Finite element method for structural dynamic

and stability analyses

1

Prof C S Manohar Department of Civil Engineering

IISc, Bangalore 560 012 India

Module-1 Approximate methods and FEM

Lecture-1 Equations of motion using Hamilton’s principle

What is this course about?

• FEM in the context of structural mechanics problems.

• What are the issues to be dealt with, when the FEM, as

developed for analysis of static problems, is extended to

deal with problems of structural dynamic and elastic

stability analyses

• New questions

• New phenomena

• New numerical tools

• Application areas

2

• This is not a first course in

• FEM

• Structural dynamics

• Structural Stability analysis

Pre-requisites • Matrix methods of static structural analysis

• A first course in theory of vibrations

• Elements of elastic stability analysis

• Matrix algebra and ode-s

The course will contain brief overviews of the main

ideas of these subjects 3

Different Facets of FEM

• A numerical method to obtain approximate solutions to

boundary value problems (ODE-s, PDE-s).

• A modeling and simulation tool

• A tool in computer aided engineering that exploits the

power of computers • Modeling (pre-processing)

• Numerical solutions

• Scrutiny of results (post processing)

• Combined with sensing and actuation, a tool for • Active control of structures

• System identification and health monitoring

• An important component in hybrid testing methods

4

Topics 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

5

Three intertwining themes

• Modeling

• Numerical solutions

• Applications

6

Target audience

• Graduate level students

• Research students

• Engineers who use FEM as a tool in their work

Books

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

Cambridge.

2. W Weaver and P R Johnston, 1987, Structural dynamics by finite

elements, Prentice-Hall, Englewood Cliffs.

3. M Geradin and D Rixen, 1997, Mechanical vibrations, 2nd Edition, Wiley,

Chichester.

4. R W Clough and J Penzien, 1993, Dynamics of Structures, 2nd Edition,

McGraw-Hill, New York.

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

Delhi.

6. I H Shames and C L Dym, 1991, Energy and finite element methods in

structural mechanics, Wiley Eastern Limited, New Delhi.

7. R R Craig, 1981, Structural dynamics: an introduction to computer

methods, Wiley, NY

8. C H Yoo and S C Lee, 2011, Stability of structures, Butterworth-Heinmann,

Burlington.

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

Elsevier, Amsterdam. 7

Additional references

1. L Meirovitch, 1997, Principles and techniques of vibrations, Prentice-Hall,

New Jersey

2. O C Zienkiewicz and R L Taylor, 1989, The finite element method, Vols-I

and II, 4th Edition, McGraw-Hill, London.

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. J N Reddy, 2006, An introduction to the finite element method, 3rd Edition,

Tata-McGraw-Hill, New Delhi.

5. S S Rao, 1999, The finite element method in engineering, 3rd Edition,

Butterworth-Heinemann, Boston.

6. T J R Hughes, 2000, The finite element method, Dover, Mineola.

7. M Paz, 1985, Structural dynamics, 2nd Edition, CBS Publishers, New Delhi

8. W McGuire, R H Gallagher, and R D Ziemian, 2000, Matrix structural

analysis, 2nd Edition, John Wiley, New York.

9. P Seshu, Textbook on finite element analysis, 2003, Prentice Hall India,

New Delhi.

10.G Strang and G J Fix, 2008, 2nd Edition, An analysis of the finite element

method, Wellesley-Cambridge Press, Wellesley.

8

9

Dynamic loads

The magnitude, direction, and (or) point of application of the load

change with time.

Under the action of dynamic loads the structure vibrates, that is,

(a) the structure develops significant level of inertia forces

(b) significant level of mechanical energy is stored as kinetic energy

Note: Not all time varying loads need to be dynamic in nature;

for example,

(a) Load on a dam due to filling of a reservoir

(b) Load on a spectator gallery as a stadium gets filled up.

