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

COMPILED BY BIBIN CHIDAMBARANATHAN, AP/AERO, TREC Page 2

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 Hookes 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 McCauleys method - Area moment method

Conjugate beam method-Principle of super position-Castiglianos theorem and its

application



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.



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



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.



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.

Area

ForceStress

Where 2/ mNStress

NeLoadorForcP

2mAreaA

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 = ngthOriginalLe

ngthChangeinle

L

le

Where 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.

A

F



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 =Area

dTensileloa

20. Tensile Strain:-

The ratio of increase in length to the original length is known as tensile

strain.

e = ngthoriginalle

lengthincreasein =

L

l



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 = Area

eloadcompressiv

A

P

22. Compressive Strain:-

The ratio of decrease in length to original length is known as compressive

strain.

Compressive strain e = gthOrginallen

glengthDecreasein

e = 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



24. Volumetric Strain:-

Volumetric strain is defined as the ratio of change in volume to the original

volume of the body.

Volumetric Strain (e) =lumeOriginalvo

lumeChangeinvo

(ev) = 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 and is the engineering stress and engineering strain respectively.

26. Hookes 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 = Youngs modulus (or) modulus of Elasticity.

27. Poissons 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.

Possions ratio (H or m

1) =

alstranlongitudin

ainlateralstr

el

et

m

1 Positions ratio of material cannot be more than 0.5



28. Longitudinal Strain:-

The ratio of axial deformation the original length of the body is known as

longitudinal (or) Linear strain

Longitudinal strain el =L

l

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) = c

c

Lateral strain (et) = d

d

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

b =

t

t

Where b decrease in breadth

t decrease in thickness.

30. Youngs 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 Youngs modules.

Youngs Modulus (E) = ainTensilestr

essTensilestr=

alstrainlongitudin

essTensilestr

E = el



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) = strainVolumetric

ssDirectstre

K = ev

=

vdv

(

v

dvev )

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 = estressPermissibl

ressUltimatest

33. Stiffness:-

The stiffness may be defined as an ability of a material to withstand high load

without major deformation.

S=

W

m

N

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.



35. Deformation of body due to force acting on it:-

Consider a body subjected to tensile stress

We know

Stress ( ) = Area

Load =

A

P

Strain (e) = ngthOriginalle

ngthChangeinle

e = L

L

We know from Hookes law

E = Strain

Stress =

e

= e

AP

E = Ae

P

e = AE

P

Substitute e value in equation

AE

P =

L

L

Change in length ( L ) = AE

PL

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



37. Relationship between three modulus:-

1) Youngs Modulus (E) and Shear modulus (G)

E = 2 G (1+m

1)

Where m

1 = Poissons ratio

II) Youngs modulus (E) & Bulk modulus (K)

E = 3K ( 1-m

2)

III) Youngs Modulus(E), Shear modulus(G) and Bulk modules(K)

E = GK

KG

3

9

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.



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) = rainShearingst

ressShearingst

G =

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 = Area

Load =

1A

P

Elongation of AB EA

PLL

1

11

Tensile stress in portion BC = 2A

P

Elongation of BC EA

PLL

2

22

Tensile stress in portion CD =3A

P

Elongation of CD EA

PLL

3

33

Total elongation L = 321 LL

L = EA

PL

EA

PL

EA

PL

3

3

2

2

1

1

L =

3

3

2

2

1

1

A

L

A

L

A

L

E

P



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

AE

LPLPLPc

....332211

47. Volumetric Strain of a rectangular bar

Volumetric strain of a rectangular bar subjected to an axial force (P) is given

by

ev = v

dv

= )2

1(mL

L



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.



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 youngs modulus of two materials in a composite bar is called

modular ratio.



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.



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:-



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.



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:--



77. Stress-Strain Curve for Ferrous Metals:--



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.



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.



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.



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.



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.



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)



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 hookes

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 - Youngs 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.



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.



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



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

hookes 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 - Youngs 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



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.



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

= 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.



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

structures

Statically indeterminate structures

1 Conditions of equilibrium are

sufficient to analyze the

structures.

Conditions of equilibrium are

insufficient to analyze the structures.

2 Bending moment and shear force

is independent of material and

cross sectional area.

Bending moment and shear force is

dependent of material and

independent of cross sectional area.

3 No stresses are caused due to

temperature change and lack of fit.

Stresses are caused due to

temperature change and lack of fit.



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 youngs 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.



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.



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. Macaulays 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 Macaulays 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.



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: Mohrs 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: Mohrs 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,



UNIT IV - TORSION

1. Write down the expression for power transmitted by a shaft

P=2NT/60

Where, N-speed in rpm

T-torque

2. Write down the expression for torque transmitted by hollow shaft

T= (/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=/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.



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



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.



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 Hookes

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 = D4

32



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=C/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=2NT/60

N-speed in rpm

T-torque

26. Write down the expression for torque transmitted by hollow shaft

T= (/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



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=/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



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



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?

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 whals 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.

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 10o Helix angle is large (>10o)



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



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



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



9. What is the radius of Mohrs circle?

Radius of Mohrs circle is equal to the maximum shear stress.

10. What is the use of Mohrs 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 (td/20)

13. What are the assumptions followed in Lames 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 hoops stress in a thick cylinder

The hoops stress is maximum at the inner circumference and minimum at

the outer circumference of a thick cylinder.

15. How can you reduce hoops stress in a thick cylinder?

The hoops stress in a thick cylinder can be reduced by shrinking one

cylinder over another cylinder.



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/t20

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.



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

longitudinal stress.

1. Only hoop stress presents.

2. It withstands low pressure than

spherical shell for the same diameter.

2. It withstands more pressure than

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.



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 h=pd/2t

Longitudinal stress l=pd/4t

p- Pressure (gauge)

d- Diameter

t- Thickness

29. What is the radius of Mohrs circle?

Radius of Mohrs circle is equal to the maximum shear stress.

30. What is the use of Mohrs 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 /4 xd2

Force due to longitudinal stress = f2 x d x t

p x /4 xd2 = f2 x d x t

f2 = pd/4t



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

= 30

Stress, = Load/Area

= 50*102/600

= 83.33 N/mm2

Normal stress, n = cos2

= 83.33*cos230

= 62.5 N/mm2

34. In case of equal like principle stresses, what is the diameter of the Mohrs

circle?

Answer: Zero



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 Hookes law.

4. Explain proportional limit and elastic limit.

5. Differentiate between elasticity modulus and rigidity modulus.

6. What is poisons 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 Mohrs circle.

21. Define principal planes and principal stresses.

22. Derive a relation for volumetric strain of a body subjected to a uniaxial stress.



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 .

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 its 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 = 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.



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 Youngs 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 its 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 Poissions

ratio of this material.



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 Youngs 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 Poissions 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.



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, c=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.



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 Poissons 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

= 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

McCaulays Method.

115983736 Solid Mechanics Short Questions and Answers

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