Strain Energy Expression Research Articles

Large Deflection Analysis of Functionally Graded Beams Resting on a Two-Parameter Elastic Foundation

This paper presents a finite element procedure for the large deflection analysis of functionally graded (FG) beams resting on a two-parameter elastic foundation. The material properties of the FG beams are assumed to vary continuously in the thickness direction by a power-law distribution. Based on the strain energy expression, a shear deformable beam element, taking the effect of the material non-homogeneity and the foundation support into account, is formulated and employed in the analysis. An incremental/iterative procedure in combination with the arc-length control method is used for solving nonlinear equilibrium equations. The numerical results show that the convergence of the formulated element is fast, and the large displacements of the beams can be accurately assessed by using a few numbers of the elements. A parametric study is carried out to highlight the effect of the material non-homogeneity and the foundation support on the large deflection behavior of the beams. The influence of the aspect ratio on the large deflection response of the beam is also examined and highlighted.

Structural Multi-Damage Identification Based on Modal Strain Energy Equivalence Index Method

In order to solve structural multi-damage identification problems, a damage detection method based on modal strain energy equivalence index (MSEEI) is presented. First, an accurate expression of modal strain energy (MSE) before and after damage occurs is given. Then, according to the energy equivalence theory that the change in MSE caused by the damage should be equivalent to the energy dissipation caused by the same damage, an energy equivalence equation is deduced. Finally, four roots of the energy equivalence equation are found and a MSEEI is obtained from the four roots. Simulation results demonstrate that the proposed MSEEI method can identify structural damage locations and extent with good accuracy. Identification precision of the proposed method is clearly better than that of the modal strain energy dissipation ratio index (MSEDRI) method.

Micro-inertia effects on the dynamic characteristics of micro-beams considering the couple stress theory

This paper presents a new model for the free transverse vibrations of an Euler–Bernoulli beam using the couple stress theory of elasticity with micro-structure. Introducing the kinematic variables, the strain and kinetic energy expressions (involving micro-inertia effect) have been obtained and the Hamilton principle has been used to derive the governing equations and the related boundary conditions of the free vibrations of fixed–fixed and simply supported beams. A numerical solution has been used to study the natural frequencies, mode shapes and free vibrations of the beams. A comparative result has shown that the bending rigidity predicted by the couple stress, is closer to the experiment result than that predicted by the modified couple stress theory. The results have shown that the bending rigidity of the beams depends on the ratio of the length scale to the beam thickness, whereas the micro-inertia term depends on the ratio of the length scale to the beam length.

Vibration of high-speed rotating rings coupled to space-fixed stiffnesses

Abstract This study investigates the vibration of high-speed rotating rings coupled to space-fixed discrete stiffnesses. The ring radial and tangential deformations are defined using space-fixed (Eulerian) coordinates, where material particles pass through fixed locations in space. Engineering strain is used in the strain energy expression. The derived nonlinear equations from Hamilton's principle are linearized about the steady non-trivial configuration that results from constant ring rotation. Comparisons are made to other models in the literature that use different assumptions. The governing equations are cast in terms of matrix differential operators that reveal the system's standard gyroscopic system structure. The natural frequencies and vibration modes are calculated over a wide-range of rotation speeds for axisymmetric free rings and a non-axisymmetric ring with a space-fixed discrete stiffness element.

Nonlinear Strain Energy Formulation of a Generalized Bisymmetric Spatial Beam for Flexure Mechanism Analysis.

Analytical load-displacement relations for flexure mechanisms, formulated by integrating the individual analytical models of their building-blocks (i.e., flexure elements), help in understanding the constraint characteristics of the whole mechanism. In deriving such analytical relations for flexure mechanisms, energy based approaches generally offer lower mathematical complexity, compared to Newtonian methods, by reducing the number of unknowns-specifically, the internal loads. To facilitate such energy based approaches, a closed-form nonlinear strain energy expression for a generalized bisymmetric spatial beam flexure is presented in this paper. The strain energy, expressed in terms of the end-displacement of the beam, considers geometric nonlinearities for intermediate deformations, enabling the analysis of flexure mechanisms over a finite range of motion. The generalizations include changes in the initial orientation and shape of the beam flexure due to potential misalignment or manufacturing. The effectiveness of this approach is illustrated via the analysis of a multilegged table flexure mechanism. The resulting analytical model is shown to be accurate using nonlinear finite elements analysis, within a load and displacement range of interest.

