Limit load analysis of thick-walled concrete structures - a finite element approach to fracture

COMPUTER METHODS IN APPLlED MECHANICS AND ENGlNEERiNG 8 NORTH-HOLLA~ PUBLISHING COMPANY

8 (1976) 215..- 243

LIMIT LOAD ANALYSIS OF THICK-WALLED CONCRETE STRUCTURES A FINITE ELEMENT APPROACH TO FRACI’URE J.H. ARGYRIS*, Institut fur Statik und Dynamik

G. FAUST and K.J. WILLAM

der Luft- und Raumfahrtkonstruktionen,

University of Stuttgart,

Germany

Received 15 June 1975 The paper illustrates the interactian of constitutive modeIting and finite element solution techniques for limit load prediction of concrete structures. On the cunstitutive side, an engineering model of concrete fracture isdeveloped in which the Mohr-Coulomb criterion is augmented by tension cut-off to describe incipient failure. Upon intersection with the stress path the failure surface collapses for brittle behaviour according to one of three softening rules - no-tension, no-cohesion, and nofriction. The stress transfer accompanying the energy dissipation during local failure is modeled by several fracture rules which are examined with regard to ultimate load prediction. On the numerjcai side the effect of finite element idealization is studied first as far as ultimate load convergence is concerned. Subsequently, incremental tangential and initial load techniques are compared together with the effect of step size. Limit load analyses of a thick-walled concrete ring and a lined concrete reactor closure conclude the paper along with engineering examples.

The objective of this paper is to iilustrate the interaction of constitutive modeling and numerical solution when analysing ultimate load behaviour of thick-walled concrete structures. In this context an engineering model of failure is developed for triaxial concrete behaviour in prcand post-failure regimes. The material law follows the fracture theory of granular solids, which is based on three constitutive postulates -- the initial failure condition, the collapse model, and the inelastic rate of change due to energy dissipation during failure. To this end the Mohr-Coulomb pyramidal surface is a~]gIneI~ted initially by tension cut-offs which collapse for brittle post-failLlr~ beha~~i(~~~r to no-tei~siol~, Ilo-cohesion or n~frictioI1 conditio~ls. The inelastic stress transfer is described by one of several fracture rules which can be proposed in analogy to non-associated flow rules of plasticity theory. For numerical solution the finite element method is used for spatial discretization, the effect of which is studied with regard to the ultimate load convergence of a thick-walled concrete ring. This simple example is also used to illustrate different forms of incremental tangential stiffness and initial load methods for the solution of brittle fracture problems. In addition, the influence of step length is examined for the same problem as far as ultimate load prediction is concerned. In conclusion, two examples are analysed for which experimental data on limit load and deformation behaviour are available for comparison, viz. a thick-walled concrete cylinder and a lined concrete reactor top closure. *

Also Professor at Imperial College of Science and Technology, University of London.

216

J. H. Argyris, G. Faust and K.J.

Willam, Limit load analysis of thick-walled

concrete

structures

2. Constitutive model for triaxial concrete behaviour As examples the load-deformation relations are shown in fig. 1 for plain concrete under uniaxial conditions. The linear elastic regime is followed by a hardening range starting at the discontinuity limit A. From here up to the maximum load bearing capacity B, internal microcracking progressively weakens the material constitution. Upon continued loading in deformation controlled tests, the concrete strength deteriorates by softening up to the point of rupture C, at which a sudden collapse is observed due to global instability of the specimen. The behaviour in tension is similar to that in compression except for reduced hardening and softening, the latter remaining negligible in the case of centrally loaded test specimens. In analogy with the uniaxial behaviour there are three response regimes under triaxial conditions -~ linear elastic, nonlinear hardening, and nonlinear softening including collapse. Clearly, all nonlinear response regimes are inelastic, i.e. internal energy will be dissipated in a load cycle because of irreversible damage due to microcracking. Considering continuum models of triaxial concrete behaviour, there are the formulations of nonlinear elasticity and plasticity. Both constitutive theories exhibit serious shortcomings when applied to model concrete behaviour. The variable moduli method [ 11, [ 21, [ 31 lacks a proper loading condition and introduces incompatibilities during neutral loading. In addition, failure conditions must be supplied to limit the deformation behaviour in the ultimate response regime. On the other side, classical plasticity theory is unable to predict softening and collapse phenomena because of the built-in notions of stability and unlimited ductility. In summary, none of the nonlinear continuum theories is applicable to concrete without major modifications; their particular shortenings are summarized in [4]. Considering fracture mechanics, crack initiation and crack propagation are determined by stress intensity at discrete cracks. In consequence, one would have to analyse numerous crack orientations and crack lengths in a three-dimensional structure and compare numerical stress concentration factors with critical values obtained from associated experiments which reproduce crack configuration and the surrounding stress field. Clearly, from a practical point of view this approach is beyond present means. From a theoretical point view it is also doubtful that linear or even nonlinear fracture

Tension

Fig. 1. Stress-strain

relationship

(uniaxial

behaviour

of plain concrete).

J.H. Argyris, G. Faust and K.J.

Willam, Limit load analysis of thick-walled

concrele

strucfures

211

mechanics properly describes the accumulation of damage during progressive microcracking due to overloading which ultimately leads to either tensile or shear type failure. Fracture mechanics is rather oriented towards brittle phenomena in the presence of localized stress concentrations. Considering limit analysis concepts one has the same dilemma as with fracture mechanics because kinematic failure modes in the form of surfaces have to be identified a priori in the three-dimensional structure; besides, the underlying theory is applicable only to perfectly plastic material behaviour, which can hardly be postulated in the case of tensile cracking. For this reason we concentrate in the following on continuum descriptions of triaxial concrete failure, in particular on three models of increasing complexity: a. Pure strength model: This formulation is based on a single scalar-valued function Of stress, e.g. the maximum load bearing capacity (point B). The failure condition (2.1)

fB (a) = 0

defines concrete strength exclusively in terms of stress; however, the failure surface can be divided into regimes with perfectly plastic and perfectly brittle post-failure behaviour. The corresponding stress-strain relation is shown in fig. 2, indicating the need for additional strength limitation in compression for rupture. Note that only strength data are required from the experimentalist, but no deformation measurements. Numerical implementation is rather simple, as shown in several prapers of the recent past [43-[9]. b. Strength and deformation model: This formulation is based on two scalar-valued functions of stress and strain which correspond to the maximum load-bearing and straining capacity of the material: f(a) = 0,

f,(e)

= 0.

