2000 - vassil k simeonov - nonlinearanalysisofstructuralframesystemsbythestat[retrieved-2015!08!04]

Computer-Aided Civil and Infrastructure Engineering 15 (2000) 76–89

Nonlinear Analysis of Structural Frame Systems bythe State-Space Approach

Vassil K. Simeonov, Mettupalayam V. Sivaselvan & Andrei M. Reinhorn*

Department of Civil, Structural, and Environmental Engineering, University at Buffalo (SUNY),Buffalo, New York 14260, USA

Abstract: This article presents the formulation and solu-tion of the equations of motion for distributed parameternonlinear structural systems in state space. The essence ofthe state-space approach (SSA) is to formulate the behaviorof nonlinear structural elements by differential equationsinvolving a set of variables that describe the state of eachelement and to solve them in time simultaneously with theglobal equations of motion. The global second-order dif-ferential equations of dynamic equilibrium are reduced tofirst-order systems by using the generalized displacementsand velocities of nodal degrees of freedom as global statevariables. In this framework, the existence of a global stiff-ness matrix and its update in nonlinear behavior, a cor-nerstone of the conventional analysis procedures, becomeunnecessary as means of representing the nodal restor-ing forces. The proposed formulation overcomes the limita-tions on the use of state-space models for both static anddynamic systems with quasi-static degrees of freedom. Thedifferential algebraic equations (DAE) of the system areintegrated by special methods that have become availablein recent years. The nonlinear behavior of structural ele-ments is formulated using a flexibility-based beam macroelement with spread plasticity developed in the frameworkof state-space solutions. The macro-element formulation isbased on force-interpolation functions and an intrinsic timeconstitutive macro model. The integrated system includingmultiple elements is assembled, and a numerical exam-ple is used to illustrate the response of a simple struc-ture subjected to quasi-static and dynamic-type excitations.The results offer convincing evidence of the potential ofperforming nonlinear frame analyses using the state-spaceapproach as an alternative to conventional methods.

* To whom correspondence should be addressed. E-mail: [email protected].

1 INTRODUCTION

The formulation and solution of nonlinear inelastic struc-tures subjected to static and dynamic loading present greatdifficulties and take a high toll on computational resources.This article presents a formulation and the solution of non-linear structural analysis in state-space form applied to ini-tial boundary value problems. The proposed solution strat-egy overcomes the limitations on the use of state-spacemodels for both static and dynamic systems with quasi-static degrees of freedom. The state variables of a discretestructural system consist of global and local element quan-tities. The global state variables are the generalized dis-placements and velocities of certain nodal degrees of free-dom (DOF) of the system. The local state variables arethe “end” restoring forces and forces and deformations atintermediate sections, which describe the local nonlineari-ties. Each state variable is associated with a state equation,which characterizes its evolution in time in relation to otherstate variables or kinematic constraints. These are usuallyfirst-order differential or algebraic equations. Global stateequations describe the motion or equilibrium of the entiresystem, whereas local state equations describe the evolutionof the state variables needed for determining the conditionsof all nonlinear elements. There is a single independentvariable, represented by the “real time” in dynamic analy-sis and “intrinsic time” in quasi-static analysis. This articlesuggests a formulation of the structural elements using aflexibility approach with spread plasticity in terms of localand global state variables, which in turn assemble differen-tial algebraic equations solved by iterative solvers.

Two alternative schemes can be considered for solvingthe equations of equilibrium of discrete nonlinear dynamicor quasi-static systems. First, the most commonly usedsolution strategy is based on implicit time-stepping meth-

© 2000 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA,and 108 Cowley Road, Oxford OX4 1JF, UK.

Nonlinear analysis of structural frame systems by the state-space approach 77

ods of numerical integration; notable are the Newmark,Wilson, and Humbolt methods.3 The response is obtainedby linearization of the governing differential equations overa sequence of time steps by using the tangent stiffness anddamping matrices of the system, effectively reducing thedynamic problem to a series of quasi-static ones. Iterationsare necessary to reduce the error introduced by the useof the tangent matrices. Alternatively, the restoring forcesin the nonlinear elements can be separated and applied asexternal loads on the elastic part of the system.15,17 Theresulting second-order equations of motion can be reformu-lated as a set of first-order differential equations by addingthe nodal velocities to the set of unknowns. The forces inthe nonlinear elements are then defined as functions of theelement deformations, deformation rates, and other vari-ables that determine the state of the element. These differ-ential equations are coupled with the equations of motionof the system and are solved simultaneously.

Unfortunately, this approach is applicable to structuralmodels involving only dynamic degrees of freedom. Thelimitation arises from the fact that the quasi-static degreesof freedom cannot be eliminated by static condensationbecause the equations are formulated in the time domainand not in an incremental form in the displacement domain.This results in a system of differential algebraic equations(DAEs) instead of ordinary differential equations (ODEs).The final set of equations is a general initial boundary valueproblem, which must be solved by an appropriate methodof integration of differential algebraic systems (DASs). Thefact that ODE numerical solvers are unable to handle DAEshas prevented the generalization and widespread use of thestate-space method. Only recently have reliable integra-tion algorithms been developed and implemented in soft-ware packages specifically directed at solving DASs, suchas DASSL,6 RADAU5,11,12 and DASPK.7 These advance-ments have incited the DAE-based state-space method forfinite-element and macro-model analysis of structural sys-tems discussed in this article.

