常用的地基沉降计算方法汇总

1、6.3常用的地基沉降计算方法这里所讲的地基沉降量是指地基最终沉降量，目前常用的计算方法有：弹性力学法、分层总和法、应力面积法和考虑应力历史影响的沉降计算法。所谓最终沉降量是地基在荷载作用下沉降完全稳定后的沉降量，要达到这一沉降量的时间取决于地基排水条件。对于砂土，施工结束后就可以完成；对于粘性土，少则几年，多则十几年、几十年乃至更长时间。6.3.1 计算地基最终沉降量的弹性力学方法Boussinesq课题的位移解为依据P时，见图6-5,表面位移w（x,y,（6-8）地基最终沉降量的弹性力学计算方法是以的。在弹性半空间表面作用着一个竖向集中力o）就是地基表面的沉降量s：P1-2s=二rE式中以一

2、地基土的泊松比；E地基土的弹性模量（或变形模量Eo）;f22x+y。对于局部荷载下的地基沉降，则可利用上式，根据叠加原理求得。如图6-6所示，设荷载面积A内N（己J）点处的分布荷载为p0（己，"），则该点微面积上的分布荷载可为集中力P=po（己J）d也”代替。于是，地面上与N点距离r=J（x"）2+（y-刃2的m（x,y）点的沉降s（x,y）,可由式（6-8）积分求得：12po(,)dds(x，y)=e22(6-9)E0a.(x-)(y-)(a)任意荷载面；(b)式中以一角点沉降影响系数，由下式确定:(6-10)1'c=一JT1Jm22mln()ln(m.m1)m(

3、6-11)式中m=l/b。矩形荷载面从式(6-9)可以看出，如果知道了应力分布就可以求得沉降；反过来，若沉降已知又可以反算出应力分布。对均布矩形荷载po(>=p0=常数，其角点C的沉降按上式积分的结果为：1一；2.s=cbpoEo利用式(6-10),以角点法易求得均布矩形荷载下地基表面任意点的沉降。例如矩形中心点的沉降是图6-6(b)中的虚线划分为四个相同小矩形的角点沉降之和，即1-'121-''2s=4c(b/2)po=0bp0(6-12)EoE。式中牝=2球一中心沉降影响系数。图6-7局部荷载作用下的地面沉降（a）绝对柔性基础；（b）绝对刚性基础以上角点法的计

4、算结果和实践经验都表明，柔性荷载下地面的沉降不仅产生于荷载面范围之内，而且还影响到荷载面之外，沉降后的地面呈碟形，见图6-7。但一般基础都具有一定的抗弯刚度，因而沉降依基础刚度的大小而趋于均匀。中心荷载作用下的基础沉降可以近似地按绝对柔性基础基底平均沉降计算，即s=s(x,y)dxdyAA式中A一基底面积，s（x,y）一点（x,y）处的基础沉降。对于均布的矩形荷载，上式积分的结果为:一2hsmbpoE0(6-13)(6-14)式中m一平均沉降影响系数。s=E02-bp。(6-15)可将式（6-10）、式（6-12）、式（6-14）统一成为地基沉降的弹性力学公式的一般形式：式中b矩形基础（荷载）

5、的宽度或圆形基础（荷载）的直径,仍一无量纲沉降影响系数，见表6-1。基础形状圆形方形矩形（l/b）基础刚度1.01.52.03.04.05.06.07.08.09.010.1100.0柔0c0.640.560.680.770.890.981.051.111.161.201.241.272.00性与01.001.121.361.531.781.962.102.222.322.402.482.544.01基础沉降影响系数切值表6-1基础mm0.850.951.151.301.521.201.831.962.042.122.192.253.70刚性基础0r0.790.881.081.221.441.

