For the beam and loading shown, (a) determine the deflection at end A, (b) evaluate $y_A$ for the following data:

W10 × 33: I = 170 in $^4$ E = 29 × 10 $^6$ psi

a = 3 ft = 36 in. L = 5.5 ft = 66 in.

w = 13.5 kips/ft = 1125 lb/in.

Question

For the beam and loading shown, (a) determine the deflection at end A, (b) evaluate [latex]y_A [/latex] for the following data:

W10 × 33: I = 170 in[latex]^4[/latex] E = 29 × 10[latex]^6[/latex] psi

a = 3 ft = 36 in. L = 5.5 ft = 66 in.

w = 13.5 kips/ft = 1125 lb/in.

Accepted Answer

(M/EI) Diagram. We first draw the bending-moment diagram. Since the flexural rigidity EI is constant, we obtain the (M/EI) diagram shown, which consists of a parabolic spandrel of area [latex]A_1[/latex] and a triangle of area [latex]A_2[/latex].

[latex]A_{1}=\frac{1}{3}\left(-\frac{w a^{2}}{2 E I} ight) a=-\frac{w a^{3}}{6 E I}[/latex]

[latex]A_{2}=\frac{1}{2}\left(-\frac{w a^{2}}{2 E I} ight) L=-\frac{w a^{2} L}{4 E I}[/latex]

Reference Tangent at B. The reference tangent is drawn at point B as shown. Using the second moment-area theorem, we determine the tangential deviation of C with respect to B:

[latex]t_{C / B}=A_{2} \frac{2 L}{3}=\left(-\frac{w a^{2} L}{4 E I} ight) \frac{2 L}{3}=-\frac{w a^{2} L^{2}}{6 E I}[/latex]

From the similar triangles [latex]A^{\prime \prime} A^{\prime} B[/latex] and [latex]C C^{\prime} B[/latex], we find

[latex]A^{\prime \prime} A^{\prime}=t_{C / B}\left(\frac{a}{L} ight)=-\frac{w a^{2} L^{2}}{6 E I}\left(\frac{a}{L} ight)=-\frac{w a^{3} L}{6 E I}[/latex]

Again using the second moment-area theorem, we write

[latex]t_{A / B}=A_{1} \frac{3 a}{4}=\left(-\frac{w a^{3}}{6 E I} ight) \frac{3 a}{4}=-\frac{w a^{4}}{8 E I}[/latex]

a. Deflection at End A

[latex]y_{A}=A^{\prime \prime} A^{\prime}+t_{A / B}=-\frac{w a^{3} L}{6 E I}-\frac{w a^{4}}{8 E I}=-\frac{w a^{4}}{8 E I}\left(\frac{4}{3} \frac{L}{a}+1 ight)[/latex]

[latex]y_{A}=\frac{w a^{4}}{8 E I}\left(1+\frac{4}{3} \frac{L}{a} ight) \downarrow[/latex]

b. Evaluation of [latex]y_{A}[/latex]. Substituting the data given, we write

[latex]y_{A}=\frac{(1125 lb / in .)(36 in .)^{4}}{8\left(29 imes 10^{6} lb / in ^{2} ight)\left(170 in ^{4} ight)}\left(1+\frac{4}{3} \frac{66 in .}{36 in .} ight)[/latex]

[latex]y_{A}=0.1650 in . \downarrow[/latex]

Question 9.S-P.12: For the beam and loading shown, (a) determine the deflection...

Related Answered Questions

The rigid bar DEF is welded at point D to the uniform steel beam AB. For the loading shown, determine (a) the equation of the elastic curve of the beam, (b) the deflection at the midpoint C of the beam. Use E = 29 × 10^6 psi. ...

Verified Answer:

For the beam and loading shown, determine the slope and deflection at point B. ...

Verified Answer:

Determine the maximum deflection of the beam of Example 9.12. ...

Verified Answer:

Determine the reactions at the supports for the prismatic beam and loading shown in Fig. 9.36. (This is the same beam and loading as in Example 9.05 of Sec. 9.5.) ...

Verified Answer:

For the uniform beam and loading shown, determine (a) the reaction at each support, (b) the slope at end A. ...

Verified Answer:

For the beam and loading shown, determine the reaction at the fixed support C. ...

Verified Answer:

For the uniform beam AB, (a) determine the reaction at A, (b) derive the equation of the elastic curve, (c) determine the slope at A. (Note that the beam is statically indeterminate to the first degree.) ...

Verified Answer:

Determine the reaction at the supports for the prismatic beam and loading shown (Fig. 9.71). ...

Verified Answer:

Determine the slope and deflection at end B of the prismatic beam of Example 9.09, drawing the bending-moment diagram by parts. ...

Verified Answer:

Determine the slope and deflection at end B of the prismatic cantilever beam AB when it is loaded as shown (Fig. 9.48), knowing that the flexural rigidity of the beam is EI = 10 MN · m². ...

Verified Answer: