A multi-attribute decision making approach of mix design based on experimental soil characterization

Abstract

The clay mineral composition is one of the major factors that governs the physical properties of silty sand subgrade. Therefore, a thorough knowledge of mineral composition is essential to predict the optimum engineering properties of the soil, which is generally characterized by different indices like maximum dry density (MDD), California bearing ratio (CBR), unconfined compressive strength (UCS) and free swelling index (FSI). In this article, a novel multi-attribute decision making (MADM) based approach of mix design has been proposed for silty sand – artificial clay mix to improve the characteristic strength of a soil subgrade. Experimental investigation has been carried out in this study to illustrate the proposed approach of selecting appropriate proportion for the soil mix to optimize all the above mentioned engineering properties simultaneously. The results show that a mix proportion containing approximately 90% silty sand plus 10% bentonite soil is the optimal combination in context to the present study. The proposed methodology for optimal decision making to choose appropriate combination of bentonite and silty sand is general in nature and therefore, it can be extended to other problems of selecting mineral compositions.

Keywords

silty sand / bentonite soil / soil mix design / multi-attribute decision making

Introduction

Soil stabilization and soil treatment gives an innovative solution to improve the engineering properties of soil by improving the bearing capacity, durability of weak soil and other important characteristics as per requirement [¹,²]. It is observed using scanning electron microscope that the swelling behavior of bentonite in buffer and backfill materials allows it to control the temperature and vapor pressure around the samples. Previous researches suggest that the swelling behavior of bentonite depends on sample specifications such as sand-bentonite mass ratio, dry density, and hydration sequence [³]. It is also observed that the voids in the materials are filled almost completely by swelling deformations of bentonite absorbing water [⁴,⁵]. Such interesting behaviors of bentonite in buffer and backfill materials have attracted huge amount attention from the concerned research community. Moreover bentonite is essentially highly plastic clay containing; having not less than 85% clay. Excellent plasticity and lubricity, high dry-bonding strength, high shear and compressive strength, low permeability and low compressibility make bentonite important for many engineering applications. Bentonite is valued in foundry sand binding, drilling mud, iron ore palletisation and as waterproofing and sealing agent in civil engineering applications. Many researchers have investigated sand-bentonite mixtures for application as clay liners in industrial waste disposal facilities. Several studies have been conducted to examine hydraulic conductivity of bentonite clay liners that are used as waterproof materials in industrial waste disposal facilities [⁶‒¹⁴].

Though considerable amount of studies have been carried out on hydraulic conductivity of materials mixture with bentonite, investigations related to bentonite mixtures on pavement construction for subgrade and foundations are very scarce in literature. Very high bentonite proportion in soil mixture is not suitable for any type of construction because of its high swelling potential and low hydraulic conductivity. In the present study, effort has been made to optimize the bentonite and silty sand mix on the basis of desirable engineering properties. Silty sand is a good construction material from swelling behavior as well as hydraulic conductivity point of view, and thereby the overall engineering properties of the soil mix is attempted to be improved. We have investigated the strength characteristics of silty sand and bentonite soil mixture based on a novel MADM approach for optimum mix design.

Multi-attribute Decision-Making [¹⁵] is a powerful approach for analysis of multi-dimensional problems because of their integral ability to adjudicate different alternatives on identified criteria for possible selection of most beneficial alternative(s). To the best of authors’ knowledge, MADM has not been applied in the realm of geotechnical engineering; even though this method could potentially be a very beneficial tool for selecting appropriate proportion of a soil mix to optimize all concerned engineering properties simultaneously. In the present study, a novel MADM based algorithm has been developed for selecting appropriate proportion of a stabilized soil mix containing bentonite and silty sand to optimize several engineering properties (such as MDD, CBR, UCS and FSI) simultaneously. This article is organized as, section 1: introduction; section 2: description of MADM based algorithm for selecting optimum mix proportion; section 3: experimental characterization of the material; section 4: results and discussion on selection of soil mix proportion; section 5: conclusion.

MADM based algorithm for selection of optimum mix proportion

The MADM [¹⁵] is an effective algorithm for analyzing multi-dimensional problems to adjudicate different alternatives on identified criteria for possible selection of most beneficial alternative(s). Previous researches suggest that the multi-criteria decision making (MCDM) problem often involve a complex decision process in which multiple requirements and fuzzy condition have to be considered and proved that this method is suitable to select possible alternatives in problems involving high degree of uncertainty [¹⁶]. There are different MADM algorithms available in scientific literature [¹⁷] among which the following three most popular algorithms have been utilized in conjunction to the present problem of selecting optimum soil mix: simple additive weighting [¹⁸] weighted product model [¹⁹] and analytic hierarchy process [²⁰]. Brief descriptions about the MADM methods used in this study are furnished next.

