Strength of Materials - 2Marks

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STRENGTH OF MATERIALS

TWO MARK QUESTIONS & ANSWERS

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Prepared by Dr. J. Raja Murugadoss Professor and Head of Civil Engineering KPR Institute of Engineering and Technology Coimbatore 641 407



CHAPTER 1 STRESSES AND STRAINS

1. Define stress? When a body is subjected to any arbitrary loading, it undergoes deformation. Consequently some amount of internal force will be set-up in the body to resist the deformation produced by the external cause. The magnitude of internal resisting force per unit area is defined as stress. The unit unit of stres stresss ( ) is N/mm N/mm . 2

2. Defi Define ne stra strain in?? When a solid bar is subjected to an axial load, it undergoes deformation both in longitudinal and transverse directions. In the longitudinal direction, strain (engineering strain) can be defined as the fractional change in length. Strain is a dimensionless quantity and it is denoted by the symbol ’’. 3. Define Hooke’s law? Hooke’s states that, when a material is loaded within the elastic limit the st ress is

linearly proportional to strain. Therefore

Stress  Strain Stress = E (Strain) 2

where, where, E = Young’s Modulus (N/mm ). ‘E’ is also called as elastic constant. 4. What What do you mean mean by limit limit of of proporti proportional onality ity or elasti elasticc limit? limit? Limit of o f proportionality or elastic limit is a point in the stress-strain curve at which the linear relation between them ceases. (i.e. the point at which the straight line changes to a curve). Thereafter the stress is not directly proportional to strain and therefore Hooke’s law is not valid after the elastic limit .



CHAPTER 1 STRESSES AND STRAINS

1. Define stress? When a body is subjected to any arbitrary loading, it undergoes deformation. Consequently some amount of internal force will be set-up in the body to resist the deformation produced by the external cause. The magnitude of internal resisting force per unit area is defined as stress. The unit unit of stres stresss ( ) is N/mm N/mm . 2

2. Defi Define ne stra strain in?? When a solid bar is subjected to an axial load, it undergoes deformation both in longitudinal and transverse directions. In the longitudinal direction, strain (engineering strain) can be defined as the fractional change in length. Strain is a dimensionless quantity and it is denoted by the symbol ’’. 3. Define Hooke’s law? Hooke’s states that, when a material is loaded within the elastic limit the st ress is

linearly proportional to strain. Therefore

Stress  Strain Stress = E (Strain) 2

where, where, E = Young’s Modulus (N/mm ). ‘E’ is also called as elastic constant. 4. What What do you mean mean by limit limit of of proporti proportional onality ity or elasti elasticc limit? limit? Limit of o f proportionality or elastic limit is a point in the stress-strain curve at which the linear relation between them ceases. (i.e. the point at which the straight line changes to a curve). Thereafter the stress is not directly proportional to strain and therefore Hooke’s law is not valid after the elastic limit .



Figure 1.1 Stress-Strain Curves Also this is the point at which material undergoes rearrangement of molecular structure, in which atoms are being shifted to some other stable configuration. 5. What do you mean by the term “necking”? When a material is being loaded to its yield point, the specimen begins to “neck” (i.e. the cross sectional area of the material start decreasing) due to plastic flow. Therefore Necking can be defined as the mode of ductile flow of material in tension. Necking usually occurs where the surface imperfections are predominant.

Figure 1.2 Necking 6. What do you mean by b y Poisson’s ratio? ratio? When an axial bar is loaded, it undergoes deformation in both the direction i.e. longitudinal and transverse directions. Therefore Poisson’s ratio can be defined as the ratio of lateral strain to the longitudinal longitudinal strain or axial strain or or linear linear strain (within the elastic limit). limit) . It is denoted by the symbol ‘’ and it is a dimensionless quantity. Table 1 Poisson’s ratio for few materials Material Rubber Lead Copper Brass Steel

Poisson’s Ratio

0.48 0.44 0.37 0.33 0.29



Figure 1.3 Poisson’s Effect 7. When a material is said to be perfectly elastic? A material is said to be perfectly elastic, when it obeys Hooke’s law. In other words the material is said to be perfectly elastic when the stress is directly proportional to the strain within the elastic limit. 8. What is meant by free body diagram? A free body diagram is a complete diagram or a simplified sketch that shows all the external forces with the direction and the point of application of external load. This includes all the reactive forces by the supports and the weight of the body due to its mass. 9. What do you mean by volumetric strain? Volumetric strain can be numerically defined as the change in volume to the original volume of the material. It is denoted by the symbol ‘V’. 10. What do you mean by “Bulk Modulus”? The ratio of change in pressure (P) to the fractional change in volume i.e. volumetric strain is called bulk modulus of a material. It is denoted by the symbol ‘K’. Note: The reciprocal of the bulk modulus is defined as the compressibility of the material.



