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Geometrically Exact Finite Element Formulations for Curved Slender Beams: Kirchhoff-Love Theory vs. Simo-Reissner Theory

Christoph Meier, Wolfgang A. Wall, Alexander Popp

arXiv:1609.00119v1cs.CE

TL;DR

The paper addresses how geometrically exact Kirchhoff-Love elements compare with Simo-Reissner formulations for highly slender beams. It proposes and evaluates four Kirchhoff-Love variants, finding that they satisfy the considered requirements and can reduce discretization errors and nonlinear solver effort at high slenderness.

  • Problem

    The paper examines which benefits Kirchhoff-Love formulations can provide over existing Simo-Reissner elements for highly slender beams.

  • Method

    The authors propose four Kirchhoff-Love finite element variants, review existing formulations, and compare them analytically and numerically using C1-continuous Hermite centerline interpolation and Lie-group time discretization.

  • Results

    The proposed Kirchhoff-Love elements fulfill the considered essential requirements, while WK elements show lower discretization errors and SK-TAN and WK-TAN require fewer Newton iterations than the investigated Simo-Reissner formulations.

  • Takeaways & Limitations

    Shear-free Kirchhoff-Love formulations can provide numerical advantages for highly slender beams compared with the investigated Simo-Reissner formulations.

  • Takeaways & Limitations

    With rigid cross-sections, the standard constitutive relations described in the paper hold generally only for ν = 0 because in-plane reaction forces are required; general ν ≠ 0 requires weakening that constraint.

Abstract

from arXiv · show

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 C1-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 or preservation of objectivity and path-independence 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 the considered 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 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.

1. Introduction

The paper addresses the limited generality of geometrically exact Kirchhoff-Love finite elements for highly slender beams and develops, reviews, and compares formulations against Simo-Reissner elements. It focuses on accuracy, robustness, essential theoretical properties, and numerical performance at high slenderness ratios.

  • Motivation: Beam theories provide efficient one-dimensional models for slender bodies while representing three-dimensional motion and deformation through reduced kinematic, kinetic, and constitutive quantities.Their modern motivation is improved numerical efficiency and well-posedness compared with three-dimensional continuum formulations.
  • Research gap: Existing geometrically exact Kirchhoff-Love elements are few and generally do not match Simo-Reissner formulations in generality or fulfillment of essential properties.The review identifies limitations in representability and properties such as objectivity and path-independence.
  • Evaluation: The formulations are evaluated analytically and numerically for general three-dimensional, large-deformation, dynamic beam problems with arbitrary initial curvature and anisotropic cross-sections.The study also compares discretization errors, convergence, and nonlinear solver performance across beam slenderness ratios.
  • Research gap: The work proposes Kirchhoff-Love formulations for highly slender beams while investigating strong and weak enforcement of the shear-free Kirchhoff constraint.The motivation is especially relevant because shear contributions can create stiff, ill-conditioned numerical problems at high slenderness ratios.
  • Contributions: Two rotation interpolation strategies and two nodal triad parametrizations are combined into four proposed finite element variants based on C1-continuous Hermite centerline interpolation.The strategies use strong or weak Kirchhoff constraint enforcement and rotation-vector or tangent-vector nodal parametrizations.
  • Contributions: A Lie-group time integration scheme with optimized numerical dissipation is applied to the proposed geometrically exact Kirchhoff-Love beam elements.The scheme is implicit, one-step, energy-stable, and second-order accurate for large rotations.

2. The Rotation Group SO(3)

The section introduces beam rotations as elements of the nonlinear Lie group SO(3) and develops rotation-vector and tangent-based parametrizations for Kirchhoff formulations.

  • SO(3) configuration: Beam cross-sections carry an orthonormal triad whose orientation is represented by Λ ∈ SO(3), mapping global and local frames.The rotation tensor is an orthogonal transformation with determinant one.
  • Lie-group structure: SO(3) has noncommutative multiplication, inverse Λ^-1 = Λ^T, identity I3, and Lie algebra so(3) of skew-symmetric tensors.The exponential map connects so(3) to SO(3).
  • Rotation-vector parametrization: The rotation-vector parametrization uses three parameters ψ, with norm ||ψ|| as rotation angle and ψ/||ψ|| as rotation axis.Multiplicative spin-vector variations describe changes of Λ.
  • Tangent-based parametrization: An alternative four-parameter representation uses a non-unit tangent vector t and twist angle ϕ to construct a tangent-aligned triad.The norm ||t|| does not affect orientation, while t arises directly from the centerline description.
  • Tangent-based parametrization: The tangent-based triad is defined by a smallest rotation followed by rotation about the tangent, yielding a nonredundant parametrization for Kirchhoff beams.Spin variations are split into components parallel and perpendicular to the first triad vector.

