Theorem Of Minimum Potential Energy Research Articles

High accuracy postbuckling analysis of box section struts

AbstractThis paper presents the theoretical developments of two finite strip methods (i.e. semi‐analytical and full‐analytical) for the post‐buckling analysis of some box section struts. In the semi‐analytical finite strip approach, all the displacements are postulated by the appropriate shape functions while in the development process of the full‐analytical approach, the von‐Kármán's equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out‐of‐plane buckling deflection modes. The investigation of struts buckling behaviour is then extended to the post‐buckling study with the assumption that the deflected form after the buckling is the combination of first, second and higher (if required) modes of buckling. Thus, the full‐analytical post‐buckling study is effectively a multi term analysis, which is attempted by utilizing the so‐called semi‐energy method. In this method the von‐Kármán compatibility equation is used together with a consideration of the total strain energy of the strut. Through the solution of the compatibility equation, the in‐plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficients in the assumed out‐of‐plane deflection function. The in‐plane and out‐of‐plane deflection functions are substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficients. It is noted that the Classical Plate Theory (CPT) is applied throughout the theoretical developments. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of these methods is significantly promoted.

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b=0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.

An Investigation on Relative Post-Buckling Stiffness Variations of Symmetrically Cross-Ply Laminates

In this paper, the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of symmetrically cross-ply laminates are presented. The so-called exact finite strip is developed based on the concept that it is effectively a plate. In the development process, the Von-Karman’s equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman’s compatibility equation governing the behavior of symmetrically laminated composite plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to investigate the relative post-buckling stiffness variations of some representative thin symmetric cross-ply laminates for which the results are also obtained through the application of a semi-analytical finite strip method.

Energy approach vibration analysis of nonlocal Timoshenko beam theory

The aim of this paper is to present an efficient numerical method to analyze the vibration behavior of nano Timoshenko beams based on Eringen's nonlocal elasticity theory. The vibration frequencies of beams are firstly obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. The present method provides an efficient and extremely accurate vibration solution. Numerical results for a variety of some micro- and nano-beams with various boundary conditions are given and compared with the available results wherever possible. The small scale effects on the vibration frequencies of beams are determined and discussed.

An exact finite strip for the initial postbuckling analysis of channel section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of channel section struts. The presented method provides an efficient and extremely accurate buckling solution. The Von-Karman’s equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the channel section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. Through the solution of the Von-Karman’s compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial post-buckling behavior of some representative channel sections for which the results were also obtained through the application of a semi-energy finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Relative Post-Buckling Stiffness Calculation of Symmetrically Laminated Composite Plates Using an Exact Finite Strip

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of symmetrically laminated composite plates. The so-called exact finite strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman’s equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman’s compatibility equation governing the behavior of symmetrically laminated composite plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial post-buckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Buckling Analysis of Micro- and Nano-Rods/Tubes Based on Nonlocal Timoshenko Beam Theory Using Chebyshev Polynomials

This paper presents the elastic buckling behavior of nonlocal micro- and nano- Timoshenko rods/tubes based on Eringen’s nonlocal elasticity theory. The critical buckling loads are obtained using the theorem of minimum total potential energy and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate buckling solution. Numerical results for a variety of some micro- and nano-rods/tubes with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted. The small scale effects on the buckling loads of rods/tubes are determined and discussed.

Alternative positional FEM applied to thermomechanical impact of truss structures

This paper presents a formulation to deal with dynamic thermomechanical problems by the finite element method. The proposed methodology is based on the minimum potential energy theorem written regarding nodal positions, not displacements, to solve the mechanical problem. The thermal problem is solved by a regular finite element method. Such formulation has the advantage of being simple and accurate. As a solution strategy, it has been used as a natural split of the thermomechanical problem, usually called isothermal split or isothermal staggered algorithm. Usual internal variables and the additive decomposition of the strain tensor have been adopted to model the plastic behavior. Four examples are presented to show the applicability of the technique. The results are compared with other authors’ numerical solutions and experimental results.

A FEM Formulation for Flexible Multi-Body System Dynamic Analysis

A simple engineering language is used to present a geometrical non-linear formulation based on position description. Velocity, acceleration and strain are achieved directly from nodal position, not displacements. The proposed methodology is based on the minimum potential energy theorem. A non-dimensional space is created and the relative curvature and fibers length are calculated for both reference and deformed configurations and used to directly compute the strain energy at general points. Damping is introduced into the mechanical system by a rheonomic energy functional. The final formulation is shown simplifying the understanding and the implementation of large deflection analysis when compared with typical FEM formulations.

