Open Educational Resources

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.