Undrained seismic bearing capacity of strip footing adjacent to a heterogeneous excavation

Abstract

The analysis of the bearing capacity of strip footings sited near an excavation is critical in geotechnics. In this study, the effects of the geometrical features of the excavation and the soil strength properties on the seismic bearing capacity of a strip footing resting on an excavation were evaluated using the lower and upper bounds of the finite element limit analysis method. The effects of the setback distance ratio (L/B), excavation height ratio (H/B), soil strength heterogeneity (kB/c_u), and horizontal earthquake coefficient (k_h) were analyzed. Design charts and tables were produced to clarify the relationship between the undrained seismic bearing capacity and the selected parameters.

Graphical abstract

Keywords

excavation / finite element limit analysis / heterogeneous soil / strip footing / undrained bearing capacity

1 Introduction

Unsupported excavations in cohesive soils are typically utilized in different civil engineering structures, such as footings, piers, water and oil tanks, raft foundations, and retaining structures. They also comprise a part of the cut-and-cover method used for constructing shallow underground structures, such as underpasses and pipelines. Sufficient undrained shear strength of cohesive soil leads to the stability of unsupported excavations under short-term undrained conditions. This eliminates the need for constructing retaining walls to resist lateral earth pressure and reduces the construction cost and time of projects. Therefore, a precise stability analysis of such excavations is necessary [1–8].

A geotechnical engineer may encounter situations where a footing must be built near a vertical excavation. This typically occurs when the basement of a multistory building is constructed [9]. These conditions can influence the ultimate load that a footing can resist. Numerous researchers have investigated the bearing capacity of footings located on the flat ground [10–14]. In addition, some researchers have considered the influence of slopes and excavations on the stability and static and seismic bearing capacities of footings [15]. Kumar and Mohan Rao [16] examined how the pseudostatic horizontal earthquake coefficient influences the bearing capacity of footings near slopes, considering various ground inclinations. Azzouz and Baligh [17] assessed the impact of strip and square loads on the stability of slopes containing cohesive soil and provided design charts and tables for slope stability analysis. In terms of methods applied to estimate the bearing capacity, the stress characteristics method has been adopted for strip footings close to cohesionless slopes [18]. Shiau and Watson [19] considered a deep excavation site and assessed the bearing capacity of a footing. Georgiadis [20] employed the finite element analysis, upper bound plasticity, and stress field methods by concentrating the load inclination impact on the bearing capacity of strip footings sited close to the slopes. Georgiadis [21] used the finite element method to determine the undrained bearing capacity of footings close to slopes. The study suggested that three failure surfaces can be formed depending on the ratio of the slope height to the footing width. Shiau et al. [9] adopted the lower and upper bounds of the finite element limit analysis (FELA) to evaluate the undrained bearing capacity of a strip footing near a slope. Experimental investigations on the bearing capacity of a strip footing close to a cohesionless slope have also shown that a direct relationship exists between the bearing capacity and the setback distance, defined as the distance between the footing and the slope crest [22,23]. In addition, the finite element lower bound approach was applied to assess the maximum load sustained by a strip footing on a slope [24]. Leshchinsky and Xie [25] computed the bearing capacities of strip footings sited near slopes consisting of cohesive-frictional soils by applying a limit analysis (LA) using discontinuity layout optimization (DLO). Halder et al. [26] assessed the bearing capacity of a strip footing close to a slope by adopting the lower bound of FELA. Zhou et al. [27] plotted design charts for strip footings resting on slopes using DLO. They reported that a direct nonlinear relationship exists between the normalized bearing capacity and the distance between the slope crest and footing.

The aim of this study was to compute the undrained seismic bearing capacity of a strip footing placed on an excavation consisting of heterogeneous soil. This study focused on isotropic soil analysis with deterministic heterogeneity without considering the superstructural inertial effect. The problem is first explained in the following section. Next, a comparison of the results of this study with those obtained by other researchers is presented. Design charts and tables are then provided. Finally, a design case is presented to clarify the solution procedure.

