constitutive model for concrete under biaxial stress state

CONSTITUTIVE MODEL FOR CONCRETE UNDER BIAXIAL STRESS STATE

E. J. Stavrakakis Lecturer, Div. of Str. Eng., Dept. of Civil Eng., Aristotle Univ. of Thessaloniki, Greece C. E. Ignatakis Lecturer, Div. of Str. Eng., Dept. of Civil Eng., Aristotle Univ. of Thessaloniki, Greece G. G. Penelis Professor, Div. of Str. Eng., Dept. of Civil Eng., Aristotle Univ. of Thessaloniki, Greece

1. ABSTRACT

A biaxial constitutive model for concrete based upon the incremental theory of plasticity with isotropic hardening is presented. In order to improve the accuracy of the model, the experimental results of Kupfer et al. are used by introducing them into a data base. The model is used in the finite element computer program "RECOFIN" for the in-plane nonlinear analysis of Reinforced Concrete (R/C) structural members up to failure which has been developed in the R/C Laboratory Aristotle Univ. of Thessaloniki. A very good agreement of the analytically predicted stress-strain curves with experimental ones is observed. The results from the analysis of three specimens of plain concrete up to failure under various loading conditions are also presented.

2. INTRODUCTION

The nonlinear behavior of concrete is only one of a series of constitutive laws necessary for the formation of a reliable analytical model for Reinforced Concrete up to failure. Moreover the dominant region of biaxial stress state is only the shaded area in the figure 1. Having in mind these remarks an effort was made in order to formulate a simple but also reliable model for the concrete. A constitutive model based on the Prandtl-Reuss theory of plasticity, originally developed by Argyris and Chan [1] for metals, is used for starting point. Trying to use this model for the concrete some modifications are necessary in order to reproduce the essential behavioral differences between met-als and the brittle concrete.

3. BRIEF PRESENTATION OF THE "ARGYRIS-CHAN" MODEL

For deformation in the elastic range, before the violation of the yielding criterion, the stress-strain relation is given by the well known Hooke's law of elasticity in the following incremental form :

{da} = [Ke] {ds} (1)

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VoL 4, No. 4,1993 Constitutive Model for Concrete Under Biaxial Stress State

/ / /

/ / Typical experimental / ^ /

failure envelope , s

/ / /

/ / Subsequent Von M i s e s

y i e l d i n g el l ipses

Figure 1. Typical failure envelope of concrete under biaxial loading and Von-Mises subsequent yielding ellipses.

where { } denotes column vector, [ ] square matrix and

{da} = {dax doy da z ^ d x ^ x/2dXyZ ^ d x ^ } (2a)

{ds} = {dsx ds y d8z i/2dsXy x/2deyz x/Tds^} (2b)

For deformation in the plastic range the stress-strain relation is considered to be given by the Prandtl-Reuss incremental theory of plasticity in conjunction with the Von Mises yielding criterion.

The total strain is decomposed into elastic and plastic components

d S i j = d8§ + dejj (3)

and the final form of constitutive law ready for use in a finite element computer program is given by the following relation :

{do} = ([K e]-[3G/(l + EP/3G)/ö2][{s}{s} t]){d8} = ([Ke]-[KP]){ds} (4)

where the equivalent stress o is given by the analytical expression of the Von Mises yielding criterion in the stress space

σ = p ~ 2 = \ / 3 / 2 ( S i j ) ( S i j ) (5)

and the corresponding incremental equivalent plastic strain is determined as follows :

dgP \ 6 / 3 ( d 8 R ) ( d e R ) (6)

The modulus of plasticity E? is given by the slope of the curve (σ-εΡ) and has the following form (see fig. 2 ) :

EP = άσ/άεΡ = ... = ( E 0 E ) / ( E Q - E ) (7)

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E.J. Stavrakakis, C.E. Ignatakis and G.G. Penelis Journal of the Mechanical Behavior of Materials

Argyris and Chan have suggested the use of a single curve (σ-εΡ) under uniaxial loading for the derivation of EP under any kind of biaxial stress state (see fig. 2).

Despite its drawbacks, which will be remarked in the next paragraph, the consitutive model has been incorporated in the first form of the RECOFIN computer program und successfully used for the analysis of R/C structural members up to failure [2], [3], [4].

4. MODIFICATIONS FOR THE APPLICATION OF THE MODEL TO CONCRETE

In plasticity modeling of concrete, in addition to the above mentioned assumptions, the ultimate strength condition or failure criterion, which sets the upper bound of the attainable states of stress, has to be defined. Moreover in the compression-compression region, as it can be seen in fig. 1, the experimental failure envelope surrounds the Von Mises external yielding ellipse (σ=ί„). Thus the equivalent stress for all the hatched area takes values greater than the uniaxial strength of the material (ö>f c) . Consequently it is impossible to use the (o-§P) curve of the figure 2 for uniaxial compression in order to determine the corresponding value of EP.

