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STRUCTURAL RESPONSE OF PRESTRESSED CONCRETE MEMBERS SUBJECTED TO ELEVATED TEMPERATURES By WILLIAM J. MBWAMBO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995 ACKNOWLEDGMENTS The author expresses sincere appreciations to his advisor and committee chairman, Dr. Fagundo F.E, for devoting much patience and guidance throughout the course of this study. He also give thanks to Dr. Hoit M.I., Dr. Cook R.A., Dr. Tia M. and Dr. Chang W. for their help in various ways and for serving as members of his committee. The author also acknowledges the help and services of Prof.Thompson P.Y., Chairman of the CE Department, UF, and Mrs. Micha M., Asst. Director, International Student Services, UF, during the critical times of his studies. He also expresses deepest gratitude to his wife Stella and his son John for their understanding, patience and encouragement during the whole period of the study. The author was awarded a scholarship to study in the United States by the University of Dar es Salaam through the German Academic Exchange Services (GTZ). TABLE OF CONTENTS page ACKNOWLEDGMENTS . . .. ii ABSTRACT . . vi CHAPTERS 1 INTRODUCTION . 1 1.1. General Remarks . 1 1.2. The Problem . 1 1.3. Research Objectives . 3 1.4. Scope of the Study . 4 1.5. Problems and Solution Schemes 5 1.5.1. Problem environment 5 1.5.2. Solution Strategy . 8 1.6. Previous Research. . ... 12 1.6.1. Material Property Behavior. 13 1.6.2. Structural ResponseExperimental 17 1.6.3. Structural ResponseNumerical 18 2 THERMAL ANALYSIS . .. 22 2.1. Introduction . .. 22 2.2. Heat Transfer Models . .. 22 2.2.1. Heat Flow Models .. 22 2.2.2. Finite Element Formulations 24 2.2.3. Matrix Equations .. 25 2.2.4. Time Integration of Temperature 28 2.3. Evaluation of Thermal Matrices and Load Vector . 30 2.3.1. Interpolation Functions .. 31 2.3.2. Conductivity Matrix .. 31 2.3.3. Thermal Capacitance Matrix 32 2.3.4. Thermal Load Vector .. 33 2.4. Solution Algorithm . .. 33 3 MODELING OF TEMPERATURE DEPENDENT MATERIAL PROPERTIES .. 35 3.1. Introduction . 35 iii 3.2. Modeling Procedures . .. 35 3.3. Concrete . 37 3.3.1. Thermal Properties .. 37 3.3.2. Mechanical Properties .. 40 3.3.3. Deformation Properties .. 44 3.4. Prestressing Steel . .. 48 3.4.1. Mechanical Properties .. 48 3.4.2. Deformation Properties .. 49 3.5. Remarks . 53 4 MODELING OF PRESTRESSED CONCRETE FRAME MEMBER 54 4.1. Introduction . 54 4.2. Basic Finite Element Equations for a 2D Beam Element . .. 54 4.3. Evaluation of Displacement Functions for a Beam Element .. 57 4.4. Idealization of a Prestressed Concrete Member . 59 4.5. Evaluation of Element Stiffness Matrix 61 4.6. Incremental Strains in Concrete and Prestressing Steel .. 64 4.6.1. Concrete Deformations .. 64 4.6.2. Prestressing Steel Strains .. 67 4.7. Load Vector due to Prestress Force 67 4.8. Remarks . . 69 5 SOLUTION METHOD FOR STRUCTURAL RESPONSE 70 5.1. Introduction . 70 5.2. Induction Forces . .. 71 5.2.1. Thermal and shrinkage strain 71 5.2.2. Strength degradation .. 72 5.2.3. Shift in stress .. 73 5.3. Effective Prestress Action .. 74 5.4. Evaluation of Action Load Vector 76 5.5. Solution of Displacement Response 77 5.7. Local and Global Values .. 82 5.8. Computation of Moment Capacity Response 84 6 COMPUTER IMPLEMENTATION . .. 85 6.1. Introduction . 85 6.2. Main Program PRECET . .. 85 6.3. Program TEDIAN . .. 85 6.4. Program STREAN . .. 85 6.5. Program MAPRAN . .. 85 7 NUMERICAL ANALYSIS . .. 92 7.1. Introduction . 92 7.2. Analysis of Prestressed Concrete Beams 92 7.2.1. Description of Test Specimen 92 7.2.2. Beam Testing . .. 93 7.2.3. Design Details .. 95 7.2.4. Numerical Model .. 96 7.2.5. Numerical and Test Result 97 7.3. Prestressed Concrete Slabs .. 108 7.3.1. Description of Test Specimen .108 7.3.2. Material Specifications .. 111 7.3.3. Testing Procedures .. 111 7.3.4. Finite Element Model 112 7.3.5. Comparison of Numerical and Test Result . 112 7.4. Evaluation of Influences of Material Properties . 119 7.4.1. Effect of Thermal Properties on Temperature Distribution 120 7.4.2. Effect of other Material Properties on Deflection 122 7.4.3. Effect of Thermal Properties on Structural Response .. 124 7.5. Method of Estimating Fire Endurance of Beams . 126 7.5.1. Effects of Exposure Conditions .126 7.5.2. Specimen for Analysis .. 126 7.5.3. The Analysis Procedure and Results . 127 8 SUMMARY AND CONCLUSIONS . .. 134 8.1. Summary . . 134 8.2. Conclusions . 136 8.3. Recommendations . .. 138 REFERENCES . . 139 APPENDICES A INSTRUCTIONS FOR THE PREPARATION OF THE INPUT DATA FILE FOR THE PRECET PROGRAM .. 144 B EXAMPLE OF INPUT DATA FILE ... .152 BIOGRAPHICAL SKETCH . ... 161 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRUCTURAL RESPONSE OF PRESTRESSED CONCRETE MEMBERS SUBJECTED TO ELEVATED TEMPERATURES By William J. Mbwambo May, 1995 Chairperson: Fernando E. Fagundo Major Department: Civil Engineering A numerical procedure is developed for the nonlinear analysis of the structural response of prestressed concrete members subjected to fire. The fire environment is simulated by the timetemperature relationship of the standard fire test (SFT). The procedure computes temperature distributions, midspan deflection and moment capacity of a member supporting its design service loads throughout the simulated fire exposure period. The formulation of the numerical model and the solution of the equilibrium equations is based on the displacement finite element method. The member cross section is idealized into triangular elements for temperature distribution analysis and for evaluation of variation in material properties within the section. Beam elements are used to model the member for both deflection and moment vi capacity analysis. The transient heat transfer mechanisms of the SFT is modeled by dividing the time scale into discrete number of time steps. At each time step equilibrium equations are set and solved iteratively for temperature distribution and structural response. Incremental deflections within a time step are added cumulatively to determine the total deflection at the end of each time step. A series of numerical analyses of prestressed concrete beams and slabs, and the evaluation of the influences of temperaturedependent properties of concrete and steel on temperature distribution, deflection and moment capacity is presented. Numerical results show good agreement with corresponding test results. Temperature distribution is more sensitive to thermal capacity properties for normal weight concretes and to thermal conductivity properties for lightweight concretes. Deflection is mostly influenced by the effects of temperature on the relaxation properties of prestressing steel. Thermal properties of concrete influence both the deflection and moment capacity. Graphs for predicting fire endurance of prestressed concrete beams exposed to fire have been developed both for normal and lightweight concretes. vii CHAPTER 1 INTRODUCTION 1.1. General Remarks The determination of the behavior of prestressed concrete members subjected to high temperatures has been mostly experimental. The experiments are based on the standard fire tests (SFT). In the United States the SFT procedures are defined by ASTM E119 [1]. Results from the standard fire tests have been used to develop numerical procedures for fire safety designs. 1.2. The Problem The main objective of the Standard Fire Test (SFT) is to predict the fire endurance of building components. The parameters measured during the test of prestressed concrete members are temperature distribution within the member, and the midspan deflection of the member [2,3]. The variables considered are concrete cover and type of concretes. These variables affect the temperature distribution within a member. Numerical methods developed for fire safety design of prestressed concrete also estimate the fire endurance time. The parameter used is the moment capacity [3,4,5]. The 1 retained moment capacity is taken as a function of the temperaturedependent tensile strength of prestressing steel. The temperature of steel is derived from temperature distribution curves for a given exposure period and concrete cover. However, since beams are exposed to fire on three sides, these curves do not give a good estimate of temperature of steel in beams [3]. The variables considered are the same as those of fire tests. Many studies show that almost all the physical and mechanical properties of concrete and prestressing steel undergo considerable changes at high temperatures [6,7,8,9,10]. The procedures of the standard fire test does allow for the evaluation of the influence of these properties in the measured temperature and deflection behavior. A similar problem is experienced with the existing numerical fire design methods. The fire tests are not designed to measure changes in moment capacity. On the other hand, the numerical methods are not developed for predicting changes in member deflection. Based on these observations, it is paramount to develop a method that will address all the above short comings. The procedures should be comprehensive enough to incorporate the changes in properties of concrete and steel, and predict the history of temperature distribution, member deflection, and moment capacity. 1.3. Research Objectives The objectives of this study are as follows: 1. Develop a numerical procedure for determining temperature distribution and structural response of prestressed concrete members exposed to fires. The procedure will incorporate the nonlinearity of temperaturedependent material properties of concrete and prestressing steel. 2. Use the procedure developed in (1) to evaluate the degree of influence of temperaturedependent properties of concrete and prestressing steel on the computed temperature distribution, and the structural response. The histories of structure deformation and strength capacity are considered as main functions for defining structure response. The numerical model developed will evaluate these parameters for a prestressed concrete structure exposed to a simulated fire environment, while supporting its designed service load. The resulting model could be used to (a) estimate temperature distribution and deformation status of a loaded prestressed concrete member exposed to fire. This application will reduce dependence on experiments, (b) evaluate strength status of a prestressed concrete member after a prolonged exposure to fire and (c) show influence of temperaturedependent material properties of concrete and prestress steel on temperature distribution and structural behaviors. 