lecture 4 - welcome to delta...

Oct-15

Dr. A. Helba CIV 416E 1

A Guide to Designing Reinforced Concrete Water Tanks

Helba AlaaDr.

Roof slab

Wall

Floor slab

Tower supp. Sys.

Lecture 4

Oct-15


Load distribution on tank walls

Cylindrical wall

Rectangular wall

Analysis of Cylindrical Walls.

Using of Portland Cement Association tables.

A tour within (PCA) tables.

Lecture 4

H

pyv pyh

Load Distribution on Wall with Fixed Base

y

py = gw y = pyh + pyv

py

pmax = gw H

At a level at distance (y) from top of wall: py= gw y py is distributed to 2 components:

Oct-15


Case of a Rectangular Wall as T.W.S

ph max=

0.75 ph

H

ph max

pv ph


L1

L2

H/4 Plan

Case of a Square Wall as T.W.S

ph max=

0.75 ph H

ph max

pv max=gwH


L

L

H/4

pyh max Hal Moment and T

pyv max B. Moment (Vertical)

Mm (at middle)= phmaxL2 / 24

Mc (at corner) = phmaxL2 / 12

T (on each wall) = phmaxL / 2

Oct-15


Analysis of Walls in R.C. Tanks Case of Wall of any regular section (L)

Mm (at middle)= phL2 / 24

Mc (at corner) = phL2 / 12

T (on each wall) = phD / 2

pL2 / 24

pL2 / 12

L D L L

Case of Cylindrical Wall

H D

t

SEC. PLAN SEC. ELEV. – Load on Wall

t

Oct-15


py= pyh + pyv

Load Distribution & Deformation of cylindrical Wall fixed at base

D

py

H y

x

pyh x

pyh Ring Tension T (Horizontal)

pyv B. Moment (Vertical)

Load Distribution & Resulting Forces and Moment

At any level (y from top):

py = pyh + pyv

pyh Ring Tension T (Horizontal) pyv B. Moment (Vertical)

Cylindrical Walls

T – Diag. B.M.D.

Oct-15


py= pyh + pyv

x is the resulting radial deformation at level y due lateral pressure py

er is the resulting radial strain due to pyh . er= x/R

Deformation of cylindrical Wall fixed at base

D

py

H y

x

pyh x

py= pyh + pyv

T is the resulting horizontal tension at level y due to radial pressure pyh where T = pyh R My is the resulting vertical moment due to pyv

Resulting External Forces and Moments of cylindrical Wall

R

pyh x

MF

My

B.M.D due to pyv

t

py

H y

x

Oct-15


Load - Deformation Relationships of cylindrical Wall

Tint = Text

Then Ec x t / R = pyh R Hal Load component pyh = Ec x t / R2

= 4 Ec x t / D2

Val Load - Deformation Relationships of cylal Wall

Vertically

Transferred

component

x q El. Ld. My

Qy pyv

Slope q = dx / dy Elastic load M = EcI dq / dy = EcI d

2x / dy2

Shear Q = EcI d3x / dy3

Val Load component pyv = EcI d4x / dy4

= Ect3 x’’’’ / 12

D i f f e r e n t i a t i o n

I n t e g r a t i o n

Oct-15


py = pyh + pyv

py = 4 Ec x t / D2 + Ec t3 x’’’’ / 12

Or: (Ect

3 / 12) x’’’’ + (4 Ect / D2) x = py

Derivation of wall governing equation:

MF

R py

H y

x

pyh x

My

B.M.D due to pyv

t

Solution of the 4th degree Diff. Eqn

(E t3 / 12) x’’’’ + (4 E t / D2) x = py The General Solution of the 4th degree Diff. Eqn is:

t D

H

Diff. Equation of the wall

x = e-by (C1 sinby + C2 cosby) + P.I putting y = a H x = e-abH (C1 sinabH + C2 cosabH) + P.I

42 2

12

D tb

putting y/H = a

py

y

x

Oct-15


x = e-ak (C1 sinak + C2 cosak) + P.I

The General Solution of the 4th degree Diff. Eqn is:

Substitute:

where k is a dimensionless factor

4244

2 2

1212

HH K

D tk b

2HK

D t

Using the Solution of the 4th deg. Diff. Eqn & using the Boundary Condition for each case, one can get T and M in wall at any a

a y / H = 0 1.0

Tables for Design of Cylindrical Tanks

For any circular tank a dimensionless

factor ktank is used to identify the tank

where:

- H is the internal height of tank

- D is the internal diameter of tank

- t is the wall thickness

k tank = H2 / D t

Oct-15


Tables for Design of Cylindrical Tanks

- use Portland Cement Association (PCA) tables

(A set of 20 tables) to analyze the elements of circular tanks (walls , roofs and floors).

k tank = H2 / D t

List of PCA 20 Tables (6/5/4/1/1/2/1)

• 6 for Tension in circular Walls. Tables I VI

• 5 for Moments in circular Walls. Tables VII XI

• 4 for Moments in circular Slabs. Tables XII XV

(1 without/3 with central support)

• 1 for Shear at base of Wall. Table XVI

• 1 for Load on central support. Table XVII

• 2 for stiffness (1 for Wall / 1 for Slab) XVIII - XIX

• 1 for Supplementary Coefficients for values of H2/Dt > 16 for Walls (13 tables 6/5/1/1).Table XX

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Tension Moments

Slab Moments

Oct-15


Range of H/D vs values of the dimensionless factor H2 / Dt

H2/Dt = (H/D)(H/t)

Assume t = H/15 to H/20 say H = 16 t

Ktank = H2/Dt 16 (H/D) H/D Ktank/16

The 20 Tables consider 4 ranges of H2/Dt as:

0.4 2 step 0.4 (5 values) Shallow

3 8 step 1 (no 7) (5 values)

10 16 step 2 (4 values)

20 56 step 4/8 table XX (6 val.) Deep

Range of H/D vs values of the dimensionless factor H2 / Dt

Ktank D/H 0.4 2 40 8

3 8 5.33 2

10 16 1.6 1

20 56 0.8 2/7

Oct-15


Example on using PCA Tables

• For a circular tank with internal diameter of 8m , internal height of 4 m , and wall thickness of 0.40 m , then ,

• For Wall :

D = 8 m

H = 4 m

t = 0.40 m

K tank = H2 / Dt = (4)2 / (8 x 0.40) = 5

K tank = 5

Table I

Oct-15


+ ve tension

- ve compression

gw = 10 kN/m3

R = D/2

Table I

coeffs. for Tension in Circular Rings / Triangular Load / Fixed base , free top.

0.0 H 0.1 H 0.2 H 0.3 H 0.4 H 0.5 H 0.6 H 0.7 H 0.8 H 0.9 H 1.0 H

Fixed Base

Free Top

Table I

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Distribution of Tension along H Example: Ktank = H2/Dt = 10 +ve sign means tension

I II


III IV

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V VI

+ ve tension outside

- ve tension inside

Table VII

coeffs. for Moment in Cylindrical Wall / Triangular Load / Fixed base , free top.

0.0 H 0.1 H 0.2 H 0.3 H 0.4 H 0.5 H 0.6 H 0.7 H 0.8 H 0.9 H 1.0 H

Fixed Base

Free Top

Oct-15


Distribution of Moment along H Ex. Ktank = H2/Dt = 10 +ve sign means tension outside

VII VIII

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Comparison between tables VII and VIII

VII VIII


IX

Oct-15



X XI

Summary

PCA

Tables of Tension

and Moments

in circular Tanks

II + IV

lecture 4 - welcome to delta...

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