Finite element method for structural dynamic

and stability analyses

1

Prof C S Manohar Department of Civil Engineering

IISc, Bangalore 560 012 India

Module-12

Closure

Lecture-40 Closure and a few topics for further study

Topics which were proposed to be covered in this course

• Approximate methods and FEM

• Dynamics of truss and planar frame structures

• Damping models and analysis of equilibrium equations

• Dynamics of Grids and 3D frames

• A few computational aspects (solution of equilibrium equations,

eigenvalue problems, model reduction, and substructuring)

• Dynamic stiffness matrix and transfer matrix methods

• Dynamics of plane stress/strain, plate bending, shell and 3d elements

• Applications (earthquake engineering and vehicle structure

interactions)

• FEA of elastic stability problems

• Treatment of nonlinearity

• FE model updating

• FEM in hybrid simulations

2

Course modules

1. Approximate methods and FEM (4)

2. Finite element analysis of dynamics of planar

trusses and frames (3)

3. Analysis of equations of motion (3)

4. Analysis of grids and 3D frames (3)

5. Time integration of equation of motion (4)

6. Model reduction and substructuring schemes (2)

7. Analysis of 2 and 3 dimensional continua (6)

8. Plate bending and shell elements (4)

9. Structural stability analysis (9)

10.FE Model updating (2)

11.Nonlinear FE Models (3)

12.Closure

3

Acknowledgement 1. M Petyt, 1990, Introduction to finite element vibration analysis, CUP, Cambridge.

2. G J Simitses and D H Hodges, 2006, Fundamentals of structural stability, Elsevier,

Amsterdam.

3. R D Cook, D S Malkus, and M E Plesha, 1989, Concepts and applications of finite

element analysis, 3rd Edition, John Wiley, New York

4. S S Rao, 1999, The finite element method in engineering, 3rd Edition, Butterworth-

Heinemann, Boston.

5. W McGuire, R H Gallagher, and R D Ziemian, 2000, Matrix structural analysis, 2nd

Edition, John Wiley, New York.

6. T Belytschko, W K Liu, and B Moran, 2000, Nonlinear finite elements for continua and

structures, Wiley, Chichester.

7. J N Reddy, 2004, An introduction to nonlinear finite element analysis, Oxford

University Press, New York.

8. J N Reddy, 2013, An introduction to continuum mechanics, 2nd edition, Cambridge

University Press, New York.

9. K J Bathe, 1996, Finite element procedures, Prentice Hall of India, New Delhi.

10. J M T Thompson and G W Hunt, 1973, A general theory of elastic stability, John

Wiley, London

11. http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html [Professor

C A Felippa, University of Colorado at Boulder)

4

2 2

1 1

2

1

2 2

Hamilton's principle: Minimize the functiona

1

2

l

t t

t t

t

t

t dt T t V t dt

mu t ku t dt

L

u t

m

k

Example-1

6

0 0, 0 & , 0AE u t AE L u L t

0 0, 0 & , 0AE u t u L t

0, 0 & , 0u t AE L u L t

0, 0 & , 0u t u L t

Geometric, forced, or kinematic boundary condition: , 0 on the boundary

Free or natural boundary condition: , 0 on the boundary

u x t

AE x u x t

2

2; : ,0 & ,0 specified

u uAE x m x u x u x

x x t

IC-s

Boundary

conditions

7

, ,L AE x m x

M1k 2k

Exercise: set up the governing equation for the system shown below

,u x t

v

2 2

0

2 22

1 2

0

Hint

1 1,

2 2

1 1 1V , 0, ,

2 2 2

L

L

T t m x u x t dx Mv

t AE x u x t dx k u t v t k u L t v t

x

8

x

y

, ,L EI x m x

2

1

2

0

22

2

0

22

2

2

0 0

0 0

1Kinetic energy: ,

2

1Potential energy:

2

1 1Lagrangian ,

2 2

, ,

L

L

L L

tL L

t

T t m x v x t dx

vV t EI x dx

x

vT V m x v x t dx EI x dx

x

F v v dx F v v dxdt

L

L

Euler-Bernoulli beam

9

The 16 classical single span beams

0& 0EIv EIv

0& 0v v

0& 0EIv v

0& 0v EIv

0

0

, 0

, 0

L

L

EIv x t

EIv x t

0 EIv mv

10

2

2

2 2

1

2 2

Discrete MDOF system

has units of (rad/s) .

