機翼機身對接結構三維斷裂分析【含圖紙、說明書】
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畢 業(yè) 設 計（論 文）任 務 書
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設計（論文）題目：
機翼機身對接結構三維斷裂分析
?
學生姓名：
張嘉歡
學??號：
1204201026
專????業(yè)：
機械設計制造及其自動化
所在學院：
機電工程學院
指導教師：
王芳麗
職????稱：
發(fā)任務書日期：年月日
任務書填寫要求
1．畢業(yè)設計（論文）任務書由指導教師根據各課題的具體情況填寫，經學生所在專業(yè)的負責人審查、系（院）領導簽字后生效。此任務書應在畢業(yè)設計（論文）開始前一周內填好并發(fā)給學生。
2．任務書內容必須用黑墨水筆工整書寫，不得涂改或潦草書寫；或者按教務處統一設計的電子文檔標準格式（可從教務處網頁上下載）打印，要求正文小4號宋體，1.5倍行距，禁止打印在其它紙上剪貼。
3．任務書內填寫的內容，必須和學生畢業(yè)設計（論文）完成的情況相一致，若有變更，應當經過所在專業(yè)及系（院）主管領導審批后方可重新填寫。
4．任務書內有關“學院”、“專業(yè)”等名稱的填寫，應寫中文全稱，不能寫數字代碼。學生的“學號”要寫全號，不能只寫最后2位或1位數字。
5．任務書內“主要參考文獻”的填寫，應按照《金陵科技學院本科畢業(yè)設計（論文）撰寫規(guī)范》的要求書寫。
?6．有關年月日等日期的填寫，應當按照國標GB/T 7408—94《數據元和交換格式、信息交換、日期和時間表示法》規(guī)定的要求，一律用阿拉伯數字書寫。如“2002年4月2日”或“2002-04-02”。
畢 業(yè) 設 計（論 文）任 務 書
1．本畢業(yè)設計（論文）課題應達到的目的：
? 本畢業(yè)設計課題的主要目的是培養(yǎng)學生綜合運用所學的基礎理論、專業(yè)知識和專業(yè)基本技能分析和解決實際問題，訓練應用ANSYS軟件對機翼與機身對接機構進行有限元建模和三維斷裂分析的能力，主要包括以下幾個方面： 1．調查研究、中外文獻檢索、閱讀與翻譯的能力； 2．綜合運用基礎理論、專業(yè)理論和知識分析解決實際問題的能力； 3．查閱和使用專業(yè)設計手冊的能力； 4．應用catia軟件和ANSYS軟件進行建模和有限元分析的能力； 5．撰寫設計說明書（論文）的能力。
2．本畢業(yè)設計（論文）課題任務的內容和要求（包括原始數據、技術要求、工作要求等）：
? （1） 熟悉并理解機翼機身對接結構與受力特點；（2） 熟悉和掌握三維斷裂理論知識；（3） 應用ANSYS軟件對機翼機身對接結構進行三維建模和斷裂分析。
畢 業(yè) 設 計（論 文）任 務 書
3．對本畢業(yè)設計（論文）課題成果的要求〔包括圖表、實物等硬件要求〕：
1.外文參考資料及譯文（附原文）； 2.畢業(yè)設計開題報告一份； 3.有限元分析結果分析說明一份；
4．主要參考文獻：
[1] 吳相憲，王正為，黃玉堂主編.實用機械設計手冊.中國礦業(yè)大學出版社，1993. [2] 王洪欣，李木，劉秉忠主編.機械設計工程學[M].中國礦業(yè)大學出版社，2001. [3] 唐大放，馮曉寧，楊現卿主編.機械設計工程學[M].中國礦業(yè)大學出版社，2001. [4] 中國紡織大學工程圖學教研室等編.畫法幾何及工程制圖.上?？茖W技術出版社，1997. [5] 史美堂主編.金屬材料及熱處理.上海科學技術出版社，1983. [6] 蘇翼林主編.材料力學.高等教育出版社，1980. [7] 顧崇銜主編.機械制造工藝學.陜西科學技術出版社，1999. [8] 詹熙達主編.CATIA V5R20曲面設計教程. 北京：機械工業(yè)出版社,2013. [9] 詹熙達主編.CATIA V5R20快速入門教程. 北京：機械工業(yè)出版社,2011. [10] 劉文珽，羅毅，童明波．概率損傷容限分析模型研究[J]．航空學報，1993，14（3）：136-139． [11] 劉文珽等．概率斷裂力學與概率損傷容限/耐久性[M]．北京航空航天大學出版社，1998. [12] 羅毅，黃培彥，劉文珽．裂紋擴展壽命安全可靠性分析模型研究[J]．北京航空航天大學學報，2002，28（1）：113-115. [13] 杜永恩．概率損傷容限分析體系及其關鍵技術的研究[D]．西安：西北工業(yè)大學，2014. [14] K.Y. Lin and A.V. Styuart. Probabilistic approach to damage tolerance design of aircraft composite structures [J]. Journal of Aircraft, 2007,44(4):1309-1317. [15] Spencer B F，Tang J. Markov Model for fatigue crack growth [J]. Journal of Engineering Mechanics，1998,114:2134-2157.
