115983736 Solid Mechanics Short Questions and Answers

AE 2203

SOLID MECHANICS [FOR THIRD SEMESTER B.E AERONAUTICAL ENGINEERING STUDENTS]

COMPILED BY

BIBIN.C ASSISTANT PROFESSOR DEPARTMENT OF AERONAUTICAL ENGINEERING THE rAJAAS ENGINEERING COLLEGE (THE INDIAN ENGINEERING COLLEGE) VADAKKANGULAM - 627 116

AE 2203 - SOLID MECHANICS ANNA UNIVERSITY: CHENNAI SYLLABUS AE2203 SOLID MECHANICS

OBJECTIVE

To give brief descriptions on the behaviour of materials due to axial, bending and torsional and combined loads.

UNIT I BASICS AND AXIAL LOADING

10+3

Stress and Strain – Hooke’s Law – Elastic constants and their relationship– Statically determinate cases - statically indeterminate cases –composite bar. Thermal Stresses – stresses due to freely falling weight.

UNIT II STRESSES IN BEAMS

10+3

Shear force and bending moment diagrams for simply supported and cantilever beams- Bending stresses in straight beams-Shear stresses in bending of beams with rectangular, I & T etc cross sections-beams of uniform strength

UNIT III DEFLECTION OF BEAMS

10+3

Double integration method – McCauley’s method - Area moment method – Conjugate beam method-Principle of super position-Castigliano’s theorem and its application

COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS UNIT IV TORSION

5+3

Torsion of circular shafts - shear stresses and twist in solid and hollow circular shafts – closely coiled helical springs.

UNIT V BI AXIAL STRESSES

10+3

Stresses in thin circular cylinder and spherical shell under internal pressure – volumetric Strain. Combined loading – Principal Stresses and maximum Shear Stresses Analytical and Graphical methods. TOTAL: 60 PERIODS TEXT BOOKS

1. Nash William – “Strength of Materials”, TMH, 1998 2. Timoshenko.S. and Young D.H. – “Elements of strength materials Vol. I and Vol. II”., T. Van Nostrand Co-Inc Princeton-N.J. 1990.

REFERENCES

1. Dym C.L. and Shames I.H. – “Solid Mechanics”, 1990.


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AE 2203 - SOLID MECHANICS UNIT I - BASICS AND AXIAL LOADING

1. Strength of material When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. This resistance by which material of the body opposes the deformation is known as strength of material.

2. Define solid mechanics Solid mechanics is the science which deals with the behaviour of solids at rest or in motion under the action of external forces.

3. Ductility Ductility is the property of the material by virtue of which it undergoes a great amount of deformation before rupture.

4. Ductile material A material which undergoes a great amount of deformation before rupture is called ductile material. E.g. Mild Steel

5. Brittleness Brittleness is the property of the material by virtue of which it can undergoes a little amount of plastic deformation before rupture.

6. Brittle material A material which undergoes a very little amount of plastic deformation before rupture is called brittle material. E.g. Cast Iron


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AE 2203 - SOLID MECHANICS 7. Elastic material The material which is capable of recovering original size and shape on removal of load

8. Elastic action. When the material is loaded, it will result in deformation. When the load is removed, the deformation will be disappeared. This behaviour of the material is known as elastic action.

9. Plastic material The material which is not capable of recovering original size and shape on removal of load.

10. Toughness The strain energy required per unit volume to rupture is called toughness of the material. Higher Toughness - Ductile material Lower Toughness - Brittle material Impact test is a measure of toughness.

11. Plastic action When the material is loaded, it will result in deformation. When the load is removed, the plastic deformation will remain in the material. This behaviour of the material is known as plastic action.

12. Malleability It is a property of the metals and alloys by virtue of which it deforms plastically under compression without rupture.


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AE 2203 - SOLID MECHANICS 13. Stress:When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude of the resisting force in numerically equal to the applied force. This internal resisting force per unit area is called stress. Mathematically stress may be defined as the force per unit area. Stress 

Force Area

F A Where   Stress  N / m 2



P  LoadorForce  N

A  Area  m 2 It uses original cross section area of the specimen and also known as engineering stress or conventional stress.

14. Strain:When a body is subjected to an external force, there is some change of dimension in the body. Numerically the strain is equal to the ratio of change in length to the original length of the body Strain = e

Where

Changeinle ngth OriginalLength

l L

e=Strain

l =change in length L=Original length

15. Simple Stress:When a body is subjected to an external force in one direction only, the stress developed in the body is called simple stress. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 16. Compound Stress:When a body is subjected to an external force in more than one direction the stress developed in the body is called compound stress

17. Types of Simple Stress:There are mainly three type of stresses a. Tensile stress b. Compressive stress c. Shear stress 18. Types of strain:There are mainly three types of strains. a. Tensile strain b. Compressive strain c. Shear strain 19. Tensile stress:When a member is subjected to equal and opposite axial pulls as shown in fig. the length of the member is increased. The stress induced at any cross section of the member is called tensile stress.

Tensile stress  =

Tensileloa d Area

20. Tensile Strain:The ratio of increase in length to the original length is known as tensile strain. e=

increasein length l = L originalle ngth


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AE 2203 - SOLID MECHANICS 21. Compressive Stress:When a member is subjected to equal and opposite axial pushes as shown in fig, the length of the member is shortened. The stress induced at any cross section of the member is called Compressive stress Compressive stress  =



compressiveload Area P A

22. Compressive Strain:The ratio of decrease in length to original length is known as compressive strain. Compressive strain e = e=

Decreasein glength Orginallen gth

l L

23. Shear Stress and shear Strain. The two equal and opposite force act tangentially on any cross sectional plane of the body tending to slide one part of the body over the other part as shown in fig. The stress induced is called Shear stress and the corresponding strain is known as Shear strain


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AE 2203 - SOLID MECHANICS 24. Volumetric Strain:Volumetric strain is defined as the ratio of change in volume to the original volume of the body. Volumetric Strain (e) = (ev) =

Changeinvolume Originalvolume

v v

25. True stress The true stress is defined as the ratio of the load to the cross section area at any instant.

Where σ and ε is the engineering stress and engineering strain respectively.

26. Hooke’s Law:It states that when a material is loaded, within its elastic limit, the stress is directly proportional to the strain ie. Stress  strain (within its elastic limit)

 e   Ee E=

 e

Where E = Young’s modulus (or) modulus of Elasticity.

27. Poisson’s Ratio:When a body is stressed, within its elastic limit, the ration of lateral strain to the longitudinal strain is constant for a given material. Possion’s ratio (H or 1 et  m el

lateralstr ain 1 )= longitudin alstran m Position’s ratio of material cannot be more than 0.5


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AE 2203 - SOLID MECHANICS 28. Longitudinal Strain:The ratio of axial deformation the original length of the body is known as longitudinal (or) Linear strain

l L

Longitudinal strain el =

l - Change in length L – Original length

29. Lateral Strain:Consider a circular bar of length (L) and diameter (d) subjected to an axial load P. The length of the bar will increase while the diameter of the bar will decrease Where c  increase in length

d  Decrease in diameter Longitudinal strain (el) = Lateral strain (et) =

c c

d d

In case of rectangular bar, Lateral Strain (ec) =

t b = t b

Where b  decrease in breadth

t  decrease in thickness.

