Mixed Variational Methods for Finite Element Analysis of Geometrically Non Linear, Inelastic Bernoulli-Euller Beams

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2003; 19:809–832 (DOI: 10.1002/cnm.622)

Mixed variational methods for �nite element analysis ofgeometrically non-linear, inelastic Bernoulli–Euler beams

K. D. Hjelmstad1;∗ and E. Taciroglu2

1Department of Civil and Environmental Engineering; University of Illinois at Urbana-Champaign;Urbana; IL; 61801; U.S.A.

2Center for Simulation of Advanced Rockets; University of Illinois at Urbana-Champaign;Urbana; IL; 61801; U.S.A.

SUMMARY

Bernoulli–Euler beam theory has long been the standard for the analysis of reticulated structures. Theneed to accurately compute the non-linear (material and geometric) response of structures has renewedinterest in the application of mixed variational approaches to this venerable beam theory. Recent con-tributions in the literature on mixed methods and the so-called (but quite related) non-linear �exibilitymethods have left open the question of what is the best approach to the analysis of beams. In this paperwe present a consistent computational approach to one-, two-, and three-�eld variational formulationsof non-linear Bernoulli–Euler beam theory, including the e�ects of non-linear geometry and inelasticity.We examine the question of superiority of methods through a set of benchmark problems with fea-tures typical of those encountered in the structural analysis of frames. We conclude that there is noclear winner among the various approaches, even though each has predictable computational strengths.Copyright ? 2003 John Wiley & Sons, Ltd.

KEY WORDS: beam theory; frame analysis; �nite elements; mixed variational principles; Hu–Washizu;Hellinger–Reissner; non-linear �exibility methods

1. INTRODUCTION

Virtually all analysis of reticulated structures done today is based upon Bernoulli–Euler beamtheory. Most commercial programs available to do this sort of analysis allow only the con-sideration of prismatic beams (i.e. the properties do not change along the length of thebeam). However, most of these programs do not include capabilities to do geometricallynon-linear analysis or inelastic analysis. There is a growing awareness among structural engi-neers that these limitations are not compatible with the limit-states philosophies, particularly

∗Correspondence to: Prof. K. D. Hjelmstad, Department of Civil and Environmental Engineering, University ofIllinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.

Contract=grant sponsor: The Department of Energy, Center for Simulation of Advanced Rockets

Received 18 July 2002Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 15 January 2003

810 K. D. HJELMSTAD AND E. TACIROGLU

performance-based design, commonly used to design structures for extreme environments inwhich non-linear response is likely.It is rather straightforward to obtain an exact solution to the Bernoulli–Euler beam equa-

tions for linear, elastic, prismatic beams. The simplicity of treating the problem with classicalmethods has probably had more to do with the ubiquity of prismatic beam analysis than hasthe actual incidence of prismatic beams. That said, the importance of correctly computing thesti�ness coe�cients of non-prismatic beams has long been recognized. The sti�ness coe�-cients for non-prismatic beams are generally obtained with the classical principle of virtualforces because that approach a�ords an exact solution [1]. Recently, the merits of the so-called�exibility approach to determining sti�ness coe�cients has turned the attention of structuralanalysts to non-linear �exibility methods [2–5]. The primary goal of these investigations hasbeen to improve the accuracy of analysis with Bernoulli–Euler beam theory when the consti-tutive equations are non-linear. Some of the methods proposed have a consistent variationalstructure while others do not. The connection between the non-linear �exibility methods andcertain mixed variational methods has been recognized [6] and yet it seems that mixed meth-ods have not been completely and systematically explored for the Bernoulli–Euler beam. Inparticular, non-linear geometric e�ects have yet to be included and the relative merits of thevarious approaches is still unclear.The sampling of methods of analysis that have appeared in the literature have included

contributions that are far from transparent and the literature is spotted with unsubstantiatedclaims of superiority of one method over another. The purpose of the present paper is toexhaust the range of possibilities of variational approaches to the Bernoulli–Euler beam andto attempt, therefore, to provide a context to make comparisons among them. We shall attempt toshow in this paper that there is no clear winner among the three viable variational possibilities(one-, two-, and three-�eld formulations) in terms of computational e�ciency, at least for thetypes of problems typically encountered in non-linear structural analysis. By that same token,we shall see that each approach has a rather predictable strength in its ability to approximatethe response of the beam.In this paper we extend our earlier work [7] in two respects. First, we extend the formulation

of mixed methods for Bernoulli–Euler beams to geometrically non-linear problems and cou-pled (moment–axial interaction) inelasticity. Second, we study the performance of the variousformulations across a set of benchmark problems speci�cally designed to excite certain fea-tures of response. We make an attempt to do some cost accounting of the methods with h- andp-re�nement to clarify the question of which formulation is the best for general use. This paperalso extends the work of Petrangeli and Ciampi [8] through the incorporation of new approachesfor three-�eld problems and a more complete discussion of the computational treatment (e.g.condensation and recovery) of the mixed variables. These extensions draw on the work in mixedmethods for three-dimensional �nite element technology [8–16]. This paper answers the questionof how to extend non-linear �exibility approaches to geometrically non-linear problems.

2. NON-LINEAR BERNOULLI–EULER BEAM THEORY

The classical equilibrium and strain–displacement equations of Bernoulli–Euler beam theory are

n′ = p; U= u′ + 12(w

′)2

m′′ − (nw′)′ = q; –=w′′ (1)

Copyright ? 2003 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2003; 19:809–832

MIXED VARIATIONAL METHODS FOR FINITE ELEMENT ANALYSIS 811

�

q

�

m

w

p

u

n

n

(a) (b)

Figure 1. Planar Bernoulli–Euler beam: (a) de�nition of position and loading; and (b) de�nition ofpositive axial and transverse displacement and positive axial force and bending moment.

where m is the bending moment �eld, n is the axial force �eld, q and p are the appliedtransverse and axial load per unit length, respectively, – is the curvature �eld, U is the axialstrain �eld, w is the transverse displacement �eld, and u is the axial displacement �eld. Eachof these �elds is a function of the axial coordinate �, which is measured from the left end ofthe beam, as shown in Figure 1. This �gure also indicates the convention for positive valuesof the �eld variables. A prime denotes di�erentiation with respect to �. Throughout this paperlowercase characters represent �eld variables that depend upon �.Let us de�ne generalized displacement, stress, strain, and load vectors, respectively, as

u≡ [u; w]T; s≡ [n;m]T; e≡ [U; –]T; p≡ [p; q]T (2)

