A Reliable Method of Analysis for Geotechnical Data

Corresponding Author: Abdolrasoul Ranjbaran Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran Email: ranjbarn@shirazu.ac.ir: aranjbaran@yahoo.com Abstract: Geotechnical engineering is the art of making decisions in the presence of uncertainty, where real world problems are treated and this is associated with uncertainties arising from various sources. The sources of uncertainty may be divided into uncertainties of nature and uncertainties of mind. The uncertainties of nature are due to variation of encountered phenomena, e.g., the shear strength in a soil. This is reduced by obtaining more reliable data via the results of tests. The uncertainties of mind are related to the modelling, which may be reduced by change of philosophy. Here, the data is considered as variable instead of random so is reliable. Based on logical reasoning and concise mathematics a reliable formulation, i.e., the change of state philosophy which is digested in the Persian curve is proposed. The Persian curve method is free of uncertainties of mind. In the presented paper the Persian curve method is used to manage the variable geotechnical data in a form that can easily be used in practical geotechnical work for decision making and design.


Introduction
Geotechnical engineering is the art of making decisions in the presence of uncertainty. Within this field, real world problems are treated and this is associated with uncertainties arising from various sources. The sources of uncertainty may be divided into uncertainties of nature (aleatory) and uncertainties of mind (epistemic). The uncertainties of nature are due to variation of encountered phenomena, e.g., the shear strength in a soil. This is reduced by obtaining more reliable data via the results of tests. Some of the pioneers in the field of geotechnical engineering, such as Karl Terzaghi, Arthur Casagrande and Ralph Peck, recognized that it was not always feasible to take fully into account all the uncertainties involved in design. Instead they proposed a structured methodology where the evaluation of the probability conditions of quantities involved in design were supplemented by evaluations based on possible unfavorable deviations from these conditions. They called it the experimental or observational method (Peck, 1969;Christian, 2004;Ang and Tang, 2007). Application of probabilistic methods in geotechnical engineering has been increased in recent years. According to (El-Ramly et al., 2002), probabilistic slope stability analysis was one of the first application of reliability-based design in geotechnical engineering and dated back to the 1970. Since then, a lot of work has been put into the uncertainty in soil properties (Lumb, 1974;Orchant et al., 1987;Phoon and Kulhawy, 1999) and the development of probabilistic calculation algorithms (Griffiths and Fenton, 2004;Xu and Low, 2006). Some published papers on the subject of characterization of the geotechnical properties and the stability analysis are in (Zhang et al., 2004;2009;2010;Cao and Wang, 2013;Ching et al., 2010;Wu, 2011). Variability and uncertainty associated with natural variation of properties and inaccuracy caused by lack of information on parameters of models led the field of geotechnical engineering to be an active area of study in the last decade (Müller, 2013;Li, 2014;Johari and Khodaparast, 2015;Allahverdizadeh Sheykhloo, 2015;Maknoon, 2016;Wolebo, 2016;Das, 2016;Ersöz, 2017;Hernvall, 2017;Kanwar, 2018;Singh, 2018;Aladejare and Wang, 2018;Johari and Mousavi, 2019;Obregon and Mitri, 2019;Masoudian et al., 2019;Johari et al., 2020;Tran, 2019). A close insight into the aforementioned and other literature, led the authors to detect a need for further research and remedy. Toward the aim an intensive and extensive research is conducted in the past 20+ years. The result of investigations of Author's Research Team (ART) concluded in a new 277 perspective of the knowledge, the so called Change of State Philosophy (CSP) which is digested in the Persian Curve (PC), where a phenomenon is defined as change in the system (Ranjbaran et al., 2008;Ranjbaran, 2010;Ranjbaran et al., 2011;Ranjbaran, 2012a;2012b;Ranjbaran, 2013;Ranjbaran and Ranjbaran, 2014;Ranjbaran, 2014;2015;Ranjbaran and Ranjbaran, 2016;2017a;2017b;2017c;Ranjbaran et al., 2020a-d;Amirian and Ranjbaran, 2020;Baharvand and Ranjbaran, 2020a;. In the conventional methods of analysis, the system change information is indirectly obtained via solution of governing equations. Consequently, the conventional methods contain epistemic (lack of knowledge) uncertainty. On the other hand, in the (CSP) the system change information is directly obtained by logical reasoning, concise mathematics and reliable data. Consequently the (CSP) is free of epistemic uncertainty. Moreover the (CSP) is a free size method, i.e., it is independent of the size, material, coordinate system and etc. Consequently, it is applicable to all naturalphenomena, in different branches of human knowledge. In the presented paper the (CSP) is applied in analysis of variable data in geotechnical engineering.

