Calculation method of static earth pressure coefficient of state-dependent soil considering particle size and density

Abstract

The application discloses a kind of state-related sand static earth pressure coefficient calculation method considering particle size and compactness.The method comprises the following steps: obtaining the state parameter of the sand sample to be measured, the state parameter includes the compactness parameter reflecting the compactness of the sand sample to be measured, and the particle size parameter reflecting the particle size;Based on the compactness parameter, the peak friction angle of the sand sample to be measured is determined by a preset friction angle growth equation;Based on the peak friction angle and the particle size parameter, the static earth pressure coefficient of the sand sample to be measured is calculated by a preset static earth pressure representation model.The application overcomes the technical limitations of traditional single-variable empirical formula, effectively couples the particle size effect and state correlation in soil mechanics response, and significantly improves the calculation accuracy of static lateral earth pressure under complex sand foundation conditions.

Description

Technical Field

[0001] This invention relates to the fields of geotechnical engineering and marine engineering, and in particular to a method for calculating the static earth pressure coefficient of sand. Background Technology

[0002] The coefficient of earth pressure at rest is the ratio of horizontal stress to vertical stress in soil under conditions of no lateral deformation. It is a key strength and deformation parameter in geotechnical engineering. This value is not only the starting boundary for all numerical simulations and mechanical analyses, but also the direct theoretical benchmark for stress calculations of deeply buried underground structures. Its accuracy is of paramount technical significance for settlement prediction and long-term structural safety assessment.

[0003] Currently, the engineering community widely uses classical empirical formulas (such as the Jaky formula) to estimate this coefficient. These methods typically simplify the coefficient of earth pressure at rest to a single static function related only to the internal friction angle of the soil, neglecting the dual influence of particle size and relative density. This leads to inherent biases in the prediction of lateral pressure on soil in coarse-grained and fine-grained sand, and loose and dense states, and has become one of the main bottlenecks in the refined design and numerical analysis of geotechnical engineering.

[0004] Specifically, in practical applications, such estimation methods often use single-variable values ​​based on the critical mechanical state of the soil, assuming that the earth pressure coefficient is independent of the physical compaction state and particle size distribution of the soil layer. However, when faced with soil conditions with large spans in coarse and fine particle sizes or highly variable compaction states, this static estimation method leads to inherent deviations between the predicted horizontal lateral pressure and the in-situ measured data, gradually becoming a major bottleneck restricting the refined design and numerical analysis of geotechnical engineering.

[0005] Existing methods struggle to take into account the combined dynamic effects of soil microstructure characteristics and macroscopic occurrence when dealing with soil mechanical responses under varying geological conditions, resulting in insufficient predictive robustness and adaptability of models under extreme conditions or complex sites.

[0006] Therefore, developing an accurate method to simultaneously quantify the influence of particle size and relative density on the coefficient of earth pressure at rest is of great significance for improving the safety of engineering projects. Summary of the Invention

[0007] Purpose of the invention: To provide a method for calculating the state-dependent static earth pressure coefficient of sand that takes into account particle size and density, so as to solve the above-mentioned problems existing in the prior art.

[0008] Technical solution: A method for calculating the state-dependent coefficient of earth pressure at rest for sand, considering particle size and density, comprising:

[0009] The state parameters of the sand sample to be tested are obtained. The state parameters include the degree of compaction parameter reflecting the compaction state of the sand sample to be tested, and the particle size parameter reflecting the particle size of the sand sample to be tested.

[0010] Based on the density parameter, the peak friction angle of the sand sample to be tested is determined by a preset friction angle growth equation.

[0011] Based on the peak friction angle and the particle size parameter, the static earth pressure coefficient of the sand sample to be tested is calculated using a preset static earth pressure characterization model.

[0012] According to one aspect of this application, the state parameters include relative density Dr and median particle size d50; wherein the density parameter is the relative density Dr and the particle size parameter is the median particle size d50.

[0013] According to one aspect of this application, the calculation of the peak friction angle φ is specifically as follows:

[0014] sinφ = a0 + a1*Dr;

[0015] In the formula, Dr is the relative density; a0 is the preset base value of the peak friction angle sine in the loosest state; a1 is the preset increase ratio of the peak friction angle sine as the relative density increases.

[0016] According to one aspect of this application, the at-rest earth pressure coefficient K0 is specifically:

[0017] K0 = b1*sinφ + b0;

[0018] In the formula, φ is the peak friction angle; b0 is the preset base value of the static earth pressure coefficient; b1 is the preset increase ratio of the static earth pressure coefficient as the sine value of the peak friction angle increases.

[0019] According to one aspect of this application, the basic value b0 of the at-rest earth pressure coefficient is specifically: b0 = c1*d50 +c0;

[0020] In the formula, d50 is the median particle size; c0 and c1 are both preset model parameters.

[0021] According to one aspect of this application, the base value a0 of the peak friction angle sine in the loosest state, and the increase ratio a1 of the peak friction angle sine with increasing relative density, are determined by a preset continuous function based on the median particle size d50.

[0022] According to one aspect of this application, the continuous function is specifically a logarithmic function, and the base value a0 and the increase ratio a1 are calculated using the following formulas:

[0023] a0 = α0 + α1 * ln(d50); a1 = α2 + α3 * ln(d50);

[0024] In the formula, ln() is the natural logarithm function; α0 is the preset reference constant for the sine value of the peak friction angle in the loosest state; α1 is the preset logarithmic influence coefficient of particle size on the basic friction angle; α2 is the preset reference constant for the density increase ratio; and α3 is the preset logarithmic influence coefficient of particle size on the density increase ratio.

[0025] According to one aspect of this application, taking into account the inhibitory effect of stress level on the dilatation of sand, the preset friction angle growth equation further includes a preset stress level correction term, and the peak friction angle φ is determined by the following formula:

[0026] sinφ = a0 + a1×Dr - a2×ln(p' / pa);

[0027] In the formula, p' is the average effective stress of the sand sample to be tested; pa is the preset atmospheric pressure reference value; and a2 is the preset stress level correction coefficient.

