22. chapter 22 - modal analyses _a4

8/11/2019 22. Chapter 22 - Modal Analyses _a4

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Chapter 22 The FEM Applied to Dynamic Analyses_______________________________

214

CHAPTER

THE FEM APPLIED TO DYNAMIC ANALYSES

In civil engineering design the dynamic parameters of structures are very

important. The natural frequencies are directly related to the overall

stiffness, as well as with the stiffness of each structural component. The

main dynamic loads are earthquakes, wind, vibrating machineries or moving

loads (convoys). For structures located in seismic regions the limitation of

lateral displacement due to earthquakes (or the flexibility limitation) is

sometimes expressed as dynamic stiffness, in terms of the basic natural

frequencies.

In order to avoid the amplification effectdue to dynamic loads (phenomenon

known as resonance), the natural frequency of a civil engineering structure

should be as far as possible from the frequency of the applied load. The

resonance between civil engineering structures and their foundation ground

should be avoided. Thus, for regions with soft soils (i.e. with low natural

frequencies) stiff structures (with high natural frequencies) are

recommended. By contrary, in regions with hard rock foundation layers,

flexible structures are more appropriate. Long-span bridge structures are

very sensible to dynamic loads due to traffic and wind. Their natural

frequencies and vibration shapes are of utmost importance in the design

process. Sometimes, the criterion for limiting the thinness of a concrete

floor (or slab) can be its natural frequency. Excessive thin floors are usually

uncomfortable.

Regarding the dynamic responseof a structure, in most cases the stress and

strain level is higher during the dynamic load. Some regions may exhibit

nonlinear material behavior due to incursions in plastic state, as well as

geometric nonlinearities.

22.1 THE MODAL ANALYSIS

The first step in assessing the vibration characteristics of a structure and

highlighting its dynamic response is the modal analysis. It is also thestarting point for other dynamic analysis, such as transient dynamic

analysis, harmonic response analysis or spectrum analysis.


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______________________Basics of the Finite Element Method Applied in Civil Engineering

215

The vibration characteristics are the natural frequencies f [Hz] and theassociated mode shapes. They are intrinsic characteristics of the systems

free vibrations. The number of natural frequencies and mode shapes equals

the number of dynamic degrees of freedom of the vibrating system.

Alternative parameters are the natural vibration period T [sec] and the

natural circular frequency [rad/sec]. The natural frequencies of the

vibrating system are called the eigenvalues, while the mode shapes are

building up the eigenvectors.

In the following figure, a simple cantilever is meshed in four 2D beam

elements. Next to the finite element model, the first 3 vibration modes and

the corresponding displacement vectors are represented.

Fig. 22.1 First three vibration modes of a cantilever

22.1.1 Model requirements and method limitations

During a modal analysis only a linear behavior of the model is taken into

account. Nonlinear material properties are neglected. Elements with usual

nonlinear behavior (as contact elements) are treated as linear. It is alsoessential to define the mass properties of the model, either by declaring the

material density the elements are made off, or by using concentrated mass

1

2

3

4

5

1,1

1,3

1,4

1,2

2,3

2,1

2,2

2,4

3,1

3,4

3,3

3,2

E1

m1

E2

m2

E3

m3

E4

m4

mode 2

f2mode 1

f1

mode 3

f3


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216

elements. The Youngs modulus value Ed should reflect the dynamicbehavior of the structure*.

The natural frequencies and the mode shapes are determined out of the free

vibration-motion equations of the structure, in which the damping term is

neglected due to its insignificant value:

0KM =+ && (22.1)

The linear system is verified by free harmonic vibrations of the form:

tiii

cos = (22.2)

where:i

- is the eigenvector representing the mode shape of the ith

natural frequency**;

i - is the i

thcircular frequency (radians/second);

t - is time (seconds).

Thus, the equation system yields:

( ) 0MK =i

2 (22.3)

This equality is satisfied if either i

= 0 or if the determinant of( )MK 2 is zero. Discarding the trivial solution, the second optionrepresents an norder equation in 2 called the characteristic equationor the

eigenvalues equation:

0MK = 2 (22.4)

The solutions of this equation are the n eigenvalues (the natural circular

frequenciesi

) of the vibrating system and the n eigenvectorsi

. The

natural frequencies yield:

* Usually, different values are assigned for the Youngs modulus in static anddynamic analyses. The difference is justified by the velocity of load appliance.

** Because the solution of (22.2) cannot determine the values of displacements

, the n vectorsi

give the proportions of the various terms.


