Structural response of prestressed concrete members subjected to elevated temperatures


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Structural response of prestressed concrete members subjected to elevated temperatures
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vii, 161 leaves : ill. ; 29 cm.
Mbwambo, William J., 1956-
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Thesis (Ph. D.)--University of Florida, 1995.
Includes bibliographical references (leaves 139-143).
Statement of Responsibility:
by William J. Mbwambo.
General Note:
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University of Florida
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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).







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 Response-Experimental 17
1.6.3. Structural Response-Numerical 18


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


3.1. Introduction . 35


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


4.1. 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


5.1. 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


6.1. Introduction . 85
6.2. Main Program PRECET . .. 85
6.3. Program TEDIAN . .. 85
6.4. Program STREAN . .. 85
6.5. Program MAPRAN . .. 85


7.1. 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


8.1. Summary . . 134
8.2. Conclusions . 136
8.3. Recommendations . .. 138






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



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 time-temperature relationship of the

standard fire test (SFT). The procedure computes

temperature distributions, mid-span 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


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

temperature-dependent 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.



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 E-119 [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 mid-span deflection of the member [2,3]. The variables

considered are concrete cover and type of concretes. These

variables affect the temperature distribution within a


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


retained moment capacity is taken as a function of the

temperature-dependent 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


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 temperature-dependent

material properties of concrete and prestressing steel.

2. Use the procedure developed in (1) to evaluate the

degree of influence of temperature-dependent 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


(b) evaluate strength status of a prestressed concrete

member after a prolonged exposure to fire and

(c) show influence of temperature-dependent material

properties of concrete and prestress steel on temperature

distribution and structural behaviors.

1.4. Scope of the Study

The temperature-dependent material properties

considered in this study are

(a) compressive and tensile strength of concrete,

(b) stress-strain 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 time-temperature relationship defined for

the SFT. The imposed service load is assumed to be constant

during exposure to fire.

Mathematical models of temperature-material 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) temperature-dependent structural displacements and

(c) temperature-dependent moment capacity. 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:


(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. 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= str-tr 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 = 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

(b) prestress load functions t

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


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


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 sub-elements. 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 time-temperature 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 sub-element and prestressing steel segments,

of the beam element section. Based on the variation of

temperature across the section and the temperature-material

properties relationship, each concrete sub-element 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 sub-elements 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


(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. 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. 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.
 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


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. 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 40-50% at

750*F, and below 30% of its room temperature value at


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


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.
 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 epoxy-coated

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. 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. Application to this study

The models of temperature-material 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 stress-strain

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 Response-Experimental

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 time-temperature 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 Response-Numerical

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], multi-degree 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 FIRES-T3 [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


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 in-plane

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


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 strength-temperature 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 strength-temperature

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----------------------- ----L-------7----- ___
I- M Mn --

Structural end point, M = M

c ) Moment diagram (after exposure)

Fig. 1.5. Moment capacity of a beam exposed to fire


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 time-temperature 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


x,y = spatial coordinates

T = temperature distribution history

t = time

k = isotropic conductivity (temperature


c = specific heat capacity (temperature


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

1lx + ay + h (T Tf) + hr(Ts T) (2.3)


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)


a = Stefan-Boltzmann constant

= 0.119 10-10 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)


fShc(, ,f) 2dS + (f ,(, ,) 2dS


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


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)


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

+ 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')


(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

Eqs.(2.16) to (2.22) are lumped into a
the following transient heat equation:

matrix form of









[C] (T') + [K] (T) = (R) (2.23)


[K] = [K1] + [K2] + [K3] (conductivity matrix,


[R] = [Rj] + [R2] (thermal load matrix, due to

pseudo fire boundary conditions,


[C] = thermal capacitance matrix, temperature-


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


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


[KC] = 1- [C] tAt + [K] t.A

and the effective thermal load vector, [RC], is


[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) + (xjyk-XkYj) + (xkyi-Xik)] (2.32)

The three shape functions are defined as follows:

