What are the assumptions of Rankine s theory of earth pressure? What is included in the scope of app

Assumptions: Assuming that any point in the foundation is in a stress state when the soil failure condition is satisfied, the back of the wall is smooth, and there is no shear stress on the horizontal plane and the vertical plane, that is, the two sides are the main stress action surface; All points in the soil are in a state of extreme equilibrium. When the soil is in the active state, the maximum principal stress action surface is the horizontal plane. When the soil is in a passive state, the maximum principal stress surface is a vertical plane.

Scope of application: Under the premise that there is no friction on the wall, calculate the stress state of the retaining wall in the elastic half-space body, and obtain the soil pressure according to the ultimate equilibrium condition of the soil.

Extended Materials. In practical engineering, Rankine's earth pressure theory can meet the accuracy required in engineering, and is applied in many engineering fields. Because Rankine's theory ignores the fact that the actual wall back is not smooth and there is friction, the calculated active earth pressure is larger, and the calculated passive earth pressure is smaller.

However, this theory has certain flaws: the hypothetical uniform forward movement of the retaining wall does not correspond to the reality; The soil wedge behind the wall is in the ultimate equilibrium state, which needs to be verified. This theory can only calculate the net force of the unearthed pressure, and its assumption that the linear distribution of the earth pressure has no theoretical basis. The hypothetical smooth surface has a large margin of error, especially in cohesive soils.

The basic assumptions of Rankine earth pressure in 1857 were:

1. The wall surface of the retaining wall is vertical and smooth;

2. The surface of the retaining wall is filled with soil surface level;

3. The wall will produce sufficient displacement and deformation under the action of pressure, so that the fill is in the ultimate equilibrium state.

The calculation of Rankine earth pressure is simple, the calculation result of active earth pressure is larger than the actual one, and the calculation result of passive earth pressure is smaller than the actual one, so the earth pressure calculated by Rankine earth pressure theory is more safe, and it is also an earth pressure calculation method stipulated by the national standard, which has been widely used in the calculation of earth pressure in various industries such as foundation pits, slopes, and foundations.

The difference lies in the basic assumptions of the theory: the structure of the retaining wall, the back of the wall is vertical and smooth, and the surface of the fill behind it is horizontal and extends indefinitely.

1. Rankine Earth Pressure Theory.

It is one of the classical theories of earth pressure based on the stress state of half-space and the ultimate equilibrium condition of the soil unit.

2. Coulomb earth pressure.

In 1776, Coulomb proposed the theory of calculating earth pressure based on the study of the closed and early static equilibrium conditions of the sliding soil wedge behind the retaining wall.

3. The principle of Coulomb earth pressure.

He assumed that the retaining wall was rigid and that the fill behind the wall was cohesive. When the back of the wall moves away from or moves to the fill, and the soil behind the wall reaches the ultimate equilibrium state, the fill behind the wall is in the form of a triangular sliding soil wedge, and slides downward through the wall heel along a certain slip plane in the back of the wall and the fill soil at the same time. According to the force equilibrium condition of the triangular soil wedge, the supporting reaction force of the retaining wall to the sliding soil wedge is obtained, so as to solve the total earth pressure on the back of the retaining wall.

4. The application range of coulomb earth pressure.

The fill behind the wall is an ideal bulk body, the sliding failure surface is an orange plane, and the sliding soil wedge is regarded as a rigid body.

Rankine's earth pressure theory and Coulomb's earth pressure theory calculate earth pressure according to different assumptions and different analytical methods, and the results are the same only in the simplest case (=0, δ=0, =0), otherwise different results are obtained. The application of Rankine earth pressure theory to stress states in half-space and the concept of limit equilibrium theory is relatively clear, the formula is simple, easy to remember, and can be directly calculated by this formula for both cohesive and non-cohesive soils, so it is widely used in engineering. However, in order to make the stress state behind the wall conform to the stress state of half-space, it must be assumed that the back of the wall is upright and smooth, and the fill behind the wall is horizontal, so that the application range is limited, and because the theory ignores the influence of friction between the back of the wall and the fill, the calculated active earth pressure is large, and the calculated passive earth pressure is small.

According to the static equilibrium condition of the sliding soil wedge behind the wall, the earth pressure calculation formula is derived, which can be used for the case of the inclination of the wall back and the filling surface, but because the theory assumes that the fill is cohesive soil, the original formula of Coulomb theory cannot be used to directly calculate the earth pressure of cohesive soil.

Coulomb's theory assumes that when the backfill fails, the fracture surface is a plane, when in fact it is a curved surface. Experiments show that when calculating the active earth pressure, the fracture surface is close to a plane only when the slope of the wall back is not large and the friction angle between the wall back and the fill is small, so the calculation results are different from those calculated by the curve sliding surface. Under normal circumstances, this deviation is 2% 10% when calculating the active earth pressure, which can be considered to have met the accuracy required by the actual project, but when calculating the passive earth pressure, because the rupture surface is close to the logarithmic spiral, the calculation result has a large error, sometimes up to 2 3 times, or even larger.

1. The basis for proposing is different:

1) Rankine's earth pressure theory is a method for calculating earth pressure based on the ultimate equilibrium conditions of the soil by studying the stress state in the elastic half-space.

