基于固有频率的错断式坠落危岩体稳定系数计算模型

张晓勇; 谢谟文; 张磊; 杜岩; 高世崇; 李双全

doi:10.6052/j.issn.1000-4750.2022.04.0343

基于固有频率的错断式坠落危岩体稳定系数计算模型

张晓勇^{1, 2, ,},
谢谟文^{1, 2,},
张磊^{1, 2,},
杜岩^1,,
高世崇^1,,
李双全^1,

1.
北京科技大学土木与资源工程学院，北京 100083
2.
北京中关村智连安全科学研究院有限公司，北京 100083

基金项目: 崩塌组合式传感智能监测预警仪器研发项目(2019YFC1509602)；国家自然科学基金项目(41572274)

详细信息

作者简介:
谢谟文(1965−)，男，湖北人，教授，博士，主要从事工程防灾减灾、岩土工程研究(E-mail: mowenxie@126.com)

张　磊(1981−)，男，青海人，高工，博士，主要从事岩土检测、探测技术研究(E-mail: 13810174099@163.com)

杜　岩(1985−)，男，河南人，副教授，博士，主要从事防灾减灾方面的应用研究(E-mail: mutulei@163.com)

高世崇(1997−)，男，河南人，硕士生，主要从事岩土灾害防治研究(E-mail: 18800192678@163.com)

李双全(1996−)，男，云南人，硕士生，主要从事城市建筑结构智能防灾减灾研究(E-mail: lsq88228882@163.com)

通讯作者:
张晓勇(1992−)，男，河北人，博士生，主要从事工程安全防灾减灾研究(E-mail: 2418203296@qq.com)

中图分类号: TU45
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出版历程
- 收稿日期: 2022-04-19
- 修回日期: 2022-07-12
- 网络出版日期: 2022-07-18
- 刊出日期: 2024-04-24

A CALCULATION MODEL OF SAFETY FACTOR OF SHEAR FRACTURED FALLING DANGEROUS ROCK MASS BASED ON NATURAL FREQUENCY

ZHANG Xiao-yong^{1, 2, ,},
XIE Mo-wen^{1, 2,},
ZHANG Lei^{1, 2,},
DU Yan^1,,
GAO Shi-chong^1,,
LI Shuang-quan^1,

1.
School of Civil and Environmental Engineering, University of Science & Technology, Beijing 100083, China
2.
Beijing Zhongguancun Insititute of Safety Sscience Co., Ltd, Beijing 100083, China

Funds: WU Bangda, WU Libo. Analog Method for the Equivalent Lumped System of Foundation Half-space Theories and Associated Field Measurements[J]. China Earthquake Engineering Journal, 2015, 37(04): 1029-1036.

摘要

摘要:
错断式坠落危岩体的安全稳定程度受后缘裂缝深度控制，基于静力学的稳定系数计算可得到后缘裂隙深度临界值，但静力学计算方法无法得到裂隙深度的实时变化。基于修正Timoshenko梁建立岩梁动力学方程，结合动力基础半空间理论对岩梁的约束刚度及边界条件做深入分析，得到不同裂缝深度条件下危岩体固有频率理论解析。基于数值计算验证固有频率算法的正确性，并得出结论：随裂隙深度增大，固有频率不断降低，固有频率对危岩体后缘裂隙深度变化较为敏感，根据危岩体的固有频率值可反算危岩体后缘裂隙深度。基于固有频率与危岩体后缘裂隙深度之间的关系建立错断式坠落危岩体稳定系数计算模型，并采用室内试验进行验证，得出结论：危岩体固有频率与其后缘裂隙深度近似为负线性相关。基于固有频率的错断式坠落危岩体稳定系数计算模型可为危岩体自动化监测提供有益借鉴。
- 错断式坠落危岩体 /
- 稳定系数 /
- 修正Timoshenko梁理论 /
- 动力基础半空间理论 /
- 固有频率
Abstract:
The safety and stability of shear fractured falling dangerous rock mass is controlled by the depth of trailing edge fracture. The critical value of the depth of trailing edge fracture can be obtained by calculating the stability factor based on statics, but the statics calculation method cannot obtain the real-time change of the fracture depth. Based on the modified Timoshenko Beam theory, the dynamic equation of rock beam was established. Combined with the half-space theory of dynamic foundation, the constraint stiffness and boundary conditions of rock beam were analyzed in depth, and theoretical analysis of natural frequency of dangerous rock mass under different fracture depths was conducted. The correctness of the natural frequency algorithm was verified through numerical calculation. It was concluded that the natural frequency decreases with the increase of crack depth, and the natural frequency is sensitive to the change of fracture depth at trailing edge of dangerous rock mass. According to the natural frequency value of dangerous rock mass, the fracture depth at trailing edge of dangerous rock mass can be inversely calculated. Based on the relationship between the natural frequency and the fracture depth of the trailing edge of dangerous rock mass, the calculation model of the stability coefficient of the dangerous rock mass was established, and the indoor test was used to verify it. It was concluded that the natural frequency of the dangerous rock mass is approximately negative-linearly correlated with the crack depth of the trailing edge. The calculation model of stability factor of shear fractured falling dangerous rock mass based on natural frequency can provide useful reference for automatic monitoring of dangerous rock mass.
- shear fractured falling dangerous rock mass /
- stability factor /
- modified Timoshenko Beam theory /
- the half-space theory of dynamic foundation /
- natural frequency

