Strength of Materials

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).

GATE ME 2022 SET-2      Euler's Theory of Column
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
A linear elastic structure under plane stress condition is subjected to two sets of loading, I and II. The resulting states of stress at a point corresponding to these two loadings are as shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress \sigma _{xx} is ________.

\frac{3\sigma }{2}
\sigma \left (1+\frac{1}{\sqrt{2}} \right )
\frac{\sigma }{2}
\sigma \left (1-\frac{1}{\sqrt{2}} \right )
GATE ME 2022 SET-2      Mohr's Circle
Question 2 Explanation: 

\sigma _x=0,\sigma _y=0,\tau _{xy}=0,\theta =-45^{\circ}
\begin{aligned} \sigma _{xx}&=\sigma +\sigma _\theta \\ \sigma _\theta&=\left [ \frac{\sigma _x +\sigma _y}{2} \right ] +\left [ \frac{\sigma _x -\sigma _y}{2} \right ] \cos 2\theta +\tau _{xy} \sin 2\theta \\ \sigma _\theta &= \frac{0+\sigma }{2} +\left [\frac{0-\sigma }{2} \right ] \cos 2(-45)+0\\ \sigma _\theta &= \frac{\sigma }{2}\\ \sigma _{xx} &=\sigma +\sigma _\theta =\sigma +\frac{\sigma }{2}=\frac{3\sigma }{2} \end{aligned}
Question 3
Which one of the following is the definition of ultimate tensile strength (UTS) obtained from a stress-strain test on a metal specimen?
Stress value where the stress-strain curve transitions from elastic to plastic behavior
The maximum load attained divided by the original cross-sectional area
The maximum load attained divided by the corresponding instantaneous crosssectional area
Stress where the specimen fractures
GATE ME 2022 SET-2      Stress-strain Relationship and Elastic Constants
Question 3 Explanation: 
Tensile Strength: The tensile strength, or ultimate tensile strength (UTS), is the maximum load obtained in a tensile test, divided by the original cross-sectional area of the specimen.
\sigma _u=\frac{P_{max}}{A_o}
where, \sigma _u= Ultimate tensile strength, kg/mm^2
P_{max}= Maximum load obtained in a tensile test, kg
A_o= Original cross-sectional area of gauge length of the test piece, mm^2
The tensile strength is a very familiar property and widely used for identification of a material. It is very easy to determine and is a quite reproducible property. It is used for the purposes of specifications and for quality control of a product. Tensile strength can be empirically correlated to other properties such as hardness and fatigue strength. For brittle materials, the tensile strength is a valid criterion for design.
Question 4
A thin-walled cylindrical pressure vessel has mean wall thickness of t and nominal radius of r. The Poisson's ratio of the wall material is 1/3. When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1% and the corresponding wall thickness became \bar{t}. The corresponding change in the wall thickness of the cylindrical portion, i.e. 100\times (\bar{t}-t)/t, is ________%(round off to 3 decimal places).
GATE ME 2022 SET-1      Thin Cylinder
Question 4 Explanation: 

\pi(d+\delta d)=1.001 \pi d
\varepsilon _1=\frac{\delta d}{d}=0.001=\frac{Pd}{4tE}(2-v)
\frac{Pd}{4tE}=\frac{0.001}{(2-v)}=6 \times 10^{-4}
Radial strain
\varepsilon _2=\frac{\delta t}{t}=\frac{1}{E}\left [ \sigma _r-v(\sigma _h-\sigma _L) \right ]
\frac{\delta t}{t}=\frac{1}{E}\left [ -v\frac{Pd}{4t}(2+1) \right ]
\frac{\delta t}{t}=-\frac{Pd}{4tE}v(3)=-6 \times 10^{-4} \times \frac{1}{3}(3)=-6 \times 10^{-4}
Therefore, The corresponding change in the wall thickness of the cylindrical portion
=100 \times \left [ \frac{\bar{t}-t}{t} \right ]=100 \times \frac{\delta t}{t}=100 \times (-6) \times 10^{-4}=-0.06 \%
Question 5
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

3P/8, and downwards
5P/8, and upwards
P/4, and downwards
3P/4, and upwards
GATE ME 2022 SET-1      Euler's Theory of Column
Question 5 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 6
Assuming the material considered in each statement is homogeneous, isotropic, linear elastic, and the deformations are in the elastic range, which one or more of the following statement(s) is/are TRUE?
A body subjected to hydrostatic pressure has no shear stress
If a long solid steel rod is subjected to tensile load, then its volume increases.
Maximum shear stress theory is suitable for failure analysis of brittle materials.
If a portion of a beam has zero shear force, then the corresponding portion of the elastic curve of the beam is always straight.
GATE ME 2022 SET-1      Stress-strain Relationship and Elastic Constants
Question 6 Explanation: 
(c) Wrong : Maximum shear stress theory good for ductile material.
(d) If shear force = 0, M = C, But elastic curve is always non-linear.
Question 7
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

