Expert Answer :Consider the 1D elasticity situation shown above.

Solved by verified expert :Consider the 1D elasticity situation shown above. The composite material bar is constrained at both ends (i.e. at x = 0 and x = 2.5 m) such that displacement at each end can be treated as zero; the cross-section area throughout is A = 1 m^2. The ?rst part of the bar, from x = 0 to x = 1 m, is made from Al and has Young’s modulus E = 70 GPa; the second part of the bar, from x = 1 m to x = 2.5 m, is made from a graded chemistry steel such that Young’s modulus in that section of the bar varies linearly as given by the expression in the ?gure above. From x = 0 to x = 1 m, a linearly increasing body force is applied to the bar as shown in the ?gure; for x > 1 m, the body force is zero. A concentrated force P = 20 kN is applied at x = 2 m. a)? Using methods of strength of materials, solve this statically indeterminate problem (i.e. determine the reaction forces applied on the bar structure by the walls). b)? Again using methods of strength of materials, compute the displacement of the bar at x = 1 m and x = 2 m. c)? Use a two node element that spans from x = 0 to x = 1 m and a three node element from x = 1 m to x = 2.5 m to model this structure (use a midpoint node for the three node element). Construct the element sti?ness matrices and assemble them into the unreduced global sti?ness matrix. d)? Construct the element body force matrices and assemble them to obtain the unreduced global force matrix. e)? Solve the ?nite element (nodal) displacements as well as the ?nite element prediction for displacement at x = 2 m. f)? Show the percentage error in the ?nite element predicted displacements.