30. Young’s modules or modules of elasticity:The ratio of tensile stress to the corresponding tensile strain is constant within its elastic limit. The ratio is known as Young’s modules. Young’s Modulus (E) = E=

Tensilestr ess Tensilestr ess = Tensilestr ain longitudin alstrain

 el


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AE 2203 - SOLID MECHANICS 31. Bulk Modulus:- (K) The ratio of direct stress to the corresponding volumetric strain is constant within its elastic limit. This ratio is known as Bulk modulus. Bulk Modulus (K) =

K=

Directstress Volumetric strain

  = ev dv

( ev 

dv ) v

v

Where   stress ev = Volumetric strain

32. Factor of safety:It is defined as the ratio of ultimate tensile stress to the permissible stress (working stress) Factor of safety =

Ultimatest ress Permissiblestress

33. Stiffness:The stiffness may be defined as an ability of a material to withstand high load without major deformation. S=

W



N m

34. Strength:When an external force acts on a body it undergoes deformation. At the same time the body resists deformation. This resistant by which material of the body opposes the deformation is known as strength of material.


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AE 2203 - SOLID MECHANICS 35. Deformation of body due to force acting on it:Consider a body subjected to tensile stress We know Stress (  ) = Strain (e) = e=

Load P = Area A

Changeinle ngth Originalle ngth

L L

We know from Hooke’s law E=

= E=

P Ae

e=

P AE

Stress  = Strain e

P

A e

Substitute e value in equation P L = AE L

Change in length ( L ) =

PL AE

36. Relationship between the elastic constants E, G, K


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AE 2203 - SOLID MECHANICS 37. Relationship between three modulus:-

1) Young’s Modulus (E) and Shear modulus (G) 1 ) m

E = 2 G (1+

1 = Poisson’s ratio m

Where

II) Young’s modulus (E) & Bulk modulus (K) E = 3K ( 1-

2 ) m

III) Young’s Modulus(E), Shear modulus(G) and Bulk modules(K) E=

9 KG 3K  G

38. Creep:When a member is subjected to a constant load over a long period of time it undergoes a slow permanent deformation and this is termed as “creep”. This is dependent on temperature. Usually at elevated temperatures creep is high.

39. Proportional limit: The highest stress at which stress is directly proportional to strain.

40. Elastic limit Elastic limit is the greatest stress the material can withstand without any measurable permanent strain after unloading. Elastic limit > proportional limit.

41. Yield strength Yield strength is the stress required to produce a small specific amount of deformation. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 42. Modulus of rigidity or shear modulus (G) It is defined as the ratio of shearing stress to the corresponding shearing strain is constant within its elastic limit and it is denoted by G or C or N. Modulus of rigidity (G) = G =

Shearingstress Shearingstrain

 

Where  - Shearing stress

 - Shearing strain

43. Stresses in bars of varying section Consider the following non-uniform cross sections of a member AB, BC and CD having cross. Sectional areas A1, A2 and A3 with length L1, L2 and L3 as shown in fig. Tensile Stress in portion AB = Elongation of AB L1 

PL1 A1E

Tensile stress in portion BC = Elongation of BC L2 

P Load = A1 Area

P A2

PL 2 A2 E

Tensile stress in portion CD = Elongation of CD L3 

P A3

PL3 A3E

Total elongation L = L1  L2   3

L =

PL1 PL 2 PL3   A1E A2 E A3E

L =

P  L1 L2 L3     E  A1 A2 A3 


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AE 2203 - SOLID MECHANICS 44. Rigid body:A rigid body consists of innumerable particles. If the distance between any two or its particles, remains constant. It is known as solid body. In actual practice, all the solid bodies are not perfectly rigid bodies. However they are regarded as such, since all the solid bodies behave more or less like rigid bodies.

45. Deformable Solids;A deformable solid body may be defined as a body which undergoes deformation due to the application of external force.

46. Principal of super position Sometimes a body is subjected to external axial forces not only at its ends, but also at some of its interior cross sections along the length of the body. In such case, the force are split-up, and their effects are considered on individual sections. The total deformation is equal to the algebraic sum of the deformation of the individual sections. The principle of finding out the result deformation is known as principle of super position. The change in length of such member given by

c 

P1 L1  P2 L2  P3 L3  .... AE

47. Volumetric Strain of a rectangular bar Volumetric strain of a rectangular bar subjected to an axial force (P) is given by ev

= =

dv v

L L

(1 

2 ) m


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AE 2203 - SOLID MECHANICS 48. What is principle of super position? The resultant deformation of the body is equal to the algebraic sum of the deformation of the individual section. Such principle is called as principle of super position.

49. Stress in Composite bar:A composite member is composed of two or more different materials, joined together in such a way that the system is elongated or compressed as a single unit. In such a case, the following two governing principles are to be followed. 1. Elongation or contraction of individual materials of a composite member are equal. So, the strains induced in those materials are also equal. II. The sum of loads carried by individual materials of a composite member is equal to the total load applied on the member. Total load P = Load carried by bar 1. + Load carried by bar 2 P = P1+P2

50. Elastic limit:-, elastic body & elasticity:When an external force acts on a body, the body tends to undergo some deformation. If the external force is removed and the body comes back to its origin shape and size, the body is known as elastic body. This property, by virtue of which certain materials return back to their original position after the removal of the external force, is called elasticity. The body will regain its previous shape and size only when the deformation causes by the external force, is within a certain a limit. Thus there is a limiting value of force up to and within 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. Elastic limit:- It is the least stress that will cause permanent deformation.


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AE 2203 - SOLID MECHANICS Elasticity:- The property of certain materials returning back to their original shape and dimension after removing the applied external force is known as elasticity.

51. Ultimate Stress:It is the maximum stress based upon original cross section, to which the material can be subjected to, in a simple tensile test.

52. Breaking stress:It is the stress at failure based upon original cross section, in a simple tensile test.

53. Allowable or Working stress:Allowable stress or working stress is the maximum stress calculated for the expected conditions of service, so that the member will have a proper margin of security against failure or damage.

54. Direct stress:The stress due to axial load in a plane which is at right angle to the line of action of force is called direct stress. It is tensile or compressive in nature.

55. Composite bar:A composite bar may be defined as a bar made of two or more materials joined together. The bars are joined in such a manner that the system extends or contracts equally as one unit, when subjected to tension or compression.

56. Modular ratio:The ratio of young’s modulus of two materials in a composite bar is called modular ratio.


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AE 2203 - SOLID MECHANICS 57. Resilience:Resilience is defined as the strain energy stored in the material within the elastic limit.

