Role of extreme vertex design approach on the mechanical and morphological behaviour of residual soil composite

Abstract

This research work reports the usability of binary additive materials known as tile waste dust (TWD) and calcined kaolin (CK) in ameliorating the mechanical response of weak soil. The extreme vertex design (EVD) was adopted for the mixture experimental design and modelling of the mechanical properties of the soil-TWD-CK blend. In the course of this study, a total of fifteen (15) design mixture ingredients’ ratios for water, TWD, CK and soil were formulated. The key mechanical parameters considered in the study showed a considerable rate of improvement to the peak of 42%, 755 kN/m² and 59% for California bearing ratio, unconfined compressive strength and resistance to loss in strength respectively. The development of EVD-model was achieved with the aid of the experimental derived results and fractions of component combinations through fits statistical evaluation, analysis of variance, diagnostic test, influence statistics and numerical optimization using desirability function to analyze the datasets. In a step further, the non-destructive test explored to assess the microstructural arrangement of the studied soil-additive materials displayed a substantial disparity compared to the corresponding original soil material and this is an indicator of soil improvement. From the geotechnical engineering perspective, this study elucidates the usability of waste residues as environmental friendly and sustainable materials in the field of soil re-engineering.

Introduction

In the course of constructing civil engineering products such as highway infrastructures, buildings, embankments etc. necessitates the use of soil materials. Soil as a construction material forms a sensitive layer in road structures and this makes it valuable for both construction and other human activities. It may interest one to know that there are cases where the in situ soil contains a substantial quantity of montmorillonite clay mineral and it is termed as black cotton soil. Such soil possesses expansive tendencies, both montmorillonite and illite have high dominance in the structure of this soil and consequently accountable for swell-shrink behaviour associated with the soil. These clay minerals are pivotal menace to foundations of uncountable civil engineering products such as road embankments, roadways etc. However, during the wet period this soil material grasps water whereas during the dry period it drops it moisture. This seasonal alterations results to stability issues which will in return stress the foundation of civil engineering products like buildings and road pavements. Although a number of soil investigations have been documented on the numerous complications of using black cotton soil during construction activities^1,2, it is almost impossible for civil engineers not to encounter these soil materials in the cause of construction, most especially in the Northern part of Nigeria. In the course of constructing infrastructural projects, the stable soils are utilised as infrastructural construction material whereas the unstable ones are replaced with soil materials of good geotechnical standing or improved via stabilization protocols by the incorporation of additives. The occurrence of these unstable soil materials has been a major setback for the construction industry most especially civil engineers, thus making the execution of infrastructural projects on unfavourable soil conditions practically unavoidable. The setbacks posed by the soil when encountered during infrastructural development are well reported in the literature^2,3. To mitigate these setbacks, preventive options including soil replacement, chemical modification and special foundation systems are employed⁴. The mitigation option often applied in soft soil is amendment with conventional or nonconventional ameliorants. The act of amending soft soil via stabilization using conventional ameliorants is becoming an old-fashioned practice coupled with some environmental concerns. The production process of these ameliorant materials coupled with the discharge of greenhouse gases is one of the numerous threats posed by them. It is important to note that the amendment of soil with non-conventional ameliorants which are sustainable and economical, has proved to be very successful due to their ability to perform anticipated outcomes in the geotechnical parameters of soft clay soil compared to conventional ameliorants. Thus, the emerging advances in construction materials have proved that diverse ameliorants rich in alumina and silica such as sawdust ash—quarry dust⁵, cement kiln dust—rice husk ash⁶, cement kiln dust—metakaolin^7,8, cement kiln dust⁹, periwinkle shell ash¹⁰, oyster shell ash^11,12, quarry dust¹³, marble dust—rice husk ash¹⁴, coconut husk ash¹⁵, corncob ash¹⁶, locust bean waste ash^17,18, groundnut shell ash^19,20,21, lime-rice husk ash²², bambara nut shell ash²³, yam peel ash²⁴, metakaolin²⁵, mine tailings²⁶, palm bunch ash²⁷, cement kiln dust-periwinkle shell ash^28,29, cow bone ash—waste glass powder³⁰, sawdust ash—lime³¹, rice husk ash—sisal fibre³², coffee husk ash³³, construction waste³⁴, sawdust³⁵, nanosilica³⁶, electric arc furnace dust³⁷, iron ore tailing³⁸ and so on have been utilised in the amendment studies of deficient soils and concrete structures. Conclusively, as a result of the positive outcomes recorded by the incorporation of solid wastes in soil amelioration studies, a comprehensive review was carried out on the trends in expansive soil stabilization using calcium-based stabilizer materials³⁹. From their all-inclusive review study, it was documented that calcium-based stabilizer materials (CSMs) display pozzolanic tendencies thereby influencing the engineering, geotechnical and microstructural properties of clayey soil materials. However, it is worth noting that the main motivation behind the use of these ameliorants/additives is the decline in greenhouse gas production and the cost of purchasing conventional binding agents (Cement and lime).

Remarkable leaps and bounds have been accomplished in the field of civil engineering construction materials using single non-conventional ameliorants. Attah et al.,²⁵ studied modelling and predicting CBR values of lateritic soil treated with metakaolin (non-conventional ameliorant) and concluded that there was an enhancement in the strength values of the soil when 20% metakaolin was added by air-dried weight of the lateritic soil. Interestingly, a different approach has been explored by soil experts, which involves the use of multi non-conventional additives thereby in some cases incorporating non-conventional ameliorants combined with a little fraction of the conventional binders for soil improvement. The essence of this recent approach is to ensure that there will be adequate pozzolana in the soil matrix reacting with silica and alumina which will activate high strength in the studied soil. Similarly, Ekpo et al.,²⁸ also combined periwinkle shell ash and cement kiln dust to strengthen selected lateritic soils. In a related development, Etim et al.,⁴⁰ improved the properties of lateritic soil by using the combination of lime and periwinkle shell ash and reported the microstructural and geotechnical behaviour of both the untreated and optimally treated soils. The enhancement in the soil properties was linked with the pozzolanic interplay as a result of the incorporation of lime-periwinkle shell ash blend into the soil.