Post microbuckling mechanics of fibre-reinforced shape-memory polymers undergoing flexure deformation

The buckling mechanics of fibre-reinforced shape-memory polymer composites (SMPCs) under finite flexure deformation is investigated. The analytical expressions of the key parameters during the buckling deformation of the materials were determined, and the local post-buckling mechanics of the unidirectional fibre-reinforced SMPC were further discussed. The cross section of SMPC under flexural deformation can be divided into three areas: the non-buckling stretching area, non-buckling compression area and buckling compression area. These areas were described by three variables: the critical buckling position, the neutral plane position and the fibre buckling half-wavelength. A strain energy expression of the SMPC thermodynamic system is developed. According to the principle of minimum energy, the analytical expressions of key parameters in the flexural deformation process is determined, including the critical buckling curvature, critical buckling position, position of the neutral plane, wavelength of the buckling fibre, amplitude of the buckling fibre and macroscopic structural strain of the composite material. The results showed that fibre buckling occurred in the material when the curvature increasing from infinitesimal to the critical value. If the curvature is greater than the critical curvature, the neutral plane of the material will move towards the outboard tensile area, and the critical buckling position will move towards the neutral plane. Consequently, the half-wavelength of the buckling fibre was relatively stabilised, with the amplitude increasing dramatically. Along with the increasing of the shear modulus, the critical curvature and buckling amplitude increase, while the critical half-wavelength of the fibre buckling decrease and the critical strain of the composite material increase. Finally, we conducted experiments to verify the correction of the key parameters describing SMPC materials under flexural deformation. The values determined by the experiments proved that the theoretical prediction is correct. Additionally, the buckling deformation of the carbon fibre generated a large macroscopic structural strain of the composite material and obtained a resulting large flexural curvature of the structure with minimal material strain of the carbon fibre.

A Closed-Form Nonlinear Model for the Constraint Characteristics of Symmetric Spatial Beams

The constraint-based design of flexure mechanisms requires a qualitative and quantitative understanding of the constraint characteristics of flexure elements that serve as constraints. This paper presents the constraint characterization of a uniform and symmetric cross-section, slender, spatial beam—a basic flexure element commonly used in three-dimensional flexure mechanisms. The constraint characteristics of interest, namely stiffness and error motions, are determined from the nonlinear load–displacement relations at the beam end. Appropriate assumptions are made while formulating the strain and strain energy expressions for the spatial beam to retain relevant geometric nonlinearities. Using the principle of virtual work, nonlinear beam governing equations are derived and subsequently solved for general end loads. The resulting nonlinear load–displacement relations capture the constraint characteristics of the spatial beam in a compact, closed-form, and parametric manner. This constraint model is shown to be accurate using nonlinear finite element analysis, within a load and displacement range of practical interest. The utility of this model lies in the physical and analytical insight that it offers into the constraint behavior of a spatial beam flexure, its use in design and optimization of 3D flexure mechanism geometries, and its elucidation of fundamental performance tradeoffs in flexure mechanism design.

일차전단변형이론을 이용한 복합재료 적층평판의 효율적 열응력 해석

In this paper, an efficient yet accurate method for the thermal stress analysis using a first order shear deformation theory(FSDT) is presented. The main objective herein is to systematically modify transverse shear strain energy through the mixed variational theorem(MVT). In the mixed formulation, independent transverse shear stresses are taken from the efficient higher-order zigzag plate theory, and the in-plane displacements are assumed to be those of the FSDT. Moreover, a smooth parabolic distribution through the thickness is assumed in the transverse normal displacement field in order to consider a transverse normal deformation. The resulting strain energy expression is referred to as an enhanced first order shear deformation theory, which is obtained via the mixed variational theorem with transverse normal deformation effect(EFSDTM_TN). The EFSDTM_TN has the same computational advantage as the FSDT_TN(FSDT with transverse normal deformation effect) does, which allows us to improve the through-the-thickness distributions of displacements and stresses via the recovery procedure. The thermal stresses obtained by the present theory are compared with those of the FSDT_TN and three-dimensional elasticity.

High accuracy postbuckling analysis of channel section struts

This paper presents the theoretical developments of a high accuracy method for the post-buckling analysis of some channel section struts. In this method, 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 channel section 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-Karman 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 the method is significantly promoted.

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.

Effects of Strain Rates on Kinetics of Elements of Repeated Pattern Structures: A Continuous Modeling Approach

Space inflatable technology is a promising solution to placing large metrology systems in space. Lighter weight, higher packaging efficiency, and easier maintenance are among a few of their advantages over mechanically deployed structures. On the other hand, their large volume after deployment makes them susceptible to disturbances in space. Therefore, vibration control is one major issue for this technology. The present work is an extension of the previous work of the author on continuum modeling of these structures for their vibrations analysis. Kinetic and strain energy expressions of the fundamental lattice elements of a structure are expanded in terms of the nodal displacement components. Certain assumptions are made to reduce the order of strain components in a three-dimensional structure in order to find the equivalent continuum model. Additionally, this work includes the effects of strain rates on the kinetics of these structures. The frequency results for various structures are compared to those of a previous model which neglects such effects. It is shown that the frequency changes are noticeable when the strain rate components are included in the kinetic energy derivations.

