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Computers and Structures 84 (2006) 964–977

Shape sensitivities in the reliability analysisof nonlinear frame structures

Terje Haukaas a,*, Michael H. Scott b

a Department of Civil Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, Canada V6T 1Z4b Department of Civil, Construction, and Environmental Engineering, Oregon State University, Corvallis, OR 97331, United States

Received 6 October 2005; accepted 15 February 2006

Abstract

A unified and comprehensive treatment of shape sensitivity that includes variations in the nodal coordinates, member cross-sectionproperties, and global shape parameters of inelastic frame structures is presented. A novelty is the consideration of geometric uncertaintyin both the displacement- and force-based finite element formulations of nonlinear beam-column behavior. The shape sensitivity equa-tions enable a comprehensive investigation of the relative influence of uncertain geometrical imperfections on structural reliability assess-ments. For this purpose, finite element reliability analyses are employed with sophisticated structural models, from which importancemeasures are available. The unified approach presented herein is based on the direct differentiation method and includes variations inthe equilibrium and compatibility relationships of frame finite elements, as well as the member cross-section geometry, in order to obtaincomplete shape sensitivity equations. The analytical shape sensitivity equations are implemented in the OpenSees software framework.Numerical examples involving a steel structure and a reinforced concrete structure confirm that geometrical imperfections may have asignificant impact on structural reliability assessments.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Shape sensitivity; Direct differentiation method; Geometrical imperfections; Structural reliability; Beam-columns; Nonlinear analysis;OpenSees

1. Introduction

A number of applications in structural engineeringrequire the computation of the gradient of structuralresponse quantities with respect to input parameters. Thisis referred to as response sensitivity analysis. The mostcommon applications of response sensitivity analysis areto optimization problems, such as the minimization ofstructural cost subject to constraints and minimization ofthe difference between measured and numerical responsefor system identification purposes. Yet another optimiza-tion problem is posed in structural reliability analysis by

0045-7949/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2006.02.014

* Corresponding author. Tel.: +1 604 827 5557; fax: +1 604 822 6901.E-mail addresses: terje@civil.ubc.ca (T. Haukaas), michael.scott@

oregonstate.edu (M.H. Scott).

the first and second-order reliability methods (FORMand SORM). These methods rely upon the determinationof the ‘‘most probable failure point’’, which is the solutionto a constrained optimization problem in the space of ran-dom variables. As a by-product, FORM analysis providesimportance measures to rank the uncertain parametersaccording to their relative influence on the structural reli-ability. Importance measures remedy the problem thatindividual response sensitivities cannot be compareddirectly due to differing units. It is also emphasized thatresponse sensitivities are useful as a stand alone productin structural design because they indicate the sensitivityof a structural response quantity to changes in the designparameters.

Three requirements are put forward by the applicationof response sensitivities in gradient-based optimizationalgorithms: efficiency, accuracy, and consistency. Efficient

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 965

computation of response sensitivities is required to makegradient-based algorithms competitive with gradient-freemethods, including response surface methods, in terms ofcomputational cost. This is particularly important whenthe optimization is performed in a high-dimensional spaceof variables, in which case repeated runs to obtain gradi-ents by finite differences is infeasible. Accuracy is impera-tive to avoid convergence problems in the optimizationalgorithms, which tend to perform poorly in the presenceof even small inaccuracies in the gradients. The consistencyrequirement stems from the utilization of approximatenumerical models to obtain the structural response.Because it is the approximate response that is employedin the optimization problem, it is the gradient of theapproximate response that is required. Consequently, it isnot of interest to pursue the ‘‘exact’’ gradient of the theo-retical boundary value problem. In fact, this would leadto inconsistency between the function value and itsgradient.

Two approaches are available to obtain response sensi-tivities: finite difference methods (FDMs) and the direct dif-ferentiation method (DDM). The finite difference approachemploys re-runs of the structural analysis with perturbedparameter values to estimate the response sensitivity. Asa result, it is a computationally inefficient approach. More-over, FDMs suffer from accuracy concerns. It is a nontriv-ial task to select the value of the parameter perturbationfor nonlinear problems. If the perturbation is too small,round-off errors are introduced; while if the perturbationis too large, local nonlinearities may lead to inaccurate esti-mates of the sensitivity. The consistency requirement,however, is satisfied by the FDMs because it is the approx-imate response that is employed in the finite differenceequations.

The DDM provides an attractive alternative to FDMs.At the one-time cost of deriving and implementing analyt-ical sensitivity equations within the finite element responsealgorithm, efficient, accurate, and consistent response sen-sitivities are obtained. No finite difference computationstake place within the DDM; instead, the response equa-tions are analytically differentiated and implemented onthe computer alongside the ordinary response computa-tions. A number of researchers have contributed to thedevelopment of such analytical equations, including Choiand Santos [2], Tsay and Arora [24], Liu and Der Kiuregh-ian [15], Zhang and Der Kiureghian [25], Kleiber et al. [13],Conte et al. [3], Roth and Grigoriu [20], Scott et al. [22],and Haukaas and Der Kiureghian [9]. The DDM is moreefficient than FDMs because repeated runs of the responseanalysis are unnecessary. Accuracy is ensured at the sameprecision as the response because the same equation solveris employed to obtain both the response and the responsesensitivity. Consistency is achieved by differentiating theresponse equations after they have been spatially and tem-porally discretized by the finite element procedures. TheDDM is thus the preferred approach to computingresponse sensitivities.

In this paper, the OpenSees software framework [16] isextended and applied for shape sensitivity analysis. Open-Sees (open system for earthquake engineering simulation)is an open-source, object-oriented, general-purpose finiteelement code specifically developed for earthquake engi-neering analysis. OpenSees began as the computationalplatform for testbed simulations in the Pacific EarthquakeEngineering Research Center (PEER) and has since beenadopted by the NSF-sponsored George E. Brown Jr. Net-work for Earthquake Engineering Simulation (NEES).Work by Haukaas and Der Kiureghian [8] extends Open-Sees with response sensitivity and reliability analysis capa-bilities which allow the analyst to characterize inputparameters as random variables and compute probabilitiesof structural response events. This is termed finite elementreliability analysis (FERA), which differs from so-calledstochastic finite element methods that focus on second-moment statistics of the response. In contrast, reliabilityanalysis and specifically FERA is suited to compute prob-abilities of rare response events. This addresses the growingdemand in performance-based engineering to assess struc-tural behavior during rare events of intense loading in aprobabilistic manner.

