Calculation power of Proctor's Theory
(1) e to the cutting of joints, the rock mass forms loose rock mass after excavation, but it still has a certain adhesive force
(2) after the excavation of the chamber, the rock mass at the top of the chamber will form a natural equilibrium arch. At the side wall of the chamber, two sliding surfaces are generated along the direction of the angle between the chamber and the side wall. The calculation diagram is shown in Figure 1. However, the surrounding rock pressure acting on the top of the tunnel is only the self weight of the rock mass in the natural equilibrium arch< (3) the strength coefficient f is used to characterize the strength of rock mass. Its physical meaning is:
but in practical application, Proctor adopts an empirical calculation formula, which can easily obtain the value of F. That is,
where RC is uniaxial compressive strength (MPA)
f-an empirical coefficient with dimension of 1. In practical application, the integrity of rock mass and the influence of groundwater should be considered at the same time
(4) the rock mass at the top of the natural balanced arch can only bear the compressive stress, but not the tensile stress< (1) determination of natural equilibrium arch axis equation
in order to obtain the surrounding rock pressure of tunnel top, the expression of natural equilibrium arch axis equation must be determined firstly, and then the distance from tunnel top to arch axis must be calculated to calculate the self weight of rock mass in equilibrium arch. First, assume that the arch circumference is a quadratic curve, as shown in Figure 2. Taking any point m (x, y) on the arch axis, according to the condition that the arch axis can not bear the tensile force, the bending moment of all external forces on point m should be zero. That is,
(a)
where q is the uniformly distributed load generated by the self weight of the rock mass above the arch axis
T -- horizontal thrust of balanced arch crown section
x, Y -- X, Y axis coordinates of m point respectively
there are two unknowns in the above equation, so it is necessary to establish an equation to get its solution. It can be seen from the static equilibrium equation that the horizontal thrust t in the above equation is equal to the horizontal thrust t 'acting on the arch foot
Fig. 2 calculation diagram of natural equilibrium arch
force t', and the direction is opposite. That is,
t = t '
because the arch foot is easy to proce horizontal displacement and change the internal force distribution of the whole arch, Proctor thinks that the horizontal thrust t' of the arch foot must meet the following requirements,
t '≤ qa1f (b)
that is, the horizontal thrust acting on the arch foot must be less than or equal to the maximum friction generated by the vertical reaction, In order to maintain the stability of the arch foot. In addition, for the sake of safety, Proctor reces the horizontal thrust by half, and then makes t = qa1f / 2, which can be substituted into formula (a) to obtain the equation of arch axis as
obviously, the equation of arch axis is a parabola. According to this formula, the height of any point on the arch axis can be obtained
when the side wall is stable, x = a, y = B, we can get
when the side wall is unstable, x = A1, y = B1, we can get
where B and B1 are the rise of the arch, that is, the maximum height of the natural equilibrium arch
A -- the span of balanced arch when the side wall is stable
A1 -- the maximum span of natural balanced arch, as shown in Figure 1. According to the above formula, the maximum surrounding rock pressure in the natural equilibrium arch can be easily calculated< (2) calculation of surrounding rock pressure
Pu believes that the surrounding rock pressure acting on the top of deep buried loose rock chamber is only the self weight of rock mass in arch. However, in engineering, for convenience, the maximum surrounding rock pressure at the top of the tunnel is usually taken as the uniform load, regardless of the change of surrounding rock pressure caused by the change of tunnel axis. According to this, the maximum surrounding rock pressure at the top of the tunnel can be calculated according to the following formula
the lateral pressure in proctor's surrounding rock pressure theory can be calculated according to the following formula
in the application of Proctor's Theory, it is the key to ensure that the tunnel has enough buried depth and a natural equilibrium arch can be formed after rock excavation; The second is the determination of the firmness coefficient F. in practical application, in addition to the calculation according to the formula, appropriate correction must be given according to the construction site, the leakage of groundwater and the integrity of rock mass, so as to make the firmness coefficient more comprehensively reflect the mechanical properties of rock mass.
