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Exact
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0.3
0.2
0.1
0
Two-term approximation
One-termapproximation
t
x
y
2c
16 Q
0
(x =
0, y
= 0
, t)
22k
L2c= 1.5
= 1
Q = Q0 e t
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y = 1
Exact(x = xy = 0) = 0.15
l
Nodal averages
Value atcentres oftriangles
y
x0.440.070.07
0.440.070.07
0.860.070.07
0.000.070.07
0.450.07
0.08
0.450.060.06
0.860.070.08
0.000.070.06
0.480.080.06
0.480.060.08
0.000.090.09
0.830.04
0.04
+
!
*
x = 1
x
y
! " " "#
1.0 1.0
3.0 2.83
x x
1.01.75y y
(a) Isotropic (b) Orthotropic
Exact solution for infinite plate
Finite element solution
$ % #&
52100
47100
49
48100
x
y
Restrained in y direction from movement
A
B
C
' "" &" () * $ ! !
,- ! ! + .+ "
( ( (
A*
5
(
(
(
2 '7
/ ' #0 $ .+ $
* / )
>
=
>A<
8 '''
8&
A
(
0.370.71
0.270.73
0.110.75
0.401.06
0.490.99
0.560.89
0.552.03
0.510.27
1.890.88
0.667.49
0.231.35
0.321.15
1.950.27
2.200.36
1.780.14
2.080.13
4.500.27
2.610.38
1.860.02
0.120.01
1.580.03
1.440.04 1.10
0.05
0.980.05
1.050.05
1.150.12
1.240.16
1.030.01
1.250.04
0.970.04
0.870.06
0.970.00
0.910.04
0.860.06
0.860.13
0.780.17
0.960.12
0.920.08
0.860.03
0.960.07
1.090.01
0.970.02
0.260.02
0.670.02 0.83
0.04 0.810.08 0.74
0.140.67
0.200.76
0.04
0.790.08
0.720.08
0.630.050.62
0.11
0.660.11
0.710.04
0.530.08
0.460.020.09
0.08
0.160.04
0.300.01
0.470.04 0.540.05
0.540.11
0.510.08
0.510.06
0.210.02
0.180.01
0.200.02
0.340.04
0.070.01 0.13
0.03
0.200.01
0.320.03
0.440.03
0.410.060.41
0.10
0.290.06
0.280.04
0.360.11
0.200.04
0.970.01
0.000.020.01
0.14
1 2 30
1
2
3
Tensile zone
10 to
nne/
m2
E2 = 1 tonne/m 2
x = 2(1 y2)
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
E1 = 10
E2 = 1
y
xu = 0 Region analysed
y = 1
! + , - ." /0
! "'' A
' ! ! "8#
4
! "%# A
8 8 ''%
! ! "8# ! "#
A
Buttress webthickness 9 ft
Constant web taper1 ft in 82 ft 6 in
6060
207
ft 0
in
A A121 ft 11 in 121 ft 11 in
10 ft
125
ft 0
in47 ft 0 in 12 ft 0 in
10 ft 6 in
Sectional plan AA(a)
Dam EC
Fault(no restraintassumed)
Altered grit E = EC110
Zero displacementsassumed
Grit EG =14 ECMudstone E = EC
12
Zero displacementsassumed
(b)
" 1 ( 23
6
1116
18 8 2237
1584
2929 39
31
60 66
41198
18
202
10
5
199
260283
2814
256
12
273
108
3965
47
71
29142
8 9
328 309
5
249
16
289
16
289 289
11 1
392
513
293288
19
317
18
327
7
337
8
355
7
375
11+17
17899
+136
485
70
+17
+884
+3792
952
849
84
13 54
8
77
22
6
4
9
33 15 66
342
817
205
282
4
43
11
14
48 52
34 25
188
18
253
15
240
8
238 2462
1846
27 78
14170
304
2
288
9
285
14
290
8
202
2
347
7
310
1721
315
13
-320
6
329
+4
211
+2010
+12
345
+23+16
30
13 +144
118
+59
9 265
+26
100
+10
358
+5
178
313
+4
7
332
879
14
333 332
23
8015
296
310
25
473
303
3954 7
5917
+3
47 1
+3
42
71
0
87
4
69
+3
62
2
61
+8
71 75
8 1667
+9
46+22
54
+17
51
+838
59+20
+31
28
+38
9729
7014 11
70
17+3
300
+200
200 lb/in2
300 lb/in2
Compression ()
Arrow indicatestension
Tensile zone
Tensile zone
300
+200
200 lb/in2
300 lb/in2
Compression ()
Arrow indicatestension
Below the foundation initial rock stresses should be superimposed
(a) (b)
# 1 % #/ + + + 3
! "
''3
''7
A
B ! "''3# C !
