structural analysis for seismic action

8/10/2019 Structural Analysis for Seismic Action

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ESDEP WG 17

SEISMIC DESIGN

Lecture 17.4: Structural Analysis for

Seismic ActionsOBJECTIVE/SCOPE

To give an overview of the methods used for the analysis of structures under seismic actions.

PREREQUISITES

Basic knowledge of structural analysis and structural dynamics

RELATED LECTURES

Lecture 17.2 : Introduction to Seismic Design - Seismic Hazard and Seismic Risk

Lecture 17.3 : The Cyclic Behaviour of Steel Elements and Connections

SUMMARY

The lecture briefly presents the methods stipulated by modern design codes for the analysis ofstructures under seismic actions. Time-domain methods are briefly described and the scope oftheir application is specified. Emphasis is given to the response spectrum method as the standard

procedure proposed by, for example, Eurocode 8 [1]. In addition, a simplified response spectrummethod for regular buildings is presented. Finally inelastic behaviour and its role in design underseismic actions is discussed.

1. INTRODUCTION

Several methods are available for the structural analysis of buildings and other civil engineeringworks under seismic actions. The differences between the methods lie (a) in the way theyincorporate the seismic input and (b) in the idealization of the structure. All methods of analysismust serve the current design philosophy for seismic actions which requires that a structure mustnot collapse and must retain its structural integrity under the so-called "strong" earthquake. Thestructure also must be protected against damage and limitations of use under the so-called"moderate" earthquake. To avoid collapse, the structure is allowed to develop plastic zones inwhich seismic energy is dissipated.
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Details of the basic requirements of seismic behaviour of structures, and the criteria needed forensuring compliance with these requirements, can be found in all modern seismic design codes,e.g. Eurocode 8 [1].

2. DIRECT METHODS OF DYNAMIC ANALYSIS (TIMEINTEGRATION)

Due to the dynamic nature of seismic excitation, the actual displacements and stresses developedin a structure are time dependent, i.e. they are functions of time (t). To analyze a structure underdynamic loads, efficient methods have been developed that discretize and solve the model of thestructure on the basis of the Finite Element Method. Within this framework there exist methodsthat can perform a linear or non-linear analysis, i.e. elastic, small deformation, or inelastic, largedeformation analysis for a given seismic excitation, expressed in the form of an accelerograma(t). The cost of such analysis is generally high, while the results correspond to a particularexcitation and, as such do not offer a reliable basis for design. To increase the reliability of the

method, a set of artificial accelerograms that represent the seismicity of a particular region isusually generated. This procedure, however, renders the method very expensive.

Eurocode 8 [1] considers the use of time domain dynamic analysis, i.e. a direct dynamic analysis performed by numerical integration of the differential equations of motion. It stipulatesconditions for the use of artificially generated accelerograms and di scusses the overall reliabilityof the method. The reliability must be at least the same as that obtained by the standard

procedure of the Code which is the response spectrum method. Although the direct dynamicmethods can perform a close-to-reality analysis, this approach is justified and can be employedeffectively only for large and complex structures. It is used where no previous experience of thestructural behaviour exists, or for detailed evaluation of the response of existing structures under

specific earthquakes.The cost of an analysis based on the finite element method can be kept reasonable by using onlyline elements and by avoiding the use of surface elements. The mass of the structure of buildingsis mainly concentrated at the floor levels. This distribution permits the treatment of all themasses of the structure as lumped at the floor levels in dynamic analysis. The dynamic degreesof freedom for which inertia forces are developed can then be reduced to a reasonable number.All the remaining kinematical degrees of freedom control the statics of the structure, and canthen be expressed in terms of the dynamic degrees of freedom. In this way the number ofdifferential equations that express the dynamic response of the system can be reduced to a smallnumber, leading to reasonable and acceptable solutions.

3. RESPONSE SPECTRUM METHOD OF ANALYSIS

The time dependent solutions discussed above express the dynamic response of the structure dueto a particular earthquake given in the form of an accelerogram. They do not offer the requiredinformation for design however, because one particular earthquake cannot be representative ofthe seismicity of the area under consideration.


