Thin-walled Cross-section Research Articles

On the stability of magnetically levitated rotating rings

The dynamic stability of rotating elastic circular rings subject to magnetic levitation and radially recentering magnetic forces is studied. A geometrically exact model of elastic rings deforming in space is formulated in the context of the special Cosserat theory of curved rods. A Lagrangian description of the motion is obtained with respect to the rotating frame. The equations of motion are transformed into a set of ODEs according to a Faedo–Galerkin discretization. The effects of the magnetic and gyroscopic forces on the equilibrium states of the ring are investigated together with the loss of stability of the low-frequency modes. Circular elastic rings with open and closed thin-walled cross sections (i.e., L-shaped and boxed cross sections) are considered. The loss of stability occurring at critical angular speeds where the critical modes become unstable is proved to depend on the ring stiffness and cross-sectional symmetry/asymmetry properties.

Solving nonlinear equilibrium equations of deformable systems by method of embedded polygons

ABSTRACTSolving of nonlinear algebraic equations is an obligatory stage of studying the equilibrium paths of nonlinear deformable systems. The iterative method for solving a system of nonlinear algebraic equations stated in an explicit or implicit form is developed in the present work. The method consists of constructing a sequence of polygons in Euclidean space that converge into a single point that displays the solution of the system. Polygon vertices are determined on the assumption that individual equations of the system are independent from each other and each of them is a function of only one variable. Initial positions of vertices for each subsequent polygon are specified at the midpoints of certain straight segments determined at the previous iteration. The present algorithm is applied for analytical investigation of the behavior of biaxially compressed nonlinear-elastic beam-column with an open thin-walled cross-section. Numerical examples are made for the I-beam-column on the assumption that its material follows a bilinear stress-strain diagram. A computer program based on the shooting method is developed for solving the problem. The method is reduced to numerical integration of a system of differential equations and to the solution of a system of nonlinear algebraic equations between the boundary values of displacements at the ends of the beam-column. A stress distribution at the beam-column cross-sections is determined by subdividing the cross-section area into many small cells. The equilibrium path for the twisting angle and the lateral displacements tend to the stationary point when the load is increased. Configuration of the path curves reveals that the ultimate load is reached shortly once the maximal normal stresses at the beam-column fall outside the limit of the elastic region. The beam-column has a unique equilibrium state for each value of the load, that is, there are no equilibrium states once the maximum load is reached.

Free flexural vibration of symmetric beams with inertia induced cross section deformations

Beams with large thin-walled cross sections are not generally following the classical beam theories such as Euler-Bernoulli and Timoshenko theories. In free vibration, the cross section is deformed by the inertia induced body loads. These deformations may have significant effect on the beam modal frequencies, especially in applications involving non-structural masses. This paper presents a method to include the effect into vibration modal results obtained by the classical beam theories. Generalized mass and stiffness of the classical results are modified according to kinetic-, and strain energies of the cross section deformation. The method is validated in typical engineering case studies against fine mesh Finite Element Method and excellent agreement is found.

Staff Radiation Exposure Reduction

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the cross-section principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

Laser beam welding of atmosphere aluminum die cast material using high frequency beam oscillation and brilliant beam sources

In the serial production of components for automotive applications such as cooling and air-conditioning systems, aluminum die-cast materials are frequently used due to their excellent castability. The aim of providing light weight components can be approached with thin walled cross sections even for complex structural parts. However, cast components are usually connected to semifinished products such as profiles or tubes. The connections have to be mostly pressure tight. The joining technique for these applications has to be highly productive to obtain high component outcome and cost-efficient. Laser beam welding techniques are especially suitable for these tasks. Die-cast components have limited or no weldability due to their manufacturing process. This is due to entrapped gases within pores or cavities under high pressure conditions. Furthermore, the mold release agents for the die-cast process are inappropriate for obtaining homogeneous and sound weld seams. Consequently, this results in a larger number of pores in the weld seam and stochastic melt pool blow-outs, which prohibit mostly the use of the component. To solve these issues, a new welding technique, remoweld®T, has been developed at Fraunhofer IWS. This unique method has been extensively tested and used for serial-production. The decisive step was to use laser sources with brilliant beam quality in combination with a high frequency beam oscillation within the melt pool. In this paper, the technological approach will be presented. With the remoweld®T method, it was possible to obtain homogeneous weld seams with low porosity and a strongly reduced distortion for the first time. Minor component tolerances and a reproducible joining technique with a high output for serial production can be achieved.