(1) e to the cutting of joints, loose rock mass is formed after excavation, but it still has a certain bond force
(2) after the excavation of the chamber, the rock mass at the top of the chamber will form a natural equilibrium arch. At the side wall of the chamber, two sliding surfaces are generated along the direction of the angle between the chamber and the side wall. The calculation diagram is shown in Figure 1. However, the surrounding rock pressure acting on the top of the tunnel is only the self weight of the rock mass in the natural equilibrium arch.
At present, continuum mechanics method is generally used in soil mechanics analysis, which is right in most cases. However, it is not right in some cases, such as in the slope and underground cavern, block collapse and loess vertical slope collapse are often seen, which can not be dealt with by continuum mechanics model. They belong to the mechanics of block crack medium. Therefore, in the analysis of soil mechanics, the mechanical medium must be analyzed according to the soil structure and the soil occurrence environment conditions. Combined with the characteristics of soil engineering, the appropriate mechanical model can be given for analysis in order to obtain the actual results. According to the soil structure and the different mechanical action modes and law types of soil when the environmental stress changes, the soil can be divided into several types of soil mechanical media. According to the author's experience and understanding, the soil can be divided into three kinds of mechanical media: 1) continuous media; 2; ② Wedge shaped bulk fracture medium; ③ Columnar block fracture medium. The classification conditions and their mechanical action rules are shown in table 4-3, which is the basic basis for soil mechanical analysis
Table 4-3 soil mechanics medium division
1. Soil foundation engineering deformation analysis method
foundation engineering deformation is a very in-depth discussion of soil mechanics. Generally speaking, the foundation deformation can be estimated by the following methods. This method is suitable for both homogeneous soil and heterogeneous soil. It is called layered total method. The specific methods are as follows:
(1) divide the deformed soil into an appropriate number of horizontal layers. For multi-layer structure soil, it can be layered according to the soil interface and stress change points (Fig. 4-8)
(2) calculate the effective additional stress of each horizontal layer. For practical purposes, the value of each layer can be taken at the depth of the center of the layer (3) calculate the average value of additional vertical stress in each horizontal layer. If the thickness of each layer is small compared with the width of the foundation, Δσ< The average value of sub > Z < / sub > can be taken as the central depth stress value of delamination. Because the stress distribution has nothing to do with the characteristics of soil, the same method can be used to calculate the internal stress of homogeneous soil and multilayer soil (4) calculate the compression of each horizontal layer thickness caused by additional vertical stress Δ H:
principles of geological engineering
(5) the settlement deformation at any depth under the foundation is equal to the sum of the settlement variables of each horizontal layer above this point, that is,
principles of geological engineering
. This method takes the influence of non-uniformity into account, It is a common method to estimate the deformation of foundation engineering
At present, there are many kinds of soil slope stability analysis methods, the most commonly used one is circular slip surface method. During 1958-1960, the author carried out a series of investigations and studies on the mechanical problems of loess slope in Northwest China, and collected a large number of slope failure data. After analyzing the collected data, we get an important understanding, that is, the mechanical process of landslide of loess slope in Northwest China is: the upper soil collapses, and part of the soil of slope is squeezed to slide. The mechanical mechanism of this process can be illustrated by figure 4-9. The upper part is the collapse stress area, the lower part is the sliding stress area, and the middle part is the transition area. Internal stress in collapse area σ< The direction of sub > 1 < / sub > is approximately perpendicular to the ground, and the internal stress in the sliding zone is relatively high σ< The sub > 1 < / sub > direction is roughly parallel to the slope. According to the theory of soil balance, the failure surface of collapse stress area is closely related to the failure surface σ< The direction of sub > 1 < / sub > is 45 - ψ/ 2 jiao, ψ Is the shear angle; The relationship between the fracture surface of the slip stress zone and the stress distribution σ< Sub > 1 < / sub > 45- ψ/ In the case of slope, the angle is 45- ψ/ 2; The transition zone is the intersection angle of