Chapter 7, Problem 2P

To determine

Calculate the horizontal deflection and vertical deflection at joint B.

Answer to Problem 2P

The horizontal deflection at joint B is $\underset{\_}{4.24\text{mm}\to }$.

The vertical deflection at joint B is $\underset{\_}{5.92\text{mm}\downarrow }$.

Explanation of Solution

Given information:

The truss is given in the Figure.

The value of E is 70 GPa and the value of A is $1,000{\text{mm}}^2$

Procedure to find the deflection of truss by virtual work method is shown below.

• For Real system: If the deflection of truss is determined by the external loads, then apply method of joints or method of sections to find the real axial forces (F) in all the members of the truss.
• For virtual system: Remove all given real loads, apply a unit load at the joint where is deflection is required and also in the direction of desired deflection. Use method of joints or method of sections to find the virtual axial forces $\left(F_v\right)$ in all the member of the truss.
• Finally use the desired deflection equation.

Apply the sign conventions for calculating reactions, forces and moments using the three equations of equilibrium as shown below.

• For summation of forces along x-direction is equal to zero $\left(\sum F_x=0\right)$, consider the forces acting towards right side as positive $\left(\to +\right)$ and the forces acting towards left side as negative $\left(\leftarrow -\right)$.
• For summation of forces along y-direction is equal to zero $\left(\sum F_y=0\right)$, consider the upward force as positive $\left(\uparrow +\right)$ and the downward force as negative $\left(\downarrow -\right)$.
• For summation of moment about a point is equal to zero $\left(\sum M_{\text{at}\text{a}\text{point}}=0\right)$, consider the clockwise moment as negative and the counter clockwise moment as positive.

Method of joints:

The negative value of force in any member indicates compression (C) and the positive value of force in any member indicates tension (T).

Condition for zero force members:

1. 1. If only two non-collinear members are connected to a joint that has no external loads or reactions applied to it, then the force in both the members is zero.
2. 2. If three members, two of which are collinear are connected to a joint that has no external loads or reactions applied to it, then the force in non-collinear member is zero.

Calculation:

Consider the real system.

Find the member axial force (F) for the real system using method of joints:

Let $A_x$ and $A_y$ be the horizontal and vertical reactions at the hinged support A.

Let $C_x$ and $C_y$ be the horizontal and vertical reactions at the hinged support C.

Sketch the free body diagram of the truss as shown in Figure 1.

Refer to Figure 1.

Find the angle made by the member AB with respect to the horizontal axis.

$\begin{array}{c}\tan \theta =\frac{3}{2}\\ \theta ={\tan }^{-1}\left(\frac{3}{2}\right)\\ =56.31°\end{array}$

Find the angle made by the member BC with respect to the horizontal axis.

$\begin{array}{c}\tan \theta =\frac{3}{4}\\ \theta ={\tan }^{-1}\left(\frac{3}{4}\right)\\ =36.87°\end{array}$

Sketch the free body diagram of joint B as shown in Figure 2.

Apply equilibrium equation to the free body diagram of joint B:

Summation of forces along y-direction is equal to 0.

$\begin{array}{l}+\uparrow \sum F_y=0\\ -100-F_{AB}\sin 56.31°-F_{BC}\sin 36.87°=0\\ -F_{AB}\sin 56.31°-F_{BC}\sin 36.87°=100\end{array}$        (1)

Summation of forces along x-direction is equal to 0.

$\begin{array}{l}+\to \sum F_x=0\\ -F_{AB}\cos 56.31°+F_{BC}\cos 36.87°+50=0\\ -F_{AB}\cos 56.31°+F_{BC}\cos 36.87°=-50\end{array}$        (2)

Solve Equation (1) and Equation (2).

$\begin{array}{l}{F}_{AB}=-50.077\text{kN}\\ F_{BC}=-97.222\text{kN}\end{array}$

Sketch the resultant diagram of the real forces as shown in Figure 3.

Consider the virtual system:

For horizontal deflection apply l k at joint B in horizontal direction.

Find the member axial force ($F_{v1}$ ) due to virtual load using method of joints:

Sketch the free body diagram of the truss due to virtual load as shown in Figure 4.

