﻿ 既有网壳结构几何缺陷分布反演算法
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 浙江大学学报(工学版)  2018, Vol. 52 Issue (5): 864-872  DOI:10.3785/j.issn.1008-973X.2018.05.006 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2018.05.006
[复制中文]
WU Jun, LUO Yong-feng, WANG Lei. Inversion algorithm for the geometric imperfection distribution of existing reticulated structures[J]. Journal of Zhejiang University(Engineering Science), 2018, 52(5): 864-872.
dx.doi.org/10.3785/j.issn.1008-973X.2018.05.006
[复制英文]

### 通信联系人

orcid.org/0000-0001-8212-5605.
Email: yfluo93@tongji.edu.cn

### 文章历史

1. 同济大学 土木工程学院, 上海 200092;
2. 上海同恩土木工程科技咨询有限公司, 上海 200092

Inversion algorithm for the geometric imperfection distribution of existing reticulated structures
WU Jun1 , LUO Yong-feng1 , WANG Lei2
1. Department of Structural Engineering, Tongji University, Shanghai 200092, China;
2. Tongen Civil Engineering Technology Consulting Co. Shanghai 200092, China
Abstract: A Markov Random Field (MRF) model of existing reticulated structures was proposed by leading into the probabilistic graph model in view of the fact that the traditional sampling method can hardly obtain the actual distribution of geometric imperfections of existing structures. The calculation unit of double-node and triple-node clique was proposed. The corresponding geometric state function was deducted based on the assumption of local Markov property. The inversion equation for the geometric imperfection distribution of existing reticulated structures were proposed by introducing the joint probability distribution function of MRF. Then, in order to determine the geometric imperfection distribution, the inversion iteration equation was deducted using iterative maximum likelihood method. An experimental model of K6 single-layer reticulated shell was designed to verify the inversion algorithm by calculating and comparing the mode of geometric imperfection distribution. When the ratio of the measured points is greater than 16.5%, the geometric imperfection mode from the inversion calculation has good similarity with the actual mode. The results can even identify the abnormal values of the measured points.
Key words: existing reticulated structure    geometric imperfection    Markov random field    geometric state function    inversion iteration equation

1 马尔可夫随机场

 图 1 马尔可夫随机场模型示意图 Fig. 1 Sketch map of MRF model
 图 2 MRF系统的团类型 Fig. 2 Clique type of MRF model system

 $P\left( X \right) = \frac{1}{T}\prod\limits_{Q \in C} {{\mathit{\Psi }_Q}\left( {{X_Q}} \right)} .$ (1)

 $\begin{array}{l} P\left( X \right) = \frac{1}{T}{\mathit{\Psi }_{125}}\left( {{x_1},{x_2},{x_5}} \right){\mathit{\Psi }_{45}}\left( {{x_4},{x_5}} \right) \times \\ \;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\Psi }_{34}}\left( {{x_3},{x_4}} \right){\mathit{\Psi }_{46}}\left( {{x_4},{x_6}} \right){\mathit{\Psi }_{23}}\left( {{x_2},{x_3}} \right). \end{array}$ (2)

 图 3 节点集A和B的条件独立 Fig. 3 Conditional independence of node A and B

2 结构位形反演算法 2.1 缺陷模型假定

 图 4 网壳结构屈曲模态的几何偏差成分分析 Fig. 4 Component analysis of Geometric imperfections of reticulated shells

 图 5 网壳结构沿曲率半径方向的几何偏差 Fig. 5 Geometric imperfections along radius of curvature

 图 6 节点几何缺陷分布模式 Fig. 6 Nodal geometric imperfection distribution mode
 图 7 网壳结构的马尔可夫随机场(MRF)模型 Fig. 7 MRF model of reticulated structure
2.2 反演迭代方程 2.2.1 一维反演迭代方程

