Computers and Structures 160 (2015) 42–55

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Computers and Structures

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Numerical limit analysis of steel-reinforced concrete walls and slabs

http://dx.doi.org/10.1016/j.compstruc.2015.08.0040045-7949/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.

A.A. Pisano ⇑, P. Fuschi, D. De DomenicoDept. PAU, University Mediterranea of Reggio Calabria, via Melissari, I-89124 Reggio Calabria, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 January 2014Accepted 3 August 2015Available online 22 August 2015

Keywords:Reinforced concrete wallsSlabsLimit analysisPeak loadCollapse mechanismLarge-scale prototypes

A limit analysis based design methodology is hereafter proposed and applied for the peak load evaluationof steel-reinforced concrete large-scale prototypes of structural walls and slabs. The methodology makesuse of nonstandard limit analysis and predicts the peak load multiplier of the analyzed structures bydetecting an upper and a lower bound to it. Some useful hints on the collapse mechanism the structurewill exhibit at its limit state is also attainable. To check the reliability of the numerically detected peakloads and failure modes a comparison with experimental laboratory findings, available for the large-scalespecimens considered, is presented.

� 2015 Elsevier Ltd. All rights reserved.

1. Research motivations and introduction

The present study follows a very recent paper by the authors[30], dealing with the possibility to predict the peak load and thefailure mechanism of steel-reinforced concrete elements. Such apossibility, explored in [30] with reference to a standard bench-mark on steel-reinforced concrete beams under bending, is investi-gated here with reference to reinforced concrete (RC) structures ofpractical and greater engineering interest, namely: walls and slabs.The addressed topic belongs to a wider ongoing research pro-gramme, started by the authors in [29] in the context of RC struc-tures, but already applied with success to structural elementsmade by different constitutive materials, such as laminates of fiberreinforced polymers, see e.g. Pisano and Fuschi [26,27], Pisano et al.[28]. The proposed methodology, giving information at an ultimate(collapse) state of the structure in terms of peak load and collapsemechanism, can be viewed as a useful preliminary design tool alsofor RC-structures of large dimensions. If necessary, more accuratestep-by-step analysis, able to follow the fracturing/damaging pro-cesses exhibited by RC-structures in the post-elastic regime, can becarried out. Such a deeper investigation can however be reservedonly to confined zones or weaker structural elements detected bya much simpler limit analysis which, as shown hereafter, gives use-ful hints on the collapse mechanism and predicts, with good accu-racy, the ultimate value of the loads acting on the located weakermembers or parts.

The numerical analysis employed here is based, essentially, onthe application of non standard limit analysis theory (Lubliner[21]). The peak load of a structure made by a non standard materialas concrete (where non associativity is postulated to account for itsdilatancy) can indeed be located between an upper and a lowerbound to it. There are many examples of limit analysis in the realmof nonstandard materials, from the pioneering works of Druckeret al. [9] and Radenkovic [33], to studies concerning geotechnicalproblems, e.g. Sloan [37], Boulbibane and Ponter [5], or thosespecifically dealing with reinforced concrete, see e.g. Liman et al.[20], Larsen et al. [16]. A comprehensive and updated review oflimit analysis methods can be found in Nielsen and Hoang [25]in the field of concrete plasticity or, in the wider context of theso-called Direct Methods, in the very recent book by Spiliopoulosand Weichert [38]. On the other hand, the application of plasticitybased approaches to reinforced concrete structures, whose ductilebehavior is assured by the presence of reinforcement, is witnessedby a number of contributions, see e.g. the monographs of Chen[7,8] and, again, Nielsen and Hoang [25] or the papers by Brisottoet al. [10], Roh et al. [34], Zhang and Li [41], Benkemoun et al.[3], Carrazedo et al. [6], Wu and Harvey [40], just to quote few veryrecent contributions on this research theme being the list far to beexhaustive.

There are two finite element (FE) based iterative procedurespromoted by the authors for limit analysis of RC-structures, the lin-ear matching method (LMM), see e.g. Ponter and Carter [31],Pisano et al. [29], and the elastic compensation method (ECM),see e.g. Mackenzie and Boyle [22], Pisano et al. [30]. In the abovecouples of references, the former paper is the one where (with ref-erence to structures made of von Mises-type materials) the

A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55 43

method was conceived and first proposed, the latter that where themethod was rephrased to deal with reinforced concrete structuralelements. In particular, in Pisano et al. [29] the LMM has beenreformulated and adapted to a Menétrey–Willam-(M–W)-typeyield criterion (Menétrey andWillam [24]) focusing all the theoret-ical aspects, the mathematical and geometrical details with refer-ence to a 3D formulation in the Haigh–Westergaard coordinates.A few examples are presented there to show the applicability ofthe method to reinforced concrete simple elements. In Pisanoet al. [30], while the effectiveness of the LMM is investigated byanalyzing a standard benchmark on steel-reinforced concretebeams under bending (Bresler and Scordelis [4], Vecchio and Shim[39]), a revisited version of the ECM suitable for the M–W-typeyield criterion is proposed. In the above paper the use of the twomethods was applied for the first time to simple reinforced con-crete structures, namely beams, showing the possibility to locatethe real (experimentally detected) value of the peak load by com-puting an upper and a lower bound to it.

In the present study, skipping the theoretical details given in[29,30] to avoid repetition, the above mentioned numerical FE pro-cedures are applied to compute the peak load as well as to predictthe failure mechanisms of large-scale RC-prototypes of walls andslabs. The relevant experimental data, concerning tests carriedout up to collapse and available for the analyzed specimens, havebeen used to validate the numerical predictions so facing realexperimental findings. The following papers/reports have beenconsidered: Lefas et al. [18], where structural walls were testedunder combined action of a constant axial and a horizontal loadmonotonically increasing to failure; El Maaddawy and Soudki[11], where simply supported one-way RC-slabs were tested tofailure under four-point bending; Sakka and Gilbert [35,36], wheresimply, continuous and corner supported square and rectangularslabs subjected to line or point loads increasing to failure weretested. Some of the results of these latter tests are also reported

Fig. 1. Adopted Menétrey–Willam-type yield surface with cap: (a) deviatoric sections atin the Rendulic plane at h ¼ 0 and h ¼ p=3, respectively; and (c) 3D sketch in the princi

in more recent publications of the same Authors to which theReader can refer [13,14].

Some information on the followed nonstandard limit analysisapproach as well as on the LMM and ECM are given in the next Sec-tion 2 where the key ideas of the iterative FE numerical schemesare explained with the aid of two geometrical sketches. Details ofthe computational steps are given in Appendices A and B forcompleteness. Section 3 addresses the geometry, the material data,laboratory fixtures and the loading conditions of the analyzedRC-prototypes. The adopted mechanical model, FE meshes,modeling hypotheses are also expounded in this Section closingwith a comparison between the obtained results and theexperimental findings. Concluding remarks are given at closurein Section 4 also outlining possible future developments.

