소성재료역학 (Metal Plasticity) Chapter 6: Plasticity in pure … 6... · 2019. 9. 5. ·...
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446.631A 소성재료역학 (Metal Plasticity) Chapter 6: Plasticity in pure bending and beam theory Myoung-Gyu Lee Office: Eng building 33-309 Tel. 02-880-1711 [email protected] TA: Chanyang Kim (30-522)
Transcript of 소성재료역학 (Metal Plasticity) Chapter 6: Plasticity in pure … 6... · 2019. 9. 5. ·...
446.631A
소성재료역학(Metal Plasticity)
Chapter 6: Plasticity in pure bendingand beam theory
Myoung-Gyu Lee
Office: Eng building 33-309
Tel. 02-880-1711
TA: Chanyang Kim (30-522)
Beam theory• Study on the deflection of a beam
• Beam: a long slender straight object under transverse loading
• Assumption: small deformation, uniform symmetric cross-section shape
• Some of geometric assumptions (on straightness and the cross-sectional shape) can be released with added complexity in the solution procedure
Cantilever beam (외팔보)
Pure bending theory• Deformation of a straight object under bending moment only
• Small deformation and a uniform symmetric cross-section shape are assumed
• An accurate solution procedure supports that vertical planes remain vertical to curved outer surfaces and inner planes (parallel to outer upper and bottom surfaces) after deformation, regardless of material properties
MM
Moment 𝑴 = 𝒓 × 𝑭r
F
Pure bending theory
• Consider the rectangular cross-section, for simplicity.
0
0
: radius of curvature, K: curvature
( )
x
R
l R
l R y
l y yyK
l R R
zMzM
2h
y
x
z
y
y
R
Neutral line
0
o
Cross-section
b
• Following relations are valid regardless of material properties
• The origin of the coordinate system is located at the middle of the neutral
plane. However, the vertical position of the neutral plane in the cross-
section is unknown.
Pure bending theory
Pure bending of elasto-perfect plastic material
MM
Pure bending theory
• Elastic range
e
h
h
y
Y
E
K
e
( )e
Y
E
Y
R
zMzM
2h
y
x
z
y
y
R
Neutral line
0
o
Cross-section
b
Kinematicsy
yKR
at neutral line (y=0) 0
Material property (constitutive law) – linear elasticity
yE E EyK
R
Pure bending theory
• Elastic range
e
h
h
y
Y
E
K
e
( )e
Y
E
Y
R
zMzM
2h
y
x
z
y
y
R
Neutral line
0
o
Cross-section
b
Equilibrium equation
0, 0F M
0
0 (neutral plane=central plane)
(( 0) neutral plane central plane)
y EF dA E dA ydA
R R
ydA
F
Pure bending theory
• Elastic range
e
h
h
y
Y
E
K
e
( )e
Y
E
Y
R
zMzM
2h
y
x
z
y
y
R
Neutral line
0
o
Cross-section
b
Moment acting on beam
32 2 2
3
h h
h h
bh I y dA by dy
I : second moment of inertia
2
32
3
E
M ydA y dAR
EI Ebh KEIK
R
Pure bending theory
• Plastic range; elastic-plasticity
e
h
h
y
Y
E
K
e
Elastic range
Plastic range
h
h
YY
y
limit line
( )e
Y
E
Y
R
Kinematicsy
yKR
Constitutive law: elasto-perfect plasticity
zMzM
2h
y
x
z
y
y
R
Neutral line
0
o
Cross-section
b
the location of the boundary between the
elastic and e-p zones
Pure bending theory
• Plastic range; elastic-plasticity
e
h
h
y
Y
E
K
e
Elastic range
Plastic range
h
h
YY
y
limit line
( )e
Y
E
Y
R
zMzM
2h
y
x
z
y
y
R
Neutral line
0
o
Cross-section
b
( )( )
e
e e
e
YK
Eh
Y hhK
E R
Curvature when yield begins
Location of elastic/plastic boundary
ee
Kfor K K
h K
Stress distribution through beam
0
Y for y h
Yy for y
• Plastic range; elastic-plasticity
Pure bending theory
Moment acting on the beam
2 10
22 2
0
22 2 2 2
2( )
2 ( ) 2 ( ) ( )( )3
1= ( ( ) ) (1 ( ) ) ( )
3 3
h
h
e ee
M ydA ydA ydA
Yy bdy Yy bdy bY h h
K KhbY h bYh K K
K K
2
* 2
2( )
3
( )
e eM K K M bYh
M K M bYh
( )Y
Eh
h
h
YY
y
1 Y
2
2
Yy
0
Y for y h
Yy for y
Equilibrium equation
0, 0F M
Pure bending theory
K
M M
2 *( )bYh M
22( )3
ebYh M
( )e
YK
Eh h
eM
*M
Relation between the bending moment and curvature
Pure bending theory bending – unbending – reverse bending
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
MM
MM
MMM M
• Sequential superposition (for a non-linear system)
Remark
• M(K) relationship between 5 and 7 is the twice stretch of the M(K) relationship between 1 and 2.
