IC

NBSIR 76-1032 0

Analysis of Reinforced

Concrete Beams Subjected to

Bruce Ellingwood

James R. Shaver

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

April 1976

Final Report

Prepared for Center for Fire Research

National Bureau of Standards

Washington, D. C. 20234
*s
NBSIR 76-1032

ANALYSIS OF REINFORCED

CONCRETE BEAMS SUBJECTED TO

FIRE

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

April 1976

Final Report

Prepared for Center for Fire Research

National Bureau of Standards

Washington, D. C. 20234

U.S. DEPARTMENT OF COMMERCE, Elliot I, Richardson. SBcntary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Bruce Ellingwood

James R. Shaver

ANALYSIS OF REINFORCED CONCRETE BEAMS SUBJECTED TO FIRE

Bruce Ellingwood

James R. Shaver

Structures, Materials and Safety Division

Center for Building Technology

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

ANALYSIS OF REINFORCED CONCRETE BEAMS SUBJECTED TO FIRE

Bruce Ellingwood

James R. Shaver

Methods for analytically predicting the behavior of simply supported reinforced

concrete beams subjected to fire are presented. This is generally a two-step process

involving a thermal analysis followed by a stress analysis. This study emphasizes the

latter, wherein the determination of moment-curvature-time relationships for the beam

cross section incorporates the temperature-dependent strength degradation in the steel and

concrete as well as thermal and creep strains. The sensitivity of the predictions to

various phases of analytical modeling is investigated to establish the parameters most

important for the prediction of beam behavior and to indicate where additional data should

be gathered. A comparison of predicted behavior with that observed in fire tests shows

excellent agreement when realistic reinforcement temperature histories are used.

Key Words: Creep; fire endurance; fire tests; reinforced concrete; sensitivity analysis;

steel; structural mechanics; uncertainty.

ACKNOWLEDGMENTS

The writers would like to thank Drs. E.V. Leyendecker and C. Culver of the Center

for Building Technology and Messrs. L. Issen and D. Gross of the Center for Fire Research

for their critical review of the manuscript.

iii

TABLE OF CONTENTS

Page

Abstract iii

Acknowledgments iii

Notation vi

1.0 Introduction 1

2.0 Methods of Analysis 2 3.0 Modeling Uncertainties 12

3.1 Creep Model 13

3.2 Analysis of Thermal Strains 16

3.3 Constitutive Relations 17 4.0 Case Studies 17

4.1 T - Beam with Grade 40 Reinforcement 19

4.2 T - Beam with Grade 60 Reinforcement 34

4.3 Prestressed T - Beam 34

5.0 Conclusions and Recommendations 38

References 41

Appendix A: Instructions for Program Usage and Input Data Preparation ... 42

Appendix B: Computer Program Documentation 50

V
NOTATION

E ,E - Young's modulus for steel and concrete, respectively,

s c n, K - strain hardening exponent, strength coefficient.

M,M T ,M - moment; working load and ultimate load moments, respectively,

wl u P - axial thrust

.

T,T..,T ,T, - temperature; temperature in discrete element i j ; initial temperature;

ij o k

temperature in bar k.

Z - Zener-Holloman creep constant.

d, dj^ - depth to reinforcement from top of beam; depth to bar k.

e ,e ,e^ - total strain; total strain at top of beam; imposed strain.

T o I

I f ,f - concrete stress, ultimate concrete strength,

c c

f ,f - steel stress; steel yield stress,

s y

p' - balanced reinforcement ratio.

D

p ,p - reinforcement ratios for T-beams defined by ACI 318-71 [2] . w f

t

_J. time.

x,y - beam cross section coordinates measured from top of beam.

a^,a^ - coefficients of thermal expansion for concrete and steel, respectively.

AH/R - activation energy of creep divided by gas constant.

a ,T ,a - constants for determining temperature dependency of a and a . m m o c s

0 - constant used to determine 0(T) ; g = AH/R.

e, a - strain; stress.

e , ,e , , - thermal strain in concrete, thermal strain in steel, creep strain in

cth sth

e^^.Epg steel, prestressing strain in steel, respectively.

£^ - mechanical concrete strain at top of beam at x . c K

e - constant related to primary creep strain,

to

e e - constants used in the constitutive relations for concrete

o. P and steel, respectively.

6 - temperature-compensated time,

(b - unit rotation (curvature) .

vi
1.0 INTRODUCTION

The provision for sufficient fire resistance and reserve load-carrying capacity for

reinforced concrete structural elements is an important problem in engineering design and

is required by most building codes. Currently, this resistance to fire is determined

primarily on the basis of the performance of an element subjected to the American Society

for Testing and Materials (ASTM E-119) [3] fire test. Although the standard fire test

may not be representative of an actual fire, it is generally recognized as being necessary

to provide a basis for comparison between various designs and to satisfy the need for

reproducibility in test data. The suitability of prospective designs is thus likely to

be judged by their performance in this standard test in the foreseeable future.

Criteria for the fire-resistant design of reinforced or prestressed members are

difficult to develop. Because of the considerable cost in conducting a standard fire test

of even a simple beam, it would generally not be economical to test a sufficiently large

number of specimens to determine experimentally fire endurance for a spectrum of possible

designs. Although it is possible to determine fire resistance with a limited experimental

data base by interpolation, this procedure relies to a great extent on judgment and experience

and, moreover, provides little indication of the sensitivity of member endurance to various

designer-controlled parameters. An alternative is to use thermomechanical models to

predict analytically the behavior of reinforced concrete members subjected to fire. In

this context, the limited data extant would be used to define experimental constants

employed in the models and to update and improve the analysis procedure itself. This

procedure furnishes a logical basis not only for interpreting such test data as is available

but also for extrapolating, within reasonable limits, to situations not covered by the

data base.

This methodology is described in the following sections. The analysis of a reinforced

concrete beam section subjected to a non-linear strain distribution produced by a timedependent temperature history is discussed. A sensitivity analysis of the analytical

Numbers in brackets indicate literature references at the end of the report.

1
models is presented in order to show the effect of inaccuracies in data reduction and to provide some guidance as to where acquisition of additional data would most readily pay

off. A computer program used to perform these analyses is documented in detail, including

the preparation of input data, flow charts of the computational algorithm, and a source listing of the computer code. The methodology described herein is believed to provide a framework for systematically developing fire-resistant design procedures in the long term.

2.0 METHODS OF ANALYSIS

At room temperature, the ultimate capacity of a properly designed reinforced concrete

beam will exceed the sustained or service load moment by a prescribed margin of safety.

During the fire, however, the ultimate capacity will degrade to the point where it is less

than the service load, at which point the beam will fail. Accordingly, the reserve moment

capacity as a function of temperature or time and the period that the beam can sustain

its working load are of particular interest. In this analysis, the moment-curvature-time

relationship for a reinforced and/or prestressed concrete beam section subjected to fire

is developed. The effects of the resulting thermal expansion of the concrete and steel,

creep in steel, and progressive deterioration of the materials at elevated temperatures

are incorporated into the strength calculations. An accompanying finite element thermal

analysis program [15] is used to determine the temperature distribution on the beam cross

section. This thermal analysis is made prior to the strength analysis.

Because of the non-linear nature of the stress-strain equations for concrete and for

steel reinforcement when stressed above the proportional limit, closed-form expressions

for the moment-curvature relations for a reinforced concrete beam section are difficult to obtain. Therefore, an iterative procedure is employed wherein a strain distribution is

assumed on the cross section, the stresses are determined from the appropriate constitutive

relations, and a resultant thrust and moment are computed from these stresses; i.e.

/CT(x,y) dA = P

Jo (x,y) y dA = M (1)

/a(x,y) X dA = 0

If the thrust and moment computed from Eq. 1 equal the applied loads on the beam, the

resulting moment and curvature are recorded; otherwise, the assumed strain distribution is

modified, and the process is repeated until the calculated and applied loads converge.

Although the above computation procedure is easily visualized in a reinforced concrete

beam at room temperature, prolonged exposure to fire induces a nonlinear thermal strain

distribution on the cross section and causes the material strengths to degrade and the

steel (and, to a lesser extent, the concrete) to creep under sustained load conditions.

2
These factors affect the load-deformation characteristics of the beam, as described In the

following

.

Fig. 1 shows the distribution of strains on reference coordinate x of a beam cross

R section which has been exposed to fire. It might be noted that the reference coordinate

X is not necessary in a room temperature analysis because the strain distribution across

R

the beam width is, or is assumed to be, constant. Under elevated temperatures, however,

the strain distribution becomes nonlinear in both the depth and width direction. Under

the assumption that plane sections remain plane after deformation, the total strain e^(y)

is defined by [10],

e^(y) = (2)

in which e is the total strain at the top of the beam and d) is the unit rotation at the

o section (y is measured from the top of the beam). In what follows, extensional strain

will have positive sign.

The total strain is the sum of two components, i.e..

e^(y) = e^(x,y) + e(x,y), (3)

Note that although e^ is a function of y only, e^ and e are functions of x and y because

of the nonlinearity in the temperature distribution across the width and deptn of the

section. The imposed strain component e^(x,y) consists of thermal strains in the concrete,

e , , and steel, e , , and the creep strain in the steel, e (to be discussed in Eqs. 7 and

cth' sth' cr

^

8). The mechanical strain e(x,y) is required to establish equilibrium, subject to the

compatibility requirement expressed by Eq. 2. The stresses on the cross section are

computed from the mechanical strains using the appropriate constitutive relationship.

Eqs. 2 and 3 show this mechanical strain is given by

e(x,y) = e^ + (J)y - e^(x,y)_ (4a)

It may be observed from Eq. 4a that two parameters, e^ and ()) , are sufficient to

describe uniquely the strain distribution on the cross section once the imposed strains

3
FIG. 1 - STRAINS ON REFERENCE COORDINATE \ OF BEAM SECTION

4
are determined. These two parameters may be adjusted using a self-accelerating method

[19] for the iterative solution of equations in order to satisfy equilibrium, Eq. 1.

Rather than to iterate with e^ directly, it is convenient to employ the more conventional

parameter which denotes the mechanical limiting concrete compressive strain at the top

of the beam and at the reference width coordinate x (see Fig. 1). R

To express e^ in terms of e^, note that at (x,y) = (x^^.O) , e(Xj^,0) = and thus

"o =

^c ^I^^R'0> (^b>

where e^CXj^.O) is the imposed (thermal) strain at the top of the beam. In general, then,

the mechanical strain is given by

e (x,y) = + <^y - [e^(x,y) - e^(x^,0)], (5)

It is convenient in performing the strength computations (and, indeed, necessary for

all but the simplest cases) to discretize the beam cross section with a series of small

elements rather than to treat it as a continuum. The forces can then be obtained by

summation. Using this approach, the strain at the centroid of a concrete element furnishes

a stress which is assumed to be constant over that particular element. In particular, the

strain in the concrete element with centroidal coordinates (x^,y_.) is

e^(x.,y.) = + *y. - [e^^^ (x.,y.) - e^^^ (x^,0)]. (6a)

The strain in the steel reinforcing bar with depth d^^ would be

S^V = ^c ^ *\ - f^sth^V + ^cr^V - ^th^^R'">^ •

If the beam is prestressed in addition to or instead of being reinforced.

S^V =

^c + *\ - t^sth^V + ^r^V - ^cth(^'0>J ^s^V

in which e is prestressing strain in steel element k. ps
The thermal strains in the concrete and steel, indicated in Eq. 6, are computed from

the temperature distribution on the cross section. The temperature within each concrete

element is assumed to be constant, and is determined by averaging the nodal temperatures

calculated during the thermal analysis. The steel temperatures may either be computed

from the adjacent nodal temperatures in the surrounding concrete mass or may be read from

cards or tape at each calculation time step.

The thermal strains for concrete and steel, respectively, are

T

e , (x.,y.) = / ij a (T) dT for concrete

(7) T

e ^u(d, ) = / a (T,d, ) dT for steel sth k o s k

where and are coefficients of thermal expansion of concrete and steel, dependent on

temperature T.

The creep strain in the steel is determined using the Harmathy-Dorn theory [9, 12]

along with a time-hardening rule for determining the creep under a variable stress history.

The procedure for computing the creep increment corresponding to a small time interval is

illustrated in Fig. 2. The primary and secondary stages of creep are described by Harmathy's

creep equation

de /de = Z coth^ (e /e ) (8a) cr ^ cr to or, in integrated form.

G /e - tanh e /e = ze/e

cr to cr to to

^^^^

in which Z and e^^ are material-dependent functions of stress, and 9(t) is the temperaturecompensated time [9],

i(t) = exp [ - AH/RT(t)] dr. (9)

Term AH is the activation energy for creep, R is the gas constant, and T(t) is the temperature

as a function of time. The time-hardening rule appears appropriate for studying creep at

elevated temperatures [6, 16], and assumes that the creep strain increment during a given

6
TEMPERATURE-COMPENSATED TIME

FIG. 2 - COMPUTATION OF CREEP STRAIN INCREMENT WITH TIME-HARDENING RULE

f

•c

e o

FIG. 3 - STRESS-STRAIN CURVE FOR CONCRETE

7
time increment at a constant stress is dependent only on the accumulated temperaturecompensated time up to the start of that increment.

Finally, the stresses on the cross section are computed from the strains defined by

Eq . 6, using the appropriate temperature-dependent constitutive relations for the materials.

In the present study, the stress-strain curve for the concrete is given by [8]

^ 2f' 'c=-^-^ ^

(10)

o 1 + (e/e ) o

which is illustrated in Fig, 3, and in which f is the compressive strength and e is a

c o constant. By-'to taking the derivative,' it may be seen that 2f'/e = E , where E is the

c o c c initial tangent modulus of elasticity, and thus f^ = • '^^^ idealized dependence

of f^ and on temperature is illustrated in Fig. 4, where it may be seen that these

properties degrade at elevated temperatures [1, 4, 5].

In similar fashion, either the common elastic-perfectly plastic curve or a strainhardening model similar to the Ramberg-Osgood relation may be used to determine the

stress-strain curve for the steel. The latter relation is given by

f = E e, E < e s s P (11)

f=EE;-K(e-e)'^, e>e s s p p

and is shown in Fig. 5; K, n, and e are experimental constants. Eq . 11 has the

P capability for modeling the "rounded" stress-strain characteristics observed in steel at

elevated temperatures. The modulus E^, and the yield strength f^, are temperature-dependent

[13, 14, 20] as shown in Fig. 6 for grade 40 reinforcement. K, n, and can be computed

knowing f and E . y s In this study, the ultimate moment capacity of the beam was given by the maximum

point on the moment-curvature relationship. The corresponding compressive strain in the

concrete, e^, was typically in the range 0.003 - 0.005 for the under-reinforced beams

considered

.

8
FIG. - NORMAL WEIGHT CONCRETE PROPERTIES AT ELEVATED TE^^PERATURES
t V

U-l

s

FIG. 5 - STRESS-STRAIN CURVE IDEALIZATIONS FOR REINFORCEf^.ENT

T^F)

6 X 10

{

1 1 1500 T(°F)

(6c) Coefficient of Thermal Expansion

FIG. 6 - REINFORCING STEEL PROPERTIES AT ELEVATED TEMPERATURES

10
The accuracy with which the temperature history in the reinforcement can be established

is especially important, since the strength of a lightly reinforced beam depends primarily

on the strength of its reinforcing steel, and both the constitutive relations and predictions

of creep in the steel are affected by the temperatures. In the present algorithm, the

steel temperatures may be either computed or may be read independently at each calculation

time stage. The latter method provides a capability for directly utilizing experimental

data obtained from thermocouples attached to the steel during a fire test in making strength

predictions. Indeed, when the measured temperatures are used in the strength analysis

described above, the resulting predictions of behavior agree quite closely with test

observations

.

The temperature distribution on the beam cross section may be determined from a

finite element thermal analysis in which the appropriate heat boundary conditions are

specified. Since an examination of thermal analyses was considered to be outside the

scope of the present study, the temperature distributions were estimated from an available

two-dimensional thermal analysis program [15] which, for simplicity, treats the beam

section as homogenous, and in which the steel and the heat flow along the member are not

explicitly considered. Temperatures are thus determined at discrete nodal points. The

temperature in a reinforcing bar is then calculated from those at the three nodes adjacent

to it, weighting each nodal temperature by the area coordinate of the bar with respect to

that particular node. Although area coordinates provide a natural way to weight the nodal

temperatures, steel temperatures which are estimated from those in the surrounding concrete

can be as much as 40 percent too high. The reasons for this discrepancy are that the

reinforcement in the actual beam acts as a heat sink and longitudinal heat conductor [5],

and that the moisture in the surrounding concrete condenses around the steel, providing a

layer of insulation around the bar. In large reinforced sections, moreover, the heat sink

caused by the steel would significantly perturb the thermal distribution in the surrounding

concrete mass. Therefore, the available capacity of the beam would be severely underestimated using the above thermal analysis without correction. This will be shown quantitatively

in a subsequent example where it will also be shown that the computed and measured temperatures

will agree if the calculated temperature increments are scaled by an appropriate constant.

However, this scaling factor depends on beam geometry, reinforcing arrangement, and temperature

history, and its empirical nature introduces additional uncertainty.

11
A thermal analysis has recently become available [4,5] that allows sections with

several different materials and thermal properties to be modeled, and which promises to

remove a substantial amount of the uncertainty in steel temperature calculation. However,

the assumption therein that perfect thermal contact is maintained between two adjacent

materials may not hold in the case of prestressed beams where shielded cables are used.

Moreover, although the section is assumed to remain uncracked, the random flexural cracking

which occurs in beams would tend to raise the temperature in the reinforcement. The

importance of accurate steel temperatures in predicting beam behavior and ultimate strength

therefore mandates additional study in this area.

Since the beam section is discretized during the finite element thermal analysis, the

same grid system is employed in the strength analysis in the current study. In general,

however, this would not be necessary. The coarseness of the mesh will decrease the accuracy

of the temperature estimates as well as the calculated stress distribution and moment

capacity. Discretization problems were outside the scope of the present study, and have

not been explored in detail. The selection of an appropriate mesh relies to a great

extent on engineering judgment. A finer mesh should be selected if the thermal or mechanical

strain gradients are expected to be sharp. For example, if the neutral axis falls within

the top two element rows of a discretized beam section, the estimate of the concrete

compressive stress block will be very crude since the stress is assumed to be constant

over any element.

Similarly, the temperature-time history must be discretized for both thermal and

strength calculations. The choice of too large a time step will cause an error in the

estimates of thermal and creep strains, particularly at higher temperatures later in the

test. For the case studies discussed in the sequel, a 10 minute interval has been used

with success . «

3.0 MODELING UNCERTAINTIES

Since the behavior of reinforced concrete beams subjected to fire load is a complex

phenomenon, it is essential to identify potential sources of solution error that may arise

as a result of the mathematical modeling. These sources would include the creep model,

thermal strain analysis, and the stress-strain curve idealization, including the rate of

12
degradation of material properties with increasing temperature. While the models employed

herein are felt to be suitable for this study, the sensitivity analysis must be sufficiently

detailed that the results obtained can be realistically interpreted. Although the ultimate

moment capacity of an under-reinforced simply supported concrete beam may not be particularly

sensitive to errors in creep or thermal strains, this is not true for its curvature

(deflection) at failure. Furthermore, if the beam is partially restrained, these factors

may be important. Therefore, these limitations and sources of uncertainty in the models

and their parameters are discussed in the following paragraphs.

3.1 Creep Model

The creep model, Eq. 8, perhaps constitutes the largest single source of uncertainty

[6,7,12,17] and warrants extended attention. All creep analyses [17] attempt to relate

the creep strain e^^. its time rate de^^/dt, applied stress a, temperature T and time t in

some parametric equation, i.e.,

(e ,de /de,a) = 0 (12a)

1 cr' cr

and when time and temperature are combined into one normalizing parameter G, the temperaturecompensated time of Equation 9, this becomes

F^ (e , de /de,o) = 0 . (12b)

2 cr cr

In addition to the state variables, these functions contain certain empirical constants,

which must be determined experimentally.

The determination of these experimental constants constitutes a significant part of

the uncertainty in the creep analysis. For example, the experimental parameter E^^in

Eq. 8 determines the amount of primary creep that occurs. However, since the test specimens

are loaded and heated to test stress and temperature over a finite time, it may be difficult

to differentiate between instantaneous recoverable inelastic deformation in the specimen,

and primary creep deformation, which is irrecoverable [7,17]. Moreover, it should be

13
expected that e^^ would depend not only on o but on da/dt as well, since the anelastic

strain is dependent on the loading rate. Thus, it should not be too surprising that e^^

has been observed to be a poorly reproducible quantity which exhibits considerable scatter

[12,13]. The parameter Z in Eq . 8 describes the steady-state secondary stage creep rate.

Inasmuch as the most reliable and reproducible test results have been generated for the

steady state portion of the creep curves more confidence may be placed in values of Z

reported in the literature than in values of e . Under certain service conditions,

to however, the material may become structurally unstable, in .which case Z would be a function

of temperature as well as stress. Moreover, the hypothesis of functional similarity

underlying Eq . 8 may not be valid for a broad range of stress in all materials [17].

The temperature-compensated time parameter 6(t) described in Eq . 9 has some theoretical

basis in statistical mechanics. The key parameter in evaluating 0(t) is 6 = AH/R which

is experimentally determined and is commonly assumed to be constant at all temperatures.

Although AH has been shown to be insensitive to material structure for pure metals at

temperatures exceeding one-half the melting point [9] , the situation for alloys is more

complicated [17]. In general, AH is dependent on the structural state of the material and

its prior deformation history; if a phase change in the metal occurs due to elevated

temperature, AH would be expected to change accordingly. It seems apparent that 3 should

actually vary over the range of temperatures encountered during the fire test, and indeed

there is some evidence [17] that d3/dT > 0. Unfortunately, there is insufficient data at

present to estimate this functional relationship.

An appreciation of the numerical effect of variations in the creep parameters on the

predicted creep strain may be gained by examining the total differential of Eq . 8b, i.e.

2 de = (coth e /e - csch e /e ) de cr cr to cr to to

+ coth^e /e (OdZ + Zd9) (13a) cr to

de = C, de + dZ + C^dO (13b) cr 1 to 2 J

•

We might observe that the coefficient is bounded by 0 < < 1, and thus a variation in

e will cause a change of the same magnitude in e . The effect of variations in Z to ^ ^ cr 14
and e is not as clear, as Z ranges from 10 to 10 /hr [13] in Grade AO reinforcement

— 18 while e varies from 0 to 10 hr during a four hour duration of a standard ASTM fire test.

If Z errs by an order of magnitude, then de will err by a similar amount, as shown by

cr ^ ^

the second term in Eq. 13b. In spite of the general predictability of the secondary creep

stage, existing data indicates that this error is a distinct possibility at certain stress

levels [13,18].

Decomposing the third term in Eq . 13b, we observe that if T is essentially constant,

e ~ T 3

For the steels and temperatures of interest in the current study, B ~ 70000 [13] and

T = 1000°F (538°C); thus i/T « 70, and

(14b)

It may be concluded that an error of 10 percent in the estimate of 3 will result in a 700

percent error in the temperature-compensated time increment which, in turn, may cause an

error of an order of magnitude in the estimate of the creep strain increment according to

Eq. 13. In connection with this sensitivity and the temperature dependency noted earlier,

it is important to note that the experimental values of 3 are usually obtained over a

fairly narrow temperature range, say 800°F - 1100°F (427°C - 593°C) , while the temperature

in the steel reinforcement may range over 1000°F (538°C) during a fire test. In some

instances, the uncertainty in 6 may tend to limit the usefulness of 0(t) as a temperaturetime normalizing parameter.

There are additional limitations and restrictions in the use of Eq . 12 itself in

predicting creep behavior under non-steady stress and temperature conditions. For example,

although it is tacitly assumed in developing Eq . 12 that the stress remains constant

during the entire creep test it is actually the load that remains constant. The continual

reduction in area during testing due to accumulations of plastic deformation causes the

stress to increase. This seemingly fine point requires that the usefulness of Eqs. 12

in predicting creep under service loads be restricted to small deformations where creep

strains do not exceed a few percent.

15
When the stress is strongly time-dependent during service, one must assume that

the creep strain increment is controlled by a time-hardening or strain-hardening mechanism

or a combination of the two to predict creep deformation under these conditions [16]

.

This requires rather broad assumptions regarding the temperature-dependent viscoelastic

properties of the material. The strain-hardening mechanism implies that the material is

structurally stable and the creep increment depends on the amount of prior deformation;

satisfactory agreement with experiment has been found under conditions of variable stress

but essentially constant temperature. The time-hardening mechanism is analogous to the

flow of a a nonlinear viscous fluid with time-dependent viscosity, and seems appropriate

at elevated temperatures when the stress levels change only slightly. It should be realized

that in a reinforced concrete beam subjected to fire, the actual state is somewhere in

between these two extremes. In any event, the assumption that the stress and temperature

are constant for a particular creep curve, upon which basis creep increments are calculated

for a variable stress and temperatures, ignores the finite loading and heating times

necessary to conduct the tests from which such curves are derived.

In sum it is apparent that additional experimental creep data is required, especially

with regard to e^^ and 3, before the same degree of confidence can be placed in the creep

analysis as in the remaining portions of the methodology.

3.2 Analysis of Thermal Strains

Thermal strains are analyzed according to Eq. 7 for both concrete and steel. The

principal source of uncertainty arises from the expression for the temperature-dependent

thermal expansion a(T). The scatter in a(T) is considerable for concrete [5], and somewhat

less for steel [20] . Typically, a(T) increases approximately linearly up to some T^

beyond which it is assumed to remain constant;

a (T) = [a + da (T - T ) ] , T < T (15)

o —

= a , T > T m m

where a is typically about 6 x 10~^/°F (3.3 x 10~^/C°) . For concrete, T = 800°F (427°C)

o m

and at this temperature, 8 x 10~^/°F < a < 22 x 10~^/°F (4.4 x 10~^/°C < a < 12.2 x 10~^/°C) m m

1 A
Using Eqs. 7 and 15, this variation in thermal expansion would result in a difference in

computed thermal strains of 0.005 and 0.0106 in those elements having temperatures of

800°F (427°C) and 1200°F (629°C) respectively. For steels, at T = 1500°F (815°C)

,

m

9 X 10~V°F < < 12 X lO'^^F (or 5 x 10"^/°C < < 6.7 x 10~^/°^), implying a difference

0.0013 in computed thermal strain when the reinforcement is at 1200°F (649°C) . However,

the effect of these variations on the ultimate (reserve) moment capacity of the beams

under the fire load considered in this study was found to be slight, since the steel was

observed to have yielded well before the beam capacity was reached.

3.3 Constitutive Relations

The reinforcement stress-strain curve idealization and the modeling of the temperaturedependent reinforcement strength degradation have a significant effect on the predicted

moment capacity of the reinforced concrete beam, as will be shown subsequently.

In particular, an examination of available data on the degradation of yield stress

with temperature [5,13,14,20] revealed that the scatter in reinforcement yield stress

increased with temperature. At 800°F (427°C), for example, yield stresses reported for

intermediate grade steels ranged from 45 percent to 85 percent of their room temperature

values. In turn, this implies that the unpredictability in moment capacity will also

increase with temperature. On the other hand, the form of the concrete stress-strain curve

and the temperature-dependent material properties of the concrete were not found to be

especially influential for the lightly reinforced beams considered in this study, although

they would be expected to be important in reinforced concrete members subjected to large

axial thrusts. Moreover, although the small tension load-carrying capacity of concrete is

included in the numerical analysis, its effect on the load-deformation behavior of the

beams is negligible, except at extremely small loads.

4.0 CASE STUDIES

A lightly reinforced T-beam section fabricated with normal weight concrete, shown in

Fig. 7, was chosen for the case studies herein. A series of simply supported beams spanning 40 ft (12.2 m) with similar cross sections were tested by Portland Cement Association

17
26"

T

:5

4"

3^

(7a) -REINFORCING CONFIGURATION

25"

1

2412

8

I 2 1

4(5) 4'

(7b)- PRESTRESSING CONFIGURATION

FIG. 7 - BEAM CROSS SECTIONS ANALYZED

18
(PCA) using the standard ASTM E-119 [3] fire. In this test series, the grade of reinforcement, concrete strength, and the effect of prestressing compared to reinforcing were

considered. The usefulness of the present analysis will be demonstrated by its ability to

predict the behavior of the above beams under fire test. It might be emphasized that the

effect of end restraint on beam strength was not considered in this study. Restricting

the scope to determining the load-carrying capacity of a cross-section of course limits

the applicability of the analysis to simply supported members, where the section investigated

is that having the maximum applied moment.

4.1 T-Beam with Grade 40 Reinforcement

In the first illustration using the section shown in Fig. 7a, the reinforcement

consists of 8 No. 10 Grade 40 deformed bars with a nominal yield strength of 40 ksi (276

2 MN/m ). The reinforcement ratio of p - p = 0.0086 is considerably less than the maximum

w t allowable [2] value of 0.75p' = 0.055. The corresponding beam tested by PCA [11] sustained

b a test moment of 4770 in-kips (0.539 MN-m) for 310 minutes, at which time the test furnace

control failed; the projected beam endurance was about 6 hours. The test moment was 54

percent of the ultimate beam capacity at room temperature. At the time of test, the

' 2 concrete compressive strength was f^ = 7230 psi (49.9 MN/m ), while the actual steel yield

2 strength was f^ = 46 ksi (317 MN/m )

.

The temperature distribution on the cross section was determined as a function of

time from a thermal analysis provided by Issen [15]. The cross section was discretized in

1-inch (2.54 cm) squares, and the strains and stresses on the cross section, as well as

its load carrying capacity, were calculated at 10 minute time increments, beginning with a

temperature of 70°F (21°C) at time zero, until the ultimate capacity fell below the sustained

load moment. (This discretization was used for all case studies considered herein.)

About 1 1/2 minutes was needed to perform the thermal analysis on an UNIVAC 1108 (Exec 8)

System, followed by 1 1/2 minutes to perform the strength calculations for the entire fire

test of this beam. Beam symmetry allowed consideration of only one-half the cross section;

this is reflected in the value of moment in the strength-time curves which follow.

19
Fig. 8 describes the temperature history in the reinforcement which is computed

directly from the temperatures of the surrounding concrete mass. As might be expected,

the calculated bar temperatures are dependent upon the amount of concrete cover provided,

reaching about 1400°F (760°C) in bar 1 after 300 minutes and about 750°F (399°C) in bar 4

which is better protected. The actual temperatures in the reinforcement at midspan which

were monitored with thermocouples for the duration of this test are shown in Fig. 9

(reproduced from Ref . 11) for comparison. The computed steel temperatures are considerably

higher than the actual values for reasons discussed earlier. After 5 hours of test, for

example, the measured temperatures range from 70 percent to 85 percent of those calculated,

with the lower percentages for the hotter bars. Moreover, the measured temperatures tend

to increase more rapidly than predicted during the early stages of the fire.

To compensate for this discrepency in a reasonably simple manner, an empirical scaling

factor is introduced by which the increments in calculated steel temperatures are multiplied

so that, on the average, the computed and measured values agree. In the present analysis,

this factor is taken as a constant for simplicity. However, it is clear from inspecting

Fig. 8 and 9 that the scaling factor is dependent on the amount of bar cover and elapsed

time and, moreover, would be expected to be influenced by the type of reinforcement and

concrete moisture content as well. - ' The effect of the method selected to determine the steel temperature history on the

estimated beam strength is shown in Fig. 10, where the degradation in ultimate moment

capacity resulting from progressive material deterioration is illustrated. In these and

all subsequent calculations, the coefficient of thermal expansion in Fig. 6c was used to

calculate the thermal strains, while in computing the creep, the following experimental [13]

parameters were substituted in Eqs . 8 and 9:

AH/R = 75000

= 1.7 X 10

-10 1.75

e to

0.026 f 4.7

f < 15000

s s Z = 1.23 X 10

-16

exp [.0003 f ], f > 15000

20
TIME (Hr.)

FIG. 8 - COMPUTED TEMPERATURES IN REINFORCEMENT

(Cross section shown in Fig. 7a)

0

UJ

TIME (Hr.)

FIG. 9 - MEASURED TEMPERATURES IN REINFORCEMENT

(Cross section shown in Fig. 7a)

21
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CD 0)>

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CO^

^ CO g

c 1 (U 1 <u CO Q) 2; g CO 4J ro o ^ «—

(

^ ^ o T3

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•H c

(U U

T) CO .-H

cfl CO

(-4 iH -a

O. CU O 4-1 u CO CO .H

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l-l o w .-tCO

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to CVJ —

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22
Using the measured temperatures (Fig, 9) and an elastoplastic stress-strain curve

with the reported [11] yield stress of = 46 ksi (317 MN/m ) the extrapolated duration

predicted for the beam is about 375 minutes (curve I) which agrees quite closely with the

6 hour duration projected in Ref. 11. When the uncorrected calculated steel temperature

history of Fig. 8 is used instead, but all other parameters remain fixed, the predicted

beam endurance (curve II) becomes about 270 minutes. An error of roughly 40 percent in

estimating temperature thus causes an error of 30 percent in the predicted test duration

for this beam. Clearly, a failure to determine the temperature history in the reinforcement

accurately may limit the usefulness of the analysis in interpreting experimental data.

A notable improvement in predicting beam behavior is obtained when an empirical

scaling factor of 0.75 is applied to the calculated steel temperature, as shown by curve

III in Fig. 10. In spite of some local irregularities, the general agreement between

curves I and III is quite close, indicating that judicious use of the scaling factor can

yield reasonable predictions when the actual temperature data is unavailable or would be

difficult to obtain. Although the factor is 0.75 for this particular beam, additional

studies would be required before this result can be generalized to other beam geometries

and reinforcement arrangements. It might be emphasized, however, that since the error

induced by using the uncorrected (scaling factor of unity) calculated steel temperature

is in the conservative direction, the resulting predictions would still be useful for

purposes of design and for parametric sensitivity studies. In the absence of any experimental

data from which the scaling factor could be deduced, a value of unity should be assigned.

