Incremental Equilibrium Equations Research Articles

Piecewise hierarchical p-version curved shell element for geometrically nonlinear behavior of laminated composite plates and shells

This paper presents a nine-node three-dimensional laminated composite curved shell finite element formulation for geometrically nonlinear (GNL) analysis of laminated plates and shells where the displacement approximation for the laminate is piecewise hierarchical and is derived based on p-version. The displacement approximation for the element is developed first by establishing a hierarchical displacement approximation for each lamina of the laminate and then by imposing interlamina continuity conditions of displacements at the interfaces between the laminas. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the interlamina continuity conditions of displacements. The element formulation ensures C 0 continuity of displacements across the interelement as well as interlamina boundaries. The lamina stiffness matrices and the equivalent nodal force vectors are derived using the principle of virtual work and the hierarchical displacement approximation for the laminas. Interlamina continuity conditions are used to construct the transformation matrices for the individual laminas which permit transformation of the lamina degrees of freedom to the laminate degrees of freedom. The interlamina behavior incorporated in this formulation is in total agreement with the physics of laminate behavior for composite plates and shells. In formulating the properties of the element, complete three-dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick laminated shells and plates. Incremental equations of equilibrium are derived and solved using standard Newton's method. The total load is divided in increments and for each increment of load equilibrium iterations are performed until each component of the residuals and the generalized nodal displacement vector are within preset tolerances. Numerical examples are presented to show the accuracy, p-convergence characteristics and overall advantages of the present formulation.

An extension of the penalty function formulation to incompressible hyperelastic solids described by general measure of strain

AbstractThe penalty function formulation for incompressible hyperelastic solids was first proposed about 30 years ago. Since then all studies have been limited to invariant type formulation of the strain energy function, although it is well known that this formulation does not correctly describe the behavior of a real material. On the other hand more realistic constitutive equations, based on general measures of the strain only, have been incorporated to mixed finite element algorithms. In this article, a penalty function formulation is proposed for the analysis of stress field in materials with constitutive equations based on the general measure of strain. The reduced integration method is used to weaken the penalty constraint in order to obtain meaningful numerical results. The incremental equilibrium equations are solved using the regular Newton–Raphson algorithm. The method is applied to evaluate the stress field in materials subjected to plane strain conditions. Satisfactory agreements have been obtained with analytical solutions when available. © 1995 John Wiley & Sons, Inc.

A free‐formulation‐based flat shell element for non‐linear analysis of thin composite structures

AbstractA flat shell element based on the free‐formulation finite element concept is developed for analysing geometrically non‐linear thin composite shells. A corotational form of the updated Lagrangian formulation is utilized. Numerical results for typical validation problems are presented in order to demonstrate the accuracy and validity of this element. These results are obtained by solving the incremental equilibrium equations through the cylindrical arc‐length method.

P-Version two-dimensional beam element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear formulation (GNL) based on the total Lagrangian approach for a three-node two-dimensional curved beam element. The hierarchical element approximation functions and the corresponding nodal variables are derived directly from the Lagrange family of interpolation functions. The resulting element displacement approximation is hierarchical and can be of arbitrary and different polynomial orders in the longitudinal and the transverse directions of the beam element and ensures C 0 continuity. The element geometry is described by the coordinates of the nodes located on the axis of the beam (middle surface) and the nodal vectors describing the top and bottom surfaces of the element. The element properties are established using the principle of virtual work and the hierarchical element displacement approximation. In formulating the properties of the element complete two-dimensional stresses and strains are considered hence the element is equally effective for very slender as well as extremely deep beams. Incremental equations of equilibrium are derived and solved using the standard Newton-Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those reported in the literature. The formulation presented here removes virtually all of the drawbacks present in existing GNL beam finite element formulations and has many additional benefits. First, the currently available GNL beam formulations are based on fixed order of approximation for the displacements and thus are not hierarchical and have no provision for changing the order of approximation for the displacements u and v. Secondly, the element displacement approximations in the existing formulations are either based on a linearized displacement field for which a true Lagrangian formulation is not possible and the incremental load step size is severely limited or are based on nonlinear nodal rotation function approach in which case even though the description of the displacement field for the element is exact but additional complications arise due to the noncummutative nature of the nonlinear nodal rotation functions. The p-version displacement approximation used here does not involve the traditional nodal rotations that have been used in the existing beam formulations, thus the difficulties associated with their use are not present in this formulation.

