Ground improvement has always been one of the major thrust areas of geotechnical engineering. It is vertically crucial in the design of any structure in weak soil. Before any development or construction work for either civil structures or mining activities, it is crucial to know the local soil type, present and future use of the land area, required strengths for holding the above structural loads, and estimated cost of the project [1] . In case the soil of the selected site does not have desired structural properties, e.g., appropriate cohesion, internal angle of friction, bearing capacity, swelling factor, etc., it becomes necessary to improve these properties using external means. The effect of soil instability can be diverse, including cases of liquefaction, heaving, swelling, and plastic deformation [2] . The effects of unstable soil are correspondingly catastrophic, ranging from slope failures and foundation sinkage to total collapse of the tunnels and mine dumps, overlying buildings, and other structures [3] .

The ground improvement as a sub-branch of the Geotechnical Engineering domain has made considerable advances since the practices began to develop in the mid-20th century. Most techniques have undergone drastic changes in terms of application and optimization.

Polyurethane (PU) is one of the four major organic polymeric binder used for soil improvement. The others are; Lignosulfonate, Epoxy resins and Polyacrylamide. There have been different researches on the use of PU for soil treatment and improvement.

Polyurethane (PU) is a polymer made up of Polyol (-OH) and Polyisocyanates (-NCO). Different types of Polyol and Polyisocyanates result in different varieties of polyurethane. Its rapid reaction time (usually between 30 to 120 seconds) and lightweight make it suitable for repairing highway pavement since it cannot be blocked for a long time. Groundwater or the presence of moisture in soil could be a limiting parameter for the use of polyurethane since the liberation of gas, which could be CO₂ due to the reaction between water and isocyanate or water vapor due to heat from the reaction of Polyol and Polyisocyanates [4] .

[5] studied the effect of polyurethane mixed with polypropylene fiber on the tensile strength of sands. They found that curing of at least 12 h is required for solidification of the specimen and about 48h for stabilization. In the study, they showed that polyurethane helps in binding polypropylene with soil particles resulting in greater tensile strength, which could be 2 to 3 times initial tensile strength depending on the compaction.

[6] showed that rock-like strength could be achieved by polyurethane soil mixture without significantly increasing the weight of soil. With a 1:7 ratio of PU:Soil by weight, they achieved a much durable sample with unconfined compressive strength of more than 4 MPa with sandy silt. [7] studied the cracking and shrinkage behavior of bentonite clay with polyurethane and found an increase in stiffness of the sample. Hydrophilic PU was more effective in controlling shrinkage distress.

[8] found PU more suitable for coarser material since it could lead to hydro- fracturing in fine-grain clayey soil at the point of injection due to its viscosity. It was found that there was a significant increase in plastic strength under cyclic loading. At the same time, a decrease in the elastic property was also observed. [9] studied well-graded angular and subangular gravel stabilized with PU and concluded that with the addition of PU, there is increase cohesion without any reduction in angle of friction, unlike in some cases of fly ash. Compared to cement and lime-based stabilizers, it does not increase the brittleness and incorporate good post-failure strength. With 0% - 8% addition of hydrophobic PU and curing for 60 min, brittleness index [(peak deviatoric stress/residual deviatoric stress) − 1] for 0.5 MPa confining pressure varied from 0.01 to 0.29 whereas, in the case of lime/cemented gravel, it varied from 0.03 to 2.36. It was also noted that the failed specimen was intact after failure, and their behavior changes from contractive to dilative with the addition of PU.

Due to the challenges posed by the nature of expansive soil to stability of structures, it becomes imperative in some cases to improve the soils before structures are placed on them. There is also the need to investigate modern trends in ground improvement techniques such as use of geosynthetic materials, cement, lime, chemical polymers and other calcium based compounds in order to determine their reliability. This research thus seeks toinvestigatethe reliability in the application of poly (vinyl alcohol) (PVA) as a polymeric binder for stabilizing soft clay soils. PVA is the largest water-soluble biodegradable polymer chain that has excellent film forming and adhesive properties. It is also resistant to grease, oil, and solvent. It is highly hydrophilic and PVA solutions can be prepared easily by dissolving PVA in water. In this study PVA was used as the main stabilizing additive along with 1,2,3,4-Butane-tetracarboxylic acid (BTCA) as crosslinking agent. Furthermore, an optimization model was developed to predict the reliability index of PVA-BTCA stabilized soils.

