Article publié par le Laboratoire de Construction en Béton de l'epfl. Paper published by the Structural Concrete Laboratory of EPFL

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1 Artile publié par le Laboratoire de Constrution en Béton de l'epfl Paper published by the Strutural Conrete Laboratory of EPFL Title: Effet of load distribution and variable depth on shear resistane of slender beams without stirrups Authors: Pérez Caldentey A., Padilla P., Muttoni A., Fernández Ruiz M. Published in: olume: Pages: Country: ACI Strutural Journal. 109 pp USA Year of publiation: 2012 Type of publiation: Peer reviewed journal artile Please quote as: Pérez Caldentey A., Padilla P., Muttoni A., Fernández Ruiz M., Effet of load distribution and variable depth on shear resistane of slender beams without stirrups, ACI Strutural Journal,. 109, USA, 2012, pp [Perez12] Downloaded by infosiene ( on :18

2 ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 109-S51 Effet of Load Distribution and ariable Depth on Shear Resistane of Slender Beams without Stirrups by Alejandro Pérez Caldentey, Patriio Padilla, Aurelio Muttoni, and Miguel Fernández Ruiz The shear resistane of elements without stirrups has mainly been investigated by test setups involving simply supported beams of onstant thikness subjeted to one- or two-point loading, and most of the formulas inluded in odes have been adjusted using this experimental bakground. It is a fat, however, that most design situations involve onstant or triangular distributed loading (suh as retaining walls or footings) on tapered members. Furthermore, there seems to be few shear tests involving antilever strutures subjeted to distributed loading. These strutures, whih are ommon in everyday pratie, fail in shear near the lamped end, where the shear fores and bending moments are maximum (ontrary to simply supported beams of tests, where shear failures under distributed loading develop near the support region for large shear fores but limited bending moments). In this paper, a speifi testing program undertaken at the Polytehni University of Madrid (UPM), Madrid, Spain, in lose ollaboration with Eole Polytehnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, is presented. It was aimed at investigating the influene of load distribution and tapered beam geometris on the shear strength. The experimental program onsists of eight slender beams without stirrups. Four speimens had a onstant depth, whereas the others had variable depths (maximum depth of 600 mm [23.6 in.]). Eah speimen was tested twie: one side was tested first under point loading, and then (after repairing) the other side was tested under either uniform loading or triangular loading. The setup allowed diret omparisons between point and distributed loading. The experimental results showed a signifiant influene of the type of loading and of tapered geometries on the shear strength. On the basis of these results, and using the fundamentals of the ritial shear rak theory, a onsistent physial explanation of the observed failure modes and differenes in shear strength is provided. Also, omparisons to urrent design provisions (ACI and EC2) are disussed. Keywords: antilevers; ritial shear rak theory; load distribution; reinfored onrete; shear strength; shear tests; stirrups; variable depth. INTRODUCTION Most one-way slabs and similar members in pratie are subjeted to distributed loading. These members are usually without shear reinforement and, in many ases, their design is governed by shear. This is the ase, for instane, for retaining walls, footings, earth-overed strutures, or silos (Fig. 1(a) to (d)), as well as the support regions of some tapered slab bridges (Fig. 1(e)) and dek slabs of girder bridges (Fig. 1(f)). These members are typially antilevers or ontinuous slabs and usually have tapered geometries. This is ontrary to most researh performed on shear in members without transverse reinforement, where speimens typially orrespond to simply supported beams with onstant thikness and are subjeted to point loading. Some researh has been performed on speimens subjeted to distributed loading. 