1 (1)

FE analysis of size effects in reinforced concrete beams withoutshear reinforcement based on stochastic elasto-plasticitywith non-local softening

E. Syroka-Korol a, J. Tejchman a,n, Z. Mrz b

a Faculty of Civil and Environmental Engineering, Gdask University of Technology, Polandb Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

a r t i c l e i n f o

Article history:Received 29 July 2013Received in revised form30 April 2014Accepted 12 May 2014

Keywords:Elasto-plasticityNon-local softeningRandom fieldsReinforced concrete beamsSize effectStrain localization

a b s t r a c t

The paper presents results of FE analysis of mechanical size effects in longitudinally reinforced concreteslender beams without shear reinforcement failing in shear mode. The simulations were performedunder plane stress conditions for three beams of different sizes and a fixed shape (height/length ratio).The attention was focused on deterministic and statistical size effects related to the nominal beam shearstrength. Concrete was assumed as an isotropic elasto-plastic material exhibiting non-local softening.The bond strength between concrete and reinforcement was assumed to depend on interface slip withboth stable and softening responses. Statistical simulations were performed for spatially correlatedGaussian random fields of tensile strength using a stratified sampling reduction method. The FEnumerical results were compared with the respective own experimental test results.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The size effect phenomenon in quasi-brittle structures isrelated to a transition from a ductile behaviour of small specimensto a totally brittle response of large ones. Thus, the nominalstrength sN decreases with increasing characteristic specimendimension D. The reasons for this behaviour are: (a) intense strainlocalization regions with a certain volume (i.e. micro-crackeddamage regions called also fracture process zones, FPZ) whichprecede discrete macro-cracks; their size related to D contributesto a deterministic size effect and (b) a spatial variability/random-ness of local material properties contributing to a statistical sizeeffect that becomes dominant with increasing D.

A strong size effect also occurs in reinforced concrete beamswithout shear reinforcement wherein diagonal sheartensile frac-ture takes place in concrete. It was experimentally observedamong others by Leonhardt and Walther [1], Kani [2,3], Bhal [4],Taylor [5], Walraven [6], Chana [7], Iguro et al. [8], Bazant andKazemi [9], Shioya et al. [10], Kim and Park [11], Grimm [12],Ghannoum [13], Kawano and Watanabe [14], Podgorniak-Stanik [15], Yoshida [16], Angelakos et al. [17], Lubell et al. [18]and Syroka-Korol [19]. The diagonal cracks at failure had in

experimental tests essentially similar paths and relative lengthsat the maximum load independently of the beam size. Therefore, thissize effect in such reinforced concrete beams could be described bythe analytical deterministic (energetic) size effect law (SEL) of Type IIaccording to Bazant [20], being valid for structures of a positivesimilar geometry possessing large stress-free cracks that grow in astable manner up to the maximum load (Fig. 1)

ND offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1D=D0p ; 1

Where N is the nominal strength, and o and D0 are the empiricalparameters depending on material properties, structure geometryand structure shape [21]. They can be determined by fitting Eq. (1) tothe experimental data. The parameter D0 separates the ductile failure(D0D) from the brittle one (D0D). For very large structures (D-1),the nominal strength approaches N-D1/2. Assuming the residualstrength R for very large sizes (D-1) due to the strength ofreinforcement and compressed concrete, Eq. (1) becomes valid if Nis replaced by the expression NR [21]. For small structures (D-0),the size effect disappears. Thus, the size effect is strong only in thelimited size range. The SEL curve (Fig. 1b) in a double-logarithmicplot represents a smooth transition from a strength (plastic) limit forsmall sizes to the solution given by the Linear Elastic FractureMechanics (LEFM) for large and very large sizes.

In spite of the ample experimental evidence, the physically basedsize effect is not taken into account in practical design rules ofengineering structures, assuring a specified safety factor with respect

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Finite Elements in Analysis and Design

http://dx.doi.org/10.1016/j.finel.2014.05.0050168-874X/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail addresses: [email protected] (E. Syroka-Korol),

[email protected] (J. Tejchman), [email protected] (Z. Mrz).

Finite Elements in Analysis and Design 88 (2014) 2541

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to the failure load [22,23]. Instead, a purely empirical approach issometimes considered in building codes which is doomed to yield anincorrect formula since physical foundations are lacking.

Our objective is to provide a quantitative assessment of a sizeeffect and a related description of a brittle failure mode in slenderreinforced concrete beams without shear reinforcement underbending. The finite element method based on an isotropic elasto-plastic model with non-local softening enhanced by a character-istic length parameter of micro-structure was used in numericalstudies. The plane stress 2D calculations were performed. Materialparameters were calibrated with conventional laboratory tests andcode recommendations. A characteristic length of microstructurewas estimated by means of displacement measurements on thebeam surface using the non-invasive Digital Image Correlation(DIC) technique [24]. Deterministic calculations were perfor-med assuming a constant value of the tensile strength. In turn,statistical analyses were carried out with spatially correlatedrandom fields according to the Gauss distribution reflecting therandom nature of a local tensile strength. In order to reduce thenumber of FE statistical simulations, a stratified sampling schemewas used belonging to a group of variance reduced Monte Carlomethods. This approach enabled us a significant reduction of thesample number without affecting the accuracy of calculations. Thepresent analysis constitutes the continuation of our earlier suc-cessful simulations of a combined deterministic-statistical sizeeffect in notched [25] and unnotched concrete beams [26].

