International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 08 | Aug 2019 www.irjet.net p-ISSN: 2395-0072 © 2019, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 745 FLEXURAL ANALYSIS OF THICK BEAMS USING TRIGONOMETRIC SHEAR DEFORMATION THEORY A.T. Kunte 1 , Dr. G.R. Gandhe 2 , D.H. Tupe 3 , Dr. S.L. Dhondge 4 1 A. T. Kunte, P. G. Student, Department of Civil Engineering, Deogiri Institute of Engineering and Management Studies, Aurangabad, Maharashtra, India. 2 Dr. G. R. Gandhe, Professor, Department of Civil Engineering, Deogiri Institute of Engineering and Management Studies, Aurangabad, Maharashtra, India. 3 D. H. Tupe, Assistant Professor, Department of Civil Engineering, Deogiri Institute of Engineering and Management Studies, Aurangabad, Maharashtra, India. 4 Dr. S. L. Dhondge, Professor, Department of First Year Engineering, Deogiri Institute of Engineering and Management Studies, Aurangabad, Maharashtra, India. ---------------------------------------------------------------------***---------------------------------------------------------------------- Abstract – In this present study, A trigonometric shear deformation theory is developed for flexural analysis of beams, in which number of variables are same as that in first-order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effect and satisfy the zero transverse shear stress condition at top and bottom surface of the beam. The Governing differential equation and boundary condition of the theory are obtained by using principle of virtual work. The fixed beam subjected to uniformly distributed load is examined using present theory. The numerical results obtained are compared with those of Elementary, Timoshenko and Higher-order shear deformation theory and the available solution in the literature. Key Words: Trigonometric shear deformation theory, Transverse shear stresses, Flexure, Principle of virtual work, Axial shear stress, Equilibrium equation, Thick beam, Displacement. 1. INTRODUCTION It is well-known that elementary theory of bending of beam based on Euler-Bernoulli hypothesis that the plane sections which are perpendicular to the neutral axis before bending remain plane and perpendicular to the neutral axis after bending, implying that the transverse shear and transverse normal strains are zero. Thus, the theory disregards the effects of the shear deformation. It is also known as classical beam theory. The theory is applicable to slender beams and should not be applied to thick or deep beams. When elementary theory of beam (ETB) is used for the analysis thick beams, deflections are underestimated and natural frequencies and buckling loads are overestimated. This is the consequence of neglecting transverse shear deformations in ETB. Bresse [1], Rayleigh [2] and Timoshenko [3] were the pioneer investigators to include refined effects such as rotatory inertia and shear deformation in the beam theory. Timoshenko showed that the effect of transverse vibration of prismatic bars. This theory is now widely referred to as Timoshenko beam theory or first order shear deformation theory (FSDT) in the literature. In this theory transverse shear strain distribution is assumed to be constant through the beam thickness and thus requires shear correction factor to appropriately represent the strain energy of deformation. To remove the discrepancies in classical and first order shear deformation theories, higher order or refined shear deformation theories were developed and are available in the open literature for static and vibration analysis of beam. Levinson [4], Bickford [5], Rehfield and Murty [6], Krishna Murty [7], presented parabolic shear deformation theories assuming a higher variation of axial displacement in terms of thickness coordinate. These theories satisfy shear stress free boundary conditions on top and bottom surfaces of beam and thus obviate the need of shear correction factor. There is another class of refined theories, which includes trigonometric function to represent the shear deformation effects through the thickness. Vlasov and Leontev [8], Stein [9] developed refined shear deformation theories for thick beams including sinusoidal function in terms of thickness coordinate in displacement field. However, with these theories shear stress free boundary conditions are not satisfied at top and bottom surfaces of the beam. Further Ghugal and Dahake [10] developed a trigonometric shear deformation theory for flexure of thick beam or deep beams taking into account transverse shear deformation effect. The number of variables in the present theory is same as that in the first order shear deformation theory. The trigonometric function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. A study of literature by Ghugal and Shimpi [11] indicates that the research work dealing with flexural analysis of thick beams using refined trigonometric and hyperbolic shear deformation theories are very scarce and is still in infancy. In this paper, a trigonometric shear deformation theory is developed for flexural analysis of thick beams. The theory is applied to a fixed beam to analysed the axial displacement, Transverse displacement, axial bending stress and transverse shear stress. The numerical results have been computed for various length to thickness ratios of the beams