1-7 However, they mostly foused on simply supported beams of onstant thikness, where shear failures develop lose to the supports. Failures thus develop with relatively large shear fores but with rather limited bending moments, whih may not be suitable to reprodue failure zones of antilevers or ontinuous members 8 (with the potential failure setions in shear, subjeted to maximum shear and bending moments simultaneously). With respet to researh performed on tapered members without shear reinforement, 9-11 it deals mostly with speimens subjeted to point loading. This fat raises the question of whether design models not based on mehanial models and alibrated on the basis of existing tests (suh as EC2 12 or ACI ) are suitable to aount for a series of signifiant effets on shear strength, suh as: The strains (or rak widths) limiting shear strength 8 and atually developing on the shear-ritial region of antilevers (whose shear and bending moments may differ signifiantly from those of simply supported beams); The amount of shear that an be arried by the inlination of the ompression hord in tapered members; The influene of loads applied near supports. As shown in Kani et al. 14 for onentrated loads applied near supports (at a distane smaller than 2.5d to 3d), the shear strength of members without transverse reinforement inreases signifiantly (Fig. 2(a)). This is due to the fat that, for short-span beams, the ritial shear rak develops mostly without disturbing the shear-ritial region (strut arrying shear 8 ). The ase of slender beams with distributed loads is potentially different, however, as the ritial shear rak (shape and loation) is not only influened by the loads applied near the support but also by the rest of the distributed loads. These effets an signifiantly influene shear strength. Some of them (diret strut ation and inlination of ompression hord) may inrease shear strength and allow one to avoid plaing unneessary shear reinforement. Others (atual rak widths and some ases of inlined tension/ompression hords) may redue shear strength with respet to design ode preditions and potentially lead to unsafe designs. In a general manner (for members with or without transverse reinforement), the shear strength an be alulated, aounting for these ontributions, as R = + s + dir + (1) hord ACI Strutural Journal,. 109, No. 5, September-Otober MS No. S R2 reeived November 17, 2011, and reviewed under Institute publiation poliies. Copyright 2012, Amerian Conrete Institute. All rights reserved, inluding the making of opies unless permission is obtained from the opyright proprietors. Pertinent disussion inluding author s losure, if any, will be published in the July-August 2013 ACI Strutural Journal if the disussion is reeived by Marh 1, ACI Strutural Journal/September-Otober

3 Alejandro Pérez Caldentey is a Professor of strutural onrete at the Civil Engineering Shool of the Polytehni University of Madrid (UPM), Madrid, Spain, and Head of the Researh, Development and Innovation Department of FHECOR Consulting Engineers, Spain. He reeived his engineering degree and his PhD from UPM in 1989 and 1996, respetively. Patriio Padilla is a Projet Engineer at FHECOR Consulting Engineers. He reeived his ivil engineering degree from the National University of Tuumán, Argentina, and his PhD from UPM. ACI member Aurelio Muttoni is a Professor and Head of the Strutural Conrete Laboratory at the Eole Polytehnique Fédérale de Lausanne (EPFL), Switzerland. His researh interests inlude the theoretial basis of the design of reinfored onrete strutures, shear and punhing shear, fiber-reinfored high-strength onrete, soilstruture interation, and the oneptual design of bridges. He reeived the ACI Chester Paul Siess Award for Exellene in Strutural Researh in Miguel Fernández Ruiz is a Leturer and Researh Sientist at EPFL. He reeived his diploma in ivil engineering and his PhD from UPM in 2001 and 2004, respetively. His researh interests inlude the servieability behavior of strutures, bond, shear, and the modeling of strutural onrete using stress fields. where R is the total shear strength; is the shear fore arried by onrete (due to aggregate interlok or onrete tensile strength 8,15 ); s is the shear arried by transverse reinforement; dir is the fration of the loads arried by diret strut ation; and hord is the shear arried by the inlination of the hords. For members with suffiient shear reinforement, the ontribution of onrete to the shear strength an be negleted 15 ( 0) or taken into aount indiretly by using flatter angles for the struts. The rest of the shear-arrying omponents an be easily assessed in this ase on the basis of a physial model as, for instane, a stress field (Fig. 2(b)) and its orresponding free-body diagram (Fig. 2()). For uniformly distributed loading, the result is dir = q a (2) where q is the uniformly distributed load; and a dir is the length where loads are arried by diret strut ation. dir