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
d
Introduction
Recall Assumptions
• Small Deformations
• Linear Elastic Behavior
K D  R
Introduction
aD  aR
Linear Behavior
Introduction
A. Small Displacements









B
E
B
dV
d

r
e
V
T
K 
Inegrations over undeformed volume
Strain-displacement matrix does not depend
on d
Introduction
B. Linear Elastic Material









B
E
B
dV
d

r
e
V
T
K 
Matrix [E] does not depend on d
Introduction
C. Boundary Conditions do not change
(Implied Assumption)









B
E
B
dV
d

r
e
V
T
Constraints do not depend on
d
Introduction
If any of the assumptions is NOT satisfied
NONLINEARITIES
Material
Geometric
Assumption B
not satisfied
Assumption 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
Kdd  rd
Cannot immediately solve for {d}
Iterative Process Required
to obtain {d} so that equilibrium is satisfied
Solution Methods
Newton-Raphson
Newton Raphson
t  t
t  t
K
i 1
i 
U 
i 
U 
t  t
U
t  t
i 1
R
t  t
 U
F
i 1
i 
With initial conditions
t  t
0 
t  t
U U
t
t  t
0 
K
F F
t
0 
K
t
Modified Newton-Raphson
SPECIAL TOPICS
Boundary Conditions
Elimination Approach
Penalty Approach
Special Type Elements
Boundary Conditions – Elimination Approach
0  Ku  F Singular, No BC Applied
Consider

1
u Ku  u F
T
2
u  u1
u2
F  F1
F2
T
T
 k11

k 21
K
 

k n1
 un 

k12

k 22


kn 2

Fn 
k1n 

k2n



k nn 
T
u1
BC
u2
u1=a
u3
P4
Pi
Pn
u4
ui
ui+1
un-1
un
Boundary Conditions – Elimination Approach
1

u Ku  u F
T
T
Boundary Conditions
2

1
2
u1=a
u1k11u1  u1k12u2    u1k1nun
 u 2 k 21u1  u 2 k 22u2    u2 k 2 n un


un k n1u1  u n k n 2u 2    un k nnu n 
 u1 F1  u 2 F2  u n F n 
Boundary Conditions – Elimination Approach
Since u1=a known, DOF 1 is eliminated from

1
u Ku  u F
T
T
2
Consequently, Equilibrium requires that

ui
0
i  2,3,  n
Boundary Conditions – Elimination Approach

u 2

0
k22u2  k23u3    k2 nun  F2  k21a
0
k32u2  k33u3    k3nun  F3  k31a
u3
………

 0 kn 2u2  kn3u3    knnun  Fn  k31a
u n
Kffuf=Pf + Kfsus
Boundary Conditions – Elimination Approach
kii kij kik kil kim
ui
Pi
kji K
kjj kjk kK
jl kjm
ff
fs
kki kkj kkk kkl kkm
uujf
PPfj
kli klj klk kll klm
Ksf
Kss
kli klj klk kll klm
ul
us
u
Pl
Ps
P
uk = Pk
m
m
Boundary Conditions – Elimination Approach
kii kij kik kil kim
ui
Pi
kji K
kjj kjk kK
jl kjm
ff
fs
kki kkj kkk kkl kkm
uujf
PPfj
kli klj klk kll klm
Ksf
Kss
kli klj klk kll klm
ul
us
um
Pl
Ps
P
Kffuf+ Kfsus=Pf
Ksfuf+ Kssus=Ps
uk = Pk
uf =
m
-1
Kff (Pf +
Kfsus)
Ksfuf+ Kssus=Ps
Boundary Conditions Penalty Approach
Boundary Conditions
u1=a
P4
Pi
Pn
k=C large stiffness
u1
u1
u2
u2
u3
u3
u4
ui
ui+1
un-1
un
P4
Pi
Pn
u4
ui
ui+1
un-1
un
Us 
1
2
C u1  a 
2
Boundary Conditions Penalty Approach
k=C large stiffness
u1
Us 
u2
u3
P4
Pi
Pn
1
2
C u1  a 
2
Contributes to 
u4
ui
ui+1
un-1
un

1
u Ku 
T
1
C u1  a   u F
2
2
Consequently, for Equilibrium

ui
0
i  1,2,3,  n
2
T
Boundary Conditions Penalty Approach
k11  C 

k 21




 k n1
k12

k 22


kn 2

k1n   u1   F1  Ca 
  

k 2 n u2   F2 

 

 

  


