Moment redistribution in cold-formed steel sleeved and overlapped two-span purlin systems

Abstract

Cold-formed steel purlin systems with overlapped or sleeved connections are alternatives to continuous two-span systems and exhibit different degrees of continuity. Both connection types are highly favourable in practice since they are both strategically placed over an interior support to provide additional moment resistance and rotational capacity where the corresponding demands are at their largest, thus improving the overall structural efficiency. Until recently, full-scale testing has been the most common way of investigating the structural behaviour of such systems. In this study, numerical modelling, capable of capturing the complex contact interactions and instability phenomena, is employed. The developed finite element models are first validated against data from physical tests on cold-formed steel beams featuring sleeved and overlapped connections that have been previously reported in the literature. Following their validation, the models are employed for parametric studies, based on which the structural behaviour of the examined systems is explored, while the applicability of conventional plastic design as well as of a previously proposed design approach is investigated. Finally, the efficiency of these systems in terms of load-carrying capacity is compared with their equivalent continuous two-span systems.

Keywords

cold-formed steel continuous purlin systems finite element modelling moment redistribution overlapped purlins sleeved purlins

Introduction

Cold-formed steel purlins, when used to support the roof cladding on industrial buildings, constitute one component in a complex structural system. Their shape that has evolved through a mix of forming possibilities and seeking improved structural efficiency is often rather intricate (see Figure 1). Thus, simulation of their resulting structural behaviour, involving combinations of bending and various forms of buckling and twisting, requires sophisticated treatment if all important features are to be captured. If the interactions with their adjacent components, including provision for bolted connections, are also to be allowed for, then a challenging structural problem results.

Figure 1.

Cross-sectional shapes of cold-formed steel purlins: (a) Zed, (b) lipped Zed, (c) inclined Zed, (d) Zeta 1 and (e) Zeta 2.

For single-span roof system arrangements, the presence of the cladding attached to the top flange of the purlin that is subject to compression as a result of downward loading normally provides sufficient lateral restraint that design may be based on the full moment capacity of the cross-section, suitably reduced to allow for loss of effectiveness due to local and distortional buckling. Since the system is statically determinate, there is no scope for redistribution of moments and design is based on the most heavily loaded cross-section, typically at midspan.

However, numerous manufacturers introduce varying degrees of continuity into this type of system by making the purlins continuous over more than one span. The most frequently employed continuous systems are two-span arrangements, achieved either by double length members that are continuous over the middle support (bare systems), or by two different purlins connected over the central support either by overlapping or by employing a sleeve; these are illustrated in Figure 2. The structural performance of these commonly used roof purlin systems has been investigated by several researchers (Ayhan and Schafer, 2017; Dubina et al., 2012; Gutierrez et al., 2011; Hancock et al., 1990a, 1990b; Papangelis and Hancock, 2005; Zhang and Tong, 2008a).

Figure 2.

Commonly used two-span purlin systems: (a) bare, (b) sleeved and (c) overlapped.

Traditional provisions in structural design codes, while able to deal with individual phenomena, for example, through the use of the effective width concept to allow for loss of effectiveness in thin plating, have difficulty in accommodating combinations of phenomena, such as the interaction of the cladding with distortional buckling. As a result, design methods tend to be oversimplified or rather complex. It was not until 2016 that Hui et al. (2016) developed a design method for two-span cold-formed steel bare purlin systems, capable of accurately determining the degree of moment redistribution, as well as the interaction with the other components of the system, yielding accurate predictions in terms of load-carrying capacity.

In this article, the findings from a numerical investigation into the degree of moment redistribution occurring within typical two-span cold-formed steel purlin systems, featuring overlapping and sleeved connections, are presented. Finite element (FE) models of sleeved and overlapped systems have been developed and validated against experimental data reported in the literature, allowing further investigation of the influence of key parameters. Based on the ensuing parametric studies, the structural behaviour of the examined systems has been explored, while the applicability of conventional plastic design as well as of the previously proposed design approach (Hui et al., 2016) has been investigated. Finally, the efficiency of these systems in terms of load-carrying capacity has been compared with their equivalent bare two-span systems.

Previous contributions: the alpha (α) framework

Structural behaviour of single-span and two-span purlin systems

For a two-span continuous purlin system under downward loading, the resulting bending moment distribution assuming elastic behaviour (i.e. no redistribution) is shown in Figure 3, with the maximum moment occurring at the internal support M_sup. There is, therefore, unused capacity available within the span regions since ‘failure’ is taken as attainment of the moment capacity of the most severely loaded cross-section.

Figure 3.

Elastic bending moment diagram for two-span setup under uniformly distributed gravity loading.

