Calculation algorithm of additional stresses in irregularly shaped foundation soils under non-uniform loading

Abstract

The classical calculation method of additional stresses in foundation soils was established based on the Flamant and Boussinesq solutions in elasticity theory. This method can only calculate the additional stresses in foundations with regular shapes and under regularly distributed base pressures. Based on the Gauss–Legendre numerical product formula, which is independent of the specific form of the product function, a suitable method for calculating the additional stresses in irregularly shaped foundation soils under non-uniform loads was established in this study. Using Simpson’s formula to perform integration over the domain, Gaussian summation calculation methods for the additional stresses in foundation soil under non-uniform loads in rectangular domains, under loads in non-rectangular domains, and under irregular loads and loads in irregular integration regions were derived. The combination of the complex product algorithm and a computer program further efficiently improved the calculation accuracy. Validation examples showed that the Gaussian product calculation method of additional stresses in foundation soil was completely consistent with classical elasticity theory, and the results of the Gaussian product calculation method converged to the classical elasticity theory prediction when the complex product formula was applied. Based on validation and application examples, the method has advantages over the finite element method in terms of modeling difficulty, computational accuracy, computational volume, and computational cost when calculating additional stresses in foundation soil. The research results provide a new method and idea for the calculation of additional stresses in irregularly shaped flexible foundation soils under non-uniform loads.

Introduction

Additional stress in foundation soils leads to deformation of the foundation soil and causes building settlement^1,2,3,4. The currently calculation of additional stress on a foundation is based on the elastic theory, which assumes that the foundation soil is a homogeneous, continuous, isotropic, semi-infinite spatial elastomer^1,3. According to the Flamant and the Boussinesq solution^5,6, the additional stresses induced in the foundation soil by different foundations under different loads (concentrated load, uniform load, etc.) can be obtained^1,3. Using the vertical stress induction diagram^1,3 developed by Newmark in 1942, it is possible to calculate the additional stress in an irregularly shaped foundation under a uniformly distributed load.

The type and shape of the load distribution are not regular in practical engineering. As an irregular load, the distribution shape is irregular, so the application of the traditional calculation method is largely limited. There is a large calculation error in the actual engineering.

The traditional elasticity theory for calculating additional stresses in foundations is only applicable to regularly shaped and regularly distributed base pressures and is limited by whether the product function is productible or not. At present, the most popular methods to solve these problems are numerical simulation methods, such as the finite element method⁷, semi-finite element–semi-differential method⁸, fluid–solid coupling method⁹, discrete element¹⁰, and particle flow coupling method^11,12. The most widely used is the finite element method. The strengths of the finite element method are that it simulates a variety of geometrically complex structures and derives their approximate solutions. It applies widely through computer programs. The mathematical treatment is relatively simple and works for structures with complex shapes.But the finite element method is limited by the setting of boundary conditions, the selection of the soil body’s intrinsic model^{13,14,15,16,17,18}, and the consideration of the effect of water in the soil^{7,8,,819,20,22,23}, so the computational pre-processing, computational process, and post-processing are very complicated, and the results of the computation are not easy to control.Lagrangian interpolation is used in the pre-processing, but in order to prevent the lunge phenomenon that leads to unstable data, low order cells (order no more than 4) are used, but this increases the amount of computation and cost.

Aiming at the limitations and deficiencies of these methods, a method for calculating the additional stresses in irregularly shaped foundation soils under non-uniform loading is proposed based on a Gaussian product formulatio.

The following symbols are used in this manuscript:

Nomenclature
${\alpha _Q}$	The coefficient of the additional vertical stress under a concentrated load.
Q	The concentrated vertical force (kN).
z	The vertical depth from the calculation point to the ground surface (m).
r	Denotes the horizontal distance from the calculation point to the point where the vertical concentrated force acts (m).
Ak	The weight coefficients of the Gauss-Legendre quadrature formula.
xk	The zero point of the orthogonal polynomial-Legendre polynomial, the Gaussian point in$\int_{{ - 1}}^{1} {f(t){\text{d}}t \approx \sum\limits_{{k=0}}^{n} {{A_k}f({x_k})} }$.
s	The ratio of the depth of the calculated point z to the width of the load b s=z/b;
t	The ratio of the length of the rectangular foundation l to the width of the load b, t=l/b.
${\alpha _c}$	The coefficient of the additional vertical stress at the corner points of the uniformly distributed rectangular load, which can be looked up in the table based on $l/b$ and $z/b$.

The above variables are described in in-text citations, see text for details.

The gaussian product method for calculating additional stresses in soil

Introduction to classical elasticity theory approach

In the elastic theory, the additional stress in foundation soil is calculated based on the basic Boussinesq solution in elastic mechanics⁵.

$${\sigma _z}={\alpha _Q} \cdot Q/{z^2}$$

(1)

where ${\alpha _Q}$ is the coefficient of the additional vertical stress under a concentrated load. Q is the concentrated vertical force (kN). z is the vertical depth from the calculation point to the ground surface (m).

r-denotes the horizontal distance from the calculation point to the point where the vertical concentrated force acts (m).

