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Handling of Constraints in Finite-Element ResponseSensitivity Analysis

Quan Gu1; Michele Barbato2; and Joel P. Conte3

Abstract: In this paper, the direct differentiation method �DDM� for finite-element �FE� response sensitivity analysis is extended to linearand nonlinear FE models with multi-point constraints �MPCs�. The analytical developments are provided for three different constrainthandling methods, namely: �1� the transformation equation method; �2� the Lagrange multiplier method; and �3� the penalty functionmethod. Two nonlinear benchmark applications are presented: �1� a two-dimensional soil-foundation-structure interaction system and �2�a three-dimensional, one-bay by one-bay, three-story reinforced concrete building with floor slabs modeled as rigid diaphragms, bothsubjected to seismic excitation. Time histories of response parameters and their sensitivities to material constitutive parameters arecomputed and discussed, with emphasis on the relative importance of these parameters in affecting the structural response. The DDM-based response sensitivity results are compared with corresponding forward finite difference analysis results, thus validating the formu-lation presented and its computer implementation. The developments presented in this paper close an important gap between FE response-only analysis and FE response sensitivity analysis through the DDM, extending the latter to applications requiring response sensitivitiesof FE models with MPCs. These applications include structural optimization, structural reliability analysis, and finite-element modelupdating.

DOI: 10.1061/�ASCE�EM.1943-7889.0000053

CE Database subject headings: Finite element method; Constitutive models; Constraints; Sensitivity analysis; Soil-structureinteractions.

Introduction

Nonlinear finite-element �FE� analysis is widely recognized as anessential tool in structural analysis and design. In mechanical en-gineering, computer simulations using complex nonlinear FEmodels are commonly used in industry during the design process,reducing dramatically the need for prototype tests. In civil engi-neering, the use of high-fidelity nonlinear models to accuratelypredict the response of a structural system is even more crucial,due to the fact that prototype testing is not an option. Designcodes are gradually accounting for the response of structures wellbeyond their linear behavior �Advanced Technology Council�ATC-55� 2005�. The state-of-the-art in computational simulationof the static and dynamic response of structural and/or geotech-nical systems lies in the nonlinear domain to account for materialand geometric nonlinearities governing the complex behavior ofsuch systems, especially near their failure range.

Recent years have seen a growing interest in the analysis of

1Geotechnical Engineer, AMEC Geomatrix, 510 Superior Ave., Suite200, Newport Beach, CA 92663. E-mail: [email protected]

2Assistant Professor, Dept. of Civil and Environmental Engineering,Louisiana State Univ., 3531 Patrick F. Taylor Hall, Nicholson Extension,Baton Rouge, LA 70803. E-mail: [email protected]

3Professor, Dept. of Structural Engineering, Univ. of California at SanDiego, 9500 Gilman Dr., La Jolla, CA 92093-0085 �corresponding au-thor�. E-mail: [email protected]

Note. This manuscript was submitted on September 3, 2008; approvedon April 21, 2009; published online on April 24, 2009. Discussion periodopen until May 1, 2010; separate discussions must be submitted for indi-vidual papers. This paper is part of the Journal of Engineering Mechan-ics, Vol. 135, No. 12, December 1, 2009. ©ASCE, ISSN 0733-9399/
2009/12-1427–1438/$25.00.
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structural response sensitivity to various geometric, mechanical,material, and loading parameters defining a structure and its load-ing environment. FE response sensitivities represent an essentialingredient for gradient-based optimization methods needed invarious sub-fields of structural engineering such as structural op-timization, structural reliability analysis, structural identification,and finite-element model updating �Ditlevsen and Madsen 1996;Kleiber et al. 1997; Franchin 2004�. In addition, stand-alone FEresponse sensitivity analysis is invaluable for gaining deeper in-sight into the effect and relative importance of system and loadingparameters in regards to structural response behavior.

FE response sensitivity analysis can be performed accuratelyand efficiently using the direct differentiation method �DDM�, atthe one-time cost of deriving and implementing response sensi-tivity computation algorithms consistently with the algorithmsused for response-only computation �Zhang and Der Kiureghian1993; Conte 2001; Conte et al. 2003�. FE response sensitivityanalysis capabilities need to be continuously enhanced to takeadvantage of the state-of-the-art in computational simulation. In-deed, this process is undergoing as shown by the active researchin this field in recent years �Conte et al. 2004; Zona et al. 2004;Scott et al. 2004; Barbato and Conte 2005; Haukaas and DerKiureghian 2005; Zona et al. 2005, 2006; Barbato and Conte2006; Haukaas and Scott 2006; Barbato et al. 2007; Haukaas andDer Kiureghian 2007; Scott and Filippou 2007; Scott and Hau-kaas 2008�.

This paper extends the use of DDM-based FE response sensi-tivity analysis to FE models with multi-point constraints �MPCs�.MPCs enforce relations among the degrees of freedom �DOFs� attwo or more distinct nodes in a FE model �Felippa 2001; Cooket al. 2002�. The use of MPCs arises very often in FE structural
analysis, e.g., to enforce �1� an equal-displacement behavior �i.e.,
OF ENGINEERING MECHANICS © ASCE / DECEMBER 2009 / 1427

translations and/or rotations at different nodes are constrained tobe equal�; �2� a rigid body behavior �i.e., displacements at differ-ent nodes are related as if connected with rigid links, e.g., rigidbody constraint, rigid diaphragm constraint, rigid plate constraint,rigid rod constraint, and rigid beam constraint�; and �3� symmetryand antisymmetry conditions. Therefore, widely used commercialFE programs �e.g., SAP2000, ADINA, and ABAQUS� have richlibraries of MPCs based on several different methods for handlingconstraints.

Up to date and to the best of the writers’ knowledge, a clearanalytical and algorithmic treatment of MPCs is missing in thecontext of DDM-based response sensitivity analysis. The problemhas been usually circumvented by using ad hoc modeling solu-tions �e.g., enforcing approximately MPCs by using very stifffictitious elements�, leading to inefficiency and/or inaccuracy thathampered the advantages of using the DDM. This paper presentsa rigorous solution to the problem of handling MPCs in DDM-based FE response sensitivity analysis, thus extending the use ofthe latter to a broader range of FE models.

Finite-Element Response and Response SensitivityAnalysis

The computation via DDM of FE response sensitivities to mate-rial, geometric and loading parameters requires the extension ofthe FE algorithms for response-only computation. If r denotes ageneric scalar response quantity, the sensitivity of r with respectto the material or loading parameter � is defined as the �absolute�partial derivative of r with respect to � evaluated at �=�0, i.e.,�r /�� =�0

, where �0=nominal value of parameter �.In the sequel, following the notation proposed by Kleiber et al.

