Chapter No.1 Material Properties

a) Define stress, strain & modulus of elasticity and write the relationship between stress & strain.

Ans }

Stress (normal)

Stress is the ratio of applied load to the cross-sectional area of an element in  tension and is expressed in pounds per square inch (psi) or kg/mm2.

Strain (normal)

A measure of the deformation of the material that is dimensionless. 

  change in length   LStrain,  =   =  

 original length  L

Modulus of elasticity

Metal deformation is proportional to the imposed loads over a range of loads.

Since stress is proportional to load and strain is proportional to deformation, this implies that stress is proportional to strain. Hooke's Law is the statement of that proportionality.

  Stress     =   =   E

   Strain   The constant, E, is the modulus of elasticity, Young's modulus or the tensile modulus and is the material's stiffness. Young's modulus is in terms of 106 psi or 103 kg/mm2.  If a material obeys Hooke's Law it is elastic. The modulus is insensitive to a material's temper. Normal force is directly dependent upon the elastic modulus.

b) Define modulus of rigidity, bulk modulus and modulus of elasticity. State relationship between them.

Ans }

4.3. Elastic ConstantsElastic constants are used to express the relationship between stresses and strains. Hooke’s law , is stress/strain = a constant, within a certain limit. This means that any stress/corresponding strain = a constant, within certain limit. It follows that there can be three different types of such constants. (which we may call the elastic constants or elastic modulae) corresponding to three distinct types of stresses and strains. These are given below.

(i)Modulus of Elasticity or Young’s Modulus (E)Modulus of Elasticity is the ratio of direct stress to corresponding linear strain within elastic limit. If p is any direct stress below the elastic limit and e the corresponding linear strain, then E = p / e.

(ii)Modulus of Rigidity or Shear Modulus (G)Modulus of Rigidity is the ratio of shear stress to shear strain within elastic limit. It is denoted by N,C or G. if q is the shear stress within elastic limit and f the corresponding shear strain, then G = q / f.

(iii) Bulk Modulus (K)Bulk Modulus is the ratio of volumetric stress to volumetric strain within the elastic limit. If pv is the volumetric stress within elastic limit and ev the corresponding volumetric strain, we have K = pv / ev.

c) Explain stress strain diagram for mild steel in tension. Define Hooke’s Law. Or d) Sketch stress strain curve showing the silent points on the curve.

Ans } A typical tensile test curve for the mild steel has been shown below

e)

Nominal stress � Strain OR Conventional Stress � Strain diagrams:

Stresses are usually computed on the basis of the original area of the specimen; such stresses are often referred to as conventional or nominal stresses.

True stress � Strain Diagram:

Since when a material is subjected to a uniaxial load, some contraction or expansion always takes place. Thus, dividing the applied force by the corresponding actual area of the specimen at the same instant gives the so called true stress.

SALIENT POINTS OF THE GRAPH:

(A) So it is evident form the graph that the strain is proportional to strain or elongation is proportional to the load giving a st.line relationship. This law of proportionality is valid upto a point A. or we can say that point A is some ultimate point when the linear nature of the graph ceases or there is a deviation from the linear nature. This point is known as the limit of proportionality or the proportionality limit.

(B) For a short period beyond the point A, the material may still be elastic in the sense that the deformations are completely recovered when the load is removed. The limiting point B is termed as Elastic Limit .

(C) and (D) - Beyond the elastic limit plastic deformation occurs and strains are not totally recoverable. There will be thus permanent deformation or permanent set when load is removed. These two points are termed as upper and lower yield points respectively. The stress at the yield point is called the yield strength.

(E) A further increase in the load will cause marked deformation in the whole volume of the metal. The maximum load which the specimen can with stand without failure is called the load at the ultimate strength. The highest point �E' of the diagram corresponds to the ultimate strength of a material. su = Stress which the specimen can with stand without failure & is known as Ultimate Strength or Tensile Strength. su is equal to load at E divided by the original cross-sectional area of the bar.

(F) Beyond point E, the bar begins to forms neck. The load falling from the maximum until fracture occurs at F. [ Beyond point E, the cross-sectional area of the specimen begins to reduce rapidly over a relatively small length of bar and the bar is said to form a neck. This necking takes place whilst the load reduces, and fracture of the bar finally occurs at point F ]

f) Explain the term laterial strain, linear strain, how are they related to each other.

ans }

3. StrainsStrain is defined a the ratio of change in dimension to original dimension of a

body when it is deformed. It is a dimensionless quantity as it is a ratio between two quantities of same dimension.