N1 = A[(Xjyk-Xkyj) (y -yk)X (Xkj)y] (2.33)

S= [(Xk i Xiyk) + (Yk-i)x + (Xi k)Y (2.34)

1 (2.35)
Nk A= XiY-Xjj) + (Yiyi)x + (x-xy] (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

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

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 i-j denote the end nodes of exposed edge. The length

Lj is defined as

Lij= /(xj-xi)2 + (y-y) (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

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)

The matrix form of Eq.(2.41) is

1= (2.42)
[C]q= 1

The approach produced results with the expected


2.3.4. Thermal Load Vector

The load matrices are evaluated using line integrals.

For an element of unit thickness with two sides, (i-j) and

(j-k), 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 Un1l-1) (i-l)] U) U,,(i-1)
-S L t+At + [K(t+At -t+At ()t+At (2.44)
+ (i-1)
+ -L[Ctat (T)t

where the superscript (i-1) denotes iteration number. The

initial values are expressed as

[,^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(i-i1)
+At TAt < Er (2.46)
llr (i) t l (i-1)
2 +At -t+At

and that of boundary condition by

(i) h (i-1)
-Ir-f-f I !E e (2.47)
S (i) h (i-)l r
2 rr,,t r*;+At


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


3.2. Modeling Procedures

Most properties of concrete and steel are affected by

high temperature. In this study the temperature-dependent

behavior of material property curves is estimated by a


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 I( T-----------------------------

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. 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)


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.



.0.7 Siliceous




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]) 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.



2 Carbonate
Co Siliceous
E 1 A lightweight

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 stress-strain relation, compressive strength and

modulus of rupture. Stress-strain 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

strain-curves. The relationship between stress-strain law

and temperature is shown in Fig. 3.4 for some selected

temperature ranges.

660 0.8
c0.7- T=<100 OF
._ 0.5
)0.4- 1020 S0.3
1560 0.1
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. Stress-strain relationship as
function of temperature
(Source: Ref. [49])

For each temperature range, modulus of elasticity E is

computed for a given range m-n. 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 stress-strain 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 stress-strain relation which include the

post peak behavior.

From reported data [13], aggregate type and strength

have no significant effect on stress-strain relation at high

temperatures. As such the set of stress-strain curves

represents both normal weight and lightweight concretes. Compressive strength

Type of aggregates, degree of loading and conditions of

loading-when hot or cooled are among the important factors

which influence the behavior of concrete strength at

elevated temperatures.

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.


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


The graphic models of the variation of concrete

strength with temperature are shown in Fig. 3.5.


200 400 600 800 1000 1200 14
Temperature (deg.F)

Fig. 3.5. Compressive strength of concrete
versus temperature
(Source: Ref. [5])


The mathematical expression for the compressive strength

ratio is given by the following expression:

CSR(Ti) = CSR(Tm) + T (3.6)


CSR = concrete strength ratio (percentage of original)

The compressive strength is then computed by

FC(Ti) = CSR(Ti) FC(T6,) (3.7) 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 non-mechanical

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 stress-strain relations. 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


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


S (T) A th(T ) (3.9)

Then the incremental and total thermal strain at temperature

Ti is computed from

AE(Tmi) = a, (Ti-T) (3.10)
th(Ti) = eth(Tm) + Ae(Tmi)



) Carbonate
E 0.006-

0.0022 Lightweight

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]) 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)

e.(T) = 0.00051 Ti-68 (3.13)


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 Stress-strain 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.

o 0.7
. 0.6
^ 0.3


Strain (in/in)

Fig. 3.7. Stress-strain curves for PS
at various temperature levels
(Source: Ref. [45])

0.12 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 temperature-strength

relationship for prestressing steel is given in Fig. 3.8.

3.4.2. Deformation Properties Thermal strain

Coefficient of expansion of prestress steel increases

with the increase of temperature. It is expressed in

following form:


a(T) = (6.667 + 0.001235T) (10-6)



: 0.7
0 0.6
o 0.4



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) 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




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-

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:



Z = 8.21(1013) e0'00010

CC = Ze aT


AH/R is given in temperature units (R)

Z = Zener-Hollomon 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.







