DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS
STOCKHOLM, SWEDEN 2018
Design of an Airport Lodging Option using a Reliability
Method
A Conceptual Framework HARIANAS DEWANG
KTH ROYAL INSTITUTE OF TECHNOLOGY
Design of an Airport Lodging Option using a Reliability
Method
A Conceptual Framework
HARIANAS DEWANG
Degree Projects in Optimization and Systems Theory (30 ECTS credits) Degree Programme in Aerospace Engineering (120 credits)
KTH Royal Institute of Technology year 2018 Supervisor at Globuzzer AB: Rami Alfasfos Supervisor at KTH: Per Enqvist
Examiner at KTH: Per Enqvist
TRITA-SCI-GRU 2018:402 MAT-E 2018:85
Royal Institute of Technology School of Engineering Sciences KTH SCI
SE-100 44 Stockholm, Sweden URL: www.kth.se/sci
Design of an Airport Lodging Option using a Reliability Method – A Conceptual Framework
Abstract
Engineering products ranging from simple components to complex systems should be designed optimally with high level of reliability. The challenge here is to ensure that manufacturing costs are reduced while achieving requirements for safety and reliability. Simulation based design plays an important role in designing almost any kind of aerospace and consumer products under the competitive conditions. Single discipline simulations used for analysis are coupled together to create complex coupled simulation tools. This investigation focuses on the safety analysis for first order reliability method in a simulation-based design environment.
Design av ett alternativ för logi flygplatser med en tillförlitlighetsmetod – Ett konceptuellt ramverk
Sammanfattning
Tekniska produkter från enkla komponenter till komplexa system bör utformas optimalt med hög tillförlitlighet. Utmaningen här är att säkerställa att tillverkningskostnaderna sänks samtidigt som krav på säkerhet och tillförlitlighet uppnås. Simulationsbaserad design spelar en viktig roll vid utformningen av nästan alla slags flyg och konsumentprodukter under konkurrensförhållandena.
Enkla disciplin-simuleringar som används för analys kopplas samman för att skapa komplexa kopplade simuleringsverktyg. Denna undersökning fokuserar på säkerhetsanalysen för första ordningens tillförlitlighet i en simuleringsbaserad designmiljö.
Table of Contents
Abstract 2
Sammanfattning 3
Table of Contents 4
Table of Figure 5
Chapter 1: Introduction 6
1.1 The Need of Airport Lodging Option 6
1.2 The Safety Problem 8
1.3 Aim and Scope 10
1.4 Structure of the Thesis 11
Chapter 2: First Order Reliability Method (FORM) 12
2.1 Reliability Method 12
2.2 First Order Reliability Method (FORM) 12
2.3 Algorithm to Find MPP 19
2.4 Inverse First Order Reliability Method 22
Chapter 3: Simulation Architecture 25
3.1 Structure of the Reliability Method Assessment 25
3.2 Review of FERUM Architecture 26
3.3 Finite Element Method 27
3.4 Lodging Pod Capsule Setup 28
Chapter 4: Result and Analysis 30
4.1 Reliability Analysis Framework 30
4.2 Structure Simplification 32
4.3 Description of the FORM Program 33
4.3.1 Validation of the Reliability Program 34
4.4 Main Reliability Computation 37
Chapter 5: Conclusion 42
5.1 Summary and Conclusions 43
5.2 Recommendation for Future Work 43
References 44
Table of Figure
Figure 1.1 The design of the proposed airport lodging option 6
Figure 1.2 The Limit State Concept 8
Figure 2.1 Probability integration 12
Figure 2.2 Probability Integration in X-Space 13
Figure 2.3 Highest Probability Density at the MPP 16
Figure 2.4 MPP is a tangent point 17
Figure 2.5 The MPP search procedure 18
Figure 2.6 The flowchart of the MPP search 20
Figure 2.7 The inverse MPP search 21
Figure 2.8 The flowchart of the MPP search for inverse FORM 23 Figure 3.1 The overall structure of Reliability Method assessment 24
Figure 3.2 The structure of FERUM program 25
Figure 3.3 The FE stress analysis of the Lodging Pod 27
Figure 3.4a The original model of the pod 27
Figure 3.4b The simplified pressure vessel model 27
Figure 4.1 Initial framework of Reliability Method 29
Figure 4.2 Boundary load of the long axisymmetric 30
Figure 4.3 Simplified framework involving analytical formulation 31
Figure 4.4 Simplified lodging pod structure 32
Figure 4.5 The Simple Cantilever Beam Problem 34
Figure 4.6 Reliability value to Young’s Modulus function 38 Figure 4.7 Reliability value to shell thickness function 39
Chapter 1 Introduction
This chapter introduces the motivation, aim and scope as well the organization of the thesis work presented.
1.1 The Need of Airport Lodging Option
In recent years, global air travels have seen fluctuated in use. The industry has expanded over the past 20 years, driven by economic growth, failing real prices of air travel and increases in international business activity and leisure time [1]. This increasing growth in air transport propagates into the need for providing more-efficient airport terminal services that can stimulate demand-led growth. The ever-growing aviation industry requires an additional feature at the airport terminal to be planned, designed and constructed in a way that should allow flexible operating conditions.
With the increase of global air travel, there is also an increase in the long-haul flight demand. It is the nature of long-haul flight that the switching of different airlines can be the norm in this type of flight as it requires more intense preparation by the airline. This raises the problem why air flight can be very comprehensive with many variations of international layovers. Moreover, switching between different airlines at the connecting airport will then require longer layover. One layover duration at the connecting airport can be as extensive as 10 hours per passenger. In this matter, airport as a transfer place of thousands of passengers every day must provide a set of facility that maximizes the passenger’s comfort of long layover.
To overcome this fast-growing demand, airport terminals go through frequent transformations to accommodate technological advancements as well as changes in regulations. One proposed solution is the design implementation of an airport lodging option as one of the airport terminal services. With the expanding number of air travel by an average 7% per annum [2], there is an opportunity for growth in lodging services across the vast-majority of airports.
The importance of developing a lodging option for airport passenger has long been recognized by the airport and airline community. Numerous lodging solution in nowadays’
Figure 1.1 The design of the proposed airport lodging option
airport are still plagued by high cost provided by different private companies. Glancing at the conventional solution like airport motel; oftentimes it is quite far from the terminal building hence concerning the accessibility issue to the passenger.
The other main problem is when the traveler is being burdened by high rental cost of the available options. In the end, it’s up to the passenger to select between resting improperly at the airport lounge (which will introduce ethical-social problems) and need to pay expensive for the airport hotel service. Perhaps surprisingly, some airport officials are not totally supportive of the airport sleeping idea.
