外文翻譯---橋梁工程的發(fā)展概況

1、<p><b>　　附錄：外文翻譯</b></p><p>　　Evolvement of bridge Engineering,brief review</p><p>　　Among the early documented reviews of construction materials and structure types are the boo

2、ks of Marcus Vitruvios Pollio in the first century B.C.The basic principles of statics were developed by the Greeks , and were exemplified in works and applications by Leonardo da Vinci,Cardeno,and Galileo.In the fifteen

3、th and sixteenth century, engineers seemed to be unaware of this record , and relied solely on experience and tradition for building bridges and aqueducts .The state of the art changed rap</p><p>　　Kuzmanovi

4、c(1977) focuses on stone and wood as the first bridge-building materials. Iron was introduced during the transitional period from wood to steel .According to recent records , concrete was used in France as early as 1840

5、for a bridge 39 feet (12 m) long to span the Garoyne Canal at Grisoles, but reinforced concrete was not introduced in bridge construction until the beginning of this century . Prestressed concrete was first used in 1927.

6、</p><p>　　Stone bridges of the arch type (integrated superstructure and substructure) were constructed in Rome and other European cities in the middle ages . These arches were half-circular , with flat arche

7、s beginning to dominate bridge work during the Renaissance period. This concept was markedly improved at the end of the eighteenth century and found structurally adequate to accommodate future railroad loads . In terms o

8、f analysis and use of materials , stone bridges have not changed much ,but the theo</p><p>　　Wooden trusses were used in bridges during the sixteenth century when Palladio built triangular frames for bridge

9、spans 10 feet long . This effort also focused on the three basic principles og bridge design : convenience(serviceability) ,appearance , and endurance(strength) . several timber truss bridges were constructed in western

10、Europe beginning in the 1750s with spans up to 200 feet (61m) supported on stone substructures .Significant progress was possible in the United States and Russia duri</p><p>　　The transition from wooden brid

11、ges to steel types probably did not begin until about 1840 ,although the first documented use of iron in bridges was the chain bridge built in 1734 across the Oder River in Prussia . The first truss completely made of ir

12、on was in 1840 in the United States , followed by England in 1845 , Germany in 1853 , and Russia in 1857 . In 1840 , the first iron arch truss bridge was built across the Erie Canal at Utica . </p><p>　　The

13、 Impetus of Analysis </p><p>　　The theory of structures </p><p>　　The theory of structures ,developed mainly in the ninetheenth century,focused on truss analysis, with the first book on bridges

14、 written in 1811. The Warren triangular truss was introduced in 1846 , supplemented by a method for calculating the correcet forces .I-beams fabricated from plates became popular in England and were used in short-span br

15、idges.</p><p>　　In 1866, Culmann explained the principles of cantilever truss bridges, and one year later the first cantilever bridge was built across the Main River in Hassfurt, Germany, with a center span

16、of 425 feet (130m) . The first cantilever bridge in the United States was built in 1875 across the Kentucky River.A most impressive railway cantilever bridge in the nineteenth century was the First of Forth bridge , bui

17、lt between 1883 and 1893 , with span magnitudes of 1711 feet (521.5m).</p><p>　　At about the same time , structural steel was introduced as a prime material in bridge work , although its quality was often po

18、or . Several early examples are the Eads bridge in St.Louis ; the Brooklyn bridge in New York ; and the Glasgow bridge in Missouri , all completed between 1874 and 1883.</p><p>　　Among the analytical and des

19、ign progress to be mentioned are the contributions of Maxwell , particularly for certain statically indeterminate trusses ; the books by Cremona (1872) on graphical statics; the force method redefined by Mohr; and the wo

20、rks by Clapeyron who introduced the three-moment equations.</p><p>　　The Impetus of New Materials</p><p>　　Since the beginning of the twentieth century , concrete has taken its place as one of t

21、he most useful and important structural materials . Because of the coMParative ease with which it can be molded into any desired shape , its structural uses are almost unlimited . Wherever Portland cement and suitable ag

22、gregates are available , it can replace other materials for certain types of structures, such as bridge substructure and foundation elements .</p><p>　　In addition , the introduction of reinforced concrete i

23、n multispan frames at the beginning of this century imposed new analytical requirements . Structures of a high order of redundancy could not be analyzed with the classical methods of the nineteenth century .The importanc

24、e of joint rotation was already demonstrated by Manderla (1880) and Bendixen (1914) , who developed relationships between joint moments and angular rotations from which the unknown moments can be obtained ,the so called

