薄膜结构抗风性能概率评估方法

李英民^1,2,宋维举¹,王肖巍¹,徐军¹

(1.重庆大学 土木工程学院, 重庆 400045;2.山地城镇建设与新技术教育部重点实验室(重庆大学), 重庆 400045)



摘要:

为研究薄膜屋盖结构不同抗风需求的超越概率评估方法,以重庆地区为例,借鉴PEER基于性能的抗震设计方法,考虑风场随机性,统计重庆地区近十年每月最大风速样本概率分布特征,获得由多项式拟合的设计风速危险性曲线.用Davenport谱表示水平风速谱,推导出风速谱危险性曲线,并用五次多项式进行拟合.考虑薄膜屋盖结构的非线性特征,近似将屋盖结构抗风需求（竖向位移限值）的平均值与风速谱值关系采用幂函数表示,推导出薄膜屋盖结构风载下不同法向变形需求值的年平均超越概率表达式.以平坦薄膜屋盖为例,采用自回归模型的线性滤波器法(AR法)模拟生成30条对应不同风压的脉动风速时程,对屋盖进行动力响应分析,通过计算,获得不同变形需求下薄膜屋盖风振响应的超越概率及最大响应重现期.结果表明:多项式拟合对于薄膜屋盖抗风需求的年平均超越概率表达式推导有较好的简化效果；基于性能的薄膜屋盖结构抗风概率评估方法通过少量结构分析可以得到结构不同位移需求的超越概率评估值.

关键词:  薄膜屋盖结构  概率评估  抗风设计  闭式表达式  风速谱危险性曲线

DOI：10.11918/j.issn.0367-6234.20181214

分类号:TU375

文献标识码:A

基金项目:国家自然科学基金(51478067)； 重庆市教委科学技术研究项目(KJ1723378)



Probabilistic assessment on wind resistant performance of membrane structures

LI Yingmin^1,2,SONG Weiju¹,WANG Xiaowei¹,XU Jun¹

(1.School of Civil Engineering, Chongqing University, Chongqing 400045, China; 2.Key Lab of New Technology for Construction of Cities in Mountainous Area (Chongqing University), Ministry of Education, Chongqing 400045, China)

Abstract:

To study different wind demand exceeding probability assessment of the membrane roof, Chongqing was taken as an example. Taking PEER performance-based seismic design method as reference and considering the randomness of wind field, the probability distribution characteristics of monthly maximum wind speed samples in recent ten years were calculated, and the designed maximum wind speed risk curve was obtained by polynomial fitting. The horizontal wind speed spectrum was represented by Davenport spectrum. The nonlinear characteristics of membrane roof structure were considered, and the relationship between resistance requirements (vertical displacement limit) and the average wind speed spectrum values was approximately represented by a power function. The expression of the mean annual frequency of exceedance for displacement demand was derived. A flat membrane roof was taken as an example, 30 pulsating wind speed time intervals were generated with the AR method, and structural response analysis was carried out. Through calculation, the transcendental probability and recurrence interval corresponding to different performance levels of the roof were obtained. The results show that polynomial fitting has a better effect on the derivation of the expression of annual average exceeding probability. Performance-based wind resistance probabilistic evaluation method for membrane roofs can obtain the transcendental probabilistic evaluation values of different displacement requirements of structures through a small number of structural analyses.

Key words:  membrane roof  probabilistic evaluation  anti-wind design  closed-form expression  velocity spectrum hazard curve


李英民, 宋维举, 王肖巍, 徐军. 薄膜结构抗风性能概率评估方法[J]. 哈尔滨工业大学学报, 2018, 50(12): 107-113. DOI: 10.11918/j.issn.0367-6234.201806084.
LI Yingmin, SONG Weiju, WANG Xiaowei, XU Jun. Probabilistic assessment on wind resistant performance of membrane structures[J]. Journal of Harbin Institute of Technology, 2018, 50(12): 107-113. DOI: 10.11918/j.issn.0367-6234.201806084.
基金项目 国家自然科学基金(51478067)；重庆市教委科学技术研究项目(KJ1723378) 作者简介 李英民(1968—)，男，教授，博士生导师 通信作者 宋维举，nimrodsong@126.com 文章历史 收稿日期: 2018-06-12



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薄膜结构抗风性能概率评估方法
李英民^1,2, 宋维举¹, 王肖巍¹, 徐军¹    
1. 重庆大学 土木工程学院, 重庆 400045;
2. 山地城镇建设与新技术教育部重点实验室(重庆大学), 重庆 400045

收稿日期: 2018-06-12
基金项目: 国家自然科学基金(51478067)；重庆市教委科学技术研究项目(KJ1723378)
作者简介: 李英民(1968—)，男，教授，博士生导师
通信作者: 宋维举，nimrodsong@126.com