Lamé's problem of gravitational viscoelasticity: the isochemical, incompressible planet

SUMMARY We consider a spherical, isochemical, incompressible, non-rotating fluid planet and study infinitesimal, quasi-static, gravitational-viscoelastic perturbations, induced by surface loads, of a hydrostatic initial state. The analytic solution to the incremental field equations and interface conditions governing the problem is derived using a formulation in terms of the isopotential incremental pressure measuring the increment of the hydrostatic initial pressure with respect to a particular level surface of the gravitational potential. This admits the decoupling of the incremental equilibrium equation from the incremental potential equation. As result, two mutually independent (4 X 4) and (2 X 2) first-order ordinary differential systems in terms of the mechanical and gravitational quantities, respectively, are obtained, whose integration is algebraically easier than that of the conventional (6 X 6) differential system. In support of various types of application, we provide transfer functions, impulse-response functions and Green’s functions for the full range of incremental field quantities of interest in studies of planetary deformations. The functional forms in the different solution domains involve explicit expressions for the Legendre degrees n = 0, n = 1 and n 2 2, apply to any location in the interior or exterior of the planet and are valid for any type of generalized Maxwell viscoelasticity and for arbitrary surface loads.

P-Version axisymmetric shell element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order p ξ and p η . This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to p ξ + 1 and p η + 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C 0 continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element. The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete axisymmetric state of stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells. The formulation presented here removes virtually all of the drawbacks present in the existing GNL axisymmetric shell finite element formulations and has many additional benefits. First, the currently available GNL axisymmetric shell finite element formulations are based on fixed interpolation order and thus are not hierarchical and have no mechanism for p-level change. Secondly, the element displacement approximations in the existing formulations are either based on linearized (with respect to nodal rotation) displacement fields in which case a true Lagrangian formulation is not possible and the load step size is severely limited or are based on nonlinear nodal rotation functions approach in which case though the kinematics of deformation is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. Such limitations and difficulties do not exist in the present formulation. The hierarchical displacement approximation used here does not involve traditional nodal rotations that have been used in the existing shell element formulations, thus the difficulties associated with their use are not present in this formulation. Incremental equations of equilibrium are derived and solved using the standard Newton method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the preset formulation. The results obtained from the present formulation are compared with those available in the literature.

Analysis of elastic-plastic interaction buckling of stiffened panels by spline finite strip method

This paper presents a theoretical study on the elastic-plastic interaction buckling of imperfect longitudinally stiffened panels under axial loads. A material and geometric nonlinear spline finite strip method is developed to analyse such interaction problems. The initial geometric imperfections and residual stresses due to welding are involved in the analysis. An iterative numerical procedure combining a modified arc-length method and Newton-Raphson iteration is employed for the solution of the nonlinear incremental equilibrium equations and a full range load deflection path is obtained. Numerical results are presented which show that the authors' method is not only an efficient tool but is also accurate in the analysis of large deflection elastic-plastic behaviour of folded plate structures.

Design Sensitivity Analysis and Optimization of Nonlinear Structural Systems With Critical Loads

A continuum formulation for design sensitivity analysis of critical loads is developed for nonlinear structural systems that are subjected to conservative loading. Both geometric and material nonlinear effects are considered. Sizing design variables such as cross-sectional areas of beam or truss design components and thicknesses of plate or membrane design components, together with their shape design variables, are treated. A continuum approach is used to obtain design sensitivity expressions in integral form. For shape design sensitivity analysis, the material derivative concept and domain method are used to find variations of the critical load due to variations in shape of the physical domain. The total Lagrangian formulation for incremental equilibrium equation and one-point linearized eigenvalue problems are utilized. A numerical method is presented to evaluate continuum design sensitivity expressions using analysis results of established finite element codes. It is found that no adjoint system is necessary for design sensitivity analysis of the critical load. Numerical results show the proposed method for design sensitivity of critical loads is accurate for both sizing and shape design variables. A numerical procedure for optimal design of nonlinear structural systems is presented, using the proposed continuum design sensitivity analysis method. An optimal design problem with a stability constraint is solved.

