Minimization Of Complementary Energy Research Articles

Design‐oriented nonlinear‐elastic buckling analysis of reinforced concrete wall structures using convex optimization

AbstractThe structural response of slender reinforced concrete (RC) structures may be highly nonlinear due to cracking, reinforcement yielding, and geometrically nonlinear effects. While advanced models can simulate the detailed response of such structures, they are generally ill‐suited for limit state verification in practical design scenarios due to a high computational and modeling effort. A design‐oriented method for evaluating the geometrically linear response of cracked RC wall structures was recently presented and demonstrated to allow the analysis of large‐scale models with more than 2400 finite shell elements within minutes on a standard personal computer. This paper proposes a design‐oriented numerical method for efficient instability analysis of slender RC wall structures, also enabling the inclusion of thermal effects. Based on a two‐step linearized buckling analysis, the method first determines the geometrically linear structural response by solving the complementary energy minimization problem, including, if relevant, thermal strains and reduced material stiffness. This solution is used to derive the sectional stiffness upon which a linearized buckling problem is formulated and subsequently solved as a linear eigenvalue problem. The model is validated using examples with exact solutions, and its applicability to large‐scale models is demonstrated through an example with a four‐story stairwell modeled using more than 3200 finite elements.

Estimating Contact Forces and Moments for Walking Robots and Exoskeletons Using Complementary Energy Methods

When walking robots and exoskeletons make multiple independent contacts, the inverse dynamics problem requires additional knowledge about the contact forces and moments. To avoid measuring the contact forces and moments, many inverse dynamics controllers for walking robots optimize an objective such as minimizing torques or contact forces. In order to get a solution closer to the real solution, the underlying physical principles need to be included. This is achieved by relying on the minimization of complementary energy, which is a well known method in structural engineering. The proposed method relies on physical properties (stiffness) to obtain the additional knowledge to solve the contact forces and moments. In addition, it has the same form as the methods used in the literature. The validation on a bilateral lower-limb exoskeleton shows that the proposed method is able to predict the contact forces and moments sufficiently well, while being robust against modeling errors. The modeling dominating flexibilities suffices to achieve adequate results, making this method especially interesting for series elastic actuated robots.

Variational analysis of free-edge stress and displacement fields in general un-symmetric and thin-ply laminates under in-plane, bending and thermal loading

A variational approach based on the minimization of complementary energy is developed to determine accurately a complete solution for both free-edge stress and displacement distributions of a laminate with arbitrary lay-ups (possibly un-symmetric and made of thin plies) under combined in-plane, bending and thermal loading. The key idea is partitioning the total stresses/displacements in a laminate with free edges into unperturbed (without free edges) and unknown perturbation stresses/displacements caused by the presence of free edges. It enables the theory of variational stress-transfer to deal easily with both applied traction and displacement boundary conditions. A methodology is introduced to obtain displacement fields for a stress-based variational approach. The resulting stress and displacement fields exactly satisfy local equilibrium equations, strain-displacement relations together with all traction/displacement boundary and continuity conditions. By comparing the results with those obtained from the finite element method, the accuracy and computational efficiency of the developed model, is confirmed.

Concurrent optimization of material spatial distribution and material anisotropy repartition for two-dimensional structures

An optimization methodology to find concurrently material spatial distribution and material anisotropy repartition is proposed for orthotropic, linear and elastic two-dimensional membrane structures. The shape of the structure is parameterized by a density variable that determines the presence or absence of material. The polar method is used to parameterize a general orthotropic material by its elasticity tensor invariants by change of frame. A global structural stiffness maximization problem written as a compliance minimization problem is treated, and a volume constraint is applied. The compliance minimization can be put into a double minimization of complementary energy. An extension of the alternate directions algorithm is proposed to solve the double minimization problem. The algorithm iterates between local minimizations in each element of the structure and global minimizations. Thanks to the polar method, the local minimizations are solved explicitly providing analytical solutions. The global minimizations are performed with finite element calculations. The method is shown to be straightforward and efficient. Concurrent optimization of density and anisotropy distribution of a cantilever beam and a bridge are presented.

