2E4: SOLIDS & STRUCTURES

Lecture 9

Dr. Bidisha GhoshNotes: http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/Solids & Structures

Hooke’s LawA 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 () will give us E.

Pl lP

AE AE

Tensile Test

http://www.youtube.com/watch?v=hD_NJaZIpT0&feature=related

Tensile Test

Check this link for tensile test movie:http://web.mst.edu/~mecmovie/

Extensometer

http://www.youtube.com/watch?v=A-nN7tnXLIM

Tensile Test1. 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. Unload-reloading creates strain hardening/work hardening

Permanent deformation

Stress-Strain Diagram

• The Load-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.

GlossaryProportionality 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 loading below yield stress.

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, GElasticity can be measured for shear loading. Generally a direct shear tests or torsion test can be used.

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

tan

G

FAG

Direct shear test on soil!http://www.youtube.com/watch?v=L1fWPypBP0g

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 n

The concept is only valid for uniaxial strain and isotropic material.In case of perfectly incompressible material, n is 0.5. For all practical cases,

0< <0.5nGenerally, between 0.25-0.35For steel, assumed to be 0.3For 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

Ve

V E

2(1 )

EG

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 xx zE E

longitudinal strain, ; lateral strain,y yx xx zE E E E

What happens when we apply 3-D stress?

‘stress and strain are not proportional any more!!’

Strain Analysis

( );

( );

( );

y zxx

y x zy

x yzz

E E

E E

E E

How much does the volume change?

3-D caseLet’s assume initial volume, abc

Final volume =

Change in volume =

Hence, strain or volumetric strain,

a

b

c

(1 ) .(1 ) .(1 )x y za 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

abcE

( )(1 2 ) y z x

v E