4 Rectangular Concrete Tanks

CO~r@tC 1

INFORMATION

Rectangular Concrete Tanks While cylindrical shapes may be structurally best for tank construction, rectangular tanks frequently are preferred for specific purposes. Special processes or operations may make circular tanks inconvenient to use. When several separate cells are required, rectangular tanks can be arranged in less space than circular tanks of the same capacity. Tanks or vats needed inside a building are therefore often made in rectangular or square shapes. For these and other reasons, breweries, tanneries, and paper mills generally use rectangular tanks. Data presented here are for design of rectangular tanks where the walls are subject to hydrostatic pressure of zero at the top and maximum at the bottom. Some of the data can be used for design of counterforted retaining walls subject to earth pressure for which a hydrostatic type of loading may be substituted in the design calculations. Data also can be applied to design of circular reservoirs of large diameter where lateral stability depends on the action of counterforts built integrally with the wall. Another article on tank construction, Circular Concrete Tanks Withouf Prestressing, has been published by the Portland Cement Association.

Moment

Coefficients

Moment coefficients were calculated for individual panels considered fixed along vertical edges, and coefficients were subsequently adjusted to allow for a certain rotation about the vertical edges. First, three sets of edge conditions were investigated, in all of which vertical edges were assumed fixed while the other edges were as follows: 1. Top hinged-bottom hinged 2. Top free-bottom hinged 3. Top free-bottom fixed* Moment coefficients for these edge conditions are given in Tables 1, 2, and 3, respectively. In all tables, a denotes height and b width of the wall. In Tables 1, 2, and 3, coefficients are given for nine ratios of b/a, the limits being b/a = 3.0 and 0.5. The origin of the coordinate system is at midpoint of the top edge; the Y axis is horizontal; the X axis is vertical and its positive direction downward. The sign convention for bending moments is based on the coordinate fiber that is being stressed. For example, A$ stresses fibers parallel to the X axis, The sign convention used here is not compatible with two other conventions-namely, that (1) the subscript is the axis 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-except where they are known to be zero-at edges, quarter points, and midpoints both in X and Y directions. The slab was assumed to act as a thin plate, for which equations are available in textbooks such as Theory ot Plates and Shells by S. Timoshenko,” but since only a small portion of the necessary calculations for moment coefficients for specific cases is available in the engineering literature, they have been made especially for this text. Table 4 contains moment coefficients for uniform load on a rectangular plate considered hinged on all four sides. The table is for designing cover slabs and bottom slabs for rectangular tanks with one cell. If the cover slab is made continuous over intermediate supports, the design can follow procedures for the design of slabs supported on four sides. Coefficients for individual panels with fixed side edges apply without modification to continuous walls provided there is no rotation about vertical edges. In a square tank, therefore, moment coefficients can be taken directly from Tables 1, 2, or 3. In a rectangular tank, however, an adjustment must be made, as was done in Tables 5 and 6, similar to the modification of fixed-end moments in a frame analyzed by moment distribution. In this procedure the common-side edge of two adjacent panels is first considered artificially restrained so that no rotation can take place about the edge. Fixededge moments taken from Tables 1,2, or 3 are usually dissimilar in adjacent panels and the differences, which correspond to unbalanced moments, tend to rotate the edge. When the artificial restraint is removed these unbalanced moments will induce additional moments in the panels, Adding induced and fixed-end moments at the edge gives final end moments, which must be identical on both sides of the common edge. Moment distribution cannot be applied as simply to continuous tank walls as it can to framed structures, because moments must be distributed simultaneously along the entire length of the side edge so that moments become equal at both sides at any point of the edge. The problem was simplified and approximated to some extent by distributing moments at four points only: quarter points, midpoint, and top. The end moments in the two intersecting slabs were made identical at these four points and moments at interior points adjusted accordingly. ‘Applicable tn cases where wall slab, counterfort, and base slab are a l l built Integrally “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

Xl

_wa

bla -

y - o xla

y = b/4

MX

MV

3.00

+0.035 +0.057 +0.051

+0.010 +0.016 +0.013

2.50

to031 +0.052 +0047

2.00

y = b/2 4

Y

MV

+0026 +0.044 +0.041

+0.011 +0.017 +0.014

-0.008 -0.013 -0.011

-0.039 -0.063 -0055

+0.011 +0.017 +0.015

+0.021 +0.036 +0.036

+0010 +0.017 +0.014

-0.008 -0.012 -0.011

-0.038 -0.062 -0.055

+0.025 +0.042 +0.041

+0.013 +0.020 +0.016

+0.015 +0.028 +0.029

+0.009 +0.015 +0.013

-0.007 -0.012 -0.011

-0.037 -0.059 -0.053

1.75

to.020 +0.036 +0.036

+0013 +0.020 +0.017

+0.012 +0.023 +0025

+0.008 +0.013 +0.012

-0.007 -0.011 -0.010

-0.035 -0.057 -0051

1.50

+0.015 +0.028 +0.030

+0.013 +0.021 +0.017

+0.008 +0.016 +0.020

+0007 +0.011 +0.011

-0.006 -0.010 -0.010

-0.032 -0.052 -0.048

1.25

+o 009 +0.019 +0.023

+0.012 +0.019 +0.017

+0.005 +0.011 +0.014

+0.005 +o 009 +0.009

-0006 -0 009 -0.009

-0.028 -0.045 -0.043

1.00

+0.005 +0011 +0.016

+0.009 +0.016 +0014

+0.002 +0006 +0.009

+0.003 +0.006 +0.007

-0.004 -0.007 -0.007

-0020 -0.035 -0.035

0.75

+0.001 to.005 +0.009

+0.006 +0.011 +0.011

0 +0.002 +0.005

+0.002 +0.003 to.005

-0.002 -0.004 -0.005

-0.012 -0022 -0.025

0.50

0 +0.001 +0.004

+0.003 + 0.005 +0.007

0 +0.001 +0.002

+0.001 +0.001 +0.002

-0.001 -0.002 -0.003

-0.005 -0.010 -0.014

Minus s,gn lndlcates

vx

bla -

y - o 4 0

y = b/4 MY

Y 0

MY

to.028 kO.049 bO.046

to.070 +0.061 +0.049 +0.030

+0.015 +0032 +0.034

+0.027 +0.028 +0026 +0.018

2.50

0 bO.024 bO.042 bO.041

+0.061 +0053 +0.044 10027

0 +0.010 +0.025 +0.030

2.00

0 bO.016 kO.033 bO.035

+0.045 +0042 +0.036 r0.024

1.75

0 IO.013 bO.028 10.031

150

!

y = b/2 MX

4

-0.034 -0.027 -0.017

-0.196 -0.170 -0.137 -0.087

+0.019 +0.022 +0.022 +0.016

0 -0.026 -0023 -0.018

-0.138 -0132 -0.115 -0.078

0 +0.006 +0.020 +0.025

+0.011 +0.014 +0.016 +0.014

0 -0.019 -0.018 -0.013

-0.091 -0.094 -0.089 -0.065

+0.036 +0.035 +0.032 +0.022

0 +0005 10017 r0.021

+0008 +0.011 co.014 10012

0 -0015 -0015 -0.012

-0.071 -0.076 -0076 -0.059

0 moo9 bO.022 10.027

+0027 +0.028 +0.027 +0020

0 +0.003 +0012 +0.017

+0.005 +0008 +0.011 +0.011

0 -0012 -0013 -0.010

-0052 -0.059 -0.063 -0.052

1.25

0 10.005 bO.017 10.021

co.017 +0020 +0023 +0.017

0 +0.002 +0.009 10013

+0.003 +0.005 +0.009 +0.009

0 -0008 -0010 -0.009

-0034 -0.042 -0.049 -0.044

1.00

0 to.002 '0 010 too15

+O.OlO +0013 +0.017 +0.015

0 +o 000 +0005 +o 009

+0.002 io.003 +0006 +0.007

0 -0005 -0.007 -0007

-0.019 -0.025 -0.036 -0036

075

0 ~0.001 to.005 ~0.010

+0005 +0008 +0011 +0.012

0 +o.ooo +0002 +0.006

+0001 +0.002 +0004 +0.004

0 -0.003 -0004 -0.005

-0.008 -0013 -0022 -0026

0.50

0 ~0.000 bO.002 a007

+0.002 +0004 +0.006 +0008

0 +o.ooo +0001 +0.002

0 +0001 +0.002 +0.002

0 -0.001 -0002 -0.003

-0.003 -0.005 -0.010 -0014

3.00

ienslon on the loaded side I” all tables

0

~ / ,"