In engineering practice, distributed loads are common, and concentrated forces are idealizations. In classical soil mechanics, it is considered that when the calculation point is much farther from the boundary of the distributed load, the load approximates as a concentrated force for the calculation of additional vertical stress. Specifically, the distance used in this approximation is the horizontal distance from the point of computation to the location of the load, r.

However, this paper investigates the calculation of additional stresses in irregularly shaped foundation soils under non-uniform loading. This needs to be considered more comprehensively without approximate computational considerations. Thus, there is no need to simplify the load distribution to point loads.

The following three figures reflect r (Figs. 1, 2, 3):

Fig. 1

Stress state under vertical concentrated force.

Fig. 2

Calculation of soil stresses under uniform loads (polar coordinate system).

Fig. 3

Calculation of additional vertical stresses under uniformly distributed rectangular loaded corner points.

When the shape and size of the distributed load are not negligible, and there is a distributed load ${p_0}(x,y)$ on foundation A, the additional stress at any point M(x,y) in the foundation can be obtained by integrating over the load distribution area A using Eq. (1)^1,3.

$$\sigma (x,y)=\iint\limits_{A} {{\alpha _Q} \cdot {p_0}(x,y)/{z^2}{\text{d}}}x{\text{d}}y$$

(2)

When ${p_0}(x,y)$ is a uniform load on a rectangular area, the double integral in Eq. (2) is integrable, and the settlement at each point can be expressed algebraically.

However, when ${p_0}(x,y)$ is irregular, the double integral in Eq. (2) is generally non-integrable, the foundation settlement under an irregular load distribution cannot be expressed directly in an algebraic form.

The gaussian product formula

Legendre orthogonal polynomials and their zeros

Based on theoretical derivations and calculations, the following recurrence relations exist for the Legendre polynomials:

${P_{n+1}}\left( x \right)=\frac{{2n+1}}{{n+1}}x{P_n}\left( x \right) - \frac{n}{{n+1}}{P_{n - 1}}\left( x \right),n=1,2,\ldots$

${P_0}(x)=1,{P_1}(x)=x,{P_2}(x)=\left( {3{x^2} - 1} \right)/2$

${P_3}(x)=\left( {5{x^3} - 3x} \right)/2,{P_4}(x)=\left( {35{x^4} - 30{x^2}+3} \right)/8$

${P_5}(x)=\left( {63{x^5} - 70{x^3}+15x} \right)/8$

$${P_6}(x)=\left( {231{x^6} - 315{x^4}+105{x^2} - 5} \right)/16$$

(3)

When n ≥ 1, it proves that.

$${P_n}(x)=0$$

(4)

There are n distinct real roots in the interval$\left[ { - 1,1} \right]$, each corresponding to a zero, for a total of n distinct real zeros.

Gauss–Legendre product formulae

The double integral$\iint\limits_{R} {p(x,y){\text{d}}A}$is the volume enclosed by the surface$z=p(x,y)$ and the plane R For a rectangular region$R=\left\{ {(x,y)\left| {a \leqslant x \leqslant b,c \leqslant y \leqslant d} \right.} \right\}$, it can be written as a superposition of two single integrals:

$$\iint\limits_{R} {p(x,y){\text{d}}x{\text{d}}y}=\int_{a}^{b} {(\int_{c}^{d} {f(x,y){\text{d}}} y)} {\text{d}}x$$

(5)

For a single integral in each layer$\int_{a}^{b} {f(x){\text{d}}x}$, the per-type interval transformation is$x=\frac{{b - a}}{2}t+\frac{{a+b}}{2}$, such that the integration interval is$( - 1,1)$,Then.

$$\int_{a}^{b} {f(x){\text{d}}x} =\frac{{b - a}}{2}\int_{{ - 1}}^{1} {f(\frac{{b - a}}{2}t+\frac{{a+b}}{2})} {\text{d}}t$$

(6)

Generally, for integrals with an interval of$( - 1,1)$, the Gauss-Legendre quadrature formula can be used:

$$\int_{{{\text{-1}}}}^{{\text{1}}} {f(t){\text{d}}t} \approx \sum\limits_{{k=0}}^{n} {{A_k}f({x_k})}$$

(7)

where: ${A_k}$ is the weight coefficients of the Gauss-Legendre quadrature formula.

${x_k}$ is the zero point of the orthogonal polynomial-Legendre polynomial, the Gaussian point in Eq. (7).

From the form of the Gaussian product Eq. (7) and Table 1, it can be seen that there are two main parameters of the Gaussian–Legendre product formula: the product weights ${A_k}$ and the Gaussian points ${x_k}$.

Table 1 Nodes and weight coefficients of Gauss-Legendre quadrature formula.

Both are given in Table 1 and are independent of the specific product function.

Although it is necessary to find the value of the product function at the Gaussian point, the Gaussian point is fixed in the interval $\left[ { - 1,1} \right]$ and is not difficult to determine. This provides insight into the construction of numerical methods for the calculation of additional stresses in foundations.

The Gauss–Legendre product formula is independent of the specific form of the product function, which solves the problem of calculating Eq. (1) when the product function is not integrable.

