沉降计算例题

1、地基沉降量计算地基变形在其表面形成的垂直变形量称为建筑物的沉降量。在外荷载作用下地基土层被压缩达到稳定时基础底面的沉降量称为地基最终沉降量。一、分层总和法计算地基最终沉降量计算地基的最终沉降量，目前最常用的就是分层总和法。（一）基本原理该方法只考虑地基的垂向变形，没有考虑侧向变形，地基的变形同室内侧限压缩试验中的情况基本一致， 属一维压缩问题。 地基的最终沉降量可用室内压缩试验确定的参数（ei、Es、a）进行计算，有：变换后得：式中： S- 地基最终沉降量（mm）；e1- 地基受荷前（自重应力作用下）的孔隙比；e2- 地基受荷（自重与附加应力作用下）沉降稳定后的孔隙比；H-土层的厚度。计算沉降

2、量时，在地基可能受荷变形的压缩层范围内，根据土的特性、应力状态以及地下水位进行分层。然后按式（ 4-9 ）或（ 4-10 ）计算各分层的沉降量Si 。最后将各分层的沉降量总和起来即为地基的最终沉降量：（二）计算步骤1）划分土层如图 4-7 所示，各天然土层界面和地下水位必须作为分层界面；各分层厚度必须满足Hi00.4B （B为基底宽度）。2）计算基底附加压力 p03）计算各分层界面的自重应力crsz和附加应力bz;并绘制应力分布曲线。4）确定压缩层厚度满足（rz=0.2 bsz的深度点可作为压缩层的下限；对于软土则应满足（rz=0.1 o-sz；对一般建筑物可按下式计算zn=B（2.5-0.4

3、lnB） 。5）计算各分层加载前后的平均垂直应力p1 =（T sz;p2=（r sz+（TZ6）按各分层的pl和p2在e-p曲线上查取相应的孔隙比或确定a、Es等其它压缩性指标7）根据不同的压缩性指标，选用公式（4-9 ）、（4-10）计算各分层的沉降量Si8）按公式（4-11 ）计算总沉降量S。分层总和法的具体计算过程可参。例题41已知柱下单独方形基础，基础底面尺寸为2.5X2.5m,埋深2m,作用于基础上（设计地面标高处）的轴向荷载N=1250kN有关地基勘察资料与基础剖面详见下图。试用单向分层总和法计算基础中点最终沉降量。解：按单向分层总和法计算( 1)计算地基土的自重应力。 z 自基底

4、标高起算。当 z=0m , b sD=19.5 X 2=39( kPa)z=, b sz1=39+19.5 x 1=58kPa)z=, b sz1=58.5+20 x 1=78kPa)z=, b sz1=78.5+20 x 1=98kPa)z=, b sz1=98.5+(2010)收108.5 (kPa)z=, b sz1=108.5+(2010) 1=118.5 (kPa)z=, b sz1=118.5+18.5 x 1=1kPa)z=, b sz1=137+18.5 x 1=155kPa)2）基底压力计算。基础底面以上，基础与填土的混合容重取0=20kN/m3。(3)基底附加压力计算。(4

5、)基础中点下地基中竖向附加应力计算。用角点法计算，L/B=1 , bZi=4Ksi ； p0附加应力系数表得 Ksi。(5)确定沉降计算深度 zn考虑第层土压缩性比第层土大，经计算后确定zn=7m,见下表。例题4-1计算表格1z (m)z B/2Kscz(kPa)csz(kPa)8 sz (%)zn (m)012345670 0.8 1.6 2.4 3.2 4.0 4.8 5.60.250 00.199 90.112 30.064 20.040 10.027 00.019 30.014 8201160.790.2951.6232.2421.7115.5211.903958.578.598.81

6、08.5118.5137155.529.7118.3211.337.6按7m计(6)计算基础中点最终沉降量。利用勘察资料中的e-p曲线，求按单向分层总和法公式计算结果见下表。例题4-1计算表格2z(m)(kPa)(kPa)H( c m )自重应力 平均值(kPa)附加应力 平均值(kPa)(kPa)e1e(kPa-1)(kPa)(cm)(cm)039201100158.5160.70.278.590.29076398.551.621144108.532.240000.00035118.521.71613715.520.871660.0002048.75180.85229.6413944184.

