用Bayes法及后验分布极限确定土力学参数

阮永芬; 魏德永; 杨均; 高骏; 刘克文; 彭栓栓

doi:10.11779/CJGE202003005

用Bayes法及后验分布极限确定土力学参数 English Version

1.
昆明理工大学建筑工程学院,云南　昆明　650500
2.
中铁四院集团西南勘察设计有限公司,云南　昆明　650031
3.
中铁二院昆明勘察设计研究院有限责任公司,云南　昆明　650200
4.
云南建投第一勘察设计有限公司,云南　昆明　650031

基金项目:

云南省重点研发计划项目 2018BC008

详细信息

作者简介:
阮永芬（1964— ）,女,教授,博士,主要从事岩土工程方面的教学和科研。E-mail：ryy64@aliyun.com

中图分类号: TU443
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出版历程
- 收稿日期: 2019-03-24
- 网络出版日期: 2022-12-07
- 刊出日期: 2020-02-29

Determination of soil mechanics parameters based on Bayes method and posterior distribution limit

1.
Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming 650500, China
2.
Southwest Nonferrous Exploration Design Co., Ltd., Kunming 650031, China
3.
Kunming Survey Design and Research Institute Co., Ltd. of Creec, Kunming 650200,China
4.
Yunnan Construction Investment First Investigation and Design Co., Ltd., Kunming 650031, China

摘要

摘要: 岩土力学参数在空间分布上具有很大随机性,现有研究都是从参数小样本去推断大样本,这难免会存在一些失误。若样本数量足够大,正态分布会变得高度集中在某个值附近,勘察失误或偶然因素导致岩土参数统计误差的影响会越来越小,据此提供给勘察设计的岩土力学参数就更准确。根据收集的滇池湖相沉积的离散性高的泥炭质土、粉土及黏土的黏聚力c和内摩擦角 $φ$ 的大量试验数据,用Bayes大样本法对几种土的力学参数进行分析,充分利用已采集样本带来的先验信息,推求土力学参数的后验信息,并得到贝叶斯估计区间。当Fisher信息量确定后,土力学参数正态分布均值 $μ$ 验分布可用一个与先验分布无关的正态分布来逼近,就可得到当样本量趋于无穷时参数的收敛取值,并用后验分布极限对土力学参数的相合性及渐近正态性进行检验分析。对滇池湖相沉积土力学指标进行大样本分析,推求土力学指标的收敛值就落在贝叶斯估值区间,充分证明了分析方法的可行性及可靠性。分析结果为地区规范编制及工程经验的积累提供参考,也为设计计算参数取值的合理性提供检验。
- 力学参数 /
- 贝叶斯（Bayes）大样本方法 /
- 相合性 /
- 渐近正态性 /
- 费希尔（Fisher）信息
Abstract: Geotechnical parameters are of great randomness in spatial distribution, and the existing researches are to infer large samples from small ones, which will inevitably lead to some errors. If the number of samples is large enough, the normal distribution will become highly concentrated near a certain value. The influences of statistical errors of geotechnical parameters will become smaller and smaller resulting from the investigation errors or accidental factors, then the more accurate parameters will be provided during prospective design. According to the collected experimental data of the cohesive force $c$ and the internal friction angle $φ$ of Dianchi Lake facies deposition peat soil, silt and clay with high dispersion, the Bayesian large sample method is used to analyze the mechanical parameters of several soils, the posterior information of mechanical parameters of soils is derived based on the prior information from the collected samples, and the Bayesian estimation interval is obtained. After the Fisher information is determined, the posterior distribution of the mean value $μ$ of the normal distribution of mechanical parameters of soils can be approximated by a normal distribution independent of the prior distribution, and the reliable value of the parameters can be obtained when the sample size tends to infinity. The consistency and asymptotic normality of mechanical parameters of soils are tested and analyzed by the posterior distribution limit. A large sample analysis of the mechanical indexes of the sedimentary soils in Dianchi Lake is carried out. The reliable value of the mechanics index of soils is estimated to fall within the Bayesian evaluation interval, which fully proves the feasibility and reliability of the analytical method. The analysis results may provide reference for the compilation of regional norms and the accumulation of engineering experience, and also a test for the rationality of the design parameters.
- mechanical parameter /
- Bayes large sample method /
- consistency /
- asymptotic normality /
- Fisher information

