最新地质岩土英文文献翻译_冶金矿山地质_工程科技_专业资料

地质岩土英文文献翻译_冶金矿山地质_工程科技_专业资料
International Journal of Rock Mechanics and Mining Sciences
Analysis of geo-structural defects in flexural toppling
failure
Abbas Majdi and Mehdi Amini Abstract
The in-situ rock structural weaknesses, referred to herein as
geo-structural defects, such as naturally induced micro-cracks, are extremely responsive to tensile stresses. Flexural toppling failure occurs by tensile stress caused by the moment due to the weight of
the inclined superimposed cantilever-like rock columns. Hence, geo-structural defects that may naturally exist in rock columns are modeled by a series of cracks in maximum tensile stress plane. The magnitude and location of the maximum tensile stress in rock columns with potential flexural toppling failure are determined. Then, the minimum factor of safety for rock columns are computed by means of principles of solid and fracture mechanics, independently. Next, a new equation is proposed to determine the length of critical crack in such rock columns. It has been shown that if the length of natural crack is smaller than the length of critical crack, then the result based on solid mechanics approach is more appropriate; otherwise, the result obtained based on the principles of fracture mechanics is more acceptable. Subsequently, for stabilization of the prescribed rock slopes, some new analytical relationships are suggested for determination the length and diameter of the required fully grouted rock bolts. Finally, for quick design of rock slopes against flexural toppling failure, a graphical approach along with some design curves are presented by which an admissible inclination of such rock slopes and or length of all required fully grouted rock bolts are determined.
In addition, a case study has been used for practical verification of the proposed approaches.
Keywords Geo-structural defects, In-situ rock structural weaknesses, Critical crack length
1.Introduction
Rock masses are natural materials formed in the course of
millions of years. Since during their formation and afterwards, they have been subjected to high variable pressures both vertically and horizontally, usually, they are not continuous, and contain numerous cracks and fractures. The exerted pressures, sometimes, produce joint sets. Since these pressures sometimes may not be sufficiently high to create separate joint sets in rock masses, they can produce micro joints and micro-cracks. However, the results cannot be considered as independent joint sets. Although the effects of these micro-cracks
are not that pronounced compared with large size joint sets, yet they may cause a drastic change of in-situ geomechanical properties of
rock masses. Also, in many instances, due to dissolution of in-situ rock masses, minute bubble-like cavities, etc., are produced, which cause a severe reduction of in-situ tensile strength. Therefore, one should not replace this in-situ strength by that obtained in the laboratory. On the other hand, measuring the in-situ rock tensile strength due to the interaction of complex parameters is impractical. Hence, an appropriate approach for estimation of the tensile strength should be sought. In this paper, by means of principles of solid and fracture mechanics, a new approach for determination of the effect of geo-structural defects on flexural toppling failure is proposed.
2. Effect of geo-structural defects on flexural toppling failure
2.1. Critical section of the flexural toppling failure
As mentioned earlier, Majdi and Amini [10] and Amini et al. [11] have proved that the accurate factor of safety is equal to that calculated for a series of inclined rock columns, which, by analogy, is equivalent to the superimposed inclined cantilever beams as shown in Fig. 3. According to the equations of limit equilibrium, the moment M and the shearing force V existing in various cross-sectional areas in the beams can be calculated as follows:
(5)
( 6)
Since the superimposed inclined rock columns are subjected to uniformly distributed loads caused by their own weight, hence, the maximum shearing force and moment exist at the v ery fixed end, that is, at x=Ψ:
(7)
(8)
If the magnitude of Ψ from Eq. (1) is substituted into Eqs. (7) and (8), then the magnitudes of shearing force and the maximum moment of equivalent beam for rock slopes are computed as follows:
(9)
(10)
where C is a dimensionless geometrical parameter that is related to the inclinations of the rock slope, the total failure plane and the dip of the rock discontinuities that exist