[0028] According to one aspect of this application, before calculating the coefficient of static earth pressure of the sand sample to be tested based on the peak friction angle and the particle size parameter using a preset static earth pressure characterization model, the method further includes:

[0029] The average effective stress p' of the sand sample to be tested is obtained;

[0030] Based on the average effective stress of the sand sample to be tested, the preset particle breakage initiation stress, and the preset particle breakage coefficient, the median particle size d50 is converted into the equivalent median particle size d50,eff.

[0031] In the subsequent parameter calculation model, the equivalent median particle size d50,eff is used to replace the median particle size d50.

[0032] According to one aspect of this application, the conversion of the median particle size d50 into an equivalent median particle size d50,eff specifically includes:

[0033] When the average effective stress p' is greater than the particle breakage initiation stress pcr, the equivalent median particle size d50,eff is calculated by the following formula: d50,eff = d50 * (p' / pcr)^(-λ);

[0034] When the average effective stress p' is not greater than the particle breakage initiation stress pcr, the equivalent median particle size d50,eff is equal to the median particle size d50; where λ is the particle breakage coefficient.

[0035] Beneficial Effects: This invention improves computational accuracy by coupling two key physical mechanisms: particle size effect and state correlation. The results show good agreement with measured data. The model is constructed based on clear mechanical principles, and all parameters have clear physical meanings, exhibiting good theoretical self-consistency. This method has a wide range of applications, effectively covering various sandy soil foundation conditions from extremely loose to extremely dense, and from fine sand to coarse sand, and demonstrates robust engineering extrapolation capabilities. Attached Figure Description

[0036] Figure 1 An implementation flow of a state-dependent method for calculating the static earth pressure coefficient of sand considering particle size and density according to an embodiment of the present invention.

[0037] Figure 2 Porosity-confining pressure relationship curve (fine sand).

[0038] Figure 3 Relationship between the sine of the peak friction angle and the relative density.

[0039] Figure 4 Relationship between the sine of the peak friction angle and the coefficient of earth pressure at rest. Detailed Implementation

[0040] Example 1: A method for calculating the state-dependent coefficient of earth pressure at rest in sand, considering particle size and density, specifically includes the following steps:

[0041] Step 101: Obtain the state parameters of the sand sample to be tested. The state parameters include a compaction parameter reflecting the compaction state of the sand sample to be tested, and a particle size parameter reflecting the particle size of the sand sample to be tested.

[0042] In this embodiment, the state parameter is a core physical quantity characterizing the mechanical properties of cohesionless soil. To accurately quantify these two properties, the density parameter is specifically set as relative density Dr, and the particle size parameter is specifically set as median particle size d50.

[0043] Specifically, the relative density Dr can be obtained by combining geotechnical tests. For example, a certain confining pressure can be applied to a sand sample before isotropic consolidation, and the initial void ratio and volume change under the current confining pressure can be recorded. A consolidation void ratio-confining pressure relationship curve can then be plotted, and the relative density Dr after consolidation can be calculated. The median particle size d50 is obtained by determining the particle size distribution curve of the sample through a standard sieve analysis after the sample has stopped undergoing volume deformation and has been disassembled. The particle size value corresponding to a cumulative passing percentage of 50% is then extracted from the curve. This method transforms the macroscopic soil condition into a quantitative index that can be directly used in subsequent mathematical models.

[0044] Step 102: Based on the compaction parameter, determine the peak friction angle of the sand sample to be tested using a preset friction angle growth equation.

[0045] In this step, the relative density Dr obtained previously is substituted as an input variable into the friction angle growth equation. This friction angle growth equation aims to characterize the nonlinear or linear evolution of the shear strength of sandy soil materials as a function of their density.

[0046] In traditional engineering practice, the Jaky empirical formula based on the critical friction angle is typically used to calculate the coefficient of earth pressure at rest. However, since the critical friction angle only reflects pure sliding friction between particles and its value does not change with soil density, traditional methods cannot distinguish the differences in lateral earth pressure evolution between loose and dense sand. In this embodiment, by using the peak friction angle instead of the critical friction angle, the confined shear dilatation effect of dense sand under one-dimensional compression conditions can be effectively captured. Specifically, the denser the sand, the stronger its particle interlocking effect, and the enhanced lateral stress transfer capacity under zero lateral deformation conditions, manifested as an increased peak friction angle, which in turn affects the subsequent earth pressure calculation results.

[0047] Step 103: Based on the peak friction angle and the particle size parameter, calculate the static earth pressure coefficient of the sand sample to be tested using a preset static earth pressure characterization model.

[0048] In this step, the peak friction angle calculated previously and the initially obtained median particle size d50 are used as joint input variables and substituted into the preset static earth pressure characterization model. The static earth pressure characterization model is a multivariate coupled calculation operator, whose internal logic mathematically integrates particle size effect and state correlation.

[0049] Furthermore, in the aforementioned static earth pressure characterization model, the median grain size d50 is typically used to determine the foundation offset of the earth pressure model, while the peak friction angle is used to control the dynamic increment of the earth pressure coefficient. Through this two-level mapping relationship, the static earth pressure coefficient output by the model is no longer a static constant determined solely by the material type, but rather a physical parameter that dynamically changes with the current compaction state of the soil and its inherent coarse-grained distribution characteristics. This calculation result can be directly used as the initial stress boundary condition for the stress analysis of large, deeply buried structures or deep foundation pit support systems.

[0050] Example 2 describes the offline acquisition process of the preset parameters required for the aforementioned calculation method.

[0051] Step 201: The preset parameters in the preset friction angle growth equation and the preset parameters in the preset static earth pressure characterization model are obtained through the following pre-calibration steps: Triaxial tests are performed on consolidated sand samples with different state parameters to obtain the relationship data between the peak friction angle and the relative density, so as to invert and obtain the preset parameters in the friction angle growth equation.

