GATE Overflow for GATE XE - Recent questions in Mechanics of deformable bodies

GATE XE 2026 | Question: 52

Tue, 24 Feb 2026 15:45:43 +0000

At the peak (also denoted as UTS) of an engineering stress vs. engineering strain curve for ductile metal, the engineering strain is $0.2$. The corresponding true stress $(\sigma)$ vs. true strain ( $\epsilon$ ) relationship follows the equation: $\sigma=K \epsilon^{n}$, where $K$ and $n$ are constants.

The engineering stress at the peak in $\mathrm{MPa}$ is $\_\_\_\_$ (rounded off to one decimal place).

Given: $K=200 \mathrm{MPa}$

GATE XE 2026 | Question: 60

Tue, 24 Feb 2026 15:44:55 +0000

Which among the following options is/are CORRECT unit(s) of stress?

$\mathrm{Nm}^{2}$
$\text{Pa}$
$\text{Nm}$
$\mathrm{N} / \mathrm{m}^{2}$

GATE XE 2026 | Question: 63

Tue, 24 Feb 2026 15:44:28 +0000

A $\text{2D}$ state of stress at a point in a body is given by $\sigma_{x x}=-40 \mathrm{~MPa}$, $\sigma_{y y}=100 \mathrm{~MPa}$ and $\tau_{x y}=50 \mathrm{~MPa}$.

The radius of the Mohr's circle for the given state of stress is $\_\_\_\_$ $\mathrm{MPa}$ (rounded off to two decimal places).

GATE XE 2026 | Question: 64

Tue, 24 Feb 2026 15:44:26 +0000

Sum of the principal stresses of a $\text{2D}$ stress state $\left[\begin{array}{cc}11 & 4 \\ 4 & 5\end{array}\right]$ is $\_\_\_\_$ (in integer).

GATE XE 2026 | Question: 67

Tue, 24 Feb 2026 15:43:44 +0000

One column with square cross section of side $r$ and another column with rectangular cross section of breadth $p$ and width $q(<p)$ are made from the same material. Both the columns have one end fixed, and the other end is free. They are subjected to axial loads along the centroidal axis.

Consider the area of cross sections of both the columns to be the same. The minimum critical Euler buckling loads of the columns with rectangular and square cross-sections are $F_{\text {rect }}$ and $F_{\mathrm{sq}}$, respectively.

Then $\dfrac{F_{\text {rect }}}{F_{\text {sq }}}$ is $\_\_\_\_$ .

$\dfrac{p}{q}$
$\dfrac{q}{p}$
$\dfrac{r}{q}$
$\dfrac{r^{2}}{p q}$

GATE XE 2026 | Question: 68

Tue, 24 Feb 2026 15:43:40 +0000

Two prismatic rods of identical lengths are designed for the same strain energy density when subjected to the same axial load. One of the rods is made of steel (Young's modulus $=210 \mathrm{GPa}$ ) and another is made of aluminum (Young's modulus $=70 \mathrm{GPa}$).

If the diameter of the aluminum rod is $70$ $\mathrm{mm}$, then which one of the following options corresponds to the diameter of the steel rod in mm?

$53.19$
$70$
$210$
$29.13$

GATE XE 2026 | Question: 70

Tue, 24 Feb 2026 15:43:02 +0000

A $\text{2D}$ stress state of pure shear is shown in the figure.

Which one of the following options is equivalent to the given stress state?

Figures are not to scale.

GATE XE 2026 | Question: 73

Tue, 24 Feb 2026 15:42:39 +0000

The bending moment diagram for a simply supported beam is piecewise linear as shown in the figure. The bending moment $M(\mathrm{x})$ at $\mathrm{x}=0.5 \mathrm{~m}$ is $5$ $\mathrm{Nm}$. The beam has a rectangular cross-section with an area of $1 \mathrm{~m}^{2}$.