in rock masses, and can be determined by means of curves shown in Fig.
Mmax and Vmax will produce the normal (tensile and compressive) and the shear stresses in critical cross-sectional area, respectively. However, the combined effect of them will cause rock columns to fail. It is well understood that the rocks are very susceptible to tensile stresses, and the effect of maximum shearing force is also negligible compared with the effect of tensile stress. Thus, for the purpose of the ultimate stability, structural defects reduce the cross-sectional area of load bearing capacity of the rock columns and, consequently, increase the stress concentration in neighboring solid areas. Thus, the in-
situ tensile strength of the rock columns, the shearing effect might be neglected and only the tensile stress caused due to maximum bending stress could be used.
2.2. Analysis of geo-structural defects
Determination of the quantitative effect of geo-structural defects in rock masses can be investigated on the basis of the following two approaches.
2.2.1. Solid mechanics approach
In this method, which is, indeed, an old approach, the loads from the weak areas are removed and likewise will be transferred to the neighboring solid areas. Therefore, the solid areas of the rock columns, due to overloading and high stress concentration, will eventually encounter with the premature failure. In this paper, for analysis of the geo-structural defects in flexural toppling failure, a set of cracks in critical cross-sectional area has been modeled as shown in Fig. 5. By employing Eq. (9) and assuming that the loads from weak areas are transferred to the solid areas with higher load bearing capacity (Fig. 6), the maximum stresses could be computed by the following equation (see Appendix A for more details):
（11）
Hence, with regard to Eq. (11), for determination of the factor of safety against flexural toppling failure in open excavations and underground openings including geo-structural defects the following equation is suggested:
（12）
From Eq. (12) it can be inferred that the factor of safety against flexural toppling failure obtained on the basis of principles of solid mechanics is irrelevant to the length of geo-structural
defects or the crack length, directly. However, it is related to the dimensionless parameter “joint persistence”, k, as it was defined earlier in this paper. Fig. 2 represents the effect of parameter k on the critical height of the rock slope. This figure also shows the
=1) with a potential of limiting equilibrium of the rock mass (F
s
flexural toppling failure.
Fig. 2. Determination of the critical height of rock slopes with a potential of flexural toppling failure on the basis of principles of solid mechanics.
2.2.2. Fracture mechanics approach
Griffith in 1924 [13], by performing comprehensive laboratory tests on the glasses, concluded that fracture of brittle materials is due to high stress concentrations produced on the crack tips which causes the cracks to extend (Fig. 3). Williams in 1952 and 1957 and Irwin in 1957 had proposed some relations by which the stress around the single ended crack tips subjected to tensile loading at infinite is determined [14], [15] and [16]. They introduced a new factor in their equations called the “stress intensity factor” which
indicates the stress condition at the crack tips. Therefore if this factor could be determined quantitatively in laboratorial, then, the factor of safety corresponding to the failure criterion based on principles of fracture mechanics might be computed.
Fig. 3. Stress concentration at the tip of a single ended crack under tensile loading Similarly, the geo-structural defects exist in rock columns with a potential of flexural toppling failure could be modeled. As it was mentioned earlier in this paper, cracks could be modeled in a conservative approach such that the location of maximum tensile stress at presumed failure plane to be considered as the cracks locations (Fig. 3). If the existing geo-structural defects in a rock mass, are modeled with a series cracks in the total failure plane, then by means of principles of fracture mechanics, an equation for determination of the factor of safety against flexural toppling failure could be proposed as follows:
（13）