畢 業(yè) 設 計（論 文）任 務 書
5．本畢業(yè)設計（論文）課題工作進度計劃：
2015.12.16-2.16.3.9 畢業(yè)實習調研，完成開題報告、中英文翻譯、論文大綱 2016.3.19-2016.4.25 提交論文草稿，4月中旬中期檢查 2016.4.26-2016.5.6 提交論文定稿 2016.5.6-2016.5.13 準備答辯 2016.5.13-2016.5.26 答辯，成績評定，修改完成最終稿
所在專業(yè)審查意見：
?通過?
負責人： ??????????? ?2016? 年??? 1 ?月???18 ?日
畢 業(yè) 設 計（論 文）開 題 報 告
設計（論文）題目：
機翼機身對接結構三維斷裂分析
?
學生姓名：
張嘉歡
學??號：
1204201026
專????業(yè)：
機械設計制造及其自動化
所在學院：
機電工程學院
指導教師：
王芳麗
職????稱：
?
?年? ?月??日 ?
開題報告填寫要求
?
1．開題報告（含“文獻綜述”）作為畢業(yè)設計（論文）答辯委員會對學生答辯資格審查的依據材料之一。此報告應在指導教師指導下，由學生在畢業(yè)設計（論文）工作前期內完成，經指導教師簽署意見及所在專業(yè)審查后生效；
2．開題報告內容必須用黑墨水筆工整書寫或按教務處統一設計的電子文檔標準格式打印，禁止打印在其它紙上后剪貼，完成后應及時交給指導教師簽署意見；
3．“文獻綜述”應按論文的框架成文，并直接書寫（或打?。┰诒鹃_題報告第一欄目內，學生寫文獻綜述的參考文獻應不少于15篇（不包括辭典、手冊）；
4．有關年月日等日期的填寫，應當按照國標GB/T 7408—94《數據元和交換格式、信息交換、日期和時間表示法》規(guī)定的要求，一律用阿拉伯數字書寫。如“2004年4月26日”或“2004-04-26”。
5、開題報告（文獻綜述）字體請按宋體、小四號書寫，行間距1.5倍。
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畢 業(yè) 設 計（論文） 開 題 報 告
1．結合畢業(yè)設計（論文）課題情況，根據所查閱的文獻資料，每人撰寫不少于1000字左右的文獻綜述：
機翼機身對接接頭設計是飛機結構設計的一個重要環(huán)節(jié)，其設計的好壞嚴重關系到飛機的飛行性能和使用安全，本文詳細地闡述了飛機結構設計載荷系數的產生和發(fā)展，提出在飛機結構設計中用可靠性安全系數替代傳統安全系數的觀點。介紹分析了機翼機身對接設計思想和接頭耳片受力特性，提出了改善機翼機身對接區(qū)域傳力特性的設計方法。
現代超音速戰(zhàn)斗機常采用中單翼布置，受機身部位安排的限制，不能有中央翼通過機身，此時機翼通過幾個集中接頭與機身在側邊相連，機翼彎矩要作用于翼身對接框上，從而機身框結構要加強，結構重量加大。我們知道現代超音速戰(zhàn)斗機的機翼很薄，機翼的相對厚度約為4％，隨著機翼面積的增加，機翼載荷大幅度上升，翼梁承擔和傳遞的載荷越來越大，機翼載荷完全由接頭耳片及連接螺栓傳遞到機身對接框，因此若直接將如此結構高度的翼梁連接到加強框上，機翼機身對接接頭耳片、連接螺栓及加強框受力都是比較嚴重的。由此，現代先進戰(zhàn)斗機在總體設計時，結合總體氣動布局、機翼機身部位安排的要求采用了多種設計方法來改善對接接頭區(qū)域的傳力特性。
