Zigzag Function Research Articles

Refined multilayered plate elements based on Murakami zig–zag functions

This paper discusses the FEM use of Murakami zig–zag function (MZZF) in the two-dimensional modeling of multilayered plates. A literature overview is first conducted on the available works in which MZZF has been applied. MZZF is used to introduce the zig–zag effect in classical and higher order theories, which are formulated using only displacements as unknowns. MZZF is also considered to introduce the zig–zag effect in those theories, which are formulated on the basis of both displacement and transverse stress assumptions (mixed formulations). The present FEM formulation is validated by comparing the results with some available papers from the literature. A very thick plate ( a/ h = 4) is studied and the results are compared with the commercial code NASTRAN. Numerical results are presented to show both the effectiveness and limitations of MZZF in the modeling of layered plates. Linear up to forth order expansion for in-plane and out-of-plane displacements, in the thickness plate direction, has been compared. It has been concluded that MZZF consists of a valuable tool to enhance the performances of both classical and advanced theories. In particular, the conducted numerical evaluations have shown that it can be more convenient to enhance a plate theory by introducing the MZZF than refining it by adding two or three higher order terms. However, in order to well approximate local effects or study thick plates, advanced models (layer-wise or three-dimensional) are required.

A unified formulation to assess theories of multilayered plates for various bending problems

This work uses a unified formulation to compare about 40 theories for multilayered, composites and sandwich plates which are loaded by transverse pressure with various in-plane distributions (harmonic, constant, triangular and tent-like). So-called equivalent single layer models (ESLMs), which preserve the number of the unknown variables to be independent by the number of layers, as well as layer-wise models (LWMs) are considered in both framework of principle of virtual displacement (PVD) and Reissner Mixed Variational Theorem (RMVT). Murakami’s Zig-Zag Function is used to introduce zig-zag (ZZ) effects while independent assumptions for transverse stresses (both shear and normal components) are used to enforce interlaminar continuity (IC) between two adjacent layers. Linear and higher order expansions (fourth-order) are introduced for displacements and stresses in the thickness plate directions. The fundamental effects of transverse strain is evaluated for most of the considered analyses. The whole modeling has been herein written by employing a unified formulation recently proposed by the first author. As a results a large number of classical and advanced theories for laminated structures are formulated and the related governing equations are written in terms of so-called fundamental nuclei with only nine terms each. Navier-type, closed form solutions of these equations are presented for orthotropic plates by expanding the applied pressure loading in Fourier series. A number of conclusions have been traced as far as performance and limitations of compared theories is concerned. Quoted results could be used as benchmarks to assess available theories not considered in this paper as well as approximated solution techniques, such as finite element applications.

On the use of the Murakami's zig-zag function in the modeling of layered plates and shells

This paper discusses the use of the Murakami zig-zag function (MZZF) in the two-dimensional modeling of multilayered plates and shells. A literature overview of the available works is first presented. A `simple use' of the MZZF is discussed: the MZZF is used to introduce the zig-zag effect in classical and higher order theories which are formulated with only displacement unknowns. An `advanced use' of the MZZF is then considered to introduce the zig-zag effect in those theories which are formulated on the basis of both displacement and transverse stress assumptions. A number of new plate/shell theories has been considered. Numerical results encompassing, static, dynamic and thermally loaded orthotropic, simply supported plates and shells are presented to show both the effectiveness and limitations of the MZZF in the modeling of layered structures. Linear up to forth-order expansions for the in-plane and the out-of-plane displacements, in the thickness plate/shell direction, have been compared. It has been concluded that the MZZF is a valuable tool to enhance the performances of both classical and advanced theories. The conducted numerical evaluations have shown in particular that multilayered plate and shell theories can be greatly improved by the use of MZZF.

On the constructions of highly nonlinear zigzag functions and unbiased functions

In this paper, a number of methods for constructing highly nonlinear zigzag functions and unbiased functions are presented.

A 3D Zig-Zag Sublaminate Model for Analysis of Thermal Stresses in Laminated Composite and Sandwich Plates

A laminated plate theory and 3D finite element model based on first-order zig-zag sublaminate approximations are presented for thermal stress analysis of composite laminates and sandwich plates. The finite element is developed with the topology of an eight-noded brick, allowing the thickness of the plate to be discretized into several elements, or sublaminates, where each sublaminate can contain more than one physical layer. The temperature field is first computed by a thermal model, where the through-thickness distribution of temperature is assumed to vary linearly within each ply, and continuity of transverse flux at ply interfaces is enforced analytically. Similarly, the in-plane displacement fields in each sublaminate are assumed to be piecewise linear functions and vary in a zig-zag fashion through the thickness of the sublaminate. The zig-zag functions are evaluated by enforcing the continuity of transverse shear stresses at layer interfaces. The formulation also enforces continuity of the transverse normal stresses, a key to accurate 3D predictions of the thermal stresses. The novel features of the formulation allow accurate and efficient prediction of the distribution of temperatures, displacements and stresses in laminated plates and sandwich plates wherein the plies have dissimilar thermal and/or structural properties.

