Study on Calculation Method of Foundation Nonlinear Settlement Based on In Situ Loading Test

. In this paper, a nonlinear deformation modulus method is proposed for foundation settlement calculation. In the proposed method, the nonlinear deformation modulus under di ﬀ erent stress levels is obtained from the load-settlement curve of in situ loading test, which are then applied to the layerwise summation method for calculating foundation settlement. On this basis and referring to the Duncan-Chang model, a variable modulus constitutive model suitable for numerical calculation of foundation settlement is further proposed. The required parameters of this model are the same as those of the nonlinear deformation modulus method and can be determined by the in situ loading test. The validity of the proposed calculation methods of foundation settlement is veri ﬁ ed by the in situ loading tests under di ﬀ erent plate sizes. The results illustrate that both the nonlinear deformation modulus method and the variable modulus constitutive model compare quite well with the test results, and the deduced results can better re ﬂ ect the nonlinearity of foundation settlement.


Introduction
The settlement calculation of foundation is a primary and significant content in the field of soil mechanics.The commonly used settlement calculation methods are mainly focused on soil sampling and laboratory compression testing, and the deduced compression curve is used for settlement calculation.Even though this method is relatively simple and practical, there exists two major unneglectable theoretical deficiencies: (i) soil sample disturbance can occur during sampling; (ii) the stress and deformation conditions of the laboratory compression test are quite different from the field conditions, which makes the calculation error estimation become difficult.Since the laboratory test cannot reflect the actual deformation characteristics of the in situ soil very well, it is difficult to deduce accurate foundation settlement results even using modern advanced numerical calculation methods.Thus, the better way is to improve the acquisition of calculation parameters, namely, calculating the foundation settlement based on parameters obtained from in situ tests.This can improve the accuracy of calculated foundation settlement, which can more accurately reflect deformation characteristics of foundations.Currently, some methods for calculating foundation settlement based on in situ test results are reported [1][2][3][4][5].For example, Li [4] proposed a calculation method for foundation settlement based on cone penetration test considering different stratum conditions.Zhang et al. [5] proposed a nonlinear settlement calculation method based on the calculation parameters obtained from the pressuremeter tests.However, most of these methods are mainly provided by experiences.
The in situ loading test is considered to be one of the most reliable methods for determining the ultimate bearing capacity of foundation.The simultaneous deduced load-settlement curve can realistically reflect the deformation characteristics of the foundation during the loading process.Moreover, the deformation modulus of the in situ soil also can be determined, which can be effectively applied to the settlement calculation of foundations [6][7][8][9][10].In addition, the actual settlement process of foundation is nonlinear, and the use of a single constant deformation modulus cannot truly reflect the nonlinear deformation characteristics of the foundation affected by loadings.Thus, in this study, the nonlinear deformation modulus under different stress levels is obtained from the load-settlement curve of in situ loading test, which are then applied to the layerwise summation method and numerical constitutive model for calculating the settlement of foundations.Based on these treatments, a calculation method of foundation nonlinear settlement based on in situ loading test is established and proposed.The validity of the proposed calculation method is verified by the in situ loading tests under different plate sizes.