It is worth mentioning, the recent experimental investigation carried out by soil experts has proved the continuous efforts towards the amelioration of deficient soil using single or multi-additives. In the course of ameliorating this deficient soil, considerable improvement in the properties of the soil has been documented. However, notwithstanding the improvement recorded by most soil experts, only a few have deployed optimization tools to arrive at an appropriate target in the properties of the soil. Thus, the latest innovations in Civil engineering practice have prompted specialists in materials and construction engineering to deploy optimization techniques with the sole aim of building precise and reliable models for elucidating engineering problems^9,41,42. This development occasioned the possibility of formulating equations to become accustomed to the technical hitches connected with modelling the behaviour of soil materials⁴³. Remarkably, numerous materials and construction engineering scientists have looked into the achievability of utilizing diverse optimization tools and the conducted experimental outcomes revealed optimistic outcomes such as: extreme vertex design (EVD)⁴⁴, response surface methodology (RSM)^34,45, artificial neural network (ANN)⁴⁶, gene expression programming (GEP)^47,48, Taguchi optimization method^49,50, Scheffe’s optimization approach⁷, artificial intelligence models⁵¹ and so on. In view of this, the EVD approach is a mixture blending system which utilises a smaller fraction within the simplex and has been widely used in Civil engineering. Also, EVD is an offline optimization measure whose objective is to optimise most engineering parameters and processes. It also aids in the design of experiment process and achieving the best possible mixture combinations during optimization protocols. Moreover, to the best of the authors’ knowledge, the major advantage of EVD compared to other optimization strategies is the flexibility in terms of the imposition of other constraints factor levels by specifying both the upper and lower bounds on the components via the designation of linear constraints for blends⁵². This has prompted numerous research experts to explore the utilization of EVD in Civil engineering materials^9,49,52.

Moving onward, recent investigation have shown the level of potency displayed by residues such CK or TWP when utilised as single additive or combined with other additives in geo-engineering protocols. Howbeit, to the best of the author’s knowledge there exist infinitesimal or no known studies documented in open literature that evaluates the role of blending CK-TWP in ameliorating the geotechnical performance of an expansive soil using the principles EVD. Hence, this research becomes necessary and crucial. In this manuscript, one of the author’s hotspots is the use of scanning electron microscopy (SEM) to elucidate mutual interference between the studied soil and additives/ameliorants thereby detecting the amelioration potentials of the additives/ameliorants. The main focus of employing SEM is to also visualise soil-ameliorants interplay at the microscale efficiently and as well provide both qualitative and quantitative reports. The motivation behind the use of SEM in this study is not unconnected with the scanty use of qualitative tests in optimization studies and as well close the gap of the existing studies in soil re-engineering. Finally, it is presumed that blending CK-TWP would be a cheaper and sustainable option compared to the conventional stabilization agents such as lime, bitumen and cement. The benefit of the present investigation will entail in two folds: enhancing the soil performance and as well re-utilising / disposing of waste. Moreover, it is believed that the upshot of this study will create an enabling strategy for waste management.

Materials and methods

Materials

A potentially fine-grained soil also known as black cotton soil engaged in this current optimization exercise was acquired from its deposit in the municipality of Deba, Gombe State, Nigeria (which lies within latitude 10° 12ʹ 42.73ʹʹ N, longitude 11° 23ʹ 13.56ʹʹ E) as seen in the GIS plot in Fig. 1. The soil understudy is known to have a distinctive mineralogical and plasticity response. After excavating the soil material at an average depth of about 1.0 m, it was stacked and transported to the laboratory for the experimentation exercise. Before the experimental tests, it was sundried, pulverised and sieved with the relevant British Standard (BS) sieves. The two additive materials include: calcined kaolin (CK) and tile waste dust (TWD). The raw kaolin was gotten from a locality along the Enugu-Port Harcourt Highway. It was subjected to a controlled thermal activation/calcination and the ash residue gotten after the calcination protocols were allowed to cool and sieved via a 75 μm sieve to obtain a finely powdered substance for experimentation. The calcined kaolin had a specific gravity of 2.58. The tile wastes were gotten from a construction site in the Uyo metropolis, having a specific gravity of 2.75 as shown in Fig. 2. The dominant clay minerals obtainable in the unaltered soil were detected via means of the X-ray diffraction strategy whereas the elemental oxide compositions of the additive materials used in the study (soil, calcined kaolin and tile waste dust) were assessed utilizing the X-ray fluorescent strategy as shown in Fig. 3.

Figure 1

Location map of unaltered expansive soil material used in the paper generated using Arc GIS 10.1 software.

Figure 2

Broken tiles obtained from construction site in Uyo metropolis, Akwa Ibom Nigeria.

Figure 3

X-ray diffraction outcome of untreated ECS.

Experimental techniques

British Standard approach was used to characterize the basic parameters of the unaltered soil which include: grain size distribution, Atterberg limits, specific gravity, natural moisture content, soaked California bearing ratio (CBR), 7 days unconfined compressive strength test (UCS) and resistance to loss in strength (RLS). In this study, the research methods involved experimental design using EVD, laboratory tests and optimization studies. They four number of constituent materials considered are as follows: CK, TWD, soil and water. The proportioning of the test materials were determined by a mix ratio using the EVD approach. The values served as the percentage by weight of the dry solid added in evaluating the considered responses for the study (CBR, UCS and RLS) as established by British Standards. However, in the course of this study it can be seen that studied soil in it natural form has poor geotechnical performance and as such it calls for soil improvement. Secondly, the utilisation of TWD and CK as additive materials was connected to some reasons such as: [i] Incorporation of both TWD and CK will ascertain that there will be enough pozzolana in the soil matrix to react with both silica and alumina components of the soil in the presence of moisture. [ii] Both TWD and CK are waste materials, reutilising them will curb environmental nuisance.

