Deflection of Beams - PracticePaper

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Question 1

For the frame shown in the figure (not to scale), all members

(A B, B C, C D, G B

, and

C H)

have the same length,

L

and flexural rigidity, El. The joints at

B

and

C

are rigid joints, and the supports

A

and

D

are fixed supports. Beams GB and

\mathrm{CH}

carry uniformly distributed loads of

w

per unit length. The magnitude of the moment reaction at

A

\mathrm{wL}^{2} / \mathrm{k}

. What is the value of

\mathrm{k}

(in integer) ? ____

A	4
B	6
C	8
D	10

GATE CE 2023 SET-2 Solid Mechanics

Question 1 Explanation:

Due to symmetry, slope at mid of

B C

is zero and can be taken as slider at point

O

Distribution factors:

\begin{aligned} \mathrm{K}_{\mathrm{BA}} & =\frac{4 \mathrm{EI} / \mathrm{L}}{4 \mathrm{EI} / \mathrm{L}+\mathrm{EI} /(\mathrm{L} / 2)}=\frac{2}{3} \\ \mathrm{K}_{\mathrm{BO}}&= 1-\mathrm{K}_{\mathrm{BA}} \\ & =1-2 / 3=1 / 3 \\ \therefore \quad \mathrm{M}_{\mathrm{BA}} & =\frac{\mathrm{WL}^{2}}{2} \times \frac{2}{3}=\frac{\mathrm{WL}^{2}}{3} \\ \mathrm{M}_{\mathrm{AB}} & =\frac{\mathrm{M}_{\mathrm{BA}}}{2}=\frac{\mathrm{WL}^{2} / 3}{2}=\frac{\mathrm{WL}^{2}}{6} \\ & \therefore \mathrm{K}=6 \end{aligned}

Question 2

When a simply-supported elastic beam of span

\mathrm{L}

and flexural rigidity

EI ( E

is the modulus of elasticity and

\mathrm{I}

is the moment of inertia of the section) is loaded with a uniformly distributed load

w

per unit length, the deflection at the mid-span is

\Delta_{0}=\frac{5}{384} \frac{w L^{4}}{\mathrm{El}}

.

If the load on one half of the span is now removed, the mid-span deflection ____

A	reduces to $\Delta_{0} / 2$
B	reduces to a value less than $\Delta_{0} / 2$
C	reduces to a value greater than $\Delta_{0} / 2$
D	remains unchanged at $\Delta_{0}$

GATE CE 2023 SET-2 Solid Mechanics

Question 2 Explanation:

As Case-I and II are mirror image of each other,.
Hence,

\quad \Delta_{\mathrm{C}, 1}=\Delta_{\mathrm{C}, \mathrm{II}}

Also,

\quad \Delta_{\mathrm{C}}=\Delta_{\mathrm{C}, 1}+\Delta_{\mathrm{C}, 1}

\Rightarrow \quad \Delta_{\mathrm{C}, \mathrm{I}}=\Delta_{\mathrm{C}, \mathrm{II}}=\frac{\Delta_{\mathrm{C}}}{2}

Question 3

A single story building model is shown in the figure. The rigid bar of mass 'm' is supported by three massless elastic columns whose ends are fixed against rotation. For each of the columns, the applied lateral force (P) and corresponding moment (M) are also shown in the figure. The lateral deflection

(\delta)

of the bar is given by

\delta=\frac{P L^{3}}{12 E I}

, where L is the effective length of the column, E is the Young's modulus of elasticity and I is the area moment of inertia of the column cross-section with respect to its neutral axis.

For the lateral deflection profile of the columns as shown in the figure, the natural frequency of the system for horizontal oscillation is

A	$6 \sqrt{\frac{E I}{m L^{3}}} \mathrm{rad} / \mathrm{s}$
B	$\frac{1}{L} \sqrt{\frac{2 E I}{m}} \mathrm{rad} / \mathrm{s}$
C	$6 \sqrt{\frac{6 E I}{m L^{3}}} \mathrm{rad} / \mathrm{s}$
D	$\frac{2}{L} \sqrt{\frac{E I}{m}} \mathrm{rad} / \mathrm{s}$