The steel stress-strain curve idealization also has an important effect in the predicted

reserve capacity above working load, M -M . The effect of the elastoplastic and strain

u wl

hardening stress-strain models on predicted strength is compared in Fig. 11 by curves I and

II, where the steel yield stress has been chosen at its nominal value of 40 ksi (276

2 MN/m ) and the uncorrected calculated temperatures in Fig. 8 have been employed. Observe

from curves I and II that while the reserve capacities of the beam prior to failure may

differ considerably, the durations range from 240 to 295 minutes. Considering the uncertainties

involved, this range should not be too surprising. It might also be noted that the elastoplastic

model gives reasonable but conservative estimates, and is much easier to apply in design

and parametric sensitivity studies, since the family of stress-strain-temperature curves

can be completely specified by two rather than four temperature-dependent material constants.

23
o

o

ro O

CVI d

>

O

a

o

o

o

o

o

m

<

o

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(0

LJ UJ > >

O

o

3

OC3 O O

o o

I- t- < <

UJ

Q.P

QlO

h- (O CO < < I o

O

UJ UJ M w g

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(qi - U!) 9-01 X 1N3W0W

24
The slight irregularity in the ultimate moment capacity computed with the strain

hardening stress-strain curve occurring at 150-160 minutes is caused by slight error in

the definition of experimental constants K(T) and n(T) in the vicinity of its slope

transition at 800°F (427°C) . Although this can easily be removed by recalculating the

constant and/or using more points to define n and K vs . T, it provides another illustration

of the sensitivity of the ultimate moment calculations to stress-strain curve modeling.

The coincidence of curves I and III in Fig. 11 at room temperature (t = 0) is fortuitous,

and results from the strain hardening modeled by the reinforcement constitutive relation

which was used to obtain curve I. Thus, while the nominal yield stress is 40 ksi (276 MN/m )

the actual reinforcement stresses at ultimate depend on the reinforcement strains. Here,

the reinforcement ratio was such that these strains were 0.0206 in/in and 0.025 in/in,

2 with corresponding stresses of 45.5 and 46.1 ksi (314 and 318 MN/m ) in the two layers of

reinforcement (see Fig. 7); hence, the equality of curves I and III at t = 0.

The strength of a lightly-reinforced beam is governed primarily by the yield stress of t

reinforcement. The consequence of using a nominal design value for f instead of the actual

y value is illustrated by comparing curves II and III in Fig. 11 in which f =40 ksi (276 MN/m

y

2 and f =46 ksi (317 MN/m ), respectively. It is well known that f exhibits some scatter

y

r J y and that nominal design values are chosen accordingly so that the likelihood of actually

encountering a strength less than nominal is quite small. In statistical terms, if the

2 mean of f^ is 46 ksi (317 MN/m )

, the nominal value might correspond approximately to the

5 percentile value of its probability distribution. This inherent variability would be

expected to limit the degree of test reproducibility observed from a series of nominally

identical beams tested under controlled conditions.

Uncertainty in the yield stress is not the only determinant of observed scatter in

experimental or field data, however. The depth to the centroid of the steel reinforcement,

d, not only influences the moment capacity but also indirectly controls the temperature

elevation in the bars for a beam with fixed geometry, since larger depths would mean less

concrete bar cover. This may be seen in Fig. 12, where the predictions for the PCA [11]

beam (curve III of Fig. 10), which had a minimum concrete cover of 2 1/2 in (6.35 cm), are

compared to those for beams with 5 3/8 in (13.7 cm) and with 1 in (2.54 cm) cover. As

would be expected, the beam with the maximum d (minimum cover) yields the largest margin

of reserve capacity at room temperature. Because of the limited fire protection provided,

23
26
however, the bars also heat more rapidly and to higher temperatures with the consequence

that after 2 1/2 hours its reserve strength is reduced to that of the beam with 2 1/2 in

(6.35 cm) cover. Conversely, the predicted response of the beam with 5 3/8 in (13.7 cm)

cover was most favorable in the long run because of its relatively lower steel temperatures.

While the uncertainty in steel yield stress is in large part unavoidable due to the

inherent variability of the material, the uncertainty in bar placement arises from workmansh

and, to a certain extent, can be controlled by good construction practice. The potential

lack of cover that would result from the tendency for d to exceed its design value due to

construction loads might adversely affect the response of the beam to fire in the long

term.

Families of curves such as those in Figs. 10, 11, and 12 may be used not only for

planning and interpreting experiments but also to construct design aids to be used in

dimensioning fire-resisting structural members. For example, one such requirement might

be that the beam still carry 25 percent in excess of its service live load after 3 hours

of fire; satisfying this criteria would entail the selection of a certain amount of concrete

cover in conjunction with a given reinforcement ratio and yield strength. However, a nondimensional analysis does not appear possible since the thermal analysis is geometrydependent, and on each of a family of cross sections deemed to be representative of most

design cases it will be necessary to first perform a thermal and then a stress analysis

prior to establishing the design aids.

The predictions of moment capacity are also sensitive to the rate of steel strength

degradation. In Fig. 13a, the scatterband for the temperature-dependent degradation in

yield strength is shown with two piecewise-linear curves that might reasonably be chosen

to model this behavior. For illustrative purposes, the uncorrected calculated steel

temperatures were used. The differences in predicted ultimate moment capacity using these

two models along with the elastoplastic stress-strain curve for the steel is apparent from

Fig. 13b, where it is observed that yield strength degradation to 85 percent at 1000°F

(538°C) instead of 75 percent at 800°F (427°C) results in approximately one extra hour of

beam endurance. Figs. 10 through 13 emphasize the need for reasonably precise knowledge

of the relations for the reinforcement if credible analytical predictions of beam behavior

are to be made.

27
0 500 1000 1500 2000

T- TEMPERATURE (*F)

(13a) Temperature-Dependent Degradation in Steel Yield Strength

100 200 300

TIME (Min.)

(13b) Degradation in Moment Capacity of Beam

FIG. 13 - EFFECT OF STEEL STRENGTH DEGRADATION ON BEAM ENDURANCE

28
The behavior of the thermal and creep strains in the steel is described in Fig.

14, using the strain hardening idealization along with the (uncorrected) calculated bar

temperatures. These strains were employed in the analysis used to obtain curve I in

Fig. 11. The selection of a scaling factor of unity serves to accentuate certain effects

of thermal and creep strains in the reinforcement on structural response that will be

discussed. The thermal strains increase quite regularly, in accordance with the temperature

increase in the bars. During the first portion of the test, the creep strains are essentially

zero, but after the bar temperatures exceed approximately 800°F (538°C) the creep strains

exceed 0.02. However, the creep analysis is only valid for small strains and, moreover,

the temperature in bar 1 exceeds 1300°F (705°C) after about 260 minutes of test, implying

that AH/R = 75000 may no longer be appropriate. Therefore, the validity of the creep

calculations beyond this stage is somewhat uncertain. While these factors will not significantly affect the calculation of ultimate strength since the beam is under-reinforced,

they will cause the deflection of the beam at working load to be overestimated. It might be

noted that the creep behavior shown in Fig. 14, where a rapid strain increase follows a

long period of very slow accumulation, has been observed in other experiments (viz. Fig. 2,

Ref . 12)

.

Anomalies in creep strains are clearly reflected in the calculated values of woricing

load stress in the reinforcement; these in turn, determine the creep increments in the

subsequent time interval. Therefore, any error in estimating creep strain or working

load stress will tend to compound with time. The working load stresses in the steel are

clearly nonconstant with time, as shown in Fig. 15. Inasmuch as bar 1 is hottest, its

strength properties degrade most rapidly, its stress at working load decreases and the

additional load is picked up by bars 2, 3, and 4. Later, at about 220 minutes, bar 2 and

3 also become quite hot and began shedding their load, with the net result that bar 4, which

remains relatively cool, is loaded into the strain-hardening range. A slight instability

in the stress-time curves has been observed to occur in a number of instances when the creep

strain in one of the bars exceeds about 0.025, and shows up here beginning at about 240 minutes.

This is typically accompanied by a sudden drop in the mechanical stress in the corresponding

bar. The exact cause for this behavior must be determined from additional studies.

29
30
31
It should be remarked here that when the corrected calculated steel temperatures (or

measured temperatures) are used, the predicted creep behavior is much less dramatic, the

creep strains at failure being on the order of 0.01 - 0.02. Nevertheless, it is necessary

to have some feeling for the consequences of high steel temperatures and creep in terms

of the analysis, since these factors cannot be precluded in all beam configurations that

might be of interest.

The thermal expansion of the concrete was ignored in some previous [11] strength

calculations, where it was argued that since the temperatures in the concrete compressive

zone were much lower than the tension reinforcement, the concrete thermal strains would be

unimportant. The effect of ignoring

^^^^i computing the beam capacity in this case study

is shown in Fig. 16. It is apparent that when the elastoplastic stress-strain idealization

is used, e^j.^ has very little influence on the calculated moment capacity, while the effect

is somewhat more pronounced with the strain hardening model. However, the predicted beam

endurance is the same regardless of whether e , is included or not. Therefore, the

cth

decision on whether to include this parameter should depend on the intent and desired accuracy

of the analysis. In view of the sensitivity of the solution to other sources of uncertainity

,

the benefits derived from including concrete thermal expansion are judged to be marginal,

at least in simply supported beams, provided that reserve capacity and endurance, rather than

beam distortion are the primary factors of interest.

Finally, an example of the variability in creep strain predictions resulting from

uncertainty in the value of 3 = AH/R is shown in Fig. 17 for bar 1, which has the highest

temperature. The concrete thermal expansion was suppressed in these calculations, and

the uncorrected steel temperatures were used for illustrative purposes. It is seen that

decreasing AH/R from 75000 to 70000 causes the creep strain increment between 230 and 240

minutes to increase from 0.0097 to 0.0487. The very large imposed strain which may result

forces the mechanical stress in the corresponding bar to go into compression in order to

maintain strain compatibility. Not only does this affect the load-deformation-time relationsh

for the beam, it also implies that a reversal of inelastic deformation should occur for

the compressively stressed bar during the subsequent time increment. Unfortunately, the

computation of creep strain reversals is still clouded with uncertainty [17], and numerical

results obtained under such circumstances should therefore be viewed with some suspicion.

32
Strain-Hardening Steel Stress-Strain Curve

O THERMAL STRAINS INCLUDED

3 THERMAL STRAINS SUPPRESSED

Elastoplastic Steel Stress-Strain Curve

A THERMAL STRAINS INCLUDED

A THERMAL STRAINS SUPPRESSED

(1 in- lb = 0.113 N-m)

> 100 200 300

TIME (MIn.)

FIG. 16 - EFFECT OF CONCRETE THERMAL EXPANSION ON MOMENT CAPACITY

TIME (Min.)

FIG. 17 - SE*ISITIVITY OF CREP STRAIN IN BAR 1 TO AH/R

33
4.2 T-Beam with Grade 60 Reinforcement

The second case study considers the section shown in Fig. 7a which contains 8 No . 9

2 Grade 60 bars with a nominal yield strength of 60 ksi (413 MN/m ) . The bar locations are

the same as in the previous case study. The corresponding beam tested by PGA [11] sustained

a test moment of 5250 in-kips (0.594 MN-m) for 373 minutes, at which time the beam failed.

' 2 At the time of test, the compressive strength of the concrete was f^ = 5910 psi (41 MN/m ),

2 while the actual yield strength of the reinforcement was f^ = 66 ksi (454 MN/m )

The time-dependent degradation in ultimate moment capacity predicted from the thermomechanica

analysis is shown in Fig. 18. An elastoplastic stress-strain curve for the reinforcement

was used, along with the following experimental creep constants [13] in Eqs. 8 and 9:

AH/R = 65000

e = 1.25 X 10 ^ f to s

Z = 267.7 f ^'^^

, f < 15000

s s —

3.69 X

10""""^ exp (0.00022f^) , f^ > 15000

The use of the uncorrected calculated reinforcement temperatures has approximately the

same effect on the calculated capacity and duration that was observed with the Grade 40

reinforced beam (cf. Figs. 10 and 18). A divergence between the two solutions occurs at

about 150 minutes, and the predicted endurance is about 100 minutes less for the beam in

which the uncorrected calculated steel temperatures were employed, being 245 minutes

instead of the 355 minutes obtained when measured temperatures were used. This again

demonstrates the ability of the analysis to accurately predict beam response, provided

that an accurate reinforcement temperature history is available, and the need for such

data if credible results are to be obtained.

4.3 Prestressed T-Beam

A prestressed beam was chosen for the final case study. The section geometry and

pre-stressing arrangement are shown in Fig. 7b. The corresponding beam tested by PGA [11]

34
- 4

u>

IO

z

UJ

Elastoplastic Steel Stress-Strain Curve

(1 in- lb = 0.113 N-m)

EXPERIMENTAL TIME

TO FAILURE

I —

A MEASURED STEEL TEMPERATURES

O CALCULATED STEEL TEMPERATURES (UNCORRECTED)

100 200 300

TIME (Min.)

400

FIG. 18 - PREDICTED BEAM CAPACITY WITH GRADE 60 REINFORCEMENT

35
contained 16 tendons of cold-drawn 1/2 in (1.27 cm) diameter 7-wire strand, with a nominal

2 2 0.2% yield stress of 225 ksi (1550 MN/m ), which were pretensioned to 175 ksi (1206 MN/m )

prior to placing the concrete. The corresponding prestrain (Eq . 6c) is e = 0.0064. The

ps

' 2 concrete compressive strength at test was f^ = 5940 psi (41 MN/m ) while the strand yield

2 stress was 236 ksi (1625 MN/m ) . The observed time to failure was 237 minutes witn a

service load moment of 5030 in-kips (0.569 MN-m)

,

The strain hardening model was chosen for the stress-strain relationship for the

strand because of its observed rounded nature. The values of K and n at room temperature

2 were K = 39743000 psi (274027 MN/m ) and n = 1.113. The yield stress degrades at a somewhat

more rapid rate at elevated temperatures for the ultra high strength steels used in strand

[13] than for ordinary steels. Rather than to use the degradation model shown in Fig.

6a, therefore, the yield stress ratio was set at 0.75 at 600°F (316°C) and 0.25 at 1000°F

(538°C). The following experimental creep constants were used [13]:

AH/R = 55000

_ „ 0.67

= 3.3 X 10 f to s

64 f^ , f < 25000

s s ^= 13 8.21 X 10 exp [0.0001 f ], f ^ 25000

The predicted time-dependent degradation in ultimate moment capacity for this prestressed beam is shown in Fig. 19. The temperature history in the steel measured during

the corresponding test [11] was used in performing the calculations. The agreement between

predicted and observed endurance again appears to be reasonable with the difference being

easily attributed to the uncertainties in material response and modeling discussed in

detail in connection with the first case study. The rather precipitous drop in capacity

that occurs at 200 minutes is caused by an acceleration in the accumulated creep strains

in tendons 1, 4, and 7 (see Fig. 7b), which have the least amount of cover and whose

temperatures approach 900°F (483°C) after 3 hours of test. Similar behavior has been

noted earlier, cf. Fig. 14 and curve I in Fig. 11. In the present case study, the predicted

creep strains in the hotter bars exceeded 0,03 at 190 minutes; the limit of applicability

of the small deformation theory, therefore, is clearly being approached.

36
TO FAILURE

100 200

TIME (Min.)

300

FIG. 19 - PREDICTED MOMENT CAPACITY OF PRESTRESSED BEAM

37
5.0 CONCLUSIONS AND RECOMMENDATIONS

In this study, methods for analytically predicting the behavior of simply supported

reinforced concrete beams subjected to fire have been discussed. The stress analysis of a

beam cross section was considered, incorporating the temperature-dependent strength

degradation of the steel and concrete, as well as the thermal and creep strains. The main

objectives of the analysis were to determine (1) the reserve strength of the beam above

working load as a function of elapsed time, and (2) the time it could sustain that load

under fire. These predictions gave reasonable agreement with experimental data [11],

provided that the material properties and temperature history in the reinforcement were

accurately defined.

It should be apparent that the ability of any analytical model to predict structural

behavior is dependent not only on the model itself but also on the accuracy with which its

parameters can be defined. The sensitivity analysis performed herein revealed that the

primary factors affecting the predicted behavior are the uncertainties in the various

temperature-dependent material parameters and in the calculation of bar temperatures.

Therefore, if accuracy in analytically reproducing the results of a fire test is of particul

interest, it is essential that the structural parameters in the analysis be the same as

those in the experiment. Such reproducibility may not always be possible because of the

inability to estimate some quantities accurately and the inherent randomness in others.

In many cases, particularly in developing design standards, it would be prudent from

a safety viewpoint to select conservative values for the parameters (thus, for example,

although the average yield strengths of intermediate grade reinforcement may be 46 ksi

2 2 (317 MN/m ) a value of 40 ksi (276 MN/Mm ) is used in design) . Naturally, such assumptions

lead to a conservative estimate of the fire endurance for the beam. The question of how

conservative to choose such design parameters may best be answered using a statistical

methodology in which the various factors contributing to possible beam unreliability can

be analyzed systematically.

38
Although the findings in this study are based on limited data and thus must be

considered as preliminary in nature, the following specific conclusions and recommendations

can be presented:

1. The single most important factor affecting the predicted beam behavior is the

estimated (or calculated) temperature history in the steel reinforcement or

prestressing. The computation of these temperatures directly from those in the

surLuuiiding concrete yields a very conservative result in terms of predicted

beam endurance.

