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T E C H N I C A L A R T I C L E

Reinforced Concrete Membranes Under Shear: GlobalBehaviourS.M.R. Lopes1, L.F.A. Bernardo2, and R.J.T. Costa3

1 Department of Civil Engineering, University of Coimbra, CEMUC, Coimbra, Portugal

2 Department of Civil Engineering, University of Beira Interior, CMADE, Covilha, Portugal

3 Department of Civil Engineering, University of Coimbra, Coimbra, Portugal

KeywordsStructural Testing, Concrete Structures,

Concrete Testing, Codes of Practice and

Standards, Beams and Girders, Slabs and

Plates

CorrespondenceS.M.R. Lopes,

Department of Civil Engineering,

University of Coimbra,

CEMUC, Coimbra, Portugal

Email: [email protected]

Received: December 21, 2011; accepted:

October 6, 2012

doi:10.1111/j.1747-1567.2012.00873.x

Abstract

The main purpose of this article is to study compressed reinforced concretemembranes under perpendicular tensile stress. The state-of-the-art is presentedand it shows that not much research has been done on this topic, probablybecause it needed very expensive laboratory equipment. In the currentinvestigation, the authors have tried a new type of equipment, less expensivethan that of previous investigations, and it proved to be adequate for the type oftests needed for studying this topic. The authors tested six reinforced concretepanels under pure shear up to failure. A parameter that was studied in thesetests was the thickness of the membranes. This is of particular importance inwebs or walls of beams under shear or torsion forces. From the experimentalmeasurements, the authors have computed the principal directions and thevalues of the stresses and strains for the tested beams. In this article, thesevalues are presented and the principal strains and stresses are compared withthose at 45◦ and the two fairly agree with each other, particularly for the caseof the stresses.

Introduction

It is well known that when reinforced concrete isunder shear, the behaviour of concrete in the strutsis significantly affected by transverse tensile stresses.For example, recent studies of the behaviour of beamsunder torsion indicate the importance of includingthis effect in mathematical models used to predict theultimate limit state.1–5 This is particularly importantin the case of box girders where the walls act asmembranes that are under shear stress (Fig. 1(a));compression in one direction and tension in theperpendicular direction. Figure 1(b) shows the neg-ative influence of the transverse tensile stresses inthe stress–strain relationship of the compressive con-crete in the struts. Such studies also show that wallthickness can affect the behaviour of pre-stressed con-crete in the struts differently to predictions made bytheoretical models. It is therefore important to studythis phenomenon, particularly with regard to changescaused by wall thickness.

In general, studies developed with the aim ofmodelling the behaviour of reinforced concrete plates

subject to stresses along the plane, treat cracked rein-forced concrete as a new material with its own stressversus strain properties and take into considerationthe different behaviours of the concrete and therebars. These approaches make use of average stressesand strains, both in the concrete and the rebars(stresses and strains are measured along a length ofmaterial sufficiently long to contain several cracks).Following these approaches, these average values arethen used to determine both the equilibrium condi-tions of the panels and the deformation compatibilityconditions. The ASCE-ACI Committee 445 on Shearand Torsion6 defined methodologies based on theseassumptions as Compression Field Approaches or SmearedApproaches. This investigation makes use of such anapproach. Therefore, it is necessary to calculate therelationship between principal average stresses andprincipal average strains both for the concrete andthe ordinary rebars.

Average tension versus strain relationships havebeen proposed for ordinary rebars under stressthat take into account the interaction between the

Experimental Techniques (2012) © 2012, Society for Experimental Mechanics 1

RC Membranes Under Shear S.M.R. Lopes, L.F.A. Bernardo, and R.J.T. Costa

(a)

(b)

Figure 1 Beam under torsion: shear in the (a) wall and (b) stress–strain

relationships for concrete.

rebar and the surrounding concrete (the stiffeningeffect). Various relationships have also been proposedbetween average tension and strain for concreteunder both tensile and compressive stress. This willbe focused in section Basic Concrete and Steel Equations.

