23. chapter 23 - transient analyses _a4

8/11/2019 23. Chapter 23 - Transient Analyses _a4

1/10

Chapter 23 The Transient Dynamic Analysis_____________________________________

230

CHAPTER 3

THE TRANSIENT DYNAMIC ANALYSIS

The transient dynamic analysis is used in civil engineering to determine the

time-varying displacements, strains, stresses and forces of a structure during

time-dependent loads, as earthquake or wind. Furthermore, when nonlinear

material properties are associated to the model, the process highlights the

regions where the linear-elastic behavior is exceeded (for frame structures,

the birth of so called plastic hinges) and also the order in which these

nonlinear regions appear over the model. When exceeding the linear-elastic

behavior, the amount of residual strains reveals the quantity of energydissipated during the plastic deformation.

The transient dynamic analysis requires generally more computer resources

because, at least, it is equivalent to a multi-step static analysis. Three

solution methods are usually available: thefull method, thereduced method

and the mode superposition method. Each one has its own advantage,

according to the analysis aims, computing time and the required

preprocessing work.

The transient dynamic equilibrium equation is the equation of motion

)(ta

FKCM =++ &&& (23.1)

where

M, Cand Kare the mass, damping and stiffness matrices;

&& is the nodal acceleration vector, & the nodal velocity vector and

the displacement vector;

)(taF is the applied load vector.

The procedure used for solving the linear equation is the time integration

method (the Newmark method). It uses finite differences expansions over

the time interval t, in which the following relationships are assumed:

( ) taa nnnn ++= ++ 11 1 &&&&&& (23.2)


2/10

______________________Basics of the Finite Element Method Applied in Civil Engineering

231

2

1121 tbbt nnnnn

+

++= ++ &&&&& (23.3)

where

a, bare the Newmark integration parameters

nn ttt = +1

n&& , n

& , n are the nodal acceleration, velocity and displacement

vectors at time nt

1+n&& , 1+n

& , 1+n are the nodal acceleration, velocity and

displacement vectors at time 1+nt

Since the first intend is to calculate the displacement 1+n and the seismic

load is usually applied as ground acceleration d&& , the governing equation at

time 1+nt is:

1111 ++++ =++ nnnn dMKCM &&&&&

(23.4)

Rearranging the equations (23.4), using some notations:

( ) nnnnn aaa &&&&& 32101 = ++

1761 ++ ++= nnnn aa &&&&&& (23.5)

20

1

tba

= ;

tb

aa

=1 ;

tba

=

12 ; 1

2

13 =

ba ; 14 =

b

aa ;

= 2

25

b

ata ; ( )ata = 16 ; taa =7

and substituting 1+n&& , the general form becomes:

( ) =++ +110 naa KCM

) )nnnnnnn aaaaaa CdM &&&&&&&& 5413201 ++++++= + (23.6)


3/10


232

Once the solution is obtained for 1+n , the accelerations and the velocitiesare updated with the equations (23.5). The solution is unconditionally stable

for

2

2

1

4

1

+ ab ;

2

1a and 0

2

1>++ ba *. The Newmark method

becomes the constant average acceleration method for2

1=a and

4

1=b .

23.1 The stiffness matrix

The procedures to calculate the global stiffness matrix of the structure were

already described in the case of static analyses. The same procedures are

available for transient dynamic analyses. As before, the stiffness matrix is amatrix of constants in which the geometrical characteristics and the material

propertiesEdand (the dynamic Youngs modulus and the Poissons ratio)

are involved. It is common to assign different values for the Youngs

modulus in dynamic analyses as for static analyses.

23.2 The mass matrix

The mass matrix assessment is explained in chapter 22. The procedure is

also available for the transient dynamic analysis.