The absolute value of the maximum shear stress on the cross-section at $x=0.75 \mathrm{~m}$ is $\_\_\_\_$ $\mathrm{N} / \mathrm{m}^{2}$ (in integer).

GATE XE 2026 | Question: 74

Tue, 24 Feb 2026 15:42:26 +0000

A thin walled cylindrical container of diameter $2$ $\mathrm{m}$ and wall thickness of $2.5$ $\mathrm{cm}$ is made of steel whose Young's modulus is $200$ $\mathrm{GPa}$ and yield stress is $450$ $\mathrm{MPa}$.

Using von Mises criteria, the maximum permissible pressure is $\_\_\_\_$ $\mathrm{MPa}$ (rounded off to the nearest integer).

GATE XE 2026 | Question: 75

Tue, 24 Feb 2026 15:42:25 +0000

A solid axial bar made of steel with Young's modulus, $200$ $\mathrm{GPa}$ and Poisson's ratio, $0.3$, is subjected to uniaxial stress of $50$ $\mathrm{MPa}$.

The absolute value of the maximum shear strain on the outer surface of the bar is $\_\_\_\_$ $\times 10^{-4}$ (rounded off to two decimal places).

GATE XE 2026 | Question: 76

Tue, 24 Feb 2026 15:42:24 +0000

$\text{EFGH}$ (solid lines) is the initial configuration and $\mathrm{E}^{\prime} \mathrm{F}^{\prime} \mathrm{G}^{\prime} \mathrm{H}^{\prime}$ (dashed lines) is the deformed configuration of an object as shown in the figure. $\mathrm{E}^{\prime}$ coincides with $\text{E}$ and $\mathrm{F}^{\prime}$ coincides with $\text{F}$.

The average normal strain along the line segment $\text{OP}$ is $\_\_\_\_$ (rounded off to three decimal places).

Figure is not to scale.
All units are in $\mathrm{mm}$.

GATE XE 2026 | Question: 77

Tue, 24 Feb 2026 15:42:21 +0000

A simply supported beam of length $l=2 \mathrm{~m}$ is subjected to a concentrated moment $M=150 \mathrm{~kNm}$ at a distance $l / 2$ from the left end as shown in the figure. The elastic-strain energy $U$ of the beam is given by the following expression:

$$U=\frac{M^{2} l}{48 E I}$$

The section modulus of the beam is $E I=25 \times 10^{5} \mathrm{Nm}^{2}$.

The absolute value of the slope of the beam at a distance $l / 2$ from the left end is $\_\_\_\_$ (rounded off to three decimal places).

GATE XE 2026 | Question: 118

Tue, 24 Feb 2026 15:32:58 +0000

A rectangular plastic specimen with a cross-sectional area of $240 \mathrm{~mm}^{2}$ is subjected to tension with a force of $15000$ $\mathrm{N}$ resulting in elastic deformation. If the Young's modulus of the specimen is $34.55$ $\mathrm{MPa}$, the value of strain is $\_\_\_\_\_\_$ (rounded off to two decimal places).

GATE XE 2025 | Question: 66

Sun, 04 May 2025 19:06:53 +0000

Consider a spring-mass system with mass $m$ and spring stiffness $k$ as shown in the illustration. At time $t=0$, the mass is displaced by $P$ units and the velocity of the mass is zero. The displacement of the mass, $x(t)$, is measured from the equilibrium position.

Which one of the following functions represent $x(t)$?

$P \cos \left(\sqrt{\frac{k}{m}} t\right)+P \sin \left(\sqrt{\frac{k}{m}} t\right)$
$P \sin \left(\sqrt{\frac{k}{m}} t\right)$
$\frac{P}{2} \cos \left(\sqrt{\frac{k}{m}} t\right)+\frac{P}{2} \sin \left(\sqrt{\frac{k}{m}} t\right)$
$P \cos \left(\sqrt{\frac{k}{m}} t\right)$

GATE XE 2025 | Question: 68

Sun, 04 May 2025 19:06:48 +0000

The Mohr's circle corresponding to an infinitesimal element is shown in the figure. The plane $\text{PQ}$ in the infinitesimal element, at an angle of $\theta$ from the $x$-axis, is in a state of pure shear.

