Numerical simulation of snowdrift on an air-supported membrane structure and response analysis under snow loads

Abstract

The air supported membrane structure is a typical nonlinear flexible long-span space structure, the wind-induced drift and the resulting accumulation and distribution of snow particles on the structure may be the primary design concern among all loads in heavy-snowfall region. Thus, an accurate prediction of snow distribution on membrane surface is vital to structural design. A numerical simulation method is used to estimate snowdrift in this paper. Based on Euler-Euler method in multi-phase flow theory, this numerical model adopted Mixture model and combined with the snow deposition and erosion model, the snowdrift on an air-supported membrane coal shed is simulated, the distribution factor for roof snow load is given under different wind speed and different directions to estimate the worst load case, snow load on the air-supported membrane structure is significantly affected by snowdrift which causes significant non-uniform snow load. Furthermore, the response analysis of the air-supported membrane structure under snow load is studied, for comparison, uniform snow load case, non-uniform snow load case, and simulated snow load case under 0° wind direction are all considered. The results show that non-uniform snow load caused by snow drifting is more dangerous and should be considered.

Keywords

Air supported membrane structure snow drifting numerical simulation uneven snow load

Introduction

Air supported membrane structure is a flexible structure which uses high-performance membrane material as the building “shell,” fills air into the membrane and is supported by the air pressure difference between inside and outside the membrane surface. With the increase of span, the membrane surface stress will gradually increase. In order to transfer and reduce the membrane surface stress, the membrane surface is covered with cable net to share the membrane surface stress and further increase the stiffness and stability of the structure. The air supported membrane structure consists of membrane, reinforced cables, and internal supporting air, as shown in Figure 1. The air supported membrane structure is a typical nonlinear flexible long-span space structure, the main loads would be the wind pressure and snow load in heavy-snowfall region, the wind-induced drift and the resulting accumulation and distribution of snow particles on the structure must be considered,¹ wind-induced non-uniform snow load may be the primary design concern among all loads.

Figure 1.

Diagram of an air supported membrane structure.

Maaskant and Roorda² studied the lateral instability of two-dimensional section of cylindrical inflatable membrane under asymmetric snow load, the behavior of cylindrical air-supported membranes subjected to a concentrated line load is studied theoretically and experimentally, the structure may bifurcate from the symmetric shape into a nonsymmetric shape at loads less than the limit load. Lukasiewicz et al.³ studied the large deformation behavior and stability of cylindrical inflatable membrane structure under asymmetric load, load deflection and load path plots for these structures, undergoing very large deflections, are obtained and the effect of initial geometry and load “eccentricity” on deflection and stability behavior is investigated. Bartko and Baskaran⁴ designed and manufactured a device to test the friction coefficient between snow and roof materials, the results showed that the TPO, PVC, and EPDM membranes, painted industrial metal profiles, and roof coverings made of modified bitumen, when tested under laboratory conditions with snow specimens made of fresh snow with no freeze-thaw history, fall under the category of slippery roofs, with static friction coefficient values of 0.27, 0.1, 0.37, 0.39, and 1.55, respectively. Nam et al.⁵ and Paek et al.⁶ conducted a snow drift experiment on a stadium with a circular membrane structure roof, through the maximum new snow drift depth tested in the experiment, it was predicted that the maximum snow drift depth accumulated on the stadium roof was in the range of 3.4 –37 cm, and pointed out that the snow drift test was an important means to estimate the snow load. Gossen et al.⁷ introduced the phenomenon that the membrane roof of Toyota exhibition hall performing arts center in Pennsylvania was damaged by snow in a snowstorm in February 2007, summarized the replacement of tensioned fabric membrane and the measures taken to alleviate the snow in the future. Xie⁸ studied the response of inflatable membrane structure under snow load and wind load, and summarized the mechanical characteristics of this kind of structure. Fang⁹ deduced the relationship between the vertical ultimate bearing capacity and the shape of the air-supported membrane, internal pressure and membrane self-weight under the action of cumulative snow load, obtained the calculation formula of the vertical ultimate bearing capacity under the given conditions; Zhang¹⁰ studied the snow load distribution of inflatable membrane structure, proposed a simplified snow load distribution model, and analyzed the mechanical response, stability and strength failure of inflatable membrane structure under snow load. Zhou et al.¹¹ proposed a quasi-steady method to simulate redistribution of snow on a flat roof, which considered the effects of temporal change on snow depth. Sun et al.¹² adopted the multiphase flow theory to simulate the snow drift of membrane structure roof, the snow shape factor is given under different wind directions to estimate the worst load case, the results of finite element analysis (FEA) show that non-uniform snow load is more dangerous and should be considered. Above all, only small number of articles involve the snowdrift on membrane roofs, not considering the influence of snowdrift on the most unfavorable snow load distribution form of membranes roofs. Thus, the snowdrift on an air-supported membrane structure is investigated, and the mechanical performance under snow load is analyzed to judge the most adverse snow load distribution case.