The state-space approach has been employed extensivelyin the solution of purely dynamic linear and nonlinear prob-lems, especially in structural control and nondeterministicanalysis.2,8,17,27 These problems had no quasi-static degreesof freedom and hence resulted in ODE systems with noalgebraic equations. The state-space approach involvingDASs has been used extensively in multibody dynamics ofaerospace and mechanical assemblies.4,13,14 In these prob-lems, the kinematic constraints are algebraic equations.The first application of the state-space approach to finite-element solution of quasi-static distributed plasticity prob-lems is probably that used by Richard and Blalock23 tosolve plane stress problems. Since this work consideredonly monotonic loading, the load itself (rather than time)served as the independent monotonically increasing vari-able. No work has been reported since that time until the

beginning of this decade. DASs were used to solve largedeformation plasticity problems arising in punch stretchingin metal-forming operations. Papadopoulos and Taylor21

presented a solution algorithm based on DAEs for J2 plas-ticity problems with infinitesimal strain. Papadopoulos andLu20 subsequently extended this strategy to a generalizedframework for solution of finite plasticity problems. Iuraand Atluri16 used the DAE-based state-space approach forthe dynamic analysis of planar flexible beams with finiterotations. Fritzen and Wittekindt10 provided the first cleardescription of the methodology of formulating problemsin nonlinear structural analysis at about the same time asShi and Babuska.24 The work of these authors provides thebasis for the approach proposed here.

This article introduces first a general procedure for iden-tifying the state variables of a system already discretizedin space by the finite-element method. The set of statevariables consists of global and local (element) responsequantities. Second, this article introduces the algorithmfor constructing the system of state equations. The pro-cedure accounts for the connectivity and boundary con-ditions of the structural model, the force-displacementrelationships and the internal degrees of freedom of allelements, and the type of excitation. In general, the for-mulation results in a system of DAEs. Some backgroundinformation on DASs is provided as well as the solutionalgorithm of DASSL. Third, a nonlinear bending element isformulated for modeling beam members in the frameworkof state-space analysis. The element falls into the categoryof flexibility-based continuum models, for its formulationis based on force (instead of displacement) interpolationfunctions. Discretization across the cross section is avoidedby using a constitutive macro model. Finally, the aforemen-tioned development is implemented in a computer platform,and a numerical evaluation of the quasi-static and dynamicresponse of a typical civil engineering frame structure iscarried out.

2 CONSTITUTIVE MODELS FORSTATE-SPACE ANALYSIS

The state-space formulation can address constitutive mod-els based on rigorous plasticity theory as well as reason-able approximations represented by algebraic or differentialrelations. The existing mathematical representations of non-linear constitutive relationships can be categorized as eithertime-independent or time-dependent. Models representa-tive of the first category, such as the Ramberg-Osgood22

bilinear and elastoplastic models, describe the general-ized stress-strain curve by algebraic equations, whereastime-dependent models5,26,28,30 employ differential equa-tions. Although the state-space approach can easily han-dle the solutions based on rigorous plasticity theory, of

78 Simeonov, Sivaselvan & Reinhorn

particular interest to this study are nonlinear constitu-tive representations based on the endochronic theory ofplasticity.29 Although involving more approximations, theendochronic theory offers simpler relations described bydifferential equations, which can allow better understand-ing of the state-space approach (SSA). The intrinsic-timemodels make them practical alternatives to regular surface-based formulations due to (1) the absence of an explicityield function, (2) smooth transition between the elasticand plastic states, and (3) a single equation for all phasesof cyclic response. The endochronic models do not requirekeeping track of the yielded state; the evolutionary char-acter of the differential equations naturally accommodateskinematic hardening. Since the constitutive equations areintegrated simultaneously with the equations of equilibriumof the system, at any given time the state of the elementsand the state of the system are determined exactly. Theerror in the solution is dictated only by the tolerance of thespecific method of integration and not by the linearizationof the generalized stress-strain relations.

A modified version of the endochronic model, originallyproposed by Bouc-Wen and extended by Sivaselvan andReinhorn,26 is used throughout this development. The totalgeneralized restoring force R is modeled as a combinationof elastic and hysteretic components (Figure 1a):

R = [aK0 + (1− a)KH ]u (1)

where a is the ratio of postyield to elastic stiffness, K0 isthe initial stiffness, KH is the hysteretic stiffness, and u

is the total generalized displacement. In this parallel-springrepresentation, the stiffness of the hysteretic spring is gov-erned by

KH = K0

{1−

∣∣∣∣R∗R∗y∣∣∣∣n[η1 sgn(R∗u)+ η2]

}(2)

where R∗ = R−aK0u is the force in the hysteretic spring,R∗y = (1− a)Ry is the yield force of the hysteretic spring,n is a parameter controlling the transition between the elas-tic and plastic range, and η1 and η2 are parameters con-trolling the shape of the hysteretic loop, which must fulfillthe condition η1 + η2 = 1 (according to Constantinou andAdnane9).

A nonlinear single-degree-of-freedom (SDOF) systemsubjected to dynamic and pseudo-static forces will be usedto illustrate the state-space formulation (see Figure 1b). Thesystem has three state variables and therefore three stateequations. Let

y1 = u y2 = u y3 = R (3)

Then

my2 + cy1 + y3 − F = 0 (4)

Fig. 1. Schematic representation of nonlinear systems: (a) Modelfor restoring force; (b) inertial nonlinear SDOF system.

y2 − y1 = 0 (5)

y3 − [aK0 + (1− a)KH ]y1 = 0 (6)

For this system, y3 is the local and y1 and y2 are theglobal state variables. Correspondingly, Equation (4) is thelocal and Equations (5) and (6) are the global state equa-tions. The constitutive relation is solved simultaneouslywith the equations of motion, giving rise to a set of ordinarydifferential equations. The hysteretic stiffness is obtained


by reordering Equation (1) and substituting the result intoEquation (2):

R∗ = y3 − aK0y1 (7)

KH = K0

{1−

∣∣∣∣y3 − aK0y1

R∗y

∣∣∣∣n× [η1 sgn(y3y2 − aK0y1y2)+ η2

]}(8)

It should be noted that the choice of state variables forthis system is not unique. For example, an alternative, andprobably more natural, formulation of the same problemcan be devised using the hysteretic component of the restor-ing force as a local state variable. Let

y1 = u y2 = u y3 = R∗ (9)

Then

my2 + cy1 + aK0y1 + (1− a)y3 − F = 0 (10)

y2 − y1 = 0 (11)

y3 −KH y1 = 0 (12)

In this case, the hysteretic stiffness is calculated directlyby Equation (2). The first version, however, is preferred.In contrast with the dynamic system in Figure 1b, a quasi-static system subjected to identical force history has onlytwo state variables and hence two state equations. Let

y1 = u y2 = R (13)

Then

y2 − F = 0 (14)

y2 − [aK0 + (1− a)KH ]y1 = 0 (15)

In this case, the constitutive equation (15) is differ-ential, but the equation of equilibrium (14) is algebraic.Therefore, a set of differential algebraic equations must besolved to obtain the quasi-static response of a SDOF systemwith nonlinear restoring force. Brenan6 describes the solu-tion of such equations in detail; however, for the sake ofunderstanding, this solution is summarized below, includ-ing some simplifying assumptions.

