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INFORMATION
Rectangular Concrete Tanks
While cylindrical shapes may be structurally best fortank construction, rectangular tanks frequently are pre-ferred for specific purposes. Special processes or oper-ations may make circular tanks inconvenient to use.When several separate cells are required, rectangulartanks can be arranged in less space than circular tanksof the same capacity. Tanks or vats needed inside abuilding are therefore often made in rectangular orsquare shapes. For these and other reasons, breweries,tanneries, and paper mills generally use rectangulartanks.
Data presented here are for design of rectangulartanks where the walls are subject to hydrostatic pres-sure of zero at the top and maximum at the bottom.Some of the data can be used for design of counter-forted retaining walls subject to earth pressure for whicha hydrostatic type of loading may be substituted in thedesign calculations. Data also can be applied to designof circular reservoirs of large diameter where lateralstability depends on the action of counterforts built inte-grally with the wall.
Another article on tank construction, Circular Con-crete Tanks Withouf Prestressing, has been publishedby the Portland Cement Association.
Moment Coefficients
Moment coefficients were calculated for individualpanels considered fixed along vertical edges, and coef-ficients were subsequently adjusted to allow for a cer-tain rotation about the vertical edges. First, three sets ofedge conditions were investigated, in all of which verti-cal edges were assumed fixed while the other edgeswere as follows:
1. Top hinged-bottom hinged2. Top free-bottom hinged3. Top free-bottom fixed*
Moment coefficients for these edge conditions aregiven in Tables 1, 2, and 3, respectively. In all tables, adenotes height and b width of the wall. In Tables 1, 2,and 3, coefficients are given for nine ratios of b/a, thelimits being b/a = 3.0 and 0.5. The origin of the coordi-nate system is at midpoint of the top edge; the Y axis ishorizontal; the X axis is vertical and its positive directiondownward. The sign convention for bending moments isbased on the coordinate fiber that is being stressed. Forexample, A$ stresses fibers parallel to the X axis, Thesign convention used here is not compatible with twoother conventions-namely, that (1) the subscript is theaxis of the moment, and (2) that the moment is in a par-
Q Portland Cement Association 1969 Revised 1961
titular principal plane. Coefficients are given-exceptwhere they are known to be zero-at edges, quarterpoints, and midpoints both in X and Y directions.
The slab was assumed to act as a thin plate, for whichequations are available in textbooks such as Theory otPlates and Shells by S. Timoshenko,” but since only asmall portion of the necessary calculations for momentcoefficients for specific cases is available in the engi-neering literature, they have been made especially forthis text.
Table 4 contains moment coefficients for uniformload on a rectangular plate considered hinged on allfour sides. The table is for designing cover slabs andbottom slabs for rectangular tanks with one cell. If thecover slab is made continuous over intermediate sup-ports, the design can follow procedures for the design ofslabs supported on four sides.
Coefficients for individual panels with fixed sideedges apply without modification to continuous wallsprovided there is no rotation about vertical edges. In asquare tank, therefore, moment coefficients can betaken directly from Tables 1, 2, or 3. In a rectangulartank, however, an adjustment must be made, as wasdone in Tables 5 and 6, similar to the modification offixed-end moments in a frame analyzed by momentdistribution.
In this procedure the common-side edge of two ad-jacent panels is first considered artificially restrained sothat no rotation can take place about the edge. Fixed-edge moments taken from Tables 1,2, or 3 are usuallydissimilar in adjacent panels and the differences, whichcorrespond to unbalanced moments, tend to rotate theedge. When the artificial restraint is removed these un-balanced moments will induce additional moments inthe panels, Adding induced and fixed-end moments atthe edge gives final end moments, which must be iden-tical on both sides of the common edge.
Moment distribution cannot be applied as simply tocontinuous tank walls as it can to framed structures,because moments must be distributed simultaneouslyalong the entire length of the side edge so that momentsbecome equal at both sides at any point of the edge. Theproblem was simplified and approximated to some ex-tent by distributing moments at four points only: quarterpoints, midpoint, and top. The end moments in the twointersecting slabs were made identical at these fourpoints and moments at interior points adjusted accord-ingly.
‘Applicable tn cases where wa l l s lab , counter fo r t , and base s lab a rea l l built I n teg ra l l y
“PublIshed by McGraw-HI11 Book Co, New York, 1959
Tables 1, 2, 3, and 4. Moment Coefficients for Slabs with Various Edge Conditions
Table 1 Table 2
Moment = Coef. x wa’ /j f-lmi_ Moment=Coef.xwa3 d mlr
-
b l a
-
3.00
2.50
2.00
1.75
1.50
1.25
1.00
0.75
0.50
x l ay - o
MX MV
+0.035 +0.010+0.057 +0.016+0.051 +0.013
to031 +0.011+0.052 +0.017+0047 +0.015
+0.025 +0.013+0.042 +0.020+0.041 +0.016
to.020 +0013+0.036 +0.020+0.036 +0.017
+0.015 +0.013+0.028 +0.021+0.030 +0.017
+o 009 +0.012+0.019 +0.019+0.023 +0.017
+0.005 +0.009+0011 +0.016+0.016 +0014
+0.001 +0.006to.005 +0.011+0.009 +0.011
0 +0.003+0.001 + 0 . 0 0 5+0.004 +0.007
_wa Xl
y = b/4 y = b/2
vx 4 Y MV
+0026 +0.011 -0.008 - 0 . 0 3 9+0.044 +0.017 -0.013 -0.063+0.041 +0.014 -0.011 -0055
+0.021 +0010 -0.008 -0.038+0.036 +0.017 -0.012 -0.062+0.036 +0.014 -0.011 -0.055
+0.015 +0.009 -0.007 -0.037+0.028 +0.015 -0.012 -0.059+0.029 +0.013 -0.011 -0.053