Simple additive weighting

Simple additive weighting (SAW) is the most commonly used decision making method [¹⁸]. It is also called scoring method. If there are p alternatives and ncriteria then, the best alternative is the one that satisfies the following expression.

(1)J(\mathrm{SAW})=\max (i),\hspace{1em}{{\displaystyle \Sigma }}_{{{\displaystyle Z}}_{ij}{{\displaystyle W}}_j=\mathrm{1}}^n,\hspace{1em}\mathrm{for}\hspace{0.17em}i=\mathrm{1},\mathrm{2},\mathrm{3...},p,

where J(SAW) is the SAW score of the best alternative, Z_ij is the actual value of the i-th alternative in terms of the j-th criterion, and W_j is the weight of importance of the j-th criterion.

Weighted product model

In weighted product model (WPM), each alternative is compared with the others by multiplying a number of ratios, one for each criterion [¹⁹]. Each ratio is raised to the power equivalent to the relative weight of the corresponding criterion as shown below.

(2)J({{\displaystyle Z}}_K/{{\displaystyle Z}}_L)={{\displaystyle \Pi }}_{j=\mathrm{1}}^n{({{\displaystyle Z}}_{Kj}/{{\displaystyle Z}}_{Lj})}^{{{\displaystyle W}}_j}.

Notations in Eq. (2) are same as SAW method (section 2.1). If the term J({{\displaystyle Z}}_K/{{\displaystyle Z}}_L) is greater than one, then alternative Z_K is more desirable than alternative Z_L. The best alternative is one which is greater than or at equal to the other entire alternatives.

Analytic hierarchy process

The analytic hierarchy process (AHP) is a structured technique for organizing and analyzing complex decisions, based on mathematics and psychology by extending the basic weighting method [²⁰]. At the core of the analytic hierarchy process lies a method for converting subjective assessments of relative importance to a set of overall scores or weights by the use of principal eigenvector method for converting the paired comparison data into attribute weights, and the results of these calculations in a heuristic check of inconsistency among paired comparisons. As clear and self-explanatory description of the AHP algorithm is very scarce to find in scientific literature, the conceptual steps of AHP are illustrated below.

Step 1: For demonstrating the algorithm of AHP, we consider a problem having n numbers of criteria and p numbers of alternatives. In this algorithm, pairwise comparisons are the primary process and uses a scale (Table 1) to rate the relative preferences for two items (relative importance among each of the two criteria). Element a_ij of the matrix in Table 2 is the measure of preference of the item in i^th row when compared to the item in j^th column, while comparison of any alternative against itself must lead to the judgement that they are equally preferred. For the ease of understanding we will discuss the AHP algorithm considering four criteria (C1, C2, C3 and C4) and three different alternatives (L1, L2 and L3), i.e., n = 4 and p = 3 (as the problem considered in the present investigation has four criteria and three possible alternatives, which are discussed in section 4). It can be extended to other values of n and p following similar way. Thus as explained above, all elements on the diagonal of the pairwise comparison matrix (A₁) become 1 (i.e., a₁₁ = a₂₂ = a₃₃= a₄₄ =1). Summations for different criteria in A₁ (column-wise) matrix are defined as: A = (a₁₁ + a₂₁ + a₃₁ + a₄₁); B= (a₁₂ + a₂₂ + a₃₂ + a₄₂); C= (a₁₃ + a₂₃ + a₃₃ + a₄₃) and D= (a₁₄ + a₂₄ + a₃₄+ a₄₄).

Step 2: Relative priority for each decision is determined as shown in Table 3. First, summations are evaluated for each column of the matrix and then each element in the matrix is divided by its column total. The resulting matrix is referred to as the normalized pairwise comparison matrix.

Finally, compute the average of the elements in each row of the normalized matrix. These averages provide an estimate of the relative priorities of the elements being compared. The A₂ matrix in Table 3 is calculated as shown in Eq. (3).