where KIC is the critical stress intensity factor. Eq. (13) clarifies that the factor of safety against flexural toppling failure derived based on the method of fracture mechanics is directly related to both the “joint persistence” and the “length of cracks”. As such the length of cracks existing in the rock columns plays important roles in stress analysis. Fig. 10 shows the influence of the crack length on the critical height of rock slopes. This figure represents the limiting equilibrium of the rock mass with the potential of flexural toppling failure. As it can be seen, an increase of the crack length causes a decrease in the critical height of the rock slopes. In contrast to the principles of solid mechanics, Eq. (13) or Fig. 4 indicates either the onset of failure of the rock columns or the inception of fracture development.
Fig. 4. Determination of the critical height of rock slopes with a potential of flexural toppling failure on the basis of principle of fracture mechanics.
3. Comparison of the results of the two approaches
The curves shown in Fig. represent Eqs. (12) and (13), respectively. The figures reflect the quantitative effect of the geo-structural defects on flexural toppling failure on the basis of principles of solid mechanics and fracture mechanics accordingly. For the sake of comparison, these equations are applied to one kind of rock mass (limestone) with the following physical and mechanical properties [16]: , , γ=20kN/m3, k=0.75.
In any case studies, a safe and stable slope height can be determined by using Eqs. (12) and (13), independently. The two equations yield two different slope heights out of which the minimum height must be taken as the most acceptable one. By equating Eqs. (12) and (13), the following relation has been derived by which a crack length, in this paper called critical length of crack, can be computed:
（14a）
where ac is the half of the average critical length of the cracks. Since ac appears on both sides of Eq. (14a), the critical length of the crack could be computed by trial and error method. If the length
of the crack is too small with respect to rock column thickness, then the ratio t/(t−2ac) is slightly greater than one. Therefore one may ignore the length of crack in denominator, and then this ratio
becomes 1. In this case Eq. (14a) reduces to the following equation, by which the critical length of the crack can be computed directly:
(14b)
It must be born in mind that Eq. (14b) leads to underestimate
the critical length of the crack compared with Eq. (14a). Therefore, for an appropriate determination of the quantitative effect of geo-structural defects in rock mass against flexural toppling failure,
the following 3 conditions must be considered: (1) a=0; (2) a<ac; (3) a>ac.
In case 1, there are no geo-structural defects in rock columns and so Eq. (3) will be used for flexural toppling analysis. In case 2, the lengths of geo-structural defects are smaller than the critical length of the crack. In this case failure of rock column occurs due
to tensile stresses for which Eq. (12), based on the principles of solid mechanics, should be used. In case 3, the lengths of existing geo-structural defects are greater than the critical length. In this case failure will occur due to growing cracks for which Eq. (13), based on the principles of fracture mechanics, should be used for the analysis.
The results of Eqs. (12) and (13) for the limiting equilibrium both are shown in Fig. 11. For the sake of more accurate comparative studies the results of Eq. (3), which represents the rock columns
with no geo-structural defects are also shown in the same figure. As
it was mentioned earlier in this paper, an increase of the crack length has no direct effect on Eq. (12), which was derived based on principles of solid mechanics, whereas according to the principles of fracture mechanics, it causes to reduce the value of factor of safety. Therefore, for more in-depth comparison, the results of Eq. (13), for different values of the crack length, are also shown in Fig. As can
be seen from the figure, if the length of crack is less than the critical length (dotted curve shown in Fig. 11), failure is considered to follow the principles of solid mechanics which results the least slope height. However, if the length of crack increases beyond the critical length, the rock column fails due to high stress concentration at the crack tips according to the principles of fracture mechanics, which provides the least slope height. Hence, calculation of critical length of crack is of paramount importance.
4. Estimation of stable rock slopes with a potential of flexural toppling failure