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In order to define an envelope of different earthquakes and also to eliminate the factor of time,the concept of the response spectrum was developed. The response spectrum provides therequired information for design purposes and, at the same time, simplifies the analysis byreducing the problem to a static problem of the estimated maximum responses. The responsespectrum is defined, on a single degree of freedom system of varying frequency excited by a

specific earthquake, as the maximum response of the system, ignoring the particular time of itsoccurrence. If the response is the displacement of the system then the displacement spectrum isformed. If the response is the velocity or the acceleration, the velocity or acceleration spectra aredeveloped. The acceleration response spectrum is of primary interest in earthquake engineering.More details about earthquake response spectra are given in Lecture 17.2 .

The response spectrum method of analysis is the standard design procedure of modern seismicdesign codes, e.g. Eurocode 8. It aims to give directly the maximum effects of the earthquake inthe various elements of the structure.

The general method, called also the multi-modal method, consists of computing the various

modes of vibration of the structure and the magnitude of the maximum response in each modewith reference to a response spectrum. A rule is then used to combine the responses of thedifferent modes. For this reason the method is also known as the superposition of modalresponses method, although the same name is used for linear dynamic analysis where the modeshapes are used to decouple the differential equations of motion and convert the n-degree offreedom coupled system to n-single degree of freedom systems. The combination rule willgenerally be a square root of the sum of squares (SRSS) of the various modal responses. Thiscombination rule must be applied to all computed quantities, i.e. bending moments, shear forces,normal forces and displacements. As a consequence, the resulting internal forces do not representan equilibrated set. Where the frequencies of a structure do not differ by more than 10%,different combination rules need to be employed. In Figure 1 the steps of such an analysis by

means of the response spectrum are briefly summarised.
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The response spectrum method is valid only for linear behaviour of a structure, i.e. only for anelastic analysis with small deformations. For this reason the term elastic response spectrum isgenerally used. However an equivalent method can be developed which results from comparativelinear and non-linear analyses. It uses a modified response spectrum such that the output internalforces from a linear analysis will be correlated with the non-linear ones. This modified spectrum

is referred to as the design response spectrum. It is derived from the elastic spectrum modified byfactors that take into account the influence of the non-linearity of the structural material, the soiland other damping characteristics. In Figure 2 the design response spectra to be used in theanalysis of structures, as given in Eurocode 8 [1], are shown schematically.


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The main advantage of using the design response spectrum is that the analysis is linear while theresults represent the non-linear response of the structure.

A more simplified procedure than the multi-modal method, is the so-called equivalent staticforce analysis, sometimes also called, e.g. in Eurocode 8 [1], the simplified dynamic analysis.

This method is a particular application of the design response spectrum method where one particular mode of vibration is predominant as compared to others. This is the case for regular buildings (regular stiffness and mass distribution over the height of the building according toEurocode rules, see Lecture 17.5 ). The system is accurately modelled by a single degree offreedom system. In essence the design spectrum method is reduced to one mode of vibration toexpress the dynamic behaviour of the system. Usually the first flexural mode shape is consideredas a primary mode of vibration which can be simplified further into a simple line. The equivalentstatic forces are computed as shown in Figure 3. A classical static analysis can then be performedunder the action of these equivalent static forces. The only prerequisite of the method is thefundamental period of vibration T of the structure. It needs to be calculated in order to find theappropriate design spectrum value (T), necessary to compute the base shear V. Alternatively, if

an accurate value of the period T is not available, the value of the fundamental period can becalculated approximately by using one of the recommended formulae.


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The equivalent static force method is an approximate method which is adequate for certain typesof structures and for the preliminary design of other structures. There may be cases where thismethod is not conservative because the contribution from higher modes of vibration may besignificant. For these cases a complete dynamic response spectrum analysis is advisable for thefinal design stage.


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In Table 1 a summary of the possible methods of structural analysis under seismic actions is presented. Moreover the following remarks can be made:

The effects of earthquake on a structure depend upon its stiffness and masscharacteristics. The forces induced in flexible structures (high fundamental period T) are

generally lower than those in stiffer structures. The effects of earthquake on a structure depend upon the distribution of the mass and thestiffness of the structure. Non-regular distribution involves the influence of morevibration modes on the response.