(2.2)

Since f refers her to total strains, the second condition encloses the initial failure surface f(o) = 0 and limits ductility (fig. 3). Note that deformation measurements as well as maximum load data are -a

Fig. 2. Pure strength behaviour).

model

(perfectly

plastic,

perfectly

brittle

Fjg. 3. Strength perfectly brittle

and deformation behaviour).

model (limited

plastic,

218

J. H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled concrete structures

Fig. 4. Combined stress-strain model (limited plastic hardening; perfectly brittle behaviour).

needed at rupture. For objectivity they are converted into maximum total internal work or its inelastic component. c. Combined stress-strain model: This formulation is described by a scalar-valued function of stress defining incipient failure, e.g. at the discontinuity limit A f*(a)

= 0.

Thereafter,

(2.3)

loading is controlled

f* (0, E) = 0.

by the laws of strain-hardening

plasticity

(fig. 4): (2.4)

Since it is difficult to construct strain-hardening for arbitrary loading under triaxial conditions, an alternative approach is used to describe the surface of initial discontinuity (point A) and an additional rigid surface corresponding to maximum loading (point B = point C), assuming that the failure surface is independent of the load path: f*

(0) = 0,

flj (0) = 0.

(2.5)

These two surfaces contain the stress space in which internal energy is dissipated due to microcracking and in which the stress path is controlled by a scalar measure of damage in the form of an intrinsic time clock which terminates at rupture. Note that this model accounts for hardening prior to the ductility limit C = B. On the other hand, it requires a comprehensive test program in which stress-strain relations are obtained under triaxial conditions. Therefore, hardly any attempt has beenmade in this direction except in reference [ lo] , in which a simple scheme is proposed for reducing the stiffness in the dissipative pre-failure regime. In the following a constitutive model is developed along the lines of the pure strength model based on a very simple engineering concept of fracture in granular solids. The theory resorts to three constitutive postulates - initial failure, subsequent collapse, and inelastic rate of change due to energy dissipation. The proposed model describes material failure exclusively on the continuum level;

J: H. Argyris, G. Faust and K.J. Willam, Limit bad anal,vsis of thick-walled concrete structures

219

Fig. 5. Triaxial failure surface for concrete.

it predicts crack orientations, but it offers no explanation of the underlying physical mechanism which could be related to oriented microcracking or internal dislocations. The scope is limited to time independent creep effects and short term failure under triaxial conditions in which nonlinear. deformation behaviour prior to failure is secondary. 2.1. Initial failure model According to the pure strength model, incipient failure is described by a scalar-valued function of stress fB(a>= 0.

(2.6)

Mydrostrtic

Sction

Dcviatofic

Section

Fig. 6. Triaxial failure surface of concrete (concrete data of Launay and Gachon).

Geometrically speaking, this condition forms a surface in stress space separating elastic from inelastic response behaviour and linear from nonlinear. Confining attention to isotropic conditions, the failure model is fully defined by three i~~dependeIlt scalar fLln~tions of stress. Because of triple symmetry only a sixth of the principal stress space has to be considered. Fig. 5 shows the geometric configuration of the initial failure condition for concrete. It is basically a cone with curved meridian and noncircular base sections. The limited tension capacity is responsible for the tetrahedral shape in tension, while in compression a circular cylinder is approached in the limit. The failure surface is ~ol~v~Iliently described by hydrostatic and deviatoric sections as shown in fig. 6 for test data reported in [ 1 11. Mathematical formulations of triaxial concrete failure were directed primarily towards refinements of the pyramidal surfaces [ 12 ] for specific concrete phenomena [ 131. [ 141 . Little work has been done on continuous failure models involving all three stress invariants; only recently a rather general five-parameter model was presented in [ 15 1 to unify triaxial failure conditions. In the follo~i~lg~ an alternative ~ngili~ering fracture theory is developed for granular media in which failure occurs due to tensile cracking or shear sliding, which arc predominant features of COIIcretc type materials. When compared with hardening and softening plasticity formulations, the physical model has the distinct advantage of simplicity and provides a unique description of failure plants which are associated with either excess tension or excess shear: u. ~~~~~ir-~~~~~~~~r~~ ~~~~~ri~r~with ~~~~s~~~~ ~~~~-t~~~~ In fig. 7 the failure surface is shown in principal stress space. The tension cut-off condition forms a tetrahedron which is intersected by the hcxahedral pyramid of Bohr-Coulomb requiring three parameters for identification: (2.7) Cracking occurs in a plane norma 1 to the principal stress exceeding

I

Tymsion

the uniaxial tensile strength 1;.

Cut -Off

- Coulomb

Fig. 7. Mohr-Coulomb parameter

criterion

model in triaxial

with tension

stress spa&x?).

cut-off

(three

Fig. 8. Mohr-Coulomb Kupfer

et al.).

failure envelope

@iaxial test data of

J. H. Argyris,

Mohr-Coulomb:

G. Faust and K.J. Willam, Limit load analysis of thick-walled

f(u) =f(o,

)

us) =

u1

concrete

221

structures

If; + u3/fcu - 1.

(2.8)

Sliding occurs if major and minor principal stresses satisfy this constraint condition. The two-parameter model is expressed here in terms of the fictitious uniaxial tensile strength f; and the uniaxial compressive strength f,, , which are related to the internal cohesion c and the angle of friction rp by f,,

= 2c cos cp/(l - sin cp),

f; = 2c cos cp/(l + sin cp).

(2.9)

Substituting these expressions, into eq. (2.8) leads to the standard Mohr-Coulomb to which normal and shear stresses at the plane of failure are governed by

form according

(2.10)

171 = c - u tan cp.

In analogy to the plane of cracking there are now two planes of differential sliding containing the intermediate axis of principal stress; their directions are inclined at an angle 6 = 45” f (p/2 with the minor axis. b. Comparison with experimental data In fig. 8 the three-parameter model is compared with biaxial test data of [ 161 using the following parameter values: f, = 0.09 f,, , fi = 0.11 f,,

and

fcu = 311 kp/cm2.

(2.11)

We note that the Mohr-Coulomb criterion provides a close fit in tension-tension and tension-compression quadrants, but in biaxial compression failure is underestimated considerably by the straight line approximation.

x

Test Data

/ - e;/‘c

tiydrostalic

Section

\4’4”

Dcviatoric Sections

Fig. 9. Mohr-Coulomb failure envelope (triaxial test data of Launay et al.).

