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Part 5:

Seismic design analysis

2 PowerFrame Manual – Part 5: Seismic Design Analysis

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PowerFrame Manual – Part 5: Seismic Design Analysis 3

1 Table of Contents 1 TABLE OF CONTENTS....................................................................................... 3

2 DESIGN OF EARTHQUAKE RESISTANT STRUCTURES .......................... 4

2.1 INTRODUCTION.................................................................................................. 4 2.2 SEISMIC ZONES .................................................................................................. 5 2.3 SUBSOIL CONDITIONS........................................................................................ 6 2.4 BASIC REPRESENTATION OF THE SEISMIC ACTION .......................................... 6

2.4.1 General...................................................................................................... 6 2.4.2 Seismic design spectrum for linear analysis ............................................ 7

2.4.2.1 Use of the design spectrum.......................................................................................................................................7 2.4.2.2 The design spectrum according to Eurocode 8 .........................................................................................................8 2.4.2.3 The design spectrum according to PS92 ...................................................................................................................9

2.4.3 Alternative representations of the seismic action................................... 10 2.5 ULTIMATE LIMIT STATE VERIFICATIONS ......................................................... 11

2.5.1 Eurocode 8 .............................................................................................. 11 2.5.2 PS92......................................................................................................... 12

2.6 SERVICEABILITY LIMIT STATE VERIFICATIONS............................................... 13 2.7 DESIGN GRAVITY LOADS................................................................................. 13

2.7.1 Eurocode 8 .............................................................................................. 14 2.7.2 P92........................................................................................................... 14

3 SEISMIC ANALYSIS.......................................................................................... 16

3.1 INTRODUCTION................................................................................................ 16 3.2 MODAL ANALYSIS............................................................................................ 17 3.3 MODAL RESPONSE ANALYSIS ......................................................................... 19 3.4 MULTI-MODAL RESPONSE ANALYSIS.............................................................. 20 3.5 SEISMIC RESPONSE ANALYSIS......................................................................... 22

4 SEISMIC DESIGN ANALYSIS USING POWERFRAME............................. 23

4.1 INTRODUCTION................................................................................................ 23 4.2 SEISMIC FUNCTIONS IN THE ‘LOADS’-WINDOW ............................................ 24

4.2.1 Load groups for seismic analysis............................................................ 24 4.2.2 Definition of design gravity loads .......................................................... 25 4.2.3 Definition of seismic load group............................................................. 26 4.2.4 Generation of loads combinations.......................................................... 28

4.3 SEISMIC FUNCTIONS DURING ELASTIC ANALYSIS.......................................... 29 4.4 SEISMIC FUNCTIONS IN THE ‘PLOT’-WINDOW ............................................... 31


2 Design of earthquake resistant structures

2.1 Introduction Structures in seismic regions shall be designed and constructed in such a way that they will be able to withstand seismic actions without a risk of local or general collapse with an adequate degree of reliability, thus retaining its structural integrity and a residual load bearing capacity after the seismic event. The design seismic action is generally selected on the basis of a chosen return period and need not coincide with the event of maximum intensity that may occur at a given site.

The above mentioned “design seismic action” provides the basis of the design analysis methods described in this manual (and implemented in PowerFrame). In order to satisfy the above fundamental requirements, the following limit states shall be checked:

� Ultimate limit states:

The structural system shall be verified as having sufficient resistance and ductility. The resistance and ductility to be assigned to the structure are related to the extent to which its non-linear response is to be exploited. In operational terms, such balance between resistance and ductility is characterized by the values of the behaviour factor q:

o q=1 for non-dissipative structures. No account is taken of any hysteretic energy dissipation during the seismic event. Displacements and internal forces can be obtained through a linear elastic analysis of the structure subject to the design seismic action.

o for dissipative structures the behaviour factor q is taken greater than 1, accounting for the hysteretic energy dissipation that occurs in specifically designed zones called dissipative zones or critical regions. This capacity of structural systems to resist seismic actions in the non-linear range generally permits their design for forces smaller than those corresponding to a linear elastic response. To avoid however explicit non-linear structural analysis in design, the energy dissipation capacity of the structure is taken into account by performing the following analysis steps:

� based on the seismic design spectrum, reaction forces and internal forces are calculated using linear elastic analysis methods


� next the displacements, calculated using a linear elastic analysis method, are multiplied by the behaviour factor q.

� Serviceability limit states:

An adequate degree of reliability against unacceptable damage shall be ensured by satisfying specific deformation limits defined in the relevant standards. Again, the structural deformations can be calculated using a linear elastic analysis method, provided the non-linear structural behaviour is accounted for through the use of the behaviour factor q.

More detailed information, complementary to this reference manual, can be found in the Eurocode 8 standard document and in below reference book concerning the French standard PS92:

Règles de construction parasismique – Règles applicables aux bâtiments – PS 92 / Edition Eyrolles, 1996, ISBN 2-212-10015-9.

