December 19b8 ST 12

Journal of the

STRUCTURAL DIVISION

Proceedings of the American Society of Civil Engineers =

BEHAVIOR OF CONCRETE UNDER COMPRESSIVE LOADINGS

By 1 Demir Karsan and James O Jirsa M ASCE

INTRODUCTION

This paper describes and evaluates an experimental study of the strength and behavior of plain concrete subjected to repetitions of compressive stress Ovarious levels A total of 46 short rectangular columns were tested under cyclically varying axial loads to establish stress-strain relations for plain concrete The characteristics of the loading and unloading stress-strain reshylationships were studied and expressions for these relationships were derived

BACKGROUND

Early research on plain concrete subjected to variable load histories was aimed toward obtaining a fatigue limit for the materil Fatigue tests of ceshyment mortar were followed by fatigue tests of plain concrete in which the reported fatigue limits were generally from 40 to 60 of the staLic cylinder strength A decrease in the tangent modulus and the Poissons ratio with inshy

bull creased number of cycles of loading was reported This early work has been reviewed in considerable detail by Nordby (5)3 Murdock and Kesler (4) con-

eluded that there was no Significant fatigue limit for plain concrete under loads of the order of millions of cycles However for a given stress level theInumhlgtr of (7(1lt nlonl1(inO hillllP (olllrl hI ohtinlrl

2545 CONCRETE BEHAVIOR 2544 December 1969

1 The stress-strain relationships of concrete under histories possess an envelope curve which may be considered identical with the stress-strain curve obtained under constantly strain

2 The stress-strain relationships of concrete subjected to cyclic possess a locus of common pOints which are defined as the point reloading portion of any cycle crosses the unloading portion Stresses the common points produce additional strains while stresses at Or these points will result in tIle stress-strain path going into a loop It Was observed that the values of the common pOints depended on the stress in the cycle ie the stress amplitude

Shah et al (7B9) reported tests of prismatic specimens subjected to peated axial compression Tests showed that the maximum stress of the of common points appeared to be approximately equal to the criticalload which the volume of the concrete uruler compression ceases to decrease the micracracking in the mortar sharply increases

Most of the experimental work to date has been aimed toward obtaining fatigue stress level for concrete The loadings were generally at high The effects of acceleration and speed on the behavior were generally eliminated

OBJECTIVES

The objectives of this investigation were twofold (1) To study experim~n1 tally the behavior of concrete under various compressive loadings in determine the factors governing the responses of concrete to repeated ings and examine the mechanism of failure under these loadings and (2) develop expressions for the stress-strain relationships of the concrete on the experimental results and to use these expressions for predicting behavior of concrete under other compressive loading histories

EXPERIMENTAL PROGRAM

Test SPecirzens--The test specimens were short rectangular COlumllS Thedimensions of the column at the critical section were 3-in X 5m~

To confine failure to the mideight of the column bath ends of the specimellSl

curve Axial load was applied with a 60-ton hydraulic ram connected Id_operated high-pressure pump which provided a nearly constant rate of oil to the ram The load was transmitted to the specimen through a

yoke resting on a spherical head on the ram The movable yoke was to a rigid base plate which distributed the load to the end face

test specimen On the other end a similar rigid plate was pin-connected 00 K load cell which was used to monitor the axial load Both ends of

specimens were grouted with a quick setting high-strength gypsum cement lIoriZontal load was applied through a manually operated screw-type meshysnt As the mechanism was rotated a horizontal thrust was developed

~st the column of the load frame Horizontal loads were applied only to aintain a uniform strain across the specimen

w oJ bl

FIG l-TEST SPECIMEN FIG 2-LOAD FRAME

Strain rates were suc that te maximum c~pac1tyof the s~ecimens su~shywere flared and reInforced WIth No5 bars jected to steadily increasIng stram was reached In 3 mm to 5 mm For cyclIc

The concrete mix proportions were constant for all the specimens The I loading the peak load on the specimens was applied in 1 min to 2 min and reshyconcrete was a blend of Type m Portland Cement and 50 Colorado Rivermiddot moved in about 12 min to 1 min sand and 50 34-in gravel by weight The specimens were cast in a vertical instrumentation -Four OB-in wire resistance gages were used to monitor position ~hree 6-in x 12-in control ~ylinders were cast with each specimen the strains on the two opposite 3-in faces at the mid~elght of ~he column The speclmens were cured m a mOIst room for 2 5 or 7 days and testedati Because the specimens were subJected to strams conSIderably In excess of 7 14 or 21 days Cylinder strengths varied from 3500 psi to 5000 psi Values 1( the normal operating range of the strain gages strains were also measured of f~ for each specimen and complete details of the experimental program over a 6-12-in gage length using two differential transformer displacement are given in Ref 3 I transducers placed at the midsections of the specimens These transducers

Loading Anangement-A rigid loading frame (Fig 2) was constructed in were placed on the opposite 5-in faces of the specimen and were supported which axial Or flexural loads could be applied simultaneously or separately between light steel frames fastened to the specimen with pOinted set screws The silffness of the loading frame was sufficient to avoid problem s associated Test Procedure - Load histories were controlled by monitoring one of two with the release of energy in the unstable portion of the concrete stress- variables (0 Incremental strain during a given cycle or (2) stress level

2545 CONCRETE BEHAVIOR 2544 December 1969

1 The stress-strain relationships of concrete under histories possess an envelope curve which may be considered identical with the stress-strain curve obtained under constantly strain

2 The stress-strain relationships of concrete subjected to cyclic possess a locus of common pOints which are defined as the point reloading portion of any cycle crosses the unloading portion Stresses the common points produce additional strains while stresses at Or these points will result in tIle stress-strain path going into a loop It Was observed that the values of the common pOints depended on the stress in the cycle ie the stress amplitude

Shah et al (7B9) reported tests of prismatic specimens subjected to peated axial compression Tests showed that the maximum stress of the of common points appeared to be approximately equal to the criticalload which the volume of the concrete uruler compression ceases to decrease the micracracking in the mortar sharply increases

Most of the experimental work to date has been aimed toward obtaining fatigue stress level for concrete The loadings were generally at high The effects of acceleration and speed on the behavior were generally eliminated

OBJECTIVES

The objectives of this investigation were twofold (1) To study experim~n1 tally the behavior of concrete under various compressive loadings in determine the factors governing the responses of concrete to repeated ings and examine the mechanism of failure under these loadings and (2) develop expressions for the stress-strain relationships of the concrete on the experimental results and to use these expressions for predicting behavior of concrete under other compressive loading histories

EXPERIMENTAL PROGRAM

Test SPecirzens--The test specimens were short rectangular COlumllS Thedimensions of the column at the critical section were 3-in X 5m~

To confine failure to the mideight of the column bath ends of the specimellSl

curve Axial load was applied with a 60-ton hydraulic ram connected Id_operated high-pressure pump which provided a nearly constant rate of oil to the ram The load was transmitted to the specimen through a

yoke resting on a spherical head on the ram The movable yoke was to a rigid base plate which distributed the load to the end face

test specimen On the other end a similar rigid plate was pin-connected 00 K load cell which was used to monitor the axial load Both ends of

specimens were grouted with a quick setting high-strength gypsum cement lIoriZontal load was applied through a manually operated screw-type meshysnt As the mechanism was rotated a horizontal thrust was developed

~st the column of the load frame Horizontal loads were applied only to aintain a uniform strain across the specimen

w oJ bl

FIG l-TEST SPECIMEN FIG 2-LOAD FRAME

Strain rates were suc that te maximum c~pac1tyof the s~ecimens su~shywere flared and reInforced WIth No5 bars jected to steadily increasIng stram was reached In 3 mm to 5 mm For cyclIc

The concrete mix proportions were constant for all the specimens The I loading the peak load on the specimens was applied in 1 min to 2 min and reshyconcrete was a blend of Type m Portland Cement and 50 Colorado Rivermiddot moved in about 12 min to 1 min sand and 50 34-in gravel by weight The specimens were cast in a vertical instrumentation -Four OB-in wire resistance gages were used to monitor position ~hree 6-in x 12-in control ~ylinders were cast with each specimen the strains on the two opposite 3-in faces at the mid~elght of ~he column The speclmens were cured m a mOIst room for 2 5 or 7 days and testedati Because the specimens were subJected to strams conSIderably In excess of 7 14 or 21 days Cylinder strengths varied from 3500 psi to 5000 psi Values 1( the normal operating range of the strain gages strains were also measured of f~ for each specimen and complete details of the experimental program over a 6-12-in gage length using two differential transformer displacement are given in Ref 3 I transducers placed at the midsections of the specimens These transducers

