Dynamics Theory Enu

TheoryDynamics

Theoretical background - Dynamics

ir. B. Resseler 04/2006


Table of contents

Introduction .................................................................................................................3

Formulation of the dynamic equilibrium equations...................................................4Background ..................................................................................................................................................... 4

Natural modes, solution of free vibration................................................................... 5Back ground .................................................................................................................................................... 5

SDOF free vibration ................................................................................................................................... 5MDOF free vibration................................................................................................................................... 5Related quantities ...................................................................................................................................... 7

Practical use.................................................................................................................................................... 7Eigenmode calculation in SCIA.ESA PT ................................................................................................... 7

Harmonic loading ........................................................................................................ 9Back ground .................................................................................................................................................... 9

SDOF harmonic loading............................................................................................................................. 9MDOF harmonic loading .......................................................................................................................... 11

Practical used................................................................................................................................................ 11Harmonic load calculation in SCIA-ESA PT............................................................................................ 11

Karman Vibration....................................................................................................... 13Background ................................................................................................................................................... 13Practical use.................................................................................................................................................. 15

Karman Vibration in SCIA-ESA PT.......................................................................................................... 15

Seismic loading ......................................................................................................... 17Back ground .................................................................................................................................................. 17

SDOF of seismic systems........................................................................................................................ 17MDOF of a seismic system...................................................................................................................... 18Modal combination methods.................................................................................................................... 20Mass in the seismic analysis ................................................................................................................... 21Predominant mode................................................................................................................................... 23Multiple eigenshapes ............................................................................................................................... 23Calculation Protocol ................................................................................................................................. 23

Practical use.................................................................................................................................................. 24Seismic load calculation in SCIA-ESA PT............................................................................................... 24

Direct time integration............................................................................................... 26Back ground .................................................................................................................................................. 26Practical use.................................................................................................................................................. 27

How to get results .................................................................................................................................... 27Setup parameters of a general load case ............................................................................................... 28

Non-uniform damping ............................................................................................... 29Introduction ................................................................................................................................................... 29Back ground .................................................................................................................................................. 29Practical use.................................................................................................................................................. 30

Annex A: Numerical Damping Values....................................................................... 32

3

Introduction

The choice to do a static or a dynamic analysis of a structure depends on the relative importance of the forces related to acceleration and velocity in the global behaviour of the structure as a result of external forces acting upon it.


Formulation of the dynamic equilibrium equationsBackground

All dynamic computations can be related to the solution of the equations of dynamic equilibrium which can be written in matrix form:

M X C X K X F t ( ) (1.1)

When this matrix equation is compared to the equation of static equilibrium:

K X F (1.2)

It is clear that the dynamic equation of equilibrium requires more data than the static equation. In fact, we have to supply sufficient information to compute:

M : The mass matrix

C : The damping matrix

F(t) : The solicitation as a function of time

The results that are obtained when solving the equation include displacements as a function of time as well as velocity, acceleration and internal forces. The amount of results can therefore become exhaustive. The methods available to solve equation (1.1) are mostly based on a modal superposition principle. The assumption is made that the behaviour of the structure can be obtained by superposing a number of natural deformation modes each multiplied by a weighing factor that depends on the solicitation. These natural modes or eigenmodes are obtained from the solution of:

M X K X 0 (1.3)

The eigenvalues and eigenvectors can be used for further dynamic studies. Four different types of dynamic computations are allowed presently in SCIA.ESA PT:

1. Harmonic solicitation

2. Spectral analysis

3. Direct time integration

4. Von Karman vortices

5

Natural modes, solution of free vibration

Back ground

SDOF free vibration

To understand better the eigenmodes, the solution of a free vibration system, a SDOF (Single Degree Of Freedom) system is regarded in detail.

Consider the following system:

A body of mass m is free to move in one direction. A spring of constant stiffness k, which is fixed at one end,is attached at the other end to the body.

The equation of motion can be written as:

0)()(

tyktym (2.1)

A solution for this differential equation is: )cos()( twAty Inserting this in (2.1) gives:

0)cos()( 2 tAkm (2.2)

This implies that:

mk

(2.3)

Where is called the natural circular frequency.

The natural period T can be written as:

2

T (2.4)

The natural frequency (or eigen frequency) f can be written as:

2

1

Tf (2.5)

MDOF free vibrationFor a general, MDOF (Multiple Degree Of Freedom) structure, equation (2.1) can be written in matrix notation:

0

UKUM (2.6)


Where:

U is the vector of translations and rotations in nodes,

U is the vector of corresponding accelerations,

K is the stiffness matrix assembled for the static calculation,

M is the mass matrix assembled during the dynamic calculation.

The K-matrix is exactly the same as the stiffness matrix that is used in static computations. In this matrix, proper account can be given to anisotropic material, springs and boundary conditions. For an overview of the possibilities, please consult the manuals which describe the different element types.From equation (2.6) it is clear that the calculation model created for a static analysis needs to be completed with additional data of masses. In SCIA.ESA PT, this data is defined and stored in mass groups and mass combinations.

