lecture 1 january 17, 2006. sudhir k. jain, iit kanpur e-course on seismic design of tanks/ january...

Lecture 1

January 17, 2006

Sudhir K. Jain, IIT Kanpur E-Course on Seismic Design of Tanks/ January 2006 Lecture 1 / Slide 2

In this lecture

Types of tanks IS codes on tanks Modeling of liquid


Types of tanks

Two categories Ground supported tanks

Also called at-grade tanks; Ground Service Reservoirs (GSR)

Elevated tanks Also called overhead tanks; Elevated Service Reservoirs

(ESR)


Types of tanks

Ground supported tanks Shape: Circular or Rectangular Material : RC, Prestressed Concrete, Steel

These are ground supported vertical tanks Horizontal tanks are not considered in this

course


Types of tanks

Elevated tanks

Two parts: Container Staging (Supporting tower)


Types of tanks

Elevated tanks

Container: Material: RC, Steel, Polymer Shape : Circular, Rectangular, Intze, Funnel, etc.

Staging: RC or Steel frame RC shaft Brick or masonry shafts

Railways often use elevated tanks with steel frame staging

Now-a-days, tanks on brick or stone masonry shafts are not constructed


Use of tanks

Water distribution systems use ground supported and elevated tanks of RC & steel

Petrochemical industries use ground supported steel tanks


Indian Codes on Tanks

IS 3370:1965/1967 (Parts I to IV) For concrete (reinforced and prestressed)

tanks Gives design forces for container due to

hydrostatic loads Based on working stress design BIS is considering its revision



IS 11682:1985 For RC staging of overhead tanks Gives guidelines for layout & analysis of

staging More about this code later

IS 803:1976 For circular steel oil storage tanks



IS 1893:1984 Gives seismic design provisions Covers elevated tanks only Is under revision More about other limitations, later

IS 1893 (Part 1):2002 is for buildings only Can not be used for tanks


Hydrodynamic Pressure

Under static condition, liquid applies pressure on container. This is hydrostatic pressure

During base excitation, liquid exerts additional pressure on wall and base. This is hydrodynamic pressure This is in additional to the hydrostatic pressure


Hydrodynamic pressure

Hydrostatic pressure Varies linearly with depth of liquid Acts normal to the surface of the container At depth h from liquid top, hydrostatic

pressure = h

Hydrostatic pressure

h

h



Hydrodynamic pressure Has curvilinear variation along wall height Its direction is opposite to base motion

Base motion




Summation of pressure along entire wall surface gives total force caused by liquid pressure Net hydrostatic force on container wall is zero Net hydrodynamic force is not zero





Base motion

Circular tanks (Plan View)

Net resultant force = zero

Net resultant force ≠ zero

Note:- Hydrostatic pressure is axisymmetric; hydrodynamic is asymmetric





Base motion

Net resultant force = zero

Net resultant force ≠ zero

Rectangular tanks (Plan View)



Static design: Hydrostatic pressure is considered Hydrostatic pressure induces hoop forces and

bending moments in wall IS 3370 gives design forces for circular and

rectangular tanks Net hydrostatic force is zero on container wall Hence, causes no overturning moment on

foundation or staging Thus, hydrostatic pressure affects container

design only and not the staging or the foundation



Seismic design: Hydrodynamic pressure is considered Net hydrodynamic force on the container is not

zero Affects design of container, staging and

foundation



Procedure for hydrodynamic pressure & force:

Very simple and elegant Based on classical work of Housner (1963a)

Housner, G. W., 1963a, “Dynamic analysis of fluids in containers subjected to acceleration”, Nuclear Reactors and Earthquakes, Report No. TID 7024, U. S. Atomic Energy Commission, Washington D.C.

We need not go in all the details Only basics and procedural aspects are

explained in next few slides


Modeling of liquid Liquid in bottom portion of the container moves

with wall This is called impulsive liquid

Liquid in top portion undergoes sloshing and moves relative to wall

This is called convective liquid or sloshing liquid

Convective liquid(moves relative to tank wall)

Impulsive liquid(moves with tank wall)


Modeling of liquid

Impulsive liquid Moves with wall; rigidly attached Has same acceleration as wall

Convective liquid Also called sloshing liquid Moves relative to wall Has different acceleration than wall

Impulsive & convective liquid exert pressure on wall Nature of pressure is different See next slide