58. Proof Resilience:Proof resilience is defined as the maximum strain energy which can be stored in the material within the elastic limit.

59. Strain energy:The mechanical energy stored up in the stressed material when the stress is within the elastic limit is called strain energy. It is equal to the work done by the external force.

60. Define modulus of resilience It is the proof resilience of the material per unit volume.

61. load ratio:The ratio of load at failure to working load is called load ratio. Load ratio = load at failure/working load

62. Steady load:It is the load which does not change in magnitude and direction

63. Varying load:It is the load which changes continuously.

64. Shock load:- (Sudden load) It is the load applied or removed suddenly.


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AE 2203 - SOLID MECHANICS 65. Impact load:The impact load is applied to a member in such a way that the load falls freely through some height before striking the member.

66. What you mean by thermal stresses? If the 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.

67. Thermal stress:Thermal stress is defined as the stress induced in the material due to change in temperature

68. Thermal strain:Thermal strain is defined as the strain induced in the material due to thermal stress. 1.The rails are connected with each other using a finish plate with a gap between rails the gap permits the expansion in summer, otherwise thermal stress will be induced is rails. 2. In thermostat temperature controllers the bimetal strip will deflect in one side due to change in temperature which will control an electrical circuit.

69. Graph of lateral strain Vs Linear Strain:-


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AE 2203 - SOLID MECHANICS 70. Stress strain diagram for ductile material:

A-proportional limit

B-Elastic material

C - Upper yield point

D – Lower yield point

E- Ultimate strength F- Breaking point

71. Proportionality limit: The point up to which the stress-strain curve is a straight line. This maximum uniaxial stress up to which the stress and strain are proportional is called proportion limit

72. Elastic limit: For every material, a limiting value of stress is found up to and within which the resulting strain entirely disappears when the load is removed. The value of this stress is known as the elastic limit.

73. Permanent set : Beyond point C if the load is removed there will be plastic or permanent deformation remained in the material. It is called permanent set.


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AE 2203 - SOLID MECHANICS 74. Yield point: When a material is loaded beyond the elastic limit, the stress increases more quickly as the stress increased, up to point C. The ordinate of point C, at which there is a slight increase in strain without increase in stress is known as the yield point of the material.

75. Stress Strain Diagram for brittle material:

76. Stress-Strain Curve for Non-Ferrous Metals:--


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AE 2203 - SOLID MECHANICS 77. Stress-Strain Curve for Ferrous Metals:--


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AE 2203 - SOLID MECHANICS UNIT II - STRESSES IN BEAMS 1. Define: Beam BEAM is a structural member which is supported along the length and subjected to external loads acting transversely (i.e) perpendicular to the center line of the beam.

2. Simple Support: It restrains movement of the beam in only one direction, i.e. movement perpendicular to the base of the support. It is also known as Roller support.

3. Hinged support: It restrains movement of the beam in two directions i.e. movement perpendicular to the base of the support and movement parallel to the base of the support.

4. Fixed support: It restrains all the three possible movements of the beam. i.e. movement perpendicular to the base of the support and movement parallel to the base of the support and the rotation at the support.

5. What is mean by transverse loading on beam? If a load is acting on the beam which perpendicular to the central line of it then it is called transverse loading.

6. Statically Indeterminate beam A beam is said to be statically indeterminate if the total no. of unknown reactions are more than the no. of conditions of static equilibrium.


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AE 2203 - SOLID MECHANICS 7. Statically determinate beam A beam is said to be statically determinate if the total no. of unknown reactions are equal to the no. of conditions of static equilibrium. Commonly encountered statically determinate beams are, a) Cantilever Beam, b) Simply Supported Beam, c) Over-hanging Beam.

8. Types of loading These beams are usually subjected to the following types of loading; a) Point Load, b) Uniformly Distributed Load, c) Uniformly Varying Load, d) Concentrated Moment.

9. What is Cantilever beam? A beam whose one end free and the other end is fixed is called cantilever beam.

10. What is simply supported beam? A beam supported or resting free on the support at its both ends is called simply supported beam.

11. What is mean by over hanging beam? If one or both of the end portions are extended beyond the support then it is called over hanging beam.

12. Fixed Beam: A beam whose both the ends are fixed or built-in in the walls or in the columns, then that beam is known as the fixed beam. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 13. Continuous Beam: A beam which is supported on more than two supports that, it is called a continuous beam.

14. What is mean by concentrated loads? A load which is acting at a point is called point load.

15. What is uniformly distributed load (udl). If a load which is spread over a beam in such a manner that rate of loading ‘w’ is uniform throughout the length then it is called as udl.

16. Concentrated Moment ( Moment Acting At Any Point):If, at a point, a couple forms a moment, then that is called Concentrated Moment. It is expressed in Nm or kNm.

17. Define point of contra flexure? In which beam it occurs? It is the point where the B.M is zero after changing its sign from positive to negative or vice versa. It occurs in overhanging beam.

18. What is mean by positive or sagging BM? The BM is said to be positive if moment of the forces on the left side of beam is clockwise and on the right side of the beam is anti-clockwise. (or) The BM is said to be positive if the BM at that section is such that it tends to bend the beam to a curvature having concavity at the top.


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AE 2203 - SOLID MECHANICS 19. What is mean by negative or hogging BM? The BM is said to be negative if moment of the forces on the left side of beam is anti-clockwise and on the right side of the beam is clockwise. (or) The BM is said to be negative if the BM at that section is such that it tends to bend the beam to a curvature having convexity at the top.

.

20. Define shear force and bending moment? SF at any cross section is defined as algebraic sum of the vertical forces acting either side of beam.

BM at any cross section is defined as algebraic sum of the moments of all the forces which are placed either side from that point.

21. Shear force Shear force is an unbalanced force, parallel to the cross-section, mostly vertical, but not always, either the right or left of the section.


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AE 2203 - SOLID MECHANICS 22. Procedure for finding the shear force Thus, the procedure to find out the shear force, at a section is to imagine a cut in the beam at the section, consider either to the left or the right portion and find the algebraic sum of all the forces normal to the axis.

23. Shear force diagram Shear force diagram is the graph showing the variation of the shear

force

throughout the length of the beam.

24. Bending moment Bending Moment is an unbalanced couple, either to the right or left of the section.

25. Procedure for finding the bending moment Thus, the procedure to find out the Bending Moment, at a section is to imagine a cut in the beam at the section, consider either the left or the right portion and find the algebraic sum of the moments due to all the forces.

26. Bending moment diagram Bending Moment diagram is the graph showing the variation of the bending moment throughout the length of the beam.

27. Shear Stresses. To resist the shear force, the element will develop the resisting stresses, Which is known as Shear Stresses.

28. Bending Stresses. To resist the Bending Moment, the element will develop the resisting stresses, Which is known as bending Stresses.


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AE 2203 - SOLID MECHANICS 29. When will bending moment is maximum? BM will be maximum when shear force change its sign.

