Differentiating under integral sign in Castigliano’s theorem

Abstract

In this contribution we are dealing with application of differentiating of function defined with determined parametrical integral. The mathematical problem was analyzed form point of view of mechanics of materials. The mathematical model of loaded beam was created. Applying the cross section method we defined the exact function of bending moment. Additional properties of cantilever joint were neglected. We showed the derivation of modified Castigliano's theorem via Leibnitz rule of differentiating under integration sign. Appling the modified Castigliano's theorem we got the exact solution of the deflection of the beam. The exact solution of beam deflection was finished in PTC Mathcad Prime software (PTC-Parametric Technology). The numerical integration of the bending moment dataset and defined deflection function was done in program written in Microsoft Visual C# 2010. The Dormand-Prince numerical integration method was used for the numerical integration. Comparing the exact and numerical solution we got the error of numerical integration solution.