25、slope</p><p>　　One of the most import important recent developments in the area of analytical procedures is the extension of design to cover the elastic-plastic range , also known as load factor or ultimate

26、design. Plastic analysis was introduced with some practical observations by Tresca (1846) ; and was formulated by Saint-Venant (1870) , The concept of plasticity attracted researchers and engineers after World War Ⅰ , ma

27、inly in Germany , with the center of activity shifting to England and the United States </p><p>　　A main step forward was the 1969 addition of the Federal Highway Adiministration (FHWA)”Criteria for Reinforc

28、ed Concrete Bridge Members “ that covers strength and serviceability at ultimate design . This was prepared for use in conjunction with the 1969 American Association of State Highway Offficials (AASHO) Standard Specifica

29、tion, and was presented in a format that is readily adaptable to the development of ultimate design specifications .According to this document , the proportioning of reinf</p><p>　　Bridge Types </p>&

30、lt;p>　　A notable bridge type is the suspension bridge , with the first example built in the United States in 1796. Problems of dynamic stability were investigated after the Tacoma bridge collapse , and this work led t

31、o significant theoretical contributions Steinman ( 1929 ) summarizes about 250 suspension bridges built throughout the world between 1741 and 1928 .</p><p>　　With the introduction of the interstate system an

32、d the need to provide structures at grade separations , certain bridge types have taken a strong place in bridge practice. These include concrete superstructures (slab ,T-beams,concrete box girders ), steel beam and plat

33、e girders , steel box girders , composite construction , orthotropic plates , segmental construction , curved girders ,and cable-stayed bridges . Prefabricated members are given serious consideration , while interest in

34、box section</p><p>　　LOADS AND LOADING GROUPS</p><p>　　The loads to be considered in the design of substructures and bridge foundations include loads and forces transmitted from the superstructu

35、re, and those acting directly on the substructure and foundation .</p><p>　　AASHTO loads . Section 3 of AASHTO specifications summarizes the loads and forces to be considered in the design of bridges (supers

36、tructure and substructure ) . Briefly , these are dead load ,live load , iMPact or dynamic effect of live load , wind load , and other forces such as longitudinal forces , centrifugal force ,thermal forces , earth pressu

37、re , buoyancy , shrinkage and long term creep , rib shortening , erection stresses , ice and current pressure , collision force , and earthquake stre</p><p>　　Permanent loads </p><p>　　Dead Load

38、 : this includes the weight DC of all bridge components , appurtenances and utilities, wearing surface DW and future overlays , and earth fill EV. Both AASHTO and LRFD specifications give tables summarizing the unit weig

39、hts of materials commonly used in bridge work .</p><p>　　Transient Loads </p><p>　　Vehicular Live Load (LL) </p><p>　　Vehicle loading for short-span bridges :considerable effort has

40、 been made in the United States and Canada to develop a live load model that can represent the highway loading more realistically than the H or the HS AASHTO models . The current AASHTO model is still the applicable load

41、ing.</p><p>　　Size Effects and the Dynamic Response of Plain Concrete</p><p>　　In the last couple of decades, there have been numerous reports Ba?ant 1984; Carpinteri and Chiaia 1997; Karihaloo

42、1999; Jenq and Shah 1985 about the specimen size effects in quasi-brittle materials. For these materials, Ba?ant states that the source of the size effect is a mismatch between the size dependence of the energy release r

43、ate and the rate of energy consumed by fractureBa?ant 2000 . Whereas a significant portion of the former in- creases as the square of the specimen size, the latter i</p><p>　　In this paper, the size effect o

44、n the impact response of concrete is presented through an assessment of recently published data by the writers and others. Familiar scaling laws developed for quasi- static loading are examined in the context of dynamic

45、stress rates. This paper discusses the interplay between the specimen size, matrix strength, stress rate sensitivity, and loading configuration.</p><p>　　Scaling Laws for Quasi-Brittle Systems</p><

46、;p>　　It is well known that the quasi-static response of plain concrete is affected by the size of the specimen. Evidence gathered over decades reveals a strong dependence on size for structural con- crete behavior und

47、er compression Sabnis and Mirza 1979 , ten- sion Ba?ant et al. 1991; van Mier and van Vliet 2002 , flexureWright 1952; Ba?ant and Li 1995; Jueshi and Hui 1997 , shear Ba?ant and Sun 1987 , and torsion Zhou et al. 1998

