Solids are often placed under stress - for example, a weight is hung from a steel cable, or a steel column supports the weight of a building or a floor.Structural engineers, who make sure that a structure such as a plane, a bridge, or a sky scraper, is safe and functional, need to study the properties of solids.

Materials

We first define stress, which is a measure of the magnitude of a load that is placed on a material.

stress = FA

Stress

There are three types of stress: tension is a lengthening stress, compression is a shortening stress and shear is a cutting or bending stress.

tension

compression

shear

FYI: All three types of stress are measured in newtons / meter2 but all have different effects on solids.

Structural engineers can measure the amount of strain, which is the amount of deformation a solid is exhibiting under the action of a stress.

strain =

Materials

change in lengthoriginal length

= LL0

Strain

Engineers study stress-strain curves, which are characteristic of each material.To create a stress-strain curve, engineers subject a material to ever-greater stress forces, and measure the strain (deformation):

L0

LA

F

FYI: Typical building materials return to their original dimensions when the stress is removed, as long as that stress does not exceed a particular limit, characteristic of the material.

Hooke’s Law states that...Hooke’s Law states that...

““within the elastic limit extension within the elastic limit extension of a body is directly proportional of a body is directly proportional to the applied load.”to the applied load.”

Hooke’s LawHooke’s Law

L (m)

F (

N) Extension Extension Force Force

AppliedAppliedL L F F F=F=k k LL kk-elastic const or spring const

TheThe elastic const or spring const is the force per unit extension and it gives how stiff a spring is.

Take the Take the gradient of gradient of the graph :the graph :

FF22 – F – F11

LL22 – L – L11

F

L

L (m)

Note:Note:Make the Make the “gradient triangle“gradient triangle” as large as possible” as large as possible

F (

N)

How can we find K experimentally?How can we find K experimentally?

L (m)

When we undertake an experiment we When we undertake an experiment we should only change one variable at a should only change one variable at a time to make it a fair test. We call time to make it a fair test. We call this the this the ““independent variable”independent variable”

Quantities we measure, (and subsequently calculate) are Quantities we measure, (and subsequently calculate) are called called “dependent variables”“dependent variables”. All other variables which . All other variables which are kept the same are called the are kept the same are called the “control variables”“control variables”

Often graphs have the independent variable along the Often graphs have the independent variable along the bottom and the dependent up the sidebottom and the dependent up the side

Hooke’s law is a notable exception Hooke’s law is a notable exception

C stops obeying Hooke’s law... C stops obeying Hooke’s law...

After the limit of proportionality the material behaves in a After the limit of proportionality the material behaves in a ductileductile fashion. fashion.

The material stretches more with a small extra forceThe material stretches more with a small extra force..

material A is material A is stiffer than B & Cstiffer than B & C

L (m)

F (

N)

Strain Energy is the energy stored in a Strain Energy is the energy stored in a body due to change of shape.body due to change of shape.

The force involved ranges from 0 to The force involved ranges from 0 to F and so the average is F/2.F and so the average is F/2.but F=k but F=k LL

Average Force=½ (k Average Force=½ (k L)L)LL

The distance moved by the force is The distance moved by the force is LL

Strain Energy

The stretched spring has elastic potential energyThe stretched spring has elastic potential energy

Work has been done because the force moves Work has been done because the force moves through a distance.through a distance.

Workdone=Average Force x extensionWorkdone=Average Force x extension

Workdone=Average Force x extensionWorkdone=Average Force x extension = ==½ (k =½ (k L)L)LL =½ k ( =½ k (L)L)22

=½ k (x)=½ k (x)22

Workdone=Strain EnergyWorkdone=Strain EnergyStrain Energy Strain Energy =½ k (x)2=½ k (x)2

This means that the strain energy is represented by the area under the line on a graph of load( y axis) This means that the strain energy is represented by the area under the line on a graph of load( y axis) against extension (x-axia)against extension (x-axia)

Elasticity: Elasticity: The ability of a solid to regain The ability of a solid to regain its shape after it has been deformed or its shape after it has been deformed or distorteddistorted

Materials :Definitions 1

TensileTensile : Deformation due to stretching : Deformation due to stretching

Compressive Deformation: Compressive Deformation: deformation due to deformation due to compressioncompression

Ductile: Ductile: The ability to be drawn into a wire The ability to be drawn into a wire (Copper is a good example)(Copper is a good example)

Brittle: Brittle: Material breaks without any “give”. Material breaks without any “give”. Cannot be permanently stretchedCannot be permanently stretched

Graphs of Stress against Strain are useful. Graphs of Stress against Strain are useful. They provide a method of comparing materials They provide a method of comparing materials of different thicknesses and original lengthsof different thicknesses and original lengths