Having the advent of the powerfull personal computers, in order to remove these difficulties anc improve the accuracy of the constitutive model, the following modifications and improvements were attempted using the Kupfer et al.[5] experimental data.

4.1. Biaxial failure criterion

Instead of the various forms of simplified failure envelopes adopted in the past by many inves-tigators, the experimental failure envelopes for the three concrete qualities (f = 19.1, 31.1, 59.4MPa, see fig.3) given by Kupfer[5] are stored in a data base. Each semi-envelope defined by the bisector ο symmetry was represented by twenty(20) points and a double parabolic interpolating process was built in. The first one produces a suitable new set of twenty points defining the failure envelope of any given concrete quality, while the second one interpolates points of failure for any given principa stress ratio (c=Q2/aj) between the two nearest points of failure of the previously defined failure en velope.

Δ σ

Figure 2. Stress-strain curve of concrete uniaxially stressed and the corresponding equivalent stress-plastic strain curve.

389


Figure 3. Experimental failure envelopes for three qualities of concrete under biaxial loading [5],

4.2. Family of equivalent stress - equivalent plastic strain curves

Instead of using the single (σ-εΡ) curve deriving from a uniaxial stress state (fig.2), the actual (σ-εΡ) curve corresponding to any given principal stress ratio c=Q2/oj was computed using the triad of stress-strain curves (σ^-ε j, ε^ e3)>(see fig-4). The steps of the computational procedure are presented below. i. Kupfer et al.[5] gives the sets of (Oj-8j, ε^, £3) experimental curves for a concrete of a strength ί ς = 32.8 MPa and for several principal stress ratios. All triads of curves for these ο^Ι^χ r a t i ° s w e r e

simulated successfully using the following equation

σ ΐ / σ 1 υ + [ 1 _ 8 i / 8 i u ] e = 1 w h e r e e = Eio( siu/CTlu) (8)

where E:Q is the initial modulus of elasticity. The values of E-Q according to the theory of elasticity are the following

E l o = E o / ( 1 - v o c ) ' E 2 o = E o / ( < ^ o ) > e 3 o ~ ~ E o / ( v o + v o c ) (9)

where vQ is the initial value of Poisson's ratio. ii. For each triad of (σ^-ε z^ ε3) curves represented by the equations (8), the corresponding pairs of equivalent stress and equivalent plastic strain (σ,εΡ) were computed for two-hundred (200) points along the curves, using the equations (5) and (6). These equations take the following form for biaxial loading by introducing the principal stresses :

σ =yoj+ U2~ a jG 2 = σ ^ Ι + c 2 - c (10)

sP=\/: 2 2 2

<2/3 ( s f r ε ξ + ) (11)

Using a curve fitting process based upon the least square method a suitable ellipsoidal equation of the following form was derived fitting best with the sequence of the points defined by the two-hundred pairs (σ,εΡ) :

[ ä / ä u ] e i + [ s P / E P ] e 2 = l (12)

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E.J. Stavrakakis, C.E. Ignatakisand G.G. Penelis Journal of the Mechanical Behavior of Materials

This equation represents the analytical expression of the equivalent stress - equiv. plastic strain curves (see fig.4) corresponding to the principal stress-strain ratios examined by Kupfer et al. iii. Finally the set of principal stress ratio c=02/o^ and the corresponding three values ( ε ^ ^ , ε Ρ ) , defining completely each curve, are stored in the data base. The built in parabolic interpolation process was activated whenever a new triad of values ( e ^ ^ , §P) was required for the definition of the (σ-εΡ) curve for any given principal stress ratio c.

4.3. Requirements and functioning of the modified constitutive model.

Both the initial and the modified constitutive models require the following mechanical charac-teristics of the concrete : i. The uniaxial prismatic (cylindrical) strength (f„). ii. The compressive strain ( ε ^ ) at the moment of failure under uniaxial loading. iii. The initial values of the Poisson ratio (vQ) and of the modulus of elasticity (EQ) under uniaxial compression.

At first the program calculates by the built in interpolation process the family of twenty points defining the failure envelope of the given material. Then, according to the given material properties, the stored values of εΡ for all the stress ratios are modified automatically multiplied by the factor ( ε £ g i v e n / ε ^ stored;. This kind of "isotropic" connection between concretes with different quality is thought to be a reasonable admission.

After these automatic alterations, necessary to fit the material data base with the given concrete, the constitutive model is ready to reproduce the stress-strain behavior of the material under monotonic biaxial loading up ίο failure.

5. VERIFICATION - APPLICATION OF THE CONSTITUTIVE MODEL

To verify the model, the analytically predicted stress-strain curves were compared with the ex-perimental ones [5] for a concrete of a prismatic strength f c=32.8 MPa under biaxial stress states for various principal stress ratios.