HTML全文

图 1 危岩体风化过程示意图

Figure 1. Schematic diagram of weathering process of dangerous rock mass

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图 2 危岩体受力示意图

Figure 2. Force diagram of dangerous rock mass

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图 3 危岩体动力学模型

Figure 3. Dynamic model of dangerous rock mass

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图 4 运动分解示意图

Figure 4. Motion decomposition diagram

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图 5 半无限空间地基与基础示意图

Figure 5. Semi-infinite space foundation bed and foundation schematic diagram

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图 6 固有频率计算示意图

Figure 6. Diagram of natural frequency calculation

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图 7 数值模型示意图

Figure 7. Diagram of numerical model

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图 8 固有频率计算

Figure 8. Natural frequency calculation

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图 9 模型示意图

Figure 9. Diagram of model

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图 10 裂隙切割示意图

Figure 10. Fracture cutting schematic

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$\omega {\text{-}} h$ 关联曲线

The correlation curve of $\omega {\text{-}} h$

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${F_{\rm s}} {\text{-}} \omega$ 关联曲线

The correlation curve of ${F_{\rm s}} {\text{-}} \omega$

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${F_1}\left( {{\alpha _1}} \right)$ 计算式

Formulas of ${F_1}\left( {{\alpha _1}} \right)$

ν′	$0$	$0.25$	$0.5$
F₂(α₁)	$4 - 0.5a_1^2$	$5.3 - a_1^2$	$8 - 2a_1^2$
注：ν′为半无限空间材料泊松比。

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表 2 材料属性表

Table 2 Material property table

材料属性	材料参数
弹性模量E/GPa	0.28
泊松比ν	0.33
密度ρ/(kg·m⁻³)	2300

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表 3 模型材料配比

Table 3 Model material ratio

成分	质量比例/kg
石英砂	50
重晶石粉	30
石膏	8
甘油	1.5
纯净水	10
缓凝剂	0.02

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表 4 模型材料力学参数

Table 4 Mechanical parameters of model materials

参数名称	参数值
密度 $\rho$ / $({\text{kg} } \cdot { {\text{m} }^{ - 3} })$	2250
弹性模量 $E$ / ${\text{GPa}}$	0.28
泊松比 $\nu$	0.33
内聚力 $c$ / ${\text{MPa}}$	3.1
抗拉强度 ${R_t}$ / ${\text{MPa}}$	0.18

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表 5 固有频率与稳定系数值

Table 5 Natural frequency and stability coefficient

裂隙深度h/cm	稳定系数	固有频率 $\omega$ / ${\text{Hz}}$
裂隙深度h/cm	稳定系数	理论值	试验值
2	72.7	225.1	222.4
4	62.7	221.5	215.3
6	53.4	197.8	194.7
8	44.9	165.0	163.9
10	37.1	133.8	133.1
12	30.0	108.1	106.8
14	23.7	87.4	84.6
16	18.2	70.7	68.6
18	13.3	56.9	55.7
20	9.3	45.0	42.3
22	5.9	34.4	32.4
24	3.3	24.5	21.2
26	1.5	15.4	13.4

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