January 23, 2006

1
Lecture 3

• January 23, 2006

2
In this lecture

• Modeling of tanks
• Time period of tanks

3
Modeling of tanks

• As seen in Lecture 1 liquid may be replaced by
  impulsive and convective mass for calculation of
  hydrodynamic forces
• See next slide for a quick review

4
Modeling of tanks
mi Impulsive liquid mass mc Convective
liquid mass Kc Convective spring stiffness hi
Location of impulsive mass (without
considering overturnig 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
Graphs and expression for these parameters are
given in lecture 1.
5
Approximation in modeling

• Sometimes, summation of mi and mc may not be
  equal to total liquid mass, m
• This difference may be about 2 to 3
• Difference arises due to approximations in the
  derivation of these expressions
• More about it, later
• If this difference is of concern, then
• First, obtain mc from the graph or expression
• Obtain mi m mc

6
Tanks of other shapes

• For tank shapes such as Intze, funnel, etc.
• Consider equivalent circular tank of same volume,
  with diameter equal to diameter at the top level
  of liquid

7
Tanks of other shapes

• Example
• An Intze container has volume of 1000 m3.
  Diameter of
• container at top level of liquid is 16 m. Find
  dimensions of
• equivalent circular container for computation of
• hydrodynamic forces.
• Equivalent circular container will have diameter
  of 16 m
• and volume of 1000 m3. Height of liquid, h can be
  obtained
• as
• ?/4 x 162 x h 1000
• ? h 1000 x 4/(? x 162) 4.97 m

8
Tanks of other shapes

• Thus, for equivalent circular container,
• h/D 4.97/16 0.31
• All the parameters (such as mi, mc etc.) are to
  be obtained using h/D 0.31

9
Effect of obstructions inside tank

• Container may have structural elements inside
• For example central shaft, columns supporting
  the roof slab, and baffle walls
• These elements cause obstruction to lateral
  motion of liquid
• This will affect impulsive and convective masses

10
Effect of obstructions inside tank

• Effect of these obstructions on impulsive and
  convective mass is not well studied
• A good research topic !
• It is clear that these elements will reduce
  convective (or sloshing) mass
• More liquid will act as impulsive mass

11
Effect of obstructions inside tank

• In the absence of detailed analysis, following
  approximation may be adopted
• Consider a circular or a rectangular container of
  same height and without any internal elements
• Equate the volume of this container to net volume
  of original container
• This will give diameter or lateral dimensions of
  container
• Use this container to obtain h/D or h/L

12
Effect of obstructions inside tank

• Example A circular cylindrical container has
  internal diameter of 12 m and liquid height of 4
  m. At the center of the tank there is a circular
  shaft of outer diameter of 2 m. Find the
  dimensions of equivalent circular cylindrical
  tank.

4 m
12 m
Hollow shaft of 2 m diameter
Plan
Elevation
13
Effect of obstructions inside tank

• Solution
• Net volume of container ?/4x(122 22)x4 439.8
  m3
• Equivalent cylinder will have liquid height of 4
  m and its volume has to be 439.8 m3.
• Let D be the diameter of equivalent circular
  cylinder, then
• ?/4xD2x4 439.8 m3
• ? D 11.83 m
• Thus, for equivalent circular tank, h 4 m, D
  11.83m and h/D 4/11.83 0.34.
• This h/D shall be used to find parameters of
  mechanical model of tank

14
Effect of wall flexibility

• Parameters mi, mc etc. are obtained assuming tank
  wall to be rigid
• An assumption in the original 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.
• RC tank walls are quite rigid
• Steel tank walls may be flexible
• Particularly, in case of tall steel tanks

