常用的地基沉降计算方法

6.3常用的地基沉降计算方法年，多则十几年、几十年乃至更长时间。计算地基最终沉降量的弹性力学方法地基最终沉降量的弹性力学计算方法是以BoussinesqPy,P 12sr

E （6-8）式中 —地基土的泊松比；；r —为地基表面任意点到集中力P作用点的距离，r x

y2。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6-6A（）点处的分布荷载为（积上的分布荷载可为集中力P=代替。于是，地面上与N点距离r=分求得：

(x)2

(y)2

（,y）（,y，可由式（8）积s(x,y)

12

p(,)dd00E A (x)2(y)2 （6-9）0图6-5 集中力作用下地基表面的沉降曲线图6-6 局部荷载下的地面沉降（任意荷载面（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。C的沉降按上式积分的结果为：cs12cE0

bp0

（6-10）c式中 —角点沉降影响系数，由下式确定：c1 1 1m2 mln( )ln(m m21)c m

（6-11） 式中 。利用式（0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cs412cE0

(b/2)

120E00

bp0

（6-12）0 式中 —中心沉降影响系数。0 图6-7 局部荷载作用下的地面沉降（）（b）绝对刚性基础6-7ss(x,y)dxdy/AA （6-13）式中 基底面积，s(x,y)—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：s12sE0

mbp0

（6-14）式中 m—平均沉降影响系数。可将式（0、式（2、式（4）统一成为地基沉降的弹性力学公式的一般形式：E12Es bp00

（6-15）式中 矩形基础（荷载）的宽度或圆形基础（荷载）的直径，—无量纲沉降影响系数，见表6-1。基础形状方形矩形圆形基础形状方形矩形圆形基础刚度柔性基础c1.01.52.03.04.05.06.07.08.09.010.1100.00.640.560.680.770.890.981.051.111.161.201.241.272.001.001.121.361.531.781.962.102.222.322.402.482.544.010m0.850.951.151.301.521.201.831.962.042.122.192.253.70刚性基础r0.790.881.081.221.441.611.72–––– 2.123.40（平均沉降）可按式（6-15）取r计算，基底倾斜的弹性力学公式如下：圆形基础：

tan612E0

Peb3

（6-16a）矩形基础：

tan8K12E0

Peb3

（6-16b）式中 基础倾斜角；P—基底竖向偏心荷载合力；e—偏心距；b—荷载偏心方向的矩形基底边长或圆形基底直径；K—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。6-8

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土，即使是均质的土层，其变形模量一般随深度而增大。因此，利用弹性力学公式计算沉降的问题，在于所用的值能否反映地基变形的真实情况。地基观测资料，以弹性力学公式反算求得，则这种数据是很有价值的。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）图6-9 地基沉降计算系数0和1算，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。Christian和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：s01

bp0E0

（6-17）式中，0取决于基础埋深和宽度之比D/b，1取决于地基土厚度H和基础形状。取泊松比时0和16-9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0.5，适应。[例题6-1] 某矩形基础底面尺寸为其基底压力埋5m解：D/b=0.5，查图6-9，0=0.94考虑上层粘土，H/b=4/2=2，l/b=2，具有Eu=40MPa查图6-9，1=0.60s因此 s

0.940.6021504.23mm40考虑二层粘土均具有6-91=0.85因此 s2

0.940.8521503.20mm75考虑第一层粘土，具有Eu=75MPa，则3s0.940.621502.26mm375因此，总的瞬时沉降为：ss1s2s34.233.202.265.17mm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图6-10 土层一维压缩p1=H/2。从土的压缩试验曲线（6-10c）可以看出，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：se1e2H1（6-18）这就是土层一维压缩变形量的基本计算公式。式（6-18）也可改写成：s a (p

)H a pH12 1

1（6-19）或 ss

a Am1vpH

（6-20）或式中 a—压缩系数

Es （6-21）mv—体积压缩系数；。（二）沉降计算的分层总和法基本原理分别计算基础中心点下地基各分层土的图6-11 分层总和法沉降计算图例压缩变形量si，认为基础的平均沉降量s等于si的总和，即snsii1

（6-22）式中 计算深度范围内土的分层数。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。计算步骤1）（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。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，同或b为基础宽度。c（应从地面起算并绘出它的分布曲线。czz c 线，取=0.2（中、低压缩性土）0.1zz c ci 求出各分层的平均自重应力和平均附加应力ci 1下)ci 2 ci ci 1下)zi 2 zi zici 式中 上、下—第ici 上、i分层土上、下层面处的附加应力。zi zi2计算各分层土的压缩量。认为各分层土都是在侧限压缩条件下压力从2pp1

