Chapter 5

What is Failure? What is Static load?

M S Dasgupta BITS Pilani

1

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 M S Dasgupta BITS Pilani 2

Ductile materials - extensive plastic deformation and energy absorption (toughness) before fracture Brittle materials - little plastic deformation and low energy absorption before failure

3 M S Dasgupta BITS Pilani

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 4 %elong > 12% are recommended.Dasgupta BITS Pilani MS

DUCTILE VS BRITTLE FAILURE

(a) Ductile: warning before fracture

M S Dasgupta BITS Pilani

(b)

(c) Brittle: No warning

5

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) theory M S Dasgupta BITS Pilani 6

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 M S Dasgupta BITS Pilani 7

Maximum-Shear-Stress Theory

Failure occurs when the maximum shear stress in any element equals or exceeds the maximum shear stress in a tension test specimen.

A B 0, 1 A , 2 B , 3 0 1 3 Sy n ,A Sy n

0 A B,

1 0, 2 A, 3 B 1 3 B Sy n Sy n ,

A 0 B, 1 A , 2 0, 3 B 1 3 M S Dasgupta BITS Pilani

Sy n

, A B

Sy n

8

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” For a general state of stress, the Distortion-Energy Theory predicts yielding when Von Mises stress 1 2

1 2 2 2 3 2 3 1 2 S y or ' S y 2 For 2D:-

A B

' 2 A

1 2 2 B

M S Dasgupta BITS Pilani

9

M S Dasgupta BITS Pilani

10

SHEAR YIELD STRENGH:

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,

3 At yield , S sy 0.577 S y

According to the MSS criterion,

3

2 xy

S y xy

Sy

0.577 S y

S sy 0.5S y

M S Dasgupta BITS Pilani 11

DE criterion predicts the shear yield strength to be 15 percent more than that predicted by the MSS criterion. Hence MSS is more conservative.

Yield Strength Method

• Uniaxial Static Stress on Ductile Materials

Ductile Material Static Load

– In tension:

ANALYSIS: DESIGN:

N –In compression:

DESIGN:

max d

S yt

N

ANALYSIS:

S yt

max

S yt

max d

S yc N

N

For most ductile materials, Syt = Syc

max

M S Dasgupta BITS Pilani

12

Maximum Shear Stress

• Biaxial Static Stress on Ductile Materials

avg, max

DESIGN:

max d

N S ys

S ys N

Sy 2N

ANALYSIS:...