Analysis of the behavior of structures under the effect of progressive rupture of a cavity

The frequent presence of underground cavities in certain developable areas poses a potential collapse risk that can be detrimental to the proper functioning of infrastructures and the safety of their users. Various studies have been conducted to investigate the issues caused by the collapse of underground cavities and the impact of these collapses on surface structures.

Table 1 summarizes the main research studies analyzing cavity collapse using experimental and numerical methods, classified chronologically.

Our work focuses on analyzing the interaction between the soil and the structure, first examining the progressive collapse of an underground cavity beneath a structure and then evaluating its impact on the stability of the structure. Finally, we conduct a parametric study to understand how cavity volume, depth, and spacing influence the stability of the structure.

To predict ground movements caused by cavity degradation, geotechnical engineers have various methods at their disposal. These methods include empirical approaches using detailed field data, analytical methods based on mechanical equations, and numerical methods. These approaches are documented in the scientific literature notably by Deck et al. (2006) and Dolzhenko (2002).

The main objectives of this study are to minimize the consequences of cavity collapse, maintain the stability of the structure, and reduce deformations observed at the structural elements level. Our methodological approach involved initially validating a numerical model, followed by calculating the collapse values using an analytical method. Test results were presented using a scaled-down model. In practice, empirical methods are often guided by analytical approaches or finite element calculations. These methods are then adjusted based on experimental curves, as highlighted by Aftes (1982).

The empirical approaches described by Peck (1969) for vertical displacements and by Lake et al. (1992) for horizontal displacements are used to predict ground surface movements after the excavation of a circular tunnel (see Figure 1). Equations 1 and 2 express vertical and horizontal displacements, respectively: (1) Vertical displacement: S(x)=Smax×e−x22i2 {\rm{Vertical}}\,{\rm{displacement}}:\,{\rm{S}}({\rm{x}}) = {{\rm{S}}_{{\rm{max}}}} \times {{\rm{e}}^{{\raise0.7ex{- {{\rm{x}}^2}} \!\mathord{\left/ {\vphantom {{- {{\rm{x}}^2}} {2{{\rm{i}}^2}}}}\right.}\!\lower0.7ex{2{{\rm{i}}^2}}}}} (2) Horizontal displacement: V(x)=S(x)×xH {\rm{Horizontal}}\,{\rm{displacement}}:\,{\rm{V}}({\rm{x}}) = {\rm{S}}({\rm{x}}) \times {{\rm{x}} \over {\rm{H}}}

Figure 1:

Schematic diagram of the empirical approach by Peck (1969).

The parameter i, representing the width of the collapse trough, can be approximated by various empirical relationships taking into account the diameter D, the cover C, and the nature of the soil (Table 2).

Caudron et al. (2006) demonstrated that with some modifications, the two expressions (eqs 1 and 2) can be used to predict surface displacements as long as the movement curves remain continuous. This is the case when the upper part of the lining is made of the granular material and the cavity is rectangular in shape. Mathematical treatment and integration of the value of the increment i depending on the soil type are presented in Table 2.

They manage to propose both equations related to vertical and horizontal displacements such that i is derived from the empirical formulas (Dolzhenko, 2002), where (i = 0.15H + 0.5D). The formulas then become as follows: (3) Vertical displacement: S(x)=0.667Vcavity2.5i×e−x22i2 {\rm{Vertical}}\,{\rm{displacement}}:\,{\rm{S}}({\rm{x}}) = {{0.667{{\rm{V}}_{{\rm{cavity}}}}} \over {2.5{\rm{i}}}} \times {{\rm{e}}^{{\raise0.7ex{- {{\rm{x}}^2} \!\mathord{\left/ {\vphantom {{- {{\rm{x}}^2}} {2{{\rm{i}}^2}}}}\right.}\!\lower0.7ex{2{{\rm{i}}^2}}}}}} (4) Horizontal displacement: V(x)=S(x)×(x1m)(H1m)Y {\rm{Horizontal}}\,{\rm{displacement}}:\,{\rm{V}}({\rm{x}}) = {\rm{S}}({\rm{x}}) \times {{({\rm{x}}/1{\rm{m}})} \over {{{({\rm{H}}/1{\rm{m}})}^{\rm{\gamma}}}}} with an optimal value of γ equal to 0.87 and V_cavity=a×b (a and b the height and width of the cavity).