Basic Formulation
The traditional formulation in the academic universe is commenced with construction of the governing differential Eq. (1), in which (ψ) is the displacement function, (n) is the core order of derivative and (ψ (n) ) is the core derivative: Next step is solution of the governing equation. The procedures for solution of the governing equation are divided into the stiffness method and the flexibility method. In the stiffness method, the working parameter is selected as a decreasing system parameter called the stiffness. The governing equation for the stiffness method is in Equation (2) and shown in Fig. 1. On the other hand an increasing parameter called the flexibility, which is inverse of the stiffness, is the working parameter of the flexibility method. The governing equation for the flexibility method is in Eq.
(3) and shown in Fig. 2. Equations (2) and (3) are symbolic equations in place of conventional differential and integral equations in the literature. In these equations, (kSS = kS-kC) is the survived stiffness, (fSF = fS + fC) is the survived flexibility, (kS) is the system-stiffness, (kC) is the changestiffness, (fS) is the system-flexibility, (fC) is the changeflexibility, (F) is force and (ψ) is displacement: It is known that, in conventional methods, indirect computation of system parameters contains epistemic uncertainty. The main aim of the presented paper is to remove this uncertainty.
Toward the aim, the (F) and (ψ) are omitted from Eq. (2) and (3) as in Eq. (4), in the first step: In the next step, in view of Eq. (4) the product of equation (kSS = 1/fSF) for the changed state and equation (1/kS = fS) for the intact state is expressed in Eq. (5), as shown in Fig. 3: Equation (5) is rearranged to obtain the (kSS) and (kC), in terms of the other parameters, in Eq. (6): in which the phenomenon functions (collection of the failure function (FR) and the survive function (SR) are defined in Eq. (7): Definition of the dimensionless phenomenon functions in a unit interval, introduced a down to earth method for human knowledge. Therefore, the proposed method is free of the common problems in the conventional methods, such as singularity, instability and etc. Consequently the authors decided to complete the work. In order to complete the formulation, the unknown parameters in Eq. (8) should be explicitly determined: The investigation for explicit definition of the aforementioned functions is continued in the next paragraph via definition and construction of the so called state functions.
Development of a functional (FR and SR) in terms of two functions (fS and fC) is not possible. Therefore, the phenomenon functions are customized for (kS = fS = 1) to define the destination function (D) and the origin function (O), which are collectively called the state functions and the state ratio (R) in Eq. (9). This is an artifice to define functions (D and O) in terms of only one variable (R): Consequently the (D) and (O) are defined in terms of the (R) in Eq. (10): The state functions may be considered as solution of the boundary value problems in Eq. (11), in which (min) and (max) denote minimum and maximum respectively: The (R) with one end in the infinity, as shown in Fig. 4, is not a good working parameter. Moreover, this ratio is itself a function, so it is not wise to be used as an independent variable. Therefor, the state ratio (ξ  [0, 1]) with a zero value (ξ = 0) at the origin and a unit value (ξ = 1) at the destination is defined. In terms of the state variable, the boundary value problems in Eq. (11) is rewritten in Eq. (12): Investigation for construction of solution for boundary value problems in Eq. (12), led the authors to use their experience in structural mechanics, finite element method, mathematics and extensive research . The results are the state functions defined in Eq.
The authors invite the readers from all over the world to propose new (better in some sense) state functions in place of Eq. (13). Now attention is paid to construction of the phenomenon functions. Via the definition of the (kSS) and the (fSF) and the crack compliance (fC) in fracture mechanics (Anderson, 2005), the authors detected a fact that, the (fC) is directly proportional to the (kS)! This detection is called "the Persian Principle of Change (PPC)". In view of this principle the (fC) is defined in Eq. (15): Note that Eq. (15) is an alternative for the whole fracture mechanics (Anderson, 2005)! Insertion of Eq. (15) into Eq. (7) concluded in the general definition for the phenomenon functions in Eq. (16): The (kS) is not explicitly known so it is not a feasible working parameter. Toward better definition, Eq. (16) is rewritten in Eq. (17) in terms of the positive control parameters (aM) and (b) . The flexibility for translation and rotation of phenomenon functions in the (11) working box, which let the experts to enforce their will, is provided by selection of two control parameters from calibration of reliable data: To this end the proposed formulation is mathematically in abstract form. Consequently, it is a universal formulation, in a sense that it is independent of geometry, coordinates, material properties, size and changing agent. Therefore it is equally applies to all natural phenomena.

