# Boundary conditions for a cantilevered Timoshenko Beam

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.

• So, what would Timoshenko say? Mar 8, 2021 at 16:41

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.

• I have used continuity of $\overline{u}_2$ and $\phi_3$, force and moment balance at $x = \alpha L$. At $x = \alpha L$, shear force from both side should be included (?), so there will be three terms in expression, two expressions for shear force and one for P. Kindly correct me if I am wrong. Mar 9, 2021 at 3:29
• The shear stress at any location in a beam occurs on both sides of the shearing plane in "opposite direction". It is can be included in the expression that either describe the segment to the left, or right of the shearing plane with a single term, but not both. Otherwise, the terms simply cancel each other out..
– r13
Mar 9, 2021 at 4:02