Effective static wind-induced force estimation for clips between purlins and metal panels of standing-seam metal roofing system

Abstract

This paper mainly focuses on the establishment of an effective static estimation method for the extreme wind-induced force for clips between purlins and metal panels of the standing-seam metal roofing system (hereinafter referred to as SMRS) of typical double-slope light-weight steel portal frame structure considering dynamic characteristics of wind and structure. First, simultaneous pressure measurement with rigid gable roof models was conducted mainly considering the length-span ratio in the boundary layer wind tunnel of Tokyo Polytechnic University, Japan. Then, finite element modeling for SMRS according to the wind load path in the roofing system was carried out to check the actual wind load of the clips based on the traditional calculation method provided in design codes, and the spatial correlation of fluctuating wind pressure on the roof surface, as well as the dynamic effect of the roof structure itself, had been considered. According to the related Chinese, American, and Japanese codes, a magnification coefficient based on the traditional method of static wind-induced force for the clips was calculated and compared. Finally, a simplified estimation method of effective wind-induced force for the clips in typical zones on the roof surface considering dynamic characteristics was proposed.

Graphical abstract

Keywords

effective static wind-induced force estimation / clips / standing-seam metal roofing system / dynamic characteristics

1 Introduction

In recent years, standing-seam metal roofing systems, SMRSs, have been widely used in modern metal roofs for residential, commercial, and even large-scale public buildings due to their advantages of great waterproof, wind-resistant, and temperature stress-releasing performance, etc., while the performance under extreme wind loading is of critical importance [1–3]. Generally, the connection detailing is given by the supplier of the roofing system according to the design parameters from structural engineers on the contract documents. Therefore, to a structural engineer, how to estimate an effective wind-induced force for designing clips between purlins and metal panels of SMRS considering dynamic characteristics, i.e., the unfavorable distribution and fluctuating characteristics of wind pressure, and the dynamic effect of the roof structure itself as well, is a key issue for the roof safety. Up to now, the tributary area of a clip, A_c, and the representative gust pressure, P_w, is used to estimate the design wind force of a clip, F, i.e., F = A_cP_w (MBMA, 2012 [4]; GB 51022-2015, 2016 [5]), and usually a standard checking test is prerequired to ensure the safety of any new roofing products. However, detachment failure of clips from roof metal panels is typical and often happens in practice. For example, in March 2013, about 200 m² of the metal roof of Beijing International Airport T3 was uplifted under a strong wind load [6]. Accordingly, further investigation into the accurate and effective wind-induced force estimation of clip connection is quite necessary. El Damatty et al. [7] conducted a theoretical study on the typical SMRS using the finite element program ANSYS based on experimental research at the University of Mississippi. It was found that under strong wind load, the clips bear 68.6% of the load transferred from the roof panels, and roof damage starts from the detachment of the clip and the seam. In 2019, Luan and Li [8] conducted the experimental research and numerical simulation on the purlins of SMRS. An initial work by Jing and Li [9] showed that a magnification coefficient to consider the dynamic vibration effect of roof panels under fluctuating wind on the design wind-induced force of clips according to typical locations on the roof surface was necessary in comparison with the widely-used tributary area method based on the existing wind tunnel test data. Therefore, an effective estimation method for the design wind force of clip by full dynamic analysis based on more reliable time-history data of wind pressure around clips in typical zones is in high demand in practice. In the current study, the dynamic characteristics of wind and structure will be fully considered by numerical simulation using wind pressure data from wind tunnel test with special denser arrangement of tapes around clips in typical zones, and a simplified estimation method of effective wind-induced force for the clips in typical zones on the roof surface considering dynamic effect was proposed in comparison with the related Chinese, American, and Japanese codes.