摘要: 为研究薄膜屋盖结构不同抗风需求的超越概率评估方法，以重庆地区为例，借鉴PEER基于性能的抗震设计方法，考虑风场随机性，统计重庆地区近十年每月最大风速样本概率分布特征，获得由多项式拟合的设计风速危险性曲线.用Davenport谱表示水平风速谱，推导出风速谱危险性曲线，并用五次多项式进行拟合.考虑薄膜屋盖结构的非线性特征，近似将屋盖结构抗风需求(竖向位移限值)的平均值与风速谱值关系采用幂函数表示，推导出薄膜屋盖结构风载下不同法向变形需求值的年平均超越概率表达式.以平坦薄膜屋盖为例，采用自回归模型的线性滤波器法(AR法)模拟生成30条对应不同风压的脉动风速时程，对屋盖进行动力响应分析，通过计算，获得不同变形需求下薄膜屋盖风振响应的超越概率及最大响应重现期.结果表明：多项式拟合对于薄膜屋盖抗风需求的年平均超越概率表达式推导有较好的简化效果；基于性能的薄膜屋盖结构抗风概率评估方法通过少量结构分析可以得到结构不同位移需求的超越概率评估值.
关键词: 薄膜屋盖结构    概率评估    抗风设计    闭式表达式    风速谱危险性曲线    
Probabilistic assessment on wind resistant performance of membrane structures
LI Yingmin^1,2, SONG Weiju¹, WANG Xiaowei¹, XU Jun¹    
1. School of Civil Engineering, Chongqing University, Chongqing 400045, China;
2. Key Lab of New Technology for Construction of Cities in Mountainous Area (Chongqing University), Ministry of Education, Chongqing 400045, China


Abstract: To study different wind demand exceeding probability assessment of the membrane roof, Chongqing was taken as an example. Taking PEER performance-based seismic design method as reference and considering the randomness of wind field, the probability distribution characteristics of monthly maximum wind speed samples in recent ten years were calculated, and the designed maximum wind speed risk curve was obtained by polynomial fitting. The horizontal wind speed spectrum was represented by Davenport spectrum. The nonlinear characteristics of membrane roof structure were considered, and the relationship between resistance requirements (vertical displacement limit) and the average wind speed spectrum values was approximately represented by a power function. The expression of the mean annual frequency of exceedance for displacement demand was derived. A flat membrane roof was taken as an example, 30 pulsating wind speed time intervals were generated with the AR method, and structural response analysis was carried out. Through calculation, the transcendental probability and recurrence interval corresponding to different performance levels of the roof were obtained. The results show that polynomial fitting has a better effect on the derivation of the expression of annual average exceeding probability. Performance-based wind resistance probabilistic evaluation method for membrane roofs can obtain the transcendental probabilistic evaluation values of different displacement requirements of structures through a small number of structural analyses.
Keywords: membrane roof    probabilistic evaluation    anti-wind design    closed-form expression    velocity spectrum hazard curve    
基于性能的结构设计理论是20世纪90年代初由美国学者首次提出并且应用于结构抗震设计中^[1]，对于地震工程而言，地震的不确定性与结构参数的随机性均会导致结构动力响应的概率性.基于这种现象，美国太平洋地震工程研究中心将结构性能指标引入到抗震设计方法中，计算结构极限状态的年平均超越概率，以此描述结构性能目标^[2].

对工程结构抗风而言，Paulotto等^[3]定义了基于目标性能的结构风工程概念；Augusti等^[4-5]提出基于性能的抗风设计方法(PBWE)，并以此为基础对人行天桥进行了抗风设计；此后，文献[5-8]采用PBWE方法对悬索桥等结构进行了抗风设计.

本文将PBWE方法引入到薄膜结构抗风设计中，建立薄膜结构抗风设计概率评估方法.以平坦薄膜屋盖为算例，考虑重庆地区风场的随机性，推导薄膜屋盖抗风需求年平均超越概率的闭式表达式，并以薄膜屋盖最大法向变形限值作为抗风性能的评判标准获得薄膜屋盖不同变形限值下的超越概率值.

1 风场模型 1.1 风速样本分析对特定场地风速的概率统计通常采用极值Ⅰ型分布进行拟合^[8].下面以重庆市实测风速资料为例，验证极值I型分布对重庆市风场的适用性，表 1列出了重庆市2002—2016年的月最大风速资料(样本1)，数据来源于Weather Underground世界天气预报网站(https://www.wunderground.com/).