2
h
YY
y
2Y2Y
② ③ ⑤
⑧
⑦
Pure bending theory bending – unbending – reverse bending
Pure bending theory: initial bending
1. Point 1~2 2( )eK K K 1~2 2 1
0
2 2
2 2
1~2
2 ( )
1 {1 ( ) }
3
1 {1 ( ) }
3
h
e
M b ydy ydy
bYhh
KbYh
KK
M
* 2M bYh
22
3eM bYh
eK
①
②
2K
2 ( )
Y
Eh
2
h
h
YY
y
1 Y
2
2
Yy
Pure bending theory: initial bending
1. At point 2 2( )eK K K 2 2
2
2
1~2
1~2
2
2
1(1 ( ) )
3
e
ee e
e
KM bYh
K
KK h K
K h
hK K
K
M
* 2M bYh
22
3eM bYh
eK
①
②
2K
2 ( )
Y
Eh
2
h
h
YY
y
1 Y
2
2
Yy
Pure bending theory: unbending
2. Point 2~5 (0 2 )e eK K K
3
2~5
2~5 2 2~5
2
3
Ebh KM
M M M
2~5 2 2~5( ) ( ) ( )
EyK
y y y
K
M
* 2M bYh
22
3eM bYh
eK
eK
2 eK
①
②
③
④
⑤
2K
K
M
Pure bending theory: unbending
2. Point 2~5 (0 2 )e eK K K
K
M
* 2M bYh
22
3eM bYh
eK
eK
2 eK
①
②
③
④
⑤
2K
K
M
1) Point 3 3( )eK K
2
3
22
3 2 3
2
2( )
3
{1 ( ) } 3
e e
e
Y bYhM M K K M
Eh
KbYhM M M
K
3 2 3( ) ( ) ( )y y y
Pure bending theory: unbending
2. Point 2~5 (0 2 )e eK K K
K
M
* 2M bYh
22
3eM bYh
eK
eK
2 eK
①
②
③
④
⑤
2K
K
M
1) Point 4 4( 0)M
4 2 4 0M M M
32 2 4
2
2
4
2
21(1 ( ) ) 0
3 3
3 1(1 ( ) )
2 3
e
e
K Ebh KbYh
K
KYK
Eh K
4 2 4( ) ( ) ( )y y y
Pure bending theory: unbending
2. Point 2~5 (0 2 )e eK K K
K
M
* 2M bYh
22
3eM bYh
eK
eK
2 eK
①
②
③
④
⑤
2K
K
M
1) Point 5
5
2 2( 2 , 2 )e e e e
Y YK K K
Eh E
3 2
5
22
5 2 5
2
5
4 4
3 3
{1 ( ) }3
( 2 )
e
e
e
Ebh K YbhM
KYbhM M M
K
M M
Pure bending theory: re-bending
3. Point 5~7 5 7 2(2 , )eK K K K h
5~7 2 10
2 2
2 2
1~2
5~7
5~7 2 5~7
5~7
5~7
2 { }
1 2 {1 ( ) }
3
1 2 {1 ( ) } 2
3
2( ) ( ) 2
h
e
e e e
e
M b ydy ydy
bYhh
KbYh M
K
M M M
YK h K hK
E
hK K
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
Pure bending theory: re-bending
3. Point 5~7
5 7 2(2 , )eK K K K h
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
1) Point 6
6 2 6
6 2
6 2 6
2 2 2 2
2 2
2 2
2
61 2
3 2
0
( 2 )
21 1 (1 ( ) ) 2 (1 ( ) )
3 3
7 (1 ( ) )
3
3 7(1 ( ) )
2 2 3
3
e
e e
e
eS
K K K
K K K
M M M
K KbYh bYh
K K
KbYh
K
M KYK
Eh KEbh
Pure bending theory: re-bending
3. Point 5~7
5 7 2(2 , )eK K K K h
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
1) Point 7
7 2 7 2 2 22M M M M M M
7 2 2( 2 , )K K
Pure bending theory: reverse bending
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
2
2
h
h
YY
y
2Y
2Y
1 2Y
2
2
( )Y
y Y
23
( )Y
Y
y
limit line
4. Point 7-82
27~8 3 2 1
0
2 2 22
2 2 2
2 7~8 7
2 ( )
1 1 2 {1 ( ) ( ) }
6 61 1
2 {1 ( ) ( ) }6 6 2
h
e e
M b ydy ydy ydy
bYhh hK K