6、611.72-2.123.40刚性基础承受偏心荷载时，沉降后基底为一倾斜面，基底形心处的沉降（即平均沉降）可按式（6-15）取缶=到计算，基底倾斜的弹性力学公式如下:J-J2Pe171:tan=6-圆形基础:(6-16a)矩形基础:taiUE0Peb3(6-16b)E0b3式中基础倾斜角；P一基底竖向偏心荷载合力；e一偏心距；b一荷载偏心方向的矩形基底边长或圆形基底直径;K一计算矩形刚性基础倾斜的无量纲系数，按l/b取值，如图6-8,其中l为矩形基底另一边长。通常按式（6-15）计算的基础最终沉降量是偏大的。这是由于弹性力学公式是按匀质线性变形半空间的假设得到的，而实际上地基常常是非均质的成层

7、土，即使是均质的土层，其变形模量Eo一般随深度而增大。因此，利用弹性力学公式计算沉降的问题，在于所用的E0值能否反映地基变形的真实情况。地基土层的Eo值，如能从已有建筑物的沉降观测资料，以弹性力学公式反算求得，则这种数据是很有价值的。此外，弹性力学公式可用来计算地基的瞬时沉降，此时认为地基土不产生体积变形，例如风或其它短暂荷载作用下，构筑物基础的倾斜可按式(6-16)计算,Christian和注意式中的E0应取为地基弹性模量，并取泊松比=0.5。在大多数实际问题中，土层的厚度是有限的，下卧坚硬土层Carrier(1978)提出了计算有限厚土层上柔性基础的平均沉降计算公式：s-21也E0(6-1

8、7)式中，出取决于基础埋深和宽度之比D/b,国取决于地基土厚度H和基础形状。取泊松比卜=0.5时飞和匕如图6-9所示。对于成层土地基，可利用叠加原理来计算地基平均沉降。式(6-17)主要用于估计饱和粘土地基的瞬时沉降，由于瞬时沉降是在不排水状态下发生的，因此，适宜的泊松比卜应取0.5,适应的变形模量E0应取不排水模量Eu。例题6-1某矩形基础底面尺寸为4m>2m,其基底压力p0=150kPa,埋深1m,地基土第一层为5m厚的粘土，不排水变形模量Eu=40MPa,第二层为8m厚的粘土，Eu=75MPa,其下为坚硬土层。试估算基础的瞬时沉降。解：D/b=0.5,查图6-9,%=0.94考虑上

9、层粘土，H/b=4/2=2,l/b=2,具有Eu=40MPa查图6-9,匕=0.602150a=0.940.60=4.23mm因此40考虑二层粘土均具有Eu=75MPaH/b=12/2=6,l/b=2,查图6-9,=0.852150rtS2=0.940.85=3.20mm因此75考虑第一层木土，具有Eu=75MPa,则2150S3=0.940.6=2.26mm75因此，总的瞬时沉降为：s-siS2-S3=4.233.202.26=5.17mm6.3.2计算地基最终沉降量的分层总和法(一)一维压缩课题在厚度为H的均匀土层上面施加连续均匀荷载p,见图6-10a,这时土层只能在竖直方向发生压缩变形，

10、而不可能有侧向变形，这同侧限压缩试验中的情况基本一样，属一维压缩问题。施加外荷载之前，土层中的自重应力为图6-10b中OBA;施加p之后，土层中引起的附加应力分布为OCDA。对整个土层来说，施加外荷载前后存在于土层中的平均竖向应力(a)(bl(c)图6-10土层一维压缩V分别为p1='H/2和p2=p1+p0从土的压缩试验曲线(图6-10c)可以看出，竖向应力从p1增加到p2,将引起土的孔隙比从e1减小为e2。因此，可求得一维条件下土层的压缩变形s与土的孔隙比e的变化之间存在如下关系:ei-e2s二H（6-18）6-18）也可改写成:这就是土层一维压缩变形量的基本计算公式。式（as(P

11、2-P1)H1e1a1e1PHas=A=mvA或1e1s二也或Es式中a压缩系数；mv一体积压缩系数；Es压缩模量；H土层厚度；A附加应力面积,A=pH。（二）沉降计算的分层总和法1基本原理分别计算基础中心点下地基各分层土的压缩变形量si,认为基础的平均沉降量s等于si的总和，即ns=、sii二式中n一计算深度范围内土的分层数。(6-19)(6-20)(6-21)图6-11分层总和法沉降计算图例(6-22)1ei计算时si,假设土层只发生竖向压缩变形，没有侧向变形，因此可按式（6-18）式（6-21）中的任何一个公式进行计算。2计算步骤1）选择沉降计算剖面，在每一个剖面上选择若干计算点。在计算

