# GATE Questions & Answers of Mechanics of Materials Mechanical Engineering

#### Mechanics of Materials 107 Question(s) | Weightage 09 (Marks)

If $\sigma_1$ and $\sigma_3$ are the algebraically largest and smallest principal stresses respectively, the value of the maximum shear stress is

For an Oldham coupling used between two shafts, which among the following statements are correct?
I.           Torsional load is transferred along shaft axis.
II.          A velocity ratio of 1:2 between shafts is obtained without using gears.
III.         Bending load is transferred transverse to shaft axis.
IV.         Rotation is transferred along shaft axis.

In a linearly hardening plastic material, the true stress beyond initial yielding

A steel column of rectangular section (15 mm × 10 mm) and length 1.5 m is simply supported at both ends. Assuming modulus of elasticity, E = 200 GPa for steel, the critical axial load (in kN) is ____ (correct to two decimal places).

The state of stress at a point, for a body in plane stress, is shown in the figure below. If the minimum principal stress is 10 kPa, then the normal stress $\sigma_y$ (in kPa) is

A carpenter glues a pair of cylindrical wooden logs by bonding their end faces at an angle of $\theta=30^\circ$ as shown in the figure.

The glue used at the interface fails if

Criterion 1: the maximum normal stress exceeds 2.5 MPa.

Criterion 2: the maximum shear stress exceeds 1.5 MPa.

Assume that the interface fails before the logs fail. When a uniform tensile stress of 4 MPa is applied, the interface

A simply supported beam of width 100 mm, height 200 mm and length 4 m is carrying a uniformly distributed load of intensity 10 kN/m. The maximum bending stress (in MPa) in the beam is __________ (correct to one decimal place).

The true stress $\left(\sigma\right)$ - true strain $\left(\varepsilon\right)$ diagram of a strain hardening material is shown in figure. First, there is loading up to point A, i.e., up to stress of 500 MPa and strain of 0.5. Then from point A, there is unloading up to point B, i.e., to stress of 100 MPa. Given that the Young’s modulus E = 200 GPa, the natural strain at point B $\left(\varepsilon_B\right)$ is _________ (correct to three decimal places).

A bar is compressed to half of its original length. The magnitude of true strain produced in the deformed bar is _________________ (correct to two decimal places).

The minimum axial compressive load, $P$, required to initiate buckling for a pinned-pinned slender column with bending stiffness $EI$ and length $L$ is

A hollow circular shaft of inner radius 10 mm, outer radius 20 mm and length 1 m is to be used as a torsional spring. If the shear modulus of the material of the shaft is 150 GPa, the torsional stiffness of the shaft (in kN-m/rad) is ________ (correct to two decimal places).

A rigid rod of length 1 m is resting at an angle $\theta=45^\circ$ as shown in the figure. The end P is dragged with a velocity of $U$ = 5 m/s to the right. At the instant shown, the magnitude of the velocity $V$ (in m/s) of point Q as it moves along the wall without losing contact is

A bar of circular cross section is clamped at ends P and Q as shown in the figure. A torsional moment $T$ = 150 Nm is applied at a distance of 100 mm from end P. The torsional reactions $\left(T_P,T_Q\right)$ in Nm at the ends P and Q respectively are

A bimetallic cylindrical bar of cross sectional area 1 m2 is made by bonding Steel (Young’s modulus = 210 GPa) and Aluminium (Young’s modulus = 70 GPa) as shown in the figure. To maintain tensile axial strain of magnitude $10^{-6}$ in Steel bar and compressive axial strain of magnitude $10^{-6}$ in Aluminum bar, the magnitude of the required force $P$ (in kN) along the indicated direction is

The true stress (in MPa) versus true strain relationship for a metal is given by

$\sigma=1020\varepsilon^{0.4}$ .

The cross-sectional area at the start of a test (when the stress and strain values are equal to zero) is 100 mm2 . The cross-sectional area at the time of necking (in mm2 ) is ________ (correct to two decimal places)

The Poisson's ratio for a perfectly incompressible linear elastic  material is

An initially stress-free massless elastic beam of length $L$ and circular cross-section with diameter $d(d\ll L)$ is held fixed between two walls as shown. The beam material has Young's moduls $E$ and coefficeint of thermal expansion $\style{font-family:'Times New Roman'}{\mathrm\alpha}$.