�1997�, and considering the case of a single sensitivity parameterwithout loss of generality, the basic equations for FE response andresponse sensitivity computation are presented following the deri-vations in Zhang and Der Kiureghian �1993�. It is assumed hereinthat the response of a structural/geotechnical system is computedusing a general-purpose nonlinear FE analysis program based onthe direct stiffness method, employing suitable numerical integra-tion schemes. After spatial discretization using the FE method, theequations of motion of the considered system can be expressed as

M��u�t,�� + C��u�t,�� + R�u�t,��,�� = F�t,�� 1�

where t=time; �=scalar sensitivity parameter considered here asa material constitutive parameter or a loading parameter; u=vector of nodal displacements; M=mass matrix; C=dampingmatrix; R=history dependent internal �inelastic� resisting forcevector; and F=applied dynamic load vector, and a superposed dotdenotes one differentiation with respect to time. In the case of“rigid-soil” earthquake ground excitation, the dynamic load vec-tor takes the form F�t ,��=−M��Lug�t ,�� in which L is an in-fluence coefficient vector and ug�t ,�� denotes the input groundacceleration history.

It is assumed that the time continuous—spatially discreteequation of motion �1� is integrated numerically in time using thefollowing general one-step integration algorithm, which containsthe Newmark-� and Wilson-� methods as special cases �seeChopra 2001�

¨ ˙ ¨
un+1�� = a1un+1�� + a2un�� + a3un�� + a4un�� 2�
1428 / JOURNAL OF ENGINEERING MECHANICS © ASCE / DECEMBER 2

un+1�� = a5un+1�� + a6un�� + a7un�� + a8un�� 3�

where ai �i=1, . . . ,8� are user defined parameters that depend onthe specific time-step integration method employed and the sub-script � . . . �n+1 indicates that the quantity to which it is attached isevaluated at discrete time tn+1. Substituting Eqs. �2� and �3� in Eq.�1� written at time tn+1 yields the following nonlinear matrix al-gebraic equation in the unknowns un+1=u�tn+1� to be solved ateach time step �tn , tn+1�:

�a1M��un+1�� + a5C��un+1�� + R�un+1��,�� − Fn+1�� = 0

�4�

where

Fn+1�� = Fn+1�� − M��a2un�� + a3un�� + a4un��

− C��a6un�� + a7un�� + a8un�� 5�

Assuming that un+1 is the converged solution �up to some itera-tion residuals� for the current time step �tn , tn+1�, and differentiat-ing analytically Eq. �4� with respect to � yields the responsesensitivity equation at the structure level

Kn+1dyn�� ·

dun+1

d�= �dF

d�

n+1

dyn

− �R�un+1��,��

un+1

�6�

where

�dF

d�

n+1

dyn

=dFn+1

d�− �a1

dM

d�+ a5

dC

d�un+1�� 7�

Kn+1dyn�� = a1M�� + a5C�� + Kn+1�� 8�

where Kn+1=static consistent tangent stiffness matrix of the struc-ture at time tn+1.

The above formulation, derived explicitly for dynamic re-sponse sensitivity analysis, contains the quasi-static case as a par-ticular case, obtained simply by discarding in Eqs. �4�–�8� allterms involving the mass and damping matrices as well as theirderivatives with respect to the sensitivity parameter �. In particu-lar, the equilibrium equations, Eq. �4�, and the sensitivity equa-tions, Eq. �6�, reduce to

R�un+1��,�� − Fn+1�� = 0 �9�

Kn+1�� ·dun+1

d�=

dFn+1

d�− �R�un+1��,��

��

un+1

�10�

Handling of Constraints

Response Computation

Constraints enforce additional conditions on the DOFs of a givenFE model. More specifically, a constraint either prescribes thevalue of a DOF �“single-point constraint” or “single freedom con-straint,” e.g., support conditions� or prescribes a relation amongtwo or more DOFs �“MPC” or “multifreedom constraint,” e.g.,rigid links, rigid elements, and rigid diaphragms� �Felippa 2001;Cook et al. 2002�. Furthermore, MPCs can be differentiated in“linear” and “nonlinear” constraints, i.e., they can prescribe linearor nonlinear relations among DOFs. In FE analysis of structural
systems, linear constraints are very common.
009

Handling of single-point constraints is an easy task and re-duces to eliminating the constrained DOFs from the equations ofmotion �dynamic case� or the equations of equilibrium �quasi-static case�. Handling of MPCs requires introducing a set of equa-tions that couple the DOFs affected by the constraints, called“constraint equations.” For linear MPCs, the constraint equationscan be expressed in the following matrix form

Aun+1�� = Q �11�

where A=constant matrix and Q=constant vector. In general, ma-trix A and vector Q may depend on some �e.g., geometric� pa-rameters that are not considered as sensitivity parameters in thispaper.

Three different methods are available in the literature for im-posing MPCs, namely: �1� the “transformation equation method”or “master-slave elimination”; �2� the “Lagrange multipliermethod” or “Lagrange multiplier adjunction”; and �3� the “pen-alty function method” or “penalty augmentation.” These threemethods are well known and widely applied for response-onlycomputation �Felippa 2001; Cook et al. 2002�. Herein, the basicideas underlying these methods are recalled for response-onlycomputation, while the full DDM-based response sensitivity algo-rithm is developed and presented in the following section for eachof these three constraint handling methods. It is noteworthy thatany type of linear MPCs �e.g., equal DOF, rigid link, and rigiddiaphragm� can be handled using the three different methods con-sidered in this paper.

The transformation equation method requires to partition theequations of motion �equilibrium� and the constraint equationsbetween retained �or master� DOFs �denoted by the subscript “r”�and DOFs to be eliminated or condensed out �denoted by thesubscript “c” and also called slave DOFs�. After some algebraicmanipulations, the slave DOFs are explicitly eliminated and theequations of motion �equilibrium� at the structure level are ex-pressed only in terms of the retained DOFs. The number of equa-tions of static/dynamic equilibrium to be solved is reduced by onefor each constraint equation introduced.

The Lagrange multiplier method requires the introduction ofadditional variables � �Lagrange multipliers�. The constraints arereplaced by a system of applied nodal forces called constraintforces, which enforce the constraints. Nodal displacements andLagrange multipliers are computed simultaneously, thus augment-ing the number of equations to be solved by one for each con-straint.

The penalty function method introduces in the equations ofmotion �equilibrium� some penalty terms which can be physicallyinterpreted as fictitious high stiffness elastic elements enforcingthe constraint approximately. Each of these fictitious elements isparameterized by a numerical weight �“penalty weight” or “pen-alty number”�, such that the constraint is satisfied exactly if theweight goes to infinity. The problem size �i.e., number of equa-tions to be solved� is unchanged, but the constraints are imposedonly approximately.

