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1 Chapter 7 Lateral Earth Pressure: At rest, Rankine and Coulomb TONG Lihong 华东交通大学土木建筑学院岩土工程研究所

2 土土压力 Earth pressure 土压力 Earth pressure 挡土墙 Retaining wall 2

3 Chapter 7 Lateral Earth Pressure: At rest, Rankine and Coulomb 1 Lateral earth pressure and earth pressure at rest 2 Rankine s theory 3 Coulomb s theory 4 Common types of active earth pressure calculations 3

4 1 Lateral earth pressure and earth pressure at rest 2 Rankine s theory 3 Coulomb s theory 4 Common types of active earth pressure calculations 4

5 1 Introduction What is Retaining structure( 挡土结构物 ) What is Earth pressure ( 土压力 ) Affecting factors drain-pipe 排水管 5

6 1 Introduction-Retaining wall A structure used to support natural or artificial slopes not to collapse and to maintain soil stability, or to transfer a part of lateral load to fillings. Support natural slope The arch abutment E E Retaining structure: Retaining wall 6

7 1 Introduction-Earth pressure Earth pressure: The pressure subjected by the retaining structure due to the selfweight, load applying to the soil or the lateral compression structure. Is there earth pressure along vertical direction? Earth pressure on retaining wall 7

8 1 Introduction Types of retaining wall Types of retaining wall : Rigid wall Flexible wall anchor rod ( 锚杆 ) Sheet pile ( 板桩 ) Deformation of sheet pile 8

9 1 Introduction Types of retaining wall Retaining structure and earth pressure Rigid wall Backfill E E E basement wall The arch abutment E 9

10 1 Introduction Types of retaining wall Rigid wall buttress 扶壁 Semi-gravity wall Cantilever wall L-Form T-Form Rigid reinforced 刚性加筋 Prestress 预应力 Counterfort retaining wall Gravity wall 10

11 1 Introduction Types of retaining wall Rigid retaining wall 12

12 1 Introduction Types of retaining wall 护坡桩形成的挡土结构 Rigid retaining wall 13

13 1 Introduction Types of retaining wall 混凝土挡土墙及复合排水管完工完工 Rigid retaining wall 14

14 1 Introduction Types of retaining wall 土工格栅加筋建成 5~6.5m 高的加筋挡土墙 Rigid retaining wall 15

15 1 Introduction Types of retaining wall Rigid retaining wall 16

16 1 Introduction Types of retaining wall 加筋土挡墙 Rigid retaining wall 17

17 1 Introduction Types of retaining wall 上海市外环过江隧道岸埋段基坑支撑 Rigid retaining wall 18

18 1 Introduction Types of retaining wall 杭州地铁一号线湘湖站基坑事故 2008 年 11 月 15 日 Rigid retaining wall 19

19 1 Introduction Key problem in this chapter Earth pressure on rigid retaining wall: calculation and distribution (Amplitude, direction, point of resultant force) + H - H E P a Movement- translation and rotation 平动转动 22

20 1 Introduction-Types of earth pressure Bridge abutment Rock 1. 静止土压力 Earth pressure at rest 2. 主动土压力 Active earth pressure 3. 被动土压力 Passive earth pressure Earth pressure types 23

21 1 Introduction-Types of earth pressure Amplitude, direction, distribution and point of resultant force Passive earth pressure Active earth pressure 24

22 1 Introduction-Types of earth pressure Affecting factors on earth pressure 1. Properties of soil 2. The moving direction of retaining wall 3. The relative displacement of wall to soil 4. The friction between wall and soil 5. Types of retaining wall 25

23 1 Introduction-Types of earth pressure Influence of displacement of wall on earth pressure 1. 墙体位移的方向和相对位移量决定着所产生的土压力的性质和土压力的大小 2. 三种特定情况下的土压力, 即静止土压力主动土压力和被动土压力 3. 挡土墙所受土压力的大小并不是一个常数, 而是随位移量的变化而变化 26

24 1 Introduction-Types of earth pressure 1. Earth pressure at rest Pressure E E p =/H=0 E Basement E=E 0 E 0 E a - = H H 1~5% 1~5%o 27

25 1 Introduction-Types of earth pressure 2. Active earth pressure E E p The wall move outwards, Earth pressure gradually decreases, When the soil fails, it reaches the critical equilibrium state and the earth pressure reaches its minimum + E 0 H - H E a 1~5% 1~5 = H E 28

