A systematic approach is presented in this paper to derive the analytical deflection function of a stepped beam using singularity functions. The discontinuities considered in this development are associated with the jumps in the flexural rigidity and the applied loads. This approach is applied to static and vibration analyses of stepped beams. The same approach is later extended to perform sensitivity analysis of stepped beams. This is done by directly differentiating the analytical deflection function with respect to any beam-related design variable. The particular design variable considered here is the location of discontinuity in flexural rigidity. Example problems are presented in this paper to demonstrate and verify the derivation process. 1. Introduction The stepped beams can be found in many engineering applications in shafts, antennae, rotor blades, gun barrels, slender structures, and so forth. The changes in the cross-sectional areas and the load distribution generate discontinuities in deriving the deflection equation of a stepped beam. Many have introduced singularity or Macaulay functions to handle these discontinuities. The singularity or Macaulay functions, made of the Dirac delta function and its derivatives, can be rigorously defined based upon theories of distributions or generalized functions [1–3]. The goal of this paper is to develop a unified, singularity function-based approach to perform static, vibration, and associated design sensitivity analysis of stepped beam problems. The approach is developed based upon an observation that the product of the bending moment and the inverse of the moment of inertia can be spanned as the algebraic sum of terms. Each of these terms contains only one singularity function of zero or higher order. These terms constitute the right-hand side of the bending moment equation of the stepped beam, which can be easily integrated to obtain the analytical expression of deflection. Such analytical function can be conveniently differentiated to obtain the analytical expression of sensitivity of beam deflection with respect to any design variables. Sensitivity analysis aims to find the derivatives of structural responses, such as deflection and frequencies, with respect to structural-related parameters. Sensitivity analysis has broad applications in support of reanalysis, design optimization, and reliability analysis [4, 5]. For example, the effects of structural degrading, crack propagation, and local damage of a stepped beam can be effectively estimated through the use of the sensitivity equations derived
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