47e4751e-61fd-4cf6-8551-09556fa910bd 198 NDTM Amarasekera 2 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

How to find the
1. Two Intersection Points (AA and CC) of a defined CENTER (x,y) of a slip circle with the SLOPE Profile .
2. Number of SLICES for a given width of slices.

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

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From Textbook
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.

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

METHOD of Analysis :
Note : Textbook method using eqn 12.9 is approximate.
Instead, an accurate method of finding Strip Areas (Program 21) and Full Curve Length (Program 22) is used

Slope Geometry

Define Array of Centers

H.bund 8 m * : (Height of Bund ≡ β.slope 12 / : (Slope of line AB with Horizontal ≡ H.hz 6 m * : (Horizontal distance from A to Center ≡ H.rock 1.6 m * : (Heigth from A (Toe) to Rock Layer ≡ WIDTH 1 m * : (Define width of strips for Stability Analysis ≡ 51 line

CENS6 1010 8.5 22 8.5 187545 1.5 5151 1.1 1112 0.5 102013 - 15 - 110 - 1 0.59 - 1 0.6 - 1182 mat m * :

1. Calculations

AB' H.bund β.slope / : 16 m *

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

AB H.bund 2^AB' 2^+ sqrt : 17.8885 m *

1.1 Define Slope Toe Cordinates

Note: Pl do not change cordinates of A, as all cordinates read from ACAD are based on this defined value of A

A 1112 mat m * :

1 m * 1 m * 12 mat m 1112 mat

1.2 Define Trial Center

Center 8.5 1812 mat m * :

1.3 Radius of Circle from Center to touch the rock layer

Distance from A to B

R Center 2 el A 2 el - H.rock + : 18.6 m *

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

X cord beyond which both the inersecting pt is in hoz line Δx.neg_cen A 2 el H.rock - : 0.6 m * -

Slope of Line AB

Intercept of Line AB using c = y - mx

SL β.slope : 0.5

SP A 12 el SL A 11 el * - : 0.5 m *

L_CA Center 1 el A 1 el - (2^Center 2 el A 2 el - (2^+ sqrt : 18.58 m *

Distance from Center to A

L_CA R > 0

L_CB Center 1 el B 1 el - (2^Center 2 el B 2 el - (2^+ sqrt : 12.38 m *

Distance from Center to B

L_CB R > 0

2 Analysis

Program 1 : To find intersection points of Circle with slope profile Pt cen# Find_aa_cc rr cen# 2 el A 2 el - (H.rock + m / : L_ca cen# 1 el A 1 el - m / (2^cen# 2 el A 2 el - m / (2^+ sqrt : L_cb cen# 1 el B 1 el - m / (2^cen# 2 el B 2 el - m / (2^+ sqrt : gs# y SL x * - SP m / - L_ca rr > (Pt A ≡ (& y SL x * - SP m / - L_cb rr > (Pt B ≡ (& y Pt 2 el m / - cases : x y f x cen# 11 el m / - (2^y cen# 12 el m / - (2^+ rr 2^- (gs# 21 mat : ans x y f x y 21 mat Pt 1 el - m / Pt 2 el m / 21 mat roots m * transpose cen# 1 el A 1 el ≤ (Pt A ≡ (& x y f x y 21 mat Pt 1 el m / Pt 2 el m / 21 mat roots m * transpose cases : 61 line :

Center 8.5 m * 18 m * 12 mat

L_CB 12.38 m *

R 18.6 m *

L_CA R > 0

L_CB R > 0

Center 1 el A 1 el < 0

Center 1 el 0 < 0

Program 2 : To deal with -ve x cord of centers Pt cen Find_Neg rr cen 2 el A 2 el - (H.rock + : cen 1 el rr - cen 2 el 21 mat transpose cen 1 el Δx.neg_cen ≤ (Pt A ≡ (& cen 1 el rr + cen 2 el 21 mat transpose cases 21 line :

Program 3 : To stack and array M# STACK j 1 M# rows range j 1 ≡ bb M# j el : bb bb M# j el stack : if 11 line for bb 21 line :

Program 4 : Pt cen Find_Any ans Pt cen Find_Neg cen 1 el 0.60 m * - ≤ Pt cen Find_aa_cc cases : 11 line :

Program 25 : Sample program to find the Number of Slices in Crtical Failure Circle A aa B cc width Find_Slices_Tot slices_AB aa 1 el B 1 el - width / abs Ceil aa 1 el A 1 el > (B 1 el aa 1 el > (& A 1 el B 1 el - width / abs Ceil A 1 el aa 1 el > (cc 1 el B 1 el > (& A 1 el cc 1 el - width / abs Ceil cases : slices_HOZ1 0 aa 1 el A 1 el > A 1 el aa 1 el - width / abs Ceil cases : slices_HOZ2 0 cc 1 el B 1 el < B 1 el cc 1 el - width / abs Ceil cases : slices_Tot slices_AB slices_HOZ1 + slices_HOZ2 + : 41 line :

