1 Introduction

Collapsible soils cover approximately 10% of the world land (Jefferson et al. 2008). In Egypt, collapsible soils are located towards the western desert which covers about 65% of the area of Egypt (Sakr et al. 2008). However, most studies focused on new urban developments such as 6th of October, Sheikh Zayed, 10th of Ramadan, Nasr City, New Maady, West Cairo, New Amerya, Borg El-Arab, and Sadat City (Mossaad et al. 2008).

Collapsible soils can be formed naturally due to various geologic and geomorphologic processes or can be the result of human activity. These processes are different in nature; however, they result in the development of an open metastable structure which is the essential prerequisite to the formation of a collapsible deposit (Dudley 1970). Regardless of the formation process, a collapsible soil has a low unit weight and a relatively stiff and strong state when dry (Jefferson and Rogers 2012). Moreover, volume changes that occur are more sudden than those experienced through consolidation processes (Houston et al. 2001). Collapse is often triggered by a combination of increased stress and the addition of water. The increase in water content results in a relatively sudden volume compression. This has serious geotechnical consequences, including loss of serviceability resulting in expensive remediation or complete failure. In Egypt, structural failures caused by damage to foundations rested on collapsible soils are noticed (Sakr et al. 2008).

Researchers has conducted extensive laboratory tests to study the influence of different soil properties, such as clay content, soil gradation, initial water content, initial dry unit weight, and wetting pressure on the collapsibility of soils (Steadman1987; Alwail 1990; Lawton et al. 1989, 1992; Basma and Tuncer 1992; Ayadat and Hanna 2007; Das and Thyagarj 2016; Alassal et al 2020). Based on these laboratory tests, many empirical equations have been proposed in the literature relating the soil collapsibility to soil index properties. However, these equations are not universally valid for use in conventional geotechnical engineering practice (Das 2011).

Attempts have also been made to use constitutive relations for unsaturated soils to predict the volume change of collapsible soil. The unsaturated soil behavior is a function of two independent stress state variables, i.e. net normal stress and matric suction. It is found that the triggering mechanism for collapse is attributed to the loss of strength due to reduction in matric suction as a result of inundation (Ramos and Valencia 2013; Li and Vanapalli 2018). In other words, collapse occurs when there is a change in stress state of the soil as it goes from unsaturated condition towards a saturated condition (Li et al. 2016). Matric Suction could be measured using direct (e.g. tensiometer) and indirect (e.g. filter paper) methods (Fredlund and Rahardjo 1993).

This paper presents the results of a comprehensive experimental program in which a wide range of sandy soils is tested. The collapsibility of these sandy soils is evaluated using the single oedometer collapse test and ASTM filter paper method. In addition, index property tests were performed including sieve analysis, hydrometer, Atterberg limits, specific gravity, and maximum and minimum void ratio. This study examined the effect of related parameters including silt content, type of fines (silt/clay), initial water content (5% to 15%), drying and wetting process, and wetting pressure (0 to 200 kPa) on the collapsibility of sandy soils. In addition, scanning electronic microscope (SEM) images were used to analyze the microstructure of selected soil samples. Next, an equation was proposed to predict collapse potential as a function of the two stress-state variables that control the behavior of an unsaturated soil. Furthermore, the matric suction measurements were used to generate soil water characteristic curves (SWCCs) based on the work of Brooks and Corey (1964), Van Genuchten (1980), Fredlund and Xing (1994), and Chin et al. (2010). Finally, a simplified relationship between initial matric suction and the degree of collapse was established.

2 Materials and Tests

Collapsible sandy soils were prepared using ten different mixtures of sand, non-plastic silt, and clay. Laboratory tests were conducted on the raw materials and the prepared collapsible soils including sieve analysis and hydrometer test, Atterberg limits, specific gravity, single oedometer collapse test, and ASTM filter paper method.