1.4. Scope of the Study The temperaturedependent material properties considered in this study are (a) compressive and tensile strength of concrete, (b) stressstrain relationship of both concrete and prestressing steel, (c) tensile strength of prestressing steel, (d) thermal strain of concrete and steel, (e) shrinkage strain of concrete and (f) accelerated relaxation of prestressing steel. Concrete members are analyzed on the assumption that the moisture content is within the equilibrium relative humidity of 70 percent or less. This state simulates the common conditions attained in service [11]. The fire environment for the numerical model is simulated by the timetemperature relationship defined for the SFT. The imposed service load is assumed to be constant during exposure to fire. Mathematical models of temperaturematerial property relationships are derived from temperature test results published in the literature. The types of construction materials considered are normal weight (silicon aggregate and carbonate aggregate) and lightweight (expanded shale) concretes and cold drawn prestressing steel (wire or tendon). 1.5. Problems and Solution Schemes 1.5.1. Problem Environment The problem involves both temperature distribution and structure response analyses. The main components of analysis are (a) temperature distribution within a member, (b) temperaturedependent structural displacements and (c) temperaturedependent moment capacity. 1.5.1.1. Temperature distribution problem The temperature distribution problem is defined graphically in Fig. 1.1. Exposure conditions for a typical beam element are shown in (a). The problem is to find the temperature distribution function T(t) under the transient heat transfer condition of the SFT shown in (b). Heat is transferred from the source to the element's exposed surfaces and then distributed within the element. Heat transfer modes involved are (a) radiation and convection (Q ,hr,): from heat source to the exposed surfaces of an element and (b) conduction (Qk, p): within the element. The quantity and rate of heat transferred has the following dependencies: S SFT (b) Temperature models I he ,hr (a) Beam section showing surfaces exposed to fire Fig. 1.1. Temperature distribution problem environment (a) Qh,c, is a function of heat transfer coefficients of convection (h,) and radiation (h ,). (b) Qk,p is a function of thermal conductivity (k) and thermal capacity ( c). Higher k and lower p values result in higher temperatures for a given heat transferred. The above variables and their interrelation have to be computed continuously in the process of solving for the temperature distribution. Detailed procedures for the solution of temperature distribution are discussed in Chapter 2. 1.5.1.2. Structure response problem Fig.1.2 (a) depicts the typical system configuration to be analyzed for structural response. The system is under constant service load. However, due to effect of temperature on material properties, there is a continuous redistribution of internal stresses within an element. The core of the problem is the determination of changes in the internal load function F, and the response function r, under the constant load function L, as shown in Fig.1.2 (b). L Cconst) P P CT. e) P  \  IA S (T. e) (a) system configurat on LEGEND L, F point of equilibrium L app l ed service load i F P = function of prestress action S stiffness function based n property es of oncr te and prestressing steel RA = reference axis T = tenperatur d distribution function S= strtr s rB tr n function r r F = Internas load function (b) Loading models r = system displacement function Fig. 1.2. Structural system configuration The hypothetical characteristics of the main parametric functions of the analysis are illustrated qualitatively in Fig.1.3. Fig.1.3 (a) shows the decline in effective prestress action with exposure time due to effects shown in Fig.1.3 (b), and Fig.1.3 (c) shows variations of stiffness parameter S and displacement r, as temperature T increases. The procedure for evaluating these parameters is discussed in detail in Chapters 4 and 5. L P L = serve ice load function S = effective prestress act ion S f(T, e, a, g ) Ca) Load functions a(Pt Tt P a T, P ( t n t n n n p g tin) T (b) prestress load functions t e S, T. r function of prestress loss due to relaxat on of steel P = tension in prestress steel S f(T, e ) number of time step function of prestress loss due to geometric changes Temperature function exposure time function stress strain function r = f( S. R) S : f(E. A E = f(T. e) At function of effective section area (c) Stiffness, displacement functions Fig. 1.3. Loading, stiffness, and deformation behavior P 1.5.2. Solution Strategy The entire problem involves both temperature distribution and structure response analysis. The problem is formulated and solved using standard finite element techniques. The techniques are then implemented in computer programs. The prestressed concrete member is idealized into finite beam elements for structural response analysis. The cross section of each beam element is modeled into triangular subelements. The cross section model is utilized for temperature distribution analysis, evaluation of the variation of material properties across the section and for summation of the overall cross section properties. This approach is illustrated graphically in Fig. 1.4. The timetemperature relation of the SFT is divided into a number of thermal time segment. The time boundaries of the segment are called time step points. Solutions of equilibrium equations are performed at these time steps for both temperature distribution and structure response. The solution process begins with solution of the temperature distribution problem. Obtained temperature values are used to compute active material properties for each concrete subelement and prestressing steel segments, of the beam element section. Based on the variation of temperature across the section and the temperaturematerial properties relationship, each concrete subelement and steel segment may have different set of properties. Fig. 1.4. Finite element idealization for temperature and structure response analysis The stiffness parameters of concrete subelements and steel segments are integrated across the cross section of each beam element to determine the overall element stiffness characteristics. The strength parameters of the prestressing steel are added across the section to determine the overall effective prestressing effect on each beam section. The computed functions for each beam element are assembled to configure the structure model. The solution process is then continued to solve for structure response. The whole procedure is implemented in a computer program named PRECET (PRestressed Concrete at Elevated Temperature). The program consists of the following subprograms: (a) TEDIAN (TEmperature Distribution ANalysis), (b) STREAN (STructure REsponse ANalysis) and (b) MAPRAN (MAterial PRoperty ANalysis). Discussion on the development of the computer implementation is presented in Chapter 6. The program is used to run a number of examples which have been tested in laboratories according to Standard Fire Tests (SFT). The numerical procedure developed in this study is validated by comparing the test results with numerical results. The example problems investigated and the comparison of results, are presented in Chapter 7. 1.6. Previous Research Much of the previous research related to structural response of prestressed concrete members subjected to building fires have been devoted to studying the effects of high temperature on the properties of concretes and steels used in prestressed concrete structures and in determining the temperatures within a concrete member during the laboratory standard fire tests [3,5]. Behavior of concrete and prestressing steel under varying temperatures have been studied extensively [5]. Results from these studies show common qualitative behaviors, though may differ quantitatively depending on the nature of materials used and the testing procedures. This section presents a summary of findings from some of these studies. More information regarding material properties behavior will be given in Chapter 3. 1.6.1. Material Property Behavior. 1.6.1.1. Tensile strength of prestressing steel Day et al. [3], Abram et al. [1] and Podolny [10] have studied the effect of temperature on the tensile strength of prestressing steel. Results show that the strength of prestressing steel decreases by more than 50% at 800 F. At 1200F the strength is down to 5% of its original value. 1.6.1.2. Loss of prestress force Abram and Cruz [1] observed that loss of prestressing amounted to 15% at 600 F, 23% at 700F and between 70% and 80% at 1000 F, depending on the initial prestressing force. Only about 12% of the loss at temperatures above 700 F is recovered on cooling. The test was performed on bare wire, that is, not embedded in concrete. The report from the study did not isolate the parameters that caused the losses. 