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ScienceDirect

Comput. Methods Appl. Mech. Engrg. 303 (2016) 101–127

www.elsevier.com/locate/cma

Nonlinear isogeometric spatial Bernoulli beam

A.M. Bauer∗, M. Breitenberger, B. Philipp, R. Wuchner, K.-U. Bletzinger

Lehrstuhl f ur Statik, Technische Universit at M unchen, Arcisstr. 21, 80333 M unchen, Germany

Received 24 August 2015; received in revised form 30 November 2015; accepted 29 December 2015

Available online 1 February 2016

Abstract

A new element formulation of a geometrically nonlinear spatial curved beam assuming Bernoulli theory including torsion

without warping is proposed. The element formulation is derived directly from the 3D-continuum by means of nonlinear

kinematics, thus accounting for large displacements. The geometric description of the proposed element is adapted from the spatial

rod of Greco and Cuomo (2013) and extended to a nonlinear element formulation. The proposed formulation can handle arbitrary

orientations of the cross section along the beam.

In this publication, NURBS are used as basis functions for discretization, since they can easily provide the required

C 1-continuity between elements. The presented element formulation has four degrees of freedom, three for displacements and one

for the rotation around the center line. In order to prove the accuracy of the developed spatial Bernoulli beam, several numerical

examples are presented and compared to analytic solutions and other element formulations.

c⃝ 2016 Elsevier B.V. All rights reserved.

Keywords: Spatial thin rod; Bernoulli Beam theory; Nonlinear isogeometric analysis; NURBS; Torsion

1. Introduction

In the last years, isogeometric analysis (IGA), introduced by Hughes et al. [1], has become a broad field of research

in computational mechanics. Its main aim is to combine design and analysis by using the basis functions of Computer

Aided Design (CAD) for approximating the solution in the context of finite element analysis (FEA). IGA is mainly

based on NURBS since they can be seen as the standard in CAD, especially for free-form geometries. A detailed

description of NURBS can be found in the work of Piegl and Tiller [ 2]. Over the past ten years many publications have

shown the good properties of NURBS for analysis purposes. Meanwhile many isogeometric element formulations for

structural mechanics have been developed. A solid element formulation is presented in [ 1] and several different shell

and membrane formulations have been introduced in [3–8], among others.

Recently beam elements gained more attention [9–11]. However, a geometrically nonlinear beam element based

on Bernoulli assumptions including torsion is not yet available. The present contribution presents such a nonlinear

∗ Corresponding author.

E-mail address: am.bauer@tum.de (A.M. Bauer).

URL: http://www.st.bgu.tum.de (K.-U. Bletzinger).

http://dx.doi.org/10.1016/j.cma.2015.12.027

0045-7825/ c⃝ 2016 Elsevier B.V. All rights reserved.

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Fig. 1. Definition of the domain and its boundaries.

isogeometric spatial beam element formulation. In contrast to IGA, a whole variety of spatial beam elements [12–14]

and particularly Kirchhoff–Love resp. Bernoulli models [15–20] have been introduced, studied and established in

classical FEA.

Even though Gontier and Vollmer have started the research on 2-dimensional beams with Bezier curves as basis

functions [21] based on the theory of Simo et al. [22] already in 1995, there are only few, recently developed spatial

elements with IGA basis functions. To the authors’ knowledge, these are

– the bending-stabilized cable of Raknes et al. [9], which neglects torsional deformations,

– the locking-free spatial Timoshenko beam of Auricchio et al. [10], which can be used for small displacements

within the context of isogeometric collocation and

– the spatial Kirchhoff–Love rod for linear kinematics of Greco et al. [ 11], similar to [16], which uses the natural

frame and curvilinear angle representation.

This work has the following outline. Section 2 reviews the mechanical and geometric background for an

isogeometric element formulation. In Section 3, the element formulation is derived straightforward from the

continuum mechanical equations of the beam. Section 4 discusses some special aspects of the proposed beam element

related to initially curved beams. In Section 5, several benchmark examples are presented in order to verify the

proposed element formulation by comparing the gained numerical results to other literature and analytical results.

Finally, conclusions are drawn in Section 6.

2. Isogeometric analysis

The aim of the isogeometric analysis (IGA) is to merge design (CAD) and analysis (FEA). This is enabled by using

the same basis functions for both processes. Thus no transformation between the models is necessary and considerableamounts of time can be saved [1].

2.1. Structural mechanics

The proposed beam formulation is derived from the Principle of Virtual Work . A virtual displacement δu is applied

to the system. The performed work of internal and external forces is zero, if the system is in equilibrium [23]:

δW = −δW int + δW ext = 0. (1)

The internal and external virtual work in the beam are defined as

δW int =

Ω S : δE d x , (2)

δW ext =

Γ N

t : δu d x + Ω

ρ0B : δu d x

, (3)

where Ω describes the domain and ∂Ω the boundary in the undeformed state (see Fig. 1). The boundary consists of

the Dirichlet boundary Γ D and the Neumann boundary Γ N. S denotes the inner stresses, which is the energetically

conjugated quantity to the virtual strains δE caused by the virtual displacements δ u. B are the body forces with the

material density ρ0 and t represent the boundary forces.

The system needs to be discretized for solving the problem with the Finite Element Method . Applying the chain

rule of differentiation Eq. (1) has to be satisfied for the variation of the discretization variables δur which defines the

components Rr of the residual force vector from

δW = ∂ W

∂ur δur = Rr δur = 0 (4)

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and – considering arbitrary variations – the set of nonlinear equilibrium equations:

Rr = ∂ W

∂ur

= 0. (5)

An iterative solution by the Newton–Raphson method requires further linearization at the current displacement u∗,

introducing the displacement increments ∆us

and the components K r s

of the tangential stiffness matrix:

LIN ( Rr ) = ∂ W

∂ur

+ ∂2 W

∂ur ∂us

∆us = Rr

u∗ +

s

∂ Rr

∂us

∆us = 0. (6)

As a conclusion, the residual force vector and stiffness matrix can be decomposed into contributions of the internal

and external forces:

Rr = ∂ W

∂ur

= ∂ W int

∂ur

+ ∂ W ext

∂ur

= −

F intr + F ext

r

(7)

K r s = ∂ Rr

∂us

= ∂ 2W

∂ur ∂us

= ∂ 2W int

∂ur ∂us

+ ∂ 2W ext

∂ur ∂us

= K intrs + K ext

r s . (8)

In the absence of displacement dependent external forces the related stiffness terms K extrs vanish which is assumedin the sequel of the paper. Finally, the linearized equilibrium equations in the standard form display as

s

K r s∆us = F intr + F ext

r . (9)

2.2. Non-uniform rational B-Splines—NURBS

As basis functions for isogeometric analysis, commonly Non-Uniform rational B-Splines (NURBS) are used. They

allow describing conic sections like parabola, elliptic entities and hyperbola. Thus, they provide a uniform description

for a large range of shapes. Therefore NURBS are widely used for the geometry description in current CAD systems.

The discrete parameters of a NURBS based geometry are the coordinates of the control points P, which are generallynon-interpolating. The NURBS curve C(ξ ) is defined as the sum over all control points Pi with their respective

NURBS basis function Ri, p:

C(ξ ) =n

i =1

Ri, p(ξ )Pi . (10)

In contrast to B-Splines, the control points are weighted, which influence the curve’s velocity and shape. A NURBS

curve is the projection of a B-Spline in R4 onto R3 with homogeneously weighted control points [2].

Ri, p(ξ ) = N i, p(ξ)wi

n

j=1

N j, p(ξ)w j

. (11)

The basis function N i, p can be computed with the Cox–deBoor recursion formula [2]. It starts with p = 0 with

N i,0(ξ ) =

1, if ξ ∈ [ξ i , ξ i +1[

0, otherwise (12)

and for p ≥ 1 it is

N i, p(ξ ) =

ξ − ξ i

ξ i+ p − ξ i N i, p−1(ξ ) +

ξ i + p+1 − ξ

ξ i+ p+1 − ξ i +1 N i+1, p−1(ξ), if ξ ∈ [ξ i , ξ i+ p+1[

0, otherwise.

(13)

For a detailed description and further quantities like derivatives, the reader is referred to Piegl and Tiller [2].

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Fig. 2. Description of the continuum of the beam for the undeformed and deformed configuration. Here illustrated for a rectangular cross section.

