Pertinent discussion will be published in the March-April 1985 ACI. JoURNAL if ..... 431-437. 4. Bedard, C., "Non-Linear Finite Element Analysis of Concrete.
ACI JOURNAL
TECHNICAL PAPER
Title no. 81-27
Behavior of Reinforced Concrete Beams with a Shear Span to Depth Ratio Between 1.0 and 2.5
by Michael D. Kotsovos A finite element analysis of under- and over-reinforced concrete beams subjected to two-point loading indicates that placing shear reinforcement in the middle rather than in the shear span results in both higher load-carrying capacity and ductility when the shear span to depth ratio is between 1.0 and 2.5. It is shown that this behavior is due to collapse of the beams by "splitting" of the compressive zone of the middle span rather than "crushing" of the loading point region as is generally thought. Splitting is caused by tensile stresses that develop in the compressive zone from the interaction of adjacent concrete elements subjected to different states of stress. In fact, for all cases investigated, this interaction is found to cause co/lapse before the ultimate strength of concrete in compression is attained anywhere within the beams. Keywords: beams (supports); bearing capacity; failure; finite element method; loads (forces); reinforced concrete; shear strength; span-depth ratio; structural analysis; tensile stress.
For values of the shear span to depth ratio aJd between approximately 1.0 and 2.5, reinforced concrete beams without shear reinforcement subjected to twopoint loading are generally considered to suffer a crushing mode of failure when the applied load increases to the level at which the diagonal crack that forms within the shear span at an earlier load stage penetrates into the compressive region towards the loading point [see Fig. 1(a)J_I It has been suggested recently, however, that a crushing mode of failure in the region of the loading point is unlikely since the multiaxial compressive state of stress that exists there will cause a local increase of concrete strength. 2 Instead, it has been proposed that the diagonal crack will branch almost horizontally toward the compressive zone of the middle span of the beam in order to bypass this highstrength region. The path of crack branching should be that of a compressive stress trajectory 3 which, as indicated by the change in the direction of the compressive force path, is characterized (for local equilibrium purposes) by the presence of a resultant tensile force at right angles to the compressive path near the tip of the diagonal crack [see Fig. 1(b)]. Failure, therefore, will occur within the middle span under compression-tension stress conditions. 2 ACI JOURNAL I May-June 1984
The implication of the proposed failure mechanism is that the presence of shear reinforcement within the shear span, although it may delay occurrence of diagonal cracking, will not necessarily lead to a substantial increase of load-carrying capacity and ductility of the beam. In contrast, shear reinforcement placed in the middle span of the beam may be more efficient since it will delay, or may even prevent, the propagation of the diagonal crack into the middle span. To this end, the present work is concerned with an attempt to verify the validity of the proposal by finite element analysis. The work is based on a comparative study of strength and deformational characteristics, as well as the fracture processes, of various reinforced concrete beams with the same geometry and loading conditions but with shear reinforcement placed either within the shear span or within the middle span or throughout the length of the beam. Research significance
The work described in this paper forms part of a research program investigating the various modes of shear failure exhibited by reinforced concrete structural members under load. It is considered that these modes of failure are associated with multiaxial stress conditions that exist in the region of the paths along which the compressive forces are transmitted to the supports, rather than with the shear capacity of critical cross sections. REINFORCED CONCRETE BEAMS INVESTIGATED
The work is concerned with an investigation of the behavior of reinforced concrete beams subjected to two-point loading. Although attention if focused on the Received May 19, 1983, and reviewed under Institute publication policies. Copyright © 1984, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion will be published in the March-April 1985 ACI JoURNAL if received by Dec. I, 1984.
279
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Michael D. Kotsovos is a senior research fellow in the civil engineering department, Imperial College of Science and Technology, London. Dr. Kotsovos has been a member of the CEB Task Group "Concrete under Multioxial States of Stress." His research activities and practical experience cover a wide range of topics related to concrete structures and technology such as fracture mechanics, constitutive relationships, finite element analysis, model testing, design procedures, etc.