In rock slopes and trenches, except for the soil and rock fills, the heights are dictated by the natural topography. Hence, the desired slopes must be designed safely. In rock masses with the potential of flexural toppling failure, with regard to the length of the cracks extant in rock columns the slopes can be computed by Eqs.
(3), (12), and (13) proposed in this paper. These equations can
easily be converted into a series of design curves for selection of the slopes to replace the lengthy manual computations as well. [Fig. 12], [Fig. 13], [Fig. 14] and [Fig. 15] show several such design curves with the potential of flexural topping failures. If the lengths of existing cracks in the rock columns are smaller than the critical length of the crack, one can use the design curves, obtained on the basis of principles of solid mechanics, shown in [Fig. 12] and [Fig. 13], for the rock slope design purpose. If the lengths of the cracks existing in rock columns are greater than the critical length of the crack, then the design curves derived based on principles of fracture mechanics and shown in [Fig. 14] and [Fig. 15] must be used for the slope design intention. In all, these design curves, with knowing the height of the rock slopes and the thickness of the rock
columns, parameter (H2/t) is computed, and then from the design
curves the stable slope is calculated. It must be born in mind that
all the aforementioned design curves are valid for the equilibrium condition only, that is, when FS=1. Hence, the calculated slopes from the above design curves, for the final safe design purpose must be reduced based on the desired factor of safety. For example, if the information regarding to one particular rock slope are given [17]:
k=0.25, φ=10°, σt=10MPa, γ=20kN/m3, δ=45°, H=100m, t=1 m, ac>a=0.1 m, and then according to Fig. 12 the design slope will be 63°, which represents the condition of equ ilibrium only. Hence, the final and safe slope can be taken any values less than the above mentioned one, which is solely dependent on the desired factor of safety.
Fig. 5. Selection of critical slopes for rock columns with the potential of flexural toppling failure on the basis of principles of solid mechanics when k=0.25.
Fig. 6. Selection of critical slopes for rock columns with the potential of flexural toppling failure based on principles of solid mechanics when k=0.75..
Fig. 7. Selection of critical slopes for rock columns with the potential of flexural toppling failure based on principles of fracture mechanics when k=0.25.
Fig. 8. Selection of critical slopes for rock columns with the potential of flexural toppling failure based on principles of fracture mechanics when k=0.75.
5. Stabilization of the rock mass with the potential of flexural toppling failure
In flexural toppling failure, rock columns slide over each other so that the tensile loading induced due to their self-weighting grounds causes the existing cracks to grow and thus failure occurs. Hence, if these slides, somehow, are prevented then the expected instability will be reduced significantly. Therefore, employing fully grouted rock bolts, as a useful tool, is great assistance in increasing the degree of stability of the rock columns as shown in Fig. 16 [5] and [6]. However, care must be taken into account that employing fully grouted rock bolts is not the only approach to stabilize the rock mass with potential of flexural toppling failure. Therefore, depending up on the case, combined methods such as decreasing the slope inclination, grouting, anchoring, retaining walls, etc., may even have more effective application than fully
grouted rock bolts alone. In this paper a method has been presented
to determine the specification of fully grouted rock bolts to
stabilize such a rock mass. It is important to mention that Eqs. (15), (16), (17), (18), (19) and (20) proposed in this paper may also be used as guidelines to assist practitioners and engineers to define
the specifications of the desired fully grouted rock bolts to be used for stabilization of the rock mass with potential of flexural
toppling failure. Hence, the finalized specifications must also be checked by engineering judgments then to be applied to rock masses. For determination of the required length of rock bolts for the stabilization of the rock columns against flexural toppling failure the equations given in previous sections can be used. In Eqs. (12)