In addition, the finite element method uses Lagrangian interpolation, where n interpolation nodes can only achieve n-powers algebraic accuracy.

By contrast, the Gaussian product formula with n interpolated nodes achieves a high algebraic accuracy of 2n + 1 powers, and the weights of the nodes in the product formula are all positive, providing a high degree of numerical stability.

Simpson’s formula

The Gauss–Legendre product formula’s weight coefficients and node positions are independent of the product function, which solves the difficult problem of the non-integrable case of the product function in Eq. (1).

For the difficult problem of an irregular integral domain, Simpson’s formula can be used. For arbitrary integrals of the form

$$I=\int_{a}^{b} {f(x){\text{d}}x}$$

(8)

where the function$f(x)$ is not cumulative.

the corresponding form of Simpson’s formula for Eq. (8) is as follows.

$$S=\frac{{b - a}}{6}\left[ {f(a)+4f(c)+f(b)} \right]$$

(9)

where, $c=\left( {a+b} \right)/{\text{2}}$, a is the lower limit of integration; b is the upper lower limit of integration; and S is the Simpson integration formula value^24,25.

Equation (9) is the famous Simpson formula. A valuable feature of this equation is that it introduces the relaxation factor$4/6=2/3$, which allows Simpson’s formula to achieve a higher algebraic accuracy than the trapezoidal formula with the same two integration points a and b^24,25 .

In Simpson integration calculations, the relaxation technique is referred to as the relaxation method, and it is a common method used in the modern analysis technique^24,25.

Equation (9) appeared in the collection of papers that Simpson published in 1743, when the relaxation technique was not yet systematically established and applied^24,25.

In multiple numerical integrations, the double integral over a non-rectangular region can be approximated as rectangular regions^24,25by simply reducing it to a cumulative integral:

$$I=\int_{a}^{b} {\int_{{c(x)}}^{{d(x)}} {f(x,y){\text{d}}y{\text{d}}x} }$$

(10)

Specifically, the explicit Simpson formula was used to transform the upper and lower limits of the inner curve of the double integral into an explicit functional expression for the integral limit of the curve, thus transforming the implicit double integral in the formula for calculating the additional stress in a non-rectangular domain into an explicit cumulative integral by explicitly removing the integral number, which is similar to the rectangular domain case used to obtain the solution. The explicit analytical algebraic formula for the approximation of additional stress is obtained in a manner similar to the rectangular domain case.

$$I \approx \int_{a}^{b} {\frac{{k(x)}}{3}} [f(x,c(x))+4f(x,c(x)+k(x))+f(x,d(x))]{\text{d}}x$$

(11)

where, $k(x)=\frac{{d(x) - c(x)}}{2}$Then, the Gaussian quadrature formula is used for each integral, and the approximation of the integral I is obtained.

The gaussian product calculation method of additional stresses in foundation soil

Case of non-uniform loading in rectangular domains

Equation (2) is used for calculating the additional stress. Assuming that the length of a rectangular foundation is l and the width is b, then Eq. (2) can be written as the superposition of two single integrals^24,25.

$$\sigma (x,y)=\iint\limits_{A} {{\alpha _Q}\frac{{{p_0}(x,y)}}{{{z^2}}}{\text{d}}}x{\text{d}}y$$

(12)

$$\sigma (x,y)=\iint\limits_{A} {{\alpha _Q}\frac{{{p_0}(x,y)}}{{{z^2}}}{\text{d}}}x{\text{d}}y=\int\limits_{{ - \frac{l}{2}}}^{{\frac{l}{2}}} {{\text{(}}\int\limits_{{ - \frac{b}{2}}}^{{\frac{{\text{b}}}{{\text{2}}}}} {{\alpha _Q}\frac{{{p_0}(x,y)}}{{{z^2}}}{\text{d}}x} {\text{)d}}y}$$

(13)

Assuming$x=\frac{b}{2} \cdot m$and$y=\frac{l}{2} \cdot n$then Eq. (13) can be transformed as follows:

$$\sigma \left( {x,y} \right)=\frac{{bL}}{4}\int_{{ - 1}}^{1} {\left( {\int_{{ - 1}}^{1} {{\alpha _Q}\frac{{{p_0}\left( {0.5bm,0.5\ln } \right)}}{{{z^2}}}} dm} \right)dn}$$

(14)

In Eq. (14), m and n denote the transformation coefficients that transform the integration limits from the intervals of $x=\left[ { - b/2,b/2} \right]$ and$y=\left[ { - l/2,l/2} \right]$ to$m=n=[ - 1,1]$.