Moreover, the current cabin option nowadays still faces some sustainable and efficiency issue. Some cabins do not utilize optimal shape design and space available, making less efficient space usage. The solution now is to develop a new conceptual lodging capsule design as a visionary solution that is buildable, affordable and maximizes space available.
A number of researchers [3] identified that incorporating flexibility in airport waiting lounges will help reducing the risk of high cost of change, both financial and material, and will reduce uncertainties in adopting new technologies.
This airport lodging option as a new solution is embracing interior design that appeals to digitally connected and style-savvy travelers from around the world, instead of the bare- bones aesthetic as found in the conventional capsule hotels. The lodging capsule will use
the efficiency of space, convenience, technological advancement, interior and architectural design, all to enhance the overall travelling experience for the passengers.
Equipped with multiple-duty furniture inside, each capsule room is fundable and deliverable without interrupting day-to-day operation and maintenance. Variety of models, tools and systems were proposed to highlight the importance on providing comfort to the airport passenger, where ultimate objective is to facilitate passenger’s waiting time and as a resting spot in between the long hours of waiting.
1.2 The Safety Problem
Modern products ranging from simple components to complex systems should be designed to be optimal and reliable. The challenge of modern engineering is to ensure that manufacturing cost are reduced while achieving requirements for performance and standard reliability. Increasingly, the modern engineering community is employing optimization as a tool for design. Optimization is used to find optimal designs characterized by lower cost while satisfying performance requirements. The basic paradigm in design optimization is to find a set of design variables that optimizes an objective function while satisfying the performance constraints [4].
The most basic criteria in the design of a structure is to make sure that the strength of the structure is greater than the impact of the loads applied. However, uncertainties are inherent in design variables and parameters such as material properties, loading and geometry parameters. It is necessary to consider all these type of uncertainties in the design of any engineering system to assure reliability and quality. Traditionally, these uncertainties have been considered through partial safety factors in structural optimization methods. The purpose of reliability method is then to incorporate the information on uncertainty into actual design problems.
The reliability of an engineering design is the probability that it meets certain demands under certain conditions [5]. In other words, reliability can be depicted as the probability that an item will perform appropriately for a specified time period under a given service condition. For the stability of a structure, it should be designed such that it satisfies certain demands towards any external load. The bearing capacity (R) of the material should exceed the total external load (S) acting on it, for the structure to be stable. This can be represented as 𝑅 > 𝑆, and mathematically expressed as
𝑍 = 𝑅 − 𝑆. (1.1)
Here 𝑍 is the performance function or limit state function of the structure. This function differentiates the unsafe and safe zones with respect to R and S. This example has two stochastic variables R and S. This equation can be generalized as [6]
𝑍 = 𝑔(𝑥).
where 𝑔(𝑥) constitutes the 𝑛 basic variables 𝐱 = (𝑥., 𝑥0, … , 𝑥2) of the performance function. Given its name, the performance function is a measure of the performance of any structure. Like any mathematical equation, the performance function could have three outcomes as the following:
• 𝑔(𝑥) > 0: Safe region
• 𝑔(𝑥) = 0: Limit state region
• 𝑔(𝑥) < 0: Failure region
Figure 1.2 The Limit State Concept
(1.2)
In Figure 1.2, the curve is the performance function. The region to the right of the curve is unsafe where 𝑔(𝑥) < 0 while the region to the left of the curve is the safe region 𝑔(𝑥) >
0. The boundary or the curve represents the combination of the variables that are on the verge of failure (at the limit state).
1.3 Aim and Scope
The objective of this master thesis is the standard safety validation of the proposed airport lodging pod design with reliability method. The 3D Model is created, and its standard geometry and material properties become the baseline parameter and input to the reliability method computation.
Once the finite element (FE) computation of the 3D model is executed, we gather all necessary parameters and state limit functions to begin the reliability computation.
However, a broader aspect of the thesis work was recognized at an early stage, that there was a gap to reduce the computational load in the overall system architecture. A proposed solution to simplify the 3D model modifies the FE analysis into an analytical calculation. In this way, the cumbersome FE analysis can be bypassed and replaced with a simple formulation to get the state limit function for the reliability analysis. Finally, the results from reliability method from the new design are commented upon.
The purpose of these analysis is to gain more insights in how to determine the standard reliability index of a given model. In addition to that, one objective is to evaluate the FERUM program as a tool in reliability assessment that is commonly used in the system engineering field. A simple example for validation is provided in Chapter 4.
The current research proposes a design framework to develop an airport lodging pod option as well as to inspect its safety reliability parameter. As the thesis is a joint-program of Mechanical Engineering and System Engineering Department at KTH, the proposed conceptual framework then brings the following two particular fields of knowledge together:
• Reliability analysis from System Engineering point of view
• Product design and 3D modelling from Mechanical Engineering point of view.
1.4 Structure of the Thesis
This thesis is organized as follows: The present chapter deals with the motivation and organization of the research work presented herein. Chapter 2 consists of a brief concept of Reliability Method which becomes the basic concept in the Reliability FERUM program.
Chapter 3 consists of a brief overview of the Reliability Method assessment; description of the FERUM simulation setup, Finite element program and the 3D model setup. Chapter 4 is set aside for presentation of the model results, including discussion of the results and a new proposed solution to tackle the reliability problem. Chapter 5 consists of a summary of the conclusion we have drawn regarding this subject and a few suggestions on further research which can be made in this area.
Chapter 2
First Order Reliability Method (FORM)
This chapter introduces reliability theory that becomes the baseline theory in the reliability assessment using the FERUM Matlab program within this thesis work. Both First Order Reliability Method (FORM) and Inverse FORM are presented.
2.1 Reliability Method
Reliability is defined as the probability of a performance function 𝑔(𝐗) greater than zero, 𝑃{𝑔(𝐗) > 0}. In other words, reliability is the probability that the random variables 𝐗 = (𝑋., 𝑋0, … , 𝑋2) are in the safe region that is defined by 𝑔(𝐗) > 0. The probability of failure is defined as the probability 𝑃{𝑔(𝐗) < 0}, or that it is the probability that the random variables 𝐗 are in the failure region that is defined by 𝑔(𝐗) < 0. If the joint pdf of 𝐗 is 𝑓;(𝐱), the probability of failure is evaluated with the integral [8]
𝑝= = 𝑃{𝑔(𝐗) < 0} = ? 𝑓;(𝐱)
@(;)AB
𝑑𝐱.