26 1 Introduction-Types of earth pressure 3. Passive earth pressure - Pressure E 支撑土坡的挡土墙 E p The wall move inwards, Earth pressure gradually increases, 填土 When 填土 the soil fails, it reaches the E 堤岸挡土墙 E critical equilibrium state and the earth pressure reaches its maximum E a 填土 E E 0 地下室地下室侧墙 - = H H 拱桥桥台填土 E 1~5% 1~5% o 29

27 1 Introduction-Types of earth pressure Earth pressure at rest at depth z According to the elastic equilibrium state under lateral confined condition to calculate s v z s h s h s v Z 30

28 1 Introduction -at-rest earth pressure At-rest earth pressure coefficient K 0 For stress under later confined condition: p 0 =s h =K 0 s v = u /(1- u)s v = u /(1- u)γz K 0 = u /(1- u) s h s v s v s h The Possion s Ratio is difficult to determine for soil, K 0 is usually taken according to the empirical equation: K0 1-sinꞌ z 31

29 1 Introduction Summary Retaining wall Earth pressure Characteristics of earth pressure Affecting factors At-rest earth pressure How to calculate the active and passive earth pressure? 32

30 1 Lateral earth pressure and earth pressure at rest 2 Rankine s theory 3 Coulomb s theory 4 Common types of active earth pressure calculations 33

2 Rankine s earth pressure theory Plastic equilibrium state in semi-finite soil mass Condition for the state of plastic equilibrium and failure

31 2 Rankine s earth pressure theory Plastic equilibrium state in semi-finite soil mass Condition for the state of plastic equilibrium and failure of the soil occurs Semi-finite soil mass 45 o +/2 s v s h z 90 o - Active equilibrium state P a K 0 s v s v Rankine s earth pressure theory 34 s

32 2 Rankine s earth pressure theory s 1 s 3 Passive equilibrium state 45 -f/2 P a K 0 s v s v =z s 1f s P p 35

33 2 Rankine s earth pressure theory Fundamental conditions and assumptions Conditions Frictionless wall Vertical wall Horizontal surface of filling Assumption Every point in a soil mass is on the verge of failure 36

34 2 Rankine s earth pressure theory (A) For cohesionless soils 1. Active earth pressure p a =s h =tan 2 (45 - /2 )z (kn/m 2 ) Coefficient of Rankine s active earth pressure K a = tan 2 (45 - /2) Earth pressure distribution Resultant force E a =1/2 K a H 2 (kn/m) Action point:distance from bottom H/3 Failure surface direction: Angle to horizontal plane 45 + /2 Pa K 0 s v s v s H H/3 H K a 37

35 2 Rankine s earth pressure theory (A) For cohesionless soils 1. Active earth pressure p a = tan 2 (45-f/2 )z (kn/m 2 K a = tan 2 (45-f/2 ) p a =K a z H H/3 H K a Distribution 45+/2 Failure surface 38

36 2 Rankine s earth pressure theory (A) For cohesionless soils 2. Passive earth pressure p p =s h =tg 2 (45 + /2 )z (kn/m 2 ) K p = tg 2 (45 + /2 ) Resultant force E p =1/2 K p H 2 (kn/m) s v Action point:distance from bottom H/3 Failure surface direction: Angle to horizontal plane 45 - /2 P p s 39

37 2 Rankine s earth pressure theory (A) For cohesionless soils 2. Passive earth pressure s v P p s H 90+ H/3 45-/2 H K p Distribution Failure surface 40

38 2 Rankine s earth pressure theory 小结问题 : 1 什么是静止主动和被动土压力? 2 朗肯土压力理论的基本条件和假定 3 请画出刚性墙后砂性土的主动和被动破坏面形状 4 给出砂性土主动和被动土压力分布及公式 41

39 2 Rankine s earth pressure theory (B) For cohesive soils 粘性土 1. Active pressure Pa K 0 s v s v s p a s h ztg 2 o o (45 2 ) 2c tg(45 2 ) zk 2c a K a 42

40 2 Rankine s earth pressure theory (B) For cohesive soils 1. Active pressure z 0 Pa K 0 s v s v s E a (H-z 0 )/3 2c K a HK a -2c K a HK a 主动区 45 o +/2 Active earth pressure distribution 43

41 2 Rankine s earth pressure theory (B) For cohesive soils 1. Active pressure z 0 Pa K 0 s v s v s z < z 0 tensile stress, crack E a (H-z 0 )/3 2c K a HK a -2c K a HK a Resultant earth pressure E a z>z 0 p zk 2c K ( 0 ) 2 c a H z K a H ch K a 2 2 a a K 2 a 44

