Problem 3.13. Timoshenko beam with different end supports

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Step 1. Give the elements of the load vector (force and moment functions along the beam’s length).

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Step 2. Determine the bending and shear stiffnesses of the solid circular cross section.

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Shear stiffness is given by Eqs.(3-59) and (3-60)

Shear correction factor is given in Figure 3.28.

$$\begin{gathered} E{I}_z=E\frac{D^4\pi }{4} \\ S=\frac{GA}{n}=G\frac{D^2\pi }{4}\frac{9}{10} \end{gathered}$$

Step 3. Write the differential equation system of the Timosenko beam.

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Eq.(3-61)

$$\begin{gathered} \left[\begin{array}{cc}–S\frac{d^2}{d{x}^2}& S\frac{d}{dx}\\ –S& S–EI\frac{d^2}{d{x}^2}\end{array}\right]\left\{\begin{array}{c}v\\ {\chi }_y\end{array}\right\}=\left\{\begin{array}{c}{p}_y\\ m\end{array}\right\} \\ \left[\begin{array}{cc}–G\frac{9D\pi ^2}{40}\frac{d^2}{d{x}^2}& G\frac{9D\pi ^2}{40}\frac{d}{dx}\\ –G\frac{9D\pi ^2}{40}\frac{d}{dx}& G\frac{9D\pi ^2}{40}–E\frac{D^4\pi }{4}\frac{d^2}{d{x}^2}\end{array}\right]\left\{\begin{array}{c}v\\ {\chi }_y\end{array}\right\}=\left\{\begin{array}{c}\frac{px}{L}\\ 0\end{array}\right\} \end{gathered}$$

Step 4. Show the boundary conditions for built-in and hinged edges.

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Step 5. Differential equation system with the given boundary conditions can be solved. Solution is not presented here.

First the load vector is given. Its elements are the force and moment functions along the beam’s length. Here no moment load acts, the linearly distributed load results in the following load vector:

$\left\{\begin{array}{c}{p}_y\\ m\end{array}\right\}=\left\{\begin{array}{c}\frac{px}{L}\\ 0\end{array}\right\}$

The bending and shear stiffnesses of the solid circular cross section are given below.

Shear stiffness is given by Eqs.(3-59) and (3-60)

Shear correction factor is given in Figure 3.28.

The differential equation system of the Timosenko beam is:

Eq.(3-61)

Finally the boundary conditions for built-in and hinged edges are shown.

At the hinged end of the beam:

$$\begin{gathered} v\left(0\right)=0, \\ M\left(0\right)=0 \to M=EI{\kappa }_z=–EI\frac{d{\chi }_y}{dx} \to \frac{d{\chi }_y\left(0\right)}{dx}=0 \end{gathered}$$

At the built-in end of the beam:

$v\left(L\right)=0, {\chi }_y\left(L\right)=0$

The differential equation system with the given boundary conditions can be solved. Solution is not presented here.