10

Examples:

1. Earthquake loads on buildings, bridges, dams, power plants etc.

2. Wind loads on long span bridges, tall chimneys etc.

3. Loads due to blast and impact.

4. Running machineries in buildings.

5. Vehicle moving on a bridge.

11

Degree of freedom (DOF)

Number of independent coordinates required to specify the

displacements at all points in the system and at all times.

u t

f t

k

m

x

y

2 2 2x y L

L

m

g

12

Particle in space

DOF=3

Rigid body in space

DOF=6

Two-dof systems

x

x

z

y

y

z

13

This chunk has mass, stiffness and damping

DOF

1

Unknown Known function functionof time of

, DOFN

n n

n

x

w x t a t x N

Two-dof system

14

Remarks:

•The DOF is not an intrinsic property of a given structure.

•It is a choice exercised by the modeler.

•SDOF systems: systems with one DOF.

•MDOF systems: systems with DOF>1

•MDOF systems could be discrete (finite DOFs) or

continuous (infinite DOFs)

•The coordinates need not have direct physical meaning

(generalized coordinates)

•DOF=a number or the coordinate

FEM: replaces a system with infinite dofs by a system with

finite dofs.

15

Entities in a simple mathematical model for a vibrating system

Whenever a structure vibrates, it not only displaces, but also

accelerates. Besides, the mechanical energy is converted to heat

and/or sound.

Stiffness: resists displacement

Inertia: resists acceleration

Damping: dissipates energy

Inputs: external forcing and/or initial conditions; supply mechanical

power

•Mass: offers resistance to acceleration, stores kinetic energy

•Damper: dissipates mechanical energy

•Spring: offers resistance to displacement, stores potential energy

•Force and/or initial displacement, velocity: supplies mechanical power

16

u t

f tm

k

c

•Mass: offers resistance to acceleration, stores kinetic energy

•Damper: dissipates mechanical energy

•Spring: offers resistance to displacement, stores potential energy

•Force and/or initial displacement, velocity: supplies mechanical power

2

2

2

1Mass (kg) KE= Force=

2

1Spring (N) PE= Force=

2

1Damper (Ns/m) DE= Force=c

2

mu mu

ku ku

cu u

Position of the mass

when the spring is not

stretched

17

m

c

k

1u t

u t

mu t

2u t

1u t 2u t

2

Force=

1KE=

2

mu t

T mu t

2 1

2

2 1

Force=

1PE=

2

k u t u t

V k u t u t

2 1

2

2 1

Force=

1

2

c u t u t

DE c u t u t

Mass kg

Spring N/m

Damper Ns/m

18

1 vector of system dof-s

Lagrangian : , , ,

, total kinetic energy of the system

, total strain energy of the system

Among all the dynamic paths which satisfy

q t n

q q T q q V q q

T q q

V q q

Hamilton's principle

L

1 2

the boundary condtions

(on prescribed displacements) at all times, and

with the actual values at two arbitrary instants of time and at

every point of the body, the actual dynamic path minimizes

t t

2

1

the functional

, ,

t

t

T q q V q q dt

How does the motion take place?

19

u t

t1t 2t

1u t

2u t

u t

20

2

1

Functional: function of functions

Domain: set of admissible functions

A functional can also be viewed as a function of infinite set of variables

, ,

Optimizatio

t

t

T q t q t V q t q t dt

Remarks

n of functionals is studied in the subject of calculus

of variations.

Presently we have not included external forces and damping forces.

:

Gain familiarity with the application of the principle

by considering equations of motion of simple oscillators and

structural elements

Immediate objective

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

22

1t 2tt

1u t

2u t

t

( )u t u t t

( )u t

( )u t

1 1 2 2

1 2

Admissible functions Unknown optimal solution Variation arbitrary real number&

arbitrary fucntion 0u t u t u t u t

t t t

u t u t t

1 2sufficiently smooth, reasonably small, & 0

but otherwise arbitrary

t t t

1 1 2 2

1 2

2

1

Admissible functions Unknown optimal solution Variation arbitrary real number&

arbitrary function 0

2 21,

2

x t x t x t x tt t t

t

t

u t u t t

mu t ku t dt u t u t

L

2

1

2

1

2

1

2 2

1 1

0

0 0 for 0

0

0

0 because is arbitrary

t

t

t

t

t

t

t t

t t

dt

d u udt

d u u

t t dtu u

dt t dt

u dt u u

dt

dt u u

L L

L L

L L L

L L

2 2

0

1; ;

2

0

d

dt u u

dmu t ku t mu mu ku

u dt u u

mu ku

L L

L L LL

The equation 0 is called Lagrange's equation.

