User:Eml4500.f08.gravy.bje/Notes6

Calculation of first term of B(x)
$$ \frac{du(x)}{dx} = \begin{bmatrix} N_i^'(x) & N_{i+1}^'(x) \end{bmatrix} \begin{Bmatrix} d_i \\ d_{i+1} \end{Bmatrix}$$

$$B(x) = \begin{bmatrix} N_i^'(x) & N_{i+1}^'(x) \end{bmatrix}$$

$$ \frac{dW(x)}{dx} = B(x) \begin{Bmatrix} W_i \\ W_{i+1} \end{Bmatrix}$$

$$B(x) = \begin{bmatrix} 1+W & ... \end{bmatrix}$$

Derivation of k(i)
$$ \underline{k}^{(i)}= \int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}} \underline{\mathbf{B}}^T(\tilde{x}) (EA)(\tilde{x}) \underline{\mathbf{B}}(\tilde{x}) d\tilde{x}$$

$$\underline{k}^{(i)}= \begin{Bmatrix} \int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}} AE \frac{1}{L^2} dx & -\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}} AE \frac{1}{L^2} dx \\ -\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}} AE \frac{1}{L^2} dx & \int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}} AE \frac{1}{L^2} dx \end{Bmatrix}$$

$$\underline{k}^{(i)}= \frac{A(\tilde{x})E(\tilde{x})}{L} \begin{Bmatrix} \int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}dx & -\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}dx \\ -\int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}dx & \int_{\tilde{x}=0}^{\tilde{x}=L^{(i)}}dx \end{Bmatrix}$$

$$\underline{k}^{(i)}= \frac{A(\tilde{x})E(\tilde{x})}{L} \begin{Bmatrix} L^{(i)} & -L^{(i)} \\ -L^{(i)} & L^{(i)} \end{Bmatrix}$$

Where:

$$ A(\tilde{x}) = N_1^{(i)}(\tilde{x})A_1 + N_2^{(i)}(\tilde{x})A_2$$

$$ E(\tilde{x}) = N_1^{(i)}(\tilde{x})E_1 + N_2^{(i)}(\tilde{x})E_2$$

Therefore:

$$\underline{k}^{(i)}_{2x2} = \frac{(N_1^{(i)}(\tilde{x})A_1 + N_2^{(i)}(\tilde{x})A_2)(N_1^{(i)}(\tilde{x})E_1 + N_2^{(i)}(\tilde{x})E_2)}{L} \begin{Bmatrix} L^{(i)} & -L^{(i)} \\ -L^{(i)} & L^{(i)} \end{Bmatrix}$$