(3)\begin{array}{l}E=\left\{\left(a_{\mathrm{11}}/A+a_{\mathrm{12}}/B+a_{\mathrm{13}}/C+a_{\mathrm{14}}/D\right)\right\}/n\hfill \\ F=\left\{\left(a_{\mathrm{21}}/A+a_{\mathrm{22}}/B+a_{\mathrm{23}}/C+a_{\mathrm{24}}/D\right)\right\}/n\hfill \\ G=\left\{\left(a_{\mathrm{31}}/A+a_{\mathrm{32}}/B+a_{\mathrm{33}}/C+a_{\mathrm{34}}/D\right)\right\}/n\hfill \\ H=\left\{\left(a_{\mathrm{41}}/A+a_{\mathrm{42}}/B+a_{\mathrm{43}}/C+a_{\mathrm{44}}/D\right)\right\}/n\hfill \end{array}.

Here n denotes the size of A₁ (i.e., number of criteria, which is 4 in the present investigation). The elements of A₃matrix in Table 3 are computed from A₁ and A₂ as shown below:

(4)\begin{array}{l}I=\left(a_{\mathrm{11}}\times E\right)+\left(a_{\mathrm{12}}\times F\right)+\left(a_{\mathrm{13}}\times G\right)+\left(a_{\mathrm{14}}\times H\right)\hfill \\ J=\left(a_{\mathrm{21}}\times E\right)+\left(a_{\mathrm{22}}\times F\right)+\left(a_{\mathrm{23}}\times G\right)+\left(a_{\mathrm{24}}\times H\right)\hfill \\ K=\left(a_{\mathrm{31}}\times E\right)+\left(a_{\mathrm{32}}\times F\right)+\left(a_{\mathrm{33}}\times G\right)+\left(a_{\mathrm{34}}\times H\right)\hfill \\ L=\left(a_{\mathrm{41}}\times E\right)+\left(a_{\mathrm{42}}\times F\right)+\left(a_{\mathrm{43}}\times G\right)+\left(a_{\mathrm{44}}\times H\right)\hfill \end{array}.

A₄ matrix is obtained by dividing A₃ matrix by the corresponding elements of A₂ matrix (i.e., M = I/E; N = J/F; O = K/G and P = L/H).

Step 3: In this step, consistency check by means of computing consistency ratio is carried out. It is an important measure to check before moving further in AHP. If the degree of consistency is acceptable, the decision process can continue, otherwise the decision maker should reconsider their judgments before proceeding any further with the analysis. Consistency ratio exceeding 0.10 are indicative of inconsistent judgments. The consistency index (CI) is calculated as

(5)CI={\displaystyle \frac{{{\displaystyle \lambda }}_{\max }-n}{n-\mathrm{1}}},

where l_max is the average of the elements of A₄ matrix (i.e., l_max = (M+N+O+P)/n).

Step 4: Consistency ratio (CR) is calculated as follows.

(6)CR={\displaystyle \frac{CI}{RI}},

where RI is the random index, which is the consistency index of a randomly generated pairwise comparison matrix. The values of RI corresponding to the size of matrix can be obtained as per Table 4.

Step 5: Based on number of alternatives, another set of pairwise comparison matrices are formed using Table 1 for each criteria. The pairwise comparison matrix (A_1c) for the first criterion (i.e., C1) is shown in Table 5.

where S = (a₁ + a₄ + a₇), T = (a₂ + a₅ + a₈) and U = (a₃ + a₆+ a₉). Pairwise comparison matrices are formed for all other criteria in a similar way. From the A_1c matrix A_2c is formed as Table 6.

where a^′₁₁ = (a₁/S+ a₂/T+ a₃/U)/3, a^′₂₁ = (a₄/S+ a₅/T+ a₆/U)/3, … so on (for the first column of A_2c). Second column of the above matrix (A_2c) is computed following similar way for the second criteria and so on. Finally score corresponding to each of the alternatives are calculated as shown in Table 7.

