Building Design Rule for Glass Structures by Persian Curve

Corresponding Author: Abdolrasoul Ranjbaran Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran Email: ranjbarn@shirazu.ac.ir aranjbaran@yahoo.com Abstract: Glass is being increasingly used as a structural material. Its favorable aesthetic qualities have made it popular with modern designers. Recently glass is used for major structural elements such as beams and columns in modern and innovative architectural applications. High slenderness ratios and brittle behavior in tension contribute to the complexity of stability and ultimate strength analysis of glass structures. Moreover the available design curves and methods of ultimate strength analysis of glass structure contain epistemic uncertainty, therefore are not reliable. In this study, based on logical reasoning, concise mathematics and reliable data, a reliable method called the change of state philosophy is developed and expressed in the Persian curve. The Persian curve is used for reliable analysis of glass structures. The validity of the work is verified via its sound basic formulation and comparison of the results with those of the others in the literature.


Introduction
Glass is being increasingly used as a structural material in particular its favorable aesthetic qualities have made it popular with modern designers. The most recent developments have seen glass being used as major structural elements such as beams and columns. From the engineering viewpoint these new applications presents a series of design problems which need to be addressed before a coherent and safe design philosophy can be achieved. Porter (2011) used a fracture mechanics approach for remedy. Feldmann and Longosch (2010) focused on the results of buckling with monolithic sections of heat strengthened and tempered glass, giving buckling curves proposed by derivations and experiments. The buckling of laminated glass composite beams in compression is investigated in (Amadio and Bedon, 2011). They used a simple analytical model developed on the basis of Newmark's theory of composite beams with deformable connections. Finally, simple design criteria have been proposed to prevent buckling failure. Ouwerkerk (2011) focused on further knowledge and understanding of the structural aspects specifically related to structural glass columns and, on the basis of these findings, to design glass column as structural element in a pavilion. An exploratory study to the design aspects of glass columns was performed by doing experiments. The load carrying behavior of in plane compress and in plane shear laminated glass elements was investigated by (Amadio and Bedon, 2012;2013) respectively. Bedon and Amadio (2015), proposed a Eurocode-based approach for buckling verification of glass columns and beams. Based on experimental and numerical results from the literature, buckling curves are proposed for laminated glass beams and columns composed of up to three glass layers. The stability of laminated glass beams and columns up to five layers, subjected to compressive load, mid-span loads, uniformly distributed loads, four-point bending and pure bending or torsion, was investigated by (Costa MSLM, 2015). The shear buckling response and actual resistance of structural glass walls with non-ideal restraints has been assessed by means of extended finite element method in (Bedon and Amadio, 2017). Bedon and Amadio (2017), proposed and analytical formulation for the resistance verification of a structural element under ultimate state variable loads. The method is assessed toward three existing analytical formulations. Evaluation of lateral-torsional buckling resistance and actual behavior of the beams due to absence of standards for design of glass load-bearing structures is done by (Pešek and Melcher, 2018). Experimental results were compared with lateral torsional buckling resistance calculated according to the buckling curve approach. Analysis and design of glass structural members is an active area of study in this days (Bedon and Amadio, 2018;Bedon, 2019;Bedon et al., 2019;Koca, 2019;Santo et al., 2020).
Current state of the art of the methods for analysis of structural glass members, is based on the so called Perry-Robertson method, in which the ultimate strength is obtained via analysis of a member with initial lateral displacement. Selection of the shape and the value of initial displacement along with using the linear governing equation, introduced epistemic (lack of knowledge) uncertainty to the work. The epistemic uncertainty can only be removed by change of philosophy behind the work.