The response sensitivity implementations in OpenSeesare based on the DDM. The implementations are dividedinto an overarching framework and object-specific imple-mentations. The latter reflect the fact that OpenSees isorganized into element, section, and material objects. Theframework for sensitivity computations, as well as specificimplementations for selected elements, sections, and mate-rials, is already in place. This includes sensitivities withrespect to nodal coordinates, which previous studies sug-gest may be an important source of uncertainty in struc-tural reliability, particularly when nonlinear structuralbehavior is considered [9].

In this paper, the DDM shape sensitivity equationsinclude response sensitivities with respect to: (1) nodalcoordinates, (2) global structural or member shape param-eters, and (3) the dimensions and details of fiber-discretizedcross-sections. Of particular significance is the developmentof unified shape sensitivity equations for beam-column ele-ments in both the displacement- and force-based formula-tions. Gradient computations that incorporate shapesensitivity at all levels (structure, element, and section)are presented and their implementation in OpenSees allowsthe inclusion of a wide range of uncertain geometricalimperfections in a reliability analysis. Two numericalexamples involving a steel structure and a reinforcedconcrete structure provide insight into the importance ofuncertain geometrical imperfections relative to other uncer-tain structural properties.

2. The application of response sensitivities in finite element

reliability analysis

The need for response sensitivities in this paper stemsfrom structural reliability analysis. To achieve accurate

966 T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977

reliability assessments, sophisticated structural models areemployed to simulate structural performance. Clearly, suchpredictions can only be made in a probabilistic sense due touncertainties in the model and the input parameters. Thismotivates the utilization of FERA to obtain probabilisticpredictions of response events. Indeed, the emerging per-formance-based engineering approach is envisioned to beimplemented in a reliability framework [6,17]. The primaryobjective in FERA is to obtain the probability of rareresponse events that are identified by user-defined perfor-mance functions. An important by-product of the analysisis parameter importance measures to rank the variablesaccording to their relative importance. Response sensitivi-ties represent an essential ingredient in the analysis, asdescribed in the following.

The reliability problem for the case of a single-performance function is formulated as the multi-foldintegral

p ¼Z� � �Z

g60

f ðHÞdH; ð1Þ

where p is the sought probability, g is the performancefunction that identifies the response event for which theprobability is sought, and f(H) is the joint probability den-sity function for the random variables, which are collectedin the vector H. In FERA the performance function is spec-ified in terms of response quantities U = U(H) from a finiteelement analysis. The random variables are commonlyspecified by marginal probability distributions and correla-tion coefficients. Analytical solutions to Eq. (1) are unavail-able; however, methods such as FORM and SORM andsampling techniques provide approximate solutions. Ofparticular interest in finite element reliability analysis isFORM, followed by efficient importance sampling to cor-rect for potential nonlinearities. This analysis strategy isbeneficial because it requires relatively few evaluations ofthe performance function and, in effect, few executions ofthe finite element analysis. Moreover, FORM analysis gen-erates the parameter importance measures that are utilizedin this paper.

In FORM, the integration boundary g = 0 in Eq. (1) isapproximated by a hyperplane in the transformed spaceY = Y(H) of uncorrelated standard normal random vari-ables. For nonlinear performance functions, the ideal pointof approximation is the point on the surface g = 0 that isclosest to the origin in the Y-space. This point, termedthe most likely failure point (MPP) and denoted Y*, isthe solution to the constrained optimization problem

Y� ¼ argmin kYk j g ¼ 0f g: ð2Þ

The most efficient algorithms available to solve this optimi-zation problem utilize the gradient of the performancefunction; namely, $g = og/oY. The chain rule of differenti-ation applied to the performance function yields

ogoY¼ og

oU

oU

oHoHoY

: ð3Þ

The derivative og/oU is readily available since g is a simplealgebraic function of the response quantities U. The matrixoU/oH signifies the need to compute response gradients,which is the focus this paper, and the matrix oH/oY isthe Jacobian of the probability transformation. The Nataftransformation [14] is applied in this work, for which therequired Jacobian matrix is already available in OpenSees.This transformation is an attractive alternative to theRosenblatt transformation [19,10] because it allows a widerrange of correlation values for a variety of probability dis-tribution types. The communication between the reliabilityalgorithm and the finite element module consists of updat-ing the finite element model with realizations of the ran-dom variables H and returning U and oU/oH each timethe performance function is evaluated.

Upon determination of Y*, the probability p accordingto FORM is determined by

p ¼ Uð�bÞ; ð4Þ

where U is the standard normal cumulative distributionfunction and b is the reliability index defined in FORMas b = kY*k. Importance sampling with the sampling distri-bution centered at Y* may subsequently be performed sinceit is an efficient scheme compared with Monte Carlo sam-pling centered at the mean realization of the randomvariables.

The developments in this paper allow the characteriza-tion of imperfections in nodal coordinates and membercross-section geometry as random variables, in additionto random material and load variables. Of particular inter-est is the investigation of the importance of these variablesrelative to other sources of uncertainty. Clearly, the com-ponents of the vectors oU/oH cannot be employed for thispurpose due to the differing dimensions of the variablesh 2 H. Instead, importance measures from FORM are uti-lized, in which the components have uniform dimensions.The basis for these measures are presented by Hohenbich-ler and Rackwitz [11] and Bjerager and Krenk [1]. Applica-tions of importance measures in FERA is presented byHaukaas and Der Kiureghian [9], where the following vec-tor ranks the random variables:

c ¼ � ogoY

JY�;h�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidiag J�1

Y�;h�J�TY�;h�

� �r; ð5Þ

where og/oY is provided by Eq. (3), JY�;h� is the Jacobianmatrix of the probability transformation at the MPP, andffiffip implies the square root of each element of the argumentmatrix. It is noted that c reduces to �og/oY, which is ascaled version of the well known ‘‘alpha-vector’’ in reliabil-ity analysis when no correlation between the random vari-ables is present. Furthermore, it is common to scale thec-vector so that kck = 1. The elements of c are interpretedas the contributions from the individual random variableson the reliability of the structure. Moreover, a negative(positive) c-component indicates that the correspondingrandom variable acts as a resistance (load) variable.

