CN104346496A - 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 that active earth pressure is made a concerted effort and the method for pressure resultant force application point

Technical field

The present invention relates to the calculating of active earth pressure, concrete finger is a kind of generally determines that active earth pressure is made a concerted effort and the method for pressure resultant force application point, this method is applicable to the designing and calculating of rigid retaining wall in the departments such as traffic, water conservancy, municipal administration, building, belongs to Geotechnical Engineering field.

Background technology

On retaining wall, the calculating of soil pressure is classical soil mechanics problem, usually adopts Rankine's earth pressure theory and Coulomb's earth pressure theory to analyze.To be French scholar Coulomb proposed in the application of the very big and minimum rule of its mechanics paper opinion in Architectural Mechanics in 1773 Coulomb earth pressure theory, apart from more than 200 year the present, because its concept is succinct, easy to use, be still used widely with engineering circles in gravity retaining structures design now.

Current multiple specification all adopts Coulomb's earth pressure theory or adopts the Coulomb's earth pressure theory revised to calculate soil pressure size and the point of resultant force of rigid retaining wall, and makes corresponding barricade designing and calculating.During Coulomb's earth pressure theory supposition ultimate limit state, after wall, the soil body forms a slip soil wedge body, tries to achieve soil pressure make a concerted effort by its static balance condition.It is clinoplane that the Coulomb soil pressure computing formula of coulomb active earth pressure and correction is only applicable to the soil body after wall domatic, and domatic upper effect has the rigid retaining wall situation of evenly load.But in Practical Project, the often domatic out-of-flatness of the soil body after wall, load on domatic is also not necessarily uniform, this situation has significant impact for the size of active earth pressure and position of action point, therefore, study a kind of realistic earth pressure computation method, for the design instructing retaining wall scientifically and rationally, have important practical significance.

Summary of the invention

For the deficiency of above-mentioned Coulomb soil pressure computing method, the object of this invention is to provide and a kind ofly generally determine that active earth pressure is made a concerted effort and the method for pressure resultant force application point, this coulomb of earth pressure computation method is suitable for the barricade wall back of the body and tilts, it is the stickiness soil body after wall, domatic fluctuating, and having the generalized cases such as non-homogeneous overload, these generalized cases run into maximum situations in our reality often.

Technical scheme of the present invention is achieved in that

Generally determine that active earth pressure is made a concerted effort and the method for pressure resultant force application point, carry out as follows,

(1) determination of soil body geometric element after retaining wall and wall; Comprise retaining walls back of the body inclined angle alpha, wall height H, the domatic fluctuating situation of the soil body after wall, with y=g (x) matching;

(2) determination of body of wall soil body physical and mechanical parameter; Comprise the severe γ of the soil body after wall, internalfrictionangleφ and cohesion c, the retaining walls back of the body and the angle of friction δ banketed between the soil body;

(3) to be in the sliding wedge body OAB of state of limit equilibrium after wall for research object, suppose that the intersection point on this sliding wedge body OAB and wall top is A, the intersection point of calling in person with wall is O, is B with domatic intersections of complex curve; Be that true origin sets up coordinate system with O, the X-axis coordinate of intersection point A is x ₂; Determine the X-axis coordinate x of sliding wedge body OAB slipping plane and domatic intersections of complex curve B ₁; X-axis coordinate x ₁following equation root:

f(x)＝m ₁x ²+m ₂x-m ₃＝0

Wherein: m ₁=[aa ₁(n ₁+ n ₂)+a ₁(1-n ₁n ₂)+γ a]/2;

m ₂＝ab ₁(n ₁+n ₂)+b ₁(1-n ₁n ₂)+c(a-n ₂)；

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

n ₁＝tanφ，n ₂＝1/tan(α-δ)；

$a=g\left(x\right)/x,a_1=\mathrm{\&gamma;}\frac{n_1-a}{1+{n_1}^2};$

b ₁＝σ(x)-a ₁x；

$\mathrm{\&sigma;}\left(x\right)=\frac{c+\mathrm{\&lambda;c}-\mathrm{\&gamma;c}{g}^{\&prime;}\left(x\right)+\mathrm{\&lambda;q}\left(x\right)}{\mathrm{\&lambda;}(1-n_1{n}_2)+g^{\&prime;}\left(x\right)(1+{\mathrm{\&lambda;n}}_1+{\mathrm{\&lambda;n}}_2)-n_1};$

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

In upper a few formula, q (x) is for acting on the un-uniformly distributed of domatic upper surface, and σ (x) is for acting on the normal stress on slipping plane.