2. Key ideas of the numerical approach and computationalschemes

The key point of the promoted nonstandard limit analysisapproach (see e.g. [33]) is to encircle the yield surface of a givennonstandard material with two surfaces, precisely an outer and aninner surface and to compute, with reference to such surfaces(referred to two standard materials), an upper and a lower boundto the real collapse load multiplier pertaining to the nonstandardmaterial structure under consideration. Concrete is herein modeledas a nonstandard material obeying to a M–W-type yield surfacewhich can play the double role of inner and outer surface in thesense specified above. The M–W-type yield surface endowed withcap in compression is shown schematically in Fig. 1. Steel rein-forcement, on the other hand, are considered of an infinitely elasticbehavior. Their presence is taken into account only for what con-cern the confinement effect they exert on concrete. Such effectinjects a ductile behavior on the RC-element that is it confers to

three generic values of hydrostatic pressure; (b) tensile and compressive meridianspal stress space.

44 A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55

the element an essential requisite for the present plasticity-basedapproach.

The LMM is an iterative procedure belonging to a kinematicapproach of limit analysis and involving one sequence of linearFE-based analysis. At each iteration say #ðk� 1Þ (i.e. at each FE-

analysis under loads Pðk�1Þ�pi, with Pðk�1Þ load multiplier and �pi

assigned reference loads), the elastic moduli as well as some giveninitial stresses are modified within the structure assumed byhypothesis as made of a linear viscous fictitious material. The adjec-tive fictitious stands for the circumstance that the material mayhave elastic parameters which assume different values at differentpoints. The latter being Gauss points (GPs) in a FE discrete model ofthe structure. By doing so the fictitious solution computed in termsof linear strain rate at the generic GP, say _e‘ ðk�1Þ, (together with the

related compatible displacement rates _u‘ ðk�1Þi ) can be interpreted as

strain rate (with related displacement rate) at collapse for the realstructure. Basically, the above updating of elastic moduli and initialstresses allows to build a collapse mechanism for the structure interms, in this context, of volumetric and deviatoric strain ratecomponents and displacement rates, say _ecv , _ecdx , _e

cdy, _uc

i , where the

apex c stands for a value at collapse. For a better understandingof the method it could be very useful to look at its geometricalinterpretation in the principal stress space. In Fig. 2, on the basisof the formal analogy between the linear viscous problemand the linear elastic problem, the fictitious linear solutioncomputed at the generic GP can be located at the stress pointPL on the complementary energy equipotential surface

W ðk�1Þ½n;q;Kðk�1Þ;Gðk�1Þ; �nðk�1Þ; �qðk�1Þx ; �qðk�1Þ

y � ¼ W ðk�1Þ ¼ const: per-taining to the fictitious material. The surface is indeed a prolate

Fig. 2. Geometrical sketch, in the principal stress space, of the matching procedure#ðk� 1Þ to #ðkÞ.

spheroid (see [29] for details) whose semiaxes, say at iteration

ðk� 1Þ, are 4Gðk�1Þ and 6Kðk�1Þ—i.e. they are related to the fictitiouselastic moduli. The spheroid has also its center at �nðk�1Þ,�qðk�1Þx ; �qðk�1Þ

y —i.e. located by given fictitious initial stresses. Here

only a portion of W ðk�1Þ ¼ W ðk�1Þ is represented for simplicity.

The updating of the fictitious moduli to values KðkÞ; GðkÞ and of

the initial stresses to values ð�nðkÞ, �qðkÞx ; �qðkÞ

y Þ is carried out in sucha way that the spheroid is modified in shape and position, or, inother words, the complementary energy surface of the fictitiousmaterial is modified so that the fictitious linear solution PL isbrought onto the M–W-type yield surface, precisely onto the pointPM of normal _e‘ ðk�1Þ. It is worth noting that if the yield surface isstrictly convex (as the one here assumed) the point PM is uniquelydetermined by the given normal _e‘ ðk�1Þ. Such outward normal at PM

can then be viewed as an _ec ðk�1Þ, i.e. as a strain rate at collapse, therelated displacement rates _uc being those pertaining to the collapse

mechanism while stress coordinates of PM; ðnY ðk�1Þ;qY ðk�1Þx ;qY ðk�1Þ

y Þare the associated stresses at yield. All the ingredients to compute

an upper bound multiplier, say PðkÞUB , are known and the following

relation holds true:

PðkÞUB ¼

RV nY ðk�1Þ _ec ðk�1Þ

v þ qY ðk�1Þx _ec ðk�1Þ

dxþ qY ðk�1Þ

y _ec ðk�1Þdy

� �dVR

@Vt�pi _uc

i d @Vð Þ : ð1Þ

As it appears from the discussed geometrical sketch the

W ðkÞ ¼ W ðk�1Þ and the M–W-type yield surface match at PM , sothe name of the method. Moreover the above stress at yield,computed through the matching, do not meet the equilibrium

fulfilled at the generic Gauss point within the current element from iteration

Fig. 3. Geometrical sketch, in principal stress space, of the ECM fulfilled within thegeneric element in the FE discrete model at iteration #ðk� 1Þ of the currentsequence.

Fig. 4. Steel-reinforced concrete walls: (a) mechanical model, geometry, boundary andembedded-truss elements for re-bars; (c) reinforcement arrangement for walls type 1 s

A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55 45

conditions with the acting loads Pðk�1Þ�pi and the procedure is car-ried on iteratively until the difference between two subsequentPUB values is less than a fixed tolerance. Convergence requires that

the W ðkÞ ¼ W ðk�1Þ matches M–W-type yield surface at PM andotherwise lies outside the yield surface (see [32]).

On the other hand, the ECM is aimed at constructing an admissi-ble stress field suitable for the evaluation of a lower bound to thepeak load multiplier, according to the static approach of limit anal-ysis theory. Also the ECM, like the LMM, acts in an iterative way,but involving many sequences of linear FE-based analysis. At eachsequence, defined by a given load value, say PðsÞ�pi, with PðsÞ loadmultiplier of the current sequence, the Young modulus is reducedwithin highly loaded regions of the structure, i.e. in those elements#e where elastic stress is greater than the yield one.