Fig. 3. Different beam geometries for the same center line characterized by a rotation of the trihedral around the center line.

3. Structural element formulation

In this section, the element formulation for a spatially curved and geometrically nonlinear beam is derived. The

element is based on Bernoulli kinematics. It follows from Bernoulli theory that the cross sections remain orthogonal

to the center line after deformation and there are no changes of the cross sectional dimensions. The cross section itself

can develop in-plane cross sectional shear deformation, the so-called torsion. In the present contribution, warping

effects will be neglected. Since we develop a spatial beam element, all possible spatial displacements of the curve

u, v , w are used as degrees of freedom (DOFs). In addition a relative rotation around the center line is used as fourth

DOF.

3.1. Geometric description

In this work, the continuum of the beam is described by a center line and a moving trihedral. In the following,

upper-case and lower-case letters refer to the undeformed and deformed configuration, respectively. The convective

contravariant coordinates are denoted as θ i with i ∈ 1, 2, 3. A second index α ∈ 2, 3 is used as well in the sequel.

The derivative ∂( ·)

∂θ i will be written as (·),i .

The position vector for the continuum of the beam is denoted as X (resp. x):

X θ 1, θ 2, θ 3 = Xc θ 1+ θ 2A2 θ 1+ θ 3A3 θ 1 (14a)

x

θ 1, θ 2, θ 3

= xc

θ 1

+ θ 2a2

θ 1

+ θ 3a3

θ 1

. (14b)

Xc (resp. xc) is the position vector of the center line, Ai resp. ai are the base vectors aligned to the moving trihedral.

The cross sections are assumed to be warping free for the development of the element formulation. The base vectors

Aα resp. aα are defined such that they are unit vectors and orthogonal to the center line. The geometric description of

the beam for both configurations is illustrated in Fig. 2.

3.1.1. Alignment of the moving trihedral in the undeformed configuration

The moving trihedral is used to describe the orientation of the cross section, since the center line is not able to

provide this information (see Fig. 3).

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(a) Λ-operation (see Section 3.1.2). (b) RT-operation (see Section 3.1.3).

Fig. 4. Alignment of the moving trihedral in the undeformed configuration in two steps.

The two components Aα of the moving trihedral, which are orthogonal to the tangent of the center line, are

described by

Aα = RT (Ψ ) Λ(T0, T)A0α , (15)

where the three vectors A0α and T0 define a reference trihedral, see Fig. 4. In a first step,Λ(T0, T) aligns the reference

trihedral to the tangent at the current position. In a second step, RT (Ψ ) rotates the moving trihedral reference

Λ(T0, T)A0α to the desired orientation Aα . Here Ψ describes the rotation along the beam, i.e.Ψ = Ψ

θ 1

. This

concept is adapted from [11] for the current element formulation.Since A1 (resp. a1) is not a unit vector, the normalized tangent of the center line T (resp. t) is computed as

T = 1

∥A1∥2

A1 and t = 1

∥a1∥2

a1, (16)

where ∥..∥2 is the Euclidean norm.

3.1.2. Mapping matrix Λ

The matrix Λ(N0, N) is defined such that it maps one vector on the other.

Λ(N0, N) · N0 = N, (17)

where N0 and N are given normalized vectors. The mapping operationΛ

will be described using the Euler–Rodriguezformula [24] which is generally written as

R = e ⊗ e + cos φ

I − e ⊗ e

+ sin φ

e × I

, (18)

where e denotes the normalized rotation axis, φ the angle of the rotation and I the identity matrix of dimension 3 × 3.

For our purpose, the variables of R are defined by the following entities:

e = N0 × N

∥N0 × N∥2, cos φ = N0 · N, sin φ = ∥N0 × N∥2. (19)

Note that the definition of the cross product between a vector v and a matrix M is:

(v × M)il = ϵi jk · v j · M kl , where ϵi jk is the Levi-Civita symbol. (20)

Thus Λ(N0, N) can be computed as

Λ(N0, N) = (N0 · N) I + ∥N0 × N∥2

N0 × N

∥N0 × N∥2× I

+

1 − (N0 · N)

∥N0 × N∥22

(N0 × N) ⊗ (N0 × N) . (21)

With

1 − (N0 · N)

∥N0 × N∥22

= 1 − cos φ

sin2 φ=

1 − cos φ

1 − cos2 φ=

1

1 + cos φ=

1

1 + N0 · N. (22)

Eq. (21) can be simplified as:

Λ(N0

, N) = (N0

· N) I + (N0

× N) × I + 1

1 + N0 · N(N

0 × N) ⊗ (N

0 × N) . (23)

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(a) Λ-operation (see Section 3.1.2). (b) Rt-operation (see Section 3.1.3).

Fig. 5. Alignment of the moving trihedral in the deformed configuration.

3.1.3. Rotation matrix R N

The RN-matrix is used to rotate a vector V, which is orthogonal to N, around N by an angle φ. The Euler–Rodriguez

formula (Eq. (18)) is again used to define this matrix. The rotation axis e is replaced by the normalized vector N in

the Euler–Rodriguez formula.

RN (φ) = I cos (φ) + sin (φ) N × I + (1 − cos (φ)) N ⊗ N. (24)

Since this operation is only used to rotate vectors V, which are orthogonal to N, around N, the following holds:

(N ⊗ N) · V = N (N · V) = 0, (25)

and thus Eq. (24) reduces to

RN (φ) = I cos (φ) + sin (φ) N × I. (26)

3.1.4. Alignment of the moving trihedral in the deformed configuration

The same two steps of mapping and rotation, as introduced in Section 3.1.1, can be adapted for describing the

alignment of the moving trihedral from the undeformed to the deformed configuration.

aα = Rt (ψ ) Λ(T, t)Aα. (27)

The operationΛ

(T, t) (see Section 3.1.2) aligns the moving trihedral Ai to the deformed tangent t (Eq. (16)). Thefinal deformed base vectors aα are obtained by applying the rotation matrix Rt(ψ ) (see Section 3.1.3). It rotates the

deformed reference Λ (T, t) Aα around the deformed center line, i.e. tangent vector t, with the angle ψ , where ψ

θ 1

is the rotational degree of freedom (see Fig. 5).

3.2. Nonlinear kinematics

The kinematics are derived straightforward in this work. The Green–Lagrange (GL) strain tensor and the ener-

getically conjugated second Piola–Kirchhoff (PK2) stress tensor are used for the nonlinear formulation. A stringent,

consistent derivation until the second variation is derived in the present contribution.

3.2.1. Variation of the base vectors

The base vectors of the continuum are defined as Gi

= X,i

and gi

= x,i

. So it follows for the configurations

defined in Section 3.1:

G1

θ 1, θ 2, θ 3

= X,1

θ 1, θ 2, θ 3

= A1

θ 1

+ θ 2A2,1

θ 1

+ θ 3A3,1

θ 1

, (28a)

G2

θ 1

= X,2

θ 1, θ 2, θ 3

= A2

θ 1

, (28b)

G3

θ 1

= X,3

θ 1, θ 2, θ 3

= A3

θ 1

, (28c)

g1

θ 1, θ 2, θ 3

= x,1

θ 1, θ 2, θ 3

= a1

θ 1

+ θ 2a2,1

θ 1

+ θ 3a3,1

θ 1

, (28d)

g2

θ 1

= x,2

θ 1, θ 2, θ 3

= a2

θ 1

, (28e)

g3 θ 1 = x,3 θ

1

, θ 2

, θ 3 = a3 θ

1 . (28f)

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Considering Eqs. (15) and (27) the derivatives Aα,1 resp. aα,1 write:

Aα,1

θ 1

= RT (Ψ ),1 Λ (T0, T) A0α + RT (Ψ )Λ (T0, T),1 A0

α (29)

aα,1

θ 1

=

Rt (ψ ),1 Λ(T, t)RT (Ψ ) Λ(T0, T) + Rt (ψ )Λ(T, t),1RT (Ψ )Λ(T0, T)

+ Rt (ψ )Λ(T, t)RT (Ψ ),1 Λ(T0, T) + Rt (ψ) Λ(T, t)RT (Ψ ) Λ(T0, T),1A0

α. (30)

From Eqs. (23) and (26) the derived terms of Eqs. (29) and (30) can be expressed as:

