Introduction

A PMT is one of the in situ tests in geotechnical engineering that is used to identify and assess the Su, the deformation parameters, the in situ σH, and the permeability coefficient of soils (Ménard 1957a). It allows engineers to design stable foundations and pavements in various conditions by obtaining several parameters (Foriero and Ciza 2016). This test was developed for the first time by Ménard in 1957b. In this test, a cylindrical balloon expands under the pressure of the fluid within the borehole and changes in volume; hence, pressures can be measured continuously. Then these data are plotted in terms of the applied pressures against the change in volume or the change in the radius of the balloon. In Fig. 1 a and b, a schematic diagram of a pressuremeter test device and a typical cavity pressure curve are shown.

Fig.1
figure 1

a Schematic diagram of a pressuremeter test device and b a typical cavity pressure curve (Soleimanbeigi 2013)

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In Fig. 1b, PL is the limit pressure and σH is the total horizontal stress. The limit pressure is defined as the maximum pressure in which the cavity wall expands indefinitely. The σH is defined as the static stress of earth. There are several methods to determine the σH from a PMT curve, all of which estimate this stress approximately (Clarke 1995). σH can be obtained from the initial part of the elastic deformation of pressure–volume diagram of PMT (as shown in Fig. 1b), representing an overcome on the σH and also the beginning of the elastic deformation of soil (Briaud 1992).

The standard method of carrying out this test on site is described in ASTM-D4719-16. There are several different types of pressuremeters. The major difference between various categories of pressuremeters lies in the method of installation of the instrument into the ground. The borehole pressuremeter is the most common and has a probe that is inserted into a preformed borehole. One of the problems with this test is that conducting drilling may cause disturbance in soil, and accurate results cannot be achieved. For this reason, Wroth and Hughes (1972) and Baguelin et al. (1972) developed a test called the SBP to reduce the effect of soil disturbance during the drilling operation. A new geotechnical in situ test technique using a self-boring in situ shear pressuremeter (SBISP) was developed by Wang et al. (2018) to evaluate the initial state (horizontal at rest pressure), deformation modulus, and strength characteristics of soil. Briaud and Gambin (1984) proposed a method for drilling borehole for PMT. Elton (1981) and Ohya et al. (1982) investigated the effect of elastic tube strength on the pressuremeter modulus. Haberfield and Johnston (1988) simulated PMT of soft rock under controlled experimental conditions. Agan (2013), Oge, (2018) and Omar et al. (2018) determined the deformation modulus and tensile strength in weak rock mass by using pressuremeter. Kincal and Koca (2019) conducted a research to comparison between the EPMT in andesitic rock mass and the values of elastic modulus of intact rock core specimens. Smith and Rollins (1997) conducted PMT in arid collapsible soil. Kayabasi and Gokceoglu (2018) used PMT in order to modify liquefaction analysis methods. Tu (2018) determined the coefficient of horizontal subgrade reaction with the PMT. Oztoprak et al. (2018) proposed a numerical methodology for capturing the complete curve of a PMT. Silvestri and Tabib (2018) analyzed field test results obtained by PMT in a sensitive clay soil of Quebec.

Some geotechnical parameters of the soil can be measured by the data taken from the PMT, so that these parameters are comparable to the results of other tests. For instance, Tarawneh et al. (2018) estimated EPMT and PL from CPT for desert sand and compared them with the results of the PMT. Firuzi et al. (2019) presented correlation between SPT, CPT, and pressuremeter data in alluvium soil. In addition to what mentioned above, many researchers were able to demonstrate the empirical relationship between the PMT and the SPT. They provided a number of relationships between the NSPT obtained from the SPT and the values of the EPMT and the PL obtained from the PMT; and they proposed to use those relationships for similar soils (Yagiz et al. 2008; Bozbey and Togrol 2010; Agan and Algin 2014; Cheshomi and Ghodrati 2015; Anwar 2016; Özvan et al. 2018; Ziaie Moayed et al. 2018).