According to the fundamentals of simple plastic design, assuming that the moment capacity at the internal support can be maintained while inelastic deformations develop, an alternative ‘failure criterion’ would be that loading could be increased until the cross-sectional moment capacity is also reached within the spans, whereupon the system becomes a mechanism – see Figure 4. However, the proportions of many cold-formed purlin sections are such that the occurrence of local and distortional buckling limit the amount of rotation that could develop at the internal support before the moment there starts to fall. A further reason for the use of simple plastic design being inappropriate for cold-formed steel purlin systems is that the moment capacity itself is influenced by local and distortional buckling so that the moment levels attained at both the support and within the span would not correspond to the plastic moment capacity of the cross-section. Revision to the traditional plastic design mechanism approach can be made by recognising certain limitations in the interior support moment–rotation relationships and accepting that full redistribution may not be possible.

Figure 4.

Plastic collapse mechanism with full moment redistribution in a two-span system.

Design approach: the alpha (α) framework

Hui et al. (2016) developed sophisticated FE models in ABAQUS (ABAQUS Inc., 2013) simulating single-span and two-span bare purlin systems, taking into consideration the interaction between the steel beam and the corrugated sheeting as well as the restraining effect of the latter on the development of buckling along the top flange of the former. Their analyses showed that, for a typical continuous two-span arrangement, after the attainment of a maximum moment at the most severely loaded cross-section, located at the central support, moment redistribution occurs, usually accompanied by a reduction of the moment in the support region depending mainly on the slenderness of the cross-section, while the moment in the span increases up until its cross-sectional capacity is attained; at this point, failure of the system occurs.

It was found that the moment capacity at the central support M_sup, in the presence of a parabolic moment gradient, can be accurately replicated by the moment capacity of a single-span configuration of a beam of the same cross-section subjected to three-point bending M₃, with its span being equal to the length of the hogging region (region under negative moment) of the continuous beam. Similarly, the moment capacity in the span regions M_sp can be approximated by that of a single-span FE model of the same cross-section subjected to four-point bending M₄, with the length of the constant moment region being sufficient to allow the unimpeded development of local and distortional buckling. Note that although the capacity of the single-span FE model subjected to four-point bending was labelled as M₁ by Hui et al. (2016), the label M₄ has been adopted herein as this directly relates to the four-point bending configuration to which this system is subjected. A typical two-span system and its corresponding reference single-span systems are shown in Figure 5.

Figure 5.

Elastic bending moment diagram for a continuous two-span system and for the approximate corresponding single-span systems.

Based on the results of the conducted parametric studies, a design approach has been introduced recognising the difference between the structural behaviour as described by the idealisation of traditional plastic design and as exhibited by cold-formed steel purlin systems. The developed method uses the results from full-range FE analyses of two-span continuous systems as the basis for a simple modification − the so-called α factor − to the application of the plastic mechanism approach. It also makes direct use of the actual moment capacities of the central support and span regions, taking into account both their inherent material and geometric nonlinearities as well as the influence of the surrounding components.

The factor α (alpha), given by equation (1), expresses the drop-off in support moment at the ultimate load of the system, with a value of α = 1 representing a system able to maintain its full moment capacity at the support region, allowing full redistribution to occur

α = \frac{M_{\sup}}{M_{3}}, 0 \leq α \leq 1

(1)

According to the design proposal presented by Hui et al. (2016), equation (2), which has been derived based on numerous FE analyses and was found to give an accurate approximation of equation (1), recognises that the degree of moment redistribution within a cold-formed steel purlin system depends on the slenderness of the cross-section ${\bar{λ}}_{cs, - ve}$ as well as on the length-to-depth (L/d) ratio of the purlin

α_{d} = [0.7 - 0.0045 (\frac{L}{d})] {\bar{λ}}_{cs, - ve}^{[- 0.003 (\frac{L}{d}) - 1.4]} \leq 1

(2)

The cross-sectional slenderness of the steel section ${\bar{λ}}_{cs, - ve}$ refers to the slenderness of the cross-section under negative bending (as is the case over the central support) and is defined as

{\bar{λ}}_{cs, - ve} = \sqrt{\frac{M_{el}}{M_{cr}}}

(3)

where M_el is the yield moment of the cross-section, while M_cr is the elastic critical buckling moment, taken as the minimum of the elastic critical local buckling moment M_cr,l and the elastic critical distortional buckling moment M_cr,d. Note that M_cr,l and M_cr,d have been calculated herein using the software CUFSM (Li and Schafer, 2010), by extracting the critical loads of the two minima points of the signature curve corresponding to local and distortional buckling, respectively (Schafer, 2002; Schafer and Adany, 2006; Yap and Hancock, 2006).

The load-carrying capacity q of a two-span system can be calculated according to equation (4), determined based on equilibrium and given in equation (10.2b) of Clause 10.1.3.2 of EN 1993-1-3 (2006)

M_{sp} = \frac{{(q L^{2} + 2 M_{\sup})}^{2}}{8 q L^{2}}

(4)

where L is the length of each span of the two-span purlin system, while M_sp and M_sup correspond to the maximum moments at the span and support regions at ultimate load and can be taken as M₄ and α_dM₃, respectively, in the proposed design approach. Note that the values of M₁ and M₃ may be calculated directly for the required cross-section by FE analysis using the statically determinate arrangements of Figure 5, by conducting laboratory tests or using predictive equations.