Program 5 : To analyze array of centers M width Find_All_AA_CC j 1 M rows range : rr j el M j 12 range el 2 el A 2 el - (H.rock + : Z1 j el A M j 12 range el Find_Any : Z2 j el B M j 12 range el Find_Any : Z3 j el A Z1 j el B Z2 j el width Find_Slices_Tot : Z Z1 Z2 Z3 augment : Z M 11 mat rr Z1 Z2 Z3 15 mat : 71 line :

3. Results for the TRIAL CENTER

Center m 8.5 1812 mat

R 18.6 m *

Center 1 el 0.60 m * - ≤ 0

AA A Center Find_Any : 0.95 m * 1 m * 12 mat

CC B Center Find_Any : 24.78 m * 9 m * 12 mat

A AA B CC WIDTH Find_Slices_Tot 25

4. Results for the ARRAY of CENTER

4.1 Define Header for the output Table

HD Center X Center Y Radius AA_X AA_Y CC_X CC_Y Slices augment :

Apply Program 5 Res1 CENS6 WIDTH Find_All_AA_CC : 10 m * 10 m * 8.5 m * 22 m * 8.5 m * 18 m * 7 m * 5 m * 4 m * 5 m * 1.5 m * 5 m * 1 m * 5 m * 1 m * 1.1 m * 1 m * 1 m * 1 m * 2 m * 0.5 m * 1 m * 02 m * 01 m * 3 m * - 1 m * 5 m * - 1 m * 10 m * - 1 m * 0.59 m * - 1 m * 0.6 m * - 1 m * 182 mat 11 mat 10.6 m * 22.6 m * 18.6 m * 5.6 m * 5.6 m * 5.6 m * 5.6 m * 1.7 m * 1.6 m * 2.6 m * 1.6 m * 2.6 m * 1.6 m * 1.6 m * 1.6 m * 1.6 m * 1.6 m * 1.6 m * 181 mat 3.03 m * 2.01 m * 12 mat 0.15 m * 1 m * 12 mat 0.95 m * 1 m * 12 mat 2.46 m * 1.73 m * 12 mat 0.08 m * 1 m * 12 mat 2.42 m * - 1 m * 12 mat 2.92 m * - 1 m * 12 mat 0.7 m * - 1 m * 12 mat 0.6 m * - 1 m * 12 mat 1.4 m * - 1 m * 12 mat 1.1 m * - 1 m * 12 mat 2.4 m * - 1 m * 12 mat 1.6 m * - 1 m * 12 mat 4.6 m * - 1 m * 12 mat 6.6 m * - 1 m * 12 mat 11.6 m * - 1 m * 12 mat 2.19 m * - 1 m * 12 mat 2.2 m * - 1 m * 12 mat 181 mat 20.55 m * 9 m * 12 mat 26.99 m * 9 m * 12 mat 24.78 m * 9 m * 12 mat 12.34 m * 6.67 m * 12 mat 9.59 m * 5.3 m * 12 mat 7.01 m * 4.01 m * 12 mat 6.45 m * 3.73 m * 12 mat 2.56 m * 1.78 m * 12 mat 2.43 m * 1.72 m * 12 mat 3.58 m * 2.29 m * 12 mat 2.02 m * 1.51 m * 12 mat 2.59 m * 1.8 m * 12 mat 1.57 m * 1.29 m * 12 mat 1.4 m * - 1 m * 12 mat 3.4 m * - 1 m * 12 mat 8.4 m * - 1 m * 12 mat 1.01 m * 1 m * 12 mat 1 m * 1 m * 12 mat 181 mat 182725151011104465649132354181 mat 15 mat

