邊坡穩(wěn)定外文翻譯

1、<p><b>　　邊坡穩(wěn)定</b></p><p>　　重力和滲透力易引起天然邊坡、開挖形成的邊坡、堤防邊坡和土壩的不穩(wěn)定性。最重要的邊坡破壞的類型如圖9.1所示。在旋滑中，破壞面部分的形狀可能是圓弧或非圓弧線。總的來說，勻質(zhì)土為圓弧滑動破壞，而非勻質(zhì)土為非圓弧滑動破壞。平面滑動和復(fù)合滑動發(fā)生在那些強度差異明顯的相鄰地層的交界面處。</p><p>　　平

2、面滑動易發(fā)生在相鄰地層處于邊坡破壞面以下相對較淺深度的地方：破壞面多為平面，且與邊坡大致平行。復(fù)合滑動通常發(fā)生在相鄰地層處于深處的地段，破壞面由圓弧面和平面組成。</p><p>　　在實踐中極限平衡法被用于邊坡穩(wěn)定分析當(dāng)中。它假定破壞面是發(fā)生在沿著一個假想或已知破壞面的點上的。土的有效抗剪強度與保持極限平衡狀態(tài)所要求的抗剪強度相比，就可以得到沿著破壞面上的平均安全系數(shù)。問題以二維考慮，即假想為平面應(yīng)變的情況。二

3、維分析為三維（碟形）面解答提供了保守的結(jié)果。</p><p>　　在這種分析方法中，應(yīng)用總應(yīng)力法，適用于完全飽和粘土在不條件排水下的情況。如建造完工的瞬間情況。這種分析中只考慮力矩平衡。此間，假定潛在破壞面為圓弧面。圖9.2展示了一個試驗性破壞面（圓心O，半徑r，長度La）。潛在的不穩(wěn)定性取決于破壞面以上土體的總重量(單位長度上的重量W）。為了達到平衡，必須沿著破壞面?zhèn)鬟f的抗剪強度表示如下：</p>

4、<p>　　其中 F 是就抗剪強度而言的安全系數(shù)．關(guān)于 O點力矩平衡：</p><p><b>　　因此</b></p><p><b>　　(9.1) </b></p><p>　　其它外力的力矩必須亦予以考慮。在張裂發(fā)展過程中，如圖9.2所示，如果裂隙中充滿水，弧長La會變短，超孔隙水壓力將垂直作用在裂隙上

5、。有必要用一系列試驗性破壞面來對邊坡進行分析，從而確定最小的安全系數(shù)。</p><p>　　基于幾何相似原理，泰勒[9.9]發(fā)表了《穩(wěn)定系數(shù)》，用于在總應(yīng)力方面對勻質(zhì)土邊坡進行分析。對于一個高度為H的邊坡，沿著安全系數(shù)最小的破壞面上的穩(wěn)定系數(shù)(Ns)為：</p><p><b>　　(9.2)</b></p><p>　　對于φu =0的情況，

6、 Ns 的值可以從圖9.3中得到。Ns值取決于邊坡坡角β和高度系數(shù) D，其中DH 是到穩(wěn)固地層的深度。</p><p>　　吉布森和摩根斯特恩[9.3]發(fā)表了《不排水強度cu(φu =0)隨深度線性變化的正常固結(jié)粘土邊坡的穩(wěn)定系數(shù)》。</p><p>　　在這種方法中，潛在破壞面再次被假定為以O(shè)為圓心，以r為半徑的圓弧。試驗性破壞面（AC）以上的土體（ABCD），如圖9.5所示，被垂直劃分

7、為一系列寬度為b的條塊。每個條塊的底邊假定為直線。對于任何一個條塊來說，其底邊與水平線的夾角為α，它的高，從中心線測量，為h。安全系數(shù)定義為有效抗剪強度(τf)與保持邊限平衡狀態(tài)的抗剪強度(τm)的比值，即：</p><p>　　每個條塊的安全系數(shù)取相同值，表明條塊之間必須互相支持，即條塊間必須有力的作用。</p><p>　　作用于條塊上的力（條塊每個單元維上法向力）如下：</p&