Principles and application rules as explained in this reference, and as implemented in PowerFrame, are applicable only to buildings.

2.2 Seismic zones For the purpose of seismic design analysis, national territories are subdivided by the National Authorities into seismic zones, depending on the local hazard. By definition, the hazard within each zone can be assumed to be constant. The hazard is described in terms of a single parameter, i.e. the value ag of the effective peak ground acceleration in rock or firm soil, henceforth called ‘design ground acceleration”. The design ground acceleration, chosen by the National Authorities for each seismic zone, corresponds to a reference return period of 475 years.

In the French standard PS92, the design ground acceleration is referred to as the nominal acceleration aN. Values for aN are specified not only as a function of the seismic zone (0, Ia, Ib, II & III), but also as a function of the so-called hazard class:

� Class A: buildings of minor importance for public safety, eg. agricultural buildings, etc

� Class B: “ordinary” buildings, of which the collapse during earthquakes establishes a normal risks for the inhabitants


� Class C: buildings whose seismic resistance is of importance in view of the civil and economic consequences associated with a collapse, eg. public buildings, …

� Class D: buildings whose integrity during earthquakes is of vital importance for civil protection, eg. hospitals, fire stations, power plants, …

2.3 Subsoil conditions The influence of local ground conditions on the seismic action shall generally be accounted for by considering the appropriate subsoil classification. Where-as Eurocode 8 specifies 3 classes (A, B and C – please refer to the EC8 standard for a detailed description), the French standard PS92 foresees in a so-called building site classification (S0, S1, S2 & S3). Despite the fact that Eurocode 8 and PS92 use different terminology and different detail criteria, both standards can be said to apply the same concept to account for the impact of subsoil conditions on the intensity of a seismic event.

2.4 Basic representation of the seismic action

2.4.1 General Within the scope of both afore mentioned seismic standards, the earthquake motion at a given point of the surface of a structure is generally represented by the so-called “elastic response spectrum”. Such a structure is considered to be subject to a uniform displacement applied to the base support. It is thus implicitly assumed that all support points are subject to the same uniform excitation. If this assumption cannot be reasonably made, a so-called spatial model of the seismic action shall be used. Spatial models for seismic action are not part of the seismic design analysis implementation within PowerFrame.

The horizontal seismic action is described by two orthogonal components considered as independent and represented by the same elastic response spectrum. Unless specific studies indicate otherwise, the vertical component of the seismic action should be represented by the response spectrum as defined for the horizontal seismic action, but with (considerably) reduced ordinates.

Application of the above described elastic response spectrum assumes a linear elastic behaviour of the structure subject to the seismic action.


However, in reality, most structures will rather show a non-linear behaviour in such circumstances, as a considerable amount of energy will be dissipated due to the ductile behaviour of the structural elements and connections. Thanks to this non-linear behaviour, such structures, subject to a seismic action, can be designed for internal forces smaller than those corresponding to a linear elastic response.

Does this imply that a seismic design analysis of building structures must necessarily be built on a non-linear type of analysis? An important question, as non-linear analyses imply a number of difficulties in terms of analysis cost and complexity. Fortunately, the seismic design Standards allow for the possibility to replace a complex non-linear analysis by a linear elastic analysis, provided that:

� the analysis is based on the use of the so-called seismic design spectrum which is derived from the elastic response spectrum through the introduction of a behaviour factor q. To a large extent, this comes down to dividing the elastic response spectrum by a factor q to obtain the seismic design spectrum.

� the internal forces are calculated based on the seismic design spectrum as a result of the excitation of the structure.

� the displacements are also calculated based on the seismic design spectrum, but are subsequently multiplied by the behaviour factor q.

Thus, the behaviour factor q plays a crucial role in seismic design analyses based on Eurocode 8 or PS92. This behaviour factor will increase proportionally with the capacity of the structure to dissipate energy during a seismic action (through local plastic behaviour, for example). Both Standards provide explicit values for the behaviour factor q for steel, concrete and timber building structures. Reference is made to the appropriate standards documents for more precise information on realistic and acceptable q-values for seismic design analysis.

2.4.2 Seismic design spectrum for linear analysis

2.4.2.1 Use of the design spectrum Without going into further details on the application of seismic design spectra, it may be quite helpful to, at least, describe its basic principles.