Su of soil is an important parameter in design of geotechnical engineering structures. There is considerable demand to obtain Su of fine-grained soils in many geotechnical problems, because it is a fundamental property (Bol et al. 2019). The Su of shallow strata is a critical parameter for a safety design in deep-water operations (Li et al. 2019). Su can be measured by a variety of in situ tests such as SPT (Sowers 1979), CPT (Lunne et al. 1997; Cheshomi 2018) flat diameter test (Młynarek et al. 2018), VST (Clayton et al. 1995), and PMT (Marshland and Randolph, 1977). In addition, uniaxial, triaxial, and direct shear tests on undisturbed samples are some of the routine laboratory tests for the determination of Su (Cheshomi 2018; Bol et al. 2019). Belkhatir et al. (2013) studied the relationship between the Su of the sand–silt mixtures. Cabalar et al. (2018) showed Su decrease with increasing sand content in clay soil. Chung et al. (2012) presented an estimation of the Su of Busan clay through a comparison of the results of field vane and laboratory shear test results at two sites in the Nakdong River Delta. Shimobe and Spagnoli (2019) presented a correlation between liquidity index and Su.

Researchers tried to develop several empirical, analytical, and numerical methods to interpret this parameter through the PMT. Although the shearing plane obtained by pressuremeter differs from those in the conventional laboratory strength (Seah and Shrestha 2006), empirical methods are based on the relationship between properties of soils (Su, the internal angle friction, the cohesion, OCR, LL, PI etc.) and parameters of pressuremeter (PL and EPMT) (Ménard 1957a; Amar and Jézéquel 1972; Nasr and Gangopadhyay 1988). Su of clay soils can be interpreted by empirical equations. However, the analytical methods are based on the plane strain condition, isotropy, soil homogeneity, and undrained conditions, and they can be used to evaluate the behavior of soil under different conditions (Palmer 1972; Ladanyi 1972; Denby 1978; Ferreira and Robertson 1992; Monnet 2007). When a more precise solution of the PMT for soils is required, numerical solutions would be used to estimate the Su of clays (Zentar et al. 2001; Monnet 2007). Tschebotarioff (1973), Parcher and Means (1968), and Terzaghi et al. (1996) based on Su of fine-grained soils divided clayey soils into several categories, including very soft, soft, medium, stiff, very stiff, and hard. Table 1 presents the classification of fine-grained soils based on their Su, proposed by different researchers.

Table 1- Types of fine-grained soils according to undrained shear strength
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Palmer (1972) provided a theoretical method for interpreting the Su of clay through the PMT. Using the results of the PMT, they plotted stress–strain graphs of saturated clay and then used the graphs to calculate the Su of clay. Komornik and Frydman (1969), Amar and Jézéquel (1972), and Marshland and Randolph (1977) conducted numerous tests to suggest a number of empirical relationships to interpret the Su of clayey soils through PMT. Baguelin et al. (1972) presented a guideline to estimate the strength of clayey soils and the relative density of sandy soils using PMT. The results of studies by Houlsby and Carter (1993) and Bowles (1996) for estimating Su by PMT showed that the values obtained using PMT was larger than those determined in the field or laboratory (e.g., triaxial compression tests).

Bahar et al. (2012) developed a theoretical model and conducted a numerical analysis on saturated clay in a PMT to determine Su, and then proposed a model to predict Su. There was a good agreement with the results of model on the pressuremeter path and the experimental data from other methods (the empirical methods proposed by Ménard 1957a, and Amar and Jézéquel 1972, and the field vane and triaxial tests). Bahar et al. (2012) carried out several pressuremeter tests in clayey soils in different places. They interpreted Su of soils by three methods. The results of their studies showed that the values of Su estimated by the Bahar and Olivari (1993) method were greater than those estimated, using the Ménard method (1957a) and Amar and Jézéquel method (1972).

Soleimanbeigi (2013) analyzed the data taken from PMT regarding consolidated organic silt and overconsolidated silty clay to estimate Su based on the traditional closed-form solution and the finite element method. After comparing the results, he found that the estimated values of Su predicted from the finite element method were lower than those estimated from the traditional method (Gibson and Anderson 1961). Isik et al. (2015) conducted a numerical study on the effects of the length (L) and radius (R) of the pressuremeter and also the depth of testing on the values of shear strength obtained by the PMT. They proposed a correction coefficient based on the ratio of L/D and observed that the values of Su obtained by the empirical relationships of pressuremeter were closer to the values obtained through the CPT, FVT, and CU tests.

