ELSEVIER

Cement & Concrete Composites 16 (1994) 9-14 O 1994 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0958-9465/94/$7.00

Constitutive Relation for Steel Fibre Concrete Under Biaxial Compression K. H. Tan, K. Murugappan & P. Paramasivam*

Department of Civil Engineering, National University of Singapore, Singapore 0511

(Received 15 January 1993; accepted 23 August 1993)

Abstract

This paper presents an analytical model for predict- ing the response of steel fibre concrete under biaxial compression up to failure. The model treats the steel fibre concrete as an orthotropic material for which the constitutive relation based on a total stress-strain formulation is derived. In the deriva- tion, the biaxial compressive strength of steel fibre concrete is first evaluated by treating it analogous to a plain concrete under triaxial compression, with a confining pressure due to the fibres in the third direction. By defining equivalent uniaxial stress-strain relations and establishing the secant moduli in the principal directions, the stress-strain relation for steel fibre concrete under biaxial load- ing is obtained by a stepwise procedure. The stress-strain relations so derived are found to com- pare well with available experimental data.

Keywords: Biaxiai compression, compressive strength, stress-strain relation, failure envelope, steel fibre concrete, steel fibres.

INTRODUCTION

Although the behaviour of steel fibre concrete under uniaxial loading is relatively well under- stood, available information on its behaviour under biaxial loading is rather limited. Recent studies indicate that, while the addition of steel fibres improves the ultimate capacity under uni- axial compression only marginally, it can improve the strength of ordinary concrete under biaxial compression significantly. L2 This is because con- crete under biaxial compression fails with splitting in the third direction which is normal to the plane

*To whom correspondence should be addressed.

of loading, 3 as shown in Fig. 1. The role of steel fibres in bridging across cracks helps to reduce tensile strains and splitting, thereby increases the biaxial compressive strength of concrete.

In the evaluation of biaxial compressive strength, steel fibre concrete can be considered as equivalent to a plain concrete under triaxial load- ing, with a confining pressure due to the fibres in the third direction. The idea of applying a con- fining pressure is to model the influence of fibres on the principal tensile strains in the third direc- tion. Based on this concept, the authors have developed the strength (or failure) envelopes for steel fibre concrete under biaxial compression? However, for a complete structural analysis, the failure envelopes need to be augmented with a stress-strain relation, which is developed in this paper and compared with available experimental data.

c5

Fig. 1. Failure modes of biaxially loaded concrete. (Adapted from Ref. 3.)

10 K. H. Tan, K. Murugappan, P. Paramasivarn

ANALYTICAL MODEL

Stress-strain relation under biaxial compression Cement-based composites exhibit highly non- linear behaviour under compression. Under bi- axial compression, it is reasonable to assume steel fibre concrete as an orthotropic material having different properties in the principal directions. Thus, with o, and 03 as normal compressive stresses (taken as negative) in the principal direc- tions, the corresponding strains (negative for com- pressive and positive for tensile) can be written as

O, V303 e , - (I) E~ E3

0" 3 ~ , O , e~ = (2)

E3 E:

where, E and v are, respectively, the secant modulus and the Poisson's ratio and subscripts 2 and 3 refer to the principal directions. Also, the shear strain e23 is related to the shear stress m_.~ by

e_,3 = oz3/G (3)

where G is given by 5

( 1 - v 2 ) G : ~ ( E , + E 3 - 2 v ~ ) (4)

in which v=,~w_v3 a n d v2Es=v3E2=v~-E2E3. The constitutive relation given by eqns ( 1 )-(3) can be rewritten in a matrix form as

I } 03 1

0"23 (1 - v 2)

o ] 0 (1 -v2)G

I ] E3

E23 (5)

in which v can be taken as 0.2 for steel fibre concrete. The relation between (r~ and e~ (i = 2,3) is illustrated in Fig. 2 with the ultimate stress and the corresponding strain as 6 i and g~, respectively. The secant moduli E 2 and E 3 in eqn (5) can be

