Probabilistic Methods in Structural Engineering - Coursework Example

WEST COAST INSTITUTE OF MANAGEMENT & TECHNOLOGY List of contents Page no Introduction 1 2. Content 2-10 3. Conclusion 10 4. Appendix 11 5. Reference 13 1. Introduction The vast domain of civil engineering and architecture designing also has a significant specialty associated with the factors like risk, reliability, failure, etc., for which, different sub-domains and techniques have been developed over the course of time. Structural engineering is one of them; it deals mainly with the areas of design, evaluation, and analysis of structures, for resisting the pressure and load exerted over their structures by internal and external forces. In this particular regard, probabilistic methods and techniques are seen as vital techniques for providing significant mathematical measures for designing reliable and secure structures, which are more oriented towards maintaining their elevation and integrity, without being distracted or disturbed by external/internal calamities. 2. Content 2.1. Structural Engineering & Systems Structural engineering is sub-domain of engineering, which deals with the evaluation and analysis of large/small structural designs, offering self-sustenance, reliability, and load resistance features. As part of the civil engineering and architecture designing discipline; this particular subject carries a very deep association with the art of mathematical modeling, geometrical designing, proportion-based physical calculations, etc. As can be witnessed practically; the two common factors between all the structural samples are integrity and reliability, which are often achieved through precise mathematical and statistical calculations. In this regard, generally, the structures over which calculations or analysis is performed, are called structural systems (more technically, filtering systems). According to Augusti et al. (1984), these systems are roughly defined by three important aspects, which shape their overall integrity. These aspects include ‘input’ to the system quantities acting over the system, system quantities which are pre-included part of any system, and ‘output’ quantities, which are resulted from a system. With these in hand, an engineer working on these structures (systems) can take necessary actions for analyzing structural properties, and set different regulatory parameters, for the sake of reliability and integrity assurance. For instance, a system (structure) can be considered, which has its foundations on a horizontal plane. This system is getting influenced by external horizontal and vertical forces on fixed and arbitrary upper ends, and is leaning slightly towards one end, which is decided by the vector product of both forces. Within this system, the horizontal and vertical forces can be stated as input quantities, the system features (length, height, etc.) can be taken as the system quantities, while its inclination or bending properties can be termed as output quantities. It is with these quantities and features (in general), that structural engineering and its techniques perform necessary manipulations, so that a balanced structure with precise input and outputs values can be constructed. 2.2. Overview of Associated Risk & Reliability Assessment Aspects like risk of damage, collapse, etc. are always associated with any structure raised on the ground. In the domain of structural engineering, it falls under the category of ‘structural reliability assessment’ – a term coined in late 20th century by scholars and architects like Pugsley (1966) and Freudenthal (1947). Starting off with their work, the science of structural reliability assessment developed itself rapidly encompassing different calculation procedures with mathematical and probabilistic theories in support. In today’s world, there is a huge variety of different structural reliability assessment methods; some of which are considered standard and generalized tests for architecture assessment. In this regard, a significant research has been conducted into the subject of probabilistic risk assessment (PRA) (also known as quantified risk assessment), which was originated from different risk assessment methods used in nuclear and aeronautical industries. Further, with respect to applications; PRA is employed by regulatory bodies operating in the industry, dealing with the assessment of risks involved in different structural installations. In this regard (of PRA), there are some major structural design assessment procedures, which are followed globally in order to achieve the factors such as ‘structural safety’ (or structural reliability). These procedures generally involve following: 1. Pre-defined ratios or characteristic values for materials. 2. Presence of safety factors and limiting ratios, associated with the used materials. 3. Presence of precise quantitative figures associated with concerned design philosophy or orientation, been applied on the employed materials. 4. Capacity assessment equations associated with the use of specific materials in particular designs. At this stage, in this study, some probabilistic design methodologies commonly employed by current world of civil engineering and structural designing are to be included first, before the analysis of reliability assessment procedures. 2.3. Probabilistic Design Methodologies in Structural Engineering In the regard of probabilistic design, there are two basic design methodologies: 2.3.1. Allowable/Permissible Stress Design Methodology Allowable/permissible stress design methodology is one of the most popular elastic design techniques which have been used worldwide in different structures and architectures. In this design format; the design code or standard equation used is: Where, Sc = effect of load or stress present in the design due to applied design loading. Rc = specified design component resistance for any failure, associated with the specified yield strength of the material. SF = safety factor related to uncertainty in calculated resistance, load, analysis method, etc. In this format, the yield stress is factorized for keeping the cumulative stress in the linear elastic material as elastic as it can be; therefore, this methodology is sometimes referred as elastic theory design methodology. Further, in this particular design, the load factor of the structure is not separately applied and the total load of design is kept equal to the characteristic load of the structure. Also, the safety factor is integrated implicitly within the format formula (as can be seen in the above formula) instead of being specified as a separate explicit entity. Usually, in this type of structural design methodology, structure needs to be checked for a number of different combinations of loading. This is because of a fact that load (over the structure) may vary with the progression of time and with the changing environmental aspects (cold, storm, earthquake, heat, etc.), and structural properties dealing with these factors should be reliable enough to carry through. 