Geometric Nonlinear Formulation Research Articles

An unsymmetric 8‐node hexahedral solid‐shell element with high distortion tolerance: Geometric nonlinear formulations

SummaryA recent distortion‐tolerant unsymmetric 8‐node hexahedral solid‐shell element US‐ATFHS8, which takes the analytical solutions of linear elasticity as the trial functions, is successfully extended to geometric nonlinear analysis. This extension is based on the corotational (CR) approach due to its simplicity and high efficiency, especially for geometric nonlinear analysis where the strain is still small. Based on the assumption that the analytical trial functions can properly work in each increment during the nonlinear analysis, the incremental corotational formulations of the nonlinear solid‐shell element US‐ATFHS8 are derived within the updated Lagrangian (UL) framework, in which an appropriate updated strategy for linear analytical trial functions is proposed. Numerical examples show that the present nonlinear element US‐ATFHS8 possesses excellent performance for various rigorous tests no matter whether regular or distorted mesh is used. Especially, it even performs well in some situations that other conventional elements cannot work.

Geometric nonlinear formulation of plate buckling structure

The analytical or exact mathematical formulation of stresses and displacements for plate buckling structure become impossible to develop if the plate geometry is so complicated. Numerical technique is one another approach to solve this problem and it is chosen in this study for plate structure under in-plane and out-plane load. The formulation of elastic stiffness matrix (ke) and geometric nonlinear stiffness matrix (kg) of the plate structure due to buckling is presented and based on virtual displacement principle. The geometric nonlinear stiffness matrix (kg) is found function of internal stresses. The direct iteration technique is applied to find nodal displacements. Under this technique, the Gauss points stresses are initialized as zeros, then the kg matrix is updated, and then a new nodal displacement vector is found for the next approximation of internal stresses. Iterative process is done until convergence of displacement is satisfied. The rectangular plate with one fixed edge supported is used to test the proposed nonlinear formulation and procedure. The compressive in-plane load and moment is considered and applied for the tested plate. The plate is discretized with appropriate number of triangular finite element mesh. It is found that, the convergence of displacement is satisfied by using direct iteration technique. The load - deflection curve shows nonlinear relationship and approach to critical load. This finding shows that the direct iteration method can be accepted for the analyzing of plate buckling by considering geometric nonlinear assumption.

New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem

In this study, a new modified multi-level residue harmonic balance method is presented and adopted to investigate the forced nonlinear vibrations of axially loaded double beams. Although numerous nonlinear beam or linear double-beam problems have been tackled and solved, there have been few studies of this nonlinear double-beam problem. The geometric nonlinear formulations for a double-beam model are developed. The main advantage of the proposed method is that a set of decoupled nonlinear algebraic equations is generated at each solution level. This heavily reduces the computational effort compared with solving the coupled nonlinear algebraic equations generated in the classical harmonic balance method. The proposed method can generate the higher-level nonlinear solutions that are neglected by the previous modified harmonic balance method. The results from the proposed method agree reasonably well with those from the classical harmonic balance method. The effects of damping, axial force, and excitation magnitude on the nonlinear vibrational behaviour are examined.

Design and Experimental Analysis of an Externally Prestressed Steel and Concrete Footbridge Equipped with Vibration Mitigation Devices

AbstractA 142-m, three-span continuous footbridge over the Esino River (Italy) is considered as a case study to illustrate a number of challenging aspects in its static and dynamic design. The adoption of an optimized steel deck with a variable cross section together with the use of external prestressing tendons in the central span allows a substantial reduction of structural weights. The resulting footbridge requires a proper model for the assessment of its behavior up to the ultimate limit state as well as attention to vibration control under pedestrian loading at the service limit state. The former issue is addressed through the use of a specifically developed material and geometric nonlinear finite-element formulation. Regarding vibration control, an original combination of two different systems is used, i.e., high damping rubber (HDR) stripes and tuned mass dampers (TMDs). The HDR stripes, applied between the steel deck and the concrete floor, increase the overall damping of the footbridge, whereas t...