In order to simplify the estimation of the strengthof reinforced concrete panels, certain assumptionshave been made. For example, it was assumed thatthe principal directions of stress within a reinforcedconcrete member coincide with the principal direc-tions of its deformation state (there is no relativeslippage between the rebars and the concrete). Thisassumption has been proved to be acceptable, as

shown by studies carried out in two orthogonal direc-tions on reinforced concrete pieces with identicalreinforcement.6,7 These studies have indicated valuesof less than 10◦ between the directions of principalstress and deformation. This subject has not beenstudied recently, and there is no real consensus onthis matter. In fact, there are very few studies in thisarea, almost certainly due to the high cost of testing.However, some recent studies8–10 have investigatedbox walls and strut and tie mechanisms but did notinvestigate the strength of concrete struts under per-pendicular tensile stresses.

This article, which constitutes the first of the series,presents the state-of-the-art with regard to concretestruts under perpendicular tensile stress. Six con-crete panels were tested under shear, with particularattention given to investigating the influence of mem-brane thickness. This article also describes the testsand presents the results with respect to the principaldirections. The corresponding tensions and strains arealso discussed. Another objective was to conceive,construct, and validate a simple and easy testingapparatus. In the second article of the series,11 theinfluence of the thickness on the concrete panels isinvestigated, new experimental stress–strain relation-ships for thin plates are proposed and a comparativestudy with previsions for the strength capacity fromsome important Codes of practice is also be presented.

Basic Concrete and Steel Equations

Concrete under compressive stress

This section briefly discusses the key relationshipsconcerning the behaviour of concrete under com-pressive stress.

The behaviour of concrete in actual structures candiffer slightly from the behaviour that lab tests seemto indicate. There are some tests specially recom-mended for the evaluation of concrete propertiesin real structures.12–15 However, to anticipate thebehaviour of a future structure, conclusions by pre-vious investigations, most of them from laboratoryworks, need to be taken into account. There are somestress–strain relationships available to model the con-crete material. In this work, the authors have usedthe proposed equation by Hognestad,16 that is:

fc = f′c

[2

(εc

ε0

)−

(εc

ε0

)2]

(1)

where f′c is the maximum compression resistance

of the concrete (measured in uniaxial compressiontests on cylindrical specimens) and ε0 is the straincorresponding to f

′c (−0.002 or −0.0022).

2 Experimental Techniques (2012) © 2012, Society for Experimental Mechanics

S.M.R. Lopes, L.F.A. Bernardo, and R.J.T. Costa RC Membranes Under Shear

When establishing stress versus strain relationshipsfor concrete cracked due to shear forces, it is importantto remember that the deformation state it experiencesis very different from what occurs in test membranesunder uniaxial compression, where stress causes sometransverse strain as a result of the Poisson effect. Whencracked reinforced concrete is subjected to shear,concrete suffers significant stress because of transversestrain, as a result of which it becomes less resistantto compression and less rigid (the softening effect) thantest specimens in uniaxial compression tests.

The search for a wide-ranging, rational method forpredicting the behaviour of reinforced concrete led tothe introduction of a model known as CompressionField Theory (CFT), which was based on experimen-tal trials with concrete slabs.17 This model was laterrefined and became known as the Modified Compres-sion Field Theory (MCFT).18,19 From this emerged anew average tension versus strain relationship forconcrete under compression (Eq. 2) which took intoaccount the softening effect using βσ (a softening param-eter) defined by Eq. 3, a function of principal strain1 (stress), to modify the Hognestad parabola. Boththe maximum uniaxial compression stress of the con-crete (f

′c ) as well as the associated strain (ε0) were

multiplied by parameter β.

fc2 = βσ f′c

[2

(εc2

ε0

)−

(εc2

ε0

)2]

(2)

βσ = 1

0.8 − 0.34 εc1ε0

≤ 1.0. (3)