The M-matrix can be computed in different ways. Two classes can be recognised:

-lumped mass matrix, discrete mass in the nodes

-consistent mass matrix, distributed mass on the members

The lumped mass matrix offers considerable advantages with respect to memory usage and computational effort because in this case, the M-matrix is a diagonal matrix. The SCIA.ESA PT programs will only use lumped mass matrix. By refining the mesh, the lumped converges to the, consistent approach. The solutions of (2.6) are harmonic functions in time having the following form:

)(sin 0TTU (2.7)

Notice that in this solution a separation of variables is obtained:

- The first part, ( ), is a function of spatial co-ordinates,

- The second part, )(sin 0TT , is a function of time.

When substituting (2.7) in (2.6), an equation is obtained which is known as the Generalised Eigenproblem Equation:

02 MK (2.8)

The solution of (2.8) yields as many eigenmodes as there are equations.Each eigenmode consists of 2 parts:

- An eigenvalue: value i

- An eigenvector: vector i , which is not fully determined. The deformation shape is

known, but the scale factor is unknown.

This scale factor can be chosen in different ways. In SCIA-ESA PT, as scale factor, a M-orthonormalisationhas been implemented. This is shown in the following relation:

1 iTi M

M-orthonormalisation have following properties:

0 iTj M , When ji (2.10)

2ii

Ti K (2.11)

The algorithm used in the ESA programs to compute the eigenvalues is the well known subspace method. This method is extremely well suited to calculate a relative small amount of eigenvectors. By using this method, N (Nis specified by the user) eigenvalues are determined simultaneously. These values will be the lowest eigenvaluesif one is using the method straightforwardly.

7

The program prints each eigenvalue, the radial frequency and the frequency in periods/second. Subsequently, the modal participation factors are given. It is displayed in the calculation protocol of eigenfrequencies. These are defined as:

jT M {1l}

with : jT the M-normalised eigenvector j

M the diagonal mass matrix

{1l} a directional matrix containing a 1 in the indicated direction l and a 0 in other directions

Furthermore, the sum of masses in the analysis is displayed for each global direction X, Y and Z.

The Mass normalised eigenvectors can be displayed both, graphically as in numeric output. The user is allowed to display the mode shapes on the structure. A small tool as available to show it as a animation. For each eigenvector, the deformation of the mesh nodes is listened. The user is allowed to filter the largest amplitude for each global direction. The results are available in the result service “Displacement of nodes”.The complete list of eigenfequencies is given. For each mode is displayed the frequencies, the velocity, the acceleration and the period. You will find it in the service “Eigen frequencies” in the result menu, which is displayed after a “free vibration” calculation.

Related quantitiesTo find other related quantities, such as the generalised weight, is easy:

GW = QT W Q

where: Q eigenvector with largest amplitude 1

W weight matrix

From the value of GW, the scale factor to determine from Q can be found. If this factor is called a, such that:

Q = /a

Then

GW = (1/a)T W (1/a)

= (1/a²)T gM gM = W

= g/a²

Similarly, the modal weight can be found:

MW = GW * (participation factor)²

Practical use

Eigenmode calculation in SCIA.ESA PT The following diagram shows the required steps to perform a Free Vibration calculation:


Activate the Dynamics functionality

Input Masses Generate Masses from Static Load cases

Create a Mass Group

Create a Mass Combination

Perform a Free Vibration Calculation

Refine the Finite-Element Mesh if required

Specify the number of Eigenmodes to be calculated

9

Harmonic loadingThe structure will now be loaded with an external harmonic load, which will cause the structure to vibrate.

A forced vibration calculation can be required to check the response of a building near a railroad or major traffic lane, to check vibrations due to machinery, to verify structural integrity of a floor loaded by an aerobics class,…

Back ground

SDOF harmonic loading

To understand what is going on during the dynamic analysis of a complex structure with frames or finite elements, the forced vibration of a SDOF (Single Degree Of Freedom) system is regarded in detail.

Consider the following system:

A body of mass m can move in one direction. A spring of constant stiffness k, which is fixed at one end, is attached at the other end to the body. The mass is also subjected to damping with a damping capacity c. An external time dependant force F(t) is applied to the mass.

The equation of motion can be written as:

)()()()( tFtyktyctym

(3.1)

When the acting force on this system is a harmonic load, equation (3.1) can be rewritten as follows:

)sin()()()( tFtyktyctym

(3.2)

With: F = Amplitude of the harmonic load

= Circular frequency of the harmonic load

A solution to this equation is the following:

222 )2()1()sin())sin()cos(()(

rrtYtBtAety SDD

t

(3.3)

With: kFYS The static deflection (3.4)


mc

2The damping ratio (3.5)

21 D The damped circular frequency (3.6)

212)tan(

rr

(3.7)

r The frequency ratio (3.8)

The angle signifies that the displacement vector lags the force vector, that is, the motion occurs after the application of the force. A and B are constants which are determined from the initial displacement and velocity.The first term of equation (3.3) is called the Transient motion. The second term is called the Steady-state motion. Both terms are illustrated on the following figure:

The amplitude of the transient response decreases exponentially (te

). Therefore, in most practical applications, this term is neglected and the total response y(t) can be considered as equal to the steady-state response (after a few periods of the applied load).