The reliability is computed by [8]
𝑅 = 1 − 𝑝= = 𝑃{𝑔(𝐗) > 0} = ? 𝑓;(𝐱)
@(;)EB
𝑑𝐱.
In this chapter, the most commonly used reliability analysis methods, the First Order Reliability Method (FORM) will be presented. The basic idea of the methods is to ease the computational difficulties through simplifying the integrand 𝑓;(𝐱) and approximating the performance function 𝑔(𝐗). With the simplification and approximation, solutions to the probability of failure and the reliability value will be easily obtained.
2.2 First Order Reliability Method (FORM)
The name of the First Order Reliability Method (FORM) comes from the fact that the performance function 𝑔(𝐗) is approximated by the first order Taylor expansion
(2.1)
(2.2)
Figure 2.1 Probability Integration [8]
(linearization) [7]. The probability integrations in Equations (2.1) and (2.2) are visualized with a two-dimensional case in Figure 2.1. The figure shows the joint pdf 𝑓;(𝐱) and its contours, which are projections of the surface of 𝑓;(𝐱) on 𝑋.- 𝑋0 plane. All the points on the contours have the same values of 𝑓;(𝐱) or the same probability density. The integration boundary 𝑔(𝐗) = 0 is also plotted on 𝑋.- 𝑋0 plane.
The probability integration in Equation (2.1) or (2.2) is the volume underneath the plane (hyperplane for higher than 2D problems) of the joint pdf 𝑓;(𝐱) in the failure region 𝑔(𝐗) <
0 or the safe region 𝑔(𝐗) > 0. As the surface of the integrand 𝑓;(𝐱) forms a ‘hill’, a partition of the hill by a curve of 𝑔(𝐗) = 0 will divide the region into two parts. Figure 2.1 depicts the left part of the region with 𝑔(𝐗) > 0. The volume on the left side is the probability integration in Equation (2.2), which represents the reliability. In other words, the reliability is the volume underneath 𝑓;(𝐱) on the side of safe region 𝑔(𝐗) > 0. Vice
Figure 2.2 Probability Integration in X-Space[8]
versa, the probability of failure will be the volume underneath 𝑓;(𝐱) on the side of failure region 𝑔(𝐗) < 0, the removed part.
Direct evaluation of the probability integration in Equation (2.1) and (2.2) is extremely difficult [8]. Since a number of random variables 𝐗are involved, the probability integration is multidimensional. The dimensionality is typically high especially for engineering applications. Moreover, the integrand 𝑓;(𝐱) is the joint pdf of 𝐗and is generally a nonlinear multidimensional function. The integration boundary 𝑔(𝐗) = 0 is also multidimensional and usually a nonlinear function.
In many engineering applications, 𝑔(𝐗) is a black-box model (or simulation model), and the evaluation of 𝑔(𝐗) is computationally expensive [8]. Examples of black-box models include finite element analysis, dynamic simulation, and computational fluid dynamics.
Because of this complexity, there is rarely an analytical solution to the probability integration, except for very special cases. It is also unpractical using numerical integration to find the solution due to the high dimensionality in most engineering applications. To this
end, approximation methods, such as the First Order Reliability Method (FORM) has been developed in the area of structural reliability.
Two steps are involved in these approximation methods to make the probability integration easy to be computed. The first step is to simplify the integrand 𝑓;(𝐱) so that its contours become more regular and symmetric. The second step is then to approximate the integration boundary 𝑔(𝐗) = 0. After the two steps, an analytical solution to the probability integration will be easily found. We will discuss the FORM procedure as described below.
Step One – Simplify the integrand
The simplification is achieved through transforming the random variables from their respective original random space into a standard normal space. A probabilistic transformation 𝐔 = 𝑇(𝐗, 𝐏) from the original space of physical random variables (𝐗, 𝐏) to the normalized space 𝐔 is needed. Usually Nataf transformation or Rosenblatt transformation are used. To make the shape of the integrand 𝑓;(𝐱) regular, all random variables 𝐗 = (𝑋., 𝑋0, … , 𝑋2) must be transformed from X-space to a standard normal space, where the transformed random variables 𝐔 = (𝑈., 𝑈0, … , 𝑈2) will follow the standard normal distribution (the transformed space is named as U-space).
The transformation from 𝐗 to 𝐔 is based on the condition that the cdfs of the random variables remain the same before and after the transformation. This type of transformation is called Rosenblatt transformation which expressed by
𝐹;K(𝑥L) = 𝜙(𝑢L)
in which 𝜙(⋅) is the cdf of the standard normal distribution. The transformed standard normal variable is then given by [8]
𝑈L = 𝜙P.Q𝐹;K(𝑋L)R.
After the transformation, the probability integration becomes
𝑝= = P{𝑔(𝐔) < 0} = ? 𝜙T(𝐮)
@(T)AB
𝑑𝐮
where 𝜙V(𝐮) is the joint pdf of 𝐔. Since all the random variables are independent, the joint pdf is the product of the individual pdfs of standard normal distribution and is then given by [8]
(2.3)
(2.4)
(2.5)
𝜙V(𝐮) = W 1
√2𝜋exp ^−1 2𝑢L0_
2
L`.
.
Therefore, the probability integration becomes
𝑝== a W 1
√2𝜋exp ^−1
2𝑢L0_ 𝑑𝑢.𝑑𝑢0… 𝑑𝑢2.
2
L`.
@(bc, bd,…, be)AB
After the transformation, the integration in Equation (2.7) in U-space is identical to that in Equation (2.1) in X-space without any loss of accuracy, but the contours of the integrand 𝜙T become concentric circles (or hyperspheres for a higher dimensional problem). At this point, the integrand 𝜙T is easier to be integrated.
Step Two – Approximate the integration boundary
In order to further make the probability integration easier to be evaluated, in addition to simplifying the shape of the integrand, the integration boundary 𝑔(𝐔) = 0 will also be approximated. FORM uses a linear approximation (the first order Taylor expansion) as the following.
𝑔(𝐔) ≈ 𝐿(𝐔) = 𝑔(𝐮∗) + ∇𝑔(𝐔 − 𝐮∗)k
where 𝐿(𝐔) is the linearized performance function, 𝐮∗ = (𝑢.∗, 𝑢0∗, … , 𝑢2∗) is the expansion point, 𝑇 stands for a transpose, and ∇𝑔(𝐮∗) is the gradient of 𝑔(𝐔) at 𝐮∗. ∇𝑔(𝐮∗) is given by
∇𝑔(𝐮∗) = l𝜕𝑔(𝐔)
𝜕𝑈. ,𝜕𝑔(𝐔)
𝜕𝑈0 , … ,𝜕𝑔(𝐔)
𝜕𝑈2 no
b∗
.
