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    擋土墻上被動土壓力的變分求解方法

    2019-06-10 00:00:00 土木建筑與環境工程2019年2期

    趙國 陳建功

    摘 要:基于滑楔體整體極限平衡方程,根據變分法原理推導了被動土壓力泛函極值的變分模型,并引入拉格朗日乘子,將等周變分模型轉化為含有兩個函數自變量的泛函極值模型。依據歐拉方程、邊界條件和橫截條件,得到了滑裂面函數和滑裂面上的應力函數,函數泛函極值模型轉化為兩個未知量的函數優化模型。算例表明,對于一般土體,在作用點位置系數下界限處,滑裂面呈現對數螺旋曲面,此時被動土壓力最小;當作用點位置上移時,被動土壓力呈非線性增長,在作用點位置系數上界限處,滑裂面為平面,被動土壓力達到最大,與庫侖土壓力理論解完全一致,但作用點在墻體的相對位置并非在墻高的1/3處。結果表明,被動土壓力大小和作用點位置受坡面的起伏和坡面超載的不均勻性影響比較明顯。

    關鍵詞:剛性擋墻;被動土壓力;變分極限平衡法;滑裂面

    中圖分類號:TU476

    文獻標志碼:A? 文章編號:2096-6717(2019)02-0029-07

    Abstract:Considering an inclined rough retaining wall under the general conditions such as curvilinear fill, cohesive soil and uneven surface load, the functional extreme-value isoperimetric model about passive earth pressure is deduced based on the force equilibrium equations of the sliding mass. Then, the model can be transferred into a functional extreme-value problem with two undetermined functions by introducing Lagrange undetermined multiplier. According to Euler equations, Logarithmic spiral slip surface and normal stress distribution along the slip surface are obtained. Combined with the boundary conditions and transversality conditions, the conditional functional extremum problem of passive earth pressure? involves searching the minimum of unconstrained optimizations of function with two unknown Lagrange multiplier. Results show that the passive earth pressure resultant force is minimal when the point of resultant force is on the lower bound and it increase nonlinearly as the point of resultant force moved up to the upper bound for general soil. Accordingly, the slip face evolves from logarithmic spiral face to plan. Although the magnitude of passive earth pressure reaches maximal that is the same with result calculated from Coulombs theory, the application point of earth pressure is not at 1/3 height of the retaining wall. In addition, curvilinear fill and uneven surface load have significant effect on both the magnitude and? location of application point of passive earth pressure.

    Keywords:rigid retaining wall; passive earth pressure; variational limit equilibrium method; slip surface

    被動土壓力的計算是擋土墻上土壓力理論研究內容之一,除了對經典的朗肯土壓力理論和庫侖土壓力理論進行擴展、補充外,學者們在考慮其他諸如土拱效應、強度理論、位移模式等方面也對被動土壓力的計算進行了深入探討。李永剛[1]、王立國等[2]考慮了土拱效應原理,按庫侖理論得到土壓力強度的計算方法。侯鍵等[3]對平移模式下剛性擋土墻和滑裂面間處于被動極限平衡狀態的楔形土體進行應力分析,推導了基于土拱原理的被動土壓力系數和滑裂面水平傾角,并提出被動土壓力分布、土壓力合力及其作用位置公式。Chang[4]、蔣波等[5]和楊泰華等[6-9]對不同變位模式下的非極限主動土壓力進行計算,提出考慮位移效應的被動土壓力計算理論。程康等[10]推導了平動模式下非飽和土填土擋墻的被動土壓力系數及被動土壓力解析解。趙均海等[11]考慮基質吸力等因素的影響,結合雙剪強度理論,建立了非飽和土庫侖被動土壓力的統一解。

    變分極限平衡法最早是由Kopácsy提出的[12-14]。李興高、劉維寧等[15-17]基于墻后滑楔體整體極限平衡方程,建立了被動土壓力計算的泛函極值模型,計算結果表明,擋墻背后土體存在兩種被動臨界狀態,兩種狀態的計算結果就是被動土壓力大小的一個區間估計。該方法所建模型為填土面水平、墻背垂直的情況,不能用于一般情況。對于承受被動土壓力的擋土墻(如橋臺),有必要研究合力作用點位置對被動土壓力大小的影響,本文考慮在一般情況下(傾斜墻背,填土表面非均勻堆載,墻后為黏性土體,坡面為曲面),利用變分法原理,提出基于合力作用點位置的被動土壓力計算方法,由此可以得到擋墻被動土壓力的大小和作用點位置的范圍。

    1 被動土壓力變分計算模型

    設擋土墻高為H,坡面形狀為y=g(x),傾斜角為α,墻后土體容重為γ,墻土間的摩擦角為δ,黏聚力為c、內摩擦角為φ,qx為作用在坡面上的豎向壓力集度。考慮極限平衡條件下墻后土體形成滑動楔體,假定其滑裂面通過擋墻墻踵,用曲線y=s(x)表示,Pp為擋土墻對滑動土體的作用合力,其值等于被動土壓力合力。σ(x)、τ(x)為滑裂面上的法向和切向應力,點B為滑裂面的上端的點,其X坐標為x1,A為墻背的上端點,其X坐標為x2點,x2=-tan α,被動土壓力作用點位置至墻踵的垂直距離與擋土墻高度的比值稱為作用點位置系數ξ,見圖1。滑楔體OAB的靜力平衡方程為

