I am trying to find out the boundary conditions for above given Timoshenko beam. The boundary conditions that I have derived are
- $x_1 = 0$ $$ \overline{u}_2^L = 0 $$
$$ \phi_3^L = 0 $$
- $x_1 = \alpha L$
$$ \overline{u}_2^L = \overline{u}_2^R $$
$$ \phi_3^L = \phi_3^L $$
$$ -K_{22} \left( \dfrac{d\overline{u}_2^L}{d x_1} - \phi_3^L \right) + P + K_{22} \left( \dfrac{d\overline{u}_2^R}{d x_1} - \phi_3^R \right) = 0 $$
$$ -H_{33}^c \dfrac{d \phi_3^L}{d x_1} + H_{33}^c \dfrac{d \phi_3^R}{d x_1} = 0$$
- $x_1 = L$
$$ K_{22} \left( \dfrac{d\overline{u}_2^R}{d x_1} - \phi_3^R \right) = 0 $$
$$ H_{33}^c \dfrac{d \phi_3^R}{d x_1} = 0 $$
where,
$x_i$: coordinate along $i$-th direction
$\overline{u}_2^L$: transverse displacement of left portion of beam
$\overline{u}_2^R$: transverse displacement of right portion of beam
$\phi_3^L$: sectional rotation of left portion of beam
$\phi_3^R$: sectional rotation of right portion of beam
$K_{22}$: Sectional shear stiffness, equal to $\dfrac{5}{6}GA$ for rectangular cross-sections
However, in the book ``Structural Analysis with Applications to Aerospace Structures'' by Bauchau and Craig, the boundary conditions at $x = \alpha L$ are
$$ \overline{u}_2^L = \overline{u}_2^R $$
$$ \phi_3^L = \phi_3^L $$
$$ -K_{22} \left( \dfrac{d\overline{u}_2^L}{d x_1} - \phi_3^L \right) + P = 0 $$
$$ H_{33}^c \dfrac{d \phi_3^L}{d x_1} = 0$$
Why are shear force and bending moment equations at $x = \alpha L$ are different than mine?
Thank you.