Behavior of Reinforced Concrete Beams Designed in

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Pertinent discussion will be published in the January-February·1991 ACI. Structural Journal ..... forced Concrete (ACI 318-83)," American Concrete Institute, De-.
,\CI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title no. 87-s 14

Behavior of Reinforced Concrete Beams Designed in Compliance with the Concept of Compressive-Force Path

· · by Michael D. Kotsovos and loannis D. Lefas Paper's intent is to verify the validity of a method developed in complilmce with the concept of the "compressive-force path" for the design of reinforced concrete beams. The method is briefly presented and used to design a number of beams whose behavior is subsequently investigated by experiment. The beams chosen were such that the design details specified by the proposed method were significantly different than those specified by current code provisions. In fact, on the basis of current design concepts, the beams are deemed inCQ[Xlble of sustaining the design load; yet the beams not only sustained safely the specified design load, but did so with an amount of tnmsverse reinforcement 70 percent less, in some cases, than specified by current methods. leywurds: beams (supports); building codes; compressive streugtb; reinforced concrete; shear properties; structunl design; tests.

It has been shown that the causes of shear failure of reinforced concrete (RC) beams are associated with the development of tensile stresses in the region of the path along which the compressive force is transmitted to the supports and not, as widely believed, the stress conditions below the neutral axis. This conclusion led to the introduction of the concept of the compressive force path, 1 which has formed the basis for the development of a design method shown to predict the shear capacity of a wide range of RC beams significantly better than do the current Code methods used to design the beams. 2• The paper provides a brief description of the preceding method, which is used to design a number of beams that would be incapable of sustaining the design load on the basis of current design concepts. This work thus not only intends to provide additional evidence verifying the validity of the proposed method but also to demonstrate the inadequacy of the concepts underlying current Code provisions. RESEARCH SIGNIFICANCE

The work forms part of a research program aimed at identifying concepts that could form a suitable basis for °Kotsovos, M. D., "RC Design Based on Compressive Force Path," in press.

ACI Structural Journal I March-April1990

the development of safe and efficient design procedures. DESIGN METHOD

The design method used is fully described elsewhere• and will therefore be discussed only briefly. It is based on the concept of the compressive force path, which stipulates that the load-carrying capacity of an RC structural member is associated with the strength of · concrete in the region of the paths along which compressive forces are transmitted to the supports. The path of a compressive force may be visualized as a "flow" of compressive stresses, with varying sections perpendicular to the path direction and with the compressive force representing the stress resultant at each section [see Fig. l(a)]. Failure is considered to be related to the development of tensile stresses in the region of the path; such stresses may develop due to a number of causes including, e.g., changes in path direction, the varying intensity of the compressive stress field along the path, bond failure at the level of the tension reinforcement between two consecutive flexural or inclined cracks in regions subjected to a combination of large bending moment and large shear force, etc. 1 Physical models

The compressive force path concept may be introduced in design to develop physical models capable of providing a realistic description of the features of various RC structural members. For example, an RC beam without transverse reinforcement, subjected to two*Kotsovos, M.D., "RC Design Based on Compressive Force Path," in press. ACI Structural Journal, V. 87, No.2, March-Aprill990. Received August S, 1988, and reviewed under Institute publication policies. Copyright © 1990, 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 January-February·1991 ACI Structural Journal if received by Sept. I, 1990.

127

Michael D. Kotsovos is a lecturer in the Department of Civil Engineering, lm· perial College of Science and Technology, London, England. His research activities cover a wide range of topics related to concrete structures and technology such as fracture mechanics, constitutive relationships, finite element analysis, model testing, and design procedures.

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A Cl member Joannis D. Lefas is a research assistant in the Department of Civil Engineering, Imperial College of Science, Technology and Medicine, London, England. He graduated from the National Technical University of Athens and obtained his MSc and PhD degrees from Imperial College, London. He is currently carrying out postdoctoral research on the nonlinear modeling of reinforced concrete structures subjected to cyclic loading. His research interests also cover a wide range of topics related to structural and earthquake engineering.