Development of FE model: basic features

Although the structural behaviour of two-span bare purlin systems has been investigated previously (Hui et al., 2016), two-span systems with continuity being achieved through sleeves or by overlapping of two purlins at the central support have yet to be systematically explored. Previously developed FE models of two-span bare purlin systems (Hui, 2014), which have already been validated against physical tests reported in the literature, were employed and further developed herein for the modelling of two-span sleeved and overlapped purlin systems; their corresponding single-span reference systems have also been developed – see Figure 5. The main features of the developed numerical models are presented herein, while a more detailed description was presented by Kyvelou (2017) and Hui (2014).

Material modelling

The Ramberg–Osgood expressions proposed by Gardner and Ashraf (2006), given by equations (5) and (6), have been employed for the material modelling of cold-formed steel, while strength enhancements of the corner regions have also been introduced, according to equation (7) which has been derived based on tensile coupon tests (Hui, 2014). equations (5) and (6) are a two-stage version of the original Ramberg–Osgood expression (Hill, 1944; Ramberg and Osgood, 1943), the development of which has been described by several researchers (Arrayago et al., 2015; Gardner and Ashraf, 2006; Mirambell and Real, 2000; Rasmussen, 2003)

ε = \frac{σ}{E} + 0.002 {(\frac{σ}{σ_{0.2}})}^{n}, σ \leq σ_{0.2}

(5)

\begin{matrix} ε = \frac{σ - σ_{0.2}}{E_{0.2}} + (ε_{1.0} - ε_{0.2} - \frac{σ_{1.0} - σ_{0.2}}{E_{0.2}}) \\ {(\frac{σ - σ_{0.2}}{σ_{1.0} - σ_{0.2}})}^{n'_{0.2, 1.0}} + ε_{0.2}, σ_{0.2} < σ < σ_{u} \end{matrix}

(6)

σ_{0.2, corner} = {\begin{matrix} 1.05 σ_{0.2} & for 0 \circ < θ < 90 \circ \\ 1.10 σ_{0.2} & for 90 \circ \leq θ \leq 180 \circ \end{matrix}

(7)

In equations (5) to (7), σ and ε are the engineering stress and strain, respectively, E is Young’s modulus of the material, σ_0.2 and σ_1.0 are the 0.2% and 1% proof stresses of the flat material, respectively, E_0.2 is the tangent modulus of the stress–strain curve at σ_0.2, ε_0.2 and ε_1.0 are the total strains corresponding to the 0.2% and 1.0% proof stresses, respectively, σ_0.2,corner is the 0.2% proof strength of the corner material, θ is the internal angle of any corner of the steel cross-section, and n and n′_0.2,1.0 are the strain hardening exponents determining the degree of roundness of the stress–strain curve. For input into the developed ABAQUS numerical models, the nominal stresses and strains from equations (5) and (6) have been converted into true stresses and strains. It should also be mentioned that residual stresses were not explicitly modelled since their dominant through-thickness component is inherently included in the stress–strain curves obtained from tensile tests on coupons extracted from cold-formed steel sections (Jandera et al., 2008).

Interaction with profiled sheeting

The interaction between the top flange of the purlins and the attached profiled sheeting has been replicated by the two-spring model shown in Figure 6, which has been introduced and validated by Hui et al. (2016). For modelling the restraint provided by the fastener connecting the sheeting to the steel beam, linear translational ‘SPRINGA’ spring elements (ABAQUS Inc., 2013) have been employed, placed at the position of the physical link between the purlin and the cladding, with their assigned stiffness set at 5 kN/mm according to Haidarali (2011). In addition, nonlinear compression-only ‘SPRINGA’ spring elements, preventing upward but allowing downward displacements, were positioned at the flange–lip junction along the length of the beam, to replicate the restraining effect the sheeting has on upward deformations of the steel flange. The compressive stiffness assigned to the nonlinear springs was set equal to 5 kN/mm while their assigned tensile stiffness was negligible.

Figure 6.

Two-spring model replicating sheeting.

The out-of-plane and longitudinal displacements (degrees of freedom (dofs) 1 and 3) of the slave nodes of the two springs were equated with the equivalent displacements of their corresponding nodes on the beam to ensure that the springs were always directly above the beam while their in-plane displacement (dof 2) was equated with the displacement of the node located at the web–flange junction to prevent mobilisation of the stiffness of the springs due to global bending – see Figure 6.

Connection between purlins: bolt characteristics

The connection at the central support of two-span systems with two purlins comprises either a short inverted length of purlin (i.e. a sleeve) bolted to both purlins in sleeved systems or one purlin inverted relative to the other, with the two purlins overlapping and being bolted together in overlapped systems; note that purlins are manufactured in such a way that they can be easily nested into each other – see Figure 7.

Figure 7.

(a) Mesh density and (b) cross-section of sleeved and overlapped two-span systems at the central support.

