Pure Elastic Bending of Prismatical Bars
MATH-4960 Numerical Computing
Prof. E. Gutierrez-Miravete
Bar analysis is an everyday problem encountered by engineers. Typically the bar analysis makes underlying assumptions to simplify the approach to that of the elementary theory. One underlying assumption is that the bar cross-section remains constant in shape. Pure bending of prismatic bars takes into account deformation of the cross-section. This analysis considers deflection in all three directions. The project purpose is to derive the deflection equations and use three different techniques to solve the system of equations. These deflection equations are then used to determine the bar stress. The three different methods of analysis will be 1) analysis, 2) numerical methods, 3) finite element modeling.
Numerous textbooks can be used to reference the results of the elementary theory for various boundary conditions. These textbooks typically provide the end result for moment, deflection, and shear. To extend the analysis beyond the elementary theory, the starting deflection equations must be revisited to include effects of three-dimensional deflection. Reference (a) describes a process to determine the deflection of a bar in three dimensions. The deflection, combined with the boundary conditions, can be used to determine the stress state of the bar. This will be the approach used in the analysis portion of the approach. The numerical methods and finite element approach will use the equations derived in the analysis to solve for the deflection.
Three approaches will be used to solve the problem shown in figure 1 with equal and opposite couples about the y-axis at both ends A and B. The beam will be fixed at end A and free at end B (Cantilever Beam). For this problem L=10 ft, a=1 inch and b=2 inches. The couples applied to the ends of the bar will be 100000 in-lb. Approach one will analytically solve the displacements and the subsequent stresses at any point on the cross-section along the length of the bar. Using certain assumptions and boundary conditions an analytical solution will be obtained and confirmed with elementary beam bending theory. Approach two will solve the stresses numerically using a selected numerical method. Differential equations and systems of equations will be solved using numerical techniques. Approach three will solve the problem using a finite element method. For all three approaches solutions will be compared along with errors created in the numerical and finite element solutions. Upon analysis of errors created in the numerical and finite element approaches, refinements to the numerical procedure and finite element model mesh refinement will be performed to optimize these solutions.
(a) Timoshenko, S.P., Goodier, J.N., "Theory of Elasticity", Third Edition, McGraw-Hill
Book Company, 1970.