Mechanics of Solid-I

The ratio of modulus of rigidity to modulus of elasticity for most of the materials is

Temperature stress is a function of

Normal stress on a plane, the normal to which is inclined at angle $ \theta $ with the line of action of uniaxial stress $ \sigma $ is given by

In a Mohr's circle, the radius gives the value of

Variation of shear force in a cantilever carrying a load the intensity of which varies uniformly from zero at the free end to w per unit run at the fixed end is given by

In a transversely loaded beam, the maximum tensile stress occurs at

Maximum deflection of a simply supported beam with a centre point load is

Maximum deflection of a fixed beam carrying a uniformly distributed load is

When a shaft is subjected to a twisting moment, every cross-section of the shaft will be under

The product of the tangential force acting on the shaft and radius of shaft known as

A steel plate, 22 mm thick and 220 mm wide at one end, tappers uniformly to 12 mm thick and 180 mm wide at the other end. Determine the elongation under a pull of 20 kN when the length of the plate is 2.4 m. Assuming E = 205 GN/m$^2$.

The steel bolt shown in the figure below has a thread pitch of 1.6 mm. . If the nut is initially tightened up by hand so as to cause no stress in the copper spacing tube, calculate the stresses induced in the tube and in the bolt if a spanner is then used to turn the nut through 90$^{\circ}$. Take, E$_C$ and E$_S$ as 100 GPa and 209 GPa respectively.

A piece of material is subjected to two perpendicular stresses as follows: (a) Tensile stresses of 100 MPa and 60 MPa (b) Tensile stress of 100 MPa and compressive stress of 60 MPa (c) Compressive stress of 100 MPa and tensile stress of 60 MPa (d) Compressive stresses of 100 MPa and 60 MPa. Determine normal and tangential stresses on a plane inclined at 30$^{\circ}$ to the plane of 100 MPa stress. Also find the resultant and its inclination with the normal stress.

An overhanging beam ABC is loaded as shown in the figure below: . Draw the shear force and bending moment diagrams. Also locate point of contraflexure. Determine maximum positive and negative bending moments.

State and prove Castigliano's first theorem.

A simply supported beam has its support 8 m apart at A and B. It carries a uniformly distributed load of 6 kN/m between A and B starting from 1 m and ending at 5 m from A. The end B of the beam has an overhang of 1 m and at the free end, a concentrated load of 8 kN is applied. Determine deflection of the free end and the maximum deflection between A and B. Assume, E = 210 GPa and I = 20 x 10$^6$ mm$^4$.

Establish the relations to find the shear stress across I-section. What is the maximum value of it?

A simply supported beam of 2 m span carries an uniformly distributed load of 140 kN per m over the whole span. The cross-section of the beam is a T-section with a flange width of 120 mm, web and flange thickness of 20 mm and overall depth of 160 mm. Determine the maximum shear stress in the beam and draw the shear stress distribution for the section.

A steel bar of circular section, 100 mm diameter, carries a longitudinal pull whose line of action is parallel to the axis of the bar. At a certain transverse section, the longitudinal stresses are measured at the surface of the bar at three points A, B and C. These points being equally spaced round the section the tensile stresses at these points are A = 90 MN/m$^2$, B = 75 MN/m$^2$ and C = 30 MN/m$^2$. Find (a) the magnitude and location of the greatest and least stresses at the section; (b) the magnitude and eccentricity of the applied pull. Make a diagram showing the stresses and their positions relative to the points A, B and C.

Derive expressions for the strain energy in a three-dimensional stress system.

A bar, 3.2 m long and 16 mm in diameter, hangs vertically and has a collar attached at the lower end. Determine the maximum stress induced when a weight of 80 kg falls from a height of 32 mm on the collar. If the bar is turned down to half the diameter along half of its length, what will be the value of the maximum stress and the extension? Assume, E = 205 GPa.

Deduce the torsion equation starting the assumption made. Deduce the expressions for the maximum stresses in solid and hollow shaft.

A solid alloy shaft of 60 mm diameter is coupled with a hollow steel shaft of the same external diameter in series. If the angle of twist of the steel shaft per unit length is 80% of that of the alloy shaft, then find the inner diameter of the steel shaft. What will be the speed to transmit 300 kW if the limiting stresses in the alloy and the steel are to be 50 MPa and 72 MPa respectively? Take, G$_{steel}$ = 2G$_{alloy}$.