30. What is maximum bending moment in a simply supported beam of span ‘L’ subjected to UDL of ‘w’ over entire span? Max BM =wL2/8

31. In a simply supported beam how will you locate point of maximum bending moment? The bending moment is max. when SF is zero. Writing SF equation at that point and equating to zero we can find out the distances ‘x’ from one end .then find maximum bending moment at that point by taking moment on right or left hand side of beam.

32. What is shear force and bending moment diagram? It shows the variation of the shear force and bending moment along the length of the beam.

33. What are the types of beams? 1. Cantilever beam 2. Simply supported beam 3. Fixed beam 4. Continuous beam 5. over hanging beam

34. What are the types of loads? 1. Concentrated load or point load 2. Uniform distributed load (udl) 3. Uniform varying load(uvl)


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AE 2203 - SOLID MECHANICS 35. Write the assumptions in the theory of simple bending?

1. The material of the beam is homogeneous and isotropic. 2. The beam material is stressed within the elastic limit and thus obey hooke’s law. 3. The transverse section which was plane before bending remains plains after bending also. 4. Each layer of the beam is free to expand or contract independently about the layer, above or below. 5. The value of E is the same in both compression and tension.

36. Write the theory of simple bending equation?

Where, M - Maximum bending moment I - Moment of inertia f - Maximum stress induced y- Distance from the neutral axis E - Young’s modulus R – Radius of neutral layer.

37. Define: Moment of resistance Due to pure bending, the layers above the N.A are subjected to compressive stresses, whereas the layers below the N.A are subjected to tensile stresses. Due to these stresses, the forces will be acting on the layers. These forces will have moment about the N.A. The total moment of these forces about the N.A for a section is known as moment of resistance of the section.


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AE 2203 - SOLID MECHANICS 38. Define: Neutral Axis The N.A of any transverse section is defined as the line of intersection of the neutral layer with the transverse section.

39. Define: Section modulus Section modulus is defined as the ratio of moment of inertia of a section about the N.A to the distance of the outermost layer from the N.A. Section modulus, Where, I – M.O.I about N.A ymax - Distance of the outermost layer from the N.A

40. What is mean by positive or sagging BM? BM is said to positive if moment on left side of beam is clockwise or right side of the beam is counter clockwise.

41. What is mean by negative or hogging BM? BM is said to negative if moment on left side of beam is counterclockwise or right side of the beam is clockwise.

42. Define shear force and bending moment? SF at any cross section is defined as algebraic sum of all the forces acting either side of beam. BM at any cross section is defined as algebraic sum of the moments of all the forces which are placed either side from that point.

43. Define point of contra flexure? In which beam it occurs? Point at which BM changes to zero is point of contra flexure. It occurs in overhanging beam.


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AE 2203 - SOLID MECHANICS 44. What is meant by transverse loading of beam? If load is acting on the beam which is perpendicular to center line of it is called transverse loading of beam.

45. When will bending moment is maximum? BM will be maximum when shear force change its sign.

46. What is maximum bending moment in a simply supported beam of span ‘L’ subjected to UDL of ‘w’ over entire span Max BM =wL2/8

47. In a simply supported beam how will you locate point of maximum bending moment? The bending moment is max. when SF is zero. Write SF equation at that point and equating to zero we can find out the distances ‘x’ from one end .then find maximum bending moment at that point by taking all moment on right or left hand side of beam.

48. What is shear force? The algebraic sum of the vertical forces at any section of the beam to the left or right of the section is called shear force.

49. What is shear force and bending moment diagram? It shows the variation of the shear force and bending moment along the length of the beam.

50.

In which point the bending moment is maximum? When the shear force change of sign or the shear force is zero


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AE 2203 - SOLID MECHANICS 51.

Write the assumption in the theory of simple bending? 1. The material of the beam is homogeneous and isotropic. 2. The beam material is stressed within the elastic limit and thus obey hooke’s law. 3. The transverse section which was plane before bending remains plains after bending also. 4. Each layer of the beam is free to expand or contract independently about the layer, above or below. 5. The value of E is the same in both compression and tension.

52.

Write the theory of simple bending equation?

M/ I = F/Y = E/R M - Maximum bending moment I - Moment of inertia F - Maximum stress induced Y - Distance from the neutral axis E - Young’s modulus R - Radius of curvature

53. State the main assumptions while deriving the general formula for shear stresses The material is homogeneous, isotropic and elastic The modulus of elasticity in tension and compression are same. The shear stress is constant along the beam width The presence of shear stress does not affect the distribution of bending stress.

54. What types of stresses are caused in a beam subjected to a constant shear force ? Vertical and horizontal shear stress COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 55. Define: Shear stress distribution Variation of shear stress along the depth of the beam is called shear stress distribution

56. What is the ratio of maximum shear stress to the average shear stress for the rectangular section? Qmax is 1.5 times the Qave.

57. What is the ratio of maximum shear stress to the average shear stress in the case of solid circular section? Qmax is 4/3 times the Qave.

58. What is the maximum value of shear stress for triangular section? Qmax=Fh2/12I h- Height F-load

59. What is the shear stress distribution value of Flange portion of the I-section? q= f/2I * (D2/4 - y) D-depth y- Distance from neutral axis

60. What is the value of maximum of minimum shear stress in a rectangular cross section? Qmax=3/2 * F/ (bd)

61. Define: Shear stress distribution The variation of shear stress along the depth of the beam is called shear stress distribution.


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AE 2203 - SOLID MECHANICS 62. What is the formula to find a shear stress at a fiber in a section of a beam? The shear stress at a fiber in a section of a beam is given by

W h e r e , F = shear force acting at a section A = Area of the section above the fiber ў = Distance of C G of the Area A from Neutral axis I = Moment of Inertia of whole section about N A b = Actual width at the fiber

63. What is the shear stress distribution rectangular section? The shear stress distribution in a rectangular section is parabolic and is given by

Where,

d - Depth of the beam y - Distance of the fiber from NA

64. State the main assumptions while deriving the general formula for shear stresses The material is homogeneous, isotropic and elastic The modulus of elasticity in tension and compression are same. The shear stress is constant along the beam width The presence of shear stress does not affect the distribution of bending stress.

65. What is the ratio of maximum shear stress to the average shear stress for the rectangular section? Qmax is 1.5 times the Qavg. 66. What is the ratio of maximum shear stress to the average shear stress in the case of solid circular section? Qmax is 4/3 times the Qavg. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 67. What is the shear stress distribution value of Flange portion of the I-section?

Where, D- depth y- Distance from neutral axis 68. Where the shear stress is max for Triangular section? In the case of triangular section, the shear stress is not max at N A. The shear stress is max at a height of h/2 69. Explain with example the statically indeterminate structures? The forces on the members of a structure cannot be determined by using conditions of equilibrium ∑X=0,∑Y=0,∑M=0, it is called statically indeterminate structures. Example: fixed beam, continuous beam.