3. Simo-Reissner Beam Theory

The Simo-Reissner theory models initially curved, anisotropic beams with rigid cross-sections, six pointwise degrees of freedom, objective deformation measures, equilibrium equations, and constitutive laws.

  • Basic kinematic assumptions: The beam configuration consists of a centerline and orthonormal material triads attached to undeformable cross-sections.The initial centerline is arc-length parametrized, and the triad’s first vector is tangent to it.
  • Basic kinematic assumptions: A material point is described by the centerline position plus convective cross-section coordinates multiplied by the current triad vectors.Figure 2 illustrates the initial and deformed kinematic quantities.
  • Kinematic measures: Spatial and material curvature and angular velocity vectors are defined from spatial and material derivatives of the rotation tensor.Their components are related through the push-forward operator Λ.
  • Equilibrium and constitutive relations: The formulation combines force and moment equilibrium with constitutive relations for stress resultants, kinetic energy, inertia, and boundary conditions.The constitutive tensors encode Young’s modulus, shear modulus, cross-sectional areas, moments of inertia, and torsional inertia.
  • Objective deformation measures: Objective deformation measures vanish in the stress-free initial configuration, while Γ contains axial tension and shear and Ω contains torsion and bending.Their objective variations are expressed through centerline and spin variations.
  • Relation between 1D and 3D constitutive laws: Consistency between geometrically exact beam and 3D constitutive theories is assured under small-strain assumptions, while rigid cross-sections require in-plane reaction forces for general ν ≠ 0.A weakened constraint allowing uniform lateral contraction resolves the constitutive mismatch.

4. Kirchhoff-Love Beam Theory

Kirchhoff-Love theory enforces vanishing shear by constraining the cross-section triad to the beam tangent, reducing the primary fields and adapting curvature, dynamics, equilibrium, and boundary conditions.

  • Kirchhoff constraint: The Kirchhoff constraint requires the cross-section vectors g2 and g3 to remain perpendicular to the tangent t = r′, equivalently g1 = t.This is appropriate as a shear-free approximation for highly slender beams.
  • Constraint enforcement: The constraint can be enforced strongly through a reduced parametrization or weakly with Lagrange-multiplier fields λ2 and λ3.The reduced approach uses four degrees of freedom consisting of centerline variables and a twist parameter.
  • Constrained kinematics: Constrained curvature and spin expressions depend on the centerline tangent, intermediate triad, and tangential rotational increment.The intermediate triad contributes through its torsion, while the centerline contributes Frenet-Serret curvature.
  • Deformation measures: The Kirchhoff constraint removes shear components from the deformation measures, leaving axial tension as the remaining translational strain component.The axial strain is ϵ = ||r′|| − 1.
  • Dynamics: The formulation supports multiple dynamic variable choices, including direct angular variables, twist-rate variables, and an independent tangential angular-velocity component.These alternatives produce corresponding time-integration schemes.
  • Equilibrium and boundary conditions: Eliminating shear forces yields four equilibrium equations for centerline and twist variables, with objective virtual-work terms and tangent-orientation boundary conditions.The boundary data prescribe position, tangent orientation, cross-section twist, force, and moment.
  • Intermediate-triad choice: The Frenet-Serret intermediate triad aids analytical treatment but is unfavorable numerically because it becomes singular on straight beam segments.Alternative intermediate-triad choices are therefore relevant for computation.

5. Temporal Discretization of Primary Fields

Temporal discretization uses a Lie-group extension of generalized-α integration for beam configurations in ℝ3 × SO(3), independently of beam theory and rotation parametrization.

