Vibration analysis of laminated anisotropic shells of revolution

An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction.The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique.The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.     

A finite element model for the nonlinear analysis of reinforced and prestressed masonry walls

A materially nonlinear layered finite element model is proposed for the analysis of reinforced and/or prestressed masonry wall panels under monotonie loadings in the plane and/or out of the plane, capable of evaluating both the serviceability load and the ultimate load. An orthotropic incrementally linear relationship and equivalent uniaxial concept are used to represent the behaviour of masonry under biaxial stresses while a uniaxial bilinear elasto-plastic model with hardening is employed for rebar and the so-called ‘power-formula’ is adopted to describe the stress-strain relationship of prestressing steel.

After cracking, the smeared coaxial rotating crack model is adopted and tension stiffening, reduction in compressive strength and stiffness after cracking, and strain softening in compression are accounted for. The modified Newton-Raphson iteration method is employed to ensure convergency of non linear solution.

The proposed finite element model has been tested by a comparison with experimental data available in literature, both for reinforced and prestressed wall panels. The analysis of results shows good agreement between the values obtained by the proposed model and those obtained experimentally.     

Solution of elastic crack problems by superposition of finite elements and singular fields

A new finite element method is presented for the solution of plane elasticity problems which contain nonremovable stress singularities. Singular stress field are combined with finite element solutions by a superposition technique; an important feature of the method is that use of the singular fields may be restricted to any specified group of elements which include the singular point. It is shown that good estimates for stress intensity factors are obtained when the method is applied to crack problems.     

Stress intensity factors for an interfacial crack between an orthotropic half-plane bonded to a dissimilar orthotropic layer with a punch

The plane strain problem of determining Stress Intensity Factors (SIF) for a moving interfacial Griffith crack between an elastic orthotropic half-plane and a dissimilar orthotropic layer with a moving punch situated along the boundary of the layer have been considered. The problem is reduced to the solution of three simultaneous singular integral equations with Cauchy-type singularities. Expressions for SIF for the case of a general loading are obtained. Numerical results for some particular cases are also presented graphically.     

求解应力强度因子的样条虚边界元交替法

为求解平面裂纹问题的应力强度因子,提出基于Muskhelishvili基本解和样条虚边界元法的样条虚边界元交替法.该方法将平面内带裂纹有限域问题分解成带裂纹无限域问题与不带裂纹有限域问题的叠加.带裂纹无限域问题利用Muskhelishvili基本解法直接得出,不带裂纹有限域问题采用样条虚边界元法求解.利用该方法对复合型中心裂纹方板和I型偏心裂纹矩形板进行分析.数值结果表明该方法精度高且适用性强.     

Hybrid-finite-element analysis of some nonlinear and 3-dimensional problems of engineering fracture mechanics

In this paper, efficient numerical methods for the analysis of crack-closure effects on fatigue-crackgrowth-rates, in plane stress situations, and for the solution of stress-intensity factors for arbitrary shaped surface flaws in pressure vessels, are presented. For the former problem, an elastic-plastic finite element procedure valid for the case of finite deformation gradients is developed and crack growth is simulated by the translation of near-crack-tip elements with embedded plastic singularities. For the latter problem, an embedded-elastic-singularity hybrid finite element method, which leads to a direct evaluation of K-factors, is employed.     

Fracture mechanics based fatigue analysis of steel bridge decks by two-level cracked models

An alternative fatigue assessment is introduced for orthotropic highway bridge decks, based on fracture mechanics. Crack growth simulation is carried out by the numerical integration of the Paris formula, using K factors obtained from two-level cracked models of the bridge deck. Prior to the simulation, unit wheels are applied to the 3D-shell model of the deck, and the translations of characteristic points are transferred to the 2D plane strain or plane stress finite element sub-models of fatigue-sensitive areas. The K factor is computed by numerical extrapolation, introducing cracks in the sub-models. Fixed values of input parameters (initial- and critical crack length, crack growth rate data, and the error of K factor computation) are used for single simulations as well as parametric studies for sensitivity analysis. Reliability assessment is performed by repeated crack growth analysis, using histograms to generate input parameters by Monte-Carlo simulation at the start of each “virtual experiment”.     

