Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

Abstract

The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff–Love type, a detailed review of existing formulations of Kirchhoff–Love and Simo–Reissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak Kirchhoff constraint enforcement, respectively, as well as two different choices of nodal triad parametrizations in terms of rotation or tangent vectors are proposed. The combination of these schemes leads to four novel finite element variants, all of them based on a $C^1$-continuous Hermite interpolation of the beam centerline. Essential requirements such as representability of general 3D, large deformation, dynamic problems involving slender beams with arbitrary initial curvatures and anisotropic cross-section shapes, preservation of objectivity and path-independence, consistent convergence orders, avoidance of locking effects as well as conservation of energy and momentum by the employed spatial discretization schemes, but also a range of practically relevant secondary aspects will be investigated analytically and verified numerically for the different formulations. It will be shown that the geometrically exact Kirchhoff–Love beam elements proposed in this work are the first ones of this type that fulfill all these essential requirements. On the contrary, Simo–Reissner type formulations fulfilling these requirements can be found in the literature very well. However, it will be argued that the shear-free Kirchhoff–Love formulations can provide considerable numerical advantages such as lower spatial discretization error levels, improved performance of time integration schemes as well as linear and nonlinear solvers and smooth geometry representation as compared to shear-deformable Simo–Reissner formulations when applied to highly slender beams. Concretely, several representative numerical test cases confirm that the proposed Kirchhoff–Love formulations exhibit a lower discretization error level as well as a considerably improved nonlinear solver performance in the range of high beam slenderness ratios as compared to two representative Simo–Reissner element formulations from the literature.

Appendices

Appendix A: Rotational Shape Function Matrices

In this appendix, the shape functions $\widetilde{\mathbf {I}}^i(\xi )$ required for the multiplicative rotation increments

$$\begin{aligned} \Delta \varvec{\theta }(\xi ) = \sum _{i=1}^{n_{\Lambda }} \widetilde{\mathbf {I}}^i(\xi ) \Delta \hat{\varvec{\theta }}^i, \quad \Delta \varvec{\theta }^{\prime }(\xi ) = \sum _{i=1}^{n_{\Lambda }} \widetilde{\mathbf {I}}^{i \prime }(\xi ) \Delta \hat{\varvec{\theta }}^i. \end{aligned}$$

(188)

associated with the triad interpolation (105) and originally derived in [83] shall be presented:

$$\begin{aligned} \begin{array}{ll} \widetilde{\mathbf {I}}^i(\xi ) &= L^i(\xi ) \varvec{\Lambda }_r \mathbf {T}^{-1}(\varvec{\Phi }_{lh}(\xi )) \mathbf {T}(\varvec{\Phi }^i_l)\varvec{\Lambda }_r^T\\ &\quad+\, \delta ^{iI} \varvec{\Lambda }_r \left[ \mathbf {I_3} - \mathbf {T}^{-1} (\varvec{\Phi }_{lh}(\xi )) \left\{ \sum \limits _{j=1}^{n_{\Lambda }} L^j(\xi ) \mathbf {T}(\varvec{\Phi }^j_l) \right\} \right] \mathbf {v}^I \varvec{\Lambda }_r^T \\ &\quad+\, \delta ^{iJ} \varvec{\Lambda }_r \left[ \mathbf {I_3} - \mathbf {T}^{-1} (\varvec{\Phi }_{lh}(\xi )) \left\{ \sum \limits _{j=1}^{n_{\Lambda }} L^j(\xi ) \mathbf {T}(\varvec{\Phi }^j_l) \right\} \right] \mathbf {v}^J \varvec{\Lambda }_r^T. \end{array} \end{aligned}$$

(189)

In (189), no summation over double indices is applied. The vectors $\mathbf {v}^I$ and $\mathbf {v}^J$ are defined as

$$\begin{aligned} \begin{array}{ll} \mathbf {v}^I &{}= \frac{1}{2}\left( \mathbf {I}_3 + \frac{1}{\Phi ^{IJ}}\tan {\left( \frac{\Phi ^{IJ}}{4}\right) }\mathbf {S}(\varvec{\Phi }^{IJ})\right) , \\ \mathbf {v}^J &{}= \frac{1}{2}\left( \mathbf {I}_3 - \frac{1}{\Phi ^{IJ}}\tan {\left( \frac{\Phi ^{IJ}}{4}\right) }\mathbf {S}(\varvec{\Phi }^{IJ})\right) ,\\ \end{array} \end{aligned}$$

(190)

with the common abbreviation $\Phi ^{IJ} = ||\varvec{\Phi }^{IJ}||$. Moreover, the arc-length derivative $\widetilde{\mathbf {I}}^{i \prime }(\xi )$ reads:

$$\begin{aligned} \begin{array}{ll} \widetilde{\mathbf {I}}^{i \prime }(\xi )&=L^{i \prime }(\xi ) \varvec{\Lambda }_r \mathbf {T}^{-1}(\varvec{\Phi }_{lh}(\xi )) \mathbf {T}(\varvec{\Phi }^i_l)\varvec{\Lambda }_r^T\\ &\quad+\,L^i(\xi ) \varvec{\Lambda }_r \mathbf {T}^{-1}_{,s} (\varvec{\Phi }_{lh}(\xi )) \mathbf {T}(\varvec{\Phi }^i_l)\varvec{\Lambda }_r^T \\ &\quad-\,\delta ^{iI} \varvec{\Lambda }_r \Bigg ( \mathbf {T}^{-1}_{,s}(\varvec{\Phi }_{lh}(\xi )) \Bigg \{ \sum \limits _{j=1}^{n_{\Lambda }} L^j(\xi ) \mathbf {T}(\varvec{\Phi }^j_l) \Bigg \}\\ & \quad+\, \mathbf {T}^{-1}(\varvec{\Phi }_{lh}(\xi )) \Bigg \{ \sum \limits _{j=1}^{n_{\Lambda }} L^{j \prime } (\xi ) \mathbf {T}(\varvec{\Phi }^j_l) \Bigg \} \Bigg ) \mathbf {v}^I\varvec{\Lambda }_r^T \\ &\quad-\,\delta ^{iJ} \varvec{\Lambda }_r \Bigg ( \mathbf {T}^{-1}_{,s}(\varvec{\Phi }_{lh}(\xi )) \Bigg \{ \sum \limits _{j=1}^{n_{\Lambda }} L^j(\xi ) \mathbf {T}(\varvec{\Phi }^j_l) \Bigg \}\\ &\quad+\, \mathbf {T}^{-1}(\varvec{\Phi }_{lh}(\xi )) \Bigg \{ \sum \limits _{j=1}^{n_{\Lambda }} L^{j \prime } (\xi ) \mathbf {T}(\varvec{\Phi }^j_l) \Bigg \} \Bigg ) \mathbf {v}^J\varvec{\Lambda }_r^T. \end{array} \end{aligned}$$

(191)

Finally, the required arc-length derivative $\mathbf {T}^{-1}_{,s}(\varvec{\Phi }_{lh}(\xi ))$ is given by (see also [40, 82]):

$$ \begin{aligned} {\mathbf{T}}_{{,s}}^{{ - 1}} ({\mathbf{\Phi }}_{{lh}} (\xi )) & = {\mathbf{\Phi }}_{{lh}}^{T} {\mathbf{\Phi }}_{{lh}}^{\prime } \frac{{\Phi _{{lh}} \sin \Phi _{{lh}} - 2(1 - \cos \Phi _{{lh}} )}}{{\Phi _{{lh}}^{4} }}{\mathbf{S}}({\mathbf{\Phi }}_{{lh}} ) \\ & \quad + \frac{{1 - \cos \Phi _{{lh}} }}{{\Phi _{{lh}}^{2} }}{\mathbf{S}}({\mathbf{\Phi }}_{{lh}}^{\prime } ) \\ & \quad + \frac{1}{{\Phi _{{lh}}^{2} }}\left( {1 - \frac{{\sin \Phi _{{lh}} }}{{\Phi _{{lh}} }}} \right)\left( {{\mathbf{S}}({\mathbf{\Phi }}_{{lh}} ){\mathbf{S}}({\mathbf{\Phi }}_{{lh}}^{\prime } ) + {\mathbf{S}}({\mathbf{\Phi }}_{{lh}}^{\prime } ){\mathbf{S}}({\mathbf{\Phi }}_{{lh}} )} \right) \\ & \quad + {\mathbf{\Phi }}_{{lh}}^{T} {\mathbf{\Phi }}_{{lh}}^{\prime } \frac{{3\sin \Phi _{{lh}} - \Phi _{{lh}} (2 + \cos \Phi _{{lh}} )}}{{\Phi _{{lh}}^{5} }}{\mathbf{S}}({\mathbf{\Phi }}_{{lh}} ){\mathbf{S}}({\mathbf{\Phi }}_{{lh}} ). \\ \end{aligned} $$

