Rotational Restraint to Purlins Provided by Standing Seam Roof Systems

The ﬁnite element model used for analyzing the rotational restraint rigidity of standing seam roof systems was developed. The inﬂuences of diﬀerent factors on the rotational restraint rigidity provided by two types of standing seam roof systems were studied. The variables include local deformation of standing seam roof panels, panel thickness, clip tab thickness, and the relative sliding of clip tab and clip base. The restraint mechanism of standing seam roof systems to the purlins was studied. It is shown that the rotational restraint rigidity provided by the two types of researching standing seam roof systems mainly depends on the slide tab thickness and the roof panel thickness. Finally, formulae for calculating rotational restraint rigidity of the LSIII and SS360 standing seam roof systems were also proposed based on parametric analysis results.


Introduction
Cold-formed purlins are widely used in metal buildings due to their economy, ease of fabrication, and high strength-toweight ratios. However, these sections are weak in the lateral direction and in torsion [1]. Previous work shows that conventional roof panels which are directly through-fastened to purlins by self-drilling screws can provide full lateral bracing and some extent of rotational restraint to the purlins by virtue of their shear rigidity and resistance to local bending at the connections [1][2][3][4][5]. In order to accurately calculate the uplift capacity of purlins with top flange through-fastened to roof panels, a series of tests were conducted by several scholars [1][2][3][4][5][6][7][8][9][10][11] to determine the rotational restraint to purlins provided by the purlin-panel systems, and the test set-up and procedure were proposed by Celebi et al. [1], improved by Pekoz et al. [3], and adopted by Eurocode 3-1-3 [12]. Katnam et al. [13,14] developed a nonlinear finite element model to estimate the rotational restraint provided by the first and second generation trapezoidal sheeting to the attached purlin. e model was applicable to trough-fixed and crest-fixed single skin purlinsheeting systems, and the performance of which was validated to be in good agreement with test results. It opened the way to the development of a design method for estimating the rotational restraint provided by the sheeting to cold-formed steel purlins. To determine the rotational restraint that sheathing provides to the flange of a cold-formed steel floor joist or stud, a series of cantilever tests on joist/ stud-sheathing assemblies were conducted by Schafer [15,16]. e tests demonstrated that the rotational stiffness may be decomposed into two parts: connector and sheathing. e connector stiffness was due to the rotation of the fastener in the flange of the cold-formed steel member and was most significantly influenced by the thickness of the cold-formed steel member itself. e sheathing stiffness was due to bending of the sheathing itself and may be highly variable.
e results formed the basis for a new design method adopted in American standards (AISI-S210-10) [17] for incorporating restraint into design strength predictions for the distortional buckling mode. Gao and Moen [18,19] proposed mechanics-based equations for predicting the rotational restraint provided by through-fastened metal panels to Z-and C-section girts or purlins. e prediction equations were validated by conducting rotational restraint experiments, and the through-fastened connection stiffness was also simulated in a finite strip elastic buckling analysis with a rotational spring to demonstrate how system effects can be included in design.
In recent years, standing seam roof systems are very prevalent since they are well adapted to the thermal expansion and contraction deformation caused by temperature changes. In these roof systems, standing seam roof panels are fastened to purlins through sliding clips, so that movement of roof panels relative to the purlins is permitted. erefore, the behavior of purlins in these roof systems is directly related to the rotational restraint provided by the standing seam roof systems. Liu et al. [20] studied the rotational restraint to purlins provided by standing seam roof systems through 28 groups of tests, and it was confirmed that the total rotational rigidity of standing seam roof systems is formed by connecting the rotational rigidity of roof panels, clips, and purlins in series. However, because there are many types of standing seam roof systems, and the roof panel profile, standing seam configuration and sliding clip details are different, and it is difficult to determine all of the rotational restraints provided by these roof systems through tests. So far, the rotational restraint rigidity to purlins provided by the LSIII and SS360 standing seam roof systems has not been specially studied. us, in this paper, the rotational restraint to purlins provided by two types of standing seam roof systems widely used in China was analyzed using finite element models, and the influence factors were also investigated and formulae for calculating rotational restraint rigidity were also proposed and verified.