[0052] In this embodiment, offline parameter acquisition is used to bind a general mathematical model to the mechanical properties of a specific sand. To cover various situations that may be encountered in engineering, it is necessary to prepare consolidated sand samples with different relative densities. Specifically, triaxial shear tests can be performed on sand samples with different gradations, such as fine sand, medium sand, and coarse sand. Preset parameter inversion refers to determining the model constants in the friction angle growth equation by acquiring discrete physical test data points and using mathematical fitting methods such as the least squares method.

[0053] Step 202: For consolidated sand samples with the same median particle size and the same relative density, obtain the failure major principal stress σ1 and failure minor principal stress σ3 under at least three different confining pressures in triaxial tests.

[0054] Specifically, for sand samples with a specific median grain size and relative density, shear failure of the material at different stress levels is triggered by changing the confining pressure applied by the test system. For example, the confining pressure can be set to 100 kPa, 300 kPa, and 500 kPa. During triaxial shearing, the relationship between the principal stress difference and axial strain is continuously monitored, and the test is stopped when the axial strain reaches 16%. The ultimate stress state corresponding to the peak point is extracted from this relationship curve and decomposed into the major principal stress and minor principal stress of failure.

[0055] Step 203: Calculate the stress Mohr circle center value s using the formula p = (σ1+σ3) / 2, calculate half of the principal stress difference q using the formula q = (σ1-σ3) / 2, and plot the stress points for each triaxial test.

[0056] After obtaining the principal stresses of failure, they are mapped to the sq two-dimensional stress plane through coordinate transformation. Using the linear transformation formula described above, each failure state is transformed into a discrete stress point on the stress plane.

[0057] s = (σ1 + σ3) / 2;

[0058] Where s is the center value of the stress Mohr circle, σ1 is the major principal stress at failure, and σ3 is the minor principal stress at failure.

[0059] q = (σ1 - σ3) / 2;

[0060] Where q is half of the principal stress difference, σ1 is the major principal stress at failure, and σ3 is the minor principal stress at failure.

[0061] By plotting multiple stress points with the same density and different confining pressures in the same coordinate system, the shear strength envelope characteristics of the soil under the current state can be presented intuitively.

[0062] Step 204: Fit a linear envelope q = M*s passing through the origin to the stress point, and extract the slope M of the linear envelope as the sine value of the peak friction angle.

[0063] Since the apparent cohesion of clean, non-cohesive sand typically tends to zero, its strength envelope should physically pass through the origin. Therefore, a linear regression method forcing the stress to pass through the origin is used to fit the discrete stress points to obtain a linear envelope.

[0064] q = M * s;

[0065] Where q is half of the principal stress difference, M is the slope of the linear envelope, and s is the center value of the stress Mohr circle.

[0066] According to the geometric relationship of the Mohr-Coulomb failure criterion in soil mechanics, in a given two-dimensional stress plane, the slope of the envelope passing through the origin is numerically strictly equal to the sine of the peak friction angle. Furthermore, in some alternative implementations, if prior physical verification confirms that the cohesion of the target sand is strictly equal to 0, as a rapid alternative, a triaxial test can be performed only under a single confining pressure, and the sine of the peak friction angle can be directly calculated from a single Mohr circle. The calculation logic involves finding the difference between the major and minor principal stresses at failure, and then dividing by the sum of the two.

[0067] Step 205: Centrifuge model tests are conducted on consolidated sand samples with different state parameters to obtain the relationship data between the measured static earth pressure coefficient and the state parameters. The preset parameters in the friction angle growth equation obtained by inversion are then used to obtain the preset parameters in the static earth pressure characterization model.

[0068] Since the strict definition of the static lateral pressure coefficient (SPC) is the ratio of horizontal stress to vertical stress under conditions of no lateral deformation, conventional small-scale indoor tests are insufficient to accurately simulate the true self-weight stress state of deep soil layers. Therefore, this step utilizes centrifugal model tests, amplifying the gravity field through centrifugal acceleration to reproduce the in-situ self-weight stress distribution of the prototype soil layer in a scaled-down physical model. By performing simultaneous regression analysis of the measured SPC at each depth with the corresponding relative density and median particle size data, the various model parameters required for the aforementioned static earth pressure characterization model can be obtained through inversion.

[0069] Step 206: A rigid container is used as a centrifuge model box to maintain the zero lateral deformation condition of the consolidated sand sample.

[0070] Specifically, the centrifuge model test is conducted on a geotextile centrifuge, and the centrifuge model box is made of high-rigidity materials such as aluminum alloy. To meet the physical prerequisite of zero lateral deformation required for at-rest earth pressure calculation, the sidewalls of the centrifuge model box must possess sufficient physical stiffness; for example, the sidewall thickness can be set to no less than 20 mm. Furthermore, throughout the entire loading process, the horizontal deformation of the sidewalls must be strictly limited to below 0.05 mm. This rigid boundary physical constraint effectively suppresses the horizontal expansion of the test soil, thereby ensuring that the measured data conforms to the legal boundary conditions for at-rest earth pressure.

[0071] Step 207: Using the sand rain method, a consolidated sand sample with the target relative density is prepared in the centrifuge model box by pre-calibrating and determining the sand drop height.

[0072] In this embodiment, the sand rain method is a standard sample preparation technique that regulates the soil deposition density by controlling the kinetic energy of sand particles in free fall. Specifically, based on pre-conducted basic calibration tests, a parameter mapping relationship is established between the sand drop height, sieve aperture size, and relative density. During formal sample preparation, the system selects the corresponding sand drop height according to the target relative density and evenly scatters the sand. After sample preparation, the total mass of sand in the centrifuge model chamber is weighed, and the actual deposition volume of the sample is measured to verify whether the actual relative density achieved meets the model's expectations. Furthermore, the sand samples used in the experiment are typically kept air-dried.