3 DIFFERENTIAL ALGEBRAICSYSTEMS (DASs)

A DAS is a coupled system of N ordinary differential andalgebraic equations that can be written in the followingform:

8(t, y, y) = 0 (16)

where 8, y, and y are N -dimensional vectors, t is the inde-pendent variable, and y and y are the dependent variablesand their derivatives with respect to t .

A consistent set of initial conditions y(t0) = y0, y(t0) =y0 must be specified such that Equation (16) is satisfied atthe initial time t0. Since the system is coupled, some of theequations in (16) may not have a corresponding componentof y. Consequently, the matrix

∂8

∂ y=[∂8i

∂yj

](17)

may be singular. A measure of the singularity is the index(defined rigorously by Brenan et al.6). In a simple formu-lation this index is equal to the minimum number of timesthe implicit system in Equation (16) must be differentiatedwith respect to t to determine y explicitly as functions ofy and t . It becomes clear that a system of ODEs written inthe standard form

y = g(t, y) (18)

is index 0. The system composed of Equations (4), (5),and (6), for example, can be converted to the standard formwithout additional differentiation:

y2 =F − cy2 − y3

m(19)

y1 = y2 (20)

y3 = aK0y2 + (1− a)K0

×{

1−∣∣∣∣y3 − aK0y1

R∗y

∣∣∣∣n× [η1 sgn(y3y2 − aK0y1y2)+ η2

]}y2 (21)

The set (14) and (15) modeling the quasi-static responseof SDOF system, however, is index 1, because the algebraicEquation (14) must be differentiated once before substitu-tion into Equation (15):

y2 = F (22)

y1 = F[aK0 + (1− a)K0

{1−

∣∣∣∣y2 − aK0y1

R∗y

∣∣∣∣n× [η1 sgn(y2y1 − aK0y1y1)+ η2

]}]−1

(23)

This equation does not provide an explicit expression of thederivative in terms of the independent and dependent vari-ables. However, the left- and right-hand sides are coupledonly through the argument of the signum function, the solepurpose of which is to indicate load reversal. Noting thatthe stiffness terms in the denominator of Equation (23) arealways positive, the sign of the displacement rate y1 fol-lows that of the rate of the applied force F , and the signumcan be evaluated from information on the latter.


The solution of DAEs is more involved than the solu-tion of ODEs. Some of the difficulties, discussed in depthby Brenan,6 are summarized next. Since the algebraic anddifferential equations have different time scales, the systemappears as extremely stiff, in which case implicit integra-tion schemes have to be used. Only the differential equa-tions require initial conditions. However, it is not possi-ble to separate the variables associated with the differen-tial equations from those which are not. Therefore, initialconditions are provided for all variables. These initial con-ditions must be consistent. Finally, DASs of index greaterthan 1 pose additional problems, and only a few solutionalgorithms can handle them. A brief summary of the inte-gration method, implemented in DASSL, is provided forthe sake of completion. This implies replacing the solutiony and its derivative y at the current time by a differenceapproximation and solving the resulting algebraic equationsusing a Newton-type method. The derivative is calculatedby a backward differentiation formula:

yn =1

hnβ0

(yn −

k∑i=1

αiyn−1

)(24)

where yn, yn, and yn−1 are the approximations of the solu-tion of Equation (16) and its derivative at time tn and tn−1,respectively, hn = tn − tn−1 is the time interval, k is theorder of the backward differentiation formula relative to yn,and αi and β0 are coefficients of the method.

Substituting Equation (24) in Equation (16) results in asystem of algebraic equations:

8

[tn, yn,

1

hnβ0

(yn −

k∑i=1

αiyn−1

)]= 0 (25)

The iteration matrix of the Newton-type method is definedas follows:

N(tn, yn, yn) =∂8(tn, yn, yn)

∂y+ 1

hnβ0

∂8(tn, yn, yn)∂ y

(26)

The process for advancing from time tn−1 to the currenttime tn is summarized by the equation

ym+1n = ymn − N−1

(tn, ymn , ymn

)F(tn, ymn , ymn

)(27)

where the superscript m is an iteration counter.Further discussion is beyond the scope of this article. For

extensive treatment of the subject of numerical solutionsof DASs and the one implemented in DASSL in particular,see Brenan.6

4 FORMULATION OF AMULTI-DEGREE-OF-FREEDOM SYSTEM

Let us consider the multi-degree-of-freedom (MDOF) ideal-ization of a distributed parameter system resulting from

a finite-element discretization. The governing equations ofequilibrium can be written as

Mu(t)+ Cu(t)+ R(t) = F(t) (28)

where M is the mass matrix, u(t) is the displacement vec-tor, C is the damping matrix, R(t) is the restoring forcevector, F(t) is the forcing vector, and the dot denotes timederivative.