In order to include high strength concrete inthe model and to correct some deviations observedin practical trials, namely in low-strength concrete,Vecchio and Collins20 proposed using a curvedeveloped by Thorenfeldt et al.,21 calibrated by Collinsand Poraz,22 and defined by Eqs. 4, 5, and 6. Thesubscript ‘‘base’’ is used in Eq. 4 to highlight that thisequation was originally recommended by Thorenfeldtet al.21 for describing uniaxial compression response

of concrete in standard tests. By changing this ‘‘base’’curve, Vecchio and Collins20 proposed two models(Models A and B).

fc2,base = −fpn

(−εc2εp

)n − 1 +

(−εc2εp

)nk(4)

n = 0.80 + fp(MPa)

17(5){

−εp < εc2 < 0 k = 1.0

εc2 < −εp k = 0.67 + fp(MPa)

62 .(6)

In Model A, where a reduction in the maximum com-pression stress of the concrete and its correspondingstrain is paramount, the reduction factor for stress(βσ ) and for strain (βε) is given by Eqs. 7, 8, and 9(Fig. 2).

β = βσ = βε = 1

1.0 + KcKf≤ 1 (7)

Kc = 0.35

(−εc1

εc2− 0.28

)0.80

(8)

Kf = 0.1825√

f ′c (MPa) ≥ 1.0. (9)

The Kc parameter accounts for strains in concreteand the Kf parameter takes into account thatin high strength concrete, the softening effect ismore pronounced as a consequence of smoothedfracture planes, resulting in the earlier onset of localcompression stability failure or in earlier crack-slipfailure.19

The curve of Model A is divided into three parts:

(i) for −εc2 < βε0 , fc2 it is calculated from Eq. 4using fp = βf

′c and εp = βε0 ;

(ii) for βε0 ≤ −εc2 ≤ −ε0, fc2 = fp = βf′c ;

(iii) for −εc2 ≥ βε0 , fc2 = βfc2,base, using fc2,base calcu-lated from Eq. 4 using fp = f

′c and εp = ε0.

Vecchio and Collins also proposed a simpler ModelB that only uses a reduction factor for compression

Figure 2 Reduction factors (εc1/εc2 = −5; f′c = −35MPa; ε0 = −0.0022; α = 45◦).



stress, given by Eqs. 10 and 11.

βσ = 1

1.0 + Kc≤ 1 (10)

Kc = 0, 27

(εc1

ε0− 0.37

)(11)

Later, Vecchio et al.23 extended Model B to includehigh resistance concretes, by using Model A andincorporating a factor Kf expressed by Eq. 12.

Kf = 2.55 − 0.2629√

f ′c (MPa) ≤ 1.11 (12)

These models were based on the results of pro-portional stress tests on concrete panels with equalpercentages of reinforcement in the two orthogonaldirections, in which the test pieces failed because theconcrete was crushed before the rebars gave way andin which a lesser softening effect was observed thanthat predicted by earlier theoretical models. There-fore, Vecchio24 proposed using the parameter β fromEq. 13 for both models.

β = 1

1 + CsCd(13)

In the previous equation Cs = 0.55 and Cd assume thevalues of Kc (Eqs. 8 and 11). Coefficients Kσ and Kε

are defined in Table 1.Belarbi and Hsu25,26 also proposed a model (Eqs.

14, 15, and 16), which was based on the Hognestadparabola but with different softening parameters forstress and strain, adjusted according to the type ofstress applied to the test panels and the inclinationangle of the fractures with respect to the rebars (α).The coefficients Kσ and Kε are defined in Table 1.⎧⎪⎪⎨

⎪⎪⎩εc2 ≥ βεε0 fc2 = βσ f

′c

[2

(εc2

βεε0

)−

(εc2

βεε0

)2]

εc2 < βεε0 fc2 = βσ f′c

[1 −

(εc2−βεε02ε0−βεε0

)2]

.

(14)

βσ = 0.9√1 + Kσ εc1

(15)

βε = 1√1 + Kεεc1

(16)

Hsu27 later proposed Eq. 17 to conservatively describethe softening effect, for use in conjunction with Eq. 14.