There are mainly three methods to estimate snowdrift, that is, field measurement, wind tunnel test, and numerical simulation. Field measurement is the most intuitive approach to obtain the snow distribution, but it relies heavily on the meteorological conditions and data are often affected by unstable weather; Wind tunnel test is more controllable in conditions but constrained by the similarity criteria (it is difficult to meet all kinds of similar criteria at the same time) and it costs a lot; Numerical simulation is now widely used for its high efficiency and full scale modeling capacity, although there is imperfection on the transport equation of drift density and turbulence model. Compared with the first two methods, numerical simulation method has more prominent advantages in comprehensive cost, parameter analysis and law research, thus, the numerical simulation method is used to estimate snowdrift in this paper.

Numerical simulation method

Based on the Euler-Euler method in multiphase flow theory, the mixture model is adopted for this simulation, both the snow particles and air are treated as continuous phases, and the relationship between air phase and snow phase is one-way coupling, that is, the action of air phase makes the snow phase drift, and the movement of snow phase has no effect on the air phase. The two-phase flow is solved by adding the concentration equation of snow phase to the control equation of wind snow flow.

Governing equations

In Mixture model, the air phase and snow phase share the same mass continuity equation and momentum conservation equation as equations (1) (2).

\frac{\partial}{\partial t} (ρ_{m}) + \nabla . (ρ_{m} \cdot {\vec{v}}_{m}) = 0

(1)

\begin{array}{l} \frac{\partial}{\partial t} (ρ_{m} {\vec{v}}_{m}) + \nabla . (ρ_{m} v_{m} {\vec{v}}_{m}) \\ = - \nabla . p + [μ_{m} (\nabla {\vec{v}}_{m} + \nabla {\vec{v}}_{m}^{T})] \\ + ρ_{m} \vec{g} + \vec{F} + \nabla . (\sum_{k = 1}^{2} f_{k} ρ_{k} {\vec{v}}_{d r, k} {\vec{v}}_{d r, k}) \end{array}

(2)

where, $ρ_{m} = \sum_{k = 1}^{2} f_{k} ρ_{k}$ —the mixing density of the air and snow phases, f_k is the volume fraction of air phase ( $k$ = 1) and snow phase ( $k$ = 2), $ρ_{k}$ is the density of air ( $k$ = 1), and snow phase ( $k$ = 2);

${\vec{v}}_{m} = \frac{\sum_{k = 1}^{2} f_{k} ρ_{k} {\vec{v}}_{k}}{ρ_{m}}$ —velocity of the mixture phase;

p—flow pressure;

$μ_{m} = \sum_{k = 1}^{2} α_{k} μ_{k}$ —hybrid dynamic viscosity of air and snow phases;

${\vec{v}}_{d r . k} = {\vec{v}}_{k} - {\vec{v}}_{m}$ —relative drift velocity.

In addition, the continuity equation for snow phase is given individually by equation (3). The finite volume method is used to solve the governing equation of wind and snow flow.

\begin{array}{l} \frac{\partial}{\partial t} (ρ_{s} f) + \nabla . (ρ_{s} f {\vec{v}}_{m}) \\ = - \nabla . (f ρ_{s} {\vec{v}}_{d r, k}) + \sum_{q = 1}^{2} (m_{q p} - m_{p q}) \end{array}

(3)

where, $m_{p q}$ —Mass of snow phase converted into air phase;

$m_{q p}$ —Mass of air phase converted into snow phase.

When the above equations are time-averaged adopting Reynolds averaged approach, the Navier Stokes equations are no longer closed, and a new turbulence model equation must be introduced to make them closed.