Geometrically nonlinear analysis of hybrid beam–column with several encased steel profiles in partial interaction

This article presents a new co-rotational finite element for the large displacement analysis of hybrid steel–concrete beam/column with several encased steel profiles. The advantage of using the co-rotational approach is that the geometrical linear finite element formulation can be reused and automatically be transformed into geometrical nonlinear formulation. The exact stiffness matrix derived from the analytical solution of the governing equations for hybrid beam with longitudinal partial interaction is used for the local formulation. As a result, internal nodes used to avoid curvature locking encountered in low order polynomial finite elements are not required. Finally, five numerical applications are presented in order to illustrate the performance of the proposed formulation.

Geometrical nonlinear formulation of a Molecular Mechanics model applied to the structural analysis of single-walled carbon nanotubes

In this paper, the post-critical behavior and buckling modes of single-walled carbon nanotubes are analyzed via a Molecular Mechanics model. The main target is to develop a general formulation for the model, which has been simplified under small strains assumption, and to implement a versatile tool for the structural analysis of carbon nanotubes in the framework of geometrical nonlinearity. For this purpose, a mechanical formulation able to reproduce any load configuration and supporting conditions has been derived by using an energy approach. Then, an incremental-iterative solution procedure has been implemented in order to trace several nonlinear equilibrium paths and to obtain the corresponding critical strains of clamped–clamped nanotubes under compressive, flexural and torsional loading distributions. The model shows a good numerical performance and results in agreement with previous atomistic works. Two interatomic potentials have been adopted in order to find out the influence of different constitutive relationships on the final nonlinear response. We have concluded that the choice of the potential function has no significant effect on the final buckling strains. Our results confirm that the final buckling response is strongly determined by geometrical imperfections in the nanotube, which can be well reproduced in the proposed model, but are much more difficult to handle in continuum models.

Geometrically Linear and Nonlinear First-Ply Failure Loads of Composite Cylindrical Shells

AbstractReview of the literature on first-ply failure of composite shells shows that research reports on first-ply failure of moderately thin, laminated composite cylindrical shell panels, using a geometrically nonlinear approach, are not available. The present paper aims to fill this deficiency. It uses a finite-element code developed using eight-noded, doubly-curved elements combined with modified Sanders’ first-approximation theory for thin shells and von Karman-type nonlinear strains. The accuracy of the present geometric nonlinear and first-ply failure formulations are verified separately through solutions of two benchmark problems. Failure loads, failure modes (for individual stress or strain failure) or tendencies (for interactive stress failures), and the locations from where the failures initiate are reported. The results are discussed critically to formulate design guidelines, suggesting practical values for factors of safety applicable to failure loads. Such suggestions also consider the servic...

曲面座標系を用いたRKPMによる初期不整を含む板の座屈解析

A geometrical nonlinear formulation is presented to simulate buckling/post-buckling behaviors of panels in ship structure employing meshfree approach. Reproducing kernel is adopted in the meshfree approximation. The author's previous study, a meshfree flat shell formulation, was developed based on Mindlin-Reissener theory. However, there were difficulties in introducing initial imperfection to the flat panel. In this study, convected coordinate system is applied to the meshfree shell formulation to reproduce complicated initial imperfection, e.g., thin-horse mode. To verify the proposed approach, buckling/post-buckling behaviors of panels are simulated using total and updated Lagrangian formulations and the results are examined.