48、 . Three approaches dominate the study of size effects in quasi-brit</p><p>　　1.The statistical theory of random strength;</p><p>　　2.The theory of stress redistribution and fracture energy re

49、lease caused by large cracks; and</p><p>　　3.The theory of crack fractality.</p><p>　　Ba?ant’s Size Effect Law ?Ba?ant 1984…</p><p>　　According to Ba?ant, the size effect in solids

50、 is a smooth transi- tion from the strength criterion of plasticity applicable to small size specimens to the crack size dependence of linear elastic fracture mechanics LEFM as seen in much larger specimens . The f

51、ailure stress of a series of geometrically similar specimens of concrete is described by the following infinite series:</p><p>　　Multifractal Scaling Law ?Carpinteri and Chiaia 1997…</p><p>　　Ca

52、rpinteri and his associates used the concept of self-similar morphologies with noninteger dimensions called fractals to de- scribe the microstructure of quasi-brittle materials such as con- crete. With an increase

53、in the scale of observation, the topological fractality is thought to vanish. As the microstructure of a hetero- geneous material remains the same regardless of size, they pro- posed that the influence of macroscopic siz

54、e on the mechanical properties was a result of the interac</p><p>　　where f t = asymptotic value of the nominal strength u at infinite sizes. As opposed to BSEL, MFSL appears to suit unnotched specimens,

55、 as they have a residual strength even for extremely large sizes.</p><p><b>　　橋梁工程的發(fā)展概況</b></p><p>　　早在公元前1世紀(jì)，Marcus Vitrucios Pollio 的著作中就有關(guān)于建筑材料和結(jié)構(gòu)類型的記載和評述。后來古希臘人創(chuàng)立了靜力學(xué)的基本原理， Leona

56、rdo da Vinci 、Cardeno和Galileo 等人在工作和應(yīng)用中也證實了這些原理的正確性。而在15世紀(jì)至16世紀(jì)期間，工程師們似乎并沒有注意到這些文字記載，只是單憑經(jīng)驗和傳統(tǒng)來建造橋梁和渡槽。到了17世紀(jì)末， 隨著Leibnitz、Newton 和Bernoulli的數(shù)學(xué)理論的創(chuàng)立，橋梁建筑技術(shù)得到了快速發(fā)展。Lahire (1695)和belidor（1729）出版的關(guān)于結(jié)構(gòu)理論分析的著作為材料力學(xué)領(lǐng)域奠定了基礎(chǔ)。<

57、;/p><p>　　Kuzmanovic （1977）指出，石材和木材是橋梁建筑最早采用的材料。在從木材到鋼材的轉(zhuǎn)變過程中，鐵作為一種過渡材料被用于橋梁建筑中。根據(jù)近期的記載。早在1840年，法國就在Grisoles 建造了一座跨度為39英尺（12米）的橫跨 Garoyne 運河的混凝土橋梁， 但鋼筋混凝土橋直到本世紀(jì)初才出現(xiàn)，而預(yù)應(yīng)力混凝土到1927年才開始使用。</p><p>　　早在中

58、世紀(jì)，羅馬和歐洲的其他一些城市開始建造集上下部結(jié)構(gòu)于一體的半圓弧石拱橋，而文藝復(fù)興時期則是坦拱逐漸占主導(dǎo)地位。這種觀念在18世紀(jì)末有了明顯的改進(jìn)，并發(fā)現(xiàn)其在結(jié)構(gòu)上能適應(yīng)后來的鐵路荷載。在材料的分析和使用上，石拱橋至今沒有發(fā)生大的變化，但是由于在17世紀(jì)70年代初期（Lahire,1965）引進(jìn)了壓力線的概念，使得拱橋的理論分析得到了改進(jìn)。通過模型試驗，有關(guān)拱結(jié)構(gòu)的主要失效形式的理論得到了證實（Frezier ,1739）。對于無鉸拱，C

59、ulmann (1851 ) 引進(jìn)了彈性中心的方法，顯示了可用三個協(xié)調(diào)方程求解三個多余參數(shù)。</p><p>　　當(dāng)palladio建造了一座跨度為10英尺的三角形木制框架橋后，16世紀(jì)開始，木桁架在橋梁中得到應(yīng)用。這些設(shè)計同樣遵循橋梁設(shè)計的三個基本原則：方便（實用性）、美觀和耐久性（強度）。18世紀(jì)50年代西歐建造了若干座支承于石制橋墩上的木桁架橋，其跨度達(dá)到200英尺（61米）。19世紀(jì)期間，美國和俄羅斯由于