Equation (3.3) can then be written in a more convenient form:

222 )2()1(1

rrYY

S

(3.9)

SYY is known as the Dynamic Magnification factor, because YS is the static deflection of the system under a

steady force F and Y is the dynamic amplitude.

The importance of mechanical vibration arises mainly from the large values of SY

Y experienced in practice

when the frequency ratio r has a value near unity: this means that a small harmonic force can produce a large amplitude of vibration. This phenomenon is known as resonance. In this case, the dynamic amplitude does

not reach an infinite value but a limiting value of 2SY

.

The second term in equation (3.1) represents the damping. The damping of the model is defined by the damping ratio. Let’s remark that that damping can be over ruled by using the non-uniform damping method, where the user is allowed to define the damping of the elements on more precise way. See for more detail the chapter “Damping”.

The logarithmic decrement is the natural logarithm of the ratio of any two successive amplitudes in the same direction. This is illustrated on the following figure:

11

11

1lnXX

(3.10)

The logarithmic decrement is related to the damping ratio by the following formula:

212

MDOF harmonic loading

When the loads applied to a structure are sinusoidal, the general equation becomes:

M U C U K U F t { } cos( )

This means that more than one node of the structure can be loaded but the frequency of all solicitations should be equal.The solution which is of interest in this case is the particular solution which describes the stationary state of the structure without taking into account the initial conditions and transient behaviour (this is in contrast with the time dependent solution where one calculates the transient behaviour and eventually when integration time is long enough, finally arrives at the stationary state behaviour).The stationary state will give a sinusoidal result in each node, with a frequency equal to the applied frequency. The resulting vibration will be in faze with the load when C=0. When C is not zero, there will be a faze shift. The solution is found once more by uncoupling the equations as before. The amplitudes of the harmonic response are determined and given in the output.

Practical used

Harmonic load calculation in SCIA-ESA PTSCIA.ESA PT supports the harmonic calculation in a general 3d environment (composed by 1d and 2d elements)The Harmonic Load can be inputted after creating a Combination of Mass Groups. This implies that the steps used to perform a Free Vibration calculation still apply here and are expanded by the properties of the Harmonic Load.


Harmonic Loads in SCIA-ESA PT are always defined as nodal forces i.e. a nodal load or a nodal moment. More than one node of the structure can be loaded in a load case, but the frequency of all solicitations is equal to the forcing frequency specified for that load case.The following diagram shows the required steps to perform a Harmonic Load calculation:



Create a Mass Group


Perform a Linear Calculation



Create a Harmonic Load case

Input Harmonic Loads

13

Karman VibrationKarman vibration handles on the transverse vibration of cylindrical structures due to wind.

First the theory is explained in which reference is made to the Harmonic load since Vortex shedding is a special case of harmonic loading.

BackgroundOne of the most important mechanisms for wind-induced oscillations is the formation of vortices (concentrations of rotating fluid particles) in the wake flow behind certain types of structures such as chimneys, towers, suspended pipelines…At a certain (critical) wind velocity, the flow lines do not follow the contours of the body, but break away at some points, and then the vortices are formed.These vortices are shed alternately from opposite sides of the structure and give rise to a fluctuating load perpendicular to the wind direction. The following figure illustrates the vortex shedding for flow past a circular cylinder. The created pattern is often referred to as the Karman vortex trail:

When a vortex is formed on one side of the structure, the wind velocity increases on the other side. This results in a pressure difference on the opposite sides and the structure is subjected to a lateral force away from the side where the vortex is formed. As the vortices are shed at the critical wind velocity alternately first from one side then the other, a harmonically varying lateral load with the same frequency as the frequency of the vortex shedding is formed.

The frequency of the vortex shedding vf is given by:

dvSfv

(4.1)

With: S = Non-dimensional constant referred to as the Strouhal Number

For a cylinder this is taken as 0,2.

d = Width of the body loaded by the wind [m]

For a cylinder this equals the outer-diameter.

v = The mean velocity of the wind flow [m/s]

The manner in which vortices are formed is a function of the Reynolds number Re, which is given by:

510687,0Re dv (4.2)

The Reynolds number characterizes three major regions:

Subcritical 510Re300

Supercritical 65 105,3Re10

Transcritical Re105,3 6 In general large Reynolds numbers stays for turbulent flow.

For chimneys with circular cross section the flow is either in the supercritical or transcritical range for wind velocities of practical interest.If the vortex shedding frequency coincides with the natural frequency of the structure (resonance) quite large across-wind amplitudes of vibration will result unless sufficient damping is present. In this case, formula (4.1) can be rewritten to calculate the critical wind velocity at which resonance occurs:

fdvcrit 5 (4.3)


With: f = Natural frequency of the structure

The across-wind forces per unit length caused by the vortex shedding can be approximated by the following formula:

tCvdtP tcritL 2

21 (4.4)

With: = Air density taken as 1,25 kg/m³

tCt = A lift coefficient that fluctuates in a harmonic or random manner

and depends of the Reynolds number. The following figure shows this relation when Ct is proportional to the mode shape.