RT (Ψ ),1 = − (Ψ ),1 sin (Ψ ) I + (Ψ ),1 cos (Ψ ) T × I + sin (Ψ ) T,1 × I, (31)

Λ (T0, T),1 =

T0 · T,1

I +

T0 × T,1

× I −

T0 · T,1

(1 + T0 · T)2 (T0 × T) ⊗ (T0 × T)

+ 1

1 + T0 · T

T0 × T,1

⊗ (T0 × T) + (T0 × T) ⊗

T0 × T,1

(32)

Rt (ψ),1 = − (ψ ),1 sin (ψ) I + (ψ),1 cos (ψ) T × I + sin (ψ ) T,1 × I, (33)

Λ (T, t),1 = T,1 · t + T · t,1 I + T,1 · t + T · t,1× I − T,1 · t + T · t,1

(1 + T · t)2

(T × t) ⊗ (T × t)

+ 1

1 + T · t

T,1 · t + T · t,1

⊗ (T × t) + (T × t) ⊗

T,1 · t + T · t,1

. (34)

Furthermore the derivatives of the normalized tangent vector are:

T,1 =

A1,1

∥A1∥2

−

A1 · A1,1

A1

∥A1∥23

and t,1 =

a1,1

∥a1∥2

−

a1 · a1,1

a1

∥a1∥23

. (35)

3.2.2. Green–Lagrange strain tensor

The GL strain tensor is defined as

E = 1

2

gi j − Gi j

Gi ⊗ G j , E i j =

1

2

gi j − Gi j

. (36)

The GL strain tensor is calculated for the curvilinear coordinate system. However for the constitutive law, the

strains corresponding to the Cartesian coordinate system are used as common in engineering literature. In the sequel,

variables with respect to the Cartesian coordinate system will be labeled with ˜(..). Since the base vectors are orthogonal

to each other, the transformation rule is derived as:

˜ E i j = E i j

∥Gi ∥2 ∥G j ∥2. (37)

The length ∥Gi ∥2 of each base vector of the reference configuration is needed:

∥G1∥2 =A1 + θ 2A2,1 + θ 3A3,1

2

=

A1 + θ 2A2,1 + θ 3A3,1

·

A1 + θ 2A2,1 + θ 3A3,1

12

=

A1 · A1 + 2θ 2A2,1 · A1 + 2θ 3A3,1 · A1

12

≈ ∥A1∥2. (38)

Two simplifications were made for slender beams with h, w ≪ L, where h, w and L are the cross sectional

dimensions and the length of the beam: (i) square order terms O

θ 22

, O

θ 32

, O

θ 2 · θ 3

are neglected and

(ii) ∥G1∥2 is assumed as ∥G1∥2 ≈ ∥A1∥2 in Eq. (38). Note that, warping effects also vanish with these simplifications.

We can write for the length of the base vectors G2 and G3:

∥Gα∥2 = ∥Aα∥2 = 1. (39)

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The first metric coefficients G 11 and g11 are given by:

G11

θ 1, θ 2, θ 3

= G1 · G1 = A11

θ 1

+ 2θ 2A2,1

θ 1

· A1

θ 1

+ 2θ 3A3,1

θ 1

· A1

θ 1

, (40a)

g11

θ 1, θ 2, θ 3

= g1 · g1 = a11

θ 1

+ 2θ 2a2,1

θ 1

· a1

θ 1

+ 2θ 3a3,1

θ 1

· a1

θ 1

. (40b)

So the first component of the GL strain tensor E 11 is expressed as:

E 11 = 1

2(g11 − G11) =

1

2(a11 − A11) + θ 2(

b2 a2,1 · a1 −

B2 A2,1 · A1) + θ 3(

b3 a3,1 · a1 −

B3 A3,1 · A1)

=

ϵ 1

2(a11 − A11) + θ 2

κ21 (b2 − B2) + θ 3

κ31 (b3 − B3) . (41)

The respective strain in the Cartesian coordinate system is

E 11 = E 11

∥G1∥

2

2

= ϵ + θ 2κ21 + θ 3κ31

∥A1∥2

2 , (42)

where ϵ is the axial strain, κα1 is the change in curvature in the direction of the base vectors aα. κα1 consists of bα and

Bα , which denote the curvature of the respective configuration in the particular direction. The individual curvatures

for each direction in both configurations in Eq. (41) are defined by using the operators introduced in Eqs. (23) and

(26) as

Bα =

RT (Ψ ),1 Λ (T0, T) A02 + RT (Ψ )Λ (T0, T),1 A0

α

· A1, (43a)

bα =

Rt (ψ ),1 Λ(T, t)RT (Ψ ) Λ(T0, T) + Rt (ψ) Λ(T, t),1RT (Ψ ) Λ(T0, T)

+ Rt (ψ )Λ(T, t)RT (Ψ ),1 Λ(T0, T) + Rt (ψ ) Λ(T, t)RT (Ψ ) Λ(T0, T),1

A0

α

· a1. (43b)

Since Bernoulli theory is applied, no change of the cross section is allowed. ∥A2∥2 and ∥A3∥2 have unit length and

do not change their length in the deformed state, i.e.∥aα∥2 = 1. This yields for E 22 and E 33:

E αα = 1

2(gαα − Gαα ) = 0. (44)

Next we consider the off-diagonal terms. The torsional shear strain is computed as follows:

G1α = G1 · Gα = A1α + θ αAα,1 · Aα + θ β Aβ,1 · Aα, (45a)

g1α = g1 · gα = a1α + θ α aα,1 · aα + θ β aβ,1 · aα , (45b)

where (α, β) ∈ (2, 3), (3, 2). This holds for all following equations, if α and β appear together.

According to the Bernoulli theory it is assumed, that the cross section remains perpendicular to the center line.

This implies:

A1 · Aα = 0, a1 · aα = 0. (46)

The derivative Aα,1(θ 1) of Aα (θ 1) in direction of θ 1 only consists of two components. One of them is parallel to

A1(θ 1) and one is parallel to Aβ (θ 1) and thus:

Aα,1 · Aα = 0, aα,1 · aα = 0. (47)

With the Bernoulli assumption the torsional shear components from Eq. (45) can be written as

G1α = θ β Aβ,1 · Aα and g1α = θ β aβ,1 · aα . (48)

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Hence, the strain E 1α can be computed as follows:

E 1α = 1

2(g1α − G1α ) =

1

2θ β (

t α aβ,1 · aα −

T α Aβ,1 · Aα ) =

1

2θ β

κβα (t α − T α) ,

(49)

where:

T α = RT (Ψ ),1 Λ (T0, T) A0β + RT (Ψ )Λ (T0, T),1 A0

β · RT (Ψ )Λ (T0, T) A0

α , (50a)

t α =

Rt (ψ),1 Λ(T, t)RT (Ψ )Λ(T0, T) + Rt (ψ )Λ(T, t),1RT (Ψ ) Λ(T0, T) + Rt (ψ) Λ(T, t)RT (Ψ ),1

× Λ(T0, T) + Rt (ψ )Λ(T, t)RT (Ψ )Λ(T0, T),1

A0

β

· Rt (ψ ) Λ (T, t) RT (Ψ )Λ (T0, T) A0

α . (50b)

The respective strain in Cartesian coordinate system E 1α is then:

E 1α = E 1α

∥G1∥2 ∥Gα∥2

= κβα

2∥A1∥2θ β . (51)

3.3. Constitutive equations

Within the Total Lagrangian Formulation, the energetically conjugated stress tensor is required, here the PK2 stress

tensor S [25]. It is elaborated as the derivative of the strain energy W int w.r.t. the GL strain tensor. Stress and strain

tensors are coupled by the material law:

S = C : E. (52)

The elasticity tensor C is a fourth order tensor. Since this work only treats isotropic elastic material, a linear relation be-

tween stress and strain is appropriate and St. Venant–Kirchhoff material is applied. The 3D continuum is dimensionally

reduced to the center line for the beam formulation. Moreover, the notation is changed to Voigt notation in the follow-

ing description. All quantities within the cross section refer to A2 and A3, which are the principal axes. This, in com-

bination with the normalized tangent vector T, also implies a change to an orthonormal coordinate system (Eq. (37)).