Investigation of the contact performance of machined surface morphology

Engineering practices have indicated that surface textures and topography greatly affect the mechanical performance of contact interfaces. The effect of machined surface morphologies on the contact performance results, including average surface gap, contact area ratio, and the volume of plastically deformed material, is investigated. Depending on the machined parameters, the studied surfaces are generated using computer simulation. The elasto-plastic contact of rough surfaces is analyzed by means of semi-analytical method (SAM) and minimization of complementary energy. Results indicate that with increasing the interval of texture, average surface gap remains invariant while contact area ratio and the volume of plastically deformed material decreases.

Variational models for shear modulus of symmetric and balanced laminates with cracks in 90-layer

Three analytical models with increasing complexity, all based on minimization of complementary energy, are compared in their ability to predict shear modulus reduction of laminates with intralaminar cracks in 90-layer. The very elegant and simple (and the less accurate) model by Hashin assumes linear out-of-plane shear stress distribution across the ply in all layers. The second model assumes exponential shape of these stresses in the constraint layer keeping linear assumption in the cracked layer. The model developed in the present paper accounts for nonlinear out-of-plane shear stress thickness distribution in all layers described by shape functions determined in the procedure of minimization. Increasing the complexity of the model the predicted shear modulus of the damaged laminate increases approaching to value obtained using finite elements (FE). Results show that for laminates with relatively thick cracked layers the stress state description in the cracked layer should be refined whereas for laminates with constraint layer thicker than the cracked layer more accurate stress description in the constraint layer is necessary. More accurate solutions could be derived using the described methodology, but the involved complexity and the numerical routines required for their application diminish their value comparing with direct FE solution.

The effect of surface roughness characteristics on the elastic–plastic contact performance

Rough surface elastic–plastic contact performance is investigated in this paper. First, a computer program is developed to generate rough surfaces with given parameters; Then, the elastic–plastic contact model is developed based on minimization of complementary energy and semi-analytical method; finally, contact analysis for rough surfaces are conducted. The results show that kurtosis and skewness have significant effects on the contact performance under light-medium load; for heavy load condition and small skewness, the contact characteristic parameters change slightly along with kurtosis. Comparing with elastic contact, the low contact pressure and large contact area are predicted in elastic–plastic contacts.

Minimization of complementary energy to predict shear modulus of laminates with intralaminar cracks

The most common damage mode and the one examined in this work is the formation of intralaminar cracks in layers of laminates. These cracks can occur when the composite structure is subjected to mechanical and/or thermal loading and eventually lead to degradation of thermo-elastic properties. In the present work, the shear modulus reduction due to cracking is studied. Mathematical models exist in literature for the simple case of cross-ply laminates. The in-plane shear modulus of a damaged laminate is only considered in a few studies. In the current work, the shear modulus reduction in cross-plies will be analysed based on the principle of minimization of complementary energy. Hashin investigated the in-plane shear modulus reduction of cross-ply laminates with cracks in inside 90-layer using this variational approach and assuming that the in-plane shear stress in layers does not depend on the thickness coordinate. In the present study, a more detailed and accurate approach for stress estimation is followed using shape functions for this dependence with parameters obtained by minimization. The results for complementary energy are then compared with the respective from literature and finally an expression for shear modulus degradation is derived.

