eciv 720 a advanced structural mechanics and analysis non-linear problems in solid and structural...

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