Design of reinforced concrete deep beamsM. D. KOtSOVOS, Dipl(Eng), PhD, CEng, DIC, DSc, MIStructE Imperial College of Science 8t Technology deepbeam at its ultimate limitstateg. The cause for suchbehaviour appears to be associated with the occurrence of inclined cracking; as the beam deflects under increasing load, the faces of an inclined crack rotate Synopsis The work formspart of a comprehensive investigation thefundamental into causes of shearfailure and the development of simple and rational design methods. A previous part of the work has shown that the shear capacity of reinforced concrete (RC) beams is associated withthe strength concrete of in the region of the pathalong which the compressiveforce is transmitted of to the supportsand not, m widely considered, the region the beam below the neutral axis. This concept is used as the basis to formulate a simple design method f o r deep RC beams. The method involves the modelling of a deep RC beam as a ‘tiedframe with inclined legs’, and it is shown that it yields realisticpredictions of loadcarrying capacity both simply for supported and continuous deep beamsf o r the range of loading conditions likely to be considered in practice. Introduction Current Code (e.g. ref. l ) provisions for thedesign of reinforced concrete (RC) deep beams are essentially those for long beams, modified so as to allow for uncertainties arising because thelack of a widely accepted theory of describing deep beam behaviour. Such a theory, which is essential for the development of a simple and rational design method, is unlikely to be developed by using concepts similar those which form thebasis of current to design provisions for long beams. This because such concepts have been is shown to be incompatible withobserved RCbehaviour atboththe material213 and the structure4-6 levels. It appears, therefore, that current methods for thedesign of deep beams are unsatisfactory not only because of the lack of a soundunderlying theory but also because they include most of the unsatisfactoryelements of the methods currently used for thedesign of long beams. It has been recently demonstrated by analysis7.* and verified by experiment”6 thatthe loadcarryingcapacity of aRC long beam is associated with the strength of concrete in the region of the path along which the compressive force is transmitted to thesupports. It is considered that, if this ‘compressive force path’ concept does indeed describe the behaviour of a long beam, it should even more so describe the behaviour of a deep beam. Theuse of the above concept for the description deep of beam behaviour is discussed in the following, and an attempt presented is to develop a simple design method compatible with this concept. validity The of the method is tested by comparing the predicted behaviour with that established by experiment in previous investigationsg-13 which have yielded most of the information published to date on deep beam behaviour. Compressive force path It has been suggested that, for a RC beam at its ultimate limit state, the compressive force at the mid-cross section is transmitted to the support by following a path which, for any practical purpose, may be considered to be bilinear (see Figl)14. For a beam subjected to two-point loading with a shear span-to-depth (a/d) ratio greater than a value of approximately 2-0, the change in the path direction appears to occur at a distance of approximately twice the beam depth,d , (see Fig l(a)), whereas for smaller ratios the change inpath direction is considered occur at the cross-section to beam is, by including theloadpoint (see Fig l(b))14. Since a deep definition, a beam with a span-to-depth (1/d)ratio smaller than a value of approximately 2.0, it should be characterised by a compressive force path similar to that of a beam with a/d <2.0 (see Fig 2). There is experimental evidence indicating that the force sustained by the tensile reinforcement within the length of the horizontal projectionof the inclined portion of the pathis constant andthis impliesthat thecompressive force acting along theinclined portion of the path is also constant6. This is in agreement with experimental work that has shown that the measured strain of the reinforcement is indeed constant throughout the span of a a >2d I l I 2d 4 c Lr’ --/- - - - I a12d Path Outline of compressive stress trajectories Fig 1. Schematic representation of path of compressive force and corresponding outline of compressive stress trajectories RC beams with for various a/d ratios Path Outline of compressive stress trajectories -%-h---- l 1 1 l T I L 2d C T - 1 Fig 2. Schematicrepresentation of path of compressive force and fo corresponding outline of compressive stress trajectories r a typical RC deep beam 28 The Structural EngineerlVolume 661No.2119 January 1988 it is considered unlikely that the presence of conventional web reinforcement in the form of vertical stirrups or horizontal bars considerably improves the strength of the inclined strut. they give rise to tensile stresses in the orthogonal direction (see Figs 1 and 2). the failure of RC beams with a/d<l is associated with the stress conditions in regions away from the location where the path changes directiond. It should be noted that a precise description of the shape of the idealised path of the compressive force in the region wherethe path changes direction is not deemed essentialfor thecase of deep beams (a/d<l). Although the compressive force