Modeling of liquid

Impulsive Convective


Base motion

Base motion


Modeling of liquid

At this point, we will not go into details of hydrodynamic pressure distribution Rather, we will first find hydrodynamic forces Impulsive force is summation of impulsive

pressure on entire wall surface Similarly, convective force is summation of

convective pressure on entire wall surface


Modeling of liquid

Total liquid mass, m, gets divided into two parts: Impulsive liquid mass, mi

Convective liquid mass, mc

Impulsive force = mi x acceleration Convective force = mc x acceleration

mi & mc experience different accelerations Value of accelerations will be discussed later First we will find mi and mc


Modeling of liquid

Housner suggested graphs for mi and mc

mi and mc depend on aspect ratio of tanks Such graphs are available for circular &

rectangular tanks See Fig. 2a and 3a of Guidelines Also see next slide

For taller tanks (h/D or h/L higher), mi as fraction of m is more

For short tanks, mc as fraction of m is more


Modeling of liquid

0

0.5

1

0 0.5 1 1.5 2h/D

0

0.5

1

0 0.5 1 1.5 2h/L

For circular tanks

For rectangular tanks

mi/m

mc /m

mi/m

mc /m

See next slide for definition of h, D, and L


Modeling of liquid

D

L

Base motion

L

Base motion

Elevation

Plan of Circular tank

Plan of Rectangular tank

h


Modeling of liquid

Example 1: A circular tank with internal diameter of 8 m, stores 3 m

height of water. Find impulsive and convective water mass.

Solution:Total volume of liquid = /4 x 82 x 3 = 150.8 m3

Total liquid mass, m = 150.8 x 1.0 = 150.8 t

Note:- mass density of water is 1000 kg/m3; weight density of water is 9.81 x 1000 = 9810 N/m3.

D = 8 m, h = 3 m h/D = 3/8 = 0.375.


0

0.5

1

0 0.5 1 1.5 2h/D

mi/m

mc /m


Modeling of liquid

From graph, for h/D = 0.375 mi/m = 0.42 and mc/m = 0.56

mi = 0.42 x 150.8 = 63.3 t and

mc = 0.56 x 150.8 = 84.5 t


Modeling of liquid

Impulsive liquid is rigidly attached to wall Convective liquid moves relative to wall

As if, attached to wall with springs

Rigid

mc

Kc/2Kc/2

mi

Convective liquid(moves relative to wall)

Impulsive liquid(moves with wall)


Modeling of liquid

Stiffness associated with convective mass, Kc Kc depends on aspect ratio of tank Can be obtained from graph

Refer Fig. 2a, 3a of guidelines See next slide


Modeling of liquid

mi/m

mc/m

Kch/mg

0

0.5

1

0 0.5 1 1.5 2h/D


Modeling of liquid

Example 2: A circular tank with internal diameter of 8 m, stores 3 m

height of water. Find Kc.

Solution:Total liquid mass, m = 150.8 t (from Example 1)

= 150.8 x 1000 = 150800 kg g = acceleration due to gravity = 9.81 m/sec2

D = 8 m, h = 3m h/D = 3/8 = 0.375. From graph, for h/D = 0.375; Kc h/mg = 0.65


Modeling of liquid

Kc = 0.65 mg/h

Kc = 0.65 x150800 x 9.81/3.0 = 320,525.4 N/m

Note: - Unit of m is kg, hence unit of Kc is N/m. If we take m in ton, then unit of Kc will be kN/m.

mi/m

mc/m

Kch/mg

0

0.5

1

0 0.5 1 1.5 2h/D


Modeling of liquid

Now, we know liquid masses mi and mc

Next, we need to know where these are attached with the wall Like floor mass in building acts at centre of

gravity (or mass center) of floor Location of mi and mc is needed to obtain

overturning effects Impulsive mass acts at centroid of

impulsive pressure diagram Similarly, convective mass


Modeling of liquid

Impulsive mass acts at centroid of impulsive pressure diagram

Location of centroid: Obtained by dividing the moment due to

pressure distribution by the magnitude of impulsive force

Similarly, location of convective mass is obtained See next slide


Modeling of liquid

hi

hi, hc can be obtained from graphs They also depend on aspect ratio, h/D or h/L Refer Fig. 2b, 3b of guidelines See next slide

hc

Resultant of impulsive pressure on wall

Resultant of convective pressure on wall


Modeling of liquid

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2h/L

hc/h

hi/h

For circular tanks


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2h/D

hc/h

hi/h


Modeling of liquid

Example 3: A circular tank with internal diameter of 8 m, stores 3

m height of water. Find hi and hc.