    設擋土墻墻高6 m,墻后為砂土,重度γ=18 kN/m3,內聚力c=0,內摩擦角φ=30°,擋土墻傾角α=70°,擋墻與土體的摩擦角δ=10°,坡面傾角β=20°,上部堆載q=10 kPa。

    計算結果見圖3,圖中Φ曲線存在一Φ=0的水平直線段,此直線段的ξ系數變化范圍就是被動土壓力合力作用點的位置系數范圍,存在上下界限值(此例中下限值ξd=0.336 9,上限值ξu=0.535 5),在此范圍內,Φ=0,也即是說,當土壓力的合力作用點位于距墻鍾2.02~3.21 m范圍內時,墻后土體可以達到極限平衡狀態。圖4為被動土壓力合力隨作用點位置系數ξ的變化曲線,圖5為滑裂面隨作用點位置系數ξ的變化情況。隨著作用點位置系數的增大,被動土壓力呈非線性增長,在作用點位置下限處,被動土壓力最小,滑裂面為對數螺旋曲面,但隨著作用點位置的上移,曲率半徑逐漸增大,到上限處變為平面,此時,被動土壓力值達到最大。

    4 與庫侖土壓力理論結果對比

    按變分法的計算結果與按庫侖公式的計算結果見表1。可以看出,當作用點位置在上界限時,即滑裂面為平面時,被動土壓力值與按庫侖公式得到的土壓力值是一致的,但合力作用點位置卻不一致,這主要是因為庫侖理論假定土壓力線性分布,且不考慮土楔體力矩平衡條件。按變分法的計算結果,當滑裂面為平面時,滑裂面上的應力分布呈非線性,相應的土壓力分布也應是非線性的。

    圖6所示為庫侖理論的擋土墻模型,取墻后部分三角形滑動土楔體ABD作為研究對象,如圖7所示。Pp(y)為墻背對滑楔體的支撐反力合力,也即被動土壓力合力;R(y)為不動土體對滑楔體的反力合力;G(y)為土體ABD的重量;Q(y)為作用在AD面上豎向壓力。

    式中:ryD=σ(xD)cos φ,σ(xD)為滑裂面上D點的正應力,可由式(23)計算得到。通過變分法計算滑面上的正應力及滑面傾角,就可通過式(30)得到被動土壓力強度的非線性分布。圖9即為算例中當滑裂面為平面時的被動土壓力強度分布圖。

    表1中的安全系數一欄是指按庫侖公式進行擋墻的抗傾覆設計,取1.5的安全系數時所對應的按本文方法計算的抗傾覆安全系數范圍。若按庫侖理論進行設計,實際的被動土壓力合力作用點位置比庫侖理論設定的要高,其安全系數達不到1.5的要求,所以,應優化擋墻的結構,控制擋墻的變位模式,使合力作用點降低,提高抗傾覆安全系數。

    5 坡面起伏和非均勻超載對被動土壓

    力的影響

    設擋墻高6 m,墻后土重度γ=18 kN/m3,內聚力c=10 kPa,內摩擦角φ=20°,擋土墻傾角α=70°, 坡面形狀函數g(x)和非均勻超載q(x)均用三角函數模擬坡面的起伏和非均勻超載。表2為計算結果,可以看出,被動土壓力合力和作用點受坡面的起伏以及坡面超載的不均勻性影響不能忽視。

    6 結論

    基于墻后滑楔體整體極限平衡方程,利用變分法原理推導了被動土壓力泛函極值的變分模型,提出了相應的計算方法,當作用點位置已知時,可有效確定被動土壓力大小和滑裂面形狀位置。算例分析表明,墻后土體的極限平衡狀態對應于作用點位置系數范圍,隨著作用點位置系數的增大,被動土壓力呈非線性增長,在作用點位置下限處,被動土壓力最小,滑裂面為對數螺旋曲面,但隨著作用點位置的上移,滑裂面曲率半徑逐漸增大,到上限處變為平面。此時,被動土壓力值達到最大,且與按庫侖理論計算的被動土壓力一致,但作用點位置位于墻背中點以上,與庫侖理論所假定的合力作用點位置不一致。

    參考文獻:

    [1] 李永剛. 擋土墻被動土壓力研究[J]. 巖土力學,2003,24(2): 273-276.

    LI Y G. Study on passive earth pressure of retaining wall [J]. Rock and Soil Mechanics, 2003, 24(2): 273-276.(in Chinese)

    [2] 王立國, 王志凱. 考慮土拱效應的擋土墻被動土壓力分析[J]. 水利與建筑工程學報, 2010, 8(4): 146-149.

    WANG L G, WANG Z K. Analysis on passive earth pressure for retaining walls considering soil arching [J]. Journal of Water Resources and Architectural Engineering, 2010, 8(4): 146-149. (in Chinese)

    [3] 侯鍵,夏唐代,孔祥冰,等. 基于土拱效應原理求解擋土墻被動土壓力[J]. 巖土力學, 2012, 33(10): 2296-3000.

    HOU J ,XIA T D,KONG X B, et al. Passive earth pressure on retaining walls calculated by principle of soil arching effect [J]. Rock and Soil Mechanics, 2012, 33(10): 2296-3000. (in Chinese)

    ?
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