carried along the path at a particular location may be easily assessed. Typical inclinedcompression failures of RC deep beams under (a) uniform and (b) two-point loading As shownlater. effect on the loadcarrying capacity of deep beams.and two-point loading. However. with a/d between 1. i. This is because. The compressive stress trajectories along the path are such that. Hence. with the intersection of the directions of the reaction and the tensile reinforcement. which indicatesthat the presence of the web reinforcement has little9112. The stress ‘flow’ is considered to have a rectangular cross-section with a width equal to the beam width. A suitable depth for theinclined stress ‘flow’ is considered to be a/3. \ \ \\\\ \ \ as a ‘flow’ of compressive stresses with varying section perpendicular to the path direction and with the compressive force representing the stress resultant at each section. the use of nominal web reinforcement is considered essential not The Structural EngineerlVolume 66/No. /-\ /’ / I /- /- I -\ l \ \ / ‘ 1 l I t t Fig 4. In this model. however. a/3 should be substituted by the width of the bearing. Because of the large compressive forces carried by deep beams. a simple method is required. as recommended elsewhere15.Paper: Kotsovos \ Fig 3. The use of such methods in design is.e. respectively. cross-section of the ‘flow’ should be chosen such that the actual maximum compressive force carried along the path remains unchanged. since the above information regarding the stress field in the region of the compressive force path is essential for assessing the maximum force that can be carried along the path. however. the distance between the applied load and the reaction closest to it. A similar model has been proposed for slender beams. if any”. and this may cause only small increase in loadcarrying capacity a the of the inclined strut. The shape of the stress ‘flow’ and the intensity of the stress field are very much dependent on the beam boundary conditions. As indicated in Fig 5 . Assuming that anchorage failure. It should be noted. that. the stress flow may have a shapesimilar to thatindicated in Fig 4(a).2/19 January 1988 29 . As an alternative. such as to satisfy the static equilibrium conditions. the compressive force path for a two-point loading may also be valid for the case of a uniform load if the equivalent two-point load is applied at the third points. This view is supported by most experimental evidence published to date. the inclined stress ‘flow’ of the path is symmetrical with respect to the line connecting the intersection of the directions of the applied load and the horizontal path of the compressive force. can be prevented. Such reinforcement may delay cracking the process.g. where a is the shear span. Based on this reasoning.0 and 2-5. as indicated in Fig 36. Fig 5 shows two such simplified compressive force paths for the case of a deep beam subjected to single. the shape of the stress ‘flow’and the intensity of the stresses are difficult to establish without resorting to sophisticated methods of analysis (e. andthis rotation causes separation of the tensile steel bars from concrete. The depth of the horizontal portion of the stress ‘flow’ of the path may be assessed such that the compressive force equals the force sustained by the tensile reinforcement. and forms part of recommendations for fire resistant design of RC structures15. on the basis of the concept of the compressive force path. Deep beam model Figs l and 2 indicate that thepath of the compressive force may be visualised \. Schematic representation of bond failure due to opening of inclined crack away from each other with respect to the crack tip. This combined compression-tensionstress state leads to the formationof an inclined crack at a location where the concrete strength is exceeded. collapse eventually occurs when the compressive force exceeds the force that can be carried along the path. for equilibrium purposes. prohibitive not only because of their high cost but also because they are not widely available and their use depends on expert advice. finite element analysis). the web reinforcement is assessed such as to sustain the total compressive force if the loadcarrying capacity of the inclined concrete strut is exceeded before the flexural capacity of the beam is reached. for a deep beam. For a simply supported deep beam subjected to a load uniformly distributed on its top face. it is realistic consider that thedifference in shape between to the above ‘flow’ and that caused by an equivalent load concentrated at the two third points affects only the location of failure initiation within a particular portion (inclined or horizontal) of the path and not the magnitude of the force that can be carried along this portion (see Fig 4). If a/3 is smaller than the effective width of the bearing. As discussed in the preceding paragraph. the above recommendations propose the use of nominal web reinforcement combined with an increase of the width of the beam such as to increase the loadcarrying capacity of the inclined strut and thus prevent inclined compression failure. as well as failure of the tensile reinforcement. it is considered realistic for design purposes to replace the actual stress ‘flow’ with uniform stress ‘flow’ of intensity a equal The to the uniaxial cylinder compressive strength VC). the occurrence of such a crack within the inclined portion of the pathdoes not interrupt the flow