表 1
表 1 重庆2002—2016年月最大风速资料 Table 1 Data of monthly maximum wind speed from 2002 to 2016 m/s

年份 1月 2月 3月 4月 5月 6月 7月 8月 9月 10月 11月 12月

2002 5.0 5.8 20.0 13.9 5.8 9.7 9.7 7.2 8.1 8.1 5.8 5.0

2003 5.8 7.2 5.0 9.7 5.8 5.0 11.9 5.8 5.8 5.0 5.8 5.0

2004 1.7 5.8 5.0 8.1 8.1 5.8 7.2 9.7 7.2 5.8 5.0 5.8

2005 5.0 5.0 8.1 9.7 7.2 5.8 11.9 7.2 8.9 8.1 7.2 5.0

2006 5.8 5.8 7.2 9.7 8.9 11.1 8.9 5.8 7.2 5.0 5.0 5.0

2007 3.9 7.2 5.8 8.1 11.9 8.1 7.2 8.9 7.2 5.0 5.0 3.9

2008 5.0 3.9 7.2 7.2 8.1 8.9 8.1 8.1 5.8 5.0 5.8 5.8

2009 7.2 7.2 8.1 8.1 7.2 7.2 9.7 8.9 5.8 5.8 8.9 8.1

2010 7.2 7.2 8.1 7.2 8.1 7.2 7.2 8.9 7.2 5.8 5.8 8.9

2011 5.8 7.2 7.2 8.1 8.1 7.2 8.1 8.9 8.1 5.8 7.2 5.8

2012 5.8 7.2 8.1 8.1 8.9 7.2 7.2 9.7 8.1 5.8 7.2 5.8

2013 5.8 7.2 8.1 9.7 8.1 8.9 8.1 8.1 7.2 7.2 8.1 7.2

2014 5.8 7.2 11.1 9.7 8.1 7.2 7.2 8.1 5.8 8.1 5.8 5.8

2015 7.2 8.1 7.2 11.9 8.1 9.7 8.1 8.9 8.1 8.1 8.1 7.2

2016 8.1 5.8 8.1 7.2 8.1 8.1 7.2 8.9 8.1 5.8 8.1 7.2



表 1 重庆2002—2016年月最大风速资料 Table 1 Data of monthly maximum wind speed from 2002 to 2016


极值Ｉ分布的概率分布函数为

$F\left( x \right) = \exp \left( { - \exp \left( { - \left( {x - \beta } \right)/\alpha } \right)} \right),$ (1)

式中:α、β分别为尺度参数和位置参数，与极值Ⅰ型分布的均值μ和σ²方差的关系为

$\mu = \beta + v\alpha ,{\sigma ^2} = \frac{{{{\rm{ \mathsf{ π} }}^2}{\alpha ^2}}}{6},$ (2)

其中v=0.577 22为欧拉常数.

假设样本(x₁, x₂, x₃, …, x_n)符合极值Ⅰ型分布，则：

$\left\{ \begin{array}{l}\mu = \frac{1}{n}\sum\limits_{i = 1}^N {{x_i} = \bar X} ,\\{\sigma ^2} = \frac{1}{{n - 1}}\sum\limits_{i = 1}^N {{{\left( {{x_i} - \bar X} \right)}^2}} = S_x^2.\end{array} \right.$ (3)

由式(2)、(3)可得参数α、β的估计值$\hat \alpha $、$\hat \beta $：

$\left\{ \begin{array}{l}\hat \alpha = \frac{{2.449\;5}}{{\rm{ \mathsf{ π} }}}{S_x},\\\hat \beta = \bar X - 0.450\;05{S_x}.\end{array} \right.$ (4)

风速样本1的平均值和标准差分别为X=7.37和S_x=1.95，因此可得参数估计值分别为${\hat \alpha }$=1.52和${\hat \beta }$=6.49，下面通过K-S检验验证样本与理论分布的拟合优度，计算$\sqrt n $D_n=$\sqrt {180} $×0.08=1.07＜1.36，其中，n为样本1风速数据数量，D_n为理论分布与经验分布的最大偏差，即D_n = max{|F_n^*(X)-F(X)|}，其中F_n^*(X)为经验分布函数，F(X)为理论分布函数.样本计算结果小于临界值，因此不拒绝原假设，样本数据服从极值Ⅰ型分布.样本数据点与极值Ⅰ型分布拟合对比见图 1、2.

Figure 1
图 1 样本点与极值Ⅰ型分布函数对比 Figure 1 Simulation and contrast of distribution function


Figure 2
图 2 样本点与极值Ⅰ型概率密度函数对比 Figure 2 Simulation and contrast of probability density function


由图 1、2可知，样本数据点与极值Ⅰ型分布函数及概率密度函数拟合对比较接近，验证了采用极值Ⅰ型分布对重庆市月最大风速分布的拟合是合理的.

