MIXED AND COMPATIBLE FINITE ELEMENTS IN THE ANALYSIS PROBLEM OF ELASTOPLASTIC STRUCTURES

The equilibrium dual discrete mathematical models of the problems of limit load analysis and optimization are investigated in the article. These models are presented in terms of static and kinematic formulation using equilibrium finite elements. In these mathematical models the possible discontinuities of displacement velocities are evaluated and the velocity of energy dissipation is estimated not only within the volume of finite elements, but at the plastic surfaces between elements. At first, on the basis of the energy principle of the maximum external power [1,2] the general mathematical models (3) and (7) of static formulation of limit load analysis and optimization problems are created. In these models the yield conditions are controlled not only within the volume, but also at the surfaces of finite elements. The equilibrium finite elements and interpolation functions of strains (9) are used for discretization of these models. The constancy of external power is taken as the optimum criterion. The discrete expressions of fundamental relationships—equilibrium and geometric equations, yield conditions (10)-(12) for finite element and (14)-(17) for the discrete model of a body are developed. The discrete expressions of yield conditions are given using the classic collocation methods: collocation at the point, collocation at the sphere (element) and Bubnov-Galiorkin's collocation method [11,12]. The equilibrium equations of discrete structure are developed on the basis of virtual displacement principle while geometrical equations are derived using virtual force principle. In contrast to the approach of other authors, yield conditions and geometrical equations are described not only within finite elements, but also at the surfaces between elements. That helps to design the dual discrete mathematical models of the problems (21)-(26), in which the discontinuities of displacement velocities and the velocity of energy dissipation in the place of those discontinuities are estimated. The mathematical models (22), (24), (26), (29) and (32) of kinematic problem formulation are developed from sensible static formulations by Lagrange's multiplier method. The modified mathematical models (27)-(29) are presented. In these models the equilibrium equations are eliminated or the geometrical equations are transformed into compatible equations of plastic stress velocities, in this way decreasing the number of equations and unknown values. The dependence of the numerical results (limit load) of the frame on the approximation degree of bending moments, as well as on the discretization method of yield conditions are illustrated. In table 1 the values of limit loading parameter F 0 and their error of calculation ΔF 0 (per cent, in comparison with the analytic solution F 0 = 0,4662M 0) are presented. They are given for the first and second order finite elements with linear and parabolic distribution of bending moments using different discrete yield conditions and a different number of finite elements. The numerical result shows, that the discretization of yield conditions by Bubnov-Galiorkin's method gives the best accuracy and stable solutions. By discretizing yield conditions using the point's collocation and collocation at the element, the accurancy of numerical results depends not only on the number of elements, but also on a more or less successful choice of finite elements net.

Involutory transformations and variational principles with multi-varoables in thin plate bending problems

Chapter 8 - Basic Equations and Solution Procedure

We prove that ground states of quantum spin systems are characterized by a principle of minimum local energy and that translationally invariant ground states are characterized by the principle of minimum energy per unit volume.

An elastic perfectly plastic structure at shakedown to given cyclić loading is under consideration. The stress-strain field of dissipative system in general is related to the history of loading. And only in a particular case, i.e. at the moment prior to the failure of an elastic perfectly plastic structure the distribution of the actual residual forces is unique for each prescribed history of loading (the safety factor of shakedown approaches unity). Nevertheless, there exist some domains where the plastic strains are equal to zero. The residual forces in the statically indeterminate parts of the structure may be non-unique: the stress field is only determined by the equilibrium equations. The extremum energy principle of minimum complementary energy allows to derive the actual residual forces out of all statically admissible residual forces at the moment prior to cyclic plastic failure. Then the stress-strain field analysis problem at the moment prior to the cyclic plastic failure is formulated as a problem of non-linear mathematical programming. Formulating the dual pair of non-linear programming problem (statical and kinematic formulation of analysis problem) the differential constraints are neglected or replaced by algebraic conditions. When the safety factor is approching a unity, the degeneracy of the statical formulation of the analysis problem often can occur. In this case a mathematical model is proposed for obtaining an upper bounds for the displacement at shakedown. It is pointed out that the known Kuhn-Tucker conditions of mathematical programming theory (i.e. compatibility equations of residual strains) in concert with restriction, limiting the maximum value of total energy dissipation, make up the adaptation conditions of the structure to given cyclic loading. Kuhn-Tucker conditions used in above—mentioned problem allow to correctly interprete the physical aspect of the degeneracy problem at shakedown. When the safety factor is larger than unity an artificial degeneracy situation for the statical formulation of analysis problem can be created. Then the mathematical models presented can be applied to the analysis of unloading elastoplastic structures. With this aim in view a fictitious equiplastic structure the behaviour of which is holonomic is derived. The displacements of the fictitious structure enclose the displacements of the actual structure subject to cyclic loading.