0.0924\; m^{-2}
0.0713\; m^{-2}
0.1261\; m^{-2}
0.1417\; m^{-2}
GATE ME 2022 SET-1      Euler's Theory of Column
Question 7 Explanation: 
Draw FBD of AB

\Sigma M_A=0
W \times 2.5=T \sin 30^{\circ} \times 2.5
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 8
A cantilever beam with a uniform flexural rigidity (EI = 200 \times 10^6 N.m^2) is loaded with a concentrated force at its free end. The area of the bending moment diagram corresponding to the full length of the beam is 10000 \;N.m^2. The magnitude of the slope of the beam at its free end is ________micro radian (round off to the nearest integer).
GATE ME 2021 SET-2      Bending of Beams
Question 8 Explanation: 

\begin{aligned} &A=10000 \mathrm{~N}-\mathrm{m}^{2}\\ &\mathrm{El}=200 \times 10^{6} \mathrm{~N}-\mathrm{m}^{2} \end{aligned}

As per moment area first theorem.
\begin{aligned} \theta_{\mathrm{B}}-\theta_{A} &=\left(\frac{A}{E I}\right) A B \\ \theta_{\mathrm{B}}-0 &=\frac{10000}{200 \times 10^{6}}=0.5 \times 10^{-4} \mathrm{radian} \\ \theta_{\mathrm{B}} &=50 \mu \text { radians } \end{aligned}
Question 9
A plane frame PQR (fixed at P and free at R) is shown in the figure. Both members (PQ and QR) have length, L, and flexural rigidity, EI. Neglecting the effect of axial stress and transverse shear, the horizontal deflection at free end, R, is
GATE ME 2021 SET-2      Bending of Beams
Question 9 Explanation: 

\begin{aligned} U&=U_{P Q}+U_{Q P} \\ U&=\frac{M^{2} L}{2 E I}+\int_{0}^{L} \frac{\left(M_{x-x}\right)^{2}(d x)}{2 E I} \\ U&=\frac{(F L)^{2} L}{2 E I}+\int_{0}^{L}\left(\frac{(F x)^{2}(d x)}{2 E I}\right) \\ U&=\frac{F^{2} L^{3}}{2 E I}+\frac{F^{2} L^{3}}{6 E I}=\frac{2 F^{2} L^{3}}{3 E I} \end{aligned}
By Castigliano's theorem:
\left(\delta_{H}\right)_{R}=\frac{\partial U}{\partial F}=\frac{4 F L^{3}}{3 E I}
Question 10
A steel cubic block of side 200 mm is subjected to hydrostatic pressure of 250 N/mm^2. The elastic modulus is 2 \times 10^5 \; N/mm^2 and Poisson ratio is 0.3 for steel. The side of the block is reduced by _____mm (round off to two decimal places).
GATE ME 2021 SET-2      Stress-strain Relationship and Elastic Constants
Question 10 Explanation: 

\begin{aligned} E &=200 \mathrm{GPa} \\ \sigma &=250 \mathrm{MPa} \\ \mu &=0.3 \\ \epsilon_{x} &=\epsilon_{y}=\epsilon_{z}=\frac{\delta a}{a} \\ \frac{1}{E}\left[\sigma_{x}-\mu\left(\sigma_{y}+\sigma_{z}\right)\right] &=\frac{(\delta a)}{a} \\ \left(\frac{-\sigma}{E}\right)(1-2 \mu)&=\frac{\delta a}{a}\\ \delta a &=-\frac{(250)(1-0.6)(200)}{200 \times 10^{3}} \\ \delta a &=(-) 0.10 \mathrm{~mm} \end{aligned}
Reduction in side of cube is 0.10mm.

There are 10 questions to complete.

9 thoughts on “Strength of Materials”

  1. Question no 8 is actually wrong.. Let me explain.. In the explanation of this question as shown in this page, we get c=2L assuming Mz=0 at x=L… Now, if the question would be correct, then we would also get c=2L assuming Mz=M at x=0 since the expression is given to be valid at 0≤x≤L . But we actually get different value of ‘c’ in this assumption. Hence, it is proved that the question is wrong.


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