Anna University, Chennai
DEPARTMENT OF MECHANICAL ENGINEERING
Sub. Code/Name: ME 2254 / Strength of Materials Year/Sem: II/III
UNIT – 1 STRESS STRAIN AND DEFORMATION OF SOLIDS
1. Define stress
The force of resistance per unit area, offered by a body against deformation is
known as stress. Mathematically stress is written as,
P= External force or load
A= Cross- Sectional area
2. Define strain
When a body is subjected to some external force, there is some change of
dimension of the body. The ratio of change dimension of the body to the original dimension is known as strain.
Strain may be:
1. Tensile strain 2. Compressive strain
3. Volumetric strain, and 4. Shear strain.
3. Define elastic limit
The limiting value of force up to and with in which, the deformation
completely disappears on the removal of the force. The value of stress corresponding to this limiting force is known as the elastic limit of the material
4. State Hokes law
Hokes’s Law states that when a material is loaded within elastic limit; the
stress is proportional to the strain produced by stress.
5. Define shear stress and shear strain
The stress induced in a body, when subjected to two equal and opposite forces which are acting tangentially across the resisting section, as a result of which the body tends to shear off across the section, is known as shear stress. The corresponding strain is known as shear strain. The shear stress acts tangential to the area. It is represented by τ.
6. Define poison’s ratio
The ratio of lateral strain to the longitudinal strain is constant for a given
material, when the material is stressed within the elastic limit. This ratio is
called Poisson’s Ratio and it is generally denoted by μ
7. Define Young’s modulus
The ratio of tensile stress or compressive stress to the corresponding strain is a
constant. This ratio is known as Young’s Modulus.
The ratio of the shear stress to the corresponding shear strain within elastic
limit is known as Modulus of Rigidity or Shear Modulus. This is denoted by C
or G or N.
It is defined as the ratio of ultimate tensile stress to the working stress.
Mathematically it is written as
Whenever a body is strained, energy is absorbed in the body. The energy,
which is absorbed in the body due to strained effect, is known as strain energy.
11. Define bulk modulus
When a body is subjected to the mutually perpendicular like and equal direct
stresses, the ratio of direct stress to the corresponding volumetric strain is found to be constant for a given material when the deformation is within a certain limit. This ratio is known as bulk modulus.
A bar, made up of two or more bars of equal lengths but of different materials
rigidly fixed with each other and behaving as one unit for extension or compression when subjected to axial tensile or compressive loads, is called a composite bar.
13. Define lateral strain
The strain at right angle to the direction of applied load is known as lateral
The Stresses induced in a body due to change in temperature are known as thermal stresses.
15. Define thermal strain
If the ends of the body are fixed to rigid supports, so that its expansion is
prevented, then compressive stress and strain will be set up in the rod. This strain is known as thermal strain.
16. What is the relation between three modules?
E = 2C (1+µ )
E = 3K (1-2µ )
Where, E= Young’s Modulus
C = Modulus of Rigidity
K = Bulk Modulus
µ = Poisson’s Ratio
17. Define Resilience.
The Total strain energy stored in a body is commonly known as resilience.
Whenever the straining force is removed from the strained body, the body is capable of doing work. Hence the resilience is also defined as the capacity of strained body for doing work on removal of straining body.
18. Define Proof Resilience
The Maximum strain energy stored in a body is known as proof resilience. The
strain energy stored in the body will be the maximum when the body is stressed upto elastic limit. Hence the proof resilience is the quantity of strain energy stored in ablody when strained up to elastic limit.
19. Define Modulus of Resilience.
It is defined as the proof resilience of a material per unit volume.
If the load is constant throughout the process of deformation of the body, such a type of loading is known as suddenly applied load.If the load is dropped from certain height, before the load commences to stretch the bar that type of loading is known as Impact load.
UNIT – II BEAMS-LOAD AND STRESSES
1. Define beam?
A beam is a structural member subjected to a system of external forces at right
angles to its axis.
2. What is mean by transverse loading on beam?
If the loads on the beam are perpendicular to the axis of the beam then it is
called transverse type of loading.
3. What is cantilever beam?
A beam which is fixed at one end and free at the other end is known as
4. What is simply supported beam?
A beam supported or resting freely on the supports at its both ends is
known as simply supported beam.
5. What meant by overhanging beam?
If the end portion of a beam is extended beyond the support, such beam is
known as overhanging beam.
6. What is mean by concentrated loads?
A concentrated load is one which is considered to act at a point, although in
practice it must really be distributed over a small area.
7. What is uniformly distributed load?
A uniformly distributed load is one which is spread over a beam in such a manner that rate of loading “w” uniform along the length. The rate of loading is expressed as w N/m run. Uniformly distributed load is, represented by U.D.L.