3.1. Linear StrainLinear strain of a deformed body is defined as the ratio of the change in length

of the body due to the deformation to its original length in the direction of the force. If l is the original length and δl the change in length occurred due to the deformation, the linear strain e induced is given by

e=δl/l.

Linear strain may be a tensile strain , et or a compressive strain ec according as δl refers to an increase in length or a decrease in length of the body. If we consider one of these as +ve then the other should be considered as –ve, as these are opposite in nature.

3.2.Lateral StrainLateral strain of a deformed body is defined as the ratio of the change in length (breadth of a rectangular bar or diameter of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular bar or diameter of a circular bar) in the direction perpendicular to the force

g) Sketch stress strain curve showing the silent points on the curve.h) Define elasticity, brittleness, malleability and hardness.

P P

l lδl δl

Hardness

Hardness refers to the ability of a metal to resist abrasion, penetration, cutting action, or permanent distortion. Hardness may be increased by working the metal and, in the case of steel and certain titanium and aluminum alloys, by heat treatment and cold-working (discussed later). Structural parts are often formed from metals in their soft state and then heat treated to harden them so that the finished shape will be retained. Hardness and strength are closely associated properties of all metals.

Brittleness

Brittleness is the property of a metal that allows little bending or deformation without shattering. In other words, a brittle metal is apt to break or crack without change of shape. Because structural metals are often subjected to shock loads, brittleness is not a very desirable property. Cast iron, cast aluminum, and very hard steel are brittle metals.

Malleability

A metal that can be hammered, rolled, or pressed into various shapes without cracking or breaking or other detrimental effects is said to be malleable. This property is necessary in sheet metal that is to be worked into curved shapes such as cowlings, fairings, and wing tips. Copper is one example of a malleable metal.

Elasticity

Elasticity is that property that enables a metal to return to its original shape when the force that causes the change of shape is removed. This property is extremely valuable, because it would be highly undesirable to have a part permanently distorted after an applied load was removed. Each metal has a point known as the elastic limit, beyond which it cannot be loaded without causing permanent distortion. When metal is loaded beyond its elastic limit and permanent distortion does result, it is referred to as strained. In aircraft construction, members and parts are so designed that the maximum loads to which they are subjected will never stress them beyond their elastic limit.

i) Define & explain laterial strain and poisions ratio.Ans }

5. Poisson’s RatioAny direct stress is accompanied by a strain in its own direction and called linear strain and an opposite kind strain in every direction at right angles to it, lateral strain. This lateral strain bears a constant ratio with the linear strain. This ratio is called the Poisson’s ratio and is denoted by (1/m) or µ.Poisson’s Ratio = Lateral Strain / Linear Strain.

Value of the Poisson’s ratio for most materials lies between 0.25 and 0.33.

3.2.Lateral StrainLateral strain of a deformed body is defined as the ratio of the change in length (breadth of a rectangular bar or diameter of a circular bar) of the body due to the deformation to its original length (breadth of a rectangular bar or diameter of a circular bar) in the direction perpendicular to the force.

j) What are temperature stresses. Derive an expression for temperature stress.

k) With a new sketch explain shear stress and shear strain.

.ans }

Shear StressConsider the section x-x of the rivet forming joint between two plates subjected to a tensile force P as shown in figure.

The stresses set up at the section x-x acts along the surface of the section, that is, along a direction tangential to the section. It is specifically called shear or tangential stress at the section and is denoted by q.q =R/A=P/A.

. Shear StrainShear strain is defined as the strain accompanying a shearing action. It is the

angle in radian measure through which the body gets distorted when subjected to an external shearing action. It is denoted by Ф.

Consider a cube ABCD subjected to equal and opposite forces Q across the top and bottom forces AB and CD. If the bottom face is taken fixed, the cube gets distorted through angle to the shape ABC’D’. Now strain or deformation per unit length isShear strain of cube = CC’ / CD = CC’ / BC = radian

l) Define volumetric strain.

