Nonlinear interaction design provisions using spectral superposition

Engineering Structures 25 (2003) 1585–1595www.elsevier.com/locate/engstruct

Nonlinear interaction design provisions using spectral superposition

Jorge Vasquez ∗

Department of Structural Engineering, Pontificia Universidad Catolica de Chile, Casilla 306, Correo 22, Santiago, Chile

Received 17 October 2001; received in revised form 7 May 2003; accepted 12 May 2003

Abstract

The problem of enforcing two-variable nonlinear interaction code provisions within a spectral superposition approach to design,is addressed. An analytical procedure based on the linearization of the interaction curve through a set of tangents or secants isdeveloped. Safety with respect to the interaction curve is approximated by requiring safety with respect to that set of straight lines.For calculating the required estimators, the cross-estimator based formula for estimators of the linear combination of variables,derived in a companion paper, is used.

The analytical method developed was shown to be equivalent to a graphical method proposed by Gupta, based on inscribing anestimator ellipse within the interaction curve. The analytical method is more straightforward and handles the directional maximumin single-component excitation, which the graphical method does not. The analytical method is much easier to apply. However, ifa study calls for actually drawing the estimator ellipse, straightforward construction methods are presented, making unnecessary acumbersome equivalent modal response approximation that had been suggested. The inherent antisymmetry of the spectral superpo-sition formula, and its implications, are discussed. The effect of static loading and of symmetry of the interaction curve is also analyz-ed.

Within the analysis of an example, the work required for the application of the procedure for nonlinear interaction for designpurposes is discussed. It is found that its implementation can be achieved through a very simple function written for any standardnumerical computation software package. The example also makes quite apparent the advantages of the analytical over the graphi-cal method.

The application example, which considers the design of a concrete column in a simple 10-storey building structure, shows theoverconservativeness of a design based only in the standard estimators. The design criterion used in the example is that of the mostunfavorable direction of a single-component earthquake. 2003 Elsevier Ltd. All rights reserved.

Keywords: Modal combination; Cross-estimators; Nonlinear interaction capacity requirements; Single direction excitation; Most unfavorable direc-tion

1. Introduction

As was discussed in the companion paper [1], spectralsuperposition modal combination formulas apply to anyresponse quantity, be it a displacement, a force, a strainor stress at a given cross-section or at a given pointwithin a structure. However, that reference presents analternative method for obtaining the estimator of aresponse variable which is a linear combination of basicvariables, as are those provided by equations of staticsor of kinematics. The alternative is a formula expressing

∗ Tel.: +56-2686-4203.E-mail address: [email protected] (J. Vasquez).

0141-0296/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0141-0296(03)00124-X

the estimator of the linearly dependent variable in termsof the standard estimators of the basic variables and oftheir cross-variable estimator.

In many cases, the choice between direct evaluationof estimators and use of the linear combination formulawill depend on the available degree of interactionbetween the analysis process and the design stage. If theanalysis engine is such that during the design stage, theindividual modal components or values of any responsequantity can be easily calculated, the direct approach isquite suitable. Otherwise, retrieving from analysis thecross-estimators of the basic variables in addition to theirordinary standard estimators will make the linear combi-nation formula operational. On the other hand, there areinstances in which though the direct approach is per-

1586 J. Vasquez / Engineering Structures 25 (2003) 1585–1595

fectly feasible, it involves such an unnecessary repetitionof calculations so as to make it impractical. One suchinstance is nonlinear interaction design requirements.This paper is devoted to that problem.

Nonlinear interaction is addressed here by linearizingthe interaction curve through a set of tangents or secants.The safety condition represented by the interaction curveis transformed into safety with respect to a discrete num-ber of straight lines. This linearizes the problem by intro-ducing safety conditions that are represented by a linearcombination of the basic variables that must be keptunder a given threshold. Approximations of this type arefound to be very accurate in obtaining the collapse loadof structures by Linear Programming, as checkedthrough “exact” nonlinear optimization methods [2,3].

Introducing safety variables associated to each of theapproximating tangents or secants, an analytical designprocedure is developed. The analytical procedure isshown to be equivalent to a more toilsome graphicalmethod that has been proposed by Gupta [4]. The ellipseassociated to the graphical method is analyzed and itsconstruction studied. In particular, the inherent antisym-metry of spectral superposition based design is dis-cussed. Including static loading is also considered.