Which one of the following values of $\theta$ (in degrees) is CORRECT?

$90$
$60$
$45$
$120$

GATE XE 2025 | Question: 69

Sun, 04 May 2025 19:06:46 +0000

The two-dimensional state of stress, in an infinitesimal element, is given by

$\sigma_{x x}=800 \mathrm{MPa}, \sigma_{x y}=300 \mathrm{MPa}$ and $\sigma_{y y}=0 \mathrm{MPa}$.

Which one of the following options is the maximum shear stress (in $\mathrm{MPa}$) in the element?

$500$
$400$
$800$
$300$

GATE XE 2025 | Question: 72

Sun, 04 May 2025 19:06:39 +0000

Consider a beam with a square box cross-section as shown in the figure. The outer square has a length of $10 \: \mathrm{mm}$. The thickness of the section is $1 \: \mathrm{mm}$.

The area moment of inertia about the $x$-axis is $\_\_\_\_\_\_ \: \mathrm{mm}^{4}$ (in integer).

GATE XE 2025 | Question: 73

Sun, 04 May 2025 19:06:38 +0000

For a certain linear elastic isotropic material, the Young's modulus is $140 \: \mathrm{GPa}$ and the shear modulus is $50 \: \mathrm{GPa}$.

The Poisson's ratio for the material is $\_\_\_\_\_\_$ (rounded off up to two decimal places).

GATE XE 2025 | Question: 76

Sun, 04 May 2025 19:06:33 +0000

Which one of the following vertical columns, of circular cross-section, sustains the highest load without buckling?

Cantilever column with a length $\text{L}$ and cross-section diameter $\text{d}$
Column with hinge at one end and roller at the other end with a length $2 \:\text{L}$ and cross-section diameter $\text{d}$.
Cantilever column with a length $\text{L}$ and cross-section diameter $2\:\text{d}$
Column with hinge at one end and roller at the other end with a length $\text{L}$ and cross-section diameter $\text{d}$.

GATE XE 2025 | Question: 79

Sun, 04 May 2025 19:06:25 +0000

A closed thin cylindrical tank with a mean diameter $\mathrm{d}=300 \mathrm{~mm}$ and thickness $\mathrm{t}=2 \mathrm{~mm}$, is subjected to a uniform internal gas pressure $p$. The allowable shear stress on the curved wall of the tank is $70 \: \mathrm{MPa}$.

Based on the Tresca criteria, which one of the following options for the maximum safe value of $p$ (in $\mathrm{MPa}$) is CORRECT?

$3.73$
$7.46$
$1.87$
$5.60$

GATE XE 2025 | Question: 80

Sun, 04 May 2025 19:06:22 +0000

An infinitesimal square element $\text{PQRS}$ is shown in the figure. The $x$ and $y$ axes are also marked in the figure. The strains on the element are given by

$\varepsilon_{x x}=500 \times 10^{-6}, \: \varepsilon_{y y}=100 \times 10^{-6} \text { and } \varepsilon_{x y}=0$

Which of the following statements is/are $\textsf{CORRECT}$?

Percentage change in length of the diagonal $\text{PR}$ is $0.03$.
Change in angle between $\text{PR}$ and $\text{QS}$ is $4 \times 10^{-4} \: \mathrm{rad}$.
Change in angle between $\text{PR}$ and $\text{QS}$ is $2 \times 10^{-4} \: \mathrm{rad}$.
Percentage change in length of the diagonal $\text{QS}$ is $0.03$.

GATE XE 2025 | Question: 81

Sun, 04 May 2025 19:06:19 +0000

The figure shows the stress distribution across an internal surface of a rectangular beam of height $30 \: \mathrm{mm}$ and depth $10 \: \mathrm{mm}$. The normal stress distribution is given by the expression $\sigma_{x x}=200 y+500 \mathrm{~N} / \mathrm{mm}^{2} ; y$ is the distance in $\mathrm{mm}$ from the centroidal axis of the beam. Assume that there is no variation in the stress distribution along the $z$ -direction.