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 967

3. Top-level response sensitivity equations

The gradient of the performance function in Eq. (3) sig-nifies the need to compute sensitivities of the structuralresponse, oU/oH. To compute the response sensitivity bythe DDM, the equations that govern the structuralresponse are differentiated. For the set of parameters Hthat describe the material, geometric, and load parametersof a structural model, the global equations of static equilib-rium have the form

PrðUðHÞ;HÞ ¼ PfðHÞ; ð6Þ

where Pr is the vector of internal resisting forces of thestructure. The internal forces may depend explicitly uponH, as well as implicitly through the nodal displacement re-sponse vector U. The vector Pf represents the externalloads applied to the structure. Inertial and damping forcesare omitted from Eq. (6) because dynamic equilibrium ef-fects are independent of the element and section formula-tions considered in this paper. The extension of Eq. (6) todynamic equilibrium and the computation of the corre-sponding response sensitivity are straightforward [5].

To formulate the equations for response sensitivities atthe structural level, Eq. (6) is differentiated with respectto any parameter h selected from the vector H:

oPr

oU

oU

ohþ oPr

oh

����U

¼ oPf

oh: ð7Þ

The explicit and implicit dependence of Pr on h are takeninto account by the chain rule of differentiation. The vectoroPr/ohjU is the derivative of the resisting forces conditionedupon fixed displacements. This vector is assembled fromelement contributions in the same manner as the resistingforce vector itself. The vector oPf/oh is the derivative ofthe external load, which is nonzero only when the parame-ter h represents a load applied to the structure. Rearrange-ment of Eq. (7) gives a linear system of equations for thenodal response sensitivity oU/oh [13]:

KToU

oh¼ oPf

oh� oPr

oh

����U

; ð8Þ

where KT = oPr/oU is the tangent stiffness matrix. For eachparameter in the vector H, assembly of the right-hand sideand solution of the factorized system of equations in Eq.(8) gives the corresponding nodal response sensitivityvector. The linear form of Eq. (8) and the reuse of the fac-torized tangent stiffness matrix contribute to the computa-tional efficiency of the DDM.

The sensitivity equation in Eq. (8) requires the assemblyof derivatives of the force vector p from each element, forfixed nodal displacements:

oPr

oh

����U

¼[

num: el:

op

oh

����u

; ð9Þ

where [ denotes the assembly procedure and u is the ele-ment displacement vector. It is noted that assembly is

required over all elements that contain inelastic material re-sponse, regardless of whether h corresponds to a parameterfor the individual elements. The one-to-one mapping thatexists between the vectors U and u is independent of hand therefore does not require differentiation. The key tocomputing response sensitivities by the DDM is to obtainop/ohju from each element in the structural model. Thecomputation of op/ohju depends upon the element formula-tion for inelastic material response, as described in the fol-lowing sections.

4. Overview of element equilibrium and kinematic equations

An overview of the equilibrium and compatibility equa-tions required to assemble the resisting force vector fromelement contributions is shown in Fig. 1, where the presen-tation follows that of Filippou and Fenves [4]. The equa-tions on the left-hand side of Fig. 1 represent equilibriumbetween the internal forces at the different levels whilethose on the right-hand side of the figure represent thecompatibility relationships between the deformations ateach level. The middle column of Fig. 1 shows the consti-tutive relationships that link the forces and deformationsat each level.

In the global system, the resisting forces and nodal dis-placements of an element are contained in the vectors p andu, respectively. It is common to formulate beam-columnelements in a basic system, free of rigid body displacementmodes, where the element deformations are collected in thevector v and the corresponding forces in the vector q. Forsmall displacements, the compatibility relationship betweennodal displacements and element deformations is linear, asdescribed by the matrix–vector product v = Au. The matrixA describes the transformation of forces and displacementsbetween the global and basic systems as defined by theelement orientation in the global coordinate system. Thecontra-gradient relationship p = ATq gives equilibriumbetween element forces in the basic and global systems.

The matrices ae and b shown in Fig. 1 describe the equi-librium and compatibility relationships within the basicsystem of the element, as discussed later in this paper.The section compatibility matrix, as, relates section defor-mations to material strain at any point on the cross-sectional area.

5. Unified approach for sensitivity derivations

As indicated in Eq. (9), response sensitivity analysis bythe DDM requires the conditional derivative of p for fixedu be computed for each element in the structural model.When the parameter h represents a material property, andstandard finite element formulations are employed, it is suf-ficient to differentiate directly the equilibrium equations andthe material constitutive law under the condition of fixeddisplacements and strains [13] in order to determine op/ohju.Furthermore, as recognized by Haukaas and DerKiureghian [7], in order to obtain correct sensitivity results

Basic system

Section

Material

Global system

Fig. 1. Governing equations for beam-column elements.

968 T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977

in cases where h represents a geometric parameter, the kine-matic relationships must also be differentiated under thecondition of fixed displacements.

Although the above approach provides correct resultsfor displacement-based finite elements, it is not applicablein the force-based formulation, where the displacementfield is unspecified. The approach taken herein to derivingthe shape sensitivity equations is to combine the completederivatives of the element force vectors with the derivativesof the equilibrium and compatibility relationships that linkeach level in Fig. 1. Originally developed by Scott et al. [22]for the case where h represents a material parameter, thederivation is extended in this paper to include cases whereh represents a geometric parameter. This approach essen-tially consists of combining four equations at each levelof Fig. 1. In summary, these equations are

(a) The derivative of the equilibrium equation.(b) The derivative of the kinematic compatibility

equation.(c) The complete derivative of the force vector at the

present level to include the constitutive law.(d) The complete derivative of the force vector at the

level below to link to the lower level.