Let us also de�ne a strain–displacement operator

U(u)≡ [u′ + 12(w

′)2; w′′]T (3)

For the variational methods described herein, it is important to maintain a distinction betweenstrain as an independent �eld, e, and strain computed as derivatives of the displacement�eld, U(u).Variational methods rely on the introduction of virtual �elds that represent variations

of the independent �elds. Let the virtual displacements, stresses, and strains be denoted�u≡ [ �u; �w]T; �s≡ [ �n; �m]T, and �e≡ [ �U; �–]T, respectively. When the virtual strain is to be com-puted from the virtual displacement then we have

�U(u)=DU(u) · �u≡ dda[U(u+ a �u)]a=0 = [ �u′ + w′ �w′; �w′′]T (4)

where DU(u) · �u stands for the directional derivative of the function U(u) in the direction �u.Let us also de�ne the matrix

�(u)≡[1 w′ 0

0 0 1

](5)

and the di�erential operator

B(u)≡ [u′; w′; w′′]T (6)

With this notation DU(u) · �u=�(u)B(�u). We will often have need of B(�u) and B(�u) in ourformulations. These vectors can be obtained by using the incremental displacement �eld andthe virtual displacement �eld, respectively, in Equation (6).



m

mo

non

f (s, �) = 0

Figure 2. The yield surface for stress resultants.

In Bernoulli–Euler beam theory the bending moment m and axial force n play the role ofthe internal stress resultants, while the curvature – and axial strain U are the correspondingstrain resultants. In general, we can express the constitutive hypothesis as g(s; e)= 0. It willbe convenient, however, to also think of the constitutive equations in a strain-driven or stress-driven format (i.e. the stress computed given the strain or the strain computed given the stress,respectively). Thus, we shall write the constitutive equations alternatively (and equivalently)as

s= s(e); e= e(s) (7)

For example, for a linear, elastic beam with centroidal axes, the constitutive function takesthe form s(e)=De, where D=diag[EA;EI] is the sti�ness of the cross section. Models ofinelasticity generally include internal variables (e.g. plastic strain and hardening variables).We shall always assume that equations for the internal variables are satis�ed pointwise (at theGauss points of numerical integration). In Section 3 we present a simple model of inelasticityto make de�nite the treatment of the constitutive equations for history dependent response.

3. SIMPLE MODEL FOR INELASTICITY IN STRESS RESULTANTS

One of the most common sources of material non-linearity is inelasticity. Inelasticity providesa good benchmark for comparing competing algorithms for beam problems because cyclicinelasticity can lead to curvature and residual stress �elds that are much more complicated thantypically arise in elastic problems. This section describes a very simple model of inelasticityand allows us to show clearly how the constitutive equations of inelasticity manifest in thevariational formulations.Let the stress be given from the elastic strains as s=D[e − ep], where ep≡ [Up; –p]T is

the plastic strain and D≡ diag[EA;EI] is the elastic modulus. Let the yield function be de-�ned as f(s; �)≡

√s ·Qs − k(�), where the metric Q is de�ned through the relationship

Q−1≡ diag[n20; m20] and where n0 and m0 are the axial yield force and the yield moment, re-spectively. Figure 2 illustrates the initial yield surface. For simplicity in the examples we willconsider a linear strain hardening function k(�)=1 + ��, where � is the hardening modulus.More general models of hardening present no additional di�culty.



States are elastic inside the yield surface. Plastic strains accrue for states on the yieldsurface as ep≡ �q(s; �), where � is the consistency parameter and � is the internal variablegoverning (isotropic) strain hardening. Assuming associative �ow, we have

q≡ @f@s=

Qs√s ·Qs (8)

The evolution of the hardening variable is given by �≡√ep ·Q−1ep = �, which is an ordinary

strain hardening law. The de�nition of the plasticity formulation is completed with the classicalcomplementarity conditions (�¿0; f(s; �)60, and �f(s; �)=0) and the consistency condition,�f(s; �)=0.

3.1. Return-mapping algorithm for strain-driven formulations

The constitutive rate equations can be discretized in time by integrating with the backward-Euler method. For the time interval [tn; tn−1], let us de�ne �tn≡ tn − tn−1 and let the discreteconsistency parameter be ��n≡�tn�n. Note that a subscript n indicates that the variable isevaluated at time tn. Let

qn≡ Qsn√sn ·Qsn

(9)

where sn is the stress at time tn. Integrating the constitutive equations we have [17]

sn = sn−1 + D[en − en−1 − ��nqn]epn = epn−1 + ��nqn

�n = �n−1 + ��n

(10)

from which it is clear that the new state depends upon the consistency parameter ��n, whichis subject to the discrete complementarity conditions

��n¿0; fn60; ��nfn=0 (11)

A trial stress can be de�ned by assuming the process for the current time interval is elastic(i.e. assuming ��n=0). To wit, s

trialn ≡ sn−1 + D[en − en−1]. The yield function, evaluated at

the trial stress is ftrialn ≡f(strialn ; �n−1). The complementarity conditions require that if ftrialn 60then the trial state is the actual state and ��n=0. The new state can then be computed fromEquation (10).On the other hand, if ftrialn ¿0 then the trial state is not the actual state. It follows from the

complementarity conditions that since ��n¿0 the actual state must necessarily satisfy fn=0.De�ning R(�)≡ k(�n−1 + �)Q−1 + �D, we can compute the following recursion:

�q�=R−1(��−1)strialn ; q�=

�q�√�q� ·Q−1 �q�

; ��=q� · strialn − k(�n−1 + ��−1)

q� · Dq� (12)

The iteration can be started with �0 = 0 and should continue until the residual of Eq. (10)afalls below a certain tolerance, i.e. |R(��)q�−strialn |¡tol, where tol is some prede�ned tolerance.Upon convergence, update the consistency parameter as ��n= ��.