Table 1 Coefficients Kσ and Kε

Proportional action Sequential action

α = 45◦α = 90◦

α = 45◦α = 90◦

Kσ 400 400 400 250Kε 160 550 160 0

According to Hsu, this is a conservative expressionthat provides a lower limit for the tests carried outduring the investigation.

βσ = βε = 0.9√1 + 600εc1

(17)

Afterwards, based on tests in which the principaldirection of stresses was at an oblique angle relativeto the rebars, Mikame et al.28 proposed Eq. 18 as thecoefficient to describe the reduction in resistance tocompression.

βσ = 1

0.27 + 0.96(

εc1−ε0

)0.167 (18)

Figure 3 represents the reduction factors calculatedfrom previous models as well as from some lesscommon models not discussed here. The deviationsbetween the curves from the several proposed modelsare notable. Therefore, more reliable models need tobe considered. None of the studied models considersthe thickness of the panels as a parameter. Theseaspects justify further investigation in this area,especially with a view to incorporating variables thathave not been considered relevant before, such as theabove-mentioned thickness of the panels.

Reinforcements under tension

In the elastic phase, the average stress versus strainof rebars surrounded by concrete, when subject totension, can be mathematically described by the curveobtained from uniaxial tensile tests on isolated rebars(free from concrete).

However, when concrete cracks, there are largedifferences in the distribution of tensile stresses andstrains occurring in the rebars. These differencesoccur because throughout the length of a crack theentire traction stress is absorbed by the rebars, whilebetween the cracks, the tensile stress is absorbed byconcrete, by adherence in the interface. This is theso-called stiffening effect.

While some models for rebars embedded inconcrete (CFT, MCFT) incorporate simplified aver-age elastic stress versus strain relationships witha perfectly plastic yield plateau, others (SoftenedTruss Model–STM, Rotating Angle Softened TrussModel–RASTM) incorporate more elaborate math-ematical expressions which take into account thespatial differences in tensions and strains mentionedabove.

Following this line of thinking, Belarbi and Hsu29

proposed two curves to model the behaviour of rebarsin concrete under tension. In the first, more complex



Figure 3 Geometry of the specimens.

model, the average stress versus average strain ismathematically described by a continuous curve.In the other model (Eqs. 19 to 23), for simplicity,the stress versus strain curve of concrete understress is modelled using two rectilinear paths thatintersect at a particular stress level f ∗

n , as given byEqs. 19 to 23. {

fs < f ∗n fs = Esεs

fs > f ∗n fs = f ∗

o + E∗pεs.

(19)

f ∗n = (0.93 − 2B)fy (20)

E∗p = (0.02 + 0.25B)Es (21)

B = 1

ρ

(fcrfy

)1.5

(22)

f ∗0 = Es − E∗

p

Esf ∗n (23)

In the above equations, Es is the modulusof elasticity, E∗

p is the tangent modulus in theplastic region, f ∗

0 is the stress corresponding to theintersection of the second rectilinear path with they-axis and fcr is the tension stress that causes theconcrete to crack, which according to the authors canbe calculated as fcr = 0.3114

√fc

′(MPa).

To more easily enable these equations to be applied,Belarbi and Hsu26 take certain values as given:

Ep = 0.025Es, f0 = 0.89fy e fn = 0.091fy, resulting inEq. 24.⎧⎪⎨

⎪⎩fs ≤ (0.93 − 2B)fy fs = Esεs

fs > (0.93 − 2B)fy fs = (0.91 − 2B)fy

+(0.02 + 0.25B)Esεs.

(24)

Codes of Practice

In this study, the following documents are analysed:CEN EN 1992-1-1,30 CEB-FIP Model Code 1990,31

ACI 318-05,32 and CSA Standard A23.3-04.33 In thisarticle, the authors are only concerned with therules for members with transversal reinforcementperpendicular to the longitudinal axis. Table 2 laysout the recommended values for accepted reductionfactors (efficiency factors: ν) for concrete undercompression when applied to reinforced concreteunder shear (transverse and twisting forces). InTable 2, α is the inclination of the transverse rebars,θ is the inclination of the diagonal compression struts(relative to the longitudinal rebars), εc1 is the averagestrain of (smeared) stress of the cracked reinforcedconcrete measured transverse to the direction of theapplied compression forces and k is a coefficient thatdepends on the roughness of the surface and thediameter of the rebars.