This approach is demonstrated for the conditionalderivative of the global element forces, op/ohju. The deriv-

atives of the equilibrium and compatibility relationshipsbetween the global configuration and the basic systemare

op

oh¼ AT oq

ohþ oAT

ohq; ð10aÞ

ov

oh¼ A

ou

ohþ oA

ohu; ð10bÞ

while the complete derivatives of the global and basicforces are

op

oh¼ op

ou

ou

ohþ op

oh

����u

; ð10cÞ

oq

oh¼ oq

ov

ov

ohþ oq

oh

����v

: ð10dÞ

First, Eq. (10a) is expanded by inserting the derivatives ofthe global and basic forces, defined in Eqs. (10c) and (10d),respectively:

kg

ou

ohþ op

oh

����u

¼ ATkb

ov

ohþ AT oq

oh

����v

þ oAT

ohq; ð11Þ

where kg = op/ou and kb = oq/ov. This equation is thencombined with Eq. (10b)

kg

ou

ohþ op

oh

����u

¼ ATkbAou

ohþ ATkb

oA

ohuþ AT oq

oh

����v

þ oAT

ohq:

ð12Þ

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 969

Then, the definition of the element stiffness matrix, kg =ATkbA, simplifies this expression by cancellation of theterms involving ou/oh, and the final expression for the con-ditional derivative of the element forces is

op

oh

����u

¼ ATkb

oA

ohuþ AT oq

oh

����v

þ oAT

ohq: ð13Þ

The first and third terms on the right-hand side of Eq. (13)represent the sensitivity of the element forces to changes inthe element length, as described by the matrix oA/oh. Thismatrix is straightforward to compute from the direction co-sines of the element and it is equal to zero for any param-eter that does not represent a nodal coordinate for theelement. The second term represents the link to the nextlevel, the basic system, in which it is necessary to computeoq/ohjv. This conditional derivative depends on the elementformulation for nonlinear material behavior, as shown inthe following sections for the displacement- and force-based beam-column formulations.

6. Gradient computations for displacement-based elements

The formulation of displacement-based beam-columnelements follows standard finite element analysis proce-dures where the element displacement field is interpolatedfrom the nodal displacements [12,26]. Compatible sectiondeformations are interpolated from the element deforma-tions, e = aev, and there is weak equilibrium between thesection and basic forces, q ¼

R L0

aTe sdx, as indicated in

Fig. 1. For the standard interpolation fields of linear axialdisplacement and cubic Hermitian polynomial transversedisplacement, the axial deformation is constant and thecurvature is linear along the element, and the matrix ae

is

ae ¼1

L

1 0 0

0 6x=L� 4 6x=L� 2

� �: ð14Þ

In the implementation of the displacement-based elementthe equilibrium relationship is evaluated by numerical inte-gration (typically two-point Gauss integration) over thenormalized domain n = [�1,1]. The coordinate transfor-mation between the x-domain and the n-domain readsx = L/2(n + 1), where L is the element length. Hence, theJacobian of the transformation is dx/dn = L/2, and thenumerical integration of the equilibrium relationship is

q ¼XNp

i¼1

aTe ðniÞsðniÞ

L2

wi; ð15Þ

where Np is the number of integration points, ni is the loca-tion of the ith integration point, and wi is the associatedintegration weight. Both the points and weights are deter-ministic for Gauss integration, thus their derivatives willbe equal to zero.

Further simplification of Eq. (15) is possible by substi-tuting the x–n coordinate transformation into the matrixae, in which case the equilibrium relationship becomes:

q ¼XNp

i¼1

~aTe ðniÞsðniÞwi; ð16Þ

where the normalized interpolation matrix is

~ae ¼1

2

1 0 0

0 3n� 1 3nþ 1

� �: ð17Þ

The form of ~ae in Eq. (17) is independent of the elementlength, thus o~ae=oh will be equal to zero for all parameters.To obtain the conditional derivative oq/ohjv according tothe procedure established in the previous section, the ele-ment equilibrium and compatibility relationships are differ-entiated with respect to h:

oq

oh¼XNp

i¼1

~aTe

os

ohwi; ð18aÞ

oe

oh¼ ae

ov

ohþ oae

ohv: ð18bÞ

The complete derivatives of the basic and section forcevectors are

oq

oh¼ oq

ov

ov

ohþ oq

oh

����v

; ð18cÞ

os

oh¼ os

oe

oe

ohþ os

oh

����e

: ð18dÞ

In Eq. (18a) use is made of the independence of ~ae and wi onh. Analogous to the derivation for the conditional derivativeof the element forces in the global system, the derivatives ofthe basic and section forces from Eqs. (18c) and (18d),respectively, are inserted in Eq. (18a). Then, Eq. (18b) iscombined with the resulting expression and the definitionof the element stiffness matrix in the basic system, kb =oq/ov, allows the cancellation of terms involving ov/oh. Thisprocess results in the following equation for the conditionalderivative in the displacement-based formulation:

oq

oh

����v

¼XNp

i¼1

~aTe ks

oae

ohvþ ~aT

e

os

oh

����e

� wi; ð19Þ

where ks = os/oe is the section stiffness matrix. The vector,os/ohje, is computed from the gradient of the section consti-tutive response, and it will be discussed later in this paper.The derivative, oae/oh, is a straightforward scaling of theinterpolation matrix in Eq. (14), where the only term thatdepends on h is the common factor of 1/L:

oae

oh¼ � 1

L2

1 0 0

0 6x=L� 4 6x=L� 2

� �oLoh¼ �ae

1

LoLoh:

ð20Þ

The derivative of the element length, oL/oh, is obtained bydifferentiating the direction cosines that describe the ele-ment orientation. This derivative is equal to zero when hdoes not correspond to a coordinate of one of the elementnodes. Further simplification of the conditional derivativeof the basic forces is possible by inserting Eq. (20) intoEq. (19):

970 T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977

oq

oh

����v

¼XNp

i¼1

~aTe

os

oh

����e

wi � kbv1

LoLoh: ð21Þ

The forms of Eqs. (20) and (21) are specific to the assump-tion of linear axial and cubic Hermitian transverse dis-placement fields, for which it is possible to normalize theinterpolation matrix to ~ae in Eq. (17). This normalizationmakes for an efficient numerical implementation becauseit requires only one term in Eq. (21) to account for shapesensitivity in the basic system. Terms involving the deriva-tive of ae will appear in the conditional derivative when dis-placement fields that are not normalized by the elementlength are assumed.