hord adir 1 M = tan( d ) 2a z adir 1 tan( d) 2z (3a) where M is the bending moment at the fae of the support; z is the inner lever arm at this setion; and d is the slope of the soffit of the member. On the basis of Eq. (3a), hord is usually approximated for the design of slender members (a dir << a, tan(d) << 1) as hord M = tan( d ) (3b) z For members without shear reinforement ( s = 0), the shear fore that an be arried by onrete ( ) depends mostly on the onrete ompressive strength and on the opening of the raks in the shear-ritial region 8 (whih an be evaluated by means of strain-based models 8 or by using empirial formulas 12,13 ). With respet to the other ontributions, most design approahes 12,13 usually provide a means of aounting for dir. In many ases, however, the alulation of hord is not learly defined (hoie of ontrol setion), or no physial model (allowing a physial understanding as the one for members with transverse reinforement shown in Fig. 2(b) and ()) is provided to do so. The authors of this paper have previously investigated the shear strength of members without transverse reinforement providing a general approah based on the ritial shear rak theory 8 (CSCT). This investigation showed the suitability and onsisteny of a strain-based model to alulate shear strength for members subjeted to point or to distributed loading. Also, the influene of onentrated loads on shear failures of tapered antilever slabs (Fig. 1(f)) was investigated on the basis of the same theory with exellent results. 16 In this study, the authors present the results of a speifi experimental program on the shear strength of Fig. 1 Examples of one-way slabs and similar members without stirrups: (a) retaining wall; (b) foundation; () ut-and-over tunnel; (d) silo; (e) tapered support region of a slab frame bridge; and (f) dek slab of box girder bridge. 596 ACI Strutural Journal/September-Otober 2012

4 slender antilevers subjeted to distributed loading (approximated by eight onentrated loads). The aim was to provide experimental data on a domain where few tests are available and develop a rational model that aounts for the influene of strains at the shear-ritial region on shear strength, the effet of diret strut ation on loads near the supports, and the ontribution of the inlined ompression hords to shear strength. The experimental results are ompared to ode provisions (EC2 12 and ACI ) and the CSCT, allowing a better explanation and understanding of the behavior of the various shear-transfer ations. RESEARCH SIGNIFICANCE Shear is the governing failure mode in many strutural members without stirrups subjeted to distributed loading, suh as retaining walls, footings, and top slabs of earthovered strutures. The design formulas of many odes of pratie, however, are based on tests of simply supported beams subjeted to point loading. In addition, most available tests have been performed on onstant-thikness speimens, whereas the depth of the members is often variable in atual strutures. For empirial models based on suh an experimental bakground, this may imply that some effets signifiantly influening shear strength (suh as the state of strains at the shear-ritial region, the diret strut ation of loads near supports, or the shear arried by the inlination of the ompression hord) may not be properly aounted for. In many ases, this leads to plaement of unneessary shear reinforement or, potentially, to unsafe designs. This study presents the results of a test series on reinfored onrete (RC) slender antilevers that allows a diret omparison between the shear strength of members subjeted to a single load or to distributed loading for both onstant-thikness or tapered beams. The experimental results are onsistently explained on the basis of the CSCT and ompared to various shear models. EXPERIMENTAL INESTIGATION Speimens Eight reinfored onrete speimens were tested under four load onfigurations (eah load onfiguration was tested using two speimens). The investigated parameters were: 1) influene of load distribution (point loading, uniform loading, and triangular loading); and 2) influene of variable depth. Figure 3 shows the geometry of the four different types tested. The speimens were named as follows: CR1/CR2: Constant depth and uniformly distributed load (Fig. 3(a)); CT1/CT2: Constant depth and triangular load (Fig. 3(b)); R1/R2: ariable depth and uniformly distributed load (Fig. 3()); and T1/T2: ariable depth and triangular load (Fig. 3(d)). To avoid bending failure prior to shear failure and to keep