Basic Formulation
In this section, the basic formulation for the change of state philosophy via a sound logical basis, is developed. All parameters are managed to be nonnegative. The traditional formulations in the academic universe are divided into the stiffness method and the flexibility method. The stiffness method, is based on the change of a decreasing system parameter, called stiffness, as shown in Fig. 1. While the flexibility method makes use of flexibility (inverse of stiffness as an increasing parameter) as shown in Fig. 2. Near the end of the phenomenon, the stiffness become very small and the flexibility become very large and hence introduce uncertainty into the work. As shown in Fig. 3 near (ξ = 1) the stiffness is small and the flexibility is big, so that the systems such as computers are not able to detect their value and fail. The governing equations for these two methods is expressed in Eq. (1) and (2), in which (kSS = kS-kC) is the survived-stiffness, (fSF = fS + fC) is the survived-flexibility, (kS) is the system-stiffness, (fS) is the system flexibility, (kC) is the change-stiffness, (fC) is the change-flexibility, (F) is the applied load and () is the resultant system displacement. In conventional methods the (kSS) and (fSF) and one of (F) or () are assumed to be known and the other is the result of analysis. The symbolic Eq. (1) and (2) are representative of all conventional differential and integral equations: Computation of the parameters that are assumed to be known and the result of analysis introduce epistemic uncertainty into the work. This is the main shortcoming of all conventional methods of analysis in different branches of human knowledge. This shortcoming is removed, in the change of state philosophy , as follows. The first step was deletion of (F) and (), from the conventional equations, as in Eq. (3). Then, in view of Eq. (3), the product of equalities (kSS = 1/fSF) for the changed state and (1/kS = fS) for the intact state, as shown in Fig. 3, is expressed in Equation (4): Equation (4) is rearranged to obtain the (kSS) and (kC), in terms of the other parameters, in Eq. (5): in which the phenomenon functions (collection of the failure function (FR) and the survive function (SR)) are defined in Eq. (6): Consideration of the fact that the dimensionless phenomenon functions are defined in a unit interval, introduces a down to earth method for human knowledge. Therefore the methods of analyzes based on these functions are free of common problems, such as singularity, instability and etc., that are available in the conventional methods in human knowledge. Consequently the authors decided to continue the investigation via looking for the correct and soft definition of the (fC) and other unknown functions.
The proposed form of phenomenon functions in Eq. (6) are defined in terms of the (kC) and the (fC), which are unknown to this end. The investigation for explicit definition of these functions is continued in the next paragraph via definition and construction of the so called state functions.
Development of functionals (SR and FR) in terms of two functions (fS and fC) is not possible. Then the phenomenon functions are customized for (kS = fS = 1) to define the destination function (D), the origin function (O) (which are collectively called the state functions) and the state Ratio (R) in Eq. (7). This is an artifice to define functions (D&O) in terms of only one variable (R): Consequently the (D&O) are defined in terms of the (R) in Equation (8): The state functions may be considered as the solution of the boundary value problems as expressed in Equation (9) and shown in Fig. 4, where (min) denotes minimum and (max) denotes maximum: The state ratio, with the far end in the infinity (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. Therefore, the state variable [0 1] with a zero value ( = 0) at the origin and a unit value ( = 1) at the destination is defined. In term of the state variable, the boundary value problems in Eq. (9) is rewritten as in Eq. (10): Investigation for construction of solution for boundary value problems in Eq. (10), led the authors to make use of their experience in structural mechanics, finite element method, mathematics and their extensive research. The results are the state functions as defined in Eq. (11) Equation (11) is an average of polynomial and trigonometric functions as defined in Eq. (12) The authors invite the readers from all over the world to propose new functions for the state functions and report to them. The authors cordially appreciates their efforts.
Via the definition of the survived stiffness (kSS) and the survived flexibility (fSF) and study of the equations for computation of crack compliance in fracture mechanics where cracking is considered as the change (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. (13): The (kS) is not explicitly known and so it is not a feasible working parameter. Therefore Equation (14) is rewritten in a unified form as in Eq. (15), in terms of the positive control parameters (aM) and (b) . The control parameters are to be determined from calibration of reliable data: To this end the proposed formulation is mathematically in an abstract form, so it is a universal one in the sense that it is independent of geometrical and material properties and the changing agent. Therefore, it applies to all natural phenomena!

Persian Curves
As demonstrated in previous sections, the proposed formulation, were derived based on logical reasoning and concise mathematical logics. There was no need for construction and solution of differential and/or integral equations, which is the paramount basis of the conventional methods of analysis in human knowledge. Therefore the proposed formulation is 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. (3).