12、基底压力和地基中附加应力时，根据基础的尺寸及所受荷载的性质（中心受压、偏心或倾斜等），求出基底压力的大小和分布；再结合地基土层的性状，选择沉降计算点的位置。2）将地基分层。在分层时天然土层的交界面和地下水位面应为分层面，同时在同一类土层中分层的厚度不宜过大。一般取分层厚hiG.4b或hi=12m,b为基础宽度。3）求出计算点垂线上各分层层面处的竖向自重应力*（应从地面起算），并绘出它的分布曲线。4）求出计算点垂线上各分层层面处的竖向附加应力灯z,并绘出它的分布曲线，取e=0.2d（中、低压缩性土）或0.1=c（高压缩性土）处的土层深度为沉降计算的土层深度。5）求出各分层的平均自重应力二ci二z

13、i°ci和平均附加应力0zi,见图6-11:1,上，下、二（；=ci,二ci）21,上上下、二（'-zizi）2上式中,、上二zi、下仃4第i分层土上、下层面处的自重应力;下仃zi第i分层土上、下层面处的附加应力。6）计算各分层土的压缩量Sio认为各分层土都是在侧限压缩条件下压力从P1=仃&增加到P2Rei+仃石所产生的变形量Si,可由式（6-18）式（6-21）中任一式计算。7）按式（6-22）计算基础各点的沉降量。基础中点沉降量可视为基础平均沉降量；根据基础角点沉降差，可推算出基础的倾斜。例题6-2某柱基础，底面尺寸l冲=4>2m2,埋深d=1.5m。传至基

14、础顶面的竖向荷载N=1192KN,各土层计算指标见例表6-1和例表6-2。试计算柱基础最土层侧限压缩试验e卞曲线例表终沉降量。假定地下水位深dw=2m土层计算指标例表6-16-2土层1(kN/m2)a(MPa1)Es(MPa)粘土19.50.394.5粉质粘土19.80.335.1粉砂19.00.375.0粉土19.20.523.4解：基1氐平均压力pPl十?G"MbG基底附加压力p。:p(kPa)土层050100200粘土0.8200.7800.7600.740粉质粘土0.7400.7200.7000.670粉砂0.8900.8600.8400.810粉土0.8500.8100.7

15、800.7401192.=+20M1.5=179kPa4M2p0=p_d=179-19.51.5=150kPa取水的重度九M0kN/m3,则有效重度10,基础中心线下的自重应力和附加应力计算结果见例图6-1。到粉砂层层底，az=14.4kPa<0.2<Tc=0.2x91.9=18.3kPa,因此，沉降计算深度取为H=2.0+4.0+1.5=7.5m,从基底起算的土层压缩层厚度为Zn=7.5T.5=6.0m。例图6-1土层自重应力和附加应力分布按hi4b=0.4>2=0.8m分层。h1=0.50m,h2h6=0.80m,h7=h8=0.75m。柱基础最终沉降量计算结果如下：e1

16、-e2si一(二)ihi(1)按公式1+ei计算。各分层土沉降量计算程序例表6-3土层分层hi(m)p1i(kPa)e1iP2i(kPa)e2isi(mm)粘土0-10.5034.20.7995180.70.743915.45120.8042.90.7228165.90.680219.78粉质粘土2W0.8050.80.7197136.60.689014.283汨0.8058.40.7166115.80.69539.934七0.8066.20.7135105.40.69847.055-60.8074.00.7104102.00.69945.146-80.7581.60.8474102.70.8

17、3923.337-80.7588.40.8446104.90.83852.48粉砂(3)按公式Esi计算因止匕，s=ISi=77.44mm土层分层hi(m)p1i(kPa)e1iozi(kPa)a(MPa1)si(mm)粘土0T0.5034.20.7995146.50.3915.85120.8042.90.7228123.018.852W0.8050.80.719785.813.17粉质粘土3汨0.8058.40.716657.40.338.834七0.8066.20.713539.26.045-60.8074.00.710428.04.326-80.7581.60.847421.13.17粉

