Original Research ArticleAnalysis of laterally restrained cold-formed C-shape purlins according to Vlasov theory
Introduction
Thanks to the modern methods of joining the lightweight cladding of building structures to the roof purlins and wall girts (e.g. [1], [2], [3]), the supporting elements of the cladding can be elastically fixed with regard to torsion while at the same time their flanges can be held down in the roof or wall surface plane, depending on the joints and type of cladding used. By exploiting this effect one can increase the critical buckling load capacity in the case of double-symmetric (double-T) elements and to rationally design monosymmetric (C-shaped) elements and pointwise symmetric or asymmetric elements (Z-shaped with identical or different flanges). This problem has been indicated or experimentally investigated in many papers (e.g. [4], [5], [6], [7]). Ways of approximately calculating cold-formed purlins with Σ and Z cross-sections are proposed in, e.g. [8], [9].
Taking into account the experimental results for channel purlins interacting with corrugated sheeting [7], a general method of calculating (as thin-walled elements satisfying the assumptions of the Vlasov thin-walled member theory [10]) subjected to simultaneous bending and torsion relative to a fixed axis of rotation was developed in the Institute of Building Engineering at Wrocław University of Technology. The method and its experimental verification were presented in [11]. The research (both theoretical and experimental) was continued for pretorsioned purlins or purlins with additional point bracings with regard to torsion along their length. The results were reported in [12]. Unfortunately, despite the universality of the proposed method, it did not attract much interest then, probably due to the rather low popularity of the Vlasov theory of thin-walled members and the complicated form of the solution (in which the purlin displacement components had to be calculated from a non-linear system of algebraic equations).
As years passed, further theoretical and experimental research mainly on purlins was conducted. One should mention here work [13] (referring to standard ENV 1993-1-3) and extensive works [14], [15] in which numerical solutions of the purlin-roof decking system were proposed. In [16] an attempt was made to solve the problem within the technical Vlasov theory of thin-walled beams, on which, among others, book [17] is based. The results reported in [18] indicate that in the case of braced cold-formed C-elements, load-bearing capacity estimates based on the Vlasov theory are more accurate than the ones based on the Winter approach. This particularly applies to cold-formed sections with small or average wall thickness and with regular Cee and Zed cross-sections. The solution presented in [16] was derived using the energy method, but there are reservations concerning the reduction of the problem to the spatial stability task (e.g. [17], [19]). This constitutes a 2nd order strength theory (simultaneous bending and torsion) problem and it should be solved accordingly.
The aim of the present paper is to explain the research reported in [11], [12] and to present modified solutions for the purlins. The modification consists in neglecting low-order terms from the system of differential equations equilibrium for purlin. Thus, the solution of non-linear systems of algebraic equation is avoided and formulas enabling one to directly calculate the components of the displacement of the considered elements are obtained. The proposed solutions are sufficiently accurate for design purposes.
Section snippets
Static-strength analysis of purlins
In the general case, a prismatic thin-walled member with additional elastic constraints with regard to linear and angular displacement of its cross-section was assumed as the calculation model. The constraints are spaced along any straight line C parallel to the longitudinal axis of the member. The purlins’ lateral load, with components: mz, qx, qy, is applied along straight line Q parallel to the member's axis. The cross-section of the member is shown in Fig. 1.
For an elementary member segment
Pretorsioned channel purlins interacting with lightweight cover
The joints between corrugated sheets and thin-walled Cee purlins, made by means of sheet metal screws, even though they offer several advantages, undergo permanent deformation after the first loading, which results in the torsion of the purlins (Fig. 6). The size of this torsion depends on the magnitude of the first load. The torsion remaining after the first load can be regarded as pretorsion (geometric imperfection) for the next load cycles or it can be eliminated by means of appropriate
Conclusion
Channel purlins interacting with lightweight sheet metal roofing were the matter of the theoretical and experimental studies presented in this paper. Besides perfect purlins, also the effect of permanent torsion and additional point bracings with regard to torsion on the displacement and strain of the purlins was studied. As confirmed by the model studies, a thin-walled member subjected to bending and torsion relative to an imposed axis of rotation constitutes a calculation model for the
References (24)
Stability of cold-formed purlins braced by steel sheeting
Thin-Walled Structures
(1996)- et al.
Modelling of cold-formed purlin-sheeting systems: Part 1. Full model
Thin-Walled Structures
(1997) - et al.
Modelling of cold-formed purlin-sheeting systems: Part 2. Simplified model
Thin-Walled Structures
(1997) - et al.
Lateral-torsional buckling analysis of partial-laterally restrained thin-walled channel-section beams
Journal of Constructional Steel Research
(2004) Spatial stability of braced thin-walled members of steel structures
Journal of Constructional Steel Research
(2003)Non-uniform torsion of stiffened open thin-walled members of steel structures
Journal of Constructional Steel Research
(2007)- et al.
Connections in thin-walled structures
- et al.
Stahltrapezprofile mit Obergurtbefestigungen
Stahlbau
(1988) - et al.
Laboratory Tests of Metal Structures
(2012) Zur Kippstabilisierung von Pfetten und Riegeln durch leichte Umhüllungselemente
Bauingenieur
(1978)
Traglastversuch an durchlaufender C-Pfetten mit Aluminium-Trapezblechen als Dacheindeckung
Bauingenieur
Versuche zur Kippsicherheit von durchlaufenden Pfetten mit dünnwandigen Stahltrapezblechen als Dacheindeckung
Bauingenieur
Cited by (8)
Lateral-torsional deformations of single-span and two-span thin-walled beams with continuous bracing
2021, Journal of Constructional Steel ResearchCitation Excerpt :Distortion of the cross section is neglected. Restrained, single-span, pinned-pinned, C-section and Z-section beams under gravity and/or uplift loading have been analyzed and/or tested in a number of investigations, including [3–20]. Pinned-fixed and fixed-fixed beams were considered in [4,5,7].
Design formulas for channels subject to combined compression, shear and major axis bending
2021, Asian Journal of Civil EngineeringDynamic instability of channel-section beams under periodic loading
2020, Mechanics of Advanced Materials and StructuresLateral-torsional deformations of C-section and Z-section beams with continuous bracing
2020, Proceedings of the Annual Stability Conference Structural Stability Research Council, SSRC 2020Experimental Research on the Shear Connectors in Foam Concrete with C-Channel Embedment
2018, International Journal of Concrete Structures and MaterialsRecent contributions to the analysis and design of cold formed steel channels
2017, Recent Patents on Engineering