Modelling of cold-formed purlin-sheeting systems-Part 2. Simplified model

A number of theoretical and experimental investigations have been made into the nature of purlin-sheeting systems over the past 30 years. These systems commonly consist of cold formed zed or channel section purlins, connected to corrugated sheeting. They have proven difficult to model due to the complexity of both the purlin deformation and the restraint provided to the purlin by the sheeting. Part 1 of this paper presented a non-linear elasto plastic finite element model which, by incorporating both the purlin and the sheeting in the analysis, allowed the interaction between the two components of the system to be modelled. This paper presents a simplified version of the first model which has considerably decreased requirements in terms of computer memory, running time and data preparation. The Simplified Model includes only the purlin but allows for the sheeting's shear and rotational restraints by modelling these effects as springs located at the purlin-sheeting connections. Two accompanying programs determine the stiffness of these springs numerically. As in the Full Model, the Simplified Model is able to account for the cross-sectional distortion of the purlin, the shear and rotational restraining effects of the sheeting, and failure of the purlin by local buckling or yielding. The model requires no experimental or empirical input and its validity is shown by its good correlation with experimental results.     

Simplified models for cross-section stress demands on C-section purlins in uplift

The objective of this paper is to provide and verify simplified models that predict the longitudinal stresses that develop in C-section purlins in uplift. The paper begins with the simple case of flexural stress; where the force has to be applied at the shear center, or the section braced in both flanges. Restrictions on load application point and restraint of the flanges are removed until arriving at the more complex problem of bending when movement of the tension flange alone is restricted, as commonly found in purlin-sheeting systems. Winter's model for predicting the longitudinal stresses developed due to direct torsion is reviewed, verified, and then extended to cover the case of a bending member with tension flange restraint. The developed longitudinal stresses from flexure and restrained torsion are used to assess the elastic stability behavior of typical purlin-sheeting systems. Finally, strength predictions of typical C-section purlins are provided for existing AISI methods and a newly proposed extension to the direct strength method that employs the predicted longitudinal stress distributions within the strength prediction.     

Stress and Buckling Analysis of Cold-formed Zed-purlins Partially Restrained by Steel Sheeting

This paper presents an analysis model for cold-formed purlin-sheeting systems subjected to wind uplitt loading in which the restraint of the sheeting to the purlin is taken into account by using two springs representing the translational and rotational restraints provided by the sheeting.The set of equations is solved by means of trigonometric series and finite strip methods in which the pre-buckling stress is calculated based on the same model used for the buckling analysis rather than taken as the "pure bending" stress. The influence of spring stiffness and fixing position of the purlin and sheeting on the stresses resulted in the cross-section of the purlin is discussed in details.     

Buckling behavior of cold-formed zed-purlins partially restrained by steel sheeting

This paper presents a study on the buckling behaviour of purlin-sheeting systems under wind uplift loading. The restraint provided by the sheeting to the purlin is modeled by using two springs representing the translational and rotational restraints. The analysis is performed using finite strip methods in which the pre-buckling stress is calculated based on the same model used for the buckling analysis rather than taken as the ‘pure bending’ stress. The results obtained from this study show that, for both local and distortional buckling, the restraints have significant influence on the critical loads through their influence on the pre-buckling stress rather than directly on the buckling modes. While for lateral-torsional buckling, the influence of the restraints on the critical loads is mainly due to their influence on the buckling modes rather than the pre-buckling stress.     

Untersuchungen zur Drehbettung von biegedrillknickgefährdeten Trägern durch Sandwichelemente

Investigation of the torsional restraint of sandwich panels against lateral torsional buckling of beams. Experimental investigations and parametric FE‐analyses show that the torsional restraint of sandwich panels against lateral torsional buckling of beams can be expressed by a tri‐linear moment‐rotation‐relationship. Formulae are given for different profile types combined with sandwich panels for roof and wall for calculation of the parameters of this relationship. From this moment‐rotation‐relationship, which depends on the lateral load of the beam, the rotational stiffness c_{ϑ A} of the connection is obtained as secant stiffness. The rotational stiffness c_ϑ required for the design against lateral torsional buckling according to DIN 18800 part 2 is governed by c_{ϑ A}. The rotational stiffness values c_{ϑ A} of the connection, which so far were only known for two types of elements, can be calculated with this method for all common types of sandwich panels and different types of constructions.     