Using the kth-order Gauss-Legendre quadrature formula with weight A_k and quadrature nodes m_i and n_j,

The additional stress is.

$$\sigma (x,y)=\frac{{bl}}{4}\sum\limits_{{i=0}}^{k} {\sum\limits_{{j=0}}^{k} {{A_i}{A_j}\left[ {{\alpha _Q}\frac{{{p_0}({\text{0}}{\text{0.5}}b{m_i},0.5 L{n_j})}}{{{z^2}}}} \right]} }$$

(15)

In Eq. (15), i is the outer Gaussian point node number, where $i=1 \cdots k$, k denotes the total number of Gaussian points, j is the inner Gaussian point node number, where$j=1 \cdots k$, and k denotes the total number of Gaussian points, ${m_i}$ denotes the k Gaussian points in$m=[ - 1,1]$, and ${n_j}$ denotes the k Gaussian points in $n=[ - 1,1]$.

where the additional pressure at the bottom of the foundation is

$${p_0}(x,y)={p_0}({\text{0}}{\text{0.5}}b{m_i},0.5 L{n_j})$$

(16)

Equation (16) can be chosen in any functional type and substituted into algebraic Eq. (15), which does not include the integral sign, and then, it can be solved analytically.

Extension of applications in non-rectangular foundations

The Gaussian product calculation method for additional stresses in foundation soil is not limited to the calculation of additional stresses in regular integral domains. For the general case of non-regular integral domains, it can be converted into cumulative integration via multiple numerical integrations in non-rectangular regions. Alternatively, it can be approximated similarly to the case of a rectangular region^24,25by dual integration, as:

$$I=\int_{a}^{b} {\int_{{c(x)}}^{{d(x)}} {f(x,y){\text{d}}y{\text{d}}x} }$$

(17)

Specifically, the explicit Simpson formula was used to transform the upper and lower limits of the inner curve of the double integral into an explicit functional expression for the integral limit of the curve, thus transforming the implicit double integral in the formula for calculating the additional stress in a non-rectangular domain into an explicit cumulative integral by explicitly removing the integral number, which is similar to the rectangular domain case used to obtain the solution. The explicit analytical algebraic formula for the approximation of additional stress is obtained in a manner similar to the rectangular domain case.

$$I \approx \int_{a}^{b} {\frac{{k(x)}}{3}} [f(x,c(x))+4f(x,c(x)+k(x))+f(x,d(x))]{\text{d}}x$$

(18)

Here, $k(x)=\frac{{d(x) - c(x)}}{2}$,Then, the Gaussian quadrature formula is used for each integral, and the approximation of the integral I is obtained.

Gauss–Simpson calculation of additional stresses in irregularly shaped foundation soils under non-uniform loading

The method proposed in this study is not limited to the calculation of the additional stress in a rectangular flexible foundation. For a non-rectangular domain basis, as long as the corresponding double integral in the equation is converted into a cumulative integral following the above method, the explicit expression for the additional stress can be obtained in a manner similar to that for the rectangular domain case.

In addition, the interpolation points of the Gaussian quadrature formula are the n zeros of the Gauss-Legendre orthogonal polynomials, which can achieve an algebraic accuracy of 2n + 1, and the weights of the nodes in the quadrature formula are all positive. Thus, the method has good numerical stability.

In engineering applications, only the location of the interpolation points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type of Simpson’s formula are needed for the field engineers to obtain the additional stress manually, without complex integration and interpolation. This is nearly impossible for traditional methods and finite element methods. Therefore, the proposed method has excellent potential for broad application.

Calculation of additional stresses in three forms of foundation soils

Irregular load size but regular shape of load distribution

The function code in MATLAB for this case is given below to reflect the function figure:

t=-4:0.01:4;

[x, y] = meshgrid(t);%form the grid point matrix.

z = sin(x) + cos(y);

Figure (1).

mesh(x, y,z).

axis([-1.5 4.5 -3 3 0 2]); title(‘z = sin(x) + cos(y)’); Fig. 1

title(‘z = sin(x) + cos(y); mesh’).

colorormap jet.

colorbar.

In the case shown in Fig. 4, the loads are irregularly distributed, but they can be expressed analytically, and the load is distributed in a regular shape that satisfies certain conditions.

Fig. 4

First case: an irregular load distribution with a regular distribution shape.

The formula for calculating the additional stresses in the foundation is given by Eq. (2). Equation (2) can be calculated using Eq. (15) if the length of the rectangular foundation is l and the width is b, as follows:

$$\sigma (x,y)=\frac{{bl}}{4}\sum\limits_{{i=0}}^{k} {\sum\limits_{{j=0}}^{k} {{A_i}{A_j}\left[ {{\alpha _Q}\frac{{{p_0}({\text{0}}{\text{0.5}}bm,0.5ln)}}{{{z^2}}}} \right]} }$$

where the additional pressure at the bottom of the foundation is${p_0}(x,y)={p_0}({\text{0}}{\text{0.5}}b{m_i},0.5 L{n_j})$

According to the quadrature formulas of different orders, the coordinates and weights of the integration points in each group are determined in the order of low to high accuracy and based on the number of integration points. For each group of integration point coordinates, the additional pressure at the bottom of the foundation at the integration point is obtained. Using the Gaussian quadrature formula, the integral values with different accuracies are derived, i.e., the additional stress values in the foundation with different accuracies. The minimum number of integration points to meet the accuracy requirement is obtained through comparison with the measurement data or the results of the traditional method. In practical engineering, we only need to ensure that the number of integration points is larger than the minimum number to obtain good results. Moreover, the more the integration points, the higher the accuracy.

Irregular load size and shape of load distribution

This figure shows the view of the 3D image formed by the following function code at different angles.

