Buckling and Postbuckling of Beams, Plates, and Shells

The chapter presents basic concepts of the structural stability under the external applied loads. The applied loads may be in form of the mechanical or thermal loads, where the latter create thermal stresses due to the temperature gradient in the structure. When the applied loads result into the compressive stresses, the structural member may fail due to lack of stability. In general, three types of static stability are recognized. These are; the classical buckling or bifurcation, the finite disturbance buckling, and the snapthrough buckling. These different types of instabilities depend upon a number of factors such as the geometry of the structure, the initial geometric imperfection, the boundary conditions, and the material property distribution. The chapter also discusses the nature of the post-buckling path, where it may be stable or unstable paths depending upon the above factors.

This chapter presents buckling and post-buckling analysis of straight beams under thermal and mechanical loads. The Euler and Timoshenko beam theories are considered and buckling and postbuckling behaviors are discussed. The buckling analysis of beams with piezoelectric layers is presented and the effect of piezo-control on the beam stability is analyzed. The vibration of thermo-electrically excited beams in the state of buckling and post-buckling is discussed and the chapter concludes with the thermal dynamic analysis of beams. The beam material in this chapter is assumed to be functionally graded, where the presented formulations may be simply reduced to the beams with isotropic/homogeneous material.

The buckling and post-buckling of curved beams under mechanical distributed and concentrated loads and thermal loads with different types of boundary conditions are discussed in detail in this chapter. The existence of bifurcation points are examined for each type of loading. For those cases that bifurcation do not occur, the limit load is discussed. The chapter ends with the discussion of buckling and post-buckling of rings under hydrostatic pressure.

The stability of rectangular plates with induced in-plane compressive stresses resulting from the mechanical or thermal loads is discussed in this chapter. The kinematical relations, constitutive law, the equilibrium equations, and the stability equations for a rectangular plate are derived and the classical boundary conditions are presented. The critical buckling loads of rectangular plates under thermal and mechanical in-plane compressive loads are derived and the existence of bifurcation load for each type of given loading condition are discussed. The effect of piezoelectric control on buckling of rectangular plates under thermoelastic loading is investigated. The rectangular plates on elastic foundation under mechanical and different types of thermal conditions namely; the uniform temperature rise and the linear and nonlinear temperature distributions across the thickness of plate, are then considered and the related thermal buckling loads are obtained. Post-buckling and the geometric imperfection of rectangular plates are then followed and the chapter concludes with the discussion on the effect of material temperature dependency on the thermal critical buckling loads.

The stability of circular and annular plates under mechanical and thermal loads are presented in this chapter. The chapter begins with the presentation of the strain-displacement relations based on the von Karman and Kirchhoff assumptions employing the classical plate theory. The linear thermoelastic constitutive relations between the stress and strain components are considered and the stress and moment resultants for a plate with general heterogeneous material property, functionally graded, are obtained in terms of the nonlinear displacement components. The nonlinear equilibrium equations are derived basis on the stationary potential energy, and the linear stability equations of an annular plate are obtained by means of the adjacent-equilibrium criterion. Employing these basic governing equations, the chapter continues to present a number of practical stability problems. Thermal buckling of circular and annular plates based on the classical and shear deformable theories, circular plates on elastic foundation, rotating plate under thermal loading, and the buckling and post-buckling of plates with geometric imperfection are discussed in detail and approximate closed form solutions for a number of cases are presented.

This chapter is devoted to the stability behavior of thin cylindrical shells. The basic governing equations of thin circular cylindrical shells employing the Donnell theory with the von-Karman geometrical non-linearity are derived. The nonlinear strain-displacement relations, the nonlinear equilibrium equations, and the linear stability equations are derived employing the variational formulations. The cylindrical shell under uniform compressive axial load is considered and the buckling load is obtained and given by closed form solution. Thermal buckling of cylindrical shell made of FGM for the uniform temperature rise, linear radial temperature, and the nonlinear radial temperature are presented and the effect of piezo-control is examined. Buckling and postbuckling of thin cylindrical shells with piezo-control under thermal loads is discussed and the chapter concludes with the stability discussion of cylindrical shells on elastic foundation. The buckling loads of cylindrical shells of isotropic/homogeneous material are derived by simply setting proper values for the power law index of the FG materials.

Spherical shells, as part of structural systems, are frequently used in many structural design problems. This type of shells is capable to stand high internal or external pressures and is especially quite stable under external pressures. The behavior of deep spherical shells, in particular, under external pressure is quite unique and the bifurcation load is far from expectation. Ancient spherical domes with wide spans in historic buildings is a good example of such structure to show its remarkable stability feature. Spherical shells used in the industrial applications are exposed to different types of mechanical or thermal loads. Under these circumstances, it is necessary to predict the critical mechanical and/or thermal buckling loads of spherical shells. Closed form solutions for the buckling loads are valuable tools for designer in the design stage. This chapter presents the methods to calculate critical buckling temperatures or pressures in spherical shell made of isotropic and functionally graded materials for both perfect and imperfect shells.

Conical shells under the mechanical and thermal loads are considered in this chapter. The basic governing equations of the shell including the nonlinear strain-displacement relations of Sanders assumption, the constitutive law, the nonlinear equilibrium equations, and the linear stability equations are derived employing the variational principle. The stability of the shell is considered and the buckling loads associated with the mechanical and thermal forces are derived. In case of thermal loading, the material temperature dependency is considered and the results are compared with the case where temperature dependency of the material properties are ignored. Effect of the piezo-control on stability of conical shells under thermal loading concludes the chapter.