地球物理學(xué)科發(fā)展規(guī)劃趙壽剛1常向前1楊小平1袁華2（1.黃河水利科學(xué)研究院，河南鄭州450003；2.黃河水利委員會(huì)水文局，河南鄭州450003）摘要：對(duì)堤防工程進(jìn)行安全評(píng)判分析,最差不多的是要進(jìn)行滲流及邊坡穩(wěn)固分析,而滲流及邊坡穩(wěn)固分析結(jié)果的是否合理與選擇的土體參數(shù)具有極大的關(guān)系；由于黃河大堤工程地質(zhì)條件復(fù)雜，土體參數(shù)具有極大的變異性，利用確定性方法將專門難得出符合實(shí)際情形的結(jié)果。將可靠度理論應(yīng)用于黃河大堤安全評(píng)判，能夠充分考慮土體參數(shù)的變異特性，將使分析結(jié)果更加符合工程實(shí)際。對(duì)可靠度理論的差不多方法進(jìn)行了分析，對(duì)其在黃河大堤安全評(píng)判分析中的具體實(shí)施方法進(jìn)行了研究，該方法具有極大的應(yīng)用價(jià)值和創(chuàng)新性。關(guān)鍵詞：可靠度理論；滲流；邊坡；安全評(píng)判；蒙特卡羅法1前言黃河下游是文明遐邇的地上懸河，專門是近些年來，來水量連續(xù)偏枯，水少沙多，水沙關(guān)系不和諧，而且黃河下游的游蕩性河段還沒有得到專門好的操縱，汛期會(huì)發(fā)生順堤行洪的現(xiàn)象，甚至發(fā)生“橫河”“斜河”的情形，嚴(yán)峻威逼黃河大堤的安全；而且黃河下游大堤情形復(fù)雜，每段大堤的地質(zhì)條件以及河勢(shì)等具有不同特點(diǎn)，因此每段堤防的安全性如何？如何對(duì)堤防的防洪安全做出客觀的符合實(shí)際情形的評(píng)判等一直是人們所關(guān)懷的咨詢題。對(duì)堤防工程進(jìn)行安全評(píng)判分析,最差不多的是要進(jìn)行滲流及邊坡穩(wěn)固分析,而滲流及邊坡穩(wěn)固分析結(jié)果的是否合理與選擇的土體參數(shù)具有極大的關(guān)系,由于黃河堤防工程地質(zhì)條件復(fù)雜，大堤堤身經(jīng)多次加高培厚，土質(zhì)均一性差，即使同一種土，其物理力學(xué)指標(biāo)也相差專門大。針對(duì)某一斷面進(jìn)行滲流及邊坡穩(wěn)固安全評(píng)判時(shí)，僅憑少量地質(zhì)勘探資料進(jìn)行運(yùn)算分析，將專門難符合實(shí)際情形。在專門多情形下，甚至分析不出咨詢題?？煽慷壤碚撃軌蚪鉀Q上述咨詢題，它是建立在概率統(tǒng)計(jì)的基礎(chǔ)上，能夠充分考慮土體參數(shù)的變異特性，使分析結(jié)果更加符合工程實(shí)際。可靠度理論的差不多點(diǎn)是將阻礙工程安全的因素視為隨機(jī)變量，建立功能函數(shù)，求解失效概率或可靠度，以此來評(píng)判工程的安全性。不言而喻，利用這一理論第一要積存大量的試驗(yàn)資料，并據(jù)此研究阻礙工程安全的各種因素的概率分布規(guī)律，其次需要進(jìn)行大量的運(yùn)算。目前黃河大堤經(jīng)多次勘探已積存了數(shù)量可觀的有關(guān)資料數(shù)據(jù)，大容量高速度運(yùn)算機(jī)也已普遍使用。因此，利用可靠度理論研究黃河大堤安全評(píng)判咨詢題的條件差不多具備。2可靠度分析方法簡(jiǎn)介2.1可靠度分析差不多方法目前常用的可靠度分析方法要緊有一次二階矩法、隨機(jī)有限元法、概率矩點(diǎn)估量法（又稱Rosenblueth法）、蒙特卡羅隨機(jī)模擬方法等。（1）一次二階矩法一次二階矩法是采納只有均值和標(biāo)準(zhǔn)差的數(shù)學(xué)模型去求解結(jié)構(gòu)可靠度的方法，具體地講確實(shí)是將工程結(jié)構(gòu)功能函數(shù)按Taylor級(jí)數(shù)展開，忽略高階項(xiàng)，僅保留線性項(xiàng)，再利用差不多隨機(jī)變量的一階矩、二階矩求取均值和標(biāo)準(zhǔn)差，從而確定工程結(jié)構(gòu)的可靠指標(biāo)。由于這一方法是將非線性的功能函數(shù)作了線性化處理，因此是一種近似運(yùn)算可靠指標(biāo)的方法，但由于其簡(jiǎn)便明了，又具有專門強(qiáng)的適用性，而在工程實(shí)際中得到了較廣泛的應(yīng)用?；谝淮味A矩的方法要緊有中心點(diǎn)法、驗(yàn)算點(diǎn)法（）、映射變換法及有用分析法等4種。（2）隨機(jī)有限元法由于工程各方面的實(shí)際咨詢題受大量隨機(jī)因素的阻礙，土體的變化關(guān)系具有專門強(qiáng)的非線性和統(tǒng)計(jì)參數(shù)的變異性，許多學(xué)者致力于隨機(jī)有限元理論在工程中應(yīng)用的研究。隨機(jī)有限元簡(jiǎn)單地講確實(shí)是應(yīng)用限元法分析隨機(jī)結(jié)構(gòu)的咨詢題，按照推導(dǎo)隨機(jī)有限元操縱方程的方法不同，隨機(jī)有限元可分為Taylor級(jí)數(shù)展開法隨機(jī)有限元、攝動(dòng)法隨機(jī)有限元、以及Neuman級(jí)數(shù)展開Monte-Carlo法隨機(jī)有限元。這幾種方法差不多上圍繞隨機(jī)算子和隨機(jī)矩陣的求遞咨詢題展開的。（3）概率矩點(diǎn)估量法墨西哥人Rosenblueth于1975年提出通過點(diǎn)估量的方式來運(yùn)算巖土工程中的可靠指標(biāo)，1981年他又對(duì)這一方法進(jìn)行了完善和理論化，因此概率矩點(diǎn)估量法通常又稱為Rosenblueth法。它要緊是按照輸入隨機(jī)變量的前三階矩（均值、方差、偏態(tài)系數(shù)）來近似地描述極限狀態(tài)函數(shù)的概率矩，不必預(yù)先明白輸入隨機(jī)變量的精確分布。Rosenblueth法要求在某幾個(gè)點(diǎn)上估量功能函數(shù)的值，這些點(diǎn)按照一定的原則由隨機(jī)變量的均值以及標(biāo)準(zhǔn)差生成，按照點(diǎn)估量的功能函數(shù)值即可通過運(yùn)算公式確定可靠指標(biāo)。（4）蒙特卡羅法蒙特卡羅法的差不多原理確實(shí)是第一對(duì)各隨機(jī)變量進(jìn)行大量抽樣，然后代入運(yùn)算模型功能函數(shù)中，運(yùn)算結(jié)構(gòu)失效次數(shù)占總抽樣次數(shù)的百分?jǐn)?shù)即為其失效概率，進(jìn)而求出可靠指標(biāo)。2.2可靠度方法比選分析基于對(duì)幾種可靠度理論差不多分析方法的分析研究，分不對(duì)每種方法的特點(diǎn)進(jìn)行扼要介紹。（1）一次二階矩法是近似運(yùn)算可靠指標(biāo)最簡(jiǎn)單的方法，但其利用泰勒級(jí)數(shù)展開式，忽略了高次項(xiàng)，因此是一種近似的運(yùn)算方法，而且還有一些應(yīng)用的限制條件，如中心點(diǎn)法僅適用于差不多變量服從正態(tài)或?qū)?shù)正態(tài)分布，且結(jié)構(gòu)可靠指標(biāo)=1～2的情形。但目前許多學(xué)者對(duì)本方法進(jìn)行了改進(jìn)，因其運(yùn)算簡(jiǎn)便，在一些精度要求不高的工程中得到了廣泛應(yīng)用。