Arbitrary Cross-sectional Shape Research Articles

Stress-Strain State of two Diagonal Cavities Weighty Inclining Layered Massif System with Slots in Terms of Elastic-Creep Deformations

This paper investigated the stress-strain state of two diagonal cavities of arbitrary cross-sectional shape and the depth of a weighty inclining massif with system slots in terms of elastic and elastic-creep deformation rocks. This study was based on the anisotropic mechanical-mathematical model of inclining multilayer massif with a doubly periodic system of slots studied numerically, the patterns of distribution of elastic-creep stresses, displacements near two diagonal cavities of derived shape, and the depth of the Finite Element Method in generalized plane strain. Calculation algorithm is a designed and compiled software package for the study of the elastic-creep state of adjacent cavities of derived depth and shapes. Multivariate numerical calculation and analysis of the effect on the components of stresses and displacements near the cavities geometrics, physical parameters, and creep properties of rocks were conducted.

Group Invariant Solutions of the Full Plastic Torsion of Rod with Arbitrary Shaped Cross Sections

Based on the theory of Lie group analysis, the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied. Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters. Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions. Moreover, physical explanations of each group invariant solution are discussed by all appropriate transformations. The methodology and solution techniques used belong to the analytical realm.

Diffraction and radiation loads on open cylinders of thin and arbitrary shapes

We study the effects of having an opening in a vertical cylinder of an arbitrary cross-sectional shape when subjected to incident waves from the outside field. Both the diffraction and radiation problems of linear potential-flow theory are addressed. The cylinders considered are bottom-mounted with vanishing thickness and the problem is formulated as a hypersingular boundary-integral method. A simple higher-order procedure is presented to handle the strong singularity. Comparisons are made between the hydrodynamic properties of open and closed cylinders, and the effects of increasing the opening size are discussed and explained. Results for square, circular and elliptical open cylindrical shells, presented as examples, indicate that wave loads on the structures could be dramatically decreased to zero (effectively) at certain frequencies and opening sizes. This leads to the surprising conclusion that directing an open structure into the incident-wave field results in lower loads on the structure. A model of an open harbour with a frontal breakwater in a wave field, inspired by the idea of the Portunus Project, is also analysed.

A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses

When analysing the structural behaviour of a thin-walled member by means of Generalised Beam Theory (GBT) – a one-dimensional folded-plate theory expressing the member deformed configuration as a linear combination of cross-section deformation modes with amplitudes varying along its length – the performance of the so-called “cross-section analysis” is the key step. Indeed, it consists of determining the deformation modes and evaluating the corresponding modal mechanical properties. However, the available procedures to perform this task are strongly dependent on the cross-section geometry type, a feature precluding its general, efficient and systematic numerical implementation. In order to overcome this difficulty, this paper presents the development and illustrates the application of a novel procedure to perform “GBT cross-section analyses” that (i) is able to handle arbitrary (flat-walled) cross-section shapes, (ii) can be numerically implemented in a systematic and straightforward fashion, and (iii) provides a rational set of deformation modes, which are hierarchically organised into several families, each with well-defined structural/mechanical characteristics. Both the analytical derivations and the underlying mechanical reasoning are explained in detail, and the selected illustrative examples cover the various types of relevant cross-section geometries.

A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods

In this work, we suggest a locking-free geometrically exact finite element formulation incorporating the modes of axial tension, torsion and bending of thin Kirchhoff beams with arbitrary initial curvatures and cross-section shapes. The proposed formulation has been designed in order to represent general load cases and three-dimensional problem settings in the geometrically nonlinear regime of large deformations. From this comprehensive theory, we not only derive a general beam model but also several reduced formulations, which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads. The advantages of these reduced models arise for example in terms of simplified finite element formulations, less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models. A second core topic of this publication is the treatment of membrane locking, which is a locking phenomenon predominantly occurring in highly slender curved structures, thus, exactly in the prime field of application for Kirchhoff theories. In order to address the membrane locking effect, we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as Assumed Natural Strains (ANS) or reduced integration. The effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples.

The kern of non-homogeneous cross sections

The state of stresses in non-homogeneous bars due to eccentric axial loads is examined. The governing formulae are written in coordinate free invariant forms by means of vector and tensor algebra. The concepts of centric and eccentric axial loads for non-homogeneous bars are introduced. A general solution is developed to obtain the kern of cross section of arbitrary shape. Examples show how the results derived can be applied to determine the kern of a given inhomogeneous cross section.

