Lateral torsional buckling of rectangular reinforced concrete beams

The study presents the results of an investigation aimed at examining the lateral stability of rectangular reinforced concrete slender beams. A total of eleven reinforced concrete beams having a depth to width ratio between 10.20 and 12.45 and a length to width ratio between 96 and 156 were tested. Beam thickness, depth and unbraced length were 1.5 to 3.0 in., 18 to 44 in., and 12 to 39.75 ft, respectively. The initial geometric imperfections, shrinkage cracking conditions and material properties of the beams were carefully determined prior to the tests.;Each beam was subjected to a single concentrated load applied at mid-span by means of a gravity load simulator that allowed the load to always remain vertical when the section displaces out of plane. The loading mechanism minimized the lateral translational and rotational restraints at the point of application of load to simulate the nature of gravity load.;Each beam was simply-supported in and out of plane at the ends. The supports allowed warping deformations, yet prevented twisting rotations at the beam ends.;In the experimental part of the study, reinforced concrete beams with initial imperfections (sweep) failed under loads lower than the critical loads corresponding to the geometrically perfect configuration of the respective beams. The maximum load carried by an imperfect beam is known as the limit load (PL). In the present study, the limit load (PL) and the critical load (Pcr) were distinguished.;In the first part of the analytical investigation, a formula was developed for determining the critical loads corresponding to the lateral torsional buckling of rectangular reinforced concrete beams. The effects of shrinkage cracking and inelastic stress-strain properties of concrete and the contribution of longitudinal reinforcement to the lateral stability are accounted for in the critical load formula. The second part of the investigation focused on developing a formula for the estimation of limit loads of reinforced concrete beams with initial lateral imperfections. The proposed limit load formula was obtained by introducing the destabilizing effect of sweep as a reduction term to the critical load equation.;Finally, the experimental results were compared to the proposed analytical solution and to various lateral torsional buckling solutions in the literature. The formulation proposed in the present study was found to agree well with the experimental results. The good correlation with the experimental results and the incorporation of the geometric and material nonlinearities into the formula makes the proposed solution, given below for a simply supported rectangular reinforced concrete beam loaded with a concentrated load at midspan, practical for design purposes: Pcru=4˙Mcr L-uto˙ 48˙Ec˙Iy sinf ult˙L 3 1 where PL is the limit load; L is the unbraced length of the beam; u to is the sweep at the top of the beam at midspan; Ec is the elastic modulus of concrete; I y is the second moment of area of the beam section about the minor axis; &phis;ult is the angle of twist of the beam at midspan corresponding to the limit load (PL). Mcr is the critical moment corresponding to the geometrically perfect configuration of the beam, obtained from Equation (2):; Mcr=4.23L ˙1-1.74˙eL ˙BoGC o ˙Bo˙GC o 2 where Bo is the lateral bending rigidity, obtained from Equation (3); (GC)o is the torsional rigidity, calculated from Equation (4); e is the vertical distance of the load application point from the centroid of the midspan cross section.; Bo=b3 ˙c12 ˙11+w˙M craMcr 2˙ch-1 ˙E sec+Ec2 3 &parl0;GC&parr0;o=Esec +Ec4˙1+n ˙b3˙h 3˙1-0.63˙b h 4 where b and h are the width and height of the beam, respectively; c is the depth of the neutral axis from the compression face; Mcra is the cracking moment; o is a constant, which has a value of 1 in the absence of restrained shrinkage cracks in concrete and a value of 2/3 in the presence of restrained shrinkage cracks and upsilon is Poisson's ratio of concrete. Esec is the secant modulus of elasticity of concrete corresponding to the extreme compression fiber strain at midspan at the instant when Mcr is reached.