Exact post‐buckling stiffness calculation of box section struts

PurposeThe purpose of this paper is to develop and apply an exact finite strip (F‐a FSM) for the buckling and initial post‐buckling analyses of box section struts.Design/methodology/approachThe Von‐Karman's equilibrium equation is solved exactly to obtain the buckling loads and deflection modes for the struts. The investigation is then extended to an initial post‐buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. Through the solution of the Von‐Karman's compatibility equation, the in‐plane displacement functions are developed in terms of the unknown coefficient. These in‐plane and out‐of‐plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient.FindingsThe F‐a FSM is applied to analyze the buckling and initial post‐buckling behavior of some representative box sections for which the results were also obtained through the application of a semi‐energy finite strip method (S‐e FSM). For a given degree of accuracy in the results, however, the F‐a FSM analysis requires less computational effort.Research limitations/implicationsIn the present F‐a FSM, only one of the calculated deflection modes is used for the initial post‐buckling study.Practical implicationsA very useful and computationally economical methodology is developed for the initial design of struts which encounter post‐buckling.Originality/valueThe originality of the paper is the fact that by incorporating a rigorous buckling solution into the Von‐Karman's compatibility equation, and solving it, a fairly efficient method for post‐buckling stiffness calculation is achieved.

Effects of In-Plane Shear Loads on Transverse Shear Deformations in Composite Panels

An analytical and experimental investigation is performed to clarify the transverse shear deformations that occur in laminated and sandwich composite plates subjected to in-plane shear loads. This is part of a larger study investigating the feasibility of an in-plane shear test as a practical procedure to determine the in-plane shear modulus and/or strength of various material systems as well as the faces and cores of sandwich plates. The theorem of minimum potential energy has been used to determine the deflections and strains in the plate, including the transverse shear deformation effects. Polynomial functions that satisfy all the boundary conditions of the material system are used as trial functions. Comparison of these solutions to those first proposed by Nadai in 1925 revealed that additional terms are added to the original analysis (which used classical plate theory) to incorporate these effects. The methods developed clearly show the effects of the transverse shear deformation and are valid for any sandwich, laminate, or monocoque plate of any aspect ratio. Subsequent finite element and experimental analysis were conducted to verify the analytical study. Both the numerical and experimental results confirm that the analytical results are very good approximations for describing the linear deformation of the panels subject to in-plane shear loads.

An exact finite strip for the calculation of relative post-buckling stiffness of isotropic plates

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of isotropic flat plates. The so-called exact finite strip is assumed to be simply supported out-of-plane at the loaded ends. The strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. It is noted that in the present method, only one of the calculated out-of-plane buckling deflection modes, corresponding to the lowest buckling load, i.e., the first mode is used for the initial post-buckling study. Thus, the postbuckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman's compatibility equation governing the behavior of isotropic flat plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial postbuckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Unconstrained Finite Element for Geometrical Nonlinear Dynamics of Shells

This paper presents a positional FEM formulation to deal with geometrical nonlinear dynamics of shells. The main objective is to develop a new FEM methodology based on the minimum potential energy theorem written regarding nodal positions and generalized unconstrained vectors not displacements and rotations. These characteristics are the novelty of the present work and avoid the use of large rotation approximations. A nondimensional auxiliary coordinate system is created, and the change of configuration function is written following two independent mappings from which the strain energy function is derived. This methodology is called positional and, as far as the authors′ knowledge goes, is a new procedure to approximated geometrical nonlinear structures. In this paper a proof for the linear and angular momentum conservation property of the Newmark β algorithm is provided for total Lagrangian description. The proposed shell element is locking free for elastic stress‐strain relations due to the presence of linear strain variation along the shell thickness. The curved, high‐order element together with an implicit procedure to solve nonlinear equations guarantees precision in calculations. The momentum conserving, the locking free behavior, and the frame invariance of the adopted mapping are numerically confirmed by examples.