Loading Anangement-A rigid loading frame (Fig 2) was constructed in were placed on the opposite 5-in faces of the specimen and were supported which axial Or flexural loads could be applied simultaneously or separately between light steel frames fastened to the specimen with pOinted set screws The silffness of the loading frame was sufficient to avoid problem s associated Test Procedure - Load histories were controlled by monitoring one of two with the release of energy in the unstable portion of the concrete stress- variables (0 Incremental strain during a given cycle or (2) stress level

2547

OJ

2546 December 1969 CONCRETE BEHAVIOR

during a given cycle Loads were applied manually and the magnitude to produce the specified seress level or strain increment as

by an X-Y recorder plot A total of 46 specimens were tested in different series which are identjfied below by the distinguishing feature load history The number of tests in each series is given in parentheses

Series AM1 Steadily Increasing Strain to Failure (13 Specimens)-Specishy e usually cast in groups of two or four The strain was steadily

to failure on one specimen in each group This test was used to the effects of other loading histories Typical stress-strain curves

the test specimens in series AM are shownin Figs 3 4 and 5 The non-A ensiona coordinates F and S will be presented later

IllSedes AC2 Cycles to Envelope Curve (9 Specimens)-The concept of an

bull II Ii

velope curve for the response of concrete has been proposed by other inshyco r Wk1 Klt fC stlgators (10) The envelope curve can be defined as the limiting curveS ttg

FIG 3-CYCLIC LOADING TO ENVELOPE CURVE

10 --I I I

I AMj) b~~=~ F~~50~

-06 ~ shy~ ylt

SMITJ-t- YOUNG - n kll ~oaD

So oc to

FIG 4-COMPARISOK OF ENVELOPE CURVES VITH TEST AC4-10

- I Ibull within which all stress-strain curves lie regardless of the load pattern To

Investigate the validity of the envelope curve the strains in a given cycle were increased until the stress-strain path reached the envelope curve A-01

shy stress-strain curve for test AC2-09 is shown in Fig 3 Series AC3Varying Strain Increments (10 Specimens)-The specimens in

series AC2 were subjected to strain cycles in which a specified strain increshy00 ) ment was added during each cycle Stress-strain curves are shown in Fig 6 1IilUt r [or AC3-10 in which strain increments of 05 x 10-3 were added during each

gt cycle Some of the specimens we re loaded to an initial specified st rain such as 1 x 10-3 and then cycled to produce a given incremental strElJrr such as

bull 01 x 10-3 in each cycle In many of these tests the results we~esimilar to 5 (

those of series AC2 because the incremental strain was large enotigh-i9 proshyduce stress-strain curves which reached the envelope curve bull I

FIG 5-COllIPARISOK OF ENVELOPE CUHVES WITH TEST AC4-13 ~tl AC4 Cyd betwbullbulln Mlmom and Minlmom 81 Lovl (14

J

01-1--shy AI I shy8 II I

I ltV r nI r I1)6 1 I

bull4 1 02

o V 1 vq-r --lt co 10 la 30

5 bull cc

FIG 6-CYCLIC LOADING PRODUCING GIVEN STRAIN INCREMENT

2548 2549

December 1969

Specimens) - In this series load cycles were applied between

e

x 12-in

and the

ched

mens

giv-en levels until the strains stabilized or until the maximum stress lev-el be sustained Maxinum stresses varied between 085 fJ and 059 JI

mum stresses vaned between 0 and 070 I~ Stress-strain Curv-es lor AC4-10 and AC4-13 are shown in Figs 4 and 5 Additional details Of til perimental program are given in flef 3

BEHAVIOR OF TEST SPECIMENS

Monotonic Loading to Failure-The specimens in which the strain steadily increased to failure were used to evaluate the behavior of subjected to other load histories In order to facilitate comparisons test results stress-strain curves are plotted in normalized coordinates stress coordinate F is normali7ed with respect to I the 6-in

10

08

_ltl 06 shy

0

0

00

00 10 30

S= poundpound1)

FIG 7-l]OPOTONIC LOADKG TO FAILURE

inder strength For the specimens under monotonic loading to failure median value of the strength was 085f and the mean wasOS6 with a dard deviation of 004 r The strain coordinate S is normalizedwith to Eo the strain corresponding to the peak stress The median value of the specimens subjected to monotonic loading was 168 10-3

was 171 10-3 with a standard deviation of 014 )lt 10-3bull Strains specimens subjected to cyclic loads were normalized with respect to the of Eo for the specimen which was cast from the sa111e batch of concrete subjected to monotonic loading (0 failure

The results of several of the tests in series AMI are shown in Figs and 5 Points from the stress-strain curves of the 13 tests in this series plotted in Fig 7 Also plotted in Fig 7 are stress-strain relationshi gesled by Smith and Young (11) and Hognestad (12) The equations for curves are expressed in terms of the coordinates F and S The smith-

HOGHESTAO F= oass Z-sl fot 511pound10

i F=OlS1-aI01SforSto

bullbullL bull t

----- - shy I -

i - shy

iMITH~YOIJ~G F= O85Se-(i Sl

2middot0

CONCRETE BEHAVIOR

appears to best fit the observed stress-strain relationship and to approximate the behavior of concrete under monotonic load-

in the age of the concrete at testing and in the strength did not have an influence on the shape of the stress- strain curves obtained the specimens under monotonic loading to failure or those under

and were not considered to be significant variables in the range this investigation

JlilJlletope Curve -The test results as represented by the curves in Fig 3 aindicate that the stress-strain paths under cyclic loading generally

eKceed the envelope curve In those cases where the envelope was the stress-strain curve for the companion specimen under monotonic to failure the comparison is excellent the specimens tested the stress-strain relationship became approxshytangent to the envelope curveasshown by the stress-strain curves

AC2-09 and AC3-10 (Exgs ~iind 6) The same behavior was in specimens AC4-l(i and AC4middot1S (Figs 4 and 5) It is significant

ulation of strain under constant maximum stress levels produced when the envelope was reached

1II Fig 8 another type of load history is shown Even though the cycles quite different than those presented previously the stress-strain curves

specimen AC2-07 remained within the envelope until very high strains

oints plotted in Fig 9 are the values of peak stress and strain from for which the stress-strain path in a given cycle became apshy

coincidental with the envelope curve Although there is some the Smith-Young expression is a good approximation of the test

the specimens tested the envelope curve may be defined as the stress-curve obtained under monotonic loading to failure and approximated by

Smith-Young equation Failure was observed when a given stress-strain exceeded the envelope however the specimen could be loaded to the

regardless of the strain accumulated prior to a given cycle Strain did not appear to reduce the strength to a level below the enshy

Itshould be remembered that the envelope and the stress strain curves be altered if the strain rate or the properties of the concrete were

man Points -Sinha GerstIc and Tulin (W ind1Catedthat the locus of oints where the reloading portion of any cycle crosses the unloading

maybe defined as a stability limit at which the strains stabilize and a hysteresis loop is formed in subsequent cycles Stresses above this

produce additional strains whUe maximum stresses at or below this cause the stress-strain history to go into a loop repeating the pre

cycle without further permanent strain Using this definition if the level corresponding to a common point as indicated in the stressshy

strain history of specimen AC2-09 (Fig 3) is not exceeded in subsequent e)eles strains should not exceed the value at the common point shy

lhebehavior of the specimens in this investigation suggesfifatNjte rigorshydefinition of the stability limit The common poiTt+ampbs~wdin21 cyclic

load tests with various maximum stress levels are plotted in Fig Hr In view Ilfthewide scatter the common pOints may have a range of values

2551

lfi1

2550 December 1969

The scatter may be explained by examining the stress-strain specimen AC2-07 [Fig 8(a)) Four specimens were cycled in this m all exhibited similar behavior The location of the common points

obtained is

in a detail of the stress-strain history [Fig 8(b)) For example after

middot0 TIlT AC -07 f~ oIUOp6i

0 bull

06

04

02

()o 8 10

( A Complete Load History (b) DetaH~Co[mnu POirtt5

FIG 8- VARIA TION OF COMMON POINTS

10

8

U

0

yshyalt

$$ teCo

02

00 20 U

s bull ( (0

FIG 9-POINTS ON ImVELOPE CURVE (J1EASURED)

20 had been carried out the specimen was reloaded until the unloading tion of cycle 20 was reached (point n) and then the specimen was unloaded This routine was continued until the common pOint stabilized at points D and

CONCRETE BEHAVIOR

pound In general the magnitude of the reduction of the point of intersection deshycreased with the number of cycles If the locus of points for the first second third bull common points are drawn a family of common point curves can be

10

O

o

01

00

o 0 10 0 10 s ~ C CQ

FIG lO-COMMON POINTS (MEASURED)

