Estimation of Critical Buckling Moments in Slender Reinforced Concrete Beams

Revathi, P

At present, there are no recommendations in codes such as ACI 318 and BS 8110 to estimate the critical buckling moment in slender concrete beams. It is assumed that if the slenderness ratios of the beams are limited to the values prescribed by the codes, the failure moment of the beam will be dictated by flexure and not by buckling. Experimental studies carried out as part of the present study, however, show that the specified slenderness limits are not reliable, and failure by lateral instability can occur in slender beams designed according to the code. It is also shown that the existing formulations to predict critical buckling moments in beams, suggested by various researchers, grossly overestimate the capacities in the case of under-reinforced beams. In this paper, a modified formulation is proposed to predict theoretical buckling moment, and it is found to agree very closely with experimental results for both under-reinforced and over-reinforced beams.

Keywords: beams; buckling; reinforced concrete.

(ProQuest-CSA LLC: … denotes formulae omitted.)

INTRODUCTION

Slender beams are occasionally encountered in reinforced concrete (RC) construction. Significant research has been carried out worldwide with regard to the behavior of slender columns, and on the basis of these research findings, appropriate provisions have been incorporated in the codes of practice with regard to the design of such members. With regard to slender RC beams, however, there are no comprehensive design provisions at present in the design codes. Some limits on slenderness have been prescribed in some codes, primarily to ensure that the failure of the beam occurs due to material failure and not due to buckling instability. But no such design provisions presently exist in the case of slender concrete beams. To develop a suitable design basis for such slender RC beams, it is necessary to predict the critical buckling moments of RC slender beams and to predict the conditions under which instability failure governs the ultimate load carrying capacity.

An ideal beam (free of imperfections), which is bent in the plane of its greatest flexural rigidity, may buckle laterally at a certain critical value of the load or applied moment (Fig. 1). As long as the applied load on such a beam is below a critical value, the beam is expected to be stable. When the critical load or moment is reached, a bifurcation state of equilibrium is possible, whereby the beam can either remain in the vertical plane (trivial solution) or suddenly bend laterally, accompanied with some twisting, as illustrated in Fig. 1(c). This buckling of the beam is associated with a loss in its lateral flexural rigidity, resulting in instability and collapse. The lowest load at which this critical condition occurs represents the critical load for the beam.1 The problem of lateral buckling has been widely studied with regard to the design of laterally unsupported steel beams with varying slenderness ratios.

The problem of lateral stability of RC beams was first studied by Marshall in 1948. He presented a theoretical study and concluded that RC beams are seldom expected to meet with the problem of lateral instability and the flexural behavior remains unaffected by high slenderness. These findings were to some extent substantiated by Hansell and Winter,2 who carried out tests on 10 slender RC beams and reported that no beam failed by instability, and the failure in each case was due to flexure in the vertical plane. Lateral deflections of the beams, however, were observed, indicating the presence of some slenderness effects. Later, in their theoretical study, they established that none of their test beams were long enough to produce instability failure. Hansell and Winter2 suggested that the slenderness limits of beams must be expressed in terms of Ld/b^sup 2^ rather than the simple L/b ratio.

Siev3 made an attempt to study the problem of lateral buckling in RC beams with initial imperfections. In his experimental study, he tested six slender RC beams having rectangular and inverted L sections. He established that the percentage of reinforcement also influences the slenderness behavior of the RC beams, in addition to their dimensions. Sant and Bletzacker4 tested 11 beams of different Ld/b^sup 2^ ratio, with a high percentage (3.85%) of tension steel. Instability failure was observed in nine of these over-reinforced beams, and only two beams failed by flexure. Although the experimental study was based on over-reinforced beams, Sant and Bletzacker4 proposed that their analytical expression to calculate the critical buckling moment is equally applicable to under-reinforced rectangular RC beams. They showed that the vulnerability of the RC beams to instability failure increases as the d/b ratio increases.

