Most previous studies on collapsible soils have demonstrated considerable variability in reliability, primarily due to variations in testing procedures and sampling methods. Additionally, it has often employed static testing as its primary method of validation. However, as development continues, a gap remains in our understanding of how collapsible soil reacts to various dynamic stresses, including mechanical equipment, power stations, trains, roadways, and other dynamic loads. Conventional studies often fail to adequately represent real dynamic loading conditions. Accordingly, it is essential to investigate the response of gypseous soils to vibration and varying moisture content. This research aims to characterize the dynamic behavior of gypseous soil under different saturation states (unsaturated and saturated), subjected to harmonic loading at a relative density of 35%, with additional consideration of foundation depth and eccentric mass. The experimental program aims to establish a database that enables reliable correlations between wave attenuation and soil damping in gypseous soils. Results showed that the dynamic characteristics of gypseous soil increased by 50-52% with settlement, 3-6% with sorption stress, 47-68% with total stresses, 42-46% with acceleration, and 44-48% with vertical displacement as frequency increased. However, they decreased by 6-7% for settlement and total loads, 2-5% for acceleration, and 6-9% for vertical displacement when gypseous soil saturation rose to 60%. Saturation levels also influenced these increases, which ranged from 60% to 100% (149-150%) for settlement, 139-173% for total stresses, 50-51% for acceleration, and 52-54% for vertical displacement. Meanwhile, suction stress increased between 45% and 457% as the gypsum soil's saturation level reached 60%, then decreased between 100% and 104% as saturation increased above 60% but before reaching 100%.
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1.Introduction
Gypseous soils are widely distributed across arid and semi-arid regions, where they form a substantial portion of the surface and near-surface deposits; such soils are typically unsaturated or partially saturated, characterized by an air-water interface and a contractile skin resulting from the presence of pore water. According to [1], soil consists of air, water, and solids. However, recent research indicates that the air-water interface (or contractile skin) plays a crucial role that must be considered independently of other physical factors. The study [2] explains that when the air phase is continuous, the contractile skin interacts with soil particles, altering the mechanical behavior of unsaturated soil.
Several researchers have investigated the influence of various variables on stiffness and the material damping ratio [3–8]. The shear strain amplitude, mean effective confining stress, soil type, and plasticity index are all key factors in determining the shear modulus. Additionally, the number of loading cycles or the loading frequency plays a more critical role than the void ratio, over-consolidation ratio, grain characteristics, and degree of saturation. Other significant factors affecting the damping ratio include soil type, plasticity index, number of loading cycles, loading frequency, and shear strain amplitude [4].
The design and construction of machine foundations represent a critical component of industrial development. National investments in infrastructure and industrial facilities provide the basis for the expansion of other economic sectors, including commerce and tourism.
The foundations of machines subjected to vertical vibrations are commonly evaluated using peak acceleration as the primary control parameter for performance. Soil particles reach equilibrium at a characteristic peak acceleration that depends on the relative density of the granular soil. Further densification occurs only when this acceleration threshold is exceeded [9].
The authors [10] investigated the effects of cyclic loading on unsaturated soils using a triaxial system equipped with strain transducers for small-strain measurements and psychrometers for suction monitoring. A fixed water ratio was maintained during testing on kaolin specimens. The results indicated that suction decreased progressively with increasing number of loading cycles. Furthermore, the resilient modulus increased with rising water content up to an optimum level, beyond which it decreased sharply with further increases in water content.
The cyclic behavior of unsaturated soils was studied by the authors [11] using a triaxial test subjected to low loads and varying temperatures. Experiments have shown that under low loads, low temperatures, and high suction, the soil becomes consistently more rigid. The results of the cyclic triaxial test indicate that the soil's plasticity increases with the number of cycles, but all the unsaturated test samples reached a stable state after 100 cycles. The total plastic strain after 100 cycles was greater at higher temperatures and lower suction levels. Each result was influenced by either suction hardening or heat softening, respectively. The study also found that suction had a significant effect on the resilient modulus. The resilient modulus can increase by up to an order of magnitude when suction rises from zero to 250 kPa at a given temperature.