1.2 设计最大风速危险性分析中国现行荷载规范定义标准高度10 m处的最大风速V₁₀为基本风速^[9]，作为各地区风速统计样本，则其概率分布函数为

$F\left( {{V_{10}}} \right) = \exp \left( { - \exp \left( { - \left( {{V_{10}} - \beta } \right)/\alpha } \right)} \right),$ (5)

其中：

$\hat \alpha = \frac{{2.449\;5}}{\pi }{S_x}$；$\hat \beta = \bar X-0.450\;05{S_x}$，带入式(5)得：

$F\left( {{V_{10}}} \right) = \exp \left( { - \exp \left( {\frac{{ - \left( {{V_{10}} - \bar X + 0.450\;05{S_x}} \right)}}{{\frac{{2.449\;5}}{{\rm{ \mathsf{ π} }}}{S_x}}}} \right)} \right).$ (6)

重现期T₀的设计最大风速V_m为

${V_{\rm{m}}} = \bar X - \frac{{2.449\;5}}{{\rm{ \mathsf{ π} }}}{S_x}\left[ {0.577\;2 + \ln \left( { - \ln {P_0}} \right)} \right],$ (7)

其中P₀=1-λ=1-$\frac{1}{{{T_0}}}$，λ为年平均超越概率，

$\lambda = \frac{1}{{{T_0}}}.$ (8)

则

$\begin{array}{l}\lambda \left( {{V_{\rm{m}}}} \right) = \\\;\;1 - \exp \left( { - \exp \left( {\frac{{ - \left( {{V_{10}} - \bar X + 0.450\;05{S_x}} \right)}}{{\frac{{2.449\;5}}{{\rm{ \mathsf{ π} }}}{S_x}}}} \right)} \right).\end{array}$ (9)

对于工程结构而言，风速大小一般用风压表示.GB 50009—2012《建筑结构荷载规范》中列出了中国各地区10 a、50 a和100 a重现期的基本风压值w，根据基本风压值，可得到相应的设计最大风速V_m.例如，重庆地区重现期为10 a的基本风压值为0.25 kN/m²，重现期为50 a的基本风压值为0.40 kN/m²，100 a的基本风压值为0.45 kN/m^2[9].由伯努利公式可得

$V = \sqrt {1\;600w} .$ (10)

由式(10)可计算得到重庆市重现期为10、50 a的对应设计最大风速V_m(10 a)=20 m/s，V_m(50 a) =25.3 m/s.将其带入式(5)，可得到V₁₀的平均值为14.591 m/s，标准差S_x=4.115 m/s.将其带入式(6)即可得到相应设计风速的危险性曲线.

$\lambda \left( {{V_{\rm{m}}}} \right) = 1 - \exp \left( { - \exp \left( {\frac{{ - \left( {{V_{\rm{m}}} - 12.74} \right)}}{{3.21}}} \right)} \right).$ (11)

运用MATLAB将式(11)用五阶多项式近似拟合，得到拟合函数：

$\lambda \left( x \right) = {c_0} + {c_1}x + {c_2}{x^2} + {c_3}{x^3} + {c_4}{x^4} + {c_5}{x^5}.$ (12)

拟合后的设计风速超越概率曲线见图 3.

Figure 3
图 3 设计风速超越概率曲线 Figure 3 Wind velocity hazard curve


1.3 脉动风速危险性分析脉动风作为具有各态历经性的零均值Gauss平稳随机过程近似考虑，用Davenport谱表示10 m高度处水平顺风向脉动风速功率谱，经验表达式为^[10]

${S_V}\left( \omega \right) = \frac{{4K{f^2}}}{{\omega {{\left( {1 + {f^2}} \right)}^{4/3}}}}\bar V_{10}^2,$ (13)

式中：$f = \frac{{1\;200\;\omega }}{{{{\bar V}_{10}}}}$，ω为脉动风频率，K为表面阻力系数，根据GB 50009—2012《建筑结构荷载规范》，取K=0.0129.

由式(13)变化可得

${V_{\rm{m}}} = \frac{{1\;200\omega }}{{\sqrt {{{\left( {\frac{{2\;{{400}^2}K\omega }}{{{S_V}\left( \omega \right)}}} \right)}^{\frac{3}{4}}} - 1} }},$ (14)

将式(14)代入式(11)得

$\lambda \left( {{V_{\rm{m}}}} \right) = 1 - {{\rm{e}}^{ - \exp \left( { - \frac{{373.831\;8\omega }}{{{{\left( {\frac{{2\;{{400}^2}K\omega }}{{{S_V}\left( \omega \right)}}} \right)}^{\frac{3}{4}}} - 1}} + 3.969} \right)}}.$ (15)

通过式(15)可得到对应脉动风速谱的年平均超越概率曲线，即脉动风速谱危险性曲线.以重庆地区为例，D类场地，基于最不利设计原则，式中脉动风频率取薄膜自振频率0.32 Hz，代入式(15)并将该曲线用多项式拟合，见图 4.

Figure 4
图 4 脉动风速谱超越概率曲线 Figure 4 Velocity spectrum hazard curve


拟合函数为

$\lambda \left( y \right) = {c_0} + {c_1}y + {c_2}{y^2} + {c_3}{y^3} + {c_4}{y^4} + {c_5}{y^5},$ (16)

经计算系数为

${c_0} = 1.386,{c_1} = 0.157\;5,{c_2} = 0.003\;77,$

${c_3} = 0.000\;357,{c_4} = - 0.000\;024\;4,$

${c_5} = 4.383 \times {10^{ - 7}}.$

由于横风向脉动风速谱和垂直风向脉动风速谱远小于纵向风速谱^[11]，本文主要工作为研究薄膜结构的抗风概率性能评估方法，为简化计算，在研究中暂不考虑垂直向和横风向脉动风速影响.