Res2 Res1 STACK vectorize : 10 m * 10 m * 8.5 m * 22 m * 8.5 m * 18 m * 7 m * 5 m * 4 m * 5 m * 1.5 m * 5 m * 1 m * 5 m * 1 m * 1.1 m * 1 m * 1 m * 1 m * 2 m * 0.5 m * 1 m * 02 m * 01 m * 3 m * - 1 m * 5 m * - 1 m * 10 m * - 1 m * 0.59 m * - 1 m * 0.6 m * - 1 m * 182 mat 10.6 m * 22.6 m * 18.6 m * 5.6 m * 5.6 m * 5.6 m * 5.6 m * 1.7 m * 1.6 m * 2.6 m * 1.6 m * 2.6 m * 1.6 m * 1.6 m * 1.6 m * 1.6 m * 1.6 m * 1.6 m * 181 mat 3.03 m * 2.01 m * 0.15 m * 1 m * 0.95 m * 1 m * 2.46 m * 1.73 m * 0.08 m * 1 m * 2.42 m * - 1 m * 2.92 m * - 1 m * 0.7 m * - 1 m * 0.6 m * - 1 m * 1.4 m * - 1 m * 1.1 m * - 1 m * 2.4 m * - 1 m * 1.6 m * - 1 m * 4.6 m * - 1 m * 6.6 m * - 1 m * 11.6 m * - 1 m * 2.19 m * - 1 m * 2.2 m * - 1 m * 182 mat 20.55 m * 9 m * 26.99 m * 9 m * 24.78 m * 9 m * 12.34 m * 6.67 m * 9.59 m * 5.3 m * 7.01 m * 4.01 m * 6.45 m * 3.73 m * 2.56 m * 1.78 m * 2.43 m * 1.72 m * 3.58 m * 2.29 m * 2.02 m * 1.51 m * 2.59 m * 1.8 m * 1.57 m * 1.29 m * 1.4 m * - 1 m * 3.4 m * - 1 m * 8.4 m * - 1 m * 1.01 m * 1 m * 1 m * 1 m * 182 mat 182725151011104465649132354181 mat 15 mat

R3 Res2 11 el Res2 12 el Res2 13 el Res2 14 el Res2 15 el m * augment m / :

R3 Res2 Res1 HD CC AA Find_All_AA_CC Find_Slices_Tot Find_Any STACK Find_Neg Find_aa_cc L_CB L_CA SP SL Δx.neg_cen B R Center A AB OO' AB' CENS6 H.rock H.hz β.slope H.bund t.0 R3 HD Find_All_AA_CC Find_Slices_Tot Find_Any STACK Find_Neg Find_aa_cc L_CB L_CA SP SL Δx.neg_cen R AB OO' AB' H.rock H.hz β.slope H.bund t.0 L_CB L_CA SP SL Δx.neg_cen R AB OO' AB' H.rock H.hz β.slope H.bund t.0

R3
10	10	10.6	3.03	2.01	20.55	9	18
8.5	22	22.6	0.15	1	26.99	9	27
8.5	18	18.6	0.95	1	24.78	9	25
7	5	5.6	2.46	1.73	12.34	6.67	15
4	5	5.6	0.08	1	9.59	5.3	10
1.5	5	5.6	-2.42	1	7.01	4.01	11
1	5	5.6	-2.92	1	6.45	3.73	10
1	1.1	1.7	-0.7	1	2.56	1.78	4
1	1	1.6	-0.6	1	2.43	1.72	4
1	2	2.6	-1.4	1	3.58	2.29	6
0.5	1	1.6	-1.1	1	2.02	1.51	5
0	2	2.6	-2.4	1	2.59	1.8	6
0	1	1.6	-1.6	1	1.57	1.29	4
-3	1	1.6	-4.6	1	-1.4	1	9
-5	1	1.6	-6.6	1	-3.4	1	13
-10	1	1.6	-11.6	1	-8.4	1	23
-0.59	1	1.6	-2.19	1	1.01	1	5
-0.6	1	1.6	-2.2	1	1	1	4
#
Center X	Center Y	Radius	AA_X	AA_Y	CC_X	CC_Y	Slices
#
#
#

--------Verified with ACAD --------------

X = 3.03 Y = 2.01

X = 20.55 Y = 9.00

X = 0.15 Y = 1.00

X = 26.99 Y = 9.00

X = 0.95 Y = 1.00

X = 24.78 Y = 9.00

X = 2.46 Y = 1.73

X = 12.34 Y = 6.67

X = 0.08 Y = 1.00

X = 9.59 Y = 5.30

X = -2.42 Y = 1.00

X = 7.01 Y = 4.01

X = -2.92 Y = 1.00

X = 6.45 Y = 3.73

X = -0.70 Y = 1.00

X = 2.56 Y = 1.78

X = -0.60 Y = 1.00

X = 2.43 Y = 1.72

X = -1.40 Y = 1.00

X = 3.58 Y = 2.29

X = -1.10 Y = 1.00

X = 2.02 Y = 1.51

X = -2.40 Y = 1.00

X = 2.59 Y = 1.80

X = -1.60 Y = 1.00

X = 1.57 Y = 1.29

X = -4.60 Y = 1.00

X = -1.4 Y = 1.00

X = -6.60 Y = 1.00

X = -3.4 Y = 1.00

X = -11.60 Y = 1.00

X = -8.4 Y = 1.00

X = -2.19 Y = 1.005

X = 1.10 Y = 1.05

X = -2.20 Y = 1.00

X = 1.00 Y = 1.00

0 time t.0 - 0.1 s *