(7)\begin{array}{l}{P}^{\text{'}}=\left({{\displaystyle a}}_{\mathrm{11}}^{\text{'}}\times E\right)+\left({{\displaystyle a}}_{\mathrm{12}}^{\text{'}}\times F\right)+\left({{\displaystyle a}}_{\mathrm{13}}^{\text{'}}\times G\right)+\left({{\displaystyle a}}_{\mathrm{14}}^{\text{'}}\times H\right)\hfill \\ Q^{\text{'}}=\left({{\displaystyle a}}_{\mathrm{21}}^{\text{'}}\times E\right)+\left({{\displaystyle a}}_{\mathrm{22}}^{\text{'}}\times F\right)\left({{\displaystyle a}}_{\mathrm{23}}^{\text{'}}\times G\right)+\left({{\displaystyle a}}_{\mathrm{24}}^{\text{'}}\times H\right)\hfill \\ R^{\text{'}}=\left({{\displaystyle a}}_{\mathrm{31}}^{\text{'}}\times E\right)+\left({{\displaystyle a}}_{\mathrm{32}}^{\text{'}}\times F\right)\left({{\displaystyle a}}_{\mathrm{33}}^{\text{'}}\times G\right)+\left({{\displaystyle a}}_{\mathrm{34}}^{\text{'}}\times H\right)\hfill \end{array},

where E, F, G and H in the above equation are the variables defined in Eq. (3). The ranking of preferences for different alternatives are made based on the scores in chronological order assuming P^′ >Q^′ >R^′.

Selection of optimum soil mix proportion based on MADM

A detail flowchart presenting the proposed MADM based approach for selecting optimum soil mix proportion is furnished in Fig. 1. The entire procedure involves laboratory determination of engineering properties of different soil mixes and thereby application of three different MADM algorithms to choose single best possible alternative from a set of mutually conflicting test results. The MADM based approach along with the critical utility of such procedures in practical situations is described in the preceding sections in conjunction with experimental results.

Experimental characterization of soil properties

Particle size distribution and physical properties of the silty sand used in the present experimental investigation are shown in Fig. 2 [²²] and Table 8 respectively. In Table 8 the symbols SM indicate that silty sand (S for Sand and M for Silt or Silty) according to classification as per Indian standard. The physical properties of bentonite soil are given in Table 9. The symbols in Table 9 indicate that the classification of soil according to unified and American Association of State Highway and Transportation Officials (AASHTO). The symbol CL represented that clay soil with low plasticity (C for Clay and L for Low plasticity) according to unified soil classification system and A-6 represented that the typical material of this group is a plastic clay soil 75% or more of which usually passes the 200 micron sieve. The behaviors of materials of this group usually have high volume change between wet and dry states. There are two types of bentonites; namely, swelling-type or sodium bentonite and non-swelling-type or calcium bentonite. Sodium bentonite is usually referred to simply as bentonite, whereas calcium bentonite is called fuller’s earth. The commercial importance of bentonite depends more on its physicochemical properties. The use of sodium type bentonite is quite difficult for any type of construction because of it swelling in nature [²³]. Therefore careful experimental investigation of the properties of bentonite is essential.

The influence of silty sand on the geotechnical characteristics of bentonite clay have been investigated by conducting various laboratory tests, such as standard proctor compaction test, free swelling index, California bearing ratio test and unconfined compressive strength test. The tests have been performed for various combinations of silty sand-bentonite mixtures as presented in Table 10. The total numbers of mix that are used is seven and each sample has been tested three times for ensuring the accuracy of results.

The properties of soil (compressive strength, CBR, etc.) are dependent upon the content of moisture and density at which the soil is compacted. Generally, a high level of compaction in soil enhances these properties of the soil. Thus proper degree of relative compaction is necessary to meet desired properties of soil. Aim of the proctor test is to determine the OMC and MDD of both untreated compacted and treated soil mixtures. In this context, please note that by untreated soil we refer to hundred percent pure soil without any replacement and treated soil is a soil mixture with certain percentage of foreign materials with the virgin soil. In order to obtain these parameters, light compaction test has been conducted for the mentioned mixture proportions as per [²⁶]. The results for OMC and MDD for untreated and treated mixture are shown in Fig. 3. The CBR test has been conducted after four days of sample soaking in water as per [²⁷]. The soaked CBR test is chosen in this investigation program because most of the designs are made based upon the soaked CBR value. The UCS test is used to measure the shearing resistance of cohesive soils (clay, silty clay etc.) which may be undisturbed or remolded specimens. An axial load is applied using either strain-control or stress-control condition. Hence, the test may provide a good measure of the in-situ strength. It has been conducted on silty sand and bentonite soils of different dosage according to [²⁸]. FSI is the increase in volume of a soil, without any external constraints, on submergence in water. This standard covers a test for the determination of free swell index of soil which helps to identify the potential of a soil to swell which might need further detailed investigation regarding swelling and swelling pressures under different field conditions. The free swelling index test was carried out following [²⁹]. Results of the above mentioned tests are furnished in Table 11 for different silty sand- bentonite mixtures.