A Numerical Elastic–Plastic Contact Model for Rough Surfaces

Elastic-plastic contacts were analyzed by means of a three-dimensional numerical model based on minimization of complementary energy, in which the effect of plastic deformation is included by superimposing the plastic residual displacement on the geometry of contacting surface. The plastic strain increments are determined through the radial return method and J 2 flow theory, which are integrated with the conjugate gradient method (CGM) algorithm and the fast Fourier transform (FFT) technique for improving numerical efficiency. The results from the present numerical model were compared to those from finite element method (FEM) for three typical contacts—i.e., those of a smooth elastic-plastic ball in contact with a rigid body of different geometry: a smooth plane, a plane superposed with a single asperity, and a sinusoidal wave on the plane—which showed persistent good agreement between our model and FEM. Then, three-dimensional elastic-plastic contacts of real engineering surfaces produced by a grinding process were analyzed. Due to roughness effect, the maximum von Mises stress and plastic region were found at the locations closer to the surface. In elastic-plastic contacts, the pressure decreases at the peaks and increases at the valleys if compared to the results from purely elastic model. Moreover, plastic flow makes the rough surface flatten or become less fluctuating.

Topology Optimization Using Frame Elements (For Cases Where Eigen-frequencies are to be Maximized)

This paper presents a new topology optimization method that maximizes multiple eigen-frequencies to obtain, at the conceptual design phase, the most stable designs of mechanical structures for dynamic problems. This method is developed based on frame elements that can consider the principal direction of the second moment of inertia in the cross-section to obtain optimal results that illustrate the relationship between the objective function and desired cross-sectional properties. For each frame element, an ellipsoidal cross-section is assumed as the simplest case. The normalized cross-sectional area, the rotational angle denoting the principal direction, and the ratio of the lengths of the major and minor axes, are considered as design variables. A method for obtaining the optimal angle is derived, based on the minimization of complementary energy. Finally, several numerical examples confirm the validity and utility of the proposed method, which should be useful for mechanical design engineers at the conceptual design phase.

416 構想設計支援を目的とした離散構造要素に基づく最適設計法(GS-12 設計工学,研究発表講演)

This paper presents the outcome of research concerning the development of a structural topology optimization methodology that can be used at the conceptual design phase of mechanical products. The objective function is formulated as the minimization of the mean compliance. The optimization method is constructed based on frame elements, which are a typical discrete structural element. For each frame element, an ellipsoidal cross-section is assumed as the simplest case. The normalized cross-sectional area, the rotational angle denoting the principal direction, and the ratio of the length of the major and minor axes, are considered as design variables. A method for obtaining the optimal angle is derived, based on the minimization of complementary energy. Finally, several numerical examples confirm the validity and utility of the proposed method, which should be useful for mechanical design engineers at the conceptual design phase.

Numerical Analysis for the Elastic Contact of Real Rough Surfaces

The elastic contact of rough surfaces and the subsurface stresses caused by the contact have been analyzed by means of a numerical model based on fast Fourier transforms (FFT) and minimization of complementary energy. The elastic contact has been modeled mathematically as a linear complementarity problem and solved by a robust algorithm, Conjugate Gradient Method, while the force-displacement relation is determined through a FFT approach. After solving for the pressure distribution, the subsurface stress field is obtained by calculating the stresses due to the application of a point force, and then integrating over the contact region. In comparison with the matrix based method published in recent years, the numerical approach presented in this study is more efficient, more stable and requires less memory. It has great potential in application to problems with general contact geometry and three-dimensional surface roughness. The results show that high frequency roughness could lead to very sharp impulses and significant oscillations in pressure distribution. The amplitude and location of the maximum shear stress in the subsurface region are constantly changing when the contacting rough surfaces are in relative motion. Presented at the 53rd Annual Meeting in Detroit, Michigan May 17–21, 1998

A closed-form solution for stresses at curved free edges in composite laminates: A variational approach

An analytical method developed for determining the interlaminar stresses at straight free boundaries is extended to predict the free-edge stresses at curved boundaries of symmetric composite laminates under inplane loading. The three-dimensional (3D) stress distribution in laminates with curved boundaries is approximately described on the basis of a zero-order approximation of the boundary-layer theory. The related stress functions are found by minimization of complementary energy and the variational principle, and satisfy zero-order equilibrium equations, boundary conditions and traction continuity at interfaces between plies. Results obtained from the present method are compared with those in the literature for laminates with both straight edges and circular holes. This method provides a computational technique with the advantage of simplicity and versatility for analyzing the 3D stresses in composite laminates of general ply orientations with curved boundaries.