(4) calculate active earth pressure to make a concerted effort; Calculate active earth pressure according to following formula to make a concerted effort E _a:

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

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

$H_0=\frac{{\&Integral;}_0^{x_1}(\mathrm{\&sigma;}\sqrt{1+a^2}\sqrt{x^2(1+a^2)}+\mathrm{\&gamma;}{\mathrm{ax}}^2-\mathrm{qx})\mathrm{dx}+{\&Integral;}_{x_2}^0{\mathrm{kx}}^2\mathrm{dx}-{\&Integral;}_{x_2}^{x_1}\mathrm{\&gamma;gxdx}}{E_a\cos \mathrm{\&delta;}/\sin \mathrm{\&alpha;}}$

H in above formula ₀for the vertical range that active earth pressure point of resultant force is called in person to wall, according to H ₀the application point of active earth pressure with joint efforts on retaining wall can be determined.

Nonlinear equation in step (3) adopts secant method to solve, and concrete steps are as follows:

1) given iterations N and precision controlling amount eps;

2) if iterations is greater than N, then terminate, otherwise perform step 3);

3) selected two initial value x ₀, x ₁, x ₀get the X-coordinate of y=tan (π/4+ φ/2) x and y=g (x) point of intersection, x ₁get

The X-coordinate of y=tan (π/4-φ/2) x and y=g (x) point of intersection;

4) calculate $x_k=x_0-\frac{x_1-x_0}{f\left(x_1\right)-f\left(x_0\right)}f\left(x_1\right);$

5) if | x _k-x ₀| < eps, then x ₁=x _k, export x ₁, a, a ₁, λ, b ₁, EOP (end of program), otherwise perform 6);

6) x is made ₀=x ₁, x ₁=x _k, turn to step 2).

The earth pressure computation method that the present invention is realistic, the size of active earth pressure and position of action point are determined accurately and reliably, for the design instructing retaining wall scientifically and rationally, are had important practical significance.

Accompanying drawing explanation

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

Embodiment

Detailed process of the present invention is as follows:

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

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

(3) soil mass sliding surface and domatic intersections of complex curve place X-coordinate x is determined ₁;

(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) comprises retaining walls back of the body inclined angle alpha, and wall height H, after wall, domatic fluctuating y=g (x) matching of the soil body, is shown in Fig. 1.In figure, under state of limit equilibrium, soil body sliding surface curve y=s (x) represents, q (x) is for acting on the un-uniformly distributed of domatic upper surface, E _afor acting on the active earth pressure of wall back, σ (x), τ (x) are respectively the normal direction acted on slipping plane, tangential stress.

By sampling and laboratory facilities or survey data in detail in step (2), determine the severe γ of the soil body after wall, internalfrictionangleφ and cohesion c, the angle of friction δ that retaining walls is carried on the back and banketed between the soil body.

Soil mass sliding surface and domatic intersections of complex curve place X-coordinate x in step (3) ₁following equation root:

f(x)＝m ₁x ²+m ₂x-m ₃＝0

Wherein: m ₁=[aa ₁(n ₁+ n ₂)+a ₁(1-n ₁n ₂)+γ a]/2

m ₂＝ab ₁(n ₁+n ₂)+b ₁(1-n ₁n ₂)+c(a-n ₂)

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

n ₁＝tanφ，n ₂＝1/tan(α-δ)

