Thin-walled Beam Structures Research Articles

Corotational isogeometric shear deformable geometrically exact spatial form beam element for general large deformation analysis of flexible thin-walled beam structures

In this paper, a new 3D generic order corotational geometrically exact beam element undergoing arbitrarily large deformations is presented. The novelty of the formulation lies in the use of the standard corotational framework, that is the decomposition into rigid body motion and pure deformation, to derive not only the objective, path-independent and singularity-free spatial exact strain measure but also the internal force and the consistent tangent stiffness matrix. The new formulation is derived by firstly conducting the consistent linearization of the weak formulation for the equilibrium equations of Reissner–Simo theory in the spatial form. By constructing a co-rotated frame that continuously rotates and translates with the element, the nodal strain-producing deformations can be extracted with the rigid-body translations and rotations removed from the element total motion. Subsequently, the corotational deformation variables are exploited to establish the distribution of deformation field within the beam element through B-spline basis functions, and derive the invariant and path independent exact strain measure in the spatial form by Lie derivative formalism. Through performing the variational operations of the compact transformation between corotational and global variables, the consistent tangent stiffness matrix are derived in closed form of the discrete variables for deformation field of any order. The proposed formulation is applicable to not only low-order two-node geometrically exact elements but also high-order multi-node straight or curved ones. The accuracy and robustness of the new formulation are demonstrated through the solutions of various benchmark problems.

Stall nonlinear aeroelastic effects and active control of thin-walled composite blade with piezoelectric patches embedded

Stall nonlinear flutter behavior and active control of anisotropic composite blade have been investigated based on piezoelectric actuation. The blade body comprises an elliptical composite host structure and a lightweight honeycomb skin structure. The composite blade with elliptical section is modeled as a thin-walled beam structure with circumferential asymmetric stiffness (CAS) configuration, exhibiting transverse shear deformation, warping restraint effect, and vertical bending, with structural tailoring implemented. The stall-induced flutter suppression and nonlinear aeroelastic effects of linearized dynamic response characteristics of blade incorporating stall nonlinear aerodynamic model are investigated. The nonlinear aeroelastic coupled equations are reduced to ordinary equations by the Galerkin method, with the aerodynamic force decomposed by strip theory. Active feedback control based on boundary moment control strategy is developed to enhance the vibrational behavior and dynamic response to aerodynamic excitation and stabilize structures that might be damaged in the absence of control. Various active feedback control schemes tuned by optimal proportional-derivative (PD) controller are implemented. The excellent control effect is not only confirmed by the time response with its vibration amplitude greatly suppressed but also confirmed by the frequency change and comparison. The research provides a way for rare study of amplitude suppression of stall nonlinear aeroelastic effect of thin-walled composite blade with CAS configuration and piezoelectric patches embedded.

Development of deep learning-based joint elements for thin-walled beam structures

This study presents a new modeling technique to estimate the stiffness matrix of a thin-walled beam-joint structure using deep learning. When thin-walled beams meet at joints, significant sectional deformations occur, such as warping and distortion. These deformations should be considered in the one-dimensional beam analysis, but it is difficult to explicitly express the coupling relationships between the beams’ deformations connected at the joint. This study constructed a deep learning-based joint model to predict the stiffness matrix of a higher-order one-dimensional super element that presents the relationships. Our proposition trains the neural network using the eigenvalues and eigenvectors of the joint's reduced stiffness matrix to satisfy the correct number of zero-strain energy modes overcoming the randomly perturbed error of the deep learning. The deep learning-based joint model produced compliance errors mostly within 2% for a given structural system and the maximum error of 4% in the worst case. The newly proposed methodology is expected to be widely applicable to structural problems requiring the stiffness of a reduction model.