Turbulence model

Many different turbulence models were employed in snowdrift simulation in previous studies, such as RSM (Reynolds Stress Equation Model) model, SST (Shear Stress Transport) k-omega model, k-epsilon model and k-kl-omega model. Uematsu et al.¹³ adopted the mixing length theory in his pioneering works on CFD simulation of snowdrift. Beyers et al.¹⁴ adopted the standard k-epsilon model and pointed out that it is necessary to critically examine the influence of inaccuracies isotropic turbulence assumption on the snow accumulation and erosion. k-kl- omega model can effectively describe the transition from laminar flow to turbulence in the boundary layer, in the process of wind-induced snow drift, the process of snow particles from jump movement to suspension movement is similar to the transition from lamina flow to turbulence, and from the comparative study in the example verification in Section 3, the k-kl-omega model shows better agreement with the simulated results by Oikawa and Tomabechi,¹⁵ so the k-kl-omega transition model is adopted in this paper.

The k-kl-omega model is a three-equation eddy-viscosity turbulence model including transport equations for turbulent kinetic energy (k_T), laminar kinetic energy (k_L), and the inverse turbulent time scale (ω), these equations are as follows:

\begin{array}{l} \frac{D k_{T}}{D t} = P_{K_{T}} + R + R_{N A T} - w k_{T} - D_{T} \\ + \frac{\partial}{\partial x_{j}} [(v + \frac{α_{T}}{α_{K}}) \frac{\partial k_{T}}{\partial x_{j}}] \end{array}

(4)

\frac{D k_{L}}{D t} = P_{K L} - R - R_{N A T} - D_{L} + \frac{\partial}{\partial x_{j}} (v \frac{\partial k_{L}}{\partial x_{j}})

(5)

\begin{array}{l} \frac{D ω}{D t} = C_{ω 1} \frac{ω}{k_{T}} P_{K_{T}} + (\frac{C_{ω} R}{f_{ω}} - 1) (R + R_{N A T}) \frac{ω}{k_{T}} - C_{ω 2} ω^{2} \\ + C_{ω 3} f_{ω} α_{T} f_{W}^{2} \frac{\sqrt{k_{T}}}{d^{3}} + \frac{\partial}{\partial x_{j}} [(v + \frac{α_{T}}{α_{k}}) \frac{\partial k_{T}}{\partial x_{j}}] \end{array}

(6)

where, R—the averaged effect of the breakdown of stream wise fluctuations into turbulence;

$D_{T}, D_{L}$ —dissipation terms;

$P_{K_{T}}, P_{K L}$ —the production term of turbulence, laminar kinetic energy respectively;

$C_{ω 1}, C_{ω 2}, C_{ω 3}$ —constant terms;

R_NAT —the natural transition production term.

Deposition and erosion model

The snow particle begins to move when the wall shear stress τ on the snow surface exceeds the threshold wall shear stress τ_t, otherwise the snow particle remains stationary. The wall friction velocity is directly related to the wall shear stress, the wall friction velocity u_* and wall shear stress τ obey the following relation:

u_{*} = \sqrt{\frac{τ}{ρ}} = \frac{κ u (z)}{\ln (z / z_{s})}

(7)

Where, $ρ$ —density of air;

$u (z)$ —wind speed at the height z;

$κ$ — $K a r m a n$ constant, taken as 0.4;

$z_{0}$ —Wall roughness height, which is set as $0.00003 m$ .

Through the conversion of equation (7), the wall shear stress τ is replaced by friction velocity u_* in the numerical simulation. Therefore, the wall friction velocity is an important parameter to determine whether the snow particles drift.

The snow mass flux of erosion $q_{e r o}$ and the deposition flux $q_{d e p}$ is calculated as follows:

q = {\begin{array}{l} q_{e r o} = A_{e r o} (u_{* t}^{2} - u_{*}^{2}), & u_{*} > u_{* t}, \\ q_{d e p} = ϕ_{s} w_{f} \frac{u_{* t}^{2} - u_{*}^{2}}{u_{* t}^{2}}, & u_{*} \leq u_{* t} . \end{array}

(8)

where, $A_{e r o}$ —Constant coefficient, which is set to be $7 \times 10^{- 4}$ ;

$w_{f}$ —the snowfall velocity which is assumed to be 0.2 m/s;

$u_{*}$ —the friction velocity;

$u_{* t}$ —the threshold friction velocity, which is chosen to be 0.15–0.4 m/s according to snow surface conditions¹⁶;

$ϕ_{s}$ —Mass concentration of snow ( $ϕ_{s} = ρ_{s} f$ , f is the volume fraction of air, $ρ_{s}$ is the density of snow particles).

The thickness of snow on the wall area will change with the erosion or deposition of snow. Then the change of snow thickness in $d t$ time is

d h = \frac{q}{ρ_{s}} d t

(9)

where, dt—Calculation time step;

The total variation of snow thickness can be obtained by integrating $d h$ with time:

Δ h = \int \frac{q}{ρ_{s}} d t

(10)

Boundary conditions

Inflow boundary:

Wind profile.