 

k nn  
un   Fn 
The only modifications
Support Reaction is the force in the spring
R1  C u1  a 
Choice of C
Rule of Thumb
C  max kij  10
4
1 i  n
1 j  n
Penalty approach is easy to implement
Error is always introduced and it depends on C
Changing Directions of Restraints
4
2
y,v
1
3
v3  u3 tan 
x,u

Changing Directions of Restraints
4
2
3
1
e.g. for truss
 K 11

K
 21
K 31

K 41
K 12
K 13
K 22
0
0
K 33
K 42
K 43
u1 
D1   
v1 
K 14   D1   R1 
   
K 24 D 2  R 2 

  
K 34  D 2  R 3 

  
K 44  
D 4  R 4 
 Fx1 
R1  

 Fy1 
Changing Directions of Restraints
u3   c
 
v3   s
s U 3 
 
c V3 
D3  TU 3
c  cos 
s  sin 
Changing Directions of Restraints
4
2
Introduce Transformation
D3  TU 3
3
1
In stiffness matrix…
 K 11

K 21

T3T K 31

 K 41
K 12
K 13T3
K 22
0
T
3
0
T K 33T3
K 42
K 43T3
K 14   D1   R1 
  

K 24 D 2   R 2 

  T

T
T3 K 34  U 3  T3 R 3 

 

K 44  
D 4   R 4 
4
Connecting Dissimilar Elements
Simple Cases
3
kd  r
5
6
2
1
d  u5
d  u2
v2
v5  z 5
u3
v3
u6
v6  z 6 
u6
v6  z 6 
Connecting Dissimilar Elements
Simple Cases
3
4
d  Td
5
a
L
6
b
T
2
6 x 7 

T5

0
0

I
1
 a
1
T5    0
L
cos 
0
b
a
0
sin 
 cos 


b

 sin  
0
Connecting Dissimilar Elements
Simple Cases
Beam
Hinge
Connecting Dissimilar Elements
Simple Cases
Beam
Stresses are not
accurately computed
Connecting Dissimilar Elements
Eccentric Stiffeners
Connecting Dissimilar Elements
Eccentric Stiffeners
1
3
2
Master
4
Slave
Use Eccentric Stiffeners
Connecting Dissimilar Elements
Eccentric Stiffeners
b
1
2
3
 u3 
 u1 




 w3   Tb  w1 
 
 
 y3 
 y1 
1

Tb   0

0
4
0
1
0
 u4 
 u2 




 w4   Tb  w2 
 
 
 y4 
 y2 
 b

0

1 

Connecting Dissimilar Elements
Eccentric Stiffeners
b
1
2
3
4
3,4 Slave
1,2 Master
k   T k T
T
r  T r
T
Tb
T  
0
0

Tb 
Connecting Dissimilar Elements
Eccentric Stiffeners
b
1
2
3
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 Elements
Rigid Elements
Generalization of Eccentric Stiffeners –
Multipoint Constraints
Rigid element is of any shape and size
Use it to enforce a relation among two or
more dof
Connecting Dissimilar Elements
Rigid Elements
e.g.
1-2-3 Perfectly Rigid
3
a
Rigid Body Motion
described by
u1, v1, u2
b
2
1
Connecting Dissimilar Elements
Rigid Elements
d  Td
 u1   1
  
v1
0
  

  0
u 2 
 
 v2    a / b
u   1
3
  

 a / b
 
 v3 
0
1
0
1
0
1
0 

0
 u 
1
1  
  v1 
a / b  
u2 

0 

a / b

Elastic Foundations
Strain Energy
RECALL
1
2
l
su
dx

2
0
Elastic Foundations

1
2
σ
ε
dV


T
V
1
2
l
su
dx

2
0
  u fdV   u TdS   u Pi
T
T
V
RECALL
S
1
2
l
su
dx

2
0
u  Nq
T
i
i
Elastic Foundations
1
2
l
su
dx

2
0
u  Hq
RECALL
Additional stiffness
Due to Elastic Support
Elastic Foundations
RECALL
+
Elastic Foundations – General Cases
Foundationz
Plate/Shell/Solid of any
size/shape/order
y
x
Winkler Foundation
Soil
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
z
y
x
Soil
Infinite
Infinite
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
or
Use conventional Shape Functions for field
variable
Use shape functions for geometry that place
one boundary at infinity
Shape functions for infinite geometry
Element in Physical Space
Mapped Element
Reasonable approximations
Shape functions for infinite geometry
x  M1 x1  M 2 x2
M1  
2
1
M2 
1
1
Node 3 need not be explicitly present