1.6.1.3. Compressive strength of concrete Three types of tests have been used to determine the relationship between the compressive strength of concrete and temperature changes. The resulting strength categories are as follows: (a) 'unstressed' strength which is determined by heating the specimen with no superimposed loads. When the desired test temperature is reached, the specimen is then tested for compressive strength. (b) 'stressed' strength which is determined by first loading the concrete specimen with a desired test load. The stressed specimen is heated to a certain temperature and then tested for compressive strength. (c) 'unstressed residual' strength which is for specimens which have been heated to any desired temperature, cooled to room temperature and then tested for compressive strength. Results from compressive strength studies by Abrams [9] and Malhorta [7] show that most concrete types retain their near full strength at temperatures below 700 F. Siliceous aggregate concretes start loosing appreciable amount of strength at temperatures above 700 *F, whereas lightweight and carbonate concrete losses increase at temperatures above 1000 F. However, most concretes will retain only about 20% of their room temperature strength at 1600 F. It was observed that type of aggregate and aggregate/cement ratio have the most significant effect on the strength behavior. Specimen tested while under stress showed higher values of compressive strength than for unstressed specimen. 1.6.1.4. Elastic properties Elastic properties of concrete have been studied by Cruz [12] and Harada [13], among others. Results from Cruz show that the elastic modulus of concrete drops to 4050% at 750*F, and below 30% of its room temperature value at 1200F. Cruz [12] shows that the modulus of elasticity of concrete is affected primarily by the same factors which influence the compressive strength. The aggregate type and concrete strength do not have significant effect. However, as for compressive strength, the elasticities from stressed tests show higher values than the unstressed ones. According to report by Harada [13], original room temperature values of elastic modulus are not regained upon cooling. The effects of temperature on the elastic modulus of prestressing steel have been reported in Ref.[5]. At lower temperatures, the modulus of elasticity decreases at a slow rate with steel temperatures. The modulus is reduced by about 6% at 400F and by approximately 20% at 600 F. Above 9000F, the modulus of elasticity decreases at higher rates to almost zero at 1400 F. 1.6.1.5. Concrete bond The effect of temperature on bond strength has been studied by Morley [14], Morley and Royles [15] and Diederichs and Schneider [16]. Results show that at 900 F the maximum bond strength is reduced to 50% and to less than 20% at 1300*F. Effective bond between concrete and prestressing steel is also reduced at high temperature resulting to the extension of development length [17,18] due to bond slip. Fagundo and Richardson [19] also studied the effects of temperature on bond slip of epoxycoated prestressing strands. Results of the study shows a bond slip of 0.02 in. when the strand temperature reached of 220*F. The strands were initially stressed to 75% of their ultimate strength. 1.6.1.6. Deformation and thermal properties Characteristics of nonmechanical deformation properties, such as concrete shrinkage and thermal strains of concrete and prestressing steel, at high temperatures have been reported in various studies. Relationship of thermal properties of concrete with temperature have also been studied. This information is presented in Chapter 3. 1.6.1.7. Application to this study The models of temperaturematerial property relationships used in this study have been derived from the results published in the literature. Strength properties, nonmechanical deformation and thermal properties of concrete and prestressing steel are used as reported. Information on loss of prestress is not used, however, the loss of prestress is derived from effects of concrete shrinkage, thermal strains of concrete and steel, and relaxation properties of prestressing steel at high temperatures. Modulus of elasticity data is also not applied directly. Elastic properties are derived from information on influence of temperature on stressstrain relation of concrete and prestressing steel. Due to lack of sufficient data on bond slip of prestressing steel in relation to temperature changes, bond properties are not included in this study. Therefore a perfect bond between concrete and prestressing steel is assumed at all temperature levels. 1.6.2. Structural ResponseExperimental Fire tests have been the main procedure for studying structural response of prestressed concrete in fire environment. Most of the test data reported come from (a) Portland Cement Association PCA), Skokie, and (b) Underwriters' Laboratories, Chicago. The main objective of these tests was to determine the fire endurance of prestressed concrete members while supporting their full service loads [2,20,21,22]. Fire tests of building components are conducted under the specifications of the ASTM Standard Methods of Fire Tests of Building Construction and Materials [1]. The test sample is placed in a furnace and loaded at the desired service stage. The applied load is the maximum permissible superimposed load as required or permitted by recognized building codes. The furnace is then fired by natural gas burners from each side of the sample. The heating of the furnace is controlled by using thermocouples so that the temperature conforms to the standard timetemperature curve shown in Fig.l.5. Temperature distribution within the specimen is measured through thermocouples located at various points in the specimen. Test on a specimen is conducted until an arbitrary selected deflection limit or a defined end point is reached. Some of the end point criteria [1] are (a) Structural end point: when collapse occurs for load bearing specimens while supporting the applied loading and (b) Heat transmission end point: when the temperature of the unexposed surface of floors, roofs or walls reach an average of 250 F or a maximum of 325 F at any one point. 1.6.3. Structural ResponseNumerical Research areas for developing numerical procedures have included heat transfer and structural response analysis [23]. Numerical methods developed for temperature distribution include the use of finite difference techniques [13,24], multidegree polynomials [25], and finite element [26,27]. Finite element techniques have also been used to develop numerical methods for structural response analysis of steel structure. Jeanes [28] developed a rational method and computer implementation, FABUS II, for determining the structural response of steel buildings subjected to fire. Temperature distribution was analyzed using FIREST3 [27]. Temperature dependent properties of steel considered in the model were the yield strength, modulus of elasticity and coefficient of expansion. Structural responses studied were deflections, beam elongation and changes in stress levels across beam sections. Anderberg et al. [29] also developed numerical procedures for predicting behavior of steel structures subjected to fire and implemented in a FE computer program STEELFIRE. Structure responses are determined by the analysis of plane steel frames subjected to inplane loading. Measured temperature values were part of the input to the model. Main response computed was deflection. Other developments of numerical methods for response of steel structures in fire include those of Bresler and Iding [30] and Nakamura et al. [31]. Numerical methods for structural response of prestressed concrete structures have been limited to the determination of fire endurance. The procedures determine the end point moment capacity of a prestressed concrete member subjected to elevated temperatures [3,4,5,21]. The residual moment capacity is computed using the rectangular beam formula given in the concrete building code [32]. The basic equation is MW = *Apfpa (d ae/2) = 4M a = f 0.85fab (1.1) fpse = fpe1 0 5 Ap f where S= capacity reduction factor for flexure Ap, = the cross sectional area of the prestressing steel, in.2 fp, = the stress in the prestressing steel at the ultimate load, ksi fp, = the ultimate tensile strength of the prestressing steel, ksi, (temperature dependent) M, = nominal moment strength, in.k d = effective depth, in. a = the depth of the equivalent rectangular stress block at ultimate load, in. 8 = indicates temperature dependence Fire safety designs aids have been developed based on the results of experimental studies on temperature distribution and material behavior. The aids are in the form of graphs for temperature distribution within beams and slabs, and for temperature and strength relationship for both concrete and prestressing steel [3,4,5,22,33,34]. Steel temperature within the concrete at certain distances from fire exposed surfaces of beams or slabs, and at various times of exposure, can be estimated from the temperature distribution curves. The steel strength, f ,, in Eq. (1.1), is then computed according to the relationship between the temperature and strength of prestressing steel given in strengthtemperature curves. The data is used to compute the moment capacity at various exposure times. Flexural failure is assumed to occur when the retained moment capacity, Mg,, is reduced to the level of the applied moment M. This condition is demonstrated in Fig. 1.5. From Eq. (1.1) it can be seen that the fire endurance depends on the applied loading, and on the strengthtemperature characteristics of the prestressing steel. Design service load a) Loaded system Moment capacity M . b) Moment diagram (before exposure) b) Moment diagram (before exposure) r L7 ___ I M Mn  Structural end point, M = M c ) Moment diagram (after exposure) Fig. 1.5. Moment capacity of a beam exposed to fire CHAPTER 2 THERMAL ANALYSIS 2.1. Introduction This chapter presents the development of the procedure for the analysis of temperature distribution history of members in a fire environment. It is based on the general approach introduced by Wilson [35,36] and Zienkiewicz [37], and simplified for the needs of this study. The members are idealized by a systems of two dimensional triangular elements. The thermal boundary conditions, represented by the timetemperature relationship of the Standard Fire Test (SFT), consists of convective and radiative heat transfer mechanisms. Development of a computer program (TEDIAN) is also discussed. 