Design study of hole positions and hole shapes for crack tip stress releasing

The method of hole drilling near or at the crack tip is often used in fatigue damage repair. From a design optimization point of view, two questions are posed: Where should the hole(s) be drilled? And is there a better shape of the hole than a circular one? For the first question, we extend earlier results for isotropic material and in general study the influence of having orthotropic material. Optimal shapes are by no means circular, and we focus on the shape of a single hole centered at (or in front of) the crack tip. It is shown that the stress field at the crack boundary can be significantly improved by noncircular shapes. As a byproduct, an alternative method for extracting the stress intensity factor from a finite element analysis is presented.     

Elastodynamic fields near running cracks by finite elements

A finite element method is developed for the computation of elastodynamic stress intensity factors at a rapidly moving crack tip. The method is restricted to bodies whose surfaces and two-material interfaces are either parallel to the direction of propagation or are sufficiently remote. The crack tip starts to move at the instant that it is struck by an incident wave. The finite element grid moves undeformed with the crack tip. The main result consists in the fact that the method of non-singular calibrated crack tip elements, in which the stress-intensity factor is determined from its statically calibrated ratio to the crack opening displacement in an adjacent node, is extended to dynamic problems with moving cracks, for both in-plane and anti-plane motions. The dependence of the calibration ratio on the crack tip velocity is established from previously developed analytical solutions for the near-tip displacement fields. Numerical results compare favorably with known analytical solutions for cracks moving in an infinite solid. The grid motion causes an apparent asymmetric additional damping matrix.     

Nonlinear finite element analysis for composite structures of axisymmetric geometry and loading

A linear strain, axisymmetric, triangular finite element is formulated for the modeling of materials which are generally orthotropic in the plane of the fibers and loaded axisymmetrically. The fibers may be oriented in planes other than the symmetric plane.This necessitates the use of three displacement degrees of freedom at each node.The element is also formulated to include geometric nonlinearity to take into account any stiffening or weakening of the structure due to deformation.The element is programmed into a finite element code, which is validated using a well-known isotropic nonlinear problem. It shows very good agreement with the results of other programs.The unique application of the element is shown in the analysis of a pneumatic tire.     

正交异性钢桥肋-桥面板焊缝裂纹的三维断裂力学分析

正交异性钢桥的肋-桥面板焊缝处的疲劳裂纹是典型的三维裂纹问题,但是现在普遍采用平面应变二维裂纹模型对其进行断裂力学分析.基于Schwartz-Neuman交替法建立正交异性钢桥肋-桥面板焊缝裂纹的局部三维断裂力学分析模型;评估焊缝处表面裂纹的形状和深度对应力强度因子的影响;采用Paris公式估算等应力幅下焊缝的疲劳寿命.计算结果表明:用平面应变二维裂纹模型进行正交异性钢桥的肋-面板焊缝的断裂力学分析会严重低估其疲劳寿命;采用三维断裂力学模型进行肋-桥面板焊缝裂纹的疲劳寿命分析十分必要.     

Physical modeling with orthotropic material based on harmonic fields

Although it is well known that human bone tissues have obvious orthotropic material properties, most works in the physical modeling field adopted oversimplified isotropic or approximated transversely isotropic elasticity due to the simplicity. This paper presents a convenient methodology based on harmonic fields, to construct volumetric finite element mesh integrated with complete orthotropic material. The basic idea is taking advantage of the fact that the longitudinal axis direction indicated by the shape configuration of most bone tissues is compatible with the trajectory of the maximum material stiffness. First, surface harmonic fields of the longitudinal axis direction for individual bone models were generated, whose scalar distribution pattern tends to conform very well to the object shape. The scalar iso-contours were extracted and sampled adaptively to construct volumetric meshes of high quality. Following, the surface harmonic fields were expanded over the whole volumetric domain to create longitudinal and radial volumetric harmonic fields, from which the gradient vector fields were calculated and employed as the orthotropic principal axes vector fields. Contrastive finite element analyses demonstrated that elastic orthotropy has significant effect on simulating stresses and strains, including the value as well as distribution pattern, which underlines the relevance of our orthotropic modeling scheme.     

Dynamic finite element analysis of laminated beams with delamination cracks using contact-impact conditions

A new finite element modeling technique is presented to investigate the static and dynamic behavior of laminated composite beams with partial delamination. In this study, a recently developed rectangular beam element is used. The element has lateral and axial displacements as degrees of freedom, but not rotation. For simplicity, linear shape functions are used for the beam element. As a result, the element has six degrees of freedom, four of which are the axial nodal displacements at the corner points and the other two are the lateral displacments at the ends. In addition, contact-impact conditions are applied to the finite element modeling to avoid overlapping of the upper and lower portions of a delaminated section. The numerical study shows that, depending on existence of an embedded delamination crack and its size, the response is different for a beam with a crack and subjected to a short impulse load. Hence, the present modeling technique may be used for detection of an embedded delamination crack.     

