基坑开挖卸荷引起下卧隧道隆起的计算方法
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关键词: 基坑开挖; 隧道隆起; Mindlin 解; 弹性地基梁
中图分类号: U451+ . 1
文献标识码: A
Calculation of Tunnel Heaving due to Unloading of Pit Excavation
CHEN Yu , LI Yong-sheng
( Department of Geotechnical Engineering , T ongj i University , Shanghai 200092, P . R. China ) Abstract: At present, more and more pit excavations lie over the built Metro tunnels. The near unloading certainly will lead to heaving of underlying tunnels. Therefore, how to predict tunnel deformations to insure its working safety becomes very important. Based on Dongfang Road Underground Crossing project in Shanghai, the additional stress in the tunnel lining due to excavation of foundation pit can be deduced by Mindlin theory. Then, a method to calculate the heaving of tunnel can be deduced by elastic foundation beam theory and the calculated max imum heaving is compared with monitoring data. From the analysis, some conclusions can be drawn. Key words: ex cavation of pit foundation; tunnel heaving; Mindlin solution; elastic foundation beam
作用下( 见图 2) , 对于埋深为 z0、纵轴线平行于均
布荷载且和荷载轴线成 A角的 隧道, 根据 Mindlin
基本解[ 2] ,
Rz =
Q 8P( 1-
M)
(
1-
2M) ( z
R
3 1
0
-
d
)
+
3(
z
0R51
d
)
3
-
( 1- 2M) ( z 0- d ) R32
+
3( 3- 4M) z0 ( z 0+
y)2 +
( z0-
d) 2
上式采用五节点高斯勒让德积分, 可得:
G=
b 2
t
2
=6
i =-2
Xi (
b 2
ti ctgH+
l 2
-x
)
b 2
(
b 2
ti-
y)2+ ( z0-
d) 2
(
b 2
ti ctgH+
l 2
-
x
)
2+
(
b 2
ti- y )2+ ( z 0- d )2
-
Xi (
b 2
t ictgH-
R
5 2
+
30dz 0( z 0 +
d)3
D
dNdG
R
7 2
( 2)
其中: R1 = ( x- N) 2 + ( y- G) 2 + ( z 0- d ) 2
R2 = ( x- N) 2 + ( y- G) 2 + ( z 0 + d ) 2
上式中各积分可分别计算如下:
k Q Q dNdG R31
=
图 2 xoy 平面的投影图
Fig. 2 The projection in xoy plan
由式( 1) 可知, 隧道上某 一点( x , y , z0 ) , 在均
布荷载中的一点( N, G) 上的力 pdNdG 作用下引起的 隧道轴线处的竖向附加应力 Rz 为,
QQ Rz =
p 8P( 1- M)
l 2
-
x)
2+
(
b 2
t i- y
) 2+
(
z0+
d)
2
Xi (
b 2
ti ctgH-
1 2
-
x
)
b 2
+
(
b 2
ti- y ) 2+ ( z 0- d) 2
(
b 2
ti ctgH-
l 2
-
x)
2+
(
b 2
t i- y) 2+ ( z0+ d ) 2
j =-2 2
6 6 [ 3( 3- 4M) z0 ( z0+ d ) 2- 3d ( z0+ d ) ( 5z 0- d) ] i=-2
Xj Xi
l 2
b 2
(
b 2
sj
ctg H+
1 2
t i- x)
2+ (
b 2
sj-
y ) 2+ (
z 0-
d)
2
5-
2
2
6 ( 1- 2) M( z 0- d ) i =-2
Xi (
b 2
ti ctgH+
1 2
-
x)
b 2
