RectangularTanks PCA

7/28/2019 RectangularTanks PCA

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Design of Rectangular Concrete Tanks

The Islamic University of GazaDepartment of Civil Engineering


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RECTANGULAR TANK DESIGN

The cylindrical shape is structurally bestsuited for tank construction, but rectangulartanks are frequently preferred for specificpurposes

Easy formwork and construction process Rectangular tanks are used where partitions or

tanks with more than one cell are needed.


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The behavior of rectangular tanks isdifferent from the behavior of circular tanks

The behavior of circular tanks is axi-symmetric.

That is the reason for the analysis to use onlyunit width of the tank

The ring tension in circular tanks was uniform

around the circumference


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The design of rectangular tanks is verysimilar in concept to the design of circulartanks

The loading combinations are the same. The

modifications for the liquid pressure loadingfactor and the sanitary coefficient are the same.

The major differences are the calculated

moments, shears, and tensions in therectangular tank walls.


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The requirements for durability are the same forrectangular and circular tanks.

The requirements for reinforcement (minimumor otherwise) are very similar to those forcircular tanks.

The loading conditions that must be consideredfor the design are similar to those for circulartanks.


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The restraint condition at the base is needed todetermine deflection, shears and bending

moments for loading conditions. Base restraint conditions considered in the publication

include both hinged and fixed edges.

However, in reality, neither of these two extremesactually exist.

It is important that the designer understand the degreeof restraint provided by the reinforcing bars that

extends into the footing from the tank wall. If the designer is unsure, both extremes should be

investigated.


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Buoyancy forces must be considered in the designprocess

The lifting force of the water pressure is resisted by theweight of the tank and the weight of soil on top of theslab


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RECTANGULAR TANK BEHAVIOR

y

x

Y-axis is the horizontal axis

X-axis is the vertical axis direction downwardMx moment per unit width stretching the fibers parallel to the x directionwhen the plate is in the x-y plane. This moment determines the steel inthe x (vertical direction).

My moment per unit width stretching the fibers parallel to the y directionwhen the plate is in the x-y plane. This moment determines the steel inthe y (horizontal direction).

The Subscript in Mx and My is not the axis of the moment

The moments are not in a particular principal plane


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Moment coefficient for Slabs with various edgeConditions


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Moment coefficient for Tanks with Walls Free atTop and Hinged at Bottom (Table 5)


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Moment coefficient for Tanks with Walls Free atTop and Hinged at Bottom (Table 5)


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Moment coefficient for Tanks with Walls Hingedat Top and Bottom (Table 6)


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Moment coefficient for Tanks with Walls Hingedat Top and Bottom (Table 6)


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Shear Coefficeient

Along vertical edges, shear in one wall is also used as axial tensionin the adjacent wall and must be combined with bending moment todetermine tensile reinforcement

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


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Shear Coefficients

The first five lines in Table 7 are shears per linear length in

terms of wa2

. The remaining four lines are total shears in kN depending on

how w is given.

Shears per linear foot are for ratios ofb/a = 1/2, 1,2, and

infinity. The difference between the shear forb/a = 2and 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.

Notes on Tables 7


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Shear Coefficeient

Total pressure on a strip 1 m wide is 0.50wa2, of which two-

thirds or0.33wa2is the reaction at the bottom support and one-

third or0.17wa2is the reaction at the top.

The shear at midpoint of the bottom edge is 0.3290wa2forb/a

= 2.0, the coefficient being very close to that of one third for

infinity. In other words, maximum bottom shear is practically constant

for all values of b/a greater than 2

At the corner, shear at the bottom edge is negative and

numerically greater than shear at midpoint.

Notes on Tables 7


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Shear Coefficients

Unit shears at the fixed edge in

Table 7 were used for plotting

the curves in Fig. 1. Maximum value occurs at a

depth below the top

somewhere between 0.6aand

0.8a. Fig. 1 is useful for

determination of shear or axial

tension for any ratio of b/a andat any point of a fixed side

edge.


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Shear Coefficients

The curves in Figs. 1 and 2

are nearly identical at the

bottom. As the top is approached

curves for the free top

deviate more and more from

those for the hinged top


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Shear Coefficients

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, whentop 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.

Notes on Tables 8 and Figure 2


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MultiCell Tank

Moment coefficients from Tables 5 and 6, designated as

L coefficients, apply to outer or L shaped corners ofmulti-cell tanks.

Corner of Multicell Tank:


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MultiCell Tank

Three wall forming T-Shape: If the continuous wall, or top of the T, is part of the long sides

of two adjacent rectangular cells, the moment in the continuous

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


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MultiCell Tank

Three wall forming T-Shape: 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 othercell is empty.

For this loading condition the magnitude of moment will be

somewherebetween the L coefficients and the F coefficients.