2 Experimental set-up of wind tunnel tests

The wind tunnel tests were carried out in the 1.8 m (height) × 2.2 m (width) × 19 m (length) Boundary Layer Wind Tunnel at Wind Engineering Research Center, Tokyo Polytechnic University (TPU). There are three test models in total, including one portal frame structure rigid model (M1) with measurement points on the roof surface, and two auxiliary models (M2, M3) without any measurement points. M2 and M3 are used as two modules by attaching them to M1 on the lateral surface, to obtain the wind pressure data on the roof surface of rigid models with L/B = 1:1, 2:1, and 3:1. The model sizes of M2 and M3 are the same as M1. The relevant parameters of 3 models are listed in Tab.1. There are 174 measurement points on the roof surface, and their distribution on the roof surface is shown in Fig.1. According to the Chinese code, Technical Specification for Steel Structure of Light-weight Buildings with Gabled Frames (GB51022-2015) [5], which is similar to the design codes for this kind of roofs in other countries [10-18], the roof is divided into three zones, namely the corner zone (Zone A), the edge zone (Zone B) and the center zone (Zone C), respectively. Three typical nodes were selected from Zone A, Zone B, and Zone C, and the set-up of measurement points around these three nodes is denser compared with other points. The total test procedure includes three cases, as shown in Tab.2, and the experiment models in the wind tunnel tests are shown in Fig.2.

For terrain type, Category II defined in Japanese code (AIJ-04) [10] was used in the test. The wind speed and turbulence profiles provided in different codes and measured during the test are compared in Fig.3, and the turbulence intensity at the eave height of models is about 0.18. The scale factors, including the model geometric scale, the design wind speed scale, the time scale, as well as the blockage ratio, are listed in Tab.3. As for wind direction, eight direction cases were considered, including 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315° (Fig.4).

3 Pressure distribution measured

Due to the symmetry of the roof surface, the distribution of average wind pressure coefficient and fluctuating wind pressure coefficient in three cases under the three wind direction angles of 0°, 45°, and 90° are analyzed. The pressure is defined as positive when the direction of wind pressure is directed to the roof surface, and vice versa. The average wind pressure coefficient, {\overline{C}}_{\mathrm{p}\mathrm{c}}, and the fluctuating wind pressure coefficient, C_{\mathrm{p}\mathrm{c}}, are defined as follows

(1){\overline{C}}_{\mathrm{p}\mathrm{c}}=\frac{\overline{P}}{q_{\mathrm{h}}},C_{\mathrm{p}\mathrm{c}}=\frac{\sigma }{q_{\mathrm{h}}},

where \overline{P}, σ, and q_h are the average value of wind pressure, the standard deviation of wind pressure, and the reference wind speed pressure, respectively [11].

The average wind pressure coefficient and the fluctuating wind pressure coefficient of M1 in three cases measured in this test are shown in Fig.5 and Fig.6.

According to Fig.5, the average wind pressure distribution in three cases is described as follows. Under wind direction angle 0°, the airflow separates at the junction between the upward wall and the upward roof, and a “separation bubble” is formed within a certain range. Therefore, a large negative pressure is generated at the eaves and the corner of the upward roof, whose absolute value gradually decreases along with the direction of the wind. The whole roof surface is under negative pressure. Under a wind direction angle of 45°, the airflow separates at the corner of the roof, where a small area of separation is formed and a pair of “conical vortexes” are formed on the sides of the separation area, so that the area around the “conical vortex” is subjected to a large negative pressure, whose absolute value gradually decreases along with the direction of the wind. The whole roof surface is under negative pressure. Under a wind direction angle of 90°, the average wind pressure coefficient is symmetrically distributed on the roof surface, and the airflow is separated at the junction of the gable and the roof. A “separation bubble” is formed within a certain range, so that a large negative pressure appears on the edge of the windward roof, and the absolute value decreases gradually along the direction of the wind. The whole roof surface is under negative pressure.

According to Fig.6, the fluctuating wind pressure distribution in 3 cases is described as follows. Under wind direction angle 0°, a large fluctuating wind pressure coefficient appears around the eave and the rooftop. Under a wind direction angle of 45°, the fluctuating wind pressure coefficient is conical and unevenly distributed due to the existence of the “conical vortex,” but there is a maximum value near the roof corner. Under a wind direction angle of 90°, a large fluctuating wind pressure coefficient appears around the roof corner.

The distribution of the average wind pressure coefficient and the fluctuating wind pressure coefficient in this paper is similar to that of Holmes’s research [12], which proves the rationality of the test data for dynamic time history analysis.