11. What is “Shear Modulus”? Shear modulus ‘G’ can be defined as the ratio of shear stress to shear strain. 12. What is strain hardening? When a material is subjected to tensile test, at a particular load corresponding to the upper yield point, the material starts flowing or becomes ductile in nature. After that the material starts taking more load greater than the load corresponding to elastic limit. This phenomenon is called strain hardening. 13. What is meant by force? Force is defined as the “interaction between bodies which gives rise to an acceleration or to the deformation of the body. 14. What is shear stress? The intensity of force per unit area acting tangentially to a particular point is called shear stress. It is dented by the symbol ‘ ’. It is also called as rigidity modulus or modulus of rigidity. Note: Shear force always act parallel to a cross section. 15. What do you mean by compound bar? A compound bar is an assembly of more than one bar having same or different cross sections made of same or different materials. 16. What do you mean by thermal stresses? If an arbitrary body is allowed to expand or contract freely, with the rise or fall of temperature no stress is developed but if free expansion is prevented the stress developed is called temperature stress or strain. 17. What do you mean by strain energy? The work done in straining the material, within the elastic limit, is known as strain energy. Note: Strain energy is scalar quantity. It is always denoted by the symbol ‘ U’. Generally it is a positive quantity. In solid deformable bodies, stresses multiplied by their respective areas of cross-section are forces and the deformations are distances. Force = stress x area



Hence the internal work is defined as the product of force and the deformations. This internal work is stored in an elastic body as the internal elastic energy of deformation (or) simply elastic strain energy. If the strain energy is within the elastic limit, the work done will be completely transformed into potential energy and can be recovered during a gradual unloading of the strained material. or The potential energy stored in a body by virtue of an elastic deformation, equal to the work that must be done to produce both normal and shear strains. The unit of strain energy is Joule or N-m. Strain energy is dependent on the length and cross-sectional area of the material.

Figure 1.4 Load-Deformation Plot 18. What is complimentary energy? The area between the load-extension curve and the vertical axis is called the complimentary energy. 19. Define elastic strain energy? If the material is loaded within the elastic limit and then unloaded to zero stress, the strain also becomes zero and the strain energy stored in the body in straining the material is recoverable. However, when the material is loaded beyond the elastic limit and then unloaded, some permanent deformations will be setup in the body even after unloading. Therefore, only the partial strain energy will be recoverable and is called elastic strain energy. 20. What do you mean by strain energy density? Strain energy density is defined as the strain energy per unit volume of the material. It is actually the area under the stress-strain curve.



21. Define Proof load. The maximum load which can be applied to a body without permanent deformation is called proof load. 22. Define resilience. Resilience is defined as the capacity of a material to absorb energy upon loading. 23. Define modulus of resilience. Modulus of resilience is defined as the energy per unit volume that the material can absorb without yielding. 24. Define toughness of a material. Toughness is defined as the maximum strain energy that can be absorbed per unit volume till rupture. The modulus of toughness is a measure of the resistance of the structure to impact loading and is dependent on the ductility of the material. 25. What are the major types of deformation?

 

Elastic deformation (deformation due to loads) Thermal deformation (deformation due to temperature variation)

26. Find the magnitude of ‘P’ of a compound bar?

100 kN

P

50 kN

100 kN

Sum of all the forces acting in left direction = Sum of all the forces acting in right direction. Therefore, 100 + P = 100 + 50 P = 50 kN. 27. How will you calculate the total elongation of a compound bar which is connected in series? The total elongation of a compound bar connected in series can be computed by the relation



= l1 + l2 + l3 +…+ ln δl 

P1 L1 A1E 1



P2 L 2 A2E2

 ... 

Pn L n AnEn

where, li is the deformation on individual bar in the system. 28. How will you find the loads taken by individual bars connected in parallel and are clamped at their ends? The loads shared by individual bars can be computed by equating the displacements of each bar. (i.e. both the displacements are equal) 29. State the principle of Superposition? When a body is subjected to a number of external forces, the forces are split up, and their effects are considered on individual sections. The resulting deformation, of the body is equal to the algebraic sum of the deformations of the individual sections. Such a principle of finding the resultant deformation is called the principle of superposition.



CHAPTER 2 PRINCIPAL STRESSES AND STRAINS 1. What is meant by principal plane? Principal plane is plane in which the stress vector will be wholly normal and there will not be any tangential or shear stress in that particular plane. Such a plane is called principal plane. The corresponding stress is called principal stress. Since the resultant stress is along the normal, tangential or shear stress is always zero. Therefore the principal plane is called shearless plane 2. What do you mean by state of stress? The totality of all stress vectors acting on every possible plane passing through a point is defined to be the state of stress at a point. 3. What is principal stress? The normal stress which is acting on the principal plane is called principal stress. 4. What do you mean by octahedral plane? A plane that is equally inclined to all the three principal axes, then that plane is called octahedral plane. Also the octahedral plane is free from normal stress. 5. What is stress invariant? A stress invariant is one whose value does not change when the frame of reference is changed. For example I1, I2, I3 are the first, second and third stress invariants of three dimensional state of stress whose value does not change. 6. Give the necessary condition for a pure state of shear. For a state of pure shear to be exist, the first stress invariant should be equal to zero or in other words

I1

σx  σy σz 0



7. What is hydrostatic or isotropic pressure? If the principal stresses are equal in magnitude, then that state is called hydrostatic or isotropic. Note: Hydrostatic or isotropic causes only a change in volume. (i.e. volumetric strain). No distortion of material will takes place. 8. What is meant by residual stresses? In reality, when materials are being manufactured, they are often rolled, extruded, forged, welded and hammered. In castings, materials may cool unevenly. These processes can setup high internal stresses called residual stresses. Note:  This process causes the development of larger normal stresses near the outer surface than in the middle.  These residual stresses are self-equilibrating. i.e. they are in equilibrium without any externally applied forces.