2 Problem definition

2.1 Model geometry

The geometric features of this study are depicted in Fig.1. The undrained seismic bearing capacity of a strip footing located close to excavation is influenced by three parameters: geometrical features of the excavation, soil strength properties, and the horizontal earthquake coefficient. Thus, the undrained seismic bearing capacity of a strip footing can be written as Eq. (1):

(1)\frac{q_{\text{u}}}{\gamma B}=f\left(\frac{L}{B},\frac{H}{B},k_{\text{h}},\frac{c_{\text{u}}}{\gamma B},\frac{kB}{c_{\text{u}}}\right),

where q_u is the average value obtained from the lower and upper bounds of FELA, B is the footing width (B = 1 m), H is the excavation height, L is the distance between the footing and the excavation, c_u is the undrained shear strength of the soil, k is the strength gradient with depth, γ is the unit weight of soil, and k_h is the horizontal earthquake coefficient.

In heterogeneous soil (Fig.1), a direct relationship exists between the undrained shear strength and depth, as expressed by Eq. (2) [28–32].

(2)c_{\text{u}}=c_{\text{u0}}+k\mathit{\text{z}},

where c_u0 is the undrained shear strength of soil at ground surface level.

2.2 Numerical analysis

The method of lower [33] and upper bounds [34], coupled with the finite element method, was used to calculate the undrained bearing capacity of a strip footing considering all influential parameters, as expressed by Eq. (1). The formulation of these methods and the general computational procedure were proposed by Sloan [33,34]. These methods have been well-described in Refs. [13,35].

OptumG2 [36], which uses FELA and a linear programming approach, was used to determine the undrained seismic bearing capacity of a strip footing adjoining an excavation. A sensitivity analysis was conducted to determine the initial number of elements, ultimate number of elements, and final dimensions of the model to ensure that the results were not influenced by the boundaries [37–39]. Consequently, the initial number of elements was set to 5000, and the ultimate number of elements reached 10000 in three iterations. In the most critical cases, L_r, L_l, and L_z are equal to 9B, 10B, and 5B, respectively (L_r, L_l, and L_z are illustrated in Fig.1). The associated Mohr−Coulomb failure criterion was assigned to the soil, and the footing was modeled by adopting a weightless plate element [29,37,40]. The interface between the footing and soil was modeled as perfectly rough (δ/\phi = 1) and perfectly smooth (δ/\phi = 0) [41–43]. The bottom of the model was constrained in vertical and horizontal directions. In contrast, the two sides could move vertically (Fig.1). The pseudostatic approach was utilized in the model to simulate horizontal earthquakes [42,44].

3 Comparison with results of previous studies

A comparison was made with the results of other studies to verify the results of this study. Shiau et al. [9] adopted finite-element lower and upper bounds and calculated the bearing capacity of strip footings on a homogenous soil excavation under undrained conditions (β = 90°). Chen and Xiao [45] proposed a solution for the undrained bearing capacity of strip footings near a slope using the upper bound of FELA. Tab.1 presents comparisons between the results of the upper bound analysis of this study and those of Shiau et al. [9] and Chen and Xiao [45]. Slight differences between the results of this study and those of the other two studies were observed.

Gourvenec and Mana [30] applied the FEM to calculate the bearing capacity factors of strip footings and incorporated strength heterogeneity in their solutions. In addition, as listed in Tab.2, the values of N_c for different kB/c_u ratios obtained in this study and those of Gourvenec and Mana [30] are consistent. The values obtained in their study are within the values obtained from the lower and upper bounds solutions of FELA.

Moreover, values of q_u/(γB) were computed using the lower and upper bounds for a smooth strip footing resting on the level ground under undrained conditions. The results were compared with those of Shiau et al. [9], Mofidi Rouchi et al. [24], and Foroutan Kalourazi et al. [46], in which the same method used in this study was adopted. The comparisons in Tab.3 show that the results are in good agreement.

Finally, the bearing capacity factor (N_γ) values calculated for the strip footing on level ground in this study and other studies using methods including LB, UB, stress characteristic (SC), and limit equilibrium (LE) are presented in Tab.4. Tab.4 shows that their results are consistent.