If .

t

t

t

n nn nt

n n

N

n n

u KuR u

u Mu

R u

Ku R u

M

R u

u y R u O

2

2

1

0

2

0

2

0

2

0

Axially vibrating bar

Euler Bernoulli beam

L

L

L

L

AE x x dx

R x

m x x dx

EI x x dx

R x

R x

m x x dx

Rayleigh’s quotient

Rayleigh’s principle

11

How to lower the value of ?R x Rayleigh - Ritz Method

2

0

2

0

1

, 1, 2, , : a set of known linearly independent functions

whic allh satisfy the boundary conditions.

, 1,2, , : a set of unknown co

L

L

N

n n

n

n

n

EI x x dx

R x

m x x dx

x a x

x n N

a n N

nstants which need

to be determined

Strategy: Select , 1,2, , such that is minimized.na n N R x

12

What MWR achieves?

A PDE governing the behavior of a continuous system

has been replaced by an equivalent set of ODE-s (IVP-s)

with a view to obtain an approximate solution.

0 0

Field equation: , , , ,

ICS: ,0 , ,0

BCS: Appropriate geometric and natural BCS

EI x v x t m x v x t c x v x t f x t

v x v x v x v x

1

,N

n n

n

v x t a t x

; 0 , 0Ma Ca Ka P t a a

Drive the residue , to be small

in some sense (MWR).

e x t

13

0

0

0

1

0

,Least squares , 0 for 1, 2, ,

Collocation , 0 for 1, 2, ,

Galerkin , 0 for 1, 2, ,

, 0 Subdomain collocation

for 1, 2, ,

Petrov-Galerkin

L

n

L

n

L

n

L

n n

n

e x te x t dx n N

a

x x e x t dx n N

x e x t dx n N

U x x U x x e x t dx

n N

x

0

, 0 for 1, 2, ,

L

e x t dx n N

14

0 Weight Residue

Method of weighted residues

, 0 for 1,2, ,

; 0 , 0

L

nw x e x t dx n N

Ma Ca Ka P t a a

Assumed mode method and Lagrange's equation

Strong (operational) form, Weighted residual form, and

weak (variational) form of governing equations

15

,u x t

Mesh

Element

, ,

BCs+ICs

OR

Min A=

Lu x t F x t

Ldt

Mesh

16

1

1

, is approximated

in terms of

, ; 1, 2, ,

;

Within an element

, is approximated

as , ,

An ap

FINITE

proximate

ELEMENT

METHOD

n

e

i

r

i i j

i

Ne e e

i i

i

u x t

u x t i N

i j

u x t

u x t N x u x t

umerical method

to obtain solutions to PDE-s or

variational problems

: Elements

: Nodes

1,2, ,

i

ix

i N

17

Element level EOM

in local coordinate system

Element level EOM; ; ;

in global coordinate system

Global EOM after

s s s s s s s

s s s s s s s

t t t

s s s s s s s s s s s

t

s s s s

M u C u K u F t

M U C U K U F t

U T u M T M T K T K T

C T C T

Summary

1 1

1 1

0

assembly

of structural matrices and

before imposing boundary

condtions;

Equations for unknown

reactions

p pt t

s s s ss ss s

p pt

ss s sss s

I I

MU CU KU F t

M A M A K A K A

C A C A F t A F t

M U C

0

0 0

0 0

Equations for unknown

displace0 ; 0

ments

I I I I

II I II I II I I

MU CU KU F

U K U F t

M U C U K U F t

t

U U U U

18

A

B

A

B

Neglect axial deformation

16 17

1 4

7 19

19 22

22 10

5 17

11 17

2 8

8 14

13 14 15

cos sin

0

u uu uu uu uu uu uu uu uu uu u u

12

6

5

43

2

1

11

109

8

7

24

18

17

16

15

14

13

23

2221

20

19

Hinge

19

Stress

x

Discontinuity introduced by FEM

Displacement based FEM introduces

discontinuities in spatial variation of

quantities which are truly continuous.

20

LTI

LTI

LTI

LTI

)(tf

t

dfthtx0

)()()(

)()()( FHX )(F

)(t )(th

)exp( ti )exp()( tiH

( ) ( )

( ) ( )

( ) ( )

f t F

x t X

h t H

Input-output relations for linear time invariant systems

Linear Damping models

21

Viscous Structural

Classical Non-Classical Classical Non-Classical

Classification into viscous and structural depends upon behavior

of energy dissipated under harmonic steady state as a function of

frequency.