（2）隨機(jī)有限元法是采納有限元與概率統(tǒng)計(jì)相結(jié)合的方法，由于有限元法本身全離散的特性，使咨詢題求解的未知數(shù)大大增加，因而不管是基于攝動(dòng)解或一次二階矩的隨機(jī)有限元，依舊基于統(tǒng)計(jì)方法的隨機(jī)有限元，都不可幸免地存在著運(yùn)算量過大和精度不易操縱的咨詢題，盡管近年一些學(xué)者如吳世偉、劉寧、龔曉南等對(duì)隨機(jī)有限元法進(jìn)行了深入的研究，但實(shí)際應(yīng)用中具有復(fù)雜性。（3）概率矩點(diǎn)估量法具有不必預(yù)先明白輸入隨機(jī)變量的精確分布，應(yīng)用方便的特點(diǎn)，但其精度和有用性還有待進(jìn)一步深入研究。（4）目前，蒙特卡羅法在可靠度分析中得到了廣泛而深入的應(yīng)用，其優(yōu)點(diǎn)是回避了結(jié)構(gòu)可靠度分析中的數(shù)學(xué)難題，不需考慮功能函數(shù)的非線性和極限狀態(tài)曲面的復(fù)雜性，為解決許多難以用傳統(tǒng)的數(shù)學(xué)方法進(jìn)行處理的復(fù)雜咨詢題提供了一條有效而又可行的途徑。與其它可靠度分析方法相比，具有相對(duì)精確的特點(diǎn)，只要模擬的次數(shù)足夠多就能夠得到一個(gè)比較精確的失效概率和可靠指標(biāo)；同時(shí)，蒙特卡羅法分析可靠度咨詢題，受咨詢題條件限制的阻礙較小，其收斂性與極限狀態(tài)方程的非線性、變量分布的非正態(tài)性無關(guān)，適應(yīng)性強(qiáng)，而且由于思路簡(jiǎn)單，易于編制運(yùn)算程序。缺點(diǎn)是運(yùn)算工作量大，效率較低，但隨著抽樣技術(shù)的改進(jìn)和運(yùn)算機(jī)硬件水平的提升，這一缺點(diǎn)正在大大弱化，該方法的應(yīng)用將越來越廣泛。按照以上分析，利用蒙特卡羅方法進(jìn)行黃河大堤滲流及邊坡穩(wěn)固咨詢題的分析具有一定的優(yōu)越性，因此選用蒙特卡羅方法進(jìn)行黃河大堤安全評(píng)判咨詢題的分析研究。3可靠度理論在黃河大堤安全評(píng)判中的實(shí)施方法3.1邊坡穩(wěn)固可靠度分析3.1.1邊坡穩(wěn)固可靠度分析原理簡(jiǎn)介在工程結(jié)構(gòu)可靠度分析中，結(jié)構(gòu)的極限狀態(tài)方程通常用功能函數(shù)來描述，當(dāng)有個(gè)隨機(jī)變量阻礙結(jié)構(gòu)的可靠度時(shí)，功能函數(shù)可用下式表示：(1)當(dāng)時(shí)，結(jié)構(gòu)處于可靠狀態(tài)；時(shí)，結(jié)構(gòu)處于極限狀態(tài)；時(shí)，結(jié)構(gòu)處于失效狀態(tài)。結(jié)構(gòu)失效概率表示為：(2)式中，為失效概率，為均值，為標(biāo)準(zhǔn)差，為可靠指標(biāo)，為標(biāo)準(zhǔn)正態(tài)函數(shù)。在邊坡穩(wěn)固分析中，其功能函數(shù)通常表示如下：(3)式中，為抗滑力矩，為滑動(dòng)力矩，為安全系數(shù)。邊坡可靠指標(biāo)表示如下：(4)式中，、分不表示安全系數(shù)的均值和標(biāo)準(zhǔn)差。3.1.2邊坡穩(wěn)固蒙特卡羅法運(yùn)算程序編制由上可知，蒙特卡羅法與其它方法相比具有編程簡(jiǎn)便、結(jié)果精確度高等更多優(yōu)點(diǎn)，且其運(yùn)算工作量大的缺點(diǎn)在今天運(yùn)算機(jī)性能大幅提升的情形下已差不多得到解決，因此我們選用了蒙特卡羅法進(jìn)行邊坡可靠度分析的運(yùn)算，同時(shí)邊坡穩(wěn)固分析模型應(yīng)用經(jīng)典的極限平穩(wěn)理論，采納了畢肖普法和瑞典法。編程步驟如下：1）輸入各隨機(jī)變量統(tǒng)計(jì)特點(diǎn)和分布類型；2）采納協(xié)方差矩陣將有關(guān)變量空間轉(zhuǎn)換為不有關(guān)變量；3）隨機(jī)產(chǎn)生一組平均數(shù)并生成服從變量分布規(guī)律的一組參數(shù)；4）通過逆變換生成初始變量互不有關(guān)的一組參數(shù)；5）分不代入畢肖普和瑞典法功能函數(shù)，重復(fù)次，統(tǒng)計(jì)失效次數(shù)，并運(yùn)算失效概率；6）檢查失效概率的穩(wěn)固性，必要時(shí)增加抽樣次數(shù)，重復(fù)運(yùn)算；7）按失效概率運(yùn)算可靠指標(biāo)。3.2滲流穩(wěn)固可靠度分析3.2.1滲流操縱設(shè)計(jì)適用原則和隨機(jī)變量的確定按照水工結(jié)構(gòu)極限狀態(tài)設(shè)計(jì)原則，水工結(jié)構(gòu)應(yīng)按承載能力極限狀態(tài)和正常使用極限狀態(tài)進(jìn)行設(shè)計(jì)。有關(guān)標(biāo)準(zhǔn)規(guī)定當(dāng)土石結(jié)構(gòu)或地基產(chǎn)生滲透失穩(wěn)時(shí)，為超過了承載能力極限狀態(tài)情形之一。因此對(duì)黃河大堤滲流穩(wěn)固性分析時(shí)對(duì)應(yīng)于承載能力極限狀態(tài)，并應(yīng)符合有關(guān)標(biāo)準(zhǔn)的要求。當(dāng)利用可靠性理論研究滲流穩(wěn)固性時(shí)，原則上是將荷載效應(yīng)或稱作用效應(yīng)以及材料特性及結(jié)構(gòu)的幾何尺寸等視為隨機(jī)變量。鑒于黃河大堤的復(fù)雜性和不確定性要緊在于堤身和堤基土壤材料特性，為使咨詢題只是于復(fù)雜，同時(shí)又抓住咨詢題的本質(zhì)，因此僅將土壤滲透系數(shù)和密度視為隨機(jī)變量，而將大堤上下游水位及其斷面尺寸視為確定值，來研究黃河大堤滲流穩(wěn)固性。3.2.2滲流穩(wěn)固性功能函數(shù)與滲透破壞概率當(dāng)采納有限單元法分析滲流穩(wěn)固性時(shí)，研究的對(duì)象是可能破壞區(qū)域內(nèi)的每個(gè)土體單元。由于黃河大堤土壤差不多上屬于非管涌土，因此按下式判定背河堤腳以外區(qū)域單元土體的滲透穩(wěn)固性：(5)式中R為作用于被研究單元底面上的有效土重；S為作用于被研究單元底面上的滲透壓力。由于土壤密度的隨機(jī)性，因此R是一隨機(jī)變量；同樣當(dāng)大堤上下游水位和斷面尺寸被確定后，S僅由大堤堤身和堤基的滲透性所確定，當(dāng)土壤滲透性參數(shù)為隨機(jī)變量時(shí)，S也為一隨機(jī)變量，盡管S與土壤滲透性參數(shù)的函數(shù)關(guān)系不能用顯式寫出。實(shí)際上上式即為研究大堤滲流穩(wěn)固性的功能函數(shù)。當(dāng)Z=R－S＞0時(shí)，土體單元處于滲流穩(wěn)固狀態(tài)；當(dāng)Z=R－S=0時(shí)，土體單元處于滲流極限狀態(tài)，此式即土體滲流極限狀態(tài)方程；當(dāng)Z=R－S＜0時(shí)，土體單元處于滲流失效狀態(tài)，由于土壤滲透失穩(wěn)時(shí)稱為滲透破壞，故本文稱為滲透破壞狀態(tài)。