Once again, a geometrical sketch is used to explain the proce-dure. In Fig. 3, at iteration ðk� 1Þ of the current sequence, an elas-tic stress response evaluated within a generic finite element #e (itis an averaged value of the stresses computed at the GPs of the ele-

ment #e) is represented by point Pe ðk�1Þ#e . In the same sketch P

Y ðk�1Þ#e

is the corresponding value at yield represented by the stress point

lying on the same meridian plane of Pe ðk�1Þ#e , i.e. at h ¼ he, located by

loading conditions; (b) typical FE-mesh of 3D solid elements for concrete and 1D-quare-shaped; and (d) reinforcement arrangement for walls type 2 rectangular.

Table 1Steel-reinforced concrete walls: specimen number; compressive and tensile concretestrengths; concrete Young modulus; constant value of the applied vertical load.

Specimen f 0c ðMPaÞ f 0t ðMPaÞ Ec ðGPaÞ FV ðkNÞSW11 44.46 2.20 34.4 –SW12 45.56 2.23 34.7 230SW13 34.51 1.94 31.9 355SW14 35.79 1.97 32.3 –SW15 36.81 2.00 32.5 185SW16 43.95 2.19 34.3 460SW17 41.06 2.11 33.6 –SW21 36.38 1.99 32.4 –SW22 43.01 2.16 34.1 182SW23 40.63 2.10 33.5 343SW24 41.06 2.11 33.6 –SW25 38.25 2.04 32.9 325SW26 25.59 1.67 29.2 –

PLB

PEXP

PUB

SW11 SW12 SW13 SW14 SW15 SW16 SW17

SW21 SW22 SW23 SW24 SW25 SW26

1.21 1.27 1.

48

1.28 1.

50 1.71

1.56 1.

802.

05

1.53

1.20

1.83

1.76

1.50

2.09

1.16 1.

23 1.32

3.30

2.60

3.81

3.00 3.

404.

03

3.21 3.30

4.15

2.23 2.

65 2.83 2.92 3.

20 3.57

3.35 3.

55 3.94

2.27 2.

472.

99

Fig. 5. Steel-reinforced concrete walls: values of the upper ðPUBÞ and lower ðPLBÞbounds to the peak load multiplier against the experimentally detected one ðPEXPÞ.

1 2 4 6 8 10 12 14 16iteration number

0

4

8

12

16

20

load

mul

tiplie

r P

3.35

3.553.94

PUB

PEXP

PLB

#SW16

1 2 4 6 8 10 12 14 16 18iteration number

0.0

0.5

1.0

1.5

2.0

2.5

load

mul

tiplie

r P

1.16

1.231.32

PUB

PEXP

PLB

#SW26

(a)

(b)

Fig. 6. Steel-reinforced concrete walls. Values of the upper ðPUBÞ and lower ðPLBÞbounds to the peak load multiplier versus iteration number: LMM prediction, solidlines with square markers; ECM prediction, solid lines with triangular markers;collapse experimental threshold (after Lefas et al. [18]) dashed lines. (a) SpecimenSW16 and (b) specimen SW26.

46 A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55

the intersection of the stress vector OP�!e ðk�1Þ

#e with the M–W-type

yield surface. By hypothesis, in the sketch of Fig. 2, PY ðk�1Þ#e is ‘‘low-

er” than Pe ðk�1Þ#e . By the ECM, at the current iteration ðk� 1Þ of the

current sequence ðsÞ, the elastic modulus of element#e has then tobe reduced according to the formula:

EðkÞ#e :¼ Eðk�1Þ

#e

OP�!Y

#e

��� ���ðk�1Þ

OP�!e

#e

��� ���ðk�1Þ

264

375

2

; ð2Þ

where EðkÞ#e is the modulus to be used at next iteration and the square

of the reducing factor accelerates the procedure (see e.g. [27]). Thedescribed operation ‘‘redistributes” the stresses within the struc-ture and allows to define a maximum admissible stress value inthe whole structure for the given load. Increased values of loadsare then considered in the subsequent sequences of analysis untilfurther load increase does not allow the maximum stress to bebrought below yield by the reduction (or redistribution) procedure.A lower bound multiplier can be eventually evaluated at ‘‘lastadmissible stress value” attained for a maximum acting load, say

PðsÞD�pi, in the shape:

PLB ¼ OP�!Y

R

��� ���ðk�1Þ PðsÞD

OP�!

R

��� ���ðk�1Þ ; ð3Þ

where referring again to Fig. 3, PR has to be intended as the pointPe

#e farthest away from the M–W-type yield surface, PYR being the

corresponding stress point at yield.The possibility for applying the discussed limit analysis proce-

dures either to concrete, obeying to a M–W-type yield criterion,and to steel-rebars, obeying to a von Mises-type yield condition,within a layered FE-based limit analysis approach is actually theobject of an ongoing research. Owing to the different FE typesadopted for the two materials within the proposed FE formulation,two simultaneous yield criteria could be considered in the iterativeprocedure, one for concrete elements and one for steel elements.This would imply a simultaneous updating of the material proper-ties of both concrete and steel elements in the iterative procedure,so enabling the prediction of possible steel bar yielding at incipientcollapse. The latter is indeed oriented to include those collapsemechanisms characterized by reinforcement yielding not treatedin the following.

3. Analyzed large scale RC-prototypes: walls and slabs

As pointed out in Section 1, the main goal of the present workis to verify the effectiveness and reliability of the promoted

A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55 47

procedure tackling two types of steel-reinforced concrete struc-tural elements of engineering interest, specifically: walls and slabs.Being aware of the limitations congenital to a plasticity-basedapproach it is worth noting that the ductile behavior, which is anessential requisite for the adoption of limit analysis, is actuallyassured by the presence of steel bars which mitigate, or evennullify, many complex post-elastic phenomena exhibited by plainconcrete at incipient failure such as localization and/or fracturing/damaging mechanisms. These phenomena, due to a mainly brittlebehavior, cannot be treated with the present procedure which hasto be confined to ductile steel-reinforced concrete structures.Indeed, the RC-structures of common use in civil engineeringapplications addressed here luckily belong to the above categoryas the selected experimental tests analyzed below.

Reference is made to experimental findings taken from the rele-vant literature; in particular thirteen large-scale walls, tested up tofailure by Lefas et al. [18], and seven large-scale slabs, tested up tofailure by Sakka and Gilbert [35,36], have been considered. Theexperimental tests, in practice, have been numerically reproducedto predict the peak load as well as the failure mechanism of eachstructure and the obtained numerical results compared with thosegiven by the laboratory tests. It is worth noting that the predictionof the failure mechanism, which for the examined cases is quite

(a)

(c)

y

xz

y

xz

Fig. 7. Steel-reinforced concrete walls. Band plots of the Cartesian strain rate componentfor specimens SW16, (a) and (b), and SW26, (c) and (d). In particular: (a) and (c) showlocalizing the plastic zone and/or the collapse mechanism; (b) and (d) show the results pe

obvious—a flexural mode of failure is mainly to be expected—hasto be intended rather as the capacity of the methodology to localizenumerically the zones where mechanism starts and spreads atcollapse.

