Introduction to Solid Mechanics

In case of a circular section the section modulus is given as:

The temperature strain in a bar is ______ proportional to the change in temperature.

The ratio of lateral strain to linear strain is known as

A beam of length 6 m carries a point load 120 kN at its centre. The beam is fixed at both ends. The fixing moment at the ends is

If the principal stresses on a plane stress problem are $ \sigma_1 = 100 $ MPa and $ \sigma_2 = 40 $ MPa, then the magnitude of shear stress (in MPa) will be

A cantilever beam AB of length $ l $ has moment M applied at free end. The deflection at the free end B is given as

Which of the following are usually considered as thin cylinders?

Two shafts in torsion will have equal strength if

Oil tanks, steam boilers, gas pipes are the examples of

Two closed coil helical springs of stiffness's $ K_1 $ and $ K_2 $ are connected in parallel. The combination is equivalent to a single spring of stiffness.

Draw stress-strain curve for brittle materials, and show its yield point is determined.

A steel rod of 30 mm diameter and 400 mm length was tested in a testing machine. At a load of 135 kN, the extension in a gage length of 50 mm was measured to be 0.045 mm and the reduction in diameter was 0.008 mm. Determine Poisson's ratio and values of three elastic modulii for the test material.

A beam of an I-section show in figure is simply supported over a span of 4 metres. Determine the load that the beam can carry per meter length, if the allowable stress in the beam is 30.82 N/mm$^2$.

What is Mohr's stress circle? What is the importance of this circle?

Define the following terms: (i) Pure bending (ii) Neutral axis (iii) Section modulus (iv) Moment of resistance

What do you understand by Strain Energy? Define and derive Castigliano's theorem.

A simply supported beam is loaded and supported as shown in the figure given below: . Draw the shear force and bending moment diagrams, and determine the magnitude and location of maximum bending moment.

Derive the bending equation: $ \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} $. What do these symbols mean? State clearly the assumptions.

What do you understand by second moment of area and moment of inertia?

With the help of suitable assumptions, deduce torsion equation for a hollow circular shaft.

A hollow circular shaft 20 mm thick transmits 294 kW at 200 r.p.m. Determine the diameters of the shaft if the shear strain due to torsion is not to exceed $ 8.6 \times 10^{-4} $. Assume modulus of rigidity as 80 GN/m$^2$.

Calculate the change in dimensions of a thin cylindrical shell due to an internal pressure. Also calculate the change in length and diameter of the cylindrical shell.

A cylindrical shell 3 m long which is closed at the ends has an internal diameter of 1 m and a wall thickness of 15 mm. Calculate the circumferential and longitudinal stresses induced and also change in dimensions of the shell if it is subjected to an internal pressure of 1.5 MN/m$^2$. Take E = 200 GN/m$^2$ and $ 1/m = 0.3 $.

A plane element in a body is subjected to a normal stress of 15 kN/m$^2$ (tensile) in the x-direction as well as a shearing stress of 5 kN/m$^2$ (clockwise along perpendicular to x-axis). Draw Mohr's circle to determine: (a) Normal and shearing stress intensities on a plane inclined at an angle of 40$^{\circ}$ to the normal stress. (b) Principal stresses and their directions. (c) Maximum shearing stress.

In case of a circular section the section modulus is given as:

The temperature strain in a bar is ______ proportional to the change in temperature.

The ratio of lateral strain to linear strain is known as

A beam of length 6 m carries a point load 120 kN at its centre. The beam is fixed at both ends. The fixing moment at the ends is

If the principal stresses on a plane stress problem are $ \sigma_1 = 100 $ MPa and $ \sigma_2 = 40 $ MPa, then the magnitude of shear stress (in MPa) will be

A cantilever beam AB of length $ l $ has moment M applied at free end. The deflection at the free end B is given as

Which of the following are usually considered as thin cylinders?

Two shafts in torsion will have equal strength if

Oil tanks, steam boilers, gas pipes are the examples of

Two closed coil helical springs of stiffness's $ K_1 $ and $ K_2 $ are connected in parallel. The combination is equivalent to a single spring of stiffness.

Draw stress-strain curve for brittle materials, and show its yield point is determined.

A steel rod of 30 mm diameter and 400 mm length was tested in a testing machine. At a load of 135 kN, the extension in a gage length of 50 mm was measured to be 0.045 mm and the reduction in diameter was 0.008 mm. Determine Poisson's ratio and values of three elastic modulii for the test material.

A beam of an I-section show in figure is simply supported over a span of 4 metres. Determine the load that the beam can carry per meter length, if the allowable stress in the beam is 30.82 N/mm$^2$.

What is Mohr's stress circle? What is the importance of this circle?