8、gt;<p>　　1.條塊總重量，W=γb h（適當(dāng)時用γsat）</p><p>　　2.作用于底邊上總法向力，N（等于σl）。總體上，這個力有兩部分：有效法向力N'（等于σ'l ）和邊界孔隙水壓力U（等于ul），其中u是底邊中心的孔隙水壓力，而l是底邊長度。</p><p>　　3.底邊上的剪力，T=τml。</p><p>　　4

9、.側(cè)面上總法向力, E1和E2。</p><p>　　5.側(cè)面上總剪力，X1 和X2</p><p>　　任何的外力也必須包含在分析之中。</p><p>　　這是一種靜不定問題，為了得到解決，就必須對于條塊間作用力E 和X作出假定：安全系數(shù)的最終解答是不準(zhǔn)確的。</p><p>　　考慮到圍繞O點的力矩，破壞弧AC上的剪力T的力矩總和，必須

10、與土體ABCD重量所產(chǎn)生的力矩相等。對于任何條塊，W的力臂為rsinα，</p><p><b>　　因此</b></p><p>　　∑Tr=∑Wr sinα</p><p><b>　　則，</b></p><p>　　對于有效應(yīng)力方面的分析：</p><p><b

11、>　　或者</b></p><p><b>　　(9.3)</b></p><p>　　其中La是弧AC的長度。公式9.3是準(zhǔn)確的，但是當(dāng)確定力N'時引入了近似。對于給定的破壞面，F(xiàn)的取值將決定于力N'的計算方法。</p><p>　　在這種解法中，假定對于任何一個條塊，條間的相互作用力為零。解答包括了解出每個

12、條塊垂直于底邊的作用力，即：</p><p>　　N'=WCOSα-ul</p><p>　　因此，在有效應(yīng)力方面的安全系數(shù)（公式9.3），由下式計算：</p><p><b>　　(9.4)</b></p><p>　　對于每個條塊，Wcosα和Wsinα可以通過圖表法確定。α的取值可以通過測量或計算得到。同樣

13、地，也必須選擇一系列試驗性的破壞面來獲得最小的安全系數(shù)。這種解法所得的安全系數(shù)：與更精確的分析方法相比，其誤差通常為5-2%。</p><p>　　應(yīng)用總應(yīng)力法分析時，使用參數(shù)Cu 和φu，公式9.4中u取零。如果φu=0，那么安全系數(shù)為：</p><p><b>　　(9.5)</b></p><p>　　因為N’沒有出現(xiàn)在公式9.5中，故得

14、到的安全系數(shù)F值是精確的。</p><p>　　在這種解法中，假定條塊側(cè)面的力是水平的，即：</p><p><b>　　Xl-X2=0</b></p><p>　　為了達到平衡，任何一個條塊底邊上的剪力為：</p><p>　　解答垂直方向上的力：</p><p><b>　　(9.6

15、)</b></p><p><b>　　很方便得到：</b></p><p><b>　　l=b secα</b></p><p>　　從公式9.3，通過一些重新整理，</p><p><b>　　(9.7)</b></p><p>　　孔隙

16、水壓力通過孔壓比，可以與任何點的與總“填充壓力”相聯(lián)系，定義為：</p><p><b>　　(9.8)</b></p><p>　　(適當(dāng)時用γsat)．對于任何條塊,</p><p>　　因此公式9.7可寫為:</p><p>　　(9.9) </p><p>　　因為安全系數(shù)出現(xiàn)在公

17、式9.9的兩邊，必須使用一系列近似，才能獲得解答，但收斂很快。</p><p>　　基于計算的重復(fù)性，需要選擇充分數(shù)量的試驗性破壞面。條分法特別適合于計算機解答?？梢砸敫鼜?fù)雜的邊坡幾何學(xué)和不同的土層。</p><p>　　在大多數(shù)問題中，孔壓力比的取值ru在整個破壞面上是不一致的，但一旦存在獨立的高孔壓區(qū)，通常在設(shè)計中采用平均值（單位面積上的荷重）。同樣的，這種方法確定的安全系數(shù)過低，但

18、誤差不超過7％，多數(shù)情況下小于2％。</p><p>　　斯班瑟 [9.8] 提出了一種分析方法，在此法中，條塊間的作用力是水平的，且滿足力和力矩平衡。斯班瑟得到了只滿足力矩平衡的畢肖普簡化解，其精確度取決于邊坡條塊間作用力力矩平衡的不敏感性。</p><p>　　基于公式9.9的勻質(zhì)土邊坡的穩(wěn)定系數(shù)，是由畢肖普和摩根斯特恩[9.2]發(fā)表的。由此可見，對于給定坡角和給定土性的邊坡，安全系數(shù)

