Nonlinear Soil Stiffness Modeling for Sustainable Foundation Design and the Green Transition of the Built Environment

Abstract

Optimizing the life cycle of buildings within the green transition must also encompass foundations, which significantly influence material consumption and the embodied carbon of structures. Accurate settlement prediction is a cornerstone of sustainable design; however, engineering practice often relies on simplifications, such as assuming a constant soil deformation modulus, that lead to oversizing of foundation elements. This paper examines two types of shallow foundations, an isolated footing and a raft, founded in sandy subsoil, and compares calculation outcomes obtained using five approaches: a code-based method, parameters derived from oedometer tests, parameters from triaxial tests, and two Janbu variants that account for stiffness increasing with depth. The results reveal substantial variability in predicted settlements, ranging from underestimation with the code-based approach to overestimation with the oedometer method. The most realistic predictions were obtained using triaxial parameters and the nonlinear model, which better capture the actual deformation mechanisms of the subsoil. Although the primary aim of this study is to improve the technical accuracy of settlement prediction, these findings also demonstrate that precise geotechnical modeling naturally supports resource efficiency and contributes to sustainable construction as a secondary, yet measurable, outcome of rational design.

1. Introduction

Accurate prediction of settlements is crucial for the design of sustainable foundation elements of the structures (foundations). Excessive conservatism in settlement assessment often leads to oversizing of foundations, which means greater consumption of concrete and steel and a larger carbon footprint of the structure. From the perspective of sustainable construction, it is therefore important to design foundations to be as materially efficient as possible while maintaining safety and functionality. Unfortunately, foundations are rarely optimized in environmental terms, even though they constitute a significant share of the overall impact of the structure. It has been found that foundations are often marginalized in sustainability analyses, despite representing an important area with high potential for design and environmental optimization [1,2,3,4]. Accurate settlement prediction would limit excessive oversizing of foundations and thus reduce negative environmental impacts while maintaining the required structural reliability. In classical geotechnical calculations, the subsoil model is often simplified by adopting a constant (depth-homogeneous) deformation modulus E for the subsoil. Although this idealization facilitates calculation, it can lead to significant differences between predicted and actual soil deformations. In particular, for non-cohesive soils (sands), the assumption of a constant modulus is equivalent to ignoring the fact that deeper soil layers beneath a foundation undergo smaller strains. In engineering practice, the soil deformation modulus is often determined from laboratory tests on small specimens (e.g., oedometer tests), which frequently results in underestimation of subsoil stiffness due to sample disturbance. For example, studies of non-cohesive soils have shown that the deformation modulus determined in situ, at macro scale and under the actual stress state, can exceed oedometer-based values by as much as 6–10 times, which follows from scale limitations and the one-dimensional loading scheme in laboratory tests. As a consequence, settlements computed on the basis of such laboratory values are often substantially overestimated relative to actual displacements measured in the field [1,5,6,7]. The consequence of such an approach is oversized foundations designed with excessive reserve, which translates into unnecessarily high economic and environmental costs. As early as in the foundational geotechnical studies, attention was drawn to the difficulties of determining reliable deformation parameters of sands from laboratory specimens. Consequently, numerous settlement-estimation methods based on in situ tests were developed, whose application in non-cohesive soils is far broader than in cohesive soils. Nevertheless, in engineering practice averaged, constant values of the modulus E are still often used, which can lead to the aforementioned excessive conservatism.

In this paper, sustainability is understood not as a separate environmental topic, but as a natural consequence of technically optimized design. Improving the accuracy of settlement prediction leads directly to safer and more economical foundations, which in turn reduces excessive material use and construction impacts. The environmental aspect therefore results from sound engineering practice rather than from policy-driven objectives.

It has long been known that soil stiffness increases with depth as geostatic loading grows. Early settlement calculation methods accounted for this dependence either indirectly or directly. For example, Janbu proposed the so-called tangent modulus method, in which the soil deformation modulus depends on the stress level and increases with the depth of the loaded layer [8,9,10]. The widely used semi-empirical Schmertmann method also employs the concept of an influence zone of strains and a bilinear distribution of strains in the subsoil, which makes it possible to account for the variation in soil stiffness with depth [10,11,12]. An extensive experimental study by Burland and Burbidge [13] confirmed the importance of parameters obtained from in situ testing. On the basis of more than 200 observations of foundation settlements, these authors developed correlations linking settlement, among other factors, to the mean Standard Penetration Test (SPT) N-value within the influence zone of loading [10,13]. Their approach likewise indicates that deeper parts of the subsoil, characterized by higher penetration resistances, exhibit greater effective stiffness affecting settlement. Despite the existence of these recognized methods, a uniform modulus is often adopted in practical applications for simplicity, which as noted can overstate predicted settlement in soils whose stiffness increases with depth.