:

Table 4

Table 3

Moment = Coef. x

wa3

ID;;

,w” 4 xla

q

Coef. x

bvaz

Xl

i~hngea 1 - y ‘_I I

M

y = b/4

y=o bla

Moment

X

m

1P”@

y=o

y = b/2

y = b/4

bla

MX

MY

Mx

4

M*

MY

0 +0.010 +0.005 -0.033 -0.126

+0025 +0.019 +0.010 -0.004 -0.025

0 +0.007 +0.008 -0.018 -0.092

to.014 +0.013 +0.010 -0 000 -0.018

0 -0014 -0.011 -0.006 0

-0.082 -0.071 -0055 -0.028 0

0 +0.012 +0.011 -0.021 -0.108

+0.027 +0.022 +0.014 -0.001 -0.022

0 +0.007 +0.008 -0.010 -0.077

+0.013 +0.013 +0010 +0.001 -0.015

0 -0.013 -0.011 -0.005 0

-0.074 -0.066 -0.053 -0.027 0

0 +0.013 +0.015 -0.008 -0.086

+0.027 +0.023 +0.016 +0.003 -0.017

0 +0.006 +0.010 -0.002 -0.059

+0.009 +0.010 +0.010 +0.003 -0.012

0 -0.012 -0.010 -0.005 0

-0.060 -0.059 -0 049 -0.027 0

0 +0.012 +0.016 -0.002 -0.074

+0.025 +0.022 +0016 +0.005 -0.015

0 +0.005 +0.010 +0.001 -0.050

+0.007 +0.008 +0.009 +0.004 -0.010

0 -0010 -0 009 -0.005 0

-0.050 -0.052 -0.046 -0.027 0

0 to.008 to.016 to.003 -0.060

+0.021 +0.020 +0.016 +0006 -0.012

0 +0.004 +0.010 +0.003 -0.041

+0.005 +0.007 +0.008 +0.004 -0.008

0 -0.009 -0.008 -0005 0

-0.040 -0.044 -0042 -0.026 0

1.25

0 to.005 +0.014 +0.006 -0.047

+0.015 +0.015 +0.015 +0.007 -0.009

0 +0.002 +0.008 +0.005 -0031

+0.003 +0.005 +0.007 +0.005 -0.006

0 -0.007 -0.007 -0.005 0

-0.029 -0.034 -0.037 -0024 0

1.00

0 to.002 +0.009 +0.008 -0.035

+0.009 +0.011 +0.013 +0.008 -0.007

0 0 +0.005 +0.005 -0.022

+0.002 +0003 to.005 +0.004 -0.005

0 -0.005 -0.006 -0.004 0

-0018 -0.023 -0.029 -0.020 0

0.75

0 +0.001 +0.005 +0.007 -0024

+0.004 +0.008 +0.010 +0.007 -0.005

0 0 +0.002 +0.003 -0.015

+0.001 +0.002 +0.003 +0.003 -0.003

0 -0.002 -0.003 -0.003 0

-0007 -0.011 -0.017 -0.013 0

0.50

0 0 +0.002 +0.004 -0.015

+0.001 +0.005 +0.006 +0.006 -0.003

0 0 +0.001 +0.001 -0.008

0 +0.001 +0.001 +0.001 -0.002

0 -0.001 -0.002 -0.001 0

-0002 -0.004 -0.009 -0.007 0

Mx

MY

MX

3.00

IO.089 to.118

to022 +o 029

+0077 +0101

to.025 to.034

2.50

bO.085 to112

+0.024 +0.032

+0.070 +o 092

+0.027 +0.037

2.00

0076 +0.100

+0.027 +0.037

+0.061 +0.078

+0.028 +0038

1.75

+0.070 +0091

+0.029 +0040

+0.054 +0.070

+o 029 +0.039

1.50

to.061 t0.076

+0.031 +0.043

+0047 +0.059

to.029 +0.040

125

to 049 +0063

+0.033 +0.044

+0038 +0.047

+0.029 +0.039

1.00

to.036 10.044

+0.033 +0044

+0.027 +0033

+0027 +0036

0.75

to.022 to.025

+0.029 +0.038

+0.016 +0.018

+0023 +0.030

to.020 +0025

+0007 +0.007

bO.015 +0.019

0.50

.O.OlO -0 009

M”

3

I

Table 5. Moment Coefficients for Tanks with Walls Free at Top and Hinged at Bottom b/a = 3.0 y = o c/a Mx

M”

M,

4

4

4

4

MZ

3.00

0 +0.028 +0.049 +0.046

+0.070 +0.061 +0.049 +0.030

0 +0.015 +0.032 +0.034

+0.027 +0.028 +0026 +0.018

0 -0.034 -0.027 -0.017

-0196 -0170 -0137 -0087

0 +0.015 +0.032 +0.034

+0027 +0026 +0.026 +0.018

2.50

0 +0.028 +0.049 +0.046

+0.073 +0.063 +0.050 +0.030

0 +0.016 +0.033 +0.037

+0033 +0033 +o 029 +0.020

0 -0030 -0.025 -0.017

-0.169 -0.151 -0.126 -0.084

0 +0.009 +0.023 +0.029

0 +0.029 +0.050 +0.046

+0.075 +0.065 f0.051 +0.031

0 +0.017 +0.035 +0.037

+o 039 +0.036 +0.032 +0.021

0 -0.027 -0.023 -0016

-0.146 -0133 -0.113 -0078

0 +0.029 +0.050 +0.046

+0.076 +0065 +0.052 +0.031

0 +0.018 +0036 +0037

+0041 +0038 +0033 +0.021

0 -0025 -0.021 -0.015

0 +0.030 +0050 +0.046

+0.077 +0.066 +0.053 +0031

0 +0.018 +0.037 +0.038

+0.043 +0.039 +0.034 +0.022

0 +0.030 +0.050 +0.047

+0.078 +0.067 +0.054 +0.032

0 +0.019 +0.038 +0.038

0 +0.030 +0.051 +0.047

+0.079 +0067 +0054 +0032

0 +0.029 +0.051 +0.047 0 +0.029 +0.050 +0.046

-

R

Moment

q

z = cl4

y =b/2

y = b/4

Coef. x wa3

1.25

$

I=0 M,

M,

0 +0028 +o 049 +0046

+0.070 +0061 to.049 +0030

to013 +0.014 +0.017 +0014

0 +0022 +0041 +0040

+0057 +0050 +0043 +0027

0 +0.002 +o.o1f3 +0.022

-0.005 -0.002 +0.005 +0008

0 +0.013 10030 +0034

+0031 +0.032 +o 029 +0020

-0.137 -0.125 -0.106 -0074

0 -0.003 +0011 +0.018

-0.018 -0.012 -0.003 +0.004

0 +0.007 +0.023 +0.027

+0014 +0.018 +0.020 +0015

0 -0.024 -0.020 -0.014

-0.129 -0.118 -0.100 -0.070

0 -0.007 +o 005 +0013

-0033 -0.024 -0.012 0

0 +0.002 +0.015 +0021

-0.006 +0.004 +0.010 +0.010

+0.045 +0.041 +0.035 +0.023

0 -0022 -0.019 -0.014

-0.122 -0111 -0 095 -0.068

0 -0.011 0 +0.008

-0.052 -0 039 -0.022 -0006

0 -0.004 +0.008 +0.016

-0.031 -0.018 -0.005 +0.001

0 +0.020 +0.038 +0.038

+0.047 +0.043 +0.036 +0.023

0 -0.021 -0.018 -0.013

-0118 -0105 -0.090 -0.065

0 -0.015 -0005 to003

-0074 -0.056 -0.034 -0.014

0 -0.010 +0001 +0.009

-0.060 -0.042 -0.022 -0.009

+o 079 +0.066 +0.053 +0.031

0 +0.020 +0.037 +0.037

+0.047 +0.042 +0.036 +0.022

0 -0.021 -0.018 -0.013

-0.120 -0107 -0 090 -0066

0 -0.020 -0.011 -0.002

-0 098 -0 079 -0.051 -0.025

0 -0.016 -0.006 +0.003

-0.092 -0.070 -0045 -0024

+0.078 +0.065 +0.053 +0.031

0 +0.019 +0.035 +0.036

+0.047 +0.042 +0.035 +0.021

0 -0.023 -0.019 -0.014

-0.130 -0.115 -0.095 -0.068

0 -0.024 -0.016 -0.007

-0.126 -0105 -0.073 -0.040

0 -0.022 -0.013 -0.004

-0123 -0101 -0071 -0042

b/a = 2.5

2.50

1.25

4

y = b/2

y = b/4 c/a

z=o

z = cl4

MX

MY

4

MY

4

M*

M”