Effects of Strain Rates on Kinetics of Elements of Repeated Pattern Structures: A Continuous Modeling Approach

Space inflatable technology is a promising solution to placing large metrology systems in space. Lighter weight, higher packaging efficiency, and easier maintenance are among a few of their advantages over mechanically deployed structures. On the other hand, their large volume after deployment makes them susceptible to disturbances in space. Therefore, vibration control is one major issue for this technology. The present work is an extension of the previous work of the author on continuum modeling of these structures for their vibrations analysis. Kinetic and strain energy expressions of the fundamental lattice elements of a structure are expanded in terms of the nodal displacement components. Certain assumptions are made to reduce the order of strain components in a three-dimensional structure in order to find the equivalent continuum model. Additionally, this work includes the effects of strain rates on the kinetics of these structures. The frequency results for various structures are compared to those of a previous model which neglects such effects. It is shown that the frequency changes are noticeable when the strain rate components are included in the kinetic energy derivations.

Twisting statics and dynamics for circular elastic nanosolids by nonlocal elasticity theory

Abstract The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion can-not be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of cir-cular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displace-ment decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.

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.

Prebuckling Enhancement of Imperfect Composite Plates Using Piezoelectric Actuators

The nonlinear response of initially imperfect composite plates with piezoelectric actuators is investigated. The nonlinearity is limited to the prebuckling regime, where higher order terms present in the strain energy expression can be neglected. The advantage of the electromechanical coupling is exploited in two ways. First, the in-plane piezoelectric stress stiffening effect is used to tailor a stress distribution that inherently increases the critical buckling loads of perfect composite plates by posing an optimization problem that efficiently handles eventual uncertainties involved in the application of mechanical loadings. Second, piezoelectric bending moments are applied in order to avoid or ameliorate the undesirable effects of initial imperfections. An actuation strategy, where the piezoelectric membrane forces and bending moments are decomposed via an appropriate selection of voltages applied to piezoelectric patches that are symmetrically bonded to the top and bottom surfaces of the plate, is proposed and shown to be effective.

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.

Single beam analysis of damaged beams verified using a strain energy based damage measure

Analytical expression of a new damage measure which relates the strain energy, to the damage location and magnitude, is presented in this paper. The strain energy expression is calculated using modes and natural frequencies of damaged beams that are derived based on single beam analysis considering both decrease in mass and stiffness. Decrease in mass and stiffness are a fallout of geometric discontinuity and no assumptions regarding the physical behavior of damage are made. The method is applicable to beams, with notch like non-propagating cracks, with arbitrary boundary conditions. The analytical expressions derived for mode shapes, curvature shapes, natural frequencies and an improved strain energy based damage measure, are verified using experiments. The improvement in the damage measure is that it is not assumed that the bending stiffness of the damaged beam is constant, and, equal to that of undamaged beam when calculating the strain energy of the entire beam. It is also not assumed that the bending stiffness of the element in which the damage is located is constant.

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.

A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation

Abstract The beam constraint model (BCM), presented previously, captures pertinent nonlinearities to predict the constraint characteristics of a generalized beam flexure in terms of its stiffness and error motions. In this paper, a nonlinear strain energy formulation for the beam flexure, consistent with the transverse-direction load-displacement and axial-direction geometric constraint relations in the BCM, is presented. An explicit strain energy expression, in terms of beam end displacements, that accommodates generalized loading conditions, boundary conditions, initial curvature, and beam shape, is derived. Using energy-based arguments, new insight into the BCM is elucidated by fundamental relations among its stiffness, constraint, and energy coefficients. The presence of axial load in the geometric constraint and strain energy expressions—a unique attribute of distributed compliance flexures that leads to the elastokinematic effect—is highlighted. Using the principle of virtual work, this strain energy expression for a generalized beam is employed in determining the load-displacement relations, and therefore constraint characteristics, of a flexure mechanism comprising multiple beams. The benefit of this approach is evident in its mathematical efficiency and succinctness, which is to be expected with the use of energy methods. All analytical results are validated to a high degree of accuracy via nonlinear finite element analysis.

Micropolar Continuous Modeling and Frequency Response Validation of a Lattice Structure

A simple approach is employed here to determine an equivalent continuum representation of a lattice type structure with flexible joints. Kinetic and strain energy expressions are written in terms of the nodal velocities and strain components of the beam members, as well as the joints stiffness values. Necessary assumptions are made to reduce the order of the strain variables while retaining the effects of the microrotations that are coupled to the primary strain terms. As a result, an equivalent one-dimensional model has been found, which takes the assumptions of a micropolar continuum into account rather than an ordinary continuum. The frequency response function of the presented model has been validated experimentally and is shown to be in good agreement with the experimental results for a planar truss with Pratt girder configuration.