In the last decade (2015–2025) there has been renewed interest in quantifying the variability of the deformation modulus with strain level and in developing new formulations of this phenomenon. The literature proposes, among other things, explicit modeling of the modulus E as a function of depth; for example, assuming a linear increase in the modulus with z (depth below foundation level) [14]. This approach has been supported by field test results that confirm that the soil deformation modulus is lowest immediately beneath the foundation base and then increases with depth due to often increasing density and growing confining stress (the so-called confinement effect—increase in lateral pressure) in deeper layers [14]. Many authors now use continuous stiffness profiles obtained from in situ CPT (Cone Penetration Test) or DPL (Dynamic Probing Light) soundings to assign appropriate moduli to different layers instead of averaging them over the entire mass [14]. In parallel, nonlinear methodologies have been developed in which the deformation modulus depends on stress or strain level, referring to hyperbolic models (Duncan–Chang [15]) or employing the tangent modulus determined at different stages of loading [16]. For example, settlement calculation methods have been developed based on tangent moduli determined layerwise for successive stress increments due to loading, which makes it possible to omit arbitrary correction factors and to better capture the actual behavior of the subsoil [17,18]. The trend toward replacing a constant stiffness with an E(z) profile or with a nonlinear model is a response to the differences observed in the literature between traditional calculation results and field observations. The new approaches generally lead to smaller settlement predictions in soils whose stiffness increases with depth, which has important implications for optimization of foundation designs.

This study computes settlements for two shallow foundations on medium-dense sand: a 2 m × 2 m footing at D = 2 m and a 20 m × 60 m raft at D = 7 m, under q = 200 kPa, using one layered-integration workflow with four stiffness models: PN-81/B-03020 (constant modulus) [19], oedometer-based E_oed, triaxial-based E_TxT, and two Janbu variants (tangent and integral) with depth/stress-dependent stiffness. The contributions are to isolate the impact of the stiffness model on predicted settlements for identical geometry and loading, and provide an implementable, spreadsheet-ready Janbu integral formulation (with optional small-strain cutoff). Quantitatively, for the raft the triaxial model reduces settlement from 33.99 mm (E_oed) to 16.73 mm (E_TxT), i.e., by ~51%; PN and Janbu II give 26.97 mm and 27.01 mm, respectively. For the footing, PN underestimates settlement relative to triaxial (3.25 mm vs. 4.46 mm, ~27% lower), whereas Janbu I and Janbu II predict 16.16 mm and 11.65 mm. These results show when constant-modulus methods are unconservative (small, shallow footing) or conservative (deep, wide raft), and how calibrated stress-/depth-dependent profiles (E_TxT, Janbu II) support leaner, lower-carbon foundation choices without compromising serviceability.

2. Materials and Methods

Two types of shallow foundations were examined on a non-cohesive, medium-dense sandy subsoil (I_D = 0.66). The first is a 2 m × 2 m isolated footing founded at a depth of 2 m; the second is a 20 m × 60 m raft founded at 7 m. The design contact pressure was taken as q = 200 kPa. For both systems, the influence zone was delineated down to the so-called active depth. Soil parameters were taken from laboratory testing: the oedometer test provided the one-dimensional deformation modulus E_oed, while the triaxial test yielded the stiffness E_TxT. Triaxial compression tests under drained conditions (CD type—Consolidated Drained) were performed on standard cylindrical samples with initial dimensions of 38 mm in diameter and 75 mm in height. Oedometric tests were performed on samples with a diameter of 65 mm and a height of 20 mm, also in CD conditions. Detailed results of triaxial compression and oedometric tests—stress–strain relationship of the tested sand and values of the obtained deformation modules are included in the Supplementary S1. The corresponding calculations were carried out in Microsoft 365 Excel in accordance with the algorithm described below. Settlements were calculated using four approaches:

• Code-based approach (PN-81/B-03020):

The stress (one-dimensional strain) method recommended in PN-81/B-03020 [19] was adopted. It assumes a constant soil deformation modulus dependent on soil type and state (per the code requirements) [20]. In the analyses, the modulus M₀ was taken from the code tables for medium-dense sand. Total settlement was obtained by layered summation of settlements of all sublayers within the influence zone, as prescribed in the code [19].

• Oedometer-based approach (E_oed):

The layered method was used with the deformation modulus equal to the oedometer modulus E_oed measured in one-dimensional compression. This modulus is defined as the stress increment divided by the strain increment on the σ–ε curve from the oedometer test [21]. In the spreadsheet, the experimentally obtained E_oed values were used; for the tested sand they ranged from 16.2 to 21.3 MPa for the footing, and from 43.1 to 72.2 MPa for the raft.

• Triaxial-based approach (E_TxT):

Analogously, the layered method was applied with the deformation modulus equal to the stiffness E_TxT obtained from triaxial testing. For the tested sand, E_TxT ranged from 54.8 to 68.0 MPa (footing) and from 105.8 to 212.7 MPa (raft), i.e., markedly higher than the oedometer-based modulus in both cases. Using the higher E_TxT naturally produces smaller settlements, consistent with the literature, which reports that triaxial moduli for sands are often several times greater than oedometer moduli (and, correspondingly, that settlements computed with E_TxT are several-fold smaller than with E_oed).