conjugate fracture surface, that is (45- ψ/ 2＋45- ψ/ 2＝90- ψ Based on this, the outline of theoretical fracture surface of soil slope can be drawn. Theoretically, the theoretical fracture surface in soil is not a line, but a group (Fig. 4-10). When one or several theoretical fracture surfaces of soil are unstable, the landslide will occur and the slope will be destroyed. Figure 4-11 is an example of this theory. There are three failure surfaces in the slope at the same time, so there are three stepped failure. Therefore, in slope stability analysis, it is not only necessary to calculate the possibility of slope failure caused by the theoretical fracture of slope toe, but also to calculate the failure possibility of each theoretical fracture surface as shown in Figure 4-10, find out the most dangerous or the lowest stability fracture surface, give the stability coefficient, and evaluate the slope stability. The following is a detailed discussion on the graphic drawing method of theoretical fracture surface. As shown in Fig. 4-12:
Fig. 4-9 mechanical mechanism sketch of landslide action of slope soil
Fig. 4-10 combination of theoretical fracture surface of loess slope
< P >
Fig. 4-11 failure of qujiatai loess slope in Baoji City (slope height 18m)
Fig. 4-12 stability calculation results of qujiatai loess slope in Baoji City
(1) make slope calculation according to proportion Geometric shape AOD
(2) based on the shear test results, the shear angles at different depths are calculated, which are noted on the elevation scale ψ It can be calculated by the formula
principle of geological engineering
or by graphic method
(3) use the shear angle noted on the height scale ψ， The theoretical fracture surfaces AB, OC and DC are made in sections AB and ab. the theoretical fracture surfaces of OB and ab are in a 45-45 relationship with the slope surface ψ/ 2. The theoretical fracture surface of OC and DC sections is 45-45 ° to the vertical direction ψ/ 2 cents. The BC is divided into several equal parts and connected with point O. from point B to point O, the BC is divided into several equal parts ψ Wrap line, cross OC line at point C, and then make DC line from point C upward. At this point, a theoretical fracture surface curve is completedfigure 4-12 shows the theoretical fracture surface drawn for stability calculation of qujiatai loess slope, and the theoretical fracture surface drawn is basically consistent with the measured results shown in Figure 4-11. The upper part of the theoretical fracture surface is 90 °， Fast transition to 80 °， 65 in the middle °， The lower part is 45 ° The upper part of the measured section shown in Figure 4-11 is 80 °～ ninety °， 65 in the middle °， The lower part is 45 ° Obviously, the above method is credible. With the above theoretical fracture surface, the stability of each theoretical fracture surface can be calculated by graphic method or algebraic method, and the slope stability can be calculated. The above is a complete structure of soil slope stability analysis method. This method is credible for intact soil, but not when there are weak layers or joint surfaces in the soil. There are two common cases of failure under the control of weak plane and joint plane:
(1) failure under the control of weak plane and joint plane as shown in figure 4-13a
(2) as shown in figure 4-13b, collapse is controlled by vertical joints or fractures
Fig. 4-13 schematic diagram of loess soil failure. The failure system controlled by structural joints and weak planes glides along the weak plane. It completely conforms to Coulomb's law and can be used to analyze the slope stability simply. The problem is to identify this geological model in the field. With the geological model, it can be easily transformed into a mechanical model. The mechanical calculation is very simple and can be carried out by formula (4-34)
the collapse condition of slope under the control of vertical cracks shown in Fig. 4-13b can be analyzed by the failure criterion of soil pressure inced tension crack at the foot of slope, namely
principle of geological engineering
, where: σ< Sub > C < / sub > is the uniaxial compressive strength of soil; γ< Sub > I < / sub >, H < sub > I < / sub > are the weight and thickness of each layer
The key to the stability analysis ofsoil slope is to understand the geological model, reasonably abstract the mechanical model and select reasonable mechanical parameters, so the calculation is not complicated. At present, there is a bias that the calculation theory is deeply studied, and the selected mechanical model and mechanical parameters do not conform to the geological reality of soil, and the results often do not conform to the reality
3. Stability analysis method of underground cavern in soil