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280
Type D - shear reinforcement throughout the span In all cases, the shear reinforcement had a cross-sectional area A, of 16.09 mm 2 (0.025 in 2), spacings of 51 mm (2 in.), and a yield stress of f>v = 417 N/mm 2 (60,600 psi). DETAILS OF ANALYSIS Type of analysis
Reinforced concrete beams loaded on their plane of symmetry with respect to their transverse direction are usually analyzed by assuming that they are subjected to plane stress conditions. This assumption underestimates the significance of the small compressive or tensile stresses that develop in the transverse direction from incompatible deformational response of consecutive concrete elements subjected to different states of stress. When the transverse stresses attain values between approximately 5 to 10 percent of leo the strength of concrete under compressive plane stress conditions may increase or decrease by approximately 50 percent of fc, depending on whether the transverse stresses are compressive or tensile, respectively, as indicated by Sections be, ef, and de of the triaxial strength envelope in Fig. 4. On the other hand, the effect of the small transverse stresses is insignificant in situations where the plane state of stress is compression-tension. This is evident when comparing Section ab of the triaxial strength envelope in Fig. 4 with Section a'b' of the biaxial strength envelope in Fig. 5. Therefore, because the maximum load-carrying capacity of the beams investigated is expected to be dependent on concrete strength under the compressiontension state of stress in the middle span, the assumption of plane stress conditions should provide a suitACI JOURNAL I May-June 1984
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able basis for the analysis procedure. Furthermore, it is assumed that the wholly compressive state of stress that should exist in the region of the loading point will not allow the diagonal crack to penetrate into this region. Nonlinear procedure The nonlinear finite element procedure used is fully described elsewhere 4 and will, therefore, be discussed only briefly here. It is essentially an iterative procedure that incorporates a linear solution technique. The iterative procedure is based on the Newton-Raphson method and the residual force concept. 5 This procedure has been incorporated into a Choleski solution finite element system, 6 and its application to structural analysis requires the use of constitutive laws describing the strength and deformational properties of concrete and steel as well as the interaction between steel and concrete. Such laws form the basis for the evaluation of residual forces and the material Matrix D used by the linear solution technique. Constitutive Jaws
Concrete The constitutive law used for concrete has been the subject of previous publications. 7•8 The deformational response of the material under load is resolved into a linear elastic (i.e., recoverable during unloading) component and a nonlinear permanent component. The latter is considered to represent the effect on deformaACI JOURNAL I May-June 1984
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tion of cracking processes occurring at the microscopic level under increasing stress. Since cracking occurs to relieve high tensile stress concentrations that exist near the tips of prexisting microcracks dispersed throughout the material body, 3 reducing tensile stress concentrations is considered equivalent to an internal compressive state of stress. The effect of the cracking process on deformation, therefore, is represented as the linear elastic response of concrete under this internal compressive state of stress. To this end, the strains e1, e 2 , and e3 corresponding to a given state of principal stresses a,, a2 , and a 3 below or at the ultimate strength level are evaluated by using Hooke's law as follows e 1 = (liE) [(a 1 + s 1) - v (a2 + S 2 + a 3 + S 3)] e 2 = (II E) [(a 2 + s 2 ) - v (a 3 + s 3 + a, + s,)]
+ s 3) - v (a 1 + S 1 + a2 + S 3)] 9KG/(3K + G) and v = (3K - 2G)/(6K
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where E = + 2G) are the modulus of elasticity and Poisson's ratio, respectively; and s,, s 2 , and s3 are principal stress components of the internal compressive state of stress. The analytical expressions at the linear K and G moduli and the internal stress components are presented in Reference 8; whereas, the analytical representation of the variation of the ultimate strength level in stress space is given in Reference 7. The analytical representation of ultimate strength also describes conditions for the occurrence of "macrocracks." When the ultimate strength level of an element of concrete subjected to a state of stress with at least one tensile stress component is exceeded, then a macrocrack is thought to form on the plane of the maximutn and intermediate principal compressive stresses, with the concrete suffering a complete and immediate loss of load-carrying capacity in the orthogonal direction. In compliance with the fracture mechanism of concrete and the mechanisms of shear failure 281
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discussed in References 3 and 1, respectively, the transfer of force across the crack surfaces is considered to be insignificant. The effect of such macrocracks on deformation is assessed by using the smeared-crack approach.9·10