Under the assumption of vanishing transverse shear forcesS 23

andS 32

and normal forcesS 22

and S 33

perpendicular

to the center line, the full constitutive equation can be reduced by means of static condensation, in order to adapt to

the element formulation of this work. This is in agreement with the classical beam theory.

In the same step, the Lame constants will be replaced by Young’s Modulus E and Poisson’s ratio ν, which are more

common in the engineering literature. The shear modulus G = 12

E 1+ν

is also introduced. The reduced elasticity matrix

D reads:S 11S 12S 13

= D ·

˜ E 11

˜ E 12

˜ E 13

, D =

E 0 0

0 G 0

0 0 G

. (53)

3.4. Internal work

The equation of the weak form of the equilibrium has been introduced in Section 2.1. The derived kinematics from

the previous section (Eqs. (42) and (51)) will now be inserted in the equilibrium expression.

δW int = −

Ω ⊂R3

S : δE d Ω = −

Ω ⊂R3

S 11δ ˜ E 11 + S 12δ ˜ E 12 + S 13δ ˜ E 13 d Ω . (54)

If the base vectors Aα are the principal axes of the cross section, the equilibrium can be written as:

δW int = −

Ω ⊂R3

E ˜ E 11δ ˜ E 11 + G ˜ E 12δ ˜ E 12 + G ˜ E 13δ ˜ E 13 d Ω

= −

L

E

∥A1∥24

·

Aϵ δϵ + I A3

κ21 δκ21 + I A2κ31 δκ31

+

G I

∥A1∥22

·

−

1

2κ32 δκ32 +

1

2κ23 δκ23

d L .

(55)

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This assumption is valid for most structural applications and provides a simplification of the derivation, since a

decomposition of normal force and bending moments is possible and some terms vanish. Obviously a derivation

for general cross sections is possible, following the same pattern as established here. The variational index of this

element formulation is m = 2. NURBS basis functions are easily set such that they are in H 2(0, L ).

3.5. Inner forces

The derivation of the inner forces is also shown for the case of base vectors being principal axes of the cross section.

If this is not the case, the interaction of the single terms in the stress components has to be considered.

3.5.1. Normal force ˜ N

With S 11 =3

i=1 C 111i ˜ E 1i = E ˜ E 11 the normal force ˜ N is defined as

˜ N =

A

S 11

θ 1, θ 2, θ 3

d A =

A

E

∥A1∥22

ϵ + θ 2κ21 + θ 3κ31

d θ 2d θ 3 =

E

∥A1∥22

Aϵ, (56)

where A is the area of the cross section and E Young’s modulus of the beam.

3.5.2. Bending moments ˜ M 2 and ˜ M 3

For the bending moments, the resulting moment will be divided into two moments around the principal axes of the

cross section. The sign convention is chosen such that a positive moment generates tension at the side of the cross

section of the positive base vector Aα .

The internal normal forces with their lever arms around the particular axis yield two defined bending moments.

The lever arms are represented by θ 2 and θ 3:

˜ M α =

A

S 11

θ 1, θ 2, θ 3

θ α d A =

A

E

∥A1∥22

ϵ + θ 2κ21 + θ 3κ31

θ α d θ 2d θ 3 =

E

∥A1∥22

I Aβκα1, (57)

with the respective moment of inertia I Aβ =

A

(θ α)2 d θ 2d θ 3.

3.5.3. Torsional moment ˜ M T

The torsional moment is computed from the shear forces and their lever arm to the center line of the beam. So it is

a cross sectional in-plane moment. In this case the constitutive law writes

S 12 =3

i=1

C 121i ˜ E 1i = E

2 (1 + ν)· 2 E 12 and S 13 =

3i =1

C 131i ˜ E 1i = E

2 (1 + ν)· 2 E 13. (58)

We can compute the torsional moment of each arbitrary cross section θ 1 ∈ [0, 1] as follows:

˜ M T =

AS 12

θ 1, θ 2, θ 3

θ 3 +

S 13

θ 1, θ 2, θ 3

θ 2 d A =

G I

∥A1∥2 −

1

2κ32 +

1

2κ23

, (59)

with the moment of inertia I =

A

θ 22

+

θ 32

d θ 2d θ 3. The changes of the respective in-plane cross sectional

curvatures κ32 and κ23 have the same absolute value due to A2,1 · A3 = −A3,1 · A2 and can thus be excluded.

3.6. Cross section values for selected cases

For selected commonly used types of cross sections – here rectangular and circular cross sections – the area and

the moments of inertia are given in Table 1.

Note that the given value I for the rectangle is not the result of the preintegration, but the approximated moment

of inertia. This is done in order to be able to compare the results to other values from the literature. For more

accuracy, membrane analogy or shear flow approximation has to be used [26]. Also warping has to be considered

in the geometric assumptions.

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Table 1

Cross sectional values for selected cases. r is the radius of a circular cross

section. h denotes the cross sectional height in θ 2-direction being larger

than w in θ 3-direction.

Cross section Rectangle Circle

A hw r 2π

I A2hw3

12r 4π

4

I A3h3w

12r 4π

4

I hw3

3r 4π

2

3.7. NURBS-based 3-dimensional beam element

The expressions used to obtain the stiffness matrix and the forces for a NURBS-discretized beam are elaborated in

this section. In the following, discrete nodal values will be denoted by (ˆ). Each node has 3 displacement DOFs u , v,

w and the rotational DOF φ around the center line. The variations of the strains ϵ and κ with respect to a global vector

of degrees of freedom u are required.

u =

u1 v1 w1 φ1 . . . un vn wn φn T

. (60)

In consequence, the internal forces and the stiffness derived from the internal work in Eq. (55) are defined as follows:

F intr = −

L

∂ W int

∂ur d L =

L

E

∥A1∥24

·

A · ϵ

∂ϵ

∂ur + I A3

κ21∂κ21

∂ur + I A2

κ31∂κ31

∂ur

+

G I

∥A1∥22

·

−

1

2κ32

∂κ32

∂ur +

1

2· κ23

∂κ23

∂ur

d L (61)

K r s = L∂2 W int

∂ur ∂usd L = L

E

∥A1

∥2

4 · A ·

∂ϵ

∂us

∂ϵ

∂ur + I A3

∂κ21

∂us

∂κ21

∂ur + I A2

∂κ31

∂us

∂κ31

∂ur +

1

2

G I

∥A1∥22

·

−

∂κ32

∂us

∂κ32

∂ur +

∂κ23

∂us

∂κ23

∂ur

+

E

∥A1∥24

·

A · ϵ

∂2ϵ

∂ur ∂us+ I A3

κ21∂2κ21

∂ur ∂us+ I A2

κ31∂2κ31

∂ur ∂us

+

1

2

G I

∥A1∥22

·

−κ32

∂ 2κ32

∂ur ∂us+ κ23

∂2κ23

∂ur ∂us

d L . (62)

The beam is reduced to the center line xc, which is defined by the control points and their respective basis function

Ri, p (see Section 2.2) as:

xc = i

Ri, pxi = i

Ri, pXi + ui . (63)

Since all parameters of the undeformed geometry are invariant to the variation, the variation of the position vector xc,r

can be written as:

xc,r =

i

Ri, p

Xi + ui

,r

=

i

Ri, pui,r . (64)

The second variation of xc vanishes, because the displacements u appear linearly in xc.

xc,r ,s = i

Ri, pui,r ,s = 0. (65)

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For the membrane strain, only the variation of the center line base vector a1 with respect to the variation parameter

r , which is the r th component of u, is necessary:

a1,r = x,1,r =

i

R(i, p),1ui,r . (66)

The variation of ϵ (Eq. (41)) is expressed as:

ϵ,r = 1

2a11,r = a1a1,r . (67)

The second variation yields:

ϵ,r ,s = 1

2a11,r ,s = a1,s a1,r . (68)

The next variations with respect to the degrees of freedom are the variations of the curvatures κα1 and κβα . The

initial configuration and consequently the base vectors of the cross section A02 and A0

3 are invariant to u. Consequently,

Bα,r and T α,r are equal to 0 and hence:

κα1,r = (bα − Bα),r = bα,r κβα,r = (t α − T α),r = t α,r (69)

κα1,r ,s = (bα − Bα),r ,s = bα,r ,s κβα,r ,s = (t α − T α ),r ,s = t α,r ,s . (70)

The variation and all subvariations are stated in the Appendix. Gauss integration was chosen as integration method.

