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Chapter 8

Q. 8.2

Draw the influence lines for the vertical reaction and the reaction moment at support A and the shear and bending moment at point B of the cantilever beam shown in Fig. 8.4 (\mathrm{a}).


Verified Solution

Influence Line for \boldsymbol A_{y}.

\begin{array}{r} +\uparrow \sum F_{y}=0 \\ A_{y}-1=0 \\ A_{y}=1 \end{array}

The influence line for A_{y} is shown in Fig. 8.4 (c).

Influence Line for \boldsymbol M_{A}.

\begin{aligned} +\curvearrowleft \sum M_{A} &=0 \\ -M_{A}-1(x) &=0 \\ M_{A} &=-1(x)=-x \end{aligned}

The influence line for M_{A}, which is obtained by plotting this equation, is shown in Fig. 8.4(d). As all the ordinates of the influence line are negative, it indicates that the sense of M_{A} for all the positions of the unit load on the beam is actually counterclockwise, instead of clockwise as initially assumed (see Fig. 8.4(b)) in deriving the equation of the influence line.

Influence Line for \boldsymbol S_{B}.

S_{B}=\left\{\begin{array}{lr} 0 & 0 \leq x<3  \mathrm{~m} \\ A_{y}=1 & 3  \mathrm{~m}<x \leq 8  \mathrm{~m} \end{array}\right.

The influence line for S_{B} is shown in Fig. 8.4(e).

Influence Line for \boldsymbol M_{B}.

M_{B}=\left\{\begin{array}{lr} 0 & 0 \leq x \leq 3  \mathrm{~m} \\ M_{A}+3 A_{y}=-x+3(1)=-x+3 & 3  \mathrm{~m} \leq x \leq 8  \mathrm{~m} \end{array}\right.

The influence line for M_{B} is shown in Fig. 8.4(f).