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Microsoft Office 2007 Crack For All Versions Indirect . EIddxqx,tdwdx1AGddxEIddx.EIfrac mathrm d varphi mathrm d xrightqx,t frac mathrm d wmathrm d xvarphi frac 1kappa AGfrac mathrm d mathrm d xleftEIfrac mathrm d varphi mathrm d xright.The Timoshenko beam theory for the static case is equivalent to the Euler Bernoulli theory when the last term above is neglected, an approximation that is valid when.EIL2. AG1displaystyle frac EIkappa L2AGll 1where.Combining the two equations gives, for a homogeneous beam of constant cross section,EI d.EIAG d. 2qdx. 2displaystyle EIcfrac mathrm d 4wmathrm d x4qx cfrac EIkappa AGcfrac mathrm d 2qmathrm d x2The bending moment Mxxdisplaystyle Mxx and the shear force Qxdisplaystyle Qx in the beam are related to the displacement wdisplaystyle w and the rotation displaystyle varphi.These relations, for a linear elastic Timoshenko beam, are MxxEI xand.Qx AG wx. displaystyle Mxx EIfrac partial varphi partial xquad textandquad Qxkappa AGleft varphi frac partial wpartial xright.Derivation of quasistatic Timoshenko beam equations.From the kinematic assumptions for a Timoshenko beam, the displacements of the beam are given by.Then, from the strain displacement relations for small strains, the non zero strains based on the Timoshenko assumptions arexxuxxz x xz1.Since the actual shear strain in the beam is not constant over the cross section we introduce a correction factor displaystyle kappa such thatxz1.The variation in the internal energy of the beam isULAxxxx2xzxz d.A d. LLAz xxxxz wx d.We give the results of large deflection experiments involving the bending and twisting of 1 mm diameter nickeltitanium alloy rods, up to 2 m in length.These results. A d.Ldisplaystyle delta Uint Lint Asigma xxdelta varepsilon xx2sigma xzdelta varepsilon xzmathrm d Amathrm d Lint Lint Aleft zsigma xxfrac partial delta varphi partial xsigma xzkappa left delta varphi frac partial delta wpartial xrightrightmathrm d Amathrm d LDefine.Mxx Az xx d. A Qx Axz d.Adisplaystyle Mxx int Azsigma xxmathrm d A Qx kappa int Asigma xzmathrm d AThenULMxxxQxwx d.Ldisplaystyle delta Uint Lleft Mxxfrac partial delta varphi partial xQxleft delta varphi frac partial delta wpartial xrightrightmathrm d LIntegration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads toULMxxxQx Qxx w d.Ldisplaystyle delta Uint Lleftleftfrac partial Mxxpartial x Qxrightdelta varphi frac partial Qxpartial xdelta wrightmathrm d LThe variation in the external work done on the beam by a transverse load qx,tdisplaystyle qx,t per unit length isWLq w d.Ldisplaystyle delta Wint Lqdelta wmathrm d LThen, for a quasistatic beam, the principle of virtual work givesUWLMxxxQx Qxxq w d.L0displaystyle delta Udelta Wimplies int Lleftleftfrac partial Mxxpartial x Qxrightdelta varphi leftfrac partial Qxpartial xqrightdelta wrightmathrm d L0The governing equations for the beam are, from the fundamental theorem of variational calculus,MxxxQx0 Qxxq0displaystyle frac partial Mxxpartial x Qx0 frac partial Qxpartial xq0For a linear elastic beam.MxxAz xx d. 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