In the general case, the set of global state variables ofthe system consists of three parts: (1) generalized displace-ments along all free nodal degrees of freedom, (2) degreesof freedom with known nonzero displacements or displace-ment time histories, and (3) velocities along degrees offreedom that have masses associated with them. The dis-placements along constrained generalized coordinates areexcluded from the solution by virtue of imposing bound-ary conditions on the model. Degrees of freedom withknown nonzero displacements, however, must be includedin the set of global state variables. This situation occursfrequently in civil engineering analysis and design, wherethe support motion, due to soil settlement, slope instability,or earthquakes, is often known a priori. Similar circum-stances exist in displacement-controlled pseudo-static lab-oratory testing of structural components. Most matrix anal-ysis or finite-element programs, research or commercial,treat the dynamic support displacement as a special anal-ysis case and require information on the ground velocity.The method proposed here addresses this common problemby designating the known displacements as state variables.However natural this treatment is, it introduces additionalalgebraic equations to the set of state equations, the impli-cations of which are discussed below.

The final segment of global state variables is com-prised of the displacement rates along degrees of freedom,which have directional or rotational masses associated withthem. In practical cases, the number of velocity state vari-ables may differ substantially from the number of theirdisplacement counterparts due to routine engineering prac-tices of using a lumped, rather than consistent, massmatrix. Furthermore, rotational and even some directionalcomponents of mass, whose effect is presumed negligi-ble, are often ignored. In this development, the globalmass matrix is assumed diagonal. The inherent damping ofthe system is modeled by mass-proportional and (elastic)stiffness-proportional contributions to the global dampingmatrix. Hysteretic damping and damping from concentratedsources (dampers) is incorporated in the analysis by solvingsimultaneously the global state equations and explicit ele-ment constitutive relations, force-displacement, and force-velocity rate equations, respectively. Because of this, thedamping matrix has no contributions from the individualdampers, as is customary in conventional finite-elementanalysis. The system-restoring force vector is assembled by


standard procedures with account of the by-node-numberordering of the global degrees of freedom and element con-nectivity at the nodes. The restoring forces of the individualelements, which are expressed in terms of the element statevariables, are transformed to the nodal coordinate system byregular local-to-global transformations to obtain their con-tributions to the global vector.

The assembly of the forcing vector of the system is gov-erned by similar considerations. External excitations actingin the form of static or dynamic forces or accelerations aredefined directly in global coordinates. Distributed-elementloads are substituted with equivalent nodal restoring forces.Their effect is imposed as initial conditions for the corre-sponding element-state variables for subsequent analysis. Itmust be understood that the dimension of the system massand damping matrices, restoring force, and forcing vectoris equal to the number of free nodal displacement coordi-nates. The constrained degrees of freedom are excluded inthe assembly stage to avoid the costly reordering operationsof enforcing these boundary conditions in a separate step.It also becomes clear that in the framework of the state-space approach, formation of the global tangent stiffnessmatrix becomes unnecessary. Equilibrium equations can bewritten in state-space format

My2 + Cy1 + R(t) = F(t) (29)

using the transformations

y1 = u(t) y2 = u(t) (30)

The three parts of the set of global state equations can besummarized as follows (the writing of the time dependencehas been omitted for clarity):

My2(1:ND) + Cy1(1:ND) + R − F = 0 (31)

y1(ND+1:ND+NDH) − d(1:NDH) = 0 (32)

y2(ND+NDH+1:ND+NDH+NV ) − y1(1:NV ) = 0 (33)

where ND is the number of active nodal DOFs, NDH isthe number of nodal DOFs with known nonzero displace-ments (displacement time histories), NV is the numberof nodal DOFs that have directional or rotational massesassociated with them, and NDOF is the total number ofDOFs. Note that the mass and damping matrices in Equa-tion (31) have been condensed from their original dimen-sions (NDOF ×NDOF ) to (ND ×ND).

The state of each of the nonlinear elements of the struc-ture must be defined by evolution equations involvingthe generalized restoring forces, generalized displacements,and the internal variables used in the formulation of theelement model. Not all these are necessarily designated aslocal state variables. The idea is to define a minimal setof independent quantities and use pertinent transformations

Fig. 2. Interaction between local and global states.

for the dependent ones. In general, the state of the elementis modeled by a system of differential equations:

Re = G(Re,ue, ue, ze, ze) (34)

ze = H(ze,Re, Re,ue, ue) (35)

where G and H are nonlinear functions, Re and Re arethe generalized restoring forces and their rates, ue and ueare the generalized displacements of the element nodes andtheir rates, and ze and ze are the internal variables and theirrates.

The element state equations also must establish a connec-tion between the local state variables (generalized restor-ing forces) and global state variables (generalized displace-ments and velocities). The schematic of this interaction ispresented in Figure 2.

The principle of virtual forces is used to obtain stateEquation (34) for the beam element, as shown below. Deter-mination of the tangent flexibility matrix requires infor-mation on the distribution of plastic strain along the ele-ment. This is achieved by using the strains at intermedi-ary integration (quadrature) points as internal variables andintroducing state equations tracking their evolution Equa-tion (35).

4.1 Formulation of a flexibility-based beammacro element

The macro element is represented by equations using vari-ables at the end of the element and at sections locatedat the intermediary quadrature points. The nonlinear func-tions G and H in Equations (34) and (35) are derivedusing both end variables and variables at quadrature sec-tions. Direct relationships between the end force and dis-placement rates are only now becoming available for lineelements subjected to a combination of bending, shear, andaxial load.25 The process of defining a capable macro ele-ment requires (1) nonlinear material stress-strain relations,(2) relations representing differences between the apparentmaterial characteristics exhibited in monotonic and cyclic


loading, and (3) information on deterioration of materialproperties, expressed as pinching of hysteretic loops orstiffness or strength degradation. Although usually morepronounced in members made of composite materials, suchas reinforced concrete, the latter occurs commonly in steelconnections and soil-foundation interfaces.

