Limitations of certain curved finite elements when applied to arches

Ashwell, D.G.; Sabir, A.B.

doi:10.1016/0020-7403(71)90017-8

Cited by 87 publications

(24 citation statements)

References 4 publications

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“…In the linear regime, from early on, Timoshenko-or Mindlin-type curved beam and arch elements reportedly su ered from what is commonly known as shear and membrane locking in the thin regime [14]. This, in turn, triggered, for last 30 years, varieties of interpolationally inadequate approaches with limited success in terms of accuracy, rate of convergence and general applicability.…”

Section: Background and Motivationmentioning

confidence: 98%

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

Ray¹

2003

Numerical Meth Engineering

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SUMMARYComputer-aided mesh generation (CAMG) dictated solely by the minimal key set of requirements of geometry, material, loading and support condition can produce 'mega-sized', arbitrary-shaped distorted elements. However, this may result in substantial cost saving and reduced bookkeeping for the subsequent ÿnite element analysis (FEA) and reduced engineering manpower requirement for ÿnal quality assurance. A method, denoted as c-type, has been proposed by constructively deÿning a ÿnite element space whereby the above hurdles may be overcome with a minimal number of hyper-sized elements. Bezier (and de Boor) control vectors are used as the generalized displacements and the Bernstein polynomials (and B-splines) as the elemental basis functions. A concomitant idea of coerced parametry and inter-element continuity on demand uniÿes modelling and ÿnite element method. The c-type method may introduce additional control, namely, an inter-element continuity condition to the existing h-type and p-type methods. Adaptation of the c-type method to existing commercial and general-purpose computer programs based on a conventional displacement-based ÿnite element method is straightforward. The c-type method with associated subdivision technique can be easily made into a hierarchic adaptive computer method with a suitable a posteriori error analysis. In this context, a summary of a geometrically exact non-linear formulation for the two-dimensional curved beams=arches is presented. Several beam problems ranging from truly three-dimensional tortuous linear curved beams to geometrically extremely non-linear two-dimensional arches are solved to establish numerical e ciency of the method. Incremental Lagrangian curvilinear formulation may be extended to overcome rotational singularity in 3D geometric non-linearity and to treat general material non-linearity.

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Section: Background and Motivationmentioning

confidence: 98%

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

Ray¹

2003

Numerical Meth Engineering

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“…Higher order beam theories have been proposed to model the cross-sectional warping and to remove the shear correction factor [7][8][9]. These early attempts [1][2][3][4][5][6] are unsuccessful and su er from an excessive bending sti ness, called membrane locking, in the limit of inextensional bending or excessive shear sti ness, called shear locking, in the limit of thin beam. To alleviate these numerical di culties, special techniques based on the most popular minimum potential energy principle are proposed, such as the selective=reduced integration technique [5,6,10], ÿeld-consistent element [11,12] and anisoparametric element [13], etc.…”

Section: Introductionmentioning

confidence: 99%

“…A great deal of e ort has been devoted in recent years to the development of suitable curved beam elements. The majority of curved beam elements are of the C 1 -continuous type based on the Bernoulli-Euler theory [1][2][3]. A number of shear exible arch elements based on the Timoshenko beam theory for thick beams have also been developed [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

An effective composite laminated curved beam element

Kim

2005

Commun. Numer. Meth. Engng.

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SUMMARYIn this study, we present a new highly accurate composite laminated hybrid-mixed curved beam element. The present element, which is based on the Hellinger-Reissner variational principle and the ÿrst-order shear deformation lamination theory, employs consistent stress parameters corresponding to cubic displacement polynomials with additional nodeless degrees of freedom in order to resolve the numerical di culties due to the spurious constraints. The stress parameters are eliminated and the nodeless degrees are condensed out to obtain the (6 × 6) element sti ness matrix. Several numerical examples conÿrm the superior behaviour of the present hybrid-mixed laminated curved beam element.

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“…For the standard finite element method, namely the h-version method, locking occurs in the thin arch limit [2]. Many efforts have been focused on overcoming the locking effects, for example, mixed methods which use mixed variational principles (see [8], [12], [15], and [20]- [22]), the PetrovGalerkin method [11] in which the test function space differs from the trial function space, and reduced integration methods (see [12], [14], [16], and [18]- [22]).…”

Section: Introductionmentioning

confidence: 99%

Locking and robustness in the finite element method for circular arch problem

Zhang

1995

Numerische Mathematik

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In this paper we discuss locking and robustness of the finite element method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order p −k and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order h −2 , and is uniformly robust with order h p−2 for p > 2 which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress. Classification (1991): 65N30, 73K05, 73V05 Mathematics Subject

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Limitations of certain curved finite elements when applied to arches

Cited by 87 publications

References 4 publications

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

An effective composite laminated curved beam element

Locking and robustness in the finite element method for circular arch problem

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