Torsional Warping Research Articles

A strain gradient quadrature element for geometrically exact beam with torsion warping

Based on the weak form quadrature element method, this paper proposes a geometrically exact strain gradient beam formulation incorporating torsion warping effects. This formulation provides a computationally efficient numerical scheme for analyzing size dependent arbitrarily cross-sectional beams undergoing large displacements and rotations. By accommodating all high order strain and strain gradient terms in the formulation, the Wagner effects that crucial to the model's accuracy are taken into account. The warping functions determined by cross-sectional governing equations involving strain gradients are obtained using a C1 quadrilateral quadrature element procedure. Five numerical tests are presented to demonstrate the validity and accuracy of the proposed formulation, while investigating the influences of size effects and Wagner effects on nonlinear behaviors of beam structures.

Application and Experimental Validation of Seven-Degree-of-Freedom Beam Element for Girder Bridges during Deck Construction

During bridge deck construction, the deck finishing machine and the fresh concrete often produce large vertical loads and torsional moments acting on the bridge girder system. In some cases, these loads can cause excessive vertical deflection and transverse rotation in the bridge girders, leading to many maintenance and safety problems, such as changes in deck thickness and local and global instabilities during construction. To minimize the potential problems caused by deck construction, the AASHOTO LRFD Bridge Design Specification requires consideration of these torsional moments during the design procedure, and a detailed three-dimensional finite element analysis may be conducted. However, for bridge girders with open-section thin-walled sections, only the solid or shell element can be used to recognize the warping of the girder since the torsional warping effect is not included in the classical beam element. In this research, a warping degree of freedom was added to a beam element, and a three-dimensional beam element with seven degrees of freedom (7-DOF) at each node was derived as an alternative method for analyzing girder bridges during deck construction. A computer program based on the 7-DOF beam element was also developed in MATLAB. To assess the 7-DOF beam element, one bridge was selected to measure the transverse rotation, vertical deflection, and stress of the exterior girder and the first interior girder during deck construction. Also, three full-scale numerical models using solid elements, classical three-dimensional beam elements, and 7-DOF beam elements were created based on the geometries and loads of the experimental bridge. A comparative study was conducted by comparing the results from the numerical models and experimental monitoring data to evaluate the 7-DOF beam element. The results showed that the 7-DOF beam element had excellent behavior in analyzing the girder bridges under construction load, especially in the torsional analysis of bridge girders. Also, unlike the solid element model, which also provided reasonable results, the 7-DOF beam element model can compute the internal forces of the cross-sections along the bridge, which allows the 7-DOF beam element to be an alternative approach for design and research requiring less modeling effort and computational complexity.

Cold-formed steel strength predictions for torsion and bending–torsion interaction

Locally slender cross-section members, such as cold-formed steel open sections, are susceptible to significant twisting and high warping torsion stresses. Torsion considerations are complicated by whether it is derived as a first-order effect from loading or a second-order effect from instability. Previous direct torsion experiments on lipped Cee sections have shown significant inelastic reserve. Furthermore, the current design for combined bending–torsion interaction has limitations, including only considering the first yield and ignoring the cross-section slenderness in torsion. Two parametric studies were conducted to predict both the torsion capacity and the combined bending–torsion interaction in locally slender cross-sections. Shell finite element analysis of lipped Cee and Zee sections for both the torsion only and the combined bending–torsion cases were created using a validated model. For the torsion-only study, various cross-section geometries, steel grades, and member lengths covering the range of practically expected torsional slenderness were investigated. A set of bimoment parameters, including yield, buckling, and plastic bimoments, were calculated and the ultimate bimoment was determined by shell finite element collapse analyses. A simple uniform equation predicting the bimoment capacity was adopted and two bimoment strength curves are proposed for local and distortional buckling-controlled cases respectively. For the combined bending–torsion case, various practical cross-sections and bracing conditions were investigated with various ratios of applied torsion and bending. Shell finite element buckling and collapse analyses were performed to determine the critical and ultimate moments and bimoments. It was found that the current AISI standard is conservative under most scenarios. Updated torsion–bending interaction equations incorporating bimoment and bending moments are proposed. The interaction equations are dependent on the cross-section, the direction of the applied torsion, and the bracing condition.