4 Results and discussion

4.1 Seismic threshold value

The seismic threshold value is the value of k_h above which the strip footing slides (Fig.2(a)). The variations in k_h × q_u/c_u versus k_h for a strip footing near an excavation are shown in Fig.2. The threshold value decreased as kB/c_u increased (Fig.2). For instance, the threshold value was 0.78 when kB/c_u = 0 (homogenous soil), whereas it reached 0.14 for soil with kB/c_u = 20 (Fig.2). Moreover, as the L/B ratio increased, the threshold value decreased to a constant value for higher ratios, and the ratio at which this value became constant increased as kB/c_u decreased. The threshold values for soils with kB/c_u = 20 and 10 remained unchanged for L/B values exceeding 0.25, and for those with kB/c_u = 2 and 5, they stabilized at L/B = 0.5. However, for soils with kB/c_u = 1 and 0, the threshold value did not change when L/B exceeded 1 and 2, respectively. Regarding the influence of H/B, when kB/c_u = 2, 5, 10, or 20, the value of k_h × q_u/c_u was not influenced by H/B. However, when kB/c_u = 0 and 1, H/B could affect the soil. At kB/c_u = 0, H/B could not influence the soil when L/B ≥ 3. At kB/c_u = 1, the effect of H/B on the soil became minimal when L/B ≥ 2.

4.2 Design charts

Design charts and tables were developed for a strip footing adjoining an excavation of homogenous or heterogeneous soils. The variation in q_u/(γB) for k_h =0, the effect of L/B and H/B on the failure pattern, and finally the variation in q_u/(γB) for k_h ≠ 0, are depicted in Fig.3 and Fig.4, Fig.5 and Fig.6, and Fig.7–Fig.10, respectively.

Under k_h = 0 conditions, the rough footing had slightly higher q_u/(γB) values than the smooth (frictionless) footing for a specific value of c_u/(γB), as expected (Fig.3 and Fig.4). For the rough and smooth footings, the most significant trend in Fig.3 and Fig.4 is the direct relationship between kB/c_u and q_u/(γB). As c_u/(γB) increased from 0.5 to 5, q_u/(γB) increased for all values of kB/c_u, reaching approximately 10 times as high as its initial value at c_u/(γB) = 0.5. For a fixed value of c_u/(γB), q_u/(γB) converged to its maximum value at a lower L/B when kB/c_u increased. That is, for a smooth footing on soil with c_u/(γB) = 0.5 and kB/c_u = 5 (Fig.3), q_u/(γB) reached its peak value at an L/B value of approximately 0.8, whereas this ratio was approximately 0.3 for soil with kB/c_u = 20. The variation in the ultimate bearing capacity of the rough footing with the L/B ratio is depicted in Fig.5 for soil with H/B = 2, c_u/(γB) = 1, kB/c_u = 0.5, and k_h = 0.25.

Furthermore, H/B can impact q_u/(γB) in soils with low kB/c_u values, regardless of whether the footing is rough or smooth (Fig.3 and Fig.4). When kB/c_u = 5, 10, and 20, only one curve appeared, indicating that for a fixed L/B ratio, q_u/(γB) did not change as H/B varied. In contrast, the H/B ratio influenced q_u/(γB) in soils with kB/c_u = 0, 1, and 2. The impact of variations in H/B was more significant in homogenous soils than in heterogeneous soils. Fig.6 shows the effect of H/B on q_u/(γB) for a rough footing with L/B = 1.5 on soil with c_u/(γB) = 3, kB/c_u = 0.5, and k_h = 0.05. At c_u/(γB) = 0, the footing can only be constructed near excavations with H/B = 0.5 or 1. However, as c_u/(γB) increases, the footing can be sited adjacent to excavations with higher H/B values, that is, H/B = 10, although the q_u/(γB) values are the lowest for such H/B.

However, when k_h ≠ 0, H/B and c_u/(γB) impact q_u/(γB), similar to the k_h = 0 conditions. At a specific k_h and kB/c_u, q_u/(γB) increased 10-fold when c_u/(γB) increased from 0.5 to 5 (Fig.7–Fig.10). In addition, the H/B ratio can influence soils with kB/c_u = 0, 1, and 2 (Fig.7–Fig.10). However, when k_h > 0.2 and c_u/(γB) = 0.5 and 1, a strip footing cannot be constructed on homogenous soil owing to stability problems (kB/c_u = 0) (Fig.9 and Fig.10). Similarly, at k_h = 0.2 and c_u/(γB) = 0.5, the excavation in homogenous soils becomes unstable (Fig.8).

