User:EML4500.f08.delta 6.ramirez/101708

Homework from 10/17/08 1. Complete f1(2)

Thus,

$$f_1^{(2)} = [k_{11}^{(2)}d_1^{(2)} + k_{12}^{(2)}d_2^{(2)} + k_{13}^{(2)}d_3^{(2)} + k_{14}^{(2)}d_4^{(2)}]$$

2. Derive the details of the following equation.

$$f_4^{(1)} + f_2^{(2)} = P$$

Where,

1. $$f_4^{(1)} = [k_{41}^{(1)}d_1^{(1)} + k_{42}^{(1)}d_2^{(1)} + k_{43}^{(1)}d_3^{(1)} + k_{44}^{(1)}d_4^{(1)}]$$

2. $$f_2^{(2)} = [k_{21}^{(2)}d_1^{(2)} + k_{22}^{(2)}d_2^{(2)} + k_{23}^{(2)}d_3^{(2)} + k_{24}^{(2)}d_4^{(2)}]$$

Thus,

$$[k_{41}^{(1)}d_1^{(1)} + k_{42}^{(1)}d_2^{(1)} + k_{43}^{(1)}d_3^{(1)} + k_{44}^{(1)}d_4^{(1)}]+ [k_{21}^{(2)}d_1^{(2)} + k_{22}^{(2)}d_2^{(2)} + k_{23}^{(2)}d_3^{(2)} + k_{24}^{(2)}d_4^{(2)}] = P$$

Observation: Last two items in 1. & first two items in 2. correspond to the 4th row in the global stiffness matrix K6x6.

Hence,

$$[(k_{43}^{(1)} + k_{21}^{(2)})(k_{44}^{(1)} + k_{22}^{(2)})]$$