Question

Element 1 is a steel bar that has a circular cross-section with a radius of 30 mm. Element 2 is an aluminum bar that has a circular cross-section with a radius of 50 mm. Element 3 is a steel bar that has a circular cross- section with a radius of 60 mm. Assume for steel, the moduļus, E, is 2.0E11 Pa, and the density, p, is 7800 kg/m3. Assume for aluminum, E-7.0E10 Pa, and p-2700 kg/m3. The rigid, massless rod at the left end is there to indicate that the left end of element 1 and the left end of element 2 have the same displacement so we can use the same node, node 1, for the left end node for both of these elements a) Write the finite element modeling equations such that any calculated displacements would be in meters. Very clearly show the values in the mass and stiffness matrices after the boundary condition at node 3 has been applied. Base the mass and stiffness matrices on the bar element formulations provided in the class notes. Use the consistent mass matrix instead of the diagonal lumped mass matrix. b) Assume the system is in static equilibrium, and calculate the static displacements (in units of meters), at nodes 1 and 2. Assume the system behaves linearly c) Calculate the displacement in Element 3 at the location 0.40 m to the right of Node 2. d) Calculate the axial direction stress and strain in each element. e) Calculate the reaction force at Node 3 Rigid Massless Rod Element 1 100 kN Element 2 200 kN Element 3 Node 1 Node 2 Node 3

Answer 1