常用的地基沉降計算方法

6.3常用的地基沉降計算方法年，多則十幾年、幾十年乃至更長時間。計算地基最終沉降量的彈性力學方法地基最終沉降量的彈性力學計算方法是以BoussinesqPy,P 12sr

E （6-8）式中 —地基土的泊松比；；r —為地基表面任意點到集中力P作用點的距離，r x

y2。對于局部荷載下的地基沉降，則可利用上式，根據(jù)疊加原理求得。如圖6-6A（）點處的分布荷載為（積上的分布荷載可為集中力P=代替。于是，地面上與N點距離r=分求得：

(x)2

(y)2

（,y）（,y，可由式（8）積s(x,y)

12

p(,)dd00E A (x)2(y)2 （6-9）0圖6-5 集中力作用下地基表面的沉降曲線圖6-6 局部荷載下的地面沉降（任意荷載面（b）矩形荷載面從式（6-9）可以看出，如果知道了應力分布就可以求得沉降；反過來，若沉降已知又可以反算出應力分布。C的沉降按上式積分的結(jié)果為：cs12cE0

bp0

（6-10）c式中 —角點沉降影響系數(shù)，由下式確定：c1 1 1m2 mln( )ln(m m21)c m

（6-11） 式中 。利用式（0例如矩形中心點的沉降是圖6-6（b）中的虛線劃分為四個相同小矩形的角點沉降之和，即cs412cE0

(b/2)

120E00

bp0

（6-12）0 式中 —中心沉降影響系數(shù)。0 圖6-7 局部荷載作用下的地面沉降（）（b）絕對剛性基礎6-7ss(x,y)dxdy/AA （6-13）式中 基底面積，s(x,y)—點（x,y）處的基礎沉降。對于均布的矩形荷載，上式積分的結(jié)果為：s12sE0

mbp0

（6-14）式中 m—平均沉降影響系數(shù)。可將式（0、式（2、式（4）統(tǒng)一成為地基沉降的彈性力學公式的一般形式：E12Es bp00

（6-15）式中 矩形基礎（荷載）的寬度或圓形基礎（荷載）的直徑，—無量綱沉降影響系數(shù)，見表6-1?；A形狀方形矩形圓形基礎形狀方形矩形圓形基礎剛度柔性基礎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剛性基礎r0.790.881.081.221.441.611.72–––– 2.123.40（平均沉降）可按式（6-15）取r計算，基底傾斜的彈性力學公式如下：圓形基礎：

tan612E0

Peb3

（6-16a）矩形基礎：

tan8K12E0

Peb3

（6-16b）式中 基礎傾斜角；P—基底豎向偏心荷載合力；e—偏心距；b—荷載偏心方向的矩形基底邊長或圓形基底直徑；K—計算矩形剛性基礎傾斜的無量綱系數(shù)，按l/b取值，如圖6-8，其中l(wèi)為矩形基底另一邊長。6-8

通常按式（6-15）計算的基礎最終沉降量是偏大的。這是由于彈性力學公式是按勻質(zhì)線性變形半空間的假設得到的，而實際上地基常常是非均質(zhì)的成層土，即使是均質(zhì)的土層，其變形模量一般隨深度而增大。因此，利用彈性力學公式計算沉降的問題，在于所用的值能否反映地基變形的真實情況。地基觀測資料，以彈性力學公式反算求得，則這種數(shù)據(jù)是很有價值的。積變形，例如風或其它短暫荷載作用下，構(gòu)筑物基礎的傾斜可按式（6-16）圖6-9 地基沉降計算系數(shù)0和1算，注意式中的E0應取為地基彈性模量，并取泊松比=0.5。在大多數(shù)實際問題中，土層的厚度是有限的，下臥堅硬土層。Christian和Carrier（1978）提出了計算有限厚土層上柔性基礎的平均沉降計算公式：s01

bp0E0

（6-17）式中，0取決于基礎埋深和寬度之比D/b，1取決于地基土厚度H和基礎形狀。取泊松比時0和16-9由于瞬時沉降是在不排水狀態(tài)下發(fā)生的，因此，適宜的泊松比0.5，適應。[例題6-1] 某矩形基礎底面尺寸為其基底壓力埋5m解：D/b=0.5，查圖6-9，0=0.94考慮上層粘土，H/b=4/2=2，l/b=2，具有Eu=40MPa查圖6-9，1=0.60s因此 s