The fully developed inlet wind velocity profile is determined from the power law profile, according to Chinese load code for the design of building structures,¹⁷ the wind speed and turbulence intensity are calculated from

U (z) = U_{10} {(\frac{z}{10})}^{α}

(11)

where $U_{10}$ —the mean wind speed in the height of 10 m;

$α$ —Roughness coefficient.

Volume fraction of snow phase.

It is considered that the inlet velocity of snow phase is consistent with that of air phase, and the flow velocity is also obtained from equation (11). There are only two phases on the inlet boundary, and the sum of the integral fractions of the two phases is 1, thus, it is necessary to give the volume fraction of one phase. The volume fraction of snow phase is given here. Because the jump motion and suspension motion of wind and snow drift are considered, the volume fraction of snow phase is different at different heights. The empirical formula proposed by Pomeroy and Gray¹⁸ and Pomeroy and Male¹⁹ is adopted here

f = {\begin{array}{l} \frac{0.68}{h_{s} u_{p} u_{*} g} u_{* t} (u_{*}^{2} - u_{* t}^{2}), & z \leq h_{s}, \\ \frac{0.8 \exp [- 1.55 (4.78 u_{*}^{- 0.544} - z^{- 0.544})]}{ρ_{s}}, & z > h_{s} . \end{array}

(12)

where, g—Gravitational acceleration;

$u_{p}$ —Velocity of snow particle;

$h_{s}$ —height of saltation layer which is calculated by $h_{s} = \frac{1.6 u_{*}^{2}}{2 g}$ .¹⁸

Outflow boundary: Zero gradient condition, the wind velocity gradient and snow fraction are set to zero; Upper and lateral surface of computational domain: Zero gradient condition for all variables, and the normal components of wind velocity with respect to the boundaries are set to zero; Building and ground surface: No-slip boundary condition, except that the wall roughness $z_{0}$ = 0.00003 for snow surface.

Solution method and convergence control

The commercial software ANSYS-FLUENT is used for calculation in fluid domain, the solution method of wind-snow two-phase flow is mainly reflected in the solution of differential equations. A differential equation is resolved into several nonlinear coupled algebraic equations by using the finite volume method, the nonlinear convection coupled algebraic equation is discretized by using the second-order forward style, the nonlinear diffusion coupled algebraic equation is discretized by using the second-order precision central difference scheme, the algorithm adopts SIMPLEC algorithm, and the convergence residual is set to 10−⁶.

Model validation

Taking the measured test of Oikawa and Tomabechi¹⁵ as an example, the snow distribution of wind snow drift around the cube is studied. The side length of the cube is H = 1 m. According to the characteristics of the cube structure, the calculation domain is set as shown in Figure 2. The x direction is along-wind direction and the y direction is across-wind direction.

Figure 2.

Computational fluid domain of cube.

Unstructured grid is used for discretization in the whole calculation domain. The minimum grid size is 0.02 m, the total number of fluid elements is 1,012,365. The settling velocity of snow particles is taken as 0.2 m/s, the diameter of snow particles is taken as $1.5 \times 10^{- 4}$ m, the snow density is taken as 150 kg/m³, the wall roughness height of snow surface is taken as $3 \times 10^{- 5}$ , and the threshold velocity of snow drift is 0.15 m/s. The turbulence model k-kl-ω, Standard k-ω, Standard k-ε are selected respectively, the k-kl-omega model is a three-equation eddy-viscosity turbulence model including transport equations for turbulent kinetic energy (k_T), laminar kinetic energy (k_L), and the inverse turbulent time scale (ω); the standard k-epsilon model is a two-equation based on the transport of turbulent kinetic energy (k), and its dissipation rate (ε); the standard k-omega is also a two-equation including transport equations for turbulent kinetic energy (k), and the specific dissipation rate (ω). The calculated results of adopting three turbulence models are compared with the measured results. Figures 3 and 4 give snow depth around the cube in along-wind direction and across-wind direction, the vertical axial denotes the shape factor (S/Sd) obtained by dividing the snow depth (S) with the ground snow depth (S_d).

Figure 3.

Comparison of upwind simulation results and measured results.

Figure 4.

Comparison of cross wind simulation results and measured results.

It can be seen from Figures 3 and 4, the tendence of snow depth distribution around the cube using different turbulence is similar, the k-kl-omega model fits the measurement data better, which ensures the accuracy of this CFD (Computational Fluid Dynamics) simulation.

