Thin-walled Box Beams Research Articles

Vibration and buckling analysis of porous metal foam thin-walled beams with closed section

This paper investigates the vibration and buckling analysis of porous metal foam thin-walled box beams. These beams exhibit a unique structural configuration with symmetrical and asymmetrical porosity distributions along their wall thickness, thereby altering the effective mechanical properties. The first-order shear deformable beam theory is employed and the governing equations are derived using the Hamilton’s principle. Numerical results are presented for porous metal foam thin-walled box beams under simply-supported, clamped–clamped and clamped-free boundary conditions. The effects of various porosity parameters, length-to-side and side-to-wall-thickness ratios on the beams’ performance are also examined. A comprehensive comparison between the porous metal foam thin-walled box beams and their counterparts in the form of equivalent homogeneous are presented.

In-plane and out-of-plane free vibration analysis of thin-walled box beams based on one-dimensional higher-order beam theory

In this paper, the accurate free vibration characteristics of thin-walled box beam are discussed and the kinematic model is established by using fine shear deformation theory. The model adopts the in-plane and out-of-plane displacement fields including extension, torsion, warping and distortion, as well as the transverse shear due to bending and warping due to torsion. One-dimensional high-order beam theory is applied to the dynamic solution of thin-walled box beam, and the analytical solution of high free vibration modes of multi-deformation coupled modes under different boundary conditions is derived. The results show that warping, distortion and shear deformation play an important role in the free vibration characteristics of thin-walled box beam, and the validity of the model is verified. In addition, finite element software (ANSYS) is used for finite element simulation. The application of vibration mode in the structure design of thin-walled box beam is summarized, especially in the case of higher natural frequency. The calculation method is in good agreement with the finite element results.

Bimetallic Thin-Walled Box Beam Thermal Buckling Response.

A beam model for thermal buckling analysis of a bimetallic box beam is presented. The Euler-Bernoulli-Vlasov beam theory is employed considering large rotations but small strains. The nonlinear stability analysis is performed using an updated Lagrangian formulation. In order to account for the thermal effects of temperature-dependent (TD) and temperature-independent (TID) materials, a uniform temperature rise through beam wall thickness is considered. The numerical results for thin-walled box beams are presented to investigate the effects of different boundary conditions, beam lengths and material thickness ratios on the critical buckling temperature and post-buckling responses. The effectiveness and accuracy of the proposed model are verified by means of comparison with a shell model. It is revealed that all of the abovementioned effects are invaluable for buckling analysis of thin-walled beams under thermal load. Moreover, it is shown that the TD solutions give lower values than the TID one, emphasizing the importance of TD materials in beams.

Thermal buckling analysis of thin-walled closed section FG beam-type structures

In this paper, thermal buckling analysis of thin-walled, functionally graded (FG) beam-type structures with closed cross-sections and various boundary conditions is investigated. One dimensional finite element model, based on Euler–Bernoulli–Vlasov​ theory, is employed under the assumptions of large rotations and small strains. Two cases of temperature rise across the thickness of the cross-section walls are considered, which are uniform and linear. Stability analysis has been performed in eigenvalue manner and in load–deflection manner using Newton–Raphson method. Numerical results for thin-walled box beams and trapezoidal beams for two different FG materials configurations are presented to investigate the effects of temperature distribution, boundary conditions, and material properties on the critical buckling temperature and global buckling behaviour. The accuracy and reliability of the beam model is verified by comparing it with several benchmark examples. Good agreement was observed, showing possibility of further application of the proposed model which requires reduced computational time and provides greater modelling flexibility compared to commercial codes.

Internal instability of thin-walled beams under harmonic moving loads

Frequent resonance may result in excessive vibrations and endanger the safety of bridges. Firstly, the governing equations of motion are derived for the vertical, lateral and torsional vibrations of a mono-symmetric thin-walled box beam under a harmonic moving load. Then the resonance conditions are derived by letting the denominator of the response of concern equal to zero. For the mono-symmetric beam, the vertical resonance is uncoupled, but the torsional and lateral resonances are coupled. When in vertical resonance, the vertical motion of the beam will diverge by growing to a maximum during the acting period of the moving load. Similar phenomenon exists for the torsional–flexural resonance. A specific case occurs when the above two resonance frequencies coincide, for which the critical length of the beam can be determined. The phenomenon of simultaneous resonance is a kind of internal instability featured by the fact that the beam can transform repetitively from the vertical mode to the torsional–flexural mode, and vice versa, with no additional energy input. To avoid the inherent internal instability, a beam should be designed with a length not equal or close to the critical length presented in this paper. All the aforementioned resonances have been extensively studied for cases involving both harmonic and random moving loads and validated by the finite element analysis.