Which of the following statements is/are $\textsf{CORRECT}$?

The net force in the $x$ direction is $150 \: \mathrm{kN}$.
The net force in the $x$ direction is $75 \: \mathrm{kN}$.
The net moment about the $z$ axis is $4500\: \mathrm{Nm}$.
The net moment about the $z$ axis is $2250 \: \mathrm{Nm}$.

GATE XE 2025 | Question: 82

Sun, 04 May 2025 19:06:16 +0000

A vertical column fixed at one end is subjected to a compressive axial load at the free end. The column's section modulus, $\text{EI}$, is $9.82 \times 10^{5} \: \mathrm{Nm}^{2}$ and the cross-section area is $7.85 \times 10^{-3} \mathrm{~m}^{2}$. The length of the column is $2 \: \mathrm{m}$. The yield stress of the material is $145 \: \mathrm{MPa}$.

If the column can fail either in buckling or by Tresca's criterion, the maximum load that the structure can safely sustain is $\_\_\_\_\_\_ \: \mathrm{kN}$ (rounded off to one decimal place).

GATE XE 2025 | Question: 83

Sun, 04 May 2025 19:06:15 +0000

A simply-supported beam, with a point load $\text{P}=150 \: \mathrm{kN}$ at a distance of $\text{L} / 3$ from the left end, is shown in the figure. The elastic-strain energy $\text{(U)}$ of the beam is given by the following expression:

$$\mathrm{U}=\frac{2}{243} \frac{\mathrm{P}^{2} \mathrm{~L}^{3}}{\mathrm{EI}}$$

where the section modulus, $\mathrm{EI}$, is $16.66 \times 10^{5} \: \mathrm{Nm}^{2}$ and the length of the beam $\mathrm{L}$ is $1 \: \mathrm{m}$.

The deflection at the loading point is $\_\_\_\_\_\_\_$ mm (rounded off to two decimal places).

GATE XE 2025 | Question: 84

Sun, 04 May 2025 19:06:13 +0000

A simply-supported beam has a circular cross-section with a diameter of $20 \: \mathrm{mm}$ , area of $314.2 \mathrm{~mm}^{2}$, area moment of inertia of $7854 \mathrm{~mm}^{4}$ and a length $\text{L}$ of $4 \: \mathrm{m}$ . A point load $\mathrm{P}=100 \mathrm{~N}$ acts at the center and an axial load $\mathrm{Q}=20 \mathrm{kN}$ acts through the centroidal axis as shown in the figure.

The magnitude of the offset between the neutral axis and the centroidal axis, at $\mathrm{L} / 2$ from the left, is $\_\_\_\_\_\_$ mm (rounded off to one decimal place).

GATE XE 2025 | Question: 87

Sun, 04 May 2025 19:06:09 +0000

A bar of length $\text{L}=1 \mathrm{~m}$ is fixed at one end. Before heating its free end has a gap of $\delta=0.1 \mathrm{~mm}$ from a rigid wall as shown in the figure. Now the bar is heated resulting in a uniform temperature rise of $10^{\circ} \mathrm{C}$. The coefficient of linear thermal expansion of the material is $20 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ and the Young's modulus of elasticity is $100 \: \mathrm{GPa}$. Assume that the material properties do not change with temperature.

The magnitude of the resulting axial stress on the bar is $\_\_\_\_\_\_ \: \mathrm{ MPa}$ (in integer).

GATE XE 2025 | Question: 122

Sun, 04 May 2025 19:04:54 +0000

During material testing, stress is applied from time $t_{i}$ to $t_{f}$ as shown below:

The corresponding strain responses for three different materials are shown in plots $\mathbf{P}, \mathbf{Q}$ and $\mathbf{R}$.