Physical, chemical, or combined types of weathering can break down rock components, but soil is among the most valuable natural resources produced by these processes. Soil formation in a specific location, or "zone," is heavily influenced by the region's geology, geography, and climate. Unsaturated soil is generally regarded as a three-phase system, comprising solid, liquid, and gaseous phases. Interactions among these parts, whether under static or dynamic stress, result in significant changes to the system's characteristics. Static loads remain constant over time and space, whereas dynamic loads change in direction, position, and/or amplitude. Many researchers are eager to explore the complexities of soil behavior under various dynamic loading conditions. The response of sedimentary soils to dynamic loading, such as strong earthquake motions, is significantly influenced by their dynamic properties, including stiffness degradation, modulus reduction, and damping [12]. Examples include shear wave velocity, changes in stiffness or decreasing modulus, material damping at strain levels, and components sensitive to liquefaction, all of which are extensively studied. The ability to predict or interpret dynamic behavior depends on understanding the soil's dynamic characteristics. Geotechnical earthquake engineering relies heavily on accurate estimates of dynamic soil parameters. Factors such as stress state, confinement, stress history, void ratio, water content, and other conditions influence dynamic soil properties [13].
The resilient modulus of subgrade soils has been widely investigated and is influenced by several factors, including the number of loading cycles, loading frequency, grain-size distribution, and stress history. The study [14] concluded that stress is the most critical parameter governing this behavior.
Two principal processes of soil deformation under dynamic loading are liquefaction and cyclic mobility. Both are associated with increases in pore water pressure, which reduces the effective stress and weakens soil resistance under repeated shear loading. In loose, non-cohesive, moisture-sensitive soils, particularly coarse silt and fine to medium sands, dynamic vibrations can induce liquefaction. Under such conditions, soil grains densify, transferring intergranular stresses to the pore water. When the pore water pressure equals the total stress, the effective stress becomes zero, and the soil loses all its frictional shear strength. Liquefaction typically leads to rapid settlement and severe degradation of track or foundation geometry. Cyclic mobility, in contrast, can occur in both loose and dense soils, although liquefaction is predominantly associated with loose deposits [15].
The authors [16] experimentally demonstrated that embedment of a square footing in medium sand reduces settlement by approximately 15.2–17.3 % at a load amplitude of 0.5 t, and by 6.7–10.5 % at 1 t. In dense sand, settlement reduction reached 25.2–42.5 % at 0.5 t, 9.7–29.1 % at 1 t, and 12.6–23.2 % at 2 t.
Similarly, the authors [17] investigated contact pressure distribution beneath circular shallow foundations subjected to vertical and rocking vibration modes. Tests on dry sands of varying densities indicated that vertical settlement decreased with increasing embedment due to enhanced vertical stiffness. Under rocking vibration, stress distribution along the base increased toward the center before diminishing outward. Lateral edge strains rose from 100 % to 150 % in the rocking direction, while vertical rocking generally induced a central depression, resulting in maximum stresses beneath the footing.
Limited research has addressed the behavior of collapsible soils under vibration and repeated loading, particularly in the case of moisture-sensitive gypseous soils. To advance understanding, this study investigates the influence of repeated loading on sandy gypseous soils, with emphasis on how varying saturation levels affect key engineering properties. Factors considered include foundation type, machine characteristics, dynamic load frequency, number of load cycles, elastic modulus variation, foundation embedment, and displacement amplitude. Furthermore, the study aims to clarify the behavior of gypseous soil by analyzing the response of unsaturated samples subjected to vertical vibrations.
2.Methods
This section presents the experimental program, including soil characterization, model preparation, instrumentation, and testing procedures. The methods used for applying cyclic loads and monitoring the response of gypseous soil are described in detail to achieve the objectives of this research.
2.1.Soil Properties
The soil experiments in this paper were conducted on gypseous, collapsible soil. The soil underwent a standard set of tests to determine its physical properties. Fig. 1 displays the soil grain size distribution, and Table 1 presents some of the physical features of gypseous soil that were examined.