Studies have shown that reliable results are obtained with p + 1 Gauss points per knot span.

4. Special aspects of initially curved beams

One of the most important parts for the element formulation is the strain definition. According to Bernoulli theory,

the reference length for the strain is the center line for the whole cross section. This assumption is valid for a straight

beam. However, if the beam is initially curved, the reference length varies over the cross section with d φ · ( R ± θ 2)

(see Fig. 6).

The application of Bernoulli theory on initially curved beams has almost no influence on the displacements (see

Section 5.1). In contrast, the application of Bernoulli theory on inner forces can generate large errors, especially for the

internal normal force. Therefore, the postprocessing of the stresses is adapted in the following. For the two curvature

planes A2 and A3, as illustrated in Fig. 6, the following equations hold [27]:

ϵ(θ α ) = du(θ α)

d S (θ α)=

du0 + d Φ · θ α

d φ · ( R + θ α)=

d ϵ0 Rd φ + d Φ · θ α

d φ · ( R + θ α)= ϵ0 +

d Φ

d φ− ϵ0

θ α

R + θ α . (71)

With the relation σ = E · ϵ the inner forces can be computed in accordance to Section 3.5:

N = A E ϵ(θ

α

) d A = EAϵ0 + E d Φ

d φ − ϵ0 A θ α

R + θ α d A (72)

M =

A

E ϵ(θ α) θ α d A = E ϵ0

A

θ α d A + E

d Φ

d φ− ϵ0

A

(θ α)2

R + θ α d A. (73)

Convergence studies have revealed only very limited influence of the initial curvature on the resulting bending

moment. This can be explained by the fact that in general the treated problems are bending dominated and the

influence of the internal normal force on the bending moment is small. So they do not have to be transferred and

M ∗ ≈ M is valid, where (..)∗ denotes the corrected value. Consequently, the remaining integral for the moment in

Eq. (73) can be transformed to the remaining integral of the normal force in Eq. (72):

A(θ α)2

( R + θ α)

d A = A θ α − Rθ α

( R + θ α)

d A = R Aθ α

R + θ α d A. (74)

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Fig. 6. Illustration of the varying reference length over the thickness for curved elements.

Thus the second term of Eq. (72) can be substituted by

E

d Φ

d φ− ϵ0

A

θ α

R + θ α d A = −

M

R. (75)

The adjusted normal force N ∗ is therefore given by

N ∗ = N − M

R. (76)

Further information concerning curvature effects can be found in [27,28]. Interaction of the two curvature planes

is neglected as an effect of higher order.

5. Numerical examples

Selected examples for different loading scenarios and geometries are presented in this section. In a first step,

different types of interaction are investigated in a linear analysis for an initially curved beam. Two in-plane examples

study the behavior for geometrically nonlinear analysis, followed by two spatial, nonlinear examples with increasingcomplexity w.r.t. the initial curvature. This will verify the nonlinear spatial Bernoulli beam formulation and its correct

implementation. In the following, clamped supports are modeled by restraining the displacements of the first two

control points perpendicular to the tangent at the respective end.

5.1. Membrane-bending interaction

The first example is a quadrant. It is clamped at one end and loaded in-plane on the tip with a force in negative

y-direction (see Fig. 7). The cross section is defined with A02 = A1

2 = [0, 0, 1] at both ends. As the cross section

base vector is always perpendicular to the main curvature, it is not rotating around its center line along the beam.

Thus no geometric torsion is activated, basically a plane deformation is activated. The reference points are the tip for

displacements and the clamped end for the forces, respectively.

This configuration is examined with a linear analysis, since the analytical solution for the displacements at the tip

can be derived easily by means of the principle of virtual forces.

utip x =

π2

0

1

EI 2(−FR · cos α)(− R · (1 − sin α)) +

1

EA(−F · cos α)(− sin α) Rd α

= 1

2

−

FR3

EI 2+

FR

EA

= −0.300147573786893 m (77)

utip y =

π2

0

1

EI 2(−F R · cos α)( R · cos α) +

1

EA(−F · cos α)(cos α) Rd α

= −FR3

EI 2+

FR

EA π

4= −0.471470706400852 m. (78)

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Fig. 7. Cantilever spatial arch with in-plane tip loading: Definition of the geometry, the cross section and the loading scenario.

Fig. 8. Quadrant for: (a) p-refinement: p = 2, p = 3, p = 4, p = 5, (b) increasing number of elements: 1, 2, 3 and 4 elements.

The reference values of the normal force and the moment at the clamped end are determined with the global

equilibrium of forces:

˜ N = −F y = −1.0 kN (79)˜ M 3 = F y R = 1.0 kN m. (80)

This exemplary study has been performed with polynomial degrees from 2 to 8 and a number of elements from 1

to 256. The mesh for the different refinements is shown in Fig. 8

Fig. 9 shows the corresponding results for the displacements at the loaded tip and the inner forces and moments at

the clamped end.

5.2. Bending–torsion interaction

The bending–torsion interaction is also tested with the quadrant from Section 5.1 (see Fig. 10). Now the point load

points in z-direction. As a consequence, the structure is loaded out-of-plane. Thus bending and torsion moments result

from this loading scenario.

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Fig. 11. Relative error of the example for bending–torsion interaction for: (a) tip- z-displacement utip z , (b) tip-rotation φtip, (c) bending moment ˜ M 2

at the clamped edge, (d) torsional moment ˜ M T at the clamped edge.

The analytical solution for the displacements of the tip can again be derived from the principle of virtual forces

as

utip z =

π2

0

1

EI 3(−F R · cos α)(− R · cos α) +

1

G I (F R · (1 − sin α))( R · (1 − sin α)) Rd α

= FR3π

4 EI 3+

FR3

G I 3π

4− 2 = 0.1385450005254 m, (81)

ψ tip =

π2

0

1

EI 3(−F R · cos α)( R · cos α) +

1

G I (F · (1 − sin α))(sin α) Rd α

= −FR2π

4 EI 3+

FR2

G I

1 −

π

4

= 0.075801200345745 rad. (82)

The reference values of the inner forces at the clamped edge are calculated with the global equilibrium of forces.

˜ M 2 = −F z R = −1.0 kN m (83)

˜ M T = F z R = 1.0 kN m. (84)

The corresponding results for the proposed element formulation are shown in Fig. 11.

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Fig. 12. Convergence of the example for bending–torsion interaction: (a) tip- z-displacement utip z , (b) tip-rotation φ tip, (c) bending moment ˜ M n at

the clamped edge, (d) torsional moment ˜ M T at the clamped edge.

Fig. 12 shows the relative error µ plotted against the number of elements for different polynomial degrees. The

relative error µ can be computed as:

µ = ∥()exac − ()num∥

∥()exac∥ . (85)

For the displacements, a relative error of 1e−10 was reached in this study. The moments reach an accuracy of

1e−09.

5.3. Nonlinear analysis of a shallow arch

In this Section, a shallow arch is examined with the displacement-controlled method as a first example for nonlinear

analysis. This geometry is referred to in several publications [29–32], and thus it represents a well-established

benchmark example. The example is defined as a segment of a circular arch, which is clamped at both ends. A single

point force is applied in the center of the beam span (see Fig. 13). The reference displacement is the displacement

of the apex. The configuration uses NURBS with polynomial degree p = 3 and 16 elements as well as p = 6 and

8 elements in order to compare the behavior of different parametrizations. The reference curves in Fig. 14 had been

computed with the half of the system using symmetry conditions. Hence 16 elements in this present contribution,

where the whole system is modeled, correspond to 8 elements from the reference curve.

The reference diagram was taken from [29] and the results of the developed beam formulation are added in

Fig. 14(a).

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Fig. 13. Initial geometry of the shallow arch with input parameters.

Fig. 14. Comparison of the deflection of the shallow arch’s apex computed using the present formulation of the spatial Bernoulli Beam (SBB) with

(a) Bathe [29] Fig. 5 [31,32], (b) Lo [30] Fig. 6.

Fig. 15. Initial geometry of the mainspring with input parameters.

The shallow arch has a nonlinear post-buckling behavior. Fig. 14(b) shows a comparison of the results of the beam

with p = 3 and 16 elements with [30]. Since the displacement throughout the snap-through of the arch nicely matches

the reference solution, one can conclude that post-buckling behavior can correctly be analyzed with the presented

beam formulation.

