List of top Solid Mechanics Questions asked in GATE XE

A uniform elastic rod of constant cross-section is fixed at its left end as shown. An axial force $P$ acts as shown. Assume plane sections remain plane. The ratio of axial displacement at point $A$ $(x=4L)$ to that at point $B$ $(x=L)$ is _________ (rounded off to one decimal place).
A thin-walled spherical pressure vessel has radius 500 mm and thickness 10 mm. Yield strength is 500 MPa. The internal pressure at which yielding occurs according to Tresca criterion is _________ MPa (rounded off to one decimal place).
A force \( F \) is applied at an angle \( \theta = 30^\circ \) on an elastic column as shown in the figure. \(E\) and \(I\) are respectively the Young's modulus and the area moment of inertia. The smallest magnitude of \(F\) needed to cause buckling is:
The shear stress due to a transverse shear force in a linear elastic isotropic beam of rectangular cross-section

The truss shown is subjected to a force \( P \). All members of the truss have the same length \( L \). The reaction at A and force in member AB are 

For a state of plane strain, the normal strains are given by \(\epsilon_{xx} = 1000 \times 10^{-6}, \epsilon_{yy} = 200 \times 10^{-6}, \) and the maximum shear strain is \( \gamma_{\text{max}} = 1000 \times 10^{-6}. \)The value of shear strain\( \, \gamma_{xy} \, \)for this strain state is

A force \( F = 40 \, \text{kN} \) is applied on the hook as shown. The equivalent force-couple system at B is

The figure shows a structure with supports. The correct free body diagram when the supports are removed is 
 

A coronavirus droplet of mass 1 microgram ejects from the mouth of a patient with a velocity of 0.7 m/s and travels through air. The gravitational force experienced by it can be neglected due to the buoyancy effect. However, the droplet experiences air drag force proportional to its velocity and the drag coefficient is given as 1.0 μN-s/m. The distance travelled by the droplet before its velocity drops to 10% of its initial velocity (in m, rounded off to two decimal places) is \(\underline{\hspace{2cm}}\).
A rigid circular disc of radius 0.2 m and mass 10 kg rolls without slip on the ground at A. The coefficient of static friction \( \mu \) between ground and disc is 0.7. A torque \( T \) of 9 Nm acts on the disc as shown. Given acceleration due to gravity \( g = 10 \, \text{m/s}^2 \). The friction force (in N) acting on the disc (in integer) is \(\underline{\hspace{2cm}}\). \includegraphics[width=0.5\linewidth]{image16.png}
A prismatic solid circular rod of diameter \( d \) is bent to introduce an offset \( s = d \) as shown. The rod is further subjected to an axial load \( P \). If the maximum longitudinal stress at a section A-B in the rod (with offset) is \( n \) times the longitudinal stress in the straight rod, the value of \( n \) (in integer) would be \(\underline{\hspace{2cm}}\). \includegraphics[width=0.5\linewidth]{image17.png}
A tube of inner radius 4 cm and outer radius 5 cm can carry a maximum torque of T. This tube is now replaced by a solid circular shaft of the same material. The minimum radius of the solid circular shaft (in cm, rounded off to two decimal places) to carry the same amount of torque \( T \) is \(\underline{\hspace{2cm}}\).
A hammer of mass 1 kg is used to break an almond shell. The velocity time graph of the hammer during the impact duration is shown in the figure. The shape of force time graph is also given, which can be approximated as a triangle. A force of 300 N is required for breaking the shell, while a force of 200 N will not be able to break it, but just introduce a crack. Which one of the following events will happen?
\includegraphics[width=0.65\linewidth]{image15.png}
A naturally curved steel beam AB having Young's modulus 208 GPa, area moment of inertia \( I = 26.7 \, \text{cm}^4 \) and radius \( R = 2 \, \text{m} \) is subjected to a vertical load \( P = 1000 \, \text{N} \) at B. The end A at \( \theta = 90^\circ \) is rigidly fixed. The bending strain energy of the beam (in Nm, rounded off to two decimal places) is \(\underline{\hspace{2cm}}\).
\includegraphics[width=0.5\linewidth]{image18.png}
In System A, a rectangular block of mass M is centrally supported on a spring of stiffness K as shown. In System B, the mass is hinged at one of its ends and is supported centrally by the spring. The ratio of natural frequency of System B to that of System A (rounded off to two decimal places) is \(\underline{\hspace{2cm}}\). \includegraphics[width=0.5\linewidth]{image21.png}
At room temperature of 25°C, a gap of 1 mm exists between the ends of the rods 1 and 2 as shown. Given the cross section area \( A \) of the rods is \( 1500 \, \text{mm}^2 \), Young's modulus \( E = 75 \, \text{GPa} \), and the coefficient of thermal expansion \( \alpha = 23 \times 10^{-6} \, \text{°C}^{-1} \). When the temperature has reached 150°C, the magnitude of normal stress in each of the rods (in MPa, rounded off to two decimal places) is \(\underline{\hspace{2cm}}\).
\includegraphics[width=0.5\linewidth]{image19.png}
For the state of plane stress shown, the components of normal and shear stresses are given in terms of stress \( \sigma \) and unknown constants \( m \) and \( n \). If the normal and shear components of stress on a 45° plane are \( 2\sigma \) and zero, the values of \( m \) and \( n \) would be: \includegraphics[width=0.5\linewidth]{imagee10.png}
A thin cylinder (closed at its ends) of radius \( r \) and thickness \( t \) (\( r \gg t \)) is subjected to internal pressure \( p \). The maximum shear stress in the wall of the cylinder is
During an earthquake, a structure vibrates and the vibration can be assumed to be in simple harmonic motion at 5 Hz. At a measurement point, the RMS value of acceleration is 10 m/s\(^2\). The approximate amplitude of motion (in mm) at this point (rounded off to two decimal places) is \(\underline{\hspace{2cm}}\).
A beam having rectangular cross section is subjected to transverse loading. The ratio of maximum shear stress developed in the beam to the average shear stress is:
A simply supported beam of length L is loaded by two symmetrically applied point loads P at \( L/3 \) from each support. Both the loads are then shifted to new points which are at a distance \( L/4 \) from each support. The bending moments at the mid-section of the beam in both the cases are same. The magnitude of \( P_1 \) in terms of P is: \includegraphics[width=0.5\linewidth]{image7.png}
If the yield point shear stress obtained from the torsion test of a cylindrical specimen is \( \tau_y \), then what is the maximum value of principal strain at yielding? (\( \mu \) is Poisson's ratio and \( E \) is Young's modulus)
If the ratio of Young's modulus to bulk modulus of a material is \( \frac{3}{2} \), then the ratio of shear modulus to the Young's modulus of the material is
A two force member in equilibrium is one in which
A rigid rod OA rotates clockwise at an angular velocity of 10 rad/s. A bead B (OB = 1 m) translates outward on the rod at a speed of 5 m/s and acceleration 2.5 m/s² (both quantities with respect to the rod). The Coriolis component of acceleration is
\includegraphics[width=0.5\linewidth]{image2.png}