Determining the deflection of indeterminate beams with nonlinearly tapered cross sections using the GDQ method

This article describes the optimal and distinctive use of the numerical Generalized Differential Quadrature method in analyzing the deflection of indeterminate beams with tapered cross sections. Due to the long time it takes to use analytical methods to solve the fourth order differential equations related to indeterminate beams with different boundary conditions, the use of numerical approaches is unavoidable. One of the relatively new numerical methods that can accurately discretize and solve partial differential equations is the Generalized Differential Quadrature method. This method is based on the idea that the partial derivative of a function is approximated at a specific point of the problem interval as the weighted algebraic sum of function values at all discretized points of the whole region. In this article, the GDQ method and the manner of calculating the weight coefficients by means of this method are explained. The application of this method in the analysis of indeterminate beams with linearly tapered cross sections under different loading conditions and the comparison of the results with those obtained by the SAP2000 software have confirmed the high accuracy of this method in analyzing these types of problems. The other advantage of this numerical method over the other numerical approaches in solving partial equations is that it converges to the solution with fewer mesh points. Therefore, by making a small modification in the analytical program of Generalized Differential Quadrature approach, the deflection of indeterminate beams with nonlinearly tapered cross sections under linear and nonlinear loadings can be obtained; a task which is either impossible to accomplish using the presently available software programs or very difficult and time consuming

Keywords:

Indeterminate beams , GDQ method , Nonlinearly tapered cross section , Partial differential equations , deflection