O-

OTI - - - -

- Es

/

0 0 E, E,u ~ E,u E

Compressive Strain

Fig. 2. Biaxial and equivalent uniaxial stress-strain curvcs.

taken as

E~= o_, ( i=2,3) (6) ~'ltl

where e,u are defined as the equivalent uniaxial strains in the respective principal directions. In this study, the values of a~ are assumed to be related to e,o according to the equation given by Sanez (quoted in Ref. 6) that is (see Fig. 2):

E.t~iu o, = (7)

1+ E , _ 2 ti~, + ei~ -

LE, J

where E 0 is the elastic modulus and E~ is the secant modulus at ultimate, computed as E~ = d~/ g~u. For a given principal stress ratio 02/03, it is seen that the equivalent uniaxial stress-strain rela- tion (eqn (7)) is uniquely defined by the values of E0, d i and Eiu" In the study, the value of E0 is taken as equal to the elastic modulus of steel fibre con- crete under uniaxial compression, while the values of 6~ and giu are obtained as detailed below.

Biaxial compressive strength 01 The biaxial compressive strength for steel fibre concrete can be obtained by considering an analo- gous plain concrete under trixial stress state. The influence of the steel fibres is modelled by apply- ing a confining pressure equal to the post-cracking tensile strength of steel fibre concrete in the o,

Steel fibre concrete under biaxial compression 11

direction. The biaxial state of stress in steel fibre concrete and the equivalent analogous state of stress in plain concrete are given in Fig. 3. The biaxial compressive strength is given by the failure envelope which is defined in terms of stress in- variants. Only relevant details of the formulation are presented here and the complete details can be found in Ref. 4.

The confining pressure in the o i direction is taken equal to - o,° (Fig. 3), where o,~ is the post- cracking tensile strength of steel fibre concrete. The value of o,o depends on the length lf, the ratio of area to the perimeter r', the orientation factor r/., the length efficiency r/, and the ultimate bond strength ro of the fibres. It can be expressed a s 7

o,,, = rl , rl,,( l,/ r') V,. rol2 (8)

The failure envelope of steel fibre concrete under biaxial compression is given in terms of the stress invariants as 4

f~2 I ' - 1 = 0 (9) f(l,,J2, cos30) = A - 2 f-~p- Bf¢--~p

where f~p is the uniaxial compressive strength of plain concrete and I,, J2 and J3 are, respectively, the first invariant of the principal stresses and the second and third invariants of the deviatoric stresses. The value of ;t is expressed in terms of cos30 a#

K, cos [~ cos-'(K_, cos

2= ' c ° s [ 3 - c ° s - ' ( - K"

for cos 30 < 0

30)] for cos 30>-0

cos 30)]

(10)

%-- ~ %

~0*e cRAr[ WI01H$ ~ [ [XA62*IAr[O &

]0" 2 $1WPtl[1O far CtNItTY

BIAXIAL COMPRESSIVE SLATE OF ANALOGUS STARE OF STRESS STRESS IN FIBRE C~CREIE IN PLAIN CONCRETE

Fig. 3. Modelling for the biaxial compressive strength of steel fibre concrete.

with the value of cos 3 0 as

34~ J3 cos30= 2 J3/2 (11)

The parameters A, B, K~ and K2 depend mainly on the ratio of the uniaxial tensile to the uniaxial compressive strength (ftff'c) of concrete. For the present study, (ft/f'~) is assumed to be 0.08, for which the corresponding values of A, B, Ki and K 2 are, respectively, 1.81, 4-10, 14.49 and 0.99. s

Equivalent uniaxial strain at ultimate £~,, The value of g~ depends on the uniaxial com- pressive strength fop and the corresponding strain ecr of the basic (plain) concrete mix and the value of 6, which takes the influence of the steel fibres into account. Darwin and Pecknold ~ suggested that the following expressions can be used to relate d~u and 6~:

di.,=lecP~O~icpR-(R-1)] fOrti>fCp

[tcp [ - 1"6 (f-~)3 + 2"25 (f-~) 2+0"35

(12a)

(12b)

By calibrating eqn (12a) with the test data of Traina and Mansour 2 and Yin et aL l for the case where the principal stress ratio 02/o 3 equals 1.0, it is found that a value of R--- 1.6 can be used.