MZ

0 +0.024 +0042 +0.041

+0.0.!31 +0.053 +0.044 to.027

0 +0010 +0025 +0.030

co.019 +0.022 +0.022 +0016

0 -0.026 -0023 -0.016

-3138 -0132 -0.115 -0.078

0 +0010 +0025 +0.030

+0.019 +0.022 +0.022 +0.016

0 +0.024 +0042 +0.041

+0.061 +0053 +0.044 +0027

0 +0.025 +0.043 +0042

+0065 +0.055 +0.046 +0.028

0 +0.012 iO.028 +0.031

+0.026 to.027 +0.025 +0.018

0 -0.023 -0.020 -0014

-0.118 -0.113 -0.102 -0.070

0 +0005 +0.018 +0023

+0.003 +0.006 +0.011 +0.011

0 +0.015 +0.032 +0034

+0.038 +0.037 +0.033 +0.022

0 +0.025 +0.044 +0043

.-.,7 +0.057 +0.047 +0.028

0 +0.013 +0.029 +0.033

+0.030 +0.030 +0.027 +0.019

0 -0.021 -0.019 -0013

-0.108 -0.104 -0.096 -0.066

0 +0.001 +0.013 +0019

-0.006 -0002 +0004 +0.008

0 +0010 to.025 +0.029

+0025 +0026 +0025 +0.019

0 +0.026 +0.045 +0043

to.068 +0.058 +0047 +0.029

0 +0.014 +0.030 +0.034

+0033 +0032 +0.029 +0.019

0 -0.019 -0.018 -0.013

-0100 -0.097 -0.089 -0.063

0 -0.003 +0.008 too15

-0018 -0.012 -0.002 +0004

0 +0004 +0017 +0024

+0.008 +0013 +0.017 +0015

0 +0.026 +0.045 +0.044

+0 069 +0.059 +0.048 +0.029

0 +0.015 +0.031 +0.034

+0035 +0034 +0031 +0.020

0 -0.018 -0.016 -0012

-0.092 -0.089 -0.082 -0 059

0 -0.006 +0.003 +0011

-0.030 -0.024 -0012 -0002

0 -0.002 +0.008 +0.018

-0010 -0.003 +0.007 +0008

0 +0.026 +0.046 +0.044

+0.070 +0.060 +0.048 +0.029

0 +0.015 +0.031 +0.033

+0037 +0036 +0.032 +0021

0 -0.017 -0.015 -0.011

-0087 -0.083 -0077 -0056

0 -0010 -0.003 +0.006

-0.045 -0.036 -0.021 -0.008

0 -0.008 -0.001 +0011

-0032 -0021 -0008 0

0 +0.025 +0.045 +0043

+0.070 +0.060 +0.047 +0.029

0 to.015 +0.030 +0.033

+0.038 +0.037 +0.032 +0.020

0 -0.016 -0.014 -0.011

-0082 -0.078 -0071 -0.054

0 -0014 -0.008 +0.002

-0062 -0053 -0.035 -0016

0 -0.014 -0.009 +o 005

-0.055 -0.042 -0.025 -0.011

0 +0.025 +0.044 +0.042

+0.069 +0.059 +0.046 +0.028

0 +0.014 +0028 +0.032

+0.039 +0.038 +0.032 +0.019

0 -0.015 -0.014 -0.010

-0080 -0075 -0.068 -0.052

0 -0.019 -0.014 -0.003

-0.081 -0.072 -0.056 -0030

0 -0019 -0017 -0002

-0.080 -0.068 -0.048 -0.026

\

t y = o

ca

2.0

=

y = b/2

y = b/4

M,

MY

2.00

0 +0016 +0033 +0.036

+0.045 +0.042 +0.036 +0.024

1.75

0 +0.017 +0.034 +0036

1.50

M, 0

M,

M, 0

z=o

z = cl4

MY

4

4

M,

0 +0.006 +0020 +0025

to011 +0.014 +0016 to014

+0.016 +0.033 +0.036

co.045 +0.042 +0.036 +0024

MZ

0

+0.006 +o.ozo +0.025

+0.011 +0.014 +0.016 +0.014

-0.019 -0.018 -0.013

-0091 -0.094 -0.089 -0.065

+0048 +0.044 +0.038 +0.024

0 +0.007 +0.021 +0.025

+0.015 +0.017 +0019 +0.015

0 -0.017 -0.017 -0.012

-0.081 -0.085 -0.083 -0061

0 +0.003 +0.015 +0.020

-0001 +0006 +0011 +0012

0 +0.012 +0.027 10.031

~0032 co.032 +0.029 +0021

0 +0018 +0.035 +0.036

+0.050 +0.046 +0.039 +0.025

0 +o.ooa +0.022 +0.026

+0019 +0.021 +0.021 +0.016

0 -0015 -0.015 -0.012

-0072 -0077 -0.076 -0.058

0 0 +o 009 +0.016

-0.010 -0002 +0004 +0.008

0 to.007 +0.020 +0.025

+0.018 '0020 co.022 10017

1.25

0 +0019 +0036 +0037

+0.052 +0.048 +0.041 +0.025

0 a009 +0.023 +0.026

+0023 +0.024 +0.023 +0.017

0 -0.014 -0.014 -0.011

-0064 -0.068 -0.069 -0054

0 -0.002 +0.005 +0.011

-0021 -0.013 -0004 +0.002

0 +0.001 +0011 +0016

0 '0005 +0012 +0.011

100

0 +0.019 +0.037 to.037

+0054 +0.050 +0.042 +0.026

0 +0.010 +0.024 +0.027

+0.027 +0.027 +0.025 +0.018

0 -0.012 -0.013 -0.010

-0.058 -0.062 -0.064 -0051

0 -0.005 0 +0006

-0037 -0025 -0.015 -0006

0 -0.005 +0.001 +0.008

-0.023 -0.013 0 to.004

075

0 +0.018 +0.038 +0.037

+0.055 +0.051 +0.043 +0.026

0 +0.011 +0.025 +0.027

+0.030 +0.029 +0026 +0.018

0 -0.012 -0.012 -0.010

-0.058 -0.062 -0.062 -0.049

0 -0.009 -0.005 +o.ooz

-0 049 -0040 -0 029 -0.015

0 -0010 -0.007 +0.001

-0.044 -0031 -0.015 -0.004

0.50

0 +0.018 +0.038 +0037

+0054 +0.052 +0.044 +0.026

0 +0.011 +0.025 +0.026

+0.030 +0.029 +0.025 +0.017

0 -0.014 -0.013 -0.010

-0.065 -0068 -0064 -0.050

0 -0012 -0010 -0.003

-0.064 -0056 -0.045 -0026

0 -0014 -0.012 -0.004

-0061 -0051 -0034 -0.018

I

”

w/a

1.5

I=

y = b/4

y-o

y = b/2

z=o

* = c/4

MY

M"

4

MX

MZ

-0.012 -0.013 -0.010

-0.052 -0.059 -0.063 -0.052

0 +0003 +0.012 +0.017

+0005 +0.008 +0.011 +0.011

0 +0.009 +0.022 +0.027

co.027 +0028 to.027 to.020

+0.008 +0.012 +0.014 +0.012

0 -0010 -0.011 -0.010

-0.045 -0.050 -0.056 -0.048

0 +0.001 +0007 60.013

-0005 -0.001 +0006 +0006

0 +0.004 +0.014 +0.018

coo11 +0015 +0.020 10016

0 +0.006 +0.015 to.019

+0013 +0.016 +0.017 +0.014

0 -0008 -0.010 -0.009

-0.038 -0.042 -0.049 -0.045

0 -0.002 +0.002 +0.008

-0.016 -0010 -0.003 +0.002

0 -0001 +0006 +o 009

-0.006 +0.001 +0010 +0.010

+0.038 +0.036 +0.033 +0.022

0 +0.007 +0.016 +0.019

+0.016 +0.018 +0.019 +0.015

0 -0.008 -0.008 -0.008

-0.034 -0.038 -0.042 -0.041

0 -0.005 -0.002 +0.003

-0024 -0.020 -0.014 -0.007

0 -0.004 -0.001 +0.002

-0.019 -0013 -0.004 +0001

+0.040 +0.037 +0.034 +0.022

0 +0.007 +0.017 +0.018

+0.017 +0.019 +0.020 +0.016

0 -0.008 -0.009 -0.008

-0.036 -0040 -0.044 -0.040

0 -0.008 -0.006 -0.002

-0030 -0031 -0.027 -0018

0 -0.007 -0.006 -0.004

-0.028 -0.027 -0020 -0010

M.