A seismic design spectrum can essentially be considered as a peak response envelope of a building structure that is subject at its base support to a seismic action with design ground acceleration ag (or aN). Whereas the design ground acceleration is applicable to rock or firm soil, the seismic design spectrum contains further parameters taking into account:


� the subsoil characteristics, through the subsoil classification (EC8) or the building site classification (PS92)

� the non-linear behaviour of the building structure, through the behaviour factor q

� the linear dynamic behaviour of the building structure, through its vibration periods. Indeed:

o if, for instance, the first vibration period T1 of a building structure is known, then the ordinate of the seismic design spectrum at abscissa T1 is a measure for the structural displacements and internal forces if the corresponding eigenmode is effectively excited by the seismic action

o in reality, a large range of vibration periods Ti and corresponding eigenmodes can be calculated for any building structure. For each eigenmode that is effectively excited by the seismic action, the structural response can be obtained through the use of the seismic design spectrum. In a next step, the response values corresponding to the individual eigenmodes should be combined in an appropriate manner to calculate the total structural response induced by the seismic action.

o this analysis flow is mostly referred to as multi-modal response analysis and is explained, in more detail, in the sections 3.2 through 3.5 of this reference manual.

2.4.2.2 The design spectrum according to Eurocode 8

The normalized design spectrum, relative to the acceleration of gravity g, is given by the formulae below:

0 � T � TB ��

��

��

��

��

��

−−−−⋅⋅⋅⋅++++⋅⋅⋅⋅⋅⋅⋅⋅==== 1q

0

BTT1S)T(dS

ββββαααα

TB � T � TC q

0S)T(dSββββ

αααα ⋅⋅⋅⋅⋅⋅⋅⋅====

TC � T � TD ααααββββ

αααα 20,0

1dk

TCT

q0S)T(dS ≥≥≥≥⋅⋅⋅⋅⋅⋅⋅⋅====

��

��


TD � T ααααββββ

αααα 20,0

2dk

TDT

1dk

DTCT

q0S)T(dS ≥≥≥≥⋅⋅⋅⋅⋅⋅⋅⋅====

��

��

��

��

where:

Sd(T) seismic design spectrum, normalized relative to the acceleration of gravity g

T vibration period of the building structure (refer to 2.4.2.1)

� normalized design ground acceleration ( = ag / g )

�0 spectral acceleration amplification factor, function of subsoil classification

S soil parameter, function of subsoil classification

q behaviour factor, function of material and type of building structure

TB, TC vibration period limits of constant spectral acceleration branch

TD vibration period value defining the beginning of the constant displacement range of the spectrum

kd1, kd2 exponents which influence the shape of the spectrum for a vibration period greater than TC, TD

Reference is made to the appropriate standards documents for more precise information on the appropriate values for the parameters above.

2.4.2.3 The design spectrum according to PS92

The normalized design spectrum, relative to the acceleration of gravity g, is given by the formulae below:

0 � T � TC q

MR)T(dS ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅==== ττττρρρραααα

TC � T � TD 1dk

TCT

qMR

)T(dS��

��

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅==== ττττρρρραααα


TD � T 2dk

TDT

1dk

DTCT

qMR

)T(dS��

��

��

��

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅==== ττττρρρραααα

where:

Sd(T) seismic design spectrum, normalized relative to the acceleration of gravity g

T vibration period of the building structure (refer to 2.4.2.1)

� normalized design ground acceleration ( = aN / g )

RM spectral acceleration amplification factor, function of building site classification & valid for a viscous damping of 5% critical damping

� correction factor, function of viscous damping �

ττττ topographic amplification factor

q behaviour factor, function of material and type of building structure

TB, TC vibration period limits of constant spectral acceleration branch

TD vibration period value defining the beginning of the constant displacement range of the spectrum

kd1, kd2 exponents which influence the shape of the spectrum for a vibration period greater than TC, TD

Reference is made to the appropriate standards documents for more precise information on the appropriate values for the parameters above. However, it should be mentioned that the above formal description is slightly different from the one given in the actual PS92 documents. This is done on purpose, to pronounce more explicitly the similarity and consistency between the Eurocode 8 and PS92 approach.

2.4.3 Alternative representations of the seismic action

Next to the seismic design spectrum used with a linear elastic analysis of the building structure, alternative methods can be used for seismic design analysis. Three major approaches can be distinguished:

� power spectrum analysis: the seismic motion at a given point on the ground surface is represented as a random process, defined by a


power spectral density function. The power spectrum shall be consistent with the elastic response spectrum used for the basic definition of the seismic action as described before. This consistency shall be observed by the design Engineer.

� time-history analysis: the seismic motion may also be represented in terms of ground acceleration time-histories and related quantities (velocity and displacement). Depending on the nature of the application and on the information actually available, the description of the seismic motion can be made by using artificial accelerograms and recorded or simulated accelerograms.

� frequency domain analysis: the seismic action input is the same as in a time-history analysis, but with each accelerogram cast in the form of a Fourier series.

However, the seismic design spectrum method remains the (p)reference method, in combination with a multi-modal response analysis. The above methods may be used, but it shall always be demonstrated that all requirements are met for those methods to be applicable.

Therefore, seismic design analysis with PowerFrame is always based on the (p)reference method, which is generally applicable.

