【正文】 Tables 5 and 6) that the differences between the measured and calculated results decreased substantially when the nonlinear behaviors of ballast and subgrade materials, in particular nonrecoverable deformation of the track sublayers were considered. Experimental results obtained in this research indicate that the railseatload is almost proportional to the square of the rail displacement at any section (Fig. 7). This means that the experimental results contradict the assumption, used in the current practice, of a linear relationship between rail deflection and pressure under the rail. This highlights the importance and urgency of considering the nonlinear behaviors of the track substructures in railway track analysis and design. The proposed model is more accurate than the current methods and its implementation will improve the accuracy of the analysis and design of railway track systems. 12 Notation The following symbols were used in this paper: 13 14 References Ahmadian, M. T., Esmailzadeh, E., and Asgari, . “Dynamical stress distribution analysis of a nonuniform crosssection beam under moving mass.”Proc., ASME International Mechanical Engineering Congress and Exposition, ASME, New York. American Railway Engineering and MaintenanceofWay Association AREMA. 2021. Manual for railway track engineering, AREMA Pub. Service, USA. Dahlberg, . “Dynamic interaction between train and nonlinear railway track model.” Proc., 5th European Conf. on Structural Dy namics , 1155–1160. Fatemi, M. J., Green, M. F., Campbell, T. I., and Moucessian, A. 1996. “ Dynamic analysis of resilient crosstie track for transit system.” Proc. Am. Soc. Civ. Eng. , 122 2, 173–180. Moravcik, M. 1995. Response of railway track on nonlinear discrete support: Interaction of railway vehicles with the track and its substructure , Swets and Zeitlinger Publishers, Prague, Czech Republic. Sadeghi, J. 2021. “Modelling of dynamic behaviour of ballast and subgrade materials using semiempirical methods.” Proc., 2nd Int. Conf. on Mechanics of Structures, Materials and System . Sadeghi, J. 2021. “Experimental Evaluation of accuracy of current prac tices in analysis and design of railway track sleepers.” Can. J. . , 35, 881–893. Sadeghi, J., and Hashemi, F. 2021. “Influences of rail support conditions on mechanical behavior of railway track system.” Trans. Can. . Eng.,323–4 , 561–573. Selig, T. E., and Waters, J. M. 1994. Track geotechnology and substructure management, Telford, Derby, . 15 Stewart, H. E., and Orourke, T. D. 1988. “Load factor method for dyamic track loadings.” Proc. Am. Soc. Civ. Eng.,114 1, 21–39. Thompson, M. R., and Tayabji, D. 1976. “Track support system parameter study,” Rep. No. FRA/OR amp。所提出的方法被認(rèn)為是準(zhǔn)確和容易適用于鐵路軌道分析。 Ahmadian et al. 2021。解決這些問(wèn)題，對(duì)這里的軌道模型應(yīng)提出如下要求： ?納入軌道的縱向和橫向截面特性 。方程得以推導(dǎo)、求解過(guò)程的模型被創(chuàng)建了出來(lái)。 橫向分析了最大的軌枕，從縱向分析反應(yīng)力在于鎮(zhèn)流器，路基上的軌枕（鐵路座位負(fù)載）。同時(shí)為了達(dá)到這個(gè)目標(biāo)，還應(yīng)用了增量加載技術(shù)。