2.2. Heat Transfer Models 2.2.1. Heat Flow Models The general transient heat transfer equation is a aT + a a C aT (2.1) x + k 9 pc x ORx ) ay y t where x,y = spatial coordinates T = temperature distribution history t = time k = isotropic conductivity (temperature dependent) c = specific heat capacity (temperature dependent) p = material density (temperature dependent) Initial condition is specified as T(x,y,t=o) = To(x,y) (2.2) where To is the specified uniform temperature for every point in the structure before analysis begins. In this study its value is taken as 68 F. Using the mathematical formulation described by Segerlind [38], the convective and radiative boundary conditions are expressed as k(2.T)1 1lx + ay + h (T Tf) + hr(Ts T) (2.3) where 1, ly = directional cosine along x and y coordinates, respectively ha = convective heat transfer coefficient hr = radiative heat transfer coefficient T, = surface temperature of concrete member Tf = furnace temperature The dependence of convective heat transfer coefficient on temperature has minimal effect on the heat transfer process of the standard fire test. As temperature increases the process becomes more and more dependent on radiation. For numerical computations h is assumed constant [39,40], and approximated as h = 4.4 (Btu/hr.ft2oF) (2.4) The radiative heat transfer coefficient is expressed as hr = ue[(Ts + 460)2 + (Tf + 460)2] (Ts + Tf + 920) (2.5) where a = StefanBoltzmann constant = 0.119 1010 Btu/hr.in2.OR) E = resultant emissivity of the flames, combustion gases and the boundary surface. For furnace conditions of the standard fire test, the value of resultant emissivity is taken as 0.5 [39,40]. 2.2.2. Finite Element Formulations Based on variational principles [36,38,41], Eqs.(2.1) and (2.3) are expressed, respectively, as 1= l+ 2 + C2 2pc oa]dv (2.6) (2.7) fShc(, ,f) 2dS + (f ,(, ,) 2dS where O(x,y,t) = unknown temperature function 0o = initial temperature field t, = temperature field of surface exposed to heat source (unknown) 9, = temperature field of medium of heat source (known) The above functional are combined so that the boundary conditions are satisfied automatically after minimization of the integral I [42]. The resulting functional is I = fi2 + Ifh a(  + f)2 + pc"f2 2pc 4o]dV f)2dS + 1f h, (, f) 2dS 2J (2.8) 2.2.3. Matrix Equations Thermal matrices are formulated using standard displacement finite element procedures. Following relations are assumed: 0(x,y, t) = N(x,y) T(t) (2.9) where T(t) = nodal temperature vector N(x,y) = interpolation function (H)T =l a0 a (2.10) ax ay [D] k o] (2.11) T[r 8N aN (2.12) [H] = [B(x,y)] (T(t) ) (2.13) Making substitutions of Eqs.(2.10) to (2.13) in Eq.(2.8) results in I = 1 (T)T[B]T[D] [B] (T) dV Jv2 + f pc (T)[N]T[N] (T)dV 2J V vpc (T)T[N]T[N] ( T) dV (2.14) + h(T)T[N]T[N] (T)dS hTf[N] (T) dS + f hTfdS where h represents the heat transfer coefficient, convective or radiative. Eq.(2.14) is differentiated with respect to temperature T, and equated to zero. The minimized function is f([BJ][D] [B] dV+ sh [N][N]dS) (T) + pc[N]T[N]((T) (To))dV hTf[N] dS = 0 For an integration step A, temperature rate is given as (T) (To) = (TtAt) (Tt) = (T') where (T.) = (T,) = initial spatial temperature The terms in Eq. (2.15) are defined as follows: [K,] = f[B]T[D] [B] dV [K2] = fh[N]T[N]dS [K3] = h, [N]T[N] dS [C] = pc[N]T[N] dV [R,] = f hT,f[N] dS [R2] = frTf [N] dS J Eqs.(2.16) to (2.22) are lumped into a the following transient heat equation: matrix form of (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) [C] (T') + [K] (T) = (R) (2.23) where [K] = [K1] + [K2] + [K3] (conductivity matrix, temperaturedependent) [R] = [Rj] + [R2] (thermal load matrix, due to pseudo fire boundary conditions, temperaturedependent) [C] = thermal capacitance matrix, temperature dependent. 2.2.4. Time Integration of Temperature Over a time interval (t+ k,t), the rate of temperature change, (T'), is approximated based on following assumptions [42,43]: Ti+UAt = T(Tt+t Tt) S ) (2.24) Tt+act = (1 a) Tt + aTt+At where 0 < a <1. Then, for a time instant (t+ 4i , Eq.(2.23) is rewritten as [ C] t.* t( T) t+aAt + [K] t+at( T) t+a t = (R)t+ t (2.25) which upon substituting in Eq. (2.24) results in S+ [K)a (T) t+aAt 1 (2.26) (R) t+GAt + [C t+aAt( T) t(2.26) = (R) t.,a + [C] (T), (1 a) [K] t.,a[T] l For conditionally stable solution of Eq.(26) implicit algorithm obtained for a 40 are more preferred. Using the algorithm of a= 1/2, [35,42], the transient heat transfer equation becomes [ C] tE t + [K] t, (T) t+A= (R) tAt (2.27) ( R) t +[ [C] t+t[ K] ,] ( T) Implementing and trying other algorithms, Eq.(2.27) required more computation effort for the same level of accuracy than for a= 1. Thus, the form of equation used in this development is St[C] t + [K] (T) A (2.28) S(R)tAt + 1 [C]t+At(T)t The more convenient form for solution is [KC] (T) t., = (RC) (2.29) where the effective conductivity matrix, [KC], is given by (2.30) [KC] = 1 [C] tAt + [K] t.A and the effective thermal load vector, [RC], is (2.31) [RC] = (R) tAt + 1 C] t ( T) A t 2.3. Evaluation of Thermal Matrices and Load Vector The evaluation of thermal matrices of Eq.(2.29) is based on the characteristics defined for the two dimension triangular element presented in Fig. 2.1. The element has one temperature degree of freedom per node. Temperature field y spatial f iel d Fig.2.1. Plane triangular element 2.3.1. Interpolation Functions The area of triangular element of Fig. 2.2. is given by Ak = [(xiy xjy) + (xjykXkYj) + (xkyiXik)] (2.32) The three shape functions are defined as follows: N1 = A[(XjykXkyj) (y yk)X (Xkj)y] (2.33) S= [(Xk i Xiyk) + (Yki)x + (Xi k)Y (2.34) 1 (2.35) Nk A= XiYXjj) + (Yiyi)x + (xxy] (235) Substituting Eqs.(2.33), (2.34) and (2.35) in Eq. (2.12) we get matrix [B]: _] 1 BI BJ BC (2.36) ] 2A[CI CJ where, BI = (y Yk) CI = (x, x3) BJ = (y, y) CJ = (x, xk) BK = (yi y3) CK = (x, xi) 2.3.2. Conductivity Matrix For an element of unit thickness, Eq.(2.17) becomes (B2 + CI2) (BIBJ+ CICJ) (BIBK+CICK) K= k (B+CJ2) (BJBK+ CJCK)I SYM (BK2 + CK2) Assuming dS = 1.dL and using area coordinate method of integration [44], Eq. (2.18) takes the following form: [K hLij [2 1 0 0 0 08 [K = 6 [ 0 + 2 1 (2.38) 0 0 0 0 1 2 where ij denote the end nodes of exposed edge. The length Lj is defined as (2.39) Lij= /(xjxi)2 + (yy) (2.39) Eq.(2.19) takes the same form as Eq.(3.38), replacing h, by h,. 2.3.3. Thermal Capacitance Matrix Applying the area coordinate method [42], the capacitance Eq.(2.20) becomes [ = pA 2 1 (2.40) [C] 12 SYM 2 When the implementation of Eq.(2.40) was used temperature values of certain nodes diminished below zero before rising. The reason for this characteristic is not clear. To avoid this problem an alternative approach is utilized in this study. The capacitance matrix is computed by delineating part of element volume to a node. For element m of unit thickness the volume contribution to node i is given by S= Amp(T)c(T) (2.41) 3 33 The matrix form of Eq.(2.41) is 1= (2.42) [C]q= 1 1 The approach produced results with the expected characteristics. 2.3.4. Thermal Load Vector The load matrices are evaluated using line integrals. For an element of unit thickness with two sides, (ij) and (jk), exposed to fire, the evaluation of Eqs. (2.21) and (2.22) result to a complete thermal load vector expressed as () CTfLij 1 + hcTfLjk 0 0 1, (2.43) 6 6 0fi 1 2.4. Solution Algorithm At time (t+At), temperature dependent thermal matrices [C], and [K], of Eq.(2.28), are not known. Thus, for each time step increment &, the following iterative scheme is used to obtain an acceptable solution: 1 r Un1l1) (il)] U) U,,(i1) S L t+At + [K(t+At t+At ()t+At (2.44) (2.44) + (i1) + L[Ctat (T)t where the superscript (i1) denotes iteration number. The initial values are expressed as 1(0) [,^t t = [C]t [ (0) (2.45) [AKfc+At [K~t (R) (o) t+t = (R)t Convergence of solution is governed by both the system temperature and the boundary condition criteria. For any two successive iterations the temperature convergence is achieved by i(i) t(ii1) +At TAt < Er (2.46) llr (i) t l (i1) 2 +At t+At and that of boundary condition by (i) h (i1) Irff I !E e (2.47) S (i) h (i)l r 2 rr,,t r*;+At CHAPTER 3 MODELING OF TEMPERATURE DEPENDENT MATERIAL PROPERTIES 3.1. Introduction Both concrete and prestress steel experience certain degree of degradation when exposed to high temperatures [6,7,8,9]. It is, therefore, important to understand the behavior of these properties at elevated temperatures. The objective of this chapter is to present the mathematical relation of material properties to temperature. Most of the models are derived directly from test data and results published in the literature and summarized in references [45,5,46]. It is expected that the data used and the models developed represent the acceptable behaviors of concrete and prestress steel as related to temperature and, therefore, provide reasonable accurate basis for evaluating the structural response of prestressed concrete elements in fire. 3.2. Modeling Procedures Most properties of concrete and steel are affected by high temperature. In this study the temperaturedependent behavior of material property curves is estimated by a 35 series of discrete points connected by by linear segments as shown in Fig. 3.1. Then the parameter value, p, at a given temperature, T is computed by p(T1) = (T,) + (Ti T,) S, S P(T,1) p(T) (3.1) n TnT Tn n+1  Tn Ti 5 Tn+1 p(T) p ( T ) p(T ,) _ p I( T Tn T Ti Tn T Fig.3.1. Mathematical Modeling of Material Properties from Test Results 3.3. Concrete 3.3.1. Thermal Properties The parameters which influence the characteristics of heat transfer in solids are the thermal conductivity and thermal capacity. 3.3.1.1. Thermal conductivity Thermal conductivity of concrete measures the ability of concrete to transfer heat by conducting heat from particles at high temperature to those at low temperature. The variation of thermal conductivity of concrete with temperature is primarily determined by the conductivity of the aggregates used [47,48]. High temperatures increases the disorder of particles in aggregates. This causes the scattering of heat in all directions, hindering effective transmission of heat towards low temperature regions. As a consequence, thermal conductivity of concrete decrease with increase in temperature. The generic model relating thermal conductivity of concrete to its temperature is expressed as kc(T) = ak + bkTo (3.2) where T, = element temperature (from thermal analysis) a, b = constants derived from experimental data (according to Fig. 3.2.) The variation of thermal conductivities with temperature for the three types of concrete is shown graphical Fig. 3.2. 