The considered DDM formulation and its extension to accountfor MPCs can be used for FE models including both material andgeometric nonlinearities �Haukaas and Scott 2006; Scott and Fil-ippou 2007; Barbato et al. 2007�. The FE response sensitivityalgorithms presented in this paper extend the DDM to linear andgeometrically/materially nonlinear FE models with linear MPCs,including a comprehensive study of several MPC handling tech-
niques.
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Response Sensitivity Computation

The DDM-based FE response sensitivity computation is based onthe same set of equations for all three methods considered, i.e.

��a1M��un+1�� + a5C��un+1�� + R�un+1��,�� − Fn+1�� = 0

Aun+1�� = Q

Aun+1�� = 0

Aun+1�� = 0�

�12�and

for the dynamic and quasi-static analysis case, respectively. In thesequel: �1� the subscript “n+1” is dropped; �2� the derivations areexplicitly carried out for the quasi-static analysis case; and �3� thefinal governing sensitivity equations are presented for the moregeneral dynamic analysis case without derivations, since theseequations can be readily obtained from the quasi-static case.

Transformation Equation MethodThe structure nodal displacement vector u, nodal resisting forcevector R, nodal applied force vector F, and tangent stiffness ma-trix K are partitioned into DOFs to be retained �subscript “r”� andDOFs to be condensed out �subscript “c”� as

u�� = ur��uc�� 14a�

R�� = Rr��Rc�� 14b�

F�� = Fr��Fc�� 14c�

K�� = Krr�� Krc��Kcr�� Kcc�� 14d�

Partitioning also the constraint matrix A as A= �TrTc�, Eq. �13�can be rewritten as

where

and Trc=−Tc−1Tr in which the inverse of matrix Tc exists if the

imposed MPCs are linearly independent. The subscript “e” de-notes equivalent �or condensed� quantities. The external forces Fe

applied along DOFs ur are equivalent �in the sense of equilib-rium� to the applied external forces Fr and Fc which produce ur

and uc satisfying the constraint equation, Eq. �13b�. The relationbetween the equivalent internal forces Re �acting at DOFs ur� andthe internal resisting forces Rr and Rc �acting at DOFs ur and uc,respectively� is similar to that between Fe, Fr, and Fc.

Differentiating Eqs. �15b� and �16a� with respect to sensitivity
parameter � yields, respectively

duc

d�= Trc

dur

d��17�

and

dRe

d�=

dRr

d�+ Trc

T dRc

d��18�

Differentiating Rr and Rc with respect to �, recognizing from Eqs.�9�, �14a�, and �14b� that Rr=Rr�ur�� ,uc�� ,�� and Rc

=Rc�ur��,uc�� ,��, using the chain rule of differentiation and thetheorem of differentiation of implicit functions yields

�dRr

d�= Krr��

dur

d�+ Krc��

duc

d�+ �Rr

��

u

dRc

d�= Kcr��

dur

d�+ Kcc��

duc

d�+ �Rc

��

u

� �19�

in which

Krr�� = �Rr

�ur

�

, Krc�� = �Rr

�uc

�

Kcr�� = �Rc

�ur

�

, and Kcc�� = �Rc

�uc

�

Substituting Eq. �17� into Eq. �19� and the resulting equation intoEq. �18� yields

dRe

d�= �Krr�� + Krc��Trc + Trc

T Kcr�� + TrcT Kcc��Trc�

dur

d�

+ � �Rr

��

u+ Trc

T �Rc

��

u �20�

Finally, differentiating Eq. �15a� with respect to � and substitutingEq. �20� in the resulting equation yields the following sensitivityequations for quasi-static analysis

�Ke�� ·dur

d�=

dFe

d�− �Re

��

u

duc

d�= Trc

dur

d�� 21�

where

Ke�� = �Re

�ur

�

= Krr�� + Krc��Trc + TrcT Kcr�� + Trc

T Kcc��Trc

�22�

�Re

��

u= �Rr

��

u+ Trc

T �Rc

��

u�23�

It is worth to point out that prior to computing response sensitivi-ties at each load/time step, the equivalent tangent stiffness matrixKe in Eq. �21� has already been computed and factorized forcalculating the response. Thus, solution of Eq. �21� is computa-tionally efficient.

Following a similar derivation as in Eqs. �17�–�23�, the sensi-tivity equations for dynamic analysis are given by �recalling that
the subscript “n+1” has been dropped�

�Ke

dyn�� ·dur

d�= �dF

d�

e

dyn

− �Re

��

u

duc

d�= Trc

dur

d�

du

d�= a1

du

d�+ a2

dun

d�+ a3

dun

d�+ a4

dun

d�

du

d�= a5

du

d�+ a6

dun

d�+ a7

dun

d�+ a8

dun

d�

� �24�

where

Kedyn�� = a1Me�� + a5Ce�� + Ke�� 25�

�dF

d�

e

dyn

=dFe

d�− �a1

dMe

d�+ a5

dCe

d�ur�� 26�

In Eqs. �24�–�26�, the following matrix partitioning and definitionof equivalent matrix are employed

M = Mrr Mrc

Mcr Mcc� and

Me = Mrr + MrcTrc + TrcT Mcr + Trc

T MccTrc �27�

in which the matrix quantities considered are M=M ,C ,dM /d� ,dC /d�. Similarly, the following vector partitioning anddefinition of equivalent vector are also used:

V = Vr

Vc� and Ve = Vr + Trc

T Vc �28�

in which the vector quantities considered are V=dF /d� ,�dF /d��dyn.

It is important to recognize that correct application of theDDM requires the updating of the complete �retained and con-densed out DOFs� vectors of nodal displacement, velocity andacceleration response sensitivities at each time step.

Lagrange Multiplier MethodEq. �13� is rewritten as �Felippa 2001�

�R�u��,�� + AT�� − F�� = 0

Au�� − Q = 0� �29�

where �=vector of Lagrange multipliers. This augmented set ofequations needs to be solved simultaneously for all variables �i.e.,u and �� in both linear and nonlinear analysis. The number ofequations to be solved is equal to the sum of the number ofunrestrained �free� DOFs and the number of imposed constraints.In the Lagrange multiplier method, the constraints are replaced bya system of applied nodal forces −AT�, called constraint forces,which enforce the constraints exactly. Thus, it is clear that theLagrange multipliers � are functions of the sensitivity parametersconsidered in this study �i.e., material and loading parameters�.

