Nonlinear Beam Finite Element Research Articles

A Nonlinear Beam Finite Element with Bending–Torsion Coupling Formulation for Dynamic Analysis with Geometric Nonlinearities

Vibration analysis of wing-box structures is a crucial aspect of the aeronautic design to avoid aeroelastic effects during normal flight operations. The deformation of a wing structure can induce nonlinear couplings, causing a different dynamic behavior from the linear counterpart, and nonlinear effects should be considered for more realistic simulations. Moreover, composite materials and aeroelastic tailoring require new simulation tools to include bending–torsion coupling effects. In this research, a beam finite element with bending–torsion coupling formulation is used to investigate the effects of the deflection of beam structures with different aspect ratios. The nonlinear effects are included in the finite element formulation. The geometrical effect is considered, applying a deformation dependent transformation matrix. Stiffness effects are introduced in the stiffness matrix with Hamilton’s Principle and a perturbation approach. The results obtained with the beam finite element model are compared with numerical and experimental evidence.

Corotational force-based beam finite element with rigid joint offsets for 3D framed structures

In numerical analysis of frame structures, modeling of the connection between structural members often requires the introduction of rigid end offsets to correctly describe the stiffness of the joints. This is typical of beam-to-column connections in civil constructions but is also common in lattice materials, where the element overlapping at the joints locally increases the stiffness of the nodes. However, when nonlinear geometric effects are included, correct simulation of such phenomena is challenging.This paper presents a geometrically nonlinear three-dimensional force-based beam finite element that efficiently accounts for the effects of rigid joint offsets. The model imposes the element equilibrium in the local reference system referring to the element deformed configuration, considering the von Kármán nonlinear terms, and exploits a corotational formulation to account for rigid large displacements and rotations of the beam. Two alternative approaches are used to introduce the rigid offsets. The first derives from an existing proposal where kinematic and static nonlinear operators are adopted to describe the behavior of the rigid portions at the element ends. The second is a novel approach, easier to implement and use, that involves modifying the integration, along the element axis, of the cross-section strains and flexibility. This can be easily defined for beam models under linear geometry assumption but requires particular attention for the analysis of frame structures under large displacements and strains. Both approaches are validated through numerical examples, including comparison with other numerical methods and experimental results available in the literature.

Simplified nonlinear analysis of doubly corrugated cold‐formed steel arches

AbstractDoubly corrugated steel arches are made by pressing transverse corrugations on trapezoidal cold‐formed steel sheets. Several studies show that these transverse corrugations may have a significant detrimental effect on the stiffness and load‐bearing capacity of the original trapezoidal profile. This paper presents an experimental campaign aimed at investigating the effects of transverse corrugations on the behaviour of arches. The campaign includes reduced tests on flat sheets and sheets with a single corrugation, as well as full tests on arches with spans ranging from 10 to 20 m. Subsequently, a simplified design procedure based on nonlinear 2D beam finite element models is proposed. The models account for the transverse corrugation effects by using intermediate rotational springs and equivalent beam section areas, which are calibrated from the results of the reduced experiments. A brief sensitivity study of the finite element results with respect to different modelling factors is conducted, and the different effects of the corrugation assessed. Finally, the simplified finite element models are validated by comparing the experimental and numerical force‐displacement curves, which show reasonably good agreement. However, further research is needed to refine and extend the proposed procedure.

Robotic check of a subassembly, and its simulation

This paper discusses a quality inspection process of a subassembly of a battery cover, which is performed with an industrial robot equipped with an intelligent end-effector. The subassembly has a plastic pin and a torsion spring part. During intended use of the unit, the necessary force to actuate the pin is determined with measurements and simulations. In the measurement a self-devised intelligent end-effector is equipped with a microcontroller and a beam type load cell in order to measure the spring force. The main aim is to make an automatic decision without operator intervention at the quality check process, which is related to the quality of the subassembly. In the simulation deformation of the spring part is modeled with geometrically nonlinear 2D beam finite elements in the course of assembly and intended use. Co-rotational approach is used to consider the large displacements and rotations with small strains. Penalty method is applied to treat the contact between the spring and pin parts.

