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Article(id=1241321990506730044, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241321979433767757, articleNumber=null, orderNo=null, doi=10.3969/j.issn.0253-6099.2024.02.006, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1698422400000, receivedDateStr=2023-10-28, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1773883825184, onlineDateStr=2026-03-19, pubDate=1711900800000, pubDateStr=2024-04-01, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773883825184, onlineIssueDateStr=2026-03-19, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773883825184, creator=13701087609, updateTime=1773883825184, updator=13701087609, issue=Issue{id=1241321979433767757, tenantId=1146029695717560320, journalId=1235980550691926019, year='2024', volume='44', issue='2', pageStart='1', pageEnd='191', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773883822544, creator=13701087609, updateTime=1773884556149, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241325056454881881, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241321979433767757, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241325056454881882, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241321979433767757, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=22, endPage=26, ext={EN=ArticleExt(id=1241321990859051613, articleId=1241321990506730044, tenantId=1146029695717560320, journalId=1235980550691926019, language=EN, title=Settlement Prediction for Pile Foundation of Vertically Loaded Slope Based on HGWO-SVR Model, columnId=1236276106018484431, journalTitle=Mining and Metallurgical Engineering, columnName=MINING, runingTitle=null, highlight=null, articleAbstract=

The key factors of pile foundation settlement were explored for the slope under vertical load by using grey relational analysis, and it is found that each factor is in the following descending order by its influence: elastic modulus > slope distance > slope gradient > internal friction angle > cohesion > soil density > poisson's ratio of soil > pile length > pile diameter. In order to optimize the parameters of support vector regression (SVR) model, a novel HGWO-SVR model was proposed by integrating the differential evolution-enhanced gray wolf algorithm (HGWO). Compared with GWO-SVR and GS-SVR models, this model presents obvious advantage in prediction, with high accuracy and minor error. A settlement prediction model for pile foundation of slope was constructed based on HGWO-SVR model, and the prediction results were compared with those values calculated with existing settlement formulas. The results show that the maximum percentage error between the prediction value of HGWO-SVR model and the calculated value is 6.55%, thus verifying that this model is feasible in settlement prediction for pile foundation of slope.

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采用灰色关联分析深入研究了竖向荷载作用下斜坡桩基沉降的关键因素，各因素影响程度由大到小排序为：弹性模量>临坡距>斜坡坡度>内摩擦角>黏聚力>土体密度>土体泊松比>桩长>桩径。为优化支持向量回归（SVR）模型参数，引入差分进化，建立混合灰狼算法（HGWO），提出了一种新的HGWO-SVR模型。该模型与GWO-SVR和GS-SVR模型相比，表现出更显著的预测优势，整体预测精度高，误差较小。基于HGWO-SVR模型构建了斜坡桩基沉降的预测模型，并将其预测结果与已有沉降计算公式计算结果进行对比，结果表明，HGWO-SVR模型预测结果与公式计算结果最大误差为6.55%，验证了该模型在斜坡桩基沉降预测方面的可行性。

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蒋冲（1977—），湖南城步人，博士，教授，主要从事岩土与地下工程教学与研究。E-mail：

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蒋冲（1977—），湖南城步人，博士，教授，主要从事岩土与地下工程教学与研究。E-mail：

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蒋冲（1977—），湖南城步人，博士，教授，主要从事岩土与地下工程教学与研究。E-mail：

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（a）三维示意图；（b）有限元模型

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（a）训练集；（b）测试集

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斜坡坡度α/（°）临坡距L/m桩长H/m桩径D/m密度ρ/（kg·m-3）弹性模量E/MPa泊松比μ黏聚力c/kPa内摩擦角φ/（°）荷载F/kN斜坡桩基沉降s/mm
303.5120.61 800100.4540306008.814
453.6120.61 800130.42353590011.285
152.5121.01 800120.4030351 40017.436
303.6100.81 800100.4540301 10036.376
153.5121.01 800100.4540301 80024.104
303.5121.01 700120.4030355006.244
302.5120.61 750120.38303190032.345
303.6100.61 700130.4235307009.820
451.8120.61 600140.30353570014.870
153.5121.01 800140.4030301 50016.694
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不同参数下斜坡桩基沉降数据（部分数据）