0.9 0.8 .0.7 Siliceous S0.6 Carbonate :0.5 0.4 Lightweight 0.3 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature (deg.F) Fig. 3.2. Conductivity of concrete as a function of temperature (Source: Ref. [49]) 3.3.1.2. Thermal capacity Thermal capacity of a material refers to the amount of heat required to raise a unit volume of the material by one degree. It is a product of material density and the specific heat of the material c. The character of the cement paste and water in the concrete have the most pronouncing effect on the behavior of thermal capacity of concrete [47]. In the case of water most of the supplied heat energy will be used to remove pore waters in temperatures near 212 F. Crystalline water is removed at higher temperatures. When removal of water takes place, the thermal capacity of concrete shows substantially higher values. The dependence of thermal capacity of concrete on temperature is shown graphically in Fig. 3.3. 300 250 t200 2 Carbonate 2150 o Co Siliceous 100 E 1 A lightweight S50 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature (deg.F) Fig. 3.3. Variation of thermal capacity of concrete with temperature (Source: Ref. [49]) The thermal capacity of the various types of concrete considered in this study are modeled for various ranges of temperatures. The models takes the following form: pc(T) = apc + bpT (3.3) 3.3.2. Mechanical Properties Mechanical properties are those describing the characteristic of material under stress. Considered in this section are stressstrain relation, compressive strength and modulus of rupture. 3.3.2.1. Stressstrain relation High temperatures increases deformation properties of concrete. Thus there is a continuous variation of stress strain relation with temperature resulting to a number of straincurves. The relationship between stressstrain law and temperature is shown in Fig. 3.4 for some selected temperature ranges. 1.0 0.9 660 c0.7 T=<100 OF S0.6 ._ 0.5 )0.4 1020 0.2 1560 0.0 ... 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 Concrete Strain Fig. 3.4. Stressstrain relationship as function of temperature (Source: Ref. [49]) For each temperature range, modulus of elasticity E is computed for a given range mn. The expression is Aoa(T) = Go(T) ao(T) AE~ = n em (3.4) S(Ta (T) Emn(T, mn The mechanical stress for time step j+1, temperature range Tk+ and mechanical strain ei is given by =mi 1 M (3.5) i+1lTk+1,i) =(m(Tk+1m) + E+'Ae1 The stressstrain curves used in this study have been reported by Lie [49]. They have been adjusted to account for creep by moving the maxima to higher strains with higher temperatures. In addition, the curves give the whole characteristic of stressstrain relation which include the post peak behavior. From reported data [13], aggregate type and strength have no significant effect on stressstrain relation at high temperatures. As such the set of stressstrain curves represents both normal weight and lightweight concretes. 3.3.2.2. Compressive strength Type of aggregates, degree of loading and conditions of loadingwhen hot or cooled are among the important factors which influence the behavior of concrete strength at elevated temperatures. 42 As temperature of concrete increases the aggregates and the cement matrix expands. At the same time pore and crystalline waters are evaporated, which causes shrinkage of concrete matrix. The resulting expansion differentials cause internal cracking of concrete which results to reduction of stiffness of concrete. The extent of this phenomenon differs considerably with the type of aggregate used in concrete. Concretes with silicious aggregates are mostly affected since at temperatures above 1000 F the aggregates also undergo physical changes accompanied by sudden expansion in volume [50]. Carbonate aggregate do not normally undergo physical changes during heating, hence carbonate aggregate concretes are free from severe internal cracks. However, at very high temperatures chemical changes take place when lime is liberated from calcium carbonate. The process is beneficial in retarding temperature rises, but during cooling the lime combines with atmospheric moisture and expands in volume causing cracks and damages to the concrete [50]. Lightweight aggregates normally undergo various heating processes during manufacture, hence they provide a better physical compatibility between the matrix and the aggregate with regard to deformation. In this study experimental data used to derive mathematical models are for 'stressed tests' in which concrete is tested hot while stressed to 40 percent of its room temperature strength. 43 These results are meaningful in structural response analysis since any prestressed concrete member will be subjected to some stress from external load, prestress or both. According to Abrams [9], the compressive strength values for specimens stressed at 25 to 55 percent of their original compressive strength show no significant variation in behavior. The graphic models of the variation of concrete strength with temperature are shown in Fig. 3.5. 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 200 400 600 800 1000 1200 14 Temperature (deg.F) Fig. 3.5. Compressive strength of concrete versus temperature (Source: Ref. [5]) 1600 The mathematical expression for the compressive strength ratio is given by the following expression: CSR(Ti) = CSR(Tm) + T (3.6) where CSR = concrete strength ratio (percentage of original) The compressive strength is then computed by FC(Ti) = CSR(Ti) FC(T6,) (3.7) 3.3.2.3. Tensile strength Reported test results of tensile strength of concrete at high temperature are very sketch. However, these reports indicate that the deterioration in tension is greater than that in compression [51]. For lack of adequate information, tensile strength of concrete is taken as a function of compressive strength. The relationship is expressed as frT = 7.5f T (3.8) rT, = 5.6 f where the equations are for normal weight and lightweight concrete, respectively [32]. 3.3.3. Deformation Properties Deformation properties considered are those causing changes in the dimension of a member due to nonmechanical means. These are thermal and shrinkage strains. The thermal strain is due to thermal expansion after making allowance for drying shrinkage. This is only an assumptionsince according to Bazant [52], it is difficult to separate thermal strain from shrinkage strain under transient temperature conditions. Creep strain, which is both stress and time dependent, has been incorporated in the derivation of stressstrain relations. 3.3.3.1. Thermal expansion Thermal expansion of concrete increases with increasing temperature. The variations are influenced by the aggregate type, cement content, water content and heating rate [51]. For unstressed specimen the increase in temperature causes the increase of thermal strain to a point of disintegration. At very high temperatures around 1100 F to 1500F range most concretes indicate no expansion. In some cases the concrete shrinks due to chemical or physical reactions in the aggregates. The coefficient of expansion is derived from test results of thermal strain versus temperature. The relation is shown in Fig. 3.6. For a temperature range T m, T,, the coefficient of thermal expansion is given by the following model: S (T) A th(T ) (3.9) AT Then the incremental and total thermal strain at temperature Ti is computed from AE(Tmi) = a, (TiT) (3.10) th(Ti) = eth(Tm) + Ae(Tmi) 0.016 0.014 Siliceous 0.012 0.01O c/ S0.008 ) Carbonate E 0.006 I 0.004 0.0022 Lightweight 0 0 200 400 600 800 1000 1200 1400 1600 1800 Temperature (deg.F) Fig. 3.6. Thermal strain of concrete as function of temperature (Source: Ref. [5]) 3.3.3.2. Shrinkage strain Shrinkage strain in concrete is due to loss of moisture in the cement paste [52]. The amount of shrinkage that occur and the rate of its occurrence are thus dependence on the moisture content in the concrete before heating. In this study the concrete members are assumed to be at normal service condition of relative humidity of 70 percent. In addition the shrinkage strain considered is that caused by loss of free water from the pores of concrete. Based on this criteria all the shrinkage is assumed to take place within 68 to 212 F temperature range. The incremental shrinkage strain within a time step n is computed by the following models: Aen(Ti) = a(Ti) (e.(Ti) es(T,)) Atn (3.11) a(T) = 0.001 + T ]2 (312) 144 e.(T) = 0.00051 Ti68 (3.13) 144 where e,(T.) = cumulative shrinkage strain (at beginning of current time step) e.(T) = total potential shrinkage due to loss of free water a(T) = shrinkage rate (in/in per hour) AEs = incremental shrinkage strain At = duration of current time step (hour) As mentioned earlier, both Eqs. (3.12) and (3.13) are valid within the temperature range of 68 F to 2120F. From Eq. (3.13) the maximum total potential shrinkage at 212 F has the value of 0.001 in/in. 3.4. Prestressing Steel Most of the descriptions of property categories given in for concrete apply to prestressing steel as well. 3.4.1. Mechanical Properties 3.4.1.1. Stressstrain relationship For a given range, the steel modulus can be obtained from Eq. (3.4) after substituting for the appropriate variable. The steel stress is then given by ai(T,e) = am(Tk) + EA (3.14) oi (7,e) =ak + Emn Emi Fig. 3.7. gives the graphical representations of stress strain law as used in this study. 1.0 0.9 0.8 o 0.7 . 0.6 0.5 S0.4 ^ 0.3 0.2 0.1 0.0 0.00 Strain (in/in) Fig. 3.7. Stressstrain curves for PS at various temperature levels (Source: Ref. [45]) 0.12 3.4.1.2. Tensile strength The tensile strength of prestressing steel decreases with the increase of temperature. According to the tests done by Abrams [6], the size of the strands and the rate of heating have no significant influence on the temperature strength behaviors. Following the definitions given in Fig. 3.1, the tensile strength ratio is determined by ATSR (T) = TSR(T,) TSR(Tm) ATm = Ti T, A TSR(T) (3.15) A Tn TSR(Ti) = TSR(Tm) + SmnATm The tensile strength is obtained from FPU(Ti) = TSR(Ti) FPU(T) (3.16) The graphical representation of the temperaturestrength relationship for prestressing steel is given in Fig. 3.8. 