M ”I

1.50

0 +0.009 +0.022 +0.027

125

4

4

M

+0.027 +0.028 +0.027 +0.020

+0.003 +0.012 +0.017

+0.005 +0.008 +0.011 +0011

0 +0.010 +0.024 +0.027

+0.031 +0.031 +0.030 +0.021

0 +0.005 +0.014 +0.018

0 +0.011 +0.025 +0.028

+0.035 +0.034 +0.032 +0.022

0 +0.011 +0.025 +0028 0 +0.010 +0.024 +0.028

0

Y

0

b/a = 1.0 c/a

Y = b/4

y-o

x/a

M”

0 +0.002 +0.010 +0015

+0.010 +0.013 +0.017 +0.015

0 +0.003 +0.011 +0.016 0 +0.003 +0.012 +0.017

0

Y = b/2 Mb

0 +0.005 +0.009

+0.002 +0.003 +0006 +0.007

+0.016 +0.017 +0.020 +0.014

0 +0.001 +0.006 +0.009

+0.020 +0.018 +0.021 +0.013

0 +0.001 +0.008 +0.010

M”

0

z=o

z = c/4 M”

-0.005 -0.007 -0.007

-0.019 -0.025 -0036 -0036

+o.oc)7 +0.008 +0.009 +0.009

0 -0.004 -0.007 -0.006

+0.011 +0.010 +0.010 +0.009

0 -0.004 -0.006 -0.006

M”

0

M,

MX 0

MZ

0 +0.005 +0.009

+0.002 +0.003 +0006 +0.007

+0002 +0.010 +0.015

+0.010 f0.013 +0017 +0.015

-0.013 -0.020 -0.033 -0.032

0 -0.001 +0.002 +0.004

-0.004 -0.005 -0.001 +0.002

0 -0.001 +0.005 +0.009

+0.003 +0.003 +0.007 fO008

-0.011 -0.018 -0.032 -0.031

0 -0002 +0.001 +0002

-0.007 -0.012 -0.009 -0.005

0 -0.003 +0.002 +0.006

-0.005 -0.007 -0.005 +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

y=o

w/a

y = b/4

.?=O

z = c/4

y = b/2

MX

MY

Mx

MY

-0 008

-0 039

+0.011 +0.017 +0.014

f

MX

MI

+0035 +0057 +0051

+0.010 +0016 +0.013

3.00

+0.035 +0.057 +0.051

+0.010 +0.016 +0.013

+0.026 +0.044 +0.041

+0011 +0.017 +0014

-0.013 -0011

-0.063 -0.055

+0026 10044 +0041

2.50

+0.035 +0057 +0.051

+0.010 +0.016 +0.013

+0.026 +0.044 +0.041

+0.011 +0.017 +0.014

-0.008 -0.012 -0.011

-0.039 -0.062 -0055

+0.021 +0036 +0036

+0010 +0.017 to014

+0031 +0.052 +0047

+0011 +0017 +0.014

2.00

+0.035 +0.057 to.051

+0.010 +0.016 +0013

+0.026 +0045 +0.042

+0.011 +0.017 +0.014

-0.008 -0.012 -0011

-0.038 -0.062 -0.054

+0.015 +0.028 +0.029

to010 +0015 +0.013

+0025 +0043 +0041

to.013 to020 +0016

1.75

+0.035 +0.057 +0051

+0010 +0.015 +0013

+0027 +0.045 +0.042

+0.011 +0017 +0.014

-0.007 -0.012 -0.011

-0037 -0.060 -0053

+0011 +0021 +0024

+0008 +0.013 '0012

+0.020 +0036 +0.036

+0013 +0020 +0016

1.50

+0.035 +0.057 +0.051

+0.010 +0.015 +0.013

+0.027 +0.045 +0.042

+0.011 +0.017 +0.014

-0.007 -0.011 -0.010

-0.035 -0.057 -0.051

+0007 to.015 +0019

to.006 +0.010 +0011

to.014 +0027 to 029

+0013 +0020 to.017

1.25

+0.035 +0.057 +0.051

+0010 +0.015 +0.013

+0027 +0.046 +0.042

+0.011 +0.017 +0.014

-0.006 -0.011 -0.010

-0.032 -0.053 -0.048

+0.003 +0.008 +0013

+0.003 to.006 +0008

+0008 10017 +0.021

+0011 a017 +0.016

1.00

+0.035 +0.057 +0.051

+0.010 +0.015 +0013

+0.027 +0.046 +0.043

+0.011 +0.017 +0.014

-0.006 -0.010 -0.009

-0 029 -0.048 -0.044

-0001 to.002 +0.007

0 +0002 +0.004

to002 +0.007 +0.013

+0.008 +oc14 +0.013

0.75

+0.035 +0.057 +0.052

+0.010 +0.015 +0.013

+o.o2a +0.046 +0043

+0.011 +0.017 +0.014

-0.005 -0.008 -0.008

-0.025 -0.042 -0.039

-0.003 -0.003 +0.002

-0005 -0005 -0002

-0002 -0.001 +0.006

+0.001 +0.007 +0.007

0.50

+0036 +0.057 +0.052

+0.010 +0.015 +0.013

+0.028 +0.047 +0.043

+0.011 +0.017 +0.014

-0.004 -0007 -0.007

-0.021 -0035 -0.033

-0004 -0007 -0.004

-0.011 -0.016 -0.010

-0.005 -0.006 -0001

-0.008 -0010 -0.004

b/a = 2.5 y = b/2

c/a

6

z=o

z = c/4

4

M”

