Stress

Stress is the ratio of applied force F and cross section A, defined as "force per area".

Direct Stress or Normal Stress

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

σ = Fn / A         (1)

where

σ = normal stress ((Pa) N/m2, psi)

Fn = normal component force (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 kilonewtons (kN)

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)

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 = parallel component force (N, lbf)

A = area (m2, in2)

Strain

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 = unitless measure of engineering strain

E = Young's modulus (Modulus of Elasticity) (Pa, psi)

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. The change of length can be calculated by transforming (3) as

 dl = σ lo / E

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

     = 0.00127 (m)

     = 1.27 (mm)

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

Most metals have deformations that are proportional with the imposed loads over a range of loads. Stress is proportional to load and strain is proportional to deformation expressed by the Hooke's law like

E = stress / strain = (Fn / A) / (dl / lo)         (4)

where

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

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

Shear Modulus

S = stress / strain = (Fp / A) / (s / d)         (5)

where

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

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

Material

Young's Modulus Shear Modulus Bulk Modulus

1010 N/m2 106 lb/in2 1010 N/m2 106 lb/in2 1010 N/m2 106 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

Young Modulus:

It is convenient to express the elasticity of a material with the ratio stress to strain, a parameter also termed the tensile elastic modulus or Young's modulus of the material. This is usually given the symbol - E.

Modulus of Elasticity for some common metals at various temperatures according ASME B31.1-1995 are indicated below:

1 psi (lb/in2) = 1 psi (lb/in2) = 144 psf (lbf/ft2) = 6,894.8 Pa (N/m2) = 6.895x10-3 N/mm2

T(oC) = 5/9[T(oF) - 32]

Young's Modulus of Elasticity - E - (106 psi)

Metal

Temperature (oC)

-200

-129

-73 21 93 149 204 260 316 371 427 482 538 593 649

Temperature (oF)

-325

-200

-100

70 200 300 400 500 600 700 800 900100

0110

0120

0

Cast iron

Gray cast iron

13.4

13.2

12.9

12.6

12.2

11.7

11.0

10.2

Steel

Carbon steel C

<= 0.3%

31.4

30.8

30.2

29.5

28.8

28.3

27.7

27.3

26.7

25.5

24.2

22.4

20.4 18.0

Carbon steel C

=> 0.3%

31.2

30.6

30.0

29.3

28.6

28.1

27.5

27.1

26.5

25.3

24.0

22.2

20.2 17.9 15.4

Carbon-moly steels

31.1

30.5

29.9

29.2

28.5

28.0

27.4

27.0

26.4

25.3

23.9

22.2

20.1 17.8 15.3

Nickel steels Ni 2% - 9%

29.6

29.1

28.5

27.8

27.1

26.7

26.1

25.7

25.2

24.6

23.0

Cr-Mo steels Cr 1/2% -

2%

31.6

31.0

30.4

29.7

29.0

28.5

27.9

27.5

26.9

26.3

25.5

24.8

23.9 23.0 21.8

Young's Modulus of Elasticity - E - (106 psi)

Metal

Temperature (oC)

-200

-129

-73 21 93 149 204 260 316 371 427 482 538 593 649

Temperature (oF)

-325

-200

-100

70 200 300 400 500 600 700 800 900100

0110

0120

0

Cr-Mo steels Cr 2 1/4% -

3%

32.6

32.0

31.4

30.6

29.8

29.4

28.8

28.3

27.7

27.1

26.3

25.6

24.6 23.7 22.5

Cr-Mo steels Cr 5% - 9%

32.9

32.3

31.7

30.9

30.1

29.7

29.0

28.6

28.0

27.3

26.1

24.7

22.7 20.4 18.2

Chromium steels Cr 12%,

17%, 27%

31.2

30.7

30.1

29.2

28.5

27.9

27.3

26.7

26.1

25.6

24.7

23.2

21.5 19.1 16.6

Austenitic steels

(TP304, 310, 316, 321, 347)

30.3

29.7

29.1

28.3

27.6

27.0

26.5

25.8

25.3

24.8

24.1

23.5

22.8 22.1 21.2

Copper and copper alloys

Comp. and

leaded-Sn

bronze (C83600,

14.8

14.6

14.4

14.0

13.7

13.4

13.2

12.9

12.5

12.0

Young's Modulus of Elasticity - E - (106 psi)

Metal

Temperature (oC)

-200

-129

-73 21 93 149 204 260 316 371 427 482 538 593 649

Temperature (oF)

-325

-200

-100

70 200 300 400 500 600 700 800 900100

0110

0120

0

C92200)

Naval brass Si

& Al bronze

(C46400, C65500, C95200, C95400)

15.9

15.6

15.4

15.0

14.6

14.4

14.1

13.8

13.4

12.8

Copper (C11000)

16.9

16.6

16.5

16.0

15.6

15.4

15.0

14.7

14.2

13.7

Copper red brass Al-bronze (C10200, C12000, C12200, C12500, C14200,  C23000,  C61400)

18.0

17.7

17.5

17.0

16.6

16.3

16.0

15.6

15.1

14.5

Nickel and Nickel Alloys

Young's Modulus of Elasticity - E - (106 psi)

Metal

Temperature (oC)

-200

-129

-73 21 93 149 204 260 316 371 427 482 538 593 649

Temperature (oF)

-325

-200

-100

70 200 300 400 500 600 700 800 900100

0110

0120

0

Monel 400

(N04400)

