Engineering at Alberta Courses

Problems: Chapter 7

1- Find the shear forces and bending moments at the sections shown. For the bending moment at S₃ consider sections right before and after S₃.

Answer

Section 1 at $x=1.25\text{ m}$ : $V(1.25)=-0.38 \text{ kN}, \ M(1.25)=0.31 \text{ kN·m}$ .

Section 2 at $x=2.5\text{ m}$ : $V(2.5)=-1.63 \text{ kN}, \ M(2.5)=-0.94 \text { kN·m}$ .

Section 3 at $x=3.75\text{ m}$ : $V(3.75)=-1.63 \text{ kN}, \ M(3.75^-)=-2.97 \text{ kN·m}, \ M(3.75^+)=2.03\text{ kN·m}$ .

2- Find the shear forces and bending moments at the sections shown.

Answer

Section 1 at $x=1.5\text{ m}$ : $V(1.5)=5.0 \text{ kN}, \ M(1.5)=-12.5 \text{ kN·m}$ .

Section 2 at $x=3.0\text{ m}$ : $V(3.0)=0.0 \text{ kN}, \ M(3.0)=-10.0 \text { kN·m}$ .

3- Find the shear forces and bending moments at the sections shown.

Answer

Section 1 at $x=1.0\text{ m}$ : $V(1.0)=2.92 \text{ kN}, \ M(1.0)=2.92 \text{ kN·m}$ .

Section 2 at $x=2.0\text{ m}$ : $V(2.0)=-0.84 \text{ kN}, \ M(2.0)=3.75 \text{ kN·m}$ .

Section 3 at $x=3.0\text{ m}$ : $V(3.0)=-2.08 \text{ kN}, \ M(3.0)=2.08 \text{ kN·m}$ .

4- Find the shear forces and bending moments at the sections shown.

Answer

Section 1 at $x=2.0\text{ m}$ : $V(2.0)=0.0 \text{ kN}, \ M(2.0)=6.67 \text{ kN·m}$ .

Section 2 at $x=3.0\text{ m}$ : $V(3.0)=-2.50 \text{ kN}, \ M(3.0)=5.00 \text{ kN·m}$ .

5- Determines the functions $V(x)$ and $M(x)$ .

Answer

$V(x)=\begin{cases}5 -\frac{5x^2}{4}\text{ kN} \quad \text{for } 0< x \le 2\\\frac{5x^2}{4}-5\text{ kN} \quad \text{for } 2\le x < 4\\0.0\text{ kN} \quad \text{for } 4< x \le 5\end{cases}$

$M(x)=\begin{cases}5x - \frac{5x^3}{12}\text{ kN.m} \quad \text{for } 0\le x \le 2\\\frac{5x^3}{12}-5x + \frac{20}{3}\text{ kN.m} \quad \text{for } 2\le x \le 4\\0.0\text{ kN.m} \quad \text{for } 4\le x \le 5\end{cases}$

6- Determines the functions $V(x)$ and $M(x)$ . Consider the support at $B$ as a roller in a slot if the support reaction becomes a pulling force there.

Answer

$\begin{split}V(x)&=x -2.5\text{ kN} \quad \text{for } 0< x < 5\\M(x)&=\frac{x^2}{2} -2.5x\text{ kN.m} \quad \text{for } 0\le x \le 5\end{split}$

7- Determine the functions $V(x)$ and $M(x)$ .

Answer

$V(x)=\begin{cases}300 - 100x\text{ N} \quad \text{for } 0< x < 2\\200 -100x \text{ N} \quad \text{for } 2< x \le 4\\-200 \text{ N} \quad \text{for } 4\le x < 5\\0.0\text{ N} \quad \text{for } 5< x \le 6\end{cases}$

$M(x)=\begin{cases}300x - 50x^2\text{ N.m} \quad \text{for } 0\le x \le 2\\200 + 200x - 50x^2\text{ N.m} \quad \text{for } 2\le x \le 4\\-200x + 1000\text{ N.m} \quad \text{for } 4\le x \le 5\\0.0\text{ N.m} \quad \text{for } 5\le x \le 6\end{cases}$

8- For the loaded beam shown, find the locations where the shear force and the bending moment attain their maximum magnitudes. Then, determine the maximum magnitudes. Hint: find the local and global extrema (maxima or minima) of the functions $V(x)$ and $M(x)$ . Use derivatives for finding local extrema. You can also plot $V(x)$ and $M(x)$ for more clarification.

Answer

The maximum magnitude of the shear force is $0.5\text { kN}$ and it occurs at two locations being the supports, i.e. $x=0.0\text{ m}$ and $x=1.0\text{ m}$ .

The maximum magnitude of the bending moment is $1.125\text { kN.m}$ occurring at $x=0.50 \text{ m}$ .

9- For the loaded beam shown, find the locations where the shear force and the bending moment attain their maximum magnitudes. Then, determine the maximum magnitudes. Hint: see the previous problem.

Answer

The maximum magnitude of the shear force is $0.333\text { kN}$ occurring at $x= 0\text { m}$ , i.e. support $A$ .

The maximum magnitude of the bending moment is $0.064\text { kN.m}$ occurring at $x= 0.423\text { m}$ .

10- For the loaded beam shown, find the locations where the shear force and the bending moment attain their maximum magnitudes. Then, determine the maximum magnitudes. Hint: see the previous problem.

Answer

The maximum magnitude of the shear force is $0.5\text { kN}$ occurring at $x= 0.0\text { m}$ and $x= 2.0\text { m}$ i.e. the supports.

The maximum magnitude of the bending moment is $0.33\text { kN.m}$ occurring at $x= 1.0\text { m}$ .

11- For the loaded beam shown, find the locations where the shear force and the bending moment attain their maximum magnitudes. Then, determine the maximum magnitudes. Hint: see the previous problem.

Answer

The maximum magnitude of the shear force is $1.0\text { kN}$ occurring at $x= 0.0\text { m}$ i.e. the support.

The maximum magnitude of the bending moment is $0.50\text { kN.m}$ occurring at $x= 0.0\text { m}$ i.e. the support.