﻿ 基于灰色理论预测再生混凝土的抗冻性寿命
 上海理工大学学报  2019, Vol. 41 Issue (4): 403-408 PDF

Prediction of Frost Resistance Life of Recycled Concrete Based on Grey Theory
YANG Lu, ZHOU Zhiyun, ZHANG Dingbo, WANG Lidan
School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: By using grey theory, the prediction model of dynamic elastic modulus GM(1,1) of recycled concrete under freeze-thaw cycles was established. Based on the existing experimental data and GM(1,1) prediction model, the dynamic elastic modulus of recycled concrete under other freezing-thawing cycles was predicted. Then, according to life adjustment formula proposed by Vesikari[1], the frost-resistant life of recycled concrete under real natural conditions was predicted. Through calculation and analysis, the results show that the variance ratio of GM(1,1) of the frost resistance model is less than 0.35, and the probability of small error is more than 0.95. With high accuracy, it can be used to predict and analyze the dynamic modulus of elasticity of the frost resistance of recycled concrete. When the content of recycled aggregate is 60%, the frost resistance of recycled concrete reaches the best. For Yichang area, its anti-freezing service life can reach up to 183 years.
Key words: recycled concrete     grey theory     GM(1,1)model     dynamic elastic modulus

1 灰色理论GM(1,1)模型 1.1 建立预测模型

GM(1,1)模型是生成数列的模型，给定序列为

 $\qquad\qquad\;\;{X^{\left( 0 \right)}} = \left( {{x^{\left( 0 \right)}}\left( 1 \right),{x^{\left( 0 \right)}}\left( 2 \right), \cdots ,{x^{\left( 0 \right)}}\left( n \right)} \right)$ (1)

$X^{(0)}$ 用1-GAO累加生成算法将试验数据逐个累加，并用 $X^{(1)}$ 表示。

 $\qquad\qquad\quad{X^{\left( 1 \right)}} = \left( {{x^{\left( 1 \right)}}\left( 1 \right),{x^{\left( 1 \right)}}\left( 2 \right), \cdots ,{x^{\left( 1 \right)}}\left( n \right)} \right)$ (2)

 ${x^{\left( 0 \right)}}\left( k \right) + a{x^{\left( 1 \right)}}\left( k \right) = b$ (3)

 $\frac{{{\rm d}{x^{\left( 1 \right)}}}}{{{\rm d}t}} + a{x^{\left( 1 \right)}} = b$ (4)

 ${x^{\left( 1 \right)}}\left( t \right) = C{{\rm e}^{ - at}} + \frac{b}{a}$ (5)

 ${{Y}} \!=\! \left[ {\begin{array}{*{20}{c}} {{x^{\left( 0 \right)}}\left( 2 \right)}\\ {{x^{\left( 0 \right)}}\left( 3 \right)}\\ \vdots \\ {{x^{\left( 0 \right)}}\left( n \right)} \end{array}} \right],{ B} \!=\! \left[ {\begin{array}{*{20}{c}} {\dfrac{1}{2}\left[ {{x^{\left( 1 \right)}}\left( 2 \right) \!+\! {x^{\left( 1 \right)}}\left( 1 \right)} \right]}&1\\ {\dfrac{1}{2}\left[ {{x^{\left( 1 \right)}}\left( 3 \right) \!+\! {x^{\left( 1 \right)}}\left( 2 \right)} \right]}&1\\ \vdots & \vdots \\ {\dfrac{1}{2}\left[ {{x^{\left( 1 \right)}}\left( n \right) \!+\! {x^{\left( 1 \right)}}\left( {n - 1} \right)} \right]}&1 \end{array}} \right]\!\!\!\!\!\!\!\!$ (6)

t=1时，取 ${x^{\left( 1 \right)}}\left( 1 \right) = {x^{\left( 0 \right)}}\left( 1 \right)$ ，代入式（6），可得