222

J. ff. Argyris, G. Faust and KJ.

Willarn, Limit load analysis of thick-walled

concrete

structures

Fig. 10. Mohr-Coulomb failure envelope (triaxial test data of Launay et al.).

In fig, 9 the three parameter model is compared with triaxial test data of [ 11 I using the following parameter values: c = 99 kp/cm2,

cp=

42.6”

and

f,,= 458 kp/cm2.

(2.12)

The polyhedral straight line approximation shows rather poor agreement along the compressive meridian of the hydrostatic section and becomes even worse if we consider the biaxial sections in fig. 10 where u2 = const. Clearly, the three-parameter Bohr-Coulomb criterion with tension cut-offs exhibits several shortcommings which are partly balanced by its outmost simplicity. Possible refinements all points towards the five-parameter model [ 151 which describes a smooth surface considering the effect of intermediate principal stress and reproduces curved meridians with deviatoric sections which are nonaffine. 2.2. Collapse model Upon intersection with the stress path the initial failure surface changes its configuration in the post-failure regime due to hardening, softening or collapse. In one limit, for perfectly plastic response, the initial failure surface does not change its position and shape. In the other limit, for perfectly brittle behaviour, local instabilities suddenly occur during which the initial failure surface partially collapses. In reality, there are no brittle or ductile materials; instead, media are subjected to environments under which they exhibit brittle or ductile post-failure response. For concrete there are several factors other than stress which influence the distinction of failure modes. Loading rate, stress gradients in complex structures, composition of concrete, and microstructural effects (such as aggregate inter-

J. H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled concrete structures

223

-i

const.

Mohr’s

Diagram

Fig. 11. No-tension collapse model (different fracture rules).

lock and steel reinforcement) heavily affect the post-failure behaviour. Therefore, brittle and ductile models form at best upper and lower bounds of the actual behaviour. In one case the failure envelope retains its initial position; in the other case it collapses suddenly to a reduced configuration. In the case of the three-parameter Mohr-Coulomb model with tension cut-offs there are three feasible configurations of partial collapse: a. Zero tension condition

In this case the failure surface collapses to a no-tension condition’as soon as the tensile regime of the initial envelope is reached. Consequently, the tensile strength diminishes to zero, while cohesion and internal friction provide further resources of strength in accordance with the Mohr-Coulomb criterion as indicated in fig. 11: ft-, 0 while

sincp=.

fcu -fl-7 fcu+ft

f

(2.13)

and c = $!-( 1-sin cp)/cos p.

b. Zero cohesion condition

In this case the failure surface collapses to a no-cohesion condition as soon as the Mohr-Coulomb regime of the initial envelope is reached. Consequently, the cohesive strength diminishes to zero, and the only resource of further strength is internal friction, as indicated in fig. 12, the tensile strength

MOWS

Diagram

Fig. 12. No-cohesion collapse model (no-volume change rule).

I,.01

224

Mohr’s Diagram

Fig. 13. No-friction collapse model (no-volume change rule).

also being zero (Coulomb criterion): c + 0, ft -+ 0

while

fcu- f;

sin ‘p ~77.

Cl1

(2.14)

t

In this case the failure surface collapses to a non-friction condition as soon as the ~~ohr-Coulomb regime of the initial envelope is reached. ConsequeIltly, frictional strength diminishes to zero, and the only resource of further strength is internal cohesion, as indicated in fig. 13, the tensile strength also being zero (modified Tresca criterion): P-+ 0, fr + 0

while

(2.15)

c=J,,/Z.

2.3. Stress transfer due to fracture Besides modeling initial and subsequent failure conditions, there is still the fundamental problem of how to redistribute stress states which violate the current failure condition. This question actually refers to the implementation of coI~stitutive coIlstraint conditions in analogy to the flow rule of elastoplasticity. In the case of tensile cracking the governing stress transfer strategy if unambiguous; it simply calls for the reduction of principal stresses violating the tension cut-off condition to their allowable limit (for perfectly brittle behaviour f, --, 0). In the case of shear failure according to lvtohr-Coulomb there are several possibilities analogous to the variety of non-associated flow rules. In the following, several “fracture rules” are developed in the form of initial stress increments which describe stress transfer from non-permissible states to the governing failure condition: Q. Tens& cracking For tensile cracking the tension cut-off criterion eq. (2.7) determines the excess stress in the direction of major principal stress according to the “~lormality” rule: for

f(oi> > 0,

Ac+=~~-f~,

i = 1,2,3.

(2.16)

J. H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled

For brittle behaviour f, + 0. The resulting initial stress increment tension involves here e.g. only the major principal stress

concrete

structures

225

Au which suppresses the excess

(2.17)

Au = {Au,, O,O,O,O,O).

b. Shear failure For shear failure the Mohr-Coulomb criterion eq. (2.8) determines the excess in the direction of major and minor principal stresses Au, , Au, moving the stress state into the permissible regime: for

f(c, , a,> > 0,

(2.18)

Au = {Au,, 0, Au,, O,O, 01.

As a consequence of the Pohr-Coulomb hypothesis the resulting stress increment has no contribution in the direction of intermediate principal stress. The necessary stress transfer can be accomplished by several “fracture rules”, four of which are illustrated in fig. 14 for ductile post-failure behaviour: (9 0, = CorWt In this case we fix the major principal stress and increase the minor, thus obtaining the following stress increments for redistribution: CT1-~ u3

Au, = 0,

(2.18)

2

Ao,=2(1+tanptanrp)

@ u, = amt.

@

un = const

@lu,+u3V2=const.

0

u, = const

Fig. 14. Stress transfer

methods

for shear failure

(different

fracture

rules).

226

J. H. Argyris, G. Faust and K. J. Willam, Limit load analysis of thick-walled concrete structures

where of is defined by the Mohr-Coulomb condition

!

of =sincp c cot 9 -

Ul +*3 2

)

(2.19)

)

From geometric considerations cu=arctan(l”oszP), (iif

(2.20)

p=90”-a+cp.

comt

u3 =

In this case we fix the minor principal stress and decrease the major, thus obtaining the following stress increments for redistribution: Ao, = --2(1 - tan p tan (9)

(

Ol 2

*3

- Go , )

Ao, = 0.

(2.21)

From geometric considerations F=90” -QI -9.

