Influence of Subsoil and Building Material Properties on Mine-Induced Soil–Structure Interaction Effect

Abstract

Soil–structure interaction (SSI) refers to the dynamic interaction between a structure and the surrounding soil on which it rests. The behavior of the soil can significantly affect the response of the building structure. In the context of civil engineering and structural analysis, SSI becomes particularly important when considering the response of structures to dynamic loads such as earthquakes or so-called paraseismic loads, e.g., mining tremors. Several factors contribute to SSI. Soil and building structure material properties, foundation type, and loading conditions are the most important parameters. The article concerns SSI in the case of mining rock bursts in Poland. The influence of changes in site material conditions and building material properties on the SSI phenomenon was investigated. A few variants of different properties of typical construction materials (brick, reinforced concrete, and cellular concrete) in the case of selected representative building structure were considered. The subsoil material properties from the wide range were also taken into account. Numerical three-dimensional finite element method (FEM) analysis was applied. The adopted models of the soil-structure system were verified by data from in situ experimental vibration measurements. A significant influence of the subgrade material and the building structure material on the SSI was demonstrated.

1. Introduction

Soil–structure interaction (SSI) refers to the dynamic interaction between a structure and the surrounding soil on which it rests. The behavior of the soil can significantly affect the response of the building structure. SSI problems are extensively studied concerning different kinds of excitations: earthquakes [1,2,3,4], so-called paraseismic sources resulting from human activity—e.g., originating from mining rockbursts belong to the most intensive such vibrations [5,6,7]. Although the intensity of mining tremors is smaller than in the case of natural earthquakes, mining-induced ground motion also can result in damage to surface structures.

Ignoring the SSI effect may lead to incorrect design, especially in the case of rigid and heavy objects, because the flexibility of the substrate may significantly affect the response of the structure. The effect of SSI can make a substantial difference in how buildings behave during seismic-type excitations.

Several factors contribute to SSI: soil and building structure material properties, foundation type, and loading conditions are the most important parameters of them.

In the studies on SSI analysis, the numerical approach dominates [8,9,10,11,12,13,14]. The first approach in the analysis of SSI effects is the direct method. In this method, the model consists of the structure and soil. The solution of governing equations of the model is performed mainly in the time domain, especially in the case of nonlinear analysis. This method requires the use of a large soil computational model despite the introduction of artificial boundaries in the form of properly selected spring-dashpot or infinite elements [15]. An illustration of a direct analysis of SSI using a finite element method (FEM) continuum modeling is shown in Figure 1. In a FEM model, the soil and structure are analyzed as a complete system of related elements. Thus, the whole FEM system consists of soil mesh elements with transmitting boundaries at the ends of the soil mesh, soil-foundation interface elements, foundation elements, and structure (building) elements. Typically, there are no specific rules that dictate the distance from a building to the transmitting boundary. Usually, a distance equivalent to five times the respective (in the direction considered) dimension of the building is used. The presence of soil-foundation interface elements is needed in order to come up with the real nature of the behavior of structures. Different types of joint (interface) elements have been introduced and formulated in the literature. Generally, interface elements employed in SSI problem FEM modeling could be grouped into two major types: nodal interfaces (e.g., zero-thickness elements and node-to-node spring elements) and continuum elements (e.g., thin-layer interface elements).

The second approach is called the indirect method or substructure method. In this approach, the structure and soil are separated. The solution is performed in the frequency or time domain. There are three steps in this method: (1) the determination of the input ground motion (termed kinematic interaction), (2) the determination of the impedance functions for the foundation, and (3) the calculation of the dynamic response of the structure supported by the springs with the impedance functions determined in the second step, subjected to the input motion determined in the first step (termed as inertial interaction) (see Figure 2) [15]. The input ground motion (step 1) is determined from the measurement of vibration in situ or measurement results obtained in similar ground properties or as artificial free-field vibrations. For each particular harmonic excitation, the dynamic impedance is defined (step 2) as the ratio between force (or moment) and the resulting steady-state displacement (or rotation) at the centroid of the base of the massless foundation. In the third step, springs and dashpots with parameters corresponding to ground properties are used.

The phenomenon of interaction may concern structures made of various types of materials. If the structures with the same layout and construction are built using different materials it can change not only the dynamic characteristics of the structure but also the SSI effect. Materials used in the design of structures are, e.g., as follows: high-performance concrete (HPC) [16] or ultra-high-performance concrete (UHPC) [17], masonry [18,19,20], wood [21], reinforced concrete (RC) [20,22,23,24], and cellular concrete [25].

HPC is a specialized type of concrete that exhibits superior mechanical and durability properties compared with conventional concrete. The development of HPC involves carefully selecting materials and adjusting mix proportions to achieve the desired characteristics. Some key features and applications of high-performance concrete include enhanced strength and toughness (making it suitable for structures subjected to dynamic loads, such as earthquake-resistant buildings), improved durability, reduced permeability, increased workability and improved aesthetics, and reduced maintenance costs and applications. It is important to note that the mix design for HPC should be tailored to the specific requirements of the project and the environmental conditions it will be exposed to. Additionally, quality control during production and construction is essential to achieve the desired performance characteristics. The specific mix design for high-performance concrete varies depending on project requirements and environmental conditions. The careful selection of materials, including cement, aggregates, and admixtures, and meticulous quality control during production and placement are crucial to achieving the desired performance characteristics. Applications of HPC include, e.g., bridges, high-rise buildings, tunnels and infrastructure, precast concrete products, marine structures, and repair and rehabilitation.

