外文翻譯--在層狀土中考慮樁與樁相互作用的群樁的一種沉降計算的簡化方法

1、<p><b>　　中文6170字</b></p><p>　　出處：Yang M, Zhang X, Zhao M. A simplified approach for settlement calculation of pile groups considering pile-to-pile interaction in layered soils[J]. Journal of

2、 Central South University of Technology, 2011, 18: 2131-2136.</p><p><b>　　附錄1</b></p><p>　　A simplified approach for settlement calculation of pile groups considering pile-to-pile in

3、teraction in layered soils</p><p>　　1 Introduction</p><p>　　Piles, generally arranged in groups, are used in various applications to support structures exposed to vertical loads. In many cases,

4、the settlement of pile groups is the controlling factor in design because the primary purpose of pile groups is to limit the deformation of structures. Therefore, many researchers</p><p>　　have proposed diff

5、erent methods to investigate the behavior of the settlement of pile groups.</p><p>　　The current methods for estimating the settlement of pile groups can be categorized as: 1) Numerical methods, such as fini

6、te element method, and boundary element method. As a very powerful technique, numerical methods can readily calculate the settlements of pile groups in terms of the nonlinearity of soils and the interaction between indiv

7、idual piles by performing full three-dimensional-models for pile groups. However, the application of numerical methods is limited in practice for the complex </p><p>　　and POULOS modified the interaction fac

8、tors which can account for the different type of piles. However, the superimposing method does not consider the reinforcing effects of pile group, i.e. the settlement reduction of soils due to the presence of the neighbo

9、ring piles. As a result, the computed settlement of pile groups is usually greater than the actual result. Therefore, it is required that</p><p>　　developing a simplified approach for estimating the settleme

10、nt of pile groups considering the reinforcing effect of piles induced by the interaction between individual piles in pile groups which can readily be used in practice.</p><p>　　This work presents a simplifie

11、d approach to carry out a load settlement analysis of pile groups subjected to vertical loads in layered soils by using two models. First, the shear-deformation model of soils deduced from the method presented by RANDOLP

12、H and WROTH, is developed to simulate the interaction between individual piles in pile groups. The load-transfer model, general used in analyzing the behaviors of single piles, is then extended to estimate the settlement

13、 of pile groups by accounting </p><p>　　the pile space and pile length on the settlement of pile groups are also discussed.</p><p>　　2 Interaction between piles</p><p>　　This work f

14、ocuses on the vertically-loaded pile groups consisting of n identical piles with the same length L, diameter d, pile space S, and elastic modulus Eembedded in layered soils, as shown in Fig.1(a).Generally, the resistance

15、 of the surrounding soils at thepile/soil interface, i.e. shaft frictional force named as τ, is mobilized once the displacement of the piles occurs. The displacement of pile groups at a given depth is different from that

16、 of single pile under the same load due to the fa</p><p>　　Fig.1 Sketch of pile groups located in layered soils</p><p>　　Considered the interaction between any two piles with the pile space S in

17、 pile group, i and j, as shown in Fig.2. For pile i, the vertical displacement of the surrounding soil at depth z, defined as w(z), is composed of three parts: the first, named as w(z), is caused by the shaft frictional

18、force of pile i itself at depth z; the second is due to the shaft frictional force of pile j at the same depth z, w(z); the third is the reduction part induced by the reinforce-effect of pile j, w (z).Likewi</p>&

19、lt;p>　　Fig.2 Interaction between two piles in pile group</p><p>　　2.1 Calculation of w(z)</p><p>　　According to the formulation presented by RANDOLPH and WROTH to estimate the shear-deformat

20、ion mechanism of surrounding soils around piles subjected to the shaft frictional force τ, the displacement of a point of soils is expressed as</p><p><b>　　(1)</b></p><p>　　where r=d

21、/2, is the radius of the pile; r is the distance from the point of soil to the center of the pile; is the shear modulus of soils around the pile shaft, and the expression can be written as below accounting for the layer

22、 characteristic of soils:</p><p><b>　　(2)</b></p><p>　　where and hare the shear modulus and the thickness of the i-th layer soil, respectively; rm is the radial distance from the pil

23、e centre to a point at which the shaft shear stress induced by the pile can be considered to be negligible. The value of rm can be taken as rm=2.5ρL(1-0.5μs), where the parameter ρ is the ratio of the shear modulus of so