(192)

Here, the abbreviations $\varvec{\Phi }_{lh} = \varvec{\Phi }_{lh}(\xi )$ and ${\Phi }_{lh} = ||\varvec{\Phi }_{lh}(\xi )||$ have been applied. The limit $\mathbf {T}^{-1}_{,s}(\varvec{\Phi }_{lh}(\xi )) \rightarrow 0.5 \mathbf {S}(\varvec{\Phi }_{lh}^{\prime }(\xi ))$ can be derived for small angles $\varvec{\Phi }_{lh}(\xi ) \rightarrow \mathbf {0}$ (see [83]).

Appendix B: Dirichlet Boundary Conditions and Joints

For many applications, the formulation of proper Dirichlet boundary conditions or joints between the nodes of different beam elements are of a high practical relevance. This appendix represents a brief summary, where the possibility of formulating some basic constraint conditions will be investigated for the SK-ROT and the SK-TAN element.

1.1 Appendix B.1: SK-ROT Element

Since the SK-ROT element simplifies the formulation of Dirichlet boundary conditions and kinematic constraints in many practically relevant cases, it will be considered first.

Dirichlet boundary conditions A simple support at element node a can be realized via

$$\begin{aligned} \hat{\mathbf {d}}^a=\hat{\mathbf {d}}_u^a=\hat{\mathbf {d}}_0^a \rightarrow \Delta \hat{\mathbf {d}}^a=\mathbf {0}. \end{aligned}$$

(193)

If a clamped end should be modeled, also the cross-section orientation has to be fixed, i.e.

$$\begin{aligned} \varvec{\Lambda }^a=\varvec{\Lambda }_u^a=\varvec{\Lambda }_0^a, \quad \text {and} \quad \hat{\varvec{\psi }}^a=\hat{\varvec{\psi }}_0^a \rightarrow \Delta \hat{\varvec{\theta }} ^a=\mathbf {0}. \end{aligned}$$

(194)

Thus, the modeling of Dirichlet boundary conditions for the employed translational and rotational degrees of freedom is similar to standard finite elements that are purely based on translational degrees of freedom. This procedure can also be extended to inhomogeneous conditions. However, the determination of $\Delta \hat{\varvec{\theta }} ^a$ requires special care in this case:

$$\begin{aligned} \begin{array}{ll} \hat{\mathbf {d}}^a&{}=\hat{\mathbf {d}}_u^a(t) \rightarrow \Delta \hat{\mathbf {d}}_{n+1}^a=\hat{\mathbf {d}}_{u,n+1}^a-\hat{\mathbf {d}}_{u,n}^a, \\ \varvec{\Lambda }^a&{}=\varvec{\Lambda }_u^a(t) \rightarrow \exp { ( \mathbf {S}( \Delta \hat{\varvec{\theta }}_{n+1}^a ) ) } =\varvec{\Lambda }_{u,n+1}^a \varvec{\Lambda }_{u,n}^{a T}. \end{array} \end{aligned}$$

(195)

The multiplicative procedure of the second line simplifies to the additive procedure according to the first line if the prescribed rotation is additive, which only holds for 2D rotations.

Connections and joints A simple (moment-free) joint between the two nodes a and b of two connected elements reads:

$$\begin{aligned} \hat{\mathbf {d}}^b=\hat{\mathbf {d}}^a, \quad \delta \hat{\mathbf {d}}^b=\delta \hat{\mathbf {d}}^a, \quad \Delta \hat{\mathbf {d}}^b=\Delta \hat{\mathbf {d}}^a. \end{aligned}$$

(196)

Thus, the degrees of freedom $\hat{\mathbf {d}}^b$ can be eliminated from the global system of equations in a standard manner by simply assembling the corresponding lines and columns of the global residual vector and stiffness matrix properly. A rigid joint between two elements prescribed at the nodes a and b additionally requires to suppress any relative rotation between the associated nodal triads. It is assumed that these nodal triads differ by some fixed relative rotation $\varvec{\Lambda }_{0}$:

$$ \begin{aligned} {\mathbf{\Lambda }}^{a} = {\mathbf{\Lambda }}^{b} {\mathbf{\Lambda }}_{0} {\text{ or }}\exp ({\mathbf{S}}(\Delta \widehat{\varvec{\theta }}^{a} )) & = \exp ({\mathbf{S}}(\Delta \widehat{\varvec{\theta }}^{b} )){\mathbf{\Lambda }}_{0} \\ \to {\mathbf{\Lambda }}_{0} & = {\mathbf{\Lambda }}^{{bT}} {\mathbf{\Lambda }}^{a} . \\ \end{aligned} $$

(197)

From (197), the following relations between the associated rotation increments can be derived:

$$\begin{aligned} \begin{array}{ll} \delta \varvec{\Lambda }^{a}=\delta \varvec{\Lambda }^{b}\varvec{\Lambda }_{0} &{}\rightarrow \mathbf {S}(\delta \hat{\varvec{\theta }}^a) \varvec{\Lambda }^{a}= \mathbf {S}(\delta \hat{\varvec{\theta }}^b) \varvec{\Lambda }^{b}\varvec{\Lambda }_{0} \\ &{}\rightarrow \delta \hat{\varvec{\theta }}^b = \delta \hat{\varvec{\theta }}^a \rightarrow \Delta \hat{\varvec{\theta }}^b = \Delta \hat{\varvec{\theta }}^a. \end{array} \end{aligned}$$

(198)

Consequently, also the rotational degrees of freedom $\hat{\varvec{\psi }}^b$ can be eliminated in a standard manner by simply assembling the corresponding lines and columns of the global residual vector and of the global stiffness matrix properly.

Remark

It is emphasized that a rigid joint according to (197) is formulated via right-translation of the rotation tensor $\varvec{\Lambda }_{0}$. This is mandatory since a rigid joint represents a fixed orientation difference between material quantities, i.e. a fixed relative rotation with respect to material axes. A left-translation via

$$\begin{aligned} \varvec{\Lambda }^{a} = \varvec{\Lambda }_{0}\varvec{\Lambda }^{b} \,\, \rightarrow \,\, \varvec{\Lambda }_{0} = \varvec{\Lambda }^{a}\varvec{\Lambda }^{bT} \,\, \rightarrow \,\, \delta \hat{\varvec{\theta }}^b = \varvec{\Lambda }_{0}^T \delta \hat{\varvec{\theta }}^a \ne \delta \hat{\varvec{\theta }}^a, \end{aligned}$$

(199)

i.e. a fixed relative rotation with respect to spatial axes, has a different physical meaning.

Remark

If additive increments $\Delta \hat{\varvec{\psi }}^a$ and $\Delta \hat{\varvec{\psi }}^b$ of the rotation vectors $\hat{\varvec{\psi }}^a$ and $\hat{\varvec{\psi }}^b$ instead of the multiplicative increments $\Delta \hat{\varvec{\theta }}^a$ and $\Delta \hat{\varvec{\theta }}^b$ were applied in the linearization process, Eq. (198) has to be replaced by:

$$\begin{aligned} \Delta \hat{\varvec{\theta }}^b = \Delta \hat{\varvec{\theta }}^a \,\,\,\rightarrow \,\,\, \Delta \hat{\varvec{\psi }}^b = \mathbf {T}(\hat{\varvec{\psi }}^b)\mathbf {T}^{-1}(\hat{\varvec{\psi }}^a)\Delta \hat{\varvec{\psi }}^a \ne \Delta \hat{\varvec{\psi }}^a. \end{aligned}$$

(200)

In this case, a direct elimination of the degrees of freedom $\hat{\varvec{\psi }}^b$ via a proper assembly of the global stiffness matrix is not possible. Instead, the corresponding columns have to be scaled with the matrix $\mathbf {T}(\hat{\varvec{\psi }}^b)\mathbf {T}^{-1}(\hat{\varvec{\psi }}^a)$.