Fine to medium siliceous sand was collected from a site in New Giza, Cairo. This sand was classified as SP (poorly graded sand) based on Unified Soil Classification System (ASTM 2017). The coefficients of curvature (cc) and uniformity (cu) equaled 0.84 and 3.67, respectively. Non-plastic silt was prepared by grinding sand into particles with sizes less than 0.075 mm. The coefficients of curvature (cc) and uniformity (cu) were 2.84 and 18.50, respectively. The liquid limit (wL) was 28%, however, plastic limit could not be measured. Hence, silt is classified as ML (non-plastic silt). Clay was obtained from a quarry located in Fostat city, Cairo. For clay, the liquid limit (wL), plastic limit (wP), and plasticity index (PI) were 88%, 29% and 59%, respectively. Hence, clay was classified as CH (high plastic clay). The specific gravity of the three soils equaled 2.64. The grain size distribution curves are shown in Fig. 1a.

Fig. 1
figure 1

Grain size distribution curves: a raw materials, b Group (A) soils, and (c) Group (B) soils (Alassal et al 2020)

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The experimental program is provided in Table 1 in which the second column indicates the code used for each soil mixture. S, M, and C refer to sand, silt, and clay contents, respectively. S60-M10-C30 meant that, this soil mixture consisted of 60% sand, 10% silt and 30% clay (by weight). The detailed physical properties of these soil mixtures were mentioned in Alassal et al 2020 . Tests were conducted on two groups of collapsible sandy soils. Group (A) was used to study the effect of silt content. In this group, sand was mixed with silt at different percentages (0% to 40%) by weight to form the reconstitute collapsible soil mixture. However, clay content of 10% was maintained in all these mixtures (Fig. 1b). On the other hand, the purpose of Group (B) was to study the effect of fines type, initial water content, drying and wetting process, and wetting pressure. In this group, fines content of 40% by weight was used in all the collapsible soil mixtures. In order to prepare different types of fines, several combinations of silt and clay were used (Fig. 1c).

Table 1 Experimental Program (Dr ≈ 35%)
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All the collapsible sandy soils were prepared at a relative density of around 35%. Hence, for each soil mixture, maximum and minimum void ratios (emax, emin) were determined based on the guidelines provided by the standard ASTM (2016a). The maximum and minimum void ratios were determined by filling a standard compaction mold. The standard compaction mold dimensions were 10.2 cm in diameter and 11.6 cm in height. For emax, soil was poured into the mold as loosely as possible from a funnel which was held in a vertical position. The height of the funnel was continuously adjusted to maintain a free fall of the soil of about 10 mm. The funnel was moved in a spiral path from the outside to the center of the mold to form uniform layers. The mold was filled with soil till 10 mm above its top and the excess soil was carefully trimmed off with a straightedge. The filling and trimming operations were done with great care to avoid excessive disturbance of the soil surface and causing rearrangement and settlement of particles. For emin, soil was placed in 5 layers and each layer was compacted using a mechanical shaker. Repeated trials were done to find values of emax and emin for each mixture as shown in Fig. 2.

Fig. 2
figure 2

Variation of emax, emin, and etested: a Group (A) and b Group (B)

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The soil specimens were prepared as follows: First, the percentages of sand, silt, and clay (by weight) were mixed. Second, the dry weight required to achieve the void ratio value corresponding to a relative density of 35% was calculated. Third, an amount of distilled water was added to soil to achieve the required initial water content. Forth, the moist soil was thoroughly mixed and kneaded by hand to form a homogenous mixture. Finally, specimens were prepared by placing the moist soil in layers into a mold. Each layer was tamped using a hammer to the required height corresponding to the specified void ratio. Specially manufactured mold and compactor were designed to obtain consistent collapsible samples. The mold was 5.3 cm in diameter and 5.2 cm in height. The preparation compactor was composed of a sleeve, a hummer, and a compactor ram. The weight of hummer and the compactor ram equaled 685 gm. The sleeve was 13.2 cm in height and was divided into 5 sections, the height of each section was equal to 2.5 cm, except the first section was equal to 3.2 (equal to hammer height). This compactor provided different free fall heights, consequently, induced different compaction energies. By trying these different energies, samples could be constructed at a predetermined relative density (Fig. 3).