California bearing ratio test (CBR)

The procedure established by British Standard was engaged to find out the CBR values of virgin soil samples as well as the soil-additives admixed with the mix ratios developed by EVD strategy. The guiding principles of the British Standard Light compaction strategy were utilized in executing these experimentation exercises on the soil materials. This test involves the use of 2360 cm³ CBR mould and compacting the soil-additives uniformly with a 2.5 kg rammer in three layers. Thereafter, the compacted soil samples were subjected to a curing duration of 6 days. On completion of the 6 days of curing, the CBR samples were submerged in water for 48 h before testing (Nigerian General Specification)⁵³.

Unconfined compressive strength test (UCS)

Expansive soil material was subjected to amelioration protocols with the mixtures of CK and TDW via the EVD method. Based on the outlines ascribed in BS 1377 part 4⁵⁴, the soil mixtures were meticulously mixed and utilised for the UCS experimentation. The soil samples were compacted using British Standard Light (BSL) compaction energy and were allowed to cure for 7 days. Before the curing, the specimens were exposed to loadings axially in the UCS machine and for each experimentation, two samples were taken and the average values were documented.

Durability

Durability assessment could as well be referred to as the resistance to loss in strength when exposing soil material to an unfriendly environment such as water. Considering tropical regions like ours, the durability assessment was conducted according to the steps outlined in Ola⁵⁵, which involves the ratio of UCS of the specimen wax-cured for 7 days and de-waxed top and bottom before being soaked for another 7 days to the UCS of the specimen cured for 14 days.

Morphological interaction

Both the unaltered and multiple-additive blended soil were morphologically and analytically visualized via scanning electron microscopy (SEM). The Phenom world electron microscopy testing equipment was engaged for conducting this test. The SEM analysis provides a series of micrographs which unravels the bonding, surface texture and geometry of the specimen⁵⁶. Lately, the deployment of qualitative analysis such as SEM in optimization studies has been the hotspot for experts in the field of materials and construction engineering^{57,58,59,60,61}.

Mixture components constraints formulation

The formulation of mixture ingredients proportions and the required number of experimental runs were obtained with the aid of relevant literature works, empirical relationships, expert judgement and practical economic considerations. These design considerations are taken for the design and formulation of the constraints of the components which are imposed on the factor levels of lower and upper regions. However, the defined boundary limits and the sum-to-one constraint would modulate the generation of mixture components’ ratios using actual coding as shown in Table 1. The derived fractions of mixture components would provide a fundamental base upon which the EVD model development can be achieved to evaluate the properties of the soil-CK-TWD blended mixture.

Table 1 Design constraints.

Design of simplex and factor space

Moreover, as a result of multiple constraints imposition on the four mixture components, the resulting design factor space takes the form of a hyper-polyhedron simplex. Through the evaluation of the constraints of the components, the feasible region where the experimental region lies within the constrained portion of the simplex is derived. Also, through design matrix computation, the degrees-of-freedom (df) evaluation is achieved using a model of special-cubic as presented in Table 2. The essence is to guarantee the validity of the lack-of-fit test where fewer degrees of freedom would result in the inability of the test to identify lack-of-fit. The simplex factor space and contour plot for the 4-component mixtures were obtained with the help of design expert software and show the positioning of the experimental points within three space types of planes, vertices and edges of the simplex as shown in Figs. 4, 5. Information matrix measures which provide the leverage and build the type of the derived experimental points for the three space types were further calculated and presented in Table 3. The results obtained implied an average leverage value of 0.9333 with one lack of fit build type for experimental run 5 at edge space type.

Table 2 Design matrix Evaluation for mixture special cubic model using L_Pseudo.

Figure 4

Contour plot of the constrained simplex.

Figure 5

Experimental factor space of the components in a 4-component mixture space.

Table 3 Measures derived from the information matrix.

Furthermore, the criterion optimality algorithm and I-optimal designs also known as IV (integrated variance) which offer the lowest average prediction of the variance across the experimental regions were carried out to evaluate the design experimental space to obtain a Condition Coefficient Matrix of 15,203.166; minimum, maximum and average mean variance of 0.23, 0.625 and 2.262 respectively; G-efficiency of 41.3% which is associated with the inverse of maximum variance; the scaled optimal criterion of 1803.097 and I-optimality value of 0.61945. The derived results implied adequate multicollinearity which would ensure acceptable model goodness of fit behaviour during model development.

Design of experimental mix proportions

Details obtained from the computed information matrix of the mixture design are utilized to formulate the components mixture ratio and the corresponding number of experimental runs. The actualization of these mixed ingredients fractions and the number of runs required were possible through the evaluation of the imposed components constraints and also from the criterion optimal algorithm computation outcome to ensure essential selection of the experimental points within the feasible factor space. Fifteen runs of experiments for the experimental exercise were obtained from the computation procedure as shown in Table 4; presenting the actual and pseudo components fractional values for the factor levels and providing the fundamental template for the evaluation of the 4-components mixture to stabilize the expansive soil using TWD-CK blends.

Table 4 Component proportion mixes for the experimental study.

Results and discussion

As detailed in Table 5, is the elementary distinctive response of the representative expansive soil in its virgin state, while the particle size analysis curve is presented in Table 6. The particle size distribution exercise of the soil depicts a minute fraction of gravel (0.475%), a sand fraction of 27% and a large fraction of fine (72.028%). Sadly, the high fine content in the soil contributes to the expansive and plastic behaviour of the soil. Also, the strength indices of the studied soil were found not to meet the minimum benchmark. The oxide composition of constituent materials in this experimental investigation is shown in Table 7 which indicates good pozzolanic behaviour of the additives to improve the expansive soil’s mechanical characteristics for foundation purposes.