SOLUTION PROCEDURE

To obtain the complete stress-strain curve for steel fibre concrete under biaxial compression, the following information is necessary: (i) uniaxial compressive strength fop and the corresponding axial strain ecp of plain concrete; and (ii) elastic modulus under uniaxial compression E0 and post- cracking tensile strength 0,, of steel fibre concrete. The required curves can be obtained by a step- wise procedure as follows.

(1) For a given principle stress ratio, compute O 2 and O 3 using eqns (9)-( 11 ).

(2) Determine e2. and e3. using eqn (12).

12

(3) (4)

(5)

(6)

(7)

K. H. Tan, K. M u r u g a p p a n , I'. P a r a m a s i v a m

Choose a value of e2o. Compute o: using eqn (7) and calculate o~ from the given principal stress ratio. Corresponding to the value of 0 3, deter- mine t'3u from eqn (7). Calculate the secant moduli E~ and E 3 using eqn (6). Substitute the values of E_~ and E.~ in eqns ( 1 ) and (2) to obtain the values of e, and t~. Repeat steps (3)-(7) for increasing values of t'2u until the complete o , - e , ( i=2,3) curves arc obtained.

ct""~; / / ? f , , , /~

b~3ei

I. If : 25iron ~ :35~ h. ~5,em v,-" )% &"? 'G'I = ! 3 :?: .C'] :13

01 00 O -~(]0 -u~0 10~ ~ 2000 _l~ STRAIN (zlO 6 ) STRAIN (xt~ 4')

7

I'O0~[~[C qJR[3

%%,~0 . . . . rX~RIM[NT

i I I I ' ~0 20O0 -~ - 817,00

STRAIN [xlO "6)

Fig. 4. Biaxial stress-strain curves for specimens tested by Traina and Mansour."

COMPARISON WITH AVAILABLE EXPERIMENTAL RESULTS

The analytical model outlined in the previous section was used to predict the response of steel fibre concrete specimens tested under biaxial compression by Yin e t al. ~ and Traina and Man- sour.-" Only specimens of which the experimental stress-strain curves are available in the references are compared. The comparison covers a wide range of variables such as the type of fibres. volume fraction and aspect ratio of fibres, and the principal stress ratio o2/Os.

Specimens tested by Traina and Mansour ~ These specimens were 76 mm steel fibre concrete cubes with a mix proportion having a cement: sand :coarse aggregate:water ratio of 1 : 1-64: 1.41:0.45 by weight. The coarse aggregate used was of a maximum size of 13 mm. The plain concrete mix had an uniaxial compressive strength f~v of 40.0 MPa and tep = 0.0017. The fibres were low carbon, smooth drawn wires 30 mm long, with hooks at each end. The volume fraction of fibres in these specimens were 0-5, 1.0 and 1"5% for which the value of ot~ was computed to be 0.84, 1"68 and 2.52 MPa, respectively. Figure 4 shows the comparison of the stress-strain relations for the test specimens with the predictions of the proposed analytical method. Good agreement between the experimental and the analytical results is obtained. For the specimen subjected to a principal stress ratio of 1.0 (Fig. 4(c)), the present model gives the same curve for e2 and e~. However, different curves were obtained experimentally for e 2 and e 3 for this specimen and this can be attributed to the ortho- tropy due to the direction of casting.