The numerical results were compared with our labora-tory experiments [19,27]. The experiments were carried out on

longitudinally reinforced concrete beams of different sizes andfixed height/length ratio: 3 small-size beams of the height of200 mm and length 1500 mm, 3 medium-size beams of the heightof 400 mm and length 3000 mm and 3 large-size beams of theheight of 800 mm and length 6000 mm (the thicknesst200 mm). The beams were geometrically similar in 2 dimen-sions to avoid differences in the hydration heat effects which areproportional to the thickness of the member [22]. They and madefrom the same concrete mix with the mean aggregate diameterequal to 9 mm. The simply supported beams were subjected to4-point bending with the constant shear span-effective heightratio equal to 3. To induce a sheartension failure mechanism inconcrete, the beams were over-reinforced without shear reinfor-cement (the reinforcement ratio was always 1%). The experi-mental results showed a significant size effect on the nominalshear strength versus the beam effective height. The meannominal shear strength of large-size beams was smaller by 40%with respect to small-size beams. In all RC beams, a combineddiagonal sheartensile (significantly more tensile) and bond fail-ure mode dominated, characterized by the development of acritical diagonal sheartensile crack connected with a horizontalsplitting crack along the top of the bottom longitudinal reinforce-ment toward the beam support (a shearcompression failuremechanism did not occur in concrete). The failure mode proceededin a brittle manner in the post-critical stage.

Numerical FE analyses of slender beams without transversereinforcement were performed among others by Kotsovos andPavlovic [28,29], Vecchio and Swim [30], Sato et al. [31] and Slobbeet al. [32]. They used a non-linear elastic-brittle model ([28,29]),a smeared rotating crack approach [30], a smeared fixed crackapproach with a sequentially linear (SL) analysis [32] and adiscrete rotating crack model [31]. In calculations, a diagonaltensile brittle failure mode was usually obtained. A critical diag-onal crack propagated towards the beam top, if the beam failurewas caused by concrete splitting in a compression zone [32].It propagated towards the bottom, if the beam failure was causedby bond splitting [31]. A deterministic size effect on the nominalshear strength of beams failing by diagonal tension was studiedonly by Sato et al. [31] for four virtual reinforced concrete beamswith the height ranging from 100 mm up to 1600 mm. Thenumerical results were overestimated as compared to an analyticalsize effect formula. According to 3D simulation results in [28,29],the size effect in slender reinforced concrete beams without shearreinforcement is mainly caused by non-symmetric cracking com-bined with the unintended out-of-plane action, the latter beingimpossible to be avoided in experiments. They have also found outthat stirrups eliminate the size effect in reinforced concrete beams(in contrast to recent outcomes by Yu and Bazant [33]).

Summarized, the novel elements in our calculations for rein-forced concrete beams failing in shear are: (a) combineddeterministic-statistical FE calculations for 3 different beams bytaking strain localization and bond into account, (b) direct com-parison between numerical and experimental results and (c)application of a stratified sampling scheme to reduce the numberof statistical calculations. To our knowledge, such calculations haveso far not been performed.

2. Constitutive models

2.1. Concrete

The concrete deformational response was simulated by assum-ing an isotropic elasto-plastic constitutive model with a non-local softening which was used in our previous calculations[19,26,3436]. This relatively simple isotropic model for concrete

Fig. 1. Size effect curve of type II with large cracks or notches by Bazant [20,22](N nominal strength, D characteristic specimen size, LEFM linear elasticfracture mechanics); (a) linear scale and (b) loglog scale.

E. Syroka-Korol et al. / Finite Elements in Analysis and Design 88 (2014) 254126

(see Appendix) consists of two yield criterions: by Rankine intension (Eq. (A1)) and by DruckerPrager in compression (Eq.(A2)). The softening under tension (Fig. 2a) was characterized bythe exponential curve by Hordijk [37] (Eqs. (A7) and (A8)). Incompression, linear hardening/softening was assumed (Fig. 2b).

The concrete model requires two elastic constants: modulus ofelasticity E and Poisson's ratio , two plastic constants: internalfriction angle and dilatancy angle , one tensile yield function stf(1) and one compressive yield function sc f(2). The disadvantagesof the model are the following: the shape of the compressive failuresurface in a principal stress space is linear (not paraboloidal as inreality). In deviatoric planes, the shape is circular (in compressionstates) and triangular (in tension states); thus it does not graduallychange from a curvilinear triangle with smoothly rounded corners tonearly circular with increasing pressure. The effect of third stressdeviator invariant is not taken into account. The strength is similarfor triaxial compression and extension, and the stiffness degradationdue to strain localization and non-linear volume changes duringloading are not taken into account.

A non-local theory was used as a regularization technique[4042]. In this approach, the principle of a local action does nottake place any more. Polizzotto et al. [43] laid down a thermo-dynamically consistent formulation of non-local plasticity. In thecalculations, the softening parameters i (i1, 2) were assumed tobe non-local (independently for both yield surfaces fi) [44]

ix 1mixmRVxidR

Vxdwith i 1; 2; 2

where i(x) are the non-local softening parameters, V denotes thebody volume, x is the coordinate vector of the considered point, is the coordinate vector of the surrounding points, denotes theweighting function and m is the additional non-local parametercontrolling the size of the localized plastic zone. As a weighting

Fig. 2. Assumed hardening/softening curves in FE-calculations: (a) tensile stressversus non-local parameter st f(1) and (b) compressive stress versus non-localparameter sc f(2).

Fig. 3. Bond stress-slip relationship (shear stress against slip displacement )between concrete and reinforcement according to Dorr [51] (a) and CEB-FIP Code[39] (b).

Fig. 4. Bond relationship: radial normal stress sr,rs and bond slip versus radialnormal strain r,rs between concrete and reinforcement during splitting failure [51].

E. Syroka-Korol et al. / Finite Elements in Analysis and Design 88 (2014) 2541 27

function , the Gauss distribution function was used

r 1lcffiffiffi

p er=lc2 3

where lc is a characteristic length of micro-structure and theparameter r is the distance between material points. The averagingin Eq. (3) was restricted to a small representative area around eachmaterial point (the influence of points at the distance of r3lcwas only of 0.01%). The softening non-local parameters nearboundaries and at both sides of a localized zone were alwayscalculated on the basis of Eqs. (1) and (2) (which satisfy thenormalizing condition) [36]. Other approaches were proposed byPolizzotto [45] and Jirsek et al. [46]. The reinforcement was nottaken into account when calculating a non-local parameter inconcrete [36]. To simplify the calculations, non-local rates werereplaced by their approximations calculated with known totalstrain increments [42].

