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Example 12.2 (Page 296) - Soil Mechanics : Principles and Practice by G.E. Barnes

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A slope is to be cut into a soft clay with undrained shear strength of 30 kN/m2 and unit weight of 18 kN/m3• The slope is 8.0 m high and its inclination is 2: 1 (horizontal:vertical). Determine the factor of safety for the trial circle shown on Figure 12.20.

Equation 12.9 is used with values of b, α and mid-slice height h determined for each slice. The weight of each slice is obtained from W = Y*b*h.

Slip Circle tangential to the ROCK Layer at depth of 1.6 m below the toe.

Analysis by cordinates and dimensions

Slope Geometry

H.bund 8 m * : (Height of Bund ≡ β.slope 12 / atan : (Angle of slope line AB with Horizontal ≡ H.vc 5 m * : (Vertical height from Top Level BD to Center ≡ H.hz 6 m * : (Horizontal distance from B to Center ≡ H.rock 1.6 m * : (Heigth from Toe to Rock Layer ≡ WIDTH 1 m * : (Define width of strips for Stability Analysis ≡ 61 line

Define Accuracy for Findroot

Accuracy ΔE 103 -^103 -^21 mat mm * :

AB' H.bund β.slope tan / : 16 m *

AB H.bund 2^AB' 2^+ sqrt : 17.89 m *

OO' AB' 6 m * - : 10 m *

Define Slope Toe Cordinates

Note: Pl do not change cordinates of A, as we may have to change the Center Cordinates to find the minimum FS in future development.

A 1 m * 1 m * 12 mat : m 1112 mat

Now define initial Center cordinates,
Pl. do not change this, as ACAD values shown in this worksheet have been derived from relevant ACAD drawing.

Center 71412 mat m * :

Radius of Circle from Center to touch the rock layer

R Center 2 el A 2 el - (H.rock + : 14.6 m *

(See he ACAD diagram below)

Calculation of other important cordinates

B AB' A 1 el + (H.bund A 2 el + (12 mat : m 17912 mat

Line AB (D/S Slope) and Intercept

Slope of D/S profile =

dsl β.slope tan : 0.5

Intercept of D/S profile
using c = y - mx

dsi A 12 el dsl A 11 el * - : 0.5 m *

Initially, find the two cutting points AA and C of the Slip Circle.

Program 1 : Condition for cutting points x Cut_cond A 12 el x A 11 el ≤ dsl x * dsi + x A 11 el B 11 el ltlt B 12 el cases 11 line :

Program 2 : To find cutting point of a line and circle of a given center. Used Program 1. Note how function is passed to this program to use Program 1 r cen gs# ξ# FF int_line_circle r F2# FF 0 ≡ cen 11 el x - (2^cen 12 el y - (2^+ r 2^≡ 21 mat : r F2# gs# transpose ξ# ξ# FindRoot 21 line :

Cordinates of cutting point of SLIP CIRCLE with horizontal line thro A AA R Center A - ΔE x Cut_cond y - int_line_circle : m 0.35 121 mat

From ACAD

X = 0.35 Y = 1.00

Cordinates of cutting point of SLIP CIRCLE with horizontal line BD CC R Center B ΔE y x Cut_cond - int_line_circle : m 20.72 921 mat

From ACAD

X = 20.72 Y = 9.00

A m 1112 mat

AA transpose m 0.35 112 mat

B m 17912 mat

CC transpose m 20.72 912 mat

Center m 71412 mat

R 14.6 m *

Program 3 : To find cutting point of a line and circle of a given center. Note Function FF is passed to this program from Program 4 r cen ξ# FF int_line_circle2 cen r F2# FF 0 ≡ cen 11 el x - (2^cen 12 el y - (2^+ r 2^≡ 21 mat : cen r F2# ξ# ξ# FindRoot 21 line :

Program 4 : Cutting points with slip circle : Calls Program 3 Z# cen# r Find_Cords_Bot j 1 Z# rows range : B# j el r cen# ΔE x Z# j 1 el - (int_line_circle2 2 el : Z# 1 col B# j el augment 31 line :

Program 5 : Called by Program 6 nn width aa aa' Find_Cords_Hoz j 1 nn range : x1 j el aa 1 el width j 1 - (* + (: y1 j el aa 2 el : ans1 x1 y1 augment : ans1 ans1 aa' transpose stack : 41 line 21 line :

