Incremental Equilibrium Equations Research Articles

Global stability analysis of spatial structures based on Eigen-stiffness and structural Eigen-curve

The global stability of a spatial structure is usually analysed via a nodal load-displacement curve (NLDC). However, the global mechanical properties of a structure can hardly be reflected comprehensively by specific NLDCs, and no criterion is available to select the representative node for getting the most reasonable NLDC. In this paper, a scalar parameter derived from the incremental equilibrium equation of nonlinear stability analysis, named Eigen-stiffness, is defined to characterize the global structural stiffness. The Structural Eigen-curve (SEC), based on Eigen-stiffness, is proposed to depict the equilibrium path. Two types of extreme points on SEC are defined to determine the critical state of the structure, including the structural limit state and the structural snap-back. Firstly, the stability of a hinge-supported planar arch is analysed to introduce the SEC concept. Then a K6 reticulated shell is designed to give further understanding. Subsequently, practical application of the SEC is illustrated in the stability analysis of a roof structure, namely, Shanghai International Conference Centre. In addition, a parametric study on a K6 reticulated shell is carried out, based on the Eigen-stiffness and the SEC, to investigate the effects of the rise to span ratio and the geometric imperfection amplitude on the structural stiffness and the structural load-carrying capacity. The results demonstrate that the ultimate load-carrying capacity obtained from the SEC is equal to that from NLDC. More importantly, unlike NLDC, the SEC — free from node selection — can efficiently capture the features of global structural behaviour and the evolution of structural stiffness.

Shape pre-adjustment of deployable mesh antennas considering space thermal loads

On-orbit space deployable antennas withstand periodic thermal loads, which will degrade the antenna surface accuracy. Thus, it is necessary to preliminarily adjust the antenna before launch to make it adapt to the space temperature environment as much as possible. Aiming at this problem, a pre-adjustment method based on min-max concept is presented in this study. First, according to the force equilibrium equation of the cable element, the incremental equilibrium equation of the cable mesh antenna is derived, and the incremental expressions of the reflector's surface displacement and cable tension with respect to cable length variation and element temperature variation are also developed. Then, the optimization model for shape pre-adjustment is established using min-max method, and the pre-adjustment process is implemented by sequence planning strategy. Finally, an AstroMesh antenna is employed to demonstrate the validity of the method.

Nonlinear buckling behaviours of thin-walled functionally graded open section beams

In this paper, nonlinear buckling responses of functionally graded (FG) thin-walled open section beams based on Euler–Bernoulli–Vlasov theory is presented. The finite element incremental equilibrium equations are developed by updated Lagrangian formulation using the non-linear displacement cross-section field that accounts for large rotation effects. Young’s modulus of FG beams are varied continuously through the wall thickness based on the power-law distribution. Numerical results are obtained for thin-walled FG beams with symmetric and mono-symmetric I-section and channel-section for various configurations such as boundary conditions, geometry, skin-core-skin ratios and power-law index to investigate the flexural–torsional and lateral buckling loads and post-buckling responses. The accuracy and reliability of proposed model are proved by comparison with previous research and analytical solutions. The importance of above-mentioned effects on buckling results is demonstrated on benchmark examples.

Modified double-control form-finding analysis for suspendomes considering the construction process and the friction of cable–strut joints

The structural configuration of slender long-span suspendomes from the construction stage to the final operational stage is critical for evaluating structural safety and resiliency. In the present study, a modified double-control form-finding (MDFF) method is proposed with consideration of the construction process and the friction of cable–strut joints based on a transient structural model of a suspendome. Based on the total Lagrangian increment formulation, the incremental equilibrium equation is built to include geometric nonlinearity. Afterwards, in the first construction step, the nodal displacement, displacement condition and the tangent stiffness matrix are built, and the finite element equations are calculated based on the existing members using the geometric nonlinear finite element method (FEM). The internal force and displacement can be obtained. In the subsequent construction stages, the nodal displacement, displacement condition and the tangent stiffness matrix are modified based on the newly added members and the friction of the cable–strut joints in present stage. Throughout the whole construction process, the inverse iteration method is used to control the structural configuration, and the initial stress increment method is used to control the cable force. After repeating the above steps, iterations are done in each key construction stage until the convergence criteria are met. The result is the final configuration of the structure including the geometry and cable forces. Based on the results of a numerical test on a suspendome, the proposed MDFF is found to be more accurate than the traditional double-control form finding method (DFF).

