Multi-cell Sections Research Articles

Development of a cross-sectional finite element for the analysis of thin-walled composite beams like wind turbine blades

A method for structural analysis of thin-walled composite beams like wind turbine blades is presented. This method is based on the Nonhomogeneous Anisotropic Beam Section Analysis (NABSA) which consists in discretizing the beam cross section using finite elements. The proposed implementation uses 3-node line cross-sectional finite elements with nodes having rotational degrees of freedom to describe the cross-sectional warping displacements. Solutions obtained using this approach were verified against the corresponding analytical or numerical solutions. Agreement was very good to excellent for the computation of cross-sectional properties and distribution of stresses, strains and warping displacements for a broad range of possible composite beam behaviors including geometric and material couplings, open sections, multicell sections, and arbitrary laminates. For thin-walled layered structures, the proposed method provides models with fewer degrees of freedom than equivalent models based on a two-dimensional discretization of cross sections using triangular or quadrilateral elements such as conventional NABSA or VABS which suggests that computation time could be reduced.

Theoretical estimation of axial crushing behavior of multicell hollow sections

This research, which is a continuation of the previously published paper, develops the idea for estimating the behavior of axially crushed multicell thin walled structures. This approach employs new crushing mechanisms or simply crushing modes for different configurations of multiweb corner elements based on simulation, geometric, and kinematic observations. Furthermore, a new model is introduced named as “Induced Curvature Model” or ICM which includes the curvature of the sheet during progressive crumpling and also the hardening effect of the material under plastic deformation, instead of simple assumptions of perfect plastic material and hinge lines for the folds in case of previously offered plastic hinge model (PHM). Both ICM and PHM are employed to capture crushing properties based on the offered crushing modes and the results suggested by them are compared to each other. Different aspects of crushing behavior were studied and discussed. To show the validity of the proposed models, FE simulation studies were also conducted. Results show an average error around 7% which indicates a good agreement between theoretical and simulation findings and the applicability of the proposed novel model to a wide range of thin walled structures.

Experimental and numerical studies on the bending collapse of multi-cell Aluminum/CFRP hybrid tubes

Wrapping the structures with composites and adopting multi-cell sections are two typical methods to improve the crashworthiness of thin-walled beams. However, the effects of combining these two methods on crashworthiness are not clear. This paper aims to investigate the bending collapse and crashworthiness of the multi-cell aluminum/carbon fiber reinforced plastic (Al/CFRP) hybrid tubes under quasi-static and dynamic loading. Three-point bending tests are firstly conducted for the CFRP, Al and Al/CFRP square tubes with single or multi-cell sections. The deformation characteristics and crushing force responses of the tubes are analyzed, and the energy absorption performances are evaluated. The bending resistance of the Al tubes is increased by up to 41% attributed to the CFRP enhancement. The non-linear finite element software ABAQUS/Explicit is then employed to simulate the tests and help analyze the deformation mechanisms. Parametric studies are performed to investigate the influence of the Al wall thickness, the number of CFRP plies, loading velocity, partial wrapping and sectional shape on the crashworthiness of Al/CFRP tubes. Results show that the Al wall thickness, partial wrapping, and sectional shape have a significant influence on the deformation pattern and force response of Al/CFRP tubes, while the number of CFRP plies and the loading velocity have a relatively small influence. The specific energy absorption of Al/CFRP tubes can be increased by 11% by introducing partial CFRP wrapping, and in all cases, the multi-cell Al/CFRP tubes outperform the single-cell counterparts in crashworthiness performances.

Static and dynamic axial crushing of Al/CRFP hybrid tubes with single-cell and multi-cell sections

Adopting multi-cell sections and composite wrapping are two effective approaches to improve the crashworthiness performance of thin-walled tubes. The axial crushing of single-cell and multi-cell aluminum/carbon fiber reinforced plastic (Al/CFRP) hybrid tubes is investigated in this paper. Quasi-static and dynamic experimental tests are performed for CFRP, Al and Al/CFRP tubes with single-cell, double-cell or quadruple-cell sections. The influences of Al wall thickness and the number of composite layers and sectional cells on the crushing response and energy absorption characteristics of the tubes are analyzed. Results show that the increase of all these factors can improve the load bearing and energy absorption capability of the tubes. The enhancement effect of composite wrapping for the Al/CFRP hybrid tubes is determined by the relative strength/weakness on the energy absorption efficiency of Al and CFRP. The interaction effect between the Al and CFRP on energy absorption of various sections is also investigated and the results show that the interaction effect decreases with the increase of the number of the sectional cells, while it is enhanced by increasing the number of CFRP layers.