$a=g\left(x\right)/x,a_1=\mathrm{\&gamma;}\frac{n_1-a}{1+{n_1}^2}$

b ₁＝σ(x)-a ₁x

$\mathrm{\&sigma;}\left(x\right)=\frac{c+\mathrm{\&lambda;c}-\mathrm{\&gamma;c}{g}^{\&prime;}\left(x\right)+\mathrm{\&lambda;q}\left(x\right)}{\mathrm{\&lambda;}(1-n_1{n}_2)+g^{\&prime;}\left(x\right)(1+{\mathrm{\&lambda;n}}_1+{\mathrm{\&lambda;n}}_2)-n_1}$

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

Nonlinear equation in step (3) adopts secant method to solve, and concrete steps are as follows:

1) given iterations N and precision controlling amount eps;

2) if iterations is greater than N, then terminate, otherwise perform 3)

3) selected two initial value x ₀, x ₁.X ₀get the X-coordinate of y=tan (π/4+ φ/2) x and y=g (x) point of intersection, x ₁get

The X-coordinate of y=tan (π/4-φ/2) x and y=g (x) point of intersection;

4) calculate $x_k=x_0-\frac{x_1-x_0}{f\left(x_1\right)-f\left(x_0\right)}f\left(x_1\right)$

5) if | x _k-x ₀| <eps, then x ₁=x _k, export x ₁, a, a ₁, λ, b ₁, EOP (end of program), otherwise perform 6);

6) x is made ₀=x ₁, x ₁=x _k, turn to 2)

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

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

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

$H_0=\frac{{\&Integral;}_0^{x_1}(\mathrm{\&sigma;}\sqrt{1+a^2}\sqrt{x^2(1+a^2)}+\mathrm{\&gamma;}{\mathrm{ax}}^2-\mathrm{qx})\mathrm{dx}+{\&Integral;}_{x_2}^0{\mathrm{kx}}^2\mathrm{dx}-{\&Integral;}_{x_2}^{x_1}\mathrm{\&gamma;gxdx}}{E_a\cos \mathrm{\&delta;}/\sin \mathrm{\&alpha;}}$

H in above formula ₀for the vertical range that soil pressure application point is called in person to wall.

The theory deduction of step (3), (4), (5) is as follows:

The sliding wedge body OAB being in state of limit equilibrium after getting wall as research object, according to equilibrium of forces equation:

Obtained by ∑ X=0:

$E_a\sin (\mathrm{\&alpha;}-\mathrm{\&delta;})+{\&Integral;}_0^{x_1}\mathrm{\&tau;dx}-{\&Integral;}_0^{x_1}\mathrm{\&sigma;}{s}^{\&prime;}\mathrm{dx}=0---\left(1\right)$

Obtained by ∑ Y=0:

$\begin{array}{c}{E}_a\cos (\mathrm{\&alpha;}-\mathrm{\&delta;})+{\&Integral;}_0^{x_1}\mathrm{\&tau;}{s}^{\&prime;}\mathrm{dx}+{\&Integral;}_0^{x_1}\mathrm{\&sigma;dx}-{\&Integral;}_0^{x_1}\mathrm{qdx}\\ -{\&Integral;}_{x_2}^0[\mathrm{\&gamma;}(g-\mathrm{kx})+q]\mathrm{dx}-{\&Integral;}_0^{x_1}\mathrm{\&gamma;}(g-s)\mathrm{dx}=0\end{array}---\left(2\right)$

S '=ds/dx, x in upper two formulas ₁for the X-coordinate of a B, x ₂for the X-coordinate of an A, x ₂=-H/tan α.Be located at normal stress σ (x) and tangential stress τ (x) on sliding surface and obey Mohr-Coulomb failure criteria namely:

τ＝c+σtanφ (3)

Investigate (1), (2), (3) formula, active earth pressure E _afor independent variable function σ (x) and the functional extreme value problem of sliding surface s (x).

