Geometrically exact isogeometric Bernoulli-Euler beam based on the Frenet-Serret frame A. Borkovi c12 M. H. Gfrerer1 and B. Marussig1

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Geometrically exact isogeometric Bernoulli-Euler

beam based on the Frenet-Serret frame

A. Borkovi´c1,2, M. H. Gfrerer1, and B. Marussig1

1Institute of Applied Mechanics, Graz University of Technology, Technikerstraße 4/II, 8010

Graz, Austria, aleksandar.borkovic@aggf.unibl.org, aborkovic@tugraz.at

2University of Banja Luka, Faculty of Architecture, Civil Engineering and Geodesy,

Department of Mechanics and Theory of Structures, 78000 Banja Luka, Bosnia and

Herzegovina

Abstract

A novel geometrically exact model of the spatially curved Bernoulli-Euler beam is de-

veloped. The formulation utilizes the Frenet-Serret frame as the reference for updating

the orientation of a cross section. The weak form is consistently derived and linearized,

including the contributions from kinematic constraints and conﬁguration-dependent load.

The nonlinear terms with respect to the cross-sectional coordinates are strictly considered,

and the obtained constitutive model is scrutinized. The main features of the formulation

are invariance with respect to the rigid-body motion, path-independence, and improved

accuracy for strongly curved beams. A new reduced beam model is conceived as a special

case, by omitting the rotational DOF. Although rotation-free, the reduced model includes

the part of the torsional stiﬀness that is related to the torsion of the beam axis. This

allows simulation of examples where the angle between material axes and Frenet-Serret

frame is small. The applicability of the obtained isogeometric ﬁnite element is veriﬁed via

a set of standard academic benchmark examples. The formulation is able to accurately

model strongly curved Bernoulli-Euler beams that have well-deﬁned Frenet-Serret frames.

Keywords: spatial Bernoulli-Euler beam; Frenet-Serret frame; rotation-free beam;

strongly curved beam; geometrically exact analysis;

1 Introduction

The aim of computational mechanics is to develop accurate and eﬃcient models of various

mechanical systems. The most successful mechanical model for the simulation of slender

bodies is beam. The ﬁrst consistent beam theories were developed in 18th century, and

the search for a formulation with optimal balance between accuracy and eﬃciency is still

ongoing. To reduce the problem domain of slender bodies from 3D to 1D, the standard

assumption is that the cross sections are rigid, which results with the Simo-Reissner (SR)

beam model. By additional assumption that the cross sections remain perpendicular to

the deformed axis, the Bernoulli-Euler (BE), also known as Kirchhoﬀ-Love, beam model

arXiv:2210.00001v1 [cs.CE] 29 Sep 2022

follows. The subject of the presented research are large deformations of an arbitrarily

curved and twisted BE beam with an anisotropic solid cross section, without warping [1].

The nonlinear SR beam model has long been the main focus for researchers, partially

because its spatial discretization requires only C0-continuous basis functions, such as the

Lagrange polynomials. As the name suggests, the SR theory was founded by Reissner [2],

and later generalized by Simo [3], who conceived the term geometrically exact beam the-

ory. The main requirement of a geometrically exact formulation is that the relationship

between the conﬁguration and the strain is consistent with the balance laws, regardless of

the magnitude of displacements and rotations. The adequate description of large rotations

is one of the principal challenges since these are not additive nor commutative and consti-

tute nonlinear manifolds. This issue has been a driving force for the formulation of various

algorithms for the parameterization and interpolation of rotation [4, 5, 6, 7, 8, 9, 10]. A

turning point in this development was the ﬁnding by Crisﬁeld and Jeleni´c that the in-

terpolation of a rotation ﬁeld between two conﬁgurations cannot preserve objectivity and

path-independence [11, 12]. The reason is that incremental material rotation vectors,

at diﬀerent instances, do not belong to the same tangent space of the rotation manifold

[13]. An orthogonal interpolation scheme that is independent of the vector parameteriza-

tion of a rotation manifold is suggested in [11, 12] and several further strategies followed

[14, 15, 16].

Although the geometrically exact formulations represent the state-of-the-art in beam

modeling, their implementation is not straightforward and several alternatives exist, such

as the corotational and the Absolute Nodal Coordinate (ANC) approaches. The main idea

of the corotational formulations is to decompose the deformation into two parts. The ﬁrst

part is due to large rotations and the second is the local part, measured with respect

to the local co-rotated frame. It resembles the strategy employed in [11, 12] and allows

accurate simulation of large deformations [17, 18, 19, 20]. The ANC method is, in essence,

a solid ﬁnite element for slender bodies. It is well-suited for the implementation of 3D

constitutive models, but has issues with engineering structural analysis, where integration

with respect to the cross-sectional area is required [21, 22].