The function code in MATLAB for this case is given below to reflect the function figure:

x = 0:0.001:0.5;

m = length(x);

y = linspace(0.5,0.5,m);

z1 = sin(2*pi*x);

Figure (2).

plot3(y, x,z1,‘’).

hold on;

x1 = 0:0.001:0.5;

y1 = sin(4*pi*x1);

n = length(x1);

z3 = linspace(0,0,n);

patch(y1,x1,z3,‘r’);

hold on;

i = 0;

for i = 1:n;

x1 = 0:0.001:0.5;

y1 = 0.001*i + sin(4*pi*x1);

z1 = 0.001*i + 100*sin(2*pi*x1);

patch(y1,x1,z1,‘b’);

end;

title(‘figure; pach’).

colormap jet.

colorbar.

In the case shown in Fig. 5, the loads are irregularly distributed, but they can be expressed analytically, and the load is distributed in a special shape that satisfies certain conditions.

Fig. 5

Second case: irregular load size and special shape of load distribution.

The specificity of the shape of the load distribution is reflected in the fact that two arbitrary curves are sandwiched by two parallel lines. For the case of a non-regular integration region, explicit algebraic formulas for calculating the additional stresses can be obtained. For the integral of Eq. (17),$I=\int_{a}^{b} {\int_{{c(x)}}^{{d(x)}} {f(x,y){\text{d}}y{\text{d}}x} }$

Transforming the implicit double integral in the formula for calculating the additional stress in a non-rectangular domain into an explicit cumulative integral by explicitly removing the integral number and then the explicit analytic algebraic Eq. (18) for solving the additional stress approximation is obtained similarly to the rectangular domain case.

$I \approx \int_{a}^{b} {\frac{{k(x)}}{3}} [f(x,c(x))+4f(x,c(x)+k(x))+f(x,d(x))]{\text{d}}x$

This is then substituted into Eq. (15). That is, we obtain the Gaussian product calculation of additional stress in irregularly shaped foundation soil.

Non-uniform distribution of load values and completely arbitrary shape of load distribution

The function code in MATLAB for this case is given below to reflect the function figure:

A=[1.486,3.059,0.1;2.121,4.041,0.1;2.570,3.959,0.1;3.439,4.396,0.1;

4.505,3.012,0.1;3.402,1.604,0.1;2.570,2.065,0.1;2.150,1.970,0.1;

1.794,3.059,0.2;2.121,3.615,0.2;2.570,3.473,0.2;3.421,4.160,0.2;

4.271,3.036,0.2;3.411,1.876,0.2;2.561,2.562,0.2;2.179,2.420,0.2;

2.757,3.024,0.3;3.439,3.970,0.3;4.084,3.036,0.3;3.402,2.077,0.3;

2.879,3.036,0.4;3.421,3.793,0.4;3.953,3.036,0.4;3.402,2.219,0.4;

3.000,3.047,0.5;3.430,3.639,0.5;3.822,3.012,0.5;3.411,2.385,0.5;

3.103,3.012,0.6;3.430,3.462,0.6;3.710,3.036,0.6;3.402,2.562,0.6;

3.224,3.047,0.7;3.411,3.260,0.7;3.542,3.024,0.7;3.393,2.763,0.7];

x = A(:,1);y = A(:,2);z = A(:,3);

Figure (3).

[X, Y,Z] = griddata(x, y,z, linspace(1.486,4.271,30)’,linspace(1.604,4.276,30),‘v4’);%插值.

surf(X, Y,Z)% 3D surface.

colormap jet.

colorbar.

In the case shown in Fig. 6, the loads are irregularly distributed, but they can be expressed analytically, and the load is distributed in an irregular shape that satisfies certain conditions.

$$I=\int_{{a(x)}}^{{b(x)}} {\int_{{c(x)}}^{{d(x)}} {f(x,y) \cdot {\text{d}}x{\text{d}}y} }$$

(19)

Fig. 6

Third case: the load size is irregular but can be represented analytically, and the shape of the load distribution is irregular but satisfies certain conditions.

For Eq. (19), where both the inner and outer layers are double integrals of the upper and lower limits of the curve and the additional pressure at the bottom of the foundation is an irregular load distribution, the problem is transformed into calculating the volume of a three-dimensional flat-bottomed surface enclosed by a curved integral domain with a height of z = f(x, y). Using Simpson’s formula and Gaussian quadrature, the double integral of an irregularly loaded, irregularly shaped region is transformed into an explicit algebraic expression.