In the present formulation, the nonlinear moment-curvature relationships of cross sections (slices) at discretelocations along the element axis are chosen to representconstitutive laws for bending response only. The formula-tion including the shear and axial effects25 is more complexand beyond the scope of introducing the element’s basicformulation. The convenient macroscopic formulation hasfound utility in modeling both material plasticity and dete-rioration phenomena. The effect of incorporating these fac-tors in a predefined model of section response propagatesto the element constitutive relations. To capture the effectsof cyclic loading on the apparent material properties, themonotonic moment-curvature curves are assumed to rep-resent the cyclic response of the section. Experimentallycalibrated rules for the deterioration events26 discussed ear-lier could then be used to degrade the monotonic envelope.The smooth transition between the elastic and plastic statescan be calibrated by considering the results of curvature-controlled analysis of a section of beam member discretizedinto a large number of fibers. In the extreme case of bilinearuniaxial stress-strain curves of the individual filaments, theresulting moment-curvature relationship has a smooth tran-sition between the elastic and plastic stages of response dueto the distributed yield of the section. The actual propertiesof materials such as steel, concrete, and aluminum, com-monly used for constructing bending elements, are multi-linear rather than bilinear, which leads to an even smoothertransition. Furthermore, aluminum and some alloys exhibitnonlinearity throughout all stages of response, includingthe initial elastic range. The formulation in Equations (1)and (2) has the ability to model smooth degrading pinch-ing constitutive relations, making it a natural choice for thedevelopment of the element presented next.

The state variables chosen for the beam macro elementare the independent end forces and the curvatures of sec-tions located at the quadrature points:

ye(1:3) = Re (36)

ye(4:3+NG) = φ (37)

where Re = {Fi Mi Mj }T are the element nodal forces,including the axial force at one end and the bendingmoments at both ends, and φ = {φ1 φ2 · · · φNG}T arethe section curvatures at NG quadrature points.

The state equations define the constitutive laws of theelement and the individual sections:

˙Re = Ke˙ue (38)

Fig. 3. End degrees of freedom and element deformations.

M = [aK0 + (I− a)KH]ϕ (39)

where Ke is the element stiffness matrix of size 3× 3, thederivation of which is described later, uc = {u θi θj }Tare the axial deformation and chord rotations at the ends,M = {M1 M2 · · · MNG}T is the total bending moment atmonitored sections at the quadrature locations, I is the iden-tity matrix of size NG × NG, a = diag[a1 a2 · · · aNG]is the ratio of postyield to elastic stiffness of mon-itored sections, K0 = diag[K0, 1 K0, 2 · · · K0, NG] isthe elastic stiffness of monitored sections, and KH =diag[KH, 1 KH, 2 · · · KH,NG] is the hysteretic stiffness(rigidity) of monitored sections.

Equation (39) expresses the decomposition of the totalmoment into elastic and hysteretic components. The hys-teretic stiffness Equation (2) of a section is a functionof both the section curvature and the hysteretic moment.The latter is obtained by subtracting the elastic componentfrom the total moment. The moment rates at the integra-tion points are found by interpolation of the end momentrates, yielding a direct relationship between the section cur-vatures and the end moments:

bGMe = [aK0 + (I− a)KH]ϕ (40)

where Me = {Mi Mj }T are the element end moments,and

bG =

x1L− 1 x1

L

x2L− 1 x2

L

· · · · · ·xNGL− 1 xNG

L

is the moment interpolation matrix.

The transformation Te maps the local end degrees offreedom to the element deformations ˙ue (Figure 3):

˙ue = Teue (41)

Te =

1 0 0 −1 0 0

0 1L

1 0 − 1L

0

0 1L

0 0 − 1L

1


where ue = {ui vi θi uj vj θj }T are the local gen-eralized displacement coordinates.

Combining Equations (38) and (41) renders a formu-lation of the element force-displacement relation suitablefor considering the connection between local and globaldegrees of freedom:

˙Re = KeTeue (42)

Transforming the global generalized displacement coordi-nates of the end nodes into their local counterparts by stan-dard global-to-local operators achieves the goal of writingEquation (42) entirely in terms of state variables. Recallthat the active nodal degrees of freedom, including thosewith known displacement time histories, have already beendesignated as system state variables:

ue = Tguge (43)

˙Re = KeTeTguge (44)

where Tg is a global-to-local transformation matrix of size6× 6.

The constitutive relations at both the element and sectionlevels have been written entirely in terms of state variablesconsidering the fact that the stress-strain responses (Equa-tion 40) of monitored sections are defined as functions ofthe end moments Me and curvatures φ. The element stateequations result from Equations (44) and (40):

ye(1:3) −KeTeTgy1, e = 0 (45)

bGye(2:3) −[aK0 + (I− a)KH

]ye(4:3+NG) = 0 (46)

Equation (44) establishes one of the necessary linksbetween local and global state variables. The restoringforces of the element, assembled in the global vector R,represent its share in resisting the load imposed on the sys-tem. The full set of end forces Re is generated by equilib-rium transformations of the independent forces Re:

Re = TTe Re (47)

where Re = {Fi Vi Mi Fj Vj Mj }T .The element contribution Rg

e to the global restoring forcevector R is obtained by local-to-global transformation ofthe end forces:

Rge = TTg TTe Re (48)

4.2 Flexibility matrix of beam macro element

The stiffness matrix Ke, which links the independent endforces and displacements, is derived by inverting the ele-ment flexibility matrix. The advantage of a flexibility-basedover a stiffness-based formulation is that the element forcefield can be established exactly from information on the end

Fig. 4. End forces and section stress resultants in the localcoordinate system.

forces. The deformation field, however, cannot be describedaccurately given the end displacements.∗ The force inter-polation functions, on the other hand, are exact becausebeam elements are internally determinate. They enforce lin-ear variation of the bending moment rate and constant dis-tribution of the shear and axial force rates such that theequilibrium conditions are satisfied (Figure 4):

Re(x) = b(x) ˙Re (49)

where Re(x) = {F(x) V (x) M(x)}T are the sectionforces (stress resultants) and

b(x) =

−1 0 0

0 − 1L− 1L

0 xL− 1 x

L

is the force interpolation matrix.