(11=arctan (lc+oiiEP),

(2.22)

(iii) (ul + 0,)/2 = const

An intermediate transfer strategy assumes that the centroid of Mohr’s circle remains fixed. In this case we obtain the following stress increments for redistribution: Au, = -Au,. (iv) on =

(2.23)

const

Another intermediate transfer strategy assumes that the normal stress does not change at the plane of failure. In this case we obtain the following stress increments for redistribution: (2.24) where A~=-..!w-

cos cp

ul

-

2

u3

- Of I

The stress paths of these fracture rules are illustrated e.g. in fig. 11. The strategies with or = const and u = const are rather unlikely to reproduce the actual failure mechanism. They involve longer strestpaths and thus require more energy dissipation during fracture than the direct strategies with (ai + 0,)/2 = const and u3 = const or the normality rule. We note that the latter agree better with the principle of least entropy production, and that the transfer lo1 + us)/2 = const implies no volume change of inelastic components. 2.4. Summary The proposed engineering model for inelastic stress transfer technique differs from elastic-plastic theories in two points - there is’no decomposition of kinematic quantities, and it does not require

J.H. Argyris, G. Faust and K.J. Willam, Limit load analysis

of thick-walled concrete structures

227

a flow rule defining inelastic strain increments; instead, it involves only static quantities in the form of fictitious initial stress increments producing stress redistributions according to a given “fracture rule”. Therefore, this constitLItive model does not lead to an incremental stress-strain law, but rather to an initial load strategy for stress transfer which is governed by a set of constitutive constraint conditions. The proposed engineering formulation of concrete fracture has two shortcomings: it does not satisfy Drucker’s stability postulare even for perfectly plastic-failure behaviour, and it does not accommodate geometric effects, i.e. crack growth dependent on crack length (the influence of stress gradients). However, its transparency and simplicity of numerica application turn it into a very useful tool for engineering purposes.

3. Numerical solution of limit load problem Finite element formulations of nonlinear boundary value problems have been the subject of numerous publications in the past. Here we only recall that the principle of virtual work is readily extended to virtual velocities for discretizing the quasistatic motion of inelastic structures. A great number of numerical solution schemes have been proposed for the integration of the ensuing first order differential equations; all of them are based on incrementation or iteration. By definition the finite element displacement method provides at each load step kinematic compatibility in the small and equilibrium in the large. The main task of limit load analysis is thus the computation of admissible stress distributions satisfying the constitutive constraint conditions at each step of the hypothetical load history. When stationary conditions of stresses and deformations are reached, a lower bound of the actual collapse load is obtained according to the static theorem of limit analysis. However, boundedness is maintained by the finite element solution only if the spatial discretization error remains negligible and if the finite form of the incremental constitutive model introduces integration errors which are of higher order. We also recall that the limit load theorems are valid only for associated plasticity, i.e. ductile behaviour where the normality condition controls stress transfer. The constitutive model of section 2 violates these assumptions; thus one part of the following discussion is concerned with this particular aspect from a numerical point of view. The limit load analysis infers in principle a search for the maximum load factor hmax at which equilibrium as well as material constraints are still satisfied. The corresponding optimization problem can be stated as follows: Find h -+ hmax for which the structural eqLIilibrium and constitutive constraint conditions K(t)b=

AM

and

f(a) < 0

(3.1)

are satisfied. Numerically this implies a constrained search for the stability limit which is pursued by tracing stationary values of the internal power in terms of elastic and total components

~+hmx

if

AU(o) + 0

and

AU(E) + 0.

(3.2)

The change of energy criterion applies directly to incremental step-by-step procedures. There is no

need to reproduce the true path of evolution as long as an equilibrium position is reached which satisfies the constitutive constraints. The static theorem simple states that a lower bound of the ultimate load is found and its accuracy depends primarily on the numerical distortions of the true path. Incremental procedures trace the inelastic response for the hypothetical load history and thereby provide a complete picture of the deformation response as well as redistribution of stresses. They furnish in addition tight estimates of the ultimate load capacity if the constitutive constraints are considered in an iterative manner and discretization errors remain negligible. In the following, two aspects of numerical im~lementatio~l are discussed and illustrated with limit load analysis of a thick-walled concrete ring for which analytical results are available. In the first part, the influence of finite element discretization on the load-deformation characteristics and limit load prediction is studied. In the second part, different numerical methods are examined for the solution of brittle fracture problems with regard to accuracy and efficiency.

Three different mesh layouts are used for spatial discretization of the thick-walled concrete ring shown in fig. 1.5. Upon increasing pressure loading, the tensile strength regime controls basically the overall response behaviour of the structure. In our case we assume brittle collapse in accordance with the no-tension model and an initial tensile strength off, = 28.4 kpfcm2. With increasing pressure the radial cracks propagate at an increasing rate from the inner surface towards outside because of excess tension in the circumferential stress components. The response characteristics are illustrated in fig. 16, in which limit load predictions are compared with the analytical value for brittle behaviour Pult = 54.98 kp/cm2. We note that the computed values of the ultimate loads of the three idealizations converge rapidly towards the theoretical result as the mesh layout is refined. The coarse idealization considerably overestimates the theoretical limit load because the severe stress gradient is smoothed and thus

Measurements in cm f

: 320000

kp/cm*

I 1.0

Y i 0.20

f

f, = 28.4 kpicm’

f

21

WF

I f79WF

fine

Fig. 15. Thick-walled

concrete

ring &geometry and mesh layouts).

J.H. Argyris, I;. Faust and ICJ, Willam, Limit load analysis of thick-walled corrcrete structures

229

4

kpfcm’ 80

coarse

60

.----

LO

20

0

Fig. 16. Ultimare load predictions of no-tension model (influence of mesh refinement).

causes too little stress redistribution. We may conclude that coarse idealizations tend to stiffen the structure such that the lower bound property of the static limit analysis theorem is normally lost due to discretization errors.

Initial

Tangential

Load

Methods

Stiffness

Methods

Fig. 17. Incremental solution methods (initial load and tangential stiffness techniques).