+0.012 +0.008 -0.007 -0.035+0.023 +0.013 -0.011 -0.057+0025 +0.012 -0.010 -0051
+0.008 +0007 -0.006 -0.032+0.016 +0.011 -0.010 -0.052+0.020 +0.011 -0.010 -0.048
+0.005 +0.005 -0006 -0.028+0.011 +o 009 -0 009 -0.045+0.014 +0.009 -0.009 -0.043
+0.002 +0.003 -0.004 -0020+0006 +0.006 -0.007 -0.035+0.009 +0.007 -0.007 -0.035
0 +0.002 -0.002 -0.012+0.002 +0.003 -0.004 -0022+0.005 to.005 -0.005 -0.025
0 +0.001 -0.001 -0.005+0.001 +0.001 -0.002 -0.010+0.002 +0.002 -0.003 -0.014
Minus s,gn lndlcates ienslon on the loaded side I” all tables
b l a
-
3.00
2.50
2.00
1.75
150
1.25
1.00
075
0.50
y - o y = b / 4
4 MY Y MY
0 to.070 0 +0.027to.028 +0.061 +0.015 +0.028kO.049 +0.049 +0032 +0026bO.046 +0.030 +0.034 +0.018
0 +0.061 0 +0.019bO.024 +0053 +0.010 +0.022bO.042 +0.044 +0.025 +0.022bO.041 10027 +0.030 +0.016
0 +0.045 0 +0.011bO.016 +0042 +0.006 +0.014kO.033 +0.036 +0.020 +0.016bO.035 r0.024 +0.025 +0.014
0 +0.036 0 +0008IO.013 +0.035 +0005 +0.011bO.028 +0.032 10017 co.01410.031 +0.022 r0.021 10012
0 +0027 0 +0.005moo9 +0.028 +0.003 +0008bO.022 +0.027 +0012 +0.01110.027 +0020 +0.017 +0.011
0 co.017 0 +0.00310.005 +0020 +0.002 +0.005bO.017 +0023 +0.009 +0.00910.021 +0.017 10013 +0.009
0 +O.OlO 0 +0.002to.002 +0013 +o 000 io.003'0 010 +0.017 +0005 +0006too15 +0.015 +o 009 +0.007
0 +0005 0 +0001~0.001 +0008 +o.ooo +0.002to.005 +0011 +0002 +0004~0.010 +0.012 +0.006 +0.004
0 +0.002 0 0~0.000 +0004 +o.ooo +0001bO.002 +0.006 +0001 +0.002a007 +0008 +0.002 +0.002
! y = b/2
MX 4
0 -0.196-0.034 -0.170-0.027 -0.137-0.017 -0.087
0-0.026-0023-0.018
0-0.019-0.018-0.013
0-0015-0015-0.012
0-0012-0013-0.010
0-0008-0010-0.009
-0.138-0132-0.115-0.078
-0.091-0.094 ~-0.089
/ ,"
-0.071-0.076-0076-0.059
-0052-0.059-0.063-0.052
-0034-0.042-0.049-0.044
0-0005-0.007-0007
0-0.003-0004-0.005
0-0.001-0002-0.003
-0.019-0.025-0.036-0036
-0.008-0013-0022-0026
-0.003-0.005-0.010-0014
:-0.065
Table 3 Table 4
Moment = Coef. x wa3 ID;;
,w” 4 X l
bla xla
1.25
1.00
0.75
0.50
y = o
MX MY
+0025
y = b/4
Mx 4
y = b/2
M* MY0
+0.010+0.005-0.033-0.126
+0.019+0.010-0.004-0.025
0 to.014 0 -0.082+0.007 +0.013 -0014 -0.071+0.008 +0.010 -0.011 -0055-0.018 -0 000 -0.006 -0.028-0.092 -0.018 0 0
0 +0.027 0 +0.013 0 -0.074
+0.012 +0.022 +0.007 +0.013 -0.013 -0.066+0.011 +0.014 +0.008 +0010 -0.011 -0.053
-0.021 -0.001 -0.010 +0.001 -0.005 -0.027-0.108 -0.022 -0.077 -0.015 0 0
0 +0.027 0 +0.009 0 -0.060+0.013 +0.023 +0.006 +0.010 -0.012 -0.059+0.015 +0.016 +0.010 +0.010 -0.010 -0 049
-0.008 +0.003 -0.002 +0.003 -0.005 -0.027-0.086 -0.017 -0.059 -0.012 0 0
0 +0.025 0 +0.007 0 -0.050+0.012 +0.022 +0.005 +0.008 -0010 -0.052
+0.016 +0016 +0.010 +0.009 -0 009 -0.046
-0.002 +0.005 +0.001 +0.004 -0.005 -0.027-0.074 -0.015 -0.050 -0.010 0 0
0 +0.021 0 +0.005 0 -0.040to.008 +0.020 +0.004 +0.007 -0.009 -0.044
to.016 +0.016 +0.010 +0.008 -0.008 -0042to.003 +0006 +0.003 +0.004 -0005 -0.026-0.060 -0.012 -0.041 -0.008 0 0
0to.005+0.014+0.006-0.047
0to.002+0.009+0.008-0.035
+0.015 0 +0.003+0.015 +0.002 +0.005+0.015 +0.008 +0.007+0.007 +0.005 +0.005-0.009 -0031 -0.006
0-0.007-0.007-0.0050
0-0.005-0.006-0.0040
0-0.002-0.003-0.0030
0-0.001-0.002-0.0010
-0.029-0.034-0.037-00240
+0.009 0+0.011 0+0.013 +0.005+0.008 +0.005-0.007 -0.022
0 +0.004 0+0.001 +0.008 0+0.005 +0.010 +0.002+0.007 +0.007 +0.003-0024 -0.005 -0.015
0 +0.001 00 +0.005 0
+0.002 +0.006 +0.001+0.004 +0.006 +0.001-0.015 -0.003 -0.008
+0.002+0003to.005+0.004-0.005
+0.001+0.002+0.003+0.003-0.003
0+0.001+0.001+0.001-0.002
-0018-0.023-0.029-0.0200
-0007-0.011-0.017-0.0130
-0002-0.004-0.009-0.0070
- yi~hngea 1Moment q Coef. x bvaz
‘_II m
1 P”@
MX
bla
3.00
2.50
2.00
1.75
1.50
125
1.00
0.75
0.50
y = o
Mx MY
IO.089to.118
to022+o 029
+0077 to.025+0101 to.034
bO.085 +0.024 +0.070 +0.027to112 +0.032 +o 092 +0.037
0076 +0.027 +0.061 +0.028+0.100 +0.037 +0.078 +0038
+0.070 +0.029 +0.054 +o 029+0091 +0040 +0.070 +0.039
to.061 +0.031 +0047 to.029t0.076 +0.043 +0.059 +0.040
to 049 +0.033 +0038 +0.029+0063 +0.044 +0.047 +0.039
to.036 +0.033 +0.027 +002710.044 +0044 +0033 +0036
to.022 +0.029 +0.016 +0023to.025 +0.038 +0.018 +0.030
.O.OlO to.020 +0007 bO.015-0 009 +0025 +0.007 +0.019
y = b/4
MX M”
3
I
Table 5. Moment Coefficients for Tanks with Walls Free at Top and Hinged at Bottom
b/a = 3.0
$y = o
Mx M”
y = b / 4
M, 4
y = b / 2
4 4
z = cl4
4 MZ
0 +0027+0.015 +0026+0.032 +0.026+0.034 +0.018
0 to013+0.009 +0.014+0.023 +0.017+0.029 +0014
0 -0.005+0.002 -0.002+o.o1f3 +0.005+0.022 +0008
0 -0.018-0.003 -0.012+0011 -0.003+0.018 +0.004
0 -0033-0.007 -0.024+o 005 -0.012+0013 0
0 -0.052-0.011 -0 0390 -0.022
+0.008 -0006
0 -0074-0.015 -0.056-0005 -0.034to003 -0.014
0 -0 098-0.020 -0 079-0.011 -0.051-0.002 -0.025
0 -0.126-0.024 -0105-0.016 -0.073-0.007 -0.040
I=0
M, M,c / a
-
3.00
2.50R
Moment q Coef. x wa3
1.25
0 +0.070 0 +0.027+0.028 +0.061 +0.015 +0.028+0.049 +0.049 +0.032 +0026+0.046 +0.030 +0.034 +0.018
0 -0196-0.034 -0170-0.027 -0137-0.017 -0087
0 +0.073 0 +0033 0 -0.169+0.028 +0.063 +0.016 +0033 -0030 -0.151+0.049 +0.050 +0.033 +o 029 -0.025 -0.126+0.046 +0.030 +0.037 +0.020 -0.017 -0.084
0 +0.075 0 +o 039 0 -0.146+0.029 +0.065 +0.017 +0.036 -0.027 -0133+0.050 f0.051 +0.035 +0.032 -0.023 -0.113+0.046 +0.031 +0.037 +0.021 -0016 -0078
0 +0.076 0 +0041 0 -0.137+0.029 +0065 +0.018 +0038 -0025 -0.125+0.050 +0.052 +0036 +0033 -0.021 -0.106+0.046 +0.031 +0037 +0.021 -0.015 -0074
0 +0.077 0 +0.043 0 -0.129+0.030 +0.066 +0.018 +0.039 -0.024 -0.118+0050 +0.053 +0.037 +0.034 -0.020 -0.100+0.046 +0031 +0.038 +0.022 -0.014 -0.070
0 +0.078 0 +0.045 0 -0.122+0.030 +0.067 +0.019 +0.041 -0022 -0111+0.050 +0.054 +0.038 +0.035 -0.019 -0 095+0.047 +0.032 +0.038 +0.023 -0.014 -0.068
0 +0.079 0 +0.047 0 -0118+0.030 +0067 +0.020 +0.043 -0.021 -0105+0.051 +0054 +0.038 +0.036 -0.018 -0.090+0.047 +0032 +0.038 +0.023 -0.013 -0.065
0 +o 079 0 +0.047 0+0.029 +0.066 +0.020 +0.042 -0.021+0.051 +0.053 +0.037 +0.036 -0.018+0.047 +0.031 +0.037 +0.022 -0.013
0 +0.078 0 +0.047 0+0.029 +0.065 +0.019 +0.042 -0.023+0.050 +0.053 +0.035 +0.035 -0.019+0.046 +0.031 +0.036 +0.021 -0.014