An application example, considering the design of aconcrete column in a simple 10-storey building structureis presented. The design criterion used is based on themost unfavorable direction of a single-component earth-quake excitation. The overconservativeness of a designbased exclusively on the standard estimators is dis-cussed.

The practical problem of the required numerical com-putation effort is addressed, and a scheme for an efficientcomputational approach is proposed.

2. Nonlinear interaction provisions

Interaction curves or surfaces can be interpreted as theboundary of the region of admissible values for thedesign variables. For instance, in Fig. 1, the interactioncurve for a reinforced concrete short column of rectangu-lar cross-section is shown as the solid thick line, markingthe boundary of the region of all points representingallowable combinations of axial force and positive bend-ing moment. Of course, in symmetrical cross-sections,the region of negative moments is a reflection of the oneshown. Furthermore, it is well known [5] that for ductilematerials and cross-sections, the admissible or saferegion has to be convex. Some code interaction curvesdo not satisfy this condition due to inclusion of loaddependent safety factors. In such cases, the analysis thatfollows should be done removing the effect of thosesafety factors, and applying them when the procedureis completed.

The interaction curve, as any continuous line, can be

Fig. 1. Bending moment—axial force interaction curve for areinforced concrete rectangular cross-section short column.

regarded as the envelope of its own tangents, as isinsinuated in the same Fig. 1. Henceforth, imposingsafety with respect to each and all of the tangents, inter-preted as representative of modes of failure, is whollyequivalent to having the design variables confined withinthe safe region. Since the safe region is convex, there isno risk of conservatively chopping part of it off whenusing the tangents to define safety, or what would beworse, of inattentively including inadmissible exteriorportions into it.

The individual tangent safety condition can be writtenas the linear relationship

m(t)mo

�p(t)po

�1

In this expression, the coefficients of the bendingmoment, m(t), and of the axial force, p(t), are respect-ively, the reciprocals of the axes intercepts mo and po,as shown for a generic tangent in Fig. 2. This suggeststhe definition of a linear combination of the axial forceand the bending moment as the new “safety” variable

r(t) �m(t)mo

�p(t)po

(1)

that certainly has an estimate that can be calculated usingEq. (21) of Ref. [1].

The result is the estimator

1587J. Vasquez / Engineering Structures 25 (2003) 1585–1595

Fig. 2. Definition of mo and po intercepts on the m and p-axes for adiscrete number of tangents approximation.

R2 �M2

m2o

� 2MPmopo

�P2

p2o

(2)

expressed in terms of the standard estimators of themoment and the axial force

M2 � �n

i � 1

�nj � 1

L iL jrijmimj

P2 � �n

i � 1

�n

j � 1

L iL jrijpipj

and of the cross-variable estimator

MP � �n

i � 1

�n

j � 1

L iL jrijmipj

This set formed by the two standard estimators andthe cross-variable estimator, allows the calculationthrough Eq. (2) of the estimator of the safety variablefor any tangent, in terms of its m and p-axes intercepts,mo and po. From a practical point of view, the interactioncurve can be approximated through a discrete number oftangents, as shown in Fig. 2, and the evaluation of esti-mators for the safety variables limited to those tangentconditions. There is of course some slight unconserva-tiveness at some points in the tangent envelope approxi-mation. If this unconservativeness is unacceptable, asecant approximation should be used instead.

3. The estimator ellipse

It can be proved that this procedure is equivalent toa graphical method proposed by Gupta in the same Ref.

[4]. In fact, the values of the bending moment mr andthe axial force, pr, synchronous with respect to themaximum of r(t), as estimated by R, can be obtainedfrom Eq. (30) of Ref. [1]. They are

Rmr � RM ; Rpr � RP (3)

But from the definition of cross-variable estimators,the following relationships for RM and RP are obtained

RM � �n

i � 1

�n

j � 1

L iL jrij�mi

mo

�pi

po�mj �

M2

mo

�MPpo

RP � �n

i � 1

�n

j � 1

L iL jrij�mi

mo

�pi

po�pj �

MPmo

�P2

po

so that mr and pr can be evaluated as

Rmr �M2

mo

�MPpo

(4)

Rpr �MPmo

�P2

po

(5)

To these two equations, a third condition should beadded, originating in the first place, in the pair mr andpr being values of m(t) and p(t) occurring at the instantof time tr at which r(t) attains its maximum value, rmax,and in the second place, in the assumption that rmax isequal to the estimator R. This additional equation, for-mally resulting from inserting mr and pr into Eq. (1), is

rmax �mr

mo

�pr

po

� R (6)

The four Eqs. (2), (4)–(6) can be regarded as a para-metric representation of a curve in the mr–pr plane. Theparameters are of course mo and po. There are four para-metric equations, one more than the necessary three forthe two parameters of the formulation, because of theinclusion of the additional variable R.