Find the reactions at the supports using equilibrium equations:

Summation of moments about C is equal to 0.

$\begin{array}{l}\sum M_C=0\\ -A_y\left(6\right)+1\left(3\right)=0\\ A_y=0.5\text{kN}\downarrow \end{array}$

Summation of forces along y-direction is equal to 0.

$\begin{array}{l}+\uparrow \sum F_y=0\\ -A_y+C_y=0\\ -0.5+C_y=0\\ C_y=0.5\text{kN}\uparrow \end{array}$

Consider AB,

Summation of moments about B is equal to 0.

$\begin{array}{l}\sum M_B=0\\ -A_y\left(2\right)-A_x\left(3\right)=0\\ -0.5\left(2\right)-A_x\left(3\right)=0\\ A_x=0.333\text{kN}\leftarrow \end{array}$

Summation of forces along x-direction is equal to 0.

$\begin{array}{l}+\to \sum F_x=0\\ -A_x-C_x+1=0\\ -0.333-C_x+1=0\\ C_x=0.667\text{kN}\leftarrow \end{array}$

Sketch the free body diagram of joint B as shown in Figure 5.

Apply equilibrium equation to the free body diagram of joint B:

Summation of forces along y-direction is equal to 0.

$\begin{array}{l}+\uparrow \sum F_y=0\\ -F_{AB}\sin 56.31°-F_{BC}\sin 36.87°=0\\ -F_{AB}\sin 56.31°-F_{BC}\sin 36.87°=0\end{array}$        (3)

Summation of forces along x-direction is equal to 0.

$\begin{array}{l}+\to \sum F_x=0\\ -F_{AB}\cos 56.31°+F_{BC}\cos 36.87°+1=0\\ -F_{AB}\cos 56.31°+F_{BC}\cos 36.87°=-1\end{array}$        (4)

Solve Equation (3) and Equation (4).

$\begin{array}{l}{F}_{AB}=0.601\text{kN}\\ F_{BC}=-0.833\text{kN}\end{array}$

Sketch the resultant diagram of the virtual system $\left(F_{v1}\right)$ as shown in Figure 6.

Consider the virtual system:

For vertical deflection apply l k at joint B in vertical direction.

Find the member axial force $\left(F_{v2}\right)$ due to virtual load using method of joints:

Sketch the free body diagram of the truss as shown in Figure 7.

Find the reactions at the supports using equilibrium equations:

Summation of moments about C is equal to 0.

$\begin{array}{l}\sum M_C=0\\ -A_y\left(6\right)+1\left(4\right)=0\\ A_y=0.667\text{kN}\uparrow \end{array}$

Summation of forces along y-direction is equal to 0.

$\begin{array}{l}+\uparrow \sum F_y=0\\ A_y+C_y+1=0\\ 0.667+C_y+1=0\\ C_y=0.333\text{kN}\uparrow \end{array}$

Consider AB,

Summation of moments about B is equal to 0.

$\begin{array}{l}\sum M_B=0\\ -A_y\left(2\right)+A_x\left(3\right)=0\\ -0.667\left(2\right)+A_x\left(3\right)=0\\ A_x=0.444\text{kN}\to \end{array}$

Summation of forces along x-direction is equal to 0.

$\begin{array}{l}+\to \sum F_x=0\\ A_x-C_x=0\\ 0.444-C_x=0\\ C_x=0.444\text{kN}\leftarrow \end{array}$

Sketch the free body diagram of joint B as shown in Figure 8.

Apply equilibrium equation to the free body diagram of joint B:

Summation of forces along y-direction is equal to 0.

$\begin{array}{l}+\uparrow \sum F_y=0\\ -F_{AB}\sin 56.31°-F_{BC}\sin 36.87°-1=0\\ -F_{AB}\sin 56.31°-F_{BC}\sin 36.87°=1\end{array}$        (5)

Summation of forces along x-direction is equal to 0.

$\begin{array}{l}+\to \sum F_x=0\\ -F_{AB}\cos 56.31°+F_{BC}\cos 36.87°=0\\ -F_{AB}\cos 56.31°+F_{BC}\cos 36.87°=0\end{array}$        (6)

Solve Equation (5) and Equation (6).