The ﬁrst BE beam models that are consistent with the geometrically exact theory are

[23] and [24]. Meier et al. have discussed the issues of objectivity and path-independence

in [25], and proposed an orthogonal interpolation scheme similar to that of Crisﬁeld

an Jeleni´c [11, 12]. Membrane locking, contact, and reduced models are considered in

subsequent publications [26, 27], followed by a comprehensive review [1]. An eﬃcient BE

beam formulation based on the Cartan frame was developed in [28], where the position

and the local frame are observed independently and subsequently related by the Lagrange

multipliers.

The emergence of the spline-based isogeometric analysis (IGA) [29] has led to the de-

velopment of a series of SR beam models [30, 31, 32, 33, 34, 35, 36, 37]. The formulation

[38] arguably represents the state-of-the-art since it employs extensible directors and mod-

els various couplings. One of the main features of IGA is the smoothness of utilized basis

functions, a property that beneﬁts the BE beam due to its C1-continuity requirement.

The ﬁrst IGA BE beam models were introduced by Greco et al. in [39, 40, 41, 42], while

the ﬁrst nonlinear BE model was developed in [43]. Due to the reduction in number of

DOFs, in comparison with the SR model, multi-patch nonlinear analysis of BE beams has

received special attention [44, 45, 46, 47]. Invariance of the geometric stiﬀness matrix in

the frame of buckling analysis is considered in [48], while the eﬀect of initial curvature on

the convergence properties of the solution procedure is considered in [49]. The ﬁrst truly

geometrically exact IGA BE model that preserves objectivity and path-independence was

developed in [50].

As emphasized in this brief literature review, the crucial issues of objectivity and path-

independence in the geometrically exact beam theory are related to the nonlinear nature

of ﬁnite rotations. In order to obtain a generally applicable formulation, the orthogonal

interpolation schemes or similar procedures must be applied [25]. An alternative approach

is to utilize the Frenet-Serret (FS) triad as the reference frame for the update of rotation.

This frame does not depend on previous conﬁgurations, and the resulting formulation

is expected to be objective and path-independent. Although a natural choice, the FS

frame is avoided for beam analysis since it is not deﬁned for straight segments of a curve.

Furthermore, the FS frame exhibits signiﬁcant rotation around the curve’s tangent vector

at inﬂection points. Due to these issues, the formulation based on the FS frame fails

for arbitrary geometries. Nevertheless, the derivation, implementation and veriﬁcation of

such a computational model is of fundamental importance due to the intrinsic relation

between the FS frame, the curve and the beam model. In this paper, we develop a

formulation of this kind. Conﬁgurations for which the FS frame is not well-deﬁned are not

generally considered. An approximation of straight initial conﬁguration will nevertheless

be considered by imposing a small curvature. Regarding the inﬂection points, it can be

shown that for a regular analytic space curve, which is not a straight line, a point with a

zero-curvature is the point of analyticity of torsion [51] . This means that the torsional

angle is deﬁned at inﬂection points. However, due to the large gradients of this angle, the

FS frame exhibits signiﬁcant twisting and poor convergence is expected [52].

The calculation of torsion of the FS frame involves the third order derivative of the

position vector, implying that a C2-continuous discretization is required to obtain the

torsion ﬁeld. IGA allows high interelement continuities, up to Cp−1, where pis the order

of the basis functions. This feature makes IGA ideally suited for the implementation

of the beam model based on the FS frame. This fact was utilized in [52] for the linear

analysis of such beam model.

There are several rotation-free formulations in the literature that model a spatial BE

beam, e.g. [53] and [26]. However, since they disregard the torsional stiﬀness, the area of

application is reduced to a few speciﬁc cases in which the torsion can be neglected, such

as cables [53], or where the torsion is not present due to the speciﬁc loading conditions

[26]. Starting from the proposed BE model that is based on the FS frame, it is possible

to obtain a speciﬁc rotation-free model that is more accurate than the existing ones. The

main feature of this model is that it contains the torsion of the FS frame.

When a spatially curved beam exhibits large deformations, axial, torsional and bending

actions become coupled due to the nonlinear distribution of strain along the cross section.

It is common to disregard these couplings when modeling the BE beam [43, 39]. Recently,

axial-bending coupling was considered in the frame of linear [52, 54] and nonlinear analysis

[55, 50]. The curviness of a beam, Kd, is introduced in [56, 57] as a measure of this

coupling. Here, Kis the curvature of beam axis and dis the maximum dimension of

the cross section in the planes parallel to the osculating plane. The curviness parameter

allows classiﬁcation of beams as small-, medium- or big-curvature beams [58]. The axial-

bending coupling is signiﬁcant for Kd > 0.1, and these beams belong to the category of

big-curvature, also known as strongly curved, beams [59]. In order to apply appropriate

beam models, the current curviness at each conﬁguration must be observed [55, 50].

The present research is based on the works [60, 52, 54, 55, 50] with the aim of tending

the linear formulation based on the FS frame to the geometrically exact setting and to im-

prove the existing strongly curved BE model. To summarize, this work makes three main

contributions. The ﬁrst is the derivation of the geometrically exact FS beam formulation.