Remark

Physically reasonable boundary conditions can be completely defined by the cross-section orientation and centroid position. For all considered types of boundary conditions, the degrees of freedom $\hat{t}^{a}$ and $\hat{t}^{b}$, which are a measure for the nodal axial force, are part of the FEM solution and must not be prescribed.

1.2 Appendix B.2: SK-TAN Element

The treatment of the translational degrees of freedom required for the subsequent boundary conditions is identical to the last section and will therefore be omitted here.

Dirichlet boundary conditions In order to model a clamped end with the SK-TAN element, the simplest case of a tangent vector that is parallel to a global base vector, e.g. $\hat{\mathbf {t}}^a \parallel \mathbf {e}_1$, is considered. Then, (193) has to be supplemented by

$$ \begin{aligned} \widehat{{\mathbf{t}}}^{{aT}} {\mathbf{e}}_{2} = \widehat{{\mathbf{t}}}^{{aT}} {\mathbf{e}}_{3} = 0 & \to \Delta \widehat{t}_{2}^{a} = \Delta \widehat{t}_{3}^{a} = 0, \\ \hat{\varphi }^{a} = \hat{\varphi }_{0}^{a} & \to \Delta \hat{\varphi }^{a} = 0. \\ \end{aligned} $$

(201)

Here, the representation $\hat{\mathbf {t}}^{a} = \hat{{t}}^{a}_i \mathbf {e}_i$ of the tangent in the global frame $\mathbf {e}_i$ has been exploited. In order to prescribe boundary conditions with arbitrary triad orientation, the tangent has to be expressed in the basis of the prescribed triad:

$$ \begin{aligned} {\mathbf{\Lambda }}^{a} = {\mathbf{\Lambda }}_{u}^{a} = {\mathbf{\Lambda }}_{0}^{a} ,\,\widehat{{\mathbf{t}}}^{a} = \hat{T}_{i}^{a} {\mathbf{g}}_{i}^{a} & \to \hat{T}_{2}^{a} = \hat{T}_{3}^{a} = \Delta \hat{T}_{2}^{a} = \Delta \hat{T}_{3}^{a} = 0, \\ \hat{\varphi }^{a} = \hat{\varphi }_{0}^{a} & \to \Delta \hat{\varphi }^{a} = 0. \\ \end{aligned} $$

(202)

Consequently, in this case, the equations of the linearized residual vector that are associated with the degrees of freedom $\hat{\mathbf {t}}^{a}$ have to be transformed by the rotation tensor $\varvec{\Lambda }_0^a$ and the Dirichlet conditions have to be formulated in this rotated coordinate system. Again, the first component $\hat{T}_1$ of the tangent vector, when expressed in the material frame, represents its magnitude and must not be prescribed. If the Dirichlet conditions are time-dependent, the prescribed evolution of the relative angle has to be adapted, since now the intermediate frame $\mathbf {\Lambda }_{M_{\hat{\varphi }}}^a$ might change in time:

$$ \begin{aligned} \exp ({\mathbf{S}}[\hat{\varphi }_{{n + 1}}^{a} {\mathbf{g}}_{{1,n + 1}}^{a} ]) & = {\mathbf{\Lambda }}_{{n + 1}}^{a} {\mathbf{\Lambda }}_{{M_{{\hat{\varphi }}} ,n + 1}}^{{aT}} \\ {\text{with }}{\mathbf{\Lambda }}_{{M_{{\hat{\varphi }}} ,n + 1}}^{a} & = {\text{sr}}({\mathbf{\Lambda }}_{{M_{{\hat{\varphi }}} ,n}}^{a} ,{\mathbf{g}}_{{1,n + 1}}^{a} ). \\ \end{aligned} $$

(203)

Thus, the required value $\hat{\varphi }^a_{n+1}$ has to be determined based on the prescribed current triad $\mathbf {\Lambda }^a_{n+1}$ and the intermediate triad $\mathbf {\Lambda }_{M_{\hat{\varphi }},n}^{a}$ of the last step (see Sect. 6.2.2). The remaining conditions remain unchanged as compared to (202).

Connections and joints Based on (197), (198) and (25), the relations between $(\hat{\mathbf {t}}^a,\hat{\varphi }^a)$ and $(\hat{\mathbf {t}}^b,\hat{\varphi }^b)$ can be stated:

$$\begin{aligned} \begin{array}{ll} \delta \hat{\mathbf {t}}^b&{}=-t^b \mathbf {S}(\mathbf {g}_1^b) \underbrace{\delta \varvec{\theta }^b}_{\dot{=}\delta \varvec{\theta }^a} + \mathbf {g}_{1}^b \delta t^b\\ &{}= -t^b \mathbf {S}(\mathbf {g}_1^b) \left( \frac{1}{t^a} \mathbf {S}(\mathbf {g}_1^a) \delta \hat{\mathbf {t}}^a + \mathbf {g}_{1}^a \delta \hat{\Theta }_1^a \right) + \mathbf {g}_{1}^b \delta \hat{t}^b, \\ \delta \hat{\Theta }_1^b&{}=\mathbf {g}_{1}^{bT} \underbrace{\delta \varvec{\theta }^b}_{\dot{=}\delta \varvec{\theta }^a}= \mathbf {g}_{1}^{bT} \left( \frac{1}{t^a} \mathbf {S}(\mathbf {g}_1^a) \delta \hat{\mathbf {t}}^a + \mathbf {g}_{1}^a \delta \hat{\Theta }_1^a \right) . \end{array} \end{aligned}$$

(204)

Combining these two relations eventually yields the following total transformation matrix:

$$ \begin{aligned} \left( {\begin{array}{*{20}c} {\delta \widehat{{\mathbf{t}}}^{b} } \\ {\delta \hat{\Theta }_{1}^{b} } \\ \end{array} } \right) & = {\mathbf{T}}_{{RC}} \left( {\begin{array}{*{20}c} {\delta \widehat{{\mathbf{t}}}^{a} } \\ {\delta \hat{\Theta }_{1}^{a} } \\ {\delta \hat{t}^{b} } \\ \end{array} } \right), \\ {\mathbf{T}}_{{RC}} : & = \left( {\begin{array}{*{20}c} { - t^{b} {\mathbf{S}}({\mathbf{g}}_{1}^{b} )\frac{1}{{t^{a} }}{\mathbf{S}}({\mathbf{g}}_{1}^{a} )} & { - t^{b} {\mathbf{S}}({\mathbf{g}}_{1}^{b} ){\mathbf{g}}_{1}^{a} } & {{\mathbf{g}}_{1}^{b} } \\ {{\mathbf{g}}_{1}^{{bT}} \frac{1}{{t^{a} }}{\mathbf{S}}({\mathbf{g}}_{1}^{a} )} & {{\mathbf{g}}_{1}^{{bT}} {\mathbf{g}}_{1}^{a} } & 0 \\ \end{array} } \right). \\ \end{aligned} $$

(205)