A 3D Zig-Zag Sublaminate Model for Analysis of Thermal Stresses in Laminated Composite and Sandwich Plates

A laminated plate theory and 3D finite element model based on first-order zig-zag sublaminate approximations are presented for thermal stress analysis of composite laminates and sandwich plates. The finite element is developed with the topology of an eight-noded brick, allowing the thickness of the plate to be discretized into several elements, or sublaminates, where each sublaminate can contain more than one physical layer. The temperature field is first computed by a thermal model, where the through-thickness distribution of temperature is assumed to vary linearly within each ply, and continuity of transverse flux at ply interfaces is enforced analytically. Similarly, the in-plane displacement fields in each sublaminate are assumed to be piecewise linear functions and vary in a zig-zag fashion through the thickness of the sublaminate. The zig-zag functions are evaluated by enforcing the continuity of transverse shear stresses at layer interfaces. The formulation also enforces continuity of the transverse normal stresses, a key to accurate 3D predictions of the thermal stresses. The novel features of the formulation allow accurate and efficient prediction of the distribution of temperatures, displacements and stresses in laminated plates and sandwich plates wherein the plies have dissimilar thermal and/or structural properties.

First-order zig-zag sublaminate plate theory and finite element model for laminated composite and sandwich panels

A refined laminated plate theory and three-dimensional finite element based on first-order zig-zag sublaminate approximations has been developed. The in-plane displacement fields in each sublaminate are assumed to be piecewise linear functions and vary in a zig-zag fashion through-the-thickness of the sublaminate. The zig-zag functions are evaluated by enforcing the continuity of transverse shear stresses at layer interfaces. This in-plane displacement field assumption accounts for discrete layer effects without increasing the number of degrees of freedom as the number of layers is increased. The transverse displacement field is assumed to vary linearly through-the-thickness. The transverse normal strain predictions are improved by assuming a constant variation of transverse normal stress in each sublaminate. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of an eight-noded brick, allowing the thickness of the plate to be discretized into several elements, or sublaminates, where each sublaminate can contain more than one physical layer. Each node has five engineering degrees of freedom, three translations and two rotations. Thus, this element can be conveniently implemented into general purpose finite element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of an interdependent interpolation scheme and consistent shear strain fields. Numerical performance of the current element is investigated for a composite armored vehicle panel and a sandwich panel. These tests demonstrate that the element is very accurate and robust.

Solution of a laminated cylindrical shell using an unconstrained third-order theory

An unconstrained third-order shear deformation theory is presented for the analysis of laminated anisotropic cylindrical shells. Based on the realistic through-thickness distribution of the in-plane displacements, a zig-zag function is used to approximate the piecewise nature of the displacements. The zero-shearing condition on the laminate surfaces and continuous conditions for the transverse shear stresses on the inter-laminar surfaces have been considered for the final stresses calculation, while the displacement functions remain to be unconstrained. This theory is very useful for finite element analysis because it requests only C° continuity for the assumed displacement field. By comparison with three-dimensional elasticity theory for laminated orthotropic cylindrical shells, the performance of the present theory is verified. The problems solved in this paper illustrate that the present theory is very accurate for thin and moderately thick shells.

A solution of laminated cylindrical shells using an unconstrained third-order theory

An unconstrained third-order shear deformation theory is presented for the analysis of laminated anisotropic cylindrical shells. Based on the realistic through-thickness distribution of the in-plane displacements, a zig-zag function is used to approximate the piece-wise nature of the displacements. The zero-shearing condition on the laminate surfaces and continuous conditions for the transverse shear stresses on the inter-laminar surfaces have been considered for the final stresses calculation, the displacement functions remain to be unconstrained. This theory is very useful for the finite element analysis because it requests only C 0 continuity for the assumed displacement fields. By comparing with three-dimensional elasticity theory for laminated orthotropic cylindrical shell, the performance of the present theory is verified. The problems solved in this paper illustrate that the present theory is very accurate for the thin and moderately thick shells.

On the Foundations of the Babylonian Column ?: Astronomical Significance of Partial Sums of the Lunar Four

A characteristic feature of the Babylonian mathematical astronomy is the use of periodically varying functions in the form of sequences of numbers (e.g. arithmetic progressions, zig-zag functions, or piecewise constant step functions) to describe periodically occurring astronomical phenomena. One major achievement of the Babylonian astronomers consists in a very precise determination of the periods of the number sequences used in their ephemeris texts. Any reconstruction of the Babylonian calculation schemes must explain how the fundamental periods or period relations can be determined empirically by such astronomical observations as were compiled in the Babylonian Diaries. This paper is concerned with the Babylonian moon ephemerides. The fundamental periods used here are the length P⊙ of the more empirically founded and less theoretically than believed until now.