Fig. 3
figure 3

Dimensions of the preparation mold: a plan view, b elevation view c preparation compactor (Alassal 2020)

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Single oedometer collapse test was performed based on the work of Jennings and Knight (1975) to determine the collapse potential. After preparing specimens, as previously explained, the cutting edge of the oedometer ring was centered at the top of specimen and was slowly pushed down while trimming the soil around the ring. When the ring fully penetrated the specimen, soil above and below ring was trimmed.

Afterwards, a seating load equal to 5 kPa is applied for 5 min. Then, load increments were applied at 1-h intervals until the wetting pressure is reached, at which the specimen was flooded with distilled water. The test ends when the dial gauge reading becomes constant for an hour or 24 h whichever is greater. Dial gauges were mounted on the top of the specimen for measuring the vertical displacement. The collapse potential (Ic) was calculated using Eq. 1.

$$I_{c} = \frac{\Delta e}{{1 + e_{o} }}$$
(1)

where Δe is the decrease in void ratio due to inundation and eo is the initial void ratio. Filter paper method was performed according to ASTM (2016b). In this method, the soil matric suction was measured using filter paper disks (Whatman No. 42, ashless). Filter paper method was adopted as a simple and reliable alternative which could be used to measure a wide range of suction values (Fredlund and Rahardjo 1993). First, a pair of nominally identical specimens was prepared (Fig. 4a). Second, a stack of filter papers was inserted between the two identical specimens. The stack was composed of three filter papers in which a filter paper was sandwiched between two protective filter papers (Fig. 4b and c). In order to avoid contaminating the central paper with soil, the two protective filter papers have a slightly larger diameter (5 mm larger). Third, the specimens were immediately sealed with a plastic sealing tape and wrapped with foil paper (Fig. 4d and e). Forth, a vertical stress of 5 kPa was applied. The applied stress enhanced the contact between the filter paper disk and the soil. Consequently, water in the liquid phase and solutes exchanged freely between the filter paper and soil (Fredlund and Rahardjo 1993). Fifth, the specimens were kept in a desiccator for 7 days (Fig. 4f). During this period, water content of the initially dry filter paper increased until equilibrium was established between filter paper and soil. Then, the water content of the filter paper was measured (Fig. 4g). Finally, soil matric suction was estimated using the appropriate filter paper calibration curve (Kim et al 2016).

Fig. 4
figure 4

Matric suction determination procedure (Alassal 2020)

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Matric suction values were measured using filter paper method at two conditions. The first condition presented matric suction of the samples before testing (initial matric suction). For each soil mixture, a sample were prepared at an initial void ratio corresponding to a relative density of 35%. The initial matric suction was used to depict the effect of silt content, fines type, and initial water content on the collapsibility of sandy soils. The second condition presents the matric suction values directly before wetting which occurs at different levels of wetting pressure. For a selected soil mixture, four samples are tested at different void ratio values (corresponding to wetting pressures of 0, 50, 10, and 200 kPa). The matric suction before wetting was used to depict the effect of wetting pressure on the collapsibility of sandy soils.

3 Effect of Silt Content

Group (A) soils were used to study the effect of silt content on the soil collapsibility as illustrated in (Table 1). All the tested samples were prepared with a clay content of 10%, an initial water content of 5%, and relative density of 35%. Figure 5 shows that as silt content increased, collapse potential (Ic) increased which agreed with the results obtained by Steadman (1987). If silt content equaled zero, collapse potential was 3.3%. If a silt content of 40% was used, collapse potential increased to 11.7% (255% increase). It was also noted that a soil with a higher percentage of silt content has a higher initial matric suction (Fig. 5). Moreover, as silt content increased, a reduction in the rate of increase of collapse was observed. Increasing silt content from 0 to 10% led to an increase in collapse potential by 84%. However, increasing silt content from 30 to 40% led to an increase in collapse potential by 6% only.

