Strength Theory - Southeast University
Transcript of Strength Theory - Southeast University
Strength Theory
• 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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• 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
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
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
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
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.
• 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.
• 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
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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(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
• 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
• 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
D
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
+
-
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.
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
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
• 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
• 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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