MODEL REDUCTION WITH GEOMETRIC STIFFENING NONLINEARITIES FOR DYNAMIC SIMULATIONS OF MULTIBODY SYSTEMS

This work investigates the implementation of nonlinear model reduction to flexible multibody dynamics. Linear elastic theory will lead to instability issues with rotating beam-like structures, due to the neglecting of the membrane-bending coupling on the beam cross-section. During the past decade, considerable efforts have been focused on the derivation of geometric nonlinear formulation based on nodal coordinates. In order to reduce the computation cost in flexible multibody dynamics, which is extremely important for complex large system simulations, modal reduction is usually implemented to a rotating flexible structure with geometric nonlinearities retained in the model. In this work, a standard model reduction process based on matrix operation is developed, and the essential geometric stiffening nonlinearities are retained in the reduced model. The time responses of a tip point on a rotating Euler–Bernoulli blade are calculated based on two nonlinear reduced models. The first reduced model is derived by the standard matrix operation from a full finite element model and the second reduced model is obtained via the Galerkin method. The matrix operation model reduction process is validated through the comparison of the simulation results obtained from these two different reduced models. An interesting phenomenon is observed in this work: In the nonlinear model, if only quadratic geometric stiffing term is retained, the two reduced models converge to the full finite element model with only one bending mode and two axial modes. While if both quadratic and cubic geometric stiffing terms are retained in the nonlinear equation, the modal-based reduced model will not converge to the finite element model unless all eigenmodes are retained, that is the reduced model has no degree of freedom reduction at all.

Geometric nonlinear formulation for thermal-rigid-flexible coupling system

This paper develops geometric nonlinear hybrid formulation for flexible multibody system with large defor- mation considering thermal effect. Different from the con- ventional formulation, the heat flux is the function of the ro- tational angle and the elastic deformation, therefore, the cou- pling among the temperature, the large overall motion and the elastic deformation should be taken into account. Firstly, based on nonlinear strain-displacement relationship, varia- tional dynamic equations and heat conduction equations for a flexible beam are derived by using virtual work approach, and then, Lagrange dynamics equations and heat conduction equations of the first kind of the flexible multibody system are obtained by leading into the vectors of Lagrange mul- tiplier associated with kinematic and temperature constraint equations. This formulation is used to simulate the thermal included hub-beam system. Comparison of the response be- tween the coupled system and the uncoupled system has re- vealed the thermal chattering phenomenon. Then, the key parameters for stability, including the moment of inertia of the central body, the incident angle, the damping ratio and the response time ratio, are analyzed. This formulation is also used to simulate a three-link system applied with heat flux. Comparison of the results obtained by the proposed for- mulation with those obtained by the approximate nonlinear model and the linear model shows the significance of con- sidering all the nonlinear terms in the strain in case of large deformation. At last, applicability of the approximate non- linear model and the linear model are clarified in detail.

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin---Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.

Geometrically Nonlinear Parametric Vibrations of a Transversely Straightened Cylindrical Shell in the Process of Its Dynamic Interaction with a Medium

The problem on the parametric vibrations of a transversely straightened cylindrical shell contacting with a viscoelastic medium under the action of the internal pressure was solved on the basis of the variation principle in the geometrical nonlinear formulation. The influence of the environment on the indicated vibrations was estimated with the use of the Pasternak dynamic model.

Nonlinear Solution Formulation for Complex Space Truss

A new geometric nonlinear formulation for large-deflection static problems involving complex space truss structure is presented. Based on the finite element method, the proposed formulation uses nodal positions rather than nodal displacements to describe the problem. The strain is determined directly from the proposed position concept, using a Cartesian coordinate system fixed in space. The proposed formulation is simple and yields good results. Numerical results indicate that the proposed formulation is validity.

An efficient algorithm for mechanical analysis of bimodular truss and tensegrity structures

A new stable algorithm is developed for the geometric nonlinear analysis of bimodular truss and tensegrity structures based on the parametric variational principle (PVP) and the co-rotational approach. The bilinear constitutive model is proposed for modeling tensegrity structures which consist of prestressed struts and cables. The PVP for bimodular truss is first introduced for small deformation analysis, and then the geometric nonlinear formulation is derived by combining the PVP linear formulation with the co-rotational approach. The Newton–Raphson (NR) scheme combined with Lemke’s algorithm is employed to solve the resulting nonlinear complementarity equations. Numerical examples are given to demonstrate the validity and efficiency of the proposed PVP method for mechanical analysis of bimodular truss and tensegrity structures. Convergence behaviors of different computational schemes for geometric linear and nonlinear analyses are also discussed in detail, which shows the good numerical stability and efficiency of the proposed method.