Functional is obtained by formula (1) $J=E_a\sin (\mathrm{\&alpha;}-\mathrm{\&delta;})={\&Integral;}_0^{x_1}{F}_0\mathrm{dx}---\left(4\right)$

Wherein F ₀=s ' σ-n ₁σ-c, n ₁=tan φ

Constraint condition is obtained by formula (2): ${\&Integral;}_0^{x_1}{F}_1\mathrm{dx}={\&Integral;}_{x_2}^0[\mathrm{\&gamma;}(g-\mathrm{kx})+q]\mathrm{dx}=\mathrm{const}---\left(5\right)$

Wherein F ₁=n ₂(s ' σ-n ₁σ-c)+s ' σ n ₁+ σ+γ s-γ g+s ' c-q, n ₂=ctan (α-δ), k=-tan (α)

Be certain value on the right of factor (5) equation, above-mentioned functional extreme value is isoperimetrical figures.

According to theorem of Euler, the functional J * be constructed as follows with method of Lagrange multipliers, makes above-mentioned constrained extremal problem be converted into unconfined functional extreme value problem:

$J^{*}={\&Integral;}_0^{x_1}\mathrm{Fdx}---\left(6\right)$

F＝F ₀+λF ₁＝s′σ-n ₁σ-c+λ[n ₂(s′σ-n ₁σ-c)+s′σn ₁+σ+γs-γg+s′c-q]

F is auxiliary function, and λ is Lagrange multiplier.According to the necessary condition that isoperimetrical figures extreme value exists, sliding surface equation y=s (x) and normal stress σ (x) along sliding surface distribution must meet the transversality condition at Euler's differential equation, boundary condition and Moving Boundary place:

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

$\frac{\&PartialD;F}{\&PartialD;\mathrm{\&sigma;}}-\frac{d}{\mathrm{dx}}\left(\frac{\&PartialD;F}{\&PartialD;{\mathrm{\&sigma;}}^{\&prime;}}\right)=0---\left(7\right)$

$\frac{\&PartialD;F}{\&PartialD;s}-\frac{d}{\mathrm{dx}}\left(\frac{\&PartialD;F}{\&PartialD;s^{\&prime;}}\right)=0---\left(8\right)$

(2) integral constraint equation: cotype (5)

(3) boundary condition:

Fixed boundary condition: s (0)=0 (9)

Moving Boundary condition: s (x ₁)=g (x ₁) (10)

X ₁for B point X-coordinate

(4) transversality condition at Moving Boundary place:

$(F-s^{\&prime;}\frac{\&PartialD;F}{{\&PartialD;s}^{\&prime;}}+g^{\&prime;}\frac{\&PartialD;F}{{\&PartialD;s}^{\&prime;}}){|}_{x=x_1}=0---\left(11\right)$

By formula (7), can obtain:

$s^{\&prime;}=a=\frac{{\mathrm{\&lambda;n}}_1{n}_2+n_1-\mathrm{\&lambda;}}{1+{\mathrm{\&lambda;n}}_1+{\mathrm{\&lambda;n}}_2}---\left(12\right)$

Above formula is because of λ, n ₁, n ₂be definite value, therefore sliding surface is a slope is a inclined-plane.Consider boundary condition, then sliding surface equation is:

y＝s(x)＝ax (13)

Wherein a is except meeting formula (12), also will meet Moving Boundary condition, namely

a＝g(x ₁)/x ₁(14)

By formula (8), can obtain:

${\mathrm{\&sigma;}}^{\&prime;}=a_1=\frac{\mathrm{\&gamma;\&lambda;}}{1+{\mathrm{\&lambda;n}}_1+{\mathrm{\&lambda;n}}_2}---\left(5\right)$

Same above formula is also certain value, and the normal stress σ along sliding surface linearly distributes, that is:

σ＝a ₁x+b ₁(16)

Can be obtained by formula (12):

$\mathrm{\&lambda;}=\frac{n_1-a}{1+n_1+n_2-n_1{n}_2}---\left(17\right)$

Above formula is brought into formula (15) can obtain:

$a_1=\mathrm{\&gamma;}\frac{n_1-a}{1+{n_1}^2}---\left(18\right)$

Can be obtained by transversality condition formula (11):

$\mathrm{\&sigma;}\left(x_1\right)=\frac{c+\mathrm{\&lambda;c}-{\mathrm{\&lambda;cg}}^{\&prime;}\left(x_1\right)+\mathrm{\&lambda;q}\left(x_1\right)}{\mathrm{\&lambda;}(1-n_1{n}_2)+g^{\&prime;}\left(x_1\right)(1+{\mathrm{\&lambda;n}}_1+{\mathrm{\&lambda;n}}_2)-n_1}---\left(19\right)$

Above formula is brought into formula (16):

b ₁＝σ(x ₁)-a ₁x ₁(20)

Above (14) ~ (20) various in, as long as determine unknown number x ₁, all the other each parameters can be determined, and x ₁depend on integral constraint conditional (5).