0.940.6021504.23mm40考慮二層粘土均具有6-91=0.85因此 s2

0.940.8521503.20mm75考慮第一層粘土，具有Eu=75MPa，則3s0.940.621502.26mm375因此，總的瞬時沉降為：ss1s2s34.233.202.265.17mm計算地基最終沉降量的分層總和法（一）一維壓縮課題在厚度為H的均勻土層上面施加連續(xù)均勻荷載p，見圖6-10a，這時土層況基本一樣，屬一維壓縮問題。施加外荷載之前，土層中的自重應力為圖6-10b中OBA；施加p之后，土層中引起的附加應力分布為OCDA。對整個土層來說，施加外荷載前后存在于土層中的平均豎向應力圖6-10 土層一維壓縮p1=H/2。從土的壓縮試驗曲線（6-10c）可以看出，增加到p2，將引起土的孔隙比從e1減小為e2。因此，可求得一維條件下土層的壓縮變形s與土的孔隙比e的變化之間存在如下關系：se1e2H1（6-18）這就是土層一維壓縮變形量的基本計算公式。式（6-18）也可改寫成：s a (p

)H a pH12 1

1（6-19）或 ss

a Am1vpH

（6-20）或式中 a—壓縮系數(shù)

Es （6-21）mv—體積壓縮系數(shù)；。（二）沉降計算的分層總和法基本原理分別計算基礎中心點下地基各分層土的圖6-11 分層總和法沉降計算圖例壓縮變形量si，認為基礎的平均沉降量s等于si的總和，即snsii1

（6-22）式中 計算深度范圍內(nèi)土的分層數(shù)。計算時si，假設土層只發(fā)生豎向壓縮變形，沒有側(cè)向變形，因此可按式（6-18）～式（6-21）中的任何一個公式進行計算。計算步驟1）（斜等，求出基底壓力的大小和分布；再結(jié)合地基土層的性狀，選擇沉降計算點的位置。將地基分層。在分層時天然土層的交界面和地下水位面應為分層面，同或b為基礎寬度。c（應從地面起算并繪出它的分布曲線。czz c 線，取=0.2（中、低壓縮性土）0.1zz c ci 求出各分層的平均自重應力和平均附加應力ci 1下)ci 2 ci ci 1下)zi 2 zi zici 式中 上、下—第ici 上、i分層土上、下層面處的附加應力。zi zi2計算各分層土的壓縮量。認為各分層土都是在側(cè)限壓縮條件下壓力從2pp1

增加到p

ci

所產(chǎn)生的變形量si，可由式（6-18）～式（6-21）中任一式計算。按式沉降量；根據(jù)基礎角點沉降差，可推算出基礎的傾斜。[例題6-2]某柱基礎，底面尺寸l×b=4×2m2，埋深d=1.5m。傳至基礎頂面的豎向荷載N=1192KN，各土層計算指標見例表6-1和例表6-2。試計算柱基礎最終沉降量。假定地下水位深dw=2m。6-2

土層計算指標 例表6-1

土層側(cè)限壓縮試驗曲線 例表土層 (kN/m2)

a1)

Es

土層

0 50 100 200①粘土

19.5

0.39

4.5

①粘土

0.8200.7800.7600.740②粉質(zhì)粘土

19.8

0.33

5.1

②粉質(zhì)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基底附加壓力p0：p0pd17919.51.5150kPaw取水的重度 ≈10kN/m3，則有效重度10，基礎中心線下的自重應wz 6-1=14.4kPa<0.2=0.2×Znz 例圖6-1 土層自重應力和附加應力分布按 ≤0.4b=0.4×2=0.8m 分層。，～，h7=h8=0.75m。柱基礎最終沉降量計算結(jié)果如下：eesi