$\begin{array}{l}{F}_{AB}=-0.801\text{kN}\\ F_{BC}=-0.556\text{kN}\end{array}$

Sketch the resultant diagram of the virtual system $\left(F_{v2}\right)$ as shown in Figure 9.

The expression to find the deflection $1\left(\Delta \right)$ is shown below:

$1\left(\Delta \right)=\sum F_v\left(\frac{FL}{AE}\right)$

Here, L is the length of the member, A is the area of the member, and E is the young’s modulus of the member.

For E and A is constant, the expression becomes,

$\left(1\right)\Delta =\frac{1}{AE}\sum F_{v}FL$        (1)

Find the length of member AB using given Figure.

$\begin{array}{c}{L}_{AB}=\frac{3}{\sin 56.31°}\\ =3.606\text{m}\end{array}$

Find the product of $F_{v1}FL$ and $F_{v2}FL$ for each member as shown in Table 1.

Member	L $\left(\text{m}\right)$	$\begin{array}{l}F\\ \left(\text{kN}\right)\end{array}$	$\begin{array}{l}{F}_{v1}\\ \left(\text{kN}\right)\end{array}$	$\begin{array}{l}{F}_{v1}FL\\ \left({\text{kN}}^2\text{-m}\right)\end{array}$	$\begin{array}{l}{F}_{v2}\\ \left(\text{kN}\right)\end{array}$	$\begin{array}{l}{F}_{v2}FL\\ \left({\text{kN}}^2\text{-m}\right)\end{array}$
AB	3.606	$-50.078$	$0.601$	$-108.53$	$-0.801$	$144.65$
BC	5	$-97.222$	$-0.833$	$405.09$	$-0.556$	$270.06$
			$\sum$	$296.56$		$414.71$

Find the horizontal deflection at joint B $\left({\Delta }_{BH}\right)$:

Substitute $296.56{\text{kN}}^2\text{-m}$ for $\sum F_{v1}FL$, $1,000{\text{mm}}^{\text{2}}$ for A, and $70\text{GPa}$ for E in Equation (1).

$\begin{array}{c}\left(1\text{kN}\right){\Delta }_{BH}=\frac{296.56{\text{kN}}^2\text{-m}}{\left(1,000{\text{mm}}^{\text{2}}\right)\left(70\text{GPa}\right)}\\ \left(1\text{kN}\right){\Delta }_{BH}=\frac{296.56{\text{kN}}^2\text{-m}\times \frac{{10}^{\text{3}}\text{mm}}{1\text{m}}}{\left(1,000{\text{mm}}^{\text{2}}\right)\left(70\text{GPa}\times \frac{1{\text{kN/mm}}^2}{1\text{GPa}}\right)}\\ {\Delta }_{BH}=4.24\text{mm}\to \end{array}$

Therefore, the horizontal deflection at joint B is $\underset{\_}{4.24\text{mm}\to }$.

Find the vertical deflection at joint B $\left({\Delta }_{BV}\right)$:

Substitute $414.71{\text{kN}}^2\text{-m}$ for $\sum F_{v1}FL$, $1,000{\text{mm}}^{\text{2}}$ for A, and $70\text{GPa}$ for E in Equation (1).

$\begin{array}{c}\left(1\text{kN}\right){\Delta }_{BV}=\frac{414.71{\text{kN}}^2\text{-m}}{\left(1,000{\text{mm}}^{\text{2}}\right)\left(70\text{GPa}\right)}\\ \left(1\text{kN}\right){\Delta }_{BV}=\frac{414.71{\text{kN}}^2\text{-m}\times \frac{{10}^{\text{3}}\text{mm}}{1\text{m}}}{\left(1,000{\text{mm}}^{\text{2}}\right)\left(70\text{GPa}\times \frac{1{\text{kN/mm}}^2}{1\text{GPa}}\right)}\\ {\Delta }_{BV}=5.92\text{mm}\downarrow \end{array}$

Therefore, the vertical deflection at joint B is $\underset{\_}{5.92\text{mm}\downarrow }$.