如果能求出R、S的概率分布函數(shù)（概率密度函數(shù)），則土壤滲透破壞概率可由下式求出：(6)式中，為S的概率分布函數(shù)；為R的概率分布函數(shù)；為土壤滲透破壞概率；為土壤滲透穩(wěn)固可靠度?？梢姡?=1。3.2.3破壞概率與安全系數(shù)的關(guān)系當(dāng)R、S為兩個(gè)隨機(jī)變量，其均值分不用、表示，標(biāo)準(zhǔn)差分不用、表示。則功能函數(shù)Z的均值與標(biāo)準(zhǔn)差分不為(7)(8)當(dāng)R、S均為正態(tài)分布隨機(jī)變量時(shí)，功能函數(shù)Z亦為正態(tài)分布隨機(jī)變量，Z＜0的概率，即破壞概率為(9)上式可變?yōu)椋ㄆ渲校?10)由上式可見，t是標(biāo)準(zhǔn)隨機(jī)變量，僅為積分上限的函數(shù)。因此可記為引入可靠指標(biāo)(11)則，能夠?qū)С隹煽恐笜?biāo)與可靠度Pr的關(guān)系：(12)由（7）、（8）式代入（11）式得：（13）以下討論可靠指標(biāo)與安全系數(shù)的關(guān)系，由安全系數(shù)的定義：（14）對(duì)（12）式進(jìn)行變換：（15）由于（16）（17）VR、VS為隨機(jī)變量R、S的變異系數(shù)。因此（13）式變?yōu)椋海?8）可見，可靠指標(biāo)不僅與安全系數(shù)有關(guān)，而且與隨機(jī)變量的變異系數(shù)有關(guān)，也即安全系數(shù)確定后，工程結(jié)構(gòu)的可靠度或破壞概率與隨機(jī)變量的分布規(guī)律有關(guān)，并不是一個(gè)常量。因此從統(tǒng)計(jì)學(xué)觀點(diǎn)看，傳統(tǒng)的安全系數(shù)存在兩個(gè)咨詢題：一是沒有考慮阻礙工程結(jié)構(gòu)安全因素的隨機(jī)性質(zhì)，而靠體會(huì)或工程判定方法取值，帶有主觀因素；二是安全系數(shù)只與R和S的均值有關(guān)，這種表達(dá)方式不能反映工程結(jié)構(gòu)實(shí)際破壞情形。應(yīng)當(dāng)指出，上述結(jié)論是在R、S為正態(tài)分布隨機(jī)變量條件下得到的，如果R或S為非正態(tài)分布隨機(jī)變量，則結(jié)論仍會(huì)有一定的近似。關(guān)于滲透穩(wěn)固分析，單元滲透壓力取決于各土層的滲透系數(shù)，它是滲透系數(shù)的復(fù)雜函數(shù)，而滲透系數(shù)為對(duì)數(shù)正態(tài)分布。因此作為這些隨機(jī)變量函數(shù)的總滲透壓力不一定是正態(tài)分布。因此在以下的研究中直截了當(dāng)用破壞概率來表達(dá)黃河大堤滲流穩(wěn)固性。3.2.4滲流有限元運(yùn)算的蒙特卡羅法由概率定義知，某事件的概率可用大量試驗(yàn)中該事件發(fā)生的頻率來運(yùn)算。因此，能夠先對(duì)阻礙滲流穩(wěn)固可靠性的隨機(jī)變量進(jìn)行大量隨機(jī)取樣，然后用這些抽樣值一組一組進(jìn)行滲流有限元運(yùn)算，得出被研究單元總滲透壓力（S），再與該單元底部的有效土壤重（R）比較，確定是否發(fā)生滲透破壞,從而運(yùn)算出滲透破壞頻率，即得到該土體單元的破壞概率。設(shè)抽樣數(shù)為N，每組抽樣值的功能函數(shù)值為Zi，若Zi≤0的次數(shù)為L(zhǎng)，則滲透破壞概率為（19）因此在蒙特卡羅法中的破壞概率等于破壞頻率。為使上式達(dá)到一定的精度，N就必須取得足夠大，按照文獻(xiàn)[2]（20）參考文獻(xiàn)中取=10-3，因此N=105。也即每完成一個(gè)斷面的滲流穩(wěn)固可靠性分析須進(jìn)行10萬次滲流運(yùn)算。用蒙特卡羅法運(yùn)算可靠度咨詢題的關(guān)鍵是求已知分布隨機(jī)變量的隨機(jī)數(shù)。一樣須分兩步進(jìn)行。第一在開區(qū)間（0，1）上產(chǎn)生平均分布隨機(jī)數(shù)，然后在此基礎(chǔ)上變換為給定分布變量的隨機(jī)數(shù)。作如下變換即得到正態(tài)分布的隨機(jī)數(shù)。設(shè)隨機(jī)數(shù)un和un+1是（0，1）上兩個(gè)平均隨機(jī)數(shù)，則（21）和是標(biāo)準(zhǔn)正態(tài)分布N（0，1）上的兩個(gè)隨機(jī)數(shù)。如果隨機(jī)變量x是一樣正態(tài)分布N（mx，σx）則（22）而xn，xn+1是兩個(gè)符合N（mx，σx）分布的隨機(jī)數(shù)。關(guān)于對(duì)數(shù)正態(tài)分布的隨機(jī)變量，則利用隨機(jī)變量取自然對(duì)數(shù)后的均值和標(biāo)準(zhǔn)差，再利用公式求得對(duì)數(shù)值。然后取反對(duì)數(shù)，即得對(duì)數(shù)正態(tài)分布隨機(jī)數(shù)。為實(shí)現(xiàn)蒙特卡羅法滲流有限元運(yùn)算，利用原有滲流運(yùn)算程序研制了該法運(yùn)算程序。4結(jié)語(1)可靠度理論是建立在概率統(tǒng)計(jì)分析的基礎(chǔ)上，充分考慮了功能函數(shù)中運(yùn)算參數(shù)的隨機(jī)變異性及其有關(guān)性，比安全系數(shù)法更能反映工程實(shí)際，在對(duì)堤防工程的安全評(píng)估方面，具有極其重要的推廣應(yīng)用價(jià)值。（2）黃河大堤的安全咨詢題一直是人們專門關(guān)懷的，但用經(jīng)典的確定性分析方法往往難以模擬。采納可靠度理論成功揭示了堤防在滲流及邊坡穩(wěn)固方面存在的咨詢題，與實(shí)際出險(xiǎn)情形較為吻合，為堤防除險(xiǎn)加固設(shè)計(jì)提供了理論依據(jù)。（3）把可靠度理論應(yīng)用到黃河大堤安全評(píng)判滲流及邊坡穩(wěn)固運(yùn)算分析中進(jìn)行了研究工作，給出了具體的實(shí)施方法，該方法具有可行性、創(chuàng)新性。（4）由于可靠度理論本身的復(fù)雜性，以及堤防工程安全評(píng)判方法還沒有形成完整的體系?？煽慷壤碚撚糜诘谭拦こ贪踩u(píng)判方面的研究，仍還處于初級(jí)時(shí)期，今后可引入隨機(jī)場(chǎng)理論，有望得出更為合理成果。因此,需要進(jìn)一步對(duì)可靠度理論在堤防工程安全評(píng)判中的應(yīng)用進(jìn)行系統(tǒng)而深入的研究。參考文獻(xiàn)：[1]李青云、張建民等.長(zhǎng)江堤防工程安全評(píng)判的理論和方法研究.全國(guó)水利水電工程安全評(píng)判及病害治理技術(shù)交流會(huì),2004.12.[2]陳祖煜.土質(zhì)邊坡穩(wěn)固分析—原理·方法·程序.北京：中國(guó)水利水電出版社，2003.[3]毛昶熙.堤防滲流與防沖.北京：中國(guó)水利水電出版社，2003.[4]趙壽剛、楊小平等.黃河堤防邊坡穩(wěn)固性的可靠度分析.建筑科學(xué),2006,(3).[5]趙壽剛、常向前等.黃河標(biāo)準(zhǔn)化堤防滲流穩(wěn)固可靠性分析.巖土工程學(xué)報(bào),2007,(5).