Stress -Strain Graphs

1.1. Linear region where Linear region where Hooke’s law is obeyedHooke’s law is obeyed

2.2. The limit of proportionalityThe limit of proportionality

3.3. Elastic Limit – point where Elastic Limit – point where the material stops the material stops returning to its original returning to its original length length

4.4. Yield point(s) where the Yield point(s) where the material ‘necks’material ‘necks’

5.5. Ultimate Tensile Stress Ultimate Tensile Stress (U.T.S.)(U.T.S.)

6.6. B. breaking pointB. breaking point

Stress -Strain Graphs

Elastic Limit: Elastic Limit: The maximum amount a material can The maximum amount a material can be stretched by a force and still return to its original be stretched by a force and still return to its original shape and size. The material has no permanent shape and size. The material has no permanent change in shape or sizechange in shape or size

Materials :Definitions 2

Yield Point: Yield Point: Beyond the elastic limit, a point is Beyond the elastic limit, a point is reached at which there is a noticeably larger reached at which there is a noticeably larger permanent change in length. This results in plastic permanent change in length. This results in plastic behaviourbehaviour

Ultimate Tensile Strength: Ultimate Tensile Strength: The maximum stress The maximum stress that can be applied without breakingthat can be applied without breaking

Plasticity: Plasticity: A plastic material does not return to its A plastic material does not return to its original size and shape when the force is removed. original size and shape when the force is removed. There is a permanent stretching and change of There is a permanent stretching and change of shapeshapeStiffness: Stiffness: A measure of how difficult it is to change A measure of how difficult it is to change the size or shape of a material.the size or shape of a material.

•Thick steel wire is stiffer than thin steel wire of the Thick steel wire is stiffer than thin steel wire of the same length.same length.•Short steel wire is stiffer than longer steel wire of the Short steel wire is stiffer than longer steel wire of the same diameter.same diameter.•Steel is stiffer than copper of the same diameter and Steel is stiffer than copper of the same diameter and length, because copper extends more per unit forcelength, because copper extends more per unit force

The linear region where Hooke’s law is obeyed is of The linear region where Hooke’s law is obeyed is of interest and allows us to compare materials.interest and allows us to compare materials.

This is known as Young’s modulusThis is known as Young’s modulus

Comparing stress/strain graphs of brittle and Comparing stress/strain graphs of brittle and ductile materials for example glass and copperductile materials for example glass and copper

Stress -Strain curves

Stress -Strain curves

YOUNG'S MODULUS - CHANGE IN LENGTH

Young’s Modulus

Both tension and compression act to change the length of a material - tension stretches, and compression shortens.The elastic modulus for tension and compression (not shear) is called Young's modulus (Y). Thus

Y =stressstrain

Y = F/A

L/L0

Y = FL0

AL

Young's Modulus(for tension and compression)

We can solve the above equation for L to get a useful relationship:

L = FL0

AYDeformation Under Compression or Tension

Stress strain curves

plastic region yield region

elastic region

elastic limit

failure region

70000

60000

50000

40000

30000

20000

10000

0 0 50 100 150 200 250 300 350

400 strain (10-8)

stre

ss (

N/m

2 )

permanent set

yield point

breaking point

FYI: If a material is stressed beyond its YIELD POINT, it will become permanently deformed. As the stress is removed, it will follow the PERMANENT SET line, as illustrated:

FYI: Structural engineers NEVER design a structural component to exceed its ELASTIC LIMIT.

FYI: Since the region of the graph up to the elastic limit is linear, we can characterize a material by the slope: stress / strain. We call this ratio the ELASTIC MODULUS.

elastic modulus =stressstrain

FYI: For this particular material the elastic modulus iselastic modulus = stressstrain

40000 n/m2

5010-8= = 8 1010

n/m2FYI: Thus tables showing properties of materials can simply show a single number for that material, rather than a graph.

YOUNG'S MODULUS - CHANGE IN LENGTH

Solids and Fluids9-1 Solids and Elastic Moduli

A 15.0-meter long steel cable with a diameter of 1 cm has a 500 metric ton mass hanging from it. How much does it stretch under this tension load? Assume Young's modulus for steel is Y = 401010 n/m2.

A = r2

LL0

L

AA

A = d2

2

A = .012

2

A = 7.85410-5 m2

F = mg

F = (500)(1000)(10)

F = 5106 n

L = FL0

AY= (5106)(15)

(7.85410-5 )(401010)

L = 2.4 m

FYI: Since the cable is under TENSION it STRETCHES so that its length under the load is 15.0 + 2. 4 = 17. 4 m.

FYI: The typical stretches for cables (and compressions for columns) is extremely small.

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