Massey5 attempted to address the problem of lateral instability with a theory that includes warping rigidity. He tested 11 overreinforced rectangular beams, both singly and doubly reinforced, made of mortar and established from the experimental results that the warping can be neglected if the ends of the beams are not restrained from warping. King et al.6,7 proposed a method of checking the lateral stability based on the equilibrium of the deformed position. Aydin and Kirac8 also developed an algorithm to generate the value of critical slenderness ratio (Ld/b^sup 2^) of any RC beam. A review of the literature available so far indicates that further analytical and experimental studies need to be done to explain the slenderness effects on the flexural capacity of the beams.

EXISTING FORMULATIONS TO PREDICT CRITICAL BUCKLING MOMENTS IN SLENDER REINFORCED CONCRETE BEAMS

Timoshenko1 and others have presented the basic equation for predicting the critical (buckling) moments in beams with homogeneous elastic material under various loading and end conditions. For such beams with rectangular cross section, the generalized equation for critical buckling moment (Mbcr) may be expressed as follows

… (1)

where B is the flexural rigidity of the beam about the minor axis (= Eb^sup 3^d/12), K is the torsional rigidity ([asymptotically =] Gb^sup 3^d/3, when b

The previous equation can be used very effectively in steel members, where the elastic modulus and flexural and torsional rigidities can be very easily evaluated. But, these equations cannot be applied directly in RC members due to nonhomogeneity of concrete, presence of reinforcement, and cracking. Suitable modifications to Eq. (1) are necessary to account for these factors. Of the many factors that affect the lateral stability of an RC beam, the major ones are:

1. Presence of flange, if any (T-beam or L-beam action);

2. Degree of cracking in concrete;

3. Contribution of longitudinal reinforcement to flexural and torsional rigidity;

4. Contribution of shear reinforcement to shear rigidities;

5. Inelastic stress-strain properties of concrete and steel; and

6. Effect of sustained loading (creep effects).

It is evident that a rigorous analysis of this problem, considering all of the above factors, would be difficult and cumbersome. This complex problem can be solved by introducing some simple and conservative assumptions relating to the flexural rigidity B = (EI)^sub eff^ and torsional rigidity K = (GC)^sub eff^. Various proposals suggested by researchers are summarized in Table 1. Hansell and Winter2 proposed the use of secant modulus E^sub sec^ of concrete and limited the effective section to the concrete in the flexural compression zone (bending with respect to major axis) for evaluating B and K. Sant and Bletzacker4 proposed the use of a reduced modulus E^sub r^ (in terms of combination of short-term elastic modulus E^sub c^ and the tangent modulus E^sub tan^ = E^sub c^/2, but advocated the use of section up to the effective depth to evaluate the rigidities. Massey5,10 suggested an improvement on Hansell and Winter2 including the contribution of longitudinal reinforcement.

It may be noted that most of the experimental studies reported in existing literature, due to studies by Sant and Bletzacker4 and Massey,5 were provided with over-reinforced slender beam sections, assuming that high tension reinforcement contributes to enhanced lateral stability of the beams. Their theoretical predictions are claimed to be validated based on these over-reinforced beam test results. Hansell and Winter2 had previously tested under-reinforced slender beams; however, these beams were not slender enough to undergo buckling failure. A comparison with the experimental results (to be described later in this paper) indicates that the problem of accurate prediction of the critical buckling moment is not yet satisfactorily resolved. Perhaps for this reason, no procedure to evaluate M^sub bcr^ is prescribed in most design codes, including ACI-318,11 BS-8110,12 IS-456,13 EC-2,14 and AS-3600.15

RESEARCH SIGNIFICANCE

The existing theoretical formulations to evaluate the critical buckling moment M^sub bcr^ in slender RC beams do not clearly distinguish between under-reinforced and over-reinforced sections. No experimental studies to date have been reported in literature to validate predictions of M^sub bcr^ for under-reinforced slender beams. Furthermore, from a design point of view, no code recommendations are available to calculate the critical buckling moment. This is also apparently not required in practical designs, which are necessarily required to be within the specified slenderness limits and also to be designed as under-reinforced sections. These provisions are intended to safeguard against brittle failures. Various slenderness limits, currently adopted in design practice, based on international code provisions, are summarized in Table 2. The variability in the specifications among different codes, however, suggests a need for further research to resolve the differences. The present study specifically addresses the above issues. Experiments have been carried out on underreinforced RC slender beams, and equations are proposed to predict the critical moment capacities of such beams. Also, an improved formulation is proposed to estimate M^sub bcr^ for over-reinforced slender beams.

