Chapter 6, Problem 6.1P

Draw the influence lines for (a) the moment at C, (b) the vertical reaction at A, and (c) the shear at C. Assume A is a fixed support. Solve this problem using the basic method of Sec. 6.1.

To determine

(a)

The influence lines for the moment at $C$.

Answer to Problem 6.1P

The influence line for the moment at $C$ is shown below.

  

Explanation of Solution

Concept Used:

The unit load as $P=1\text{kN}$ is placed at different position for $x$ and the corresponding value of moment is calculated.

Calculation:

The following figure shows the free body diagram of the beam.

  

  Figure-(1)

Write the equilibrium equation for the moment about point $C$.

   $\begin{array}{l}\sum M_{\text{C}}=0\\ M_{\text{C}}-1\times \left(2-x\right)=0\\ M_{\text{C}}=2-x\end{array}$   ......... (I)

Here, the net moment about the point $C$ is $\sum M_{\text{C}}$, the moment at point $C$ is $M_{\text{C}}$ and the distance of the point load from the right end is $x$.

When the point load is at $B$ ; $x=0\text{m}$.

Substitute, $0\text{m}$ for $x$ in Equation (I).

   $\begin{array}{c}{M}_{\text{C}}=2-0\\ =2\text{kN}\cdot \text{m}\end{array}$

When the point load is at $C$, $x=2\text{m}$.

Substitute, $2\text{m}$ for $x$ in Equation (I).

   $\begin{array}{c}{M}_{\text{C}}=2-2\\ =0\text{kN}\cdot \text{m}\end{array}$

Conclusion:

According to the value of $M_{\text{C}}$ for different position of point load, draw the influence line diagram as shown in Figure-(2).

  

  Figure-(2)

To determine

(b)

The influence line for the vertical reaction at $A$.

Answer to Problem 6.1P

The influence line for the vertical reaction at $A$ is shown below.

  

Explanation of Solution

Concept Used:

The unit load as $P=1\text{kN}$ is placed at different position for $x$ and the corresponding value of the vertical reaction is calculated.

Calculation:

The following figure shows the free body diagram of the beam.

  

  Figure-(3)

Write the equilibrium equation for the forces acting in the vertical direction.

   $\begin{array}{l}\sum F_{\text{Y}}=0\\ 1-R_{\text{A}}=0\\ R_{\text{A}}=1\text{kN}\end{array}$

Here, summation of the vertical forces is $\sum F_{\text{Y}}$, and the vertical reaction at point $A$ is $R_{\text{A}}$.

Conclusion:

According to the value of the vertical reaction at $A$, draw the influence line diagram as shown in Figure-(4).

  

  Figure-(4)

To determine

(c)

The influence line for the shear at $C$.

Answer to Problem 6.1P

The influence line for the shear at $C$ is shown below.

  

Explanation of Solution

Concept Used:

The unit load as $P=1\text{kN}$ is placed at different position for $x$ and the corresponding value of the shear force is calculated.

Calculation:

Consider the right segment of the beam as shown in Figure-(5).

  

  Figure-(5)

Write the equilibrium equation for the forces acting in the vertical direction.

   $\begin{array}{l}\sum F_{\text{Y}}=0\\ -V_{\text{C}}+1=0\\ V_{\text{C}}=1\text{kN}\end{array}$

Here, summation of the vertical forces is $\sum F_{\text{Y}}$, the shear force at $C$ is $V_{\text{C}}$.

Now, consider the left segment of the beam as shown in Figure-(6).

  

  Figure-(6)

Write the equilibrium equation for the forces acting in the vertical direction.

   $\begin{array}{l}\sum F_{\text{Y}}=0\\ 1-1-V_{\text{C}}=0\\ V_{\text{C}}=0\text{kN}\end{array}$

Conclusion:

According to the values of $V_{\text{C}}$, draw the influence line diagram as shown in Figure-(7).

  

  Figure-(7)