(
b 2
ti- y) 2+ ( z 0- d ) 2
(
b 2
t ict gH+
dG
Gctg H-l / 2
b/2
Q-b/2 [ ( G- y) 2 + ( z 0- d ) 2]
Gctg H+
1 2
-
x
Gctg H+
l 2
- x)2 +
( G-
y)2 +
( z0-
d) 2
[ ( G- y) 2 + ( z0- d) 2]
GctgH-
1 2
-
x
Gctg H-
l 2
-
x)2 +
( G-
y ) 2+ (
z0+
d)
2
7 2
故有:
6 Rz =
8P(
p 1-
M)
#
2
( 1- 2M) ( z 0- d)
i =-2
Xi (
b 2
t ict gH+
1 2
-
x
)
b 2
(
b 2
t i- y) 2+ ( z0- d ) 2
(
b 2
t i ctgH+
l 2
-
x
)
2+
(
b 2
ti- y ) 2+ ( z 0-
Xj Xi
l 2
b 2
(
b 2
sj ctgH+
1 2
ti- x ) 2+ (
b 2
sj- y) 2+ ( z 0+ d ) 2
5 2
22
6 6 +30dz 0( z 0+ d) 3 j = 2 i =-2
Xj Xi
l 2
b 2
(
b 2
sj ctgH+
1 2
ti- x ) 2+ (
b 2
sj- y) 2+ ( z 0+ d ) 2
d )2
(
b 2
ti ctgH-
l 2
-
x
)
2+
(
b 2
ti- y) 2+ ( z 0+ d) 2
k 6 6 D
dNdG
R
5 1
=
j =-2 2 i =-2
XjXi
l 2
b 2
(
b 2
sj ctgH+
l 2
t i- x) 2+ (
b 2
sj-
y ) 2+ (
z0-
d)
2
5 2
k 6 6 D
dNdG
( 1- 2M) ( z 0- d)
D
dNdG
R
3 1
+
QQ 3( z 0- d ) 3 D
dRNd51G- ( 1- 2M) ( z 0- d )
QQD
dNdG
R
3 2
+
[ 3( 3- 4M) z 0 ( z 0 +
d) 2- 3d ( z 0 +
d) #
QQ QQ ( 5z0- d) ] D
dNdG
5 2
( 3)
上式即为求算作用在隧道轴线处的附加应力
计算公式。
3 隧道的纵向位移
当隧道附加作用广义荷载引起垂直于隧道的
附加应力时, 可以将隧道纵向看作是分布荷载下的
Winkler 弹性地基无限长梁[ 3] , 因此, 隧道与地层相
互作用的力学方程可写为:
92
地下空间与工程学报
第 1卷
体重力强度, 方向向上;
中图分类号: U451+ . 1
文献标识码: A
Calculation of Tunnel Heaving due to Unloading of Pit Excavation
CHEN Yu , LI Yong-sheng
( Department of Geotechnical Engineering , T ongj i University , Shanghai 200092, P . R. China ) Abstract: At present, more and more pit excavations lie over the built Metro tunnels. The near unloading certainly will lead to heaving of underlying tunnels. Therefore, how to predict tunnel deformations to insure its working safety becomes very important. Based on Dongfang Road Underground Crossing project in Shanghai, the additional stress in the tunnel lining due to excavation of foundation pit can be deduced by Mindlin theory. Then, a method to calculate the heaving of tunnel can be deduced by elastic foundation beam theory and the calculated max imum heaving is compared with monitoring data. From the analysis, some conclusions can be drawn. Key words: ex cavation of pit foundation; tunnel heaving; Mindlin solution; elastic foundation beam
作用下( 见图 2) , 对于埋深为 z0、纵轴线平行于均
布荷载且和荷载轴线成 A角的 隧道, 根据 Mindlin
