Strength of Materials Theory Sample

STRESS-STRAIN AND

PROPERTIES OF METALS

ASSUMPTIONS MADE IN STRENGTH OF
MATERIAL :
a) Material is continuous (no cracks & voids)
b) Material is homogenous & isotropic.
c) There are no internal forces in a body prior to loading
d) Principle of superposition is valid
(It states that effect of number of forces is equal to
algebraic sum of individual forces)
e) Self weight is ignored:
f) St. Venants principle is valid (It states that except in the
region of extreme ends of a bar carrying direct loading the
stress distribution over the cross section is uniform)

STRESS

* It is the internal resistance offered by the body against

external loading or deformation. Stresses are of following

types :

i) Direct or normal stresses which may be tensile or

compressive

  1
2

 = Resisting force against deformation

A = Area on which resistance is acting

P = Applied load

Unit - N/m2 or Pa

N/mm2 or MPa

ii) Shear or tangential stresses - Force acting tangential to

surface is called shear force and corresponding stress is

called shear stress

  Area Resisting shear force acting
on which shear force is

iii) Transverse or bending stress

iv) Tensional or twisting stresses.

Note : ** For direct stresses, if area under consideration

is original area, then it is known as Engineering stress or

nominal stress. But, if area taken is actual area then stress

is known as true stress.

Engg. or Nominal stress = P (A0 = original area)
A0

True stress = P (A1 = instantaneous area)
A1

**Stress is a tensor quantity i.e. it has one magnitude and

two directions.

STRAIN : -

It is defined as change in dimension per unit original
dimension

 = change in dimension
original dimension

a) Normal or axial strain or direct strain : It may be
tensile or compressive

b) Shear or tangential strain : It is angular distortion

between two planes at right angle (). Expressed in
radians.

c) Volumetric strain : Change in volume per unit
original volume

v = change in volume  V
original volume V

Unit : Strain is a unit less quantity

Note :
* Due to normal stress, there will be changes in
dimensions & volume without distortion in shape.
* Due to shear, there will be only distortion, no change in
dimension & volume

HOOKE’S LAW
Within elastic limit, Stress  strain

or stress = const = E
strain

i) Young’s modulus of elasticity (E)

E = t or cc
t

Elastic Constants for different materials:

Esteel = 200 GPa - for all steels irrespective of carbon
Ediamond = 1200 GPa
EAl = 70 GPa
ECI = 100 GPa
Ebrass = 100 GPa
Etimber = 10 GPa
Ecu = 120 Gpa
EBronze = 80 GPa
Erubber = 10 GPa

Note :
* Slope of  -  diagram upto proportional limit is E
* More E - More elasticity
* Diamond is more elastic than steel, steel is more elastic
than rubber.
* E is constant for a material under all circumstances and
its value can be calculated by using slope within
proportional limit
ii) Modulus of rigidity or shear modulus (C, N or G) :-

 G = shear stress = 
shear strain 

* G is also constant. For any material G < E

Significance : More G  less deflection  more
stiffness

iii) Bulk modulus (K)

K = Normal stres or spherical stress = n
volumetric strain

Significance : More K  less change in dimension or less
compressible
*V is also called dialation. K is called dialation const.
* E > K > G (for any isotropic, homogenous mtl)

LATERAL STRAIN : It is defined as change in lateral
dimension per unit to original lateral dimension

lateral strain = d = change in diameter
d original diameter

POISSONS RATIO () or  1 
 m 
 

 Lateral strain = d /d = (for cylindrical rod)
Linear strain l /l

Note : Value of  for any material varies from 0 to 0.5

Significance : more  more ductility

Value of  for different materials

i) Cork (almosto)  = 0
ii) Incompressible fluid  = 0.5
iii) Clay  = 0.5
iv) Paraffin wax  = 0.5
v) Rubber  = 0.5
vi) Isotropic martial  = 0.25 to 0.33
vii) Metals  > 0.25
viii) Non metals  < 0.5
ix) steel  = 0.33
x) Concrete  = 0.15

Note : There is no normal stress in transverse direction yet
there is strain. This is due to Poisson effect.

RELATIONSHIP BETWEEN E, K & G

* E = 3K (1 - 2)

* E = 2K (1 + )

* E = 9KG
3K G

*  = 3K  2G
6K  2G

Total number of elastic constants for different
materials :-

Material Total Elastic Independent
constant Elastic constant
i) Isotropic mtl 4. (E,K,G, )
ii) Orthotropic 12 2 (E, )
iii) Aleotropic or  9
anisotropic 21

ELONGATION OF BARS
1) A bar of uniform cross section area

2) Non-uniform bar

l  p  l1  l2  l3 
 A1E1 A2E2 A3E3 
 
 
 

3) Tapering bar of circular cross section whose
diameter changes from d1 to d2

l   4Pl
d1d1E

4) Elongation due to self weight :

a) Bar of uniform section :

l  Wl
2AE

(w = weight of bar)

b) Bar of tapering section : (Conical bar)

l  Wl
6AE

5) Bar of uniform strength :
* For a bar to have constant strength, the stress at any
section due to external load & weight of the portion below
it should be constant

l
A1 A2e
( = specific weight,  = constant stress)

