# Euler’s Theory of Column

 Question 1
A rigid beam AD of length 3a = 6 m is hinged at frictionless pin joint A and supported by two strings as shown in the figure. String BC passes over two small frictionless pulleys of negligible radius. All the strings are made of the same material and have equal cross-sectional area. A force F = 9 kN is applied at C and the resulting stresses in the strings are within linear elastic limit. The self-weight of the beam is negligible with respect to the applied load. Assuming small deflections, the tension developed in the string at C is _________ kN (round off to 2 decimal places). A 2.57 B 3.65 C 1.45 D 4.65
GATE ME 2022 SET-2   Strength of Materials
Question 1 Explanation:
F.B.D of beam \begin{aligned} \delta l_1:\delta l_2:\delta l_3&=T_1:T_2:T_3\\ 1:2:3&=T_1:T_2:T_3\\ \text{Let }T_1&=T\\ \therefore \; T_2&=2T,T_3=3T\\ \Sigma M_A&=0\\ \therefore \; -T \times a-2T &\times 2a-3T \times 3a+F \times 2a=0\\ \therefore \;\; F \times 2a &=14 Ta\\ 2F&=14T\\ T&=\frac{F}{7}=\frac{9}{7}kN\\ T_2&=2T=\frac{18}{7}=2.57kN \end{aligned}
 Question 2
An L-shaped elastic member ABC with slender arms AB and BC of uniform cross-section is clamped at end A and connected to a pin at end C. The pin remains in continuous contact with and is constrained to move in a smooth horizontal slot. The section modulus of the member is same in both the arms. The end C is subjected to a horizontal force P and all the deflections are in the plane of the figure. Given the length AB is $4a$ and length BC is $a$, the magnitude and direction of the normal force on the pin from the slot, respectively, are A 3P/8, and downwards B 5P/8, and upwards C P/4, and downwards D 3P/4, and upwards
GATE ME 2022 SET-1   Strength of Materials
Question 2 Explanation: No vertical deflection allowed
$\delta _{VC}=\delta _{VB}=0 \text{ (Vertical deflection) }$ \begin{aligned} \delta _B&=\frac{ML^2}{2EI}-\frac{NL^2}{3EI}=0\\ \frac{M}{2}&\frac{NL}{3}\\ N&=\frac{3M}{2L}\;\;(\because L=4a)\\ N&=\frac{3Pa}{2 \times 4 a}\\ N&=\frac{3}{8}P \text{ (downward)} \end{aligned}

 Question 3
A uniform light slender beam AB of section modulus EI is pinned by a frictionless joint A to the ground and supported by a light inextensible cable CB to hang a weight W as shown. If the maximum value of W to avoid buckling of the beam AB is obtained as $\beta \pi ^2 EI$, where $\pi$ is the ratio of circumference to diameter of a circle, then the value of $\beta$ is A $0.0924\; m^{-2}$ B $0.0713\; m^{-2}$ C $0.1261\; m^{-2}$ D $0.1417\; m^{-2}$
GATE ME 2022 SET-1   Strength of Materials
Question 3 Explanation:
Draw FBD of AB $\Sigma M_A=0$
$W \times 2.5=T \sin 30^{\circ} \times 2.5$
$T=2W$
Compressive load acting on AB $=T \cos 30^{\circ}=2W \times \frac{\sqrt{3}}{2}=\sqrt{3}W$
Buckling happens when $\sqrt{3}W=P_{cr}=\frac{\pi ^2 EI}{L_e^2}$
$\sqrt{3}W=\frac{\pi ^2 EI}{L^2} \;\;(\because L_e=L \text{as both ends hinged})$
$W=\frac{1 \times \pi ^2 EI}{\sqrt{3} \times (2.5)^2}=0.0924 \pi^2 EI$
$W0.0924 \pi^2 EI=\beta \pi ^2 EI$
$\beta =0.0924 m^{-2}$
 Question 4
A column with one end fixed and one end free has a critical buckling load of 100 N. For the same column, if the free end is replaced with a pinned end then the critical buckling load will be ________N (round off to the nearest integer).
 A 800 B 450 C 1020 D 680
GATE ME 2021 SET-2   Strength of Materials
Question 4 Explanation:  For a given material in 0.1 and length,
$\mathrm{Pe} \propto \frac{1}{\alpha^{2}}$
where, $\quad \alpha=$ length fixity coefficient
\begin{aligned} \frac{(P e)_{I I}}{(P e)_{I}} &=\left(\frac{\alpha_{I}}{\alpha_{I I}}\right)^{2}=\left(\frac{2 L}{1 / \sqrt{2}}\right)^{2}=8 \\ \left(P_{e}\right)_{I I} &=8\left(P_{e}\right)_{I}=800 \mathrm{~N} \end{aligned}
 Question 5
A right solid circular cone standing on its base on a horizontal surface is of height H and base radius R. The cone is made of a material with specific weight w and elastic modulus E. The vertical deflection at the mid-height of the cone due to self-weight is given by
 A $\frac{wH^2}{8E}$ B $\frac{wH^2}{6E}$ C $\frac{wRH}{8E}$ D $\frac{wRH}{6E}$
GATE ME 2021 SET-1   Strength of Materials
Question 5 Explanation: \begin{aligned} P_{\mathrm{x}-\mathrm{x}}&=\frac{(-) \mathrm{W} A_{x-x}(x)}{3}\\ \text{Contraction of small strip }&=(\delta l)_{\text {strip }}\\ &=\frac{P_{x-x} d x}{(A E)_{x-x}} \\ &=\frac{w\left(A_{x-x}\right)(x) d x}{3\left(A_{x-x}\right) E}=\frac{w x}{3 E} d x \end{aligned}
Contraction of conical bar at mid height ( i.e. $x=\frac{H}{2}$ )
\begin{aligned} &=\int_{H / 2}^{H}(\delta l)_{\operatorname{stn} p} \\ &=\frac{W}{3 E} \int_{H / 2}^{H} x d x \\ &=\frac{w}{6 E}\left[H^{2}-\frac{H^{2}}{4}\right]=\frac{w H^{2}}{8 E} \end{aligned}

There are 5 questions to complete.