Theoretical Analysis Of Beams With Fixed Ends

In this dissertation, the plane bending problem for homogeneous isotropic,orthotropic and functionally graded orthotropic beams with fixed ends is studied. The main work and conclusions are as follows:(1) Based on the two types of fixed boundary conditions introduced by Timoshenko and Goodier, a new type of fixed boundary condition for the fixed end is proposed. By using the Airy stress function method and the new type of fixed boundary condition, the elasticity solution of stress and displacement for four kinds of homogeneous isotropic beams with fixed ends is derived and compared with the existed elasticity solution and the finite element solution. The comparison shows that the accuracy of elasticity solution can be improved efficiently by employing the new type of fixed boundary condition.(2) We improved the fixed boundary condition proposed by Dai and Ji.Employing the Airy stress function method and the improved fixed boundary condition,the elasticity solution of stress and displacement for four kinds of homogeneous isotropic beams with fixed ends is derived. Comparison with the existed elasticity solution and the finite element solution shows that the accuracy of elasticity solution can also be increased efficiently by using the improved fixed boundary condition.(3) The plane bending problem for four kinds of homogeneous orthotropic beams with fixed ends is studied by using the Airy stress function method. The above two types of new simplified fixed boundary conditions proposed in this dissertation are employed, and the elasticity solutions of stress and displacement are obtained,respectively. The comparison reveals that the above two type of new simplified fixed boundary conditions are equivalent.(4) The plane bending problem for the homogeneous orthotropic beams with two fixed ends and arbitrary thickness is investigated by using the state space method. The beam may be subjected to arbitrary loads on its upper or/and lower surface. The boundary displacement functions are introduced in the assumed expression of displacement and defined as the state variables, then the homogeneous state equation about the state variable vector is constructed and the state space solution is obtained.Especially, the state space solution for the beam acted by the symmetric loads on its upper or/and lower surface is also presented. Three examples are analyzed to verify the efficiency of the present method.(5) The functionally graded orthotropic cantilever beam with arbitrary thickness is studied by using the state space method. The beam may be subjected to arbitrary loads on its upper or/and lower surface, and the material properties may be vary along the thickness arbitrarily. The assumed expressions of stress and displacement are refined, and the boundary displacement functions are introduced too, then the homogeneous state equation about the state variable vector is obtained. Employing the approximate laminate model, the state space solution is derived. Three examples are presented to verify the efficiency of the present method, in which the beam is subjected to a transverse force, uniform load or linearly distributed load on its upper surface,respectively.(6) Based on the classical beam theory and the plane cross-section assumption, a cantilever beam composed of three-layers and a statically indeterminate beam composed of multi-layers are analyzed. The calculating formulae of displacement and stress are obtained. The influence of the layer thickness and the distance between two layers is discussed. The results can be used as the basis for optimizing design.