Koiter's Theory Research Articles

Koiter–Newton Reduced-Order Method Using Mixed Kinematics for Nonlinear Buckling Analysis

The Koiter–Newton method improves the computational efficiency of nonlinear buckling analysis; however, the construction of reduced-order models using fully nonlinear kinematics is still a tedious and time-consuming work. In this paper, the Koiter–Newton reduced-order method using mixed nonlinear kinematics is presented for the geometrically nonlinear buckling analysis of thin-walled structures. Strain energy variations up to the fourth order were achieved using mixed kinematics for the improved Koiter theory. Corotational kinematics, which is inconvenient for high-order variations, was applied to calculate the first- and second-order variations for the internal force and tangent stiffness, respectively, whereas the third- and fourth-order strain energy variations were facilitated by explicit algebraic formulations using updated von Kármán kinematics. A reduced-order model with 1+m degrees of freedom was established, of which m perturbation loads were considered to make the method applicable for buckling problems. The geometrically nonlinear response was traced using a predictor–corrector strategy by combining the nonlinear prediction solved by the reduced-order model and the correction using Newton iterations. Numerical examples of structures with various buckling behaviors demonstrate that the performance of the proposed method is not obviously affected by using simplified kinematics, and sometimes it even exhibits a superior capability for path-following analysis.

A reduced-order solution for critical buckling temperature of thin-walled structures

This work presents a reduced-order solution for the thermoelastic geometrically nonlinear response of simply-supported thin-walled structures subjected to the purely thermal load. Previously, the reduced-order method was only applicable to the buckling problem with a fixed temperature value. Now, the method is reformulated to achieve the critical buckling temperature from the thermoelastic geometrically nonlinear buckling analysis. The thermal buckling problem is represented using the thermal expansion model and its equivalent mechanical model, respectively. For the equivalent mechanical model, the simply-supported constraints need to be partially released and the temperature field is converted to be a thermal load. The reduced-order model is constructed based on the improve Koiter theory for thermal buckling analysis of the equivalent mechanical model. The thermoelastic geometrically nonlinear response is traced using the reduced-order solution together with a predictor-corrector process. The reduced-order solution of the equivalent mechanical model is validated by fully nonlinear solution of the thermal expansion model. Various plates and shells are selected to demonstrate the accuracy and highly efficiency of the proposed method for thermal buckling analysis.

A graphical method for undrained analysis of cavity expansion in Mohr–Coulomb soil

A rigorous analytical solution is developed for undrained cavity expansion problems in non-associated Mohr–Coulomb soil, based on a novel graphical analysis approach and on the Lagrangian description through tracking only the responses of a soil particle at the cavity surface. The mathematical difficulties involved with the flow rate calculation when the stress state lies on the corner/edge of two adjacent (Mohr–Coulomb) yield surfaces, for both cylindrical and spherical cases, are tackled by using the generalised Koiter theory for non-associated plasticity. In particular, through the unique geometrical formulation, the effective stress path pertaining to the cylindrical cavity problem can be very conveniently directly determined, and is found to consist of simple, piecewise straight lines in the deviatoric stress plane with the orientations dependent of the relative magnitude of Poisson's ratio and the friction angle. This thus renders possible the removal of the stringent intermediacy assumption for the vertical stress that is commonly adopted in previous formulations, and hence the development of a complete cylindrical cavity expansion solution in Mohr–Coulomb soil under undrained loading conditions. The stress and deformation responses of the cavity, including the typical pressure–expansion curves and limiting cavity pressure, are finally analytically obtained along with the Lagrangian form of the radial equilibrium equation in completely explicit forms. The closed-form solution provided in this paper is the first rigorous one of its kind, thus completing the analytical analysis of the cavity expansion problem with the classical Mohr–Coulomb model; this is deemed to be essential for the interpretation of in situ test results pertaining to cohesive-frictional soils.

A highly efficient and accurate FE-ROM method for thermal-mechanical buckling of heat transfer panels

Thin-walled panels are commonly used in high-speed aircrafts and space launcher vehicles. These applications often serve in a complex operating environment with the thermal-mechanical coupling loading. The steady-state temperature field eventually produced by the aerodynamic heating should be taken into account when solving the structural elastic deformations. This work proposes a highly efficient and accurate finite element based reduced-order modeling (FE-ROM) method for the thermal-mechanical buckling of heat transfer panels. The classical Koiter asymptotic expansion is reformulated to be applicable for thermal-mechanical buckling problems. The thermal-mechanical reduced-order model is constructed based on the novel Koiter theory. The complex temperature field produced by the heat transfer is skillfully considered in the reduced system. The predictor-corrector scheme based on the reduced system is developed to trace the thermoelastic geometrically nonlinear response. Various flat/curved panels with isotropic/orthotropic materials, are selected to validate the high efficiency and accuracy of the proposed method, considering different thermal-mechanical coupling loads, heat resource locations, geometrical shapes and lamination configurations.