The ratio of modulus of rigidity to modulus of elasticity for most of the materials is

Temperature stress is a function of

Normal stress on a plane, the normal to which is inclined at angle $ \theta $ with the line of action of uniaxial stress $ \sigma $ is given by

In a Mohr's circle, the radius gives the value of

Variation of shear force in a cantilever carrying a load the intensity of which varies uniformly from zero at the free end to w per unit run at the fixed end is given by

In a transversely loaded beam, the maximum tensile stress occurs at

Maximum deflection of a simply supported beam with a centre point load is

Maximum deflection of a fixed beam carrying a uniformly distributed load is

When a shaft is subjected to a twisting moment, every cross-section of the shaft will be under

The product of the tangential force acting on the shaft and radius of shaft known as

A steel plate, 22 mm thick and 220 mm wide at one end, tappers uniformly to 12 mm thick and 180 mm wide at the other end. Determine the elongation under a pull of 20 kN when the length of the plate is 2.4 m. Assuming E = 205 GN/m$^2$.

The steel bolt shown in the figure below has a thread pitch of 1.6 mm. . If the nut is initially tightened up by hand so as to cause no stress in the copper spacing tube, calculate the stresses induced in the tube and in the bolt if a spanner is then used to turn the nut through 90$^{\circ}$. Take, E$_C$ and E$_S$ as 100 GPa and 209 GPa respectively.

A piece of material is subjected to two perpendicular stresses as follows: (a) Tensile stresses of 100 MPa and 60 MPa (b) Tensile stress of 100 MPa and compressive stress of 60 MPa (c) Compressive stress of 100 MPa and tensile stress of 60 MPa (d) Compressive stresses of 100 MPa and 60 MPa. Determine normal and tangential stresses on a plane inclined at 30$^{\circ}$ to the plane of 100 MPa stress. Also find the resultant and its inclination with the normal stress.

An overhanging beam ABC is loaded as shown in the figure below: . Draw the shear force and bending moment diagrams. Also locate point of contraflexure. Determine maximum positive and negative bending moments.

State and prove Castigliano's first theorem.

A simply supported beam has its support 8 m apart at A and B. It carries a uniformly distributed load of 6 kN/m between A and B starting from 1 m and ending at 5 m from A. The end B of the beam has an overhang of 1 m and at the free end, a concentrated load of 8 kN is applied. Determine deflection of the free end and the maximum deflection between A and B. Assume, E = 210 GPa and I = 20 x 10$^6$ mm$^4$.

Establish the relations to find the shear stress across I-section. What is the maximum value of it?

A simply supported beam of 2 m span carries an uniformly distributed load of 140 kN per m over the whole span. The cross-section of the beam is a T-section with a flange width of 120 mm, web and flange thickness of 20 mm and overall depth of 160 mm. Determine the maximum shear stress in the beam and draw the shear stress distribution for the section.

A steel bar of circular section, 100 mm diameter, carries a longitudinal pull whose line of action is parallel to the axis of the bar. At a certain transverse section, the longitudinal stresses are measured at the surface of the bar at three points A, B and C. These points being equally spaced round the section the tensile stresses at these points are A = 90 MN/m$^2$, B = 75 MN/m$^2$ and C = 30 MN/m$^2$. Find (a) the magnitude and location of the greatest and least stresses at the section; (b) the magnitude and eccentricity of the applied pull. Make a diagram showing the stresses and their positions relative to the points A, B and C.

Derive expressions for the strain energy in a three-dimensional stress system.

A bar, 3.2 m long and 16 mm in diameter, hangs vertically and has a collar attached at the lower end. Determine the maximum stress induced when a weight of 80 kg falls from a height of 32 mm on the collar. If the bar is turned down to half the diameter along half of its length, what will be the value of the maximum stress and the extension? Assume, E = 205 GPa.

Deduce the torsion equation starting the assumption made. Deduce the expressions for the maximum stresses in solid and hollow shaft.

A solid alloy shaft of 60 mm diameter is coupled with a hollow steel shaft of the same external diameter in series. If the angle of twist of the steel shaft per unit length is 80% of that of the alloy shaft, then find the inner diameter of the steel shaft. What will be the speed to transmit 300 kW if the limiting stresses in the alloy and the steel are to be 50 MPa and 72 MPa respectively? Take, G$_{steel}$ = 2G$_{alloy}$.