2 结构危险性分析为实现薄膜结构抗风概率化评估，在完成风场分析后，需求解不同风场强度下的结构非线性响应，统计获得响应均值与标准差，通过结构危险性分析得到薄膜结构抗风性能需求的年平均超越概率值.

本文理论推导过程不考虑结构响应的极限状态DM随机特性影响，性能需求参数为EDP，EDP选取膜面最大法向变形值，即λ(DM)=λ(EDP).则薄膜结构抗风性能需求参数EDP年平均超越概率为

$\lambda \left( {{\rm{DM}}} \right) = \lambda \left( {{\rm{EDP}}} \right) = \int {G\left( {{\rm{EDP}}\left| {{\rm{IM}}} \right.} \right){\rm{d}}\lambda \left( {{\rm{IM}}} \right)} ,$ (17)

其中IM为风场强度.

在理论推导过程中做如下假定：

1) 考虑薄膜结构非线性振动特性，根据频域分析过程，结构位移响应的根方差为功率谱在频域内的积分的1/2次幂^[12]，而功率谱与风速谱值相关.文献[13]试验结果表明，结构响应均值与风速谱值存在幂级数关系，因此，可近似将结构抗风位移需求表示成风速谱值的幂函数关系，即η_EDP=α(S_V)^b.

2) 假定脉动风为零均值平稳高斯随机过程，在相同的风速谱下，薄膜结构抗风位移需求满足对数正态分布，并且假定标准差σ_EDP为常量，不随风速谱值而变化.

3) 由于脉动风速谱中同时包含平均风速和脉动风速信息，本文风场强度采用脉动风速谱超越概率曲线进行描述，并进行多项式拟合，简化理论推导过程.

由式(17)可知：

$\begin{array}{l}\lambda \left( d \right) = \int {G\left( {d\left| {{S_V}} \right.} \right){\rm{d}}\lambda \left( {{S_V}} \right)} = \\\;\;\;\;\;\;\;\;\;\;\;\int {P\left( {D > d\left| {{S_V}} \right.} \right){\rm{d}}\lambda \left( {{S_V}} \right)} = \\\;\;\;\;\;\;\;\;\;\;\;\int {\frac{{{\rm{d}}P\left( {D > d\left| {{S_V}} \right.} \right)}}{{{\rm{d}}{S_V}}}\lambda \left( {{S_V}} \right){\rm{d}}{S_V}} ,\end{array}$ (18)

式中D为抗风需求值.

因为：

$\begin{array}{l}P\left( {D > d\left| {{S_V}} \right.} \right) = 1 - P\left( {D \le d\left| {{S_V}} \right.} \right) = \\\;\;\;\;\;\;\;1 - P\left( {\ln D \le \ln d\left| {{S_V}} \right.} \right) = \\\;\;\;\;\;\;\;1 - P\left( {\frac{{\ln D - \ln {\eta _D}}}{{{\sigma _D}}} \le \frac{{\ln D - \ln {\eta _D}}}{{{\sigma _D}}}\left| {{S_V}} \right.} \right) = \\\;\;\;\;\;\;\;1 - \Phi \left[ {\frac{{\ln \left( {d/{\eta _D}} \right)}}{{{\sigma _D}}}} \right],\end{array}$ (19)

其中Φ[·]表示标准正态分布函数.

则

$\begin{array}{l}\frac{{{\rm{d}}P\left( {D > d\left| {{S_V}} \right.} \right)}}{{{\rm{d}}{S_V}}} = \frac{{\rm{d}}}{{{\rm{d}}{S_V}}}\left\{ {1 - \Phi \left[ {\frac{{\ln \left( {d/{\eta _D}} \right)}}{{{\sigma _D}}}} \right]} \right\} = \\\;\;\;\;\;\; - \frac{{\rm{d}}}{{{\rm{d}}{S_V}}}\left\{ {\Phi \left[ {\frac{{\ln d - \ln {\eta _D}}}{{{\sigma _D}}}} \right]} \right\} = \\\;\;\;\;\;\; - \frac{{\rm{d}}}{{{\rm{d}}{S_V}}}\left\{ {\Phi \left[ {\frac{{\ln d - \ln \left( {a{{\left( {{S_V}} \right)}^b}} \right)}}{{{\sigma _D}}}} \right]} \right\}.\end{array}$ (20)

整理式(20)得

$\frac{{{\rm{d}}P\left( {D > d\left| {{S_V}} \right.} \right)}}{{{\rm{d}}{S_V}}} = \frac{b}{{{S_V}{\sigma _D}}}\Phi \left[ {\frac{{\ln d - \ln \left( {a{{\left( {{S_V}} \right)}^b}} \right)}}{{{\sigma _D}}}} \right].$ (21)