The bolts employed for tying the two overlapping purlins or sleeves have been replicated with ‘SPRING2’ nonlinear spring elements (ABAQUS Inc., 2013), while their assigned load-deformation behaviour, presented in equation (8) and shown in Figure 8, has been derived by Ho and Chung (2006) based on experimental data for bolted connections obtained from lapped shear tests (Ho and Chung, 2004)

F_{b} = a_{b} dt f_{u}

(8)

where t is the thickness of the steel section, d is the bolt diameter, f_u is the ultimate tensile strength of the material and a_b is a strength coefficient, expressed by equation (9)

a_{b} = {\begin{matrix} 30 δ_{o} \\ 1.25 (δ_{o} - 0.02) + 0.6 \\ 0.85 \ln (\frac{δ_{o} - 0.05}{0.35}) + 1.075 \end{matrix} \begin{matrix} for δ_{o} \leq 0.02 mm \\ for 0.02 \leq δ_{o} \leq 0.40 mm \\ for δ_{o} > 0.40 mm \end{matrix}

(9)

where δ_o is the deformation of the fastener. Note that for the numerical simulations presented herein d = 16 mm and f_u = 800 MPa.

Figure 8.

Load–deformation relationship assigned to nonlinear springs.

The in-plane and longitudinal displacements of the connected nodes were controlled by the spring characteristics, while their out-of-plane displacements were equated using the command *EQUATION (ABAQUS Inc., 2013).

The contact interaction between the two purlins in the overlapping length has been modelled using surface-to-surface hard contact with the Coulomb friction coefficient μ set equal to 0.3, while the optional parameter ADJUST = 0.0 has been used in conjunction with the *CONTACT PAIR command to prevent over-closure of the two surfaces after the inclusion of geometric imperfections in the model.

Element type and mesh

The general purpose four-noded three-dimensional S4R shell elements with reduced integration and hourglass control (ABAQUS Inc., 2013) were chosen for the modelling of the cold-formed steel purlins as this type of elements has been found capable of yielding accurate replication of the observed physical behaviour in previous studies (Haidarali and Nethercot, 2011; Hui et al., 2016; Natario et al., 2014a). The mesh density, shown in Figure 7 and chosen according to the principles described by Yu (2005), Ren (2012) and Natario et al. (2014b), was sufficiently fine to limit the propagation of hourglass modes securing accurate replication of the observed physical phenomena while keeping the computational time within reasonable limits. Note that the longitudinal mesh size was kept constant at 10 mm.

Geometric imperfections

The thin-walled nature of cold-formed steel members, making them susceptible to local instabilities, renders modelling of the initial geometric imperfections necessary for the accurate prediction of their capacity and post-buckling behaviour. Hence, for the numerical simulations described herein, geometric imperfections were generated by directly specifying the deformed geometry in the FE models. The local and distortional buckling mode shapes were first identified from the signature curve of CUFSM (Li and Schafer, 2010) and then the two modes were distributed longitudinally, through sinusoidal functions with periods equal to the corresponding critical wavelengths along the member length and superposed. The amplitudes employed for scaling the local and distortional buckling mode shapes were 0.1t and 0.3t respectively, as illustrated in Figure 9, where t is the thickness of the nongalvanised steel section, according to previously performed experimental (Kyvelou et al., 2017) and industrial measurements (Boutell and Hui, 2013). Slightly larger mean imperfection values have been measured in other studies (Schafer and Pekoz, 1998).

Figure 9.

Amplitudes of (a) local and (b) distortional buckling mode shapes.

Loading and solution scheme

The adaptive automatic stabilisation scheme of ABAQUS (ABAQUS Inc., 2013) has been employed for the conducted analyses since the highly unstable post-buckling behaviour of cold-formed steel members in conjunction with the wide contact surface between the two purlins rendered the response of the model numerically unstable and, hence, the use of the typically employed modified Riks solver ineffective.

A five-tier whiffletree beam system has been modelled, distributing the load P applied to its top to 32 equally spaced points along the length of the purlin system (P′= P/32), approximating with high accuracy a uniform distributed load – see Figure 10. The whiffletree system was modelled with B31 linear beam elements, which were assigned a high stiffness – by setting an artificially high value of Young’s modulus – to avoid excessive bending deformations, while the joints between the elements were modelled as pins (*MPC, PIN) to ensure an even distribution of loading along the levels of the whiffletree and, ultimately, on the purlins.

Figure 10.

Whiffletree developed for loading of the sleeved and overlapped two-span purlin systems.