15
Effect of wall flexibility

• Wall flexibility affects impulsive pressure
  distribution
• It does not substantially affect convective
  pressure distribution
• Refer Veletsos, Haroun and Housner (1984)
• Veletsos, A. S., 1984, Seismic response and
  design of liquid storage tanks, Guidelines for
  the seismic design of oil and gas pipeline
  systems, Technical Council on Lifeline Earthquake
  1Engineering, ASCE, N.Y., 255-370, 443-461.
• Haroun, M. A. and Housner, G. W., 1984, Seismic
  design of liquid storage tanks, Journal of
  Technical Councils of ASCE, Vol. 107, TC1,
  191-207.
• Effect of wall flexibility on impulsive pressure
  depends on
• Aspect ratio of tank
• Ratio of wall thickness to diameter
• See next slide

16
Effect of wall flexibility

• Effect of wall flexibility on impulsive pressure
  distribution

h/D 0.5
tw is wall thickness
From Veletsos (1984)
Impulsive pressure on wall
17
Effect of wall flexibility

• If wall flexibility is included, then mechanical
  model of tank becomes more complicated
• Moreover, its inclusion does not change seismic
  forces appreciably
• Thus, mechanical model based on rigid wall
  assumption is considered adequate for design.

18
Effect of wall flexibility

• All international codes use rigid wall model for
  RC as well as steel tanks
• Only exception is NZSEE recommendation (Priestley
  et al., 1986)
• Priestley, M J N, et al., 1986, Seismic design
  of storage tanks, Recommendations of a study
  group of the New Zealand National Society for
  Earthquake Engineering.
• American Petroleum Institute (API) standards,
  which are exclusively for steel tanks, also use
  mechanical model based on rigid wall assumption
• API 650, 1998, Welded storage tanks for oil
  storage, American Petroleum Institute,
  Washington D. C., USA.

19
Effect of higher modes

• mi and mc described in Lecture 1, correspond to
  first impulsive and convective modes
• For most tanks ( 0.15 lt h/D lt 1.5) the first
  impulsive and convective modes together account
  for 85 to 98 of total liquid mass
• Hence, higher modes are not included
• This is also one of the reasons for summation of
  mi and mc being not equal to total liquid mass
• For more information refer Veletsos (1984) and
  Malhotra (2000)
• Malhotra, P. K., Wenk, T. and Wieland, M., 2000,
  Simple procedure for seismic analysis of
  liquid-storage tanks, Structural Engineering
  International, 197-201.

20
Modeling of ground supported tanks

• Step 1
• Obtain various parameters of mechanical model
• These include, mi, mc, Kc, hi, hc, hi and hc
• Step 2
• Calculate mass of tank wall (mw), mass of roof
  (mt) and mass of base slab (mb)of container
• This completes modeling of ground supported tanks

21
Modeling of elevated tanks

• Elevated tank consists of container and staging

Roof slab
Wall
Container
Floor slab
Staging
Elevated tank
22
Modeling of elevated tanks

• Liquid is replaced by impulsive and convective
  masses, mi and mc
• All other parameters such as hi, hc, etc, shall
  be obtained as described earlier
• Lateral stiffness, Ks, of staging must be
  considered
• This makes it a two-degree-of-freedom model
• Also called two mass idealization

23
Modeling of elevated tanks
mc
Kc/2
Kc/2
mc
Kc
hc
mi
hi
mi ms
hs
Ks
Two degree of freedom system OR Two mass
idealization of elevated tanks
Spring mass model
24
Modeling of elevated tanks

• ms is structural mass, which comprises of
• Mass of container, and
• One-third mass of staging
• Mass of container includes
• Mass of roof slab
• Mass of wall
• Mass of floor slab and beams

25
Two Degree of Freedom System

• 2-DoF system requires solution of a 2 2 eigen
  value problem to obtain
• Two natural time periods
• Corresponding mode shapes
• See any standard text book on structural dynamics
  on how to solve 2-DoF system
• For most elevated tanks, the two natural time
  periods (T1 and T2) are well separated.
• T1 generally may exceed 2.5 times T2.