Ultra-high-performance concrete (UHPC) is a cementitious composite material composed of an optimized gradation of granular constituents, the ratio of a water-to-cementitious material of less than 0.25, and a high percentage of discontinuous internal fiber reinforcement.

A masonry structure is a building or construction made primarily from individual units or elements of masonry, which are usually bonded together using mortar. Masonry is one of the oldest and most enduring construction methods, and it involves the assembly of materials like bricks, concrete blocks, natural stone, or clay tiles to create walls, columns, arches, and other architectural elements. Some key aspects of masonry structures refer to construction methods, strength and durability, architectural features, thermal mass, fire resistance, maintenance, applications, reinforced masonry to enhance their load-bearing capacity and resistance to seismic forces, seismic considerations to enhance their earthquake resistance, and restoration and preservation.

Wooden structures refer to buildings or constructions primarily constructed using wood as the main building material. Wood has been used in construction for centuries due to its abundance, versatility, and renewable nature. Wooden structures are valued for their aesthetic appeal, sustainability, and ability to create warm and inviting living or working spaces.

Reinforced concrete (RC) is a composite material. It comprises different constituent materials with very different properties that complement each other. In the case of reinforced concrete, the component materials are almost always concrete and steel. The steel is the reinforcement. Reinforced concrete is a versatile construction material that combines the compressive strength of concrete with the tensile strength of steel, resulting in a highly durable and structurally efficient building system. Some key aspects of RC structures refer to concrete composition, steel reinforcement, design and engineering (resistance to various forces, including gravity, wind, and seismic forces), types of RC structures, strength and durability, flexibility in design, construction techniques, reinforced concrete elements for carrying specific loads and fulfill structural and architectural requirements, seismic design in regions prone to earthquakes involving specialized engineering techniques and materials to ensure the safety of the building and its occupants during an earthquake, maintenance, sustainability, and fire resistance.

Cellular concrete (also called autoclaved aerated concrete, porous concrete, and aerated concrete) combines cement, water, and preformed foam. All of the production technologies developed for this material use standard components—cement, sand, water, lime, gypsum, and an expanding agent (powder or paste of aluminum)—to foam the cellular concrete. The purpose of the foam is to supply a mechanism by which a relatively high proportion of stable air voids can be induced into the mixture and produce a cellular or porous solid upon curing the mixture.

Seismic-type ground motion can be induced not only by earthquakes but also by so-called paraseismic sources, such as mining tremors. Mining-related surface vibrations constitute an essential problem in mining areas. Underground exploitation results in induced seismicity and, as a consequence, a negative paraseismic vibration impact on the environment. Damage to surface structures (residential or office buildings, among others) may occur as a result. Mining tremors are similar to natural earthquakes due to their randomness, lack of human control of the time and place of their occurrence, and also due to the energy magnitude of the strongest mining rock bursts. However, some parameters could be different, such as vibration duration, depths of hypocentre, and predominant frequencies of vibrations. Thus, the analysis of the influence of mining-associated vibrations on buildings is very important.

The study concerns the analysis of the SSI phenomenon in the case of mining rock bursts in Poland. This article focuses on one very important problem of SSI, especially in designing and constructing buildings—transmission of mining vibrations from the ground to the foundation of the building. The influence of changes in site material conditions (subsoil properties) and building material properties on the SSI phenomenon was investigated. A few variants of different properties of typical construction materials (brick, reinforced concrete, and cellular concrete) in the case of selected representative building structure were considered. The subsoil material properties from the wide range were also taken into account. Numerical three-dimensional finite element method (3D FEM) analysis was applied. The adopted models of the soil–structure system were verified using data from in situ experimental vibration measurements. A significant influence of the subgrade material and the building structure material on the SSI was demonstrated.

Additionally, similarly to SSI, tests were conducted on the ground material and building material influence on an important parameter that affects SSI—the natural frequency of the building.

In particular, the significance of the work comes from its new contributions: (a) experimentally (using in situ measurements) validated 3D FEM modeling of the soil-building system with three variants of subsoil models (nonflexible, flexible without damping taken into consideration, flexible taking into account damping); (b) numerical evaluation of the influence of both subsoil and building material properties on the natural frequencies of building with typical structure with load-bearing walls; (c) numerical evaluation of the mine-induced building foundation vibration dependence on the material properties of subsoil and the material of building structure.

2. 3D FEM Model

2.1. Introduction Remarks

Over recent decades, computational technology advancements have revolutionized geotechnical and structural engineering, enabling more complex analyses. The finite element method (FEM), with its flexibility and ability to model complex material behaviors and interactions, became indispensable in SSI research. Today, 3D FEM in SSI analysis not only allows for realistic load scenario modeling but also integrates geotechnical research data and building construction. This tool enables detailed analysis of unique construction geometries, diverse soil conditions, and their interactions. It also facilitates exploring new building materials and construction techniques.