24、ils at the depth L/2 to that of the soil at L, and μs is the Poisson ratio of the soil.</p><p>　　So, the expression of w(z) is</p><p><b>　　(3)</b></p><p>　　and the relat

25、ive equivalent stiffness of the springs is</p><p><b>　　(4)</b></p><p>　　2.2 Calculation of w(z)</p><p>　　For pile j, there is also a shaft frictional force at the depth

26、z to resist the vertical load at the pile top, τ.Likewise, according to the shear-deformation formulation, w(z) can be written as</p><p><b>　　(5)</b></p><p>　　Obviously, accounting f

27、or all the action of the other piles, the relative equivalent stiffness of springs can be written as</p><p><b>　　(6)</b></p><p>　　2.3 Calculation of w(z)</p><p>　　This p

28、art of displacement of the soil around the pile i is induced by the reinforce-effect of pile j. The value of the stress of pile j at the depth z caused by the spread of τ, can be expressed as</p><p><b>

29、;　　(7)</b></p><p>　　For pile j, τcan be taken as a negative frictional force which pulls pile j down, whereas pile j generates a counter force with the same value but opposite direction namely , which

30、reduces the vertical displacement of the soil around the pile i. Hence, the value of w(z) is</p><p><b>　　(8)</b></p><p>　　Accounting for all the other piles action, the relative equi

31、valent stiffness of springs can be written as</p><p><b>　　(9)</b></p><p>　　So, the total equivalent stiffness of springs along pile i can be readily obtained:</p><p><

32、;b>　　(10)</b></p><p>　　3 Procedure for calculating settlement of pile group</p><p>　　3.1 Developing load-transfer function for individual pile in pile groups</p><p>　　The a

33、nalysis method, proposed originally by COYLE and REESE , is an efficient method to predict the load settlement relationship for single piles subjected to vertical load for its simplicity and capability of incorporating t

34、he nonlinear behavior of soils. However, due to the emission of influence of pile-to-pile interaction on the deformation of the soil surrounding the pile, it is rather difficult to be extended to pile-group analysis. In

35、this work, a load-transfer function is developed based </p><p>　　interaction between individual piles in pile group.</p><p>　　Pile i, supported by a series of nonlinear springs along pile shaft

36、or pile bottom to resist the vertical load Pat the pile top, is taken out to be analyzed separately, as shown in Fig.3(a). The stiffness of spring at the pile bottom can be conveniently expressed using the following equa

37、tion suggested in Ref.:</p><p><b>　　(11)</b></p><p>　　where Gand μare the shear modulus and Poisson ratio of the soil at the pile bottom.</p><p>　　Fig.3 Load settlement

38、analysis of individual pile in pile-group:</p><p>　　Load analysis of pile i; (b) Load analysis of pile j</p><p>　　Considering one element with the finite length dz of pile i at the depth z, all

39、the loads exerting on the finite element can be described as two parts: vertical load located at the top and bottom, P(z) and P(z)+dP(z), and shaft frictional force τ(z). The relative differential equation can be establi

40、shed according to the equilibrium condition in vertical direction as</p><p><b>　　(12)</b></p><p>　　where u is the perimeter of the pile. Besides, the elastic compresstion of the fini

41、te element can be expressed as</p><p><b>　　(13)</b></p><p>　　where Ais the area of the cross-section of pile.</p><p>　　If it is assumed that there is no slipping in the

42、pile/soil interface, substitute Eq.(12) into Eq.(13) so that the load-transfer function of pile i subjected to vertical load can be written as</p><p><b>　　(14)</b></p><p>　　3.2 Solut

43、ion for settlement of pile group</p><p>　　If the axial load on the top of pile i is assumed as P, the boundary condition of Eq.(14) can be easily expressed as</p><p><b>　　(15)</b><

44、;/p><p>　　So, the solution of Eq.(14) may be simplified as</p><p><b>　　(16)</b></p><p><b>　　where</b></p><p>　　Obviously, the value of Eq.(16) when

45、 z=0 is the settlement of pile i, which can also be regarded as the settlement of the n-pile group, because the rigid cap makes all the individual pile in pile group deform simultaneously, which can be expressed as</p