将式(21)带入式(18)得

$\begin{array}{l}\lambda \left( d \right) = \int {\frac{b}{{{S_V}{\sigma _D}}}\Phi \left[ {\frac{{\ln d - \ln \left( {a{{\left( {{S_V}} \right)}^b}} \right)}}{{{\sigma _D}}}} \right]\lambda \left( {{S_V}} \right){\rm{d}}{S_V}} = \\\;\;\;\;\;\;\int {\frac{b}{{{S_V}{\sigma _D}}}\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}{{\rm{e}}^{ - \frac{1}{2}{{\left[ {\frac{{\ln d - \ln \left( {a\left( {{S_V}} \right)b} \right)}}{{{\sigma _D}}}} \right]}^2}}}\lambda \left( {{S_V}} \right){\rm{d}}{S_V}} .\end{array}$ (22)

将式(16)代入式(22)，并整理得

$\begin{array}{*{20}{c}}{\lambda \left( d \right) = \int {\frac{b}{{{S_V}{\sigma _D}}}\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}{{\rm{e}}^{ - \frac{1}{2}{{\left[ {\frac{{\ln d - \ln \left( {a\left( {{S_V}} \right)b} \right)}}{{{\sigma _D}}}} \right]}^2}}}} \sum\limits_{i = 0}^5 {{c_i}S_V^i{\rm{d}}{S_V}} = \sum\limits_{i = 0}^5 {{c_i}\int {\frac{b}{{{S_V}{\sigma _D}}}\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}{{\rm{e}}^{ - \frac{1}{2}{{\left[ {\frac{{\ln d - \ln \left( {a\left( {{S_V}} \right)b} \right)}}{{{\sigma _D}}}} \right]}^2}}}} {{\rm{e}}^{i\ln {S_V}}}{\rm{d}}{S_V}} = }\\{\sum\limits_{i = 0}^5 {{c_i}{{\rm{e}}^{\left[ {\frac{1}{2}{i^2}{{\left( {\frac{{{\sigma _D}}}{b}} \right)}^2}} \right]}}{{\rm{e}}^{i\ln {{\left( {\frac{d}{a}} \right)}^{\frac{1}{b}}}}}} \cdot \int {\frac{b}{{{S_V}{\sigma _D}}}\frac{1}{{\sqrt {2\pi } }}{{\rm{e}}^{ - \frac{1}{2}{{\left[ {\left( {\ln {S_V} - \left[ {\ln {{\left( {\frac{d}{a}} \right)}^{\frac{1}{b}}} + i{{\left( {\frac{{{\sigma _D}}}{b}} \right)}^2}} \right]} \right)/\frac{{{\sigma _D}}}{b}} \right]}^2}}}{\rm{d}}{S_V}} .}\end{array}$ (23)

令

$\psi = \left[ {\left( {\ln {S_V} - \left[ {\ln {{\left( {\frac{d}{a}} \right)}^{\frac{1}{b}}} + i{{\left( {\frac{{{\sigma _D}}}{b}} \right)}^2}} \right]} \right)/\frac{{{\sigma _D}}}{b}} \right],$

则${\rm{d}}\psi {\rm{ = }}\frac{b}{{{S_V}{\sigma _D}}}{\rm{d}}{\mathit{S}_V}$, 将其带入式(23)得

$\begin{array}{l}\lambda \left( d \right) = \sum\limits_{i = 0}^5 {{c_i}{{\rm{e}}^{\left[ {\frac{1}{2}{i^2}{{\left( {\frac{{{\sigma _D}}}{b}} \right)}^2}} \right]}}{{\rm{e}}^{i\ln {{\left( {\frac{d}{a}} \right)}^{\frac{1}{b}}}}}} \cdot \int {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}{{\rm{e}}^{ - \frac{1}{2}{\psi ^2}}}{\rm{d}}\psi } = \\\;\;\;\;\;\;\;\sum\limits_{i = 0}^5 {{c_i}{{\left[ {{{\left( {\frac{d}{a}} \right)}^{\frac{1}{b}}}} \right]}^i}{{\rm{e}}^{\left[ {\frac{1}{2}{i^2}{{\left( {\frac{{{\sigma _D}}}{b}} \right)}^2}} \right]}}} .\end{array}$ (24)

式中：c_i由式(16)给出，d为薄膜屋盖结构允许最大竖向变形值，a、b为η_D的拟合曲线参数，σ_D为薄膜结构响应标准差.

3 薄膜结构抗风性能概率评估薄膜结构抗风性能概率评估分析步骤为：1)风场分析，获得λ(IM)；2)结构分析，获得不同IM下结构非线性响应；3)结构危险性分析，获得λ(EDP).以薄膜屋盖结构为例，屋盖膜材E₁=1 400 MPa，E₂=900 MPa，ρ=1.226 kg/m³，h=0.001 m.屋盖结构长宽分别为a=b=20 m.屋盖模型见图 5.