PROPOSED FORMULATION FOR FLEXURAL AND TORSIONAL RIGIDITIES

Effective flexural rigidity B

In the present study, it is proposed to adopt a modified version of flexural rigidity B = (EI)^sub eff^, originally proposed by Branson16 to calculate RC beam deflections at service loads. Branson’s equation gives the value of B, which is effectively a weighted average of gross flexural rigidity (EI)^sub gr^ and cracked flexural rigidity (EI)^sub cr^, and takes the following form

… (2)

where E^sub c^ = elastic short-term modulus of concrete; M = bending moment at service loads; M^sub R^ = cracking moment; I^sub gr^ = second moment of area of gross section; and I^sub cr^ = second moment of area of the cracked transformed section (including the contribution of longitudinal steel). Equation (2) needs to be modified in the present context to account for the fact that lateral instability: a) is associated with bending with respect to the minor axis; and b) occurs under loads that are usually well beyond service loads.

It is proposed to limit the effective section to the uncracked compression zone, ignoring concrete in the tension zone below the neutral axis for bending in the vertical plane, as suggested by Hansell and Winter2 and Massey.5 Massey also included the contribution of longitudinal steel in his proposed expression for B (refer to Table 1). However, this appears to be meaningful only if the steel stress level is well below yield, as in the case of over-reinforced beams. In the case of under-reinforced beam sections whose tension reinforcement is likely to yield when the loading approaches the collapse load, the contribution of this steel to the flexural stiffness of the beam is expected to be negligible. Hence, while applying Branson’s equation (Eq. (2)), the expression for I^sub cr^ may taken as b^sup 3^c^sub u^/12 for under-reinforced beams2 and as

…

for over-reinforced beams.5 It is also seen that the contribution of the compression steel to flexural stiffness is not generally significant, and hence may be ignored in under-reinforced beams.17 Also, the value of the bending moment M to be considered in Eq. (2) cannot be taken as the bending moment at service loads. Based on an analysis of experimental studies, it is taken as 80% of the ultimate flexural moment capacity M^sub u^ of the beam section (at midspan). Hence, the modified expression for flexural rigidity may be proposed as follows (Eq. (3)) for slender rectangular beams, for the purpose of estimating the critical buckling moment

… (3)

…

where b = breadth of the section; h = overall depth of the section; c^sub b^ = depth of neutral axis of the balanced section; c^sub u^ = the depth of neutral axis at ultimate load; E^sub s^ = elastic modulus of steel; and I^sub sy^ = second moment of area of longitudinal steel about the minor axis.

Effective torsional rigidity K

According to Saint Venant’s theory,18 the torsional rigidity K of an elastic solid rectangular member may be expressed as Gβcb^sup 3^h, where G = shear modulus of rigidity; β^sub c^ = St. Venant’s torsional constant; b = smaller dimension of rectangular section; and h = larger dimension of rectangular cross section. In the case of reinforced concrete, however, following cracking of concrete, the torsional rigidity reduces significantly. In the present study, Tavio’s torsional rigidity equation,19 which is a modified form of Hsu’s equation,20 is adopted. This expression includes the contribution of the longitudinal bars as well as the shear stirrups to estimate the cracked torsional rigidity of any RC rectangular section. The effective torsional rigidity K is given as

… (4)

where µ = rigidity multiplier taken as 1.2 for under-reinforced sections and 0.8 for over-reinforced sections; E^sub s^ = elastic modulus of steel; A^sub o^, A^sub c^, and po are cross-section properties of the beam section (see notations). ρ^sub l^ = A^sub l^/A^sub c^ and ρ^sub t^ = A^sub t^ × p^sub o^/A^sub c^ × s denotes the ratio of reinforcement in longitudinal and transverse directions, respectively, in which A^sub l^ = total cross-sectional area of longitudinal steel; A^sub t^ = crosssectional area of one leg of transverse stirrups; and s = spacing of stirrups.