Classification into classical and non-classical depends

upon

orthogonality (or lack of orthogonality) of damping matrix

with respect to undamped normal modal matrix.

22

ResponseQuantity

Receptance

Displacement Admittance Dynamic stiffness

Dynamic compliance

Dynamic flexibility

Velocity Mobility Mechanical impedance

Acceleration Accelerance Apparent mass

R FR F R

Nomenclature for FRF

23

12

FRF calculations (valid for both viscous and structural damping models

(a) Viscously damped system

exp

b Structurally damped system

MU CU KU F i t

M i C K

MU K i

S

Direct calcul

umma

t n

ry

a io

12

2 21

2

exp

c Viscously damped system with classical damping

2

; ; ; Diag

Nrn sn

rs

n n n n

t t

i

D U F i t

M K iD

i

K M M I K

Calculation based on mode superposition

24

2 21

2

d Structurally damped system with classical damping

; ; ; Diag

e Viscously damped system with nonclas

njk rk

jr

k k k

t t

i

iD

K M M I K

Calculation based on mode superposition (continued)

* *

*1

* *

*

*

1 2

2 21

2

sical damping

; ; ; Diag

Diag

f Structurally damped system with nonclassical damping

;

nrk jk jk rk

jr

k k k

t t

n

njk rk

jr

k k

t

i i

B A A I A

s

K iD s M

2 2 2

1 2; Diag t

nM I K iD s s s

25

Dynamic stiffness matrix for an Euler-Bernoulli beam

Focus : Steady state behavior

1 1,u t p t

2 2,u t p t 4 4,u t p t

3 3,u t p t

exp1,2,3,4

exp

k k

k k

u t i tk

p t P i t

1 2 1 2, , , , , ,EI m l c c h h

26

State vector and transfer matrix

2 exp i t 1 exp i t

2 expP i t 1 expP i t

, ,AE m l

,u x t

x

2 2

State vector=

Transfer matrix

R L

P

TP P

T

27

P t

1 1,u t P t

3 3,u t P t

4 4,u t P t

5 5,u t P t 2 2,u t P t

6 6,u t P t, , , ,EI AE m l c

2D beam (frame)

element

4 4,u t P t

3 3,u t P t

6 6,u t P t

5 5,u t P t

1 1,u t P t 2 2,u t P t

2D beam (grid)

element

28

2 2,u t P t

2-noded element with 3 dofs per node

1 1 4 2

3 1 2 2 6 3 5 4

,

,

x t u t x u t x

v x t u t x u t x u t x u t x

3 3,u t P t

5 5,u t P t

6 6,u t P t

1 1,u t P t 4 4,u t P t

29

2 2

2 2

3 2 2

2 2

2 2

2 2

0 0 0 0

0 4 6 0 2 6

0 6 12 0 6 12

0 0 0 0

0 2 6 0 4 6

0 6 12 0 6 12

140 700 0 0 0

0 4 22 0 3 13

0 22 156 0 13 54

70 1404200 0 0 0

0 3 13 0 4 22

0 13

m m

m m

JGl JGl

EI EI

l l l l

l lEIK

l JGl JGl

EI EI

l l l l

l l

I I

m m

l l l l

l lmlM

I I

m m

l l l l

54 0 22 156l l

30

3D beam element

1

11

10

9

8

7

6

5

43

2

12, , , , , , , ,x y zE G I I I A J l

z

y

x

Translation in mForce in N

Rotation in radForce in Nm

, , , , , , ,u x t v x t w x t x t

31

2 2

0 0

Axial deformation Twisting

2 22 2

2 2

0 0

Bending@z Bending@y

2

1 1

2 2

1 1

2 2

with =

L L

L L

z y

uU AE dx GJ dx

x x

v wEI dx EI dx

x x

J y zx z

2

2 2 2 2

0 0 0 0

Axial deformation Twisting Bending@z Bending@y

2 2

1 1 1 1 + +

2 2 2 2

with

A

L L L L

m

m

A

dA

T mu dx I dx mv dt mw dt

I y z dA

0 0

Consider a -dof system

, ,

0 ; 0

This equation constitutes a set of semi-discretized system of

coupled

Numerical integration of equations of equillibrium

Remark

sec

s:

ond

N

MU CU KU R U t U t t F t

U U U U

order ode-s. That is, these equations have been

obtained after discretizing the spatial variables.