For a given phenomenon, the lifetime is truncated at a workable interval ([o T]) and is mapped onto the state variable in Eq. (16), where (o) is the origin point (O) and (T) is the end point (T) of the lifetime:
As functions of the lifetime, the (FR) is renamed as Persian-Fasa-curve (PF) and the (SR) is renamed as Persian-Shiraz-curve (PS) and the two collectively called the Persian-curves (PC). In comply with vocabulary of human knowledge, the (PS) is the unified equation for the capacity and reliability and the (PF) is the unified equation for the probability and fragility. The reliable (capacity and reliability) and the reliable (probability and fragility) data are managed in decreasing and increasing order respectively. For a reliable data, (PS) for increasing data and (PF) for decreasing data, is defined as the Persian-curves (PC) in Eq. (17) 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 the phenomenon functions with respect to the state variable (), in Eq.
are the derivatives of (FR, SR and D) with respect to the () respectively. Note that, in spite of the paramount role of the probability density in conventional probability theory, in the (CSP) it has no permanent role and is only used for comparison with the conventional one: For a reliable decreasing data as shown in Fig. 6 and for a reliable increasing data as shown in Fig. 7, the positive control parameters (aM) and (b) are obtained, in terms of the coordinates of the Key-Points (KPS), in Eq. (19), see the Appendix: The  Eq. (19). Finally, via logical reasoning, a unified equation for capacity (design) and fragility (assessment) curve is proposed as follows. Extensive research of the author's team  concluded into the fact that the Persian curve for complete failure is defined in Eq. (21) (We looked for reliable capacity data in different codes, theses, dissertations, papers and reports . Analysis of these data shown that the values of (aM = 2 and b = 1) satisfied all requirements. Then substitution of the (aM = 2 and b = 1) into Eq. (17) and (18) Persian curve is managed as a bridge between the abstract mathematical concepts and the real world. Therefore it is applied for stability analysis of glass structures in the following section. Perhaps these are the most beautiful (simple, explicit and accurate) equations in the literature. Here is the end of the proposed logical formulation. Note that the question ((kC) and (fC) = ?) in Equation (4) is logically answered.

Persian Curve Interpretation
For the case of lifetime as value of real world data, 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).

Persian-Curves for Glass Structure Capacity
The proposed Persian-curves are used in load bearing capacity of glass structures as follows. In comply with the literature the verification of the work is done via comparison of the results with that of the others in the following examples. As explained in previous sections, the unified Persian curves from Equation (21) is to be used for determination of the ultimate strength of structural members. The capacity curve (PSU), the fragility curve (PFU) and the density distribution (PZU) for glass members are compared with the results of experiments in the following examples, where (ξ = λ/3). The data is scanned from the addressed references.
Example 1 Feldmann and Langosch (2010), conducted tests for buckling capacity of glass planes under axial loading. The buckling loads of column specimens without eccentricity (FELD), is compared with the unified Persian curves in Fig. 8. Close agreement of the results verified the work.
Example 2 Amadio and Bedon (2012) focused on the load carrying behavior of in plane compressed laminated glass elements. They proposed some analytical formulation and verified their results via comparison of the results with the results of test selected from open literature. Their numerical and experimental results, for lateral torsional buckling of beams in compression (AMAD) is scanned and compared with the unified Persian curves in Fig. 9. Excellent agreement of the results is a flag for reliability of the unified Persian curves.

Example 3
Monolithic and laminated glass elements, are typically brittle and slender, thus frequently subjected to buckling phenomena. Amadio and Bedon (2013) focused on the load carrying behavior of 2-layyer and 3-layyer simply supported laminated glass panels subjected to in-plane shear load. They verified their work via comparison with the numerical analysis and with the experimental results. The numerical and the test results (AMAD) for a 2-layyer laminated glass panel under in plane shear load is compared with the unified Persian curves in Fig. 10. The results expressed the validity of Persian curves for ultimate strength analysis of laminated panels under in plane shear load.

Example 4
Structural glass is frequently used in modern buildings. The verification of load-carrying elements composed of traditional materials, analytical procedures and consolidated verification methods are available in standards, this is not the case for glass. A Eurocode based approach is proposed for the buckling verification of glass columns and beams in (Bedon and Amadio, 2015). They used numerical and experimental results for verification of their proposed method. The experimental and numerical results for flexural torsional buckling load for monolithic and laminated glass beams (BEDON) is compared with the unified Persian curves in Fig. 11. The feasibility of the unified Persian curve as a design curve for glass structure is apparent from this example.

Example 5
Pešek and Melcher (2018), conducted experimental research of stability behavior of laminated structural glass beams. The purpose of their research was evaluation of lateral-torsional buckling resistance and actual behavior of beams due to absence of standards for design of glass load bearing structures. Their experimental results (PESEK) is compared with the unified Persian curves in Fig. 12.
At a point in the loaded system the released Energy Density (ER) over the available Energy Density (EA) is equal to (R = D/O) (Ranjbaran et al., 2020c). Then at (ξ = 0.5) the (ER = EA) and the system become unstable. That is a reason for better agreement between the theory and experiments for (ξ ≥0.5).

Conclusion
The following conclusions are obtained from the present study.
The author's research team conducted an intensive research in the past two decades on analysis of natural phenomena. Via logical reasoning and concise mathematics they proposed the so called change of state philosophy which is expressed in the Persian curve. The Persian curve is then used to derive the universal capacity design curve for all structures. The proposed method is simple, explicit and exact and needs no nonlinear analysis. The validity of the work is verified via concise mathematics and comparison of the results with the results of experiments for stability analysis of structural glass members.
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.

:
Lifetime parameter

O:
Lifetime origin