-Ie

00 I ZIl C (0

FIG n-EFFECT OF 1I1INIMUM STRESS LEVEL ON COMMON POINTS ~

test results plottedin Fig 10 and the behavior exhibged tnmiddot tests such as AC2-07 (Fig 8) show that intersecting points of load cyclesttftije envelope Curve constituded an upper limit on the common points (hereafter called comshymon point limit) As cycles with lower stress levels were introduced the

10

2551

lfi1

2550 December 1969

The scatter may be explained by examining the stress-strain specimen AC2-07 [Fig 8(a)) Four specimens were cycled in this m all exhibited similar behavior The location of the common points

obtained is

in a detail of the stress-strain history [Fig 8(b)) For example after

middot0 TIlT AC -07 f~ oIUOp6i

0 bull

06

04

02

()o 8 10

( A Complete Load History (b) DetaH~Co[mnu POirtt5

FIG 8- VARIA TION OF COMMON POINTS

10

8

U

0

yshyalt

$$ teCo

02

00 20 U

s bull ( (0

FIG 9-POINTS ON ImVELOPE CURVE (J1EASURED)

20 had been carried out the specimen was reloaded until the unloading tion of cycle 20 was reached (point n) and then the specimen was unloaded This routine was continued until the common pOint stabilized at points D and

CONCRETE BEHAVIOR

pound In general the magnitude of the reduction of the point of intersection deshycreased with the number of cycles If the locus of points for the first second third bull common points are drawn a family of common point curves can be

10

O

o

01

00

o 0 10 0 10 s ~ C CQ

FIG lO-COMMON POINTS (MEASURED)

-Ie

00 I ZIl C (0

FIG n-EFFECT OF 1I1INIMUM STRESS LEVEL ON COMMON POINTS ~

test results plottedin Fig 10 and the behavior exhibged tnmiddot tests such as AC2-07 (Fig 8) show that intersecting points of load cyclesttftije envelope Curve constituded an upper limit on the common points (hereafter called comshymon point limit) As cycles with lower stress levels were introduced the

10

2553

stress level

~~1yen

2552 December 1969

paint of intersection was reduced but stabiltzed at a lower bound limit)

The effect of the minumum stress level on the common points is iIIull trated in Fig 11 Specimens AC4-12 and AC4-13 were cast from the salll batch of concrete Both were subjected to the same maximum but the minimum stress levels were different The common points for both specimens were identical The same behavior was noted in other specimens On this basis it can be assumed that the common points were independent Of the minimum stress levels in a particular load history

The dependency of the common pOints on the maximum stress level is shown in Fig 12 for 5 tests with cyclic loadings between a zero minimUJll stress level and different maximum stress levels In test AC4-12 the maxi mum stress level 079 f was higher than the peak value of the common

nST FmQIIi(

bull AC4- n 071

middot At 10 U ~middotn bulln 0

AC ~ 03 4040

bull Ae4 -01 ~ I us)

01 ~

ui lo- =t_

+-f-----J----

When

0 1-----

liMn

~ - G

bullbull I I bull JO1000

s bull C Ie

FIG 12-COMMON POINTS FOR TESTS WITH CONSTANT MAXIMUM STRESS LEVEL

point limit and as a result the points of i~tersection formed a smooth curve located approximately on the common paint limit The maximum stresslevel for test AC4-1O was 076 f ~ which was about equal to the peak value of the common point limit The points of intersection for this speCimen followed the common point limit initially then formed an approximately horizontal until the strain accumulation reached the common point limit This trend also apparent in test AC4-11 Although the strain accumulation was than expected this can be explained by the higher envelope curve m for the companion specimen under monotonic loading to failure maximum stress was reduced to 063 nand 055 f in tests AC4-03 and 01 the cOmmon points gradually increased but approached the stability

and strainaccuinulation ceased under continued The observed behavior may be summarized as

1 The stress and strain at the peak of the load cycle were the prime

CONCRETE BEHAVIOR

abIes in determining the location of the common point Minimum stress levels did not appear to have a significant effect on the common points

2 Peak stress-strain values above the common point limit produced points of intersection very near this limit With lower peak values the points of intersection fell between the common pOint limit and the stability limit

0 11--__

I I 1111111 H H i

10 +--------- ---shy

lt

oU

04

02

00 J I q rl 00 200 0

s tt

FIG 13-LOADINGCURVES

10

O

-v 06

shy ~

04

01

00

00 10 20 0 SmiddotpoundO

FIG 14-UNLOADINGCURVES

3 If the stress and strain at the peak of the load cycle wataoove the stashybility limit strains accumulated until failure occurred oruntif strain accushymulations reached the stability limit At this point strains stabilized and formed a closed hysteresis loop for subsequent cycles

2555

lE~1ttlli)iI1

2554 December 1969

Note that if the effects of time were considered creep strains would

there wasthe observed behavior With reduced strain rates the stress-strain would shift toward the strain axis and it would be difficult to define a limit (6)

Nonrecoverable Strains -Nonrecoverable or plastic strains are the strains corresponding to a zero stress level on loading or unloading stress-strain curves The changes observed in the slopes of the stress-strain curves sug gest a relationship between the plastic strain ratio Sp and the nature of the loading curves

Loading curves from a number of spec1mens subjected to different lOad histories are plotted in Fig 13 Each group of curves originated from a similar plastic strain ratio It is apparent that the slope of the curves gradu ally decreased with increasing values of Sp The common point limit (the locus of common points of load cycles to the envelope curve) is also shown in Fig 13 It can be seen that the common point limit corresponds approxishymately to the point at which the slope of loading curves changes significantly Previous investigations (79) have shown that the change in slope can be atshytributed to a significant increase in microcracking

Unloading curves from a number of specimens in which the unloading portion of the cycle started at or near the envelope are plotted in Fig 14 In each case the minimum stress level was zero These plots show that the plasshytic strain ratio was a major variable in determining the shape of the lOading and unloading curves The load history preceding a given value of Sp did not Significantly alter the curves originating at that value of S p

PREDICTION OF FAILURE

Derivation oj Expressions jor Stress-Strain Curves- Using the observed response of the specimens expressions were developed for loading and unshyloading stress-strain curves in order to duplicate the observed response analytically and predict failure under load histories other than those actually imposed on the specimens As shown in Figs 13 and 14 the shapes of the loading and unloading curves appear to be functions of the nonrecoverable or plastic strain ratio In order to develop expressions for these curves various polynomials were compared with the experimental curves and a second deshygree parabola was selected to represent the shape of the curves Better apshyproximations with higher order or transcendental expressions might have been obtained However considering the accuracy of the test results the advantages of a simple stress-strain relation outweigh the small gain in acshycuracy derived using higher order approx1mations

To account for the changing shape of the loading and unloading curves with increasing plastic strains the stress-strain curves were developed as funcshytions of the plastic strain ratio For a given plastic strain ratio relationships between the strain at which a loading curve will intersect the previous unshyloading curve (common point) and the envelope curve were obtained

Envelope Curve-The equation for the envelope curve the expression deshyveloped by Smith and Young (12) has been presented previously (Fig 7 and 9) and is repeated below in terms of the nOrmalized parameter FE and SE points on the envelope

(I-SE) FE = 0S5 SE e bullbull (1)

CONCRETE BEHAVIOR

common Points-The experimental results (Fig 10)tndicated that although a variation in the location of the common points a common point

lilllitand a stability limit could be established Analysis of the common points produced exponential expressions of a form similar to the envelope curve

_ Se [t-S e(O315+0n(3)] Fe - 3 0315 + 0773 e bullbullbull (2)

The common point limit 3 076 and the stability limit 3 = 063 are plotted in Fig 10 The variation in 3 accounts for the change in the maxima of the lilllits Since the common point for a given cycle of load was dependent on the Illagnitude of stress in the previous cycle the following distinction must be Illade as to the value of 3 for the applicable common point

1 If the peak lies above the common point limit 3076 the common point is on the common point limit

2 If the peak lies in the regiOn between themiddot common point and stability limits 3 varies betweenO 76 and 063

3 If the peak lies below the stability limit the common point corr~sponds to the peak and the stress-strain curves form a closed hystereSiS loop Note that this criterion implies that if stresses do not exceed 063 n cyclic loadshyings will not produce failure

The values of 3 for the common point and stability liroits are in the range of critical stresses reported by other investigators Shah et al (79) have reported the onset of major microcracking at 70 to 90 of the ultimate load The value of 3 at the stability limit is 063 n (74 of the specimen strength) andat the common point limit is D76 n (90 of the specimen strength) which indicates that the behavior of the concrete is controlled primarily by microshycracking It is also interesting to note that RUsch (6) has reported that the sustained load strength of concentrically loaded specimens is 75 to SO of the static strength which corresponds with the value 6f 3 for the stability limit