The definition of the flexibility-based beam element isbased on the principle of virtual forces. The relationshipbetween the end force and displacement rates is obtainedby equating the internal and external virtual work:

δRTe ue =

∫ L

0δRe(x)

T ε(x) dx (50)

where ε(x) = {ε(x) γ (x) φ(x)}T are the section defor-mations, and δRe are virtual forces.

The rates of deformation at a particular location x arerelated to the respective stress resultant rates through thetangent matrix of section flexibility distributions f(x):

ε(x) = f(x)Re(x) (51)

where f(x) =

1EA

0 0

0 1GAs

0

0 0 1[aK0+(1−a)KH ]

∗ The shape functions needed for direct generation of the element stiff-

ness matrix by the virtual displacement method yield correct interpolationof the displacements (and, therefore, their second-order spatial derivatives)only for the case of elastic prismatic members.


The tangent flexibility matrix F of the element is derivedby substituting the equilibrium condition (49) and the con-stitutive relation (51) in the virtual work equation (50):

Fe =∫ L

0b(x)T f(x)b(x) dx (52)

All changes of material behavior along the element axisare reflected in the matrix of flexibility distributions. Theaxial, shear, and bending constitutive equations (51) arecoupled if a section is predicted to respond inelasticallyby a predefined yield criterion.25 The matrix of sectionflexibility has off-diagonal terms in the plasticized regionsof the element. The axial force-bending moment inter-action has been studied extensively, and numerous mod-els, the majority of which are based on the classic the-ory of plasticity, have been developed over the years. Theeffect of the combined action of shear and moment on therespective capacities and flexibilities has received its shareof attention, resulting in empirical formulas. In practicalsituations, however, shear-moment interaction, other thanthe inherent coupling by equilibrium, is rarely considered,and the shear rigidity is considered elastic until failure. Itshould be noted that this article aims at presenting a gen-eral algorithm for defining a flexibility-based beam elementin the framework of state-space analysis. The plastic cou-pling of the section constitutive relations is contained inEquations (40) and (52), which express the essence of themethod. Detailed treatment of axial force-bending momentinteraction in two dimensions and biaxial moment interac-tion in three-dimensional space25 is beyond the scope ofthis article.

The Gauss-Legendre quadrature formulas are commonlyused to integrate Equation (52) numerically. Unfortunately,this standard scheme does not use integration points at theends of the interval of integration. The Lobatto quadraturerule samples the ends of the element in order to detectimmediately the onset of nonlinear effects and provides asuitable solution to this problem. Its n integration points,symmetrically placed with respect to the center of the inter-val, integrate exactly a polynomial of order (2n−3). In thisdevelopment, the nodes of the Lobatto quadrature controlthe pattern of intraelement discretization of the nonlinearbending element. The coordinates of the monitored sectionsdirectly follow the abscissas list of the rule.

5 NUMERICAL APPLICATIONS

To illustrate the method, the system of state equations ofthe example structure in Figure 5 is assembled and solvedfor two different types of excitations: (1) quasi-static and(2) dynamic. The simple portal frame was selected toillustrate explicitly the approach; however, complex struc-tures can be handled easily. The selected loading histo-ries induce strong inelastic response with multiple excur-sions in the nonlinear range. The quasi-static analysis uses

Fig. 5. Example frame.

displacement-controlled loading, commonly employed inlaboratory testing of components. The dynamic analysisuses a ground-acceleration record from the 1994 Northridgeearthquake.

The response of the state-space macro-element model iscompared with finite-element solutions of the same prob-lem using standard time-stepping algorithms in combina-tion with Newton-Raphson iterations. The ANSYS1 anal-ysis platform is chosen among the available programsbecause it hosts a nonlinear bending element representativeof an alternative trend in modeling the effect of materialplasticity on the element constitutive relations.

The span, height, section characteristics, and materialproperties of the model structure are derived from an actualdesign. Wide-flanged steel I-sections are used for the framemembers. The uniaxial stress-strain curve of the material(A36) is assumed bilinear with 3 percent hardening and thefollowing properties: E = 199,955 kN/mm2 (29,000 ksi),σy = 248.2 kN/mm2 (36 ksi), and ET = 5999 kN/mm2

(870 ksi).The section constitutive model, mathematically expres-

sed by Equations (1) and (2), requires definition of (1) theinitial bending rigidity K0, (2) the postyield bending rigid-ity aK0, (3) the parameter n controlling the smoothness oftransition, and (4) a discrete yield point My . An assump-tion for the latter is clearly needed, although the marginbetween the yield and plastic moment of wide-flanged I-sections is relatively narrow. The bending moment, whichcauses full plastification of both flanges, is considered areasonable idealization of the yield point. It is also straight-forward to calculate for any level of axial load. Figure 6presents a comparison between the moment-curvatureresponse obtained by deformation-controlled loading of afiber model of the column cross section and the solutionof the differential equation (1) in combination with Equa-tion (2) for the parameters in Table 1. The function of thecurvature variation with time, needed for the integration, isassumed sinusoidal; however, it could be any smooth func-tion, since time is only an auxiliary variable in the case ofquasi-static loading.

In all analysis cases, the macro-element model for theproposed state-space solution consists of three bending ele-ments. The number of internal degrees of freedom needed


Fig. 6. Moment-curvature response of column cross sections.

Table 1Cross-sectional properties and parameters of sections in example

A I α ≈ ET/E My

Element (cm2) (cm4) (%) n (kNm)

Columns 57.99 4504.98 3.03 5 or 10 117.05

Beam 73.96 12535.21 3.03 5 or 10 215.67

for accurate representation of the response of the corre-sponding member of the structure is established in thefollowing paragraph. To simplify the examples, we haveassumed (1) elastic axial force–average strain and shearforce–shear angle relations, (2) no interaction between axialforce and moment, and (3) infinite shear rigidity of thecross section of the element. Under these assumptions, theflexibility matrix Equation (52) can be evaluated numeri-cally as follows:

F1. 1 =L

EA0(53)

F2. 2 =NG∑n=1

(xnL− 1)2 1

[anK0 n + (1− an)KHn]ωnL

2(54)

F2. 3 = F3. 2 =NG∑n=1

(xn

L− 1

)

× xnL

1


2(55)

F3. 3 =NG∑n=1

(xn

L

)2 1


2(56)

where xn, an, K0n, KHn, and ωn are the x coordinate in theelement coordinate system, the ratio of postyield to initialbending rigidity, the initial bending rigidity, the hystereticbending rigidity, and the integration weight of a section atthe nth Lobatto quadrature point, respectively.