Analysis of elasto-plastic postbuckling and energy release rate for delaminated fiber metal laminated beams in thermal environment

The elasto-plastic postbuckling of fiber metal laminated beams with delamination and the energy release rate along the delamination front are discussed in this paper. Considering geometrical nonlinearity, thermal environment and geometrical initial imperfection, the incremental nonlinear equilibrium equations of delaminated fiber metal laminated beams are established, which are solved using the differential quadrature method and iterative method. Based on these, according to the J-integral theory, the elasto-plastic energy release rate is studied. The effects of some important parameters on the elasto-plastic postbuckling behavior and energy release rate of the aramid reinforced aluminum laminated beams are discussed in details.

Diffusion Effects Induced by Dislocations in Crystalline Materials Subjected to Large Strains

Abstract. We reconsider here the FEM-algorithm for solving the initial and boundary value problems performed within the viscoplastic constitutive framework and proposed in our paper [1]. The problems concerning the deformation of a sheet composed of a single fcc-crystal, generated by different slip systems simultaneously activated, are solved numerically for an in-plane stress state. The variational formulation is associated to the incremental equilibrium equations and is coupled with an update procedure for the state variables, which are described by the differential type equations, as well as for the non-local evolution equations of the dislocation densities. The length scale parameter is introduced into the model through the diffusion-like parameter which enters the evolution equations for dislocation densities. For more accuracy of the simulation, the shape functions have been chosen polynomials with higher than one degree. We do not consider that once a slip system was activated it remains active for the rest of simulation. The activation condition is a key point in the numerical algorithm. As a numerical example, we perform a tensile test of a rectangular and non-rectangular metallic sheet, comparring the results of the simulation when two, respectively eight slip systems are considered.

Slip systems and flow patterns in viscoplastic metallic sheets with dislocations

The work addresses the rate dependent crystal plasticity with hardening dependent on the dislocation density, when the local and non-local (diffusion-like) evolution equations for the dislocation densities are considered. A new procedure is proposed for solving initial and boundary value problems based on the incremental equilibrium equations in terms of variational results, coupled with a rate-type constitutive model. The problems concerning the deformation of a sheet composed of a single fcc-crystal, generated by different slip systems simultaneously activated, are solved numerically for an in-plane stress state. Necessary and sufficient conditions for obtaining the in-plane stress state are proved under the assumptions of multislip flow rule, and cubic elastic symmetry. The variational problem which defines the velocity field in the actual configuration at time t is solved by the finite element method (FEM) and the current state in the sheet is defined via an update algorithm when either local (differential type) or non-local (diffusion-like) evolution equations describe the dislocation densities. Boundary value problems resulting from the simulation of compressive and tensile tests are solved by considering all eight activated slip systems together with an activation condition. Finally, a comparison between the numerical results obtained by using the non-local (diffusion-like) evolution equations and local (differential type) equations which describe the dislocation densities is carried out.

A finite element model to simulate long-term behavior of prestressed concrete girders

The development of a finite element model for time-dependent analysis of bonded prestressed concrete girders at service conditions is presented. The effects of creep and shrinkage of concrete and relaxation of steel tendons are taken into account. The concrete creep is modeled based on the Dirichlet series creep compliance with efficiency in simulating the stress history. In addition, the interaction between different time-dependent effects is fully considered in the numerical procedure. The numerical method is formulated based on the layered Euler–Bernoulli beam theory. In the constructed incremental equilibrium equations, the equivalent nodal load increments consist of four components contributed by external loads, concrete creep, concrete shrinkage and tendon relaxation, while the stiffness matrix is composed of the material and geometric stiffness matrices. Numerical examples show that the proposed model can well predict the long-term behavior of prestressed concrete beams, and that the time-dependent effects have important influence on the structural behavior.