A primary challenge in implementing 3D FEM models is ensuring accurate input data, which is crucial for analysis reliability. This includes precise material property modeling of soil and structures. In the context of our research on the transmission of vibrations from the ground to buildings, the choice of the Ansys Mechanical APDL [26] software as the simulation tool is pivotal. This software stands out for its advanced finite element modeling (FEM) capabilities, which are essential for accurately depicting the complex geometries of buildings and the heterogeneous nature of soil. The analyzed numerical models were crafted using ANSYS APDL (ANSYS Parametric Design Language), a versatile tool for finite element analysis. ANSYS APDL is particularly suitable for this type of modeling due to its powerful parametric capabilities, which allow for the customization and automation of the simulation process. Its ability to accurately simulate the physical behavior of structural elements under various loads and constraints makes it an excellent choice for predicting the performance of building structures.

2.2. Building Model

The numerical model of a building is based on detailed architectural drawings of an actual building, as shown in Figure 3. The base plans include a ground floor plan (a), a first-floor plan (b), and a cross-section of the foundations (c). These plans serve as the blueprint for constructing a precise and representative model of the structure.

Figure 4 displays a visualization of a three-dimensional finite element building model (FEM) created in ANSYS, with a finite element mesh applied to the structure.

Additionally, Figure 5 illustrates the individual components of the numerical model, broken down layer by layer to show the complexity and detail captured in the FEM analysis. Starting from the top, the model includes the roof (a), knee walls and roof supports (b), first-floor slab (c), the entirety of the first floor including structural elements (d), ground-floor slab (e), the detailed ground floor (f), the surface level or ground level (g), foundation walls (h), and finally, the foundation footings and tie beams (i).

By starting with the roof, the building’s topmost structure was composed of prefabricated troughed plates, modeled as a shell using the four-node element SHELL181. This element had six degrees of freedom at each node and was suitable for both linear and nonlinear applications, as well as for analyzing thin to moderately thick shell structures. The supporting knee walls, constructed from brick, were also modeled with SHELL181 elements, characterized by a thickness of 0.12 m. For the ceilings of both the ground floor and the first floor, originally cast from reinforced channel slabs, an equivalent thickness of 0.132 m for the first-floor slab and 0.2 m for the ground-floor slab was calculated. These slabs were modeled using SHELL181 elements. The ceiling ties around the floor plates were represented using BEAM188 elements, capturing the dimensions of 0.24 m by 0.24 m. The stairwell, a reinforced concrete structure, was modeled with SHELL181 elements, with a stair thickness of 0.1 m. For the load-bearing walls, three variants of typical materials used in buildings were considered: brick walls with a thickness of 0.25 m, considering an additional mass for the external insulation; reinforced concrete walls with a thickness of 0.15 m; and cellular concrete walls with a thickness of 0.25 m. The partition walls of 0.12 m thickness were built of brick, and the monolithic reinforced concrete foundation walls were 0.25 m thick. All of the walls (load-bearing, partition, and foundation walls) were modeled with SHELL181 elements. Whereas the monolithic reinforced concrete foundation footings (with cross-sections of 0.4 m by 0.7 m or 0.4 m by 0.5 m) were modeled using two-node 3D BEAM188 elements, each of which had six degrees of freedom. The foundation tie rods, square in cross-section at 0.3 m by 0.3 m, were introduced to stiffen the foundation due to the building being subject to mining-induced vibrations. This is particularly important in ensuring structural integrity under such conditions.

The model assumed a linearly elastic material behavior, a reasonable approximation given that the actual building had shown no signs of damage, suggesting that stress and strain levels had not reached critical limits. As written above, the majority of the elements were modeled using SHELL181 elements, except for the foundation footings, foundation tie rods, and ceiling ties, which employed BEAM188 elements due to their specific structural functions. A complete overview of the building material properties used in the carried-out FEM simulations is outlined in

Table 1. Material parameters of structural building elements.

Structural Element	Material	Elastic Modulus [GPa]	Density [kg/m3]	Poisson’s Ratio [–]
strip footing	concrete	20.8	2500	0.25
foundation tie rods	reinforced concrete	19.7	2500	0.25
foundation walls	reinforced concrete	23.1	2500	0.25
ceilings	reinforced concrete	27.0	2500	0.25
ceiling ties	reinforced concrete	27.0	2500	0.25
load-bearing walls	brick	2.85	1800	0.25
	reinforced concrete	31.0	2500	0.25
	cellular concrete	1.80	600	0.25
partition walls	brick	2.85	1800	0.25
lintels	reinforced concrete	27.0	2500	0.25
stairs	reinforced concrete	27.0	2500	0.25
supporting knee walls	brick	2.85	1800	0.25
prefabricated roof panels	reinforced concrete	27.0	2500	0.25

. The values from Table 1 are based on the technical documentation of the modeled building and building standards related to masonry, reinforced concrete, and cellular concrete structures.