19、隨γu 線性變化，因此可以表示為：</p><p>　　F=m-γu (9.10)</p><p>　　其中m和n是穩(wěn)定系數(shù)。系數(shù) m 和 n 是β,φ’， c'/γ及深度系數(shù) D的函數(shù)。</p><p>　　假定潛在破壞面與邊坡面平行,所在深度與邊坡長度相比很小。那么,邊坡可以看作無限長,忽

20、略端部效應(yīng)。邊坡與水平線成β角,破壞面深度為z如圖9.7中所示。水位線在破壞面以上高度mz (0<m<1)處,與邊坡平行。假定穩(wěn)定滲流發(fā)生在與邊坡平行的方向上。任何垂直條塊側(cè)面上的力是等值反向的,且破壞面上任意一點的應(yīng)力狀態(tài)是相同的. </p><p>　　應(yīng)用有效應(yīng)力法,沿著破壞面上的土的抗剪強度為:</p><p><b>　　安全系數(shù)為:</b>&l

21、t;/p><p><b>　　σ,τ和μ表達為:</b></p><p>　　接下來的特殊情況是需要引起注意的。如果 c’=0 和 m=0 (即坡面與破壞面間的土是不完全飽和的),那么: </p><p><b>　　(9.11)</b></p><p>　　如果c’=0 和m=1(即水位線與邊坡面一致

22、) ，那么：</p><p><b>　　(9.12)</b></p><p>　　應(yīng)當(dāng)注意的是，當(dāng)c’=0 時，安全系數(shù)是與深度無關(guān)的。如果c’ 大于零，那么安全系數(shù)就是z 的函數(shù)，如果z 比規(guī)定值還小的話，β可能會超過φ’ 。 </p><p>　　應(yīng)用總應(yīng)力分析法，需使用抗剪強度參數(shù)cu 和φu ，而u取值為零。</p>&

23、lt;p>　　摩根斯特恩和普萊斯[9.4]提出了一般分析法，此法滿足所有的邊界條件和平衡條件，破壞面可以是任何形狀，圓弧，非圓弧或符合型。破壞面以上的土體被劃分為一系列垂直的平面，問題通過假定每部分之間垂直邊界上的作用力E 和X的關(guān)系 而轉(zhuǎn)化為靜定。這個假定的形式為</p><p>　　X=f(x)E (9.13)</p><

24、p>　　其中f(x)是描述隨土體而變化的比值X/E 的形式的任意函數(shù)，而λ是尺寸效應(yīng)系數(shù)。λ的值是在解安全系數(shù)F時一同獲得的。在每個垂直邊界上能夠確定作用力E 和X的值及作用點。對于任意的假定函數(shù) f(x) ，有必要仔細地檢查解答，以確定其在物理學(xué)上的合理性（即破壞面以上土體中沒有剪切破壞或張力）。函數(shù)f(x)的選擇對于F的計算值的影響不能超過 5% ，通常假定f(x)=l。</p><p>　　這種分析包

25、含了λ和F值相互作用的復(fù)雜過程，如摩根斯特恩和普萊斯[9.5]所描述的那樣，計算機的運用是必不可少的。</p><p>　　貝爾[9.1] 提出了一種滿足所有平衡情況，假定破壞面可能是任何形狀的分析方法。土體被劃分成一系列垂直的條塊，通過沿著破壞面上的法向作用力的假想分配，轉(zhuǎn)化為靜定問題。</p><p>　　薩爾瑪 [9.6] 基于條分法發(fā)展了一種方法，在此法中，產(chǎn)生極限平衡所要求的臨界

26、地震加速度是確定的。這種分析方法在分析中假定了條塊間垂直作用力的分配。同樣的，滿足所有的平衡條件，破壞面可以是任何形狀。靜安全系數(shù)是土的抗剪強度必須減小，以致于臨界加速度為零時的系數(shù)。</p><p>　　計算機的使用對于貝爾法和薩爾瑪法來說，是必不可少的。所有的解答必須要檢查，以確保它們在物理學(xué)上是可以接受的。</p><p>　　Stability of Slopes</p>

