Part I - Theoretical Calculations Theoretical calculations or back-of-the-envelope calculations are very important to do before running a finite element analysis. They give us a sense of what to expect out

of the analysis and how to set it up, i.e., how to choose an appropriate model, how to apply loads & boundary conditions, and how to interpret the results of the simulation. (i) Consider a cross section at (a) the fixed support, and (b) at 100 mm from the fixed support. Draw the free body diagrams for these isolations and show the internal resultant loads on these sections (using the sign convention discussed in class). Calculate the bending stress and transverse shear stress distribution on these 2 cross sections. Also calculate the deflection of the free end of the beam using beam theory from your mechanics class (you do not need to derive anything; you may use the relevant equations from the appendix of the Mechanics of Materials textbook by Hibbeler)./nPart II - 3D Solid Model (iii) Model the structure in 3D - you can create the cross section on the YZ plane and extrude it in the positive X direction for the appropriate length. This will ensure that the coordinate system is oriented in the same way that we consider in examples solved in class. (iv) Fix the back face of the beam using a fixed support. Recognize that this leads to a stress singularity at the corners of the back face of the beam. Explain why this stress singularity is seen and how you can deal with it using Saint Venant's principle. (v) Apply the loading to the cantilever beam in ANSYS. Mesh the model with a mesh element size of 50 mm. Determine the maximum deflection of the beam and the maximum normal and shear stress magnitudes at the fixed support. Compare these with the results of Part I above. What do you observe? Then mesh the model with a mesh element size of 35 mm, 25 mm, 12.5 mm, and 6.25 mm (let's call these iterations 2, 3, 4, and 5). Plot the maximum normal and shear stress magnitudes vs. the iteration number for the different values of the mesh element size - this is called a convergence analysis. What do you observe? Explain why you see this behavior. (vi) Insert a surface at the cross section (along the YZ plane) at a distance of 100 mm from the fixed support. Show figures of the variation of the bending stress and shear stress along this surface. (vii) Repeat the analysis above looking at the stress magnitudes at the cross section 100 mm from the fixed support. What do you observe? Explain this behavior. (viii) Determine the effect of the depth-to-span ratio of the beam on the ratio of the maximum transverse shear to the maximum bending stresses. Do this by varying the length of the beam - choose lengths of 1 m, 0.75 m, 0.5 m, 0.375 m, 0.25 m, and 0.125 m. Use the smallest mesh size (6.25 mm) to ensure convergence of the solution. Repeat the analysis/nPart III - 2D Plane Stress Model (ix) Model the structure as a 2D plane stress model by sketching a rectangle of dimensions 1m x 100 mm on the X-Y plane. Use the "Surfaces from Sketches" tool to create a surface body and set its thickness to 50 mm. (x) Repeat the analysis of (v) to (vii) from Part II above using the 2D plane stress model. Note that to repeat part (vi), you will need to create a path/line instead of a surface at 100 mm from the support for the 2D model. Compare results and comment on which model is more accurate (calculate a percentage difference of numerical results), as well as which is more efficient (compare the size of the model - i.e., the number of nodes and elements in the model).