2. The temperature-dependent material properties of the reinforcement are

significant in establishing the reserve capacity of the beam as a function

of time and in predicting beam endurance. Concrete properties do not appear to

be especially important, however, provided that the section does not carry axial

load. In fact, the exclusion of concrete thermal expansion seemed to have

little effect insofar as fire endurance calculations were concerned.

3. The Inherent (random) variability of the reinforcement strength and

uncertainty in placing the reinforcement arising from careless workmanship would

tend to limit the reproducibility of beam fire tests and the predictability of

member behavior in service.

4. The creep model is the weak link in the analysis. This is not because of its

concept, which is as easily justified and defended as any available alternative,

but because of the uncertainty in the definition of the model parameters and the

sensitivity of the creep predictions to them. One must bear in mind the limitations

of the creep model ^nd the conditions under which its parameters were derived.

39
The elastoplastic curve appears to be appropriate and sufficient for modeling

the stress-strain relation for reinforcing steel. The reinforcement strength

properties at all temperatures can thus be completely specified with two temperatur

dependent material constants. There appears to be little advantage in using the

strain hardening model with its two additional required constants, unless strain

hardening commences immediately upon yielding.

It appears feasible to use this analysis for developing criteria for the design

of fire-resistant beams, provided that reasonable temperature and material

property estimates can be obtained. Families of curves derived from analyses

similar to those illustrated in Figures 10, 11, 12, 18 and 19, could be employed

to determine the effect of such factors as bar cover and reinforcement ratio on

fire resistance. Since the present analysis tacitly assumes that the parameters

are s ingle-valued , a thorough analysis of modeling and statistical uncertainties

should be performed before specific design aids can be advanced.

40
REFERENCES

^' Abraras, M.S., "Compression Strength of Concrete at Temperatures to 1600°F," in

Temperature and Concrete , American Concrete Institute SP 25, 1971.

2. ACI Committee 318-71, Building Code Requirements for Reinforced Concrete, American

Concrete Institute, Detroit, Michigan, 1971.

3. ASTM Designation: E 119-67, Standards Methods of Fire Tests of Building Construction

and Materials, American Society for Testing and Materials, Philadelphia, Pennsylvania

1967 .

4. Becker, J., et . al. , "Reinforced Concrete Frames in Fire Environments," preprint

2250, ASCE National Structural Engineering Meeting, Cincinnati, Ohio, 1974.

5. Bizri, H., "Structural Capacity of Reinforced Concrete Columns Subjected to Fire

Induced Thermal Gradients," Report No, UC-SESM-73-1, University of California,

Berkeley, 1973.

6. Boresi, A. P., Sidebottom, O.M., "Creep of Metals Under Multiaxial States of

Stress," Nuclear Engineering and Design , Vol. 18, September 1972.

7. Coffin, L.F., Goldhoff, R.M., "Prediction Testing in Elevated Temperature

Fatigue and Creep: Status and Problesm," in Testing for Prediction of Material

Performance in Structures and Components , ASTM STP 515, 1972.

8. Desayi, P., Krishnan, S., "Equation for the Stress-Strain Curve of Concrete,"

ACI Journal , Proc. Vol. 61, No. 3, March 1964.

9. Dorn, J.E., "Some Fundamental Experiments on High Temperatures Creep," Journal

Mech. and Physics of Solids , Vol. 3, 1954.

10. Gatewood, B., Thermal Stresses , McGraw-Hill , NY, 1957.

11. Gustaferro, A.H., et . al . , "Fire Resistance of Prestressed Concrete Beams, Study C: Structural Behavior During Fire Tests" RD009.01B, Portland Cement Association, 1971.

12. Harmathy, T.Z., "A Comprehensive Creep Model," Journal of Basic Engineering ,ASME

,

September 1967.

13. Harmathy, T.Z., Stanzak, W. , "Elevated-Temperature Tensile and Creep Properties of

Some Structural and Prestressing Steels" in Fire Test Performance , ASTM STP 464, 1970.

14. Holt, J.M., "Short-Time Elevated Temperature Properties of USS Cor-Ten, USS Man-Ten,

USS Tri-Ten and ASTM A36 Steel," USS. Steel Applied Research Laboratory Project

57.19-AOl (1) Report, 1963.

15. Issen, L.A., "Analytic Model for Calculating Fire Resistance of Simply Supported

Prestressed and Reinforced Concrete Beams," NBS Special Publication 411, Fire Safety

Research, Nov. 1974.

16. Nickell, R.E., "Thermal Stress and Creep," in Structural Mechanics Computer Programs , ed. by Pilkey, ed. al.. University Press of Virginia, 1974.

17. Rabotnov, Yu. N. , Creep Problems in Structural Members , John Wiley, NY, 1969.

18. Thor, J., "Deformations and Critical Loads of Steel Beams Under Fire Exposure Conditions,"

National Swedish Building Research Document D 16:1973.

19. Traub, J.F., Iterative Methods for the Solution of Equations , Prentice-Hall, Inc., 1964.

20. Uddin, T., Culver, C, "Effects of Elevated Temperature on Structural Members," Journal

of Structural Division , ASCE. Vol. 101, No. ST7 , July, 1975.

USCOMM NBS DC 41
APPENDIX A

li'lSTRUCTIOiMS FOR PROGRAM USAGE AND INPUT DATA PREPARATION

This appendix contains the specific instructions for preparation of the card input needed

to use this program. The required data is broken into (4) major blocks:

' I. Beam description and analysis control parameters;

II. Cross-section geometry;

III. Material property parameters;

IV. Time and/or temperature increments and output requests.

These input blocks are read by the program in the sequence given. It should be noted at the

outset that the program was written in FORTRAN V and executed on a UNIVAC 1108 (Exec 8) system.

A.l Beam Description and Analysis Control Parameters

The input block containing beam description and analysis control parameters requires

input for the following four (4) subgroups.

A) Beam description: (2 cards required)

Each card may contain up to 78 alphanumeric characters used to describe the

beam being analyzed in Cols. 1-78 on each card.

B) Control Words: (1 card required)

Two alphanumeric words containing six characters each are located in columns

1-13 with a space between each word in column 7.

1) Control Word 1: (Cols. 1-6)

' REINFO - means that the beam contains reinforcing steel;

PRESTR - means that the beam contains prestressed steel;

REPRST - means that the beam contains both reinforcing and prestressed steel.

2) Control Word 2: (Cols. 8-13)

RAMOSG - means that the steel stresses are to be computed using the RambergOsgood idealization

FLATT - means that the steel stresses are to be computed using the elastoplastic idealization

C) Working Load Moment: (1 card required)

The working load moment in (in-lb) is given as a floating point number in columns

1 to 10.

42
D) Convergence Criteria: (1 card required)

Two numeric words are required and are located in columns 1-20.

1) Criteria Word 1: (Cols, 1-10)

An integer which specifies the maximum number of iterations allowed for

convergence. The minimum value is recommended as 20 since the initial

iteration cycle is generally long.

2) Criteria Word 2: (Cols. 11-20)

A floating point number which specifies the tolerance within which the

resultant axial thrust must fall for convergence. One (1.0) pound has been

used with usually less than four cycles required for convergence within

this tolerence.

A. 2 Cross-Section Geometry

The input block for cross-section geometry contains two subgroups. The cross-section

discretization used by this program is the one generated and used by the finite element

thermal analysis in determining the temperature distribution on the cross-section. This

discretization is made available to the program via an output tape from the thermal analysis '

which also contains the temperature distribution on the cross-section at given time increments.

In order to properly read the grid from the output tape two (2) parameters are required.

A) Cross-Section Geometry: (1 card required)

This card contains four (4) integers located in columns 1-40 which specify the

cross-section geometry parameters.

1) Control Word 1: (Cols. 1-10)

This integer specifies the number of reinforcing and/or prestressed steel

elements in the cross-section.

2) Control Word 2: (Cols. 11-20)

An integer which is the minimum row number for the grid input to the thermal

analysis

.

3) Control Word 3: (Cols. 21-30)

An integer which is the maximum row number for the grid input to the thermal

analysis.

43
4) Control Word 4: (Cols. 31-40)

An integer which specifies the reference line Xd for computation of the

R imposed (thermal) strain. As an example, in the case of a symmetric crosssection this reference line could be the axis of symmetry and the reference

integer would then be the column number used by the thermal analysis to specify

the grid line which lies along the axis of symmetry,

B) Steel Geometry: (1 card required for each reinforcing bar and/or prestressed

strand)

Each bar or strand card contains three (3) floating point numbers in columns 1 to

30 with an integer in columns 31-40. The first number in Cols. 1-10 is the

x-coordinate in inches for the bar with reference to the coordinate axis used to

input the grid to the thermal analysis. The second number located in columns 11

to 20 specifies the y-coordinate in inches and the third floating point number in

Cols. 21-30 gives the area of the bar in square inches. The integer located in

Cols. 31-40 specifies whether the steel is a reinforcing bar or a prestressing

strand. A zero (0) is input for a reinforcing bar and a one (1) for prestressed

steel.

' A. 3 Material Property Parameters

The input block for the material property parameters is broken into two (2) subgroups

with one for the concrete material properties and the other for the properties of the

reinforcing and/or prestressed steel. Each subgroup in turn has two (2) parts: (1)

Input for the material properties at the base (room) temperature and (2) Input for the

degradation or change in material properties as a function of temperature. A third data

group for the creep properties of the steel and prestressing strains is also required.

The degradation or change in all material properties as a function of temperature for

both steel and concrete is modeled by a piece-wise linear curve in the program. Input

data required by the program for handling the material properties in this manner are the

total number of end points for linear segments plus the ordinate and abcissa values for each

end point. Fig. 4a exhibits a typical piece-wise linear curve which in this case shows the

degradation of concrete strength with temperature. This curve is comprised of three (3)

segments and requires information at four (4) end points for its complete description.

44
In the event that a material property is required at a temperature which is greater

than the maximum temperature value input, an error message is written and the value at the

maximum temperature is used.

In addition, seven constants are required to compute the stress-dependent primary

creep parameter, £j.q> and the secondary creep rate, Z, which appear in Eq . 8. Specifically,

^to = ^1 (^s)

Z = C3 iff\ <

Z = exp (C^f ) , f > 5 6 s s 7

The expression for Z depends on whether the steel stress exceeds a transition stress C^.

Finally, the computation of temperature-compensated time Eq , 9 requires definition of

C„ = AH/R

o

A) Concrete Material Properties:

1) Base Temperature Properties: (1 card required)

This card contains three (3) floating point numbers in Cols. 1-10, 11-20 and

21-30 which are, respectively, the maximum compressive strength for the

concrete at the base temperature, its modulus of elasticity, and the constant

0.85, which is defined in ACI 318-71, Sect. 10.2.7 [2],

2) Degradation or change in material properties with temperature. The following

required sequence of input is used for concrete:

a) Compressive Strength

b) Tensile Strength

c) Modulus of Elasticity

d) Thermal Expansion of Concrete

45
Each input consists of one or more cards with the following format:

Card No. 1 (required): Contains an integer in columns 1 to 10 which gives

the number of end points required to specify the piece-wise linear curve.

The x-y coordinates for the first three end points are placed on the card

beginning in column 21. Floating point numbers are used to specify the x and

y coordinates of the end points in that order. Each coordinate is contained

in a field of ten (10) columns.

Card Nos. 2, 3, etc. (if necessary): These cards contain the x-y coordinates

for the remaining end points. Again, floating point numbers in fields of

ten (10) columns beginning in column 1 are used to specify the x and y

coordinates, respectively. If the y coordinate for the final point on the

curve is in the field given by columns 71-80 on any card, then a blank card

must follow before the next input group.

B) Reinforcing and/or Prestressed Steel Properties

1) Base Temperature Properties: (1 card required)

This card contains two (2) floating point numbers in Cols. 1-10 and 11-20

which are respectively, the yield strength of the steel and the modulus of

elasticity for the steel at the base temperature.

2) Degradation or change of material properties with temperature. The following

required sequence of input is used for steel:

a) Yield strength

b) Strain hardening exponent for strain hardening constitutive model

c) Strength coefficient for strain-hardening constitutive model

d) Modulus of Elasticity

e) Coefficient of Thermal Expansion for Steel

The input format for each material property change is the same as that given in the

concrete segment. The information for the strain-hardening constitutive model is not

required if the elasto-plastic idealization is used to model the steel stress-strain

relationship

.

46
3) Creep Properties and Prestressing Strains

a) Creep Properties (2 cards required)

The eight (8) coefficients are placed, four (4) to a card, in fields on

10 columns each using the Fortran E format. The input order of these

coefficients is the same as that given by the subscripts attached to the

coefficients presented in the introduction to this section.

b) Prestressing Strains

Prestressing strains, when required, are input in the same order as used

for the steel coordinates. They are input as floating point numbers in

fields of 10 columns with eight (8) strains to a card. In the case

where both reinforced and prestressed steel are present in the beam, zero

(0) strain must be input for the reinforcement bars. This information is

not required for a beam containing only reinforcing steel.

A. 4 Time and/or Temperature Increments and Output Requests

The input block for the time and/or temperature increments and output requires data

for three (3) subgroups.

Four types of output are available during each computation at a specified time. As a

minimum, the elapsed time from the beginning computation and the ultimate moment capacity

for the cross-section at that time increment are always printed. The other three types

of output are optional and returned only during the computation at a given time if an

integer one (1) is specified in the appropriate input field. If the optional output is

not wanted then an integer (0) is input on the card. The following is a description of

the optional output.

Output A) The steel temperatures and steel material properties at that temperature

including the accumulated creep of the steel from time zero.

Output B) The assumed top concrete strain at the reference line, the curvature which

satisfies equilibrium and compatibility for the assumed strain, the moment

capacity for that strain and the number of iterations needed to satisfy the

convergence criterion are given for each strain. In addition, the stress and

strains in the steel are given at working load and ultimate moment.

47
Output C) This includes all of the information from Output B plus the total stress

and strain in the concrete and steel and the mechanical stress and strain

in the concrete and steel for each assumed top concrete strain. It should

be noted that a request for Output C gives Output B and thus Output B

should not be requested on the input card.

A) Temperature and time parameters: (one card required)

This card contains: (1) a floating point number in columns 1-10 which gives

the base (room) temperature for the initial calculation, (2) a six character

alphanumeric word in columns 15-20 which specifies whether the steel

temperatures are calculated from the temperature of the surrounding concrete

(word must be CALCST) or read from cards (word must be READST) , (3) a floating

point number in columns 21-30 which allows for adjustment of the steel

temperatures when they are determined from the temperature of the surrounding

concrete and (4) an integer in columns 31-40 which specifies the number of

time increments starting with the base temperature for which the moment-curvature

of the beam cross-section is to be determined.

B) Time and Output Requests: (one card required for each time a computation is wantec

Each card contains a floating point number in columns 1-10 which specifies the

time of the calculation and three (3) integers in columns 11-20, 21-30 and 31-

40, which specify the optional output (A, B, or C, respectively) wanted with

each time calculation.

C) Steel Temperature: (one or more cards is required for each time a computation

is wanted if READST is input)

Each card contains a maximum of eight (8) bar or strand temperatures as floating

point numbers in fields of ten (10) columns beginning with column 1. If there

are less than 8 bars or strands than only one (1) card is needed to input the

steel temperatures at each time increment

.

The following is a sample of the input data that was used to obtain curve I in Fig. 10 of

the main body of the report.

48
2ERM G4 - 4CiY PEItiFDRCEnENT - UriIT:;= IN- HIH - HFGREE:]: -F> PEINFD FLRTT

233501:10.

2 0 1. 4 1 26 14

-•> . X C -> X . iJ . 1.1

J. J. . t 3.125 1.270

. J. Lit! . J . J X . l: p U

"> l"! -1 "j"' fl M fl I'l

1 J v 1.1 'J . . J

* 1 U. X . U • J IJ IJ ^ X . 0 X U U . IJ t \ "> < Ti i"i 'J . d. J 1 iJ

.

X . L>

* ~ f1 f1 X c U U

.

. C. -I

~^ ^ f1 f1 d. X LI LI

.

J -7

fl i5l_linrv .ilKi.'

1 A 1 n f1 "•' f1 fl •'1 'J b J'

3

fl ! fl fl f1 iJ J U U -J 0 0

I'l fl J iJ . fl f1 f1 f1 f1 . J L' -J L'C U

' fl fl LI X >J U .

t*i Ti fl I'l "*V Ti « IJ IJ JJ 'J i_ '

J

A

-' M fl t"l 4o U U 'J . —

*• "j Ti Ti fl Ti ri M * U IJ 'J IJ 1.1 '.1

.

4 7 0. 1 . 0 3 0 0

.

.75 1 \ 0 0

.

.20

d X U !J . •J '

J

4 . 0. X . Ul:. t.l

.

J LI -J . 1 . 01 3 1 2 0 0

.

X . IJLI5li 3

210 0. 1.0

4 7 0 . 03 03 00 0. 3 00. 2559100 0. 1 2 0 0

.

1 333 0 0 0 0

.