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Fig. 1 - (a) Compressive force path; (b) proposed "comb/ike" model point loading, may be modeled as a "comblike" structured tied by the tension reinforcement, as shown in Fig. l(b). The figure shows that the "comb" comprises a frame with inclined legs, providing a simplified but realistic description of the shape of the compressive force path, and a number of "teeth" representing the concrete cantilevers, which form between consecutive flexural or inclined cracks occurring within the beam web under increasing load. The shape of the cross section of the model is that of the actual beam. The dependence (indicated in Fig. 2) of the beam behavior on the shear span-depth ratio (a/d) 3 is reflected on the relative lengths of the inclined and horizontal members of the model frame; the horizontal projection of the inclined legs of the frame may be given a value equal to either the shear span, for aid < 2, or twice the beam depth, for a/d > 2. On the other hand, the width of the teeth may be taken to be equal to half the depth of the horizontal member of the frame, since there is comprehensive experimental evidence4 that indicates crack spacing at the ultimate limit state is approximately equal to half the depth of the neutral axis. The same model may also be used to represent an RC beam subjected to uniformly distributed loading. A 128

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Fig. 2 - Types of behavior exhibited by reinforced concrete beams without shear reinforcement subjected to two-point loading beam span > 6d may be modeled as for a beam subjected to two-point loading with aid > 2. For smaller beam spans, the uniformly distributed loading is considered to be equivalent to concentrated loading applied at the third points. In compliance with the compressive-force path concept, strength is considered to be predominantly provided by the tied frame, with the concrete "teeth" making a small contribution to shear resistance through the bond forces that develop between concrete and tension reinforcement. Although (in contrast to Kani's hypothesis 3) these forces are considered insufficient to cause flexural failure of the teeth, their presence is essential for the overall beam strength since (as discussed elsewhere') bond failure causes a significant stress redistribution within the compressive zone (see Fig. 3), which may lead to collapse.

Failure criteria

To implement the preceding model in design, it is essential to complement it with a failure criterion. Such a failure criterion cannot be unique, since failure may occur due to a number of causes. As discussed elsewhere, 1 these can be classified in four categories, each yielding a particular type of structural behavior characterized by a specific mode of failure (see Fig. 2). The ACI Structural Journal I March-April 1990

(f V.a • AT. z

before bond fnilurt

V.11 • T• .t.z after bond fnilurt

Fig. 3 - Effect of bond failure on stress conditions in compressive zone

discussion of Types I and IV behavior (indicated in Fig. 2) is beyond the scope of this paper. These behavior types, which correspond to flexural and deep-beam behavior, have been discussed in depth elsewhere; 5 •6 therefore, only the failure criteria relevant to Types II and III behavior are discussed in the following. Type II behavior - This characterizes beams in which the horizontal projection of the inclined portion of the compressive force path (inclined leg of frame of proposed model) is greater than or equal to twice the beam depth. Under the combined action of bending moment and shear force, such beams may fail due to a number of causes before flexural capacity is attained. 1 An analytical description of the combination of these internal actions causing failure has already been derived empirically, 2 and is as shown, with a slight modification, in Appendix A. Type III behavior-characterizes beams in which the horizontal projection of the inclined portion of the compressive force path (inclined leg of frame of model) is equal to the shear span a, the latter attaining values between d and 2d. As indicated in Fig. 2, a = 2d is taken to correspond to the lowest point of Kani's valley, whereas a = d corresponds to flexural capacity, provided Type IV behavior is prevented. The combination of internal actions causing failure for a = d and a = 2d may be assessed by using the failure criteria for Types IV (Reference 6) and II behavior. For intermediate values of the shear span (i.e., d < a < 2d), it is considered sufficient for design purposes to establish the combination of internal actions by linear interpolation. The resulting combination of internal actions may be used in design as indicated in Appendix A for Type II behavior.

Provision of transverse reinforcement As indicated in Appendix A, if the conditions for failure are fulfilled before flexural failure occurs, one ·of the design solutions that will allow the beam to attain its flexural capacity involves the provision of transverse reinforcement. Such reinforcement is provided to sustain that portion of the internal tensile actions (developing in the region of the compressive force path) that cannot be sustained by concrete alone. For Type II behavior, significant internal tensile actions may develop, for equilibrium purposes, within both the region where the compressive force path changes direction and, for the case of point loading, the ACI Structural Journal I March-April 1990

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Fig. 4 - Schematic representation of additional internal actions developing due to presence of transverse reinforcement in region where path changes direction

horizontal portion of the path in the region of point loads. 1 The transverse reinforcement required to sustain the portion of the first of the actions just described, in excess of that which can be sustained by concrete alone, will slightly modify the comblike model [shown in Fig. l(b)], as indicated in Fig. 4. This figure shows the region where the inclined and horizontal members of the model join and indicates that the transverse reinforcement not only sustains the action of the vertical component [ V as defined in Section (a) of Appendix B] of the inclined compression but also subjects the shaded concrete block of the beam web where it is anchored, to a compressive force D. This force balances the shear force V acting at the right-hand side of the above block. Note, however, that the transverse reinforcement will be activated only when the capacity of concrete to sustain alone the action of the internal tensile force is attained. When this occurs, the excess tensile force will be sustained by the reinforcement, which may be assessed as described in Appendix B. Following current. practice, the spacing of the transverse reinforcement should be smaller than the beam effective depth d, which is considered to be equal to the width (in the longitudinal direction) of the shaded block of the web (see Fig. 4). As discussed earlier, transverse reinforcement may also be required (within the horizontal portion of the path in the region of point loads) to sustain tensile stresses that may develop when bond failure occurs between two consecutive flexural or inclined cracks. 1 Bond failure will increase the depth of the right-hand crack, thus causing stress redistribution as indicated in Fig 3. For simplicity, consider a rectangular cross sec129

