CN104346496B - Method for determining resultant force and resultant force application point of active earth under common conditions - Google Patents

Abstract

The invention discloses a method for determining the resultant force and the resultant force application point of active earth under common conditions. The method comprises the following steps: 1), determining the geometric elements of a retaining wall and an earth mass behind the wall; 2) determining the physical and mechanical parameters of the wall earth mass; 3) determining the coordinate x1 of the intersection point X of an earth mass sliding surface and a slope curve; 4) calculating the resultant force of the active earth pressure according to the formula Ea=[a1(a-n1)x＜2＞1/2+b1x1-cx1]/sin(alpha-delta); 5), calculating the pressure resultant force application point of the active earth pressure according to the formula which is defined as the specification. The method conforms with an actual active earth pressure calculating method, accurately and reliably determines the value and the application point position of the active earth pressure and has significant practical significance in the scientific and reasonable guidance of the design of the retaining wall.

Description

Generally determine active earth pressure make a concerted effort and pressure resultant force application point method

Technical field

The present invention relates to the calculating of active earth pressure, refers specifically to a kind of generally determination active earth pressure and makes a concerted effort and press The method of power point of resultant force, the design meter of this method rigid retaining wall suitable for the departments such as traffic, water conservancy, municipal administration, building Calculate, belong to Geotechnical Engineering field.

Background technology

The calculating of soil pressure on retaining wall is classical soil mechanics problem, generally using Rankine's earth pressure theory and coulomb soil Pressure theory is being analyzed.Coulomb earth pressure theories are that French scholar Coulomb discussed pole in 1773 in its mechanics paper Propose in application of the maximum and minimum rule in Architectural Mechanics, away from more than 200 years the present, due to its concept succinctly, user Just, still neutralize engineering circles in gravity retaining structures design now to be used widely.

At present multiple specifications calculate rigidity gear using Coulomb's earth pressure theory or using the Coulomb's earth pressure theory of amendment The soil pressure size of cob wall and point of resultant force, and make corresponding barricade design calculating.Coulomb's earth pressure theory assumes limit shape During state, the soil body forms a slip soil wedge body after wall, tries to achieve soil pressure by its static balance condition and makes a concerted effort.Coulomb active earth pressure And the Coulomb soil pressure computing formula of amendment is only applicable to that the soil body after wall is domatic for clinoplain, domatic upper effect has evenly load Rigid retaining wall situation.But in Practical Project, the often domatic out-of-flatness of the soil body after wall, it is domatic on load it is also not necessarily equal Cloth, such case has significant impact for the size and position of action point of active earth pressure, therefore, research one kind meets reality Earth pressure computation method, for the design for scientifically and rationally instructing retaining wall, have important practical significance.

The content of the invention

For the deficiency of above-mentioned Coulomb soil pressure computational methods, it is an object of the invention to provide one kind generally determines Active earth pressure make a concerted effort and pressure resultant force application point method, this coulomb of earth pressure computation method is suitable for barricade wall carries on the back and inclines Tiltedly, it is the stickiness soil body after wall, domatic fluctuating, and have the ordinary circumstances such as non-homogeneous overload, these ordinary circumstances are often that we are real Most situations are run in border.

The technical scheme is that what is be achieved in that：

Generally determine active earth pressure make a concerted effort and pressure resultant force application point method, carry out as follows,

(1) after retaining wall and wall soil body geometric element determination；Inclined angle alpha, the high H of wall, soil after wall are carried on the back including retaining walls The domatic fluctuating situation of body, is fitted with y=g (x)；

(2) determination of body of wall soil body physical and mechanical parameter；Including severe γ of the soil body after wall, internalfrictionangleφ and cohesiveness C, the angle of friction δ that retaining walls are carried on the back and banketed between the soil body；

(3) the sliding wedge body OAB with after wall in state of limit equilibrium is as object of study, it is assumed that the sliding wedge body OAB and wall top Intersection point be A, with wall heel intersection point be O, with domatic intersections of complex curve be B；Coordinate system, the X of intersection point A are set up by zero of O Axial coordinate is x₂；Determine the X-axis coordinate x of sliding wedge body OAB sliding surfaces and domatic intersections of complex curve B₁；X-axis coordinate x₁It is below equation Root：

F (x)=m₁x²+m₂x-m₃=0

Wherein：m₁=[aa₁(n₁+n₂)+a₁(1-n₁n₂)+γa]/2；

m₂=ab₁(n₁+n₂)+b₁(1-n₁n₂)+c(a-n₂)；

n₁=tan φ, n₂=1/tan (α-δ)；

A=g (x)/x,

b₁=σ (x)-a₁x；

Q (x) is the un-uniformly distributed for acting on domatic upper surface in upper several formulas, and σ (x) is to act on the normal direction on sliding surface Stress.

