Answer Happy

The figures below illustrate an axial loaded composite memberthat is comprised of fourdifferent materials defined as materials 1 through 4. Note thereare two rigid end caps atthe LHS and RHS of the member. The LHS end cap is adhered tomaterials 2 and 4, andthe RHS end cap is adhered to materials 3 and 4 as shown. A totalforce Fa2 is applied tothe LHS rigid end cap. The LHS end of the material 1 part and theRHS end cap are subjectto zero fixed displacement conditions, so that u1 = u4 = 0. Developa model containing foursprings (called k1,k2,k3 and k4) and four nodes (defined as 1,2,3and 4 shown below).IMPORTANT NOTE: The end caps are modeled as nodes 1 and 4 and notwith springs!
d) Derive the required nodal equilibrium equations and write thenode number at the leftof each equation.
e) Re-arrange your equations developed in d) into the standardform. Note this will helpyou in part f) described below
f) Define [K], {U} and {f} (in terms of the element stiffnesses,unknown nodaldisplacements, and the applied force) for the nodal equilibriumequations expressed inthe standard matrix form:{f} = [K] {U}
g) Define the checks that you can make on the stiffness matrix[K]:Check 1)_________________________________________Check 2)_________________________________________Check 3)_________________________________________
h) Does your stiffness matrix [K] pass the above tests? Explainand circle your answersCheck 1): YES/NO __________________________________Check 2): YES/NO __________________________________Check 3): YES/NO __________________________________
Applied force Fza u=0 Zero fixed displacement at LHS of the member and at the RHS end cap. Applied force = F2a at LHS end cap Fixed displacement Node 1 X Node 2 2 3 Material sections Node 3 Node 4 Axial section showing the four sections of different materials u=0 Fixed displacement