Table 2 Reduction factors proposed in standards and codes

Shear TorsionCE

NEN

1992

-1-1 0.6

(1 − fck (MPa)

250

)If the design value of the stress of the

shear reinforcement is lower than 80%

of that of the yield characteristic

value:

CEB

-FIP

MC

90

{0.6 if fck ≤ 60 MPa

0.9 − fck (MPa)200 > 0.5 if fck ≥ 60 MPa.

0.6(

1 − fck (MPa)250

)0.6

(1 − fck (MPa)

250

)(simplified equation)

Cracking parallel to the direction of the compressive

tensions:: 1/(

1 + k εc1ε0

)FIB, CEB-FIP (1999): 1/

(0.8 + 0.34 εc1

ε0

)

ACI

318-

05 2/3

(1/2)√

f′c (MPa)0.9(1+cot(α))

2/3√f′c (MPa) 0.9 cot(θ ) (sin(θ ))2

CSA

23.3

-04 1/(0.8 + 170εc1) ≤ 1

Simply put, in order to consider the reducedresistance of concrete cracked by compression inmembers submitted biaxially to compression andtension, as is the case for lattice type struts, theregulatory documentation recommends reductionfactors for the concrete’s resistance to compression.This reduction in resistance is attributed essentially tostress that appears in the concrete between the crackscaused by the stress transferred via adherence fromthe rebars to the concrete.

Experimental Programme

Testing options

The literature refers to tests of the behaviour ofconcrete under compressive stress in the presence oftransverse strain in which reinforced concrete panelswere subjected to shear stresses. The equipment usedfor these tests is very expensive. Despite offering theflexibility to accurately vary certain parameters (suchas the state of stress or deformation to which thepanels are submitted) the equipment did not easilyallow the geometry of the panels to be varied.17

Therefore, for this study, a different apparatus wasdeveloped. It allows the investigation of stress statessimilar to those generated by pure shear in reinforcedconcrete panels of different thicknesses. It was alsomuch less expensive than previous test equipment.

The biggest difference between this study and thoseusing the more expensive apparatus is that instead ofsubjecting the entire panel to the required stress (pureshear) only a small area in the centre of the test pieceis subjected to pure shear.

Table 3 Reinforcement steel of the specimens

Distributed reinforcement Tie

Series Name Name φ (mm) Spacing (mm) Name φ (mm) n.◦

I SIE4 A500NR 6 7 A500NR 10 5SIE7 A500NR 8 7 A500NR 10 10

SIE10 A500NR 10 7.5 A400NR 12 10II SIIE4 A500NR 6 7 A500NR 10 5

SIIE7 A500NR 8 7 A500NR 10 10SIIE10 A500NR 10 7.5 A400NR 12 10

Testing models

Figure 3 illustrates the geometry adopted for thepanels tested. Two equal series of panels were tested(Series I and II), each series having three panels.The test pieces were plates measuring 860 × 860 mmwith a thickness of e (variable during testing) fixed to apiece measuring 200 × 350 × 860 mm for attachmentto a reaction wall.

The panel section of each test piece was reinforcedwith an orthogonal rebar mesh (identical in everycase) in order to limit strain by limiting cracking inthe concrete. The reinforcement ratio was 1% This is ausual value for panels. As for instance, Nakachi et al.34

published a work on cycle loads on shear panelsand they have constructed panels with reinforcementratios between 0.7 and 1.2%. Horizontal rebars werealso used in order to resist bending forces (upperrebars: tie). The reinforcement of the test specimensis illustrated in Fig. 3 and defined in Table 3 (referringonly to the reinforcement in the panel section). Thecover adopted for the rebars was 2 cm. For practicalreasons, the concrete was poured into the panels withthese ones horizontally placed in the floor (Fig. 3). Inother words, the concrete was poured in a directionperpendicular to the plane of the load.

