2-lectures lec 19 modifications of the mohr theory for brittle materials

8/12/2019 2-Lectures LEC 19 Modifications of the Mohr Theory for Brittle Materials

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Slide #Chapter 16: Failure Resulting from Static Loading

2


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3


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6-9 Modifications of the Mohr Theory forBrittle Materials

Brittle-Coulomb-Mohr (BCM) Theory

Modified I-Mohr Theory


4

-


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Brittle Coulomb-Mohr (BCM) Theory

Same as discussed before for Coulomb-Mohr Theory for ductilematerials. For

0utS

=

A B


5

10

0

A BA B

B A B

n

nut uc

uc

n

S S

S

=

=


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Graphically

0

10

A A B

A BA B

ut

n

n

S

S S

=

=


6

0B A B

uc

n

S =

Figure 6-27

Biaxial fracture data ofgray cast iron comparedwith various failurecriteria.


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7


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Modified I-Mohr (M1M) Theory

Observed experimental data resulted in modification of BrittleCoulomb Mohr (BCM) Theory.

Similar to the Maximum-Normal-Stress (MNS) Theory and theBrittle Coulomb-Mohr (BCM) Theory in the First and Thirdquadrants.


8

In the fourth quadrant, it is less conservative than the BCMTheory but more conservative than the MNS theory.


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0

0 1and

A B

BA B


9

10 1

0

and

A

BA B

A

A B

nc

= >


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Fail

Safe

Safety factor:

i. 1st and 4th Quadabove Pure Shear Line

ii. 4th Quad belowPure

1&0

0

|| >>

>>=

A

BBA

BAut

An

S


10

ABuc

BnS

>>= 0

ear ne

nSSS

SS

uc

B

utuc

Autuc 1)(=

1&0 || >>>A

BBA


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11


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Modified II-Mohr (M2M) Theory

Similar to the previous theories but different in the fourthquadrant.

Data lying in the fourth quadrant were found to be outside theextended region.


12

ME 307 MACHINE DESIGN I12

The Modified II-Mohr (M2M) extends the region in the fourthquadrant where using a parabolic relation.

The parabola is of the form such that itis tangent to the vertical at

1A B

21B B Aa b c + + =

A B ut

S = =


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: 0

0 1and

Case A A A B

BA B

utn

S

=

Similar toprevious case(M1M)


13

2

1 0 1

0

anCase B:

Case C:

d

BA B

A B

A

B A B

nn ut

ut ut uc

uc

n

S

S S S

S

+ + = >

=


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Modified II-Mohr

(M2M) Theory

Safety factor:

4th Quad below Pure ShearLine.

Safe


14

ME 307 MACHINE DESIGN I14

Figure 6-28

Fourth quadrant

data for a grade 40 cast iron,compared with a modified IIMohr fracture locus.

( ),A B

Fail


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6-10 Failure of Brittle Materials Summary

All theories agree in the first and third quadrant.

In the fourth quadrant:

MNS fails.

CM Conservative.


15

Mod. I Mohr less conservative. Mod. II Mohr matches experimental data better.


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Figure 6-29

A plot of experimentaldata points obtainedfrom tests on Cast Iron.Shown also are thegraphs of three failure


16

usefulness for brittlematerials. Notice pointsA, B, C and D. To avoidcongestion in the firstquadrant, points havebeen plotted for

As well as for theopposite sense.

A B >


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6-11 Selection of Failure Criteria

For Ductile Materials:

The preferred criterion is the DE theory, although somedesigners also apply the MSS theory because of its simplicityand conservative Nature.


17

For Brittle behavior:Refer to the flowchart of Figure 6-30


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Ductile Materials and Brittle Material


18


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Ductile Materials and Brittle Material

Slide #Chapter 16: Failure Resulting from Static Loading 19

Figure 6-30

Failure Theoryselection flowchart.


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6-12Static or Quasi Static Loadingon a Shaft

In machinery, the general

term shaft refers to amember, usually of circularcross-section, whichsupports gears, sprockets,


20

, , .,

which is subjected totorsion and to transverse oraxial loads acting singly orin combination.

An axle is a non-rotatingmember that supportswheels, pulleys, andcarries no torque.

Quasi static loading on a shaft means

shaft rotating at a very low speed.