Choose the option(s) where the strain response is correctly mapped to its material class.

$\text{P}$- purely elastic; $\text{Q}$- purely viscous; $\text{R}$- viscoelastic
$\text{P}$- purely elastic; $\text{Q}$- viscoelastic; $\text{R}$- purely viscous
$\text{P}$- purely viscous; $\text{Q}$- purely elastic; $\text{R}$- viscoelastic
$\text{P}$- purely viscous; $\text{Q}$- viscoelastic; $\text{R}$- purely elastic

GATE XE 2024 | Question: 35

Sun, 21 Jul 2024 16:42:18 +0000

Mechanical behaviour of a crystalline ceramic material is best described as

ductile
brittle
viscoelastic
Viscoplastic

GATE XE 2024 | Question: 56

Sun, 21 Jul 2024 16:42:01 +0000

The engineering stress $(\sigma)$ vs. engineering strain $(\varepsilon)$ curve obtained by conducting uniaxial tension test on a steel specimen is shown in the figure $\text{(the sketched curve is not to the scale)}$. The specimen exhibits cup-and-cone failure within its gage length. Which point on the curve corresponds to the beginning of necking in the test specimen?

GATE XE 2024 | Question: 59

Sun, 21 Jul 2024 16:41:58 +0000

A mass $M$ is hung from a frictionless, massless pulley. The pulley is suspended by using an inextensible, massless rope of which one end is directly fixed to a support, and the other end is connected to the support through a linear spring of stiffness constant $k$ $\text{(see figure)}$. The natural frequency of this system is

$\sqrt{\frac{4 k}{M}}$
$\sqrt{\frac{2 k}{M}}$
$\sqrt{\frac{k}{M}}$
$\sqrt{\frac{k}{2M}}$

GATE XE 2024 | Question: 60

Sun, 21 Jul 2024 16:41:57 +0000

A simply-supported beam of rectangular cross-section $\text{(width $w$ and height $h$)}$ is subjected to the loads as shown in the figure $\text{(i)}$. The enlarged view of the beam cross-section is shown in figure $\text{(ii)}$. The coordinate system is indicated in the figures. Assuming Euler-Bernoulli beam approximation, the shear stress $\tau_{x z}$ and normal stress $\sigma_{x x}$ at the origin, $\text{O}$ are respectively given by

$\frac{3 P}{2 w h}, \frac{3 P L}{2 w h^{2}}$
$0, \frac{3 P L}{2 w h^{2}}$
$\frac{3 P}{2 w h}, 0$
$0,0$

GATE XE 2024 | Question: 62

Sun, 21 Jul 2024 16:41:55 +0000

A solid circular shaft of diameter $100$ mm is subjected to a torque $3 \pi \mathrm{kNm}$. Which of the following statements about the state of stress in the shaft is/are correct?

The maximum shear stress is $48$ MPa
The maximum tensile stress is $48$ MPa
The magnitude of maximum compressive stress is $48$ MPa
The magnitude of shear stress is $48$ MPa at all points in the shaft

GATE XE 2024 | Question: 63

Sun, 21 Jul 2024 16:41:55 +0000

A spring is connected to an elastic bar as shown in the figure. The spring has a stiffness constant of $10^{7} \mathrm{~N} / \mathrm{m}$. The bar is 70 mm long, and has an area of crosssection $10 \mathrm{~mm}^{2}$. The Young's modulus of the bar material is $70,000 \mathrm{MPa}$. A force $F=5000 \mathrm{~N}$ is applied at point $O$ along the axis of the bar and the spring. The resulting deflection of Point $O$ in mm $\text{(rounded to one decimal place)}$ is ___________.