Specimens tested by Yin et al. Yin et al. ~ have reported test results of steel fibre concrete plate specimens (152 x 152 x 38 ram) loaded under biaxial compression. The plain concrete in this case had a uniaxial compressive strength J~p of 37.6 MPa and t'cp=0.0020. The basic concrete mix used was 1:2.16:1.88 by weight of cement, sand and coarse aggregate (9.5 mm maximum size), with a water-to-cement ratio of 0.6. The coarse aggregate consisted of quartz, flint and with some feldspar. Smooth straight slit carbon steel fibres were used in the specimens. For such fibres, the value of ultimate bond strength t u is 4-15 MPa as reported by Swamy and Mangaff and Mansur et al. ~ Thus, the value of o,,, was calculated as 1.153 MPa and 2.306 MPa for ! or 2% of 25.4 mm (1") steel fibres respectively and 0"756 MPa for 1% of 19-05 mm (]") fibres. Figure 5 shows that the predicted stress-strain curves compared very well with the test results for these specimens.

Effect of volume fraction and aspect ratio From the biaxial compressive stress strain rela- tions in Figs 4, 5(b) and 5(c), it is seen that for a given principal stress ratio o2/o 3, an increase in fibre content Vf results in higher ultimate strength cL or d3 and the corresponding strains g2 or g3. The effects are summarized in Fig. 6 together with the test values. An increase of about 25% in bi- axial compressive strength and 35% in the corre- sponding strains was observed with the addition of 1% of fibres. The results of Figs 5(a) and (b) also indicated an increase in both the biaxial com- pressive strength and the corresponding strain with increasing aspect ratio of fibres although the increase is less considerable when compared to the effect of volume fraction of fibres. Figure 7 shows that increasing the aspect ratio of the fibres

Steel fibre concrete under biaxial compression 13

:I ,o

7 / ~=

~ os ~ lb~ .~ ~, 1.o

t~R[S S l ~ tibet S Flirts ~ u~ " 0~, If =tgme VI=I% ,1'~ l l ,2$tmm ~1~

- - - - - IIRIUM[IT ~ - - f£mfll] l(~t #=: I . . . . ' ' ' ' ' ' -~e ' 30mm - 2 0 ~ ~ 1000 0 -2000 -~,000 1000 -6000

Sl'RA ~(,~ IxlO "~ ) SIRAIIIIH Ix'g4l ST~III Ic~(:l~ ~ 25./,Straight mm fibres

'°r- --- -! ' I ' % sb 6'0 v, : l ' / . ~0 8'0 ,~ , ~ ~, . ~spect ratio I f / d

u , , , I __ X ( :

lill _:o . . . . . o, . : i

STRAIN /zl0 "6) SlRl ] l l (x lO~l .c~_ - ] 0 0 0

idl (,I ~ -500 Fig. 5. Biaxial stress-strain curves for specimens tested by 0 Yin etalJ t,O 50

Hook ended fibres//~nal.~ E!pt.

2.0

~ 1 0

RI

0%' I

• , . . . . . . . . . . , , . . . - " " " ' " " ' " . . . . . . - . * -

Anal E x_p_t Hook ended fibres • 30ram

L 25Straight/, mm fibres i - - - - "

oflo-a: 1 0 I I I

0.5 1.0 1.5 20

vf (%) (a)

-3500

.,o -3000

>, -2500

I ~ -2000

i ~ -1500

.~ -1000

~', -500

/ .o _

Hook ended fibres[ o~I *~3 30ram I Straight fibres I ! I a [ ~ l g 3 25 t, mm L

t J i ~/°'3 =10 O~ 0-5 1.0 1.5 2.0

vf {%1 (b)

Fig. 6. Ef fect o f vo lume f rac t ion o f f ibres on biaxia] compressive strength and strain of steel fibre concrete.

90 100

100

Fig. 7. Effect of aspect ratio of fibres on biaxial com- pressive strength and strain of steel fibre concrete.

from 40 to 100 leads to an increment in the bi- axial compressive strength and the corresponding strain by about 10 and 15%, respectively.

Effect of principal stress ratio The principal stress ratio is found to influence the biaxial compressive strength and the correspond- ing strain of steel fibre concrete significantly, as can be seen from Figs 5(b), (d) and (e) and shown in Fig. 8. For 02/03 < 1, the values of 63 and E3 increase with the principal stress ratio up to a value of about 0.2-0.4 after which they decrease with increase in the ratio of o2/03. The values of 6, and t? 2 increase with the 02/03 ratio and the principal strain g2 drifts towards the principal compressive strain g3, as o2/o s approaches 1. It is also seen that the value of g2 changed from tensile to compressive when o2/o3 is approximately equal to the Poisson's ratio, that is, 0"2.