2.3. Subsoil Model

Three variants of the subsoil model allowing various properties of the subsoil to be taken into account were considered. The first variant of the subsoil model, marked as SM1, assumed nonflexible subsoil (foundations of the building are fixed in the soil with infinite stiffness). In the second one, noted as SM2, the flexibility of the subsoil was represented through the use of spring elements that allow for movement in both vertical and horizontal planes. In the third variant of the subsoil model, marked as SM3, damping was incorporated into springs to account for energy dissipation.

The SM2 subsoil model, using elastic springs, allows for the consideration of the soil’s capacity to deform elastically under seismic-type loads. Therefore, the COMBIN14 element from the ANSYS element library was utilized to achieve this modeling. The springs were aligned along the horizontal x- and y-axes and the vertical z-axis, reflecting the soil’s capacity to deform elastically under seismic loads. The COMBIN14 spring elements in ANSYS were adept at representing the three-dimensional interaction between the soil and the structure. Figure 6 illustrates the implementation of a spring-based soil model (SM2) within the ANSYS simulation environment and shows the transverse, longitudinal, and vertical placement of spring elements that emulate the soil’s resistance.

The stiffness values for the spring elements that facilitate both horizontal (in transverse and longitudinal directions) and vertical movement were determined by employing the Savinov soil model and the soil reaction coefficient Cz, which is determined by the ratio of the vertical unit base pressure to the elastic strain [27]. The Savinov method for determining the dynamic elastic properties of the subsoil is applicable to rectangular foundations. It is assumed in the Savinov method that the subsoil is an unlimited elastic plane on which a homogeneous membrane is placed and stretched in all directions (there are no friction forces between the plane and the membrane). The values of the Cz coefficient depend on the geotechnical category of the soil. The spring stiffness along all directions was parametrically varied, reflecting diverse soil conditions as characterized by the modulus coefficient Cz, with evaluations conducted across a wide stiffness range from 35 MPa to 65 MPa.

The third variant of the subsoil model (SM3), in addition to soil flexibility, also took damping into account. Damping coefficients were calculated, considering the soil density equal to 1800 kg/m³ (for medium-dense sands), the shear wave velocity (200 m/s), and the total contact area of the foundation. This information was collated from geotechnical surveys and corroborated by peer-reviewed literature on regional seismicity.

Numerical analysis of the transmission of the free-field wave to the building foundation was based on the Newmark algorithm with Rayleigh damping model [28] and a damping ratio equal to 5%.

3. Experimental Validation of the Models

The proposed models were validated using in situ measurements performed in the Upper Silesian Coalfield (USC) region [29,30]. The USC area is one of the mining regions in Poland with the most intensive mining tremors.

The actual building used to validate the models was a two-story masonry building, which is used for office purposes. This building, which does not have a basement, is founded on concrete strips, and the building subsoil consists of medium and fine sand with yellow dust inclusions. According to the geological documentation, dynamic soil parameter Cz = 50 MPa corresponds to such site (ground) conditions. The load-bearing masonry walls form a transverse-longitudinal system. The building is 12.7 m × 29.9 m in plan and a height of 7.3 m. Figure 7 shows a photo of this building and horizontal directions parallel to the transverse (x) and longitudinal (y) axis of the building. For a detailed description of the structure and subsoil layers, please refer to articles [29,30].

Since this article focuses on the transmission of mining vibrations from the free-field to the foundation of the building, therefore simultaneously registered courses of horizontal components x and y of free-field and building foundation vibrations were taken into account at the model verification stage. Additionally, based on the recorded courses of vibration components, the dominant vibration frequencies were calculated using the FFT (Fourier spectra).

From hundreds of mining shocks registered during the full-scale tests [29,30], two representative and different mining tremors were considered for validating models. Those mining tremors (rock bursts) were marked as No. 1 and No. 2. The analyzed vibration components x, y (accelerations denoted as a) and corresponding FFT results are shown in Figure 8 (direction x) and Figure 9 (direction y).

Figure 8 and Figure 9 confirm that records of vibration accelerations induced by mining tremors measured at the same time on the free field near the building and on the building foundation level can differ significantly [6,7,29,30]. For instance, in the cases of both representative mining shocks, the maximum acceleration values (a_max) of the horizontal components x and y of vibrations at the level of the building foundations were several tens of percent smaller than on the free field. For mining tremor No. 1, for the horizontal components x of vibrations, the results of the FFT analysis indicated the dominant frequencies of the recorded courses on the ground in the range of 10.5–14 Hz and from the higher band from 15.5 to 18 Hz. These frequencies were attenuated by the building acting as a low-pass filter. The lower frequencies, from the range of 4 to 8 Hz, are practically transmitted from the ground to the building foundation without changes. This is particularly well visible in the case of mining tremor No. 2—comp. Figure 8b. For the component x of vibrations for mining tremor No. 2, the FFT results show a wide band of dominant frequencies. The frequency characteristic of this component was different compared to mining tremor No. 1. The low-frequency band in the course of building foundation vibrations overlapped with the dominant vibration band for the x component. This band was in the range of 3.2 to about 8.2 Hz. Higher vibration frequencies in the range from 8.2 to 16 Hz were attenuated by the building—comp. Figure 8b. In the case of the component y of vibrations, the frequency characteristic differed from that of the horizontal component x. For mining tremor No. 1, the dominant frequencies up to 10 Hz were transmitted almost without changes. Ground vibration frequencies from 10 to 15 Hz were significantly attenuated by the building. In the case of mining tremor No. 2, vibration frequencies were transmitted without changes up to about 7.8 Hz. Above a frequency of about 8 Hz in the frequency characteristic of foundation vibrations, the influence of the building in attenuating the dominant frequencies occurring in the ground was visible—comp. Figure 9b.