The elastic analysis performed within the two summarized iter-ative procedures have been carried out using the FE code ADINA,(ADINA [1]), with meshes of 3D-solid 8-nodes elements with2� 2� 2 GPs per element for modeling concrete and embedded2-nodes, 1-GP, truss elements utilized for modeling re-bars andstirrups. Such embedded truss elements (refer to ADINA fordetails) are 1D FEs connecting the intersections of the rebar axeswith the faces of the 3D solids concrete elements. Such intersec-tions are ‘‘generated nodes” on the 3D solid FEs faces constrainedto the three closest corner nodes of the 3D element itself. A perfectbond between concrete and steel reinforcement, assumed as said ofan indefinitely elastic behavior, has been considered. The numberof finite elements is different for each specimen type and is chosenafter a preliminary mesh sensitivity study to assure an accurate FEelastic solution. Nodal loads, equivalent to the distributed loadexerted by rigid plates in the laboratory fixture, are considered.Boundary conditions (constraints) consistent with those of theexperimental tests will be specified next for each specimentype. For what concerns the ‘‘choices” related to the assumed 3D

(b)

(d)

y

xz

y

xz

_ecy in the deformed configurations at the ultimate value of the horizontal load PUB FH

the results obtained at last converged solution of the LMM on the fictitious structurertaining to an elastic solution of the real structure, i.e. with the real elastic parameters.

48 A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55

concrete constitutive model, the uniaxial compressive strength, f 0c ,has been taken from the quoted references; the uniaxial tensilestrength, f 0t , when not available, has been assumed as

f 0t ¼ 0:33ffiffiffiffif 0c

qas suggested by Bresler and Scordelis [4]; the value

of the eccentricity parameter e of the M–W-type yield surfacehas been evaluated by the expression e ¼ ½2þ f 0t=f

0c�=½4� f 0t=f

0c� as

suggested in Balan et al. [2], the f 0t=f0c ratio being assumed as a mea-

sure of the material brittleness. Other three values have beenfinally fixed to locate the M–W-type yield surface in the principalstress space, namely: nv assumed as nv ¼

ffiffiffi3

pf 0c=m with m given by

m :¼ 3 ðf 02c � f 02

t Þe=f 0c f 0tðeþ 1Þ, see Pisano et al. [29]; na ¼ 0:7923 f 0cand nb ¼ 1:8964 f 0c as suggested by Li and Crouch [19].

Finally, a Fortran main program has been used to drive both theiterative procedures here proposed updating, at each GP of eachelement, the fictitious elastic parameters when performing theLMM or realizing the redistribution procedure within the ECM.

3.1. Walls

The large-scale wall models, addressed by Lefas et al. [18], weretested under the combined action of a constant axial (vertical) anda horizontal load monotonically increasing to failure. The experi-ments were oriented to investigate the effect on wall behavior ofthe height-to-width ratio, the axial load, the concrete strengthand the amount of web horizontal reinforcement also performinga critical examination of some concepts underlying the ACI Build-ing Code provisions for the design of reinforced concrete structuralwalls in force in the early eighties. The attention, for obvious rea-sons, is hereafter focused exclusively on the experimental dataand results given in the above quoted paper where details on thetest set up are given.

The assumed mechanical model reporting geometry, loadingand boundary conditions of the analyzed walls is sketched inFig. 4(a) where, besides the geometrical dimensions L; t and H

(a)

(b)

yLy

x

z

3L

1P p

2L

P p

2P p

yst 15 mm

barx15 mmx

z

Fig. 8. Steel-reinforced concrete simply-supported slab #1 (which is the control specimSakka and Gilbert [35]): (a) mechanical model, geometry, boundary and loading conarrangement at cross-section.

specified for the two types of walls shown in Fig. 4(c) and (d), FH

denotes the horizontal reference load, assumed equal to 100 kNand amplified by the load multiplier P; FV is the vertical constant(fixed) load whose value is given, for each specimen, in Table 1(�f V being the equivalent distributed load). As specified in the refer-enced experimental paper all the walls were monolithically con-nected to an upper and a lower reinforced concrete beam bothworking as cages for the anchorage of the vertical bars. The upperbeam was also the rigid element through which axial and horizon-tal loads were applied to the wall specimens, the lower beam, sim-ulating a rigid foundation, was used to fix the specimen to thefloor.

The model of Fig. 4(a) takes into account such experimental fix-ture assigning zero displacements to the points belonging to theshaded bottom cross-section of the wall and zero relative horizon-tal displacements (x and z directions) to the points belonging to theshaded top cross-section. The horizontal load PFH is then applica-ble to any point (or FE node) of such horizontal (rigid) top cross-section. Fig. 4(c) and (d) specify, as said, specimens type 1 and type2, respectively and, besides their geometrical dimensions, give thereinforcement arrangement made of vertical and horizontalhigh-tensile steel bars of diameters equal to 8 and 6.25 mm,respectively. Stirrups of mild steel bars of 4 mm diameter werealso present as additional horizontal reinforcement to confine thewall edges. Table 1 completes the needed data giving the concretematerial parameters for each specimen labeled borrowing fromLefas et al. [18]. In particular, for concrete the Young modulus, Ec ,

has been evaluated as Ec ¼ 22;000 ðf 0c=10Þ0:3

MPa, following Euro-code 2 [12] while a Poisson ratio of m ¼ 0:2 has been assumed forall the specimens. E ¼ 200 GPa has also been assumed for the steelreinforcement. Finally, in Fig. 4(b) the FE mesh scheme adopted inthe analysis is shown with 3D-solids for concrete (616 and 624 forspecimens type 1 and type 2, respectively) and 1D-embeddedtrusses (686 and 718 for specimens type 1 and type 2, respec-tively). The number of truss elements reduced to 446 and 502 for

(c)

t

3L2L

1L1L

1

xL

xst

15 mm 15 mmbary

yL

z

y

en of El Maaddawy and Soudki [11]) and slab #2 (coincident with specimen SS4 ofditions; (b) reinforcement arrangement along x direction; and (c) reinforcement

A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55 49

specimens SW17 and SW26, respectively, where a lower percent-age of horizontal bars was used.

The obtained results are summarized in Fig. 5 where for eachanalyzed wall specimen, are given: the peak load multiplier valuesdetected experimentally (after Lefas et al. [18]) and the predictions,in terms of upper and lower bound values to such peak, furnishedby the present approach.