the stability problems of underground cavern in soil, such as tunnel and soil reservoir, have been studied for a long time. The starting point of these studies is to take the collapse soil of the tunnel top as the external load of the support, thus forming the concept of load support system in underground engineering buildings. It seems that the main problem of soil mechanics in underground engineering construction is to find out the collapse height of the soil at the top of the tunnel. Therefore, many people are studying the calculation formula of the collapse height of the tunnel top soil. Among these results, the most famous one is proctor's collapse arch theory, which was controlled for half a century. The main contents of Proctor's theory are as follows
The basic points of his theory are as follows:
Fig. 4-14 mechanical model of Proctor collapse arch
(1) Proctor defines soil shear angle as soil strength coefficient, which is commonly called Proctor coefficient, that is,
principle of geological engineering
(2) the width of cavern is 2B < sub > 1 < / sub >, and the height of cavern is h, The width of the collapse arch is 2B < sub > 2 < / sub >, and the soil supporting the arch foot is in line with the tunnel wall_ 0062_ 0155. JPG "> < title > < / Title >
< / picdesc > < / imagedata > < / imageobject > < / inlinemediaobject > corner, Then the half width of collapse arch is B < sub > 2 < / sub >:
principle of geological engineering
(3) the mechanical equilibrium condition of collapse arch is
principle of geological engineering
principle of geological engineering
, where t is horizontal reaction; F is the additional shear force
(4) at that time, x = B < sub > 2 < / sub > y = h < sub > G < / sub >, Then formula (4-41) becomes
principles of geological engineering
substituting the above results into formula (4-43) to get
principles of geological engineering
(5) the extreme value of H < sub > G < / sub >
< Principles of geological engineering (6) is known from formula (4-47), The earth pressure at any point is
principle of geoengineering
and the maximum earth pressure is
principle of geoengineering
in the design of underground engineering, the σ< Sub > Vmax < / sub > as the earth pressure, the lining thickness is designed
What are the advantages and disadvantages of this theory? Can it be used in underground engineering design? In the author's opinion, first of all, it should be affirmed that this theory has merits. Because in the construction of underground caverns in soil, whether it is artificial or natural, the stable cavern shape of the roof is arch. This provides a practical basis for the collapse arch theory. This proves that it is feasible to use proctor's theory in the stability calculation of underground caverns, but the application of Proctor's theory in rock mechanics is not in line with the reality. In addition, this alone is not enough. When the buried depth of underground cavern is large, flow deformation often occurs in the construction process, that is, non-stop deformation. This is why Platts theory can not answer this question. This problem is related to the stress in the soil. We will discuss this problem below Thestress limit equilibrium theory is shown in Fig. 4-15. P < sub > 0 < / sub > is the vertical stress in soil, λ P
In karst area, when the karst cave has been formed in the foundation, the pressure arch theory can be used to determine the stability of the cave. Due to the stress redistribution of rock and soil around the karst cave, when the stress is greater than the allowable strength of rock and soil, some rock or soil will lose balance and slide downward (karst cave or soil cave). Engineering practice and experimental results show that the sliding collapse of rock or soil is not endless. When the sliding collapse reaches a certain degree, it will not move downward, Therefore, the whole top surrounding rock mass composed of rock blocks is in a new equilibrium state, which is similar to arch, also known as pressure arch
In 1907, Russian scholar prototchyakonov put forward the theory of natural equilibrium arch, that is, proctor's pressure arch theory. The theory of pressure arch is widely used in mines, tunnels and underground caverns. At present, there are various hypotheses for calculating the shape and stress of pressure arch. Due to the different assumptions about the shape and stress of pressure arch, the calculated surrounding rock pressure is different. However, the discriminant formula recommended by the Handbook of Engineering Geology (Fourth Edition) is widely used in karst foundation evaluation. Its principle and formula are based on the calculation formula of Proctor's pressure arch theory, but it has the following defects: the influence of horizontal stress is not considered in the stress condition, the cohesive force of rock and soil is not considered, and the safety factor has been doubled in the derivation process. Considering the above factors, the calculation formula of karst foundation stability based on pressure arch theory is deced as follows1.8.1 pressure arch
in order to determine the arch, in Figure 1.4, take the arc length om segment to analyze the force balance conditions