26
Two Degree of Freedom System

• Hence the 2-DoF system can be treated as two
  uncoupled single degree of freedom systems
• One representing mi ms and Ks
• Second representing mc and Kc

27
Modeling of elevated tanks
mc
Kc
mi ms
mi ms
mc
Ks
Ks
Kc
Two uncoupled single degree of freedom systems
Two degree of freedom system
when T1 2.5 T2
28
Modeling of elevated tanks

• Priestley et al. (1986) suggested that this
  approximation is reasonable if ratio of two time
  periods exceeds 2.5
• Important to note that this approximation is done
  only for the purpose of calculating time periods
• This significantly simplifies time period
  calculation
• Otherwise, one can obtain time periods of 2-DoF
  system as per procedure of structural dynamics.

29
Modeling of elevated tanks

• Steps in modeling of elevated tanks
• Step 1
• Obtain parameters of mechanical analogue
• These include mi, mc, Kc, hi, hc, hi and hc
• Other tank shapes and obstructions inside the
  container shall be handled as described earlier
• Step 2
• Calculate mass of container and mass of staging
• Step 3
• Obtain stiffness of staging

30
Modeling of elevated tanks

• Recall, in IS 18931984, convective mass is not
  considered
• It assumes entire liquid will act as impulsive
  mass
• Hence, elevated tank is modeled as single degree
  of freedom ( SDoF) system
• As against this, now, elevated tank is modeled as
  2-DoF system
• This 2-DoF system can be treated as two uncoupled
  SDoF systems

31
Modeling of elevated tanks

• Models of elevated tanks

m Total liquid mass
m ms
mi ms
Ks
mc
Ks
Kc
As per the Guideline
As per IS 18931984
32
Modeling of elevated tanks

• Example An elevated tank with circular
  cylindrical container has internal diameter of
  11.3 m and water height is 3 m. Container mass is
  180 t and staging mass is 100 t. Lateral
  stiffness of staging is 20,000 kN/m. Model the
  tank using the Guideline and IS 18931984
• Solution
• Internal diameter, D 11.3 m, Water height, h
  3 m.
• Container is circular cylinder,
• ? Volume of water ?/4 x D2 x h
• ? /4 x 11.32 x 3
  300.9 m3.
• ? mass of water, m 300.9 t.

33
Modeling of elevated tanks

• h/D 3/11.3 0.265
• From Figure 2 of the Guideline, for h/D 0.265
• mi/m 0.31, mc/m 0.65 and Kch/mg 0.47
• mi 0.31 x m 0.31 x 300.9 93.3 t
• mc 0.65 x m 0.65 x 300.9 195.6 t
• Kc 0.47 x mg/h
• 0.47 x 300.9 x 9.81/3 462.5 kN/m

34
Modeling of elevated tanks

• Mass of container 180 t
• Mass of staging 100 t
• Structural mass of tank, ms
• mass of container 1/3rd mass of staging
• 180 1/3 x 100
• 213.3 t
• Lateral stiffness of staging, Ks 20,000 kN/m

35
Modeling of elevated tanks
m ms
mi ms
mc
Ks
Ks
Kc
mi 93.3 t, ms 213.3 t, mc 195.6 t, Ks
20,000 kN/m, Kc 462.5 kN/m
m 300.9 t, ms 213.3 t, Ks 20,000 kN/m
Model of tank as per IS 18931984
Model of tank as per the Guideline
36
Time period

• What is time period ?
• For a single degree of freedom system, time
  period (T ) is given by

• M is mass and K is stiffness
• T is in seconds
• M should be in kg K should be in Newton per
  meter (N/m)
• Else, M can be in Tonnes and K in kN/m

37
Time period

• Mathematical model of tank comprises of impulsive
  and convective components
• Hence, time periods of impulsive and convective
  mode are to be obtained

38
Time period of impulsive mode

• Procedure to obtain time period of impulsive mode
  (Ti) will be described for following three cases
• Ground supported circular tanks
• Ground supported rectangular tanks
• Elevated tanks

