Orthotropic Structures Research Articles

Initial Investigation of Energy Finite Element Validation on High‐Frequency Flexural Vibration of Stiffened Thin Orthotropic Plates

Energy finite element analysis (EFEA) has unique advantages in solving high‐frequency dynamic responses of orthotropic structures, due to its ability to obtain detailed local response information. In order to accurately predict high‐frequency vibration response of the stiffened orthotropic plate, EFEA theory on the propagation of bending wave in the orthotropic structure and the energy transfer coefficient which express the energy transfer at the stiffener was investigated. Based on the EFEA theory presented, high‐frequency dynamic responses of a stiffened orthotropic plates were predicted. Furthermore, tests were done for the same problem, and differences between prediction and test were discussed. Finally, the future works were pointed out.

Steady heat transfer analysis of orthotropic structure based on Element-Free Galerkin method

The mathematical model of steady heat transfer analysis for orthotropic structure is established based on Element-Free Galerkin (EFG) method and the essential boundary conditions are enforced using penalty method. The reliability of heat transfer model and MATLAB programs have been verified through three engineering examples in practice. The effects of nodes distribution pattern, scaling factors, penalty factors and weight functions on the EFG temperature results of different orthotropic structures are evaluated. A comparison of the CPU computing time between EFG and FEM is also studied. By comparison with finite element temperature and temperature gradient, it is revealed that the temperature based on EFG has higher calculation precision for steady heat transfer problem, and the temperature is not affected by nodes distribution pattern. The rational values of scaling factors and penalty factors are provided respectively. In addition, the computational relative error increases with the decrease of the orthotropic factor during various weight functions, and the cubic spline and parabola weight functions are the preferred choices in the heat transfer analysis of the orthotropic structure.

Computational contact mechanics analysis of laterally graded orthotropic half-planes

PurposeFrictional sliding contact problems between laterally graded orthotropic half-planes and a flat rigid stamp are investigated. The presented study aims at guiding engineering applications in the prediction of the contact response of orthotropic laterally graded members.Design/methodology/approachThe solution procedure is based on a finite element (FE) approach which is conducted with an efficient FE analysis software ANSYS. The spatial gradations of the orthotropic stiffness constants through the horizontal axis are enabled utilizing the homogeneous FE approach. The Augmented Lagrangian contact algorithm is used as an iterative non-linear solution method in the contact analysis.FindingsThe accuracy of the proposed FE solution method is approved by using the comparisons of the results with those computed using an analytical technique. The prominent results indicate that the surface contact stresses can be mitigated upon increasing the degree of orthotropy and positive lateral gradations.Originality/valueOne can infer from the literature survey that, the contact mechanics analysis of orthotropic laterally graded materials has not been investigated so far. In this study, an FE method-based computational solution procedure for the aforementioned problem is addressed. The presented study aims at guiding engineering applications in the prediction of the contact response of orthotropic laterally graded members. Additionally, this study provides some useful points related to computational contact mechanics analysis of orthotropic structures.

Bending and buckling of a metal seven-layer beam with crosswise corrugated main core – Comparative analysis with sandwich beam

The study is devoted to thin-walled simply supported sandwich beams: three and seven-layer beams. Seven-layer beams are composed of a trapezoidally crosswise corrugated main core and two three-layer facings with trapezoidally lengthwise corrugated cores. Essential difference between such a structure and classical three-layer beams consists in the fact that the facings are of three-layer structure too. Hence, the structure is a seven-layer beam, the layers of which are distinguished by different properties. The thin-walled beam with three trapezoidally corrugated cores is an innovatory orthotropic structure, not referred to in the literature. Characteristic feature of the beam consists in differentiation of shear effects in particular layers, according to the core corrugation direction. Deformation of the cross section of the beam depends on the direction too. An original mathematical model is formulated, which includes the hypothesis of deformation of the cross section of the beam, inclusive of the displacement and strain fields, and rigidities of the layers in particular directions. The adequate model of three-layer beam with steel foam core is also formulated. Basing on the principle of the total potential energy the equations of equilibrium are derived for three and seven-layer beams, respectively. Then the equations are analytically solved. In case of seven-layer beams the deflections and the critical axial force are determined for different values of the trapezoidal corrugation pitch of the main core. The results are compared with adequate for three-layer beam and presented in Tables. It is concluded that such seven-layer beam – plate band with three-layered facings is more resistant to the load than the classical three-layered beams of the same mass with regard to strength and stability. Additional effect of the research consists in generalization of classical theories of sandwich plates.