[0073] Step 208: Arrange multiple lateral earth pressure sensors at equal intervals along the depth direction on the inner surface of the side wall of the centrifuge model box.

[0074] To obtain the lateral stress distribution pattern that continuously varies along the vertical depth, multiple monitoring nodes are pre-set on the side wall of the centrifuge model box. Specifically, the lateral earth pressure sensor can be a miniature earth pressure cell, whose sensing surface needs to be in direct and flush contact with the sand sample to avoid localized stress concentration or the introduction of additional boundary friction interference errors. For example, four sensors can be arranged at equal intervals along the depth direction, and each miniature earth pressure cell must undergo independent load calibration and data zeroing before installation.

[0075] Step 209: Control the centrifuge to rise to the target centrifugal acceleration, and obtain the readings of each of the lateral earth pressure sensors in a steady state as the measured lateral earth pressure.

[0076] During the loading phase of the experiment, the geocentrifuge was started and its acceleration was gradually and uniformly increased from 1g to the set target centrifugal acceleration. For example, this target centrifugal acceleration could be set to 50g, which corresponds to the vertical stress at a depth of 1 cm in the scaled-down model, strictly equivalent to the gravitational stress at a depth of 50 cm in the prototype site. To prevent damage to the sand particle interlocking structure from instantaneous inertial impacts, the acceleration rate should be constrained within a reasonable tolerance range, for example, set to no more than 5g per minute. After the geocentrifuge has been running stably at the target centrifugal acceleration level for a period of time, the sensor output is continuously monitored until the rate of change of all lateral earth pressure sensor readings over time is lower than a preset fluctuation tolerance threshold. The current stable reading is then recorded as the measured lateral earth pressure data at that specified depth.

[0077] Example 3 provides a statistical algorithm for calculating and maintaining the robustness of earth pressure at rest based on linear mapping. It specifically describes the hierarchical linear mapping relationship between state parameters and the coefficient of earth pressure at rest, as well as a statistical algorithm to improve the physical robustness of measured data.

[0078] Step 301, in determining the peak friction angle of the sand sample to be tested based on the compaction parameter and using a preset friction angle growth equation, the peak friction angle φ is specifically determined using the following formula:

[0079] sinφ = a0 + a1*Dr;

[0080] In the formula, Dr is the relative density; a0 is the preset base value of the peak friction angle sine in the loosest state; a1 is the preset increase ratio of the peak friction angle sine as the relative density increases.

[0081] In this embodiment, relative density is used as an input variable to quantify the degree of pore filling in sand. The formula establishes a linear growth mapping mechanism between the sine value of the peak friction angle and the relative density. The basic value of the sine value of the peak friction angle in the loosest state defines the theoretical lower limit of the shear friction resistance characteristics of the sand at extremely low density. The increase in the sine value of the peak friction angle with increasing relative density reflects the efficiency of increasing the shear strength of the soil by one unit increase in density. Typically, both parameters are positive. Furthermore, according to statistical analysis of offline triaxial test data, when the median particle size increases, the basic value of the sine value of the peak friction angle in the loosest state usually shows a decreasing trend. This is because the overall specific surface area of ​​coarse-grained sand is relatively small, resulting in a reduction in the total number of friction contact points between particles.

[0082] Step 302, in the calculation of the static earth pressure coefficient of the sand sample to be tested based on the peak friction angle and the particle size parameter using a preset static earth pressure characterization model, the static earth pressure coefficient K0 is specifically calculated using the following formula:

[0083] K0 = b1*sinφ + b0;

[0084] In the formula, φ is the peak friction angle; b0 is the base value of the static earth pressure coefficient preset based on the particle size parameter; b1 is the preset increase ratio of the static earth pressure coefficient as the sine value of the peak friction angle increases.

[0085] In this step, the sine of the peak friction angle is configured as an intermediate coupling variable to transmit the influence of density. In the model parameters, the percentage increase in the coefficient of earth pressure at rest as the sine of the peak friction angle increases is configured as a positive parameter. This positive property physically indicates that, under strictly zero lateral deformation conditions, the interlocking of particles within dense sand induces a confined dilatation effect, resulting in a higher efficiency of horizontal stress transmission compared to loose sand. This mechanism explains the objective law of a positive correlation between the coefficient of earth pressure at rest and density, overcoming the technical deficiency of traditional methods that fail to reflect the influence of density when estimating based on the critical friction angle.

[0086] The friction angle in the existing Jaky formula is the critical friction angle, whose value does not change with density. Therefore, the Jaky formula cannot distinguish between loose and dense sand. The friction angle in this invention is the peak friction angle, whose value increases with increasing density, thus incorporating the influence of density into the earth pressure calculation framework. The K0 benchmark offset caused by differences in particle size between different sand types is characterized separately by the particle size parameter in the aforementioned base value b0.

[0087] Step 303, the basic value b0 of the static earth pressure coefficient is calculated based on the particle size parameter using the following formula:

[0088] b0 = c1*d50 + c0;

[0089] In the formula, d50 is the median particle size; c0 and c1 are both preset model parameters.

[0090] Through this linear equation, the system incorporates the median particle size, which characterizes particle size, into the overall calculation framework. The constant term in the equation provides the basic offset, while the model parameter multiplied by the median particle size is responsible for dimensional transformation and scaling adjustment, with units in millimeters to the power of negative one. Substituting this basic value equation into the aforementioned formula for calculating the coefficient of earth pressure at rest yields a comprehensive output formula that couples particle size and compaction.

[0091] In some embodiments, normalization can also be used, such as b0 = c1*(d50 / dref) + c0.

[0092] Step 304: In obtaining the relationship data between the measured at-rest earth pressure coefficient and the state parameter, the measured at-rest earth pressure coefficient is calculated using the least squares regression method through the origin; specifically, it is calculated using the following formula:

[0093] K = ∑(σi * hi) / (γ * ∑(hi 2 ));

[0094] In the formula, K is the measured static earth pressure coefficient; i is the measuring point number; σi is the measured lateral earth pressure at the i-th measuring point; hi is the depth of the i-th measuring point; and γ is the measured unit weight of the consolidated sand sample.