4 Finite element analysis

4.1 Finite element model

The ABAQUS finite element model of typical SMRS based on wind load transfer path, i.e., wind load acting on the roof panels is transferred by the clips to the purlins supported on the steel portal frame of the buildings, was established in this paper. The length of the roof is 18 m and the slope angle is 10°. Considering the roof slope, the actual width of the roof is 9/cos10° = 9.14 m. The roof slab is composed of ribbed steel panels, which are relatively stiff along the seam. The cross-section of the roof panel, the connection details of the seam, and the typical geometry of the clips and purlins are shown in Fig.7. The general construction of this roofing system is: the roof panel is vertically raised along the length direction using special occlusal equipment, which is buckled on the clip through the double folding seam, and the clip is fixed on the purlin by self-tapping screws. To ensure calculation accuracy, and to improve calculation efficiency, the steel panels are simplified into flat steel panels according to the principle of equivalent bending stiffness. The length, width, and thickness of the panel are 1523, 400, and 3.5 mm, respectively, which is simulated by 3D shell element. The clip mainly plays the role of load transfer element in the roofing system, as a result, it can be simulated by a 3D beam element. A total of 322 clips are arranged on the four corners of the roof panels, so they are arranged at a spacing of 400 mm along the purlin direction and at a spacing of 1523 mm along the vertical purlin direction. There are seven purlins arranged along the roof width direction, each with 3 spans. The local deformation and the stress state of the purlin have little effect on the finite element calculation results, so the purlin is also simulated by a 3D beam element with a spacing of 1523 mm and a span of 6 m. Z-shaped steel with edge stiffener [13] is adopted in the section of purlin.

The damage of the element itself is not considered in clip design, most of the elements are in an elastic state under wind load. Therefore, the ideal elastic model is adopted as the constitutive model. The specific material information is shown in Tab.4.

The specific mechanical properties of the roof lock seam are not studied in depth, and the seam can be partially rotated but not detached under the wind load. Therefore, a coupling connection is applied to the seam connection, in which the 3 translational degrees of freedom in each seam between the roof panels are restricted, while the 3 rotational degrees of freedom are released. To ensure that the wind load from the roof panel can be transferred to the clip, the clip is coupled with the corresponding node of the roof panel, and 6 degrees of freedom are restricted. The self-tapping screws still firmly fix the clips on the purlin under the wind load, so the displacement of the clips and the nodes corresponding to the purlin is directly coupled in the finite element model.

In practice, the light portal frame structure supports the roofing system. Since the stiffness of the roofing system is much smaller than the main frame structure, the solid-web purlin can be designed as a simply supported beam and the hinge constraint is applied at the end of the purlin, which can ensure that the wind load can be transferred in the roof components.

The ABAQUS finite element model for time history analysis is shown in Fig.8, and the clip numbering is summarized in Fig.9. The size of the roof is 18 m × 9 m and the slope angle is 10°. The roof slab is composed of ribbed steel plates, which is equivalent to flat steel plates according to equivalent bending stiffness. The plate element width is 400 mm and the thickness is 3.5 mm, which is simulated by 3D shell element. The clip mainly plays the role of load transfer element in the roofing system, as a result, it can be simulated by 3D beam element. The purlin is also simulated by 3D beam element, with the spacing of 1.5 m and the span of 6 m. Z-shaped steel with edge stiffener is adopted in the section of purlin. Since the damage of the element itself is not considered in clip design, most of the elements are in elastic state under wind load. Therefore, the ideal elastic model is adopted as the constitutive model. It should be noted that, the effect of meshing was checked by using half sizes in current study with a good convergence, since the dynamic deflection effect of the panels to the clips is just cared about in current study, not the stress distribution in the panels.

The time-domain method is applied in the calculation of the wind-induced response of SMRS, and the relevant parameters are as follows.

1) Basic wind pressure, 0.55 kPa (the return period is 50 years, considering the Shanghai area as an example).

2) Time period step, 0.002 s (the sampling frequency of the wind tunnel test is 500 Hz).

3) Structural damping ratio, 0.02 [14].

4) Rayleigh damping model, C = αM + βK, is used in SMRS, which is a classic damping system. Coefficients α and β can be calculated by Eq. (2) [15].

(2)\alpha =\frac{2w_1{w}_2({\xi }_2{w}_1-{\xi }_1{w}_2)}{w_1^2-w_2^2},\beta =\frac{2({\xi }_1{w}_1-{\xi }_2{w}_2)}{w_1^2-w_2^2},

where ξ₁ and ξ₂ are the damping ratios corresponding to the 1st and 2nd natural frequency w₁ and w₂, respectively.