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2

i

if = (cycles per time unit)

To enhance the numerical precision each eigenvector is normalized

according to the largest component such that

1=i

T

i M (22.5)

and the larges component becomes 1 (unity). The eigenvectors are

orthogonal relative to the mass matrix and to the stiffness matrix:

0=

n

T

m

M

, m

n(22.6)

0=n

T

m K , mn

22.1.2 The mass matrix

Now it is appropriate to discuss about the mass matrix, which was

disregarded on purpose during the chapters dedicated to static analyses (in

order to avoid making matters worse). The details regarding the mass matrix

formulation where postponed because dynamic analyses are based on the

inertia loads evaluation, meaning forces that are proportional with the mass.

Such inertia loads, when applied on certain directions, may be much more

effective then the usual gravity loads. It is obvious that the mass property

is used also in static analyses, when the structures own weight is taken into

account.

In the FEM the mass property can be assigned in two ways: firstly, as the

real mass, computed by multiplying the elements volume V with the

materials density (declared as one of its properties) and secondly, by

using the concentrated mass element, out of the computer codes library.

This element is a dimensionless finite element, defined by a single node (a

local concentrated mass with a prescribed value).

For the first category, the total mass matrix M is assembled by the usualrule, based on element sub-matrices:


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218

= VeT

e dVNNM (22.7)

where Me is known as the element mass matrix. The mass matrix M is a

square table with the same dimension as the stiffness matrix (i.e. depending

on the number of nodal DOF). M can be used in computations as a

consistent mass matrix (with all terms with nonzero values) or as a

lumped, diagonal matrix, even if no concentrated masses exist. For many

computational processes the lumped matrix is more convenient and

economical. However, the lumping methodology is not easy for higher order

elements.

For the triangular, 2D plane stress or plain strain element, with nodes l, m

and nthe shape functions matrix Nis defined as

[ ]nmle

NNN= IN (22.8)

in which

=

10

01I and

++=

2

ycxbaN iii

i, etc, where is the area of the

triangle. The displacement component uof the 2D triangular element can be

expressed in terms of nodal displacements ul, umand unas

( ) ( ) ( )[ ]nnnnmmmmllll

uycxbauycxbauycxbau ++++++++

=2

1

(22.9)

in which the coefficients are obtained by circular permutations

mnnmlyxyxa =

mnnmlyyyb == (22.10)

nmmnlxxxc ==

and

( )lmnyx

yx

yx

nn

mm

ll

area2

1

1

1

det2 == (22.11)


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219

As the same approximation is available for the other Cartesian direction y,the displacement vector in terms of the shape functions yields:

[ ] dIIIdN ==

=

nmlNNN

v

u (22.12)

If the thickness of the element t is constant over its area, the mass matrix

yields:

= dxdytT

eNNM or = dxdyNNt sre IM (22.13)

Substituting the shape functions, the value of the integral will be:

( )( ) sr

srdxdyNN

sr =

=

when

when

6/1

12/1 (22.14)

Fig. 22.2 Elemental mass matrix triangular element

With the notationM= tthe consistent mass matrix becomes

210

021

410

041

410

041

410

04

1

210

02

1

410

04

14

10

04

1

410

04

1

210

02

1

3

Me=M (22.15)

x

y

l

m

n

, t,


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221

Using the notation iT

Lx= the special eigenvalue relationship (22.17) isreached, in which

11 = TMLLS is a symmetric form.

After determiningi

(usually a few selected values corresponding to the

largest vibration periods T) the x vectors are found and hence the

eigenvectorsi

. If the mass matrix is lumped, the procedure of determining

the eigenvalues is simplified, which is an important advantage of the

diagonalization.

22.1.4 Master and slave DOF

For solving an eigenvalue problem the computer effort is one order ofmagnitude higher than for the equivalent static analysis. A technique applied

to simplify the solution is to select only a set of nodes (with the

corresponding DOFs), called masternodes(and masterdegrees of freedom

respectively), which are significant for the dynamic response of the

structure. The process is similar with sub-structuring, being based on matrix

partitioning. It is assumed that all the other DOF of the structure (called

slaveDOFs) depend in some way on the masterDOF:

ms T = (22.21)

where Tis the dependence matrix.

The total displacement vector is divided into 2 parts:

mm T

T

I

s

m *=

=

= (22.22)

The dynamic equation of the whole system (disregarding the damping

effect) can be reduced by applying previous assumption

0KM ** =+mm

&& (22.23)

where *** KTTK T= and *** MTTM T= . This is the so called reducedproblem, with a smaller number of unknowns.