• Janbu approach:

The Janbu method [22] was applied in two computational variants used for layered evaluation of settlements in non-cohesive soils whose stiffness depends on stress level. In both approaches, the settlement increment in a soil layer of thickness $\Delta z$ depends on the increment of vertical effective stress $\Delta \sigma \left(z\right)$ at the layer midpoint and on the compressibility (constrained) modulus M, which varies with effective stress, Equation (1):

$$M{\left({\sigma }^{\prime }\right)=m·{{\sigma }_a}^{1-j}·\left({\sigma }^{\prime }\right)}^j$$

(1)

where m is the modulus number, j ∈ (0,1] is the stress exponent (for sands most commonly j = 0.5), and σ_a = 100 kPa is the reference stress. The initial effective stress in the layer is typically taken as σ₀′(z) = (D + z)·γ′, whereas the load-induced increment is Δσ(z) = q·I_z(B,L,D,z), with I_z the influence factor (e.g., Newmark/Steinbrenner) and q the contact pressure. The total settlement is obtained by summing the increments over all layers.

• -

  Janbu I—tangent-modulus method

In the Janbu I variant, the strain increment in a layer is computed by treating the modulus as constant and equal to its value at the initial state ${{\sigma }^{\prime }}_0$. This yields a linear approximation of the constitutive curve segment for the given load step, Equation (2):

$$M\left({{\sigma }^{\prime }}_0\right)=m·{{\sigma }_a}^{1-j}·{\left({{\sigma }^{\prime }}_0\right)}^j;\Delta s={\displaystyle \frac{\Delta \sigma \left(z\right)·\Delta z}{M\left({{\sigma }^{\prime }}_0\right)}}$$

(2)

and then s = ∑Δs over the layers. This approach is computationally very simple (one modulus evaluation per layer) and, through Δσ(z) = q I_z, it correctly accounts for the decay of stress increments with depth and for foundation geometry. Because for j > 0 (sands) the actual stiffness increases as loading proceeds, keeping M(${{\sigma }^{\prime }}_0$) constant typically overestimates strains; the Janbu I variant is therefore moderately conservative. The approximation error diminishes with a finer layer discretization Δz and if needed with splitting Δσ into smaller load steps. In the immediate vicinity beneath the foundation base, where ${{\sigma }^{\prime }}_0$is small, it is common practice to introduce a lower stiffness bound E_min to represent small-strain stiffness and to avoid artificial softening of the first layers.

• -

  Janbu II—integral approach

In the Janbu II variant, the constitutive relation is integrated with respect to stress within each layer, from ${{\sigma }^{\prime }}_0$to ${{\sigma }^{\prime }}_1={{\sigma }^{\prime }}_0+\Delta \sigma ,$thereby capturing the full nonlinear evolution, Equation (3):

$$\epsilon ={\int }_{{{\sigma }^{\prime }}_0}^{{{\sigma }^{\prime }}_1}{\displaystyle \frac{d{\sigma }^{\prime }}{M\left({\sigma }^{\prime }\right)}}={\displaystyle \frac{{\left({{\sigma }^{\prime }}_1\right)}^{1-j}-{\left({{\sigma }^{\prime }}_0\right)}^{1-j}}{(1-j)·m·{{\sigma }_a}^{1-j}}};\Delta s=\epsilon ·\Delta z;s_d=\sum \Delta s$$

(3)

For the case typical of sands, j = 0.5, this reduces to a simple closed form, Equation (4):

$$\epsilon ={\displaystyle \frac{1}{5m}}\left(\sqrt{{{\sigma }^{\prime }}_1}-\sqrt{{{\sigma }^{\prime }}_0}\right);\Delta s=\epsilon ·\Delta z;s_d=\sum \Delta s$$

(4)

The Janbu II variant reflects the stiffness increase during loading within each layer and therefore usually yields somewhat smaller and more physically realistic settlements than the tangent-modulus variant. At the same time, it requires the same input ($\Delta \sigma \left(z\right)=q\hspace{0.17em}·I_z({{\sigma }^{\prime }}_0,m,j{,\sigma }_a$)) and, provided units are consistent (kPa), is just as easy to implement in a spreadsheet. In both approaches, correct specification of $\Delta \sigma \left(z\right)=q\hspace{0.17em}·I_z\left(B,L,D,z\right)$ and unit consistency are essential: with γ′ in kN/m³ and depth in meters, σ′ is in kPa without any additional factor 9.81; M, σ′, and Δσ should all be in kPa. The integration depth should be commensurate with foundation scale (typically 2–3B for small footings and 3–4B for large rafts), verifying stabilization of the cumulative settlement s as the profile is extended. In practice, it is advisable to report the Janbu II result as the reference value (more faithful nonlinear representation) and to use Janbu I as a conservative upper-bound check; both should be computed on the same Δσ(z) profile and layer mesh. Calibration of the modulus number m (and, where appropriate, a small-strain cutoff E_min) against laboratory and/or in situ data enables tuning of predictions to the actual serviceability strain range.

3. Results of Calculation and Discussion

Settlements of a 2 m × 2 m isolated footing embedded at D = 2 m and a 20 m × 60 m raft embedded at D = 7 m were computed using four deformability schemes (see Supplementary Files S2–S9): (1) the code-based PN-81/B-03020 method [19], (2) an oedometer-based modulus E_oed, (3) a triaxial modulus E_TxT, and (4) a nonlinear Janbu formulation in which stiffness increases with depth, implemented in two variants: Janbu I based on $M\left({{\sigma }^{\prime }}_0\right)=m·{{\sigma }_a}^{1-j}·{\left({{\sigma }^{\prime }}_0\right)}^j$ with j = 0.5 and Janbu II with respect to stress within each layer, from ${{\sigma }^{\prime }}_0$to ${{\sigma }^{\prime }}_1={{\sigma }^{\prime }}_0+\Delta \sigma$.