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斜坡坡度α/（°）临坡距L/m桩长H/m桩径D/m密度ρ/（kg·m-3）弹性模量E/MPa泊松比μ黏聚力c/kPa内摩擦角φ/（°）荷载F/kN斜坡桩基沉降s/mm
303.5120.61 800100.4540306008.814
453.6120.61 800130.42353590011.285
152.5121.01 800120.4030351 40017.436
303.6100.81 800100.4540301 10036.376
153.5121.01 800100.4540301 80024.104
303.5121.01 700120.4030355006.244
302.5120.61 750120.38303190032.345
303.6100.61 700130.4235307009.820
451.8120.61 600140.30353570014.870
153.5121.01 800140.4030301 50016.694
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sx1x2x3x4x5x6x7x8x9
0.5561.1251.1321.191.0201.1021.0030.9090.9200.556
1.1110.4821.1321.190.9071.1020.7521.0611.0741.111
0.5561.2860.7550.9521.0201.0241.0531.0611.0740.556
0.5561.1251.1321.1901.0200.7871.1281.2120.9200.556
1.1110.8041.1320.7140.9920.9450.9520.9090.9511.111
0.5561.4471.1320.7140.9631.0241.0531.0610.9200.556
1.1110.4821.1321.1901.0200.9451.0030.9091.0741.111
1.1111.1580.9430.7141.0201.0241.0531.0611.0741.111
1.6670.8040.7550.9521.0201.1021.0030.9090.9201.667
1.6671.2860.7551.1901.0200.9451.0030.9091.0741.667
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归一化处理后的数据（部分数据）

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sx1x2x3x4x5x6x7x8x9
0.5561.1251.1321.191.0201.1021.0030.9090.9200.556
1.1110.4821.1321.190.9071.1020.7521.0611.0741.111
0.5561.2860.7550.9521.0201.0241.0531.0611.0740.556
0.5561.1251.1321.1901.0200.7871.1281.2120.9200.556
1.1110.8041.1320.7140.9920.9450.9520.9090.9511.111
0.5561.4471.1320.7140.9631.0241.0531.0610.9200.556
1.1110.4821.1321.1901.0200.9451.0030.9091.0741.111
1.1111.1580.9430.7141.0201.0241.0531.0611.0741.111
1.6670.8040.7550.9521.0201.1021.0030.9090.9201.667
1.6671.2860.7551.1901.0200.9451.0030.9091.0741.667
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评价项单位子数列关联度排名
弹性模量EPax60.7671
临坡距Lmx20.7342
斜坡坡度α（°）x10.7313
内摩擦角φ（°）x90.7284
黏聚力cPax80.7275
密度ρkg/m3x50.7226
泊松比μx70.7147
桩长Hmx30.6968
桩径Dmx40.6879
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灰色关联度

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评价项单位子数列关联度排名
弹性模量EPax60.7671
临坡距Lmx20.7342
斜坡坡度α（°）x10.7313
内摩擦角φ（°）x90.7284
黏聚力cPax80.7275
密度ρkg/m3x50.7226
泊松比μx70.7147
桩长Hmx30.6968
桩径Dmx40.6879
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模型名称罚参数C核参数g
HGWO-SVR6.860 60.508 41
GWO-SVR53.286 40.507 3
GS-SVR17.512 30.464 16
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模型参数

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模型名称罚参数C核参数g
HGWO-SVR6.860 60.508 41
GWO-SVR53.286 40.507 3
GS-SVR17.512 30.464 16
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模型名称样本空间R2RMSEMAE
HGWO-SVR训练集0.9990.5570.031
测试集0.9841.3590.232
GWO-SVR训练集0.9782.3870.133
测试集0.9033.5160.574
GS-SVR训练集0.9593.2660.186
测试集0.9422.7050.489
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不同模型预测结果对比

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模型名称样本空间R2RMSEMAE
HGWO-SVR训练集0.9990.5570.031
测试集0.9841.3590.232
GWO-SVR训练集0.9782.3870.133
测试集0.9033.5160.574
GS-SVR训练集0.9593.2660.186
测试集0.9422.7050.489
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材料弹性模量/Pa泊松比密度/（kg·m-3）黏聚力/kPa内摩擦角/（°）
桩3.0×10100.102 500——
土1.2×1070.401 8003.0×10435
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验证工况

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材料弹性模量/Pa泊松比密度/（kg·m-3）黏聚力/kPa内摩擦角/（°）
桩3.0×10100.102 500——
土1.2×1070.401 8003.0×10435
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基于HGWO-SVR模型的竖向受荷斜坡桩基沉降预测
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蒋冲 1, 2 , 施泽雄 2
矿冶工程杂志 | 采矿 2024,44(2): 22-26
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矿冶工程杂志 | 采矿 2024, 44(2): 22-26
基于HGWO-SVR模型的竖向受荷斜坡桩基沉降预测
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蒋冲1, 2 , 施泽雄2
作者信息
  • 1.水能资源利用关键技术湖南省重点实验室，湖南　长沙　410014
  • 2.中南大学　资源与安全工程学院，湖南　长沙　410083