21 00. 0 . 0 1. 7E - 1 0 1 . 75E 0 2.3E -02 4 . "^E 0 1. 23E13 3.0E-04 1 . 5E 04 7. 5E04

3 7 0. BLRNF CnRIi

1. 0 30 0. . 35 210 0. 0. 0

3 7 0. 0 0 0 0 03 150 0. . ririfinri'it 21 00. . 00 0012

92.95

0. 10.

20.

30.

40.

50.

3 0

.

BL^^NK CRRIi

RERDST 1.0 39

Additional time and output request data,

as indicated by the above.

150.2 121.3 107.2 92.95

223.3 135.9 171.6 107.2

273.3 257.4 223.3 123.7

233.0 234.3 257.4 150.2

271.7 257.4 257.4 173.3

234.3 257.4 250.2 214.5

Additional bar temperature data corresponding

to each time step, as indicated by the above.

END OF

49
APPENDIX B

COMPUTER PROGRAM DOCUMENTATION

This appendix contains the source listing for the computer program used to perform the

analysis of a reinforced concrete beam section subjected to a non-linear temperature distri

bution described in Section 2 of this report. In addition to the source listings for the

main program and subroutines, flow diagrams are given for the main program, which controls

the main input and sequencing of calculations, and subroutine MOMCUR, which controls the

computation of the moment-curvature relationship for a given temperature distribution. The

program which is written in FORTRAN V is restricted in use since it requires input from a

thermal analysis of the cross-section under investigation. Those source statements in the

main program which require information from the thermal analysis are located at line 44

through line 50 and line 138. These statements would have to be modified in order to accep

information (cross-section discretization and nodal point temperatures) from a thermal

analysis different from the one used in this study [15]

.

50
MAIN PROGRAM

Controls Input and General Sequence of Calculations

\ Title & Options /

I \ Wor king Load Moment M^ /

Iteration Control Parameters

Geometry of Cross-section

1. Input Control Parameters

2. Steel Coordinates

3. Concrete Element Coordinates

from thermal analysis AMGJ65

Compute Element Location

area, depth to centroid,

etc

.

(SECTN)

C oncrete Material Properties

Steel Material Properties,

prestressing included,

if required

Temperature Control Parameters

and Output Requests

Initialize Calculation

51
Set Output Requests

Initialize Temperature-dependent Material

Properties, Determine Increments of

Thermal Expansion and Creep.

(INITAL)

Moment-Curvature Relationship

Steel Stress at Working Load

Ultimate Moment Capacity of Beam

(MOMCUR)

52
C MlilN PPDi5P«M CDNTPOLS IliPUTt CRLCULmTIDN SEQEHCE HMD OUTPUT

c

r I mem: I DM TITLE ' c'6> . TCDDE < ?li.50> . TIMEP < 1 00* . IMDUT U 00»4>

COMMDM bcDM^JPl=(Mi:-.E. IMIM'^O) » IMRX<50:» > JM I M. JM8X . MEQ <50> >

1 XX> JU'^O) •YYt.p.O.^O^ 'Xi (clO^ .Vnv50-> » BPYC 'i20»3* »MLS<c:0.4> >YDEP

CDMMDM i:TL HSSi. SrEPT(20^ .STEMP<£0'^ . STEMl - aO ' .MSMmT <£0> jFYcSO) »

1 FiK <c'0> .RM(£0> .RnorS (20> .RLPHhS <:c0> « EPST (^O .EPSPST <20> » SITPM' c'O) »TSTS (aO > » SHPEt=t<£0^ » STRESS .EPSW' c'0> .FSW<:20) » 3 EPSULT <20> .FSIJLT<20>

CDMriDM • CDMC- CDEPT (£9. 49> . CRPER < 29 . 49"^ . CTEMP (£9? 49> >CTEM1 ';29»49>»

1 FPC ';£9. 49) » FPT 'ri-9«49 ) -.RMOIiC (c,"?? 49^ t RLPHRC <£9' 49) .CEPT <£9» 49> » £ hLDTCT'JI STPM •:£9.49) . TSTC • £9. 49 > . CSTPES < £9 • 49')

CDMMDM' CPpvCPEEP (£0) . TCMPT ';£0:> » Z <£0> • EPTD • £0:) f EPSCl <:20> » 1EPSC£ '; £0> • CZ <8» £>

i: MMOh ' MRT I N •• FC • EC » NFC . PFCT U 0 . £> . MF T . PFT < 1 0 . £^ < MEC » RECT (1 0 f 2> >

1 MRLC « RLPHC 1 0 . £> < FYS > ES - MYS ? PYS •: 1 0 • £ > • MRM . RMT < 1 0 2> > £ MK? RKT • 1 0» £> » FYSPS » MYSP - PYSPS <in,£.;. , MRMPS > RMPST < 1 0. 2> » 3 MK PS . RKPST a 0 » £> » MES » FES a 0 . £> MRLS - RLPHS ( 1 0 » £>

CDMMDH '- TTEMP.' T I ME ? T I MP . TEMP - S 0 . 5 0> - TEMP I M . TFCTR

CDMMQM.' IMIiEX.'MT IMC. IPEF » TDL . M I TER » S I HERL . STCRL » IEPSCj ISTRSR* ICDMV

DQuBLE PRECISIOM TCMPT

REMIMD 10

C

C IMPUT FROGPRM comtrdl prprmeters

c PERU ( 5 » I £5 Ci> a I TLE < I .) » I = I » 26:)

10 URITE <t.'l2^0> aiTLE ;!:). 1 = 1. £6>

C REnli Ui FP.UGF.nr. nPTIDMS PS HLPHRMUMERIC DRT<i

PERD f. > 1 £5 1 ) EMTYPE . S I HERL

PERD ' 5. 1 194!) EMWL

MPITE <.6f i:30u> BMWL

1 :- 0 0 FDRMRT •:; 1 X .

' WORK I m LORL MDMEMT = ' , E 1 0 . 5 »

' I M-LB ' > RERIKSj l£2i:) MITER» TDL

C

C IMPUT BERM GEOMETRY RESCRIPT I DM

C

RERri<:5. 1£52") MSS. JMIM» JMRX? IREF» IGEOM

JPRMbE = JMRX - JMIM

REftD ' 5. 1 19Ci;» <XS <.I> » Y3 '• I> J SRRER <I> » 1 = 1 » MSS>

'.'PITE 130i;)

1 3 0 1 FDRMRT 1 OX . STEEL CDDRD I MRTES / 1 3X > XS '

. 1 3X » YS ' > 1 OX »

" RRER ' > I.IPITE'.6» 13 02 ) <XSa:) »YS<I> > SRRER < I!) » I = 1»MSS>

13 0£ FDRMRT >: 3E1 5. 4>

C PERU Ih THE MDDRL PT.DRTR FRDM RMGT65

JPRM= JMRX- JMIM+1

RERD '10) aMRX<:j:> » imim<:j> jMEQcj.) >.j=i, jrrm:) »em no 690 J=1..JRRM

IRRH=hEQ':.J:)

PERU 1 0> r XX •:. I » J.) » YY < I • J;) » TCDDE I > .J> . I CDDE » I = 1 » I RRM>

690 CDMTIMUE

CRLL SECTM

IF':.IGEDM .EQ. \> CRLL GEDMDT

53
c

C INPUT CONCRETE MhTEPIhL PPDPERTIES

c PERIiC?. 1 190> FC» EC» XF

MPITE 130 3"> FC»EC

l?':!? FDPMRT ("sv. 'PC =• !.E15.5»5X. 'EC =''>E15.5>

PC = XP PC

PERD t^. I l-?!) MFC* < (PFCT < I > J> »J=1 •.2;) » 1 = 1 .NFO

PERD ' 5. 1 191> NPT* V (PPT <I ...!> . .1=1 »£> » 1 = 1 »hPT>

PERD 5 . 1 1 9 1 > HEC » < vRECT ( I , J> » .1= 1 . , I = 1 . MEO

PERD (5!. 1191> MRLCf <iHLPHC<I. J> » J=l»2;' » I = 1-.MRLC>

C

C INPUT STEEL MRTERIRL PRPRMETERS

C

IP'BMTYPE .EQ. PPESTPO hO TD 12

C PEIH<^OPCING JTEEL PROPERTIES

PERU t ?!. 1 l'r'4> FY>» ES

PERU < 5 . 1 1 9 1 .) MVS . <: <;PYS < I . J ) » J= 1 » 2> . I = 1 » MYS>

PERU 5 . 1 1 9 1 > MRN . •. RNT < I • J> • J= 1 • 2> « I = 1 > NRM>

PERri':5' 1191> NK:» :. •. RKT < I . J > > J=l.d> » I = 1>HK>

C'ERr ' 5 5 1 1 8b> >::CZ C I . 1 > ? I = 1 . 3>

MRITE(6. 13ri4> FYS»ES

1 3 04 PDPMRT c:Y.X »

- FYS = •> E 1 5 . = - 5X .- ES = > E 1 5 . 5>

IP<BMTYPE .EQ. PEIMFDO hD TO 11

C hSriRT TELLS WHICH STEEL ELEMENTS RRE PRESTRESSED

PERU '.5 . 1 1 39;) 'rMSMRT U > » I = 1 » MSS.)

GO TO 13

12 m 1183 I ^ 1j MSS

1138 hSMRTa) = 1 C PRESTPESSIMG CRBLE ONLY

13 PERD 1 194:' FYSPS. ES

RERD':5. 1191> MYSP, < <RYSPS(I» J> > .J=l>a> > I = 1»NYSP>

PERD • S - 1 1 9 1 > NRf lPS - •- • RNPST < I ? J") > J= 1 ? £> . I = 1 . MRNPS>

PERD 5 1191:-' MKPS. 'iRKPSTCI. j;:' » J=l»2> , I = 1»NKPS)

PERD ' 5 ' 1 1 36 ) ( CZ I n <=:> » I = 1 . 3>

l.lPITE';r,» 1305> PYSPSfES

13 05 PDPMRT r-SX. ' FYS -iPS) = ' » E 1 5 . 5 > 5X > '^ES CPS) =' >E15.5>

GO TO 14

11 DO 1137 I = 1» NSS

1137 HSMRTa:) = n C ES ' T;- 'ES (TO;. RhD RLPHR^S) RPE THE SRME FDR

C BOTH PEINFDPCIMG RhD PPESTPESSIhG

14 PERD':^. 119i:) MES? oPESa* J> « J=l»3:) . 1 = 1. MES)

PERD ;5. 1 19i:> MRLSj < fRLPHS U > J> • J=l « £> > 1 = 1 . hRLS)

C ITPRIhS IM PPESTPESSIhG CRBLE

IP'BMTYPE .EQ. 'PEINFD'> GO TD 15

PERD ' S. 121 D - EPSPST a;) » 1 = 1 .NSS!)

DO 16 1=1 » NSS

IP •HSMRTCI) .EQ. 0:> GO TD 16

MP I TE ' 6 . 1 3 06 ' I . EPSPST < I

1306 PDPMRT <5X. 'BRP' , 12, 2X, PPESTPESSED TD STPRIN',E15.

16 CONTINUE

54
c

C IMFUT TIME hNH/DR TEMPEPRTURE PRPFiMETEPS hND DUTPUT PEOUESTS

c

15 PEfiD'S. 1 1^2> TEMPIH.:STrftL»TFCTP,MCRLC

MPITE'.t.. 1:3 07> TEnP IM. NCRLC

M-iri? FDRMRT ( 1X-. •• STRPTIMG TEf^P = ^ F8 . 5X .

' MCRLC =' '.I5> IFcSTCRL .EG). CRLCST^^ MP I TE < 6 » 1 I-: Ci;?> TFCTR

IFt'STCRL .EO. -PERDST'^ UP I TE ( t> » 1 5 09 > M-iOy FDRMRT ( IX. ' STEEL TEMPEPRTUPE IS CRLCULRTEIi PPDM SUPPDUND IMG'

/

1 CDMCPETE TEMPEPhH.'PE mITH RPJUSTMEMT FRCTHP OF- jFi^.S)

1309 FDPMRT ax? 'STEEL TEMPEPRTUPE IS IMFUT RT ERCH TIME IMCPEMEMT "> DO i:00 0 I = 1,.MCRLC

£000 PERU '..59 1170.) TIMER < I> - vIMOUT (I . J) . ..1=1 ,4>

C

C CRLCULRTIOM SEQUEMCE < MOMEMT- CURVRTUPE-T IME PELRTIDMSHIPS > C

MTIMC = 1 TIME =0.0

ISTEMP = 0

lEPSC = 0

ISTPSR = 0 C SET OUTPUT REQUESTS

1U91 IF aMGUTCMTIMC* 1) .EQ. i:> ISTEMP = 1 IF aHDUT <:MTINC»2> .EQ. 1' lEPSC = 1 IF<IMnUT'::MTIMCp3:) .EQ. 1> ISTRSR = 1 CRLL IMITRL

I F< ISTEMP .EQ. i:> CRLL TEMPDT

CRLL MDMCUR •: BMWL . « 1 09 0> IF (MTIMC .EQ. MCRLO GD TO 1090

HTIl^C = MTIMC + 1 TIMP = TIME

C TIMER IS TIME RT WHICH CRLCULRTIOM IS PEPFDRMEIif CDRRESC POMIUMG TO SDME» BUT MOT MECESSRR ILY RLL» OF THE TIMES

C OUTPUT FROM THE THERMRL RMRLYZER

109£' PERHaO:? TIME» TEMP

IF':.RBSaiMER<MTIMC>-TIME:> .GE. . 05> GO TO 1092

ISTEMP = 0 lEPSC = 0 ISTRSR = 0 GO TO 1091

109 0 PERIK^. 1£:50:' <TITLE •: I ) » 1 = 1 .£6:) IF (TITLE .EQ. ' EMU DF' .» GO TO 115 0 GO TO 10

C

C EMD OF CRLCULRTIOM SEQUEMCE

C

115 0 STOP

ld5M FDPMRT a3Rie./13R6>

lc51 FDPMRT <2 <R<:.» 1X> » I6» IX.RiS)

1170 FOPMRT(F10. 0»4I10>

ladl FOPMRT ' I 1 O.Fl 0. 0> l£5c' FDPMRT <5 1 lO!) llr.M FDPMRT CI 61 5>

1^11 FDPMRT CSFl 0. 0)

1190 FDPMRT (3F1 0. O)

1191 FDPMRT ( I 1 0 > 1 OX > t-F 1 0 . 0/ • 8F 1 0 . 0> > 1194 FDPMRT (£F 10. 0>

llx':. FDPMhT (4E 16. 4>

lly-:" FDPMRT CaO I £>

1191 FDPMRT (Fl 0. 0. 4X.R6.F1 0. O? 1 1 0> EMD

55
::l.lI;RDUTiriE SECTh

lDMMDM hEDM JPHlSbE. IfllNi^.O . IMPXiSO) « JM I N. JMnX . MEQ <50>

1 XX ?.U. 50;' .YYi ?0'50> • X S i.dO> >YS<cO> »ERYC«:cO>3) »ISLS'::£0»4> » YDEP

i: DririnM ''TL HIS* i DEFT (2<)>

r DMMDM ' •: DhC DEPT < 29. 49;) . C FiRER < 29» 49>

C FIMD DEPTH OF ERCH BRP FPDM TOP

73 0 Yf1RX = 0. YMIN=10000.