The thermal-mechanical buckling and postbuckling design of composite laminated plate using a ROM-driven optimization method

The high temperature environment makes the buckling of composite laminated plate happen earlier. Further weight savings can be achieved once the plate has a favorable postbuckling stiffness. This study aims to address this issue by developing a ROM-driven lamination optimization to maximize the buckling and postbuckling performance of compressed plates under initial temperature field. The reduced-order model(ROM) constructed using the improved Koiter theory is reformulated to be suitable for thermal-mechanical buckling problems. A linear subspace of the force space is represented to be the span of the mechanical load, the thermal load and a set of predefined perturbation loads. The thermal load related to the initial temperature field is constant during the increase of in-plane compressive load. An independently additional degree of freedom corresponding to the thermal load is implemented into the construction of the reduced-order model. The buckling and postbuckling performance can be extracted from the solution of reduced-order model, and then subsequently selected as inputs to achieve the optimal laminate. The main novelty of this work is to achieve the thermal-mechanical buckling and postbuckling optimization, benefitting from the highly efficient single run of thermoelastic geometrically nonlinear analysis. Various numerical examples have been selected to validate the performance of the proposed method, considering the influences of length-to-width ratio, α 1/α 2, ply number and nonuniform temperature field.

A novel and highly efficient strategy to determine the ‘Worst’ imperfection shape for buckling of cylindrical shell panels

The cylindrical shell panels subjected to in-plane compression experience more complex snap-back behavior associated with sharp turning around the limit-point load. The initial geometric imperfections play a significant role in the evaluation of the limit load along the nonlinear buckling response. Due to the highly computational cost for imperfection sensitivity analyses, the determination of a ‘worst’ imperfect shape that leads to the minimum limit load becomes a challenge. In this work, the initial geometrical imperfection shape is assumed to be the combination of numbers of closely-spaced buckling modes with different participation coefficients. The reduced order model based on the improved Koiter theory is constructed using all the closely-spaced modes, only once for ‘perfect’ structure and then applicable for any imperfections by just updating the independent imperfection term. Independent imperfection terms with gradient and Hessian forms are designed to be suitable for imperfections with small and relatively large magnitudes, respectively. The optimization strategy is developed to find out the optimal participation coefficients of closely-spaced modes that give the ‘worst’ imperfect shape. Two imperfection sensitivity indices, the limit load under a certain imperfection magnitude and the reduction factor of limit load under a certain interval of imperfection magnitudes, are defined and applied as the objective functions, respectively. The genetic algorithm is selected to find the optimal solutions efficiently benefiting from the a posteriori accounts of imperfections in function calls. Cylindrical shell panels with various configurations are applied to validate the good performance of the proposed strategy.

The reformulated Koiter–Newton method for thermal–mechanical buckling and postbuckling analysis of thin‐walled structures

AbstractThe reduced‐order modeling method, termed as the Koiter–Newton method, is reformulated to be applicable for geometrically nonlinear thermal–mechanical analysis of thin‐walled structures. The thermal load is treated as the independently unchanged load corresponding to the initial temperature field. The internal force space is expanded using the mechanical load, the thermal load, and the predefined perturbation loads. The thermal–mechanical reduced‐order model is constructed using the first to fourth‐order derivatives of strain energy with thermal effects in terms of the degrees of freedom (DOFs). An additional DOF related to the thermal load appears in the construction of reduced‐order model based on the novel Koiter theory. The path‐following scheme is proposed to make the method able to trace the entire geometrically nonlinear thermoelastic response. A much larger step size can be achieved benefiting from the favorable prediction of the reduced‐order model, compared to the classical Newton‐like methods. Both the temperature‐independent and temperature‐dependent material properties are considered. The thermal–mechanical buckling and postbuckling behaviors obtained by the proposed method are validated and discussed using various numerical examples.