Nonlinear Deformation Modulus Method for
Foundation Settlement Calculation 2.1.Nonlinear Deformation Modulus Method.Assuming a soil layer with a thickness Δh j located at a depth h j is subjected to a small incremental load Δp i , the deformation of this soil layer can be approximately viewed as linear process and given as follows.
where E ij is the deformation modulus of the soil layer at a depth h j under an external loading p i ; α is the stress distribution coefficient, and Δp i α defines the stress increment generated by the incremental load Δp i .Then, according to the layerwise summation method, the total settlement of all layers under incremental load Δp i can be summarized as where n refers to the total numbers of soil layers.
The key aspect of the above calculation method is the reasonable determination of the deformation modulus E ij .It can be seen from previous analysis that the loadsettlement curve obtained by the in situ loading test can more accurately reflect the deformation characteristics of foundations during the loading process, and the deformation modulus of the foundation soil can be determined accordingly.Previous researches [11][12][13][14] indicate that the load-settlement (p-s) curve obtained by the in situ loading test is generally expressed by a hyperbolic function.
where a and b are unknown coefficients that need to be determined.According to the characteristics of hyperbolic function and the theory of elastic mechanics [11] where D is the side length or diameter of the in situ loading test plate; μ is the Poisson's ratio of the soil; ω is the coefficient that reflects the shape and stiffness of the plate, for rigid square and circular plates, and the values of ω respective are equal to 0.88 and 0.79; E 0 is the initial deformation modulus of the soil; while p u is the ultimate load.
In addition, the foundation settlement caused by the in situ loading test can be calculated approximately by the Boussinesq solution of elastic mechanics: where E is the deformation modulus of soil at the bottom of the plate corresponding to the applied loading p.
Transformation from the Equation ( 5) can be found that From the Equation (3), the tangent derivative of p-s curve, deduced from the in situ loading test, at any point can be expressed as Assuming that Δp/Δs = dp/ds and substituting Equations ( 7) and ( 4) into Equation ( 6), the deformation modulus of 2 Geofluids the soil at the bottom of the loading plate is In the above equation, p/p u is the ratio of the load p to the ultimate load p u of foundation soil at the bottom of the loading plate, which reflects the influence of stress level on the deformation modulus E of the soil.For soils at different depths, the additional load p after diffusion decreases, and the corresponding ultimate load p u increases with an increase in the depth, which leads to an increase in the deformation modulus E according to Equation ( 8).It conforms to the nonlinear deformation characteristics of foundation.Then, applying the deformation modulus of soil at different depths to the layerwise summation method, namely, substituting Equation ( 8) into Equations ( 1) and ( 2), the settlement calculation of the foundation is performed, which is called as nonlinear deformation modulus method in this paper.

Parameter Determinations Based on In Situ Loading Tests.
From above section, it can be concluded from Equation (8) that the nonlinear deformation modulus method needs to determine the ultimate load p u of the soil and the initial deformation modulus E 0 , wherein p u can be calculated according to the cohesion c and internal friction angle φ of soil and the plate size.Therefore, three soil parameters of soil (i.e., c, φ, and E 0 ) are required to be determined for the nonlinear deformation modulus method.In order to overcome the shortcomings of the parameters determination in the laboratory test, in this study, these three parameters are obtained based on the in situ loading tests.
Note that the load-settlement (p-s) curve can be readily obtained from the in situ loading tests.Using the hyperbolic function as an input to fit the p-s curve, the coefficients a and b can be easily deduced, which can be directly applied to inverse calculation of parameters c, φ, and E 0 for the nonlinear deformation modulus method.
The c and φ values can be inversely calculated from Vesic's ultimate bearing capacity equation.
where N c , N q , and N γ are the bearing capacity coefficients and can be obtained according to the φ value; S c , S q , and S γ are the plate shape factors of the in situ loading test; q is the surcharge loading applied on both sides of the plate; γ is the unit weight of the soil, which equals to effective weight when below the groundwater level.
It should be pointed out that, in Equation ( 9), there are only two unknown parameters, namely, c and φ.For sand, c = 0 and φ can be inversely calculated from the ultimate bearing capacity (p u ) of foundation soils obtained from the in situ loading tests.For clay, then, one of these two parameters is assumed empirically, and the value of the other parameter can be inversely calculated from the ultimate bearing capacity (p u ).
The initial deformation modulus E 0 of the soil is determined by the Boussinesq solution of the elastic mechanics.When the hyperbolic function coefficient a is fitted according to the ps curve of the in situ loading tests, the expression for E 0 value is