 ${{C}} = {{\rm e}^a}\left[ {{x^{\left( 0 \right)}}\left( 1 \right) - \frac{b}{a}} \right]$ (7)

 ${x^{\left( 1 \right)}}\left( {t \!+\! 1} \right) \!=\! \left( {{x^{\left( 0 \right)}}\left( 1 \right) \!-\! \frac{b}{a}} \right){{\rm e}^{ - at}} \!+\! \frac{b}{a},\;\;\;t \!=\! 0,1,2, \cdots ,n\!\!\!\!\!\!\!\!\!$ (8)

 $\begin{split}&\quad{{\hat x}^{\left( 0 \right)}}\left( k \right) = {\alpha ^{\left( 1 \right)}}{{\hat x}^{\left( 1 \right)}}\left( {t + 1} \right) = {{\hat x}^{\left( 1 \right)}}\left( {t + 1} \right) - {{\hat x}^{\left( 1 \right)}}\left( t \right),\qquad\qquad\qquad\qquad\\ &\qquad k = 1,2 ,\cdots ,n\\[-10pt]\end{split}$ (9)
1.2 模型精度检验

 $\qquad\qquad{\rm{\varepsilon }}\left( k \right) = {x^{\left( 0 \right)}}\left( k \right) - {{\hat x}^{\left( 0 \right)}}\left( k \right),\;\;\;k = 1,2, \cdots ,n$ (10)

 $q = \frac{{\varepsilon \left( k \right)}}{{{x^{\left( 0 \right)}}\left( k \right)}} \times 100{\text{%}}$ (11)

 $\bar x = \frac{1}{n}\sum\limits_{k = 1}^n {{x^{(0)}}} \left( k \right)$ (12)
 $\mathop S\nolimits_1^2 = \frac{1}{n}\sum\limits_{k = 1}^n {{{[{x^{(0)}}(k) - \bar x]}^2}}$ (13)

 $\bar \varepsilon = \frac{1}{n}\sum\limits_{k = 1}^n {\varepsilon (k)}$ (14)
 $\mathop S\nolimits_2^2 = \frac{1}{n}\sum\limits_{k = 1}^n {{{[\varepsilon (k) - \bar \varepsilon ]}^2}}$ (15)

 ${{C}} = {S_2}/{S_1}$ (16)

 $p = \left\{ {\left| {\varepsilon \left( k \right) - \bar \varepsilon } \right| < 0.674\;5{S_1}} \right\}$ (17)

2 应用案例 2.1 试验概况与其预测试验数据

2.2 预测模型GM(1,1)的建立

 ${X^{\left( 0 \right)}} = \left( {48.77,45.48,42.55,25.99,21.19,17.80} \right)$

 ${X^{\left( 1 \right)}} = \left( {48.77,94.25,136.8,162.79,183.98,201.78} \right)$

 ${ B} = \left[ {\begin{array}{*{20}{c}} { - 71.51}&1\\[1.5pt] { - 115.53}&1\\[1.5pt] { - 149.80}&1\\[1.5pt] { - 173.39}&1\\[1.5pt] { - 192.88}&1 \end{array}} \right],\quad{ Y} = \left[ {\begin{array}{*{20}{c}} {45.48}\\[1.5pt] {42.55}\\[1.5pt] {25.99}\\[1.5pt] {21.19}\\[1.5pt] {17.80} \end{array}} \right]$

 ${\hat x^{\left( 1 \right)}}\left( {t + 1} \right) = - 221.54{{\rm e}^{ - 0.24t}} + 270.31,\;t = 0,1,2, \cdots ,n$

2.3 预测模型GM(1,1)的结果分析

3 寿命预测与分析

 $t = \frac{{kN}}{M}$ (18)

 $\begin{split} & \quad{{\hat X}^{\left( 1 \right)}} = (48.77,44.38,46.82,45.84,44.38,42.92,40.48,\qquad\qquad\qquad\qquad\\ & \qquad 39.02,35.11,33.16,29.75,27.31) \end{split}$