23.3 The damping matrix

The damping matrix of a finite element is expressed in terms of a damping

force which is proportional with its mass and also in terms of the

deformation velocity. The first dependence is expressed by the parameter

as a force vector in the integral expression of the functional, while the

second one corresponds to a stress vector &dd E= which is due to

elements stiffness. In a general form, the element damping matrix is:

( )dVVe

d

TT

e += BEBNNc (23.7)

The total damping matrix of the structure Cis obtained following the sameassembling procedures as for the global stiffness matrix.

*Zienkiewicz, O. C.- The Finite Elements Method, Mc Graw Hill, 1979


4/10


233

Using the notations = and EE d = the general form of the Rayleigh

damping model outcomes:

KMC += (23.8)

where and are scalar multipliers.

The values of and are generally not known directly, but are calculated

from the modal damping ratios i , which are the ratios of actual damping to

critical damping for each vibration mode i. If i is the natural circular

frequency of mode i, and satisfy the relation:

22

i

i

i

+= (23.9)

In practical structural problems the mass damping may be ignored (= 0).

Because only one value of can be prescribed during a load step, the

dominant frequency should be chosen. When using both and multipliers,

it is commonly assumed that the sum of their terms are nearly constant over

a frequency range. Therefore, knowing the damping ratio and a circular

frequency range i to j , two simultaneous equations can be solved for

and .

A more complete expression of the damping matrix takes into account the

damping properties assigned to materials and the element damping matrices

available for some elements in the element library:

==

+++=nelem

k

kj

nmat

j

j11

CKKMC (23.10)

where j is the stiffness matrix multiplayer for materialj, Kjthe structures

region with assigned materialj, nmatthe number of materials with dampingproperties, Ck the element damping matrices and nelem the number of

elements with specified damping.


5/10


234

23.4 Transient dynamic solution methods

The most applied solution methods for solving equation (23.6) are the full

solution method, the reduced solutionmethod and the mode superposition

method.

Thefull solution methodsolves the equation directly without any additional

assumptions. The initial (or start) values of 0 , 0& , 0

&& must be known. If

non-zero initial conditions are required, they are assigned by performing a

static analysis load step.

The initial displacements are:

if there is no previous load step

displacement vector from the static load step

The initial velocities are:

in the case of an initial static load case; t is the timeincrement

otherwise

The initial acceleration is 0&& = 0.

The total force applied on each node is computed as a sum of inertia,

damping and static loads over all elements connected at that location. The

element inertia load is computed by eem

e &&MF = , where eM is the element

mass matrix and e&& the element acceleration vector. While the acceleration

of each nodal DOF is given by equation (23.3) for time 1+nt , e&& is an

average acceleration between 1+nt and nt , since it is assumed that the

average acceleration is the true one.

The damping load part of the element is computed by eec

e &CF = , where

eC is the element damping matrix and e&

the element vector velocity, givenby equation (23.5).

=s

00

=0t

s

0&


6/10


235

The nodal reaction loads are computed as the negative of the sum of inertia,damping and static effects over all elements which are connected to a given

fixed displacement node.

A faster solution can be obtained applying the reduced method. In the

general motion equation (23.4) the characteristic matrices M, Cand Kare

computed considering two main assumptions:

- only a selected set of masterDOF are taking into account (the DOF

considered essential to characterize the response of the structure);

- the matrices are constant during the loading time interval, thus

nonlinear effects are suppressed (plasticity, large deflections, etc).

Supplementary assumptions as a constant time step size and the use of only

concentrated loads applied on master DOF are also considered.

The reduced motion equation system yields:

( ) =++ +110 naa KCM))))

nnnnnnn aaaaaa CdM)&&

)&

)))&&

)&

))&&

)

5413201 ++++++= + (23.11)

where the ^ symbol denotes the reduced matrices and vectors.

The reduced solution is obtained by inverting the left-hand side of the

equation and performing a matrix multiplication at each time step. The full

solution is computed by expandingthe reduced solution to the other DOF of

the model, which are calledslaveDOF. Once the expansion pass is done by

computing all nodal displacements, the elements stresses are evaluated.