Variable Modulus Constitutive Model for Foundation Settlement Calculation
Currently, the determinations of constitutive model parameters of soils are mainly focused on laboratory tests.However, for strong structural soils, the results from laboratory tests may be greatly influenced due to the stress release and disturbance caused by sampling, which leads to a quite different results compared with the actual ones.Thus, it is difficult to ensure the accuracy of calculated results according to laboratory test parameters.Meanwhile, as the constitutive properties of soils are quite complicate, it is almost impossible to establish a constitutive model that combines all characteristics of the soils.Under such specific case, establishing a practical constitutive model that reflects the main characteristics and parameters that are easy to measure is more effective and of great significance.
It should be mentioned that the main feature of foundation settlement problem is that the deformation of soils is nonlinear [15][16][17].The Duncan-Chang model based on the generalized Hooke's law is widely used in practical 3 Geofluids engineering because it is easy to determine the required parameters and can further reflect the most prominent deformation characteristics (i.e., nonlinearity) of soils.However, as these require parameters are usually determined by the conventional triaxial compression test in the laboratory, it also has certain limitations especially for some strong structural soils.In order to combine the advantages and make up for the limitations in parameter determinations of Duncan-Chang model, a simplified Duncan-Chang model, called variable modulus constitutive model in this paper, is proposed on the basis of the nonlinear deformation modulus method and Duncan-Chang model.Hence, the constitutive where dσ x , dσ y , dσ z , dτ xy , dτ yz , and dτ zx are the incremental stress components; dε x , dε y , dε z , dγ xy , dγ yz , and dγ zx are the incremental strain components; ½D is the elastic matrix and given by where E and μ define the deformation modulus and Poisson's ratio of the soil, respectively.In the variable modulus constitutive model, the simplified expressions of these two parameters are as follows: In the above two equations, E 0 and μ 0 , respectively, refer to the initial deformation modulus and Poisson's ratio of the soil.μ f is the Poisson's ratio of the soil under ultimate failure state and generally equal to 0.49.ðσ 1 − σ 3 Þ f is the ultimate strength of soils based on the Mohr-Coulomb failure criterion and expressed as From above analysis, it can be concluded that the basic parameters of the variable modulus constitutive model include cohesion c, internal friction angle φ, initial deformation modulus E 0 , and initial Poisson's ratio μ 0 .In general, μ 0 is approximately set to 0.3, and other three soil parameters of soil (i.e., c, φ, and E 0 ) can be determined by the in situ loading tests.There-fore, a variable modulus constitutive model based on in situ test parameters can readily be established as a simple but practical constitutive model for foundation settlement calculations.

In Situ Tests and Verifications
4.1.In Situ Tests.In order to verify both the nonlinear deformation modulus method and variable modulus constitutive model for foundation settlement calculation, the representative strong structural granite residual soil in Guangzhou (China) was selected, and several series of in situ loading tests with the size of square plates equal to 1 m 2 (1 m × 1 m ) and 2 m 2 (1:4 m × 1:4 m) were carried out.The obtained ps curves of in situ plate loading tests are shown in Figure 1.It can be seen that test 1# was loaded by 50 kPa each stage, and tests 2# and 3# were loaded by 60 kPa each stage, and the ps curves of the two in situ load tests 1# and 2# with identical plate size (1 m 2 ) are basically the same.In addition, a rapid increase in the settlement indicates that the loading applied to the test 2# almost reaches the ultimate bearing capacity of the granite residual soil.
For comparison, laboratory tests, including soil physical property tests and conventional triaxial compression tests, also carried out soil samples.The sampling depths are 1.0 m and 2.5 m, respectively.The deduced test results are shown in Table 1 and Figure 2   5 Geofluids obtained from the in situ loading test results.In present study, the parameters c, φ, and E 0 of granitic residual soil are determined based on the results of in situ loading test 2# (as shown in Figure 1).The nonlinear deformation modulus method is then adopted to calculate the ps curves, which are compared with the results of in situ loading tests 1# and 3# and to validate the rationality of the nonlinear deformation modulus method.
From the results of the in situ loading test 2# (as shown in Figure 1), a rapid increase in the settlement was observed when the applied loading increases from 780 kPa to 840 kPa, which indicates that this applied loading almost reaches the ultimate bearing capacity of the granite residual soil.Taking the average value of the two-stage loadings as the ultimate bearing capacity, thus the value of p u equals to 815 kPa, and the parameters c and φ can be inversely calculated after substituting the p u values into Equation (9).According to the empirical value of the soil parameters in Guangzhou, it is assumed that the frictional angle of the granite residual soil (φ) is equals to 25 °; then, the calculated cohesion of soils (c) is 25 kPa.
After rewriting the Equation ( 3), the simplified expression of the form is Using Equation (15) to fit the ps curve of the in situ loading test 2#, the fitted relationship is y = s p = 0:0011s + 0:0261: Figure 3 displays the fitted relationship of results of in situ loading test 2#.As can be seen from Figure 3, these results can be extremely well expressed as hyperbolic func-tions, and the fitted hyperbolic function coefficient α is 0.0261.Thus, the initial deformation modulus of the granite residual soil can be obtained from Equation (10) and given as follows.
Table 2 presents the calculation parameters c, φ, and E 0 of the nonlinear deformation modulus method that are obtained according to the results of the in situ loading test 2#.The calculated results of the ps curves with identical loading conditions as in situ tests 1# and 3# by the nonlinear deformation modulus method are shown in Figure 4.The associated in situ test results are also included.It can be seen that the ps curves calculated by the nonlinear deformation modulus method compare very well with the in situ test results, which primarily verifies the rationality of the nonlinear deformation modulus method.