 In real world problems, such residual stresses may be large and should be carefully investigated and then added to the calculated stresses for the initially stress-free material. 9. What is meant by Spherical and deviatoric state of stress? If a generate state of stress is decomposed into two components, it falls into categories, i.e. Hydrostatic state of stress (or) Spherical state of stress and deviatoric state of stress. Let P = 1/3 I 1, I1 is the first stress invariant.

σ x τ x y τ xz   P 0    τ σ τ  yx y yz   0 P τ τ σ  0 0  zx zy z 

(σ x  p )τ x y τ xz     0  τ yx (σ y  p ) τ yz   P  τ zx τ zy (σ z  p )    0

Here the first term represents the spherical state of stress and the other tangential or shear stress will be completely zero. The second term is called deviatoric state of stress or state of pure shear or simply stress deviator. Deviatoric state of stress at a point is derived by subtracting the mean of the normal stress components of the stress matrix (i.e. diagonal components) from each diagonal term of the stress matrix.



10. What is meant by cross-shear? Out of nine stress components

i.e.

x, y, z, xy, xz, yz, yx, zx, zy,

six components are independent

components. These are known as cross-shears.



CHAPTER 3 THEORIES OF FAILURE 1. List out any five important theories of failure.

    

Maximum Principal Stress Theory (or) Rankine’s Theory Maximum Shear Stress Theory (or) Tresca Yield Theory Maximum Elastic Strain Theory (or) St. Vanant’s Maximum Elastic Energy Theory Energy of Distortion Theory

2. State the theory of “Energy of Distortion Theory”. According to this theory, the energy absorbed during the distortion of an element is responsible for failure, not the total energy absorbed. Note: The energy of distortion can be obtained by subtracting the energy of volumetric expansion from the total energy. 3. State the theory of “Maximum Elastic Strain Energy”. According to this theory, failure occurs at a point in a body when the maximum strain at that point exceeds the value of the maximum strain in a uniaxial test of the material at yield point. 4. State the theory of “Maximum Elastic Energy”. According to this theory, failure at any point in a body, subject to a state of stress begins only when the energy per unit volume absorbed at the point is equal to the energy absorbed per unit volume of the material when subjected to the elastic limit under a uniaxial state of stress. 5. State the theory of “Maximum Shear Stress Theory”. The maximum shear stress theory (or) simply the maximum shear theory, results from the observation that in a ductile material, slip occurs during yielding along critically oriented planes. It is assumed that yielding (plastic state) of the material depends only on the maximum shear stress that is attained within an element. 6. State the theory of “Maximum Principal Stress Theory” According to this theory, the maximum principal stress in the material determines the failure regardless of what the other two principal stresses are, so long as they are algebraically smaller. Note:



This theory is not much supported by experimental results. 7. Why theories of failure are necessary? When a material is subjected to any state of stress or strain, failure or fracture occurs due to many factors or causes. It may be due to principal stress, maximum shear stress at a point or octahedral shear stress. To understand, to improve the performance of material under loading condition and to prevent the failure or fracture, knowledge of theories of failure is much essential. 8. Define dilatation. In the infinitesimal (small) strain theory, dilatation is defined as the change in volume per unit volume, is referred to as dilatation.

e

 ε x  ε y  ε z

Note: The shear strains cause no change in volume.



CHAPTER 4 SHEAR FORCE AND BENDING MOMENT

1. What is a beam? A beam is a horizontal member which is always loaded in the transverse direction, perpendicular to the longitudinal axis of the beam. 2. What are the different types of beams?

   

Simply supported beam (one end hinged and other being roller supported) Cantilever beam (one end fixed and other end being free) Fixed beam (both the ends are clamped or fixed) Over-hanging beam (some loaded portion of the beam extends beyond the support)

3. What are the types of loads?

    

Point Load with zero inclination Point Load with some inclination to the transverse axis Uniformly Distributed Load (U.D.L) Varying Load Moment

4. What is meant by transverse loading of beam? If the load is acting perpendicular to longitudinal axis of the beam then it is called transverse loading of beam. 5. List the various types of support.

  

Simple support (it resist loads which are acting perpendicular to the longitudinal axis of the beam) Fixed support (it resists forces in all direction and also restrict the rotation of the beam) Hinged support (it can resist forces in two directions but allows rotation about the axis of the pin, example, hinge)

6. Define shear force. Shear force is defined as the internal force developed in the material of the beam to balance the externally applied loads in order to achieve equilibrium of all parts of the beam.