-0.120-0107-0 090-0066
-0.130-0.115-0.095-0.068
0 +0.070+0028 +0061+o 049 to.049+0046 +0030
0 +0057+0022 +0050+0041 +0043+0040 +0027
0 +0031+0.013 +0.03210030 +o 029+0034 +0020
0 +0014+0.007 +0.018+0.023 +0.020+0.027 +0015
0 -0.006+0.002 +0.004+0.015 +0.010+0021 +0.010
0 -0.031-0.004 -0.018+0.008 -0.005+0.016 +0.001
0 -0.060-0.010 -0.042+0001 -0.022+0.009 -0.009
0 -0.092-0.016 -0.070-0.006 -0045+0.003 -0024
0-0.022-0.013-0.004
-0123- 0 1 0 1
b/a = 2.5
c/ay = b/4 y = b/2 z=oz = cl4
4 M*
0 +0.019+0010 +0.022+0025 +0.022+0.030 +0.016
0 +0.003+0005 +0.006+0.018 +0.011+0023 +0.011
0 -0.006+0.001 -0002+0.013 +0004+0019 +0.008
0 -0018-0.003 -0.012+0.008 -0.002too15 +0004
0 -0.030-0.006 -0.024+0.003 -0012+0011 -0002
0 -0.045-0010 -0.036-0.003 -0.021+0.006 -0.008
0 -0062-0014 -0053-0.008 -0.035+0.002 -0016
0 -0.081-0.019 -0.072-0.014 -0.056-0.003 -0030
MX MY 4 MY M” MZ0 +0.0.!31 0 co.019
+0.024 +0.053 +0010 +0.022+0042 +0.044 +0025 +0.022+0.041 to.027 +0.030 +0016
0 -3138-0.026 -0132-0023 -0.115-0.016 -0.078
0 +0065 0 +0.026 0 -0.118+0.025 +0.055 +0.012 to.027 -0.023 -0.113+0.043 +0.046 iO.028 +0.025 -0.020 -0.102+0042 +0.028 +0.031 +0.018 -0014 -0.070
0 .-.,7 0 +0.030 0 -0.108+0.025 +0.057 +0.013 +0.030 -0.021 -0.104+0.044 +0.047 +0.029 +0.027 -0.019 -0.096+0043 +0.028 +0.033 +0.019 -0013 -0.066
0 to.068 0 +0033 0 -0100+0.026 +0.058 +0.014 +0032 -0.019 -0.097+0.045 +0047 +0.030 +0.029 -0.018 -0.089+0043 +0.029 +0.034 +0.019 -0.013 -0.063
0 +0 069 0 +0035 0 -0.092+0.026 +0.059 +0.015 +0034 -0.018 -0.089+0.045 +0.048 +0.031 +0031 -0.016 -0.082+0.044 +0.029 +0.034 +0.020 -0012 -0 059
0 +0.070 0 +0037 0 -0087+0.026 +0.060 +0.015 +0036 -0.017 -0.083+0.046 +0.048 +0.031 +0.032 -0.015 -0077+0.044 +0.029 +0.033 +0021 -0.011 -0056
0 +0.070 0 +0.038 0 -0082+0.025 +0.060 to.015 +0.037 -0.016 -0.078+0.045 +0.047 +0.030 +0.032 -0.014 -0071+0043 +0.029 +0.033 +0.020 -0.011 -0.054
0 +0.069 0 +0.039 0 -0080+0.025 +0.059 +0.014 +0.038 -0.015 -0075+0.044 +0.046 +0028 +0.032 -0.014 -0.068+0.042 +0.028 +0.032 +0.019 -0.010 -0.052
0 +0.061+0.024 +0053+0042 +0.044+0.041 +0027
0 +0.038+0.015 +0.037+0.032 +0.033+0034 +0.022
0 +0025+0010 +0026to.025 +0025+0.029 +0.019
0 +0.008+0004 +0013+0017 +0.017+0024 +0015
0 -0010-0.002 -0.003+0.008 +0.007+0.018 +0008
0 -0032-0.008 -0021-0.001 -0008+0011 0
0 -0.055-0.014 -0.042-0.009 -0.025+o 005 -0.011
0 -0.080-0019 -0.068-0017 -0.048-0002 -0.026
2.50
1.25
4
\-0071-0042
t
y = o y = b/4
M, MY M, M,
0 +0.045 0 +0.011+0016 +0.042 +0.006 +0.014+0033 +0.036 +o.ozo +0.016+0.036 +0.024 +0.025 +0.014
0 +0048 0 +0.015+0.017 +0.044 +0.007 +0.017+0.034 +0.038 +0.021 +0019+0036 +0.024 +0.025 +0.015
0 +0.050 0 +0019+0018 +0.046 +o.ooa +0.021+0.035 +0.039 +0.022 +0.021+0.036 +0.025 +0.026 +0.016
0 +0.052 0 +0023+0019 +0.048 a009 +0.024+0036 +0.041 +0.023 +0.023+0037 +0.025 +0.026 +0.017
0 +0054 0 +0.027+0.019 +0.050 +0.010 +0.027+0.037 +0.042 +0.024 +0.025to.037 +0.026 +0.027 +0.018
0 +0.055 0 +0.030+0.018 +0.051 +0.011 +0.029+0.038 +0.043 +0.025 +0026+0.037 +0.026 +0.027 +0.018
0 +0054 0 +0.030+0.018 +0.052 +0.011 +0.029+0.038 +0.044 +0.025 +0.025+0037 +0.026 +0.026 +0.017
= 2.0
c a
2.00
1.75
1.50
1.25
100
075
0.50
z = cl4y = b/2
M, MY
-00910-0.019-0.018-0.013
-0.094-0.089-0.065
z=o
4 4to0110
+0.006+0020+0025
+0.014+0016to014
M, MZ0 co.045
0-0.017-0.017-0.012
0-0015-0.015-0.012
0-0.014-0.014-0.011
0-0.012-0.013-0.010
0-0.012-0.012-0.010
0-0.014-0.013-0.010
-0.081 0 -0001-0.085 +0.003 +0006-0.083 +0.015 +0011-0061 +0.020 +0012
-0072 0 -0.010-0077 0 -0002-0.076 +o 009 +0004-0.058 +0.016 +0.008
-0064 0 -0021-0.068 -0.002 -0.013-0.069 +0.005 -0004-0054 +0.011 +0.002
-0.058 0 -0037-0.062 -0.005 -0025-0.064 0 -0.015-0051 +0006 -0006
-0.058-0.062-0.062-0.049
-0 049-0040-0 029-0.015
-0.065-0068-0064-0.050
0-0.009-0.005+o.ooz
0-0012-0010-0.003
-0.064-0056-0.045-0026
+0.016 +0.042+0.033 +0.036+0.036 +0024
0 ~0032+0.012 co.032+0.027 +0.02910.031 +0021
0 +0.018to.007 '0020+0.020 co.022+0.025 10017
0 0+0.001 '0005+0011 +0012+0016 +0.011
0 -0.023-0.005 -0.013+0.001 0+0.008 to.004
0 -0.044-0010 -0031-0.007 -0.015+0.001 -0.004
0 -0061-0014 -0051-0.012 -0034-0.004 -0.018
I I = 1.5
y = b/4 y = b/2
4 M Y 4 MY
0 +0.005 0 -0.052+0.003 +0.008 -0.012 -0.059+0.012 +0.011 -0.013 -0.063+0.017 +0011 -0.010 -0.052
0 +0.008 0 -0.045+0.005 +0.012 -0010 -0.050+0.014 +0.014 -0.011 -0.056+0.018 +0.012 -0.010 -0.048
0 +0013 0 -0.038+0.006 +0.016 -0008 -0.042+0.015 +0.017 -0.010 -0.049to.019 +0.014 -0.009 -0.045
0 +0.016 0 -0.034+0.007 +0.018 -0.008 -0.038+0.016 +0.019 -0.008 -0.042+0.019 +0.015 -0.008 -0.041
0 +0.017 0 -0.036+0.007 +0.019 -0.008 -0040+0.017 +0.020 -0.009 -0.044+0.018 +0.016 -0.008 -0.040
”1.50
w/a* = c/4 z=o
M" 4 MX MZ
0 +0005 0 co.027+0003 +0.008 +0.009 +0028+0.012 +0.011 +0.022 to.027+0.017 +0.011 +0.027 to.020
0 -0005 0 coo11+0.001 -0.001 +0.004 +0015+0007 +0006 +0.014 +0.02060.013 +0006 +0.018 10016
0 -0.016 0 -0.006-0.002 -0010 -0001 +0.001+0.002 -0.003 +0006 +0010+0.008 +0.002 +o 009 +0.010
0 -0024 0 -0.019-0.005 -0.020 -0.004 -0013-0.002 -0.014 -0.001 -0.004+0.003 -0.007 +0.002 +0001
0 -0030 0 -0.028-0.008 -0031 -0.007 -0.027-0.006 -0.027 -0.006 -0020-0.002 -0018 -0.004 -0010
y - o
M. M”I
0 +0.027+0.009 +0.028+0.022 +0.027+0.027 +0.020
0 +0.031+0.010 +0.031+0.024 +0.030+0.027 +0.021
0 +0.035+0.011 +0.034+0.025 +0.032+0.028 +0.022
0 +0.038+0.011 +0.036+0.025 +0.033+0028 +0.022
0 +0.040+0.010 +0.037+0.024 +0.034+0.028 +0.022
125
b/a = 1.0
c/a x/ay - o Y = b / 4 Y = b / 2 z = c/4 z=o
M” Mb M” M” M” M,
0 +0.010 0 +0.002 0 -0.019 0 +0.002+0.002 +0.013 0 +0.003 -0.005 -0.025 0 +0.003+0.010 +0.017 +0.005 +0006 -0.007 -0036 +0.005 +0006+0015 +0.015 +0.009 +0.007 -0.007 -0036 +0.009 +0.007
MX MZ
0 +0.010+0002 f0.013+0.010 +0017+0.015 +0.015
0 +0.016 0 +o.oc)7 0 -0.013 0 -0.004 0 +0.003+0.003 +0.017 +0.001 +0.008 -0.004 -0.020 -0.001 -0.005 -0.001 +0.003+0.011 +0.020 +0.006 +0.009 -0.007 -0.033 +0.002 -0.001 +0.005 +0.007+0.016 +0.014 +0.009 +0.009 -0.006 -0.032 +0.004 +0.002 +0.009 fO008