M,

MZ

4

MZ

2.50

+0.031 +0.052 +0.047

+0.011 +0.017 +0.015

+0.021 +0.036 +0.036

+0.010 to.017 +0.014

-0.008 -0.012 -0.011

-0.038 -0.062 -0.055

+0.021 +0036 co.036

+0010 +0017 +0.014

+0031 +0.052 +0.047

+0.011 +0.017 +0.015

2.00

+0.031 +0.052 +0.047

+0.011 +0.017 +0.015

+0.021 +0.036 +0.036

+0.010 to.017 +0.014

-0.008 -0.012 -0.011

-0.038 -0.061 -0.054

+0.015 +0.028 +0.029

+o 009 +0015 +0.013

+0025 +0042 +0.041

+0.012 +0.020 +0.016

1.75

+0.032 +0.052 +0.047

+0.011 +0.018 +0.015

+0.021 +0.036 +0.036

+0.010 +0.017 +0.014

-0.007 -0.012 -0.011

-0.037 -0.059 -0.053

+0011 +0.022 +0.024

+0.008 +0.013 +0.012

+0.020 +0035 +0.035

+0.012 +0021 +0.017

1.50

+0.032 to.052 +0.047

+0.011 +0.018 +0.015

+0.022 +0.037 +0036

+0.010 +0.017 +0.014

-0.007 -0.011 -0.010

-0.035 -0057 -0.051

+0.007 +0.015 +0019

+0.006 +0.010 +0.010

'0.014 +0.027 +0.029

+0013 +0.021 +0.017

1.25

+0.032 +0.052 +0.048

+0.011 +0.018 +0.015

+0.022 +0.038 +0.037

+0010 +0.017 +0.014

-0.006 -0.011 -0.010

-0.032 -0.053 -0.048

+0003 +0.008 +0.014

+0.004 +0.007 +0.008

+0.007 co.018 +0022

+0.012 +0019 to.018

100

+0.032 +0.053 +0.048

+0.011 +0.018 +0.015

+0.023 +0038 +0.038

+0.011 +0.017 +0.015

-0.006 -0.010 -0.009

-0028 -0048 -0044

-0.001 +0.002 to.007

0 +0002 +a004

+0.002 to007 +0013

+0.008 +0014 +0013

0.75

+0.033 +0.054 +0.049

+0.011 +0.018 +0.015

+0024 +0.039 +0.038

co.011 +0.017 +0.015

-0005 -0.008 -0.008

-0.024 -0041 -0.039

-0003 -0.003 0

-0.005 -0005 -0002

-0.002 0 10006

-0.002 +0005 +0006

0.50

+0.033 +0.054 +0.049

+0.012 +0.018 +0.015

+0.024 +0040 +o 039

+0.011 +0.017 to015

-0004 -0.007 -0.007

-0.021 -0035 -0034

-0004 -0007 -0.004

-0.011 -0016 -0.010

-0.005 -0.006 -0.001

-0.008 -0010 -0004

b/a = 2.0 c/a

y - o

l-

y = b/4

Mx

4

y = b/2

z=o

z = cl4

Mx

4

4

MY

4

4

M,

Mz

2.00

+0.025 +0.042 +0.040

+0.013 +0.020 +0.016

+0.015 +0.028 +0.029

+0.009 +0.015 +0.013

-0007 -0012 -0.011

-0.037 -0.059 -0.053

to015 +0028 +0.029

+0.009 +0.015 +0013

+0.025 +0.042 +0.040

+0.013 +0020 +0.016

1.75

+0.025 +0.042 +0.040

+0.013 +o.ozo +0.016

+0.015 +0.028 +0.029

+0.009 +0.015 +0.013

-0.007 -0.012 -0.010

-0.036 -0.058 -0052

+0.011 +0.022 +0.024

+0.008 +0.013 +0012

+0.020 +0.035 +0.035

+0.013 +0.021 +0.017

1.50

+0.025 +0.043 +0.041

+0.013 +0.020 +0.016

+0.016 +0.028 +0.029

+0.009 +0.015 +0.013

-0.007 -0.011 -0.010

-0034 -0.056 -0.050

+0.007 +0.015 +0.019

+0.006 +0.011 +0010

+0.014 +0027 +0.029

+0.013 +0.021 +0.017

125

+0.0?6 +0.043 +0.041

+0.013 +0.020 +0.016

+0.016 +0.029 +0.030

+0.010 +0.015 +0.013

-0.006 -0.010 -0.010

-0.032 -0052 -0.048

+0.003 +0.008 +0.013

+0003 +0007 +0008

+0.007 +0.018 +0.021

+0.011 +0.019 +0.016

1.00

+0.026 +0.044 +0.041

+0.013 +0.020 +0.016

+0.017 +0.030 +0.031

+0.010 +0.016 +0.014

-0.006 -0.009 -0.009

-0.028 -0.046 -0.044

-0.001 +0.002 +0.007

0 +0.002 +0.004

+0.002 +0.007 +0.013

+0.008 +0.014 +0.013

0.75

+0.027 +0.045 +0.042

+0.013 +0.020 +0.016

+0.018 +0031 +0.032

+0.010 +0.016 +0.014

-0.005 -0.008 -0.008

-0.024 -0.040 -0.041

-0.003 -0002 +0.002

-0.004 -0004 -0002

-0.001 0 to.005

+o.m2 +0.005 to.008

0.50

+0.027 +0.046 +0.042

+0.013 +0.020 +0.016

+0.019 +0.033 +0.032

+0.010 +0.017 +0.015

-0.004 -0.007 -0.007

-0.021 -0.034 -0.037

-0.004 -0.006 -0.003

-0010 -0015 -0010

-0.004 -0.006 -0.002

-0.007 -0.009 -0.003

-

-

b/a = 1.5 c/a

1

y - o

y = b/4

y = b/2

z = c/4

z=o

4

MY

4

MY

4

MY

4

MZ

4

4

150

+0015 +0.028 +0.030

+0013 +0.021 +0.017

+0.008 +0.016 +0.020

+0.007 +0.011 +0.011

-0.006 -0.010 -0.010

-0.032 -0.052 -0.048

+0008 +0.016 +0.020

+0.007 +0.011 +0.011

+0.015 +0.028 +0.030

+0.013 +0.021 +0.017

125

+0.016 +0.029 +0.030

+0.013 +0.021 +0.017

+0.009 +0.017 +0.020

+0.008 to.012 +0.012

-0.006 -0.010 -0.009

-0.029 -0.049 -0.045

+0.004 +0.009 +0.014

+0.004 +0008 +o 009

+0.009 +0.018 +0023

+0.012 +0.019 +0.016

+0.016 +0.030 to.031

+0.013 +0.021 to.017

+0.010 +0.019 +0.021

+0.009 +0.012 +0.013

-0005 -0.009 -0.008

-0.025 -0.043 -0.041

0 to.003 +0008

+0.001 +0.003 +0.005

+0.003 +0.008 +0.014

+0008 +0014 +0.014

+0.018 +0.032 +0.032

+0.014 +0.022 +0.018

+0.011 +0.021 +0.022

+0.010 +0.014 +0.014

-0.004 -0.007 -0.007

-0.021 -0036 -0.036

-0002 -0.002 +0.002

-0.003 -0.004 0

-0.001 +0.001 +0.006

to.002 +0.005 +0.008

+0.020 +0035 +0034

+0.016 +0.024 +0.020

+0.013 +0.023 +0.024

+0.012 +0.018 +0.016

-0.003 -0.006 -0.007

-0.017 -0.031 -0.033

-0.003 -0.006 -0003

-0 009 -0.014 -0.008

-0004 -0.005 -0.001

-0006 -0007 -0001

b/a = 1.0 c/a

y=o

y = b/4 4

z=o

I = c/4

y = b/2

4

M”

MY

M.

M”

M,

M.x

4

+0.005 +0.011 +0016

+0.009 +0.016 +0.015

+0.002 +0.006 +0.009

+0.003 +0.006 +0.007

-0.004 -0.007 -0.007

-0.020 -0.035 -0.035

+0.002 +0.006 +0.009

+0.003 +0.006 +0.007

+0.005 +0.011 +0.016

+0.009 +0.016 +0.015

+0.006 +0.013 +0.017

+0010 +0.017 +0016

+0.003 +0.008 +0.010

+0.004 +0.008 +0.008

-0003 -0.006 -0.006

-0.016 -0.029 -0.031

0 +0.001 +0.004

0 +0.001 +0.003

+0.001 +0.005 +0.008

+0.005 +o 009 +0010

to.007 +0.015 +0.018

+0.011 +0018 +0.016

+0.005 +0.010 +0.012

+0.006 +0.010 +0.010

-0.002 -0.004 -0.005

-0.010 -0.021 -0.026

-0.002 -0.003 -0.001

-0.005 -0.007 -0.004

-0.003 -0.003 0

-0.002 -0.002 +0.001

M”

7

triangle with the same area as the trapezoid representing the actual load distribution. The intensity of load is the same at middepth in both cases and when the wall is supported at both top and bottom edges, the discrepancy between triangle and trapezoid has relatively little effect at and near the supported edges

In this manner, moment coefficients were computed and are tabulated in Tables 5 and 6 for top and bottom edge conditions as shown for single-cell tanks with a large number of ratios of b/a and c/a, b being the larger and c the smaller of.the horizontal tank dimensions. Moments in vertical and horizontal directions equal the coefficients times wa3, in which w is the weight of the liquid. Note that the loading term is wa3 for all wall slabs subject to hydrostatic pressure but is wa2 for the floor slab in Table 4, which has uniformly distributed load. In the first case, w is weight per cubic foot, but in the latter it is weight per square foot. There is a peculiarity about the horizontal end moments in the slabs at the free top edge. Calculations of such moments by means of the trigonometric series used result in a value of zero, whereas these moments actually have finite values and may even be comparatively large. Horizontal end moments at the free edge were therefore established by extrapolation. The consistency of extrapolated moment coefficients was checked by plotting and studying curves. This gave reasonably good results, although coefficients thus determined are probably not quite as accurate as the coefficients that were computed. A condition prevails at the quarter point of the free edge, similar to that at the end point but to a lesser degree. At the midpoint of the free edge the coefficients were computed, extrapolation being used only for checking purposes. When a tank is built underground, the walls must be investigated for both internal and external pressure. The latter may be due to earth pressure or to a combination of earth and groundwater pressure. Tables and other data presented can be applied in ‘the case of pressure from either side but the signs are opposite. In the case of external pressure, actual load distribution may not necessarily be triangular as assumed in the tables. Consider for illustration a tank built below ground with earth covering the roof slab and causing a trapezoidal distribution of lateral earth pressure on the walls. In this case it gives a fairly good approximation to substitute a

