Chapter 12, Problem 54AP

Figure P12.38 shows a light truss formed from three struts lying in a plane and joined by three smooth hinge pins at their ends. The truss supports a downward force of $\stackrel{\to }{F}=1000\text{N}$ applied at the point B. The truss has negligible weight. The piers at A and C are smooth. (a) Given θ₁ = 30.0° and θ₂ = 45.0°, find n_A and n_C. (b) One can show that the force any strut exerts on a pin must be directed along the length of the strut as a force of tension or compression. Use that fact to identify the directions of the forces that the struts exert on the pins joining them. Find the force of tension or of compression in each of the three bars.

Figure P12.38

To determine

The reaction force at $A$ and $C$.

Answer to Problem 54AP

The reaction force at $A$ is ${\overrightarrow{n}}_{\text{A}}=366\text{N}$, and the reaction force at $C$ is ${\overrightarrow{n}}_{\text{C}}=634\text{N}$.

Explanation of Solution

Section 1:

To determine: The reaction force at $A$.

Answer: The reaction force at $A$ is ${\overrightarrow{n}}_{\text{A}}=366\text{N}$.

Given information: The magnitude of the downward support is $1000\text{N}$, the angle ${\theta }_1=30°$ and the angle ${\theta }_2=45°$.

The following figure shows the force diagram of the three beams.

Figure-(I)

Formula to calculate the ratio of reaction forces at $A$ and $C$ using trigonometric relation is,

$\frac{{\overrightarrow{n}}_{\text{C}}}{{\overrightarrow{n}}_{\text{A}}}=\frac{\tan {\theta }_2}{\tan {\theta }_1}$

• ${\overrightarrow{n}}_{\text{A}}$ is the reaction force at $A$.
• ${\overrightarrow{n}}_{\text{C}}$ is the reaction force at $B$.
• ${\theta }_1$ is the angle made by the beam $AB$ with horizontal.
• ${\theta }_2$ is the angle made by the beam $BC$ with horizontal.

Substitute $45°$ for ${\theta }_2$ and $30°$ for ${\theta }_1$ in the above equation to find ${\overrightarrow{n}}_{\text{C}}$.

$\begin{array}{c}\frac{{\overrightarrow{n}}_{\text{C}}}{{\overrightarrow{n}}_{\text{A}}}=\frac{\tan 45°}{\tan 30°}\\ {\overrightarrow{n}}_{\text{C}}=\left(1.732\right){\overrightarrow{n}}_{\text{A}}\end{array}$ (I)

Formula to calculate the net forces acting in vertical direction is,

${\overrightarrow{n}}_{\text{A}}-\overrightarrow{F}+{\overrightarrow{n}}_{\text{C}}=0$

• $\overrightarrow{F}$ is the vertical downward force.

Substitute $1000\text{N}$ for $\overrightarrow{F}$ and $\left(1.732\right){\overrightarrow{n}}_{\text{A}}$ for ${\overrightarrow{n}}_{\text{C}}$ in the above equation to find ${\overrightarrow{n}}_{\text{A}}$.

$\begin{array}{c}{\overrightarrow{n}}_{\text{A}}-\left(1000\text{N}\right)+\left(\left(1.732\right){\overrightarrow{n}}_{\text{A}}\right)=0\\ {\overrightarrow{n}}_{\text{A}}=366\text{N}\end{array}$

Conclusion:

Therefore, the reaction force at $A$ is $366\text{N}$.

Section 2:

To determine: The reaction force at $C$.

Answer: The reaction force at $C$ is $634\text{N}$.

Given information: The magnitude of the downward support is $1000\text{N}$, the angle ${\theta }_1=30°$ and the angle ${\theta }_2=45°$.

Substitute $366\text{N}$ for ${\overrightarrow{n}}_{\text{A}}$ in the equation (I) to find ${\overrightarrow{n}}_{\text{C}}$.

$\begin{array}{c}{\overrightarrow{n}}_{\text{C}}=\left(1.732\right){\overrightarrow{n}}_{\text{A}}\\ =\left(1.732\right)\left(366\text{N}\right)\\ =633.912\text{N}\\ \approx 634\text{N}\end{array}$

Conclusion:

Therefore, the reaction force at $C$ is $634\text{N}$.

To determine

The direction of forces that the struts exert on pin joints, and the forces acting on the each of the three beams.

Answer to Problem 54AP

The direction of force that the strut $AB$ exert on joint $A$ is along $AB$ from $B$ to $A$, direction of force that the strut $AB$ exert on joint $B$ is along $AB$ from $A$ to $B$, direction of force that the strut $BC$ exert on joint $B$ is along $BC$ from $C$ to $B$, direction of force that the strut $BC$ exert on joint $C$ is along $BC$ from $B$ to $C$, direction of force that the strut $AC$ exert on joint $A$ is along $AC$ from $C$ to $A$, direction of force that the strut $AC$ exert on joint $C$ is along $AC$ from $A$ to $C$, force of tension exert on the beam $AB$ is $732\text{N}$, the force of tension exert on the beam $BC$ is $896\text{N}$, the force of tension exert on the beam $BC$ is $896\text{N}$ and the force of tension exert on the beam $AC$ is $634\text{N}$.