It is natural to expand the performance function 𝑔(𝐔) at a point that has the highest contribution to the probability integration. In another words, it is preferable to expand the function at the point with the largest value of the integrand, namely the highest probability density. With the integration going away from the expansion point, the integrand function values quickly diminished. The point that has the highest probability density on the performance function will be approximated at the MPP. Maximizing the joint pdf 𝜙V(𝐮) at (2.6)
(2.7)
(2.8)
(2.9)
the limit state of 𝑔(𝐔) = 0 gives the location of the Most Probable Point. The mathematical model for locating the MPP is then given by [11]
p
minb ‖𝐮‖
𝑠. 𝑡. 𝑔(𝐔) = 0
where ‖⋅‖ represents the norm (length or magnitude) of a vector, namely [11]
‖𝐮‖ = w𝑢.0+ 𝑢00+ ⋯ + 𝑢20 = y 𝑢L0
2
L`.
.
The solution of this optimization problem 𝐮∗ is the minimum distance of a point 𝐮 on the failure surface 𝐺(𝐝, 𝐮) = 0 from the origin of the standard normal space and is called the Most Probable Point (MPP) or 𝛽-point, as 𝛽 = ‖𝐮∗‖. The solution to the model is the MPP and is denoted by 𝐮∗ = (𝑢.∗, 𝑢0∗, … , 𝑢2∗). As shown graphically, the MPP is the shortest distance point from the limit state 𝑔(𝐔) = 0 to the origin O in U-space. The minimum distance 𝛽 = ‖𝐮∗‖ is called reliability index.
Figure 2.3 Highest Probability Density at the MPP [8]
(2.10)
(2.11)
If we let the vector of 𝛼L be [11]
𝐚 = (𝛼., 𝛼0, … , 𝛼2) = 𝛻𝑔(𝐮∗)
‖𝛻𝑔(𝐮∗)‖. The probability of failure can also be written as
𝑝= ≈ ϕ •y 𝛼L𝑢L∗
2
L`.
‚ = ϕ(𝐚𝐮∗k)
in which 𝐚𝐮∗k is the inner (dot) product of the unit vector 𝐚and the vector of the MPP𝐮∗. As shown in Figure 2.4, since the MPP 𝐮∗ is the shortest distance point from the origin to the performance function curve 𝑔(𝐔) = 0, the MPP is the tangent point of the curve 𝑔(𝐔) = 0 and the circle with the radius of 𝛽. Therefore, the MPP vector 𝐮∗ is perpendicular to the curve 𝑔(𝐔) = 0. The direction of the MPP can be represented by the
Figure 2.4 MPP is a tangent point [11]
unit vector 𝐮∗/‖𝐮∗‖ = 𝐮∗/𝛽 . On the other hand, the direction of the gradient is also perpendicular to the curve at the MPP, and its direction can be represented by the unit vector 𝐚. Therefore,
𝐮∗ 𝛽 = 𝐚
(2.12)
(2.13)
(2.14)
or,
𝐮∗= −𝛽𝐚.
Therefore, the probability of failure is evaluated by
𝑝= = 𝑃{𝐿(𝐔) < 0} = 𝜙(𝛽).
As a summary, the procedure of the FORM is as follows.
1. Rosenblatt transformation of the original random variables from X-space to U-space.
2. Search of the MPP in U-space and Calculation of the reliability index 𝛽.
3. Calculation of Reliability 𝑅 2.3 Algorithm to Find MPP
From the above discussion, it is noted that the key in calculating the probability of failure (and reliability) lies on the MPP location in U-space. Since it is very difficult and almost impossible to solve the MPP search model in Equation (2.10) analytically, many numerical methods have been developed for the MPP search. Here in the following section author introduces MPP search algorithm that will be later used as the foundation to FERUM computation.
The MPP search algorithm uses a recursive formula and is based on the linearization of the performance function. The procedure is demonstrated in Figure 2.5.
Figure 2.5 The MPP search procedure [11]
(2.15)
(2.16)
Let the MPP in kth iteration be 𝐮„. The linearized function is given by 𝑔(𝐮) = 𝑔(𝐮„) + ∇𝑔(𝐮„)(𝐮 − 𝐮„)k.
Let the linearized function be zero, then MPP 𝐮„…. in the next iteration will be on the queue, namely,
𝑔(𝐮„….) = 𝑔(𝐮„) + ∇𝑔(𝐮„)(𝐮„….− 𝐮„)k = 0.
We also know that
𝐮„ = −𝛽„𝐚„
As shown in Fig 8, since 𝐮„….is the shortest distance point from the origin to the line, vector 𝐮„….is perpendicular to the line and is directed from the origin O to 𝐮„….. The unit vector of the gradient 𝐚„ is also perpendicular to the line and in the opposite direction to 𝐮„….. Because the magnitude of 𝐮„…. is the distance from the origin to 𝐮„…. (the reliability index), then
𝐮„….= −𝛽„….𝐚„. Substituting 𝐮„ and 𝐮„…. yields,
𝑔(𝐮„) + ∇𝑔(𝐮„)(𝐚„)k(𝛽„− 𝛽„….) = 𝑔(𝐮„) + ‖∇𝑔(𝐮„)‖(𝛽„− 𝛽„….) = 0 Rearranging gives,
𝛽„…. = 𝛽„+ 𝑔(𝐮„)
‖∇𝑔(𝐮„)‖
Therefore, the updated point is given by
𝐮„…. = −𝐚„†𝛽„+ 𝑔(𝐮„)
‖∇𝑔(𝐮„)‖‡
To use these recursive formulas of 𝛽„…. and 𝐮„…., a starting point 𝐮B is required. Usually, the origin 𝐮B = 0 is set as the starting point. The algorithm runs until |𝛽„….− 𝛽„| ≤ 𝜀 to stop the recursion (𝜀 is a very small quantity). The flowchart of the MPP search is shown as follows.
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
Figure 2.6 The flowchart of the MPP search
The above MPP search algorithm is easy to use and program, hence becomes the basic idea behind the open source program Reliability Method with Matlab that has been used during the thesis work. Due to its good feature, it is widely used in the field of structural reliability and probabilistic engineering design.