By inspection of the numerical findings, in three over thethirteen examined cases (precisely walls SW11, SW24, SW25)the experimental peak load value lies outside the range numeri-cally predicted, such cases are wrong predictions of the adoptedmethodology. However, it is worth noting that the error on wallSW25 could indeed be due to a trouble which occurred duringthe laboratory test and which had been reported by the Experi-menters, that might have invalidated the experimentally detectedpeak load value. To be precise, an unintended eccentricity of thevertical and horizontal loads occurred due to an unexpected shiftof the fixture transferring the axial load on the wall. By excludingspecimen SW25 the percentage of wrong predictions on the peakload was reduced from about 23.08% to about 16.67% which isan acceptable value from an engineering point of view whendealing with practical (real) structures. On the other hand, thepercentages of error between the experimental peak load valueand the predicted lower bound to it for walls SW11, SW24 andSW25 range from a maximum of 27.5% to a minimum of 17.3%.If it is taken into account that the promoted procedure can beviewed as a preliminary design tool at ultimate limit states, the

Fig. 9. Steel-reinforced concrete continuous-supported slab #3 (coincident with slab CS5boundary and loading conditions; (b) reinforcement arrangement at roller support and aand at mid-span at cross-section; (d) reinforcement arrangement at interior support alonmidspan); and (e) reinforcement arrangement at interior support in the cross-section.

above percentages are fully compensated by the safety factorson acting loads and material strengths required by design rules(see e.g. Eurocode 2). In all other cases the predicted interval,within which the experimentally detected value falls, is quite nar-row showing a good performance of the proposed methodology.Fig. 6 shows, for two of the analyzed specimens (i.e. one for eachtype, precisely: SW16 and SW26), the plots of the upper and thelower bounds to the peak load multiplier versus the iterationsnumber.

Analogous results are obtained for all the other cases but areomitted for sake of brevity. In all cases a monotonic and rapid con-vergence of both methods is exhibited.

A deeper comprehension of the mechanical behavior of thewalls at collapse can be gained by the prediction of the walls fail-ure modes. As said, the LMM ‘‘builds” the collapse mechanism thestructure exhibits when the loads attain their peak value or, moreexactly, they reach the evaluated upper bound value to such peak.Moreover, such a mechanism is built on a fictitious structure i.e. it islocated within the analyzed structure made, by hypothesis, of amaterial endowed with a fictitious spatially varying distributionof elastic parameters and initial stresses.

A possibility to predict the walls failure mechanism is thengiven by the possibility to point out the plastic zone (collapsemechanism) at ‘‘last converged solution” of the LMM. To this aimthe plots of the displacement rates (i.e. the final deformed configu-ration), as well as those of the Cartesian strain rate component _ecyhave been considered on the walls loaded by PUB FH (plus the

of Sakka and Gilbert [35]): (a) mechanical model, geometry (all dimension in mm),t mid-span along longitudinal axis; (c) reinforcement arrangement at roller supportg longitudinal axis (top bars along x have a length of 1800 mm and are centered at

Fig. 10. Steel-reinforced concrete two-way corner-supported slabs #4; #5; #6 and #7, coincident with specimens S2S-5, S2S-6, S2R-4 and S2R-5 of Sakka and Gilbert [36]:(a) mechanical model, geometry (all dimensions in mm), boundary and loading conditions; (b) reinforcement arrangement along x direction; and (c) reinforcementarrangement along y direction.

50 A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55

pertinent constant FV ) and at the ‘‘last” distribution of fictitiousparameters and initial stresses.

Fig. 7(a) and (c) show, again only for specimens SW16 andSW26, the Cartesian strain rate component _ecy distribution in thedeformed (final) configuration attained by LMM at convergence.The plasticized zones so located appear sufficiently confined andreasonably close to the damaged zones experimentally detected.Also the deformed shapes, particularly that of the slender (type2) specimen (see again Fig. 7(c)), show clearly how around suchplasticized zones the remainder structure rotates rigidly exhibitinga collapse/failure mechanism. The distribution of _ecy on thedeformed configuration located on the fictitious structure of theLMM, as the ones given by Fig. 7(a) and (c), that is the ‘‘collapsemechanism” the method builds seems indeed to give some reliablehints on the expectable failure mode of the structure but, obvi-ously, only from a qualitative point of view. The level of detail indescribing the state of incipient collapse even if not exhaustivecan however be useful to localize critical zones or weaker membersfor example within reinforced concrete structures of larger dimen-sions. A prediction, even if qualitative, of the plasticized zoneslocating the collapse mechanism cannot obviously be obtained ifa simple elastic solution of the real wall (i.e. with the real materialparameters) loaded by PUB FH is carried out, as shown by the plotsgiven in Fig. 7(b) and (d).

3.2. Slabs

Three types of large-scale steel-reinforced concrete slab speci-mens have been analyzed for an overall number of seven speci-mens: two simply-supported, one continuous-supported and fourcorner-supported. The analyzed specimens have been selected from

a wider survey of experimental tests. The simply-supported slabs,hereafter named slab #1 and slab #2 as well as the continuoussupported, slab #3, are those tested by El Maaddawy and Soudki[11] and Sakka and Gilbert [35] (see also [13,14]). The first paperwas oriented to verify the use of mechanically-anchored unbondedfiber reinforced polymer system to upgrade reinforced concreteslabs. This goal is out of the present study but, among the six slabsthere tested to failure under four-point bending, one was used as a‘‘control specimen”, i.e. without any fiber reinforced polymerstrengthening system and it has been here chosen as slab #1.The second quoted paper/report analyzes eleven slabs, four slabswere simply supported and seven were continuous over two equalspans; two classes of steel reinforcement were also considered:low ductility and normal ductility. The experimental work was car-ried on to investigate on the effect of reinforcement ductility on thestrength and failure modes of such eleven one-way reinforced con-crete slabs. The slabs with low ductility reinforcement showed tofail by ‘‘brittle fracture of the reinforcement” (see e.g. [35]). Suchnon conventional flexural mode of failure is out of the predictivecapacities of the present approach.

The only two samples, among the eleven, reinforced withnormal-ductility steel bars (exhibiting a ductile mode of failure)have then been chosen: the simply supported SS4, named slab#2 hereafter, and the continuous supported CS5, named slab #3in the following. Finally the four corner-supported specimenshereafter analyzed are taken from the paper/report by Sakka andGilbert [36], where the strength and ductility of two-way corner-supported reinforced concrete slab panels containing low- andnormal-ductility bars were investigated by laboratory tests up tofailure. Once again, among eleven slabs, six square and five rectan-gular, the four containing normal-ductility bars have been chosen,namely the square shaped S2S-5 and S2S-6, here denoted as slab

Table 2Steel-reinforced concrete simply-supported slabs #1 and #2 sketched in Fig. 8: specimen number; geometrical data, bars’ diameters and spacing, value of the applied referenceload.