Moreover, the effect of L/B on q_u/(γB) decreased with an increase in k_h. In soils with kB/c_u = 20 and 10, q_u/(γB) attained its highest value over the range of L/B when k_h > 0.1 and k_h > 0.2, respectively (Fig.8–Fig.10). This implies that in such soils, q_u/(γB) is not influenced by L/B. Similarly, for soils with kB/c_u = 0, 1, 2, and 5, the value of L/B at which q_u/(γB) converged to its maximum value decreased as k_h increased, with the lowest values at k_h = 0.4.

The increase in k_h resulted in a decreased q_u/(γB) (Fig.7–Fig.10); that is, for the soil with c_u/(γB) = 0.5 and kB/c_u = 20, the maximum value of q_u/(γB) was almost 5 at k_h = 0.1 (Fig.7), whereas it decreased to more than 1.2 at k_h = 0.4 (Fig.10). In addition, as k_h increased, the differences between the values of q_u/(γB) for soils with different kB/c_u values decreased such that the values converged for all cases of kB/c_u and c_u/(γB) when k_h increased to 0.4.

Values of q_u for different H/B, c_u/(γB), and kB/c_u values were obtained for the most critical conditions of L/B (i.e., L/B = 0) and various k_h (Tab.5 and Tab.6) for perfectly smooth and perfectly rough footings. As the perfectly smooth footing slides when subjected to seismic loads owing to the lack of friction, this footing was only analyzed statically (k_h = 0). The NaN in the tables denoting “not a number” indicates that q_u was not calculated because of the instability of the excavation.

The analysis results suggest that when kB/c_u < 2, q_u is influenced by H/B, whereas for higher values of kB/c_u (2, 5, 10, and 20), H/B minimally influenced q_u. Hence, for each value of kB/c_u, c_u/(γB), and k_h, the values of q_u are presented for different H/B ratios in Tab.5. In contrast, the values of q_u obtained for the different H/B values and each kB/c_u, c_u/(γB), and k_h are averaged in Tab.6, and the standard deviation is presented.

5 Example

A 1 m width and perfectly rough strip footing is to be constructed 2 m away from an excavation at a height of 5 m. Design charts should be used to evaluate the bearing capacity of the strip footing. The undrained geotechnical conditions are as follows: c_u = 40 kPa, k = 400 kN/m²/m, γ = 20 kN/m³, k_h = 0.1.

Solution:

c_{\mathrm{u}}/(\gamma B)=40/(20\times 1)=2,

kB/c_{\mathrm{u}}=400/40=10,

L/B=2/1=2,

H/B=5.

Using Fig.7(d) and considering c_u/(γB) = 2 and kB/c_u = 10 for L/B = 2 and H/B = 5,

q_{\mathrm{u}}/(\gamma B)=\mathrm{17.3.}

Finally, the bearing capacity is obtained as q_u = 17.3 × 20 × 1 = 346 kPa.

6 Conclusions

Lower and upper bounds of the FELA were adopted to compute the undrained seismic bearing capacity of a strip footing sited close to a heterogeneous excavation. The effects of several variables, including the setback distance ratio (L/B), excavation height ratio (H/B), soil strength heterogeneity (kB/c_u), and horizontal earthquake coefficient (k_h), on the normalized bearing capacity (q_u/(γB)) were evaluated. The conclusions of this study are as follows.

1) The results of the proposed method were compared with those of previous studies and demonstrated a good agreement.

2) The threshold value depends on kB/c_u, L/B, and H/B. However, the first two parameters had a more significant influence than the latter.

3) In all cases, q_u/(γB) increased with an increase in L/B and stabilized at a specific L/B value. Thus, an optimum distance exists from the excavation at which the footing can be placed safely because the bearing capacity is maximum and does not change at further distances.

4) kB/c_u significantly influenced the normalized bearing capacity. In contrast, H/B was the least influential variable on the bearing capacity.

5) An increase in k_h resulted in a decrease in the normalized bearing capacity. In addition, at a high horizontal earthquake coefficient, the effect of heterogeneity on the maximum bearing capacity disappears, indicating that both heterogeneous and homogenous soils have equal maximum bearing capacities.

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