For the purpose of response sensitivity analysis, the governingequilibrium equations for the response, Eq. �29�, are differentiatedwith respect to � to obtain the following response sensitivity
equations �in matrix form� in the case of quasi-static analysis:
009

K AT

A 0��

du

d�

d�

d�� = �dF

d�− �R

��

u

0� �30�

In the dynamic analysis case, the response sensitivity equationsbecome �recalling that the subscript “n+1” has been dropped�

� Kdyn AT

A 0��

du

d�

d�

d�� = ��dF

d�dyn

− �R

��

u

0�

du

d�= a1

du

d�+ a2

dun

d�+ a3

dun

d�+ a4

dun

d�

du

d�= a5

du

d�+ a6

dun

d�+ a7

dun

d�+ a8

dun

d�

� �31�

Penalty Function MethodThe following equation is constructed from Eq. �13� �Felippa2001�

R�u��,�� + �AT�A�u�� − F�� − AT�Q = 0 �32�

where �=diagonal matrix of penalty weights, with �ii�0 and�ij =0 for i� j �i , j=1,2 , . . . ,nDOF in which nDOF=number of un-restrained DOFs�. For conservative systems, Eq. �32� is obtainedfrom minimization of the potential energy of the considered struc-tural system augmented by the penalty function 1 /2�Au−Q�T��Au−Q�, also called Courant quadratic penalty function�see Felippa 2001�, i.e., equating to zero the derivative of theaugmented potential energy with respect to u. The penalty termsintroduced in the governing equilibrium equations, Eq. �32�, canbe physically interpreted as forces acting in fictitious high stiff-ness elastic elements enforcing the constraint approximately. Theterm AT�A is commonly referred to as “penalty matrix.” Thestiffnesses of these fictitious elements are parameterized by thepenalty weights �ii so that the constraints are satisfied exactly ifthese weights go to infinity, in which case the solutions of Eqs.�13� and �32� coincide. In the case of structural systems withnonlinear hysteretic material behavior �i.e., nonconservative sys-tems�, Eq. �32� is obtained through the introduction of the penaltyterms by analogy with conservative systems.

The number of equations to be solved in Eq. �32� is the sameas for the unconstrained structural system, but the constraints areimposed only approximately for finite values of the penaltyweights �ii. Particular care is needed in the choice of the penaltyweights to balance the trade-off between reducing the constraintviolation, which requires to increase the penalty weights, and lim-iting the solution error due to ill-conditioning with respect toinversion of the modified tangent stiffness matrix, K+ �AT�A�, ata given iteration of a given load step. This ill-conditioning in-creases for increasing values of the penalty weights.

For response sensitivity analysis, the governing equilibriumequations in Eq. �32� are differentiated with respect to � yieldingthe following response sensitivity equations for quasi-static analy-
sis:
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�K + �AT�A�� ·du

d�=

dF

d�− �R

��

u�33�

In the dynamic analysis case, the response sensitivity equationsare given by �recalling that the subscript “n+1” has beendropped�

��Kdyn + �AT�A�� ·

du

d�= �dF

d�dyn

− �R

��

u

du

d�= a1

du

d�+ a2

dun

d�+ a3

dun

d�+ a4

dun

d�

du

d�= a5

du

d�+ a6

dun

d�+ a7

dun

d�+ a8

dun

d�

� �34�

In order to obtain consistent response sensitivities �Conte et al.2003�, the penalty weights in the sensitivity equations, Eqs. �33�and �34� must be equal to the penalty weights used for responsecomputation in Eq. �32�. Clearly, the choice of the penalty weightvalues affects the accuracy of both response and response sensi-tivity, but in a consistent way.

Validation Examples

The DDM-based response sensitivity computation algorithms forhandling MPCs presented above were implemented by the writersin the finite-element analysis software framework OpenSees�Mazzoni et al. 2005; Gu 2008�. OpenSees is an open sourceobject-oriented software framework written in C�� program-ming language for static and dynamic, linear and nonlinear FEanalysis of structural and/or geotechnical systems. OpenSees pro-vides capabilities for response, response sensitivity and reliabilityanalyses of structural, geotechnical and soil-foundation-structureinteraction �SFSI� systems. Algorithms for response-only analysisbased on FE models with MPCs were already implemented inOpenSees for all three constraint handling methods considered inthis paper. Extension of these algorithms to enable response sen-sitivity analysis while using FE models with MPCs greatly en-hances the existing capabilities of OpenSees. To date, the MPCsavailable in OpenSees are: �1� “equal DOF,” which enforces equaldisplacements/rotations at different DOFs of the FE model; �2�“rigid link,” which imposes a rigid connection between differentDOFs; and �3� “rigid diaphragm,” which imposes a rigid behaviorfor the in-plane motion of nodes belonging to the same plane.

The methodology presented in this paper for DDM-based FEresponse sensitivity computation and its software implementationwas validated by comparing DDM and forward finite difference�FFD� analysis results for several application examples using allthree constraint handling methods considered. Due to space limi-tation, only selected results from two benchmark applicationsanalyzed using the transformation equation method are presentedin the following sections. It is noteworthy that, for each applica-tion example considered, FE response and response sensitivityresults obtained using different constraint handling methodspresent only negligible differences, which are related to the con-vergence tolerance used for response analysis.

Two-Dimensional Soil-Foundation-Structure InteractionSystem

The first application example consists of a two-dimensional SFSI
system subjected to earthquake excitation. The structure is a two-

story, two-bay, reinforced concrete frame, as shown in Fig. 1. Thefoundations consist of reinforced concrete square footings beloweach column. The foundation soil is a layered clay with materialproperties varying along the depth.

The frame structure of this SFSI system is modeled by usingdisplacement-based Euler-Bernoulli frame elements with distrib-uted plasticity and four Gauss-Legendre integration points. Everyphysical beam and column is discretized into three and two frameelements, respectively, of equal length. Section stress resultants atthe integration points are computed by using fiber sections withconcrete and reinforcing steel material layers. The concrete ma-terial is modeled using the Kent-Scott-Park model with no tensionstiffening �Scott et al. 1982�. The concrete constitutive parametersare: fc=concrete peak strength in compression; fu=residualstrength; �0=strain at peak strength; and �u=strain at which theresidual strength is reached. Different material parameters areused for the confined �identified by the subscript “core”� and un-confined �identified by the subscript “cover”� concrete in the col-umns. The constitutive behavior of the reinforcing steel ismodeled using the one-dimensional J2 plasticity model with lin-ear kinematic and isotropic hardening �Conte et al. 2003�. Thematerial parameters defining the J2 plasticity model are: E=Young’s modulus; fy =yield strength; Hkin=kinematic hardeningmodulus; and Hiso= isotropic hardening modulus.