The consistent tangent matrix can be computed knowing the consistency parameter. Let%n≡ ��n=kn, where kn≡ k(�n−1 + ��n), and let Dn≡ [D−1 + %nQ]−1. Then the tangent sti�nessmatrix is given by the expression [17]

Dn≡ Dn − �nDnqn ⊗ Dnqn (13)

where, with k ′n≡ k ′(�n−1 + ��n), the parameter �n has the expression

�n≡ 1− k ′n%nk ′n + (1− k ′n%n)�n

; �n≡ qn · Dqn (14)

Note that, if ��n=0 (i.e. an elastic state) then @s=@e=D. If ��n¿0, then the consistent tangentis @s=@e=Dn. The method to obtain the stress and the consistent tangent described in thissection is apropos to both the classical sti�ness and the mixed-enhanced formulations.

3.2. Return-mapping algorithm for stress-driven formulations

The constitutive update procedure for stress-driven problems is slightly di�erent from thestrain-driven case. Given the stress vector sn (from the interpolation), we can evaluate a trialyield function ftrialn ≡f(sn; �n−1). If ftrialn 60 then the trial state is the actual state and thestate can be updated as

en= en−1 + D−1[sn − sn−1]; �n= �n−1; epn = epn−1 (15)

On the other hand, if ftrialn ¿0 then the complementarity conditions imply that ��n¿0 and thatf(sn; ��n)=0. This equation can be solved iteratively as

��+1n = ��n −1

k ′(�n−1 + ��n)(√sn ·Qsn − k(�n−1 + ��n)) (16)

Now, with ��n known, we can easily obtain the strain as

en= en−1 + ��nqn + D−1[sn − sn−1] (17)

Note that, if ��n = 0 (i.e. an elastic state) then @e=@s=D−1. If ��n¿0, then the consistenttangent is @e=@s=D−1

n (see Equation (13)). This format is apropos to the Hellinger–Reissnerformulation.

4. VARIATIONAL APPROACHES TO NUMERICAL APPROXIMATION

Within the context of variational approaches to the problem of determining the unknown �eldsu; s, and e, given the loading p, there are three basic formulations [18, 19]. The classicalsti�ness approach allows variations of u and satis�es the equilibrium equations weakly. Thestrain–displacement and constitutive equations are strongly enforced. The Hellinger–Reissnerapproach allows variations of u and s and satis�es the equilibrium and strain–displacementequations weakly. The constitutive equations are strongly enforced. Finally, the mixed-enhanced (Hu–Washizu) approach allows variations of u, s, and e and satis�es all of the�eld equations weakly. This section outlines the approach to computation based upon thesedi�erent variational formulations.



4.1. The one-�eld (Classical Sti�ness) approach

The classical sti�ness approach is based upon the displacement-based functional

S(u; �u)≡∫ ‘

0[ �U(u)Ts(U(u))− �uTp] d� (18)

By the fundamental theorem of the calculus of variations, equilibrium is satis�ed if S(u; �u)=0for all variations �u [20]. Let ui represent a con�guration of the beam (not necessarily anequilibrium con�guration). This functional can be linearized at the state ui to give L[S]=Si +�Si, where Si= S(ui ; �u) and

�Si≡ dd�[S(u+ ��u; �u)]u=u i =

∫ ‘

0BT(�u)[�Ti Di�i +Gi]B(�ui) d� (19)

where �i=�(ui);Di= @ s=@e is the tangent sti�ness evaluated at the state ui, and the matrixGi=G(si) is the geometric sti�ness

G(s)=

0 0 0

0 n 0

0 0 0

(20)

evaluated at the state si. Note that the argument of the matrix G is the stress s. It should beunderstood that in the case of a displacement-based formulation the stress is evaluated fromthe displacement as s= s(U(u)). For the mixed formulations the stress s will be an independent�eld and is used in G directly.In Newton’s method we set L[S]= 0 for all �u to provide a means of estimating an in-

cremental state �ui that, when added to ui, will yield a state that comes closer to satisfyingS(u; �u)=0. The process can be repeated to convergence.In the classical sti�ness method, we interpolate the displacement �eld as a linear com-

bination of known base functions h (which are functions of �) and unknown displacementparameters U as u= hTU. The incremental displacement and virtual displacement �elds areinterpolated similarly as �u= hT�U and �u= hT �U, respectively, where �U and �U are thediscrete incremental and virtual displacement parameters, respectively. The matrix h interpo-lates both the axial displacement u and the transverse displacement w. The speci�c form ofthis matrix is

hT≡[g1 g2 · · · gM 0 0 · · · 0

0 0 · · · 0 h1 h2 · · · hN

](21)

The set of functions {g1; : : : ; gM} interpolate the axial displacement �eld, where M is thetotal number of terms in the interpolation. The set of functions {h1; : : : ; hN} interpolate thetransverse displacement �eld, where N is the number of terms in the interpolation. Thisorganization of h implies that the displacement unknowns are organized in the array U with theM axial displacement parameters followed by the N transverse displacement parameters. Theminimal interpolation is two terms for the axial displacement, e.g. g1 = �=‘ and g2 = 1− �=‘(to assure continuity of the displacement), and four terms for the transverse displacement,e.g. the cubic hermitian polynomial functions (to assure continuity of the displacement and



its �rst derivative). Both interpolations can be augmented with additional functions as shownin Section 5.Substitution of the interpolated �elds into the linearized classical sti�ness functional yields

the discrete version L[S]= �UT[Ki�Ui − ru(ui)]. The condition L[S]= 0 for all �U implies

the discrete equation

Ki�Ui − ru(ui)= 0 (22)

where the tangent sti�ness Ki can be computed as

Ki≡∫ ‘

0a[�Ti Di�i +Gi]a

T d� (23)

and the residual force ru(ui) is given from the general expression

ru(ui)=∫ ‘

0[a�Ti s(U(ui))− hp] d� (24)

The matrix a is simply the result of interpolating B(·) and has the particular form

aT≡

g′1 g′2 · · · g′M 0 0 · · · 0

0 0 · · · 0 h′1 h′2 · · · h′N

0 0 · · · 0 h′′1 h′′2 · · · h′′N

(25)

The sti�ness matrix and residual force can be assembled in the standard manner from elementcontributions to give the linearized (incremental) global equilibrium equations. Equilibrium isestablished for the non-linear case by iteratively solving the equation Ki�Ui= ru(ui) andupdating the estimate of the displacements with the equation Ui+1 =Ui +�Ui until the forceresidual is small enough, i.e. ‖ru(ui)‖¡tol. The iterative computation must be started with anestimate U0 and the tolerance tol on the satisfaction of equilibrium must be speci�ed a priori.