Stress Analysis for the Orthotropic Pressurized Structure of Cylindrical Shell with Nozzle

The structure of cylindrical shell with nozzle has been widely used in pressure equipment. The analysis and design for the structure of cylindrical shell with nozzle are generally based on the isotropy of metal at present, which may cause certain error and risk. Because of the approaches of rolling, heat treatment, and so on, the materials’ properties are different along each direction in practical engineering. In this paper, based on anisotropic thin shell theory, the analysis for orthotropic pressurized structure of thin-walled cylindrical shell with nozzle was conducted by using finite element software Ansys. The influences of thickness ratio and elastic modulus ratio were comprehensively considered, and also compared with the stress results under isotropic structure. The analysis results show that the values of stress and stress concentration factor for the structure of cylindrical shell with nozzle decrease with elastic modulus ratio, but increase with thickness ratio. Compared with the influence of thickness ratio, the elastic modulus ratio has significant effect on stress level. The rational use of the material anisotropic property can optimize the structure of thin-walled cylindrical shell with nozzle. The results in present paper provided a reference for the design of orthotropic pressurized structure of cylindrical shell with nozzle.

The influence of fiber-crack angle on the crack tip parameters in orthotropic materials

Identification of the critical angle between fiber and crack direction in orthotropic materials for avoiding catastrophic failure is necessary. Recognition of the optimum regions for creating notches in orthotropic materials for creation of maximum load-bearing capability is an important parameter in structural design. In this paper, the critical angles between crack and fiber direction are predicted, extracting crack tip parameters in orthotropic materials using stress series expansion and numerical method. The variations of crack tip parameters with respect to the angle between crack and fiber for modes I and II are presented. The presented functions for these variations can be used in the optimum design procedure of orthotropic structures. The obtained results are validated using experimental study.

Extension of the variational theory of complex rays to orthotropic shallow shell structures

Nowadays, the interest of aerospace and automotive industries on virtual testing of medium-frequency vibrational behavior of shallow shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e., FEM, SEA) are not fully suitable for the medium-frequency bandwidth. In this context the Variational Theory of Complex Rays (VTCR) is taking place as an ad-hoc technique to address medium-frequency problems. It is a Trefftz method based on a weak variational formulation. It allows great flexibility because any shape function that satisfies the governing equations can be used. This work further develops such theory. In particular, orthotropic materials are introduced in the VTCR formulation for shallow shell structures. A significant numerical example is proposed to show the strategy.

Approach to Critical Crack Opening Displacement Modeling of Damage in Metal Sheets after Composite Patch Bonded Repair

Abstract The paper proposes a method of calculating the maximum displacement in the aircraft metal structure repaired by CPBR (bonded composite patch repair). The calculations were made based on specimens. The specimens were prepared according to the current requirements used in aviation. The 2024-T3 alloy metal sheet was a structure. To repair the structure used the boron-epoxy composite patch in prepreg technology was used. The metal structure was modeled as an isotropic elastic body. The metal structure coated with the composite patch was modeled as an orthotropic structure. Based on this, the stress was determined in the metal structure. The size opening displacement in the metal structure was determined based on the model of linear elastic fracture mechanics for the plane stress state. The calculation results were verified by measuring the displacement in laboratory conditions. The laboratory tests made it possible to demonstrate the accuracy of the proposed approach.

The orthotropic elastic properties of fibrolamellar bone tissue in juvenile white-tailed deer femora.