From discussing above, integral constraint conditional (5) is with a unknown quantity x ₁equation, order:

$f\left(x_1\right)={\&Integral;}_0^{x_1}{F}_1\mathrm{dx}-{\&Integral;}_{x_2}^0[\mathrm{\&gamma;}(g-\mathrm{kx})+q]\mathrm{dx}=0$

Launch:

$G\left(x_1\right)=m_1{x}_1^2+m_2{x}_1-m_3=0---\left(21\right)$

Wherein

m ₁＝[aa ₁(n ₁+n ₂)+a ₁(1-n ₁n ₂)+γa]/2

m ₂＝ab ₁(n ₁+n ₂)+b ₁(1-n ₁n ₂)+c(a-n ₂)

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

Obtain x ₁after, by formula (14), formula (18) and formula (20) obtain sliding surface function s (x) and and along sliding surface distribute normal stress σ (x), finally by formula (4) obtain active earth pressure make a concerted effort.

After determining active earth pressure size, active earth pressure point of resultant force position can be obtained according to the equalising torque of slide mass OAB.

By ∑ M _o=0:

$H_0=[{\&Integral;}_0^{x_1}(\mathrm{\&sigma;}\sqrt{1+a^2}\sqrt{x^2(1+a^2}+\mathrm{\&gamma;}{\mathrm{ax}}^2-\mathrm{qx})\mathrm{dx}+{\&Integral;}_{x_2}^0{\mathrm{kx}}^2\mathrm{dx}-{\&Integral;}_{x_2}^{x_1}\mathrm{\&gamma;gxdx}]\sin \mathrm{\&alpha;}/E_a\cos \mathrm{\&delta;}---\left(22\right)$

H in above formula ₀for the vertical range that soil pressure application point is called in person to wall.

The present invention is further illustrated below in conjunction with example:

In order to verify rationality of the present invention and correctness, get wall high 4 meters, banket severe 18kN/m ³, respectively the coulomb formulae discovery result of the result of calculation in the situation of c=0 and c ≠ 0 and above-mentioned improvement is contrasted, in table 1.As can be seen from the contrast of result of calculation, as c=0, the result of calculation of soil pressure distribution result and Coulomb theory is on all four, when c ≠ 0, result of calculation be also basically identical by the coulomb formulae discovery result improved.But trail force load position based not always acts on 1/3 high place of wall.

The domatic fluctuating of the soil body after consideration wall, g (x) adds a sine function with an inclination oblique line and simulates, and non-homogeneous overload q (x) sine function adds a constant to simulate, in table 2.Other parameters be c=8, α=70 °, δ=15 ° as can be seen from result of calculation, the unevenness of domatic fluctuating and domatic overload is made a concerted effort on active earth pressure and the impact of application point can not be ignored.

The result of calculation of table 1 context of methods and Coulomb theory contrasts

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

Finally it should be noted that, above embodiment is only in order to illustrate technical scheme of the present invention and unrestricted, although applicant's reference preferred embodiment is to invention has been detailed description, those of ordinary skill in the art is to be understood that, technical scheme of the present invention is modified or equivalent replacement, only otherwise depart from aim and the scope of the technical program, all should be encompassed in the middle of right of the present invention.