Table 1. Calculated total settlements of the analyzed foundations.

Method	Total Settlement–sd (mm)
	Isolated Footing Embedment Depth–2 m bgl	Raft Foundation Embedment Depth–7 m bgl
PN-81/B-03020 [19]	3.89 mm	64.78 mm
Oedometer	18.44 mm	81.64 mm
Triaxial	5.36 mm	40.18 mm
Janbu I	14.72 mm	90.34 mm
Janbu II	11.34 mm	81.03 mm

Legend: bgl—below ground level.

compiles the total settlements s_d (mm) for both foundations. A direct comparison shows that the predictions differ markedly across methods, with substantial discrepancies arising from the chosen deformability model.

• Case: 2 m × 2 m isolated footing (embedment depth 2 m bgl)

The settlement computed according to PN-81/B-03020 [19] is approx. 3.89 mm, which is the lowest among all the methods. This value is significantly lower than the prediction based on the triaxial modulus (5.36 mm), underestimating settlement by approximately 27%. When using the oedometer-based modulus (E_oed), the predicted settlement reaches 18.44 mm, which is about 3.4 times higher than that from the code-based method. Applying the nonlinear Janbu formulation with stress-dependent stiffness, the calculated settlement is 14.72 mm for Janbu I (tangent stiffness variant) and 11.34 mm for Janbu II (secant stiffness, averaged over stress path). The Janbu I value is about 20% lower than the oedometer result, while Janbu II falls between the triaxial and oedometer predictions, approximately +111% higher than the triaxial outcome, and -38% lower than the oedometer based estimate.

• Case: 20 m × 60 m raft foundation (embedment depth 7 m bgl)

For the raft foundation, PN-81/B-03020 [19] predicts the settlement 64.78 mm, while the triaxial-based approach yields 40.18 mm. The oedometer-based method gives ≈ 81.64 mm, and is therefore more conservative. The Janbu methods produce 90.34 mm (Janbu I) and ≈ 81.03 mm (Janbu II), both of which are approximately double the triaxial-based estimate and broadly similar to the oedometer result.

These findings highlight the significant impact of the adopted deformability model on settlement prediction. For the shallow footing, the PN-81/B-03020 [19] method underestimates settlement (3.89 mm) compared to results based on moduli derived from triaxial (5.36 mm), Janbu (11.34–14.72 mm), or oedometer tests (18.44 mm). Otherwise, for the raft foundation, the triaxial method again predicts the lowest settlement (40.18 mm), significantly below the values obtained using more realistic stiffness models: PN (64.78 mm), Janbu II (81.03 mm), oedometer (81.64 mm), and Janbu I (90.34 mm). These outcomes confirm the high sensitivity of settlement predictions to the selected deformability model. The Janbu approach, with the parameters used here (m = 250, j = 0.5), yields results that fall between the triaxial and oedometer estimates for the footing, and closely match or exceed oedometer-based values for the raft. The resulting ranking observed in Table 1 is therefore: for footing: PN < Triaxial < Janbu II < Janbu I < Oedometer and for raft: Triaxial < PN < Janbu II ≈ Oedometer < Janbu I.

The ranking is consistent with expectations: the oedometer modulus (1D constrained straining of reconstituted sand) tends to be the softest; the triaxial modulus is higher; and the Janbu model, by accounting for stress-dependent stiffness, can bracket the laboratory methods depending on the chosen parameters m and j. If closer agreement with the triaxial outcomes is desired, the parameter m can be calibrated (and, near the base, a small-strain cut-off Eₘᵢₙ introduced) without changing the overall methodology. The accompanying plots (footing and raft) reflect these trends clearly: the oedometer and Janbu I curves exhibit the largest settlements; Janbu II lies below Janbu I due to its integral treatment of nonlinearity; PN-81/B-03020 [19] closely follows TxT for the footing, but overestimates settlement compared to triaxial predictions for the raft (Figure 1).

The results in Table and the accompanying plots underscore the strong sensitivity of computed settlements to the adopted subsoil deformability model. The code method PN-81/B-03020 [19], which relies on elastic coefficients and a single constant (homogeneous) modulus M₀, underestimates settlement for the small, shallow footing on sand: for the 2 m × 2 m footing, the code yields 3.89 mm, i.e., lower than all other methods based on laboratory-derived stiffnesses. This outcome aligns with the stress state at the base (tens of kPa), where the in situ stiffness of medium-dense sand is relatively low. The applied moduli for the tested sand (E_oed ≈ 5.4 MPa, E_TxT ≈ 18.5 MPa) are well below typical assumed values of M₀ (e.g., ~80 MPa), so applying an overly stiff, constant modulus inevitably leads to underpredicted settlements. In the case of the large 20 m × 60 m raft founded at 7 m bgl, the same constant-modulus assumption becomes conservative: PN gives 64.78 mm, whereas the triaxial-based method predicts 40.18 mm. This reversal is attributed to increasing soil stiffness with depth and effective stress, particularly in deeper layers (approximately 15–30 m bgl). While the PN method uses a single modulus, both the triaxial and Janbu methods capture this increase via stress-dependent stiffness, explaining the underestimation of deformation by the code in the raft case. In summary, a code method based on a uniform modulus can produce divergent results depending on the foundation case: overly optimistic for lightly loaded, shallow foundations, and overly conservative for deeper or larger foundations.