7. Gradient computations for force-based elements

In the force formulation [23], it is the compatibility rela-tionship rather than equilibrium that is stated in integralform. The equilibrium and compatibility equations ares = bq and v ¼

R L0

bTedx, respectively, as indicated inFig. 1. The matrix b interpolates section forces from theelement end forces based on static equilibrium in the basicsystem:

b ¼1 0 0

0 x=L� 1 x=L

� �: ð22Þ

Due to the normalization of the x-coordinate by the ele-ment length in Eq. (22), the force interpolation matrix doesnot depend on any parameter, therefore its derivative,ob/oh, is equal to zero. The compatibility relationship isevaluated by numerical integration

v ¼XNp

i¼1

bTðniÞeðniÞL2

wi: ð23Þ

Neuenhofer and Filippou [18] developed a state determina-tion procedure for force-based elements that bypasses theinternal iterations required to satisfy the compatibility rela-tionship in Eq. (23) while enforcing equilibrium at each sec-tion along the element. Gauss–Lobatto quadrature isstandard for the implementation of force-based elementsbecause it places integration points at the element endswhere bending moments are known to be largest in the ab-sence of element loads.

To establish the conditional derivative oq/ohjv in theforce-based formulation, the equilibrium and compatibilityrelationships are differentiated with respect to h:

os

oh¼ b

oq

oh; ð24aÞ

ov

oh¼XNp

i¼1

bT oe

ohL2þ bTe

1

2

oLoh

� wi: ð24bÞ

The complete derivatives of the basic and section forcevectors are

oq

oh¼ oq

ov

ov

ohþ oq

oh

����v

; ð24cÞ

os

oh¼ os

oe

oe

ohþ os

oh

����e

: ð24dÞ

The independence of b and wi on h is utilized in Eqs. (24a)and (24b). The process of combining equations to arrive atan expression for oq/ohjv is conceptually similar to that forthe displacement-based formulation, but slightly more in-volved. First, the derivatives of the basic and section forcesfrom Eqs. (24c) and (24d) are inserted in Eq. (24a). Theresulting expression is rearranged to give an equation foroe/oh:

oe

oh¼ fsbkb

ov

ohþ fsb

oq

oh

����v

� fs

os

oh

����e

; ð25Þ

where fs ¼ k�1s is the section flexibility matrix. This expres-

sion is combined with Eq. (24b), then from the definition ofthe element flexibility matrix, fb ¼

R L0

bTfsbdx, and theidentity fbkb = I, the terms involving ov/oh cancel, andthe final expression for the conditional derivative is

oq

oh

����v

¼ kb

XNp

i¼1

bTfs

os

oh

����e

L2� bTe

1

2

oLoh

� wi: ð26Þ

From the element compatibility relationship of Eq. (23),further simplification of the conditional derivative ispossible

oq

oh

����v

¼ kb

XNp

i¼1

bTfs

os

oh

����e

L2

wi � kbv1

LoLoh: ð27Þ

It is important to note the functional equivalence of Eqs.(21) and (27) for the displacement- and force-based formu-lations, respectively, where only one term is required to ac-count for shape sensitivity of the element basic system.

Additional terms involving ob/oh appear in Eq. (27)when the interpolation of section shear forces is present;however, for the common case where shear effects areignored, the form of Eq. (27) leads to an efficient numericalimplementation because ob/oh is zero. The conditionalderivative of the section forces, os/ohje, depends on theconstitutive model at each integration point along the ele-ment, as discussed in the following section.

8. Gradient computations at the section and material levels

The response at every cross-section along the element isdefined in terms of the section deformations, e, and thecorresponding section forces, or stress resultants, s, asindicated in Fig. 1. Regardless of the element formulation,the conditional derivative of the section forces, os/ohje, isrequired to determine the element contribution to the gra-dient of the global resisting force vector, as seen in Eqs.(21) and (27). This derivative can be obtained by eitherdirect differentiation of a closed-form stress-resultant plas-ticity relationship or by numerical integration of the mate-rial stress over the cross-section. In the former case, the

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 971

problem reduces to deriving analytic gradient equations fora section constitutive law; while in the latter case, thesection compatibility and equilibrium equations, e = ase

and s ¼R

A aTs rdA, respectively, must be differentiated.

The cross-section integral is evaluated by numerical inte-gration over a user-defined number of fibers, Nf:

s ¼XN f

i¼1

aTs riAi; ð28Þ

where the section compatibility matrix for the assumptionof plane sections remain plane, as = [1 � yi], contains thefiber locations, yi. The material stress at the ith fiber loca-tion is ri, and Ai is the corresponding fiber area. Followingthe same procedure as in previous sections, the conditionalderivative of the section forces is obtained by differentiat-ing the section equilibrium and compatibility equationswith respect to h:

os

oh¼XN f

i¼1

oaTs

ohriAi þ aT

s

ori

ohAi þ aT

s rioAi

oh

� ; ð29aÞ

oei

oh¼ as

oe

ohþ oas

ohe: ð29bÞ

The complete derivatives of the section force vector and thematerial stress are

os

oh¼ os

oe

oe

ohþ os

oh

����e

; ð29cÞ

ori

oh¼ ori

oei

oei

ohþ ori

oh

����ei

; ð29dÞ

where ei is the strain at the ith fiber location. The substitu-tion of the section force and material stress derivativesfrom Eqs. (29c) and (29d), respectively, into Eq. (29a)and subsequent combination of Eqs. (29a) and (29b) fol-lowed by cancellation of terms involving oe/oh via the sec-tion stiffness matrix, ks, gives the conditional derivative ofthe section forces

os

oh

����e

¼XN f

i¼1

oaTs

ohriAi þ aT

s kmoas

oheAi þ aT

s

ori

oh

����ei

Ai þ aTs ri

oAi

oh

!;

ð30Þwhere km = or/oe is the material stiffness. The derivativesoas/oh and oAi/oh correspond to variations in the locationand size, respectively, of the ith fiber in the cross-section.These terms are important in computing the structural re-sponse sensitivity to the dimensions and details of membercross-sections. The computations of oas/oh and oAi/oh aredemonstrated in Appendix I for a fiber-discretized wideflange section. A similar approach is performed for rein-forced concrete sections in which it is particularly impor-tant to determine the response sensitivity to the amountand placement of steel reinforcement, as demonstrated inthe numerical examples at the end of this paper.

