Assessment of spread foundation settlement using statistical determination of characteristic values of subsoil properties

The European draft standard EN 1997-1:2004 recommends that the characteristic value of the geotechnical parameter X_k should be selected as a conservative estimate of the value that determines the occurrence of a limit state. The characteristic value of the geotechnical parameter X_k is the most probable value of a given parameter at which the considered limit state will occur. Conservative estimation of the mean value involves selecting the mean value from a limited set of geotechnical parameter values with a confidence level of 95%. The characteristic value of parameter X_k is determined on the basis of mean values X_mean derived from laboratory or field tests, alternatively based on derived values using formulas or empirical relationships.

The European standard EN 1997-1:2022-09 recommends that the mean X_mean value of the geotechnical parameter is calculated according to formula (1): (1) Xmean=1n∑Xi {X_{{\rm{mean}}}} = {1 \over n}\sum {{X_i}} where X_i is the result of the determination of the parameter concerned and n is the number of determinations.

The measure of variability of a geotechnical parameter in a given soil layer is the coefficient of variability V_x calculated according to formula (2): (2) Vx=sxxmean {V_x} = {{{s_x}} \over {{x_{{\rm{mean}}}}}} where S_x is the standard deviation, most often from a sample, in the case of a limited number of test results. The standard deviation for sample S_x is expressed by formula (3): (3) Sx=Σi=1n(Xi−Xmean)2n−1 {S_x} = \sqrt {{{\Sigma_{i = 1}^n{{({X_i} - {X_{{\rm{mean}}}})}^2}} \over {n - 1}}}

Assessment of a parameter in classical mathematical statistics is based on a random sample taken from the population. In an alternative approach, derived from Bayes' theorem (Alén, 1998; Alén, Sällfors, 1999), the assessment may be based not only on a random sample, but also on the so-called a priori information. A priori knowledge may be either expert knowledge or a result from previous research. Population parameters to be estimated include parameter θ, for example, the mean, and the standard deviation from the population; in Bayesian analysis, they are treated as random variables. Bayes' theorem for random variables with a continuous probability distribution can be expressed as formula (4): (4) f(θ|x)=f(x|θ)⋅f(θ)∫Ωf(x|θ)⋅f(θ)dθ f(\theta |x) = {{f(x|\theta ) \cdot f(\theta )} \over {\int\limits_\Omega {f(x|\theta ) \cdot f(\theta )d\theta}}} where f(θ|x) is the a posteriori density function of parameter θ, after the sample's result x has been observed, f(θ) is the a priori distribution density function of parameter θ, f(x|θ) is the credible function, that is, the density function of the conditional observation's result x with a given value of θ, and Ω is the set of possible values of parameter θ. Therefore, on the basis of Bayes' theorem, the a priori density function of parameter θ is actualized with the use of information from a sample.

A common case is the estimation of an unknown parameter θ, which is the mean in a normal population for which the standard deviation σ₀ is known. The a priori knowledge about the mean θ of this population can be used, which shows that parameter θ is a random variable with a normal distribution with parameters m₁ and σ₁, while the mean of the drawn n-element sample is m₂. Therefore, the a posteriori distribution of the random variable θ is also normal, with the mean m and standard deviation s calculated as follows (formulae 5 and 6): (5) m=(1/σ12)⋅m1+(n/σ02)⋅m2(1/σ12)+(n/σ02) m = {{(1/\sigma_1^2) \cdot {m_1} + (n/\sigma_0^2) \cdot {m_2}} \over {(1/\sigma_1^2) + (n/\sigma_0^2)}} (6) σ=1(1/σ12)+(n/σ02) \sigma = {1 \over {(1/\sigma_1^2) + (n/\sigma_0^2)}} where m and σ are the mean and standard deviation, respectively, of the a posteriori distribution of the random variable θ, m₁ and σ₁ are the mean and standard deviation, respectively, of the a priori distribution of the random variable θ, m₂ is the mean of the drawn n-element sample, σ₀ is the known standard deviation, and n is the number of elements in the sample.

Using Bayesian analysis, the mean values and standard deviation for cone resistance q_c and constrained modulus M from CPT, as well as for dilatometer modulus E_D and constrained modulus M from DMT were calculated based on formulae (5) and (6). Calculated mean values of constrained modulus M were used to calculate the settlements shown in Table 4.

Schneider (1997; 1999) took into account the need for a careful estimation of geotechnical parameters and proposed the often used formula (7) for assessing the characteristic value X_k of geotechnical parameters (Schneider, 1997; 1999; Schneider, Fitze, 2013; Puła, 2014; Sulewska, Lechowicz, 2024): (7) Xk=Xmean−0.5⋅Sx {X_k} = {X_{{\rm{mean}}}} - 0.5 \cdot {S_x}

The draft standard prEN 1997-1:2022-09 introduces the concept of a representative value of ground properties X_rep. Representative values of ground properties X_rep are specific geotechnical properties of a given subsoil layer.