1e2)ih

計算。1 i1各分層土沉降量計算程序 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土層粘土①分層hi(m)土層粘土①分層hi(m)e1ie2isi(mm)粉質(zhì)粘土②粉砂③h

( a 1ei

zi

i計算i各分層土沉降量計算程序 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土層粘土①分層hi(m)土層粘土①分層hi(m)e1iσzi(kPa)a(MPa–1)si(mm)粉質(zhì)粘土②粉砂③6–87–80.750.7581.688.40.84740.844621.116.50.373.172.48

E

i計算s146.50.5085.857.439.228.0)0.804.5 5.1(21.116.5)0.7516.2852.305.6474.22mm5.06.3.3地基沉降量計算的應力面積法《建筑地基基礎設計規(guī)范》所推薦的地基最終沉降量計算方法是另一種形式的分層總和法。該方法采用了“應力面積”的概念，因而稱為應力面積法。(一）土層壓縮變形量Δs的計算及應力面積的概念假設地基是均勻的，即土在側(cè)限條基底至地基任意深度z范圍內(nèi)的壓縮量為（見圖2：圖6-12 應力面積的概

sz10

z0

dz A

（6-23）z z式中—土的側(cè)限壓縮應變， /Ez zA—深度z范圍內(nèi)的附加應力面積，0A0

z2dz。因為zKsp0KszA為：A0

dzp0z0

z0

sdz

（6-24）為便于計算，引入一個豎向平均附加應力（面積）系數(shù)A/(p0Z)。則式（6-23）改寫為：zp0s zp0sE （6-25）s式（6-25）就是以附加應力面積等代值引出一個平均附加應力系數(shù)表達的從基底至任意深度zi范圍內(nèi)地基沉降量的計算公式。由此可得成層地基沉降量的計算公式（3in Assn A

i1 0z z )npinpi1

Esi Esi

i1

（6-26）0ii式中 pz0ii

和p

zi1

和

深度范圍內(nèi)豎向附加應力面積Ai

和的等0代值。0因此，用式（6-26）計算成層地基的沉降量，關鍵是確定豎向平均附加應力系數(shù)。由豎向平均附加應力系數(shù)的定義，均布荷載作用于矩形土表，相對其形心，在坐標處，從土表到z深處，可導出圖6-13 成層地基沉降計算的概念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的整數(shù)部分） （6-28a）Xxa Yyjb （6-28b）i i6-2；利用式也可以6-36-5，以供查用。為了提高計算準確度，地基沉降計算深度圍內(nèi)的計算沉降量s，尚須乘以一個沉降計算經(jīng)驗系數(shù)s?！督ㄖ鼗A設計規(guī)范》規(guī)定s的確定方法為：ss/s （6-29）式中 利用地基沉降觀測資料推算的最終沉降量。各地區(qū)宜按實測資料制定適合于本地區(qū)各類土的s值，而《建筑地基基礎設計規(guī)范》提供了一個采用表值，見表6-6。綜上所述，《建筑地基基礎設計規(guī)范》推薦的地基最終沉降量s的計算公式如下：ss

ni1

p(z0 i

i1

zi1

)/Esi

（6-30）式中，n―地基沉降計算深度范圍內(nèi)所劃分的土層數(shù)。均布矩形荷載角點下的平均豎向附加應力系數(shù)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載角點下的平均豎向附加應力系數(shù) 表6-3l/b0.2 l/b0.4 l/b0.6 l/b0.8 l/b1.0z/b

角點1角點2角點1角點2角點1角點2角點1角點2角點1角點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角點2角點1角點2角點1角點2角點1角點2角點1角點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