39
Ti for ground-supported circular tanks

• Ground supported circular tanks
• Time period of impulsive mode, Ti is given by

• Mass density of liquid
• E Youngs modulus of tank material
• t Wall thickness
• h Height of liquid
• D Diameter of tank

40
Ti for ground-supported circular tanks

• Ci can also be obtained from Figure 5 of the
  Guidelines

C
h/D
41
Ti for ground-supported circular tanks

• This formula is taken from Eurocode 8
• Eurocode 8, 1998, Design provisions for
  earthquake resistance of structures, Part 1-
  General rules and Part 4 Silos, tanks and
  pipelines, European Committee for
  Standardization, Brussels.
• If wall thickness varies with height, then
  thickness at 1/3rd height from bottom shall be
  used
• Some steel tanks may have step variation of wall
  thickness with height

42
Ti for ground-supported circular tanks

• This formula is derived based on assumption that
  wall mass is quite small compared to liquid mass
• More information on time period of circular tanks
  may be seen in Veletsos (1984) and Nachtigall et
  al. (2003)
• Nachtigall, I., Gebbeken, N. and Urrutia-Galicia,
  J. L., 2003, On the analysis of vertical
  circular cylindrical tanks under earthquake
  excitation at its base, Engineering Structures,
  Vol. 25, 201-213.

43
Ti for ground-supported circular tanks

• It is important to note that wall flexibility is
  considered in this formula
• For tanks with rigid wall, time period will be
  zero
• This should not be confused with rigid wall
  assumption in the derivation of mi and mc
• Wall flexibility is neglected only in the
  evaluation of impulsive and convective masses
• However, wall flexibility is included while
  calculating time period

44
Ti for ground-supported circular tanks

• This formula is applicable to tanks with fixed
  base condition
• i.e., tank wall is rigidly connected or fixed to
  the base slab
• In some circular tanks, wall and base have
  flexible connections

45
Ti for ground-supported circular tanks

• Ground supported tanks with flexible base are
  described in ACI 350.3 and AWWA D-110
• ACI 350.3, 2001, Seismic design of liquid
  containing concrete structures, American
  Concrete Institute, Farmington Hill, MI, USA.
• AWWA D-110, 1995, Wire- and strand-wound
  circular, prestressed concrete water tanks,
  American Water Works Association, Colorado, USA.
• In these tanks, there is a flexible pad between
  wall and base
• Refer Figure 6 of the Guideline

46
Ti for ground-supported circular tanks
Types of connections between tank wall and base
slab

• Such tanks are perhaps not used in India

47
Ti for ground-supported circular tanks

• Impulsive mode time period of ground supported
  tanks with fixed base is generally very low
• These tanks are quite rigid
• Ti will usually be less than 0.4 seconds
• In this short period range, spectral
  acceleration, Sa/g has constant value
• See next slide

48
Ti for ground-supported circular tanks
Impulsive mode time period of ground supported
tanks likely to remain in this range
Sa/g
49
Ti for ground-supported circular tanks

• Example A ground supported steel tank has water
  height, h 25 m, internal diameter, D 15 m and
  wall thickness, t15 mm. Find time period of
  impulsive mode.
• Solution h 25 m, D 15 m, t 15 mm.
• For water, mass density, ? 1
  t/m3.
• For steel, Youngs modulus, E
  2x108 kN/m2.
• h/D 25/15 1.67. From Figure 5,
  Ci 5.3

50
Ti for ground-supported circular tanks
Time period of impulsive mode,
0.30 sec

• Important to note that, even for such a slender
  tank of steel, time period is low.
• For RC tanks and other short tanks, time period
  will be further less.

