Design of reinforced concrete deep beams M. D. KOtSOVOS, Dipl(Eng), DIC, PhD, DSc, CEng, MIStructE Imperial College of Science 8t Technology

Synopsis The work formspart of a comprehensive investigation into the fundamental causes of shear failure 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 with the strength of concrete in the region of the path along which the compressive force is transmitted to the supports and not, m widely considered, the region of the beam below the neutral axis. This concept is used as the basis to formulate a simple design method for deep RC beams. The method involves the modelling of a deep RC beam as a ‘tied frame with inclined legs’, and it is shown that it yields realistic predictions of loadcarrying capacity for both simply supported and continuous deep beams for the range of loading conditions likely to be considered in practice.

Introduction Current Code (e.g. ref. l ) provisions for the design of reinforced concrete (RC) deep beams are essentially those for long beams, modified so as to allow for uncertainties arising because of the lack of a widely accepted theory 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 to those which form the basis of current design provisions for long beams. This is because such concepts have been shown to be incompatible with observed RC behaviour at both the material213 and the structure4-6 levels. It appears, therefore, that current methods for the design of deep beams are unsatisfactory not only because of the lack of a sound underlying theory but also because they include most of the unsatisfactory elements of the methods currently used for the design of long beams.

It has been recently demonstrated by analysis7.* and verified by experiment”6 that the loadcarrying capacity of a RC long beam is associated with the strength of concrete in the region of the path along which the compressive force is transmitted to the supports. 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. The use of the above concept for the description of deep beam behaviour is discussed in the following, and an attempt is presented to develop a simple design method compatible with this concept. The validity 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 Fig l)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 in path direction is considered to occur at the cross-section including the load point (see Fig l(b))14. Since a deep beam is, by 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 projection of the inclined portion of the path is constant and this implies that the compressive force acting along the inclined 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

deep beam at its ultimate limit stateg. The cause for such behaviour appears to be associated with the occurrence of inclined cracking; as the beam deflects under increasing load, the faces of an inclined crack rotate

a >2d I l

2d 4 c

I

Lr’

--- /----

Pat h

Outline of compressive stress trajectories

I a12d

Fig 1. Schematic representation of path of compressive force and corresponding outline of compressive stress trajectories for RC beams with various a/d ratios

-%-- Path h---- Outline of compressive

stress trajectories

l l 1 1

T LC 2d T I - 1

Fig 2. Schematic representation of path of compressive force and corresponding outline of compressive stress trajectories fo r a typical RC deep beam

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Paper: Kotsovos

Fig 3. Schematic representation of bond failure due to opening of inclined crack

away from each other with respect to the crack tip, and this rotation causes separation of the tensile steel bars from concrete, as indicated in Fig 36.

The compressive stress trajectories along the path are such that, for equilibrium purposes, they give rise to tensile stresses in the orthogonal direction (see Figs 1 and 2). This combined compression-tension stress state leads to the formation of an inclined crack at a location where the concrete strength is exceeded. It should be noted, however, that, for a deep beam, the occurrence of such a crack within the inclined portion of the path does not interrupt the flow of the compressive stresses and thus the beam is capable of sustaining a significantly higher load. Assuming that anchorage failure, as well as failure of the tensile reinforcement, can be prevented, collapse eventually occurs when the compressive force exceeds the force that can be carried along the path.

Deep beam model Figs l and 2 indicate that the path of the compressive force may be visualised 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. Although the compressive force carried along the path at a particular location may be easily assessed, such as to satisfy the static equilibrium conditions, the shape of the stress ‘flow’ and the intensity of the stresses are difficult to establish without resorting to sophisticated methods of analysis (e.g. finite element analysis). The use of such methods in design is, however, prohibitive not only because of their high cost but also because they are not widely available and their use depends on expert advice. Hence, a simple method is required, 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.

The shape of the stress ‘flow’ and the intensity of the stress field are very much dependent on the beam boundary conditions. For a simply supported deep beam subjected to a load uniformly distributed on its top face, the stress flow may have a shape similar to that indicated in Fig 4(a).

/-\ /- I -\

/- l \ /’ \

/

/ I 1 ‘ l I

t t Fig 4. Typical inclined compression failures of RC deep beams under (a) uniform and (b) two-point loading

\ \ \, \ \ \ \ \

\ \

As shown later, it is realistic to consider that the difference in shape between 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). Based on this reasoning, it is considered realistic for design purposes to replace the actual stress ‘flow’ with a uniform stress ‘flow’ of intensity equal to the uniaxial cylinder compressive strength VC). The cross-section of the ‘flow’ should be chosen such that the actual maximum compressive force carried along the path remains unchanged.

Fig 5 shows two such simplified compressive force paths for the case of a deep beam subjected to single- and two-point loading, respectively. As discussed in the preceding paragraph, 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. The stress ‘flow’ is considered to have a rectangular cross-section with a width equal to the beam width. 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. As indicated in Fig 5 , 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, with the intersection of the directions of the reaction and the tensile reinforcement. A suitable depth for the inclined stress ‘flow’ is considered to be a/3, where a is the shear span, i.e. the distance between the applied load and the reaction closest to it. If a/3 is smaller than the effective width of the bearing, a/3 should be substituted by the width of the bearing, as recommended elsewhere15.