27、;<p>　　Gravitational and seepage forces tend to cause instability in natural slopes, in slopes formed by excavation and in the slopes of embankments and earth dams. The most important types of slope failure are il

28、lustrated in Fig.9.1.In rotational slips the shape of the failure surface in section may be a circular arc or a non-circular curve．In general，circular slips are associated with homogeneous soil conditions and non-circula

29、r slips with non-homogeneous conditions．Translational and compound slips o</p><p>　　Translational slips tend to occur where the adjacent stratum is at a relatively shallow depth below the surface of the slop

30、e:the failure surface tends to be plane and roughly parallel to the slope.Compound slips usually occur where the adjacent stratum is at greater depth，the failure surface consisting of curved and plane sections．</p>

31、<p>　　In practice, limiting equilibrium methods are used in the analysis of slope stability. It is considered that failure is on the point of occurring along an assumed or a known failure surface．The shear streng

32、th required to maintain a condition of limiting equilibrium is compared with the available shear strength of the soil，giving the average factor of safety along the failure surface．The problem is considered in two dimensi

33、ons，conditions of plane strain being assumed．It has been shown that a two</p><p>　　This analysis, in terms of total stress，covers the case of a fully saturated clay under undrained conditions, i.e. For the c

34、ondition immediately after construction．Only moment equilibrium is considered in the analysis．In section, the potential failure surface is assumed to be a circular arc. A trial failure surface(centre O，radius r and lengt

35、h La)is shown in Fig.9.2. Potential instability is due to the total weight of the soil mass(W per unit Length) above the failure surface．For equilibrium the</p><p>　　where F is the factor of safety with resp

36、ect to shear strength．Equating moments about O：</p><p><b>　　Therefore</b></p><p><b>　　(9.1) </b></p><p>　　The moments of any additional forces must be taken

37、into account．In the event of a tension crack developing ，as shown in Fig.9.2，the arc length La is shortened and a hydrostatic force will act normal to the crack if the crack fills with water．It is necessary to analyze th

38、e slope for a number of trial failure surfaces in order that the minimum factor of safety can be determined．</p><p>　　Based on the principle of geometric similarity，Taylor[9.9]published stability coefficient

39、s for the analysis of homogeneous slopes in terms of total stress．For a slope of height H the stability coefficient (Ns) for the failure surface along which the factor of safety is a minimum is</p><p><b&

40、gt;　　(9.2)</b></p><p>　　For the case ofφu =0，values of Ns can be obtained from Fig.9.3.The coefficient Ns depends on the slope angleβand the depth factor D，where DH is the depth to a firm stratum．</

41、p><p>　　Gibson and Morgenstern [9.3] published stability coefficients for slopes in normally consolidated clays in which the undrained strength cu(φu =0) varies linearly with depth．</p><p>　　In thi

42、s method the potential failure surface，in section，is again assumed to be a circular arc with centre O and radius r．The soil mass (ABCD) above a trial failure surface (AC) is divided by vertical planes into a series of sl

43、ices of width b, as shown in Fig.9.5.The base of each slice is assumed to be a straight line．For any slice the inclination of the base to the horizontal isαand the height, measured on the centre-1ine,is h. The factor of

44、safety is defined as the ratio of the available shear </p><p>　　The factor of safety is taken to be the same for each slice，implying that there must be mutual support between slices，i.e. forces must act betw

45、een the slices．</p><p>　　The forces (per unit dimension normal to the section) acting on a slice are：</p><p>　　1.The total weight of the slice，W=γb h (γsat where appropriate)．</p><p&g

46、t;　　2.The total normal force on the base，N (equal to σl)．In general this</p><p>　　force has two components，the effective normal force N'(equal toσ'l ) and the boundary water force U(equal to ul )，whe

47、re u is the pore water pressure at the centre of the base and l is the length of the base．</p><p>　　3.The shear force on the base，T=τml.</p><p>　　4.The total normal forces on the sides, E1 and E

48、2.</p><p>　　5.The shear forces on the sides，X1 and X2.</p><p>　　Any external forces must also be included in the analysis．</p><p>　　The problem is statically indeterminate and in or

49、der to obtain a solution assumptions must be made regarding the interslice forces E and X：the resulting solution for factor of safety is not exact．</p><p>　　Considering moments about O，the sum of the moments