To verify and assess the suitability of the proposed models, a comparison of the measurement results of the horizontal components of vibration at the foundation level with those calculated for the three analyzed variants of the subsoil model was carried out in terms of the vibration waveforms in the time domain, the dominant frequencies, and the maximum values of acceleration vibrations. These comparisons are shown in Figure 10 and Figure 11 and in

Table 2. Measured and calculated maximum values of foundation vibration (a [m/s²]) in the case of actual masonry building (brick load-bearing walls).

Measurement or Calculation	Maximum Value of a [m/s2]
	Direction x		Direction y
	No. 1	No. 2	No. 1	No. 2
measured	0.22	0.65	0.31	0.54
SM1	0.51	1.31	0.81	1.76
SM2	0.26	0.97	0.28	0.77
SM3	0.21	0.82	0.26	0.75

.

In the case of mining tremor No. 1, for the horizontal component x of vibrations, larger maximum values of vibrations at the foundation level were obtained for the SM1 model than from measurements. This proves that the SM1 model does not correspond well with the real building. A similar result was obtained for mining tremor No. 2. Good agreement between the calculated records of the x component of vibrations was obtained for the SM2 and SM3 models, with slightly larger values calculated for the SM2 model, which results from taking into account the damping in the SM3 model.

In the calculated FFT course for the horizontal component x for mining tremor No. 1, low components were visible in the range of 4.4–5.2 Hz and from 9.3 to 10.4 Hz. Higher vibration components were 12 to 17.5 Hz. In the FFT course for the waveform of the x component of vibrations from measurements, vibration components at frequencies of 6–8.5 Hz were also visible, which did not appear for the SM2 and SM3 models.

In the calculated FFT course for the horizontal component x for mining tremor No. 2, low components were smaller than for mining tremor No. 1 and remained in the range of 3.5–5.2 Hz and from 8.7 to 9.7 Hz. The dominant low frequencies in the vibration waveforms calculated at the level of the foundations for the analyzed models corresponded well with those obtained from in situ vibration measurements. Additionally, in the SM1 model, higher dominant frequencies from the range of 13.5 to 15 Hz were visible.

In the case of the y-component of vibrations for mining tremors No. 1 and No. 2, significantly higher maximum vibration values at the foundation level were obtained in the case of a fully fixed model (SM1, nonflexible subsoil) compared with measured vibrations. This is due to the greater stiffness of the fixed model compared with the actual building. The calculated maximum values of the horizontal component y of vibrations for the SM2 and SM3 models did not show significant differences. This proves the negligible effect of damping included in the SM3 model.

A representative record of the horizontal component y of vibrations (mining tremor No. 1) measured at the foundation level was characterized by a wide band of dominant low frequencies from 6.3 to 9 Hz. However, a dominant low frequency of about 6.3 Hz could be distinguished. In the case of the calculated waveform of the horizontal component y of vibrations at the foundation level, for the SM1 model (full fixations), the low dominant frequency was about 7.5 Hz, so it was higher than the measured frequency. This indicates the influence of the susceptibility of the substrate on the measured record of the y component of vibrations. The y component of vibrations also had a wide band of dominant frequencies, ranging from 12.5 to 19 Hz. A higher dominant frequency of about 19 Hz could also be distinguished. This frequency did not show up in the measured vibration record. By comparing the FFT curves calculated based on the representative component in the horizontal y direction at the level of the foundations of the SM2 and SM3 models, one can conclude that there were no significant differences in the dominant frequencies in these models. This may indicate a minor influence of the damping assumed in the subsoil in the form of dashpots. In the calculated vibration curves of the foundations for the SM2 and SM3 models, one can distinguish one lower dominant frequency band of about 5 Hz. Similar ranges of dominant frequencies in the y component could be distinguished for mining tremor No. 2, with the difference that for this component, there was a wider band of higher dominant frequencies, ranging from 12.5 to 19 Hz. In the case of the horizontal x component, the band of dominant higher frequencies was 15 to 20 Hz—see Figure 11.

The results of the above analysis led to the conclusion that the accuracy of the proposed 3D FEM models of soil-building systems was successfully verified by full-scale experimental results. Therefore, the validated models could be employed to analyze the transfer of the mine-induced free-field vibration to the building foundation.

4. Influence of Subsoil and Building Material Properties on the Natural Frequencies of Building

The nature of the SSI phenomenon depends mainly on the properties of the subsoil and the properties of the building structure. Natural frequencies of buildings, especially fundamental natural frequencies, are the parameters that allow you to simply identify one of the most important characteristics of the structure—building stiffness.