If the vortex shedding is taken as harmonic, equation (4.4) can be written as:

vtcritvL fCvdtPtP 2sin21)sin( 2

0 (4.5)

According to reference, assuming a constant wind profile, the equivalent modal force due to the fluctuating lift force of equation (4.5) is given by:

H

tvcritvL dzzzCfvdtPtP0

2 )}(){(2sin21)sin( (4.6)

With: )(z = Modal shape at height z

H = Total height of the structure

The dynamic amplitude Y at resonance can be written as:

2SYY (4.7)

The static deformation YS is given by:

200

MP

KPYS (4.8)

M is the equivalent modal mass of a prismatic member given by:

dzzzmMH

2

0

)}({)( (4.9)

With: m(z) = The mass per unit height

When combining formulas (4.7) and (4.8) the maximum response of a SDOF system subjected to a harmonic excitation may be written as:

15

21

2 M

PY L (4.10)

It follows that when the vortex shedding occurs with the same frequency as the natural frequency of the structure, the maximal amplitude is given by:

21

)}({)(

)}(){(21

2

0

2

0

2

dzzzm

dzzzCvdY H

H

tcrit (4.11)

When it is assumed that the mass per unit height is constant and that the lift coefficient is proportional to the mode shape, formula (4.11) can be simplified to the following:

mS

CdY t22

3

16 (4.12)

This equation may be used as a first estimate of likely response of the structure.

Practical use

Karman Vibration in SCIA-ESA PTIn SCIA-ESA PT, the Vortex Shedding was implemented according the Czech loading standard.The effect is only taken into account if the critical wind velocity calculated by formula (4.3) is between a minimal and maximal value. These two extremes can be defined by the user. These values are taken as 5 m/s and 20 m/s.

In addition to formula (4.11), in SCIA-ESA PT it is possible to specify the length of the structure where the Von Karman effect can occur. For each geometric node of the structure, it is possible to relate a length of the cylinder to the node. This implies that, in order to obtain precise results, the structure should be modelled with sufficient geometric nodes.

By default the effect can occur over the entire height of the structure however, when there are specific obstacles on the surface of a chimney for example, these obstacles will hamper the formation of the vortices and thus reduce the Von Karman effect. In practice, this is exactly the solution to suppress vortex-induced vibration, the fitting of special ribs on the surface of the cylinder.


The following diagram shows the required steps to perform a Vortex Shedding calculation:



Create a Mass Group





Create a Karman Vibration Load case

Input Karman Loads (lengths)

17

Seismic loadingDuring an earthquake, the subsoil bearing a structure moves. The structure tries to follow this movement and as a result, the masses in the structure begin to move. Subsequently, these masses subject the structure to inertial forces. The method which is implemented in SCIA.ESA PT is based on modal superposition. The user defines the loading by spectrums in the 3 global directions X, Y and Z (in a 3d model).The program computes the modal deformation and modal forces.

Two methods are implemented to consider the missing mass into the seismic analysis.

Depending on the selected Modal combination method, the modal results are summed to obtain the final results.The results (deformations, internal forces, reactions) are displayed on the standard way. A dedicated menu was implemented in order the display the modal and summed deformation and acceleration.

Back ground

SDOF of seismic systems

Consider the SDOD system shown in figure bellow which illustrates the displacement of a system that is submitted to a ground motion.

Where: yg(t) : is the ground displacement,

y(t) : is the total displacement of the mass

u(t) : is the relative displacement of the mass

The total displacement can thus be expressed as follows:

)()()( tutyty g (5.1)

Since yg is assumed to be harmonic, it can be written as:

)sin()( tYty gg (5.2)

The equilibrium equation of motion can now be written as:

0)()()(

tuktuctym (5.3)

Since the inertia force is related to the total displacement (y) of the mass and the damping and spring reactions are related to the relative displacements (u) of the mass.When (5.1) is substituted in (5.3) the following is obtained:

0)()())()((

tuktuctytum g ;

or

)()()()( tymtuktuctum g (5.4)

This equation is known as the General Seismic Equation of Motion. This equation can be used to illustrate the behaviour of structures that are loaded with a seismic load:

y(t)

k

c

m

yg(t) u(t)


Substituting (5.2) in (5.4) gives the following:

)sin()()()( 2 tYmtuktuctum g

This equation can be compared with equation (3.2) of the chapter Harmonic loading. As a conclusion, the ground motion can also be replaced by an external harmonic force with amplitude:

2 gYmF

When a structure has to be designed to withstand an earthquake, spectral analysis is often used because the earthquake loading is given under the form of a response spectrum. This response spectrum can be either a displacement or a velocity or an acceleration spectrum. The relation of an earthquake given by an acceleration time history and the corresponding spectrum is given by:

max

)( ))(sin()(1),(

dTeyS Tgd (5.5)

With: )(gy

: Ground acceleration in function of time

: Damping factor

T: The period 2

The integral between the square brackets is known as the Duhamel integral. This integral is the solution of the differential equation (5.4).