42 2 Rankine s earth pressure theory (B) For cohesive soils 1. Passive pressure s s v P p H E p p p s h ztg p 2 (45 p ) 2 p zk 2c 2c tg(45 K p ) 2 2c K p HK p Passive pressure distribution Resultant pressure 1 2 E p H K p 2cH 2 K p 45

43 2 Rankine s earth pressure theory 基本条件和假定极限应力分析破坏形式主动和被动砂土和粘性土主动土压力系数被动土压力系数静止土压力系数 K K a K p tg 2 tg 2 K0 1 (45 (45 K 1 f / 2) Summary P a f / 2) sin f a 0 K p s v 1 sin f 1 sin f 1 sin f 1 sin f P p 46 s

44 1 Lateral earth pressure and earth pressure at rest 2 Rankine s theory 3 Coulomb s theory 4 Common types of active earth pressure calculations 库仑 C. A. Coulomb ( ) 47

45 3 Coulomb s earth pressure theory Back face of retaining wall is sloping with a constant sloping angle ; Back face of retaining wall is frictional, Friction angle is ; Surface of backfill is arbitrary How to determine the earth pressure? 48

46 3 Coulomb s earth pressure theory Assumptions: 1. Failure surface is a plane 2. Failure wedge is rigid 3. Only two surfaces reach critical equalibirium state Coulomb s earth pressure theory Active critical equilibrium Passive critical equilibrium 49

47 3 Coulomb s earth pressure theory (A) Active pressure-sandy soil Take a trial failure surface to determine E To determine the critical for maximum E, we have de/d =0, to determinie,then: E B W C 1 C 2 C 3 C 4 C 5 1 E H 2 K a 2 a K a cos 2 cos cos( )[1 2 E p ( f ) sin( f )sin( f ) ] cos( )cos( ) E 0 = H E a - H 1~5% 1~5% o E A R W 1 R 1 E 1 50

48 3 Coulomb s earth pressure theory E K a 1 H 2 2 a K a cos 2 cos cos( )[1 2 ( f ) E E a sin( f )sin( f ) ] cos( )cos( ) -Coulomb s active earth pressure coefficient 2 HK a Particularly : ===0,Rankine s theory is recovered E a 1 3 H Distribution:triangular distribution 51

49 3 Coulomb s earth pressure theory (B) Passive earth pressure-sandy soils The solving method is same to that used for determining active earth pressure C E 1 H 2 K p 2 p A W E 库仑 K p E cos 2 cos cos( )[1 B 2 ( f ) sin( f )sin( f ) ] cos( )cos( ) R 2 R 52 W

50 3 Coulomb s earth pressure theory Passive earth pressure distribution H - E p E p H/3 HKp p p 1 d z 2 K de 2 p p zk dz dz p 53

51 3 Coulomb s earth pressure theory 三. 图解法 - 主动土压力 C 2 C 3 C 4 C 5 W 1 O C 3 C 1 B W 2 O R 1 W 3 E 1 W 4 E A W 5 54

52 3 Coulomb s earth pressure theory 2. (C. Culmann s graphical method) 库尔曼图解法在图中使力三角形顶点 o 与墙底 A 重合,R i 方向与 AC i 方向一致 C 2 C 1 C 2 C 3 C 4 C 5 B R 1 - W R E 旋转 90+ A 1 M 55

53 3 Coulomb s earth pressure theory 3. graphical method for cohesive soils C B z 0 z 0 2c K a D C AE E W C AD C AD C AE R W R E a A W 中包括 BCDE E a 56

54 3 Coulomb s earth pressure theory Comparison between Rankine s and Coulomb s earth pressure theories (1) Analysis method 极限平衡状态区别朗肯库仑土体内各点均处于极限平衡状态极限应力法刚性楔体, 滑面处于极限平衡状态滑动楔体法 57

55 3 Coulomb s earth pressure theory (2) Application conditions 朗肯 1 墙背光滑垂直填土水平库仑 2 坦墙坦墙 3 墙背垂直填土倾斜墙背填土无限制粘性土一般用图解法 58

56 3 Coulomb s earth pressure theory (3) calculation error Rankine s theory 墙背垂直填土水平墙与填土摩擦系数 >0 E 朗肯填土为砂土 c=0 E 库仑 R E 库仑 W 郎肯主动土压力偏大 R E 朗肯 W 郎肯被动土压力偏小 59