Slab specimen L1 (mm) L2 (mm) L3 (mm) Lx (mm) Ly (mm) t (mm)

#1 250 500 150 1800 500 100#2 500 500 250 2500 850 106

barx (mm) sx (mm) bary (mm) sy (mm) �p1 (N/mm) �p2 (N/mm)

#1 11.3 235 – – 100 –#2 12 410 12 200 – 117.65

Table 3Steel-reinforced concrete corner-supported slabs #4, #5, #6, and #7 sketched in Fig. 10: specimen number; geometrical data, bars’ spacing and diameters, value of the appliedreference load.

Slab specimen Lx (mm) L1x (mm) L2x (mm) Ly (mm) L1y (mm) L2y (mm) t (mm)

#4 2400 790 250 2400 790 250 106.1#5 2400 790 250 2400 790 250 100#6 2400 520 520 3600 820 820 100#7 2400 790 250 3600 1390 250 101.6

sx (mm) sy (mm) dx (mm) dy (mm) barx (mm) bary (mm) �p1 (N/mm2) �p2 (N/mm2)

#4 200 200 80 80 10 10 0.278 –#5 300 300 130 130 12 12 0.278 –#6 300 300 130 130 12 12 – 4.444#7 200 200 80 80 10 10 0.278 –

Table 4Steel-reinforced concrete slabs: specimen number; compressive and tensile concretestrengths; elastic concrete properties.

Slab specimen f 0c (MPa) f 0t (MPa) Ec (GPa) m

#1 25.00a 1.65b 28.96e 0.2#2 38.00c 3.68d 27.47c 0.2#3 37.80c 3.17d 26.47c 0.2#4 32.20f 3.09f 27.97f 0.2#5 26.70f 2.88f 25.59f 0.2#6 58.00f 4.68f 34.19f 0.2#7 44.00f 3.56f 28.29f 0.2

a After El Maaddawy and Soudki [9].b Computed as f 0t ¼ 0:33

ffiffiffiffiffif 0c

qas suggested by Bresler and Scordelis [3].

c After Sakka and Gilbert [35].d Derived by the flexural strength as f 0t ¼ f 0cf =1:2 according to Eurocode 2 [10].e Derived by the compressive strength as E ¼ 22ðf 0c=10Þ

0:3GPa according to

Eurocode 2 [10].f After Sakka and Gilbert [36].

Table 5Steel-reinforced concrete slabs: specimen number; number of 3D-Solid elements and1D-Embedded Truss elements used for the FE analyses.

Slab specimen Number of 3D-solid Number of 1D-Emb. trusses

#1 660 60#2 960 162#3 912 290#4 768 336#5 768 336#6 576 232#7 1056 492

PLB

PEXP

PUB

Slab1

0.35 0.41 0.

46

0.30 0.

34 0.39 0.44 0.45 0.

51

1.30

1.33 1.

45

1.05 1.

15 1.31

0.58 0.

67 0.73 0.77 0.84

1.06

Slab2 Slab3 Slab4 Slab5 Slab6 Slab7

Fig. 11. Steel-reinforced concrete slabs: values of the upper ðPUBÞ and lower ðPLBÞbounds to the peak load multiplier against the experimentally detected one ðPEXPÞ.

A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55 51

#4 and slab #5; the rectangular S2R-4 and S2R-5 here named slab#6 and slab #7, respectively. The tests set up, the material dataand many other details are given in the above quoted papers towhich reference is made; in the following the essential informationare summarized focusing the considered slab specimens.

The mechanical model, the geometry, loading and boundaryconditions, as well as the reinforcement arrangement of the sevenslabs analyzed are sketched in Fig. 8(a)–(c) for slab #1 and #2,

Fig. 9(a)–(e) for slab #3, Fig. 10(a)–(c) for slabs #4–#7. Tables 2and 3, which refer to Figs. 8 and 10 respectively, complete theneeded data specifying also the values of the applied referenceloads. Finally, Table 4 gives, for all specimens, the concrete mate-rial properties while the steel Young modulus has been assumedequal to 205 GPa for all samples. No FE-meshes are shown forthe slabs being worth the general remarks given at the beginningof Section 3 for what concerns the adopted finite elements,namely: 3D-solids are used to model concrete and 1D-embeddedtrusses to model re-bars. In Table 5, for completeness, are giventhe numbers of elements used for the FE analyses of eachspecimen.

Fig. 11 shows the results obtained for the seven slabs analyzedand given, as before, in terms of experimentally detected peak loadmultipliers (after the quoted paper) and upper and lower boundpredictions to them. In all cases the predicted interval of boundingmultipliers embraces the real peak load multiplier value and is alsoquite narrow confirming a good performance of the proposedmethodology also for this type of steel-reinforced concrete ele-ment. Also in this case a monotonic and rapid convergence isexhibited (less than ten iterations are required to stop both proce-dures). Plots of the type shown in Fig. 6(a) and (b) for the walls areobtained for the slabs and are here omitted for brevity.

(a) (b)

(c) (d)

z

x

yx

z

yx

z

y

Fig. 12. Steel-reinforced concrete simply-supported slab #2. Band plots of the Cartesian strain rate component _ecx in the deformed configurations at the ultimate value of theacting loads: (a) results obtained at last converged solution of the LMM on the fictitious structure localizing the plastic zone and/or the collapse mechanism; (b) results pertainingto an elastic solution of the real structure, i.e. with the real elastic parameters; (c) comparison of the deformed shapes given in (a) and (b) in the plane x—z; and (d) photographof slab #2 at failure after Sakka and Gilbert [35].

(a) (b)

(c) (d)

yx

z

yx

z

y

z

x

Fig. 13. Steel-reinforced concrete corner-supported slab #7. Band plots of the Cartesian strain rate component _ecy in the deformed configurations at the ultimate value of theacting loads: (a) results obtained at last converged solution of the LMM on the fictitious structure localizing the plastic zone and/or the collapse mechanism; (b) results pertainingto an elastic solution of the real structure, i.e. with the real elastic parameters; (c) comparison of the deformed shapes given in (a) and (b) in the plane y—z; and (d) photographof slab #7 at failure after Sakka and Gilbert [36].

