 5 has a significantly larg er depthwidth ratio and also demonstrates a better comparison in coupling terms than Footing 6. Observing differences between modes, vertical impedance function predictions appear to match measured values more than horizontal, rocking, and coupled predictions Much of the behavior exhibited by measured PAGE 170 170 and predicted impedance functions were consiste nt with what was generally expected from Gazetas (1991). 6.3 Assessment Noticeable differences appear in the comparis on between predicted and measured values. It is necessary to address these inconsistencies as they are observed in both footings, in all modes, and are of many degrees of severity. It seems appropriate to approach these differences in three categories, measurement error, pr ediction parameter accuracy, and cone model assumptions. The following paragraphs discuss pot ential reasons for these inconsistencies in each of the three stated categories. 6.3.1 Measurement Error To study the effects of systematic bias on measured impedance functions, Figures 69 through 612 present measured impedance functions with 5% bias added to all voltages read during testing. It was determined that an er ror band of artificial error deemed larger than reasonably expected would indicate potential measurement error if it encompassed a large percentage predicted values on a plot. Voltages from the load cell were increased by 5% and decreased by 5% to calculate th e +5% and 5% impedance values respectively. Similarly, the velocity transducer voltages are decreased by 5% and increased by 5% to calculate +5% and 5% impedance values respectively. Adjusting all vo ltages by 5% more than accounts for equipment manufacturers stated error. Also, adjusting load cell and transducer values in opposite directions produces the largest discrepanc y between unadjusted and adjusted values. Observing each error plot, it seems clear that predicted impedance f unction values rarely fall within the error band of the measured values. In instances that predicted curves cross into the error band it appears attribut able to the measured values possessing a trend that crosses the curves at few frequencies then proceeds on its steep change. A few error bands may contain PAGE 171 171 predicted curves for a frequency range of up to 15 Hz but this appears to be coincidental with a majority of the predicted curve lying significan tly outside the error bands. After accounting for potential error that was determined to be large, it seems unlikely that measurement error from testing or calculation is sufficient to explain the differences between measured and predicted impedance functions. 6.3.2 Prediction Parameters The second potential source for the differences between measured and predicted values may be found in the cone model soil profile para meters. Errors due to input parameters were likely minimized through thorough characterization of the site. Many parameters have been measured well and, in cases where assumptions were necessary, input va lues possessed limited influence on predictions. A parametric assessment was made to determine the influence that input values had on cone models Modeling parameters were varied and predictions made via CONAN as part of a sensitivity analysis. Thes e predictions were compared to cone models produced with unadjusted values and the si gnificance of each para meter was assessed. The comparison demonstrated a narrow change in impedance functions corresponding to relatively large variations in most parameters. In fact, changes in Poissons ratio, material damping, and soil density produced minimal differences when compared to unadjusted predictions. Typical difference s were significantly less dramatic than those seen in the measurement error bands of Section 6.3.1. The largest influence seemed to be that of the shear modulus. The calculation of shear modulus util izes soil density and shear wave velocity. Though density is easily measured or assumed with a fair degree of accuracy, characterization of the shear wave velocity required complex geophy sical tests. For the calculation of shear modulus, surface