Influence of Radial Slots on the Vibration Characteristics of Circular Saw Blade

In order to improve thin circular saw blade’s cutting stability, it’s nonlinear dynamic incremental equilibrium equation was obtained according to nonlinear vibration theory and D’Alembert principle. It could be concluded from the above equation that cutting several radial slots on the circumference of the saw blade will cause significant change of dynamic characteristics of the saw blade. This thesis, employing the finite element method and matrix perturbation principle, calculated and studied respectively the effect of number, length and width of the radial slots on natural frequencies of the saw blade. Results show that number and length have distinct influence on natural frequencies but hardly does width have. So proper choice of number, length and width of the radial slots can improve effectively the dynamic characteristics of the saw blade.

Axisymmetric bifurcations of thick spherical shells under inflation and compression

Incremental equilibrium equations and corresponding boundary conditions for an isotropic, hyperelastic and incompressible material are summarized and then specialized to a form suitable for the analysis of a spherical shell subject to an internal or an external pressure. A thick-walled spherical shell during inflation is analyzed using four different material models. Specifically, one and two terms in the Ogden energy formulation, the Gent model and an I1 formulation recently proposed by Lopez-Pamies. We investigate the existence of local pressure maxima and minima and the dependence of the corresponding stretches on the material model and on shell thickness. These results are then used to investigate axisymmetric bifurcations of the inflated shell. The analysis is extended to determine the behavior of a thick-walled spherical shell subject to an external pressure. We find that the results of the two terms Ogden formulation, the Gent and the Lopez-Pamies models are very similar, for the one term Ogden material we identify additional critical stretches, which have not been reported in the literature before.

Floating stress-point integration meshfree method for large deformation analysis of elastic and elastoplastic materials

SUMMARY A new meshfree formulation of stress-point integration, called the floating stress-point integration meshfree method, is proposed for the large deformation analysis of elastic and elastoplastic materials. This method is a Galerkin meshfree method with an updated Lagrangian procedure and a quasi-implicit time-advancing scheme without any background cell for domain integration. Its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation, which is not generally used in meshfree formulations. Hence, this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved. The details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method. Copyright © 2012 John Wiley & Sons, Ltd.

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.

Spectral analysis of nonlocal regularization in two‐dimensional finite element models

SUMMARYStrain‐softening in geomaterials often leads to ill‐posed boundary‐valued problems (BVP), which cannot be solved with finite element methods without introducing some kind of regularization such as nonlocal plasticity. Hereafter we propose to apply spectral analysis for testing the performance of nonlocal plasticity in regularizing ill‐posed BVP and producing mesh‐independent solutions when local plasticity usually fails. The spectral analysis consists of examining the eigenvalues and eigenvectors of the global tangential stiffness matrix of the incremental equilibrium equations. Based on spectral analysis, we propose a criterion for passing or failing the test of constitutive regularization in the context of BVP. If the eigenvalues of the tangential operator are all positive then the regularization succeeds, otherwise it fails and may not prevent artificial mesh‐dependent solutions from appearing. The approach is illustrated in the particular case of a biaxial compression with strain‐softening plasticity. In this particular case, local softening plasticity is found to produce negative eigenvalues in the tangential stiffness matrix, which indicates ill‐posed BVP. In contrast, nonlocal softening plasticity always produces positive eigenvalues, which regularizes ill‐posed BVP. The dominant eigenvectors, which generate localized deformation patterns, have a bandwidth independent of mesh size, provided that the mesh is fine enough to capture localization. These mesh‐independent eigenmodes explain why nonlocal plasticity produces numerical solutions that are mesh‐independent. Copyright © 2011 John Wiley & Sons, Ltd.

An efficient Galerkin meshfree formulation for shear deformable beam under finite deformation

This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam. The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection, large deformation as well as finite rotation. The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation. The Lagrangian meshfree shape function is utilized to discretize the variational equation. Subsequently to resolve the shear and membrane locking issues and accelerate the computation, the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation. Numerical results reveal that the present formulation is very effective.