Load-deflection curves for steel tubular struts associated with varying material effects

A nonlinear finite element model for tracing the inelastic pre- and post-buckling load-deformation path of tubular struts has been developed. The analysis accounts simultaneously for both the geometrical and material nonlinearities. The influence of strain-unloading is included. In particular, the model takes into account a generalised stress-strain relationship and the influence of strain hardening and the Bauschinger effect. The structure tangent stiffness matrix is obtained by using a series of transformation matrices to update the element geometry. A so-called arc-length iterative numerical procedure is combined with a modified Newton-Raphson method to obtain a solution to the incremental equations of equilibrium. The procedure has been applied to trace out the inelastic load-deflection paths of pin-ended struts with varying slenderness ratios and initial geometrical imperfections for both as-received and prestrained struts. By comparing the theoretical results with the experimental results of the as-received struts and the struts prestrained in tension, both fully aged and unaged, it is shown that the developed theoretical model can predict with reasonable accuracy the experimental maximum loads and the post-buckling load-deflection paths of steel tubular struts.

The criteria for identifying the type of critical points

It is well known that the buckling of a structure includes two kinds of different physical phenomena. One is called snap-through buckling and the other is called bifurcation. In this paper, the criteria to identify the two different unstable types are developed. The analysis in the present paper shows that the type of critical points can be directly determined by the incremental equilibrium equations in the general finite element analysis and that it only depends on whether or not the load increment does work through the unstable mode associated with the critical load.

Force recovery procedures in nonlinear analysis

Conventionally, incremental-iterative schemes have been used in solving nonlinear problems. There are two phases involved in such analyses. The first or ‘predictor’ phase relates to solution of the structural displacement increments from the total incremental equations of equilibrium, while the second or ‘corrector’ phase is concerned with recovery of the element forces from the element displacement increments obtained in the first phase. In this paper, it will be demonstrated that accuracy of the numerical solutions depends primarily on the corrector or procedure for recovering the element forces. The expression used in the predictor can only affect the number of iterations required at each incremental step, but not the final shape of the load-deflection curves. A key factor in selecting the procedure for force recovery is that higher order nonlinear effects must be included, based on rigorous continuum mechanics formulations.

On kinematical relations of spatial framed structures

In three-dimensional space the non-commutative nature of finite rotations complicates the formulation of incremental equilibrium equations, capable for handling large rotational increments. The rotation matrix can be expressed in concise and elegant form as an exponential of a skew-symmetric matrix, a result which can be deduced by arguments based on Lie's theory of groups as is done in quantum mechanics. Keeping in mind that the kinematical relations for an updated incremental finite element formulation are sought, the rotation matrix can be expressed as a series expansion in which higher order terms than quadratic can be omitted, if small incremental rotations are assumed. In this paper the shortcomings of using a linear rotation matrix in deriving the equilibrium equations of spatial beam structures are discussed. Stability problems are considered especially. The resulting discretized non-linear equilibrium equations are solved using Fried's orthogonal trajectory method, which allows easy handling of bifurcation and snap-through instability points. Numerical examples of lateral buckling analyses of beams are presented

Design sensitivity analysis of critical load factor for nonlinear structural systems

A continuum formulation of sizing design sensitivity analysis of the critical load factor is developed for nonlinear structural systems that are subject to conservative static loadings. Both geometric and material nonlinear effects are considered. Sizing design variables, such as thickness and cross-sectional areas of structural components, are treated. The total Lagrangian formulation for an incremental equilibrium equation and one- and two-point linearized eigenvalue problems are utilized to estimate the critical load factor. A distributed parameter approach and the adjoint variable method are used to obtain design sensitivity expressions in continuum mechanics formulation. A numerical method is presented to evaluate design sensitivity expressions using established finite element codes. The proposed method is tested using several examples. The numerical results show that the proposed method for design sensitivity of the critical load factor is accurate. It is found that, as the applied load approaches the critical load, the design sensitivity of the critical load factor does not converge to the design sensitivity of the actual critical load, even though the estimated critical load converges to the actual critical load and the design sensitivity of the critical load factor is correct.