Steel The deformational response of reinforcing steel bars have been represented by the trilinear stress-strain relationship of Fig. 6. This relationship is similar to that recommended in CP 11011 except for the final branch, which has been given a slight inclination to avoid the numerical instabilities associated with a zero stiffness. Concrete-steel interaction Use of the smeared-crack approach may lead to an underestimate of the stiffness of concrete in the tensile zone of the beam, because it does not allow for the stiffening effect of the uncracked concrete. Such an approach would be incompatible with a detailed modeling of the interaction between steel and concrete. As a result, this interaction is considered to be adequately described by the assumption of perfect bond. Finite element mesh The finite element mesh typical of all four types of beams investigated is shown in Fig. 7. I so parametric elements were used to model both concrete and steel. The position of the steel elements is indicated by the dotted lines, which coincide with the boundaries of the adjacent concrete elements (see Fig. 7, 11 to 13). Eightnode plane elements were used for concrete and threenode bar elements, with zero stiffness in the transverse direction, for both tension and shear reinforcement. Use of such elements for steel implies zero dowel ac282
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tion, and this is compatible with the mechansims of shear failure discussed elsewhere. 2
Verification of analysis procedure Verification of the finite element analysis procedure used in this work is the subject of another publication in which the predicted behavior of a wide range of structural forms under increasing load is shown to correlate closely with experimental values. 4 Predicted and experimental load-deflection relationships of a beam without shear reinforcement but geometrically similar to the beams investigated in this work are shown in Fig. 8. It is interesting to note that the relationships correlate very closely for loads increasing to a level close to the maximum load-carrying capacity of the beam. Beyond this level it is not feasible to predict by plane stress analysis the near-plastic deformational behavior of the beam, since it was shown experimentally that such behavior is not, as generally accepted, due to the strain-softening characteristics of concrete but due to triaxial stress conditions developing within the compressive zone when a load level close to the maximum sustained load is attainedY These triaxial stress conditions, however, appear to affect the deformation rather than strength characteristics of the beams. RESULTS OF ANALYSIS The main results of the investigation are summarized in Table 1 and Fig. 9 to 13. Table 1 gives the predicted maximum load that can be sustained by the beam together with values calculated on the basis of the ACI Standard, 13 whereas the predicted load-deflection relationships are shown in Fig. 9 and 10. It should be noted ACI JOURNAL I May-June 1984
Table 1 - Maximum load-carrying capacity in kN (kip) predicted by finite element analysis (FEA) and ACI 318 for the beams investigated a,/d
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135.0 (0.21) FEA ACI 24 10.2 (5.40) (2.3) 26 27 (5.85) (6.07) 10.2 24 (5.40) (2.3)
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that predicted values of both the maximum load and maximum deflection should be considered as lowerbound values since, as discussed in the preceding section, any triaxial compressive stress conditions that may develop when the ultimate strength capacity is approached are ignored by the plane stress analysis employed. Fig. II and I2 indicate typical patterns of macrocracking and deflected shape corresponding to the maximum load sustained by the beams, whereas Fig. I3 indicates the deflected shape and various stages of the macrocracking process typical for all types of beams investigated under increasing load. DISCUSSION OF RESULTS Strength characteristics Table I indicates that, for beams with aJd = 2.27, the maximum load-carrying capacity of the Type C beams (shear reinforcement within middle span) is higher than that of the Type B beams (shear reinforcement within shear span), irrespective of the amount of tension reinforcement. Such behavior supports the proposal that collapse of these beams is caused by failure of the compressive zone of the middle span under compression-tension stress conditions and not by failure of the region of the loading point under a wholly compressive state of stress. Using shear reinforcement, therefore, for preventing failure of the middle span
283
Table 1 also includes the values of the maximum load-carrying capacity calculated by using the procedures recommended by ACI 318. 13 For beams with shear reinforcement within the shear span (Beams B and D), these values were calculated on the basis of the flexural capacity. On the other hand, the maximum load-carrying capacity of the beams without shear reinforcement within the shear span (Beams A and C) was assessed on the basis of the shear capacity. It is interesting to note that while the table indicates a close correlation between the calculated values and those predicted by the analysis for Beams B and D, the values calculated on the basis of the shear capacity are significantly smaller than the analytical predictions. This deviation is attributed to the inadequacy of the concepts that form the basis of the shear design procedures.