A similar relation can also be formulated for the iterative increments. Since the multiplicative rotation increment components $\Delta \Theta _1 = \mathbf {T}_{\Theta _{M1} \mathbf {t}} \Delta \mathbf {t} + \Delta \varphi$ [see (20)] have to be expressed by additive increments $\Delta \mathbf {t}$ and $\Delta \varphi$ for the chosen linearization, an additional transformation is required compared to (205):

$$ \begin{aligned} \left( {\begin{array}{*{20}c} {\Delta \widehat{{\mathbf{t}}}^{b} } \\ {\Delta \hat{\varphi }^{b} } \\ \end{array} } \right) & = \widetilde{{\mathbf{T}}}_{{RC1}} {\mathbf{T}}_{{RC}} \widetilde{{\mathbf{T}}}_{{RC2}} \left( {\begin{array}{*{20}c} {\Delta \widehat{{\mathbf{t}}}^{a} } \\ {\Delta \hat{\varphi }_{1}^{a} } \\ {\Delta \hat{t}^{b} } \\ \end{array} } \right), \\ \widetilde{{\mathbf{T}}}_{{RC1}} : & = \left( {\begin{array}{*{20}c} {{\mathbf{I}}_{3} } & {\mathbf{0}} \\ { - {\mathbf{T}}_{{\Theta _{{M1}} {\mathbf{t}}}} } & 1 \\ \end{array} } \right),\,\widetilde{{\mathbf{T}}}_{{RC2}} : = \left( {\begin{array}{*{20}c} {{\mathbf{I}}_{3} } & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{T}}_{{\Theta _{{M1}} {\mathbf{t}}}} } & 1 & 0 \\ {{\mathbf{0}}^{T} } & 0 & 1 \\ \end{array} } \right). \\ \end{aligned} $$

(206)

Equations (205) and (206) allow to transform the corresponding lines and columns of the global residual vector and of the global tangent stiffness matrix properly and to eliminate the degrees of freedom $(\hat{\mathbf {t}}^b,\hat{\varphi }^b)$ from the global system of equations. Again, the magnitude of the tangent vector $\hat{t}^b$ is not influenced by the rigid joint and enters the system of equations as new degree of freedom. While in the last section, the motion of the rigid joint was completely determined by the set $(\hat{\mathbf {d}}^a, \hat{\varvec{\psi }}^a,\hat{t}^a,\hat{t}^b)$, in this section the alternative set $(\hat{\mathbf {d}}^a,\hat{\mathbf {t}}^a,\hat{\varphi }^a,\hat{t}^b)$ is employed.

All in all, it can be concluded that the realization of clamped ends with arbitrary orientation or of rigid joints between beams is simpler for the SK-ROT formulation based on nodal rotation vectors. While for these elements such conditions can directly be formulated in the global coordinate system, the tangent vector-based SK-TAN formulation requires an additional transformation of the corresponding lines and columns of the global residual vector and stiffness matrix. In Sect. 11, some properties of the tangent vector-based variant will become apparent which make this type of formulation favorable for many applications. If certain element nodes require Dirichlet conditions of the type considered here, it is still possible to apply a hybrid approach, and to replace the nodal tangents by nodal rotation vectors as primary variables only at the specific nodes where such conditions are required. All the results derived so far apply in a similar manner to the WK-TAN and WK-ROT elements.

Appendix C: Linearization of SK-TAN Element

Before deriving the linearization of the SK-TAN element, some former definitions are repeated:

$$\begin{aligned} \begin{array}{ll} \mathbf {t} :=&{}\mathbf {r}^{\prime }, \quad \mathbf {g}_1 := \frac{\mathbf {r}^{\prime }}{||\mathbf {r}^{\prime }||}, \quad \,\,\,\, \widetilde{\mathbf {t}} := \frac{\mathbf {r}^{\prime }}{||\mathbf {r}^{\prime }||^2}, \\ \mathbf {g}_1^{\prime } =&{} \frac{\mathbf {\mathbf {r}^{\prime \prime }}}{||\mathbf {r}^{\prime }||} - \frac{(\mathbf {r}^{\prime T}\mathbf {r}^{\prime \prime })\mathbf {r}^{\prime }}{||\mathbf {r}^{\prime }||^3}, \quad \widetilde{\mathbf {t}}^{\prime } = \frac{\mathbf {r}^{\prime \prime }}{||\mathbf {r}^{\prime }||^2} - \frac{2(\mathbf {r}^{\prime T}\mathbf {r}^{\prime \prime })\mathbf {r}^{\prime }}{||\mathbf {r}^{\prime }||^4}. \end{array} \end{aligned}$$

(207)

These quantities will be required for later derivations. Linearization of (207) yields:

$$ \begin{aligned} \Delta {\mathbf{g}}_{1} & = \frac{1}{{||{\mathbf{r}}^{\prime } ||}}\left( {{\mathbf{I}}_{3} - {\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} } \right)\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}}, \\ \Delta \widetilde{{\mathbf{t}}} & = \frac{1}{{||{\mathbf{r}}^{\prime } ||^{2} }}\left( {{\mathbf{I}}_{3} - 2{\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} } \right)\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}}, \\ \Delta {\mathbf{g}}_{1}^{\prime } & = - \frac{{({\mathbf{r}}^{{\prime T}} {\mathbf{r}}^{{\prime \prime }} )}}{{||{\mathbf{r}}^{\prime } ||^{3} }}\left( {{\mathbf{I}}_{3} - {\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} } \right)\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}} \\ & \quad - \frac{1}{{||{\mathbf{r}}^{\prime } ||}}\left( {{\mathbf{g}}_{1}^{\prime } \otimes {\mathbf{g}}_{1}^{T} + {\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{{\prime T}} } \right)\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}} \\ & \quad + \frac{1}{{||{\mathbf{r}}^{\prime } ||}}\left( {{\mathbf{I}}_{3} - {\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} } \right)\mathsf{H}^{{\prime \prime }} \Delta \widehat{\mathsf{d}}, \\ \Delta \widetilde{{\mathbf{t}}}^{\prime } & = - \frac{{2({\mathbf{r}}^{{\prime T}} {\mathbf{r}}^{{\prime \prime }} )}}{{||{\mathbf{r}}^{\prime } ||^{4} }}\left( {{\mathbf{I}}_{3} - 2{\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} } \right)\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}} \\ & \quad - \frac{2}{{||{\mathbf{r}}^{\prime } ||^{2} }}\left( {{\mathbf{g}}_{1}^{\prime } \otimes {\mathbf{g}}_{1}^{T} + {\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{{\prime T}} } \right)\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}} \\ & \quad + \frac{1}{{||{\mathbf{r}}^{\prime } ||^{2} }}\left( {{\mathbf{I}}_{3} - 2{\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} } \right)\mathsf{H}^{{\prime \prime }} \Delta \widehat{\mathsf{d}}. \\ \end{aligned} $$

(208)

In the following, the linearization of the SK-TAN element based on the residual vector (165) will be derived. The linearization of (165) obeys the following general form:

$$ \begin{aligned} \Delta \mathsf{r}_{{\widehat{{\mathbf{d}}}}} & = \int\limits_{{ - 1}}^{1} {\left( {\Delta \mathsf{v}_{{\theta _{ \bot } }}^{\prime } {\mathbf{m}} + \mathsf{v}_{{\theta _{ \bot } }}^{\prime } \Delta {\mathbf{m}} + \Delta \overline{\mathsf{v}} _{\epsilon{}} \bar{F}_{1} + \overline{\mathsf{v}} _{\epsilon{}} \Delta \bar{F}_{1} } \right)Jd\xi } \\ & \quad - \int\limits_{{ - 1}}^{1} {\left( {\mathsf{H}^{T} \Delta {\mathbf{f}}_{\rho } + \Delta \mathsf{v}_{{\theta _{ \bot } }} \widetilde{{\mathbf{m}}}_{\rho } + \mathsf{v}_{{\theta _{ \bot } }} \Delta {\mathbf{m}}_{\rho } } \right)Jd\xi - [\Delta \mathsf{v}_{{\theta _{ \bot } }} {\mathbf{m}}_{\sigma } ]_{{\varGamma_{\sigma } }} } \\ \Delta \mathsf{r}_{{{\mathbf{\hat{\Theta }}}_{1} }} & = \int\limits_{{ - 1}}^{1} {(\Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }}^{\prime } {\mathbf{m}} + \mathsf{v}_{{\theta _{{\parallel \Theta }} }}^{\prime } \Delta {\mathbf{m}} - \Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }} \widetilde{{\mathbf{m}}}_{\rho } } \\ & \quad - \mathsf{v}_{{\theta _{{\parallel \Theta }} }} \Delta {\mathbf{m}}_{\rho } )Jd\xi - [\Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }} {\mathbf{m}}_{\sigma } ]_{{\varGamma_{\sigma } }} . \\ \end{aligned} $$