the reinforement ratio within reasonable limits for elements designed without shear reinforement, high-strength steel bars with a nominal harateristi yield stress of 835 MPa (121 ksi) were used as tensile reinforement. To avoid anhorage failures, steel plates were provided at both ends of the main reinforing bars (refer to Fig. 4). The reinforement of all speimens was idential, onsisting of two 26.5 mm (1.04 in.) bars on the tensile fae. On the ompression fae, two 12 mm (0.47 in.) ordinary bars onsisting of mild reinforement steel with a nominal harateristi yield stress of 500 MPa (72.5 ksi) were plaed (refer to Fig. 4). Fig. 2 Influene of slenderness and inlination of ompression hord on shear strength: (a) Kani s valley for members without transverse reinforement; (b) stress field at support region of tapered members with shear reinforement; and () free-body diagram orresponding to previous stress field. Conrete over for all speimens was 25 mm (1 in.) with an effetive depth d at the lamped setion equal to 562 mm (22.1 in.) and a reinforement ratio r (= A s /(b w d)) equal to 0.79% for all speimens. The geometry of the speimens was seleted so that the enter of gravity of the applied loads remains the same in all ases, with a onstant ratio between the resultant of the applied fores and the effetive depth (a/d) equal to Eah speimen was tested on both sides, allowing for a diret omparison of the shear strength of onentrated and distributed loading. The speimen was loaded on the long antilever side with eight fores simulating a distributed load (applied through 160 x 80 mm [6.3 x 3.2 in.] neoprene pads) and supported on a steel plate (supporting surfae equal to 210 x 250 mm [8.3 x 9.8 in.]). As a result, a point load reation developed at the short antilever (introdued through a steel plate of 200 x 250 x 20 mm [7.87 x 7.87 x 0.78 in.]). For tests under uniform load, the weight of the loading arrangement in the long antilever was 4.0 kn (0.9 kips), whereas for tests under triangular load, the weight of the loading arrangement in the long antilever was 5.9 kn (1.33 kips). All speimens developed shear failure within the short antilever in the first phase of the test. Thereafter, the failure zone was repaired using three sets of two steel plates (one plaed on the top and the other on the bottom side of the beam and tied together by means of four mild steel 12 mm [0.47 in.] bars) and the long antilever was loaded again until failure developed within the longer antilever. The distane between the eight loads on the long antilever as well as their ACI Strutural Journal/September-Otober

5 Fig. 3 Tested speimens: (a) test setup for Speimens CR (uniform/point load-onstant depth); (b) test setup for Speimens CT (triangular/point load-onstant depth); () fores and dimensions of Speimens R (uniform/point load-tapered); and (d) fores and dimensions for Speimens T (triangular/point load-tapered). (Note: Dimensions in mm [in.].) Fig. 4 Reinforement layout. (Note: Dimensions in mm [in.].) load inrement, rak development on the speimen was marked and reorded. Eah test took approximately 3 hours to be performed. Both jaks transferred the same load in tests with uniform load (CR1-2/R1-2), whereas in the tests with triangular loading, the jak losest to the entral support arried 3/4 of the load while the other jak arried the remaining fourth. Figure 3 desribes the way the load was applied for both types of loading. Fig. 5 Evolution of onrete strength with time. magnitude was hosen to simulate a onstantly distributed load or a triangular load (refer to Fig. 3). The load was introdued by means of two 500 kn (112 kip) jaks, applied on the long antilever. Load was inreased by inrements of 25 kn (5.6 kips) until 100 kn (22.5 kips) and by inrements of 10 kn (2.25 kips) thereafter. After eah Conrete properties All speimens were onreted at the same time with onrete oming from the same bath with a target 28-day speified strength of 25 MPa (3600 psi). The ement ontent of onrete was 320 kg/m 3 (539 lb/yd 3 ) with a water-ement ratio (w/) of The maximum aggregate size of onrete was 20 mm (0.79 in.). Compression tests were arried out on ylinders 150 mm (5.91 in.) in diameter and 300 mm (11.81 in.) high the day after eah test (a total of 13 onrete tests). Figure 5 plots the evolution of the onrete strength with time. Also plotted is the EN predition of strength inrease with time for the ement lass used (this urve is fitted on the basis of the measured ompressive strength and is used later in the analysis of test results to determine the ompressive strength of the speimens [Table 1]). 598 ACI Strutural Journal/September-Otober 2012

