Chapter 5

Failures Resulting from Static Loading

Chapter 5

Failures Resulting from Static Loading

Asst. Prof. Dr. Supakit Rooppakhun

School of Mechanical Engineering, Institute of Engineering,

Suranaree University of Technology

School of Mechanical Engineering, Institute of Engineering,

Suranaree University of Technology

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 Important Design Equations

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• A static load is a stationary force or couple applied to a member.

(unchanging- magnitude, point or applied point, direction)

• Failure can mean a part has separated into two or more pieces; has become permanently distorted, thus ruining its geometry; has had its reliability downgraded; or has had its function compromised, whatever the reason.

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Failures

Failure of truck driveshaft spline due to

corrosion fatigue

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Example of Failures

Impact failure of a lawn-mower blade

driver hub

Failure of an overhead-pulley retaining

bolt on a weightlifting machine

brittle fracture initiated by stress

concentration

The fractures exhibit the classic 45

degree shear failure

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Static Strength

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

• More often than not it is necessary to design using only published values of yield strength, ultimate strength, percentage reduction in area, and percentage elongation.

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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

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• There is no universal theory of failure for the general case ofmaterial properties and stress state. Instead, over the yearsseveral hypotheses have been formulated and tested, leading totoday’s accepted practices most designers do.

• 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)

Failure Theories

True strain fracture

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Failure TheoriesTrue strain fracture

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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 that specimen begins to yield”.

• The MSS theory is also referred to as the Tresca or Guest theory.

2 3 0

simple tension test

or

It can be modified to incorporate

a factor of safety, n by

or

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Maximum-Shear-Stress Theory for Ductile Materials

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

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

σ

Case 1:

AAA

Case 2:

Case 3:

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Distortion-Energy Theory for Ductile Materials

• Originated from observation that ductile materials stressed hydrostatically (equal principal stresses) exhibited yield strengths greatly in excess of expected values.

• Theorizes that if strain energy is divided into hydrostatic volume changing energy and angular distortion energy, the yielding is primarily affected by the distortion energy.

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Distortion-Energy Theory for Ductile Materials

• The distortion-energy theory predicts that..

“yielding occurs when the distortion strain energy per unit volumereaches or exceeds the distortion strain energy per unit volume for yieldin simple tension or compression of the same material”.

• Deriving the Distortion Energy

Hydrostatic stress is average of principal stresses

Strain energy per unit volume,

Substituting Eq. (3–19) for principal strains into strain energy equation,

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Distortion-Energy Theory for Ductile Materials

• Deriving the Distortion Energy

Strain energy for producing only volume change is obtained by substituting av for 1, 2, and 3

Substituting av from Eq. (a),

Obtain distortion energy by subtracting volume changing energy, Eq. (5–7), from total strain energy, Eq. (b)

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Distortion-Energy Theory for Ductile Materials

• Deriving the Distortion Energy

Tension test specimen at yield has 1 = Sy and 2 = 3 = 0 Applying to Eq. (5–8), distortion energy for tension test specimen is

DE theory predicts failure when distortion energy, Eq. (5–8), exceeds distortion energy of tension test specimen, Eq. (5–9)

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Left hand side is defined as von Mises stress or effective stress σ′ as

For plane stress simplifies to (principal stress A , B , 0)

Using xyz components of three-dimensional stress, the von Misesstress can be written as

for plane stress

Distortion-Energy Theory for Ductile Materials

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(σz = xz = yz =0)

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Distortion-Energy Theory for Ductile Materials

• Consider a case of pure shear τxy ,where for plane stress σx = σy = 0. For

yield

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

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*** The distortion-energy theory is also called• The von Mises / von Mises-Hencky theory

• The shear-energy theory

• The octahedral-shear-stress theory

3 1

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Coulomb-Mohr Theory for Ductile Materials

• Not all materials have compressive strengths equal to their corresponding tensile values.

• The idea of Mohr is based on three “simple” tests: tension, compression, andshear, to yielding if the material can yield, or to rupture.

• The practical difficulties lies in the form of the failure envelope.

• A variation of Mohr’s theory, called the Coulomb-Mohr theory or the internal-friction theory, assumes that the boundary is straight.

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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

Case 1: σA ≥ σB ≥ 0.

For this case, σ1 = σA

and σ3 = 0. Equation

(5–22) reduces to a

failure condition of

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

σ

Case 1:

AAA

Case 2:

Case 3:

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Failure of Ductile Materials Summary

• Either the maximum-shear-stresstheory or the distortion-energytheory is acceptable for design andanalysis of materials that would failin a ductile manner.

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

• If the problem is to learn why a partfailed, then the distortion-energytheory may be the best to use.

• For ductile materials with unequalyield strengths, Syt in tension and Syc

in compression, the Mohr theory isthe best available.

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Suranaree University of Technology

School of Mechanical Engineering, Institute of Engineering,

Suranaree University of Technology

School of Mechanical Engineering, Institute of Engineering,

Suranaree University of Technology

School of Mechanical Engineering, Institute of Engineering,

Suranaree University of Technology

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Suranaree University of Technology

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Suranaree University of Technology

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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 -ultimate tensile and Suc -ultimate compressive strengths,

given as positive quantities.