Program 6 : To find TOP Cordinates of Strips on Horizontal Surfabe: Calls Program 5 width aa c TC_horiz nn1 A 1 el aa 1 el - width / abs Ceil : nn2 c 1 el B 1 el - width / abs Ceil : ans1 nn1 width - A aa Find_Cords_Hoz : ans2 nn2 width B c Find_Cords_Hoz : ans1 ans2 21 mat 31 line A 1 el aa 1 el ≥ (c 1 el B 1 el > (& ans1 nn1 width - A aa Find_Cords_Hoz : 11 line A 1 el aa 1 el ≥ (c 1 el B 1 el < (& ans2 nn2 width B c Find_Cords_Hoz : 11 line cases 31 line :

Result may be a Nested Array or not depending on the Center Cords

From ACAD

TC_hoz WIDTH AA CC TC_horiz : m 11 0.35 122 mat 179189199209 20.72 952 mat 21 mat

X = 1.00 Y = 1.00

A 1 el AA 1 el ≥ (CC 1 el B 1 el ≥ (& 1

X = 0.35 Y = 1.00

X = 17.00 Y = 9.00

A 1 el AA 1 el ≥ (CC 1 el B 1 el < (& 0

X = 19.00 Y = 9.00

A 1 el AA 1 el < (CC 1 el B 1 el ≥ (& 0

X = 20.72 Y = 9.00

Program 7 : Top Cords on Sloping Surface AB : Called by Program 8 width nn aa bb ββ# rem# Find_AB_Cords j 1 nn (range : x1 j el aa 1 el width j * + (: x1 aa 1 el x1 stack : y1 j el aa 2 el width j * ββ# tan * + (: y1 aa 2 el y1 stack : x1 y1 augment 51 line rem# 0 ≡ x1 j el aa 1 el width j * + : x1 aa 1 el x1 bb 1 el stack : y1 j el aa 2 el width j * ββ# tan * + (: y1 aa 2 el y1 bb 2 el stack : x1 y1 augment 51 line cases 21 line :

Program : 8 Top Cords of strips on Slope AB : Calls Program 7 width# a b β# Find_TC_AB ΔX a 1 el b 1 el - (abs : nn ΔX width# Floor (: rem# ΔX nn - : nn1 nn width# / : width# nn1 a b β# rem# Find_AB_Cords 51 line :

A 1 el AA 1 el ≥ (CC 1 el B 1 el ≥ (& 1

A 1 el AA 1 el ≥ (CC 1 el B 1 el < (& 0

A 1 el AA 1 el < (CC 1 el B 1 el ≥ (& 0

Program 9 : To find TOP Cordinates of Strips on Sloping Surface AB Calls Program 8 width aa c β# TC_AB width A B β# Find_TC_AB A 1 el aa 1 el ≥ (c 1 el B 1 el ≥ (& width A c β# Find_TC_AB A 1 el aa 1 el ≥ (c 1 el B 1 el < (& width aa B β# Find_TC_AB cases 11 line :

To obtain BC_AB, directly use Find_Cords_Bot(() function.
Program 4.

Top Cords of Strips on Slope AB Program 9.

TC_AB WIDTH AA CC β.slope TC_AB :

BC_AB TC_AB Center R Find_Cords_Bot :

From ACAD

TC_AB m 112 1.5 324 2.5 536 3.5 748 4.5 9510 5.5 11612 6.5 13714 7.5 15816 8.5 179172 mat

X = 1.00 Y = 1.00

BC_AB m 1 0.69 2 0.28 3 0.04 - 4 0.29 - 5 0.46 - 6 0.57 - 7 0.6 - 8 0.57 - 9 0.46 - 10 0.29 - 11 0.04 - 12 0.28 13 0.69 14 1.19 15 1.79 16 2.5 17 3.36 172 mat

X = 1.00 Y = 0.69

X = 2.00 Y = 1.50

X = 3.00 Y = -0.04

X = 5.00 Y = 3.00

X = 5.00 Y = -0.46

X = 7.00 Y = -0.60

X = 13.00 Y = 0.69

X = 15.00 Y = 1.79

X = 16.00 Y = 8.50

X = 17.00 Y = 9.00

X = 17.00 Y = 3.36

Program 10 : To find Bottom Cordinates of Top Strip Lines cutting the Slip Circle - Calls Program 4 M cen r a2 c Find_All_BC_Hoz2 Z1# M 1 el cen r Find_Cords_Bot : Z2# M 2 el cen r Find_Cords_Bot : Ans# Z1# Z2# 21 mat : 31 line A 1 el a2 1 el ≥ (c 1 el B 1 el ≥ (& Ans# M cen r Find_Cords_Bot : A 1 el a2 1 el ≥ (c 1 el B 1 el < (& Ans# M cen r Find_Cords_Bot : cases :