An extended GBT formulation for elastoplastic analyses of perforated thin-walled members

A physically and geometrically nonlinear beam finite element modelling of thin-walled members with arbitrarily complex holes is presented. The beam finite element approximation of the member configuration space is achieved by using the extended generalized beam theory kinematic description, which covers discontinuities in member cross-sections due to holes or cut-outs of various shapes. The proposed beam finite element deals with perforated members in a manner of treating them as non-perforated ones by filling the perforations with much softer materials. Then the level-set functions are used to represent the bi-material interfaces and also used to construct the enrichment functions which describe the discontinuities in the displacement fields. The enrichment functions are then integrated into the standard GBT-based beam finite element approximation by the Partition of Unity Method. The J2-plasticity model combined with isotropic hardening and associated flow rule for isotropic materials (e.g., soft steel) is used in the model. To show the potential of the proposed approach, four illustrative examples are provided, where results obtained by the presented model are compared with the shell finite element results. The proposed GBT model shows significant superiorities in the computational efficiency and structural clarity over the shell model.

Topology optimization of structural frames based on a nonlinear Timoshenko beam finite element considering full interaction

AbstractThis article presents an approach for the topology optimization of frame structures composed of nonlinear Timoshenko beam finite elements (FEs) under time‐varying excitation. Material nonlinearity is considered with a nonlinear Timoshenko beam FE model that accounts for distributed plasticity and axial–shear–moment interactions through appropriate hysteretic interpolation functions and a yield/capacity function, respectively. Hysteretic variables for curvature, shear, and axial deformations represent the nonlinearities and evolve according to first‐order nonlinear ordinary differential equations (ODEs). Owing to the first‐order representation, the governing dynamic equilibrium equations, and hysteretic evolution equations can thus be concisely presented as a combined system of first‐order nonlinear ODEs that can be solved using a general ODE solver. This avoids divergence due to an ill‐conditioned stiffness matrix that can commonly occur with Newmark–Newton solution schemes that rely upon linearization. The approach is illustrated for a volume minimization design problem, subject to dynamic excitation where an approximation for the maximum displacement at specified nodes is constrained to a given limit, that is, a drift ratio. The maximum displacement is approximated using the p‐norm, thus facilitating the derivation of the analytical sensitivities for gradient‐based optimization. The proposed approach is demonstrated through several numerical examples for the design of structural frames subjected to sinusoidal base excitation.

Nonlinear Aeroelastic Analysis for Highly Flexible Flapping Wing in Hover

Aeromechanics of highly flexible flapping wings is a complex nonlinear fluid–structure interaction problem and, therefore, cannot be analyzed using conventional linear aeroelasticity methods. This paper presents a standalone coupled aeroelastic framework for highly flexible flapping wings in hover for micro air vehicle (MAV) applications. The MAV-scale flapping wing structure is modeled using fully nonlinear beam and shell finite elements. A potential-flow-based unsteady aerodynamic model is then coupled with the structural model to generate the coupled aeroelastic framework. Both the structural and aerodynamic models are validated independently before coupling. Instantaneous lift force and wing deflection predictions from the coupled aeroelastic simulations are compared with the force and deflection measurements (using digital image correlation) obtained from in-house flapping wing experiments at both moderate (13 Hz) and high (20 Hz) flapping frequencies. Coupled trim analysis is then performed by simultaneously solving wing response equations and vehicle trim equations until trim controls, wing elastic response, inflow and circulation converge all together. The dependence of control inputs on weight and center of gravity (cg) location of the vehicle is studied for the hovering flight case.

Method to Evaluate Effect of Spudcan Penetration on Adjacent Jacket Piles

When a jack-up drilling rig operates near an adjacent jacket platform, the effect of the large-diameter spudcan penetration on adjacent platform piles should be evaluated if the spudcan-pile clearance is less than one spudcan diameter. In this study, a method is developed to calculate the response of a pile subjected to spudcan penetration-induced load and pile head load transmitted from upper structure. The spudcan penetration-induced lateral pile deflection is first determined by large deformation finite element calculation. The spudcan penetration-induced load along pile shaft is then determined by transforming the pile lateral deflection into equivalent nodal loads in a nonlinear foundation beam finite element model. The pile response is finally calculated using an isolated pile finite element model subjected to penetration-induced loads and pile head loads transmitted from the upper structure or an overall finite element model which includes the upper structure and piles. Centrifuge model tests under 50g condition are conducted to verify the validity of the proposed method. To further investigate the applicability of the proposed method, the effects of pile head constraint, lateral pile head load magnitude and its direction, spudcan-pile clearance and undrained shear strength are studied. The effect of spudcan penetration on an adjacent pile is found to increase with rotation stiffness at the pile head. Spudcan penetration affects the adjacent piles most when the lateral pile head load is facing back to the spudcan. The effect of spudcan penetration decreases significantly when the spudcan-pile clearance is larger than the diameter of the spudcan and decreases with increasing undrained shear strength. Furthermore, the spudcan penetration has more influence on the pile deflection than the bending moment.

