No abstract

No abstract

The analysis of the large deformation of a non-linear cantilever functionally graded material (FGM) beam is made. When subjected to an end moment, explicit expressions for deflection and rotation are derived for a functionally graded beam with work hardening of power law. The effects of the gradient distribution of Young’s modulus and the material non-linearity parameter on the deflections of the FGM beam 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 a FGM beam can bear larger applied load than a homogeneous beam. Moreover, to determine an optimal gradient distribution, an optimum design of a beam of a lighter weight and larger stiffness is given. The influence of the geometric non-linearity of the beam is also studied. Large and small deformation theories predict nearly the same deflections with 5% error when rotation is less than 45°, and the predictions based on the small deformation theory are overestimated to exceed 10% when rotation is greater than 60°.

Nomenclature A = constant of integration a = inner radius B = constant of integration b = outer radius c m = constant coefficients E, Er, E r = Young's modulus (dependent on r) E i = Young's modulus at the inner surface Fr; t = kernel of normalized Fredholm integral equation fr = known function defined by Eq. (10) Kr; t = kernel of Fredholm integral equation Lr; t = kernel of Fredholm integral equation P n x = Legendre polynomials q i , q o = pressure at the inner and outer surfaces r = radial coordinate t = normalized radial variable u r = radial displacement x = normalized radial variable = ratio of Young's modulus at the outer surface to that at the inner surface = gradient index = polar angle coordinate , r = Poisson's ratio (dependent on r) r = radial stress = circumferential stress ' = azimuth-angle coordinate

A tall building may be modeled as a shear beam since its bending is mainly induced by relative sliding of parallel floor slabs. Classical shear beams do not consider the presence of bending moment. Since the bending moment of a building 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 cooperating with bending moment is needed. This paper 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 lumped 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.

No abstract