0 +0.020 0 +0.011 0 -0.011 0 -0.007 0 -0.005+0.003 +0.018 +0.001 +0.010 -0.004 -0.018 -0002 -0.012 -0.003 -0.007+0.012 +0.021 +0.008 +0.010 -0.006 -0.032 +0.001 -0.009 +0.002 -0.005+0.017 +0.013 +0.010 +0.009 -0.006 -0.031 +0002 -0.005 +0.006 +0.001
5
Table 6. Moment Coefficients for Tanks with Walls Hinged at Top and Bottom
Moment = Coef. x wa3
b/a = 3.0
C/a
3.00
2.50
2.00
1.75
1.50
1.25
1.00
0.75
0.50
f
w/ay = o y = b/4 y = b/2 z = c/4 .?=O
MX MY Mx MY MX MI+0.035 +0.010 +0.026 +0011+0.057 +0.016 +0.044 +0.017+0.051 +0.013 +0.041 +0014
+0.035 +0.010 +0.026 +0.011+0057 +0.016 +0.044 +0.017+0.051 +0.013 +0.041 +0.014
+0.035 +0.010 +0.026 +0.011+0.057 +0.016 +0045 +0.017to.051 +0013 +0.042 +0.014
+0.035 +0010 +0027 +0.011+0.057 +0.015 +0.045 +0017+0051 +0013 +0.042 +0.014
+0.035 +0.010 +0.027 +0.011+0.057 +0.015 +0.045 +0.017+0.051 +0.013 +0.042 +0.014
+0.035 +0010 +0027 +0.011+0.057 +0.015 +0.046 +0.017+0.051 +0.013 +0.042 +0.014
+0.035 +0.010 +0.027 +0.011+0.057 +0.015 +0.046 +0.017+0.051 +0013 +0.043 +0.014
+0.035 +0.010 +o.o2a +0.011+0.057 +0.015 +0.046 +0.017+0.052 +0.013 +0043 +0.014
+0036 +0.010 +0.028 +0.011+0.057 +0.015 +0.047 +0.017+0.052 +0.013 +0.043 +0.014
-0 008-0.013-0011
-0.008-0.012-0.011
-0.008-0.012-0011
-0.007-0.012-0.011
-0.007-0.011-0.010
-0.006-0.011-0.010
-0.006-0.010-0.009
-0.005-0.008-0.008
-0.004-0007-0.007
-0 039-0.063-0.055
-0.039-0.062-0055
-0.038-0.062-0.054
-0037-0.060-0053
-0.035-0.057-0.051
-0.032-0.053-0.048
-0 029-0.048-0.044
-0.025-0.042-0.039
-0.021-0035-0.033
+0026 +0.01110044 +0.017+0041 +0.014
+0.021 +0010+0036 +0.017+0036 to014
+0.015 to010+0.028 +0015+0.029 +0.013
+0011 +0008+0021 +0.013+0024 '0012
+0007 to.006to.015 +0.010+0019 +0011
+0.003 +0.003+0.008 to.006+0013 +0008
-0001 0to.002 +0002+0.007 +0.004
-0.003 -0005-0.003 -0005+0.002 -0002
-0004 -0.011-0007 -0.016-0.004 -0.010
+0035 +0.010+0057 +0016+0051 +0.013
+0031 +0011+0.052 +0017+0047 +0.014
+0025 to.013+0043 to020+0041 +0016
+0.020 +0013+0036 +0020+0.036 +0016
to.014 +0013+0027 +0020to 029 to.017
+0008 +001110017 a017+0.021 +0.016
to002 +0.008+0.007 +oc14+0.013 +0.013
-0002 +0.001-0.001 +0.007+0.006 +0.007
-0.005 -0.008-0.006 -0010-0001 -0.004
b/a = 2.5
c/a
2.50
2.00
1.75
1.50
1.25
100
0.75
0.50
y = b/2
4 M”-0.008 -0.038-0.012 -0.062-0.011 -0.055
-0.008 -0.038-0.012 -0.061-0.011 -0.054
-0.007 -0.037-0.012 -0.059-0.011 -0.053
-0.007 -0.035-0.011 -0057-0.010 -0.051
-0.006 -0.032-0.011 -0.053-0.010 -0.048
-0.006 -0028-0.010 -0048-0.009 -0044
-0005 -0.024-0.008 -0041-0.008 -0.039
-0004 -0.021-0.007 -0035-0.007 -0034
z = c/4
M, MZ
+0.021 +0010+0036 +0017co.036 +0.014
+0.015 +o 009+0.028 +0015+0.029 +0.013
+0011 +0.008+0.022 +0.013+0.024 +0.012
+0.007 +0.006+0.015 +0.010+0019 +0.010
+0003 +0.004+0.008 +0.007+0.014 +0.008
-0.001 0+0.002 +0002to.007 +a004
-0003 -0.005-0.003 -00050 -0002
-0004 -0.011-0007 -0016-0.004 -0.010
z=o
4 MZ+0.031 +0.011 +0.021 +0.010+0.052 +0.017 +0.036 to.017+0.047 +0.015 +0.036 +0.014
+0.031 +0.011 +0.021 +0.010+0.052 +0.017 +0.036 to.017+0.047 +0.015 +0.036 +0.014
+0.032 +0.011 +0.021 +0.010+0.052 +0.018 +0.036 +0.017+0.047 +0.015 +0.036 +0.014
+0.032 +0.011 +0.022 +0.010to.052 +0.018 +0.037 +0.017+0.047 +0.015 +0036 +0.014
+0.032 +0.011 +0.022 +0010+0.052 +0.018 +0.038 +0.017+0.048 +0.015 +0.037 +0.014
+0.032 +0.011 +0.023 +0.011+0.053 +0.018 +0038 +0.017+0.048 +0.015 +0.038 +0.015
+0.033 +0.011 +0024 co.011+0.054 +0.018 +0.039 +0.017+0.049 +0.015 +0.038 +0.015
+0.033 +0.012 +0.024 +0.011+0.054 +0.018 +0040 +0.017+0.049 +0.015 +o 039 to015
+0031 +0.011+0.052 +0.017+0.047 +0.015
+0025 +0.012+0042 +0.020+0.041 +0.016
+0.020 +0.012+0035 +0021+0.035 +0.017
'0.014 +0013+0.027 +0.021+0.029 +0.017
+0.007 +0.012co.018 +0019+0022 to.018
+0.002 +0.008to007 +0014+0013 +0013
-0.002 -0.0020 +000510006 +0006
-0.005 -0.008-0.006 -0010-0.001 -0004
6
b/a = 2.0l-
y - o y = b/4 y = b/2
Mx 4 Mx 4 4 MY+0.025 +0.013 +0.015 +0.009 -0007 -0.037+0.042 +0.020 +0.028 +0.015 -0012 -0.059+0.040 +0.016 +0.029 +0.013 -0.011 -0.053
+0.025 +0.013 +0.015 +0.009 -0.007 -0.036+0.042 +o.ozo +0.028 +0.015 -0.012 -0.058+0.040 +0.016 +0.029 +0.013 -0.010 -0052
+0.025 +0.013 +0.016 +0.009 -0.007 -0034+0.043 +0.020 +0.028 +0.015 -0.011 -0.056+0.041 +0.016 +0.029 +0.013 -0.010 -0.050
+0.0?6 +0.013 +0.016 +0.010 -0.006 -0.032+0.043 +0.020 +0.029 +0.015 -0.010 -0052+0.041 +0.016 +0.030 +0.013 -0.010 -0.048
+0.026 +0.013 +0.017 +0.010 -0.006 -0.028+0.044 +0.020 +0.030 +0.016 -0.009 -0.046+0.041 +0.016 +0.031 +0.014 -0.009 -0.044
+0.027 +0.013 +0.018 +0.010 -0.005 -0.024+0.045 +0.020 +0031 +0.016 -0.008 -0.040+0.042 +0.016 +0.032 +0.014 -0.008 -0.041
+0.027 +0.013 +0.019 +0.010 -0.004 -0.021+0.046 +0.020 +0.033 +0.017 -0.007 -0.034+0.042 +0.016 +0.032 +0.015 -0.007 -0.037
- -
c/az=o
M, Mz+0.025 +0.013+0.042 +0020+0.040 +0.016
+0.020 +0.013+0.035 +0.021+0.035 +0.017
+0.014 +0.013+0027 +0.021+0.029 +0.017
+0.007 +0.011+0.018 +0.019+0.021 +0.016
+0.002 +0.008+0.007 +0.014+0.013 +0.013
-0.001 +o.m20 +0.005
to.005 to.008
-0.004 -0.007-0.006 -0.009-0.002 -0.003
z = cl4
4 4
to015 +0.009+0028 +0.015+0.029 +0013
+0.011 +0.008+0.022 +0.013+0.024 +0012
+0.007 +0.006+0.015 +0.011+0.019 +0010
+0.003 +0003+0.008 +0007+0.013 +0008
-0.001 0+0.002 +0.002+0.007 +0.004
-0.003 -0.004-0002 -0004+0.002 -0002
-0.004 -0010-0.006 -0015-0.003 -0010
2.00
1.75
1.50
125
1.00
0.75
0.50
1b/a = 1.5
c/az=o
4 4
+0.015 +0.013+0.028 +0.021+0.030 +0.017
+0.009 +0.012+0.018 +0.019+0023 +0.016
+0.003 +0008+0.008 +0014+0.014 +0.014
-0.001 to.002+0.001 +0.005+0.006 +0.008
-0004 -0006-0.005 -0007-0.001 -0001
z = c/4
4 MZ
+0008 +0.007+0.016 +0.011+0.020 +0.011
+0.004 +0.004+0.009 +0008+0.014 +o 009
0 +0.001to.003 +0.003+0008 +0.005
-0002 -0.003-0.002 -0.004+0.002 0
-0.003 -0 009-0.006 -0.014-0003 -0.008
y = b/4 y = b/2
4 MY 4 MY