简支Z型檩条在风吸力作用下应力分布及极限状态研究

轻钢工业厂房中,Z型檩条在其截面的非对称平面内承受风荷载,上翼缘受压型钢板连续约束作用而下翼缘自由,应力分布比较复杂,存在局部或整体稳定问题,是围护结构抗风设计的主要控制因素。基于薄壁杆件结构力学以及经典梁理论,分析了简支Z型檩条在风吸力作用时考虑压型钢板对檩条约束作用的应力分布,并依据相关试验资料,验证了分析方法的正确性。然后,针对檩条横截面应力分布、侧向位移及极限状态等力学性能进行定性参数分析。研究表明,该方法可以精确计算檩条横截面应力分布,压型钢板对檩条的扭转及侧翼约束作用分别影响檩条上、下翼缘的峰值应力,使得构件局部屈曲应力增大,而畸变屈曲应力降低,从而改变檩条局部失稳模式,使得檩条由局部屈曲失稳转变为畸变屈曲失稳。     

Effect of loading on the rotational restraint of cold-formed purlins

Cold-formed purlins are restrained in plane of the top flange by connected sheeting both laterally and rotationally. The stiffness of the rotational restraint C_D is influenced by external loading of the purlin. Though, simple tests (recommended, i.e. by valid standards) do not cover the influence of loading. It is shown in this paper that the effect of loading on the rotational restraint is different for cold-formed and for hot-rolled purlins. An analytical model was proposed and solved for Z-shaped and C-shaped purlins and for both gravity and uplift loading, and appropriate formulas were derived. The model was based on study of redistribution of contact forces as function of loading q. It is demonstrated that the effect of loading on the stiffness C_D is different for the listed cases. Model results are verified with results of numerical simulation.     

Design of thin-walled purlins for distortional buckling

Roof purlins and sheeting rails are generally loaded through the cladding members that they support and this provides both rotational and translational restraint. This restraint reduces the tendency to lateral torsional buckling and thus increases the importance of distortional buckling in the design procedure. This paper shows that the fundamental behaviour of restrained purlins under both downward and uplift load can be best understood with the aid of ‘Generalized Beam Theory’ (GBT). GBT also provides a yardstick by which approximate design methods can be assessed. The existing approximate methods are evaluated and an improvement is proposed. The proposed design approach is then validated by comparison with test results.     

Modelling of cold-formed purlin-sheeting systems- Part 1: Full model

Purlin-sheeting systems used for roofs and walls commonly take the form of cold formed channel or zed section purlins, screw-connected to corrugated sheeting. These purlin-sheeting systems have been the subject of numerous theoretical and experimental investigations over the past three decades, but the complexity of the systems has led to great difficulty in developing a sound and general model. This paper presents a non-linear elasto-plastic finite element model, capable of predicting the behaviour of purlin-sheeting systems without the need for either experimental input or over simplifying assumptions. The model incorporates both the sheeting and the purlin, and is able to account for cross-sectional distortion of the purlin, the flexural and membrane restraining effects of the sheeting, and failure of the purlin by local buckling or yielding. The validity of the model is shown by its good correlation with experimental results. A simplified version of this model, which is more suitable for use in a design environment, is presented in a companion paper.     

Structural analysis of assemblages of thin-walled members

A matrix method for the analysis of structural systems composed of thin-walled members is presented. The matrix displacement analysis includes the effects of thin-walled non-uniform torsion theory, cross-section asymmetry, eccentric restraint as well as joint types peculiar to thin-walled members. The method is used for a prediction of the elastic behaviour of a set of representative test frames. The test frames were pitched-roof portals constructed from channel sections bent about their major-axis and supported by eccentric restraints simulating purlins and girts.     

蒙皮支撑C型截面檩条的静力性能分析

将蒙皮体系对檩条的约束作用简化为一个侧向弹簧和一个扭转弹簧,并将此模型用ANSYS进行分析,分别考虑侧向弹簧和扭转弹簧对檩条的蒙皮作用,并对其静力性能进行了分析。     

Thermal buckling of rotationally restrained steel columns

The in-plane elastic buckling of a steel column under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or an elastic foundation. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain-displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the buckling of columns in three different thermal loading cases is studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The thermal gradient plays a positive role in improving the stability of columns. Furthermore, the effect of thermal gradients decreases while increasing the rotational restraint stiffness and decreasing the slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the rotational stiffness of thermal restraints will increase the critical temperature.     

Numerical Studies of the Rotational Stiffness of Purlin–Sheeting System

In light-gauged steel purlin-to-sheeting roof systems, the attached sheeting can provide rotational restraints to the purlin. The magnitude of the additional rotational stiffness offered by the sheeting will affect the load bearing capacity of the purlin. The current design method in Eurocode3 (EC3) is less accurate in predicting the purlin–sheeting rotational stiffness as it neglects the effect of wall thickness of the purlin. An integral model is introduced based on the finite element method. Comparisons are made between numerical results and existing experiments and a good agreement is observed. Parametric studies are conducted based on the validated model to investigate the influences of geometric dimensions on the rotational stiffness. Two modified coefficients are proposed for calculating the rotational stiffness based on the codified formulae in EC3, where the effect of the wall thickness and the flange width of the purlin are both considered.     