目前對接頭問題的研究有很多，孫庚茂、丁惠粱等采用釘“超元”模擬復合材料的連接，宋恩鵬等采用梁元和彈簧元分別模擬螺栓連接，張永杰等采用體單元對螺栓連接進行剛度分析。采用螺栓、彈簧元或釘元模擬螺栓連接都是從整體結構承載出發(fā)，給出了有限元仿真中螺栓連接的簡化方法，但不能解決螺栓的預緊力、孔邊接觸應力、接頭的屈服極限等方面的問題。而大部分接頭的出現都會伴隨著孔結構的產生．孔邊的受力由于要考慮摩擦和接觸問題，使結構孔邊周圍呈現出非線性特征，有時為了更大的挖掘接頭結構的承載能力，甚至還要考慮結構局部進入塑性區(qū)應力分布．這就更增加了非線性程度以及求解接頭接觸問題的復雜性。隨著有限元軟件的發(fā)展，接觸問題的數值方法得到了很大的提高，通過不斷細化網格可以得到逼近精確解的數值解。
本次設計的主要研究工作是探討傳統安全系數的發(fā)生和發(fā)展，分析傳統安全系數的利弊，建立可靠性安全系數觀念；對機翼機身對接接頭形式和布局以及接頭耳片的構型和受力特性進行分析和研究；對原型飛機橫梁結構受力特性進行有限元應力分析；根據大改飛機設計要求，優(yōu)化結構設計參數，給出接頭加強型和改進型設計方案；對橫梁結構疲勞危險部位進行損傷容限評定。
本次設計采用實體建模三維有限元方法，對飛機、翼身對接主承力接頭進行傳力特性分析，對機身半框模型、機翼主梁一機身橫梁組合模型在CATIA和ANSYS下計算結果進行了評價。根據結構設計要求和結構限制條件，優(yōu)化結構設計參數，給出了飛機機翼機身對接主承力接頭的加強型和改進型設計方案。從結構尺寸、重量、疲勞危險部位應力水平、對氣動力的影響以及裝配工藝性等方面對加強型和改進型設計方案進行了比較分析。疲勞斷裂起始于結構細節(jié)，對疲勞危險關鍵部位進行了損傷容限評定，提出了提高結構抗疲勞斷裂能力的措施。
畢 業(yè) 設 計（論文） 開 題 報 告
2．本課題要研究或解決的問題和擬采用的研究手段（途徑）：
本課題所要研究及解決的問題：
本課題是機翼機身對接結構三維斷裂分析，需要在給出和查閱到的設計基本資料上完成該裝置需要的結構布置，需要研究及解決的問題如下：
（1）在了解現有幾種三維斷裂的原因，需要應用有限元分析軟件對斷裂原因進行仿真計算；
（2）需要利用有限元軟件研究分析初始裂紋尺寸的變化，裂紋擴展速率，無損檢查，飛行載荷和使用頻譜。
擬采用的研究手段（途徑）：
1、文獻收集
廣泛收集與永磁磁懸浮技術相關的資料；
2、實踐與實習
通過實驗室內建設基本模型進行測試分析，通過大量的數據來進行設置的計算機控制系統建立。
3、運用CATIA和ANSYS設計軟件進行三維分析；
4、結合指導老師的指點，分進度，分階段實施，并對相關問題展開研究。
畢 業(yè) 設 計（論文） 開 題 報 告
指導教師意見：
1．對“文獻綜述”的評語：
通過文獻綜述，該生對機身機翼對接接頭三維斷裂國內外研究現狀有了較清晰的認識，下一步可以通過Ansys軟件對對接接頭進行有限元建模和三維斷裂分析研究。
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2．對本課題的深度、廣度及工作量的意見和對設計（論文）結果的預測：
本課題深度和工作量適中，具有一定的工程應用價值，相信通過該生對各種機身機翼對接接頭三維斷裂的研究，在飛機設計時對對接接頭設計時具有一定的參考價值。
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3.是否同意開題：√ 同意 □ 不同意
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???????????????????????????????????? ??指導教師：??????????????
