﻿ 层间位移约束下高层框架-支撑结构的单步优化法
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (11): 1494-1500, 1593.  DOI: 10.11908/j.issn.0253-374x.2018.11.004 0

### 引用本文

SUN Feifei, MA Zhidong, JIA Ruizi. Single-step Optimization Method Under Inter-story Drift Constraints for High-rise Braced-frame Structures[J]. Journal of Tongji University (Natural Science), 2018, 46(11): 1494-1500, 1593. DOI: 10.11908/j.issn.0253-374x.2018.11.004

### 文章历史

1. 同济大学 土木工程防灾国家重点实验室，上海 200092;
2. 同济大学 土木工程学院，上海 200092

Single-step Optimization Method Under Inter-story Drift Constraints for High-rise Braced-frame Structures
SUN Feifei 1, MA Zhidong 2, JIA Ruizi 2
1. State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China;
2. College of Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: The multi-constrained optimization problem of regular high-rise structures based on inter-story displacement constraints was transformed into a single constraint problem. On this basis, the single-step optimization method(SOM) based on virtual work was proposed. The optimization results of SOM were compared with the optimization results of SAP2000. The results show that SOM based on virtual work can reduce the mass of material by 10%.
Key words: regular high-rise structures    inter-story displacement constraints    single-step optimization method

1 高层建筑规则结构的层间位移机制

1.1 算例的结构信息

 图 1 框架支撑结构算例 Fig.1 Example of braced-frame structure
1.2 各楼层对目标楼层层间位移角的贡献规律

 ${\theta _t} = \sum\limits_j {{\theta _{t,j}}}$ (1)

 图 2 第j层对第t层层间位移角的贡献 Fig.2 Contribution of jth floor to the inter-story drift of tth floor

1.3 某一楼层对各层层间位移角的贡献规律

 图 3 整十层对层间位移角的贡献 Fig.3 Contribution of each 10th floor to inter-story drift
1.4 整体转动对层间位移角的贡献规律

 ${\theta _t} = {\theta _1} + {\theta _2} + {\theta _3}$ (2)

 图 4 层间位移角分离曲线 Fig.4 Separation curve of inter-story drift
1.5 不同构件对结构顶部转动的贡献规律

 图 5 层间位移角按构件分离曲线 Fig.5 Separation curve of inter-story drift by elements

1.6 高层建筑规则结构的变形机制

2 单步优化法的建立

2.1 层间位移约束下结构优化问题

 $\left\{ \begin{array}{l} \min W = \sum\limits_{i = 1}^n {{\rho _i}{A_i}{L_i}} \\ {\rm{s}}.\;{\rm{t}}.\;\;{\theta _j} \le \left[ \theta \right]\\ \;\;\;\;\;\;{A_i} \ge \underline {{{\rm{A}}_{\rm{k}}}} \\ \;\;\;\;\;\;i = 1,2, \cdots ,n,j = 1,2, \cdots ,m \end{array} \right.$ (3)

2.2 基本假定

 ${\theta _j} = \sum\limits_{i = 1}^n {{\delta _i}}$ (4)
 ${\delta _i} = \frac{{{F_i}{f_i}}}{{{E_i}{A_i}}}{L_i} + \int\limits_L {\frac{{{M_i}{m_i}}}{{{E_i}{I_i}}}{\rm{d}}x}$ (5)

 ${{\delta '}_i} = \frac{{{A_i}}}{{{{A'}_i}}}{\delta _{Ni}} + \frac{{{I_i}}}{{{{I'}_i}}}{\delta _{Mi}}$ (6)

2.3 虚功准则法

 ${\delta _i} = \frac{{{\tau _i}}}{{{A_i}}} + \frac{{{\upsilon _i}}}{{{I_i}}}$ (7)
 ${\theta _j} = \sum\limits_i {\frac{{{\tau _i}}}{{{A_i}}}} + \sum\limits_i {\frac{{{\upsilon _i}}}{{{I_i}}}}$ (8)

 $\left\{ \begin{array}{l} \min W = \sum\limits_{i = 1}^n {{\rho _i}{A_i}{L_i}} \\ {\rm{s}}.\;{\rm{t}}.\;\;{\theta _j} = \sum\limits_i {\frac{{{\tau _i}}}{{{A_i}}}} + \sum\limits_i {\frac{{{\upsilon _i}}}{{{I_i}}}} \le \left[ \theta \right]\\ \;\;\;\;\;\;{A_i} \ge \underline {{{\rm{A}}_{\rm{k}}}} ,i = 1,2, \cdots ,n \end{array} \right.$ (9)

 $- \frac{{\partial \theta }}{{\partial {A_k}}}/\frac{{\partial W}}{{\partial {A_k}}} = \frac{1}{\lambda }$ (10)

 $\left( {\frac{{{\tau _k}}}{{{A_k}}} + \frac{{{\upsilon _k}}}{{{I_k}}}} \right)/{\rho _k}{L_k}{A_k} = {\gamma _k} = \frac{1}{\lambda }$ (11)

2.4 单步优化法的提出

2.4.1 约束条件的转换

2.4.2 单步优化问题

 $\left\{ \begin{array}{l} \min W = \sum\limits_{i = 1}^n {{\rho _i}{A_i}{L_i}} \\ {\rm{s}}.\;{\rm{t}}.\;\;{\theta _{{\rm{rtop}}}} \le \left[ {{\theta _{{\rm{rtop}}}}} \right]\\ \;\;\;\;\;\;{A_i} \ge \underline {{{\rm{A}}_{\rm{k}}}} ,i = 1,2, \cdots ,n \end{array} \right.$ (12)