  • Lie-group time integration: The time integrator applies to Reissner and Kirchhoff configurations consisting of position and rotation fields in ℝ3 × SO(3).Its construction is independent of the chosen rotation parametrization.
  • Time-stepping procedure: The Lie-group generalized-α scheme evaluates the weak form at the endpoint tn+1 and updates rotations multiplicatively.Translational quantities use a standard Newmark update.
  • Modified quantities: Modified and physical accelerations are related through αm and αf, while analogous relations are used for angular accelerations.The formulation also provides relations for translational and material angular velocities and accelerations.
  • Numerical properties: The integration scheme provides second-order accuracy, unconditional stability in the linear regime, controllable high-frequency damping, and minimized low-frequency damping.The optimal parameter choice matches the standard generalized-α method.
  • Applicability: Its simplicity and flexibility allow direct application across Reissner or Kirchhoff theories, spatial interpolations, and nodal rotation parametrizations.No further adaptations are required for these choices.
  • Related work: The review notes that energy-momentum schemes are abundant for Simo-Reissner formulations but limited for Kirchhoff-Love formulations, especially anisotropic dynamic cases.One cited scheme handles only temporal discretization of the beam centerline.

6. Spatial Discretization Methods for Primary Fields

The spatial discretization methods combine triad interpolation, spin-vector interpolation, objectivity checks, and anti-locking strategies for geometrically exact Kirchhoff-Love beam elements.

  • Nodal parametrization: Rotation-vector nodal parametrization simplifies complex Dirichlet boundary conditions and joint modeling compared with tangent-vector parametrization.Among the reviewed straight formulations, only those using rotation-vector-based triad parametrization provide this simplification.
  • Triad interpolation: The proposed triad interpolation constructs an orthonormal field from the centerline tangent and nodal triads, independently of the chosen nodal primary variables.Using a material triad as reference also makes the element formulation symmetric and extends the maximal orientation difference.
  • Spin-vector interpolation: The Bubnov-Galerkin interpolation can represent arbitrary constant spin-vector distributions, whereas the Petrov-Galerkin variant generally cannot.For the Petrov-Galerkin interpolation, exact representation is restricted to 2D problems or the zero spin-vector case, affecting angular-momentum conservation analysis.
  • Objectivity: The proposed interpolation is objective because its relative-angle field remains unchanged under rigid-body motion, yielding consistent rotated deformation measures and torsion.The derivation states that unchanged interpolation variables and transformation properties establish the required objectivity.
  • Membrane locking: The minimally constrained strain method avoids membrane locking while using the minimal constraint count needed to prevent zero-energy modes in straight and general 3D configurations.Its optimal constraint ratio is reported as r_h = r = 4 in the general 3D case; alternative underconstrained variants can produce singular stiffness matrices.
  • Membrane locking: For arbitrarily curved configurations, the MCS method cannot exactly represent constant curvature with vanishing axial tension, although numerical results show membrane locking is avoided.The curved configuration is slightly over-constrained despite retaining the optimal constraint ratio for the cited MCS variant.

7. Simo-Reissner Beam Element

The CJ Simo-Reissner element uses reduced integration to avoid locking at high slenderness ratios and is assessed for conservation properties and exact pure-bending representation.

  • Reference formulation: The CJ element is used as the reference formulation for numerical comparisons, including investigations of locking and mechanical conservation.Its residual vector is derived from the space-continuous Simo-Reissner problem using discrete trial and weighting functions.
  • Avoidance of locking: Reduced Gauss integration with nn − 1 integration points is proposed to avoid shear and membrane locking in highly slender beams.The scheme integrates the internal-force contribution of an nn-noded element using nn − 1 points.
  • Avoidance of locking: The constraint ratio r = 2 is identified as optimal for the CJ formulation, so no locking effects are expected.The element can also exactly represent the internal energy of three-dimensional pure-bending states through its triad interpolation.
  • Avoidance of locking: The reduced integration scheme exactly integrates the shear and axial contributions occurring in the energy integral for the examined formulation.Exact representation of discrete hyperelastic energies for pure bending in two and three dimensions is verified through numerical test cases.
  • Conservation properties: The CJ element guarantees conservation of linear and angular momentum because its discrete fields exactly represent the required constant translation and rotation variations.This holds for both spin-vector discretizations and the discretized centerline variation.
  • Conservation properties: Only the Bubnov-Galerkin variant exactly represents hyperelastic and kinetic energy rates, whereas the Petrov-Galerkin variant is not variationally consistent with the triad interpolation.The stated consequence concerns exact energy conservation for the spatially discretized, time-continuous problem.