60、其跨越主要河流的需要，而且兩國都具有豐富的適用于建橋的木材資源，因此木制橋梁在美、俄兩國有可能取得更為顯著的成績。木制橋梁具有良好的經(jīng)濟性，因為其初期投資較低，施工速度較快。</p><p>　　盡管有文獻(xiàn)記載，早在1734年，在普魯士就修建了第一座橫跨Oder河的鐵鏈橋，但從木橋到鋼橋的過渡大概開始于1840年。美國于1840年建成了第一座全鐵桁架橋，其后，英格蘭、德國和俄羅斯分別于1845年、1853年和18

61、57年也建成了鐵桁架橋。1840年，第一座鐵桁架拱橋出現(xiàn)在Utica的Erie運河上。</p><p>　　理論分析的推動作用 </p><p>　　主要從19世紀(jì)發(fā)展起來的機構(gòu)分析理論著重于桁架的分析，首部關(guān)于橋梁工程的著作于1811年出版。1846年出現(xiàn)了一種Warren 三角形桁架和計算這種桁架精確內(nèi)力的分析方法。用板件組合而成的工字形梁在英國逐漸普及并在小跨度橋梁中得到應(yīng)用。<

62、;/p><p>　　1866年Culmann闡述了懸臂桁架橋的原理，一年后在德國的Hassfurt的Main 河上就建造了首座主跨跨度達(dá)425英尺（130米）的懸臂梁橋。美國的首座懸臂梁橋于1875年建于Kentucky河上。19世紀(jì)最引人注目的鐵路懸臂梁橋要數(shù)Firth of Forth橋，此橋建于1883年至1890年間，跨度達(dá)1，711英尺（521.5米）大約就在這一時期，結(jié)構(gòu)鋼在橋梁工程中作為一種主要材料被推

63、廣應(yīng)用，盡管此時鋼材的性能大都較差。幾個早期的工程實例是：（1）St.Louis的Eads 橋；（2）New York的Brooklyn 橋；（3）Missouri 的 Glasgow 大橋，這些橋都建于1874年至1883年間。</p><p>　　談起對結(jié)構(gòu)分析河設(shè)計理論的改進(jìn)特別應(yīng)該提到：Maxwell 所作的貢獻(xiàn)，尤其是他在超靜定桁架方面的工作；Cremona 關(guān)于圖解靜力學(xué)的著作（1872）；由Mohr

64、 重新定義的力法以及Clapeyron 提出的三彎矩方程</p><p><b>　　新材料的推動作用</b></p><p>　　自從20世紀(jì)初起，混凝土就是一直是最有效和最重要的建筑材料之一。由于混凝土可以較容易地澆注成各種形狀的結(jié)構(gòu)物，因此它在建筑上的使用價值幾乎是無限的。只要有普通水泥和合適的骨料混凝土就可以替代其他材料建造某些類型的結(jié)構(gòu)，諸如橋梁下部結(jié)構(gòu)及基

65、礎(chǔ)等。</p><p>　　另外，在本世紀(jì)初，鋼筋混凝土在多跨框架結(jié)構(gòu)中的應(yīng)用對結(jié)構(gòu)分析提出了新的分析要求用19 世紀(jì)的古典分析方法不能用來分析高次靜定結(jié)構(gòu)。Manderla (1880)和Bendixen (1914)論證了節(jié)點轉(zhuǎn)角的重要性，提出了節(jié)點彎矩和轉(zhuǎn)角之間的關(guān)系，從而可求解未知的節(jié)點彎矩，這種方法被稱為轉(zhuǎn)角－撓度法。 Calisev (1923)的工作使得框架結(jié)構(gòu)的分析有可能進(jìn)一步簡化，他利用逐步近似

66、的方法將方程組的求解簡化為一個簡單表達(dá)式的迭代計算。Cross (1930)進(jìn)一步改進(jìn)和歸納了這種方法，從而形成了彎矩分配法。</p><p>　　在結(jié)構(gòu)分析領(lǐng)域的近期發(fā)展中最重要的改進(jìn)之一是將設(shè)計的范圍延伸到彈塑性范圍，即所謂的荷載因子法或極限狀態(tài)設(shè)計法。Tresca (1846) 根據(jù)一些世紀(jì)觀察結(jié)果提出了塑性分析法，Saint-Venant (1870)系統(tǒng)地闡述了這種分析方法。第一次世界大戰(zhàn)以后，塑性的概