Variation of MDD, CBR, UCS and FSI with dosage of bentonite is presented in Fig. 4, which reveals that the proportion of bentonite and silty sand in a soil mixture has high sensitivity to different engineering properties of the soil. The MDD, CBR, UCS and FSI values for a particular soil mix are all quite important parameters to determine the physical as well as engineering character of the soil. The experimental results, as presented in Table 11, reveal that M5, M6 and M2 mixes have the maximum value of MDD, CBR and UCS respectively, while M6 mix is the best for FSI point of view. Such confounding outcomes of experimental results often raise a critical question about which mix should be chosen to simultaneously satisfy (best possible alternative) all the requirements in an optimal manner. In this article, we have proposed a novel MADM based approach to choose the best possible alternative (among M2, M5 and M6 in the present study) in this situation.

Selection of soil mix proportion based on MADM approach

In this section, the MADM based approach for selecting best soil mix proportion (refer to Fig. 1) is described in conjunction with the experimental results presented in section 3. As discussed in section 3, there are three different mix proportions (M2, M5 and M6) that are desirable according to different criteria such as MDD, CBR, UCS and FSI. From practical design perspective, it is always essential to arrive at a single and best possible alternative among such non-unique experimental outcomes. We have applied three most powerful MADM algorithms such as SAW, WPM and AHP to choose the best possible alternative among the above mentioned three mix proportions.

Simple additive weighting

As described in section 2.1, SAW method is applied on the experimental outcomes to choose the best possible alternative. Table 12 shows the experimental results for four different selection criteria (MDD, CBR, UCS and FSI) corresponding to the three best possible alternatives and the respective weights, which are determined based on preference factor among the attributes. Direct Weight Elicitation Technique and Rank-Order Centroid method are used to assign the weights. The weights in this study have been assigned using the following equation.

(8){{\displaystyle W}}_j=\mathrm{1}/n{{\displaystyle \Sigma }}_{j=i}^n\mathrm{1}/j

where n is the number of criteria and W_j is the weight for j^th criterion. For example, in the present study, the criterion ranked first, is weighted (1+ 1/2+ 1/3+ 1/4) / 4= 0.52, the second criterion is weighted (1/2+ 1/3+ 1/4) / 4= 0.27, and so on. In order to avoid complexity, weights are rounded off to nearest decimal value.

The SAW scores corresponding to different mixes are furnished in Table 13 and the best possible alternative has been ranked as 1 (refer to Eq. (1)). Thus according to simple additive weighting method M6 is the best mix proportion among the three alternatives.

Weighted product model

For calculating the WPM score corresponding to each of the mix proportion, same preference matrix is used as shown in Table 12. The WPM scores and ranking of different alternatives (calculated following Eq. (2)) are presented in Table 14, wherein it is quite evident that M6 is the best possible soil mix proportion. Please note that the result following WPM method is consistent with the outcomes of SAW method.

Analytic hierarchy process

Analytic hierarchy process is applied further on the experimental outcomes to make a decision among the three mutually conflicting best potential alternatives. The algorithm for AHP described in section 2.3 is numerically applied in context to the present experimental investigation. Saaty’s [²¹] preference scale with numerical ratings corresponding to different degree of preferences is shown in Table 1. A pair-wise comparison matrix A₁ is constructed based on relative importance among each of the two criteria as presented in Table 15. The values assumed in Table 15 are needed to be checked further in a later step of the procedure for their consistency. If the consistency criterion does not get satisfied, these values are needed to be changed and the entire procedure is to be repeated. From Table 15, the matrix A₂ is calculated (refer section 2.3) and thereby other two matrices A₃ and A₄ are calculated as described in Table 16. To check the validity of the assumptions made in Table 15, consistency ratio (CR) is calculated as CR = CI / RI. CI is consistency index (refer Eq. (5), where n is the matrix size of A₁ and l_max is calculated as the average of A₄ matrix, i.e., l_max = 4.1226263). RI is random index that depends on the number of elements being compared, i.e., the size of A₁ matrix. Value of RI can be obtained from Table 4.

In the present analysis CI = 0.0408, as described above and RI = 0.9. Therefore, value of CR becomes 0.0454 (≤0.1). As CR is less than 0.1, the degree of consistency exhibited in the pair wise comparison matrix for comfort is acceptable for further analysis.