8. Motivation for ”shear-free” beam theories

Shear-free Kirchhoff-Love formulations are motivated by the increasingly stiff and ill-conditioned behavior of highly slender shear-deformable beams, while also offering fewer degrees of freedom and smooth centerline geometry.

  • Motivation: Simo-Reissner elements are efficient and accurate, but their shear contribution decreases as beam slenderness increases.Avoiding shear-mode stiffness motivates shear-free Kirchhoff-Love formulations for thin beams.
  • Improved stability: At slenderness ratio ζ = 100, removing shear modes increased the time step by ≈100, while additionally removing axial tension increased it by a further factor of ≈5.The comparison used full Simo-Reissner, Kirchhoff-constrained, and additionally inextensible beam models.
  • Linear-solver performance: Shear and axial stiffness contributions are expected to increase the tangent-stiffness condition number quadratically with slenderness ratio ζ.The text therefore identifies an additional inextensibility constraint as potentially beneficial for iterative linear-solver conditioning, although it is outside this contribution’s focus.
  • Nonlinear-solver performance: Numerical examples show Reissner nonlinear-solver performance deteriorating drastically with slenderness, while Kirchhoff formulations require an almost unchanged total number of Newton iterations.The paper notes that the relation between nonlinear-solver performance and conditioning is less direct than for linear solvers.
  • Reduced system size: Kirchhoff formulations omit shear-deformation degrees of freedom and can therefore achieve comparable approximation and discretization error with fewer degrees of freedom.This expectation is stated to hold provided convergence-deteriorating phenomena such as locking do not occur.
  • Smooth geometry representation: C1-continuous centerline interpolation yields smooth beam-to-beam contact kinematics for efficient and robust contact algorithms.The proposed elements use this smooth geometry representation as a design feature.
  • Scope boundary: Adding an inextensibility constraint could improve iterative linear-solver performance, but direct enforcement is not straightforward while preserving boundary-node interpolation.For a straight element, the two boundary nodal positions must satisfy a constraint rather than vary independently.

9. Kirchhoff-Love Beam Element Based on Strong Constraint Enforcement

The strong-constraint Kirchhoff element uses C1 Hermite centerline interpolation with tangent- or rotation-vector-based nodal triad parametrizations. Its variants preserve equilibrium and selected conservation properties, while the tangent-based formulation has optimal constraint counting but cannot exactly represent pure-bending energy.

  • Formulation: The strong-constraint formulation combines C1 Hermite centerline interpolation with either tangent-vector or rotation-vector nodal triad parametrization.The tangent-based variant is developed first, followed by a rotation-vector reparametrization.
  • Formulation: The tangent-based element uses quadratic rotation interpolation because its triad orientation depends on the relative angle and quadratic tangent field.Interpolation order nΛ > 3 does not improve approximation quality, whereas nΛ < 3 reduces the convergence rate.
  • Reparametrization: The tangent- and rotation-based residual formulations are equivalent at mechanical equilibrium when the tangent-to-rotation transformation is nonsingular.The transformation depends nonlinearly on the primary degrees of freedom and must therefore be included in consistent linearization.
  • Locking and representation: The strong tangent-based element has optimal constraint ratio r = 4, so membrane locking is not expected.A straight-beam state with arbitrary axial-tension distribution can be represented exactly, with no associated zero-energy modes.
  • Locking and representation: The strong tangent-based element cannot exactly represent pure-bending hyperelastic energy, producing a slightly increased discretization error.This limitation is verified numerically for pure bending in both two and three dimensions.
  • Conservation properties: Both spin-interpolation variants exactly conserve linear momentum, but only the Bubnov-Galerkin variant exactly represents rigid-body rotation rates and guarantees time-continuous energy conservation.The Petrov-Galerkin variant is not variationally consistent with the triad interpolation and cannot guarantee energy conservation.

10. Kirchhoff-Love Beam Element Based on Weak Constraint Enforcement

The weak-constraint Kirchhoff elements derive from a C1-continuous Simo-Reissner formulation and enforce vanishing shear through collocation at three nodes. Their tangent and rotation parametrizations avoid membrane locking and retain momentum conservation, with energy conservation requiring Bubnov-Galerkin spin interpolation.

