MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 1Name: ______________________ You are allowed one sheet of notes. 1. For the one-dimensional problem shown, calculate: a. The global stiffness matrix before the application of boundary conditions. b. The reduced stiffness matrix after the application of boundary conditions. K1 = 10,000 N/mm K2 = 5,000 N/mm K3 = 10,000 N/mm F = 500 N 10 u1 u2 u3 u4 F 1 3 2MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 2 2. Give the correct order for the following FEA tasks, considering both how NX Nastran works and the current best practices. a. Add Boundary Conditions b. Solve for displacements c. Visualize the results d. Solve for strains e. Create the mesh f. Create geometry g. Anticipate physical behavior, possibly using check calculations h. Determine whether FEA is warranted i. Solve for stresses 3. In NX Nastran, explain what each of the following is for: a. Part file (.prt) b. Fem file (.fem) c. Sim file (.sim) d. Solution Navigator e. Post-Processing toolbar 3 5 Step 1 2 3 4 5 6 7 8 9MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 3 4. To what do DOF 1, DOF 2, … DOF 6 refer, when applying user-defined restraints in the Lab Assignments? 5. What is the difference between rod (or bar) elements and beam elements? 6. What does the FEA software do when the yield stress is exceeded in a linear static analysis? 7. If an element has a quadratic shape function in x, will the strain also be a quadratic function of x? 2 2 2 2MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 4 8. Given the truss structure shown, calculate the stress and strain in truss element 1 if: A1 = 0.0004 m2 E1 = 200x109 Pa L1 = 2 m 10 1 3 2 1 2 F X Y φ=150° ×−×−==−−333322111021010000vuvuvuDMAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 5 9. Give the equivalent nodal forces and moments that represent the distributed force on the beam shown. Give your answer in the form of the global force vector R. 5 2 m q = 100 N/m 4 m 1 6 m 8 m x 2 3 4 5 ==555444333222MFFMFFMFFMFFMFFyyyyyxxxx111xRMAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 6 10. For the example on the right: (i) Solve for the two elemental stiffness matrices. (ii) Assemble the global stiffness matrix. (iii) Compute the global applied force vector (R) considering only the gravitational force acting on the rod elements. (iv) After applying the appropriate restraint condition(s), solve for the nodal displacements. (v) Solve for the reaction force(s) at the restraint(s). (vi) Solve for the element strains. (vii) Solve for the element stresses. L1 L2 u2 u3 A1, E1, ρ1 1 2 A2, E2, ρ2 L1= L2 = 1 m A1 = 0.001 m2 A2 = 0.004 m2 E1 = E2 = 70x109 Pa ρ1 = ρ2 = 2700 kg/m3 u1 30MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 7MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 8MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 9 11. In Question 10, what is the displacement at the middle of element 1 (i.e., at 0.5 m from the top)? 12. Plot the displacement of both elements as a function of the distance from the top. 13. In Question 10, what is the strain at the middle of element 1 (i.e., at 0.5 m from the top)? 14. Plot the strain of both elements as a function of the distance from the top. 4 2 2 2MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 10 15. In Question 10, what is the stress at the middle of element 1 (i.e., at 0.5 m from the top)? 16. Plot the stress of both elements as a function of the distance from the top. 17. In the above questions, will the answers be the exact answers? If your answer is no, what aspect of the problem makes it so the FEA answer is not fully correct? 18. Consider the horizontal BAR element with cantilevered support conditions. Is the stiffness matrix singular (i.e., would you be able to solve for the displacements)? 2 L F A, E 2 2 2MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 11 19. For the beam elements shown (with shape functions given below), the nodal displacements have been calculated in meters and radians as: −−==0076.000010.00076.00332211θθθvvvD a) Plot (sketch) the vertical displacement v(x) for the entire beam (both elements). b) Plot the angle of the beam θ(x) for the entire beam. c) Plot the moment M(x) carried by the beam for the entire beam. d) Plot the stress σ(x) at the bottom of the beam. e) Plot the stress σ(x) at the middle of the beam cross-section. • In the plots, write actual values at node positions. • Indicate the order of the polynomials for each plot. Recall that for a beam element v(x) = Nd, where: +−−+−+−=23233222323322232231LxLxLxLxLxLxxLxLxN and dxdvx =)(θ BdEIdxvdEIM ==22, where: . 15 2 m 4 m 1 x 2 3 y 150 mm 200 mm E = 70 GPa I = 0.0001 m4 =2211θθvvd+−−+−+−=2322326212664126LxLLxLLxLLxLBMAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 12Question 3 cont’d.MAE 456 FINITE ELEMENT ANALYSIS EXAM 1 Practice Questions 13Question 3