???????????????????????????????????????? 2016 年?? 03 月?? 08 日
所在專業(yè)審查意見：
同意
???????????????????????????????????????? 負責人：??????????????
?????????????????????????????????????????? 2016 年??? 03 月?? 09 日
26th ICAF Symposium – Montreal, 1-3 June 2011
*Challenges in Damage Tolerance Approach for
Dynamic Loaded Rotorcraft Components – From
Risk Assessment to Optimal Inspection Planning
Jack Zhao1 and David Adams2
1 Structural Methods and Prognostics
2 Ground Test
Sikorsky Aircraft Corporation
Stratford, CT 06516
USA
jzhao@sikorsky.com
Abstract. The use of Crack Growth Damage Tolerance as a substantiation methodology for helicopter dynamic components is receiving increased attention as a logical and viable improvement in fatigue reliability and structural integrity. It
has seen only limited use in helicopters because the addition of difficult periodic inspections was seen as a significant burden to the operator. However the certifying agencies are moving towards the simultaneous use of both Safe-Life and Damage Tolerance methodologies on each component. In order to mitigate the cost issue, a means to optimize the inspection protocol using a risk-informed damage tolerance based fatigue reliability model and maintenance optimization tool is evaluated in this paper. It was desired to maintain the same “6-9’s” level of
structural reliability for Damage Tolerance that is now the standard practice for safe-life substantiations. The newly developed fatigue reliability methodology incorporates the variabilities in initial crack size, crack growth rate, nondestructive inspections, flight loads, and the usage spectrum. The reliability model is further integrated with optimization technique for inspection planning. An example case using the crack propagation test result from a helicopter main rotor spindle is evaluated with the reliability model. The concept of DT risk assessment and optimal inspection planning, impact of NDI detection capability and repair quality on risk reduction, and importance of incorporating CBM logistic requirement are demonstrated. It is concluded that a fatigue reliability model for
Damage Tolerance was successfully demonstrated and that it can be used to determine an optimized inspection protocol that reduces the operator’s inspection burden while providing the required 6-9’s level of fatigue reliability.
1 Introduction
Damage Tolerance, specifically Crack Growth Damage Tolerance, has been successfully applied in a limited number of helicopter fatigue substantiations for * Oral presentation.
928 J. Zhao and D. Adams more than 50 years, although it was originally called “Fail-Safe” methodology. The number of applications is now increasing, driven by an increased emphasis on Damage Tolerance by civil and military certifying agencies. The FAA’s Amendment 28 to FAR 29.571 in 1989 provided that Fail-Safety (Damage Tolerance) was an equal-choice option to Safe Life as a substantiation methodology. And a pending new 29.571 will require implementation of both methods on every substantiated component. Damage Tolerance methodology relies on the assumption that the component exhibits some initial damage that subsequently grows progressively over a period of time prior to catastrophic failure. A successful damage tolerance design must be capable of: 1) predicting crack initiation; 2) accurate modelling of subsequent crack growth; and 3) adequate NDI methodology with suitable inspection schedule. The advantage of a Crack Growth Damage Tolerance method over Safe Life is that the cause of an initial crack or damage does not matter since the inspection program will detect the presence of whatever crack occurs before it becomes catastrophic, with a significant safety margin. The disadvantage is the cost of the
inspection program in terms of the intrusive down time, man-hours, training, and equipment required. Damage Tolerance will not be accepted as a viable and desirable methodology unless its benefits are perceived to be worth its cost. There is, therefore, an opportunity to employ a reliability approach to determine an optimum inspection methodology – one that provides a required level of structural reliability but does not require unnecessary or too-frequent inspections.