18、砂7-80.7588.40.844616.50.372.48例表6-4hi因止匕s=2si=72.44mm二zisi二/asi=()i：-zihi(2)按公式1+8计算各分层土沉降量计算程序146.5s=4.50.50(123.085.857.439.228.0)0.805.1月誓h近HE图6-12应力面积的概念任意7度z范围内的压缩量为(见图6-12):0.75(21.116.5)=16.2852.305.64=74.22mm5.06.3.3地基沉降量计算的应力面积法建筑地基基础设计规范所推荐的地基最终沉降量计算方法是另一种形式的分层总和法。该方法采用了“应力面积”的概念，因而称为应力面积法

19、。(一)土层压缩变形量As的计算及应力面积的概念假设地基是均匀的，即土在侧限条件下的压缩模量Es不随深度而变，则从基底至地基zs=0dz=1z.AF0二zdz=Es0Es(6-23)式中名一土的侧限压缩应变，名=Qz/EA深度z范围内的附加应力面积，s；zA=；：.-2dz。因为。z=Ksp0,积A为：Ks为基底下任意深度z处的附加应力系数。因此，附加应力面A=0二zdz=P00Ksdz(6-24)为便于计算,（6-23）改写为:引入一个竖向平均附加应力（面积）系数&=A/（poZ）。则式s二贮Es(6-25)式（6-25）就是以附加应力面积等代值引出一个平均附加应力系数表达的从基底至

20、任意深度z范围内地基沉降量的计算公式。由此可得成层地基沉降量的计算公式（见图6-13):AiA1='、'(苗zi-:izi)-Esi(6-26)式中p0可zi和p0«izi和zi-1深度范围内竖向附加应力面积Ai和Aiv的等因此,代值。用式（6-26）计算成层地基的沉降量，关键是确定竖向平均附加应力系数由竖向平均附加应力系数0的定义，均布荷载po作用于夕!形（2a>2b）土表，相对其形心，在坐标（x,y）处，从土表到z深处，可导出&的解析式如下：其中/rp天然地面标高图6-13成层地基沉降计算的概念T(4)11zarctan,，丫,_,i女2Xi2Yi2

21、z2(X：Y；z2/)(X：Y；Yj)-XiIn:22222(XiYjzYj)(XiYjYj)-YjIn(Xi2Yj2z2-Xi)(Xi2Yj2-Xi)(Xi2Yj2z2Xi)(X：Yj2-Xi)j=intt，一，一八2（表小取2的整数部分）Xi=x(-1)'aY=y(-1)jb(6-27)(6-28a)(6-28b)为便于计算，均布矩形荷载角点下的反列于表6-2;利用式（6-24）也可以求出均布三角形荷载角点、圆形面积均布荷载中心点和周边点下的0值，并列于表6-3至表6-5,以供查用。为了提高计算准确度，地基沉降计算深度围内的计算沉降量s',尚须乘以一个沉降计算经验系数风。建

22、筑地基基础设计规范规定吸的确定方法为：5s=sc/s（6-29）式中s无一利用地基沉降观测资料推算的最终沉降量。各地区宜按实测资料制定适合于本地区各类土的外值，而建筑地基基础设计规范提供了一个采用表值，见表6-6o综上所述，建筑地基基础设计规范推荐的地基最终沉降量s的计算公式如下:ns=：s"Po(:izi-)/Esii=1(6-30)式中，n地基沉降计算深度范围内所划分的土层数l/bz/b1.01.21.41.61.82.02.42.83.23.64.05.010.C0.00.25000.25000.25000.25000.25000.25000.25000.25000.25000

23、.25000.25000.25000.2500.20.24960.24970.24970.24980.24980.24980.24980.24980.24980.24980.24980.24980.2490.40.24740.24790.24810.24830.24830.24840.24850.24850.24850.24850.24850.24850.2480.60.24230.24370.24440.24480.24510.24520.24540.24550.24550.24550.24550.24550.2450.80.23460.23720.23870.23950.24000.240

24、30.24070.24080.24090.24090.24100.24100.2411.00.22520.22910.23130.23260.23350.23400.23460.23490.23510.23520.23520.23530.2351.20.21490.21990.22290.22480.22600.22680.22780.22820.22850.22860.22870.22880.2281.40.20430.21020.21400.21640.21900.21910.22040.22110.22150.22170.22180.22200.2221.60.19390.20060.2