Shear Coefficients Shear values along the edges of a tank wall are needed for investigation of shear and development stresses. Along vertical edges, shear in one wall is also used as axial tension in the adjacent wall and must be combined with bending moment to determine tensile reinforcement. Various data for shear were computed and are given in Table 7. The wall is considered fixed at the two vertical edges while top and bottom edges are assumed to be hinged. The wall panel with width b and height a is subject to hydrostatic pressure due to a liquid weighing w lb per cubic foot. The first five lines in Table 7 are shears per linear foot in terms of wa*. The remaining four lines are total shears in kips or pounds depending on how w is given. Shears per linear foot are for ratios of b/a = %, 1,2, and infinity. The difference between the shear for b/a = 2 and infinity is so small that there is no necessity for computing coefficients for intermediate values. When b/a is large, a vertical strip of the slab near midpoint of the b dimension will behave essentially as a simply supported one-way slab. Total pressure on a strip 1 ft wide is 0.50waz, of which two-thirds or 0.33wa2 is the reaction at the bottom support and one-third or 0.17wa2 is the reaction at the top. Note in Table 7 that shear at midpoint of the bottom edge is 0.3290waz for b/a = 2.0, the coefficient being very close to that of onethird for infinity. In other words, maximum bottom shear is practically constant for all values of b/a greater than

Table 7. Shear at Edges of Slabs Hinged at Top and Bottom

‘h

bla

1

2

5

10

lnfmtty

Midpoint of bottom edge Corner at bottom edge

+o 1407wa’ - 0 2575wa”

+o 2419we -0 4397wa’

+o 3290w.a’ - 0 5633w.F

+o 3333waz -0 6000wa’

M,dpo,“t of flxed side edge Lower third-pant of side edge Lower quarter-pant of side edge

+o 1260wa’ *o 173&v@ ‘0 1919wP

+O 2562wa’ +o 3113wa’ +o 3153w.e

+0.3604waz ‘0 4023wa’ ‘0 3904w.3’

‘0 3912w.a’ +0 4116wa’ ‘0 39t30wa.

Total at top edge Total at bottom edge Total at one foxed side edge Total at all four edges ‘Negatwe s,gn lndlcates tEsbmated

8

reaction

0 0 0 0

OOOOwa’b 0460wa-‘b 2260wazb 5000wa’b

acts I ” darectlon

0 0 0 0

0052wa’b 0960wa’b 1994wa’b 5000waJb

of load

0 0 0 0

0536w.+b 1616wa’b 1322wa’b 5000wa’b

0 0 0 0

,203~~b 2715wa’b 0541wa.b 5000wa.b

0 1435wa’b 0 3023wa’b 0 0271 wa’b 0 5000wa’b

0 0 0 0

1667wa’b 3333wa.b 275wav 5000wa’b

\a J

As will be shown, this is correct only when the top edge is supported, not when it is free. At the corner, shear at the bottom edge is negative and numerically greater than shear at midpoint. The change from positive to negative shear occurs approximately at the outer tenth points of the bottom edge. These high negative values at the corners arise because deformations in the planes of the supporting slabs are neglected in the basic equations and are therefore of only theoretical significance. These shears can be disregarded in checking shear and development stresses. Unit shears at the fixed edge in Table 7 were used for plotting the curves in Fig. 1. There is practically no change in shear curves beyond b/a = 2.0. Maximum value occurs at a depth below the top somewhere between 0.6a and 0.8a. Fig. 1 is useful for determination of shear or axial tension for any ratio of b/a and at any point of a fixed side edge. Total shear from top to bottom of one fixed edge in Table 7 must equal the area within the corresponding curve in Fig. 1, and this relationship was used for checking the curves ‘Total shears computed and tabulated for a hanged top were also used in making certain adjustments to determine approximate values of shear for walls with free too-recorded in Table 8. For b/a = % in Table 7, total shear at the top edge is so small as to be practically zero, and for b/a = 1 .O total shear, 0.0052, is only 1% of total hydrostatic pressure, 0.5000.’ Therefore, it is reasonable to assume that removing the top support will not materially change total shears at any of the other three edges when b/a = Y2 and 1. At b/a = 2.0, there is a substantial shear at the top 2.

‘Loading

term is omitted here.

9" 0.3 g ij a4 f g 0.5 % a6 p B a7

1.0

0

0.1

0.2

0.3

Shear per lin. ft. = coef x wa2

Fig. 1.

Table 8. Shear at Edges of Slabs Free at Top and Hinged at Bottom*

tJa

Mldpolnt

1

2

3

‘ 0 141wa: -0 258wa-$

*o 242w.F -0 440-a.

‘ 0 3awa7 - 0 583~9

‘ 0 45wet

Top of flxed side edge Mldpomf of flxed side edge Lower third-wont of side edae

0 ooowa. +O 128wa‘

‘0 olowa’

*o 100wa

*O 258wa’ *0311wa-

.o 375wa: *o 406W%

-0 165wa-

*o 174ws

Lower quarter-WI”, of side edge Total at bottom edge Total at one faxed s,de edge Total al all four edges

‘0

*o 315w.T

*o 390wa

Corner

of bottom edge at

bottom

edge

192-a-

0 048wa b 0 226wa b

0 096wa.b 0 202wa.b

0 500w.s.

0 500wa-b

b

0 204wa.b 0 148wa:b 0 500wa. b

- 0 sowa ‘ 0 406WW *o 416w.F ,O 398wa

0 286wa.b 0 107wa.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 total shears on the other three sides is only 0.4462. If the top support is removed, the other three sides must carry a total of 0.5000. A reasonable adjustment is to multiply each of the three remaining total shears by 0.5000/ 0.4462 = 1 .12, an increase of 12%. This was done in preparing Table 8 for b/a = 2.0. A similar adjustment was made for b/a = 3.0, where the increase is 22%. Total shears recorded in Table 8 were used to determine unit shears also recorded in that table. Consider for illustration the shear curves in Fig. 1 and imagine the top is changed from hinged to free. As already stated, for b/a = % and 1 it makes little difference in total shearthe area within shear curves-whether the top is supported or not. Consequently, curves for b/a = % and 1 remain practically unchanged. They were transferred almost without modification to Fig. 2, which covers the case with top free. For b/a = 2 an adjustment was made. A change in the support at the top has little effect upon shear at the bottom of the fixed edge. Consequently, the curves in Figs. 1 and 2 are nearly identical at the bottom. Gradually, as the top is approached curves for the free top deviate more and more from those for the hinged top, as in Fig. 2. By trial, curve for b/a = 2 was so adjusted that its area equals the total shear for one fixed edge for b/a = 2.0 in Table 8. A similar adjustment was made for b/a = 3.0, which is the limit of moment coefficients given. One point of interest stands out In a comparison of Figs. 1 and 2. Whereas for b/a = 2.0 and 3.0 total shear is increased 12% and 22%, respectively, when top is free instead of hinged, maximum shear is increased but slightly, 2% at most. The reason is that most of the increase in shear is near the top where shears are relatively small. The same general procedure was applied, but not illustrated, for adjustment of unit shear at midpoint of bottom, but in this case the greatest change resulting from making the top free is at midpoint where shear is

9

Fig. 3.

0

0.1

@.2

0.3

04

Shear per lin ft. - coef Y wa2

Fig. 2. large for the hinged-top condition. For illustration, for b/a = 3.0, unit shear at midpoint of the bottom is0.33wa2 with hinged top but 0.45wa2 with free top-an increase of approximately one-third. Shear data were computed for wall panels with fixed vertical edges. They can be appli$d with satisfactory results to any ordinary tank wall even if vertical edges are not fully fixed.

Open-Top Single-Cell Tank The tank in Fig. 3 has a clear height of a = 16 ft. Horizontal inside dimensions are b = 40 ft and c = 20 ft. The tops of the walls are considered free and the bottom hinged. The tank contains water weighing 62.5 lb per cubic foot. Coefficients for moment and shear are selected from tables or diagrams for b/a = 40/16 = 2.50 and c/a = 20/l 6 = 1.25. Moments are in foot-kips if coefficients are multiplied by wa3/1000 = 62.5 x 163/1000 = 256; and shears are in kips if coefficients are multiplied by waz/lOOO = 62.5 x 16z/1000 = 16. Moment coefficients taken from Table 5 for b/a = 2.50 and c/a = 1 .25 are tabulated below. Coefficients for x = a (bottom edge), being equal to zero, are omitted.

