Ss16MCE 321-Chapter 5 Failure-Theories2

Mechanical Design I (MCE 321) L. Romdhane, Summer 2015, 7:31 AM -- 1--

Summer 2016

Mechanical Design 1 (MCE 321)

Chapter 5 Failures Resulting from

Static Loading

Dr. Lotfi Romdhane [email protected]

5 Failures Resulting from Static Loading

Chapter

Outline

5-1 Static Strength

5-2 Stress Concentration

5-3 Failure Theories

5-4 Maximum-Shear-Stress Theory for Ductile Materials

5-5 Distortion-Energy Theory for Ductile Materials

5-6 Coulomb-Mohr Theory for Ductile Materials

5-7 Failure of Ductile Materials Summary

5-8 Maximum-Normal-Stress Theory for Brittle Materials

5-9 Modifications of the Mohr Theory for Brittle Materials

5-10 Failure of Brittle Materials Summary

5-11 Selection of Failure Criteria

5-12 Introduction to Fracture Mechanics

5-13 Stochastic Analysis

5-14 Important Design Equations

3

Failures

crack

4

Failures

Attributes of Failure? Part separation into two or more pieces

Permanent distorted with significant change in geometry

Downgraded reliability

Compromised function

Static Load? Stationary force or torque applied to a member with fixed

magnitude, point of application and direction.

Objective Establish a relationship between strength, static loading

and failure

5

Static Strength

• Ideally, in designing any machine element, the engineer should have access to the results of a great number of strength tests of the particular material chosen.

• It is necessary to design using only published values of yield strength, ultimate strength, percentage reduction in area, and percentage elongation.

Design Categories

• Failure of the part would endanger human life

• Large quantities are produced

• Small quantities or rapid prototyping

• The part was produced already but failed

6

Stress Concentration

• Stress concentration is a highly localized effect.

• Geometric (theoretical) stress-concentration factor for normal stress Kt and shear stress Kts is defined as

Stress concentration is more consequential in brittle materials than ductile ones.

7

Need for Static Failure Theories

• Uniaxial stress element (e.g. tension test)

• Multi-axial stress element

One strength, multiple stresses

How to compare stress state to single strength?

Strength Sn

Stress

8

Failure Theories

• Events such as distortion, permanent set, cracking, … are

among the several ways in which a machine element fails.

• What is the mechanism of failure? Is the failure mechanism

simple?

• What is the critical parameter—the critical stress, strain or

energy?

In today’s design practice:

• There is no universal theory of failure for the general case of

material properties and stress state.

• Failure theories propose appropriate means of comparing

multi-axial stress states to single strength

• Usually based on some hypothesis of what aspect of the

stress state is critical

• Some failure theories have gained recognition of usefulness

for various situations

9

Failure Theories

• Structural materials: Ductile (failure strain > 5 %)

OR

Brittle (failure strain < 5 %) [Rule of Thumb]

• The generally accepted theories are:

Ductile materials (yield criteria)

Maximum shear stress (MSS)

Distortion energy (DE)

Ductile Coulomb-Mohr (DCM)

Brittle materials (fracture criteria)

Maximum normal stress (MNS)

Brittle Coulomb-Mohr (BCM)

Modified Mohr (MM)

10

Maximum-Shear-Stress Theory for

Ductile Materials

The maximum shear stress theory predicts that yielding begins

whenever the maximum shear stress in any element equals or

exceeds the maximum shear stress in a tension test specimen

of the same material when the specimen begins to yield.

[Also known as the Tresca or Guest theory]

1 2 3

1 3max

1 3

2 2

y

y

S

S

Simple tension:

1 2 3

max

, 0, 0

2 2

0.5

y

sy y

S

S S

11

• Assuming a plane stress problem with σA ≥ σB, there are three cases to consider

Maximum-Shear-Stress Theory for

Ductile Materials

Case 1: σA ≥ σB ≥ 0. For this case, σ1 = σA and σ3 = 0. Equation (5–1) reduces to a yield condition of

Case 2: σA ≥ 0 ≥ σB . Here, σ1 = σA and σ3 = σB , and Eq. (5–1) becomes

Case 3: 0 ≥ σA ≥ σB . For this case, σ1 = 0 and σ3 = σB , and Eq. (5–1) gives

12

Distortion-Energy Theory for Ductile Materials

The Distortion-Energy theory predicts that yielding occurs

when the distortion strain energy per unit volume reaches or

exceeds the distortion strain energy per unit volume for yield

in simple tension or compression of the same material

[Also known as the von Mises-Hencky, shear-energy or octahederal-shear-stress theory]