3.4.2. Deformation Properties 3.4.2.1. Thermal strain Coefficient of expansion of prestress steel increases with the increase of temperature. It is expressed in following form: (3.17) a(T) = (6.667 + 0.001235T) (106) 1.0 0.9 0.8 : 0.7 0 0 0.6 0.5 0 o 0.4 20.3 S0.2 0.1 0.0 0 200 400 600 800 1000 1200 1400 1600 Temperature (deg.F) Fig. 3.8. Variation of prestress steel strength with temperature (Source: Ref. [5]) Variation of coefficient of thermal expansion of prestress steel with temperature is shown in Fig.3.9. Within a time step the constant value of coefficient of thermal expansion used in computations is given by an + an.1 (3.18) s 2 The corresponding thermal strain is expressed as S(T = + AT (3.19) 3.4.2.2. Relaxation/creep strain The relaxation property of prestressing steel can be defined as the decrease in stress with time under constant strain. However, most of the reported tests have measured 51 9.0 8.5 Sa S7.5 S 7.0 0 200 400 600 8o0 1000 1200 1400 1600 18002000 Temperature (deg.F) Fig. 3.9. Expansion coefficient of prestress steel as function of temperature (Source: Ref. [45]) it in terms of rate of loss of stress at given stress, that is, creep. At high stresses and temperatures creep rate become substantial. According to tests by Yakovlev et al. [53], a prestressing of about 128,000 1b/in 2 in a wire was completely lost when heated to 572 *F in 40 minutes at a rate equal to that used during fire resistance tests of a E E 7.5 % 7.0 6.5 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature (deg.F) Fig. 3.9. Expansion coefficient of prestress steel as function of temperature (Source: Ref. [45]) it in terms of rate of loss of stress at given stress, that is, creep. At high stresses and temperatures creep rate become substantial. According to tests by Yakovlev et al. [53], a prestressing of about 128,000 lb/in 2 in a wire was completely lost when heated to 572 F in 40 minutes at a rate equal to that used during fire resistance tests of a prestressed concrete element. According to the information given by Harmathy [54], the creep characteristics for prestress steel are described as follows: AH 54000 R (3.20) Z = 8.21(1013) e0'00010 CC = Ze aT where AH/R is given in temperature units (R) Z = ZenerHollomon parameter (h 1) T = absolute temperature of prestressing steel (R) a = stress in prestressing steel (lb/in 2) ecr = creep rate (in/in.h). Creep rates for some selected stress levels are presented in Fig. 3.10. 0.1 0.01 0.001 0.0001 1E05 1E06 500 650 700 750 800 Temperature (deg.F) Fig. 3.10. Creep rate for prestress steel as function of temperature and stress (Source: Eq. (3.20)) 3.5. Remarks The models presented in this chapter have been formulated by approximating the test result curves by piecewise linear segments. Similar technique was used by Franklin [55] to model stressstrain relationship of concrete in his study of reinforced concrete frames. This approach was used in this study because of lack of mathematical models for most of the material properties considered. However, generalized empirical models have been used whenever possible. CHAPTER 4 MODELING OF PRESTRESSED CONCRETE FRAME MEMBER 4.1. Introduction This chapter discusses the modeling of prestressed concrete members. The formulation is based on direct stiffness method of structural analysis using two dimensional beam elements. Using the idealization of Fig. 1.4 the stiffness of an element is determined by integrating the temperaturedependent subelement properties. 4.2. Basic Finite Element Equations for a 2D Beam Element Figure 4.1 presents a finite planar beam element with six nodal displacements, r 1 to r6. These displacement components can be expressed in a generic form as u = {u, V}T (4.1) where u represents displacement components associated with axial action in the xaxis and v represents both translation in yaxis and rotation due to flexural action about the zaxis. These components are shown in Fig. 4.2. Relationship between the generic displacement u and the nodal displacements [56] is given by U = Nr (4.2) where N represents the displacement shape function, and r the nodal displacements as shown in Fig. 4.1. Fig.4.1. Displacement components of a 2D beam element u r ) r x U 2 ( r4 ) A 4  U v 1(r 2 r ) x 3 v 2 ( r 5 4( e2 r (b) Fig. 4.2. Axial and flexural elements Straindisplacement relationship is expressed as (4.3) e = du where E = strain vector d = linear differential operator By Substituting Eq. (4.2) in Eq. (4.3), the strain equation becomes e = dNr (4.4) resulting to the definition of straindisplacement matrix B B = d(4.5) The corresponding stress is given by a = Ee (4.6) = EBr where E = stressstrain matrix EB = stressdisplacement matrix The relationship between the nodal displacements and the external load is expressed by Kr = R (4.7) where K is the element stiffness matrix and R represents the load vector. The stiffness matrix is formulated as K = f BEBdv (4.8) Jv The corresponding load vector F, due to strain energy of internal stresses in the element is computed from F = fB'rEdv (4.9) 4.3. Evaluation of Displacement Functions for a Beam Element Based on the actions associated with the nodal displacements shown in Fig. 4.1, the beam element is decomposed into: axial and flexural elements. The generic displacement equation for the axial element is expressed as = u (4.10) where r = {r, r}T= {u, u2}T (4.11) Corresponding displacement function is given by u = c c + c2x (4.12) where c, and c2 are constants corresponding to the nodal displacements. Expression for the shape function is = xL xx (4.13) Straindisplacement matrix B evaluates to 1 (4.14) S= [1 1]i For the flexural element, displacement relation is given by (4.15) The components are represented by r = {r2 r3, r5, rT ={v1, 01, V2, 2)T (4.16) where Sdv1 dx (4.17) dv9 02 dv2 dx Based on plain section theorem, translation in x direction of any point on the crosssection of the element is determined from dv (4.18) u = y dx Substituting Eqs.(4.18) into (4.3), flexural strain becomes =du = d2v = (4.19) dx dx2 where 4 represents the curvature. Differential operator takes the following form: d= d2 (4.20) dx2 The displacement function is expressed as v = + c2x + c3x2 + c4x3 (4.21) Shape functions are then given by 2X3 3x2L + L3 T 1 xL 2x2L2 + XL3 (4.22) L3 2x3 + 3x2L x3L x2L2 By substituting Eqs.(4.20) and (4.22) in Eq.(4.5), the straindisplacement matrix B becomes 12x 6L BT_ y 6xL 4L2 (4.23) L3 12x + 6L 6xL 2L2 4.4. Idealization of a Prestressed Concrete Member A prestressed concrete member is modeled into a series of beam elements and tendon segments as shown in the Fig.4.3(a). A steel segment is assumed to be straight, has a constant tension force and spans the beam element. The interaction between the prestress steel and concrete is assumed to occur at the ends only. The profile and the location of a steel segment in a beam element is defined by two end eccentricities and a reference plane as shown in Fig. 4.3(b). It is determined by the expression given in Eq.(4.24). (4.24) 1 yp = (ei + e,) The reference plane is used to estimate steel segment temperature, incremental strains and contribution to the element stiffness. beam node interaction point between and steel tendon 'Tendon segments a) Finite Element Model of Beam Specimen and Steel Tendon y RA = reference axis for the sect ion .x RP = reference plane for steel b) A Beam element with straight tendon segment and a constant prestress force Fig. 4.3. FE idealization of a PC member 4.5. Evaluation of Element Stiffness Matrix The strain displacement matrices of Eqs. (4.14) and (4.23) are modified to accommodate all six displacement components shown in Fig. 4.1. The matrices become Ba = [1 0 0 1 0 0] L (4.25) for the axial and y yBb =  L 0 12x 6L 6xL 4L2 0 12x + 6L 6xL 2L2 for the flexural element. Eq.(4.8) is then rewritten as K = (Ba ~ yb) a yBb)]dv or as the following summation: K = Kaa + Kab + Kba + hbb where the components are expressed as follows: Kaa = f BEBadv Kab = f yBSEBbdv K^ ( yB EB~dv Kbb= fvy bESadv b = f y2 TEBdv J (4.26) (4.27) (4.28) (4.29) For convenience of integration, each part of Eq.(4.29) is decomposed in two integral terms as follows: aa = fvBEBadv = fAEdA. fLBBadx Ka= f yBEBbdv= ydA. BBadx (4.30) K v T A JL Kb= S yBb dv = fEy2dA. B Bbdx Kba = Eab The variables in Eq. (4.30) have following properties: Matrix B, is constant as shown in Eq. (4.25). Eq. (4.26) shows that matrix B, is a variable of x only. These matrices will be integrated analytically. The stressstrain matrix E is temperature dependent and a function of both x and y. In this study each subelement, from the idealization of the beam element in thermal analysis, is assumed to have a uniform temperature. Thus for a given temperature E is constant for each subelement. The variable A is the section area of each subelement and is assumed constant along the element length L. Variable y gives the location of a subelement or the projected plane of the prestressing steel segment with respect to the reference axis of the element. Based on the assumptions described above the stiffness matrix is evaluated term by term. The contribution of the prestressing steel to the element stiffness matrix is obtained by considering the steel segment as a subelement located at its projected reference plane defined in Fig. 4.3. and given by Eq.(4.24). The first terms of Eq. (4.30) are integrated numerically as follows: EA = fEdA = EcAc + r ElpyAp r L n, n., (4.31) EQ = fEydA = EciiAci Eypsyi A .1=1 2=1 El = fEy2 = E i (,0 +yiAci) + EpsiypsiApsj A 2=1 =1 where no, np, = number of concrete subelements and steel segments respectively within an element, Ap, = crosssection area of prestressing steel, loI = second moment of area of a subelement about its centroid axis. Integration of the second terms results in 1 0 0 1 0 0 0 0 0 0 0 fLBa x 0 0 00 (4.32) aL = a L SYM 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 fLrBb l o 0 0 0 0 0 (4.33) dx L= O0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 BLBbdx 1 JLbL 0 0 0 0 0 12 6L 0 12 6L 4L2 0 6L 2L2 0 0 0 SYM 12 6L 4L2 The composite element stiffness matrix is obtained by substituting Eqs. (4.32), (4.33) and (4.34) back into Eq. (4.28). The resulting matrix is given by Eq.