(4) calculate active earth pressure to make a concerted effort；Active earth pressure is calculated according to following formula to make a concerted effort E_a：

(5) active earth pressure point of resultant force is calculated；Active earth pressure point of resultant force is calculated according to following formula：

Wherein k=-tan α；

H in above formula₀For the vertical dimension that active earth pressure point of resultant force to wall is called in person, according to H₀Can determine that active soil pressure Power application point with joint efforts on retaining wall.

Nonlinear equation in step (3) is solved using secant method, is comprised the following steps that：

1) iterationses N and precision controlling amount eps are given；

2) if iterationses are more than N, terminate, otherwise execution step 3)；

3) two initial value x are selected₀, x₁, x₀Take the X-coordinate of y=tan (π/4+ φ/2) x and y=g (x) point of intersection, x₁Take y The X-coordinate of=tan (π/4- φ/2) x and y=g (x) point of intersection；

4) calculate

If 5) x_k-x₀| ＜ eps, then x₁=x_k, export x₁、a、a₁、λ、b₁, 6) EP (end of program) otherwise perform；

6) x is made₀=x₁, x₁=x_k, turn to step 2).

The present invention meets actual earth pressure computation method, and the size of active earth pressure and position of action point determine accurate It is really and reliable, for the design for scientifically and rationally instructing retaining wall, have important practical significance.

Description of the drawings

Fig. 1-earth pressure computation illustraton of model of the present invention.

Specific embodiment

Detailed process of the present invention is as follows：

(1) after retaining wall and wall soil body geometric element determination；

(2) determination of body of wall soil body physical and mechanical parameter；

(3) X-coordinate x at soil mass sliding surface and domatic intersections of complex curve is determined₁；

(4) calculate active earth pressure to make a concerted effort；

(5) active earth pressure point of resultant force is calculated；

Geometric element in step (1) includes that retaining walls carry on the back inclined angle alpha, and y=g is used in the high H of wall, the domatic fluctuating of the soil body after wall X () is fitted, see Fig. 1.Soil body sliding surface is represented with curve y=s (x) under state of limit equilibrium in figure, and q (x) is to act on domatic The un-uniformly distributed on surface, E_aTo act on the active earth pressure of wall back, σ (x), τ (x) are respectively and act on sliding surface Normal direction, tangential stress.

Pass through sampling and laboratory facilities in step (2) or survey data in detail, determine severe γ of the soil body after wall, internal friction Angle φ and cohesiveness c, the angle of friction δ that retaining walls are carried on the back and banketed between the soil body.

X-coordinate x at soil mass sliding surface and domatic intersections of complex curve in step (3)₁It is the root of below equation：

F (x)=m₁x²+m₂x-m₃=0

Wherein：m₁=[aa₁(n₁+n₂)+a₁(1-n₁n₂)+γa]/2

m₂=ab₁(n₁+n₂)+b₁(1-n₁n₂)+c(a-n₂)

n₁=tan φ, n₂=1/tan (α-δ)

A=g (x)/x,

b₁=σ (x)-a₁x

Nonlinear equation in step (3) is solved using secant method, is comprised the following steps that：

1) iterationses N and precision controlling amount eps are given；

2) if iterationses are more than N, terminate, otherwise perform 3)

3) two initial value x are selected₀, x₁。x₀Take the X-coordinate of y=tan (π/4+ φ/2) x and y=g (x) point of intersection, x₁Take y The X-coordinate of=tan (π/4- φ/2) x and y=g (x) point of intersection；

4) calculate

If 5) | x_k-x₀| ＜ eps, then x₁=x_k, export x₁、a、a₁、λ、b₁, 6) EP (end of program) otherwise perform；

6) x is made₀=x₁, x₁=x_k, turn to 2)

Step (4) calculates active earth pressure and makes a concerted effort according to following formula：

Step (5) calculates active earth pressure point of resultant force according to following formula：

H in above formula₀For the vertical dimension that soil pressure point of force application to wall is called in person.

Step (3), (4), (5) theoretical derivation it is as follows：

The sliding wedge body OAB after wall in state of limit equilibrium is taken as object of study, according to the equilibrium equation of power:

Obtained by ∑ X=0：

Obtained by ∑ Y=0：

S '=ds/dx, x in upper two formula₁For the X-coordinate of point B, x₂For the X-coordinate of point A, x₂=-H/tan α.It is located on sliding surface Normal stress σ (x) and tangential stress τ (x) obey Mohr-Coulomb failure criteria：

τ=c+ σ tan φ (3)

Investigate (1), (2), (3) formula, active earth pressure E_aFunctional extreme value for independent variable function σ (x) and sliding surface s (x) is asked Topic.