Application of the substructuring technique to non-linear dynamic structural analysis

The use of the substructuring technique for the solution of two-dimensional non-linear problems of dynamic response with the direct time integration method is examined. The object is to develop schemes that can considerably reduce the computational expense in analyses of locally and fully non-linear dynamic problems as compared with a traditional analysis. After a review of the principle of the incremental equilibrium equations of motion, some practical illustrations of the technique are dealt with in detail. A specific flow chart is proposed and a number of numerical examples are presented.

Geometrically Nonlinear Analysis in the Vicinity of Critical Points by the Generalized Inverse

In the geometrically nonlinear analysis for equilibrium paths, the determinant of the tangent stiffness matrix at the critical point becomes zero so that a numerically unstable situation appears in the vicinity of critical points. To avoid such a situation, many numerical methods including the arc-length method and the perturbation method have been developed. In this paper, an analytical method to pursue the geometrically nonlinear equilibrium paths in the vicinity of critical points such as limit point and bifurcation point is presented by using the generalized inverse. In the first part, the perturbation equations for the incremental equilibrium equations are derived. Then, critical points on the equilibrium path are classified into limit point and bifurcation point by using the existence condition of solution. In the second part, an analytical method for post-critical paths beyond critical points is presented by means of the generalized inverse. In the final part, the application of the present method to the post-buckling analysis of a shallow arch and cable domes subjected to the symmetrical loads is shown.

Analysis of forming rate sensitive materials

To study how the rate of deformation effects forming rate sensitive materials, the Bodner-Partom elastic-viscoplastic constitutive law is incorporated into a finite element program. This law postulates both elastic and plastic components of deformation at any stress level. The stress is a Hookean function of the elastic strain, while the plastic deformation rate is a function of the deviatoric stress and an internal state variable defining the load history. The finite element derivation adopts the small strain assumption with updated coordinates. The equilibrium rate equation is formulated using total velocities with the nonlinearities incorporated into an equivalent plastic load vector depending upon the current stress. The resulting equation explicitly includes time and is a true rate equation. The program calculates the current stress field using the incremental equilibrium equation. An iterative technique is used to ensure that the assumed current load rate used to calculate the current stress field is correct. Convergence of the iteration procedure needs only be monitored at the velocity specified nodes. To demonstrate the applicability of this method, two plane strain problems, a tensile bar and strip rolling, with rate sensitive materials are investigated.

Material rate dependence and mesh sensitivity in localization problems

The role of material rate dependence in setting the character of governing equations is illustrated in the context of a simple one-dimensional problem. For rate-dependent solids, the incremental equilibrium equations for quasi-static problems remain elliptic and wave speeds for dynamic problems remain real, even in the presence of strain-softening. The pathological mesh sensitivity associated with numerical solutions of localization problems for rate-independent solids is eliminated. In effect, material rate dependence implicity introduces a length scale into the governing equations, although the constitutive description does not contain a parameter with the dimensions of length. Numerical results are presented that illustrate the localization behavior of slightly rate-dependent solids under both quasi-static and dynamic loading conditions.

Nonlinear dynamic analysis of frame structures

A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida et al., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.

NONLINEAR ANALYSIS OF THIN-WALLED STRUCTURES BY A COUPLED FINITE ELEMENT METHOD

This paper presents a finite element method which enables to analyze a large displacement behavior of elasto-plastic thin-walled structures which fail by overall, local or interactive instability. An incremental equilibrium equation for a thin-walled beam-column is derived in a stiffness matrix form by using a moving element coordinate system and an incremental variational principle. So called multipoint constraints technique is used to connect plate elements with a beam element at the coupling nodal point in the cross section. Numerical results of the present method are compared with the exact solutions and experimental results. Validity and efficiency of the present method are confirmed.

On the Comparison Between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites

To investigate the relation between the macroscopic and microscopic instability predictions for certain microstructured media, we study the stability of an axially stretched fiber-reinforced composite under plane strain conditions. The microstructural instability is attributed to a bifurcation buckling of the fibers while the corresponding macroscopic one occurs at the loss of ellipticity in the homogenized incremental equilibrium equations of the material. The effects of geometry and material properties on those phenomena are analyzed. The macroscopic approach consistently and sometimes considerably overestimates the stable region of the composite. The attractive feature in this work is that all the investigations can be done by using analytical solutions instead of the numerical ones employed in similar investigations so far.