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Deformational characteristics Fig. 9 shows the load-deflection relationships predicted for various types of beams with As = 84.83 mm 2 (under-reinforced concrete beams). The figure indicates that, for both values of avid investigated, Type A beams (i.e, the beams without shear reinforcement) exhibited a brittle load-deflection relationship considered to be indicative of shear failure. Except for the beams with a/d = 3.4 and shear reinforcement within the middle span, shear reinforcement either delayed or prevented such failure and resulted not only in an increase in load-carrying capacity but also in ductile behavior. In contrast to the previously mentioned behavior, all beams with As = 135 mm 2 (over-reinforced concrete beams) exhibited a brittle load-deflection relationship irrespective of the presence of shear reinforcement. However, as discussed in the preceding section, the maximum deflection predicted for the beams with shear reinforcement throughout the span should be considered as a lower-bound limit since the plane stress analysis employed does not allow for the confining effect of the shear reinforcement.
rather than delaying the occurrence of the diagonal crack within the shear span leads to a higher load-carrying capacity of the beams. In contrast to the behavior of these beams, those with a/ d = 3.4 sustain a higher load when the shear reinforcement is placed within the shear span rather than the middle span. Such behavior was not unexpected because beams with values of a/d greater than approximately 2.5 are known to collapse due to failure of their shear span. It is also interesting to note in Table 1 that placing shear reinforcement throughout the span generally improved the load-carrying capacity of the beams in spite of the fact that the plane stress analysis employed did not allow for any increase in the compressive strength of the concrete in the compressive zone of the middle span due to the transverse restraint induced by shear reinforcement.
Fracture processes The macrocracking patterns predicted for the beams with a/ d = 2.27 under their maximum sustained load indicate that Type A beam (beam without shear reinforcement) failed at the next load increment since it could not sustain the extension of diagonal cracking through the compressive zone [see Fig. 11 (a)]. Placing shear reinforcement within the shear span (Type B beam) resulted in an increase of the load level essential to cause extension of diagonal cracking [Figure 11(b)]. For Type C beams (shear reinforcement within the middle span only), diagonal cracking extended inside the compressive zone towards the loading point at an earlier load stage equivalent to that which caused collapse of the Type A beam. However, biaxial compressive stress conditions in the region of the loading point combined with the presence of shear reinforcement prevented extension of the crack within the compressive zone of the middle span. Furthermore, when the
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the loading point and led to a higher load-carrying capacity of the beam. Fig. 13 indicates the process of microcracking of the Type C under-reinforced beam with a,/ d = 2.27 under increasing load. This process is typical for all the beams investigated. The figure also indicates the deflected shape of the beam, the deflection being magnified by a factor of 10. From the figure, it can be seen that macrocracking initiates in the middle span and progressively spreads towards the supports with the increasing load. Concurrently, the macrocracks extend toward the compressive zone, and as expected, those within the shear span exhibit an inclination towards the loading point. The macrocrack pattern remains unchanged for load increasing from 20 to 27 kN, which is the maximum load sustained by the under-reinforced beam with same a,ld but without shear reinforcement (Type A). Further load increments result in macrocrack branching within the compressive zone of the shear span towards the loading point and eventually lead to splitting of the compres285