Functional is obtained by formula (1)

Wherein F₀=s ' σ-n₁σ-c, n₁=tan φ

Constraints is obtained by formula (2)：

Wherein F₁=n₂(s′σ-n₁σ-c)+s′σn₁+σ+γs-γg+s′c-q,K=-tan α factors (5) equation the right is certain value, and above-mentioned functional extreme value is isoperimetrical figures.

According to Euler's theorem, with method of Lagrange multipliers following functional J is constructed^*, convert above-mentioned condition extreme-value problem For unconfined functional extreme value problem：

F=F₀+λF₁=s ' σ-n₁σ-c+λ[n₂(s′σ-n₁σ-c)+s′σn₁+σ+γs-γg+s′c-q]

F is auxiliary function, and λ is Lagrange multiplier.According to the essential condition that isoperimetrical figures extreme value is present, sliding surface equation y =s (x) and normal stress σ (x) along sliding surface distribution must are fulfilled for the horizontal stroke at Euler's differential equation, boundary condition and Moving Boundary The condition of cutting：

(1) the Euler differential equations of auxiliary function F：

(2) integral constraint equation：Same formula (5)

(3) boundary condition：

Fixed boundary condition：S (0)=0 (9)

Moving Boundary condition：s(x₁)=g (x₁) (10)

x₁For B point X-coordinate

(4) transversality condition at Moving Boundary：

By formula (7), can obtain：

Above formula is because of λ, n₁、n₂Definite value is, therefore it is a inclined-planes that sliding surface is a slope.Consider boundary condition, then sliding surface equation is：

Y=s (x)=ax (13)

Wherein a will meet Moving Boundary condition in addition to meeting formula (12), also, i.e.,

A=g (x₁)/x₁ (14)

By formula (8), can obtain：

Same above formula is also certain value, and the normal stress σ along sliding surface is linearly distributed, i.e.,：

σ=a₁x+b₁ (16)

Can be obtained by formula (12)：

Substituting the above to formula (15) can obtain：

Can be obtained by transversality condition formula (11)：

Substitute the above to formula (16)：

b₁=σ (x₁)-a₁x₁ (20)

During above (14)～(20) are various, as long as determining unknown number x₁, remaining each parameter can determine, and x₁Depending on product Divide constraint equation (5).

From the foregoing discussion, it should be apparent that integral constraint conditional (5) is with a unknown quantity x₁Equation, order：

Launch：

Wherein

m₁=[aa₁(n₁+n₂)+a₁(1-n₁n₂)+γa]/2

m₂=ab₁(n₁+n₂)+b₁(1-n₁n₂)+c(a-n₂)

Obtain x₁Afterwards, sliding surface function s (x) is obtained and and along the normal direction of sliding surface distribution by formula (14), formula (18) and formula (20) Stress σ (x), finally obtains active earth pressure and makes a concerted effort by formula (4).

After active earth pressure size is determined, active earth pressure point of resultant force position can be according to the power of slide mass OAB Square balance is obtained.

By ∑ M_O=0：

H in above formula₀For the vertical dimension that soil pressure point of force application to wall is called in person.

The present invention is further illustrated with reference to example：

In order to verify the reasonability and correctness of the present invention, it is high 4 meters to take wall, severe 18kN/m of banketing³, respectively to c=0 and Result of calculation in the case of c ≠ 0 is contrasted with above-mentioned improved coulomb formula result of calculation, is shown in Table 1.From the right of result of calculation Than as can be seen that as c=0, soil pressure distribution result is on all four with the result of calculation of Coulomb theory, when c ≠ 0, Result of calculation with improved coulomb formula result of calculation be also basically identical.But trail force load position is simultaneously not always acted on Wall it is high 1/3 at.

Consider the domatic fluctuating of the soil body after wall, g (x) adds a SIN function to simulate with an inclination oblique line, non-homogeneous overload q X () SIN function adds a constant to simulate, be shown in Table 2.Other specification is c=8, α=70^°, δ=15^°Can from result of calculation Go out, active earth pressure is made a concerted effort for domatic fluctuating and the inhomogeneities of domatic overload and the impact of application point is to ignore 's.