4.2. The two-�eld (Hellinger–Reissner) approach

Consider the following the two-�eld (Hellinger–Reissner type) functional for the Bernoulli–Euler beam

H (u; s; �u; �s)≡∫ ‘

0[ �U(u)Ts − �uTp− �sT(e(s)− U(u))] d� (26)

This functional depends upon the two �elds s and u (and their virtual counterparts) and hasthe property that if H =0 for all variations �u and �s then the state (u; s) satis�es the classicalequilibrium and strain–displacement equations. This functional can be linearized about thestate (ui ; si) to give L[H ]=Hi +�Hi, where Hi=H (ui ; si ; �u; �s) and

�Hi≡ dd�[H (u+ ��u; s+ ��s; �u; �s)](u=u i ; s=s i) (27)



is the directional derivative of the functional in the direction of an increment of the two �elds.The quantity �Hi has the particular expression

�Hi=∫ ‘

0[BT(�u)(�Ti �s+GiB(�u))− �sT(D−1

i �s − �iB(�u))] d� (28)

where �i and Gi are given by Equations (5) and (20), respectively, and D−1i = @e=@s is the

tangent compliance evaluated at the state si. As in the classical sti�ness formulation, the termswith B(·) can be interpolated with the displacement interpolation h given in Equation (21),with the result being the matrix a. The stresses s must be interpolated independently. Thisinterpolation will be expressed as s= bTS, where S is the vector of force parameters. We willorganize the interpolation as

bT≡[b1 b2 · · · bA 0 0 · · · 0

0 0 · · · 0 b1 b2 · · · bB

](29)

The incremental and virtual stress �elds are also interpolated with b as �s= bT�S and�s= bT �S, respectively. Although it is not required, we shall interpolate the axial force n andthe bending moment m with the same base functions bi. The number of terms in the axialinterpolation is A and the number in the bending interpolation is B. As will be discussed later,these functions will be selected as orthogonal polynomials, starting with the constant function.Again, we substitute the interpolated �elds into the linearized functional L[H ]. The con-

dition L[H ]= 0 for all �U and �S implies the discrete equations

�Gi�Ui + Bi�Si + ru(ui ; si) = 0

BTi �Ui − Fi�Si + rs(ui ; si) = 0(30)

where the matrix Bi, the geometric sti�ness matrix �Gi and the �exibility matrix Fi are givenby the expressions

BTi ≡∫ ‘

0b�iaT d�; �Gi≡

∫ ‘

0aGiaT d�; Fi≡

∫ ‘

0bD−1

i bT d� (31)

The residuals are de�ned as

ru(ui ; si)≡∫ ‘

0[a�Ti si − hq] d�; rs(ui ; si)≡

∫ ‘

0b[e(si)− U(ui)] d� (32)

There are basically three choices for implementing the global solution scheme for the mixedformulation. First, one can retain both �Ui and �Si as global variables. One would then forman element sti�ness and residual for the combined unknowns as

Ki≡[ �Gi Bi

BTi Fi

]; r≡

[ru(ui ; si)

rs(ui ; si)

](33)

and assemble a combined global sti�ness and residual in the standard manner, taking careto include the stress ‘degrees of freedom’ in the global equation numbering scheme. The



drawback of this choice is a larger global system of equations and potential ill-conditioningof the system matrix.The second alternative involves elimination of the stress variables at the element level.

Provided that Fi is invertible, we can express the incremental generalized stress in terms ofthe incremental displacement using Equation (30). To wit,

�Si=F−1i [rs(ui ; si) + BTi �Ui] (34)

This result can be substituted into the �rst equation to produce and incremental equations forthe displacement alone as

�Ki�Ui + �ri=0 (35)

where the reduced sti�ness and residual (of dimension equal to the number of displacementdegrees of freedom) are given by

�Ki≡BiF−1i BTi + �Gi ; �ri≡ ru(ui ; si) + BiF−1

i rs(ui ; si) (36)

As usual, the equations would be assembled into global equations for the incremental displace-ment. Upon solving the global equations for �Ui, the incremental stresses can be computedfrom Equation (34). Note that the computation of the increment in stress requires either thestorage of Fi ;Bi, and rs(ui ; si) or their recomputation. Finally, the state is updated in the usualmanner as Ui+1 =Ui +�Ui and Si+1 =Si +�Si.The third alternative was suggested by Simo et al. [14]. The displacement state ui+1 is

known upon solving the global equations and updating. Now the functional H can be regardedas a single-�eld functional

H (s; �s)≡∫ ‘

0�sT(e(s)− U(ui+1)) d� (37)

Using the same procedure of linearization, but viewing only the �eld s as a variable, weobtain the incremental equation

F��S�i+1 = rs(s�i+1) (38)

where the �exibility matrix F� is the same as before (computed at the stress state s�i+1), andthe revised residual is

rs(s)≡∫ ‘

0b[e(s)− U(ui+1)] d� (39)

Equation (38) is solved repeatedly and the stress state is updated in the usual manner asS�+1i+1 =S

�i+1 + �S

�i+1. The iteration can start with S

0i+1 =Si. Upon convergence to ‖rs‖¡tol,

the stress state can be updated as Si+1 =S�i+1. Note that this �nal stress state is di�erent fromthe ones generated by the �rst two methods. In particular, it exactly forces the residual inEquation (39) to zero. If the function U(·) is linear, then a single iteration of Equation (38)is equivalent to the previous two methods. For a linearly elastic beam, the stress convergesexactly in one iteration. Because the updated stress is di�erent from the Newton estimate, thisthird approach will a�ect the convergence of Newton’s method. Because the new algorithmis consistent with the original one, the quadratic asymptotic rate of convergence is preserved.



Formation and solution of Equation (38) requires repeated computation of F� and the newresidual rs(s�i+1). Thus, it is not clear which of the three implementations is the most eco-nomical.