Description of materials

The concrete mix used is summarised in Table 4. Eachseries I and II were cast using the same concrete batch.

The compressive strength of concrete was measuredin uniaxial compression tests on cubic test specimens

Table 4 Concrete mix

Components Concrete mix (per m3)

Sand (kg) 841Gravel 1 (kg) 960Normal Portland cement Type II / 32.5 (kg) 350Water (L) 175A/C 0.5Density (kg/m3) 2326



Table 5 Compressive stress in concrete

Series e (cm) f′cm (MPa)

I 4 25.27 26.0

10 24.5II 4 26.5

7 24.919 26.5

Table 6 Results from tension tests of steel bars

φ (mm) fym (MPa) εym (×10−6)

6 595.2 29768 576.3 288110 570.18 285112 469.10 2346

(15 cm per side) cured in conditions similar to thoseof the panels. Table 5 lists the average compressionstrength values for the concretes. For each plate, fourconcrete cubic test specimens were tested.

The rebars used in the test pieces were ordinarycommercial rebars of heat-laminated steel A500 NRand A400 NR with diameters of 6, 8, 10, and 12 mm.Tensile tests on steel test specimens (six specimen perdiameter) were carried out to find the average yieldstress, fym, and yield strain corresponding to the start ofthe yield plateau, εym (Table 6). The εym parameter wascalculated from the measured fym and assuming thestandard value usually indicated for ordinary rebarsfor the elasticity modulus: Es = 200 GPa.

Testing apparatus

The testing apparatus (Fig. 4), designed to enablethe reinforced concrete test pieces to be subjectedto a shear force, incorporates a reinforced concretereaction wall. A metallic bracket fixed to thereaction wall supports an electromechanical actuator(1000 kN). The test piece is basically tested as if itwere a large slim corbel, with a concentrated forceapplied at its extremity.

The loading plates are designed to distribute theload across the entire height of the concrete, therebyavoiding the load being concentrated at the top ofthe piece which could lead to localised cracking inthe finer panels. It should be noted that the testingprocedure was firstly validated with an elastic FiniteElement Analysis (FEA). A plane model composedwith shell elements (stress plane state) was created,loaded, and supported according to Fig. 4. Thisnumerical model showed that, in the central zone ofthe plate, the direction of principal stresses were 45◦

inclined with respect to a horizontal line. These resultsshowed that a pure shear stress state was achieved inthis area by using the referred testing procedure.

Instrumentation

Strategically positioned digital strain gauges were usedto measure the global deformation states of the testpanels (LVDTs) (Fig. 5).

According to the approach adopted for this study,it is necessary to know the average strains in thecentral zone of the specimens. These strains shouldbe measured in an area large enough to contain

Figure 4 Test setup.



(a) (b)

(c) (d)

Figure 5 Instrumentation.

Figure 6 Crack patterns.



several cracks. Therefore, a network of demec targetswas placed on either side of the panel as well as in thecentral zone under pure shear force (demec targets aresmall discs to be stuck to the concrete surface; each ofthese discs have a hole into which an extremity of astrain measuring device will be adjusted; a second discis needed for the adjustment of the other extremityof the device and subsequent length reading). Thesetargets were offset by 20 mm and served as a referenceto measure average strain throughout the test (Fig. 5).

Additionally, resistance extensometers (straingauges) were glued to the surface of the stiffeningrods in both directions (at the central area of the testspecimen) and in the rebars of the tie, respectivelyto investigate the yielding load and to control theflexural effect induced by the loading (Fig. 5). Thevertical force applied to the models was measured bya load cell (500 kN) placed between the actuator andthe steel plate used to transfer the load. A data loggerwas used to read and record the data.