Fibrolamellar bone is a transient primary bone tissue found in fast-growing juvenile mammals, several species of birds and large dinosaurs. Despite the fact that this bone tissue is prevalent in many species, the vast majority of bone structural and mechanical studies are focused on human osteonal bone tissue. Previous research revealed the orthotropic structure of fibrolamellar bone, but only a handful of experiments investigated its elastic properties, mostly in the axial direction. Here we have performed for the first time an extensive biomechanical study to determine the elastic properties of fibrolamellar bone in all three orthogonal directions. We have tested 30 fibrolamellar bone cubes (2×2×2mm) from the femora of five juvenile white-tailed deer (Odocoileus virginianus) in compression. Each bone cube was compressed iteratively, within its elastic region, in the axial, transverse and radial directions, and bone stiffness (Young's modulus) was recorded. Next, the cubes were kept for 7days at 4°C and then compressed again to test whether bone stiffness had significantly deteriorated. Our results demonstrated that bone tissue in the deer femora has an orthotropic elastic behavior where the highest stiffness was in the axial direction followed by the transverse and the radial directions (21.6±3.3, 17.6±3.0 and 14.9±1.9 Gpa, respectively). Our results also revealed a slight non-significant decrease in bone stiffness after 7days. Finally, our sample size allowed us to establish that population variance was much bigger in the axial direction than the radial direction, potentially reflecting bone adaptation to the large diversity in loading activity between individuals in the loading direction (axial) compared with the normal (radial) direction. This study confirms that the mechanically well-studied human transverse-isotropic osteonal bone is just one possible functional adaptation of bone tissue and that other vertebrate species use an orthotropic bone tissue structure which is more suitable for their mechanical requirements.

An explanation of warp knitted fabric deformation under uni-axial tension by a novel index as shape ratio

Although the Poisson’s ratio (PR) can predict the deformation of engineering materials under uniaxial tension, but seems that it cannot predict well the deformation of structures such as fabrics. It can be attributed to the lack of continuity among constituent elements that leads to some distortion of them during tension. Therefore, the PR is not a suitable tool to explain precisely the deformation of structures such as fabrics. For this reason, a new geometrical index was defined to quantify the deformation of fabrics in order to use instead of PR. This index named shape ratio is obtained by division of the variation of squared perimeter to the area of a printed square on the fabric under uniaxial tension. To check the defined index, warp-knitted fabrics with various structures such as Tricot, Locknit, and Three- and Four-needle satin were produced and tested using tensile tester. The results show that shape ratio can predict the deformation of warp-knitted fabrics as an orthotropic structure rather than the PR.

直交異方性シェル構造の材料配向角と形状の最適化

直交異方性シェル構造の材料配向角と形状の最適化

Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement

This paper presents a damping loss factor estimation method of two-dimensional orthotropic structures. The method is based on a scanning laser vibrometer measurement. The dispersion curves of the studied structures are first estimated at several chosen angles of propagation with a spatial Fourier transform. Next the global damping loss factor is evaluated with the proposed inverse wave method. The method is first tested using numerical results obtained from a finite element model. The accuracy of the proposed method is then experimentally investigated on an isotropic aluminium panel and two orthoropic sandwich composite panels with a honeycomb core. The results are finally compared and validated over a large frequency band with classical methods such as the half-power bandwidth method (3dB method), the decay rate method and the steady-state power input method. The present method offers the possibility of structural characterization with a simple measurement scan.

Thermal Stress Analysis of an Embedded Crack in a Graded Orthotropic Coating-Substrate Structure

An analysis of a coupled plane thermoelastic problem for a graded orthotropic coating-substrate structure is performed under thermomechanical loading conditions. The crack direction is parallel to the free surface. Applying the superposition principle and Fourier integral transform, the heat conduction and plane elasticity equations lends themselves to the derivation of two sets of Cauchy-type singular integral equations. The thermal stress intensity factors are defined and evaluated. In the numerical results, the effects of the orthotropy parameters, thermoelastic non-homogeneity parameters, and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors (TSIFs) are studied. The obtained results can be used to design graded orthotropic coating-substrate structures under thermomechanical loading.