The proposed expressions to calculate the critical buckling moment for under-reinforced beams are validated through a set of tests, which is discussed in the sections that follow. The analytical predictions for over-reinforced beams are also validated using the test results reported in existing literature.

EXPERIMENTAL INVESTIGATION

The experimental study involved the testing of seven under-reinforced RC rectangular beams of identical concrete mixtures with different depths and lengths. The ends of the beams were simply supported in the vertical and horizontal planes, but torsionally restrained with freedom to warp. The beams were loaded symmetrically with two point loads, sufficiently apart to produce uniform bending moment over a considerable length of the specimen, to increase the vulnerability of the beams to buckling. Figure 2 shows the layout of loading scheme.

The test setup (Fig. 3) was essentially the same as adopted by previous researchers, and is originally due to Hansell and Winter.2 Two load assemblies were set up to transmit the applied concentrated load with minimal lateral restraint, using rollers placed in between two steel boxes. A preliminary test was conducted with the load assembly to ensure that the applied loading restrains the lateral deflection of the beam only marginally (the lateral restraining force was about 0.5% of the vertical load). Except for these marginal restraining forces, the beams were laterally and torsionally unrestrained over the entire span.

Specimens

The dimensions of the beams were selected such that their slenderness was in the region of the limits as proposed by various codes. Considering the practical difficulties in handling very long beams, the spans were limited to 5.0 and 6.0 m (16.40 and 19.68 ft) and the breadth of the beams to 80 and 100 mm (3.15 and 3.93 in.) to achieve the required slenderness. The L/b ratio of the test beams was varied from 50 to 75 and Ld/b^sub 2^ ratio from 200 to 328, with 1.2% of tension reinforcement. Table 3 lists the dimensions and the reinforcement details of the seven test beams. All the beams were designed to fail by flexure in the vertical plane.

Materials

All the beams were cast using concrete batched in the laboratory. The design mixture proportion was 1:1.6:2.9 by weight, with a water-cement ratio of 0.5. Ordinary portland cement of 53 grade was used along with locally available river sand as fine aggregate. Crushed granite aggregate of 20 mm (0.78 in.) nominal maximum size was used as coarse aggregate. Reinforcing steel with a characteristic yield stress of 415 MPa (60,190 psi) was used for the longitudinal bars and 250 MPa (36,259 psi) for the stirrups. Six test cubes were cast during each casting of the beams, and the specific cube strength was found to be 45 MPa (6526 psi).

Specimen instrumentation and test procedure

Dial gauges of 0.01 mm (0.00039 in.) accuracy were used to measure the lateral as well as the vertical deflections at midspan and at quarter-span. The beams were placed in position and the loading and support assemblies were assembled as shown in Fig. 3. The initial lateral imperfections (8 to 13 mm [0.31 to 0.51 in.]) were recorded as indicated in Table 3. Care was taken to transmit the load without any eccentricity. The beams were loaded at 0.5 kN (0.11 kips) intervals up to the first crack and subsequent load steps were adjusted so that failure would occur after 15 to 20 load steps. The beams were loaded to failure and a complete set of deflection readings were taken at each load increment, using dial gauges.