Validation of FE models

Prior to undertaking parametric studies, the single-span and two-span sleeved FE models were validated against a library of reported physical tests (Bryan and Davies, 1979, 1980a, 1980b; Bryan and Deakin, 1989; Davies and Deakin, 1992; Deakin, 1991a, 1991b; Deakin and Melbourne, 1997; Leach, 1990), covering Zeta I, Zeta II and Zed sections – see Figure 1. Additional validation against tests on overlapped single-span systems comprising Zed sections reported by Ho and Chung (2004) was also carried out. A summary of the obtained results in terms of the mean FE/test capacity and coefficient of variation (COV) for each set of tests is presented in Table 1. The overall validation results indicate a good agreement of the numerical predictions with the test data with minimal scatter. The buckling behaviour and the deflected shapes also match the observed failures modes; a typical comparison is presented in Figure 11 based on tests reported by Leach (1990). Comparison of the load–deformation response of two typical specimens as obtained by numerical simulations and as observed in the corresponding physical tests is also shown in Figure 12.

Table 1.

Summary of comparisons between finite element and test results.

Test type	Section type	Number of tests	Mean M_u,FE/M_u,test	COV of M_u,FE/M_u,test
Single-span sleeved	Zeta 1, Zeta 2, Zed	34	1.00	0.07
Two-span sleeved	Zeta 1	9	0.97	0.07
Single-span overlapped	Zed	19	1.01	0.08

COV: coefficient of variation.

Figure 11.

Comparison of typical observed failure mode of two-span sleeved system from physical test (Leach, 1990) and numerical simulation.

Figure 12.

Comparison of load–displacement response from FEM and physical test: (a) sleeved system and (b) overlapped system.

Parametric studies and results

Examined systems

Following validation of the numerical models, a series of parametric studies have been performed on two-span sleeved and overlapped purlin systems in order to examine their structural behaviour and investigate the degree of moment redistribution present. The examined two-span systems comprised cold-formed steel sections of four different depths and six different thicknesses, while, for each section, three alternative lengths have been examined: 4, 6 and 8 m; the corresponding single-span reference systems (see Figure 5) have also been analysed. A summary of the examined systems is presented in Table 2, while a typical modelled cross-section is illustrated in Figure 13. The identification system of the examined specimens begins with the letter S or L for sleeved or overlapped systems, respectively, followed by the number corresponding to the height of the steel section, then the thickness of the section and finally the length of each span. For instance, the system denoted S12516-8 refers to a sleeved two-span system comprising a steel beam of 125 mm height and 1.6 mm thickness with a length of 8 m for each span. Details of the overlapping lengths and bolt arrangements employed for the examined sleeved and overlapped systems are presented in Figure 14 and Table 2.

Figure 13.

Typical cross-section of examined purlin systems.

Figure 14.

Bolt arrangements of sleeved and overlapped systems at the central support.

Table 2.

Summary of examined purlin systems.

Specimen	h (mm)^a	t (mm)^a	b_t (mm)^a	b_b (mm)^a	l_o (mm)^b	Bolt code^b	Cross-sectional slenderness ${\bar{λ}}_{cs, - ve}$
L12520	125	2.0	60	50	900	A	0.521
L12516	125	1.6	60	50	900	A	0.594
L20020	200	2.0	72	65	1400	B	0.626
L17518	175	1.8	72	65	1200	A	0.663
L15015	150	1.5	72	65	1300	A	0.732
L20013	200	1.3	72	65	1400	B	0.801
S12520	125	2.0	60	50	700	A	0.521
S12516	125	1.6	60	50	700	A	0.594
S12513	125	1.3	60	50	700	A	0.669
S17516	175	1.6	72	65	980	A	0.710
S20016	200	1.6	72	65	1000	C	0.712
S15015	150	1.5	72	65	820	A	0.732
S20013	200	1.3	72	65	1000	C	0.801

Refer to Figure 13 for details on cross-sectional dimensions.

Refer to Figure 14 for details on bolt arrangements.

For two-span sleeved and overlapped systems, the cross-sectional slenderness of the purlin located at the central support (subjected to negative moment) can be calculated according to equation (10), accounting for the double thickness of the section as well as for the cross-sectional slendernesses of both the standard orientation and inverted purlin or sleeve

{\bar{λ}}_{cs, sl, - ve} = {\bar{λ}}_{cs, lp, - ve} = \frac{1}{\sqrt{2}} average ({\bar{λ}}_{cs, - ve, 1}; {\bar{λ}}_{cs, - ve, 2})

(10)

where ${\bar{λ}}_{cs, - ve, 1}$ and ${\bar{λ}}_{cs, - ve, 2}$ are the cross-sectional slendernesses of the purlin in the standard and inverted orientations, respectively.

The material properties employed for the cold-formed steel are presented in Table 3 and are the same as those employed by Hui et al. (2016) in order to allow a direct comparison of the results. In Table 3, ν is Poisson’s ratio of the steel material, E is Young’s modulus, σ_0.2 is the yield (0.2% proof) strength for the flat parts of the cross-section, σ_u is the ultimate tensile strength and n and n′_0.2,1.0 are the strain hardening exponents for the two-stage Ramberg–Osgood material model, presented in equations (5) and (6) of this article.

Table 3.

Material characteristics of cold-formed steel.