Table 5 Geochemical properties of untreated soil specimen.

Table 6 Particle size fraction of study soil.

Table 7 Oxide composition of constituent materials in this paper.

Presented in Fig. 6 is the Differential scanning calorimetry (DSC) of the raw kaolin. The endothermic peak reactions took place between 552 and 953.9 °C this is connected with dihydroxylation of minerals⁶³. This is a pointer towards the formation phase of calcined kaolin (CK). Interestingly, this DSC outcome is comparable to that documented by⁶⁴ who highlighted that the formation phase of metakaolin sourced from Ajebo kaolin is about 550 °C.

Figure 6

Differential scanning calorimetry (DSC) result of raw kaolin used in the paper.

Impact of an optimal blend of ameliorants on California bearing ratio (CBR)

Table 8 presents the soaked CBR values for soil-calcined kaolin mixtures with tile waste dust. It shows that by adding the additives into the soil the CBR values of the treated soil mixtures yield prominent enhancement in terms of strength. The value of the CBR increased from an initial value of 9.41% to an optimal value of 61.4% as shown in a contour plot in Fig. 7. By comparing the CBR outcomes of the natural soil with the optimally treated soil sample, the rate of increment is about 552%. This remarking influence of an increase in CBR values results from some reasons. Primarily, the combination of calcium ions and water facilitates chemical interplay within the soil mixtures which thereby activates the establishment of new products such as CSH and CAH which has the affinity of influencing bonding properties. Another promising reason may be attributed to the high dominance of cementitious content in the soil matrix which would need hydration to ensure robust pozzolanic interplay as a result of the flocculation and agglomeration of clay particles that occurred. This chemical reaction forms a soil matrix that could efficiently enhance the non-bonded soil particles speedily and as well promote higher strength. A similar phenomenon was also documented by^9,13,59,60.

Table 8 Test outcomes.

Figure 7

Contour plots showing effects of TWD-CK interaction on the mechanical strength response.

Impact of an optimal blend of ameliorants on unconfined compressive strength

The outcomes of unconfined compressive strength (UCS) carried out on the soil samples in the course of this experimental work are presented in Table 8. By comparing the UCS upshots of the unaltered soil to that of additive-altered soil, an amazing increase was witnessed with increasing calcined kaolin (CK) inclusion level irrespective of tile waste powder (TWP) content. The inclusion of these binary additives instigated an increase in UCS from 102 to 755 KN/m² as shown in Table 3 and this translate to approximately seven times increment as shown in a contour plot in Fig. 7. The causes for this UCS improvement mechanism due to additive inclusion can be accredited to three reasons and they include: [1] Filling the voids within expansive soil grains by the binary additives during compaction which thereby imparts higher density to the soil, [2] alteration of the soil physical structure due to the higher specific gravity of the additives (i.e. CK and TWP having 2.58 and 2.5, respectively) in comparison with that of the BCS (2.40), and [3] formation of pozzolanic cementitious compounds between CK, TWP and BCS. In the presence of moisture, the soil reacts with the additives thereby releasing the calcium ions for replacement and exchange in the diffuse double layer of clay particles. The result aligns with that obtained by^9,62.

Impact of an optimal blend of ameliorants on resistance to loss in strength (RLS)

In the course of experimenting durability of the soil specimens, the obtained upshot is shown in Table 3. The durability value of 33% (which implies a loss in strength of 67%) was noticed to be the least possible value compared to 59% (which implies a loss in strength of 41%) being the utmost value. The least value of 33% and peak value of 59% were achievable as a result of mix combination ratios of 3.24:6.48:0.81:5.67 and 1.62:3.65:5.26:5.67 for water, tile waste dust, calcined kaolin and soil respectively. Consequently, the initial RLS value of 33% could be linked with the high dominance of swelling minerals in the parent soil whereas the peak value of 59% might not be unrelated with the presence of additives. It is important to note that the additives considered in this study are rich in the oxides of alumina and silica which are responsible for pozzolanic activities within the soil matrix and onward increment in mechanical strength. The peak RLS value in this study is low compared to the 80% requirement specified in⁵⁵. Howbeit, considering the soaking duration of 7 days against the 4 days as earlier specified by Ola⁵⁵, the peak RLS of 59% was recorded at mixture ratios of 1.62:3.65:5.26:5.67 for water, tile waste dust, calcined kaolin and soil respectively, looks promising. This experimental outcome is not far from the upshots of previous investigators who deployed other solid waste materials^7,65.

Impact of an optimal blend of ameliorants on micro-morphological test

Scanning electron microscope (SEM) observation

Lately, researchers with specialty in the field of materials and construction engineering have deployed a scanning electron microscope (SEM) to unravel the microstructural response of both soil and concrete materials^7,60,61,66. The main motivation for the authors to subject both the unstructured and additive structured soil samples to the SEM testing was to establish the morphological amendment of the tested soil based on the optimization exercise. Figure 8 (a and b) shows the SEM micrograph of the raw soil sample at 300X and 500X magnifications. The unstructured soil material unveils a lumpy clay particle with a blackish appearance because of the absence of hydration products. As shown in the micrographs of the additive structured soil (Fig. 9), it results in a fairly whitish surface appearance with staggered bonding. On a second note, the inconsistencies noticed between the unstructured and additives-structured soil could be related to the incorporation of additives. The presence of these additives activated the dissolution of silicates thereby reacting with calcium and as well enhance the pozzolanic protocols within the soil matrix and the production of cemented materials. It is believed that the cemented composite bonds the aggregates and improves the strength response of the additive-structured soil^67,68,69,70.

Figure 8

SEM micrographs of unstructured expansive soil at magnification levels of 300 × and 500x.

Figure 9

SEM micrographs of additive structured expansive soil at magnification levels of 300 × and 500x.