Crack path bifurcation at a tear strap in a pressurized shell

Bifurcation of an initially longitudinal through crack in an internally pressurized cylindrical shell at a circumferential stiffener is investigated using a finite element analysis. The finite element model is developed from a fracture test of an aluminum shell having a 22.9 cm radius, a 1.02 mm wall thickness, and stiffened by two externally bonded circumferential straps spaced 40.6 cm apart. After initial stable crack growth in the longitudinal direction with increasing pressure, the crack propagated dynamically toward the strap, bifurcated near the strap into circumferential branches running parallel to the straps. Stable and unstable crack growth curves of pressure versus half-crack length are determined from the nonlinear analysis using a critical value of the crack tip opening angle as the criterion to predict crack growth. Although the crack growth curves are determined from a static analysis, they corroborate the test results for the location of crack path bifurcation. Also, the principal stress criterion for predicting crack turning is consistent with the test.     

Post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load is presented in this paper. The results for radial load ratios are presented in the form of empirical formulae in terms of the central deflection of the annulus to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.     

A finite element analysis of an axisymmetrically loaded orthotropic shell of revolution

The objective of this study was to develop a finite element matrix method of analysis for symmetrically loaded orthotropic shells of revolution using closed form elasticity solutions for the element. A computer program for structural analysis was developed based on this method.

The program was used to analyze orthotropic cylindrical shells with edge loads, orthotropic spherical shells with edge loads, and pressurized ellipsoidal shells.

For the ellipsoidal shells, the ratio of the major to minor axis (a/b) varied from 0.2 to 1.8. The orthotropic materials used had ratios of Young's modulus in the meridional direction to Young's modulus in a direction tangent to a parallel circle (E₁/E₂) that ranged from 0.2 to 1.8.

For the structures and orthotropic materials studied, it was found that the edge effect, as signified by the meridional moment, was affected by the Young's moduli ratio E₁/E₂, the radius of curvature R₂ in the plane containing a normal to the shell surface and a tangent to a parallel circle, and Poisson's ratio v₂, the latter being more prominent for large E₁/E₂ values. The range of the E₁/E₂ ratio caused the meridional edge moment to double, increasing as the E₁/E₂ ratios increased from 0.2 to 1.8, for pressurized ellipsoidal shells. The meridional edge moment more than doubled as the ellipsoidal axes ratio, a/b, ranged from 0.2 to 1.8.     

Dynamic response of orthotropic curved bridge decks due to moving loads

The dynamic response of horizontally curved bridge decks simply supported along the radial edges under the action of the moving vehicle is investigated. The bridge deck is idealised as a number of finite strips with orthotropic elastic properties. The stiffness and mass matrix of an individual element were derived using a homogeneous differential equation of an orthotropic plate in polar co-ordinates. The vehicle is idealized as a sprung mass moving at a constant speed in a circular path parallel to the central line of the bridge. The unsprung mass of the vehicle is assumed to be always in contact with the bridge surface during its motion. Viscous damping is taken into account for both bridge and vehicle. Dynamic deflections and moments are presented for the mid-point of the bridge deck and the values have been compared with the available analytical solution.     

Post-buckling behaviour of cylindrically orthotropic annular plates

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates is presented in detail. The results for radial load ratios are presented in the form of empirical formulae in terms of the deflection at the inner edge to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.     

Dynamic finite element analysis of nonaxisymmetric structures

A dynamic finite element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite nonaxisymmetric features in the circumferential direction. The purpose of the analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometries will be related to the corresponding axisymmetrie solution. This analysis is an extension of previously published work in which a similar approach was developed for static structural problems. The analysis is developed in terms of a cylindrical coordinate system r, θ and z. As the first step of the analysis, the geometry is divided into several segments in the r-θ plane. Each segment is then divided into a set of quadrilateral elements in the r-z plane. The axisymmetric displacements are obtained for each segment by solving a related axisymmetric configuration. A perturbation analysis is then performed to match the solutions at certain points between the segments, and obtain the perturbation displacements for the total structure. The total displacement is then the axisymmetric displacement plus the perturbation displacement. The analysis allows for elastic-plastic materials with orthotropic yield criterion based on Hill's yield function and kinematic strain hardening. The finite element dynamic equations are solved by finite differences by dividing the time domain into incremental steps. The solution has been programmed on a computer and applied to a number of examples.