增加到p

ci

所产生的变形量si，可由式（6-18）～式（6-21）中任一式计算。按式沉降量；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×b=4×2m2，埋深d=1.5m。传至基础顶面的竖向荷载N=1192KN，各土层计算指标见例表6-1和例表6-2。试计算柱基础最终沉降量。假定地下水位深dw=2m。6-2

土层计算指标 例表6-1

土层侧限压缩试验曲线 例表土层 (kN/m2)

a1)

Es

土层

0 50 100 200①粘土

19.5

0.39

4.5

①粘土

0.8200.7800.7600.740②粉质粘土

19.8

0.33

5.1

②粉质粘土0.7400.7200.7000.670③粉砂19.00.375.0③粉砂0.8900.8600.8400.810④粉土19.20.523.4④粉土0.8500.8100.7800.740p Nlb

1192b4

201.5179kPa基底附加压力p0：p0pd17919.51.5150kPaw取水的重度 ≈10kN/m3，则有效重度10，基础中心线下的自重应wz 6-1=14.4kPa<0.2=0.2×Znz 例图6-1 土层自重应力和附加应力分布按 ≤0.4b=0.4×2=0.8m 分层。，～，h7=h8=0.75m。柱基础最终沉降量计算结果如下：eesi

1e2)ih

计算。1 i1各分层土沉降量计算程序 例表6-30–10.5034.20.7995180.70.743915.451–20.8042.90.7228165.90.680219.782–30.8050.80.7197136.60.689014.283–40.8058.40.7166115.80.69539.934–50.8066.20.7135105.40.69847.055–60.8074.00.7104102.00.69945.146–80.7581.60.8474102.70.83923.337–80.7588.40.8446104.90.83852.48土层粘土①分层hi(m)土层粘土①分层hi(m)e1ie2isi(mm)粉质粘土②粉砂③h

( a 1ei

zi

i计算i各分层土沉降量计算程序 例表6-4i0–10.5034.20.7995146.50.3915.851–20.8042.90.7228123.018.852–30.8050.80.719785.813.173–40.8058.40.716657.40.338.834–50.8066.20.713539.26.045–60.8074.00.710428.04.32土层粘土①分层hi(m)土层粘土①分层hi(m)e1iσzi(kPa)a(MPa–1)si(mm)粉质粘土②粉砂③6–87–80.750.7581.688.40.84740.844621.116.50.373.172.48

E

i计算s146.50.5085.857.439.228.0)0.804.5 5.1(21.116.5)0.7516.2852.305.6474.22mm5.06.3.3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。(一）土层压缩变形量Δs的计算及应力面积的概念假设地基是均匀的，即土在侧限条基底至地基任意深度z范围内的压缩量为（见图2：图6-12 应力面积的概

sz10

z0

dz A

（6-23）z z式中—土的侧限压缩应变， /Ez zA—深度z范围内的附加应力面积，0A0

z2dz。因为zKsp0KszA为：A0

dzp0z0

z0

sdz

（6-24）为便于计算，引入一个竖向平均附加应力（面积）系数A/(p0Z)。则式（6-23）改写为：zp0s zp0sE （6-25）s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任意深度zi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（3in Assn A

i1 0z z )npinpi1

Esi Esi

i1

（6-26）0ii式中 pz0ii

和p

zi1

和

深度范围内竖向附加应力面积Ai

和的等0代值。0因此，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的定义，均布荷载作用于矩形土表，相对其形心，在坐标处，从土表到z深处，可导出图6-13 成层地基沉降计算的概念1 3 X

(X2Y2z2Y

)(X2Y2Y) (1)ii(z

ii Xln i

j i j j2z

2X2Y2z2 i (X2Y2z2

)(X2Y2Y)i0

i i i

j i j j(X2Y2z2X)(X2Y2X)Yln i

i i j i)j (X2Y2z2X)(X2Y2X)

(6-27)i j i i j iji j其中 int[ ]2（表示取2的整数部分） （6-28a）Xxa Yyjb （6-28b）i i6-2；利用式也可以6-36-5，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s。《建筑地基基础设计规范》规定s的确定方法为：ss/s （6-29）式中 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s值，而《建筑地基基础设计规范》提供了一个采用表值，见表6-6。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：ss

