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ECIV 720 A Advanced Structural
Mechanics and Analysis
Non-Linear Problems in Solid and Structural Mechanics
Special Topics
Introduction
Nonlinear Behavior:
Response is not directly proportional to the action that produces it.
P
Introduction
Recall Assumptions
• Small Deformations• Linear Elastic Behavior
RDK
Introduction
RD aa Linear Behavior
Introduction
A. Small Displacements
eV
T dV rdBEB K
Inegrations over undeformed volume
Strain-displacement matrix does not depend on d
Introduction
B. Linear Elastic Material
eV
T dV rdBEB K
Matrix [E] does not depend on d
Introduction
C. Boundary Conditions do not change
(Implied Assumption)
eV
T dV rdBEB Constraints do not depend on d
Introduction
If any of the assumptions is NOT satisfied
NONLINEARITIES
MaterialAssumption B not satisfied
GeometricAssumption A & or C not satisfied
Classification of Nonlinear Analysis
Small Displacements, small rotations Nonlinear stress-strain relation
Classification of Nonlinear Analysis
Large Displacements, large rotations and small strains – Linear or nonlinear material behavior
Classification of Nonlinear Analysis
Large Displacements, large rotations and large strains – Linear or nonlinear material behavior
Classification of Nonlinear Analysis
Change in Boundary Condition
Classification of Nonlinear Analysis
Nonlinear Analysis
drddK
Cannot immediately solve for {d}
Iterative Process Required
to obtain {d} so that equilibrium is satisfied
Solution Methods
Newton-Raphson
Newton Raphson
11 ittttiitt FRUK
iittitt UUU 1
UU ttt 0 KK ttt 0
FF ttt 0
With initial conditions
Modified Newton-Raphson
SPECIAL TOPICS
Boundary ConditionsElimination ApproachPenalty Approach
Special Type Elements
Boundary Conditions – Elimination Approach
Consider FuKuu TT 2
1 B C u1
u2u3
u4
uiui+1
un-1
un
P4
Pi
Pn
u1=a nT uuu 21u
nT FFF 21F
nnnn
n
n
kkk
kkk
kkk
21
22221
11211
K
FKu 0 Singular, No BC Applied
Boundary Conditions – Elimination Approach
FuKuu TT 2
1
nn
nnnnnnnn
nn
nn
FuFuFu
ukuukuuku
ukuukuuku
ukuukuuku
2211
2211
2222221212
1121211111
2
1
Boundary Conditions
u1=a
Boundary Conditions – Elimination Approach
n2,3,i 0
iu
Consequently, Equilibrium requires that
FuKuu TT 2
1
Since u1=a known, DOF 1 is eliminated from
Boundary Conditions – Elimination Approach
akFukukuk nn 2122323222 02
u
akFukukuk nn 3133333232 03
u
akFukukuk nnnnnn 313322 0
nu
………
Kffuf=Pf + Kfsus
Boundary Conditions – Elimination Approach
kii kij kik kil kim ui
uj
uk
ul
kji kjj kjk kjl kjm
kki kkj kkk kkl kkm
kli klj klk kll klm
kli klj klk kll klm um
=
Pi
Pj
Pk
Pl
Pm
Kff
Kfs
Ksf Kss
uf
Pf
us Ps
Boundary Conditions – Elimination Approach
kii kij kik kil kim ui
uj
uk
ul
kji kjj kjk kjl kjm
kki kkj kkk kkl kkm
kli klj klk kll klm
kli klj klk kll klm um
=
Pi
Pj
Pk
Pl
Pm
Kff
Kfs
uf
Pf
us
Kffuf+ Kfsus=Pf
Ksf Kss Ps
Ksfuf+ Kssus=Ps Ksfuf+ Kssus=Ps
uf = Kff (Pf + Kfsus)-1
Boundary Conditions Penalty Approach
u1
u2u3
u4
uiui+1
un-1
un
P4
Pi
Pn
u1
u2u3
u4
uiui+1
un-1
un
P4
Pi
Pn
k=C large stiffness
2
1 21 auCU s
Boundary Conditionsu1=a
Boundary Conditions Penalty Approach
FuKuu TT auC 212
1
2
1
u1
u2u3
u4
uiui+1
un-1
un
P4
Pi
Pn
k=C large stiffness
2
1 21 auCU s
n2,3,,1i 0
iu
Consequently, for Equilibrium
Contributes to
Boundary Conditions Penalty Approach
nnnnnn
n
n
F
F
CaF
u
u
u
kkk
kkk
kkCk
2
1
2
1
21
22221
11211
The only modifications
Support Reaction is the force in the spring
auCR 11
Choice of C
Rule of Thumb
