Boundary conditions for a cantilevered Timoshenko Beam

Question

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.

Answer 1

I don't recognize your equation, but, as you are interested in "boundary condition", at which only one face exist, so you shall eliminate one of the shear deformation terms as indicate below.