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.     

Nonlinear axisymmetric static and transient analysis of orthotropic thin tapered circular plates

This study deals with the geometrically nonlinear axisymmetric static and transient analysis of cylindrically orthotropic elastic thin tapered circular plates subjected to uniformly distributed and discrete central loads. Differential equations in terms of transverse displacement w and stress function ψ have been employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in space domain and Newmark-β scheme in time domain have been used. Step function dynamic loads are considered. Static and dynamic results have been presented for isotropic and orthotropic immovable clamped and simply supported plates with linearly varying thickness for three values of taper ratios and the effect of varying thickness has been investigated. A simple approximate method is used to predict the maximum dynamic response to step load from the results for static loads and is found to yield sufficiently accurate results.     

An h-hierarchical adaptive scaled boundary finite element method for elastodynamics

A posteriori h-hierarchical adaptive scaled boundary finite element method (ASBFEM) for transient elastodynamic problems is developed. In a time step, the fields of displacement, stress, velocity and acceleration are all semi-analytical and the kinetic energy, strain energy and energy error are all semi-analytically integrated in subdomains. This makes mesh mapping very simple but accurate. Adaptive mesh refinement is also very simple because only subdomain boundaries are discretised. Two 2D examples with stress wave propagation were modelled. It is found that the degrees of freedom needed by the ASBFEM are only 5%–15% as needed by adaptive FEM for the examples.     

Compatibility Condition in Theory of Solid Mechanics (Elasticity,Structures, and Design Optimization)

The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This shortcoming prevented the formulation of a direct method to calculate stress and strain, which are currently obtained indirectly by differentiating the displacement. We have researched and understood the compatibility condition for linear problems in elasticity and in finite element structural analysis. This has lead to the completion of the “method of force” with stress (or stress resultant) as the primary unknown. The method in elasticity is referred to as the completed Beltrami-Michell formulation (CBMF), and it is the integrated force method (IFM) in the finite element analysis. The dual integrated force method (IFMD) with displacement as the primary unknown had been formulated. Both the IFM and IFMD produce identical responses. The IFMD can utilize the equation solver of the traditional stiffness method. The variational derivation of the CBMF produced the existing sets of elasticity equations along with the new boundary compatibility conditions, which were missed since the time of Saint-Venant, who formulated the field equations about 1860. The CBMF, which can be used to solve stress, displacement, and mixed boundary value problems, has eliminated the restriction of the classical method that was applicable only to stress boundary value problem. The IFM in structures produced high-fidelity response even with a modest finite element model. Because structural design is stress driven, the IFM has influenced it considerably. A fully utilized design method for strength and stiffness limitation was developed via the IFM analysis tool. The method has identified the singularity condition in structural optimization and furnished a strategy that alleviated the limitation and reduced substantially the computation time to reach the optimum solution. The CBMF and IFM tensorial approaches are robust formulations because both methods simultaneously emphasize the equilibrium equation and the compatibility condition. The vectorial displacement method emphasized the equilibrium, while the compatibility condition became the basis of the scalar stress-function approach. The tensorial approach can be transformed to obtain the vector and the scalar methods, but the reverse course cannot be followed. The tensorial approach outperformed other methods as expected. This paper introduces the new concepts in elasticity, in finite element analysis, and in design optimization with numerical illustrations.     

The use of optimization techniques in the analysis of cracked members by the finite element displacement and stress methods

Mathematical programming is applied to the two-dimensional stationary crack problem of a body composed of nonlinear elastic incompressible material. Fully admissible displacement as well as stress formulations are used to discretize the problem. Crack tip singularity is introduced in the displacement formulation by enriched elements for plane stress and, in certain cases, by superposition for plane strain. Pointwise incompressibility is obtained through constrained displacement functions. For three crack geometries Rice's J integral is evaluated by the energy difference method for different values of the hardening index. The numerical results, which are also applicable to secondary creep problems, appear to suggest a bounding character.     

Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first order ordinary differential equations. The streamwise shear stress function θ and the cross-flow velocity component V are represented as suitable functions of the streamwise velocity component U. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30° incidence. The results are compared with corresponding finite difference solutions.     

Time step integration algorithms with predetermined coefficients for second order equations

In this paper, the effect of using the predetermined coefficients in constructing time step integration algorithms suitable for linear second order differential equations based on the weighted residual method is investigated. The second order equations are manipulated directly. The displacement approximation is assumed to be in a form of polynomial in the time domain and some of the coefficients can be predetermined from the known initial conditions. The algorithms are constructed so that the approximate solutions are equivalent to the solutions given by the transformed first order equations. If there are m predetermined coefficients (in addition to the two initial conditions) and r unknown coefficients in the displacement approximation, it is shown that the formulation is consistent with a minimum order of accuracy m+r. The maximum order of accuracy achievable is m+2r. This can be related to the Padé approximations for the second order equations. Unconditionally stable algorithms equivalent to the generalized Padé approximations for the second order equations are presented. The order of accuracy is 2r−1 or 2r and it is required that m+1⩽r. The corresponding weighting parameters, weighting functions and additional weighting parameters for the Padé and generalized Padé approximations are given explicitly.     