(209)

In order to identify the element stiffness matrix ${{\mathbf {\mathsf{{k}}}}}_{SK-TAN},$ (209) has to be brought in the form

$$\begin{aligned} \begin{array}{ll} \Delta {{\mathbf {\mathsf{{r}}}}}_{SK-TAN}=:{{\mathbf {\mathsf{{k}}}}}_{SK-TAN} \Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{TAN}. \end{array} \end{aligned}$$

(210)

The vector $\Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{TAN}$ has already been defined in Sect. 9.1. The moment-related terms yield:

$$ \begin{aligned} \Delta \mathsf{v}_{{\theta _{ \bot } }}^{\prime } {\mathbf{m}} & = \mathsf{H}^{{\prime \prime T}} {\mathbf{S}}({\mathbf{m}})\Delta \widetilde{{\mathbf{t}}} + \mathsf{H}^{{\prime T}} {\mathbf{S}}({\mathbf{m}})\Delta \widetilde{{\mathbf{t}}}^{\prime } , \\ \Delta {\mathbf{m}} & = - {\mathbf{S}}({\mathbf{m}})\Delta \varvec{\theta } + {\mathbf{c}}_{m} \Delta \varvec{\theta }^{\prime } , \\ \Delta \mathsf{v}_{{\theta _{ \bot } }} \widetilde{{\mathbf{m}}}_{\rho } & = \mathsf{H}^{{\prime T}} {\mathbf{S}}(\widetilde{{\mathbf{m}}}_{\rho } )\Delta \widetilde{{\mathbf{t}}}, \\ \Delta \mathsf{v}_{{\theta _{ \bot } }} {\mathbf{m}}_{\sigma } & = \mathsf{H}^{{\prime T}} {\mathbf{S}}({\mathbf{m}}_{\sigma } )\Delta \widetilde{{\mathbf{t}}}, \\ \Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }} \widetilde{{\mathbf{m}}}_{\rho } & = (\mathsf{L}_{\parallel }^{T} \otimes \widetilde{{\mathbf{m}}}_{\rho }^{T} )\Delta {\mathbf{g}}_{1} , \\ \Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }} {\mathbf{m}}_{\sigma } & = (\mathsf{L}_{\parallel }^{T} \otimes {\mathbf{m}}_{\sigma }^{T} )\Delta {\mathbf{g}}_{1} , \\ \Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }}^{\prime } {\mathbf{m}} & = (\mathsf{L}_{\parallel }^{{\prime T}} \otimes {\mathbf{m}}^{T} )\Delta {\mathbf{g}}_{1} + (\mathsf{L}_{\parallel }^{T} \otimes {\mathbf{m}}^{T} )\Delta {\mathbf{g}}_{1}^{\prime } . \\ \end{aligned} $$

(211)

Here, many of the relations already derived in Sect. 6.2.4 could be re-used. The field of multiplicative rotation vector increments $\Delta \varvec{\theta }$ follows directly from Eq. (122):

$$ \begin{aligned} \Delta \varvec{\theta } & = \mathsf{v}_{{\theta _{{\parallel \Theta }} }}^{T} \Delta \widehat{\varvec{\Theta }}_{1} + (\mathsf{v}_{{\theta _{ \bot } }}^{T} + \mathsf{v}_{{\theta _{{\parallel d}} }}^{T} )\Delta \widehat{\mathsf{d}}, \\ \mathsf{v}_{{\theta _{{\parallel \Theta }} }} & = \mathsf{L}_{\parallel }^{T} \otimes {\mathbf{g}}_{1}^{T} ,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \mathsf{v}_{{\theta _{ \bot } }} = - \mathsf{H}^{{\prime T}} {\mathbf{S}}(\widetilde{{\mathbf{t}}}), \\ \mathsf{v}_{{\theta _{{\parallel d}} }} & = \left( {\sum\limits_{{i = 1}}^{{n_{\Lambda } }} {L^{i} } {\mkern 1mu} \mathsf{v}_{1} (\xi _{i} ) - \mathsf{v}_{1} (\xi )} \right) \otimes {\mathbf{g}}_{1}^{T} , \\ \mathsf{v}_{1} (\xi ) & = \frac{{\mathsf{H}^{{\prime T}} (\xi )({\mathbf{g}}_{1}^{I} \times \widetilde{{\mathbf{t}}}(\xi )) - \mathsf{H}^{{\prime T}} (\xi _{I} )({\mathbf{g}}_{1} (\xi ) \times \widetilde{{\mathbf{t}}}^{I} )}}{{1 + {\mathbf{g}}_{1}^{{{\kern 1pt} T}} (\xi ){\mathbf{g}}_{1}^{I} }}. \\ \end{aligned} $$

(212)

In a similar manner, the associated arc-length derivative $\Delta \varvec{\theta }^{\prime }$ follows from Eq. (123):

$$ \begin{aligned} \Delta \varvec{\theta}^{\prime} & = \mathsf{v}_{{\theta _{{\parallel \Theta }} }}^{{\prime T}} \Delta \widehat{{\mathbf{\Theta }}}_{1} + (\mathsf{v}_{{\theta _{ \bot } }}^{{\prime T}} + \mathsf{v}_{{\theta _{{\parallel d}} }}^{{\prime T}} )\Delta \widehat{\mathsf{d}}, \\ \mathsf{v}_{{\theta _{{\parallel \Theta }} }}^{\prime } & = \mathsf{L}_{\parallel }^{{\prime T}} \otimes {\mathbf{g}}_{1}^{T} + \mathsf{L}_{\parallel }^{T} \otimes {\mathbf{g}}_{1}^{{\prime T}} , \\ \mathsf{v}_{{\theta _{ \bot } }}^{\prime } & = - \mathsf{H}^{{\prime \prime T}} {\mathbf{S}}(\widetilde{{\mathbf{t}}}) - \mathsf{H}^{{\prime T}} {\mathbf{S}}(\widetilde{{{\mathbf{t^{\prime}}}}}), \\ \mathsf{v}_{{\theta _{{\parallel d}} }}^{\prime } & = \left( {\sum\limits_{{i = 1}}^{{n_{\Lambda } }} {L^{{i\prime }} } \mathsf{v}_{1} (\xi _{i} ) - \mathsf{v}_{1}^{\prime } (\xi )} \right) \otimes {\mathbf{g}}_{1}^{T} \\ & \quad + \left( {\sum\limits_{{i = 1}}^{{n_{\Lambda } }} {L^{i} } \mathsf{v}_{1} (\xi _{i} ) - \mathsf{v}_{1} (\xi )} \right) \otimes {\mathbf{g}}_{1}^{{\prime T}} , \\ \mathsf{v}_{1}^{\prime } (\xi ) & = \frac{{\mathsf{H}^{{\prime T}} (\xi )\left( {{\mathbf{g}}_{1}^{I} \times \widetilde{{{\mathbf{t^{\prime}}}}}(\xi )} \right) + \mathsf{H}^{{\prime \prime T}} (\xi )\left( {{\mathbf{g}}_{1}^{I} \times \widetilde{{\mathbf{t}}}(\xi )} \right)}}{{1 + {\mathbf{g}}_{1}^{T} (\xi ){\mathbf{g}}_{1}^{I} }} \\ & \quad - \frac{{\mathsf{H}^{{\prime T}} (\xi _{I} )\left( {{\mathbf{g}}_{1}^{\prime } (\xi ) \times \widetilde{{\mathbf{t}}}^{I} } \right) + \left( {{\mathbf{g}}_{1}^{{\prime T}} (\xi ){\mathbf{g}}_{1}^{I} } \right){\mathbf{v}}_{1} (\xi )}}{{1 + {\mathbf{g}}_{1}^{T} (\xi ){\mathbf{g}}_{1}^{I} }}, \\ \end{aligned} $$