Snowdrift simulation

The coal shed is an air-supported membrane structure (Figure 5), it is located in Harbin, northeast China, which is a heavy-snowfall region. This air-supported membrane structure has dimensions of 389 m long, 101 m wide, and 40 m high. Arrangement of cable net is plotted in Figure 5, each grid of the steel cable net has dimensions of 2.5 m × 2.5 m.

Figure 5.

The pneumatic coal shed structure: (a) plan view and (b) side view.

The provisions for snow load in the Chinese Load code¹⁷ mainly based on empirical judgment and limited observation data, only the snow load distribution of some simple structures is given, for structures like this membrane structure, there are insufficient references to determine the snow load.

Geometrical modeling

Full scale model of the air-supported membrane shed is established in numerical simulation, for the symmetry properties of this structure, only three wind directions (i.e. 0°, 45°, and 90°) are considered (Figure 6(a)).

Figure 6.

Sketch of wind direction angle and computational region: (a) wind direction angle and (b) computational region.

The dimension of the computational region is selected as X × Y × Z = 15L × 21B × 7H, where L, B, H refers to the shed length, shed width, and the shed height respectively. The blockage ratio of the building in the computational domain is less than 3% which is generally accepted in CFD applications, the fluid field size is large enough to neglect the influence of the computational region boundary on interior flow field according to China’s standard for wind tunnel test methods of Construction Engineering (JGJ/T338-2014).

Two mesh sizes were used to investigate the effect of mesh insensitive, the denser meshes close to shed surface are double times the number in the initial mesh, and both of the meshes predicted the same snow load distribution, which suggests that the solution becomes mesh-independent after a particular level of refinement. At last, the denser meshes is selected.

The computational domain is meshed containing 1,772,083 cells, an unstructured grid is used, a finer mesh is arranged out on the building wall to adapt to the change of the fluid field, relatively coarse mesh is arranged at the periphery (Figure 6(b)), close to shed surface, the mesh consists of an inflation layer with gradually increasing cell size with a minimum height of 0.1 m, the maximum dimensionless wall distance y⁺ is less than 150.

Simulation parameters

The inlet wind velocity is a key parameter for snowdrift simulation. A relatively low wind velocity may not induce a significantly snowdrift, while a relatively high velocity may blow the snow away from the roof. In this paper, the distribution factor for inflatable membrane snow load under wind speeds of 3, 4, 5, and 6 m/s (10 min average velocity at 10 m height) is simulated respectively, the worst distribution factor for snow load is determined through the responses analysis of the structure under different snow load cases caused by different wind speeds, the snowdrift duration Δt in equation (10) is assumed to be 8 h, properties of snow particle are the same as Section 3, the initial snow depth is set as 0.3 m corresponding to the uniform snow load of 0.45 kN/m².

Data processing

In most current load codes and standards, the curved roof snow load is calculated using the following formula:

S = μ_{r} * S_{d}

(13)

$S_{d}$ is the ground snow load, ground snow load determination for sites is based on an extreme value statistical analysis of data available in the vicinity of the site using a value with a 2% annual probability of being exceeded (50-year mean recurrence interval).

µ _r is roof slope factor, this parameter represents the combination effect of roof slope, snowdrift, sliding snow etc.

Hence, the simulated snow load is given as contour maps of the roof slope factor µ _r to estimate the effect of snowdrift (see Figures 7 –9).

Figure 7.

Roof slope factor under 0° wind direction: (a) 3 m/s wind speed, (b) 4 m/s wind speed, (c) 5 m/s wind speed, and (d) 6 m/s wind speed.

Figure 8.

Roof slope factor under 45° wind direction: (a) 3 m/s wind speed, (b) 4 m/s wind speed, (c) 5 m/s wind speed, and (d) 6 m/s wind speed.

Figure 9.

Roof slope factor under 90° wind direction: (a) 3 m/s wind speed, (b) 4 m/s wind speed, (c) 5 m/s wind speed, and (d) 6 m/s wind speed.

In addition, portions of curved roofs having a slope exceeding 60° shall be considered free of snow load.

Simulation and results

Roof slope factor at four different wind speed under 0° wind direction were shown in Figure 7, with the increase of wind speed, the coverage area of snow load on the surface of membrane structure decreases gradually, roof slope factor on the windward side and roof top area gradually decreases to zero with the increase of wind speed, while roof slope factor on the leeward area increases at first and then decreases with the wind speed from 3 to 6 m/s, when the wind speed is 5 m/s, roof slope factor in the leeward area is the largest. This non-symmetrical load may bring about adverse effect on the membrane structure which will be shown in next section.