If the beam is slowly and uniformly heated, the tempreature rise required to cause the beam to buckle is prportional to

The state of stress at a point is MPa. The maximum normal stress (in MPa) at that point is__________

For a loaded cantilever beam of uniform cross-section, the bending moment (in N mm) along the length is M(x)=5x2+10x, where x is the distance (in mm) measured from the free end of the beam. The magnitude of shear force (in N) in the cross-section at x=10 mm is__________

A cantilever beam of length L and flexural modulus EI is subjected to a point load P  at the free end. The elastic strain energy stored in the beam due to bending (neglecting transverse shear) is

A steel bar is held by two fixed supports as shown in the figure and is subjected to an increase of temperature $\triangle T=100\;^\circ\mathrm C$. If the coefficient of thermal expansion and Young’s  modulus of elasticity of steel are and 200 GPa, respectively, the magnitude of thermal stress (in MPa) induced in the bar is__________

The cross sections of two hollow bars made of the same material are concentric circles as shown in the figure. It is given that r3 > r1 and r4 > r2 , and that the areas of the cross-sections are the same. $J_1$ and $J_2$ are the torsional rigidities of the bars on the left and right, respectively. The ratio $J_2/J_1$ is

A cantilever beam having square cross-section of side $\alpha$ is subjected to an end load. If $\alpha$ is increased by 19%, the tip deflection decreases approximately by

A horizontal bar with a constant cross-section is subjected to loading as shown in the figure. The Young's moduli for the sections AB and BC are 3E and E, respectively.

For the deflection at C to be zero, the ratio P/F is ____________

The Figure shows cross-section of a beam subjected to bending. The area moment of inertia $(\mathrm{in}\;\mathrm{mm}^4)$ of this cross-section about its base is _________ .

A simply-supported beam of length 3L is subjected to the loading shown in the figure.

It is given that $P = 1 N, L = 1$ m and Young’s modulus $E=200\;\mathrm{GPa}.$ The cross-section is a square with dimension 10 mm × 10 mm. The bending stress (in Pa) at the point A located at the top surface of the beam at a distance of $1.5L$ from the left end is _____________

(Indicate compressive stress by a negative sign and tensile stress by a positive sign.)

A shaft with a circular cross-section is subjected to pure twisting moment. The ratio of the maximum shear stress to the largest principal stress is

A thin cylindrical pressure vessel with closed-ends is subjected to internal pressure. The ratio of circumferential (hoop) stress to the longitudinal stress is

A machine element XY, fixed at end X, is subjected to an axial load P, transverse load F, and a twisting moment T at its free end Y. The most critical point from the strength point of view is

The value of true strain produced in compressing a cylinder to half its original length is

A rigid horizontal rod of length 2L is fixed to a circular cylinder of radius R as shown in the figure. Vertical forces of magnitude P are applied at the two ends as shown in the figure. The shear modulus for the cylinder is G and the Young’s modulus is E.

The vertical deflection at point A is

A simply supported beam of length 2L is subjected to a moment M at the mid-point $x = 0$ as shown in the figure. The deflection in the domain $0 ≤ x ≤ L$ is given by

$w=\frac{-Mx}{12\;EIL}\left(L-x\right)\left(x+c\right)$ ,

where E is the Young’s modulus, I  is the area moment of inertia and c is a constant (to be determined) .

The slope at the center $x = 0$ is

In the figure, the load P = 1 N, length L = 1 m, Young’s modulus E = 70 GPa, and the cross-section of the links is a square with dimension 10 mm × 10 mm. All joints are pin joints.

The stress (in Pa) in the link AB is ___________

(Indicate compressive stress by a negative sign and tensile stress by a positive sign.)

A circular metallic rod of length 250 mm is placed between two rigid immovable walls as shown in the figure. The rod is in perfect contact with the wall on the left side and there is a gap of 0.2 mm between the rod and the wall on the right side. If the temperature of the rod is increased by $200^o\mathrm C$, the axial stress developed in the rod is __________ MPa.

Young’s modulus of the material of the rod is 200 GPa and the coefficient of thermal expansion is $10^{−5}$ per ${}^\circ\mathrm C$.

The cross-sections of two solid bars made of the same material are shown in the figure. The square cross-section has flexural (bending) rigidity $I_1$, while the circular cross-section has flexural rigidity $I_2$. Both sections have the same cross-sectional area. The ratio $I_1/I_2$ is

The state of stress at a point on an element is shown in figure (a). The same state of stress is shown in another coordinate system in figure (b).