Figure 1. Grain size in the gypseous soil.
Table 1. The characteristics of the gypseous soil.
Property
Value
Reference
Specific gravity
Gs
2.43
ASTM D854[1]
Distribution of particle size, %
Fines
20
ASTM D422[2]
Sand
79
Gravel
1
Atterberg limits, %
L.L
20
ASTM D8541
P.L
16
ASTM D4318[3]
P.I
4
ASTM D43183
Classification of soil
U.S.C.S
SM
ASTM D2487[4]
Min. of dry density
rd-min, g/cm3
1.20
ASTM D4254[5]
emax
1.03
Max. of dry density
rd-max, g/cm3
1.71
ASTM D4253[6]
emin
0.42
Additionally, the B.S. 1377[7] specification was followed to conduct a standard set of experiments necessary to determine the soil's chemical properties. The various chemical characteristics of soil are shown in Table 2 [18].
Table 2. A summary of gypseous soil's chemical characteristics.
Property
Rate of value
Gypsum content (CaSO4), %
45.0
Carbonate content (CaCo3), %
22.5
Total sulphate content (SO3), %
21.07
Organic matters (O.M), %
0.72
Total soluble salts (T.S.S), %
40.1
pH value (pH)
7.23
2.2.Model tests
To investigate the behavior of gypseous soil at different saturation levels under harmonic loading and to replicate the dynamic properties of the physical model for the machine's foundation, a small-scale model was created at a 1:100 scale. Each component was carefully calibrated, and built-in measuring sensors were included. At a relative density of 35 %, 54 models were tested. Fig. 2 shows the detailed testing schedule.
Figure 2. Testing program.
2.3.The Model and Instruments
The prototype container box has a steel plate thickness of 6 mm, with external dimensions of 500 mm by 500 mm by 570 mm. Two layers of absorbent material (each 10 mm thick) were attached to the inside surfaces of the iron box to reduce the reflection of vibration waves at the box boundaries. Each absorption layer is made from different materials (rubber and polystyrene). The steel container box used in the tests is shown in Fig. 3.
Figure 3. The steel container box.
For this purpose, a compact oscillator was built for lab testing. Also known as a 2-mass oscillator, this device is used to generate harmonic vibrations. The speed regulator panel enables the user to adjust the voltage supplied to the motor, which in turn alters the motor's speed and the oscillator's vibration frequency, ranging from 5.0 Hz to 40.0 Hz, depending on the required cyclic frequency for testing. The machine that produces the harmonic vibrations is shown in Fig. 4.
Soil mechanics relies heavily on stress sensors, which measure total soil pressure, typically with a capacity of 100 kPa. Soil properties, including strength, compressibility, and stability, can be evaluated with their help [19]. The shape of the stress sensor is illustrated in Fig. 5.
Figure 4. Harmonic vibration generator (mechanical oscillator).
The accelerometer sensor records acceleration along three axes (X, Y, and Z) with adjustable ranges and resolutions, capturing both dynamic accelerations induced by motion or impact, as well as static acceleration due to gravity. The sensor has a measurement capacity of up to 16 g. Data collected by the accelerometer and its recorder were processed using SeismoSignal software to compute displacement, velocity, and acceleration time histories for each axis and at sensor locations within or around the soil model. The accelerometer is illustrated in Fig. 6.
Figure 5. Stress sensors.
Figure 6. Accelerometer sensor.
Measuring pore water pressure is crucial for assessing soil behavior. In this study, a suction pressure sensor with a range of −100 kPa to +100 kPa was used to monitor the pressure difference (i.e., Ua-Uw) between pore air and pore water. Negative values indicate suction (tensile state), while positive values show compression.
The Linear Variable Differential Transformer (LVDT) is an electromechanical transducer that converts linear mechanical displacement into a proportional electrical signal. LVDTs are widely used as displacement meters due to their high precision, which enables them to measure movements as small as a few micrometers.
Soil moisture, pH, and light intensity were measured using portable soil meters. Two different devices were employed to assess moisture conditions in both unsaturated and saturated soil specimens. The models of the meters and their specifications are presented in Fig. 7.
a. Model of (SKU-6281) b. Model of (BAF-2.0)
Figure 7. The soil moisture meters.