JRHM=Jf*1RX-JMIM+l

DO 790 .J=1. JPRM

IRRW=NEQ<J>

DD 790 1=1. IRRN

YMhv=RMRX 1 ( YMRX . YY CI » J> > Yn I H=HM I N 1 '.YM I N . YY a » J.) > 790 CDHTIMUE

VDEP=YMRX-YMIM

I.IP I TE <6 . 1 £6 O YMRX . YM I N » YDEP

l£v.O FDRMRT'.- YMRX = £10.5-' YMIH = .El 0.5^' YDEP = -.EIO.S)

DD 8 00 M=1>MSS

SDEPT rN) =YMRX-YS <N> 800 CDHTIMUE

C ChLCULRTE the RRER DF ERCH QURD RCRDSS ERCH HDRIZDMTRL STRIP RND

C FlhD DEPTH DF ERCH CDMCRETE ELEMENT

DD 83 0 .1=1. JRRMGE

IP = IMRX - imim< j:) DD 33 0 1=1, IR

I U = I + I M I h •: J : - 1 M I M < J + i ) v'h= ( Y y : I , J> +YY I + 1 . J> + Y Y C I U . .J + 1 > + YY •: I IJ+ 1 > J+ 1 > > ^4 . 0

CDEPTvI. J::'=YMRX-YM

CRLL TRRER • XX < I f J> > YY <. I j J> » XX C I + 1 > J) > 't Y < I + 1 > J> » XX < I IJ+ 1 » J+

£1> >YYaiJ+l J J+l> .Rl>

CRLL TRRER ( XX •: I » J ) n YY < I . J;' . XX < I U-»- 1 » J+ 1 > . YY I IJ+ 1 . J+ 1 > > XX C

clU . J+ 1 .:• . 't Y :; I U . J + 1 > y fid)

CRPER CI . J::' =R1+R£

830 CDMTIMUE

RRER=0

DD 360 J=l . JPRMbE

IRh=IMRX':.J>-IMIMC.J>

DD 35 0 1=1 . IRR

HPER=RPER+CRRER C I » J)

850 CDHTIMUE

86 0 CDHTIMUE

MRITE •::6.37 0> RRER

87 0 FDRMRT •: / DERM HRER= ' . E10.5>

IFCSTCRL .EQ. 'RERD'> hO TD 380

C LDChTE THE 4 HERREST MDDES TD ERCH PRP

CRLL CQURD

83 0 CDHTIMUE

RETURM

EMD

SUERDUTIME TRRER c X 1 f Y 1 - X3 . Y3 . X3 » Y3 . RRER:)

C THi: CRLCULRTES THE RRER DF R TRIRHGLE .IF PT-I 1.3>RMD 3 RRE

C lUPPLIED IH R CDUHTER CLDCKWISE SEQUEHCE THE RRER MILL BE PDSITIVE

RPER= 0 . 5* < X 1 ':.Yc:-Y3> -'/'.d* Y 1 - Y3 ) -•X3* CY 1 -\d> > RETURM

EHD

56
cc

c

c

c

c

cc

c

lUEPDUTINE CQUFID

THIx iUBRDUTIME IDEMTIFIE? WHERE EhCH BRP IS LGCmTED I.IPT NDBE DP

G'lJHnPILRTEPML ELEHEMT

CDMriDN-GEDn JPRhGE* lMIM'5n;-' . IMRXiSn^ . JM I h JMRX , HEQ < 5C0 »

1 XX :;0« =;0:) .YYCSOtSC) tXSi.cO^ fYi-icLO) » ERYC >: c! 0 > 5;' » NLS <£ 0 » 4> • YDEP

li I MEr^S I DN X (5^ » Y <:5> . X I < 4:) > Y I <4>

hL:::>:;iS>K> LRBLES STEEL N =BhP MUMFEP

K: = 1 BRP NUMBER

=i I OF PEFEPENC

=3 J OF REFEPENi:

=4 ERR LDCRTIOrj

IF NLS (H. 4) =1 BRP I =a BRP I =3 BRP I TEST FDR BRR LDCRTIDH IN CDhCRETE ELEMENT

no ^^o NN=i.hss

sx=xs (:nm>

SY=YS <iW>

TEST EACH QURHPILRTEPRL IN SEQUENCE

DO 190 J=l<JPRNbE

IR = IMRX (J::- - ININ<:.J> DD ISO 1=1, IR

IT=I + IMIN CJ; -iniN c.J+l>

X ( 1} =VY f I , J)

X <£> =XX (I + l , J!) X<.3>=XXaT+l » J+l>

X (;4> =xx I T ? J+ 1 > x<5;>=xci>

Y •! 1> =YY < I > J:> Y<:2>=YY<i + i> J";" Y (3:' =YY aT+1 » J+l>

Y C 4 :> = Y Y < I T , J + 1 :> Y <Z ':> =Y >'A ? DO cO M=1.4

CRLL TRRER CX (M? .

NODE

NODE

RT

ININ

NODE I». LH SIDE

PH SIDE

OF

OF

QURD

QURD

£0

C

30

BRP IS I.I I THIN THIS

IF<flES(SX-X(L>)

IF ' RBS CSY-Y a •

)

M j . X ( M+ 1 y , Y < M+ 1 ;5 J SX . SY J RPER

hO TD 18 0

IF RT R NODE PDINT

i: IDENTIFY

CNOTE THRT

40

50

IF (RPER. LT. 0^ CONTINUE

QURDPILRTEPRL. CHECK

DO :-;0 L=1.4

.i?E. l.E-06> bO TO 3 0 .6E. l.E-Ob.) GO TO 3 0 bO TO ' 4 0 > ^ 0 , iS 0 , 7 0) t L CONTINUE

GO TO 80

NODE WHERE BRP IS LDCRTED

NODE LRBELS RPE TRUE LRBELS RND NOT NDPMRLIZED LREELS

60

70

80

NL : • NN. = i + ir«iiN cj .) -1

NL -:r<N!. 3;' =J+JMIN-1

GO TO 8 0 NL S ':NN. a> =I+IMIN<

J

NL i: - NN, 3:' =J+ JMIN-1

GO TO 80

r*L : • T\fu ? > = 1 T+iriIN < j+n

nt : ' uri. 3 • = J+JMIN

bU TD All UL ; NN. = IT+IMIN j+i) -1

rn. : fiUn :-: > =J+ JMIN

: ' MM. 1

)

= NN

r<L :

' nri. 4 ' = 1 Gn in ic'i".

57
C IltEhTIFY I.IHICH SIDE GF "(ihi, fnP IS IM

90 ThLL fHk t-H < X (• 1 > , Y <' 1 > » X <-3> » Y C'3:> > ?:X » SY » fiREfl) IF Ht f n. . n, i^q jq j fjf, C EhP is IN PH SIDE DF 'Ml'f' UhTFPhL

i3D TO 11"

100 ML I ' MH* -»

' 110 riLZ 'fM. 1 ' =riM MLS 'MM..;' • = I •IMIM( J> -1

MLS ' MM, iniM- 1 i£:o mxy=ml: 'ru(.4.

GD 10 ' I 1^"' 1 --O) , NXY

130 ' BPYC':MN,1> = 1 BRYC '::MM.2)=0

ERYC (MM? 3) =0

TD 220

^"^^ '-"^LL TFlPER - X 1 > , Y < 1 > ? X Cli) . Y <3) . X c 4-) , Y < 4^ , PPEfl)

hREh1=hPEFi

XI (i) =x <:3;) Yi <i:) =Y <:3> XI <2:-> =x (:4> Yl Cd) =Y 'i:4> XI <3> =X<1>

Yl (3:) =Y CD

XI <A.> =X <3>

YI •:4> =Y 03

"3D TD 160

'-'='LL TRPER ^XCl ) J Ya> jXCc!) , Y<2> ,X';3) ,Y<3) ,FlREft'> HREh1=FiRER

XI <l> =X :£>

Yia>=Y';2>

YI «;:£> =Y <3>

xi<3>=x<i:)

YI <3:> =Y (i:)

^

, XI '::4;) =X <2> " YI <4:> =Y CR> DD 170 11 = 1,3

. . '-i^i-L TRPER 'Xi ai:) ,Yi<ii> ,xi ai + r- ,Yi <ii + rM c» SY, RRER>

BPYC ':MM, 1 1 > =RPER.- RPER

l^f'

_ CDMTIMUE

hO TD EldO

1=3 0 CDMTIMUE

I'?'-' ' -• CDMTIMUE

MPITE :>:.. £10) MM

^1'-' fdpmrt : 0 e r r d r crmmdt ldcrte brp' , i-^.^

chll exit

22 0 ldmtimue

PETUPM

EMD

58
SUEPDUTIME IMITftL

C

CDMMDN'- TTEMP/TIhE

l.lRITE<6. 10£>

102 FDPMRKIHO

MR I TE (I?. J 103) TIME

lO:-: FDPMRT a OX. ' ELflPSEr TIME =' .F8.,5» MIMUTESO

C INITIRLIZE-. COMPUTE MRTEPIRL PRPRMETEPS. TEMPEPRTUPES IN

C CQHCPETE fiNP STEEL ELEMEMTS. RtHD MnN-EOUIL I BRRT IMG STPRIhS

C IN iECTlOM. THIS SET DF H DMPUTRT I DhS MEED BE PERFORMED

C nMLY OMCE EVERY TIME IMCPEMEMT.

C

C TEMPEPRTURE nSTRIEUTIDN DM CROSS SECTIDM

CRLL TEMSET

C MRTEPIRL PROPERTIES FDR THIS TEMPEPRTURE DISTRIBUTIDM

CRLL MRTPPP

C IF TEMPEPRTURE IS ELEVRTED» COMPUTE THERMRL EXPRMSION

C DF STEEL RMD CONCRETE . RMD CREEP STRRIMS IM PE I MFDRCEMEMT

C THERMRL STRRIMS

CRLL THSTRM

C CREEP STRRIMS IN THE STEEL

CRLL CREEP

101 CONTINUE

RETURN

END

59
vlJERDUTINE TEM^ET

C TEM'.ET lETi UP THE TEMFEPHTUPEr IN 5TEEL hND CDhCPETE ELEMEMTS

CnriMDM'i^EOM'JPRhiSE. It^IN' SO^ . iriMX<':.0>

CDMnON STL fiiS- SDEPT <2fO . STEMP -'^O:) • STEMl <20>

CGMMDM CDMC CDEPT' c'9.49:' . CRRER < c"? » 4'?" .CTEMP ».E:9^49> fCTEhl ';29.49>

CDMMDM TTEMP - T IME • T I . TEhP ^ :-:0. ^iV' , TEMPIM. TFCTR

CDMnOM IMDEX - riTIMi: . IPEF- TDL» MITEP. i IDERL. STCflL

C IF hTIML = 1. PDDh tEMPEPhTURE ChLCULmT IONS

IF 'NT INC .EQ. 1> bD TD 4 0 DO 10 I»= 1. NSS

1 0 i TENl >: i> = STEMP < !> IF-;iTCRL .EQ. PERriST'> GO TD 20

C CRLT CRLCULRTES THE TEMPERRTURE OF ERCH BRR

CRLL CRLT

GO TD 70

C IFCSTCRL = RERIO . STEEL TEMPERRTUPES RPE PERD SEPRRRTELY

20 PERU <S, :]:0> -iSTEMP ^N;) > N= 1 > NSS>

30 FDPNRT a6E10.5>

7 0 no 71 J = 1. JRRNGE

IRRhGE = imh».j> -iriiN<: J> no 71 I = 1? IPRNGE

CTEMl a. J> = CTEnP':I» j:) IH = I+IMIN( J'-IMIN' J+l)

71 CTEMP-:I>j;;' = TEMP CI » .J> +TEMP < 1 + 1 . J) -t-TEMP OR » J+l>

lTEMPaR+1 vJ+l.))

RETLiRN

C

C ROOM TEMPEPRTUPE RSSIGNMEMT

40 DO 5 0 r-i=l,NSS

STEMP'.N>=TEMPIN

50 CONTINUE

no 6 0 .1=1. JRRNGE

NI = IMRXCJ.'-ININ<J>

no 60 I=1»NI

CTEMPa.J:- = TEMP IN

to CONTINUE

RETURN

END

iUEPOUTINE CRLT

C THIS ELEMENT CRLCULRTES THE STEEL TEMPERRTUPES FROM CRLCULRTED

C TEMPERRTUPES

COMMON GEOM--.JRRNGE » IMIN<50"> » IMRXt:50> » JMIN. JMRXt NEQ (50) » 1 XX' :-:0'50> >YY<30.50) >XS':i:0> »YS'£'0,> ? BRYC <:c:0> 3.> >NLS<20>4>

CaMMaN - STL.--NSS. SDEPT (SO) . STEMP <£0>

COMMON.' TTEMP.' TIME. T IMP, TEMP (30»50> »TEMPIM» TFCTR

DO 6 0 N=l,NSS

NXYiNLS<N'4>

J=NLS'::N.3>-.JMIN+1

I=NLS'N,£:)-IMIN':.J>+1

I T = I + I M I N < J> - 1 M I N < J+ 1

)

GO TO a 0,2 0, 3 0) , NXY

10 stemp<n:)=temp<i,.j>

GO TD 40

20 STEMP <K> =ERYC <Nj l> TEMP <I , J) +BRYC (N. 2> TEMP CIT+l , J+l> +BRYC <M»3

3> TEMP <lT. .J-»-l> GO TD 40

30 STEMP <:N> =ERYC <N . 1 > TEMP < I . j:) +BRYC (N , 2> TEMP CI + 1 , J> +BRYC CN » 3> T

3EMP': IT+1, J+l!)

40 STEMP <N> = CSTEMP CN) -TEMPIN) TFCTR+TEMP IN

60 CONTINUE

RETURN

EMD

60
lUtPDUTiriE MRTPPP

C rUFRnUTINE CDMPUTEt RMODC* FPC . RLPHRC FOP COHCPETE

C RND RriDDS. YIELD S TPE(^i?.TH» ITPEhGTH L DEFF IC I EMT . HRPDENGIM

C EXPDMEMT» RMD RLPHRS FDP STEEL BRPS RMD STPRIiH.

C

rnnMDN''HEDM.''JPRNi3E. IM1M<50> . IMRX(:50>

CDMMOM' STL.'fSSS» SIiEPT - c'Ci) , STEMP <3 0) ,STEH1 <2(.0 <.hSMRT(20) >FY<20> j 1 RK '. £ 0 > 'J^ M • £ 0 > • R D D S < d U > • RL PHRS < 2 0 CnriMDN.- CDhC -"CDEPT CS"?- 49> • CRPER <29f49'> • CTEMP (29. 49:> . CTEMl <:29» 49> » 1 FPC ' 29. 49;' . PPT (29. 49;' . RMDDC (29. 49> « RLPHRC <29. 49>

CDMMDM ' MRT I M •• PC . EC . HFC . FFCT ( 1 u . 2> . hPT • PFT C 1 U . 2:) » NEC » PECT < 1 0» 2> » 1 HHLC. RLPHC < 1 0. 2> « FYi • E S . MYS . PY? CI 0. 2> . MRM. RNT CI 0. 2)

2 MK . RKT C 1 fi , 2) . FYSPS , MYSP . PYSPS C 1 0 - 2:> » HRhPS -> RNPST C 1 0 » 2) » I NKPS. RKPSTciri.2:) . MEi . PES CI 0. 2> » MRLS » RLPHS C 1 0» 2>

CDMMDrf.' I hUEX'-NT I hC

IF(MTIMC .i5T. n 60 TD 101

C TIME = U — POOM TEMPEPRTHPE — SECT I DM PPDPERTIES CDMSTfiMT

FT = 5.*SQPTCFC>

DD 110 J = 1> JPRNGE

IPRMGE = IMRXC.J> - IMINf.J)

IiQ 110 I = 1. IRRHGE

RtiDDC C I . J> = EC

FPCCI..J> = FC

FPT<I,J> = FT

110 RLPHRC CI »J> = 0. HQ 111 I = If HSS

IFCNSMRTd) .EQ. 1) hQ TD 112

c ZMRT ci:> = 0 - DPDINRRY PE-BRP

FYCI> = FYS

RK C I

':> = RKTC1.2>

RH C I > = RMTC1»2>

GO TD 113

C H : MRT C I

= 1 - PPESTPESSING CRBLE

112 FYCI> = FYSPS

RK C I > = RKPSTCLS)

RN' I> = RhPST<:i»2>

1 1 ? RMDHS

(

I> = ES

Ill RLPHRS <l> = RLPHSC1»2>

RETURM

61
C TIME .ME. 1 - EbfiVRTEIi TEMPEPftTUPE - ChLCULhTE MRTERIRL

C PPDPEFTIEi: FDP EhCH CDMCPETE ELEMEHT RND EhCH STEEL ERR

101 CaMTINUE

C FPDFERTIES FDP CDhCRETE

DG liCi J = 1' JPRMbE

IRRMGE = iriRX ': J:' - IMIMCJ.)

DD laO I = 1> IRRhHE

l TR = O: TEflP '. I » J ' +1: TEM 1 ' I t J:> > RMDDCCI. J) = EC*VHLIJE ':nEC>PECT>i::Tfl)

FPC ;: I . J :' = FC ^VRL UE ( MFC , RFC T . CTR>

PP r a J J;' = FT^VRLUE MFT . RFT > CTR>

RLPHRC': I»J> = VRLUE CMRLC . RLPHC > CTfl>

liCi CDMTIMUE

C PPDPERTIES FGP STEEL - BOTH REIMFDPCING RMD PRESTRESSIhG

DD 121 I = 1» NSS

i:TR = CiTEMPa:>+STEMl<:i>:)./2.