Plastic Strain Ratio (Sp)-Fig 15 shows the relationship between the plasshytic strain ratioSp and the strain ratio at the common point Se The expression for the curves passing through these pOints is the following

Sp (176 - fl)(0160 S( + 0133 Se) (3)

in which 063 S (3 S 076 Fig 16(a) shows the relation between the plastic strain ratio Sp and the strain ratio at the point where a given loading curve starting at Sp intersects the envelope curve SE Fig 16(b) shows the relation between the plastiC strain ratiO S p and the strain ratio at the point where a given unloading curve starts on the envelope curve The equations for these relationships are the following

Loading Sp 0093 Sj + 091 SE (4)

- ~~tUnloading Sp 0145 Sf + 013 SF bullbullbullbullbullbullbullbullbullbull ~~c~ bullbull

Loading Curves -The expressions for loading curves are secolld degree parabolas which pass through the following three points (1) The point at which the reloading curve or its extension starts (Sp F =0) (2the centommonpoint

2557

lWMiyenlflifV)p ~

2556 December 1969

u~r----

~ e

~I

=1pound

U+------~ ~--

ID (b) poundIt

FIG 15-RELATIONSHIP BETWEEN PLASTIC STRAIN AND STRAIN AT COMMON POINTS

~ ~L

U

1

7r

$p o_onsE O-lUUf

UtoIIJlAOfiG FII~O ENVELOPE TO Sp_

101 eE 1lt

FIG IS-RELATIONSHIP BETWEEt ENVELOPE STRAINS AND PLASTIC STRAINS

CONCRETE BEHAVIOR

and (3) the point at which the reloading curve or its extension(Sc Fc) aches the envelope curve (SE FE)

re Unloading Curves - The three points through which the second degree pashyrabola unloading curves pass are as follows (1) The pOint at which the unshyloading curve or its extension to the envelope originates (SE FE) (2) the ollllllon point (Se Fe) and (3) the plastic strain ratio (Sp 0) at which the ~nloading curve or its extension terminates

For cycle ABCD[Fig 17(a)] the three pOints through which the curves pass are determined in the following manner Thevalue of Sp at point B is found using Eq 5 (unloading from A) point C is found using Eq 3 for Se and Eq 2 for FC point D is found by solving Eq 4 for SE and Eq 1 for FE

10

__-1bullbull NV 0 -shy -- shy0 bullbull

0

COMMON POINT LIMIT0 (ll=o7ol

00 r (e 00 -nl 0

0

0

(514111 NVltOPbull -~~ 0 __ - jt~ lSI f$mQJ~-

A G - 06

1

I

---

0

0-1 ~ I

oor (- 10 10bull fbi

FIG n-LOADING AND UNLOADING CURVES (COMPUTED)

For cycle EFGH [Fig 17(b)] between specified values of FmaxandFmin some modification of the preceding procedure is necessary The unloading curve EGF is part of curve E GF and the value of S p for curve E GF is found by trial and error so that the curve will pass through point E when Eqs 1 2 3 and 5 are satisfied The unloading curvemiddot is terminated at F when F min Is reached The loading portion FGH passes through points ($~Fmin) (Se Fe) which was found in the preceding step and point H (SE F~determined using the value of at pOint F inEq 4 The curve is terminat-edwhen Fmax is reached A similar approach may be used to determine stressstrain paths if a given strain increment is to be added

A computer program was written to solve the various cases presented

2561 2560 December 1969

0lt15 4lt~--lt lt-lt ~~poundASUMO-

[ _0 ltT-laquolt060

Ij 0lt75

lt I IEXPfRIENTAL (FNlCURVEI

01lt I 1 1- bull

IL_ Il ~ --- --0__--___

0 I j I

--I middot I I _ 01$ Pshy FOtiQuolmlOltU - __lt_

- ~ Imiddot I_lt-middotI---middot~middotii-I - l 060

00 Ht N Numb of Cyclbullbull

FIG 22-NUMBER OF CYCLES TO FAILURE (Fmax CONSTANT Fmin 0)

~~I

J K

0middot8

l~~ 015

01 0amp 0lt 0-3

_ 02 01+ 0

0middot0

f= ___ --A~LllM ___ _____

Omiddot I-------------r-------shy0lt1 ZOO 00

N =Number 01 Cycles

FIG 23-NUlIBEH OF CYCLES TO FAILURE(Fmax CONSTANT Froin I 0)

CONCRETE BEHAVIOR

and the two curves should intersect at some point below D [Fig 19(b) J In adshydition the plastic strain ratios at point G differ considerably On the basis of the observed results the assumption of uniqueness would not appear to be fllrranted

Fig 20 shows the computed and measured response for specimen AC4-10 which was cycled between stress levels of F max = 077 and F min O The speeimen failed in cycle 21 and failure was predicted in cycle 25 In Fig 21 the response of specimen AC4-13 is shown This specimen was subjected to cyclic loads between stress levels of Fmax 079 and Fmin == 040 The specshylInen failed in cycle 28 and failure was predicted in cycle 34 If uniqueness of the loading and unloading curves was assumed failure would be predicted after only three cycles

The computed number of cycles to fallure for tests in which the load is varied between a given maximum stress level Fmax and a minimum stress of zero is shown in Fig 22 Both measured and computed values are ltplotted Since the observed maximum of the stability limit was at a stress ratio of 063 the experimental curves shown in Fig 22 will become asymptotic to F ~ 063 Fig 23 shows the computed nuinber of cycles to failure for loadings between given maximum and minimum stress levels The maximum stresS ratio is plotted along the ordinate For example the number of cycles to failure with Fmax == 080 and F min == 0040 is approximately 25 Using these curves (Figs 22 23) the number of cycles to failure may be estimated

SUMMARY AND CONCLUSIONS

A series of 46 short rectangular test specimens (Fig I) were subjected to repetitions of compressive stress to various levels to obtain expressions for the response of plain concrete The expressions developed are functions of the ultimate stress and strain values of standard 6 x 12-in control cylinders and the loading history Using these expressions the response of plain conshycrete subjected to varying load histories can be estimated

The following conclusions were obtained

1 For the specimens tested the envelope curve coincided with the stressshystrain curve for a specimen under monotonic loading to failure (Fig 7) The stress-strain path reached the envelope regardless of the strain accumulated prior to a particular cycle

2 The location of the common pOints was dependent primarily on the magshynitude of the maximum stress and strain of the previous load cycle The comshymon points for loading from nonzero levels were identical to the common points corresponding to load cycles starting at a stress lev~l of zero (Fig 11)

3 Examination of the location of the common points shows that failure would be produced under repeated loads with stresses exceeding about 0 63 f ~ the maximum of the stability limit This limit was independent of the minishymum stress levels in the cycles

middot4 Loading and unloading curves starting from a pOint wtt11tli tke stressshystrain domain were not uriique-andthe value of stress and strain at the peak of the previous loading cycle must be known to estimate the relptmse

5 The analytical expressions obtained for the envelope curve the common point and the stability limits and the loading and unloading stress-strain reshy

2563

shy

2562 December 1969

lations produce results that compare well with the experimental results (Fi~s 3 1B 20 21) The formulation of these expressions provides a general lUethi od for estimating the number of cycles to failure under repeated loads (FigS 22 23) with strairi rates similar to those considered in the investigation

APPENDIX I-REFERENCES

I Hognestad E A Study of Combined Bending and Axial load in Reinforced Concrete Mein bers University of Illinois Engineering Experimental Station Bulletin Series No 399 195L

2 Hognestad E Hanson N W and McHenry Dbull Concrete Stress Distribution in Ultimate Strength Design Journal of the American Concrttl1lnstitute Vol 52 No4 December 1955 pp455-479 i

3 Karsan I D Behavior of Plain Concrete under Variable load Histories thesis presented to Rice University at Houston Texas in 1968 in partial fulfillment of the requirements ror the degree of Doctor of Philosophy

4 Murdock J W and Kesler C E Effect of Range of Stress on Fatigue Strength or Plain Concrete Beams Journal of the American Concrete Institute Vol 55 No2 August 1958 Pp 221-231

5 Nordby G M Fatigue of Concrete-A Review of Research Journal of the American Con crete Institute Vol 55 No2 August 1958 pp 191-219

6 Rusch H Researches toward a General Flexural Theory for Structural Concrete Journal oJ the American Concrete Institute Vol 57 No I July 1960 pp 128

7 Shah S P Sturman G M and Winter G Microcnicking and Inelastic Behavior of Con crete Flexural Mechanics of Reinforced Concrete ASCE 196550 The International Sym posium Miami Florida 1964