Figure 4. Typical stress-strain curves of concrete under biaxial loading and the corresponding equivalent stress-plastic strain curve.

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The fitting of predicted stress-strain curves (σ^-ε j) with the experimental ones was exellent, while for the (σ^-ε^) and (σ^-ε^) curves some deviations are observed. In the figure 5a the ex-perimental and analytical stress-strain curves for biaxial compression (02/0 j = 0.52) are compared, while in the figure 5b the corresponding input and output equiv. stress - equiv. plastic strain curves are presented. In the figure 5c and 5d the exper. and anal, stress - strain curves for uniaxial compres-sion and tension - compression (09/0^= 0.052) are also compared respectively.

In order to demostrate the ability of the RECOFIN program, equiped with the modified con-stitutive model, the following plain concrete models, having a uniaxial unconfined compressive strength fc=32.8 MPa, were analyzed until failure. i. A model of a plain concrete cubic specimen was analyzed under uniaxial compression between fixed plattens. In the figure 6 the cracking sequence until failure is shown. The first vertical cracks are observed at the center of the model under a mean compressive stress of 30.0 MPa. The final failure takes place under a stress of 36.0 MPa. The previously cracked elements at the center of the model start crushing and inclined cracking fronts rapidly propagate from the center towards the edges of the loaded faces, forming the well known pair of buttended truncated pyramids [6]. It is wor-thy to be noted that the local direction of cracking of each one element on the inclined splitting plane is almost vertical. The ultimate cubic strength of the model (f =36.0 MPa) resulted 10% greater than the unconfined compressive strength of the material, while the reported experimental values for the f ^ / f j . ratio lay between 1.10 and 1.20 for the concrete quality under consideration [6]. This underestimation of the cubic strength must be attributed to the biaxial nature of the RECOFIN program. Consequently the confining effect of the friction at the transverse direction detween the steel plattens and the loaded faces of the model is not taken into account. ii. A model af a plain concrete prismatic specimen, having a height to width ratio equal to 4.0, was analyzed under uniaxial compression between fixed plattens. The failure was brittle. The begining and propagation of cracking up to failure of the model has taken place at the same loading step un-der a mean compressive stress of 32.5 MPa. The first cracks and the final damage pattern at failure are shown in the figure 7. The inclined splitting planes, at the upper (and lower) part of the model (see figure at failure) are formed by propagating cracking fronts connecting the first cracked ele-ments (see the figure at the begining of cracking). Afterwards a wider cracking front propagates ver-tically to the center of the model while the previously cracked elements of the inclined splitting planes start crushing progressively. Finaly the collapse is indicated by the numerical instability of the computational process. This cracking pattern is typical for prismatic specimens under uniaxial com-pression between fixed plattens [6]. It is noteworthy that the local direction of cracking of all the damaged elements is almost vertical and the strength of the model found to be almost identical with the unconfined uniaxial compressive strength of the material. iii. Finally a plain concrete cubic specimen was analyzed under concentrated compressive forces along the median lines of opposite faces (Brazilian test). The well known variation of stresses over the center cross section of the model at the first loading step (elastic response) is shown in the figure 8. Overlooking the premature damage of a pair of elements at the 90% and 95% of the ultimate load (probably owed to the rather arbitrary distribution of the loading forces), the failure was brittle. The beginning of cracking and the damage pattern at failure are also shown in the figure 8. Starting from the center, the model splitted allong the vertical cross section. The local direction of cracking of the damaged elements is vertical at the center while inclines towards the loaded point as the cracking zone propagates towards the loaded edges. Under the loaded area the cracking zone widens, sur-rounds the loaded area and reaches the surface, while the local cracking direction inclines rapidly towards the loaded points. With the complete formation of the splitting plane the model becomes unstable indicating the physical collapse. The principal stresses at the center of the model under the ultimate load (Fu = 190 KN) found to be:

Oj = σχ = f c s = +2.94 MPa = 0.090fc, o 2 = σ = -9.11 MPa (13)

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(d)

Figure 5. Comparison of analytically predicted stress-strain curves with the experimental ones [5] for concrete under biaxial loading.

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Cracked e lement

Crushed e lement

Crack pattern at fa i lure fcc= Ou = 36.0 Mpa

f c= 32.8 M p a /Concrete

cube

20 cm

I Steel , platen

/

F.E.Mode l

Figure 6. Analytical simulation with the RECOFIN pro-gram of a plain concrete cubic specimen under uniaxial compression between fixed plattens until failure.

