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# 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 the ratio of applied force F to a cross section area - defined as "force per unit 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 (lbf/in2))

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

A = area (m2, in2)

• a kip is an imperial unit of force - it equals 1000 lbf (pounds-force)
• 1 kip = 4448.2216 Newtons (N) = 4.4482216 kilo Newtons (kN)

A normal force acts perpendicular to 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 a plane is usually denoted as "shear stress" and can be expressed as

τ = Fp / A                               (2)

where

τ = shear stress (Pa (N/m2), psi (lbf/in2))

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

A = area (m2, in2)

A shear force lies in the plane of an 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".

• Normal strain - elongation or contraction of a line segment
• Shear strain - change in angle between two line segments originally perpendicular

Normal strain 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 - unit-less

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

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

Note that strain is a dimensionless unit since it is the ratio of two lengths. But it also common practice to state it as the 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) to

dl = σ l/ E

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

= 0.00127 m

= 1.27 mm

#### Strain Energy

Stressing an object 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 of Elasticity - or Modulus of Rigidity

G = stress / strain

= τ / γ

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

where

G = Shear Modulus of Elasticity - or Modulus of Rigidity (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)

### Bulk Modulus Elasticity

The Bulk Modulus Elasticity - or Volume Modulus - is a measure of the substance's resistance to uniform compression. Bulk Modulus of Elasticity is the ratio of stress to change in volume of a material subjected to axial loading.

### Elastic Moduli

Elastic moduli for some common materials:

Stress, Strain and Young's Modulus for some common Materials
MaterialYoung's Modulus
- E -
Shear Modulus
- G -
Bulk Modulus
- K -
(GPa)
(106 psi)
(GPa)
(106 psi)
(GPa)
(106 psi)
Aluminum 70 24 70
Brass 91 36 61
Copper 110 42 140
Glass 55 23 37
Iron 91 70 100
Steel 200 84 160
• 1 GPa = 109 Pa (N/m2)
• 106 psi = 1 Mpsi = 103 ksi

## Related Topics

• ### Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.
• ### Statics

Loads - forces and torque, beams and columns.

## Related Documents

• ### Engineering Materials - Properties

Some typical properties of engineering materials like steel, plastics, ceramics and composites.
• ### Hooke's Law

Hooke's law - force, elongation and spring constant.
• ### Malleability vs. Brittlenes vs. Ductility

Plastic deformation properties.
• ### Metals and Alloys - Bulk Modulus Elasticity

The Bulk Modulus - resistance to uniform compression - for some common metals and alloys.
• ### Metals and Alloys - Young's Modulus of Elasticity

Elastic properties and Young's modulus for metals and alloys like cast iron, carbon steel and more.
• ### Modulus of Rigidity

Shear Modulus (Modulus of Rigidity) is the elasticity coefficient for shearing or torsion force.
• ### Poisson's Ratio

When a material is stretched in one direction it tends to get thinner in the other two directions.
• ### Poisson's Ratios Metals

Some metals and their Poisson's Ratios.
• ### Process Pipes - Allowable Stress vs. Temperature

Allowable wall stress in pipes according ASME M31.3.
• ### Restricted Thermal Expansion - Force and Stress

Stress and force when thermal expansion a pipe, beam or similar is restricted.
• ### Rotating Bodies - Stress

Stress in rotating disc and ring bodies.
• ### Steam Boiler Shells - Stress vs. Boiler Pressure

Calculate the stress in steam boiler shells caused by steam pressure.
• ### Steels - Endurance Limits and Fatigue Stress

Endurance limits and fatigue stress for steels.
• ### Stress

Stress is force applied on cross-sectional area.
• ### Stress in Thick-Walled Cylinders or Tubes

Radial and tangential stress in thick-walled cylinders or tubes with closed ends - with internal and external pressure.
• ### Stress in Thin-Walled Cylinders or Tubes

Hoop and longitudinal stress thin-walled tubes or cylinders.
• ### Structural Lumber - Properties

Properties of structural lumber.
• ### Threaded Bolts - Stress Area

Weight rating of threaded hanger rods.
• ### Young's Modulus, Tensile Strength and Yield Strength Values for some Materials

Young's Modulus (or Tensile Modulus alt. Modulus of Elasticity) and Ultimate Tensile Strength and Yield Strength for materials like steel, glass, wood and many more.

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## Citation

• The Engineering ToolBox (2005). Stress, Strain and Young's Modulus. [online] Available at: https://www.engineeringtoolbox.com/stress-strain-d_950.html [Accessed Day Month Year].

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9.19.12