Roof slope factor at four different wind speed under 45° wind direction were shown in Figure 8. with the increase of wind speed, the coverage area of snow load on the surface of membrane structure decreases gradually, roof slope factor on the windward side and roof top area gradually decreases to zero with the increase of wind speed, while roof slope factor on the leeward area increases at first and then decreases with the wind speed from 3 to 6 m/s, when the wind speed is 4 –5 m/s, roof slope factor in the leeward area is the largest. This non-symmetrical load may bring about adverse effect on the membrane structure which will be shown in next section.

Roof slope factor at four different wind speed under 90° wind direction were shown in Figure 9, with the increase of wind speed, the coverage area of snow load on the surface of membrane structure decreases gradually, roof slope factor on the windward side gradually decreases to zero with the increase of wind speed, roof slope factor on the roof top area unevenly decreases from left to right with the increase of wind speed, while roof slope factor on the leeward area increases at first and then decreases with the wind speed from 3 to 6 m/s, when the wind speed is 4–5 m/s, roof slope factor in the leeward area is the largest.

Seen from wind-induced snowdrift simulation results under 0°, 45°, and 90° wind direction, when the wind speed is 3 m/s, only a small amount of erosion at the membrane surface, snow load presents a slightly asymmetrical distribution; when the wind speed of 4 –5 m/s, large erosion occurs on the windward side and top, at the same time, a lot of snow accumulation on the leeward side, the maximum roof slope factor occurs in the leeward part which reaches 1.8–1.9, snow load presents serious asymmetrical distribution; when the wind speed increases to 6 m/s, erosion area further expands, but due to the excessive wind speed, more snow on the membrane surface is blown off, and the accumulated snow load on the lee side gradually decreases.

Mechanical performance under snow load

The wind-induced snow load on the air-supported structure at different wind speed is obtained from CFD simulation in Section 4.4, structure performance under redistributed snow load was analyzed to estimate the worst load case.

Load cases

Consider the limit of the paper, only the mechanical performance analysis under 0° wind direction is performed. For comparison, the uniform snow load case and un-uniform snow load case, which are suggested in Chinese Technical specification for membrane structure²⁰ under 0° wind direction, are also considered.

The uniform snow load and un-uniform snow load corresponding to distribution factor for roof snow load µ _r specified in the technical specification²⁰ is shown in Figure 10.

Figure 10.

Uniform and non-uniform distribution factor for snow load.²⁰

All the load cases are indicated in Table 1, the internal pressure and self-weight should also be a necessary part of all load cases, for simplicity, not listed.

Table 1.

Load cases for analysis.

Case 1	Whole-span uniform snow load suggested²⁰
Case 2	Non-uniform snow load suggested²⁰
Case 3	Simulated snow load when wind speed is 3 m/s
Case 4	Simulated snow load when wind speed is 4 m/s
Case 5	Simulated snow load when wind speed is 5 m/s
Case 6	Simulated snow load when wind speed is 6 m/s

Material properties

The PVC (with PVDF) fabric is used to form the facades, material density is 1200 kg/m³, the elastic modulus in warp direction and in weft direction are 2000 kN/m, Poisson’s ratio is 0.25, the thickness of material is 1 mm; The cable net system is made of wire-rope with sheathed HDPE (High Density Polyethylene), the diameter of the cable net is 20 mm, the elastic modulus is 1.6 × 10¹¹ N/m², the Poisson’s ratio is 0.3, and the material density is 7850 kg/m³. The internal pressure of the structure is set to 700 Pa when resisting strong snow load.

Analysis and results

ANSYS software is adopted to perform the nonlinear finite element analysis, cables are discretized as Link 10 units with tension only, and the film is discretized as Shell 41 units with three nodes, the tension option is set to ignore the bending stiffness of the film surface.

Load cases indicated in Table 1 are analyzed, the stress and deformation analysis are conducted to judge which one is the most adverse load case, and to investigate the snowdrift effect on its load bearing capacity.