70. Differentiate

the

statically

determinate

structures

and

statically

indeterminate structures?

S. No

Statically determinate

Statically indeterminate structures

structures 1

Conditions of equilibrium are

Conditions of equilibrium are

sufficient to analyze the

insufficient to analyze the structures.

structures. 2

3

Bending moment and shear force

Bending moment and shear force is

is independent of material and

dependent of material and

cross sectional area.

independent of cross sectional area.

No stresses are caused due to

Stresses are caused due to

temperature change and lack of fit.

temperature change and lack of fit.


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AE 2203 - SOLID MECHANICS 71. Define continuous beam? A continuous beam is one which is supported on more than two supports. For usual loading on the beam hogging (- ive) moments causing convexity upwards at the supports and sagging (+ ive) moments causing concavity upwards occur in mid span.

72. What are the advantages of continuous beam over simply supported beam? 1. The maximum bending moment in case of continuous beam is much less than in case of simply supported beam of same span carrying same loads. 2. In case of continuous beam, the average bending moment is lesser and hence lighter materials of construction can be used to resist the bending moment.

73. Define Flexural Rigidity of Beams? The product of young’s modulus (E) and moment of inertia (I) is called Flexural Rigidity of Beams. The unit is N mm2

74. Define fixed beam? A beam whose both ends are fixed is known as a fixed beam. Fixed beam is also called as beam. In case of fixed beam both its ends are rigidly fixed and the slope and deflection at the fixed ends are zero.

75. What are the advantages of fixed beam? i. For the same loading, the maximum deflection of a fixed beam is less than that of a simply supported beam. ii. For the same loading the fixed beam is subjected to lesser maximum bending moment. iii. The slope at both ends of a fixed beam is zero. iv. The beam is more stable and stronger.


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AE 2203 - SOLID MECHANICS 76. What is meant by propped cantilever? Propped cantilevers means cantilevers supported on a vertical support at a suitable point. The vertical support is known as prop.

77. What are disadvantages of a fixed beam? i.

Large stresses are setup by temperature changes.

ii.

Special care has to be taken in aligning supports accurately at the same level.

iii.

Large stresses are set if a little sinking of one support takes place.

iv.

Frequent fluctuations in load ingrender the degree of fixity at the ends very uncertain.

78. Define shear force diagram and bending moment diagram? A shear force diagram is one which shows the variation of the shear force along the length of the beam. And a bending moment diagram is one which shows the variation of the bending moment along the length of the beam.

79. What do you mean by point of contra flexure? At some point the bending moment is zero after changing its sign from positive to negative or vice – versa. That point is known as the point of contra flexure or point of inflexion.


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AE 2203 - SOLID MECHANICS UNIT III - DEFLECTION OF BEAMS 1. What are the methods for finding out the slope and deflection at a section? The important methods used for finding out the slope and deflection at a section in a loaded beam are 1. Double integration method 2. Moment area method 3. Macaulay’s method 4. Conjugate beam method

2. Why moment area method is more useful, when compared with double integration? Moment area method is more useful, as compared with double integration method because many problems which do not have a simple mathematical solution can be simplified by the moment area method.

3. Explain the Theorem for conjugate beam method? Theorem I : “The slope at any section of a loaded beam, relative to the original axis of the beam is equal to the shear in the conjugate beam at the corresponding section”

Theorem II: “The deflection at any given section of a loaded beam, relative to the original position is equal to the Bending moment at the corresponding section of the conjugate beam”

4. Define method of Singularity functions? In Macaulay’s method a single equation is formed for all loading on a beam, the equation is constructed in such a way that the constant of Integration apply to all portions of the beam. This method is also called method of singularity functions. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 5. What are the points to be worth for conjugate beam method? 1. This method can be directly used for simply supported Beam 2. In this method for cantilevers and fixed beams, artificial constraints need to be supplied to the conjugate beam so that it is supported in a manner consistent with the constraints of the real beam.

6. Define: Mohr’s Theorem for slope The change of slope between two points of a loaded beam is equal to the area of BMD between two points divided by EI. Slope,

7. Define: Mohr’s Theorem for deflection The deflection of a point with respect to tangent at second point is equal to the first moment of area of BMD between two points about the first point divided by EI. Slope,


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AE 2203 - SOLID MECHANICS UNIT IV - TORSION 1. Write down the expression for power transmitted by a shaft P=2πNT/60 Where, N-speed in rpm T-torque

2. Write down the expression for torque transmitted by hollow shaft T= (π/16)*Fs*((D4-d4)/d4 Where, T-torque q- Shear stress D-outer diameter d- Inner diameter

3. 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. T=π/16 * Fs*D3 where, T-torque q - Shear stress D – diameter

4. Define torsional rigidity The torque required to introduce unit angle of twist in unit length is called torsional rigidity or stiffness of shaft.

5. What is composite shaft? Sometimes a shaft is made up of composite section i.e. one type of shaft is sleeved over other types of shaft. At the time of sleeving, the two shafts are joined together, that the composite shaft behaves like a single shaft. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 6. What is a spring? A spring is an elastic member, which deflects, or distorts under the action of load and regains its original shape after the load is removed.

7. State any two functions of springs. 

To measure forces in spring balance, meters and engine indicators.



To store energy.

8. What are the various types of springs? i. Helical springs

ii. Spiral springs

iii. Leaf springs

iv. Disc spring or Belleville springs

9. Classify the helical springs. 1. Close – coiled or tension helical spring. 2. Open –coiled or compression helical spring.

10. What is spring index (C)? The ratio of mean or pitch diameter to the diameter of wire for the spring is called the spring index.

11. What is solid length? The length of a spring under the maximum compression is called its solid length. It is the product of total number of coils and the diameter of wire. Ls = nt x d Where, nt = total number of coils.

12. Define pitch. Pitch of the spring is defined as the axial distance between the adjacent coils in uncompressed state. Mathematically Pitch=free length n-1 COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 13. Define spring rate (stiffness). The spring stiffness or spring constant is defined as the load required per unit deflection of the spring. K= W/y Where , W - load y- Deflection

14. 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.

15. What are the differences between closed coil & open coil helical springs? Closed coil spring The spring wires are coiled very closely, each turn is nearly at right angles to the axis of helix . Helix angle is less (70 to 10o) Open coil spring The wires are coiled such that there is a gap between the two consecutive turns. Helix angle is large (>10o)

16. Write the assumptions in the theory of pure torsion. 1. The material is homogenous and isotropic. 2. The stresses are within elastic limit 3. C/S which are plane before applying twisting moment remain plane even after the application of twisting moment. 4. Radial lines remain radial even after applying torsional moment. 5. The twist along the shaft is uniform 17. Define : Polar Modulus Polar modulus is defined as the ratio of polar moment of inertia to extreme radial distance of the fibre from the centre. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 18. Write the equation for the polar modulus for solid circular section

19. Define Torsion When a pair of forces of equal magnitude but opposite directions acting on body, it tends to twist the body. It is known as twisting moment or torsional moment or simply as torque. Torque is equal to the product of the force applied and the distance between the point of application of the force and the axis of the shaft.