Input starting point
𝐮 = 𝐮B, 𝛽 = |𝐮|
𝐚 = 𝛻𝑔(𝐮)
‖𝛻𝑔(𝐮)‖
𝐮2‹Œ = −𝐚𝛽
|𝛽2‹Œ− 𝛽| ≤ 𝜀 𝐮 = 𝐮2‹Œ
𝛽 = 𝛽2‹Œ
Stop No
Yes 𝛽•Ž• = 𝛽 + 𝛻𝑔(𝐮)
‖𝛻𝑔(𝐮)‖
𝐮B
2.4 Inverse First Order Reliability Method (Inverse FORM)
The following section describes how to estimate p-percentile value 𝑔• using FORM.
The evaluation of the percentile value of the performance function is an inverse problem of the reliability analysis. The problem can be stated as: Find the p-percentile value 𝑔• given the probability
𝑃{𝑔(𝐗) < 𝑔•} = 𝑝.
The above equation indicates that the probability that the performance function is less than the p-percentile value 𝑔• is equal to p.
To make use of FORM algorithm author has discussed, let a new function be 𝑔‘(𝐗) = 𝑔(𝐗) − 𝑔•.
And the MPP for 𝑃{𝑔′(𝐗) < 𝑔•} = 𝑃{𝑔(𝐗) < 𝑔•} be 𝐮∗. From FORM, if the probability p is known, the reliability index is given by
𝛽 = |𝜙P.(𝑝)|.
Since the reliability index is a distance (always non-negative), the absolute value is used in the above equation. As illustrated in Fig 15, the MPP 𝐮∗is a tangent point of the circle with radius 𝛽 and the performance function 𝑔‘(𝐗) = 𝑔(𝐗) − 𝑔•= 0 is also a point that has the minimum value of 𝑔(𝐗) on the circle. Therefore, the MPP search for an inverse reliability analysis problem becomes ‘To find the minimum value of 𝑔(𝐗) on the 𝛽-circle (or 𝛽- hypersphere for higher dimensional problems)’.
Figure 2.7 The inverse MPP search [11]
(2.24)
(2.25)
(2.26)
The mathematical model for the MPP search is then stated as: Find the MPP 𝐮∗ where the performance function 𝑔(𝐔) is minimized while 𝐮∗ remains on the surface of the 𝛽-circle.
p
min b 𝑔(𝐔) 𝑠. 𝑡. ‖𝐮‖ = 𝛽
Since at the MPP, 𝐮∗ = − ‖𝐮∗‖𝐚∗ = −𝛽𝐚∗, in the kth iteration 𝐮„….= −𝛽𝐚„
Where
𝐚„ = 𝛻𝑔(𝐮„)
‖𝛻𝑔(𝐮„)‖
The algorithm runs until |𝐮„….− 𝐮„| ≤ 𝜀 to stop the recursion (𝜀 is a very small quantity).
The MPP search algorithm for the inverse reliability problem has the same features as the MPP search algorithm in the last section.
After the MPP 𝐮∗ is found, the p – percentile value 𝑔• is calculated at the MPP as 𝑔• = 𝑔(𝐮∗)
The flowchart of the MPP search for the reverse reliability problem is drawn in Figure 2.8 It has been discussed how to estimate the reliability, which is defined as the probability that a performance function (performance) is safe. The most commonly used reliability analysis method FORM has been discussed.
Reliability analysis methods are used to evaluate a specific probability at the limit state, they are not intended for generating a complete distribution of a performance function or its statistical moments. However, if different values that cover the range of distribution of the performance function are used as limit states, FORM is applicable to generating the whole distribution of the performance function.
Since FORM approximates a performance function at the MPP, the accuracy of the methods depends upon how accurate the approximated performance function is in U- space. If the performance function in U-space is close to a linear function when FORM is used, this method will produce accurate reliability estimations. If the performance function (2.27)
(2.28)
(2.29)
(2.30)
is highly nonlinear in U-space, FORM may generate a larger error. Even though a performance function is close to linear in X-space, it may become highly nonlinear in U- space because the normal to non-normal transformation from X-space to U-space is nonlinear. Only under very special cases, for example, the random variables are normally distributed, the transformation is linear.
As for efficiency, FORM is efficient since it uses only the first order derivative. It is noted that a single reliability analysis needs to perform a number of deterministic analysis on the performance function for the MPP search. For many engineering problems, a performance function is expensive to evaluate, and no analytical derivative exists. When the derivative has to be evaluated numerically, the computational effort will be approximated proportional to the number of random variables. In such case, FORM may not be applicable for large scale problem, and an approach such as Monte Carlo simulation will be more suitable for such large problem.
Figure 2.8 The flowchart of the MPP search for inverse FORM Input starting point, 𝛽
𝐮 = 𝐮B
𝐮2‹Œ = a𝛽
|𝐮„….− 𝐮„| ≤ 𝜀 𝐮 = 𝐮2‹Œ
Stop No
Yes 𝐮B, 𝛽
𝐚„ = 𝛻𝑔(𝐮„)
‖𝛻𝑔(𝐮„)‖
Chapter 3
Simulation Architecture
This chapter introduces the general structure of the thesis work by explaining the different simulation architecture and its setup from both Finite Element Analysis (FEA) and Reliability Method computation. The FEA analysis is done through Autodesk Fusion 360 and the reliability computation is done through FERUM Matlab Program.
3.1 Structure of the Reliability Method Assessment
A selection of Reliability assessments have been implemented for the course of this thesis work. The work is based of initial Finite Element (FE) computation in the first loop and the Reliability computation on the outer loop.
For the FE computation, analysis is done through a built-in feature inside Autodesk Fusion 360 after the 3D model of the lodging pod capsule has been exported. General input such as geometry and dimension, material properties as well as load and disturbance are taken into account. Finally, the FE computation yields outputs (subsequent state limit functions) such as maximum displacement and stresses, which will be further iterated into the next part of the reliability computation.
Reliability computation analysis is done with the Finite Element Reliability Using Matlab (FERUM) program. Taking the input from the previous FE computation, the program outputs the reliability index (β) as one of the results. This number will then be compared to the target β-value and if any difference (error) exists, several iterations are made until the target reliability index is achieved. Figure 3.1 clearly depicts this computation scheme.
Figure 3.1 The overall structure of Reliability Method assessment
3.2 Review of FERUM Architecture
The Finite Element Reliability Using Matlab (FERUM) software is a collection of Matlab functions used to run Reliability Analysis. The first version of this toolbox was released in 1999 at the Structural Engineering Mechanics and Material Division of the Department of Civil & Environmental Engineering in the University of California at Berkeley (UCB). Last release from UCB is the version 3.1 in 2002 [12]. A few years later, a new version of the FERUM code has been released. This version is based on the work carried out at the Institut Français de Mécanique Avancée (IFMA) in Clermont-Ferrand, France and is named FERUM 4.1. This software includes additional functions and improved capabilities [13].