2.5 Ultimate limit state verifications Based on the seismic combinations described below, a verification of member resistance and stability shall be made, in addition to the verifications to be made for the fundamental combinations.

It should also be mentioned that the seismic design standards not only impose requirements in terms of member resistance and stability in the ultimate limit states, but also specify a number of additional requirements on the level of ductility, global equilibrium, horizontal displacements, … Reference is made to the appropriate standards documents for more precise information on those requirements.

2.5.1 Eurocode 8 Ultimate limit state verifications must be performed for the accidental (or seismic) type of combination given below, in addition to the fundamental combination:

ΣΣΣΣj>=1 Gk,j + �1 AEd + ΣΣΣΣi>=1 ψ ψ ψ ψ2,i Qk,i


where

…k: characteristic value

G: permanent action

AEd: design value of seismic action, based on seismic design spectrum

�1: importance factor. This factor varies between 0.8 and 1.4 as a function of the importance category to which the building structure belongs

• Category IV, �1=0.8 : buildings of minor importance for public safety, eg. agricultural buildings, etc

• Category III, �1=1.0: “ordinary” buildings, of which the collapse during earthquakes establishes a normal risk for the inhabitants

• Category II, �1=1.2: buildings whose seismic resistance is of importance in view of the civil and economic consequences associated with a collapse, eg. public buildings, …

• Category I, �1=1.4: buildings whose integrity during earthquakes is of vital importance for civil protection and national defense, eg. hospitals, fire stations, power plants,

Remarks: in case PS92 is used, the design ground acceleration already takes into account the “importance” of a building structure in terms of public safety. In case Eurocode 8 is used, this is not the case, but the “importance” is accounted for through the introduction of �1.

Q: variable action

ψ2: combination coefficient for quasi-permanent value of the variable action

2.5.2 PS92 Ultimate limit state verifications must be performed for the accidental (or seismic) type of combination given below, in addition to the fundamental combination:

ΣΣΣΣj>=1 Gk,j + E + ψ ψ ψ ψ1,1 Qk,1 + ΣΣΣΣi>=2 ψ ψ ψ ψ2,i Qk,i


where


G: permanent action

E: design value of seismic action, based on seismic design spectrum

Q: variable action

Q1: most unfavourable variable action

ψ1: combination coefficient for frequent value of variable action

ψ2: combination coefficient for quasi-permanent value of variable action

2.6 Serviceability limit state verifications

Serviceability limit state verifications must be performed with respect to the horizontal deflections induced by the seismic action. Eurocode 8 specifies limits on the so-called “interstorey drift” – being the average relative horizontal displacement at each level of the building structure. In addition, PS92 also imposes limits on total lateral deflection of the complete building structure.

2.7 Design gravity loads From the above, ( section 2.4.2.1 ) it is clear that the design analysis of building structures subject to a seismic event, requires a modal analysis. Such modal analysis calculates eigenfrequencies and corresponding eigenmodes. The results of this analysis strongly depend on the amount of gravity loads that are considered during the calculations. Such gravity loads are of course not only related to the structure’s self-weight, but also to the permanent or dead loads and (to a lesser extent) to the variable or live loads.

Then the question arises how permanent and variable loads are to be considered to derive the design gravity loads to be used during the modal analysis and the subsequent seismic design analysis. Both seismic design standards discussed in this reference manual provide the necessary specifications.


2.7.1 Eurocode 8 To derive design gravity loads to be used for modal analysis of the building structure, gravity loads related to following combination of loads must be considered:

ΣΣΣΣj>=1 Gk,j + ΣΣΣΣi>=1 ψ ψ ψ ψE,i Qk,i

where


G: permanent action

Q: variable action

ψE: � . ψ2

�: correlation coefficient accounting for the degree to which different stories are occupied simultaneously. Values for � need to be taken from the seismic standard


2.7.2 P92 To derive design gravity loads to be used for modal analysis of the building structure, gravity loads related to following combination of loads must be considered:

ΣΣΣΣj>=1 Gk,j + ψψψψE1,1 Qk,1 + ΣΣΣΣi>=2 ψ ψ ψ ψE2,i Qk,i

where


G: permanent action

Q: variable action

Q1: most unfavourable variable action

ψEx: � . ψx (x = 1, 2)

�: correlation coefficient accounting for the degree to which different stories are occupied simultaneously. Values for � need to be taken from the seismic standard


ψ1: combination coefficient for frequent value of variable action



3 Seismic analysis 3.1 Introduction Let us consider the example of a column, fixed at its base where it is subjected to an horizontal acceleration. It is assumed that this horizontal acceleration varies as a function of time such that vibrations are induced in the column. It can be shown that these column vibrations can be described as a linear superposition of the column’s eigenmodes, in which the eigenmodes react completely independent of each other to the imposed base excitation:

= � * + � * + γγγγ * + � * + …

The combination factors �, �, γγγγ , �, … are a priori unknown, but can be calculated as a function of the imposed base acceleration and as a function of the structure’s damping properties. The eigenmodes, on the other hand, are independent of the imposed base excitation and are mostly also independent of the damping properties of the structure. They can be calculated starting from the structure’s stiffness and mass properties. This calculation is usually referred to as modal analysis.