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MultiCell Tank

Three wall forming T-Shape:

If the unloaded third wall of the unit is disregarded, orits

stiffness considered negligible, moments in the loaded walls

would be the same L coefficients. If the third wall is assumed to have infinite stiffness, the

corner is fixed andthe F coefficients apply.

The intermediate value representing more nearly the truecondition can be obtained by the formula.

where n: number of adjacent unloaded walls

2

nEnd Moments L L F

n


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MultiCell Tank


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MultiCell Tank

Intersecting Walls:

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 applybecause 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 side of the wall under consideration

are filled, and again the F coefficients apply.


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MultiCell Tank

Intersecting Walls:

Maximum moments in the

shorter walls adjacent to

the intersection occur

when diagonally opposite

cells are filled, and for this

condition the LCoefficients apply.


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Example 1 (Open-Top Single-Cell Tank)

A

C

E

BFD

The tank shown has a clear height of a = 4 m.

horizontal inside dimensions are b = 10.0 m and

c = 5.0 m. The tops of the walls are considered

free and the bottom hinged.


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Coefficients for moment and shear are selected from

tables or diagrams for:

b/a = 10/4.0 = 2.50 and c/a =5/4 = 1.25. Moments are in ton.m if coefficients are multiplied

by wa3 = 1.0 x 64= 64

Shears are in ton if coefficients are multiplied bywa2 = 16.

Moment coefficients taken from Table 5 forb/a = 2.5

andc/a = 1.25 are tabulated below.



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x/a y=0 y=b/4 y=b/2 z=c/4 z=0Mx My Mx My Mx My Mx Mz Mx Mz

0 0 0.069 0 0.035 0 -0.092 0 -0.030 0 -0.010

1/4 0.026 0.059 0.015 0.034 -0.016 -0.089 -0.006 -0.26 -0.002 -0.003

1/2 0.045 0.048 0.031 0.031 -0.016 -0.062 0.003 -0.012 +0.008 +0.007

3/4 0.044 0.029 0.034 0.020 -0.012 -0.059 0.011 -0.002 +0.018 +0.008



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The largest moment occurs in the horizontal direction at the

top of the corner common to both walls (line AB) and equals

MAB=-0.092wa3

= -0.092 x 64 = -5.9 ton-m. The negative sign simply indicates that tension is on the inside

of the wall.

Maximum horizontal moment at midpoint of the longer wall

(line CD) is: MCD= +0.069wa3 = 0.069 x 64 = +4.42 ton-m.

The positive sign shows that tension is in the outside of the

wall.

There is also some axial tension on this section that can be

taken equal to end shear at the top of the shorter wall.



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For use in connection with Fig. 2, ratio of b/a for the shorter

wall is 5/4 = 1.25. The shear is 0.03wa2 = 0.03 x 16 =0.48

ton.

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 the axial tension taken from Fig. 2 for

b/a = 1.25 is equal to N = -0.30wa2 = -0.30 x 16 =-4.80 ton per

linear meter, which is not negligible.

Moment is M = 0.048wa3 = 0.048 x 64 = 3.06 ton-m



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In the shorter wall,positive moments are all relatively small.

Maximum positive moment is vertical: 0.018wa3 = 0.018 x 64

=1.15 ton-m.

Maximum Mx in the vertical strip at midpoint oflonger panel

0.045wa3 = 0.045 x 64 = 2.88 ton-m.



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A

C

E

BFD

The tank in this example 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 aconcrete slab with dowels extending from the wall into

the slab without moment reinforcement across the

bearing surface.

Example 2 (Closed Single-Cell Tank)


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Coefficients for moment and shear are selected from tables or

diagrams for:

b/a = 10/4.0 = 2.50 andc/a =5/4 = 1.25.

Moments are in ton-m if coefficients are multiplied by wa3 =

1.0 x 64= 64

Shears are in ton if coefficients are multiplied by wa2 = 16.

Moment coefficients taken from Table 6 forb/a = 2.5 andc/a =

1.25 are tabulated below.



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x/ay=0 y=b/4 y=b/2 z=c/4 z=0

Mx My Mx My Mx My Mx Mz Mx Mz

1/4 0.032 0.011 0.022 0.010 -0.006 -0.032 0.003 0.004 0.007 0.012

1/2 0.052 0.018 0.038 0.017 -0.011 -0.053 0.008 0.007 0.018 0.019

3/4 0.048 0.015 0.037 0.014 -0.010 -0.048 0.014 0.008 0.022 0.016

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.



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With a free top, maximum Mx= +0.045wa3and maximum My

= -0.092wa3.

With a hinged top, maximum Mx=+0.052wa3and 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% less forwall with free top than with hinged top.

The maximum My 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 aty=0.



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Maximum moment is My=-0.053wa3=-0.053 x 64=-3.4 ton-m.

