Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 138 Structural elements The Lagrangian is L - I Ees Eeb dx FoZ L Ls Lb where Ls - y Ees dx FữZs LỴ Lb - ị Eeb dx F0Zb L Since the Lagrangian can be split into the sum of a shear term Ls and a bending term Lb it is anticipated that shear and bending deflections are uncoupled from each other. Furthermore bending is still governed by the Bernoulli-Euler model. Hence the variational calculation is detailed here for the shear term only SLs - KGsf W H jdx FQ8Zs L 0 t Q dx J dx J Once more expression is integrated by parts and the cofactors of ỗZs are set to zero. ỗLs F0ỖZs - KGs dZ 8Zs icGsi -iAsZ dx 0 9x JQ Jo dx2 The deflection due to elastic shear is thus governed by the equation Ed2ZS kGS - - 0 V dx2 J with the boundary conditions 9Zs Zs 0 0 KGsULZ F0 dx J L The system is consistent with equations and except the shear weighting factor K. The total deflection is Z x Zs x Zb x fJ 3x2L - x3 kGS 6Ei This result may be used to discuss the relative importance of shear and bending deflections in relation to the slenderness ratio of the beam. Let us consider for Straight beam models Hamilton s principle 139 Figure . Transverse deflection due to elastic shear and bending for two cantilever rods upperplot L R 10 lower L R 2 instance a circular cylindrical rod of radius R and Poisson s ratio V K 1. We obtain Z x FoL nR2E The result is evidence that the relative contribution of bending to the total deflection increases as L R 2 validating thus the Bernoulli-Euler model for slender beams as illustrated in Figure . The Rayleigh-Timoshenko dynamic model The same problem as in the last subsection is studied now in dynamics. Focusing on the case of beams of small slenderness ratios for which shear effects are 140 Structural elements important the kinetic energy due to the rotation of the cross-sections with respect to the bending axis can no longer be neglected. The kinetic energy of a .