J.H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled concrete structures

230

3.2. Incremental

solution method

Incremental solution methods have been detailed in numerous publications of the recent past. Here we only compare the influence of tangential stiffness and initial load techniques on the ultimate load prediction of the thick-walled concrete ring above. Fig. 17 shows four incremental solution schemes which involve updating either the stiffness properties at each load step (tangential stiffness approach TS) or the equivalent residual load corrections at constant stiffness (initial load methods DIM and NIM). Detailed discussions of these techniques have been presented previously; here we only refer to [ 171 and [ 181 for the initial load method, the iteration cycles of which may be interpreted physically as viscous behaviour in time [191. Fig. 18 illustrates numerical predictions of these four solution techniques using the fine mesh layout; these are compared with the analytical ultimare load value. Clearly, the direct incremental method DIM considerably stiffens the system since the tensile stresses are actually not reduced to zero by iteration. The residual stresses delay the brittle action and infer physically numerical softening in a gradual manner. Even when combined with the tangential stiffness method without iteration, TS + DIM, the prediction is still much too stiff because the excess tensile stresses are in reality not reduced to zero. Only the normal iterative method NIM and its combination with the tangential stiffness method, TS + NIK, yield acceptable results, meaning that numerical distortions of the actual constitutive model remain within a given tolerance (combination of discretization and numerical integration error due to finite step length). This example indicates how the solution method introduces another ingredient into the overall model of the physical phenomenon. In this paiticular case the ultimare load capacity is very sensitive with regard to numerical distortions. In conclusion, the influence of the step length is indicated in fig. 19 for the fine mesh layout and

TS + DIM

TS+NIM J’@l-

Fig. 18. Ultimate

load predictions

of no-tension

model

(influence

5L 98

of numerical

solution

scheme).

J. H. Argyris,

G. Faust and K.J. Willam, Limit load analysis of thick-walled

concrete

structures

231

80

60

LO

20

0

1

1

2

I

I

I

6

8

I

IO

12

14

" 10-31nn7

Fig. 19. Ultimate load predictions of no-tension model (influence of step size).

the iterative initial load scheme NIM which is used in the remainder of this paper. In one case the pressure is incremented from the elastic limit p, = 27.5 kp/cm’ in load steps of Api = 2.5 kp/cm* reaching failure between p, = 55 to 57.5 kp/cm* due to cracking of the whole section and lack of convergence in correspondance to stationary values of elastic and total energies. In the other case the pressure is incremented from the elastic limit in a single load step Ap = 27.5 kp/cm’ reaching failure also between p, = 55 to 57.5 kp/cm2. Clearly the nonlinear deformation characteristics in this case are approximated by a secant, but the ultimate load prediction is hardly affected by the enormous increase of step length. The total number of iterations are 85 in the first case versus 18 in the latter. This suggests that rather large load increments can be used at least initially if the ultimate load capacity is of primary interest. For the case where safety with regard to prescribed overload levels is the main goal no incrementation is required at all and the load can be applied in a single step except near the actual ultimate value, where the rate of convergence deteriorates rapidly. 3.3. Summary Several observations can be made when comparing tangential stiffness versus initial load methods aside from considerations with regard to programming ease and effectiveness of the finite element software:

a. Tangential stiffness solution First we note that the tangential stiffness method requires the formulation of instantaneous stressstrain relations according to hypoelastic or elastic-plastic formulations, thus it can not be applied directly to the fracture model of section 2. For the case that the tangential stiffness method ian be used its success depends on the effective solution of the following five tasks: (i) Incremental stress-strain laws are linear in the differential sense only; thus for finite load in-

232

J.H ArKyris, C. Faust and K.J. Willam, Limit load analysis oj’thick-walled concrete structures

crements the linearization errors accumulate and lead in general to considerable distortions of the true path of evolution if no corrective measures are taken in the form of tangential stiffness or initial load iterations. (iii For non-proportiol~al loading the state of inelastic loading is unknown at the beginning of the load increment. It depends on the stress path within the finite increment; thus corrective measures in form of iterations are required if wrong assumptions are made by extrapolating results of previcus load steps. (iii) For finite load steps the material failure surface is normally penetrated within the load increment, i.e. the material transits from the elastic into the inelastic regime. The proper decomposition into an elastic and inelastic portion is a nonlinear problem by itself, thus requiring again corrective measures in the form of iterations and quadrature within the increment. (iv) For strain-softening behaviour or cracking, the tangential material model loses its positive definite character. As a matter of fact the ultimate load capacity of the structure is reached if these local instabilities propagate through the structure until the structural stiffness becomes singular; its “proper” detection is a rather difficult problem. fv) Non-associated flow or fracture rules lead to nonsymmetric elastic-plastic models. In this case the underlying finite element software has to be modified for nonsymmetric matrices. Clearly, storage requirements and computational effort would vastly increase if the tangential stiffness method is applied to this type of constitutive model. b. Initial bad sohtion Iterative techniques pose two fundamental problems which have been responsible for the sensitivity of the initial load method with constant stiffness: (i) Most iterative techniques are locally convergent only; thus their success strongly depends on the numerical ilnpleme~ltation, e.g. initial strain versus initial stress formulatio~is [ 173 . (ii) The rate of convergence depends largely on the ratio of tangent modulus to initial modulus, on the accuracy of the first guess and on the occurence of unloading; thus optimal underrelaxation factors and extrapolation techniques for improving the initial guess are crucial for the effectiveness of the iterative method [ 181. Fxperience has shown that the tangential stiffness method yields results which are on the stiff side in problems with degrading stiffness properties; thus a lower bound of the ultimate load value is normally lost even if discretization errors remain negligible. In contrast, the initial load results are more sensitive with regard to instabilities and thus maintain better boundedness of the solution. In the case of non-associated flow rules, transition with curved failure surfaces, and unloading problems, the initial load method with constant stiffness offers distinct advantages over tangential stiffness schemes. In addition, softening effects and constitutive collstrai~~ts in the form of stress transfer methods can be implemented directly via initial loads; in this case the detour over equivalent tangential material laws is often not even possible.

4. Examples

of ultimate

load analysis

In the following the influence of different constitutive postulates is examined in two examples with regard to limit load and nonlinear deformation characteristics. In the first problem crack propagation of a thick-walled concrete cylinder is studied under increasing pressure for which the limit

J.H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled

concrete

233

structures

load capacity is known from experiments. The engineering fracture theory is applied using brittle, ductile and intermediate collapse models and different fracture rules for stress transfer. In the second problem nonlinear deformation and limit load response of a lined concrete reactor top closure are investigated for which experimental results are available for comparison. 4. I. Thick-walled

concrete

cylinder

This rather academic example is selected because (i) the simple geometric configuration introduces severe stress gradients, (ii) analytic solutions are readily computed for the limiting cases of perfectly brittle and perfectly ductile behaviour, and (iii) experimental values are available for the limit load of an eqL~ivalent pressurized concrete tube 1201. Geometric data and finite element mesh layout of the thick-walled concrete cylinder are shown in fig. 20. Pressurization of the specimen introduces severe stress concentrations at the inner surface as obtained by Levy’s solution for linear elastic response. The stress path is predominantly biaxial in tension-compression where it is constrained ‘by tension cut-off and Bohr-Co~llomb. The particular parameter values are indicated in fig. 21. The ratio of uniaxial compressive strength to uniaxial ten-. sile strength is selected in analogy to the biaxial Griffith criterion fm/ft= S/l. With!, = 44 kp/cm’ a rather high value was chosen for tensile strength to account somehow for the effect of stress concentration (compare bending tensile strength versus values obtained from centrally loaded uniaxial tests). Unfortunately, no experimental data are available for this particular concrete; thus only qualitative statements can be made from a comparison of experimental and numerical results.