EXPERIMENTAL RESULTS AND DISCUSSION

The test results relating to failure loads are summarized in Table 4. The various failure modes to be expected based on an interpretation of ACI,11 BS,12 and IS13 code provisions (Table 2) are also included in Table 4. All the measured deflection data are furnished in Fig. 4 through 6, so that this is readily accessible to researchers. The lateral deflection component in these figures includes the observed initial imperfection. All the seven beams failed due to lateral instability before reaching their ultimate flexural load capacity, although in three cases the beam sizes were within the slenderness limits proposed by design codes as indicated in Table 4. The ultimate load W^sub u^ corresponding to flexural moment capacities of the various beam sections (excluding safety factors) are also included in Table 4, and it is seen that the actual failure load W^sub test^ varied in the range 64.7 to 90.9% of W^sub u^. The reduced failure loads induced by buckling cannot be presently predicted by any recommended procedure in the prevailing ACI,11 BS,12 and IS13 codes.

Figure 7 shows load-deflection behavior for a typical beam in terms of deflections at midspan in the vertical and horizontal planes. It is seen that with gradual increase in loading, the beam undergoes not only vertical deflection but also lateral deflection, with the top portion (compression zone) deflecting more in relation to the bottom region, resulting in twisting. It is seen that the load deflection behavior becomes increasingly nonlinear at higher loads, and at the critical load W^sub test^, the phenomenon of buckling takes place suddenly, with the lateral deflections becoming unbounded. Details of vertical deflection, lateral deflection at top and lateral deflection at bottom for all the seven test beams are indicated in Fig. 4 through 6, respectively. In all cases, flexural cracks were initially observed in the soffit region, gradually propagating vertically on both sides due to bending in the vertical plane. However, with increasing lateral deflections, the cracking was seen to be more pronounced on one side (convex side), with the cracks closing on the concave (compression) side.

COMPARISON OF EXPERIMENTAL RESULTS WITH ANALYTICAL PREDICTIONS

For the simply supported, rectangular test beams subject to two concentrated loads as shown in Fig. 2, the constants C1 and C^sub 2^ in Eq. (1) take the values of 3.33 and 1.0, respectively.9 Hence, the critical buckling moment for the test beams may be expressed as

… (5)

where in the proposed formulation the flexural rigidity B and torsional rigidity K are obtainable from Eq. (3) and (4), respectively. These rigidity calculations involve the elastic short-term modulus E^sub c^ and modulus of rupture f^sub r^ of the concrete, for which various empirical formulations are available. In the present study, the following equations (in MPa units) given in ACI 31811 are adopted

E^sub c^ = ρ^sup 1.5^^sub c^0.043 [the square root of]f’^sub c^ (6)

f^sub r^ = 0.622 [the square root of]f’^sub c^ (7)

Considering an equivalent16 f’^sub c^ = 0.8f^sub ck^ average cylinder strength; f’^sub c^ = 36 MPa (5221 psi); and ρ^sub c^ = 2500 kg/m^sup 3^ (156 lb/ft^sup 3^), the values of E^sub c^ and f^sub r^ are obtained as 32,250 and 3.7 MPa (4,677,467 and 536 psi), respectively.

Table 5 shows the critical moments for the test beams, calculated as per the existing formulations given by previous researchers (Table 1) as well as the proposed formulation. It may be noted that al the seven test beams are expected to have, by the existing theories, higher critical moments than their flexure failure moment capacities, in spite of their high slenderness. However, these predictions were disproved by all the seven tests, in which the failure occurred by lateral instability at load levels below the values corresponding to failure by flexure (W^sub test^

It can be seen from Table 5 that the proposed formulation is a significant improvement over the existing theories. The predicted critical buckling loads are invariably conservatively estimate (W^sub bcr^

VALIDATION OF PROPOSED FORMULATION FOR OVER-REINFORCED BEAMS

The proposed formulation for over-reinforced beams is validated through the results obtained from the experimental studies done by previous researchers. Sant and Bletzacker4 tested three pairs of RC rectangular beams simply supported with a concentrated point load applied at its midspan. The critical buckling moment equation for Sant and Bletzacker4 test beams may be expressed as

… (8)

Similarly, the critical buckling moment for the 11 beams tested under uniform bending moment by Massey5 is given by

… (9)

Using Eq. (3) and (4) for B and K, respectively, the critical buckling moments for over-reinforced beams are calculated from Eq. (8) and (9) and given in Table 6, using the data reported by the authors.4,5 The magnitude of errors in the calculated critical moments by Sant and Bletzacker4 and Massey5,10 are found to be in the range of 17 to 27% and 6.2 to 20%, respectively. The critical buckling moments calculated by the proposed formulation is in agreement with experimentally observed failure moments and the errors are within 2.5 to 10%.