Numerical vibration correlation technique for thin-walled composite beams under compression based on accurate refined finite element

This paper investigates the virtual vibration correlation technique for the evaluation of the variations of the natural frequencies for highly flexible thin-walled composite beams. In this regard, refined finite element based on the Carrera Unified Formulation (CUF) is developed. Thin-walled composite beams with various cross-sections of box, I-shaped and channel-shaped types, are investigated. Additionally, a comparison of the CUF results with the implemented shell models and the available literature is presented in order to evaluate the accuracy and efficiency of the proposed method. It is indicated that classical beam theories such as Euler–Bernoulli and Timoshenko beam theories failed to have an accurate prediction of the natural frequencies and dynamic response of thin-walled beam structures. Therefore, the necessity of employing higher-order and refined beam theories capable of capturing cross-sectional deformations is highlighted. Furthermore, to fully demonstrate the capabilities of the CUF-1D method with efficient Lagrange expansion, a more complex structural problem of a channel-shaped composite beam with the different numbers of transverse stiffeners is studied, and conclusions about the buckling behavior and variations of natural frequencies are drawn. It is shown that the numerical assessments by the proposed efficient CUF method in this paper correlate well with the shell results which are more computationally expensive.

Thin-walled Beam Bending Quartic Deplanation Analytical Function for Improved Experiment Matching

Ensuring the accuracy of an analytical solution is important in modeling real engineering structures. For determining the stress-deformation condition of a thin-walled beam structure in bending, inadequately, the simple beam formula can only provide uniform average stress-deformation distribution at specific cross-sectional elevations. The recently developed analytical solution for determining stress-deformation conditions with consideration of the shear lag effect of a prismatic thin-walled box beam subjected to transverse load causing bending using classical quadratic deplanation function did not provide an accurate but rather a good estimate when compared with finite element analysis results. In this paper, the objective of the study was to see if the accuracy can be improved. The same Vlasov’s method was used. The method was developed using Calculus of Variations based on a Stress-form stationary condition complementary energy which included the shear lag effect. All calculations were computerized using the software MAPLE 18 which assisted in getting quick results (especially for preliminary design studies) for various geometries, material properties, and loading conditions. Several deplanation functions were introduced. Two variants of the quadratic functions, i.e. x 2 y 3 and x 2 y, and the quartic functions, i.e. x 4 y 3 and x4 y were used to find the best match against empirical data and FEA (finite element analysis) results. The finding was that the quartic variant of the deplanation functions provided improved matching with experimental data as well as FEA results. Noteworthy to point out that the characteristic of the functions with quartic-x or x4 of having almost flat variation in the center part enhances matching with the experimental data. Moreover, the characteristic of the function with cubic-y or y3 of having steeper slopes enhances matching with the FEM results at the edge.

General Stiffness Matrix for a Thin-Walled, Open-Section Beam Structure

This paper is to review the theory of thin-walled beam structures of the open cross-section. There is scant information on the performance of structures made from thin-walled beam elements, particularly those of open sections, where the behavior is considerably complicated by the coupling of tensile, bending and torsional loading modes. In the combined loading theory of thin-walled structures, it is useful to mention that for a thin-walled beam, the value of direct stress at a point on the cross-section depends on its position, the geometrical properties of the cross-section and the applied loading. This applies whether the thin-walled section is closed or open but this study will be directed primarily at the latter. Theoretical analyses of structures are fairly well established, considered in multi-various applications by many scientists. However, due to the present interest in lightweight structures, it is necessary to specify where the present theory lies. It does not, for example, deal with compression and the consequent failure modes under global and local buckling. Indeed, with the inclusion of strut buckling failure and any other unforeseen collapse modes, the need was perceived for further research into the subject. Presently, a survey of the published works has shown in the following: 1) The assumptions used in deriving the underlying theory of thin-walled beams are not clearly stated or easily understood; 2) The transformations of a load system from arbitrary axis to those at the relevant centre of rotation are incomplete. Thus, an incorrect stress distribution may result in; 3) Several methods are found in the recent literature for analyzing the behaviour of thin-walled open section beams under combined loading. These reveal the need appears for further study upon their torsion/flexural behaviour when referred to any arbitrary axis, a common case found in practice. This review covers the following areas: 1) Refinement to existing theory to clarify those observations made in 1 - 3 above; 2) Derivation of a general elastic stiffness matrix for combined loading; 3) Calculation of the stress distribution on the cross-section of a thin-walled beam. A general transformation matrix that accounts for a load system applied at an arbitrary point on the cross-section will be published in a future paper.