(4.35). EA L 0 EQ L EA L 0 EQ L 0 12EI L3 6E1 L2 0 12EI L3 6EI L2 EQ L 6EI L2 4EI L EQ L 6EI L2 2EI L SEA L 0 EQ L EA L 0 EQ L 0 12EI L3 6EI L2 0 12EI L3 6ElI L2 _EQ' L 6EI L2 2EI L EQ L 6EI L2 4EI L (4.35) 4.6. Incremental Strains in Concrete and Prestressing Steel 4.6.1. Concrete Deformations Fig. 4.4 (a) shows the incremental displacements of the end nodes of a beam element after two successive operations. These displacements are with respect to the reference axis (RA). The centroid axis (CA) changes its location continuously due to changes in section properties. The incremental displacements are transformed to relate to the (4.34) 65 Ar 3 r r (a) RA = reference axis CA = centroid axis ....... V d stance of CA A    *" " A .r form PA I = Iteration number 3(b) Fig. 4.4. Incremental nodal displacements current location of the CA before computing the strains. The transformation is shown in Fig. 4.4 (b) and achieved by following expression: Ar? = Ar ycAr3 (4.36) Ar = Ar ycAr6 Using the transformed values the effective incremental displacements translations and rotation are given by Eqs. (4.37) and (4.38) respectively. 66 Ax = Ar4 Ar1 (4.37) Ay = Ar, Ar2 AO = Ar6 Ar3 (4.38) Curvature of the element local axis is expressed as A A (4.39) Lo Incremental axial strain components are computed by S= (4.40) for axial action and Ae = yA4 (4.41) for bending action. L is the original element length and y is the distance from beam element centroid axis to the centroid of the subelement in considerations. The total incremental strain is obtained from summation of Eqs. (4.40) and (4.41), and expressed as AE = Aa + AEb (4.42) Cumulative incremental strain of a subelement for time step i and iteration j is given by Eq. (4.43). j + A (4.43) 4.6.2. Prestressing Steel Strains Steel strain is first computed for the projected plane and then transformed to the correct profile. The procedure uses the following models: Aeb = yp eaa = Aa + AEb (4.44) Ae = AEA aa cos a The cumulative strain for a steel segment is similarly computed from Eq.(4.43). 4.7. Load Vector due to Prestress Force Element nodal forces resulting from prestress action are shown in Fig. 4.5. Following the definitions given in the figure the length of the steel segment and its orientation with reference to the xaxis of the element are computed as follows: AX = XJ XI AY= ei ei Lo = yAX2 + AY2 (4.45) cosa AX Lo sina A Lo where XI and XJ are the updated xcoordinates of element nodes i and j, respectively, and L is the original element length. R 4 e 7  Ri (1 = 1, 6) = element forces due to prestress action P = tension in the tendon segment P P = components of P acting on concrete at i interaction points Fig.4.5. Idealization of Prestress Action into Beam Element Nodal Forces Components of the prestress force along local axes are P, = Pcosa Py = Psina y (4.45) Translation of the components to element nodes i and j result in the following end moment expressions: R3 M = Pei (4.47) R = M = Pxe The element load vector due to prestress action is given by p p piT (4.48) Rp = {Px Py Pxei Px Py Pxe}T 4.8. Remarks The formulation of the beam equations in this chapter is based on the standard displacement stiffness method. The evaluation of the stiffness matrix is performed term by term across the section element. This approach is applied in order to incorporate the variations of stiffness properties across the section due to the state of temperature distribution and stressstrain relationship within different parts of the section. CHAPTER 5 SOLUTION METHOD FOR STRUCTURAL RESPONSE 5.1. Introduction This chapter presents the technique for solving structural responses of prestressed concrete members subjected to high temperatures. The basic equilibrium equations to be solved are given as follows: (a) At the beginning of a solution cycle (a) Kr = R (5.1) (5.1) (b) KAr = AR where the stiffness function K and the applied load function R or AR are known. The task is to solve for the displacement function r or Ar. (b) At the end of a solution cycle (a) Kr = F (5.2) (b) rAr = AF In this case the stiffness function K is an update based on the displacement function r or Ar computed in Eq. (5.1). The internal load function F is the unknown to be determined. Techniques for evaluating the stiffness function K have been discussed in Chapter 4. The evaluation of the load 70 functions R and F requires the continuous determination of the following parameters in all solution cycles: (a) Induction forces and (b) effective prestress action These parameters are discussed below. 5.2. Induction Forces Induction forces are defined here as those forces which are caused by stress induced in the system as a result of the effects of temperature on properties of concrete and prestressing steel. The induction force vector AG is given by A 0 A yA (5.3) L A 0 yA Ao = o aT + AaD + AaS where Au is the vector of the induced stress. The main contributing factors to the induced stress function are (a) thermal and shrinkage strain of concrete ( Au'), (b) strength degradation ( Ao) and (c) shift in stressstrain relation ( Aus). 5.2.1. Thermal and Shrinkage Strain Stresses induced to the system due thermal and shrinkage strains are the equivalent mechanical stresses required to effect equal strains if the system was free to undergo the deformation. Thermal strains of concrete are computed using Eq. (3.10) and shrinkage strains by Eq. (3.11). For each section subelement the induced thermal and shrinkage stresses at time step i are given by AUa = EiAE (5.4) where Auo = induced stresses Aci = thermal and shrinkage strain Ej = valid modulus Since the temperature of a subelement does not change within a time step stresses due to thermal and shrinkage strains remain constant in all operations within a time step. They are computed at the beginning of the time step only. 5.2.2. Strength Degradation Based on the material propertytemperature relationship both concrete and prestressing steel degrade as temperature increases. Degradation is defined here as the deterioration in strength and stiffness properties. A concrete subelement is considered degraded when (a) crushed: au f'c(Ti) determined by Eq. (3.7) (b) cracked: ua k ft(Tj) determined by Eq. (3.8) (c) Ee s 0 as determined by Eq. (3.4). Criteria for prestress steel degradation are (a) tension failure: f p, fpu(T) determined by Eq.(3.16) (b) E, < 0 as determined by Eq. (3.4). Concrete subelements and steel segments flagged as degraded are excluded in the computation of section properties. The stress which was previously carried by a degraded concrete subelement is redistributed into the system. The above parameters controlling the material degradation are both temperature and strain dependent. Since total strain may change within a time step, these induced stresses are computed for each iteration. The stresses induced into the system due to degradation are expressed as Aoa = Aaj0 (5.5) where i is the time step and j is the iteration number. Superscript D is an indicator for degradation induced function. For prestressing steel, the degraded steel segments do not contribute to the prestress action of the containing beam element. 5.2.3. Shift in stress At the beginning of a time step the active stress strain curve may change due to new temperature range. The resulting effect is that a concrete subelement will support less stress though the strain remains the same. This phenomenon is illustrated in Fig. 5.1. The stress induced in the system due to this concrete behavior is given by S= o oa (5.6) where S indicates source as stress shift, and j is the last iteration of the previous time step. 6 66E ( T ) A 6/6 6 11 ,1 . Fig. 5.1. Shift of stressstrain curve 5.3. Effective Prestress Action Prestress action in a beam element is dependent on the effective tension in the contained steel segments. The tension in a steel segment is a function of total steel strain for a particular temperature range. Since strain changes with each iteration, steel tension has to be updated continuously within each time step. The theoretical steel tension is subject to reduction due to partial loss of stress. The function considered in this study as the main sources of loss of prestress are stress relaxation, thermal and shrinkage strains of concrete and thermal strain of steel. The strains caused by these parameters are assumed to remain constant within a time step hence are determined at beginning of each time step. Though steel modulus and stress may change within a time step, for simplicity the partial loss of stress is held constant within a time step. For a time step i the loss of stress is expressed as Aoi = E,,(AEpsi + Aei + AEh,i + e,i) (5.7) oi = Oi + A.i where Epo = thermal strain of steel, from Eq. (3.19) Ec = thermal strain of concrete, from Eq. (3.10) eh = shrinkage strain, from Eq. (3.11) E6 = creep strain of steel, from Eq. (3.20) Ep, = modulus of steel at the beginning of time step, from Eq. (3.4) The effective prestress force for time step i and iteration j is computed by following expressions: fps, fpsi ai (5.8) Pi = fjs, iA where Ap, = section area of prestress steel, a = cumulative stress loss, from Eq. (5.7) 5.4. Evaluation of Action Load Vector The action load vector is defined as the effective loading of the system. It is derived from externally superimposed node and element loads, equivalent element node loads due to prestress action and induction forces discussed in Section 5.2. The transformation of the imposed element load into equivalent node loads is shown in Fig. 5.2. Equivalent nodal load vector is then expressed as R = (R1, R2, R3, R4, R5, R6} (5.9) byL = {O, 6, L, 0, 6, L} 12 where by is the uniform superimposed load acting in the positive ydirection. The action load vector is the summation of results of Eqs. (4.48), (5.1), (5.9) and external node loads. Thus for the time step i H2 R5 b x R6 R i L Fig. 5.2. Equivalent nodal forces and iteration j the action load vector R is given by R ,1 = Rn + R i+ RJ,. 