IF(MSMRTa> .EQ. i:- bD TD 1££

C M'MRTa;.' = 0 - nPDIMRRY RE-ERR

FYa> = FT:S*VRLlJE'.MYS?RV5.STfi:)

RK a .:• = VRLUE MK » RKT , STR)

RM •:: I = VRLUE - MRM • RMT > STR>

bD TD 123

C MIMRTa> = 1 - BRP I IS PRESTRESSIhG CRELE

Vd.d FYd;' = FYSPS*VRLUE ' MYSP.RYSPSjSTR)

HK': I> = VRLUE (MK PS jRKPST^STR:)

RM - I ' = VRLUE : NRflPS • RMPST » STR>

C FDTH PEIIMFDPCIMb RMD PRESTPESS HRVE SRME EXPRESS lOMS

C FDP MDDULUi!. CDEFFICIEMT DP THERhRL EXPRMSIGN

12 3 HhDDS •: I :• = ES VRLUE < MES . PES . STR>

RLPHRSa:- = VRLUE (MRLS»RLPHS»STR)

121 CDMTIMUE

PETURM

EMD

FUMCTIDM VRLUE -MP. PRP» T> C FUMCTIDM riETERMIHES MRTERIRL PPDPEPTY VRLUE FDR TEMPERATURE T

D I MEMS I DM PRRa0.2>

1 = 1 1 Ci4 IF • PRR a . 1 > -T:) 101,1 02 » 1 n:? 102 VRLUE = PRP<:i»2>

RETURN

1 03 VRLUE = •PRR •; I - 1 . 2> + < PRR < I , 2> -PRR < I - 1 j £> > <T-PRR < I - 1 . I ) > /

1 'PRR - 1. 1;> -PRR <;i-i , i:>> RETURN

101 1 = 1 + 1 IF -I .LE. MR) GD TD 104

VRLUE = PRR<.MP.2>

MRITE •6? 1 U5:> T. PRRCMP-l)

lOS FDPMRT •:;5X. • EPRDR - EDUMDS DP CURVE DESCRIBIMG MRTERIRL

IPRPRMETER EXCEEIiEIi. T = . E 12. 5» 5X . TEMP <:NP> ='-E12.5>

RETURN

END

62
SUBROUTINE THSTRN

C THIS PDUTINE COMPUTES THE THEPMRL STPHINS IM COhCRETE 1- STEEL

COntlDN 'I5E0M' JPHNGEf iniri<;50> . IMhX'..50:)

COMriDM- STL - hSS. SDEPT <d(.0 . STEMP - diVj »STEM1 <aO> jMSMfIT <;20> » FY «20> »

1 HK > d 0> « Hh •: £ 0> • »=lhODS 0> ? MLPHmS ( ^ 0> > EPST <2 CO i:;0MNDh' C0MC ''CDEPT': c:9«4¥:> . CRPER •: 29. 4-?^ . CTEMP f £9. 49> .CTEMl <29»49;) »

1 FPC ' £9. 49.) . FPT <£9!. 49> . RMODC (29» 49> » MLPHmC •:;29» 49.'> CEPT <29» 49> » 2 RLIiTCT

COMMDM • I HUEX'-MT I NC » IPEF

C SINCE RT PDDM TEMPEPRTUPE THSTPM IS NOT CRLLED*

C CEPT Rhli EPST MUST BE IMITIRLIZEH WHEN NT INC = 1 IF - NT INC .i3T. !> GO TO 72

RLIiTCT = 0.0 no 63 J = 1. JPRNGE

IP = imr:>:<.j>-imin<j)

DO t.8 I = 1» IR

e.s CEPT a » .J) = 0. no t.9 N = 1» nss

69 EPSTCN!) = 0. RETURH

C

C THEPMRL EXPRNSION IN CONCRETE

72 no 7i:i J = 1, JPRNGE

IRRNGE = I MRX •;..!> - IMIN<.J> no 7 0 I = 1, IRRNGE

nCEPT = RLPHRC C I . J> "i CTEMP < I . J> -CTEM 1 < I • .J> > 70 CEPT(i»j:> = cept<i».j:) + ncEPT

C THEPMRL EXPRNSION RT TOP OF BERM

11 = IPEF - IM IN (JPRNGE)

12 = IPEF - IMIN < JPRNGE- 1> SL = CEPT ( 1 2 » JPRNGE- 1 > -CEPT < 1 1 . JPRNGE)

SL = SL -' Mi nEPT a 2 (. JPRNGE- 1 > -CnEPT C 1 1 » JPRNGE) ) RLBTCT = CEPT < 1 1 » JPRNGE) -SL*CnEPT (.11, JPRNGE)

C

C THEPMRL EXPRNSION IN THE REINFORCEMENT

no 71 N = 1» NSS

nSEPT = RLPHRS (:N>*< STEMP <N> -STEMl (N))

71 EPSTCN) = EPSTCN) + nSEPT

C

RETURN

ENn

63
-IJPPDUTIHE CREEP

C*THE lFEE«=' RMhLYII^ IS BRSED DM THE DDPH-HRPMRTHY THEDRY RMD ON THE

TIME HHRDEMiriG RULE

c CDnnDN iTL' NSS* SPEPT «:E'i:f> • STEMP ' 20^ . I TEhl (20> » MSMRT (d<)> . FY <.£0>

1 hK ' c O ' «Rh<;E'0) >RhnDS<:20> . RL PHRS < S 0 ) .EPST' c'O •EPSP:S:T<£0> » 2 I i TPh ' c'O . nJZl-HaO} .SRRER^SO) » STRESS <£0) fEPSUK^O^ .FSWC^O)

CDMnnh 'CPP.'CREEP raO) » TiIMPT <2 0> » Z (£0) . EPTD (2 0) , EPSCl (20> » lEPSL^t^Ci) ,CZ<:;8.2>

CDMMDN - TTEMP.' TIME* T IMP

fDriMDh - INDEX' NT IMC

DiriENSIDh IiELH<£0>

DDUELE PRECISIDM TCMPT. DELH. TflV» HEP

C MDPKIMi? STRESS IN REINFDPCEMEMT IS REQUIRED FDR THIS RDUTIME

C MHEN NTINC^= 1- PRRRMETERS MUST BE IMITIRLIZED

IF ' NT IMC .GT. i:.' GO TO 16 no 17 I = 1. MSS

TCMPTCn =0.0

CREEP a> =0.0

EPSCl a:) = 0. K = MSMflTa> + l 17 riELHa> = cz(:3»K>

RETURN

C TETERMINE HRRMRTHY-HOPM COMSTflMTS

16 DD Id I = 1. MSS

RFSI.I = RBS<FSI.I<I>>

K = NSMRT<I>+1

EPTDti::' = CZa.K:)*HFSW**CZ<2>K>

I F CRFSW - CZ (7 J k::> > 1 2 » 1 2 » 1

3

12 Za::' = CZ <3? K> RFSW^^CZ <:4»K> GD TD 1

0

13 Z I = CZ (5 > K > EXP <CZ <.h > K) RFSW>

10 CONTINUE

C INITIRLIZE EPSCl WHEN NT IMC .ME. 1 !! 13 N = 1, MSS

Y = Z':m>*tcmpT'::m:^.'EPTO(;m>

IF':;Y .GT. .2?> GQ TD 19

X = CERT Ct'*^. 038>

GD TD 13

19 X = -2.*a.+Y)

X = Y+ 1.-2. EXP <X> 15 EPSCl '^M:' = X^EPTD'::M>^hBSc;FSI.KMV)/FSI.I<:M>

C Cnr-IPUTE CREEP INCREMENT FDR THIS TINE INTEPVRL HMD

C TDTRL RCCUMULRTEIi CREEP STRRIN

no 22 0 M=1.MSS

13 0 TRV=.5^' STEMP CM) +STEM1 •:M>>+46 0. riEP=DELH':N:'.-TRV

TCMPT (m:^ =tcmpt <n;:' +riEXP •-riEP) <:time-timp> .'-6 0. 150 Y=Z cm ' TCMPT <.N> . EPTD <n> IF • Y.GT. .29> GD TD 160

X=CBRT (:Y^5. 033>

GD TD 170

16 0 X = -2.^a.+Y>

X = Y+1 , -2. EXP <:x;) 170 EPSC2f M> = X^EPTD ':M> RBS <FSU CM) ) /FSW <;M> HEPSC = EPSC2':M;) - EPSCl rN> IFvDEPSC .LT. C\.:> HEPSC = 0.0

22 0 CREEP 'N' = CREEP <M> + BEPSC

RETURN

END

64
MOMCUR

Determines Moment-Curvature Relation for Fixed Time Temperature Using Concrete Strain, e

no

Select Concrete Strain e

ITER ^ 1

Concrete Strains,

(CEARC,

Stresses, Forces

FCONC)

1 Steel Strains, S (SEARC,

tresses. Forces

FSTEE)

yes

(J) c|> + A((> ITER ^ ITER + 1

_r Define Moment M

Output

Steel Stresses at Working Load

65
^ Return to Main

SUBPDUTIME hDMCUP<IiMWL»*>

C

C lUFPOUT IhE CFiLCULmTES MEr HHM I C RL EQU I L I BPhT I Ml-. STPBINS

C Ih r DM": PETE PMH STEEL HT R GIVEN TEMPEPRTUPE. RND

C ChLCULRTES nOr^EMT-CUPVRTUPE PELRTION FPOM R SET OF

C TRFULRTED CDNCPETE STPRIMS

C

PI MEMS I DM EMC^O)

CDHrinM. GEOM .JPRMGE» IMIM' SlT) , IMRXC50> • JM I M^ JMRX . MEQ < 5 0> >

1 :>::: '30 • TiLi> « yy ::-;n,^o?> «xsv£o.> »ys<£Ci) 'Ppyc ';£:o«3:> «mls<.c'0»4> >ydep

CDMMDM' STL MSS»SDEPT':£Ci> , STEMP <£• 0) »STEM1 <2<i} .MShRTC£Ci> .FY(20> »

1 RK <20> .ftM<£0> >PMDDS<£0:> »flLPHRS<.£0> jEPSTi'SO »EPSPST<'20> » i

" STPM • ffO) . TSTS f.cO::' , SRPER <;20) » STRESS <:20:> » EPSW <.£0> » FSW <:20> » 5 EPSULT •:2fi> - FSULT <2 0)

CDhMDh ' TTEMP.'TIME

'"MNDM IMHEX' NTINC. I PEF • TDL » M I TEP <• S I PERL » STCRL » IEPSCj ISTRSfl. ICOMV

DOUBLE PPEC I S I DM PH I . PH I ML • PH I L . KELT • GMMR . 6ML . bMML

IF'MTIMC .ME. !> GO TO 1069

PHII.IL = 0. 0 0 04 .' YDEP

lOfcV? MSTEP =1

PHIL = PHIWL

BMP =0.0

MM = 0 30 0 0 ICDMV = 0 1074 EPSC = EPS rNTIMC.NSTEP.MMi.EPSWL)

PHI = PHIL

ITER = 1 L HiSUME PHI RMD ITEPRTE THROUGH ITER .LE. MITER CYCLES TO

C FIND PHI •::THPUST=0>

1072 CRLL CERRC -iEPSCjPHD

CRLL FCOMC (SUMCj SUMT» SUMM>

CRLL SERPC '-EPSC!. PHD

CRLL FSTEE : SUMC > SUMT . SUMM)

THRUST = SUMC + SUMT

IF - RBS aHPUST> .LE. TDL> GO TO 1071

C HDPIZDMTRL FORCES DO MOT BRLRMCE - RDJUST PHI RMD TRY RGRIM

IF (ITER .GE. MITER) GD TO 1073

PHI = PHI - DELT'::iCDMV.MM. ITEP.EPSC»PHI.THRUST5PHIL»GML»

1GMI.IL'GMMR>

ITER = ITER > 1 GD TO 1072

C SOLUTION FOP PHI EXCEEDS NITER TRIRLS RLLDl.iRBLE.

1073 I.IPITE Cbi 1 OOl!)

1001 FDPMRT ax. • SOLUTION FOP PHI DDES MOT CONVERGE'!)

IFdCONV .EQ. l!) CRLL EXIT

i.lPITE <6' 1 1 oi:> 1101 FDPMRT : 1 OX .

' PH I' » 9X .' P ' » 3X ?' PH I R

' . 8X >

' PR ' » 9X > GMR ' j 9X . ' DPH I

' > ICOMV = 1 GD TO 1074

66
C DEFIhE BEMriNi? MDMEMT

1071 EM':MITEP') = SUtlM

C HiSUnED Pm FDP EPS «;hSTEP*l)

PHIL = PHI

GML = GMMfl

IPaiTPtR .EQ. 1> CRLL JTSRDT (HSTEP. EPSC. PHI . SUMM. ITER)

I F a EPS C . EQ . 1 > CRLL EP i H DT (m < NSTEP . EPSC , PH I , -i UMM > I TER>

C CHECK IF CRLCULRTED MDMEhT RPPPOXIMRTES WDRKING LORD

C MDMENT BMML

IF ':Bri<HSTEP> .LT. Bf1l.lL> GO TO 1077

C CHECK FDP FIRST RSSUMED STPRIh -MSTEP = 1> IM TIME IMCREMEMT

IF':NiTEP .ME. 1> TD 1090

riM = -

!? TD 1074

1090 IF ' BMP .GE. BMWL!) GD TD 1075

C COMPUTE I.IDPKIMG STRESSES IM STEEL RT BMWL

CBMI.IL = ( BMI.IL-BMP;) ' - EM < NSTEP> -BMP.")

iiD 1076 N = 1j rsss

EPSl.l':M> = EPSI.KH) + CSSTRM rM;! -EPSW > CEMWL

1 076 FSW (M) = STESTR •: EPSW th) , FY ':H:> » RK <.H> , RN (h:> j RMDDS CM) » S IDEAL)

PHI ML = PHII..IL + CBMWL CPHI - PHIWL)

EPS1..IL = EPSC

bMl.iL = 6MMR

IF a EPSC .EO. 1 .DR. ISTRSft .EQ. 1) CRLL STWLDT

GD TD 1075

1077 LD 1075 M = 1, NSS

107S EPSIilCrO = SSTRM<M)

PHI ML = PHI

MM = 1 1075 IF'::BM(MSTEP) .GE. BMP .DP. BM CMSTEP) .LE. O.O^ GD TD 1079

MRITE ' Aj 1 OO:-:) TIME. BM':MSTEP-1)

lOO:-: FDPMRT ::5X. • TIME = '

•> F 1 0 . 4 . 5X .

- ULT I MRTE MDMEMT CRPRCITY ='»E15.

IF a EPSC .EQ. 1 .DR. ISTPSR .EQ. 1) CRLL STULDT

IF (BM f MSTEP-1) .GT. BMI.IL> RETURN

lilRITE >Ji.f 1 fyih)

FDRMRT (:5X. ' BERM CRMMDT CRRRY WDPKIMG LDRD*****'')

RETURN 2

1079 BMP = BMCMSTEP)

DD 1081 H = 1» NSS

EPSULTan:- = sstrh<:n)

1081 FSULTCM ) = STRESS CM)

r^iTEP = rtSTEP + 1 GD TD 80 00

END

67
FUMCTIDM EPS 'HTIMC.h^ITEP'Mri'EPSI.IL)

C THi: FUHCTIOM PETEPnihES THE IMCPEMEhT FOR EPSC

C Bh:ED dm THE PREVIDUi EPSCML Rf^D MHETHER THE

C CDMPUTED BM I S . GT . DP .LT. EMML '^Mfl = -1 OR MM = 1> C

C DMCE EPiCl.iL IS LDCRTED EPSC IS IMCEPEMEMTED UMTIL

C THE ULTIMRTE BM IS PERCHED

C

IF (MM .ME. LT) GD TO 2 0 0 IFtHTIlHi* .EQ. 1 .: GD TO 100

IF'MiTEP .EQ. 1> EPSL = EPSWL

GD TD £00

lOu IFiMSTEP .EQ. I) EPSL = -0.0004

£00 IF 'MM .GT. 0> GD TD 300

EPS = EPSL + 0. 0002

EPSL = EPS

PETUPM

300 IF (EPSL .LE. -0. 00 09;) GD TD 400

EPS = EPSL - 0. 0002

EPSL = EPS

RETURIi

4 0 0 EPS = EPSL - 0. 0005

EPSL = EPS

RETURM

EMD '

'

SUFPnUT I ME C ERPC '. EPSC » PH I > C THIS ELEMEMT FIMDS THE STRESS IM ERCH ELEMENT RTTRE LDCRTIMG THE

C RPPPDPRIRTE TEMPEPRTUPE LEVEL

rnUELE PPECISIDM PHI

CDMMDM GEDM '- JPRMGE» IMIM«;50> . IMRX<:50>

CDMMDM. CDMC • CDEPT (£•?. 49) . CRRER < £9 • 49> »CTEMP <£9»49") »CTEM1 <29»49) »

1 FPC ' £9« 49> • FPT ':29« 49 > » RMDDC ',£9. 49> • RLPHRC (:29> 49> »CEPT (29* 49.') » £ RLDTCT . CSTPM ':£9' 49'' ? TSTC •:£9« 49) « iSSTRES <£9« 49>

C RLL DISTRMCES EXCEPT YY RRE MERSURED FPDM THE TDP DF

C THE BERM

DD ££0 J = 1, JRRMGE

IRRMGE»= IMRX<.J:) - IMIM<J>

DD ££0 I = 1, IPRNGE

CSTRM-:!. J) = EPSC + PHI CDEPT<:i,J> - CCEPT U » .0 -RLDTCT)

TSTC-I»J> = CSTRM 'I . J) + CEPT<I»J>

CSTPES <; I J J.' = CDMSTR CCSTRM •: I , J) , FPC < 1 » J> > FPT C I > J} . RMDDC H » J"> "> ££0 CDMTIMUE

RETURM

EMD

68
FUriCTIOM i:DNSTP<X.FC»FT»EMDrC)

c

C FUMCTIDM CDMPUTES CDMCRETE ELEMENT STPESSf i^IVEM STPflIM

C

P = FC'-EMDDC

ECULT = 1.75*R

IF •;>: .LT. EC.LILT> GO TD 13

IF <X .hi . 0. IV) <SD TO 1

0

IF-:X) 11.12,12

1 1 CDNSTP = EMDDC^X/ < 1 . + < X.'R> g)

RETURN

12 CDNSTR * EMDDC^X

RETURN

13 RE = ( ECULT.'- R>**2

= EMDIiC^ECULT.--- a . +RE>

:lh = EriDri:*"::i.-RE>.'- ';i.+RE>**2

CDNSTR = R + SLR* (X-ECULT)