7. Define bending moment. Bending moments are internal moments developed in the material of a beam to balance the tendency for external forces to cause rotation of any part of the beam. 8. What is mean by positive or sagging bending moment? Bending moment is said to positive if moment on left side of beam is clockwise or right side of the beam is counter clockwise. 9. What is mean by negative or hogging bending moment? Bending moment is said to negative if moment on left side of beam is counterclockwise or right side of the beam is clockwise. 10. Define shear force diagram. A SFD is a diagram that indicates how a force applied perpendicular to the axis (i.e. parallel to the cross section) of a beam is transmitted along the length of the beam. 11. Define bending moment diagram. A BMD is a diagram that indicates how the applied loads to a beam create a moment variation along the length of the beam. 12. What is meant by point of contra-flexure? Point of contra-flexure or point of contrary flexure in point in the cross section of beam at which the shear force becomes zero (i.e. it changes the sign) and the bending moment is maximum. It is otherwise called as point of inflexion. 13. List out some properties of shear force diagram and bending moment diagram. 

The S.F.D. will consist of rectangles or series of rectangles if the beam is loaded with point loads.  The S.F.D. will consist of inclined lines for the portion of the beam on which the U.D.L. is acting.  The S.F.D. will consist of parabolic lines for the portion of the beam over which triangular load distribution is acting.  The B.M.D. will consist of inclined lines, if the beam is loaded with point loads.  The B.M.D. will consist of parabolic lines for the portion over which U.D.L is acting.  The B.M.D will consist of “cubic or third degree polynomials” if the load distribution is triangular.  The B.M.D. will consist of fourth degree polynomial if the load distribution is parabolic.



14. Write the bending equation or classic flexure formula and state its significance. M I

f

E

y

R

 

Here, M is the bending moment, I is the moment of inertia, f is the maximum bending stress, y is the distance of the fiber from the neutral axis, E is the elastic modulus and R is the radius of curvature. 15. What is isotropic and orthotropic? In isotropic, the material property will be same in all mutually perpendicular directions. In orthotropic, the material property will be different in all mutually perpendicular directions 16. What is meant by homogeneous and heterogeneous? Homogeneous means: the material property remains same in all discrete points. Heterogeneous means: the material property will vary from one point to other point. (i.e. the material property will not be same in all points) 17. What is meant by simple bending? When a beam is subjected to a transverse load in such a way that it develops only bending. Other actions will be absent. For example  

When a beam is subjected to equal and opposite couples, the beam will be subjected to bending alone. When a beam is subjected to two point loading, the beam will be subjected to bending alone.

18. What is meant by section modulus or modulus of section? The term I/y is called section modulus or modulus of section. The strength or load carrying capacity of beam depends on this section modulus. It is usually denoted by the letter ‘Z’. Z=

bd 2 6

(for rectangular sections)



Z=

Z=

π

πd 3 32

(D 4 - d 4 ) 32 D

(for solid circular sections)

(for hollow circular sections)

19. What is meant by modulus of rupture? The bending stress at failure or rupture is called modulus of rupture. Note: In theory of simple bending we have assumed that the beam is loaded within the elastic limit. And when the beam is loaded beyond its proportionality the beam will fail. The modulus of rupture is used to compare the bending strength of different beams made of various sizes and materials. 20. What are all the various assumptions made in the theory of simple bending?     

Transverse section of the beam remain plane before and after bending It obeys Hooks law. The material is homogeneous and isotropic The beam is initially straight and of constant cross-section The radius of curvature of the beam is very large when it is compared to the other dimensions of the beam.

The beam consists of infinite number of longitudinal fibres which is free to expand or contract during bending 21. Comment on Load Carrying capacity of beams? The strength of the section or the load carrying capacity of a beam does not depend upon the sectional area provided but upon the disposition of that area in relation to its neutral axis. In other words, the strength of beam directly depends on the sectionmodulus ‘Z’ of the beam.

22. What do you mean by shear flow? Shear flow is defined as the longitudinal force per unit length transmitted across the section at level ‘y1’ from the neutral axis. If the shear stress is multiplied by the corresponding width of the section, the quantity obtained is known as shear flow. It is denoted by ‘q’ and is given by q = τ. z (here ‘z’ denotes the width of the section corresponding to that layer)



23. How will you calculate the value of shear stress at a particular distance from the neutral axis?



V.A '.y I.Z

Here, ‘V’ is the corresponding shear force at a particular distance from the neutral axis, A’ is the partial area of the section, ‘ y ’is the moment arm of this partial area with respect to neutral axis, ‘I’ is the moment of inertia of the section and ‘Z’ is the corresponding width of the layer or fiber and ‘  ’ is the shear stress at a particular distance from the neutral axis. 24. Prove that the maximum shear stress in a rectangular beam is 1.5 times the average shear stress. [university two mark question] 25. Draw the bending stress variation of a simply supported beam. 2

The value of bending stress (N/mm ) is zero at the level of neutral axis and maximum at the extreme fiber of the cross-section of the beam. The bending stress is always proportional to the distance of the fiber from the neutral axis. The value of bending stress increases as the distance of the fiber increases. Above the neutral axis, the beam experiences compressive stress and at the same time it is subjected to tensile stress below the neutral axis. The bending stresses always cause the member to bend in the transverse direction.