The non-local model was implemented in the commercial finiteelement code ABAQUS [47] with the aid of subroutine UMAT (userconstitutive law definition) and UEL (user element definition) forefficient computations [42]. The calculations were carried outusing a large-displacement analysis available in the ABAQUScode [47]. According to this method, the current configuration ofthe body was taken into account. The Jaumann rate of the Cauchystress was taken. The conjugate strain rate was the rate ofdeformation. The rotation of the stress and strain tensor wascalculated with the HughesWinget method [48]. It is known thatthe stress and strain rate are not work conjugate in this formula-tion and erroneous results occur, in particular for highly compres-sible materials [49]. Our calculations showed however that theeffect of large displacements on the failure force in reinforcedbeams was negligible (less than 1%). Thus the effect of a workconjugacy error was not important. The non-local averaging wasperformed in a current configuration. This choice was governed bythe fact that element areas in this configuration were automati-cally calculated by ABAQUS [47]. The edge and vertex in theRankine yield function were taken into account by the interpola-tion of 23 plastic multipliers according to Koiter's rule [50]. Thesame procedure was adopted in the case of combined tension(Rankine criterion) and compression (DruckerPrager criterion).

2.2. Reinforcement

To simulate the reinforcement behaviour, an elastic-perfectlyplastic material with the von Misses criterion was assumed

f ssij q f y; 4where fy is the yield stress for steel bars.

2.3. Bond between concrete and reinforcement

To describe the interaction between concrete and reinforce-ment, a bond relationship was defined. In general, two differentbond-failure mechanisms may appear connected to a pull-out orsplitting mode. The pullout bond failure takes place when the baranchorage length is insufficient to carry tensile stresses in the

longitudinal reinforcement, whereas the splitting bond failureoccurs when the concrete cover thickness is insufficient to resistradial cracks caused by local forces transmitted through bar ribs.

To consider bond model, an interface with a zero thickness wasassumed along a contact surface, Since the bond model was crucialfor describing numerically the experimental failure mechanism,three different bond stressslip definitions were tested. Initially,the simple and well-known models proposed by Dorr [51] andCEB-FIP Model Code [39] were used which described the pull-outfailure by a relationship between the bond shear stress and slipdisplacement . The first bond-slip model neglects softening andassumes a yield plateau when the pullout failure begins (Fig. 3a).The model needs solely 2 parameters: the tensile strength ft andslip displacement u at which perfect slip occurs (usuallyu0.06 mm). In turn, the model in CEB-FIP Model Code [39]assumes softening and residual yield (Fig. 3b). The 6 modelparameters 1, 2, 3, , max and f depend on the concreteconfinement and bond conditions.

In order to describe the splitting bond failure along reinforce-ment, the model by Akkerman [52] was also used, being amodification of the original model proposed by den Uijl and Bigaj[53]. The model takes into account the evolution of the radialstress sr,rs versus the radial strain r,rs and is divided into 3 phases(Fig. 4). The first phase (0rr,rsrr,rs,max) characterizes a non-linear material behaviour caused by cracks, the second one(r,rs,maxor,rsrr,rs,res) includes linear softening and the thirdone (r,rs,resor,rs) described the residual behaviour

sr;rsr;rs sr;rs; max k

2

1k2 0rr;rsrr;rs: maxsr;rs; max 1 1br;rs r;rs: maxr;rsr;rs: max

r;rs: maxor;rsrr;rs:res

sr;rs;res r;rs:resor;rs

8>>>>>:

5with

k Err;rs; maxsr;rs; maxand r;rs

r;rs; max: 6

The empirical parameter b determines the ratio between themaximum and residual stress (bsr,rs,res/sr,rs,max); it was assumedas b0.2. The initial radial stiffness (Eq. (6)) is

Er Ecef f 0:5b20:252bcef f 0:5b20:252b

!1

; 7

where b is the bar diameter, ceff is the effective concrete cover andE and are the modulus of elasticity of concrete and Poisson'sratio, respectively.

The limit radial normal stresses and strains were defined asfollows:

sr;rs; max 2f tcef fb

0:88; r;rs; max 4:2

f tE

cef fb

1:088

sr;rs;res b sr;rs; max; r;rs;res f tE

2cef fb

c0b

; 9

where c0 is the empirical parameter influencing softening in thesecond phase (r,rs,maxor,rsrr,rs,res). The radial strains were

Fig. 5. Random field areas included in calculations of sampling parameter.


computed as follows:

r:rs

0:5btan b; ; 10

where b is the cone angle between the bar axis and cone-shapedcracking surface starting from ribs. The fictitious value wasassumed as b[o]0.1fc [MPa]. In turn, the bond stress was

related to the radial normal stress sr,rs by the fictitious friction rule

sr;rs cot b 11with cot b1. The model needs 8 parameters: 3 depending onconcrete properties E, and ft, 2 depending on the specimen andreinforcement geometry ceff and b and 3 empirical ones b, c0,and b.