Boundary-integral-based process for calculating stiffness matrices of space frame elements with axially varying cross section

This paper presents a strategy to directly compute the stiffness matrix of 3D (space) frame elements having arbitrary cross sections and generic rigidity variation along their axes. All the necessary section properties are determined by means of formulations based purely on boundary integrals. To determine the torsional constant and the torsion center, this strategy applies the Boundary Element Method (BEM). To model thin-walled cross-sections, the strategy calls for activating integration algorithms devised specifically to deal with the nearly singular integrals involved. To express all other section properties (i.e. area, first and second moments of area, and the shear form factors) in terms of boundary integrals, the strategy employs Green's theorem. The existing boundary-element meshes, used to determine the torsion constants, are employed to evaluate the corresponding boundary integrals. In applying the proposed strategy – the pure boundary-integral-based process (PBIP) – we consider space frame elements with geometrically complex cross-sections varying along their axes.

Finite element modeling of operation for thin-walled open cross section bars to analyze plate-rod systems

We solve the problem of increasing the accuracy of finite-element analysis for the strains of thin-walled open cross section bars eccentrically reinforcing the flat sheet.

Mechanical and Parametric Analysis of Cracks in Polypropylene Fiber Concrete U-Shaped Girder

The U-shaped girder is a type of open thin-walled structure, which is used in urban rail transit engineering. Although it employs polypropylene fiber concrete to avoid cracks, the girder is still easier to crack than other traditional structures owing to its special open thin-walled cross section. In this study, a cracking accident of a U-shaped girder, which happened during the prestressed steel tensioning, was studied by field investigation and mechanical analysis through the finite element (FE) method. An outline of the cracks was presented. The nonlinear material properties of the polypropylene fiber concrete and steel were discussed and used in the finite element model. The effects of the main design parameters, such as the flange slab thickness, anchorage position, and prestressed steel layout, were evaluated based on the results of the FE analysis. The results indicate that the extremely low rigidity of the web and oversize of the web longitudinal prestressed steels are the main reasons for the cracks. The risk of cracks can be reduced by increasing the thickness of the flange slab and changing the anchorage position and prestressed steel layout. Some suggestions are provided for avoiding cracks based on the results of the research.

Bending of thin-walled beams of open section with influence of shear—Part II: Application

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the cross-section principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

WITHDRAWN: Boundary-integral-based process for calculating stiffness matrices of space frame elements with axially varying cross section

Abstract This paper presents a strategy to directly compute the stiffness matrix of 3D (space) frame elements having arbitrary cross sections and generic rigidity variation along their axes. All the necessary section properties are determined by means of formulations based purely on boundary integrals. To determine the torsional constant and the torsion center, this strategy applies the Boundary Element Method (BEM). To model thin-walled cross-sections, the strategy calls for activating integration algorithms devised specifically to deal with the nearly singular integrals involved. To express all other section properties (i.e. area, first and second moments of area, and the shear form factors) in terms of boundary integrals, the strategy employs Green's theorem. The existing boundary-element meshes, used to determine the torsion constants, are employed to evaluate the corresponding boundary integrals. In applying the proposed strategy – the pure boundary-integral-based process (PBIP) – we consider space frame elements with geometrically complex cross-sections varying along their axes.

Design variation of thin-walled composite beam cross-section properties

Purpose The purpose of this paper is to present a design sensitivity analysis continuum formulation for the cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, on shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. Design/methodology/approach In order to determine its properties, the cross-section geometry is modeled by quadratic isoparametric finite elements. For design sensitivity calculations, the cross-section is modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the design elements may coincide with the laminates that constitute the cross-section. Findings The developed formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties depending on warping. The lamina orientation and the laminate thickness are selected as design variables. Originality/value The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections.