Static and dynamic axial crushing of self-locking multi-cell tubes

A new type of self-locking multi-cell structure fabricated by assembling four C-shaped open sections is proposed in this paper. Quasi-static and dynamic experimental tests are performed to investigate the axial crush resistance and energy absorption characteristics of this type of structures. The experiment results show that the SEA of the assembled multi-cell sections is 35–40% and 40–50% higher than that of the constituent C-shaped elements for quasi-static and dynamic loading, respectively. Numerical analyses are carried out to simulate the experiment and perform the parametric study on the influences of geometric parameters, boundary condition and load speed on the crush resistance of structures. The behavior and performance of single C-shaped open sections are also studied, and the interaction effects originated from the self-locking features are investigated quantitatively. In addition, theoretical expressions are derived to predict the crush resistance of the assembled self-locking multi-cell sections. Comparisons show that the theoretical expressions can predict the static and dynamic mean crushing forces of the self-locking multi-cell structures with errors below 5% in most cases.

Energy absorption and optimization design of multi-cell tubes subjected to lateral indentation

The local indentation of thin-walled beams is a key mechanism for the collapse of beams under lateral loads. The indentation behavior of multi-cell tubes is investigated in this work. Quasi-static experimental tests are conducted first for multi-cell tubes with three different sections. The deformed shapes and force responses are obtained by indenting the tubes resting on a rigid surface. Numerical simulations are then performed to analyze the static and dynamic responses of the tubes, and the accuracy of the numerical model is validated by the experimental results. The crushing force responses and deformation characteristics of the multi-cell sections are analyzed in detail and the energy absorption performances of them are evaluated. Results reveal that the cross-sectional shapes of the tubes have great influence on their deformation process and crushing force responses. Moreover, to further increase the energy absorption efficiency, sequential response surface method (SRSM) is employed to achieve the optimal designs of the multi-cell sections. Results show that the energy absorption performances of the sections are greatly improved by the SRSM optimization.

Axial crushing of tubes fabricated by metal sheet bending

Energy absorption characteristics of thin-walled tubes fabricated by bending of metal sheets are studied in this work. Square tubes are prepared by folding aluminum alloy 3003 H12 sheets and quasi-static axial crushing tests are then performed with universal material testing machine. The energy absorption capacity and deformation of these folded tubes are analyzed and compared with traditional square tubes. Nonlinear finite element code LS-DYNA is employed to simulate the crushing tests and the numerical results are compared with the experiment. The influences of various factors, including the boundary condition, loading speed, fillet radius and sectional shape, on the crushing force and deformation of the folded tubes are investigated numerically. In addition, folded tubes with multi-cell sections are also analyzed and compared with traditional tubes. Some suggestions are provided for the design of folded single or multi-cell tubes. Since the folded tubes are easily prepared, cost-effective for small-scale production and flexible in both sectional shape and geometric parameters, they are quite promising to be applied widely in various engineering fields.

The crush resistance of four-panel angle elements

Four-panel angle elements are frequently encountered in multi-cell sections. To predict the energy absorption of such sections under axial loading, theoretical models have to be established for possible collapse modes of these elements. In this paper, four-panel angle elements are classified into four types: corner element, K-shaped, X-shaped and tree-shaped element. The influence of geometric parameters and type of triggers on collapse modes and crush resistance of these elements was firstly investigated numerically by using LS-DYNA. Theoretical models were then established for most commonly developed collapse modes of each type of four-panel angle elements. Elements with different width, thickness and angles were analyzed to validate the proposed models and the theoretical predictions by these models showed good agreement with numerical results. In addition, axial crushing test of two multi-cell sections with K-shaped elements was carried out quasi-statically and the experimental results validated the good accuracy of the present numerical and theoretical models.

Comparative analysis of energy absorption capacity of simple and multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections

Energy must dissipate during a collision to prevent damage and injury. To reduce loss from collision, energy absorbers are used that dissipate energy upon deformation and folding to prevent damage to critical parts of a structure. In this paper, simple and multi-cell thin-walled tubes made from aluminum with triangular, square, hexagonal and octagonal sections were subjected to quasi-static loading. The experimental results were then compared with numerical simulations. The results showed that the energy absorption capacity of multi-cell sections is greater than for that of simple sections. Also, hexagonal and octagonal sections in a multi-cell configuration absorbed the greatest amounts of energy per unit of mass.