Instead of Sd (displacement response spectrum), Sv (velocity response spectrum) or Sa (acceleration response spectrum) can be used. These spectra are related by :

Sa = .Sv = ².Sd (5.6)

The three spectra are normally given on the same figure, using different scale-axes. (See picture of spectral density bellow)

MDOF of a seismic systemFor MDOF (Multiple Degree Of Freedom) systems, equation (5.4) can be written in matrix notation as a set of coupled differential equations:

19

gYMUKUCUM

1 (5.7)

The matrix {1} is used to indicate the direction of the earthquake. For a two-dimensional structure (three degrees of freedom) with an earthquake that acts in the x-direction, the matrix is a sequence like {1,0,0,1,0,0,1,0,0,…}

The resulting set of coupled differential equations is reduced to a set of uncoupled differential equation by a transformation U = Z.Q, where Z is a subset of (the eigenvectors) and Q is a vector, which is time-dependent.

gYMQZKQZCQZM

1 or

gTTTT YMZQZKZQZCZQZMZ

1

This can be simplified to a set of uncoupled differential equations:

gT YMZQQCQ

12*(5.8)

Where *C is a diagonal matrix containing terms like ii2 .

Each equation j has a solution of the form:

t

jT

gT

j dTeYMZQ j

0

)( ))(sin()(11

(5.9)

To obtain the maximum displacements, the displacement response spectrum Sd of equation (5.5) can be substituted:

),(1max, jjdT

j SMZQ (5.10)

And:

),(1max, jjdT

j SMZZU ; or

Or:

),(max, jjdj SZU (5.11)

Where 1 MZ T is known as the modal participation factor.

One obtains in each node a value of (Uj)max, j=1,m (m<n). The maximal global displacement is the sum of the absolute values of the displacements. Often, the RMS (root mean square) is used as maximal displacement:

Umax = (Uj)²max (5.12)

The SCIA.ESA PT program supports 3 combination methods:

1. SRSS (Square Root of Sum of Squares)2. CQC(Complete Quadratic Combination)3. MAX (max method)

The detail is described in next chapter, “Modal combination method”


Modal combination methods

Modal combination methods are used to calculate the response R of a seismic analysis. The term "response" (R) refers to the results obtained by a seismic analysis, i.e. displacements, velocities, accelerations, member forces and stresses.

Because the differential equations were uncoupled, a result will be obtained for each mode j.To obtain the global response Rtot of the structure, the individual modal responses R(j)have to be combined.

The modal combination methods that are used in SCIA-ESA PT are:

1. SRSS-method (Square Root of Sum of Squares)

N

jjtot RR

1

2)(

(5.13)

With:

)( jR : The response of mode j

()

2. CQC-method (Complete Quadratic Combination)

N

i

N

jjjiitot RRR

1 1)(,)(

, (5.14)

With:

)()( , ji RR : The response of mode i and j

ji, : Modal Cross Correlation coefficients.

222222,

4141

8 23

rrrr

rr

jiji

jijiji

(5.15)

i

jr

i , j : Damping ratio for mode i and j.

This method is based on both modal frequency and modal damping. The CQC-method requires the input a Damping Spectrum.

Let’s remark that the defined spectrum is overruled, in case non-uniform damping method is used. See for detail at chapter “damping”.

3. MAX-method

N

jjjtot RRR

MAX1

2)(

2)(

, (5.16)

With:

SRSS;

21

)( jR : The response of mode j

)( MAXjR : The maximum response of all modes

Remarks: the short info at the seismic load case displays what is selected. Following format is used:

Short info = {Modal combination method}& “/” &{Coeff X} & “X” & “;” {Coeff Y} & “Y” & “;” {Coeff Z} & “Z” & “/”{acceleration coefficient}Where

{Modal combination method} : SRSS or CQC or MAX

{coeff X, Y, Z} : The coefficient of spectrum X, Y and Z

X,Y, Z : The spectrum X in the global X

{acceleration coefficient} : The assumed acceleration coefficient:

Mass in the seismic analysisThe mass can be taken in tree ways in the seismic analysis. In the menu “General seismic load”, the group “Mass in the analysis” contains 3 options:

1. Participation mass only2. Missing mass in modes3. Residual mode

In case “Participation mass only” is selected, then only the participation mass from the selected number of modes is taken into account in the seismic analysis.

In case “Missing mass in modes” is selected, then the missing mass is assigned to the known modes (E.G. modes selected by the user in the analysis).In case “Residual mode” is selected, then the missing mass is taken in the seismic analysis as an extra mode which represents the weight of the missing mass. The modal result of this mode is computed by a static equivalent load case. Remark: The used method is displayed in the linear calculation protocol

Redistribute missing mass to the known modes

The aim of the method is to smooth the missing mass to the known modes and then compute modal

deformations and then the modal forces. Afterwards its summed depending the selected rule SRSS, CQC,

MAX

Missing mass is assigned to known eigenmodes. Suppose we have determined k eigenmodes, where

, , ,( ) ,1

nFreq

ef k ef k j total kj

M M M

(5.17)

k i direction

,ef kM is effective mass

Missing mass can now be written as

, ,miss

k total k ef kM M M (5.18)

Ration between effective mass and missing mass is


,

missk

aef k

MrM

(5.19)

Now we can write these formulas

, ,( ) , ,( )miss

ef k j k ef k jM r M

( ), ( ), , ,( ) , ,( ) ( ),sgnmiss missj k j k ef k j ef k j j kM M (5.20)

( ),sgn 1j k

Then (5.21)

( ), , ( ) ( ), ,

, , , ,2( ) ( )( )

( , ) missj k a k j j k a cut off

k x y z k x y zj j

j

S S

u

Residual mode

The aim of the method is to evaluate the missing mass as an extra mode which is computed as an equivalent static load case. The static load represents the weight of the missing mass under the cut-off acceleration. Afterwards its summed depending the selected rule SRSS, CQC, MAXThe missing mass is computed in each node as difference between total mass and effective mass.

nFreq

1ji,k,total)j(,i,k,efi,k,ef MMM (5.22)

k is direction

i is node

j is mode

i,k,efM is effective mass, direction k, node i

Missing mass can now be written as

i,k,effi,k,total,kimiss MMM (5.23)

A static load case of weight in computed, which is handled as a “real” mode.