57 3 Coulomb s earth pressure theory (3) calculation error-coulomb s thoery 由于实际滑裂面不一定是平面主动土压力偏小不一定是最大值被动土压力偏大不一定是最小值滑动面滑动面 60

58 3 Coulomb s earth pressure theory (3) Calculation error Compared with theoretical arithmetic =0 c=0 滑裂面是直线, Ka, Kp 与理论值相同 0 K a K p 朗肯偏大 10% 左右, 工程偏安全库仑偏小一些 ( 可忽略 ), 工程可用朗肯偏小可达几倍, 应用不可靠库仑偏大可达几倍, 应用不可靠实际工程问题中, 土压力计算非常复杂 61

59 1 Lateral earth pressure and earth pressure at rest 2 Rankine s theory 3 Coulomb s theory 4 Common types of active earth pressure calculations 62

60 4 Common types of active earth pressure calculations in Engineering project 1. Load applied on backfill (Key) 2. Stratified backfill (key) 3. Water in the soil (key) 63

61 4 Common types of earth pressure calculations A. Backfills subject to load 1. From Rankine s theory s z sh Z H s 1 =z+q zk a p a = s 3 =qk a +zk a qk a HK a 64

62 4 Common types of earth pressure calculations A. Backfills subject to load 2. From Coulomb s theory l q similar triangles A W C W R H E a R qk a HK a B 1 2 cos Ea H Ka qhk a 2 cos G 65

63 4 Common types of earth pressure calculations A. Backfills subject to load 3. Local load Rankine s theory 45 o +/2 q 45 o +/2 HK a qk a 66

64 4 Common types of earth pressure calculations B. Stratified backfills (Key) A 1 f 1 c 1 H 1 Rankine s theory 1 Continuity of p a at point B B B 2 f 2 c 2 H 2 2 Continuity of d p a /d z at point B p a s C h ztg 2 o o (45 2 ) 2c tg(45 2 ) 69

65 4 Common types of earth pressure calculations B. Stratified backfills (Key) f 1 = f 2 c 1 = c 2 =0 2 > 1 2 < 1 A B 1 f 1 c 1 H 1 2 f 2 c 2 H 2 C 70

66 4 Common types of earth pressure calculations B. Stratified backfills (Key) 过顶点虚拟 q=h 1 A B 1 f 1 c 1 H 1 2 f 2 c 2 H 2 C 1 = 2 c 1 = c 2 =0 f 2 > f 1 f 2 < f 1 71

67 4 Common types of earth pressure calculations B. Stratified backfills (Key) A B 1 f 1 c 1 H 1 2 f 2 c 2 H 2 C 1 = 2 f 2 = f 1 c 1 >0 c 2 =0 c 1 =0 c 2 >0 72

68 4 Common types of earth pressure calculations C. Water exists in backfills 1. 构造要求 : 一般用砂性土, 墙内设排水孔及反滤层, 填土表面设沟堤等截流 2. 水下部分土压力 p a = K a s z 水压力 p u =u ( 静水压力渗流压力超静孔压 ) 土工织物反滤砂砾石料排水管排水孔 73

69 4 Common types of earth pressure calculations 四. 填土中有水砂土朗肯理论墙基不透水 A f H 1 土压力 K a H 1 水压力 B f H 2 C 不透水层 K a (H 1 +H 2 ) w H 2 74

70 4 Common types of earth pressure calculations 四. 填土中有水填土渗透系数比地基土小得多垂直有效应力 = 总应力孔压土压力 H 1 H 1 u=0 K a H 1 H 2 透水地基 sa H 2 K a (H 1 + sa H 2 ) 75

71 4 Common types of earth pressure calculations 五. 墙背形状有变化带卸荷台的挡土墙土压力 A 45 o +/2 H 1 D C B C H 1 K a H 2 E E (H 1 +H 2 )K a 76

72 本章完 81

#4 ~ #5 12 m m m 1. 5 m # m mm m Z4 Z5

#4 ~ #5 12 m m m 1. 5 m # m mm m Z4 Z5 2011 6 6 153 JOURNAL OF RAILWAY ENGINEERING SOCIETY Jun 2011 NO. 6 Ser. 153 1006-2106 2011 06-0014 - 07 300142 ABAQUS 4. 287 mm 6. 651 mm U455. 43 A Analysis of Impact of Shield Tunneling on Displacement