bYhK K K
27~8
2
+Y=E K
( )
Y Y
YE K
E
Pure bending theory: reverse bending
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
2
2
h
h
YY
y
2Y
2Y
1 2Y
2
2
( )Y
y Y
23
( )Y
Y
y
limit line
4. Point 7-8
7~8
2
( )
Y
YE K
E
7 7~8 77~8 7~8
2
7 2 2 2
7~8 2 7~8
2
2( ) 2( )2
( 2 ))
e e e
e
K K K
Y Kh K KK K
E
YK K K K h
E
M M M
2
e
Ywhile h
EK
Pure bending theory: reverse bending
(1) Point 7
K
M
* 2M bYh
22
3eM bYh
1
SK
eK
* 2
8( )M M bYh
eK
2 eK
2K7 2( )K K
8( )K
①
②
③
④
⑤
⑥⑦
7 2 2( 2 , )K K
2 2
7 2
2
12 {1 ( ) } 2
3
(Consistent)
eKM bYh M
K
8( , 0)K (2) Point 8
2 2
8
2
2
8 2 8
8 2 8
12 (1 ( ) )
6
eKM bYh
K
M M M bYh
K K K
2
2
h
h
YY
y
2Y2Y
⑧②⑧’
Pure bending theoryM
2
cM bYh
22
3eM bYh
eK
①②
③
④
⑤
2K
2
h
YY
y
2Y2Y
② ③' ⑤'
⑥
⑥'
⑦* 2
8( )M M bYh
7 2( )K K
K
⑧
4K 2
SK
6 2
3 3
6 26
32 2 2
6 2 6
2
62 2
2 2 43 2
2 2
3 3
21( 0) {1 ( ) }
3 3
3 1{1 ( ) }
2 2 3
3
e
eS
K K
Ebh K Ebh KM
K Ebh KM M M bYh
K
M KYK K K K
Eh KEbh
Full recovery of the initial bending curvature(point 6) occurs during elastic unloading from the initial bending
2 2e eK K K
Beam theory• Study on the deflection of a beam
• Long slender straight object under transverse loading
• Small deformation and a uniform symmetric cross-section shape
cf) Timoshenko beam theory
• Shear deformation is considered (especially for a thicker beam)
Euler- Bernoulli beam theory
• Shear deformation is neglected (especially for a thin beam)
Bending of thick book
Mid-surface orientation and shape
Traffic light
Long & slender shape
Images : https://en.wikipedia.org/wiki/Timoshenko_beam_theoryhttps://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory
Beam theory
• Beam : 1-D • Plate : 2-D
• Shell : 3-D
Equilibrium in beam – distributed load
( ) w x per length
M ( )M x dx
V ( )V x dxdx
x
y
z
x
y
z
( )w x
xx
xy
xz
o
Symmetric stress distribution
on a beam with symmetric cross-section
Bending moment and shear force
distribution on a beam
Equilibrium in beam – distributed load
( ) w x per length
M ( )M x dx
V ( )V x dxdx
x
y
z
( ) dV
w xdx
( ) dM
V xdx
• Force equilibrium
• Moment equilibrium
(reference at the center)
( ) ( )2 2
( ) ( ( ) ) ( )2 2
( ) ( )
dx dxV x V x dx
dx dxV x V x dV V x dx
M x dx M x dM
( ) ( ) ( )V x dx wdx V x dV wdx V x
Kinematics : infinitesimal deformation theory
( ) w x per length
M ( )M x dx
V ( )V x dxdx
x
y
z
x
y
z
( )w x
xx
xy
xz
o
2
22
3 22 2
( ){1 ( / ) }
d vd vdxKdxdv dx