6 Table 1 Conrete age and ompressive strength at loading of tested speimens (alulated aording to EN fitted on basis of experimental values; refer to Fig. 5) Test Conrete age, days f, MPa (psi) CR (4500) CR (4870) R (5110) R (5130) T (5170) T (5200) CT (5230) CT (5240) Table 2 Shear fore at failure (due to external fores and self-weight) at fae of support R, kn (kips) Point load R,po Distributed load R,di R,di / R,po CR1 132 (29.7) 174 (39.1) 1.32 CR2 154 (34.6) 190 (42.7) 1.23 R1 148 (33.3) 236 (53.1) 1.59 R2 144 (31.4) 240 (54.0) 1.67 CT1 114 (25.6) 241 (54.2) 2.11 CT2 149 (33.5) 308 (69.2) 2.07 T1 118 (26.5) 300 (67.4) 2.54 T2 141 (31.7) 250 (56.2) 1.77 Measurements and load introdution The reations were measured by means of load ells loated at eah support (two under the entral support and two at the bak stay [short antilever]). The load applied at the jaks was reorded by the entral hydrauli pressure and was onsistent with the measured reations (differenes remaining less than 3% in all ases). The defletions of the antilever were also reorded by means of extensometers loated at midspan, antilever tip, and at the bak stay ross setion. Test results The antilever beams were tested at different ages, as shown in Table 2, performing both tests on the same speimens in the same day. All tests failed in a brittle manner by shear prior to yielding of the tensile reinforement. Table 2 summarizes the results of the tests in terms of the total shear fore ating at the fae of the support region at failure. The total shear fore aounts for self-weight of the member and of the weight of the loading arrangement (Table 2). As an be seen, the test results show a signifiant influene of the type of loading. For onstant depth members, the shear fore is approximately 27.5% higher (32% and 23%) for members subjeted to uniformly distributed loading than for members subjeted to point loading, and 109% higher (107% and 111%) for members subjeted to triangular loading, also with respet to point load results. Figure 6 shows the loation and shape of the ritial shear raks of the various tests. For tests subjeted to distributed loading, it an be noted that in some ases the ritial shear Fig. 6 Shear-ritial raks for eah beam. rak developed between the first and the seond applied loads, whereas for others it developed between the seond and the third or even third and fourth loads (influening the number of loads that ould be arried by diret strut ation). DISCUSSION OF TEST RESULTS Two important effets were investigated with the tests performed: the variable depth and the diret strut ation of loads. Diret strut ation of the load ours near the support region, where the struts arrying loads are not interepted by the shear rak, leading to failure, and thus do not onur to the shear fore that has to be arried through it (refer to Fig. 7(a)). Typially this is aounted for in design by negleting or by reduing loads near supports. 12 With respet to the effet of variable depth, its influene an be understood with the help of Fig. 7(a) and (b). Aording to the standard beam theory, and for antilevers with an inlined soffit, the shear fore resisted by the web an be redued aounting for the vertial omponent of the ompression hord fore 17 hord. This term has been experimentally aknowledged by some researhers. 9,11 It should be noted that, aording to the proposed model, it aounts only for the loads applied outside the diret support region. This is justified beause the fore on the ompression hord is not influened by diretly strutted loads (refer to Fig. 7(a)). The influene of these two effets (diret strut ation and inlination of ompression hord) on the strength of the tested beams an be learly observed. With respet to diret strut ation of the loads, it an be noted that for members with onstant thikness, the failure load signifiantly inreased when uniformly distributed or triangular loads were applied (refer to the previous setion and Table 2). This learly onfirms that loads near the support region do not onur to the shear that has to be arried by the ritial shear rak and an thus be redued or negleted. ACI Strutural Journal/September-Otober