5.4. Mainspring

As a last plane benchmark, the main-spring example provides a pure bending example. A simple cantilever isloaded by a bending moment at the tip (see Fig. 15). Loaded by a moment M = π

L EI , the beam bends to a half

circle. The closed circle is reached for M = 2π L

EI . Fig. 16 shows the deformed geometry for several load steps. The

cantilever is modeled with 10 elements and a polynomial degree of p = 6.

The analytical result of the deflection at the tip is determined. The deformed configurations depict circular arches

respectively segments of these. The radius R is defined by the applied moment and the stiffness as R = EI M

. The beam

maintains its length since there is no normal stress. Thus the arc length of the circular arch is defined. The analytical

displacements can be computed out of the length L and the radius R.

u = L − sin

L

R

· R , v =

1 − cos

L

R

· R . (86)

Fig. 17 shows that the numerical results nicely match the analytical reference solution.

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Fig. 16. Deformed geometry of the mainspring for 8 load steps: (a) without control points, (b) with control points.

Fig. 17. Numerical results of the mainspring compared to analytic results.

5.5. Nonlinear analysis of a 45

-bending-beam

A fully spatial, well-established example in the literature for nonlinear analysis is the 45 -bending-beam, treated

e.g. in [33,30,29,34]. The beam is also a segment of a circular arch with a radius of 100 m, but only clamped at one end

(see Fig. 18). At the other end, a point load is applied out-of-plane similar to the example in Section 5.2. The system

is modeled with a polynomial degree of p = 3 and 16 elements. In Fig. 19(a) the non-dimensional tip-displacements

are compared to [29]. The computed deflection nicely matches the reference solution. The significance of the fully

nonlinear formulation is demonstrated clearly by a comparison to the results gained by a linear beam formulation (see

Fig. 19(a)).

5.6. Spatial helicoidal spring

The final example is a completely three-dimensional structure, a helicoidal spring with one loop (see Fig. 20). The

center line is defined by:

Xc(θ 1) =

10 sin(θ 1 2π )

10 cos(θ 1 2π )

20 θ 1

, θ 1 ∈ [0, 1] . (87)

The control points are obtained by Rhinoceros 5 [35]. The modeling uses 24 elements with p = 3. Since there is

no reference solution available in the literature, another classical FEM-model, namely a co-rotational beam element

formulation presented in [14] is taken as reference solution. Thus the results of the proposed formulation are compared

to a completely independent element formulation and it can be concluded that two matching results verify the element

formulation (see Fig. 21).

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Fig. 18. Initial geometry of the 45-bending-beam with input parameters.

Fig. 19. 45-bending-beam: (a) comparison to Bathe [29] Fig. 9, (b) deformed geometry for several load parameters k .

6. Conclusions and outlook

A new element formulation for a nonlinear isogeometric spatial Bernoulli beam including torsion has been derived

from the 3D-continuum. The geometric description is adapted from [11] and used for geometrically nonlinear

kinematics. For the post-processing of the inner forces, a correction term has been introduced to obtain reliable results

for the normal force in case of initially curved beams.The developed element has been implemented in the structural analysis code at the authors’ Chair. The formulation

has been tested with selected benchmark examples against analytic solutions if available or other nonlinear beam

element implementations. These benchmarks cover linear examples with membrane-bending resp. bending–torsion

interaction. Nonlinear examples with straight resp. curved initial configurations show the element’s behavior for large

deformations. From these benchmarks it can be concluded that the element provides accurate results and that the

above mentioned correction factor has successfully been applied.

For future research, one could extend the beam theory to account for further torsion related effects, e.g. warping,

and material assumptions. Furthermore the consideration of the curvature effects in the stiffness matrix could be

further investigated. The application in a multipatch scenario has to be derived. The spatial beam is also interesting

for dynamic applications, particularly if it is coupled to other elements like membranes or shells. In this case it could

be applied in a B-Rep edge element formulation along trimming curves [36].

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Fig. 20. Initial geometry of the spatial helicoidal spring with tip load.

Fig. 21. (a) Comparison of numerical results for the spatial helicoidal spring of the present work and the co-rotational beam element of Krenk [14],

(b) deformed geometry for several load parameters k .

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the SPP project “LeichtBauen

mit Beton” (BL 306/23-1). The support is gratefully acknowledged.

Appendix. First and second variations

A.1. Curvature terms

bα,r =

Rt (ψt ),1,r Λ (T, t) RT (ψ0)Λ (T0, T) A0α + Rt (ψt ),1 Λ (T, t),r RT (ψ0)Λ (T0, T) A0

α

+ Rt (ψt ),r Λ (T, t),1 RT (ψ0)Λ (T0, T) A0α + Rt (ψt ) Λ (T, t),1,r RT (ψ0) Λ (T0, T) A0

α

+ Rt (ψt ),r Λ (T, t) RT (ψ0),1 Λ (T0, T) A0α + Rt (ψt ) Λ (T, t),r RT (ψ0),1 Λ (T0, T) A0α

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+ Rt (ψt ),r Λ (T, t) RT (ψ0) Λ (T0, T),1 A0α + Rt (ψt ) Λ (T, t),r RT (ψ0) Λ (T0, T),1 A0

α

· a1

+

Rt (ψt ),1 Λ (T, t) RT (ψ0) Λ (T0, T) A0α + Rt (ψt )Λ (T, t),1 RT (ψ0)Λ (T0, T) A0

α

+ Rt (ψt ) Λ (T, t) RT (ψ0),1 Λ (T0, T) A0α + Rt (ψt )Λ (T, t) RT (ψ0) Λ (T0, T),1 A0

α

· a1,r (A.1)

bα,r ,s = Rt (ψt ),1,r ,s Λ (T, t) RT (ψ0) Λ (T0, T) A0α + Rt (ψt ),1,s Λ (T, t),r RT (ψ0) Λ (T0, T) A0α

+ Rt (ψt ),r ,s Λ (T, t),1 RT (ψ0) Λ (T0, T) A0α + Rt (ψt ),s Λ (T, t),1,r RT (ψ0) Λ (T0, T) A0

α

+ Rt (ψt ),r ,s Λ (T, t) RT (ψ0),1 Λ (T0, T) A0α + Rt (ψt ),s Λ (T, t),r RT (ψ0),1 Λ (T0, T) A0

α

+ Rt (ψt ),r ,s Λ (T, t) RT (ψ0) Λ (T0, T),1 A0α + Rt (ψt ),s Λ (T, t),r RT (ψ0) Λ (T0, T),1 A0

α

+ Rt (ψt ),1,r Λ (T, t),s RT (ψ0) Λ (T0, T) A0α + Rt (ψt ),1 Λ (T, t),r ,s RT (ψ0) Λ (T0, T) A0

α

+ Rt (ψt ),r Λ (T, t),1,s RT (ψ0) Λ (T0, T) A0α + Rt (ψt )Λ (T, t),1,r ,s RT (ψ0) Λ (T0, T) A0

α

+ Rt (ψt ),r Λ (T, t),s RT (ψ0),1 Λ (T0, T) A0α + Rt (ψt ) Λ (T, t),r ,s RT (ψ0),1 Λ (T0, T) A0

α

+ Rt (ψt ),r Λ (T, t),s RT (ψ0) Λ (T0, T),1 A0α + Rt (ψt ) Λ (T, t),r ,s RT (ψ0) Λ (T0, T),1 A0

α

· a1

+

Rt (ψt ),1,r Λ (T, t) RT (ψ0) Λ (T0, T) A0

α + Rt (ψt ),1 Λ (T, t),r RT (ψ0) Λ (T0, T) A0α

+ Rt (ψt ),r Λ (T, t),1 RT (ψ0) Λ (T0, T) A0α + Rt (ψt )Λ (T, t),1,r RT (ψ0) Λ (T0, T) A0α

+ Rt (ψt ),r Λ (T, t) RT (ψ0),1 Λ (T0, T) A0α + Rt (ψt )Λ (T, t),r RT (ψ0),1 Λ (T0, T) A0

α

+ Rt (ψt ),r Λ (T, t) RT (ψ0) Λ (T0, T),1 A0α + Rt (ψt )Λ (T, t),r RT (ψ0)Λ (T0, T),1 A0