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图 1 3种土的样本频率直方图及P-P图

Figure 1. Frequency histogram and P-P diagram of three soil samples

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图 2 力学参数均值μ的后验分布图

Figure 2. Posterior distribution of mean μ of mechanical parameters

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图 3 粉土φ和黏土c的样本频率直方图与P-P图

Figure 3. Frequency histogram and P-P diagram of silt φ & clay c

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图 4 力学参数的 μ 的后验相合性检验图

Figure 4. Posterior consistency tests on μ of mechanical parameters

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表 1 土的力学参数的样本数

Table 1 Number of samples for mechanical parameters of soils

土类	泥炭质土	粉土	黏土
黏聚力c/个	269	230	540
内摩擦角φ/个	294	248	476

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表 2 3种土的力学参数均值和方差

Table 2 Mean values and variances of mechanical parameters of three soil samples

泥炭质土				粉土				黏土
$\bar{c}$ /kPa	$σ_{c}^{2}$	$\bar{φ}$ /(°)	$σ_{φ}^{2}$	$\bar{c}$ /kPa	$σ_{c}^{2}$	$\bar{φ}$ /(°)	$σ_{φ}^{2}$	$\bar{c}$ /kPa	$σ_{c}^{2}$	$\bar{φ}$ /(°)	$σ_{φ}^{2}$
29.4	9.8	5.9	1.9	16.0	2.6	17.4	0.4	27.8	12.1	9.16	1.79
24.9	10.0	6.7	2.5	15.9	3.7	17.6	0.4	30.9	9.7	8.79	1.65
19.6	7.8	6.6	2.0	17.6	3.5	17.2	0.5	29.6	11.5	9.27	2.08
32.1	11.3	6.5	3.0	18.7	4.5	17.6	0.6	33.1	9.5	9.41	1.96
27.5	8.0	5.9	2.4	17.4	4.3	17.5	0.5	31.7	7.7	8.26	1.76
26.7	9.6	6.2	2.7	15.4	4.7	17.3	0.8	29.5	9.8	8.87	1.95
32.9	7.9	6.9	2.9	15.9	3.5	17.6	0.6	27.5	10.0	8.11	1.73
26.3	9.2	5.8	2.8	17.1	3.4	17.7	0.7	27.2	8.5	9.20	1.54
24.4	8.6	6.4	2.5	17.4	3.6	17.5	0.6	27.7	10.4	9.32	1.40
23.9	9.9	6.5	2.6	18.0	4.3	17.3	0.3	27.5	11.0	8.71	1.75
25.1	10.2	6.4	2.8	14.5	3.3	17.2	0.1	29.4	11.3	9.77	1.47
27.8	11.2	6.7	2.4	17.3	3.5	17.6	0.5	28.3	11.1	8.29	1.58
28.8	7.6	6.5	2.0	16.1	4.3	17.3	0.6	33.2	10.2	8.70	1.94
31.2	7.6	5.9	2.1	18.0	2.6	17.7	0.7	30.1	11.8	8.80	1.78
29.3	8.0	5.6	2.3	18.1	3.0	17.2	0.5	29.5	10.4	8.49	1.38
27.0	8.7	6.4	3.3	14.8	2.0	17.4	0.6	25.5	10.8	8.78	1.63
25.3	8.1	5.9	2.4	17.8	2.4	17.7	0.6	32.9	12.5	8.05	1.95
30.4	9.8	7.6	1.4	16.1	3.9	17.5	0.6	27.9	9.7	8.31	1.80
26.6	7.1	6.3	2.3	17.8	3.0	17.5	0.7	31.4	9.6	8.60	2.26
31.3	8.4	8.0	2.7	16.8	2.1	17.5	0.4	27.3	10.9	8.76	1.58
26.4	7.4	7.3	2.4	18.6	4.0	17.4	0.5	29.3	7.0	9.48	1.91
25.3	6.6	7.1	3.1			17.4	0.2	29.8	10.1	8.55	1.53
23.7	9.3	5.3	1.7					28.4	7.0	8.43	1.76
26.3	8.5	6.7	2.9					31.6	10.1	8.77	1.82
27.7	10.0	6.7	2.6					25.5	11.1
27.4	7.6	7.6	2.7					28.0	10.4
26.8		5.7	2.5					32.9	11.7
		7.4	2.4
		5.9	2.6
		7.5	1.8