Alzubaidi (2015a) evaluated the horizontal at rest pressure by five different methods of interpretation (the inflection point method, the numerical iteration method developed by Gibson and Anderson 1961, the graphical iteration method developed by Marsland and Randolph 1977, the up side-down curve method developed by Van Wambeke and Hericourt 1975, the stress relief method developed by Alzubaidi 2015b).

Alzubaidi (2015b) evaluated the Su in sandy silt soil by three methods for interpretation (Gibson and Anderson 1961; Palmer 1972; Ladnayi 1972) and five methods of analysis for deducing the horizontal at rest pressure. The results of that study showed that the values of Su had a considerable difference, when using different values of the horizontal at rest pressure estimated by those five methods.

In Table 2, some relationships proposed by different researchers to calculate the Su of clayey soils are presented.

Table 2 Relationships provided by different researchers to obtain Su by pressuremeter test
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Cassan (1972) and Amar and Jézéquel (1972) defined a relationship between Su and PL presented in Eq. (1) in Table 2. The effect of total horizontal stress as PL − σH was considered in their equation. Ménard (1957b) proposed a factor known as the pressuremeter constant (β). This factor can be obtained using the values of PL * = PL − σH and Su using this equation β = PL*/Su. Many researchers proposed a number of values for β. There are several factors that can make a difference in the values of β, such as disturbance of soil, anisotropy, lack of sufficient precision in measurement total horizontal stress at rest, and difference in the reference strength (Clarke1995).

In this study, to develop empirical relationships between Su and PL for very stiff to hard saturated clayey soils in one of the central cities of Iran, 44 pressuremeter and uniaxial test results were considered to develop proper relationships between Su and PL. Then, the effect of σH and some key physical properties of soil (such as LL, PI, and ω) on the proposed empirical equations were investigated. In the next step, the proposed relationships were compared with some equations, proposed by other researchers.

Materials and method

Site specification

Qom city is located 148 km south west of Tehran, Iran (34.6416° N, 50.8746° E). For construction of subway, an extensive geotechnical studies were conducted in line with a length of 15 km. This research was carried out using geotechnical information obtained from subway project in Qom city. For this purpose, 18 boreholes with depth between 25 and 40 m were drilled. The study area is underlain by recent alluvium. Figure 2 presents the location area and subsurface soil condition in the route of the study. Based on this figure five layers as follows Fig. 1c are separated. These soils were composed from sandy gravel and gravelly sand, silty clayey sand, silty clay, and clayey silt. In this study, mainly clayey silt and silty clay soils (layer nos. 2 and 3) were taken into account.

Fig. 2
figure 2

a Location of site and subway rout. b Subsurface soil condition in the subway rout. c Soil layers’ description

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Uniaxial test

During drilling of boreholes, pressuremeter tests were performed according to the standard (ASTM D4719-16), and undisturbed samples were taken from soils in the sections where the pressuremeter tests were carried out and the undisturbed samples were transferred to the laboratory for testing. Thin-walled tube sampler was used for obtaining undisturbed sample based on ASTM D1587/D1587M-15.

Uniaxial tests were performed according to the standard (ASTM D2166-16) on undisturbed samples. An array of tests was performed to determine the Su parameter of the soil. Although there is a shortcoming in the uniaxial test due to the lack of confining pressure, its application in geotechnical studies of projects is common, because of its simplicity, low cost, and high efficiency. In this study, since clayey soils were saturated and the loading rate was relatively high, the shear strength values obtained from the uniaxial tests were regarded as Su. In Fig. 3a, b and c, uniaxial test apparatus with a sample photo before and after test with an unconfined compressive strength (qu) versus axial strain curve for the sample is shown respectivly.