The above proposed 3D FEM models were used for the numerical evaluation of the influence of both subsoil and building material properties on the natural frequencies of typical low-rise buildings with load-bearing walls.

The fundamental natural frequencies of masonry (with brick walls) building vibrations depending on the dynamic parameter Cz describing the material of subsoil (for SM3 subsoil model) are listed in

Table 3. Fundamental natural frequencies of masonry (with brick walls) building vibrations depending on the dynamic parameter Cz describing the material of subsoil (for SM3 subsoil model).

Cz [MPa]	Fundamental Natural Frequencies of Building Vibrations [Hz]
	fx	fy	ft
35	3.96	4.39	4.58
40	4.12	4.54	4.75
45	4.27	4.68	4.89
50	4.40	4.80	5.02
55	4.52	4.91	5.14
60	4.63	5.01	5.25
65	4.73	5.10	5.35
$\infty$	9.45	9.36	10.45

. The results obtained in the case of nonflexible subsoil (SM1 subsoil model, Cz = $\infty$ MPa) are also presented in Table 3. Additionally, this relationship for the SM2 subsoil model is shown graphically in Figure 12. The frequency of horizontal transverse vibration is denoted by fx, the frequency of horizontal longitudinal vibration is marked by fy, and the frequency of torsional vibration is denoted by ft.

A notable influence of ground material properties on the fundamental natural horizontal (fx and fy) and torsional (ft) vibration frequencies can be observed. The values of appropriate frequencies of the same building could significantly vary depending on the parameter Cz. The differences in frequency values are particularly visible in the case of the comparison of the impact of weak or moderate (Cz = 35–65 MPa) and nonflexible (Cz = ∞ MPa) subsoil—they can reach more than twice. However, even in the case of substrates with a much smaller difference in properties, this influence is also very clear. The frequency values (fx, fy, ft) are reduced as the substrate stiffness decreases. These observations apply to both the SM2 and SM3 subsoil models.

The influence of material properties of load-bearing walls on the natural frequencies of the low-rise building is presented in the case of example three typical construction materials: brick, reinforced concrete, and cellular concrete. For numerical analysis using SM2 and SM3 subsoil models, the moderate stiffness of the subsoil was assumed, and Cz = 50 MPa was taken into account. The results of the computations for the SM2 and SM3 subsoil models, as well as the case of the SM1 subsoil model (nonflexible subsoil, fixed base model), are presented in

Table 4. Fundamental natural frequencies of building vibrations depending on the material of bearing walls in the case of a nonflexible subsoil (SM1 subsoil model) and the subsoil with the parameter Cz = 50 MPa (SM2 and SM3 subsoil models).

Material of Bearing Walls	Subsoil Model	Fundamental Natural Frequencies of Building Vibrations [Hz]
		fx	fy	ft
brick	SM1	9.45	9.36	10.45
	SM2	4.39	4.78	5.00
	SM3	4.40	4.80	5.02
reinforced concrete	SM1	18.86	21.48	23.14
	SM2	4.92	5.44	5.74
	SM3	4.93	5.45	5.75
cellular concrete	SM1	8.89	8.77	9.95
	SM2	4.70	5.09	5.39
	SM3	4.72	5.11	5.41

.

The results obtained lead to the conclusion that the fundamental natural frequencies of a building with a fixed type of structure, founded on the same subsoil, clearly depend on the material of the load-bearing walls used. This effect is especially visible in the case of nonflexible subsoil (SM1 subsoil model).

As could be expected, the stiffest structure of the building is made of reinforced concrete load-bearing walls, the appropriate computed frequencies of which are the highest. Nevertheless, the differences in frequencies regarding the other two typical building construction materials (brick and cellular concrete) are also significant and could be important in the analysis of the SSI effect.

The values of the considered frequencies computed for the building with a given wall material were practically the same using the SM2 and SM3 subsoil models. The differences were smaller than 0.5%.

5. Transmission of Mining Vibrations from the Ground to the Foundation of the Building Depending on the Material Properties of the Subsoil and Building

Numerical analyses concerned the evaluation of vibration transmission from the free-field to the foundations of a building with typical load-bearing wall construction. Two main problems were considered.

In the first one, differences in building foundation vibrations depending on the material properties of the subsoil were investigated in the case of an example model masonry building in which the load-bearing walls were built of bricks. The impact on the vibration transmission of two types of subsoil models, SM2 and SM3, was analyzed for different values of the dynamic, elastic, uniform vertical deflection Cz. The values of the Cz parameter varied from 35 MPa to 65 MPa. The Cz parameter corresponding to the foundation on the rock (Cz = $\infty$ MPa, nonflexible subsoil, SM1 subsoil model) was also taken into account.

The second main issue concerned the influence of the material of the building’s load-bearing walls on the transmission of mining vibrations from the ground to the foundation of the building. Three typical building materials were considered: brick, reinforced concrete, and cellular concrete. Using the SM2 and SM3 subsoil models, the calculations were performed using the determined subsoil properties (Cz = 50 MPa), corresponding to the actual subsoil from the USC region.