8 Shah S P and Winter G Inelastic Behavior and Fracture of Concrete Journal of the American Concrete Institute Vol 63 No9 September 1966 pp 925-930

9 Shah S P and Winter G Response of Concrete to Repeated Loadings RlLEM Internamiddot tional Symposium on the Effects of Repeated Loading on Materials and Structural Elements Mexico City 1966

10 Sinha B P Gefstle K H and Tulin l G Stress-Strain Relations for Concrete under Cyclic Loading Journal of the American Concrete Institute VoL 61 No2 February 1964 pp 195-211

Smith G M and Young l E Ultimate Theory in Flexure by Exponential Function Jour nal of the American Concrete Institute Vol 52 No3 November 1955 pp 349 359

APPENDIX II -NOTATION

The following symbols are used in this paper

f~ = ultimate compressive strength ofstandard 6-in x 12-incylinder f = concrete stress

f max = maximum compressive stress reached in a given cycle F = I In = stress ratio

CONCRETE BEHAVIOR

FaX = maximum stress ratio in a given cycle in minimum stress ratio in a given cycleFe = stress ratio at the common point FE = stress ratio on the envelope curve

S = tlto strain ratio maximum strain ratio in a given cycle

SJUllC minimum strain ratio in a given cycleSIIlin strain ratio at the common point Se euro pi euro 0 = plastic or residual strain ratioSp strain ratio on the envelope curve a factor relating the common point with the stress and strain ratios of the peak of the previous load cycle

( = concrete strain at I (0 concrete strain at I ~ and (p plastic strain

2555

lE~1ttlli)iI1

2554 December 1969

Note that if the effects of time were considered creep strains would

there wasthe observed behavior With reduced strain rates the stress-strain would shift toward the strain axis and it would be difficult to define a limit (6)

Nonrecoverable Strains -Nonrecoverable or plastic strains are the strains corresponding to a zero stress level on loading or unloading stress-strain curves The changes observed in the slopes of the stress-strain curves sug gest a relationship between the plastic strain ratio Sp and the nature of the loading curves

Loading curves from a number of spec1mens subjected to different lOad histories are plotted in Fig 13 Each group of curves originated from a similar plastic strain ratio It is apparent that the slope of the curves gradu ally decreased with increasing values of Sp The common point limit (the locus of common points of load cycles to the envelope curve) is also shown in Fig 13 It can be seen that the common point limit corresponds approxishymately to the point at which the slope of loading curves changes significantly Previous investigations (79) have shown that the change in slope can be atshytributed to a significant increase in microcracking

Unloading curves from a number of specimens in which the unloading portion of the cycle started at or near the envelope are plotted in Fig 14 In each case the minimum stress level was zero These plots show that the plasshytic strain ratio was a major variable in determining the shape of the lOading and unloading curves The load history preceding a given value of Sp did not Significantly alter the curves originating at that value of S p

PREDICTION OF FAILURE

Derivation oj Expressions jor Stress-Strain Curves- Using the observed response of the specimens expressions were developed for loading and unshyloading stress-strain curves in order to duplicate the observed response analytically and predict failure under load histories other than those actually imposed on the specimens As shown in Figs 13 and 14 the shapes of the loading and unloading curves appear to be functions of the nonrecoverable or plastic strain ratio In order to develop expressions for these curves various polynomials were compared with the experimental curves and a second deshygree parabola was selected to represent the shape of the curves Better apshyproximations with higher order or transcendental expressions might have been obtained However considering the accuracy of the test results the advantages of a simple stress-strain relation outweigh the small gain in acshycuracy derived using higher order approx1mations

To account for the changing shape of the loading and unloading curves with increasing plastic strains the stress-strain curves were developed as funcshytions of the plastic strain ratio For a given plastic strain ratio relationships between the strain at which a loading curve will intersect the previous unshyloading curve (common point) and the envelope curve were obtained

Envelope Curve-The equation for the envelope curve the expression deshyveloped by Smith and Young (12) has been presented previously (Fig 7 and 9) and is repeated below in terms of the nOrmalized parameter FE and SE points on the envelope

(I-SE) FE = 0S5 SE e bullbull (1)

CONCRETE BEHAVIOR

common Points-The experimental results (Fig 10)tndicated that although a variation in the location of the common points a common point

lilllitand a stability limit could be established Analysis of the common points produced exponential expressions of a form similar to the envelope curve

_ Se [t-S e(O315+0n(3)] Fe - 3 0315 + 0773 e bullbullbull (2)

The common point limit 3 076 and the stability limit 3 = 063 are plotted in Fig 10 The variation in 3 accounts for the change in the maxima of the lilllits Since the common point for a given cycle of load was dependent on the Illagnitude of stress in the previous cycle the following distinction must be Illade as to the value of 3 for the applicable common point

1 If the peak lies above the common point limit 3076 the common point is on the common point limit

2 If the peak lies in the regiOn between themiddot common point and stability limits 3 varies betweenO 76 and 063

3 If the peak lies below the stability limit the common point corr~sponds to the peak and the stress-strain curves form a closed hystereSiS loop Note that this criterion implies that if stresses do not exceed 063 n cyclic loadshyings will not produce failure

The values of 3 for the common point and stability liroits are in the range of critical stresses reported by other investigators Shah et al (79) have reported the onset of major microcracking at 70 to 90 of the ultimate load The value of 3 at the stability limit is 063 n (74 of the specimen strength) andat the common point limit is D76 n (90 of the specimen strength) which indicates that the behavior of the concrete is controlled primarily by microshycracking It is also interesting to note that RUsch (6) has reported that the sustained load strength of concentrically loaded specimens is 75 to SO of the static strength which corresponds with the value 6f 3 for the stability limit

Plastic Strain Ratio (Sp)-Fig 15 shows the relationship between the plasshytic strain ratioSp and the strain ratio at the common point Se The expression for the curves passing through these pOints is the following

Sp (176 - fl)(0160 S( + 0133 Se) (3)

in which 063 S (3 S 076 Fig 16(a) shows the relation between the plastic strain ratio Sp and the strain ratio at the point where a given loading curve starting at Sp intersects the envelope curve SE Fig 16(b) shows the relation between the plastiC strain ratiO S p and the strain ratio at the point where a given unloading curve starts on the envelope curve The equations for these relationships are the following

Loading Sp 0093 Sj + 091 SE (4)

- ~~tUnloading Sp 0145 Sf + 013 SF bullbullbullbullbullbullbullbullbullbull ~~c~ bullbull

Loading Curves -The expressions for loading curves are secolld degree parabolas which pass through the following three points (1) The point at which the reloading curve or its extension starts (Sp F =0) (2the centommonpoint

2557

lWMiyenlflifV)p ~

2556 December 1969

u~r----

~ e

~I

=1pound

U+------~ ~--

ID (b) poundIt

FIG 15-RELATIONSHIP BETWEEN PLASTIC STRAIN AND STRAIN AT COMMON POINTS

~ ~L

U

1

7r

$p o_onsE O-lUUf

UtoIIJlAOfiG FII~O ENVELOPE TO Sp_

101 eE 1lt

FIG IS-RELATIONSHIP BETWEEt ENVELOPE STRAINS AND PLASTIC STRAINS

CONCRETE BEHAVIOR

and (3) the point at which the reloading curve or its extension(Sc Fc) aches the envelope curve (SE FE)

re Unloading Curves - The three points through which the second degree pashyrabola unloading curves pass are as follows (1) The pOint at which the unshyloading curve or its extension to the envelope originates (SE FE) (2) the ollllllon point (Se Fe) and (3) the plastic strain ratio (Sp 0) at which the ~nloading curve or its extension terminates

For cycle ABCD[Fig 17(a)] the three pOints through which the curves pass are determined in the following manner Thevalue of Sp at point B is found using Eq 5 (unloading from A) point C is found using Eq 3 for Se and Eq 2 for FC point D is found by solving Eq 4 for SE and Eq 1 for FE

10

__-1bullbull NV 0 -shy -- shy0 bullbull

0

COMMON POINT LIMIT0 (ll=o7ol

00 r (e 00 -nl 0

0

0

(514111 NVltOPbull -~~ 0 __ - jt~ lSI f$mQJ~-

A G - 06

1

I

---

0

0-1 ~ I

oor (- 10 10bull fbi

FIG n-LOADING AND UNLOADING CURVES (COMPUTED)

For cycle EFGH [Fig 17(b)] between specified values of FmaxandFmin some modification of the preceding procedure is necessary The unloading curve EGF is part of curve E GF and the value of S p for curve E GF is found by trial and error so that the curve will pass through point E when Eqs 1 2 3 and 5 are satisfied The unloading curvemiddot is terminated at F when F min Is reached The loading portion FGH passes through points ($~Fmin) (Se Fe) which was found in the preceding step and point H (SE F~determined using the value of at pOint F inEq 4 The curve is terminat-edwhen Fmax is reached A similar approach may be used to determine stressstrain paths if a given strain increment is to be added