Conventional Approach to Crack Growth Substantiations
Sikorsky’s methodology for the substantiation of flight-critical fatigue-loaded components is entirely empirical and was initially developed in the early 1960’s for aluminium spar main rotor blades. This substantiation, called “Blade Inspection Method”, or BIM, is still in use today on thousands of rotor blades. It is based on sensing a loss of internal gas pressure in the event of a spar crack, with the inspection interval based on a full-scale fatigue test program that fully characterized the crack growth behaviour under conservative maximum flight loads and severe usage. Sensing of the pressure loss is done by a special visual indicator at the blade root. The inspection interval is essentially a pre-flight visual inspection that was set at minimum 3 to 1 reduction in the test crack growth time from detection to failure. Inspections start at zero time. This method – conservative full-scale test determination of crack growth, demonstration of a field inspection method, determination of a failure point, and an inspection interval based on a fraction of the test time – is still in use today with a few developments. We now require a static test demonstration for critical crack size, we avoid the inclusion of any blunting effect in metals due to high
fatigue test loads, we have employed the method in composites, and we have standard methodology for number of fatigue test specimens and the inspection interval reduction factor. The basic method is accepted by all of our civil and military certifying agencies as illustrated in the figure below from the FAA’s AC
29-2C MG-11. Damage Tolerance Approach for Dynamic Loaded Rotorcraft Components 929
Fig. 1 Potential-to-Functional Failure Curve from NAVAIR 25-403.
A reliability determination has not been part of the current crack growth damage tolerance method. Because of the conservative treatments of the flight loads, the usage, and the test-based crack growth characteristics in the substantiation, the current method meets the generic requirement that failure is “extremely remote”, and this criteria has been achieved in 50 years of service. There has been a methodology development, Reference [3], called “Empirical Damage Tolerance”, which allows the determination of an inspection interval for a different load spectrum than was applied in the full-scale test program. This development is also useful in the reliability studies that follow and is described in
more detail later.
Reliability-Based Approach to Helicopter Damage Tolerance
The work done to show the reliability of a Damage Tolerant approach for helicopter dynamic component fatigue is not extensive. One early effort did show that a 6-9’s level of reliability was achieved for a multiple load path case, Reference [9]. However a good starting point for reliability-based approach is Reliability-Centered Maintenance (RCM) as described in NAVAIR 25-403, Reference [5]. The figure below illustrates the key points of RCM. This is a much more general methodology, referring to the decline in a functional capability to the point where the functionality is declared failed. The figure is generally known as a P-F curve. The P-F interval is the age interval (in flight hours, cycles, or calendar time) between the Potential Failure (some loss of functionality) becoming detectable (P) to the point of the defined functional failure (F). The inspection interval (I) is a defined fraction of the PF interval. 930 J. Zhao and D. Adams
Fig. 2 Potential-to-Functional Failure Curve from NAVAIR 25-403.