Usually, the governing equations of the numerical manifold method (NMM) are derived from the minimum potential energy principle. For many applied problems it is difficult to derive in general outset the functional forms of the governing equations. This obviously strongly restricts the implementation of the minimum potential energy principle or other variational principles in NMM. In fact, the governing equations of NMM can be derived from a more general method of weighted residuals. By choosing suitable weight functions, the derivation of the governing equations of the NMM from the weighted residual method leads to the same result as that derived from the minimum potential energy principle. This is demonstrated in the paper by deriving the governing equations of the NMM for linear elasticity problems, and also for Laplace’s equation for which the governing equations of the NMM cannot be derived from the minimum potential energy principle. The performance of the method is illustrated by three numerical examples.

Summary A problem of elastic structures stress-strain field determination is considered in this article. A theoretical background and computation algorithms of equilibrium finite element method are presented there. The different external effects are estimated, namely: load, prestressing, inicial strains and support settlements. The dual relationships (equilibrium and geometrical equations, stiffness and flexibility equations) of equilibrium element, also the expressions of stiffness and flexibility matrices are given. These relationships describe the stress-strain field of the finite element and allow to transforme the flexibility matrix to the stiffness matrix and on the contrary. There are presented direct and variational formulations of the problem. The algorithms of the forces method and displacements method are made for their solution. Easier is the algorithm of displacements method, because making equations of the forces method needs to solve the system of equilibrium equations. But the formulation and solving of the displacements equations can be changed by direct formulation of global flexibility matrix of the structure. In these cases, when the degree of static indeterminacy of the discrete model of structure is quite less than the number of freedom degrees, suth the change can be effective. The computational results illustrate the high accuracy of equilibrium elements not only with respect to forces, but also to displacements. The direct stiffness method in equilibrium elements makes it possible not only to simplify significantly implementation of the equilibrium element method, but also to include equilibrium elements in libraries of geometrically compatible elements.

The advanced Automated Fiber Placement (AFP) technologies allow the fiber trajectories to be curvilinearly steered over the panel platform, which brings out a novel type of cylindrical panel structure, termed as Variable Angle Tow (VAT) composite cylindrical panel. So far, the thermal postbuckling problem of VAT composite cylindrical panels under temperature loadings remain unexplored. The aim of this article is to cover this lack by developing a methodology based on a generalized Rayleigh–Ritz formulation in the framework of the principle of thermoelastic complementary energy. The proposed panel model is based on Donnell’s shallow shell theory and accounts for von Kármán’s geometrical nonlinearity. With the aid of the Lagrange multiplier method and the introduction of Airy’s stress function, a mixed variational formula that governs the nonlinear thermoelastic problem of VAT composite cylindrical panels is derived from the principle of thermoelastic potential energy. The Rayleigh-Ritz formulation combined with the mixed variational formula is then applied to derive the simultaneous nonlinear thermoelastic equations, and the Crisfield’s arc-length method, which can cope well with snap-through or snap-back problems, is employed to trace the equilibrium path that characterizes the thermal postbuckling features. Two different modeling strategies, namely, semi-inverse method and Lagrangian multiplier method, are adopted to formulate the nonlinear thermoelastic problem of VAT composite cylindrical panels undergoing temperature change. The application of Lagrangian multiplier method overcomes some limitations inherent in semi-inverse method in terms of constructing Rayleigh-Ritz formulation, and thus provides generality to model general in-plane boundary constraint. The accuracy and robustness of the proposed Rayleigh-Ritz model are validated against finite element solutions and previously published results. Effects of geometric imperfection, curvature radius, aspect ratio and fiber orientation angle on thermal postbuckling responses of VAT cylindrical plates are discussed by several examples. The study presented here may be helpful for the preliminary design of VAT curved panels in thermal environments.

, 요소 경계에서의 연속성이 기존의 모델의 합성력에 비해 개선됨을 제시하였다 최종적으로 수렴한 비선형해에 . 대한 유효성을 보이기 위하여 기존 모델의 비선형 수렴해와 현재 모델의 비선형 수렴해를 비교하여 제시하였다. Abstract: A mixed finite element model was developed using the classical plate theory to analyze the nonlinear bending of a plate. The appropriate weight functions for the constraints integrated over the domain were determined by the Lagrange multiplier method by using the principle of minimum virtual energy; which provides the constitutive relations between force-like variables and strains. All of detail terms of element wise coefficient matrices and associate tangent matrices to be used in the Newton iterative method are presented. Then, the linear solutions of the current model and those of the traditional displacement model under the SS (simple support) boundary conditions were compared with the existing analytical solution. The post-processed images of the nonlinear results of the force-like variables are presented to show the continuity of the solutions at the joint of the element boundaries. Finally, the converged nonlinear finite element solutions of the current model are compared with those of existing traditional displacement model.