如果在每個(gè)節(jié)點(diǎn) u和 v代表在 X和 Y方向的位移，并代表形狀函數(shù)，結(jié)點(diǎn)位移，位移函數(shù)可以寫(xiě)成 (3) 18 (4) (5) 其中ξ = X / A， = Y / B，其中 A =矩形元素的長(zhǎng)度的二分之一， B =矩形元素的寬度的一半。t =元素厚度隨深度的增加以模擬三維應(yīng)力的分布， 39。[電子 ]=應(yīng)力應(yīng)變關(guān)系矩陣。{K}=剛度矩陣，這是根據(jù)下列公式定義的： (2) 其中， [A]=應(yīng)變 位移變換矩陣， [E]中應(yīng)力應(yīng)變矩陣。在這種技術(shù)中，元素的厚度隨深度是多種多樣的裝載在垂直平面上的的擴(kuò)散與擴(kuò)散。代表對(duì)彈簧支撐連續(xù)梁鐵路軌枕子系統(tǒng)。納入軌道下部結(jié)構(gòu)參數(shù)的壓力相關(guān)的屬性，鎮(zhèn)流器和路基材料的彈性模量，被認(rèn)為是散裝應(yīng)力的函數(shù)。然而，在文學(xué)的非線性模型，不考慮主要由累計(jì)負(fù)荷造成的永久性軌道偏轉(zhuǎn)，和三個(gè)軌道系統(tǒng)的三維屬性的效果。大多數(shù)這些模型假定軌道下部結(jié)構(gòu)是線性建模為彈性基礎(chǔ)或一系列離散線性彈簧和阻尼器在垂直方向(Stewart and Orourke 1988。全面的實(shí)地測(cè)試被實(shí)施，以評(píng)估該模型的準(zhǔn)確性和可靠性。 []? = strain matrix at the end of the i th loading step。 d= length of the sleeper。 E = Young’s modulus； v= Poisson’s ratio。 published online on May 11, 2021. Discussion period open until May 1, 2021。 Fatemi et al. 1996。 Models。 ? Be cost efficient。 {U} resulting nodal displacement for the system。t is calculated as follows: where Z = depth of the element and ? = angle of distribution as shown in Fig. 3. In this research, the angle of distribution is assumed to be constant. where l = length of the element。 1? = major principal stress。 D78256, Federal Railroad Administration, DOT, Washington, ., 232–231. Turek, J. 1995. Interaction of railway vehicles with the track and its substructure , Swets and Zeitlinger Publishers, Prague, The Czech Republic. Zhu, J. . “Analysis of dynamic behavior of low vibration track under wheel load drop by a finite element method algorithm.” Proc. Inst. Mech. Eng., F J. Rail Rapid Transit , 222 2, 217–223. 中文譯文 非線性軌道模型改進(jìn)的平面應(yīng)變技術(shù)的開(kāi)發(fā)應(yīng)用 Javad Sadeghi1and Hossein Askarinejad2 摘要 ：鎮(zhèn)流器和路基層行為對(duì)路軌性能的貢獻(xiàn)在當(dāng)前鐵路軌道模型中給予了充分考慮。 DOI： CE數(shù)據(jù)庫(kù)標(biāo)題：鐵軌 。 Zhu 2021)。 ?成本效益 。通過(guò)全面實(shí)地的調(diào)查實(shí)驗(yàn)，并與先前的模型結(jié)果進(jìn)行比較，對(duì)該模型的準(zhǔn)確性進(jìn)行了評(píng)估。矩形平面應(yīng)變?cè)卮淼能壵砗玩?zhèn)流器和路基。在這種技術(shù)中，應(yīng)用負(fù)載為等額遞增而且每個(gè)增量的問(wèn)題得以解決。應(yīng)變矢量列的矩陣表示： (6) (7) 其中 x? 、 y? ：正常應(yīng)變 。t 的計(jì)算公式如下： (10) 其中 Z=元素的深度， ? 為角度的分布如圖 3所示。 xy? =剪應(yīng)力， E=楊氏模量； v=泊松比 。{U}：導(dǎo)致系統(tǒng)的節(jié)點(diǎn)位移 。矩形平面應(yīng)變?cè)貏偠染仃嚨募{入使得修改后的平面應(yīng)變技術(shù)更加完善?？v向分析認(rèn)為單一的鐵路軌枕鎮(zhèn)流器路基系統(tǒng)上的點(diǎn)負(fù)載（輪載）。 為了盡量減少計(jì)算成本 ，同時(shí)考慮立體軌道屬性，一個(gè)修改后的可以模擬三維荷載的平面應(yīng)變方法與傳播兩個(gè)階段的二維有限元（ FE）模型得以研發(fā)。 軌道元件的非線性特性，一定程度上在軌道建模的過(guò)程中，已經(jīng)被一些學(xué) 者，如湯普森和塔雅姬（ 1976年），塔瑞克（ 1995年），莫拉捷（ 1995），達(dá)爾伯格（ 2021年）所考慮。 介紹 多年的鐵路工程的研究一些鐵路軌道模型，但軌道結(jié)構(gòu)的特性和荷載條件沒(méi)有得到充分代表。這項(xiàng)技術(shù)使計(jì)算時(shí)間和成本降至最低。 []i? = stress matrix obtained due to thei th loading increment。 b = height of the sleeper。 xy? = shear stress。 approved on April 20, 2021。 Material nonlinearity. Introduction Years of research in railway engineering has resulted in several railway track models, yet the properties and loading conditions of the track structure have not been adequately represented. Majority of these models have assumed the track substructure to be linear and modeled it as an elastic foundation or a series of discrete linear sp