Fig. 5
figure 5

Effect of silt content on Collapse potential and initial matric suction

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All the tested samples were prepared at the same relative density, nevertheless, silt content has a pronounced effect on collapse potential. Moreover, the use of initial void ratio was unsuccessful to provide an explanation as shown in Fig. 6. Hence, the skeleton void ratio (es) was proposed to be used instead of the void ratio (e). es is defined as the void ratio of soil if fines are considered as voids rather than solid particles. As shown in Fig. 6, as the silt content increased, the skeleton void ratio (es) increased, consequently, collapse potential (Ic) increased. This phenomenon could be explained based on the work of Ni et. al (2004). The non-plastic silt particles acted as voids rather than binding agents. Therefore, increasing non-plastic silt content (while sustaining the same clay content) led to increase in the total volume of voids (voids + space occupied by silt), accordingly, collapse potential increased.

Fig. 6
figure 6

Collapse potential vs void ratio (e) and skeleton void ratio (es) (Dr = 35%—wi = 5%)

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4 Effect of Fines Type

Six soil mixtures were used to study the effect of fines type (Group B, Table 1). For all these soil mixtures, sand content of 60% was adopted and the total percentage of fines is maintained at 40% while using different percentages of silt and clay. Plasticity index (PI) was used as an indication of the fines type. The plasticity index (PI) increased upon using higher clay contents. For clay content = 0, 10, 20, 30, 38, and 40%, PI = 0, 24, 35, 52, 57 and 59%, respectively. Soils containing higher clay contents (higher PI values) experienced higher magnitudes of collapse (Fig. 7). Fines type has also a pronounced effect on the measured values of initial matric suction. The initial matric suction was 135 kPa, if no clay was added (PI = 0), however, it reached a value of 36,000 kPa if clay content of 40% was used (PI = 59%). According to Alwail (1990), clay fines were found to take different forms between sand grains among which are “clay balls”. In the current study, SEM micrographs of two different mixtures revealed that clay balls are formed (Fig. 8). Most of these balls were expected to disintegrate upon inundation, consequently, reduction in volume took place. It was noticed that for clay content of 30%, bigger clay balls were formed compared to soil with clay content of 10%.

Fig. 7
figure 7

Effect of fines type on collapse potential and initial matric suction

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Fig. 8
figure 8

SEM showing micro-structure of collapsible soil specimens at dry condition [1] sand particle, [2] void, [3] clay balls, [4] silt for: a S60-M30-C10 and b S60-M10-C30

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5 Effect of Initial Water Content

The initial water content (wi) has a significant effect on the behavior of collapsible soils. When the samples were prepared at higher values of water content, larger values of volumetric strains (Δe /[1 + eo]) were induced before wetting. On the other hand, the measured volumetric strains upon wetting decreased. Thus, the total volumetric strains (before + at wetting) decreased, as wi increased (Fig. 9). As shown in Fig. 10, as wi increased, lower values of initial matric suction were induced. As a result, weaker bonds were formed, and the soil micro-structure became unstable. Consequently, higher values of compression were depicted under applied pressure before wetting. This conclusion was supported by Basma and Tuncer (1992), Das and Thyagarj (2016), Phanikumar et al. (2016) and Zamani and Badv (2019).

