CE 305 : Mechanics of Solids I
Conducted by,
Niloy Karmoker
Lecturer, Dept. of CE
Barishal Engineering College
Email: Srabonkarm@gmail.com
5.1 Classifying Loads on Materials
• Normal Load (Axial load): Load is perpendicular to the
supporting material.
- Tension Load: As the ends of material are pulled apart
to make the material longer, the load is called a tension
load.
- Compression Load: As the ends of material are pushed in
to make the material smaller, the load is called
a compression load.
Tension
Compression
Classifying Loads on Materials
• Shear Load : Tangential load
pulling apart
Cargo
Pressure
Classifying Loads on Materials
•Torsion Loads: Angular distortion on a component, such as a
shaft, when a moment is applied. (Twisting)
•Thermal Loads: Distortion caused be heating or cooling a
material. A normal load is created when the material is
constrained in any direction in the plane that is constrained.
5.2 Stress and Strain
In order to compare materials, we must have measures.
• Stress : load per unit Area
F
σ
A
F : load applied in pounds
A : cross sectional area in in²
σ : stress in psi
A
F F
Stress and Strain
• Strain:
- Ratio of elongation of a material to the original length
- unit deformation
Lo e
e
ε
Lo L
e : elongation (ft)
Lo : unloaded(original) length of a material (ft)
ε : strain (ft/ft) or (in/in)
Elongation:
e L Lo
L : loaded length of a material (ft)
Baldwin Hydraulic Machine for Tension & Compression test
Stress-Strain Diagram
• A plot of Strain vs. Stress.
•The diagram gives us the behavior of the material and
material properties.
• Each material produces a different stress-strain
diagram.
Stress-Strain Diagram
ultimate
tensile
strength 3 necking
UT S
Strain
yield Fracture
strength Hardening
y 5
2
Elastic region
Plastic slope=Young’s(elastic) modulus
Region yield strength
Plastic region
ultimate tensile strength
Elastic strain hardening
σ Eε Region
4
fracture
σ 1
E
ε E
σy
Strain ( ) (e/Lo)
ε 2 ε1
Stress-Strain Diagram
• Elastic Region (Point 1 –2)
- The material will return to its original shape
after the material is unloaded( like a rubber band).
- The stress is linearly proportional to the strain in
this region.
σ
σ Eε or E
ε
σ : Stress (psi)
E : Elastic modulus (Young’s Modulus) (psi)
ε : Strain (in/in)
- Point 2 : Yield Strength : a point at which permanent
deformation occurs. ( If it is passed, the material will
no longer return to its original length.)
Stress-Strain Diagram
ultimate
tensile
strength 3 necking
UT S
Strain
yield Fracture
strength Hardening
y 5
2
Elastic region
Plastic slope=Young’s(elastic) modulus
Region yield strength
Plastic region
ultimate tensile strength
Elastic strain hardening
σ Eε Region
4
fracture
σ 1
E
ε E
σy
Strain ( ) (e/Lo)
ε 2 ε1
Stress-Strain Diagram
The ELASTIC Range Means:
- The strain, or elongation over a unit length, will behave linearly (as in
y=mx +b) and thus predictable.
-The material will return to its original shape (Point 1) once an applied load
is removed.
- The stress within the material is less than what is required to create a
plastic behavior (deform or stretch significantly without increasing stress).
Stress-Strain Diagram
Plastic Region (Point 2 –3)
- If the material is loaded beyond the yield strength,
the material will not return to its original shape
after unloading.
- It will have some permanent deformation.
- If the material is unloaded at Point 3, the curve will
proceed from Point 3 to Point 4. The slope will be
the as the slope between Point 1 and 2.
- The distance between Point 1 and 4 indicates the
amount of permanent deformation.
Stress-Strain Diagram
ultimate
tensile
strength 3 necking
UT S
Strain
yield Fracture
strength Hardening
y 5
2
Elastic region
Plastic slope=Young’s(elastic) modulus
Region yield strength
Plastic region
ultimate tensile strength
Elastic strain hardening
σ Eε Region
4
fracture
σ 1
E
ε E
σy
Strain ( ) (e/Lo)
ε 2 ε1
Stress-Strain Diagram
Strain Hardening
- If the material is loaded again from Point 4, the
curve will follow back to Point 3 with the same
Elastic Modulus(slope).
- The material now has a higher yield strength of
Point 4.
- Raising the yield strength by permanently straining
the material is called Strain Hardening.
Stress-Strain Diagram
ultimate
tensile
strength 3 necking
UT S
Strain
yield Fracture
strength Hardening
y 5
2
Elastic region
Plastic slope=Young’s(elastic) modulus
Region yield strength
Plastic region
ultimate tensile strength
Elastic strain hardening
σ Eε Region
4
fracture
σ 1
E
ε E
σy
Strain ( ) (e/Lo)
ε 2 ε1
Stress-Strain Diagram
Tensile Strength (Point 3)
- The largest value of stress on the diagram is called
Tensile Strength(TS) or Ultimate Tensile Strength
(UTS)
- It is the maximum stress which the material can
support without breaking.
Fracture (Point 5)
- If the material is stretched beyond Point 3, the stress
decreases as necking and non-uniform deformation
occur.
- Fracture will finally occur at Point 5.