Testing

The test panels were subject to controlled deformation(a constant velocity of 0.015 mm/s). Figure 6 showsthe crack patterns for some of the panels. From Fig. 6,it is possible to verify that the cracking pattern in thecentral area of the specimens is compatible with apure shear stress state. This observation validates thetesting procedure and confirms the numerical results.The lateral bracing system of panel SIIE10 did notfunction as desired and results from this panel havetherefore been disregarded.

Analysis of Experimental Results

When calculating the compression field angle of theconcrete (θ) based on Compression Field Approaches orSmeared Approaches, it is assumed that the principalstress directions coincide with the principal directionsof the deformation state. This hypothesis has been

Figure 7 Principal direction of

compression of the deformation state.



experimentally tested on specimens with rebars intwo orthogonal directions.29

Deformation state

The principal direction 2 of the deformation state wasdetermined from the demecs targets readings. Usingaverage readings in the three directions for eachload step (horizontal, x; vertical, y, and 45◦), and byusing Mohr’s circle for strain, simple arithmetic allowsthe calculation of the distortion γxy/2 by knowingεα = εc45, εx, εy and α = 45◦. Using γxy/2 for each loadstep, the angles that the principal directions of thedeformation state make with the x axis, as well astheir deviation from the predicted (45◦) angle, weremeasured.

Figure 7 shows the variation in angle of deforma-tion state (principal direction 2) relative to the x axisin the test panels, as well as its deviation from thepredicted (45◦) angle.

Figure 7 shows that for smaller loads, deformationstate (principal direction 2) shows large percentagedeviations from the expected angle (45◦). Thisbehaviour may have its origins in the low level straincaused by such low loading and the inherent errorsin registering such small changes.

These deviations from the expected angle aresignificantly reduced once the concrete starts tocrack, which occurs at between 15 to 25% of theload necessary for failure. Cracking also leads tothe reduction and stabilization of the angle thatdeformation state principal direction 2 makes withthe x axis.

Using deformation state principal direction 2 (θ2ε),principal strain 2 (εc2) was calculated for each set ofreadings from Mohrıs circle for strain. This value wascompared to the one obtained experimentally at 45◦

to the x-axis (εc45) in Fig. 8, which also shows thepercentage by which εc2 deviates from εc45.

Figure 8 εc2 versus εc45.



Table 7 Summary of results 1

SIE4 SIE7 SIE10 SIIE4 SIIE7

X(θ2ε ) (◦) 44.1 44.3 42.1 41.9 45.4X(|θ2ε − 45|) (◦) 2.7 2.3 5.3 3.3 1.6X(|(θ2ε − 45)/45|) (%) 5.9% 5.2% 11.7% 7.3% 3.5%X(|(εc2 − εc45)/εc45|) (%) 1.0% 1.2% 6.2% 2.0% 0.5%

Figure 8 shows that, although deformation state(principal direction 2) of panel (θ2ε) deviatessignificantly from the expected (45◦), the strain inthis direction (εc45) shows little significant deviationfrom principal strain 2 of panel (εc2), in other wordsε45 ≈ εc2.

Table 7 shows the average angle that deformationstate principal direction 2 makes with the x axis foreach test, the average absolute deviation of this anglerelative to the expected 45◦, as well as the correspond-ing absolute average percentage, and the experimen-tally measured average absolute percentage deviation

of the strain from 45◦ to the x axis relative to principalstrain 2 of the deformation state.

The average values reported in Table 7 do notinclude readings taken before cracking, due to thepercentage errors observed.

Table 7 again shows that despite the percentagedeviation between deformation state principal direc-tion 2 (θ2ε) and 45◦, the difference between the strainin these directions is not significant. It can be seenthat the 10-cm thick panel from Series I has an aver-age absolute percentage deviation of extension, indeformation state principal direction 2 relative to thestrain at an angle of 45◦ to the larger x axis, higher6.2% when compared to the values recorded for theother panels.

Concrete stress

The test pieces were reinforced concrete panels witha uniformly distributed dense orthogonal lattice.

Figure 9 Principal direction of

compression of the stress state.