67、念吸引著研究人員和工程師們的注意力，開始主要是在德國。二次世界大戰(zhàn)后，隨著科研學(xué)術(shù)重心的轉(zhuǎn)移，英國和美國的科研人員對此進(jìn)行了廣泛的研究。概率設(shè)計法是一種新的設(shè)計方法，這種方法有望替代傳統(tǒng)的確定性方法。</p><p>　　一個主要的進(jìn)步是1969版的美國聯(lián)邦公路管理局（FHWA）的“鋼筋混凝土橋梁勾踐設(shè)計準(zhǔn)則”中包括了強度和正常使用的極限狀態(tài)設(shè)計法。這本設(shè)計準(zhǔn)則是與“美國各州公路工作者協(xié)會（AASHO）”1969

68、年的設(shè)計規(guī)范聯(lián)合使用的，它的表達(dá)方式使其很容易適應(yīng)極限狀態(tài)設(shè)計規(guī)范的發(fā)展。根據(jù)這本設(shè)計準(zhǔn)則，鋼筋混凝土勾踐（包括柱）的配料可以通過其各個階段的工作性能來限定：彈性的、帶裂縫工作的極限狀態(tài)的。設(shè)計是荷載作用效應(yīng)，所有根據(jù)作用荷載計算所得的量叫做設(shè)計值，如：設(shè)計彎矩、設(shè)計軸載或或設(shè)計剪力。結(jié)構(gòu)的承載力被認(rèn)為是結(jié)構(gòu)抗力方面的參數(shù)，所有根據(jù)材料的理論強度計算得來并經(jīng)過修正得強度計算值叫做結(jié)構(gòu)抗力值，如：彎矩抗力值（抵抗彎矩），軸力抗力值或剪力抗

69、力值。在正常使用極限狀態(tài)下，需驗算構(gòu)件得撓度、最大裂縫寬度和疲勞強度。</p><p><b>　　橋型</b></p><p>　　一種值得注意得橋型是吊橋，首座吊橋1796年建于美國。隨著Tacoma大橋得跨塌，動力穩(wěn)定被作為問題來研究，并取得了顯著得理論成果。Steinman (1929) 總結(jié)了全世界建于1741年至1928年間得大約250座吊橋。</p

70、><p>　　隨著州際體系得建立和結(jié)構(gòu)等級分類的需要，某些橋型在橋梁界占有重要的地位。這些橋型包括混凝土上部結(jié)構(gòu)（板橋、T梁橋、混凝土箱梁橋）、鋼梁橋、鋼箱梁橋、組合界哦故、正交異性板結(jié)構(gòu)、分段施工的結(jié)構(gòu)、曲線梁橋和斜拉橋。預(yù)制構(gòu)件受到了足夠的重視，箱型截面梁也占有重要的地位。</p><p><b>　　荷載及荷載組合</b></p><p>　

71、　在橋梁下部結(jié)構(gòu)和基礎(chǔ)設(shè)計中要考慮的荷載包括：從上部結(jié)構(gòu)傳下來的荷載和直接作用于下部結(jié)構(gòu)的基礎(chǔ)的荷載。</p><p>　　AASHTO荷載 AASHTO規(guī)范第三部分總結(jié)了橋梁設(shè)計（上、下部結(jié)構(gòu)）要考慮的荷載和作用力。主要有：恒載、活載、活載沖擊力或動力作用、風(fēng)荷載以及其他力——如縱向力、離心力、溫度力、土壓力、浮力收縮及徐變、拱肋縮短、安裝應(yīng)力、冰及水流壓力、沖撞力及地震應(yīng)力。除了這些通常能夠量化大的典型荷載外

72、，AASHTO同樣認(rèn)識到諸如活動支座處產(chǎn)生的摩擦以及由于橋梁勾踐的沉降差而產(chǎn)生的應(yīng)力等間接荷載效應(yīng)。</p><p>　　LRFD規(guī)范將荷載劃分為截然不同的兩種：長期荷載和短期荷載。</p><p><b>　　長期荷載</b></p><p>　　荷載：包括所有橋梁構(gòu)件、器件及輔助設(shè)備、道路面層的凈重及未來鋪裝重量、填土恒載。AASHTO及L