Development, formulation and validation of the extreme vertex design (EVD) model

For data processing of the derived experimental responses which is essential for the development of the EVD model in the 4-components mixture optimization, quadratic (square-root, λ = 0.5) power transformation was adopted. The experimental responses obtained for CBR, UCS and RLS indicated 11–42%, 410–755 kN/m² and 33–59% with max-to-min ratios of 3.81818, 1.84146 and 1.78788 respectively. Statistical analysis was carried out through fits summary, diagnostic computation and influences, numerical and graphical optimization to ascertain the optimal 4-components mixture ratio to improve the engineering behaviour of problematic soil. After the development of the model coefficients, Post analysis, point prediction, confirmation and model simulation were further achieved to validate the prediction performance of the EVD model with the aid of a design expert and Minitab 18 software.

Fit-summary statistics

This statistical assessment is first executed on the system datasets to derive their viability and adequacy which would enable the actualization of the requirements for the EVD modelling measures using a mixture coding of L_Pseudo. The fit-summary computation provides important information which would aid the choice of starting point suitable for the development of the EVD model. The essential collection of statistical indicators such as lack-of-fit, coefficient-of-determination (R-sqd), sum-of-squares (SS) and predicted-sum-of-squares (PRESS) is calculated as presented in Tables 9, 10, 11, 12, 13, 14, 15, 16, 17. The computed statistical fit results indicated a preference for the quadratic model for CBR response with R-sqd. of 0.9929 while linear models were suggested for both UCS and RLS responses with R-sqd. of 0.8844 and 0.9463 respectively. From the sequential sum of squares, p-value of 0.0040, 0.0001 and 0.0001 were calculated for CBR, UCS and RLS responses respectively.

Table 9 Model summary statistics for CBR response.

Table 10 Lack of fit for CBR response.

Table 11 sequential Model sum of squares [Type I] for CBR response.

Table 12 Model Summary Statistics for UCS Response.

Table 13 Lack of Fit for UCS Response.

Table 14 Sequential model sum of squares [Type I] for UCS response.

Table 15 Model summary statistics for RLS Response.

Table 16 Lack of Fit for RLS response.

Table 17 Sequential Model Sum of Squares [Type I] for RLS Response.

Analysis of variance (ANOVA) statistical results

Analysis of variance statistical computation was carried out for the preferred model to assess the mixture ingredients fractions’ significance levels on the CBR, UCS and RLS response variables using L_Pseudo coded units. The ANOVA computation of partial sum-of-squares and R-sqd. Statistics calculation results are presented in Tables 18, 19, 20, 21, 22, 23. From the statistical analysis results, the model p-value of 0.0001 was derived for the three response variables while F-value of 77.67, 28.04, and 64.61 were obtained for CBR, UCS and RLS responses respectively. The indication of this statistical result showed that the derived model terms are statistically significant.

Table 18 ANOVA for quadratic mixture model for CBR response.

Table 19 R-sqd. calculations for CBR response.

Table 20 ANOVA for linear mixture model for UCS response.

Table 21 R-sqd. calculations for UCS response.

Table 22 ANOVA for Linear Mixture model for RLS Response.

Table 23 R-sqd. calculations for RLS response.

The calculated 0.9110 result for R-Sqd. Pred. showed satisfactory agreement with the R-Sqd. Adjtd. of 0.9801 which indicates a difference of less than 0.2 when both indices were compared. Adeq.-Prec. evaluates the signal-to-noise ratio and a ratio value of 27.804 was obtained which signified an adequate signal with a computed ratio outcome > 4.

The computed 0.7928 result for R-Sqd. Pred. implied suitable agreement with the R-Sqd. Adjtd. of 0.8528 which indicates a difference of < 0.2 when both statistical indices were compared. Adeq.-Prec. measures the signal-to-noise ratio and a ratio value of 14.730 was obtained which denoted an adequate signal with a computed ratio outcome > 4.

The calculated 0.9019 results for R-Sqd. Pred. implied suitable agreement with the R-Sqd. Adjtd. of 0.9317 which indicates a difference of < 0.2 when both statistical measures were compared. Adeq.-Prec. assesses the signal-to-noise ratio and a ratio value of 22.772 was derived which denoted an adequate signal with a computed ratio outcome greater than 4.

Coefficient estimates and model equations

After ANOVA computation which involved thorough statistical assessments were executed to achieve suitable estimates of the model coefficients for the three response parameters carried out with the aid of Design Expert software. The calculations show low and high confidence intervals for the model estimates, variance inflation factor (VIF), standard error and derived model numerical coefficients as presented in Tables 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34. The lack of orthogonality impact on the variances of the EVD model coefficients developed is assessed through VIF which is proportional to the square of the model’s standard error. The model coefficients for the CBR responses which were derived using quadratic polynomials and UCS and RLS derived using linear polynomials are presented in Tables 25, 30 and 34 respectively. The coefficients for the UCS and RLS responses were linear polynomial, hence, base point terms and constraint region bounded factor level effects for Piepel and Cox direction were analyzed as presented in Tables 27, 28, 29, 32, 33. The derived results for UCS responses in Piepel direction showed Prob >|t| of 0.2004, 0.0005, < 0.0001 and 0.0268, for components A, B, C and D respectively and 0.4912, 0.2406, < 0.0001 and 0.0554 for components A, B, C and D respectively in Cox direction. Therefore, results for RLS responses in Piepel direction indicated Prob >|t| of 0.0026, < 0.0001, < 0.0001 and 0.0069 for components A, B, C and D respectively while 0.0148, 0.2503, < 0.0001, 0.0368 for components A, B, C and D respectively in Cox direction.

Table 24 Quadratic Model Coefficients Calculation Results for CBR Response.

Table 25 Derived final equation for CBR response.

Table 26 Linear model coefficients calculation results for UCS response.

Table 27 UCS and RLS responses base points in terms of pseudo and real components.