For the plane stress, with σA ≥ σB,

It can be converted to design equations

as:

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Modifications of the Mohr Theory for Brittle Materials

Brittle-Coulomb-Mohr

Modified Mohr

Case 1:

Case 2:

Case 3:

Case 1:

*Case 2:

Case 3:

Biaxial fracture data of gray cast iron with various

failure criteria

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Failure of Brittle Materials Summary

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

In the 1st 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 4th quadrant the modified Mohr theory represents the data best.

In the 3rd quadrant the points A, B, C, and D are too few to make any suggestion concerning a fracture locus.

Experimental data point obtained from

test on cast iron

Selection of Failure Criteria

First determine ductile vs. brittle

For ductile

◦ MSS is conservative, often used for design where higher

reliability is desired

◦ DE is typical, often used for analysis where agreement with

experimental data is desired

◦ If tensile and compressive strengths differ, use Ductile

Coulomb-Mohr (DCM)

For brittle

◦ Mohr theory is best, but difficult to use

◦ Brittle Coulomb-Mohr (BCM) is very conservative in 4th

quadrant

◦ Modified Mohr (MM) is still slightly conservative in 4th

quadrant, but closer to typical

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Selection of Failure Criteria in Flowchart Form

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Fig. 5−21

Introduction to Fracture Mechanics

Linear elastic fracture mechanics (LEFM) analyzes crack growth

during service

Assumes cracks can exist before service begins, e.g. flaw,

inclusion, or defect

Attempts to model and predict the growth of a crack

Stress concentration approach is inadequate when notch

radius becomes extremely sharp, as in a crack, since stress

concentration factor approaches infinity

Ductile materials often can neglect effect of crack growth, since

local plastic deformation blunts sharp cracks

Relatively brittle materials, such as glass, hard steels, strong

aluminum alloys, and steel below the ductile-to-brittle

transition temperature, benefit from fracture mechanics

analysis

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Quasi-Static Fracture

Though brittle fracture seems instantaneous, it

actually takes time to feed the crack energy from the

stress field to the crack for propagation.

A static crack may be stable and not propagate.

Some level of loading can render a crack unstable,

causing it to propagate to fracture.

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Quasi-Static Fracture

Foundation work for fracture mechanics established by Griffith

in 1921

Considered infinite plate with an elliptical flaw

Maximum stress occurs at (±a, 0)

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Fig. 5−22

Quasi-Static Fracture

Crack growth occurs when energy release rate from applied

loading is greater than rate of energy for crack growth

Unstable crack growth occurs when rate of change of energy

release rate relative to crack length exceeds rate of change of

crack growth rate of energy

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Crack Modes and the Stress Intensity Factor

Three distinct modes of crack propagation

◦ Mode I: Opening crack mode, due to tensile stress field

◦ Mode II: Sliding mode, due to in-plane shear

◦ Mode III: Tearing mode, due to out-of-plane shear

Combination of modes possible

Opening crack mode is most common, and is focus of this text

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Fig. 5−23

Mode I Crack Model

Stress field on dx dy element at crack tip

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Fig. 5−24

Stress Intensity Factor

Common practice to define stress intensity factor

Incorporating KI, stress field equations are

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Stress Intensity Modification Factor

Stress intensity factor KI is a function of geometry, size, and

shape of the crack, and type of loading

For various load and geometric configurations, a stress

intensity modification factor b can be incorporated

Tables for b are available in the literature

Figures 5−25 to 5−30 present some common configurations

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Stress Intensity Modification Factor

Off-center crack in plate in

longitudinal tension

Solid curves are for crack tip

at A

Dashed curves are for tip at B

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Fig. 5−25

Stress Intensity Modification Factor

Plate loaded in

longitudinal tension

with crack at edge

For solid curve there

are no constraints to

bending

Dashed curve obtained

with bending

constraints added

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Fig. 5−26

Stress Intensity Modification Factor

Beams of rectangular

cross section having an

edge crack

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Fig. 5−27

Stress Intensity Modification Factor

Plate in tension containing circular hole with two

cracks

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Fig. 5−28

Stress Intensity Modification Factor

Cylinder loaded in axial tension having a radial crack of

depth a extending completely around the circumference

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Fig. 5−29

Stress Intensity Modification Factor

Cylinder subjected to

internal pressure p, having

a radial crack in the

longitudinal direction of

depth a

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Fig. 5−30

Fracture Toughness

Crack propagation initiates when the stress intensity

factor reaches a critical value, the critical stress

intensity factor KIc

KIc is a material property dependent on material,

crack mode, processing of material, temperature,

loading rate, and state of stress at crack site

Also know as fracture toughness of material

Fracture toughness for plane strain is normally lower

than for plain stress

KIc is typically defined as mode I, plane strain fracture

toughness

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Typical Values for KIc

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Brittle Fracture Factor of Safety

Brittle fracture should be considered as a failure

mode for

◦ Low-temperature operation, where ductile-to-

brittle transition temperature may be reached

◦ Materials with high ratio of Sy/Su, indicating little

ability to absorb energy in plastic region

A factor of safety for brittle fracture

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Example 5−6

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Example 5−6 (continued)

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Fig. 5−25

Answer

Example 5−6 (continued)

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Example 5−7

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Answer

Example 5−7 (continued)

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Example 5−7 (continued)

Shigley’s Mechanical Engineering DesignFig. 5−26

Example 5−7 (continued)

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