4.2 Analysis method

The wind force of the clip is described in the equation below.

(3)F_{\mathrm{u}}=\eta P_{\mathrm{w}}{A}_{\mathrm{c}},

where \eta is the magnification coefficient considering the spatial correlation of fluctuating wind pressure on the roof surface and the dynamic effect of the roof structure itself. P_w is the standard value of wind pressure, defined in GB51022-2015 [5], which is taken as 0.55 kN/m² in this paper considering the Shanghai area as an example. A_c is the dependent area of one clip, as defined in GB51022-2015 [5].

The wind pressure time history of the measurement point in the wind tunnel test is calculated as follows.

(4)\mathit{F}(\mathit{t})=0.5{\rho \mathit{v}}_{\mathrm{H}}^2{\mathit{C}}_{\mathrm{p}}(\mathit{t}){\mathit{A}}_{\mathrm{c}},

where ρ is the air density, v_H is the reference wind speed, and C_p(t) is the wind pressure coefficient time history of the measurement point.

An interpolation method based on the proper orthogonal decomposition (POD) method was applied to obtain the wind pressure time history of each node based on the current wind pressure data of limited nodes [16]. First, the POD interpolation method was applied based on the limited control points whose number is not locally increased, and the wind pressure data of the clips based on the POD analysis result was separately input to the finite element model for dynamic time history analysis to obtain the extreme wind-induced force of the clips. Then, the POD analysis time history data of the clips within a certain range of area A_c, 4A_c, and 9A_c was replaced by the actual wind tunnel test data, respectively, and the wind pressure data of the clips based on the updated data was separately input to the finite element model for dynamic time history analysis. Finally, a convergence result will be obtained, which can be taken as the final estimated value of ŋ.

4.3 Analysis results

Considering the most unfavorable case for clip design, 3 wind direction cases (0°, 45°, and 90°) were analyzed in this paper. Tab.5 gives the finite element estimation result of wind-induced force for typical clips in 3 zones (Clip 1, 19, and 172 in Zone A, B, and C, respectively) based on GB 51022-2015 [5], and the calculated values in bold are the results of the last iteration with 9A_c. It can be concluded that.

1) In Zone A and Zone B, the fluctuating wind pressure is large and the wind vortex is remarkable, so as the number of iterations and the analysis area increase gradually, more actual wind pressure data can be utilized compared with the POD interpolation method, the calculation result gradually increases.

2) In Zone C, the fluctuating wind pressure and the average wind pressure are rather small, so as the number of iterations and the analysis area increase gradually, some small data can be replaced by the interpolation value in the POD interpolation method, and the calculation result gradually decreases.

3) The effective load-bearing area of the clips is limited, so when the analysis area is increased to a certain extent, the value of ŋ is no longer increased, and the iterative result gradually converges. The last iteration result (values in bold) is the final estimation result.

There are 20 clips in Zone A, 122 clips in Zone B, and 180 clips in Zone C. Due to space limitations, 8 typical clips from Zone A, 14 typical clips from Zone B, and 15 typical clips from Zone C were selected for analysis. Tab.6–Tab.8 give the statistical result of the last iteration for typical clips from 3 zones in Cases 1–3. The maximum value of ŋ in 3 cases is summarized in Tab.9. According to Tab.9, it’s obvious that due to the spatial correlation of the wind pressure on the roof, the calculated value of ŋ for some clips is larger than 1.0. Based on Tab.9, it can be concluded that the length-span ratio has little effect on the values of ŋ in Zone A and Zone B. The maximum values of ŋ for Case 1 and Case 2 are close to each other in Zone C while it’s smaller in Case 3, which indicates that the change of the value of ŋ with the length-span ratio is not proportional but the overall trend is the same.