作者簡(jiǎn)介:趙壽剛（1971-），男，河北南皮人，高級(jí)工程師，要緊從事堤防安全及災(zāi)難防治方面的研究。聯(lián)系地址：河南省鄭州市順河路45號(hào)黃河水利科學(xué)研究院工程力學(xué)所郵政編碼：450003StudyonApplicationofReliabilityTheorytoSafetyEvaluationoftheYellowRiverDikeZhaoShou-gang1,ChangXiang-qian1,YangXiao-ping1,Yuanhua2(1.YellowRiverInstituteofHydraulicResearch,Zhengzhou,450003,China;2.HydrologyBureauofYellowRiverConservancyCommission,Zhengzhou,450003,China)Abstract:Forevaluatingthesafetyofthedike,itisnecessarytocarryouttheseepageandslopestabilityanalysis.Theresultsofseepageandslopestabilityanalysisaremuchrelatedtothegeotechnicalparameters.OnaccountofthecomplexgeologicalconditionsandthegreatvariabilityofsoilparametersoftheYellowRiverdike.Itisveryhardtocomeupwithactualresultsbydeterministicmethods.IfreliabilitytheoryisappliedforsafetyevaluationoftheYellowRiverdike,thevariabilityoftheparametersofsoilpropertiescanbefullyconsidered.Theresultswillbemoreinaccordingwiththeengneeringpractice.ByanalyzingthebasicmethodsofreliabilitytheoryandstudyingthespecificimplementationmethodsforthesafetyevaluationoftheYellowRiverdike,it’smuchvaluableandinnovativeforapplyingthemethod.Keywords:Reliabilitytheory;Seepage;Slope;SafetyEvaluation;MonteCarlomethod1IntroductionTheYellowRiverisaworld-famoussuspendedchannel.Andespeciallyinrecentyears,continuingwatervolumeisbecomingdry,andtherearelesswaterandmoresandintheriver.Relationshipbetweenwaterandsedimentareuncoordinated.AsthewanderingreachofthelowerYellowRiverhasnotbeenwellcontrolled,floodembankment,evensomehorizontalandslantingstreamswillhappeninthefloodseason.ItwillbeseriousthreattothesafetyoftheYellowRiverdike.AsconditionsofthelowerYellowRiverdikearecomplex,what’smore,thegeologicalconditionsofeachsectiondikeandriverregimehavedifferentcharacteristics.Thereforehowissafetyofeachdike?Andhowtorealizeobjectiveevaluationoftheproblemisalwaysaconcernedforpeople.Forevaluatingthesafetyofthedike,itisnecessarytoapplytheseepageandslopestabilityanalysis.Whethertheresultsoftheseepageandslopestabilityanalysisarereasonable,it’sgreatrelatedtothesoilparameters.OnaccountofcomplexgeologicalconditionsoftheYellowRiverdikeandseveralHeighteningthickofthelevee,homogeneousnatureofthesoilispoor.Evenifthesametypeofsoil,itsphysicalandmechanicalindicatorsarealsomuchdifference.Inviewofsafetyevaluationoftheseepageandslopestabilityofacertainsectionfromthesmallnumberofgeologicalexploringdataanalysisalone,itwillbedifficulttobefittotheactualsituation.Inmanycases,evennothingcanbeanalyzed.Reliabilitytheorycansolvetheaboveproblem.Basedonstatisticalprobability,itcanfullyconsiderthevariationofsoilparameters.Theresultofanalysiswillbemuchcorrespondedtothereality.Thebasicpointsofreliabilitytheoryaretomakethefactorsaffectingsafetyasrandomvariables,formingthefunctionandsolvingreliabilityorfailureprobabilityandevaluatingthesafetyoftheproject.Itgoeswithoutsayingthatifusingthetheory,firstlyalargeamountoftestdataneedtobeaccumulated.Accordingtothedata,theprobabilitydistributionofthevariousfactorsaffectingprojectsafetyarestudied,followedbytheneedforalotofcomputation.NowithasaccumulatedalargequantityofdatabyseveralexplorationfortheYellowRiverdike.Large-capacityandhigh-speedcomputerhasbeenwidelyused.Therefore,forapplyingreliabilitytheorytosafetyevaluationoftheYellowRiverdike,theconditionshavebeenmatured.2Introductionofreliabilityanalysismethods2.1ReliabilityanalysismethodsAtpresent,reliabilityanalysismethodsincludemainlyLinearsecondordermomentmethod,stochasticfiniteelementmethod,Probabilitymomentpointestimationmethod(alsoknownRosenblueth),Monte-Carlosimulationmethod.