Hasofer-Lind method which is also called first order reliability method was used for reliability analysis in this study. Geotechnical engineers deal with materials in which loads and resistances are combined and whose distributions and properties are not well known thereby bringing in uncertainties in design. These uncertainties can in geotechnical materials be tackled by an observational method [10] which is widely accepted and successful. This reliability method proposes a new definition for reliability index using geometric interpretation Statistical parameters which are normally described by their means, variances and covariance must include the properties of the geotechnical materials as well as their relationships. The determination of the statistical moment of performance function is basically the calculation of its mean and variance while the determination of the probability of failure can be less rigorous if the performance function has a well-defined probabilistic description like the normal distribution. For a geotechnical system with a load Q and resistance R the safety margin which is the performance function of the system M is expressed as:

M = R − Q (1)

This method of reliability is centered on the assumption that both the load, Q and the resistance, R are normalized thereby leading to the normality of the safety margin, M. According to [10] , reliability index β is given by the expression:

β = μ M σ M (2)

where:

μ M = μ R − μ Q (3)

And similarly, the variance of M ( σ M 2 ) is

σ M 2 = σ R 2 + σ Q 2 − 2 ρ R Q ⋅ σ R ⋅ σ Q (4)

Combining Equations (3) and (4), gives:

β = μ R − μ Q σ R 2 + σ Q 2 − 2 ρ R Q ⋅ σ R ⋅ σ Q (5)

If the load and the resistance are not correlated, it then means the correlation coefficient ρ_RQ is zero and Equation (5) is reduced to:

β = μ R − μ Q σ R 2 + σ Q 2 (6)

From these mathematical expressions it can be deduced that as the Reliability index β increases, the probability of failure decreases thereby making the Reliability index β similar in behavior to the factor of safety.

R e = 1 − P f (7)

In estimating and predicting the reliability of using PVA-BTCA in ground improvement, the Scheffe’s optimization technique is employed. Several authors [11] [12] [13] [14] have carried out concrete mixture researches with development of mathematical models, most of which were based on Scheffe’s Simplex theory.

[15] defined a simplex as a structural representation (shapes) of lines or planes joining assumed points of constituent materials of a mixture and which such points are equidistant from each other. According to [16] , a (q, m) mixture, with q being the number of factors and m being the degree of assumed polynomial, the simplex coordinate system, X_i, and the number of design space points in the simplex lattice, N is defined by Equation (8) and Equation (9) respectively;

X i = 0 , 1 m , 2 m , ⋯ , 1 (8)

N = ( q + m − 1 ) ! m ! ( q − 1 ) ! (9)

According to [17] , mixture proportions are being represented in pseudo (theoretical) mix ratios. Pure substance exist at the vertices points and the method rely on the condition that the summation of all pseudo mix ratios at any point must be equal to 1. Mathematically:

∑ i = 1 q X i = 1 (10)

To achieve the condition of Equation (10), actual mix ratios must be converted to pseudo mix ratios. The relationship between pseudo and actual mix ratios, according to [17] is given by;

Z = [ A ] X (11)

where: Z = column matrix of real component ratio.

X = column matrix of pseudo component ratio.

[A] = coefficient matrix which is the transpose of the permutation matrix [P].

The permutation matrix is obtained from experience derived from reviewed literatures and/or intelligent guesses of the mixture proportions of the factors or mix components. For a (q, m) mixture, the general form of the polynomial model is [17] ;

Y = b 0 + ∑ b i x i + ∑ b i j x i x j + ∑ b i j k x i x j x k + ⋯ + ∑ b i 1 , i 2 , ⋯ , i m x i 1 x i 2 x i m (12)

where; 1 ≤ i ≤ q , 1 ≤ i ≤ j ≤ q , 1 ≤ i ≤ j ≤ k ≤ q

b₀ is a constant coefficient.

This study employed this technique for the development of optimization or prediction models to predict the reliability indices based on some performance parameters of PVA-BTCA stabilized soft soil.

For (4, 2) mixtures, as employed in this study, X_i becomes 0, 1/2 and 1 while N becomes 10 for treatment procedures on application of Equations (8) and (9) respectively. This gives rise to the (4, 2) and simplex lattice presented in Figure 1. PVA was limited to 0.1% - 2% by weight of the dry soil and 1,2,3,4 BTCA was limited to 0.1% - 0.5% by weight of the dry soil. The water content was varied in the range of 10% - 20% by weight of stabilizer-soil mix for all stabilization processes. These range of values were used in the development of the permutation matrix [P] resulting to; (0.998; 0.001; 0.001; 0.1), (0.99034; 0.00733; 0.00233; 0.133), (0.98266; 0.01367; 0.00367; 0.167), and (0.975; 0.02; 0.005; 0.2) for the PVA-BTCA stabilized soil to form Table 2 and Table 3. Table 2 and Table 3 are

matrix mix design tables showing pseudo and actual components for the (4, 2) simplex lattice for trial and control mixes of PVA-BTCA stabilized soil. The number of the design space points N, translates to the minimum number of experimental runs required for development of optimization model of the modified soft soil. These actual mix components are arranged in the format, (soft soil; PVA; BTCA; water).