Generalized strain-based finite element for non-linear stability analysis of beams with thin-walled open cross-section

Based on the generalized strain theory a shear-deformable finite element is developed for nonlinear stability analysis of thin-walled open-section beams in multibody systems. In this formulation a finite number of deformations, characterized as generalized strains, is defined which are related to dual stress resultants in a co-rotational frame. The stiffness formulation is based on a second-order approximation of the local elastic displacement field. Timoshenko’s beam theory and Vlasov’s modified thin-walled beam theory are used to include the shear strain effects due to non-uniform bending and restrained warping torsion and their mutual coupling effects. Axial shortening associated with the Wagner Hypothesis is taken into account such that the nonlinear behaviour of the beam is predicted accurately, especially under large torsion. Coupling between bending and torsional deformation due to non-coincident centroid and shear centre is modelled using a second-order cross-section transformation matrix. Cubic Hermitian polynomials are used as shape functions for the lateral displacements and twist rotation to derive the elastic and geometric stiffness matrices. Geometric nonlinearities associated with axial elongation and bending curvatures are described by additional torsion, bending and warping-related quadratic deformation terms yielding a set of modified deformations. The inertia properties of the beam are described using both consistent and lumped mass formulations. The latter is used to model rotary and warping inertias of the beam cross-section. The accuracy and the computational efficiency of the new beam element is demonstrated in several static and dynamic examples.

Comparison between numerical models for unbraced multiple bay pallet racks

AbstractUnbraced pallet racks are cold‐formed steel structures intended for the storage of products. They are subjected to severe lateral actions, and are particularly instability prone in the down aisle direction. Therefore, their design is compromised and needs of an accurate modelling. In current design practice, models using simplified beam formulations are regularly considered (such as Euler‐Bernoulli or Timoshenko), being them not able to reproduce torsional warping, or cross‐sectional distortion; these phenomena are accounted for indirectly via reduced sectional properties, and in the post‐process. This paper presents a comparison between three single column models, intended for the modelling of racks with a significant number of bays. Three models are presented: –Two 2D beam models using gross and equivalent section properties –A 3D beam model using equivalent section properties. Additionally, the results of these analyses are compared with those of a 3D model using shells and beams, presented in a previous research paper. Two rack configurations with different level heights are considered. The aim of this study is to capture the influence perforations of the uprights, warping, eccentricities and cross‐sectional distortion in the nonlinear response of racks subject to severe lateral actions.

Weak form quadrature element analysis of spatial geometrically exact composite beams with torsional warping

Spatial thin-walled laminated composite beams under large deformations with torsional warping effects are studied. A novel two-dimensional cross-section model for composite beams with shear and torsion-warping deformations is proposed wherein the three-dimensional warping and coupling effects caused by inhomogeneous anisotropic materials are considered. A weak form quadrature element formulation is established for nonlinear analysis of geometrically exact composite beams with torsional warping and shear deformations. By combining the two-dimensional cross-section model and the nonlinear one-dimensional beam model, the three-dimensional stress and strain fields can be obtained. Numerical examples are provided to investigate the torsional warping effects and verify the effectiveness of the present formulation. Results of the present beam formulation which are achieved at a significantly lower computational cost are in excellent agreement with those of shell and solid models of commercial codes.

A one-dimensional model for mechanical coupling metamaterials using couple stress theory

A new model for a one-dimensional (1D) metamaterial model is formulated based on the couple stress theory. Different point groups of crystal will affect the mechanical couplings in the 1D metamaterial model. We find that when the material belongs to D2 point group, an unusual set of equations is decoupled from governing equations: axial force–torsion–warping (FTW) metamaterial model, which describes an unusual mechanical coupling that cannot occur in traditional materials. The deformation behaviors of the current model under a different axial force are analyzed by using the FTW metamaterial model. The numerical results show that the middle part of the current model has maximum torsional deformation when subjected to sine-type axial force. Since the end of the current model is fixed, the current model does not have compressional deformation globally. Moreover, when the current model is subjected to cosine-type axial force, the FTW coupling is successfully achieved. Then, we study a rod-like metamaterial model under an axial end force. The results show that the compressional deformation occurs at the free end, with a counterclockwise torsion. The current work provides guidance for the structural design and theoretical analysis of the rod-like metamaterial when using the couple stress theory.