3. Random fields in statistical calculations

In the first step, the material tensile strength ft was the solerandom parameter in our statistical analyses. The spatially corre-lated random fields of the tensile strength were described by aGauss distribution function and the homogenous squared expo-nential auto-covariance correlation function C

Cx1; x2 exp x1x22

lcor2

!; 12

where x1 and x2 are the co-ordinate points and l is the correlationlength. The auto-covariance function had the following spectraldecomposition:

Cx1; x2 1

i 1if ix1f ix2; 13

where the eigenvalues i and eigenfunctions fi(x) were the solutionof the Fredholm integral equation of the second kindZDCx1; x2f ix1dx1 if ix2 14

The spatially correlated random fields H(x,) (here with the zeromean and unit variance) were defined according to the KarhunenLove expansion [54,55] by an infinite linear combination oforthogonal functions with random coefficients

Hx; 1

i 1

ffiffiffiffii

pif ix; 15

where i() is the vector of uncorrelated random variables sampledfrom N(0,1) distribution. The approximated solution Hx; wasobtained by truncating the series in Eq. (17) after M terms. The

Fig. 6. Stratified sampling scheme used in statistical FE analyses (cdf cumulativeprobability function, sampling parameter, Pi probability intervals).

Table 1Geometry of reinforced concrete slender beams of Figs. 7 and 8.

Dimension Small-size beamSL20

Medium-size beamSL40

Large-size beamSL80

L [mm] 1500 3000 6000Leff [mm] 1200 2700 5620H [mm] 200 400 800D [mm] 160 360 750a [mm] 480 1080 2250b [mm] 240 540 1120

Fig. 7. Cross-section of slender reinforced concrete beams with horizontal bars used in calculations and experiments: (a) small-size SL20, (b) medium-size SL40 and(c) large-size SL80.


resulting error was minimized by sorting eigenvalues in a decreas-ing order and controlling the sum of i for i1,, M.

An analytical solution of Eq. (17) is available with e.g. anexponential autocorrelation function (Ghanen and Spanos [56]).In other cases, Eq. (17) has to be solved numerically. In our study,the wavelet-Galerkin method proposed by Phoon et al [57,58] wasused wherein conventional bases (e.g. trigonometric or polyno-mials) were replaced by Haar wavelets. A family of the orthogonal

Haar wavelets was created from a so called mother function

x 1 xA 0;0:5 1 xA 0:5;1 0 otherwise

8>: 16

by scaling and shifting it according to formula

j;kx j2jxk; j; kAN; 17

where j and k are the positive integer constants responsible forscaling and shifting and j is the function amplitude. In the currentstudy j1, which forms an orthogonal basis. The extended set ofi, which creates the complete set of orthogonal functions over thedomain [0, 1], was defined after [58] as

0x 1 and ix 2jxk; 18where i2jk; j0, 1,m1, k0, 1,,2j 1 (m the waveletlevel). Each eigenfunction was approximated by a truncated seriesof the N Haar wavelets

f kx N1

i 0dki ix T xDk; 19

where N2m and dik are the wavelet coefficients.In order to reduce the number of statistical simulations, the

so-called stratified sampling scheme originally proposed byNeyman [59] was used [60,61]. Initially 2000 random fields ofthe local tensile strength were generated for each beam size. Next,the generated samples were classified according to the sampling

Fig. 8. Geometry and loading scheme of reinforced concrete slender beams without stirrups.

Fig. 9. Experimental crack pattern on front (solid lines) and back (dashed lines) side of medium-size reinforced concrete beam.

Fig. 10. FE mesh for small-size beam SL20 (a) and large-size beam SL80 (b) (note that beams are not proportionally scaled).

Fig. 11. Calculated vertical force P versus deflection u with various bond definitionscompared with experiments (small-size reinforced concrete beam SL20).


parameter, chosen as the mean tensile strength in two shearregions of beams (Fig. 5) according to their failure mode. Thus,we assumed that the effect of the beam mid-region on thestatistical failure force was negligible (the effect of the samplingregion size on results merits further investigations). For the leftand right shear region of Fig. 5, the mean value was separatelycalculated and afterwards the lower value was used. Next, therandom fields were classified and arranged in an increasing orderaccording to the calculated mean value. Based on the samplingparameter, the cumulative probability of an initially generated setof the random fields was calculated (Fig. 6). Next, the cumulativeprobability function was divided into a finite number n of theequal intervals (Pi, i1,,n), where n corresponded to the numberof samples. From each subset of samples included in the intervalsof the equal probability, only one sample (nearest to the mid-point) was chosen for further FE analyses. Thus, the chosenrepresentative set of n samples included random fields of themean value in a shear region from the lowest to the highest one.We used the sampling procedure after the random fields weregenerated instead of sampling of on random vectors (Eq. (15))during a random field generation [62] because of the influence ofthe input random vector on a random distribution. The samplingon output random fields enabled us to choose the samples in acritical localized zone of the beam shear region, being approxi-mately representative for the beam failure force. Our methodallows for a fast convergence to the mean failure force [60,61].

4. FE input data

The FE-analyses of longitudinally reinforced concrete slenderbeams without shear reinforcement were performed with 3 differ-ent beam sizes of a similar geometry from experiments by SyrokaKorol [19]. The beams had the same dimensions H L as inthe experiments: 2001500 mm2 (small-size beam SL20),

4003000 mm2 (medium-size beam SL40) and 8006000 mm2(large-size beam SL80) (Table 1). All specimens had the constantthickness t200 mm and constant reinforcement percentage1% (Fig. 7). The beams were subjected to 4-point bending(Fig. 8) under the constant shear span to the effective depth ratioa/D3. The numerical calculations were carried out under planestress conditions since experimental crack patterns were verysimilar on both sides of all beams (Fig. 9 shows these patterns ina medium-size beam).

The specimens were cast from concrete C35/45 of a maximumaggregate size 32 mm (the characteristic compressive strengthfck35 MPa, the characteristic tensile strength fctk2.2 MPa,Young modulus E34 GPa and Poisson's ratio 0.2). The internalfriction angle was 141 with rbcs 1.2 (Eq. (A2) and (A4)) and thedilatancy angle 81 [34]. The calculations were carried out withthe characteristic length lc5 mm based on the experimentallymeasured mean width of localized tensile zones wloc by means ofthe DIC technique equal to 20 mm. The non-locality parameterwas m2 [42]. The ultimate non-local softening parameter intension was mainly 1u0.0207 (Gfgf4lcE180 N/m) and in com-pression 2u0.0057 (Gcgc4lcE2700 N/m) based on initial calcu-lations. The steel behaviour was specified by the yield stressfy500 MPa, the elastic modulus Es200 GPa and the Poisson'sratio s0.3.