A reliability-based optimal inspection interval would provide a required predetermined level of structural reliability while minimizing the cost of conducting inspections too frequently. One simplified approach to the reliability is discussed in NAVAIR 25-403, where the Inspection Interval was initially determined by requiring that the projected probability of failure be reduced to less than or equal to the acceptable probability of failure. The interval of on-condition task, denoted as I, can be estimated by:
where PF is the Potential-to-Functional failure interval and n denotes the number of inspections during P-F interval. In general, n can be determined by either safety requirements or cost optimization. For flight-critical components, the total risk
considering the inspections shall not exceed the maximum acceptable risk, Therefore,
where acc P is the maximum acceptable level of probability of function failure and is probability of detecting a potential failure in one inspection assuming it exists. The equation above implicitly assumes the failure will always occur in the P-F interval and a constant detectability which is independent to the size of damage. The extreme condition satisfying the risk constraint occurs if the total risk equals to the maximum acceptable level. Accordingly, the number of inspections can be determined by
Damage Tolerance Approach for Dynamic Loaded Rotorcraft Components 931 The approach outlined in Eq. 1-3 is based on assumption that a potential failure always exists within the P-F interval and is independent between inspections. As a result, the inspection interval may be too conservative, meaning too-frequent inspections, which does not meet our minimized cost objective. The basic RCM approach does not consider the failure mechanics or the scatter of failure progression. Often, the potential failure mode under consideration exhibits inherent randomness. This is particularly important for the failure modes
associated with progressive damage accumulation such as crack initiation and growth, corrosion, and mechanical wear. To effectively address variability and uncertainty of damage progression and understand their impact on P-F interval, it is highly desirable to incorporate stochastic characterization of failure progression into the RCM process. In this paper, a new approach is proposed to establish a risk-based interval for on- condition tasks by incorporating a baseline probability of failure and a characteristic detectability for inspection capability. Generally, the probability of failure for a component under scheduled inspections can be expressed as the probability of a sequence of events, such as:
Where, pF0 is the probability of failure before the first inspection due to excessive damage progression; G i p is the probability that damage will grow to a detectable limit right before the ith inspection (i =1,2,??, n); ND i p is the conditional probability that inspection will not be able to detect damage at the ith inspection given that damage exists, and F i p is the conditional probability that un-detected damage at the ith inspection will further grow to failure before the next inspection [(i+1)th] or end of intended service life. Clearly, the probability of failure of these events depends on the probability of damage progress, inspection capability, the timing of inspection, and the number of inspections. Therefore, a more rigorous risk assessment of inspection planning requires comprehensive understanding of the physics of damage initiation, progression and associated randomness, as well as the mathematical model representing inspection capability, and advanced probabilistic methodology capable of performing complicated numerical simulation and assessment. Due to its simplicity for further implementation, the concept of P-F interval and procedure outlined in NAVAIR 00-25-403 serve as a good starting point for establishing a rough estimate of inspection interval. For the purpose of addressing inherent randomness of failure progression and to further facilitate quantitative risk assessment and management for CBM, a more rigorous approach incorporating physics-based damage accumulation model, inspection capability, and advanced probabilistic methodology is needed urgently.932 J. Zhao and D. Adams
2 Challenges in a Damage Tolerance Approach
In damage tolerance approach, structural integrity is ensured through a predictive crack growth model representing the true nature of damage progression, nondestructive inspections to eliminate excessive damaging, and proper repair and maintenance actions. Many factors affects the effectiveness, robustness, an accuracy of the damage tolerance approach, including validation of crack growth model, qualify capability of desirable NDI methods, developing optimal inspection plan, establishing repair limits and criteria for proper maintenance
actions, and setting up rational level of target reliability for risk management. This paper discusses some of the aforementioned technical challenges associated with rotorcraft components and presents a stochastic methodology for predicting rotorcraft component fatigue lifetimes and optimal inspection intervals and assessing underlying risk.
Prediction of fatigue crack growth behaviour
Damage tolerance approach relies heavily on capability of a fracture mechanics (FM) model to accurately predict potential damage progression initiated at preidentified locations. Several commonly used FM software packages are available for such purpose, including NASGRO, AFGRO, and FASTRAN. They are developed based on linear elastic fracture mechanics and possess a rich library of stress intensity solutions for the commonly encountered structural configuration and geometric profile for anticipated crack growth. From time to time, more
advanced fracture mechanics may be employed for more complicated crack growth behaviour, structural layups, and loading, if there is the stress intensity solutions do not exist. These advanced fracture mechanics tools, such as BEASY and FRANC-3D, engage boundary element based numerical procedure and simulation. Occasionally, the crack behaviour will also be observed and derived directly from crack growth testing at full component level, such as the empirical damage tolerance approach reported in Reference [3]. These approaches represent various levels of modelling and numerical simulation efforts to ensure adequacy of the fracture mechanics model building and accuracy of the predictive capability. For the purpose of qualifying a crack growth model for further DT application, model validation is critical important. There are several ways to achieve the goal. One engages seeded fault testing and the other is to compare the predicted results against the fielded cracking data for further correlation.