Nonlinear stress and displacement analysis of rigid plane frames by total complementary energy minimization

New and powerful methods for stress and displacement analysis of rigid plane frames subjected to material nonlinearities are proposed on the basis of the principle of minimum complementary energy, the Crotti-Engesser theorem and mathematical programming techniques. The stress analysis problem is formulated as a minimization of the total complementary energy subject to the equilibrium equations at the free nodes. The unknown member end forces are determined by solving the energy minimization problem using a modified sequential quadratic programming and linear programming algorithms. The displacement at an arbitrary point i is determined as the first partial derivative of the minimized total complementary energy with respect to the load P i acting at the point i. The partial derivatives are numerically calculated using a finite difference formula which is embedded in the stress analysis algorithm. The problem formulation and analysis algorithms of the proposed methods are applicable to rigid frame structures with any type of material nonlinearity. The accuracy, reliability and efficiency of the proposed methods are demonstrated by comparing their results with those of the displacement method of analysis for several statically indeterminate rigid frame structures with three types of material nonlinearities.

Method for calculating the interlaminar stresses in symmetric laminates containing a circular hole

An efficient approximate solution for three-dimensional stress distributions around a circular hole in symmetric laminates under a set of far-field in-plane stresses is presented. Stress functions for each ply are assumed according to the boundary-layer equilibrium equations. In addition, all the boundary conditions for each ply and traction continuity at the ply interface are exactly satisfied. Eventually, the unknown parameters in stress functions are determined by the minimization of complementary energy of the whole laminate. Numerical examples are presented in comparison with given literature which shows that the present method is efficient. I. Introduction D UE to stiffness discontinuity between plies, it is easy to cause interlaminar stress concentrations near the freeedge region in composite laminates. Such stresses will commence delamination, matrix crackings, and failure of laminates, especially under fatigue loading. Since 1970, numerous investigations15 have used various methods to determine the interlaminar stresses at the straight free edge of composite laminates. Pipes and Pagano1 adopted the anisotropic elasticity theory in conjunction with the finite difference method to analyze a simple four-ply laminate. Wang and Crossman2 used a finite element approach to investigate this same problem. Wang and Choi3'4 derived an analytical solution based on Lekhnitskii's stress potentials and the theory of anisotropic elasticity to determine the exact order of stress singularity at the free edges of laminate. In an effort to develop an efficient method to deal with thick laminates (say 100 plies), Kassapoglou and Lagace5 used the force balance method in conjunction with the principle of minimum complementary energy to obtain an analytical solution for interlaminar stresses at straight free edges. Owing to the complicated geometry for the curved free edges as compared with the straight free edges, little work has been done for composite laminates with curved free edges. Basically, the analysis of straight free edge may be assumed as a problem with two-dimensional stress and strain variations. However, the curved free edge is a typical three-dimensional problem. This difference has rendered the analysis for the curved free edge more difficult than the straight free edge. A boundary-layer theory based on the perturbation technique for isotropic elastic plates with a circular hole developed by Reiss6 has been extended to composite laminates by Tang.7-8 This approach is based on identifying the boundary-layer problem as two equivalent problems, namely, a modified torsion problem and a modified plane strain problem. However, Tang's solution can only satisfy part of boundary conditions in an average sense, which may result in unreliable stress results near the free edges.9 In addition, Tang's solution will be very tedious for laminates with numerous plies. Zhang and Ueng10 proposed a simplified method to investigate the effect of the ratio of hole radius to laminate thickness on the interlaminar stress distributions around a hole in a (0/90 deg)s laminate under far-field tensile or shear stress; however, it is assumed that the order of stress singularity is prescribed and