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6-12 Static or Quasi Static Loadingon a Shaft

The design of a shaft involvesthe study of


21

. a c an a ue s ress

and strength

2. Bending and TorsionalDeflection and rigidity

3. Critical Speed


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The fundamentalkinematic component ofour mechanical universeis the wheel and axle.An essential part of this


22

revolute joint is theshaft. It is a goodexample of a static andquasi-static loading, anddynamically loadedapplication.

chapter


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The stress at an element located

on the surface of a solid roundshaft of diameter dsubjected tobending, axial loading, andtwisting is


23

3 232 4

xM Fd d

= +

3

16

xy

T

d

=

Normal stress

Shear stress

Non-zero principal stresses

1 22

2,

2 2

x y x y

A B xy

+ = +


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( )

1/2 1/22 2 2 2

1/22 2

3

' 3

4' 8 48

A A B B x xy

M Fd Td

= + = +

= + +

Von Mises stress


24

( )

( )

1 22 2

max

1/22 2

max 3

14

2 2

2

8 64

A Bx xy

M Fd Td

= = +

= + +

Maximum Shear

Stress Theory


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( )

1/2 1/22 2 2 2

1/22 2

3

' 3

4' 8 48

A A B B x xy

M Fd Td

= + = +

= + +

Von Mises stress (6-37)


25

( )

( )

1 22 2max

1/22 2

max 3

1 42 2

28 64

A Bx xy

M Fd T

d

= = +

= + +

Maximum Shear

Stress Theory(6-38)

The above equations permit an estimate of and whendiameter is given, or an estimate of when an allowable valueof or is given.

' maxd d

' max


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(6-39)

For a design factor of ,the distortion energy theory of ductile failure

gives an allowable stress of

dn

' y

all

S =


26

d

For a design factor of ,the maximum shear theory of ductile failuregives an allowable shear stress of

dn

2

sy y

ll

d d

S Sn n

= = (6-40)


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Static or Quasi Static Loading on a Shaft-bendingand Torsion

1/22 2

3

16' 4 3M T

d

= + Von Mises stress (6-41)

Under many conditions, the axial force F in Eqs. (6-37) and (6-38) iseither zero or so small that its effect may be neglected. With F = 0,

Eqs. (6-37) and (6-38) become


27

1/22 2

max 316 M T

d

= + Maximum Shear

Stress Theory

(6-42)


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Static or Quasi Static Loading on a Shaft-bendingand Torsion

( )1/3

1/22 2

1/2

16 4 3y

nd M TS

= +

Von Mises stress

(6-43)

Substitution of the allowable stresses from Eqs. 6-39 and 6-40 we find


28

3

4 3y

M Tn d S

= +

( )

( )

1/3

1/22 2

1/22 2

3

32

1 32

y

y

nd M T

S

M Tn d S

= +

= +

Maximum Shear

Stress Theory

(6-44)

(6-45)

(6-46)


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Example 6-5

Consider the wrench in Ex. 6-3, Fig. 6-24, as made of cast

iron, machined to dimension.The force F required tofracture this part can beregarded as the strength of


2929

.

material is ASTM grade 30 castiron, find the force Fwith

a) Coulomb-Mohr failuremodel

b) Mod. I-Mohr failure modelc) Mod. II-Mohr failure model

Figure. 6-24


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Example 6-5(Contd)

Assumptions:1. Lever DC strong enough

and is not part of theproblem.

2. Since Grade Cast Iron is abrittle material and cast


3030

iron, the stressconcentration factorsand are set to unity.

3. Table A-24 =

Figure. 6-24

tK

tsK

31109

KpsiKpsi

ut

uc

SS

=

=


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Example 6-5(Contd)

Assumptions:

4. Stresses on element at A

Tensile bending stress

Torsional stress.


3131

5. The location at A is theweakest location and itgoverns the strength ofthe assembly.

Figure. 6-24


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Example 6-5 (Contd)

( ) ( )

( )

( )3332 1432

1 142.61x t t

FM M

K K FI C d

= = = =

Shear stress

Normal stress


3232

( ) ( )( )

33

16 1516 1 76.41

xy ts tsFT r TK K F

J d

= = = =

1 22

2,

2 2

x y x y

A B xy

+ = +



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Example 6-5 (Contd)


( )

1 22

142.6 0 142.6 0

, 76.42 2

175.8

A B

A

F F

F

F

+ = +

=


3333

.B =

Case II

It locates the principal stresses in

the fourth quadrant of theplane ( )

,A B


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Example 6-5

(Contd)

a) Brittle Coulomb-Mohr (BCM) Theory

10A B A B

nut uc

S S

=

Slide # 3434Chapter 16: Failure Resulting from Static Loading

34

FourthQuadrant( )

( )( )3 3

32.2175.8 131 10 109 10

FF =

Solving for F yields

167lbfF =


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Example 6-5

(Contd)

b) Modified I MohrTheory

The slope of the load line is

32.2

Slide # 3535Chapter 16: Failure Resulting from Static Loading

35

( )3175.8

131 10

A Fut

S

= =

FourthQuadrant

176lbfF =

.

175.8B A

= =

2-lectures lec 19 modifications of the mohr theory for brittle materials

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