The parameters can be eliminated by solving for themthe pair of Eqs. (4) and (5), linear if the reciprocals ofmo and po are considered as the unknowns. The resultof this solution is

1mo

� RP2mr�MPpr

M2P2�MP2 ;1po

� RM2pr�MPmr

M2P2�MP2

Replacing these results into Eq. (6), the followingrelationship is derived

RP2mr�MPpr

M2P2�MP2mr � RM2pr�MPmr

M2P2�MP2 pr � R

which after simplification by R adopts the form

P2m2r�2MPmrpr � M2p2

r � M2P2�MP2 (7)

The curve representing Eq. (7) in the m–p plane, orthe equivalent parametric Eqs. (2), (4)–(6), is an ellipse.


Gupta called it the interaction ellipse. However, to avoidusing the word interaction both for this ellipse and forthe curve limiting the capacity of the cross-section, thename estimator ellipse will be preferred. In a way it isa better name, since the ellipse has its origin in the tensorcharacter of spectral superposition estimation rather thanin the interaction condition that m and p must satisfy.

It can be easily shown that the estimator ellipse isinscribed within a rectangle with half-sides the standardestimators M and P. Indeed, by implicitly differentiatingEq. (7) with respect to pr, and then making dmr/dpr equalto zero, the following relationship between the coordi-nates mr and pr for a maximum mr,max of mr, is estab-lished

(pr)mr=mr,max�

MPM2 mr,max

so that after substitution into the same Eq. (7), the result

m2r,max � M2

is obtained. A similar procedure will give the limits ±Pfor pr.

Gupta’s graphical safety condition is expressed bystating that the estimator ellipse has to lie entirely withinthe safe region. In the limiting situation, both the ellipseand the interaction curve share a common tangent. Fig.3 shows how the design ellipse can be superposed ontop of a design chart of interaction curves to choose therequired cross-section.

Obviously, the graphical criterion can be visualized asensuring that all (mr, pr) points lie at the same side asthe origin with respect to each and all of the tangents tothe interaction curve. The consequence of this is thatfor each and every pair of parameters, mo and po, thecorresponding R will be equal to, or less than, one. Theequivalence of the graphical and the proposed tangentsafety analytical criterion is thus established.

The estimator ellipse may be drawn by finding its axes

Fig. 3. Estimator ellipse design procedure: finding the required inter-action curve

from Eq. (7), both with regard to magnitude and pos-ition. Indeed, the axes of the ellipse can be obtained fromthe eigenvectors of the coefficient matrix

1P2M2�MP2�P2 �MP

�MP M2 �and its semi-axes, as the reciprocals of the square rootsof the corresponding eigenvalues.

The estimator ellipse curve can also be drawn pointby point. Gupta proposes an approximation based on thedefinition of what he calls an “equivalent modalresponse”. However the approximation is not necessary.With less numerical work, a point by point constructionof the curve can be directly derived from Eq. (7). Eventhe parametric Eqs. (2), (4)–(6) could be used for thesame purpose.

It can be easily recognized that for checking thecapacity of a member, the analytical procedure is morestraightforward and easier to use than the graphicalmethod. At the design stage, however, the graphical pro-cedure depicted in Fig. 3, might seem preferable. But itis not really so. The analytical procedure proposed here,notwithstanding the need of testing for different interac-tion curves, is definitely simpler than actually drawingthe ellipse in the m–p interaction plot. Furthermore, atsome expense, though, the analytical approach can beextended to three variable interactions, as would be thecase of axial force plus biaxial bending interaction, whilethe graphical method would involve an improbable vis-ualization of an estimator ellipsoid within a three-dimen-sional interaction surface.

An additional advantage of the analytical method withrespect to the graphical construction of the estimatorellipse is its ability to combine with the design criterionbased on the most unfavorable direction of a single-component earthquake excitation. Indeed, there is nodirectional maximum estimator ellipse, while the tablesfor the analytical method can be easily combined formost unfavorable direction design.

