Singularities in polynomial representations of transverse shear in finite elements

AIAA Journal ◽  
1997 ◽  
Vol 35 ◽  
pp. 1248-1250
Author(s):  
James M. Greer ◽  
Anthony N. Palazotoo
Keyword(s):  
Finite Elements ◽  
Transverse Shear ◽  
Polynomial Representations

Singularities in Polynomial Representations of Transverse Shear in Finite Elements

AIAA Journal ◽  
10.2514/2.233 ◽  
1997 ◽  
Vol 35 (7) ◽  
pp. 1248-1250
Author(s):  
James M. Greer ◽  
Anthony N. Palazotto
Keyword(s):  
Finite Elements ◽  
Transverse Shear ◽  
Polynomial Representations

Mechanics and Finite Elements of Shells

1989 ◽  
Vol 42 (5) ◽  
pp. 129-142 ◽  
Author(s):  
Gerald Wempner
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
General Theory ◽  
Transverse Shear ◽  
Three Dimensional ◽  
Discrete Models ◽  
Thick Shell ◽  
Dimensional Approximation ◽  
Transverse Strains

This article begins with a brief review of the foundations: The classical theory of Love is described with attention to the underlying hypothesis and consequent limitations. A more general theory is described which removes the constraints of Love; the inclusion of transverse strains admits simpler finite elements, accommodates the thick shell via layers and even a transition to the three-dimensional approximation. The concept of the finite element is reviewed in the context of the discrete approximation of shells. Specific attention is given to those problems which are peculiar to shells: The predominant roles of flexural and extensional deformations, the lesser role of transverse shear, can lead to excessive stiffness (“locking”). Origins and procedures are described to circumvent these problems. The review is intended to bridge some chasms between the mechanics of the continua and the discrete models of finite elements. As such, the emphasis is upon those mechanical attributes of shells and elements which play key roles in forming practical models. Since the limitations of space, time and the author’s knowledge, preclude a full expose, the review includes only commentaries on some topics, such as inelasticity, nonlinearity and instability. Citations include original sources and some recent works which provide entree to contemporary developments.

Effect of Assumed Tow Architecture on Predicted Moduli and Stresses in Plain Weave Composites

1995 ◽  
Vol 29 (16) ◽  
pp. 2134-2159 ◽  
Author(s):  
Clinton Chapman ◽  
John Whitcomb
Keyword(s):  
Finite Elements ◽  
Cross Section ◽  
Transverse Shear ◽  
Material Properties ◽  
Three Dimensional ◽  
Textile Composites ◽  
Unit Cell ◽  
Accurate Prediction ◽  
Constant Cross Section ◽  
Plain Weave

This paper examines the effect of assumed tow architecture on the predicted moduli and stresses in plain weave textile composites. In particular, the effect of how a constant cross-section is assumed to sweep-out the volume of a tow is explored. Two architectures are examined which have a sinusoidal tow path and a lenticular cross-section. Three-dimensional finite elements are employed to model a T300/Epoxy plain weave composite with symmetrically stacked mats. Macroscopically homogeneous in-plane extension and shear and transverse shear loadings were considered. Symmetries are exploited which permitted modeling of only 1/32nd of the unit cell. Accounting for the variation of material properties throughout each element is determined to be necessary for accurate prediction of stresses in the composite. For low waviness, the two tow architectures examined are very similar. At high waviness, the stress predictions are much more sensitive to the assumed tow geometry.

A General Theory of Shells and the Complementary Potentials

1986 ◽  
Vol 53 (4) ◽  
pp. 881-885 ◽  
Author(s):  
G. Wempner
Keyword(s):  
Finite Elements ◽  
General Theory ◽  
Linear Function ◽  
Transverse Shear ◽  
Surface Deformation ◽  
Reference Surface ◽  
Shear Deformations ◽  
Theory Of Shells