The determination and enhancement of compliant modes for high-amplitude actuation in lattices

This paper details the nonlinear design of adaptive lattices by determination and enhancement of compliant modes and optimizing the designed structure for delivering high amplitude actuation. The particular focus is the kagome lattice geometry—a pattern with some unique and useful actuation properties. Developing a novel design tool, the stiffness matrix of the beam assembly is calculated using a developed second-order geometrically nonlinear beam finite element formulation allowing large rotations. Based on this formulation in conjunction with singular value decomposition of the stiffness matrix, the modal optimization technique reduces the continuous structure with many degrees of freedom to a small number of low energy modes, which form the basis of designing the adaptive structure. For delivering high-amplitude actuation, the designed structure needs to be re-optimized due to changes in the nonlinear stiffness matrix under large deformation. This is performed via Bayesian optimization and by removing some internal members of the lattice. The integrity and feasibility of the optimum design is guaranteed via defining some constraints on removed members.

Topology optimization of structural systems based on a nonlinear beam finite element model

A method for the topology optimization of structures composed of nonlinear beam elements based on a hysteretic finite element modeling approach is presented. In the context of the optimal design of structures composed of truss or beam elements, studies reported in the literature have mostly considered linear elastic material behavior. However, certain applications require consideration of the nonlinear response of the structural system to the given external forces. Particular to the methodology suggested in this paper is a hysteretic beam finite element model in which the inelastic deformations are governed by principles of mechanics in conjunction with first-order nonlinear ordinary differential equations. The nonlinear ordinary differential equations determine the onset of inelastic deformations and the approximation of the signum function in the differential equation with the hyperbolic tangent function permits the derivation of analytical sensitivities. The objective of the optimization problem is to minimize the volume of the system such that a system-level displacement satisfies a specified constraint value. This design problem is analogous to that of seismic design where inelastic deformations are permitted, yet sufficient stiffness is required to limit the overall displacement of the system to a specified threshold. The utility of the method is demonstrated through numerical examples for the design of two structural frames and a comparison with the solution from the topology optimization assuming linear elastic material behavior. The comparisons show that the nonlinear design is either comprised of a lower volume for a given level of performance, or offers better performance for a given volume in comparison with optimized linear design.

A nonlinear cross-section deformable thin-walled beam finite element model with high-order interpolation of warping displacement

In this paper, a new finite element model is developed for nonlinear analysis of thin-walled beams by introducing a high-order interpolation for the warping displacement field to better simulate actual cross-section deformation. It is discovered that, due to the significant coupling between bending and membrane deformations in each wall, it is crucial for a nonlinear thin-walled beam finite element to be capable to describe complex warping displacement distribution over the cross-section. In the geometrically exact beam element model developed, the warping displacement is expressed by using a higher-order interpolation scheme, while the distortion displacements are independently described by using a number of cross-section in-plane deformation modes. Numerical results are presented to demonstrate the effectiveness of the proposed model and investigate possible impacts of interpolation schemes for the warping displacement distribution on the solution accuracy under different problem settings. The results show that with an appropriate high-order warping displacement interpolation scheme, the proposed beam model can produce very accurate results even in the case of small wall thickness, outperforming many of the existing models. It is therefore concluded that special attention is needed on the warping displacement field for a finite element model to accurately simulate the cross-section distortional deformation of thin-walled beams. • A geometrically exact cross-section deformable thin-walled beam finite element model is developed. • The distortion and warping displacements are described by distortion modes and high-order interpolation, respectively. • Numerical examples are presented for a comprehensive investigation of the present finite element model. • The importance of warping displacement field for ensuring solution accuracy is pointed out.