 $\left( 1\!,0.91\!,0.96\!,0.94\!,0.91\!,0.88\!,0.83\!,0.80\!,0.72\!,0.68\!,0.61\!,0.56 \right)$

 图 1 使用时间与相对动弹性模量之间的关系（宜昌地区） Fig. 1 Relationship between service time and relative dynamic modulus of elasticity (Yichang area)

 图 2 再生骨料掺量对混凝土使用寿命的影响（宜昌地区） Fig. 2 Effect of recycled aggregate content on service life of concrete (Yichang area)
4 结　论

a. 将再生混凝土的动弹性模量作为混凝土耐久性的主要评价指标，在试验数据的基础上，建立了灰色GM(1,1)预测模型，其计算结果与试验数据符合度较高，说明GM(1,1)的抗冻性模型可用于再生混凝土抗冻性能的动弹性模量预测和分析。因此，灰色理论可为再生混凝土的耐久性能研究提供新的思路，为实际工程中再生混凝土的抗冻性破坏预测提供参考。

b. 基于灰色预测模型得出的预测值进行再生混凝土抗冻性寿命预测发现，掺入再生骨料对混凝土的抗冻性确有显著影响，随着再生骨料的增加，有效寿命逐渐减小。60%的再生骨料掺量为该类再生混凝土的最优配掺比，此时，再生混凝土的抗冻性能达到最佳。从我国宜昌地区来看，其抗冻使用寿命可达183年。

 [1] 李秋义, 高嵩, 薛山. 绿色混凝土技术[M]. 北京: 中国建材工业出版社, 2014. [2] 李秋义, 全洪珠, 秦原. 混凝土再生骨料[M]. 北京: 中国建筑工业出版社, 2011. [3] 魏应乐. 再生混凝土的耐久性及控制措施研究[J]. 混凝土, 2010(1): 81-85. DOI:10.3969/j.issn.1002-3550.2010.01.027 [4] 周宇, 郑秀梅, 李广军, 等. 再生骨料混凝土抗冻性能试验研究[J]. 低温建筑技术, 2013(12): 14-16. DOI:10.3969/j.issn.1001-6864.2013.12.006 [5] 刘曙光, 王志伟, 张菊, 等. 基于灰色理论的PVA-FRCC抗冻性寿命预测[J]. 混凝土与水泥制品, 2013(8): 43-47. DOI:10.3969/j.issn.1000-4637.2013.08.011 [6] 王立久, 汪振双, 崔正龙. 基于冻融损伤抛物线模型的再生混凝土寿命预测[J]. 应用基础与工程科学学报, 2011(1): 29-35. DOI:10.3969/j.issn.1005-0930.2011.01.004 [7] 刘思峰, 杨英杰, 吴利丰, 等. 灰色系统理论及其应用[M]. 北京: 科学出版社, 2014. [8] 邓聚龙. 灰色系统基本方法[M]. 4版, 武汉: 华中工学院出版社, 1996. [9] LI B X, YUAN X L, CUI G, et al. Application of the grey system theory to predict the strength deterioration and sevice life of concrete subjected to sulfate environment[J]. Journal of the Chinese Ceramic Society, 2009, 37(12): 2112-2117. [10] 姜曼骁. 灰色理论在混凝土耐久性研究中的应用[D]. 南京: 南京理工大学, 2012. [11] 阎岩, 张明义, 王家涛. 灰色理论在快速载荷试验数据处理中的应用[J]. 岩土力学, 2006, 27(5): 799-806. DOI:10.3969/j.issn.1000-7598.2006.05.024 [12] 王丽丹, 周志云, 叶林飞, 等. 复合微粉和聚丙烯纤维对再生混凝土抗冻性研究[J]. 上海理工大学学报, 2017, 39(3): 301-306. [13] 李金玉, 邓正刚, 曹建国, 等. 混凝土抗冻性的定量化设计[J]. 混凝土, 2000(12): 61-65.