9EE406.13.ppt strength of material sh sdx hv czn h c

xzvcxvc

1. Recap
In the last class, we have learnt
about solving
• Problems to calculate different elastic
constants
based on their relationship
1

2. Objective
On completion of this class, you
would be able to know about
• Deriving the formula for
volumetric strain
2

3.  Rectangular bar subjected to axial pull
• Original Volume = l b t
• Final volume = ( l+ δl) (b- δb) (t-δt)
= l b t + b t δl -lb δt – l t δb
• Change in Volume, δV = b tδl – lbδt – l tδb

Volumetric Strain
3

4.  Rectangular bar subjected to axial pull

Longitudinal strain = δl / l = f / E

Lateral strain = δb / b = μ (f/E)
= δt / t = μ (f/E)

Volumetric Strain (Contd..)
4

5. Volumetric strain, e
v
= δV/V

= (δl / l) – (δb / b) - (δt / t)
e
v
= e- e
l
-e
l


= f/E - f/mE - f/mE


e
v = f/E (1-2/m )
Volumetric Strain (Contd..)
 Rectangular bar subjected to axial pull

5

6. Longitudinal strain =δl / l = f / E
Lateral strain =δd / d= 1/m. (f/E)
Original volume of the bar V =
Final volume of the bar V
1
=
Volumetric Strain (Contd..)
2
(d d )(l l)
4

  
2
dl
4

 Circular bar subjected to axial pull
6

7. 7
Volumetric Strain (Contd..)
 Circular bar subjected to axial pull
Change in volume, δV = V
1 - V
2
(d l 2 l d d)
4

    
Volumetric Strain, e
V
V
V


l d
2
l d
 
 
= e-e
1
e
V
=
2
(1 )
E m



8. 8
Volumetric Strain (Contd..)
 Rectangular bar subjected to tri-axial pull


f
X
Z
Y
X
f
z
f
y
Fig 1

9. 9
Volumetric strain , δV/V = e
x
+ e
y
+ e
z
where e
x
= strain in x- direction


Volumetric Strain (Contd..)
 Rectangular bar subjected to tri-axial pull

x y z
x
( ( )
e
E
    

yx z
E mE mE
 
  

10. 10
Volumetric Strain (Contd..)
 Rectangular bar subjected to tri-axial pull


Similarly ;
z y x
z
( ( )
e
E
    

y x z
y
( ( )
e
E
    

and
x y zV
(1 2 )
V E
  
  
x y zV 2
(1 )
V E m
  
 

or

11. 11
Summary
In this class we have learnt about
 The volumetric strains of different cross
sections namely
•Rectangular bar subjected to axial pull
•Circular bar subjected to axial pull
•Rectangular bar subjected to tri-axial
stresses

12. 12EE 402. 13 12
Quiz
1. When ever a rectangular body is subjected to
an axial pull the volumetric strain is
a) σ/E(1-2/m)
b) σ/m(1-2/E)
c) σ/E(1-2/m)
d) None of the above

13. 13
Quiz
2. Volumetric strain is the ratio of
a. Lateral strain to linear strain
b. Shear stress to shear strain
c. Change in volume to original volume
d. Change in diameter to original diameter