+0.008 +0.007 -0.006 -0.032+0.016 +0.011 -0.010 -0.052+0.020 +0.011 -0.010 -0.048
+0.009 +0.008 -0.006 -0.029+0.017 to.012 -0.010 -0.049+0.020 +0.012 -0.009 -0.045
+0.010 +0.009 -0005 -0.025+0.019 +0.012 -0.009 -0.043+0.021 +0.013 -0.008 -0.041
+0.011 +0.010 -0.004 -0.021+0.021 +0.014 -0.007 -0036+0.022 +0.014 -0.007 -0.036
+0.013 +0.012 -0.003 -0.017+0.023 +0.018 -0.006 -0.031+0.024 +0.016 -0.007 -0.033
y - o
4 MY
+0015 +0013+0.028 +0.021+0.030 +0.017
+0.016 +0.013+0.029 +0.021+0.030 +0.017
+0.016 +0.013+0.030 +0.021to.031 to.017
+0.018 +0.014+0.032 +0.022+0.032 +0.018
+0.020 +0.016+0035 +0.024+0034 +0.020
150
125
b/a = 1.0
y = b/2 I = c/4 z=oc/a
M.x 4+0.005 +0.009+0.011 +0.016+0.016 +0.015
+0.001 +0.005+0.005 +o 009+0.008 +0010
-0.003 -0.002-0.003 -0.0020 +0.001
y = o y = b/4
4 M” 4 MY
+0.005 +0.009 +0.002 +0.003+0.011 +0.016 +0.006 +0.006+0016 +0.015 +0.009 +0.007
+0.006 +0010 +0.003 +0.004+0.013 +0.017 +0.008 +0.008+0.017 +0016 +0.010 +0.008
to.007 +0.011 +0.005 +0.006+0.015 +0018 +0.010 +0.010+0.018 +0.016 +0.012 +0.010
M. M” M” M,
+0.002 +0.003+0.006 +0.006+0.009 +0.007
-0.004 -0.020-0.007 -0.035-0.007 -0.035
-0003 -0.016-0.006 -0.029-0.006 -0.031
-0.002 -0.010-0.004 -0.021-0.005 -0.026
0 0+0.001 +0.001+0.004 +0.003
-0.002 -0.005-0.003 -0.007-0.001 -0.004
7
In this manner, moment coefficients were computedand are tabulated in Tables 5 and 6 for top and bottomedge conditions as shown for single-cell tanks with alarge number of ratios of b/a and c/a, b being the largerand c the smaller of.the horizontal tank dimensions. Mo-ments in vertical and horizontal directions equal thecoefficients times wa3, in which w is the weight of theliquid. Note that the loading term is wa3 for all wall slabssubject to hydrostatic pressure but is wa2 for the floorslab in Table 4, which has uniformly distributed load. Inthe first case, w is weight per cubic foot, but in the latterit is weight per square foot.
There is a peculiarity about the horizontal end mo-ments in the slabs at the free top edge. Calculations ofsuch moments by means of the trigonometric seriesused result in a value of zero, whereas these momentsactually have finite values and may even be compara-tively large. Horizontal end moments at the free edgewere therefore established by extrapolation. The con-sistency of extrapolated moment coefficients waschecked by plotting and studying curves. This gavereasonably good results, although coefficients thusdetermined are probably not quite as accurate as thecoefficients that were computed. A condition prevails atthe quarter point of the free edge, similar to that at theend point but to a lesser degree. At the midpoint of thefree edge the coefficients were computed, extrapolationbeing used only for checking purposes.
When a tank is built underground, the walls must beinvestigated for both internal and external pressure. Thelatter may be due to earth pressure or to a combinationof earth and groundwater pressure. Tables and otherdata presented can be applied in ‘the case of pressurefrom either side but the signs are opposite. In the caseof external pressure, actual load distribution may notnecessarily be triangular as assumed in the tables.Consider for illustration a tank built below ground withearth covering the roof slab and causing a trapezoidaldistribution of lateral earth pressure on the walls. In thiscase it gives a fairly good approximation to substitute a
triangle with the same area as the trapezoid represent-ing the actual load distribution. The intensity of load isthe same at middepth in both cases and when the wallis supported at both top and bottom edges, the discrep-ancy between triangle and trapezoid has relatively littleeffect at and near the supported edges
Shear Coefficients
Shear values along the edges of a tank wall are neededfor investigation of shear and development stresses.Along vertical edges, shear in one wall is also used asaxial tension in the adjacent wall and must be combinedwith bending moment to determine tensile reinforce-ment.
Various data for shear were computed and are givenin Table 7. The wall is considered fixed at the two verti-cal edges while top and bottom edges are assumed tobe hinged. The wall panel with width b and height a issubject to hydrostatic pressure due to a liquid weighingw lb per cubic foot.
The first five lines in Table 7 are shears per linear footin terms of wa*. The remaining four lines are total shearsin kips or pounds depending on how w is given. Shearsper linear foot are for ratios of b/a = %, 1,2, and infinity.The difference between the shear for b/a = 2 and infinityis so small that there is no necessity for computing co-efficients for intermediate values.
When b/a is large, a vertical strip of the slab near mid-point of the b dimension will behave essentially as a
0.50waz, of which two-thirds or 0.33wa2is the reaction at the bottom support and one-third or0.17wa2 is the reaction at the top. Note in Table 7 thatshear at midpoint of the bottom edge is 0.3290waz forb/a = 2.0, the coefficient being very close to that of one-third for infinity. In other words, maximum bottom shearis practically constant for all values of b/a greater than
Table 7. Shear at Edges of Slabs Hinged at Top and Bottom
b l a ‘h 1 2 5 10 lnfmtty
Midpoint of bottom edge +o 1407wa’ +o 2419weCorner at bottom edge -0 2575wa” -0 4397wa’
M,dpo,“t of flxed side edge +o 1260wa’ +O 2562wa’Lower third-pant of side edge *o 173&v@ +o 3113wa’Lower quarter-pant of side edge ‘0 1919wP +o 3153w.e
Total at top edge 0 OOOOwa’b 0 0052wa’bTotal at bottom edge 0 0460wa-‘b 0 0960wa’bTotal at one foxed side edge 0 2260wazb 0 1 9 9 4 w a ’ bTotal at all four edges 0 5000wa’b 0 5000waJb
‘Negatwe s,gn lndlcates reaction acts I” darectlon of loadtEsbmated
+o 3290w.a’- 0 5633w.F
+0.3604waz‘0 4023wa’‘0 3904w.3’
0 0536w.+b 0 ,203~~b0 1616wa’b 0 2715wa’b0 1322wa’b 0 0541wa.b0 5000wa’b 0 5 0 0 0 w a . b
+o 3333waz-0 6000wa’
‘0 3912w.a’+0 4116wa’‘0 39t30wa.