Table 28 Constraint region bounded component effects for piepel direction for UCS response.

Table 29 Constraint region bounded component effects for cox direction for UCS response.

Table 30 derived Final equation for UCS response.

Table 31 Linear Model Coefficients Calculation Results for RLS Response.

Table 32 Constraint egion bounded component effects for Piepel direction for RLS response.

Table 33 Constraint region bounded component effects for cox direction for RLS response.

Table 34 Derived final equation for UCS response.

Diagnostics plots

To achieve the corroboration of the regression model assumptions and to elucidate if the experimental observations utilized for the EVD-model development were significantly unjustifiable and at the same time pose a noticeable influence on the analysis results. Studentized residuals were adopted to accomplish this statistical analysis which is used to detect outliers which significantly differ from regular observations as a result of a verifiable error in the mixture design. Moreover, varying normal distributions observed are mapped to a unit standard normal-distribution using the studentized residuals for the CBR, UCS and RLS responses respectively. Statistical-diagnostic tests carried out include predicted vs. residual, normal probability test, predicted vs. actual, run plot vs. residuals, and box-cox power transformation.

Graph of normal-probability

The graphical results evaluating the normal probability help to confirm that the derived errors also known as residuals follow or assume normal distribution positioned nearest to the regression line. The plot illustrates the relationship between the externally studentized residuals on the x-axis against the normal probability (%) on the y-axis for the CBR, UCS and RLS response variables as shown in Fig. 10. Details such as definite configurations like curvilinear shapes are examined carefully to ascertain the possibility of deploying suitable model transformation of the response parameters. The results derived from the plots indicate the majority of the points from the horizontal axis are within the range of -1 to + 1 studentized residuals and also within 10–90% normal probability for the three response cases under study.

Figure 10

Graphical presentation of residuals normal Probability.

Graph of predicted versus studentized residuals

The constant variance assumption of the residuals is validated for the three response parameters through the residuals vs. predicted statistical diagnostic test. The plot presents the model-predicted values on the x-axis against the externally studentized residuals on the y-axis of the graph as shown in Fig. 11. The presented graphical results indicated clustering of the derived model-predicted values at the zero studentized point for the majority of the observations. The plot for CBR response was observed to be positioned at the limits of ± 4 within the boundary points of ± 6.254, the responses for UCS and RLS were thus observed to show a similar trend as they are situated at ± 2 limits within the boundary points of ± 3.8273.

Figure 11

Diagnostic statistical graphs of residual versus predicted results.

Experimental run versus studentized residuals Plot

The examination for plundering factors which pose a significant impact on the response variables during the statistical calculations is achieved using this diagnostic computation graph. The plot presents the experimental run’s effects on the target responses on the x-axis against the externally studentized residuals on the y-axis as shown in Fig. 12. From the graphical results similar to the residual vs. predicted diagnostic plots, the observed design points were situated at the boundary points of ± 4 for the CBR response and ± 2 for both the UCS and RLS response which signifies a time-dependent lurking variable in the background of the presented scattered plot.

Figure 12

Residual versus experimental run diagnostic graphs.

Predicted versus experimental result graphical plots

This diagnostic statistical analysis plot essentially helps to obtain a unitary or group of values which are difficult to accurately estimate by the generated EVD-model in a straight line graph of the laboratory outcomes on the x-axis and the predicted values on the y-axis as shown in Fig. 13. The essence of this diagnostic computation is to assess the relationships between the square root transformed model predicted and actual values by how well the designed experimental data fit the regression line. The details derived from the graphical plots for the three responses indicated a good and acceptable correlation between the model estimated and laboratory values.

Figure 13

Diagnostic graphs of actual versus model predicted.

Power transformation using box-cox plot

This analytical computation plot offers suitable guidelines for the selection of an adequate power transformation model to examine the significant impacts on the response parameters by the factor levels. The results derived from this diagnostic assessment help to attain the best lambda outcomes on the horizontal axis of the plot which are derived at the minimum point of the parabolic curve corresponding to the sum of squares residuals’ natural logarithm on the vertical axis as shown in Fig. 14. From the presented graphs for CBR response showed current lambda of 0.5, confidence intervals (C.I.) of -0.07 and 1.41 lambda for low and high limits respectively, and best lambda of 0.6. Also, the UCS response showed current and best lambda of 0.5 and 2.06 respectively, low and high C.I. of -0.38 and 4.55 respectively while the RLS response showed current and best lambda of 0.5 and -0.02 respectively, low and high C.I. of -2.2 and 1.98 respectively.

Figure 14

Box-Cox plots for power transformation.

Diagnostic-influence plots

The statistical diagnostic influence evaluates the significant impact of runs of the experiment on the outcome of the analysis. The influence plots present a suitable statistical perspective in the experimental design when multiple or unitary cases stands out from other groups and the statistical influence computation is achieved using the following analytical tools; leverage vs. the experimental run, cook’s distance and statistical difference in fits (DFFITS) vs. run of experiments.

The cook’s distance

The Cook’s distance is adopted in regression analysis to ascertain the influence of data points or outliers in explanatory variables concerning the target response variables. This computation also helps to determine the feasible experimental planes where negative effects are achieved and also where acceptable correlation outcomes are derived. The plot presents runs of experiments on the x-axis against the calculated cook’s distance result as shown in Fig. 15. From the derived results, the experimental runs were mostly positioned at the cook distance zero point, however, for the CBR response, the experimental points lie within the cook’s distance limits of 0–2 and 0–1 for both the UCS and RLS responses.

Figure 15

Cook’s distance.