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222

Although reasonably good eigenvalues are obtained using a lower numberof DOF, for further computations the solution for theslaveDOF should also

be known. The Tdependency can be assumed so that the general pattern of

displacement should follow the masterDOF displacements. The easiest way

is to prescribe them

displacements on the unloaded structure in static

conditions:

=

=

mm

s

mmsm

smss

f

0

KK

KKK (22.24)

As slave nodes are unloaded

0KK =+ msmsss or msmsss KK1= (22.25)

Thus

smssKKT

1= . (22.26)

Finally, it is up to the user to choose how master and slave DOF are

assigned. It is obvious that DOF with no attached mass (or with small

attached mass) are the first ones to be declared slave DOF, because their

contribution to the dynamic response of the structure is insignificant. The

FEA computer codes have an automatic option for masterDOF selection.

The selection procedure is based on the ratio value between the diagonal

terms of the stiffness and mass matrices, iiii MK / . The DOF correspondingto the largest values of the ratio are selected until the desired number of

masterDOF is reached.

22.1.5 The structural response

The total response of a structural system with several dynamic DOF is

defined as the sum of shape functions multiplied by the generalized

coordinates. This representation is based on the free vibrations eigenvectors.

Each eigenvector represents one of the n pattern displacements. By their

amplitudes, considered as generalized coordinates, any deformed shape can

be represented. The combination of first three vibration shapes into a

displacement shape is shown in figure 22.3. For each modal component n

the displacement is computed as the product between the eigenvector and

the modal amplitude Yn:


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223

nnn Y= (22.26)

The total displacement is computed as the sum of its modal components:

nnn YYY +++= ...2211 (22.27)

According to relationship 22.27, the modal matrix is used to transform

the generalized coordinates Yinto geometrical coordinates .

Fig. 22.3 Displacement of a 4 dynamic DOF system, as sum of its modal

components

An example of modal analysis is represented below. The structure of a three

storey building is made of concrete diaphragm walls, spatial frames and

concrete floors. The 3D finite elements model is shown in figure 22.4.a.

Although the columns and beams are meshed with line-type elements the

representation is made emphasizing their real cross section. The equivalent

mass was evaluated by assigning the material density. The masses of the

faade and the partition walls are taken into account using an artifice, by

melting them in the floors density. The constraints are applied at the

bottom nodes, by suppressing all DOF.

12,1 3,11,1

1,3

1,4

1,2

2,4

2,2

2,3

4

3,2 2

3,4

3,3

+ =

3

111 Y= 333 Y=222 Y= Y=


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224

For a building with a limited height and a reasonably regular structure, thefirst three vibration modes (and frequencies) define with sufficient accuracy

its dynamic behavior. The first vibration shape occurs usually along the

transversal direction, as the most flexible one, while the second vibration

shape occurs along the longitudinal direction. The third vibration shape

represents the torsion of the structure, due to the different locations of the

mass centroid and the rotational stiffness centroid, leading to an overall

torsion moment.

Although the presented structure has only longitudinal symmetry, the

expected results are achieved. The natural vibration modes as well as the

corresponding calculated frequencies are shown in figures 22.4.b, cand d.

22.2 THE SPECTRUM ANALYSIS

The spectrum analysis represents the next step in the dynamic response

assessment of a structure. Usually the deterministic spectrum method is

used. It must be preceded by a modal analysis, finished with the expansion

step of the required vibration modes.

The aim of the analysis is to calculate the displacement, strain and stress

fields as components of the dynamic response of the structure. The

assessment is based on the use of various response spectra, followed by a

probabilistic combination of maximum effects.

The main assumptions of the spectrum method are:

- the structure has a linear-elastic behavior;

- in case of the single-point response spectrum, the structure is excited

by a spectrum of known frequency components applied

simultaneously on all constrained points (or on specified master

DOF);

- in case of the multi-point response spectrum, the structure may be

excited by different input spectra at different points.

The maximum response values of a simple oscillator to a variable excitation

applied on the support point - such as an acceleration )(tu&& - are called


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spectral response values. The corresponding graphical representations ofthese values versus a frequency range are called response spectra.

Fig. 22.4 Finite elements model (a); first, second and third vibration shapes

(b, c, and d respectively)

a. b.

c. d.