Constitutive law & beam analysis
( ) w x per length
M ( )M x dx
V ( )V x dxdx
x
y
z
x
y
z
( )w x
xx
xy
xz
o
M EIK
• Linear elasticity
2 4
2 4( )( )
dV d M d vw x EI
dx dx dx
• Elasto-perfect plastic behavior2 2
2 2
2 2
2 2
( )( ) ( ( ))
( ( ))
dV d M d fw x K x
dx dx dx
d f d vx
dx dx
Statically determinate vs indeterminate
• Statically determinate • Statically indeterminate
Figures : Shames, Introduction to solid mechanics
Solution procedure of the SD (statically determinate) problem
• Find from the equilibrium condition
• Utilizing the moment-curvature relationship, find out the curvature distribution
• Integrate the curvature to find the deflection
( ), ( )V x M x
M
K0
2
2
d vK v
dx
Example : Beam with elasto-plastic behavior
x
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
, 0
V wx C
at x l V wl C
V wx
( ) dV
w xdx
( ) dM
V xdx
• Equilibrium equations
Example : Beam with elasto-plastic behaviorx
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
, 0
V wx C
at x l V wl C
V wx
2
202
202
2 20
2
, 02
( )2
M Vdx C
wx C
wat x M C
wM x
Example : Beam with elasto-plastic behaviorx
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
, 0
V wx C
at x l V wl C
V wx
2
202
22 2
2
22 2
2
0
0
2
0,
2 4, ,
2 3 3
2 3, ,
2 2
0
-
ee e
cc c e
e
e c
M Vdx C
wx C
at x
w bYhM bYh therefore w
w bYhM bYh therefore w w
where
w w Elastic range
w w w Elasto plastic range
:
:
• Find
1
3
1
3
0
1
h
x
l
2
3
*
h
*x
l
1
1
x relation
2 2
2 2 2 2
2( ) 1
2 3
1 1(1 ( ) ) or (1 ( ) )
3 3
c
e
wM l x
KM bYh bYh
h K
Example : Beam with elasto-plastic behavior
• Find
1
3
1
3
0
1
h
x
l
2
3
*
h
*x
l
1
1
x relation
2 2 2 2 2 2
2
2
2 2
2 2
2 2
2 2
1( ) ( ) ( ) (1 ( ) )
2 2 3
2 21
3
1 (1 ( ) ) (1 ( ) )
3
1 ( ) 1 ( )
3
1 ( ) ( ) 1
3
1 ( ) ( ) 1 3(1 ) 1
c
c
wwM x l x l x bYh
h
bYhwith and w
l
x
l h
x
l h
x
h l
xHyperbola equation
h l
Example : Beam with elasto-plastic behavior
• Find
1
3
1
3
0
1
h
x
l
2
3
*
h
*x
l
1
1
x relation 2 2
*
2*
*
2 2 * *
2
1 ( ) ( ) 1 3(1 ) 1
) 0, 3(1 )
1 1 3 2) 1, ( ) ( 1 )( )
3(1 ) 3
3 2
3
2 , ( ) 2( ) 1, 0, 1.0
3
1, ( ) 3(
xHyperbola equation
h l
xi
l h
xii
h l
x
l
Further more
xxat
h l l h
xat
h l
2 *
*
1 1) 0, 3 , 0.6
3 1.732
60%0
tan
xx
h l l
Plastic range is about
theelastic property is impor t h
Example : Beam with elasto-plastic behavior
Example : beam with elasto-plastic behavior
Basic equations
* 3 2
3
xx
22 2
2( )
2
cw d vM x EIK EI
dx
2
2( ) , ( ) ,
dV dM d vw x V x K