The remaining task in computing the structural responsesensitivity is to obtain ori=ohjei

for the material response at

each fiber location. A number of references providedetailed derivations of the response sensitivity for particu-lar constitutive laws, including Zhang and Der Kiureghian[25], Kleiber et al. [13], Roth and Grigoriu [20], andHaukaas and Der Kiureghian [8]. Implementations inOpenSees include the J2 plasticity model and a numberof uniaxial material models for inelastic behavior of steeland concrete.

9. Response gradients with respect to global shape

parameters

The derivations in the previous sections include elementresponse sensitivities with respect to nodal coordinates.However, it is frequently of interest to obtain sensitivitieswith respect to global shape parameters, which includethe end coordinates of frame members that are discretizedinto several finite elements, as well as parameters thatdescribe the global geometrical imperfection of a structure.Thus, the perturbation of a global shape parameter willperturb several nodal coordinates.

To obtain response sensitivities with respect to a globalshape parameter, an explicit relationship between thedependent nodal coordinates, eH, and the shape parametersis required. For the case of one shape parameter, denoted~h, which may coincide with a nodal coordinate of a mem-ber, this relationship is the expressed aseH ¼ f ð~hÞ; ð31Þwhere eH is the set of nodal coordinates that depend on theglobal shape parameter ~h.

As an example, consider a straight multi-element mem-ber with n nodes in 2-D space. The coordinates of the mem-ber ends are denoted (x1,y1) and (xn,yn). Any of these fourparameters can be the parameter ~h for which the responsesensitivity is sought. To establish the explicit form of Eq.(31), the other nodal coordinates of the member areexpressed in terms of the end coordinates:

xi ¼ x1 þ ðxn � x1Þi� 1

n� 1; i ¼ 1; 2; . . . ; n; ð32aÞ

yi ¼ y1 þ ðyn � y1Þi� 1

n� 1; i ¼ 1; 2; . . . ; n; ð32bÞ

where n � 1 is the number of elements into which the mem-ber is discretized.

To obtain response sensitivities with respect to the glo-bal shape parameter, the chain rule of differentiation isapplied to the nodal response sensitivity:

oU

o~h¼ oU

o eH o eHo~h

: ð33Þ

The matrix oU=o eH is available from the derivations in pre-vious sections, while the vector o eH=o~h is obtained by differ-entiation of Eq. (31). For the example in Eq. (32), when thesensitivity of a response quantity u is sought with respect tothe x-coordinate of node 1 of the member, Eq. (33)becomes

972 T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977

ouox1

¼Xn

i¼1

ouoxi

1� i� 1

n� 1

� : ð34Þ

When ~h represents a global structural shape imperfection,the vector o eH=o~h is established by computing the imperfec-tion at each node due to a unit global imperfection. Thus,the vector o eH=o~h is interpreted as a vector of influencecoefficients.

Global structural shape imperfections, as well as the endcoordinates of multi-element members are uncertain quan-tities. Thus, they should be considered as random variablesin a structural reliability analysis, as demonstrated in thefollowing numerical examples of a steel and a reinforcedconcrete frame.

10. Numerical examples

The response sensitivity equations derived in this paperhave been implemented in OpenSees and verified by finitedifference calculations. To investigate the importance ofuncertain geometrical imperfections relative to other uncer-tain structural parameters, static pushover reliability anal-yses are performed for two types of structures: a steel frameand a reinforced concrete frame. Static pushover analysis isthe prevalent analysis approach for capacity assessment inearthquake engineering since it assesses the ability of thestructure to reach a target displacement demand. Hence,the analyses in this paper are intended to address probabi-listic assessments of structural capacity.

10.1. Finite element reliability analysis of steel structure

The first structure considered is the three-bay, three-story steel frame in Fig. 2. Each member is discretized intofour displacement-based elements to represent the nonlin-ear distribution of curvature along the member length. Afiber-discretization represents the response of the wide-flange steel cross-sections of the frame members, as shown

Fig. 2. Steel frame structure. Node numbers and

in Fig. 2. There are two fibers in each flange and ten fibersin the web. The stress–strain behavior of each steel fiber isrepresented by the uniaxial material model shown inFig. 5a, for which there are three material parameters: (1)elastic modulus, E; (2) yield strength, fy; and (3) second-slope stiffness ratio, a.

All material and geometric parameters of the structuralmodel are considered uncertain. The dimensions d, tw, bf,and tf of the cross-section of each member are modeledas uncorrelated normal random variables with means250, 20, 250, and 20 mm, respectively, and 2% coefficientof variation (cov). The elastic modulus, E, of each memberis a lognormal random variable with mean 200,000 MPa,5% cov, and correlation coefficient 0.6 with the elastic mod-ulus of the other members. The steel yield strength, fy, ofeach member is a lognormal random variable with mean300 MPa, 10% cov, and correlation coefficient 0.6 with fy

of the other members. The stiffness ratio, a, of each mem-ber is a lognormal random variable with mean 0.02, 10%cov, and correlation coefficient 0.6 with a of the othermembers. In total, 21 members · 7 parameters = 147 ran-dom variables represent the material and cross-sectiongeometry parameters. Additionally, the two coordinatesof each of the 16 connection nodes are considered to beuncorrelated normal random variables, giving a grand totalof 179 random variables. The standard deviation of thevertical coordinates is 10 mm, while the standard deviationof the horizontal coordinates varies from 10 mm at thebase to 25 mm at the roof. The variation of the standarddeviation with the height is due to the potential for a globalsway of the building due to geometrical imperfection. It isnoted that each member is assumed to remain straight, thatis, the location of the internal member nodes is describedby Eq. (32).