Complete photonic band gaps and tunable self-collimation in the two-dimensional plasma photonic crystals with a new structure

In this paper, the properties of complete photonic band gaps (CPBGs) and tunable self-collimation in two-dimensional plasma photonic crystals (2D PPCs) with a new structure in square lattices, whose dielectric fillers (GaAs) are inserted into homogeneous and nomagnetized plasma background are theoretically investigated by a modified plane wave expansion (PWE) method with a novel technique. The novel PWE method can be utilized to compute the dispersion curves of 2D PPCs with arbitrary-shaped cross section in any lattices. As a comparison, CPBGs of PPCs for four different configurations are numerically calculated. The computed results show that the proposed design has the advantages of achieving the larger CPBGs compared to the other three configurations. The influences of geometric parameters of filled unit cell and plasma frequency on the properties of CPBGs are studied in detail. The calculated results demonstrate that CPBGs of the proposed 2D PPCs can be easily engineered by changing those parameters, and the larger CPBGs also can be obtained by optimization. The self-collimation in such 2D PPCs also is discussed in theory under TM wave. The theoretical simulations reveal that the self-collimation phenomena can be found in the TM bands, and both the frequency range of self-collimation and the equifrequency surface contours can be tuned by the parameters as mentioned above. It means that the frequency range and direction of electromagnetic wave can be manipulated by designing, as it propagates in the proposed PPCs without diffraction. Those results can hold promise for designing the tunable applications based on the proposed PPCs.

Rational design of capillary-driven flows for paper-based microfluidics.

The design of paper-based assays that integrate passive pumping requires a precise programming of the fluid transport, which has to be encoded in the geometrical shape of the substrate. This requirement becomes critical in multiple-step processes, where fluid handling must be accurate and reproducible for each operation. The present work theoretically investigates the capillary imbibition in paper-like substrates to better understand fluid transport in terms of the macroscopic geometry of the flow domain. A fluid dynamic model was derived for homogeneous porous substrates with arbitrary cross-sectional shapes, which allows one to determine the cross-sectional profile required for a prescribed fluid velocity or mass transport rate. An extension of the model to slit microchannels is also demonstrated. Calculations were validated by experiments with prototypes fabricated in our lab. The proposed method constitutes a valuable tool for the rational design of paper-based assays.

Stiffness Constants of Homogeneous, Anisotropic, Prismatic Beams

This paper presents a complete set of analytical expressions for the stiffness constants of a generalized Euler–Bernoulli beam theory for homogeneous, anisotropic, prismatic beams with arbitrary cross-sectional shapes. These expressions are extracted from exact solutions of the linear equations of three-dimensional elasticity for the cases of loading by axial forces, torques, and bending moments about two orthogonal directions. Closed-form expressions are derived for the extensional stiffness and the extension-related coupling terms. Expressions for the remaining stiffness constants are derived in terms of the torsional stiffness: the expression of which is in terms of a function that needs to be obtained. The resulting expressions reveal both similarities and differences from its isotropic and orthotropic counterparts. For elliptical, anisotropic cross sections and rectangular, orthotropic cross sections, all stiffness constants are known in closed form. These closed-form expressions constitute a standard with which the ability of two-dimensional beam cross-sectional analyses to model material anisotropy may be assessed. The calculated stiffness constants, from one such cross-sectional analysis, are successfully validated in this manner.

Nonlinear performance of continuum mechanics based beam elements focusing on large twisting behaviors

In this paper, we present the nonlinear formulation and performance of continuum mechanics based beam elements, in which fully coupled 3D behaviors of stretching, bending, shearing, twisting, and warping are automatically considered. The beam elements are directly degenerated from assemblages of 3D solid elements under the assumptions of Timoshenko beam theory. Therefore, cross-sectional discretization is possible and the elements can model complicated 3D beam geometries including curved and twisted geometries, varying cross-sections, eccentricities, and arbitrary cross-sectional shapes. In particular, the proposed nonlinear formulation can accurately predict large twisting behaviors coupled with stretching, bending, shearing, and warping. Through various numerical examples, we demonstrate the geometric (and material) nonlinear performance of the continuum mechanics based beam elements.

Bending Method for Forming Curvature and Various Cross-Section Shapes

Previous studies on the bending process of extruded aluminum alloys have focused on a high-precision bending process that controls various undesirable phenomena. In contrast, this paper proposes a new bending method in which undesirable deformations are intentionally occurred and can be deformed into an arbitrary cross-sectional shape. By bending the extruded section, we sought to establish a novel bending process, in which a curvature and cross-sectional shape are produced in a single process. Undesirable deformations that occur by bending channel materials with a removed tensile flange include both the outward and inward deformations of the webs. Based on such deformations, this report proposes several types of cross-sectional shapes that can be formed through simulations and press bending experiments.

An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods

The objective of this work is the development of a new finite element formulation for beams according to the Kirchhoff theory of thin rods, which includes the deformation states of axial tension, torsion and bending. The proposed formulation accounts for large deformations in three-dimensional problem settings consisting of slender, prismatic beams with arbitrarily curved initial geometries and arbitrary cross-section shapes. Like in geometrically exact Reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations, rotations and strains. A novel orthogonal interpolation strategy is applied to the triad field representing the cross-section orientation in order to fulfill the Kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem. The continuity requirements resulting from the corresponding weak form are fulfilled by a cubic Hermite interpolation, thus leading to a C1-continuous representation of the beam centerline. Fundamental properties such as objectivity, path-independence, consistency and accuracy of the developed beam element are verified by means of suitable numerical examples.