4.3. The three-�eld (mixed-enhanced) approach

Consider the following three �eld (Hu–Washizu type) functional for the Bernoulli–Euler beam

J (u; s; e; �u; �s; �e)≡∫ ‘

0[ �U(u)Ts − �uTp− �sT(e − U(u))− �eT(s − s(e))] d� (40)

This functional depends upon the three �elds u, s and e (and their virtual counterparts) andhas the property that if J =0 for all variations �u, �s, and �e then the state (u; s; e) satis�esthe classical equilibrium, the strain-displacement equations, and the constitutive equations.This functional can be linearized about the state (ui ; si ; ei) to give L[J ]= Ji + �Ji, whereJi= J (ui ; si ; ei ; �u; �s; �e) and

�Ji≡ dd�[J (u+ ��u; s+ ��s; e+ ��e; �u; �s; �e)](u=ui ; s=si ; e=ei) (41)

is the directional derivative of the functional in the direction of an increment of the two �elds.The quantity �Ji has the particular expression

�Ji=∫ ‘

0[BT(�u)(�Ti �s+GiB(�u))− �sT(�e − �iB(�u))− �eT(�s − Di�e)] d� (42)

where �i, and Gi are given by Equations (5) and (20), respectively, and Di= @s=@e is thetangent sti�ness evaluated at the state ei. As in the classical sti�ness formulation, the termswith B(·) can be interpolated with the displacement interpolation h given in Equation (21),with the result being the matrix a. The stresses s can be interpolated as s= bTS, as in theprevious section.The strains must also be interpolated in this mixed formulation. The idea behind the mixed-

enhanced formulation is to let the strain be interpolated as e= bTE0 + cTEe. The strain pa-rameters E0 represent the part of the strain that is interpolated identically to the stresses andEe are the enhanced strains. The enhanced strain interpolation is organized as

cT≡[c1 c2 · · · cA 0 0 · · · 0

0 0 · · · 0 c1 c2 · · · cB

](43)

Again, although it is not required, we shall interpolate the axial strain ” and the curvature –with the same base functions ci. The number of terms in the axial interpolation is A and thenumber in the bending interpolation is B (not necessarily the same number as in the stressinterpolation).If we substitute our interpolations of displacement, stress, and strain into the linearized

functional we get the discrete version of the functional. The condition L[J ]= 0 for all �U, �S, �E0,



and �Eeimplies the following set of equations:

�Gi�Ui + Bi�Si + ru(ui ; si ; ei) = 0

BTi �Ui − P�E0i −C�Eei + rs(ui ; si ; ei) = 0

−P�Si + J11i �E0i + J12i �Eei + r0(ui ; si ; ei) = 0

−CT�Si + J21i �E0i + J22i �Eei + re(ui ; si ; ei) = 0

(44)

where the matrices P and C are integrals of only interpolation functions and have the expres-sions

P≡∫ ‘

0bbT d�; C≡

∫ ‘

0bcT d� (45)

The (sti�ness) matrices J11i ;J12i =[J21i ]T, J22i are given by the expressions

J11i ≡∫ ‘

0bDibT d�; J12i ≡

∫ ‘

0bDicT d�; J22i ≡

∫ ‘

0cDicT d� (46)

and the residuals are de�ned as

ru(ui ; si ; ei) ≡∫ ‘

0[a�Ti si − hq] d� rs(ui ; si ; ei) ≡

∫ ‘

0b[ei − ”(ui)] d�

r0(ui ; si ; ei) ≡∫ ‘

0b[si − s(ei)] d� re(ui ; si ; ei) ≡

∫ ‘

0c[si − s(ei)] d�

(47)

The base functions bi and ci can be chosen to be orthogonal on [0; ‘]. If this choice is made(and it will be in the sequel) then the matrix C is identically zero.The same three choices are available for implementing the global solution scheme for the

three-�eld mixed formulation as were available for the two-�eld mixed formulation. First, onecan retain all of the parameters �Ui, �Si, �E0i , and �Eei in the global solution. One wouldthen form an element sti�ness and residual for the combined unknowns as

Ki≡

�Gi Bi 0 0

BTi 0 −P 0

0 −P J11i J12i

0 0 J21i J22i

; ri≡

ru(ui ; si ; ei)

rs(ui ; si ; ei)

r0(ui ; si ; ei)

re(ui ; si ; ei)

(48)

and assemble a combined global sti�ness and residual in the standard manner. The drawbackof this choice, again, is a larger global system of equations. Numerical conditioning of thoseequations is also an issue as the unknowns can have vastly di�erent scales.The second alternative is to condense the system to only displacement unknowns. Pro-

vided that J22i is invertible, the incremental generalized stress and strain can be eliminated



from Equation (44) in favour of the displacement increment. De�ning, the matrix �i≡BiP−1

we �nd

�E0i = �Ti �Ui − P−1rs

�Eei = [J22i ]−1[re − J21i �E0i ]

�Si = P−1[J11i �E0i + J

12i �E

ei − r0]

(49)

These results can be substituted into the �rst equation to produce and incremental equationsfor the displacement alone as

��Ki�Ui + �ri= 0 (50)

where the reduced sti�ness and residual (of dimension equal to the number of displacementdegrees of freedom) are given by

��Ki≡�i �Ji�Ti + �Gi ; ��ri≡ ru + �i[r0 + �JiP−1rs − J12i [J22i ]−1re] (51)

where �Ji≡J11i −J12i [J22i ]−1J21i . As usual, the equations would be assembled into global equa-tions for the incremental displacement. Upon solving the global equations for �Ui, the incre-mental stresses and strains can be computed from Equation (49). Again, the computation ofthe increment in stress and strain requires either the storage or recomputation of the elementmatrices and residuals. Finally, the state is updated in the standard fashion.The third alternative is similar to the one described in the previous section. However, the

presence of more �elds complicates the recovery of the additional �elds. This recovery canbe done sequentially, somewhat like the sequence implicit in Equation (49). First, we recog-nize that the displacement state can be updated upon solving the global equations obtainedby assembling the element contributions from Equation (50). We can determine the strainparameters E0i+1 by recognizing that the residual rs should go to zero at the new state. To wit,

rs=∫ ‘

0b(ei+1 − U(ui+1)) d�= 0 (52)

Substituting ei+1 = bTE0i+1 + cTEei+1, recognizing the orthogonality of b and c we �nd that

E0i+1 =P−1

∫ ‘

0bU(ui+1) d� (53)

By similar reasoning, noting that we now know Ui+1 and E0i+1, the equations re= 0 leads usto the local iteration

[Eei+1]�+1 = [Eei+1]

� − [J22� ]−1∫ ‘

0cs(e�i−1) d� (54)

where e�i+1 = bTE0i+1 + c

T[Eei+1]� is the strain associated with the current estimate of the en-

hanced strain Eei+1. This local Newton iteration can be started with [Eei+1]

0 =Eei and cancontinue until the residual (the integral term in Equation (54)) reduces in absolute value towithin an acceptable tolerance. Now the enhanced strain Eei+1 is known and hence the strain



Table I. Base functions for interpolation.