46、><p><b>　　(17)</b></p><p>　　where Q is the total load applied at the center of the cap.</p><p>　　4 Verification by model test</p><p>　　A model test of 3×3

47、 pile group subjected to axial loads in a two-layer soil system is carried out to verify the proposed approach discussed above. A layout of the foundation is shown in Fig.4. Each of the concrete testing pile has an elast

48、ic modulus (E) of 20 GPa with a diameter (d) of 62.5 mm and a length (L) of 2 000 mm. All the piles are placed at an identical space of 4d (d is the pile diameter) and connected at the pile top by a rigid cap made of hig

49、h-strength organic glass with a elastic m</p><p>　　layer and clay layer with the basic properties listed in Table 1, where w, γ, c, , μ represent water content, unit weight, internal cohesion, friction angle

50、, and Poisson ratio, respectively.</p><p>　　Fig.4 Experimental model of 3×3 pile-group: </p><p>　　Plan view; (b) Cross-sectional view</p><p>　　Table1 Basic properties of soils

51、in model test</p><p>　　The vertical displacements of the cap also defined as the settlement of pile group are measured by four dial indicators located at each corner of the cap in the loading process, and th

52、e average value is considered as the settlement of pile group. A series of tests for pile group are conducted under various vertical loads applied at the center of the cap. Figure 5 shows the comparison between the measu

53、red settlements of pile group and the predicted settlements calculated based on the previously p</p><p>　　Fig.5 Comparison of computed and measured load settlement</p><p>　　curve for pile group

54、in test</p><p>　　5 Parametric studies</p><p>　　In the design process of pile group, pile space and pile length are two important parameters that determine the cost and the construction difficult

55、y. In order to gain a foundational understanding of the effects of these two parameters on the settlement of pile group, a parametric study for the settlement of pile group is conducted. The basic parameters required are

56、 as follows: the piles with pile length L=25 m, diameter d=1.2 m, elastic modulusE=27 GPa, are placed in an isotropic and homogeneous</p><p>　　5.1 Influence of pile space</p><p>　　Figure 6 shows

57、 the settlements of two kinds of pile groups, 3×3 and 2×2 pile groups, with different pile spaces subjected to the same vertical load. The pile spaces vary from 3d to 9d. It can be seen from the both cases that

58、 the settlement of pile group decreases with the increase of pile space if the amount of the piles in pile group is a constant. The reason is that the interaction between individual piles is weakened with the increase of

59、 pile space, which indicates that the settlement of pile</p><p>　　Fig.6 Influence of pile space on settlement of pile-group</p><p>　　5.2 Influence of pile length</p><p>　　The settle

60、ments of a 2×2 pile group with different pile lengths are obtained by using the presented approach, as illustrated in Fig.7, where the slenderness of pile is defined as the ratio of the length to diameter of pile. O

61、bviously, the relative settlement decreases with the increase of pile length because more shaft frictional force is mobilized. However, only slight changes can be observed as the pile length reaches a certain value, whic

62、h indicates that there exists a critical pile length of</p><p>　　Fig.7 Influence of pile length on settlement of pile group</p><p>　　6 Conclusions</p><p>　　By developing two models

63、to simulate the load-transfer behavior of pile groups in both vertical and lateral directions, a simplified approach for estimating the settlement of pile groups considering the pile-to-pile interaction is presented. The

64、n, pile-group loading test is conducted to verify the proposed approach. Two conclusions can be drawn from the parametric study:</p><p>　　1) The settlement of pile groups decreases with the increase of the p

65、ile space when the total amount of the individual piles is kept as a constant.</p><p>　　2) There exists a critical pile length in a fixed arrangement pile group under a certain load. The shaft frictional for

66、ce pile beyond the critical pile length cannot be mobilized.</p><p><b>　　附錄2</b></p><p>　　在層狀土中考慮樁與樁相互作用的群樁的一種沉降計算的簡化方法</p><p><b>　　1 引言</b></p><p

67、>　　樁，通常是以群樁的形式存在，使用在各種應用中來支持承受豎向荷載的結構。在許多情況下，群樁的沉降是設計控制因素，因為群樁最初的目的是限制結構的變形。因此，許多研究者建議了不同的方法來研究群樁的沉降性質(zhì)。</p><p>　　目前估算群樁沉降的方法可歸納為：1)數(shù)值方法，例如有限元法和邊界元法。作為一項強有力的技術，數(shù)值方法根據(jù)土的非線性和基樁的相互作用，通過對群樁實行全三維模型，可以很容易的計算出群樁的