For discrete systems, with dofs, Lagrange's equation can

be generalized as

0; 1,2, ,

where , ; 1,2,

i i

i i

d

dt u u

n

di n

dt u u

q t q t i

L L

L L

L L , .n

25

2

1

2

1

2 2

0

, , is a functional

1 : one parameter family of functions

2

The condition 0 is not a sufficient condtion

for minimum of . The discussi

t

t

t

t

T u t u t V u t u t dt

mu t ku t dt

d

d

Remarks

on on whether this condition

implies minimum, maximum, or or stationary value of is not

considered here. We will refer to (0) as being the extremum and

the optimizing function as the extermizing fun

ction.

Even when (0) is a minimum, it need not be the absolute minimum.

Refer, for a further discussion on these issues,

R Weinstock, 1974, Calculus of variations, Dover, NY.

26

2

1

(continued)

The condition for stationarity

of the functional ,

is

The condition for stationarity

of the

0

function , is

0 & 0

t

t

u t u t dt

d

dt u

F u v

F F

uu

v

Remarks

L

L L

27

1u t 2u t

1m

1k 2k2m

2 2

1 1 2 2

22

1 1 2 2 1

22 2 2

1 2 1 2 1 1 2 2 1 1 2 2 1

1 1 1 1 2 1 2 1 1

2 22 2 2 2 1

1

2

1

2

1 1, , ,

2 2

0; 1,2

0 0

00

i i

T t m u m u

V t k u k u u

u u u u m u m u k u k u u

di

dt u u

m u k u k u u m u

m um u k u u

L

L L

1 2 2 1

2 2 2

0k k k u

k k u

1 2

2;

DOFs: &

n

u t u t

Example-2

28

M

m

k

g

Y

L

X

u

2;

DOFs: &

n

u t t

2 2 2

2

1 1 1( )

2 2 2

1

2

T t Mu mX mY

V Ku mgY

2; ,

sin ; 1 cos ;

cos ; sin

n q u

X u L Y L

X u L Y L

Example-3

2 2 2

2

2 22

2

22

1 1 1( )

2 2 2

1

2

2; ,

sin ; 1 cos ;