This set of equations constitutes a set of initial value problems.

We consider solution of the ab

0 1 2 1

ove equation at a set of

discrete time instants with .

The basic idea is to replace the derivatives appearing

in the above equations by finite difference approximations

an

n n n nt t t t t t t

d then solve the resulting algebraic equations. 32

Explicit method with first order accuracy

Implicit method with first order accur

Forward Euler

Backward

acy

Explicit method with second order accurac

Euler

Cen a

y

tr

Discussion on following methods

l difference

Newmark's family of methods

HHT- method and genera

Implicit method with second order accuracy

Explicit-Implicit method

lization

HHT- with operator sp

s

litt ing

33

34

At least second order accuracy

Unconditional stability when applied to LTI systems

Controllable algorithmic damping in higher modes

Investiga

Desirable features of numerical integration schemes

No overshoot

te spectral radius as frequency

Spectral radius 1 as driving frequency 0

Excessive oscillations during the first few steps

Self-starting

No more than one set of implicit

equations to be solved at each step

35

0 0

1

2

; 0 & 0

m

m

s

t t t t

m m m

r m r m r m r

ss sm

ss ss

MX CX KX F t X X X X

X tX t X t

X t

M X t C X t K X t F t

M X C X K X F t

I

K K

I

K M

Recall : model reduction

Static condensation

Dynamic condensation 12

1

sm sm

m t t

m m m

s

K M

SEREP

Coupling techniques

• Spatial coupling method

• Modal coupling method

– Fixed interface (component mode synthesis)

– Free interface

36

37

x

y

z

transverse load

x

y

z

longitudinal load

Membrane action

Bending action

Membrane action can be analysed using plane stress elements

Bending action: topic for the study

38

Linear triangular plane stress element

12

3

1 1,x y

2 2,x y

3 3,x y

1u

2u

3u2v

1v

3v

, , ,x u x y t

, , ,y v x y t

Field variables

, ,

, ,

u x y t

v x y t

1 2 3

4 5 6

, ,

, ,

u x y t t t x t y

v x y t t t x t y

Thickness=h

39

, , ,x u x y t

, , ,y v x y t

,x y

1 11 ,x y

4 44 ,x y

3 33 ,x y

2 22 ,x y

1, 1 1, 1

1,1 1,1

,

Linear quadrilateral element

t

e

A

t

e

A

K hB DBdA

M h N NdA

: transform the element domain to square

to facilitate evaluation of the integrals.

Idea

4 3

21

A

B

40

8

54 3

21

7

6

Rectangular hexahedron

Pentahedron

Isoparametric hexahedron

Tetrahedron

41

Isoparametric hexahedron element

1

2

1

3

23

4

4

5

5

6

7

68

78

x

z

y

8 noded element with 3dofs/node

42

Axisymmetric problems

3D axisymmetric solid

Not necessarily prismatic

Not necessarily thin or thick

Geometry

Surface tractions , , = ,

Body force: , , 0,

, , = , , , , = ,r r z z

f r z f r z

F r z

F r z F r z F r z F r z

Loads

, , 0

, , ,

, , ,

v r z

u r z u r z

w r z w r z

Displacements

Linear, homogeneous

elastic, isotropic

Material

Rotational

Symmetry

about an axis

,r u

,z w,v

43

Four noded linear quadrilateral thick plate bending element

4

3

2

1

3 3,x y 3 4,x y

2 2,x y

1 1,x y

x

y

1,1

1, 1 1, 1

0,0

1,1

4

1

4

1

, ,

, ,

i i

i

i i

i

x x N

y y N

31 1

2 12 2

t t s

A A

hV D dA kh D dA

2a

c

, , yy v

e

,z w

, , xx u

Examples: Bridge deck, building floors, ship hulls,

Beam centroidal axis is placed eccentrically to the middle surface of the plate.