Define the following terms: (i) Pure bending (ii) Neutral axis (iii) Section modulus (iv) Moment of resistance

What do you understand by Strain Energy? Define and derive Castigliano's theorem.

A simply supported beam is loaded and supported as shown in the figure given below: . Draw the shear force and bending moment diagrams, and determine the magnitude and location of maximum bending moment.

Derive the bending equation: $ \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} $. What do these symbols mean? State clearly the assumptions.

What do you understand by second moment of area and moment of inertia?

With the help of suitable assumptions, deduce torsion equation for a hollow circular shaft.

A hollow circular shaft 20 mm thick transmits 294 kW at 200 r.p.m. Determine the diameters of the shaft if the shear strain due to torsion is not to exceed $ 8.6 \times 10^{-4} $. Assume modulus of rigidity as 80 GN/m$^2$.

Calculate the change in dimensions of a thin cylindrical shell due to an internal pressure. Also calculate the change in length and diameter of the cylindrical shell.

A cylindrical shell 3 m long which is closed at the ends has an internal diameter of 1 m and a wall thickness of 15 mm. Calculate the circumferential and longitudinal stresses induced and also change in dimensions of the shell if it is subjected to an internal pressure of 1.5 MN/m$^2$. Take E = 200 GN/m$^2$ and $ 1/m = 0.3 $.

A plane element in a body is subjected to a normal stress of 15 kN/m$^2$ (tensile) in the x-direction as well as a shearing stress of 5 kN/m$^2$ (clockwise along perpendicular to x-axis). Draw Mohr's circle to determine: (a) Normal and shearing stress intensities on a plane inclined at an angle of 40$^{\circ}$ to the normal stress. (b) Principal stresses and their directions. (c) Maximum shearing stress.

The maximum stress produced in a bar of tapering section is at

The energy stored in a body when strained within elastic limit is known as

Strain energy is the

A vertical column has two moments of inertia (i.e. $ I_{xx} $ and $ I_{yy} $). The column will tend to buckle in the direction of the

The neutral axis of the cross-section of a beam is that axis at which the bending stress is

Euler's formula holds good

The object of caulking in a riveted joint is to make the joint

In the torsion equation $ \frac{T}{J} = \frac{\tau}{R} = \frac{C\theta}{L} $, the term $ J/R $ is called

Strain resetters are used to

If the depth is kept constant for a beam of uniform strength, then its width will vary in proportional to (where M = Bending moment)

Two cylindrical rods one of steel (Est = 200 GPa) and other of brass (Ebr = 105 GPa) are joined together at point C shown in Fig. 1 and are restrained by rigid supports A and E. Determine (i) reactions at supports A and E, (ii) stresses in both rods and (iii) displacement at point C.

A 250 mm long aluminium tube (E = 70 GPa) of 36 mm outer diameter and 28 mm inner diameter may be closed at both ends by means of single-threaded screw of 1.5 mm pitch on covers. A solid brass rod (Ebr = 105 GPa) of 25 mm diameter is placed inside the tube with one cover screwed on tight while the second cover is not placed. The rod is slightly longer than tube and thus cover must be forced against rod by rotating it one quarter of a turn to close it. Determine stresses and deformations in the tube and the rod shown in Fig. 2.

Fig. 3 shows assembly of a circular aluminium shell (Eal = 70 GPa, $ \alpha_{al} $ = 23.6 x 10$^{-6}$/$^{\circ}$C) with fully bonded steel circular core rod inside (Est = 200 GPa, $ \alpha_{st} $ = 11.7 x 10$^{-6}$/$^{\circ}$C) which is unstressed at a temperature of 20$^{\circ}$C. Determine the stresses in the aluminium shell and the steel core when the temperature is raised to 180$^{\circ}$C. Consider only axial deformations.

The 45$^{\circ}$ strain rosette is mounted on a machine element shown in Fig. 4 and provides the following readings from each gauge: $ \epsilon_a = -650 \times 10^{-6} $, $ \epsilon_b = -300 \times 10^{-6} $, $ \epsilon_c = -480 \times 10^{-6} $, $ \nu = 0.33 $. (i) Determine in-plane principal strains. (ii) Determine maximum in-plane shear strain and associated average normal strain. (iii) Represent the deformed element.

A steel pipe (E = 200 GPa) of length L is held by two fixed supports shown in Fig. 5. When mounted, the temperature of the pipe was 20$^{\circ}$C. In use, however the cold fluid moves through the pipe causing it to cool considerably. The pipe has a uniform temperature of -15$^{\circ}$C in use. The coefficient of linear expansion of the material is 12 x 10$^{-6}$/$^{\circ}$C for the working temperature range. Determine the state of stress and strain at central portion of the pipe as a result of this cooling. Neglect the local end effects near the end supports, body force and drag forces in the pipe.