The work of Caudron et al. (2007), which particularly focused on soil-structure interaction phenomena during the formation of a cavity near a surface frame, was used as a reference as it perfectly aligns with the concept of this research. The authors chose a real case (the Malakoff limestone quarry in the Paris region) involving a cavity of 10 meters wide and 2 meters high with a depth of 8 meters formed by 9 laminated soil layers (see Fig. 2) (Caudron, 2007; Caudron et al., 2006; Caudron et al., 2004).

Figure 2:

Real model of the cavity.

The characteristics of the actual materials are detailed in Table 3.

The main source of problems and complexity in designing a physical model lies in the importance of adhering to similarity rules with respect to the original phenomenon. Obviously, for a full-scale model, this is not limiting. The main limitation in this case comes from the cost and feasibility of tests.

This is why it is common to resort to reduced models, which present a number of advantages: speed, reproducibility, and the possibility of working until failure. However, to ensure that the phenomenon obtained in the reduced model exhibits behavior similar to that observed in full scale, it is necessary to ensure compliance with a number of rules. These are the laws of similarity, as presented by Dehousse and Arnould (1971) and by Bazant (2004). Since then, Garnier has specified their application to the field of geotechnics (Garnier, 2001a; Garnier, 2001b).

Each letter accompanied by an asterisk (*) represents the scale factor associated with the change in scale for the respective quantity. Table 4 provides the meanings of each quantity.

Caudron et al. (2006) extensively presented the small-scale physical model: design and limitations of the model. The first step in defining the small-scale physical model was the laws of similarity. The assumption is made that the thermal effect has no influence on the series of tests, which makes the problem somewhat simpler. Then, three scale factors must be determined in order to establish the entire relationship between the full-scale case study and the small-scale model. These scale factors concerned gravity, density, and length. The tests are carried out under normal gravity, so the corresponding scale factor is 1.

The analogous soil has a unit weight of 65 kN/m³, so the scale factor on density is 3. The final scale factor concerns length. It was set at 1/40 in order to have a test bench with practical dimensions. From this point, all other scale factors can then be deduced from the laws of similarity and these three values. Therefore, it is not possible to adhere to all similarity rules, particularly those concerning stress states. The results of small-scale tests will then be qualitative rather than quantitative. The value 1/40 is chosen, which remains within the limits of the commonly accepted range of values for the Schneebeli material used (Ovesen, 1979) (see Table 5).

The experimental reduced model is simpler, represented at a scale of 1/40, with a soil mass of 1000 mm width for a covering height of 200 mm above the cavity (see Fig. 3).

Figure 3:

Experimental scale model.

The Schneebeli analog material, used to represent the soil, consists of a mixture of assemblies of steel rods with diameters of 3, 4, and 5 mm in precise proportions (see Fig. 4) (Schneebeli, 1956; Schneebeli, 1957). The lining consists of a cohesive bench with a thickness equivalent to 50 mm located at the top of the cavity. This is topped with a layer of pulverulent material equivalent to 150 mm in thickness. The cavity, with a height equivalent to 50 mm, is gradually created up to a maximum width corresponding to 250 mm in five steps (Caudron et al., 2004; Caudron et al., 2006; Caudron et al., 2007).

Figure 4:

Overview of Schneebeli rolls.

The structure used for the study of soil-structure interaction is of the steel beam-post type. The load considered, uniformly distributed over the beam elements, corresponds to 10 kPa, comparable to permanent loads and service loads. The stiffness (ES and Eiz with E the Young's modulus, S the section, and Iz the moment of inertia) of the structural elements respects the scaling: the behavior obtained is purely elastic. The rules of similarity govern the relationship between a full-scale object and its scale model. In the context of a similarity that must account for the mechanical behavior of the object, a number of scale factors must be considered. Fig. 5 shows a schematic view of the full-size building as well as its scale model.