The theoritically predicted maximun tensile stress at the same point according to the theory of elas-ticity is given by the following formula [6]:

2F f_. = — = 3.024MPa = 0.092 L

The calculated relation between f c s and f c is in agreement with relative experimental results [6],

6. CONCLUSIONS

Argyris and Chan developed a constitutive model suitable for metals using the Prandtl-Reuss plasticity theory. For the application of the model on concrete it is necessary to define a failure criterion, while the use of a single equivalent stress - equiv. plastic strain curve (σ-εΡ) is proved to be incorrect. The experimental failure envelopes of Kupfer for concrete under biaxial stress state are in-troduced into the model and the failure envelope for any given concrete quality is defined by a parabolic interpolation. The experimental stress - strain curves of Kupfer for concrete under biaxial loading are successfuly aproximated by a family of ellipsoidal equations. After a suitable mathemati-

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E.J. Stavrakakis, C.E. Ignatakisand G.G. Penelis Journal of the Mechanical Behavior of Materials

Cracked element

Crushed element

First damages FU=1300KN ^ Ou=32.5mp»

Figure 7. Analytical simulation with the RECOFIN pro-gram of a plain concrete prismatic specimen under uniaxial compression between fixed plattens until failure.

Prismatic Concrete specimen

F.E.Model

Ε U O.

1 • 20.0-

uz. ψ

-fc=32.8Mpa

Steel " -ζ platen

i X X >< i J

X X \ i 3 χ X i

i Ε υ 8

i

Jök

-̂8«2.5cm —I

cal process the equiv. stress - equiv. plastic strain curves (σ-εΡ) for any given principal stress ratio are defined automatically in order to compute the value of the plastic modulus E? which is the essen-tial parameter of the constitutive model.

These modifications proved to be successful. The model was introduced to the finite element program RECOFIN capable for nonlinear analysis of R /C structural members up to failure. The ac-curate analysis of plain concrete models under various loading conditions proves the effectiveness of the constitutive model.

Final Crack pattern

FU=1300KN JL Ou=32.5Mpa

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Vol 4, No. 4,1993 Constitutive Model for Concrete Under Biaxial Stress State

Concrete , _ _ Cube fc=32.8Mpa

F.E. Model F h

λκ Cracked ΛΙΙΙΙγκ element

Crushed element

< χ 2.25 —I k 1.0 cm

Stress diagrams at the central cross section : F=400KN

950(KN) fcs = 0x = 3.024 MPa Crack initiation B π Α

Figure 8. Analytical simulation with the RECOFIN program of the "Brazilian test" on a plain concrete cubic specimen.

396

04 0.51

0.58 Q60 0.6 1<

0.62'

rtheor xmax = σ Χ β = \

^ = 0 . 6 3 7 M P a j

oca KM Fu/2=950KN First damages / Crack pattern at f a i l u r e ^

(900KN)


7. NOTATION

{ds} = incremental strain column vector {do} = incremental stress column vector sx = components of deviatoric stress tensor ν = Poisson's ratio (vQ = initial value) Ε = tangent modulus of elasticity E q = initial value of Ε under uniaxial loading G = shear modulus J9 = second invariant of deviatoric stress tensor [K] = stiffness matrix e = superscript denoting elastic quantity i,j = subscripts for coordinate axis (summation convention holds) ρ = superscript denoting plastic quantity t = superscript denoting the transposed tensor u = subscript for ultimate stress or strain 1,2,3 = subscripts for principal stresses or strains f c = uniaxial unconfined compressive strength of concrete f ^ = uniaxial cubic compressive strength of concrete f t = strength of concrete under uniaxial direct tension f c s = splitting tensile strength of concrete (Brazilian test)

8. REFERENCES

1. Argyris, I.H., and Chan, A.S., 'Static and Dynamic Elastoplastic Analysis by the method of Finite Element in Space and Time', Proc. of the Symposium on Foundation of Plasticity. Leyden, 1973, pp. 147-175.

2. Penelis, G. and Stavrakakis, Ε., Ά Finite Element Model for analysing in-plane R/C structural members until failure', in Greek, Proc. of the 2nd Greek Concrete Conf.. Thessaloniki, Greece, May 1975.

3. Ignatakis, C., Stavrakakis, E., and Penelis, G., 'The behavior of R/C Corbels using the Finite Element method', in Greek, Proc. of the 7th Greek Concrete Conf.. Patra, Greece, October, 1985.

4. Ignatakis, C., Stavrakakis, E., and Penelis, G., 'Parametric Analysis of R/C columns under Axial and Shear Loading using the Finite Element method', ACI Journal. Proc. V. 86, No. 4, July-August 1989, pp. 413-418.

5. Kupfer, Η., Hilsdorf, Η., and Rusch, Η., 'Behavior of Concrete under Biaxial Stresses', ACI Journal. Proc. V.66, No. 8, Aug. 1969, pp. 656-666.

6. Avram, C. et al., 'Concrete Strength and Strains'. Elsevier Scient. Publ. Co., ISBN 0-444-99733-4, 1981.

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constitutive model for concrete under biaxial stress state

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