Assuming that the boundary of the curved integral domain$\left\{ {\begin{array}{*{20}{c}} {a(x)\sim b(x)} \\ {c(x)\sim d(x)} \end{array}} \right.$in the y-direction is [$\left[ {{c_{\hbox{min} }},{d_{\hbox{max} }}} \right]$], where ${c_{\hbox{min} }}$, and ${d_{\hbox{max} }}$ are constants, nodes c and d exist in the y-direction of the integral domain, between $\left[ {c,d} \right]$]and $\left[ {{c_{\hbox{min} }},c} \right]$,$\left[ {d,{d_{\hbox{max} }}} \right]$. The integral limits $a(x)\sim b(x)$ in the x direction in Eq. (19) can be represented analytically by the same function. Such integral domains are common in the widely adopted in circular and square art architectures. Under this premise, Eq. (19) can be converted to

$$\begin{gathered} I=\int_{{a(x)}}^{{b(x)}} {\int_{{c(x)}}^{{d(x)}} {f(x,y) \cdot {\text{d}}x{\text{d}}y} } \hfill \\ {\text{=}}\int_{{a(x)}}^{{b(x)}} {\int_{{{c_{\hbox{min} }}}}^{c} {{f_{1 - 2}}(x,y) \cdot {\text{d}}x{\text{d}}y} } +\int_{{a(x)}}^{{b(x)}} {\int_{c}^{d} {{f_{2 - 3}}(x,y) \cdot {\text{d}}x{\text{d}}y} } +\int_{{a(x)}}^{{b(x)}} {\int_{d}^{{{d_{\hbox{max} }}}} {{f_{3 - 4}}(x,y) \cdot {\text{d}}x{\text{d}}y} } \hfill \\ =\int_{{{c_{\hbox{min} }}}}^{c} {\int_{{a(x)}}^{{b(x)}} {{f_1}(x,y)} \cdot {\text{d}}x{\text{d}}y} +\int_{c}^{d} {\int_{{a(x)}}^{{b(x)}} {{f_2}(x,y)} \cdot {\text{d}}x{\text{d}}y} +\int_{d}^{{{d_{\hbox{max} }}}} {\int_{{a(x)}}^{{b(x)}} {{f_3}(x,y)} \cdot {\text{d}}x{\text{d}}y} \hfill \\ {\text{=}}{I_1}+{I_2}+{I_3} \hfill \\ \end{gathered}$$

(20)

For

$$\begin{gathered} {I_1}=\int_{{{c_{\hbox{min} }}}}^{c} {\int_{{a(x)}}^{{b(x)}} {{f_1}(x,y)} \cdot {\text{d}}x{\text{d}}y} \hfill \\ {I_2}{\text{=}}\int_{c}^{d} {\int_{{a(x)}}^{{b(x)}} {{f_2}(x,y)} \cdot {\text{d}}x{\text{d}}y} \hfill \\ {I_3}{\text{=}}\int_{d}^{{{d_{\hbox{max} }}}} {\int_{{a(x)}}^{{b(x)}} {{f_3}(x,y)} \cdot {\text{d}}x{\text{d}}y} \hfill \\ \end{gathered}$$

(21)

Simpson’s formula is used to achieve explicitation of the inner integral variable limit integrals, and then the Gauss–Legendre formula is used to deal with each integral’s constant limit integral.

Complex the Gaussian product method for calculating additional stresses in soil

Based on the Gauss–Legendre product formula for the calculation of additional stresses in soil, the zeros of the orthogonal polynomials are calculated by Eq. (3), from which it can be seen that when$n \geqslant 6$ in Eq. (4), ${P_n}(x)=0$. Thus, it is necessary to solve algebraic equations with one variable more than six powers, and although the Gaussian points are fixed, considering the rounding error and numerical stability, it is uneconomical to rely on improving the algebraic accuracy by raising the order of n in Eq. (4) with${P_n}(x)=0$ to improve the algebraic accuracy. Furthermore, the Gaussian points are generally no longer added, and the complexified The Gaussian product formula is used for the calculation.

This is illustrated by a practical arithmetic example.

The additional stress at the corner points of a rectangular foundation under a uniform load p₀ has an analytical solution in traditional elastic theory (Zhang et al., 2006):

$$\begin{gathered} {\sigma _z}=\frac{{3{p_0}}}{{2\pi }}\int_{0}^{1} {\int_{0}^{b} {\frac{{{z^3}}}{{{{\left( {{x^2}+{y^2}+{z^2}} \right)}^{\frac{5}{2}}}}}dxdy} } \hfill \\ =\frac{{{p_0}}}{{2\pi }}\left[ {arc\tan \frac{t}{{s\sqrt {1+{s^2}+{t^2}} }}+\frac{{st}}{{\sqrt {1+{s^2}+{t^2}} }}\left( {\frac{1}{{{s^2}}}+\frac{1}{{1+{s^2}}}} \right)} \right]+{t^2}={\alpha _c}{p_0} \hfill \\ \end{gathered}$$

(22)

Here,

$\:s$ is the ratio of the depth of the calculated point z to the width of the load b $s=z/b$;

s is the ratio of the length of the rectangular foundation l to the width of the load b, $t=l/b$;

${\alpha _c}$is the coefficient of the additional vertical stress at the corner points of the uniformly distributed rectangular load, which can be looked up in the table based on $l/b$ and $z/b$.

In Eq. (8), the numerical expression for the coefficient of the additional stress is:

$${\alpha _c}=\frac{{bl}}{4} \cdot \sum\limits_{{i=0}}^{k} {\sum\limits_{{j=0}}^{k} {{A_i}{A_j}\frac{{{\alpha _Q}}}{{{z^2}}}} }$$

(23)

The nodes and coefficients of the Gauss–Legendre quadrature formula when n = 2, 3, 4, and 5 are used to form Eq. (13). The values of the corner point influence coefficients are calculated when l/b takes various values. Then, the result is compared with the analytical solution.