Calculation and Prediction of Crushing Process of Al–Mg–Si Alloy Thin-Walled Components

Crushing performance is being widely investigated because it is the key performance indicator of the thin-walled beam structure in automobile safety components. In this study, five kinds of ageing state with different yield strengths were prefabricated with a self-developed KHC63 alloy. Using a rectangular tube profile, the physical relationship between crushing properties and mechanical properties, and the structure of the profile, were studied. According to variation characteristics, the crushing curve was divided into four typical stages: elastic stage, bending stage, compaction stage and folding stage. In the elastic stage, the peak load of the component is related to material elasticity and structure elasticity, and the relationship is approximately linear. In the bending stage, the driving factor of plastic hinge deformation is the bending moment, and it is always constant. In the compaction stage, the crushing curve correlates well with the cosine function. In the folding stage, the crushing process begins to lose stability and is hard to predict. The mathematical relationships between force and displacement were established according to the characteristics of each stage, and the calculated results were close to the measured results except for the folding stage. The deviation of the calculated energy absorption from the measured data in the first three stages is only 4.3%, but it is 10.6% in the folding stage. The calculation method used for the first three stages is, therefore, suitable.

Optimization of thin-walled beam structures: Monolithic versus staggered solution schemes

This contribution compares the performance of a monolithic and two staggered numerical update schemes for a coupled shape and topology optimization method for thin-walled beam structures. In order to limit the computational time of the optimization method, the thin-walled beam structures are modeled as 2.5D configurations. In this approach, 1D beam elements are used to simulate the beam response in the longitudinal direction, while the cross-sectional properties of the beam elements are calculated from additional 2D finite element method analyses. The sensitivities with respect to the shape and topology design variables are derived in closed form in order to take advantage of a computationally efficient, gradient-based optimization algorithm. The numerical examples concern basic circular and square thin-walled cantilever beam structures, as well as a more complex, non-prismatic thin-walled beam structure representative of a rotor blade used in a horizontal-axis wind turbine. For each numerical example the computational time and the solution computed by the monolithic and staggered update schemes are compared, which provides clear insight into the numerical efficiency of the solution procedure and the uniqueness of the computational result. Although the three different update schemes for the cases examined result in comparable optimized design concepts, the computational efficiency and the specific minimal compliance value found turn out to be rather sensitive to the algorithmic details of the numerical update scheme applied. Since the monolithic update scheme typically navigates a larger design space than the two staggered update schemes, for most cases examined it provides the lowest structural compliance. In the specific case whereby shape optimization (virtually) has no influence on the final, optimized beam configuration, the structural compliance computed by the monolithic scheme may relate to a local minimum that is less optimal compared to the value calculated by a staggered update scheme.

A nonlinear dynamical model for rotating composite thin-walled beams subjected to hygrothermal effects

A rotating composite thin-walled beam structure usually works in environment with strong hygrothermal characteristics. Temperature variation and moisture absorption can influence physical properties of composite, and furthermore change mechanical behavior of the structure. Hygrothermal influences depend on composite, layup scheme, geometry and nonlinearity of the beam. However, a comprehensive analytical model including these factors is lacking. This work aims to establish such a model for rotating composite thin-walled beams subjected to hygrothermal effects so that relevant problems can be solved analytically, which can aid numerical simulation and experimental test. Many important factors including hygrothermal effects, nonlinearity, mode coupling, arbitrary layup schemes, rotation, external forces, transverse shear, non-uniformity of cross section, constrained torsion, warping, setting angle and pretwist angle are considered. Governing equations of the coupled vibration are established by using the generalized Hamiltonian principle. Reduction verification, simplification principles and reduced models are discussed. The solution of the model is illustrated and some examples are given. Finally, the influences of temperature, moisture absorption and ply angle on hygrothermal internal forces of constrained thin-walled beams with CUS and CAS configurations are analyzed.