20. What are the assumptions made in Torsion equation a. The material of the shaft is homogeneous, perfectly elastic and obeys Hooke’s law. b. Twist is uniform along the length of the shaft c. The stress does not exceed the limit of proportionality d. The shaft circular in section remains circular after loading e. Strain and deformations are small.

21. Define polar modulus It is the ratio between polar moment of inertia and radius of the shaft.

£ = polar moment of inertia = J Radius

R

22. Write the polar modulus for solid shaft and circular shaft. £ = polar moment of inertia = J Radius

R J = π D4 32


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AE 2203 - SOLID MECHANICS 23. 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.

24. Write torsional equation T/J=Cθ/L=q/R T-Torque J- Polar moment of inertia C-Modulus of rigidity L- Length q- Shear stress R- Radius

25. Write down the expression for power transmitted by a shaft P=2πNT/60 N-speed in rpm T-torque

26. Write down the expression for torque transmitted by hollow shaft T= (π/16)*Fs*((D4-d4)/d4 T-torque q- Shear stress D-outer diameter D- inner diameter

27. Write the polar modulus for solid shaft and circular shaft It is ratio between polar moment of inertia and radius of shaft


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AE 2203 - SOLID MECHANICS 28. 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 shaft. T=π/16 * Fs*D3 T-torque q Shear stress D diameter

29. Define torsional rigidity Product of rigidity modulus and polar moment of inertia is called torsional rigidity

30. What is composite shaft? Sometimes a shaft is made up of composite section i.e. one type of shaft is sleeved over other types of shaft. At the time of sleeving, the two shaft are joined together, that the composite shaft behaves like a single shaft.

31. What is a spring? A spring is an elastic member, which deflects, or distorts under the action of load and regains its original shape after the load is removed.

32. State any two functions of springs. To measure forces in spring balance, meters and engine indicators. To store energy.

33. What are the various types of springs? i.

Helical springs

ii.

Spiral springs

iii.

Leaf springs

iv.

Disc spring or Belleville springs


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AE 2203 - SOLID MECHANICS 34. Classify the helical springs. 1. Close – coiled or tension helical spring. 2. Open –coiled or compression helical spring.

35. What is spring index (C)? The ratio of mean or pitch diameter to the diameter of wire for the spring is called the spring index.

36. What is solid length? The length of a spring under the maximum compression is called its solid length. It is the product of total number of coils and the diameter of wire. Ls = nt x d Where, nt = total number of coils.

37. 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 + Ymax + 0.15 Ymax

38. Define spring rate (stiffness). The spring stiffness or spring constant is defined as the load required per unit deflection of the spring. K= W/y Where

W -load Y – deflection

39. Define helical springs. The helical springs are made up of a wire coiled in the form of a helix and is primarily intended for compressive or tensile load COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 40. Define pitch. Pitch of the spring is defined as the axial distance between the adjacent coils in uncompressed state. Mathematically Pitch=free length n-1 41. What are the differences between closed coil & open coil helical springs? The spring wires are coiled very The wires are coiled such that closely, each turn is nearly at right there is a gap between the two angles to the axis of helix

consecutive turns.

Helix angle is less than 10o

Helix angle is large (>10o)

42. What are the stresses induced in the helical compression spring due to axial load? 1. Direct shear stress 2. Torsional shear stress 3. Effect of curvature

43. What is whal’s stress factor? C = 4C-1 + 0.615 4C-4

C

44. 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.

45. 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.


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AE 2203 - SOLID MECHANICS 46. 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.

47. 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.

48. What are the different kinds of end connections for compression helical springs? The different kinds of end connection for compression helical springs are Plain ends Ground ends Squared ends Ground & square ends


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AE 2203 - SOLID MECHANICS UNIT V - BI AXIAL STRESSES

1. . Define thin cylinder? If the thickness of the wall of the cylinder vessel is less than 1/15 to 1/20 of its internal diameter, the cylinder vessel is known as thin cylinder.

2. What are types of stress in a thin cylindrical vessel subjected to internal pressure? These stresses are tensile and are known as 

Circumferential stress (or hoop stress )



Longitudinal stress

. 3. What is mean by circumferential stress (or hoop stress) and longitudinal stress? The stress acting along the circumference of the cylinder is called circumferential stress (or hoop stress) whereas the stress acting along the length of the cylinder is known as longitudinal stress.

4. What are the formula for finding circumferential stress and longitudinal stress? Circumferential stress, f1 = pd / 2t longitudinal stress, f2

= pd / 4t

5. What are maximum shear stresses at any point in a cylinder? Maximum shear stresses at any point in a cylinder, subjected to internal fluid pressure is given by (f1 –f2) / 2 = pd / 8t


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AE 2203 - SOLID MECHANICS 6. What are the formula for finding circumferential strain and longitudinal strain? The circumferential strain (e1) and longitudinal strain (e2) are given by

7. What are the formula for finding change in diameter, change in length and change volume of a cylindrical shell subjected to internal fluid pressure p?

8. Define principle stresses and principle plane.

Principle stress: The magnitude of normal stress, acting on a principal plane is known as principal stresses. The intensity of stress on the Principal Planes are known as “ Principal Stresses ”

Principle plane: The planes which have no shear stress are known as principal planes. The planes which carry only Direct Stresses and no Tangential Stresses are called “ Principal Planes ” COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 9. What is the radius of Mohr’s circle? Radius of Mohr’s circle is equal to the maximum shear stress.

10. What is the use of Mohr’s circle? To find out the normal, resultant and principle stresses and their planes.

11. List the methods to find the stresses in oblique plane? 1. Analytical method 2. Graphical method

12. Define thick cylinders Thick cylinders are vessels, containing fluid under pressure and whose wall thickness is not small (t≥d/20)

13. What are the assumptions followed in Lame’s equation 1. The material of the column is homogenous. 2. Plane section is perpendicular to the longitudinal axis of the cylinder remain plane after the application of internal pressure. 3. The material is stressed within the limit. 4. All the fibers of the material are free to expand or contract independent without being constrained by the adjacent fibers.

14. State the variation of hoop’s stress in a thick cylinder The hoop’s stress is maximum at the inner circumference and minimum at the outer circumference of a thick cylinder.

15. How can you reduce hoop’s stress in a thick cylinder? The hoop’s stress in a thick cylinder can be reduced by shrinking one cylinder over another cylinder.


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AE 2203 - SOLID MECHANICS 16. What is middle third rule? For a rectangular section b/6 , h/6 is from the centre with four points a safe zone for loading is obtained. This section is a rhombus of a diamond shape is known as core or kernel of the section. So , for loading and no tension will occur in the section the size rhombus is b/3 , h/3. This is called as middle third rule of the safe zone.

