A total Lagrangian Timoshenko beam formulation for geometrically nonlinear isogeometric analysis of spatial beam structures

Vo, Duy; Nanakorn, Pruettha; Bui, Tinh Quoc

doi:10.1007/s00707-020-02723-6

Cited by 22 publications

(6 citation statements)

References 49 publications

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“…With regard to geometrically nonlinear analysis, the Lagrangian formulations are commonly used in the form of total Lagrangian, updated Lagrangian, and corotational formulations. In total Lagrangian formulations [14][15][16][17], the system equation terms are defned in terms of the fxed global frame that does not change through the analysis. Tis generates relatively large strains, displacements, and rotations that need special procedures to handle.…”

Section: Introductionmentioning

confidence: 99%

Corotational Finite Element Dynamic Analysis of Space Frames with Geometrically Nonlinear Behavior Based on Tait–Bryan Angles

Elerian,

Shebl,

Elkaranshawy

2023

Shock and Vibration

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The aim of this study is to compose a corotational finite element formulation for space frames with geometrically nonlinear behavior under dynamic loads. Using a moving frame through three successive rotations similar to Euler angles is one of the oldest techniques; however, there are still some gaps that require attention, mainly due to singularity. Hence, alternative techniques had been developed, sometimes elusive and computationally expensive. In this paper, we went back to the old technique and filled the gaps. Three-coordinate systems are used, i.e., the fixed global coordinate system, the fixed local coordinate system that is attached individually to every element, and the corotational local frame for each element that moves and rotates with the element. The deformation is always small relative to the corotational frame. The successive rotations between different coordinate systems are expressed using Tait–Bryan angles. Lagrange’s equation is used to derive the equation of motion, and the stiffness and mass matrices are obtained using the Euler–Bernoulli beam model. A MATLAB code is developed based on the Newton–Raphson method and the Newmark direct integration implicit method. In traditional techniques, singularity is attained when any rotation angle in the fixed local frame approaches π / 2 , and if any is greater than π / 2 , the techniques could fail to specify the location of the element. In this paper, each case is treated with a proper procedure, and special handling of trigonometric formulations prevents singularity and correctly specifies the location of elements in all situations. Different examples of beams and frames are analysed. While the method is not intricate, it is timesaving, is highly effective, provides more stable and robust analysis, and gives sufficiently accurate results. Compared to the parametrization of the finite rotations technique, the method has a significant reduction in the convergence rate because it avoids the storage of joint orientation matrices.

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Section: Introductionmentioning

confidence: 99%

Corotational Finite Element Dynamic Analysis of Space Frames with Geometrically Nonlinear Behavior Based on Tait–Bryan Angles

Elerian,

Shebl,

Elkaranshawy

2023

Shock and Vibration

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“…Several works consider nonlinear shear-deformable spatial beams in the framework of IGA, [14,15,16,17,18,19,20]. In the context of the BE theory, a torsion-and rotation-free spatial cable formulation is developed in [21] with attractive nonlinear dynamic applications.…”

Section: Introductionmentioning

confidence: 99%

Geometrically exact static isogeometric analysis of arbitrarily curved plane Bernoulli-Euler beam

Borković,

Marussig,

Radenković

2021

Preprint

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We present a geometrically exact nonlinear analysis of elastic in-plane beams in the context of finite but small strain theory. The formulation utilizes the full beam metric and obtains the complete analytic elastic constitutive model by employing the exact relation between the reference and equidistant strains. Thus, we account for the nonlinear strain distribution over the thickness of a beam. In addition to the full analytical constitutive model, four simplified ones are presented. Their comparison provides a thorough examination of the influence of a beam's metric on the structural response. We show that the appropriate formulation depends on the curviness of a beam at all configurations. Furthermore, the nonlinear distribution of strain along the thickness of strongly curved beams must be considered to obtain a complete and accurate response.

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“…The isogeometric technique for the spatial discretization has attracted much attention [28]. The nonlinear shear-deformable spatial beam is readily considered in the framework of isogeometric analysis (IGA) [29,30,31,32,33,34,35]. Regarding the nonlinear analysis of BE beams in the context of IGA, a torsion-and rotation-free spatial cable formulation is developed in [36] with attractive nonlinear dynamic applications.…”

Section: Introductionmentioning

confidence: 99%

Geometrically exact static isogeometric analysis of an arbitrarily curved spatial Bernoulli-Euler beam

Borković,

Marussig,

Radenković

2021

Preprint

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The objective of this research is the development of a geometrically exact model for the analysis of arbitrarily curved spatial Bernoulli-Euler beams. The complete metric of the beam is utilized in order to include the effect of curviness on the nonlinear distribution of axial strain over the cross section. The exact constitutive relation between energetically conjugated pairs is employed, along with four reduced relations. The isogeometric approach, which allows smooth connections between finite elements, is used for the spatial discretization of the weak form. Two methods for updating the local basis are applied and discussed in the context of finite rotations. All the requirements of geometrically exact beam theory are satisfied, such as objectivity and path-independence. The accuracy of the formulation is verified by a thorough numerical analysis. The influence of the curviness on the structural response is scrutinized for two classic examples. If the exact response of the structure is sought, the curviness must be considered when choosing the appropriate beam model.

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A total Lagrangian Timoshenko beam formulation for geometrically nonlinear isogeometric analysis of spatial beam structures

Cited by 22 publications

References 49 publications

Corotational Finite Element Dynamic Analysis of Space Frames with Geometrically Nonlinear Behavior Based on Tait–Bryan Angles

Corotational Finite Element Dynamic Analysis of Space Frames with Geometrically Nonlinear Behavior Based on Tait–Bryan Angles

Geometrically exact static isogeometric analysis of arbitrarily curved plane Bernoulli-Euler beam

Geometrically exact static isogeometric analysis of an arbitrarily curved spatial Bernoulli-Euler beam

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