A computer program was written to solve the various cases presented

2561 2560 December 1969

0lt15 4lt~--lt lt-lt ~~poundASUMO-

[ _0 ltT-laquolt060

Ij 0lt75

lt I IEXPfRIENTAL (FNlCURVEI

01lt I 1 1- bull

IL_ Il ~ --- --0__--___

0 I j I

--I middot I I _ 01$ Pshy FOtiQuolmlOltU - __lt_

- ~ Imiddot I_lt-middotI---middot~middotii-I - l 060

00 Ht N Numb of Cyclbullbull

FIG 22-NUMBER OF CYCLES TO FAILURE (Fmax CONSTANT Fmin 0)

~~I

J K

0middot8

l~~ 015

01 0amp 0lt 0-3

_ 02 01+ 0

0middot0

f= ___ --A~LllM ___ _____

Omiddot I-------------r-------shy0lt1 ZOO 00

N =Number 01 Cycles

FIG 23-NUlIBEH OF CYCLES TO FAILURE(Fmax CONSTANT Froin I 0)

CONCRETE BEHAVIOR

and the two curves should intersect at some point below D [Fig 19(b) J In adshydition the plastic strain ratios at point G differ considerably On the basis of the observed results the assumption of uniqueness would not appear to be fllrranted

Fig 20 shows the computed and measured response for specimen AC4-10 which was cycled between stress levels of F max = 077 and F min O The speeimen failed in cycle 21 and failure was predicted in cycle 25 In Fig 21 the response of specimen AC4-13 is shown This specimen was subjected to cyclic loads between stress levels of Fmax 079 and Fmin == 040 The specshylInen failed in cycle 28 and failure was predicted in cycle 34 If uniqueness of the loading and unloading curves was assumed failure would be predicted after only three cycles

The computed number of cycles to fallure for tests in which the load is varied between a given maximum stress level Fmax and a minimum stress of zero is shown in Fig 22 Both measured and computed values are ltplotted Since the observed maximum of the stability limit was at a stress ratio of 063 the experimental curves shown in Fig 22 will become asymptotic to F ~ 063 Fig 23 shows the computed nuinber of cycles to failure for loadings between given maximum and minimum stress levels The maximum stresS ratio is plotted along the ordinate For example the number of cycles to failure with Fmax == 080 and F min == 0040 is approximately 25 Using these curves (Figs 22 23) the number of cycles to failure may be estimated

SUMMARY AND CONCLUSIONS

A series of 46 short rectangular test specimens (Fig I) were subjected to repetitions of compressive stress to various levels to obtain expressions for the response of plain concrete The expressions developed are functions of the ultimate stress and strain values of standard 6 x 12-in control cylinders and the loading history Using these expressions the response of plain conshycrete subjected to varying load histories can be estimated

The following conclusions were obtained

1 For the specimens tested the envelope curve coincided with the stressshystrain curve for a specimen under monotonic loading to failure (Fig 7) The stress-strain path reached the envelope regardless of the strain accumulated prior to a particular cycle

2 The location of the common pOints was dependent primarily on the magshynitude of the maximum stress and strain of the previous load cycle The comshymon points for loading from nonzero levels were identical to the common points corresponding to load cycles starting at a stress lev~l of zero (Fig 11)

3 Examination of the location of the common points shows that failure would be produced under repeated loads with stresses exceeding about 0 63 f ~ the maximum of the stability limit This limit was independent of the minishymum stress levels in the cycles

middot4 Loading and unloading curves starting from a pOint wtt11tli tke stressshystrain domain were not uriique-andthe value of stress and strain at the peak of the previous loading cycle must be known to estimate the relptmse

5 The analytical expressions obtained for the envelope curve the common point and the stability limits and the loading and unloading stress-strain reshy

2563

shy

2562 December 1969

lations produce results that compare well with the experimental results (Fi~s 3 1B 20 21) The formulation of these expressions provides a general lUethi od for estimating the number of cycles to failure under repeated loads (FigS 22 23) with strairi rates similar to those considered in the investigation

APPENDIX I-REFERENCES

I Hognestad E A Study of Combined Bending and Axial load in Reinforced Concrete Mein bers University of Illinois Engineering Experimental Station Bulletin Series No 399 195L

2 Hognestad E Hanson N W and McHenry Dbull Concrete Stress Distribution in Ultimate Strength Design Journal of the American Concrttl1lnstitute Vol 52 No4 December 1955 pp455-479 i

3 Karsan I D Behavior of Plain Concrete under Variable load Histories thesis presented to Rice University at Houston Texas in 1968 in partial fulfillment of the requirements ror the degree of Doctor of Philosophy

4 Murdock J W and Kesler C E Effect of Range of Stress on Fatigue Strength or Plain Concrete Beams Journal of the American Concrete Institute Vol 55 No2 August 1958 Pp 221-231

5 Nordby G M Fatigue of Concrete-A Review of Research Journal of the American Con crete Institute Vol 55 No2 August 1958 pp 191-219

6 Rusch H Researches toward a General Flexural Theory for Structural Concrete Journal oJ the American Concrete Institute Vol 57 No I July 1960 pp 128

7 Shah S P Sturman G M and Winter G Microcnicking and Inelastic Behavior of Con crete Flexural Mechanics of Reinforced Concrete ASCE 196550 The International Sym posium Miami Florida 1964

8 Shah S P and Winter G Inelastic Behavior and Fracture of Concrete Journal of the American Concrete Institute Vol 63 No9 September 1966 pp 925-930

9 Shah S P and Winter G Response of Concrete to Repeated Loadings RlLEM Internamiddot tional Symposium on the Effects of Repeated Loading on Materials and Structural Elements Mexico City 1966

10 Sinha B P Gefstle K H and Tulin l G Stress-Strain Relations for Concrete under Cyclic Loading Journal of the American Concrete Institute VoL 61 No2 February 1964 pp 195-211

Smith G M and Young l E Ultimate Theory in Flexure by Exponential Function Jour nal of the American Concrete Institute Vol 52 No3 November 1955 pp 349 359

APPENDIX II -NOTATION

The following symbols are used in this paper

f~ = ultimate compressive strength ofstandard 6-in x 12-incylinder f = concrete stress

f max = maximum compressive stress reached in a given cycle F = I In = stress ratio

CONCRETE BEHAVIOR

FaX = maximum stress ratio in a given cycle in minimum stress ratio in a given cycleFe = stress ratio at the common point FE = stress ratio on the envelope curve

S = tlto strain ratio maximum strain ratio in a given cycle

SJUllC minimum strain ratio in a given cycleSIIlin strain ratio at the common point Se euro pi euro 0 = plastic or residual strain ratioSp strain ratio on the envelope curve a factor relating the common point with the stress and strain ratios of the peak of the previous load cycle

( = concrete strain at I (0 concrete strain at I ~ and (p plastic strain

2557

lWMiyenlflifV)p ~

2556 December 1969

u~r----

~ e

~I

=1pound

U+------~ ~--

ID (b) poundIt

FIG 15-RELATIONSHIP BETWEEN PLASTIC STRAIN AND STRAIN AT COMMON POINTS

~ ~L

U

1

7r

$p o_onsE O-lUUf

UtoIIJlAOfiG FII~O ENVELOPE TO Sp_

101 eE 1lt

FIG IS-RELATIONSHIP BETWEEt ENVELOPE STRAINS AND PLASTIC STRAINS

CONCRETE BEHAVIOR

and (3) the point at which the reloading curve or its extension(Sc Fc) aches the envelope curve (SE FE)

re Unloading Curves - The three points through which the second degree pashyrabola unloading curves pass are as follows (1) The pOint at which the unshyloading curve or its extension to the envelope originates (SE FE) (2) the ollllllon point (Se Fe) and (3) the plastic strain ratio (Sp 0) at which the ~nloading curve or its extension terminates

For cycle ABCD[Fig 17(a)] the three pOints through which the curves pass are determined in the following manner Thevalue of Sp at point B is found using Eq 5 (unloading from A) point C is found using Eq 3 for Se and Eq 2 for FC point D is found by solving Eq 4 for SE and Eq 1 for FE

10

__-1bullbull NV 0 -shy -- shy0 bullbull

0

COMMON POINT LIMIT0 (ll=o7ol

00 r (e 00 -nl 0

0

0

(514111 NVltOPbull -~~ 0 __ - jt~ lSI f$mQJ~-

A G - 06

1

I

---

0

0-1 ~ I

oor (- 10 10bull fbi

FIG n-LOADING AND UNLOADING CURVES (COMPUTED)