    • 蒋冲（1977—），湖南城步人，博士，教授，主要从事岩土与地下工程教学与研究。E-mail：

Settlement Prediction for Pile Foundation of Vertically Loaded Slope Based on HGWO-SVR Model
Chong JIANG1, 2 , Zexiong SHI2
Affiliations
  • 1.Hunan Provincial Key Laboratory of Hydropower Development Key Technology, Changsha 410014, Hunan, China
  • 2.School of Resources and Safety Engineering, Central South University, Changsha 410083, hunan, China
出版时间: 2024-04-01 doi: 10.3969/j.issn.0253-6099.2024.02.006
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摘要
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采用灰色关联分析深入研究了竖向荷载作用下斜坡桩基沉降的关键因素，各因素影响程度由大到小排序为：弹性模量>临坡距>斜坡坡度>内摩擦角>黏聚力>土体密度>土体泊松比>桩长>桩径。为优化支持向量回归（SVR）模型参数，引入差分进化，建立混合灰狼算法（HGWO），提出了一种新的HGWO-SVR模型。该模型与GWO-SVR和GS-SVR模型相比，表现出更显著的预测优势，整体预测精度高，误差较小。基于HGWO-SVR模型构建了斜坡桩基沉降的预测模型，并将其预测结果与已有沉降计算公式计算结果进行对比，结果表明，HGWO-SVR模型预测结果与公式计算结果最大误差为6.55%，验证了该模型在斜坡桩基沉降预测方面的可行性。

斜坡桩基  /  沉降预测  /  灰色关联分析  /  改进灰狼算法

The key factors of pile foundation settlement were explored for the slope under vertical load by using grey relational analysis, and it is found that each factor is in the following descending order by its influence: elastic modulus > slope distance > slope gradient > internal friction angle > cohesion > soil density > poisson's ratio of soil > pile length > pile diameter. In order to optimize the parameters of support vector regression (SVR) model, a novel HGWO-SVR model was proposed by integrating the differential evolution-enhanced gray wolf algorithm (HGWO). Compared with GWO-SVR and GS-SVR models, this model presents obvious advantage in prediction, with high accuracy and minor error. A settlement prediction model for pile foundation of slope was constructed based on HGWO-SVR model, and the prediction results were compared with those values calculated with existing settlement formulas. The results show that the maximum percentage error between the prediction value of HGWO-SVR model and the calculated value is 6.55%, thus verifying that this model is feasible in settlement prediction for pile foundation of slope.