Two cases can be considered, depending on the sensitivity of the limit state of the ground to the spatial variability of a given property: -

case A – if the verified limit state of the ground is insensitive to the spatial variability of a given ground property in the volume of soil involved, the representative value of a given parameter X_rep is its nominal value X_nom, that is, the mean, according to formula (8): (8) Xrep=Xnom {X_{{\rm{rep}}}} = {X_{{\rm{nom}}}}

The characteristic value of the geotechnical parameter X_k may be calculated according to formula (10): (10) Xk=Xmean1∓knVx=Xmean1∓knSxxmean {X_k} = {X_{{\rm{mean}}}}\left[ {1 \mp {k_n}{V_x}} \right] = {X_{{\rm{mean}}}}\left[ {1 \mp {{{k_n}{S_x}} \over {{x_{{\rm{mean}}}}}}} \right] where k_n is the factor depending on the number n, ∓ means that k_nV_X should be subtracted when the lower value of X_k is required or added when its upper value is required.

In general, with the exception of the 2–3 m thick surface layer, the tested subsoil consists of Quaternary moraine deposits underlain by fine sands. The moraine deposits consist of two layers: layer III comprising brown boulder clay of the Wartanian glaciation and layer IV comprising gray boulder clay of the Odranian glaciation. The large variation in the values of geotechnical parameters in the tested cohesive subsoil results from the heterogeneity of the soil caused by sedimentation conditions and diagenetic processes that the soil has been subject to in its geological history. The analyzed cohesive soils of post-glacial origin (Pleistocene), deposited as moraine sediments of the Riss glaciation, originating from two stages, that is, cooling within one glaciation, that is, formed in two-time stages and differing visually in color (brown and gray boulder clays) and large spatial variability of geotechnical properties characteristic for boulder clays.

The index properties of boulder clays and the thicknesses of layers and sublayers are presented in Table 1. The tested soils can be classified as preconsolidated low-plasticity sandy clay saCl or silty sandy clay sasiCl clays (according to EN ISO 14688–2:2018). The building was founded below the surface of brown boulder clay. Because of the difference in thickness of boulder clays below the foundation level and the properties of gray boulder clay, differential settlements were expected. Compression curves obtained from incremental loading oedometer tests conducted up to 30 MPa (Lechowicz et al., 2017) indicate that the preconsolidation stress σ′_p = 800–1000 kPa and is much higher than the effective stresses induced by loading of building construction. In building settlement calculations, the secondary constrained modulus values were only used.

To determine the geotechnical parameters of the subsoil under building No. 34, four CPTs and four DMTs were carried out. The profiles of the cone resistance q_c, sleeve friction f_s, and friction ratio R_f from CPT are shown in Figure 3. The profiles of the material index I_D, horizontal stress index K_D, and dilatometer modulus E_D from DMT are shown in Figure 4. CPT and DMT test results indicate that the gray boulder clay is softer than the brown boulder clay.

Figure 3:

Profiles of cone resistance q_c, sleeve friction f_s, and friction ratio R_f from CPT.

CPTs = cone penetration tests

Figure 4:

Profiles of material index I_D, horizontal stress index K_D, and dilatometer modulus E_D from DMT.

DMTs = dilatometer tests

Variability of the thickness of foundation sublayers was determined based on the profiles of cumulative values of cone resistance Σq_c from CPTs and cumulative values of dilatometer modulus ΣE_D from DMTs (Figure 5). In the case of the gray boulder clay (layer IV), three sublayers (IVa, IVb, and IVc) were distinguished as shown in Figure 5.

Figure 5:

Profiles of cumulative values Σq_c from CPTs and cumulative values ΣE_D from DMTs below the foundation level.

CPTs = cone penetration tests, DMTs = dilatometer tests

To evaluate the value of the constrained modulus M from CPT, the empirical correlations proposed by Senneset et al. (1989) were used as follows: (11) M=αqn M = \alpha {q_n} where α is the empirical coefficient, q_n = (q_c − σ_v0) is the net cone resistance, σ_v0 = Σ(γ · h) is the total overburden stress, γ is the unit weight, and h is the layer thickness. Values of the constrained modulus M from CPT were calculated based on the empirical correlation (11) with the empirical coefficient α = 8.25 proposed by Kulhawy and Mayne (1990). The empirical coefficient α proposed by Kulhawy and Mayne concerns static penetration tests with measurement of pore water pressure (CPTUs). Experience shows that in the case of preconsolidated cohesive soils, the pore water pressure measured during penetration is close to zero, so cone resistance q_c from CPT can be assumed in formula (11) instead of total cone resistance qt (Młynarek et al., 2023).