73、RFD規(guī)范都給出了表格，總結(jié)了橋梁工程重常用才兩的單位重量。</p><p><b>　　短期荷載</b></p><p>　　汽車荷載 小跨度橋梁的汽車荷載：美國和加拿大已致力于發(fā)展一種比H或HS AASHTO模型更實際的代表高速公路活荷載的模型。到目前為止，AASHTO模型仍被廣泛采用。</p><p>　　素混凝土的尺寸效應(yīng)及動態(tài)響應(yīng)&l

74、t;/p><p>　　在過去的幾十年中，在過去的幾十年中，有許多關(guān)于準(zhǔn)脆性材料試件尺寸效應(yīng)的報告。針對這些材料，Ba?ant認(rèn)為源于裂紋能量吸收率和釋放率的不匹配。能量吸收率的增加的很大一部分隨著尺寸平方的增加而增加，然而釋放率的增加確實線性的。因此，通過減少標(biāo)本的能源釋放率，減少名義應(yīng)力被看作是補償差額的一種手段。</p><p>　　不同于準(zhǔn)靜態(tài)荷載，在動力學(xué)領(lǐng)域樣本尺寸效應(yīng)的研究還沒有獲

75、得很大的重視。這種嘗試是主要限于纖維增強聚合物。關(guān)于水泥基材料的數(shù)據(jù)極為稀缺并且近期已開始關(guān)注沖擊速率。設(shè)計規(guī)范及試驗標(biāo)準(zhǔn)方法的缺乏阻止我們賦予建筑材料抵抗沖擊和爆炸的能力。此外，沖擊試驗介紹一些不相干的影響，諸如慣性和試驗機的影響。也許最嚴(yán)重的障礙是水泥基復(fù)合材料的固有的應(yīng)力變化率敏感性。Morton說建立一個精確的速率敏感性材料的規(guī)模模型是不太可能的。此外，在動力作用下尺度模式的適宜性仍在研究中在本文中，我們將著重研究在高頻應(yīng)力作用

76、下水泥基材料的尺寸問題。在本文中, 關(guān)于混凝土沖擊反應(yīng)的尺寸效應(yīng)將通過對其他學(xué)者發(fā)表的數(shù)據(jù)的估計來闡明。為我們所熟悉的用于準(zhǔn)靜態(tài)荷載的標(biāo)度率在動應(yīng)力作用的背景下將被再度審視。這篇文章討論試樣尺寸、強度矩陣、應(yīng)力速率敏感性及加載形式之間的相互作用。</p><p><b>　　準(zhǔn)脆性系統(tǒng)的標(biāo)度律</b></p><p>　　眾所周知，素混凝土準(zhǔn)靜態(tài)響應(yīng)受試樣尺寸的影響。

77、幾十年來收集的資料的顯示了結(jié)構(gòu)用混凝土在壓縮、拉伸、彎曲、剪切、扭轉(zhuǎn)作用下其性能對尺寸強烈的依賴性。三種方法主導(dǎo)了對準(zhǔn)脆性材料尺寸效應(yīng)的研究：</p><p>　　1. 隨機強度的統(tǒng)計理論;</p><p>　　2.寬裂縫導(dǎo)致應(yīng)力重分布和斷裂能的釋放的理論3. 裂紋分形性理論。</p><p>　　Ba?ant尺寸效應(yīng)律（Ba?ant1994）據(jù)Ba?ant ，

78、固體的尺寸效應(yīng)是從適用于小尺寸試樣的塑性強度準(zhǔn)則向大尺寸試樣中常見線彈性斷裂力學(xué)裂縫尺寸依賴性的一種平穩(wěn)過渡。一系列幾何相似的混凝土試樣的破壞應(yīng)力通過下列無窮級數(shù)來描述：</p><p><b>　　} (1)</b></p><p>　　多邊標(biāo)度律（Carpinteri 和Chiaia 1997）</p><p>　　Carpinteri和他

79、的同事通過使用自相似形態(tài)學(xué)的概念和被稱作分形的非整數(shù)尺寸來描述諸如混凝土這樣的準(zhǔn)脆性材料的微觀結(jié)構(gòu)。隨著探測程度的加深，拓?fù)浞中涡员徽J(rèn)為將要消失。由于這一混合材料依然不受尺寸影響，因此他們假定宏觀尺寸對力學(xué)性能的影響是試樣尺寸b和特征長度相互作用的結(jié)果。在此假設(shè)基礎(chǔ)之上，建議如下所示的多分形標(biāo)定律（MFSL）：</p><p><b>　?。?）</b></p><p&g