Modification of Block Theory and its Application by Fu Helin, Li Liang, Liu Baochen, Civil Engineering College, Changsha Railway University, China K-H. Lux Clausthal Technical University, Clausthal, Germany |

Abstract

Among the slope slide problems in rock engineering, joints and cracks in rock-mass influence most of the slide problems. The joints and cracks deeply concern with sliding plane. The study of joints and cracks becomes main content of rock mechanics. The joints and cracks formed in long geological history, and became somewhat trench, somewhat scale, somewhat shape, which influencing the rock-mass character. Block Theory is one of the useful tools to analyze slope stability of crack rock mass, but it has some shortcomings. In this paper the Authors make modification to block theory and apply the modification in analyzing slope stability.

Keywords: Block theory, Modification, Application, In situ

Rock Mass Structural Analysis by Block Theory

Hard and semi-hard rock masses are cut into different inlaying blocks. In natural state, these blocks are in stable condition. When the rock-mass is excavated, some blocks will develop sliding according to structural planes, and then a chain reaction may occur, which may lead the whole slope to collapse. The block theory is a new method to analyze rock-mass stability with Graph Theory, Set Theory, geometry and Vector algebra. The block theory has first been put forward by Genhua and Goodman (Refs. 3 and 4). The basic assumptions of the theory are (1) The structural planes are planes through the analyzing rock-mass (i.e. joints and cracks, faults), (2) The structural bodies are rigid, (3) The reason for the rock mass collapse is that the structural blocks slide over each other at discontinuities. On the basis of these assumptions, the structural planes and approaching space plane are regarded as planes in space, the structural bodies as convex bodies, different forces as space vectors, and then determine the quantity of block types and their sliding probability with geometrical theory. The characteristics of the block theory are (1) The analysis of block theory is in three dimensions; (2) The aim of block theory is to find key blocks on excavated planes. If the rock-mass must be kept safe, before the key blocks exposure, the rock mass must be supported; (3) the block theory considers only the shear strength of structural planes and the deformation of the blocks themselves are not considered (4) The analysis result relies first on the mechanical parameters *C*, f, and then on shape of structural plane.

Types of blocks and rhombus awls

Types of blocks

Types of blocks can be divided into two types, i.e. limited blocks and unlimited blocks, limited blocks into stable and unstable blocks, the unstable blocks into safe blocks, potential sliding blocks and key blocks, shown in Figure 1.

*Block:* the rock-mass is cut by different structural planes and approaching planes.

*Unlimited block:* not an isolated block by different structural planes and approaching planes.

*Limited block:* an isolated block by different structural planes and approaching planes.

*Stable block:* the block is supported in each direction by nearby blocks. If the nearby blocks dont slide, the block takes place no movement.

*Safe block:* under the pressure of gravity and from nearby blocks, even the shear strength are null, the block also keep safe.

*Potential sliding block* under the pressure of gravity and from nearby blocks, when the shear strength decreases, the block may undergo a slide.

*Key block:* under the pressure of gravity and from nearby blocks, and then shear strength is not enough, when the supported measure is not taken, the block will slide.

**Figure 1.** Types of blocks in two dimensions

(a) unlimited block (b) reverse block (c) stable block

(d) potential slide block (e) key block

The basic approach of block theory is through geometrical analysis, to get rid of the unlimited and stable blocks, and then though movement analysis, to determine all potential slide blocks under the pressure of gravity and from nearby blocks, in the end, according to the mechanical characters, to determine the key block on excavated plane, and then to take supported measure.

Types of rhombus awls

*Rhombus awls:* if the structural and approaching planes make parallel movement to original point, then the planes will form a series of rhombus awls.

*Crack rhombus awl:* formed by structural planes in half space, signed as JP.

*Excavated rhombus awl:* formed by approaching and structural planes in half space, signed as EP.

*Space rhombus awl*: the rhombus awls out of the excavated rhombus awls, signed as SP.

*Block rhombus awl*: formed by one or more approaching and structural planes in half space, signed as BP.