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may be designed as indicated in Appendix C; their presence beyond the critical section is deemed essential, since it has been shown experimentally that reinforcing with stirrups only to the critical section does not safeguard against brittle failure. 7 This type of failure may be prevented by extending the transverse reinforcement beyond the crack tip to a distance approximately equal to the neutral axis depth. EXPERIMENTAL PROGRAM

Fig. 5- (a) and (b) Cross-sectional characteristics and (c) reinforcement details of end zones of beams tested in the program (I mm = 0.0394 in.)

tion. The resulting transverse tensile stresses, with the corresponding stress resultant and the amount of reinforcement required to sustain it, may easily be assessed as described in Appendix B. Such reinforcement should be placed beyond the region where the compressive force path changes direction within the shear span. For Type III behavior, failure is associated with the large reduction of the neutral axis depth in the region of the tip of the main inclined crack. 1 Such a reduction in depth will lead to the development of tensile stresses within the compressive zone, for the reasons described elsewhere.' Therefore, failure will occur when the strength of the compressive zone under the combined action of compressive and tensile stresses is exceeded. This type of failure may be prevented by providing stirrups throughout the lengtq of the horizontal projection of the inclined crack (which for the case of twopoint loading equals the shear span a). The stirrups

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The experimental work to be described was intended not only to provide evidence to verify the proposed design method but also to demonstrate the inadequacy of current shear-design procedures. It was considered essential, therefore, that the T -beams used in the program were sized such that they complied with this objective, rather than to be representative of beams used in practice. Beam details The type of beams tested in the program were 3200 mm long and simply supported with a span of 2600 mm (see Fig. 5). In all cases, the distance between the beam ends and the support was 300 mm, and within this length the beams had a rectangular cross section 200 mm wide x 290 mm high [see Fig. 5(b)). The beams were under-reinforced with two 20 mm diameter high-yield deformed bars. To eliminate the possibility of anchorage failure, the steel bars were extended to about 280 mm beyond the support and welded onto steel end plates, as indicated in Fig. 5(c). Furthermore, to prevent splitting along the interface ACI Structural Journal I March-April 1990

between the steel bars and concrete, the end zones of the beams were reinforced with eight 6 mm diameter high-yield steel links [see also Fig. 5(c)]. In all cases, the transverse reinforcement was assessed by using the procedure described in Appendix B. Depending on the transverse reinforcement, the beams tested were classified as follows (see Fig. 6): Beams A had transverse reinforcement comprising two single-legged straight links with 6 mm diameter and 120 mm spacing extending from the top face of the beam to the level of the tension reinforcement throughout the shear span. Beams B had the following types of transverse reinforcement (also comprising links with 6 mm diameter and 120 mm spacing): (a) links as for Beam A but only within the portion of the shear span between the support and the section lying at a distance 2d from the support, and (b) two single-legged links within the remaining portion of the shear span, extending from the level of the tension reinforcement to the top face and then, following the perimeter of the flange, ending up within the beam web. Beams C had the following types of transverse reinforcement comprising links with 1.6 mm diameter and 25 mm spacing: (a) links as for Beam A, and (b) additional flange reinforcement beyond the section at a distance of 2d from the support, as indicated in Fig. 6. The yield stress /y and the ultimate strength fu of the 20 mm diameter steel bars were 500 and 670 MPa, respectively, whereas the yield stress of the 6 and 1.6 mm diameter links were 570 and 360 MPa, respectively. Full details of the concrete mix used for the tests are given in Table 1. Testing

The loading configurations to which the beams were subjected are shown in Fig. 7. Beams A and B were subjected to two-point loading, whereas Beam C was subjected to four-point loading. Two of each beam type · were tested under a given loading configuration. The two-point loading was applied through a hydraulic ram and a spreader beam supported by 250 x 50 mm loading plates placed with the smaller dimension along the length of the beam. The four-point loading was applied using two hydraulically connected rams and two spreader beams supported as for the case of two-point loading. The precise location of the load points is indicated in Fig. 7. At each load point, the load was applied in increments of 3 kN. At each increment, the load was maintained constant for at least 2 min to measure the deformation response of the beams, mark the cracks, and take photographs of the beam crack pattern. The load was measured by using a load cell and the deformation response by displacement transducers (L VDTs) measuring the deflection at (a) the midspan section (in all cases), and (b) either the section under the load point, for two-point loading, or the. section halfway between the two consecutive load-points closest to the supports, for four-point loading. The load and deflections were ACI Structural Journal I March-April1990

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3.18 2.00 0.91 1.00 0.68

Cube strength at 28 days, N/mm' Cube strength at time of testing, N/mm' Cylinder strength at time of testing, N/mm' Average age at testing, days 1 N/mm'

45.2 48.4 31.7 50

= 145 psi.