Fig. 3
figure 3

a Uniaxial test apparatus. b A sample before and after test. c Unconfined compressive strength (qu) versus axial strain for a sample

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Pressuremeter test

Pressuremeter tests were performed according to the standard (ASTM D4719-16). The pressuremeter, used in this research, was a PBP of type GC. The reason for choosing GC pressuremeter was its suitability for dense and hard soils (Baguelin et al. 1972). It can be noted that soils in the study area were very stiff to hard clays. In addition, it is widespread and more acceptable to use this type of pressuremeter in Iran. Depending on the type of employed pressuremeter, the shear modulus, the EPMT, and the PL can be obtained directly from results of pressuremeter tests. In Fig. 4, PMT apparatus with a pressure versus volume curve is shown. Equation (1) proposed for determining the EPMT (Murthy 2008; Agan 2013):

$$ {E}_{\mathrm{PMT}}=\frac{2\left(1+u\right)\left({V}_0+{V}_m\right)\varDelta P}{\varDelta V} $$
(1)

where EPMT (kPa) is the pressuremeter modulus, ν is the Poisson’s ratio (equal to 0.33), and V0 is the volume of the uninflated probe at the ground surface; ΔP, ΔV, and Vm in Fig. 4b are presented.

Fig. 4
figure 4

a PMT apparatus. b Pressure versus volume curve. c Pressure versus 1/V to calculate the PL. d The clay layer that has been tested

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One derived parameter from the PMT is the limit pressure (PL, pressure at which failure occurs); one of the ways to determine a limit pressure is to extrapolate the pressure–volume curve (Baguelin et al. 1972). In this method PL is defined as the pressure where the probe volume reaches twice the original soil cavity volume (V0 + 2Vi), where Vi is the corrected volume reading at the pressure where the probe makes contact with the borehole. Based on ASTM D4719-16, if the test was performed to read sufficient plastic deformation, PL can be determined by a 1/V versus pressure plot, as shown in Fig. 4c. Points from the plastic range of the test generally fall in an approximate straight line. The extension of this line to twice the original probe volume gives the PL on the plot. In this research, this method is used to determine the PL. Figure 5 illustrates the flowchart of the present research along in situ and laboratory variables for proposing empirical equations.

Fig. 5
figure 5

The flowchart of the present study along with the measured parameters

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Test results

In this study, 44 pressuremeter tests and 44 corresponding uniaxial tests were performed. Su values of soil samples were obtained by the uniaxial tests. Using the results of pressuremeter tests, PL and σH were calculated. PL and σH were used to interpret the Su of clayey soils.

Limit pressure

Using pressure–volume curves acquired from the pressuremeter tests (as shown in Fig. 4c as a sample curve), the PL was calculated for all 44 pressuremeter tests. The variation of PL values with depth and frequency are shown in Fig. 6 a and b. Upper and lower bounds for PL–depth graph have been drawn, indicating an increasing trend of PL values with depth. As can be seen, most of the data are gathered between 2000 and 6000 kPa. The upper and lower bounds were drawn as visually so that most of the data fall within the range between the two lines.

Fig. 6
figure 6

a Variation of PL with depth based on 44 pressuremeter tests. b Frequency range of PL values

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Total horizontal stress

Values of σH were determined by the initial part of the elastic deformation of pressure–volume curve of the PMT (as shown in Fig. 1b). In Fig. 7 a and b, variations of σH with depth and frequency of σH are presented. As shown in Fig. 7a, the upper bound and the lower bound of σH are plotted and the trend of σH with depth is similar to that found in PL; with increasing in depth, the σH increase, and based on Fig. 7b most of the data are located between 200 and 600 kPa.

Fig. 7
figure 7

a Variation of σH with depth based on 44 pressuremeter tests. b Frequency range of σH values

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Undrained shear strength

The undisturbed samples were taken at the same location, where the pressuremeter tests were conducted. The samples were sent to the laboratory for conducting uniaxial tests. Since the clay samples were saturated and not allowed to be drained in the uniaxial test apparatus, the angle of internal friction angle was assumed to be zero, similar to UU triaxial tests. Therefore, in this case, the Su can be obtained by Eq. (2):

$$ {S}_u=\raisebox{1ex}{${q}_u$}\!\left/ \!\raisebox{-1ex}{$2$}\right. $$
(2)

where Su is the undrained shear strength and qu is the uniaxial compressive strength (UCS), which is the maximum axial compressive stress that a upright-cylindrical sample of material can withstand before failure.