Mathematical description of space blocks

In analytical geometry, the normal function of structural plane of rock-mass can be described as

| (1) |

The structural plane divides the whole space into two half space; a half space can be described as

(2) |

If a block is composed of *n* half spaces, its mathematical description is

(3) |

So the block with *n* planes can be described with *n* functions, where it is the intersection set of *n* half spaces.

In (1)-(3), each function includes a constant *D*, whose meaning is the position of planes, half spaces and blocks in space. If each plane moves parallel towards the origin (point), then the function will take place changes, i.e.

The description of the plane through the origin (point) is

(4) |

The description of the half space through the origin is

(5) |

If all boundary planes are through origin point, then the planes form rhombus awls, it can be designated with inequalities

(6) |

i.e. rhombus awls are intersection sets of boundary planes through the origin.

The principle of limited blocks

If a rock-mass is cut by structural planes, then it will form unlimited and limited blocks. The limited block type is the object of analysis. The principle of limited blocks is

_{}
---null set.

**Figure 2.** Unlimited blocks in two dimensions

**Figure 3.** Limited blocks in two dimensions

The mathematical determination rule of limited blocks: If (6) has only a solution (0,0,0), then rhombus awls of block are null set, the block is limited, otherwise, rhombus awls of block are not null set, the block is unlimited.

The principle of potential sliding block

If the blocks composed by structural and approaching planes are limited, and the crack blocks are unlimited, then the block may potentially slide. If the blocks composed by structural and approaching planes are limited, and the crack blocks are also limited, then the block is stable, i.e.

(7) |

Description method of block or rhombus awl

There are three approaches of description method of block or rhombus awl, i.e. direct method, number method and mark method.

Direct method

*U _{i}* is upper space of

Number method

0, 1, 2 and 3 are replaced the interconnection of structural planes, approaching planes, and blocks.

0 as block site in upper half space of this plane;

1 as block site in under half space of this plane;

2 as this plane is not a boundary plane of block;

3 as parallel planes of the block.

This method can disclose the possible connection of structural planes, approaching planes.

Mark method

For calculation by computer, +1 is assigned for upper half space, -1 for under space, _{} for parallel boundary planes of block.

Vector determination rule of potential sliding block

Determination of quantity of crack blocks

After Rock-mass being cut, it forms crack blocks, most of them are unlimited blocks, only a few are limited blocks. Total number of calculation formula of crack blocks, unlimited blocks and limited blocks under different condition are shown in table 1.

**Table 1.** Total number of calculation formula of crack blocks, unlimited blocks and limited blocks

The number of parallel planes | Total number of crack blocks | Total number of unlimited blocks | Total number of limited blocks | Condition |

No parallel structural plane | ||||

A parallel plane | ||||

A arbitrary parallel structural plane | ||||

Two determined parallel structural planes | 2 | |||

Two arbitrary parallel structural planes | ||||

M determined structural planes | 0 | |||

M arbitrary parallel structural planes | 0 | |||

Vector determination rule of crack blocks

The steps of vector determination rule of crack blocks are following :

There are *n* structural planes, the inkling angle and tendency of _{} are _{} and _{} so the upper normal vector of _{} is where

(8) |

(2) The intersection of different structural planes, i.e. verge vector _{} is

(9) |

(10) |

So the number of verge vectors is _{} i.e. _{}.

(3) To different possible intersection of structural planes, it is essential to determine ob _{} as a real verge of a rhombus awl. When the multiplication of verge vector and upper normal vector is bigger then null, then the vector it in the upper half space. Otherwise, the vector is in the lower half space.

For calculation, _{}is as trend parameter, i.e.

(11) |

in the formula, when

(12) |

So trend parameter includes:

The trend parameters form a matrix _{}. The elements of the matrix [*D*] are composed of the mark of a block _{},

(13) |

So the determination matrix of block is [T],

(14) |

According to (11) then

(15) |

The determination matrix of a BP [T] is shown in Table 2.