In the present work a function of First Order Reliability Method has been tested to carry out the Reliability Method computation. First Order Reliability Method (FORM) Matlab Program is a method of estimating the reliability index and failure function approximated by its linear form. The FERUM Program allows user to estimate invariant reliability index irrespective to the limit state function (safety margin) formulation. This index is also known as the Hasofer and Lind reliability index [14]. Stochastic variables that constructed limit state function are transformed to the standard normal stochastic variables. Based on these variables, the limit state function is transformed to the standardized domain.
In standardized domain, the reliability index is defined as the shortest distance from the origin to the limit state function, and this solution is termed as a design point in standardized domain. To find the shortest distance, the numerical differentiation with iterative algorithm is applied [15]. This method is incorporated by ‘ReliabilityByFORM.m’.
The probability of failure is estimated approximately using the reliability index by: 𝑃 ≈ 𝜙(−𝛽).
Figure 3.2 The structure of FERUM program
Within Matlab code, the Inverse FORM is based on the FORM logic with some modification to incorporate the search direction [15]. Both analytical and structural problems can be solved. An example of a simple structural beam load is included to verify the overall FERUM results. All the Reliability Methods implemented have been applied and the results have been analyzed.
3.3 Finite Element Method
The finite element method in this thesis work mainly deals with the validation of the analytical solution of the pressure vessel problem. Both the results from analytical pressure vessel problem and the FE analysis from the same structure under the same disturbance model will be compared and commented upon.
The original idea was to perform the analysis with a finite element software analysis to obtain the state limit function. As previously described, since this approach is computationally heavy to map different scenario case when other iteration needs to be performed, an analytical solution to the problem is chosen to proceed with the FE simulation. In this way the FE calculation can be replaced with a simple yet representative design model of a pressure vessel problem.
The original FE model of the lodging pod is more accurate than the simplified model because of its comprehensive characterization and geometry, but it requires higher computational resources. For optimization purposes it is non-viable to use the original FE model because a very long process is needed to converge to the optimum solution. For this reason, a simplified model was proposed to adjust a less accurate but more computationally-feasible model, the latter is used in the optimization procedure of FORM as stated previously in this work.
Figure 3.3 The FE stress analysis of the Lodging Pod
3.4 Lodging Pod Capsule Setup
The methodology described in previous subchapter is applied to the real simplified case of the 3D lodging pod capsule. A side view of the finite element model representing the design region is shown in Figure 3.4. The model has R = 0.5 m cylindrical radius and variable thickness h throughout the design with mean of 0.5 cm and standard deviation of 0.05 cm.
The external disturbance is constrained to affect the external sides of the capsule with equally distributed pressure.
Figure 3.4a The original model of the pod Figure 3.4b The simplified pressure vessel model
The maximal deflection of the capsule is set to be 0.01 cm ± 0.001 cm as this is the standard safety of pressure vessel design [16]. This number then becomes the state limit function and acts as one of the input to the FERUM program.
Chapter 4
Result and Analysis
This chapter deals with the simplification of the problem and its proposed solution using a proven analytical framework of the similar ellipsoidal pressure-vessel problem. On the whole it gives the particular Reliability method approach using First Order Reliability Method.
4.1 Reliability Analysis Framework
Ideally, the Finite Element (FE) analysis is used to give the subsequent state limit function of the lodging pod model under certain properties and external disturbances. However, a broader aspect of this framework was recognized at an early stage. The author recognizes that within each iteration to find the designated reliability value, there must be a slight alteration in the FE computation. While the result of involving FE in the analysis this way might be exact, it is rather cumbersome and computationally heavy.
With this reasoning, author concludes that It is not efficient to create a function from continuous iteration of FE software to Matlab interlinking. The efficiency here concerns more in the impractical software usability, where FE each iteration of Reliability Method requires another complicated FE computation inside the loop before passing the result to the Matlab Reliability computation.
The author then finds another solution to bypass the complicated FE analysis by replacing this method with a far simpler analytical formulation of the similar ellipsoidal pressure- vessel problem[17].
Figure 4.1 Initial framework of Reliability Method
Far simpler analytical procedure for calculating pressure vessel strength can be developed on the basis of approximate solutions of axisymmetric shell theory equation system by Baličević et all [17].
Using this theory, we get formulas for internal forces and displacements that occur due to boundary effect in the walls of ellipsoidal and cylindrical part of the vessel. The three important formulas are stated as follows.
𝑢” = — ˜ ™• –d (1 + 𝛾) 𝑒P; cos x (Displacement function) 𝑀 = −¡ ¢•
£d (1 + 𝛾) 𝑒P; sin x (Bending moment function) 𝐹” = •√0
¡ ¢£ (1 + 𝛾) 𝑒P; cos (x +¤—) (Normal force distribution) where 𝑥 = ∫ 𝛽B¦ B 𝑑𝑠, and 𝛽B = ª§¨ (.P©√– « d).
Moreover, 𝑃 is the external disturbance pressure, E is the Young’s Modulus and h is the shell thickness. Using this analytical formulation allows author to replace the FE computational block with a simplified formulation and each iteration now can be performed inside Matlab. The simplified framework becomes as the following.
Figure 4.2 Boundary load of the long axisymmetric
(4.1) (4.2) (4.3)
Figure 4.3 Simplified framework involving analytical formulation
This framework allows users to analyze the particular state limit function parameter to be computed (element displacement, bending moment or internal force). Applying this analytical framework of the ellipsoidal pressure vessel problem requires some criterion to be met.
1. External pressure has the biggest influence on the amount of stress, all other types of loads are considered to be less important
2. At membrane stress state, it is assumed that only normal forces occur in the wall, while shear forces and bending moments are not considered.
3. In the design phase, pressure vessels should be formed so that the real stress is approximately the same as the membrane state in order to avoid bending of the walls and high stresses due to bending.
4.2 Structure Simplification
In the following subchapter, we elaborate the global simplification goals by proposing a number of related design simplification, which cover the many aspects of simplicity after considering the analytical feasibility through ellipsoidal pressure-vessel problem. This simplification allows author to focus particularly on the reliability analysis more and also maintain an overview of the standard safety requirements.
Because we are mainly dealing with complex system, some of the components must be simplified in the lodging pod structure to make light computational load and put more emphasis on the reliability analysis. Continuing the explanations described in the previous chapter, some of the simplification and assumption are to be made. Below, author briefly presets each simplification assumption that’s being considered.