  • Formulation: The weak-constraint formulation first derives a C1-continuous Simo-Reissner element, then enforces vanishing shear strains to obtain Kirchhoff variants.The Reissner formulation serves as an intermediate step rather than the final element.
  • Constraint enforcement: The weak Kirchhoff constraint is enforced exactly at three collocation points without additional Lagrange multipliers.The formulation retains the term “weak constraint enforcement” because collocation is used as the discrete basis of the space-continuous formulation.
  • Parametrization: WK-TAN uses tangent-based nodal variables, while WK-ROT transforms the same weak-constraint residual into rotation-vector-based boundary variables.The rotation-vector parametrization replaces nodal tangents by tangent magnitudes and rotation vectors.
  • Locking: The WK-TAN element has optimal constraint ratio r = rh = 4, so membrane locking effects are not expected.Its constraint counts are identical to those of the strong tangent-based element.
  • Energy representation: WK-TAN and WK-ROT can exactly represent the internal energy of a three-dimensional pure-bending state.For pure bending, the axial-tension energy contribution vanishes and the remaining energy is split into torsion and bending contributions.
  • Conservation properties: The weak-constraint elements exactly conserve linear and angular momentum, while exact time-continuous energy conservation requires Bubnov-Galerkin rather than Petrov-Galerkin spin interpolation.This conservation conclusion follows from the corresponding interpolation and strong-constraint investigations.

11. Numerical Examples

Numerical examples assess objectivity, convergence, locking, discretization errors, energy errors, and nonlinear solver performance for the investigated beam formulations. The results show exact rigid-body behavior for the proposed Kirchhoff formulations, optimal convergence, and advantages of weak Kirchhoff formulations in several highly slender cases.

  • Numerical setup: The numerical study evaluates the proposed formulations using in-house finite-element implementations, with convergence assessed from displacement increments and residual norms.Typical tolerances include δX = 10^-8 and δR values from 10^-7 to 10^-13 across slenderness ratios ζ = 10, 100, 1000, 10000.
  • 11.1. Example 1: Verification of objectivity: Under 20π rigid-body rotation of an initially stress-free quarter circle, the proposed formulations preserve zero internal energy up to machine precision, whereas the SR formulation accumulates energy and deforms the beam.Within 10 rotations, the SR formulation reaches normalized internal energy near Πint,r/4.
  • 11.2.1. Comparison of discretization errors: The MCS method eliminates locking across the investigated slenderness ratios, whereas reduced integration and ANS are less effective for the more general combined moment-and-force load case.Reduced integration is sufficient for the special pure-moment case but reaches its limits when the deformation produces non-symmetric curvature distributions; ANS collocation points become deformation-dependent.
  • 11.2.1. Comparison of discretization errors: For the pure-moment load case, all element formulations exhibit the expected fourth-order convergence, while the Kirchhoff WK-TAN element reaches comparable discretization errors with fewer degrees of freedom than the Reissner element.Because no shear deformation is present, Reissner and Kirchhoff elements converge toward the same analytic solution.
  • 11.2.2. Comparison of discretization errors: The Kirchhoff solution differs from the Reissner solution by less than 10^-7 at ζ = 10000, supporting WK-TAN as the numerical reference for the remaining examples.For ζ = 100, the relative difference is below 10^-3, and all formulations exhibit fourth-order convergence in the cited nonlinear example.
  • 11.2.2. Comparison of discretization errors: For the combined load case, the WK-TAN element has a lower energy error than SK-TAN while both retain fourth-order energy convergence.The reported explanation is that WK-TAN has matching numbers of unknowns and equations for energetically representing pure bending states.

SK-TAN element.

The SK-TAN formulation is evaluated across three-dimensional bending, path-independence, slenderness, and complex helix and dynamic cases. It shows fourth-order convergence, robust nonlinear performance, and consistent behavior for highly slender beams.