Simplified analysis methods, such as static equivalent force analysis, generally can beapplied to regular structures, but in some cases may give unsafe results.

Non-regular structures require more sophisticated analysis, such as the response spectrumor modal superposition method.

Large complex structures with special features of behaviour should be analysed by moreelaborate methods such as non-linear dynamic analysis.

The designer should always keep in mind that in all the above-mentioned methods of

analysis, many uncertainties have been rationalized. The control of the uncertaintiesrequires compliance with the rules of "good practice" mentioned in Lecture 17.5 . Theuncertainties relate to behaviour of the structural material under cycling loading,discrepancy of the earthquake characteristics, real damping factor, effects of soil-structure interaction etc.

It is clear from the above discussion that the design of an earthquake resistant structure is acomplex task which requires engineering judgement. It must be performed by experiencedengineers. The blind use of computer software as blackboxes may result in inadequate design.

4. INELASTIC BEHAVIOUR AND ITS ROLE IN DESIGN

The elastic design of an earthquake resistant structure leads to very expensive structures.Moreover it is not consistent with the current design philosophy which seeks to establishcontrolled dissipative zones in the structure where seismic energy can be dissipated by means ofductile hysteretic behaviour. The principal dissipative zones in steel structures are plastic hinges(in bending), sheared web panels and members under plastic tension (Figure 4).


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In Figure 5 the difference in energy dissipation between the elastic and inelastic concept is presented. The energy input E i of an earthquake is counterbalanced inside the structure by thefollowing sum of terms:

E i = E e + E d + E ye + E kin

where

Ee is the energy of elastic strain

Ed is the energy dissipated in a viscoelastic way

Eye is the energy dissipated by yielding

Ekin is the kinetic energy.


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To obtain a stable earthquake resistant structure, either the energy input is minimized by meansof special techniques, such as base isolation of the building, or the dissipative terms in the righthand side of the equation are increased. The term E ye must be increased as much as possible. Itshould be noted that by taking into account elastoplastic energy dissipation, a considerableweight reduction of the structure is achieved. In Figure 6 the moment rotation diagram of two

equivalent beam elements is considered from the point of view of energy dissipation. Theresisting moment M 1 required to resist an earthquake elastically, is 3 times greater than theresisting moment M 2 of the elastoplastic element with a ductility of 2. Expressed in terms ofweight, beam 2 is only equivalent to 0,6 of beam 1. Thus the ductile behaviour allows forsubstantial economy in the size of the elements of a structure. This economy is even moresubstantial since the local ductility can be higher than 2. In steel structures the value of localductility can be as high as 10.


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In order to design structures with dissipative behaviour by employing an elastic analysis which iseasy for the design office, certain rules have to be followed. They assure the safe formation of asmany as possible local dissipative zones, avoiding local failure mechanisms.

To approximate the results of a non-linear dynamic analysis by performing an elastic analysis,the conventional response spectrum method is modified by reducing the spectrum in some wayto account for the inelastic energy dissipation of the real structure under the earthquake action.


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This reduction is accomplished by using the structural behaviour factor q. It can generally bedefined as the ratio between the maximum accelerogram that a structure can withstand withoutfailure and the accelerogram for which yielding appears somewhere in the structure. Thedefinition is general and can be applied to different quantities of interest. In steel structures, oneway to establish the correlation between a conventional elastic analysis and the real inelastic

behaviour is as follows:

For a given structure under a specific earthquake action a(t), a series of computations of the non-linear dynamic response is performed by applying actions (t), where is a multiplier. Byincreasing the value of the following successive situations emerge (Figure 7) [2, 3]:


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values are such that all sections of the structure remain elastic. In these cases, if d is adisplacement that characterizes the deformation of the structure, e.g. storey drift, then dwill be proportional to .

The particular value of which corresponds to the phase where yield stress is reached in

one section of the structure is called e. In the next phase, the values are such that the real d's are smaller than the d's calculated

by the elastic analysis, i.e. supposing unlimited elastic behaviour, because of the energydissipation by yielding.