and (13), if the factor of safety is replaced by an allowable value, then the calculated parameter t will indicate the thickness of the combined rock columns which will be equal to the safe length of the rock bolts. Therefore, the required length of the fully grouted rock bolts can be determined via the following equations which have been proposed in this paper, based on the following cases.
Fig. 9. Stabilization of rock columns with potential of flexural toppling failure with
fully grouted rock bolts.
Case 1: principles of solid mechanics for the condition when (a<a c):
(15)
Case 2: principles of fracture mechanics for the condition when
(a>a c):
(16)
Where FSS is the allowable factor of safety, T is the length of the fully grouted rock bolts, and Ω is the angle between rock bolt longitudinal axis and the line of normal to the discontinuities of rock slope.
Eqs. (15) and (16) can be converted into some design curves as shown in Fig. In some cases, one single bolt with a length T may not guarantee the stability of the rock columns against flexural toppling failure since it may pass through total failure plane. In such a case, the rock columns can be reinforced in a stepwise manner so that the thickness of the sewn rock columns becomes equal to T [11].
Eq. (17) represents the shear force that exists at any cross-sectional area of the rock bolts. Therefore, both shear force and shear stress at any cross-sectional area can be calculated by the following proposed equations:
（17）
（18）
where V is the longitudinal shear force function, τ is the
shear stress function, and Q(y) is the first moment of inertia.
According to the equations of equilibrium, in each element of a beam, at any cross-sectional area the shear stresses are equal to
that exist in the corresponding longitudinal section [18]. Hence, the total shear force S in the longitudinal section of the beam can be calculated as follows:
The inserted shear force in the cross-sectional area of the rock bolt is equal to the total force exerted longitudinally as well. Therefore,
the shear force exerted to the rock bolt's cross-section can be computed as follows:
7. Conclusions
In this paper, geo-structural defects existing in the in-situ rock columns with the potential of flexural toppling failure have been modeled with a series of central cracks. Thereafter on the basis of principles of both the solid and fracture mechanics some new equations have been proposed which can be used for stability analysis and the stabilization of such rock slopes. The final outcomes of this research are given as follows:
1. Geo-structural defects play imperative roles in the stability of rock slopes, in particular, flexural toppling failure.
2. The results obtained on the basis of principles of solid mechanics approach indicate that the length of cracks alone has no influence on the determination of factor of safety, whereas the value of joint persistence causes a considerable change in its value. On the other hand, the factor of safety obtained based on principles of fracture mechanics approach is strongly influenced by both the length of existing cracks in rock columns and joint persistence as well.
3. The critical length of cracks represents the equality line of the results obtained from both approaches: solid mechanics and fracture mechanics.
4. If the length of the crack is less than the critical length, failure is considered to follow the principles of solid mechanics. However, if the length of crack increases beyond the critical length, the rock column fails due to high stress concentration at the crack tips, according to the principles of fracture mechanics.
5. The present proposed equations are also converted into some design graphs that can be used for ease of application and to reduce manual lengthy calculations for determining the critical height of rock slopes with the potential of flexural toppling failure.
6. In this paper, on the basis of principles of both solid mechanics and fracture mechanics some equations are proposed to determine the safe length and the diameter of the fully grouted rock bolts for stabilization of rock slopes with the potential of flexural toppling failure.
7. For simplicity of computations, some design graphs for determination of the length of the fully grouted rock bolts for stabilization of rock slopes with the potential of flexural toppling failure are also presented.
8. Slope stability analysis of the Galandrood mine shows the new approach is well suited for the analysis of flexural toppling failure.
国际岩石力学与工程学报
地质结构缺陷对弯曲倾倒破坏的影响
作者：Abbas Majdi and Mehdi Amini
摘要
原位岩石弱点，在此统称为地质结构缺陷，如自然诱发的微裂纹，对拉应力有很大影响。