GATE XE 2024 | Question: 69

Sun, 21 Jul 2024 16:41:48 +0000

A critical point on a component is subjected to the state of stress $[\sigma]$ as given in the following. The yield strength of the material is $400$ MPa . By considering maximum shear stress $\text{(Tresca)}$ theory, the possible value(s) of $\sigma_{0}$ at the onset of yielding is/are,
\[
[\sigma]=\left[\begin{array}{ccc}
280 & 0 & 0 \\
0 & \sigma_{o} & 0 \\
0 & 0 & -60
\end{array}\right] \mathrm{MPa}
\]

$340$
$680$
$-120$
$-460$

GATE XE 2024 | Question: 70

Sun, 21 Jul 2024 16:41:48 +0000

A plane passing through a point $\text{Q}$ inside a body is shown. The unit normal of the plane is $\hat{n}=0.6 \hat{\imath}+0.8 \hat{\jmath}$, as shown in the figure. The traction (stress) vector on the plane at point $\text{Q}$ is given by $\vec{t}=(50 \hat{\imath}+20 \hat{\jmath}) \mathrm{MPa}$. Given that at point Q , $\sigma_{x x}=\sigma_{y y}$, the shear stress component $\tau_{x y}$ $\text{(in MPa, rounded off to two decimal places)}$ is _________.

GATE XE 2024 | Question: 72

Sun, 21 Jul 2024 16:41:46 +0000

A beam of rectangular cross-section, as shown in the figure, is made of two perfectly bonded layers of different materials and equal thickness. The Young's moduli of the two materials are $E_{1}$ and $E_{2}$, where $E_{1}=2 E_{2}$. The beam is subjected to pure bending. If $t=1 \mathrm{~mm}$, the distance of the neutral plane from the top surface of the beam is __________ $\text{(in mm, rounded off to two decimal places)}$.

GATE XE 2024 | Question: 73

Sun, 21 Jul 2024 16:41:45 +0000

A stepped beam is made of a material whose Young's modulus is $E$. The dimensions of the two stepped sections are such that the sectional moments of inertia, $I_{1}$ and $I_{2}$, are related as $I_{1}=8 I_{2}$. The beam is fixed at one end and a load of $F$ is applied at the free end as shown in the figure. Under this loading condition, if the strain energy of the stepped beam is written as $U=\beta \frac{F^{2} L^{3}}{E I_{1}}$, then the value of $\beta$ is ___________ $\text{(rounded off to two decimal places)}$.

GATE XE 2024 | Question: 74

Sun, 21 Jul 2024 16:41:44 +0000

A cylindrical pressure vessel is constructed by bolting two symmetric halves of flanged semi-cylindrical shells. A cross-sectional view of the vessel is shown in the figure. The inner diameter of the vessel is $2$ m and the length is $10$ m. Each row comprises $100$ bolts along the length of the vessel. If the vessel is pressurized to a net pressure of $6 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}$, and assuming the end caps of the vessel do not take any load in the radial direction, then the load borne by each bolt is _________ $\text{(in kN, rounded off to onc decimal place)}$.

GATE XE 2023 | Question: 40

Wed, 14 Feb 2024 18:09:44 +0000

\begin{tabular}{c|c}
Q. 50 & The tensile true stress $(\sigma)-$ true strain $(\epsilon)$ curve follows the Hollomon equation: \\
$\qquad \sigma=500 \epsilon^{0.15} \mathrm{MPa}$ \\
At the maximum load, the work-hardening rate $\left(\frac{d \sigma}{d \epsilon}\right)$ is (in $\left.\mathrm{MPa}\right):$ \\
\hline
\end{tabular}

GATE XE 2023 | Question: 58

Wed, 14 Feb 2024 18:09:27 +0000

\begin{tabular}{|c|c|}
\hline Q. 68 & \begin{tabular}{l}
Two equivalent descriptions of the state of stress at a point are shown in the \\
figure. The normal stresses $\sigma_{1}$ and $\sigma_{2}$ as shown on the right must be, respectively,
\end{tabular} \\
\hline & \\
\hline