The bar diameters were b10 mm, 16 mm and 20 mm inthe beams SL20, SL40 and SL80, respectively, and b10 mm forthe second reinforcement layer in the large-size beam SL80. In themodel by Dorr [51] Eq. (7), the slip displacement was u0.06 mm(as originally proposed by Dorr [51]). When using the bond modelby CEB-FIP [39], for the unconfined concrete and good bondconditions case, the material parameters were: 10.6 mm,20.6 mm, 31 mm, 0.4, max2.0fck11.83 MPa andf0.15max1.77 MPa. In the bond model by Akkermann [52](Eqs. (5)(11)), we assumed ceff22 mm, b0.2 [39] andc00.18 mm (based on own initial FE simulations).

Fig. 12. Contours of non-local tensile softening parameter 1 in small-size reinforced concrete beam SL20 at failure with various bond-slip models: (a) perfect bond, (b) byDorr [51], (c) by CEB-FIP [39] and (d) by Akkermann [52] from FE analyses compared to experimental crack pattern (single lines).


FE meshes of quadrilateral elements composed of four diag-onally crossed triangles were used (Fig. 10). The finite elementshad mainly the area of 7.510 mm2 (widthheight) which wasequal to (1.52) lc to obtain mesh-independent results [24,26].In the case of the large-size beam, the fine mesh covered the beammid-part of 4410 mm only. Totally, 1306001690344 finite elementswere used depending upon the beam size. The steel bars weremodelled as 2D elements with the width of 1.5 mm and height of2.02.5 mm depending on the bar diameter b. Since our FEanalyses were two-dimensional, the bar height was taken as0.5b to obtain a similar contact surface as in experiments(20.5b). The influence of the mesh size on FE results waschecked by comparing 2 different meshes in the small-sizereinforced concrete beam SL20 assuming the bond model byAkkermann [52].

5. Effect of bond-slip model

Fig. 11 shows the calculated load-displacement diagrams withdifferent bond models of Section 3 for the small size-beam SL20,and Figs. 12 and 13 the calculated evolution of the non-localtensile softening parameter 1.

Using the model by Akkermann [52] only, a satisfactoryagreement was achieved with experiments (Figs. 12 and 13). Theultimate vertical force from deterministic simulations was 91.4 kNwhich was by 1% lower only than the average failure force from

experiments (91.9 kN). With the remaining bond models, themaximum vertical force was too high due to the different mainfailure mechanism assumed that was not consistent with theexperimental failure mode (steel yielding instead of diagonaltension). The number of localized zones varied from 5 with themodel by Dorr (1980) [51] and by CEB-FIP [39] (Fig. 13c) up to8 with the perfect bond (Fig. 12a). Thus, the pattern of localizedzones differed from the experimental one.

The evolution of localized zones was similar to the experimen-tally observed cracks with the bond model by Akkermann [52](Fig. 13). First, vertical tensile bending localized zones appearedat the mid-region, next inclined shear localized zones developedin the shear span region, and finally a diagonal sheartensilezone occurred connected to the splitting failure along reinforce-ment, where bond radial stresses sr,rs reached their residualvalue (marked as dashed line in Fig. 13e). A critical inclinedlocalized zone occurred at the distance of dc270 mm fromthe support with the ratio of dc/a0.56 being identical as theaverage experimental value from 3 tests. In total, 9 localizedzones were calculated whose average spacing was 87 mm(the average experimental crack spacing was also similar91 mm).

We investigated also the effect of plastic parameters , , Gf andGc on FE results. The effect of (with rbcs 1.151.25) and (with812o) was negligible. The lower Gf, less localized zones appearedand a critical diagonal localized zone occurred at a higher distancefrom the support. In turn, the effect of Gc on a localized zones

Fig. 13. Evolution of non-local tensile softening parameter 1 in small-size reinforced concrete beam SL20 with bond-slip definition by Akkermann [52] from FE analysis atdifferent vertical resultant force: (a) P30 kN, (b) P50 kN, (c) P85 kN, (d) Pmax91.4 kN and experimental crack pattern after failure [19] (e). (For interpretation of thereferences to color in this figure, the reader is referred to the web version of this article.)


pattern and load capacity was negligible. The assumed softeningcurves of Fig. 2 with Gf180 N/m and Gc2700 N/m provided themost realistic FE results with respect to experiments.

The effect of the mesh size on the results is demonstrated inFig. 14. The results show that the FE results with a fine and a coarse

mesh are similar (Fig. 14A and B). The failure force for the coarsemesh (element size 1520 mm2) was 93 kN while for the finemesh (element size 7.510 mm2) was 91 kN (Fig. 14A). Based onthese results, we assumed that there was no need to decreasemore the size of finite elements (less than 7.510 mm2).

Fig. 14. Comparative deterministic FE results with coarse (element size 1520 mm2) and fine mesh (element size 7.510 mm2) for small-size reinforced concrete beamSL20: (A) load-deflection diagram and (B) contours of non-local tensile softening parameter 1 at failure (a) coarse mesh and (b) fine mesh.

Fig. 15. Evolution of diagonal sheartensile localized zone with marked points: (a) at P76 kN, (b) at P86 kN, (c) at peak and (d) at failure (small-size reinforced concrete beam).