Energy absorption of multi-cell stub columns under axial compression

Multi-cell metal columns were found to be much more efficient in energy absorption than single-cell columns under axial compression. However, the experimental investigations and theoretical analyses of them are relatively few. In this paper, the quasi-static axial compression tests are carried out for multi-cell columns with different sections. The significant advantage of multi-cell sections over single cell in energy absorption efficiency is investigated and validated. Numerical simulations are also conducted to simulate the compression tests and the numerical results show a very good agreement with experiment. Theoretial analyses based on constitutive element method are proposed to predict the crush resistance of multi-cell columns and the theoretical predictions compare very well with the experimental and numerical results.

Analysis of Thin-Walled Straight Beams with Generally Shaped Closed Sections Using Numerically Determined Sectional Deformation Functions

This investigation presents one-dimensional static and eigenvalue analyses of thin-walled straight beams with generally shaped closed single-cell or multicell sections. For accurate beam analysis, sectional warping and distortional deformations should be considered in addition to the standard Timoshenko displacement field, but it is difficult to obtain the deformation functions analytically for arbitrarily shaped sections. Thus, a numerical method is proposed to obtain sectional deformations for any arbitrarily shaped sections. Once the deformations are identified, they can be integrated over a cross section to yield one-dimensional higher order beam equations. For the numerical determination, the cross section of a thin-walled beam is modeled as a beam frame, where the warping and distortional deformation functions of the section are identified as the eigenmodes of the frame model; the lowest few energy mode sets of in-planar and out-of-planar modes are selected as the distortional and warping deformation functions, respectively. The validity of this approach is checked by comparing the present results with shell finite-element results. For numerical tests, several thin-walled closed sections, including those with flanges or varying wall thicknesses, are considered. The effect of the number of selected warping and distortion sets on solution convergence is also investigated.

The torsion of the multicell sections

The paper is focused on the stress analysis of thin-walled multicell sections subjected to pure torsion. The shear flow and stiffness characteristics of the cross section for torsion are given. Example: aircraft wing section. The theory for thin-walled closed sections used in this paper was developed by Bredt (3). The shear flows obtained are used in the design of skins and interior webs, ribs and fasteners at skin splices, skin web junctions and the joints where the ribs meet the skin or webs.

Multiobjective optimization of multi-cell sections for the crashworthiness design

Plastic deformation of structures absorbs substantial kinetic energy when impact occurs. For this reason, energy-absorbing components have been extensively used in the structural design of vehicles to intentionally absorb a large portion of crash energy to reduce the severe injury of occupants. On the other hand, high peak crushing force may to a certain extent indicate the risk of structural integrity and biomechanical damage of occupants. For this reason, it is of great significance to maximize the energy absorption and minimize the peak force by seeking for optimal design of these components. This paper aims to design the multi-cell cross-sectional thin-walled columns with these two crashworthiness criteria. An explicit finite element analysis (FEA) is used to derive higher-order response surfaces for these two objectives. Both the single-objective and multi-objective optimizations are performed for the single, double, triple and quadruple cell sectional columns under longitudinal impact loading. A comparative analysis is consequently given to explore the relationship between these two design criteria with the different optimization formulations.

An unconditionally convergent algorithm for the evaluation of the ultimate limit state of RC sections subject to axial force and biaxial bending

AbstractWe present a numerical procedure, based upon a tangent approach, for evaluating the ultimate limit state (ULS) of reinforced concrete (RC) sections subject to axial force and biaxial bending. The RC sections are assumed to be of arbitrary polygonal shape and degree of connection; furthermore, it is possible to keep fixed a given amount of the total load and to find the ULS associated only with the remaining part which can be increased by means of a load multiplier. The solution procedure adopts two nested iterative schemes which, in turn, update the current value of the tentative ultimate load and the associated strain parameters. In this second scheme an effective integration procedure is used for evaluating in closed form, as explicit functions of the position vectors of the vertices of the section, the domain integrals appearing in the definition of the tangent matrix and of the stress resultants. Under mild hypotheses, which are practically satisfied for all cases of engineering interest, the existence and uniqueness of the ULS load multiplier is ensured and the global convergence of the proposed solution algorithm to such value is proved. An extensive set of numerical tests, carried out for rectangular, L‐shaped and multicell sections shows the effectiveness of the proposed solution procedure. Copyright © 2007 John Wiley & Sons, Ltd.