2.3.2. Partial Factor Design Methodology Partial factor design methodology employs different partial factors associated with the collective or individual resistances offered by several materials incorporated within the structure; along with the partial factors offered by the load types included within the structure. Although, every individual structure carries distinct and unique load characteristics but on a general scale, four major load categories can be made: 1. Enduring load (e.g. gravity) 2. Current load (e.g. pressure, temperature, moisture, etc.) 3. Dynamic load (e.g. sudden shocks, architectural swing, etc.) 4. Ecological load (e.g. snow, wind, etc.) In this regard, the partial factor design methodology reflects the uncertainty level linked with both constancy and variation in the magnitudes of materials involved; both on individual and combined levels. A formal format of partial factor design methodology can be presented as: Where, Rd = component design resistance (failure mode) Sd = stress on the component γi = load factor L1 = load type based on characteristic loading γ = component resistance factor Rk = nominal resistance The basic advantage associated with the partial factor methodology is reflection of structural uncertainty in both strength and loading terms instead of a single safety factor. On the other hand, the limitation associated with it is in the form of format confusion, since design load effect reflected in the above stated format formula is quite ambiguous in its nature. Further, in some types of structures, the load effects have to be analyzed, evaluated, and determined by dividing the loads associated with different materials, with in a structure. This particular aspect is missing in the partial factor design methodology. 2.4. Structural Analysis through Probabilistic Methods Modern architectural theory suggests the probabilistic analysis of structures made in the built-environment. These probabilistic methods are surrounded by a philosophical approach of calculating all the risks associated with the structure, and determining its reliability and integrity through these methods. According to Ang & Tang (1984); these probabilistic methods, in general, follow some basic steps to ensure the reliability and integrity features of any structure. These steps include: 1. Identification of all significant failure modes (of any structure or operations) and defining them with respect to their probability of occurrence (i.e. P(F) = X%). 2. Standardizing a failure step or level – the level at which the structure should be perceived as failed. 3. Identification of all the failure originating sources (events, physical and non-physical forces, etc.) included within and outside the surrounding of the structure. 4. Calculation of overall structural and component reliability, through assessment of risk associated with overall structure, and components used within. 5. Evaluation of collected risk evidences (i.e. either genuine or not). Actually, each technique of probabilistic risk assessment finds its mathematical foundations with the ‘probability of failure’ equation, outlined by Ang & Tang (1984) and Augusti et al. (1984) as: This particular equation finds its analytical solution in some cases; however, on practical basis, using this equation again results in non-deterministic results, which are the very reason that his mathematical presentation of failure probability is not utilized in its original form (Ang & Tang, 1984). Therefore, there are number of other probabilistic techniques (which are actually based on the equation above) to state the probability of failure associated with the components or structure in which they are employed. Given below is a brief description of each probabilistic method, along with the probabilistic items used, respectively: 2.4.1. Mean Value Estimation: In this technique of probabilistic failure (structural) assessment, a failure function is considered which can be approximated by using the combination of uncertainties of two variables: Z = R – S Where, Z = Failure Function R = Uncertainty in structural resistance S = Uncertainty in structural loading The above defined dependent variables (R and S), if are assumed and calculated to be independent in nature, then the reliability index (β) can be evaluated as: Where, E [ ] = Expected mean value Var [ ] = Expected variance value Further, if both variables (R and S) are considered (or assumed to be) log-normally distributed, then the reliability index can be estimated as: Where, VR and VS can be stated as the coefficients of variation in the variables. Further, a complete and accurate formula derived from the above two formulas can be made as: The main limitation associated with mean value estimation techniques is of invariance i.e. they are not invariant to formulate the failure function. For instance, the failure function should be defined Z = 0 (as a surface), therefore, there should be a ground to transform the failure function in any form, for example, in logarithmic states like this: Z = R – S Z’ = loge (R) – loge (S) In the above stated example, however, the mean reliability value estimate is not observed equal, therefore; the technique do not results in invariant estimates. 2.4.2. First Order Reliability Methods - FORM To overcome this invariability (invariance) difficulty inherited in the mean value estimation technique with the failure function, it was realized that it is obligatory to change (transform) the basic variables into normal variables, which are independent and standard in their nature. In this manner, the defined space reflected by independent standard normal variable is termed as U – space, while X – space is designated to serve as the basic variable space. Further, the transformation activity of independent variables can be guaranteed from the cumulative probability of the distribution, i.e. from the collective characteristics. Actually, the first order technique involves estimating the probability of failure by linearizing the variable of failure surface at nearest point to origin, in a standard normal space, or U-space. Although the failure surface in standardized normal space is not usually planar at curvature, therefore, the point near origin surface is so small and standard that the first-order linearization rarely do not works. Figure 1: Transformation from basic variable space to U-space and first-order reliability estimate 2.4.3. Second Order Reliability Methods (SORM): The second order reliability methods are just used to fine tune the accuracy of first order methodical estimates. This is done by including curvature information to the beta point, followed by the approximation of the failure surface by a quadratic surface. In general, the basic difference between first and second order reliability methods is in the form of probabilistic estimation i.e. estimation of probability to give indication of curvature of the failure surface. 