CONCLUSIONS

A review of the existing codes such as ACI 318, BS 8110, IS 456, EC 2, and AS 3600 shows that there is no codal formulation available presently to predict the critical buckling moment in RC slender beams. This is apparently not required when the beams are within the slenderness limits prescribed by the codes. All the codes invariably specify the slenderness limits for RC beams, in order to ensure that the beams have sufficient lateral stability against lateral buckling. The codal recommendations on slenderness limits are based on experimental and theoretical studies carried out by Hansell and Winter,2 Sant and Bletzacker,4 and Massey.5 The experimental data, however, is limited and does not include under-reinforced beams. In the present study, tests were carried out on seven under-reinforced slender beams, which all failed by buckling at critical loads that were substantially less than those predicted by the formulations due to Hansell and Winter,2 Sant and Bletzacker,4 and Massey.5 Also, the tests show that the present limits on slenderness ratios do not guaranty that failure will not occur due to lateral instability.

The analytical prediction of critical buckling moment of RC beams requires proper prediction of flexural and torsional rigidities of the RC beams. A simple formulation is proposed in this study for these rigidities. The proposed formulation for critical buckling moment yields values of M^sub bcr^ that agree very closely with the experimental results carried out in this study for under-reinforced sections. They also agree well with respect to the experimental results reported in literature for over-reinforced sections.

NOTATION

A^sub c^ = gross area of concrete

A^sub l^ = area of longitudinal steel

A^sub t^ = cross-sectional area of one leg of transverse stirrups

A^sub o^ = area enclosed by centerline of longitudinal steel

B = effective flexural rigidity

b = breadth of beam

b^sub 1^ = spacing between longitudinal bars along breadth

C^sub 1^ = constant to account for type of loading

C^sub 2^ = constant to account for support condition

c = distance from extreme compression fiber to neutral axis

c^sub u^ = depth of neutral axis at ultimate load

d = effective depth of beam

d^sub 1^ = spacing between longitudinal bars along depth

d^sub s^ = diameter of longitudinal steel

E^sub c^, E^sub s^ = elastic modulus of concrete and steel, respectively

E^sub r^ = reduced modulus of concrete

E^sub sec^ = secant modulus of concrete

E^sub tan^ = tangent modulus of concrete

f’^sub c^ = characteristic compressive strength of concrete cylinder, MPa

f^sub r^ = flexural tensile strength of concrete

G^sub c^, G^sub s^ = rigidity modulus of concrete and steel, respectively

h = overall depth of beam

I^sub cr^ = cracked moment of inertia of beam at ultimate

I^sub eff^ = effective moment of inertia of beam

I^sub gr^ = gross moment of inertia of beam

I^sub sy^ = second moment of area of longitudinal steel about minor axis

K = effective torsional rigidity

L = span of beam

M = bending moment

M^sub bcr^ = critical buckling moment

M^sub R^ = flexural cracking moment

M^sub u^ = ultimate flexural bending moment

p^sub o^ = perimeter of centerline of longitudinal steel

s = spacing of shear stirrups

W^sub bcr^ = critical buckling load

W^sub test^ = failure load from experiment

W^sub u^ = ultimate flexure failure load

β^sub c^ = St. Venant’s torsional constant

µ = rigidity multiplier

υ^sub c^,υ^sub s^ = Poisson’s ratio of concrete and steel, respectively

ρ^sub l^ = longitudinal steel ratio

ρ^sub t^ = transverse steel ratio

REFERENCES

1. Timoshenko, S. P., Theory of Elastic Stability, 2nd Edition, McGraw-Hill Book Co., New York, 1961, 354 pp.

2. Hansell, W., and Winter, G., “Lateral Stability of Reinforced Concrete Beams,” ACI JOURNAL, Proceedings V. 56, No. 5, Sept. 1959, pp. 193-214.