1. The lodging pod has zero stress concentration factor (sharp edges such as door and other hollow parts are removed).
2. Material properties are uniform throughout. This further implies that the mechanical properties, such as Young’s Modulus, Tensile properties, and safety factors have the same characteristic throughout the structure.
3. With negligible wall thickness and the curvature radius ratio, the structure satisfies the general criteria for a pressure vessel with ellipsoidal heads problem.
Figure 4.4 Simplified lodging pod structure
4.3 Description of the FORM Program
First Order Reliability Method (FORM) is a method of estimating the reliability index and failure function approximated by its linear form. FORM allows estimating invariant reliability index irrespective to the limit state function (safety margin) formulation. This index is also known as the Hasofer and Lind reliability index [14]. Stochastic variables that construct limit state function are transformed to the standard normal stochastic variables.
Based on these variables, the limit state function is transformed to standardized domain.
In standardized domain, the reliability index is defined as the shortest distance from the origin to the limit state function, and this solution is termed the design point in standardized domain.
As previously described in Chapter 3, the reliability program that becomes the point of interest is FERUM Software. This program allows users to estimate reliability index and failure function approximated by First Order Reliability Method (FORM). Prior to any analysis being made using this program, a simple beam problem is to be considered. Similar to the original problem being solved, a reliability computation is calculated analytically and
its results are to be compared within the same configuration of the numerical Matlab computation.
To find the shortest distance, the numerical differentiation with iterative algorithm is applied, this method is incorporated by ‘ReliabilityByFORM.p’. The probability of failure is estimated approximately using the reliability index by: 𝑃®¯–° ≈ ϕ(−𝛽®¯–°). Moreover, inverse FORM is based on the FORM logic with some modification to incorporate search direction. Output of the algorithm is a parameter under the interest. If more than one parameter under interest was set up (indicating multiple solutions), then the program estimates the shortest distance point in parameter space and reports those values.
4.3.1 Validation of the Reliability Program
This verification and validation process is intended to discover the relative error between the analytical and numerical computation using Matlab, as early as possible during the development process. This section describes a framework that showcases a simple beam problem, which its solution to the reliability problem can be computed/tracked manually each step.
Analytical computation
A cantilever beam is illustrated in Fig 4.5. One of the failure modes is that the tip displacement exceeds the allowable value, 𝐷B. The performance function is the difference between 𝐷B and the tip displacement, and the function is given by
𝑔 = 𝐷B− 4𝐿¨ 𝐸𝑤𝑡µ^𝑃¶
𝑡0_
0
+ ^𝑃; 𝑤0_
0
where 𝐷B = 3”, 𝐸 = 30 x 106 psi is the modulus of elasticity, 𝐿 = 100” is the length, 𝑤 and 𝑡 are width and height of the cross section, respectively, and 𝑤 = 2” and 𝑡 = 4”. 𝑃; and 𝑃¶ are external forces with normal distributions 𝑃; ~ 𝑁(500, 50) lb and 𝑃¶ ~ 𝑁(1000, 100) lb.
(4.4)
Figure 4.5 The Simple Cantilever Beam Problem
The probability of failure is defined as the probability of the allowable value less than the tip displacement, i.e.
𝑝== 𝑃 p𝑔 = 𝐷B− 4𝐿¨ 𝐸𝑤𝑡µ^𝑃¶
𝑡0_
0
+ ^𝑃; 𝑤0_
0
≤ 0¹.
First, the normally distributed variables 𝑃; and 𝑃¶ are transformed into the standard normal variables
𝐔 = º𝑈;, 𝑈¶» = l𝑃; − 𝜇•½
𝜎•½ ,𝑃¶− 𝜇•¿ 𝜎•½ n or
𝐗 = º𝑃;, 𝑃¶» = À𝜇•½+ 𝑈;𝜎•½ , 𝜇•¿+ 𝑈¶𝜎•¿Á.
The transformed performance function in U-space becomes
𝑔 = 𝐷B− 4𝐿¨
𝐸𝑤𝑡µl 𝜇•¿ + 𝑈¶𝜎•¿
𝑡0 n
0
+ ^𝜇•½+ 𝑈;𝜎•½
𝑤0 _
0
.
The gradient of 𝑔(𝐔) is given by
∇𝑔(𝐔)
= 4𝐿¨ 𝐸𝑤𝑡
⎝
⎜⎜
⎛ (𝜇;+ 𝑈;𝜎;)𝜎;
𝑤—µ^ 𝜇¶+ 𝑈¶𝜎¶
𝑡0 _
0
+ À𝜇; + 𝑈;𝜎; 𝑤0 Á0
, (𝜇¶+ 𝑈¶𝜎¶)𝜎¶ 𝑡—µ^ 𝜇¶ + 𝑈¶𝜎¶
𝑡0 _
0
+ À𝜇; + 𝑈;𝜎; 𝑤0 Á0
⎠
⎟⎟
⎞
The starting point of the MPP is set to 𝐮B = (0,0).
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
Iteration 1
At 𝐮B = (0,0), 𝐠B = 0.67076, ∇𝑔(𝐮B) = (−0.37268, −0.046585),
‖∇𝑔(𝐮B)‖ = §(−0.37268)0+ (−0.046585)0= 0.3756, 𝐚B =‖∇@(𝐮∇@(𝐮££))‖ = (−0.9923, −0.1240), and βB = ‖𝐮B‖ = 0.
This produces a new point,
𝐮. = −𝐚BÐβB +‖∇@(𝐮@º𝐮££»)‖Ñ = −(−0.9923, −0.1240)B.ÒÓBÓÒB.¨ÓÔÒ = (1.7722, 0.22152).
Iteration 2
At 𝐮. = (1.7722,0.22152), 𝐠(𝐮.) = −0.015931, ∇𝑔(𝐮.) = (−0.38984, −0.036775),
‖∇𝑔(𝐮.)‖ = §(−0.38984)0+ (−0.036775)0 = 0.3916, 𝐚B =‖∇@(𝐮∇@(𝐮cc))‖ = (−0.9956, −0.0939), and β. = ‖𝐮.‖ = 1.7859.
This produces a new point,
𝐮0 = −𝐚.Ðβ.+‖∇@(𝐮@º𝐮cc»)‖Ñ = −(−0.9956, −0.0939) À1.7859 +PB.B.ÔÕ¨.
B.¨Õ.Ò Á
= (1.7375, 0.16391)
The process continues until the solution converges. The search determines after 4 iterations because the solutions in iteration 4 are very close to those in iteration 3. The complete convergence history is shown in Table 4.1.