Refined shear deformation theory of laminated shells

This paper analyzes the effect of shear deformation in thick laminated anisotropic shells using a mixed formulation based on the functional proposed by Jing and Liao. The displacement field uses a zigzag function in addition to the Reissner-Mindlin type in-plane displacements and a constant transverse deflection. The effect of transverse shear deformation is included through an independently assumed transverse shear stress field. The initial curvature effect is included in the strain-displacement relations, stress resultants, and the assumed shear stress field

Diagrams Venn and How

The fascinating drawing by Boyd [1] shows a Venn diagram consisting of five rectangles. It suggested to us the desirability of finding aesthetically pleasing Venn diagrams with arbitrarily many sets, possessing moreover the property that the appropriate distribution of regions is obvious on sight. It is easy to see that Boyd's method cannot work for more than five sets. Instead, we propose here another simple construction of Venn diagrams-one that may be of particular interest to students taking a course in finite mathematics. Our presentation makes use of base 2 arithmetic, the notion of absolute value, and a simple mathematical induction; on the other hand, each of these devices can be eliminated either by appropriate explanations, or by simple handwaving. A general discussion of Venn diagrams, including historic and bibliographic references, can be found in [2] and [3]. It is generally (though not universally) agreed that in a Venn diagram for n sets a) the boundary of each set should be a simple closed curve in the plane, and b) these curves should determine precisely 2 n regions, each one a connected set of the form X1 n X2 n .. X nXn where X is chosen to be either the interior or the exterior of the jth set. We shall understand Venn diagrams in this sense, adding to these conditions the requirement that the boundary of each set should be easily distinguishable by sight from the boundaries of the n 1 other sets. We require further that the boundaries be easily constructed polygons. We begin with a rule for labelling the regions in Venn diagrams. First the sets of a given Venn diagram are labelled by integers from 1 to n. (In the illustrative example shown in FIGURE 1 we have n = 6 and have used Roman numerals for the set labels.) Then each, of the 2 n regions is labelled by the binary number an ... a3a2a1, where a= 1 if the region is in the set labelled j and aj =0 if the region is in the complement of the set j. For brevity, we write this binary label in the decimal notation; thus 9 stands for 001001, and is the label of the region contained in the sets 1 and 4 and outside the sets 2, 3, 5, and 6. Note that with this labelling the difference between labels of two regions adjacent along the boundary of the jth set is 2i'. Such a labelling is useful in various situations (cf. [4]). Now for the promised construction of a Venn diagram for n sets. First let's define the zigzag function y = C(x) as a periodic function with period 2 by setting C(x) = Ix-11-1/2 for 0 < x < 2. For j = 1,2,3... let the set V, be bounded by x = 0, x = 1, y-1/2 and the zigzag

Length of the Day in Jaina Astronomy

According to post-Vedānga pre-Siddhāntic Jaina canonical texts, 18 muhūrtas (1 muhūrta = 48 minutes) and 12 muhūrtas are respectively the greatest and the smallest lengths of daylight. The length of any day (daylight) of the year can be computed therefrom through a simple linear zigzag function. However, the ratio 3: 2 between the greatest and the smallest length of daylight holds good for a latitude 35° north very near to that of Babylon. This ratio is also propounded in Jyotisa Vedāngam. It is exposed here that the ratio 3: 2 holds equally good for Gandhāra, a renowned seat of ancient Indian culture. Gandhāra might have been used for purposes like those of a standard station for purposes of time-reckoning in ancient India. However, by applying a correction for the variable rate of flow of water through the orifice of a water-clock, it has been exposed that the actual ratio of maximum and minimum lengths of daylight measured in time-units is different from the ratio of the amounts of water to be poured into the water-clock on the respective days. It is revealed that the ratio 3:2 of the amounts of water to be poured into the water-clock on the longest and shortest days (daylights) is equivalent to the ratio √3: √2 of the actual time-lengths of the longest and the shortest days respectively. It refers to a latitude 19° 6 north, very near to that of Ujjainī, another renowned seat of ancient Indian culture.

Kinetische Analyse von komplexen Desorptionsreaktionen

The kinetic parametersĒ and\(\bar N\) can be determined with the aid of fractional thermodesorption. The application of a zig-zag temperature-time function provides the possibility to measure\(\left( {\frac{{\partial \ln ( - \dot \theta )}}{{\partial 1/T}}} \right)_{\theta ,stat}\) at an approximately constant surface coverage. The stationarity of the system will be guaranteed if the desorption rates for the increasing and decreasing parts of a temperature zig-zag are nearly the same. The stationarity can be controlled during the experiment and may be corrected by changing the temperature-time programme. The experimental fractional thermodesorption procedure is demonstrated using the system methanol/zeolite 13X as example.