Figure 7 shows the comparison between the Gaussian quadrature formulas of different orders and the traditional elastic theory method in terms of the coefficient of the additional stress at the corner points of a rectangular foundation. It can be seen that as the number of quadrature nodes increases, the accuracy of the Gaussian quadrature formula increases accordingly and approaches the analytical solution for the elastic theory.

Fig. 7

Contrast of calculated value by Gaussian quadrature method of different order of influence coefficient of settlement.

As can be seen from Fig. 7, the approximation value calculated using Gaussian quadrature is smaller than the analytical solution of the elasticity theory, which converges continuously from the lower limit of the analytical solution.

The advantage of Eq. (23) is that the formula can be expressed analytically, and the calculation process can be performed algebraically, which can be done by hand. The disadvantage is that the calculation accuracy can only reach the accuracy that can be achieved by the fifth-order Gauss–Legendre product formula. Then, the order must be increased to improve the calculation accuracy, which will increase the rounding error, leading to numerical destabilization, and will greatly increase the number of calculations.

Usually, to further improve the calculation accuracy of Eq. (2), the complex quadrature formula can be used. That is, the interval can be divided into several equal parts, and each part can be solved separately using the Gaussian quadrature formula, which effectively improves the calculation accuracy. Figure 8 illustrates the calculation of the coefficient of the additional stress using the complex Gaussian quadrature formula with n = 5 and 20 intervals. Compared with the regular Gaussian quadrature formula and the traditional elastic theory method, the complex quadrature formula does improve the calculation accuracy. The curve basically overlaps with that of the traditional elastic theory method.

The complex Gaussian quadrature formula is computationally intensive and can be implemented using computer programming. In this study, MATLAB was used to program the solving process for the dual complex Gauss-Legendre quadrature formula^24,25.

Fig. 8

Contrast of calculated value by complex Gaussian quadrature method and 5 order Gaussian quadrature method.

The meaning of the figure in the figure is the difference in the additional stress coefficients between the conventional elasticity theory method and the Gauss–Legende quadratic formula when $l/b$ takes different values.

The maximum relative divergence between the conventional elasticity theory method and the Gauss–Legende quadratic formula is calculated as:

${\text{Relative Divergence=}}\frac{{\left. {\left| {{E_{{\text{elasticity}}}} - {E_{{\text{method}}}}} \right.} \right|}}{{{E_{{\text{elasticity}}}}}} \times 100\%$

Based on the comparison of the different curves, this maximum divergence is more pronounced in the low-order Gaussian quadrature method curves, and is most prominent at $l/b$ ratios greater than 5 in terms of the horizontal coordinates.

Validation and application of methods

Field applications

The Dongrong mine is located in the northern succession area of the Shuangyashan Mining Bureau, Heilongjiang Province, China, and has an overall design capacity of 5.1 million t/a. It is situated on the first-grade terrace of the Songhua River and belongs to the river floodplain phase. The soil is composed of a clay layer (0.5–16 m thick) and a sand layer (about 40 m) composed of Holocene (Q4) alluvium, which is a typical binary structure. The groundwater is phreatic water, and the water in the lower sand layer is pressurized water and is abundant. The upper phreatic water is just 0.5–1 m below the surface, and the water is not harmful to any kind of concrete. The standard freezing depth is 2.20 m.

The Quaternary and Tertiary strata directly overlie the coal strata, so a special freezing method has to be used in the new mine to construct and drill the shaft. The first completed mine in the Dongrong mining area, Dongrong mine No. 2, has a design capacity of 1.5 million t/a. The main shaft tower of the mine was built on special artificial freeze-thaw soil (general soil and sandy soil from the quaternary alluvium) in the frozen shaft.

The proposed method was used to calculate the additional stress in the foundation of the main shaft tower of Dongrong mine No. 2. The tower’s foundation is rectangular, with a length of l = 200 cm and a width of b = 400 cm. The foundation is buried 100 cm deep, and the designed bottom is 60 cm from the natural bottom. The modulus of the deformation of the foundation soil is ${E_0}=5{\text{MPa}}$, the Poisson’s ratio is $\upsilon =0.25$, and the unit weight is $\gamma =19.8{\text{kN}}/{{\text{m}}^3}$. The load is equivalent to$z={\text{cos}}(\pi /400 \cdot x)$(MPa).

The following region is considered: ($\:-\text{200\:cm}\le\:x\le\:200\:\text{cm}-\text{200\:cm}\le\:x\le\:200\:\text{cm}$, $- {\text{100 cm}} \leqslant y \leqslant {\text{100 cm}}$). A program was written to solve this problem, and the result was 19.26499813 MPa.

Validation of finite element numerical simulations

Based on the field data of Dongrong mine, the foundation length was selected appropriately. Theoretical and finite element modeling calculations were carried out simultaneously to compare the calculation results of the two methods. The Adina software was used for the finite element simulation. Software information:

Adina Headquarters in the United States.