All the analyses were conducted for the two representative mining tremors No. 1 and No. 2 in the horizontal x and y directions.

The influence of the Cz coefficient value on the transmission of transverse x vibrations from the ground to the foundations (time history and corresponding Fourier spectrum) using the SM2 subsoil model and forcing with mining tremors No. 1 and No. 2 is shown in Figure 13. At the same time, an analogous relationship regarding the longitudinal y direction is shown in Figure 14.

Similar analyses were performed for the subsoil model SM3. The influence of the Cz coefficient value on the transmission of vibrations from the free-field to the foundations in the horizontal x and y directions using the SM3 subsoil model and forcing with mining tremors No. 1 and No. 2 is shown respectively in Figure 15 and Figure 16.

Additionally, maximum values of foundation vibration (a) in the case of masonry building (brick load-bearing walls) depending on the material properties of subsoil (Cz) are compared in

Table 5. Maximum values of foundation vibration (a [m/s²]) in the case of masonry buildings (brick load-bearing walls) depending on the material properties of subsoil (Cz [MPa]).

Cz [MPa]	Maximum Value of a [m/s2]
	SM2 Model				SM3 Model
	Direction x		Direction y		Direction x		Direction y
	No. 1	No. 2	No. 1	No. 2	No. 1	No. 2	No. 1	No. 2
35	0.19	0.81	0.23	0.63	0.18	0.70	0.22	0.65
40	0.21	0.87	0.25	0.68	0.19	0.74	0.24	0.69
45	0.23	0.93	0.26	0.73	0.20	0.78	0.25	0.72
50	0.26	0.97	0.28	0.77	0.21	0.82	0.26	0.75
55	0.29	1.00	0.29	0.80	0.22	0.85	0.27	0.78
60	0.32	1.02	0.30	0.84	0.23	0.88	0.28	0.81
65	0.34	1.04	0.31	0.86	0.24	0.90	0.29	0.83
$\infty$ (SM1)	0.51	1.31	0.81	1.76	0.51	1.31	0.81	1.76

.

By comparing the vibration courses of the foundations from Figure 13 and Table 5 (SM2 subsoil model, x direction), the significant influence of the Cz coefficient value on the calculated maximum vibration acceleration values is visible. It can be noticed that the larger the value of the Cz coefficient, the greater the maximum value of vibration acceleration. The maximum values of foundation vibrations are greater in the case of mining tremor No. 2, which is related to the greater intensity of this tremor compared with tremor No. 1. For example, in the SM2 model, the difference between the maximum vibration acceleration values related to mining tremor No. 1 and calculated for the Cz coefficient values of 35 MPa and Cz = 65 MPa was 44.1%. For the mining tremor No. 2, this difference was smaller and equaled 22.1%.

In the case of y horizontal direction (see results for SM2 subsoil model presented in Figure 14 and Table 5), particularly the significant influence of the Cz coefficient value on the calculated maximum vibration acceleration and phases of the calculated records were observed for mining tremor No. 2—comp. Figure 14b. The larger the value of the Cz coefficient, the greater the maximum value of vibration acceleration. For example, in the SM2 model and mining tremors No. 1 and No. 2, the difference between the maximum vibration acceleration values calculated for the Cz coefficient values of 35 MPa and Cz = 65 MPa was, respectively, 25.8% and 26.7%—comp. Figure 14 and Table 5.

Comparing the FFT courses from the calculated foundation vibration courses induced by mining tremors No. 1 and No. 2, regardless of the value of the Cz coefficient, differences in the transmission of dominant frequencies in the band up to 17 Hz were observed, see, e.g., Figure 13. For mining tremor No. 1, the horizontal x component of vibrations with a frequency of approx. 5.3 Hz, corresponding to Cz values from 45 MPa to 60 MPa, were best transmitted from the ground to the foundations. Similarly, higher frequencies from the 9.7 Hz and 10.5 Hz bands transmitted without disturbances from the ground to the foundations for the whole assumed Cz values from 35 MPa to 60 MPa—comp. Figure 13a. For mining tremor No. 2, the horizontal x component of vibrations with a frequency of approx. 4.5 Hz, 7.75 Hz, 8.7 Hz, and 10.3 Hz, corresponding to Cz values from 35 MPa to 60 MPa, were best transmitted from the ground to the foundations—comp. Figure 13b. Results of the calculated FFT course in the horizontal y direction at foundation vibrations induced by mining tremor No. 1 indicate dominant frequencies in the range of about 5 Hz and 7.5 Hz regardless of the value of the Cz coefficient. Mining tremor No. 2 indicates slightly greater dominant frequencies in the calculated FFT course, referring to the transmitted vibration records from free-field to foundations—comp. Figure 14.