Young’s modulus E (GPa)	Poisson’s ratio ν	Yield strength σ_0.2 (MPa)	Exponent n	Exponent n′_0.2,1.0
210	0.3	390	20	2

Observed structural behaviour

The observed structural behaviour of the examined systems, which was generally found to be in accordance with the descriptions given by Hui et al. (2016), is presented herein. The behaviour of a two-span system in which the purlin cross-section is sufficiently stocky for considerations of local and distortional buckling not to significantly impair its ability to maintain its cross-sectional capacity is presented in Figure 15. With increasing load, the moment at the central support increases to its maximum value M_sup, which can be approximated by its reference moment capacity M₃. With the moment capacity having being attained at the central support, the moment there remains essentially constant, with the effect of increasing load causing redistribution, whereby the span moment continues to increase until it reaches its cross-sectional capacity, approximated by M₄. It is only after the maximum capacity has been attained within the span that unloading of the system is initiated. The progressive development of the support and span moments with increasing load is shown in Figure 15, where M_sup and M_sp are the moments at the support and in the span, respectively, M₃ and M₄ are the cross-sectional capacities of the support and span regions, respectively, as determined by the reference single-span models, P is the total load applied to the purlins through the top of the whiffletree and δ_u is the midspan displacement corresponding to the failure load P_ult. Note that P = q L, where q is the uniformly distributed load applied to the two-span system, while L is the length of each span.

Figure 15.

Typical support and span moment–midspan displacement responses for systems comprising stocky cross-sections (L12516-6).

If, however, the cross-sectional behaviour is influenced by considerations of local and distortional buckling, the behaviour of the system is affected. As shown in Figure 16, for the support region the full moment capacity of the cross-section is almost reached but, having attained a maximum value, it gradually decreases while the moment in the span builds to its maximum value, which can be approximated by the moment capacity of the single-span beam under uniform bending, M₄. Thus, while eventual failure (i.e. the peak load of the system) still corresponds to attainment of the full moment capacity within the span, this is not accompanied by a level of support moment equal to the full moment capacity of the support region.

Figure 16.

Typical support and span moment–midspan displacement responses for systems comprising slender cross-sections (L20013-6).

Finally, in some cases of overlapped two-span systems, it has been observed that the enhanced capacity in the overlapping region exceeds the higher moment that it attracts due to the system continuity, causing the span moment to become critical. In this case, as shown in Figure 17, with increasing load, the moment within the span M_sp reaches its cross-sectional capacity M₄ before M_sup reaches its reference moment capacity M₃. Hence, redistribution does not occur, with the system failing at the elastically distributed moment levels illustrated in Figure 18. Note that the described span-critical behaviour was exhibited only by the overlapped systems as, for the sleeved systems, failure always occurs at the central support due to the single thickness of the sleeve locally – see Figure 19.

Figure 17.

Typical support and span moment–midspan displacement responses for span-critical systems (L17518-4).

Figure 18.

Support and span moments at ultimate load for span-critical scenarios.

Figure 19.

Single cross-sectional thickness of sleeve locally at central support.

A summary of the observed failure modes is shown in Figures 20 to 22. All the sleeved systems exhibited the same type of failure, with local and distortional buckling initially developing at the web and bottom flange of the sleeve located at the central support (region A of Figure 20), followed by distortional buckling occurring at the top flange of the purlin located within the span (region B of Figure 20), where failure was reached. Similarly, for the overlapped systems with their critical cross-section being located near the central support, distortional buckling was initially observed at the bottom flange of the purlins located towards the edge of the overlapping region, within the purlin length subjected to negative moment (region A of Figure 21). Ultimately, the load-carrying capacity of the system was reached with distortional buckling occurring at the top flange of the purlin located within the span of the system and subjected to sagging moment – see Figure 21. However, for the span-critical overlapped systems presented in Figure 17, failure of the system occurred with distortional buckling developing at the top flange of the purlin located within the span with buckling not having yet been initiated along the parts of the purlins under negative moment – see Figure 22.

Figure 20.

Typical failure mode of sleeved systems (S20016-8).

Figure 21.

Typical failure mode of overlapped support-critical systems (L12516-6).

Figure 22.

Typical failure mode of overlapped span-critical systems (L17518-4).

Design method – the alpha framework

The applicability of the design method described in section ‘Design approach: the alpha (α) framework’, which has been devised for bare two-span systems under gravity loading (Hui et al., 2016), is examined herein for sleeved and overlapped two-span purlin systems. In order to use this approach, it is necessary to know

An estimate of the moment capacity of the support region, M₃;

An estimate of the moment capacity of the span region, M₄;

The reduction factor for the support moment at the ultimate load of the system, α_d.

