International Journal of Engineering Trends and Technology (IJETT) Volume 39 Number 2- September 2016

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1 Study of Structural Behaviour of RC Slender Beams Alfina Abdul Samad 1, Ramadass.S 2, Mervin Ealiyas Mathews 3 1 M. Tech in SECM, Department of Civil Engineering, JBCMET Ernakulam- India 2 Associate Professor, Department of Civil engineering, SOE CUSATL 3 Assistant Professor, Department of Civil Engineering, JBCMET Ernakulam India Abstract Slender beams are occasionally encountered in reinforced concrete constructions. The behavior of slender reinforced concrete beams is different from that of normal proportioned beams. In case of slender beams there are limitations in existing theoretical formulations to estimate the failure moment capacity. Highly slender beams are prone to sudden instability mode of failure. Moderately slender beams are susceptible to slenderness effects and they may undergo flexural failure at moment values less than flexural capacity corresponding to material failure. This reduction in moment capacity of RC slender beams is not mentioned in IS code. This paper presents the results of some rectangular slender beams on the basis of theoretical formulations proposed by the researchers analytically using ANSYS software. The critical buckling moment is found for a verity of cases and comparing with analytical test results. Reduced moment capacities of RC slender beams are find out the basis of slenderness ratio. Keywords RC Slender Beam, Critical Buckling Moment, slenderness ratio I. INTRODUCTION Slender beams are occasionally encountered in reinforced concrete (RC) construction. Slenderness is the important factor which largely depends on stability of the RC beams. The present code specification says slenderness is related to beam dimensions only. But several studies and researchers conclude that Slenderness in beams is depending on many factors not only in beam dimensions. They largely depend on reinforcement ratio. Slenderness can adversely influence the behavior of RC beams, by inducing a sudden instability failure or causing an appreciable reduction in flexural strength. The behavior of slender reinforced concrete (RC) beams is different from that of normal proportioned beams. Highly slender beams are prone to sudden instability mode of failure. Moderately slender beams are also susceptible to slenderness effects and they may undergo flexural failure at moment values less than flexural capacity corresponding to material failure (M uf ).This behavior is well recognized and accounted for in steel design, but not adequately in concrete design. A. General II. METHODOLOGY The methodology adopted is as follows: Collecting the experimental test results Analysis of slender RC beams using ANSYS Comparison of experimental and analytical test results Parametric study of RC slender beams Finding out the critical buckling moment Finding out the normalized slenderness ratio Finding out the moment reduction factor Finding out the modes of failure B. Theory and formulation The behavior of slender reinforced concrete (RC) beams is different from that of normal proportioned beams. Highly slender beams are prone to sudden instability mode of failure. Moderately slender beams are also susceptible to slenderness effects and they may undergo flexural failure at moment values less than flexural capacity corresponding to material failure (M uf ). Proper estimation of critical buckling moment (M bcr ) is necessary to predict the behavior more accurately as the slenderness depends on relative values of M uf and M bcr. The dimensions of the beam where taken on the basis of limiting slenderness ratio. Where, C1 and C2 are factors that account for type of loading and beam end conditions respectively; E c is the elastic modulus of concrete R is the flexural resistance factor for the given beam section (with width b and effective depth d); and α and β are respectively the effective flexural rigidity and torsional rigidity coefficients that account for cracking and tension stiffening. (1) (2) ISSN: Page 88

2 C. Critical Buckling Moment The generalized expression for calculating the critical buckling moment [3] is (1) coefficient. is the elastic modulus of concrete and is the rigidity modulus of concrete. The general equation (5) Where, C 1 is a factor that accounts for type of loading (having a value of π under pure bending, 1.35π under central point loading,1.09π under point loads at onethird span,3.54 for uniformly distributed loads),c 2 accounts for the end conditions of the beam (having a value of unity for simply supported end conditions and 0.5 for fixed condition) and C 3 accounts for the location of the load with respect to the centroidal axis of the rectangular beam (having a value of unity when the load is applied at the centroid).the general equation for calculating the constant C 3 is [2] (2) Where, Percentage of tension reinforcement Longitudinal reinforcement ratio Transverse reinforcement ratio Rigidity multiplier taken as 1.5 Elastic modulus of steel (6) The equation for calculating elastic modulus of concrete and rigidity modulus of concrete from [13] (7) Where, and are the effective flexural rigidity and effective torsional rigidity, is height of load above shear centre, L is the total length of the beam. Considering an equivalent cylinder strength, Assuming an average value of ratio of concrete average (8) =0.15 for Poisson s (9) (10) (11) Fig 3.1 position of transverse load above shear centre The general equations for calculating,, α and β is from [3] Area of longitudinal reinforcement Area of tension reinforcement Area of compression reinforcement Longitudinal reinforcement ratio Transverse reinforcement ratio Cross sectional area of one leg of transverse stirrups Gross concrete area Spacing of stirrups (3) (4) Where, and are the effective flexural rigidity coefficient and effective torsional rigidity ISSN: Page 89

