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
Example 1: Ductile Failure
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
Example 2: Ductile Failure
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
Example 3: Ductile Failure
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
Example 3: Ductile Failure
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
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