Maximum moment in a vertical strip is Mx= 0.052wa3=0.052

x 64 = 3.3 ton-m.

Axial compression (N) on the section subject to this moment,and loads per linear meter can be taken as follows:

4 meter high wall: 2 x 0.30 x 2.5 = 1.5 ton

30 cm. top concrete slab: 0.30x 2.5 x5/2 = 1.9 ton 1-meter fill on top of slab: 1x1.75 x 5/2 = 4.4 ton

Live load on top of fill: 0.50 x 5/2 = 1.25 ton

_____________

Total = 9.05 ton


T d B Sl b


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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 500 kg/m2 and an earthfill weighing 1.75

t/m2.

Estimating slab thickness as 30 cm. gives a total design load of

0.50 + 1.75 + 2.5(0.3) = 3.0 t/m2.

From Table 4, for a ratio of 10/5 = 2, select maximumcoefficient of 0.100, which gives maximum M=0.100wa2 =

0.100 x 3.0 x 25 = 7.5 ton-m.

Top and Base Slabs


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Top and Base Slabs

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, say 0.005bd cm2 in each direction.

Length of these bars can be taken as = a= x 5 = 1.25 m.

d l b


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Assume the closed single-cell tank has a base slab of reinforced

concrete.

Weight of base slab and liquid does not create any bending orshearing stresses in concrete provided the subsoil is uniformly

well compacted.

Weight transferred to the base through the bottom of the wall is:

Top slab: 3.0 x 5.6 x 10.6 = 178.1 ton.

Walls: 4x0.3x2.5(2x10.3 +2x5.3) = 93.6 ton

---------------

Total = 271.7 ton

Top and Base Slabs

T d B Sl b


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If the base slab extends 20 cm. outside the walls, its area is 11

x 6 = 66 m2.

The average load of w = 271.7/66 = 4.1 t/m2 is used for design

of the base slab just as w = 3.0 t/m2 was used for design of thetop slab.

Total average load on the subsoil is water weight (4 x 1.0) +

4.1 + weight of base slab, say 4.0 + 4.1+1 = 9.1 ton/m2

, whichthe 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

Top and Base Slabs

Two Cell Tank Long Center Wall


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Two-Cell Tank, Long Center Wall

The tank in Figure consists of two adjacent cells, each with

the same inside dimensions as the open-top single cell tank

(a clear height of a = 4 m. Horizontal inside dimensions are

b = 10.0 m and c = 5.0 m). The top is considered free.



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The tank consists of four L-shaped and two T-shaped units.

L coefficients 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 in the following tables:




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Two-Cell Tank Long Center Wall


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Note that F coefficients in this tabulation are used only for

calculation of coefficients L-(L-F)/3 that are to be used for

design at the intersection of the center and outer walls as

shown.


Two-Cell Tank Long Center Wall


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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(in PCA report) can be applied both to center and outer walls.


Two-Cell Tank Short Center Wall


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Two Cell Tank, Short Center Wall

The tank in Figure consists of two cells with the same

inside dimensions as the cells in the two-cell tank with

the long center wall. The center wall is 5 m wide in this

example.



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Design procedure is identical for both two-cell tanks, but

the schedule of coefficients is different because the

longer side of the cell in this example is continuous

instead of the shorter side. Note from the following tabulation that the long wall

must be designed for a maximum My coefficient that

occurs at the center wall of -0.138 instead of -0.092 at

the corner in the tank in previous example.

Maximum moment is My=-0.1 38wa3 = -0.138 x 64 = -

8.83 ton-m.



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Two-Cell Tank, Short Center Wall


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Two-Cell Tank, Short Center Wall


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6m8m

Counterforted Tank Walls


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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 counterfortedwalls.

The slab shown in Figure is free at

the top and may be considered fixedat 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.



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Cou e o ed a a s

Consider for illustration a wall panel of a counterforted wall in

which spacing of counterforts is b=10m and height is a = 5 m.

From Table 3, for b/a = 10/5 = 2, select the following

coefficients 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.

Details at Bottom Edge


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g

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 wallsupported on a relatively narrow continuous wall footing,



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g

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.

The base slab in Fig. 9 is placed on top of the wall footing andthe 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 2.5 cm at the bottom is considered

adequate.

A waterstop may not be needed in the construction joints whenthe vertical joint is made watertight



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g

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 tendsto rotate the wall base in Fig. 10 in a counterclockwrse

direction.



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g

The wall therefore is not fixed at the bottom edge and it is

difficult to predict the degree of restraint

The waterstop must then be placed off center as indicated.

Provision for transmitting shear through direct bearing can bemade by inserting a key as in Fig. 9 or by a shear ledge as in

Fig. 10.

At top of wall the detail in Fig. 10 may be applied except thatthe 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.

RectangularTanks PCA

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