Measurements

E:

in cm

320000 kplcm’

No - Tension

Y I 0.20

-Tension Cut-Off 4 = 44 kpicm2

-Mohr -Coulomb

c = 78.5 kpf’cm2 (p = ll.E”

Fig. 20. Thick-walled concrete cylinder (geometry and mesh

layout).

Fig. 21. Biaxial failure model of concrete (Mohr-Coulomb condition with tension cut-off).

J. H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled

234

concrete

structures

% kplcm’

4 kp/cm* Crack

Zones

Units

0

10

20

30

I

u 10m3mm

Fig. 22. Ultimate load predictions (influence of collapse hypothesis).

kp/cm*

Orr

kplcm’

Fig. 23. Stress distribution and crack zones (brittle no-tension model).

a. Comparison with test data Fig. 22 illustrates the nonlinear deformation and ultimate tile and an intermediate collapse model which are compared 141 kp/cm’ [201.

load characteristics of a brittle, a ducwith the experimental value of pult =

The brittle no-tension model predicts a maximum pressure of pl = 89 kp/cm* at which splitting fracture propagates through the cylinder wall. This numerical value has been confirmed by independent analytical solutions; however, it is rather low when compared with the experimental limit load. Radial and circumferential stress distributions are shown in fig. 23 for different levels of internal pressure. The no-tension collapse mechanism leads to large stress redistributions since the tensile stresses are all reduced to zero in the zone of cracking. At p, = 40 kp/cm* radial cracks initiate from the inner surface towards the outside. Upon continued loading the cracks propagate at an increasing rate, causing sudden collapse if the crack zone extends up to half of the outer radius. Immediately before the limit pressure of pI = 89 kp/cm* is reached the auter part of the cylinder is still intact transforming the pressure in form of an elastic cylinder of reduced thickness. The ductile model predicts a maximum pressure of pl = 230 kp/cm*, at which plastic flow propagates through the total cylinder wall. This numerical value has been confirmed by analytical limit load calculations; however it is much too high when compared with the experimental limit load. The corresponding radial and circumferential stress distributions are shown in fig. 24 for different levels of internal pressure. The ductile flow mechanism redistributes the tensile stress concentrations, freezing the stress distribution at the tensile strength value off, = 44 kp/cm* ., At p, = 40 kp/cm* plastification initiates from the inside and propagates upon continued loading very slowly towards the outside.

J.H. Argyris, G. Faust and K.J. Willam, Limit load analysis of thick-walled concrete structures

235

! 1

Ott

I

kplcm2

j--pv.,.j

:.::..:.,:.:::: iji’

:: ::,so::::

::“”

.:: ::y:.,

““”

:c ::$ 2:5 ( 1

1 ,, 11

._:::’

Units

/

]

Plastic 1 Zones

kp/cm2

% kplcm2

Fig. 24. Stress distribution

and plastic

zones (ductile

model).

We note that at pi = 150 kp/cm’ the circumferential stresses drop even below ft at the inner surface because the lv?ohr-Coulomb condition now controls the stress path rather than tension cut-off. Collapse is approached if the plastic front reaches the outer surface in contrast to brittle fracture above. The large differences between numerical limit load predictions emphasize the importance of degrading tensile strength in variable stress envirnoments. The perfectly brittle and perfectly ductile collapse models provide bounds for the actual limit response behaviour, which are very wide because the axisymmetric idealization does not account for differences in crack propagation along the circumference. The intermediate collapse model of fig. 22 withf, = 22 kp/cm’ is an attempt to account for additional strength reserves because of radial cracks which are discrete rather than smeared along the circumference. b. Influence of fracture rule A study of fracture rules and their effect on limit load prediction concludes the thick-walled cylinder example. In fig. 14 four possibilities were shown for stress redistribution during fracture in analogy to the choice of non-associated flow rules in plasticity. These four fracture rules are now applied to the ultimate load analysis of the thick-walled con-

236

J. H. Argvris, G. Faust and K.J. Willam, Limit loadanalysis

0

0

01 0

10

10

20

3.0

20

30

Fig. 25. Ultimate load predictions

60

40

5.0

5.0

of thick-walled

60

6.0

concrete

structures

70

u

70

InternahWork

10 'cm

kpcm

(influence of fracture rule: ductile model).

Crete cylinder above assuming ductile behaviour. In addition, a normality rule oI is introduced in analogy to the normality principle of associated plasticity. Fig. 25 shows the resulting deformation characteristics together with the evolution of internal work for increasing pressure. It is interesting that the same limit load value is reached in all cases, suggesting that for ductile models the ultimate load response is independent of the fracture rule. The (03 = const)-rule, the normality rule and the no-volume change rule are considerably stiffer, i.e. they involve less stress redistributions or energy dissipation than the (ol = const)-rule and the (on = const)-rule. From the principle of least entropy production they are most likely to reproduce the actual behaviour, and from a computational point they provide more rapid convergence than the other stress transfer models. 4.2. Lined concrete reactor top closure As a second example a 1:ll scale model of an actual PCRV top closure is analysed (see fig. 26). This complex engineering problem was tested by the Danish atomic energy authorities who supplied a few test results for comparison [ 2 1 ] . The lined concrete lid was pressurized in several cycles of increasing amplitude during which the load was transferred to the inclined supports by the combined action of concrete and steel. The composite behaviour was controlled by bending and she?:,

J H. Argyris, G. Faust and K. J. Willam, Limit load analysis of thick-walled concrete structures

supports \ * 26.5’

231

Supports #’

P--i

Liner

ttt

ttt .

