Strength Theory - Southeast University

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Strength Theory [email protected]

Transcript of Strength Theory - Southeast University

Page 1: Strength Theory - Southeast University

Strength Theory

[email protected]

Page 2: Strength Theory - Southeast University

• Strength Condition for Simple Stress States(简单应力状态的强度理论回顾)

• Fracture Criteria for Brittle Materials(脆性材料强度理论)

• Yield Criteria for Ductile Materials(塑性材料强度理论)

• Summary of Four Strength Theories(四大强度理论)

• Remarks on Strength Theory(强度理论补充说明)

• Measurement of Strain & Strain Rosettes(应变计与应变花)

Contents

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Page 3: Strength Theory - Southeast University

• Tensile/compressive stress:

• Bending normal stress:

• Torsional shearing stress:

• Bending shearing stress:

• Strength condition at locations under complicated stress state:

Strength theory

x

y

zdx

dz

dyFF

• Preliminary: find the three principal stress components σa, σb and σc.

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Strength Condition for Simple Stress States

max

max

NF

A

max

maxP

T

W

*

Smax

max

Z

z

F S

I b

max

max

Z

z

M

W

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1. Maximum tensile stress criteria

• Brittle materials fail suddenly in tensile tests. The failure condition is

characterized by the ultimate strength σU.

1 U Failure criteria:

Strength condition: 1 U

• Accurate for materials

primarily under tensile stress

state.

• No influences from σ2, σ3 are

taken into consideration.

• σ1 must be tensile stress.

2. Maximum tensile strain criteria

1 U Failure criteria:

Strength condition:

11 2 3

1 2 3

U

U

E E

• Only applicable to linearly

elastic brittle materials (until

failure)

• Only accurate for few brittle

materials4

Fracture Criteria for Brittle Materials

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3. Maximum shearing stress (Tresca) criteria

max1D at yielding 2: Y Y

For σa and σb with the same sign,

max max , ; max ,2 2 2

a b Ya b Y

For σa and σb with opposite signs,

max ;2 2

a b Ya b Y

• Conservative criteria and

widely adopted.

• No influences from σ2 is

taken into consideration.

1 3max 1 3;

3 0, 0, 0

2 2

:

Y

a c

Y

bD

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Yield Criteria for Ductile Materials

2 0, 0, 0 :a b cD

Page 6: Strength Theory - Southeast University

4. Maximum distortion energy (von Mises)

criteria:

2

2 2

2 2 2

2 2 2

2 2 2

1D at yieldi

1

3

1

3

1

ng , 0 :

2 0, 0, 0 :

3 0, 0,

2

:

1

6

0

Y Y

d a

a Y b

d a b b c

c

a

a b b

d Y a

b

c a

c

a b c

a b b Y

d Y

a b b c c a

uE

D

D

uE

uE

u u

u u

Y

• Economic criteria.

• All three principal

stress components are

taken into

consideration.

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Yield Criteria for Ductile Materials

Page 7: Strength Theory - Southeast University

1. Maximum tensile stress criteria: 1 1r

2. Maximum tensile strain criteria: 2 1 2 3( )r

3. Maximum shearing stress: 3 1 3r

4. Maximum distortion energy criteria:

2 2 2

4

1

2r a b b c c a

All of the four types of strength theory can be written in a universal

form, in terms of an effective stress σr:

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Summary of Four Strength Theories

Note: the limit stress has been replaced by the allowable stress.

Page 8: Strength Theory - Southeast University

• Failure mechanism depends on not only mechanical behavior but

also the stress states.

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Remarks on Strength Theory

• Under most stress states, select the maximum tensile stress / strain

criteria for brittle materials and the maximum shearing stress /

distortion energy criteria for ductile materials.

• Experiments demonstrate: all materials fail by rapture under tri-

axial tensile stress state and yielding under tri-axial compressive

stress state. Hence, brittle or ductile failure criteria should be

selected accordingly.

• For dangerous points under uni-axial stress state, maximum tensile

stress criteria is typically selected.

• For dangerous points under pure shearing stress state, maximum

shearing stress criteria is typically selected.

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• For the plane stress state shown, analyze the

strength condition based on the four types of

strength theory.

2

2

2 2

2 2

14

2 2

x y x ya

xy

b

a

b

2 2 2 2

1 2 3

1 14 , 0, 4

2 2 2 2

• Solution

1. Maximum tensile stress criteria:

2 2

1 1 1

14

2 2r

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Sample Problem

X

Y

xyx

Page 10: Strength Theory - Southeast University

2. Maximum tensile strain criteria:

2 1 2 3

2 2

2

( )

1 14

2 2

r

r

3. Maximum shearing stress:

2 2

3 1 3 3 4r r

4. Maximum distortion energy criteria:

2 2 2

4

2 22

2 2 2 2 2 2

4

2 2

4

1

2

1 1 14 4 4

2 2 2 2 2

3

r a b b c c a

r

r

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Page 11: Strength Theory - Southeast University

(a)

σ

σ (a)(a)

σ

σ

(c)

σ

σ(c)(c)

σ

σ

(b)

σ

(b)(b)

σ

(d) σ

σ

(d)(d) σ

σ

• Based on the maximum shearing stress criteria, which one of the

following stress states is most dangerous?