Persian Curve
As observed, the proposed formulation is derived based on logical reasoning and concise mathematics. Moreover, there was no need for construction of differential and integral equations, which is the paramount basis of the conventional methods of analysis in human knowledge. Consequently, the proposed formulation is reliable and free of epistemic uncertainty, because it is based on obvious and certain basis, for example the definition of flexibility as inverse of stiffness in Eq. (4).
For a given phenomenon, the lifetime is truncated at a workable interval (λ [λO, λT]) and is mapped onto the state variable as in Eq. (18), where (λO) is the origin point (O) and (λT) is the end point of lifetime respectively: In terms of the lifetime, the (FR) is renamed as Persian-Fasa-curve (PF), the (SR) is renamed as Persian-Shiraz-curve (PS) and the two collectively called the Persian curve (PC), defined in Eq. (19), in which (PO) is the ordinate of the origin point (O) and (PT) is the ordinate of the truncated (end) point (T). Note that insertion of (PO = 1 and PT = 0) and (PO = 0 and PT = 1) into Eq. (19) conclude into (FR) and (SR) respectively as in Eq. (17): In comply with the vocabulary of human knowledge, the (PS) is the unified equation for capacity and reliability representing a decreasing data and the (PF) is the unified equation for the probability and fragility representing an increasing data. Moreover, in comply with the common practice in stochastic analysis the (probability) density distribution, here called the Persian-Zahedan-curve (PZ), is defined as the derivative of phenomenon functions with respect to the (ξ), in Eq. (20), in which (FR (1) , SR (1) and D (1) ) are derivatives of (FR, SR and D) with respect to (ξ) respectively. In spite of the paramount role of the density distribution in stochastic theory, it has no role in the (CSP) and it is used only for comparison with the conventional one: Toward determination of control parameters, Eq. (19) is rearranged as in Eq. (21): In view of Eq. (21) and noting that at (M) (DM/OM = 1) then (aM), (aN) and (b), for a reliable decreasing data as shown in Fig. 6 or for a reliable increasing data as shown in Fig. 7 are obtained in terms of the Key Points coordinates (KPS), in Eq. (22), see Appendix:  22). The (PC) solves quite nonlinear problems without any iteration but with simple and accurate artifice. Note that the (PC), similar to a natural phenomenon, is independent of, any coordinate system and any man made principle.

Persian Curve Interpretation
For the case of lifetime as value of real world data (as geotechnical data in this study), the Persian Curves are interpreted as follows. The value of (PS) for a data point (λA) denotes the ratio of number of data (nS) greater than (λA) over the total number of data (nt). It is a weight for data in the upper region. Similarly, the value of (PF) for a data point (λA) denotes the ratio of number of data (nF) less than (λA) over the total number of data (nt). It is a weight for data in the lower region.
For the case of lifetime as a parameter of the system (such as relative slenderness ratio of structures), the Persian Curves are interpreted as follows. The value of (PS) for a data point (λA) denotes efficiency of the system, while the system-deficiency is denoted by the value of (PF).