7 where b is the width of the member; d is its effetive depth; f is the ompressive strength of onrete measured in the ylinder; w is the opening at the shear-ritial region of the rak; and d g is the maximum aggregate size. By adopting the hypothesis 8 that the width of the ritial shear rak is proportional to the deformation e at a ontrol fiber (loated at 0.6d from the tension reinforement and alulated assuming a linear-elasti behavior of onrete in ompression and no tensile strength 8 ) times the effetive depth of the member (w e d), the following expression results 8 bd bd f f 1 2 = 6 e d d g [ ] SI units: MPa, mm or 2 = 2 U.S. ustomary: psi, in. e d d g [ ] (5) Fig. 7 Shear-transfer ations in slender beam without stirrups: (a) load-transfer ation and region for alulation of diret support ontribution ( dir ); (b) setion for alulation of ontribution of ompression hord ( hord ); and () setion and loads for alulation of. With respet to the inlination of the ompression hord, speimens subjeted to a single onentrated load (at approximately 3d) showed no signifiant influene for tapered geometries. This is explained by the fat that the load was rather lose to the support and the inlined ompression zone ould only be partly ativated (justifying values of parameter a dir up to 3d). On the ontrary, for the antilevers subjeted to uniform loading, the inrease on the strength with the inlination of the ompression hord an be learly observed (an approximately 30% inrease with respet to onstant thikness members [refer to Table 2]). The members subjeted to triangular loading, however, are less influened by this fat, as the largest loads were applied near the supports and the diret strut ation was mostly governing. SHEAR STRENGTH OF MEMBERS SUBJECTED TO DISTRIBUTED LOADING ACCORDING TO CSCT The inrease in the failure load observed in the tests an be understood onsidering the priniples of the CSCT, whih are thoroughly presented in Muttoni and Fernández Ruiz. 8 Aording to this theory, failure in shear develops when a rak (originated by flexure) develops through the inlined theoretial strut arrying shear, limiting its strength and thus not allowing the member to reah its flexural apaity. The shear strength therefore depends on the width of the ritial shear rak (evaluated through the bending moments of the member), as well as on the roughness of the rak surfae (evaluated through the maximum aggregate size) Aording to this expression, the shear strength depends on the strains (opening of the raks) of the member. Thus, it allows for a onsistent aount of the different strengths between antilevers and simply supported beams subjeted to distributed loading, whose bending moments at the shearritial setions are rather different for the same level of applied shear. Appliations of this expression as well as a derivation of design formulas on its basis an be found elsewhere for members failing prior to yielding of the flexural reinforement 8 as well as for members failing after development of plasti strains in the flexural reinforement 18 (whih is pertinent for statially redundant members, suh as those shown in Fig. 1() to (f)). The approah of the CSCT an also be onsistently used for tapered members. To do so, the strain of Eq. (5) is to be evaluated aounting for the moment at the shear ontrol setion due to the loads that are not arried by diret strut ation (only loads ontributing to the opening of the ritial shear rak [refer to Fig. 7()]). With respet to the other shear-arrying ontributions ( dir and hord [refer to Eq. (1)]), they an also be evaluated on the basis of the mehanial model shown in Fig. 7(a). The shear fore arried by diret strut ation (thus not ontributing to the shear that has to be arried by the ritial shear rak) an be diretly alulated by integrating loads applied between the support and the distane a dir dir a dir = q dx (6) 0 where q refers to the distributed load on the beam. With respet to the shear arried by the inlination of the ompression hord, it an also be alulated by evaluating hord at the same setion (a distane a dir from the border of the support plate [refer to Fig. 7(b)]), where diret strut ation of the loads starts to bd = f f( wd, ) g (4) 1 hord = tan( d ) (7) z1 600 ACI Strutural Journal/September-Otober 2012 M