The foundation footings and soil layers are modeled usingisoparametric four-node quadrilateral finite elements with bilineardisplacement interpolation. The foundation footings are modeledas linear elastic with Young’s modulus E=20,000 MPa andPoisson’s ratio �=0.20. The soil mesh is shown in Fig. 1. The soildomain is assumed to be under plane strain condition with a con-stant soil thickness of 4 m, corresponding to the interframedistance. The soil materials are modeled using a pressure-inde-

SFSI system model (unit: m)

7.2

3.6

7.0 7.0

AB

AB

B

B

C C

C C

32.4

3.6

[email protected]

4 @ 2.0 2 @ 2.3

[email protected]

u4u3

u2

u1

u5

u6

∆1 = u1- u2∆2 = u2- u3∆3 = u3- u4∆4 = u4- u5∆5 = u5- u6∆6 = u6

z

IP1

S1

2 @ 1.2

Fig. 1. 2D model of SFSI system: geometry, cross-sectio

ε

σ

fc

fu

ε0εuu

2fc/ε0

kinp

kin

E (HEE H

⋅ +=+ +

Par. UnitConfined Unconfinedε0εufcfu

0.004 0.0020.014 0.008

--

MPaMPa

34.520.7 0.0

27.6

E [MPa]fy [MPa]Hkin [MPaHiso [MPa

Par.

Concrete Steel

Fig. 2. Material constitutiv


pendent multi-yield-surface J2 plasticity model �Prevost 1977; Q.Gu et al. 2009� specialized for plane strain condition. Each of thefour soil layers is characterized by a different set of materialparameters, namely, Gi=low strain shear modulus, Bi=bulkmodulus, �i=shear strength, with i=1,2 ,3 ,4 corresponding to thenumbering of the soil layers �from top to bottom�.

The three types of material constitutive models used in thisapplication example are shown in Fig. 2, together with the valuesof the constitutive parameters. The inertia properties of the framestructure are modeled through translational masses lumped at thenodes, taking into account the structure own weight, as well asdead and live loads. The inertia properties of the soil mesh arerepresented through lumped element mass matrices computedfrom the soil mass density taken as 2 ,000 kg /m3 for all soillayers.

A crucial point in modeling SFSI systems such as the oneconsidered here is the use of MPCs in order to: �1� enforce dis-placement compatibility at the interface between structural�frame� elements and continuum �quadrilateral� elements�column-footing connections�, and �2� impose a simple shear con-dition for the soil �see Fig. 3�. These modeling requirements aresatisfied by using the equal DOF type of MPC.

FE response and response sensitivity analyses are performedusing the model described above. After quasi-static application ofthe gravity loads �modeled as vertical forces concentrated at thenodes of the FE mesh�, earthquake excitation is applied by im-posing a total acceleration time history at the base of the compu-tational soil domain. The ground acceleration time historyconsidered is the balanced 1940 El Centro earthquake recordscaled by a factor of 5 �Fig. 1�. Response sensitivities to all ma-terial parameters �four parameters for the confined concrete, fourfor the unconfined concrete, four for the reinforcing steel, three

0 5 10 15 20−20

0

20

time [sec]

s-section properties (unit: cm) Cover Core

6 bars # 9 6 bars # 8 4 bars # 9

u·· gm

s2⁄

[] 5 x El Centro Earthquake (1940)

z

y

30 25z

y

20 25

2520

Bz

y

20

2520

C 20

operties of structural members, and input ground motion

ε

E

00.2.9

e

131σ ′

231σ ′ 33

1σ ′Deviatoric plane

mfNYSf 2/mH

rG

τNYSτmτ

maxτ

γmγOctahedral shearstress-strain

Hyperbolic backbonecurve

Soil

Layer # Gi [kPa]54450338006125096800

160100180290

1234

τi [kPa]Bi [MPa]332635

44

els used in the SFSI model

Cros

14.4

x

2520

A

nal pr

σ

fy

fy

E

Ep

iso

iso

H )H

200024816120

]]

Valu

e mod

009

for each of the four soil layers, and two for the elastic foundationfootings for a total of 26 material parameters� are computed usingthe DDM.

Figs. 4 and 5 show the time histories of the interstory drifts �atthe middle column� and soil interlayer drifts �at the nodes belowthe middle column�, respectively. In Fig. 4, it is observed that thetime histories of the two interstory drifts are very similar, sincethe overall rotation �rocking� of the structure due to soil flexibilityproduces a major contribution to the floor horizontal displace-ments. Fig. 6 displays the moment-curvature hysteretic responseat the left-most Gauss-Legendre integration point of the left beamat the first story, identified in Fig. 1 as Section 1 �S1�. Fig. 7provides the shear stress-strain hysteretic response at IntegrationPoint 1 �IP1, see Fig. 1� near the bottom of the soil mesh. FromFigs. 5–7, it is observed that, during the earthquake considered,

Nodes on the boundaries at same elevation z are tied

Soil Beam Nodes tied togetherFooting

Equal DOF: ui = uj;

together (both horiz. and vert.)

i, j: DOF numbers

Fig. 3. Multipoint constraints used in the SFSI model: equal DOF

0 1 2 3 4

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Tim

Interstorydrifts[m]

Fig. 4. Response of the 2D SFSI s

0 1 2 3 4

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

∆3

∆4

∆5

∆6

Tim

Soilinterlayerdrifts[m]

Fig. 5. Response of the 2D SFSI sys

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the SFSI system undergoes significant inelastic behavior in boththe structure and the foundation soil. In particular, large inelasticand residual interlayer drifts are induced in the bottom soil layer�see Fig. 5� due to �1� the high intensity earthquake excitationconsidered �peak ground acceleration at the base of the computa-tional soil domain=1.59 g� and �2� the low postyield shear stiff-ness in the clay soil model.