(1)LinearsecondordermomentmethodLinearsecondordermomentmethodusesonlymathematicalmodelofthemeanandstandarddeviationtosolvethestructuralreliability.SpecificallystructurefunctionistoworkaccordingtoTaylorwithneglectinghigh-levelitemandonlyretaininglinearelement.Themeanandstandarddeviationareobtainedbyusingfirstordermomentandsecondordermomentofbasicrandomvariables,thendeterminingthereliableindexoftheprojectstructure.Becausethisapproachistodealwiththenonlinearfunctionforlinearfunction,itisanapproximatecalculationmethodofreliableindex.Butbecauseofitsconcisionandstrongapplicability,ithasbeenwidelyusedinengineeringpractice.Thefourmethodsarebasedonthelinearsecondordermoment.(2)StochasticfiniteelementBecausealargenumberofpracticalprojectproblemsareinfluencedbyrandomfactors.Therelationshipbetweenthesoilandstatisticalparametershasgreatnonlinearvariation.Manyscholarswerecommittedtotheapplicationoffiniteelementtheoryinengineeringresearch.Itsayssimplythatstochasticfiniteelementisamethodofapplyingthefiniteelementmethodtoanalyzetheproblemofrandomstructure.Accordingtothedifferentmethodsofderivingtheequations,stochasticfiniteelementmethodcanbedividedintoTaylorstochasticfiniteelement,finiteelementrandomperturbationmethod,therandomfiniteelementofNeumanSeriesMonte-Carlomethod.Theseveralmethodsareexpendedaroundthisstochasticoperatorandrandommatrixinthedeliveryissue.(3)ProbabilitymomentpointestimationmethodMexicansRosenbluethbroughtforwardtocalculatethegeotechnicalengineeringreliablityindexbypointestimatemethodin1975.In1981heimprovedthismethodandtheory.Therefore,theprobabilitymomentpointestimationmethodusuallycalledRosenbluethmethod.Itisprimarilybasedonfrontthreemomentoftherandomvariable(mean,variance,skewnesscoefficient)todescribetheprobabilitymomentoflimitstatefunction.Itneednotknowtheprecisedistributionoftherandomvariableinadvance.Rosenbluethmethodrequiressomecertainnumberofpointstoestimatevalueofthefunction.Thesepointsaregeneratedfromthemeanandstandarddeviationofrandomvariablesbysomecertainprinciples.Accordingtothefunctionvalueofpointestimatecandeterminereliableindexbycalculatingformula.(4)MonteCarlomethodThebasicprincipleofMonteCarlomethodisfirstlytotakeoutalarge-numbersample,thentakingintothemodel-function,thecalculatingfailuretimesarethepercentageofthetotalsamples,consequentlythereliableindexcanbeobtained.2.2AnalysiselectionofreliabilitymethodBasedonstudyofsomebasicreliabilitytheoryanalysismethods,thecharacteristicsofeachmethodwerediscussedbriefly.(1)Linearsecondordermomentmethodisasimplemethodofapproximatecalculationreliableindex.ButusingTaylorseriesexpandingtoignorethehigh-power,soitisanapproximatecalculationmethod,andtherearesomerestrictionconditions.Forexample,centralpointmethodadaptsonlytotheconditionofnormalorlog-normaldistributionandthestructuralreliabilityindex=1～2.Butmanyscholarshaveimprovedthemethod.Becausethecalculationissimple,themethodhasbeenwidelyusedforsomelowrequiredprojects.