Torsional warping analysis of arbitrary beam cross-sections by weak form quadrature elements

Torsional warping is essential for modern beam theories. The commonly used finite element method generally suffers from low convergence rates. A weak form quadrature element formulation for torsional warping analysis of beams with arbitrary cross-section is established. Two numerical examples are given to verify the effectiveness of the present formulation. Results demonstrate that the present formulation is of high accuracy and high efficiency.

Study of Practical Analysis Method for Shear Warping Deformationof Composite Box Girder with Corrugated Steel Webs

Shear warping deformation is an important part of the flexural and constrained torsion analysis of composite box girder with corrugated steel webs (CBG-CSWs), which is also the main reason for the complex force analysis of box girders. A new practical theory for analyzing shear warping deformations of CBG-CSWs is presented. By introducing shear warping deflection and corresponding internal forces, the flexural deformation of CBG-CSWs is decoupled to the Euler-Bernoulli beam (EBB) flexural deformation and the shear warping deflection. On this basis, a simplified method for solving shear warping deformation using the EBB theory is proposed. According to the similarity of the governing differential equations of constrained torsion and shear warping deflection, a convenient analysis method for the constrained torsion of CBG-CSWs is derived. Based on the decoupled deformation states, a beam segment element analytical model applicable to EBB flexural deformation, shear warping deflection, and constrained torsion deformation is proposed. A variable section beam segment analysis program considering the variation of section parameters is developed for CBG-CSWs. Numerical examples of constant and variable section continuous CBG-CSWs show that the stress and deformation results obtained by the proposed method are in good agreement with the 3D finite element results, verifying the effectiveness by the proposed method. Additionally, the shear warping deformation has a great influence on the cross-sections near the concentrated load and middle supports. This impact along the beam axis decays exponentially, and the decay rate is related to the shear warping coefficient of the cross-section.

FINITE ELEMENT ANALYSIS FOR THIN-WALLED MEMBER SUBJECTED TO COMBINED LOADING

Thin-walled structures are widely used in various structural engineering applications due to their advantage of high bearing strength when compared to self-weight and used in a complex loading situation where subjected to combined loadings. When a thin-walled section is subjected to a combined load with restrained torsion, they are ineffective at resisting, resulting in a reduction in beam capacity due to torsion and additional warping stresses. A finite element calculation can be used to analyze a 3D bar of thin-walled structural sections. Different commercial software and studies commonly consider six degrees of freedom at each node of a member for a space frame without considering the effect of warping restraint at the member's ends. This paper presents a finite element calculation for thin-walled sections with restrained torsion using the 14x14 member stiffness matrix, which includes warping as an additional degree of freedom and is commonly used for open thin-walled sections. In this study, we considered two different methods for including the additional degree of freedom for the stiffness matrix, which are very close to each other for small values of characteristics number.

Analysis of vibration characteristics of stators of electrical machines under actual boundary

This paper presents a general method for the studies of the vibration characteristics of an electrical machine stator with encased construction under practical boundary conditions. In this paper, artificial springs are introduced at both ends of the stator, and arbitrary boundary conditions can be simulated by adjusting the spring stiffness, including the support conditions in practical applications of the stator. In order to cooperate with the solution of the vibration characteristics of the stator with arbitrary edges, the Gram-Schmidt method is employed to construct characteristic orthogonal polynomial series to simulate the mode shape functions under arbitrary boundary conditions. The analysis is based on the Novozhilov shell theory, which is more applicable to the construction and actual operating environment of stators of electric vehicle drive motors, and takes into account moment of inertia neglected in the past. Further, the effects of teeth, windings, frame and cooling ribs are incorporated in the analysis, and the teeth with complicated shapes are considered as longitudinal ribs of unlimited cross-section, with bending and torsional coupling motion and warping deformation considered. The Rayleigh-Ritz method is used to build a unified dynamic analysis model of the stator of an electric machine, which can provide information on the radial, circumferential and axial vibrations of the stator with higher time efficiency, and a series of comparison studies and experiments are performed to demonstrate the general validity and accuracy of the model. On this basis, the possibility of construction optimization of the stator is discussed to improve the vibration and noise performance while reducing the cost.

