1. FYI: All three types of stress are measured in newtons / meter2 but all have different effects on solids.
Materials
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.
We first define stress, which is a measure of the magnitude of a load that is placed on a material.
Stress F A
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.
© 2006 By Timothy K. Lund
shear
tension
compression

2. Materials
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.
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
change in length
original length L L0
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): F
© 2006 By Timothy K. Lund A L L0

3. Hooke’s Law Hooke’s Law states that...
“within the elastic limit extension of a body is directly proportional to the applied load.”
Extension  Force Applied
L  F
F=k L
k-elastic const or spring const
F (N)
The elastic const or spring const is the force per unit extension and it gives how stiff a spring is.
L (m) 3

4. Take the gradient of the graph :
How can we find K experimentally?
Take the gradient of the graph :
F2 – F1
L2 – L1
F (N) F L
L (m)
Note:
Make the “gradient triangle” as large as possible 4

5. Hooke’s law is a notable exception
When we undertake an experiment we should only change one variable at a time to make it a fair test. We call this the “independent variable”
L (m)
Quantities we measure, (and subsequently calculate) are called “dependent variables”. All other variables which are kept the same are called the “control variables”
Often graphs have the independent variable along the bottom and the dependent up the side
Hooke’s law is a notable exception 5

6. material A is stiffer than B & C
F (N)
L (m)
C stops obeying Hooke’s law...
After the limit of proportionality the material behaves in a ductile fashion.
The material stretches more with a small extra force. 6

7. Strain Energy
Strain Energy is the energy stored in a body due to change of shape.
The stretched spring has elastic potential energy Work has been done because the force moves through a distance.
Workdone=Average Force x extension
Workdone=Average Force x extension ==½ (k L)L =½ k (L) =½ k (x)2
The force involved ranges from 0 to F and so the average is F/2. but F=k L
Average Force=½ (k L)L The distance moved by the force is L
Workdone=Strain Energy Strain Energy =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) against extension (x-axia) 7

8. Materials :Definitions 1
Elasticity: The ability of a solid to regain its shape after it has been deformed or distorted
Tensile : Deformation due to stretching
Compressive Deformation: deformation due to compression
Ductile: The ability to be drawn into a wire (Copper is a good example)
Brittle: Material breaks without any “give”. Cannot be permanently stretched 8

9. Stress -Strain Graphs
Graphs of Stress against Strain are useful. They provide a method of comparing materials of different thicknesses and original lengths 9

10. Stress -Strain Graphs Linear region where Hooke’s law is obeyed
The limit of proportionality
Elastic Limit – point where the material stops returning to its original length
Yield point(s) where the material ‘necks’
Ultimate Tensile Stress (U.T.S.)
B. breaking point 10

11. Materials :Definitions 2
Elastic Limit: The maximum amount a material can be stretched by a force and still return to its original shape and size. The material has no permanent change in shape or size
Yield Point: Beyond the elastic limit, a point is reached at which there is a noticeably larger permanent change in length. This results in plastic behaviour
Ultimate Tensile Strength: The maximum stress that can be applied without breaking 11

12. Plasticity: A plastic material does not return to its original size and shape when the force is removed. There is a permanent stretching and change of shape
Stiffness: A measure of how difficult it is to change the size or shape of a material.
Thick steel wire is stiffer than thin steel wire of the same length.
Short steel wire is stiffer than longer steel wire of the same diameter.
Steel is stiffer than copper of the same diameter and length, because copper extends more per unit force 12

13. This is known as Young’s modulus
The linear region where Hooke’s law is obeyed is of interest and allows us to compare materials.
This is known as Young’s modulus 13

14. Stress -Strain curves
Comparing stress/strain graphs of brittle and ductile materials for example glass and copper 14

15. Stress -Strain curves 15

16. (for tension and compression)
Young’s Modulus
YOUNG'S MODULUS - CHANGE IN LENGTH
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
stress
strain
Y =
Young's Modulus
(for tension and compression)
F/A
L/L0
Y =
© 2006 By Timothy K. Lund
FL0
AL
Y =
We can solve the above equation for L to get a useful relationship:
Deformation Under Compression or Tension
FL0 AY
L =

17. 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.
Stress strain curves
FYI: Thus tables showing properties of materials can simply show a single number for that material, rather than a graph.
FYI: For this particular material the elastic modulus is
stress
strain
40000 n/m2
5010-8
elastic modulus = =
= 8 1010 n/m2
FYI: Structural engineers NEVER design a structural component to exceed its ELASTIC LIMIT.
elastic region
yield region
failure region
plastic region
70000
60000
50000
40000
30000
20000
10000
strain (10-8)
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:
yield point
elastic limit
breaking point
stress (N/m2)
permanent set
© 2006 By Timothy K. Lund
stress
strain
elastic modulus =

18. Solids and Fluids 9-1 Solids and Elastic Moduli
FYI: Since the cable is under TENSION it STRETCHES so that its length under the load is = m.
FYI: The typical stretches for cables (and compressions for columns) is extremely small.
YOUNG'S MODULUS - CHANGE IN LENGTH
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 = 401010 n/m2. A A
A = r2
F = mg 2 d 2 L0 L
F = (500)(1000)(10)
A = 
F = 5106 n 2
.01 2
A = 
© 2006 By Timothy K. Lund
A = 7.85410-5 m2 L
FL0 AY
(5106)(15)
(7.85410-5 )(401010)
L = =
L = 2.4 m

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