Design and Application of Dynamic Impact Test System for Auto Body Structures

This paper presents a specialized test system used for the dynamic impact of auto body structures. The detailed structure of the host machine, the anti-secondary impact system, and the human-computer interaction interface for data acquisition and processing are designed and integrated with the control system to form a complete dynamic impact test system. The typical single-hat thin-walled beam structure is token as an example, and the functional test of the system is carried out. The results show that the test system has comprehensive functions, high test accuracy and convenient operation, and can meet the requirements of auto body structures dynamic impact test.

Analysis of warping and distortion transmission in mixed shell\u2013GBT (generalized beam theory) models

Warping and distortion are relevant kinematic features of thin-walled beam structures, which have a non-trivial analysis. On this basis, this paper not only evaluates the possible kinematic transmissions involving high-order warping and distortion, but also presents a procedure to analyze structures using mixed models based on shell and Generalized Beam Theory (GBT) elements. In this mixed beam-shell structure, the traditional shell elements are applied at structural detailing points, such as joints, and GBT elements are used to model the beams/columns. Such a modeling technique uses the benefits of both elements. Shell elements can easily simulate different types of geometry conditions and details, such as stiffeners and holes; meanwhile, for the beams and columns, GBT can provide high performance, accuracy, and an easy modeling approach with clear results. The numerical formulation is based on multi-freedom constraint techniques. Special attention is given to the Master–Slave method, which is developed based on GBT kinematic assumptions. Furthermore, there is a discussion concerning the choice of the master degrees of freedom and its implications in numerical performance. An example of a thin-walled hollow circular cross section illustrates the proposed approach and is compared with fully shell element models.

Monitoring of Real-Time Complex Deformed Shapes of Thin-Walled Channel Beam Structures Subject to the Coupling Between Bi-Axial Bending and Warping Torsion

Structural health monitoring (SHM) is a research focus involving a large category of techniques performing in-situ identification of structural damage, stress, external loads, vibration signatures, etc. Among various SHM techniques, those able to monitoring structural deformed shapes are considered as an important category. A novel method of deformed shape reconstruction for thin-walled beam structures was recently proposed by Xu et al. [1], which is capable of decoupling complex beam deformations subject to the combination of different loading cases, including tension/compression, bending and warping torsion, and also able to reconstruct the full-field displacement distributions. However, this method was demonstrated only under a relatively simple loading coupling cases, involving uni-axial bending and warping torsion. The effectiveness of the method under more complex loading cases needs to be thoroughly investigated. In this study, more complex deformations under the coupling between bi-axial bending and warping torsion was decoupled using the method. The set of equations for deformation decoupling was established, and the reconstruction algorithm for bending and torsion deformation were utilized. The effectiveness and accuracy of the method was examined using a thin-walled channel beam, relying on analysis results of finite element analysis (FEA). In the analysis, the influence of the positions of the measurement of surface strain distributions on the reconstruction accuracy was discussed. Moreover, different levels of measurement noise were added to the axial strain values based on numerical method, and the noise resistance ability of the deformation reconstruction method was investigated systematically. According to the FEA results, the effectiveness and precision of the method in complex deformation decoupling and reconstruction were demonstrated. Moreover, the immunity of the method to measurement noise was proven to be considerably strong.

B-spline finite element approach for the analysis of thin-walled beam structures based on 1D refined theories using Carrera unified formulation

In the current study, 1D refined beam theories on the basis of Carrera Unified Formulation (CUF) are combined with Isogeometric approach (IGA) for the static and free vibration analysis of thin-walled beam structures. The B-spline basis functions utilized in IGA are employed to approximate displacement field due to their interesting attributes in analysis. N-Order Taylor-like expansions are utilized in the framework of CUF which presents finite element matrices in the form of fundamental nuclei that is independent of the type and order of expansion. Higher-order B-spline basis functions attenuate the effect of shear locking properly and higher-order theories presented by CUF are free from Poisson locking phenomenon and utilizations of shear correction factor. Several numerical results including both solid and thin-walled structures are investigated and it is shown that coupling IGA and CUF ends in a suitable methodology to analyze beam structures.

