Section B – Analytical calculation of the beam (10%)
For the shape of cross section that you have selected in A.8 above, select dimensions from catalogues and draw a sketch showing the beam under a three-point bending test (use the given point load). Supports and external loads must be clearly shown and justified. Sketch the Free Body Diagram of the beam and calculate analytically (hand calculations) the maximum shearing force, the maximum bending moment, the maximum deflection and the maximum normal stress due to bending. The hand calculations must be clear and complete. Also, fully define all the necessary information for these calculations and justify your selections. Initially, ignore the self-weight and then repeat the calculations including the self-weight.
Section C – Finite Element Analysis of the beam (by hand) (20%)
For the purposes of the current section, model the entire beam as an assembly of two beam elements and, after carrying all calculations by hand, get a solution for the vertical deflection. More particularly, for each one of the two elements, write the stiffness matrix and, based on that information, update / form the global stiffness matrix of the structure. In the sequel, apply proper boundary conditions and solve the so formed system of equations. For this very last step (solution of system of equations), any software (e.g. MatLab, MS Excel, etc.) may be used. After getting a solution for the nodal displacements, estimate the maximum deflection, the maximum bending moment, the maximum shear force and the maximum normal stress due to bending. Also, estimate the reaction forces and check for force equilibrium. Furthermore, estimate the reaction moments and check for moment equilibrium. Initially ignore the self-weight and then repeat the calculations including the self-weight.