Sandwich‐Dachbauteile für Flachdächer

Sandwich panels for roof covering. Sandwich panels for roof covering, which are currently used, are in principle always equipped with a trapezoidal profiled face. It is because of this geometry that the longitudinal joints of the panels are easy made by overlapping the external metal faces. The bending stiffness of these profiled faces also largely avoid the increasing in deflection caused by creep effects of the core as a result of long‐time loading from snow and self weight. In particular for flat roofs of industrial buildings there are some disadvantages caused by the profiling of the outer faces, e. g. the sealing of openings in the roof. Furthermore the mounting of the profiled panels always have to be realized in the direction of apex to cullis because of the rain water flow conditions. This requirement premises a substructure with purlins. In the described research project sandwich panels with flat faces were developed, where penetrations can be sealed in an easy way and mounting of the panels perpendicular to the roof pitch, from frame to frame (without purlins) is possible. All necessary research was conducted, in particular with regards to creeping effects, sealing of the panel joint and special fixings. On the base of the obtained research results, practice oriented solutions are proposed.     

压型钢板屋面对檩条的扭转约束

压型钢板屋面对檩条施加扭转约束,从而提高檩条的稳定性。针对杭萧钢构股份有限公司的HXY-478屋面压型钢板与檩条采用夹板连接的屋面体系,对檩条施加的扭转约束进行研究。通过ANSYS分析,对上下翼缘受压的扭转约束提出了计算公式。     

Numerical and experimental investigation of rotational stiffness of zed-purlins connection with sandwich panels

A rotational resistant stiffness of the zed-purlins connection with sandwich panels is investigated. A simple finite element method model of the connection is proposed. The numerical analysis of the model performed by ABAQUS software, in physically linear and geometrically nonlinear ranges, leads to the rotational resistant stiffness sought. The numerical solution obtained is verified experimentally. Two variants of distribution of screws connecting the purlin with the sandwich panel are taken into investigation. A good agreement of both, theoretical and experimental, results should be noticed. Conclusion related to possible application in design practice is formulated.     

Behavior of beam web panel under opposite patch loading

Elastic buckling is studied for a panel with various boundary conditions including simple supports, fixed supports and elastic restraints. The panel is subjected to opposite patch loading. Following a review of existing work on the effects of localized compression, also known as patch loading, a study is conducted to take into account the restraints provided by the flanges of the I beam in a realistic manner. This study is based on a finite element model implemented in the CAST3M software. A new equation is proposed to calculate the buckling critical coefficient for a beam web panel considering the rotational stiffness provided by the flanges. The model is then applied to longitudinally stiffened web panels which are subjected to opposite patch loading.A parametric analysis is performed to determine the transition from a global buckling mode to a local buckling mode where the sub-panels on each side of the stiffener behave separately. The numerical results show that the flexural rigidity of the stiffener is the appropriate parameter that governs the buckling mode. From these results, a formula is proposed to calculate the buckling critical coefficient of stiffened web panels.     

蒙皮支撑C型檩条的有限元分析

本文借助有限元软件ANSYS对蒙皮支撑简支C型檩条在重力荷载和风荷载作用下的受力性能进行了有限元分析,分析时将蒙皮体系对檩条的约束作用采用两个弹簧：一个侧向弹簧和一个扭转弹簧来代替,对其静力性能进行了分析。     

有约束梁的整体稳定

探讨两种约束对梁整体稳定的影响,并提出用于设计工作的实用方法。其一是梁在平面内的端部约束,以框架梁为代表。现行钢结构设计规范没有这方面的规定,需要加以补充;其二是梁跨度范围内的侧向支承点对梁稳定提供的约束。计算这类梁的稳定,需要考虑各段之间的相互约束。现行规范只给出支承点为等间距时的等效临界弯矩系数,缺少不等间距时的计算方法。所推荐的用计算长度表现约束效应的计算方法既具有普遍性,又概念清晰。还论述了承受端弯矩梁段的非弹性影响,并提出相应的计算公式。     

Modelling of axial restraints for limiting temperature calculation of steel members in fire

At ambient temperature, design actions calculated for a beam under lateral loading may remain unchanged irrespective of whether axial restraints at the beam’s ends are present or not. However, at elevated temperatures, axial restraints may induce significantly different design actions from those calculated at ambient temperature, mainly due to thermal restraining reactions. As a result, the limiting temperature may be reduced. In this paper, a method is presented for limiting temperature calculation of steel beams in fire in multi-storey buildings. The method takes into account the fact that the axial force generated in a steel beam in fire by axial restraints may increase to such a magnitude that the strength criterion, in terms of axial-bending interaction, may be breached. In addition, it has been recognised that the degree of axial restraints provided at the ends of a steel beam varies, depending on the axial restraint stiffness provided by neighbouring members. A simple technique to model the axial restraints provided to a steel beam by neighbouring members and the procedure for its limiting temperature calculation in a multi-bay situation in a building will be presented. The approach caters for different end conditions of the columns below and above the beam under consideration. Catenary action due to large deflection effect for beams in fire will also be discussed.