What are complementary shear stresses? Explain with diagram.

In a stressed body, at a point, on two perpendicular planes, normal stresses are +100 MPa and +60 MPa and the shear stress is on these planes. If the maximum principal stress at the point is 136 MPa, calculate the maximum shear stress at the point.

The modulus of rigidity of material is 39 GPa. A 10 mm diameter rod of the material is subjected to an axial tensile force of 5 kN and the change in its diameter is 0.002 mm. Calculate the Poisson's ratio of the material.

In a strained material at a point, the strains are $ \epsilon_{xx} = 600 \mu $ strain, $ \epsilon_{yy} = 200 \mu $ strain and $ \epsilon_{xy} = 300 \mu $ strain. What is the maximum principal strain at the point?

Define Hooke's law.

A solid circular shaft is subjected to a bending moment of 3 kN-m and a torque of 1 kN-m. The shaft is to be made in carbon steel for which the yield strength in tension and in shear is 480 MPa and 265 MPa respectively. Calculate the diameter of the shaft using distortion energy theory.

A bar of 2 m length and 10 mm x 10 mm section is subjected to a bending moment of 20 kN-m. Find the strain energy stored, if E = 200 GPa.

Write the maximum strain energy theory.

Explain the terms section modulus and polar modulus.

Using moment-area method, find the maximum deflection of a simply-supported beam loaded with UDL.

Determine the deflection at a point 1 m from the left-hand end of the beam loaded as shown in the following figure using Macaulay's method. Take EI = 0.65 MN-m$^2$.

A beam of T-section is subjected to a shear force of 50 kN. Draw the shear stress profile of the section having following three dimensions: Flange = 100 mm x 20 mm, Thickness of web = 20 mm, Overall depth = 100 mm.

A torque of 4 kN-m is applied on a shaft of diameter 60 mm. Calculate the shearing stress at a point just below the surface and at another point which is at distance of 20 mm from the axis. Consider the cylindrical region of radius 15 mm and calculate the torque carried by this cylinder.

A timber beam 80 mm wide and 160 mm deep is reinforced with two steel plates 5 mm thick and 60 mm wide on top and bottom. If bending moment of 800 N-m acts at section of this beam, calculate the magnitude of maximum fiber stresses in tensions and compression in wood and steel. Assume E$_{s}$/E$_{w}$ = 15.

An element in a stressed material has tensile stress of 500 MN/m$^2$ and a compressive stress of 350 MN/m$^2$ acting on two mutually perpendicular plane and equal shear stress of 100 MN/m$^2$ on these plane. Find principal stresses, maximum shearing stresses and position of principal plane using Mohr circle diagram.

A 20 mm diameter bolt is subjected to a pull of 20 kN and a shear force of 5 kN. Calculate the maximum direct and shear stresses induced in the section and specify the position of the plane carrying these stresses with reference to the axis of the bolt. Also calculate the stress which acting alone will produce same maximum strain. Take $ \mu $ = 0.25.

A simply-supported beam of length 9 m coupling two point loads 210 kN and 125 kN at 2 m and 6 m from left support respectively. The self-weight of beam is 26 kN/m. Find the slope at 4 m, deflection at centre and maximum deflection. Take EI constant.

Derive and discuss Castigliano's theorem.

Resilience and proof resilience.

Strain energy.

Poisson's ratio.

What are complementary shear stresses? Explain with diagram.

In a stressed body, at a point, on two perpendicular planes, normal stresses are +100 MPa and +60 MPa and the shear stress is on these planes. If the maximum principal stress at the point is 136 MPa, calculate the maximum shear stress at the point.

The modulus of rigidity of material is 39 GPa. A 10 mm diameter rod of the material is subjected to an axial tensile force of 5 kN and the change in its diameter is 0.002 mm. Calculate the Poisson's ratio of the material.

In a strained material at a point, the strains are $ \epsilon_{xx} = 600 \mu $ strain, $ \epsilon_{yy} = 200 \mu $ strain and $ \epsilon_{xy} = 300 \mu $ strain. What is the maximum principal strain at the point?

Define Hooke's law.

A solid circular shaft is subjected to a bending moment of 3 kN-m and a torque of 1 kN-m. The shaft is to be made in carbon steel for which the yield strength in tension and in shear is 480 MPa and 265 MPa respectively. Calculate the diameter of the shaft using distortion energy theory.