 $G\left( {Y,Z} \right) = T \cdot P\left( X \right) = \prod\limits_{Q \in C} {{\mathit{\Psi }_Q}\left( {{X_Q}} \right)} .$ (3)

 ${\mathit{\Psi }_Q}\left( {{X_Q}} \right) = \exp \left( { - {\mathit{H}_Q}\left( {{\mathit{\boldsymbol{X}}_Q}} \right)} \right).$ (4)

 ${\mathit{H}_Q}\left( {{X_Q}} \right) = \frac{{{{\left( {{x_i} - {x_j}} \right)}^2}}}{{l_Q^2}}.$ (5)

 $G\left( {Y,Z} \right) = \prod\limits_{Q \in C} {\exp \left( { - \frac{{{{\left( {{x_i} - {x_j}} \right)}^2}}}{{l_Q^2}}} \right)} .$ (6)

 $L\left( {y;Z} \right) = \prod\limits_{Q \in C} {\exp \left( { - \frac{{{{\left( {{x_i} - {x_j}} \right)}^2}}}{{l_Q^2}}} \right)} .$ (7)

L$\hat Z$处达到极大值, 则称$\hat Z$为参数集Z的极大似然估计量.求解极大似然估计量的问题即为求解似然函数的极值问题, 通常可由对数似然方程组来表达, 即

 $\frac{{\partial \ln L}}{{\partial {x_k}}} = 0,{x_k} \in Z.$ (8)

 $- 2\sum\limits_{Q \in {C^ * }} {\frac{{\left( {{x_k} - {x_j}} \right)}}{{l_Q^2}}} = 0.$ (9)

 图 8 一维迭代方程的团集合 Fig. 8 Clique list of 1-D iterative equation
2.2.2 二维反演迭代方程

 $Ax + By + Cz + D = 0.$ (10)

 $\cos \theta = \frac{{\left| {\mathit{\boldsymbol{m}} \cdot \mathit{\boldsymbol{n}}} \right|}}{{\left| \mathit{\boldsymbol{m}} \right| \cdot \left| \mathit{\boldsymbol{n}} \right|}} = \frac{C}{{\sqrt {{A^2} + {B^2} + {C^2}} }} = C.$ (11)

 $\left. {\begin{array}{*{20}{c}} {D = 0}\\ {A{x_2} + B{y_2} + C\left( {{z_2} - {z_1}} \right) = 0,}\\ {A{x_3} + B{y_3} + C\left( {{z_3} - {z_1}} \right) = 0.} \end{array}} \right\}.$ (12)

 $C \propto \frac{1}{{{{\left( {{z_1} - {z_2}} \right)}^2}{{\left( {{z_2} - {z_3}} \right)}^2}{{\left( {{z_3} - {z_1}} \right)}^2}}}.$ (13)

 ${\mathit{H}_Q}\left( {{X_Q}} \right) = \frac{{{{\left( {{x_{i1}} - {x_{i2}}} \right)}^2}{{\left( {{x_{i2}} - {x_{i3}}} \right)}^2}{{\left( {{x_{i3}} - {x_{i1}}} \right)}^2}}}{{S_Q^3}}.$ (14)

 $\begin{array}{l} G\left( {Y,Z} \right) = \\ \;\;\;\prod\limits_{Q \in C} {\exp \left( { - \frac{{{{\left( {{x_{i1}} - {x_{i2}}} \right)}^2}{{\left( {{x_{i2}} - {x_{i3}}} \right)}^2}{{\left( {{x_{i3}} - {x_{i1}}} \right)}^2}}}{{S_Q^3}}} \right)} . \end{array}$ (15)

 $\sum\limits_{Q \in C_1^ * } {\frac{{ - {{\left( {{x_{i2}} - {x_{i3}}} \right)}^2}}}{{S_Q^3}}\left[ {2x_k^3 + {\zeta _b}x_k^3 + {\zeta _c}{x_k} + {\zeta _d}} \right]} = 0.$ (16)

 ${\xi _b} = - 3\left( {{x_{i2}} + {x_{i3}}} \right).$ (17)
 ${\zeta _c} = \left( {x_{i2}^2 + x_{i3}^2 + 4{x_{i2}}{x_{i3}}} \right).$ (18)
 ${\zeta _d} = - \left( {{x_{i2}}x_{i3}^2 + x_{i2}^2{x_{i3}}} \right).$ (19)

C1*为包含节点k的所有三节点团集合, 如图 9中三角形面所示.