(213)

with $\widetilde{\mathbf {t}}^{\prime } = \mathbf {r}^{\prime \prime }/||\mathbf {r}^{\prime }||^2 - 2(\mathbf {r}^{\prime T}\mathbf {r}^{\prime \prime })\mathbf {r}^{\prime }/||\mathbf {r}^{\prime }||^4$. The remaining linearizations in (211) have already been derived in (208). In contrast to the spin vector field $\delta \varvec{\theta }$, the increment field $\Delta \varvec{\theta }$ has to be expressed via additive increments $\Delta \hat{\varphi }^i$. The required relation is given by (129) and repeated here:

$$ \begin{aligned} \Delta \widehat{\varvec{\Theta }}_{1} & = (\hat{\Theta }_{1}^{1} ,\hat{\Theta }_{1}^{2} ,\hat{\Theta }_{1}^{3} )^{T} , \\ \Delta \hat{\Theta }_{1}^{i} & = - \frac{{\overline{{\mathbf{g}}} _{1}^{{iT}} {\mathbf{S}}({\mathbf{g}}_{1}^{i} )}}{{1 + {\mathbf{g}}_{1}^{{iT}} \overline{{\mathbf{g}}} _{1}^{i} }}\frac{{\mathsf{H}^{\prime } (\xi ^{i} )\Delta \widehat{\mathsf{d}}}}{{||{\mathbf{t}}^{i} ||}} + \Delta \hat{\varphi }^{i} . \\ \end{aligned} $$

(214)

The linearization of the element residual terms associated with axial tension results in:

$$ \begin{aligned} \Delta \bar{F}_{1} & = EA\Delta {\bar{\epsilon}} = EA\overline{\mathsf{v}} _{\epsilon{}}^{T} \Delta \widehat{\mathsf{d}}, \\ \Delta \overline{\mathsf{v}} _{\epsilon{}} & = \sum\limits_{{i = 1}}^{3} {L^{i} } (\xi )\Delta \mathsf{v}_{\epsilon{}} (\xi ^{i} ), \\ \Delta \mathsf{v}_{\epsilon{}} & = \frac{{\mathsf{H}^{{\prime T}} }}{{||{\mathbf{r}}^{\prime } ||}}({\mathbf{I}}_{3} - {\mathbf{g}}_{1} \otimes {\mathbf{g}}_{1}^{T} )\mathsf{H}^{\prime } \Delta \widehat{\mathsf{d}}. \\ \end{aligned} $$

(215)

Based (93), the linearization of the inertia forces reads:

$$\begin{aligned} \begin{array}{ll} -{{\mathbf {\mathsf{{H}}}}}^T \Delta \mathbf {f}_{\rho } = \rho A c_{\ddot{\mathbf {r}}1} {{\mathbf {\mathsf{{H}}}}}^T {{\mathbf {\mathsf{{H}}}}} \Delta \hat{{{\mathbf {\mathsf{{d}}}}}}, \quad c_{\ddot{\mathbf {r}}1} = \frac{1 - \alpha _m}{(1 - \alpha _f)\beta \Delta t^2}. \end{array} \end{aligned}$$

(216)

The time integration factor $c_{\ddot{\mathbf {r}}1}$ of the modified generalized-$\alpha$ scheme according to Sect. 5 slightly differs from the corresponding factor of the standard generalized-$\alpha$ scheme. The linearization of the inertia moments yields:

$$ \begin{aligned} - \Delta {\mathbf{m}}_{\rho } & = {\mathbf{S}}({\mathbf{m}}_{\rho } )\Delta \varvec{\theta } \\ & \quad + {\mathbf{\Lambda }}[c_{{\mathbf{W}}} \{ {\mathbf{S}}({\mathbf{W}}){\mathbf{C}}_{\rho } - {\mathbf{S}}({\mathbf{C}}_{\rho } {\mathbf{W}})\} + c_{{\mathbf{A}}} {\mathbf{C}}_{\rho } ]\Delta \widetilde{{\mathbf{\Theta }}}_{{n + 1}} , \\ \Delta \widetilde{{\mathbf{\Theta }}}_{{n + 1}} & = \Lambda _{n}^{T} \Delta \widetilde{\varvec{\theta }}_{{n + 1}} = \Lambda _{n}^{T} {\mathbf{T}}(\widetilde{\varvec{\theta }}_{{n + 1}} )\Delta \varvec{\theta }, \\ c_{{\mathbf{A}}} & = \frac{{1 - \alpha _{m} }}{{(1 - \alpha _{f} )\beta \Delta t^{2} }},\quad c_{{\mathbf{W}}} = \frac{\gamma }{{\beta \Delta t}}. \\ \end{aligned} $$

(217)

For clarity, the indices $n + 1$ and n of the current and previous time step have explicitly been noted for some of the quantities occurring in (217). All the other quantities are evaluated at $t_{n + 1}.$ As already introduced in Sect. 5, the fields $\widetilde{\varvec{\Theta }}_{n+1}$ and $\widetilde{\varvec{\theta }}_{n+1}$ are the material and spatial multiplicative rotation increments relating the current configuration and the converged configuration of the previous time step $t_n.$ The two vectors are related by the transformation

$$ \begin{aligned} \widetilde{{\mathbf{\Theta }}}_{{n + 1}} & = {\mathbf{\Lambda }}_{{n + 1}}^{T} \widetilde{\varvec{\theta }}_{{n + 1}} {\mathbf{ = \Lambda }}_{n}^{T} \widetilde{\varvec{\theta }}_{{n + 1}} \\ \to \Delta \widetilde{{\mathbf{\Theta }}}_{{n + 1}} & = {\mathbf{\Lambda }}_{n}^{T} \Delta \widetilde{\varvec{\theta }}_{{n + 1}} . \\ \end{aligned} $$

(218)

The second step in (218) is valid since $\widetilde{\varvec{\theta }}_{n+1}$ is an eigenvector with eigenvalue one of the rotation tensor $\varvec{\Lambda }_{n+1}\varvec{\Lambda }^T_{n}$ between the configurations n and $n + 1,$ thus $\varvec{\Lambda }_{n+1}\varvec{\Lambda }^T_{n}\widetilde{\varvec{\theta }}_{n+1} = \widetilde{\varvec{\theta }}_{n+1}.$ Furthermore, $\Delta \widetilde{\varvec{\Theta }}_{n+1}$ and $\Delta \widetilde{\varvec{\theta }}_{n+1}$ represent the fields of additive increments of $\widetilde{\varvec{\Theta }}_{n+1}$ and $\widetilde{\varvec{\theta }}_{n+1}$ between two successive Newton iterations, whereas $\Delta \varvec{\theta }$ as given by (212) represents the field of multiplicative rotation increments between two successive Newton iterations.

Appendix D: Linearization of WK-TAN Element

The residual vector of the WK-TAN element is given in Eq. (184). The linearization of (184) has the general form:

$$ \begin{aligned} \Delta \mathsf{r}_{{\widehat{{\mathbf{d}}}}} & = \int\limits_{{ - 1}}^{1} {\left( {\Delta \overline{\mathsf{v}} _{{\theta _{ \bot } }}^{\prime } {\mathbf{m}} + \overline{\mathsf{v}} _{{\theta _{ \bot } }}^{\prime } \Delta {\mathbf{m}} + \Delta \overline{\mathsf{v}} _{\epsilon{}} \bar{F}_{1} + \overline{\mathsf{v}} _{\epsilon{}} \Delta \bar{F}_{1} } \right)Jd\xi } \\ & \quad - \int\limits_{{ - 1}}^{1} {\left( {\mathsf{H}^{T} \Delta {\mathbf{f}}_{\rho } + \Delta \overline{\mathsf{v}} _{{\theta _{ \bot } }} \widetilde{{\mathbf{m}}}_{\rho } + \overline{\mathsf{v}} _{{\theta _{ \bot } }} \Delta {\mathbf{m}}_{\rho } } \right)Jd\xi } - [\Delta \overline{\mathsf{v}} _{{\theta _{ \bot } }} {\mathbf{m}}_{\sigma } ]_{{\varGamma_{\sigma } }} \\ \Delta \mathsf{r}_{{{\mathbf{\hat{\Theta }}}_{1} }} & = \int\limits_{{ - 1}}^{1} {(\Delta \overline{\mathsf{v}} _{{\theta _{{\parallel \Theta }} }}^{\prime } {\mathbf{m}} + \overline{\mathsf{v}} _{{\theta _{{\parallel \Theta }} }}^{\prime } \Delta {\mathbf{m}} - \Delta \overline{\mathsf{v}} _{{\theta _{{\parallel \Theta }} }} \widetilde{{\mathbf{m}}}_{\rho } } \\ & \quad - \overline{\mathsf{v}} _{{\theta _{{\parallel \Theta }} }} \Delta {\mathbf{m}}_{\rho } )Jd\xi - [\Delta \overline{\mathsf{v}} _{{\theta _{{\parallel \Theta }} }} {\mathbf{m}}_{\sigma } ]_{{\varGamma_{\sigma } }} . \\ \end{aligned} $$