52 A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55

For what concerns the slabs’ failure mode prediction, all theremarks made for the walls in the previous subsection hold true.In Figs. 12(a)–(d) and 13(a)–(d) the predicted failure modes areshown, for sake of brevity, only for slabs #2 (simply-supported)and #7 (corner-supported), respectively. Once again the failuremodes predicted by the LMM on the fictitious structure at conver-gence are compared with the elastic solutions obtainable on thereal structure at ultimate load level. It is clear how the LMM locatesa reliable collapse mechanism, a sort of plastic hinge arises atmid-span spreading to the whole slab width while the remainder

portions of the slab (till the supports) rotate rigidly around suchhinge. Quite impressive is the comparison between the numeri-cally predicted mechanisms and the ones documented by the pho-tographs at failure of the experimental tests. Peculiar are the _ecyconcentrations at the corner supports as well as the prominentflexural deformations, in the x—z plane, showed by the meaning-less elastic solution on the real slab at ultimate load for specimen#7, see Fig. 13(b). Such effects are far from the real state of the slabat incipient collapse and, in facts, they do not appear in the pre-dicted mechanism shown in Fig. 13(a). The latter exhibits rigid

A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55 53

rotations around a narrow plasticized zone at mid span and almostno flexural deformation in the x—z plane, despite the punctual sup-ports, that is it predicts almost exactly the collapse mechanismdetected experimentally, see Fig. 13(d). It is worth noting thatthe predictions on the failure mechanisms are qualitative ratherthan quantitative. Once again, the reliable information lie indeedeither on the _ecy distribution and concentration evidenced by the col-ored contour maps which locate the plasticized zones or on theshape of the deformed configuration, both predicting the collapsemechanism the slab will exhibit.

4. Concluding remarks and future developments

Large-scale prototypes of steel-reinforced concrete walls andslabs have been analyzed to validate a nonstandard limit analysisnumerical approach recently proposed by the authors. With thisaim detailed experimental laboratory tests carried out up to failureand available in the literature have been considered as a bench-mark. The combination of two FE-based procedures, the formeraimed to search for an upper bound the latter able to give a lowerbound to the peak load multiplier, is the key-idea of the promotedplasticity-based methodology oriented to practical reinforced con-crete structures. Indeed, the confining effect of the steel bars andthe ductile behavior injected by their presence in such structures,a circumstance shared by many RC-structures of common use inthe civil engineering applications, allows to apply a nonstandardlimit analysis approach as the one pursued here.

With all the limitations of a plasticity based approach for con-crete, the obtained numerical results, as witnessed by comparisonwith the experimental findings, appear to give useful and quitereliable information on the peak load, failure modes and criticalzones of the analyzed reinforced concrete structural elements. Afully 3D formulation has been employed that is at structural level,by 3D-solid FEs, as well as at constitutive level, with a M–W-typeconcrete model implemented in the principal stress space. To thisconcern it is worth remarking that a formulation based on princi-pal stress components instead of generalized stress variables, suchas membrane forces, shear forces and bending moments, oftenemployed for limit analysis of RC-structures (see e.g. [15,17,23])and requiring section-forces-based yield criteria strictly relatedto geometry and loading conditions of the mechanical problemunder study, can be easily applied to structural RC elements of gen-eral shapes suffering general loading conditions. It is also worthnoting that the utilized FE numerical procedures are both basedon sequences of elastic analysis so resulting easy to apply withany commercial FE code. Moreover, the whole methodology canbe easily rephrased with reference to Drucker–Prager or Mohr–Coulomb or other criteria specifically oriented to concrete. Thechoice of M–W-type model being justified by the strict convexityof such yield surface which is an essential requisite to achievethe matching within the LMM. Straight meridians, if present asin the above mentioned widely used criteria, it should be approx-imated by smooth curves, a drawback which can be overcomewithout much effort (see e.g. [32]).

Finally, two possible future steps can be envisaged: (i) the appli-cation of the presented limit analysis approach to existing RC-framed structures for the evaluation of their actual load bearingcapacity. To this concern the material parameters and strengths,detected in situ, could be used to calibrate the adopted M–W-type concrete model and (ii) the simultaneous application of thepresent FE-based limit analysis either to concrete, governed bythe M–W-type criterion, or to steel bars, handled by a von Mises-type criterion. Such improvement would allow to take into accountthe bars’ yielding often exhibited by RC-structures at a state ofincipient collapse. Both goals are the object of an ongoing research.

Appendix A. Linear matching method

The sequence starts assigning to all FEs of the discrete structuralmodel an initial set of fictitious elastic parameters, say Eð0Þ; mð0Þ, andinitial stresses �nð0Þ; �qð0Þ

x ; �qð0Þy . To initialize the iterative procedure,

say for k ¼ 1, set Pðk�1ÞUB ¼ Pð0Þ

UB ¼ 1; where capital P denotes the loadmultiplier of assigned reference loads �pi , the subscript UB stands

for ‘‘upper bound” value and where, for k ¼ 1; Pð0ÞUB can be any arbi-

trary value to start the sequence (it can indeed be assumed equalto unity as suggested above). At the beginning compute also thebulk and shear modulus, Kð0Þ ¼ Eð0Þ=3ð1� 2mð0ÞÞ andGð0Þ ¼ Eð0Þ=2ð1þ mð0ÞÞ, respectively. The following operative stepscan then be envisaged:

Step #1: Perform a fictitious elastic analysis with elastic parame-

ters Kðk�1Þ; Gðk�1Þ; initial stresses �nðk�1Þ; �qðk�1Þx ; �qðk�1Þ

y ;

loads Pðk�1ÞUB

�pi. Compute a fictitious kinematic linear solu-

tion (at each Gauss point on each FE), namely: _e‘ ðk�1Þv ,

_e‘ ðk�1Þdx

, _e‘ ðk�1Þdy

, _u‘ ðk�1Þi together with the corresponding

stress values n‘ ðk�1Þ; q‘ ðk�1Þx ; q‘ ðk�1Þ

y (the apex ‘ standingfor a value computed by the fictitious linear analysis).Compute also the (constant) value of the complemen-tary energy potential pertaining to the fictitious solu-tion, namely:

� �

W ðk�1Þ ¼ 1

2n‘ ðk�1Þ _e‘ ðk�1Þ

v þ q‘ ðk�1Þx

_e‘ ðk�1Þdx

þ q‘ ðk�1Þy

_e‘ ðk�1Þdy

:

It is worth noting that n‘ ðk�1Þ; q‘ ðk�1Þx ; q‘ ðk�1Þ

y , identify astress point, say PL, on the equipotential surface

W n;q;Kðk�1Þ;Gðk�1Þ; �nðk�1Þ; �qðk�1Þx ; �qðk�1Þ

y

h i¼ W ðk�1Þ whose

outward normal has components _e‘ ðk�1Þv , _e‘ ðk�1Þ

dx, _e‘ ðk�1Þ

dy.