Uncertainty modeling and quantification for DT approach
Primarily, probabilistic uncertainty analysis and risk assessment involves modeling all of the fundamental quantities entering the problem, and also all uncertainties that arise from lack of knowledge in these quantities, which may affect failure of the component or system. These terms are referred to as basic variables including quantities of structural dimensions and material properties, yield stress and other ultimate response limitations, operating conditions and degradations, environmental and loading factors, etc. The sources of uncertainty in Damage Tolerance Approach for Dynamic Loaded Rotorcraft Components 933 probabilistic analysis can be mainly classified into two categories as aleatory and epistemic uncertainties. Aleatory uncertainties refer to the natural randomness associated with an uncertain quantity, which is inherent in time, in space and measurements. This kind of uncertainty is quantified through the collection and analysis of data to fit to theoretical distributions and, since it is inherent, it cannot be reduced. Epistemic uncertainties reflect a lack of knowledge or information about a quantity, which can be considered in either model or statistical uncertainty-subdivisions. Modal uncertainties arise from simplifications and idealizations that are necessary to model the behavior in a reliability analysis, or from an inadequate understanding of the physical causes and effects. Statistical uncertainties are only due to a shortage of information, and originate from a lack of sufficiently large samples of input data. Statistical uncertainties can be reflected
through either parameters of a distribution with a limited set of data or the type of a theoretical distribution to be chosen to fit to data. Since epistemic uncertainty is associated with a lack of knowledge and/or information it follows that it can be reduced through an increase in knowledge by gathering data for a longer period, taking more measurements or carrying out further tests, doing research, and by expert judgment. In order to consider these uncertainties in a structural analysis, appropriate uncertainty models are essential for performing reliability methods to estimate the probability of failure. As one of the key building blocks of a damage tolerance risk assessment and design process, all the sources of uncertainty and their statistical characteristics related to the key design variables must be identified, quantified and further integrated into probabilistic damage tolerance design system. It is well recognized that fatigue initiation and its subsequent crack growth is a random phenomenon. As depicted in Figure 3, various sources of uncertainties contribute to random fatigue and fracture process, including fatigue initiation time, micro-crack initiation and propagation, stress intensity threshold, crack growth rate, usage and loads, and inspection capability for product/in-service inspection.
Fig. 3 Uncertainty Identification for Damage Tolerance Approach.
It is beyond the scope of this paper to provide comprehensive review of the statistical procedure for modelling of aforementioned sources of variability associated with DT assessment, details of statistical procedure, methodologies, and practices for DT uncertainty identification and modelling can be found in
reference [10].
934 J. Zhao and D. Adams
Probabilistic risk assessment methodology
Probabilistic methodologies have been widely applied for uncertainty quantification and associated risk assessment. Among the procedures developed for structural reliability assessment and failure probability prediction, a prominent position is held by simulation methods. The Monte Carlo simulation technique, as the basis of all simulation-based techniques, is the most widely applied numerical tool in probabilistic analysis. The convergent rate of the Monte Carlo estimator is appropriately measured by the coefficient of variation of the estimated probability of failure. In general, the basic Monte Carlo technique requires a large sample size to achieve accurate estimate of probability of failure. This becomes a major limitation for the practical application of basic Monte Carlo simulation in structural reliability applications involved in a small probability of failure. To address the challenge, the Importance Sampling technique has been developed and becomes the most prevalent approaches in the context of simulation-based methods for probabilistic analysis. In importance sampling scheme, instead of drawing random samples arbitrarily as the way implemented in a basic Monte Carlo simulation, the majority of the random samples are drawn from the region that contributes the most for the probability of failure. Several approaches can be employed to identify the important region; including 1) MPP obtained through first order reliability methods (FORM) or second order reliability methods
(SORM) solution; 2) a priori estimate from pre-sampling; and 3) Markov Chain Monte Carlo simulation. In general, the efficiency of the Importance Sampling technique improves significantly with a large reduction of the variance of estimator, once the appropriate Importance Sampling density function is
identified. In general, DT risk assessment requires generating and repeatable drawings of short-life samples. This requirement dictates the utilization of sampling based methodology in risk a
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