Fig. 9
figure 9

Relationship between (Δe/[1 + eo]) and initial water content (wi): before wetting, at wetting (Ic), and total volumetric strains (S60-M02-C38)

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Fig. 10
figure 10

Effect of initial water content on initial matric suction

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Natural collapsible soils are formed through wetting or drying processes. The effect of the preparation process on the collapsibility of soil mixture S60-M30-C10 was examined. The wetting process was adopted for the entire experimental program. For the drying process, sample were prepared at high values of water content (15% to 12%). Afterwards, these samples were air-dried for a period ranging between 3 and 48 h, until the required water content was achieved. It was noted from Fig. 11 that the measured collapse potential was independent of the preparation process which was different from the conclusions drawn by Fredlund et al. (2011). This was attributed to the narrow range of water content values which was selected to prepare samples during both the drying and wetting processes (wi = 2.5% to 12%).

Fig. 11
figure 11

Effect of formation process on collapse potential (Ic) at various initial water content (wi) values (S60-M30-C10)

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6 Effect of Wetting Pressure

For each soil mixture from Group (B), four samples were prepared. Each sample was tested at a different wetting pressure (Pw). Figure 12 shows that collapse potential (Ic) increased with increasing wetting pressure (Pw). Noticeable values of collapse occurred upon inundation at Pw = 0 as PI increased (higher clay content). Large void spaces were developed between sand grains due to the existence of clay balls (Fig. 8) which was expected to disintegrate upon inundation even under low stress levels. However, the increasing rate of Ic declined as Pw increased until Pw = 100 kPa. Beyond this value, Ic remained almost constant or slightly decreased. The same trend was noticed by Rabbi et al. (2014) which could be attributed to two reasons. First, there was a maximum degree of densification which could be triggered by the collapse mechanism and was achieved at a certain pressure level. Any further increase in pressure beyond this level caused, subsequently, little or no change in collapse potential. Second, applying high levels of pressure led to the partial breakage of the binding agents before inundation. Figure 13 shows the results of single oedometer collapse tests performed on one of the soil mixtures. The dry loading curve presents the results of loading the sample at its natural water content. The wet loading curve presents the results of applying load on the inundated sample. The slope of the dry loading curve increased as higher stress levels were applied which was a strong indication of breakage of the binding agents before inundation. On the other hand, the slope of the wet loading part of the curve was independent of the stress level.

Fig. 12
figure 12

Effect of wetting pressure (Pw) on Collapse potential (Ic)

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Fig. 13
figure 13

Single Oedometer test results at different wetting pressures (Pw) for S60-M30-C10

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For soil mixture S60-M30-C10, the void ratio values (e) at Pw = 0, 50, 100, and 200 kPa were obtained from the dry loading curve in Fig. 13. Four samples were prepared at these values of void ratio (e) and at water content (wi) of 5%. Then, the matric suction of each sample was determined using the filter paper technique. The results were used to plot a relationship between matric suction before wetting and collapse potential (Ic). Figure 14 depicted that as matric suction before wetting increased, collapse potential decreased. A large value of matric suction before wetting was an indication of a more stable soil structure; hence, soil experienced a considerably lower amount of collapse. Moreover, at Pw = 100 kPa and 200 kPa, the matric suction values before wetting were almost the same which explains why approximately the same values of collapse potential were measured under the two different stress levels. These results reflected the major role of matric suction and wetting pressure in collapse mechanism. Therefore, a relationship was established between collapse potential (Ic) and the unsaturated two independent stress variables, namely matric suction before wetting and net normal stress. Based on the test loading conditions, the pore air pressure equaled to atmospheric pressure, therefore, the net normal stress equaled the total applied stress (= wetting pressure). The same procedure was repeated for three additional soil mixtures (S60-M02-C38, S60-M10-C30, and S60-M20-C20). Afterwards, the relationship between collapse potential (Ic) and the unsaturated two independent stress variables was plotted in 3D space as shown in Fig. 15. The 3D surface was presented by Eq. 2, which predicted values of collapse potential (Ic) with best fit R2 value equaled 0.98. Equation 2 could be used for loose sandy soil with fines content of 40%, low water content, and matric suction before wetting values larger than 1,000 kPa. Equation 2 was developed based on the results obtained from the tested reconstituted samples. Thus, it provided a qualitative indication of collapse. It should be noted that the tested samples reach saturation levels that are not normally encountered in the field. In addition, the estimated collapse potential values are significantly affected by soil structure and the type of bonding agent. Therefore, field assessment is necessary for a full collapse evaluation (Jefferson and Rogers 2012).