IFOIDNSTR .LE. O.) RETURN

C ITPESi IN ELEMENT IS ZERO IF IT EXCEEDS CRUSHINi^ STRRIM

C OR TENSILE STRENGTH

1 0 CDNSTR = 0.0

RETURN

END

SUERDUT I r<E FCDNC (SUMC , SUNT , SUMM)

C THIS ELEMENT FINDS THE FORCE RND MOMENT FOP EfiCH ELEMENT

COMMON-' "3EOM - .IRRNI5E. IMIN-Sij) , IMhX<50>

COMMON' CDNC ^CDEPT (29» 49) . CRRER C29. 49:) > CTEMP (29. 49> , CTEMl <29» 49) »

1 RFC (29. 4'=''' . FPT (29. 49) . RMODC ( 29. 49:) . RLPHRC (29. 49:) , CEPT <29. 49!) f 2 RLDTCT.CSTRN(29.49> , TSTC C29 . 49::' » CSTRES (29 » 49>

SUNC = 0. SUNT = 0. SUMM = 0. DO 'Z'O Jil.JRRNGE

IIRRN = IMRX(J:)-IMIN<J>

DO 4i:i 1 = 1.1 IRRN

FDRCE=CSTRES ( I . J> CRRER < I , J> IF (FORCE) 10.40.20

10 SUMC=SUMC+FDRCE

GO TD 30

20 SUMT=SUMT+FDPCE

30 SUMM=SUMM+FDRCE*CDEPT <I » J)

40 CONTINUE

50 CONTINUE

RETURN

END

69
SUPPOUTIME SERPC - EPSCPHn

C DETEPMINES iTPESS Iti EACH STEEL ELEMENT DF BEFtM SECTION

C

IiDUBLE PPECISIDN PHI

CDMMQN- STL ' NSS . SDEPTcan^ .STEMP' SO .STEMl - ^O) ,NSMRT(2 0") »FY<:20> »

1 hK ' £0 ^ • MM • c O> . HMOriS ' SO'' . RLPHR v iaO > . EPST k20) ? EPiPST (20}

2 SSTPN • ) . TSTS •.:2fr> « SFlPEH •.20> • STRESS <r20"> CDMMDM.'CDMC'CriEPT '29. 4?) . CRPER <29*49> CTEMP < 29 . 49> » CTEMl (29f 49> »

1 FPC 29. 49;> . FPT <:29. 49> - RMODC f:29» 49> > RLPHRC <:29. 49:> • CEPT <:29» 49> » 2 RLDTCT*

CDMMDM' C pp.' CREEP <20>

CDMMDM I MHEX hT I hC . I PEF . TDL » N I TEP . S I DERL

C PDDM TEMPERRTUPE - ND IMPOSED STRRIHS OTHER THRN

C PPESTRESS» IF RhY

C ELEVRTED TEMPERRTURE CRLCULRTIONS REQUIRE THERMRL EXPRMSIDN RMD CREEP

STPRINS

DO 102 I = 1. MSS

SSTPNa") = EPSC + PHI SDEPT<I> - 'EPSTU) + CREEP<I> - RLDTCT>

IF<MSMRT< I> .EQ. ff> GD TO 101

SSTPM<I> = SSTRM<I> + EPSPSKI)

101 TSTS'I) = SSTPNa> + EPSTCD +CREEPa>

102 STRESS <: I > =STESTP <SSTRH < I > ,FY<I> >«K<i:>.» flM<I> jRMODS<I> ,SIDEflL>

RETURM

END

FUNCT I DN STESTR '.X . YS . RK > RN . ES « S I DEflL"> C

C FUNCTION COMPUTES REINFORCEMENT STRESSES. GIVEN

C NECHRNICRL STRRIN.

C

IF^SIDERL .EQ. 'FLRTTO bO TO 16

RX = X

EPL = 0.75 YS /- ES

IF^RXMCj 10» 11

10 RX = -RX

11 EU = -ES • •::RK*RN>>**a..'';RN-l.>> + EPL

IF ifiK .hi. EU:> GO TO 13

IF .::RX .hT. EFL> GO TO 12

STESTR = RX ES

GO TO 15

13 RX = EU

12 STESTR = ES*RX - RK> CRX-EPL;) RN

15 IF'X .LE. 0.> STESTR = -STESTR

RETURN

C

C THIS PORT ION INSERTED FDR NSITIVITY STUDY OF STRESS-STRRIN

C IDERLIZRTION

C

16 RX = RBS •::X> EY = YS .' ES

IF< RX .GE. EY : GO TO 17

STESTR = RX ES

GO TO 1:3 17 STESTR = YS

18 IF •:: X .LT. 0.0 : STESTR = - STESTR

RETURN

C

END

70
SUFPDUTINE FSTEE :. SUMC > i UNT . vlJMM-)

C THT: ELEMEMT CRLCULFiTES the FDPCE in the steel HMD THE MDMEfiT

C CfHpFIED BY THE STEEL

CDriMDh STL- MSS. SDEFT <.20> , STEMP <20 > » STEMl <dO> . hShHT »:c:0) » FY <20:) » 1 RK £ U> li liN • £ IV) . HMDDS 0:> • RLPHR i Cd Ci> ? EPST £ 0> > EPSPST 0) » £ S STPM •;£ri:. , TSTS (:£0> . vRPER •.dy}> . STPESS i£0>

C ILIMM. iUMTi. RMIi SUnM HRVE BEEN IMITIRLIZED Ili FCDhC

C

DD 4 0 M = 1, NSS

FORCE = STRESS ':N> SRPER CN)

IF vFDPCE) 10.4i:i!ic:0 10 :UMC=SUMC+FDPCE

GD TO 3 0 20 SUNT=SIJMT+FOPCE

30 SUMM=SUriM+FDRi:.E*SriEPT <M> 40 CDhTIMUE

RETURN

END

DDUBLE PPECISIDN FUNCTION DELT ICONV* MM» IE . EPSC?. PHI . P,

IDELTMX. i5riL» bMl..iL» bRhMR>

C

C PPDhRRM DETEPNINES OPTIMRL INCREMENT FOP ZEROING- THRUST

C ON BERN USING SELF-RCCELLERRT I NG ITEPRTIDN.

C REFERENCE - J. TPRUB? ITEPRTIVE METHODS FDR SOL'N OF EQNS» P. 186

C

C IF FIRST STRRIN. SELECT INITIAL VRLUES

COriMDN ' INDEX ' NT INC

DDUBLE PRECISION PHI » RPHI « GRhMR. BETR- BTP» DELTMX , GML GMWL

IF -IE .^J. 1> GO TO 101

C DEFINE INITIRL VRLUE DF RCCELLERRT I NG PRRRhETER BETR»

C UTILIZING FRCT THRT DP-'D ':PHI ) .GT. 0

IF'MN .EQ. 0> GO TO 104

BETR = -1. rGNL

GD TO 102

104 IF (NT INC .EQ. i:> GO TO 103

BETR = -1. / GMWL

GD TD 102

103 BETH =-1.0

GO TO 102

101 BETR = -l..'GRMMR

102 BTP = BETR P

HPHI = PHI + BTP

CRLL CERRC C EPSC » RPH I

CRLL FCDNC SUNC • SUNT » SUMM>

CRLL SERRC (EPSCiifiPHn

CRLL FSTEE (.SUMC . SUNT . SUMM)

PR = SUMC + SUNT

GRMMR = (.PR - P) /BTP

DELT = P .-' GRMMR

IF .:DRBS'::DELT> .GT. DELTMX> DELT = DELTMX DRBS^DELT) DELT

I F (. I CDNV . EQ . 1 > MP I TE - 6 . 1 0 0 ) PH I , P , RPH I » PR j GRMMR » DELT

1 0 0 FDRMRT (D 1 2 . 5 » E 1 2 . 4 » D 1 £ . 5 . E 1 £ . 4 , 2D 1 2 . 4:) RETURN

END

71
rUFPDUTIME OUTPUT

CnnnDM-r-EDM JPl=lHGE'. IMIh)<50> . IMRX<.50> , JM I M» JMhX « MEQ <50> .

1 X;-; -;ri, SO' . YY C.:-iCi.50> »Xi ':20> -Y: >;30;> .BPYC ^iO. ?> »MLS<20»4>

CGMMDH- STL - N-?S. SBEPT (20> . STEMP <20> . iTEMl <£0) .hSMFtT <£0> .FY<20>

I .Hh<£0> .RMDIiStc'O .RLPHMi <£0) ,EP?T (SO -.EPSPST<20>

c' : "TPH • :£Li:> . tsts - cO • srper <ro> » stress t£0) .epsm- sco .psuuao

:•! ep : ULT <£ ij;-' • fsult <aio

CDMMDN - CDNC • I- DEPT ( 29» 49 '> , CRPER <£9 « 49> . C TEMP < 29 ' 49) »CTEM1 < £9»49>»

1 FPL '29»49.' .FPT(29'49;' • mMOPC c'9 » 49) ? RLPHRC >: 29> 49) » CEPT <.29» 49) » 2 RLDTi:T«CSTPM':29.49> . TSTC '.:29' 49) . CSTPES '.29» 49)

CDMriDh tPP ''CREEP ':2 0) .TCMPT •::2 0) *Z<aO:> »EPTa<20)

i:.aririDh''TTEMP'-TiME

double precision tcmpt»phi

c ehtry geomot

c JRRh = JMRX - JMIN + 1 MRITE-:6>710)

71 0 FGRMRT v9X!i - J .^X* ' IMRX' !i6X« ' IMIN'' »7X» 'NEQ') no 730 J =1 . JPRM

M = J + JhIM - 1 73 0 i.iR I TE '^A 72 U) tl » I MRX < J) » I M I M <. J) > MEQ < J>

72 0 FDRMRT <:41 10)

WRITE tb? 740)

74 0 FDRMRT •: 4X ?

' I ' « 4X >

' J ' ? 1 3X » XX < 1 3X » YY " > HD 75 0 -1= l.JRRIi

IRRM = NEQCJ)

DO 750 I =1 ' IRRM

J J = J + JMIN - 1 II = I + iriiii<j) - 1

' - / 75 0 WRITE <6. 7fE.O) I I > JJ > XX < I » J) » YY <: I » J)

760 FORMRT •::£I5>2E15.5) no 24 0 h=l.MSS

WRITE <6»230) CH. K » NLS CH? K) > K=l > 4)

23 0 FDRMRT •:;4 < ' MLS : . 1 2 ?

' >' > 1 2 >

' ) = ' > 1 2> > 24 0 CDMTIMUE

WRITE •:.6?250)

25 0 FORMRT '') no 27 0 H=1.MSS

WRITE 26 0) -rM? K > BRYC ':M' K) > K= 1 » 3)

260 FDRMRT <.3 </ BRYC ••' ' . 1 2 1

' » » I 1 » ) = ' » E 1 0. 5) > 27 0 CDMTIMUE

WRITE <6»250>

RETURN

C

ENTRY TEMPDT

C

WRITE «;:6> 200)

20 n FDRMRT ':.7X> ' BRR' •5X» ' TEMPO

WRITE •;6j201) (M. STEMP <M) » M=l.NSS)

2 0 1 FDRMRT • I 1 0 > F 1 0 . 4)

lilRITE ';6» 30 0)

3 0 0 FDRMRT < 7X .

' BRR ' !• 1 OX .

' F

Y ' - 7X -

' RMOPS ' « 1 OX .

' RK ' » 1 OX .

' RN ' ) WR I TE < 6 ? 3 0 1 ) M . FY • ti) . RMDDS M) « RK ^M) • RN ^ N) . M= I » NSS)

301 FDRMRT-:! 10.4E12.4)

WRITE (6' 40 0)

4 0 0 FDRMRT <7X< ' BRP ' » 7X «

' EPSTH'

)

I.IR I TE 6 » 4 0 1 ) • N » EPST r M) » M= 1 » NSS)

401 FDRMRT 'I 1 0- El 2. 5)

WRITE ':6.2S0)

dx ft FDRMRT < 7X »

' BRR ''11 X f

' Z ' • 3X »

' EPTO '' . 9X , TCT '' , 7X >

' CPP ' ) WRITE ':6j271) 'iM.Z'N- ttPTD-N- , TCMPT rN) » CREEP tN) ,M=1»MS::)

27 1 FDRMRT :: I 1 0 . 2E 1 2 . 4 . B 1 2 . 5 . E 1 2 . 4)

RETURN

72
EMTPY vTSRDT MSTEP » EPSC » PH I j SUMM. ITER)

C

Ulp I T E • » 1 1 S6> T I ME . EPSC

113.^. FDFMHT <

• 0' 1 OX, ' TDTflL STPftlNS RT TIME'.F6.1,' EPSC ' <• E 1 2 . 4> nn 413 J = ItJPRMGE

IP = IMRXC.l) - IMIM<J>

41? MPITE^'£.'.500> Jj aSTCa. J> , 1 = 1, IR>

l.JRITE 1 187>

lis? FDPMRT <:7X, ERP- ,4X!. TDTRL STEEL STRRIH'>

MR I TE :; 6 , 1 1 33> <N , T S T S t M> , N= 1 » NS S>

1133 FDPMRT' I10,E12.4:>

i.lRITE (6, 1 189:)

1133 FDPMRT Ci' - l nx, ' MECHRMICRL CDNCPETE STPRIMS')

no 414 J = l,JPRNi5E

IP = IMRX-rj' - IMIIiCJ)

414 I..IRITE -r^, 5U0> J, (CSTRMCI, J> , 1 = 1, IR>

I.lRITE '6, 1 034>

1034 FDPMRT ClHO.' lOX, CDNCPETE STRESSES')

DD 412 J = 1, JPRNI5E

IP = IMRX':J> - IMIN<.J>

412 l.lRITE'::6,50 0::' J, -rCSTPES < I , .j:> , I = 1 , IR>

5 0 0 FDPMRT 15, (TE 1 0 . 4) ) I.lRITE 1 190>

1190 FDPMRTf' O' -lOX, 'MECHRMICRL STEEL STRRIMS STRESSES')

I.lRITE 1 19i:)

1191 FDPMRT <7y. ,

' ERR •

, t.X , STRR I M

' , 6X •

•- STRESS

)

l.iRITE':6. 1033 ' <:h , SSTRM CM) , STRESS CM:) ,h=l»MSS)

I.lRITE '::6, 1 192)

1192 FDPMRT <'• O' ) GD TD 30 00

C

EMTPY EPSCDT CMM, MSTEP, EPSC, PHI , SUMM, ITER)

C

IF<MM .hE. 0") bD TD 4000

30 0 0 MPITE Ce, 2 0 02)

2 0 02 FDPMRT 1 1 X ,

' EPSC ' , 1 2X ,

- PHI', 9X , - MDMEHT ' , 4X , ' CYCLES ' > 4 0 0 0 I.IR I TE 6 , 1 0 02> EPSC , PH I , SUMM , I TER

1002 FDPMRT : E 1 Dl 5. 5 , E 1 5 . 5 , 1 1 0> RETUPM

C

EMTRY STWLDT

C

MPITE (•;.,101>

101 FDPMRT a OX,' MDPKIMG STRESSES STRRIMS IM STEEL')

WRITE '-6? 1 02)

1 02 FDPMRT < 7X . ' BRP ' , SX , ' FSWL ' , 7X , ' EPSWL '

)

MPITE f 6. 1 033) CH-FSMir^ •EPSM.rM) , N=1»HSS)

MRITE <b» 2 002)

PETURM

C

ENTRY STULDT

C

I.lRITE - A- 1 032)

1032 FDPMRT (7X. • BRP' ,6X, 'EPSULT' ,7X, 'FSULT')

MPITE t., 1 033) CM, EPSULT CM) , FSULT CM) » M=l ,NSS)

103 5 FnPMRTCI10,2E12.4)

RETURN

FhO

73
NBS-n4A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. miMI.K A I ION OK Ki:i'ORT NO. NBSIR 76-1032

2. (xiv't Accession

No.

3. Recipient's Accession No.

4. T1TI.I-: AND siiin iTi.i-; Analysis of Reinforced Concrete Beams Subjected to Fire

5. Publication Date April 1976

6. Performing Organization Code

7. AUTHOR(S)

Bruce Ellingwood and James R. Shaver

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAMF AND ADDRFSS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

same as 9

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary ol most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.) Methods for analytically predicting the behavior of simply supported reinforced

concrete beams subjected to fire are presented. This is generally a two-step process

involving a thermal analysis followed by a stress analysis. This study emphasizes

the latter, wherein the determination of moment-curvature-time relationships for the

beam cross section incorporates the temperature-dependent strength degradation in the

steel and concrete as well as thermal and creep strains. The sensitivity of the

predictions to various phases of analytical modeling is investigated to establish the

parameters most important for the prediction of beam behavior and to indicate where

additional data should be gathered. A comparison of predicted behavior with that

observed in fire tests shows excellent agreement when realistic reinforcement

temperature histories are used.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Creep; fire endurance; fire tests; reinforced concrete; sensitivity analysis; steel;

structural mechanics; uncertainty.

18. AVAILABILITY ("7 Unlimited

fx"' For Official Distribution. Do Not Release to NTIS

1

^ Order From Sup. of Doc, U.S. Government Printing Office Washington. D.C. 20402. SD Cat. No. CI 3

1 1 Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

79

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

USCOMM-DC 29042-P74