Figure 11 compares the vertical displacement under each loading case, it is shown that the vertical deformation under uniform snow load is the smallest (dmax = −0.92 m), the largest deformation (dmax = −7.74 m, this deformation value has exceeded the allowable value of the specification) occurs in the simulated snow load when wind speed is 5 m/s (case 5), although the maximum roof slope factor for snow load determined by the formula given in Figure 10 in case 2 reaches 2.0, which is larger than the maximum roof slope factor 1.9 in case 5, the maximum vertical displacement (dmax = − 5.09 m, this deformation value has exceeded the allowable value of the specification) in case 2 is smaller than that in case 5 due to different positions and forms of snow distribution. It can be inferred that the non-uniform snow load by wind must be considered because it may be one of the control load cases in structural design. The recommended roof slope factor values are given in Chinese Technical specification for membrane structure,²⁰ but this roof slope factor for cylindrical structures is not fully applicable to three-dimensional air supported membrane structures.

Figure 11.

Vertical displacement under different snow load cases: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, and (f) Case 6.

It can be seen from Figure 11, when wind speed is 3, 4, and 6 m/s, the maximum vertical displacement of the structure is −1.79, −1.37, and −3.69 m respectively, which are less than the vertical displacement when the wind speed is 5 m/s, so the case 5 is the most unfavorable among the four simulated snow load cases (case 3~case 6). The responses of the structure correspond to the snowdrift distribution at the membrane surface, a relatively low wind velocity (3 or 4 m/s) does not induce a significantly snowdrift, while a relatively high velocity (6 m/s) blows the more snow away from the roof. Therefore, the wind speed that may cause the most disadvantageous snowdrift distribution to the structure needs to be obtained through simulation or test in combination with local meteorological conditions.

Figure 12 gives the stress distribution in membrane under all load cases, although the total snow load is very different in different case, the stress distribution is very similar, the maximum stress in membrane located at the middle of the membrane in uniform snow load case, the maximum stress in membrane located at the middle of the membrane and the corner of the membrane with less non-uniform snow load in non-uniform snow load cases. Due to many reinforced cables on the membrane, the stiffness is greatly strengthened, external loads acting on membrane can be transferred to the cable net. The maximum stress in case 2 and case 5 is almost the same, and the maximum stress in case 3, case 4, and case 6 is almost the same.

Figure 12.

Stress distribution in membrane under different snow load cases: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, and (f) Case 6.

Through response analysis of the air-supported membrane structure under snow load, it can be concluded that severe non-uniform snow load does not cause excessive stress in the membrane, but it causes excessive deformation of the membrane.

Snow from the upper level of roofs made of thermoplastic materials (TPO, PVC) will slide at 15° as defined in Sentence 6 of Article 4.1.6.2 of NBCC (National Building Code of Canada).²¹ So, in the case of such large deformation of the membrane structure, it is very easy to produce sliding accumulation of snow on the membrane surface, forming “Matthew effect,” resulting in further snow aggravation on local membrane, resulting in membrane damage.

But, the sliding between snow and PVC membrane roof and the sliding between the cable-mesh and the membrane are not considered in this paper, which will be considered in subsequent research. Results can be provided to the specification²⁰ to limit the local deformation of the membrane structure to prevent this kind of snow sliding. If the structural design cannot control the deformation within this limit, the snow removal measures should be adopted for the membrane structure to reduce the accumulation of snow in heavy snowfall weather.

Conclusions

In this study, the wind-induced snowdrift on an air-supported membrane roof is investigated. The comparison of the mechanical performance between uniform snow load, non-uniform snow load, and simulated snow load is also analyzed. Conclusions are as follows:

(1) CFD method is adopted to simulate the snowdrift around a cube, the simulated snow distribution tendency of snow depth distribution around the cube adopting different turbulence is similar, and the k-kl-ω model shows better agreement with observation data, thus the k-kl-ω turbulence model is adopted in this paper and this CFD method can be used to estimate the snow drift on the air-supported membrane structure.

(2) Snow load on the air-supported membrane structure is significantly affected by snowdrift which causes significant non-uniform snow load. Under 0°, 45°, and 90° wind direction, when the wind speed is 3 m/s, only a small amount of erosion at the membrane surface, snow load presents a slightly asymmetrical distribution; when the wind speed is 4 –5 m/s, large erosion occurs on the windward side and top, at the same time, a lot of snow accumulation on the leeward side, snow load presents serious asymmetrical distribution; when the wind speed increases to 6 m/s, erosion area further expands, but due to the excessive wind speed, more snow on the membrane surface is blown off, and the accumulated snow load on the lee side gradually decreases.