(2)Stochasticfiniteelementmethodusesthefiniteelementmethodwiththecombinationofprobabilityandstatistics.Duetothewholediscretecharacteristicoffiniteelementmethoditself,unknownfactorsofsolvingtheproblemaregreatlyincreased,thusnotonlythestochasticfiniteelementbasedontheperturbationsolutionorlinearsecondordermoment,butalsothestochasticfiniteelementbasedonstatisticmethodhaveallinevitablytheproblemofcontrolaccuracyandmorecalculationlabour.AlthoughinrecentyearssomescholarssuchasWuShiwei,LiuNingandGongXiaonanetc.allstudydeeptothestochasticfiniteelementmethod.However,it’sstillcomplexityfortheactualapplication.(3)Rosenbluethmethodhasthefeaturesthatcannotknowinadvancetherandomvariableandtheexactdistributionofconvenient,but,itsaccuracyandpracticalitystillneedfurtherstudy.(4)NowtheMonteCarlomethodhasbeenusedwidelyanddeeplyinreliabilityanalysis.Theadvantageisthatitavoidesthemathematicalproblemsofstructuralreliabilityanalysiswithoutconsideringthenonlinearfunctionandthecomplexityoflimitstatesurface.Forsolvingmanydifficultproblemsthatcan’tbedealedwithbyusingtraditionalmathematicalmethods,itprovideaneffectiveandfeasibleway.Reliabilityanalysiscomparedwithothermethods,itisrelativelyprecise.Itcanobtainmoreaccurateandreliableindexofthefailureprobabilitybyonlysufficientnumberofsimulation.Meanwhile,byapplyingMonteCarlotoanalyzethereliabilityproblems,theinfluencerestrictedbyismoresmaller,Itsconvergenceandultimatestateofnonlinearequationsaren’trelatedtothevariablesofthenon-normaldistribution.Ithastheadvantageofthehighadaptabilityandsimpleideasandeasypreparationprogram.Thedrawbackishighworkloadandlessefficient.Butwiththeimprovementofsamplingtechniqueandcomputerhardwarelevel,thisshortcomingisgreatlyweakened.Themethodwillbeusedwidely.Basedontheaboveanalysis,ithassomeadvantagesforapplyingMonteCarlomethodtoanalyzeseepageandslopestabilityoftheYellowRiverdike.MonteCarlomethodshouldbeselectedforthesafetyevaluationanalysisoftheYellowRiverdike.3ImplementationmethodofreliabilitytheorytosafetyevaluationoftheYellowRiverdike3.1Reliabilityanalysisofslopestability3.1.1PInthereliabilityanalysisoftheengineeringstructure,thestructurelimitingconditionequationcanusuallybedescribedbyfunction,whennrandomvariableinfluencestructurereliability,thefunctionequationcanbeindicatedasfollows(1)When,thestructureisatthereliablecondition;When,thestructureisatthelimitingcondition;When,thestructureisatthefailuremode.Structuralfailureprobabilityisexpressedas:(2)Whereisthefailureprobability;istheaveragevalue;isthestandarddeviation;isthereliableindex;isthestandardnormalmodefunction.Inslopestabilityanalysis,anditsfunctioncanbeindicatedasfollows:(3)Whereistheglide-resistancemoment;istheglidemoment;isthesafetycoefficient.Slopereliableindicatorcanbeindicatedasfollows:(4)3.1.2MonteCarloCalculationProgramofSlopestabilityFromtheaboveresult,comparedtheMonteCarlomethodwithothermethods,ithastheprogrammingsimpleandtheresultsofhigherprecision.Although,ithastheshortcomingsofworkloadsubstantially,withimprovingtheperformanceoftoday'scomputer,theproblemhasbasicallybeensolved.Therefore,wechoosetheMonteCarlomethodforreliabilityanalysisoftheslope.MeanwhileItappliesclassicallimitequilibriumtheory(theBishopandSweden)forslopestabilityanalysismodel.ThefollowingstepsofProgrammingaredescribedasfollows:1)Inputtingthedistributiontypeandstatisticcharacteristicofeachrandomvariable.2)Relatedvariableswillbeconvertedtonon-space-relatedvariablesbycovariancematrixmethod.3)Generatingauniformrandomnumberandagroupofvariableparametersofobeyingdistribution.4)Obtainingagroupofparametersofindependentinitialvariablesbyinversetransform.5)IncorporatingintotheBishopandSwedenfunction,repeatingntimes,calculatingstatisticalfailurerateandthefailureprobability.6)CheckingfailureProbabilitystability,ifnecessary,increasingthesamplenumberandcalculatingrepeatedly.7)Calculatingreliabilityindexasthefailureprobability.3.2Reliabilityanalysisofseepagestability3.2.1PrincipleofseepagecontroldesignandrandomvariablesdefinedAccordingtolimitstatedesignprincipleofhydraulicstructure,hydraulicstructureshouldbedesignedasthelimitstateandnormallimitcarryingcapacity.Whenitoccurredunstabletoduetoseepageinthesoil-rockstructureorfoundationastherelatedstandardsregulation,itisregardedasexceedingthelimitcarryingcapacity.ThereforeanalyzingseepageoftheYellowRiverdike,thelimitcarryingcapacitystateshouldbeconsideredandcanbeconformtotherelevantstandards.Whenusingthereliabilitytheorytostudytheseepagestability,theloadingeffect,materialpropertiesandstructuregeometryareallregardedasrandomvariables.AsthecomplexityanduncertaintyoftheYellowRiverleveearemainlycausedbythesoilmaterialpropertiesofdikereachesandfoundation.Formakingtheproblemeasierandgraspingtheessenceoftheproblem,sothesoildensityandpermeabilityareonlyregardedasrandomvariables.Butthedownstreamwaterlevelandthecross-sectionaldimensionsaredeterminedascertainvalueinordertostudytheseepagestabilityoftheYellowRiverdike.3.2.2SeepagestabilityfunctionandseepagefailureprobabilityWhenusingthelimitedunitmethodforanalyzingseepagestability,theresearchobjectmaybeeachsoilunitofdestroyedarea.BecausethemainsoiloftheYellowRiverdikebasicallybelongstothenon-pipingearth,thereforetheseepagestabilityoftheunitsoilbodyoutsidethebackstreambankfootarejudgedasthefollowingequation:(5)whereRistheeffectivesoilweightloadedontheunitbottomsurface;Sistheosmoticpressureontheunitbottomsurface.Duetotherandomnessofsoildensity,Risarandomvariable;Whenupstreamanddownstreamwaterlevelandcross-sectionaldimensionsofthedikeweredetermined,theSvalueisonlyobtainedbythepermeabilityofthefoundationandreachesoflevee.Whenthesoilpermeabilityparametersarerandomvariables,theSvalueisrandomvariabletoo.AlthoughfunctionalrelationbetweenSandsoilpermeabilityparameterscannotwrittenasexplicitformula.Infact,theaboveformulaisthefunctionofstudyingseepagestabilityoflevee.WhenZ=R-S>0,thesoilunitsareinsteadystateofseepage;WhenZ=R-S=0,thesoilunitsareinlimitstate,theformulaistheequationoflimitstateofsoilseepage;WhenZ=R-S<0,thesoilunitsareinstateoffailure.Asresultsofinstabilityinsoilinfiltration,itcanbecalledthestateofinfiltrationdamage.IfRandS,theprobabilitydistributionfunction(probabilitydensityfunction)arecalculated,thesoilinfiltrationdamageprobabilitycanbeobtainedasfollows:(6)WhereistheprobabilitydistributionfunctionofS;istheprobabilitydistributionfunctionofR;istheprobabilityofsoilinfiltrationdamage;isthereliabilityofsoilinfiltrationstability.Obviously,+=1.3.2.3Relationbetweenfailureprobabilityandsafetyfactor(7)(8)WhentheRandSarerandomvariablesofnormaldistribution,Z-functionisalsorandomvariableofnormaldistribution.TheprobabilityofZ<0isnamelythedamageprobability(9)Theaboveequationcanbechanged（）(10)Fromtheaboveformula,thevalueoftisastandardrandomvariable；isaceilingintegralfunction.ThereforeitcanbecreditedasIntroducingreliableindicator(11)ThenTherelationbetweenreliableindexandreliability(Pr)canbededuced:(12)Putting(7)and(8)typeintotheformulaof(11),obtained:(13)Discussingtherelationbetweenreliableindexandsafetyfactor,anddefinitingforsafetyfactor:(14)Transformingthetypeof(12):(15)As(16)(17)VRandVSarevariationcoefficientofrandomvariableRandS.Therefore,(13)theformulaisgiven:(18)Itshowsthatthereliabilityindexisnotonlyrelatedtothesafetyfactor,butalsovariationcoefficientoftherandomvariable.Afterthesafetyfactorisdetermined,thereliabilityofengineeringstructureordamageprobabilityarerelatedtothedistributionofrandomvariables.Itisn’taconstant.Sofromstatisticalview,therearetwoproblemsinthetraditionalsafetyfactor.Firstly,itdoesn’tconsiderrandampropertiesaffectingthesafetyfactorsofengineeringstructure,andonlydependingonexperienceorengineeringjudgment.Ithassubjectivefactors.Secondly,thesafetyfactorisonlyrelatedtotheaveragevalueRandS.Thismethodcan’treflecttheactualstructuraldamage.ItshouldbenotedthattheaboveconclusionsareobtainedwhenRandSaretherandomvariablesofnormaldistribution.IfRorSisrandomvariableofnon-normaldistribution,theconclusionsarestillapproximate.Forseepagestabilityanalysis,theunitosmoticpressuredependsonthesoilpermeabilitycoefficient.Itisacomplexfunction.Butthepermeabilitycoefficientislog-normaldistribution.Sofunctionoftotalosmoticpressureisnotnecessarilynormaldistribution.Therefore,failureprobabilitydirectyexpressseepagestabilityoftheYellowRiverdikeinthefollowingresearch.3.2.4MonteCarlofiniteelementmethodofseepagecalculationKnownfromthedefinitionofprobability,theprobabilityofsomeincidentscanbecalculatedbythefrequencyoftheincidentfromlargenumberoftests.Thus,alargenumberofrandomsamplesmayfirstlybechosefromrandomvariablesofaffectingtheseepagestabilityreliability,Thenanalyzingseepagecalculationoffiniteelementmethodforeachgroupofrandomvalue,thetotalosmoticpressure(S)ofunitcanbeobtainedbytheresearch.Thencomparingwiththebottomeffectivesoilweight(R),determiningwhetherseepagefailurewilloccur.Andcalculatingthedamagefrequency,lastlythefailureprobabilitycanbeobtained.Supposingthesamplingnumber(N),functionvalueofeachgroupofsamplingisZi,iftimeofZi≤0isL,thentheseepagefailureprobabilityis(19)Therefore,thefailureprobabilityoftheMonteCarlomethodisequivalenttofailurefrequency.Formakingaboveformulahascertainaccuracy,Nmustbegivenenoughlarge.Accordingtoreference[2](20)Given=10-3,soN=105..whenfinishingcompletereliabilityanalysisofeachsection,theseepagestabilitymustbecalculatedfor100,000times.TherandomnumberofknowndistributionisthekeytoMonteCarlomethodforcalculatingreliability.Generallyitisdividedintotwosteps.Firstlygeneratingrandomnumberofuniformlydistributedisintheinterval(0,1).onthebasis,transformingintotherandomnumberofgivendistribution.Therandomnumberofnormaldistributioncanbeobtainedbythefollowingtransformation.Supposingrandomnumberunandun+1aretwoevenrandomnumbersin(0,1),then(21)andaretworandomnumbersofstandard