Analysis of laterally restrained cold-formed C-shape purlins according to Vlasov theory

Highlights

• The results of research on purlins interacting with a cover made from corrugated sheets.

• Ideal cold-formed purlins and with geometric imperfections were investigated.

• Solutions subjected to bending and torsion were derived using the Bubnov–Galerkin method.

• The theoretical results were validated by experiments on models of a natural-scale.

Abstract

This paper presents the results of theoretical and experimental research on channel purlins interacting with a cover made from corrugated sheets. Perfect (without pretorsion) cold-formed purlins and cold-formed purlins with geometric imperfections (permanent pretorsion or pointwise bracing with regard to torsion) were investigated. Appropriate solutions for the purlins subjected to simultaneous bending and torsion relative to a fixed axis of rotation were derived using the Bubnov–Galerkin method in both its full version and simplified (but sufficiently accurate for design purposes) version. As a result, direct relations for the linear and angular displacements of the purlins were obtained. The relationships can be used to calculate the forces and the strain in the purlins. The theoretical results were validated by experiments on a natural-scale roof fragment. Finally, practical conclusions were drawn from the research.

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.