In theory, an infinite number of eigenmodes can be calculated for any building structure. In practice however, it will be largely sufficient to consider only the N lowest eigenmodes for further use during a dynamic analysis. This explains the advantages of such an approach as compared to a direct dynamic analysis, during which the structure’s response to the imposed excitation is obtained through a direct integration of the equations of movement as a function of time:

� using modal analysis techniques, the N lowest eigenfrequencies and eigenmodes are calculated, independent of the imposed excitation

� next, the structural response to the imposed excitation is calculated using a multi-modal response analysis as a combination (or superposition) of the N lowest eigenmodes, having the benefit that


only a limited number ( N ) of equations must be solved. Furthermore, the previuously calculated eigenmodes can be re-used when other excitation types are considered.

3.2 Modal analysis The objective of a modal analysis is to calculate the N lowest eigenfrequencies fi (expressed in Hertz) of a structure, along with the corresponding eigenmodes �i.

A number of remarks:

� very often, different terminology is used for eigenfrequency:

o eigenperiod Ti, being the inverse of the eigenfrequency fi (expressed in seconds).

o eigenpulsation �i = 2�* fi

� eigenmodes �i cannot be interpreted in absolute terms: only the eigenmode shape can be interpreted, not its amplitude – at least not without additional information

� the additional information needed for absolute interpretation is the so-called modal mass mi corresponding to an eigenmode �i. Modal mass can most easily be explained as the fraction of the total mass of the structure which effectively participates in the displacements described by the eigenmode �i. For example, it can be derived from the figure below that for the first eigenmode �1 all nodes of the column always move in-phase and that consequently, the distributed mass of the structure will globally move in-phase. Only the amplitude of movement of the distributed mass will be variable (zero at the column basis, maximum at top of column). With the second eigenmode �2 , not all nodes will move in-phase any more. As a consequence, part of the distributed mass will move in one sense while the remaining part of the distributed mass will move in the opposite sense. As a whole, less mass is effectively “mobilized” by this mode shape, resulting in a lower modal mass for the second eigenmode (assuming the maximum displacement of both eigenmodes �1 and �2 to be equal). In general, it can indeed be stated that with increasing eigenfrequency fi, the wave length of the eigenmodes �i will decrease and modal mass mi will also decrease (again assuming that maximum displacement of all eigenmodes �i is equal).


�1 �2 �3 �4

The above explanations allow to understand that the contribution of the higher eigenmodes will decrease quickly in the context of a dynamic analysis, as an increasingly smaller mass will be mobilized by the eigenmodes with increasing eigenfrequency.

Taking into account all those considerations, the introduction of this section should be modified as follows:

the objective of a modal analysis is to calculate the N lowest eigenfrequencies fi (expressed in Hertz) of a structure, along with the corresponding eigenmodes �i and modal masses mi

The knowledge of the eigenmodes �i and corresponding modal masses mi allows for an unambiguous and absolute interpretation of the response of a building structure subjected to a dynamic type of excitation.

In case a modal analysis is performed as the basis for a seismic analysis, mostly the concept of effective modal mass is being used. The effective modal mass differs from modal mass in the sense that effective modal masses take into account the type of excitation that is applied to the structure. Effective modal masses can thus only be calculated once a seismic action has been defined, more in particular once the directions in which a seismic action is to be considered, are known.

The importance of effective modal massa relates to the fact that the sum of all effective modal masses (in case an infinite number of eigenmodes is considered) equals the total mass of the structure. Furthermore, both Eurocode 8 and PS92 specify the criterium below to determine the number of eigenmodes N that should at least be used for seismic analysis:

the number of eigenmodes N to be used for a seismic analysis can be limited to n, provided the sum of effective modal masses for the n lowest eigenmodes equals at least 90% of the total mass of the building structure, for each direction in which a seismic action is considered


For additional criteria with respect to the number of eigenmodes N that must be taken into account for a seismic analysis, reference is made to the appropriate standards documents.

Structures which have a high degree of symmetry can have 2 (or more) eigenmodes at the same eigenfrequency. Such modes are referred to as double modes (or triple modes, …). According to modal analysis theory, the corresponding mode shapes are mutually orthogonal. In practice (definitely in view of a correct multi-modal response analysis) it is crucial that eigenvalue solver of an analysis program is capable to calculate such double modes while respecting their orthogonality properties. The PowerFrame analysis core is perfectly capable to deal with this situation and thus guarantees at any time a correct basis for a multi-modal response analysis.