The finite-element model in ANSYS1 was created usingthe thin-walled plastic beam element BEAM 24 (ANSYS1).

Table 2Differences between beam element formulations

ANSYS Proposed

Element for-mulation

Stiffness-based Flexibility-based

Cross-sectionaldiscretiza-tion

Fiber model None

Constitutivemodel

Bilinear σ -ε curve Smooth M-φ curve

Postyieldslope

ET = 3%E EIT = 3.03%EI0

Variation ofstiffness

Linear variation of ETbetween the endsand the midpoint

Polynomialinterpolationbetween quadraturepoints

The latter belongs to the class of stiffness-based fiber ele-ment models. The cross sections of the frame memberswere divided into 10 layers parallel to the bending axis, themaximum that could be produced with the default num-ber of points. Three strips of equal thickness were definedin each flange, two in the upper and lower eighths of theweb height and two in the remainder of the web. Onlythe axial stresses and strains were used for determining theeffect of material nonlinearity; the shear components wereneglected. The constitutive relation is assumed bilinear withproperties (E, σy, ET ) identical to those used to obtain themoment-curvature relationship for the flexibility-based ele-ment. To have the effect of material nonlinearity accumu-lated through the thickness and length, the fiber moduli arefirst integrated over the cross-sectional area, and then theintegral for the tangent stiffness matrix is evaluated alongthe element length. This integration uses a standard two-point Gauss rule, while the integration through the depthdepends on the user definition of the cross-sectional layers.A note of interest is that the quadrature points along thelength are interior, while the stress evaluation and, there-fore, the calculation of the individual fiber moduli are per-formed at the two ends and the middle of the element. Thevalues at the integration points are computed by assum-ing linear distribution of the tangent moduli between adja-cent stress evaluation points. The differences between theANSYS1 beam model and the proposed element are sum-marized in Table 2.

5.1 Quasi-static displacement analysis

The node and element numbering scheme are shown inFigure 7. Table 3 maps (1) the generalized nodal displace-


ments of the structure (Figure 7) into the global state vari-ables and (2) the independent generalized end forces andcurvatures at the quadrature points of each element into thelocal state variables. The indices indicate position in thesolution vector.

State variables y11 to y20, y26 to y35, and y41 to y50, notshown in the table, represent curvatures of sections locatedat the interior quadrature stations. The compatibility of thedisplacements between the ends of connected elements isaccounted for implicitly by denoting them by the sameglobal displacement state variables, since the beam-columnconnections are assumed rigid. For example, at joint 2,

x-displacement|element 1

= x-displacement|element 2 = u4 = y1

y-displacement|element 1

= y-displacement|element 2 = u5 = y2

Rotation|element 1

= rotation|element 2 = u6 = y3

Fig. 7. Node and element numbering and activedisplacement DOF.

Table 3Global and local state variables for quasi-static

displacement analysis

7 Global u4 u5 u6 u7 u8 u9

State variable y1 y2 y3 y4 y5 y6

8 Element 1 Fi Mi Mj φ1 · · · φ12

State variable y7 y8 y9 y10 · · · y21





In the general case of a flexible joint, explicit compat-ibility state equations could be written. For example, therotational flexibility of joint 2 could be incorporated easilyusing the following compatibility state equation:

rotation|element 1 − rotation|element 2

= rotation|joint 2(joint flexibility)

and introducing new functions for the resulting additionalstate variables. In the examples presented here, however,the beam-column connections are assumed rigid.

The equations of static equilibrium are obtained byremoving the inertial and damping forces from Equa-tion (31). The global state equations (57) involve onlyactive DOFs and DOFs with known displacement histories.

y4

−y7 + (y23 + y24)/L2

y9 + y23

−y22 + (y38 + y39)/L3

−(y23 + y24)/L2 − y37

y24 + y38

−

d(t)

0

0

0

0

0

= 0 (57)

where d(t) is the displacement history applied at node 2,and L1, L2, and L3 are the lengths of the respective ele-ments.

The state of each element is defined by a pair of equa-tions; see Equations (58) to (63) below, in which the paren-thesized superscript refers to the element number.

{y7 y8 y9}T −[F(1)e

]−1T(1)e T(1)g

× {0 0 0 y1 y2 y3}T = 0 (58)

b(1)G {y8 y9}T −[a(1)K(1)

0 +K(1)H

]× {y10 y11 · · · y21}T = 0 (59)

{y22 y23 y24}T −[F(2)e

]−1T(2)e T(2)g

× {y1 y2 y3 y4 y5 y6}T = 0 (60)

b(2)G {y23 y24}T −[a(2)K(2)

0 +K(2)H

]× {y25 y26 · · · y36}T = 0 (61)

{y37 y38 y39}T −[F(3)e

]−1T(3)e T(3)g

× {y4 y5 y6 0 0 0}T = 0 (62)

b(3)G {y38 y39}T −[a(3)K(3)

0 +K(3)H

]× {y40 y41 · · · y51}T = 0 (63)

It is noteworthy that performing all matrix operations inthese equations yields the exact form of the system ofDAEs assembled for DASSL.


Fig. 8. Quasi-static displacement analysis: shear versus driftof column 1.