Step #2: Locate on the M–W-type yield surface the ‘‘stress pointat yield”, say PMðnM;qM; hMÞ, whose outward normal has

components _e‘ ðk�1Þv , _e‘ ðk�1Þ

dx, _e‘ ðk�1Þ

dy. The uniqueness of such

point is assured by the strict convexity of the M–W-typeyield surface.

Step #3: Impose the so-called ‘‘matching conditions”, i.e. findthose values of the fictitious moduli and initial stresses,

say: GðkÞ; KðkÞ; �nðkÞ; �qðkÞx ; �qðkÞ

y , (to be used, if necessary, atnext iteration) in such a way that the equipotential sur-

face W n;q;KðkÞ;GðkÞ; �nðkÞ; �qðkÞx ; �qðkÞ

y

h i¼ W ðk�1Þ matches the

M–W-type yield surface at PL � PM . The components_e‘ ðk�1Þv , _e‘ ðk�1Þ

dxand _e‘ ðk�1Þ

dycan in this way be interpreted

(together with the related compatible displacementrates) as a collapse mechanism.

Step #4: Set _ec ðk�1Þv ¼ _e‘ ðk�1Þ

v , _ec ðk�1Þdx

¼ _e‘ ðk�1Þdx

, _ec ðk�1Þdy

¼ _e‘ ðk�1Þdy

,

_uci ¼ _u‘ ðk�1Þ

i and evaluate the upper bound multiplierwith the standard format of the kinematic approach oflimit analysis, namely:

R � �

PðkÞUB ¼

V nY ðk�1Þ _ec ðk�1Þv þqY ðk�1Þ

x _ec ðk�1Þdx

þqY ðk�1Þy _ec ðk�1Þ

dydVR

@Vt�pi _uc

i d @Vð Þ ;

where nY ðk�1Þ; qY ðk�1Þx ; qY ðk�1Þ

y are the stresses at yield givenby the detected PM on the M–W-type surface and wherethe integrals have to be performed numerically on the FEmesh. At current iteration, the work done by constant loads(i.e. not amplified by the load multiplier), if any, is subtractedfrom the numerator.

54 A.A. Pisano et al. / Computers and Structures 160 (2015) 42–55

Step #5: The above stresses at yield nY ðk�1Þ; qY ðk�1Þx ; qY ðk�1Þ

y do not

satisfy the equilibrium conditions with the loads Pðk�1ÞUB

�pi

and a new fictitious analysis is performed with the

updated values ð�ÞðkÞ and loads PðkÞUB

�pi. The iterations(analyses) start and they carry on till the equilibriumis fulfilled. This latter circumstance is verified by evalu-ating the difference between two subsequent PUB values.

Namely: if jPðkÞUB � Pðk�1Þ

UB j is less than or equal to a fixedtolerance stop the iterations, else, set k ¼ k� 1 and goto step #1.

Appendix B. Elastic compensation method

The procedure starts with a first sequence, say sequence s ¼ 1,

carried out for given loads PðsÞD�pi ¼ Pð1Þ

D�pi, where: �pi denote, as

before, assigned reference load; PðsÞD is the design load multiplier

pertaining to the sequence s (at start, for s ¼ 1, Pð1ÞD can be assumed

equal to unity). Denoting with ð�ÞðkÞ a quantity ð�Þ pertaining to thek-th iteration, or k-th elastic analysis, within the current sequences, the following steps can be envisaged:

Step #1: Set k ¼ 1 and assign to all FEs the known (real) mate-

rial elastic parameters, namely: Eðk�1Þ ¼ Eð0Þ,mðk�1Þ ¼ mð0Þ (the latter is kept constant during the anal-ysis) or, if necessary, assign the bulk and shear modu-lus, Kð0Þ and Gð0Þ, respectively.

Step #2: Perform an elastic analysis with material parameters

Eðk�1Þ; mðk�1Þ and loads PðsÞD�pi computing the principal

stresses at each Gauss point of each FE. Let refer tothe element #e for simplicity. Average also such stres-ses inside the element so obtaining an ‘‘averaged elas-tic solution” within the e-th element. The latter, aftertransformation in the H–W coordinates, locates inthe principal stress space a stress point, say

Pe ðk�1Þ#e ðne;qe; heÞ.

Step #3: Compute, in the Rendulic plane at h ¼ he, the H–Wcoordinates of the corresponding stress point at yield

within the element #e, say PY ðk�1Þ#e ðnY ;qY ; hY Þ, with

hY � he. That is locate on the M–W-type yield surface

the point on the direction OP�!e ðk�1Þ

#e

.OP�!e

#e

��� ���ðk�1Þ.

Step #4: Update the Young moduli within the current element#e according to the formula:

EðkÞ#e :¼ Eðk�1Þ

#e OP�!Y

#e

��� ���ðk�1Þ�OP�!e

#e

��� ���ðk�1Þ� 2. For PðsÞ

D – 1,

i.e. after the first sequence, the above updating formulais applied only in the highly loaded elements where

OP�!e

#e

��� ���ðk�1Þ> OP

�!Y#e

��� ���ðk�1Þ.

Step #5: Detect the ‘‘maximum stress” in the whole FE mesh, or,

equivalently, locate the stress point, say Pðk�1ÞR ,

‘‘farthest away” from the M–W-type yield surfaceand evaluate the pertinent stress point at yield,

PYðk�1ÞR . (The latter being the intersection between the

M–W-type yield surface and the straight line having

direction OP�!ðk�1Þ

R

.OP�!

R

��� ���ðk�1Þ:

Step #6: Check if the stress point Pðk�1ÞR just reaches (is below) its

corresponding yield value:

(i) if OP�!

R

��� ���ðk�1Þ< OP

�!YR

��� ���ðk�1Þthen compute a lower bound

multiplier as:

Pðk�1ÞLB :¼ OP

�!YR

��� ���ðk�1Þ PðsÞD

OP�!

R

��� ���ðk�1Þ ;

set Pðsþ1ÞD > Pðk�1Þ

LB , i.e. increase the intensity of the actingloads, set also s ¼ sþ 1 and go to step #1 to perform anew sequence of elastic analyses.

(ii) if OP�!

R

��� ���ðk�1ÞP OP

�!YR

��� ���ðk�1Þtry ‘‘to redistribute”, that is set

k ¼ k� 1 and go to step #2 to start a new analysis with

the current PðsÞD , i.e. within the current sequence but with

the updated Young moduli of step #4.When inside the current sequence s the ‘‘maximumstress” detected at step #5 of analysis k-th is such that

OP�!

R

��� ���ðkÞ P OP�!

R

��� ���ðk�1Þit means that the stress redistribu-

tion (or compensation) failed and the searched PLB valuecoincides with the load multiplier of the previous

sequence, i.e. PLB � Pðs�1ÞD ; the procedure ends.

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