# Stress, Strain and Young's Modulus

## Stress is force per unit area - strain is the deformation of a solid due to stress

### Stress

Stress is "force per unit area" - the ratio of applied force F to cross section area - defined as "force per area".

• tensile stress - stress that tends to stretch or lengthen the material - acts normal to the stressed area
• compressive stress - stress that tends to compress or shorten the material - acts normal to the stressed area
• shearing stress - stress that tends to shear the material - acts in plane to the stressed area at right-angles to compressive or tensile stress

#### Tensile or Compressive Stress - Normal Stress

Tensile or compressive stress normal to the plane is usually denoted "normal stress" or "direct stress" and can be expressed as

σ = Fn / A                                    (1)

where

σ = normal stress ((Pa) N/m2, psi)

Fn = normal force acting perpendicular to the area (N, lbf (alt. kips))

A = area (m2, in2)

• a kip is a non-SI unit of force - it equals 1,000 pounds-force
• 1 kip = 4448.2216 Newtons (N) = 4.4482216 kilo Newtons (kN)

The normal force acts perpendicular to the area and is developed whenever external loads tends to push or pull the two segments of a body.

#### Example - Tensile Force acting on a Rod

A force of 10 kN is acting on a circular rod with diameter 10 mm. The stress in the rod can be calculated as

σ = (10 103 N) / (π ((10 10-3 m) / 2)2)

= 127388535 (N/m2

= 127 (MPa)

#### Example - Force acting on a Douglas Fir Square Post

A compressive load of 30000 lb is acting on short square 6 x 6 in post of Douglas fir. The dressed size of the post is 5.5 x 5.5 in and the compressive stress can be calculated as

σ = (30000 lb) / ((5.5 in) (5.5 in))

= 991 (lb/in2, psi)

#### Shear Stress

Stress parallel to the plane is usually denoted "shear stress" and can be expressed as

τ = Fp / A                               (2)

where

τ = shear stress ((Pa) N/m2, psi)

Fp = shear force in the plane of the area (N, lbf)

A = area (m2, in2)

The shear force lies in the plane of the area and is developed when external loads tend to cause the two segments of a body to slide over one another.

### Strain (Deformation)

Strain is defined as "deformation of a solid due to stress" and can be expressed as

ε = dl / lo

= σ / E                              (3)

where

dl = change of length (m, in)

lo = initial length (m, in)

ε = strain - unitless

E = Young's modulus (Modulus of Elasticity) (N/m2 (Pa), lb/in2 (psi))

• Young's modulus can be used to predict the elongation or compression of an object

Note that the normal strain is a dimensionless unit since it is a ratio of two lengths. But it also common to practice to state it as a ratio of two length units - like m/m or in/in.

#### Example - Stress and Change of Length

The rod in the example above is 2 m long and made of steel with Modulus of Elasticity 200 GPa (200 109 N/m2). The change of length can be calculated by transforming (3) as

dl = σ l/ E

= (127 106 Pa) (2 m) / (200 109 Pa)

= 0.00127 (m)

= 1.27 (mm)

#### Strain Energy

Stressing a bar stores energy in it. For an axial load the energy stored can be expressed as

U = 1/2 Fn dl

where

U = deformation energy (J (N m), ft lb)

### Young's Modulus - Modulus of Elasticity (or Tensile Modulus) - Hooke's Law

Most metals deforms proportional to imposed load over a range of loads. Stress is proportional to load and strain is proportional to deformation as expressed with Hooke's law

E = stress / strain

= σ / ε

= (Fn / A) / (dl / lo)                                     (4)

where

E = Young's modulus (N/m2) (lb/in2, psi)

Modulus of Elasticity, or Young's Modulus, is commonly used for metals and metal alloys and expressed in terms 106 lbf/in2, N/m2 or Pa. Tensile modulus is often used for plastics and is expressed in terms 105 lbf/in2 or GPa.

### Shear Modulus

S = stress / strain

= τ / γ

= (Fp / A) / (s / d)                                    (5)

where

S = shear modulus (N/m2) (lb/in2, psi)

τ  = shear stress ((Pa) N/m2, psi)

γ = unit less measure of shear strain

Fp = force parallel to the faces which they act

A = area (m2, in2)

s = displacement of the faces (m, in)

d = distance between the faces displaced (m, in)

### Elastic Moduli

Elastic moduli for some common materials:

MaterialYoung's ModulusShear ModulusBulk Modulus
1010 N/m2106 lb/in21010 N/m2106 lb/in21010 N/m2106 lb/in2
Aluminum 7.0 10 2.4 3.4 7.0 10
Brass 9.1 13 3.6 5.1 6.1 8.5
Copper 11 16 4.2 6.0 14 20
Glass 5.5 7.8 2.3 3.3 3.7 5.2
Iron 9.1 13 7.0 10 10 14
Lead 1.6 2.3 0.56 0.8 0.77 1.1
Steel 20 29 8.4 12 16 23

## Related Topics

• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more
• Statics - Loads - force and torque, beams and columns

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