pile foundation of slope  /  settlement prediction  /  grey relational analysis (GRA)  /  improved gray wolf algorithm
蒋冲, 施泽雄. 基于HGWO-SVR模型的竖向受荷斜坡桩基沉降预测. 矿冶工程杂志, 2024 , 44 (2) : 22 -26 . DOI: 10.3969/j.issn.0253-6099.2024.02.006
Chong JIANG, Zexiong SHI. Settlement Prediction for Pile Foundation of Vertically Loaded Slope Based on HGWO-SVR Model[J]. Mining and Metallurgical Engineering, 2024 , 44 (2) : 22 -26 . DOI: 10.3969/j.issn.0253-6099.2024.02.006
正文
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目前，对于桩基沉降计算大多集中于平地桩基沉降的理论计算，然而，由于斜坡的存在，斜坡桩基沉降与平地桩基沉降有很大不同。目前对于斜坡桩基沉降计算的研究较少，尚未形成系统的理论，已有的研究成果大都集中于水平荷载作用下斜坡桩基的横向响应[1]、横向变形[2]和横向承载力，或竖向受荷斜坡桩基的荷载传递规律[3]。因此，有必要研究竖向受荷斜坡桩基沉降计算方法。
随着机器学习的飞速发展，已有大量学者将其引入土体参数[4]、地表沉降[5-6]、边坡可靠性、单桩荷载-位移曲线、嵌岩桩沉降、极限承载力等岩土工程预测。关于桩基沉降预测模型的研究主要有高阶神经网络、基因表达式编程、支持向量机、LM算法等。由于斜坡桩基的特殊性，应用上述模型进行沉降预测时存在如下问题：①支持向量回归模型超参数选取较为困难，模型泛化能力较差；②使用传统智能优化算法寻找模型超参数时易陷入局部最优值，降低预测精度。
本文首先使用有限元软件建立100个斜坡桩基受荷沉降模型，获取不同模型参数下斜坡桩基沉降数据，引入改进的灰狼算法优化支持向量回归模型，并基于此模型预测竖向荷载作用下斜坡桩基沉降，探讨该模型在竖向受荷斜坡桩基沉降预测中的适用性，对比结果证实了该模型的可靠性和适用性。
1　沉降预测模型建立
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1.1　数据获取及分析
1.1.1　有限元模型的建立
为获取竖向荷载作用下斜坡桩基沉降数据，建立了三维有限元模型。建模过程中，假设斜坡土体为单层土体，斜坡土体和桩体材料参数保持不变。模型如图1所示。模型中X、Y、Z方向土体尺寸分别为10 m、20 m、20 m，其中斜坡部分根据斜坡坡度切削土体，桩体所处位置关于YZ平面对称，为土体中间位置。桩体采用线弹性模型；土体变形分为弹性阶段和塑性阶段：弹性变形阶段本构模型为弹性模型，塑性变形阶段本构模型为摩尔-库伦模型。参考工程经验以及《工程地质手册》[7]选取模拟桩土材料参数。
建立基本模型之后，通过调整斜坡坡度、泊松比、黏聚力、内摩擦角等参数，建立100个有限元模型，计算得出桩基沉降随荷载变化的数据。采用逐级加载模式，为增加数据量，每级加载20 kN，逐级加载至2 000 kN。大部分研究建议“沉降变形控制的沉降量一般取桩径的3%～6%”[8]，因此最大沉降量统一取0.06倍桩径，即删除超过0.06倍桩径的沉降数据。处理异常数据之后，100个数值模拟试验共获得5 083组数据。部分数据见表1。
1.1.2　输入输出数据分析
为研究各项输入参数与输出参数之间的关系，选用灰色关联分析（GRA）对获取的数据进行分析。GRA主要通过搜索灰色关联度来描述每个变量属性之间的关系[9]，再据此确定每个变量对目标影响的关联度，计算公式为：
式中s为母序列，即斜坡桩基沉降；xi为子数列，即各项输入参数；s（k）为母序列的第k个值；xi（k）为第i个子序列的第k个值；ζi（k）为第i个子序列第k个值的关联度；ρ为分辨系数，分辨系数越小，分辨率越好，ρ通常取0.5[10]；ri为第i个子序列与母序列的关联度。
模拟获取的数据为不同竖向荷载作用下斜坡桩基沉降值，因此进行灰色关联分析时去除荷载这一因素，同时为消除荷载对沉降的影响，统一取荷载为1 000 kN时的数据，将各个变量分别进行归一化处理，部分数据如表2所示。将其代入式（1）～（2），计算结果如表3所示。
由表3可得，土体弹性模量是斜坡桩基沉降的首要影响因素，同时，各项影响因素之间差距不大，可知选取的输入变量与输出变量之间关系密切。
1.2　HGWO-SVR斜坡桩基沉降预测模型
为建立斜坡桩基沉降预测模型，使用支持向量回归模型训练数据。支持向量机（SVM）能够在最小化采样点误差的同时，在不受数据维数限制的情况下提高模型的泛化能力[11]。为了解决SVM的回归拟合问题，Vapnik等人在支持向量机分类的基础上引入了不敏感损失函数ε，通过适当的非线性函数变换，将输入变量映射到高维特征空间，并在此空间寻求线性回归最优超平面（见式（3））[12]。
灰狼算法（GWO）[13]在进行狩猎行为时易陷入局部最优解。因此本文引入差分进化算法（DE）[14]优化灰狼算法狩猎过程，将DE算法与GWO算法结合，建立混合灰狼算法（HGWO），灰狼种群通过变异加强种群多样性，提高其全局搜索能力。在解决土木工程中的复杂问题时，需要优化罚参数和核函数参数，以提高预测精度。本文采用HGWO优化算法寻找SVR模型中罚参数C、核参数g的最优值，以此来提高模型的精度和鲁棒性。
