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Chapter 5: Failures Resulting from Static Loading

The concept of failure is central to the design process, and it is by thinking in terms of obviating failure that successful designs are achieved. Henry Petroski Design Paradigms

5.1 Static Strength •  Yield Strength

– Depending on heat treatment, surface finish, size – Use in multi-axial loading conditions

Sy

5.2 Stress Concentration

nomts

nomt

KKττ

σσ

=

=

max

max

See Table A-15 and 16 Walter D. Pilkey, Peterson’s Stress Concentration Factors,2nd ed. John Wiley & Sons, New York, 1997. FEA analysis can be used.

Stress concentration factors for round bar with fillet. (a) Axial load and (b) Bending..

Round Bar with Fillet

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Stress concentration factors for round bar with groove. (a) Bending and (b) Torsion.

Round Bar with Groove

Stress concentration factors for round bar with hole.

Round Bar with Hole

Figure 6.8 Flat plate with fillet axially loaded showing stress contours for (a) square corners; (b) rounded corners; (c) small grooves; and (d) small holes.

Stress Contours Failure Prediction for Multiaxial Stresses

•  The limit of the stress state on a material – Ductile Materials - Yielding – Brittle Materials - Fracture

•  In a tensile test, Yield or Failure Strength of a material.

•  In a multiaxial state of stress, how do we use Yield or Failure Strength?

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A. Maximum Shear Stress (MSS) Criterion (Tresca Criterion)

2maxyστ =

yBA

yB

yA

S

S

S

=−

=

=

σσ

σ

σFor Plane Stress: σA, σB have same signs.

σA, σB have opposite signs.

In a tension specimen: τmax

σy

(σ3=0)

The diameter of the Mohr circle = σy

Ductile Material

[ ] ( )[ ]31232123

22

21332211 2

21

21

σσσσσσνσσσεσεσεσ ++−++=++=E

u

For the three principal stresses;

in an uniaxial tension test. 2

31

Yd SE

u ν+=

For plane stress; [ ]22

31

BBAAd Eu σσσσ

ν+−

+=

222yBBAA S=+− σσσσ

σ2

σ1

MSST

DET

After taken out the hydrostatic stress (σave=(σ1+σ2+σ3)/3) ( ) ( ) ( )aveaveave σσσσσσ −−− 321 ,,Now substitute σ1, σ2 and σ3 with

B. Distortion Energy (DE) Criterion (von Mises) Ductile Material

Ductile Material •  Use of von Mises Stress •  New Form: No Failure

( ) ( ) ( )

( ) ( ) ( ) ( )[ ]

( )( ) 2/1222

2/122

2/122222

2/1213

232

221

3

StressPlaneFor

621

2

StressesaxialMultiFor

xyyyxx

BBAA

xzyzxyxzzyyx

τσσσσ

σσσσσ

τττσσσσσσ

σσσσσσσ

++−=

+−=

+++−+−+−=

⎥⎦

⎤⎢⎣

⎡ −+−+−=

−

yS≤σ

FStresses acting on octahedral planes. (a) General state of stress; (b) normal stress; (c) octahedral stress.

Octahedral Stresses

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Ductile Material C. Coulomb-Mohr (DCM) theory Some metals have a higher compressive strength than tensile strength. –  Tensile, Shear and compression

1

22

22

22

22

31

31

31

13

1133

12

1122

=−

+

−=

+−

−−

−

−=

−

−

ct

tc

tc

t

t

SS

SS

SS

S

SOCOCCBCB

OCOCCBCB

σσ

σσ

σσ

τ

σ σ3 σ1 -Sc St C1 C2 C3

B1 B2

B3

O

cBBA

c

B

t

ABA

tABA

SSS

S

−≥≥≥

≥−≥≥

≥≥≥

σσσ

σσσσ

σσσ

0:3Case

10:2Case

0:1Case

St

St

Sc

Sc σA

σB

Graphical representation of maximum-shear-stress theory for biaxial stress state.

Graphical representation of distortion energy theory for biaxial stress state.

MSS and DE for Biaxial Stress State

Three-dimensional yield locus for MSS and DE.

Three-Dimensional Yield Locus Brittle Materials A. Maximum Normal Stress (NMS) Theory

ucut SS −≤≥ 31 or σσ

B. Modification to Mohr Theory

BAuc

B

BAuc

B

ut

A

BAut

A

nS

nSS

nS

σσσ

σσσσ

σσσ

≥≥−=

≥≥=−

≥≥=

0

01

0

σ1

σ2

NMS

B

Sut

Sut

Suc

Suc

ucButA SS −≤≥ σσ orFor Plane Stress

Brittle-Columb-Mohr