cos ; sin

1 1 1cos sin

2 2 2

11 cos

2

, , ,

1 1 1cos sin

2 2 2

T t Mu mX mY

V Ku mgY

n q u

X u L Y L

X u L Y L

T Mu m u L m L

V Ku mgl

u u

Mu m u L m L

L L

L 2

211 cos

2Ku mgL

30

2 22 2

2

2

1 1 1 1cos sin 1 cos

2 2 2 2

0 cos sin 0

0 cos sin sin 0

Mu m u L m L Ku mgL

dM m u mL ku

dt u u

dmL mLu mLu mgL

dt

L

L L

L L

Remark

The approach can handle nonlinear systems

31

1x

2x

1k

M

r

1x

2k

m

2x

Example-4

2 1

2 1

No slip =r x x

x x

r

2

2

MrJ

2 2 2

1 2

22 2 2

1 2

222 12 2

1 2 2

22

1 1 2 2 1

1 1 1+ +

2 2 2

1 1= +

2 2 4

1 1= +

2 2 4

1 1

2 2

T mx Mx J

Mrmx Mx

x xMrmx Mx

r

V k x k x x

32

2222 12 2 2

1 2 1 1 2 2 12

1 1 2 1 1 2 1 2

1 1

2 2 1 2 2 1

2 2

1

2

1 1 1 1+

2 2 4 2 2

0 02

0 02

2 2

3

2 2

x xMrmx Mx k x k x x

r

d Mmx x x k x k x x

dt x x

d MMx x x k x x

dt x x

M Mm

x k

xM M

L

L L

L L

1 2 2 1

2 2 2

0

Note

1

2

1

2

t

t

k k x

k k x

T t x t Mx t

V t x t Kx t

33

Distributed parameter systems : Axially vibrating rod

x

, ,L AE x m x

2

0

2

0

2 2

0 0

0

1Kinetic energy: ,

2

1Potential energy: ,

2

1 1Lagrangian , ,

2 2

, , ,

L

L

L L

L

T t m x u x t dx

V t AE x u x t dx

T V m x u x t dx AE x u x t dx

F u x t u x t dx

L

L

,u x t

34

1 1 2 2

Admissible Unknown optimal Variationfunction solution

2 1

Let at , , & , ,

Admissible functions: satisfy the above conditions

, , ,

, is such that , , 0.

We will

t t u x t u x t t u x t u x

u x t u x t x t

x t x t x t

2

1

1 2

2 2

0 0

shortly come to the question of what condtions

, must satisfy at 0 and for & .

1 1, ,

2 2

t L L

t

x t x x L t t t t

m x u x t dx AE x u x t dx dt

35

2

1

2

1

2

2

1

1

0

0

00

0 at 0

[ , , , , ] 0

Consider , ,

, , , ,

Similarly , ,

, , , ,

t L

t

t

t

tt

tt

L

LL

d

d

m x u x t x t AE x u x t x t dxdt

m x u x t x t dt

m x u x t x t m x u x t x t dtt

AE x u x t x t dx

AE x u x t x t AE x u x t x t dxx

36

1

2

1

2

1

2

0

2

2

0

0

0 at 0

, , 0

Since , is chosen arbitrarily, we take that each of these terms are

separately equal

, , , ,

to

tL

t

t L

t L

t

tm x u x t x t dx AE x u x t x t d

d

d

uAE x u x t m x x t dxdt

x

t

t

x t

2

1

2

1

2

1

2

2

0

0

0

0. That is

, , 0

, , 0

, , 0

t L

t

L

t

t

t

L

t

uAE x u x t m x x t dxdt

m x u x t x t dx

AE x u x t x t d

x t

t

37

2

1

2

2

0

2

2

Consider

, , 0

, is arbitrary , 0

Valid over the length of the bar

t L

t

uAE x u x t m x x t dxdt

x t

ux t AE x u x t m x

x t

2

1

2

1

2 2 1 1

2 1

Consider

, , , , , ,

, , 0 , , 0

t

t

t

t

m x u x t x t m x u x t x t m x u x t x t

x t x t m x u x t x t

38

0

Consider the term

, , , , 0 0, 0,

The two terms on the RHS need to be zero individually

(again, because the variation is arbitrary)

Two situations arise:

L

AE x u x t x t AE L u L t L t AE u t t

Boundary condtions

The unknown function, that is, , , is not specified at the boundaries

(as in the case of a free end of the rod). Here it is "natural" to expect that the term

, or 0 0, is zero.

When the unk

u x t

AE L u L t AE u t

nown function, that is, , , is specified to be zero (as in the

case of a clamped end), the variation, in order that it confirms with the stipulated

geometric constraints, needs to be zero.

u x t

39

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

40

, at 0 & represents the axial thrust at 0 &

respectively. From a physical stand point, it is clear that if the

displacement is specified to be zero, that is the end is clamped, ther

AE x u x t x L x L

Remarks

e

would be an axial thrust, representing the reaction, at the boundaries.

For a free end, displacement at the end is not specified.

The boundary conditions that we obtain at ends where the unknown

fi

eld variable is not specified are called the natural boundary conditions

(also called additional or dynamical boundary conditions).

41

The boundary conditions that we obtain at ends where the unknown

field variable is specified are called the geometric boundary conditions

(also called the essential or imposed boun

Remarks (continued)

dary condtions).

The variational method identifies the required boundary condtions

consistent with the physics of the problem along with the governing

field equation.