The components $\style{font-family:'Times New Roman'}{\left(\tau_{xx},\tau_{yy},\tau_{xy}\right)}$ are given by

A square plate of dimension L × L is subjected to a uniform pressure load p = 250 MPa on its edges as shown in the figure. Assume plane stress conditions. The Young’s modulus E = 200 GPa.
The deformed shape is a square of dimension $L-\;2\;\delta$. If $L=\;2\;\mathrm m$ and $\delta=0.001\;\mathrm m$, the poisson's ratio of the plate material is _________

Two circular shafts made of same material, one solid (S) and one hollow (H), have the same length and polar moment of inertia. Both are subjected to same torque. Here, ${\theta }_{s}$ is the twist and ${\tau }_{s}$ is the maximum shear stress in the solid shaft, whereas is the twist and ${\tau }_{H}$ is the maximum shear stress in the hollow shaft. Which one of the following is TRUE?

A beam of length L is carrying a uniformly distributed load w per unit length. The flexural rigidity of the beam is EI. The reaction at the simple support at the right end is

Consider a stepped shaft subjected to a twisting moment applied at B as shown in the figure. Assume shear modulus , G = 77 GPa. The angle of twist at C (in degrees) is _____

Consider a steel (Young’s modulus E = 200 GPa) column hinged on both sides. Its height is 1.0 m and cross-section is 10 mm × 20 mm. The lowest Euler critical bucking load (in N) is _______.

Which of the following types of stress strain relationship best describes the behaviour of brittle materials, such as ceramics and thermosetting plastics, ($\sigma$ stress and $\epsilon$ = strain)?

A cantilever beam with flexural rigidity of 200 N.m2 is loaded as shown in the figure . The deflection (in mm) at the tip of the beam is ____.

A rod is subjected to a un-axial load within linear elastic limit. When the change in the stress is 200 MPa, the change in the strain is 0.001. If the Poisson’s ratio of the rod is 0.3, the modulus of rigidity ( in GPa) is______

A gas is stored in a cylindrical tank of inner radius 7 m and wall thickness 50 mm. The gage pressure of the gas is 2 MPa. The maximum shear stress (in MPa) in the wall is

A cantilever beam OP is connected to another beam PQ with a pin joint as shown in figure. A load of 10 kN is applied at the midpoint of PQ. The magnitude of bending movement (in kN-m) at fixed end O is

A cantilever beam with square cross section of 6 mm side is subjected to a load of 2 kN normal to the top surface as shown in figure. The Young’s modulus of elasticity of the material of the beam is 210 GPa. The magnitude of slope ( in radian) at Q (20 mm from the fixed end) is_____.

In a plane stress condition, the components of stress at a point are , and . The maximum shear stress (in MPa) at the point is

A hollow shaft of 1 m length is designed to transmit a power of 30 kW at 700 rpm. The maximum permissible angle of twist in the shaft is 1o. The inner diameter of the shaft is 0.7 times the outer diameter. The modulus of rigidity is 80 GPa. The outside diameter (in mm) of the shaft is _______.

For the given fluctuating fatigue load, the value of stress amplitude and stress ratio are respectively.

A cylindrical tank with closed ends is filled with compressed air at a pressure of 500 kPa. The inner radius of the tank is 2 m, and it has wall thickness of 10 mm. The magnitude of maximum in-plane shear stress (in MPa) is____________

For the overhanging beam shown in figure, the magnitude of maximum bending moment (in kN-m) is _____.

A circular rod of length ‘L’ and area of cross-section ‘A’ has a modulus of elasticity ‘E’ and coefficient of thermal expansion ‘α’. One end of the rod is fixed and other end is free. If the temperature of the rod is increased by ΔT, then

A metallic  rod of 500 mm  length and 50mm diameter when subjected to a tensile force of 100KN at the ends,experinces an increse an its length by 0.5mm and a reduction in its diameter by 0.015mm.The poission's ratio of the road material is__________

The state of sttress at a point is given by ${\sigma }_{x}=-6$ MPa, ${\sigma }_{y}=4$ MPa, and ${\tau }_{xy}=-8$ MPa.The maximum tensiile stress (in MPa) at the point is ________

A 200 mm long, stress free rod at room temperature is held between two immovable rigid walls. The temperature of the rod is uniformly raised by 250°C. If the Young’s modulus and coefficient of thermal expansion are 200 GPa and 1×10−5 /°C, respectively, the magnitude of the longitudinal stress (in MPa) developed in the rod is _______

A metal rod of initial length L0 is subjected to a drawing process. The length of the rod at any instant is given by the expression, L(t)=L0(1+t2), where t is the time in minutes. The true strain rate (in min−1) at the end of one minute is _______

A steel cube, with all faces free to deform, has Young’s modulus, E, Poisson’s ratio, ν, and coefficient of thermal expansion, α. The pressure (hydrostatic stress) developed within the cube, when it is subjected to a uniform increase in temperature, ΔT, is given by

A thin plate of uniform thickness is subject to pressure as shown in the figure below

Under the assumption of plane stress, which one of the following is correct?