5 Discussion of analysis results

5.1 Comparison with Jing and Li’s research

In Jing and Li’s research work [9], the POD interpolation method was also applied to obtain the dynamic distribution of wind pressure on the whole surface based on the information from the limited measurement points. The efficiency of the analysis result depends on the number of measurement points around the analyzed node in 3 typical zones, which is quite limited in Jing and Li’s model, however. In this paper, the number of measurement points around the typical clip nodes in the corner zone, edge zone, and center area is largely increased (Fig.1) to get a more accurate simulation result when determining the wind-induced force for clips of SMRS. As a result, further research in this paper is quite necessary to verify the accuracy and practicability of the magnification coefficient, ŋ, focusing on the measurement point distribution method of the rigid model. The analysis result of Jing and Li’s research is based on CECS102-2002 [17]. Therefore, to obtain a better comparison result with Jing and Li’s research, the magnification coefficient, ŋ, was recalculated based on CECS102-2002 [17], and the comparison result is shown in Tab.10.

According to Tab.10, the value of ŋ in this paper is 3.7%–14.8% smaller than that of Jing and Li’s research in Zone B and Zone C under various wind direction cases. In Zone A, the calculation result in this paper is 23.4%–31.4% larger than Jing and Li’s result, which is probably because of the number of measurement points in the corner area. As mentioned in Section 3, there is an obvious “separation bubble” generated in Zone A under three wind direction angles, 0°, 45°, and 90°, resulting in a significant increase of wind pressure in this area. In this paper, the number of measurement points in Zone A is largely increased, and the description of the increase of local wind pressure is more accurate compared to the result of the POD method, so a larger magnification coefficient was obtained.

5.2 Comparison with different codes

The extreme wind force of the clips in 3 cases was calculated based on Chinese code (CECS102-2002 [17], GB 51022-2015 [5]), American code (ASCE/SEI 7-16) [18], and Japanese code (AIJ-04) [10]. For American code, the simplified method proposed in ASCE/SEI 7-16 [18] was used in this section. It’s worth noting that the time interval for the basic wind speed defined in Chinese code is 10 min, while it’s 3 s for American code. Based on the same time interval, the basic wind speed of American code is about 1.42 times that of Chinese code [19].

5.2.1 Roof zoning in different codes

The relevant regulations about roof zoning of low-rise gable roofs with a slope no more than 10° in four codes are shown in Fig.10.

In CECS102-2002 [17] and GB 51022-2015 [5], the roof is divided into 3 zones, and the edge zone width is the smaller value between 40% of the average roof height and 10% of the minimum horizontal dimension, but not less than 4% of the minimum house dimension or 1.0 m.

In ASCE/SEI 7-16 [18], the roof is divided into 6 zones, and the ridge areas are considered in the corner and edge zones. The edge zone width is the smaller value between 40% of the eave height and 10% of the minimum horizontal dimension, but not less than 4% of the minimum house dimension or 0.9 m.

In AIJ-04 [10], the roof is divided into 7 zones considering different roof slopes. However, the value of the extreme wind coefficient for the roof with a slope of no more than 10° is the same in some zones, as a result, there are only 3 roof zones for the roof with a slope of no more than 10°. The edge zone width is the smaller value between 40% of the average roof height and 10% of the minimum horizontal dimension.

5.2.2 Comparison results in Zone A, B, and C

The values of ŋ in Zone A, B, and C defined in GB 51022-2015 [5] were calculated and compared according to the finite element analysis result. The comparison results are shown in Fig.11–Fig.13, and the maximum values of ŋ based on 4 codes are listed in Tab.11. According to Fig.11–Fig.13 and Tab.11, the following conclusions can be obtained.

1) In Zone A and Zone B, the calculated value of ŋ based on AIJ-04 [7] is the smallest among the 4 codes, while in Zone C, the calculated value of ŋ based on ASCE/SEI 7-16 [15] is the smallest. The calculated value of CECS102-2002 [14] is the largest in 3 zones. Generally speaking, the maximum value of ŋ in each roof zone exceeds 1.0, indicating that the safety of these 4 codes regarding the extreme wind force of the clips cannot be guaranteed.

2) In Zone A, the value of ŋ for clips 1–5 is larger than that of clips 6–8, because clips 1–4 are close to the upwind eaves and clip 5 is close to the gable where the wind pressure is relatively large.

3) In Zone B, the value of ŋ for clips 1–7 is slightly larger than that of clips 8–14, because clips 1–7 are close to the upwind eaves where the wind pressure is relatively large. Generally speaking, the calculated value of ŋ in Zone B fluctuates little.