For cycle EFGH [Fig 17(b)] between specified values of FmaxandFmin some modification of the preceding procedure is necessary The unloading curve EGF is part of curve E GF and the value of S p for curve E GF is found by trial and error so that the curve will pass through point E when Eqs 1 2 3 and 5 are satisfied The unloading curvemiddot is terminated at F when F min Is reached The loading portion FGH passes through points ($~Fmin) (Se Fe) which was found in the preceding step and point H (SE F~determined using the value of at pOint F inEq 4 The curve is terminat-edwhen Fmax is reached A similar approach may be used to determine stressstrain paths if a given strain increment is to be added

A computer program was written to solve the various cases presented

2561 2560 December 1969

0lt15 4lt~--lt lt-lt ~~poundASUMO-

[ _0 ltT-laquolt060

Ij 0lt75

lt I IEXPfRIENTAL (FNlCURVEI

01lt I 1 1- bull

IL_ Il ~ --- --0__--___

0 I j I

--I middot I I _ 01$ Pshy FOtiQuolmlOltU - __lt_

- ~ Imiddot I_lt-middotI---middot~middotii-I - l 060

00 Ht N Numb of Cyclbullbull

FIG 22-NUMBER OF CYCLES TO FAILURE (Fmax CONSTANT Fmin 0)

~~I

J K

0middot8

l~~ 015

01 0amp 0lt 0-3

_ 02 01+ 0

0middot0

f= ___ --A~LllM ___ _____

Omiddot I-------------r-------shy0lt1 ZOO 00

N =Number 01 Cycles

FIG 23-NUlIBEH OF CYCLES TO FAILURE(Fmax CONSTANT Froin I 0)

CONCRETE BEHAVIOR

and the two curves should intersect at some point below D [Fig 19(b) J In adshydition the plastic strain ratios at point G differ considerably On the basis of the observed results the assumption of uniqueness would not appear to be fllrranted

Fig 20 shows the computed and measured response for specimen AC4-10 which was cycled between stress levels of F max = 077 and F min O The speeimen failed in cycle 21 and failure was predicted in cycle 25 In Fig 21 the response of specimen AC4-13 is shown This specimen was subjected to cyclic loads between stress levels of Fmax 079 and Fmin == 040 The specshylInen failed in cycle 28 and failure was predicted in cycle 34 If uniqueness of the loading and unloading curves was assumed failure would be predicted after only three cycles

The computed number of cycles to fallure for tests in which the load is varied between a given maximum stress level Fmax and a minimum stress of zero is shown in Fig 22 Both measured and computed values are ltplotted Since the observed maximum of the stability limit was at a stress ratio of 063 the experimental curves shown in Fig 22 will become asymptotic to F ~ 063 Fig 23 shows the computed nuinber of cycles to failure for loadings between given maximum and minimum stress levels The maximum stresS ratio is plotted along the ordinate For example the number of cycles to failure with Fmax == 080 and F min == 0040 is approximately 25 Using these curves (Figs 22 23) the number of cycles to failure may be estimated

SUMMARY AND CONCLUSIONS

A series of 46 short rectangular test specimens (Fig I) were subjected to repetitions of compressive stress to various levels to obtain expressions for the response of plain concrete The expressions developed are functions of the ultimate stress and strain values of standard 6 x 12-in control cylinders and the loading history Using these expressions the response of plain conshycrete subjected to varying load histories can be estimated

The following conclusions were obtained

1 For the specimens tested the envelope curve coincided with the stressshystrain curve for a specimen under monotonic loading to failure (Fig 7) The stress-strain path reached the envelope regardless of the strain accumulated prior to a particular cycle

2 The location of the common pOints was dependent primarily on the magshynitude of the maximum stress and strain of the previous load cycle The comshymon points for loading from nonzero levels were identical to the common points corresponding to load cycles starting at a stress lev~l of zero (Fig 11)

3 Examination of the location of the common points shows that failure would be produced under repeated loads with stresses exceeding about 0 63 f ~ the maximum of the stability limit This limit was independent of the minishymum stress levels in the cycles

middot4 Loading and unloading curves starting from a pOint wtt11tli tke stressshystrain domain were not uriique-andthe value of stress and strain at the peak of the previous loading cycle must be known to estimate the relptmse

5 The analytical expressions obtained for the envelope curve the common point and the stability limits and the loading and unloading stress-strain reshy

2563

shy

2562 December 1969

lations produce results that compare well with the experimental results (Fi~s 3 1B 20 21) The formulation of these expressions provides a general lUethi od for estimating the number of cycles to failure under repeated loads (FigS 22 23) with strairi rates similar to those considered in the investigation

APPENDIX I-REFERENCES

I Hognestad E A Study of Combined Bending and Axial load in Reinforced Concrete Mein bers University of Illinois Engineering Experimental Station Bulletin Series No 399 195L

2 Hognestad E Hanson N W and McHenry Dbull Concrete Stress Distribution in Ultimate Strength Design Journal of the American Concrttl1lnstitute Vol 52 No4 December 1955 pp455-479 i

3 Karsan I D Behavior of Plain Concrete under Variable load Histories thesis presented to Rice University at Houston Texas in 1968 in partial fulfillment of the requirements ror the degree of Doctor of Philosophy

4 Murdock J W and Kesler C E Effect of Range of Stress on Fatigue Strength or Plain Concrete Beams Journal of the American Concrete Institute Vol 55 No2 August 1958 Pp 221-231

5 Nordby G M Fatigue of Concrete-A Review of Research Journal of the American Con crete Institute Vol 55 No2 August 1958 pp 191-219

6 Rusch H Researches toward a General Flexural Theory for Structural Concrete Journal oJ the American Concrete Institute Vol 57 No I July 1960 pp 128

7 Shah S P Sturman G M and Winter G Microcnicking and Inelastic Behavior of Con crete Flexural Mechanics of Reinforced Concrete ASCE 196550 The International Sym posium Miami Florida 1964

8 Shah S P and Winter G Inelastic Behavior and Fracture of Concrete Journal of the American Concrete Institute Vol 63 No9 September 1966 pp 925-930

9 Shah S P and Winter G Response of Concrete to Repeated Loadings RlLEM Internamiddot tional Symposium on the Effects of Repeated Loading on Materials and Structural Elements Mexico City 1966

10 Sinha B P Gefstle K H and Tulin l G Stress-Strain Relations for Concrete under Cyclic Loading Journal of the American Concrete Institute VoL 61 No2 February 1964 pp 195-211

Smith G M and Young l E Ultimate Theory in Flexure by Exponential Function Jour nal of the American Concrete Institute Vol 52 No3 November 1955 pp 349 359

APPENDIX II -NOTATION

The following symbols are used in this paper

f~ = ultimate compressive strength ofstandard 6-in x 12-incylinder f = concrete stress

f max = maximum compressive stress reached in a given cycle F = I In = stress ratio

CONCRETE BEHAVIOR

FaX = maximum stress ratio in a given cycle in minimum stress ratio in a given cycleFe = stress ratio at the common point FE = stress ratio on the envelope curve

S = tlto strain ratio maximum strain ratio in a given cycle

SJUllC minimum strain ratio in a given cycleSIIlin strain ratio at the common point Se euro pi euro 0 = plastic or residual strain ratioSp strain ratio on the envelope curve a factor relating the common point with the stress and strain ratios of the peak of the previous load cycle

( = concrete strain at I (0 concrete strain at I ~ and (p plastic strain

2561 2560 December 1969

0lt15 4lt~--lt lt-lt ~~poundASUMO-

[ _0 ltT-laquolt060

Ij 0lt75

lt I IEXPfRIENTAL (FNlCURVEI

01lt I 1 1- bull

IL_ Il ~ --- --0__--___

0 I j I

--I middot I I _ 01$ Pshy FOtiQuolmlOltU - __lt_

- ~ Imiddot I_lt-middotI---middot~middotii-I - l 060

00 Ht N Numb of Cyclbullbull

FIG 22-NUMBER OF CYCLES TO FAILURE (Fmax CONSTANT Fmin 0)

~~I

J K

0middot8

l~~ 015

01 0amp 0lt 0-3

_ 02 01+ 0

0middot0

f= ___ --A~LllM ___ _____

Omiddot I-------------r-------shy0lt1 ZOO 00

N =Number 01 Cycles

FIG 23-NUlIBEH OF CYCLES TO FAILURE(Fmax CONSTANT Froin I 0)

CONCRETE BEHAVIOR

and the two curves should intersect at some point below D [Fig 19(b) J In adshydition the plastic strain ratios at point G differ considerably On the basis of the observed results the assumption of uniqueness would not appear to be fllrranted