C

T



CHAPTER 5 COLUMNS

1. Define Column. Column is a vertical, long and slender member which is subjected to only compressive load. Every discrete cross section of the column will be in a state of axial compression. 2. What are the major classifications of a column? Columns are classified into  

Long columns and Short Columns based on the slenderness ratio of the column.

3. What do you mean by slenderness ratio? Slenderness ratio is defined as the ratio of effective length of the column to its smallest radius of gyration. It usually denoted by the symbol ‘’

λ 

EffectiveLength (le ) radius of gyration (r)

4. What do you mean by the term effective length of a column? Effective length of a column is defined as the distance between the adjacent points of inflexion in a column. Point of inflexion is the point at which the column experiences lateral bending. In other words, it may be defined as the product of actual length of a column and the end-fixity factor. The end fixity factor is usually denoted by the symbol ‘K’. 5. What do you mean by radius of gyration? Radius of gyration is a measure of slenderness of the column. It can be defined as r

I A

where, I is the moment of inertia of the column and A is the cross sectional area of the column. I and A are geometric properties of column’s cross section. The value of ‘I’ changes with respect to axis and therefore if ‘r’ is minimum at a particular axis, then the failure of the column will be likely in that axis.



6. What are the failure modes of columns? Columns are classified into long columns and short columns based on the slenderness ratio. In general long columns always fails by buckling and short columns always fails by crushing of materials in the column. 7. What do you mean by buckling and buckling load? Buckling is a failure mode of column at which the straight configuration of the column changes to some other deformed configuration. The minimum load at which the stable equilibrium transforms to another deformed stable configuration is called buckling load. Buckling is also called as lateral bending and this phenomenon is referred as elastic instability. Buckling load is also called as   

Critical load Crippling load Bifurcating load

8. What do you mean by real column? Real column is one which is practically exists in the real world. All these columns have  

Initial eccentricity i.e. the load is not concentric at every cross section of the column. Initial crookedness i.e. the column is not perfectly straight

9. Mention any one method of finding the critical load of a long column. Euler has established one empirical equation to determine the buckling load or critical load or load carrying capacity of long columns based on certain assumptions. This buckling load is called Euler’s critical load (Pcr).

Pcr 

π 2 EI le

10. What are the assumptions involved in the derivation of Euler’s critical load?   

Column is perfectly straight and there is no crookedness or imperfections in the member Load is acting concentric at every cross section of the column. It is valid only for long columns



11. What is the influence of the assumptions of Euler on the load carrying capacity of a real column? Euler has assumed that the column is initially straight and there is no eccentricity in the column. But in actual practice there is no such real columns which have zero eccentricity and perfect geometry. Therefore the Euler’s critical load is always higher than the actual critical load of a long column. 12. What do you mean by eccentricity? Eccentricity is defined as the perpendicular distance between the point of application of load to the longitudinal axis of the column. It is usually denoted by the symbol ‘e’. 13. Draw the relation between slenderness ratio of the column against the critical stress of the column. 14. How the slenderness ratio of the column, affect the strength of column? The strength of column depends upon many parameters. But the length of column plays a major role in determining the strength of column. If the slenderness ratio of the column increases, the strength of the member generally decreases. Also the critical stress of the column also decreases. 15. List out the effective length (s) of column for different boundary conditions. 

Both ends being pinned or hinged:



Both ends being fixed:



One end fixed and other end free:



One end fixed and other being pinned:

e

e

=

= 0.5 e

=2 e

= 0.707

16. Discuss the effect of initial imperfections and eccentricity in the column. In the derivation of Euler Buckling load for long columns, the member is assumed to be straight and loading is assumed to be concentric at every cross section. However, in real world engineering practice, the members are not perfectly straight and moreover the load is not concentric at every cross section. Unlike the perfect column, which remains straight up to the Euler load, the initially deformed member begins to bend as soon as the load is applied. The deflection increases slowly at first and then rapidly increases. In eccentric loaded columns, bending begins as soon as the load is applied. The deflection increases slowly in the beginning and then rapidly increases.



Columns with large eccentricities deflect considerably at loads well below the Euler load, whereas the column with small eccentricities of loading do not bend significantly until the load is fairly close to the Euler load. Excessive bending of the column sometimes leads to complete collapse of the member. Column with small eccentricities of loading can therefore be expected to support loads only slightly less than the Euler loads. The load carrying capacity of a real column is always less than that of Euler ‘s column due to the presence of initial imperfections and eccentricities.



CHAPTER 6 TORSION OF CIRCULAR SHAFT & SPRINGS 1. What is power? Power can be defined as the rate of transferring energy. It is calculated as P = T x ‘n’ where, P is the power, T is the torque and ‘n’ is the rotational speed. 2. What do you mean by Torsion? Torsion refers to the loading of a circular or non-circular member that tends to cause it to rotate or twist. Such a load is called torque, torsional moment, rotational moment, twisting moment or simply couple. 3. What are the assumptions made in Torsion equation     

The material of the shaft is homogeneous, perfectly elastic and obeys Hooke’s law. Twist is uniform along the length of the shaft The stress does not exceed the limit of proportionality The shaft circular in section remains circular after loading Strain and deformations are small.