Nonlinear Collapse of General Thin-Walled Cross-Sections Under Pure Bending

Thin-walled beams exhibit a nonlinear response to bending moments due to the progressive flattening of the cross-section, a behavior commonly referred to as the Brazier effect. Most approaches to model this effect are limited to either circular cross-sections or to cross-sections made of isotropic materials. This article proposes an efficient two-step method of predicting the nonlinear collapse of thin-walled cross-sections of arbitrary geometry with isotropic and orthotropic materials. The procedure relies on representing the cross-section by two-dimensional nonlinear corotating beam elements with imposed in-plane loads proportional to the curvature, combined with a finite strip buckling analysis based on the deformed cross-section. By comparison with existing analytical and numerical modeling approaches, it is demonstrated that the present method can capture the cross-section flattening and critical moment for buckling of thin-walled structures commonly found in the industry.

Computation of Thin-Walled Cross-Section Resistance to Local Buckling with the Use of the Critical Plate Method

Abstract Thin-walled bars currently applied in metal construction engineering belong to a group of members, the cross-section res i stance of which is affected by the phenomena of local or distortional stability loss. This results from the fact that the cross-section of such a bar consists of slender-plate elements. The study presents the method of calculating the resistance of the cross-section susceptible to local buckling which is based on the loss of stability of the weakest plate (wall). The “Critical Plate” (CP) was identified by comparing critical stress in cross-section component plates under a given stress condition. Then, the CP showing the lowest critical stress was modelled, depending on boundary conditions, as an internal or cantilever element elastically restrained in the restraining plate (RP). Longitudinal stress distribution was accounted for by means of a constant, linear or non-linear (acc. the second degree parabola) function. For the critical buckling stress, as calculated above, the local critical resistance of the cross-section was determined, which sets a limit on the validity of the Vlasov theory. In order to determine the design ultimate resistance of the cross-section, the effective width theory was applied, while taking into consideration the assumptions specified in the study. The application of the Critical Plate Method (CPM) was presented in the examples. Analytical calculation results were compared with selected experimental findings. It was demonstrated that taking into consideration the CP elastic restraint and longitudinal stress variation results in a more accurate representation of thin-walled element behaviour in the engineering computational model.

LINEAR AND NONLINEAR THERMOELASTIC ANALYSES OF BEAMS UNDER TEMPERATURE GRADIENT

In a structure, beams are often connected with other members such as columns, which provide considerable restraints against the rotation and extension of the beam ends. When a beam is subjected to an in-plane temperature gradient field, the temperature gradient tends to change the curvature of the beam in the transverse direction and expand the beam in the axial direction. The restrained actions will produce bending moments and compressive forces in the beam, which increase with an increase of the temperature differential and average temperature of the temperature gradient field. When these actions reach critical values, the elastically restrained beam may bifurcate from its primary in-plane equilibrium state to a lateral-torsional buckled equilibrium configuration. This paper carries out linear and nonlinear thermoelastic analyses of an elastically restrained beam of doubly symmetric open thin-walled cross-section that is subjected to a linear temperature gradient field over its cross-section. It is found that geometric nonlinearity influences the thermoelastic responses of the beam to the temperature changes significantly. The influence decreases with an increase of the stiffness of the elastic restraints.

Linearized buckling analysis of isotropic and composite beam-columns by Carrera Unified Formulation and dynamic stiffness method

This article introduces a one-dimensional (1D) higher-order exact formulation for linearized buckling analysis of beam-columns. The Carrera Unified Formulation (CUF) is utilized and the displacement field is expressed as a generic N-order expansion of the generalized unknown displacement field. The principle of virtual displacements is invoked along with CUF to derive the governing equations and the associated natural boundary conditions in terms of fundamental nuclei, which can be systematically expanded according to N by exploiting an extensive index notation. After the closed form solution of the N-order beam-column element is sought, an exact dynamic stiffness (DS) matrix is derived by relating the amplitudes of the loads to those of the responses. The global DS matrix is finally processed through the application of the Wittrick-Williams algorithm to extract the buckling loads of the structure. Isotropic solid and thin-walled cross-section beams as well as laminated composite structures are analyzed in this article. The validity of the formulation and its broad range of applicability are demonstrated through comparisons of results from the literature and by using commercial finite element codes.

