Elastic Static Analysis Of Beams And Plates With Variable Thickness

This thesis studies the static properties of beams and plates with variable thickness, based on the small-strain linear elasticity theory which does not rely on any artificial hypotheses. Firstly, for the simple-supported boundary conditions, the two-dimensional elasticity solution of simple-supported beams with variable thickness and the three-dimensional elasticity solution of simple-supported rectangular plates with variable thickness under arbitrary loads are presented by using the Fourier series expansion method. Then, for the non-simply-supported boundary conditions, the clamped boundary condition is taken as an example. The elasticity solution of varying thickness beams with one end clamped and the other end simply supported under static loads are presented by introducing the unit pulse functions and using the boundary relaxation method. Finally, the Fourier series expansion method is extended to study the bending problem of beams with variable thickness and rectangular plates with variable thickness, subjected to thermo-mechanical load where the beams and the plates are, respectively, made of isotropic materials, orthotropic materials, functionally graded materials and piezoelectric materials.The detailed contents of the thesis are given as follows:(1) On the basis of the two-dimensional plane elasticity theory, the general expressions of displacements, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at two ends of the beam, have been deduced. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams.(2) On the basis of three-dimensional elasticity theory, the general expressions for displacements and stresses of the rectangular plate under static loads, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at four edges of the plate, are analytically derived. The unknown coefficients in the stress solutions are approximately determined by using the double Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the plates.(3) For the varying thickness beams with one end clamped and the other end simply-supported, by introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. Then the exact analytical solution is obtained. The unknown coefficients can be determined by using the Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge.(4) For the functionally graded materials, the Young's modulus is graded through the thickness following the exponential-law and the Poisson's ratio keeps constant. Firstly, the analytical solution of the governing differential equations can be obtained. Then using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the structures, the bending problem of simply-supported functionally graded beams with variable thickness and simply-supported functionally graded rectangular plates with variable thickness are studied.(5) For the piezoelectric materials, the general expressions of displacement fields and piezoelectric field, which exactly satisfy the governing differential equations and the simply-supported boundary conditions, are derived firstly. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the structures. The bending problem of simply-supported piezoelectric beams with variable thickness and simply-supported transversely isotropic piezoelectric rectangular plates with variable thickness are studied.(6) For the simple-supported beam with variable thickness and simple-supported rectangular plate with variable thickness under the temperature field, we need to solve the temperature distributions on beam and plate by using Fourier sinusoidal series expansions according to the temperature boundary condition at first. Then the temperature load is exerted to the beam and the plate. The two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads and the three-dimensional thermoelastic analysis of rectangular plates with variable thickness subjected to thermo-mechanical loads are presented.(7) For the multi-span plates, the exact expressions of the displacements, which satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived firstly. The reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients are then determined by the boundary conditions on the upper and lower surfaces of the plate. Three-dimensional elasticity solution of simple-supported rectangular plate on point supports, line supports and elastic foundation are studied and the simply-supported functionally graded rectangular plates with internal elastic line supports are also presented.