3 Failur Moment (knm) International Journal of Engineering Trends and Technology (IJETT) Volume 39 Number 2- September 2016 The experimental program reported in [3] has been considered. Three beams are selected from the experimental tests beams. The first beam of 5000 mm long, 120 mm wide, 650mm deep and the second and third beams of 6000 mm long, 100mm wide and 450mm deep. Fig 2: cross-sectional properties of the beam Bea m label Limiting slendern ess ratio λ Percentag e tension steel pt Stirrup diameter (mm) Stirrup spacing (mm) Average cube strength (MPa) B B B D. Slenderness Limits and Moment Reduction Factor Slenderness limits are specified in design codes to avoid instability failure. Revathi and Menon[12], proposed a limit λ to the slenderness ratio Ld/b 2, incorporating the influence of various design variables (P t, P c, ρ tr, f ck and f y ). The expression was derived by equating M uf and M bcr. The normalized slenderness ratio (12) For values of < 1, the flexural mode of failure is expected to prevail, and for > 1, the instability mode of failure is expected. However, some interaction between these two modes of failure can be expected, along with reduced moment capacity, in the neighborhood of = 1 Slenderness effects do not manifest when is less than 0.5, and such non slender beams are expected to have a flexural strength given by M u = M uf (i.e., η = 1). Similarly, in the case of extremely slender beams with > 1.27, failure is expected to occur by buckling, with M u = M bcr. In the intermediate range, 0.5 < > 1.27, a reduction in moment capacity is observed, along with the tendency for the beam to deflect laterally and twist with increasing. In this transition zone, clearly slenderness (instability) effects manifest, although the sudden (brittle) instability mode of failure is not likely to occur for values of < 1. The moment reduction factor can be calculated using the equation given below [3] E. Validation (13) TABLE 1: PRELIMINARY DETAILS OF BEAMS [1] The beams shown in the above table are analyzed using ANSYS software and find out the failure moment and load deflection behaviors. The analytical results and experimental results are compared. This comparative result is shown below. TABLE.2: COMPARISON OF EXPERIMENTAL AND ANALYTICAL RESULTS Beam label Experiment al Failure Moment M exp (knm) Analytic al failure moment M u (knm) Experimental Maximum load (kn) B1 B2 B3 Fig 3: Analytical and experimental failure moments The failure moments of three beams B1, B2 and B3 obtained analytically is compared with experimental values. For beam B1, the failure moment values obtained from analytical and Analytical maximum load (kn) B B B EXPERIMENTAL ANALYTICAL ISSN: Page 90

4 experimental test are same. For beam B2 failure moments obtained from experimental test is 118.5kNm. The failure moment of beam B2 from analysis is obtained as kNm. There is no large variation from experimental and analytical results. The failure moment of beam B3 obtained from experimental test is 126.9kNm and that obtained analytically is kNm.there is no large variation between analytical and experimental results. Fig 4: Load-deflection curve for B1 Fig 5: Load-deflection curve for B2 III. RESULTS AND DISCUSSION A. Calculation of Critical Buckling Moment and Ultimate Flexural Failur Moment The behaviors of slender reinforced concrete beams are different from that of normally proportioned beams. Highly slender beams are prone to sudden instability mode of failure. Moderately slender beams are also susceptible to slenderness effects and they may undergo flexural failure at moment values less than flexural capacity corresponding to material failure (M uf ). Proper estimation of critical buckling moment (M bcr) is necessary. BEAM Table 3 Loading and support conditions CASE B1 1.1 B2 B CONDITION CONDITION CASE B eff K eff M bcr M uf Fig 6: Load-deflection curve for B E E E E E E E E E E E E E E E E E E E E E E E E ISSN: Page 91