The mode superposition methodsums factored mode shapes obtained from a

modal analysis to calculate the dynamic response. The mass and stiffness

matrices are constant during the analysis, as well as the time step size. The

element damping matrices are neglected.

The equations of motion may be expressed as:

snd sFFKCM +=++ &&& (23.12)


7/10


236

wherend

F are the time varying nodal forces (which can be also expressedas

nddM && when inertia forces), sF the load vector from the modal analysis

andsa load scale factor.

The displacement vector is defined in terms of modal coordinatesyi, so that:

=

=n

i

ii

1

y (23.13)

where i is the mode shape of mode iand n the number of mode shapes

taking into account. By substituting into (23.12):

sndn

i

ii

n

i

ii

n

i

ii sFFyKyCyM +=++ === 111

&&& (23.14)

Multiplying with Tj

and applying the orthogonality conditions of the

natural mode shapes (22.6), the outcome is:

( )sndTjjjTjjjTjjjTj sFFyKyCyM +=++ &&& (23.15)

The coefficients of this equation are substituted as follows:- according to the normality condition 1= j

T

j M ;

- it can be demonstrated that iijT

j 2= C , where i is the

fraction of critical damping and i the natural circular frequency of

modej;

- the third term is22

jj

T

jjj

T

j == MK ;

- the right-hand side term is conveniently noted as fj.

The motion equation system in modal coordinates yields:

jjjjjjj fyyy =++

2

2 &&&

(23.16)


8/10


237

Since j is an arbitrary mode, the previous system represents a set of nuncoupled equations with the nunknowns yj. Using relationship (23.13), the

modal coordinates are back-converted into geometric displacements . If the

initial modal analysis is performed on a reduced model (using only master

DOF) the matrices and load vectors would be expressed in terms of these

DOF. In this case an expansion pass is necessary to compute the

displacements of theslaveDOF.

23.5 Practical aspects of performing a transient dynamic analysis

Since the solution of equation (23.4) depends on the external excitation,

represented by the ground acceleration

d (variable in time), the acceleration

vector has to be expressed as a constant value for each time step. Thus, a

continuous acceleration recording (or an artificial one) has to be transformed

into a set of discrete values, according to the total duration of the dynamic

load and the chosen time step, t. Figure 23.1 represents the discretization

of a recorded accelerogram lasting 0.5 seconds into 10 equal intervals, the

constant values corresponding to each time step being listed below.

Time step 1 2 3 4 5 6 7 8 9 10 11

a (m/s2) 0 1 3 2 0 -2 -1 0 2 1 3

Fig. 23.1 Discretization of an accelerogram

Regarding the response acceleration of the structure, it can be assumed to beconstant or with a linear variation over the time step. If the acceleration is

constant, the response velocities have a linear variation over the load step

0

1

2

3

-1

-2

0,05 0,10 0,15 0,20 0,25 0,30 0,40 0,500,35 0,45 s

)(2sma

t t

t t


9/10


238

and the response displacements is a quadratic one (see figure 23.2). Theinitial conditions may be chosen as:

t= 0 0 = 0.

=..

0

Fig. 23.2 Acceleration, velocity and displacement evolutions over the time step

The time step size t should be chosen small enough for acquiring an

acceptable numerical precision and to avoid numerical instabilities. The

precision of the numerical integration is essentially related to the time step

size. The main factors that should be taken into account are:

- the magnitude of the lowest natural vibration period which is

significant for the structural response; the time step size should notexceed 1/10 of its value, in order to achieve a good representation of

the corresponding mode shape;

Constant accelerations

Linear velocities

Quadratic displacements

t

t&

t&&

tt +

tt +&

tt +&&

t

( )ttt ++= &&&&&&2

1)(

tttt +=+ )( &&

..

tttt +=+ )( &&..

t+t


10/10

23. chapter 23 - transient analyses _a4

Documents