$$\begin{aligned}I_{c} &= 0.206\left( {P_{w} } \right) + { }0.0016\left( {u_{a} { } - u_{w} } \right)_{i} - 0.0007{ }\left( {P_{w} } \right)^{2} { }\\&\quad - 2.7{*}\left( {10} \right)^{ - 8} {*}\left( {u_{a} { } - u_{w} } \right)_{{}}^{2} { } + { }X \end{aligned}$$
(2)

where:Ic =collapse potential of soil (%),Pw =wetting pressure in kPa, (ua - uw)= matric suction before wetting in kPa, X = factor (= -15.15 for Dr ≈ 35%, wi = 5%, fines content = 40%).

Fig. 14
figure 14

Collapse potential (Ic) versus matric Suction before wetting for S60-M30-C10

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Fig. 15
figure 15

General relationship between collapse potential (Ic), net normal stress and matric suction before wetting

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7 Soil Water Characteristics Curves (SWCCs)

Soil water characteristic curves (SWCCs) presents the variation in volumetric water content υw) with respect to matric suction (ua-uw). The SWCC is expressed as a continuous sigmoid function. The main points on the SWCC are the air-entry value and residual suction value. The air-entry value (ψaep) is defined as the matric suction at which air enters the largest pores. The residual water content (θw)r is the water content at which water phase is discontinuous; hence, further increase in matric suction does not result in any significant decrease in the volumetric water content. The suction value corresponding to (θw)r is the residual suction (ψr) (Satyanaga et al. 2017).

The soil water characteristic curves (SWCCs) were developed for Group (B) soils. Matric suction values were measured at different water content values for each soil mixture. Next, equations used to predict SWCCs based on the work of Brooks and Corey (1964), Van Genuchten (1980), Fredlund and Xing (1994) and Chin et al. (2010) were adopted. The equations introduced by these methods consisted of basic soil parameters and fitting variables. The fitting variables were selected via a statistical program “SigmaPlot” which was used to find the best fitting equation.

The method proposed by Brooks and Corey (1964) predicted SWCCs using basic soil parameter (\(\uptheta_{s}\)) and fitting variables (\(\lambda\) and \(\uppsi_{aep}\)) as shown in Eqs. 3 and 4. The residual water content (θw)r was assumed as the smallest value of volumetric water content \((\uptheta_{w}\)) at which matric suction was measured using ASTM filter paper technique.

Table 2 presents a summary of the basic soil parameters, the fitting variables, and best fit R2 values for the four soil mixtures and the proposed SWCCs are shown in Fig. 16. It was noted that as clay content increased, smaller pores were formed, consequently, air-entry value increased which agreed with the results obtained by Gitirana and Fredlund (2004) and Santyanaga et al. (2017). ψaep equaled 1.94 kPa for clay content of 10% and equaled 3.73 for clay content of 38% (92.3% increase). Van Genuchten (1980) proposed a similar approach using fitting variables (α, N, M) as presented in Eq. 5. The results suggested that Eqs. 3, 4, 5 provided good fit to the measured data, with R2 equaled or higher than 0.93 (Table 2).

$${{\uptheta}}_{w} = (\theta _{w} )_{r} + [{{\uptheta}}_{s} - ({{\uptheta}}_{w} )_{r} ](\frac{{{{\uppsi}}}}{{{{\uppsi}}_{{aep}} }})^{{ - \lambda }} \;for\;{{ \uppsi}} \ge {{ \uppsi}}_{{aep}}$$
(3)
$$\uptheta_{w} = \uptheta_{s} \; for\; \uppsi < \uppsi_{aep}$$
(4)

where:\({\uppsi }\) = Matric suction head = (ua-uw)/ γw,\({\uppsi }_{{{\text{aep}}}}\) = Air-entry pressure head, \((\uptheta_{w} )_{r}\) = Residual water content, \(\uptheta_{s}\) = Porosity = (Vv/Vt), \(\lambda\) = Pore-Size distribution index.