倾斜叠加悬臂式岩柱的自重会引起拉应力，该拉应力会造成岩柱弯曲倾倒破坏。

因此，天然的存在于岩柱中的地质结构缺陷可以被一系列最大主应力平面的裂隙模拟。

首先，有潜在弯曲倾倒破坏的岩柱最大主应力的大小和位置是确定的。

其次，岩柱的最小安全系数是可以通过固体和断裂力学原理单独地计算出来。

目前，一个新的公式被提出来确定这些岩柱的临界裂隙长度。

该公式指出：如果天然裂缝的长度小于临界裂纹长度，基于固体力学原理的结果是更合适的；否则，基于断裂力学原理获得的结果更可以接受。

随后，为了所设计岩质边坡的稳定性，一些新的解析关系被提出去测定完全灌浆锚杆的长度和直径。

最后，为抵抗岩石弯曲倾倒破坏而进行的快速设计，根据一些设计曲线，一个形象化的方式被提出，通过这种方法岩质边坡的倾角和所有完全灌浆锚杆的长度是可以确定的。

此外，一个案例研究已经被用于实际验证所提出的方法。

关键词地质结构缺陷，原位岩石结构的弱点，临界裂隙长度
第一章简介
岩体是在亿万年中形成的天然物质。

由于它们形成过程中和形成之后一直受到大的变化的纵横向压力，通常，它们是不连续的并包含许多裂隙和裂缝，
受到施加压力的作用，有时产生节理群。

有时这些压力不足够高以创建单独的节理结构面，但它们能产生微缝和微裂纹，这种结果不能被视为产生独立联合集。

虽然这些微裂纹影响并不明显与大型联合集比较，但它们可能会导致原位岩体地质力学性能的急剧变化。

此外，在许多情况下，由于原位岩体的解散、微小的泡状空腔等产生，这会造成了原位拉伸强度严重下降。

因此，一方面不应该用在实验室获得的强度取代原位强度；另一方面，由于复杂参数的相互作用测量原位岩石抗拉强度是不切实际的。

因此，一种去估算拉伸强度的适当方法被寻求。

本文根据过固体和断裂力学原理，在测定地质结构缺陷对弯曲倾倒破坏的影响上一个新的方法被提出。

第二章 地理结构缺陷对弯曲倾倒破坏的影响
2.1弯曲倾倒破坏的临界断面
如前所述，Majdi 、Amini （研究[10] ）和 Amini （研究[11]）等，已经证明准确的安全系数接近一系列斜岩柱依次类推，就等于叠加倾斜的悬臂梁。

根据极限平衡方程，存在于各种截面梁中的弯矩M 和剪力V 可如下计算出：
δγcos 5.02)(x t M x -= (5)
δγcos )(x t V x -= ( 6)
由于倾斜叠加岩柱受到其自身的重量造成均匀分布载荷，因此，最大剪切力和力矩存在于非常固定端，即在x=Ψ：
δγψcos 5.02max t M = (7)
δγψcos max t V = (8)
如果方程（1）中的Ψ代入方程（7）及（8），等效梁的剪切力和最大弯矩的大小计算如下：
δγcos 5.02max CH t M = (9)
C H t V δγcos max = (10)
其中C 是一个无量纲几何参数，与岩质边坡的倾向，总破坏面和存在于岩体中不连续岩石的倾角相关。

Mmax 和Vmax 将分别产生法向（拉伸，压缩）的应力和临界面上的剪应
力。

然而，它们共同作用将导致岩柱破坏。

岩石受拉应力和最大剪切力共同作用与单独受拉伸应力作用的影响相比是可以忽略不计的，这很容易理解。

因此，结构上的缺陷减少了岩柱截面承担负荷的能力，为了达到有效的最终目的，增加坚硬区域的应力集中。

因此，岩柱原位拉伸强度和剪切的共同作用可能被忽略，只有由于最大弯曲应力引起的拉应力可以被使用。

2.2地质结构缺陷的分析
岩体地质结构缺陷定量影响的测定可以基于以下两种方法调查。

2.2.1固体力学方法
在这种方法中，这的确是一个老办法：当薄弱环节的负荷被删除时，他将转移到邻近地区的坚固区域。

因此，岩柱中的坚固部分由于超载和应力集中，最终因相互作用而过早失效。

在本部分，为了分析弯曲倾倒破坏上的地质结构缺陷,一系列横截面上的裂缝已被设为蓝本。

通过方程（9）和把薄弱环节的负载转移到有较高的承载能力区域的假设，最大应力可以通过以下公式：
()
k t CH -=1cos cos 32max ϕδγσ （11）
因此关于方程(11)，为了测定在露天挖掘和有地质结构缺陷的地下洞室抵抗弯曲倾倒破坏的安全系数，提出下列公式：
()ϕ
δγσcos cos 3121CH k t F s -= （12） 从方程（12）可以直接的推断：以固体力学原理获得的抵抗弯曲倾倒破坏的安全系数与地质结构缺陷的长度或裂缝的长度无关。