MATERIAL NONLINEAR ANALYSIS OF RIGID PLANE FRAMES BY TOTAL COMPLEMENTARY ENERGY MINIMIZATION

A new and powerful material nonlinear analysis method for rigid plane frames is proposed on the basis of the principle of minimum complementary energy and mathematical programming algorithms. The analysis problem is formulated as the total complementary energy minimization problem subjected to the equilibrium equations at the free nodes in terms of the unknown forces acting at the ends of member elements. The unknown member end forces are determined by solving the energy minimization problem using a modified sequential quadratic programming algorithm.The problem formulation and analysis algorithm of the proposed method are applicable for rigid frame structures with any types of nonlinear materials. The power, reliability, efficiency and practical usefulness of the method presented are demonstrated by comparing the results obtained with ones by the displacement method of analysis for several statically indeterminate rigid frame structures with three types of nonlinear materials

ANALYSIS OF BEAM STRUCTURES WITH NONLINEAR MATERIALS BY TOTAL COMPLEMENTARY ENERGY MINIMIZATION

A new analysis method for straight and continuous beam structures with nonlinear materials is proposed on the basis of the principle of minimum complementary energy and mathematical programming algorithm.The bending moment distribution of a continuous beam structure is expressed in terms of the unknown redundant bending moments acting at the supports and the analysis problem is formulated as an unconstrained total complementary energy minimization problem, Then the redundant bending moments are determined by solving the energy minimization problem with the aid of a sequential quadratic approximation algorithm. In the process of the energy minimization, pre-arranged bending moment-complementary energy relations for the cross-sections of beam elements made of given nonlinear materials are used very effectively for calculations of the total complementary energy and the sensitivities of beam structure.The problem formulation and analysis algorithm of the proposed method are quite simple and applicable for any types of linear and nonlinear material beam problems. The reliability and efficiency of the method are confirmed by comparing the results obtained with those by the finite element method for several statically indeterminate continuous beams with linear and three types of nonlinear materials.

Nonlinear analysis of truss by energy minimization

New analysis methods for trusses with material and geometric nonlinearity are presented using the energy principles and modified sequential quadratic programming (SQP) algorithms. The analysis problem of a truss with material nonlinearity is formulated as the total complementary energy minimization problem and the member forces are determined by using modified SQP algorithms. The truss subject to nonlinearities both in material properties and finite displacements is solved by the combination of complementary and potential energy minimization algorithms. The problem formulations on the basis of energy principles are quite simple and the proposed analysis methods can be applicable for any type of nonlinear material problem. The efficiency and reliability of the proposed method are clarified by giving numerical results for 3–33-bar statically indeterminate trusses with three types of nonlinear materials.

ANALYSIS OF TRUSSES WITH NONLINEAR MATERIALS BY USING ENERGY PRINCIPLES AND SEQUENTIAL QUADRATIC PROGRAMMING

New analytical methods for trusses with nonlinear materials are presented by using energy principles and sequential quadratic programming (SQP) algorithms. The analytical problem of nonlinear truss is formulated as the total potential or complementary energy minimization problem and minimized with respect to the displacements of panel points or member forces respectively by using modified SOP algorithms. The formulations of the energy methods are quite simple and the methods can be applicable for any types of nonlinear material problems. Computational results for 3-31 bar statically indeterminate trusses with several types of nonlinear materials show that convergency and reliability of the method based on the complementary energy minimization is most excellent and the final results of nonlinear truss analyses can be obtained less than 2/3-1/3 of computational efforts compared with the incremental load method.

Analysis of stiffness reduction of cracked cross-ply laminates

Stiffness reduction of cracked [ 0° m 90° n ] s laminates is analyzed by variational methods on the basis of the principle of minimum complementary energy. For this purpose admissible stress systems are constructed which satisfy equilibrium and all boundary and interface conditions. The optimal stress field is then determined by minimization of complementary energy. The analysis allows for crack interaction and random crack distribution. Results are given for Young's modulus, shear modulus and Poisson's ratio. Young's modulus results are in excellent agreement with experimental data for [[ 0° 90° 3 ]] s glass/epoxy laminate.