1 (5.10) where vector R~ is induction load vector given by Eq.(5.1) and R~ is the equivalent prestress load vector given by Eq.(4.48) which have been adjusted by Eq.(5.8). Ra is constant and includes elements of Eq.(5.9) and external point loads. Vector R, is computed for each iteration due to the changing vectors Rg and I. 5.5. Solution of Displacement Response The equilibrium equations to be solved for structural displacement responses are Eqs. (5.1) and (5.2). Action load vector R is computed by Eq. (5.10). Fig. 1.2. shows that the internal load function F attains a different curve for each time step as the system attains a new deformed equilibrium state r. This characteristic is caused by changes in prestress load function P and stiffness function S as depicted in Fig. 1.3. The shifting behavior of the function F curve presents complications in the accurate determination of the new value of F at the beginning of each time step. To avoid the problems of tracing a new curve of the function F in each time step, a method of unbalanced load adjustment is utilized. The strategy in this approach is to determine an equivalent incremental load function which will cause increment in the structure deformation from the equilibrium deformation attained in the previous time step. The computation techniques for the first time step are therefore different from the other time steps. The solution process in the first time step is shown in Fig. 5.3. The initial operations involved in solving Eqs. (5.1) and (5.2) in the first cycle of the first time step are (a) r = 0 (b) K 0 = f(r() (5.11) (c) R = f(L, P(r) (d) KArr = R where letter f indicates functional dependence. Operations in subsequent iterations are expressed as follows: (a) r = r + Ar1 (b) Kj = f(r ) (c) AF = K A(5.12) (d) R = f(L, P(rP)) (e) Uj = R> Fj (f) K:Ar1 = Uj R, F R 1=0 R =0 Rt F [OT ) A 0 t t =0 F t 0 t=O 1 F F ARR F T t=F0 t_ 0__ t= t=0 8r 2 Sr S0 t=O t= 0 rO rt r2 r t=O tO t=O t=O Fig. 5.3. Solution of equilibrium equations: 1st time step where U is the unbalanced load vector to be determined. The solution process for other time steps involves determination of incremental action load, and the incremental internal load vector. The process is presented graphically in Fig. 5.4. The incremental loads to be computed at the beginning of each time step are (a) induction forces due to thermal and shrinkage strain, material degradation and shift in stressstrain relation law and (b) changes in the prestress action. U U I+ 1 R, F AGD G i AR 0 1 + 1 U +1 2 U 1+1 0r r +*1 1+1 1+1 r I+I Fig. 5.4. Solution process at subsequent time steps The operations required in this stage to compute the incremental loads are given by the following expressions: (a) = r. (b) K.+K = f(r2+1) (c) ACo+1 = a (TiEi) oi+1(Ti+1,,i) (d) Gij = f(Aoa,+ Acj+, 'Ao) (5.13) (e) ARi+ = + (L, P+) I (L, P]) (f) ARo+1 = AR,+1 + Gi+, + Uf (g) Ki+Ari+ = ARio+ where U = unbalanced load function carried forward from the previous time step, S = indicates induced stress due to shift in stress, T = indicate induced stress due to thermal effects (shrinkage and expansion) D = indicate induced stress due to material degradation (crushing or cracking) Parameters to be computed in the subsequent iteration include (a) current deformation of the system, (b) update of stiffness matrix, (b) induction force due to material degradation, (c) update of prestress action and (d) incremental internal load function. Mathematically, these operations are described in Eq. (5.14). The total incremental displacement within a time step is given by Eq. (5.15). For both equations j (a) r1 = rfi+ + 6ri+r (b) Kji = f(rij+) (c) Firl (d) AFj+ = AFij, + 68Fj1 (e) j = f(L, P'+) (5.14) (g) G = f(a ,o= (rfj+) ff j+, (T 1)) (h) AR = ARi1 + j,+ + 8G/l (i) Ui = &Ail AF= i (j) KiJ6rj 1 = Uj+1 A]+1 = (5.15) 1 represents the number of iterations and 6ri+, is the displacement increment for each iteration. Total structure displacement response vector at the end of iteration j for the time step i+1 is determined by rj r+ Ar+ (5.16) .i+l = r'*.i+l + rl At the end of each iteration the system is tested for load convergence. The acceptable load tolerance level ER is determined by the following expression: (Aiz ++ AF/1) 5.7. Local and Global Values The computations above involve transformation of matrices and vectors from local element to global structural coordinates, and vice versa. The transformation matrix a is expressed as csO s c 0 0 0 0 1 (5.18) cs O 0 sc 0 0 0 1 where c = cos 6 s = sin 8 9 = angle between global and local xaxes Thus the element stiffness matrix K., given by Eq.(4.35), is transformed to structure matrix K, by = aTea (5.19) The element load vector ( R) due to superimposed uniform load given by Eq. (5.9), and the equivalent prestress nodal action given by Eq. (4.48), is transformed to global load vector R, by the following expression: R = aTRe (5.20) R, = a "e The global nodal displacements given by Eq. (5.16) are transformed to element nodal displacements by re = ars (5.21) 5.8. Computation of Moment Capacity Response The basic moment capacity equation for flexural members is given by Eq. (1.1). The moment equation is rewritten as R= 1 rc(0.85 ffb) 0. 5pf) ra = 1 (5.22) M = ,[rF,(di 0.5 Rr.F)] 2=1 where r, = temperature coefficient of compressive strength of concrete, r, = coefficient for steel stress at ultimate load n = number of steel segments contained in a beam element Within the limits of the heat transmission end point criteria the average loss of compressive strength of concrete in the compression zone is 2% [6]. Therefore the coefficient rc equals 0.98. At fire temperatures the coefficient r, is taken as equal to 0.98 [57]. CHAPTER 6 COMPUTER IMPLEMENTATION 6.1. Introduction The numerical procedures presented in Chapters 2 to 5 have been incorporated in the computer program PRECET (PREstressed Concrete subjected to Elevated Temperatures). This chapter describes the structure of the program. The instructions for preparation of input data file are given in Appendix A. 6.2. Main Program PRECET The program PRECET consists of all the routines for reading and initial processing of input data. As the primary executor PRECET transfers initial controls to data modules and then to executors for temperature analysis, TEDIAN and for response analysis, STREAN. The hierarchical structure of the program PRECET is presented in Fig. 6.1. 6.3. Program TEDIAN TEDIAN, TEmperature DIstribution ANalysis, is the executor for temperature distribution. It incorporates procedures discussed in Chapter 2. In each time step TEDIAN is called first to determine temperature distribution across the beam crosssection. Temperature values for the subelement are saved for use in computing temperaturedependent material properties. The flow chart of the program TEDIAN is shown in Fig. 6.2. 6.4. Program STREAN The program STREAN, STructure REsponse ANalysis is the main executor of response analysis. It incorporates procedures described in Chapters 4 and 5. STREAN is activated at each time step after temperature analysis. It calls the material property routines to compute relevant properties of each subelement based on its temperature. The flow chart of the execution logic of program STREAN is presented graphically in Fig. 6.3. 6.5. Program MAPRAN MAPRAN, MAterial PRoperty ANalysis, consists of several independent modules each of which computes a specific property. The formulation of the routines is based on the material properties and procedures presented in Chapter 3. Thermal property routines are called from TEDIAN while deformation and mechanical property modules are called from program STREAN. Fig. 6.1. Flow Chart of Program PRECET Fig. 6.2. Flow Chart for Temperature Distribution Analysis Program (TEDIAN) Fig. 6.2. Flow Chart for Temperature Distribution Analysis Program (TEDIAN) (cont'd). Fig. 6.3. Flow Chart for Structure Response Program (STREAN) Fig. 6.3. Flow Chart for Structure Response Program (STREAN) (cont'd). CHAPTER 7 NUMERICAL ANALYSIS 7.1. Introduction This chapter presents the results of the application of program PRECET in predicting the structural response of prestressed concrete members subjected to elevated temperatures. The structural configurations analyzed have been subjected to standard fire tests. The results of the numerical model are compared with those obtained from the fire tests. Two sets of problems analyzed are prestressed concrete beams and prestressed concrete slabs. Results presented include temperature distribution, midspan deflection and moment capacity ratio for beams, midspan deflection and moment capacity ratio for slabs. 7.2. Analysis of Prestressed Concrete Beams 7.2.1. Description of Test Specimen The beam configurations used in the analysis are based on the fire tests conducted on eighteen prestressed concrete beams at the Portland Cement Association's Beam Furnace [2]. All the beams had identical crosssection and of the same span of 20 ft. Types of concrete used were normal weight and lightweight aggregates. Three cover thicknesses of 1, 2, and 3 in. were investigated for each concrete type. The specimens were divided in six groups each of three identical beams. The first three groups were of beams made of normal weight, silicious aggregate concrete and the rest were made of lightweight expanded shale concrete. The three groups in each concrete type were categorized in terms of the concrete covers described above. 7.2.2. Beam Testing Fig. 7.1 (a) shows a schematic of the test setup for the beam fire test. During the test steel temperatures at selected locations in concrete were measured at five sections, A, B, C, D and E along the beam length. The beams were loaded with a design service load distributed along the beam length at five locations as shown in Fig. 7.1 (a). Reported test results for each group have been averaged from results of the three beams in each group. In addition temperatures averages were determined from readings along the beam length. Temperature readings from the extreme locations, A and E, were lower than at sections B, C and D. For this reason only temperature values from sections B, C and D were used in computing the averages. 