Research Object
Two types of standing seam roof systems widely used in China were taken as the research objects. One type was the LSIII standing seam roof system consisted of LSIII standing seam roof panels fastened to the purlins with LS003 clips. Another type was SS360 standing seam roof system consisted of SS360 standing seam roof panels fastened to the purlins with S3PC-1 clips. Both of these roof panels, although from different manufacturers, had very similar profiles. e panel sidelaps were mechanically seamed in the field to form the 450°and 360°seams for LSIII and SS360 roof systems, respectively. e yield stress of both roof panels as reported by the manufacturers was 345 MPa, and the mean measured thickness including coating was 0.6 mm. e LS003 clip was composed of a clip base 2.0 mm in thickness and a slide tab 1.0 mm in thickness, the slide tab was 110 mm in length and attached to the clip base with an interlocking design that permits relative movement between them. e S3PC-1 clip consisted of an L-shaped clip base 2.6 mm in thickness and a thinner slide tab (0.8 mm in thickness), the L-shaped clip base was 115 mm in length with a rectangular slot on the vertical plane, the slide tab was attached to the clip base at the slotted interface, which allows the movement of the slide tab within the confines of the slot length (91 mm). Both the clip bases were fastened to purlins with two self-drilling screws, and the dimensions and connection details are shown in Figures 1-3. Z-purlins were used in the experimental program. e angle between the lip and flange is 45°for all the Z-sections. e cross-section dimensions of the purlins (height, flange width, and thickness) varied between tests.

Finite Element Analysis
Using the general purpose finite element analysis (FEA) program ANSYS, finite element models used for calculating the rotational restraint rigidity provided by standing seam roof systems were developed according to [20], nonlinear analysis was performed, and the rotational restraints on purlins provided by the two types of standing seam roof systems were also analyzed. Figure 4, the finite element model consisted of 3 pieces of roof panels supported by two lines of purlins at ends with one row of clips connected at the midspan of the roof panels. A pair of concentrated forces with same value and opposite direction was applied at two screw connection points of each clip base to simulate the torque transmitted from the purlin. e standing seam roof panels and clips were modeled using the SHELL 181 element, which is a 4-node shell element with six degrees of freedom at each node. And based on reasonable consideration of the stress stiffening, large rotation, and large strain, the SHELL 181 element is well suited for analyzing thin to moderately thick shell structures. e finite element mesh size of the models was investigated to provide both accurate and time-efficient results. e standing seam roof panels and clips were all modeled as nonlinear materials using the ideal elastic-plastic model, the yield stress of which was obtained from product reports provided by the manufacturers. e full-scale tests on wind uplift capacity of purlins supporting the standing seam roof systems presented by Luan and Li [21,22] showed that the mechanical seams of the panel sidelaps were fixed tightly, and no failure or separation were observed. erefore, the translation freedoms of the corresponding nodes between the panels and the slide tabs were also coupled as shown in Figure 5. Considering that the movement distance of the slide tab relative to the clip base was generally not very large in the tests, to simplify the finite element model, it is assumed that no relative movement had occurred between the slide tab and clip base; therefore, along the longitudinal direction of the roof panel, the translation freedom of the corresponding nodes was also coupled. e influence of the relative movement was also specially studied. e standing seam roof panels were supported by two C-purlins at the ends, and the roof panels were connected with purlins through sliding clips, so the translation freedoms of the corresponding nodes between the clip base and supporting purlins were also coupled. e purlins were assumed to be simply supported, to simulate the boundary condition, and at both ends of each purlin, the translation in the vertical direction of the central point and the translations in the lateral direction of the web line points were constrained. e translation in the longitudinal direction of the central point at one end of the purlin was also constrained to avoid rigid body displacement. e element types, material models, and boundary conditions were exactly the same as that being used in the full model presented by Luan and Li [21,22], in which the accuracy of the full model was verified by a comparison between the experimental results and the finite element results, and it is shown that good agreement is achieved between the experimental results and finite element results for the flexural strength and the failure modes. e difference of purlin flexural strength obtained from tests and finite element analysis was smaller than 5%. e load-displacement curves obtained from the finite element analysis were all in great agreement with the test results. erefore, the finite element models are reliable and can be used for further researches.

Calculation of Rotational Restraint
Rigidity. Using the finite element analysis results, the rotational restraint rigidity (K) provided by the standing seam roof system can be calculated as follows:   where T is the torque applied on the clip base, θ is the corresponding rotation angle of the clip base, F is the concentrated force applied at the screw points, s is the space between the screws, δ 1 and δ 2 are the corresponding displacements at two screw connection points. e rotational restraint rigidity on the unit length of purlin (K tor ) can be calculated by where w rf is the width of the standing seam roof panel.

Finite Element Analysis
Results. Using the FE model, the rotational restraints' rigidity to purlins provided by the two types of standing seam roof systems studied in this paper was analyzed. e thickness of the roof panel in both types of roof systems was 0.6 mm, and the panel dimensions and connection details were exactly the same as shown in Figures 1-3. Figure 6 shows the deformation and stress distribution in the roof panel. Figure 7 shows the deformation and stress distribution of two types of clips. As can be seen, the bending stiffness of the two types of standing seam roof panels is large enough that the clips always fail before roof panels, and the S3PC-1 clip is much easier to deform because the slide tab of which is thinner.