Table 4.1 The result of each reliability index iteration
Iteration 𝑔 ∇𝑔 𝑈;, 𝑈¶ β
0 0.67076 (-0.37268, -0.046585) (1.7722, 0.22152) 0 1 -0.015931 (-0.38984, -0.036775) (1.7375, 0.16391) 1.7859 2 -0.00032102 (-0.38986, -0.036758) (1.7367, 0.16375) 1.7453 3 -2.6 x 10-9 (-0.38986, -0.036761) (1.7367, 0.16376) 1.7444
The MPP is found at 𝐮∗= (1.7367, 0.16376), and the reliability index is β = 1.7444. The probability of failure is
𝑝= = Φ(−1.7444) = 0.04054.
The reliability is
𝑅 = 1 − 𝑝= = 1 − 0.040541 = 0.9595
FORM Computation Result
Using the same material properties, external load and exact state limit function, through the FORM Matlab program we get the following results.
Table 4.2 The comparison of analytic and numerical reliability calculation Reliability Index
(Analytic)
Reliability Index
(Numeric)
Error
(Analytic vs Numeric)
1.74440 1.744737 1.932 x 10-4
Given the relatively small error between the analytical and computational Matlab FORM result, the FORM program is proven to be reliable enough to handle reliability analysis.
4.4 Main Reliability Computation
The simulation configuration has been previously discussed in Chapter 3. Initially for the first iteration, the material that becomes the point of interest is Carbon Reinforced Fiber Plastic (CRFP). CRFP are materials consisting of several components: a basic or carrier substance known as the matrix, as well as a reinforcing second component – carbon fiber – which is embedded in the matrix. Depending on the matrix material and fiber type, these properties can vary widely and be optimized for use in the requested applications.
With Young’s modulus 150 GPa, this combination results in high-performance materials with new properties. The properties of CRFP materials bring to bear precisely when other materials reach their limits. Because of its high strength and good rigidity, CRFP are indispensable in high-tech applications such as the lodging pod structure, thus allowing author to pick this particular material for the first iteration.
With the simulation setup given in Chapter 3, the Matlab Reliability computation yields the following result.
Table 4.3 The simulation setup and its result
Pressure Young’s Modulus Shell Thickness Reliability Index
500 atm 150 GPa 0.5 cm 4.8847
Another simulation tests were performed multiple times to observe the physical representation which happened in this setup. For example, it can be seen that as the Young’s Modulus is increased, the stronger the material will be and the bigger the reliability index value. Moreover, an increase in the structure deflection will allow the structure to be less flexible thus lowering the reliability value. Also, external pressure plays an important role, but since we set it as a fixed parameter throughout, there’s no need to alter its value.
Table 4.3 shows that this reliability index clearly exceeds the minimal reliability index requirement (𝛽 = 3.0). Despite that result lies in the safe area, it is important to trade-off the other free parameter to achieve the minimum reliability index. This is in a sense that with non-optimal reliability index there comes a consequence of non-optimal material selection. In this case, perhaps material CFRP can be replaced with a less stronger material to achieve the required reliability index. In the practical application, the lighter and stronger the material the more expensive the production cost will be. This at least gives the basic logic concerning the problem at hand.
To get the exact number of reliability index, we will implement the Inverse FORM analysis, which was previously described in Chapter 2. Firstly, we set 𝛽 as the fixed parameter and Young’s Modulus (E) as the free parameter. On the second iteration, we set maximum deflection as the free parameter. The curve that depicts the variation in the Young’s Modulus respect to the change in the reliability index is shown in Figure 4.6 below.
Figure 4.6 Reliability value to Young’s Modulus function
Also, the curve that depicts the variation in shell thickness respect to the change in the reliability index is shown in Figure 4.7 below.
Figure 4.7 Reliability value to shell thickness function Finally, the three results can be seen from the table below.
Table 4.4 Different simulations result
Pressure Young’s Modulus Shell Thickness Reliability Index
500 atm 150 GPa 0.5 cm 4.8847
500 atm 98.289 GPa 0.5 cm 3.00001
500 atm 150 GPa 0.32763 cm 3.00001
Following this result, we can imply that with the fixed shell thickness of 0.5 cm, the required material’s Young’s Modulus is equal to 98.289 GPa. This would roughly equivalent to Cast Iron Grade-20 material with the Young’s modulus 100 GPa, just a slight stiffer from the material S-Glass Fibre that has the maximum Young’s Modulus 96 GPa. Moreover, if the
material setting is unchanged (still CRFP with 150 GPa), author find a compromise in shell thickness, with only 0.33 cm required to reach the standard reliability index.
It is noted that the reliability analysis methods are used to evaluate a specific probability at the limit state, and they are not intended to generate a complete distribution of a performance function or its statistical moments. However, if different values that cover the range of distribution of the performance function are used as limit states, FORM is applicable to generate the whole distribution of the performance function.
For many engineering problems, a performance function is expensive to evaluate, and no analytical derivatives exists. When the derivative has to be evaluated numerically, the computational effort will be approximated proportional to the number of design variables.
In this way, FORM may not be applicable for large scale problems. An approach such as Monte Carlo simulation or other approximation methods are needed to resort the problem in the future work.
We have discussed how to estimate the reliability, which defined as the probability that a performance function is safe. The most commonly used reliability analysis method FORM has been discussed.
Chapter 5 Conclusion
This chapter presents an overview and general conclusions related to the work developed in this thesis. The general topic of research is to analyze the safety problem of a lodging pod structure based on reliability method. In chapters 3 and 4, the simulation architecture of FERUM with First Order Reliability Method has been described to perform the analysis.
For simplicity purpose, a block of Finite Element simulation has been replaced with analytical computation of a pressure vessel problem to get the appropriate state limit function. The main objective was to bypass the cumbersome FE simulation with the analytical formulation such that both Reliability and state limit function calculations operate under the same simulation setup.
Another focus in this thesis work was to find different solution to the structural reliability problem. A simple approach has been performed to fix one parameter and let the other becomes the free variable, and vice versa.
5.1 Summary and Conclusions
In chapter 4, an analytical formulation to get the displacement state limit function is analyzed. Traditionally, reliability method of this kind requires thorough finite element computation inside the reliability loop. Using this analytical formulation, the basic idea is to integrate the result of the analytical formulation directly into the reliability computation block. Reliability results here yield two different solution of material properties and shell thickness, as summarized in the table below.
Table 5.1 Different simulations result
Pressure Young’s Modulus Shell Thickness Reliability Index
500 atm 150 GPa 0.5 cm 4.8847
500 atm 98.289 GPa 0.5 cm 3.00001
500 atm 150 GPa 0.32763 cm 3.00001