50、 of the shear forces T on the failure arc AC must equal the moment of the weight of the soil mass ABCD．For any slice the lever arm of W is rsinα,</p><p><b>　　therefore</b></p><p>　　∑

51、Tr=∑Wr sinα</p><p><b>　　Now,</b></p><p>　　For an analysis in terms of effective stress，</p><p><b>　　Or</b></p><p><b>　　(9.3)</b><

52、/p><p>　　where La is the arc length AC．Equation 9.3 is exact but approximations are introduced in determining the forces N'．For a given failure arc the value of F will depend on the way in which the forces

53、N' are estimated． </p><p>　　In this solution it is assumed that for each slice the resultant of the interslice forces is zero．The solution involves resolving the forces on each slice normal to the base，

54、i.e.</p><p>　　N'=WCOSα-ul</p><p>　　Hence the factor of safety in terms of effective stress (Equation 9.3) is given by</p><p><b>　　(9.4)</b></p><p>　　The

55、 components WCOSαand Wsinαcan be determined graphically for each slice．Alternatively，the value of αcan be measured or calculated．Again，a series of trial failure surfaces must be chosen in order to obtain the minimum fact

56、or of safety．This solution underestimates the factor of safety：the error，compared with more accurate methods of analysis，is usually within the range 5-2%.</p><p>　　For an analysis in terms of total stress th

57、e parameters Cu andφu are used and the value of u in Equation 9.4 is zero．If φu=0 ,the factor of safety is given by</p><p><b>　　(9.5)</b></p><p>　　As N’ does not appear in Equation 9

58、.5 an exact value of F is obtained．</p><p>　　In this solution it is assumed that the resultant forces on the sides of the</p><p>　　slices are horizontal，i.e.</p><p><b>　　Xl-X2

59、=0</b></p><p>　　For equilibrium the shear force on the base of any slice is</p><p>　　Resolving forces in the vertical direction:</p><p><b>　　(9.6)</b></p>&

60、lt;p>　　It is convenient to substitute</p><p><b>　　l=b secα</b></p><p>　　From Equation 9.3，after some rearrangement，</p><p><b>　　(9.7)</b></p><p

61、>　　The pore water pressure can be related to the total ‘fill pressure’ at any</p><p>　　point by means of the dimensionless pore pressure ratio，defined as</p><p><b>　　(9.8)</b><

62、/p><p>　　(γsat where appropriate)．For any slice,</p><p>　　Hence Equation 9.7 can be written:</p><p>　　(9.9) </p><p>　　As the factor of safety occurs on both sides of E

63、quation 9.9,a process of successive approximation must be used to obtain a solution but convergence is rapid．</p><p>　　Due to the repetitive nature of the calculations and the need to select an adequate numb

64、er of trial failure surfaces，the method of slices is particularly suitable for solution by computer．More complex slope geometry and different soil strata can be introduced．</p><p>　　In most problems the valu

65、e of the pore pressure ratio ru is not constant over the whole failure surface but，unless there are isolated regions of high pore pressure，an average value(weighted on an area basis) is normally used in design．Again，the

66、factor of safety determined by this method is an underestimate but the error is unlikely to exceed 7％and in most cases is less than 2％．</p><p>　　Spencer [9.8] proposed a method of analysis in which the resul

67、tant Interslice forces are parallel and in which both force and moment equilibrium are satisfied．Spencer showed that the accuracy of the Bishop simplified method，in which only moment equilibrium is satisfied, is due to t

68、he insensitivity of the moment equation to the slope of the interslice forces．</p><p>　　Dimensionless stability coefficients for homogeneous slopes，based on Equation 9.9，have been published by Bishop and Mor

69、genstern [9.2].It can be shown that for a given slope angle and given soil properties the factor of safety varies linearly with γu and can thus be expressed as</p><p>　　F=m-nγu

70、 (9.10)</p><p>　　where，m and n are the stability coefficients．The coefficients，m and n are</p><p>　　functions ofβ,φ’，the dimensionless number c'/γand the depth factor D.</p><p&

71、gt;　　Using the Fellenius method of slices，determine the factor of safety，in terms of effective stress，of the slope shown in Fig.9.6 for the given failure surface．The unit weight of the soil，both above and below the water