4. Symmetry and static loading

It should be realized that the R estimator of Eq. (2)is antisymmetric, as shown by the antisymmetry of theestimator ellipse with respect to the origin (Fig. 3).Indeed, Eq. (7) is that of an ellipse centered at the origin.This antisymmetry is a consequence of the reversiblenature of the spectral representation of earthquake exci-tation. When the interaction curve is also antisymmetricwith respect to the origin, the antisymmetry of the esti-mators takes care at the same time of both the conditionfor the direct variable of Eq. (1) and for the sign-reversedvariable in the opposite quadrant

s(t) �m(t)�mo

�p(t)�po

� �r(t)


Indeed, the S and R estimators are identical, and thereis no need for a second condition.

On the contrary, symmetry of the interaction curvewith respect to a single axis does require an additionalcondition. If there is symmetry with respect to the p-axis, as in the case of the interaction curve of Fig. 1, foreach secant approximation line defined by intercept pairs(mo, po), the line symmetric with respect to the p-axismust also be considered. This implies the condition thata second response variable

s(t) �m(t)�mo

�p(t)po

must also be less or equal to one. The estimator for thisvariable, as obtained from Eq. (21) of Ref. [1], is

S2 �M2

m2o

�2MPmopo

�P2

p2o

Of course, at estimator level, the capacity condition forthe (�mo, po) symmetric secant approximation is S�1.

When relevant, allowance must be made to considerthe effect of static forces, generating a moment mstat andan axial force pstat. For a given secant approximation,this is achieved by replacing out r(t) in the expressionfor the capacity condition

mstat

mo

�pstat

po

� r(t)�1

by substituting it by its estimator R. Introducing thenotation

R� � R �mstat

mo

�pstat

po

,

the capacity condition continues to be the same unit of1, but now written as R��1.

Of course, when the interaction curve is symmetricwith respect to the p-axis, the condition S��1, with

S� � S �mstat

�mo

�pstat

po

will also be needed. And when the antisymmetry of theinteraction curve is used to reduce the number of secantapproximations to be considered, the definition of R� hasto be modified to include the absolute value of the staticforces contribution, to read

R� � R � |mstat

mo

�pstat

po | (8)

The absolute value takes into account the fact thatgiven the reversible nature of the earthquake responseestimation, the effect of the static loading on the anti-symmetric secant approximations will always be anincrease of the demand.

In the graphical scheme, making allowance for static

Fig. 4. Estimator ellipse design procedure for a case with superposedstatic loading.

loading mstat and pstat, requires moving the estimatorellipse so as to have its center displaced in the m–p planeto the position (mstat, pstat). Fig. 4 is a variation of Fig.3 that illustrates the center displacement.

5. An application example

The problem of the example is proportioning thereinforcement for a RC column in the sixth storey of asimple 10-storey building structure. All storey plans areidentical, with the structural layout shown in Fig. 5. The

Fig. 5. Typical floor plan of 10-storey model.


storeys are 3.6 m high. Four exterior columns have0.5 × 0.5 m cross-sections; the three columns of the cen-tral line are 0.8 × 0.5 m; the two exterior columns in theupper corners are 0.5 × 0.8 m. Biaxial bending interac-tion will be considered, but the variation of the axialforce due to lateral deformation will be neglected. Theparticular column chosen as the example for the dis-cussion is the one in the left-lower corner of the plan.

The mass of the storeys is assumed equal to uniformlydistributed weight forces of 10 and 7.5 kN/m2 distributedover the floor and roof diaphragms, respectively. With-out much loss of generality, for the purpose of this illus-tration, an ideal shear behavior of the storeys will beassumed, as would result from very stiff girders con-necting the top of the columns. The modulus of elasticityof the concrete was taken as 20 000 MPa.

The first three modal periods are found to be 0.868,0.648, and 0.615 s. Modes are strongly coupled, it isimpossible to assign an x-, y- or torsional direction pre-dominance to any one of them.

The excitation used is the UBC design response spec-tra. Two cases are considered. Case A, a Zone 4, soilprofile type SB: Ca = 0.40, Cv = 0.40 at a distance fromknown seismic sources greater than 10 km, and Case B,a site-specific soil profile type SF: Ca = 0.34, Cv =0.42. The code factor R is taken as 8.5, for a specialmoment-resisting frame.

The values of CQC estimators and cross-estimators,U2 = M2

x, V2 = M2y, and UV = MxMy, were obtained for

the x, y and 45° direction excitations, as calculated fromthe generic Eqs. (22)–(24) of Ref. [1], respectively. Theyare given in Table 1.

The biaxial moment interaction curves for the column0.5 × 0.5 m for five different reinforcements are givenin Fig. 6. The reinforcements are (a) eight #9 bars; (b)four #10 bars and four #9 bars; (c) eight #10 bars; (d)four #11 bars and four #10 bars; and (e) eight #11 bars.The nominal diameter of #9, #10, and #11 bars are 28.6,32.2, and 35.8 mm, respectively. The yield strength ofsteel was taken as 275 kPa, and the compression strengthof concrete as 27.5 kPa. Table 2 contains the Mxx andMyy axes intersects of six-secant approximations to the

Table 1Calculated estimators and cross-estimators (kN m)2

M2x M2

y MxMy

Case Ax 120 568 112 631 �53 022y 14 449 106 861 14 84745° 51 615 145 174 39 965Case Bx 132 836 124 105 �58 406y 15 922 117 760 16 36545° 56 872 159 987 44 048

Fig. 6. Biaxial moment interaction curves for cross-section shown.Reinforcement: (a) eight #9 bars; (b) four #10 plus four #9 bars; (c)eight #10 bars; (b) four #11 plus four #10 bars; (e) eight #11 bars.

five interaction curves. The curves of Fig. 6 correspondto a zero axial load, which is conservative.

For the two cases, A and B, and for each direction,x, y and 45°, estimators of the set of secant safety con-dition r(t) of Table 3 variables will have to be evaluated.Of course, given the symmetry of the interaction curveswith respect to either axis, say with respect to the y-axis,the additional set of secant safety condition variables

s(t) �mx

Myy

�my

Mxx

would give rise to a table similar to Table 3. Their esti-mators will also have to be evaluated. Symmetry withrespect to the x-axis is ensured, as was previously dis-cussed, through the antisymmetry of the estimators.

The R and S estimators of these r(t) and s(t) responsevariables are to be calculated as

R2 �M2

x

M2yy

�2MxMy

MxxMyy

�M2

y

M2xx

(9)

S2 �M2

x

M2yy

�2MxMy

MxxMyy

�M2

y

M2xx

(10)

where the Mxx and Myy intercepts are those pertaining tothe particular cell of Table 3, each cell corresponding toa given reinforcement curve and to an individual secantin the piece-wise linear approximation of Table 2.

For Case A, the corresponding M2x, M2

y, and MxMy esti-mators from Table 1 are used, Table 4 is built for thex-direction excitation, Table 5 for the y-direction exci-tation, and Table 6 for the 45° direction excitation.

The top five rows in these tables are the values of theR estimators of Eq. (9), while the bottom five rows, writ-


Table 2Intercepts for secant approximation (kN m)

Curve Axis Line 1 Line 2 Line 3 Line 4 Line 5 Line 6

a Mxx 7114 1192 566 371 322 298Myy 298 324 406 625 913 10387

b Mxx 9068 1697 595 410 350 330Myy 331 349 466 681 1118 8019

c Mxx 9001 1951 725 449 383 361Myy 363 380 470 687 1065 9945

d Mxx 9915 2118 754 482 418 397Myy 400 418 531 776 1207 8150

e Mxx 22100 2460 1009 509 442 430Myy 434 451 518 818 1382 6151

ten with a different typeset, correspond to the S esti-mators of Eq. (10).

It is quite clear from the tables that if only y-directionexcitation were considered, reinforcement (b) would bestrong enough. The limit is given by the R requirementsof the Line 6 secant approximation. However, the x-direction is more demanding. The S requirement of theLine 4 secant shows that even the (c) reinforcement isno longer enough, and cross-section reinforcement (d)has to be chosen. The demand placed by the 45° direc-tion is even larger. Reinforcement (d) fails to meet theR2

45� = 1 limit for the Line 5 secant approximation, andreinforcement (e) has to be provided. The estimatorellipse drawn in Fig. 7 for this case illustrates the actualmaterialization of the capacity requirement.