6. Calculated size effect on shear strength

6.1. Deterministic calculations

The deterministic simulations were carried out with the con-stant tensile strength ft2.2 MPa. The results showed a significanteffect of a beam effective depth on the nominal shear strength.In analyses, the failure mechanism was similar in all beam sizes:after appearance of several localized zones in a pure bending

region, some inclined localized zones started to develop in a shearregion at the support which gradually propagated along a beamheight during loading until they reached a compressive zone. Thecritical inclined localized zones (preceding failure), whichoccurred at the nearest distance x to the support (xEa/2), werethe last ones which occurred in the shear region. They initiatedhorizontal bond splitting failure zones along the reinforcementpropagating simultaneously towards both supports (Figs. 1719). Asimilar failure mode happened during laboratory tests, however asingle horizontal splitting crack always appeared there.

Table 2 presents the calculated local normal (un) and tangen-tial (ut) displacement increments across a diagonal localizedzone during its propagation path (Fig. 15) in the small beamSL20. The dominant mechanism in a diagonal localized zone wasnormal extension. Both displacement increments un and utgradually increased during deformation (e.g. un increased from0.02 mm up to 0.39 mm, whereas ut increased from 0.13 mm upto 0.25 mm across EF). However, the tangential displacementincrement ut reflected mainly the beam deflection.

The evolution of the vertical force versus the deflection wasconsistent with the experimental measurements for all beam sizes(Fig. 16). The calculated maximum vertical forces were: 91.4 kN(small-size beam), 167.9 kN (medium-size beam) and 263.9 kN(large-size beam). Hence, the nominal shear strength, expressedby NV/tD with V0.5Pmax, was N1.43 MPa, N1.17 MPa andN0.88 MPa, respectively it decreased strongly with the beamheight. The calculated width of all localized zones was aboutwloc4lc, i.e. was similar as in our other FE simulations ofreinforced concrete elements [34,36,63]. It was also similar as inour experiments (measured with the DIC technique using thelength resolution of 100 pixel/mm), which was 16.519.6 mm [19].

A direct comparison between the experimental and numericalfracture pattern is demonstrated in Fig. 20. The critical diagonallocalized zone occurred at the distance dc458 mm fromthe support (dc/a0.424) in the beam SL40 and dc1161 m(dc/a0.523) in the beam SL80. The corresponding experimentalaverage values of the ratio dc/a were 0.58 (SL40) and 0.54 (SL80).Thus, the numerical outcome was too small in the beam SL40 andrealistic in the beam SL80. The average spacing of vertical localizedzones s at the mid-region was equal to: 87 mm (SL20), 150 mm(SL40) and 324 mm (SL80), i.e. it was close to the experimentalone, which was 76 mm, 155 mm and 300 mm.

The localized zones at failure in the beam SL20 covered 86% ofits height (0.86H), while in beams SL40 and SL80: 0.773H and0.784H, respectively. The mean cracked region height in theexperiment was 0.864H (SL20), 0.816H (SL40), 0.836H (SL80),indicating a good agreement with numerical results with thesmall discrepancy of 6% in the beam SL40 and SL80.

6.2. Statistical calculations

The effect of a variability of the local concrete tensile strength fton the nominal shear strength was investigated using spatiallycorrelated random fields. The random fields were generated withthe mean value of ft2.2 MPa and variation coefficient covft0.12(covftsft/ft) using the Gauss distribution [26]. The range ofcorrelation was assumed to be lcor100 mm5wloc (Eq. (16))based on calculations in unnotched concrete beams [26]. Usingthe stratified sampling method (Section 4), 12 samples were solelychosen for FE numerical analyses. Fig. 21 presents one exemplaryrandom distribution of the local tensile strength for eachbeams size.

In the calculations, a diagonal sheartensile failure mechanismalways occurred (as in deterministic simulations). The 12 verticalforcedeflection curves from statistical calculations compared tothe deterministic curve are shown in Fig. 22. The mean failure

Fig. 16. Vertical force P deflection u diagrams from deterministic simulationscompared to mean experimental results from 3 tests: (a) small-size beam SL20,(b) medium-size beam SL40 and (c) large-size beam SL80.


vertical forces from statistical calculations were as follows:83.0 kN (74.0 kN), 146.0 kN (76.1 kN) and 234.5 kN (712.7 kN)in the small-size, medium-size and large-size beams, respectively.The corresponding mean nominal shear strengths NV/tD(V0.5Pmax) were: 1.30 MPa, 1.01 MPa and 0.78 MPa, respectively.They were solely by 9%, 13% and 11% lower than the deterministicoutcomes. Thus, a statistical size effect was practically negligiblethat was in agreement with the theory by Bazant and Planas [22].

The contours of a non-local tensile softening parameter (withthe marked sections where the bond radial strain was higher thanthe limit value r,rsZr,rs,max) are shown in Figs. 2325.

The statistical width of localized zones and statistical height ofvertical localized zones in a constant bending moment region wassimilar to the deterministic values, whereas the statistical heightof a critical diagonal localized zone was slightly higher than indeterministic calculations: 0.92H (small-size beam SL20), 0.87H(medium-size beam SL40) and 0.83H (large-size beam SL80),respectively. The calculated normalized position of a criticalinclined localized zone dc/a in all beams during statistical simula-tions was higher on average than in corresponding deterministicanalyses and in experiments (i.e. the localized zone developed at alarger distance from the support). It was dc/a0.58 (Fig. 23a) 0.64 (Fig. 23c), dc/a0.46 (Fig. 24a) 0.79 (Fig. 24c) and dc/a0.43(Fig. 25a) 0.70 (Fig. 25c) in the beams SL20, SL40 and SL80,respectively. There exists a clear relationship between the beambearing capacity and the position of a critical inclined localizedzone from which the bond failure developed. Generally, when acritical inclined zone appeared at a larger distance from theloading point (i.e. at a smaller distance from the support), thebeam bearing capacity was higher. It was, in particular, visible in

large-size beams (Fig. 25) where the maximum vertical forcestrongly varied. The highest vertical force (Fig. 25a) was at acritical inclined localized zone at the distance of 1.28 m from theloading point, and the lowest one (Fig. 25c) at the distance of0.68 m. The average spacing of vertical tensile zones in the small-size beams was s94 mm (86105 mm), in the medium-sizebeams s170 mm (137198 mm) and in the large-size beamss355 mm (291463 mm).