(219)

In order to identify the element stiffness matrix ${{\mathbf {\mathsf{{k}}}}}_{WK-TAN},$ (219) has to be brought in the form

$$\begin{aligned} \begin{array}{ll} \Delta {{\mathbf {\mathsf{{r}}}}}_{WK-TAN}=:{{\mathbf {\mathsf{{k}}}}}_{WK-TAN} \Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{TAN}. \end{array} \end{aligned}$$

(220)

The linearization of vectors of the form $\bar{{{\mathbf {\mathsf{{v}}}}}}_{...}$ and $\bar{{{\mathbf {\mathsf{{v}}}}}}_{...}^{\prime }$ as originally defined in (184) yields:

$$ \begin{aligned} \Delta \overline{\mathsf{v}} _{{\theta _{ \bot } }} & = - \sum\limits_{{i = 1}}^{3} {L^{i} } (\xi )\Delta \mathsf{v}_{{\theta _{ \bot } }} (\xi ^{i} ), \\ \Delta \overline{\mathsf{v}} _{\epsilon} & = \sum\limits_{{i = 1}}^{3} {L^{i} } (\xi )\Delta \mathsf{v}_{\epsilon} (\xi ^{i} ), \\ \Delta \overline{\mathsf{v}} _{{\theta _{{\parallel \Theta }} }} & = \sum\limits_{{i = 1}}^{3} {L^{i} } (\xi )\Delta \mathsf{v}_{{\theta _{{\parallel \Theta }} }} (\xi ^{i} ), \\ \end{aligned} $$

(221)

with $\Delta \bar{{{\mathbf {\mathsf{{v}}}}}}^{\prime }_{...} = \sum _{i=1}^{3} L^i{,\xi }(\xi )/J(\xi ) \Delta {{\mathbf {\mathsf{{v}}}}}_{...} (\xi ^i)$. The linearization of the vectors ${{\mathbf {\mathsf{{v}}}}}_{...}$ and ${{\mathbf {\mathsf{{v}}}}}_{...}^{\prime }$ has already been stated in the last section. Also the linearization of the moment stress resultant has the same form as in the last section:

$$\begin{aligned} \begin{array}{ll} \Delta \mathbf {m}&= -\mathbf {S}(\mathbf {m}) \Delta \varvec{\theta } + \mathbf {c}_m \Delta \varvec{\theta }^{\prime }. \end{array} \end{aligned}$$

(222)

However, the fields $\Delta \varvec{\theta }$ and $\Delta \varvec{\theta }^{\prime }$ originally defined in Sect. 6.2.3 are this time given by

$$\begin{aligned} \begin{array}{ll} \Delta \varvec{\theta } = \sum\limits _{i=1}^{3} \widetilde{\mathbf {I}}^i(\xi ) \Delta {\varvec{\theta }}(\xi ^i), \,\,\,\, \Delta \varvec{\theta }^{\prime } = \sum\limits _{i=1}^{3} \frac{1}{J(\xi )}\widetilde{\mathbf {I}}_{,\xi }^i(\xi ) \Delta {\varvec{\theta }}(\xi ^i). \end{array} \end{aligned}$$

(223)

Due to the Kirchhoff constraint, the nodal increments $\Delta {\varvec{\theta }}(\xi ^i)$ can be expressed according to:

$$ \begin{aligned} \Delta \varvec{\theta }(\xi ^{i} ) & = \Delta \hat{\Theta }_{1}^{i} {\mathbf{g}}_{1} (\xi ^{i} ) + \mathsf{v}_{{\theta _{ \bot } }}^{T} (\xi ^{i} )\Delta \widehat{\mathsf{d}}, \\ \Delta \hat{\Theta }_{1}^{i} & = - \frac{{\overline{{\mathbf{g}}} _{1}^{{iT}} {\mathbf{S}}({\mathbf{g}}_{1}^{i} )}}{{1 + {\mathbf{g}}_{1}^{{iT}} \overline{{\mathbf{g}}} _{1}^{i} }}\frac{{\mathsf{H}^{\prime } (\xi ^{i} )\Delta \widehat{\mathsf{d}}}}{{||{\mathbf{t}}^{i} ||}} + \Delta \hat{\varphi }^{i} . \\ \end{aligned} $$

(224)

The linearization of the inertia forces and moments is identical to the corresponding results of the last section given in (216) and (217). However, for the WK-TAN element, the rotation increment field $\Delta \varvec{\theta }$ is given by Eq. (223).

Appendix E: Linearization of SK/WK-ROT Elements

The nodal primary variable variations of the SK/WK-TAN and the SK/WK-ROT elements read:

$$ \begin{aligned} \delta \widehat{\mathsf{x}}_{{TAN}} : & = (\delta \widehat{{\mathbf{d}}}^{{1T}} ,\delta \widehat{{\mathbf{t}}}^{{1T}} ,\delta \hat{\Theta }_{1}^{1} ,\delta \widehat{{\mathbf{d}}}^{{2T}} ,\delta \widehat{{\mathbf{t}}}^{{2T}} ,\delta \hat{\Theta }_{1}^{2} ,\delta \hat{\Theta }_{1}^{3} )^{T} , \\ \delta \widehat{\mathsf{x}}_{{ROT}} : & = (\delta \widehat{{\mathbf{d}}}^{{1T}} ,\delta \widehat{\varvec{\theta }}^{{1T}} ,\delta \hat{t}^{1} ,\delta \widehat{{\mathbf{d}}}^{{2T}} ,\delta \widehat{\varvec{\theta }}^{{2T}} ,\delta \hat{t}_{1} ,\delta \hat{\Theta }_{1}^{3} )^{T} . \\ \end{aligned} $$

(225)

In a similar manner, the set of iterative nodal primary variable increments have been defined as:

$$ \begin{aligned} \Delta \widehat{\mathsf{x}}_{{TAN}}: & = (\Delta \widehat{{\mathbf{d}}}^{{1T}} ,\Delta \widehat{{\mathbf{t}}}^{{1T}} ,\Delta \hat{\varphi }^{1} ,\Delta \widehat{{\mathbf{d}}}^{{2T}} ,\Delta \widehat{{\mathbf{t}}}^{{2T}} ,\Delta \hat{\varphi }^{2} ,\Delta \hat{\varphi }^{3} )^{T} , \\ \Delta \widehat{\mathsf{x}}_{{ROT}}: & = (\Delta \widehat{{\mathbf{d}}}^{{1T}} ,\Delta \widehat{\varvec{\theta }}^{{1T}} ,\Delta \hat{t}^{1} ,\Delta \widehat{{\mathbf{d}}}^{{2T}} ,\Delta \widehat{\varvec{\theta }}^{{2T}} ,\Delta \hat{t}^{2} ,\Delta \hat{\varphi }^{3} )^{T} . \\ \end{aligned} $$

(226)

The transformations between these primary variable variations and increments is given by:

$$\begin{aligned} \begin{array}{ll} \delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{TAN} = \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{{{\mathbf {\mathsf{{x}}}}}}} \delta \hat{\mathbf {x}}_{ROT} \quad \text {and} \quad \Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{TAN} = {{\mathbf {\mathsf{{T}}}}}_{M\hat{{{\mathbf {\mathsf{{x}}}}}}} \Delta \hat{\mathbf {x}}_{ROT}. \end{array} \end{aligned}$$

(227)