5 Table 4. Calculation of Critical buckling moment B. Normalized Slenderness Ratio and Moment reduction Factor TABLE 5 NORMALIZED SLENDERNESS RATION AND MOMENT REDUCTION FACTOR M bcr M uf M u η BEAM (knm) (knm) After analyzing the table 5 which found out by analytical method is compared with the results of the experimental results. In the case 1.1, Muf = Mu and. So 1.1 will face flexural failure. While comparing the failure moment of B2, with three different loading conditions shows similar results, that is the value of Ultimate fluxeral failure moment is greater than critical buckling mome ( Muf>Mbcr). The value of normalized slenderness ratio lies in the region between 0.5 and 1.27 (0.5< <1) from [1]. That means the beam will face flexural tension failure with reduced capacity. In the three different loading cases of B3, the first two conditions show similar mode of failure with experimental results. That is ultimate moment capacity is greater than critical buckling moment, but the normalized slenderness ratio is greater than 1.27 and it will face sudden instability mode of failure. But in the third case 3.3, the value of normalized slenderness ratio is in the region.5< <1.27. So at this case, the beam will face flexural mode of failure with reduced capacity. TABLE 6 MODES OF FAILURE CASE MODE OF FAILURE FLEXURAL TENSION FLEXURAL TENSION FLEXURAL TENSION FLEXURAL TENSION SUDDEN INSTABILITY SUDDEN INSTABILITY FLEXURAL TENSION IV. CONCLUSION An analytical study is conducted on slender beams and found out that reduction in moment capacity is occurred in moderately slender beams with slenderness ratio. The factors depends on slenderness ratio is studied and found out that it depends on beam dimensions, longitudinal reinforcements, transverse reinforcement, grade of concrete and loading and support conditions The equation for critical buckling moment is validated using the analytical data and found that it is not applicable for fixed boundary conditions. Normalized slenderness ratio and moment reduction factors are found out for different support conditions. From the normalized slenderness ratio, different modes of failure of beams are obtained to measure the structural stability of RC beams. The moment reduction factor can be multiplied with the ultimate flexural failure moment obtained from IS code for design of moderately slender beams in future. ACKNOWLEDGMENT We express our sincere gratitude to Dr. T. G. Santhosh Kumar, Principal, JBCMET and Prof. K. Soman, Head of the Civil Engineering Department, for all the facilities provided to successfully complete this work. We are also very thankful to all the faculty members of the department, especially Structural Engineering specialization for their constant encouragement during the project. We would like to take the opportunity to thank our family members and all our friends who have directly or indirectly helped us in our work and in the completion of this paper. REFERENCES 1] Bureau of Indian standards IS456. Code of practice for plain and reinforced concrete for general building construction. New Delhi;2000 2] Girija K., Behaviour of slender reinforced concrete beams, PhD thesis, Indian Institute of Technology Madras, Chennai, ] Girija K and Menon D (2011) Reduction in flexural strength in rectangular RC beams due to slenderness Engineering Structures, 33 (8): ] Girija K and Menon D (2011) Improved design guidelines for slender rectangular RC beams session1-5:new model code ISSN: Page 92

6 5] Girija K and Menon D (2014) Load deflection behaviour of slender rectangular reinforced concrete beams the Indian concrete joural 6] Hansell W, Winter G. Lateral stability of reinforced concrete beams. ACI J Proc 1959;31(3): ] Hsu TTC. Post-cracking torsional rigidity of reinforced concrete sections. ACI J Proc 1973;70(5): ] Massey C. Lateral instability of reinforced concrete beams under uniform bending moments. ACI J Proc 1967;64(3): ] Massey C, Walter KR. The lateral stability of a reinforced concrete beam supporting a concentrated load. Build Sci 1969;3: ] Pillai S.U. and Menon D., Reinforced Concrete Design, Tata McGraw Hill, New Delhi, ] Revathi P. and Menon D, Estimation of critical buckling moments inreinforced concrete slender beams, ACI Structural Journal, March/April 2006 Vol. 103, No.2, pp ] Revathi P. and Menon D., Slenderness effects in reinforced concrete beams, ACI Structural Journal, July/August 2007, Vol.104, No.4, pp ] Revathi P. and Menon D., Assessment of flexural strength of slender RC rectangular beams, Indian Concrete Journal, May 2009, Vol.83, No.5, pp ] Siev A. The lateral buckling of slender reinforced concrete beams. Mag Concr Res 1960;12(36): ] Sant JK, Bletzacker RW. Experimental study of the lateral stability of RC beams. ACI J Proc 1961;33(6): ] Tavio TengS. Effective torsional rigidity of reinforced concrete members. ACI Struct J 2004;101(2): ISSN: Page 93