In the next step, another pair wise comparison matrix (A_1c) is formed separately for each of the four criteria (as shown in Table 17) based on Saaty’s [²¹] preference scale (Table 1). A_2c matrix in Table 18 is computed from Table 17 by calculating relative priority vector of each matrix as described in section 2.3. Finally score corresponding to each of the alternatives (Table 19) are calculated by matrix multiplication (A_2c × A₂), wherein it is evident that M6 is the best design alternative as per AHP. Thus all the three MADM algorithms suggest that M6 is the best suitable alternative mix designation among the prospective three alternatives on the basis of four different desirable engineering criteria of soil mix.

A novel attempt has been made in this article for soil mix design based on experimental soil characterization in a deterministic framework. However, variability in the input parameters is often associated with experimental characterization of soil properties due to the inherent uncertainty of the systems. Quantification of such uncertainties using probabilistic and non-probabilistic [³⁰–⁵¹] methods has attracted immense attention from different fields of science and engineering in the recent past due to inevitable consequences. Thus future research in conjunction to the present work will definitely follow the path of integrating the effect of such uncertainties with the proposed MADM based approach for optimum soil mix design. The present deterministic investigation will serve as a valuable reference for such studies in future.

Conclusions

A novel multi-attribute decision-making based approach for selecting most suitable soil proportion of silty sand and bentonite soil has been proposed in this article for soil subgrade construction. Pure bentonite (also known as virgin soil) is not suitable for direct civil engineering applications. Thus an attempt has been made to choose optimum proportion of bentonite and silty sand to achieve an ideal soil mix for construction based on four desirable engineering properties such as, maximum dry density, California bearing ratio, unconfined compressive strength and free swelling index. Seven soil mix proportions consisting of silty sand and bentonite soil have been chosen in a systematic manner and the above mentioned four engineering properties have been experimentally determined for each of the mixes. Direct experimental outcomes reveal that three mix proportions can be desirable for four different criteria. However for practical engineering solutions, designers always need to select one best alternative. We have proposed the multi-attribute decision-making based approach relying on a strong mathematical foundation to choose the best alternative among these three preferable soil mix proportions. Three most powerful multi-attribute decision-making algorithms (simple additive weighting, weighted product model and analytic hierarchy process) have been utilized and the results show that the mix proportion containing 90% silty sand plus 10% bentonite soil is the best alternative in the present investigation. Accuracy of the results can be further improved by considering more number of prospective mix proportions at initial stage. However, this will increase the experimental expenses at the same time. The proposed multi-attributedecision making based approach can be extended further for other problems in material mix designs dealing with selection of single most preferable alternative from multiple prospective alternatives.

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[44] Dey S, Mukhopadhyay T, Spickenheuer A, Gohs U, Adhikari S. Uncertainty quantification in natural frequency of composite plates—An Artificial neural network based approach. Advanced Composites Letters, 2016d, 25(2): 43–48

[45] Dey S, Naskar S, Mukhopadhyay T, Gohs U, Spickenheuer A, Bittrich L, Sriramula S, Adhikari S, Heinrich G. Uncertain natural frequency analysis of composite plates including effect of noise—A polynomial neural network approach. Composite Structures, 2016e, 143: 130–142

[46] Dey S, Mukhopadhyay T, Adhikari S. Metamodel based high-fidelity stochastic analysis of composite laminates: A concise review with critical comparative assessment. Composite Structures, 2017a, 171: 227–250

[47] Dey S, Mukhopadhyay T, Naskar S, Dey T K, Chalak H D, Adhikari S. Probabilistic characterization for dynamics and stability of laminated soft core sandwich plates. Journal of Sandwich Structures & Materials, 2016, doi: 10.1177/1099636217694229

[48] Naskar S, Mukhopadhyay T, Sriramula S, Adhikari S. Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties. Composite Structures, 2017, 160: 312–334

[49] Metya S, Mukhopadhyay T, Adhikari S, Bhattacharya G. System reliability analysis of soil slopes with general slip surfaces using multivariate adaptive regression splines. Computers and Geotechnics, 2017, 87: 212–228

[50] Dey S, Mukhopadhyay T, Sahu S K, Adhikari S. Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading. European Journal of Mechanics/A Solids, 2018, 67: 108–122

[51] Mukhopadhyay T, Adhikari S, Batou A. Frequency domain homogenization for the viscoelastic properties of spatially correlated quasi-periodic lattices. International Journal of Mechanical Sciences, 2017, in press. doi: 10.1016/j.ijmecsci.2017.09.004

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