For the examined systems, the values of M₃ and M₄ have been determined by FE analyses using the statically determinate arrangements of Figure 5, while the drop-off of the moment at the support α_d has been calculated according to equation (2). M₃ and M₄ could alternatively be obtained through physical testing or design calculations. Moment values of α_dM₃ for the support region and M₄ for the span region have been then used directly with equation (4), which is the basic equation of moment equilibrium for the system, to determine its load-carrying capacity q_d,α, assuming that moment redistribution can occur

M_{4} = \frac{{(q_{d, α} L^{2} + 2 α_{d} M_{3})}^{2}}{8 q_{d, α} L^{2}}

(11)

To consider the span-critical cases, equation (4) is used with the moment at the support and span regions taken as 16M₄/9 and M₄, respectively, according to the elastic moment distribution of Figure 18 for a bare two-span system under a uniformly distributed load, to determine the load-carrying capacity of the system q_d,sp when no moment redistribution occurs. Note that, although the ratio of the support to span moments M_sup/M_sp depends on the distribution of stiffness along the member length, which will vary in the support region due to the sleeve/overlapping arrangement, for the examined systems, the M_sup/M_sp ratio was found to be approximately equal to the one assuming that the cross-section is of uniform thickness along the length of the beam (within a 2% difference on average). It was therefore decided that the elastic ratio M_sup/M_sp = 16/9 yields reasonable predictions for the moment distribution within the examined systems and can therefore be adopted. The ratio of moments could alternatively be determined based on the elastic response of the sleeved or overlapped system from the FE simulation

M_{4} = \frac{{(q_{d, sp} L^{2} + 2 (16 M_{4} / 9))}^{2}}{8 q_{d, sp} L^{2}}

(12)

Hence, the load-carrying capacity of the system q_d can be taken as the minimum value between q_d,α and q_d,sp

q_{d} = min (q_{d, α}; q_{d, sp})

(13)

Table 4 presents the comparisons between the ultimate load-carrying capacities q_FE obtained from the conducted FE analyses of sleeved two-span systems and those derived in accordance with equation (13), with a mean q_FE/q_d ratio of 1.01 and a COV of 0.04. The equivalent comparisons for the examined overlapped two-span systems are presented in Table 5, with a mean q_FE/q_d ratio of 1.01 and a COV of 0.04. The moment at the support M_sup and at the span M_sp at ultimate load, the reduction factor alpha accounting for the drop-off in moment at the support region α_d calculated according to equation (2) as well as the single-span reference moment capacities M₃ and M₄ are also reported in Tables 4 and 5. The span-critical systems are marked with an asterisk (*) in Table 5.

Table 4.

Numerical results and comparison with the proposed design method for sleeved two-span systems.

Specimen	M_sup (kN m)	M_sp (kN m)	M₃ (kN m)	M₄ (kN m)	q_FE (kN/m)	α _d	q_d (kN/m)	q_FE/q_d
S12520-4	−9.44	8.67	−9.32	8.98	6.38	1.00	6.61	0.96
S12520-6	−8.88	8.74	−9.14	8.98	2.89	1.00	2.92	0.99
S12520-8	−9.40	8.71	−9.11	8.98	1.65	1.00	1.64	1.00
S12516-4	−7.15	6.79	−6.57	6.98	4.92	1.00	5.00	0.98
S12516-6	−6.41	6.66	−6.59	6.98	2.17	1.00	2.22	0.98
S12516-8	−6.72	6.70	−6.51	6.98	1.24	0.94	1.23	1.01
S12513-4	−4.55	5.18	−4.83	5.38	3.72	1.00	3.80	0.98
S12513-6	−4.72	5.14	−4.77	5.38	1.66	0.90	1.64	1.01
S12513-8	−4.83	5.20	−4.73	5.38	0.95	0.78	0.89	1.07
S17516-4	−11.91	11.02	−13.00	11.52	8.34	0.99	8.67	0.96
S17516-6	−12.62	11.45	−12.93	11.52	3.87	0.91	3.76	1.03
S17516-8	−12.14	11.26	−12.91	11.52	2.15	0.84	2.06	1.04
S20016-4	−14.20	12.19	−15.92	13.85	9.80	1.00	10.53	0.93
S20016-6	−15.16	12.82	−15.96	13.85	4.54	0.94	4.59	0.99
S20016-8	−14.53	12.61	−15.95	13.85	2.46	0.87	2.53	0.97
S15015-4	−8.71	7.79	−8.97	8.67	6.32	0.92	6.23	1.01
S15015-6	−8.76	8.52	−8.88	8.67	2.83	0.84	2.69	1.05
S15015-8	−9.01	8.50	−8.80	8.67	1.60	0.75	1.47	1.09
S20013-4	−10.72	9.10	−11.60	10.55	7.26	0.84	7.52	0.97
S20013-6	−10.97	9.74	−11.64	10.55	3.46	0.79	3.28	1.05
S20013-8	−10.45	9.87	−11.59	10.55	1.86	0.73	1.81	1.03
							Average	1.01
							COV	0.04

COV: coefficient of variation.

Table 5.

Numerical results and comparison with the proposed design method for overlapped two-span systems.