Note that if the analysis was limited to the traditionalseparate x- and y-direction excitations, with the obtentionof only the standard estimators, the design criterionwould have no other possibility than providing capacityfor a pair of moments mx and my obtained from thosestandard estimators. For Case A, where the x-directionexcitation is clearly dominant, the values must be mx

= Mx = √120 568 = 347, my = My = √112 631 = 336.Table 3, Line 4 test for cross-section e would give anapparently unacceptable

r �347818

�336509

� 1.084

This would result in an unnecessary use of heavierreinforcement.

If the design criterion is safety with respect to themost unfavorable direction of single-component exci-tation, the cross-directional estimators RxRy and SxSy willbe needed. Its calculation can be easily done by trans-forming the 45° excitation response. Indeed, the rosetteEq. (13) of Ref. [1]

RxRy � R245��

12(R2

x � R2y)

conveniently expresses the RxRy cross-estimator in termsof the R2

45�, R2x, and R2

y estimators. This results in Table 7.

From the values for R2x, R2

y and RxRy (and S2x, S2

y andSxSy) of Tables 4, 5 and 7, the most unfavorable directionresponse estimator of Eq. (9) of Ref. [1]

R2M �

12

(R2x � R2

y) �12�(R2

x�R2y)2 � 4RxRy

2

can be tabulated as in Table 8, showing that design isstill controlled by Line 5 in the secant approximation.The overstrength of reinforcement (e), which continuesto be the reinforcement design choice, has changedthough. It has dropped from 0.099 = (1�0.901) asrequired by Line 5 in Table 6, to 0.079 = (1�0.921) asrequired by Line 4 in Table 8. Of course, the near coinci-dence of the directional maximum with the demand pre-dicted by the 45° excitation is due to the fact that inCase A, the most unfavorable direction for this columnhappens to be very near to 45°.

6. Practical computational considerations

Building the tables that are needed for these verifi-cations may be thought to be a tedious and laborioustask. And indeed, for hand calculations, it would be verymuch so. However, with the use of a numerical compu-tation software package, or even with a spreadsheet,work is quite minimal.

Furthermore, there is no need to write or even toinspect the whole table. Actually, only the controllingvalue has to be checked. A table has 2nc rows and ns

columns, where nc is the number of cross-sections, andns is the number of secant approximations. The first nc

rows contain the R2 values, and the last nc rows, the S2

values. The controlling value is found as the largest R2 orS2 in the first row in which all entries are less than one.

For instance, in verifying Case B and using the valuesof the M2

x, M2y, and MxMy estimators for that case, read

from Table 1, a set of tables similar to Tables 4–6, 8 ofCase A could be built. However, they can be replacedby the single tabular result of the controlling R2 or S2


Tab

le3

Seca

ntsa

fety

cond

ition

vari

able

s

Cur

veL

ine

1L

ine

2L

ine

3L

ine

4L

ine

5L

ine

6

a�m

x

298�+

�my

7114�

�mx

324�+

�my

1192�

�mx

406�+

�my

566�

�mx

625�+

�my

371�

�mx

913�+

�my

322�

�mx

1038

7�+�m

y

298�

b�m

x

331�+

�my

9088�

�mx

349�+

�my

1697�

�mx

466�+

�my

595�

�mx

681�+

�my

410�

�mx

1118�+

�my

350�

�mx

8019�+

�my

330�

c�m

x

363�+

�my

9001�

�mx

380�+

�my

1951�

�mx

470�+

�my

725�

�mx

687�+

�my

449�

�mx

1065�+

�my

383�

�mx

9945�+

�my

361�

d�m

x

100�+

�my

9915�

�mx

118�+

�my

2118�

�mx

531�+

�my

754�

�mx

776�+

�my

482�

�mx

1207�+

�my

418�

�mx

8130�+

�my

397�

e�m

x

434�+

�my

2210

0��m

x

154�+

�my

2460�

�mx

518�+

�my

1009�

�mx

818�+

�my

509�

�mx

1382�+

�my

142�

�mx

6151�+

�my

130�


Table 4Case A. R2

x’s and S2x’s for secant safety condition variables (kN m)2

Curve Line 1 Line 2 Line 3 Line 4 Line 5 Line 6

a 1.306 0.951 0.621 0.670 0.871 1.235b 1.068 0.848 0.491 0.549 0.743 0.993c 0.882 0.723 0.448 0.470 0.613 0.836d 0.727 0.597 0.360 0.402 0.516 0.684e 0.629 0.515 0.357 0.360 0.466 0.572a 1.406 1.499 1.543 1.585 1.593 1.304b 1.139 1.206 1.255 1.308 1.284 1.073c 0.947 1.009 1.070 1.158 1.133 0.895d 0.780 0.837 0.890 0.969 0.936 0.750e 0.651 0.706 0.763 0.869 0.813 0.652