Analysis of Reliable Data
The (CSP) has great similarity with the probability theory. This is taken for granted to increase the reliability of analysis. The (PF) is equivalent to the Cumulative Distribution Function (CDF) and the (PZ) is equivalent to the Probability Density Function (PDF). In probability theory the (PDF) plays the paramount role. 282 Different probability distributions are classified according to their (PDF's). Determination of the (CDF) from the (PDF) is a difficult process. Moreover limited number of (PDFs) adds to the uncertainty of the results and necessitates the development of new (PDF) for new problems in all branches of human knowledge. That is not the case for the (CSP), where the (PF) as a pseudo (CDF) is directly determined from the reliable data. So it is simple, cheap and accurate. There is no need for the (PDF) at all. It is added for completeness and comparison with the results of the others in the literature.
Within the field of geotechnical engineering, real world problems are treated and this is inevitably associated with uncertainties arising from various sources. The sources of uncertainty may be related to uncertainties of nature and uncertainties of mind, often referred to aleatory and epistemic uncertainties respectively. In view of the logical reasoning leading to the Persian curve, the uncertainty of mind is omitted. There remained the uncertainty of nature which is gradually reduced by providing more reliable data. Therefore in this study the data is treated as varying instead of uncertain. Analysis of data is conducted in the following steps.
Step I: Divide the data between its minimum and maximum, into several intervals (preferably more than 10).
Step II: Determine the frequencies (histogram) for the selected intervals (use the frequency function in Excel software).
Step III: Cumulate the frequencies in Step II to obtain the cumulative data. Scale the data and cumulative data into unit intervals.
Step IV: Select the key points as in Eq. (23) and compute the control parameters as in Eq. (22).
Step V: Compute the Persian curves from Eq. (19) and the density curve from Eq. (20).
In the following examples varying reliable data are managed in a form to prepare for reduction of uncertainties in the art of decision making in practical geotechnical engineering.

Solution
The measured values of live load on the warehouse floor is shown in Table 1a. The corresponding density and the cumulative distribution (cum) are computed and shown in Table 1b The same data is analyzed by (Kanwar, 2018). The proposed formulation is exact and is more than hundred times simpler than that of the (Kanwar, 2018).

Example 2
The data of basalt rock uniaxial compressive strength parameter (S) are taken from (Cui et al., 2017) and shown in Table 2a. Analyze the data by the Persian curve method.

Solution
For the given data, the frequencies and cumulative data (cum) (BASALT) computed, as in Table 2b and shown in Fig. 9. Then the key points are selected as in Eq. (26). The selected key points are used to calculate the control parameters in Eq. (27). Finally, the Persian-Fasacurve (PFR), the Persian-Shiraz-curve (PSR) and the Persian-Zahedan-curve (PZR) are shown in Fig. 9

Example 3
The Nipigon River landslide occurred on the morning of April 23rd 1990, at the north area of the town of Nipigon, Ontario, Canada. The results of 121 corrected un-drained shear strength (V) at the Nipigon river slope is shown in Table 3a (Kanwar, 2018). Analyze the data by the Persian curve method.

Solution
For the given data, the frequencies and the cumulative data (cum) (NIPIG) are computed, as in Table 3b and shown in Fig. 10. Then the key points are selected as in Eq. (28). The selected key points are used for determination of the control parameters as expressed in Eq. (29). Finally the Persian-Fasa-curve (PFR), the Persian-Shiraz-curve (PSR) and the Persian-Zahedancurve (PZR) are shown in Fig. 10. Simplicity and accuracy of the presented formulation for the case of the Nipigon river slope is a good index for verification of the proposed work:

Example 4
The uniaxial compressive strength (S) of rock from an open-pit slope of China (Deng et al., 2004), is shown in Table 4a. Analyze the data by the Persian curve method.