Both laboratory based approaches use moduli determined for the investigated sand, yet they lead to markedly different predictions. The oedometer modulus Eoed is measured on (usually reconstituted) specimens under a constrained one-dimensional strain path, which does not reproduce the three-dimensional stress state beneath a foundation. Here, it gives the largest settlements: 18.44 mm (footing) and 81.64 mm (raft). The triaxial modulus E_TxT (strains of order 2–5‰) produces the smallest settlements: 5.36 mm and 40.18 mm, respectively. Thus, for the sands analyzed, using E_oed leads to less favorable (larger) settlements than using E_TxT. It should also be stressed that both laboratory approaches treat each layer with a constant modulus over the entire stress increment up to q, i.e., they assume implicitly linear behavior. In reality, soils are nonlinear and exhibit high small-strain stiffness (G₀, E₀) that degrades with increasing stress and strain; the single value E adopted in a layer is therefore an effective average between the initial state and the service stress. For small footings, the imposed stresses are low, and the effective modulus should be closer to E₀, which explains why the triaxial-based settlements are lower than the oedometer-based ones. In practice, designers sometimes select a modulus corresponding to about half the contact stress (approx. 0.5q) as a heuristic way of accounting for nonlinearity, but for sands this is often insufficient to capture the full deformation response [23]. The most elaborate (and debated) framework is the Janbu method, which assumes that the constrained modulus increases with depth, i.e., with effective stress σ’. The empirical form $M\left({{\sigma }^{\prime }}_0\right)=m·{{\sigma }_a}^{1-j}·{\left({{\sigma }^{\prime }}_0\right)}^j$ (with j = 0.5 for sands) reflects in situ experience that deeper layers under greater overburden are less compressible [24]. In our calculations, this stress-dependent stiffness leads to:

• Janbu I (tangent modulus): settlements of 14.72 mm (footing) and 90.34 mm (raft).
• Janbu II (integral modulus): 11.34 mm and 81.03 mm, respectively.

Two observations follow: The integral variant (Janbu II) systematically yields smaller settlements than the tangent-modulus variant (Janbu I), because it accounts for the progressive stiffening that occurs within the load step as stress increases in a given layer. For the raft, Janbu II remains slightly below the oedometer method, while Janbu I yields the highest settlement overall. For the footing, Janbu I is close to the oedometer prediction, whereas Janbu II falls between the triaxial and oedometer results. The cumulative settlement plots reinforce these tendencies: the Janbu curves maintain a steeper slope at depth, activating deeper strata, whereas the triaxial curve (constant ETxT) flattens with depth. A well-known limitation of the basic Janbu law is the lack of a small-strain cut-off: because $M{\propto \left({{\sigma }^{\prime }}_0\right)}^j$, the modulus tends to very low values as ${\sigma }^{\prime }\to 0$, immediately beneath the base where the relative increment $\Delta \left({\sigma }^{\prime }/{{\sigma }_o}^{\prime }\right)$ is largest, this may underestimate the effective stiffness and amplify computed strains. This may underestimate the effective stiffness and amplify computed strains. In reality, sands possess structure and a finite preconfinement that provide appreciable initial stiffness [25]. As emphasized by Bjerrum, strains beneath well-designed foundations are typically very small (on the order of per mille or smaller) [23]. To reconcile the model with observed behavior, practice often introduces a minimum modulus Emin (or a threshold ${{\sigma }_o}^{\prime }$) below which stiffness no longer diminishes. Building on this idea, two-stage nonlinear stiffness models have been proposed that incorporate a high small-strain modulus E₀ = Eₘₐₓ and a transition to a conventional nonlinear branch (Janbu-type or hyperbolic) [26,27,28]. Such enhancements improve agreement with reality: small strains generate small settlements due to high E₀, while larger loads trigger stiffness degradation and more pronounced settlement increments. From a design perspective, laboratory-based methods are credible when specimens are representative and reflect in situ density and moisture. However, sampling and reconstitution of sands are intrinsically difficult. Consequently, in situ penetration tests (e.g., CPT) are widely used, and numerous correlations between CPT indices and deformation moduli exist [29,30,31]. Recent procedures for deriving modulus E from CPTU (Piezocone penetration test) and DMT (Flat dilatometer test) provide local stiffness profiles and reduce reliance on arbitrary parameter choices [32]. At the modeling end, approaches that combine high initial stiffness with stress-dependent degradation (e.g., the Hardening-Soil model in finite element analysis [33,34,35,36,37] or the two-level frameworks of Atkinson and Sällfors [25,26,27,38]) offer a balanced path between safety and material efficiency. Overall, the present dataset confirms the following ranking (from smallest to largest settlements):

-

For raft: Triaxial < PN < Janbu II < Oedometer < Janbu I,

-

For footing: PN < Triaxial < Janbu II < Janbu I ≈ Oedometer.

These patterns are consistent with both the physical interpretations discussed above and the broader evidence in the literature [19,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38].