Similar results analysis regarding the influence of the Cz subsoil parameter on the transmission of vibrations from the free field to the foundations, as in the case of the SM2 subsoil model, were also performed for the SM3 subsoil model. It can be stated, for example, that the difference between the maximum vibration acceleration values calculated for the Cz coefficient values of 35 MPa and Cz = 65 MPa in the x direction induced by mining tremor No. 1 and No. 2 was equal, respectively, to 25% and 22.2%—comp. Figure 15 and Table 5. The difference between the maximum vibration acceleration values calculated for the Cz coefficient values of 35 MPa and Cz = 65 MPa in the y direction induced by mining tremor No. 1 and No. 2 was equal, respectively, to 24.1% and 21.7%—comp. Figure 16 and Table 5. By analyzing the SM3 ground substrate model and comparing the FFT courses calculated on the building foundations based on the recorded free-field vibration courses induced by mining tremors No. 1 and No. 2, it could be stated that in the horizontal direction x for mining tremor No. 1, two clear bands of dominant frequencies could be distinguished, amounting to 5.3 Hz and 10.25 Hz—comp. Figure 15.

In the case of forcing No. 2, the bands of dominant frequencies were smaller than the dominant frequency bands induced by mining tremor No. 1 and were 3.7 Hz, 4.7 Hz and 5.7 Hz. In the case of the horizontal direction y, the bands of dominant frequencies were almost the same for the assumed values of the coefficient Cz. The character of the vibrations transmitted from the free field to the fixed model (Cz = $\infty$ MPa) was different compared with the courses for the case with the included Cz = 35–65 MPa (comp. Figure 16).

As mentioned above, further numerical analyses were focused on assessing the impact of the building’s construction material on the transmission of vibrations from the free field to the foundations. The influence of three types of construction material, i.e., brick wall, reinforced concrete, and cellular concrete, on the ground-foundation vibration transfer in the case of two subsoil models (assuming the real soil properties under the building corresponding to Cz equal 50 MPa) is shown in Figure 17, Figure 18, Figure 19 and Figure 20 and

Table 6. Maximum values of foundation vibration (a [m/s²]) depending on the material of the building structure.

Material of Building Structure	Maximum Value of a [m/s2]
	SM2 Model				SM3 Model
	Direction x		Direction y		Direction x		Direction y
	No. 1	No. 2	No. 1	No. 2	No. 1	No. 2	No. 1	No. 2
brick	0.26	0.97	0.28	0.77	0.21	0.82	0.26	0.75
reinforced concrete	0.40	1.11	0.35	0.88	0.25	0.97	0.34	0.85
cellular concrete	0.34	1.15	0.34	0.94	0.28	0.97	0.32	0.90

.

For forcing No. 1, in the case of the SM2 subsoil model, the highest acceleration values in the transverse direction x were obtained for the building model made of reinforced concrete material, whereas for the two remaining materials, the level of foundation accelerations was similar—comp. Figure 17 and Table 6. From the mining tremor No. 1 and the three analyzed materials, in the vibration courses of the foundation, two frequency bands dominated, the lower one around 5.5 Hz and the higher one in the range from 9.3 Hz to 10.3 Hz—comp. Figure 17. From forcing No. 2, in the level of the building foundation, a low dominant frequency band of 4.7 Hz could be distinguished for the three adopted materials of building models, and additionally, for the model with reinforced concrete material, a band of 5.7 Hz. Additionally, for models with three different materials, a common higher band of dominant frequencies 8.7–10.2 Hz could be distinguished—comp. Figure 17.

Based on the analysis of the SM3 subsoil model with the three applied materials and using mining tremor No. 1, we found that the maximal acceleration at the foundation level was almost the same in the horizontal y direction. This is shown in Figure 20 and Table 6. We also established that the dominant frequencies of the vibrations in the foundation occur in the wide band of 5.3–8.7 Hz for the assumed materials, as seen in the FFT vibration courses of the foundation. Similarly, by using forcing No. 2, the values of maximal acceleration at the foundation level for the three materials were almost the same, as shown in Figure 20, and the dominant frequencies remained in the band of 5.3–10 Hz. The model made of cellular concrete had higher values of dominant frequencies of the vibrations at the foundation, and they were equal to 12.5– 3.8 Hz, as shown in Figure 20.

The transmission of vibrations from the ground to the building foundation may also differ significantly depending on the building structure material, as can be seen, for example, in Figure 18 and Figure 19.

6. Conclusions

This paper presents an application of proposed variants of 3D FEM models for the estimation of the influence of typical building materials, as well as material properties of typical subsoils on one of the SSI effects—vibration transmission from the free-field to the building foundation in the case of mining tremors. The adopted models of soil-structure system were carefully verified using data from in situ experimental vibration measurements.

The results of the carried out 3D FEM analysis show a significant influence of the subsoil material properties on the transmission of the mine-induced free-field vibration to the building foundation and also on the building’s natural frequencies of vibration. The impact of the construction materials used in the building is also important.

The results obtained using the SM2 and SM3 subsoil models are very close. Therefore, the SM2 subsoil model, as the simpler one, could be recommended as sufficiently accurate in practical applications. Whereas, the SM1 subsoil model should not be considered in further mine-induced SSI effect analyses. This model variant, which assumed rigid subsoil, served as a control, enabling a baseline comparison for the impact of introducing the actual susceptibility of subsoil in subsequent models.

In summary, the numerical three-dimensional finite element method analysis carried out allowed for the successful evaluation of the influence of subsoil and building material properties on mine-induced soil–structure interaction effect.