72、 table，is 20 kN／m 3 and the relevant shear strength parameters are c’=10 kN/m2 andφ’=29°.</p><p>　　The factor of safety is given by Equation 9.4.The soil mass is divided into slices l.5 m wide. The weig

73、ht (W) of each slice is given by</p><p>　　W=γbh=20×1.5×h=30h kN／m</p><p>　　The height h for each slice is set off below the centre of the base and the</p><p>　　normal and

74、tangential components hcosαand hsinαrespectively are determined graphically，as shown in Fig.9.6.Then</p><p>　　Wcosα=30h cosα</p><p>　　W sinα=30h sinα</p><p>　　The pore water pressur

75、e at the centre of the base of each slice is taken to beγwzw，where zw is the vertical distance of the centre point below the water table (as shown in figure)．This procedure slightly overestimates the pore water pressure

76、which strictly should be) γwze，where ze is the vertical distance below the point of intersection of the water table and the equipotential through the centre of the slice base．The error involved is on the safe side．</p

77、><p>　　The arc length (La) is calculated as 14.35 mm．The results are given in</p><p><b>　　Table 9.1</b></p><p>　　∑Wcosα=30×17.50=525kN／m</p><p>　　∑W sinα=

78、30×8.45=254kN／m</p><p>　　∑(wcos α-ul)=525—132=393kN／m</p><p>　　It is assumed that the potential failure surface is parallel to the surface of the slope and is at a depth that is small compa

79、red with the length of the slope. The slope can then be considered as being of infinite length，with end effects being ignored．The slope is inclined at angle βto the horizontal and the depth of the failure plane is z．a(chǎn)s s

80、hown in section in Fig.9.7.The water table is taken to be parallel to the slope at a height of mz (0<m<1)above the failure plane．Steady seepage is assumed</p><p>　　In terms of effective stress，the shea

81、r strength of the soil along the failure plane is</p><p>　　and the factor of safety is</p><p>　　The expressions forσ,τandμare:</p><p>　　The following special cases are of interest．I

82、f c’=0 and m=0 (i.e. the soil</p><p>　　between the surface and the failure plane is not fully saturated),then</p><p><b>　　(9.11)</b></p><p>　　If c’=0 and m=1(i.e. the wa

83、ter table coincides with the surface of the slope)，then：</p><p><b>　　(9.12)</b></p><p>　　It should be noted that when c’=0 the factor of safety is independent of</p><p>

84、　　the depth z．If c’ is greater than zero，the factor of safety is a function of z, and βmay exceedφ’ provided z is less than a critical value．</p><p>　　For a total stress analysis the shear strength parameter

85、s cu andφu are used with a zero value of u.</p><p>　　Morgenstern and Price[9.4]developed a general analysis in which all boundary and equilibrium conditions are satisfied and in which the failure surface may

86、 be any shape，circular，non-circular or compound．The soil mass above the failure plane is divided into sections by a number of vertical planes and the problem is rendered statically determinate by assuming a relationship

87、between the forces E and X on the vertical boundaries between each section．This assumption is of the form</p><p>　　X=λf(x)E (9.13)</p><p>　　where f(x)is an arbi

88、trary function describing the pattern in which the ratio X/E varies across the soil mass andλis a scale factor．The value ofλis obtained as part of the solution along with the factor of safety F．The values of the forces E

89、 and X and the point of application of E can be determined at each vertical boundary．For any assumed function f(x) it is necessary to examine the solution in detail to ensure that it is physically reasonable (i.e. no she

90、ar failure or tension must be implied wi</p><p>　　The analysis involves a complex process of iteration for the values ofλ and F，described by Morgenstern and Price[9.5]，and the use of a computer is essential.

91、</p><p>　　Bell [9.1] proposed a method of analysis in which all the conditions of equilibrium are satisfied and the assumed failure surface may be of any shape．The soil mass is divided into a number of verti

92、cal slices and statical determinacy is obtained by means of an assumed distribution of normal stress along the failure surface．</p><p>　　Sarma [9.6] developed a method，based on the method of slices，in which

93、the critical earthquake acceleration required to produce a condition of limiting equilibrium is determined．An assumed distribution of vertical interslice forces is used in the analysis．Again，all the conditions of equilib

94、rium are satisfied and the assumed failure surface may be of any shape．The static factor of safety is the factor by which the shear strength of the soil must be reduced such that the critical acceleration is ze</p>