Bending degradation of thin-walled box beams made of aluminum omega profile and GFRP panel connected by mechanical fasteners

The paper presents the results of laboratory tests and numerical simulations for thin-walled beams subjected to 3-point bending (3-PB). The beams were made of two parts: an aluminum (7075) omega profile and a GFRP (glass fiber reinforced plastic) panel, which were connected by rivets. The paper focuses on the analysis of deformation states and gradual degradation in these elements. The modeling is described by parameters as the number of rivets, the height of the aluminum profile, and the stiffness of the composite which strongly influence the failure force. The experiments were carried out for two settings of beams on supports, in which the composite panel was located on the side of the supports or on the side of the load acting on the beam axis. The laboratory tests allowed to confirm the correctness of the numerical model. As a result of numerical simulations, the most optimal material and geometric configuration were determined, taking into account the maximum carrying criterion force at 3-PB. The effort of the composite material was also analyzed based on the Tsai-Hill hypothesis.

Shear lag including axial balance of box beams by finite segment model

A finite segment element including axial balance is formulated to describe shear lag in thin-walled box beams having constant or variable cross sections made from steel or other materials. The axial balance neglected in the conventional finite segment element model (CFSM) is enforced by adding the nodal longitudinal displacements, while shear lag and shear deformation are incorporated using the nodal shear lag functions and rotations, respectively. The homogeneous solutions deduced by the analytical method are utilized for constructing the element shape functions. By invoking the minimum potential energy theorem, the stiffness matrix and the equivalent nodal force vector are then derived for the element. The precision of the proposed finite segment model (PFSM) is verified against the results yielded from the solid finite element model (SFEM), the finite strip model (FSTM), and the experiments. A continuous box beam having varying cross sections is chosen for comparing the neutral axis depth to the centroidal axis depth. Subsequently, the influence of the axial balance on the mechanical behavior is evaluated. Moreover, the effects of three major geometric parameters are discussed for stress analysis. The results reveal that the proposed finite segment model is capable of reproducing the mechanical behavior of box beams having constant or varying cross sections, and that the stress analysis concerning the continuous box beam with variable cross sections is substantially affected by the axial balance condition.

Xiong'an Railway Station: A Supersized Railway Station in a High Seismic Intensity Zone

Xiong'an Railway Station is the first super-large elevated railway station located in a high seismic intensity zone. A reinforced concrete frame structural system is adopted for the rail-bearing floor and the lower floors, while a steel structural system is adopted for the large-span roof. H-shaped steel frame beams are adopted for the elevated waiting hall, while double flange H-shaped steel beams are adopted in areas subject to great forces. Plane trusses are arranged bidirectionally between the frame beams to facilitate the laying and maintenance of equipment pipelines. The main span of the elevated waiting hall is 78 m, and variable-height box beams are adopted. The platform canopies are supported by specially shaped steel pipe columns, and the roof frame beams are connected to the tops of columns through spherical bearings. Structural innovations have been carried out in terms of semi-embedded column bases, stepped wall thickness steel pipe columns with special section shapes, large-span stiffened thin-walled box beams, and bidirectional large displacement bearings.

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.