wave analysis and crosshole su rveys were conducted. The results of each survey allowed for the calculation of a set of impedance function predic tions. Crosshole over PAGE 172 172 predicted impedance functions significantly wher eas MASW produced more accurate results. Based upon these findings, cone model impedan ce function predictions employing MASW were deemed more suitable than those of crosshole testing and thus, represented predicted values when compared to measured values. Inconsistencies between pr edicted and measured impedance functions do not appear to result from material parameters inserted into the cone model. 6.3.3 Model Assumptions The third potential source for differences may be found in cone modeling assumptions. Multiple factors may account for inconsistencies, among the most obvious seem to be the assumptions of a wave increasing its size via a one dimensional cone and one wave type being propagated. In reality, waves do not spread in a cone, but instead in all directions. The cone may only account for a limited dissipation of ener gy due to geometric influences not accounting for lateral dissipation. Also, many wave type s emanate from the foundation. For example, vertical and rocking modes account for only Pwave behavior when calculating the wave front of the cone. Displacement in these directions may primarily cause Pwaves by pushing into the supporting soil but, the side of the f ooting also displaces in shear with respect to the soil which is neglect by the cone models. Similarly, though shear waves may be the primary waves experienced in predicting horiz ontal motion, compression waves are created between the vertical soilfooting interface as we ll as horizontal shear waves that go unaccounted for. In fact these phenomena may increase significantly with an increas e in the depth width ratio. To truly capture the dynamic behavior of the system it seems th at Rayleigh, shear, and compression waves need to be included in any model. Soil homogeneity may be an inaccurate assumption. Cone models assume a layers material properties are consistent throughout when, in fact, th at is rarely if ever the case. There exists stress strain variability induced solely by the depth a material occupies PAGE 173 173 meaning the composition of a material may be cons istent but its depth in the layer may influence its behavior. Lateral variation is not accounted for when using c one models. Layer assumptions may influence cone model accuracy. CONAN assume s the only variation in the model exists at perfectly flat infinite layer interfaces. This assumption is rarely accurate in practice. Characteristics such as voids, trenches, adjacent structures etc. potentially influence cone model accuracy. It appears likely that fundamental shortcomings in cone model assumptions may explain the differences between measured and cone predicted impedance functions. Cone models appear to lack the ability to predict impedance functions with a high degree of accuracy in complex three dimensional problem s. Vertical cone model predictions tend to characterize the impedance functi ons of the two available foot ings fairly well by presenting curves that match measured values in magnitude, shape, and slope. To a lesser extent, horizontal and rocking impedance function predictions seem to possess a relationship with measured values by demonstrating similar magnitudes, slopes, a nd shapes. Coupling terms demonstrate some similarities to measured values though s eem dependent on a higher depthwidth ratio. Inconsistencies do not seem to be caused by m easurement error or parametric assessments; rather, they appear to be the result of cone model assumptions that do not accurately reflect conditions and behaviors typica lly experienced in the fiel d. Based upon the apparent shortcomings of cone models, it seems an approach capable of addressing these more complicated wave propagation problems is nece ssary to more accurately predict impedance functions. PAGE 174 174 Table 61. Free Field Soil Proper ties for Footing 5 and Footing 6. Layer No. Layer Interface Depth (m) MASW Shear Wave Velocity (ft/s) Shear Modulus (Pa) Poissons Ratio Density (kg/m3) Damping (%) 1 1.52 438 2.775E+07 0.333 1558 0.03 2 1.80 712 7.332E+07 0.333 1558 0.03 3 4.91 712 7.506E+07 0.333 1595 0.03 4 8.38 712 9.167E+07 0.333 1948 0.03 5 10.52 999 1.805E+08 0.333 1948 0.03 6 13.72 999 1.747E+08 0.333 1886 0.03 7 Half Space 1344 3.162E +08 0.333 1886 0.03 PAGE 175 175 Table 62. CONAN Input File fo r Footing 5 Vertical Model. Layer Type Foundation Radius (m) Corrected Shear Modulus (Pa) Poissons Ratio Density (kg/m3) Damping (%) Thickness (m) F 0.559 L 0.559 2.77E+07 0.333 1558 0.03 0.237 L 0.559 2.77E+07 0.333 1558 0.03 0.237 L 0.559 2.77E+07 0.333 1558 0.03 0.237 L 0 4.32E+07 0.333 1558 0.03 0.813 L 0 8.10E+07 0.333 1558 0.03 0.274 L 0 7.58E+07 0.333 1595 0.03 3.109 L 0 9.18E+07 0.333 1948 0.03 3.475 L 0 1.81E+08 0.333 1948 0.03 2.134 L 0 1.75E+08 0.450 1886 0.03 3.2 H 0 3.16E+08 0.450 1886 0.03 PAGE 176 176 Table 63. CONAN Input File for Footi ng 6 Horizontal and Rotational Model. Layer Type Foundation Radius (m) Corrected Shear Modulus (Pa) Poissons Ratio Density (kg/m3) Damping (%) Thickness (m) F 1.74 L 1.74 2.77E+07 0.333 1558 0.03 0.304804 L 1.74 2.77E+07 0.333 1558 0.03 0.304804 L 1.74 2.77E+07 0.333 1558 0.03 0.304804 L 0 3.26E+07 0.333 1558 0.03 0.609589 L 0 8.24E+07 0.333 1558 0.03 0.274 L 0 7.73E+07 0.333 1595 0.03 3.109 L 0 9.21E+07 0.333 1948 0.03 3.475 L 0 1.81E+08 0.333 1948 0.03 2.134 L 0 1.75E+08 0.45 1886 0.03 3.2 H 0 3.16E+08 0.45 1886 0.03 PAGE 177 177 A B C Figure 61. ANCO vertical compar ison for Footing 5. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imaginary Compone nt. C) Vertical Da shpot Coefficient. PAGE 178 178 A B C Figure 62. ElectroSeis horiz ontal comparison for the oned egreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft. A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component. C) Horizontal Dashpot Coefficient. PAGE 179 179 A B C Figure 63. ElectroSeis horiz ontal comparison for the oned egreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft. A) Rocking Impedance, Real Component. B) Rocking Impedance, Imaginary Component. B) Rocking Dashpot Coefficient. PAGE 180 180 A B C D E F Figure 64. ElectroSeis horiz ontal comparison for the oned egreeoffreedom at 4 ft, independent coupling twodegreeoffreedom at 2 ft and 4 ft, the equal coupling twodegreeoffreedom analyses of Footing 5 at elevations 2 ft and 4 ft. A) Coupling Impedance, Real Component. B) Coupling Impedance, Imaginary Component. C) Coupling Impedance, Real Component Adjusted. D) Coupling Impedance, Imaginary Component Adjusted. E) Coupling Dashpot Coefficient. F) Coupling Dashpot Coefficient Adjusted. PAGE 181 181 A B C Figure 65. Thumper and TRex vert ical comparison for Footing 6. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imag inary Component. C) Vertical Dashpot Coefficient. PAGE 182 182 A B C Figure 66. ANCO horiz ontal comparison for the onedeg reeoffreedom flush mount, independent coupling twodegreeoffree dom, the equal coupling twodegreeoffreedom analyses of Footing 6. A) Hori zontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Component C) Horizontal Da shpot Coefficient. PAGE 183 183 A B C Figure 67. ANCO horiz ontal comparison for the onedeg reeoffreedom flush mount, independent coupling twodegreeoffree dom, the equal coupling twodegreeoffreedom analyses of Footing 6. A) Rocki ng Impedance, Real Component. B) Rocking Impedance, Imaginary Component. C) Rocking Dashpot Coefficient. PAGE 184 184 A B C Figure 68. ANCO horiz ontal comparison for the onedeg reeoffreedom flush mount, independent coupling twodegreeoffree dom, the equal coupling twodegreeoffreedom analyses of Footing 6. A) C oupling Impedance, Real Component. B) Coupling Impedance, Imaginary Component. C) Coupling Dashpot Coefficient. PAGE 185 185 A B C D E F Figure 69. Footing 5 Vertical and 1DoF Error Band. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imaginary Component C) Horizontal Impedance 1DoF, Real Component. D) Horizontal Impedance, 1DoF, Imaginary Component. E) Rocking Impedance, 1DoF, Real Component. F) Rocking Impedance, 1DoF, Imaginary Component. PAGE 186 186 A B C D E F Figure 610. Footing 5 2DoF Error Band. A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Com ponent. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imagin ary Component. E) Coupled Impedance, Real Component. F) Coupled Impedance, Imaginary Component. PAGE 187 187 A B C D E F Figure 611. Footing 6 Vertical and 1DoF Error Band. A) Vertical Impedance, Real Component. B) Vertical Impedance, Imaginary Component C) Horizontal Impedance 1DoF, Real Component. D) Horizontal Impedance, 1DoF, Imaginary Component. E) Rocking Impedance, 1DoF, Real Component. F) Rocking Impedance, 1DoF, Imaginary Component. PAGE 188 188 A B C D E F Figure 612. Footing 6 2DoF Error Band. A) Horizontal Impedance, Real Component. B) Horizontal Impedance, Imaginary Com ponent. C) Rocking Impedance, Real Component. D) Rocking Impedance, Imagin ary Component. E) Coupled Impedance, Real Component. F) Coupled Impedance, Imaginary Component. PAGE 189 189 CHAPTER 7 CLOSURE 7.1 Summary of Findings A comparison between cone models to meas ured values of impedance functions was conducted to determine the effectiveness of pred ictions for shallow foundations when applied to conditions found in the field. Two typical conc rete foundations were tested. The site was located on the Riverside Campus of Texas A& M and chosen for the extensive geotechnical testing previously conducted and an acceptable foundation already existing. One footing was constructed to meet the requirements of testi ng. Cone models for multiple modes and each foundation were calculated through CONAN utilizing previous geotechnical investigation data, physical properties of the re spective foundation, and geophysical surveys performed for this study. Each foundation was dynamically loaded during vertical and ho rizontal testing via multiple shakers and various equipment arra ngements. Dynamic foundation displacement responses were measured with respect to loadin g. Displacements were analyzed to calculate measured impedance functions in accordance with available literature. One singledegreeoffreedom and two twodegreeoffreedom approach es produced impedance functions in multiple modes for each foundation. Results for both pred icted and measured impedance functions were presented together. The fi ndings are presented here: When compared to more rigorous numerical methods and limited field data previously available, cone models seem to be capable of predicting impedance functions accurately. Though data from numerical techniques and measured impedance functions lack some characteristic typically enc ountered in industry, cone m odel evaluations demonstrate a positive relationship with each on a promising scale. This relationship was strong enough to initiate field testing. Full scale field experiments were designed to measure and record displacements of footings in all modes of interest and under multiple loading arrangements. Load and displacement data was successfully analyzed to produce measured impedance functions via a reversecalculation program. A revers ecalculation program was developed which executed three techniques that produce three se ts of measured impedance functions from PAGE 190 190 the same data set. With few exceptions, thes e impedance functions ag ree fairly well with each other. The comparison between cone model predicted and measured impedan ce functions reveal important similarities and differences. Both agree somewhat with respect to real and imaginary magnitudes. Expected behavior is generally recognized in predicted and measured values, in both foo tings, and in all modes. Foo ting 5 exhibits good agreement between predicted and measured values when observing trends with respect to frequency while Footing 6 demonstrates consistency in magnitude between predicted and measured values. When comparing modes, predicted vertical impedance functions appear to compare well to measured values. To a lesser extent horizontal and vertical predictions compare fairly well with some exceptions. Perhaps the most important differences between predicted and measured impedance f unctions are the consistent under prediction of damping and the increased dropoff of measured real va lues observed