Unconfined Compressive Strength (UCS): this experiment was conducted in accordance to [18] . Readings of force (F) were taken from the proving ring dial

⁺Where; X₁, Z₁ = pseudo and actual component of soft soil; X₂, Z₂ = pseudo and actual component of PVA; X₃, Z₃ = pseudo and actual component of BTCA; X₄, Z₄ = pseudo and actual component of water.

⁺Where; X₁, Z₁ = pseudo and actual component of soft soil; X₂, Z₂ = pseudo and actual component of PVA; X₃, Z₃ = pseudo and actual component of BTCA; X₄, Z₄ = pseudo and actual component of water.

gauge and the stress applied to the ends of the sample (major principal stress) is computed according to Equation (13).

σ 1 = F A (13)

where: A is the cross-sectional area of the sample. The unconfined compressive strength was measured as the maximum value σ₁, which may or may not coincide with the maximum force measurement.

For (4, 2) simplex problem (PVA-BTCA stabilization), the reduced second degree polynomial form of Equation (12)becomes;

Y = α 1 X 1 + α 2 X 2 + α 3 X 3 + α 4 X 4 + α 12 X 1 X 2 + α 13 X 1 X 3       + α 14 X 1 X 4 + α 23 X 2 X 3 + α 24 X 2 X 4 + α 34 X 3 X 4 (14)

where; Y = Expected response, α i , α i j = Coefficients of the quadratic polynomial, X_i, X_j = Pseudo proportion of factors considered

According to Scheffe’s simplex principle, the coefficients above can be determined from Equations (15).

α i = Y i α i j = 4 Y i j − 2 Y i − 2 Y j } (15)

Reliability index models developed were subjected to F-test for validation. The F-value is given as the ratio of variance between the predicted response value and experimental value.

Mathematically, the F-test is represented by Equation (16):

F = S 1 2 S 2 2 (16)

where; S 1 2 = Larger of both variances

S 2 2 = Smaller of both variance

S² is obtained from Equation (17)

S 2 = 1 n − 1 [ ∑ ​ ( Y − Y ¯ ) 2 ] (17)

where: Y ¯ = Average mean of response, Y

Y = Means of response

The model developed was declared adequate if the F-value calculated in accordance to Equation (16) is less than tabulated value.

Given the foregoing, constant mean load of 75 kPa was used in the UCS test, this becomes μ_Q. This constant load application results to a constant load deviation of 27.39 kPa, which becomes σ_Q. This results to a reliability index value of 2.07

⁺Where; X₁, Z₁ = pseudo and actual component of soft soil; X₂, Z₂ = pseudo and actual component of PVA; X₃, Z₃ = pseudo and actual component of BTCA; X₄, Z₄ = pseudo and actual component of water.

by application of Equation (11). Similarly, other reliability indices were obtained as shown in Figure 2.

With the aid of Figure 2 in conjunction with Equation (15), the model coefficients for the reliability index of PVA-BTCA stabilized soil based on UCS were derived and model obtained as Equation (18):

β PVA-BTCA ( UCS ) = 2.07 X 1 + 2.13 X 2 + 3.10 X 3 + 2.39 X 4 + 0.16 X 1 X 2 + 1.54 X 1 X 3       + 3.2 X 1 X 4 + 1.66 X 2 X 3 + 4.28 X 2 X 4 − 0.66 X 3 X 4 (18)

Figure 3 presents the reliability index values of PVA-BTCA stabilized soil based on UCS for the control mixes. Table 5 presents the F-statistics for validation of Equation (18) where the developed model was tested for adequacy at 5% level of significance. With the aid of Table 5 and Equation (16) the F-value was obtained as 2.47. Because F-cal (2.47) is less than F-tab (3.18), the model is considered adequate.

Microsoft excel solver was used to optimize or combine components to yield the most reliable outcome. In optimization, there must be an objective function subjected to a set of constraints.

Objective function;

Maximize; Equation (18)

Subjected to the following constraints;

X 1 + X 2 + X 3 + X 4 = 1 (19a)

X 1 , X 2 , X 3 , X 4 ≥ 0 (19b)

Using the constraints (Equations 19(a) & 19(b)), the pseudo proportions of PVA-BTCA soil components were obtained as; X₁ = 0; X₂ = 0.207831; X₃ = 0.792169, X₄ = 0; with Max(β) = 3.17. On application of the transformation equation, the actual or real components were obtained as: 98.4256% for soil, 1.2352% for PVA, 0.3392% &BTCA, and 15.9934% for water giving a reliability index value of 3.17.Using normal distribution table, this value of reliability index translates to a reliability of 0.99936 (99.936%).