Higher-order Vlasov torsion theory for thin-walled box beams

Non-negligible sectional deformations, such as warping and distortion, occur in thin-walled beams under a twisting moment. For accurate analysis, these deformations need to be considered as additional kinematic degrees besides the degrees of freedom used in the classical St. Venant torsion theory. Vlasov pioneered to develop a higher-order beam theory for torsion that incorporates warping and distortion, but more sectional deformation modes than those considered in the Vlasov theory are needed to improve solution accuracy. Several theories were developed towards this direction, but no higher-order beam theory for torsion appears to allow explicit F-U and σ-F relations (U: kinematic variables, F: generalized forces, σ: stresses) as established by the Vlasov theory. In that the explicit relations are useful to interpret the physical significance of the generalized forces and can be critical in deriving explicit equilibrium conditions among the generalized forces at a joint of multiple thin-walled beams, a theory allowing the explicit relations needs to be developed. In this study, we newly propose a higher-order Vlasov torsion theory that not only includes as many torsion-related modes as desired but also provides the explicit F-U and σ-F relations that are fully consistent with those by the Vlasov theory. Towards this direction, we show that expressing the σ-U relation only with sectional mode shapes orthogonal to each other is critical in establishing explicit F-U and σ-F relations. We then establish new recursive relations that can be used to express each of derivatives for the sectional mode shapes involved in the σ-U relation as a linear combination of other orthogonal sectional mode shapes. In the developed theory, even stresses at off-centerline positions of the beam cross-section are explicitly related to F. The validity and accuracy of the proposed theory are confirmed by examining displacements, stresses, and eigenfrequencies for several torsion problems. The numerical results by the proposed theory are in good agreement with those by the shell analysis.

Nonlinear Buckling Analysis of Thin-Walled Box Beams considering Shear Lag

In this work, a general geometric nonlinear model of straight thin-walled box beams (STBBs) under combined eccentric and axial loads is established. In order to accurately reflect the behavior of STBB, the additional shear lag warping is added to enrich the displacement field. It is necessary to define the section shape function to describe the local section deformation. Therefore, extension, bending, torsion, distortion, and shear lag effects are expressed by the generalized coordinate method. Based on the stability of transverse unconstrained box beam theory, meaningful higher-order solutions can be obtained by defining a set of coupled deformation modes. The equilibrium equation is discretized by the Galerkin method, and the Newton–Raphson incremental method is used to derive and solve the nonlinear governing equations. On this basis, the analytical expression of stiffness matrix is established. For solving the stability problem, the effectiveness of the proposed method is verified by comparing the calculation results of shell element (Ansys) with other theories. Numerical examples even show that the proposed method can not only get the influence of shear lag but also obtain the variation of lateral buckling of the beam model.

Consistent higher-order beam theory for thin-walled box beams using recursive analysis: Edge-bending deformation under doubly symmetric loads

Thin-walled box beams generally exhibit complex sectional deformations that are not significant in solid beams. Accordingly, a higher-order beam theory suitable for the analysis of thin-walled box beams should include degrees of freedom representing sectional deformations. In a recent study, a recursive analysis method to systematically derive sectional membrane deformations has been proposed to establish a consistent higher-order beam theory. In this study, another recursive analysis method is proposed that is suitable for the closed-form derivation of new sectional bending deformations representing the bending of edges (or walls) of the cross-section shown in a box beam under doubly symmetric loads. A consistent 1D higher-order beam theory appropriate to include these additional deformation modes as beam degrees of freedom is also established. The proposed theory provides explicit formulas that relate stresses to generalized forces including self-equilibrated forces such as bimoments. Furthermore, sectional modes are hierarchically derived so that the level of solution accuracy can be effectively and systematically controlled. Thus, the accuracy for static displacement/stress calculations and eigenfrequencies can be adjusted to be fully comparable with plate/shell results. When general doubly symmetric loads are applied to a box beam, the edge membrane modes derived in an earlier study can also be used as additional degrees of freedom besides the edge-bending modes derived in this study. The validity of the proposed beam approach is verified through the analyses of static displacements and stresses as well as eigenfrequencies for free vibration problems.