Specimen	M_sup(kN m)	M_sp(kN m)	M₃(kN m)	M₄(kN m)	q_FE(kNm)	α _d	q_d(kNm)	q _FE q _d
L12520-4	−17.61	9.62	–	8.98	8.65	1.00	–	–
L12520-6	−14.41	8.77	−13.12	8.98	3.36	1.00	3.29	1.02
L12520-8	−12.83	8.86	−11.90	8.98	1.82	1.00	1.79	1.02
L12516-4	−13.42	7.34	–	6.98	6.60	1.00	–	–
L12516-6	−11.30	6.87	−10.10	6.98	2.63	1.00	2.55	1.03
L12516-8	−10.01	6.45	−9.06	6.98	1.36	0.94	1.35	1.00
L20020-4	−36.02	17.19	−38.18	18.58	16.36	1.00	16.51*	0.99
L20020-6	−35.15	18.03	−33.41	18.58	7.37	1.00	7.34*	1.00
L20020-8	−28.83	16.40	−28.84	18.58	3.63	1.00	3.92	0.93
L17518-4	−25.95	12.95	−28.02	13.56	12.09	1.00	12.05*	1.00
L17518-6	−25.00	13.72	−22.76	13.56	5.47	0.97	5.24	1.05
L17518-8	−21.07	12.03	−19.95	13.56	2.66	0.86	2.73	0.97
L15015-4	−16.53	8.05	−17.66	8.60	7.59	0.92	7.65*	0.99
L15015-6	−16.66	8.68	–	8.60	3.54	0.84	–	–
L15015-8	−14.11	7.81	−13.13	8.60	1.75	0.75	1.63	1.07
L20013-4	−18.41	8.95	−20.96	10.55	8.45	0.84	9.16	0.92
L20013-6	−18.89	10.39	−19.90	10.55	4.14	0.79	3.89	1.07
L20013-8	−16.80	9.64	−16.40	10.55	2.13	0.73	2.00	1.07
							Average	1.01
							COV	0.04

COV: coefficient of variation.

Span-critical systems

The proposed design method was found to be capable of accurately predicting the load-carrying capacity of sleeved and overlapped two-span purlin systems and could potentially be expanded to systems comprising alternative cross-sectional shapes, provided that the cross-sectional slenderness can be accurately evaluated.

Comparison of bare, sleeved and overlapped systems

In Table 6, comparisons of the load-carrying capacities of the examined bare, sleeved and overlapped systems (q_br, q_sl and q_lp, respectively) normalised by the corresponding volume of the material used (V_br, V_sl and V_lp, respectively) are made. Note that the capacities of the bare two-span purlin systems q_br employed for the presented comparisons were reported by Hui et al. (2016). The systems achieving continuity between the two spans through overlapping arrangements were found to be significantly more efficient than the equivalent bare and sleeved two-span systems, with increased capacities (normalised by material usage) of 12% and 19%, respectively. This can be explained with reference to the double thickness of the cross-section located along the central support due to the overlapping purlins being able to withstand more load than the cross-section with single thickness. Furthermore, overlapping requires fewer components than the sleeved systems and hence may be more efficient from a practical/construction viewpoint.

Table 6.

Comparison of efficiency of bare, sleeved and overlapped purlin systems.

Specimen	q_br (kN/m)	q_sl (kN/m)	q_lp (kN/m)	$\frac{q_{lp} / q_{br}}{V_{lp} / V_{br}}$	$\frac{q_{lp} / q_{sl}}{V_{lp} / V_{sl}}$
12520-4	6.95	6.38	8.65	1.12	1.36
12520-6	2.93	2.89	3.36	1.07	1.16
12520-8	1.58	1.65	1.82	1.09	1.11
12516-4	–	4.92	6.60	–	1.34
12516-6	–	2.17	2.63	–	1.21
12516-8	–	1.24	1.36	–	1.10
15015-4	–	6.32	7.59	–	1.20
15015-6	–	2.83	3.54	–	1.25
15015-8	–	1.60	1.75	–	1.09
20013-4	7.02	7.26	8.45	1.08	1.16
20013-6	3.09	3.46	4.14	1.25	1.20
20013-8	1.76	1.86	2.13	1.14	1.14
			Average	1.12	1.19
			COV	0.05	0.07

COV: coefficient of variation.

Concluding remarks

The findings from a numerical investigation into the degree of moment redistribution occurring within typical two-span cold-formed steel purlin systems have been presented in this article. Two-span purlin systems achieving continuity by employing sleeved and overlapped arrangements have been modelled, their structural behaviour has been investigated and the applicability of plastic design to these systems has been examined. It was found that moment redistribution is possible within these systems, usually accompanied by a reduction in moment capacity at the central support. Overlapping of purlins along the central support was found to be the most efficient way of ensuring continuity for a two-span purlin system, yielding increased capacities over continuous bare and sleeved systems. Finally, based on the conducted parametric studies, a previously devised method for the design of continuous bare purlin systems, making direct use of cross-section capacities at key locations, together with a factor to allow for the fall-off in moment at the central support, has been assessed and advanced for application to sleeved and overlapped systems.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The authors are grateful to Ayrshire Metal Products for their financial and technical contributions to the project.

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