[0095] When processing discrete data obtained from centrifuge model tests, robust measured representative values ​​need to be calculated to ensure the accuracy of the inversion of the preset parameters for the static earth pressure coefficient. Based on the surface boundary conditions, when the depth is 0, both the effective vertical self-weight stress and the lateral earth pressure are strictly zero. Therefore, the statistical regression line must physically pass through the origin of the coordinate system. Furthermore, the unit weight of the consolidated sand sample is kilonewtons per cubic meter; therefore, the unit weight is used instead of density to eliminate the order-of-magnitude calculation bias caused by gravitational acceleration. Using the depth squared term in the formula as the weighting denominator assigns higher statistical weight to the data from deeply buried measuring points, effectively suppressing the random measurement errors caused by boundary effects at shallow lateral earth pressure measuring points.

[0096] In some alternative implementations, for conditions where shallow boundary effects are insignificant or where all measuring points are located in deep homogeneous soil layers, an alternative simplified data processing scheme can be to directly calculate the measured at-rest earth pressure coefficient using the arithmetic mean method. Specifically, for each measuring point, the ratio of the measured lateral earth pressure to the corresponding vertical effective self-weight stress is calculated. Then, the ratios for all measuring points are arithmetically summed and divided by the total number of measuring points. This alternative scheme reduces the computational overhead of on-site data processing.

[0097] Example 4 provides an optimal algorithm for parameter continuity and complete model closure, which describes the optimal calculation process for expressing the friction angle evolution-related parameters as a continuous logarithmic function of the median particle size.

[0098] Step 401, the base value a0 of the peak friction angle sine value in the loosest state, and the increase ratio a1 of the peak friction angle sine value with the increase of relative density, are determined by a preset continuous function based on the median particle size d50.

[0099] In the basic algorithm provided in the aforementioned embodiments, the base value of the peak friction angle sine in the loosest state and the increase ratio of the peak friction angle sine with increasing relative density are configured as discrete material constants for a specific type of sand. When faced with a novel, unverified sand material, the computational system must reacquire offline test data to independently fit the above two discrete parameters, which leads to a generalization obstacle in the overall computation process. To solve this technical problem, this embodiment optimizes and upgrades the basic mapping algorithm. Specifically, by introducing a continuous function mapping mechanism, the originally fixed material constants are reconstructed into continuous physical variables that are dynamically calculated based on the median particle size. Through this continuous function mapping mechanism, after configuring a universal global reference constant once, the system only needs to receive the median particle size data of the sand sample to be tested to automatically calculate the friction angle evolution parameters corresponding to the specific sand, eliminating the computational dependence of repeatedly acquiring offline test data for new sand types.

[0100] Step 402, the continuous function is specifically a logarithmic function, and the base value a0 and the increase ratio a1 are calculated by the following formulas respectively: a0 = α0 + α1 * ln(d50) a1 = α2 + α3 * ln(d50) Where, ln() is the natural logarithmic function; α0 is the preset reference constant of the sine value of the peak friction angle of the loosest state; α1 is the preset logarithmic influence coefficient of particle size on the base friction angle; α2 is the preset reference constant of the density increase ratio; α3 is the preset logarithmic influence coefficient of particle size on the density increase ratio.

[0101] It's important to note that d50 is the median particle size in millimeters. In some cases, standardization can be applied, for example:

[0102] a0 = α0 + α1 * ln(d50 / dref); a1 = α2 + α3 * ln(d50 / dref); where dref is the preset reference particle size, and in engineering, the dimensional unit 1 can also be used.

[0103] Furthermore, the global reference constants α0, α1, α2, and α3 can be obtained through the following calibration method: Perform the triaxial test procedure described in Example 2 on at least three different median particle sizes of sand to obtain the baseline value a0 of the peak friction angle sine under the loosest state for each sand type and the increase ratio a1 of the peak friction angle sine with increasing relative density; subsequently, using the natural logarithm of the ratio of the median particle size to the reference particle size for each sand type as the independent variable, perform linear regression on a0 and a1 respectively to obtain the intercept and slope, which are the four global reference constants mentioned above. These parameters can be obtained using existing fitting methods.

[0104] In this embodiment, a logarithmic function is configured as the core continuous function form for parameter evolution. The shear friction mechanical properties of sand are controlled by the combined effects of particle specific surface area and microscopic contact mechanical state. Soil mechanics spatial relationships show that particle specific surface area exhibits a non-linear decreasing trend with increasing median particle size, and shows a linear correlation in a logarithmic coordinate system. Therefore, configuring a logarithmic function can capture the physical mechanism of parameter variation with median particle size. Specifically, the logarithmic influence coefficient of particle size on the foundation friction angle is configured as negative, indicating that as the obtained median particle size increases, the relatively small specific surface area of ​​coarse particles leads to a reduction in the total number of shear friction contact points in the loosest state. Conversely, the logarithmic influence coefficient of particle size on the proportion of increase in compaction is configured as positive, indicating that as the obtained median particle size increases, the geometric interlocking effect and constrained shear dilatation effect produced by coarse particles in a densely packed state are more pronounced.

[0105] Furthermore, considering the mathematical extrapolation divergence characteristic of the natural logarithmic function when the independent variable approaches 0, to prevent the underlying algorithm logic from outputting negative results that lose physical and mechanical meaning, it is necessary to configure boundary constraint intervals for the input parameters of the aforementioned logarithmic function model. For example, the effective calculation range of the median particle size can be constrained to between 0.1 mm and 2.0 mm. This constraint interval fully covers the distribution range of fine to coarse sand in conventional engineering geological classifications. When the obtained median particle size is lower than 0.1 mm or higher than 2.0 mm, the system can trigger a parameter exceeding-limit anomaly and suspend the execution of subsequent coefficient mapping processes.