2.4.Preparation of the Model for the Test
The subsequent assessment method may be demonstrated:
To ensure uniform density, the soil sample was placed in successive layers. The net surface area of the model box was approximately 480 × 480 mm, with each layer compacted to a thickness of 50 mm. The gypseous soil was first weighed to determine its target density (loose or medium). It was then placed in the box and compacted manually using a steel or wooden tamper until a uniform 50 mm layer was achieved (Fig. 8a). Figs. 8b and 8c demonstrate the ideal locations for stress and accelerometer sensors inside the model; thus, it is important to follow the instructions in paragraph (2) above until the desired model height (about 500 mm) is obtained. A bubble level was used to make sure the surface was even after the earth was compacted and placed in the steel box. As shown in Fig. 8e, the foundation model, including the harmonic vibration system, was positioned at the plan center (X-Y center) of the box. A single LVDT-equipped magnetic holder was fastened to the sides of the container. The zero reading was obtained by touching the LVDT to the wings at the footing's specified center. Fig. 8e depicts how the models were saturated with water to the desired level: a 50-liter water tank was placed on a wooden table at a height of 500 mm, and a plastic tube with a diameter of 6 mm was connected from the tank to the steel box via the valve at the box's base. After filling the model to capacity with water, the air was forced out of the soil by leaving it uncovered for one day under a double nylon cover, resulting in a fully saturated sample of granular gypseous soil. A piece of plastic pipe (as a cover) with an inner diameter of 102 mm was placed on the soil's surface and around the foundation in order to prevent collapse while preparing a sample with a foundation depth of 0.5 B and 1.0 B. The soil was then sprinkled with a little water using a small water sprinkler until the gypseous soil granules adhered to a small degree and protected it from collapsing. As shown in Fig. 8f, the ring was cut transversely to allow for easy insertion and removal of the pipe section from the foundation system. Additionally, the granular soil sample had already absorbed the necessary amount of water before the test commenced. A harmonic vibration system was used to apply dynamic loads. Fig. 8g displays tension, pressure, accelerometer, and LVDT sensor data over time.
As shown in Fig. 8h, the cyclic frequencies are maintained until failure occurs or the target strain ratio is reached.
The case numbers, model configurations, and experimental variables are summarized in Table 3. The main parameters considered were:
Saturation degree: three levels were tested (6 %, 60 %, and 100 %); Depth of foundation: foundations were embedded at 0.5 B and 1.0 B below the soil surface to assess the effect of embedment on dynamic response; Depth of sensors: stress and accelerometer sensors were installed at 0.5 B and 1.0 B to evaluate vibration penetration with depth; Eccentric mass: two eccentric masses (28 g and 44.8 g) were applied using the harmonic vibration machine, corresponding to operating frequencies of 9 Hz, 12 Hz, and 15 Hz.
a. Compaction of soil at 5 cm layer b. Placing sensors inside a model
c. The distribution of sensors d. Placing of foundation at a center
e. The harmonic vibration and soil in the model box with equipment
f. Preparing a model at depth foundation (Df = 0.5 B or 1.0 B)
g. Beginning of the test h. The model at failure state
Figure 8. A summary of model preparatory procedures up until their failure.
Table 3. Case studies and models with explanatory variables.
Model details and variable factors
At Df (0.0B)
At Df (0.5B)
At Df (1.0B)
Sr (%)
me (gm)
fn (Hz)
Df
(mm)
Case No.
Df
(mm)
Case No.
Df (mm)
Case No.
Dry (Natural)
6%
28.0
9
0.0 B
(0.0)
mm
1
0.5 B
(50.0)
mm
19
1.0 B
(100)
mm
37
12
2
20
38
15
3
21
39
44.8
9
4
22
40
12
5
23
41
15
6
24
42
Unsaturated
60%
28.0
9
7
25
43
12
8
26
44
15
9
27
45
44.8
9
10
28
46
12
11
29
47
15
12
30
48
Saturated
100%
28.0
9
13
31
49
12
14
32
50
15
15
33
51
44.8
9
16
34
52
12
17
35
53
15
18
36
54
Three frequencies 9 Hz, 12 Hz, and 15 Hz were used in this study to calculate and for each frequency, as shown in Equations (1) to (12) [20].
