# ECIV 720 AA dvanced Structural Mechanics and Analysis - PowerPoint

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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

d
Introduction

Recall Assumptions

• Small Deformations
• Linear Elastic Behavior

KD  R
Introduction

aD  aR
Linear Behavior
Introduction

A. Small Displacements

V B EBdV d  re 
T

K 
Inegrations over undeformed volume

Strain-displacement matrix does not depend
on d
Introduction

B. Linear Elastic Material

V B EBdV d  re 
T

K 
Matrix [E] does not depend on d
Introduction

C. Boundary Conditions do not change
(Implied Assumption)

V B EBdV d  re 
T

Constraints do not depend on   d
Introduction

If any of the assumptions is NOT satisfied

NONLINEARITIES

Material                  Geometric
Assumption B              Assumption A & or C
not satisfied             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       i 1           i     t  t         t  t       i 1
K            U                      R             F
t  t    i        t  t        i 1             i 
U                   U              U
With initial conditions
t  t       0                                      0 
U U         t                     t  t
K K     t

t  t     0 
F F        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

1 T
Consider   u Ku  uT F            BC            u1
2                                     u2
u  u1 u2  un 
T                                 u1=a           u3
P4
FT  F1 F2  Fn                                 u4
Pi      ui
 k11 k12       k1n                           ui+1
k    k 22      k2n                           un-1
K  21                                  Pn
                                            un

                     
k n1 k n 2     k nn 
Boundary Conditions – Elimination Approach

1 T                    Boundary Conditions
  u Ku  u F     T

2                             u1=a

  u1k11u1  u1k12u2    u1k1nun
1
2
 u2 k 21u1  u2 k 22u2    u2 k 2 nun

 un k n1u1  un k n 2u2    un k nnun 
 u1 F1  u2 F2  un F n 
Boundary Conditions – Elimination Approach

Since u1=a known, DOF 1 is eliminated from

1 T
  u Ku  u F
T

2
Consequently, Equilibrium requires that


 0 i  2,3, n
ui
Boundary Conditions – Elimination Approach

        k22u2  k23u3    k2nun  F2  k21a
0
u2
        k32u2  k33u3    k3nun  F3  k31a
0
u3
………
       kn 2u2  kn3u3    knnun  Fn  k31a
0
un

Kffuf=Pf + Kfsus
Boundary Conditions – Elimination Approach

kii kij kik kil kim   ui    Pi
k
kji Kjj kjk kjl kjm   u
ujf   P
Pfj
ff    Kfs
kki kkj kkk kkl kkm   uk = Pk
kli klj klk kll klm   ul    Pl
Ksf        Kss     us    Ps
kli klj klk kll klm   um    P m
Boundary Conditions – Elimination Approach

kii kij kik kil kim    ui   Pi
kji K kjk kjl kjm
kjj     Kfs       u
ujf   Pj
Pf
ff
kki kkj kkk kkl kkm    uk = Pk
kli klj klk kll klm    ul   Pl
Ksf        Kss      us   Ps
kli klj klk kll klm    um   Pm

-1
Kffuf+ Kfsus=Pf       uf =   Kff (Pf +   Kfsus)
Ksfuf+ Kssus=Ps         Ksfuf+ Kssus=Ps
Boundary Conditions Penalty Approach
Boundary Conditions
u1=a                   k=C large stiffness
u1                  u1
u2                  u2
U s  C u1  a 
u3                  u3
1           2
P4                    P4               2
u4                  u4
Pi     ui             Pi   ui
ui+1                ui+1

un-1                un-1
Pn     un
Pn   un
Boundary Conditions Penalty Approach

k=C large stiffness
u1
U s  C u1  a 
1           2
u2                2
u3
P4                  Contributes to 
u4

  u Ku  C u1  a   u F
Pi    ui
1 T    1           2   T
ui+1
2      2
un-1
Pn           Consequently, for Equilibrium
un

 0 i  1,2,3, n
ui
Boundary Conditions Penalty Approach

k11  C  k12     k1n   u1   F1  Ca
 k                        u   F 
 k2n   2  
           k 22                           
 
21                                2

                            
                           
 k n1      kn 2    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          x,u
3

v3  u3 tan 
Changing Directions of Restraints
4               2

3
1         K 11 K 12   K 13   K 14   D1   R1 
K                         D  R 
K 24   2   2 
 21 K 22      0
  
K 31   0     K 33   K 34  D2  R 3 
                            
K 41 K 42    K 43   K 44  D4  R 4 

u1               Fx1 
e.g. for truss         D1             R1   
v1               Fy1 
Changing Directions of Restraints

u3   c s U 3 
          
v3   s c V3 

D3  TU 3

c  cos     s  sin 
Changing Directions of Restraints

4       2     Introduce Transformation

3                             D3  TU 3
1
In stiffness matrix…

 K 11    K 12 K 13T3     K14   D1   R1 
 K                               D   R 
K 24   2   2 
 21 K 22        0
  T 
T3T K 31 0 T3T K 33T3   T3 K 34  U 3  T3 R 3 
T

                                  
 K 41 K 42    K 43T3     K 44  D4   R 4     
Connecting Dissimilar Elements
Simple Cases
4
3

5
kd  r
6

2

d  u5
1
v5  z 5 u6 v6  z 6 
d  u2 v2    u3 v3 u6 v6  z 6 
Connecting Dissimilar Elements
4                    Simple Cases
3
d  Td
a         5
6
L
b
2                       T5 0
T   0 I
6 x 7      

1
 a      0      b       0 
1
T5    0       a      0       b   
L
cos  sin   cos   sin  
                            
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
1
b
2
3

4
 u3          u1                u4          u2 
                                              
 w3   Tb  w1                w4   Tb  w2 
                                        
 y3          y1                y4          y2 

1   0        b
0
Tb       1          
0 
0
    0        1 

Connecting Dissimilar Elements
Eccentric Stiffeners
1
b
2
3

4
3,4 Slave

1,2 Master

k   T k T
T

Tb     0
r  T r
T
T            
0      Tb 
Connecting Dissimilar Elements
Eccentric Stiffeners
1
b
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
b             u1, v1, u2
2
1
Connecting Dissimilar Elements
Rigid Elements

d  Td
 u1   1         0     0 
v                         
  1     0       1     0 
 u1 
u 2   0
                 0     1  
                           v1 
 v2    a / b   1    a / b  
u3   1          0     0    u 2 
                          
 v3   a / b
                1    a / b

Elastic Foundations

Strain Energy

RECALL
l
1

2
su dx
20
Elastic Foundations

l
1 T       1
   σ εdV   su dx
2

2 V       20
  u fdV   u TdS   u Pi
T             T           T
i
V             S
i

RECALL        1
l


2
su dx
20
u  Nq
Elastic Foundations
l
1
 su dx     u  Hq
2

20
RECALL

Additional stiffness
Due to Elastic Support
Elastic Foundations

RECALL            +
Elastic Foundations – General Cases

Plate/Shell/Solid of any          Foundationz
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
2             1
M1               M2 
1             1
Node 3 need not be explicitly present

```
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