[0106] To more clearly explain the forward inference logic of parameter continuity and complete model closure in this embodiment, a simplified numerical calculation example using dimensionless variables is provided. Assume that, after calibration, the pre-configured baseline constant α0 for the sine of the peak friction angle in the loosest state is 0.40, and the logarithmic influence coefficient α1 of particle size on the basic friction angle is -0.05. The system obtains the median particle size d50 of a certain sand sample after normalization as 1.0. Substituting this input into the logarithmic equation of the basic value, the natural logarithm ln(1.0) is calculated to be 0, thus deducing the baseline value a0 of the sine of the peak friction angle in the loosest state as 0.40. Assume that the system simultaneously calculates, using another set of constants, the increase ratio a1 of the sine of the peak friction angle with increasing relative density is 0.20. When the system receives an input relative density Dr of 0.8, substituting this into the friction angle growth equation yields the peak friction angle sine value for this state: sinφ = 0.40 + 0.20 * 0.8 = 0.56. Finally, combining the preset parameters of the at-rest earth pressure characterization model, such as configuring the amplification ratio b1 to 1.3 and the base value b0 to -0.30, the final at-rest earth pressure coefficient K0 is calculated to be: 1.3 * 0.56 - 0.30 = 0.428. Through the above calculation process, it can be seen that, relying on the logarithmic continuous function module, the system can complete the closed-loop output calculation of the at-rest earth pressure coefficient using only the two acquired basic state parameter inputs.

[0107] Example 5 describes an alternative calculation method for the static earth pressure coefficient, used to consider the inhibitory effect of high confining pressure on the shear dilatation of sand particles in scenarios of deep burial or large stress changes.

[0108] Step 501, considering the inhibitory effect of stress level on the dilatation of sand, the preset friction angle growth equation also includes a preset stress level correction term, which is specifically determined by the following formula to replace the peak friction angle φ:

[0109] sinφ = a0 + a1×Dr - a2×ln(p' / pa)

[0110] In the formula, p' is the average effective stress of the sand sample to be tested; pa is the preset atmospheric pressure reference value; and a2 is the preset stress level correction coefficient.

[0111] In this embodiment, for engineering scenarios such as caisson foundations or deep foundation pit support, the soil depth varies greatly, resulting in significant differences in effective stress at different depths. In the aforementioned basic embodiment, the peak friction angle is only characterized as a function of relative density, thus treating the coefficient of earth pressure at rest as a constant independent of depth. However, soil mechanics theory shows that under high confining pressure, the dilatational effect between sand particles is physically suppressed, causing the peak friction angle to decrease with increasing stress level. To eliminate the calculation bias caused by a single density variable in deep scenarios, this embodiment constructs an alternative model considering the three-dimensional coupling of stress, density, and particle size.

[0112] In practice, the calculation system obtains the average effective stress representing the current burial depth stress state, divides it by the atmospheric pressure reference value to achieve dimensionlessness, and then calculates the natural logarithm. This logarithmic term is multiplied by a preset stress level correction coefficient to form a negative penalty term representing the dilatation suppression effect. This correction coefficient is always greater than 0 to ensure that the reduction in the calculated peak friction angle increases monotonically with the increase of the average effective stress. By executing this calculation logic, the final calculated static earth pressure coefficient is transformed into a variable that dynamically varies with depth, suitable for lateral stress extrapolation in engineering scenarios with large depths.

[0113] Furthermore, regarding the physical extraction process of the preset parameters, the stress level correction coefficient needs to be inverted using offline triaxial test data. During the calibration phase, the relative density of the sand samples is kept constant, and the peak friction angle of the samples is measured under at least three different confining pressure conditions. Subsequently, a multiple linear regression statistical algorithm is invoked to perform matrix fitting on the acquired stress state dataset, simultaneously outputting the base value under the loosest state, the density increase ratio, and the stress level correction coefficient. Typically, the system configures the atmospheric pressure reference value to 101.325 kPa.

[0114] As a backward-compatible alternative to the above calculation scheme, when the engineering project has a shallow structure and the stress level variation is extremely small, the calculation system can reset the stress level correction coefficient to 0, or forcibly lock the input average effective stress to an atmospheric pressure reference value. Under this parameter setting, the natural logarithm term or correction product in the alternative formula is calculated to zero, and the alternative formula is mathematically strictly degenerated into a basic friction angle growth equation containing only the relative density variable. Through this mechanism design, the system achieves parameterized switching between the deep-buried high-stress model and the conventional shallow benchmark model without changing the main calculation framework.

[0115] Example 6 describes a pre-correction algorithm for fragile sand with equivalent particle size. This algorithm addresses the pre-correction of the original median particle size by performing an equivalent reduction transformation when fragile sand undergoes particle breakage under high stress conditions.

[0116] Step 601: Based on the obtained average effective stress of the sand sample to be tested, the preset particle breakage initiation stress, and the preset particle breakage coefficient, the median particle size d50 is converted into an equivalent median particle size d50,eff; in the subsequent parameter calculation model, the equivalent median particle size d50,eff is used to replace the median particle size d50.

[0117] In this embodiment, marine engineering often involves brittle soil materials such as calcareous sand, coral sand, or weathered rock sand. These materials are prone to significant particle breakage in high-stress environments such as the deep seabed or high dam foundations, causing the actual particle size of the sample to gradually decrease with increasing stress levels. If the original median particle size of the soil under initial uncompressed conditions is used to calculate the static earth pressure coefficient, the theoretical calculation model will become disconnected from the actual physical state. To enable the model to cover such extreme physical conditions, the system intervenes with pre-correction logic for the initial state parameters before executing the aforementioned basic calculation process. Specifically, the average effective stress characterizing the current environmental load is obtained and compared numerically with the preset particle breakage initiation stress characterizing the material's fracture resistance. Combined with the preset particle breakage coefficient, the updated equivalent median particle size d50,eff is calculated and output. Further, after completing the above conversion calculation, the system uses the equivalent median particle size d50,eff carrying the reduction attribute as a new state parameter, which is then passed over to the subsequent basic static earth pressure characterization model for correlation calculation.