8. What are the different types of beams?
1. Cantilever beam 2. Simply supported beam 3. Over hanging beam
4. Fixed beam 5. Continuous beam.
9. Define point of contra flexure? In which beam it occurs?
At some point, the B.M. is zero after changing its sign from positive from
negative or vice-versa. That point is known as the point of contra flexure or point of inflexion
10. What is meant by positive or sagging BM?
The bending moment at a section is considered positive if the bending
movement at that section is such that it tends to bend the beam to a curvature having concavity at the top as often called sagging moment
11. What is meant by negative or hogging BM?
Similarly the bending movement(B.M.) at a section is considered negative if the bending movement at that section is such that it tends to bend the beam to a curvature having convexity at the . The positive B.M. is often called sagging moment and negative B.M. as hogging moment.
12. Define shear force and bending moment?
The algebraic sum of vertical forces at any section of a beam to the right or left of the section is known as shear force.
The algebraic sum of moments of all the forces to the right or left of the section is known as bending moment.
13. When will bending moment is maximum?
The bending moment will be maximum at a point where the shear force is minimum.
14. What is maximum bending moment in a simply supported beam of span
‘L’subjected to UDL of ‘w’ over entire span?
The maximum bending moment in a simply supported beam will be at the
point where the shear force line crosses the base line.
16. What is shear force and bending moment diagram?
The shear force diagram is one which shows the variation of the shear force
along the length of the beam.
The bending moment diagram is one which shows the variation of bending moment along the length of the beam.
17. What are the types of loads?
The different types of loads are
1. Concentrated load or point load
2. Uniformly Distributed load
3. Uniformly Varying load.
18. What is bending stress?
The stresses introduced by bending moment are known as bending stresses
19. Define pure bending.
If a length of beam is subjected to a constant bending moment and no shear
force(Zero shear force) then the stresses setup in that length of the beam due to bending moment only and that length of the beam is said to be in pure bending or simple bending.
20. Write the theory of simple bending equation?
21. State any three assumptions of Theory of simple bending.
1. The material of the beam is homogeneous and isotropic
2. The value of young’s modulus of elasticity is the same in tension and
3. The transverse sections which were plane before bending remain plane after
22. What is neutral layer of a beam?
If a beam is subjected to simple bending the top layer of beam is subjected to
compression and the bottom layer is subjected to tension. At a level between top and bottom, there will be a layer which is neither shortened nor elongated. This layer is known as Neutral layer.
24. Define Moment of Resistance.
Due to pure bending, the layers above NA are subjected to compressive
stresses where as the layers below NA are subjected to tensile stresses. Due to these stresses, the forces will be acting on the layers. These forces will have moment about the NA. The total moment of these forces about the NA for a section is known as Moment of Resistance of that section.
25. Define Section Modulus.
Section modulus is defined as the ratio of moment of inertia of a section about
the NA to the distance of the outer most layers from the NA. It is denoted by the symbol Z. Mathematically,
UNIT – III TORSION
1. Define Torsion?
A shaft is said to be in torsion, when equal and opposite torques are applied at
the two ends of shaft. The shaft is subjected to a twisting moment. This causes the shear stresses and shear strains in the material of the shaft.
2. What are the assumptions made in torsion equation.
Torsion is based on the following assumptions:
§ The material of the shaft is uniform throughout.
§ The twist along the shaft is uniform.
§ The shaft is uniform circular section throughout.
§ Cross-sections of the shaft, which are plane before twist, remain plain after twist.
§ All radii which are straight before twist remain straight after twist.
3. Define polar modulus
Polar modulus is defined as the ratio of the polar moment of inertia to the
radius of the shaft.
4. Write down the expression for power transmitted by a shaft.
Power(Watts) = ----------------- Where,
N = Speed of the shaft, rpm
T = Mean Torque transmitted, Nm.
5. Write the polar modulus for solid shaft and hollow shaft. π D3
Zp = ----------- where D = Diameter of solid shaft
Zp = --------------------- where, Do = Outer Diameter of hollow shaft
16 Do Di = Inner diameter of Hollow shaft
6. Why hollow circular shafts are preferred when compared to solid circular
For the same power transmission and Material, the weight of hollow shafts are less compared to solid shaft.
7 . Write torsional equation
T τ GӨ
------ = ------ = ------ where,
J r l
T = Torque Transmitted, Nm
J = Polar Moment of Inertia, m4
r = Radius of shaft, m
G = Modulus of rigidity, N/m2
Ө = Angle of Twist, Radians, l = Length of Shaft, m
8. Write down the expression for torque transmitted by hollow shaft.
Π Do4 – Di4
Torque transmitted by a hollow shaft – T = -------τ[-----------------]
Where Do = Outer Diameter of hollow shaft
Di = Inner diameter of Hollow shaft
Τ = Shear stress induced in the outer surface of the shaft
9. Write down the equation for maximum shear stress of a solid circular section in diameter D when subjected to torque T in a solid shaft.