Adiabatic eigenflows in a vertical porous channel

AbstractThe existence of an infinite class of buoyant flows in a vertical porous channel with adiabatic and impermeable boundary walls, called adiabatic eigenflows, is discussed. A uniform heat source within the saturated medium is assumed, so that a stationary state is possible with a net vertical through-flow convecting away the excess heat. The simple isothermal flow with uniform velocity profile is a special adiabatic eigenflow if the power supplied by the heat source is zero. The linear stability analysis of the adiabatic eigenflows is carried out analytically. It is shown that these basic flows are unstable. The only exception, when the power supplied by the heat source is zero, is the uniform isothermal flow, which is stable. The existence of adiabatic eigenflows and their stability analysis is extended to the case of spanwise lateral confinement, viz. in the case of a vertical rectangular channel. A generalisation of this study to a vertical channel with an arbitrary cross-sectional shape is also presented.

A theoretical model to predict gas permeability for slip flow through a porous medium

Based on slip flow at pore level a theoretical model is presented to predict the gas permeability and thereby the overall pressure drop for flow through a porous medium. The model maps the porous structure to a number of parallel micro-channels of arbitrary but constant cross-sectional shapes which remains uniform along the flow path. The gas permeability is found to follow Klinkenberg's equation. The Klinkenberg's slip factor is obtained as a function of matrix porosity and no-slip permeability as well as gas properties. Results are generalized by assuming an arbitrary polygonal shape for the pores. The proposed methodology is simple to follow and easy to implement. Theoretical predictions are then compared to existing experimental data in the literature to observe good agreement.

Torsion of an elastic rod whose cross-section is a parallelogram, trapezoid, or triangle, or has an arbitrary shape by the method of transformation to a rectangular domain

A coordinate transformation is used to take the domain of the rod cross-section to a rectangular domain for which the spectra of eigenfunctions and eigenvalues are known. The torsion function is represented as a generalized Fourier series to reduce the problem to solving a closed linear system of algebraic equations for the expansion coefficients. It is shown that these Fourier series converge absolutely, because the expansion coefficients decrease by a cubic law depending on the term number. We prove that the approximate solution in the form of a finite sum of the Fourier series converges to the exact solution. This theorem is generalized to the case of a rod cross-section of arbitrary shape.

Elastic instability and free vibration analyses of tapered thin-walled beams by the power series method

In this paper, the flexural-torsional buckling and free vibration of tapered thin-walled beam-columns with arbitrary cross-section shape are extensively investigated. The governing equilibrium equations and motion equations are obtained from the stationary condition of the potential energy. The strain energy is derived in presence of initial stresses. In the work of the applied forces, effects of load eccentricities from shear and centroid centerlines are taken into account. Free vibration is considered in presence of harmonic excitations. In presence of arbitrary boundary conditions and variable cross-section properties, a semi-analytical approach based on power series method is adopted in solution. According to this method, displacements and geometric constants are approximated by polynomial functions up to a certain order, where accurate results are reached. The flexural-torsional buckling loads or natural frequencies are determined by solving an eigenvalue problem. In order to measure the accuracy and to check the validity of the present method, several examples including flexural-torsional behavior and free vibration analysis of non-prismatic thin-walled members with web and flange tapering and various boundary conditions are considered. The obtained results are compared to the finite element simulations and other available solutions.

A positivity preserving central scheme for shallow water flows in channels with wet-dry states

We present a high-resolution, non-oscillatory semi-discrete central scheme for one- dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow- water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

Quantification of wall shear stress using finite-element interpolations in multidimensional phase contrast MR data of the thoracic aorta

Background Different methods have been proposed to estimate Wall Shear Stress (WSS). Morgan and co-workers (J Thorac Cardiovasc Surg and Ann Biomed Eng 1998) used a finite difference scheme to quantify the WSS tensor. However, it is well known that finite-difference methods cannot effectively handle complex geometries, as those found in the cardiovascular system. To account for arbitrary cross-section shapes, Stalder et al., MRM 2008, used B-spline (BS) interpolations to smoothly describe the lumen contours. In this work, we propose and validate a new method for calculating the WSS distribution based on Finite-Element (FE) interpolations.

A computer method for nonlinear inelastic analysis of 3D composite steel–concrete frame structures

This paper presents an efficient computer method for nonlinear inelastic analysis of three-dimensional composite steel–concrete frameworks. The proposed formulation is intended to model the geometrically nonlinear inelastic behaviour of composite frame elements using only one element per physical member. The behaviour model accounts for material inelasticity due to combined bi-axial bending and axial force, gradual yielding is described through basic equilibrium, compatibility and material nonlinear constitutive equations. In this way, the states of strain, stress and yield stress are monitored explicitly during each step of the analysis, the arbitrary cross-sectional shape, various stress–strain relationships for concrete and steel and the effect of material imperfections such as residual stresses are accurately included in the analysis. Tangent flexural rigidity of cross-section is derived and then using the flexibility approach the elasto-plastic tangent stiffness matrix and equivalent nodal loads vector of 3-D beam-column element is developed. The method ensures also that the ultimate strength capacity of the cross-section is nowhere exceeded once a full plastified section develops. The proposed nonlinear analysis formulation has been implemented in a general nonlinear static purpose computer program. Several computational examples are given to validate the effectiveness of the proposed method and the reliability of the code to approach large-scale spatial frame structures.

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.