Solution:

D = 8 m, h = 3m

h/D = 3/8 = 0.375.


0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2h/D

hc/h

hi/h


Modeling of liquid

From graph, for h/D = 0.375;

hi/h = 0.375

hi = 0.375 x 3 = 1.125 mand hc/h = 0.55

hc = 0.55 x 3 = 1.65 m

Note :- Since convective pressure is more in top portion, hc > hi.


Modeling of liquid

Hydrodynamic pressure also acts on base Under static condition, base is subjected to

uniformly distributed pressure Due to base motion, liquid exerts nonuniform

pressure on base This is in addition to the hydrostatic pressure on the

base See next slide


Modeling of liquid

Hydrostatic pressure on base

Base motion

Hydrodynamic pressure on base


Modeling of liquid

Impulsive as well as convective liquid cause nonuniform pressure on base Nonuniform pressure on base causes

overturning effect This will be in addition to overturning effect of

hydrodynamic pressure on wall See next slide


Modeling of liquid

Overturning effect due to wall

pressure

Overturning effect due to base

pressure

hi

Note:- Both the overturning effects are in the same direction


Modeling of liquid

Total overturning effect of wall and base pressure is obtained by applying resultant of wall pressure at height, hi

* and hc*.

• In place of hi and hc discussed earlier For overturning effect due to wall pressure

alone, resultant was applied at hi For hi and hi

*, see next slide


Modeling of liquid

hi

h*i

Location of resultant of wall pressure when effect of base pressure is not included

Location of Resultant of wall pressure when effect of base pressure is also included


Modeling of liquid

Similarly, hc and hc* are defined

hc

h*c

Location of resultant of wall pressure when effect of base pressure is not included

Location of Resultant of wall pressure when effect of base pressure is also included


Modeling of liquid

hi and hi* are such that

Moment due to impulsive pressure on walls only = Impulsive force x hi

Moment due to impulsive pressure on walls and base = Impulsive force x hi

*

hc and hc* are such that

Moment due to convective pressure on walls only = Convective force x hc

Moment due to convective pressure on walls and base = Convective force x hc

*


Modeling of liquid

hi* is greater than hi

hc* is greater than hc

Refer Fig. C-1 of the Guidelines hi

* & hc* depend on aspect ratio

Graphs to obtain hi, hc, hi*, hc

* are provided Refer Fig. 2b & 3b of guidelines Also see next slide Please note, hi

* and hc* can be greater than h


Modeling of liquid

hi/h

hi*/hhc/h

hc*/h

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2h/D


Modeling of liquid

Example 4: A circular tank with internal diameter of 8 m, stores 3 m height of

water. Find hi* and hc

*.

Solution: D = 8 m, h = 3m h/D = 3/8 = 0.375. From graph, for h/D = 0.375;


Modeling of liquid

hi*/h = 1.1

Hence hi* = 1.1 x 3 = 3.3 m

Similarly, hc*/h = 1.0

Hence, hc* = 1.0 x 3 = 3.0 m

hi/h

hi*/hhc/h

hc*/h

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2h/D


Modeling of liquid

This completes modeling of liquid Liquid is replaced by two masses, mi & mc

This is called mechanical analogue or spring mass model for tank

See next slide


Modeling of liquid

Rigid

mc

Kc/2Kc/2

mi

hi (hi

*)

hc (hc

*)

mi = Impulsive liquid mass

mc = Convective liquid mass

Kc = Convective spring stiffness

hi = Location of impulsive mass (without considering overturning caused by base pressure)

hc = Location of convective mass (without considering overturning caused by base pressure)

hi* = Location of impulsive mass

(including base pressure effect on overturning)

hc* = Location of convective mass

(including base pressure effect on overturning)

Mechanical analogue or

spring mass model of tank


Modeling of liquid

mi, mc, Kc, hi, hc, hi* and hc

* can also be obtained from mathematical expressions: These are given in Table C 1 of Guidelines These are reproduced in next two slides


h

Dh

D

m

mi

866.0

866.0tanh

D

h

h

mgK c 68.3tanh836.0 2

D

hD

h

m

mc

68.3tanh

23.0

375.0h

hi

Dh /

09375.05.0

for 75.0/ Dh

75.0/ Dhfor

D

h

D

h

D

h

h

hc

68.3sinh68.3

0.168.3cosh

1

For circular tanks

1250 -86602

8660.