α

· a1,s

+

Rt (ψt ),1,s Λ (T, t) RT (ψ0) Λ (T0, T) A0α + Rt (ψt ),1 Λ (T, t),s RT (ψ0)Λ (T0, T) A0

α

+ Rt (ψt ),s Λ (T, t),1 RT (ψ0) Λ (T0, T) A0α + Rt (ψt )Λ (T, t),1,s RT (ψ0) Λ (T0, T) A0

α

+ Rt (ψt ),s Λ (T, t) RT (ψ0),1 Λ (T0, T) A0α + Rt (ψt )Λ (T, t),s RT (ψ0),1 Λ (T0, T) A0

α

+ Rt (ψt ),s Λ (T, t) RT (ψ0)Λ (T0, T),1 A0α

+ Rt (ψt ) Λ (T, t),s RT (ψ0)Λ (T0, T),1 A0α

· a1,r (A.2)

t α,r = Rt (ψt ),1,r Λ (T, t) RT (ψ0)Λ (T0, T) A0β + Rt (ψt ),1 Λ (T, t),r RT (ψ0) Λ (T0, T) A

0β

+ Rt (ψt ),r Λ (T, t),1 RT (ψ0)Λ (T0, T) A0β + Rt (ψt ) Λ (T, t),1,r RT (ψ0) Λ (T0, T) A0

β

+ Rt (ψt ),r Λ (T, t) RT (ψ0),1 Λ (T0, T) A0β + Rt (ψt ) Λ (T, t),r RT (ψ0),1 Λ (T0, T) A0

β

+ Rt (ψt ),r Λ (T, t) RT (ψ0)Λ (T0, T),1 A0β + Rt (ψt ) Λ (T, t),r RT (ψ0) Λ (T0, T),1 A0

β

·

Rt (ψt ) Λ (T, t) RT (ψ0) Λ (T0, T) A0α

+

Rt (ψt ),1 Λ (T, t) RT (ψ0)Λ (T0, T) A0β + Rt (ψt )Λ (T, t),1 RT (ψ0)Λ (T0, T) A0

β

+ Rt (ψt )Λ (T, t) RT (ψ0),1 Λ (T0, T) A0β + Rt (ψt )Λ (T, t) RT (ψ0) Λ (T0, T),1 A0

β

· Rt (ψt ),r Λ (T, t) RT (ψ0) Λ (T0, T) A0

α + Rt (ψt )Λ (T, t),r RT (ψ0)Λ (T0, T) A0α (A.3)

t α,r ,s = Rt (ψt ),1,r ,s Λ (T, t) RT (ψ0) Λ (T0, T) A0β + Rt (ψt ),1,s Λ (T, t),r RT (ψ0)Λ (T0, T) A0

β

+ Rt (ψt ),r ,s Λ (T, t),1 RT (ψ0) Λ (T0, T) A0β + Rt (ψt ),s Λ (T, t),1,r RT (ψ0)Λ (T0, T) A0

β

+ Rt (ψt ),r ,s Λ (T, t) RT (ψ0),1 Λ (T0, T) A0β + Rt (ψt ),s Λ (T, t),r RT (ψ0),1 Λ (T0, T) A0

β

+ Rt (ψt ),r ,s Λ (T, t) RT (ψ0) Λ (T0, T),1 A0β + Rt (ψt ),s Λ (T, t),r RT (ψ0) Λ (T0, T),1 A0

β

·

Rt (ψt )Λ (T, t) RT (ψ0) Λ (T0, T) A0α

+

Rt (ψt ),1,r Λ (T, t) RT (ψ0) Λ (T0, T) A0β + Rt (ψt ),1 Λ (T, t),r RT (ψ0)Λ (T0, T) A0

β

+ Rt (ψt ),r Λ (T, t),1 RT (ψ0) Λ (T0, T) A0β + Rt (ψt ) Λ (T, t),1,r RT (ψ0) Λ (T0, T) A0

β

+ Rt (ψt ),r Λ (T, t) RT (ψ0),1 Λ (T0, T) A0β + Rt (ψt ) Λ (T, t),r RT (ψ0),1 Λ (T0, T) A

0β

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+ Rt (ψt ),r Λ (T, t) RT (ψ0) Λ (T0, T),1 A0β + Rt (ψt ) Λ (T, t),r RT (ψ0) Λ (T0, T),1 A0

β

·

Rt (ψt ),s Λ (T, t) RT (ψ0) Λ (T0, T) A0α + Rt (ψt ) Λ (T, t),s RT (ψ0)Λ (T0, T) A0

α

+

Rt (ψt ),1,s Λ (T, t) RT (ψ0) Λ (T0, T) A0β + Rt (ψt ),s Λ (T, t),1 RT (ψ0) Λ (T0, T) A0

β

+ Rt (ψ

t )

,sΛ (T, t) R

T (ψ

0)

,1Λ (T

0, T) A0

β + R

t (ψ

t )

,sΛ (T, t) R

T (ψ

0)Λ (T

0, T)

,1A0

β+ Rt (ψt ),1 Λ (T, t),s RT (ψ0) Λ (T0, T) A0

β + Rt (ψt ) Λ (T, t),1,s RT (ψ0) Λ (T0, T) A0β

+ Rt (ψt ) Λ (T, t),s RT (ψ0),1 Λ (T0, T) A0β + Rt (ψt ) Λ (T, t),s RT (ψ0)Λ (T0, T),1 A0

β

·

Rt (ψt ),r Λ (T, t) RT (ψ0)Λ (T0, T) A0α + Rt (ψt ) Λ (T, t),r RT (ψ0) Λ (T0, T) A0

α

+

Rt (ψt ),1 Λ (T, t) RT (ψ0) Λ (T0, T) A0β + Rt (ψt ) Λ (T, t),1 RT (ψ0) Λ (T0, T) A0

β

+ Rt (ψt ) Λ (T, t) RT (ψ0),1 Λ (T0, T) A0β + Rt (ψt ) Λ (T, t) RT (ψ0)Λ (T0, T),1 A0

β

·

Rt (ψt ),r ,s Λ (T, t) RT (ψ0)Λ (T0, T) A0α + Rt (ψt ),s Λ (T, t),r RT (ψ0) Λ (T0, T) A0

α

+ Rt (ψt ),r Λ

(T, t),s RT (ψ0)Λ

(T0, T) A0

α + Rt (ψt )Λ

(T, t),r ,s RT (ψ0)Λ

(T0, T) A0

α . (A.4)

A.2. Rotation matrix RN

Rt (ψt ),r = −ψt ,r sin ψt I3 + ψt ,r cos ψt t × I3 + sin ψt t,r × I3 (A.5)

RT (ψ0),r = 0 (A.6)

Rt (ψt ),1,r = −ψt ,1,r sin ψt I3 − ψt ,1ψt ,r cos ψt I3 + ψt ,1,r cos ψt t × I3 − ψt ,1ψt ,r sin ψt t × I3

+ ψt ,1 cos ψt t,r × I3 + ψt ,r cos ψt t,1 × I3 + sin ψt t,1,r × I3 (A.7)

RT

(ψ0

),1,r

= 0 (A.8)

Rt (ψt ),r ,s = −ψt ,r ,s sin ψt I3 − ψt ,r ψt ,s cos ψt I3 + ψt ,r ,s cos ψt t × I3 − ψt ,r ψt ,s sin ψt t × I3

+ ψt ,r cos ψt t,s × I3 + ψt ,s cos ψt t,r × I3 + sin ψt t,r ,s × I3 (A.9)

RT (ψ0),r ,s = 0 (A.10)

Rt (ψ),1,r ,s = −ψt ,1,r ,s sin ψt I3 − ψt ,1,r ψt ,s cos ψt I3 − ψt ,1,s ψt ,r cos ψt I3

− ψt ,1ψt ,r ,s cos ψt I3 + ψt ,1ψt ,r ψt ,s sin ψt I3

+ ψt ,1,r ,s cos ψt t × I3 − ψt ,1,r ψt ,s sin ψt t × I3

+ ψt ,1,r cos ψt t,s × I3 − ψt ,1,s ψt ,r sin ψt t × I3

− ψt ,1ψt ,r ,s sin ψt t × I3 − ψt ,1ψt ,r ψ,s cos ψt t × I3

− ψt ,1ψt ,r sin ψt t,s × I3 + ψt ,1,s cos ψt t,r × I3

− ψt ,1ψt ,s sin ψt t,r × I3 + ψt ,1 cos ψt t,r ,s × I3+ ψt ,r ,s cos ψt t,1 × I3 − ψt ,r ψt ,s sin ψt t,1 × I3