ni1

p(z0 i

i1

zi1

)/Esi

（6-30）式中，n―地基沉降计算深度范围内所划分的土层数。均布矩形荷载角点下的平均竖向附加应力系数表6-2z/b 1.0 1.2 1.4 1.6 1.8

l/b2.4 2.8 3.2 3.6 4.0 5.00.00.25000.25000.25000.25000.25000.25000.25000.25000.25000.25000.25000.25000.0.20.24960.24970.24970.24980.24980.24980.24980.24980.24980.24980.24980.24980.0.40.24740.24790.24810.24830.24830.24840.24850.24850.24850.24850.24850.24850.0.60.24230.24370.24440.24480.24510.24520.24540.24550.24550.24550.24550.24550.0.80.23460.23720.23870.23950.24000.24030.24070.24080.24090.24090.24100.24100.1.00.22520.22910.23130.23260.23350.23400.23460.23490.23510.23520.23520.23530.1.20.21490.21990.22290.22480.22600.22680.22780.22820.22850.22860.22870.22880.1.40.20430.21020.21400.21640.21900.21910.22040.22110.22150.22170.22180.22200.1.60.19390.20060.20490.20790.20990.21130.21300.21380.21430.21460.21480.21500.1.80.18400.19120.19600.19940.20180.20340.20550.20660.20730.20770.20790.20820.2.00.17460.18220.18750.19120.19380.19580.19820.19960.20040.20090.20120.20150.2.20.16590.17370.17930.18330.18620.18830.19110.19270.19370.19430.19470.19520.2.40.15780.16570.17150.17570.17890.18120.18430.18620.18730.18800.18850.18900.2.60.15030.15830.16420.16860.17190.17450.17790.17990.18120.18200.18250.18320.2.80.14330.15140.15740.16190.16540.16800.17170.17390.17530.17630.17690.17770.3.00.13690.14490.15100.15560.15920.16190.16580.16820.16980.17080.17150.17250.3.20.13100.13900.14500.14970.15330.15620.16020.16280.16450.16570.16640.16750.3.40.12560.13340.13940.14410.14780.15080.15500.15770.15950.16070.16160.16280.3.60.12050.12820.13420.13890.14270.14560.15000.15280.15480.15610.15700.15830.3.80.11580.12340.12930.13400.13780.14080.14520.14820.15020.15160.15260.15410.4.00.11140.11890.12480.12940.13320.13620.14080.14380.14590.14740.14850.15000.4.20.10730.11470.12050.12510.12890.13190.13650.13960.14180.14340.14450.14620.4.40.10350.11070.11640.12100.12480.12790.13250.13570.13790.13960.14070.14250.4.60.10000.10700.11270.11720.12090.12400.12870.13190.13420.13590.13710.13900.z/b 1.0 1.2 1.4 1.6 1.8

l/b2.4 2.8 3.2 3.6 4.0 5.05.00.09350.10030.10570.11020.11390.11690.12160.12490.12730.12910.13040.13250.5.20.09060.09720.10260.10700.11060.11360.11830.12170.12410.12590.12730.12950.5.00.09350.10030.10570.11020.11390.11690.12160.12490.12730.12910.13040.13250.5.20.09060.09720.10260.10700.11060.11360.11830.12170.12410.12590.12730.12950.5.40.08780.09430.09960.10390.10750.11050.11520.11860.12110.12290.12430.12650.5.60.08520.09160.09680.10100.10460.10760.11220.11560.11810.12000.12150.12380.5.80.08280.08900.09410.09830.10180.10470.10940.11280.11530.11720.11870.12110.6.00.08050.08660.09160.09570.09910.10210.10670.11010.11260.11460.11610.11850.6.20.07830.08420.08910.09320.09660.09950.10410.10750.11010.11200.11360.11610.6.40.07620.08200.08690.09090.09420.09710.10160.10500.10760.10960.11110.11370.6.60.07420.07990.08470.08860.09190.09480.09930.10270.10530.10730.10880.11140.6.80.07230.07790.08260.08650.08980.09260.09700.10040.10300.10500.10660.10920.7.00.07050.07610.08060.08440.08770.09040.09490.09820.10080.10280.10440.10710.7.20.06880.07420.07870.08250.08570.08840.09280.09620.09870.10080.10230.10510.7.40.06720.07250.07690.08060.08380.08650.09080.09420.09670.09880.10040.10310.7.60.06560.07090.07520.07890.08200.08460.08890.09220.09480.09680.09840.10120.7.80.06420.06930.07360.07710.08020.08280.08710.09040.09290.09500.09660.09940.8.00.06270.06780.07200.07550.07850.08110.08530.08860.09120.09320.09480.09760.8.20.06140.06630.07050.07390.07690.07950.08370.08690.08940.09140.09310.09590.8.40.0601.06490.06900.07240.07540.07790.08200.08520.08780.09890.09140.09430.8.60.05880.06360.06760.07100.07390.07640.08050.08360.08620.08820.08980.09270.8.80.05760.06230.06630.06960.07240.07490.07900.08210.08460.08660.08820.091209.20.05540.05990.096370.06970.07210.07610.07920.08170.08370.08530.08820.08130.9.60.05330.05770.06140.06720.06960.07340.07650.07890.08090.08250.08550.07380.10.00.05140.05560.05920.06490.06720.07100.07390.07630.07830.07990.08290.07190.10.40.04960.05370.05720.06270.06490.06860.07160.07390.07590.07750.08040.06820.10.80.04790.05190.05530.06060.06280.06640.06930.07170.07360.07510.07810.06490.11.20.04630.05020.05350.05630.05870.06090.06440.06720.06950.07140.07300.07590.11.60.04480.04860.05180.05450.05690.05900.06250.06520.06750.06940.07090.07380.12.00.04350.04710.05020.05290.05520.05730.06060.06340.06560.06740.06900.07190.12.80.04090.04440.04740.04990.05210.05410.05730.05990.06210.06390.06540.06820.13.60.03870.04200.04480.04720.04930.05120.05430.05680.05890.06070.06210.06490.z/b 1.0 1.2 1.4 1.6 1.8