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表 3 先验参数取值与后验参数计算表

Table 3 Values of a priority parameter and calculation of posterior parameter

土名	参数	n	μ₀	$σ_{0}^{2}$	$θ$	$τ^{2}$	r/2	$λ$ /2	k	v_n	k_n	$μ (\bar{x})$	v_n $σ_{n}^{2}$	$σ_{n}^{2}$
泥炭质土	c/kPa	27	27.18	9.30	27.18	2.93	52.34	0.170	3.17	131.68	30.17	27.18	242.14	1.83
泥炭质土	φ/(°)	6.52	2.75	6.48	0.67	30.48	0.080	4.10	90.96	34.10	6.52	79.92	0.88
粉土	c/kPa	22	16.92	3.73	16.97	1.21	19.26	0.180	2.78	60.52	24.78	16.93	78.70	1.30
粉土	φ/(°)	24	17.45	0.54	17.45	0.15	8.41	0.060	3.60	40.82	27.60	17.45	12.54	0.04
黏土	c/kPa	27	19.37	10.84	29.40	2.23	56.37	0.190	4.86	139.74	31.86	29.40	695.73	4.97
黏土	φ/(°)	24	8.78	1.84	8.78	0.46	67.93	0.025	4.00	159.86	28.00	8.78	42.37	0.26

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表 4 参数的可信区间统计表

Table 4 Statistical table of confidence interval of parameters

土类	指标	Bayes可信区间	经典统计学可信区间
泥炭质土	c/kPa	[26.6,27.8]	[24.3,30.04]
泥炭质土	φ/(°)	[6.4,6.7]	[6.1,7.2]
粉土	c/kPa	[16.6,17.2]	[15.6,18.6]
粉土	φ/(°)	[17.41,17.48]	[16.2,18.4]
黏土	c/kPa	[29.2,30.6]	[28.7,32.1]
黏土	φ/(°)	[8.7,9.0]	[6.9,10.3]

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表 5 不同区间站点参数指标选取值

Table 5 Parameter values for different interval sites

站点名称	泥炭质土		粉土		黏土
站点名称	c/kPa	φ/(°)	c/kPa	φ/(°)	c/kPa	φ/(°)
河尾村站	27.0	6.4	17.8	17.4	29.2	9.0
迎海路站	26.3	6.5	16.6	17.6	29.8	8.3
滇池学院	25.6	5.9	15.8	17.1	29.6	8.5

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表 6 c和φ值样本总数及分组情况

Table 6 Total number and grouping of c and φ samples

土类	粉土		黏土		泥炭质土
指标	c	φ	c	φ	c	φ
样本总数/个	230	248	540	476	269	294
第一组/个	100	100	200	200	100	100
第二组/个	200	200	400	400	200	200
第三组/个	230	248	540	476	269	294

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表 7 力学参数均值和相合中心

Table 7 Means and coincidence centers of mechanical parameters

土类	样本量	正态分布峰值		相合中心
土类	样本量	c/kPa	φ/(°)	$ρ_{c}$ /kPa	$ρ_{φ}$ /(°)
黏土	200	28.7		29.4
	400	29.1		29.4
	540	29.4		29.4
粉土	100		17.50		17.50
	200		17.47		17.50
	248		17.50		17.50

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