2.4.4. Monte Carlo Simulation Method Monte Carlo simulation method presents an easy method to estimate the probability of failure associated with a component or the structural system. In general, this technique asserts including a set of values for sampling the basic variables (at random) from the probability density function, and to observe the occurrence of failures through evaluation of failure function. In this manner, a large number of samples are generated, through which the probability density is calculated through simulation, and the ratio between number of samples and total number sample trends is observed for obtaining the exact value of failure probability. In this method, the basic limitation involved is the number of samples which are needed to be generated, since this number may go to million or above, which often leads to unavoidable complexity in the analysis and evaluation. However, there are number of techniques present, which are often used by structural engineers to reduce the number of samples associated with any failure probability analysis. 2.5. Applications of Probabilistic Methods in Structural Engineering As discussed above, probabilistic methods of failure, risk, and reliability analysis find their applications in many of the structural designing methodologies, and operations. These methods have been used significantly for years, in order to assess the commercial risk involved within the structures, and are still employed by structural engineers at all levels of architecture designing and civil engineering. Below is the presentation of few major applications, according to the scope of this study: 2.5.1. Applications in Safety factor Calibration This is by far the most versatile and common application of probabilistic methods of reliability, failure, and risk assessment. In this application, the design of major structures or bridges is analyzed where the economic losses or loss of life would be significant. Further, where a limit state code is introduced as a direct replacement to an existing working stress code the choice of the target reliability is relatively straightforward, provided that the existing code is considered to produce designs with acceptable reliability and economy. The target reliability is then derived by defining the objective of calibration (calibration classes), selecting a set of structural components for the reflection of component code and assembly, designing of a WSD, calculating the probability of failure (component and structural), and finally evaluating the reliability through the failure ratio analysis (Augusti et al., 1984). 2.5.2. Applications in General Probabilistic Designing The general probabilistic designing involves three basic levels i.e. level 1, 2, and 3. At level 1, a rough estimate regarding the probabilistic aspects of structure is made, which helps in the development of a limit state design. Further, at the second level, each component used in the structure (on selective basis) is evaluated with respect to its geometrical and proportional properties, leading to a probabilistic estimate of component reliability, and load bearing abilities at micro level. This is, though, a hectic process, but finds its significance in almost all the structural engineering projects. Finally, at the third level, a probabilistic model of entire model, with all the probabilistic details integrated within, is prepared. In this regard, probabilistic methods are usually or extensively used at second and third levels, since main difficulty is the choice of appropriate target reliabilities for the various limit states, which seems to be fulfilled if probabilistic methods of reliability assessment are applied at the last two levels. 3. Conclusion In this study, presented is some basic information regarding the discipline of structural engineering, its historical significance, and particular usage of mathematical and probabilistic methods in its domain. According to the scope of this study, there are a lot of applications of probabilistic methods in structural engineering; all of them backed up by a philosophy that, in order to calculate the probability of occurrence of an event associated with the structure, there needs to be a calculated account of each and every component involved in the structure, along with the design on which the structure is raised. This, along with several other aspects, presents an idea of how reliable, integrated, secure, and protected is the structure in its basis. In this regard, there are few probabilistic design methodologies (like allowable/permissible stress design methodology and partial factor design methodology) present in the literature, which are again analyzed and evaluated for the involvement of failures through different probabilistic methods like FORM, SORM, mean value estimation, and Monte Carlo simulation techniques. Further, these techniques find their mere applications in structural engineering processes like safety factor calibration, probabilistic designing, etc. Appendix: This appendix section is dedicated for the pictorial representation of FORM and SORM methods: In the diagram above, it can be seen that both diagrams represent SORM and FORM, and in both diagrams, it can be observed that beta function is limited by the progression of E (or S in this study). Further, it can also be noted in the second diagram that when the load effect and resistance are made statistically independent variables with arbitrary distribution, the exact probability of failure is obtained by the determining the probability of integration. This depiction has been made in the above diagram, where, dx represents the point of interjection of E (load factor) and R (resistance). References Augusti, G., Baratta, A. & Casciati, F. (1984): Probabilistic Methods in Structural Engineering, Taylor & Francis. Ditlevsen, O. & Madsen, O.H. (1996): Structural Reliability Methods, John Wiley & Sons Inc. Pugsley, A.G. (1966): The safety of structures journal, Arnold, London. Freudenthal, A.M. (1947): The safety of structures journal, ASCE Transactions (112). Ang, A.H.S. & Tang, W.H. (1984): Probability concepts in engineering planning and design decision, risk and reliability, John Wiley & Sons. Melchers, R.E. (1999): Structural reliability, analysis and prediction, Ellis Horwood, Chichester. Read More