Static and Dynamic Analyses of Rectangular Plates with Stepped Thickness

Abstract

This chapter presents the analytical methodology for the static and dynamic problems of rectangular plates with stepped thickness in bending state subjected to vertical loads. The discontinuous variation in the rigidity and mass of the plates due to the voids is also expressed as a continuous function by means of the extended Dirac function. First, the general governing equations for rectangular plates with stepped thickness are proposed on the basis of the Kirchhoff-Love hypothesis. Second, the analytical methodologies for the static and dynamic problems are presented by means of the Galerkin method. Third, for practical use, the approximate solutions for the static and dynamic problems are proposed in closed-form solution.

Appendix: Calculation Including Extended Dirac Function

The integral calculation including the extended Dirac function $D\left( {x - x_{i} } \right)$ can be written as

$$ \int\limits_{0}^{{l_{x} }} {D(x - x_{i} )f(x){\text{d}}x} = \int\limits_{{x_{{i - (b_{xi} /2)}} }}^{{x_{{i + (b_{xi} /2)}} }} {\left[ {\int\limits_{0}^{{l_{x} }} {\delta (x - \xi )f(x){\text{d}}x} } \right]} {\text{d}}\xi = \int\limits_{{x_{{i - (b_{xi} /2)}} }}^{{x_{{i + (b_{xi} /2)}} }} {f(\xi ){\text{d}}\xi } $$

(8.52)

in which $\xi$ is a supplementary variable of x. The n-th derivatives of the extended Dirac functions can therefore be expressed as

$$ \int\limits_{0}^{{l_{x} }} {D^{(n)} (x - x_{i} )f(x){\text{d}}x} = \int\limits_{{x_{{i - (b_{xi} /2)}} }}^{{x_{{i + (b_{xi} /2)}} }} {( - 1)^{n} f^{(n)} (\xi ){\text{d}}\xi } $$

(8.53)

in which the superscripts enclosed within parentheses indicate the differential order.