in Footing 6. Attempts to uncover possible reasons for di fferences between predicted and measured impedance functions reduced the likelihood that they were due to error in measurements. An analysis demonstrated that predicted valu es typically fell well outside a conservative error band around measured impedance functions. An assessment of cone model parameters decreased the po ssibility that input parame ters were responsible for inconsistencies between predicte d and measured values. It wa s demonstrated that a modest change in most input parameters influences impedance function values minimally. Shear modulus seemed to possess the greatest infl uence over impedance functions but was also well characterized indicating that differences we re not a result of inaccurate input data. Cone model prediction assumptions could not be disqualified as a source of difference between predicted and measured values. Potential aspects that may account for inconsistencies of cone models were discussed. 7.2 Conclusions Reliable techniques appear to have been develo ped to dynamically excite full scale in situ shallow foundations. Two foundations were tested via multiple shakers, arrangements, and in multiple modes. The responses of the footings se em to have been measured accurately. Multiple reversecalculation techniques were executed and produced seemingly reliable measured impedance functions. Predicted impedance functions were produced ba sed on an array of soil tests. Preliminary predictions exhibited a significan t level of agreement with the limited measurements previously available. These predictions also demonstrat e considerable agreement with more rigorous prediction methods there by indicating good potential fo r in situ prediction. PAGE 191 191 Cone models appear capable of providing an a pproximation of in situ impedance functions. The accuracy of the prediction seems related to but not limited to the mode of excitation and geometric properties of the footing. The most significant differences between predicted and measured impedance functions are the under predic tion of damping and a l ack of a considerable drop in stiffness with an increase in frequency. These inconsistencies appear to be the result of inadequacies in cone modeling assumptions. 7.3 Recommendations After a review of the findings and conc lusions, the following recommendations are presented: Further indentify reasons for in consistencies. Upon identify ing these reasons, select new 2dimensional or 3dimensional soil/structure interaction models that better represent in situ conditions. Testing to this point is still somewhat limite d. There are very few measured impedance function results available. More experimentation at the same site, and then at several sites, would allow additional insight into aspects of shallow foundations that may influence any prediction method of interest. PAGE 192 192 LIST OF REFERENCES Apsel, R. J. and Luco, J. E. (1987), Im pedance Functions for Foundations Embedded in a Layered Medium: An Integr al Equation Approach, Earthquake Engineering and Structural Dynamics, Vol. 15, pp. 213231. Briaud, J., Gibbons, R. M., (1994), Data and Pr edictions Request for the Spread Footing Prediction Event, Predicted and Measured Behavior of Five Spread Footings On Sand: Proceedings of a Prediction Symposium, Geotechnical Special Publication No. 41, June, pp. 1185. Crouse, C. B., Hushmand, B., Luco, J. E., and Wong, H. L. (1990), Foundation Impedance Functions: Theory versus Experiment, Journal of Geotechnical Engineering, Vol. 116, No. 3, March, pp. 432449. De Barros, F. C. P. and Luco, J. E. (1995), Identification of Founda tion Impedance Functions and Soil Properties from Vibration Test s of the Hualien Containment Model, Soil Dynamics and Earthquake Engineering, Vol. 14, pp. 229248. Fry, Y. (1963), Development and Evaluation of Soil Bearing Capacity Foundations of Structures. Field Vibratory Tests Data, U.S. Army Waterways Experiment Station, Corp of Engineers, Vicksbur g, Mississippi, 18 pp. Gazetas, G. (1991), Foundation Vibrations, Chapter 15 in Foundation Engineering Handbook, 2nd ed., Edited by H.Y. Fang, Van