7、09068.50125.50194000.000268611.835.92188.5070.96159.46.1177490.926.840103.541.93145.43660.000268480.617.450113.526.98140.48323843930.618.061127.518.62146.12500.000331470.598.6500.717155.511.900146.2513.71159.96.60.000523040.599.24162370300.000700.291.60620300.建筑地基基础设计规范推荐的沉降计算法下面计算沉降量的方法是建筑地基基础设计规范（

8、GBJ7-89）所推荐的，简称规范推荐法，有时也叫应力面积法。（一）计算原理应力面积法一般按地基土的天然分层面划分计算土层，引入土层平均附加应力的概念，通过平均附加应力系数，将基底中心以下地基中zi-1-zi深度范围的 附加应力按等面积原则化为相同深度范围内矩形分布时的分布应力大小，再按矩形分布应力情况计算土层的压缩量，各土层压缩量的总和即为地基的计算沉降 量。理论上基础的平均沉降量可表示为式中：S-地基最终沉降量（mm ;n-地基压缩层（即受压层）范围内所划分的土层数；p0-基础底面处的附加压力（kPa）;Esi-基础底面下第i层土的压缩模量（MPa ;zi、zi-1-分别为基础底面至第i层

9、和第i-1层底面的距离（mm ;ai、a i-1-分别为基础底面计算点至第i层和第i-1层底面范围内平均 附加应力系数，可查。表4-1矩形面积上均布荷载作用下，通过中心点竖线上的平均附加应力系数z/ BL/B1.01.21.41.61.82.02.42.83.23.64.05.0100.1.001.001.001.001.001.001.001.001.001.001.001.001.00000000000000000.0.990.990.990.990.990.990.990.990.990.990.990.990.99178888888888880.0.980.990.990.990.99

10、0.990.990.990.990.990.990.990.99270122233333330.0.960.970.970.970.970.970.980.980.980.980.980.980.98373689900111120.40.9360.9470.9530.9560.9580.9650.9610.9620.9620.9630.9630.9630.9630.0.900.910.920.920.930.930.930.930.930.940.940.940.94505493579900000.0.850.870.890.890.900.900.910.910.910.910.910.91

11、0.91688083602344550.0.810.840.850.860.870.870.880.880.880.880.880.880.88760551614567780.0.770.800.810.830.830.840.850.850.850.850.850.860.86851919415789000.0.730.760.780.790.800.810.820.820.820.830.830.830.83954476316901231.0.690.720.740.760.770.780.790.790.800.800.800.800.80083945328134671.0.660.69

12、0.710.730.740.750.760.770.770.770.770.780.78134734341579021.0.630.660.680.700.710.720.730.740.740.750.750.750.75213635574924681.0.600.630.650.670.680.690.710.710.720.720.730.730.73313748819580351.0.570.600.620.640.660.670.680.690.700.700.700.710.71435981276158141.0.540.580.600.620.630.640.660.670.67

13、0.680.680.690.69580427346936031.0.520.550.580.590.610.620.640.650.650.660.660.670.67646093511836051.0.500.530.550.570.590.600.620.630.630.640.640.650.65723871301836161.0.480.510.520.550.570.580.600.610.610.620.620.630.63823761301949381.0.460.490.510.530.550.560.580.590.600.600.610.610.62933761313160

14、622.0.440.470.490.510.530.540.560.570.580.590.590.600.60065983535404062.0.420.450.480.500.510.520.540.550.560.570.570.580.59199205869748512.0.410.440.460.480.490.510.530.540.550.550.560.570.57243649103283072.0.400.420.450.460.480.490.510.520.530.540.540.550.56308194658748642.40.3870.4140.4360.4540.4

15、690.4810.5000.5130.5230.5300.5350.5430.5512.0.370.400.420.440.450.460.480.500.500.510.520.530.53541315860962092.0.360.380.410.420.440.450.470.480.490.500.500.510.52629082537649882.0.350.370.390.410.430.440.460.470.480.490.490.500.51717860214427672.0.340.360.380.400.410.430.440.460.470.480.480.490.50

16、816748093206562.0.330.350.370.390.400.410.430.450.460.460.470.480.49916737981195563.0.320.340.360.380.390.400.420.440.450.450.460.470.48026637971195473.0.310.330.350.370.380.390.410.430.440.440.450.460.47137737870084473.0.300.320.340.360.370.380.400.420.430.430.440.450.46258847970195583.0.290.320.33

17、0.350.360.370.390.410.420.420.430.440.46370958971196603.0.280.310.330.340.350.370.380.400.410.420.420.430.45492169182207723.0.280.300.320.330.350.360.380.390.400.410.410.420.44524381203328943.0.270.290.310.330.340.350.370.380.390.400.410.420.43667503425530163.0.260.290.300.320.330.340.360.370.380.39

18、0.400.410.42790835647752393.0.260.280.300.310.320.330.350.360.370.380.390.400.42834168969984523.0.250.270.290.300.320.330.340.360.370.380.380.390.41977491292207854.0.250.270.280.300.310.320.340.350.360.370.370.390.40011824525539184.0.240.260.280.290.300.310.330.340.360.360.370.380.40165268858862424.

