User:Eml4500.f08.delta 6.ramirez/globalstiffmat

The following variables are known: E(1) = 2 A(1) = 3 L(1) = 5 θ(1) = 30° E(2) = 4 A(2) = 1 L(2) = 5 θ(2) = -30° E(3) = 3 A(3) = 2 L(3) = 10 θ(3) = -135° The following variables can be determined:

$$k_{(1)} = \frac{E^{(1)}A^{(1)}}{L^{(1)}} = \frac{6}{5} = 1.2$$ $$k_{(2)} = \frac{E^{(2)}A^{(2)}}{L^{(2)}} = \frac{4}{5} = 0.8$$ $$k_{(3)} = \frac{E^{(3)}A^{(3)}}{L^{(3)}} = \frac{6}{10} = 0.6$$

$$l^{(1)} = cos\theta ^{(1)} = 0.866$$ $$m^{(1)} = sin\theta ^{(1)} = 0.5$$ $$l^{(2)} = cos\theta ^{(2)} = 0.866$$ $$m^{(2)} = sin\theta ^{(2)} = -0.5$$ $$l^{(3)} = cos\theta ^{(3)} = -0.707$$ $$m^{(3)} = sin\theta ^{(3)} = -0.707$$ Note: l(3) = m(3) The following are the element stiffness matrices, which are needed to compute the global stiffness matrix.

$$k^{(e)} = k_{(e)}\begin{bmatrix} l^{(e)2} & l^{(e)}m^{(e)} & -l^{(e)2} & -l^{(e)}m^{(e)}\\ l^{(e)}m^{(e)} & m^{(e)2} & -l^{(e)}m^{(e)} & -m^{(e)2} \\ -l^{(e)2} & -l^{(e)}m^{(e)} & l^{(e)2} & l^{(e)}m^{(e)}\\ -l^{(e)}m^{(e)} & -m^{(e)2} & l^{(e)}m^{(e)} & m^{(e)2} \end{bmatrix}$$ Thus,

$$k^{(1)} = \begin{bmatrix} 0.9 & 0.52 & -0.9 & -0.52\\ 0.52 & 0.3 & -0.52 & -0.3 \\ -0.9 & -0.52 & 0.9 & 0.52\\ -0.52 & -0.3 & 0.52 & 0.3 \end{bmatrix}$$

$$k^{(2)} = \begin{bmatrix} 0.6 & -0.346 & -0.6 & 0.346\\ -0.346 & 0.2 & 0.346 & -0.2 \\ -0.6 & 0.346 & 0.6 & -0.346\\ 0.346 & -0.2 & -0.346 & 0.2 \end{bmatrix}$$

$$k^{(3)} = \begin{bmatrix} 0.3 & 0.3 & -0.3 & -0.3\\ 0.3 & 0.3 & -0.3 & -0.3\\ -0.3 & -0.3 & 0.3 & 0.3\\ -0.3 & -0.3 & 0.3 & 0.3 \end{bmatrix}$$

Now the global stiffness matrix can be computed.

$$K = \begin{bmatrix} 0.9 & 0.52 & -0.9 & -0.52 & 0 & 0 & 0 & 0\\ 0.52 & 0.3 & -0.52 & -0.3 & 0 & 0 & 0 & 0\\ -0.9 & -0.52 & 1.8 & 0.474 & -0.6 & 0.346 & -0.3 & -0.3\\ -0.52 & -0.3 & 0.474 & 0.8 & 0.346 & -0.2 & -0.3 & -0.3\\ 0 & 0 & -0.6 & 0.346 & 0.6 & -0.346 & 0 & 0\\ 0 & 0 & 0.346 & -0.2 & -0.346 & 0.2 & 0 & 0\\ 0 & 0 & -0.3 & -0.3 & 0 & 0 & 0.3 & 0.3\\ 0 & 0 & -0.3 & -0.3 & 0 & 0 & 0.3 & 0.3 \end{bmatrix}$$