式中si为输出向量；αi、、αj、均为拉格朗日乘子；C为罚参数；K（xi，xj）为高斯径向基函数（RBF）。在高维非线性空间中，与其他核函数相比，RBF具有良好的泛化性、非线性预测性能，且需要调整的参数较少，因此本文选择RBF作为核函数：
为验证HGWO优化SVR模型的有效性及实用性，额外训练一个GWO-SVR模型以及一个网格搜索（GS）优化的SVR模型进行对比。HGWO优化算法中参数设定为：NP=20，N=100，SFmax=0.8，SFmin=0.2，CR=0.3，罚参数C的范围设为[10-3，103]，核宽度g的范围设为[10-3，103]，以便HGWO等优化算法可以在类似于正方形的二维平面上搜索最优值，采用五折交叉验证。
初始化HGWO-SVM模型的各个参数，对输入数据进行模型训练，采用HGWO算法、GWO算法、GS算法对支持向量机模型参数寻优后得到最终模型参数如表4所示。
2　预测结果评估与分析
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2.1　模型预测结果评估
模型的精度通过确定系数（R2）、均方根误差（RMSE）、平均绝对误差（MAE）进行评估。R2的值介于0和1之间，R2越接近1，说明模型预测能力越好。RMSE、MAE表示真实数据与预测数据之间的偏差，值越小，表明模型的预测能力越好。计算公式为：
式中si为真实沉降；为真实沉降的平均值；为模型输出的斜坡桩基沉降预测值。
根据获取的斜坡桩基在不同竖向荷载作用下的桩顶沉降数据建立模型训练数据库。数据集分为两部分，其中80%为训练集，20%为测试集。各模型经过对数据集的训练和学习，预测结果对比见表5。
综合分析表5可知，3种模型对竖向荷载作用下斜坡桩基沉降变形预测结果均很好，HGWO-SVR模型预测精度高于其他两种模型，HGWO-SVR在训练集及测试集中R2最高、MSE最低，可见HGWO-SVR具有更高的预测精度以及良好的预测稳定性。测试集中，SVR模型R2最低，可能因为罚参数C设置略高，模型处于过拟合状态，但HGWO-SVR模型在测试集仍表现良好，体现了差分进化算法改进灰狼算法的全局寻优能力，不易陷入局部最优解。
2.2　模型预测结果对比
图2为3种模型沉降真实值与预测值的对比。由图2可知，各模型沉降真实值与预测值之间误差较小，其中HGWO-SVR模型预测精度远优于其他两种算法。各模型在训练集均有较好的表现，但在测试集中，GWO-SVR、GS-SVR模型预测偏差明显大于HGWO-SVR模型。由图2也可以看出，经过HGWO算法优化的SVR模型在预测精度及鲁棒性上均有提高，进一步证明了优化算法的有效性。
3　预测效果验证
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以文献[15]提出的计算斜坡桩基沉降计算方法为例，选取两种工况（详见表6），与本文建立的机器学习模型进行对比，结果如图3所示。
由图3可知，HGWO-SVR模型预测沉降值与公式计算沉降值在趋势以及数值上都十分接近，说明HGWO-SVR模型对于竖向荷载作用下斜坡桩基沉降预测具有较高的精度。其中工况1理论计算与模型预测结果最大误差为6.55%，工况2理论计算与模型预测结果最大误差为5.33%，两者趋势基本一致，进一步证明了HGWO-SVR模型预测斜坡桩基沉降是可行的。
4　结论
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1）通过灰色关联分析法分析各项输入参数对竖向荷载作用下斜坡桩基沉降的影响，各因素影响程度由大到小排列顺序为：弹性模量>临坡距>斜坡坡度>内摩擦角>黏聚力>土体密度>土体泊松比>桩长>桩径。
2）通过差分进化改进灰狼算法，将其用于SVR模型惩罚因子及核函数参数寻优中优化支持向量回归模型，提出HGWO-SVR模型，并与已有模型进行对比，结果表明，HGWO-SVR模型整体预测精度高、误差小，预测效果明显优于GWO-SVR、GS-SVR模型。
3）通过HGWO-SVR模型建立斜坡桩基沉降预测模型，并将其与现有沉降计算公式进行比较，模型与理论计算结果最大误差为6.55%，证明了HGWO-SVR模型应用于斜坡桩基沉降的可行性。
基金
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  • 水能资源利用关键技术湖南省重点实验室开放研究基金项目(PKLHD202103)
文献
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doi: 10.3969/j.issn.0253-6099.2024.02.006
  • 接收时间：2023-10-28
  • 首发时间：2026-03-19
  • 出版时间：2024-04-01
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  • 收稿日期：2023-10-28
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水能资源利用关键技术湖南省重点实验室开放研究基金项目(PKLHD202103)
作者信息
    1.水能资源利用关键技术湖南省重点实验室，湖南　长沙　410014
    2.中南大学　资源与安全工程学院，湖南　长沙　410083
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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