6) Elongation of composite beam

l = l1 + l2 + l3 (algebraically)
7) Elongation of Compound bar :
P = P1 + P2
l1 = l2

Load shared by part 1

P1  PA1E1
A1E1 A2E2

Load shared by part 2

P2  PA2E2
A1E1 A2E2

Volumetric Strain for a rectangular bar sub to 3
mutually perpendicular tensile stress

  V  1 ( x  y  z) (1 - 2)
V E

When x = y = Z = 



V  3 (1 2)
E



Volumetric strain of cylindrical rod sub to axial load :

v  l  2 d
l d

Volumetric strain of sphere subjected to tensile force

V  3v  3 d
V d

d = strain in dia

Volumetric strain of rectangular body subjected to
axial force :

V  P (1 2)
V btE

STRESS-STRAIN DIAGRAMS :

1) For Mild Steel (Ductile Material)

A - Limit of proportionality
B - Elastic limit point
C - Upper yield point
D - Lower yield point - yielding begins at this point
DE - Strain hardening region (The mtl in this region
undergoes change in its atomic & cyrstalline structure
resulting in increased resistance to further drformation.
This portion is not used for structural design)
E - Ultimate stress point
E F - Necking region (strain softening)
F - Fracture point

Note : * The magnitude of stress corresponding to upper
yield points depends on cross section area, shape of
specimen & type of equipment used to perform the test. It
has no practical significance so lower yield pt is
considered as true characteristic yield stress for M.S.
** Materials that undergo large strains before failure are
classified as ductile. Advantage of ductility is that visible
deformation can occur before failure hence remedial
action can be taken.

2) For Brittle Material

Generalized -  curve

Residual strain : Loading beyond elastic limit causes
residual strain or permanent set

L = Residual strain or permanent set

Stress Stain Diagram for different materials :

Engineering and True Stress Strain Curve

PROOF STRESS :
When a material such as Aluminum which does not have
an obvious yield point & yet undergoes large strains after
proportional limit, the yield stress is determined by offset
method.

A line parallel to initial linear part is drawn, which is
offset by some standard amount of strain such as 0.2%.
The intersection of the offset point (A) defines the yield
stress which is slightly above proportionality limit and is
called proof stress.



PROPERTIES OF METALS

i) Ductility : It is the property by which material can be
stretched
Eg. : M.S., Al, Cu, Mn, Lead, Brass etc.
ii) Brittleness : It is lack of ductility i.e. material cannot
be stretched. For brittle materials, fracture point &
ultimate point are same. Materials with strain less than
5% at fracture point are regarded as brittle & those having
strains > 5% at fracture pt are called ductile
iii) Malleability : The property by which material can be
uniformly extended in a direction without rapture.
iv) Toughness : This property enables material to absorb
energy without fracture. This property is very desirable in
case of cyclic loading or shock loading.

Modulus of toughness : Area under entire stress -strain
curve and is the energy absorbed by material of the
specimen / unit vol. up to fracture.

Mod. of toughness =   y u  f
 
 2 

y = yield tensile strength
u = ultimate tensile strength
f = strain at fracture pt
** Material having higher modulation of toughness will
be very ductile.
Modulus of resilience : Maximum elastic energy / unit
vol. that can be absorbed without attaining plastic stage.

Mod. of resilience ( u) = sy2
2E

** Higher u - Higher yield strength
** Higher toughness is desirable for gears, chains, crane
hooks, etc, Higher resilience is desirable in springs
v) Hardness : Resistance to indentation or scratching or
surface abrasion.
vi) Fatigue : The behavior of material under variable
loads is referred as fatigue.
vii) Creep : Additional strains over a long peiod of time
is called creep.
viii) Relaxation : A wire attached between two rigid
supports after sometime stress in wire diminishes &
reaches constant value called relaxation of material.

fig : creep in bar under constant load Fig : Relaxation

ix) Tenacity : ultimate strength in tension is called
Tenacity.

Note : * Ductile materials are tough & brittle materials are
hard. As carbon content increases ductility
decreases but ultimate strength increases.
* Ductile materials are strong in tension, weak in shear,
moderate in compression
* Brittle materials are : weak in tension, strong in
Compression, moderate in shear
* Theoretically, ductile materials are equally strong in
tension & in compression but practically due to buckling
this materials are weak in compression & very weak in
shear.

Factor of Safety (N)

Used to determine permissible stress. Permissible stress is
used to get safe dimensions of a component under
strength criteria

N  Permissible Failure stress critical or limiting stress or safe stress
stress or allowable or working or Design

 per  Failure stress
FOS

For ductile  per = y
N

For brtile  per = u
N

Reasons for FOS :
i) Unknown loading conditions
ii) Imperfect workmanship
iii) Unknown environmental conditions
iv) Unreality of assumptions made in STM equation

v) Effect of true stress
vi) Effect of stress concentration

Margin of Safety = N - 1
Note : Margin of safety is reduced to 0 or less, structure
fails.