Table 2-Predicted and measured load-carrying capacity of beams tested

Beam Load type (1) (2) A

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c

Four-point

Total sustained load Predicted, kN ACI British Building Proposed code Code method Measured, kN (3) (4) (5) (6) 120.0

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45.6

32.2

223.6

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recorded by using a computer-logger capable of measuring to a sensitivity of ± 0.1 N and ± 0.0001 mm, respectively. TEST RESULTS

The main results of the tests are given in Table 2 and Fig. 8 through 10. Table 2 provides information related to the load-carrying capacity of the beams tested in the program, whereas Fig. 8 depicts the load-deflection curves obtained from the tests. Fig. 9 shows typical crack patterns of the beams tested at various load stages. Finally, Fig. 10 indicates the crack pattern in the region where failure initiated for each type of beam tested. 131

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M. only nominal stirrups would be needed. If M, < M., either increase area of tensile steel (thus increasing M, to a level greater than or equal to M,), or increase the cross section; the alternative is to provide stirrups in accordance with the requirements described in Appendixes B and C.

APPENDIX B

Assessment of transverse reinforcement for Type II behavior - (a) Region where path changes direction (see Fig. 5) Excess tensile force T,. = V = V,- V, V, = M,

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APPENDIX D Design calculations for Beams C in Fig. 9 Flexural capacity

Transverse reinforcement over a length d to sustain Tw A,= 628 mm'

f, = 560 N/mm' where f" is the characteristic strength of transverse reinforcement. (b) Horizontal portion of path (see Fig. B1) Using information in Fig. B1, the following steps are used:

1. Az = (Vx)/(2T); v = v. - v" 2. x' = 2 (d - z - A z) > 0. If x' < 0, increase cross section. 3. Assess nominal triaxial compressive stress u: = Cl(bx') 4. Assuming 0.8f", describes uniaxial condition, assess confining pressure u,."' required for 0.8f", to increase to u:, from expression u: = 0.8f", + 5u,.,1 , u,.,1 = (u: - 0.8f,,,)/5. 5. Assume transverse tensile stress u, = - u,."' (compression + ve) 6. Tensile force over length owill be

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o to sustain

APPENDIX C Assessment of transverse reinforcement for Type Ill behavior (see Fig. C1) The moment equilibrium condition of the free body, shown in Fig. C1, can be expressed as follows

where M 1 = R·a.

138

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r:::> T = 351,680 N J;,, = 32.6 N/mm', u, = (0.8)(32.6) = 26 N/mm' Since C = T, A, = 351,680/26 = 13,526 mm'

Thus, x = 76 mm and x, = 33 mm. Lever arm z = 240-33 = 207 mm. Hence, flexural capacity Mf = 351,680·207 = 72,797,760 N-mm. Maximum sustained four-point load as indicated in Fig. 9 is 4·52,000N.

Shear force sustained by concrete There are two possible shapes of path as indicated in Fig. Dl. Path /-Shear span a = 300 mm ( < 2d = 480 mm and > d = 240 mm) as shown in Fig. D1(a). Thus we assess M, by assessing M, for a = 480 mm and a = 240 mm; M, for s = 300 mm is obtained by linear interpolation. For s = 480 mm from design equation in Appendix A, M, = 25,500,000 N-mm For s = 240 mm, M, = M1 = 72,797,760 N-mm. Thus, for s = 300 mm, M, = 72,797,760 - (72, 797,760 25,500,000)601240 = 60,973,320 N-mm However, applied bending moment at s = 300 mm is M, 31,020,000 N-mm 37,500 N. Thus, the path is more critical and transverse reinforcement is required. Tranverse reinforcement (i) For excess tension due to change in path direction [see Appendix B(a)] Tw = 52,000-37,500 = 14,500 N Aw = 14,500/360 = 40 mm'

ACI Structural Journal I March-April1990

Provide 2 0 1.6- 1 - 25 ( = 39 mm') (ii) For excess tension due to bond failure [see Appendix B(b)J

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Tensile stress resultant over length of 240 mm T~ = 0.5·200·240 = 24,000 N Reinforcement required is A~ = 24,000/360 = 67 mm' Additional flange reinforcement 0 1.6- 3-25 (see Fig. 6).

ACI Structural Journal I March-April 1990

Fig. Dl

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