Figure 5:

Real and scale model of the structure (Caudron et al, 2007).

Table 6 shows the characteristics obtained from the predimensioning of the structure with the assumed data and the equivalent characteristics for the scale model with the similarity laws. However, the use of a coherent material is a prerequisite for representing the resistant bench needed to create the cavity.

All conducted characterization tests by researchers on the Schneebeli material show that its behavior is identical to that of dense sand (Dolzhenko, 2002; Kastner, 1982). Therefore, the material was modified by Caudron (2006) to represent a cohesive material.

Cohesion was achieved using an aqueous adhesive. Laboratory tests were conducted to determine the mechanical characteristics of the modified material, and the improvement concerns the friction angle, cohesion, and volumetric behavior from a pulverulent soil to a cohesive one. The biaxial tests, therefore, began with different concentrations of adhesive at C/2, C/4, and C/8 (C: glue concentration; C/2, C/4, and C/8: concentration fractions), gradually diluting it until the desired behavior was obtained. Two concentrations of adhesive, those at C/4 and C/8, seem to allow obtaining a material that exhibits mechanical characteristics close to those desired. The modified material exhibits characteristics close to those desired, so they are more finely characterized in Table 7.

The finite element method is employed to model the soil-structure interaction during the formation of a cavity. Once the model is established, a comparison between the numerical results and the experimental data from (Caudron, 2007), as well as those from the analytical method, will be performed. The modeling involves several successive and distinct steps, such as data input, definition of boundary conditions, meshing, calculation phases, simulation startup, and results analysis.

To model the structure, it is important to present all the data related to the different materials: building geometry, material properties (powdery and cohesive soil, air, column-beam structure, footings), load, cavity dimension, and depth. The schematic representation of the model is shown in Figure 6. It consists of a soil block with a height of 25 m and a width of 40 m. Comprising two superimposed soil layers, the upper layer is of powdery type with a height of 6 m overlying a cohesive soil layer with a height of 19 m. A two-level structure is embedded in the upper layer at a depth of 1.2 m. The lower layer encompasses a cavity with a height of 2 m and a length of 10 m in a rectangular shape.

Figure 6:

Model geometry: (a) global geometry, (b) structure, and (c) cavity diagramming.

Based on the values of the properties of the various materials used in the experimental study, the same values are incorporated into the material database of the numerical model. Tables 8 and 9, respectively, present the properties of the soils and structural elements. Among the problems encountered during the modeling is the representation of the rupture process and the simulation of the void present during the progression of the cavity. We chose the same properties for air to simulate the void existing between the structural elements. Air is represented as a material with low physical properties.

For the calculations, we assume that the interfaces between the different soil layers are perfectly adherent, implying continuity of vertical stresses and vertical displacements. Boundary conditions are ensured both by embedding the soil at the base and on the sides and by positioning the structure at a sufficiently distant distance from the edges to allow for good stress distribution in the soil.

The soil mass has been discretized entirely by 15-node triangular finite elements. The same type of elements has been adopted for meshing both the soil body and the structure to ensure correct assembly. The mesh consists entirely of 561 elements and 4631 nodes. Local mesh refinement has been performed in areas where strong gradients are likely to appear, i.e., around the cavity, to obtain a good estimation of stress and displacement fields (see Fig. 7).

Figure 7:

Model meshing.

The reasoning process adopted for the calculation of such a model led us to establish 7 phases:

Phase 0: Initiation of stresses (K0 procedure) to determine initial effective stresses.

This phase involves activating the structure (footings and framework) and backfilling to ensure the stability of the structure. At this stage, the soil is loaded only under its own weight.

Phase 3: Loading.

Phases 4, 5, 6, 7 represented in Figure 10 express the process of rupture numerically simulated according to the collapse mode of the cavity.

Figure 8:

Calculation phases.