Damage of thin-walled box beams with omega cross-section joined by adhesive layers subjected to bending deformation process

The subject of this study is aluminum box beams of omega cross-section, consisting of an omega profile and connected to the metal by the adhesive interface of the area 120 cm2. In the laboratory tests were used specimens made from 7075 aluminum with dimensions 300 × 84 × 36 mm. To connect the components of the specimens an adhesive layer Loctite 9545 was used.The samples were subjected to 3-point bending (3-PB). Laboratory tests were conducted using a MTS (25kN) and a Digital Image Correlation (DIC) system, in order to accurately determine states of deformation of the box beams. With the load increase the profiles deform elastically at the beginning and then in the final stage of deformation process, a collapse of the side walls and local plastic deformation occured. This causes a sudden drop of the load capacity of the entire profile.To complete the experimental analysis, the numerical model of the thin-walled box beam was created. The model allows for observation of stress and displacement fields variation in the bending element, as well as places of their concentration. Moreover, a detailed analysis of damage growth and the final destruction of the connecting elements associated with the load increase was performed. Numerical simulations were done using ABAQUS code. The FEM analysis results confirm correctness of the assumptions in the numerical model and allow for a more detailed description of the process decohesion adhesive layer and the damage of aluminium elements of the thin-walled box beams.

Consistent higher-order beam theory for thin-walled box beams using recursive analysis: Membrane deformation under doubly symmetric loads

We propose a consistent higher-order beam theory in which cross-sectional deformations defining degrees of freedom are derived in the framework consistent with the mechanics of the proposed one-dimensional beam theory. This approach contrasts with earlier methods in which the procedure used to derive sectional deformations and the final beam theory are based on models of different levels. An advantage of the proposed consistent approach is that the generalized force-stress relation even for self-equilibrated forces such as bimoments can now be explicitly written. Also, sectional deformations can be systematically derived in closed form by the recursive and hierarchical approach. Accordingly, the accuracy in both displacement and stress can be adjusted so that obtained results are fully comparable with plate/shell results. We mainly conduct analysis of membrane deformations occurring in thin-walled box beams subjected to doubly symmetric loads such as axially-loaded forces. This case is elaborately chosen to better explain the fundamental concepts of our newly proposed approach. A brief description is also provided to show that these concepts are applicable to other types of loads such as bending and torsion. We confirm the accuracy of the theory proposed here by calculating stress and displacement in several examples.

Analysis of flexural natural vibrations of thin‐walled box beams using higher order beam theory

SummaryIn order to study the influence of high‐order shear deformations and shear lag on the dynamic characteristics of thin‐walled box beams (TWBBs), this paper takes the Hamilton principle as a basis to consider multiple factors such as high‐order shear deformations, shear lag, and cross section rotary inertia of the TWBBs. The vibration differential equations and natural boundary conditions of TWBBs are deduced. On the basis of eight examples of TWBBs with different boundary conditions and span–width ratios, analytical results of this paper are compared with those of the ANSYS finite element method. Both results are in good agreement with each other, and the validity of the calculation method is verified. The effects of higher order shear deformations and shear lag on natural vibration characteristics of TWBBs are analyzed. And some meaningful conclusions are drawn: This theory shows the capability to accurately describe both higher order deformations and shear lag; when the span–width ratio is small, neglecting higher order web deformations will produce a large calculation error; under the action of shear lag, the natural vibration frequency of TWBBs decreases greatly, which cannot be neglected; and both the high‐order shear deformations and shear lag effect increase with increasing mode order and increase with decreasing span–width ratio.

Buckling analysis of thin-walled box beams under arbitrary loads with general boundary conditions using higher-order beam theory

When a higher-order or generalized beam theory is used for the buckling analysis of thin-walled beams, the analysis accuracy critically depends on the number and shapes of the cross-sectional modes associated with warping and distortion. In the study, we propose to use the hierarchically-derived cross-sectional modes consistent with the higher-order beam theory for the analysis of pre-buckling stress and buckling load. The proposed formulation is applicable to any box beams subjected to arbitrary loads and general boundary conditions. We demonstrate the effectiveness of the proposed method by performing buckling analyses for axial, bending, torsional, and general loadings. Length-to-height ratios of the beams are also varied from 1 to 100. If up to fifty cross-sectional and rigid-body modes are employed, the calculated buckling loads are found to match favorably those predicted by the shell finite element analysis. In that a unified buckling analysis under general loads is developed for box beams, the present study is expected to contribute towards new possibilities for the efficient buckling analysis of more general box beam structures involving several joints.