Numerical solutions of strain localization with nonlocal softening plasticity

Softening plasticity, when it is deprived of a characteristic length, often leads to boundary value problems that are ill-posed and produce unreliable numerical solutions. As meshes are refined, plastic strains localize into bands that become narrower and narrower, and generate load–displacement responses with excessive softening, which may even degenerate into unrealistic snapbacks. In the case of discrete systems, the underlying reasons for this poor numerical performance can be examined using a spectral analysis of the tangential stiffness matrix derived from incremental equilibrium equations. In one-dimensional boundary-valued problems, the introduction of a weak softening element into a mesh of elasto-plastic elements produces a dominant eigenvector that clearly exhibits a strain localization width directly related to the weak element size. The dominant eigenvector that controls the incremental solution varies spuriously when the mesh is refined, which results in mesh dependency. Over-nonlocal softening plasticity introduces a length scale, forces more elements to become plastic, and preserves the width of the plastic zone when the mesh is refined. When one-dimensional meshes are refined, spectral analysis shows that the dominant eigenvector does not vary erratically, and that its deformation patterns are smooth and display a constant localization width. This explains why the spatial distribution of plastic strains and the global load–displacement responses converge to analytical solutions in one dimension. The generalization to higher dimensions will be the object of future studies.

NONLINEAR ELASTO-DYNAMIC ANALYSIS OF I-BEAMS CURVED IN-PLAN

The elastic response of curved beams subjected to moving vertical loads and dynamic loads is investigated. Incremental dynamic equilibrium equations are derived by using the principle of virtual work. Newmark's step-by-step procedure is adopted to discretise the dynamic equilibrium equations and obtain the time history response. Geometric nonlinearities due to large deflections and rotations are taken into account. A total Lagrangian finite element formulation is developed. The numerical models are compared with the existing analytical solutions and employed to show the effects of geometric nonlinearities as well as the initial curvature on the dynamic behaviour of curved I-beams. It is shown that the geometric nonlinearities are significant even for service loads. The nonlinear behaviour of a curved beam is substantially different from the nonlinear behaviour of a straight beam when the initial curvature is not small.

Elasto-plastic buckling analysis of laminated plates including interfacial damage

Elasto-plastic buckling of orthotropic laminated plates, which include interfacial damage, is analyzed in detail. Firstly, a novel mixed hardening yield criterion, as an improvement of Hill’s counterpart, is proposed for the orthotropic materials on the basis of the plastic theory. And differing from Hill’s theory, the present yield criterion is related to the spherical tensor of stress. Then, the incremental elasto-plastic constitutive relations of the mixed hardening orthotropic materials are presented. Secondly, the incremental static equilibrium equations for laminated plates including interfacial damage are established based on Von-Karman type theory and the principle of minimum potential energy. Finally, the elasto-plastic buckling of laminated plates are solved by adopting the Galerkin method and iteration scheme. The numerical results show that buckling of the plate occurs easier due to the existence of interfacial damage, and the critical load trends to constant when the interfacial damage approaches a certain degree. Also, the effect of anisotropy on buckling is obvious and the analysis of elasto-plastic buckling is necessary.

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.

Asymptotic results for bifurcations in pure bending of rubber blocks

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length η, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of strain are reduced to a fourth-order linear eigenproblem that displays a multiple turning point. It is found that for 0 < η < ∞, the block experiences an Euler-type buckling instability which in the limit η → ∞ degenerates into a surface instability. Singular perturbation methods enable us to capture this transition, while direct numerical simulations corroborate the analytical results.

Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure

In this paper, we consider bifurcation from a circular cylindrical deformed configuration of a thick-walled circular cylindrical tube of incompressible isotropic elastic material subject to combined axial loading and external pressure. In particular, we examine both axisymmetric and asymmetric modes of bifurcation. The analysis is based on the three-dimensional incremental equilibrium equations, which are derived and then solved numerically for a specific material model using the Adams–Moulton method. We assess the effects of wall thickness and the ratio of length to (external) radius on the bifurcation behaviour.