**Table 2.** Determination matrix of BP

in [T] | Determination of |

All 0or include +1 and -1 at the same time | _{} is not real verge of BP, the block may be slide |

All 0 and +1 |
_{} is a real verge of BP, the block is stable |

All 0 and -1 |
_{} is a real verge of BP, the block is stable |

_{} is upper normal vector of boundary plane _{ } is normal vector of boundary plane _{} to inner BP.

Determination of key block

The mechanical model of block

The forces applied on blocks are shown in Figure 4.

**Figure 4.** The forces applied on a block

Active combination load _{} is active combination of gravity of block, ground water pressure, strengthening load of bolts.

(2) Normal anti-load of sliding surface _{},

(16) |

_{} --- Normal anti-load of sliding surface _{} because the structural plane has no tension strength, _{}

_{}---Unit normal vector to block of structural plane _{}.

(3) Shear combination of friction of sliding plane _{},

| (17) |

_{}---Friction angle of structural plane _{}

_{}---Direction of movement of block.

The determination of the state of a block

Assumption: there exists a force

(18) |

According to (18), it can be determined that

If F > 0 then block is a key block

If F =0 then block is in critical state;

If F < 0 the block is in stable state.

Movement analysis of block

There are three movement forms of block, i.e. fall from rock-mass, slide according to single plane and slid according to two planes, shown in Figure 5.

1. Fall from rock-mass

If the movement direction of potential slide block _{} is not parallel to each structural plane, then according to (18), the complete and necessary condition is that the direction of movement of block is the same as the direction of active combination of loads i.e.

(19) |

2. Slide according to single plane

If the movement direction of potential slide block _{} is parallel to structural plane i, then according to (18), the complete and necessary condition is

, and | (20) |

_{} is reflection of _{} in plane *i*.

3. Slide according to two planes

If the movement direction of potential slide block _{} is parallel to structural plane *i* and *j* at same time then according to (18), the complete and necessary condition is

(21) |

and

(22) |

_{} and _{} are reflections of _{} in plane *i* and plane *j*.

Application of the block theory

to the analysis of slope stability

Collapse types for a rock-mass slope

Collapse types of rock-mass slope are shown in Figures 6 through 9.

Shear slide (Figures 6 and 7)

Fall collapse (Figure 8)

Fractal collapse (Figure 9)

In this paper, only collapse of crack slope is considered.

Analysis of the shear collapse mode of a cracked rock-mass

Slide according to Plane

The Plane collapse mode includes brief plane mode, step mode and two slide blocks etc., shown in Table 3.

**Figure 5.** Movement of a block

**Figure 6.** Sliding of a block

**Figure 7.** Complex sliding of a block

**Figure 8.** Fall collapse of a block

**Figure 9.** Fractal collapse of block

**Table 3.** Shear collapse mode of cracks in a rock-mass

Collapse mode | Collapse figure | Character | |

Sliding, according to plane | The direction, inkling are similar to the slope, the tendency angle is smaller than slope angle, but bigger than friction angle of structural plane | A slide plane and a slide block | |

A slide plane and a open crack | |||

some slide planes and some horizontal joints | |||

A main slide plane and an active and passive pair of slide blocks | |||

Collapse mode of no-open-crack

The collapse mode of without open crack is shown in Figure 10 its stability factor _{} is,

(23) |

_{}---natural density of rock-mass, kN/cm^{3}

_{}---friction angle of structural plane (_{})

C--- cohesion of structural plane, kPa

**Figure 10.** Collapse mode of no-open-crack

Collapse mode of with open crack on the top of slope

The collapse mode of with open crack on top of slope is shown in Figure 11, its stability factor _{} is,

(24) |

(25) |

**Figure 11.** Collapse mode of with-open-crack on the top of slope

(1) Collapse of step mode

Collapse of step mode is shown in Figure 12. Under pressure of outer loads and ground water, the slope stability factor is:

(26) |

W---weight of slide block, (kN)

