Consider a 2D rectangular cantilever with unit thickness shown in figure below. Its half height c=100mm, and length and l=1000mm. It is subjected to a down-ward shear force P=-1000 N. The cantilever is made of aluminum with youngs modulus of the material is E=68 GPa and shear modules of G=28 GPa. Using the same stresses obtained in the lecture, but considering displacement boundary conditions at point A (x=l, y=0): u=0, v=0, and uy=0, to perform the following studies.

1. Analytical expressions. Derive the analytical expressions for both displacement u and v; plot the distribution of both u and v along the boundary x=l, and discuss the differences between these results and the results obtained in the lecture.
2. Compute the displacements numerically. Use N_e triangular elements to discretize the 2D cantilever. This may result in N_n nodes. Each node will have 2 displacement components, and hence there will be a total of 2N_n nodal displacements over the mesh. Establish the relationship between the strains in each of the elements with the displacements at the nodes of the element (known as the strain and nodal-displacement relation using the strain matrix B given in Eq.(7.38) in the reference book: The Finite Element Method, A Practical Course, by GR Liu and SS Quek). Each elements will produce 3 equations (because there are 3 strain components) that relates 6 nodal displacements. Using the same stresses obtained in the lecture to compute all the nodal displacements. Compare the numerical results with the analytical results obtained in 1), and study the effects of the mesh density on the numerical solution for the displacements.

(Hint: you would need to write a code in a computer language of your choice to do this work. The number of the equations should be 3N_e, which will be more than the total unknowns (that is 2N_n nodal displacements). Hence, you would need to use a least-square solver to compute numerical solution of these 2N_n nodal displacements.)

3. After the nodal displacements are obtained, use the strain and nodal-displacement relations and the constitutive equations to compute numerical stresses. Compare the numerical stresses with the original analytical stresses.

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