Geometrically Nonlinear Exact Mathematical Model And Numerical Solution Of Timoshenko Sandwich Beams

On the basis of accurately considering axial extension and the first-order transverse shearing, geometrically nonlinear governing equations for Timoshenko sandwich beams, subjected to both thermal and mechanical loads, are formulated. By using a shooting method, numerical solutions to the thermal elastic stability of the Timoshenko sandwich beams are obtained and the equilibrium configurations and equilibrium paths of the beams with thermoelastic nonlinear large deformation under thermomehcanical loads are presented. The main contents of this thesis are as follows:1. By considering both the transverse shearing and the axial extension, governing equations for the Timoshenko sandwich beams with large elastic deformations under termomechanical loads are established, which consisting of a system of strong nonlinear ordinary differential equations with seven independent unknown functions. By using shooting method to numerically solve the two-point boundary value problem numerical solutions are obtained. The numerical results show that when the material properties and the thicknes of the skin layers are symmetric about the geometrically middle plane and the temperature rise is uniform the sandwhich beam undergoes thermal postbuckling deformation, which means that the thermal postbuckling equilibrium paths of the central deflection versus the temperature rise are bifurcation form. When the beam is subjected to transversely non-uniform temperature rise, equilibrium paths of the beam with pinned-pinned ends show the characteristic behaviors of thermal bending but for the beam with fixed-fixed ends they are still bifurcation forms.2. Based on the above theory and numerical method, thermal elastic stability of of Timoshenko sandwich beam with the pinned-fixed ends, subjected to uniform temperature rise and supported by an elastic foundation, are investigated. When the material properties and the thickness of the skin layers are symmetric, transition of the buckling modes produced by the stiffness of the elastic foundation is examined. Numerical results show that the buckling modes of the uniform heated beam will transform and the signs of the bending moment (or the curvature) at the fixed end will change at the transition values of the stiffness parameter. Different stiffness transition values indicate different transition characteristics between the buckling modes. Curves of critical temperature rise depending continuously on the foundation stiffness parameter are plotted, from which it can be found that ranks of the critical buckling modes of the beam increase with the increase in the value of the stiffness parameter. Then, for the case that the beam is constrained by end rotational springs, effects of the variation in the values of the thickness and the material properties of the skin layers on the post-buckling behaviors of the beam are analyzed. Finally, thermal post-buckling characteristics of the sandwich beams constrained by both the elastic foundation and the end rotational springs are studied quantitatively.3. Thermal post-buckling of the fixed-fixed sandwich beams with FGM top skin and homogenous bottom skin under transversly non-uniform temperature rise is studied. It is assumed that the material proiperty changes continuously in the power law along the thickness derection of the top skin. Temperature distribuation of the beam under non-uniform heating is determined by the heatconduction equation independently. By using shooting method to solve the corresponding nonlinearboundary value problem thermal post-bucklinmg responses of the sandwich beam with FGM skinlayer are arrived at. Influences of parameters of the material gradient and the nonuniform temperaturerise on the thermal post-buckling behavior. Numerical results show that for the fixed-fixed sandwichbeam the post-buckling behavior is bifurcation form even under transversly non-uniform temperaturerise and with unsymmetric material distribution about the geometric middle plane. When the volumefraction of the ceramic is specified the cirtical buckling temperature decreases with the increase of thenon-uniform temperature rise parameter. For the same mean temperature parameter, the deformationsof beam become more significant when the non-uniform temperature parameter increases.