_{}---tendency angle of main joint, (_{})

_{}---collapse angle, (_{})

U-float load on slide plane, (kPa)

V---water pressure in open crack, (kPa)

E---horizontal coefficient of earthquake

T---outer load

_{}---tendency angle of outer angle, (_{})

_{}---average friction angle of main joint, (_{})

_{}---Total length of main joint along step curve. (m)

_{}---cohesion of main joint (kPa)

**Figure 12.** Collapse of step mode

Collapse modes of two-slide-blocks are:

Composed of three joints, there are many the possible two slide blocks, shown in Figure 13 is a typical collapse mode. Its collapsing condition is _{} > _{}, _{} > _{}, _{} > _{}, _{} < 90_{}, the joint 2 must exposure in bottom or in slope.

On the basis of limit balance theory, the stability factor of passive slide block is:

(27) |

_{}---tendency angle of structural plane 2 _{}

_{}---friction angle of structural plane 2 _{}

_{}---Total cohesion structural plane 2 (kPa)

_{}---Anti-pressure structural plane 2 (kPa)

**Figure13.** Collapse mode of two slide blocks

Collapse of quadrangle mode

As shown in Figure 14, inkling of two structural planes of quadrangle body are anti inkling of intersection is similar to inkling of slope, the inkling angle of intersection is smaller than slope angle, but bigger than combining friction angle.

1. Determination of intersection of structural plane A - B

(*a*) Inkling of intersection A - B, _{},

(28) |

_{} - _{}---are inkling and tendency angle of structural plane A

_{} - _{}--- are inkling and tendency angle of structural plane B

(*b*) Inkling angle of intersection A - B, _{}

_{}

(*c*) Included angle of intersection A - B, _{},

(29) |

2. Hydraulic pressure

There are many types of hydraulic pressure, shown in Fig 15 is a type of hydraulic pressure after rain.

(30) |

3. The forces applied on quadrangle body

The forces applied on quadrangle body include:

W---weight of quadrangle body (KN)

_{}---effective normal anti pressure of structural plane A

_{}--- effective normal anti pressure of structural plane B

_{}---hydraulic pressure of structural plane A

_{}--- hydraulic pressure of structural plane B

S---potential slide load in the direction of sliding line

Stability factor of slope is:

(31) |

**Figure 14.** Inkling and Tendency angle of Combination

**Figure 15.** Hydraulic pressure intersection of quadrangle body

**Figure 16.** Forces applied on quadrangle body

The modification of block theory

Shown in above, the assumptions in block theory is, the joints are unlimited long, this doesnt correspond the reality. In situ engineer, the disclosed joints are limited long. Because of the limited length of joints, some of blocks are not really cut from the mother rock-mass, they can not form crack bodies, some of limited blocks by block theory will become unlimited blocks, in situ engineer, the number of key blocks will be reduced. Otherwise, the key blocks from parallel structural planes, though, their volumes are different, the stability factors are same by block theory. Actually, while excavated, some blocks will slide or fall, but some in stable state. These phenomenon demonstrate that the block theory has these shortcomings. In order to do correct analysis, it is necessary to do some modification to block theory.

To joints of rock-mass, much research work has been made.

On the basis of the research results of Chowdhury one can conclude:

(1) The joint traces distribute in the rule of negative index,

(2) The positions of joints are equal to the position of verges of blocks.

The joint trace distribute in the rule of negative index, its distributing density function:

(32) |

where _{} is the end density of joint trace, reversing value of length of trace.

Its cumulative probability is:

(33) |

For example, for a three-rhombus awl, its sliding probability is analyzed.

If a rhombus awl is composed of three verges, the length of a verge II is a so the length of the structural plane that including verge II must longer than a otherwise it will not form a rhombus awl. So the rhombus awl exposure probability is the probability of each structural plane bigger than a, and exposure at same time. In the same way, the probability of the slid key block is multiplier of the probability of each structural plane bigger than a.