0 1435wa’b0 3023wa’b0 0271 wa’b0 5000wa’b
0 1667wa’b0 3 3 3 3 w a . b0 275wav0 5000wa’b
8
\asimply supported one-way slab. Total pressure on a Jstrip 1 ft wide is
2. As will be shown, this is correct only when the topedge is supported, not when it is free.
At the corner, shear at the bottom edge is negativeand numerically greater than shear at midpoint. Thechange from positive to negative shear occurs approxi-mately at the outer tenth points of the bottom edge.These high negative values at the corners arise be-cause deformations in the planes of the supportingslabs are neglected in the basic equations and aretherefore of only theoretical significance. These shearscan be disregarded in checking shear and developmentstresses.
Unit shears at the fixed edge in Table 7 were used forplotting the curves in Fig. 1. There is practically nochange in shear curves beyond b/a = 2.0. Maximumvalue occurs at a depth below the top somewhere be-tween 0.6a and 0.8a. Fig. 1 is useful for determination ofshear or axial tension for any ratio of b/a and at anypoint of a fixed side edge.
Total shear from top to bottom of one fixed edge inTable 7 must equal the area within the correspondingcurve in Fig. 1, and this relationship was used for check-ing the curves ‘Total shears computed and tabulated fora hanged top were also used in making certain adjust-ments to determine approximate values of shear forwalls with free too-recorded in Table 8.
For b/a = % in Table 7, total shear at the top edge is sosmall as to be practically zero, and for b/a = 1 .O totalshear, 0.0052, is only 1% of total hydrostatic pressure,0.5000.’ Therefore, it is reasonable to assume that re-moving the top support will not materially change totalshears at any of the other three edges when b/a = Y2 and1. At b/a = 2.0, there is a substantial shear at the top
‘Loading term is omitted here.
9" 0.3
gij a4
fg 0.5
% a6p
B a7
1.00
Fig. 1.
0.1 0.2 0.3
Shear per lin. ft. = coef x wa2
Table 8. Shear at Edges of Slabs Free at Topand Hinged at Bottom*
tJa 1 2 3
Mldpolnt o f b o t t o m e d g e ‘ 0 141wa: *o 242w.F ‘ 0 3awa7 ‘ 0 45wetCorner at bottom edge -0 258wa-$ -0 440-a. - 0 583~9 - 0 sowa
Top of flxed side edge 0 ooowa. ‘0 olowa’ *o 100wa -0 165wa-
Mldpomf of flxed side edge +O 128wa‘ *O 258wa’ .o 375wa: ‘ 0 406WWLower third-wont of side edae *o 174ws *0311wa- *o 406W% *o 416w.F
Lower quarter-WI”, of side edge ‘0 192-a- *o 315w.T *o 390wa ,O 398waTotal at bottom edge 0 0 4 8 w a b 0 0 9 6 w a . b 0 2 0 4 w a . b 0 286wa.bTotal at one faxed s,de edge 0 226wa b 0 2 0 2 w a . b 0 148wa:b 0 107wa.bTotal al all four edges 0 500w.s. b 0 500wa-b 0 500wa. b 0 500~4 b
‘Data dewed by modlfymg values compufed for waifs hanged fop and boflom
tThls value could not be esflmated accurately beyond two decimal places
Wegat~ve s!gn lndlcates react~o” acts I” d,recfwn of load
edge when hinged, 0.0538, so that the sum of totalshears on the other three sides is only 0.4462. If the topsupport is removed, the other three sides must carry atotal of 0.5000. A reasonable adjustment is to multiplyeach of the three remaining total shears by 0.5000/0.4462 = 1 .12, an increase of 12%. This was done in pre-paring Table 8 for b/a = 2.0. A similar adjustment wasmade for b/a = 3.0, where the increase is 22%.
Total shears recorded in Table 8 were used to deter-mine unit shears also recorded in that table. Considerfor illustration the shear curves in Fig. 1 and imagine thetop is changed from hinged to free. As already stated, forb/a = % and 1 it makes little difference in total shear-the area within shear curves-whether the top is sup-ported or not. Consequently, curves for b/a = % and 1remain practically unchanged. They were transferredalmost without modification to Fig. 2, which covers thecase with top free. For b/a = 2 an adjustment was made.A change in the support at the top has little effect uponshear at the bottom of the fixed edge. Consequently, thecurves in Figs. 1 and 2 are nearly identical at the bottom.Gradually, as the top is approached curves for the freetop deviate more and more from those for the hingedtop, as in Fig. 2. By trial, curve for b/a = 2 was so ad-justed that its area equals the total shear for one fixededge for b/a = 2.0 in Table 8. A similar adjustment wasmade for b/a = 3.0, which is the limit of moment coeffi-cients given.
One point of interest stands out In a comparison ofFigs. 1 and 2. Whereas for b/a = 2.0 and 3.0 total shearis increased 12% and 22%, respectively, when top isfree instead of hinged, maximum shear is increased butslightly, 2% at most. The reason is that most of the in-crease in shear is near the top where shears are rela-tively small.
The same general procedure was applied, but notillustrated, for adjustment of unit shear at midpoint ofbottom, but in this case the greatest change resultingfrom making the top free is at midpoint where shear is
9
0
Fig. 2.
0.1 @.2 0.3
Shear per lin ft. - coef Y wa2
04
large for the hinged-top condition. For illustration, forb/a = 3.0, unit shear at midpoint of the bottom is0.33wa2with hinged top but 0.45wa2 with free top-an increaseof approximately one-third.
Shear data were computed for wall panels with fixedvertical edges. They can be appli$d with satisfactoryresults to any ordinary tank wall even if vertical edgesare not fully fixed.
Open-Top Single-Cell Tank
The tank in Fig. 3 has a clear height of a = 16 ft. Hori-zontal inside dimensions are b = 40 ft and c = 20 ft. Thetops of the walls are considered free and the bottomhinged. The tank contains water weighing 62.5 lb percubic foot.
Coefficients for moment and shear are selected fromtables or diagrams for b/a = 40/16 = 2.50 and c/a =20/l 6 = 1.25. Moments are in foot-kips if coefficientsare multiplied by wa3/1000 = 62.5 x 163/1000 = 256;and shears are in kips if coefficients are multiplied bywaz/lOOO = 62.5 x 16z/1000 = 16.
Moment coefficients taken from Table 5 for b/a = 2.50and c/a = 1 .25 are tabulated below. Coefficients for x =a (bottom edge), being equal to zero, are omitted.
0 0 +0069 0 +0035 0 - 0 . 0 9 2 0 - 0 . 0 3 0'A +0.026 GO59 +0015 +0034 - 0 . 0 1 6 - 0 . 0 8 9 -0006 - 0 . 0 2 4 -:002 I:::;;H +0.045 +0046 +0.031 +0.031 - 0 . 0 1 6 - 0 . 0 6 2 ' 0 . 0 0 3 - 0 . 0 1 2 +O.@X3 +OOO,% +0.044 to.029 ' 0 . 0 3 4 +0020 - 0 . 0 1 2 -0059 +001, - 0 . 0 0 2 +o,o,i3 +0008
The largest moment occurs in the horizontal direc-tion at the top of the corner common to both walls andequals -0.092wa3 = -0.092 x 256 = -23.6 ft kips. Thenegative sign simply indicates that tension is on the in-
10
Fig. 3.
side and need not be considered In subsequent calcu-lations.
Maximum horizontal moment at midpoint of the longerwall is +0.069wa3 = +0.069 x 256 = +17.7 ft kips. Thepositive sign shows that tension is in the outside of thewall. There is also some axial tenslon on this sectionthat can be taken equal to end shear at the top of theshorter wall. For use in connection with Fig. 2, ratio ofb/a for the shorter wall is 20/ 16 = 1.25. Shear is 0.03wa2= 0.03 x 16 =0.48 kips. The effect of axial tension is neg-ligible in this case and the steel area can be determinedas for simple bending.
Horizontally at x = a/2, axial tension taken from Fig. 2for b/a = 1.25 is equal to N = -0.30wa2 = -0.30 x 16 =-4.80 kips per linear foot, which is not negligible. Mo-ment is M = 0.048wa3 = 0.048 x 256 = 12.3 ft kips.
In the shorter wall, positive moments are all relativelysmall. Maximum positive moment is vertical: 0.01 8wa3
x 256 = 4.6 ft kips.Maximum Mx in the vertical strip at midpoint ot longer
0.045wa3 = 0.045 x 256 = 11.5 ft klps.Maximum shear at the bottom taken from Table 8 is
V = 0.42wa2 = 0.42 x 16 = 6.72 kips.