& $\tau_{\mathrm{o}}$ and $-\tau_{\mathrm{o}}$ \\
\hline
& $-\tau_{\mathrm{o}}$ and $\tau_{\mathrm{o}}$ \\
\hline
& $\frac{\tau_{0}}{\sqrt{2}}$ and $-\frac{\tau_{0}}{\sqrt{2}}$ \\
\hline
& $-\frac{\tau_{0}}{\sqrt{2}}$ and $\frac{\tau_{0}}{\sqrt{2}}$ \\
\hline
\end{tabular}

GATE XE 2023 | Question: 59

Wed, 14 Feb 2024 18:09:26 +0000

\begin{tabular}{l|l}
Q.69 & The state of strain at a point in a machine component is given as \\
$\varepsilon_{x x}=2.5 \times 10^{-4}, \varepsilon_{y y}=2.0 \times 10^{-4}, \varepsilon_{z z}=-1.5 \times 10^{-4}, \varepsilon_{x y}=2.5 \times 10^{-4}$, \\
$\varepsilon_{y z}=-0.5 \times 10^{-4}, \varepsilon_{z x}=-1.0 \times 10^{-4}$. The volumetric strain at this point is \\

& $4 \times 10^{-4}$ \\
\hline
& $3 \times 10^{-4}$ \\
& $-5 \times 10^{-4}$ \\
& $-3 \times 10^{-4}$
\end{tabular}

GATE XE 2023 | Question: 60

Wed, 14 Feb 2024 18:09:25 +0000

\begin{tabular}{|l|l|}
\hline Q.70 & A thin walled, closed cylindrical vessel of inside diameter $d$ and wall thickness $t$ \\
contains a fluid under pressure $p$. The figure below shows a part of the cylindrical \\
vessel; end caps are not shown. Consider the small element shown with sides \\
parallel and perpendicular to the axis of the cylinder. The stresses $\sigma_{1}$ and $\sigma_{2}$ are \\
\hline

& $\sigma_{1}=\frac{p d}{2 t} ; \sigma_{2}=\frac{p d}{4 t}$ \\
& $\sigma_{1}=\frac{p d}{t} ; \sigma_{2}=\frac{p d}{2 t}$ \\
& $\sigma_{1}=\frac{p d}{4 t} ; \sigma_{2}=\frac{p d}{2 t}$ \\
& $\sigma_{1}=\frac{p d}{2 t} ; \sigma_{2}=0$ \\
\hline
\end{tabular}

GATE XE 2023 | Question: 64

Wed, 14 Feb 2024 18:09:21 +0000

\[
\text { Q. } 74
\]

For a plane stress problem, the principal stresses are $100 \mathrm{MPa}$ and $50 \mathrm{MPa}$. The magnitude of maximum shear stress (in $\mathrm{MPa}$ ) in the material is (rounded off to one decimal place).

GATE XE 2023 | Question: 66

Wed, 14 Feb 2024 18:09:19 +0000

\begin{tabular}{|l|l|}
\hline Q.76 & A cylinder made of rubber (length $=L$ and diameter $=d)$ is inserted in a rigid \\
container as shown in the figure. The rubber cylinder fits snugly in the rigid \\
container. There is no wall friction. The modulus of elasticity of the rubber is $E$ \\
and its Poisson's ratio is $v$. The cylinder is subjected to a small uniform pressure $p$ \\
as shown in the figure. The resulting axial strain $\left(\varepsilon_{z z}\right)$ is \\
\hline

\\
& $-\frac{p}{E}$ \\
\hline
\\
\\
$-\frac{p}{E}\left[\frac{(1+v)(1-2 v)}{(1-v)}\right]$ \\
\hline
\end{tabular}