68、沉降。然而，數(shù)值方法的應用是有限的，它在復雜的建模程序和較高的計算要求，尤其是在大量群樁的實踐中受到限制。2)等效墩的方法。這種方法把群樁看成一個整個的墩來簡化計算群樁沉降的步驟，這等同于單樁的荷載傳遞函數(shù)的方法。等效墩方法的明顯的缺陷是計算出的沉降只與等效墩的面積相關，忽略了群樁中樁的數(shù)量和樁的間距的影響。3)疊加的方法。這種方法，最初由POULOS提出，最近被廣泛的應用，通過把任意兩個基樁的相互作用因素進行疊加來計算群樁的沉降。LE

69、E發(fā)明了一種計算剛性和柔性群樁的相互作用因素的程序。COSTANZO和LANCELLOTA提出了一種近似的解決方法來評估考慮了樁周土的非線性性質(zhì)的相互作用因素。WONG和 POULOS修改了能夠考慮不同樁型的相互作用因素。然而，疊加方法沒有考慮群樁的加強效應，即，由于相鄰樁的存在使土體的沉降減少。因此，計算出的群樁沉降通常要比實際結果大。因此，提出一種能夠在實踐中容易使用的計</p><p>　　這項工作通過使用

70、兩個模型介紹了一種簡化方法，對在層狀土中受到豎向荷載的群樁進行荷載沉降分析。首先，土的剪切變形模型是從RANDOLPH和WROTH提出的方法中推斷出來的，這種方法是用來模擬群樁中基樁的相互作用的。荷載傳遞模型，通常用在分析單樁的性質(zhì)中，后來被擴展到計算考慮了基樁的相互作用的群樁的沉降。因此，群樁的沉降和豎向荷載之間的關系得到了發(fā)展。在群樁上的某項試驗結果用來確認在這項工作中所提出的方法。也討論了樁間距和樁長對群樁沉降的影響。</p

71、><p><b>　　2 樁間的相互作用</b></p><p>　　這項工作的重點是埋深在層狀土中由相同長度、相同直徑、相同彈性模量的n個相同的樁所組成的豎向荷載群樁，如圖Fig.1(a)所示。通常，樁土界面上周圍土的阻力，即，樁身摩擦力命名為，一旦樁產(chǎn)生位移，就會產(chǎn)生。在給定深度處的樁群的位移不同于在相同荷載下的單樁的位移，由于由許多相鄰樁的相互作用所引起的加強效應局

72、限了沿樁土的位移。因此，在計算群樁的沉降時考慮基樁之間的相互作用是非常必要的。土被假設為連接在樁身上來模擬承受樁身摩阻力的土的性質(zhì)的一系列非線性彈簧，如圖Fig.1(b)所示。明顯地，彈簧的剛度，命為樁身摩阻力和土的位移比，與群樁中基樁的相互作用有關。</p><p>　　圖1 層狀土上的群樁</p><p>　　考慮在群樁中樁間距為的任意兩個樁之間的相互作用，i和j，如圖Fig.2所示。

73、對于i樁來說，在深度z處周圍土的豎向位移，命為w(z)，被分為三部分：第一，命名為w(z)，是由在深度z處i樁本身的樁身摩阻力所產(chǎn)生的；第二是由于在相同深度z處j樁的樁身摩阻力產(chǎn)生，w(z)；第三是由j樁的加強效應所誘發(fā)的減少部分，w(z)。同樣，彈簧的等效剛度值也是由相同的三部分組成的。計算每一部分的土的豎向位移和彈簧的等效剛度的步驟介紹如下。</p><p>　　圖2 群樁中兩樁之間的相互作用</p&g

74、t;<p>　　2.1 w(z)的計算</p><p>　　根據(jù)由RANDOLPH 和WROTH提出的公式來估算承受樁身摩阻力τ的樁周土的剪切變形機制，一點處土的位移表示為：</p><p><b>　　(1)</b></p><p>　　式中r=d/2，是樁的半徑；r是從土的一點到樁中心的距離；G是樁周土的剪切模量，這個表達式考