Solution
For the given data, the frequencies and the cumulative data (cum) (CST) are computed as in Table 4b and shown in Fig. 11. The key points are selected as in Eq. (30). The selected key points are used for determination of the control parameters in Eq. (31). Then the Persian-Fasacurve (PFR), the Persian-Shiraz-curve (PSR) and the Persian-Zahedan-curve (PZR) are shown in Fig. 11. In order to see the excellence of the proposed work, the interested reader may refer to Example 1 in chapter 3 of (Singh, 2018) Example 5 The measurement from the field Vane tests (V) (Singh, 2018), is casted in Table 5a. Analyze the data by the Persian curve method.

Solution
The frequencies and the cumulative data (cum) (VST) for the vane shear data are computed as in Table 5b and shown in Fig. 12. Then the key points are selected as in Eq. (32). The selected key points are used for determination of the control parameters in Eq. (33). Finally the Persian-Fasa-curve (PFR), the Persian-Shirazcurve (PSR) and the Persian-Zahedan-curve (PZR) are shown in Fig. 12. In order to see the beauty and excellence of the proposed work, the interested reader may refer to section 3.3.2 of (Singh, 2018)

Example 6
The cohesion data of fine grained alluvial soils at the Paglia River alluvial plain in Central Italy (Di Matteo et al., 2013) is casted in Table 6a. Analyze the data by the Persian curve method.

Solution
The frequencies and the cumulative data (cum) (CPT) for cohesion (C) are computed as in Table 6b and shown in Fig. 13. Then the key points are selected as in Equation (34). The selected key points are used for determination of the control parameters in Equation (35). At this stage the Persian-Fasa-curve (PFR), the Persian-Shiraz-curve (PSR) and the Persian-Zahedan-curve (PZR) are shown in Fig. 13. In order to see the excellence of the proposed work, the interested reader may refer to Example 2 in chapter 3 of (Singh, 2018)

Example 7
The shear angle data of fine grained alluvial soils at the Paglia River alluvial plain in Central Italy, (Di Matteo et al., 2013), is casted in Table 7a. Analyze the data by the Persian curve method.

Conclusion
Geotechnical engineering is the art of making decisions, where the real world problems are treated and this is associated with uncertainties arising from various sources. The sources of uncertainty may be divided into uncertainties of nature (aleatory) and uncertainties of mind (epistemic). The uncertainty of nature is due to variation of encountered phenomena, e.g., the soil properties as shear strength. The uncertainties of mind are related to the modelling and may be reduced by change of philosophy. A close insight into the open literature, led the authors to detect a need for further research and remedy. Toward the aim an intensive and extensive research is conducted in the past 20+ years. The result of investigations of Author's Research Team (ART) concluded in a new perspective of the knowledge, the so called Change of State Philosophy (CSP) which is digested in the Persian Curve (PC), where a phenomenon is defined as change in the system. In the conventional methods of analysis, the system change information is indirectly obtained via solution of governing equations. Consequently, the conventional methods contain epistemic (lack of knowledge) uncertainty. On the other hand, in the (CSP) the system change information is directly obtained by logical reasoning, concise mathematics and reliable data. Consequently the (CSP) is free of epistemic uncertainty. Moreover the (CSP) is a free size method, i.e., it is independent of the size, material, coordinate system and etc. Consequently it is applicable to all natural phenomena, in different branches of human knowledge. In the presented paper the (CSP) is applied in analysis of variable data in geotechnical engineering. Via application of the (PC) to seven set of reliable data the validity of the work is verified.
Substitution of Equation (38) Moreover the control parameter (b) is determined by substitution of Equation (39) and coordinates of the next key point (N) into Equation (19), as in Equation (40) Since the direction of (T to C) and (C to O), for a point (C) on both of the increasing and decreasing data, is the same, then the control parameters (aM and b) are always positive.

aM:
Control parameter at M aN: Control parameter at N A: Sign parameter A = -1: For decreasing data A = +1: For increasing data b: Control parameter (power) CSP: Change of state philosophy