Collar D is released from rest and slides without friction downward from a distance of h = 300 mm where it strikes a head fixed to the end of compound rod ABC shown in Fig. 6. Rod segment AB is made of aluminium (E1 = 70 GPa) and it has a length of L1 = 800 mm and a diameter of d1 = 12 mm. Rod segment (BC) is made of bronze (E2 = 105 GPa) and it has a length of L2 = 1300 mm and a diameter of d2 = 16 mm. What is the allowable mass for collar D if the maximum normal stress in the aluminium rod segment must be limited to 200 MPa?

A bar ABC revolves in a horizontal plane about a vertical axis at the midpoint C shown in Fig. 7. The bar, which has length 2L and cross-sectional area A, revolves at constant angular speed $ \omega $. Each half of the bar (AC and BC) has weight W1 and supports a weight W2 at its end. Derive the following formula for the elongation of one-half of the bar (that is, the elongation of either AC or BC): $ \frac{L^2 \omega^2}{3 g E A} (W_1 + 3W_2) = \delta_{AC} = \delta_{BC} $, where E is the modulus of elasticity of the material of the bar and g is the gravitational acceleration.

A rectangular plate of dimensions 250 mm x 100 mm is formed by welding two triangular plates shown in Fig. 8. The plate is subjected to a compressive stress of 2.5 MPa along the long dimension and a tensile stress of 12 MPa along the short dimension. Determine the (i) normal stress acting perpendicular to the line of the weld and the shear acting parallel to the weld; (ii) maximum principal stress and the orientation of principal plane; (iii) maximum absolute shear stress and the orientation of plane.

A cylindrical tank with hemispherical heads is constructed of steel sections that are welded circumferentially. The tank diameter is 1.25 m, the wall thickness is 22 mm and the internal pressure is 1750 kPa, shown in Fig. 9. Determine the (i) maximum tensile stress in the heads of the tank; (ii) maximum tensile stress in the cylindrical part of the tank; (iii) tensile stress acting perpendicular to the welded joints; (iv) maximum shear stress in the heads of the tank; (v) maximum shear stress in the cylindrical part of the tank.

Consider the state of stress in a bar subjected to compression in the axial direction. Lateral expansion is restrained to half the amount it would ordinarily be if the lateral faces were load free. Find the effective modulus of elasticity.

Compare typical stress-strain curves of mild steel and aluminium (using appropriate figures). Frame your discussion using the following characteristics: Proportional limit, yield point, strain hardening, ultimate strength and breaking stress.

Using Mohr's circle, determine the principal stress and the planes. Show the same on element separately.

Derive an expression for deformation of tapering bar (circular cross-section).

State the assumptions and derive general torsional equation.

A machine belt is threaded through a tubular sleeve of length 15 cm and the nut is turned up just tight by hand. Using wrenches, the nut is then turned further, the bolt being put in tension and the sleeve in compression. If the bolt has 5 threads per cm, and the nut is given an extra quarter turn 90$^{\circ}$ by the wrenches, estimate the tensile force in both the bolt and sleeve which are of steel and the cross-sectional areas are: Bolt area = 6 cm$^2$, sleeve area = 4 cm$^2$, E = 210 GPa.

Define thick and thin cylinder. Also derive an expression for circumferential stress in a thin cylinder.

The maximum stress produced in a bar of tapering section is at

The energy stored in a body when strained within elastic limit is known as

Strain energy is the

A vertical column has two moments of inertia (i.e. $ I_{xx} $ and $ I_{yy} $). The column will tend to buckle in the direction of the

The neutral axis of the cross-section of a beam is that axis at which the bending stress is

Euler's formula holds good

The object of caulking in a riveted joint is to make the joint

In the torsion equation $ \frac{T}{J} = \frac{\tau}{R} = \frac{C\theta}{L} $, the term $ J/R $ is called

Strain resetters are used to

If the depth is kept constant for a beam of uniform strength, then its width will vary in proportional to (where M = Bending moment)

Two cylindrical rods one of steel (Est = 200 GPa) and other of brass (Ebr = 105 GPa) are joined together at point C shown in Fig. 1 and are restrained by rigid supports A and E. Determine (i) reactions at supports A and E, (ii) stresses in both rods and (iii) displacement at point C.

A 250 mm long aluminium tube (E = 70 GPa) of 36 mm outer diameter and 28 mm inner diameter may be closed at both ends by means of single-threaded screw of 1.5 mm pitch on covers. A solid brass rod (Ebr = 105 GPa) of 25 mm diameter is placed inside the tube with one cover screwed on tight while the second cover is not placed. The rod is slightly longer than tube and thus cover must be forced against rod by rotating it one quarter of a turn to close it. Determine stresses and deformations in the tube and the rod shown in Fig. 2.