A benchmark is established using the ANSYS1 finite-element model by reducing the mesh size until any furtherrefinement does not change the solution. This approach rec-ognizes the fact that the error due to inaccurate interpola-tion of the strain field in stiffness-based nonlinear bendingelements could be minimized by finer discretization.18

The quasi-static displacement history in Figure 8 wasapplied at the top of column 1. After several analyses, itwas established that doubling the number of elements inthe columns from 40 to 80 causes no apparent changes inthe (rotation) solution. The beam remains elastic and doesnot necessitate subdivision. The response of a model with80 elements in each column will serve as a benchmark forverification in all analysis cases considered in this article.A note of interest about the ANSYS1 solution is that theNewton-Raphson iterations are performed until the normof the force imbalances falls within the tolerance limit of0.1 percent of the norm of the applied load. A line-searchtechnique is activated to accelerate convergence.

The macro-element model (Figure 7) for the proposedstate-space solution requires only one nonlinear bendingelement for each frame member, in contrast to the solutionby finite elements (ANSYS1). The number of quadraturepoints is increased to achieve accuracy comparable withthe benchmark model. This alternative method of refine-ment is applicable because the element model is flexibility-based and introduces only numerical integration error in theglobal solution. Results of analyses with gradually increas-ing numbers of quadrature points (not shown here for thesake of brevity) reveal a consistent trend of convergenceto the benchmark solution. A comparison of quasi-staticresponse with the densely meshed ANSYS1 model showsthat 12 integration points per element yield adequate accu-racy of the proposed state-space model. The same numberof quadrature points will be used in all subsequent analyses.

The minor differences between the two solutions in thenonlinear range are mostly due to the different section con-stitutive models. As expected, the transition between the

elastic and plastic phases is better approximated by the fibermodel. The smooth macro model, however, provides anadequate representation with far less computational effort.The quality of the fit could be improved by adjusting theestimate for the yield moment My of the I-section (seeTable 1) to include some web yielding. However, we feelthat the fully plastic moment overestimates the yield point(as defined in the constitutive macro model) and prefer theengineering way of calculating it.

5.2 Dynamic acceleration analysis

The ground acceleration at the north abutment of theSR14/I5 bridge in California, which collapsed during the1994 Northridge earthquake, was used as input for a non-linear dynamic analysis. The inherent damping of the struc-ture is assumed to be 5 percent of critical, modeled by amass-proportional damping matrix. The elastic periods ofvibration are estimated at 0.75 and 0.05 s. The mass of thestructure is lumped at the joints of the frame. For clarity,only horizontal directional components of mass are consid-ered. Correspondingly, the horizontal velocities of nodes 2and 3 (see Figure 7) are added to the global state variablesof the system (Table 4).

Due to the insertion of the mass degrees of freedom,two new state equations (65) are appended to the global set(64):

m1 y7

0

0

m2 y8

0

0

+

α m1 y1

0

0

α m2 y4

0

0

+

−(y10 + y11)/L1 + y24

−y9 + (y25 + y26)/L2

y11 + y25

−y24 + (y40 + y41)/L3

−(y25 + y26)/L2 − y39

y26 + y40

−

−m1 ug

0

0

−m2 ug

0

0

= 0 (64)

{y7 y8}T − {y1 y4}T = 0 (65)

Table 4Ground acceleration analysis: Global and local state variables

Global u4 u5 u6 u7 u8 u9 u4 u7

State variable y1 y2 y3 y4 y5 y6 y7 y8

Note: Element state variables same as Table 3 but translated by the ruley dn+2 = y sn, for n ≥ 7.


Fig. 9. Ground acceleration analysis: shear versus drift column 1.

where m1 = m2 = 24.9626 kN·s/m2 are the lumpedmasses; α = 0.8378 is the mass-proportional dampingcoefficient, and ug(t) is the base acceleration.

As expected, the local state equations (Equations 58through 63) are identical to those for the quasi-static forcecase, except for the translated position indices of the statevariables in the solution vector. The analysis results inFigure 9 show good agreement with the benchmark. Thedifferences are less than 1 percent at major peaks and some-what higher at low-stressed excursions.

6 REMARKS AND CONCLUSIONS

This article presents a general formulation for macro-element analysis in state space. The state variables of a typ-ical discrete system are separated into global and local. Thestate of each of the elements of the structure is determinedby evolution equations involving the end forces, deforma-tions, deformation rates, and other internal variables thatare specific to each element type. The fundamental idea ofthe state-space approach is to solve in time the global equa-tions of motion (equilibrium) simultaneously with all evo-lution equations of all nonlinear elements. The problems,which involve also quasi-static degrees of freedom, can beresolved in state space by using an appropriate method forintegration of the resulting system of differential algebraicequations.

This article presents the formulation and solution ofa flexibility-based nonlinear bending macro element forstructural analysis in the framework of the state-spaceapproach (SSA). The local state variables are the indepen-dent end forces and the curvatures at intermediary integra-tion (quadrature) sections. The formulation employs force-interpolation functions, which strictly satisfy equilibrium,regardless of the geometry. The stress-strain relations atthe quadrature locations are represented by a differentialconstitutive macro model based on summation of the elas-

tic and hysteretic components of the section moment. Theelement flexibility matrix is evaluated by numerical integra-tion. The accuracy can be controlled by intraelement dis-cretization, i.e., changing the number of sampling points.Unlike element subdivision, which implies increasing thenumber of elements, this operation involves less computa-tional effort.

The proposed solution strategy is compared with conven-tional finite-element method, based on time-stepping inte-gration algorithms in combination with Newton-Raphsoniterations. The general nature of the formulation was illus-trated with two numerical examples dealing with the anal-ysis of the same typical structure to different types ofexcitation: quasi-static displacements and ground acceler-ations. These cases are solved with a computer programimplementing the state-space approach and a finite-elementplatform (ANSYS). All results show excellent agreementand demonstrate that the state-space analysis can providean alternative to conventional methods.

ACKNOWLEDGMENTS

We gratefully acknowledge the financial support bythe Multidisciplinary Center for Earthquake EngineeringResearch (MCEER), which is supported by National Sci-ence Foundation, the State of New York, and the FederalHighway Administration.

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