On the free vibration problem of thin shallow and non-shallow shells using tensor formulation

Linear free vibration of thin shells with rectangular base such as parabolic conoids, elliptic paraboloids and hyperbolic paraboloids are investigated. These shells are described by non-orthogonal curvilinear coordinates hence, Koiter theory, which uses a tensor formulation, is applied to accurately describe the strain–displacement relations. Four different boundary conditions are considered: simply supported with movable edges, simply supported with immovable edges; fully clamped with movable edges and fully clamped with immovable edges. Using an energy approach, Rayleigh–Ritz method is applied to obtain a set of ordinary differential equations of motion and Fourier series and Legendre polynomials are used to describe the displacements fields of the shell. Both, shallow and non-shallow shells were studied and obtained results of natural frequencies and vibration modes were compared with those from Maguerre theory and FEM using Abaqus, as can be seen, a good agreement was observed with FEM showing the efficacy of the applied theory. When comparing natural frequencies to those from Marguerre theory, there are substantial differences, indicating that lack of use of the tensor formulation which is not appropriate to shells with non-orthogonal coordinates.

Buckling response of composite cylindrical shells with various stiffener layouts under uniaxial compressive loading

This research presents a study on the buckling behaviour of the stiffened cylindrical shells made of laminated glass fibre-reinforced polymer (GFRP) with arbitrary stiffeners under axial compression by using the curved finite strip method. The stiffeners can be positioned at the inner and outer surfaces of the shell, and is no need to be located on the nodal lines. The governing equations of motion are extracted from Koiter's theory based on the first-order shear deformation theory and solved employing the principle of the minimum potential energy. Then, both the critical buckling load and the buckling load coefficient are calculated by solving characteristic equations. Boundary conditions considered are simple-simple, clamped–clamped, and clamped-simple supports. A relatively high-order polynomial function for the displacement components in the transverse direction is assumed for the shell and the stiffeners. A displacement model is assumed with twenty-four degrees of freedom arranged at four nodal lines for each strip. To validating the proposed method, the results of this investigation are compared with the finite element, experimental and other published numerical results. The benefits of this method are that number of elements, and therefore time consumed for analysis is much less, and the mesh refinement process is much more convenient than the conventional finite element method. Finally, the effects of the different parameters such as Poisson's ratio, boundary condition, thickness variations and geometrical parameters of shell and stiffeners, angle, eccentricity, torsional stiffness, cross-sectional area, number of stiffeners on the buckling are studied. The results obtained can be employed as a significant benchmark for researchers to verify their analytical and numerical approaches.

Dynamics and Stability Analysis of Rotating Cylindrical Shells in Annular Fluid Medium

Stability and dynamics of rotating coaxial cylindrical shells conveying incompressible and inviscid fluid are investigated. The interior shell is assumed to be flexible while the exterior cylinder is rigid. Using Sander’s–Koiter theory assumptions and following Hamilton’s principle, governing equations of motion are determined in their integral form. Employing the extended Galerkin method of solution, the integral equations of motion are projected to their equivalent system of algebraic equations. Fluid equations are fundamentally based on the linearized inviscid Navier–Stokes equations. Impermeability condition on the fluid and structure interface as well as the zero radial velocity component on the exterior shell give the coupled equations of motion governing the dynamics of fluid-loaded coaxial cylindrical shells. Using the coupled fluid–structural model, stability boundaries are determined for the rotating interior shell. Various parameter studies are conducted and effects of mass ratio, gap distance between the interior and exterior shells, boundary conditions of the interior shell, length to radius ratio on the stability margins are thoroughly investigated and reported.

An accurate and efficient implementation of initial geometrical imperfections in the predictor–corrector reduced-order modeling method

A predictor–corrector reduced-order modeling method based on Koiter theory is proposed for nonlinear buckling analysis of thin-walled structures with initial geometric imperfections. The imperfections are implemented into the von Kármán kinematics and are considered using an a priori and a posteriori manners, respectively, in buckling analysis. Since the nonlinear predictor obtained from the reduced order model with imperfections provides a fairly large step size in path-tracing, the a priori account of imperfections still shows advantages in computational efficiency. The advantages are more significant when considering a posteriori the effects of imperfections in the reduced order model of the structure without imperfections, which means building once and for all, by simply adding some independent imperfection terms in the reduced system. The a posteriori accounts of imperfections with the first/second-order accuracies, are developed, respectively, to deal with the linear/nonlinear prebuckling behaviors and small/large imperfection amplitudes. The good performance of the proposed method in terms of reliability, accuracy and computational effort is demonstrated with several examples using the linear buckling modes or measured geometric shapes as whole-field geometric imperfections.