Fig. 3 shows assembly of a circular aluminium shell (Eal = 70 GPa, $ \alpha_{al} $ = 23.6 x 10$^{-6}$/$^{\circ}$C) with fully bonded steel circular core rod inside (Est = 200 GPa, $ \alpha_{st} $ = 11.7 x 10$^{-6}$/$^{\circ}$C) which is unstressed at a temperature of 20$^{\circ}$C. Determine the stresses in the aluminium shell and the steel core when the temperature is raised to 180$^{\circ}$C. Consider only axial deformations.

The 45$^{\circ}$ strain rosette is mounted on a machine element shown in Fig. 4 and provides the following readings from each gauge: $ \epsilon_a = -650 \times 10^{-6} $, $ \epsilon_b = -300 \times 10^{-6} $, $ \epsilon_c = -480 \times 10^{-6} $, $ \nu = 0.33 $. (i) Determine in-plane principal strains. (ii) Determine maximum in-plane shear strain and associated average normal strain. (iii) Represent the deformed element.

A steel pipe (E = 200 GPa) of length L is held by two fixed supports shown in Fig. 5. When mounted, the temperature of the pipe was 20$^{\circ}$C. In use, however the cold fluid moves through the pipe causing it to cool considerably. The pipe has a uniform temperature of -15$^{\circ}$C in use. The coefficient of linear expansion of the material is 12 x 10$^{-6}$/$^{\circ}$C for the working temperature range. Determine the state of stress and strain at central portion of the pipe as a result of this cooling. Neglect the local end effects near the end supports, body force and drag forces in the pipe.

Collar D is released from rest and slides without friction downward from a distance of h = 300 mm where it strikes a head fixed to the end of compound rod ABC shown in Fig. 6. Rod segment AB is made of aluminium (E1 = 70 GPa) and it has a length of L1 = 800 mm and a diameter of d1 = 12 mm. Rod segment (BC) is made of bronze (E2 = 105 GPa) and it has a length of L2 = 1300 mm and a diameter of d2 = 16 mm. What is the allowable mass for collar D if the maximum normal stress in the aluminium rod segment must be limited to 200 MPa?

A bar ABC revolves in a horizontal plane about a vertical axis at the midpoint C shown in Fig. 7. The bar, which has length 2L and cross-sectional area A, revolves at constant angular speed $ \omega $. Each half of the bar (AC and BC) has weight W1 and supports a weight W2 at its end. Derive the following formula for the elongation of one-half of the bar (that is, the elongation of either AC or BC): $ \frac{L^2 \omega^2}{3 g E A} (W_1 + 3W_2) = \delta_{AC} = \delta_{BC} $, where E is the modulus of elasticity of the material of the bar and g is the gravitational acceleration.

A rectangular plate of dimensions 250 mm x 100 mm is formed by welding two triangular plates shown in Fig. 8. The plate is subjected to a compressive stress of 2.5 MPa along the long dimension and a tensile stress of 12 MPa along the short dimension. Determine the (i) normal stress acting perpendicular to the line of the weld and the shear acting parallel to the weld; (ii) maximum principal stress and the orientation of principal plane; (iii) maximum absolute shear stress and the orientation of plane.

A cylindrical tank with hemispherical heads is constructed of steel sections that are welded circumferentially. The tank diameter is 1.25 m, the wall thickness is 22 mm and the internal pressure is 1750 kPa, shown in Fig. 9. Determine the (i) maximum tensile stress in the heads of the tank; (ii) maximum tensile stress in the cylindrical part of the tank; (iii) tensile stress acting perpendicular to the welded joints; (iv) maximum shear stress in the heads of the tank; (v) maximum shear stress in the cylindrical part of the tank.

Consider the state of stress in a bar subjected to compression in the axial direction. Lateral expansion is restrained to half the amount it would ordinarily be if the lateral faces were load free. Find the effective modulus of elasticity.

Compare typical stress-strain curves of mild steel and aluminium (using appropriate figures). Frame your discussion using the following characteristics: Proportional limit, yield point, strain hardening, ultimate strength and breaking stress.

Using Mohr's circle, determine the principal stress and the planes. Show the same on element separately.

Derive an expression for deformation of tapering bar (circular cross-section).

State the assumptions and derive general torsional equation.

A machine belt is threaded through a tubular sleeve of length 15 cm and the nut is turned up just tight by hand. Using wrenches, the nut is then turned further, the bolt being put in tension and the sleeve in compression. If the bolt has 5 threads per cm, and the nut is given an extra quarter turn 90$^{\circ}$ by the wrenches, estimate the tensile force in both the bolt and sleeve which are of steel and the cross-sectional areas are: Bolt area = 6 cm$^2$, sleeve area = 4 cm$^2$, E = 210 GPa.

Define thick and thin cylinder. Also derive an expression for circumferential stress in a thin cylinder.