3.3 Modal response analysis The objective of a modal response analysis is to calculate the response of a structure in case 1 specific eigenmode �i (with corresponding eigenfrequency fi and modal mass mi) is excited by the applied dynamic load.

It can be shown that such an eigenmode �i actually behaves as an equivalent mass-spring-system with (modal) mass mi and (modal) stiffness ki. The eigenfrequency fi of such a mass-spring-system is given by

i

iii m

k21

2f ⋅⋅⋅⋅========

ππππππππωωωω

In case such a mass-spring-system is subjected at its base to a seismic design spectrum Sd(T) in a specific direction, it can be shown that this system will undergo a peak displacement given by the relationship below:

i2i

id

i

ii

)T(Sgmlv ΦΦΦΦ

ωωωω⋅⋅⋅⋅��

��

�� ⋅⋅⋅⋅⋅⋅⋅⋅��

��

��====

where

vi peak displacement vector

Ti eigenperiod of the eigenmode �i ( = 1 / fi )

li effective modal mass corresponding to the eigenmode �i, for the direction in which the seismic design spectrum is applied


Once the peak displacement is known, the corresponding peak values for the internal forces (moments, shear forces, compression or tensile forces, …) can easily be derived by making use of the stiffness characteristics of all structural elements.

3.4 Multi-modal response analysis The objective of a multi-modal response analysis is to calculate the response of a structure in case several eigenmodes �i (with corresponding eigenfrequencies fi and modal masses mi) are excited by the applied dynamic load.

In case a structure is subjected at its base to a seismic design spectrum Sd(T) in a specific direction, several eigenmodes �i will, in principle, be excited. For each eigenmode, the peak displacement vector vi can be calculated as explained in section 3.3. The total peak displacement vector can then be obtained through a quadratic combination of the contributions of all N eigenmodes �i :

====

±±±±====N

1i

2ivv

A completely similar combination formula can be applied to calculate the peak values for the internal forces, based on the contributions of all N eigenmodes:

====

±±±±====N

1i

2iEE

Is should be noted that ±±±± in the above formulae accounts for the fact that, although the direction in which the seismic design spectrum is applied is fully determined, this design spectrum always needs to be considered in positive & negative sense along this direction.

The number of eigenmodes N to be considered is not known “a priori”. As has already been mentioned in section 3.2, the required number of eigenmodes N can be determined on the basis of the effective modal masses. Indeed, it is known that the sum of all effective modal masses (in case an infinite number of eigenmodes is considered) equals the total mass


of the structure. Furthermore, both Eurocode 8 and PS92 specify the criterium below to determine the number of eigenmodes N that should at least be used for seismic analysis:

the number of eigenmodes N to be used for a seismic analysis can be limited to n, provided the sum of effective modal masses for the n lowest eigenmodes equals at least 90% of the total mass of the building structure, for each direction in which a seismic action is considered

For additional criteria with respect to the number of eigenmodes N that must be taken into account for a seismic analysis, reference is made to the appropriate standards documents.

The question remains however what needs to be done in case the sum of effective modal masses does not reach the above described threshold of 90% for the N available eigenmodes, for at least one principal direction of the seismic action. In this case, 2 possibilities remain open:

• an entirely new modal analysis is performed, increasing the number N of eigenmodes to be calculated. Of course, such an analysis requires computing time which increases stongly as a function of the requested number eigenmodes. This may make such an approach rather uneconomical.

• alternatively, quasi-static correction techniques can be used with the available set of eigenmodes. With this approach, it is still assumed that the eigenfrequencies of the unavailable mode shapes are sufficiently high with respect to the most relevant frequencies covered by the seismic design spectrum. In this case, it can readily be assumed that such eigenmodes respond statically (and not dynamically) to the seismic excitation. The static response of the unavailable eigenmodes can be derived relatively easy using the set of calculated eigenmodes and considering a load distribution that is the static equivalent to the seismic excitation. This static response is then applied as a correction term to the results of the previously performed multi-modal response analysis.

During seismic design analysis, PowerFrame will in all cases apply a quasi-static correction (no intervention is required from the user), which ensures maximum accuracy of the analysis results in all cases (even when the sum of effective modal mass is below 90% of total mass for one or more principal directions of the seismic action).


3.5 Seismic response analysis The objective of a seismic response analysis is to calculate the response of a structure subjected simultaneously to a design spectrum in a number of mutually orthogonal directions.

According to the seismic standards EC8 and PS92, each seismic action must consider design spectra applied in 2 orthogonal horizontal directions (further referred to as X’ and Z’). For both directions, the same design spectrum is to be used, but the directions in which the design spectrum is to be applied are not “a priori” known. In principle, those directions must be defined such that the corresponding seismic action has the most unfavourable effect on the building structure.

