Analysis of Transverse Loading on a Beam Utilizing the Beltrami-Michell Equations

The stress field of a beam with a circular cross-section have been developed in the current work. One end of the beam is fixed while the other end is under traverse load at its center. The Beltrami-Michell compatibility equations were utilized to obtain coefficients in an assumed stress function which can be used to derive the stress field. To visualize the stress distribution in the beam, one can use MATLAB to generate surface plots and contour plots for the developed stress field. According to the plots, the maximum can be found at the center area of the cross-section while the minimum is captured at the same area. The maximum shear stress in the section occurs at points along the perimeter of the section. Moreover, the goal of this paper is to prove that any stress function with higher order terms always converge to the same stress solution for the beam utilizing lower order terms.