l/b2.4 2.8 3.2 3.6 4.0 5.014.40.03670.03980.04250.04480.04680.04860.05160.05400.05610.05770.05920.06190.15.20.03490.03790.04040.04260.04460.04630.04920.05150.05350.05510.05650.05920.16.00.03320.03610.03850.04070.04250.04420.04690.04920.05110.05270.05400.05670.18.00.02970.03230.03450.03640.03810.03960.04220.04420.04600.04750.04870.05120.20.00.02690.02930.03120.03300.03450.03590.03830.04020.04180.04320.04440.04680.三角形分布的矩形荷载角点下的平均竖向附加应力系数 表6-3l/b0.2 l/b0.4 l/b0.6 l/b0.8 l/b1.0z/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20.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01120.21610.01400.23080.01480.23330.01510.23390.01520.23410.40.01790.18100.02450.20840.02700.21530.02800.21750.02850.21840.60.02070.15050.03080.18510.03550.19660.03760.20110.03880.20300.80.02170.12770.03400.16400.04050.17870.04400.18520.04590.18831.00.02170.11040.03510.14610.04300.16240.04760.17040.05020.17461.20.02120.09700.03510.13120.04390.14800.04920.15710.05250.16211.40.02040.08650.03440.11870.04360.13560.04950.14510.05340.15071.60.01950.07790.03330.10820.04270.12470.04900.13450.05330.14051.80.01860.07090.03210.09930.04150.11530.04800.12520.05250.13132.00.01780.06500.03080.09170.04010.10710.04670.11690.05130.12322.50.01570.05380.02760.07690.03650.09080.04290.10000.04780.10633.00.01400.04580.02480.06610.03300.07860.03920.08710.04390.09315.00.00970.02890.01750.04240.02360.04760.02850.05760.03240.06247.00.00730.02110.01330.03110.01800.03520.02190.04270.02510.046510.00.00530.01500.00970.02220.01330.02530.01620.03080.01860.03360.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01530.23420.01530.23430.01530.23430.01530.23430.01530.23430.40.02880.21870.02890.21890.02900.21900.02900.21900.02900.2191z/b l/b1.2 l/b1.4 l/b1.6 l/b1.8 l0.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01530.23420.01530.23430.01530.23430.01530.23430.01530.23430.40.02880.21870.02890.21890.02900.21900.02900.21900.02900.2191z/b

l/b0.2 l/b0.4 l/b0.6 l/b0.8 l/b1.01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20.60.03940.20390.03970.20430.03990.20460.04000.20470.04010.20480.80.04700.18990.04760.19070.04800.19120.04820.19150.04830.19171.00.05180.17690.05280.17810.05340.17890.05380.17940.05400.17971.20.05460.16490.05600.16660.05680.16780.05740.16840.05770.16891.40.05590.15410.05750.15620.05860.15760.05940.15850.05990.15911.60.05610.14430.05800.14670.05940.14840.06030.14940.06090.15021.80.05560.13540.05780.13810.05930.14000.06040.14130.06110.14222.00.05470.12740.05700.13030.05870.13240.05990.13380.06080.13482.50.05130.11070.05400.11390.05600.11630.05750.11800.05860.11933.00.04760.09760.05030.10080.05250.10330.05410.10520.05540.10675.00.03560.06610.03820.06900.04030.07140.04210.07340.04350.07497.00.02770.04960.02990.05200.03180.05410.03330.05580.03470.057210.00.02070.03590.02240.03790.02390.03950.02520.04090.02630.0403l/b3.0 l/b4.0 l/b6.0 l/b8.0 l/b10.00.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01530.23430.01530.23430.01530.23430.01530.23430.01530.23430.00.00000.25000.00000.25000.00000.25000.00000.25000.00000.25000.20.01530.23430.01530.23430.01530.23430.01530.23430