To investigate the importance of geometrical imperfec-tions relative to other structural parameters deterministicloads are applied to the structure. The gravity loads are50 kN at the external connections and 100 kN at internal

element numbers (in parentheses) are shown.

Table 1Ranking of the 25 most important parameters in the steel frame example

Object Parameter c-Value

Member 5 fy �0.38034Member 7 fy �0.37828Member 3 fy �0.32439Member 2 fy �0.32253Member 10 fy �0.27291Member 6 fy �0.27279Member 9 fy �0.26853Member 4 fy �0.22111Member 1 fy �0.2168Member 8 fy �0.13287Member 11 fy �0.13264Member 5 d �0.11495Member 7 d �0.11403Member 3 d �0.10055Member 2 d �0.09987Member 10 d �0.08305Member 9 d �0.08215Member 6 d �0.08194Member 5 bf �0.07104Member 7 bf �0.07054Member 1 d �0.07005Member 4 d �0.06978Member 3 bf �0.0614Member 2 bf �0.061Member 5 tf �0.05928

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 973

connections. The lateral loads vary with the height asshown in Fig. 2, with maximum value 400 kN at the rooflevel.

To assess the lateral displacement demand on the struc-ture, a finite element reliability analysis is undertaken inorder to obtain the probability that the total drift of theroof exceeds 3%. The performance function for thisresponse event is

g ¼ 3%� 12 m� u13; ð35Þwhere u13 is the horizontal displacement at node 13. TheMPP in the FORM analysis is obtained after five evalua-tions of the performance function and its gradient. Thatis, five runs of the finite element analysis with different real-izations of the random variables are required. The resultingreliability index, b, is 2.01, which implies a 0.022 probabil-ity of exceeding the 3% target drift. Fig. 3 shows the dis-placement response at node 13 versus the load factor atthe mean realization and the MPP realization of the ran-dom variables. As expected, moderate nonlinearity isobserved at the mean, while significant yielding occurs atthe failure displacement.

Of particular interest in this paper is the ranking of therandom variables according to the importance measure inEq. (5). The 25 most important variables as the 179 ran-dom variables are shown in Table 1. The yield strengthsof the column members, except that of member 12, whichranks 34th, rank among the most important parameters.This emphasizes the fact that yielding takes place in the col-umns in this example. The web depth, d, and flange width,bf, of several column members rank 12 through 25 in Table

0 50 100 150 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Displace

Load

fact

or

Mean realizaMPP realizat

Fig. 3. Load–displacement response at the mean and MPP re

1, indicating the importance of imperfections in the cross-section geometry, even with the low cov of 2%. The hori-zontal coordinates of nodes 4, 8, and 12 in the right columnline rank among the 50 most important random variables,

00 250 300 350 400

ment [mm]

tionion

3% drift

alizations of the random variables for the steel structure.

974 T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977

thus indicating a relatively high importance of imperfectionin the global geometry of the structure.

10.2. Finite element reliability analysis of reinforced concrete

structure

It is of interest to investigate whether the observationsmade for the steel frame are also valid for a reinforced con-crete frame. For this purpose, a reliability analysis of thetwo-bay, two-story structure in Fig. 4 is performed. Eachframe member is represented by a single force-based ele-ment to capture the variation in curvature that results fromthe interaction of axial and moment forces. Each fiber in

Fig. 4. Reinforced concrete frame structure. Node numb

Fig. 5. Material models for (a) steel, (b) unconfined concrete in girders and

the cross-sections shown in Fig. 4 is modeled by a uniaxialmaterial model. The core and cover concrete material fibersare described by a uniaxial model with a modified Kent–Park backbone curve [21] with zero stress in tension andlinear unloading/reloading, as shown in Fig. 5b and c,respectively. The bilinear model used in the previous exam-ple represents the stress–strain response of the reinforcingsteel.

All material and geometry parameters are considereduncertain. The cross-sectional dimensions b and h of themembers are uncorrelated normal random variables withthe mean values given in Fig. 4 and 5% cov. The area ofeach reinforcing bar is a normal random variable with

ers and element numbers (in parentheses) are shown.

column cover regions, and (c) confined concrete in column core regions.

-20 0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Displacement [mm]

Load

fact

or

Mean realizationMPP realization

2% drift

Fig. 6. Load–displacement response at the mean and MPP realizations of the random variables for the reinforced concrete structure.

Table 2Ranking of the 25 most important parameters in the reinforced concreteframe example

Object Parameter c-Value

Member 3 h �0.34697Member 7 h �0.34339Member 4 h �0.32343Member 8 h �0.31364Member 4 fy �0.2853Member 5 h �0.24275Member 8 fy �0.23581Member 7 fy �0.23421Member 3 fy �0.17896Member 3 Cover depth 0.15903Member 1 h �0.15483Member 4 Cover depth 0.14254Member 7 Cover depth 0.1311Member 5 Cover depth 0.12401Member 4 �cover

cu 0.12336Member 8 Cover depth 0.12045Member 5 fy �0.12021Member 1 fy �0.09227Node 2 x-coordinate �0.0902Member 1 Cover depth 0.07958Node 5 x-coordinate 0.07674Member 10 fy �0.07497Member 4 f 0core

c 0.07103Member 9 fy �0.06721Node 3 x-coordinate �0.06562

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 975

the mean value shown in Fig. 4 and 2% cov. The thicknessof the cover concrete in each member is a normal randomvariable with mean 75 mm and 10% cov.