71 Elton Avenue.

Watertown, MA 02472, USA.

Tel: (617) 926–5199.

Fax: (617) 926 − 0238.

www.adina.com; www.adina.com.cn; info@adina.com.cn.

Full name of ADINA and Version number:

ADINA (Automatic Dynamic Incremental Nonlinear Analysis) Version 9.0.0.

Three dimensional solid elements were used for the model, and the bottom surface of the model was fixed. The load was applied using the spatial function, which defined a point array of two-dimensional spatial distribution surfaces and was used to define the load used to generate the cosine surface load with a rectangular base that is parallel to the y axis. Figure 9 shows the finite element model and the load function.

Fig. 9

The finite element numerical simulation model.

The modeling parameters for the numerical simulation are similar to the data applied in the field. The short side b of the foundation was set to be the x axis, the long side l of the foundation was set to be the y axis, and the mean load was taken to be a three-dimensional surface:$z={\text{cos}}(\frac{\pi }{{{\text{400}}}}x)$

on the base ($b \times l$), and ($- 200 \leqslant x \leqslant {\text{200}}$, $- l \leqslant y \leqslant l$). For$b={\text{400}}$, the maximum value of the load $z=\cos (\pi /400 \cdot {\text{0}})={\text{1}}$MPa occurs at x =0.

The mesh size is a square with sides of 20. The Poisson’s ratio of the soil beneath the foundation is $\nu ={\text{0}}{\text{0.25}}$, and ${E_0}={\text{5 MPa}}$.

When$l/b$ has different values, a finite element model is established to investigate the difference between the simulation results of the additional stress and the results of the method proposed in this study.

Fig. 10

Contrast of settlement calculated by Gaussian quadrature method and numerical simulation method under the action of non uniform cosine loading.

As can be seen from Fig. 10, the numerical simulation results are generally consistent with those of the Gauss-Simpson Quadrature Algorithm in terms of the magnitude, but there are localized differences in the curve patterns. The curve direction and the shape of the Gaussian quadrature formula method are very similar to the curve for a uniform load. The numerical simulation curve is initially smaller than or similar to that of the Gaussian quadrature, and then it becomes larger than the Gaussian quadrature. This is due to the boundary conditions of the numerical simulation model. During the simulation, the boundary conditions were applied on the bottom surface of the 3D model, and no constraints were applied on the sides. Therefore, when the long side l is small, the loading element is at the center of the top surface of the 3D model, which is constrained by the surrounding elements, i.e., the completely confined condition. Therefore, the numerical simulation results are similar to or smaller than the results of the Gaussian quadrature. When the long side l is large, the model size and the number of elements are limited by the computer’s memory, the rectangular base corner points are closer to the model boundary, and since there are no constraints on the sides of the 3D model, i.e., the unconfined condition, the additional stress becomes very large.

According to Fig. 10, the advantages of the Gaussian product calculation method were mainly reflected in three points: (1) the trends of the calculation curves under a variety of working conditions were close to the trends of the curves obtained by classical elasticity theory, (2) the calculation curves did not have large fluctuations, and thus, the results were more stable than the results of the finite element method, and (3) the Gaussian product method was not affected by the boundary conditions of the finite element method.

This comparison shows that the Gaussian quadrature formula is more concise and practical, its results are more reasonable, and the calculation is more controllable. In contrast, the FE simulation is computationally intensive, complex to implement and operate; the calculation results are limited by the model, load, and boundary conditions; and it is hard to control the reasonableness of the calculation results.

Conclusions

To calculate the additional stresses of flexible foundations under non-uniform loads, a numerical method based on elasticity theory and a Gaussian product formula was proposed and compared with classical elasticity theory. The specific conclusions are as follows:

• (1) The nodes and weights of the Gaussian product formula are determined by Legendre orthogonal polynomials, and the calculation is independent of the specific form of the product function. This approach has superb algebraic accuracy and can be effectively applied to the calculation of additional stresses in foundation soils under non-uniform loads.

• (2) By combining Simpson’s formula, which makes the implicit limits of integration explicit, with the Gauss–Legendre product formula, an explicit formula for the additional stresses in a flexible foundation of arbitrary shape was obtained.

• (3) A comparison with the results of classical elasticity theory showed that there was a small amount of calculated data for the Gaussian product formula method, and with the increase in the number of product nodes, the calculated values converged to the analytical solution of elasticity theory.

• (4) The Gaussian product formula method shows no obvious accuracy improvement after more than 5 orders. Then, the order must increase to improve the calculation accuracy, which increases the rounding error and leads to numerical destabilization. It also greatly increases the number of calculations. The product formula can be compounded to continue improving the accuracy of the calculation. The complex Gaussian quadrature formula is computationally intensive and can be implemented using computer programming.

• (5) The validation and application of the method showed that the Gaussian product calculation method has distinct advantages, which were mainly reflected in three points: (1) the trends of the calculation curves under a variety of working conditions were close to the trends of the curves of classical elasticity theory, (2) there were no large fluctuations in the calculation curves, and thus, the results were more stable than those of the finite element calculations, and (3) the Gaussian product calculation method was not affected by the boundary conditions of the finite element method.