Giáo trình Strength of Materials - Chapter 5: Failure theories

CHAPTER 5: FAILURE THEORIES 
Failure – A part is permanently distorted (bị bóp méo) 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 = 0.67 Su 
CHAPTER 5: FAILURE THEORIES 
A 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) 
(Thuyết bền: ứng suất tiếp lớn nhất – TB3) 
Yield strength of a material is used to design components made of 
ductile material 
 = Sy 
 = Sy 
 =Sy 
(max )component > ( )obtained from a tension test at the yield point Failure 
(max )component < 
Sy 
2 
To avoid failure 
max = 
Sy 
2 n 
n = Safety factor 
Design equation 
 Sy 
2 
=  
FAILURE THEORIES – DUCTILE MATERIALS 
SPECIAL CASES 
A special planar state of stress 
σx= σ 
τxy= τ 
Purely shear state of stress 
τxy= τ 
 
n
S y
TB
22
4 223 


 
n
S y
22



FAILURE THEORIES – DUCTILE MATERIALS 
• Distortion energy theory (von Mises-Hencky) 
(Thuyết bền: Thế năng biến đổi hình dáng lớn nhất – TB4) 
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 → failure 
FAILURE THEORIES – DUCTILE MATERIALS 
The area under the curve in the elastic region is called the Elastic Strain Energy. 
U = ½ ε 
3D case 
UT = ½ 1ε1 + ½ 2ε2 + ½ 3ε3 
ε1 = 
1 
E 
2 
E 
3 
E 
v v 
ε2 = 
2 
E 
1 
E 
3 
E 
v v 
ε3 = 
3 1 
E 
2 
E 
v v 
Stress-strain relationship 
E 
UT = (1
2 + 2
2 + 3
2) - 2v (12 + 13 + 23) 
2E 
1 
FAILURE THEORIES – DUCTILE MATERIALS 
Ud = UT – Uh 
Distortion strain energy = total strain energy – hydrostatic strain energy 
Substitute 1 = 2 = 3 = h 
Uh = (h
2 + h
2 + h
2) - 2v (hh + hh+ hh) 
2E 
1 
Simplify and substitute 1 + 2 + 3 = 3h into the above equation 
Uh = (1 – 2v) = 
2E 
3
h
2 
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 = (1
2 + 2
2 + 3
2) - 2v (12 + 13 + 23) 
2E 
1 
(1) 
(2) 
FAILURE THEORIES – DUCTILE MATERIALS 
Strain energy from a tension test at the yield point 
1= Sy and 2 = 3 = 0 Substitute in equation (2) 
3E 
1 + v 
(Sy)
2 
Utest = 
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 
FAILURE THEORIES – DUCTILE MATERIALS 
½ 
2D case, 3 = 0 
(1
2 – 12 + 2
2) < Sy =  
Where  is von Mises stress 
′ = 
Sy 
n 
Design equation 
(1 – 2)
2 + (1 – 3)
2 + (2 – 3)
2 
2 
½ 
< Sy 
FAILURE THEORIES – DUCTILE MATERIALS 
Pure torsion,  = 1 = – 2 
(1
2 – 2 1 + 2
2) = Sy
2
3
2
 = Sy
2 Sys = Sy / √ 3 → Sys = .577 Sy 
Relationship between yield strength in 
tension and shear 
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) 
σx= σ 
τxy= τ 
τxy= τ 
 
n
S y
TB
33
3
2
1
2
14 


 
n
S y
33


or 
FAILURE THEORIES – DUCTILE MATERIALS 
′ = 
Sy 
n 
max = 
Sy 
2n 
Maximum shear stress theory Distortion 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 Cost Weight Size 
FAILURE THEORIES – DUCTILE MATERIALS 
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. 
FAILURE THEORIES – DUCTILE MATERIALS 
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 
n 
max = 
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 
 
 
Sut 3 1 Stress 
state 
FAILURE THEORIES – BRITTLE MATERIALS 
Sut 
Suc 
Sut 
Suc 
Safe 
Safe 
Safe Safe 
Cast iron data 
Modified Coulomb-Mohr theory 
(Thuyết Bền Morh – TB5) 
1 
3 or 2 
Sut 
Sut 
Suc 
-Sut 
I 
II 
III 
Three design zones 
3 or 2 
FAILURE THEORIES – BRITTLE MATERIALS 
Modified Coulomb-Mohr theory 
1 
2 
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 Sut 
1 ( 
1 
Sut 
1 
Suc 
 – ) – 
2 
Suc 
= 
1 
n 
Design equation 
1 = 
Sut 
n 
Design equation 
1 = 
Sut 
n Design equation