An isogeometric formulation of the Koiter’s theory for buckling and initial post-buckling analysis of composite shells

Numerical formulations of the Koiter theory allow the efficient prediction, through a reduced model, of the behavior of shell structures when failure is dominated by buckling. In this work, we propose an isogeometric version of the method based on a solid-shell model. A NURBS interpolation is employed on the middle surface of the shell to accurately describe the geometry and the high continuity typical of the displacement field in buckling problems and to directly link the CAD model to the structural one. A linear interpolation is then adopted through the thickness together with a modified generalized constitutive matrix, which allows us to easily eliminate thickness locking and model multi-layered composites. Reduced integration schemes, which take into account the continuity of the shape functions, are used to avoid interpolation locking and make the integration faster. A Mixed Integration Point strategy makes it possible to transform the displacement model into a mixed (stress–displacement) one, required by the Koiter method to obtain accurate predictions, without introducing stress interpolation functions. The result is an efficient numerical tool for buckling and initial post-buckling analysis of composite shells, characterized by a low number of DOFs and integration points and by a simple and quick construction of the reduced model.

Commutativity of the strain energy density expression for the benefit of the FEM implementation of Koiter's initial postbuckling theory

SummaryThe concept of full commutativity of displacements in the expression for strain energy density for the geometrically nonlinear problem has been introduced for the first time and fully established in this paper. Its consequences for the FEM formulation have been demonstrated. As a result, the strain energy, equilibrium equation, and incremental equilibrium equation for the geometrically nonlinear problem can all be presented in a unified manner involving various stiffness matrices that are all symmetric, unique, and explicitly expressed. As an important application, the framework has been employed in the FEM implementation of Koiter's initial postbuckling theory, which has been handicapped by its mesh sensitivity in evaluating one of the initial postbuckling coefficients. This has largely prevented it from being incorporated in mainstream commercial FEM codes. Based on the outcomes of this paper, the mesh sensitivity problem has been completely resolved without the need to use any specially formulated element. As a result, Koiter's theory can be practically and straightforwardly implemented in any FEM code. The results have been verified against those found in the literature.

Viscoelastic modelling and dynamic characteristics of CNTs-CFRP-2DWF composite shell structures

Abstract The carbon nanotubes (CNTs) are well known for its application in the areas of advanced composite materials for their improved elastic properties. The large specific interfacial surface area owing to tiny dimensions of CNTs may greatly escalation the interfacial sliding which may promote the dissipation of energy in a dynamic situation that makes them very ideal for damping applications in engineering structures/systems. The present article investigates the viscoelastic modelling and dynamic responses of the CNTs – based carbon fiber reinforced two dimensional woven fabrics (CNTs-CFRP-2DWF) composite spherical shell panels where CNTs are reinforced in the polymer matrix phase. The Mori – Tanaka (MT) micromechanics in conjunction with weak interface (WI) theory has been developed for the mathematical formulations of the viscoelastic modelling of CNTs-based polymer matrix phase. Further, strength of material (SOM) method has been employed to formulate viscoelastic material behavior of the yarn and finally the viscoelastic properties of the representative unit cell (RUC) is established based on the unit cell method (UCM). An eight-node shell element with five degree of freedom per node has been formulated to study the vibration damping characteristics of spherical shell structures made by CNTs-CFRP-2DWF composite materials. The shell finite element formulation is based on the transverse shear effect as per the Mindlin's hypothesis, and stress resultant-type Koiter's shell theory. Frequency and temperature dependent material properties of such CNTs-CFRP-2DWF composite materials have been obtained and analysed. Impulse and frequency responses of such structures have been performed to study the effects of various important parameters such as volume fraction of CNTs, interfacial condition, agglomeration, temperature, geometries of shell panel on the material properties and such dynamic responses. Obtained results demonstrate that quick vibration mitigation may be possible using such CNTs-CFRP-2DWF composite material which is desirable to overcome the drawbacks of conventional CFRP woven fabric composite materials.

The effect of a weak nonlinearity on the lowest cut-off frequencies of a cylindrical shell

The plane strain problem for a thin circular cylindrical shell is considered within the framework of the Sanders–Koiter theory. The relative shell thickness and displacement amplitude are chosen to be of the same asymptotic order. The leading nonlinear correction to the lowest cut-off frequencies is derived using the method of multiple scales. In contrast to the traditional two-mode Galerkin expansions assuming inextensibility of the shell transverse cross section, the developed fourth-order asymptotic scheme operates with five angular modes. The obtained results reveal asymptotic inconsistency of previous approximate solutions to the problem.