3.3. Volumetric StrainVolumetric strain of a deformed body is defined as the ratio of the change in

volume of the body to the deformation to its original volume. If V is the original volume and δV the change in volume occurred due to the deformation, the volumetric strain ev induced is given by ev =δV/V

Consider a uniform rectangular bar of length l, breadth b and depth d as shown in figure. Its volume V is given by,

X XP

P

XX

PP P

A

X

q q

X

Q

QD

A B

C C’D’

V = lbdδV = δl bd + δb ld + δd lbδV /V = (δl / l) + (δb / b) + (δd / d)ev = ex +ey +ez

This means that volumetric strain of a deformed body is the sum of the linear strains in three mutually perpendicular directions.

m) Explain the term elastic limit & working stress.

4.1. Elasticity and Elastic LimitElasticity of a body is the property of the body by virtue of which the body regains its original size and shape when the deformation force is removed. Most materials are elastic in nature to a lesser or greater extend, even though perfectly elastic materials are very rare. The maximum stress upto which a material can exhibit the property of elasticity is called the elastic limit. If the deformation forces applied causes the stress in the material to exceed the elastic limit, there will be a permanent set in it. That is the body will not regain its original shape and size even after the removal of the deforming force completely. There will be some residual strain left in it.

Working stressWorking stress= Yield stress/Factor of safety.

n) Explain the principle of complementary shear stress

Complementary Strain Consider a rectangular element ABCD of a body subjected to simple shear of intensity q as shown. Let t be the thickness of the element.

Total force on face AB is , FAB = stress X area = q X AB X t.Total force on face CD is, FCD = q X CD Xt = q XAB Xt.

Z

Y

X

l

b

d

qq

q

q'

q'

q

A B

CD

t

FAB and FCD being equal and opposite, constitute a couple whose moment is given by,M =FAB X BC = q XAB X BC X tSince the element is in equilibrium within the body, there must be a balancing couple which can be formed only by another shear stress of some intensity q’ on the faces BC and DA. This shear stress is called the complementary stress.FBC = q’ X BC X t FDA = q’ X DA X t = q’ X BC XtThe couple formed by these two forces is M’ = FBC X AB = q’ X BC X tFor equilibrium, M’ = M.Therefore q’ = qThis enables us to make the following statement.In a state of simple shear a shear stress of any intensity along a plane is always accompanied by a complementary shear stress of same intensity along a plane at right angles to the plane.

o) Differentiate between elastic & plastic deformation.

Chapter No.2 Principles Plane & Stresses

a) Using sketches differentiate between tangential & normal stress.b) What are principle planes. How do the stresses occurs in the principle planes.

Chapter No.3 Strain Energy

a) Explain in brief graduial, sudden & impact loading.b) Explain strain energy & modulus of resilence.c) Derive an expression for strain energy stored in a material when load is suddenly

applied.

Chapter No.4 Shear Force & Bending Moment

a) Explain with a neat sketch various types of beams load and support.b) What do you interpret with a shear force diagram is

1) Zero or change in sign2) A straight line parallel to reference line.3) Inclined straight line

c) Define shear force & bending moment.

Chapter No.5 Moment of Inertia

a) State & explain parallel axis theorem.b) State & explain perpendicular axis theorem.

c) What is section modulus.

Chapter No.6 Simple Bending

a) Write the bending equation on the theory of simple bending. Explain the various parameters involved.

b) State the assumptions made on theory of pure bending.c) Explain the theory of simple bending.

d) On the equation . Explain each term & derive the expression for

e) What are the practical applications of bending equation.f) Explain the term Neutral Axis & Flexure.

Chapter No.7 Torsion

a) What are assumptions made in theory of pure torsion.

Chapter No.8 Thin cylinder

a) Write the expression for two kinds of stresses induced in thin cylinder.b) Explain the difference between thin and thick cylinder with neat sketches.c) When is cylinder is classified as thin cylinder.d) Explain Hoop Stress & Longitudinal Stress

Chapter No.9 Column & Struts

a) What is equivalent length of a column. State the various end conditions of the column.

b) State the assumptions made in Eulers theory of column.c) For different end condition show how bending occurs in case of long column.d) Define 1) Buckling Load 2) Slenderness Ration 3) Long Columne) With the help of sketches show different end condition of column.f) What is eccentric loading.g) Discuss the type of failure of column.