T1= 0.48 sec

f1= 2.09 Hz

T2= 0.36 sec

f2= 2.78 Hz

T3= 0.18 sec

f3= 5.37 Hz


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According to the meaning of these representations, a response spectrum maybe a displacement spectrum, a velocity spectrum or an acceleration

spectrum. An example is shown in figure 22.5, each graph corresponding to

a different dumping value.

The maximum response values for an oscillator with several dynamic DOF

are calculated based on the spectral representations, as it is shown in figure

22.3. The spectra are distinguished on each excitation direction. They are

assigned as tabular data, according to an accepted frequency range or natural

vibration periods range. Also, each vibration mode which is taken into

account has a differentparticipation factorto the overall result.

Fig. 22.5 Acceleration spectra for three different dumping factors.

The participation factor of the ith

mode, for a given excitation direction is

defined as:

MDT

ii = for the support excitation option, or

FT

ii = for the force excitation option,

where i - the normalized eigenvector;D - a vector describing the excitation direction

F - the input force vector.

T(s)

Sa(m/s2)

1

2

3

2.5

1.5

0.5

00.5 1 1.5 2 2.5

= 0.01

= 0.02

= 0.05

T1T2

Sa,2

Sa,1


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The excitation direction vector has the general form

.....321kkk DDD=D

where kjD is the excitation at DOFjin the direction k.

The displacement vector for each mode is computed from the eigenvector

by using the mode coefficients i:

ui= ii (22.28)

The mode coefficient is computed in different ways, depending on theexcitation type. For a velocity excitation of the support points,

i

iiv

i

S

,= (22.29)

with Sv,i the spectral velocity for the ith

mode, as input value from the

velocity spectrum at frequencyfi, and ithe ith

natural circular frequency.

For an acceleration excitation of the support points,

2,

i

iiai S

= (22.30)

with Sa,i the spectral velocity for the ith


velocity spectrum at frequencyfi., and

iii S ,= (22.31)

with S,i the spectral velocity for the ith


velocity spectrum at frequencyfi.

The spectral values Sv,i, Sa,iand S,ibetween the ones defining the spectrumvertexes are calculated by interpolation.


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The modal displacements are finally combined in order to obtain themaximum response of the structure. It includes strains, stresses and reaction

forces as well, if they where computed in the expansion step. The most used

combination methods are the following:

- The Square Root of the Sum of Squares method; the total modal

response (displacement, stress or reaction) is calculated with the relationship

==

n

iiRR

1

2 (22.32)

withRithe modal response of mode i.

- The Naval Research Laboratory Sum method calculates the

maximum modal response as

+==

n

iiRRR

2

2

1 (22.33)

where1R is the absolute value of the largest modal response at the peculiar

point andRithe modal response of the same point from other modes.

- The Grouping method; the maximum modal response yields

== =

n

i

n

jjiij RRR

1 1

(22.34)

where

>

=

1.0if0

1.0if1

i

ji

i

ji

ij

The main disadvantage of using the modal analysis results in the design

process is due to the fact that the maximum modal response values are

always positive (being obtained from square roots). In this circumstance, thecombination of these results with other load cases (as the results of a static

analysis, due to gravity load) should be made with care.


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22.3 PERFORMING MODAL AND SPECTRUM ANALYSES

In order to perform a modal analysis, the following steps should be

followed:

1. Build the finite element model. All the modeling rules shown in chapters

25 and 26 are available, with the subsequent remarks:

- only the linear behavior is valid in a modal analysis; if nonlinear

elements or material properties are specified, they are treated as linear;

- it is compulsory to define the mass of the model, either as materials

density or by using concentrated mass elements;

- no loads are applied during a spectrum analysis (only prestresseffects if the case);

- the DOF where the base excitation spectrum will be applied should

be constrained.

2. Select the vibration modes extraction method and the number of modes to

be extracted. Frequency ranges may be used in order to filter unnecessary

vibration modes. Enough modes should be chosen to cover the frequency

range to characterize the structure's response. The accuracy of the solution

depends on the number of modes used: the larger the number, the higher the

accuracy.

3. Select the master DOF (user defined or automatically, using the computercode options).

4. Calculate the natural frequencies and the corresponding mode shapes.

5. Define the spectrum (spectral value versus frequency curve) spanned in

the range of expected natural frequencies. Define the dumping coefficients.

6. Expanding the reduced solution to the full DOF set.

7. Combine the modes in a separate solution phase, selecting the appropriate

combination method.

22. chapter 22 - modal analyses _a4

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