For comparison across methods, average effective deformation moduli Eeff were also back-calculated to be consistent with the settlements obtained in the spreadsheets (

Table 2. Average effective deformation modulus values corresponding to the computed settlements for the selected methods and foundation types.

Method	${\mathit{s}}_{\mathit{d}}\left(\mathbf{m}\mathbf{m}\right)\to$ Effective Modulus (MPa)
	Isolated Footing Embedment Depth—2 m bgl H = 4 m	Raft Foundation Embedment Depth—7 m bgl H = 80 m
PN	4.00 mm → 200.0 MPa	70.32 mm → 227.5 MPa
Oedometer	18.22 mm → 43.9 MPa	110.16 mm → 145.2 MPa
Triaxial	5.48 mm → 146.0 MPa	118.34 mm → 135.2 MPa
Janbu I	15.16 mm → 52.8 MPa	95.28 mm → 167.9 MPa
Janbu II	11.77 mm → 68.0 MPa	85.94 mm → 186.2 MPa

Legend: For the raft, the settlement influence depth was taken as H = 4B = 80 m. Legend: bgl—below ground level.

). The values were computed from the inverse relation, Equation (5):

$$E_{eff}={\displaystyle \frac{q·H}{s_d}}$$

(5)

where $E_{eff}$ is the effective deformation modulus (kPa), $q$ = 200 kPa is the adopted service load, $H$ is the depth of the influence zone (taken as 2B for the footing and 4B for the raft), and s_d is the calculated settlement (m).

The values in Table 2 reveal several consistent patterns regarding the effective deformation modulus under different foundation types and deformability models.

First, the triaxial-based effective modulus clearly exceeds its oedometer-based counterpart in the case of footing. For the shallow footing, the effective modulus from triaxial testing is 146 MPa, compared to just 43.9 MPa from the oedometer test, a factor of ~3.3 with corresponding settlements of 5.48 mm and 18.22 mm, respectively. For the raft, the values are 135.2 MPa (TxT) and 145.2 MPa (oedometer), yielding settlements of 118.34 mm and 110.16 mm. While the moduli are close, the slightly lower triaxial stiffness results in marginally larger settlements. These observations align with well-established trends in geotechnical literature: stiffness obtained from tests simulating realistic three-dimensional stress states (or in situ) tends to be significantly higher than that from one-dimensional oedometer tests [5,7].

Second, the depth and scale effect is clearly evident. Except for the triaxial method, all effective modulus values are higher for the raft than for the shallow footing. This reflects the increase in effective stress with depth and the corresponding stiffening of the sandy subsoil. For instance, in the PN method, E_eff increases from 200 MPa (footing) to 227.5 MPa (raft), while in Janbu II it rises from 68.0 MPa to 186.2 MPa. These differences correspond to larger deformations in the shallow footing and a more rigid response under the deeper raft foundation, in line with geotechnical expectations.

Third, the Janbu methods fall between the oedometer and triaxial results and offer a more nuanced interpretation. For the footing, Janbu I and Janbu II produce effective moduli of 52.8 MPa and 68.0 MPa, with settlements of 15.16 mm and 11.77 mm greater than the triaxial result, but more favorable than oedometer. In the raft case, the effective moduli from Janbu I and II are 167.9 MPa and 186.2 MPa, yielding settlements of 95.28 mm and 85.94 mm, respectively. This places them between the PN (70.32 mm) and oedometer or triaxial outcomes. The integral variant (Janbu II) always yields lower settlements than the tangent modulus variant (Janbu I) due to its inclusion of modulus increase within the stress path during loading. These trends are quantitatively consistent with the fundamental relationship $s_d\propto {\displaystyle \frac{1}{E_{eff}}}$. For example, in the raft case, increasing the modulus from PN’s 227.5 MPa to Janbu II’s 186.2 MPa leads to a proportional increase in settlement from 70.32 cm to 85.94 mm. The ratio of moduli (227.5/186.2 = 1.22) matches the observed settlement increase (85.94/70.32 = 1.22), confirming the expected inverse dependency. Likewise, for the footing, PN yields a very stiff E_eff of 200 MPa and a low settlement of 4.00 mm, while the triaxial modulus of 146 MPa corresponds to a higher settlement of 5.48 mm again, matching the modulus ratio (200/146 = 1.37) and the settlement increase (0.548/0.400 ≈ 1.37). Such findings support broader literature evidence that stiffness moduli derived from in situ or three-dimensional stress paths can be 6–10 times higher than those derived from classical oedometer tests [5,7]. For example, in the footing case, PN and triaxial moduli are roughly 4–5 times higher than the oedometer-based estimate. From a design perspective, these differences are not just academic, they directly affect the economy, material efficiency, and environmental footprint of a foundation system. Overly conservative moduli (as in oedometer tests or Janbu I without adjustments) may lead to overestimated settlements, prompting unnecessarily large foundations, thicker concrete sections, and increased steel reinforcement. This escalates construction costs and embodied carbon. For instance, relying on the oedometer modulus for the raft results in settlements exceeding 110 mm, compared to 86 mm with Janbu II and just 70 mm using PN-81/B-03020 [19]. Conversely, assuming a too-stiff constant modulus (as in PN for shallow footings) may lead to underestimated settlements and potential serviceability limit state failures. For the 2 m × 2 m footing, PN estimates a settlement of just 4 mm, while the more realistic triaxial-based estimate is 5.48 mm, suggesting PN may be ~27% too low in this case. In light of these insights, the most balanced and rational strategy is to adopt a stress- and depth-dependent stiffness profile E(z), preferably calibrated to triaxial-based settlements. For the analyzed cases, this corresponds to: 146 MPa for the footing (TxT) and 135 MPa for the raft (TxT). To mitigate unrealistic stiffness softening near the foundation base in Janbu-type models, it is advisable to introduce a minimum small-strain modulus E_min. This stabilizes the settlement profile and prevents artificially high strain concentrations just beneath the structure. Ultimately, accurate settlement prediction enables smarter design decisions: optimized foundation size and depth, minimized earthworks, reduced concrete and reinforcement use, and improved cost predictability. From a sustainability perspective, this leads to lower emissions, less material waste, and better performance over the structure’s life cycle.