 图 6 结构整体转动曲线 Fig.6 Rigid body rotation curve of the structure
 ${\theta _{\rm{d}}} = \left( {{\theta _{\max }} - \left[ \theta \right]} \right)\frac{{{\theta _{{\rm{rtop}}}}}}{{{\theta _{\max ,n}}}}$ (13)

 $\left[ {{\theta _{{\rm{rtop}}}}} \right] = {\theta _{{\rm{rtop}}}} - {\theta _{\rm{d}}}$ (14)

 $\left[ {{\theta _{{\rm{rtop}}}}} \right] = {\theta _{{\rm{top}}}} - {\theta _{\rm{d}}}$ (15)

2.4.3 单步优化过程

 ${\beta _i} = \sqrt[2]{{\frac{{{\gamma _i}}}{{{\gamma _g}}}}}$ (16)

βi一旦确定，初始结构的构件尺寸需要被加强的程度也就确定，以此实现单步优化，使结构以较高的效率满足层间位移约束条件.具体实现过程如下所示：

(1) 确定初始结构，构件尺寸取满足强度要求的最小截面.

(2) 在结构顶部加单位弯矩虚荷载.

(3) 提取初始结构的层间位移角曲线和整体转动曲线，计算构件比虚应变能γi、结构平均比虚应变能γave.

(4) 计算结构顶部转角的目标位移角减小量θd和结构放大系数α(α的推导过程见第3.2节)，确定结构的目标比虚应变能γg.

(5) 由比虚应变能γiγg计算构件修正系数βi，那么构件面积修正为βiAi.若构件修正面积Ai大于截面最大限值[Amax]，取Ai=[Amax].

(6) 根据构件修正后面积，在构件库中选择新截面.

(7) 重新运行结构分析，检查强度和位移约束.

3 单步优化法的关键算法 3.1 构件修正系数βi的计算方法

 ${{\delta '}_i} = \frac{{{A_i}}}{{{{A'}_i}}}{\delta _{Ni}} + \frac{{{I_i}}}{{{{I'}_i}}}{\delta _{Mi}}$ (17)

 $I = 0.390\;9A$ (18)
 图 7 AISC型钢与GB型钢截面I与A线性拟合 Fig.7 Linear fitting of inertia and area of steel section in AISC and Chinese code

W24系列拟合方程为

 $I = 0.845\;6A$ (19)

 $I = 1.094\;0A$ (20)

HN系列拟合方程为

 $I = 0.379\;4A$ (21)

 $\frac{{{{I'}_i}}}{{{I_i}}} = \frac{{\eta {{A'}_i}}}{{\eta {A_i}}} = {\beta _i}$ (22)
 ${{\delta '}_i} = \frac{{{\delta _{Ni}}}}{{{{A'}_i}/{A_i}}} + \frac{{{\delta _{Mi}}}}{{{{I'}_i}/{I_i}}} = \frac{1}{{{\beta _i}}}{\delta _i}$ (23)

 ${{\gamma '}_i} = \frac{{{{\delta '}_i}}}{{{{V'}_i}}} = \frac{{\frac{1}{{{\beta _i}}}{\delta _i}}}{{{\beta _i}{V_i}}} = \frac{1}{{\beta _i^2}}{\gamma _i}$ (24)

 ${\beta _i} = \sqrt[2]{{\frac{{{\gamma _i}}}{{{\gamma _g}}}}}$ (25)

3.2 结构放大系数α的计算方法

 $\sum {{\gamma _i}{A_i}{L_i}} + \sum {\gamma AL} = {\theta _{{\rm{top}}}}$ (26)

 $\sum {\frac{{{\gamma _{{\rm{ave}}}}}}{\alpha }{\beta _i}{A_i}{L_i}} + \sum {\gamma AL} = {\theta _{{\rm{top}}}} - {\theta _{\rm{d}}}$ (27)

 $\alpha = {\left( {\frac{{\sum {\sqrt {{\gamma _{{\rm{ave}}}}{\gamma _i}} {A_i}{L_i}} }}{{\sum {{\gamma _i}{A_i}{L_i}} - {\theta _{\rm{d}}}}}} \right)^2}$ (28)

4 算例分析 4.1 算例

4.2 单步优化法的优化结果

 图 8 单步优化法优化结果 Fig.8 Results of single-step optimization method

4.3 不同优化方法优化结果对比

SAP2000作为强大的结构设计软件，为用户提供了以位移或者周期为目标的钢结构自动优化功能.对于位移优化，SAP2000软件预测哪个构件需要增加尺寸，以控制基于构件内单位体积能量的位移.单位体积能量更多的构件比能量更少的构件需要增加更大的比例尺寸.只要所考虑的强度允许，有些单位体积能量小的构件是可以减小尺寸的[9].在SAP2000软件“设计”菜单中，通过对“钢框架设计”设置风荷载工况下的侧向位移目标.基于上述位移目标，SAP2000软件可以提供对结构的构件截面自动校核与设计.

 图 9 优化结果质量分布 Fig.9 Mass distribution of optimized structures

 图 10 单步优化法与SAP2000软件优化结果对比 Fig.10 Comparison of optimization results between single-step optimization method and SAP2000 software
5 结论

(1) 通过将高层建筑规则结构层间位移的多约束问题转化为单约束问题实现结构的单步优化.

(2) 在高层建筑规则结构中，最大层间位移角往往出现在结构的中上部，而在这些部位，层间位移以结构的整体转动为主，这是实现结构单步优化的基础.

(3) 基于虚功准则法的单步优化法根据结构整体转动比虚应变能调整构件截面尺寸，使结构整体转动比虚应变能趋于均匀分布并满足层间位移约束，算法物理意义明确，易于掌握.

(4) 将单步优化法的优化结果与SAP2000软件的优化结果进行对比，在都能达到位移限值优化目标下，基于虚功的单步优化法用钢量降低约10%.

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