A 1 el AA 1 el ≥ (CC 1 el B 1 el ≥ (& 1

A 1 el AA 1 el ≥ (CC 1 el B 1 el < (& 0

A 1 el AA 1 el < (CC 1 el B 1 el ≥ (& 0

From ACAD

Result may be a Nested Array or not depending on the Center Cords Calls Program 10

X = 1.00 Y = 0.69

BC_HOZ TC_hoz Center R AA CC Find_All_BC_Hoz2 : m 1 0.69 0.35 122 mat 17 3.36 18 4.4 19 5.68 20 7.35 20.72 952 mat 21 mat

X = 0.35 Y = 1.00

X = 17.00 Y = 3.36

X = 19.00 Y = 5.68

X = 20.72 Y = 9.00

Program 11 : To Find Mid Cordinates ZZ# Find_Mid_Cords j 1 ZZ# rows 1 - (range : x# j el 0.5 ZZ# j 1 el ZZ# j 1 + (1 el + (* : y# j el 0.5 ZZ# j 2 el ZZ# j 1 + (2 el + (* : x# y# augment 41 line :

Program 12 : To Find Mid Cordinates between vertical strip lines on Horizontal Surface - Calls Program 9 M a2 c Find_TMC_Hoz Z1# M 1 el Find_Mid_Cords : Z2# M 2 el Find_Mid_Cords : Ans# Z1# Z2# 21 mat : 31 line A 1 el a2 1 el ≥ (c 1 el B 1 el ≥ (& Ans# M Find_Mid_Cords : 11 line A 1 el a2 1 el ≥ (c 1 el B 1 el < (& Ans# M Find_Mid_Cords : 11 line cases :

A 1 el AA 1 el ≥ (CC 1 el B 1 el ≥ (& 1

A 1 el AA 1 el ≥ (CC 1 el B 1 el < (& 0

A 1 el AA 1 el < (CC 1 el B 1 el ≥ (& 0

TMC_HOZ (Top Mid Cords on Horizontal lines AA / BC)

BMC_HOZ (Bottom Mid Cords on Horizontal lines AA / BC))

TMC_HOZ TC_hoz AA CC Find_TMC_Hoz :

BMC_HOZ TMC_HOZ Center R AA CC Find_All_BC_Hoz2 :

From ACAD

TMC_HOZ m 0.68 112 mat 17.5 9 18.5 9 19.5 9 20.36 942 mat 21 mat

X = 0.68 Y = 1.00

BMC_HOZ m 0.68 0.84 12 mat 17.5 3.86 18.5 5.01 19.5 6.46 20.36 8.11 42 mat 21 mat

X = 0.68 Y = 0.84

X = 17.50 Y = 3.86

X = 17.50 Y = 9.00

X = 18.50 Y = 5.01

X = 19.50 Y = 9.00

X = 19.50 Y = 6.46

X = 20.36 Y = 9.00

X = 20.36 Y = 8.11

TMC_AB (Top Mid Cords on Slope AB )

BMC_AB (Bottom Mid Cords on Slope AB)

TMC_AB TC_AB Find_Mid_Cords :

BMC_AB TMC_AB Center R Find_Cords_Bot :

From ACAD

TMC_AB m 1.5 1.25 2.5 1.75 3.5 2.25 4.5 2.75 5.5 3.25 6.5 3.75 7.5 4.25 8.5 4.75 9.5 5.25 10.5 5.75 11.5 6.25 12.5 6.75 13.5 7.25 14.5 7.75 15.5 8.25 16.5 8.75 162 mat

X = 1.50 Y = 1.25

BMC_AB m 1.5 0.48 2.5 0.11 3.5 0.17 - 4.5 0.38 - 5.5 0.52 - 6.5 0.59 - 7.5 0.59 - 8.5 0.52 - 9.5 0.38 - 10.5 0.17 - 11.5 0.11 12.5 0.48 13.5 0.93 14.5 1.47 15.5 2.13 16.5 2.91 162 mat

X = 1.50 Y = 0.48

X = 3.50 Y = 2.25

X = 3.50 Y = -0.17

X = 5.50 Y = 3.25

X = 5.50 Y = -0.52

X = 12.50 Y = 6.75

X = 12.50 Y = 0.48

X = 14.50 Y = 7.75

X = 14.50 Y = 1.47

X = 16.50 Y = 8.75

X = 16.50 Y = 2.91

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