+ ψt ,r cos ψt t,1,s × I3 + ψt ,s cos ψt t,1,r × I3 + sin ψt t,1,r ,s × I3 (A.11)

RT (ψ0),1,r ,s = 0. (A.12)

A.3. Mapping matrix Λ

Λ (T, t),r =

T · t,r

I3 +

T × t,r

× I3 −

T · t,r

(1 + T · t)2 (T × t) ⊗ (T × t)

+

1

1 + T · t T × t,r⊗ (T × t) + (T × t) ⊗ T × t,r (A.13)

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124 A.M. Bauer et al. / Comput. Methods Appl. Mech. Engrg. 303 (2016) 101–127

Λ (T0, T),r = 0 (A.14)

Λ (T, t),1,r =

T,1 · t,r + T · t,1,r

I3 +

T,1 × t,r + T × t,1,r

× I3

+2

T,1 · t + T · t,1

T · t,r

(1 + T · t)3

(T × t) ⊗ (T × t)

−

T,1 · t,r + T · t,1,r

(1 + T · t)2 (T × t) ⊗ (T × t)

− T,1 · t + T · t,1

(1 + T · t)2

T × t,r

⊗ (T × t) + (T × t) ⊗

T × t,r

−

T · t,r

(1 + T · t)2

T,1 × t

+

T × t,1

⊗ (T × t) + (T × t) ⊗

T,1 × t

+

T × t,1

+

1

1 + T · t

T,1 × t,r

+

T × t,1,r

⊗ (T × t) +

T,1 × t

+

T × t,1

⊗

T × t,r

+

T × t,r

⊗

T,1 × t

+

T × t,1

+ (T × t) ⊗

T,1 × t,r

+

T × t,1,r

(A.15)

Λ (T0, T),1,r = 0 (A.16)

Λ (T, t),r ,s = T · t,r ,s I3 + T × t,r ,s× I3 +2

T · t,r

T · t,s

(1 + T · t)3

(T × t) ⊗ (T × t)

− T · t,r ,s

(1 + T · t)2 (T × t) ⊗ (T × t) −

T · t,r

(1 + T · t)2

T × t,s

⊗ (T × t)

+ (T × t) ⊗

T × t,s

+ −

T · t,s

(1 + T · t)2

T × t,r

⊗ (T × t) + (T × t) ⊗

T × t,r

+

1

1 + T · t

T × t,r ,s

⊗ (T × t) +

T × t,r

⊗

T × t,s

+

T × t,s

⊗

T × t,r

+ (T × t) ⊗

T × t,r ,s

(A.17)

Λ (T0, T),r ,s = 0 (A.18)

Λ (T, t),1,r ,s =

T,1 · t,r ,s + T · t,1,r ,s

I3 +

T,1 × t,r ,s + T × t,1,r ,s

× I3

−

6 T,1 · t + T · t,1 T · t,r T · t,s(1 + T · t)4 (T × t) ⊗ (T × t)

+2

T,1 · t,s + T · t,1,s

T · t,r

+ 2

T,1 · t + T · t,1

T · t,r ,s

(1 + T · t)3

(T × t) ⊗ (T × t)

+2

T,1 · t + T · t,1

T · t,r

(1 + T · t)3

T × t,s

⊗ (T × t) + (T × t) ⊗

T × t,s

+

2

T,1 · t,r + T · t,1,r

T · t,s

(1 + T · t)3

(T × t) ⊗ (T × t)

− T,1 · t,r ,s + T · t,1,r ,s

(1 + T · t)2 (T × t) ⊗ (T × t)

− T,1 · t,r + T · t,1,r (1 + T · t)2 T × t,s⊗ (T × t) + (T × t) ⊗ T × t,s

+2

T,1 · t + T · t,1

T · t,s

(1 + T · t)3

T × t,r

⊗ (T × t) + (T × t) ⊗

T × t,r

−

T,1 · t,s + T · t,1,s

(1 + T · t)2

T × t,r

⊗ (T × t) + (T × t) ⊗

T × t,r

−

T,1 · t + T · t,1

(1 + T · t)2

T × t,r ,s

⊗ (T × t)

+

T × t,r

⊗

T × t,s

+

T × t,s

⊗

T × t,r

+ (T × t) ⊗

T × t,r ,s

+

2

T · t,r

T · t,s(1 + T · t)3 T,1 × t+ T × t,1⊗ (T × t)

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+ (T × t) ⊗

T,1 × t

+

T × t,1

−

T · t,r ,s

(1 + T · t)2

T,1 × t

+

T × t,1

⊗ (T × t) + (T × t) ⊗

T,1 × t

+

T × t,1

−

T · t,r

(1 + T · t)2

T,1 × t,s

+

T × t,1,s

⊗ (T × t) +

T,1 × t

+ T × t,1⊗ T × t,s+

T × t,s

⊗

T,1 × t

+

T × t,1

+ (T × t) ⊗

T,1 × t,s

+

T × t,1,s

−

T · t,s

(1 + T · t)2

T,1 × t,r

+

T × t,1,r

⊗ (T × t) +

T,1 × t

+

T × t,1

⊗

T × t,r

+

T × t,r

⊗

T,1 × t

+

T × t,1

+ (T × t) ⊗

T,1 × t,r

+

T × t,1,r

+

1

1 + T · t

T,1 × t,r ,s

+

T × t,1,r ,s

⊗ (T × t)

+

T,1 × t,r

+

T × t,1,r

⊗

T × t,s

+ T,1 × t,s+ T × t,1,s⊗ T × t,r + T,1 × t+ T × t,1⊗ T × t,r ,s+ T × t,r ,s⊗ T,1 × t+ T × t,1+ T × t,r ⊗ T,1 × t,s+ T × t,1,s+

T × t,s

⊗

T,1 × t,r

+

T × t,1,r

+ (T × t) ⊗

T,1 × t,r ,s

+

T × t,1,r ,s

.

(A.19)

A.4. Normalized tangent t resp. T

t,1,r = a1,1,r

∥a1∥2

−

a1 · a1,r

a1,1

∥a1∥32

+3

a1 · a1,1

a1

a1 · a1,r

∥a1∥5

2

− a1 · a1,1,r

+

a1,r · a1,1

a1

∥a1∥32

− a1 · a1,1

a1,r

∥a1∥32

(A.20)

t,1,1 = a1,1,1

∥a1∥2

−

a1 · a1,1

a1,1

∥a1∥32

+3

a1 · a1,1

2a1

∥a1∥52

−

a1 · a1,1,1

+

a1,1 · a1,1

a1

∥a1∥32

−

a1 · a1,1

a1,1

∥a1∥32

(A.21)

t,r ,s = −a1,r

a1 · a1,s

∥a1∥3

2

−a1,s

a1 · a1,r

+ a1

a1,s · a1,r

∥a1∥3

2

+3

a1 · a1,r

a1

a1 · a1,s

∥a1∥5

2

(A.22)

t,1,r ,s = −a1,1,r

a1 · a1,s

∥a1∥

3

2

− a1 · a1,r

a1,1,s +

a1,s · a1,r

a1,1

∥a1∥

3

2

+3

a1 · a1,r

a1,1

a1 · a1,s

∥a1∥

5

2

+ 3a1,s · a1,r

a1 · a1,1 a1 + a1 · a1,r

a1,s · a1,1 + a1 · a1,1,s a1

∥a1∥52

+ 3

a1 · a1,r

a1 · a1,1

a1,s

∥a1∥52

− 15

a1 · a1,r

a1 · a1,1

a1

a1 · a1,s

∥a1∥7

2

−

a1,s · a1,1,r

+

a1,1,s · a1,r

a1

∥a1∥32

−

a1 · a1,1,r

+

a1,1 · a1,r

a1,s

∥a1∥32

+ 3

a1 · a1,1,r

+

a1,1 · a1,r

a1

a1 · a1,s

∥a1∥5

2

− a1,s · a1,1

+

a1 · a1,1,s

a1,r

∥a1∥32

+ 3 a1 · a1,1

a1,r a1 · a1,s∥a1∥52

. (A.23)

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