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ME422 FEM Homework #8 Distributed: January 31, 2013 Due: February 7, 2013 Consider a thin slab which is ﬁxed to a wall and has a pressure, P , on its free edge: w L E ν pressure, P 2L Problem 1: Using standard Mechanics of Materials and elasticity equations and concepts, and assuming that the wall doesn’t have any eﬀect except to keep the slab from moving sideways, (a) approximately what is the τxx stress in the slab? (b) approximately what is the deﬂection of the end of the slab? (c) approximately what is the strain xx in the slab? (d) approximately what is the strain yy in the slab due to the Poisson’s ratio eﬀect? Use the approximation ν = 1/3. (e) approximately what is the change in height of the slab due to the Poissson’s ratio eﬀect? (ν = 1/3.) 1 Problem 2: Discretize the top half of the slab with a single 4-node isoparametric ﬁnite element. There will be three degrees of freedom in the system. v1 2 1 3 4 u1 pressure, P u4 (a) Find the 3×3 global stiﬀness matrix for the system. Again, use ν = 1/3, and assume that the system is in plane stress. (It is.) You may ﬁnd Maple useful for some of these calculations, although actually there are only 4 unique values in the 3×3 matrix. (b) Find the forcing vector for the system (3 entries, 2 of which are non-zero). (c) Solve for the displacements of the nodes. Sketch the new shape of the element. Compare the displacements to the answer you got for 1b and 1e above. I would expect that the 1b answer would agree quite closely, and the 1e answer is surprisingly close as well. (d) Solve for the three components of stress in the element at the element center (s=0, t=0). Compare the appropriate stress to the answer you found in 1a, and discuss any diﬀerences you ﬁnd. Problem 3: No problem 3. Projects! 2