Figure 9:

Phases 1, 2, and 3: (a) Phase 1 (excavation), (b) Phase 2 (soil + structure), and (c) Phase 3 (loading).

Figure 10:

Cavity rupture process. (a)

Phase 4 (initial cavity rupture)

(b)

Phase 5 (2nd cavity rupture)

(c)

Phase 6 (3rd cavity rupture)

(d)

Phase 7 (total cavity rupture)

We triggered the rupture by initiating a thin layer of air void, which we then gradually extended until complete collapse.

After several attempts and corrections, the sequence of calculation phases has been established without interruption, and we have reached the phase of result exploitation (launch of calculations) as shown in Figure 11:

Figure 11:

Calculation launch.

According to the experimental results (Fig. 12a), we observed that the rupture of the stiff bench occurs shortly after degradation. The latter fall into the cavity, and the soil reduced to powder on the surface follows their movement. The foundations of the structure are then evaluated, revealing some smaller voids in the fracture areas in the stiff bench and significant detachment of the right footing of the structure. Figure 12b presents the visualization of the last phase of the numerical model (total cavity rupture). In this phase, the structure is tilted to the right, and consequently, the maximum displacement at footing 1 is recorded, as it is located at the cavity axis.

Figure 12:

Final phase of rupture.

To demonstrate the validity of our model, we compared the results of vertical and horizontal collapses of different footings with the vertical and horizontal displacements at the surface of the cavity. The validation of the numerical model relies on comparing the results with those derived from experimental data and compared analytical results. Figures 13 and 14 display the results of the structure collapse comparison using analytical, numerical, and experimental methods. A good correlation is obtained between numerical and experimental results.

Figure 13:

Vertical displacements (numerical, experimental, and analytical) of the footing.

Figure 14:

Horizontal displacements (numerical, experimental, and analytical) of the footings.

A slight deviation of 5 mm from the analytical results is observed. The maximum collapse occurs at footing 1, which is the most affected due to its positioning with the axis passing through the center of the cavity, unlike the other footings, which are slightly inclined.

The maximum displacement is observed on the right side of the structure, corresponding to the cavity axis (Position x = 20 m) (see Figure 13). Horizontal displacements are depicted in the graph (Figure 14), showing three curves with two types of positive and negative displacements. A significant influence of the structure is observed on the left side.

There is a good agreement between the curves of numerical, experimental, and analytical collapses, where the maximum value of numerical collapse is almost equal to that of experimental collapse (with a difference of a few millimeters). The maximum slope of collapse is located at the axis of the cavity. Gradients or breaks in the curve of the numerical method are attributed to the simulation of cavity rupture (see Figure 10c). In the numerical method, the cavity is subdivided into small squares, and the higher the number of squares, the more the gradient of the curve attenuates.

Figure 15 shows that the vertical displacements of the footings are identical for the experimental and numerical methods, although there is a slight difference for the analytical method (8.3 mm as the highest estimate for footing 4). It can also be observed that there is a slight difference in the horizontal displacements of the footings using the three numerical, experimental, and analytical methods (5.67 mm as the highest estimate for footing 2).

Figure 15:

Differences in displacements of each footing among the three methods.

To study the gradual degradation of cavities and their impact on structure stability, a comparative method was employed to develop influential parameters such as volume, spacing, and depth of cavities. This approach will evaluate the effect of these parameters using variable ratios, including a ratio of b/a ranging from 0.5 to 3, a ratio of H/B varying from 0.5 to 3, and a ratio of L/B also ranging from 0.5 to 3. The dimensions B, H, a, b, and L are shown in Figure 12 (Djamel Saadi et al., 2020).

Figure 17 illustrates the behavior of the soil and the soil-structure interaction concerning the increase in cavity volume and structure instability. After analyzing the behavior of the footings, it is observed that those aligned with the cavity axis undergo more deformations than those away from this axis. Stress levels peak as the cavity volume increases. Vertical stresses are evident on the sides of the cavity, with a compression value of −450 kN/m².

Figure 16:

Models used in this study: (B) width of the footing, (H) depth of the cavity, (L) cavity spacing, (a) cavity height, and (b) cavity width.