13

Distortion-Energy Theory for Ductile

Materials

• Consider a case of pure shear

Thus, the shear yield strength predicted by the distortion energy theory is

• The von Mises stress can be written as

• The von Mises stress for plane stress (x-y) plane is

21

22' ABAA

14

Example 1: Ductile Failure

Consider a material with 𝑺𝑦𝑡 = 𝑺𝑦𝑐 = 700 𝑀𝑃𝑎 and a true strain of

𝜺𝑓 = 0.55. Estimate the factor of safety for the following principle stresses,

using MSS and DE theories.

a. 490, 490, 0 𝑀𝑃𝑎

b. 210, 490, 0 𝑀𝑃𝑎

c. 0, 490, −210 𝑀𝑃𝑎

d. 210, 210, 210 𝑀𝑃𝑎

15


The state of plane stress shown occurs at a critical point of a

steel machine component. As a result of several tensile tests,

it has been found that the tensile yield strength is Sy= 250

MPa for the grade of the used steel.

Determine if the material has yielded using:

a) Max. Shear Stress Theory

b) Distortion Energy Theory

16

Coulomb-Mohr Theory for Ductile

Materials

• For plane stress, when the two nonzero principal stresses are σA ≥ σB , we have a situation similar to the three cases given for the MSS theory

𝑆𝑡

𝑆𝑡

−𝑆𝑡

−𝑆𝑡

−𝑆𝑐

−𝑆𝑐

Case 2: σA ≥ 0 ≥ σB . Here, σ1 = σA and σ3 = σB , and Eq. (5–22) becomes

Case 3: 0 ≥ σA ≥ σB . For this case, σ1 = 0 and σ3 = σB , and Eq. (5–22) gives

Case 1: σA ≥ σB ≥ 0.

For this case, σ1 = σA and σ3 = 0. Equation

(5–22) reduces to a

failure condition of

17

Failure of Ductile Materials Summary

• Either the maximum-shear-stress theory or the distortion-energy theory is acceptable for design and analysis of materials that would fail in a ductile manner.

• For design purposes the maximum-shear-stress theory is easy, quick to use, and conservative.

• If the problem is to learn why a part failed, then the distortion-energy theory may be the best to use.

• For ductile materials with unequal yield strengths, Syt in tension and Syc in compression, the Mohr theory is the best available.

18


The shaft consists of a solid segment AB and a hollow segment BC, which are

rigidly joined by the coupling at B.

Determine the allowable torque using-- a) Max. Shear Stress Theory

b) Distortion Energy Theory

Use a factor of safety of 1.5 against yielding. The tensile yield strength, 𝑺𝑦𝑡 = 250 𝑀𝑃𝑎

0

0

State of Plane Stress

19


The steel pipe shown in `the figure has an inner diameter of 60 𝑚𝑚 and an outer

diameter of 80 𝑚𝑚. It is subjected to loads as illustrated. Determine if these loads

cause failure using the Distortion Energy theory. 𝑺𝑦𝑡 = 250 𝑀𝑃𝑎

𝑥

𝑦

𝑧

𝑥

𝑦

𝑧

20


101.9 0 116.4

0 0 0

116.4 0 0

101.9 116.4

116.4 0

State of Plane Stress (XZ Plane)

21

Example 4: Ductile Failure: M-C Theory

22

Ductile and brittle Failure

23

Maximum-Normal-Stress Theory for

Brittle Materials

• The maximum-normal-stress (MNS) theory states that failure occurs whenever one of the three principal stresses equals or exceeds the strength.

• For a general stress state in the ordered form σ1 ≥ σ2 ≥ σ3. This theory then predicts that failure occurs whenever

where Sut and Suc are the ultimate tensile and compressive strengths, respectively, given as positive quantities.

24

Example :Maximum-Normal-Stress

Theory A short concrete cylinder having a diameter of 50 mm is subjected

to a torque of 500 Nm and an axial compressive force of 2kN.

Determine if it fails according to maximum-normal-stress theory.

Sut=28 MPa

25

Modifications of the Mohr Theory for

Brittle Materials

Brittle-Coulomb-Mohr

Modified Mohr

𝑆𝑡

𝑆𝑡 −𝑆𝑡

−𝑆𝑡

−𝑆𝑐

−𝑆𝑐

26

Modifications of the Mohr Theory for

Brittle Materials

Brittle-Coulomb-Mohr

Modified Mohr

Example 5−5

Shigley’s Mechanical Engineering Design Fig. 5−16

Example 5−5 (continued)

Example 5−5 (continued)

30

Failure of Brittle Materials Summary

Brittle materials have true strain at fracture is 0.05 or less.

In the first quadrant the data appear on both sides and along the failure curves of maximum-normal-stress, Coulomb-Mohr, and modified Mohr. All failure curves are the same, and data fit well.

In the fourth quadrant the modified Mohr theory represents the data best.

In the third quadrant the points A, B, C, and D are too few to make any suggestion concerning a fracture locus.

Cast Iron

31

Selection of failure Criteria

Ss16MCE 321-Chapter 5 Failure-Theories2

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