(3) Through finite element analysis, deformation and stress distribution are obtained in uniform snow load case, non-uniform snow load case and simulated snow load case under 0° wind direction. It is shown that the simulated snow load is the control load case when wind speed is 5 m/s, which indicates that the non-uniform snow load should be considered in structural design, and the inlet wind velocity is a key parameter for snowdrift simulation, the wind speed that may cause the most unfavorable snowdrift distribution to the structure needs to be obtained through simulation or test in combination with local meteorological conditions.

(4) Through response analysis of the air-supported membrane structure under snow load, it can be concluded that severe non-uniform snow load may not cause excessive stress in the membrane, but it causes excessive deformation of the membrane. Therefore, some snow melting or snow removal techniques must be adopted for the air-supported membrane structure to reduce the accumulation of snow in heavy snowfall areas.

Non-uniform distribution snow load on curved roofs is very complicated and affected by many factors, in current numerical simulation, only the influence of roof slope and wind speed is considered, a complex process with multiple dependent variables is simplified. Further studies should focus on thermal factor, sliding and snowdrift on roofs which changes with the large deformation of the structure simultaneously for practical application.

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 work has been financially supported by the National Natural Science Foundation of China (Grant No. 51878014), which are gratefully acknowledged.

ORCID iD

HE Yanli

References

Majowiecki

. Snow and wind experimental analysis in the design of long-span sub-horizontal structures. J Wind Eng Ind Aerodynamics 1998; 74-76: 795–807.

Maaskant

Roorda

. Stability of cylindrical air-supported structures. J Eng Mech 1985; 111(12): 1487–1501.

Lukasiewicz

Balas

Glockner

. Stability and large deformation behavior of nonsymmetrically loaded cylindrical membranes. J Struct Mech 1986; 14(2): 229–255.

Bartko

Baskaran

. Snow friction coefficient for commercial roofing materials. J Cold Reg Eng 2018; 32(1): 06017005.

Nam

You

, et al. A study on the types of snowdrifts in domed stadium using snow wind tunnel. In: Proceedings of the 8th Asia-Pacific conference on wind engineering, Chennai, India, 2013.

Paek

Nam

You

, et al. Snow wind tunnel experiment for snowdrift prediction of domed stadium. Adv Mater Res 2014; 919-921: 216–221.

Gossen

Chen

Shan

, et al. Snow induced ponding of Toyota pavilion canopy, montage mountain. In: Proceedings of the 2008 structures congress - 2008, Vancouver, BC, Canada, 2008.

Xie

. Shape state and static analysis for pneumatic membrane. Master’s Thesis. Guangzhou, China: South China University of Technology, 2014. (in Chinese).

Fang

. Study on load effect of the cylindrical air-supported structure. Master’s Thesis, Xi’an University of Architecture and Technology, Xi’an, China, 2013. (in Chinese).

10.

Zhang

. Mechanical response analysis and failure mechanism of inflatable membrane structure under snow load. Master’s Thesis, Xi’an University of Architecture and Technology, Xi’an, China, 2016. (in Chinese).

11.

Zhou

Kang

, et al. Numerical simulation and wind tunnel test for redistribution of snow on a flat roof. J Wind Eng Ind Aerodynamics 2016; 153: 92–105.

12.

Sun

. Numerical simulation of snowdrift on a membrane roof and the mechanical performance under snow loads. Cold Reg Sci Technol 2018; 150: 15–24.

13.

Uematsu

Nakata

Takeuchi

, et al. Three-dimensional numerical simulation of snowdrift. Cold Reg Sci Technol 1991; 20(1): 65–73.

14.

Beyers

Sundsbø

Harms

. Numerical simulation of three-dimensional, transient snow drifting around a cube. J Wind Eng Ind Aerodynamics 2004; 92(9): 725–747.

15.

Oikawa

Tomabechi

. Formation processes of the deposition and erosion of snow around a model building. J Jpn Soc Snow Ice 2003; 65(3): 207–218.

16.

Kind

. Mechanics of aeolian transport of snow and sand. J Wind Eng Ind Aerodynamics 1990; 36(2): 855–866.

17.

GB 50009-2012. Load code for building structures. Beijing: China Construction Industry Press, 2012 (in Chinese).

18.

Pomeroy

Gray

. Saltation of snow. Water Resour Res 1990; 26(7): 1583–1594.

19.

Pomeroy

Male

. Steady-state suspension of snow. Hydrol J 1992; 136(1–4): 275–301.

20.

CECS158-2015. Technical specification for membrane structures. Beijing: China Planning Press, 2015 (in Chinese).

21.

NBCC (National Building Code of Canada). Canadian commission on building and fire codes. Ottawa: National Research Council Canada, 2015.