From (32), is accumulation of probability that is probability of length smaller than . If probability of length bigger than is then

(34) |

To three joints, their bottom densities are the probability of a three joints rhombus awl, i.e. the probability of three joints bigger than at the same time is,

(35) |

From (35), the probability of a three-vertex rhombus awl can be calculated.

(35) can be inferred to four or more verges rhombus awls, if bottom density of each joint are _{} so the probability of a rhombus awl that composed of *n* structural planes (each verge bigger than _{}) sliding probability is

(36) |

From (36) it can be obtained (1) the longer the structural plane, i.e. the smaller is _{}, the bigger the probability _{}of the key block slide; (2) if _{} is unchanged, i.e. the length of structural plane unchanged, the smaller the length of _{} the bigger the probability _{} of the key block slide, i.e. the smaller the volume of key block, the bigger the probability _{} of the key block slide; and (3) The fewer the number of structural planes to a block, the bigger the probability of the key block slide.

Application of modified block theory in an *in-situ* case

Xiangqian Railway is a main railway in China. Its geologic condition is very complicated, especially in Dizhuang area. When it rains, a slope slide normally takes place. It often hinders traffic, leads to great loss to our economy. In order to solve the slope stability problem, first a geological survey has been done. On the basis of geological survey results, the authors modified block theory to analyze the slope stability. The geological survey results are shown in table 4, 5, 6 and Figure 17. The geological survey content includes the area geological survey, the state of structural plane, the condition of structural plane, etc.

**Table 4.** Geological survey result in north slope in k303+600

Number | type | Inkling angle |
Tendency angle |
Width of joints (cm) |
Roughness | fillings |

fault | 296 | 40 | 2-5 | rough | cohesive soil | |

fault | 252 | 40 | 2-5 | rough | cohesive soil | |

joint | 34 | 48 | 0 | rough | ||

joint | 330 | 70 | 0 | |||

The rock-mass of slope is red sand stone, the inkling angle of slope plane is _{}, the inkling angle of slope top is _{}

**Table 5.** Geological survey result in north slope in k303+800

Number | type | Inkling angle |
Tendency angle |
width of joints (cm) |
Roughness | fillings |

fault | 96 | 75 | 2-5 | rough | cohesive soil | |

fault | 278 | 40 | 2-5 | rough | cohesive soil | |

joint | 134 | 48 | 0 | rough | ||

joint | 330 | 70 | 0 | |||

The rock-mass of slope is red sand stone, the inkling angle of slope plane is _{}, the inkling angle of slope top is _{}

**Table 6.** Geological survey result in north slope in k303+850

Number | type | Inkling angle | Tendency angle |
width of joints (cm) |
Roughness | fillings |

fault | 130 | 40 | 2-5 | rough | cohesive soil | |

fault | 252 | 40 | 2-5 | rough | cohesive soil | |

joint | 222 | 75 | 0 | rough | ||

joint | 270 | 48 | 0 | |||

joint | 50 | 70 | ||||

The rock-mass of slope is red sandstone, the inkling angle of slope plane is _{}, the inkling angle of slope top is _{}

**Figure 17.** Geological survey result of north slope in

k303+600, k303+800, k303+850

_{} While the blocks composed by structures are too many, here only the joints in north slope in k303+600 are made detailed analysis, others are omitted. The joints consisting blocks in north slope in k303+600 are _{}, _{}, _{} and slope plane. Here _{} as 1, _{} as 2, _{} as 3, slope plane as 4.

According to formula (8), we obtain

_{} = (0.415, 0.616, 0.669

_{} = (-0.577, 0.281, 0.766)

_{} = (-0.469, 0.813, 0.342)

_{} = (0.606, 0.673,0.422)

So the numbers of total verges is _{}

_{} = (0.283, -0.703, 0.472)

_{} = (-0.333, -0.455, 0.626)

_{} = (0.710, 0.230, -0.652)

_{} = (-0.526, -0.162, -0.337)

_{} = (0.634, 0.707, 0.218)

_{} = (0.573, 0.405, -0.177)

The matrix of trend parameter of structural planes and space planes is

_{}

To JP0011, its _{} = [1, 1, -1, -1], according to formula (13), we obtain

_{}, so the determination matrix is

_{}

Since determination matrix [*T*] includes +1 and -1 at same time, _{}, and also _{}, so JP001 is a potential slide block.