Figure 17:

Stress in the yy plane.

The impact on the structure is negligible as the stress approaches zero, but the footings are exposed to stresses ranging from −150 kN/m² to −300 kN/m² depending on their position relative to the cavity.

The relative exploitation of results is presented in Fig. 18.

Figure 18:

Vertical and horizontal displacements according to the three ratios.

From observing the curves of vertical and horizontal displacements (Fig. 18), it can be seen that each curve is symmetrical with respect to the cavity axis.

For the H/B ratio: The displacement value increases as the cavity depth decreases.

For the L/B ratio: The displacement value increases as the distance between the cavities decreases.

For the b/a ratio: We also notice that the displacement value increases with the increase in dimensions of cavity. This led us to search for an equation expressing the relationship between the cavity volume and the vertical and horizontal displacements that occur in the soil.

The different profiles were analyzed based on the parameters a, b, H, B, and L, with trend lines plotted. The correlation coefficients R², ranging from 0.90 to 0.98, demonstrate a good fit of the data. The equations obtained from the curves in Figure 19 allow for accurate prediction of the horizontal or vertical collapse value based on the characteristics of the void beneath the structure (volume, depth, and spacing), as well as the initial displacement (without void), as illustrated in Equations 5 and 6. (5) Uy=Uy0(0.347ba+0.9383+4.0803HB−0.766+4.0803LB−0.17) {{\rm{U}}_{\rm{y}}} = {{\rm{U}}_{{{\rm{y}}_0}}}(0.347{{\rm{b}} \over {\rm{a}}} + 0.9383 + 4.0803{{{\rm{H}} \over {\rm{B}}}^{^{- 0.766}}} + 4.0803{{{\rm{L}} \over {\rm{B}}}^{- 0.17}}) (6) Ux=Ux0(0.3525ba+0.9514+8.3075HB−1.016+1.8262LB−0.34) {{\rm{U}}_{\rm{x}}} = {{\rm{U}}_{{{\rm{x}}_0}}}(0.3525{{\rm{b}} \over {\rm{a}}} + 0.9514 + 8.3075{{{\rm{H}} \over {\rm{B}}}^{^{- 1.016}}} + 1.8262{{{\rm{L}} \over {\rm{B}}}^{- 0.34}})

Figure 19:

The variation in displacements under the three ratios.

Equations 5 and 6 obtained are specific to the type of soil used in this study and provide an initial prediction. They can be used to anticipate the risks to which structures are exposed due to the rupture of cavities beneath the foundations.

According to Figure 20, we notice that the most influential ratio on the stability of the structure is the depth ratio of the cavity. This prompts us to focus our analysis on the significance of the void depth and its impact on structural instability. We also observe that the significance of volume, depth, and dimensions is more significant when the ratio lies between 0 and 2.5. Beyond a ratio of 2.5, all three parameters will have an equivalent impact.

Figure 20:

Displacement assembly according to the three ratios.

In this study, we verified the validity of the numerical model by relying on both the results from an experimental model conducted by one of the researchers and analytical laws. We furthered our investigation through a parametric study based on varying the ratio between the volume, depth, and distance between cavities. –

The obtained results are reliable, providing the developed model with a solid foundation for future case simulations.

In this regard, this research was primarily proposed to analyze the cavity failure process leading to structural instability due to the effect of differential displacement. This results in high stresses on load-bearing elements and the consequent appearance of cracks and fractures. Through the use of finite element methods, other parameters such as the position of the structure, anchoring of the footings above the void, and shape can be addressed to better understand the behavior of structures facing abrupt void collapses. These findings necessitate, in a subsequent study, proposing technical solutions to reduce potential disasters and analyzing their effectiveness in mitigating settlement and maintaining maximum structural stability until human intervention.

The next phase of this study will involve predicting the bearing capacity of the soil based on the three parameters, dimensions, depth, and spacing, and establishing their relationship with footing design principles. This risk management approach aims to prevent any significant damage and strengthen the margin of safety.