nj
ni
kC ij
1
1
10max 4
Error is always introduced and it depends on C
Penalty approach is easy to implement
Changing Directions of Restraints
1
2
3
4
x,u
y,v
tan33 uv
Changing Directions of Restraints
4
3
2
1
4
2
2
1
44434241
343331
242221
14131211
0
0
R
R
R
R
D
D
D
D
KKKK
KKK
KKK
KKKK1
2
3
4
1
1
1y
x
F
FR
1
11 v
uDe.g. for truss
Changing Directions of Restraints
3
3
3
3
V
U
cs
sc
v
u
sincos sc
33 TUD
Changing Directions of Restraints
1
2
3
4 Introduce Transformation
4
33
2
1
4
3
2
1
443434241
3433333313
242221
143131211
0
0
R
RT
R
R
D
U
D
D
KTKKK
KTTKTKT
KKK
KTKKK
TTTT
33 TUD In stiffness matrix…
Connecting Dissimilar ElementsSimple Cases
rdk
666555 zz vuvu d1
2
34
5
6
6663322 zvuvuvu d
1
2
34
5
6
Connecting Dissimilar ElementsSimple Cases
dTd a
bL
I0
0T5T76x
sincossincos
T5 ba
ba
L00
001
Connecting Dissimilar ElementsSimple Cases
Hinge
Beam
Connecting Dissimilar ElementsSimple Cases
Beam
Stresses are not accurately computed
Connecting Dissimilar ElementsEccentric Stiffeners
Use Eccentric Stiffeners
Connecting Dissimilar ElementsEccentric Stiffeners
1
23
4Slave
Master
Connecting Dissimilar ElementsEccentric Stiffeners
1
1
1
3
3
3
y
b
y
w
u
w
u
T
100
010
01 b
bT
b 1
23
4
1
23
4
2
2
2
4
4
4
y
b
y
w
u
w
u
T
Connecting Dissimilar ElementsEccentric Stiffeners
TkTk T
b
b
T0
0TT
b 1
23
4
1
23
4
rTr T
3,4 Slave
1,2 Master
Connecting Dissimilar ElementsEccentric Stiffeners
b 1
23
4
1
23
4
The assembly displays the correct stiffness in states of pure stretching and pure bending
The assembly is too flexible when curvature varies – Use finer mesh
Connecting Dissimilar ElementsRigid Elements
Rigid element is of any shape and size
Generalization of Eccentric Stiffeners – Multipoint Constraints
Use it to enforce a relation among two or more dof
Connecting Dissimilar ElementsRigid Elements
e.g.
1
2
3 a
b
1-2-3 Perfectly Rigid
Rigid Body Motion described by
u1, v1, u2
Connecting Dissimilar ElementsRigid Elements
dTd
2
1
1
3
3
2
2
1
1
1
001
1
100
010
001
u
v
u
baba
baba
v
u
v
u
v
u
//
//
Elastic Foundations
l
dxsu0
2
2
1
Strain Energy
RECALL
Elastic Foundations
ii
TiS
T
V
T
l
V
T
dSdV
dxsudV
PuTufu
εσ0
2
2
1
2
1
l
dxsu0
2
2
1
Nqu
RECALL
Elastic Foundations
l
dxsu0
2
2
1Hqu
Additional stiffness
Due to Elastic Support
RECALL
Elastic Foundations
+RECALL
Elastic Foundations – General Cases
Soil
xy
zFoundationPlate/Shell/Solid of any
size/shape/order
Winkler Foundation
Elastic Foundations – General Cases
Winkler Foundation Stiffness Matrix
• s is the foundation modulus• H are the Shape functions of the
“attached element”
Winkler Foundations
• Resists displacements normal to surface only• Deflects only where load is applied• Adequate for many problems
Other Foundations
• Resists displacements normal to surface only• They entire foundation surface deflects• More complicated by far than Winkler• Yields full matrices
Elastic Foundations – General Cases
Soil
xy
z
InfiniteInfinite
Infinite
Infinite
Infinite Elements
Infinite Elements
Infinite Elements
Use Shape Functions that force the field variable to approach the far-field value at infinity but retain finite size of element
Use conventional Shape Functions for field variable
Use shape functions for geometry that place one boundary at infinity
or
Shape functions for infinite geometry
Element in Physical Space Mapped Element
Reasonable approximations
Shape functions for infinite geometry
2211 xMxMx
1
21M
1
12M
Node 3 need not be explicitly present