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 stress-strain 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.


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 temperature-dependent 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 x-axis and v represents both translation

in y-axis and rotation due to flexural action about the

z-axis. These components are shown in Fig. 4.2.

Relationship between the generic displacement u and the

nodal displacements [56] is given by

U = Nr


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


U 2 ( r4 )
-------------------------------A -----4

- U

v 1(r 2

r ) x

v 2 ( r 5

4( e2 r


Fig. 4.2. Axial and flexural elements

Strain-displacement relationship is expressed as


e = du


E = strain vector

d = linear differential operator

By Substituting Eq. (4.2) in Eq. (4.3), the strain equation


e = dNr (4.4)

resulting to the definition of strain-displacement matrix B

B = d(4.5)

The corresponding stress is given by

a = Ee (4.6)
= EBr


E = stress-strain matrix

EB = stress-displacement 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)

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)


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)

Strain-displacement matrix B evaluates to

1 (4.14)
S= [-1 1]i

For the flexural element, displacement relation is given by


The components are represented by

r = {r2 r3, r5, rT ={v1, 01, V2, 2)T (4.16)


dx (4.17)

02 dv2

Based on plain section theorem, translation in x direction

of any point on the cross-section of the element is

determined from

dv (4.18)
u = -y

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)

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

strain-displacement 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).


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


RA = reference axis
for the sect ion
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]


for the axial and

-yBb = -

12x 6L
6xL 4L2
-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





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 stress-strain

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


no, np, = number of concrete subelements and steel

segments respectively within an element,

Ap, = cross-section 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 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

0 0 0 0 0
12 6L 0 -12 6L
4L2 0 -6L 2L2
0 0 0
SYM 12 -6L

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).






















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



Ar 3 r


RA = reference axis

CA = centroid axis

.......- V d stance of CA
-A ----- ---- ------- --*" "

A .r
form PA
I = Iteration number


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
Ar = Ar ycAr6

Using the transformed values the effective incremental

displacements translations and rotation are given by

Eqs. (4.37) and (4.38) respectively.


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)

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
ea-a = Aa + AEb (4.44)

Ae = AEA a-a
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 x-axis of the element are

computed as follows:

AY= ei ei

Lo = yAX2 + AY2 (4.45)

cosa AX

sina A

where XI and XJ are the updated x-coordinates of element

nodes i and j, respectively, and L is the original

element length.

R 4
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


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 stress-strain relationship within different

parts of the section.


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)
(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
(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


Techniques for evaluating the stiffness function K have

been discussed in Chapter 4. The evaluation of the load


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 yA (5.3)
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 stress-strain 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)


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 property-temperature 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


(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 66-E ( T )

A 6/6
6 11 ,1 .

Fig. 5.1. Shift of stress-strain 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)
oi = Oi- + A.i


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


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}
= -{O, 6, L, 0, 6, -L}

where by is the uniform superimposed load acting in the

positive y-direction. 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
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()
(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
t=F0 t_ 0__ t=


8r 2 Sr

S0 t=O t= 0

rO rt r2 r
t=O t-O 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 stress-strain

relation law and

(b) changes in the prestress action.

U I+ 1

R, F


G i

AR 0
1 +

U +1

U 1+1

0r r
+*1 1+1 1+1 r

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+


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


(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'+)

(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)

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

-s c 0 0
0 0 1 (5.18)
cs O
0 -sc 0
0 0 1


c = cos 6

s = sin 8

9 = angle between global and local x-axes

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)]


r, = temperature coefficient of compressive strength of


r, = coefficient for steel stress at ultimate load

n = number of steel segments contained in a beam


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].


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 cross-section.

Temperature values for the subelement are saved for use in

computing temperature-dependent 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


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).


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, mid-span deflection and

moment capacity ratio for beams, mid-span 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 cross-section 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.