A simplified-but-general nonlinear thin-walled beam finite element model with independent interpolated axial displacement

This paper presents a study performed on the simplified thin-walled beam finite element model that takes into account the effects of shear deformation and higher-order axial displacement. Without using the torsion-related warping function derived from classical theories, the axial displacement field over the cross-section is constructed by using independent Langrage interpolation polynomials. Then, the effect of shear stress distribution can be reasonably included and the beam model is dedicated to thin-walled members with arbitrary cross-section. The expressions of the Green-Lagrange strain components are derived. The element elastic stiffness matrix and geometrical stiffness matrix are obtained to perform static and buckling analyses. Numerical examples are employed to demonstrate the effectiveness of the proposed model and discuss the probable accuracy-loss from different deformation assumptions and nonlinear strain definitions. As revealed by the results, the transverse shear deformation and higher-order axial displacement could exert significant effects on solution accuracy for static and buckling analyses under the circumstances of small slenderness and unsymmetrical cross-section. In addition, the nonlinear strain terms related to torsion and quadratic terms of the transverse shear deformation are found to be crucial to stability analysis, for which they ought to be included in a simplified strain definition. • A thin-walled beam finite element with interpolated axial displacements is developed. • The expressions of the Green-Lagrange strain components are derived. • Effects of shear deformation and higher-order axial displacement are discussed. • A simplified strain definition for flexural-torsional buckling analysis is suggested.

A branch-switching procedure for analysing instability of steel structures subjected to fire

The paper describes the development of a two-dimensional (2D) co-rotational nonlinear beam finite element that includes advanced path-following capabilities for detecting bifurcation instability in ...

Finite element modeling of temperature load effects on the vibration of local modes in multi-cable structures

Understanding thermal effects on the vibration of local (cable-dominant) modes in multi-cable structures is a complicated task. The main difficulty lies in the modification by temperature change of cable tensions, which are then undetermined. This paper applies a finite element procedure to investigate the effects of thermal loads on the linear dynamics of prestressed self-weighted multi-cable structures. Provided that boundary conditions are carefully handled, the discretization of cables with nonlinear curved beam elements can properly represent the thermoelastic behavior of cables as well as their linearized dynamics. A three-step procedure that aims to replace applied pretension forces with displacement continuity conditions is used. Despite an increase in the computational cost related to beam rotational degrees of freedom, such an approach has several advantages. Nonlinear beam finite elements are usually available in commercial codes. The overall method follows a thermoelastic geometrically non-linear analysis and hereby includes the main sources of non-linearities in multi-cable structures. The effects of cable bending stiffness, which can be significant, are also naturally accounted for. The accuracy of the numerical approach is assessed thanks to an analytical model for the vibration of a single inclined cable under temperature change. Then, the effects of thermal loads are investigated for two cable bridges, highlighting how natural frequencies can be affected by temperature. Although counterintuitive, a reverse relative change of natural frequency may occur for certain local modes. This phenomenon can be explained by two distinct mechanisms, one related to the physics intrinsic to cables and the other related to the thermal deflection of the superstructure. Numerical results show that cables cannot be isolated from the rest of the structure and the importance of modeling the whole structure for a quantitative analysis of temperature effects on the dynamics of cable bridges.

NONLINEAR NUMERICAL MODELLING OF BASALT REBAR REINFORCED CONCRETE STRUCTURES

The ever increasing tendency of more complex architecture and increasing use of basalt fibers in concrete, mainly due to corrosion resistance, requires a suitable, accurate and computationally efficient numerical method for modelling of mechanical behavior. A novel numerical modelling methodology for basalt fiber reinforced concrete structures is proposed. In this paper, the main focus is on modelling concrete beams with basalt longitudinal rebars and steel shear rebars. The proposed method is based on two step simulation method. On the first step a database of flexural stiffness depending on stress-strain state is created using non-linear simulations with continuum finite elements. The database of flexural stiffness is used in second step by performing nonlinear beam finite element simulation of frame structures. The numerical method showed good agreement with experimental results. The use of pre-computed database of flexural stiffness significantly accelerate non-linear simulations and whole building can analyzed by taking into account material non-linearity.