4) In Zone C, the value of ŋ for is sorted from large to small as clips 1–5 > clips 6–10 > clips 11–15, because clips 1–5 are located on the row of purlin closest to the upwind eaves while clips 11–15 are close to the rooftop.

5) The calculation result differs little between the 3 cases, indicating that the length-span ratio (L:B) has little effect on the extreme wind force of the clips in each zone.

5.2.3 Extra analysis for Japanese code and American code

For AIJ-04 [10], the analyzed clips selected in Section 4 are located in Zone R_b, Zone R_a, and Zone R_f, respectively, so additional analysis on the wind-induced force of the clips in Zone R_c, Zone R_d, Zone R_e, and Zone R_g is necessary. Similarly, for ASCE/SEI 7-16 [18], the analyzed clips selected in Section 4 are located in Zone 3e, Zone 2e, and Zone 1, respectively, while the wind-induced force of the clips in Zone 3r, Zone 2n, and Zone 2r are different and need addition analysis. According to the previous analysis result, the length-span ratio of the house has little effect on the calculation results of ŋ. Therefore, taking only Case 1 as an example, 8, 9, and 14 representative clips in Zone 3r (R_d), 2n (R_c), and 2r (R_e, R_g) are selected for analysis, respectively. The analysis result is shown in Tab.12.

Comparing the results in Fig.12–Fig.13 and Tab.12, it can be seen that, in general, the extreme wind-induced force of the clips in Zone 3r (R_d) and Zone 2n (R_c) is equivalent to that of Zone B, and the extreme wind-induced force of the clips in Zone 2r (R_e, R_g) is equivalent to that of Zone C. In summary, the roof zoning method in ASCE/SEI 7-16 [18] is relatively conservative in terms of the wind-induced force for clips in the roof ridge area and underestimates the wind-induced force for clips in the windward eaves and corner areas.

5.3 Recommended value of ŋ based on different codes

To ensure the safety and applicability of engineering design, the maximum value of A_e/A_c for the clips in each zone is taken as the magnification coefficient, ŋ (Eq. (4)). The research by Ginger and Holmes [20] shows that the wind pressure of the upward roof surface increases with the increase of the length-span ratio, while the wind pressure of the windward roof surface is less affected by the length-span ratio. Therefore, for roofs with a length-span ratio larger than 3, the value of ŋ can be taken as case 3 (L:B = 3). According to the summary result in Tab.11 and Tab.12, the recommended values of ŋ based on CECS102-2002 [17], GB 51022-2015 [5], ASCE/SEI 7-16 [18] and AIJ-04 [10] are concluded in Tab.13.

6 Conclusions

In this paper, considering the actual influence area of the bearing force for clips of SMRS and the spatial correlation of the fluctuating wind on the roof, an effective estimation method for extreme wind-induced force for clips of SMRS is proposed. Based on the wind tunnel test data, the finite element model was established to obtain the dynamic time history analysis results. The calculation results of wind-induced force for the clips based on GB51022-2015 [2] were compared with the time history analysis results. Accordingly, the magnification coefficient, ŋ, was introduced into the estimation of the effective static wind force of the clips. The following conclusions were reached.

1) For double-slope roofs with a slope of 10° under various wind direction angles, the whole roof is under negative wind pressure. In the case of wind direction 0°, the eaves and the corner are subjected to a large negative pressure; in the case of wind direction 45°, the wind cone area around the roof corner is subjected to a large negative pressure; in the case of wind direction 90°, the eaves of the upward roof are subjected to a large negative pressure.

2) Due to the actual influence range of the bearing force for the clip, and the spatial correlation of the fluctuating wind on the roof, the extreme wind-induced force of the clips is significantly larger than the calculation result of the Chinese Code.

3) When calculating the extreme wind force of the clips using finite element analysis, the POD interpolation method homogenizes the wind pressure of some areas (such as the corner zone and the edge zone) to some extent. Therefore, increasing the number of measurement points performed in these areas and replacing the POD analysis result with the actual wind pressure time history can result in more accurate analysis results.

4) For double-slope roofs with a slope of 10°, the roof zoning in various codes is introduced, and the values of the magnification coefficient, ŋ, based on various codes are recommended, as shown in Tab.13.

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