Verification of the Variable Modulus Constitutive
Model.For the variable modulus constitutive model, the three parameters c, φ, and E 0 are also required, and they also can be obtained according to the in situ loading test results.Similarly, the values of c, φ, and E 0 of the granite residual soil are determined based on the results of in situ loading test 2# and presented in Table 3.In this section, the rationality of the variable modulus constitutive model is verified by comparisons of numerical results using the FLAC 3D and the results of in situ loading tests 1# and 3#.The calculation mesh is shown in Figure 5, and the associated comparisons  6 Geofluids are shown in Figure 6.The ps curves calculated by numerical analysis based on the variable modulus constitutive model match well with the in situ test results, which verify the rationality of the proposed variable modulus constitutive model.It is known that the Duncan-Chang model can reflect the nonlinear deformation characteristics of foundation soils, but the required parameters are primarily obtained from conventional triaxial tests.In order to compare with the calculated results of the variable modulus constitutive model, the associated numerical results of the Duncan-Chang model are also included in Figure 6.The calculated parameters of the Duncan-Chang model are shown in Table 4.
As can be seen from Figure 6, the numerical results of the Duncan-Chang model based on conventional triaxial test parameters are quite different from in situ loading test results and numerical results of the variable modulus constitutive model.The main reason may be that the granite residual soil has strong structural characteristics [18][19][20][21], and the effects of stress release and disturbance caused by sampling   7 Geofluids lead to large differences between laboratory test parameters and actual ones.Even though the variable modulus constitutive model is a simplified model based on the Duncan-Chang model, it can reflect the nonlinear deformation of the foundation soil, and its parameters are obtained from in situ test.Comparing the numerical results with the in situ loading test results, the variable modulus constitutive model based on the in situ test parameters for calculation of foundation nonlinear settlement has some advantage over the Duncan-Chang model based on the conventional triaxial test parameters.

Conclusions
In this study, a nonlinear deformation modulus method and a variable modulus constitutive model for foundation settlement calculation based on in situ loading test are proposed.The validity of these two calculation models are verified by the comparisons of calculated results with in situ loading test results under different plate sizes.The main conclusions can be summarized as follows: (1) The calculation parameters, which can reflect the undisturbed nature and the deformation nonlinearity of the soil under the actual stress state, are the key factors that directly affect the calculation accuracy of foundation settlement.The nonlinear deformation modulus method and variable modulus constitutive model proposed based on in situ loading test in this paper can better take these two key factors into account (2) The results calculated from both the nonlinear deformation modulus method and the variable modulus constitutive model proposed in this study compare very well with the in situ loading test results under different plate sizes, which can also better reflect the nonlinearity of foundation settlement (3) The required parameters of both the nonlinear deformation modulus method and the variable modulus constitutive model are the cohesion c, the internal friction angle φ, and the initial deformation modulus E 0 of soils, which are common parameters for practical engineering and can be readily deduced

Figure 1 :
Figure 1: The ps curves of in situ plate loading test.

4. 2 . 2 Figure 3 :
Figure 3: The fitting relationship between s/p and s based on the results of in situ loading test 2#.

Figure 4 :
Figure 4: Comparison of calculated results of the nonlinear deformation modulus method and in situ test results.

Figure 5 :
Figure 5: Mesh divisions for numerical analysis using the variable modulus constitutive model.

Figure 6 :
Figure 6: Comparison of numerical results using the variable modulus constitutive model and in situ test results.

Table 1 :
Physical property test results of the laboratory tests.

Table 2 :
Calculated parameters of the nonlinear deformation modulus method.

Table 3 :
Calculated parameters of the variable modulus constitutive model.

Table 4 :
Calculated parameters of the Duncan-Chang model.