In conjunction with the FE response analysis, a FE responsesensitivity analysis is performed. Asymptotic convergence of FFDanalysis results �i.e., convergence for increasingly smaller pertur-bations of the sensitivity parameter� to DDM results has been

5 6 7 8 9 10

∆1

∆2

: time histories of interstory drifts

5 6 7 8 9 10

ime histories of soil interlayer drifts

−0.015 −0.01 −0.005 0 0.005 0.01 0.015−150

−100

−50

0

50

100

150

Moment[kN-m]

χ [m-1]

Fig. 6. Response of the 2D SFSI system: moment-curvature hyster-etic response at section S1 �see Fig. 1�

e [s]

ystem

e [s]

tem: t


verified for several global and local response quantities as well asfor all sensitivity parameters considered. Due to space limitation,only few of these results are presented here. Fig. 8 shows theasymptotic convergence of FFD-based to DDM-based sensitivi-ties of the first interstory drift �global response quantity� to theyield stress of the reinforcing steel. Fig. 9 provides a zoom viewof this convergence trend. Response sensitivities are given in nor-malized form, i.e., multiplied by the nominal value of the corre-sponding sensitivity parameter, and can thus be interpreted as onehundred times the change in the response quantity due to onepercent change in the sensitivity parameter. These normalizedsensitivities directly provide the relative importance of the differ-ent parameters on the response quantity of interest. The conver-gence studies performed validate the analytical formulation andthe computer implementation of the DDM algorithms for han-dling MPCs in regards to the two-dimensional SFSI system con-sidered here as first application example. Fig. 10 displays the timehistories of the sensitivities of the frame first interstory drift to thefollowing material parameters: �1� fc,core=strength of core con-crete; �2� fy =yield stress of reinforcing steel; and �3� �3=shearstrength of the third soil layer �second layer from the bottom�.Each of these three parameters, fc,core, fy, and �3, represents theparameter to which the first interstory drift time history is mostsensitive for each of the three material types �concrete, reinforc-ing steel, and soil, respectively�. It is noteworthy that, in general,

−0.1 −0.05 0 0.05 0.1−60

−40

−20

0

20

40

60

τ[kPa]

γ

Fig. 7. Response of the 2D SFSI system: shear stress-strain hyster-etic response at integration point IP1 of the soil �see Fig. 1�

0 1 2 3 4

−0.15

−0.1

−0.05

0

0.05

0.1

Ti

d∆2

dfy

---------

f y[m]

⋅

Fig. 8. Validation of DDM results for the 2D SFSI system through Fof reinforcing steel


the strength related material parameters become predominant inaffecting the system response when the system undergoes signifi-cant yielding �Barbato 2007; Gu 2008� as in the present case. Asshown in Fig. 10, �3 is the parameter with the largest peak abso-lute value �over the entire time history� of the normalized sensi-tivities. The sensitivity of 2 with respect to �3 remains positiveover most of the earthquake duration, indicating that an increasein �3 produces an increase in 2. In fact, the value of �3 sets alimit on the seismic load that can be transferred from the base ofthe soil domain to the base of the foundations and therefore to thesuperstructure. If the earthquake excitation is strong enough toyield the soil, an increase in �3 produces larger ground surfaceacceleration. Fig. 10 also shows that fy is the second most impor-tant parameter in regards to 2. The sensitivity of 2 with respectto fy is negative over most of the earthquake duration, consis-tently with the intuitive expectation that an increase in fy reduces2, everything else remaining the same.

Three-Dimensional Three-Story Reinforced ConcreteFrame Structure with Rigid Diaphragm Behavior atEach Floor

The second application example consists of a three-dimensional,one-bay, three-story reinforced concrete frame building with con-

5 6 7 8 9 10

DDM

∆fy/f

y=1e−1

∆fy/f

y=1e−3

∆fy/f

y=1e−4

zoom view

alysis: normalized sensitivity of first interstory drift to yield stress fy

5.6 5.65 5.7 5.75 5.80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

DDM∆f

y/f

y= 1e−1

∆fy/f

y= 1e−3

∆fy/f

y= 1e−4

Time [s]

d∆2

dfy

---------

f y[m]

⋅

Fig. 9. Validation of DDM results for the 2D SFSI system throughFFD analysis: zoom view of normalized sensitivity of first interstorydrift to yield stress fy of reinforcing steel

me [s]

FD an

009

crete slabs at each floor, as shown in Fig. 11. Each story is ofheight h=12 ft �h=3.66 m� and the bay is of span L=20 ft �L=6.10 m� in both horizontal directions. All columns are ofsquared cross-sectional shape with side d=18 in �0.457 m�, eight#8 rebars of longitudinal reinforcement and reinforcement coverc=1.5 in �0.038 m�. All beams have a rectangular cross section 24in �0.610 m� deep and 18 in �0.457 m� wide.

The floor concrete slabs are modeled through a rigid dia-phragm MPC at each floor level. Column and beam members are

0 1 2 3 4−0.1

−0.05

0

0.05

0.1

0.15

fc,core

fy

τ3

Ti

d∆2

dθ---------

θ[m]

⋅

Fig. 10. Response sensitivities of the 2D SFSI system: normalized

0 5 10 15 20 25−1

0

1

0 5 10 15 20 25−1

0

1

12 ft

12 ft

12 ft

15 1619

1110

18

1312

17

7

14

985 6

(1 foot = 0.3048 m)

20 ft 20 ft

x

y

z

∆i1 = ui,9

∆i2 =ui,14-ui,9

∆i3 =ui,19-ui,14

(i = x, y)

34

1 2

Ground acceleration

Ground acceleration

Time [s]

Time [s]

u·· g[g]

u·· g[g]

(1 g = 9.81 m/s2)

in x-direction

in y-directionS1

Fig. 11. 3D one-bay three-story reinforced concrete building: struc-tural model and input ground motion

0 5

−0.04

−0.02

0

0.02

0.04

0.06

0.08

∆x1

∆x2

∆x3

Interstorydrifts[m]

Fig. 12. Response of 3D building: time

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modeled using two and one, respectively, displacement-basedEuler-Bernoulli frame elements with four Gauss-Legendre inte-gration points each. Column cross sections are discretized in fi-bers of confined concrete, unconfined concrete and steelreinforcement. Beam cross sections are assumed to have a linearelastic behavior. A detailed description of the FE model of thisstructure is presented elsewhere �Barbato 2007; Gu 2008�. Thematerial constitutive models used are �1� the Kent-Scott-Parkmodel for concrete, with parameters fc,core=34.5 MPa, fu,core

=24.1 MPa, �0,core=0.005, �u,core=0.020 for confined concrete,and fc,cover=27.6 MPa, fu,cover=0 MPa, �0,cover=0.002, �u,cover

=0.006 for unconfined concrete, and �2� the one-dimensional J2

plasticity model for reinforcing steel, with parameters E=210 GPa, fy =248 MPa, Hkin=4.29 GPa, and Hiso=0 Pa.

After quasi-static application of the gravity loads, the structureis subjected to a bidirectional earthquake excitation with groundacceleration time histories taken as the two horizontal compo-nents of the 1978 Tabas earthquake �see Fig. 11�. Figs. 12 and 13plot the time histories of the interstory drifts in the x- andy-directions, respectively. As expected, the interstory drift is larg-est for the first story and decreases with story elevation. The peakfirst interstory drift in the x-direction �0.080 m� is slightly largerthan that in the y-direction �0.069 m�. These peak interstory driftscorrespond to significant interstory drift ratios of 2.20% and1.89%, respectively. Fig. 14 shows the moment-curvature hyster-etic response in the x-direction at the integration point identifiedas Section 1 �S1 in Fig. 11�. Significant inelastic flexural behavioris observed at this specific location. The irregular �“wavy”�moment-curvature hysteresis loops are due to the interaction be-tween axial and bidirectional bending behaviors.

