Chapter 5

Failure resulting from static load

What is Failure?

What is Static load?

1M S Dasgupta BITS Pilani

Stress Concentration

Changes in cross section causes localized stress concentrations and

severity depends on the geometry of the discontinuity and nature of

the material.

2

Stress concentration factor

Kt = max/ o

– max, maximum stress at discontinuity and o, nominal stress.

– Kt, value depends only on geometry of the part.

3

4

Design improvement to reduce stress concentration

5

Stress concentration & Notch

Sensitivity factor• Kt = stress concentration factor

• Kf = fatigue stress concentration factor

• q = notch sensitivity factor

M S Dasgupta BITS Pilani 6

Q. A steel shaft (ultimate strength 600 MPa) with a shoulder fillet

radius 3mm connects a 32mm radius dia side with 38mm dia.

Estimate Kt and Kf .

Terminologies

1. Failure theory (FT) to use depends on material (ductile or brittle) and type

of loading (static or dynamic).

2. Terminology:

• Su (or Sut) = ultimate strength in tension

• Suc = ultimate strength in compression

• Sy = yield strength in tension

• Sys = 0.5*Sy = yield strength in shear

• Sus 0.75*Su = ultimate strength in shear

• Se = endurance strength 0.5*Su or get from S-N curve

• S’e = estimated actual endurance strength = Se(ka) (kb) (kc) (kd) - - -

• S’se 0.577* S’e = estimated actual endurance strength in shear

7

Ductile materials - extensive plastic deformation and

energy absorption (toughness) before fracture

Brittle materials - little plastic deformation and low energy

absorption before failure

8

Ductility and % Elongation

• Ductility is the degree to which a material

will deform before ultimate fracture.

• Percent elongation is used as a measure

of ductility.

• Ductile Materials have %elong. 5%

• Brittle Materials have %elong. < 5%

• For machine members subject to repeated

or shock or impact loads, materials with

%elong > 12% are recommended. 9

Ductile:

warning before

fracture

Brittle:

No

warning

DUCTILE VS BRITTLE FAILURE

(a) (b) (c)

10

Failure Prediction Methods

• Ductile materials are designed based on

yield criteria – Maximum shear stress (MSS) theory

– Distortion energy (DE) theory

– Ductile Coulomb-Mohr (DCM) theory

• Brittle materials are designed based on

fracture criteria– Maximum normal stress (MNS) theory

– Brittle Coulomb-Mohr (BCM) theory11

Maximum-Normal-Stress Theory

• The maximum-normal-stress theory states that

failure occurs whenever one of the three

principal stresses equals or exceeds the

strength.

• For principal stress

• σ1 ≥ σ2 ≥ σ3

σ1 ≥ Sut or σ3 ≤ −Suc

12

Maximum-Shear-Stress Theory

n

S

n

S y

A

y

BA

BA

,

0,,

,0

31

321

n

S

n

S y

BA

y

BA

BA

,

,0,

,0

31

321

n

S

n

S

y

B

y

B

A

BA

,

,

,0

,0

31

3

2

1

Failure occurs when

the maximum shear

stress in any element

equals or exceeds the

maximum shear

stress in a tension

test specimen.

13M S Dasgupta BITS Pilani

Distortion-Energy (DE) Theory

“Failure 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”

yy SorS

'2

12

13

2

32

2

21

2

For a general state of stress, the Distortion-Energy Theory predicts

yielding when Von Mises stress

21

22'

BBAA For 2D:-14

15

According to DE (von Mises) criterion, substituting the pure

shear state of stress in the 2-D DE criterion, the two

normal stresses being zero,

SHEAR YIELD STRENGH:

ysy

y

y

xyyxy

SSyieldAt

SS

S

577.0,

577.03

3 2

ysy SS 5.0According to the MSS criterion,

DE criterion predicts the shear yield strength to be 15 percent more than that

predicted by the MSS criterion. Hence MSS is more conservative.16

Yield Strength Method

• Uniaxial Static Stress on Ductile Materials

In tension:

In compression:

For most ductile materials, Syt = Syc

Static

LoadDuctile Material

N

S yt

d max

N

S yc

d max

DESIGN:

ANALYSIS:

DESIGN:

ANALYSIS:

max

ytSN

max

ytSN

17

Maximum Shear Stress

• Biaxial Static Stress on Ductile Materials

Ductile materials begin to yield when the maximum shear stress in a load-carrying

component exceeds that in a tensile-test specimen when yielding begins.

N2

S

N

S yys

dmax

avg, max

max

ysSN

DESIGN:

ANALYSIS:

18

von Mises Stress

• Alternate Form

For uniaxial stress when y = 0,

• Triaxial Distortion Energy (1 > 2 > 3)

222 3 xyyxyx'

22 3 xyx'

2

)()()('

2

32

2

31

2

21

19

Summary Static Failure Theories:

• Ductile materials fail on planes of max

shear stress:

– Max shear stress theory

– Distortion energy theory

• Brittle materials fail on planes of max

normal stress:

– Max Normal Stress Theory

– Modified Mohr Theory

20

Brittle failure or ductile failure? Key: is the fracture surface

on a plane of max shear or max normal stress.

TORQUE:

DUCTILE BRITTLE21

Real Shaft

22

M S Dasgupta BITS Pilani 23

The figure shows a shaft mounted in bearings at A and D and

having pulleys at B and C. The forces shown acting on the

pulley surfaces represent the belt tensions. The shaft is to be

made of AISI 1035 CD steel using a design factor of 2.

Based on MSS & DE What diameter should be used for the

shaft?