Gauss-Simpson Quadrature Algorithm for Calcaulting Additional Stress in Foundation Soils

: 10 The additional pressure at the bottom of a building ’s foundation produces an additional stress in 11 the foundation soils under the building ’s foundation. In order to overcome the limitations of 12 traditional elastic theory methods and the finite element method when calculating the additional 13 stress in foundation soils, we use the Gauss-Simpson formula to derive the Gauss-Simpson 14 Quadrature Algorithm based on the elasticity theory. The Gauss-Simpson Quadrature Algorithm 15 is a method designed to calculate the additional stress in foundation soils under an irregularly 16 shaped foundation and an irregular load distribution. This new method is based on the fact that the 17 Gaussian quadrature formula and the Simpson formula are independent of the specific type of 18 integrand. The finite element method with n interpolation points can only achieve an algebraic 19 accuracy of n. The interpolation points of the Gaussian quadrature formula are n zeros of 20 orthogonal polynomials, which can achieve an algebraic accuracy of 2n+1. Moreover, the weights 21 of the nodes in the quadrature formula are all positive, and thus, it has a high numerical stability. 22 In the proposed method, the Simpson formula is necessary. The Simpson formula is used to 23 transform the implicit additional stress formula with the integral sign into an explicit cumulative 24 integral, which can be considered similar to the rectangular domain case to obtain an explicit 25 analytical algebraic formula for solving the additional stress approximation. In engineering 26 applications, we only need to provide the field engineers with the locations of the interpolation 27 points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type 28 of Simpson's formula, and then, the results of the additional stress can be calculated manually, 29 which is nearly impossible using the traditional methods and finite element methods. From the 30 point of view of academic rigor and theoretical completeness, it is possible to use the compound 31 Gauss-Simpson Quadrature Algorithm in conjunction with the looping function in computer 32 programs. Under standard conditions, the proposed Gauss-Simpson Quadrature Algorithm is in 33 good agreement with the results of the traditional elasticity theory.

orthogonal polynomials, which can achieve an algebraic accuracy of 2n+1. Moreover, the weights 21 of the nodes in the quadrature formula are all positive, and thus, it has a high numerical stability. 22 In the proposed method, the Simpson formula is necessary. The Simpson formula is used to 23 transform the implicit additional stress formula with the integral sign into an explicit cumulative 24 integral, which can be considered similar to the rectangular domain case to obtain an explicit 25 analytical algebraic formula for solving the additional stress approximation. In engineering 26 applications, we only need to provide the field engineers with the locations of the interpolation 27 points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type 28 of Simpson's formula, and then, the results of the additional stress can be calculated manually, 29 which is nearly impossible using the traditional methods and finite element methods. shaped foundation under a uniformly distributed load [5][6]. 48 In practical engineering, there are numerous cases with irregular load distributions, and the 49 uniform load assumption is a simple and rough averaging method, which is an expedient way to 50 deal with the problem. In traditional methods, the most widely used stress distributions are 51 rectangular and circular, which have difficulty meeting the requirements of modern architecture 52 and artistic aesthetics. 53 The type and shape of the load distribution are not regular in practical engineering [7][8]. The 54 traditional elastic theory used to calculate the additional stress of the foundation is limited by the 55 integrability of the integrand, and it can only be used to calculate the additional stress on a 56 foundation with a regular shape and a uniformly distributed load. Therefore, it is of major 57 theoretical and engineering significance to develop a method of calculating the additional stress in 58 an irregularly shaped foundation under an irregular load distribution. 59 From the perspective of the additional pressure at the bottom of the foundation, it is an 60 irregular load distribution, and its distribution shape is irregular [9][10][11][12]. Therefore, the application 61 of the traditional calculation method is largely limited. The uniform load assumption in practical 62 engineering is a simple averaging method, and it is a limitation of traditional calculation methods, 63 including the Newmark chart method. 64 In addition, the most widely used types of stress distribution are rectangular and circular, which cannot meet the requirements of modern architecture with artistic aesthetics. Currently, the 66 most widely used mainstream method used to solve the problem of an irregular load distribution 67 and an irregular distribution shape is FE simulations. Therefore, the method proposed in this study 68 must have significant advantages over the finite element method in order to have value for practical 69 applications.

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The method proposed in this study has something in common with the finite element method.

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In the elastic theory, the additional stress in foundation soil is calculated based on the basic 100 Boussinesq solution in elastic mechanics [3]: (2) 113 When 0 ( , ) P x y is a uniform load on a rectangular area, the double integral in Equation (2) 114 is integrable, and the settlement at each point can be expressed algebraically. However, when 115 0 ( , ) P x y is irregular, the double integral in Equation (2) , it can be written as a 124 superposition of two single integrals: (3) 126 For a single integral in each layer  b a x x f d ) ( , the per-type interval transformation is , such that the integration interval is (−1,1). Then, (4) 129 Generally, for integrals with an interval of (−1,1), the Gauss-Legendre quadrature formula 130 can be used: Here, k A is the weight coefficients of the Gauss-Legendre quadrature formula ( From the Gaussian quadrature formula and Table 1, it can be seen that the quadrature weights    (7) 167 Using the kth-order Gauss-Legendre quadrature formula with weight Ak and quadrature nodes mi and nj, the additional stress is data or the results of the traditional method. In practical engineering, we only need to ensure that 182 the number of integration points is larger than the minimum number to obtain good results.

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Moreover, the more the integration points, the higher the accuracy. stress is given: Here, Here, . Then, the Gaussian quadrature formula is used for each integral, and 217 the approximation of the integral I is obtained.  In the case shown in Fig. 3, the loads are irregularly distributed, but they can be expressed 225 analytically, and the load is distributed in an irregular shape that satisfies certain conditions.

Irregular Load Distribution and Irregular Distribution Shape
For Equation (13)

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Simpson's formula is used to achieve explicitation of the inner integral variable limit integrals, 243 and then the Gauss-Legendre formula is used to deal with each integral's constant limit integral.

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The additional stress at the corner points of a rectangular foundation under a uniform load    h is the length of the outer complex interval.

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% t is the Gauss-Legendre integration points; and% A is the Gauss-Legendre integration 378 weight.

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The following is the specific function field: The solution command is  The result is 0.01626499813 (MPa), which is smaller than the value we measured at the 383 construction site. it is not suitable for application in practical engineering.

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(4) For an irregular load distribution in a rectangular domain, the additional stress can be 410 directly obtained using Table 1, that is, the explicit analytical algebraic expression of the integral 411 value. The higher the order, the more terms in the algebraic equation. In general, a satisfactory 412 accuracy can be achieved when n<5. Moreover, the nodal positions and weight coefficients in 413   Table 1 are independent of the type of integrand, i.e., irregular load distributions. we studied the calculation accuracy of the Gaussian quadrature formula beyond the 5th order. Since 433 the improvement of the calculation accuracy was not obvious in this case, we introduced the 434 complex quadrature formula. A simple computer program was written to implement this method, 435 which may not be suitable for direct field applications.

Data Availability Statements 437
The data that support the findings of this study are available from the corresponding author upon 438 reasonable request.