Figure 5
图 5 平面膜屋盖 Figure 5 Planar membrane roof


初始预张力为2 kN.屋盖结构自振频率为0.32 Hz. CECS 158:2015《膜结构技术规程》中5.3.4条规定：根据正常使用极限状态设计的膜结构的法向位移不得超过规定限值^[14].实际工程中，由于使用功能要求，对于薄膜屋盖结构法向位移在满足规程要求的前提下存在不同性能水准限值，本文主要工作为研究概率评估方法，未对薄膜屋盖不同性能水准限值进行研究，因此在进行基于性能概率的薄膜屋盖抗风评估时，工程需求参数EDP取屋盖膜面的最大变形值U_m.寻求不同薄膜法向位移限值下的超越概率.

3.1 风场分析用脉动风速谱年平均超越概率λ(S_V)表示风的危险性，将重庆市风速谱年平均超越概率用五次多项式拟合，多项式系数分别为

${c_0} = 1.386,{c_1} = 0.157\;5,{c_2} = 0.003\;77,$

${c_3} = 0.000\;357,{c_4} = - 0.000\;024\;4,$

${c_5} = 4.383 \times {10^{ - 7}}.$

3.2 结构分析计算不同强度风下的薄膜结构响应，本文利用MATLAB采用Newmark-β法对薄膜屋盖进行时域分析.取重庆地区设计重现期分别对应10、50、100 a的3种计算工况进行分析，水平风向沿x向，风向角为0°.各工况对应风荷载信息见表 2.

表 2
表 2 重庆风荷载基本信息 Table 2 Wind load information of Chongqing T₀/a λ=1/T₀ w/(kN·m^-2) V_m/(m·s^-1)

10 0.10 0.25 20.00

50 0.02 0.40 25.30

100 0.01 0.45 26.83



表 2 重庆风荷载基本信息 Table 2 Wind load information of Chongqing


将薄膜屋盖均匀划分为36个区域，对应不同重现期风荷载工况，每工况取10条随机样本数据，对薄膜屋盖结构进行风振时程分析.本文模拟点的对应编号见图 6.基本风压为0.25，地面粗糙度类别为D类.采用AR自回归模型模拟生成脉动风速时程，截止频率为100 Hz，初始频率为0.001 Hz，采样时间间隔取△t=0.1 s，确定自回归阶数p=4，时间步数N=1 024.

Figure 6
图 6 薄膜屋盖表面风速模拟点位置 Figure 6 Position number of wind speed simulation points


以11号点为例，考虑时间及空间相关性，采用MATLAB随机生成风速时程曲线，见图 7(a).图 7(b)对比节点模拟风速功率谱密度曲线与目标功率谱密度曲线，通过对比可以看出，本文模拟生成的脉动风速统计概率与理论概率吻合较好.

Figure 7
图 7 11号点处模拟风速时程曲线与风速功率谱密度曲线 Figure 7 Time history curve of simulated wind speed and simulated wind speed power spectral density of 11 point


作用在节点i位置上的风荷载可表示为

${P_i}\left( t \right) = {P_{i1}}\left( t \right) + {P_{i2}}\left( t \right).$ (25)

考虑平均风和脉动风对薄膜屋盖的作用影响，则

${P_{i1}}\left( t \right) = \frac{1}{2}{C_{\rm{p}}}\rho A{\left( {\bar V + V\left( t \right)} \right)^2},$

式中：C_p为薄膜屋盖平均风压分布系数，风荷载模拟中采用的体型系数参考文献[15]中薄膜平屋盖的分压分布系数；ρ为空气密度，取ρ=1.226 kg/m³；A为节点对应面积；V为平均风速，V(t)为脉动风速.

式(25)中，P_i2(t)为薄膜屋盖内侧附加空气质量产生的惯性荷载，可取

${P_{i2}}\left( t \right) = - {m_a}\frac{{{\partial ^2}U}}{{\partial {t^2}}},$

其中m_a为附加空气质量，具体计算公式为^[15]

${m_a} = - \frac{{\int_0^b {\int_0^a {\frac{{\partial h\left( {x,y,t} \right)}}{{\partial t}} \cdot p \cdot {\rm{d}}x{\rm{d}}y} } }}{{\int_0^b {\int_0^a {\frac{{\partial h\left( {x,y,t} \right)}}{{\partial t}} \cdot \frac{{{\partial ^2}h\left( {x,y,t} \right)}}{{\partial t}} \cdot {\rm{d}}x{\rm{d}}y} } }},$

其中p为作用在薄膜屋盖上的气动力，h(x, y, t)为薄膜屋盖法向位移.