(1) r is the horizontal left supporting force of the right part of the arch on the arc length OM, along the tangent direction of o point (2) s is the inclination of the left lower part of the arch m n to the arc length OM, pointing to the upper right supporting force, along the tangent direction of m point3 σ< sub>v= γ· Z is the vertical downward compressive stress of overlying rock and soil directly above the arc OM, that is, the self weight stress of overlying rock and soil; σ< sub>h＝ λ·γ· Z is the lateral pressure of rock and soil (ignoring the change of their height along the y-axis)
Figure 1.4 diagram for pressure archthe coordinates of point m are (x, y). If the arc segment OM is in the state of force balance, the sum of the four forces to m point torque should be 0, That is:
theory and practice of geotechnical engineering in Guilin Karst Area
Simplified:
theory and practice of geotechnical engineering in Guilin Karst Area
or
theory and practice of geotechnical engineering in Guilin Karst Area
formula (1.32) shows that the shape of pressure arch is ellipse, The axis ratio of ellipse is < inlinemediaobject > < imageobject > < imagedata role = 3 "fileref =. / image / figure-0041-0001. JPG > < title > < / Title >
< / picdesc > < / imagedata > < / imageobject > < / inlinemediaobject >, and the center is (0, < inlinemediaobject > < imageobject > < imagedata role = "3" fileref =. / image / figure-0041-0002. JPG "> < title > < / Title >
< / picdesc > < / imagedata > < / imageobject > < / inlinemediaobject >)
1.8.2 pressure arch height
in order to determine the arch height, the equilibrium condition of the force on the left half of the arch is taken as follows:
(1) r is the horizontal left supporting force of the right half of the arch on the arc length, along the tangent direction of o point
(2) V is the vertical upward supporting force of the arch foot (n point) on the left half of on (3) H is the horizontal right thrust of the arch foot (n point) on the left half of on4 σ< sub>v＝ γ· Z is the vertical downward compressive stress of overlying rock and soil directly above the arc om to the left half of on, that is, the self weight stress of overlying rock and soil (ignoring the change of downward height along the Y axis)
5 σ< sub>h＝ λ·γ· Z is the lateral pressure of rock and soil, and the direction is horizontal to the right (ignoring the change of the downward height along the y-axis)
When the on section of the left half of the arch is in equilibrium, there are:
theory and practice of geotechnical engineering in Guilin Karst Area
when the arch is in limit equilibrium, there are:
H = V · f < sub > k < / sub > (1.36)
where: F < sub > k < / sub > - consolidation coefficient of rock and soil,
< / picdesc > < / imagedata > < / imageobject > < / inlinemediaobject >
R < sub > C < / sub > -- Ultimate uniaxial compressive strength of rock mass
According to equation (1.33), equation (1.34), equation (1.35) and equation (1.36), the expression of pressure arch height h < sub > U < / sub > in limit state is obtained as follows:
theory and practice of geotechnical engineering in Guilin Karst area λ Is the lateral pressure coefficient, i.e σ< sub>h/ σ< sub>v
Therefore, the recommended pressure arch height h in engineering practice should be:
theory and practice of geotechnical engineering in Guilin Karst Area
where k is the safety factor; Generally, it can be 2-3
If k = 2, the expression (1.38) becomes < inlinemediaobject > < imageobject > < imagedata role = "3" fileref =. / image / figure-0041-0008. JPG > < title > < / Title >< / picdesc > < / imagedata > < / imageobject > < / inlinemediaobject >, Compared with the calculation formula of Proctor's pressure arch < inlinemediaobject > < imageobject > < imagedata role = "3" fileref =. / image / figure-0041-0009. JPG > < title > < / Title >
< / picdesc > < / imagedata > < / imageobject > < / inlinemediaobject >, it can be found that the lateral pressure is considered σ< Under the influence of sub > H < / sub >, the pressure arch height h is smaller than that of Proctor
1.8.3 example
the red clay foundation of resial slope in Guilin Karst area is covered near the surface of limestone (d < sub > 3 < / sub > R) of Upper Devonian Rongxian formation, and there is often a layer of soft and flow plastic clay, in which soil caves are often developed. It is assumed that the diameter of the soil cavity is 1.0 m, the cohesion c of the soft flow plastic red clay is 20 kPa, and the internal friction angle is 1.0 M φ＝ ten °， Lateral pressure coefficient λ= 6, soil weight γ＝ 18 kN/m 3
substituting the relevant parameters into equation (1.38), ignoring the effect of cohesion, k = 2.5. In order to make the foundation stable, the minimum thickness of the roof clay is: H < sub > 1 < / sub > = 1.29 M
it's better to see the quality requirements of your collection files. Generally speaking, if it's civil level, the general collection card is OK. If it's high-end, it's better to use FPGA