12+
( (
( (
)
* ( 2 ' (
Upstream
Tensile zone
Compression ( )
200lb/i
n2
400lb/in 2
Tension ( + )
Stresses exclude theoriginal rock stressesin foundation
Arrow indicatestension
Cracked zone(materialconsidered)as E = 0
Tensile zone buttension less thanprobable initialcompression in rock
1 ( 4)5 6 7 % #"8
Compression ( )
200 lb/in2Tension ( + )
Arrow indicatestension
+200
100
100 lb/in2
Temperature in this region 15F
Temperature dropslinearly here
No temperature charge in foundation
1 ( & % ! "9 : 9 "9%
5 31
2
49842
380
8
251257
62
13
3251
17
59
9
98
8
124
135
217
355 37
9
2 10
6
5
1
7
2
62
753
3
291
52
1
56
3
19
49
6
425
685
6
56 71
59 8
851
98
3
75
112
770
13
48 40
3
29
10
10
11
727
8
4
61
3
871
113
1
105
108
0
1
149
2
198
25
117
12
116
21
130
20
111101
1610
88
1
8926
1521
35
3222
54
4
352
2
61 1 48
6 498
52
49
389
7
11
92103
18
0
79
6 53
54
5
114
8
1612
27
12
67
121
7
28
1534 2614
429
14715
8
29323520374
105
40794955
5470
311440
4 x 106 lb
1.3 x 106 lbLoad effectof prestressing
Prestressing cables and gate reactions
60'
135
25' 25'10'Tailwater level
0 20 40 60 80 100 ft
0 10 20 30 40 ft
Water level
155'
+
+ ;
250000 lb/ft2
Indicates tension
x
y
+
* (
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7 = = + J() H ! "H ;#
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6
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6
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z
rz (rz)
r (r)
z (z)
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$ 3
6
6
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6
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,
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#
0.85
2.492.06 2.09
1.93
2.21 2.26
0.010.70
0.46
0.340.16
0.76 0.560.32 0.01
1.932.082.242.33
2.35 2.16 2.01
0.770.40 0.12
0.77
0.40
0.112.36
2.172.01
2.33
2.20
2.04
0.13
0.37
0.80
Unitpressure
2.0
1.0
0
1.0
Exact
(a)
(b)
(c)
r
Stresses on section AAAxis of revolution
(a) (b) (c)
A
A
! " # $ % ! " # "% ! & "" "
% &
1/ &7 $ 1/ #7 4
6 6 #4$
% &
0
C " #&
100
400
200
0
200
400
600
0
50
100
r
r
t
T (C)
(a)
t
100 C 0 C32 51
6
23
340
(b)ExactTriangular averageQuadrilateral average
r
422
$ +, - . / . ! ! % ! 0& "
2p
Stress scale(arrow for tension) (a)(b)
Axis of revolution
Internalpressure p
q
! 01 % , " ! ! & " -)!
$ " #7 0
' &
" #2 A $ " #8 / C
% 2
Axis of symmetry (a)
2000 1000500
0
500
500
100020003000
1000
20003000
1000
1000
2000
3000
2000
3000CL
0
(Zero contourcoincides with
boundary)
! 2 $ . & 3 * ./ ./ ) -4. 5)6 -4 5 -)
(
" #;
, 9 '/(
"#
$ A # 7 % ;
D
$
$ A ) " =* 1/ &8> E $ 1/ ##
6
0 10 20 30 40 50 60
Distance scale (ft)
Finite element subdivision
Bottom of pile
100
ft
90 ft
p = 162000 lb
E pile = 600 pile = 0.25
E1 = 11 = 0.35
E2 = 402 = 0.30
Radius of pile = 1.5 ft
CL
p38
000
4294
052
200
0 5 10 15 20 25
Distance scale (ft)
010
0 20
0 30
0 40
0
Str
ess
scal
e (lb
/ft2)
E1
E2
Vertical stresses onhorizontal sections
Finite element solution for pile problem
Finite element solution forBossinesq problem
Exact solution for Boussinesq problem
% ! 3" ! 7 8& 9"- 5 / ,
$
&'
?5 % % 2
.% F)* G! + 5;7#
?5 % H? ? "
1@ 2 ;7#4 ! ) I: G @G+ ;#& @ $ % '!, ( $? ?C 0J
? #7E2 ;7;# 1B 5 ! 7;E2& ;7#7 .% F)* 4 @G+ ;22
!