17. Distinguish b/w thick and thin cylinders?

Thick cylinder

Thin cylinder

d/t<20

d/t>20

Radial stress is important.

Radial stress is negligible.

18. Define compound cylinder? To increase the pressure bearing capacity of the cylinder and to reduce the hoop stress across the thickness of the cylinder, two cylinders are combined together in one cylinder as shrink fitted over one another cylinder. This type of arrangement is called compound cylinder.

19. 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 1/20.

20. In a thin cylinder will the radial stress vary over the thickness of wall? No, in thin 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 cylinder.


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AE 2203 - SOLID MECHANICS 21. What are the types of stresses setup in the thin cylinders? 1. Circumferential stresses (or) hoop stresses 2. Longitudinal stresses

22. Distinguish between cylindrical shell and spherical shell.

Cylindrical shell

Spherical shell

1. Circumferential stress is twice the 1. Only hoop stress presents. longitudinal stress. 2.

It withstands low pressure than 2. It withstands more pressure than

spherical shell for the same diameter.

cylindrical shell for the same diameter.

23. What is the effect of riveting a thin cylindrical shell? 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.

24. In thin spherical shell, volumetric strain is -------- times the circumferential strain. Three.

25. What do you understand by the 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 sometimes pre stressed by winding with a steel wire under tension.

26. Define – hoop stress? The stress is acting in the circumference of the cylinder wall (or) the stresses induced perpendicular to the axis of cylinder.


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AE 2203 - SOLID MECHANICS 27. Define- longitudinal stress? The stress is acting along the length of the cylinder is called longitudinal stress.

28. A thin cylinder of diameter d is subjected to internal pressure p . Write down the expression for hoop stress and longitudinal stress. Hoop stress σh=pd/2t Longitudinal stress σl=pd/4t p- Pressure (gauge) d- Diameter t- Thickness

29. What is the radius of Mohr’s circle? Radius of Mohr’s circle is equal to the maximum shear stress.

30. What is the use of Mohr’s circle? To find out the normal, resultant stresses and principle stress and their planes.

31. List the methods to find the stresses in oblique plane? 1. Analytical method 2. Graphical method

32. Derive an expression for the longitudinal stress in a thin cylinder subjected to a uniform internal fluid pressure. Force due to fluid pressure

= p x П/4 xd2

Force due to longitudinal stress = f2 x Пd x t p x П/4 xd2 = f2 x Пd x t f2 = pd/4t


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AE 2203 - SOLID MECHANICS 33. A bar of cross sectional area 600 mm^2 is subjected to a tensile load of 50 KN 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° Stress, σ = Load/Area = 50*102/600 = 83.33 N/mm2 Normal stress, σn = σ cos2θ = 83.33*cos230° = 62.5 N/mm2

34. In case of equal like principle stresses, what is the diameter of the Mohr’s circle? Answer: Zero


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AE 2203 - SOLID MECHANICS IMPORTANT 2 MARKS AND 16 MARKS QUESTIONS PART A 1. Explain stress and strain. Differentiate between strain and elongation. 2. What is the difference between shearing and tearing? 3. State and explain Hooke’s law. 4. Explain proportional limit and elastic limit. 5. Differentiate between elasticity modulus and rigidity modulus. 6. What is poison’s ratio? 7. Explain working stress and factor of safety. 8. Establish a relation between longitudinal strain and volumetric strain. 9. Explain simple shear and complementary shear. 10. Explain gradual, sudden, impact and shock loading. 11. Explain the various points on the stress-stain curve of an elastic material. 12. What is the difference between shrinking on and press fit? 13. What is the difference between uniformly distributed load and uniformly varying load? 14. Find the torque which a shaft of 250mm can safely transmit, if the shear stress is not exceed 46N/ mm2. 15. State one moment area theorem. 16. Prove that the maximum Bending Moment occurs where shear force is either zero or change in sign. 17. A cantilever projecting 2.5 m from a wall is loaded with UDL of total load 80 kN. Determine the M.I. of the beam section, if the deflection of the beam at free end be 10mm. E= 205kN/mm2. 18. Write two relations related to Elastic constants. 19. Explain the term polar modulus. 20. Write significance of Mohr’s circle. 21. Define principal planes and principal stresses. 22. Derive a relation for volumetric strain of a body subjected to a uniaxial stress. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 23. Draw stress-strain curve for mild steel in tension. 24. What do you mean by beam of uniform stress? 25. Define ‘torsional stiffness’. 26. In a line sketch of an open coiled helical spring, mark the angle of helix ‘α’. 27. Define ‘principal planes’. 28. Draw conjugate beam for a simply supported beam with central point load ‘W’. 29. What is the length of a uniformly loaded (loaded to full length) cantilever if the deflection and slope at the free end are 25 mm and 0.01 radians respectively? 30. State one moment area theorem.

PART B 1. A bar of length 300mm is 50 mm square for 120mm of it’s length, 25 mm diameter for 80 mm length and 40 mm diameter for remaining length. If a tensile force of 100 kN is applied to the bar, calculate the maximum and minimum stresses produced and the total elongation. 2. The Modulus of Elasticity of a round bar is 110 GPa and shear Modulus is 45 GPa. Find the Bulk modulus and lateral contraction of the bar 40 mm diameter and 3 m long when stretched by 3mm. 3. A reinforced concrete column 300 x 300mm has four reinforcing steel bars of 25mm diameter in each corner. Find the safe axial load on the column when the concrete is subjected to a stress of 5 N/mm2 . What is the corresponding stress in steel ? Take Es / Ec = 18. 4. A metallic bar 250mm x 100mm x 50mm is subjected to loads along X,Y and Z directions. The load along X direction is 400 kN (Tension) and act over the face 100mm x 50mm of the bar. The load acting along Y direction is 4000 kN (Compression) and act over the face 250mm x 100mm of the bar and a load of 2000 kN (Tension) act along Z direction over the face 250mm x 50mm. Modulus of Elasticity is 200 GPa and

γ = 0.25 . Find the change in volume. Also find the change

that should be considered for the load in Y direction so that change in volume is zero. COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC

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AE 2203 - SOLID MECHANICS 5. A rectangular bar made up of steel is 3m long and 15mm thick. The rod is subjected to axial load of 40 kN. The width of the rod varies from 75mm at one end to 30mm at other end. Find the extension of the bar if E = 200GPa. 6. Steel plate of 20mm thickness tapers uniformly from 100mm to 50mm in a length of 400mm. What is elongation of the plate, if an axial force of 80 kN acts on it? Take E = 200GPa. 7. A short hollow cast iron cylinder of wall thickness 10mm is to carry a compressive load of 600kN. Determine the outside diameter of the cylinder if the ultimate crushing stress for the material is 540 MN/m2. Take Factor Of Safety as 6. 8. A reinforced concrete column 40cm x 40 cm is reinforced with six steel rods of diameter 20mm. Calculate the safe load that the column can carry if the allowable stress in concrete is 4 MPa and Young’s Modulus for steel is 15 times that of concrete. If the column supports an axial load of 600 KN, what is the compressive stress in: a. Concrete. b. Steel. 9. A rectangular bar of cross section 30mm x 60mm and length 200mm is restrained from expansion along it’s 30mm x 200mm sides by surrounding material. Find the change in dimension and volume when a compressive force of 180 KN acts in axial direction. Take E = 200GPa and 10. A steel flat plate tapers uniformly 200mm to 100mm width in a length of 500mm and uniform thickness of 20mm. Determine the elongation of the plate, if it is subjected to an axial pull of 40 KN. Take Take E = 2 x 105 N/mm2. 11. The Modulus of Rigidity of a material is 4 x 104 N/mm2. A 10 mm diameter rod of thin material is subjected to an axial pull of 5 KN and the change in diameter is observed to be 0.002 mm. Calculate the Modulus of Elasticity and the Poission’s ratio of this material.