dx dx dx
Elastic zone
2 22 2
2 2
3( ) (1 )
2 2
cwd v Y xx
dx EI Eh
x
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
• Calculate M with K
Example : beam with elasto-plastic behavior
Basic equations
* 3 2
3
xx
2
2( ) , ( ) ,
dV dM d vw x V x K
dx dx dx
Plastic zone
2 2 2 21( ) (1 ( ) )
2 3
c ew KM x bYh
K
2
2
2 2
=
3(1 (1 ( ) )) 3(1 (1 ( ) ))
eKd v Y
dx x xEh
• Calculate M with Kx
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
Example : beam with elasto-plastic behavior
• Boundary conditionsx
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
( ) 0, ( 0) 0dv
v x xdx
• Continuity of shear load
dv
v and are continuousdx
*
22 4 4
3
3 2)
3
3 3 1 5
2 2 12 123
3 11 3 2ln 3 2 ( )
33 1
xxi
Yv x x
Eh
x
• Solutions
Example : beam with elasto-plastic behavior
• Boundary conditionsx
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
( ) 0, ( 0) 0dv
v x xdx
• Continuity of shear load
dv
v and are continuousdx
*
2 2 22 2 2
2
3 2) 0
3
( )ln
3 1
3 11 3 2 1 2ln 3 2 3
3 23 1
xxii
xx xYv x
Eh
• Solutions
Example : beam with elasto-plastic behavior
• Boundary conditionsx
y
( )w x w
wl wl
2l
V
wl
wl
M
ll
ll0
0
eM
cM
x
ll 1
3 l
1
3l0
h
*x
1 *
h
2
3 h
( ) 0, ( 0) 0dv
v x xdx
• Continuity of shear load
dv
v and are continuousdx
• Solutions
2
2
1 1 3 2ln
3 3 1
3 1 1 23 2 3
3 2
Yv
Eh0for x
2 5 2
0, 2 / 3123
e
Ywith v x
Eh
Example : beam with elasto-plastic behavior
• Deflection at the center of the beam
(0)
(0)e
v
v
1
2
31
Beam collapses with α=1
Example : unloading of beam( w(x)=0)
M
K'K
M
K'K
'M
'K
• Elastic recovery of curvatures during unloading
• The new curvature distribution after elastic unloading
2
2
' ( )( ) '( ) ( ) ( )
d v M M xK x K x K x K x
dx EI EI
Elastic zone *xx
1
3
1
3
0
1
h
x
l
2
3
*
h
*x
l
1
1
'K K 2
20
d v
dx
Plastic zone *0xx
2 2' ( )
2
cwMK x
EI EI
2 22 2
2 2
2 2
1 3( ) (1 )
2 23(1 (1 ( ) )) 3(1 (1 ( ) ))
cwd v Y Y xx
dx EI Ehx xEh
Example : unloading of beam( w(x)=0)
M
K'K
M
K'K
'M
'K
• Elastic recovery of curvatures during unloading
• The new curvature distribution after elastic unloading
2
2
' ( )( ) '( ) ( ) ( )
d v M M xK x K x K x K x
dx EI EI
1
3
1
3
0
1
h
x
l
2
3
*
h
*x
l
1
1
Plastic zone *0xx
2 2' ( )
2
cwMK x
EI EI
2 22 2
2 2
2 2
1 3( ) (1 )
2 23(1 (1 ( ) )) 3(1 (1 ( ) ))
cwd v Y Y xx
dx EI Ehx xEh
Example : unloading of beam( w(x)=0)• Schematic view of residual stress in plastic zone after unloading
YY
y
2Y2Y
② ④'④
2
h