[0118] Step 602: When the average effective stress p' is greater than the particle breakage initiation stress pcr, the equivalent median particle size d50,eff is calculated using the following formula:

[0119] d50,eff = d50 * (p' / pcr)^(-λ);

[0120] When the average effective stress p' is not greater than the particle breakage initiation stress pcr, the equivalent median particle size d50,eff is equal to the median particle size d50; where λ is the particle breakage coefficient; ^ represents exponential operation.

[0121] In some cases, standardization is required to convert the above data into d50,eff / dref. In this step, the system incorporates a segmented attenuation judgment logic based on a stress threshold. When the obtained average effective stress p' > pcr, the particle breakage attenuation calculation module is triggered. In the above formula, the ratio of the average effective stress p' to the particle breakage initiation stress pcr is used as the dimensionless stress base, and the particle breakage coefficient λ is used as the nonlinear attenuation exponent. Since the particle breakage coefficient is usually positive, the entire power term constitutes a reduction multiplier less than 1, resulting in a monotonically decreasing equivalent median particle size. When the obtained average effective stress p' <= pcr, the system determines that no particle breakage has occurred that causes a change in macroscopic physical parameters. In this case, the exponential attenuation calculation logic is bypassed, and the original median particle size d50 is directly assigned to the equivalent median particle size d50,eff.

[0122] The methods for obtaining the preset particle breakage initiation stress (pcr) and particle breakage coefficient (λ) need to be completed through a specific offline physical calibration program. Specifically, at least five isotropic compression tests under different confining pressure gradients are conducted on the target easily breakable sand to ensure that the set stress test range fully covers the expected breakage interval. After the sample compression reaches a stable state, a sample disassembly and sieving operation is performed, and the actual median particle size under the current confining pressure is measured. By plotting the relationship curve between the median particle size and the confining pressure, the inflection point representing the breakage triggering state on the curve is extracted, and the confining pressure value corresponding to this inflection point is set as the particle breakage initiation stress (pcr). At the same time, the descending segment of the curve after the inflection point is extracted, and linear regression calculation is performed in a double logarithmic coordinate system. The slope of the obtained regression line is set as the dimensionless particle breakage coefficient (λ).

[0123] As a specific material implementation method of the above technical solution, for conventional hard structural materials such as quartz sand, particle breakage is usually considered insignificant within the range of conventional engineering stress not exceeding 1 MPa. In this case, the system's input condition consistently satisfies p' <= pcr, the piecewise function automatically locks into the calculation branch that does not perform reduction, and the equivalent median particle size is always equal to the initial median particle size. This mechanism design ensures that the pre-correction algorithm can achieve backward compatibility and seamless switching of parameters with the aforementioned basic hard sand example.

[0124] Example 7: A method for calculating the state-dependent coefficient of earth pressure at rest in sand, considering particle size and relative density. This method can more accurately predict the actual coefficient of earth pressure at rest in sand, and includes the following steps:

[0125] (1) Preparation of consolidated sand samples with different state parameters: Confining pressure (σ3) was applied to the triaxial sand samples, and the consolidation void ratio-confining pressure relationship curve and median particle size-confining pressure relationship curve of the samples were obtained. The state parameters of the samples in the subsequent formal test were calculated based on these two curves.

[0126] The state parameter refers to the median particle size d. 50 Relative density D r ;

[0127] The porosity-confining pressure relationship curve of the sample is obtained as follows: after equivalent consolidation of the sample, σ3 is applied, the current σ3 value and the volumetric strain of the sample at the current σ3 are recorded, and the relative density D is obtained by combining the initial porosity of the sample. r ;

[0128] The method for obtaining the median particle size-confining pressure relationship curve of the sample is as follows: after the volumetric deformation of the sample stops, the current σ3 value is recorded and the sample is disassembled. The gradation curve of the sample is determined by sieving, and the median particle size d is obtained. 50 ;

[0129] (2) Triaxial tests were conducted on consolidated sand samples with different state parameters to obtain different median particle sizes d. 50 The peak friction angle φ and relative density D r Based on the relationship, establish the equation for the growth curve of the peak friction angle;

[0130] The triaxial test conditions include at least three confining pressures and three relative densities.

[0131] The equation for the growth curve of the peak friction angle is in the form of:

[0132] sinφ = a0 + D r × a1 (1)

[0133] In the formula, a0 is the base value of the peak friction angle sine, that is, the peak friction angle sine in the loosest state; a1 is the increase ratio of the peak friction angle sine as the relative density increases.

[0134] (3) Conduct centrifugation model experiments to obtain different median particle sizes d. 50 The coefficient of earth pressure at rest K and the relative density D r Based on the relationship, establish the characterization equation for the coefficient of earth pressure at rest;

[0135] In the centrifugal model test, the lateral earth pressure at different median particle size, different relative density, and different depths needs to be measured, including at least three relative densities, three depths, and three median particle sizes.

[0136] The characterization equation for the coefficient of earth pressure at rest is established as follows:

[0137] 1) First, establish the characterization equations for the coefficient of earth pressure at rest K and the peak friction angle φ, in the form:

[0138] K = b1 sinφ + b0 (2)

[0139] In the formula, b0 is the basic value of the coefficient of earth pressure at rest; b1 is the percentage increase of the coefficient of earth pressure at rest as the sine value of the peak friction angle φ increases.

[0140] It should be noted that the at-rest earth pressure coefficient K in formula (2) is calculated based on the lateral earth pressure measured at different depths. As a simplified data processing scheme, the calculation formula is as follows:

[0141] (3)

[0142] In the formula, σ i h represents the lateral earth pressure at the i-th measuring point. i Let be the depth of the i-th measuring point; γ be the unit weight of the soil; and n be the number of measuring points. Alternatively, the least squares regression method through the origin described above can be used.