Theoretical prediction and numerical simulation of multi-cell square thin-walled structures

The axial crushing of square multi-cell columns were studied analytically and numerically. Based on the Super Folding Element theory, a theoretical solution for the mean crushing force of multi-cell sections were derived by dividing the profile into 3 parts: corner, crisscross, and T-shape. Numerical simulations of square multi-cell sections subjected to dynamic axial crushing were conducted and an enhancement coefficient was introduced to account for the inertia effects for aluminum alloy AA6060 T4. The analytical solutions show an excellent agreement with the numerical results. It was found that the crisscross part was the most efficient component for energy absorption and the energy absorption efficiency of a single-cell column can be increased by 50% when the section was divided into 3×3 cells. Finally, the proposed method was extended to analyze the plateau stress of square cell honeycomb subjected to out-plane axial crushing and to some extent validate the mechanical insensitivity of honeycomb to cell size.

Structural Behavior of Thin- and Thick-Walled Composite Blades with Multicell Sections

A mixed beam approach that combines both the stiffness and the flexibility formulation in a unified manner has been performed to model and analyze coupled composite blades with closed, multiple-celled cross sections. The analysis model includes the effects of elastic couplings, shell wall thickness, transverse shear deformation, torsion warping, and constrained warping. Reissner's semicomplementary energy functional is used to derive the beam force-displacement relations. The influence of the shell bending strain measures as well as the membrane strain measures are incorporated in the formulation. For completeness required in a rigorous beam theory, four separate continuity conditions are imposed on each cell of the closed, multicelled sections. The theory is validated against experimental test data, detailed finite element analysis results, and other analytical results found in the literature for coupled composite beams and blades with various cross sections. These include two-cell box beams with bending-torsion and/or extension-torsion couplings and bending-torsion coupled composite blades with two-cell airfoils. The correlation between the present theory and other methods is found to be good, dependent on the geometries and material distributions adopted in the blades. Numerical results showing the effects of including the shell bending strain measures are examined. The effects of inappropriate treatment of the direction of integration for two-celled composite beams are also investigated in the current framework of the analysis.

A Study on Assessment of Composite Couplings for Helicopter Rotor Blades with Multi-cell Sections

In this work, a closed-form analysis is performed for the structural response of coupled composite blades with multi-cell sections. The analytical model includes the effects of shell wall thickness, transverse shear, torsion warping and constrained warping. The mixed beam approach based on Reissner's semi-complementary energy functional is used to derive the beam force-displacement relations. The theory is validated against experimental test data and other analytical results for coupled composite beams and blades with single-cell box-sections and two-cell airfoils. Correlation of the present method with experimental results and detailed finite element results is found to be very good.

Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption

The axial crushing of hollow multi-cell columns were studied analytically and numerically. A theoretical solution for the mean crushing force of multi-cell sections were derived, and the solution was shown to compare very well with the numerical predictions. Numerical studies were also carried out on foam-filled double-cell and triple-cell columns. Based upon the numerical results, closed-form solutions were derived to calculate the mean crushing strength of these sections. It was found that the interaction effects between the foam core and the column wall contribute to the total crushing resistance by the amounts equal to 140% and 180% of the direct foam resistance for double cell and triple cell respectively. Finally, the relative merits of single-cell, multi-cell and foam-filled sections were discussed.

Automatic analysis of multicell thin-walled sections

An automatic procedure is outlined for the determination of the shear centre and the evaluation of the overall state of stress in multicell thin-walled sections subject to axial force, bending moment, shearing force and torque. Graph theory is shown to be the rationale to establish a topological model of the section which is preliminary to a computer implementation of the shear stress analysis. Specifically, we exploit the main features of the Depth-First-Search graph algorithm in order to automatically determine a number of independent circuits equal to the degree of connection m of the section. The algorithm also localize the m slits which make the section open, a preliminary step for the analysis of multicell sections subject to a shearing force. Further, the evaluation of the first elastic area moment at any point of the open section is addressed by means of the Open-Section-Cut algorithm elaborated in this paper. The outlined procedure entails a considerable simplification of the analysis, since the geometrical data which need to be assigned are only the strictly necessary ones, namely the coordinates of the vertices, the branches connecting them and their thickness. A numerical example, carried out for a ship's hull by means of a computer program written in Mathematica, shows the effectiveness of the proposed approach.

Network Analysis of Thin-Walled Sections in Uniform Torsion

Network theory is applied to the problem of Saint-Venant torsion in thin-walled multi-cell sections by treating the section as a network of interconnected limbs. The method has considerable advantages over traditional methods of calculation since it exploits the topological independence of a network from its physical characteristics. Only routine matrix operations are required to set up and solve a system of linear algebraic equations. Four distinct network methods are identified—circuit, cutset, node and face analyses. Each require the solution of far fewer equations than the total number of problem variables. The appropriate method may be selected to minimize the number of equations.