Claims (2)

1. generally determine that active earth pressure is made a concerted effort and the method for pressure resultant force application point, it is characterized in that: carry out as follows,

(1) determination of soil body geometric element after retaining wall and wall; Comprise retaining walls back of the body inclined angle alpha, wall height H, the domatic fluctuating situation of the soil body after wall, with y=g (x) matching;

(2) determination of body of wall soil body physical and mechanical parameter; Comprise the severe γ of the soil body after wall, internalfrictionangleφ and cohesion c, the retaining walls back of the body and the angle of friction δ banketed between the soil body;

(3) to be in the sliding wedge body OAB of state of limit equilibrium after wall for research object, suppose that the intersection point on this sliding wedge body OAB and wall top is A, the intersection point of calling in person with wall is O, is B with domatic intersections of complex curve; Be that true origin sets up coordinate system with O, the X-axis coordinate of intersection point A is x ₂; Determine the X-axis coordinate x of sliding wedge body OAB slipping plane and domatic intersections of complex curve B ₁; X-axis coordinate x ₁following equation root:

f(x)＝m ₁x ²+m ₂x-m ₃＝0

Wherein: m ₁=[aa ₁(n ₁+ n ₂)+a ₁(1-n ₁n ₂)+γ a]/2;

m ₂＝ab ₁(n ₁+n ₂)+b ₁(1-n ₁n ₂)+c(a-n ₂)；

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

n ₁＝tanφ，n ₂＝1/tan(α-δ)；

$a=g\left(x\right)/x,a_1=\mathrm{\&gamma;}\frac{n_1-a}{1+{n_1}^2};$

b ₁＝σ(x)-a ₁x；

$\mathrm{\&sigma;}\left(x\right)=\frac{c+\mathrm{\&lambda;c}-\mathrm{\&lambda;c}{g}^{\&prime;}\left(x\right)+\mathrm{\&lambda;q}\left(x\right)}{\mathrm{\&lambda;}(1-n_1{n}_2)+g^{\&prime;}\left(x\right)(1+\mathrm{\&lambda;}{n}_1+\mathrm{\&lambda;}{n}_2)-n_1};$

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

In upper a few formula, q (x) is for acting on the un-uniformly distributed of domatic upper surface, and σ (x) is for acting on the normal stress on slipping plane;

(4) calculate active earth pressure to make a concerted effort; Calculate active earth pressure according to following formula to make a concerted effort E _a:

$E_a=[a_1(a-n_1)x_1^2/2+b_1{x}_1-c{x}_1]/\sin (\mathrm{\&alpha;}-\mathrm{\&delta;})$

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

$H_0=\frac{{\&Integral;}_0^{x_1}(\mathrm{\&sigma;}\sqrt{1+a^2}\sqrt{x^2(1+a^2)}+\mathrm{\&gamma;a}{x}^2-\mathrm{qx})\mathrm{dx}+{\&Integral;}_{x_2}^{0}k{x}^2\mathrm{dx}-{\&Integral;}_{x_2}^{x_1}\mathrm{\&gamma;gxdx}}{E_a\cos \mathrm{\&delta;}/\sin \mathrm{\&alpha;}}$

H in above formula ₀for the vertical range that active earth pressure point of resultant force is called in person to wall, according to H ₀the application point of active earth pressure with joint efforts on retaining wall can be determined.

2. according to claim 1ly determine that active earth pressure is made a concerted effort and the method for pressure resultant force application point, it is characterized in that: the nonlinear equation employing secant method in step (3) solves, and concrete steps are as follows:

1) given iterations N and precision controlling amount eps;

2) if iterations is greater than N, then terminate, otherwise perform step 3);

3) selected two initial value x ₀, x ₁, x ₀get the X-coordinate of y=tan (π/4+ φ/2) x and y=g (x) point of intersection, x ₁get the X-coordinate of y=tan (π/4-φ/2) x and y=g (x) point of intersection;

4) calculate $x_k=x_0-\frac{x_1-x_0}{f\left(x_1\right)-f\left(x_0\right)}f\left(x_1\right);$

5) if | x _k-x ₀| <eps, then x ₁=x _k, export x ₁, a, a ₁, λ, b ₁, EOP (end of program), otherwise perform 6);

6) x is made ₀=x ₁, x ₁=x _k, turn to step 2).