Solutions for rectangular cross-section

The $F$ matrix defined in (2.80) may be presented most conveniently in rectangular coordinates by expressing each entry as the multiplication of two terms:

$${F_{nm}(\beta_x,\beta_y) = \frac{1}{S_1}\int\limits_{S_1}\psi_{n}^{(1)}\psi_{m}^{(2)}dS}$$

$$= \frac{1}{2a_1}\int\limits_{-a_1}^{a_1}dx\phi_{n_x}^{(1)}\phi_{m... ...2)}\frac{1}{2b_1}\int\limits_{-b_1}^{b_1}dy\sigma_{n_y}^{(1)}\sigma_{m_y}^{(2)}$$

$$= X_{n_x m_x}Y_{n_y m_y}$$ (2.89)

$$X_{n_x m_x}(\beta_x) = \left\{ \begin{array} {r@{\quad:\qua... ... \beta_x}{m_x \beta_x + n_x} & n_x>0. \\ \end{array} \right.$$ (2.90)

$$Y_{n_y m_y}(\beta_y) = \left\{ \begin{array} {r@{\quad:\qua... ... \beta_y}{m_y \beta_y + n_y} & n_y>0. \\ \end{array} \right.$$ (2.91)

where $\beta_x = \frac{a_1}{a_2}$ and $\beta_y = \frac{b_1}{b_2}$. When both $\beta_x$ and $\beta_y$ tend to 1 we get

$$X_{n_x m_x}(\beta_x) \approx \left\{ \begin{array} {r@{\qua... ...2 - n_x^2} & n_x \neq m_x, n_x > 0. \\ \end{array} \right.$$ (2.92)

$$Y_{n_y m_y}(\beta_y) \approx \left\{ \begin{array} {r@{\qua... ...2 - n_y^2} & n_y \neq m_y, n_y > 0. \\ \end{array} \right.$$ (2.93)

where $\epsilon_x = (a_2 - a_1)/a_1$ and $\epsilon_y = (b_2 - b_1)/b_1$. The $V$ matrix will have entries given by

$$V_{nm} = F_{nm}(1/\beta_x,1/\beta_y).$$ (2.94)