5

32.5 -A

Measurements in cm Fig. 26. Top closure

of concrete

reactor

vessel (1: 11 scale model

LM-3).

whereby the latter was ultimately responsible for a wedge type failure at a pressure level of pulr = 377 kp/cm2. The idealization of the structure is shown in fig. 27 in which axisymmetric finite elements are used for the discretization of concrete and steel components. The failure behaviour of concrete is described by the fracture model of section 2. The Mohr-Coulomb’ condition with tension cut-offs is augmented by ductile and no-cohesion post-failure hypotheses. The strength parameters of the initial failure surface are

638 Degrees Fig. 27. 1 :ll

scale top closure

of Freedom

model LM-3 (axisymmetric

finite element

idealization).

238

J.H. Argyris, G. Faust and R J. Wiliam, Limit Ioad analysis of thick-walled concrete structures

f, = 458 kp/cm'

ft = 43 kp/cm2,

,

fi = 87.5 kp/cm2,

(4.1)

or in terms of cohesion and friction c = 100 kp/cm*

,

‘p =

42.8 kp/cm2.

(4.2)

The modulus of elasticity and Poisson’s ratio for concrete are EC = 255 000 kp/cm* ,

Yc =

0.15.

(4.3)

In all cases the no volume change rule is used for stress transfer due to fracture, (ui + us)/2 = const. The failure behaviour of the steel components is described by an associated plasticity formulation after von Iv?iseswithout strain hardening using the following yield strengths: liner:

f, = 2 100

kp/cm'

,

(4.4) flange: f, = 2600 kp/cm2. The modulus of elasticity and Poisson’s ratio for steel are Es = 2 100000 kp/cm2,

v, =

0.30.

(4.5)

For numerical solution the incremental initial iteration method NIM with constant stiffness is utilized. The load is applied in the form of pressure increments of Api = 10 kplcm’ ; at each stage

SXJ*

400-

300-

200.

loo-

20

4.0

6.0

8.0

Fig. 28. 1 :I 1 scale top closure model LM-3 (axial displacements versus pressure).

" mm

239

J. H. Argyris, G. Faust and K.J. WiNam, Limit load analysis of thick-walled concrete structures

i.0

i.0

60

6'0

Id.0

A,

i06 k&n

Fig. 29. 1 :ll scale top closure model LM-3 (internal work versus pressure).

an equilibrium position is computed iteratively up to relative tolerance of IIAull/ll uII < 10m3 in the displacements. Fig. 28 shows the computed displacements at the top surface together with the resulting test data. The experimental collapse load p,[, = 377 kp/cm2 compares favourably with the numerical predic-

o cohesion

0

“a

-3

0

3

Model

6

Elf 7..

Fig. 30. 1: 11 scale top closure model LM-3 (tangential strains

versus

pressure).

240

/ Radial

I

Crack Zones (q,)

Radial

Crack

Zones (4,)

t

I2 I

Circumferential

Mohr -Coulomb

Fig. 31. collapse,

1: 11 s&e

top

Crack Zones

Failure

closure

pi = 400 kp/cm*f.

( qr, azz)

Zones f~~,cr__~,

ci,f

model LM-3 (no cohesion

Circumferential

Crack Zones

Mohr - Coulomb Failure

Fig. 32. 1: 11

s&e

t&p

closure

Zones

( or,, o;,l

(a,, C1;,U&)

model LM-3 (ductile

model,

p1 = 590 kp/cm2).

tion of the no-cohesion model prlcoh = 400 kp/cm*, while the ductile formulation approaches failure only at an ultimate pressure of PJdti = 590 kpfcm2. Both analytical deformation characteristics are too flexible in the nonlinear regime up to pi = 250 kpjcm2 ,* thereafter the brittle and the ductile model bound the actual response behaviour. Near ultimate load the total internal work approaches stationary values asymptotically, as shown in fig. 29 for the two post-failure formulations. Note that the ductile model fractures before a horizontal tangent is reached, indicating that elastic and total internal work criteria are only sufficient

J. H. Argyris, G. Faust and K.J. Willam, Limit load analysis

of thick-walled

concrete

structur<‘s

241

conditions for collapse. The no-cohesion prediction of limiting pressure agrees rather well with the experimental ultimate load. Fig. 30 shows the corresponding circL~mferentia1 strain predi~tio~ls at point (r, z) = (15.0, 10.5) together with the corresponding test data. In the case of local strain considerable discrepancies must be expected since the numerical results are based on smearing individual cracks. In this light the agreement between the no-cohesion model and experimental data is surprisingly good while the ductile model simply doesnot capture the weakening of the material constitution near ultimate pressure. The growth of failure zones is illustrated in figs. 31 and 32 for the brittle no-cohesion model and the ductile hypothesis. Cracking is distinguished according to excess tension of the circumferential components u, (radial cracks) or the Y - z components (circumferential cracks). Moreover, failure occurs due to Mohr-Coulomb type sliding, for which the zones are shown in the bottom figure. In both post-failure models radial and cir~u~~ferential cracks develop at a very early stage the outside and penetrate the center of the concrete slab. Another zone of sliding type failure develops at a second stage in the zone of high shear and extends rapidly towards the inclined supports. In a third stage this type of material failure penetrates the compressive zone at the pressurized bottom surface leading ultimately’ to collapse of the top closure structure.

5, Concluding remarks In connection with ultimare load behaviour of thick-walled concrete structures two topics were discussed - (i) an engineering approach to triaxial concrete fracture and (ii) the numerical computation of limit load. In summary: fi) The engineering model of triaxial concrete fracture is based on three consitutive postulates initial failure, subsequent collapse and stress transfer. Conceptually the formulation is a pure strength approach in which no statements are made as far as stress-strain relations are concerned and thus only strength data are required from the experimentalist. The Mohr-Coulomb criterion with tension cut-offs provides information on the orientation of failure planes (crack directions); it also furnishes a transparent picture of ductile post-failL~re behaviour or brittle collapse in the form of no-tension, no-cohesion or no-friction models. Energy dissipation during failure is responsible for accompanying stress redistribution; its direction is controled by a fracture rule in analogy to non-associated plasticity. The transfer rule with the minimum energy dissipation corresponds to the least entropy production principle, in addition to providing the most effective convergence rates in the iterative solution process. In the present examples the utlimate load predictions were rather insensitive with regard to the choice of fracture rule but very responsive as far as collapse hypothesis is concerned. (ii) The numerical solution of ultimate load poses a central problem, i.e. the bounding property of the predicted value. We recall here that the bounding theorems of limit analysis are only valid for associated plasticity; thus the fracture model above would need to be restricted to ductile bchaviour and stress transfer according to the normality rule. On the other hand, the finite element idealization strongly affects the ultimate load prediction in such a way that coarse mesh layouts predict fracture loads which are too high, thus losing boundedness due to spatial discretization. As far as incremental solution is concerned, iteration within each load step is imperative in order to avoid numerical distortions of the constitutive model, particularly near collapse. The choice of step length bears little relevance to the limit load prediction, but convergence characteristics are strongly affected near ultimate load.