25、0490.20790.20990.21130.21300.21380.21430.21460.21480.21500.2151.80.18400.19120.19600.19940.20180.20340.20550.20660.20730.20770.20790.20820.2082.00.17460.18220.18750.19120.19380.19580.19820.19960.20040.20090.20120.20150.2012.20.16590.17370.17930.18330.18620.18830.19110.19270.19370.19430.19470.19520.1

26、952.40.15780.16570.17150.17570.17890.18120.18430.18620.18730.18800.18850.18900.1892.60.15030.15830.16420.16860.17190.17450.17790.17990.18120.18200.18250.18320.1832.80.14330.15140.15740.16190.16540.16800.17170.17390.17530.17630.17690.17770.1783.00.13690.14490.15100.15560.15920.16190.16580.16820.16980

27、.17080.17150.17250.1733.20.13100.13900.14500.14970.15330.15620.16020.16280.16450.16570.16640.16750.1683.40.12560.13340.13940.14410.14780.15080.15500.15770.15950.16070.16160.16280.1633.60.12050.12820.13420.13890.14270.14560.15000.15280.15480.15610.15700.15830.1593.80.11580.12340.12930.13400.13780.140

28、80.14520.14820.15020.15160.15260.15410.1554.00.11140.11890.12480.12940.13320.13620.14080.14380.14590.14740.14850.15000.1514.20.10730.11470.12050.12510.12890.13190.13650.13960.14180.14340.14450.14620.1474.40.10350.11070.11640.12100.12480.12790.13250.13570.13790.13960.14070.14250.1444.60.10000.10700.1

29、1270.11720.12090.12400.12870.13190.13420.13590.13710.13900.1414.80.09670.10360.10910.11360.11730.12040.12500.12830.13070.13240.13370.13570.137表6-2均布矩形荷载角点下的平均竖向附加应力系数日z/bl/b1.01.21.41.61.82.02.42.83.23.64.05.010.05.00.09350.10030.10570.11020.11390.11690.12160.12490.12730.12910.13040.13250.1345.20.09

30、060.09720.10260.10700.11060.11360.11830.12170.12410.12590.12730.12950.1325.40.08780.09430.09960.10390.10750.11050.11520.11860.12110.12290.12430.12650.1295.60.08520.09160.09680.10100.10460.10760.11220.11560.11810.12000.12150.12380.1265.80.08280.08900.09410.09830.10180.10470.10940.11280.11530.11720.11

31、870.12110.1246.00.08050.08660.09160.09570.09910.10210.10670.11010.11260.11460.11610.11850.1216.20.07830.08420.08910.09320.09660.09950.10410.10750.11010.11200.11360.11610.1196.40.07620.08200.08690.09090.09420.09710.10160.10500.10760.10960.11110.11370.1176.60.07420.07990.08470.08860.09190.09480.09930.

32、10270.10530.10730.10880.11140.1146.80.07230.07790.08260.08650.08980.09260.09700.10040.10300.10500.10660.10920.1127.00.07050.07610.08060.08440.08770.09040.09490.09820.10080.10280.10440.10710.1107.20.06880.07420.07870.08250.08570.08840.09280.09620.09870.10080.10230.10510.1097.40.06720.07250.07690.0806

33、0.08380.08650.09080.09420.09670.09880.10040.10310.1077.60.06560.07090.07520.07890.08200.08460.08890.09220.09480.09680.09840.10120.1057.80.06420.06930.07360.07710.08020.08280.08710.09040.09290.09500.09660.09940.1038.00.06270.06780.07200.07550.07850.08110.08530.08860.09120.09320.09480.09760.1028.20.06

34、140.06630.07050.07390.07690.07950.08370.08690.08940.09140.09310.09590.1008.40.0601.06490.06900.07240.07540.07790.08200.08520.08780.09890.09140.09430.0988.60.05880.06360.06760.07100.07390.07640.08050.08360.08620.08820.08980.09270.0978.80.05760.06230.06630.06960.07240.07490.07900.08210.08460.08660.088