The results of the analysis clearly indicate that the choice of deformability model has a direct and substantial impact on the predicted settlements and, consequently, on foundation design decisions. For the 2 m × 2 m isolated footing, the most reliable and realistic predictions are obtained using the triaxial modulus or the Janbu II method calibrated with appropriate parameters. These approaches offer a balanced representation of sand stiffness, accounting for the stress–strain response more accurately than overly simplistic constant-modulus assumptions. In the case of the 20 m × 60 m raft foundation, both the triaxial method and the Janbu II approach again deliver consistent and realistic settlement estimates. These methods reflect the natural increase in soil stiffness with depth, which is essential for capturing the behavior of deep and heavily loaded foundations. Meanwhile, the PN-81/B-03020 [19] method, which assumes a constant, depth-invariant modulus, may still be suitable for preliminary assessments or conservative estimates, but lacks the resolution and realism required for refined analysis. Overall, the confirmed values of the effective modulus E_eff across all methods underscore a key conclusion: the selection of a deformability model is not a minor detail, it is a critical design decision. Methods that incorporate stress- and depth-dependent stiffness, such as triaxial testing or nonlinear Janbu models enhanced with a lower-bound small-strain modulus, provide a robust and rational framework. They help reduce both overdesign, which leads to excessive material use, and underdesign, which risks exceeding serviceability limits. Adopting such approaches ultimately improves the durability, efficiency, and sustainability of geotechnical solutions.

4. Engineering and Sustainability Implications of Accurate Settlement Prediction

The accuracy of predicting settlements of shallow foundations translates directly into the degree of environmental and economic sustainability of their design. When calculations account for the variability of soil stiffness with depth, especially in sandy soils, where stiffness increases with increasing effective stress, the resulting design becomes inherently more efficient and environmentally friendly. The key aspects of sustainable development affected by the precision of foundation-settlement forecasts are discussed below:

• Optimization of material use and structural sizing

The results of the comparative analyses clearly demonstrate that incorporating a realistic, stress-dependent increase in soil stiffness with depth, as captured by models such as Janbu’s nonlinear E(z)) formulation, substantially reduces predicted settlements compared to conventional approaches that rely on oedometer-based values or assume a constant subsoil modulus, as in code-based methods like PN-81/B-03020 [19]. This reduction is particularly evident for large and deep foundations, such as the 20 × 60 m raft, where the triaxial method predicted settlements nearly 40% lower than the PN-based estimate, and almost 50% lower than those derived using oedometer values. Such differences are directly tied to the proportional relationship $s_d\propto {\displaystyle \frac{1}{E_{eff}}}$, which highlights how the choice of modulus significantly influences the predicted settlement. By better reflecting actual subsoil behavior, models that capture stiffness variation with depth and stress, such as Janbu II with a calibrated E_min, allow designers to avoid excessive conservatism. In practical terms, this means the potential to reduce foundation dimensions and material usage without compromising safety or serviceability. Designers can also optimize reinforcement quantities due to improved control over settlement predictions, and in some cases, reduce the necessary embedment depth, thereby limiting earthworks and the volume of excavated spoil. Taken together, these improvements directly support key goals of sustainable construction: minimizing environmental impact, improving material efficiency, reducing embodied carbon, and enhancing structural longevity in line with life-cycle design principles [39,40].

• Reduction of carbon footprint and construction costs

An overly conservative approach to selecting soil parameters is not only uneconomic but also environmentally unfavorable. For example, the production of cement, the primary constituent of concrete, accounts for approx. 8% of global anthropogenic CO₂ emissions [41]. Owing to more accurate settlement forecasts, it is possible to reduce the required amount of concrete in the foundation, which directly translates into a decrease in the “embodied” carbon footprint of the structure. Life-cycle assessments (LCA) of buildings indicate that foundations may account for 15–20% of total material consumption in typical building projects [42]. Reducing this share by 20–30% at the foundation-design stage can therefore result in a significant improvement of the environmental balance over the entire life cycle of the building. Moreover, avoiding oversizing reduces both initial capital expenditures and construction-related environmental loads, especially in remote regions or those with limited access to raw materials, where transporting concrete and steel entails high energy use [43]. As a result, a more precise approach to foundation design simultaneously brings economic benefits to the investor and helps reduce greenhouse gas emissions.