In Fig. 26, a comparison of the maximum shear strength,expressed in analogy to an elastic beam theory as N1.5 V/(tD),between FE deterministic and experimental outcomes togetherwith the calibrated SEL curve by Bazant (Eq. (1)) and two solutionsfor a rigid-perfectly plastic body is given.

The original plastic solution for the shear failure mechanism witha diagonal crack given by Eq. (20) [64,65] assumes an inclinedstraight yield line starting from the support, whereas the improvedplastic solution (Eqs. (21) and (22)) by Zhang [66] considers aninclined straight yield line starting at a certain distance from thesupport x (x0.74D(a/D-2). The shear strengths by Nielsen andBrstrup [64,65] and by Zhang [66], which create the upper boundof the load bearing capacity, based on the equation of the internaland external work, are calculated as follows:

0:5f nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 a

D

2r aD

" # 0:5cf ck

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 a

D

2r aD

" #20

and

0:5f nnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ax

D

2rax

D

" # 0:5scf ck

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ax

D

2rax

D

" #21

Fig. 17. Contours of non-local tensile softening parameter 1 (a) compared to experimental crack pattern (b) and its distribution along beam bottom (c) and along mid-height(d) from deterministic simulations with small-size reinforced concrete beam.


Fig. 18. Contours of non-local tensile softening parameter 1 (a) compared to experimental crack pattern (b) and its distribution along beam bottom (c) and along mid-height(d) from deterministic simulations with reinforced concrete medium-size beam.

Fig. 19. Contours of non-local tensile softening parameter 1 (a) compared to experimental crack pattern (b) and its distribution of along beam bottom (c) and along mid-height (d) from deterministic simulations with large-size beam.


with

c 1:63:5ffiffiffiffiffiffif ck

p 0:27 1 1ffiffiffiffiD

p

0:150:58; ; 22

where cfck denotes the effective compressive strength of un-crackedconcrete whereas scfck is the effective compressive strength of

cracked concrete (by taking into account the decreasing concretecohesion, expressed through the additional reduction factor sE0.6).

The FE deterministic nominal shear strengths match very wellthe experimental findings and experimentally calibrated SEL curveby Bazant (with o2.87 MPa and D0213 mm in Eq. (1) using theLevenbergMarquardt non-linear regression method). Both FE and

Table 2Calculated normal un and tangential ut displacement increments between points of Fig. 15 across diagonal localized zone during its evolution (small-size reinforcedconcrete beam).

Vertical force u [mm] Distance ab Distance cd Distance ef Distance gh Distance ik Distance ln Distance mo Distance pr

76 kN un 0.01 0.01 0.02 0.04 0.09 0.01 0.02 0.15ut 0.12 0.12 0.13 0.13 0.11 0.09 0.12 0.12

86 kN un 0.03 0.05 0.10 0.15 0.22 0.11 0.04 0.27ut 0.15 0.15 0.17 0.18 0.14 0.07 0.13 0.12

At peak un 0.09 0.14 0.23 0.31 0.40 0.29 0.16 0.40ut 0.16 0.17 0.21 0.22 0.18 0.01 0.11 0.09

After failure un 0.16 0.26 0.39 0.49 0.61 0.46 0.29 0.48ut 0.19 0.19 0.25 0.28 0.20 0.06 0.11 0.07

Fig. 20. Contours of non-local tensile softening parameter 1 from deterministic simulations compared to experimental crack pattern (lines) with: (a) small-size,(b) medium-size and (c) large-size beam.

Fig. 21. Exemplary random fields of tensile strength ft in: (a) small-size, (b) medium-size and (c) large-size beam used in FE analyses (note that beams are not proportionallyscaled).


experimental results lie obviously below solutions predicted bythe upper bound plastic theory (Eqs. (20) and (21)) and above thelower elastic bound (1). The improved plastic solution (Eq. (21))is closer to experimental results. The upper horizontal asymptotesfor small structures by Eqs. (20) and (21) are at 3.59 MPa and3.13 MPa, respectively, whereas for the experimentally cali-brated Eq. (1) at 2.87 MPa (Fig. 26).

Considering statistical simulations, all mean values underesti-mate the experimental shear strength by 10% on average (thisresult is probably caused by the assumption of a deterministicbond model in statistical computations or a sampling regionlimited to beam shear regions). The mean shear strength reductionin statistical computations with respect to deterministic resultswas close to the assumed coefficient of variation covft of the tensilestrength. Substituting into the experimentally calibrated SEL for-mula (Eq. (1)) the reduced value of o, expressed by o(1covft)2.87(10.12)2.53 MPa, a perfect match between the new SELcurve and the statistical results was obtained (Fig. 27). Therein, thebrittleness number defined by the ratio D/D0 was not modified.This indicates that the size effect on the nominal shear strength ispredominantly of a deterministic type.

With respect to FE numerical results on a size effect by Satoet al. [31], our deterministic results indicate a stronger reduction ofthe nominal shear strength with similar increasing beam size (38%between the small- and large-size beam against 20%).

7. Concluding remarks

The following conclusions can be drawn from our plane stressnon-linear FE analyses with a constant and spatially correlatedrandom distribution of the local tensile strength in geometricallysimilar longitudinally reinforced concrete beams without shearreinforcement failing in shear:

An isotropic elasto-plastic constitutive model with non-local softening enhanced by a characteristic length of micro-structure was able to realistically reproduce both a verycomplex experimental failure mode in beams (combineddiagonal sheartensile failure and horizontal bond splittingfailure) and a pronounced reduction of the nominal shearstrength with increasing beam size by taking a bond modelinto account.