Figure 10 fc2 versus fc45.

During testing, they were subjected to known levelsof perpendicular and tangential stresses that wereused to calculate the principal stress directions forvarious loads. The general state of stresses can be splitin the stresses applied to the concrete in the principaldirections of stresses or in the stresses applied to theconcrete facets in the x, y directions (x horizontal),and in tensile stresses of the reinforcement bars. Bybalancing the stresses in the vertical and horizontaldirections (case of pure shear stress), the compressivestresses in concrete in horizontal (fcnx) and vertical(fcny) directions are computed. By using the normalstresses fcnx and fcny as well as the tangential stressesfctxy, the principal directions of the stress state for thezone under study is known from Mohrıs circle forplain stress.

Figure 9 shows the variation in principal stressdirection 2 in the concrete of the test pieces aswell as the deviation from the expected (45◦)angle.

Generally, the angle that principal stress direction2 makes with the x axis increases slightly withincreasing load. This is in contrast to what wasobserved for principal deformation state direction 2(Fig. 7).

Additionally, the way in which θ2ε diverges fromthe expected value (45◦)at the beginning of the testdoes not hold true for θ2σ .

Using the values of actuating tensions in the faces ofthe concrete parallel to the x − y axes, it was possibleto calculate principal stress 2 of the concrete (fc2) foreach set of readings from Mohrıs circle for plain stress.This value was compared to the stress in the concretein a direction at 45◦ from the axis x (fc45). Figure 10shows the values recorded for these quantities as wellas their percentage deviation.

Figure 10 shows that the principal stress 2 of theconcrete (fc2), despite the deviation of its direction(θ2σ ) from 45◦, does not deviate significantly fromthe stress value fc45.



Table 8 Summary of results 2

SIE4 SIE7 SIE10 SIIE4 SIIE7

X(θ2σ ) (◦) 46.5 46.2 48.3 47.4 44.6X(|θ2σ − 45|) (◦) 2.1 1.7 3.7 2.5 1.0X(|(θ2σ − 45)/45|) (%) 4.6% 3.8% 8.2% 5.4% 2.1%X(|(fc2 − fc45)/fc45|) (%) 0.2% 0.2% 0.6% 0.3% 0.0%

Table 8 presents the average angle that principaldirection 2 of the concrete’s stress state (θ2σ ) makeswith the x axis for each test, the average absolutedeviation of this angle relative to the expected angle(45◦) as well as the corresponding average absolutepercentages, and the average absolute percentagedeviation of stress in principal stress direction 2 inthe concrete (fc2) relative to the compression stress atan angle of 45◦ to the axis x (fc45). Just as for Table 7,readings taken before fracturing were not used.

Comparing the average values of θ2σ from Table 8with the average values of θ2ε from Table 7, there issome divergence between the principal directions ofthe deformation state and the principal directions ofthe stress state in the concrete.

It can also be seen that although principal direction2 of the stress in the concrete (θ2σ ) shows significantdeviation from the predicted 45◦, the difference in thestress values in the two directions is insignificant.

Conclusions

Experimental readings of strains in the central zone ofthe panels showed that the actual principal directionsof the panels were approximately coincident tothe predictions of numerical computations (45◦).Therefore, the dimensions of the panels and thetesting set up are appropriate to ensure a stabilisedarea of stresses in the middle zone of the panel.As a consequence, the test procedure presented inthis article can be used to study the behaviour ofreinforced concrete panels under pure shear. Strainvalues were highly dispersed for low loads. Dispersionwas significantly reduced for loads between 15 and25% of the failure load and over.

With regard to the deformation state of the walls,the expected value of 45◦ from the principal directionwas generally not achieved, as the results deviated 3.5and 11.7% from this angle. The errors that resultedfrom assuming strain at 45◦ from the principaldirection instead of parallel to it varied between 0.5and 6.2%.

With regard to the stress in the concrete, theerror resulting from assuming stress at 45◦ from the

principal direction instead of parallel to it was verysmall (between 0 and 0.6%).

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