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
where, = Damping ratio; = Modified mass ratio; = Coefficient of damping; = Coefficient of critical damping in (N-sec/m); = Total machinery and foundation masses in (kg) and (kN), respectively; = Soil density in (kg/m3) and (kN/m3), respectively; = Ground acceleration ( = 9.81 m/sec2); = Poisson's ratio; = Radius of the circular foundation (radius of the loaded area) in (m); = Dynamic shear modulus of the soil in (kPa); = Static Spring constant (N/m); = Rotating masses' natural circular frequency in (rad/sec); = Frequency of natural circular in (Hz); = Resonant frequency in (Hz); = Amplitude of the vibration at frequency resonance in (mm); = The mass of eccentricity in (kg); = The distance of eccentricity in (mm); = The maximum of vertical load (N); = Total force in (N); = Displacement in (mm); = Velocity in (m/sec); and = Acceleration in (m/sec2).
3.Results and Discussion
Fig. 9 presents representative time histories of settlement, acceleration, velocity, displacement, total stresses, and suction pressure obtained from the model tests. For clarity, one case model was selected for each saturation condition (natural, partially saturated, and fully saturated) to illustrate the general response, since the overall shapes of the dynamic curves were similar, differing mainly in average magnitudes. Figs. 10–21 further illustrate the relationships between dynamic soil properties and operating frequency (fn) or degree of saturation (Sr), considering additional variables such as foundation depth (Df) and eccentric mass (me).
In general, Figs. 10, 12, 14, 16, 18, and 20 illustrate the variation of gypseous soil dynamic properties with operating frequency at different foundation depths (0.0 B, 0.5 B, and 1.0 B) and eccentric masses (28 g and 44.8 g). In contrast, Figs. 11, 13, 15, 17, 19, and 21 illustrate the impact of saturation degree on the same dynamic properties under identical foundation depths and eccentric masses.
a. Settlement–Time curve b. Suction pressure–Time curves
c. Total stress–Time curves at points number 1 to 4
d1. Acceleration–Time curve at point number 1
d2. Velocity–Time curve at point 1 d3. Displacement–Time curve at point 1
e1. Acceleration–Time curve at point number 2
e2. Velocity–Time curve at point 2 e3. Displacement–Time curve at point 2
f1. Acceleration–Time curve at point number 3
f2. Velocity–Time curve at point 3 f3. Displacement–Time curve at point 3
g1. Acceleration–Time curve at point number 4
g2. Velocity–Time curve at point 4 g3. Displacement–Time curve at point 4
Figure 9. Typical dynamic characteristics at 100 percent saturation.
a. Settlement–Frequency curves b. Suction stress–Frequency curves
c. Damping ratio–Frequency curves d. Vertical stresses–Frequency curves
e. Acceleration–Frequency curves f. Displacement–Frequency curves
Figure 10. The behavior of gypseous soil's dynamic properties at different operating frequencies for case study numbers 1, 2, 3, 7, 8, 9, 13, 14, and 15, with me = 28.0 g, Df = 0.0 B, and Dr = 35%.
a. Settlement–Saturation degree curves b. Suction stress–Sr curves
c. Damping ratio–Sr curves d. Vertical stresses–Sr curves
e. Acceleration–Sr curves f. Displacement–Sr curves
Figure 11. The behavior of gypseous soil's dynamic properties at different saturation levels with me = 28.0 g, Df = 0.0 B, and Dr = 35% for case study numbers 1, 2, 3, 7, 8, 9, 13, 14, and 15.
a. Settlement–Frequency curves b. Suction stress–Frequency curves
c. Damping ratio–Frequency curves d. Vertical stresses–Frequency curves
e. Acceleration–Frequency curves f. Displacement–Frequency curves
Figure 12. The behavior of gypseous soil's dynamic properties with varying operating frequencies at me = 44.8 g, Df = 0.0 B, and Dr = 35% for case study numbers 4, 5, 6, 10, 11, 12, 16, 17, and 18.