GBT deformation modes for curved thin-walled cross-sections based on a mid-line polygonal approximation

This paper addresses the determination of deformation modes for curved thin-walled cross-sections through the polygonal approximation of the cross-section mid-line, in the framework of Generalised Beam Theory (GBT). A new GBT cross-section analysis procedure is proposed, which discretises the geometry independently of the number of cross-section DOFs considered to obtain the deformation modes. This procedure is more efficient than the classic GBT one, since curved geometries may be accurately described without increasing the number of deformation modes. In particular, polygonal sections with rounded corners can be easily handled. For illustrative purposes, the procedure is applied to several cross-sections with curved walls and it is shown that it leads to accurate results with coarse cross-section DOF discretisations.

薄肉複合部材の動的崩壊挙動 -充填効果係数による低変形速度域におけるエネルギ吸収量の推定-

In the traffic machines such as the automobile, many thin wall closed section members made of steel are used for the improvement in the collision safety. In these members, the local buckling under compressive stress is mentioned as a main problem. In order to improve the mechanical performance of the member, the technique which fills the inside of closed section with the low-density foaming material is used. In this study, axial compression test and 3-point bending test at the various low deformation velocities from elastic region to plastic collapse were carried out on the composite material which filled the hat type thin-walled cross section member with 2 kinds of epoxy resin foaming material in which the foaming rate (density) is different. Characteristics of the energy absorption for composite member were estimated from the stress contribution ratio of the constructional member. This technique is effective as a simplified estimation method for characteristics of composite member in the design.

Free vibration analysis of simply supported beams with solid and thin-walled cross-sections using higher-order theories based on displacement variables

Solutions for undamped free vibration of beams with solid and thin-walled cross-sections are provided by using refined theories based on displacement variables. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. Subsequently, the differential equations of motion and the natural boundary conditions are formulated in terms of fundamental nuclei by using CUF and the strong form of the principle of virtual displacements. The second-order system of ordinary differential equations is then reduced into a classical eigenvalue problem by assuming simply supported boundary conditions. The proposed methodology is extensively assessed for different solid and thin-walled metallic beam structures and the results are compared with those appeared in published literature and also checked by finite element solutions. The research demonstrates that: (i) the innovative 1D closed form CUF represents a reliable and compact method to develop refined beam models with solely displacement variables; (ii) 3D-like numerically exact solutions of complex structures can be obtained with ease; and (iii) the numerical efficiency of the present method is uniquely robust when compared to other methods that provide similar accuracies.

Accurate static response of single- and multi-cell laminated box beams

This paper is devoted to the static analysis of laminated beams with both compact and thin-walled cross-sections. The kinematic models are obtained by means of the Carrera Unified Formulation (CUF), which is a hierarchical formulation leading to very accurate and computationally efficient finite element (FE) models. According to the latest developments in the framework of CUF, it is possible to easily adopt both equivalent-single-layer and layer-wise approaches, by expanding the unknown kinematic variables on the beam cross-section with either Taylor-like or Lagrange-like polynomials, respectively. A number of laminated beam structures are analysed and particular attention is given to laminated single- and multi-cell cross-section beams with open and closed contours. Moreover, in order to demonstrate the effectiveness of the proposed refined elements, the results in terms of displacements and stresses are compared with solid FE solutions and, when possible, with the results available from the research literature.

An efficient 3D numerical beam model based on cross sectional analysis and Ritz approximations

A 3D beam model, i.e. a beam that may deform in space and experience longitudinal and torsional deformations, is developed considering Timoshenko's theory for bending and assuming that the cross section rotates as a rigid body but may deform in longitudinal direction due to warping. The cross sectional properties are firstly calculated and then inserted at the equation of motion. The beam is assumed to be with an arbitrary cross section, with linearly varying thickness and width, and with an initial twist. The model is appropriate for open and closed thin-walled cross sections, and also for solid cross sections. The objective of the current research is to demonstrate that complex beam structures can be modeled accurately with reduced number of degrees of freedom.