FEM: Nonlinear Beam Deflection
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Introduction to the Finite
Element Method
Nonlinear Beam Deflection
FEM: Nonlinear Beam Deflection
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Nonlinear Strain Displacement
Relation
FEM: Nonlinear Beam Deflection
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Nonlinear Strain Displacement
Relation • The von Karman large deflection strain-
displacement relation for the deflections u,
and w can be written as follows
2
22
2
1
x
wz
x
w
x
ux
zm
FEM: Nonlinear Beam Deflection
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Using the relations …
}{
xx1 32
aH
axw
w
}{
1
bH
bxu
u
bwbbw wNwTHw 1
mummu wNwTHu 1
FEM: Nonlinear Beam Deflection
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For the derivatives …
bbbw wBwT
dx
dH
dx
dw
1
bbbbw wBwT
dx
Hd
dx
wd
1
2
2
2
2
mmmmu wBwT
dx
dH
dx
du
1
FEM: Nonlinear Beam Deflection
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Giving the strain relation …
• Where:
• Is a scalar function of x
bbbmm
m
wzww
z
BBB
2
1
}{
x
w
FEM: Nonlinear Beam Deflection
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Inplane Forces and Bending
Moments
FEM: Nonlinear Beam Deflection
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Inplane Forces and Bending
Moments
• For homogeneous symmetric cross sections:
T
T
M
N
D
A
M
N
0
0
EAQhA EIQh
D 12
3
TEAdzzyxTQNh
hT
2/
2/),,(
0),,(2/
2/
h
hT zdzzyxTQM
FEM: Nonlinear Beam Deflection
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Expanding equation
Tm
Tbmm
Tm
NNN
NwAwA
NAN
BB2
1
][
bb wDDM B][}]{[
FEM: Nonlinear Beam Deflection
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The Potential Energy
Volume height
TT
ijij dzMNdVU }{}{
dz
wDw
NwAwA
ww
Uheight
bb
T
b
T
b
Tbmm
TTT
b
T
m
T
m
BB
BB
BB
][
2
1
*
FEM: Nonlinear Beam Deflection
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Skipping the details …
• We get the element equation as:
m
b
mb
bmnm
TN
m
b
W
W
n
n
nn
kT
k
k
00
02
3
1
01
11
2
1
00
0
0
0
Tm
b
p
p 0
0
FEM: Nonlinear Beam Deflection
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Where …
• The linear stiffness matreces
dzDkheight
b
T
bb BB ][][
dzAkheight
m
T
mm BB][
FEM: Nonlinear Beam Deflection
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Where …
• The nonlinear stiffness matreces
dzAnnheight
T
m
T
bmmb BB]1[]1[
dzBNnheight
m
T
nm B1
dzAnheight
TT
BB2
3]2[
FEM: Nonlinear Beam Deflection
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Where …
• The thermal effect terms
dzBNkheight
T
T
TN B][
dzNpheight
T
T
mTm B
FEM: Nonlinear Beam Deflection
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In a compact form …
TTN PPWNNKK
2
3
11
2
1
FEM: Nonlinear Beam Deflection
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Note
• For details for the derivation, follow the link
on the pages: • https://wikicourses.wikispaces.com/Topic+Nonlinear+Solid+Mechani
cs
• https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics
FEM: Nonlinear Beam Deflection
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Solving the Equations for
Post-Buckling Deflection
FEM: Nonlinear Beam Deflection
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When no external loading exists
• We may write the equations in the form:
• Introducing a new function:
TTN PWNNKK
2
3
11
2
1
023
11
2
1
TTN PWNNKKW
FEM: Nonlinear Beam Deflection
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Using truncated Taylor
expansion • Linear approximation:
• Where:
WdW
WdWWW
tan21 KNNKK
dW
WdTN
FEM: Nonlinear Beam Deflection
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Using Newton Algorithm:
Repeat, until:
TiiiTNi PWNNKKW
2
3
11
2
1
iiiWWK 1tan
ii WKWi
1
tan1
11 iii WWW
toliW 1max