4. Write the governing equation for torsion of circular shaft? T



τ

J R



Gθ L

where, T-Torque; J- Polar moment of inertia; G-Modulus of rigidity; L- Length of the shaft; - Shear stress; R- Radius of the shaft.

5. What is the type of stress induced in a structural member subjected to torsional loading? Shear Stress. The variation of shear stress is linear and it vary from zero at the neutral axis and reaches the maximum value at the extreme fiber of the shaft. i.e. shear stress  radius



6. Define polar moment of inertia and establish the equations for a solid and hollow circular shaft. Polar moment of inertia can be defined as

J J

πD 4 32

(solid circular shaft)

π (D 4

 d4 )

32

(hollow circular shaft)

7. Define polar modulus? Polar modulus can be defined as

Zp  Zp 

πD3 16

(solid circular shaft)

π (D 4 - d 4 ) 16D

(hollow circular shaft)

8. Why the shear stress is maximum at the outer surface of the shaft than the inner core? When the circular shaft is subjected to torsional loading, the shear stress is maximum at the extreme fiber of the shaft. This is due to the reason that, the extreme fibers are much strained than the inner surface near centroidal axis of the member. This is the reason why the shear stress is maximum at the extreme fiber of the shaft. Also the materials inside the shaft are not that much utilized at the time of torsional loading. Also it this is the reason why hollow circular shafts are preferred rather than the solid one for practical use. 9. Why hollow circular shafts are preferred when compared to solid circular shafts?  

The torque transmitted by the hollow shaft is greater than the solid shaft. For same material, length and given torque, the weight of the hollow shaft will be less compared to solid shaft.

10. What is torsional stiffness? The measure of torsional stiffness is the angle of twist of one part of a shaft relative to another part when a certain torque is applied.



11. What are various types of rigidity modulus?   

Flexural rigidity (EI) Torsional rigidity (GJ) Plate rigidity

12. Define spring? A spring is an elastic member, which deflects under the action of load and regains its original shape after the removal load. 13. What are the various types of springs?    

Disc spring (or) Belleville spring Leaf spring Spiral spring Helical spring

Helical springs can be again classified into  

Open coil helical spring Closed coil helical spring

14. State any two major functions of a spring.  

To absorb the shock energy To measure forces in spring balance and engine indicators

15. Define pitch? Pitch of the spring is defined as the axial distance between the adjacent coils in uncompressed state. Mathematically it can be calculated as Pitch = (length/ (n-1)) where, n is the number of turns available in the coil. 16. What is spring index (C)? The ratio of pitch or mean diameter to the diameter of wire for the spring is called the spring index.



17. What is solid length? The length of a spring under its maximum compression is called its solid length. It is the product of total number of coils and the diameter of wire. It is usually denoted by the symbol L s. 18. Define free length. Free length of the spring is the length of the spring when it is free or unloaded condition. It is equal to the solid length plus the maximum deflection or compression plus clash allowance. Lf = solid length + Y max + 0.15 Ymax 19. Define stiffness of spring or spring rate. The spring stiffness or spring constant is defined as the load required per unit deflection of the spring 20. Define helical springs. The helical springs are made up of a wire coiled in the form of a helix and are primarily intended for compressive or tensile load. Closed coil springs are meant for taking tensile load (springs balance) and the other one is for taking compressive load (Shock observer). 21. What are the differences between closed coil & open coil helical springs?

Closed coil helical spring

Open coil helical spring

Meant for tensile load

Meant for compressive load

The spring wires are coiled very closely, each turn is nearly at right angles to the axis of helix

The wires are coiled such that there is a gap between the two consecutive turns.

Helix angle is less than 10

o

o

Helix angle is large (>10 )

22. What are the various stresses induced in the open coil helical spring?   

Torsional shear stress Direct shear stress Stress arises due to curvature



23. What is buckling of springs? The helical compression spring behaves like a column and buckles at a comparative small load when the length of the spring is more than 4 times the mean coil diameter 24. What is buckling of springs? The helical compression spring behaves like a column and buckles at a comparative small load when the length of the spring is more than 4 times the mean coil diameter. 25. What is surge in springs? The material is subjected to higher stresses, which may cause early fatigue failure. This effect is called as spring surge. 26. Define active turns. Active turns of the spring are defined as the number of turns, which impart spring action while loaded. As load increases the no of active coils decreases. 27. Define inactive turns. An inactive turn of the spring is defined as the number of turns which does not contribute to the spring action while loaded. As load increases number of inactive coils increases from 0.5 to 1 turn



CHAPTER 7 DEFLECTION OF BEAMS 1. Why deflection of beams is needed for engineering applications like mechanical engineering? The spindle of a lathe or drill press and the arbor of a milling machine carry cutting tools for machining metals. Therefore the deflection of the spindle would have an adverse effect on the accuracy of the machine output. The manner of loading and support of these machine elements behave like that of a real beam. This is the reason why deflection of beams is necessary for engineering applications like mechanical engineering. 2. Name the various methods of determining slope and deflection of beams.    