基本解[ 2] ,
Rz =
Q 8P( 1-
M)
(
1-
2M) ( z
R
3 1
0
-
d
)
+
3(
z
0R51
d
)
3
-
( 1- 2M) ( z 0- d ) R32
+
3( 3- 4M) z0 ( z 0+
y)2 +
( z0-
d) 2
上式采用五节点高斯勒让德积分, 可得:
G=
b 2
t
2
=6
i =-2
Xi (
b 2
ti ctgH+
l 2
-x
)
b 2
(
b 2
ti-
y)2+ ( z0-
d) 2
(
b 2
ti ctgH+
l 2
-
x
)
2+
(
b 2
ti- y )2+ ( z 0- d )2
-
Xi (
b 2
t ictgH-
R
5 2
+
30dz 0( z 0 +
d)3
D
dNdG
R
7 2
( 2)
其中: R1 = ( x- N) 2 + ( y- G) 2 + ( z 0- d ) 2
R2 = ( x- N) 2 + ( y- G) 2 + ( z 0 + d ) 2
上式中各积分可分别计算如下:
k Q Q dNdG R31
=
图 2 xoy 平面的投影图
Fig. 2 The projection in xoy plan
由式( 1) 可知, 隧道上某 一点( x , y , z0 ) , 在均
布荷载中的一点( N, G) 上的力 pdNdG 作用下引起的 隧道轴线处的竖向附加应力 Rz 为,
QQ Rz =
p 8P( 1- M)
l 2
-
x)
2+
(
b 2
t i- y
) 2+
(
z0+
d)
2
Xi (
b 2
ti ctgH-
1 2
-
x
)
b 2
+
(
b 2
ti- y ) 2+ ( z 0- d) 2
(
b 2
ti ctgH-
l 2
-
x)
2+
(
b 2
t i- y) 2+ ( z0+ d ) 2
j =-2 2
6 6 [ 3( 3- 4M) z0 ( z0+ d ) 2- 3d ( z0+ d ) ( 5z 0- d) ] i=-2
Xj Xi
l 2
b 2
(
b 2
sj
ctg H+
1 2
t i- x)
2+ (
b 2
sj-
y ) 2+ (
z 0-
d)
2
5-
2
2
6 ( 1- 2) M( z 0- d ) i =-2
Xi (
b 2
ti ctgH+
1 2
-
x)
b 2
(
b 2
ti- y) 2+ ( z 0- d ) 2
(
b 2
t ict gH+
dG
Gctg H-l / 2
b/2
Q-b/2 [ ( G- y) 2 + ( z 0- d ) 2]
Gctg H+
1 2
-
x
Gctg H+
l 2
- x)2 +
( G-
y)2 +
( z0-
d) 2
[ ( G- y) 2 + ( z0- d) 2]
GctgH-
1 2
-
x
Gctg H-
l 2
-
x)2 +
( G-
y ) 2+ (
z0+
d)
2
7 2
故有:
6 Rz =
8P(
p 1-
M)
#
2
( 1- 2M) ( z 0- d)
i =-2
Xi (
b 2
t ict gH+
1 2
-
x
)
b 2
(
b 2
t i- y) 2+ ( z0- d ) 2
(
b 2
t i ctgH+
l 2
-
x
)
2+
(
b 2
ti- y ) 2+ ( z 0-
Xj Xi
l 2
b 2
(
b 2
sj ctgH+
1 2
ti- x ) 2+ (
b 2
sj- y) 2+ ( z 0+ d ) 2
5 2
22
6 6 +30dz 0( z 0+ d) 3 j = 2 i =-2
Xj Xi
l 2
b 2
(
b 2
sj ctgH+
1 2
ti- x ) 2+ (
b 2
sj- y) 2+ ( z 0+ d ) 2
d )2
(
b 2
ti ctgH-
l 2
-
x
)
2+
(
b 2
ti- y) 2+ ( z 0+ d) 2
k 6 6 D
dNdG
R
5 1
=
j =-2 2 i =-2
XjXi
l 2
b 2
(
b 2
sj ctgH+
l 2
t i- x) 2+ (
b 2
sj-
y ) 2+ (
z0-
d)
2
5 2
k 6 6 D
dNdG
( 1- 2M) ( z 0- d)
D
dNdG
R
3 1
+
QQ 3( z 0- d ) 3 D
dRNd51G- ( 1- 2M) ( z 0- d )
QQD
dNdG
R
3 2
+
[ 3( 3- 4M) z 0 ( z 0 +
d) 2- 3d ( z 0 +
d) #
QQ QQ ( 5z0- d) ] D
dNdG
5 2
( 3)
上式即为求算作用在隧道轴线处的附加应力
计算公式。
3 隧道的纵向位移
当隧道附加作用广义荷载引起垂直于隧道的
附加应力时, 可以将隧道纵向看作是分布荷载下的
Winkler 弹性地基无限长梁[ 3] , 因此, 隧道与地层相
互作用的力学方程可写为:
92
地下空间与工程学报
第 1卷
体重力强度, 方向向上;