Weight minimisation of displacement-constrained truss structures using a strain energy criterion

This paper discusses a new method for the solution of the general truss weight-minimisation problem under simultaneous stress and displacement constraints. The method introduces a novel strain-energy-density criterion that refers to the ratio of the virtual strain energy per unit volume in each structural member to its average value on the whole structure. The virtual strain energy comes from the unit-load theorem and it is proportional to the product of the axial member forces due to both the actual loads and a virtual unit load that is applied at the node with the maximum displacement. A simple recursive formula for updating the cross-sectional areas, based on displacement constraints, is presented. A general subsequent algorithm applicable to both single and multiple load cases follows this formula. The results are encouraging since in all test cases the method was found to be robust and generally led to the same weight level as the literature, in both small and large structures.     

A finite-difference scheme for mixed boundary value problems of arbitrary-shaped elastic bodies

This paper presents a program based on a finite-difference technique, which solves plane stress and plane strain problems of arbitrary shaped elastic bodies with mixed boundary conditions. A new formulation of governing equations in terms of the displacement potential function ψ, as introduced by Uddin (Finite difference solution of two-dimensional elastic problems with mixed boundary conditions, MSc Thesis, Carleton University, Canada, 1966), has been used. This formulation has the capability to handle problems of mixed boundary condition, which is beyond the ability of the conventional formulations in terms of Airy's stress function φ. Results found with this program for classical problems are in very good agreement with known solutions. This program can handle practical boundary conditions very efficiently.     

Planar crack occupying a finite doubly connected region inside an infinite elastic solid

A method is proposed for the approximate solution of the problem of an embedded pressurized planar crack occupying a finite doubly connected region inside an infinite elastic solid. The formulation of the problem produces a system of two integral equations for determining the unknown normal stresses on the plane of the crack outside the crack region, which can be solved using numerical procedures. The proposed method has been applied to obtain the opening mode stress intensity factors K_I, along the boundary lines of an annular crack subjected to a uniform internal pressure.     

Nonlocal large deformation and localization behavior of metals

The present paper deals with a nonlocal continuum plasticity model which includes the dependence of the yield function on a nonlocal equivalent plastic strain measure. Particular attention is focused on the formulation of a generalized I₁–J₂ yield criterion to describe the effect of hydrostatic stress on the plastic flow properties of metals, and the nonlocal equivalent plastic strain is defined as a weighted average of the corresponding local measure taken over the neighboring material points of the body. The nonlocal yield condition leads to a partial differential equation which is solved using the finite difference method at each iteration of a loading step. Since this requires no additional boundary conditions, the displacement-based finite element procedure is governed by the standard principle of virtual work, and the associated linearized variational equations are obtained in the usual manner from a consistent linearization algorithm. Numerical simulations of the elastic–plastic deformation behavior of ductile metal specimens show the influence of the various model parameters on the deformation and localization prediction. The proposed nonlocal theory preserves well-posedness of the governing equations in the post-localization regime and prevents pathological mesh sensitivity of the numerical results. The internal length scale incorporated in the model determines the size of the localized shear bands.     

Approximate finite-element method of stress analysis of non-axisymmetric configurations

A finite-element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite non-axisymmetric features in the circumferential direction. The purpose of the present analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometires will be related to the corresponding axisymmetric solution. To analyze these structures, the geometry is divided into several segments in the r-θ 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 stresses and strains are then calculated at any desired point once the total displacements are known. The method is applied to a number of examples to illustrate the accuracy of the method. The results for these examples are presented and discussed. Some of these examples illustrate the difference between the present approach and the well known substructural analysis and it is shown that the present method is much more accurate.     

A singular ES-FEM for plastic fracture mechanics

The stress and strain fields around the crack tip for power hardening material, which are singular as r approaches zero, are crucial to fracture and fatigue of structures. To simulate effectively the strain and stress around the crack tip, we develop a seven-node singular element which has a displacement field containing the HRR term and the second order term. The novel singular element is formulated based on the edge-based smoothed finite element method (ES-FEM). With one layer of these singular elements around the crack tip, the ES-FEM works very well for simulating plasticity around the crack tip based on the small strain formulation. Two examples are presented with detailed comparison with other methods. It is found that the results of the presented singular ES-FEM are closer to reference solution, which demonstrates the applicability and the effectiveness of our method for the plastic field around the crack tip.     