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AE 2203 - SOLID MECHANICS 12. A circular rod of 100mm diameter and 500mm long is subjected to a tensile force of 1000KN. Determine the Modulus of Rigidity, bulk modulus and change in volume if Poission ratio = 0.3 and Young’s Modulus = 200GPa. 13. Two vertical rods one of steel and the other of copper are rigidly fixed at the top and 50 cm apart. Diameters and lengths of each rod are 2 cm and 4 cm respectively. A cross bar fixed to the rods at the lower end carries a load of 5000 N such that the cross bar remains horizontal even after loading. Find the stress in each rod and the position of the load on the bar. Take E for steel = 2 x 105 N/mm2 and E for copper 1 x 105 N/mm2. 14. A bar of cross-section 8mm x 8mm is subjected to an axial pull of 6KN. The lateral dimension of the bar is changed to 7.9975mm x 7.9975mm. If the Modulus of Rigidity of the material is 9 x 105 N/mm2. Determine the Poission’s ratio and Modulus of Elasticity. 15. A wooden tie 3m long 75mm wide and 100mm thick is subjected to an axial pull of 4500 kg and the stretch is 4mm.Find the value of E for timber. 16. The rod of a hydraulic lifts 12m long and 4cm in diameter. It is attached to a plunger 11cm in diameter working under a pressure of 500kg/cm2.If E equals 2*106kg/cm2 find the change in length of the rod. 17. A tie bar 25mm diameter carries a load which causes a stress of 1200 kg/cm2.If it is attached to a cast iron bracket by means of 4 holes which can be stressed upto 900 kg/cm2, find the diameter of the bolts. 18. A steel punch can be worked to a compressive stress of 8 tons/cm2.Find the least diameter of the hole which can be punched through a steel plate of 12mm thickness if its ultimate shear strength is 3.2 tons/cm2. 19. A mild steel flat 12cm wide by 2cm thick and 6m long carries an axial pull of 30 tons.E =2000tons/cm2, 1/m = 0.26.Calculate the change in dimensions and volume. 20. A straight bar of steel 3m long has rectangular section which varies uniformly from 10cm x 12mm at one end to 25mm x 12mm at the other end . What is the change in length and a pull of 2300kg. E= 2*106 kg/cm2.


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AE 2203 - SOLID MECHANICS 21. A weight of 25 kg is dropped into a collar at the end of a vertical bar 1.8m long and 25mm dia from a height of 10 cm. Calculate the maximum instantaneous extension and stress produced in the section. E=2x106 kg/cm2. 22. A wrought iron bar 5cm dia has to transmit a shock energy of 8Kg-m. Calculate the maximum instantaneous stress and the elongation produced. Assume E=2x106 kg/cm2. 23. Find the stresses in steel for the following data: Reinforced concrete column size 30mmX300mm, steel bars 4 numbers of 28mm diameter. Es/Ec=18, σc=stress in concrete 5 N/mm2. Find also the safe axial load. 24. A straight rectangular bar 3 m long 12 mm thick tapers uniformly from 100 mm at one end to 25 mm at the other. Find the extension of the bar under a load of 25 kN. E=200 kN/mm2. 25. A girder 9m long is loaded with a UDL of 1.8 kN/m over a length of 4m from left end. Draw B.M and S.F diagrams for the girder and calculate the magnitude and position of the maximum B.M. 26. A straight rectangular bar 3 m long 12mm thick tapers uniformly from 100mm at one end to 25mm at the other. Find the extension of the bar under a load of 25kN. E = 200 kN/mm2. 27. A T-shaped cross-section of a beam is to a vertical shear force of 100 kN. Calculate the shear stress at the neutral axis and at the junction of the web and the flange. Moment of inertia about the horizontal neutral axis is 11340 cm4. 28. Obtain a relation for the slope and deflection at the free end of a cantilever beam AB of span ‘l’ and flexural rigidity EI when it is carrying a point load ‘W’ at free end. 29. Obtain a relation for the slope and deflection at the free end of a cantilever beam AB of span ‘l’ and flexural rigidity EI when it is carrying a uniformly distributed load ‘w’ over the entire length. 30. Derive the torsion relation making necessary assumptions. 31. Derive an expression for the stress on an oblique section of a rectangular body when it is subjected to direct stresses in two mutually perpendicular directions.


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AE 2203 - SOLID MECHANICS 32. Show that in the case of a thin cylindrical shell subjected to an internal fluid pressure the tendency to burst length wise is twice as great as a transverse section. 33. A hollow shaft (D = 440 mm, d= 200mm) is of length 12m. Find the maximum torque it can transmit if the angle of twist is not to exceed 1.50 for the full length. 34. (i) Derive a relation for elongation of a circular bar of uniformly tapering section subjected to an axial tensile load. 35. ii) The modulus of rigidity of a material is 4X104 MPa. A 10mm diameter rod of this material is subjected to an axial pull of 5 kN and the change in diameter is observed to be 0.002 mm. Calculate the modulus of elasticity and the Poisson’s ratio of this material. 36. (i) Derive a relation for change in length of a bar hanging freely under its own weight. (ii) A tapered bar, 100 mm diameter at one end and 200 mm diameter at the other, and 1000 mm long, is initially free of stress. If the temperature of the bar drops by 200C, determine the maximum stress in the bar, take E = 2X105 Mpa and α = 12.5X10-6/C. 37. An I section has top flange of 360mmX30mm thick, a bottom flange of 90mmX30mm thick, and a web of 30mm thickness and 360mm depth. The overall depth is 420mm. It has a vertical axis of symmetry. Calculate the maximum shear stress for a shear force of 100 kN. 38. Derive relations for slope at the supports and maximum deflection for a simply beam AB with a bending couple M of clockwise nature at A. Use moment area method. 39. A simply supported beam of span L is subjected to equal loads W/2 at each 1/3rd span points. Find the expressions for deflection under the load and at mid span. Use McCaulay’s Method.


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115983736 Solid Mechanics Short Questions and Answers

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