[0143] 2) Secondly, establish the basic value b0 of the coefficient of earth pressure at rest and the median particle size d. 50 Characterization equation:

[0144] b0 = c1d 50 +c0 (4)

[0145] In the formula, c0 and c1 are model parameters.

[0146] 3) Finally, by substituting formulas (1) and (4) into formula (2), we obtain:

[0147] K=b1(a1D r + a0) + c1d 50 +c0 (5)

[0148] To address the technical problem that existing technologies struggle to simultaneously consider both microscopic components and macroscopic conditions, leading to inherent biases in predictions, this invention constructs a graded mapping model by acquiring the relative density and median particle size of sand. This model objectively reflects the effects of particle interlocking and confined dilatation, and transforms the calculation of the static earth pressure coefficient from a static, single-variable constant system to a dynamic, bivariate coupled system, significantly improving the simulation accuracy of initial stress boundary conditions for conventional sandy soil foundations.

[0149] To address the technical problem of poor site adaptability caused by discrete model parameters, this invention transforms the originally fixed friction angle constant into a continuous logarithmic function that depends on the particle size, thereby achieving complete parameter closure of the calculation model and completely eliminating the need for repeated calibration when the system faces unknown new sand types.

[0150] To address the issue of insufficient robustness in predictions under extreme working conditions and complex sites, this invention implements technical breakthroughs from three dimensions: Firstly, it introduces a shear dilatation suppression correction penalty term for deep-buried high-stress scenarios, enabling the lateral earth pressure prediction to dynamically and adaptively adjust with depth; secondly, it introduces a particle breakage equivalent particle size conversion mechanism for fragile marine sands, avoiding model distortion caused by physical reduction in particle size under high confining pressure; and thirdly, it configures a least-squares regression statistical algorithm through the origin in the bottom-level data calibration stage, effectively suppressing random error interference from shallow physical boundary effects, thus solidifying the physical robustness of static earth pressure prediction for large-scale engineering projects from the data source.

[0151] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A method for calculating the state-dependent coefficient of earth pressure at rest in sand, considering particle size and density, characterized in that, include: The state parameters of the sand sample to be tested are obtained. The state parameters include the degree of compaction parameter reflecting the compaction state of the sand sample to be tested, and the particle size parameter reflecting the particle size of the sand sample to be tested. Based on the density parameter, the peak friction angle of the sand sample to be tested is determined by a preset friction angle growth equation. Based on the peak friction angle and the particle size parameter, the static earth pressure coefficient of the sand sample to be tested is calculated using a preset static earth pressure characterization model.

2. The method according to claim 1, characterized in that, The state parameters include relative density Dr and median particle size d50; wherein, the density parameter is the relative density Dr, and the particle size parameter is the median particle size d50.

3. The method according to claim 2, characterized in that, The calculation of the peak friction angle φ is as follows: sinφ = a0 + a1*Dr; In the formula, Dr is the relative density; a0 is the preset base value of the peak friction angle sine in the loosest state; a1 is the preset increase ratio of the peak friction angle sine as the relative density increases.

4. The method according to claim 3, characterized in that, The at-rest earth pressure coefficient K0 is specifically: K0 = b1*sinφ + b0; In the formula, φ is the peak friction angle; b0 is the preset base value of the static earth pressure coefficient; b1 is the preset increase ratio of the static earth pressure coefficient as the sine value of the peak friction angle increases.

5. The method according to claim 4, characterized in that, The basic value of the static earth pressure coefficient, b0, is specifically: b0 = c1*d50 + c0; In the formula, d50 is the median particle size; c0 and c1 are both preset model parameters.

6. The method according to claim 3, characterized in that, The base value a0 of the peak friction angle sine in the loosest state, and the increase ratio a1 of the peak friction angle sine with increasing relative density, are determined by a preset continuous function based on the median particle size d50.

7. The method according to claim 6, characterized in that, The continuous function is specifically a logarithmic function, and the base value a0 and the growth rate a1 are calculated using the following formulas: a0 = α0 + α1 * ln(d50); a1 = α2 + α3 * ln(d50); In the formula, ln() is the natural logarithm function; α0 is the preset reference constant for the sine value of the peak friction angle in the loosest state; α1 is the preset logarithmic influence coefficient of particle size on the basic friction angle; α2 is the preset reference constant for the density increase ratio; and α3 is the preset logarithmic influence coefficient of particle size on the density increase ratio.

8. The method according to claim 3, characterized in that, Considering the inhibitory effect of stress level on the dilatation of sand, the preset friction angle growth equation also includes a preset stress level correction term, and the peak friction angle φ is determined by the following formula: sinφ = a0 + a1×Dr - a2×ln(p' / pa); In the formula, p' is the average effective stress of the sand sample to be tested; pa is the preset atmospheric pressure reference value; a2 is the preset stress level correction factor.

9. The method according to claim 2, characterized in that, Before calculating the coefficient of static earth pressure of the sand sample based on the peak friction angle and the particle size parameter using a preset static earth pressure characterization model, the method further includes: The average effective stress p' of the sand sample to be tested is obtained; Based on the average effective stress of the sand sample to be tested, the preset particle breakage initiation stress, and the preset particle breakage coefficient, the median particle size d50 is converted into the equivalent median particle size d50,eff. In the subsequent parameter calculation model, the equivalent median particle size d50,eff is used to replace the median particle size d50.

10. The method according to claim 9, characterized in that, The step of converting the median particle size d50 into an equivalent median particle size d50,eff specifically includes: When the average effective stress p' is greater than the particle breakage initiation stress pcr, the equivalent median particle size d50,eff is calculated by the following formula: d50,eff = d50 * (p' / pcr)^(-λ); When the average effective stress p' is not greater than the particle breakage initiation stress pcr, the equivalent median particle size d50,eff is equal to the median particle size d50; where λ is the particle breakage coefficient.

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