Fig 20 shows the computed and measured response for specimen AC4-10 which was cycled between stress levels of F max = 077 and F min O The speeimen failed in cycle 21 and failure was predicted in cycle 25 In Fig 21 the response of specimen AC4-13 is shown This specimen was subjected to cyclic loads between stress levels of Fmax 079 and Fmin == 040 The specshylInen failed in cycle 28 and failure was predicted in cycle 34 If uniqueness of the loading and unloading curves was assumed failure would be predicted after only three cycles

The computed number of cycles to fallure for tests in which the load is varied between a given maximum stress level Fmax and a minimum stress of zero is shown in Fig 22 Both measured and computed values are ltplotted Since the observed maximum of the stability limit was at a stress ratio of 063 the experimental curves shown in Fig 22 will become asymptotic to F ~ 063 Fig 23 shows the computed nuinber of cycles to failure for loadings between given maximum and minimum stress levels The maximum stresS ratio is plotted along the ordinate For example the number of cycles to failure with Fmax == 080 and F min == 0040 is approximately 25 Using these curves (Figs 22 23) the number of cycles to failure may be estimated

SUMMARY AND CONCLUSIONS

A series of 46 short rectangular test specimens (Fig I) were subjected to repetitions of compressive stress to various levels to obtain expressions for the response of plain concrete The expressions developed are functions of the ultimate stress and strain values of standard 6 x 12-in control cylinders and the loading history Using these expressions the response of plain conshycrete subjected to varying load histories can be estimated

The following conclusions were obtained

1 For the specimens tested the envelope curve coincided with the stressshystrain curve for a specimen under monotonic loading to failure (Fig 7) The stress-strain path reached the envelope regardless of the strain accumulated prior to a particular cycle

2 The location of the common pOints was dependent primarily on the magshynitude of the maximum stress and strain of the previous load cycle The comshymon points for loading from nonzero levels were identical to the common points corresponding to load cycles starting at a stress lev~l of zero (Fig 11)

3 Examination of the location of the common points shows that failure would be produced under repeated loads with stresses exceeding about 0 63 f ~ the maximum of the stability limit This limit was independent of the minishymum stress levels in the cycles

middot4 Loading and unloading curves starting from a pOint wtt11tli tke stressshystrain domain were not uriique-andthe value of stress and strain at the peak of the previous loading cycle must be known to estimate the relptmse

5 The analytical expressions obtained for the envelope curve the common point and the stability limits and the loading and unloading stress-strain reshy

2563

shy

2562 December 1969

lations produce results that compare well with the experimental results (Fi~s 3 1B 20 21) The formulation of these expressions provides a general lUethi od for estimating the number of cycles to failure under repeated loads (FigS 22 23) with strairi rates similar to those considered in the investigation

APPENDIX I-REFERENCES

I Hognestad E A Study of Combined Bending and Axial load in Reinforced Concrete Mein bers University of Illinois Engineering Experimental Station Bulletin Series No 399 195L

2 Hognestad E Hanson N W and McHenry Dbull Concrete Stress Distribution in Ultimate Strength Design Journal of the American Concrttl1lnstitute Vol 52 No4 December 1955 pp455-479 i

3 Karsan I D Behavior of Plain Concrete under Variable load Histories thesis presented to Rice University at Houston Texas in 1968 in partial fulfillment of the requirements ror the degree of Doctor of Philosophy

4 Murdock J W and Kesler C E Effect of Range of Stress on Fatigue Strength or Plain Concrete Beams Journal of the American Concrete Institute Vol 55 No2 August 1958 Pp 221-231

5 Nordby G M Fatigue of Concrete-A Review of Research Journal of the American Con crete Institute Vol 55 No2 August 1958 pp 191-219

6 Rusch H Researches toward a General Flexural Theory for Structural Concrete Journal oJ the American Concrete Institute Vol 57 No I July 1960 pp 128

7 Shah S P Sturman G M and Winter G Microcnicking and Inelastic Behavior of Con crete Flexural Mechanics of Reinforced Concrete ASCE 196550 The International Sym posium Miami Florida 1964

8 Shah S P and Winter G Inelastic Behavior and Fracture of Concrete Journal of the American Concrete Institute Vol 63 No9 September 1966 pp 925-930

9 Shah S P and Winter G Response of Concrete to Repeated Loadings RlLEM Internamiddot tional Symposium on the Effects of Repeated Loading on Materials and Structural Elements Mexico City 1966

10 Sinha B P Gefstle K H and Tulin l G Stress-Strain Relations for Concrete under Cyclic Loading Journal of the American Concrete Institute VoL 61 No2 February 1964 pp 195-211

Smith G M and Young l E Ultimate Theory in Flexure by Exponential Function Jour nal of the American Concrete Institute Vol 52 No3 November 1955 pp 349 359

APPENDIX II -NOTATION

The following symbols are used in this paper

f~ = ultimate compressive strength ofstandard 6-in x 12-incylinder f = concrete stress

f max = maximum compressive stress reached in a given cycle F = I In = stress ratio

CONCRETE BEHAVIOR

FaX = maximum stress ratio in a given cycle in minimum stress ratio in a given cycleFe = stress ratio at the common point FE = stress ratio on the envelope curve

S = tlto strain ratio maximum strain ratio in a given cycle

SJUllC minimum strain ratio in a given cycleSIIlin strain ratio at the common point Se euro pi euro 0 = plastic or residual strain ratioSp strain ratio on the envelope curve a factor relating the common point with the stress and strain ratios of the peak of the previous load cycle

( = concrete strain at I (0 concrete strain at I ~ and (p plastic strain

2563

shy

2562 December 1969

lations produce results that compare well with the experimental results (Fi~s 3 1B 20 21) The formulation of these expressions provides a general lUethi od for estimating the number of cycles to failure under repeated loads (FigS 22 23) with strairi rates similar to those considered in the investigation

APPENDIX I-REFERENCES

I Hognestad E A Study of Combined Bending and Axial load in Reinforced Concrete Mein bers University of Illinois Engineering Experimental Station Bulletin Series No 399 195L

2 Hognestad E Hanson N W and McHenry Dbull Concrete Stress Distribution in Ultimate Strength Design Journal of the American Concrttl1lnstitute Vol 52 No4 December 1955 pp455-479 i

3 Karsan I D Behavior of Plain Concrete under Variable load Histories thesis presented to Rice University at Houston Texas in 1968 in partial fulfillment of the requirements ror the degree of Doctor of Philosophy

4 Murdock J W and Kesler C E Effect of Range of Stress on Fatigue Strength or Plain Concrete Beams Journal of the American Concrete Institute Vol 55 No2 August 1958 Pp 221-231

5 Nordby G M Fatigue of Concrete-A Review of Research Journal of the American Con crete Institute Vol 55 No2 August 1958 pp 191-219

6 Rusch H Researches toward a General Flexural Theory for Structural Concrete Journal oJ the American Concrete Institute Vol 57 No I July 1960 pp 128

7 Shah S P Sturman G M and Winter G Microcnicking and Inelastic Behavior of Con crete Flexural Mechanics of Reinforced Concrete ASCE 196550 The International Sym posium Miami Florida 1964

8 Shah S P and Winter G Inelastic Behavior and Fracture of Concrete Journal of the American Concrete Institute Vol 63 No9 September 1966 pp 925-930

9 Shah S P and Winter G Response of Concrete to Repeated Loadings RlLEM Internamiddot tional Symposium on the Effects of Repeated Loading on Materials and Structural Elements Mexico City 1966

10 Sinha B P Gefstle K H and Tulin l G Stress-Strain Relations for Concrete under Cyclic Loading Journal of the American Concrete Institute VoL 61 No2 February 1964 pp 195-211

Smith G M and Young l E Ultimate Theory in Flexure by Exponential Function Jour nal of the American Concrete Institute Vol 52 No3 November 1955 pp 349 359

APPENDIX II -NOTATION

The following symbols are used in this paper

f~ = ultimate compressive strength ofstandard 6-in x 12-incylinder f = concrete stress

f max = maximum compressive stress reached in a given cycle F = I In = stress ratio

CONCRETE BEHAVIOR

FaX = maximum stress ratio in a given cycle in minimum stress ratio in a given cycleFe = stress ratio at the common point FE = stress ratio on the envelope curve

S = tlto strain ratio maximum strain ratio in a given cycle

SJUllC minimum strain ratio in a given cycleSIIlin strain ratio at the common point Se euro pi euro 0 = plastic or residual strain ratioSp strain ratio on the envelope curve a factor relating the common point with the stress and strain ratios of the peak of the previous load cycle

( = concrete strain at I (0 concrete strain at I ~ and (p plastic strain