Closed Single-Cell Tank
The tank in this section differs from the preceding oneonly in that the tops of the walls are considered hingedrather than free. This condition exists when the tank iscovered by a concrete slab with dowels extending fromthe wall into the slab without moment reinforcementacross the bearing surface.
Moment coefficients taken from Table 6 are givenbelow. All coefficients for x = 0 (top edge) and x = a (bot-tom edge), being equal to zero, are omitted.
With a free top, maximum M, = +0.045wa3 and maxi-mum My = -0.092wa3. With a hinged top, maximum n/l,= +0.052wa3 and maximum My = -0.053wa3. It is to beexpected that a wall with hinged top will carry more loadvertically and less horizontally, but it is worth noting thatmaximum coefficient for vertical moment is only 13%
A= 0.018
,’panel is
less for wall with free top than with hinged top.Another noteworthy point is that maximum M, coeffi-
cient at y = 0 is +0.069 for a free top but +0.018 for ahinged top. Adding top support causes considerablereduction in horizontal moments, especially at y = 0.
Maximum moment is -0.053~~1~ = -0.053 x 256 =-13.6 ft kips.
Maximum moment in a vertical strip is M = 0.052~~1~ =0.052 x 256 = 13.3 ft kips. Axial compression (N) on thesection subject to this moment, and loads per linear footcan be taken as follows:
8-ft-high wall: 8 x 1 .08 x 0.150 = 1.3 kips12-in. top concrete slab: 0.150 x20/2 = 1.5 kips’
3-ft fill on top of slab: 0.300 x 20/2 =3.0 kips’Live load on top of fill: 0.100 x 20/2 = 1
forN =6.8 kips and to design tensile steel for N = 1 .3 + 1.5 = 2.8kips, in which fill and live load are disregarded.
Top and Base Slabs
The closed single-cell tank is covered with a concreteslab. Assume the slab is simply supported along all foursides and has a live load of 100 psf and an earthfillweighing 300 psf.
Estimating slab thickness as 12 in. gives a total designload of 100 + 300 + 150 = 550 psf. From Table 4, for aratio of 40/20 = 2, select maximum coefficient of 0.100,which gives maximum M = 0.1 00wa2 F 0.100 x 0.550 x20.02 = 22.0 ft kips.
At the corners, a two-way slab tends to lift off thesupports; and if this tendency is prevented by dowelingslab to support, cracks may develop in the top of theslab across its corners. Nominal top reinforcementshould therefore be supplied at the corners, say0.005bdsq in. per foot in each direction. Length of these bars canbe taken as %a = l/4 x 20 = 5 ft.
Assume the closed single-cell tank has a base slab ofreinforced concrete. Weight of base slab and liquid doesnot create any bending or shearing stresses in concreteprovided the subsoil is uniformly well compacted. Weighttransferred to the base through the bottom of the wall is
Top slab: 0.550 x 22 x 42 = 510 kipsWalls: 16x0.162(2x41.1 +2~21,1)=320kips
830 kipsIf the base slab extends 9 in. outside the walls, its area
is 43.7 x 23.7 = 1035 sq ft. The average load of w =830,000/ 1035 = 800 psf is used for design of the baseslab just as w = 550 psf was used for design of the topslab.
Total average load on the subsoil is 16 x 62.5 + 800 +weight of base slab, say 1000 + 800 + 200 = 2000 psf,which the subsoil must be able to carry.
If there is an appreciable upward hydrostatic pressureon the base slab, the slab should also be investigated forthis pressure when the tank is considered empty.
\ -‘ProportIons of tank being deslgned are such that for determlning
axial compression In sde walls, all the top load may be consideredcarned the short way
Multicell Tank
Multicell tanks do not lend themselves readily to mathe-matically accurate stress analysis It is possible, how-ever, with the tables presented here for single-cell tanksand for individual wall panels with fixed vertical edges toestimate moment coefficients for symmetrical multicelltanks with sufficient accuracy for design purposes. Whileresults obtained by the following procedure are approxi-mate and should therefore be considered as a guide toengineering judgment, the procedure does give a con-servative design.
Because a rotation of one corner has comparativelylittle effect on moments at adjacent corners in atankwithwall panels supported on three or four sides, moments inthe walls of a multicell tank are essentially the same asin single-cell tanks-except at corners where more thantwo walls intersect. Moment coefficients from Tables 5and 6, designated as L coefficients, apply to outer or L-shaped corners of multicell tanks (see Fig. 4a) as well asto interior sections in all walls, that is, sections desig-nated as y = b/4, y = 0, z = c/4, and z = 0. Moment coeffi-cients for design sections at corners where more thantwo panels intersect depend on the loading conditionproducing maximum moment and on the number of inter-secting walls.
In Fig. 4b, three walls form a T-shaped unit. If the con-tinuous wall, or top of the T, is part of the long sides oftwo adjacent rectangular cells, the moment in the con-tinous wall at the intersection is maximum when bothcells are filled. The intersection is then fixed and mo-ment coefficients, designated as F coefficients, can betaken from Tables 1, 2, or 3, depending on edge condi-tions at top and bottom. These three tables cover panelswith fixed side edges. If the continuous wall is part of theshort sides of two adjacent rectangular cells, momentat one side of the intersection is maximum, when the cellon that side is filled while the other cell is empty. Like-wise the end moment in the center wall is maximumwhen only one cell is filled. For this loading condition themagnitude of moment will be somewhere between theLcoefficients and the F coefficients. If the unloaded thirdwall of the unit is disregarded, or its stiffness considerednegligible, moments in the loaded walls would be thesame as in Fig. 4a, that is, the L coefficients apply. If thethird wall is assumed to have infinite stiffness, the corneris fixed and the f coefficients apply. The intermediatevalue representing more nearly the true condition canbe obtained by the formula:
End moments = L -nG2(i -F)
(4 b)
Fig. 4.
411
.O kips’
6.8 kips
tIt is conservative to check compressive stress
in which n denotes number of adjacent unloaded walls.This formula checks for n equal to zero and infinity. In anL-shaped unit n equals 0 and the end moments equalL - O(L - F) = L. Inserting n equal to infinity will givenl(n + 2) = 1 and the end moments equal L - 1 (L - F) = f,which also checks.
In Fig. 4c, two continuous walls form a cross. If inter-secting walls are the walls of square cells, moments atthe intersection are maximum when any two cells arefilled and the F coefficients in Tables 1,2, or 3 apply be-cause there is no rotation of the joint. If the cells are rec-tangular, moments in the longer of the intersecting wallswill be maximum when two cells on the same sideof thewall under consideration are filled, and again the F coef-ficients apply. Maximum moments in the shorter wallsadjacent to the intersection occur when diagonallyopposite cells are filled, and for this condition the L coef-ficients apply.
Fig. 5 shows moment coefficients at wall intersec-itions in two- and four-cell tanks. Where coefficients arenot shown, L coefficients of Tables 5 and 6 apply.
Two-Cell Tank, Long Center Wall
The tank in Fig. 6 consists of two adjacent cells, eachwith the same inside dimensions as the open-top single-cell tank and the closed single-cell tank. The top is con-sidered free.
In accordance with the types of units in Fig. 4, the tankconsists of four L-shaped and two T-shaped units. L co-eff icients from Table 5 for b/a = 2.50 and c/a = 1.25, andF coefficients from Table 2, for b/a = 2.50 and 1.25, aretabulated as follows:
Long outer walls
L = coeffuents from Table 5 forbla = 2 . 5 0 a n d cla = 1 . 2 5
x / ay = b / 2
Iy = b / 4
Iy = o
M” Mv M. Mv Mx Mv
0 I 0 -0.092 I 0 +0035 I 0 to.069% I - 0 0 1 8 - 0 . 0 8 9 to.015 to.034 +0.026 +0.059‘h - 0 0 1 6 - 0 0 8 2 to031 to.031 +0.045 +o.o4a% -0.012 -0 059 +0034 +0.020 I +0.044 +0.029
Short outer walls
L = coefficients from Table 5 for I F =Tca~fetTfz Ib l a = 2 . 5 0 a n d c/a = 1 2 5 b/a = 1 25
L .L&.? = c/2 L = Cl4 z=o 3
x/a .? =c/2Mx Ml 4 Mz MN Mz M.Y M, 4 MZ
0 0 -0.092 0 -0.030 0 -0.010 0 - 0 0 3 4 0 -0.073% -0.018 -0.089 -0.006 - 0 0 2 4
I
- 0 0 0 2 -0.003 -0.008 -0.042 -0.015 -0.073‘h - 0 0 1 6 -0.082 +0.003 -0.012 ‘0.008 +0.007 -0.010 -0.049 -0.014 -0.071% -0.012 -0.059 +0.011 -0.002 +0.018 +0.008 -0.009 - 0 0 4 4 -0.011 -0.054
Center wall
F = coef f r o mL = coefficients from Table 5 for Table 2 for
bla = 2.50 and c/a = 1.25 bla = 2.50 L L-F
XIBy = o y = b / 4 y = b/2 y = b / 2 3
4 MY 4 MY MI MY 4 MY 4 MY0 0 +0.069 0 +0.035 0 -0.092 0 -0.138 0 -0.107
‘74 +0.026 to.059 to.015 to.034 -0.018 -0.089 -0.026 -0.132 -0.021 -0.103‘h +0.045 +0.048 io.031 +0.031 -0.016 -0.082 -0.023 -0.115 -0.018 -0.093% +0.044 +0.029 +0034 +0.020 -0.012 -0.059 - 0 0 1 6 -0.078 -0.013 -0.065
‘VLLgFig. 5.
