List of top Solid Mechanics Questions on Stresses and strains asked in GATE XE

An infinitesimal square element \(PQRS\) is shown in the figure. The x and y axes are also marked in the figure. The strains on the element are given by: \[ \varepsilon_{xx} = 500 \times 10^{-6}, \varepsilon_{yy} = 100 \times 10^{-6}, \varepsilon_{xy} = 0 \]

Which of the following statements is/are CORRECT?

The two-dimensional state of stress in an infinitesimal element is given by: \[ \sigma_{xx} = 800 \, \text{MPa}, \sigma_{yy} = 0 \, \text{MPa}, \sigma_{xy} = 300 \, \text{MPa} \] Which one of the following options is the maximum shear stress (in MPa) in the element?

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

The state of strain at a point in a machine component is given as \[ \varepsilon_{xx} = 2.5 \times 10^{-4}, \varepsilon_{yy} = 2.0 \times 10^{-4}, \varepsilon_{zz} = -1.5 \times 10^{-4}, \] \[ \varepsilon_{xy} = 2.5 \times 10^{-4}, \varepsilon_{yz} = -0.5 \times 10^{-4}, \varepsilon_{zx} = -1.0 \times 10^{-4}. \] The volumetric strain at this point is .............

A stepped rod of length 2 m is fixed at both ends A and C. The area of cross-section of AB is 200 mm$^{2}$ and that of BC is 100 mm$^{2}$. A force $F$ is applied at section B causing a displacement of 0.1 mm in the direction of the force. Young’s modulus of the rod is 200 GPa. The applied force $F$ in N is _________ (round off to the nearest integer).

A cantilever beam (span 1 m) carries a UDL of $q=1250$ N/m over 0.4 m. A force $F=1000$ N acts at a distance $L$ from the fixed end. The distance $L$ is such that the bending moment at the fixed end is zero. The beam has a rectangular cross-section of depth 20 mm and width 24 mm. Find the maximum bending stress in MPa (round off to nearest integer).

The beam in the figure is subjected to a moment \( M_0 \) at mid span as shown. Which of the following is the vertical reaction at B?

A circular shaft is rigidly connected to a wall at one end. The shaft has a solid portion and a hollow portion as shown in the figure. The length of each portion is $L$ and the shear modulus of the material is $G$. The polar moment of inertia of the hollow portion is $J$ and that of the solid portion is $50J$. A torque $T$ is applied at the rightmost end as shown. The rotation of the section $PQ$ is

A rectangular plate of uniform thickness having initial length \( a \) and width \( b \) is placed between two rigid immovable walls. The temperature of the plate is increased by \( \Delta T \). The plate is free to expand along the \( y \) and \( z \) directions. The mid-surface of the plate remains in the \( xy \)-plane. The Poisson’s ratio is \( \nu \) and the coefficient of thermal expansion is \( \alpha \). Assuming that the plate is initially free of stresses, the change in length of the plate after the increase in temperature is given by:

The frame shown is subjected to a uniformly distributed load of 1000 N/m over 0.6 m. Neglecting the frame weight, the maximum shear force (in N) in the region between supports A and B is _________.

A massless semicircular rod held fixed at end A is in the xy-plane as shown. A force P along the negative z direction is acting at point B on the rod. The unit vectors along x, y and z directions are denoted by i, j and k. Due to the applied force P, the cross-section of the rod at point D will be subjected to

In the truss shown in the figure, all the members are pin jointed to each other. The members AB, BD, DE and DC have the same length. For the given loading, which of the following is the correct statement?

End B of the 2 m long rigid rod $AB$ is constrained to move horizontally in the slot as shown in the figure and has a velocity of $1.0\,\mathbf{i}$ m/s. The angular velocity of the rod at the instant shown is $2$ rad/s. The unit vectors along $x$ and $y$ directions are denoted by $\mathbf{i}$ and $\mathbf{j}$. The velocity of point A in m/s is then given by:

The assembly of four masses connected by rigid mass-less rods is kept on a smooth horizontal floor as shown in the figure. Under the applied force $2F$, the magnitude of angular acceleration of the assembly at the instant shown is:

A particle is constrained to move at a constant speed on an inclined plane (ABCD) along the curved path shown in the figure. Edges AD and BC are parallel to the y-axis. The inclined plane makes an angle $\theta$ with the $xy$-plane. The velocity vector of the particle makes an angle $\phi$ with the dotted line which is parallel to edge AB. If the speed of the particle is 2 m/s, $\phi = 30^\circ$, and $\theta = 40^\circ$, then the z-component of the velocity of the particle in m/s is

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