For each direction k, selected in the “General seismic load” interface, the amplitude of static load is computed

as:

i,kmiss

)cutoff(,k MSa (5.24)

Sak is acceleration of “cut off” frequency in direction k (i.e. last calculated frequency)

MissMk,i missing mass in direction k, node i

Afterwards, the extra mode is summed depending the selected rule SRSS, CQC, MAX

Remark1: In case of CQC, we do not assume correlation with the other modes (i.e. absolute value is added)

Remark2: The cut off frequency is the frequency of the latest modes in the analysis. It is of the responsibility of the user to select the correct number of modes

23

Predominant modeThe aim of the method is to take the sign of the mode in the seismic analysis, which computes by default only positive results. In menu General seismic load, the user selects for “Signed results”. When the property is "checked", then a combo is displayed wherein the use selects a mode shape. The list items displayed in the combo box are: “default”; 1; 2; ..; N ( N the number of frequencies selected by the user).

The default stays for modes shape with biggest mass participation is used (direction X, Y and Z together). Beside the “default”, the user is allowed to select a mode shape, which will determine the sign of the modal combination.

Multiple eigenshapesThe tool can be used in the seismic analysis in case SRSS.Modes are taken together in case the precision is fulfilled

Classic SRSS

2 2 2 2 2(1) (2) (3) (4) (5) ...R R R R R R

SRSS with multiple eigen shapes

If

( )( ) ( )

( )

1 , ;ii j

j

precision where i j

Mode (i) and (j) are multiple. Then for example

22 2 2(1) (2) (3) (4) (5) ...R R R R R R (5.25)

Where mode 2 and 3 are multiple.

Calculation ProtocolIn the calculation protocol of SCIA-ESA PT the intermediate results that were determined while calculating the global effect of a spectral loading can be found.

This paragraph describes the formulas that have been used to determine those intermediate results.

Natural circular frequency and modal shape

Mass matrix DM ][

Mass vector }1{][}{ DMmNatural circular frequency of mode shape j )( j

Natural normalized modal shape )(}{ j , With 1}{][}{ )()()( jjD

Tj MM

Total mass in kth direction totkM ,

Acceleration response spectrum )(,, jkaS

Direction kTotal number of directions NK

Participation factor of the mode shape j in direction k


Participation factor }{}{}{}{

)()(, m

Mm T

kj

Tk

jk

Effective mass 2)(,)(

2)(,)(,, jkjjkjefk MM

Participation mass ratio totk

jefkjk M

ML

,

)(,,)(,

Mode coefficient for mode j

Mode coefficient in kth

direction 2)(

)(,)(,,)(,

j

jkjkajk

SG

Total mode coefficient 2

)(

1)(,)(,,

)(j

NK

kjkjka

j

SG

Response of mode shape j

Displacement)()()( }{}{ jjj Gu

)()(,)( }{}{ jkjkjk Gu

Acceleration)()(

2)()( }{}{ jjjj Gu

)()(,)(,,)()(,2

)()( }{}{}{ jkjkjkajkjkjjk SGu

Lateral force in node i for direction k )(,,)(,)(,,)(,,)(,, jkijkjkajkijki SmF

Shear force in direction k

}{}{}{}{ )()(,)(,,)()(,,)(, mSmuFF Tjkjkjka

Tjk

ijkijk

2)(,)(,,)(, jkjkajk SF

Overturning moment in node i for direction k ijkijkjkakijki zSmM )(,,)(,)(,,,)(,,

Overturning moment in direction k

i

ijkijkjkakii

jkijk zSmMM )(,,)(,)(,,,)(,,)(,

i

ijkikijkjkajk zmSM )(,,,)(,)(,,)(,

Practical use

Seismic load calculation in SCIA-ESA PT

In SCIA-ESA PT, a Seismic Load can be inputted after creating a Combination of Mass Groups. This implies that the steps used to perform a Free Vibration calculation still apply here and are expanded by the properties of the Seismic Load.

The Design Spectra are defined for a damping ratio of 5%. If the structure has another damping ratio, the spectrum has to be adapted with a damping correction factor .

25

Activate the functionalities - Dynamics- Seismic


Create a Mass Group





Create a Seismic Load case

Define a Seismic Spectrum


Direct time integration

Back groundThe title may be misleading because normally in the literature, this name is used for a dynamic computation without modal superposition. In the software, the eigenmodes are determined first and are used to uncouple the equilibrium equations (1.1) into a set of m uncoupled second order differential equations which are solved one by one by direct time integration. The uncoupling is based on the properties given by equations (2.10) and (2.11). In equation (1.1) a solution for x is assumed to be of the form:

x = Q (6.1)

Where is the matrix of eigenvectors (n*n) and Q is a vector which is time dependent.

