Question 9.SP.11: For the prismatic beam and loading shown, determine the slop......

MODELING and ANALYSIS:
M∕EI Diagram. From a free-body diagram of the beam (Fig. 1), determine the reactions and then draw the shear and bending-moment diagrams. Since the flexural rigidity of the beam is constant, divide each value of M by EI and obtain the M/EI diagram shown.

Reference Tangent. In Fig. 2, since the beam and its loads are symmetric with respect to the midpoint C, the tangent at C is horizontal and can be used as the reference tangent. Referring to Fig. 2 and since \theta_C=0 ,

\theta_E=\theta_C+\theta_{E / C}=\theta_{E / C}          (1)

y_E=t_{E / C}-t_{D / C}            (2)

Slope at E. Referring to the M/EI diagram shown in Fig. 1 and using the first moment-area theorem,

\begin{aligned} & A_1=-\frac{w a^2}{2 E I}\left(\frac{L}{2}\right)=-\frac{w a^2 L}{4 E I} \\ & A_2=-\frac{1}{3}\left(\frac{w a^2}{2 E I}\right)(a)=-\frac{w a^3}{6 E I} \end{aligned}

Using Eq. (1),

\theta_E=\theta_{E / C}=A_1+A_2=-\frac{w a^2 L}{4 E I}-\frac{w a^3}{6 E I}

\theta_E=-\frac{w a^2}{12 E I}(3 L+2 a) \quad \theta_E=\frac{w a^2}{12 E I}(3 L+2 a) ⦪

Deflection at E. Use the second moment-area theorem to write

Use Eq. (2) to obtain

\begin{aligned} y_E & =t_{E / C}-t_{D / C}=-\frac{w a^3 L}{4 E I}-\frac{w a^4}{8 E I} \\ & =-\frac{w a^3}{8 E I}(2 L+a) \quad y_E=\frac{w a^3}{8 E I}(2 L+a) \downarrow \end{aligned}