Leverage versus run

Leverage computation is a statistical diagnostic influence tool which is utilized to examine the extents to which each experimental point affects the developed model’s goodness-of-fit. It is also deployed to evaluate the distance with which the explanatory variables from a particular observation are far apart from the set of other observations in the mixture design experiments. This diagnostic influence plot presents the derived leverage values from 0 to 1 on the vertical axis against the run of experiments on the horizontal axis as shown in Fig. 16. The presented plots for the CBR response lie within leverage values of 0.5–0.9 with the boundary line drawn at 0.6667. However, the UCS and RLS responses exhibited similar trends and lie within the leverage points of 0.2–0.4 with the boundary lines observed at 0.53333–0.26667.

Figure 16

Diagnostic influence plot of leverage versus run.

Statistical difference in fits (DFFITS) versus runs

Statistical Difference in Fits (DFFITS) is a statistical influence technique which evaluates the variation of the EVD-model predicted response for the ith observation in the experimental mixture design. This diagnostic test helps to clarify for a particular design point the estimated model variability when it is barred for the model fitting processes as shown in Fig. 17. The presented analytical plots which indicate the fifteen design experimental runs on the horizontal axis against the computed DFFITS outcomes on the y-axis. From the results, the majority of the data points were observed to be positioned within the boundary points of ± 2.4449 except runs number 1, 8, 12, 13 and 14 for the CBR response. However, the data points for both CBR and RLS were observed to be situated within similar boundary points of ± 1.54919 except experimental run number 1 for RLS response. The diagnostic statistical summary and influences for the analytical computations carried out in this experimental investigation and EVD model development for the CBR and UCS responses are shown in Tables 35, 36, 37. The results present the predicted vs. actual values, the computed residuals, leverages, internally and externally studentized residuals, cook’s distance and influence on fitted value.

Figure 17

Diagnostic influence plot of dffits versus experimental run.

Table 35 Diagnostics case statistics for CBR response.

Table 36 Diagnostics case statistics for UCS response.

Table 37 Diagnostics case statistics for RLS response.

Optimization overview

The numerical and graphical optimization exercise is executed after the model fit computation, ANOVA, statistical diagnostics and influence calculations were carried out. The mixture experiment design optimization is achieved through the desirability function which examines the imposed components criteria or constraints upon the model variables to maximize the target responses. The criteria also known as the goal of the optimization can be minimum, maximum or in-range factors to constrain the variables based on the perceived objectives for the exercise. The objective functions are systematically adjusted using assigning adaptive weight functions in line with the preset criteria of the model variables. These multi-collinearity conditions would enable the attainment of favourable conditions to obtain a desirability score of 1.0 through a scale of $0 \le d\left( {y_{i} } \right) \le 1$ boundary conditions. The optimization measures for this 4-component mixture design explore the varying formulated components’ fractions combinations in the factor space to sort after response that the imposed criteria or components constraints for the factor levels and the corresponding optimized dependent variables simultaneously. The expected goal of this optimization exercise is set to maximize the target responses while the four mixture components are set at the in-range option to ascertain the optimal proportion of the factor levels to produce maximum responses bounded by the derived upper and lower limits from the experimental details as shown in Table 38.

Table 38 Variables constraints for optimization.

The computed optimization solution derived from analytical proceedings of the 4-component mixture experiment design showing seventeen optimized solutions is presented in Table 39. From the derived results, solution number 13 was preferentially selected showing an optimal desirability score of 1.0 at a combination ratio of 0.100:0.097:0.450:0.0353 for water, TWD, CK and soil respectively to produce maximized CBR, UCS and RLS response of 44.611%, 810.687kN/m² and 63.448% respectively.

Table 39 Computed mixture design optimization solutions.

Graphical optimization plots of ramps and bar

The analytical outputs derived from the 4-components mixture design optimization are presented graphically through ramps plot and bar charts for clarity and understanding of the obtained results. The ramp and bar chart illustrates the optimal solution for the factor levels in red colours while the target responses are in blue colouration to elucidate the obtained desirability function computation outputs as shown in Figs. 18–19. The graphical optimization results presented indicated the explanatory and response variables’ desirability score of 1.0, while the combined optimal desirability score of 1.0 was calculated which signifies adequate performance taking into consideration the imposed factors criteria.

Figure 18

Ramps graph for the desirability function.

Figure 19

Bar Graph for the desirability function.

Optimization trace plot

Trace plots are deployed to assess all mixture components’ effects in the feasible experimental space considering the derived optimal responses and it’s similar to perturbation plots adapted for analysis in non-mixture designs. The referenced optimized mixture blend is derived using this graphical assessment tool which is aimed at obtaining the target response sensitivity when related to the formulation deviation near the reference blend. The trace plot adapted for this evaluation study is through the Piepel direction along which the fractions of varying amounts of all the other mixture components are held constant. The plot presents the pseudo-coded units ranging from -1 to + 1 on the x-axis to indicate significant locations relative to the coded scale as shown in Fig. 20. The results in the plot indicate that components C and B were positioned between − 1.0 and − 0.5 and 0.5–1.0 respectively on the coded units’ axis.

Figure 20

Optimization trace piepel plot.

Optimization contour plot

The contour plot is a two-dimensional graphical analysis computational tool that helps the constrained experimental space visualization to observe the numerical factor points against the mixture components within the simplex through an iterative optimization solution as shown in Fig. 21. The presented graphical plots indicated a three-edged triangular plot with three of the four mixture components simplex showing the optimized ratio of 0.54718, 0.497318 and 0.497318 for water, TWD and CK respectively.

Figure 21

Contour plots for the optimal solution.

3D surface plots

3-dimensional surface plots are graphical illustrations of the interactions and connectivity behaviour between the factor levels to the response variables. The 3-dimensional graphs for the mixture components concerning the dependent variables and desirability function optimal solution and presenting the corresponding experimental design points’ response surface is shown in Fig. 22. The plotted results indicated optimal responses of 44.611%, 810.687 kN/m² and 63.448% for CBR, UCS and RLS respectively.

Figure 22

3D plot for the optimal solution.