As part of a seismic response analysis, two multi-modal response analyses will be performed: one for each horizontal direction (X’, Z’). This delivers following results (based on the principes as described in 3.4):

� EX’ - peak values of displacement & internal forces for a seismic design spectrum applied along X’

� EZ’ - peak values of displacement & internal forces for a seismic design spectrum applied along Z’

The results of both multi-modal response analyses can then be combined as follows to

� EX’ “+” 0.30 EY’ “+” 0.30 EZ’

� 0.30 EX’ “+” EY’ “+” 0.30 EZ’

� 0.30 EX’ “+” 0.30 EY’ “+” EZ’

The “+” in those expressions must be interpreted as “to be combined with”, such that the above expressions effectively represent response envelopes describing the effects of a seismic action along X’ and Z’.


4 Seismic design analysis using PowerFrame

4.1 Introduction This section of the manual on seismic design analysis describes in more detail the practical use of PowerFrame’s seismic design capabilities. As a starting point, it can be said that the seismic design analysis is an integral part of the entire structural design analysis. In other words, seismic design analysis does not represent an extra step to be taken at the end of the structural design analysis, but is completely embedded in the full design analysis process.

Therefore, seismic design analysis is part of the standard elastic analysis procedure within PowerFrame. From a user point of view, there will be little difference between elastic analyses with and without seismic actions. The major differences related to the introduction of seismic actions, are:

� in the ‘Loads’-window:

o the explicit definition of a loads group in which the design gravity loads are managed (as a function of the permanent loads and the variable loads), as they are to be used for any type of dynamic analysis (in particular for any type of seimic analysis)

o the explicit definition of a seismic loads group

o the generation of seismic loads combinations for the ultimate limit states, next to the fundamental loads combinations

� during the elastic analysis:

o when the elastic analysis is launched, a multi-modal reponse analysis will automatically be performed. The results of this analysis will then be combined with the effects of the static loads, consistent with the definition of all loads combinations

� in the ‘Plot’-window

o of course, you now have access to all analysis results for seismic loads combinations. Apart from this, no major differences will be observed in comparison with a traditional static type of analysis. Code checks (for steel, concrete or timber) will of course take into account all available combinations:

� ULS FC (fundamental combinations)

� ULS SC (seismic combinations)


� SLS QP (quasi-permanent combinations)

� SLS RC (rare combinations)

4.2 Seismic functions in the ‘Loads’-window

4.2.1 Load groups for seismic analysis Through the icon “Load factor” in the icon toolbar of the

‘Loads’-window, 2 load groups can be defined that are specific to a modal and/or seismic analysis:

� Gravity loads for vibration analysis: this group includes all gravity loads which must be taken into account during any type of dynamic analysis. Those design gravity loads can quickly be derived from the permanent loads and the variable loads that have defined in several load groups,

through the use of the icon . At any time, it is also possible to add discrete masses (or gravity loads) manually at nodes by simply selecting those nodes and

then use the icon .

� Seismic: this group contains the definition of the seismic action. The definition itself can be done through the icon

. At any time, the load group Seismic will include a visualization of the design gravity loads to be used for vibration analysis. This information always remains consistent between both load groups Gravity loads for vibration analysis and Seismic.

The load groups described above can be configured using the dialogue window shown below. It should be noted that the icon shown in the column at the right, will automatically adapt itself to the selected type of load group. Where-as it is possible to change this icon manually in case of static loads (to define how the loads that are part of the load group should be applied to the structure), this is not possible for the gravity loads and the seismic load group.

Also remember to select the appropriate seismic design standard in the dialogue window shown below. If Eurocode 8 is selected, then the definition of the seismic load group requires the specification of one of 4 categories (I


up to IV, corresponding with the importance category of the building structure as described in section 2.5.1). Depending on the category that is selected, the corresponding importance factor is automatically assigned to the safety coefficients in the table below.

Final remark: in case any of the French design standards is selected (either CM66 or BAEL91), only the French seismic design standard PS92 can be used.

4.2.2 Definition of design gravity loads Through the icon of the icon toolbar, the dialogue window as shown below is launched. This dialogue window enables the definition of the correlation coefficients to be used for the calculation of design gravity loads (refer to 2.7 for more information on correlation coefficients). For each load group, a value for � can be defined manually, based on the specifications provided by the selected design standard.

Based on the correlation coefficients, PowerFrame will automatically calculate design gravity loads to be used during any type of dynamic analysis (modal, seismic, …) based on the static type of loads that are part of the


selected load groups. Those design gravity loads are visualized in the ‘Loads’-window on the geometry of the analysis model, in case either “Gravity loads for vibration analysis” or “Seismic” is selected as the active load group.

4.2.3 Definition of seismic load group The icon enables the user to completely define the seismic design action through the dialogue window shown below. Depending on the seismic design standard that has been selected, a number of parameters must be defined to characterize the seismic event.