The stability analysis of rock slopes has been a prominent topic in the field of rock mechanics, primarily due to the widespread occurrence of discontinuous structural planes in rock masses. Based on this complex characteristic of rock slopes, this paper proposes a novel numerical method, the Partitioned-Rigid-Element and Interface-Element (PRE-IE) method. In the PRE-IE method, the structure is modeled as several rigid bodies and discontinuous structural planes, which are, respectively, divided into partitioned rigid elements and interface elements. Taking the contact force of node pairs and the displacement of the rigid body centroid as mixed variables, according to the principle of minimum potential energy, the governing equations of PRE-IE can be established using the Lagrange multiplier method and then solved using the nonlinear contact iterative method and the incremental method. A classic case study demonstrates that using the failure of all contact node pairs as the criterion for slope failure is appropriate. This criterion is objective and avoids the potential impact of personal bias on safety factor calculations. Two numerical examples of differently shaped slopes are provided to verify the correctness and validity of the PRE-IE method. By comparing the safety factor calculated using the PRE-IE method with those obtained from other different methods, as well as comparing the computational time, it is shown that the PRE-IE method, in combination with the SRM, can accurately and efficiently analyze the stability problems of rock slopes.

Buckling and vibration analysis of axially functionally graded nanobeam based on local stress- and strain-driven two-phase local/nonlocal integral models

The minimum energy principle is the most fundamental principle in physical systems, which is that the evolution of physical systems always tends toward the minimum energy states. Coupled mode theory in metamaterials describes the energy interaction between structures and modes in metamaterials. In this paper, we investigate why the coupled mode theory arises from the minimum energy principle. We theoretically demonstrate our finding from the coupled mode theory and illustrate the full-wave simulations and experiments from the transmission spectra.

A new flexible sheet metal forming method of incremental bending

A distressed material can be considered as a two-phase composite material with the distressed areas (volumes) as one phase and the original material as the other phase. A parallel diffusion model was developed in which the damaged phase and the original phase are arranged parallel to the direction of diffusion. The parallel model is associated with the isogradient principle in that the concentration gradients in the damaged phase and in the original phase are the same. A serial diffusion model was developed in which the two phases are arranged perpendicular to the direction of diffusion. The serial model is associated with the isoflux principle in that the fluxes in the two phases are the same. The two-phase composite models were further extended into general multiphase composite models to evaluate the effect of multilevel damage on the effective diffusivity of distressed materials. A principle of minimum complementary “chemical-flux energy” and a principle of minimum “potential chemical energy” were established similar to the commonly used principles of minimum complementary energy and minimum potential energy. Based on the two new minimum energy principles, it was shown that the parallel model is the lower bound and the serial model is the upper bound for effective diffusivity of distressed materials. Diffusivity of distressed concrete was used as an example. The experimental data agreed quite well with predictions of the theoretical models.

We present a method for verifying the contact stress of elastic bodies. An explicit formulation of the total contact force, a fraction function with the numerator as a linear function and the denominator as a quadratic convex function, is derived by using the surrogate model of quadratic optimization for the contact problem. Then a bound formulation is obtained for the sum of the nodal contact forces, which is an explicit formulation of matrices of the finite element model, derived by maximizing the fraction function under the constraint that the sum of the normalized nodal contact forces is one. The bound is solved with the problem dimensions being only the number of contact nodes or node pairs, which are much smaller than the dimension for the original problem. Next, a scheme for constructing an upper bound on the contact stress is proposed that uses the bound on the sum of the nodal contact forces obtained on a fine finite element mesh and the nodal contact forces obtained on a coarse finite element mesh. Finally, the proposed method is verified through an example to demonstrate its feasibility and robustness. Keywords-contact stress; bound; surrogate; verification; finite elements I. INTRODUCTION Based on the principle of minimum potential energy, an elastic contact problem is essentially an optimization problem with the contact condition as a constraint. In the Karush-Kuhn- Tucker (KKT) conditions for these optimization problems, the Lagrange multipliers that represent the nodal forces are in an unbounded positive space. It is fortunate that when aggregating the constraints with a so-called surrogate constraint (1,2), we obtain an explicit formulation of the sum of the nodal contact forces, and the variable — the normalized nodal contact force — is only constrained in a bounded simplex. Therefore, it is possible to find an upper bound on the sum of nodal contact forces, and when this is done, we can obtain a bound on the contact stress with the information provided by the normalized nodal contact forces solved with a coarse mesh. For construction of the bounded feasible region, we have to solve an optimization problem with an objective being a fraction function. We first prove that the objective is pseudo concave in a neighborhood of the optimum, and then, with a suitable initial solution, optimization methods can be used to solve the fractional programming problem to obtain the optimum value. In this paper, we construct a bound that is explicitly formulated in terms of matrices regarding the finite element model, by maximizing the fraction function with a larger constraint field.