裂缝边坡三维极限上限拓展分析

饶平平; 吴健; 崔纪飞; 赵琳学

doi:10.11779/CJGE202109005

裂缝边坡三维极限上限拓展分析 English Version

1.
上海理工大学土木工程系,上海　200093
2.
悉尼大学土木工程学院,澳大利亚　悉尼　2008

基金项目:

国家自然科学基金项目 42077435

详细信息

作者简介:
饶平平(1984— ),男,博士,副教授,主要从事岩土力学教学与科研工作。E-mail：raopingping@usst.edu.cn

中图分类号: TU431
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出版历程
- 收稿日期: 2020-12-14
- 网络出版日期: 2022-12-02
- 刊出日期: 2021-08-31

Extended three-dimensional analysis of cracked slopes using upper-bound limit method

1.
Department of Civil Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
School of Civil Engineering, The University of Sydney, Sydney 2008, Australia

摘要

摘要: 裂缝的存在会降低边坡稳定性,而在边坡宽度受到限制时三维效应更加显著,因此有必要对裂缝边坡稳定性进行评估。针对裂缝边坡三维稳定性研究,基于极限分析上限定理,在三维破坏机构中引入一条垂直张拉裂缝,并引进机构参数拓展裂缝边坡三维破坏模式,包括坡面破坏和坡底破坏,建立能量平衡方程并通过优化算法求解裂缝边坡稳定系数上限解。根据g-line图像法绘制裂缝边坡稳定性图表以便读取安全系数。分析了边坡宽高比、坡角以及土体内摩擦角对裂缝边坡破坏模式和裂缝深度及位置的影响规律。结果表明：对于确定的边坡几何形态以及土体参数,存在最小边坡宽高比B/H^*,当边坡宽高比小于B/H^*,边坡发生坡面破坏且坡顶裂缝的影响可以忽略；内摩擦角φ小于5°时,边坡发生坡底破坏,而对于裂缝边坡,仅在φ=1°左右发生坡底破坏；随着边坡宽高比的增大,裂缝深度逐渐增加,裂缝位置逐渐远离坡肩,但对于坡角为75°的边坡裂缝深度先增大后减小。
- 边坡稳定性 /
- 极限分析 /
- 裂缝 /
- 安全系数 /
- 图表法
Abstract: The existence of cracks will reduce the stability of a slope and the three-dimensional effect is more significant when its width is limited. Therefore, it is necessary to evaluate the stability of cracked slopes. In order to study the three-dimensional stability of cracked slopes, based on the upper-bound theorem of limit analysis, a vertical tensile crack is introduced into the three-dimensional failure mechanism, and the mechanism parameters are introduced to extend the three-dimensional failure mode of slopes, including face failure and base failure. The energy balance equation is established, and the upper bounds of stability number of cracked slopes are obtained by the optimization algorithm. The stability charts of cracked slopes are established based on the g-line graphical method to read the factor of safety conveniently. The influences of slope width-to-height ratio, slope angle and internal friction angle of soils on the failure mechanism of cracked slopes and the crack depth and location are analyzed. The results show that for the specific slope geometry and soil parameters, there is a minimum slope width B/H^*. When the slope width is less than B/H^*, the failure surface passes above the slope toe, and the influences of the crack on the upper surface of the slope can be ignored. When the internal friction angle φ is less than 5°, the failure surface passes below the slope toe; for the cracked slope, only when φ=1°, failure surface passes below the slope. As the width of the slope increases, the crack depth gradually increases, and the crack location gradually moves away from the slope crest, while the crack depth of the slope with inclination 75° increases first and then decreases.
- slope stability /
- limit analysis /
- crack /
- factor of safety /
- graphical method

