Minimum Principle of Complementary Energy for Nonlinear Elastic Cable Networks with Geometrical Nonlinearities

The static analysis of elastic cable networks submitted to generic, conservative loads and prescribed dislocations is considered in this paper. The cables are assumed as stress-unilateral (only tensile stresses are admitted), and represented according to a Lagrangian standpoint, under the customary large displacements hypothesis. A variational formulation of the problem is given, as the stationary of a saddle functional with respect to displacements and (signconstrained) tractions. Uniqueness properties for the solution are derived from this statement, together with two complementary (constrained) minimum formulations, which correspond to the well-known extremum principles of the total potential energy and the complementary energy. The case of a network loaded only at the nodes is exposed as a specialization.

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.

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function , we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

The present paper concerns the mathematical study and the numerical treatment of the problem of semirigid connections in bolted steel column base plates by taking into account the possibility of appearance of separation phenomena on the contact surface under certain loading conditions. In order to obtain a convenient discrete form to simulate the structural behaviour of a steel column base plate, the continuous contact problem is first formulated as a variational inequality problem or, equivalently, as a quadratic programming problem. By applying an appropriate finite element scheme, the discrete problem is formulated as a quadratic optimization problem which expresses, from the standpoint of Mechanics, the principle of minimum potential energy of the semirigid connection at the state of equilibrium. For the numerical treatment of this problem, two effective and easy-to-use solution strategies based on quadratic optimization algorithms are proposed. This technique is illustrated by means of a numerical application.

SummaryThe derivation of a non-linear theory on bending of orthotropic sandwich plates is carried out by the principle of complementary energy from elasticity. The governing differential equations and natural boundary conditions are obtained. The assumptions used are, namely, the facings are orthotropic thin elastic plates with negligible bending rigidities and are made of the same material; the orthotropic core can take the transverse shear only; and the transverse shortening of the core may be ignored. The geometrical non-linearities are equivalent to von Kármán’s theory for single-layered plates. Through the introduction of a “shear function”, the number of differential equations is reduced to three and the equations are in rather simple form. It appears that the equations obtained here would require, comparatively, the least amount of work for analysing the finite deflection sandwich plate problems having a wide range of properties of practical interest.

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

The aim of this note is to analyse the results in the paper by D.Y. Gao and G. Strang [Geometric nonlinearity: Potential energy, complementary energy, and the gap function, Quart. Appl. Math. 47 (1989), pp. 487–504] where, according to D.Y. Gao, the complementary gap function was first introduced, and several flaws, mistakes, and inconsistencies are present, and to discuss the mathematical manner in which the mentioned paper presents itself, especially from the point of view of convex analysis.

The suspension cable structure with a small rise-span ratio (less than 1/30) is adopted in the flexible photovoltaic support, and it has strong geometric nonlinearity. Based on the principle of energy, the increment of cable force and the change of cable displacement under concentrated force are derived for the suspension cable in an equilibrium state under uniform distributed load, and the equivalent bending stiffness of simply supported beam under bending is given. Considering the strain energy generated by cable force variation, the method presented in the paper has higher calculation accuracy for suspension cable structures with a small rise-span ratio, and includes the special case of a large rise-span ratio. An engineering example of flexible photovoltaic support with a span of 15m is calculated and analyzed, and then compared with the finite element calculation results. The results show that the calculation error is less than 1% when the rise is 0.4 m, and the calculation error is less than 3% when the rise is 1.5m. The calculation formula in the paper is simple and accurate, which can provide a reference for static analysis and structural design of flexible photovoltaic support.

Large-amplitude free vibration analysis of uni-axially single stiffened rectangular plates subjected to transverse loading with simply supported boundary conditions has been presented. The mathematical formulation is based on energy principle. Geometric nonlinearity is accounted for by consideration of nonlinear strain-displacement relations. The static problem is solved through an iterative scheme and the dynamic problem is solved with the static displacement field as an initial guess. Influence of stiffener position, plate aspect ratio, and stiffener to plate thickness ratio on the large amplitude dynamic behavior has been studied. The dynamic behavior has been furnished in the form of backbone curves in a dimensionless frequency-amplitude plane.

Chapter 8 - Finite-Element Time Domain Method

The stable operation of the integrated energy system (IES) is affected by the fluctuation of heat or electricity loads, energy or communication problems, and the interaction of different energy networks. To overcome above challenges, firstly, the integrated dynamic models with three-time scales are established for control design, fully considering dynamic behaviors, energy coupling, and constraints of power flow including electrical, heat and gas on network stability. Then, a novel distributed model predictive control (DMPC) consensus algorithm with process control and trajectory control is proposed to achieve stable operation for IES based on second-order cone programming (SOCP). A unified distributed cooperative control mode is formed in IES by implementing the proposed stability control method, which is suitable for different forms of energy. The control design takes into account the different dynamic characteristics of energy equipment and the energy coupling feature, which realizes the multi-scale time cooperation and dynamic process control for IES by tuning the control parameters. The optimal solution is obtained by rolling optimization for achieving frequency/voltage/temperature/pressure/flow regulation and consensus objectives. Finally, the proposed model and method are validated in various simulation scenarios, such as load variation, energy flow variation, nodes removed, and topology change.

A note on the contributions of Cotterill, Castigliano, Crotti and Engesser to an energy principle of structures

The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stiffness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

In this article, a numerical solution method for the finite-strain rate-independent Cosserat theory of crystal plasticity is developed. Based on a time-incremental minimization problem of the mechanical energy, a limited-memory Broyden–Fletcher–Goldfarb quasi-Newton method applied to a finite-difference discretization is proposed. First benchmark tests study the convergence to an analytic solution. Further simulations focus on the investigation of rotation localization zones, the bending of a rod, and a torsion experiment.

A new and computationally efficient implementation of the variable charge method of Streitz and Mintmire (1994 Phys. Rev. B 50 11996) is presented. In particular a local chemical potential approach that optimizes the charge on only those atoms expected to be ionic is developed. By doing so, the charge fluctuation problem experienced in regions far from any oxygen is solved, leading to a linear minimization problem of the electrostatic energy. In the dilute oxygen limit, such an approach can lead to at least an order of magnitude saving in computation.