19、0.240.260.270.290.300.310.320.340.350.350.360.370.39210602281296764.0.230.250.270.280.290.300.320.330.340.360.350.370.39365046625539104.0.230.250.260.270.290.300.310.320.330.340.350.360.38410580069973544.0.220.240.260.270.280.290.310.320.330.340.340.350.37565035403317984.0.220.240.250.260.270.280.30

20、0.310.320.330.340.350.37620589957751334.0.210.230.250.260.270.280.290.310.320.320.330.340.36785034492196774.0.210.230.240.250.260.270.290.300.310.320.330.340.36841589946640224.0.210.220.240.250.260.270.280.300.310.310.320.330.35907135491195775.0.200.220.230.240.260.260.280.290.300.310.320.330.350637

21、9094663022（二）规范推荐公式由（4-12）式乘以沉降计算经验系数 巾s,即为规范推荐的沉降计算公 式：式中：巾s -沉降计算经验系数，应根据同类地区已有房屋和构筑物实测最终沉 降量与计算沉降量对比确定，一般采用的数值；表4-2沉降计算经验系数ips基底附加压力 p0（kPa）压缩模里Es（MPa）2.54.07.015.020.0p0=fk1.41.31.00.40.2p0 0.75fk1.11.00.70.40.2注：表列数值可内插；当变形计算深度范围内有多层土时，Es可按附加应力面积 A的加权平均值采用，即（三）地基受压层计算深度的确定计算深度zn可按下述方法确定：1）存在相邻荷

22、载影响的情况下，应满足下式要求:式中：Sn-在深度zn处，向上取计算厚度为的计算变形值；Az查;表4-3 Az取值B(m)22V B44V B88V B 1515V B30 z(m)0.30.60.81.01.21.5 Si-在深度zn范围内，第i层土的计算变形量。2）对无相邻荷载的独立基础，可按下列简化的经验公式确定沉降计算深度 zn:规范法的具体计算过程可参。【例题42】已知柱下单独方形基础，基础底面尺寸为 2.5X2.5m,埋深2nl 作用于基础上（设计地面标高处）的轴向荷载 N=1250kN有关地基勘察资料与 基础剖面详见下图。试用规范法计算基础中点最终沉降量。解：按建筑地基基础设计规

23、范计算，采用下式，计算结果详见下表。例题4-2计算表格z(m)L/Bz/BEsi(kP a)(cm)(cm)000.250001.00.80.23460.23460.234644184.274.272.01.60.19390.38780.153268611.806.073.02.40.15780.47340.085677490.896.964.03.20.13100.52400.050668480.597.555.04.00.11140.55700.03343930.608.156.04.80.09670.58020.023231470.598.747.05.60.08520.59640.01

24、6223040.579.317.66.080.08040.61100.0146350000.039.34按规范确定受压层下限，zn=2.5（2.5-0.4ln2.5）=5.3m ;由于下面土层仍软弱，在层粘土底面以下取Az厚度计算，根据的要求，取Az=0.6m,则zn=7.6m,计算得厚度Az的沉降量为0.03cm,满足要求。查得沉降计算经验系数Ws=1.1%那么，最终沉降量为:三、按粘性土的沉降机理计算沉降根据对粘性土地基在局部（基础）荷载作用下的实际变形特征的观察和分析， 粘性土地基的沉降S可以认为是由机理不同的三部分沉降组成 （图4-8）,亦即:式中：Sd-瞬时沉降（亦称初始沉降）； S

25、c-固结沉降（亦称主固结沉降）； Ss-次固结沉降（亦称蠕变沉降）。瞬时沉降是指加载后地基瞬时发生的沉降。由于基础加载面积为有限尺寸,加载后地基中会有剪应变产生，剪应变会引起侧向变形而造成瞬时沉降固结沉降是指饱和与接近饱和的粘性土在基础荷载作用下，随着超静孔隙水压力的消散，土骨架产生变形所造成的沉降（固结压密）。固结沉降速率取决于 孔隙水的排出速率。次固结沉降是指主固结过程（超静孔隙水压力消散过程）结束后，在有效应 力不变的情况下，土的骨架仍随时间继续发生变形。这种变形的速率取决于土骨 架本身的蠕变性质。（一）瞬时沉降计算瞬时沉降没有体积变形，可认为是弹性变形，因此一般按弹性理论计算，按 式（4-17）求解。式中：沉降系数，可从中查用；p0-基底附加应力；N-泊松比，这时是在不排水条件下没有体积变形所产生的变形量，所以应取产0.5;Eu-不排水变形模量，常根据不排水抗剪强度Cu和Eu的经验关系式（4-18）求得。上式中的低值适用于较软的、高塑性有机土，高值适用于一般较硬的粘性土表4-4沉降系数,值受衲圆形