Double – Integration method Macaulay’s method Moment – Area method Conjugate Beam method.

3. Describe the boundary conditions that can be used for finding out the values of the constants of integration in case of common type of beams. Support Fixed end Free end Roller (i.e. pinned or hinged)

Deflection Zero Yes Zero

Slope Zero Yes Zero

Moment Yes Zero Zero

4. What do you mean by flexural rigidity? Flexural rigidity is defined as the product of Young’s Modulus and the moment of Inertia (I) of the section 5. Define the term slope. Slope is defined as the rotation of the beam axis from its original position. 6. Define deflection. The displacement of a particular point located in the longitudinal axis of the beam in the vertical direction is called deflection. Deflection may be either upward or downward depending upon the direction of the load which is acting on the beam



7. Write down the moment – curvature relationship?

EI

d2y dx

2

M

where M is the bending moment, EI is the flexural rigidity and ‘y’ is the deflection of the beam. 8. Explain the procedure of finding the slope and deflection of a beam using Macaulay’s method?      

Find the reaction at the supports Take a section at a distance ‘x’ from the left support such that it covers all the loads in the beam. Form the moment – curvature expression that relates the bending moment Integrate the moment curvature expression twice to obtain the expressions for slope and deflection. Apply the boundary conditions and the find the constants involved in the moment – curvature expression. Find the slope and deflection at various points by substituting the value for ‘x’.

9. List out the relationship exists between slope, deflection, bending moment and the load. Slope 

dy dx

Bending Moment

Shear Force 

Load 

 EI

2

d y dx 2

d3y

dx 3 d4y

dx

4

10. Write down the two Moment – Area theorems? The angle between the tangents at two points A and B of a deflection curve is equal to the area of the M/EI diagram between A and B. The displacement of B from the tangent at A is equal to the moment of the M/EI diagram between A and B about the point B.



11. State the principle involved in finding the slope and deflection of beams using Moment-Area theorem. Moment – Area method uses the elastic curve equation or moment curvature expression, but the integration is carried out by doing so, the kinematic boundary conditions are not considered. 12. What is conjugate beam? Conjugate beam is a fictitious beam which has the same length as the real beam, but supported in such a manner that when it is loaded with M/EI diagram of the real beam, the shear and bending moment at a section in the conjugate beam give the slope and deflection at the corresponding section of a real beam. 13. Explain, how the load is applied in Conjugate beam method and its applicability to different types of beam? In conjugate beam method, the beam is loaded with elastic weight M/EI corresponding to the actual load. For cantilever beams, fixed beams and continuous beams, if this method is applied, the fixed ends behave like that it is subjected to rotations and translations. Hence for this type of beams some artificial restraints have to be applied to the conjugate beam, so that it is supported in a manner consistent with the constraints of the real beam. 14. Why deflection of beams is needed for engineering applications like mechanical engineering? The spindle of a lathe or drill press and the arbor of a milling machine carry cutting tools for machining metals. Therefore the deflection of the spindle would have an adverse effect on the accuracy of the machine output. The manner of loading and support of these machine elements behave like that of a real beam. This is the reason why deflection of beams is necessary for engineering applications like mechanical engineering. 15. Give the conjugate beam for the cantilever shown below.



CHAPTER 8 UNSYMMETRICAL BENDING

1. Define unsymmetrical bending? Unsymmetrical bending is defined as the bending caused by the loads that are inclined to the principal planes of bending. Note: 



For the analysis of unsymmetrical bending, the applied forces must be resolved at the shear centre parallel to the principal axes of the crosssection. In unsymmetrical sections, the neutral axis does not pass through the geometrical centre of the section.

Example: purlin of a roof truss. 2. Define Shear Centre? Shear centre is defined as the point of intersection of the bending axis and the plane of the transverse section. (or) Shear centre of a section can be defined as the point about which the applied forces is balanced by the set of shear forces obtained by summing the shear stresses over the section. (or) Note:   

Shear centre is also known as centre of twist. In case of unsymmetrical section the shear centre does not coincide with the centroid of the given section. When the load passes through the shear centre then there will be only bending and no twisting will be there.

3. What are the two reasons for unsymmetrical bending?  

The section is symmetrical (rectangular, circular, I-section) but the load line is inclined to both the principal axes. The section itself is unsymmetrical (angle section, channel section) and the load line is along the centriodal axis.



4. What do you mean by stress concentration? In real world problems of practical importance, when there is an abrupt change in the cross-section such as at the roots of threads of a bolt, at a section of a beam (or) plate having a hole, the stresses are no longer uniformly distributed. The stresses will be higher in magnitude, usually occur at the discontinuities. Such stresses are called stress concentration. Note:  

It is often referred as “Localized Stresses” In theory of elasticity, it is called “St. Venant’s principal” and according to this principle the stress will be higher near the vicinity of the hole or discontinuities or the point of application of the load and it will diminish or vanish as distance from the hole increases.