Dynamic Analysis of Cable-Stayed Bridges Affected by Accidental Failure Mechanisms under Moving Loads

The dynamic behavior of cable-stayed bridges subjected to moving loads and affected by an accidental failure in the cable suspension system is investigated. The main aim of the paper is to quantify, numerically, the dynamic amplification factors of typical kinematic and stress design variables, by means of a parametric study developed in terms of the structural characteristics of the bridge components. The bridge formulation is developed by using a geometric nonlinear formulation, in which the effects of local vibrations of the stays and of large displacements in the girder and the pylons are taken into account. Explicit time dependent damage laws, reproducing the failure mechanism in the cable system, are considered to investigate the influence of the failure mode characteristics on the dynamic bridge behavior. The analysis focuses attention on the influence of the inertial characteristics of the moving loads, by accounting coupling effects arising from the interaction between girder and moving system. Sensitivity analyses of typical design bridge variables are proposed. In particular, the effects produced by the moving system characteristics, the tower typologies, and the failure mode characteristics involved in the cable system are investigated by means of comparisons between damaged and undamaged bridge configurations.

Geometric nonlinear formulation for curved beams with varying curvature

Based on exact Green strain of spatial curved beam, the nonlinear strain-displacement relation for plane curved beam with varying curvature is derived. Instead of using the previous straight beam elements, curved beam elements are used to approximate the curved beam with varying curvature. Based on virtual work principle, rigid-flexible coupling dynamic equations are obtained. Physical experiments were carried out to capture the large overall motion and the strain of curved beam to verify the present rigid-flexible coupling formulation for curved beam based on curved beam element. Numerical results obtained from simulations were compared with those results from the physical experiments. In order to illustrate the effectiveness of the curved beam element methodology, the simulation results of present curved beam elements are compared with those obtained by previous straight beam elements. The dynamic behavior of a slider-crank mechanism with an initially curved elastic connecting rod is investigated. The advantage of employing generalized-α method is pointed out and the special nonlinear dynamic characteristics of the curved beam are concluded.

Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation

This paper presents a geometric nonlinear analysis formulation for beams of functionally graded cross-sections by means of a Total Lagrangian formulation. The influence of material gradation on the numerical response is investigated in detail. Two examples are given that illustrate the main features of the formulation, in which the behavior of beams of graded cross-sections is compared with homogeneous material beams. A motivation for this work is the potential development of functionally graded risers for the offshore oil exploration industry.

A FEM procedure based on positions and unconstrained vectors applied to non-linear dynamic of 3D frames

This study presents an alternative three-dimensional geometric non-linear frame formulation based on generalized unconstrained vector and positions to solve structures and mechanisms subjected to dynamic loading. The formulation is classified as total Lagrangian with exact kinematic description. The resulting element presents warping and non-constant transverse strain modes, which guarantees locking-free behavior for the adopted three-dimensional constitutive relation, Saint-Venant–Kirchhoff, for instance. The application of generalized vectors is an alternative to the use of finite rotations and rigid triad's formulae. Spherical and revolute joints are considered and selected dynamic and static examples are presented to demonstrate the accuracy and generality of the proposed technique.

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.

Nonlinear Analysis of Composite Beams with Partial Interaction in Steel Frame Structures at Elevated Temperature

Flooring systems containing steel-concrete composite beams are common in steel frame structures and it is widely recognized that their behavior under fire loading is profoundly different to that of simply supported composite beams under fire loading. This difference in behavior is due to the presence of restraints provided by cooler members in a compartment fire in a composite frame structure and there is extensive evidence from fire tests that composite beams with unprotected steel components in steel frames perform much better than simply supported composite beams with unprotected steel components. In order to model the structural response of a composite beam restrained in this way at elevated temperature, recourse is needed to a geometric nonlinear formulation, since the transverse beam deflections are large and interact with the substantial axial compressive force in the member at the early stages of the fire. This paper presents such a formulation, which incorporates partial interaction between the concrete slab and steel component, as well as the degradation of the stiffnesses of the components of the composite beam prior to yield at elevated temperature. The generic technique that is developed is shown to agree with solutions reported elsewhere and provides a structural model for the response of a composite beam in fire with the potential for inclusion in prescriptive code rules for rational fire engineering based designs.