Non-linear lateral buckling analysis of unequal thickness thin-walled box beam under an eccentric load

A non-linear theory for straight thin-walled box beam (STBB) subjected to an eccentric force, considering the distortional deformation and unequal thickness of the webs, is developed to determine the buckling capacity in this paper. Due to the coupling of warping, torsion and distortion of the cross-section, the kinematic model becomes considerably complicated. The theory considers the geometric non-linear coupling caused by the non-linear terms in each deformation mode under the generalized displacement coordinate. The non-linear differential equations is discretized by Galerkin's method, which has been shown that the interaction effects can be successfully captured between the pre-buckling deflections and geometric non-linearity. The post-buckling response of the present beam model is compared with the results of the finite element simulation (Ansys15.0) and F.Mohri's method (2002), which appears to be very conservative. The formulation is solved exactly by checking the effect of the variable parameters: the width-to-height ratio of the cross-section, the load height and the thickness of the webs. The viability and effectiveness of this method has been established by comparing with other existing numerical solutions.

Effect of Shear and Distortion Deformations on Lateral Buckling Resistance of Box Elements in the Framework of Eurocode 3

This paper treats the distortional and shear deformation effects on the elastic lateral torsional buckling of thin-walled box beam elements, under combined bending and axial forces. For the purpose, a nonlinear kinematic model based on higher order theory is used applicable to both short and long thin-walled box beams. Because in the kinematic model of the higher order theory integrates additional flexibility terms related to shear, distortion and warping effects, it accurately predicts the lateral torsional buckling of the straight box beams. Ritz’s method is adopted as solution strategy in order to obtain the nonlinear governing equilibrium equations, then the buckling loads are computed by solving the eigenvalue problem basing on the singularity of the tangential stiffness matrix. Owing to flexural–torsional and distortional couplings, new matrices are obtained in both geometric and initial stress parts of the tangent stiffness matrix. The proposed method with the new stiffness terms, is efficient and accurate in lateral torsional buckling predictions, when compared with the commercial FEM code ABAQUS results. Based on the existing European guidelines EC3, an extensive numerical investigation is performed to demonstrate the effects of both shear and distortional deformations on the moment carrying capacity. The convenience of the model is outlined and the limit of models developed without shear and distortion deformation effects on lateral buckling loads evaluation is discussed.

Higher-order beam theory for static and vibration analysis of composite thin-walled box beam

A higher-order beam theory suitable for accurate analysis of composite thin-walled box beams is developed. Because anisotropic and laminate effects in composite beams produce deformation patterns that do not appear in isotropic beams, accurate analyses for composite beams require elaborately defined sectional shape functions that describe local cross-sectional deformations. Here, we present a new systematic method to define these functions and establish a one-dimensional finite element based on a higher-order beam theory for composite thin-walled box beams. The validity of the developed approach is checked by solving static and eigenvalue problems with composite thin-walled box beams, and by comparing the obtained numerical results with those obtained by ABAQUS shell elements.

Experimental and numerical studies of a cracked thin-walled box-beams

This paper discusses the problem of detection and location of cracks in a cantilever beam. To solve this problem, a damage index based on the modal displacement method is applied in both experimental and numerical studies. Thin-walled composite box-beams with rectangular cross-section are tested. Two types of beam configuration are examined: circumferentially asymmetric stiffness (CAS) and circumferentially uniform stiffness (CUS). The CAS configuration is characterized by the coupling between bending in a flexible direction and torsion, while the CUS configuration is characterized by coupled bending in two perpendicular directions (flexible and stiff). Numerical models of the tested beams are made by the finite element method. A series of simulations are performed for different damage locations and ply orientations. The Lanczos method is used to solve the eigenvalue problem. Natural frequencies and free vibration modes of the beams are determined. To validate the numerical models, a test stand is build. The test specimens are made of unidirectional carbon-epoxy prepreg by the autoclave technique. The damage of the reinforcement fibres is induced by making a crack in the beam using a circular saw. Selected numerical results are validated with two different experimental methods. First, the experimental test is conducted using a non-contact laser vibrometer provided with a scanning head and a data acquisition-visualization unit. Next, a modal analysis is performed by the contact method using the LMS analyzer with a modal hammer and an acceleration sensor. All numerical and experimental results of the beams with cracks are compared with the findings obtained for the specimens without any damage.