It is geometrically exact in a sense that it can model arbitrary large deformations involv-

ing ﬁnite, but small, strains. The restriction is that the beam must have a well-deﬁned

FS frame during the deformation, meaning that straight segments and inﬂection points

should not occur. It turns out that many academic examples satisfy this requirement.

The second contribution is the consideration of a special case for the proposed formulation

that is obtained by omitting the rotational DOF. In contrast to existing rotation-free BE

models, this one includes one part of the torsion. The resulting rotation-free BE beam

model can give approximate results for speciﬁc deformation cases. The third contribution

is a rational constitutive model for strongly curved beams. The nonlinear terms of total

strain with respect to the cross-sectional axes are taken into account and simpliﬁed models

are deduced. This approach improves upon the strongly curved beam models considered

in [55, 50] where only the linear terms of incremental strain were considered.

The paper is organized as follows. The next section presents the basic relations of the

beam metric and kinematic, while the strain and stress measures are deﬁned in Section

3. The ﬁnite element formulation is elaborated in Section 4 and numerical examples are

presented in Section 5. The conclusions are delivered in the last section.

2 Conﬁguration of Bernoulli-Euler beam

A spatial BE beam at an arbitrary reference conﬁguration Cis deﬁned by the position of

its axis and the orientation of cross sections. The position of a spatial curve is a vector,

while the orientation of the rigid cross section is, in general, described with the rotation

tensor. Since we are dealing with the BE beam, the cross sections are perpendicular

to the beam axis at each conﬁguration, which leaves us to determine only the rotation

in the cross-sectional plane. Therefore, the metric of deformed conﬁguration is deﬁned

analogously to the metric of an arbitrary reference conﬁguration.

Boldface letters are used for the notation of vectors and tensors. An asterisk sign is

used to designate a current, unknown, conﬁguration. Greek index letters take the values

of 2 and 3, while Latin ones take the values of 1, 2 and 3. Partial and covariant derivatives

with respect to the mth coordinate of the convective frame (ξ, η, ζ) are designated with

(•),m and (•)|m, respectively. Time derivative is marked as ˙

(•). Other speciﬁc designations

will be introduced as they appear in the text.

The details on the NURBS-based IGA modeling of curves are skipped for the sake of

brevity since they are readily found elsewhere [29, 61].

2.1 Metric of the beam axis

The beam axis is a spatial curve, deﬁned with its position vector:

r=r(ξ)=xm(ξ)im=xmim,(x1=x, x2=y, x3=z),(1)

where imare the base vectors of the Cartesian coordinate system, Fig. 1. A curve can be

parameterized with either the arc-length coordinate s∈[0, L] or some arbitrary paramet-

ric coordinate ξ∈[0,1], where Lis the length. For every C1continuous curve, we can

Figure 1: Reference and current conﬁgurations of a spatial BE beam. A conﬁguration is

deﬁned with the position vector of beam axis and the orientation of cross sections.

uniquely deﬁne a tangent vector g1:

g1=r,1=dr

dξ=xm

,1im=dr

dξ=ds

dξt=√gt,√g=√g1·g1,(2)

where tis the unit-length tangent of the beam axis.

There is an inﬁnite set of local vector bases that can be deﬁned to frame a curve, FS

and Bishop frames being the most prominent ones. The unique feature of the FS frame is

that one of its base vectors is aligned with the curvature of a line, while the Bishop frame

is characterized with zero torsion [62]. One approach to the BE beam modeling using the

ANC formulation and the Bishop frame can be found in [63].

Let us now focus on the FS triad that consists of the tangent, normal and binormal

(t,n,b). The normal vector is the unit vector of curvature, while the binormal is per-

pendicular to the osculating plane and completes the orthonormal FS triad. Due to its

intrinsic connection to the curvature, the FS frame cannot be deﬁned for straight lines

and has sudden changes near the inﬂection points. These issues are readily discussed in

the context of beam formulations [25, 52]. The derivatives of FS base vectors are deﬁned

with the well-known formulae:





t,s

n,s

b,s



=



0K0

−K0τ

0−τ0





b

,(3)

where Kis the curvature of a line while τis the torsion of the FS frame.

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GeometricallyexactisogeometricBernoulli-EulerbeambasedontheFrenet-SerretframeA.Borkovic1,2,M.H.Gfrerer1,andB.Marussig11InstituteofAppliedMechanics,GrazUniversityofTechnology,Technikerstrae4/II,8010Graz,Austria,aleksandar.borkovic@aggf.unibl.org,aborkovic@tugraz.at2UniversityofBanjaLuka,FacultyofAr...

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Geometrically exact isogeometric Bernoulli-Euler beam based on the Frenet-Serret frame A. Borkovi c12 M. H. Gfrerer1 and B. Marussig1

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