Table 5Case A. R2

y’s and S2y’s for secant safety condition variables (kN m)2


a 0.178 0.289 0.550 0.942 1.150 1.213b 0.143 0.206 0.475 0.772 0.957 0.990c 0.120 0.168 0.356 0.657 0.813 0.828d 0.099 0.140 0.313 0.564 0.679 0.688e 0.080 0.115 0.216 0.505 0.603 0.590a 0.150 0.136 0.292 0.686 0.948 1.194b 0.124 0.105 0.261 0.560 0.806 0.967c 0.102 0.088 0.182 0.464 0.667 0.812d 0.084 0.073 0.165 0.405 0.562 0.669e 0.074 0.062 0.102 0.363 0.506 0.567

Table 6Case A. R2

45�’s and S2

45�’s for secant safety condition variables (kN m)2


a 0.620 0.700 1.113 1.532 1.736 1.661b 0.500 0.608 0.936 1.259 1.427 1.360c 0.417 0.504 0.744 1.088 1.230 1.137d 0.344 0.419 0.637 0.925 1.023 0.947e 0.282 0.350 0.488 0.830 0.901 0.817a 0.545 0.386 0.418 0.843 1.192 1.609b 0.447 0.339 0.359 0.687 1.019 1.300c 0.368 0.288 0.275 0.570 0.838 1.092d 0.303 0.238 0.239 0.498 0.706 0.898e 0.266 0.206 0.182 0.446 0.639 0.756

(Table 9) directly obtained from the correspondingevaluation functions.

The meaning of the inadmissible S2M = 1.015 value

obtained for the most unfavorable direction is that eventhe stronger reinforcement under consideration is insuf-ficiently strong. However, an (f) reinforcement, obtainedby adding to the eight #11 bars of reinforcement (e) twoextra #9 bars should be enough. The interaction curve forthis reinforcement is linearized also through six secantapproximations. The intercepts for these secant lines aregiven in Table 10.

Processing with the R2M function for the (f) reinforce-

ment will give as result a controlling value as low as0.777 for R in the Line 5 secant approximation. For laterreference, the whole part of the matrix of estimators cor-responding to reinforcement (f) is given in Table 11.

A relevant question at this stage is to know if the (f)reinforcement would continue to be acceptable if staticmoments were present. For instance, assume thatmoments mx = 35 kN m, my = 22.5 kN m, originatingin gravity loading, were to be considered. The answerrequires calculating the R� of Eq. (8) to find


Table 7Case A. RxRy’s and SxSy’s for secant safety condition variables (kN m)2


a �0.122 0.179 0.527 0.727 0.726 0.437b �0.106 0.081 0.453 0.599 0.577 0.369c �0.084 0.059 0.342 0.525 0.517 0.305d �0.069 0.050 0.301 0.442 0.425 0.261e �0.072 0.034 0.202 0.397 0.366 0.236a �0.233 �0.432 �0.499 �0.293 �0.079 0.361b �0.184 �0.317 �0.399 �0.247 �0.026 0.280c �0.156 �0.260 �0.351 �0.241 �0.062 0.239d �0.129 �0.217 �0.289 �0.190 �0.042 0.188e �0.097 �0.179 �0.250 �0.170 �0.020 0.146

Table 8Case A. R2

M’s and S2M’s for secant safety condition variables (kN m)2


a 1.320 0.996 1.114 1.545 1.749 1.661b 1.080 0.858 0.936 1.270 1.437 1.360c 0.891 0.729 0.747 1.096 1.239 1.137d 0.734 0.602 0.639 0.933 1.031 0.947e 0.638 0.518 0.500 0.836 0.907 0.817a 1.448 1.625 1.718 1.672 1.603 1.614b 1.171 1.291 1.396 1.382 1.285 1.305c 0.975 1.078 1.192 1.233 1.141 1.096d 0.803 0.894 0.991 1.027 0.941 0.902e 0.666 0.753 0.847 0.921 0.815 0.762

Table 9Case B. Controlling maxima for x-, y-, 45° and most unfavorable direc-tion