An equilibrium method for stress calculation using finite element displacement models

The stress computation concept described in [1]is extended here to arbitrary meshes and elements — in particular to triangular elements. After calculating the nodal displacements — using complete and conforming displacement models — we assume linear stress distributions and corresponding virtual displacements at element peripheries. The nodal stress values are then determined by the principle of virtual work. The right-hand side of the resulting system of algebraic equations consists of the work done by the known nodal stress resultants acting along the virtual displacements. In general, the system of equations is nonquadratic on the structural level. Gauss's transformation produces a symmetric, positive definite band matrix. This kind of stress calculation is called the equilibrium method.A dual node method is also given. It involves the inversion of element matrices instead of the master matrix.Various examples of plane stress, plate bending and shell problems show much better accuracy of stresses in comparison with conventional methods. Furthermore, these techniques improve the computational efficiency considerably. There is also a special advantage in the possibility of choosing arbitrary subregions of the structure for stress calculation.     

Laminated glass plates: revealing of nonlinear behavior

A laminated glass unit which consists of two glass plies and an interlayer polyvinyl butyral (PVB) shows very complex behavior in the range of service loads. Complex behavior is due to the effect of geometry that undergoes large deflection and order difference in modulus of elasticity of glass and PVB. The nonlinear behavior of laminated glass plate is represented by five coupled nonlinear partial differential equations for lateral and in-plane displacements obtained through the variational approach. Linear algebraic equations are obtained by writing the differential equations at discrete points of glass plates by using finite difference method with central differences. Since the algebraic equations are nonlinear, the iterative technique is employed to predict the displacements through the banded matrix solver for lateral displacement and strongly implicit method for in-plane displacements. Solution will only be reached when variable underrelaxation parameter is used; otherwise solution diverges. Results denote that complex stress fields develop, and maximum stress is traveling; starting at the center, following x-axis, then moving on the diagonal and settling at the region closer to the corner at high nonlinearity level.     

Axisymmetric static and dynamic buckling of orthotropic truncated shallow conical caps

This investigation deals with the axisymmetric static and dynamic buckling of a cylindricaliy orthotropic truncated shallow conical cap with clamped edge. The cases of conical caps with a free central circular hole and with a hole plugged by a rigid central mass have been considered. The governing equations are formulated in terms of normal displacement w and stress function Ψ. The orthogonal point collection method is used for spatial discretisation and the Newmark-β scheme is used for time-marching. Analysis has been carried out for a uniformly distributed conservative load normal to the undeformed surface and a central axial ring load at the hole. Dynamic load is taken as a step function load. The influence of orthotropic parameter β and annular ratio on the buckling loads has been investigated. New results for static and dynamic buckling loads have been presented for the isotropic and orthotropic truncated conical caps. Dynamic buckling loads obtained from static analysis have been found to agree well with the dynamic buckling loads based on transient response.     

Mixed meshless formulation for analysis of shell-like structures

An efficient mixed meshless computational method based on the Local Petrov–Galerkin approach for analysis of plate and shell structures is presented. A concept of a three-dimensional solid is applied allowing the use of complete three-dimensional constitutive equations, and exact shell geometry can be described. Discretization is carried out by using both the moving least square approximation and the polynomial functions. Independent field variables are the strain and stress tensor components expressed in terms of the nodal values, which are then replaced by the nodal displacements by using the independent displacement interpolation. A closed global system of equations with only nodal displacements as unknown variables is derived. The undesired locking phenomena are fully suppressed. The proposed mixed formulation is numerically more efficient than the available meshless fully displacement approach, as demonstrated by the numerical examples.     

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.     

Analysis of continuum by the integrated force method

The novel formulation termed the integrated force method (IFM) has been established for finite element discrete analysis. In this paper we have extended the IFM for the analysis of continuum taking circular plate as the example. The primary variables of the analysis are moments. All the continuum equations (equilibrium equations and compatibility conditions) in the field and on the boundary are obtained in moments from the stationary condition of the variational functional of the IFM. A new stress function required for the functional is defined. The variational functional yields the known equations along with the novel boundary condition identified as the boundary compatibility condition. The moment solution for the plate problem is obtained without any recourse to displacements either in the field or on the boundary. From moments, displacements are obtained by integration and boundary displacement continuity conditions. The IFM solution and boundary compatibility conditions are verified using Timoshenko's work and finite element displacement method.     

Sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric, fully-coupled, thermoviscoplastic response of shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of large-strain, transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be viscoplasticity with strain hardening and thermal hardening, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. The energy-balance equation is solved using a Galerkin procedure, with the temperature as the fundamental unknown.Spatial discretization is performed in one dimension (meridional direction) for the momentum and constitutive equations of the shell, and in two dimensions (meridional and thickness directions) for the energy-balance equation. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) for the momentum equation; a predictor-corrector version of the trapezoidal rule is used for the energy-balance equation; and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to the thickness, mass density, Young's modulus, two of the material parameters characterizing the viscoplastic response and the three parameters characterizing the thermal response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of some of these quantities at selected times.