Effects of warping shear deformation on the torsional load-bearing behaviour of assembled half-shell structures for wind energy towers

This paper deals with an extended analytical calculation model for the torsional load-bearing behaviour of assembled half-shell structures. The objective is to transfer the complex torsional load-bearing behaviour of segmented tower constructions into a simplified and mechanically comprehensible bar model. The presented approach determines the proportional load-bearing effects of shear force and torsion of a cut out single half-shell for a global torsional load based on the deformation equilibrium. The previous model approach – including the principle of load redistribution and the flexibility method – leads to a slenderness-independent determination of the load-bearing components. The developed method enables the specific isolation of the geometric influencing variables by separating the deformation-depending partial internal forces. In addition, the deformation approach can be used to investigate the influence of warping shear softness on the load-bearing behaviour of compact reinforced concrete segments with vertical joints according to the extended warping theory. Compared to the extended first-order warping torsion, the significantly simplified deformation calculation based on the second-order tension rod analogy leads to sufficiently accurate results. In general, the internal force calculation is considered to be extremely sensitive to the smallest changes, so that the improved and much more accurate approach finds its application. The numerical validation leads to very good matches of the internal forces as well as the deformations and stresses.

A comparison of the mechanical properties of curved box girders with corrugated steel webs to curved box girders with concrete webs

Curved composite box girder bridges with corrugated steel webs (CSWs) have already been constructed around the world. However, limited work has been done on the comparisons of mechanical properties between curved box girders (CBGs) with CSWs and CBGs with traditional concrete webs. This can be attributed to the difficulty separating flexure, torsion, and distortion using experimental or finite element analysis methods. In this paper, first, a practical method that can solve deflections, torsional angles, distortional angles, stresses, and internal forces of simply supported and continuous CBGs with intermediate diaphragms is introduced. Then, a parametric analysis is conducted to study the difference in mechanical properties of CBGs with CSWs and CBGs with concrete webs. The results show that: (1) Compared with traditional concrete box girders, the torsional and distortional resistances of box girders with CSWs are significantly reduced. Under external loading, all deflections, torsional angles, distortional angles, flexural normal stresses, restrained torsional and distortional warping normal stresses are generally larger. (2) For the example presented in this paper, the torsional warping normal stresses can reach 30% of the flexural normal stresses, and the distortional warping normal stresses can reach 80% of the flexural normal stresses. For this reason, they cannot be neglected in the design. (3) By increasing the number of intermediate diaphragms, drawbacks of using CSWs due to the increase in distortional angles and distortional warping normal stresses can be mitigated, closing the gap between the 2 different web types. In this regard, it is suggested that at least 1 diaphragm should be installed at midspan and if the diaphragms are arranged by equal spacing, their number is suggested to be odd.

On the restrained torsional warping of thin‐walled elastic shafts

AbstractThis paper studies the torsion of thin‐walled shafts when warping is restrained. It focuses on the treatment of the warping–twist constraint (whereby the twisting of the shaft is constrained to control the warping of its cross sections) by different formulations of warping torsion. This constraint underlies the kinematics of the classical TWKV formulation of Timoshenko‐Wagner‐Kappus‐Vlasov, whereas it is relaxed in the widely‐used RBV formulation of Reissner‐Benscoter‐Vlasov and the new mixed formulation recently proposed by the author. These two unconstrained formulations are still able to enforce the constraint in a penalty‐type limit process depending on specific combinations of the section's torsional constants, some new to the literature. Here we obtain rigorous estimates of these torsional constants and penalty parameters in the thin‐wall limit, thus providing an alternative understanding of the physical response of these sections. In particular, we prove that the constrained limit is attained in the thin‐wall limit for open sections but not for closed (multiply‐connected) cross sections. These analyses also provide an understanding of how those three formulations model the torsion of elastic shafts. To this purpose, we also undertake a detailed numerical evaluation of these considerations. The results presented here show the improved performance gained by the new mixed formulation of warping torsion, avoiding the anomalies observed for the TWKV formulation and the incorrect (not in equilibrium) RBV stresses.

