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## Q. 8.7

Draw the influence lines for the vertical reactions at supports $A$ and $E$, the reaction moment at support $A$, the shear at point $B$, and the bending moment at point $D$ of the beam shown in Fig. $8.10 (\mathrm{a})$.

## Verified Solution

Influence Line for $\boldsymbol A_{y}$. To determine the general shape of the influence line for $A_{y}$, we remove the restraint corresponding to $A_{y}$ by replacing the fixed support at $A$ by a roller guide that prevents the horizontal displacement and rotation at $A$ but not the vertical displacement. Next, point $A$ of the released structure is given a small displacement $\Delta$, and a deflected shape of the beam is drawn as shown in Fig. 8.10(b). Note that the deflected shape is consistent with the support and continuity conditions of the released structure. The end $A$ of the beam, which is attached to the roller guide, cannot rotate, so the portion $A C$ must remain horizontal in the displaced configuration. Also, point $E$ is attached to the roller support; therefore, it cannot displace in the vertical direction.

Thus, the portion $C F$ rotates about $E$, as shown in the figure. The two rigid portions, $A C$ and $C F$, of the beam remain straight in the displaced configuration and rotate relative to each other at the internal hinge at $C$, which permits such a rotation. The shape of the influence line is the same as the deflected shape of the released structure, as shown in Fig. 8.10(b).

By recognizing that $A_{y}=1 \mathrm{k}$ when a $1 \mathrm{k}$ load is placed at $A$, we obtain the value of $1 \mathrm{k} / \mathrm{k}$ for the influence-line ordinate at $A$, The ordinates at points $C$ and $F$ are then determined from the geometry of the influence line. The influence line for $A_{y}$ thus obtained is shown in Fig. 8.10(b).

Influence Line for $\boldsymbol E_{y}$. The roller support at $E$ is removed from the given structure, and a small displacement, $\Delta$, is applied at $E$ to obtain the deflected shape shown in Fig. 8.10(c). Because of the fixed support at $A$, the portion $A C$ of the released beam can neither translate nor rotate as a rigid body. The shape of the influence line is the same as the deflected shape of the released structure, as shown in the figure.

By realizing that $E_{y}=1 \mathrm{k}$ when the $1 \mathrm{k}$ load is placed at $E$, we obtain the value of $1 \mathrm{k} / \mathrm{k}$ for the influence-line ordinate at $E$. The ordinate at $F$ is then determined from the geometry of the influence line. The influence line thus obtained is shown in Fig. 8.10(c).

Influence Line for $\boldsymbol M_{A}$. To remove the restraint corresponding to the reaction moment $M_{A}$, we replace the fixed support at $A$ by a hinged support, as shown in Fig. $8.10$ (d). Next, a small rotation $\theta$ in the positive (counterclockwise) direction of $M_{A}$ is introduced at $A$ in the released structure to obtain the deflected shape shown in the figure. The shape of the influence line is the same as the deflected shape of the released structure.

Because the ordinate of the influence line at $A$ is zero, we determine the ordinate at $C$ by placing the $1 \mathrm{k}$ load at $C$ on the original beam (Fig. 8.10(d)). After computing the reaction $E_{y}=0$ by applying the equation of condition $\sum M_{C}^{C F}=0$, we determine the moment at $A$ from the equilibrium equation:

$+\curvearrowleft \sum M_{A}=0 \quad M_{A}-1(10)=0 \quad M_{A}=10 \mathrm{k}-\mathrm{ft}$

Thus, the value of the influence-line ordinate at $C$ is $10 \mathrm{k}$-ft $/ \mathrm{k}$. The ordinate at $F$ is then determined by considering the geometry of the influence line. The influence line thus obtained is shown in Fig. 8.10(d).

Influence Line for $\boldsymbol S_{B}$. To remove the restraint corresponding to the shear at $B$, we cut the given beam at $B$ to obtain the released structure shown in Fig. $8.10(\mathrm{e})$. Next, the released structure is given a small relative displacement, $\Delta$, to obtain the deflected shape shown in the figure. Support $A$ is fixed, so portion $A B$ can neither translate nor rotate as a rigid body. Also, the rigid portions $A B$ and $B C$ must remain parallel to each other in the displaced configuration. The shape of the influence line is the same as the deflected shape of the released structure, as shown in the figure.

The numerical values of the influence-line ordinates at $B$ are determined by placing the $1 \mathrm{k}$ load successively just to the left and just to the right of $B$ (Fig. 8.10(e)) and by computing the shears at $B$ for the two positions of the unit load. The ordinates at $C$ and $F$ are then determined from the geometry of the influence line. The influence line thus obtained is shown in Fig. 8.10(e).

Influence Line for $\boldsymbol M_{D}$. An internal hinge is inserted in the given beam at point $D$, and a small rotation $\theta$ is applied at $D$ to obtain the deflected shape shown in Fig. 8.10(f). The shape of the influence line is the same as the deflected shape of the released structure, as shown in the figure.

The value of the influence-line ordinate at $D$ is determined by placing the $1 \mathrm{k}$ load at $D$ and by computing the bending moment at $D$ for this position of the unit load (Fig. 8.10(f)). The ordinate at $F$ is then determined from the geometry of the influence line. The influence line thus obtained is shown in Fig. 8.10(f).