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2.5 subjected to two point load has indicated that placing shear reinforcement within the middle span rather than the shear span results in a significant improvement of both load-carrying capacity and ductility of the beams. Such behavior supports the view that, unlike beams with avid greater than 2.5, failure of the beams is due to branching of the diagonal crack within the shear span toward the compressive zone of the middle span and not due to crushing of the compressive region of the loading point. It has also been found that collapse of the beams always occurs before the compressive strength of concrete is exceeded anywhere within the beams. Even in the compressive zone concrete fails under combined compressive and tensile stresses, the tensile stresses result from the interaction of adjacent concrete elements subjected to different states of stress. REFERENCES I. Kong, F. K., and Evans, R. H., Reinforced and Prestressed
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sive zone of the middle span and complete collapse of the beam. Failure mechanism It is interesting to note that, in all cases investigated, collapse of the beams occurs before the strength of concrete under a wholly compressive state of stress is attained anywhere within the structure. This is because under a compressive state of stress close, but not beyond, the ultimate strength level concrete undergoes expansion at right angles with the direction of the larger compressive stress, which induces in adjacent concrete elements tensile stresses sufficient to cause macrocracking or failure of these elements. Such macrocracking or failure results in a dramatic volumetric expansion of the elements that, in turn, transforms the adjacent wholly compressive state of stress into a state of stress with at least one stress componenet being tensile. It apprears, therefore, that this interaction between adjacent concrete elements is unlikely to allow the concrete to attain ultimate strength in compression. Such behavior is similar to that predicted for plain concrete structural forms subjected to concentrations of load 14 • 15 and observed in under-reinforced concrete beams subjected to flexure. 12
CONCLUSIONS A nonlinear finite element analysis of under- and over-reinforced concrete beams with a/ d smaller than
286
Concrete, 2nd Edition, The English Language Book Society and Nelson, London, 1980, 412 pp. 2. Kotsovos, Michael D., "Mechanisms of 'Shear' Failure," Magazine of Concrete Research (London), V. 35, No. 123, June 1983, pp. 99-106. 3. Kotsovos, M. D., "Fracture Processes of Concrete under Generalised Stress States," Materials and Structures/Research and Testing (RILEM, Paris), V. 12, No. 72, Nov.-Dec. 1979, pp. 431-437. 4. Bedard, C., "Non-Linear Finite Element Analysis of Concrete Structures," PhD thesis, University of London, 1983, 286 pp. 5. Zienkiewicz, 0. C., The Finite Element Method in Engineering Science, 2nd Edition, McGraw-Hill Book Co. Limited, London, 1971, pp. 369-412. 6. Hitchings, D., FINEL Programming Manual, Department of Aeronautics. Imperial College, 1980, 96 pp. 7. Kotsovos, Michael D., "A Mathematical Description of the Strength Properties of Concrete under Generalised Stress," Magazine of Concrete Research (London), V. 31, No. 108, Sept. 1979, pp. 151-158. 8. Kotsovos, M. D., "Concrete-A Brittle Fracturing Material," Materials and Structures/Research and Testing (RILEM, Paris), V. 17, No. 98, Mar.-Apr. 1984. 9. Argyris, J. H.; Faust, G.; and William, K. J., "Finite Element Modelling of Reinforced Concrete Structures," Introductory Report, IABSE Colloquium on Advanced Mechanics in Reinforced Concrete (Delft, 1981), International Association for Bridge and Structural Engineering, Zurich, 1981, V. 33, pp. 85-106. 10. Gogate, Anand B., and Bishara, Alfred G., "Finite Element Analysis of Deep Concrete Beams," Indian Concrete Journal (Bombay), V. 54, No. 12, Dec. 1980, pp. 326-334. I I. "Code of Practice for the Structural Use of Concrete, Part I. Design, Materials and Workmanship," (CP IIO:Part I :November 1972), British Standards Institution, London, 1972, !54 pp. 12. Kotsovos, M. D., "A Fundamental Explanation of the Behaviour of Reinforced Concrete Beams in Flexure Based on the Properties of Concrete under Multiaxial Stress," Materials and Structures! Research and Testing (RILEM, Paris), V. 15, No. 90, Nov.-Dec. 1982, pp. 529-53 7. 13. ACI Committee 318, "Building Code Requirements for Reinforced Concrete (ACI 318-77)," American Concrete Institute, Detroit, 1977, 102 pp. 14. Kotsovos, M.D. "An Analytical Investigation of the Behaviour of Concrete under Concentrations of Load," Materials and Structures/Research and Testing (RILEM, Paris), V. 14, No. 83, Sept.-Oct. 1981, pp. 341-348. 15. Kotsovos, Michael D., and Newman, John B., "Effect of Boundary Conditions upon the Behaviour of Concrete under Concentrations of Load," Magazine of Concrete Research (London), V. 33, No. 116, Sept. 1981, pp. 161-170.
ACI JOURNAL I May-June 1984
