Static/Dynamic Mechanical Behavior Analysis Of Functionally Graded Beams

Beams are one of the most common engineering structures. In the field of aerospace, mechanical and civil engineering, many components subjected to applied loading can be understood as beams or columns. Functionally graded materials (FGMs) possess continuously varying material properties. This feature can effectively remove abrupt stress jump when across the interface between two bonded dissimilar materials. Therefore, due to this feature they have been widely used in a variety of engineering fields. In this thesis, the mechanical behaviors of a non-linear functionally graded material cantilever beam subjected to an end force or moment are investigated by using large and small deformation theories. Free vibration of bidirectionally functionally graded beams and axially inhomogeneous beams is analyzed. In addition, the natural frequencies of nonclassical shear beams standing on an elastic foundation and carrying a mass is studied.This thesis is composed of seven chapters. Chapter one is the introduction, in which the development of relative subjects is outlined. Chapter two to six are the main chapters, and some novel results and conclusions are drawn. The final chapter gives a simple summary and expectation of future researches. Here, main results are emphasized as follows:(1) The analysis of the large deformation of a non-linear cantilever functionally graded beam is made. For an FGM beam of power-law hardening, the location of the neutral axis is determined. When subjected to an end moment, explicit expressions for deflection and rotation are derived for an FGB. The effects of the gradient distribution of Young's modulus and the material non-linearity parameter on the deflections of the FGB are analyzed. Our results show that depth-dependent Young's modulus and material non-linearity have a significant influence on the deflections of the beam, and an FGB can bear larger applied load than a homogeneous beam. Moreover, to determine a optimal gradient distribution, an optimum approach for choosing a beam with a lighter weight and larger stiffness is given.(2) The mechanical behaviors of a non-linear functionally-graded-material cantilever beam subjected to an end force are investigated by using large and small deformation theories. The Young's modulus is assumed to be depth-dependent. The effects of the depth-dependent Young's modulus and nonlinearity parameter on the deflections and rotations of the FGM beam are analyzed. Our results show that different gradient indexes may change bending stiffness of the beam so that the FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.(3) The free vibration of bidirectionally functionally graded beams is studied. For a slender beam with axially varying and depth-dependent material properties, a governing equation is derived from two-dimensional elasticity theory. By converting it to a Fredholm integral equation, numerical results of the natural frequencies for free vibration are obtained for simply-supported beams with symmetrically-distributed material properties. The effects of gradient variation and rotary moment of inertia on the natural frequencies are discussed. It is found that the natural frequencies are sensitive to the gradient variation. Rotary inertia plays an important role in determining the natural frequencies.(4) Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler-Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited. For some gradient index, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating inhomogeneous beam structures.(5) The natural frequencies of a shear beam standing on an elastic foundation and carrying a mass are obtained. Tall buildings may be modelled as shear beams since the corresponding bending is mainly induced by relative sliding of parallel floor slabs. The classical shear beams do not consider the presence of bending moment. Since the bending moment of buildings can be originated from tension-compression force couple due to combined effects of strong wall-to-wall and wall-to-slab interactions, a modified shear beam theory with consideration of bending moment is needed. The final chapter gives a theoretical analysis of a refined shear beam model. Emphasis is placed on the determination of the natural frequencies of a shear beam on an elastic base such as soil and carrying a mass. The characteristic equation for free vibration of a modified shear beam is obtained. The influences of translational and rotational spring stiffnesses, axially compressive loads and attached mass on the natural frequencies are illuminated.