Membrane and bending action gets

Plate stiffened by beam elements

coupled

A Steady state of

rest

periodic motion

random motion

An externally imposed

disturbance

perturbation

The state is stable if

response to perturbation

dies and original state

is restored

Original state

is not restored

Motion grows

without limit

The state is

unstable

Motion neither

grows nor decays:

The state is

unstable or stable?

q x

P P

PP

A

B

C

D

q x

q x

q x

EIy M x

EIy Py M x

ivEIy q x

ivEIy Py q x

3 2

A

B cos sin

C

D cos sin

y ax b PI

y a x b x PI

y ax bx cx d PI

y a x b x cx d PI

P

EI

47

0 03

0 02

0 0

Summary

3 tan=

2 1 cos0

cos

tan

2

u uu

u

uu

u u

l uM M M u

u

P P

Q

B

Q l cR

l

A

QcR

l

BA

cl c

0 0.5 1 1.5 210

-1

100

101

102

u

Sta

bili

ty f

un

ctio

ns

chi

epsilon

xi

48

y x

0y xP P

P P

e e

P P

q x

These three problems are mathematically equivalent.

The

• transverse load

• eccentrically applied axial load

• initial imperfections

are manifestations of departures from an ideal situation.

How about the study of the ideal situation itself?

49

What should be such that a adjoining equillibrium position becomes

possible?

Assume: an adjoining equillibrium position is indeed possible.

P

x

y x

P P

2

20; 0 0; 0

This is an eigenvalue problem.

For what values of does this equation admit a nontrivial solution?

d yEI Py y y L

dx

P

50

, ; ,

Equillibrium points

*, * 0

*, * 0

* ; *

exp

x f x y y g x y

f x y

g x y

x t x t y t y t

f f f f

t t tx y x yst

t t tg g g

x y

Stability of equillibrium points

If 0, lim the fixed point *, * is unstable

If 0, lim 0 the fixed point *, * is stable

t

t

sg

x y

s a ib

ta x y

t

ta x y

t

51

Consider a system with generalized cordinates.

Focus attention on statically loaded structures.

A stationary value of the total potential energy with res

n

Energy methods for stability analysis

Axiom -1

pect to the

generalized coordinates is necessary and sufficient condition for the

equillibrium state of the system.

A complete relative minimum of the total potential energy with respect

to the g

Axiom - 2

eneralized coordinates is necessary and sufficient for the stability

of an equillibrium state of the system.

J M T Thompson and G W Hunt, 1973, A general theory of elastic stability, John Wiley, London

52

Buckling due to bifurcation of equillibrium

x

y x

P P

sinL LL

P

k

cosL cosL

-1.5 -1 -0.5 0 0.5 1 1.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

theta

P/4

kl

1P

3P

2P

st1 buckling load

rd3 buckling load

nd2 buckling load

Limit load buckling

53

2 2 2

0 0

0 0 0

0 0 0

1

2

0 01

0 0 with =2

0 0

xx zx

V

e e e

xx xy xz

t

xy yy yz

V

u v w u u v v w wU dV

x x x y z y z y z

u

u

v N u N u G u

w

s

U s dV s

s

0 0 0

0 01

0 02

0 0

0 01

with 0 0 Geometric stiffness matrix2

0 0

xz yz zz

tt t

e e e e

V

t

V

s

U u G s G u dV u K u

s

s

K G s G dV

s

54

Imperfection sensitive structures

P PP

1k

2k

3k

A AA

B BB

l

mP

mP

55

Parametrically excited systems

P t

,f x t

P t

gx t

gy t

gm g y t d

d

y

h

x

y

A

,a v

vk

sm

vc

um

u t

, , ,EI m c l

,y x t

PP

P

P

Line of action of remains unaltered as beam deforms. Static analysis canbe used to find cr

P

P

Line of action of remains tangential to the deformed beam axis. Static analysis does not leadto correct value of .cr

P

P

“Follower”

forces

57

0 0

How to characterize resonances in systems governed by equations of the form

0; 0 ; 0

when the parametric excitations are periodic.

How to arrive at FE models for P

M t X C t X K t X X X X X

Problem 1

Problem 2

DE-s with time varying coefficients?

Are there any situations in statically loaded systems, wherein one needs

to use dynamic analysis to infer stability conditions?