System modeling and instability control of wind turbine blade based on hydraulic pitch system and radial basic function neural network proportional–integral–derivative controller

System modeling and aeroservoelastic control for divergent instability of stall-induced composite wind turbine blade modeled as thin-walled symmetric layup beam analysis have been investigated based on hydraulic pitch system and radial basic function neural network control. The blade is modeled as single-cell thin-walled beam structure with the circumferentially asymmetric stiffness design, exhibiting flap/lead-lag bending coupling deformation. The stall flutter and aeroservoelastic control of composite blade are investigated based on dynamic stall Beddoes–Leishman aerodynamic model and radial basic function neural network proportional–integral–derivative controller, with pitch actuator performed by hydraulic system. The system motion equations consist of the aeroelastic equations and the six-order pitch equation. The nonlinear aeroelastic responses, including both flap/lead-lag responses and pitch angle responses under different parameters, are solved by Galerkin method and the nonlinear time integration scheme with nonlinear residual analysis. To verify the effectiveness of the control scheme and realize visualized display of large thin-walled blade in the laboratory, experimental platform based on hardware-in-the-loop simulation is built to realize real-time control and virtual simulation. The platform structure consists of PLC hardware, monitoring interface in configuration software, and simulation environment that is connected by the OPC server with PLC system. The platform lays the foundation for vibrational behavior research on visualization of large wind turbine blade under divergent stall situation and verifies the real-time feasibility of the control algorithm proposed.

Deflection monitoring of thin-walled wing spar subjected to bending load using multi-element FBG sensors

This paper presents a measurement technology based on the multi-element fiber Bragg grating (FBG) sensors for monitoring the deflection of thin-walled wing spar. A finite element model of the tested thin-walled beam structure subjected to bending load is developed in order to analyze stress status and to derive the convenient position of the optical fiber sensors for a correct monitoring of the wing spar beam. FBG rosette has been designed between the shear web and spar cap as a new installation position of bonding optical fiber sensors for estimation the deflection. An experimental campaign is carried out in order to verify the proper behavior of the deflection of wing spar beam and sensing strategy. The loading test results revealed that multi-element FBG rosette was successfully implemented on the tested thin-walled wing spar beam as well as providing reliable strain and deflection monitoring data from the FBG sensors. The results show that a good correspondence between the input load and output strain of the FBG sensing system with good reproducibility, and the comparative analysis between FBG and resistance strain gage have been achieved, which measurement results of both are consistent. These data and monitoring method are useful for fatigue life prediction of the potential thin-walled I-beam structure.

Vibration control of rotating piezo-composite blade beam with CUS configuration based on optimal LQG controller

Vibration control of rotating composite blade beam with piezoelectric patch embedded is investigated. Stall flutter of piezo-composite blade driven by nonlinear aerodynamic forces is analyzed based on anisotropic circumferentially uniform stiffness (CUS) configuration. The blade is modeled as single-cell thin-walled beam structure, exhibiting the couplings among three displacements of vertical bending, lateral bending and transverse shear deformation, with structural tailoring implemented. The transversely piezoelectric actuating element is embedded in a manner such that its surface is parallel to the mid-surface of the blade beam. Piezoelectric damping ratios of rotating piezo-composite blade are described, with influences of different ply angles and rotating speeds illustrated. The flutter suppression for stall aeroelastic behavior based on an optimal LQG controller (OLC) with a dynamic regulator is highlighted with obvious effects demonstrated. In contrast with conventional LQG controller, the superiority of OLC controller is apparently demonstrated by time response and piezoelectric feedback voltage. Analytical proof of the structural modeling and feasibility analysis of the physical realization of the OLC algorithm are also investigated by comparisons of different modeling theories, and demonstrated by experimental platform.

