Chapter 5, Problem 152RP

Draw the shear and bending-moment diagrams for the beam and loading shown, and determine the maximum absolute value (a) of the shear, (b) of the bending moment.

Fig. P5.152

To determine

Draw: The shear force diagram for the beam and loading.

Find the maximum absolute value of the shear.

Answer to Problem 152RP

The maximum absolute value of the shear force is $\underset{\_}{\left|V_{\max }\right|=85\text{N}}$.

Explanation of Solution

Show the free-body diagram of the entire beam as in Figure 1.

Determine the vertical reaction at point B by taking moment about point A.

$\begin{array}{l}{\displaystyle \sum M_A=0}\\ -75\left(250\right)+3750-75\left(500\right)+3750+B_y\left(750\right)=0\\ -18750+3750-37500+3750+750B_y=0\\ B_y=65\text{N}\end{array}$

Determine the vertical reaction at point A by resolving the vertical component of forces.

$\begin{array}{l}{\displaystyle \sum F_y=0}\\ A_y-75-75+B_y=0\\ A_y-150+65=0\\ A_y=85\text{N}\end{array}$

Show the free-body diagram of the sections as in Figure 2.

Section AC (1-1):

Determine the shear force at the section by resolving the vertical component of forces.

$\begin{array}{l}{\displaystyle \sum F_y=0}\\ 85-V=0\\ V=85\text{N}\end{array}$

Section CD (2-2):

Determine the shear force at the section by resolving the vertical component of forces.

$\begin{array}{l}{\displaystyle \sum F_y=0}\\ 85-75-V=0\\ V=10\text{N}\end{array}$

Section DB (3-3):

Determine the shear force at the section by resolving the vertical component of forces.

$\begin{array}{l}{\displaystyle \sum F_y=0}\\ V+65=0\\ V=-65\text{N}\end{array}$

Show the calculated shear force values as in Table 1.

Location (x) mm	Shear force (V) N
A (0 mm)	85
C (1-1) (250 mm)	85
C (2-2) (250 mm)	10
D (2-2) (500 mm)	10
D (3-3) (500 mm)	–65
B (750 mm)	–65

Plot the shear force diagram as in Figure 3.

Refer to the Figure 3;

The maximum absolute value of the shear force is $\underset{\_}{\left|V_{\max }\right|=85\text{N}}$.

To determine

Draw the bending moment diagram for the beam and loading.

Find the maximum absolute value of the bending moment.

Answer to Problem 152RP

The maximum absolute value of the bending moment is $\underset{\_}{\left|M_{\max }\right|=21.3\text{N-m}}$.

Explanation of Solution

Show the free-body diagram of the entire beam as in Figure 4.

Determine the vertical reaction at point B by taking moment about point A.

$\begin{array}{l}{\displaystyle \sum M_A=0}\\ -75\left(250\right)+3750-75\left(500\right)+3750+B_y\left(750\right)=0\\ -18750+3750-37500+3750+750B_y=0\\ B_y=65\text{N}\end{array}$

Determine the vertical reaction at point A by resolving the vertical component of forces.

$\begin{array}{l}{\displaystyle \sum F_y=0}\\ A_y-75-75+B_y=0\\ A_y-150+65=0\\ A_y=85\text{N}\end{array}$

Show the free-body diagram of the sections as in Figure 5.

Section AC (1-1):

Determine the bending moment at the section by taking moment about the section.

$\begin{array}{l}{\displaystyle \sum M=0}\\ M-85\left(x\right)=0\\ M=85x\end{array}$

Section CD (2-2):

Determine the bending moment at the section by taking moment about the section.

$\begin{array}{l}{\displaystyle \sum M=0}\\ M-85\left(x\right)+3750+75\left(x-250\right)=0\\ M-85x+3750+75x-18750=0\\ M=10x+15000\end{array}$

Section DB (3-3):

Determine the bending moment at the section by taking moment about the section.

$\begin{array}{l}{\displaystyle \sum M=0}\\ -M+65\left(750-x\right)=0\\ -M+48750-65x=0\\ M=48750-65x\end{array}$

Show the calculated bending moment values as in Table 2.

Location (x) mm	Bending moment (V) N-mm
A (0 mm)	0
C (1-1) (250 mm)	21250
C (2-2) (250 mm)	17500
D (2-2) (500 mm)	20000
D (3-3) (500 mm)	16250
B (750 mm)	0

Plot the bending moment diagram as in Figure 6.

Refer to the Figure 6;

The maximum absolute value of the bending moment is,

$\begin{array}{c}\left|M_{\max }\right|=21250\text{N-mm}\times \frac{\text{1}\text{m}}{\text{1000}\text{mm}}\\ =21.3\text{N-m}\end{array}$

The maximum absolute value of the bending moment is $\underset{\_}{\left|M_{\max }\right|=21.3\text{N-m}}$.