0 'A H %

0 +0.026 +0.045 +0.044

+0069 GO59 +0046 to.029

0 +0015 +0.031 '0.034

+0035 +0034 +0.031 +0020

0 -0.016 -0.016 -0.012

-0.092 -0.089 -0.062 -0059

0 -0006 '0.003 +001,

-0.030 -0.024 -0.012 -0.002

-:002 +O.@X3 +o,o,i3

I:::;; +OOO, +0008

The largest moment occurs in the horizontal direction at the top of the corner common to both walls and equals -0.092wa3 = -0.092 x 256 = -23.6 ft kips. The negative sign simply indicates that tension is on the in-

10

side and need not be considered In subsequent calculations. Maximum horizontal moment at midpoint of the longer wall is +0.069wa3 = +0.069 x 256 = +17.7 ft kips. The positive sign shows that tension is in the outside of the wall. There is also some axial tenslon on this section that can be taken equal to end shear at the top of the shorter wall. For use in connection with Fig. 2, ratio of b/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 negligible in this case and the steel area can be determined as for simple bending. Horizontally at x = a/2, axial tension taken from Fig. 2 for b/a = 1.25 is equal to N = -0.30wa2 = -0.30 x 16 = -4.80 kips per linear foot, which is not negligible. Moment is M = 0.048wa3 = 0.048 x 256 = 12.3 ft kips. In the shorter wall, positive moments are all relatively small. Maximum positive moment is vertical: 0.01 8wa3 = 0.018 x 256 = 4.6 ft kips. Maximum Mx in the vertical strip at midpoint ot longer panel is 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 one only in that the tops of the walls are considered hinged rather than free. This condition exists when the tank is covered by a concrete slab with dowels extending from the wall into the slab without moment reinforcement across the bearing surface. Moment coefficients taken from Table 6 are given below. All coefficients for x = 0 (top edge) and x = a (bottom edge), being equal to zero, are omitted.

With a free top, maximum M, = +0.045wa3 and maximum My = -0.092wa3. With a hinged top, maximum n/l, = +0.052wa3 and maximum My = -0.053wa3. It is to be expected that a wall with hinged top will carry more load vertically and less horizontally, but it is worth noting that maximum coefficient for vertical moment is only 13%

A

,’

less for wall with free top than with hinged top. Another noteworthy point is that maximum M, coefficient at y = 0 is +0.069 for a free top but +0.018 for a hinged top. Adding top support causes considerable reduction 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 the section subject to this moment, and loads per linear foot can be taken as follows: 8-ft-high wall: 8 x 1 .08 x 0.150 = 1.3 kips 12-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 .O kips’ 6.8 kips It is conservative to check compressive stress forN = 6.8 kips and to design tensile steel for N = 1 .3 + 1.5 = 2.8 kips, in which fill and live load are disregarded.

t

Top and Base Slabs The closed single-cell tank is covered with a concrete slab. Assume the slab is simply supported along all four sides and has a live load of 100 psf and an earthfill weighing 300 psf. Estimating slab thickness as 12 in. gives a total design load of 100 + 300 + 150 = 550 psf. From Table 4, for a ratio of 40/20 = 2, select maximum coefficient of 0.100, which gives maximum M = 0.1 00wa2 F 0.100 x 0.550 x 20.02 = 22.0 ft kips. At the corners, a two-way slab tends to lift off the supports; and if this tendency is prevented by doweling slab to support, cracks may develop in the top of the slab across its corners. Nominal top reinforcement should therefore be supplied at the corners, say0.005bd sq in. per foot in each direction. Length of these bars can be taken as %a = l/4 x 20 = 5 ft. Assume the closed single-cell tank has a base slab of reinforced concrete. Weight of base slab and liquid does not create any bending or shearing stresses in concrete provided the subsoil is uniformly well compacted. Weight transferred to the base through the bottom of the wall is Top slab: 0.550 x 22 x 42 = 510 kips Walls: 16x0.162(2x41.1 +2~21,1)=320kips 830 kips If 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 base slab just as w = 550 psf was used for design of the top slab. 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 pressure on the base slab, the slab should also be investigated for this pressure when the tank is considered empty. \

Multicell Tank Multicell tanks do not lend themselves readily to mathematically accurate stress analysis It is possible, however, with the tables presented here for single-cell tanks and for individual wall panels with fixed vertical edges to estimate moment coefficients for symmetrical multicell tanks with sufficient accuracy for design purposes. While results obtained by the following procedure are approximate and should therefore be considered as a guide to engineering judgment, the procedure does give a conservative design. Because a rotation of one corner has comparatively little effect on moments at adjacent corners in atankwith wall panels supported on three or four sides, moments in the walls of a multicell tank are essentially the same as in single-cell tanks-except at corners where more than two walls intersect. Moment coefficients from Tables 5 and 6, designated as L coefficients, apply to outer or Lshaped corners of multicell tanks (see Fig. 4a) as well as to interior sections in all walls, that is, sections designated as y = b/4, y = 0, z = c/4, and z = 0. Moment coefficients for design sections at corners where more than two panels intersect depend on the loading condition producing maximum moment and on the number of intersecting walls. In Fig. 4b, three walls form a T-shaped unit. If the continuous wall, or top of the T, is part of the long sides of two adjacent rectangular cells, the moment in the continous wall at the intersection is maximum when both cells are filled. The intersection is then fixed and moment coefficients, designated as F coefficients, can be taken from Tables 1, 2, or 3, depending on edge conditions at top and bottom. These three tables cover panels with fixed side edges. If the continuous wall is part of the short sides of two adjacent rectangular cells, moment at one side of the intersection is maximum, when the cell on that side is filled while the other cell is empty. Likewise the end moment in the center wall is maximum when only one cell is filled. For this loading condition the magnitude of moment will be somewhere between theL coefficients and the F coefficients. If the unloaded third wall of the unit is disregarded, or its stiffness considered negligible, moments in the loaded walls would be the same as in Fig. 4a, that is, the L coefficients apply. If the third wall is assumed to have infinite stiffness, the corner is fixed and the f coefficients apply. The intermediate value representing more nearly the true condition can be obtained by the formula: End moments = L -nG2(i -F)

-

‘ProportIons of tank being deslgned are such that for determlning axial compression In sde walls, all the top load may be considered carned the short way

(4

b)

4

Fig. 4.

11

in which n denotes number of adjacent unloaded walls. This formula checks for n equal to zero and infinity. In an L-shaped unit n equals 0 and the end moments equal L - O(L - F) = L. Inserting n equal to infinity will give nl(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 intersecting walls are the walls of square cells, moments at the intersection are maximum when any two cells are filled and the F coefficients in Tables 1,2, or 3 apply because there is no rotation of the joint. If the cells are rectangular, moments in the longer of the intersecting walls will be maximum when two cells on the same sideof the wall under consideration are filled, and again the F coefficients apply. Maximum moments in the shorter walls adjacent to the intersection occur when diagonally opposite cells are filled, and for this condition the L coefficients apply. Fig. 5 shows moment coefficients at wall intersecitions in two- and four-cell tanks. Where coefficients are not 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, each with the same inside dimensions as the open-top singlecell tank and the closed single-cell tank. The top is considered free. In accordance with the types of units in Fig. 4, the tank consists of four L-shaped and two T-shaped units. L coeff icients from Table 5 for b/a = 2.50 and c/a = 1.25, and F coefficients from Table 2, for b/a = 2.50 and 1.25, are tabulated as follows: Long outer walls L = coeffuents from Table 5 for bla = 2 . 5 0 a n d cla = 1 . 2 5 y = b/2 x/a

%

Mv

-0018 -0016 -0.012

-0.092 -0.089 -0082 -0 059

0 I 0 ‘h %

I

y =b/4

I

M”

I

y = o

I

M.