By increasing the values further, a max value is computed which corresponds to thesame elastic and inelastic displacement. This coincidence is due to the increasing role ofP- effects, which increase the displacements.

The behaviour factor q, is then defined as:

q = max / e

Thus the existence of a meeting point between the two forms of behaviour, allows a direct link between the linear and non-linear computations. The equivalence states that, for a givenaccelerogram a(t) and a known value of q, the usual linear analysis under the action a(t)/q and theusual checks on stresses, give the same safety level as the dynamic non-linear calculations underthe action of a(t). This equivalent is due to the counteraction of the yielding effect which reducesthe displacements, and the P- effect on the structure which increases the displacements.

The real displacements of the structure d s are given as q times the elastic displacements d e calculated by using the reduced forces, i.e.

ds = q d e

The values of the factor q for various types of steel buildings are given in Lecture 17.5 . Allrecent design codes use a similar approach with slightly different values for the q factor. Thesediscrepancies are justified by the fact that q factors are not only functions of the shape of thestructure, but they depend also on the accelerograms a(t) considered. The accelerograms differfrom one part of the world to the other. Other points of difference may be due to the selected

parameter characterizing the behaviour, which may be the equal energy dissipation rather thanthe displacements, and due to the safety factors used for the elastic analysis, which usually arehigher than those used for the inelastic analysis. Thus the appropriate q factors involve atheoretical approach but also an engineering judgement.

It should be noted also that the analysis using a q reduction factor for an earthquake action isconventional. Safety in the various structural elements is assured by requiring the computedcomparison stresses to be less than or equal to the yield stress. For the design of connections,under a real earthquake, the real comparison stresses are equal to f y in dissipative zones. It is forthis reason that connections close to dissipative zones must be designed to transmit the plasticdesign resistance of elements and not the elastic internal forces computed on the basis of anelastic analysis using a q reduction factor.


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5. CONCLUDING SUMMARY The design philosophy for structures to resist seismic actions requires that the

structure must not collapse and must retain its structural integrity under a "strong"earthquake. The structure must also not be damaged or limited in use under a"moderate" earthquake. To avoid collapse, the structure is allowed to develop

plastic zones in which seismic energy is dissipated. Methods given by modern design codes for the analysis of structures under

seismic actions assess their behaviour against these performance requirements. Time-domain methods are used but their application is expensive. The response spectrum method is the standard procedure of modern seismic

design codes, e.g. Eurocode 8. A simplified response spectrum method for regular buildings is available.

The elastic design of an earthquake resistant structure leads to very expensivestructures. Consequently the current design philosophy uses controlled dissipativezones in the structure where seismic energy can be dissipated by means of ductilehysteretic behaviour.

6. REFERENCES

[1] Eurocode 8: "Structures in Seismic Regions - Design", CEN, (in preparation).

[2] Ballio, G. (1985) ECCS Approach for the Design of Steel Structures to Resist Earthquakes.Symposium on Steel in Buildings, Luxembourg. IASE-AIPC-IVBH Report Volume 48 pp 313-380.

[3] Ballio, G. (1990) European Approach to Design of Steel Structures. 1990, Proc of HongKong Fourth World Congress - Tall Buildings: 2000 and Beyond, pp 935-946.

Table 1: Methods of analysis for structures under seismic actions

Data needed Type of analysis Use - Design Codes

DIRECT DYNAMIC ANALYSIS

(Time domain)

Accelerogram a(t)

(real or artificial)

Characteristics of the structure,elastic & inelastic (e.g. M- curvesfor connections)

Linear or non-linear

Direct Integration

procedure permitted byCodes but not for design

Use only for large and

complex structures

Use for evaluation ofresponse of existingstructures under a specificearthquake


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RESPONSE SPECTRUMANALYSIS

Design Response Spectrum

Characteristics of the structure,elastic only

Modal analysis (linear)

Mode shape superposition

Standard design procedurein Seismic Codes

No limitations of use

EQUIVALENT STATICFORCE ANALYSIS

Design Response Spectrum

Characteristics of the structure,elastic only

Static analysis

First vibration mode is predominant

Procedure permitted byCodes for buildings withspecific limitations ofregularity

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