The elastic modulus, E, of the reinforcing steel of eachmember is a lognormal random variable with mean200,000 MPa, 5% cov, and correlation coefficient 0.6 withE of the other members. The reinforcing steel yieldstrength, fy, for each member is a lognormal random vari-able with mean 420 MPa, 10% cov, and correlation coeffi-cient 0.6 with fy of the other members. The stiffness ratio, a,of the reinforcing steel of each member is a lognormal ran-dom variable with mean 0.05, 10% cov, and correlationcoefficient 0.6 with a of the other members. All concretematerial parameters in Fig. 5b and c are lognormal randomvariables with the mean values shown in the figures and10% cov. The concrete strength parameters, f 0c and f 0cu,are inter-correlated by 0.6, as are the corresponding strains,ec and ecu. The two coordinates of each node are consideredto be uncorrelated normal random variables. The verticalcoordinates are assigned a standard deviation of 10 mm,while the horizontal coordinates are assigned standarddeviations that vary from 10 mm at the base to 20 mm atthe roof. There are a total of 142 random variables for thisreliability analysis. The gravity loads are G4 = G6 = G8 =850 kN, G5 = 1700 kN, and G7 = G9 = 430 kN, while thelateral loads are P4 = 450 kN and P7 = 900 kN. All ofthe applied loads are deterministic.

The performance function for this example is defined todetermine the probability the roof displacement will exceed2% drift:

g ¼ 2%� 8:3 m� u7; ð36Þ

where u7 is the lateral displacement of node 7. A FORMfinite element reliability analysis converges in nine itera-tions to the reliability index 2.76 with corresponding prob-ability 0.0029. Significant nonlinearity occurs at the mostlikely failure realization of the random variables, as shown

976 T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977

in Fig. 6. The 25 most important random variables arelisted in Table 2. Remarkably, it is found that the cross-sec-tion height h of the members adjacent to the inner connec-tion are the most important parameters, followed by theyield strength of the reinforcing steel of the same members.These results indicate a high influence of geometricalimperfections on the reliability of the reinforced concretestructure. It is also noted that the depth of the cover con-crete ranks high in importance. This finding may justifyfurther investigation of the dispersion in the amount ofcover in reinforced concrete structures. As in the previousexample, the global structural shape imperfection repre-sented by the horizontal nodal coordinates also ranks highin importance with nodes 2, 3, and 5 listed among the 25most important random variables.

Fig. 7. Wide flange section dimensions and fiber discretization.

11. Conclusions

A comprehensive and unified treatment of response sen-sitivity equations by the direct differentiation method wasdeveloped. Geometric, material, and load parameters areincluded in both the force-based and the displacement-based formulations of inelastic beam-column response. Sen-sitivity equations for global shape parameters account forgeometric imperfections of structural members discretizedinto multiple finite elements. The analytical equations havebeen implemented and verified in the OpenSees software.The finite element reliability analysis of a three-bay, three-story steel frame demonstrated the member cross-sectionaldimensions, particularly the section depth and the flangewidth, rank high in importance. For the two-bay, two-storyreinforced concrete structure, the cross-section depth, aswell as the thickness of the cover concrete, are importantparameters relative to the material parameters. In eachexample, the importance ranking of geometrical imperfec-tions relative to other structural parameters indicates a sig-nificant influence of uncertain geometrical parameters onreliability assessments, even when the dispersion in theprobability distribution is small.

Acknowledgments

This work and the corresponding software implementa-tion in OpenSees have been supported in part by the PacificEarthquake Engineering Research Center under grant no.EEC-9701568 from the National Science Foundation(NSF) and by a Discovery Grant from the Natural Sciencesand Engineering Research Council of Canada (NSERC).Their support is gratefully acknowledged.

Appendix I. Wide flange section shape sensitivity equations

As indicated in Eq. (30), it is necessary to compute thederivatives oas/oh and oAi/oh at each fiber location in orderto account for variations in the geometric parameters thatdefine a fiber-discretized cross-section. The derivative of

the section compatibility matrix at the ith fiber location isoas=oh ¼ ½ 0 �oyi=oh �.

The section is defined by the following geometric param-eters, as shown in Fig. 7 the overall section depth, d; theweb thickness, tw; the flange width, bf; and the flange thick-ness, tf. The number of fibers through the depth of the webis Nfw, and Nff is the number of fibers through the thicknessof each flange. Each of the derivatives, od/oh, otw/oh,obf/oh, and otf/oh is equal to either one or zero dependingon the property h represents in the structural model. BothNfw and Nff are deterministic parameters.

I.1. Web fibers

From the section dimensions and the number of fibers,the area of each fiber in the web is the web area dividedby the number of web fibers:

Aw ¼dwtw

N fw

; ð37Þ

where the depth of the web, dw is

dw ¼ d � 2tf : ð38ÞThe distance from the reference axis to the centroid of theith web fiber is

yi ¼d2� tf

� � i� 1

2

� dw

N fw

; i ¼ 1; . . . ;N fw: ð39Þ

The differentiation of Eq. (37) with respect to h gives thesensitivity of the size of the web fibers to the parameter

oAw

oh¼ 1

N fw

dw

otw

ohþ odw

ohtw

� ; ð40Þ

where the sensitivity of the web depth is

odw

oh¼ od

oh� 2

otf

oh: ð41Þ

Similarly, the differentiation of Eq. (39) gives the sensitivityof the web fiber centroid locations

T. Haukaas, M.H. Scott / Computers and Structures 84 (2006) 964–977 977

oyi

oh¼ 1

2

odoh� otf

oh

� � i� 1

2

� 1

N fw

odw

oh; i ¼ 1; . . . ;N fw:

ð42Þ

I.2. Flange fibers

For the fibers in the flange regions, the size of each fiber is

Af ¼bf tf

N ff

; ð43Þ

and the centroid location of the ith fiber is

yi ¼d2� i� 1

2

� tf

N ff

; i ¼ 1; . . . ;N ff : ð44Þ

The sensitivity of the size of each flange fiber to h is

oAf

oh¼ 1

N ff

bf

otf

ohþ obf

ohtf

� ; ð45Þ

while the sensitivity of the ith flange fiber centroid location is

oyi

oh¼ 1

2

odoh� i� 1

2

� 1

N ff

otf

oh; i ¼ 1; . . . ;N ff : ð46Þ

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