hD

.tanh

hD

.

h

*hi

for

33.1/ Dh

45.0 33.1/ Dh

for

D

h

D

h

D

h

h

hc

68.3sinh68.3

01.268.3cosh

1*

Modeling of liquid



125.0

866.0tanh2

866.0*

h

Lh

L

h

hi

for

33.1/ Lh

45.0 33.1/ Lh

for

L

h

L

h

L

h

h

hc

16.3sinh16.3

01.216.3cosh

1*

375.0h

hi for 75.0/ Lh

Lh /

09375.05.0 75.0/ Lhfor

L

h

L

h

L

h

h

hc

16.3sinh16.3

0.116.3cosh

1

L

h

h

mgK c 16.3tanh833.0 2

h

Lh

L

m

mi

866.0

866.0tanh

L

hL

h

m

mc

16.3tanh

264.0

Modeling of liquid


Modeling of liquid

Note, in Table C-1 of the Guideline, there are two typographical errors in these expressions For circular tank, first expression for hi/h shall

have limit as “for h/D 0.75” For circular tank, in the expression for hi

*/h, there shall be minus sign before 0.125

These two errors have been corrected in the expressions given in previous two slides


Modeling of liquid

mi and mc are needed to find impulsive and convective forces Impulsive force, Vi = mi x acceleration Convective force, Vc = mc x acceleration

Rigid

mc

Kc/2Kc/2

mi Vi

Vc


Modeling of liquid

Vi and Vc will cause Bending Moment (BM) in wall


Modeling of liquid

BM at bottom of wall BM due to Vi = Vi x hi

BM due to Vc = Vc x hc

Total BM is not necessarily Vi X hi+ Vc X hc

More about this, later

Rigid

mc

Kc/2Kc/2

mi Vi

Vc

hi

hc


Modeling of liquid

Overturning of the container is due to pressure on wall and base Pressure on base does not cause BM in wall

Overturning Moment (OM) at tank bottom OM is at bottom of base slab Hence, includes effect of pressure on base Note the difference between bottom of wall

and bottom of base slab


Modeling of liquid

OM at bottom of base slab OM due to Vi = Vi x hi

*

BM due to Vc = Vc x hc*

Vi

Vc

Rigid

mc

Kc/2Kc/2

mi

hi*

hc*


Modeling of liquid

mi and mc will have different accelerations We yet do not know these accelerations

ai = acceleration of mi

ac = acceleration of mc

Procedure to find acceleration, later

Use of mi, mc, hi, hc, hi* and hc

* in next example Acceleration values are assumed


Modeling of liquid

Example 5:

A circular tank with internal diameter of 8 m, stores 3 m height of water. Assuming impulsive mass acceleration of 0.3g and convective mass acceleration of 0.1g, find seismic forces on tank. Solution:Geometry of tank is same as in previous examples. D = 8 m, h = 3mFrom previous examples:mi = 63.3 t mc = 84.5 t

hi = 1.125 m hc = 1.65 m

hi* = 3.3 m hc

* = 3.0 m

Impulsive acceleration, ai = 0.3g = 0.3 x 9.81 = 2.94 m/sec2

Convective acceleration, ac = 0.1g = 0.1 x 9.81 = 0.98 m/sec2


Modeling of liquid

Example 5 (Contd..)

Impulsive force, Vi = mi x ai = 63.3 x 2.94 = 186.1 kN

Convective force, Vc = mc x ac = 84.5 x 0.98 = 82.8 kN

Bending moment at bottom of wall due to Vi = Vi x hi

= 186.1 x 1.125 = 209.4 kN-mBending moment at bottom of wall due to Vc = Vc x hc

= 82.8 x 1.65 = 136.6 kN-mOverturning moment at bottom of base due to V i = Vi x hi

*

= 186.1 x 3.3 = 614.1 kN-mOverturning moment at bottom of base due to Vc = Vc x hc

*

= 82.8 x 3.0 = 248.4 kN-m


At the end of Lecture 1

In seismic design, mechanical analogue of tanks are used, wherein, liquid is replaced by impulsive & convective masses

These masses and their points of application depend on aspect ratio

Graphs and expressions are available to find all these quantities These are based on work of Housner (1963a)

lecture 1 january 17, 2006. sudhir k. jain, iit kanpur e-course on seismic design of tanks/ january...

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