"
! # $ %& '() * *+
! ,
! - &. &. ).. /.. 0! 1 ! &. 02 &.1 ).
! !
' &. &. &. /... ! &) ... , &. &. ).. $&..
# 3
3 4$. ! 5 6
, 4$
z
y
x
i
m
p
j
! !
4$
7 6 0)(1
$ & ( ) 4&6
$ & ( ) 4( $ )
' 6 0)+1
$4
4)
4
$
$
$
$
4+
8
$
$
$
$
$
$
$
$
$
4+
! , 4$
! $&
44
9
4:
4 4/
;
!
!
404/1
4$.
. .
.
.
. .
.
.
.
$4
. .
. .
. .
.
.
.
4$$
.
$
$
$
.
.
.
4$&
"
9 &$ 0 )&(1
. . 4$(
' "
! 01 0? 1
$ $ &
$ . . .$ . . .
$ . . .$ && . .
$ && .$ &&
4$)
! 3 0&$.1
!
3
! 4$+
! 6 0)()1
! . 4$4
@ A
B 6 0)(41C
2 ) ! "
z
y
x
(a)
z
x
y
(b)
!
! " - ' 0 D18 , 4&
, , 4( 0 1
9 2 $)
, 4) 5 E 0
each dividedthus
Two parts
"
+ 1 F , 4& ! 4
' , 4+ 44 5 8 =
- , 4+$& ' - , 44 2 5 (:+ % + $& , 4: $. ... 2 / "
B C
D
H
G
E
F
z
y
x
A
10 ft
Boundary conditions
u = v = w = 0 on ABCD
u = 0 on AEHDv = 0 on AEFB
All other boundaries free
symmetry
! # $%
45tons
5
10
5 10x
z = 0.5 ft
Section
z (T/ft2)
10
20
30
45tons
10
20
10
z = 1.75 ft
z = 3.75 ft
wE (ft2)
Computed value
Exact value
(a) (b)
5
# $% & '
9 ft (2.74 m)
3 ft (0.91 m)
R
22 ft (6.71 m)
15 ft (4.57 m)
40 ft (12.2 m)
1.5 ft (0.46 m)
Z
8 ft (2.43 m)
Prestressingsystem E = 5 x 10
6 lb/in2
3.52 x 106 kg/cm2
Concrete
= 0.15
rr
zz rz
400
500700
300
100
300
0
0300
0
400
100
700 500
200
200
400
600
700
800
1000
1200
1400
1000
800
600
500
100
200
300
400
00
200
300
100
0
0
0
100
200
200
2008001200
500
1000
200
0
0
200
600
900
1000
900
400700
800
900
1000
600
1400
1000
700
400
200
1400
1000
600
2000
700
500
300
100
" ( )
' - -
! $( 2 $+ "
? $) , 4/
(a)
(b)
# ! "
(. ... !
$+$: '
$%
$ *5 # 7 ? ?? 8F :..>: $&( '$ ((>++ $
! "
#$ % % &
' (
) )* + , % - "
%
%
. /0 1
% 2 ! + ! ,
!
!
" + ( 3 4#,
)
% 5#
+ , )
1 53
5
% 1 567 )
%
) /
58
9 %
: :! *
% . ; +58, +56,
% 1 5$
; +5$, %
% 1 5##
0 + /9 7, !
+55,&
%
1 5#3
% +55,
! " ##
9 .
#3
%
#
3
3
5#
0 ?@ ; +55,A ; +5$, +55,
% ( 4
9
! " #
%
9
5#6 ! ; +5#3, ; +5#, . !
5#8
7! 0
( ; +5#8, +5#3, %
1
# 5#$
5#5
" #
%
2
.
9
% )9
5#4
#6
$ #
! $# % %
. 9
% !
% ! + , 9 9 .
: 5#
y
x
y'
x '
Stratification
"
9 = . 9 = 9 ?; +5#5,A
% . 9
! &% #'
%
!
