EXACT SHAPE FUNCTIONS AND DISPLACEMENT FIELDS FOR NON-PRISMATIC EULER-BERNOULLI BEAMS DERIVED FROM GOVERNING DIFFERENTIAL EQUATIONS

Abstract

This study presents an exact analytical formulation for the deflection response of non-prismatic Euler-Bernoulli beams with continuously varying flexural rigidity. Starting from the classical beam assumptions, the governing differential equation with variable coefficients is derived and solved directly for a general power-law representation of stiffness variation along the beam axis. A load-potential decomposition is introduced to decouple the contribution of external loading from the homogeneous displacement field, allowing a unified and exact treatment of uniformly distributed loads, concentrated loads, and partially applied loads within a single analytical framework. Based on the resulting homogeneous solution space, exact shape functions are constructed by enforcing nodal compatibility and boundary conditions, ensuring pointwise satisfaction of equilibrium without resorting to assumed polynomial interpolation. Closed-form expressions for displacements are obtained for both cantilever and simply supported beams. The analytical solutions are verified through numerical comparisons with finite-element results obtained using a discretized representation of the non-prismatic beam, showing excellent agreement across all loading cases. The proposed formulation provides a rigorous analytical benchmark for non-prismatic beam behavior and offers a reliable reference for the verification and development of numerical and finite-element beam models with variable stiffness.