Nostrand Reinhold, New York, pp. 553593. Kausel, E., (1981), An Explicit Solution for the Green Func tions for Dynamic Loads in Layered Media, Research Report R8113, MIT. Kausel, E. (1974), Forced Vibration of Circular Foundations on Layered Media, Research Report R7411, MIT. Luco, J. E. (1976), Vibrations of a Rigid Disc on a Viscoelastic Medium, Nuclear Engineering an Design, 36, pp. 325340. Luco, J. E. and Mita, A. (1987), Response of a Circular Foundation on a Uniform HalfSpace to Elastic Waves, Earthquake Engineering and Structural Dynamics, Vol. 15, pp. 105118. Luco, J. E., Trifunac, M. D., and Wong, H. L. (1988), Isolation of SoilStructure Interaction Effects by FullScale Forced Vibration Tests, Earthquake Engineering and Structural Dynamics, Vol. 16, pp. 121. Luco, J. E. and Wong, H. L. (1990), Forced Vibration of Lotung Containment Model: Theory and Observations, Journal of Engineering Mechanics, Vol. 116, No. 4, April, pp. 845861. PAGE 193 193 Lysmer, J., Tabatabaie, R. M., Tajirian, F., Vahndani, S., and Ostadan, F. (1981), A System for Analysis of SoilStructure Interaction. Research Report GT8102, University of California, Berkely. Lysmer, J., Udaka, T., Tsai, C.F., and Seed, H. B. (1975), FLUSHA Computer Program for Approximate 3D Analysis of Soil Structure Interaction Problems, Report No. EERC 7530, University of California, Berkley. Mita, A. and Luco, J. E. (1987), Dynamic Response of Embedded Foundations: A Hybrid Approach, Computational Methods in Applied Mechanics and Engineering, Vol. 63, pp. 233259. Mita, A. and Luco, J. E. (1989), Impedance Func tions and Input Motions for Embedded Square Foundations, Journal of Geotechnical Engineering, Vol. 115, No. 4, April, pp. 491503. Nii, Y. (1987), Experimental HalfSpace Dynamic Stiffness, Journal of Geotechnical Engineering, Vol. 113, No. 11, November, pp. 13591373. Prakash, S. P. and Puri, V. K. (1988), Foundations for Machines:Analysis and Design, John Wiley & Sons, Inc., New York, 656 pp. Richart, F. E., Jr., Hall, J. R., and Woods, R. D. (1970), Vibration of Soils and Foundations, PrenticeHall International, Inc., New Jersey, N.J. Tassoulas, J. L., (1981), Elements for Numerical Analysis of Wave Motion in Layered Media. Research Report R812, MIT. Tran, K. T. and Hiltunen, D. RJ. (2008), An Appraisal of Surfa ce Wave Methods for Soil Characterization, Submission for 2009 Annual Meeti ng of Transportation Research Board., Washington, D.C. Waas, G.. (1972), Analysis Method for Footi ng Vibrations through Layered Media, Ph.D. Thesis, University of California, Berkley. Wolf, J. P. and Deeks, A. J. (2004), Foundation Vibration Analysis : A StrengthofMaterials Approach, Elsevier, 218 pp. Wong, H. L. and Luco, J. E. (1976), Dynami c Response of Rigid Foundations of Arbitrary Shape, Earthquake Engineering and Stru ctural Dynamics, Vol. 4, pp. 579587. Wong, H. L., Trifunac, M. D., and Luco, J. E. (1988), A Comparison of SoilStructure Interaction Calculations with Results of FullScale Forced Vibration Tests, Soil Dynamics and Earthquake Engineering, Vol. 7, No. 1, pp. 2231. Wong, H. L. and Luco, J. E. (1978), Tables of Impedance Functions and Input Motions for Rectangular Foundations, Report CE7815, Department of Civil Engineering, University of Southern California, Lo s Angeles, December, 92 pp. PAGE 194 194 Wong, H. L. and Luco, J. E. (1985), Tables of Impedance Functions for Square Foundations on Layered Media, Soil Dynamics and Earthquake Engineering, Vol. 4, No. 2, pp. 6481. PAGE 195 195 BIOGRAPHICAL SKETCH Patrick William Dunn was born in State Coll ege Pennsylvania and raised in Ledyard Connecticut. After high school he attended the University of Connecticut and supplemented his education in geotechnical e ngineering by working for CTL Thompson in Denver Colorado during his last two summers. Afte r receiving his Bachelor of Scie nce he worked for Great Lakes Dredge and Dock Company until August of 2002 when he began his studies at the Pennsylvania State University. He received his Master of Sc ience in December of 200 5 in civil engineering and accepted a fellowship at the University of Florida to study civil engineering. 