Dynamic Optimization of Functionally Graded Thin-Walled Box Beams

This paper introduces a mathematical model for optimizing the dynamic performance of thin-walled functionally graded box beams with closed cross-sections. The objective function is to maximize the natural frequencies and place them at their target values to avoid the occurrence of large amplitudes of vibration. The variables considered include fiber volume fraction, fiber orientation angle and ply thickness distributions. Various power-law expressions describing the distribution of the fiber volume fraction have been implemented, where the power exponent was taken as the main optimization variable. The mass of the beam is kept equal to that of a known reference beam. Side constraints are also imposed on the design variables in order to avoid having unacceptable optimal solutions. The mathematical formulation is carried out in dimensionless quantities, enabling the generalization to include models with different cross-sectional types and beam configurations. The optimization problem is solved by invoking the MatLab optimization ToolBox routines, along with structural dynamic analysis and eigenvalue calculation routines. A case study on the optimization of a cantilevered, single-cell spar beam made of carbon/epoxy composite is considered. The results for the basic case of uncoupled bending motion are given. Conspicuous design charts are developed, showing the optimum design trends for the mathematical models implemented in the study. It is concluded that the natural frequencies, even though expressed in implicit functions, are well-behaved, monotonic and can be treated as explicit functions in the design variables. Finally, the developed models can be suitably used in the global optimization of typical composite, functionally graded, thin-walled beam structures.

Cross-sectional shape design of automobile structure considering rigidity and driver's field of view

Thin-walled beam structure can be used to efficiently predict the performances of automobile frame for conceptual design. However, it is an open issue to acquire an accurate cross-sectional shape of A-pillar structure considering both rigidity and driver's field of view (DFOV). This paper proposes an approach to calculate the cross-sectional rigidity and DFOV. Firstly, formulations of cross-sectional properties, including open, single-cell, double-cell, three-cell, four-cell section, are summarized. Secondly, the obstruction angle is introduced to describe DFOV, which is acquired by a nonlinear optimization model. Finally, a A-pillar example of Toyota RAV4, solved by the developed software – CarFrame, proves that the proposed method can be completely applied at the conceptual design of the automobile structure.

Vibration and aeroelastic control of wind turbine blade based on B-L aerodynamic model and LQR controller

Vibration and aeroelastic control of anisotropic composite wind turbine blade modeled as symmetric layup beam analysis have been investigated based on Beddoes-Leishman (B-L) dynamic stall aerodynamic model and linear quadratic controller. The blade is modeled as single-cell thin-walled beam structure, exhibiting flap bending-lag bending-twist coupling deformation, with constant pitch angle set. The stall flutter and aeroelastic control of composite blade are investigated based on some structural and dynamic parameters, with structural damping computed. The aeroelastic partial differential equations are reduced by Galerkin method, with the nonlinear aerodynamic forces computed by the method of fitting static elastic coefficients. Linear quadratic controller is applied to enhance the vibrational behavior in stall situation under divergent conditions and stabilize displacements that might be unstable in the absence of control.

Nonlinear modal interactions in composite thin-walled beam structures with simultaneous 1:2 internal and 1:1 external resonances

Nonlinear dynamic characteristics of a composite aircraft wing structure modeled by a geometrically nonlinear anisotropic thin-walled beam in the presence of simultaneous 1:2 internal and 1:1 external resonances are investigated. Some prominent non-classical effects such as of transverse shear strain, warping inhibition, and three-dimensional strain are considered in the beam model. Moreover, circumferentially asymmetric stiffness lay-up configuration is adapted to generate the transverse bending-twisting elastic coupling. The solution methodology is based on the Extended Galerkin’s Method, and the method of multiple scales is applied to the system in order to obtain the equations of amplitude and modulation. Steady-state solutions and their stability are investigated. The peculiarity of the internal resonances and the conditions for saturation and jump phenomenon during the modal interactions are discussed and the commercial code ABAQUS is used to validate the theoretical results we have obtained. Finally, the prominent features of modal interactions in composite thin-walled beam structures are summarized and pertinent suggestions concerning safe design of the wing structures are given.