Substitution in equation (1.1) gives:

M Q C Q F (6.2)

When the equation is pre-multiplied with T, and equations (2.10) and (2.11) are taken into account, one obtains :

²Q C Q Q FT T (6.3)

This set of equations is still coupled because of the damping term. If however C-orthogonality is assumed (this means that T.C. reduces to only diagonal terms), then the equations are uncoupled and can be solved separately. The global results are obtained by superposition of the individual results (6.1) is also the exact solution if the assumption of C-orthogonality holds. If however, only a few eigenvectors (m<n) are used in instead of all the eigenvectors, then the system of equations and the superposition of the solutions gives a solution x which is an approximation of the exact solution.

In SCIA.ESA PT, C-orthogonality is assumed and it is also assumed that all modal damping factors are constant. This means that:

Ti ijC 2 (6.4)

The value of is one of the input data and is called damping factor.

The number of eigenvectors that is taken into account is also specified by the user. This value is equal to the number of eigenvectors computed in the eigenvalue computation.The method used to solve each uncoupled second order differential equation is the Newmark-method. This method is unconditionally stable but the accuracy depends on the time step. This time step has to be given by the user. However, to help him in his choice, a value determined by the program will be used if the user does not specify a value. This proposed value is computed as:

0.01 T

Where T smallest period of all the modes which have to be taken into account

This proposed value guarantees accuracy better than 1% over each period of integration of this highest mode. In most cases, a larger time step can be used because the contribution of this last mode is small.

This brings us to the question about the number of modes that should be used. When the time dependent terms on the left hand side of equation (6.3) are neglected, the solution for qj (a term of Q) is:

q Fj j jT 1 2/ (6.5)

27

This indicates that the lowest eigenmodes (j small) will contribute more than the highest modes (wj large), if dynamic terms are neglected. This can give a first idea on how many modes to use.A second criterion is the periodicity of F. Any mode which coincides with the loading frequency should be taken into account.Modal weight is a third criterion that can be used. If you add all modal weights in a particular direction together and divides this result by 9.81*sum of nodal masses in the same direction, you obtain a value smaller than 1. If this value is close to 1, it means that the higher modes will not contribute anymore. If, on the contrary, the value is smaller than 0.9, one can doubt about the value of a subsequent modal superposition.

Practical use

How to get resultsIn project data the user selects the functionality “General dynamics”The user defined the load case Dead load

The user defines a mass combination

The user defines a load case of type Dynamic with specification General dynamics and setup the parameters of menu “Dynamic load case data” (for example total time 1.5s, step 0.1s, log decrement 5%)

The user de start the linear analysis

The program generated load cases for each time step where the results of that time step are stored.

The program generated result class which maces it easy to view the results in graphic form. Bellow an example of the horizontal displacement of a node is displayed for the general dynamic load case BG3.

Remarks

1. When the user modifies the parameters of the dynamic load case and the starts the calculation again, then the generated load cases are updated.

2. Then master load case doesn’t contain results. 3. The generated are stored in an exclusive group and can be used in combination with other load

cases


Setup parameters of a general load case

Total time : The total time of the dynamic analysis

Integration step Auto, when it is checked, then 1/100 of smallest period is take.When Auto isn’t checked, then the user is allowed to select an integration step value.

Output step Step for generating the load cases. The value need to be smaller or equal at the integration step

Log. Decrement Damping defined as logarithmic decrement

29

Non-uniform damping

Introduction

Damping is present in all oscillatory systems and is a measure of a vibration structure to dissipate energy.

The dynamic tools of ESA take account of damping. Sometimes it is defined directly by the damping ration, sometimes by a damping spectrum. Please consult the specific chapters for detail. In general we can say that by default a mean value of damping is taken for the complete system for a mode.

With non-uniform damping, the user is allowed to define damping on the members and on the supports. The program computed based on energy consideration a mean value for each node.The non-uniform damping is supported for:

1. Harmonic load2. Seismic load

Back groundSCIA.ESA PT supports two types of damping:

Dumping Inside the materials “ ” is defined by a percentage from critical damping

(Logarithmic decrement, another quantity frequently used to defined damping, is defined by21

2

)

Geometrical damping from soil, defined by a coefficient at the level of the supports.

Both kind of damping are taken into account by an equivalent damping ratio. For each mode, it is computed as following:

appuis

2iss

inoeuds

2ijjj2

ii )(C

4)(k1

(7.1)

With:

j The damping ration in node j

jk The stiffness in node j

ij The modal displacement in j for mode i

i The frequency of mode i

sC The damping constant of the supports “s” (kg/s)

is The model displacement in support node s for mode i

ca user defined parameter (>0, default =0.5)

Note that:

On supports, damping is defined for the 3 global directions X, Y and Z

Proof :

The equivalent model damping i is obtained by ponderation of the potential energies ijE used by the

mode in part j of the structure where the damping is j .


jij

jijj

i E

E

Case1: damping of the material

j

ij

iji

ij

ijiji MKEE ][][ 2

With [K] and [M] respectively rigidity matrix and mass matrix.

j

ijjii mE 22 )( With jm the mass at node j

We assume that j j

ijjjiijj mE 22 )( .