35、20.09120.9519.20.05540.05990.096370.06970.07210.07610.07920.08170.08370.08530.08820.08130.0939.60.05330.05770.06140.06720.06960.07340.07650.07890.08090.08250.08550.07380.09010.00.05140.05560.05920.06490.06720.07100.07390.07630.07830.07990.08290.07190.08810.40.04960.05370.05720.06270.06490.06860.0716

36、0.07390.07590.07750.08040.06820.08510.80.04790.05190.05530.06060.06280.06640.06930.07170.07360.07510.07810.06490.08311.20.04630.05020.05350.05630.05870.06090.06440.06720.06950.07140.07300.07590.08111.60.04480.04860.05180.05450.05690.05900.06250.06520.06750.06940.07090.07380.07912.00.04350.04710.0502

37、0.05290.05520.05730.06060.06340.06560.06740.06900.07190.07712.80.04090.04440.04740.04990.05210.05410.05730.05990.06210.06390.06540.06820.07313.60.03870.04200.04480.04720.04930.05120.05430.05680.05890.06070.06210.06490.07014.40.03670.03980.04250.04480.04680.04860.05160.05400.05610.05770.05920.06190.0

38、67l/bz/b1.01.21.41.61.82.02.42.83.23.64.05.010.C15.20.03490.03790.04040.04260.04460.04630.04920.05150.05350.05510.05650.05920.06516.00.03320.03610.03850.04070.04250.04420.04690.04920.05110.05270.05400.05670.06218.00.02970.03230.03450.03640.03810.03960.04220.04420.04600.04750.04870.05120.05720.00.026

39、90.02930.03120.03300.03450.03590.03830.04020.04180.04320.04440.04680.052三角形分布的矩形荷载角点下的平均竖向附加应力系数值表6-3l/b=0.2i/b=0.4l/b=0.6l/b=0.8l/b=1.0z/b角点1角点2角点1角点2角点1角点2角点1角点2角点1角点20.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01120.21610.01400.23080.01480.23330.01510.23390.01520.23410.40.

40、01790.18100.02450.20840.02700.21530.02800.21750.02850.21840.60.02070.15050.03080.18510.03550.19660.03760.20110.03880.20300.80.02170.12770.03400.16400.04050.17870.04400.18520.04590.18831.00.02170.11040.03510.14610.04300.16240.04760.17040.05020.17461.20.02120.09700.03510.13120.04390.14800.04920.15710.

41、05250.16211.40.02040.08650.03440.11870.04360.13560.04950.14510.05340.15071.60.01950.07790.03330.10820.04270.12470.04900.13450.05330.14051.80.01860.07090.03210.09930.04150.11530.04800.12520.05250.13132.00.01780.06500.03080.09170.04010.10710.04670.11690.05130.12322.50.01570.05380.02760.07690.03650.090

42、80.04290.10000.04780.10633.00.01400.04580.02480.06610.03300.07860.03920.08710.04390.09315.00.00970.02890.01750.04240.02360.04760.02850.05760.03240.06247.00.00730.02110.01330.03110.01800.03520.02190.04270.02510.046510.00.00530.01500.00970.02220.01330.02530.01620.03080.01860.0336z/bl/b=1.2l/b=1.4l/b=1

43、.6l/b=1.8l/b=2.00.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01530.23420.01530.23430.01530.23430.01530.23430.01530.23430.40.02880.21870.02890.21890.02900.21900.02900.21900.02900.2191z/bl/b=0.2i/b=0.4i/b=0.6i/b=0.8i/b=1.0角点1角点2角点1角点2角点1角点2角点1角点2角点1角点20.60.03940.20390.03970.2043

44、0.03990.20460.04000.20470.04010.20480.80.04700.18990.04760.19070.04800.19120.04820.19150.04830.19171.00.05180.17690.05280.17810.05340.17890.05380.17940.05400.17971.20.05460.16490.05600.16660.05680.16780.05740.16840.05770.16891.40.05590.15410.05750.15620.05860.15760.05940.15850.05990.15911.60.05610.14430.05800.14670.05940.14840.06030.14940.06090.15021.80.05560.13540.