This theory incorporates the attributes which are essential to the approximation of shells by finite elements. It is limited only by one assumption: Displacement is a linear function of distance along the normal to a reference surface. Deformation is decomposed into rotation and strain; the rotation carries elements of the reference surface to the same surface in any subsequent state. Transverse-shear deformations accommodate simple elements. The theory is couched in the potential Pv and in the complementary potential Pc; these have the property, Pv + Pc= 0 for all admissible (equilibrated) states. The theory is also cast in the complementary functional P¯c of stress and displacement, and the functional P¯v of displacement, strain and stress; P¯c and P¯v are akin to the functionals of Hellinger-Reissner and Hu-Washizu. These alternate functionals provide the means to develop various hybrid elements.

scholarly journals DEFORMATION TRANSVERSE SHEAR BENDING STATE OF A THIN PLATE LAYER OF AN ANISOTROPIC GEOLOGICAL MEDIUM FROM THE ACTION OF CONCENTRATED ENERGY IMPULSES

2021 ◽  
Vol 1 ◽  
pp. 117-121
Author(s):  
Ilya Kolesnikov ◽  
Viktor Tatarinov ◽  
Tatiana Tatarinova
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Transverse Shear ◽  
Earth’S Crust ◽  
Real Problem ◽  
Geological Medium ◽  
Deformation State ◽  
Earth's Crust ◽  
Shear Bending

A method is proposed for study the structural stability of the deformation state of structural blocks of the earth's crust, approximated in the form of plate layers of the geological medium when transverse shear bending from the action of concentrated energy impulses. Advances here are carried out in the two directions. First, in contrast to the previous article, the physical and mechanical model of the geological medium is endowed with anisotropic properties, which makes it possible to increase the adequacy of the obtained numerical results to the specifics of the real problem. Secondly, instead of the simplest bilinear 4-node finite elements, the special spectral non-algebraic 8-node finite iso-parametric finite elements are used, the use of which significantly increases both the accuracy of calculations and their reliability in the sense of ensuring the robustness of calculations for relatively small values of the plate thickness. It should be noted that the Finite Element Method uses exclusively only algebraic finite elements (power polynomials in the h-version and orthogonal polynomials in the p-version). It is known from approximation theory that the use of spectral non-algebraic approximations improves the quality of approximations. Therefore, their introduction into the structure of finite element calculations can improve the quality of modeling in the study of the strain-stress-state (SSS) of the geological medium. A structural block (SB) is understood as a plate layer with plan dimensions exceeding the thickness by more than 10 times. The identification of hazardous zones in the rock massive due to stress concentration is complemented by the development of mechanical, mathematical and computational tools for modeling the curvature of the earth's crust during bending based on the classical theory of Kirchhoff and refined Reissner-Mindlin theory. Test calculations have shown that the accuracy of the calculation and the quality of geometric modeling of fragments of an anisotropic geological environment based on the refined 8-node spectral finite element is significantly better than for the 8-node algebraic finite

Nonlinear Through-Thickness Behavior of a Toroidal Shell Using 2-D Finite Elements

2001 ◽  
Author(s):  
James M. Greer ◽  
Anthony N. Palazotto
Keyword(s):  
Finite Elements ◽  
Transverse Shear ◽  
Strain Energy ◽  
Shell Thickness ◽  
Three Dimensional ◽  
Toroidal Shell ◽  
Two Dimensional ◽  
Qualitative And Quantitative ◽  
Pipe Elbow ◽  
State Of Stress

Abstract The use of two-dimensional shell finite elements is explored for finding the three-dimensional state of stress in a toroidal shell. The torus under study represents a 90-degree pipe elbow with a pressure load on a portion of its surface. Layer-wise polynomials are used to represent the transverse shear and normal stretch deformations in the shell. These functions are chosen such that displacements and stresses (but not strains) are continuous at the ply interfaces. Both isotropic and composite (cross-ply) versions of the shell are investigated, and the thicknesses of each are varied to see the effect on through-thickness behavior. Significant qualitative and quantitative differences in these behaviors are observed, particularly in the important direct through-thickness (peeling) stress. The contribution of the transverse deformations to strain energy is investigated and, in most of the shells studied, the thickness stretch component is found to be a greater contributor to strain energy than the transverse shear, though the transverse shear contribution is seen to vary more dramatically with changes in shell thickness.

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