The transformation matrices $\widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{{{\mathbf {\mathsf{{x}}}}}}}$, originally defined in (169), and ${{\mathbf {\mathsf{{T}}}}}_{M\hat{{{\mathbf {\mathsf{{x}}}}}}}$ have the following form:

$$\begin{aligned} \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}} = \left( \begin{array}{ccccc} \mathbf {I}_3 &{}&{}&{}&{}\\ &{}\widetilde{\mathbf {T}}^1&{}&{}&{}\\ &{}&{}\mathbf {I}_3&{}&{}\\ &{}&{}&{}\widetilde{\mathbf {T}}^2&{}\\ &{}&{}&{}&{}1 \end{array} \right) , \,\, {{\mathbf {\mathsf{{T}}}}}_{M\hat{\mathbf {x}}} = \left( \begin{array}{ccccc} \mathbf {I}_3 &{}&{}&{}&{}\\ &{}\mathbf {T}_M^1&{}&{}&{}\\ &{}&{}\mathbf {I}_3&{}&{}\\ &{}&{}&{}\mathbf {T}_M^2&{}\\ &{}&{}&{}&{}1 \end{array} \right) . \end{aligned}$$

(228)

These two different matrices are required, since the primary variable variations of the SK/WK-TAN elements are based on the multiplicative quantities $\delta \hat{\Theta }_{1}^i$, whereas the corresponding iterative primary variable increments are based on the additive quantities $\Delta \hat{\varphi }^i$. The matrices $\widetilde{\mathbf {T}}^i$ and $\mathbf {T}_M^i$ (25, 24) are evaluated at the element boundary nodes:

$$\begin{aligned} \widetilde{\mathbf {T}}^i := \widetilde{\mathbf {T}}(\xi ^i) \quad \text {and} \quad \mathbf {T}_M^i := \mathbf {T}_M(\xi ^i) \quad \text {for} \quad i=1,2. \end{aligned}$$

(229)

In Sect. 9.2, it has already been shown that the following residual transformation is valid:

$$\begin{aligned} {{\mathbf {\mathsf{{r}}}}}_{ROT} = \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}^T {{\mathbf {\mathsf{{r}}}}}_{TAN}. \end{aligned}$$

(230)

In a similar manner, also the linearized element residual vector can be transformed:

$$\begin{aligned} \begin{array}{ll} \Delta {{\mathbf {\mathsf{{r}}}}}_{ROT} =&{} \Delta \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}^T {{\mathbf {\mathsf{{r}}}}}_{TAN} + \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}^T \underbrace{\Delta {{\mathbf {\mathsf{{r}}}}}_{TAN}}_{={{\mathbf {\mathsf{{k}}}}}_{TAN}\Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{TAN}} =: \ {{\mathbf {\mathsf{{k}}}}}_{ROT} \Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{ROT}, \\ {{\mathbf {\mathsf{{k}}}}}_{ROT} :=&{} \left( \widetilde{{{\mathbf {\mathsf{{H}}}}}}_{\hat{\mathbf {x}}} ({{\mathbf {\mathsf{{r}}}}}_{TAN}) + \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}^T {{\mathbf {\mathsf{{k}}}}}_{TAN} {{\mathbf {\mathsf{{T}}}}}_{M\hat{\mathbf {x}}}\right) . \end{array} \end{aligned}$$

(231)

Here, the matrix $\widetilde{{{\mathbf {\mathsf{{H}}}}}}_{\hat{\mathbf {x}}} ({{\mathbf {\mathsf{{r}}}}}_{TAN})$ has been introduced, given by

$$\begin{aligned} \widetilde{{{\mathbf {\mathsf{{H}}}}}}_{\hat{\mathbf {x}}} ({{\mathbf {\mathsf{{r}}}}}_{TAN}) = \left( \begin{array}{ccccc} \mathbf {0} &{}&{}&{}&{}\\ &{}\widetilde{{{\mathbf {\mathsf{{H}}}}}}^1&{}&{}&{}\\ &{}&{}\mathbf {0}&{}&{}\\ &{}&{}&{}\widetilde{{{\mathbf {\mathsf{{H}}}}}}^2&{}\\ &{}&{}&{}&{}0 \end{array} \right) . \end{aligned}$$

(232)

This matrix is used for representing the linearization of $\widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}$:

$$\begin{aligned} \widetilde{{{\mathbf {\mathsf{{H}}}}}}_{\hat{\mathbf {x}}} ({{\mathbf {\mathsf{{r}}}}}_{TAN}) \Delta \hat{{{\mathbf {\mathsf{{x}}}}}}_{ROT} := \Delta \widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}^T {{\mathbf {\mathsf{{r}}}}}_{TAN}. \end{aligned}$$

(233)

After calculating the derivative of $\widetilde{{{\mathbf {\mathsf{{T}}}}}}_{\hat{\mathbf {x}}}$ and re-ordering the result, the submatrices $\widetilde{{{\mathbf {\mathsf{{H}}}}}}^i$ can be stated:

$$\begin{aligned} \widetilde{{{\mathbf {\mathsf{{H}}}}}}^i = \left( \begin{array}{cc} \mathbf {S}(\mathbf {r}_{ T A N ,\hat{\mathbf {t}}^{i}} ) \mathbf {S}(\mathbf {g}_1^{i}) - r_{ T A N ,\hat{\Theta }_1^{i}} \mathbf {S}(\mathbf {g}_1^{i}) \,\,\,&{}\,\,\, \mathbf {S}(\mathbf {g}_1^{i})\mathbf {r}_{ T A N ,\hat{\mathbf {t}}^{i}} \\ -\mathbf {r}_{ T A N ,\hat{\mathbf {t}}^{i}}^T\mathbf {S}(\mathbf {g}_1^{i}) &{} 0 \end{array} \right) \end{aligned}$$

(234)

for $i=1,2.$ From (231), the following transformation rule for the the element stiffness matrix can be stated:

$$ \begin{aligned} \Delta {\mathbf{r}}_{{ROT}} & = {\mathbf{k}}_{{ROT}} \Delta \widehat{{\mathbf{x}}}_{{ROT}} \\ {\text{with }}{\mathbf{k}}_{{ROT}} & = {\mathbf{\widetilde{H}}}_{{\widehat{{\mathbf{x}}}}} ({\mathbf{r}}_{{TAN}} ) + {\mathbf{\widetilde{T}}}_{{\widehat{{\mathbf{x}}}}}^{T} {\mathbf{k}}_{{TAN}} {\mathbf{T}}_{{M\widehat{{\mathbf{x}}}}} . \\ \end{aligned} $$

(235)

In order to apply this transformation, the components of the element stiffness matrices ${{\mathbf {\mathsf{{k}}}}}_{TAN}$ and ${{\mathbf {\mathsf{{k}}}}}_{ROT}$ have to be arranged in the same order as the element residual vectors:

$$ \begin{aligned} \mathsf{r}_{{TAN}}: & = ({\mathbf{r}}_{{TAN,\widehat{{\mathbf{d}}}^{1} }}^{T} ,{\mathbf{r}}_{{TAN,\widehat{{\mathbf{t}}}^{1} }}^{T} ,r_{{TAN,\hat{\Theta }_{1}^{1} }} , \\ & \quad {\mathbf{r}}_{{TAN,\widehat{{\mathbf{d}}}^{2} }}^{T} ,{\mathbf{r}}_{{TAN,\widehat{{\mathbf{t}}}^{2} }}^{T} ,r_{{TAN,\hat{\Theta }_{1}^{2} }} ,r_{{TAN,\hat{\Theta }_{1}^{3} }} )^{T} , \\ \mathsf{r}_{{ROT}}: & = (\mathsf{r}_{{ROT,\widehat{{\mathbf{d}}}^{1} }}^{T} ,\mathsf{r}_{{ROT,\widehat{\theta }^{1} }}^{T} ,r_{{ROT,\hat{t}^{1} }} , \\ & \quad \mathsf{r}_{{ROT,\widehat{{\mathbf{d}}}^{2} }}^{T} ,\mathsf{r}_{{ROT,\widehat{{\theta ^{2} }}}}^{T} ,r_{{ROT,\hat{t}^{2} }} ,r_{{ROT,\hat{\Theta }_{1}^{3} }} )^{T} . \\ \end{aligned} $$

(236)