Problem 3

58

,a v

vk

sm

vc

um

u t

nl

1

1

1

n

n i

i

x l

1

n

n i

i

x l

0

1

n

i

i

x l

1

1

1

0 0

1

0 1

position of the wheel in local coordinate system

n

i

i

n

i

i

x l

x l

1

1

1 0

1 1 0

1

Element level equation of

Let &

0

mo

0

tion

n

n

n n

n

n

s

t

n n v

n n v

t

n n

V

t Md F

u t V

M

tM M t

u tm

tC C t I t t t c

u tI t t t c

K K t I t t t

0

1 1 0

1 0

00

Further steps: assembly, imposition of BCs

v

n n v

tnn n u s

tk

u tI t t t k

FI t t t m m g

60

Mathematical

models

Experimental

models

Can we combine them?

What are the issues?

Studies on existing structures

Finite element

model updating

61

To be determined To be determined from Unknownexperimentally FE model I updating

parameters

Experimental

determination

of

S S

S

Tasks

Analytical determination of elements of the matrix

Sensitivity analysis

Solution of the( often overdetermined) set of equations

Iterations

Pseudo - inverse

Singular value decomposition

Forward proble

Tikhonov Regula

m of design sen

rization

sitivty

Sources of nonlinearity

• Nonlinear strain-displacement relations

(geometric nonlinearity)

• Nonlinear constitutive laws (nonlinear

stress-strain relations)

• Nonlinearity associated with boundary

conditions

• Nonlinear energy dissipation mechanisms

62

63

Stress

StrainMaterial nonlinear; small displacementsand small strains

Material linear/nonlinear; large rotationand small strains

Material linear/nonlinear; large rotationand large strains

Nonlinear boundaryconditions Stress

Strain

65

Nonlinearly elastic systems and systems with hereditary

nonlinearities

, , , ;0 ;

(0) & 0 specified

, , Nonlinear function of instantaneous values of &

mx cx kx g x t x t t h x x t f t

x x

g x t x t t x t

.

, ;0 Nonlinear function of response time histories up to time .

x t

h x x t t

-5 -4 -3 -2 -1 0 1 2 3 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Restoring Force Vs. Displacement

Displacement,mm

Re

sto

rin

g F

orc

e, kN

66

Base & referenceconfigurations coincide(Taken to remain fixed)

Currentconfiguration

Total Lagrangian

approach

67

Baseconfiguration

Reference configurationupdated at each incrementwhile solving the equilibrium equation

Currentconfiguration

Updated Lagrangian

approach

68

Baseconfiguration Co-rotated configuration

(base configuration undergoesrigid motion; cg-s overlap)

Currentconfiguration

Co-rotational

formulation

• Base configuration is used as reference to measure rotations.

• Co-rotated configuration is used as reference to measure

current stress and strains

Strain measures

• Infinitesimal strains: for body under rigid

body rotations, the strains would not be

zero.

• New measures needed:

– Rigid body motions imply zero strains

– For small strains, the infinitesimal strain

definitions are to be restored.

69

Stress measures

• Cauchy stress tensor: defined with respect to deformed

geometry. This would not be known in advance.

• Two alternatives:

– Stress as a measure which conjugates with a measure of strain

to produce internal energy

– As a quantity which produces a traction vector in conjunction

with a normal vector defined with respect to a surface element

70

71

2 2 2

2

2 2 2 2 2 2

2

Sum of virtual external work done on a body and the virtual

work stored in the body should be zero.

Consider configuration C

W : V V S

e d V f u d V t u d S

Principle of virtual displacements

2 2 2

2 2 2 2 2 2

2

0

W 0ij ij i i i i

V V S

e d V f u d V t u d S

72

2 0

2 0

2 0

0

0

2 2 2 2 0

2 0 0

2 2 2 0

0

2 2 2 0

0

2 2

0 0

All quantities reckoned with respect to undeformed configuration (C ).

ij ij ij ij

V V

i i i i

V V

i i i i

S S

ij ij

V

e d V S E d V

f u d V f u d V

t u d S t u d S

S E

Total Lagrangian approach

0 0

0 2 0 2 0

0 0 0i i i i

V S

d V f u d V t u d S

73

2 1

2 1

2 1

1

2 2 2 2 1

2 1 1

2 2 2 1

1

2 2 2 1

1

2

1

All quantities reckoned with respect to the latest known configuration (C ).

ij ij ij ij

V V

i i i i

V V

i i i i

S S

ij

e d V S d V

f u d V f u d V

t u d S t u d S

Updated Lagrangian approach

1

1 1

2

1 1

2

1 1

2 2 1 2

1 1 1

2 2 1 2 1

1 1 1

updated Green-Lagrange strain tensor

body force refered to in C .

surface traction refered to in C .