Structural dynamics of electric machine stators: Modelling guidelines and identification of three-dimensional equivalent material properties for multi-layered orthotropic laminates

Simulating the dynamic behaviour of heterogeneous finite-element structures such as electric motors often requires to homogenise the models in the first place. Current homogenisation methods do not always imply computing an equivalent homogeneous material׳s elasticity matrix and are often restrained to specific uses. In this document, a novel approach of equivalent material identification is developed for multi-layered orthotropic structures. A finite-element model of a 3D stratified structure is created, as well as its equivalent homogeneous medium. The dynamic behaviour of the homogeneous structure with the equivalent material identified by the new method is compared at low frequencies to the reference stack and to equivalent materials created using other existing homogenisation techniques. It is shown that this approach is more accurate than existing reference homogenisation methods. Applied to the magnetic core׳s finite-element model of a real laminated electric machine stator, the method enables simulating the experimental behaviour with good accuracy, without need of time-consuming model updating procedures.

Vibration Modeling of Multilayer Composite Structures with Viscoelastic Layers

This work deals with the vibration of orthotropic multilayer sandwich structures with viscoelastic core. A finite element model is derived from a classical zigzag model with shear deformation in the viscoelastic layer. The aim of the present work is to establish numerical models and develop numerical tools to design multilayer composites structures with high damping properties. To fulfill this purpose, a finite element model has been developed for vibration analysis of a sandwich plate (elastic orthotropic)/(viscoelastic orthotropic)/(elastic orthotropic). A numerical study from the variation of the damping properties of the structures was performed according to the faces materials fibers orientation.

A Trefftz approach for medium-frequency vibrations of orthotropic structures

The Variational Theory of Complex Rays (VTCR) was developed in order to calculate the vibrational response of structures in the medium-frequency range. It leads to a numerical approximation of this response through the resolution of a small system of equations which, contrary to element-based methods, does not result from a refined spatial discretization. This strategy has already been validated for assemblies of isotropic plates. In this paper, we extend it to the case of orthotropic planar structure. The method is illustrated through various orthotropic plate vibration problems under several types of harmonic loading in the medium-frequency range.

Strengthening the Orthotropic Steel Deck Structure of the Movable Bridge across the Hartelkanaal, The Netherlands

This paper focuses on the fatigue assessment and the strengthening of the orthotropic steel deck structure of the movable bridge across the Hartelkanaal. The chosen strengthening method involves adhesively bonding an additional steel plate to the existing deck plate. Prior to carrying out this bonding, the existing fatigue cracks are repaired. The bonding procedure is carried out in situ using a vacuum infusion process.

Mode I Crack Problem for a Functionally Graded Orthotropic COATING-Substrate Structure

In this paper, the Fourier integral transform-singular integral equation method is presented for the Mode I crack problem of the functionally graded orthotropic coating-substrate structure. The elastic property of the material is assumed vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the boundaries of the strip. Numerical examples are presented to illustrate the effects of the crack length, the material nonhomogeneity and the thickness of coating on the stress intensity factors.

Three-Dimensional Exact Free Vibration Analysis of Spherical, Cylindrical, and Flat One-Layered Panels

The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate two-dimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).

Computing the broadband vibroacoustic response of arbitrarily thick layered panels by a wave finite element approach

A robust procedure for the prediction of the dynamic response of layered panels within a SEA wave-context approach is proposed hereby. The dispersion characteristics of two dimensional composite orthotropic structures are predicted using a Wave Finite Element method. By manipulating the mass and stiffness matrices of the modelled structural segment a polynomial eigenvalue problem is formed, the solutions of which correspond to the propagation constants of the waves travelling within the structure. The wavenumbers and group velocities for waves comprising out of plane structural displacements can then be calculated. Using the numerically extracted wave propagation data the most important SEA quantities of the structure are calculated, namely the modal density and the radiation efficiency of each wave type. The vibroacoustic response of the structure under a broadband diffused excitation is then computed within a SEA approach. The impact of the symmetric and the antisymmetric vibrational motion of the panel on its Sound Transmission Loss is exhibited and the approach proves robust enough for thin as well as for thick layered structures.