._r
12
Fig. 6.
Note that f coefficients in this tabulation are used
only for calculation of coefficients L-L-that are to be3
used for design at the intersection of the center andouter walls as shown in Fig. 5a.
Coefficients for the center wall are for one cell filled,the negative sign indicating tension on the loaded side.All signs must be reversed when the other cell is filled.
Shear coefficients in Tables 7 and 8 as well as in Figs.1 and 2 can be applied both to center and outer walls.
Two-Cell Tank, Short Center Wall
M, coefficient thatoccurs at the center wall of -0.138 instead of -0.092 atthe corner in the tank in Fig. 6. Maximum moment is M,, =-0.1 38wa3 = -0.138 x 256 = -35.3 ft kips.
Short outer walls
L = coefflcents from Table 5 forb/a = 2 50 and c/a = 1 25
Long outer walls
F = coel fromL = coeffxwXs from Table 5 for Table 2 for
b a = 2 50 and cia = 1 25 b/a = 2 50y = b/2 y = b / 4 y=o y = b/2
0 lo -0 138
I
-0092 I 0 +0035 I 0 10069 1 0‘1 -0016 - 0 069
I
+ o 015 +o 034 *O 026 to 059 -0 026 -0 1 3 2I -0 016 -0 062 +o 0 3 1 *o 0 3 1 ‘ 0 045 ‘ 0 046 -0 023 - 0 115
J/r -0 012 -0 059 +o 034 -0 020 ‘ 0 044 *o 029 - 0 0 1 6 -0 078
Center wall
F = cod fromL = coefflclents from Table 5 for Table 2 for
b/a = 2.50 and c a = 1 25 ba= 1 2 5L-F
z=o I = Cl4 z = Cl2 .? = Cl2 L-T
maMx Mz Mx Mz Mx Mz 4 MZ Mx Mz
0 0 -0.010 0 -0030 0 - 0 0 9 2 0 -0.034 0 - 0 0 7 3% - 0 0 0 2 - 0 0 0 3 -0.006 -0 024 - 0 016 -0 089 - 0 0 0 6 -0.042 -0.015 -0 073% +0 OQ6 10007 to.003 - 0 . 0 1 2 -0.016 -0 062 - 0 0 1 0 -0 049 -0.014 -0.071y. +0018 +ooo6 +0.011 -0002 - 0 0 1 2 - 0 0 5 9 -0.009 - 0 0 4 4 - 0 0 1 1 -0.054
I t I t I
Counterforted Tank Walls
In a tank or reservoir with large horizontal dimensions,say three or four times the height, and without a rein-forced concrete cover slab, it becomes necessary todesign walls as cantilevers or, when they are quite high,as counterforted walls. The slab in Fig. 8 is free at the topand may be considered fixed at the bottom. If counter-forts are spaced equidistantly, the slab may also betaken as fixed at counterforts. For this type of construc-tion, coefficients in Table 3 apply.
Fig. a.Consider for illustration a wall panel of a counterfort-
ed wall in which spacing of counterforts is b = 40 ft andheight is a = 20 ft. From Table 3, for b/a = 40/20 = 2,select the following coefficients.
r I ty = o y = b/4 I y = b/2
Mx MY 4 MV f"% MV
Fig. 7.
1 3
‘.
The tank in Fig. 7 consists of two cells with the sameinside dimensions as the cells in the two-cell tank withthe long center wall. The difference is that the centerwall is 40 ft wide in the previously discussed tank, but20 ft wide in this example.
Design procedure is identical for both two-cell tanks,but the schedule of coefficients is different because thelonger side of the cell in Fig. 7 is continuous instead ofthe shorter side as in Fig. 6.
Note from the following tabulation that the long wallmust be designed for a maximum
Procedure for using these coefficients to determinemoments and design of the wall is similar to that illus-trated for the open-top single-cell tank shown in Fig. 3.
Details at Bottom Edge
Note that all tables except one are based on the as-sumption that the bottom edge is hinged. It is believedthat this assumption in general is closer to the actualcondition than that of a fixed edge. Consider first thedetail in Fig. 9, which shows the wall supported on arelatively narrow continuous wall footing, and then Fig.10 in which the wall rests on a bottom slab.
Fig. 9.
In Fig. 9 the condition of restraint at the bottom of thefooting is somewhere between hinged and fixed butmuch closer to hinged than to fixed. Resultant of pres-sure on the subsoil lies well within the edge of the foot-ing, and the product of resultant and its eccentricity isusually much smaller than the moment at the bottom ofthe wall when it is assumed fixed. Furthermore, thefoot-ing must rotate about a horizontal axis in order to pro-duce eccentric loading on the subsoil and rotation itselfrepresents a relaxation of restraint.
When the wall footing is not capable of furnishingmuch restraint, it is not necessary to provide for hingeaction at the construction joint in Fig. 9. The dowels areclose to the surface, leaving the center of the joint freefor insertion of a shear key. Area of steel in the dowelsalong each face may be taken as not less thanO.O025bd,and extension of the dowels above the constructionjoint may be made not less than say 3 ft.
The base slab in Fig. 9 is placed on top of the wall foot-ing and the bearing surface is brushed with a heavy coatof asphalt to break the adhesion and reduce frictionbetween slab and footing. The vertical joint betweenslab and wall should be made watertight. A joint width of1 in. at the bottom and 1% in. at the top is consideredadequate. As indicated in Fig. 9, the bottom of the jointmay be filled with oakum, the middle with volcanic clayof a type that expands greatly when moistened, and theupper part sealed with mastic. Any leakage will makethe clay penetrate into fissures and expand, pluggingthe leak. Mortar mixed with iron powder has been usedextensively for joints such as in Fig. 9, and so has lead
joint filler, but both iron powder and lead are not alwaysreadily available. A waterstop may not be needed in theconstruction joints when the vertical joint in Fig. 9 ismade watertight.
In Fig. 10 a continuous concrete base slab is providedeither for transmitting the load coming down through thewall or for upward hydrostatic pressure. In either case,the slab deflects upward in the middle and tends to ro-tate the wall base in Fig. 10 in a counterclockwrse direc-tion. The wall therefore is not fixed at the bottom edge.It is difficult to predict the degree of restraint. The rota-tion may be great enough to make the bottom edgehinged or may be even greater. Under the circum-stances it is advisable to avoid placing moment rein-forcement across the joint and to cross the dowels atthe center. The waterstop must then be placed off cen-ter as indicated. Provision for transmitting shear throughdirect bearing can be made by inserting a key as in Fig. 9or by a shear ledge as in Fig. 10.
The waterstop in Fig. 10 may be galvanized steel,copper, preformed rubber, or extruded plastic.
At top of wall the detail in Fig. 10 may be applied ex-cept that the waterstop and the shear key are not essen-tial. The main thing is to prevent moments from beingtransmitted from the top of the slab into the wall becausethe wall is not designed for such moments.
Fig. 10.
14
Metric Conversion Factors
To convert from To Multiply byinch (in.) meter (m) 0.0254
feet (ft) meter (m) 0.3048
square feet (sq ft) square meter (m2) 0.0929
pound (lb) kilogram (kg) 0.4536
kip (1000 lb) kilogram (kg) 453.6
Ib/lin ft kg/m 1.488
kip/lin ft kg/m 1488.
Ib/sq ft kg/m2 4.88
Ib/cu ft kg/m3 16.02
ft-kips newton-meter (Nm) 1356.
ft-kips kilogram-meter (kgm) 138.2
The prefixes and symbols listed are commonly usedto form names and symbols of the decimal multiplesand submultiples of the SI units.
Multiplication Factor Prefix Symbol
1 000 000 000 = 109 giga G1 oooooo= 106 mega M
1000=10~ kilo kl=l -
0.001 = 10-3 milli m0.000 001 = 10-6 micro I-10.000 000 001 = 1 o-9 nano n