The maximum shear stress induced at the outer surface of the solid shaft.
10. Define torsional rigidity
Tensional rigidity or stiffness of the shaft is defined as the product of modulus
of rigidity (C) and polar moment of inertia of the shaft (J).
Tensional rigidity = C x J.
11. What is composite shaft?
A shaft made up of two or more different materials and behaving as a single shaft is known as composite shaft.
12. What is a spring? Name two important types of spring.
Springs are elastic bodies which absorb energy due to resilience. The absorbed energy may be released as and when required. The two important types of springs are
1. Laminated or leaf springs
2. Helical springs
13. State any two functions of springs.
1. It will absorb energy when it is loaded and releases the absorbed energy as and when required
2. It will act as cushion
14. Define helical springs specify the type of helical springs Helical springs are the thick spring wires coiled in to a helix The two types of helical springs are
(i) Close coiled helical springs
(ii) Open coiled helical springs.
15. What is solid length of a spring?
The solid length of spring means the distance between the coils when the coils are touching each other. There is no gap between the coils. The soilid length is given by
Solid length = Number of coils x Diameter of the wire = nX d
16. Define spring rate (stiffness).
The stiffness of the spring is defined as the load per unit deflection.
S = --------- δ
Where W = Load and δ = Deflection
17. Define pitch of the spring.
The distance between to consecutive coils is known as pitch of the spring.
18. What is spring index (C)?
The term Spring index in relation to springs it means the ratio of the mean diameter of the spring to the diameter of the wire. The preferred index range is 4 to 12.
19. What is free length of spring?
This is the length over all of the spring in the fully unloaded position (free position).
21. What are the differences between closed coil & open coil helical springs?
The wire is coiled so close that the plane containing each turn is nearly at right angle to the axis of helix
The spring wire is coiled in such away that, there is a large gap between the two consecutive turns and takes compression load also.
The helix angle is less than 10o
The helix angle is greater than 10o
Due to loading, spring will elongate
Due to loading, spring will contract
22. What is the use of wahl’s factor?
(i) The whals factor is used to account the stress due to direct shear and curvature stress of spring.
(ii) It is used to calculate shear stress induced in a helical spring bt considering shear stress due to twisting.
23. What kind of stress is introduced when an axial load acts on a closed coil spring?
Shear stress is introduced when an axial load acts on a closed coil spring.In addition direct shear stress due to load is introduced. Stress due to curvature of the wire is also introduced.
24. What are the assumptions made ina closed coil helical spring subjected to axial load?
The assumptions are
1. An element of an axially loaded helical spring, behaves essentially as a straight bar in pure tension.
2. The planes perpendicular to the bar axis do not warp or distort during deformation.
25. List out the uses of helical springs.
Automobiles and railway carriages
Reciprocating mechanisms and clock springs.
UNIT – IV
1.What are the methods for finding out the slope and deflection at a section? (i) Double Integration Method
(ii) Moment Area method
(iii) Macaulay’s method
(iv) Conjucate beam method
2. State mohr’s theorems.
(i) The change of slope between any two points is equal to the net area of the
B.M Diagram between these points divided by EI
(ii)The total deflection between any two points is equal to the moment area of
B.M Diagram between the two points about the last point divided by EI.
3. What is the relation between slope and deflection?
If y= deflection
Slope = dy/dx
4. State the equation for bending moment and shear force.
BM = EI d2y/dx2
SF = EI d3y/dx3
5. Give expression for determining slope at the supports of a simply supported beam carrying a point load at the centre.
6. Give expression for determining deflection of a simply supported beam carrying a point load at the centre.
7. Give expression for determining slope and deflection of a simply supported beam carrying a UDL over the entire span.
8. What are the steps to be followed in determining slopes and deflections at any point of a beam using Macaulay’s method?
(i) Brackets are to be integrated as a whole.
(ii) Constants of integrations are written after the first term
(iii) The section, for which B M equation is to be written, should be taken in the last part of the beam.
9. What is the condition for maximum deflection in a beam.?
For maximum deflection, the slope dy/dx must be equal to Zero.
10. State the expression for determining slope in area moment method.
Area of BMD Өb = ------------------
11. State the expression for determining deflection in area moment method.
y = --------- EI
12. What is a column?
A vertical member of a structure, which is subjected to axial compressive load and is fixed at both of its ends is known as a column.