of the compressive stresses and thus the beam is capable of sustaining a significantly higher load. may lead to steel congestion and. (b) Considering that thetension reinforcementyields before the loadcarrying capacity of the horizontal portion of the stress ‘flow’ is attained. the external load carried by the ‘flow’ to the support. however. assess the amount of tension reinforcement required to satisfy the equilibrium condition of the horizontal internal actions. A typical procedure for the case of twopoint loading (see Fig 5(b)) may be formulated as follows (see Fig 6): (a) Assuming the beam depth and width. should bethe sizing of the abovemembers such as to sustain these actions. This load can be easily transferred to the top face of the beam by using stirrups designed so as to withstand the loading as indicated in Fig 816. which can be achieved by following recommendationsof current Code the provisions for deep beam design. I 1 t Fig 7. If the above condition cannot be satisfied with the given values of d and b. adjust b and repeat ( b) = P Fig 6. Design method The conceptsdescribed in the preceding section indicate that a deep beam will withstand the actionof an applied load if the resulting internal actions can be safely sustained by members ofthe proposed model. d assess the depth of the horizontal portion of thestress ‘flow’ by satisfying the moment equilibrium condition with respect to the intersection of the directions of the reaction and the tension reinforcement. Schematic representation of members of proposed model designed as %olumns’ 30 The Structural EngineerNolume 66/No. inclined compression members ofthe model as ‘columns’ (see Fig Such an approach. or equal to. a 4 / \ C (a) Moment equilibrium Cz = Pa yields x (b) Horizontal force equilibrium T = C yields A . adjust d and b accordingly.If not. are given. respectively.2/19 January 1988 . It may be interesting to note that the proposed design method may be easily extended to apply for deep beams subjected to loading applied to their bottom face. therefore. Proposed models for deep beams under (a) single-point and (b) twopoint and/or uniform loading only for crack control purposes but also because it reduces the likelihood of‘instability’ failures due to the out-of-planeactions related to the heterogeneous nature of concrete.Paper: Kotsovos (c) Check whetherthe vertical component of the compressive force carried by the inclined portion of the stress ‘flow’ isgreater than. The abovedesign procedure should be complemented by good detailing. and 6 . The objective the of a design procedure. adjust thebeam width. It should be noted that an alternative to adjusting the cross-sectional dimensions d and b of a deep beam may be designing the horizontal and 7). (c) Check whether a / 3 satisfies vertical force equilibrium C p sin p If not. may be followed only in circumstances that require fixed values of d and b . and repeat the process. Proposed method for designing a RC deep beam Fig 5. thus. b . kN Fig 9. Theslight overestimate of loadcarrying capacity in certain cases is due to the larger variability the results obtained for the of beams without web reinforcement.300 U c 0. Placing nominal web reinforcement considerably reduces the variability. As indicated in the figures. 10. kN 600 o Inclinedcompression failure + 500 6 Flexural failure -90% confidence limits 400 / I 5 L g.2119 January 1988 31 . No distinction has been drawn between beams with and without web reinforcement since. the results obtained in a recently published work on continuous deep beams13 have also been included. single-point. 100 (b) 200 300 400 500 Experimentalload. with the predicted values always being on the safe side. g L Q 200 100 Inclinedcompression failure Flexural failure 1-4 90% confidence limits for means o I I I I + I 600 100 (c) 200 300 400 500 Experimental load. The investigationcovers a wide range of loading conditions including uniform. the above moment will be the first to reach its ultimatevalue. the effectof such reinforcement on loadcarrying capacity appears to be insignificantg911J2.- 2 200 2 100 a Fig 8. Fora continuous beam with a uniform flexural capacity throughout its length. Fig 10 indicates an equally good correlation between predicted and experimental values for the case of deep beams subjected to uniform and single-pointloading. Schematic representation method of transfer of load from bottom of to top face of deepbeam Verification of design method The above procedure has been used to assess the loadcarrying capacity of a large number of deep beams whose behaviour already been established has by experiment elsewhereg-13. kN Simply supported deep beams Fig 9 indicates a sufficientlyclose correlation for practical purposes between predicted and experimental values for the case of deep beams subjected to two-point loading. However. Correlation of predicted loadcarrying capacity of RC deep beams under two-point loading with experimental values reported in refs. Thecorrelations between predicted and measured values are shown in Figs 9. the observed modes of failure appearto be in agreement with the predictions. and 12. and two-point loading. although thoseof the observed modes of failure characterisedby inclined cracking are usually reported in the literature as ‘shear’ failures. as discussed earlier. Continuous deep beams The loadcarrying capacity of the continuous RC deep beams maybe calculated by assuming that the indeterminate