  • Convergence and locking: All investigated Kirchhoff elements achieved the expected fourth-order convergence in both L2- and energy-error norms.With the MCS method, element slenderness did not influence the discretization error, confirming avoidance of membrane locking.
  • 3D pure bending: The 3D pure-bending solution is a helix aligned with the applied end-moment direction, and SK-TAN exhibits the expected fourth-order relative L2-error convergence.The test uses initially straight beams with torsion-inducing three-dimensional end moments and slenderness ratios ζ = 100 and ζ = 10000.
  • Path-independence: The proposed formulations yield path-independent discrete solutions when the analytic beam solution is path-independent.The comparison uses relative L2-error between solutions obtained from different loading paths for the same centerline discretization.
  • Nonlinear solver performance: For slenderness ratio ζ̃ = 100, SK/WK-TAN discretizations solve the problem in one load step and eight Newton iterations.For ζ̃ = 10000, they again require one load step and eight iterations, while Reissner discretizations require 30–60 load steps and 350–450 iterations.
  • Nonlinear solver performance: At high slenderness, the proposed Kirchhoff formulations are reported as robust and efficient relative to many Reissner alternatives, while comparisons based on isolated lucky shots are cautioned against.The literature comparison is difficult because procedures for determining minimal Newton-iteration counts are not uniform.
  • General 3D helix: The twisted-helix test with three-dimensional initial curvature, anisotropic cross-sections, and initial twist shows consistent convergence, with no visible difference between SK-TAN Petrov-Galerkin and SK-TAN+CS Bubnov-Galerkin variants.The test covers ζ = 100 and ζ = 10000 and compares initial and final helix shapes under axial force.

12. Conclusion

The work proposes and evaluates geometrically exact Kirchhoff-Love elements for highly slender beams, comparing them with existing Kirchhoff-Love and Simo-Reissner formulations. The proposed elements satisfy the considered theoretical requirements and show accuracy and solver-performance advantages in high-slenderness tests.

  • Conclusion: The study proposes four Kirchhoff-Love element variants using alternative rotation interpolation schemes and nodal triad parametrizations.The variants extend earlier formulations to improve accuracy, practical applicability, and dynamic-problem capability.
  • Conclusion: Only a few existing Kirchhoff-Love formulations support general 3D, large-deformation problems, and they typically satisfy only some essential requirements.The review categorizes these formulations as isotropic, straight, or anisotropic.
  • Conclusion: The proposed elements theoretically and numerically satisfy objectivity, path-independence, spatial convergence, high-slenderness locking avoidance, and conservation properties.The evaluation also examines Bubnov-Galerkin and Petrov-Galerkin discretizations and compares locking treatments including MCS and Assumed Natural Strains.
  • Conclusion: Below ζ =100, the Kirchhoff-Love and Simo-Reissner model difference typically remained below 0.1% in relative L2-error, while the difference decreased quadratically with increasing slenderness.The proposed WK elements had lower discretization errors than the investigated Simo-Reissner formulation; SK elements performed worse in some examples.
  • Conclusion: At ζ =10000, Simo-Reissner formulations required up to two orders of magnitude more Newton iterations than the proposed SK-TAN and WK-TAN elements.At ζ =100, all formulations required iteration counts in at least the same order of magnitude, whereas Simo-Reissner iteration counts increased considerably with slenderness.

Appendix A. Definition of rotational shape function matrices

Appendix A presents the rotational shape-function matrices and the derivative relations needed for multiplicative rotation increments and triad interpolation.

  • Definition of rotational shape function matrices: The appendix defines shape functions ˜Ii(ξ) required for multiplicative rotation increments associated with the triad interpolation.These shape functions were originally derived in reference.
  • Definition of rotational shape function matrices: The vectors vI and vJ are defined for the rotational shape-function construction, with ΦIJ abbreviated as its norm.The appendix also introduces the arc-length derivative ˜Ii′(ξ).
  • Definition of rotational shape function matrices: The notation Φlh = Φlh(ξ) and Φlh = ||Φlh(ξ)|| is used when presenting the shape-function expressions and their limiting form.The limiting behavior is noted as originating in the original work.

Appendix B. Modeling of Dirichlet boundary conditions and joints

Appendix B explains how the SK-ROT and SK-TAN elements impose boundary conditions and connect beam elements through moment-free or rigid joints. SK-ROT permits more direct constraint handling, while SK-TAN requires coordinate transformations for some conditions.

