Part A

Learning Goal:

To use the moment$-$area theorems for determining slopes and deflections at specific point on a beam.

There are two moment$-$area theorems. Theorem $1$ is used for finding the change in slope between two tangent lines, and Theorem $2$ is used for finding the vertical distance between two tangent lines. Below are the two moment$-$area theorems.

Theorem $1$: The change in slope between any two points on the elastic curve equals the area of the $\frac{M}{E}$I diagram between these two points.

Considering $($Figure $1),$ the area under the M$/$EI diagram is ${}_{\frac{B}{A}}={\displaystyle \underset{A}{\overset{B}{}}}\frac{M}{EI}dx.$ The notation ${}_{\frac{B}{A}}$ is the angle of the tangent at $B$ measured with respect to the tangent at $A.$

Theorem $2$: The vertical deviation of the tangent at a point $(A)$ on the elastic curve with respect to the tangent extended from another point $(B)$ equals the "moment" of the area under the $\frac{M}{E}$I diagram between the two points. This moment is computed about point $A($the point on

igure

$2$ of $3$

Use the moment$-$area theorems to determine the slope at $B$ for the beam shown in $($Figure $2),$ where $a=16ft$ and $w=760l\frac{b}{f}t.$ Take $E=2\mathrm{.}4{10}^4$ksi and $I=550i{n}^4.$

Express your answer in appropriate units to three significant figures.

View Available Hint$($s$)$

Part B

Use the moment$-$area theorems to determine the displacement at $C$ for the beam shown in $($Figure $2),$ where $a=16ft$ and $w=760l\frac{b}{f}t.$ Take $E=2\mathrm{.}4{10}^4$ksi and $I=550i{n}^4.$

Express your answer in appropriate units to three