Figure 8 a shows the variation of Su obtained by the laboratory tests against the depth. After drawing the upper and the lower bounds of the graph, it can be observed that with increasing the depth, Su increases too, but with a slight slope. The frequency of Su values is shown in Fig. 8b. The frequency range of Su data used in this study was between 56 and 618 kPa.

Fig. 8
figure 8

a Variation of Su with depth based on 44 uniaxial compressive strength tests. b Frequency range of Su values

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As seen in the histogram shown in Fig. 8b, the values of Su for most tested samples were more than 100 kPa. Therefore, with regard to the values of Su and also according to classifications presented in Table 1, it can be concluded that the target clays were very stiff to hard clay.

Physical properties (PI, LL, and ɷ)

Using samples, taken during drilling, the plasticity index (PI), the liquid limit (LL), and the water content (ω) of soil samples were determined. The variation of PI, LL, and ω (%) with depth are shown in Fig. 9 a, b, and c, respectively. Upper and lower bounds for all 3 graphs are plotted in Fig. 9 a, d and c. As can be seen, PI, LL, and ω have an increasing trend with depth.

Fig. 9
figure 9

Variation of soil properties with depth. a Plasticity index. b Liquid limit. c Water content

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Empirical relationships between Su and PL

In Fig. 10, based on the data presented in previous section, the variation of Su against the PL is shown for all 44 tests.

Fig. 10
figure 10

Variation of Su against the PL for very stiff to hard clays

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Referring to Fig. 10, Eq. (3), a linear relationship between PL and Su, can be obtained with a determination coefficient of 0.68.

$$ {S}_u=0.55\ {P}_L+58\ n=44\ R2=0.68 $$
(3)

Su and PL are in kilopascal. This equation is valid for very stiff to hard clays with a PL greater than 2000 kPa.

Many researchers have considered σH as a variable in the proposed relationship between the PL and the Su (Ménard 1957a; Amar and Jézéquel 1972; Lukas and De Bussy 1976; Marshland and Randolph, 1977; Martin and Drahos 1986). They used PL–σH instead of just PL. In this study, in order to investigate the effect of σH, the value of this parameter was determined according to the results of the pressuremeter tests. Similar to the work of previous researchers, the values of σH were contracted from the values of PL and their effects were captured in the proposed empirical equation. Figure 11 shows the variation of P*L = PL–σH against Su.

Fig. 11
figure 11

Variation of Su against the P*L = (PL-σH) for very stiff to hard clays

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According to Fig. 11, Eq. (4) is obtained between the P*L = (PL–σH) and Su with a determination coefficient of 0.68,

$$ {S}_u=0.58\ \left({P}_L-{\sigma}_H\right)+75\ n=44\ R2=0.68 $$
(4)

The units of Su and PL and σH are in kilopascal. This equation is valid for very stiff to hard clays with P*L greater than 1600 kPa.

The determination coefficients of Eqs. (3) and (4) showed that σH had insignificant effect on the relationship between the Su and PL, since the soils of the studied area were very stiff to hard clays. This finding was similar to the one previously proposed by Clarke (1995). Therefore, it seems that it is not necessary to implement σH in the relationships obtained.

In order to study the effect of physical parameters of soil such as the PI, LL, and ω, a comparison between Su and PL, Su, PL, PI, LL, and ω were carried out through a multivariate analysis in SPSS statistical software. In this regard, Su was considered as a dependent variable and other parameters including PL, PI, LL, and ω were considered as independent variables. Equation (5) can be used to interpret Su values for very stiff to hard clays.

$$ {S}_u=0.031+2.29\ LL+0.54\ PI-1.77\ \omega +0.0593\ {P}_L\ n=44\ R2=0.73 $$
(5)

It is worth mentioning that Su and PL are in kilopascal and LL, PI, and ω are in percent. According to the observed determination coefficient, using physical parameters such as PI, LL, and ω of the soil can have a positive impact on results because the engineering properties of fine-grained soils depend on their physical properties. As can be seen in Eq. (5), the determination coefficient in multivariate method is 4% higher than that of the single variable method. Due to these enhanced determination coefficients obtained in the case of a single variable and a multivariate mode, Su can be determined using either Eq. (3) or Eq. (5).

