The use of Timoshenko's exact solution for a cantilever beam in adaptive analysis

Augarde, Charles E.; Deeks, Andrew

doi:10.1016/j.finel.2008.01.010

Cited by 63 publications

(31 citation statements)

References 18 publications

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“…For the FEM approximation for the displacement u of the 2D plane stress model, we use eight node quadrilateral elements, see, for example, the work of Fish and Belytschko . This choice was also used in the works of Labuschagne et al and Aguarde and Deeks, which consider similar problems to the one considered in this paper. For the FEM approximation of the variables ( v T , θ ) of the Timoshenko beam model, we employ the two node Timoshenko beam elements as shown in the work of Friedman and Kosmatka that provides a stiffness matrix, which is exactly integrated and is free of shear‐locking.…”

Section: Forward Problem and Damage Modelmentioning

confidence: 99%

Modeling errors due to Timoshenko approximation in damage identification

Castello

Kaipio

2019

Numerical Meth Engineering

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Summary The use of accurate computational models for damage identification problems may lead to prohibitive costs. Damage identification problems are often characterized as inverse ill‐posed problems. Thus, the use of approximate models such as simplified physical and/or reduced‐order models typically yields misleading results. In this paper, we carry out a preliminary study on a particular simplified physical model, the Timoshenko beam model in the context of damage identification. The actual beam is a two‐dimensional relatively high aspect ratio (thickness/length) beam with a distributed damage that is modeled as a spatially varying Young modulus. We state the problem in the Bayesian framework for inverse problems and carry out approximative marginalization over the related modeling errors. The numerical experiments suggest that the proposed approach yields more stable results than using the Timoshenko beam model as an accurate model. Due to the severity of the Timoshenko approximation, however, the posterior error estimates of the proposed approach are not always feasible in the probabilistic sense.

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Section: Forward Problem and Damage Modelmentioning

confidence: 99%

Modeling errors due to Timoshenko approximation in damage identification

Castello

Kaipio

2019

Numerical Meth Engineering

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“…The clamped cantilever beam is a standard benchmark for 2D elasticity, consisting of a rectangular beam, which is subject to Dirichlet boundary conditions at one end and to a shear load at the opposite one. Depending on how the boundary condition and the loading are defined, the obtained solution with exhibit a very different regularity . In order to produce a smooth solution, we here take the shear load g to be parabolic on the loaded edge and the essential boundary conditions along the clamped edge are applied according to the analytical solution given by Timoshenko and Goodier:

u_{x} = \frac{P y}{6 Ē I} ((), false(6 L - 3 x false) x + false(2 + \overset{ν}{true} false) y^{2} - \frac{3 D^{2}}{2} false(1 + \overset{ν}{true} false)),

u_{y} = \frac{P}{6 Ē I} ((), 3 \overset{ν}{true} y^{2} false(L - x false) + false(3 L - x false) x^{2}),

where

Ē = E false/ false(1 - ν^{2} false)

\overset{ν}{true} = ν false/ false(1 - ν false)

, and I is the second‐area moment of the beam section.…”

Section: Numerical Experimentsmentioning

confidence: 99%

kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains

Quaglino

Krause

2018

Numerical Meth Engineering

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Summary We propose a novel finite‐element method for polygonal meshes. The resulting scheme is hp‐adaptive, where h and p are a measure of, respectively, the size and the number of degrees of freedom of each polygon. Moreover, it is locally meshfree, since it is possible to arbitrarily choose the locations of the degrees of freedom inside each polygon. Our construction is based on nodal kernel functions, whose support consists of all polygons that contain a given node. This ensures a significantly higher sparsity compared to standard meshfree approximations. In this work, we choose axis‐aligned quadrilaterals as polygonal primitives and maximum entropy approximants as kernels. However, any other convex approximation scheme and convex polygons can be employed. We study the optimal placement of nodes for regular elements, ie, those that are not intersected by the boundary, and propose a method to generate a suitable mesh. Finally, we show via numerical experiments that the proposed approach provides good accuracy without undermining the sparsity of the resulting matrices.

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“…We considered the indeterminate beam problem shown in Figure 3, which is a fixed end supported beam applied by a Dirac delta function [12,13]. The exact integration and reduced integration were used to analyze the problems.…”

Section: The Fixed End Supported Beam Applied By a Dirac Delta Functimentioning

confidence: 99%

1st‐Order Shear Deformable Beam Formulation Based on Meshless Wavelet Galerkin Method

Lee

2017

Mathematical Problems in Engineering

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This paper examined and discussed a Meshless Wavelet Galerkin Method (MWGM) formulation for a first-order shear deformable beam, the properties of the MWGM, the differences between the MWGM and EFG, and programming methods for the MWGM. The first-order shear deformable beam (FSDB) consists of a pair of second-order elliptic differential equations. The weak forms of two differential equations are deduced using Hat wavelet series. The exact integration and reduced integration were used to analyze the problems. Some indeterminate beam problems are considered. Condition numbers of the stiffness matrix were analyzed with exact integration and reduced integration for two cases of these problems. Consequently, the results were converged on the analytic solutions. The shear-locking phenomenon also occurred in the MWGM as it occurs in the conventional FEM. The stiffness matrix calculated from the reduced integration causes a similar numerical error to the stiffness matrix calculated from the exact integration in the MWGM. The MWGM showed desirable results in the examples.

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The use of Timoshenko's exact solution for a cantilever beam in adaptive analysis

Cited by 63 publications

References 18 publications

Modeling errors due to Timoshenko approximation in damage identification

Modeling errors due to Timoshenko approximation in damage identification

kFEM: Adaptive meshfree finite‐element methods using local kernels on arbitrary subdomains

1st‐Order Shear Deformable Beam Formulation Based on Meshless Wavelet Galerkin Method

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