0

i

i

ij ij

V

i i i i

V S

f

t

S d V R

R f u d V t u d S

Follow up

• Material nonlinearity

• Stability analysis: inclusion of nonlinearity

at different levels

• Hybrid testing

• Bayesian filtering

• Uncertainty modelling and fem

• Thermal loads: fire

• Anisotropy

74

Application of FE models in

structural testing • Pseudo-dynamic tests

– How to handle complicated inelastic

behavior?

• Real time sub-structure testing

– How to test interacting primary and secondary

system?

75

76

Hybrid simulations in earthquake qualification testing

Hybrid simulations

Shake table test Effective force test

(Reaction wall and strong floor

based)

Experimental partTest structure

Numerical part

Geometry Less payload

Time Slow down; less stringent requirements on hardware

Geometry and time

Scaling

Traditional

methods

77

Numerical solution

gMx Cx R x M x t

ComputedDisplacements x t

Earthquake excitations

MeasuredReactions

R x t

Pseudo Dynamic

Testing

-5 -4 -3 -2 -1 0 1 2 3 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Restoring Force Vs. Displacement

Displacement,mm

Re

sto

rin

g F

orc

e, kN

78

Experimentally explicit & numerically implicit scheme

Step size adjustment via a variational equation

Bending-torsion coupled piecewise nonlinear system.

mod

Inertial force + Dissipation force + Stiffness force = Applied forceExperimental modelNumerical model Numerical el

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Restoring Force Vs. Displacement

Displacement,mm

Re

sto

rin

g F

orc

e, kN

0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1.5

-1

-0.5

0

0.5

1

1.5

Displacement along X-axis Vs. Time

Time,s

Am

plit

ud

e, m

m

EFT

PSD

79

50kN, ±75 mm, Fatigue rated (2) 100kN, ±75 mm, Fatigue rated (1) 300kN, ±75 mm, Fatigue rated (1) Frequency up to ~50 Hz

2370MS, 32-Bit,

TMS320C5502, USB2,

16 DAQ, 5 kHz, 8 DAC

Reaction wall: L-Shape, 6x5x5x1 m Strong Floor: 200 PCC, 900 RCC

Hybrid Simulation Lab

96 Channels data acquisition system

Understood(numerical)

Not understood(experimental)

gx t

1x t

2x t 1x t

1m

1k

1c

2x t

2m

f x

2c

gx t

2x t

2m

f x

2c

1x t

1m

1k

1c

gx t

1x t

Measure reaction transferred to the support cf t

cf t

E

N

Real time interaction between N and E

Time delays

Noisy measurements

No need for scaling in space or time

Greater demands on hardware and FEA

What is new and challenging?

• FE modeling enters control algorithm of

the shake table/actuators • Accuracy, stability, speed,…

• Compulsions of real time operations:

modeling of time delays; need to have

“fast” solvers.

81

Frameworks for modeling uncertainties

• Probability theory

• Interval analysis

• Convex sets

• Fuzzy set

• Hybrid models

82

Challenge

How to combine these tools with structural analysis methods?

Note

In understanding structural failures one needs to model

structural nonlinearities as well as uncertaitnies.

83

185 190 195 200 205 210 215220

240

260

280

300

320

340

x1

x2

185 190 195 200 205 210 215220

240

260

280

300

320

340

x1

x2

185 190 195 200 205 210 215220

240

260

280

300

320

340

x1

x2

185 190 195 200 205 210 215220

240

260

280

300

320

340

x1

x2

No new

data are

likely to be

available

84

Interval models: ; 1,2, ,i i ix x x i n

1x 1x

2x

2x

1x

2x

85

Convex models:

: positive definite matrix; 0.

tx x a

a

1x

2x

The region need not be ellipsoidal.

It needs to be convex.

86

X

, ,0 1A AA x x x

Fuzzy models

A x

x x

1

x

The membership function needs to be convex with max value=1.

:

min :

max :

AF x X x

x f f F

x f f F

87

Fuzzy interval

At membership value ,

0 1

x x x

Fuzzy convex set

At membership level ,

0 1

tx x a

Structural

systems

Uncertain actions

Uncertain

System parameters

Uncertain outputs

Propagation of uncertainties must be consistent with the laws of

mechanics