75、慮土的分層性質(zhì)，被寫成以下形式：</p><p><b>　　(2)</b></p><p>　　式中G和h分別是第i層土的剪切模量和厚度；r是從樁中心到一點的徑向距離，在這一點上由樁所誘發(fā)的樁身剪應力可以忽略不計。r的值可視為r= 2.5ρL(1-0.5μ)，式中參數(shù)ρ是在L/2深度處土的剪切模量和在L深度處土的剪切模量之比，μ為土的泊松比。</p>

76、<p>　　所以，w(z)的表達式是：</p><p><b>　　(3) </b></p><p>　　彈簧的相對等效剛度為：</p><p><b>　　(4)</b></p><p>　　2.2 w(z)的計算</p><p>　　對于j樁來說，在深度z處也有

77、一個樁身摩阻力來抵抗在樁頂?shù)呢Q向荷載τ。同樣，根據(jù)剪切變形公式，w(z)可被寫為：</p><p><b>　　(5)</b></p><p>　　明顯地，考慮其他樁的所有行為，彈簧的相對等效剛度可被寫為：</p><p><b>　　(6)</b></p><p>　　2.3 w(z)的計算<

78、;/p><p>　　在i樁周圍的土的這部分位移是由j樁的加強效應所誘發(fā)的。在深度z處j樁的應力值是由τ的傳播所引起的，可被表達為：</p><p><b>　　(7)</b></p><p>　　對于j樁來說，τ可看成是使j樁下降的一個負摩擦阻力，而j樁產(chǎn)生一個名為的大小相等、方向相反的一個反力，這個反力減少了i樁周圍土的豎向位移。因此，w(z)的

79、值為：</p><p><b>　　(8)</b></p><p>　　考慮其他所有樁的行為，彈簧的相對等效剛度可被寫為：</p><p><b>　　(9)</b></p><p>　　所以，沿著i樁的彈簧的總的有效剛度可以很容易的獲得：</p><p><b>

80、　　(10)</b></p><p>　　3 計算群樁沉降的步驟</p><p>　　3.1 群樁中基樁的荷載傳遞函數(shù)的發(fā)展</p><p>　　這種分析方法，最初是由COYLE和REESE提出的，因為它的簡便性和結合土的非線性性質(zhì)的能力，是一種預測受豎向荷載單樁的荷載與沉降的關系的有效方法。然而，由于樁與樁之間的相互作用對樁周土變形的影響的釋放，將其擴

81、展到群樁分析方法是比較困難的。在這項工作中，在群樁中基樁的上述相互作用分析的基礎上，發(fā)展了一個荷載傳遞函數(shù)。</p><p>　　用來抵抗作用在樁頂?shù)呢Q向荷載P的，被沿著樁身或樁底的一系列非線性彈簧所支持的i樁，被拿出來單獨分析，如圖Fig.3(a)所示。樁底彈簧的剛度用Ref建議的以下公式可以很方便的表達：</p><p><b>　　(11)</b></p&

82、gt;<p>　　式中G和μ是樁底土的剪切模量和泊松比。</p><p>　　圖3 群樁中基樁荷載沉降分析方法</p><p>　　i樁荷載分析；(b) j樁荷載分析</p><p>　　考慮到在深度z處i樁的有有限長度dz的一個元素，作用在這個有限元素上的所有荷載可被分為兩部分：作用在樁頂和樁底的豎向荷載，P(z) 和P(z)+dP(z)，樁身摩阻力

83、τ(z)。在豎直方向上，根據(jù)平衡條件，相對差分方程可被建立為：</p><p><b>　　(12)</b></p><p>　　式中u是樁的周長。除此之外，有限元素的彈性壓縮量可被表達為：</p><p><b>　　(13)</b></p><p>　　式中A是樁的橫截面積。</p>

84、<p>　　如果假設在樁土界面上沒有滑移，用Eq.(13) 替換Eq.(12)，因此，承受豎向荷載的i樁的荷載傳遞函數(shù)可被寫為：</p><p><b>　　(14)</b></p><p>　　3.2 群樁沉降的計算方法</p><p>　　如果i樁樁頂?shù)妮S向荷載假設為P，Eq.(14)的邊界條件可被很容易的表達為：</p