The relationship between true strain (${\epsilon }_{T}$) and engineering strain (${\epsilon }_{E}$) in a uniaxial tension test is given as

The flexural rigidity (EI) of a cantilever beam is assumed to be constant over the length of the beam shown in figure. If a load P and bending moment PL/2 are applied at the free end of the beam then the value of the slope at the free end is

A cantilever beam of length, L, with uniform cross-section and flexural rigidity, EI, is loaded uniformly by a vertical load, w per unit length. The maximum vertical deflection of the beam is given by

Consider the two states of stress as shown in configurations I and II in the figure below. From the standpoint of distortion energy (von-Mises) criterion, which one of the following statements is true?

Two solid circular shafts of radii R1 and R2 are subjected to same torque. The maximum shear stresses developed in the two shafts are ${\tau }_{1}$ and ${\tau }_{2}$. If R2/ R2=2, then ${\tau }_{2}$/${\tau }_{1}$ is _______

Consider a simply supported beam of length, 50h, with a rectangular cross-section of depth, h, and width, 2h. The beam carries a vertical point load, P, at its mid-point. The ratio of the maximum shear stress to the maximum bending stress in the beam is

A force P is applied at a distance x from the end of the beam as shown in the figure. What would be the value of x so that the displacement at ‘A’ is equal to zero?

If the Poisson's ratio of an elastic material is 0.4, the ratio of modulus of rigidity to Young's modulus is _______

The number of independent elastic constants required to define the stress-strain relationship for an isotropic elastic solid is _______

A thin gas cylinder with an internal radius of 100 mm is subject to an internal pressure of 10 MPa. The maximum permissible working stress is restricted to 100 MPa. The minimum cylinder wall thickness (in mm) for safe design must be ____

A rod of length L having uniform cross-sectional area A is subjected to a tensile force P as shown in the figure below. If the Young’s modulus of the material varies linearly from E1 to E2 along the length of the rod, the normal stress developed at the section-SS is

Two threaded bolts A and B of same material and length are subjected to identical tensile load. If the elastic strain energy stored in bolt A is 4 times that of bolt B and the mean diameter of bolt A is 12 mm, the mean diameter of bolt B in mm is

A long thin walled cylindrical shell, closed at both the ends, is subjected to an internal pressure. The ratio of the hoop stress (circumferential stress) to longitudinal stress developed in the shell is

A simply supported beam of length L is subjected to a varying distributed load sin (3$\pi$ x/L) Nm-1, where the distance x is measured from the left support. The magnitude of the vertical reaction force in N at the left support is

A thin walled spherical shell is subjected to an internal pressure. If the radius of the shell is increased by 1% and the thickness is reduced by 1%, with the internal pressure remaining the same, the percentage change in the circumferential (hoop) stress is

A cantilever beam of length L is subjected to a moment M at the free end. The moment of inertia of the beam cross section about the neutral axis is I and the Young’s modulus is E. The magnitude of the maximum deflection is

For a long slender column of uniform cross section, the ratio of critical buckling load for the case with both ends clamped to the case with both ends hinged is

The homogeneous state of stress for a metal part undergoing plastic deformation is

$T=\left(\begin{array}{ccc}10& 5& 0\\ 5& 20& 0\\ 0& 0& -10\end{array}\right)$,

where the stress component values are in MPa. Using von Mises yield criterion, the value of estimated shear yield stress, in MPa is

The state of stress at a point under plane stress condition is

and ${\tau }_{xy}=40MPa$

The radius of the Mohr’s circle representing the given state of stress in MPa is

A solid steel cube constrained on all six faces is heated so that the temperature rises uniformly by $\Delta$T. If the thermal coefficient of the material is $\alpha$, Young’s modulus is E and the Poisson’s ratio is $\nu$ , the thermal stress developed in the cube due to heating is

A simply supported beam PQ is loaded by a moment of 1 kN-m at the mid-span of the beam as shown in the figure. The reaction forces RP and RQ at supports P and Q respectively are

A column has a rectangular cross-section of 10mm x 20mm and a length of 1m. The slenderness ratio of the column is close to

A thin cylinder of inner radius 500mm and thickness 10mm is subjected to an internal pressure of 5 MPa. The average circumferential (hoop) stress in MPa is

A torque T is applied at the free end of a stepped rod of circular cross-sections as shown in the figure. The shear modulus of the material of the rod is G. The expression for d to produce an angular twist θ at the free end is