GATE XE 2023 | Question: 67

Wed, 14 Feb 2024 18:09:17 +0000

\begin{tabular}{|c|}
\hline \begin{tabular}{l}
The state of stress at the critical location in a structure is $\sigma_{x x}=420 \mathrm{MPa}$, \\
$\sigma_{y y}=100 \mathrm{MPa}, \sigma_{z z}=\sigma_{x y}=\sigma_{y z}=\sigma_{z x}=0$. The yield stress of the material in \\
uniaxial tension is $400 \mathrm{MPa}$. Select the correct statement among the following.
\end{tabular} \\
\hline \begin{tabular}{l}
The structure is safe by both Tresca (maximum shear stress) theory and von-Mises \\
(distortion energy) theory.
\end{tabular} \\
\hline \begin{tabular}{l}
The structure is safe by Tresca (maximum shear stress) theory and unsafe by \\
von-Mises (distortion energy) theory.
\end{tabular} \\
\hline \begin{tabular}{l}
The structure is unsafe by Tresca (maximum shear stress) theory and safe by \\
von-Mises (distortion energy) theory.
\end{tabular} \\
\hline \begin{tabular}{l}
The structure is unsafe by both Tresca (maximum shear stress) theory and \\
von-Mises (distortion energy) theory.
\end{tabular} \\
\hline
\end{tabular}

GATE XE 2023 | Question: 68

Wed, 14 Feb 2024 18:09:16 +0000

Q. 78
The figure shows a column of rectangular cross section $100 \mathrm{~mm} \times 80 \mathrm{~mm}$. It
carries a load of $60 \mathrm{kN}$ at a point $30 \mathrm{~mm}$ from the edge $P Q$. The values of stress
component $\sigma_{z z}$ on surfaces $P Q Q^{\prime} P^{\prime}$ and $S R R^{\prime} S^{\prime}$, at points far away from both
ends of the column, are respectively
(C)
(D)

GATE XE 2023 | Question: 69

Wed, 14 Feb 2024 18:09:15 +0000

\begin{tabular}{|l|l}
Q.79 & Consider an electric pole with dimensions as shown in the figure. Let the end $R$ be \\
subjected to a vertical force $F$. The flexural rigidity of both vertical and horizontal \\
bars is $E I$. Neglect the axial deflection of the vertical bar, and all effects of \\
self-weight. The vertical deflection at end $R$ is \\
(A) & $\frac{7 F L^{3}}{3 E I}$ \\
(B) & $\frac{10 F L^{3}}{3 E I}$ \\
(C) & $\frac{5 F L^{3}}{3 E I}$ \\
\hline
\end{tabular}

GATE XE 2023 | Question: 70

Wed, 14 Feb 2024 18:09:14 +0000

Q.80 A uniform cantilever beam has flexural rigidity $E I$ and length $L$. It is subjected to
a concentrated force $F$ and moment $M=2 F L$ at the free end as shown. The
deflection $(\delta)$ at the free end is

$\frac{11 F L^{3}}{12 E I}$
$\frac{8 F L^{3}}{9 E I}$
$\frac{4 F L^{3}}{3 E I}$
$\frac{7 F L^{3}}{6 E I}$

GATE XE 2023 | Question: 74

Wed, 14 Feb 2024 18:09:11 +0000

Q. 84
A thin steel plate is loaded in the $x$-y plane as shown in the figure. Take the
Poisson's ratio of steel to be 0.3 and the modulus of elasticity of steel to be 200
$\mathrm{GPa}$. The strain along the $z$-direction is $\varepsilon_{\mathrm{zz}}=-3 \times 10^{-4}$. The value of $\sigma_{y y}$ (in
$\mathrm{MPa}$ ) is___unded off to one decimal place).

GATE XE 2023 | Question: 76

Wed, 14 Feb 2024 18:09:09 +0000

Q. 86 A slender uniform elastic rod of length $1 \mathrm{~m}$ and of solid circular cross-section of diameter $50 \mathrm{~mm}$ is originally straight. It is then loaded by equal and opposite end moments as indicated in the figure. The resulting lateral displacement of the mid-point of the rod is $10 \mathrm{~mm}$ (displacements are exaggerated in the figure). The maximum longitudinal strain in the rod is $p \times 10^{-3}$, where $p$ is (rounded off to one decimal place).