Stress-Strain Diagram
ultimate
tensile
strength 3 necking
UT S
Strain
yield Fracture
strength Hardening
y 5
2
Elastic region
Plastic slope=Young’s(elastic) modulus
Region yield strength
Plastic region
ultimate tensile strength
Elastic strain hardening
σ Eε Region
4
fracture
σ 1
E
ε E
σy
Strain ( ) (e/Lo)
ε 2 ε1
Stress-Strain Diagram
A36 Steel
Material Properties
Characteristics of Material are described as
• Strength
• Hardness
• Ductility
• Brittleness
• Toughness
Material Properties
Strength:
- Measure of the material property to resist deformation
and to maintain its shape
- It is quantified in terms of yield stress y or ultimate
tensile strength ult .
- High carbon steels and metal alloys have higher strength
than pure metals.
- Ceramic also exhibit high strength characteristics.
Material Properties
Hardness:
- Measure of the material property to resist indentation,
abrasion and wear.
- It is quantified by hardness scale such as Rockwell and
Brinell hardness scale that measure indentation /
penetration under a load.
- Hardness and Strength correlate well because both
properties are related to inter-molecular bonding. A
high-strength material is typically resistant to wear
and abrasion.
A comparison of hardness of some typical materials:
Material Brinell Hardness
Pure Aluminum 15
Pure Copper 35
Mild Steel 120
304 Stainless Steel 250
Hardened Tool Steel 650/700
Hard Chromium Plate 1000
Chromium Carbide 1200
Tungsten Carbide 1400
Titanium Carbide 2400
Diamond 8000
Sand 1000
Material Properties
Ductility:
- Measure of the material property to deform before failure.
- It is quantified by reading the value of strain at the
fracture point on the stress strain curve.
- Ductile materials can be pulled or drawn into pipes, wire,
and other structural shapes
- Examples of ductile material :
low carbon steel
aluminum
copper
brass
Material Properties
Brittleness:
- Measure of the material’s inability to deform before failure.
- The opposite of ductility.
- Example of ductile material : glass, high carbon steel,
ceramics
Brittle
Ductile
Strain
Material Properties
Toughness:
- Measure of the material ability to absorb energy.
- It is measured by two methods.
a) Integration of stress strain curve
- Slow absorption of energy
- Absorbed energy per unit volume
unit : (lb/in²) *(in/in) =lb·in/in³
b) Charpy test
- Ability to absorb energy of an impact without
fracturing.
- Impact toughness can be measured.
Material Properties
Fatigue:
• The repeated application of stress typically produced by
an oscillating load such as vibration.
• Sources of ship vibration are engine, propeller and waves.
MAXIMUM stress decreases as the number of loading cycles increases.
Endurance Limit : A certain threshold
Steel stress which will not cause the fatigue
failure for the number of cycles.
Aluminum Aluminum has no endurance limit
Cycles N at Fatigue Failure
Factors effecting Material Properties
Temperature :
Increasing temperature will:
- Decrease Modulus of Elasticity
(As Long as Structure Does Not Change)
- Decrease Yield Strength
- Decrease Ultimate Tensile Strength
- Decrease Hardness
- Increase Ductility
- Decrease Brittleness
Environment:
- Sulfites, Chlorine, Oxygen in water,
Radiation, Pressure
Ways to Effect / Alter Material Properties
Alloying (Adding other elements to alter the molecular properties):
- Steel: Carbon, chromium, molybdenum, nickel, tungsten,
manganese
- Aluminum: Copper, manganese, silicon, zinc, magnesium
Thermal Treatments (Application of heat over varying time):
Annealing:
- Heating higher than its critical temperature then
cooling slowly.
- Improves hardness, strength, and ductility.
- Ship’s hulls are annealed.
Hardening:
- Heating higher than its critical temperature then
cooling rapidly.
- Improves hardness.
- Increases internal stresses, may cause cracking.
Ways to Effect / Alter Material Properties
Thermal Treatments:
Tempering:
- Steel is heated below the critical temperature and
cooled slowly.
- Used with hardening to reduce the internal stresses.
Hot-Working:
- Forming of shapes while material is hot.
- Less internal stresses due to annealing (change in
the molecular structure).
Cold-Working:
- Forming shapes while material is cold.
- Causes internal stresses, resulting in a stronger shape.
Corrosion & Corrosion Protection
Corrosion is the destruction of metals due to oxidation or
other chemical reactions.
Corrosion Protection:
- Design to eliminate conditions favorable to corrosion
- You, a wire brush, and paint
- Cathodic Protection
- Charging the metal to slow/ stop reaction
with other elements
- Providing a sacrificial metal to give up ions
instead of the structure giving up ions (and
corroding)
Example:
Mooring line length =100 ft
diameter=1.0 in
Axial loading applied=25,000 lb mooring line
Elongation due to loading=1.0 in
1) Find the normal stress.
F 25,000 lb loading
2
31,800 psi
A 0.785 in
A r 2 (0.5in)2 0.785 in2
2) Find the strain.
e 1in
0.00083 (in / in)
Lo 100 ft 12 in
1 ft
Example:
y 60,000 psi
- Salvage crane is lifting an object of 20,000 lb.
- Characteristics of the cable UT 70,000 psi
diameter=1.0 in, length prior to lifting =50 ft E 35 106 psi
1) Find the normal stress in the cable.
F 20,000 lb
2
25,478 psi
A 0.785 in
(A r 2 (0.5 in)2 0.785 in2 )
2) Find the strain.
25,478 psi
0.000728 (in / in)
E 35 10 psi
6
3) Determine the cable stretch in inches.
e
Lo
12in
e Lo (0.000728 in / in) (50 ft ) 0.44 in
1 ft
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