Explanation of Solution

Section 1:

To determine: The direction of forces that the struts exert on pin joints.

Answer: The direction of force that the strut $AB$ exert on joint $A$ is along $AB$ from $B$ to $A$, direction of force that the strut $AB$ exert on joint $B$ is along $AB$ from $A$ to $B$, direction of force that the strut $BC$ exert on joint $B$ is along $BC$ from $C$ to $B$, direction of force that the strut $BC$ exert on joint $C$ is along $BC$ from $B$ to $C$, direction of force that the strut $AC$ exert on joint $A$ is along $AC$ from $C$ to $A$, and the direction of force that the strut $AC$ exert on joint $C$ is along $AC$ from $A$ to $C$.

Given information: The magnitude of the downward support is $1000\text{N}$, the angle ${\theta }_1=30°$ and the angle ${\theta }_2=45°$.

The following figure shows the forces exerted by strut on each joint.

Figure-(II)

From the shown Figure-(II), the direction of the direction of force that the strut $AB$ exert on joint $A$ is along $AB$ from $B$ to $A$, direction of force that the strut $AB$ exert on joint $B$ is along $AB$ from $A$ to $B$, direction of force that the strut $BC$ exert on joint $B$ is along $BC$ from $C$ to $B$, direction of force that the strut $BC$ exert on joint $C$ is along $BC$ from $B$ to $C$, direction of force that the strut $AC$ exert on joint $A$ is along $AC$ from $C$ to $A$, and the direction of force that the strut $AC$ exert on joint $C$ is along $AC$ from $A$ to $C$.

Conclusion:

Therefore, the direction of force that the strut $AB$ exert on joint $A$ is along $AB$ from $B$ to $A$, direction of force that the strut $AB$ exert on joint $B$ is along $AB$ from $A$ to $B$, direction of force that the strut $BC$ exert on joint $B$ is along $BC$ from $C$ to $B$, direction of force that the strut $BC$ exert on joint $C$ is along $BC$ from $B$ to $C$, direction of force that the strut $AC$ exert on joint $A$ is along $AC$ from $C$ to $A$, and the direction of force that the strut $AC$ exert on joint $C$ is along $AC$ from $A$ to $C$.

Section 2:

To determine: The force exert on the beam $AB$.

Answer: The force of tension exert on the beam $AB$ is $732\text{N}$.

Given information: The magnitude of the downward support is $1000\text{N}$, the angle ${\theta }_1=30°$ and the angle ${\theta }_2=45°$.

Formula to calculate the force exert on the beam $AB$ is,

$F_{\text{AB}}\sin {\theta }_1=n_{\text{A}}$

• $F_{\text{AB}}$ is the force exert on the beam $AB$.

Substitute $30°$ for ${\theta }_1$ and $366\text{N}$ for $n_{\text{A}}$ in the above equation to find $F_{\text{AB}}$.

$\begin{array}{c}{F}_{\text{AB}}\sin \left(30°\right)=\left(366\text{N}\right)\\ F_{\text{AB}}=732\text{N}\end{array}$

Conclusion:

Therefore, the force of tension exert on the beam $AB$ is $732\text{N}$.

Section 3:

To determine: The force exert on the beam $BC$.

Answer: The force of tension exert on the beam $BC$ is $896\text{N}$.

Given information: The magnitude of the downward support is $1000\text{N}$, the angle ${\theta }_1=30°$ and the angle ${\theta }_2=45°$.

Formula to calculate the force exert on the beam $BC$ is,

$F_{\text{BC}}\sin {\theta }_2=n_{\text{C}}$

• $F_{\text{BC}}$ is the force exert on the beam $BC$.

Substitute $45°$ for ${\theta }_2$ and $634\text{N}$ for $n_{\text{C}}$ in the above equation to find $F_{\text{BC}}$.

$\begin{array}{c}{F}_{\text{BC}}\sin \left(45°\right)=\left(634\text{N}\right)\\ F_{\text{BC}}=896.36\text{N}\\ \approx 896\text{N}\end{array}$

Conclusion:

Therefore, the force of tension exert on the beam $BC$ is $896\text{N}$.

Section 4:

To determine: The force exert on the beam $AC$.

Answer: The force of tension exert on the beam $AC$ is $634\text{N}$.

Given information: The magnitude of the downward support is $1000\text{N}$, the angle ${\theta }_1=30°$ and the angle ${\theta }_2=45°$.

Formula to calculate the force exert on the beam $AC$ is,

$F_{\text{AC}}=F_{\text{AB}}\cos {\theta }_1$

• $F_{\text{AC}}$ is the force exert on the beam $AC$.

Substitute $30°$ for ${\theta }_1$ and $732\text{N}$ for $F_{\text{AB}}$ in the above equation to find $F_{\text{AC}}$.

$\begin{array}{c}{F}_{\text{AC}}=\left(732\text{N}\right)\cos 30°\\ =633.93\text{N}\\ \approx 634\text{N}\end{array}$

Conclusion:

Therefore, the force of tension exert on the beam $AC$ is $634\text{N}$.