可是，它与无量纲参数“节理持久”相关，k 在本部分前面已经定义。

图（2）代表参数k 对岩质边坡临界高度的影响。

这些数据也显示了有潜在弯曲倾倒破坏岩体（
1）的
极限平衡。

图2：在固体力学原理上测定的有潜在弯曲倾倒破坏岩质边坡的临界高度
2.2.2断裂力学方法
1924年Griffith 通过完成全面的玻璃试验，得出的结论是脆性材料的断裂是由于裂纹尖端产生的高应力集中造成裂痕扩大产生的（图3）。

Williams 在1952和1957年和Irwin 在1957年曾提出了一些关系，单端裂纹周围的应力在受无限拉伸载荷条件下的研究[14]，[15]和[16]。

他们在方程中介绍了一个新的参数 “应力强度因子”，它表明在裂纹尖端的应力状态。

因此，如果这个参数在实验室里能定量的确定，那么相应的安全系数（基于断裂力学原理的破坏判据）可能被计算出来。

图3：拉伸载荷作用下单端裂纹尖端的应力集中
同样的，有潜在弯曲倾倒破坏岩柱中存在的地质结构缺陷可效仿计算出。

正如本部分前面提到的，可用一个保守的方法仿效裂缝，就是假定破坏面的最大拉应力处是裂缝位置。

如果岩体中存在的地质结构缺陷，用总破坏面内一系列裂缝模拟，通过断裂力学原理，一个测定抵抗弯曲倾倒破坏的安全系数的方程被提出如下：
()()
k a a t CH K t F IC si ππϕδγsec 2cos cos 322-= （13） 其中是临界应力强度因子。

方程（13）表明基于断裂力学原理计算的抵抗弯曲倾倒破坏的安全系数与“联合持久”和“裂缝长度”直接相关。

因此存在于岩柱中的裂缝的长度在应力分析中有重要的作用。

图10显示了裂纹长度对岩质边坡临界高度的影响。

这些数据代表了有潜在弯曲倾倒破坏岩体的极限平衡方程。

由此可以看出，一个裂纹长度的增加导致岩石边坡临界高度的下降。

相反的是根据固体力学原理方程（13）或图4表明：岩柱开始破坏或者裂缝开始发育。

图4：在断裂力学原理的基础上对潜在弯曲倾倒破坏岩石边坡临界高度的测定
第三章 两种方法结果的比较
图4的曲线分别代表了方程（13）。

这些数据基于固体力学和断裂力学原理定量的反映了地质结构缺陷对弯曲倾倒破坏的影响。

为了方便应用，这些方程被应用于以下物理和力学性能条件下的研究： m MPa K IC 38.1=，MPa 47.81=σ，γ=20 kN/m3，k=0.75。

在任何研究实验中，一个安全和稳定的边坡高度均可以单独地用方程
（12）和（13）确定。

两个方程得到两个不同的坡度高度，其中必须采取最小高度作为一个最可以接受的值。

通过方程（12）和（13），推导出了裂纹长度，在此部分被称为裂纹临界长度，可以计算如下：
()()k k K a t t a IC c c ππσsec 1122
12⎥⎦⎤⎢⎣⎡-⎪⎪⎭⎫ ⎝⎛-= （14a ）
c a 是裂缝平均临界长度的一半。

由于c a 出现在方程（14a ）两边，裂纹的临界长度可通过试验和误差的方法计算。

如果相对于岩柱的厚度裂纹长度过
小，t/(t −2ac)的比值会稍大一点。

因此，如果忽略分母中裂缝的长度，这个比值将变为1。

在这一部分方程（14a ）简化为下面的公式，其中裂纹的临界长度可直接计算：
()()k k K a IC c ππσsec 112
1⎥⎦⎤⎢⎣⎡-= (14b) 我们必须牢记方程（14b ）与方程（14a ）相比将导致裂纹临界长度偏小。