%5#>
! )9
1
1
531
/ 0 9
%536
)9
1
1
538
0 ; +5#$, 9
3
8 ( 838
% &' (
9 ! 2
: 5 0
! . 2 5
L
s
r
i
j
k
m
y
x
qn
"
6 ' C = . 2 9
8 :
% 9
% . : 58 %
2
#
#
3 1 53$
(a) (b)
2042(2085)
3041(3095)
3352(3400)
3123(3190)
4793(4850)
5317(5380)
3656(3695)
5684(5745)
6332(6405)
3818(3855)
5957(6027)
6644(6715)
1921 3132 3251
3214 4712 5397
3556 5764 6247
3914 5876 6728
" $
!" !# $ "%
600
1200
1400
1800
229 444 547 608631 623
563495 475 438
397 350 290 218 99
217
289
347
391
421
436
43587113001740
1292
855
806
725
621
480
366609 4807358329651113116011451063888707 895445
351
654
671
676
670
666 1240 1605 1743
1617
1235
1216
1170
1073
8901285 1485 1550 1540 1463 1378 1347 1260 1143 989
765
983
1132
1231
1740
14891740
1430
1740
1740
14321150
1741174317431743
1743
1743
1743
1611
1775
1580
1505
1610
17651712
200
400
8001000
1600
1743
L
L
G2/G1 = 3
G2 G1
L/2
CL
= 0 on external boundary
" %
" !&'
.
#1 % . 2 4
/ 9 ) : 5$
'
1 535
= +, 9 %
9 %
0 9
)
$# #
% . ) %
; +535, '
B =
: 55
$() *&
% 9 ) +536, +538, B
: 54
#
+)%%) #
"
)
>
%
=
H = 0H = 100
Computed values
Equipotentials of exact solution
Stratificationkx = 4ky
94.1
94.0
93.694.2
93.794.4
89.389.2
89.3
91.6
89.2
89.6
89.4
85.1
85.3
85.2
85.0
80.5
85.2
81.0
76.3
71.775.0
76.5
80.9
76.3
80.6
76.0
80.8
80.5
76.0 70.7
70.4
70.2
69.7
69.762.1
71.7
66.7
63.0
62.9
62.5
61.9
61.5
60.350.3
50.4
49.9
50.3
50.8
50.7
51.2
58.3 52.8
50.0
41.7
35.233.3
35.3
35.0
35.0
34.8
33.2
32.6
34.0
18.618.4
79.6
20.2
19.8
20.2
19.8
20.2
25.0
16.7
8.4
7.7
8.2
7.6
8.68.9
8.27.8
ku
" ) (
) * +
9
"
3 1
# 5
% +63 , ' + ( ,
DamH = 100 H = 0
95.6 90.5
77.767.5
56.548.9
36.734.726.1
19.2
11.291.1 83.5
70.6
71.1 59.5
49.441.7
42.3
35.929.2
22.5
16.9
6.79.7 8.9
7.27.7
10.015.817.2
13.1
18.6
21.5
24.322.8
20.0
30.030.0
31.4
26.7
27.829.0
24.329.4
35.7
32.7
31.2
32.7 33.3
34.2
40.833.1
33.9
36.2
33.7
36.340.0
35.1
47.640.8
55.5
54.345.3
41.6
39.3
35.634.3
34.1
34.235.0
37.1
39.2
40.0
43.9
47.741.5
45.4
50.4
56.5
58.8
53.9
50.7
60.7
70.1
79.4
71.8
79.3
81.087.3
92.2
90.791.2
83.5
84.3
83.2
90.0
89.1
80.0
77.5
73.1
71.1
76.0
69.1
61.7
63.768.770.0
66.0
60.050.0
k = 4
k = 1
k = 2
k = 1
k = 1
k = 4
k = 9
k = 3
ImperviousEquipotentials
" ( +
CL0 0 0 0 0 0
0
0 0
0
0 0
0
0 0
0
0
0
0100
100
100
100
100
100
100
100
100
100
100100100
31.0 31.229.7
41.4
30.6
13.6
22.4
30.7
14.2
13.3
34.6
10.8
8.3
36.5
8.434.3
29.2
30.263.4
65.2
64.7
64.3
65.6
64.7
62.4
54.9
62.7
63.5
63.864.464.5
Axi
s of
rot
atio
nal s
ymm
etric
al
1020
40
6080
90
6.7
23.6
13.8
37.3
" * + *+
H
Moving wall
14 21 28 35 42 49 56
5554
5352
5150
Element subdivision
1234567
11111
12
12
L = H/6x (an) L
" + ,
-
+
%
: 5#1%
# ! 9
#1
:
+ #, 1
: 5#1
)
; +534, %
#>
: 5## 6 '
!