$$\uptheta_{w} = (\uptheta_{w} )_{r} + \frac{{\uptheta_{s} - (\uptheta_{w} )_{r} }}{{[1 + ( \propto \uppsi )^{N} ]^{M} }}$$
(5)

where:\({\uppsi }\) = Matric suction head = (ua-uw)/γw, \(\propto\) = Coefficient related to air-entry pressure, \((\uptheta_{w} )_{r}\) = Residual water content, \(\uptheta_{s}\) = Porosity = (Vv/Vt), N = Coefficient related to pore-size distribution, M = 1 – (1/N).

Table 2 Summary of Parameters used to predict SWCCs
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Fig. 16
figure 16

SWCCs a) S60-M30-C10, b) S60-M20-C20, c) S60-M10-C30, d) S60-M02-C38

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Fredlund and Xing (1994) have also used porosity as a basic soil parameter. However, four fitting variables (a, b, m and \(\uppsi_{r}\)) should be calculated as shown in Eqs. 6 and 7. To determine these variables, empirical equations reported by Chin et al. (2010) are used (Eqs. 8, 9, 10, 11 for fine grained soils and Eqs. 12, 13, 14, 15 for coarse grained soils).

$$\uptheta_{w} = C(\uppsi )\frac{{\uptheta_{s} }}{{\left[\ln (e + (\frac{\uppsi }{a})^{n}\right]^{m} }}$$
(6)
$${\text{C}}\left( {\uppsi } \right) = \left[1 - \frac{{{\text{ln}}\left( {{1 + }\frac{{\uppsi }}{{{\uppsi }_{{\text{r}}} }}} \right)}}{{{\text{ln}}\left( {{1 + }\frac{{{1000000}}}{{{\uppsi }_{{\text{r}}} }}} \right)}}\right]$$
(7)

where: \({\uppsi }\). = Matric suction = (ua-uw), \({\uppsi }_{{\text{r}}}\) = Matric suction corresponding to residual water content,

\({\uptheta }_{{\text{s}}}\) = Porosity = (Vv/Vt), e = Natural number = 2.718281, a, n, m = Curve fitting parameters.

For fine grained soil (% Passing sieve #200 > 30%):

$${\text{a = - 2}}{.4}\left( {\text{x}} \right){ + 722} \left( {{\text{kPa}}} \right)$$
(8)
$${\text{n = 0}}{\text{.07(x)}}^{{{0}{\text{.4}}}}$$
(9)
$${\text{m = 0}}{\text{.015(x)}}^{{{0}{\text{.7}}}}$$
(10)
$${\uppsi }_{{\text{r}}} {\text{ = 914 exp[ - 0}}{.002}\left( {\text{x}} \right){]} \left( {{\text{kPa}}} \right)$$
(11)

For coarse grained soil (% Passing sieve #200 < 30%):

$${\text{a = 0}}{\text{.53 (D}}_{{{50}}} {)}^{{{ - 0}{\text{.96}}}}$$
(12)
$${\text{n = x}}$$
(13)
$${\text{m = - 0}}{\text{.23ln(x) + 1}}{.13}$$
(14)
$${\uppsi }_{{\text{r}}} {\text{ = 100 kPa}}$$
(15)

where (D50) is the sieve diameter corresponding to 50% passing, in this research D50 = 0.1 mm and (x) is an adjustable variable. For soil mixtures: S60-M02-C38, S60-M10-C30, and S60-M20-C20, equations proposed for fine grained soils (Eqs. 8, 9, 10, 11 ) are used to determine the values of the fitting variable. For soil mixture: S60-M30-C10, equations proposed for coarse grained soils (Eqs. 12, 13, 14, 15) provided better results. Table 2 shows that R2 ranges between 0.79 and 0.90.