  • Modeling of Dirichlet boundary conditions and joints: The appendix investigates basic Dirichlet boundary conditions and joints for the SK-ROT and SK-TAN elements.The SK-ROT element is treated first because it simplifies many practically relevant constraints.
  • Dirichlet boundary conditions: Clamped ends require fixing cross-section orientation, while simple supports constrain selected tangent components without prescribing tangent magnitude.For arbitrary orientations, SK-TAN conditions are formulated in a rotated coordinate system.
  • Connections: Multiplicative rotation increments permit direct elimination of connected rotational degrees of freedom, whereas additive rotation-vector increments require stiffness-column scaling by T(ˆψb)T−1(ˆψa).The latter prevents direct elimination through standard assembly alone.
  • Connections: Rigid joints use right-translation because they preserve a fixed orientation difference relative to material axes; left-translation represents a different physical constraint.This distinction changes the physical meaning of the joint condition.
  • Dirichlet boundary conditions: The nodal axial-force measures ˆta and ˆtb remain FEM solution variables and must not be prescribed when defining physically reasonable boundary conditions.Boundary conditions can instead be defined completely through cross-section orientation and centroid position.
  • Comparison of SK-ROT and SK-TAN: SK-ROT formulates clamped ends and rigid joints directly in global coordinates, while SK-TAN requires transforming residual and stiffness lines and columns.This makes constraint treatment simpler for SK-ROT in the considered cases.

Appendix C. Linearization of SK-TAN element

Appendix C derives the SK-TAN element’s residual linearization and stiffness contributions, including translational, rotational, axial, moment, and inertia terms. It relates additive and multiplicative rotation increments within the Newton linearization and time-integration framework.

  • Linearization of SK-TAN element: The appendix introduces the required definitions and then linearizes the underlying residual and its individual contributions for the SK-TAN element.The sequence covers the general residual linearization before specializing force, moment, inertia, and rotational terms.
  • Linearization of SK-TAN element: The SK-TAN linearization starts from the element residual vector with strain re-interpolation and brings it into the form needed to identify the element stiffness matrix.The vector ∆ˆxTAN is used in this formulation.
  • Rotational terms: The derivation reuses earlier relations for moment terms and obtains the multiplicative rotation-increment field and its arc-length derivative from equations (121) and (122).These relations provide the rotational contributions required by the SK-TAN tangent matrix.
  • Force and inertia contributions: The linearization includes residual terms associated with axial tension, inertia forces, and inertia moments.The inertia-moment contribution uses the modified generalized-α time-integration factor c¨r1.
  • Rotation increments: Material and spatial multiplicative rotation increments relate the current configuration to the converged configuration at the previous time step.Their additive Newton-iteration increments are distinguished from multiplicative increments between Newton iterations.
  • Rotation increments: The transformation between material and spatial rotation increments follows from the spatial increment being an eigenvector of the relative rotation tensor with eigenvalue one.This property is used in the derivation of the rotation-increment relation.

Appendix D. Linearization of WK-TAN element

Appendix D presents the WK-TAN element residual and its linearization, including rotation-increment fields, vector and moment-stress-resultant terms, and Kirchhoff-constraint relations.

  • The WK-TAN element residual vector is specified through equation (180).
  • The element residual linearization is organized into a general form used to identify the WK-TAN element stiffness matrix.
  • The linearization includes the vectors originally defined in equation (180) and the moment stress resultant.
  • For WK-TAN, the rotation increment fields ∆θ and ∆θ′ are defined separately, with nodal increments constrained by the Kirchhoff condition.
  • The inertia-force and inertia-moment linearizations follow the corresponding results from the preceding section.

Appendix E. Linearization of SK-ROT and WK-ROT elements

Appendix E develops the linearization and variable transformations for SK-ROT and WK-ROT elements, distinguishing multiplicative variations from additive iterative increments.

  • The appendix defines nodal primary-variable variations and iterative increments for the SK/WK-TAN and SK/WK-ROT elements.
  • Transformations relate primary-variable variations to iterative primary-variable increments.
  • Separate transformation matrices are required because variations use multiplicative quantities δ ˆΘi, while iterative increments use additive quantities ∆ˆϕi.
  • The transformation matrices are evaluated at the element boundary nodes and are used to transform residuals and linearized residual vectors.
  • The derivative of ˜Tˆx is calculated and reordered to obtain submatrices, while a stiffness-matrix transformation rule is stated and requires consistent component ordering.
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