则薄膜屋盖的非线性振动控制方程为

$\left[ M \right]\left\{ {\ddot U} \right\} + \left[ C \right]\left\{ {\dot U} \right\} + \left[ K \right]\left\{ U \right\} = \left\{ P \right\}.$ (26)

采用Newmark-β法对式(26)进行时域分析^[16].分析结果如下：

统计表 3数据可得薄膜屋盖位移峰值响应平均值的拟合曲线，拟合参数a=0.024，b=0.67.

表 3
表 3 薄膜屋盖位移峰值响应 Table 3 Maximum displacement response of the membrane roof m

工况 10 a 50 a 100 a

1 0.144 9 0.231 1 0.280 6

2 0.166 8 0.234 3 0.308 2

3 0.158 0 0.253 9 0.261 8

4 0.150 0 0.251 5 0.292 5

5 0.158 6 0.217 2 0.318 9

6 0.173 0 0.246 1 0.292 6

7 0.153 5 0.223 1 0.203 8

8 0.162 5 0.271 0 0.274 6

9 0.146 9 0.242 9 0.289 1

10 0.153 9 0.251 7 0.312 0

平均值 0.154 7 0.234 2 0.270 5

标准差 0.008 2 0.013 5 0.031 4



表 3 薄膜屋盖位移峰值响应 Table 3 Maximum displacement response of the membrane roof


$\begin{array}{l}\lambda \left( d \right) = 1.386 - 0.157\;5{\left( {\frac{d}{{0.024}}} \right)^{\frac{1}{{0.67}}}}{{\rm{e}}^{\left[ {\frac{1}{2}{{\left( {\frac{{0.038\;5}}{{0.67}}} \right)}^2}} \right]}} + 0.003\;77{\left[ {{{\left( {\frac{d}{{0.024}}} \right)}^{\frac{1}{{0.67}}}}} \right]^2}{{\rm{e}}^{\left[ {2{{\left( {\frac{{0.038\;5}}{{0.67}}} \right)}^2}} \right]}} + 0.000\;357 \times \\{\left[ {{{\left( {\frac{d}{{0.024}}} \right)}^{\frac{1}{{0.67}}}}} \right]^3}{{\rm{e}}^{\left[ {\frac{9}{2}{{\left( {\frac{{0.038\;5}}{{0.67}}} \right)}^2}} \right]}} - 0.000\;024\;4{\left[ {{{\left( {\frac{d}{{0.024}}} \right)}^{\frac{1}{{0.67}}}}} \right]^4}{{\rm{e}}^{\left[ {8{{\left( {\frac{{0.038\;5}}{{0.67}}} \right)}^2}} \right]}} + 4.383 \times {10^{ - 7}} \times {\left[ {{{\left( {\frac{d}{{0.024}}} \right)}^{\frac{1}{{0.67}}}}} \right]^5}{{\rm{e}}^{\left[ {\frac{{25}}{2}{{\left( {\frac{{0.038\;5}}{{0.67}}} \right)}^2}} \right]}}.\end{array}$ (27)

由式(27)可得薄膜屋盖结构不同位移需求的年平均超越概率.年平均超越概率见表 4.

表 4
表 4 薄膜屋盖对应位移需求的超越概率 Table 4 Exceeding probability assessment for the membrane roof under different displacement demands 位移性能点 年平均超越概率/% 响应重现期/a

L/300 65.89 1.52

L/250 51.24 1.95

L/200 33.51 2.98

L/150 16.97 5.89

L/125 11.58 8.64

注：其中L取薄膜屋盖结构长边尺寸



表 4 薄膜屋盖对应位移需求的超越概率 Table 4 Exceeding probability assessment for the membrane roof under different displacement demands


重庆场地风荷载下，薄膜屋盖结构最大法向位移的年平均超越概率曲线见图 8.

Figure 8
图 8 位移需求危险性曲线 Figure 8 Displacement demand hazard curve


上述研究可知，采用本文推导得到的闭式表达式，经过少量计算，即可获得薄膜屋盖对应不同变形要求的年平均超越概率值及最大响应重现期，本文提出的方法给后续薄膜屋盖结构对应不同内力需求、加速度需求等指标的超越概率求解提供了理论基础, 从而实现薄膜屋盖抗风设计概率化评估.

4 结论1) 以极值Ⅰ型分布函数对重庆市月平均最大风速进行估算，通过K-S检验表明极值Ⅰ型分布函数可以较好拟合重庆地区风速.

2) 五次多项式能够很好拟合设计最大风速与脉动风速谱年平均超越概率曲线，多项式拟合对于薄膜屋盖抗风需求的年平均超越概率表达式推导有较好的简化效果.

3) 推导得到薄膜屋盖抗风需求年平均超越概率表达式，获得薄膜屋盖对应不同变形要求的超越概率值.从最大程度上减少了计算工作量.

4) 提出薄膜屋盖结构抗风性能概率评估框架，通过少量结构分析可以得到薄膜屋盖结构不同位移需求的超越概率评估值.


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