In case of the modal displacement is normed by the mass, then the mean internal modal damping of mode I is given by:

j

ijjj

i

ijjj

jijj

ji

j

ijjji

jij

jijj

i kmm

m

E

E2

22

22

22

)()(

1)()(

)(

For practical use we take a same value of damping at all the elements,

We obtain i

Case2: geometrical damping

For a simple oscillator, the damping is defined by:

mc

2 With c the coefficient of damping

By analogy with the material damping, we can state:

s s s

iss

iiss

is

siiss Cm

mCE 222 )(

2)(

2

s

iss

is

is

siss

i CE

E2)(

21

We obtain in total:

ports

iss

inodesj

ijjj

ii Ck

sup

222 )(

2)(1

It is recommended by the Septem for choosing 5.0 (then 1 )

Practical use

Damping on members is related with the material.

Sometimes the used want one value assigned at the whole structures

31

Sometimes the user have mixed structures

The user is able to over rule easily the damping for some members

The user is able to easily define and correct the damping for the whole structure

Damping at supports is only valid for:

Flexible nodal support Support orientated in global direction XYZ only

Damping groups

The user defines damping groups (similar to mass groups). In order to activate the non-uniform damping:

1. The functionality “Non-proportional damping” is selected at menu “Project data” 2. A damping group is defined and connected at a mass combination3. A mass combination is defined and connect at a dynamic load case (harmonic, seismic)

Two types of default damping data is available for members (1d/2d). The user is allowed to select the proper default foe each damping group:

One value for all the structure, the “Global default” One value per material used in the projects, the “Material default”

The default damping settings are stored in the set-up of the damping service.

In case of global, one value is displayed {default 5%} In case of material defaults:

The damping is stored at the materials in the material library of ESA.

The set-up of the damping service contains also

Max modal damping: one value {used to limit the calculated damping per mode; default 30% } Alfa {factor for supports; >0; default 1}

Add data for damping at members and supports

In each group the user is able to assign damping at the elements (similar to additional mass on members in mass groups). Damping can be added at a:

A 1d member A 2d member A nodal supports

Following data is available:

In case of 1d and 2d members, tree possibilities are available

Logarithmic decrement: one value {default = 0.001} Relative damping: one value {default =0.001} Rayleight: two value { Alfa and beta; default both = 0}

In case of nodal support: tree values, one value per global direction X, Y and Z {C value; default = 0}

Note that the user is not able to add support dampers

1. If the support in not flexible2. If the support is rotated

In this case, a massage is displayed and the input is cancelled

In case the user adds damping in a specific group, then the default is over ruled for that member in that group.


Annex A: Numerical Damping Values In this annex, some numerical values for structural damping are given.

A. EC 8 - Part 6 (ENV 1998-6:2003 Annex B) suggest the following values for the damping ratio:Structural material Damping ratio Steel elements 1% - 4%

Concrete elements 2% - 7%

Ceramic cladding 1.5% - 5%

Brickwork lining 3% - 10%

B. Other values for damping are suggested by EC1 - Part 2-4 (ENV 1991-2-4:1995 Annex C):

The fundamental logarithmic decrement d is given by:

das dddd Where:

sd : Fundamental structural damping,

ad : Fundamental aero dynamical damping

dd : Fundamental damping due to special devices

The structural damping is given by:

min

111

s

s

dbnad

Where:

1n : Fundamental flexural frequency,

min11 ,, ba : Parameters given in the following table for different structural types.

Structural type1a 1b min

Reinforced concrete buildings 0.045 0.030 0.080Steel buildings 0.045 0 0.050Mixed structures: concrete + steel 0.080 0 0.080Reinforced concrete towers 0.050 0 0.025Lattice steel towers 0 0.030 0Reinforced concrete chimneys 0.075 0 0.030Prestressed steel cable 0 0.010 0Unlined welded steel stacks 0 0.015 0Steel stack with one liner or thermal insulation 0 0.025 0Steel stack with two or more liners 0 0.030 0Steel with brick liner 0 0.070 0Coupled stacks without liner 0 0.015 0Guyed steel stack without liner 0 0.040 0

Welded 0 0.020 0High resistance bolts 0 0.030 0

Steel bridges

Ordinary bolts 0 0.050 0Prestressed without cracks

0 0.040 0Concrete bridges

With cracks 0 0.100 0Parallel cables 0 0.006 0Bridge cablesSpiral cables 0 0.020 0

33

E.g.: for a steel building with first frequency of 3Hz, the logarithmic decrement is:

0.045 x 3 + 0 = 0.135 (> 0.05)

C. Other values for the logarithmic decrement are suggested by:

Structural material Logarithmic decrementSteel (welded) 0,025

Reinforced or Prestressed Concrete 0,056

Brickwork 0,25

Wood 0,13

In this reference, preliminary formulas can also be found for aerodynamic damping and damping caused by the foundation.

Documents

Dynamics Theory Enu