A Triangular Plate Bending Element Based on Discrete Kirchhoff Theory with Simple Explicit Expression

A Simple three-node Discrete Kirchhoff Triangular (SDKT) plate bending element is proposed in this study to overcome some inherent difficulties and provide efficient and dependable solutions in engineering practice for thin plate structure analyses. Different from the popular DKT (Discrete Kirchhoff Theory) triangular element, using the compatible trial function for the transverse displacement along the element sides, the construction of the present SDKT element is based on a specially designed trial function for the transverse displacement over the element, which satisfies interpolation conditions for the transverse displacements and the rotations at the three corner nodes. Numerical investigations of thin plate structures were conducted, using the proposed SDKT element. The results were compared with those by other prevalent plate elements, including the analytical solutions. It was shown that the present element has the simplest explicit expression of the nine-DOF (Degree of Freedom) triangular plate bending elements currently available that can pass the patch test. The numerical examples indicate that the present element has a good convergence rate and possesses high precision.

DYNAMIC BIAXIAL STRESS ANALYSIS OF FLAT LAYERED CERAMIC COMPOSITES

We study theoretically the biaxial bending of symmetric, flat layered ceramic composites (laminates) due to external loading. We focus on three-layered alumina/zirconia laminates. We compare the principal stresses in the samples in the case of static and harmonic dynamic loading. The dynamic equation within the Kirchhoff theory for thin homogeneous plates is first generalized to the case of multilayered plates. It is solved numerically with the relaxation method, which we have developed for this purpose.

On the applicability limits of refined theories in describing of the flexural edge wave in plates

Исследуются пределы применимости уточненных теорий изгиба пластины при описании дисперсии изгибной краевой волны и амплитуды её возбуждения парой сосредоточенных скручивающих моментов, приложенных на торце. Методом численного сравнения с решением трехмерной задачи показано, что теория типа Тимошенко пригодна для описания краевой волны на частотах, не превосходящих 30% от первой частоты запирания. Уточненная теория изгиба пластин с приведенной инерцией в сочетании с классическими граничными условиями позволяет уточнить скорость волны по сравнению с теорией Кирхгофа, но значительно искажает амплитуду. The applicability limits of refined plate bending theories in describing of the flexural edge wave dispersion and its excitation amplitude are investigated. The wave is excited by a pair of twisting couples applied to the edge of the plate. Numerical comparison with the solution of 3D problem shows that Uflyand-Mindlin theory is applicable at the frequencies up to 30% of the first cut-off. The higher order asymptotic theory of plate bending with modified inertia and classical boundary conditions allows to improve the describing of the velocity comparing to Kirchhoff theory, but leads to a considerable error in describing of the amplitude.

The Transverse Shear Effect of a New Strain Based Sector Finite Elements for Plate Bending Problems

In this paper, we present the transverse shear effect on the plate bending. The element used is a sector finite element called SBSP (Strain Based Sector Plate-Kirchhoff Theory-), it used for the numerical analysis of circular thin plate bending., and it based on the strain approach. This element has four nodes and three degrees of freedom per node. Through the numerical applications with different loading cases and boundary conditions; This makes the present element robust, better suitable for computations.

Numerical Simulation of Bending Stiffness Analysis for Spring Linkage Applied to In-Pipe Robot

Spring linkage can be applied to in-pipe robots for connecting different modules together and can make it pass through elbows more easily. However, its stiffness cannot be set to be too hard or too soft. This paper tries to make a balance between the compressive stiffness and the bending stiffness of the spring. After a brief introduction to the construction mechanism and some assumptions, the mathematical representation of the spring bending stiffness was deduced based on the Kirchhoff theory which describes the spatial curve with displacement rather than time. Then, some simulations aiming at verifying the correctness of the deduced bending stiffness expression were carried out. Finally, the relationship between the two rigidities was found out, which helps to find a way to decrease the bending stiffness of spring while keeping its compressive stiffness strong enough.

Physical modeling and geometry configuration simulation for flexible cable in a virtual assembly system

Purpose In this paper, a novel and unified method for geometry configuration simulation of flexible cable under certain boundary conditions is presented. This methodology can be used to realize cable assembly verification in any computer-aided design/manufacturing system. The modeling method, solution algorithm, geometry configuration simulation and experimental results are presented to prove the feasibility of this proposed methodology. The paper aims to discuss these issues. Design/methodology/approach Considering the gravity, bending and torsion, modeling of cable follows the Kirchhoff theory. For this purpose, Euler quaternions are used to describe its spatial geometry configuration by a carefully chosen set of coordinates. Then the cable is discretized by the FEM, and the equilibrium condition per element is computed. In this way, the global static behavior is independent of the discretization. The static evolution of the cable is obtained by numerical integration of the resulting Kirchhoff equations. Then the manner is demonstrated, in which this system of equations can be decoupled and efficiently solved. Geometry configuration simulation examples with different boundary conditions are presented. Finally, experiment validation are given to describe the effectiveness of the models and algorithms. Findings The method presented in this paper can be adapted to computer-aided assembly verification of flexible cable. The experimental results indicate that both of the model and algorithm are efficient and accurate. Research limitations/implications The method should be extended to flexible cables with multiple branches and more complex constraints (holes, curved surfaces and clamps) and non-circular sections. Dynamic assembly process simulation based on the Kirchhoff theory must be considered in the future. Originality/value Unlike in previous approaches, the cable behavior was independent of the underlying discretization, and the finite element approach enables physically plausible cable assembly verification.

Computation of elastic equilibria of complete Möbius bands and their stability

Determining the equilibrium configuration of an elastic Möbius band is a challenging problem. In recent years numerical results have been obtained by other investigators, employing first the Kirchhoff theory of rods and later the developable, ruled-surface model of Sadowsky–Wunderlich. In particular, one such strategy used does not deliver an equilibrium configuration for the complete unsupported strip. Here we present our own systematic approach to the same problem for each of these models, with the ultimate goal of assessing the stability of flip-symmetric configurations. The presence of point-wise constraints considerably complicates the latter step. We obtain the first stability results for the problem, numerically demonstrating that such equilibria render the total potential energy a local minimum. Along the way we introduce a novel regularization for the singular Wunderlich model that delivers equilibria for complete strips having sufficiently narrow widths, which can then be tested for stability.