Minimal displacement interpolation functions

Axial Transverse Legendre polynomials

g1 = x h1 = 1− 3x2 + 2x3 p1 = 1g2 = 1− x h2 = x2(3− 2x)‘ p2 =

√3(1− 2x)

h3 = x(1− 2x + 2x2) p3 =√5(1− 6x + 6x2)

h4 = x2(x − 1)‘ p4 =√7(1− 12x + 30x2 − 20x3)

ei+1 is completely known. Finally, recognizing that r0 = 0 must be satis�ed, we can determinethe stress parameters from

Si+1 =P−1∫ ‘

0bs(ei+1) d� (55)

The economies of the third method should be evident. Assuming that no element level in-formation is saved (except, of course, the state variables), then the recovery requires theevaluation of residuals and evaluation and inversion of the matrices P and J22� (repeatedly).

RemarkNodeless (bubble) displacement degrees of freedom can appear in any of the three types ofbeam elements. These unknowns are not bound by any inter-element continuity requirementsand therefore need not appear as unknowns in the global equations. They can easily becondensed out and recovered at the element level much like the stress and curvature parametersin the mixed formulations.

5. CHOICE OF INTERPOLATION FUNCTIONS

For the classical sti�ness method and the mixed methods, the minimal interpolation of the axialdisplacement �eld has two terms and the minimal interpolation of the transverse displacementhas four terms. These interpolations are accomplished with the two linear and four cubicHermitian polynomials in Table I, in which –≡ �=‘. This interpolation is necessary becauseit is the simplest means of enforcing the interelement continuity of the displacement �eld.Enhancement of the displacement �eld is possible for all formulations. This enhancement

can be accomplished using the so-called bubble functions for the axial and transverse basefunctions. Let

�g≡ x(1− x); �h≡ x2(1− x)2 (56)

A sequence of enhanced displacement interpolation functions can then be de�ned as

g2+i=�gpi; h4+i=�hpi (57)

where the functions pi are the (orthonormal) Legendre polynomials, the �rst four of whichare given in Table I.



For the mixed methods the stress �eld is interpolated with the Legendre polynomials, i.e.,bi=pi. Because the Legendre polynomials are orthogonal on [0; 1], they are the right choiceto interpolate the enhanced curvatures in the mixed-enhanced method. The functions ci mustbe orthogonal to the functions bi. Hence, the �rst enhanced strain interpolation function is theLegendre polynomial following the last one used for stress interpolation. For example, if weuse a two term approximation for the stress and a three term approximation for the enhancedstrain then the appropriate interpolation matrices are

bT≡[p1 p2 0 0

0 0 p1 p2

]; cT≡

[p3 p4 p5 0 0 0

0 0 0 p3 p4 p5

](58)

Note that one of the rami�cations of selecting the orthonormal Legendre polynomials isthat the matrix P= I. This choice leads to savings in computation as this matrix must beinverted in both of the mixed methods.In the classical sti�ness method there is one choice that needs to be made—the number

of base functions to use for the displacement �eld (above the minimum required to enforcecontinuity). We will refer to the classical sti�ness methods as CSi with the subscript i indicat-ing the number of transverse displacement interpolation functions used (e.g. CS4 is the usualmethod used in structural analysis). The method based upon the Hellinger–Reissner type offunctional has two choices—the number of displacement base functions and the number ofstress base functions. We will refer to the Hellinger–Reissner type methods as HRij with thesubscript i indicating the number of transverse displacement interpolation functions and thesubscript j indicating the number of stress interpolation functions. Similarly, we will refer tothe mixed-enhanced methods as MEijk where i, j, and k indicate the number of transversedisplacement, stress, and enhanced strain interpolation functions, respectively. Note that theaxial and transverse �elds can be interpolated with a di�erent number of base functions.Hjelmstad and Taciroglu [7] have shown that the best mixed elements have i= j+2 for the

transverse �elds and i= j+1 for the axial �elds, where i is the number of displacement param-eters and j is the number of stress parameters. These elements are all stable and convergent.Throughout the examples selection of the number of axial parameters follows this convention.

6. ELEMENT PERFORMANCE IN BENCHMARK EXAMPLES

In this section, we compare the performance of the classical sti�ness, Hellinger–Reissner andthe mixed-enhanced beam formulations through a set of benchmark problems. Each problemis solved for di�erent levels of mesh re�nement and order of interpolation. Gauss–Lobattoquadrature is used throughout for numerical integration of the �nite element matrices. Gauss–Lobatto quadrature uses N sampling points to exactly integrate polynomial integrands ofdegree up to 2N − 3. The computational cost of each element (type and order) are reported,in an approximate sense, for a comparison of their performance.

6.1. Transversely loaded non-prismatic cantilever beam

A cantilever beam of length ‘=1 is transversely loaded at its free end with a loading ofP=1, H =0. The linearly varying depth gives rise to a bending modulus that varies asD(�)= (1 + �)3. The beam and its loading are shown in Figure 3.



Figure 3. Cantilever beam with a transverse load and variable bending sti�ness.

Table II. Normalized displacement for the HR elementsfor the cantilever beam problem.