Nonlinear numerical modelling of basalt rebar reinforced concrete structures

The ever increasing tendency of more complex architecture and increasing use of basalt fibers in concrete, mainly due to corrosion resistance, requires a suitable, accurate and computationally efficient numerical method for modelling of mechanical behavior. A novel numerical modelling methodology for basalt fiber reinforced concrete structures is proposed. In this paper, the main focus is on modelling concrete beams with basalt longitudinal rebars and steel shear rebars. The proposed method is based on two step simulation method. On the first step a database of flexural stiffness depending on stress-strain state is created using non-linear simulations with continuum finite elements. The database of flexural stiffness is used in second step by performing nonlinear beam finite element simulation of frame structures. The numerical method showed good agreement with experimental results. The use of pre-computed database of flexural stiffness significantly accelerate non-linear simulations and whole building can analyzed by taking into account material non-linearity.

Nonlinear kinematics Reissner’s beam with combined hardening/softening elastoplasticity

In this work, we present geometrically non-linear beam finite element with embedded discontinuity which can represent elastoplastic constitutive behavior with both hardening and softening response. The constitutive equations are presented in rate form by using the multiplicative decomposition of deformation gradient. Formulation of elastoplastic response is presented in terms of stress resultants including the interaction between axial force, shear force and bending moment appropriate for metallic materials. The softening response is used to model the failure in connections, introducing displacement field discontinuity or a rotational hinge. The hinges or displacement discontinuity are presented in the framework of incompatible modes that can handle three different failure modes dealing with bending, shearing or axial deformation. With several numerical simulations, the FEM implementation is proven very robust for solving the problems of practical interest, such as push-over analysis.

A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

In this paper, a new computational framework is presented for the analysis of nonlinear beam finite elements subjected to large strains. Specifically, the methodology recently introduced in Bonet et al. (Comput Methods Appl Mech Eng 283:1061---1094, 2015) in the context of three dimensional polyconvex elasticity is extended to the geometrically exact beam model of Simo (Comput Methods Appl Mech Eng 49:55---70, 1985), the starting point of so many other finite element beam type formulations. This new variational framework can be viewed as a continuum degenerate formulation which, moreover, is enhanced by three key novelties. First, in order to facilitate the implementation of the sophisticated polyconvex constitutive laws particularly associated with beams undergoing large strains, a novel tensor cross product algebra by Bonet et al. (Comput Methods Appl Mech Eng 283:1061---1094, 2015) is adopted, leading to an elegant and physically meaningful representation of an otherwise complex computational framework. Second, the paper shows how the novel algebra facilitates the re-expression of any invariant of the deformation gradient, its cofactor and its determinant in terms of the classical beam strain measures. The latter being very useful whenever a classical beam implementation is preferred. This is particularised for the case of a Mooney---Rivlin model although the technique can be straightforwardly generalised to other more complex isotropic and anisotropic polyconvex models. Third, the connection between the two most accepted restrictions for the definition of constitutive models in three dimensional elasticity and beams is shown, bridging the gap between the continuum and its degenerate beam description. This is carried out via a novel insightful representation of the tangent operator.

Formulation of a geometrically nonlinear 3D beam finite element based on kinematic-group approach

A relatively simple finite element with 12 degrees of freedom is proposed for geometrically nonlinear analysis of spatial shear deformable beams and rods. The finite-element formulation is based on the concept of kinematic group, which is a geometrical object comprising two nodes on the rod axis and two adjoined vectors (directors) which define orientation of the cross section at each node. Results of sample problems are given to show the applicability of the element to study nonlinear deformation and stability of three-dimensional elastic rods undergoing finite rotations.

Force-Based Beam Finite Element (FE) for the Pushover Analysis of Masonry Buildings

A simplified approach for analyzing the nonlinear response of masonry buildings, based on the equivalent frame modeling procedure and on the nonlinear equivalent static analyses, is presented. A nonlinear beam finite element (FE) is formulated in the framework of a force-based approach, where the stress fields are expanded along the beam local axis, and introduced in a global displacement-based FE code. In order to model the nonlinear constitutive response of the masonry material, the lumped hinge approach is adopted and both flexural and shear plastic hinges are located at the two end nodes of the beam. A classical elastic-plastic constitutive relationship describes the nonlinear response of the hinges, the evolution of the plastic variables being governed by the Kuhn-Tucker and consistency conditions. An efficient element state determination procedure is implemented, which condenses the local deformation residual into the global residual vector, thus avoiding the need to perform the inner loops for computing the element nonlinear response. The comparison with some relevant experimental and real full-scale masonry walls is presented, obtaining a very good agreement with the available results, both in terms of global pushover curves and damage distributions.