1. MESF593 Finite Element Methods HW #4 11:00pm, 10 May, 2010

2. Prob. #1 (10%) A structure consists of 3 different materials and is modeled by 3D brick elements (8-nodes per element) as shown in Figs. (a)/(b). According to the discussion in the class, for heat conduction analysis, the sphere-like joints can be replaced by an equivalent model using bar elements (2-nodes per elements) as shown in Figs. (c)/(d). Such replacement can give quite similar results. However, if you perform thermal-mechanical analysis, you will find the model in Figs. (a)/(b) having stresses due to the mismatch of CTEs while the model in Figs. (c)/(d) with no stress at all. Why?  1, CTE 1, E 1  2, CTE 2, E 2 T1T1 T2T2  3, CTE 3, E 3 (a) Front View x z (b) Edge View y z T1T1 T2T2  1, CTE 1, E 1  2, CTE 2, E 2  3, CTE 3, E 3 (c) Front View x z (d) Edge View y z  =  /  C: Thermal Diffusivity CTE: Coefficient of Thermal Expansion E: Young’s Modulus T 1 : Temp. at the Top Surface of Material 1 T 2 : Temp. at the Bottom Surface of Material 2 T 1 > T 2, CTE 1 > CTE 2, CTE 3 = 0

3. Prob. #2 (20%) x = 2x = 18 F(x) A quadratic function F(x) is defined as shown above in the range between x = 2 and x = 18. Use the 1-point, 2-point, and 3-point, respectively, Gauss- Legendre quadrature method to evaluate the area underneath the blue curve and compare these numerical integration values to the analytical solution obtained from calculus.

4. Prob. #3 (35%) A structure is modeled with 3 linear bar elements (2-nodes per element). All 3 bars have the same Young’s modulus (E), density (  ), and area of cross-section (A). The bar lengths are L, L/2 and L/2, respectively, as shown above. The 2 red nodes are rigidly tied together. Also, the 2 orange nodes are rigidly tied together as well. Use the consistent mass formulation to estimate the natural frequencies and the harmonic vibration mode shapes. (Note: the vibration is only allowed in the horizontal direction) I III II L L/2 13 2

5. Prob. #4 (35%) A green circular plate has a triangular hole inside. There is a heat source, denoted by the red dot (with a total power generation of P), at the center of the triangular hole and it is connected to the green plate through 3 bars. The thermal conductivity, area of cross-section, and length of each bar are given as shown above. Assuming there is a uniform temperature T g over the whole green plate, find the temperature T r at the red dot and the heat flux through each bar. TgTg P III III