N HR42 HR53 HR64

4 0.910 2.792 1.5705 0.995 1.041 1.4696 1.000 1.002 1.0557 1.000 1.000 1.0118 1.000 1.000 1.001Exact 1.000 1.000 1.000

The number of quadrature points required to fully integrate the �nite element matrices andvectors di�ers among the three approaches. Because the bending sti�ness varies cubicallyalong the length of the beam, the full integration of the element matrices of the CSi elementsrequire N¿i sampling points with the Gauss–Lobatto quadrature. The ME421, ME422, ME431,ME432, ME531, and ME532 elements require, respectively 5, 6, 6, 7, 6, and 7 point quadraturefor exact integration of the system matrices for this problem.The elements of the �exibility matrix F in the HR elements involve rational fractions. Thus,

it is not possible to determine the number of sampling points required for exact integration.Table II shows the normalized tip displacement for HR42, HR53, and HR64 elements obtainedwith di�erent orders of quadrature. The values in this table are normalized by the exact tipdisplacement.The error measure used throughout the rest of this paper is de�ned as follows. Let us

denote the square integral of a function u over the length of the domain as

I(u)≡∫ ‘

0u2 d� (59)

The error in the �eld u is then de�ned to be

Err(u)≡√I(u− u)=I(u) (60)

where u is the exact value of the �eld and u is the �nite element approximation. The errorcan be computed for any of the computed �elds.Figure 4 shows how the numerical errors vary with mesh re�nement for the cantilever beam

problem. The displacement errors are essentially identical across all three approaches whenthe number of displacement parameters is equal. The two- and three-�eld mixed methods



51

Number of Elements

10 51 10 51 10

100

100

100

10-8

10-8

10-7

Err

(w

)E

rr (

m)

Err

(x)

(a) (d)

(e)

(f)

(g)

(h)

(i)(c)

(b)

Figure 4. Displacement, moment, and curvature errors for the cantilever beam under meshre�nement (a, b, c) Classical Sti�ness [ , �, +, and ◦ for CS4, CS5, CS6 and CS7], (d,e, f) Hellinger–Reissner [ , �, + for HR42, HR53, and HR64], (g, h, i) mixed-enhanced

[ , �, +, and ◦ for ME421, ME422, ME531 and ME532].

capture the moment exactly, as expected, while the CS elements do not. The curvature errorsare identical for CS and HR elements with the same number of displacement parameters.Evidently, the performance of the ME elements with i displacement parameters and k enhancedstrain parameters is identical to the CS and HR elements with i+ k displacement parameters.

6.2. Linearized buckling analysis of non-prismatic cantilever beam

A second interesting class of problems are the matrix eigenvalue problems that arise from lin-earized buckling. The non-prismatic cantilever beam of Figure 3, with P=0, H = � illustratesthe performance of the three approaches on buckling eigenvalue problems. Table III displaysthe two lowest buckling eigenvalues obtained for di�erent elements for a single element mesh.The CS elements converge from above, as is well-known. The HR and ME elements appearto converge from below, except for known ‘bad’ elements like ME431 and ME432 [7]. Theperformance in linearized buckling appears to be dominated by the displacement interpolationindicating that the primary di�culty is getting the geometric sti�ness right.



Table III. Approximations of the lowest two of the buckling eigenvalues forthe CS, HR and ME elements.

CS4 CS5 CS6 CS7 CS8 CS12

�1 5.0149 4.6347 4.6128 4.6121 4.6121 4.6121�2 178.98 67.882 62.592 62.592 62.438 62.397

HR42 HR53 HR64 HR75 HR86

�1 4.4413 4.6005 4.6118 4.6121 4.6121�2 151.16 57.681 61.868 62.536 62.393

ME421 ME422 ME431 ME432

�1 4.4944 4.4453 4.9973 4.9940�2 154.01 151.40 178.38 178.27

ME531 ME532 ME642 ME643 ME752 ME753

�1 4.6028 4.6006 4.6118 4.6118 4.6121 4.6121�2 58.669 57.759 61.872 61.868 62.537 62.536

0 0.75

P

t

-200

200

w0.75 0t^

P^

w^

P^

5.0

1

P (t)�

21

2.5

(a)

(b) (c)

Figure 5. Cyclically loaded two-span beam example: (a) problem geometry; (b) transverse load versustime; and (c) the transverse displacement under the load versus load.

6.3. Continuous beam with non-linear material behaviour

A two-span continuous prismatic beam of length ‘=4 is transversely loaded at the middle ofits �rst span with a transverse point load as shown in Figure 5(a). The beam is elasto-plasticand behaves in accord with the model presented in Section 3. The uniform bending and axialelastic sti�nesses are EI=3000 and EA=3000, respectively. The isotropic hardening modulusfor the linear hardening rule is �=0:2. The yield moment and the yield axial force are ofequal magnitude and are chosen as m0 = n0 = 20. The continuous beam is discretized with3 beam �nite elements, two elements of unit length for the left span, and a single elementfor the right span. Figure 5(b) displays the variation of the load with respect to time given



4

Err (κ)

0

40

-40

�

Err (w)0.15

-0.15

Err (m)

40 40

CS

Ele

men

tsH

R E

lem

ents

ME

Ele

men

tsFi

ne M

esh

Solu

tion

40

-40

40

-40

75

-75

0.15

-0.15

0.15

-0.150.2

-0.2

0.015

-0.015

0.015

-0.015

0.015

-0.0150.05

-0.05

Figure 6. Errors in the transverse displacement, curvature, and moment �elds in the two-span continuousbeam example at the end of the cyclic loading regime along with the �ne mesh (exact) solution

(.....CS4, HR42, ME421, - - - CS5, HR53, ME531, —— CS7, HR64, ME532).

by the formula P(t)=200 sin(t=4) sin(2t). Figure 5(c) displays the load versus transversedisplacement as computed with (three) CS4 beam elements.The response of the beam is computed with di�erent types of elements at time t=3:95 s

when the load reaches P(t)= P=167. The ‘exact’ solution was obtained using a very �nemesh consisting of 40 CS, HR or ME elements (all gave the same result). For the threeelement mesh, the displacement, moment and curvature error diagrams are shown in Figure 6.



Figure 7. Geometrically nonlinear response: (a) geometry and loading; (b) response for 20CS4 elements; (c) convergence for CS elements [ , �, and + for CS4, CS5, CS6]; (d)convergence for HR elements [ , �, + for HR42, HR53, and HR64]; and (e) convergence

for ME elements [ , �, +, and ◦ for ME421, ME422, ME531 and ME532].

Again, as this �gure indicates, the moment �eld is captured almost exactly by even the lowestorder mixed elements, but not by the CS elements. Also, the displacement �elds captured byall the elements, with the exception of CS4, have similar accuracy. The complex curvature�eld, however, proves di�cult for all elements to capture with this coarse mesh. However,the HR and ME elements seem to do better than the CS elements.