8 Table 3 Comparison between theoretial models and test results ACI EC2 12 CSCT 8 Test Shape Load dir / R hord / R test / al dir / R hord / R test / al dir / R hord / R test / al CR1 Prismati Point 0.0% 0.0% % 0.0% % 0.0% 0.99 CR2 Prismati Point 0.0% 0.0% % 0.0% % 0.0% 1.13 CT1 Prismati Point 0.0% 0.0% % 0.0% % 0.0% 0.81 CT2 Prismati Point 0.0% 0.0% % 0.0% % 0.0% 1.06 R1 Tapered Point 0.0% 25.4% % 25.4% % 0.0% 1.09 R2 Tapered Point 0.0% 25.4% % 25.4% % 0.0% 1.06 T1 Tapered Point 0.0% 25.4% % 25.4% % 0.0% 0.87 T2 Tapered Point 0.0% 25.4% % 25.4% % 0.0% 1.04 Point load average (CO) 0.88 (0.17) 1.01 (0.17) 1.01 (0.11) CR1 Prismati Uniform 20.3% 0.0% % 0.0% % 0.0% 0.91 CR2- Prismati Uniform 20.3% 0.0% % 0.0% % 0.0% 0.98 R1- Tapered Uniform 20.3% 24.6% % 24.6% % 8.5% 1.18 R2- Tapered Uniform 20.3% 24.6% % 24.6% % 8.5% 1.20 Uniform load average (CO) 1.14 (0.03) 1.31 (0.04) 1.07 (0.13) CT1 Prismati Triangular 25.2% 0.0% % 0.0% % 0.0% 1.18 CT2 Prismati Triangular 25.2% 0.0% % 0.0% % 0.0% 1.51 T1 Tapered Triangular 25.2% 24.9% % 24.9% % 10.3% 1.46 T Tapered Triangular 25.2% 24.9% % 24.9% % 10.3% 1.21 Triangular load average (CO) 1.40 (0.16) 1.65 (0.17) 1.34 (0.12) All tests presented in this paper average (CO) 1.07 (0.25) 1.25 (0.26) 1.11 (0.17) 3 Prismati Point 0.0% 0.0% % 0.0% % 0.0% Tapered Point 0.0% 35.3% % 35.3% % 11.8% Tapered Point 0.0% 42.2% % 42.2% % 10.9% Tapered Point 0.0% 56.5% % 56.5% % 23.2% R Tapered Point 0.0% 56.6% % 56.6% % 23.3% Tapered Point 0.0% 77.8% % 77.8% % 41.5% 1.03 All tests (0.13) 0.77 (0.13) 0.95 (0.06) All series 1.09 (0.22) 1.12 (0.31) 1.06 (0.17) Prismati beams 1.18 (0.23) 1.30 (0.26) 1.06 (0.19) Tapered beams 1.02 (0.20) 0.99 (0.32) 1.07 (0.15) where z 1 and M 1 are the lever arm and ating bending moment at the ontrol setion where diret strut ation starts to develop (Fig. 7(b)), respetively. This is justified beause the loads between the edge of the ontrol setion and the distane a dir do not ontribute to inreasing the fore in the ompression hord (refer to Fig. 7(a)). It an be noted that Eq. (7) (derived on the basis of the mehanial model in Fig. 7(a) for members without shear reinforement) differs from Eq. (3a) (derived on the basis of the mehanial model of Fig. 2(b) for members with shear reinforement) in the setion at whih the moment and lever arm are to be evaluated as well as on the influene between the a dir -to-a ratio. SHEAR STRENGTH PREDICTIONS ACCORDING TO ACI , EC2, AND CSCT Comparisons of tests with design odes (ACI , EC2) and with the ritial shear rak theory are given in Table 3 for the tests presented in this paper as well as similar tests from the literature. 10 In this table, the perentage of the total shear fore arried by diret strut ation and by the inlination of the ompression hord are also given. All models onsider a redution of the loads applied near supports (diret strut ation). With respet to EC2, loads are redued between 0 and 2d to aount for diret strut ation aording to the load redution oeffiient x shown in Fig. 8. The ACI approah presents a stepwise funtion negleting (in terms of ating shear) all loads within the fae of the support and a distane equal to d (refer to ACI , Setion R ). With respet to the CSCT, and aording to its priniples, all loads between the support and the distane at whih the ritial shear rak interepts the flexural reinforement an, in priniple, be negleted (x = 0) for the alulation of the ating shear fore at the ritial setion. From the test results, it has been observed that the ACI Strutural Journal/September-Otober

9 Fig. 8 Load redution fator x aounting for diret strut ation aording to EC2, ACI , and proposed law for CSCT for slender members. value of parameter a dir depends on the flexural reinforement ratio and the angle of the tapered beam. For analysis or design purposes, however, adopting a onstant value for this length (a dir = 2.75d) is suffiiently aurate (Fig. 8). It should be noted that this approah is valid for slender beams (a/d > 2.5 to 3) with only a fration of the load applied near the support. With respet to the shear arried by the ompression hord, it an be alulated on the basis of Fig. 7(b). For the CSCT, the value a dir = 2.75d is again adopted. In addition, the fore in the ompression hord is alulated assuming that the onrete behaves linearly in ompression and arries no stress in tension (in aordane to CSCT hypotheses 8 ). For ACI and EC2, this ontribution has been alulated (in bending and shear) for the setion at the fae of the support (where the tip of the ritial shear rak is loated over the ompression zone [Fig. 6], indiating that the ompression hord an arry a fration of the shear fore due to its inlination). The results of Table 3 show that all models provide relatively good estimates of the strength for failures under point loading of prismati beams. For tapered beams under point load, however, ACI and EC2 overestimate the strength. This is mostly due to the hoie of the alulation of hord at the setion at the fae of the support (with a minimum value of measured-to-alulated strength ratio