The context of methods of table 1 is contrasted with the result of calculation of Coulomb theory

Result of calculation under the domatic fluctuating of table 2 and non-homogeneous overload

It is last it should be noted that above example is only unrestricted to illustrate technical scheme, although Shen Ask someone to be described in detail the present invention with reference to preferred embodiment, it will be understood by those within the art that, to this Bright technical scheme is modified or equivalent, without departing from the objective and scope of the technical program, all should be covered In the middle of scope of the presently claimed invention.

Claims (1)

1. generally determine active earth pressure make a concerted effort and pressure resultant force application point method, it is characterised in that：By following step Suddenly carry out,

(1) after retaining wall and wall soil body geometric element determination；Inclined angle alpha, the high H of wall, soil body slope after wall are carried on the back including retaining walls Face fluctuating situation, is fitted with y=g (x)；

(2) determination of body of wall soil body physical and mechanical parameter；Including severe γ of the soil body after wall, internalfrictionangleφ and cohesiveness c, gear The angle of friction δ that cob wall wall is carried on the back and banketed between the soil body；

(3) the sliding wedge body OAB with after wall in state of limit equilibrium is as object of study, it is assumed that the friendship on the sliding wedge body OAB and wall top Point is A, is O with the intersection point of wall heel, is B with domatic intersections of complex curve；Coordinate system is set up by zero of O, the X-axis of intersection point A is sat It is designated as x₂；Determine the X-axis coordinate x of sliding wedge body OAB sliding surfaces and domatic intersections of complex curve B₁；X-axis coordinate x₁It is the root of below equation：

F (x)=m₁x²+m₂x-m₃=0

Wherein：m₁=[aa₁(n₁+n₂)+a₁(1-n₁n₂)+γa]/2；

m₂=ab₁(n₁+n₂)+b₁(1-n₁n₂)+c(a-n₂)；

$m_3={\∫}_{x_2}^{x_1}\[\mathrm{\γ}g\left(x\right)+q\left(x\right)\]dx-{\∫}_{x_2}^0\mathrm{\γ}kxdx;$

n₁=tan φ, n₂=1/tan (α-δ)；

A=g (x)/x,

b₁=σ (x)-a₁x；

$\mathrm{\σ}\left(x\right)=\frac{c+\mathrm{\λ}c-{\mathrm{\λcg}}^{\′}\left(x\right)+\mathrm{\λ}q\left(x\right)}{\mathrm{\λ}(1-n_1{n}_2)+g^{\′}\left(x\right)(1+{\mathrm{\λn}}_1+{\mathrm{\λn}}_2)-n_1};$

$\mathrm{\λ}=\frac{n_1-a}{1+n_1+n_2-n_1{n}_2};$

Q (x) is the un-uniformly distributed for acting on domatic upper surface in upper several formulas, and σ (x) is that the normal direction acted on sliding surface should Power；

(4) calculate active earth pressure to make a concerted effort；Active earth pressure is calculated according to following formula to make a concerted effort E_a：

$E_a=\[a_1(a-n_1)x_1^2/2+b_1{x}_1-{\mathrm{cx}}_1\]/\sin (\mathrm{\α}-\mathrm{\δ})$

(5) active earth pressure point of resultant force is calculated；Active earth pressure point of resultant force is calculated according to following formula：

$H_0=\frac{{\∫}_0^{x_1}\[\mathrm{\σ}\left(x\right)\sqrt{1+a^2}\sqrt{x^2(1+a^2)}+{\mathrm{\γax}}^2-q\left(x\right)x\]dx+{\∫}_{x_2}^0{\mathrm{kx}}^{2}dx-{\∫}_{x_2}^{x_1}\mathrm{\γ}g\left(x\right)xdx}{E_a\cos \mathrm{\δ}/\sin \mathrm{\α}}$

Wherein k=-tan α；

H in above formula₀For the vertical dimension that active earth pressure point of resultant force to wall is called in person, according to H₀Can determine that active earth pressure is closed Application point of the power on retaining wall；

Nonlinear equation in step (3) is solved using secant method, is comprised the following steps that：

1) iterationses N and precision controlling amount eps are given；

2) if iterationses are more than N, terminate, otherwise execution step 3)；

3) two initial value x are selected₀, x₁, x₀Take the X-coordinate of y=tan (π/4+ φ/2) x and y=g (x) point of intersection, x₁Take y=tan The X-coordinate of (π/4- φ/2) x and y=g (x) point of intersection；

4) calculate

If 5) | x_k-x₀| ＜ eps, then x₁=x_k, export x₁、a、a₁、λ、b₁, 6) EP (end of program) otherwise perform；

6) x is made₀=x₁, x₁=x_k, turn to step 2).