According to formula (31), calculate the stability factor of this potential slide block (in water saturated state), _{}. The appearing lengths of verges on the slope are: _{} so _{} the sliding probability is _{}.

With the same method, we obtain the determination matrix of JP0010 [T],

_{}

While in array 1 and 4, the values are bigger than 0 or equal 0 at same time, the block is unlimited, the block is stable.

On the basis of the above calculation, all blocks (in saturated condition) stability and sliding probability are calculated, the analysis results shown in Tables 7 and 8.

**Table 7.** All blocks (in saturated condition) stability

Position | Block | Slide type | Slide plane | Stability factor | ||

K303+600 | inclined four planes body | 1.17 | ||||

K303+800 | Inclined six planes body | 1.08 | ||||

K303+850 | Inclined five planes body | 1.14 | ||||

**Table 8.** All blocks (in saturated condition) sliding probability

Position | Block | Slide type | Slide plane | Sliding probability |

K303+600 | inclined four planes body | 0.614 | ||

K303+800 | Inclined six planes body | 0.545 | ||

K303+850 | Inclined five planes body | 0.609 | ||

From the results in Tables 7 and 8, it is observed that:(1) According to block theory, unstable blocks exist in north slope in k303+600,k303+800 and k303+850, in saturated condition, the stability factors are all smaller 1.20, and according to modified block theory, all slide probabilities of unstable blocks are about 0.6, this means, the three unstable possible take place slide.

According to the values of stability factors, the slide array is k303+800,k303+850, k303+600, but according to the values of slide probabilities, the slide array is k303+600, k303+850, k303+800.

The real situation is that in the unstable block in north-slope in k303+600 a slide has taken place in July 1997; in the unstable block in north slope in k303+850 a slight slide has also taken place in July 1997, but till now the unstable block in north-slope in k303+800 no slide has yet taken place. This means that to analyze stability of cracked rock-mass, the modified block theory is better than the conventional block theory, as its analysis results better predict the reality.

Conclusion

From above analysis, the following useful conclusions can be obtained:

(1) To analyze the stability of a crack in a rock-mass, block theory is a useful tool to first determine the block types, i.e. unlimited block, limited block, stable block, safe block, potential sliding block, key block.

(2) Because of the limited length of joints, some of blocks are not really cut from the mother rock-mass, they can not form crack bodies, some of limited blocks by block theory will become unlimited blocks, in situ engineer, the number of key blocks will be reduced. Otherwise, the key blocks from parallel structural planes, though, their volumes are different, and the stability factors are same by block theory. Actually, while excavated, some blocks will slide or fall, but some in stable state. These phenomena demonstrate that the block theory has some shortcomings. It is necessary to do some modification to block theory.

On the basis of a case analysis results, the modified block theory better represents the reality compared with the old block theory.

This analysis method can also be applied to the analysis of tunnel stability.

REFERENCES

- Helin, Fu, Theoretical Analysis Model for Block Cracked Rock-mass Slope Stability and its Application, Dr. Degree Dissertation, South Central University, China, 2000.9
- Arr J., Fractal Character of Joint Surface Roughness in Welded Ruff at Yucca Mountain, Nevada, 30
^{th}US Symposium on Rock Mechanics, Balkema, Rotterdam, pp. 193-200, 1989 - Liu Jinghua, Block Theory and Its Application, Chin. Publ., pp. 31-49, 1988 (
*Ed.*-Block theory references can be found in, for example, http://www.pantechnica.com/resources/references.htm PanTechnica References.) - Goodman R.E
*Method of Geological Engineering in Discontinuous Rock*, West Publishing Company, St. Paul, New York, pp 77-86,1978

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