6.4. Geometrically nonlinear analysis

A prismatic cantilever beam of length ‘=1 is loaded with transverse and axial loads of equalmagnitude at its free end, as shown in the cartoon in Figure 7(a). The axial and the bendingsti�ness of the beam are EI=200, EA=100, respectively. The beam’s response, obtainedby a geometrically non-linear analysis made with 20 CS4 beam �nite elements, is shown inFigure 7(b). The results of the mesh re�nement study for this problem is presented in Figure7(c)–(e). Here, we see that most of the elements perform well, even for the coarse meshes.

7. WHICH ELEMENT IS BEST?

From the example problems it should be evident that there is no clear winner among thethree di�erent formulations. Each has merit in certain circumstances. All converge with mesh



Table IV. Floating point operation counts fordi�erent beam elements.

NQ CS4 CS5 CS6 CS7

10 7500 33100 46900 64700Min 2900 16100 28100 45900

NQ HR42 HR53 HR64

10 8900 45500 72200Min 5400 32700 59300

NQ ME421 ME422 ME531 ME532

10 12200 13000 55400 58000Min 5700 7200 32600 39300

re�nement. Each has a di�erent relative computational cost. In this section we attempt tomake an approximate accounting for the computational e�ort associated with each approach.We present a computational cost analysis for all the elements based on the error trends wehave observed in the examples and the operation counts required to compute the �nite elementmatrices for each element.Assuming that we use 10-point Gauss–Lobatto quadrature, three plastic iterations and, when-

ever present, three Newton iterations for recovering both the bubble degrees of freedom andthe mixed �eld variables each, we obtain the (�oating point) operation counts for variousformulations as presented in the �rst row(s) of Table IV. In the second row(s), the opera-tion counts are computed assuming that the material behavior is linear elastic (i.e. 1 plasticiteration) and the beam element has a cubic variation of bending sti�ness. Thus, we use theminimum number of quadrature points to exactly integrate the element matrices. The operationcounts presented in Table IV are approximate.Using the operation counts presented in Table IV, we can compare the cost versus per-

formance of di�erent element formulations. For example, for a given mesh and element for-mulation (say, two CS4 elements) we can compute the total error de�ned as etot ≡Err(w) +Err(m) + Err(–) and compute the e�ort de�ned as the product of the number of elementsin the mesh and the operation count (2× 2900=5800 for two CS4 elements with minimumorder of quadrature). We consistently ignore the additional e�ort required for solving a largersystem of equations for �ner meshes for all types of elements. Using the data in Table IVand the total errors, we can generate a graph containing total error versus the correspond-ing e�ort. Figure 8 contains such graphs generated for the �rst benchmark problem whichinvolves the transverse loading of a non-prismatic cantilever beam (see Figure 3). We cansee that the smallest total errors are attained by the CS7 and ME532 type elements but atgreat cost. Among the two types however, CS7 is more costly. On the other hand, whileattaining similar total error levels, HR64, CS6 and ME531 require respectively less e�ort. Thetrend at higher amounts of total error (and correspondingly lower levels of e�ort) is about thesame. Considering the approximate nature of the operation counts and the trends of the resultsin Figure 8, we conclude that, at least for the �rst benchmark problem, one cannot clearlyidentify a preferred element type. In other words, to attain a speci�c level of accuracy, all



Eff

ort (

log)

10-7 10-5 10-3 10-1

Total Error

106

105

104

103

106

105

104

103

106

105

104

103

(a)

(b)

(c)

Figure 8. E�ort versus Total Error for bending of the nonprismatic cantilever beam benchmark problem:(a) CS elements [ , �, +, and ◦ for CS4, CS5, CS6 and CS7]; (b) HR elements [ , �, + for HR42,

HR53, and HR64]; (c) ME elements [ , �, +, and ◦ for ME421, ME422, ME531 and ME532].

element types (CS, HR, or ME) require similar amounts of computational e�ort. The apparentadvantage of low-order elements (of any variety) with re�ned meshes will be o�-set by theadditional cost of solving the global equations (which is not re�ected in these costs).

8. CONCLUSIONS

The formulations of various methods for geometrically nonlinear planar Bernoulli–Euler beam�nite elements have been presented. In particular, one-�eld (CS), two-�eld (HR), and three-�eld (ME) variational formulations have been formulated and studied. We have examinedstrategies for condensing and recovering the nodeless variables at the element level and the



choice of interpolation (we have con�ned our attention to convergent and stable mixed meth-ods). Extending these ideas to three dimensional Bernoulli–Euler beams is straightforward,as is the extension to Timoshenko beams. For Timoshenko beams the mixed formulationshave the additional merit of curing shear locking (which is not an issue for Bernoulli–Eulerbeams).Simple benchmark problems that are representative of those commonly encountered in the

structural analysis of frames have been used to compare the performance and e�ciency ofeach of the di�erent formulations. Mixed methods and p-re�nement for the classical sti�nessmethod clearly provide high coarse-mesh accuracy and, with proper enrichment of the mixed�elds the accuracy can easily be increased, though with added computational cost. Similarimprovements in accuracy can also be attained with h-re�nement with a comparable computa-tional cost. The elements with high coarse-mesh accuracy may be attractive for general frameanalysis when adaptive mesh re�nement procedures are not available.In certain circumstances, one of the response �elds may be more important to capture than

the others. In problems where an accurate resolution of the curvature �eld is essential (e.g.inelastic response) ME elements are best. In problems, where an accurate resolution of themoment �eld is necessary, HR elements are best. On the other hand, if only an accuratedisplacement �eld is needed (e.g., geometrically nonlinear but materially linear analysis ofnonprismatic beams) p-re�ned CS elements are best.In contrast to other recent papers on mixed methods for beam elements, we conclude

that, despite certain di�erences in performance, there is no clear advantage over all to mixedformulations over the classical displacement-based methods for Bernoulli–Euler beams.

ACKNOWLEDGEMENTS

The research reported herein was supported by the Department of Energy through the Center for Sim-ulation of Advanced Rockets. This support is gratefully acknowledged. The opinions expressed in thispaper are those of the authors and do not necessarily re�ect those of the sponsor.

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Mixed Variational Methods for Finite Element Analysis of Geometrically Non Linear, Inelastic Bernoulli-Euller Beams