• Modern soil testing in support of sustainability

Traditional reliance solely on oedometer test results may prove insufficient in sandy soils, where triaxial tests better reflect the stress-dependent nature of deformability. Wider application of triaxial tests conducted under conditions of consolidation with full drainage of pore water (so-called CD—consolidated drained-tests), which better reflect actual loading paths, enables more precise forecasting of long-term settlements and reduces the need for empirical correction factors in design calculations. It also facilitates fuller use of test results in advanced numerical modeling of the subsoil. The International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE), in its guidelines, emphasizes that parameters obtained from properly conducted triaxial tests are more representative for modern design and should become standard for key, critical infrastructure projects, particularly from the perspective of their sustainable character [43].

• Support for the Sustainable Development Goals

The presented approach is consistent with the broader goals of responsible resource use and climate-conscious engineering. These outcomes stem naturally from technically optimized and economically efficient design methods rather than from independent sustainability policies, and they align with previous research showing that design optimization at the foundation stage can yield long-term environmental and economic benefits throughout a structure’s life cycle [44].

• Prospects for implementation in design standards

The results obtained constitute an argument for developing geotechnical design standards toward accounting for depth-dependent subsoil stiffness. Although contemporary standards (such as Eurocode 7 [45], together with annexes enabling the use of advanced methods of analysis) allow the use of nonlinear soil models, design practice still often rests on overly simplified assumptions and outdated code recommendations. The routine use of deformation modulus profiles that increase with depth in calculations, supported by appropriate in situ and laboratory tests, promotes a shift away from conservative, prescriptive approaches towards sustainable, performance-based design. Such a shift increases the competitiveness of sustainable engineering solutions and provides designers with tools to balance safety requirements, costs, and environmental impact already at the concept stage. It is therefore advisable that future design guidance explicitly provide for the use of nonlinear E(z) functions (with parameter calibration to local ground conditions and protocols for validating test results), analogous to the emerging provisions for unsaturated soils or time-dependent phenomena.

5. Conclusions

The study clearly confirms the substantial influence of the depth-dependent variability of the soil deformation modulus on the prediction of foundation settlements, particularly in sandy soils, where stiffness increases with effective stress. By analyzing two representative cases, a 2 m × 2 m shallow footing and a 20 m × 60 m raft at 7 m depth, using laboratory test results and parametric modeling, several key conclusions emerge:

• Depth-dependent stiffness is essential for reliable settlement prediction

For wide and/or deeply embedded foundations, constant modulus assumptions fail to capture the increase in soil stiffness with depth. For instance, effective moduli for the raft ranged significantly e.g., E_eff = 135.2 MPa (triaxial) to 186.2 MPa (Janbu II), highlighting the inadequacy of homogeneous stiffness models. Applying stress-dependent functions like Janbu’s E(z) better reflects the actual deformation mechanisms, especially under service loads.

• Traditional methods (PN-81/B-03020 and oedometer) are conservative and often imprecise

These methods either underestimate settlements for small footings (e.g., PN: 3.9 mm vs. 5.5 mm from triaxial data) or overestimate them for rafts (e.g., oedometer: 110 mm vs. triaxial: 118 mm), often resulting in oversized foundations, increased material demand, and inflated construction costs. They neglect nonlinear stress–strain behavior and the increase in stiffness with depth.

• Triaxial testing and nonlinear modeling (e.g., Janbu II) are preferred

Consolidated-drained triaxial tests provide moduli under realistic stress states (e.g., E_eff = 146 MPa for the footing), and when coupled with calibrated nonlinear functions and a lower-bound stiffness (E_min), enable precise and rational design. Even spreadsheet-based layered models using E(z) offer a reliable and practical solution for routine design tasks.

• Advanced stiffness modeling supports sustainable geotechnical design

More accurate settlement prediction reduces the need for conservative safety margins, enabling slimmer foundations, shallower embedment, and optimized reinforcement. These changes directly support Low-Impact Materials and Construction Strategies, reducing concrete and steel usage, limiting earthworks, and minimizing spoil. This leads to:

• • a lower carbon footprint of the foundation system;
  • more efficient use of natural and industrial resources;
  • enhanced durability of structures and long-term performance reliability;
  • full alignment with sustainable construction principles.

Overall, the link to sustainability is presented only as a practical outcome of accurate engineering analysis, not as a separate research objective.

• Recommendation for engineering practice and design codes

Geotechnical design methodologies should be systematically updated to incorporate stress- and depth-dependent stiffness models as standard practice. Design codes ought to facilitate and promote nonlinear constitutive modeling, particularly with the inclusion of small-strain stiffness thresholds, and support the expanded application of triaxial testing, especially in the context of critical infrastructure or resource-efficient construction. This shift not only enhances predictive accuracy and structural reliability but also aligns geotechnical engineering with broader environmental and economic sustainability objectives.

The integration of laboratory-based triaxial testing with depth- and stress-dependent stiffness models, such as Janbu II, provides a robust and sustainability-oriented framework for geotechnical design. The findings clearly demonstrate the technical accuracy, material efficiency, and environmental benefits of replacing conventional constant-modulus assumptions with more realistic, nonlinear approaches.