The size effect on the nominal shear strength of slender RCbeams without shear reinforcement was of a deterministic typeonly. The nominal shear strength from deterministic simula-tions strongly decreased with increasing specimen size. Thedeterministic size effect was caused by localized zones with aconstant width and a linearly varying height. It was in agree-ment with the energetic size effect by Bazant. Thus, it was notcaused by non-symmetric cracking combined with the unin-tended out-of-plane beam action. The mean shear strengths instatistical computations were solely lower by 10% than corre-sponding deterministic outcomes. The mean statistical shearstrength was proportional to the mean tensile strength reducedby the standard deviation.

The effect of the bond definition between concrete and rein-forcement was pronounced and led to the different failuremode, load capacity, beam stiffness and number of localizedzones. The application of the bond model by Akkermann [52]contributed to a sheardiagonal tensile failure mechanism innumerical simulations as in experiments.

A position of a critical diagonal localized zone in deterministiccalculations was close to experimental outcomes. In statisticalcalculations, a critical diagonal localized zone was located at ahigher distance from the beam support, hence the meanbearing capacity was lower than the corresponding determi-nistic one. The height of a cracked section in experiment wassuccessfully reproduced by both deterministic and statisticalanalyses. The height of a critical diagonal localized zone wasslightly larger in statistical calculations than in deterministicones. The width of localized zones was similar in deterministicand statistical calculations.

Fig. 22. Vertical force P deflection u diagrams (12 curves) for reinforced concretebeams from statistical FE simulations (solid lines) compared to one deterministiccurve (dashed line): (a) small-size beam, (b) medium-size beam, and (c) large-size beam.


Fig. 23. Contours of non-local tensile softening parameter 1 from statistical simulations of small-size reinforced concrete beam corresponding to: (a) highest failure force,(b) mean failure force, and (c) lowest failure force.

Fig. 24. Contours of non-local tensile softening parameter 1 from statistical simulations of medium-size reinforced concrete beam corresponding to: (a) highest failureforce, (b) mean failure force, and (c) lowest failure force.

Fig. 25. Contours of non-local tensile softening parameter 1 from statistical simulations of large-size reinforced concrete beam corresponding to: (a) highest failure force,(b) mean failure force, and (c) lowest failure force.


Acknowledgements

The research work has been carried out within the project:Innovative ways and effective methods of safety improvementand durability of buildings and transport infrastructure in thesustainable development financed by the European Union(POIG.01.01.02-10-106/09-01) and the project Experimental andnumerical analysis of coupled deterministic-statistical size effectin brittle materials financed by National Research Centre NCN(UMO-2013/09/B/ST8/03598).

Appendix

In order to describe the concrete behaviour, two failure criteriawere assumed. In tension regime, the Rankine criterion was usedwith the yield function f1 using isotropic softening and associatedflow rule and in compression, the DruckerPrager yield surface f2with isotropic hardening/softening and non-associated flow rulewas used [36]

f 1si; 1 max s1;s2;s3;f gst1r0; A1

f 2sij; 2 qp tan c2 qp tan 113tan

sc2r0

A2

qffiffiffiffiffiffiffiffiffiffiffiffi32sijsij

r; p 1

3skk; A3

tan 31rsbc

12rsbc; A4

g1 f 1 A5

g2 qp tan A6where, si the principal stress (i1, 2, 3), st the uniaxial tensileyield stress, 1 the softening parameter equal to the maximumprincipal plastic strain 1p, q the Mises equivalent deviatoricstress, p the mean stress, the internal friction angle in themeridional stress plane (pq plane), c the cohesion related touniaxial compression strength, sij the deviator of the stresstensor sij, (sijsijijp) sc the uniaxial compression yield stress,2 the hardening/softening parameter corresponding to theplastic vertical normal strain during uniaxial compression, gi flow potential function, rbcs the ratio between the biaxialcompressive strength and uniaxial compressive strength(rbcsE1.2) and the dilatancy angle (a). The last term inEq. (A3) results from the yield condition qp tan c0 foruniaxial compression with qsc and p1/3sc.

The exponential function by Hordijk [37] was always taken as anon-linear softening function in tension (Fig. 4a)

st1 f t 1A113expA21A31 A7with

A1 c1u; A2

c2u

and A3 1u1c31expc2; A8

wherein ft is the uniaxial tensile strength, u is the ultimatesoftening parameter and ci are the empirical parameters (c13and c26.93).

The compressive strength of concrete fc was assumed to beequal to the characteristic compressive strength fck [38] and tensilestrength ft to the characteristic tensile strength fctk [28], respec-tively

f ctk 0:7f ctm 0:7ffiffiffiffiffiffif 2ck

3q

with b 0:3 A9The elastic modulus was Ec9500(fck8)1/3 [38] and the Poisson0sratio 0.2. The ultimate value of the softening parameter undertension u was estimated from the assumed concrete fractureenergy Gfgfwloc (gf area under the softening curve st(1),wlocE4lc the width of a localized zone), wherein Gf wascalculated following CEB-FIP [39] as

Gf Gfof cm=f cmo0:7; A10where fcm fckfck is the mean compressive strength withfck8 MPa, the reference value is fcmo10 MPa and Gfo denotesthe parameter depending on the maximum aggregate size(Gfo0.058 N/mm with the aggregate diameter of 32 mm). Theultimate softening parameter under compression 2 was calculatedsimilarly as in tension (GcE4lcgc) from the compressive fractureenergy based on initial computations [35].

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FE analysis of size effects in reinforced concrete beams without shear reinforcement based on stochastic...IntroductionConstitutive modelsConcreteReinforcementBond between concrete and reinforcement

Random fields in statistical calculationsFE input dataEffect of bond-slip modelCalculated size effect on shear strengthDeterministic calculationsStatistical calculations

Concluding remarksAcknowledgementsAppendixReferences

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strong size effect

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