of 0.66 for ACI and 0.76 for EC2). For speimens subjeted to uniform or triangular loading, the preditions are somewhat oarse, espeially for EC2 and ACI Calulations for ACI were performed using Eq. (11-3) of that ode. It should be noted that Eq. (11-5) provides very similar results, with differenes smaller than 4%. The CSCT provides the best results, only slightly underestimating the strength of tapered members subjeted to uniform and triangular loading. CONCLUSIONS This study presents the results of an experimental investigation on the shear strength of reinfored onrete slender antilevers with prismati and tapered shapes, subjeted to point loading, uniform loading, and triangular loading. Based on the results of this experimental investigation, the following onlusions an be drawn: 1. There is a signifiant effet of the type of loading on the shear apaity of slender RC members without shear reinforement. 2. The behavior and shear strength of antilevers are very different from that of simply supported beams subjeted to distributed loading due to the fat that the strains and rak width at the shear ritial regions are very different (maximum shear and bending moments in antilevers, maximum shear but limited bending moments in simply supported beams). This fat, however, is not aknowledged by many odes of pratie. 3. For the onstant-depth antilevers tested, the same elements arried 27% more load for uniformly distributed loading than for point loading, and more than 100% for triangular loading than for point loading. In the ase of variable depth, the inrease on the load that an be arried is 63% for distributed loading and, again, more than 100% for triangular loading. 4. Regarding the positive effet of variable-depth members (inlination of ompression hord arrying shear), this phenomenon plays a signifiant role in members without shear reinforement only for loads applied at a ertain distane to the support (approximately 2.5 to 3 times the effetive depth of the member). 5. A omparison of test results to design approahes (ACI , EC2, and the CSCT) shows that best preditions are obtained using the CSCT. This theory onsistently aounts for the various shear-transfer ations (diret strut ation and inlination of ompression/tension hords) and for the strains developing in the shear-ritial region. The signifiane of this topi in strutural onrete enourages further investigations that provide more experimental results (ompleting those of this paper) to improve and refine the theoretial and mehanial models disussed in this study. ACKNOWLEDGMENTS The authors wish to express their gratitude and sinere appreiation to the following people, ompanies, and offiial entities that have made this researh possible: The Spanish Ministry of Eduation for funding this program as part of researh projet BIA ; J. Sánhez (DRAGADOS) for providing the formwork for the beams; F. Martínez (DYWIDAG) for providing the high-strength bars; H. Ortega (CELSA) for providing the mild reinforement; F. Garía (HYMPSA) for providing the onrete mixture; H. Corres (UPM) for his help and enouragement; J. Torrio (UPM) for preparing the test setup; and S. Lips (EPFL) for drawing the figures. NOTATION a = shear span (for distributed load, distane between support, and resultant of loads) a dir = distane of diret support region b = width of member d = effetive depth d 2 = effetive depth at shear-resisting ontrol setion d g = maximum aggregate size f = ompressive strength of onrete measured in ylinder M = bending moment at fae of support M 1 = bending moment at setion where diret support starts to develop q = distributed load = shear fore at fae of support of speimen (aounting for external loads, self-weight, and weight of loading arrangement) = shear fore arried by onrete al = alulated shear strength hord = shear fore arried by inlination of ompression hord dir = shear fore arried by diret strut ation R = shear strength R,di = shear strength of speimen subjeted to distributed loading R,po = shear strength of speimen subjeted to point loading s = shear fore arried by transverse reinforement test = measured shear strength w = opening at shear-ritial region of rak x = oordinate z = inner lever arm at fae of support z 1 = inner lever arm at setion where diret support starts to develop d = inlination of soffit of antilever e = strain at ontrol depth r = longitudinal reinforement ratio x = load redution fator aounting for diret strut ation 602 ACI Strutural Journal/September-Otober 2012

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