Theory and application of orthogonal effects in a

3-D response spectrum analysis

K. Anastassiadis, I.E. Avramidis

Department of Civil Engineering, Aristotle University of

Thessaloniki, 54006 Thessaloniki, Greece

Abstract

In this paper, exact relations and formulae based on a general methodologywithin the framework of the response spectrum method are given, which allowcalculation of the critical earthquake direction and evaluation of the peakprobable values of any typical response quantity along with the simultaneousvalues of other related quantities. The theory covers unidirectional as well asbidirectional (isotropic or orthotropic) seismic input and can be directly used tosatisfy modern seismic design codes' requirements. A numerical exampleclarifies and confirms the presented theoretical results.

1 Introduction

According to the research work of Penzien/Watabe [1], the seismic input atthe base of any given structural system can be considered as consisting of threesimultaneously occuring and statistically independent components directed alonga set of principal axes : a vertical and two orthogonal horizontal excitations. Inthe initial phase of the ground motion, the orientation of these principal axes ischanging with time. However, after a short period of time it becomes practicallystable with the major principal axis (strong excitation) along the epicentraldirection "a" and the minor principal axis (weak excitation) along the verticaldirection "c". Consequently, the three earthquake components correspond ingeneral to three different response spectra R%, R% and R^, where "b" indicatesthe horizontal direction which is orthogonal to the major (epicentral) seismicexcitation.

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650 Earthquake Resistant Engineering Structures

A simplified form of this seismic input model has been already accepted bymany modern earthquake design codes. The simplification usually consists inprescribing the same response spectrum R^=R^ for both horizontal componentsand setting R =(2/3)R for the vertical one.

In recent years, progresses have also been made on the field of theresponse spectrum method of analysis of structures subjected to the describedmulti-directional seismic excitation. However, the results of the related researchwork are not yet widely known and, despite their simplicity, are not yetincorporated neither in seismic design codes nor in the professional softwarepackages for structural analysis [2]. As an example, clause 3.3.3.1 of theEuropean Seismic Code EC8/94, Part 1-2 demands the determination of themost unfavourable orientation of the bi-directional seismic excitation, thusignoring the fact that the building's response to bi-directional seismic input isindependent of the input's orientation [3, 4, 5, 6]. Furthermore, the spatialcombination of response quantities resulting from two (or three) seismiccomponents is still based on the empirical "percentage combination rules" of the100-30-30 or 100-40-40 type, thus overlooking the exact formulae for theextreme value of a given quantity as well as for the concurrent values of two (ormore) quantities [5].

In this paper, the tensorial properties of an arbitrary response quantity of astructure subjected to bi-directional (or uni-directional) seismic excitation aresynopsised. Also, exact relations and formulae are given, which can be used fora straightforward calculation of the probable simultaneous values of two or threestress quantities in a cross section of a building's structural member. Anumerical example clarifies and confirms the presented theoretical results.

2 Notation

The earthquake motion is defined by spectrum R^ for the epicentraldirection and by spectrum Ry for the direction normal to that. A fixed globalorthogonal reference system Oxyz is used for the structure. The spectra R^ orRy are applied individually in the direction of the x- or y-axis according toFigure la. The corresponding peak probable values of a typical responsequantity X (force or displacement) are symbolized as X,%% , X,^ , X,%y andX,ya , where the first subscript refers to motion in direction x or y and thesecond (a or b) to the input earthquake spectrum (R% or Ry). An analogousnotation is used for the variable orthogonal reference system O Tjt according toFigure Ib.

In Figure 2a, the simultaneously applied input spectra of a two component(bidirectional) design earthquake are shown. The epicentral component Racoincides either with axis x or axis y. The corresponding peak probable valuesof a response quantity X are symbolized by X,% and X,y respectively, where thesubscript indicates the actual epicentral direction. An analogous notation is usedfor the variable orthogonal reference system O n (Figure 2b).

Transactions on the Built Environment vol 20, 1996 WIT Press, www.witpress.com, ISSN 1743-3509

Earthquake Resistant Engineering Structures 651

Figure 1: Response quantity notation for unidirectional seismic excitations

Figure 2: Response quantity notation for bidirectional seismic excitations

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652 Earthquake Resistant Engineering Structures

3 Transformation of response quantities

Assuming that the epicentral and minor principal axes of the seismicexcitation (in the Penzien-Watabe's sense of the term) coincide with axes (,T|),the following transformation relations for the response quantities are valid :

= X, X,

, = X,% sin {3 + X,y cos (3 -

sin20

sin2(B

where

(la)

db)

(1C)

(2a)

X,y - (2b)

(2c)

It is important to note that relations (1) are similar to the transformationrules for the components of a symmetric second order tensor. Consequently, thefour quantities X,% , X,y and X%y = Xy% can be considered as componentsof a symmetric second order tensor, expressed analytically by matrix

r 2 Xxy

x,.

in the Oxyz reference system and by matrix

in the O r|t reference system. Due to its tensorial character, the arbitraryresponse quantity X is characterized by the following properties which arecommon to all symmetric second order tensors :

(a) The trace and the determinant of the above matrices are not dependent on theorientation of the earthquake excitation :

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Earthquake Resistant Engineering Structures 653

'y

2 2 2

(3a)

(3b)

(b) There is a specific earthquake orientation defined by the axes (1,11) for which

the correlation factor X^ vanishes. This specific orientation is defined by acute

angle {3^ which satisfies the relation

tan - 2 X%y / ( X^ - X,y (4)

according to equation (Ic) and the indications in Figure 3. For the bidirectionalearthquake of Figure 4a, as well as for its transposition in Figure 4b, theresponse quantity X takes the following maximum and minimum values

correspondingly :

max X = X,

\ (5a)

2 2min X = X,jj

(5b)

(b)

y

Figure 3: Determination of unfavourables seismic directions

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654 Earthquake Resistant Engineering Structures

fa)

fl

Figure 4: Favourables and unfavourables seismic directions

The form of these relations allows for a graphical calculation of X,j , X,,/ andangle p, using Mohr's circle (Figure 5).

Figure 5: Graphical