8 Matric Suction versus Collapse Potential

The current study revealed that initial matric suction has a major role in determining the collapsibility of sandy soils. The effect of fines (type and content) and initial water content was reflected on the measured matric suction of the samples before inundation (Li and Vanapalli 2018). The initial matric suction and collapse potential were measured for 20 soil samples prepared at different values of initial water content (2.5% to 15%), fines type (PI = 0% to 59%), silt content (0% to 40%) and sand content (50% to 90%) were used. All the sample were prepared at a relative density of about 35% and inundated at wetting pressure (Pw) of 100 kPa. The results were plotted in Fig. 17 and a simplified relationship between initial matric suction and collapse potential (Ic) was established as presented in Eq. 16. In general, soil collapsibility increased with the increase of the measured initial matric suction (Silveira and Rodrigues 2020).

$$I_{c} = 4.8 *\log \left( {u_{a} - u_{w} } \right)_{i} - 3.66$$
(16)

where:Ic = Collapse potential of soil (%), (ua - uw)i = Initial matric suction in kPa.

Fig. 17
figure 17

Degree of collapse based on initial matric suction

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According to Jennings and Knight (1975), soils were classified based on the degree of collapse that a particular soil was expected to exhibit under specified conditions. The degree of collapse depended on collapse index (Ie) for soils inundated at 200 kPa. The current study showed that the soil inundated at 100 kPa and 200 kPa exhibited almost the same values of collapse potential (Ie = Ic). Therefore, the limits reported in Jennings and Knight (1975) were plotted on Fig. 17 from which Table 3 was concluded. Soils were expected to experience no collapsibility problems when initial matric suction was less than 9 kPa. The problem became moderate if the initial matric ranged between 9 to 60 kPa. If the soil initial matric ranged from 60 to 700 kPa, collapsibility problem was expected. When initial matric suction exceeded 700 kPa, the problem became severe to very severe.

Table 3 Degree of collapse based on initial matric suction
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9 Conclusion

This paper presents the results of an extensive experimental work conducted on reconstituted collapsible sandy soils. The non-plastic silt particles acted as voids rather than binding agents. Therefore, increasing non-plastic silt content led to increase in the total volume of voids, accordingly, collapse potential increased. The study showed that sandy soils mainly containing non-plastic silt exhibited lower values of collapse compared to sandy soils mainly containing clay. It revealed that collapse potential increased with increasing clay content up to 40%. In this case, the dominant mechanism responsible for collapse (upon inundation) was the disintegration of clay balls. For the tested soils, collapse potential increased with increasing wetting pressure (Pw) up to a certain value (Pw = 100 kPa), after which collapse potential remained the same or slightly decreased with further increase in wetting pressure, because of partial breakage of bonding material during dry loading. Using higher values of initial water content weakened the soil micro-structure which led to a lower magnitude of collapse. Meanwhile, the collapse potential of soil (Ic) was not influenced by the preparation process for the tested range of initial water contents.

Soil matric suction was measured using the ASTM filter paper method. The variation in the measured suction values were found to be directly influenced by the change in the soil related properties and wetting pressure. Therefore, a relationship was established between collapse potential (Ic) and the unsaturated two independent stress variables, namely matric suction before wetting and net normal stress. The ASTM Filter paper method has also been used as a feasible alternative to generate SWCCs for collapsible soils. The parameters used to predict SWCC’s for four different mixtures were estimated based on the work of Brooks and Corey (1964), Van Genuchten (1980), Fredlund, and Xing (1994), and Chin et al. (2010). Finally, a simplified relationship between initial matric suction and the degree of collapse was established.