242

J.H. Argyris, G. Faust and K.J. Willam, Limit load an&s&

of‘ thick-walledconcrete structures

In conclusion, neither bounds nor uniqueness can be assured from a theoretical viewpoint, but the limit load predictions of the engineering fracture model are rather accurate for engineering purposes if all irlgredients in the numerical solution process are properly considered. However, one should be fully aware of the shortcomings of the simple proposal which raises several questions re-, quiring further examination. Here we only mention two fundamental problems of softening materials exhibiting partial collapse in the form of brittle post-failure behaviour: uniqueness of limit load prediction, boLl]idedness of limit load prediction, Besides these theoretical considerations, a proper formulation of triaxial stress-strain relations in the inelastic response regime is lacking. To this end the internal variable theory together with damage accumulation concepts according to the time-to-fracture hypothesis provide promising guide lines for alternative form~llations. Another question is the proper strength simulation in variable stress fields. This poses a rather serious problem with regard to failure in tension. In order to account for size effects and geometric effects in form of stress concentrations, stochastic fracture models provide a promising alternative for predicting crack initiation and crack propagation not only in terms of stress but also in terms of stress gradients.

Acknowledgements

The present study was sponsored by the “BundesministeriL~m fur Forschung und Technologie, Forderungsvorhaben SBB4” under the liaison of Dr. N. Bunke, whose cooperation we gratefully acknowledge.

References [ 1 ] R. Palaniswamy and S.P. Shah, Fracture and stress-strain relationship of concrete under triaxial compression, J. Strut. Div., ASCE 100 (1974) 901-916. [2]1. Nelson and M.L. Baron, Application of variable moduli models to soil behaviour, Int. J. Sol. Strut. 7 (1971) 399-417. [ 31 H.B. Kupfer, Das Verhalten des Betons unter mehrachsiger Kurzzeitbelastung unter besonderer Berucksichtigung der zweiachsigen Beanspruchung, Deutscher Ausschuss fur Stahlbeton 229 (Berlin, 1973) l-131. [4] J.H. Argyris, G. Faust, J. Szimmat, E.P. Warnke and K.J. Willam, Recent developments in the finite element analysis of prestressed concrete reactor vessels, Nuclear Eng. Design 28 (1974) 42-75. [5] J.H. Argyris, G. Faust, .i. Szimmat, E.P. Warnke and K.J. Willam, Finite element ultimate load analysis of three-dimensional concrete structures, Colloquium on nonlinear mechanics, Jablonna, Poland, Sept. 1974; also ISD report no. 174 (1975). [6] S. Valliappan and T.F. Doolan, Nonlinear stress analysis of reinforced concrete, J. Strut. Div., ASCE, 98 (1972) 885-898. [7] M. Suidan and W.C. Schnobrich, Finite element analysis of reinforced concrete, J. Strut. Div., ASCE, 99 (1973) 2109-2122. [8] J. ColviUeand J. Abbasi, Plane stress reinforced concrete finite elements, J. Strut. Div., ASCE, 100 (1974) 1067-1081. (91 C.S. Lin and A.C. Scordelis, Nonlinear analysis of RC shells of general form, 3. Strut. Div., ASCE 101 (197.5) 523-538. [lo] K.M. Romstad, M.A. Taylor and L.R. Herrmann, Numerical biaxial characterization for concrete, J. Eng. Mech. Div., ASCE 100 (1974) 935-948. [ 111 P. Launay and H. Gachon, Strain and ultimate strength of concrete under triaxial stress, AC1 Special Publ. SP-34 (1972) 269-282. [ 121 B. Paul, Macroscopic criteria for plastic flow and brittle fracture in: H. Liebowitz (ed.), Fracture, an advanced treatise 2 (Academic Press 1968) 313-491.

J, H. Argyris, G. Faust and K. J. Willam, Limit load analysis of thick-walled concrete structures

243

[13] H. Reimann, &it&he Span~u~s~ustande des Betons bei me~achsider, ruhender Kurzzeitbe~stu~, Deutscher Ausschuss fur Stahlbeton 175 (Berlin, 196.5) 35-63. [14] D.J. Hannant and C.O. Frederick, Failure criteria for concrete in compression, Mag. Concrete Res. 20 (1968) 137-144. 1151 K.J. Willam and E.P. Warnke, Constitutive model for the triaxial behaviour of concrete, IABSE Seminar on Concrctc Structures Subjected to Triaxial Stresses (Bergamo, 1974). 1161 H.B. Kupfer, B.K. Hilsdorf and Il. Rusch, Behaviour of concrete under biaxial stresses, J. Amer. Concrete inst. 66 (1969) 656-666. [17] J.H. Argyris and D.W. Scharpf. Method of elastoplastic analysis, proceedings of the ISD-ISSC Symposium on Finite Element Techniques (Institut fur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, Stuttgart, 1969) 381-416. Also ZAMP 23 (1972) 517-551. fi8) H. Balmer, J.St. Doltsinis and M. Konig, Elastoplastic and creep analysis with the ASKA program system, Camp. Meth. Appl. Mech. Eng. 3 (1974) 87-104. 1191 OC. Zienkiewicz and LC. Cormeau, Visco-plasticity-plasticity and creep in elastic solids, a united numerical solution approach, Int. J. Numer. Meth. Eng. 8 (1974) 821-845. [ZO] H. Martin, Zusammenhang zwischen Ober~chenbesc~ffenheit, Verbund und Sprengwirkung von Bewehrungsstahlen unter Kurzzeitbelastung 228, Deutscher Ausschuss fur Stahlbeton (Berlin, 1973) l-50. [ 211 S.I. Anderson and N.S. Ottosen, Ultimate load behaviour of PCRV top-closures, theoretical and experimental investigations, Preprints of 2nd Int. Conf. SMiRT, Berlin, Paper H 4/3 (19’73).