13. What is a strut?
Strut is a member of a structure which is not vertical or whose one or both of its ends are hinged or pin jointed.
14. Specify how the long and short column will fail?
All short columns fail due to crushing where as long column will fail due to buckling or crushing.
15. What is buckling load?
The load at which the column just buckles is known as buckling load or critical load or crippling load.
16. Give the expressions for a column by Euler’s formula for different end conditions.
17. What is effective length of a column?
The effective length of a given column with given end conditions is the length of an equivalent column of the same material and cross section with hinged ends, and having the value of crippling load equal to that of the given column.
18. What is slenderness ratio?
Slenderness ratio is the ratio of the effective length of the column to the least radius of gyration.
19. State the limitation of Euler’s formula.
If the slenderness ratio is small, the crippling stress will be high. But for the column material, the crippling stress cannot be greater than the crushing stress. Hence when the slenderness ratio is less than a certain limit, Euler’s formula gives a value of crippling stress greater than the crushing stress. In this limiting case, we can find the value of l/k for which crippling load equal to crushing stress.
20. Specify the expression for Rankine’s formula.
P = Crippling load by Rankine’s formula
Pc = Crushing load
PE = Crippling load by Euler’s formula
UNIT – V Analysis of Stresses in Two Dimensions
1. When will you call a cylinder as thin cylinder?
A cylinder is called as a thin cylinder when the ratio of wall thickness to the diameter of cylinder is less than 1/20
2. In a cylinder will the radial stress vary over thickness of wall?
No, in the cylinders radial stress developed in its wall is assumed to be constant since the wall thickness is very small as compared to the diameter of the cylinder.
3. What is the ratio of circumferential stress to longitudinal stress of a thin cylinder?
4. What is the maximum principal stress in a thin cylindrical?
The maximum principal stress in a thin cylinder is circumferential stress.
10. For a thin cylinder, write down the expression for volumetric strain.
internal presser ‘P’?
14. In a thin cylindrical shell if hoop strain is 0.2×10-3 and longitudinal strain is 0.005×10-3, find out volumetric strain.
Volumetric strain, ev = 2ec ea
= 0.2 × 10-3 +0.005 × 10-3
ev = 0.25 × 10-3
15. Different between thin cylinder and thick cylinder.
1. Ratio of wall thickness to the
diameter of cylinder is less that
2. Circumferential stress is assumed to be constant throughout the all thickness.
1. Ratio of wall thickness to the
dia., of cylinder is more than
2. circumferential stress varies from inner to outer wall thickness
16. Distinguish between cylindrical shell and spherical shell.
1. Circumferential stress is
twice the longitudinal stress.
2. It withstands low pressure than spherical shell for the same diameter.
1. Only hoop stress presents.
2. It withstands more pressure than cylindrical shell for the same diameter.
17. Write the equation for the change in diameter and length of a thin cylinder shell, when subjected to an internal pressure.
Riveting reduces the area offering the resistance. Due to this, the circumferential and longitudinal stresses are more. It reduces the pressure carrying capacity of the shell.
19. What do you understand by term wire winding of thin cylinder?
In order to increase the tensile strength of a thin cylinder to withstand high internal pressure without excessive increase in wall thickness, they are some times pre-stressed by winding with a steel wire under tension.
20. Define principal stresses and principal plane.
Principal stress: The magnitudes of normal stress, acting on a principal plances are known as principal stresses.
Principal planes: The planes which have no shear stress are known as principal planes.
21. Give the expression for maximum shear stress in a two dimensional stress system.
22. What is the radius of Mohr’s circle?
Radius of Mohr’s circle is equal to the maximum shear stress.
23. A bar of cross sectional area 600mm2 is subjected to a tensile load of
50KN applied at each end. Determine the normal stress on a plane inclined at 30° to the direction of loading.
A = 600 mm2
Load, p = 50KN
θ = 30°
24. Give the expression for stresses on an inclined plane when it is subjected to a axial pull.
Normal stress, σn = p cos2θ
25. What is the use of Mohr’s circle?
To find out the normal, tangential, resultant stresses and principal stress and their planes.
26. What are the planes along which the greatest shear stresses occur?
Greatest shear stress occurs at the planes which is inclined at 45° to its normal
27. Write the expression for a normal stress on an inclined plane in a block which is subjected to two mutually perpendicular normal stresses and shear stresses?
28. At a point in strained material is subjected to a compressive stress of
100N/m2 and shear stress of 60 N/mm2.determine graphically or otherwise the principal stresses.
σ = -100 N/mm2
29. In case of equal like principal stresses, what is the diameter of the Mohr’s
30. What is the value of maximum shear stress when the principal stresses are