85、><p><b>　　(15)</b></p><p>　　所以Eq.(14)的計算方法可被簡化為：</p><p><b>　　(16)</b></p><p><b>　　式中</b></p><p>　　明顯地，當z=0時Eq.(16)的值是i樁的沉降

86、，這也可被看作為第n個樁群的沉降，因為剛性樁頂使群樁中所有基樁同時變形，可被表達為：</p><p><b>　　(17)</b></p><p>　　式中Q是樁頂中心的所有荷載。</p><p><b>　　4 驗證模型試驗</b></p><p>　　進行一個在兩層土中承受軸向荷載的3×

87、;3群樁的模型試驗來驗證上面討論的方法。一個基礎的布局如圖Fig.4所示。每一個混凝土試驗樁的彈性模量E為20 GPa，直徑為62.5 mm，長度為2000 mm。所有樁都具有相同的間距4d(d為樁的直徑)，在樁頂被彈性模量Ec=60 GPa的高強度有機玻璃所制成的剛性帽所連接。土包括粉砂性黏土層和黏土層，其基本性質(zhì)列于表1，其中w，γ，c，，u分別代表含水量、重度、內(nèi)部黏聚力、摩擦角和泊松比。</p><p>

88、　　圖4 3×3群樁的試驗模型</p><p>　　平面圖；(b)橫截面圖</p><p>　　表1 模型試驗中土的基本性質(zhì)</p><p>　　樁帽的豎向位移可被定義為群樁的沉降量，它可被在加載過程安裝在樁帽中心的四個位移千分表所測得，其平均值可被認為是群樁的沉降量。在樁帽中心上的多樣的豎向荷載下做了一系列群樁的試驗。圖5為群樁的測量沉降值和在先前提出的

89、方法上計算出來的預測沉降值之間的比較。由圖可以看出，計算值和測量值符合的很好。發(fā)現(xiàn)在它們之間有很小的不同，特別是在小的荷載水平下，它類似于實際工程中的群樁的工作荷載。</p><p>　　圖5 試驗中群樁的計算沉降和測量沉降的比較</p><p><b>　　5 影響因素分析</b></p><p>　　在設計群樁的過程中，樁間距和樁長是確定造

90、價和施工難度的兩個很重要的參數(shù)。為了獲得影響群樁沉降量的這兩個參數(shù)的基本理解，開展了群樁沉降量的參數(shù)研究。需要的基本參數(shù)如下：樁長L=25 m，直徑d=1.2 m，彈性模量E=27 GPa，被放置在剪切模量G=6 MPa，泊松比μ=0.35的各向同性、均質(zhì)土中，而作用在樁頂?shù)暮奢d是一個常量P=1000 kN。</p><p>　　5.1 樁間距的影響</p><p>　　圖6為受到相同的豎

91、向荷載、具有不同樁間距的3×3 和2×2兩種不同群樁的沉降量。樁間距在3d和9d之間變化。從這兩種情況可以看出，當群樁中樁的數(shù)量是常量時，群樁的沉降量隨樁間距的增加而降低。原因是隨著樁間距的增加，基樁之間的相互作用降低，這表明被其他樁所誘發(fā)的群樁的沉降降低了。因此，在群樁的設計中，基樁的樁間距應當被適當保持在一個較高的值。</p><p>　　圖6 樁間距對群樁沉降的影響</p>

92、<p><b>　　5.2 樁長的影響</b></p><p>　　一個具有不同樁長的2×2群樁的沉降可用提出的方法獲得，如圖Fig.7所示，其中樁的長徑比可定義為樁的長度和樁的直徑的比值。明顯地，隨著樁長的增加相應的沉降量減少，因為產(chǎn)生了更多的樁身摩阻力。然而，當樁長達到某個值時，只能看到很小的變化，這表明群樁存在一個臨界的樁長。換句話說，超過臨界長度的那部分樁對群樁

93、承載力的貢獻可以忽略不計。因此，合理的樁長應比臨界長度小，它與荷載、樁的性質(zhì)、樁周土的性質(zhì)和群樁的排列有關。</p><p>　　圖7 樁長對群樁沉降的影響</p><p><b>　　6 結論</b></p><p>　　通過模擬豎向和側向的群樁的荷載傳遞性質(zhì)的兩個模型，提出了考慮了樁與樁相互作用的群樁沉降的計算的簡化方法。然后，做了群樁的荷