Symplectic analysis of thin-walled curved box girders with torsion, distortion and shear lag warping effects

Shear lag effect and torsional and distortional warping can significantly affect the performance of thin-walled curved box girders used in modern bridge engineering. The structural behaviour of these bridges exhibits complexity due to coupled bending and torsion together with warping effects of non-uniform torsion, distortion and shear lag. A practical method of analysis, based on the symplectic approach, with the same perspective as the Higher Order Beam Theories, is presented for overcoming the difficulties of numerical approaches via the Finite Element Method. In this paper, the Hamiltonian Structural Analysis method implements the analysis of the shear lag effect together with non-uniform torsion and distortion, for curved box girder bridges. The examples given can allow engineers to evaluate the effects of the negative shear lag phenomenon and the distribution of axial stresses in box slabs and webs, due to different warping effects, which strongly influence thin-walled box sections subjected to point loads and restrained areas.

On the modeling of restrained torsional warping: an analysis of two formulations

On the modeling of restrained torsional warping: an analysis of two formulations

Exact solutions for thin-walled composite open section beams using a unified state space coupled field formulation

Thin-walled open section composite beams (TWOSCB) used extensively in aerospace structures exhibit several non-classical effects such as torsional warping, warping shear, material coupling, and shear deformation. Arbitrary cross-section and material coupling make the governing equations highly coupled and difficult to solve analytically. In this paper, a unified coupled-field formulation is developed to analyze TWOSCB accommodating all non-classical effects. Using the state-space method to derive analytical expressions for displacements and forces, Vlasov solutions of isotropic beams are recovered. Results obtained for arbitrarily laminated mono-symmetric composite channel and I-section beams subjected to bending and torsion loads show excellent agreement with available results in the literature.

Derrick crane robustness scenarios

The paper deals with derrick cranes (derricks) that are equipment typically used in the mining industry and focuses on the analysis of few robustness scenarios. Derricks are composed of built-up steel members which are widely and efficiently used in hoisting applications due to their high payloads at relevant outreaches. During the in-service use, local damages might occur and as such, the aim of the paper is to study how such damages influence the overall structural performance. In particular, reference was made to a derrick for which six geometrical configurations and five different damage scenarios have been analysed. Owing it to the extensive use of angles in the built-up component of each derrick, structural analyses have been carried out by using a commercial refined finite element analysis package (FEAP) offering the warping torsion as an additional degree of freedom for each beam node. Research outcomes allow for a clear identification of the parts of a derrick that should be protected and well-designed to guarantee a robust structure for its entire in-service life.

Exact finite beam element for open thin walled doubly symmetric members under torsional and warping moments

Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with those based on exact solution and ABAQUS finite beam element solution at a small fraction of the computational and modelling cost involved.

Modal Analysis of a Thin-Walled Box-Girder Bridge and Railway Track Using Finite Element Framework

Modal analysis has received widespread acceptance in past few decades for a wide range of applications. Bridges and buildings are two of the most popular structures that use this application in the context of civil engineering. The current study aims to apply finite element technique to estimate the free vibration characteristics of a railway track and a box-girder bridge. The curved bridge is numerically modeled using thin-walled box-beam finite elements that take into account torsional warping, distortion, and distortional warping, all of which are important characteristics of thin-walled box-girders. A commercially available finite element software ANSYS is used to simulate the railway track in three dimensions. The study is restricted to the initial design stage of a thin-walled box-girder bridge decks, in which a full three-dimensional finite element model is not required. For the thin-walled box-girder bridge, a MATLAB code has been built that yields the corresponding modal parameter results, whilst the modal parameters of the railway track system are acquired using ANSYS software.