A bar of 2 m length and 10 mm x 10 mm section is subjected to a bending moment of 20 kN-m. Find the strain energy stored, if E = 200 GPa.

Write the maximum strain energy theory.

Explain the terms section modulus and polar modulus.

Using moment-area method, find the maximum deflection of a simply-supported beam loaded with UDL.

Determine the deflection at a point 1 m from the left-hand end of the beam loaded as shown in the following figure using Macaulay's method. Take EI = 0.65 MN-m$^2$.

A beam of T-section is subjected to a shear force of 50 kN. Draw the shear stress profile of the section having following three dimensions: Flange = 100 mm x 20 mm, Thickness of web = 20 mm, Overall depth = 100 mm.

A torque of 4 kN-m is applied on a shaft of diameter 60 mm. Calculate the shearing stress at a point just below the surface and at another point which is at distance of 20 mm from the axis. Consider the cylindrical region of radius 15 mm and calculate the torque carried by this cylinder.

A timber beam 80 mm wide and 160 mm deep is reinforced with two steel plates 5 mm thick and 60 mm wide on top and bottom. If bending moment of 800 N-m acts at section of this beam, calculate the magnitude of maximum fiber stresses in tensions and compression in wood and steel. Assume E$_{s}$/E$_{w}$ = 15.

An element in a stressed material has tensile stress of 500 MN/m$^2$ and a compressive stress of 350 MN/m$^2$ acting on two mutually perpendicular plane and equal shear stress of 100 MN/m$^2$ on these plane. Find principal stresses, maximum shearing stresses and position of principal plane using Mohr circle diagram.

A 20 mm diameter bolt is subjected to a pull of 20 kN and a shear force of 5 kN. Calculate the maximum direct and shear stresses induced in the section and specify the position of the plane carrying these stresses with reference to the axis of the bolt. Also calculate the stress which acting alone will produce same maximum strain. Take $ \mu $ = 0.25.

A simply-supported beam of length 9 m coupling two point loads 210 kN and 125 kN at 2 m and 6 m from left support respectively. The self-weight of beam is 26 kN/m. Find the slope at 4 m, deflection at centre and maximum deflection. Take EI constant.

Derive and discuss Castigliano's theorem.

Resilience and proof resilience.

Strain energy.

Poisson's ratio.

For mild steel, the ratio of modulus of elasticity in tension and compression is equal to

For an isotropic elastic material, the number of independent elastic constant is

The Plane of maximum shear stress at any point are inclined to the principal planes through that point at an angle of

In the same loading Condition, if the diameter of the circular sectional beam is doubled, its deflection is reduced by

A simply supported beam carries a couple at a point on its span, the shear force

The stress in a beam is less if its section modulus is

For two shafts joined is parallel, the _____ is each shaft in the same.

An element of a stressed body is subjected to only two normal stresses of equal value but opposite sign in two perpendicular directions. The maximum shear stress in the element is

Complementary shear stresses are _____ is magnitude and are of _____ sign.

A prismatic bar of length $ l $, Cross-sectional area A and Young's modulus E is subjected to an axial load P, its strain energy is:

Differentiate between Engineering stress and true stress.

A reinforced Concrete column 200 mm is diameter is designed to carry on axial compressive load of 300 kN, Determine the required area of the reinforcing steel if the allowable stresses are 6 MPa and 120 MPa for the concrete and steel respectively. $ E_{concrete} $ = 14 GPa, $ E_{steel} $ = 200 GPa.

A steel rail is 32 m long and is laid at a temperature of 24$^{\circ}$C. Determine: (i) the stress in the rails at 80$^{\circ}$C when there is no allowance for expansion, (ii) the stress in the rails at 80$^{\circ}$C when there is an expansion allowance of 8 mm per rail, (iii) the expansion allowance for no stress in the rails at 80$^{\circ}$C and (iv) the maximum temperature for not stress in the rails when expansion allowance is 8 mm. Co-efficient of linear expansion $ \alpha = 11 \times 10^{-6} / ^{\circ}C $ and E = 205 GPa.

In a biaxial stress system, the stresses at a point are shown in fig. 1. Find: (i) Principal stresses and their position. (ii) Maximum shear stresses and their position. and (iii) The stresses on a plane inclined at 30$^{\circ}$ to the vertical.