2) Coulomb earth pressure: Coulomb proposed the theory of calculating earth pressure in 1776 based on the study of the static equilibrium conditions of the sliding soil wedge behind the retaining wall.

2. The assumptions are different

1) Rankine earth pressureRankine assumes that any point in the foundation is in a stress state when the soil failure condition is satisfied, and in this case, the calculation formula of active earth pressure and passive earth pressure is derived. In his basic theoretical derivation, Rankine made the following assumptions: the wall is rigid, and the back of the wall is straight; The surface of the fill behind the wall is horizontal; The back of the wall is smooth, and there is no friction between the back of the wall and the fill.

2) Coulomb earth pressure theory: It is based on the study of the static equilibrium conditions of the sliding wedge behind the retaining wall, which assumes that the fill is a uniform sandy soil, and the sliding surface is through two sets of planes of the wall toe, one along the back of the wall, and the other is generated in the soil, and the sliding soil wedge between the two sets of planes is a rigid body.

According to the static equilibrium condition of the soil wedge, the earth pressure acting on the retaining wall is solved according to the plane problem. Coulomb theory has assumed that the filler behind the wall is non-cohesive soil, so for the case of cohesive soil, the Coulomb earth pressure theory cannot be directly applied to calculate the earth pressure, and methods such as the equivalent internal friction angle method or ** method need to be adopted to calculate the earth pressure of the retaining structure when cohesive soil.

3. The actual application situation is different

1) Rankine earth pressure: In practical engineering, because they can meet the accuracy required by engineering, they are used in many engineering fields. Because Rankine's theory ignores the fact that the actual wall back is not smooth and there is friction, the calculated active earth pressure is larger, and the calculated vertical moving earth pressure is smaller.

2) Coulomb earth pressure: A large number of laboratory experiments and field observation data show that the active earth pressure calculated by Coulomb theory is very close to the measured results, but the error between the passive earth pressure and the measured value is large.

What Rankine earth pressure and Coulomb earth pressure have in common: both are used to calculate earth pressure.

What they have in common: Both make use of the Mohr-Coulomb strength theory.

Differences: 1. Different properties: In 1776, Coulomb proposed the theory of calculating earth pressure based on the study of the static equilibrium conditions of the sliding soil wedge behind the retaining wall. Rankine's earth pressure theory is used to calculate the active or passive earth pressure of the soil on a retaining wall.

2. Different characteristics: Rankine earth pressure theoryWhen the soil is in the active state, the maximum principal stress action surface is the horizontal plane. Coulomb earth pressure theory is in the form of a triangular sliding soil wedge, which slides downward through the wall heel along a certain slip plane in the back of the wall and the filled soil.

3. The release time is different: Rankine's classical theory of earth pressure proposed in 1857. In 1776, the French scholar Coulomb proposed the theory of Coulomb earth pressure according to the force equilibrium condition when the soil wedge behind the wall was in the ultimate equilibrium state.

1. Rankine's theory of earth pressure:

1. Advantages: The basic assumption of Rankine's earth pressure theory is that the retaining wall structure has a vertical and smooth back of the wall, and the surface of the fill is horizontal and infinitely extended.

2. Disadvantages: It is considered that the earth pressure is parabolic along the height of the wall, and then the active earth pressure distribution equation is solved.

2. Coulomb earth pressure theory:

1. Advantages: The basic assumption of the Coulomb earth pressure theory is that the retaining wall and the sliding soil body are regarded as rigid bodies, and the soil behind the wall is filled with non-cohesive sand, and when the wall body is offset forward or backward, the sliding soil body behind the wall is sliding along the back of the wall and a plane through the heel of the wall.

2. Disadvantages: The moment balance method is used to calculate the position of the resultant force application point, instead of direct integration.

Similarities: both require retaining walls.

The movement is to make the shear force behind the wall fill.

The shear strength (ultimate limit state) earth pressure is reached. All use Mohr Coulomb.

Strength theory. Differences: The Langken theory directly calculates the earth pressure at each point on the back of the wall according to the stress conditions of each point in the soil in equilibrium.

The back of the wall is required to be smooth, the surface of the fill is horizontal, and the calculation results are wild. The Coulomb theory is based on the static equilibrium condition in which the wedge type between the back of the wall and the sliding surface is in the ultimate equilibrium state to calculate the total earth pressure. The back of the wall can be tilted, the rough fill surface can be tilted, and the calculation results of the active pressure meet the requirements, while the passive pressure error is large.

Rankine's theory is to consider the failure of the fill behind the wall at every point to reach the limit state; Coulomb's theory considers the limit equilibrium of a rigid body of sliding soil.

Advantages of Rankine earth pressure theory: the formula is simple and easy to use; Disadvantages: It is not applicable to the slope of the wall and the slope of the fill surface behind the wall.

Advantages of Coulomb earth pressure theory: it is applicable to the inclination of the wall and the inclination of the filling surface behind the wall; Disadvantages: It is inconvenient to consider the situation of cohesive soil.

The basic assumptions of Rankine's earth pressure theory are that the back of the wall is upright and smooth, and the fill surface behind the wall is horizontal.

The basic assumptions of the Coulomb earth pressure theory are:

1. The filling behind the wall is uniform sandy soil;

2. Two sets of planes of the East China fracture surface through the corner of the wall;

3. The sliding soil is a wedge-shaped rigid body.