Efficient finite difference formulation of a geometrically nonlinear beam element

Jirásek, Milan; Ribolla, Emma La Malfa; Horák, Martin

doi:10.1002/nme.6820

Cited by 16 publications

(28 citation statements)

References 26 publications

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“…The present paper extends the geometrically exact formulation presented in [1] to curved beams undergoing large displacements and rotations. The theoretical framework is developed in Section 2 and the corresponding numerical procedures are described in Section 3.…”

Section: Introductionmentioning

confidence: 80%

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mentioning

confidence: 77%

“…The approach developed in [1] will now be extended to initially curved beams. Consider that the centerline of the undeformed beam is a planar curve of length L. An auxiliary curvilinear coordinate s is defined as the arc length measured along the undeformed centerline, with s ∈ [0, L].…”

Section: Kinematic Descriptionmentioning

confidence: 99%

“…The equilibrium equations in their differential form could be derived from the stationarity conditions of the total potential energy functional using the standard variational approach. This procedure is described in detail in [1] and it will not be repeated here because the resulting equations…”

Section: Equilibrium Equationsmentioning

confidence: 99%

“…The geometrically exact beam theory is still attracting researchers, with recent developments in the isogeometric approach (e.g., [14,15]) or in computational procedures related to the parameterisation of rotations using the rotation vector ( [16,17]). In a recent work [1] we have presented a numerical formulation for two-dimensional straight beams under large sectional rotations based on the shooting method: The boundary value problem is converted into an initial value problem handled by a finite difference scheme, and the estimated values used in artificially added initial conditions are iteratively adjusted until the boundary conditions on the opposite beam end are satisfied. On the global (structural) level, the governing equations are assembled in the same way as for a standard FE beam element with six degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Efficient formulation of a two-noded curved beam element under finite rotations

Hořák¹,

Ribolla²,

Jirásek³

2021

Preprint

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The paper extends the formulation of a 2D geometrically exact beam element proposed in our previous paper [1] to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The paper develops the theoretical framework based on the Navier-Bernoulli hypothesis but the approach could be extended to shear-flexible beams. The initial shape of the beam is captured with high accuracy, for certain shapes including the circular one even exactly. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by five examples that cover circular and parabolic arches and a spiral-shaped beam. It is also shown that, for initially curved beams, a cross effect in the relations between internal forces and deformation variables arises, i.e., the bending moment affects axial stretching and the normal force affects the curvature.

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Section: Introductionmentioning

confidence: 80%

“…

…”

mentioning

confidence: 77%

Section: Kinematic Descriptionmentioning

confidence: 99%

Section: Equilibrium Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient formulation of a two-noded curved beam element under finite rotations

Hořák¹,

Ribolla²,

Jirásek³

2021

Preprint

Self Cite

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Efficient formulation of a two‐noded geometrically exact curved beam element

Horák

Ribolla

Jirásek

2022

Numerical Meth Engineering

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The article extends the formulation of a 2D geometrically exact beam element proposed by Jirásek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier-Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant.The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral-shaped beam, and a spring-like beam with a zig-zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix.

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Newton vs. Euler–Lagrange approach, or how and when beam equations are variational

Babilio,

Lenci

2024

Meccanica

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There is a clear and compelling need to correctly write the equations of motion of structures in order to adequately describe their dynamics. Two routes, indeed very different from a philosophical standpoint, can be used in classical mechanics to derive such equations, namely the Newton vectorial approach (i.e., roughly, sum of forces equal to mass times acceleration) or the Euler–Lagrange variational formulation (i.e., roughly, stationarity of a certain functional). However, it is desirable that whichever derivation strategy is chosen, the equations are the same. Since many structures of interest often consist of slender and highly flexible beams operating in regimes of large displacement and large rotation, we restrict our attention to the Euler-Bernoulli assumptions with a generic initial configuration. In this setting, the question that arises is: What conditions must the constitutive assumptions satisfy in order for the equations of motion obtained by Newton’s approach to be identical to the Euler–Lagrange equations derived from an appropriate Lagrangian, natural or virtual, for any arbitrary initial configuration? The aim of this paper is to try to answer this basic question, which indeed does not have an immediate and simple answer, in particular as a consequence of the fact that bending moment could be related to two different notions of flexural curvature.

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Efficient finite difference formulation of a geometrically nonlinear beam element

Cited by 16 publications

References 26 publications

Efficient formulation of a two-noded curved beam element under finite rotations

Efficient formulation of a two-noded curved beam element under finite rotations

Efficient formulation of a two‐noded geometrically exact curved beam element

Newton vs. Euler–Lagrange approach, or how and when beam equations are variational

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