Mv

0 to.015 to031 +0034

+0035 to.034 to.031 +0.020

Mx

Mv

I

0 +0.026 +0.045 I +0.044

to.069 +0.059 +o.o4a +0.029

._r

Short outer walls L = coefficients from Table 5 for b l a = 2 . 5 0 a n d c/a = 1 2 5 x/a 0

%

‘h %

.? = c/2 Mx

L = Cl4 Ml

0 -0.018 -0016 -0.012

I

-0.092 -0.089 -0.082 -0.059

F

=Tca~fetTfz b/a = 1 25

z=o

.? =c/2

4

Mz

MN

Mz

M.Y

0 -0.006 +0.003 +0.011

-0.030 -0024 -0.012 -0.002

0 -0002 ‘0.008 +0.018

-0.010 -0.003 +0.007 +0.008

0 -0.008 -0.010 -0.009

I

M, -0034 -0.042 -0.049 -0044

I L .L& 3 4

MZ

0 -0.015 -0.014 -0.011

-0.073 -0.073 -0.071 -0.054

Fig. 5.

Center wall F = coef f r o m Table 2 for bla = 2.50

L = coefficients from Table 5 for bla = 2.50 and c/a = 1.25 XIB 0 ‘74 ‘h %

12

y=o 4 0 +0.026 +0.045 +0.044

MY +0.069 to.059 +0.048 +0.029

y = b/4 4 0 to.015 io.031 +0034

y = b/2 MY

+0.035 to.034 +0.031 +0.020

MI 0 -0.018 -0.016 -0.012

MY -0.092 -0.089 -0.082 -0.059

y =b/2 4 0 -0.026 -0.023 -0016

MY -0.138 -0.132 -0.115 -0.078

L L-F 3 4 0 -0.021 -0.018 -0.013

MY -0.107 -0.103 -0.093 -0.065

‘VLLg

Short outer walls L = coefflcents from Table 5 for b/a = 2 50 and c/a = 1 25

Long outer walls

Fig. 6. y = b/2

Note that f coefficients in this tabulation are used only for calculation of coefficients L-L-that are to be 3 used for design at the intersection of the center and outer 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

‘.

The tank in Fig. 7 consists of two cells with the same inside dimensions as the cells in the two-cell tank with the long center wall. The difference is that the center wall is 40 ft wide in the previously discussed tank, but 20 ft wide in this example. Design procedure is identical for both two-cell tanks, but the schedule of coefficients is different because the longer side of the cell in Fig. 7 is continuous instead of the shorter side as in Fig. 6. Note from the following tabulation that the long wall must be designed for a maximum M, coefficient that occurs at the center wall of -0.138 instead of -0.092 at the corner in the tank in Fig. 6. Maximum moment is M,, = -0.1 38wa3 = -0.138 x 256 = -35.3 ft kips.

F = coel from Table 2 for b/a = 2 50

L = coeffxwXs from Table 5 for b a = 2 50 and cia = 1 25

0

-0092 I 0 069 + o 015 -0 062 +o 0 3 1 -0 059 I +o 034

lo

‘I1

I

J/r

y = b/4

-0016 -0 016

+0035 +o 034 *o 0 3 1 -0 020

-0

-0 012

y=o

I

0 *O 026 ‘ 0 045 ‘ 0 044

y = b/2

10069 to 059 ‘ 0 046 *o 029

1 0 -0 026 -0 023 -0016

-0 -0 -0 -0

138 132 115 078

Center wall F = cod from Table 2 for ba= 1 2 5

I = Cl4

.? = Cl2

z=o

ma 0 % % y.

L = coefflclents from Table 5 for b/a = 2.50 and c a = 1 25

I

Mx

Mz

0 -0002 +0 OQ6 +0018

-0.010 -0003 10007 +ooo6

Mx

t

z = Cl2

Mz

0 -0030 -0.006 -0 024 to.003 - 0 . 0 1 2 +0.011 -0002

I

L-F L-T

Mx

Mz

4

MZ

Mx

0 - 0 016 -0.016 -0012

-0092 -0 089 -0 062 -0059

0 -0006 -0010 -0.009

-0.034 -0.042 -0 049 -0044

0 -0.015 -0.014 -0011

t

I

Mz -0073 -0 073 -0.071 -0.054

Counterforted Tank Walls In a tank or reservoir with large horizontal dimensions, say three or four times the height, and without a reinforced concrete cover slab, it becomes necessary to design walls as cantilevers or, when they are quite high, as counterforted walls. The slab in Fig. 8 is free at the top and may be considered fixed at the bottom. If counterforts are spaced equidistantly, the slab may also be taken as fixed at counterforts. For this type of construction, coefficients in Table 3 apply.

Fig.

a.

Consider for illustration a wall panel of a counterforted wall in which spacing of counterforts is b = 40 ft and height is a = 20 ft. From Table 3, for b/a = 40/20 = 2, select the following coefficients. r

I

y=o Mx

MY

t

y = b/4 4

y = b/2

I MV

f"%

MV

Fig. 7.

13

Procedure for using these coefficients to determine moments and design of the wall is similar to that illustrated 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 assumption that the bottom edge is hinged. It is believed that this assumption in general is closer to the actual condition than that of a fixed edge. Consider first the detail in Fig. 9, which shows the wall supported on a relatively narrow continuous wall footing, and then Fig. 10 in which the wall rests on a bottom slab.

joint filler, but both iron powder and lead are not always readily available. A waterstop may not be needed in the construction joints when the vertical joint in Fig. 9 is made watertight. In Fig. 10 a continuous concrete base slab is provided either for transmitting the load coming down through the wall or for upward hydrostatic pressure. In either case, the slab deflects upward in the middle and tends to rotate the wall base in Fig. 10 in a counterclockwrse direction. The wall therefore is not fixed at the bottom edge. It is difficult to predict the degree of restraint. The rotation may be great enough to make the bottom edge hinged or may be even greater. Under the circumstances it is advisable to avoid placing moment reinforcement across the joint and to cross the dowels at the center. The waterstop must then be placed off center as indicated. Provision for transmitting shear through direct bearing can be made by inserting a key as in Fig. 9 or 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 except that the waterstop and the shear key are not essential. The main thing is to prevent moments from being transmitted from the top of the slab into the wall because the wall is not designed for such moments.

Fig. 9.

In Fig. 9 the condition of restraint at the bottom of the footing is somewhere between hinged and fixed but much closer to hinged than to fixed. Resultant of pressure on the subsoil lies well within the edge of the footing, and the product of resultant and its eccentricity is usually much smaller than the moment at the bottom of the wall when it is assumed fixed. Furthermore, thefooting must rotate about a horizontal axis in order to produce eccentric loading on the subsoil and rotation itself represents a relaxation of restraint. When the wall footing is not capable of furnishing much restraint, it is not necessary to provide for hinge action at the construction joint in Fig. 9. The dowels are close to the surface, leaving the center of the joint free for insertion of a shear key. Area of steel in the dowels along each face may be taken as not less thanO.O025bd, and extension of the dowels above the construction joint may be made not less than say 3 ft. The base slab in Fig. 9 is placed on top of the wall footing and the bearing surface is brushed with a heavy coat of asphalt to break the adhesion and reduce friction between slab and footing. The vertical joint between slab and wall should be made watertight. A joint width of 1 in. at the bottom and 1% in. at the top is considered adequate. As indicated in Fig. 9, the bottom of the joint may be filled with oakum, the middle with volcanic clay of a type that expands greatly when moistened, and the upper part sealed with mastic. Any leakage will make the clay penetrate into fissures and expand, plugging the leak. Mortar mixed with iron powder has been used extensively for joints such as in Fig. 9, and so has lead

14

Fig. 10.

Metric Conversion Factors To convert from inch (in.) feet (ft) square feet (sq ft) pound (lb) kip (1000 lb) Ib/lin ft kip/lin ft Ib/sq ft Ib/cu ft ft-kips ft-kips

To meter (m) meter (m) square meter (m2) kilogram (kg) kilogram (kg) kg/m kg/m kg/m2 kg/m3 newton-meter (Nm) kilogram-meter (kgm)

Multiply by 0.0254 0.3048 0.0929 0.4536 453.6 1.488 1488. 4.88 16.02 1356. 138.2

The prefixes and symbols listed are commonly used to form names and symbols of the decimal multiples and submultiples of the SI units. Multiplication Factor

Prefix

Symbol

1 000 000 000 = 109 1 oooooo= 106 1000=10~ l=l = 10-3 0.001 = 10-6 0.000 001 0.000 000 001 = 1 o-9

giga mega kilo

G M k m I-1 n

milli micro nano