Discussion

In order to evaluate the proposed empirical relationships between Su and PL, a comparison between the proposed relationships in this research and the equations provided by other researchers (presented in Table 2) was made. For this purpose, the values of PL obtained from the pressuremeter tests in this research were placed in relationships 1 to 7 and the values of Su were estimated. Then the values of Su estimated through the empirical equations were compared with the values measured from the laboratory tests. Figure 12 shows the data points and regression lines for comparison of proposed relations 1 to 7 by previous researchers with the proposed relationships of the present research (Eq. 3).

Fig. 12
figure 12

Comparison of the proposed equations in present research to interpret Su by pressuremeter tests with those equations proposed by other researchers

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In Fig. 12, the line Y = X (the black solid line) representing the determination coefficient of 1 is drawn up to determine the range of variations of values obtained from Eqs. 1 to 7 and the proposed equation of the present study. Comparison showed that all equations have the same trend, but slopes of the lines increased with softening of soil. The estimated values of Su from the proposed equation of this study were within the range of values estimated by Eq. 3 that proposed by Cassan 1972 and Amar and Jézéquel 1972 for stiff to very stiff clays. The proposed relationship in this research is also for very stiff to hard clays. Therefore, it can be concluded that the relationship proposed in this study is closer to Eq. 3 due to similarity in their consistency. Thus, clay consistency has a significant impact on the proposed empirical relationship; hence, for clay with different consistency, practitioners should use distinct experimental relationships. Accordingly, stiffness of clayey soil is dependent on physical properties of soil, especially the moisture content, as shown in Eq. (5). In other words, taking into account the physical properties (PI, LL, ω) may increase the determination coefficient.

The values of pressuremeter constant (β) in this study are plotted in Fig. 13a. The rang of β values was between 7 and 35 with an average of 12.6. In Fig. 13b, the β value in the present study was compared with the values presented by previous researchers in clay with different consistencies.

Fig. 13
figure 13

a Histogram of Pressuremeter constant (β). bβ value in this study and previous research

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Cassan (1972) and Amar and Jézéquel (1972) proposed β values of 5.5 for soft to firm clays, 8 for firm to stiff clays, and 15 for stiff to very stiff clays. Marsland and Randolph 1977 recommended β value of 6.8 for stiff clays, and Lukas and De Bussy (1976) suggested the value of 5.1 for all clay soils. Martin and Drahos (1986) proposed 10 as a value for β when the soil is stiff clay. The value of β, obtained in the present study, has a good agreement with those proposed by Cassan (1972) and Amar and Jézéquel (1972) for stiff and very stiff clays. Therefore, as discussed, the value of β is a function of clay consistency. Anisotropy, lack of sufficient precision in measurement, the total horizontal stress, and difference in the reference strength can affect the value of pressuremeter constant (Clark 1995).

Conclusions

This research was carried out based on 44 pressuremeter and uniaxial tests on very stiff to hard saturated clay samples. The key conclusions drawn from the findings of this study can be summarized as follows:

An empirical relationship between Su–PL for very stiff to hard saturated clay with R2 = 0.68 was suggested. The estimated Su from the PMT was less than the measured Su obtained from laboratory tests. To evaluate impact of total horizontal stress, an empirical equation between Su–P*L = PL–σH with R2 = 0.68 was proposed. By comparing the relationship between Su–PL and Su–P*L it was found that σH had an insignificant impact on the proposed relationship.

In order to evaluate the effect of PI, LL, and ω on Su and PL, a multivariate equation was proposed. The finding revealed that incorporation of these variables can affect the proposed relationship and would increase the coefficient of determination.

The proposed equation in this research was compared with a number of equations proposed by other researchers. Those relationships were functions of consistency of soil, so that obtained values of Su from the proposed equation developed in this study were within the range of values estimated by the equations proposed by Cassan 1972 and Amar and Jézéquel 1972 due to the similarity in their consistency.

The values of the pressuremeter constant (β) were dependent on clay consistency. In this study the average of β was 12.6, and it had a good agreement with those proposed by Cassan (1972) and Amar and Jézéquel (1972) for stiff and very stiff clays.

The applicability of proposed relationships in this research is for very stiff to hard clayey soils, with PL between 2000 and 6000 kPa.