$
.
=
9###
! % #'
.
% $56
. " +: 5#3,
: 5#
9 . $ #>$$ % ! !
#6#5
28 35 42 49 560
0.2
0.4
0.6
0.8
14 21
Exact solution(Westergaard)
Constantacceleration
Linearly varyingacceleration
p/ Ha0a0
a0
1
2
3
4
5
6
7
" , ,
%
2
0 #431
Non-conducting
Non-conducting
Typical volume element
q Specified on this face
(a)
R = 1.27 H
6748'
= constant1.25 H
H
4 H
CL
= 0
= 1922'
= 3844'
= 586'
= 586'
= 6748'
ZH
p1a1 gH
(b)
Present solution
Electrolytic tank solution
Excess pressure
Relative acceleration
Density
p1a1
" , .
. D9
% 151
. . 0 . . )9 + , 0 . /9 :
90
70
50
3010
a
ce
90
70
50
30
10
f
b
d
" '
% . 9 ; +51, # % . &
#6
% 5#
1
!
; +51,
% 153
.
5
" ! ( $ /#%
. . ; +5$, .
%% 1 56 1 . %
?; +5$,A .
/ = 9
9
' . 0 ; +5#$, +5#5,
% 58
%
%
5$
/ % 1
!
: 5#6 ' ! 4 #4
9
3 3 3
55
%
E 3 . 9
>
, -' #'
; +54, @
D
+0 .,
!
333>
CL
CL
Coil
Element subdivision
B'
A'A
B
5.260.21
0.85
5000.0
4000.0
3000.0
2000.0
1000.0
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0x (cm)
2.04.0
6.0
8.010.0
12.0
y (cm)
Tota
l fie
ld s
tren
gth
HA
-t/m
(a)
Field due tocurrent HsTotal field H(with magnetization)
300.0
200.0
100.0
0.0 4.0 cm
y
x
z
Sca
lar
pote
ntia
l A
mp-
t
(b)
" $ $% ( $% '&
.
% *&
%
" ) ) ) ) 2 % 9 9 : 9 D1 #$
2 9 )
1 5>
% 9
561
)
? ; +53,A
.
% 1 56#
L
O
CL CL
Inlet
U
2.625 in 2.375 in1.
182
in
Contours of ph21/6UL
h = 116 in116 in
" % ,
% 1 563/ ) .
56
. %
% 1 566
) ) /
9
9 E
% )
54
/
) ) %
9 ) !
%
. 9 C ) @?: 5#$+,A ) ?: 5#$+,A
!
. : ; +535,
(a)
(b)
p = 0
p = 0
p = 0
x
y
" ) +
+ & $% 0 . $%
%
1 568
.
0
9 /
56$
) . )
+0,
565% (
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/ 61C64
- .
9 9 F
.
/(
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3 E
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$8
88
/(
4 J: ; $1
> $
#1 'D
! " !
6#4C #>## C5# ( #>$>
#3 F; B :
% % + F' 7 J% 58
# // -! /
F * F% ( %
3#5C64 #>$>#6 ( DEI B . * B 5 8C4> #>51
#8 ( D( ' : 9 . #>5#
#$ -' D* / 9 : .
D%%31 B #3 #>53#5 ; D . " ' ( 346 /
#>5
#4 C8$ #>55
#> J (! % 5>31 J (! % % .
-+$, #>41
3# *E % F ! + F B * #>$$
33 DD F :
" ! +( :, 836 #>$>3 DD F %H :
" ! +( :, 6>8 #>5136 J' / * ( %
# $ "
! #166C$ #>$>38 J: -! I' ' / . &
# , " ! +( :, # #>53
3$ I' ' / . # * #>C$4 #>56
35 (D F I
3 J' / 7 D0! * ( % ) . # $ " ! >$#C6 #>$>
"J * / . ) # * 36C83 #>51
6 7 E *' B % .
) # " " ! >54C413 #>5#8 (%I -; "! "F ' : ) ) .
" # 'H$ #>5
$ -; "! J 00 # % 65>C4 #>$>
5 ( * : ) % %+ F' 7, E # #85C43 #>58
4 J / / . ) ! 36#C8# #>$4
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#>85
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66 JD (
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83
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8+" 46
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