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图 1 裂缝边坡三维旋转破坏机构

Figure 1. Three-dimensional failure mechanism of cracked slopes

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图 2 裂缝边坡三维旋转破坏拓展机构

Figure 2. Extended three-dimensional failure mechanism of cracked slopes

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图 3 平面应变机制插入示意图

Figure 3. Schematic diagram of insert portion of plane-strain mechanism

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图 4 读取安全系数示意图

Figure 4. Schematic diagram of reading factor of safety

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图 5 几何约束对三维破坏机构的影响

Figure 5. Influences of geometric constraints on 3D failure mechanism

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图 6 不考虑坡顶裂缝情况下边坡三维稳定性图表

Figure 6. Stability charts of 3D slopes without cracks on upper surface

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图 7 考虑坡顶裂缝情况下边坡三维稳定性图表

Figure 7. Stability charts of 3D slopes with cracks on upper surface

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图 8 边坡宽度对三维破坏机构的影响

Figure 8. Influences of slope width on 3D failure mechanism

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图 9 三维效应下边坡坡角对裂缝深度的影响

Figure 9. Influences of slope inclination on crack depth under 3D condition

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图 10 三维效应对裂缝深度的影响

Figure 10. Influences of three-dimensional effects on crack depth

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表 1 不考虑坡顶裂缝情况下三维边坡稳定系数对比

Table 1 Comparison of 3D stability factors of slopes without cracks

β/(°)	φ/(°)	B/H	θ ₀/(°)	θ _h/(°)	r ₀′/r₀	γH/c
β/(°)	φ/(°)	B/H	θ ₀/(°)	θ _h/(°)	r ₀′/r₀	本文解答	文献[19]解答	文献[20]解答
30	15	0.5	53.69	97.25	0.941	73.106^a	73.156^a	73.120^a
45	15	0.5	30.39	89.54	0.888	32.184^a	32.371^a	32.198^a
	30	0.8	57.27	95.34	0.856	62.121	62.213	62.121
60	15	0.5	14.43	80.14	0.843	20.889^a	21.512^a	20.805^a
	30	0.8	42.14	87.18	0.739	26.464	26.486	26.464
注：^a坡面破坏。

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表 2 考虑坡顶裂缝情况下三维边坡稳定系数对比

Table 2 Comparison of 3D stability factors of slopes with cracks

β/(°)	φ/(°)	B/H	γH/c
β/(°)	φ/(°)	B/H	本文解答	文献[17]解答
30	10	0.8	28.00	27.74
	20	3.0	47.81	47.75
45	10	0.8	15.82	15.68
	20	3.0	17.54	17.53
60	10	0.8	11.15	11.07
	20	3.0	10.60	10.59
70	10	0.8	8.33	8.26
	20	3.0	7.02	7.01

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表 3 边坡稳定系数(φ=15°)

Table 3 Stability factors of slopes (φ=15°)

B/H		β/(°)
		30			45			60			75
		γH/c	δ/H	x/H	γH/c	δ/H	x/H	γH/c	δ/H	x/H	γH/c	δ/H	x/H
0.5	PC	73.155^a	0.003	0.043	31.978^a	0.126	0.041	19.903^a	0.331	0.041	14.085^a	0.477	0.070
0.5	IN	73.106^a	—	—	32.184^a	—	—	20.889^a	—	—	15.834^a	—	—
0.6	PC	61.199^a	0.015	0.075	26.685	0.145	0.043	16.670	0.389	0.057	11.713	0.599	0.082
0.6	IN	61.002^a	—	—	26.725	—	—	17.482	—	—	13.322	—	—
0.8	PC	46.051	0.021	0.107	20.913	0.186	0.111	13.524	0.392	0.117	9.598	0.587	0.126
0.8	IN	46.071	—	—	21.220	—	—	14.306	—	—	10.936	—	—
1.0	PC	38.698	0.046	0.123	18.230	0.211	0.136	11.965	0.386	0.148	8.505	0.571	0.150
1.0	IN	38.795	—	—	18.648	—	—	12.786	—	—	9.797	—	—
1.5	PC	30.874	0.084	0.145	15.298	0.234	0.173	10.190	0.382	0.186	7.217	0.553	0.182
1.5	IN	31.126	—	—	15.855	—	—	11.087	—	—	8.503	—	—
3.0	PC	25.124	0.114	0.170	12.994	0.248	0.207	8.722	0.379	0.220	6.111	0.539	0.211
3.0	IN	25.540	—	—	13.686	—	—	9.714	—	—	7.433	—	—
注：^a坡面破坏；PC考虑坡顶存在裂缝情况；IN考虑坡顶无裂缝情况；x/H裂缝位置与坡肩B的距离。

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表 4 边坡稳定系数(φ=30°)

Table 4 Stability factors of slopes (φ=30°)

B/H		β/(°)
		45			60			75
		γH/c	δ/H	x/H	γH/c	δ/H	x/H	γH/c	δ/H	x/H
0.5	PC	96.560^a	0.001	0.040	40.219^a	0.125	0.041	23.673^a	0.389	0.029
0.5	IN	96.417^a	—	—	40.590^a	—	—	25.446^a	—	—
0.6	PC	80.718^a	0.021	0.061	33.511^a	0.187	0.042	19.361^a	0.552	0.016
0.6	IN	80.287^a	—	—	33.935^a	—	—	21.226^a	—	—
0.8	PC	61.978	0.045	0.067	25.882	0.263	0.061	15.137	0.523	0.081
0.8	IN	62.121	—	—	26.464	—	—	16.534	—	—
1.0	PC	53.724	0.068	0.069	22.644	0.250	0.092	13.261	0.491	0.107
1.0	IN	54.066	—	—	23.398	—	—	14.647	—	—
1.5	PC	45.243	0.088	0.076	19.260	0.251	0.115	11.186	0.466	0.134
1.5	IN	45.836	—	—	20.216	—	—	12.652	—	—
3.0	PC	39.061	0.098	0.084	16.677	0.252	0.132	9.516	0.452	0.156
3.0	IN	39.864	—	—	17.830	—	—	11.120	—	—
注：^a坡面破坏；PC考虑坡顶存在裂缝情况；IN考虑坡顶无裂缝情况；x/H裂缝位置与坡肩B的距离。

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