CONCLUSIONS

An analytical model to predict the response of steel fibre concrete under biaxial compression up

60 70 80 91] Aspect ratio t f /d

(bl

Anal. Expt. Hook ended fibres I o~2 eE 3 30ram Straight fibres l ~ ,-% l~ 3 25~mm

14 K. H. Tan, K. Murugappan, P. Paramasivam

20

I i

~,. ci.

'-' Iio"

O0 O0

/ r ( ~ / % ) / / D~-(

I H ~ fibres I - oC j , - !!Omm I i_Sltaight fibres I----o0

/ . . . . . . l / , ..... ,

o2 oL 06 oe ~o

-~000

,~-3000/~~ -2000 ~ '

I ~ g7 - " ' t~' .J.~"" ,, -looo / . . ; I " "

- 00 ./j../h"o 0 _ . . . . . ~ " I Anat. Expt,

[Hook ended f ihres~- - o ~ " - - o 500 '; 30ram : "~3

10001 I Straight fibres i-- ' - - o~:2 t 25 t, mm l - - . mmg~

150Q, t , i t Vt=l°/' (r2 o t 06 o e ~ o

Fig. 8. Effect of principal stress ratio on biaxial com- pressive strength and strain of steel fibre concrete.

to failure is presented. The stress-strain relations are obtained by considering equivalent uniaxiai stress-strain relations and using a stepwise proce- dure. The biaxial stress-strain relations so derived are found to compare well with available experi- mental data. Analytical predictions indicated that

the biaxial compressive strength and the corre- sponding strains of concrete can be increased by the addition of steel fibres in particular or by using fibres with higher aspect ratio. They are found to be greatly influenced by the principal stress ratio.

ACKNOWLEDGEMENTS

The work upon which the paper is based was supported by National University of Singapore research grant RP880646.

REFERENCES

I. Yin, W. S., Su, E. C. M., Mansur, M. A. & Hsu, T. T. C., Biaxial tests of plain and fibre concretc. AC'! Mater. J., 86 (3)(1989) 236-43.

2. Traina, L. A. & Mansour, M. A., Biaxial strength and deformational behaviour of plain and steel fiber con- crete. ACT Mater. J., 88 (4)( 1991 ) 354-62.

3. Nelissen, L. J. M., Biaxial testing of normal concretc. Heron (Delft), 18 (1) 45.

4. Murugappan, K., Paramasivam, P. & Tan, K. H., Failure envelope for steel fiber concrete under biaxiai compres- sion. ASCEJ. Mater. Civil Engng, 5 (4) (1993) 436-46.

5. lnoue, N., Koshika, N. & Suzuki, N., Analysis of shear wall based on Collins panel test. Finite Element Analysis of Reinforced Concrete Structures. ASCE, pp. 288-99.

6. Darwin, D. & Pecknoid, D. A., Non-linear biaxial stress- strain law for concrete. J. Engng Mechanics Day. AS('E. 1 0 3 (EM2) (1977) 229-41.

7. Lim, T. Y., Paramasivam, P. & Lee, S. L., Analytical model for the tensile behaviour of steel fiber concrete. ACI Mater. J., 84 (4) (1987) 286-98.

8. Ottosen, N. S., A failure criterion for concrete. J. Engng Mechanics Div. ASCE, 1 0 3 (EM4) (1977) 527-35.

9. Swamy, R. N. & Mangat, P. S., A theory for the flexural strength of steel fibre reinforced concrete. Cement and Concrete Res., 4 (2)(1974) 313-25.

10. Mansur, M. A., Natagaki, S., Lec, S. H. & Oosummi- moto, Y., Torsional response of reinforced fibrous concrete beams. A ( 7 Smwt. J., 86 ( 1 ) (1989) 36-44.