因此，对于适当的确定岩体中地质结构缺陷对弯曲倾倒破坏的定量影响时，以下三个条件必须加以考虑：(1) a=0； (2) a<c a ；(3) a>c a 。

在第一种情况下，岩柱中没有地质结构缺陷，因此方程(3) 将用于弯曲倾倒的分析。

在第二种情况下，地质结构缺陷的长度远远小于裂纹临界长度，岩柱由于拉应力产生破坏，故在固体力学原理基础上的方程（12）将被应用。

在第三种情况下，地质结构缺陷的长度远远大于裂纹临界长度，岩柱因裂缝变大产生破坏，在断裂力学原理基础上的方程（13）将被用于分析。

方程（12）和（13）对极限平衡方程的结果如图11所示。

为了更准确地比较方程（3）的研究结果，这代表了没地质结构缺陷的岩柱也以相同的数据被显示。

由于一个裂纹长度的增加并没有直接影响方程（12），这是本文前面提到的，它是根据固体力学原理而非断裂力学原理推导出的，因而它会导致安全系数值的降低。

为了更深入的比较，裂纹长度为不同的值时，方程（13）的结果也显示在图11中。

从图中可以看出，如果裂纹长度小于临界长度时，基于固体力学原理将产生最小边坡高度。

但是，如果裂纹长度的增加超出临界长度，根据断裂力学原理裂纹尖端的高应力集中将导致岩柱破坏，这推导出最小边坡高度。

因此，裂纹临界长度计算是至关重要的。

第四章 有潜在弯曲倾倒破坏稳定岩质边坡的估计
岩石边坡和沟道，除了土壤和岩石填入的高度取决于自然地形，因此必须考虑理想的安全的边坡设计。

有潜在弯曲倾倒破坏的岩体，相对于岩质边坡中的裂缝长度可以通过本文之前提到的方程（3）、（12）和（13）计算出来。

这
些方程可以转换成一系列设计曲线能很容易地选择斜坡，以及取代冗长的手工计算。

图12、13、14和15显示了几种有潜在弯曲倾倒破坏的设计曲线。

如果岩柱现有裂缝长度小于裂纹临界长度，基于固体力学原理可以使用设计曲线得到如图12和13所示，这是岩质边坡的设计目的。

如果岩柱现有裂缝长度大于裂纹临界长度，基于断裂力学原理派生的设计曲线（如图7和8所示）必须用于边坡设计的目的。

总之，为了去诶的那个岩质边坡的高度、岩柱的厚度和参数(H2/t)，这些设计曲线被设计，通过设计曲线设计稳定的边坡。

我们必须牢
F=1前提条件下是有效的平衡状态。

因此，根据上记，上述所有设计曲线仅在s
述设计曲线设计边坡时，为了达到最终设计安全的目的，必须降低基于安全所需的因素。

例如，如果一个特殊岩质边坡被赋予以下条件（研究17）：
k=0.25，φ=10°，σt=10MPa,，γ=20kN/m3，δ=45°，H=100 m，t=1 m，ac>a=0.1 m，通过图12设计坡度将是63 °它仅代表了临界平衡条件。

因此，最后安全坡度可以取比上面提到值小的任何一个值，这完全是较少依赖所需的安全因素。

图5：基于固体力学原理有潜在弯曲倾倒破坏的岩柱在k=0.25是临界坡度的选择
图6：基于固体力学原理有潜在弯曲倾倒破坏的岩柱在k=0.75是临界坡度的选择
图7：基于断裂力学原理有潜在弯曲倾倒破坏的岩柱在k=0.25是临界坡度的选择
图8：基于断裂力学原理有潜在弯曲倾倒破坏的岩柱在k=0.75是临界坡度的选择
第五章有潜在弯曲倾倒破坏岩体的稳定性
在弯曲倾倒破坏过程中，岩柱间彼此滑动，以至于在拉伸载荷作用下它们自身重量增加导致现有的裂缝扩展和产生破坏。

因此，如果这些滑动不知何故。