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Ken Youssefi Mechanical Engineering Dept., SJSU 2

Failure Theories – Static Loads

Static load – a stationary load that is gradually applied having an unchanging magnitude and direction

Failure – A part is permanently distorted and will not function properly.

A part has been separated into two or more pieces.

Material Strength

Sy = Yield strength in tension, Syt = Syc

Sys = Yield strength in shear

Su = Ultimate strength in tension, Sut

Suc = Ultimate strength in compression

Sus = Ultimate strength in shear = .67 Su




Ductile and Brittle MaterialsA ductile material deforms significantly before fracturing. Ductility is measured by % elongation at the fracture point. Materials with 5% or more elongation are considered ductile.

Brittle material yields very little before fracturing, the yield strength is approximately the same as the ultimate strength in tension. The ultimate strength in compression is much larger than the ultimate strength in tension.




Failure Theories – Ductile Materials

• Maximum shear stress theory (Tresca 1886)

Yield strength of a material is used to design components made of ductile material

= Sy

Sy

2=

= Sy

=Sy

(max )component > ( )obtained from a tension test at the yield point Failure

(max )component < Sy

2

To avoid failure

max = Sy

2 nn = Safety factor

Design equationadmission.edhole.com




• Distortion energy theory (von Mises-Hencky)

Hydrostatic state of stress → (Sy)h

h

h

h

t

t

Simple tension test → (Sy)t

(Sy)t(Sy)h >>

Distortion contributes to failure much more than change in volume.

(total strain energy) – (strain energy due to hydrostatic stress) = strain energy due to angular distortion > strain energy obtained from a tension test at the yield point → failureadmission.edhole.com




The area under the curve in the elastic region is called the Elastic Strain Energy.

Strain energy

U = ½ ε

3D case

UT = ½ 1ε1 + ½ 2ε2 + ½ 3ε3

ε1 = 1

E

2

E

3

Evv

ε2 = 2

E

1

E

3

Evv

ε3 = 3 1

E

2

Evv

Stress-strain relationship

E

UT = (12 + 2

2 + 32) - 2v (12 + 13 + 23) 2E

1admission.edhole.com




Ud = UT – Uh

Distortion strain energy = total strain energy – hydrostatic strain energy

Substitute 1 = 2 = 3 = h

Uh = (h2 + h

2 + h2) - 2v (hh + hh+ hh) 2E

1

Simplify and substitute 1 + 2 + 3 = 3h into the above equation

Uh = (1 – 2v) =2E

3h2

6E

(1 – 2v)(1 + 2 + 3)2

Ud = UT – Uh = 6E

1 + v(1 – 2)

2 + (1 – 3)2 + (2 – 3)

2

Subtract the hydrostatic strain energy from the total energy to obtain the distortion energy

UT = (12 + 2

2 + 32) - 2v (12 + 13 + 23) 2E

1 (1)

(2)admission.edhole.com



Strain energy from a tension test at the yield point

1= Sy and 2 = 3 = 0 Substitute in equation (2)

3E

1 + v(Sy)

2Utest =

To avoid failure, Ud < Utest

(1 – 2)2 + (1 – 3)

2 + (2 – 3)2

2

½ < Sy

Ud = UT – Uh = 6E

1 + v(1 – 2)

2 + (1 – 3)2 + (2 – 3)

2 (2)




½

2D case, 3 = 0

(12 – 12 + 2

2) < Sy = Where is von Mises stress

′ = Sy

nDesign equation

(1 – 2)2 + (1 – 3)

2 + (2 – 3)2

2

½ < Sy



Pure torsion, = 1 = – 2

(12 – 2 1 + 2

2) = Sy2


32 = Sy2

Sys = Sy / √ 3 → Sys = .577 Sy

Relationship between yield strength in tension and shear

(x)2 + 3(xy)

2 =

Sy

n

1/2

If y = 0, then 1, 2 = x/2 ± [(x)/2]2 + (xy)2

the design equation can be written in terms of the dominant component stresses (due to bending and torsion)



Design Process

′ = Sy

nmax =

Sy

2n

Maximum shear stress theoryDistortion energy theory

• Select material: consider environment, density, availability → Sy , Su

• Choose a safety factor

The selection of an appropriate safety factor should be based on the following:

Degree of uncertainty about loading (type, magnitude and direction)

Degree of uncertainty about material strength

Type of manufacturing process

Uncertainties related to stress analysis

Consequence of failure; human safety and economics

Codes and standards

n CostWeightSize



Design Process

Use n = 1.2 to 1.5 for reliable materials subjected to loads that can be determined with certainty.

Use n = 1.5 to 2.5 for average materials subjected to loads that can be determined. Also, human safety and economics are not an issue.

Use n = 3.0 to 4.0 for well known materials subjected to uncertain loads.



Design Process

• Formulate the von Mises or maximum shear stress in terms of size.

• Optimize for weight, size, or cost.

• Select material, consider environment, density, availability → Sy , Su

• Choose a safety factor

• Use appropriate failure theory to calculate the size.

′ = Sy

nmax =

Sy

2n



Failure Theories – Brittle Materials

One of the characteristics of a brittle material is that the ultimate strength in compression is much larger than ultimate strength in tension.

Suc >> Sut

Mohr’s circles for compression and tension tests.

Compression test

Suc

Failure envelope

The component is safe if the state of stress falls inside the failure envelope.

1 > 3 and 2 = 0

Tension test

Sut3 1Stress

state




1

Sut

Suc

Sut

Suc

Safe

Safe

Safe Safe

-Sut

Cast iron data

Modified Coulomb-Mohr theory

1

3 or 2

Sut

Sut

Suc

-Sut

I

II

III

Three design zones

3 or 2




1

3

Sut

Sut

Suc

-Sut

I

II

III

Zone I

1 > 0 , 2 > 0 and 1 > 2

Zone II

1 > 0 , 2 < 0 and 2 < Sut

Zone III

1 > 0 , 2 < 0 and 2 > Sut 1 (1

Sut

1

Suc

– ) –2

Suc=

1n

Design equation

1 =Sut

nDesign equation

1 =Sut

nDesign equation


Admission in india

Education

elastic strain energy

stressstrain relationship1

theyield strength

theultimate strength

yield point failure

hydrostatic stress

yield strength intension

appropriate failure