Direction Value Curve Line Row

x 0.958 e 4 10(S)y 0.913 c 6 3(R)45° 0.900 e 6 5(R)M 1.015 e 4 10(S)

Table 10Intercepts for secant approximations (kN m)

Curve Axis Line 1 Line 2 Line 3 Line 4 Line 5 Line 6

f Mxx 16720 2276 1024 591 500 476Myy 503 540 678 957 1609 3090

Table 11Case B. R2

M’s and S2M’s for secant safety condition variables (kN m)2


f 0.519 0.394 0.405 0.681 0.777 0.774f 0.555 0.618 0.641 0.748 0.694 0.678

R� � �0.777 �35

1609�

22.5500

� 0.948

in which the Mxx = 500 and Myy = 1609 Line 5 interceptsfor the secant approximation of the (f) reinforcement areused. The result is acceptable. However, capacity withrespect to other secant approximations may become criti-cal, so that they also have to be checked. The R� associa-ted to Line 6 is

R� � �0.774 �35

3090�

22.5476

� 0.938


Fig. 7. Estimator ellipse for case A, 45° direction excitation.

and it does check. Obviously the rest of the R�’s are lessdemanding. The S�’s involve a minus Mxx intercept inthe static force contribution, so they will not control. Theconclusion is that the (f) reinforcement is sufficient.

7. Conclusions

A solution to the problem of enforcing two-variablenonlinear interaction code provisions within a spectralsuperposition approach to design was devised. It is ananalytical procedure based on the linearization of theinteraction curve through a set of tangents or secants,in which safety with respect to the interaction curve isapproximated by requiring safety with respect to that setof straight lines. A linearized safety condition isexpressed by stating that a normalized variable, that isa linear combination of the basic variables, must notexceed the value of one. The analytical procedure estab-lishes that the spectral estimators for those normalizedvariables, associated to each and all of the tangents orsecants, must also be less than one.

For calculating the required estimators, the cross-esti-mator based formula derived in the companion paper [1]for estimators of linear combination of variables, isavailable. For computational economy, use of this for-mula is suitable even in situations in which the modalcomponents of all response variables are easily retriev-

able, so that direct application of the spectral superpo-sition formula is feasible.

The analytical method was shown to be equivalent toa graphical method proposed by Gupta [4], based oninscribing an estimator ellipse within the interactioncurve. The analytical method is more straightforwardand handles the directional maximum in single-compo-nent excitation, which the graphical method does not.If notwithstanding the computational advantages of theanalytical method, a study calls for actually drawing theestimator ellipse, straightforward construction methodsare made available. Indeed, methods are presented whichmake unnecessary the use of the cumbersome equivalentmodal response approximation suggested in the sameRef. [4].

The inherent antisymmetry of the spectral superpo-sition formula and its implications were discussed. Inparticular, a study was made of the implications withrespect to the effect of static loading. Symmetry in theinteraction curve was also analyzed, showing how tomake full use of that symmetry.

Through an example, the computational aspects of theapplication of the design procedure for nonlinear interac-tion provisions, were discussed. It was found that therequired work can be done using a very simple functionwritten for any standard numerical computation softwarepackage. The example also makes quite apparent theadvantages of the analytical over the graphical method.

The application example, which considers the designof a concrete column in a simple 10-storey buildingstructure, shows the overconservativeness of a designbased only in the standard estimators. The design cri-terion which considers the most unfavorable directionsingle-component earthquake excitation was used. Therequired cross-direction estimator was obtained from theadditional information supplied by the 45° estimator,through the rosette formula derived in Ref. [1].

References

[1] Vasquez J. Spectral modal combination of linearly related vari-ables, companion paper.

[2] Vasquez J. Use of mathematical programming in earthquake analy-sis. In: Savidis SA, editor. Earthquake resistant construction anddesign. Rotterdam: Balkema; 1994. p. 865–72.

[3] de la Llera JC, Vasquez J, Chopra AK, Almazan JL. A macro-element model for inelastic building analysis. Earthquake EngStruct Dyn 2000;29:1725–57.

[4] Gupta AK. Response spectrum method in seismic analysis anddesign of structures. Boston: Blackwell Scientific Publications,1990.

[5] Heyman J. Plastic design of frames. Cambridge: Cambridge Uni-versity Press, 1971.

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Nonlinear interaction design provisions using spectral superposition