Chapter 17, Problem 17.1P

To determine

The stresses at the corners of the cross-section of given (S4S) timber beam at the point of maximum moment.

Explanation of Solution

Given information:

For the given $\text{100mm by 100mm}$ timber beam, we have a diagram as below:

We have following data,

Given beam is $\text{100mm by 100mm}$ timber

$\text{a = 100mm }\times \left|\frac{1\text{ m}}{{10}^3\text{ mm}}\right|$

Total length of the beam, $L=4m$

Point loads acting of the beam, $P_x=200N{\text{ and P}}_y=450N$

For the given beam, we have free body a diagram as below:

We know that if the point loads are acting over the center of the simply supported beam then the maximum bending moment occurs at the mid span of it.

So the maximum moment at the mid span about x-axis would be,

$\begin{array}{l}{M}_x=\frac{P_y\times L}{4}\\ M_x=\frac{450\times 4}{4}\\ M_x=450\text{ N-m}\end{array}$

In the same manner, the maximum moment at the mid span about y-axis is,

$\begin{array}{l}{M}_y=\frac{P_x\times L}{4}\\ M_y=\frac{200\times 4}{4}\\ M_y=200\text{ N-m}\end{array}$

For the cross-section of the beam, we know that the second moment of inertia about centroidal x and y axes must be same. This can be determined as follow:

  $\begin{array}{l}{I}_x=\frac{a^4}{12}\\ I_x=\frac{{\left[100\times \left|1/{10}^3\right|\right]}^4}{12}\text{ (Substituting value of a =}100mm\times \left|1m/{10}^{3}mm\right|\\ I_x=8.333\times {10}^{-6}{\text{ m}}^4\end{array}$

We can obtain bending stress by use of below mentioned equation:

$\begin{array}{l}s=\pm \frac{M_{x}y}{I_x}\pm \frac{M_{y}x}{I_y}\\ s=\pm \frac{M_x\left(a/2\right)}{I_x}\pm \frac{M_y\left(a/2\right)}{I_y}\end{array}$

In the above equation, putting values,

$\begin{array}{l}s=\pm \frac{M_x\left(a/2\right)}{I_x}\pm \frac{M_y\left(a/2\right)}{I_y}\\ s=\pm \frac{450\left(\left[100\times \left|1/{10}^3\right|\right]/2\right)}{8.333\times {10}^{-6}}\pm \frac{200\left(\left[100\times \left|1/{10}^3\right|\right]/2\right)}{8.333\times {10}^{-6}}\\ s=\pm 2700108\pm 1200048{\text{ N/m}}^2\\ s\approx \pm 2.7\pm 1.2\text{ MPa}\end{array}$

Finally calculating the stress at points 1,2 , 3 and 4.

$\begin{array}{l}{s}_1=-2.7-1.2\\ s_1=-3.9\text{ MPa}\\ s_2=-2.7+1.2\\ s_2=-1.5\text{ MPa;}\\ s_3=+2.7+1.2\\ s_3=3.9\text{ MPa};\\ s_4=+2.7-1.2\\ s_4=+1.5\text{ MPa}\end{array}$

Conclusion:

The stresses at the corners of the cross-section of given (S4S) timber beam are $-3.9\text{ MPa}$ , $-1.5\text{ MPa}$ , $3.9\text{ MPa}$ and $+1.5\text{ MPa}$ at the point of maximum moment.