a. Settlement–Saturation degree curves b. Suction stress–Sr curves
c. Damping ratio–Sr curves d. Vertical stresses–Sr curves
e. Acceleration–Sr curves f. Displacement–Sr curves
Figure 13. The behavior of gypseous soil’s dynamic properties with varying degrees of saturation for case study numbers 4, 5, 6, 10, 11, 12, 16, 17, and 18 at me = 44.8 g, Df = 0.0 B, and Dr = 35%.
a. Settlement–Frequency curves b. Suction stress–Frequency curves
c. Damping ratio–Frequency curves d. Vertical stresses–Frequency curves
e. Acceleration–Frequency curves f. Displacement–Frequency curves
Figure 14. The behavior of gypseous soil's dynamic properties at various operating frequencies with me = 28.0 g, Df = 0.5 B, and Dr = 35% for case studies numbered 19, 20, 21, 25, 26, 27, 31, 32, and 33.
a. Settlement–Saturation degree curves b. Suction stress–Sr curves
c. Damping ratio–Sr curves d. Vertical stresses–Sr curves
e. Acceleration–Sr curves f. Displacement–Sr curves
Figure 15. The behavior of gypseous soil's dynamic properties with varying degrees of saturation at me = 28.0 g, Df = 0.5 B, and Dr = 35% for case study numbers 19, 20, 21, 25, 26, 27, 31, 32, and 33.
a. Settlement–Frequency curves b. Suction stress–Frequency curves
c. Damping ratio–Frequency curves d. Vertical stresses–Frequency curves
e. Acceleration–Frequency curves f. Displacement–Frequency curves
Figure 16. The behavior of gypseous soil's dynamic properties under varying operating frequencies at me = 44.8 g, Df = 0.5 B, and Dr = 35% for case study numbers 22, 23, 24, 28, 29, 30, 34, 35, and 36.
a. Settlement–Saturation degree curves b. Suction stress–Sr curves
c. Damping ratio–Sr curves d. Vertical stresses Sr curves
e. Acceleration–Sr curves f. Displacement–Sr curves
Figure 17. The behavior of gypseous soil's dynamic properties with varying degrees of saturation at me = 44.8 g, Df = 0.5 B, and Dr = 35% for case study numbers 22, 23, 24, 28, 29, 30, 34, 35, and 36.
a. Settlement–Frequency curves b. Suction stress–Frequency curves
c. Damping ratio–Frequency curves d. Vertical stresses–Frequency curves
e. Acceleration–Frequency curves f. Displacement–Frequency curves
Figure 18. The dynamic properties of gypseous soil behavior with varying operating frequencies at me = 28.0 g, Df = 1.0 B, and Dr = 35% for case study numbers 37, 38, 39, 43, 44, 45, 49, 50, and 51.
a. Settlement–Saturation degree curves b. Suction stress–Sr curves
c. Damping ratio–Sr curves d. Vertical stresses–Sr curves
e. Acceleration–Sr curves f. Displacement–Sr curves
Figure 19. The behavior of gypseous soil's dynamic properties with varying degrees of saturation at me = 28.0 g, Df = 1.0 B, and Dr = 35% for case study numbers 37, 38, 39, 43, 44, 45, 49, 50, and 51.
a. Settlement–Frequency curves b. Suction stress–Frequency curves
c. Damping ratio–Frequency curves d. Vertical stresses–Frequency curves
e. Acceleration–Frequency curves f. Displacement–Frequency curves
Figure 20. The behavior of gypseous soil under dynamic properties with varying operating frequencies at me = 44.8 g, Df = 1.0 B, and Dr = 35% for case study numbers 40, 41, 42, 46, 47, 48, 52, 53, and 54.
a. Settlement–Saturation degree curves b. Suction stress–Sr curves
c. Damping ratio–Sr curves d. Vertical stresses–Sr curves
e. Acceleration–Sr curves &am