In conjunction with the FE response analysis, a DDM-basedresponse sensitivity analysis is performed, considering as sensi-

5 6 7 8 9 10

vities of first interstory drift to most important material parameters

10 15[s]

ies of interstory drifts in the x-direction

me [s]

sensiti

Time

histor


tivity parameters the material parameters characterizing the con-fined concrete �four parameters�, unconfined concrete �fourparameters�, reinforcing steel �four parameters� and the stiffnessproperties of the elastic beams �one parameter: Young’s modulusE=24.9 GPa�, for a total of 13 material parameters. Fig. 15shows the asymptotic convergence of FFD-based to DDM-basedsensitivities of the first interstory drift �a global response quantity�in the x-direction, considering as sensitivity parameter the com-pressive strength of the confined concrete fc. Fig. 16 provides a

0 5

−0.04

−0.02

0

0.02

0.04

0.06

0.08∆

y1

∆y2

∆y3

Interstorydrifts[m]

Fig. 13. Response of 3D building: time

−0.04 −0.03 −0.02 −0.01 0 0.01 0.02−400

−300

−200

−100

0

100

200

300

Moment[kN-m]

χ [m-1]

Fig. 14. Response of 3D building: moment-curvature hysteretic re-sponse about the x-axis at Section 1 �S1 in Fig. 11�

0 5−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04DDM

∆fc/f

c= 5e−1

∆fc/f

c= 1e−1

∆fc/f

c= 1e−5

du9

dfc

--------

f c[m]

⋅

Fig. 15. Validation of DDM results for 3D building through FFD anaconcrete strength fc


zoom view of these convergence results. Fig. 17 displays theasymptotic convergence of FFD-based to DDM-based responsesensitivity results, in which the moment about the x-axis at Sec-tion 1 �a local response quantity� is considered as response quan-tity and the Young’s modulus of the reinforcing steel is consideredas sensitivity parameter. A zoom view of the above convergencetrend is given in Fig. 18. The results of the convergence studiesperformed validate the analytical formulation and computerimplementation of the DDM algorithms for handling MPCs in thecontext of this second application example. Fig. 19 plots the sen-sitivity time histories of the first interstory drift in the x-directionto the Young’s modulus of the reinforcing steel E, strength of theunconfined concrete fc,cover, and yield stress of the reinforcingsteel fy. These parameters �in order of decreasing relative impor-tance� are the ones to which the considered response parameter,x1, is most sensitive among all 13 sensitivity parameters consid-ered. In this case, a stiffness related sensitivity parameter �E� isdominant in regards to the first interstory drift x1, suggesting thatthe plastic deformations experienced by the building frame arelimited and the elastic properties control significantly the seismicresponse behavior of the system.

Conclusions

This paper presents the extension of the direct differentiationmethod �DDM� to finite-element �FE� models with multi-pointconstraints �MPCs�. The DDM is an accurate and efficient methodfor computing FE response sensitivities to material, geometric

10 15[s]

ies of interstory drifts in the y-direction

10 15[s]

zoom view

ormalized sensitivity of first interstory drift in the x-direction to core

Time

histor

Time

lysis: n

009

and loading parameters. Response sensitivity algorithms for bothquasi-static and dynamic FE analyses are developed for three dif-ferent constraint handling techniques, namely: �1� the transforma-tion equation method; �2� the Lagrange multiplier method; and �3�the penalty function method.

The response sensitivity algorithms developed in this paper areimplemented into a general-purpose nonlinear FE analysis frame-work. The methodology is illustrated through two applicationexamples: �1� a two-dimensional soil-foundation-structure inter-action system subjected to earthquake excitation, and �2� a three-dimensional, one-bay by one-bay, three-story reinforced concreteframe structure with floor slabs modeled as rigid diaphragms sub-jected to bidirectional earthquake excitation. DDM-based re-sponse sensitivities are compared with the corresponding resultsobtained through forward finite difference �FFD� analyses withdecreasing values of the parameter perturbations. Asymptoticconvergence of FFD-based to DDM-based response sensitivityresults validates the algorithms presented and their computerimplementation.

The developments presented in this paper close an importantgap between FE response-only analysis and FE response sensitiv-ity analysis through the DDM, extending the latter to applicationsrequiring response sensitivities using FE models with MPCs.

12.4 12.45 12.5 12.55 12.6 12.65 12.7−12

−10

−8

−6

−4

−2

0

2

4

6x 10

−3

DDM∆f

c/f

c= 5e−1

∆fc/f

c= 1e−1

∆fc/f

c= 1e−5

Time [s]

du9

dfc

--------

f c[m]

⋅

Fig. 16. Validation of DDM results for 3D building through FFDanalysis: zoom view of normalized sensitivity of first interstory driftin the x-direction to core concrete strength fc

0 5−5000

0

5000

DDM

∆ E/E =5e−1

∆ E/E =1e−1

∆ E/E =1e−3

dMx

dE-----------

E[kN-m]

⋅

Fig. 17. Validation of DDM results for 3D building through FFD ana1 �S1 in Fig. 11� to Young’s modulus E of reinforcing steel

JOURNAL

Such applications include structural optimization, structural reli-ability analysis, and finite-element model updating.

Acknowledgments

The writers gratefully acknowledge the support of this researchby �1� the National Science Foundation under Grant No. CMS-0010112; �2� the Pacific Earthquake Engineering Research�PEER� Center through the Earthquake Engineering ResearchCenters Program of the National Science Foundation under AwardNo. EEC-9701568; and �3� the Louisiana Board of Regentsthrough the Pilot Funding for New Research �Pfund� Program ofthe National Science Foundation Experimental Program to Stimu-late Competitive Research �EPSCoR� under Award No.NSF�2008�-PFUND-86. The writers wish to thank Dr. FrankMcKenna for his invaluable help in implementing the responsesensitivity algorithms for handling multi-point constraints in Op-enSees. Any opinions, findings, conclusions or recommendationsexpressed in this publication are those of the writers and do notnecessarily reflect the views of the sponsors.

