# 2E4: SOLIDS & STRUCTURES

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29 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

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2E4: SOLIDS & STRUCTURES

Lecture 9

Dr.
Bidisha

Ghosh

Notes:
http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/So
lids & Structures

Hooke’s Law

A material which regains its shape when the external load
is removed is considered as ‘perfectly elastic’.

From tensile tests, it can be seen within the range of elastic behaviour of a
material the elongation is proportional to both the external load and the
length of the bar.

For linearly elastic materials, this Stress is proportional to
strain.

The factor of proportionality between stress and strain is
called, ‘Modulus of Elasticity’ or Young’s modulus.

E has the dimension of stress

1
Pl Pl
E A AE

 
E
 

We already know Hooke’s law, but what
does it tell us?

It tells us that how a material is going to behave and change size
(length/width/height).

How do we know E?

E is always found out from experiments. So, we have to stretch or
compress things to know that what is the value of E for any material.

The relationship between stress and strain is defined by E.

And, actually it is the relation between load and deformation.

So, for a material of known length and area a graph of load (
P) vs.
deflection (
𝛿
) will give us E.

Pl l
P
AE AE

 
 
 
 
Tensile Test

Tensile Test

Check this link for tensile test movie:

http://web.mst.edu/~mecmovie/

Extensometer

-
nN7tnXLIM

Tensile Test

1.
Linear elastic region. Slope
of
this linear part is the young’s
modulus.

2.
The proportional limit is the
stress when stress
-
strain
relationship is starts to become
nonlinear.
(Beyond this limit the
material is not elastic)

3.
Yielding (strain hardening)

4.
Ultimate strength

5.
necking

6.
Fracture Stress

7.
-
hardening/work hardening

Permanent
deformation

Stress
-
Strain Diagram

-
deformation plot does not provide material properties.

But, when converted to stress
-
strain plot it provides all the information needed.

Notice elastic limit and proportionality limits
are different! Some materials are still elastic
beyond the linear (proportional) section of
the curve.

But in all practical cases they are same.

Notice ultimate stress is
higher than fracture stress.
This is because this graph
do not plot the true stress
accounting for the
reduction in area due to
necking. This is called
engineering stress. The true
stress actually is higher at
fracture.

Glossary

Proportionality Limit: The point till which the stress
-
strain curve is linear.

Elastic Limit: The point beyond which the material will no longer go back to its
original shape when the load is removed.

Yield Point: It is the point at which the material will have an appreciable
elongation or yielding without any increase in load.

Ultimate Strength: The maximum ordinate in the stress
-
strain diagram is the
ultimate strength or tensile strength.

Fracture Strength: It is the strength of the material at rupture. This is also known
as the breaking strength.

Residual Strain: In the plastic region, after unloading the material does not go
back to its original shape and the remaining strain in the material is called
residual strain and the elongation is called permanent set.

Work Hardening: Also known as strain hardening, after yielding occurs the
material can withstand increase amount of stress, showing increase in strength.

True stress
-
strain & engineering stress
-
strain: The engineering strain is calculated
using the initial cross
-
sectional area of the specimen.

Creep: A solid material deforms permanently under the influence of continuous

Stress
-
Strain Diagram

Ductile materials are those which can yield and undergo significant
deformation in normal temperature.

Brittle materials rupture with little deformation.

Concrete

Concrete is very weak in tension (10% of its compressive strength) and
very strong in compression.

Concrete behaves like a brittle material when assumed homogenous.

compression testing

of concrete

Properties of Typical Materials

Material

Young's Modulus
(Modulus of Elasticity)
(GPa)

Ultimate
Strength
(MPa)

Yield
Strength
(Mpa)

Aluminum

69

110

95

Bone (compression)

9

170

Concrete (high
strength)(compression)

30

40

Diamond (C)

1220

Wood (compression)

9
-
13

40
-
50

Glass

50
-

90

50

Steel

200

400

250

Hooke’s Law: Shear Modulus

shear modulus

or

modulus of
rigidity,
G

Elasticity can be measured for
shear tests or torsion test can be
used.

Using Hooke’s law for the linear
elastic part of the stress
-
strain
diagram,

tan
G
F
A
G
 

 

Direct shear test on soil!

Poisson’s Ratio

In elastic range, the ratio of lateral strain to elastic strain is
constant.

The lateral strain caused due to Poisson's ratio do not result/create any
stress in lateral direction.

lateral strain
axial strain
and,
y
x
y x z x

   
   
   
dx

dy

dz

Values of

The concept is only valid for uniaxial strain and isotropic
material.

In case of perfectly incompressible material,

is 0.5. For all practical
cases
,

0<

<0.5

Generally, between 0.25
-
0.35

For steel, assumed to be 0.3

For concrete, assumed to be 0.1

For incompressible material, 0.5 (may be, water)

Relation between elastic moduli:

0
(1 2 )
unit volume change or dialation, (1 2 )
x
x
V
e
V E



   
2(1 )
E
G
v

Strain Energy

The external work done on an elastic
body in
causing it to
distort/deform
from its
original
state is
stored in the body
as strain energy. For perfectly elastic body no dissipation of
energy occurs and this energy is recoverable on unloading.

Strain energy is the area
under the linear part of
stress strain curve

Strain Analysis

What happens when we apply 1
-
D stress?

What happens when we apply 2
-
D stress?

longitudinal strain,;lateral strain,
x x
x z
E E
 
  
  
longitudinal strain,;lateral strain,
y y
x x
x z
E E E E
 
 
   
   
What happens when we apply 3
-
D stress?

‘stress and strain are not proportional

any more!!’

Strain Analysis

( )
;
( )
;
( )
;
y z
x
x
y
x z
y
x y
z
z
E E
E E
E E
 

 

 
 
 

 

 

 

 
How much does the volume change?

3
-
D case

Let’s assume initial volume,
abc

Final volume

=

Change in volume

=

Hence, strain or volumetric strain,

a

b

c

(1 ).(1 ).(1 )
x y z
a b c
  
  
(1 )(1 )(1 ) 1
1
[ ( ) ( ) ( )]
x y z
x y z x y y z z x z x y
x y z y x z z x y
abc
abc
abc
E
  
      
        
 
   
 
 
      
 
        
( )
(1 2 )
y z x
v
E
  
 
 
 