Post-Buckling Analysis of Curved Beams

Stretchability of the stretchable and flexible electronics involves the post-buckling behaviors of internal connectors that are designed into various shapes of curved beams. The linear displacement–curvature relation is often used in the existing post-buckling analyses. Koiter pointed out that the post-buckling analysis needs to account for curvature up to the fourth power of displacements. A systematic method is established for the accurate post-buckling analysis of curved beams in this paper. It is shown that the nonlinear terms in curvature should be retained, which is consistent with Koiter's post-buckling theory. The stretchability and strain of the curved beams under different loads can be accurately obtained with this method.

Vibration damping characteristics of carbon nanotubes-based thin hybrid composite spherical shell structures

ABSTRACTThe present work deals with the evaluation of elastic properties and dynamic analyses of thin hybrid composite shell structures, which consist of conventional carbon fiber as the reinforcing phase and multiwalled carbon nanotubes-based polymer as the matrix phase. The Mori-Tanaka and strength of material method has been implemented to determine the elastic properties of such hybrid composite structures without and with considering agglomerations. An eight-noded shell element, which considers stress resultant-type Koiter's shell theory and transverse shear effect as per Mindlin's hypothesis having five degrees of freedom at each node, has been utilized for discretizing and analysis of such hybrid shell structures. The Rayleigh damping model has been implemented in order to study the effect of carbon nanotubes (CNTs) on damping capacity of such hybrid composite shell structures. Different types of spherical shell panels have been analyzed in order to study the time and frequency responses. Results show that the elastic properties as well as damping properties of such hybrid composite structures improved with the addition of CNTs as compared to conventional carbon fiber reinforced composites laminates; effects of some important parameters on the vibration characteristic of such hybrid composite shell structures have also been presented. The effects of agglomeration parameters on the elastic properties and their influences on the dynamic responses considering different layers and stacking sequences have also been investigated.

Nonlinear vibrations and energy exchange of single-walled carbon nanotubes. Circumferential flexural modes

In this paper, the nonlinear vibrations and energy exchange of single-walled carbon nanotubes (SWNTs) are studied. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The SWNT deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported, clamped and free boundary conditions are considered. The circumferential flexural modes (CFMs) are investigated. Two different approaches based on numerical and analytical models are compared. In the numerical model, an energy method based on the Lagrange equations is used to reduce the nonlinear partial differential equations of motion to a set of nonlinear ordinary differential equations, which is solved by using the implicit Runge–Kutta numerical method. In the analytical model, a reduced form of the Sanders–Koiter theory assuming small circumferential and tangential shear deformations is used to get the nonlinear ordinary differential equations of motion, which are solved by using the multiple scales analytical method. The transition from energy beating to energy localization in the nonlinear field is studied. The effect of the aspect ratio on the analytical and numerical values of the nonlinear energy localization threshold for different boundary conditions is investigated.

Postbuckling of Hyperelastic Plates

Curvature is simply expressed as the second derivative of the plate deflection in prior studies of post-buckling of plates. It is shown in this paper that the higher-order terms in curvature should be retained, consistent with Koiter's post-buckling theory. This paper also solves the dilemma whether the increase of post-buckling load is proportional to the square of the ratio of the post-buckling deflection w to the plate thickness t, (w/t)2, as in most prior studies, or to the characteristic in-plane length L of the plate, (w/L)2, as discovered in some recent studies.

Local buckling and post-buckling of composite channel-section beams – Numerical and experimental investigations

Abstract Local buckling and post-buckling of thin-walled composite channel-section beams under pure bending are described. Profiles were subject to bending in the plane of the lowest second moment of area. Thin-walled beams were made of an eight-layer GFRP composite with six different arrangements of plies. An asymptotic analytical–numerical method was used in the investigations, whereas the ANSYS software based on the finite element method and the experimental test results were employed in the numerical simulations. The analytical–numerical method is based on asymptotic Koiter's theory for conservative systems, modified by Byskov and Hutchinson. Two different finite element models were prepared: the first one with boundary conditions closer to the analytical–numerical method model and the second one with boundary conditions close to those present on the test stand. A four-point bending test was used in the experimental test. The results obtained in the above-mentioned numerical methods are compared to these obtained experimentally. The advantages and disadvantages of the applied methods and models are presented and discussed.