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Exercise

Page 12: Strength Theory - Southeast University

• For the beam shown below: (a) Sketch the stress state at points A, B

and C; (b) Sketch the Mohr’s circle for points A, B and C; (c) Sketch

the stress state along principle directions for points A, B and C; (d) By

satisfying the maximum tensile stress criteria, determine the

orientation of the fracture slot at points A, B and C.

BA C

B1

BC

AC

B

C 1

3

C 1

3

A

1

3

A

1

3

F F aa

AB

C

A

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Sample Problem

Page 13: Strength Theory - Southeast University

• For the thin-walled pressure vessel shown below, D = 1000 mm, t =

10 mm, p = 3.6 MPa and [σ] = 170 MPa. denote the inner diameter,

wall thickness and pressure respectively. Check the strength

condition according to the maximum distortion energy criteria.

p

t

pD

t2

pD

t4

l

circ

axial

radial

2 2 2

4

180 MPa2

90 MPa4

3.6 MPa 0

1156 MPa 170 MPa

2

a

b

c

r a b b c c a

pD

t

pD

t

p

Sample Problem

Page 14: Strength Theory - Southeast University

+

-

640

640

600

600

120

120

FS (kN)

• For the beam shown, q = 40 kN/m, F = 480 kN, and [] = 160 MPa.

Check the strength condition along longitudinal lines 1, 2 and 3

according to the maximum distortion energy criteria.

B

1 m 6 m 1 m

AD

F Fq

800 mm

y

z840 mm

240 mm

12 mm

1

3

2

• Solution

1

2

3

+

620 620800

+++

M (kNm)

2. Stress states along

Lines 1, 2, and 3.

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Sample Problem

1. Diagram of internal forces.

Page 15: Strength Theory - Southeast University

3. Strength condition at Line 1: 1

max max

max

3 3 9 3 3 9 3 4

3 3

4 3

, 800 , 420

240 10 840 10 12 (240 12) 10 800 10 12 2.126 10

800 10 420 10160 [ ]

2.126 10

z

z

D

z

r

M yy

I

I

M M

kNm mm

m

MPa

4. Strength condition at Line 2:

maxmax

max

3 3 3

3 3 3 3 3

3 3

max 3 3

1 2 3

, 640 , 12

[(240 10 20 10 ) 410 10 ]

[(12 10 400 10 ) 200 10 ] 2.96 10

640 10 2.96 1073.5

12 10 2.126 1073.5 , 0, 73.5

s sA ss z

B

z

z

F Sb

I b

M a

F F

P

F

S

kN mm

m

MPa

MPa

2 2 214 1 2 2 3 3 12

[( ) ( ) ( ) 127 160 [ ]r MPa MPa

2

Page 16: Strength Theory - Southeast University

3 3

3

* 3 3 3 3 3

* 3 3

3 3

2 2

4

620 10 400 10116.7 [ ]

2.126 10

(240 10 20 10 ) 410 10 1.968 10

600 10 1.968 1046.3

12 10 2.126 10

3 141.6 160 [ ]

z

z

s z

z

r

My

I

S

F S

bI

MPa

m

MPa

MPa MPa

5. Strength condition at Line 3:

• Critical cross-section is

identified by both large

shearing stress and bending

moment (x = 1 m)!

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+

-

640

640

600

600

120

120

FS (kN)

+620 620

800++

+M (kNm)

3

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• Strain gages indicate normal strain

through changes in resistance.

yxOBxy 2

• With a 45o rosette, εx and εy are measured

directly. γxy is obtained indirectly with,

1 1 1

2 2 2

3 3 3

cos 2 sin 22 2

c

2

2os 2 sin 2

2 2

cos 2 sin 22 22

x y x y xy

x y x y xy

x y x y xy

• Normal and shearing strains may be

obtained from normal strains in any three

directions,

cos2 sin22 2

x y x y

x xy

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Measurement of Strain & Strain Rosettes

Page 18: Strength Theory - Southeast University

• Strength Condition for Simple Stress States(简单应力状态的强度理论回顾)

• Fracture Criteria for Brittle Materials(脆性材料强度理论)

• Yield Criteria for Ductile Materials(塑性材料强度理论)

• Summary of Four Strength Theories(四大强度理论)

• Remarks on Strength Theory(强度理论补充说明)

• Measurement of Strain & Strain Rosettes(应变计与应变花)

Contents

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