It should be noted that a precise description of the shape of the idealised path of the compressive force in the region where the path changes direction is not deemed essential for the case of deep beams (a/d<l). This is because, on the basis of the concept of the compressive force path, 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.

A similar model has been proposed for slender beams, with a/d between 1.0 and 2-5, and forms part of recommendations for fire resistant design of RC structures15. In this model, 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. As an alternative, 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.

Because of the large compressive forces carried by deep beams, 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. Such reinforcement may delay the cracking process, and this may cause only a small increase in the loadcarrying capacity of the inclined strut. This view is supported by most experimental evidence published to date, which indicates that the presence of the web reinforcement has little9112, if any”, effect on the loadcarrying capacity of deep beams. However, the use of nominal web reinforcement is considered essential not

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a 4

/ C \

( b) Fig 5. Proposed models for deep beams under (a) single-point and (b) two- point and/or uniform loading

only for crack control purposes but also because it reduces the likelihood of ‘instability’ failures due to the out-of-plane actions related to the heterogeneous nature of concrete.

Design method The concepts described in the preceding section indicate that a deep beam will withstand the action of an applied load if the resulting internal actions can be safely sustained by the members of the proposed model. The objective of a design procedure, therefore, should be the sizing of the above members such as to sustain these actions. A typical procedure for the case of two- point loading (see Fig 5(b)) may be formulated as follows (see Fig 6): (a) Assuming the beam depth and width, d and 6 , respectively, are given, assess the depth of the horizontal portion of the stress ‘flow’ by satisfying the moment equilibrium condition with respect to the intersection of the directions of the reaction and the tension reinforcement. If the above condition cannot be satisfied with the given values of d and b, adjust d and b accordingly. (b) Considering that the tension reinforcement yields before the loadcarrying capacity of the horizontal portion of the stress ‘flow’ is attained, assess the amount of tension reinforcement required to satisfy the equilibrium condition of the horizontal internal actions.

(c) Check whether the vertical component of the compressive force carried by the inclined portion of the stress ‘flow’ is greater than, or equal to, the external load carried by the ‘flow’ to the support. If not, adjust the beam width, b, and repeat the process.

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 inclined compression members of the model as ‘columns’ (see Fig 7). Such an approach, however, may lead to steel congestion and, thus, may be followed only in circumstances that require fixed values of d and b.

The above design procedure should be complemented by good detailing, which can be achieved by following the recommendations of current Code provisions for deep beam design.

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. 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.

(a) Moment equilibrium Cz = Pa yields x

(b) Horizontal force equilibrium T = C yields A ,

(c) Check whether a/3 satisfies vertical force equilibrium C p sin p = P

If not, adjust b and repeat

Fig 6. Proposed method for designing a RC deep beam

I 1

t Fig 7. Schematic representation of members of proposed model designed as %olumns’

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/ I I \

Fig 8. Schematic representation of method of transfer of load from bottom to top face of deep beam

Verification of design method The above procedure has been used to assess the loadcarrying capacity of a large number of deep beams whose behaviour has already been established by experiment elsewhereg-13. The correlations between predicted and measured values are shown in Figs 9, 10, and 12. The investigation covers a wide range of loading conditions including uniform, single-point, and two-point loading. In most cases, the beams considered are simply supportedg-12; however, the results obtained in a recently published work on continuous deep beams13 have also been included.

No distinction has been drawn between beams with and without web reinforcement since, as discussed earlier, the effect of such reinforcement on loadcarrying capacity appears to be insignificantg911J2. However, it should be noted that the measured values of the beams without web reinforcement exhibit a significantly larger variability.

As indicated in the figures, the predicted modes of failure are classified into two types: (a) those characterised by 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). For the latter case, yielding of the tension steel is assumed to have always preceded collapse for the cases considered. In general, the observed modes of failure appear to be in agreement with the predictions, although those of the observed modes of failure characterised by inclined cracking are usually reported in the literature as ‘shear’ failures.

o Inclined compression failure + Flexural failure 1-4 90% confidence limits for means

100 200 300 4 Experimental load, kN

Fig 9. Correlation of predicted loadcarrying capacity of RC deep beams under two-point loading with experimental values reported in refs. (a) 9 and 12, (b) IO, and (c) I1

500

400

Y 300

U 0 0

U

V U

d

2 200

a 2 .-

100

(b)

600

500

400

5 g- 300 L

U 0, c

g 200 Q L

100

(c)

1 0 0 200 300 400

7

500 Experimental load, kN

o Inclined compression failure + Flexural failure 6 -90% confidence limits /

I I I I I I 100 200 300 400 500 600

Experimental load, kN

Simply supported deep beams Fig 9 indicates a sufficiently close correlation for practical purposes between predicted and experimental values for the case of deep beams subjected to two-point loading. The slight overestimate of loadcarrying capacity in certain cases is due to the larger variability of the results obtained for the beams without web reinforcement. Placing nominal web reinforcement considerably reduces the variability, and the predicted values appear always to be on the safe side. Fig 10 indicates an equally good correlation between predicted and experimental values for the case of deep beams subjected to uniform and single-point loading, with the predicted values always being on the safe side.