 图 9 二维迭代方程的团集合 Fig. 9 Clique list of 2-D iterative equation
2.2.3 混合迭代方程

 $\begin{array}{l} G\left( {Y,Z} \right) = \prod\limits_{Q \in C} {\exp \left( { - \frac{{{{\left( {{x_i} - {x_j}} \right)}^2}}}{{l_Q^2}}} \right)} \times \\ \;\;\;\prod\limits_{Q \in C} {\exp \left( { - \frac{{{{\left( {{x_{i1}} - {x_{i2}}} \right)}^2}{{\left( {{x_{i2}} - {x_{i3}}} \right)}^2}{{\left( {{x_{i3}} - {x_{i1}}} \right)}^2}}}{{S_Q^3}}} \right)} . \end{array}$ (20)

 $\begin{array}{l} \sum\limits_{Q \in {C^ * }} {\frac{{ - {{\left( {{x_{i2}} - {x_{i3}}} \right)}^2}}}{{S_Q^3}}\left[ {2x_k^3 + {\zeta _b}x_k^3 + {\zeta _c}{x_k} + {\zeta _d}} \right]} - \\ \;\;\;\;2\sum\limits_{Q' \in {C^ * }} {\frac{{\left( {{x_k} - {x_j}} \right)}}{{l_{Q'}^3}}} = 0. \end{array}$ (21)

 图 10 既有网壳结构几何缺陷反演计算流程图 Fig. 10 Geometric imperfection inversion calculation flow chart of reticulated structures
3 试验分析 3.1 结构信息

 图 11 试验单层球面网壳模型 Fig. 11 Experimental single layer reticulated shell model
 图 12 网壳节点编号图 Fig. 12 Node numbering of reticulated shell model
3.2 节点几何缺陷推演

 图 13 网壳几何缺陷分布云图 Fig. 13 Nephogram of geometric imperfection of experimental mode

 图 14 节点集Z的几何缺陷计算值(标记点均为计算节点) Fig. 14 Computational geometric imperfection curves of node set Z

 ${\rm{PCC}} = \frac{{N\sum {XY} - \sum X \sum Y }}{{\sqrt {N\sum {{X^2}} - {{\left( {\sum X } \right)}^2}} \sqrt {N\sum {{Y^2}} - {{\left( {\sum Y } \right)}^2}} }}.$ (22)

 图 15 测点分布图(节选) Fig. 15 Distribution of measuring points (excerpt)

 图 16 计算几何缺陷的PCC趋势 Fig. 16 PCC trends of computational geometric imperfection
3.3 整体稳定承载力分析

 图 17 网壳试验模型的极限荷载 Fig. 17 Ultimate loads of sample shell

4 结论

(1) 通过试验分析表明, 当测点数量占总比约55%时, 反演迭代方程的计算结果与实测值的Pearson相关系数(PCC)为0.77, 计算值与实测值具有很高的相似度.

(2) 当实测值出现漂移异常时, 反演迭代方程的计算结果可识别节点偏移实测值的异常区域.

(3) 通过将测点数量逐级递增并计算对应的PCC值, 得出当测点数量递增至16.5%的过程中, PCC值迅速达到0.6；而后当测点数量继续增加时, PCC值增加缓慢, 说明测点数量占总比16.5%时, 可由最小的测点数量得到满意的计算相似度.

(4) 依据反演结果计算的网壳稳定承载力与真实结构相当, 略偏安全.

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