bending moment of the internalsupport is equal tothat obtained by elastic analysis. an under-reinforced beam should behave in a ductile manner in 100 300 200 4 Experimental load. it should be noted that the measured values of the beams without web reinforcement exhibit a significantly larger variability.Paper: Kotsovos /I I\ 500 7 400 Y 300 U d 0 0 U U V . For the latter case. the predicted modes of failure areclassified into two types: (a) those characterisedby failure of the inclined concrete member of the model (inclined compression failure) and (b) those characterised by failure of the horizontal concrete member of the model (flexural failure). In general. the beams considered are simply supportedg-12. (a) 9 and 12. however. (b) IO. In most cases. yielding of the tension steel is assumed to have always preceded collapse for the cases considered. When this occurs. and thepredicted values appear always to be on the safe side. and (c) I 1 The Structural EngineerNolume 661No. Bethlehem. forms a suitable basis for a design procedure. M. M. 13 References for 1. kN Fig 12. and Ananthanarayana. Journal. 6. therefore. idealising the path as a frame with inclinedlegs simplifies the design procedurefor deep beams into a procedure for designing the members of the frame such as to (a) suatain a given load.6 2 3 14. 81.: ‘Fracture mechanics and concrete behaviour’. Kotsovos. pp103-112 “shear” failure’. pp99-106 5 .M. CEB/FIP Model code concrete structures. M. pp32-38 J. D.5’. 79. RILEM. 1.: ‘Fracture processes of concrete under generalised Materials and Structures..6 January 1986. A. V. 65B. 35. Since failure is considered to occur due to the loss of loadcarrying capacity of concrete in the region of the path along which the compressive force is transmitted to the supports.D. July-August 1986. ACI Journal. g W 200 U W . ASCE. andOng. 72. No. . andEibl. Design and detailing concrete structures of for fire resistance. No. Interim Guidance by aJointCommittee of theInstitution of Structural Engineers and the Concrete Society. On the basis ofthe above. October 1965. Ramakrishnan.:‘Mechanismsof Concrete Research. pp1026-1034 9.: ‘Ultimate strength of ACI 65. May-June 1984. Y. Proceedings..: ‘Strength and behaviour of deep beams in shear’. Councilon Tall Buildings.. and Siess. Proposed model for continuous RC deep beam + 90 ‘lo confidence limits for means 500 1000 Experimental Load.87-98 11. pp19-41 10.. D. F. Kotsovos. Kotsovos. and Vantsiotis. Heft 178 32 The Structural EngineerlVolume 66/No. April 1979 16. P. ACI Journal. Magazine of Concrete Research. F. G. and Cole.M. pp431-437 3.and Newman. Bobrowski. 1. a I a13 Fig 11.- W U L 100 + + + / ” I I I 100 200 300 G( Experimental load. the close correlation between predicted and experimental values shown in Fig 12 is obtained. ACI Journal. Engineering Structures. Lehigh University. Kotsovos. London. English ed. Magazine of. B. Kotsovos. S.5’. S. with an adequate margin of safety. Suchbehaviour allows load redistribution and the ultimate limit state is reached when the flexural capacity at another section away from the supports is attained. Rogowski. C. as indicated in Fig 11. andWalther. 67.: ‘Behaviour of reinforced shear-span-to-depthratio between 1. The Structural Engineer. Kotsovos.. No. Correlation of predicted loadcarrying capacity of continuous and simply-supported RC deep beams under single-point loading with experimental values reported in ref.0and 2. pp83-115 15. K. 9. No. June 1981. 4. November-December 1986. D. Proceedings. 4.: ‘Web reinforcement effects on deepbeams’. P. J. 3. The apparent good correlation between predicted experimental values and is attributed to the validity of the concepts that form the basis of the proposed model. USA. No.: ‘Shear strength of deep beams’. deep beams in shear’. 123. 1968 pp.ACI Journal. ACI Journal. Leonhardt. K .. pp279-286 8. Using this modelto predict the loadcarrying capacity of continuous deep beams tested elsewherel’. 12. Smith. H.: ‘Shear considerations’. D. N. Kotsovos.M. June 1983. kN Concluding remarks The work presented indicates that modelling a deep beam as a tied frame with inclined legs appears to yield realistic predictions of loadcarrying capacity and. December 1970. No. M. Y.: Wandartige Trager.. Proc. No. No. MacGregor. ~ ~ 6 1 4 .: ‘Shear failure ofreinforced concrete beams’. ST5. J.: ‘Testsof reinforced concrete deep beams’. 6.12. D. No. J.Council on Tall Buildings& Urban Habitat. R. Proceedings. ppl-l0 concrete beams with a 7. 91. and (b) prevent any undesirable-because of its sudden and catastrophic occurrence-brittle failure. A. Kong. 3. 33. stress states’. Pa.: ‘Behaviour of reinforced concrete beams with shearspan-to-depth ratios greater than 2. D. Journal of the Structural Division. Rawdon de Paiva. January 1987. No. M. 115. No. F. D. Committee 21D.: ‘Behaviour of reinforced concrete T-beams in shear.. Proceedings. M. the model proposedfor simply supported deep beams can easily be extended to describe the ultimate limit state of a continuous deep beam. Proceedings. pp1010-1017 12. November-December 1979..2/19 January 1988 .D.. Proceedings. Cement & Concrete Association. workshop. Third International Con$ on Tall Buildings. May-June 1982. Robins. Kotsovos. 2.Paper: Kotsovos 400 o X+ Inclined compression failure Flexural failure Uniform load +o 300 Y z U d the region of the support. pp201-213 13.84. February.. March 1987. Proc. No.. 1978 2. D. Deutscher Ausschuss fur Stahlbeton. J.83.