Analysis of Influence Factors.
eoretically, the rotational restraint rigidity provided by the standing seam roof systems is mainly related to the bending stiffness of roof panels and the sliding performance of clips. e bending stiffness of roof panels generally depends on the thickness and the cross-section profile of the roof panel, and the sliding performance of clips is mainly determined on the thickness of the slide tab. Considering that significant outof-plane bending deformation of the flat part of the roof panel was observed in the tests, the out-of-plane deflection should also be seen as a factor causing the change of the panel cross-section profile. erefore, using the finite element model, the influence of different factors on the rotational restraint rigidity provided by the two types of standing seam roof systems was studied. e variables include the relative movement between the slide tab and clip base (S tb ), the out-of-plane deflection at mid-point of the roof panel cross-section (D rf ), the roof panel thickness (t rf ), and slide tab thickness (t st ). Analysis results are shown in Figure 8. It is observed that the rotational restraint rigidity mainly depends on the slide tab thickness and the roof panel thickness because the slide tab is the weakest link in the rotational restraint transmission path of the standing seam roof system, and the performance of the mechanical seams connecting the slide tab and the roof panels is closely related to the thickness of roof panel and slide tab. e out-of-plane bending deformation of roof panel and the relative movement between the slide tab and clip base almost have no influence on the rotational restraint rigidity, except for LS003 clip, when the relative movement distance is greater than 40 mm, and half of the slide tab separates from the clip base; thus, the rotational restraint rigidity sharply decreases, however, which is not expected to occur in the actual engineering.

Formulae for Rotational Restraint Rigidity
Considering the significant influence on rotational restraint rigidity from the thickness of roof panel and slide tab, further parametric finite element analysis was conducted. Based on the parametric analysis results, formulae for calculating rotational restraint rigidity of the LSIII and SS360 standing seam roof systems were generalized as follows: where t rf is the thickness of the roof panel, t st is the thickness of the slide tab, and E is Young's modulus.      Figure 8: Correlation between the rotational restrain rigidity K tor and factors: (a) between K tor and S tb , (b) between K tor and D rf , (c) between K tor and t rf , and (d) between K tor and t st .

Advances in Civil Engineering
To verify the accuracy of the formulae, the rotational restraint rigidity calculated by the proposed formulae was compared with the test results of similar standing seam roof systems by Liu et al. [20]. However, the rotational restraint rigidity obtained from the tests by Liu et al. [20] is total rigidity provided by roof panel, clip, and purlin (K z ), which can be calculated by where K z1 is rotational restraint rigidity provided by the roof system, which is the same as K tor in this paper and K z2 is rotational rigidity of purlin, which is given by where E is Young's modulus, t is purlin thickness, v is Poisson's ratio, b 1 is the distance from the inside selfdrilling screw to purlin web, b 2 is the space between two self-drilling screws, h is purlin height, and α is the rigidity reduction factor of the top flange of purlin, for LSIII roof system, it is 0.4 and for HX-478 roof system, it is 0.5. e calculation results of Equations (3) and (4) cannot be compared with the test results directly. erefore, based on the rotational restraint rigidity of roof system calculated by Equation (3) or Equation (4), using Equations (5) and (6), the total rotational restraint rigidity was calculated and compared with the test results (K zt ) by Liu et al. [20]. Table 1 shows the comparison results of LSIII roof system, and it is observed that the calculation results are in good agreement with the test results. e comparison results of SS360 roof system used in this paper and HX-478 roof system tested by Liu et al. [20] are listed in Table 2. It can be seen that the largest difference between the calculation results and the test results reaches about 30%, which is because although the configuration and working mechanism of SS360 roof system and HX-478 roof system are exactly the same, there are still some differences between them, such as the width of roof panel and the dimensions of clips. And, in general, the comparison shows that the rotational restraint rigidity of standing seam roof systems calculated by the formulae proposed in this paper is reliable.

Conclusion
In order to study the rotational restraint rigidity provided by two types of standing seam roof systems widely used in China, the finite element model used for analyzing the rotational restraint rigidity of standing seam roof systems was developed based on a test verified full finite element model. e influences of different factors on the rotational restraint rigidity provided by two types of standing seam roof systems were also analyzed. e variables include local deformation of standing seam roof panels, panel thickness, clip tab thickness, and the relative sliding of clip tab and clip base. e analyses showed that the rotational restraint rigidity of standing seam roof systems mainly depended on the roof panel thickness and slide tab thickness. Formulae for calculating rotational restraint rigidity of the LSIII and SS360 standing seam roof systems were also proposed, and the comparison between formulae calculating results and test results showed that the rotational restraint rigidity of standing seam roof systems calculated by the formulae proposed in this paper is reliable.
Data Availability e datasets generated during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.