In case Eurocode 8 has been selected, following data are required:

� sub-soil class (A, B or C)

� behaviour factor q. Appropriate values for q are given in the EC8 documents, as a function of the material (steel, concrete, timber) and as a function of the type of construction

� ground acceleration ag for the seismic zone that is considered

In case PS92 has been selected, following data are required:

� building site class (S0, S1, S2 or S3)


� behaviour factor q. Appropriate values for q are given in the PS92 documents, as a function of the material (steel, concrete, timber) and as a function of the type of construction

� relative damping factor �, expressed as a % of critical damping. By default, a value of 5% is considered. The PS92 document provides useful indications for realistic values for different types of structures.

� nominal acceleration aN for the seismic zone that is considered

� topographic amplification factor ττττ , to be determined from the PS92 document

According to the seismic standards EC8 and PS92, each seismic action must consider design spectra applied in 2 orthogonal horizontal directions (further referred to as X’ and Z’). For both directions, the same design spectrum is to be used, but the directions in which the design spectrum is to applied are not “a priori” known. In principle, those directions must be defined such that the corresponding seismic action has the most unfavourable effect on the building structure. Those directions are chosen by the parameter “Main direction” in the above dialogue window.

Once the definition of the seismic action has been completed and confirmed, the visualization shown below will become available in the ‘Loads’-window. This visualization includes following data:


� the directions in which seismic actions will be applied (represented by red arrows for the horizontal components, and by a green arrow for the vertical component).

� the design gravity loads to be used for modal analysis (and subsequent seismic analysis)

4.2.4 Generation of loads combinations It should not be overlooked that with any design analysis that includes seismic actions, it is necessary to generate fundamental combinations as well as seismic combinations for the ultimate limit states (see dialogue window below).


4.3 Seismic functions during elastic analysis

A design analysis in which seismic actions should be accounted for, is launched in the same way as a design analysis related to static loads only. If however, the user is only interested in the eigenfrequencies and eigenmodes of a structure, then he should use the tab-page “Modal analysis” rather than “Elastic analysis” in the dialogue window below.


When an elastic analysis is launched after seismic combinations have been generated, PowerFrame will, by itself, automatically take the necessary extra steps that are required in this situation. Indeed, as a first step PowerFrame will perform a multi-modal analysis in order to calculate the response of the structure corresponding to the seismic load group. Next, this response will be included in the appropriate way in all ULS – SC combinations.

If the user decides to perform a second-order analysis or takes into account the effects of global structural imperfections, then those options will only be applicable to the static type of loads and to their contribution in the seismic load combinations. The seismic response will always be evaluated according to a linear first-order theory without consideration of structural imperfections, and will be combined with the afore-mentioned results for static type of loads.

The multi-modal analysis will start with a calculation of the N lowest eigenfrequencies of the building structure using the subspace iteration method. As a user, you will specify the number N along with the maximum number of iterations to be used for the calculation of eigenfrequencies and corresponding eigenmodes. Note that the number of eigenfrequencies N is limited to an absolute maximum of 40. Depending on the effective number of DOFs (degrees of freedom) in the analysis model, a stricter limit may be applicable:

� in case the number of DOFs (#dof) is lower than 16, the maximum number of eigenfrequencies is limited (#dof)/2.

� in case the number of DOFs (#dof) is larger than or equal to 16, the maximum number of eigenfrequencies is limited (#dof – 8), with an absolute maximum of 40.

Up front however, it is not known how many eigenfrequencies and eigenmodes are really needed for a high-quality seismic analysis. Therefore, it is recommended to start the analyses with a relatively small number of eigenmodes and then evaluate the sum of the effective modal masses corresponding to the calculated eigenmodes, for both directions defined with the seismic action. In case this sum exceeds 90% of the total mass of the structure for both directions (to be derived from the table below which is


presented by PowerFrame upon completion of the modal analysis), the specified number of eigenmodes N is sufficient for a seismic analysis that complies with the requirements of the seismic standards.

In case the sum of the effective modal masses does not comply with the above criterion, the number of eigenfrequencies N to be calculated should be increased until the target value of 90% is achieved. If for one or more principal directions of the seismic event this sum remains well below 90% for a relatively high number of eigenmodes N, then PowerFrame will still ensure maximum analysis accuracy through the automatic automatische application of a quasi-static correction.

4.4 Seismic functions in the ‘Plot’-window

The icon toolbar of the ‘Plot’-window enables you to visualize all familiar types of analysis results (displacements, internal forces, stresses, reaction forces) for all types of load groups and load combinations, including the seismic ones.


The following items should not be overlooked during the interpretation of the results:

� results for a seismic action are always the result of a linear elastic analysis in which the non-linear behaviour of the building structure is accounted for through the introduction of a behaviour factor q.

� results for a seismic action are always a combination of independent design spectra applied to mutually orthogonal (horizontal) directions. Furthermore, the seismic action is considered in positive and negative sense for each direction. Results for a seismic action are thus always presented as envelopes.

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