27.8

27.3

26.8

26.0

25.4

25.0

24.7

24.3

24.1

23.7

23.1

22.6

22.1 21.7 21.2

Titanium

Unalloyed titanium grades 1, 2, 3 and

7

15.5

15.0

14.6

14.0

13.3

12.6

11.9

11.2

Aluminum and aluminum alloys

Grades 443,

1060, 1100, 3003, 3004, 6063

11.1

10.8

10.5

10.0

9.6 9.2 8.7

1 psi (lb/in2) = 6,894.8 N/m2 (Pa) T(oC) = 5/9[T(oF) - 32]

Strength:

To describe elastic properties of linear objects like wires, rods, or columns which are stretched or compressed, a convenient parameter is the ratio of the stress to the strain, a parameter called the "Young's modulus" or "Modulus of Elasticity" of the material. Young's modulus can be used to predict the elongation or compression of an object as long as the stress is less than the yield strength of the material.

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

ABS plastics   2.3 40

Acrylic   3.2 70

Aluminum 10.0 69 110 95

Aluminium Bronze 120

Antimony 11.3      

Aramid 70 - 112

Beryllium (Be) 42  287    

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

Bismuth 4.6      

Bone   9170

(compression)

Boron   3100

Brass   102 - 125 250

Brass, Naval 100

Bronze   96 - 120

Cadmium 4.6      

Carbon Fiber Reinforced Plastic   150

Cast Iron 4.5% C, ASTM A-48   170

Chromium 36      

Cobalt 30      

Concrete, High Strength (compression)

  3040

(compression)

Copper 17 117 220 70

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

Diamond (C)   1220

Douglas fir Wood   1350

(compression)

Fiberboard, Medium Density 4

Flax fiber 58

Glass   50 - 9050

(compression)

Glass reinforced polyester matrix

17

Graphene 1000

Grey Cast Iron 130

Gold 10.8      

Hemp fiber 35

Iridium 75      

Iron 28.5      

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

Lead 2.0      

Magnesium metal (Mg) 6.4 45

Manganese 23      

Marble   15

Mercury        

Molybdenum (Mo) 40  329    

Nickel 31      

Niobium (Columbium) 15      

Nylon   2 - 4 75 45

Oak Wood (along grain)   11

Osmium (Os) 80  550    

Phosphor Bronze 116

Pine Wood (along grain)   9 40

Platinum 21.3      

Plutonium 14      

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

Polycarbonate   2.6 70

Polyethylene HDPE (high density)

  0.8 15

Polytehylene, LDPE (low density)

0.238

Polyethylene Terephthalate, PET

  2 - 2.7 55

Polyimide   2.5 85

Polypropylene   1.5 - 2 40

Polystyrene   3 - 3.5 40

Potassium        

Rhodium 42      

Rubber   0.01 - 0.1

Selenium 8.4      

Silicon 16  130 - 185    

Silicon Carbide   450 3440

Silver 10.5      

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

Sodium        

Stainless Steel, AISI 302   180 860 502

Steel, Structural ASTM-A36   200 400 250

Steel, High Strength Alloy ASTM A-514

  760 690

Tantalum 27      

Teflon. PTFE 0.5

Thorium 8.5      

Titanium 16      

Titanium Alloy   105 - 120 900 730

Tungsten (W)   400 - 410

Tungsten Carbide (WC)   450 - 650

Uranium 24      

Vanadium 19      

Wrought Iron   190 - 210    

Material

Young's Modulus (Modulus of Elasticity)

- E -

Ultimate Tensile

Strength- Su -

(106 N/m2, MPa)

Yield Strength

- Sy -(106 N/m2,

MPa)(106  psi)(109 N/m2,

GPa)

Zinc 12  

1 N/m2 = 1x10-6 N/mm2 = 1 Pa = 1.4504x10-4 psi 1 psi (lb/in2) = 144 psf (lbf/ft2) = 6,894.8 Pa (N/m2) = 6.895x10-3 N/mm2

Note! Use the pressure unit converter on this page to switch the values to other units.

Strain

Strain can be expressed as

strain = dL / L         (1)

where

strain = (m/m) (in/in)

dL = elongation or compression (offset) of the object (m) (in)

L = length of the object (m) (in)

Stress

Stress can be expressed as

stress = F / A         (2)

where

stress = (N/m2) (lb/in2, psi)

F = force (N) (lb)

A = area of object (m2) (in2)

Young's Modulus (Tensile Modulus)

Young's modulus or Tensile modulus can be expressed as

E = stress / strain = (F / A) / (dL / L)         (3)

where

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

Elasticity

Elasticity is a property of an object or material which will restore it to its original shape after distortion.

A spring is an example of an elastic object - when stretched, it exerts a restoring force which tends to bring it back to its original length. This restoring force is in general proportional to the stretch described by Hooke's Law.

Hooke's Law

One of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law which can be expressed as

Fs = -k dL         (4)

where

Fs = force in the spring (N)

k = spring constant (N/m)

dL = elongation of the spring (m)

Yield strength

Yield strength, or the yield point, is defined in engineering as the amount of stress that a material can undergo before moving from elastic deformation into plastic deformation.

Ultimate Tensile Strength

The Ultimate Tensile Strength - UTS - of a material is the limit stress at which the material actually breaks, with sudden release of  the stored elastic energy.