Questions & Answers of Mechanics of Materials

Question No. 19

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

Question No. 42

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 ____________

Question No. 43

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 _________ .

 

Question No. 44

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.)

Question No. 117

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

Question No. 118

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

Question No. 121

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

 

Question No. 133

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

Question No. 142

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

Question No. 143

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

Question No. 144

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.)

Question No. 217

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

 

 

Question No. 218

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

 

Question No. 243

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 _________

Question No. 244

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, θs is the twist and τs is the maximum shear stress in the solid shaft, whereas θis the twist and τH is the maximum shear stress in the hollow shaft. Which one of the following is TRUE?

Question No. 245

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

 

Question No. 25

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 _____

Question No. 27

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

Question No. 30

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

Question No. 54

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 ____.

Question No. 117

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______

Question No. 118

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

Question No. 141

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

Question No. 143

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_____.

Question No. 144

In a plane stress condition, the components of stress at a point are σx=20 MPa, σy=80 MPa and τxy=40 MPa. The maximum shear stress (in MPa) at the point is

Question No. 147

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 _______.

Question No. 211

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

Question No. 227

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____________

Question No. 256

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

Question No. 16

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

Question No. 17

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__________

Question No. 40

The state of sttress at a point is given by σx=-6 MPa, σy=4 MPa, and τxy=-8 MPa.The maximum tensiile stress (in MPa) at the point is ________

Question No. 50

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 _______

Question No. 61

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 _______

Question No. 116

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

Question No. 121

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?

Question No. 134

The relationship between true strain (εT) and engineering strain (εE) in a uniaxial tension test is given as

Question No. 140

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

Question No. 141

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

Question No. 148

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?

Question No. 217

Two solid circular shafts of radii R1 and R2 are subjected to same torque. The maximum shear stresses developed in the two shafts are τ1 and τ2. If R2/ R2=2, then τ2/τ1 is _______

Question No. 242

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

Question No. 245

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?

Question No. 317

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

Question No. 319

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

Question No. 341

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 ____

Question No. 4

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

Question No. 5

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

Question No. 21

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

Question No. 41

A simply supported beam of length L is subjected to a varying distributed load sin (3π 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

Question No. 10

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

Question No. 22

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

Question No. 23

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

Question No. 26

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

T=1050520000-10,

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

Question No. 34

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

σxx=40MPa, σyy=100MPa and τxy=40MPa

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

Question No. 42

A solid steel cube constrained on all six faces is heated so that the temperature rises uniformly by ΔT. If the thermal coefficient of the material is α, Young’s modulus is E and the Poisson’s ratio is ν , the thermal stress developed in the cube due to heating is

Question No. 4

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

Question No. 10

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

Question No. 23

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

Question No. 26

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

Question No. 52

A triangular-shaped cantilever beam of uniform–thickness is shown in the figure. The Young’s modulus of the material of the beam is E. A concentrated load P is applied at the free end of the beam.

The area moment of inertia about the neutral axis of a cross-section at a distance x measured from the free end is

Question No. 53

A triangular-shaped cantilever beam of uniform–thickness is shown in the figure. The Young’s modulus of the material of the beam is E. A concentrated load P is applied at the free end of the beam.

The maximum deflection of the beam is

Question No. 8

The state of plane-stress at a point is given by σx =−200MPa, σy = 100MPa and τ= 100MPa . The maximum shear stress in MPa is

Question No. 52

A massless beam has a loading pattern as shown in the figure. The beam is of rectangular cross-section with a width of 30mm and height of 100mm.

The maximum bending moment occurs at

Question No. 53

A massless beam has a loading pattern as shown in the figure. The beam is of rectangular cross-section with a width of 30mm and height of 100mm.

The maximum magnitude of bending stress (in MPa) is given by

Question No. 11

If the principal stresses in a plane stress problem are σ1 = 100 MPa,σ2 = 40 MPa, the magnitude of the maximum shear stress (in MPa) will be

Question No. 35

A solid shaft of diameter, d and length L is fixed at both the ends. A torque, T0 is applied at a distance, L/4 from the left end as shown in the figure given below.

The maximum shear stress in the shaft is

Question No. 40

A frame of two arms of equal length L is shown in the adjacent figure. The flexural rigidity of each arm of the frame is EI. The vertical deflection at the point of application of load P is

Question No. 7

The transverse shear stress acting in a beam of rectangular cross-section, subjected to a transverse shear load, is

Question No. 8

A rod of Length L and diameter D is subjected to a tensile load P. Which of the following is sufficient to calculate the resulting change in diameter?

Question No. 30

The strain energy stored in the beam with flexural rigidity EI and loaded as shown in the figure is

Question No. 31

For the component loaded with a force F as shown in the figure, the axial stress at the corner point P is

Question No. 32

A solid circular shaft of diameter 100 mm is subjected to an axial stress of 50 Mpa. It is further subjected to a torque of 10 kNm. The maximum principal stress experienced on the shaft is closest to

Question No. 36

The rod PQ of length L with flexural rigidity EI is hinged at both ends. For what minimum force F is it expected to buckle?

Question No. 46

A two dimensional fluid element rotates like a rigid body. At a point within the element, the pressure is 1 unit. Radius of the Mohr’s circle, characterizing the state of stress at the point, is

Question No. 76

A cylindrical container of radius R = 1 m, wall thickness 1 mm is filled with water up to a depth of 2 m and suspended along its upper rim. The density of water is 1000kg/m3 and acceleration due to gravity is 10 m/s2 The self-weight of the cylinder is negligible. The formula for hoop stress in a thin – walled cylinder can be used at all points along the height of the cylindrical container

The axial and circumferential stress (σa, σc)  experienced by the cylinder wall at mid-depth (1 m as shown) are

(A) (10, 10) MPa (B) (5, 10) MPa (C) (10, 5) MPa (D) (5, 5) MPa

Question No. 77

A cylindrical container of radius R = 1 m, wall thickness 1 mm is filled with water up to a depth of 2 m and suspended along its upper rim. The density of water is 1000kg/m3 and acceleration due to gravity is 10 m/s2 The self-weight of the cylinder is negligible. The formula for hoop stress in a thin – walled cylinder can be used at all points along the height of the cylindrical container

If the Young’s modulus and Poisson’s ratio of the container material are 100GPa and 0.3, respectively, the axial strain in the cylinder wall at mid-depth is

Question No. 10

In a simply – supported beam loaded as shown below, the maximum bending moment in Nm is

Question No. 13

A steel rod of length L and diameter D, fixed at both ends, is uniformly heated to a temperature rise of ΔT. The Young’s modulus is E and the co efficient of linear expansion is α. The thermal stress in the rod is

Question No. 42

A uniformly loaded propped cantilever beam and its free body diagram are shown below. The reactions are

Question No. 44

A 200 × 100 × 50 mm steel block is subjected to a hydrostatic pressure of 15 MPa. The Young’s modulus and Poisson’s ratio of the material are 200 GPa and 0.3 respectively. The change in the volume of the block in mm3 is

Question No. 45

A stepped steel shaft shown below is subjected to 10 Nm torque. If the modulus of rigidity is 80 GPa, the strain energy in the shaft in N mm is

Question No. 80

A machine frame shown in the figure below is subjected to a horizontal force of 600 N parallel to z – direction.

The normal and shear stresses in MPa at point P are respectively

Question No. 81

A machine frame shown in the figure below is subjected to a horizontal force of 600 N parallel to z – direction.

The maximum principal stress in MPa and the orientation of the corresponding principal plane in degrees are respectively