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Numerical and Experimental Analysis of Solder Joint Self-Alignment in Fiber Attachment Soldering

Fiber attachment soldering is low cost and high-precision technology in direct-coupling optoelectronic packaging. For accurate alignment, it is crucial to understand the self-alignment behavior of solder joint. In this research, the self-alignment method by using surface tension of molten solder and by adopting specific pad shape was proposed. First, the self-alignment model of solder joint in fiber attachment soldering was developed by using the public domain software called SURFACE EVOLVER and the three-dimensional geometry of solder joint with different solder volume was analyzed. Then, the self-alignment behavior of solder joint with an initial yaw misalignment was discussed and the theoretical equilibrium positions of ellipse and square pad were calculated. Next, based on the minimum potential energy theorem and data from geometry simulation, the influences of design and material parameters on the standoff height (SOH) were analyzed. Furthermore, experiments were done to examine the theoretical equilibrium positions of ellipse and square pad and the SOHs of solder joints were measured by using confocal scanning laser microscope. The numerical results show that the theoretical equilibrium positions of ellipse and square pad are the major axis of ellipse and the diagonal of square, respectively. SOH can be controlled by adopting proper solder volume, which is above the critical value for specific pad. The experimental results show that the solder joint with initial yaw angle can be self-aligned to the theoretical equilibrium position of pad and solder joint with ellipse pad substrate demonstrates smaller alignment error than those with square pad substrate. The measurement results of SOH are in agreement with the simulation results.

Effective design of a sandwich beam with a metal foam core

The subject of the paper is a simply supported sandwich beam with a metal foam core. Mechanical properties of the core vary through its depth. A nonlinear hypothesis of deformation of a plane cross section of the beam is assumed and described. The elastic potential energy and the work of the load are formulated. The system of differential equations of the equilibrium is derived on the basis of the theorem of minimum total potential energy. The system of equations is analytically solved. The stress state and the critical force for the beam are determined. The optimal dimensionless parameters of the beam are calculated. Results of the numerical investigations are presented in figures.

Cohesive zone size of microcracks in brittle materials

Abstract Consideration of cohesive microcracks in continuum micromechanics is a challenging task since a lot of applications (such as, e.g., estimation of the stiffness of a microcracked solid) require a priori knowledge of the size of the cohesive zone. The latter, however, can be determined analytically only for the special case of Barenblatt–Dugdale cracks, i.e. for cracks with spatially constant cohesive tractions. Herein, we deal with the general case of spatially non-constant cohesive tractions: Generalizing the Barenblatt–Dugdale approach, we consider that each crack is surrounded by a plane annular cohesive zone characterized by a constitutive softening law (introduced as a power law) relating the vector of cohesive tractions to the displacement discontinuity. The size of this cohesive zone is then estimated using the theorem of minimum potential energy, based on a class of kinematically admissible displacement fields.

20506 統一エネルギ原理に基づいた骨組構造解析プログラムの開発(計算力学)

Framed Structural Analysis was established by Slope -Deflection Method in the early 20th Century. With the progress of Electric Computer, it was completed by the Displacement Method. But, Force Method has more advantage in structural design. It is well known that the minimum potential energy theorem is proposed with respect to the displacements u_i of a given solid and theoretically it gives the upper bound of the stiffness evaluation while the minimum complementary energy theorem gives the lower bound so that the latter method may give the more conservative results of the stress and displacement calculations that the former. In this paper, a Framed Structural Analysis based on the mixed Method is reported.

Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin’s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved. The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.

Positional FEM formulation for flexible multi-body dynamic analysis

This paper presents a simple formulation to deal with flexible multi-body dynamic systems by the finite element method. The proposed methodology is based on the minimum potential energy theorem written regarding nodal positions. Velocity, acceleration and strain are achieved directly from positions, not displacements. A non-dimensional space is created and the relative curvature and fibers length are calculated for both reference and deformed configurations and used to calculate the strain energy at general points. The classical Newmark equations are used to integrate time. Damping is introduced into the mechanical system by a rheonomic energy functional. The final formulation has the advantage of being simple and easy to teach, when compared to classical counterparts. The behavior of a bench-mark problem (spin-up maneuver) is studied regarding the influence of mass representation on its overall transient and steady-state behavior. Three other examples are presented to show the applicability of the technique, namely, a flexible slider–crank mechanism, a flexible beam flight and a Peaucellier-type mechanism. The results are compared with other authors’ numerical solutions.