Draw the shear force and Bending moment diagram for the beam shown in figure (2). Also locate the point of contra flexure if any?

The tension flange of a Cast iron I-section beam is 240 mm wide and 50 mm deep, the compression flange is 100 mm and 20 mm deep whereas the web is 300 mm x 30 mm. Find the load per m run which can be carried over a 4 m span by a simply supported beam if the maximum permissible stresses are 90 MPa in compression and 24 MPa in tension.

A beam, simply supported at ends A and B is loaded with two point loads of 60 kN and 50 kN at a distance 1 metre and 3 metre respectively from end A. Determine the position and magnitude of maximum deflection and slope.

The solid circular shaft as shown in fig.3 is used to transmit power of given values. Compute the maximum shear stress and angle of twist between two ends. At point A 50 hp is input and at B and C 30 hp and 20 hp respectively are taken off. (Neglect bending effect) $ G = 8.5 \times 10^{10} N/m^2 $, N = 530 rpm, Diameter of shaft = 40 mm.

State and prove Castiglione's first theorem.

Find the slope and deflection at the free end of a cantilever which carries a uniformly distributed load and a point at the free end.

For mild steel, the ratio of modulus of elasticity in tension and compression is equal to

For an isotropic elastic material, the number of independent elastic constant is

The Plane of maximum shear stress at any point are inclined to the principal planes through that point at an angle of

In the same loading Condition, if the diameter of the circular sectional beam is doubled, its deflection is reduced by

A simply supported beam carries a couple at a point on its span, the shear force

The stress in a beam is less if its section modulus is

For two shafts joined is parallel, the _____ is each shaft in the same.

An element of a stressed body is subjected to only two normal stresses of equal value but opposite sign in two perpendicular directions. The maximum shear stress in the element is

Complementary shear stresses are _____ is magnitude and are of _____ sign.

A prismatic bar of length $ l $, Cross-sectional area A and Young's modulus E is subjected to an axial load P, its strain energy is:

Differentiate between Engineering stress and true stress.

A reinforced Concrete column 200 mm is diameter is designed to carry on axial compressive load of 300 kN, Determine the required area of the reinforcing steel if the allowable stresses are 6 MPa and 120 MPa for the concrete and steel respectively. $ E_{concrete} $ = 14 GPa, $ E_{steel} $ = 200 GPa.

A steel rail is 32 m long and is laid at a temperature of 24$^{\circ}$C. Determine: (i) the stress in the rails at 80$^{\circ}$C when there is no allowance for expansion, (ii) the stress in the rails at 80$^{\circ}$C when there is an expansion allowance of 8 mm per rail, (iii) the expansion allowance for no stress in the rails at 80$^{\circ}$C and (iv) the maximum temperature for not stress in the rails when expansion allowance is 8 mm. Co-efficient of linear expansion $ \alpha = 11 \times 10^{-6} / ^{\circ}C $ and E = 205 GPa.

In a biaxial stress system, the stresses at a point are shown in fig. 1. Find: (i) Principal stresses and their position. (ii) Maximum shear stresses and their position. and (iii) The stresses on a plane inclined at 30$^{\circ}$ to the vertical.

Draw the shear force and Bending moment diagram for the beam shown in figure (2). Also locate the point of contra flexure if any?

The tension flange of a Cast iron I-section beam is 240 mm wide and 50 mm deep, the compression flange is 100 mm and 20 mm deep whereas the web is 300 mm x 30 mm. Find the load per m run which can be carried over a 4 m span by a simply supported beam if the maximum permissible stresses are 90 MPa in compression and 24 MPa in tension.

A beam, simply supported at ends A and B is loaded with two point loads of 60 kN and 50 kN at a distance 1 metre and 3 metre respectively from end A. Determine the position and magnitude of maximum deflection and slope.

The solid circular shaft as shown in fig.3 is used to transmit power of given values. Compute the maximum shear stress and angle of twist between two ends. At point A 50 hp is input and at B and C 30 hp and 20 hp respectively are taken off. (Neglect bending effect) $ G = 8.5 \times 10^{10} N/m^2 $, N = 530 rpm, Diameter of shaft = 40 mm.

State and prove Castiglione's first theorem.

Find the slope and deflection at the free end of a cantilever which carries a uniformly distributed load and a point at the free end.