竖曲线习题

竖曲线练习题

1、设在桩号K2 ＋600 处设一竖曲线变坡点，高程100.00 m . i1 =1%, i2 = －2%,竖曲线半径3500 m试计算竖曲线个点高程（20m整桩即能被20整除的桩号）

解：ω = i2 － i1 = -2% －1% = -3% 为凸曲线。

曲线长L = R∙ω = 3500×0.03 = 105m . 切线长T = L/2 = 105÷2 = 52.5 m

竖曲线起点桩号= (K2 ＋600 ) －52.5 = K2 ＋547.50

竖曲线终点桩号= ( K2 ＋600) ＋52.5 = K2 ＋652.50

竖曲线起点高程= 100.00 －52.5×0.01 = 99.45 m

竖曲线终点高程= 100.00 －52.5×0.02 = 98.95 m

各20 m整桩

K2＋560 X1 = （K2 + 560）－( K2 +547.5) = 12.5 m

h1 =X²/2R = 12.5 X12.5 ÷7000 = 0.022 m

切线高程:100.00 －[(K2 + 600) －(K2 + 560)] X 0.01 = 99.60 m

设计高程99.60 －0.022 = 99.578 m

K2＋580 X1 = （K2 + 580）－( K2 +547.5) = 32.5 m

h1 =X²/2R = 32.5 X32.5 ÷7000 = 0.151 m

切线高程:100.00 －((K2 + 600) －(K2 + 580)) X 0.01 = 99.80 m

设计高程99.80 －0.151 = 99.649 m

K2＋600 X1 = T =（K2 + 6000）－( K2 +547.5) = 52.5 m

h1 =X²/2R = 52.5 X52.5 ÷7000 = 0.394 m

切线高程:100.00 m

设计高程100.00 －0.394 = 99.606 m

K2＋620 X1 = （K2 + 652.5）－( K2 +620) = 32.5 m

h1 =X²/2R = 32.5 X32.5 ÷7000 = 0.151 m

切线高程:100.00 －((K2 + 620) －(K2 + 600)) X 0.02 = 99.60m

设计高程99.60 －0.151 = 99.49 9m

K2＋640 X1 = （K2 + 652.5）－( K2 +640) = 12.5 m

h1 =X²/2R = 12.5 X12.5 ÷7000 = 0.022 m

切线高程:100.00 －[(K2 + 640 －(K2 + 600)] X 0.02 = 99.2 m

求。竖曲线长度不小于500 m 。试确定竖曲线最小半径值并计算K1 ＋800 、K1 ＋840、K1 ＋860 设计高程。

解：ω = i2 － i1 = 1.5% －（－2.5）% = 4% 为凹曲线。

T = L/2 推出R = 2T/ω = 100/0.04 = 2500 m

E = T²/2R =2500/5000 = 0.5 m

竖曲线起点桩号= (K1 ＋840 ) －50 = K1 ＋790

竖曲线终点桩号= ( K1 ＋840) ＋50 = K1 ＋890

K1＋800 X1 = （K1 + 800）－( K1+790) = 10 m

h1 =X²/2R = 100 ÷5000 = 0.02 m

切线高程:200.50 －[(K1+ 840) －(K1 + 800)] X 0.025 = 201.50 m

设计高程201.50 ＋0.02 = 201.52m

K1＋860 切线高程:200.50 m

设计高程200.50 ＋0.50 = 201.00m

K1＋860 X1 = （K1 + 890）－( K1+860) = 30 m

h1 =X²/2R = 900 ÷5000 = 0.18 m

切线高程:200.50 －[(K1+ 860) －(K1 + 800)] X 0.015 = 200.80 m

设计高程200.80 ＋0.18 = 200.98m

3、某公路连续的三个变坡点桩号分别为：K8 + 700、K9 + 100、K9 + 380.对应连续设计标高分别为：77.567 m、65.356 m、68.717 m。在变坡点K9 + 100处竖曲线半径为3000 m。求（1）该竖曲线要素及起始点桩号（2）桩号K8 + 980、K9 + 100 K9 + 060、K9 + 150、K9 + 220设计高程.

解：（1）i1 =（65.356－77.756）÷ 400 =－0.031.i2 =（68.717－65.356）÷ 280 =0.012 ω = i2 － i1 =0.012 + 0.031 = 0.043 为凹曲线

曲线长L = R∙ω = 3000×0.043 = 129m . 切线长T = L/2 = 129÷2 = 64.5 m

中心竖距E = T²/2R =64.5 X64.5÷6000 = 0.69 m

竖曲线起点桩号= (K9 + 100 ) －64.5 = K9 + 035.5

竖曲线终点桩号= (K9 + 100) ＋64.5 = K9 + 164.5

（2）

K9 + 060X1 = （K9 + 060）－( K9 + 035.5) = 24.5 m

h1 =X²/2R = 24.5 X 24.5 ÷ 6000 = 0.1 m

切线高程:65.356 ＋[(K9+ 100) －(K9 + 060)] X 0.031 = 66.59 m

设计高程66.59 ＋0.1 = 66.69m

K9 + 150X1 = （K9 + 164.5）－( K9 +150) = 14.5 m

h1 =X²/2R = 14.5 X 14.5 ÷ 6000 = 0.035 m

切线高程:65.356 ＋[(K9+ 150) －(K9 + 100)] X 0.012 = 65.956 m

设计高程:65.956 ＋0.035 = 65.991m

K8 + 980 设计高程:65.356 ＋[(K9+ 100) －(K8 + 980)] X 0.031 = 69.076 m

K9 + 220 设计高程:65.356 ＋[(K9+220) －(K9 + 100)] X 0.012 = 66.796 m