3. Siev, A., “The Lateral Buckling of Slender Reinforced Concrete Beams,” Magazine of Concrete Research, V. 12, No. 36, Aug. 1960, pp. 155-164.

4. Sant, J. K., and Bletzacker, R. W., “Experimental Study of Lateral Stability of Reinforced Concrete Beams,” ACI JOURNAL, Proceedings V. 58, No. 6, Dec. 1961, pp. 713-736.

5. Massey, C., “Lateral Instability of Reinforced Concrete Beams Under Uniform Bending Moments,” ACI JOURNAL, Proceedings V. 64, No. 3, Mar. 1967, pp. 164-172.

6. King, G.; Pauli, W.; and Sprey, W., “Lateral Stability of Prestressed and Ordinary Reinforced Concrete Beams,” Anniversary Publication, F. Levi, ed., Politecnico di Torino, Dipartmento di Ingegneria Srtucturale, 1989, pp. 238-243.

7. King, G.; Pauli, W.; and Sprey, W., “Lateral Stability of Prestressed and Ordinary Reinforced Concrete Beams,” FIP Brochure, 1990, pp. 151-155.

8. Aydin, R., and Kirac, N., “Lateral Buckling of Reinforced Concrete Beams with Lateral Support,” Structural Engineering and Mechanics, V. 6, No. 2, Mar. 1998, pp. 161-172.

9. Allen, H. G., and Bulson, P. S., Background to Buckling, McGraw Hill, UK, 1980, 582 pp.

10. Massey, C., and Walter, K. R., “The Lateral Stability of a Reinforced Concrete Beam Supporting a Concentrated Load,” Building Science, V. 3, No. 1, Sept. 1969, pp. 183-187.

11. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-02) and Commentary (318R-02),” American Concrete Institute, Farmington Hills, Mich., 2002, 443 pp.

12. British Standards Institution, BS 8110, “Code of Practice for Structural Use of Concrete,” London, 1985, pp. 3, 8.

13. Bureau of Indian Standards, IS 456, “Code of Practice for Plain and Reinforced Concrete for General Building Construction,” New Delhi, 2000, 39 pp.

14. European Committee of Standardization, EC-2, “Design of Concrete Structures,” Brussels, 1997, 160 pp.

15. Council of Standards Australia, AS 3600, “Concrete Structures,” Sydney, 2001, 92 pp.

16. Pillai, S. U., and Menon, D., Design of Reinforced Concrete Structures, Tata McGraw Hill, New Delhi, 2002, 366 pp.

17. Warner, R. F.; Rangan, B. V.; Hall, A. S.; and Faulkes, K. A., Concrete Structures, Addison Wesley Longman, Melbourne, 164 pp.

18. Timoshenko, S. P., and Goodier, J., Theory of Elasticity, McGraw Hill, New York, 1985, 251 pp.

19. Tavio, and Teng, S., “Effective Torsional Rigidity of Reinforced Concrete Members,” ACI Structural Journal, V. 101, No. 2, Apr. 2004, pp. 252-260.

20. Hsu, T. T. C., “Post-Cracking Torsional Rigidity of Reinforced Concrete Sections,” ACI JOURNAL, Proceedings V. 70, No. 5, May 1973, pp. 352-360.

P. Revathi is a doctoral research scholar at the Department of Civil Engineering, Indian Institute of Technology, Madras, India. She received her master’s degree from Pondicherry Engineering College, Pondicherry, India, in 2002.

Devdas Menon is a professor of structural engineering at the Department of Civil Engineering, Indian Institute of Technology. His research interests include reinforced and prestressed concrete, structural reliability, structural dynamics, wind, and earthquake engineering.

Copyright American Concrete Institute Mar/Apr 2006

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