10 15e [s]

view

ormalized sensitivity of moment response about the x-axis at Section

7.75 7.8 7.85 7.9 7.95 8

−4000

−3000

−2000

−1000

0

1000

2000

DDM∆ E/E =5e−1∆ E/E =1e−1∆ E/E =1e−3

Time [s]

dMx

dE-----------

E[kN-m]

⋅

Fig. 18. Validation of DDM results for 3D building through FFDanalysis: zoom view of normalized sensitivity of moment responseabout the x-axis at Section 1 �S1 in Fig. 11� to Young’s modulus E ofreinforcing steel

Tim

zoom

lysis: n


References

Advanced Technology Council �ATC-55�. �2005�. Evaluation and im-provement of inelastic analysis procedure, Advanced TechnologyCouncil, Redwood City, Calif.

Barbato, M. �2007�. “Finite element response sensitivity, probabilisticresponse and reliability analyses of structural systems with applica-tions to earthquake engineering.” Ph.D. dissertation, Dept. of Struc-tural Engineering, Univ., of California at San Diego, La Jolla, Calif.

Barbato M., and Conte J. P. �2005�. “Finite element response sensitivityanalysis: a comparison between force-based and displacement-basedframe element models.” Comput. Methods Appl. Mech. Eng., 194�12–16�, 1479–1512.

Barbato, M., and Conte, J. P. �2006�. “Finite element structural responsesensitivity and reliability analyses using smooth versus non-smoothmaterial constitutive models.” Int. J. of Reliability and Safety, 1�1–2�,3–39.

Barbato, M., Zona, A., and Conte, J. P. �2007�. “Finite element responsesensitivity analysis using three-field mixed formulation: generaltheory and application to frame structures.” Int. J. Numer. MethodsEng., 69�1�, 114–161.

Chopra, A. K. �2001�. Dynamics of structures: Theory and applicationsto earthquake engineering, 2nd Ed., Prentice-Hall, Upper SaddleRiver, N.J.

Conte, J. P. �2001�. “Finite element response sensitivity analysis in earth-quake engineering.” Earthquake engineering frontiers in the new mil-lennium, Spencer & Hu, Swets & Zeitlinger, Lisse, The Netherlands,395–401.

Conte, J. P., Barbato, M., and Spacone, E. �2004�. “Finite element re-sponse sensitivity analysis using force-based frame models.” Int. J.Numer. Methods Eng., 59�13�, 1781–1820.

Conte, J. P., Vijalapura, P. K., and Meghella, M. �2003�. “Consistent finiteelement response sensitivity analysis.” J. Eng. Mech., 129�12�, 1380–1393.

Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. �2002�. Con-cepts and applications of finite element analysis, 4th Ed., Wiley, NewYork.

Ditlevsen, O., and Madsen, H. O. �1996�. Structural reliability methods,Wiley, New York.

Felippa, C. A. �2001�. “Introduction to finite element method.” Ebook,�http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/Home.html� �May 10, 2009�.

Franchin, P. �2004�. “Reliability of uncertain inelastic structures underearthquake excitation.” J. Eng. Mech., 130�2�, 180–191.

Gu, Q. �2008�. “Finite element response sensitivity and reliability analy-sis of soil-foundation-structure-interaction systems.” Ph.D. disserta-tion, Dept. of Structural Engineering, Univ. of California at San

0 5

−0.1

−0.05

0

0.05

0.1

0.15

E

fc,cover

fy

d∆x1dθ-----------

θ[m]

⋅

Fig. 19. Response sensitivities of 3D building: normalized sensitivparameters

Diego, La Jolla, Calif.


Gu, Q., Conte, J. P., Elgamal, A., and Yang, Z., �2009�. “Finite elementresponse sensitivity analysis of multi-yield-surface J2 plasticity modelby direct differentiation method.” Comp. Meth. Appl. Mech. Eng.198�30–32�, 2272–2285.

Haukaas, T., and Der Kiureghian, A. �2005�. “Parameter sensitivity andimportance measures in nonlinear finite element reliability analysis.”J. Eng. Mech., 131�10�, 1013–1026.

Haukaas, T., and Der Kiureghian, A. �2007�. “Methods and object-oriented software for FE reliability and sensitivity analysis with ap-plication to a bridge structure.” J. Comput. Civ. Eng., 21�3�, 151–163.

Haukaas, T., and Scott, M. H. �2006�. “Shape sensitivities in the reliabil-ity analysis of nonlinear frame structures.” Comput. Struc., 84�15–16�, 964–977.

Kleiber, M., Antunez, H., Hien, T. D., and Kowalczyk, P. �1997�. Param-eter sensitivity in nonlinear mechanics: Theory and finite elementcomputations, Wiley, New York.

Mazzoni, S., McKenna, F., and Fenves, G. L. �2005�. “OpenSees Com-mand Language Manual.” Pacific Earthquake Engineering Center,University of California, Berkeley, �http://opensees.berkeley.edu/��May 10, 2009�.

Prevost, J. H. �1977�. “Mathematical modelling of monotonic and cyclicundrained clay behaviour.” Int. J. Numer. Analyt. Meth. Geomech.,1�2�, 195–216.

Scott, B. D., Park, P., and Priestley, M. J. N. �1982�. “Stress-strain be-havior of concrete confined by overlapping hoops at low and high-strain rates.” J. Am. Concr. Inst., 79�1�, 13–27.

Scott, M. H., and Filippou, F. C. �2007�. “Response gradients for nonlin-ear beam-column elements under large displacements.” J. Struct.Eng., 133�2�, 155–165.

Scott, M. H., Franchin, P., Fenves, G. L., and Filippou, F. C. �2004�.“Response sensitivity for nonlinear beam-column elements.” J. Struct.Eng., 130�9�, 1281–1288.

Scott, M. H., and Haukaas, T. �2008�. “Software framework for parameterupdating and finite element response sensitivity analysis.” J. Comput.Civ. Eng., 22�5�, 281–291.

Zhang, Y., and Der Kiureghian, A. �1993�. “Dynamic response sensitivityof inelastic structures.” Comput. Methods Appl. Mech. Eng., 108,23–36.

Zona, A., Barbato, M., and Conte, J. P. �2004�. “Finite element responsesensitivity analysis of steel-concrete composite structures.” Rep. No.SSRP-04/02, Dept. of Structural Engineering, Univ. of California, SanDiego, Calif.

Zona, A., Barbato, M., and Conte, J. P. �2005�. “Finite element responsesensitivity analysis of steel-concrete composite beams with deform-able shear connection.” J. Eng. Mech., 131�11�, 1126–1139.

Zona, A., Barbato, M., and Conte, J. P. �2006�. “Finite element responsesensitivity analysis of continuous steel-concrete composite girders.”

10 15[s]

first interstory drift in the x-direction to most important material

Time

ity of

Steel Compos. Struct., 6�3�, 183–202.

009

handling of constraints in finite-element response ...jaguar.ucsd.edu/sensitivity/gu_2009_b.pdf ·...

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