Based on the comprehensive analysis of the results of DMTs and oedometer tests of the heavily preconsolidated boulder clays prevailing in the Warsaw region, the values of constrained modulus M obtained from oedometer tests were compared with the values of dilatometer modulus E_D to obtain factor R_M = M/E_D. The empirical correlation to evaluate the constrained modulus M from DMTs was as follows (12): (12) M=RMED M = {R_M}{E_D} where R_M is the empirical factor related to the horizontal stress index K_D.

The empirical correlation between factor R_M and horizontal stress index K_D proposed by Lechowicz et al. (2017) shown in Figure 6 is (13): (13) RM=0.14+1.6 log KD {R_M} = 0.14 + 1.6\;\log \;{K_D}

Figure 6:

Relationship between factor R_M and horizontal stress index K_D. R² = determination coefficient.

Settlement calculations of spread foundations were carried out taking into account changes in stresses with depth and different values of the constrained modulus for the distinguished layers. Due to that, the preconsolidation stress σ'_p is much higher than the stresses induced by loading, therefore the settlements of spread foundations were calculated as for preconsolidated soils (Lechowicz et al., 2017). For a given layer, settlement s was calculated using the following formula (14): (14) S=ΔσZ⋅hM S = {{\Delta {\sigma_Z} \cdot h} \over M} where Δσ_z is the increase in vertical stress in a given layer caused by building loading, h is the thickness of a given layer, and M is the constrained modulus for a given layer.

To determine the distribution of vertical stress beneath uniformly loaded footing, the Fadum nomogram published by Poulos and Davis (1974) was used. In the case of entire cohesive subsoil (layers III + IV, together), the total settlement was calculated assuming a shared value of constrained modulus. For a cohesive subsoil divided into two layers (III and IV), the total settlement was calculated assuming separate values of constrained modulus for each layer. In the case of separation of three sublayers in layer IV, the different values of constrained modulus for each layer (III, IVa, IVb, and IVc) were used.

Profiles of cone resistance q_c from CPT and constrained modulus M were subjected to statistical analysis. Statistical analysis was performed for the entire cohesive subsoil, for the subsoil divided into two layers, that is, brown boulder clay (layer III) and gray boulder clay (layer IV), as well as for a layer of gray boulder clay subdivided into three sublayers (IVa, IVb, and IVc). For each dataset, the mean, minimum, and maximum values with standard deviation were calculated according to formula (3) and the coefficient of variation was calculated according to formula (2). Results of statistical analysis of the cone resistance q_c and the constrained modulus M are shown in Table 2.

Statistical analysis of the results from CPT carried out for the entire cohesive subsoil indicates large variability of the constrained modulus M with the coefficient of variation V_x = 0.335. Large variability of the constrained modulus M was particularly obtained for layer III with the coefficient of variation V_x = 0.448. The separation of three sublayers within layer IV resulted in the reduction of the coefficient of variation from V_x = 0.305 to V_x = 0.155–0.233.

Profiles of the dilatometer modulus E_D from DMTs and the constrained modulus M were subjected to statistical analysis. For each dataset, the mean, minimum, and maximum values with the standard deviation and the coefficient of variation were calculated. Results of statistical analysis of the dilatometer modulus E_D and the constrained modulus M are shown in Table 3.

Statistical analysis of the results from DMTs carried out for the entire cohesive subsoil indicates a large variability of the constrained modulus M with the coefficient of variation V_x = 0.452. Compared to the CPT results, less variability of the constrained modulus M was obtained for layer III with the coefficient of variation V_x = 0.257. The separation of three sublayers in layer IV reduced the coefficient of variation from V_x = 0.370 to V_x = 0.170–0.268.

Calculations of the characteristic values of the constrained modulus M_k from CPTs and DMTs were carried out using Schneider's formula (7) and according to standard prEN 1997-1:2022-09.

The following assumptions were made for the calculation of the characteristic values of the constrained modulus M according to standard prEN 1997-1:2022-09: –

The serviceability limit state is sensitive to the spatial variability of the subsoil, that is, settlement calculations constituting case B.

Characteristic values of the constrained modulus M were determined from CPT and DMT (both separately and jointly) for the entire cohesive subsoil, for subsoil divided into two layers of brown boulder clay and gray boulder clay, as well as for a layer of gray boulder clay subdivided into three sublayers. Table 4 shows the characteristic values of the constrained modulus M. Analysis of the calculation results indicates that the lowest characteristic values of the constrained modulus M were obtained using the draft standard prEN 1997-1:2022-09.