Continuous deep beams The loadcarrying capacity of the continuous RC deep beams may be calculated by assuming that the indeterminate bending moment of the internal support is equal to that obtained by elastic analysis. For a continuous beam with a uniform flexural capacity throughout its length, the above moment will be the first to reach its ultimate value. When this occurs, an under-reinforced beam should behave in a ductile manner in

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400

300

z Y

U g 200 d

U W W U

W .-

100 L

o Inclined compression failure X + Flexural failure

+o Uniform load

+ +

+

I I 100 200 300 G(

+ +

+

/ ” I I I

100 200 300 G( Experimental load, kN

a I

a13

Fig 11. Proposed model for continuous RC deep beam

+ 90 ‘lo confidence limits for means

500 1000 Experimental Load, kN

Fig 12. Correlation of predicted loadcarrying capacity of continuous and simply-supported RC deep beams under single-point loading with experimental values reported in ref. 13

the region of the support. Such behaviour allows load redistribution and the ultimate limit state is reached when the flexural capacity at another section away from the supports is attained.

On the basis of the above, the model proposed for simply supported deep beams can easily be extended to describe the ultimate limit state of a continuous deep beam, as indicated in Fig 11. Using this model to predict the loadcarrying capacity of continuous deep beams tested elsewherel’, the close correlation between predicted and experimental values shown in Fig 12 is obtained.

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, therefore, forms a suitable basis for a design procedure.

The apparent good correlation between predicted and experimental values is attributed to the validity of the concepts that form the basis of the proposed model. 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, idealising the path as a frame with inclined legs simplifies the design procedure for deep beams into a procedure for designing the members of the frame such as to (a) suatain a given load, with an adequate margin of safety, and (b) prevent any undesirable-because of its sudden and catastrophic occurrence-brittle failure.

References 1.

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CEB/FIP Model code for concrete structures, English ed., London, Cement & Concrete Association, 1978 Kotsovos, M. D.: ‘Fracture processes of concrete under generalised stress states’, Materials and Structures, RILEM, 12, No. 72, November-December 1979, pp431-437 Kotsovos, M. D., and Newman, J. B.: ‘Fracture mechanics and concrete behaviour’, Magazine of Concrete Research, 33, No. 115, June 1981, pp103-112 Kotsovos, M. D.: ‘Mechanisms of “shear” failure’, Magazine of. Concrete Research, 35, No. 123, June 1983, pp99-106 Kotsovos, M. D.: ‘Shear failure of reinforced concrete beams’, Engineering Structures, 9, No. 1, January 1987, pp32-38 Kotsovos, M. D., Bobrowski, J., and Eibl, J.: ‘Behaviour of reinforced concrete T-beams in shear; The Structural Engineer, 65B, No. 1, March 1987, ppl-l0 Kotsovos, M. D.: ‘Behaviour of reinforced concrete beams with a shear-span-to-depth ratio between 1.0 and 2.5’, ACI Journal, Proceedings, 81, No. 3, May-June 1984, pp279-286 Kotsovos, M. D.: ‘Behaviour of reinforced concrete beams with shear- span-to-depth ratios greater than 2.5’, ACI Journal, Proceedings, 84, No. 6, November-December 1986, pp1026-1034 Rawdon de Paiva, H. A., and Siess, C. P.: ‘Strength and behaviour of deep beams in shear’, Journal of the Structural Division, Proc. ASCE, 91, No. ST5, October 1965, pp19-41 Ramakrishnan, V., and Ananthanarayana, Y.: ‘Ultimate strength of deep beams in shear’, ACI Journal, Proceedings, 65, No. 2, February,

Kong, F. K . , Robins, P. J., and Cole, D. F.: ‘Web reinforcement effects on deep beams’, ACI Journal, Proceedings, 67, No. 12, December 1970, pp1010-1017 Smith, K. N., and Vantsiotis, A. S.: ‘Shear strength of deep beams’, ACI Journal, Proceedings, 79, No. 3, May-June 1982, pp201-213 Rogowski, D. M., MacGregor, J. G., and Ong, S. Y.: ‘Tests of reinforced concrete deep beams’, ACI Journal, Proceedings, 83, No.

Kotsovos, M. D.: ‘Shear considerations’, Council on Tall Buildings, Committee 21D, Proc. workshop, 6 January 1986, Third International Con$ on Tall Buildings, Council on Tall Buildings & Urban Habitat, Lehigh University, Bethlehem, Pa., USA, pp83-115 Design and detailing of concrete structures for fire resistance, Interim Guidance by a Joint Committee of the Institution of Structural Engineers and the Concrete Society, April 1979 Leonhardt, F., and Walther, R.: Wandartige Trager, Deutscher Ausschuss fur Stahlbeton, Heft 178

1968 pp.87-98

4, July-August 1986, ~ ~ 6 1 4 - 6 2 3

32 The Structural EngineerlVolume 66/No.2/19 January 1988