5. What is stress concentration factor (k)? The ratio of average nominal stress ( max) to the maximum stress ( n) is called stress concentration factor. It is generally denoted by the symbol ‘k’. 6. For unsymmetrical bending. Write the equation to find the inclination of Neutral axis if the load is acting in a plane inclined at an angle of ‘’ to vertical. tan  = - ( Ivv / Iuu) tan  Note:    

The equation of the Neutral Axis (N.A.) can be found by finding the locus of the points about which the resultant stress is zero. The maximum stress will occur at a point which is at the greatest distance from the neutral axis. All the points of the section on one side of the N.A. will carry stresses of the same nature and on the other side of its axis, of opposite sign. In addition to bending stress if there is any direct stress, the N.A. will be a straight line but it will not pass through centre of Gravity ‘G’. Then for finding the equation of the N.A. the resultant stress which is the algebraic sum of direct and bending stresses will be equated to zero.

7. What do you mean by Curved beams? Curved beams are the structural members or beams which are having some initial curvature. Note: 

A simple flexure formula may be used for curved beams for which the radius of curvature is more than five times the beam depth.





For deeply curved beams, the N.A. and the centroidal axes do not coincide and hence the simple bending formula is not applicable. That’s why the Winkler-Bach formula is widely used.

8. What are the basic assumptions in the derivation of bending stress equation of Winkler-Bach formula?    

Plane transverse sections before bending remain plane after bending Limit of proportionality is not exceeded. Radial strain is negligible. The material considered is isotropic and obe ys Hooke’s Law.

9. Define Fatigue? A phenomenon loading to fracture under repeated (or) fluctuating cyclic stresses below the tensile strength of the material is called Fatigue. Note: Fatigue fractures are progressive starting as minute crack developing under the action of fluctuating stresses. 10. What is Fatigue life? The number of cycles of stress that can be sustained prior to failure of a specified nature for a stated stress condition. 11. What is Fatigue or Endurance Limit? The maximum stress below which a material can presumably endure an infinite number of stress cycles. If the stress is not completely reversed, the value of the mean stress or the maximum stress or the stress ratio, should be mentioned. 12. What is Fatigue or endurance ration? The ratio of the fatigue limit to the tensile strength is defined as endurance limit. 13. What is Fatigue Strength? The limiting stress below which a material will withstand a specified number of cycles of stress without fracture. 14. What is Overstressing? The damage to fatigue properties of a material by cycling for a time at a stress above the fatigue limit.



15. What is coaxing? (or) under-stressing? An improvement in the fatigue behavior of a material as a result of initial cyclic stressing below the normal fatigue limit and then subjected gradually to the designed stress range. Apparently due to strain aging of effects it may double the number of reversals to failure in the case of steels, above their normal fatigue limit. 16. What is Corrosion Fatigue? The Fatigue caused (or) aggravated by corrosion. 17. What is cumulative damage law (or) linear damage (or) Miner’s law? In a fatigue test employing various stress ranges and frequencies sometimes with rest periods between successive application of stress on a single specimen, failure occurs when the sum of damage ratios (cycle ratios) attains unity, that is

 n/N = 1 where, n = number of cycles at stress ‘’ actually performed N = average number of cycles to failure at the stress ‘’. 18. What is meant by Laminated Spring or Leaf Spring? Laminated Spring or Leaf Spring is defined as the “a structural system which is made of a number of plates which are attached one over the other”. Laminated springs are supported at the middle instead of ends. In Laminated beams the load is subjected at the ends of the beam. Note: Laminated springs are fabricated with some initial curvature, so that the central deflection will disappear when the spring is loaded with its full load carrying capacity. 19. What is cycle ratio? The ratio of the number of cycles (n) of stress applied to a specimen (or structure), to the number of cycles to failure (N) at the same stress, that is n/N



20. What is Damage ratio? The ratio of the difference between the fatigue life ‘N’ of the material at a particular stress stepped-off from -N curve and the number of reversals (n) actually endured, to the fatigue life (N) is called damage ratio. Therefore, Damage Ratio = (N-n)/N 21. What are the different types of Fatigue Stress?     

Direct Stress Plane bending Rotating bending Torsion Combined Stress



CHAPTER 9 PRESSURE VESSELS

1. What do you mean a pressure vessel? Pressure vessel is usually a spherical or cylindrical container intended for the storage of liquids and gases under high internal pressure. 2. What are the types of stresses induced in a pressure vessel due to its internal pressure?  

Longitudinal stress i.e. stress acting in the direction of longitudinal axis of the pressure vessel Hoop stress (Circumferential stress or tangential stress) i.e. the stress developed in the circumferential or radial direction

3. What are major classifications of a pressure vessel? Pressure vessels are classified into  

Thin walled pressure vessels Thick walled pressure vessels

If the mean radius (average of outer and inner radius) to the thickness of the pressure vessel is greater than or equal to 10, it is called thin walled pressure vessels otherwise it is called thick walled pressure vessels. 4. Explain the variation of stress over the thickness of wall of a thin walled pressure vessel. In case of thin walled pressure vessel, the thickness of the wall is very small compared to the radius of the vessel. Also there is no variation of stress and it is only a constant. 5. What are the general shapes of pressure vessels in practice?  

Cylindrical Spherical


Strength of Materials - 2Marks

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