EVD-Model post-analysis and simulation

Post-analysis and model simulation was carried out after the EVD-model development, numerical and graphical optimization to validate the derived factor values, point prediction and confirmation to achieve sample means and coefficient table. Also, a simulation exercise of the EVD-model generated to assess its applicability and performance is further achieved.

Point prediction computation

This post-analysis tool essentially provides an avenue for the factor values to be reviewed which utilizes the fit statistical computation of the developed model and the prescribed settings for the factor tools to determine interval estimates after the analysis is successfully carried out as presented in Table 40. The tabulated outputs showed standard deviation of 1.877, 51.757 and 2.999, low to high confidence interval (CI) values of 39.553–50.506, 734.737–881.168 and 58.893–67.381 for CBR, UCS and RLS response parameters respectively.

Table 40 Point prediction.

Coefficient-table

The coefficient table presents the magnitude combination ratio of the factor levels which evaluates the 4-components TWD-CK-soil mixture blend aimed at improving the engineering behaviour of expansive soil for pavement construction as shown in Table 41. From the presented multi-criteria optimization analysis, quadratic was preferred for CBR response and linear models for both UCS and RLS responses to obtain a generalization of experimental data. The legend column was used to assess the p-values for every term of the models where p-values < 0.01 signifies the best measures for strong significance presented in red, p-values ≤ 0.05 and ≥ 0.01 indicates a moderate level of significance presented in green, p-values ≤ 0.10 and ≥ 0.05 signifies a slight level of significance presented in blue while p-values ≥ 0.10 indicate a level of non-insignificant displayed in black.

Table 41 Coefficient table.

EVD model simulation

The simulation of the EVD-model is carried out after the derivation of the coefficient table and post-analysis process. It is also the last phase of the statistical analysis of the 4-components mixture design optimization and is executed to validate the necessary statistical assessments carried out to develop the EVD-model to simulate real-life conditions which provide adequate guidance for contractors, field operators, designers and consultants on the applicability and performance of the developed EVD-model. The simulation procedure involves comparing statistically the experimental or actual values with the simulated model results as shown in Fig. 23 using analysis of variance (ANOVA) at a 95% confidence interval. The statistical evaluation of the simulated vs actual results was achieved using Microsoft excel software and the outputs are presented in Tables 42, 43, 44. From the computed results, P(F <  = f) two-tail of 0.9965, 0.9888 and 0.9997 was computed which is greater than 0.05 for CBR, UCS and RLS target responses respectively. The derived statistical results indicate that there is no significant difference between the laboratory or actual and model predicted results which implies acceptable model estimation accuracy in line with the research findings of Aju et al.⁷¹.

Figure 23

Actual versus EVD-model predicted responses.

Table 42 t-test Two-sample assuming equal variances for CBR response.

Table 43 t-Test two-sample assuming equal variances for UCS response.

Table 44 t-Test two-sample assuming equal variances for RLS response.

Comparison with previous studies

The recent hotspot in the field of materials and construction engineering is the utilization of waste residue in conjunction with a suitable optimization technique in soil amelioration protocols. The use of waste residue is a result of the high cost of cement, depletion of the ozone layer due to cement production and as well as promoting sustainable environment whereas the deployment of optimization technique is to aid in model building and as well reduce the process of the random number of experiments experienced in the trial and error approach. Thus, this necessitated the deployment of extreme vertex design of experiment approach in the current study. Notably, in this current experimental study, the CBR upshots of soil-calcined kaolin mixtures with tile waste dust demonstrated an appreciating trend with the addition of additive dosages. In contrast to the study of Alaneme et al.⁹ who engaged EVD in the optimization of a single additive for the amelioration of expansive soil, they reported an increasing CBR with the incorporation of additive material. However, both studies used black expansive clay soils but in their study, a single additive was considered whereas in this current study binary additives were considered and this might not be unrelated to the rate of strength gained. Ikeagwuani et al.⁵⁹ established coherently that optimization of multi-additives via means of Taguchi mixture experimental design in a certain way influenced the mechanical response (differential free swell, California bearing ratio, and unconfined compressive strength) of expansive soil. Surprisingly, the durability response recorded in this study increased with an increase in the dose of additives to a peak value of 59%. The durability outcome for the optimal soil additive mixture fell short of the 80% requirement and it seems to be consistent with those of⁶⁶. In summary, the extreme vertex design optimization strategy used in this experimental study endorses the explorations of^9,44 who disclosed the effectiveness and efficiency of EVD in civil engineering materials.

Conclusion

This study documents the upshots of successful soil re-engineering protocols of a weak soil ameliorated with the combination of TWD and CK based on the principles of EVD strategy. Based on AASHTO classification and USCS scheme, the soil in its original form falls under the class of A-7–6 material with group index of 14 and CH, respectively. The upshot of this optimization study indicates that using the mixture ratio of 0.10:0.23:0.32:0.35 for water, TWD, CK and soil respectively, optimally enhanced the key parameters including CBR, UCS and RLS. The documented rate of enhancement in the soil’s mechanical might not be far from the pozzzolanic interplay within the soil structure as a result of the introduction of additive materials, which was later confirmed qualitatively via microstructural experiments such as scanning electron microscopy (SEM). From the geo-engineering point of view, the benefit of the present investigation entails in numerous folds: [i] the integration of waste materials as ameliorants promotes a sustainable environment thereby regulating the depletion of natural resources and possible occurrence of global warming, [ii] it will create a cost-effective strategy in the eradication of mammoth amount of waste materials are generated into the society thereby reducing nuisance and cleaner environment and [iii] in the course of infrastructural developments, the practice of waste materials reutilization in soil re-engineering as either cement or lime surrogate materials facilitates lessening in cost of Civil engineering projects. The use of EVD strategy have shown a huge success thus, the use of other optimization techniques such as Taguchi, response surface methodology etc. would be a welcome development as it will create room for further studies and as much a more efficient optimal ratio could be achievable.