51
Ti for ground-supported circular tanks

• In view of this, no point in putting too much
  emphasis on evaluation of impulsive mode time
  period for ground supported tanks
• Recognizing this point, API standards have
  suggested a constant value of spectral
  acceleration for ground supported circular steel
  tanks
• Thus, users of API standards need not find
  impulsive time period of ground supported tanks

52
Ti for ground-supported rectangular tanks

• Ti for ground-supported rectangular tanks
• Procedure to find time period of impulsive mode
  is described in Clause no. 4.3.1.2 of the
  Guidelines
• This will not be repeated here
• Time period is likely to be very low and Sa/g
  will remain constant
• As described earlier
• Hence, not much emphasis on time period evaluation

53
Ti for Elevated tanks

• For elevated tanks, flexibility of staging is
  important
• Time period of impulsive mode, Ti is given by

OR

• mi Impulsive mass of liquid
• ms Mass of container and one-third mass of
  staging
• Ks Lateral stiffness of staging
• Horizontal deflection of center of gravity of
  tank when a
• horizontal force equal to (mi ms)g is
  applied at the
• center of gravity of tank

54
Ti for Elevated tanks

• These two formulae are one and the same
• Expressed in terms of different quantities
• Center of gravity of tank refers to combined mass
  center of empty container plus impulsive mass of
  liquid

55
Ti for Elevated tanks

• Example An elevated tank stores 250 t of water.
  Ratio of water height to internal diameter of
  container is 0.5. Container mass is 150 t and
  staging mass is 90 t. Lateral stiffness of
  staging is 20,000 kN/m. Find time period of
  impulsive mode
• Solution h/D 0.5, Hence from Figure 2a of the
  Guideline, mi/m 0.54
• ? mi 0.54 x 250 135 t
• Structural mass of tank, ms
• mass of container 1/3rd
  mass of staging
• 150 90/3 180 t

56
Ti for Elevated tanks

• Time period of impulsive mode

0.79 sec.
57
Lateral stiffness of staging, Ks

• Lateral stiffness of staging, Ks is force
  required to be applied at CG of tank to cause a
  corresponding unit horizontal deflection

CG
Ks P/ ?
58
Lateral stiffness of staging, Ks

• For frame type staging, lateral stiffness shall
  be obtained by suitably modeling columns and
  braces
• More information can be seen in Sameer and Jain
  (1992, 1994)
• Sameer, S. U., and Jain, S. K., 1992,
  Approximate methods for determination of time
  period of water tank staging, The Indian
  Concrete Journal, Vol. 66, No. 12, 691-698.
• Sameer, S. U., and Jain, S. K., 1994, Lateral
  load analysis of frame staging for elevated water
  tanks, Journal of Structural Engineering, ASCE,
  Vol.120, No.5, 1375-1393.
• Some commonly used frame type staging
  configurations are shown in next slide

59
Lateral stiffness of staging, Ks
Plan view of frame staging configurations
4 columns
6 columns
8 columns
60
Lateral stiffness of staging, Ks
24 columns
52 columns
61
Lateral stiffness of staging, Ks

• Explanatory handbook, SP22 has considered braces
  as rigid beams
• SP22 1982, Explanatory Handbook on Codes for
  Earthquake Engineering, Bureau of Indian
  Standards, New Delhi
• This is unrealistic modeling
• Leads to lower time period
• Hence, higher base shear coefficient
• This is another limitation of IS 18931984
• Using a standard structural analysis software,
  staging can be modeled and analyzed to estimate
  lateral stiffness

62
Lateral stiffness of staging, Ks

• Shaft type staging can be treated as a vertical
  cantilever fixed at base and free at top
• If flexural behavior is dominant, then
• Its stiffness will be Ks 3EI/L3
• This will be a good approximation if height to
  diameter ratio is greater than two
• Otherwise, shear deformations of shaft would
  affect the stiffness and should be included.

63
Time period of convective mode

• Convective mass is mc and stiffness is Kc
• Time period of convective mode is

64
Time period of convective mode

• mc and Kc for circular and rectangular tanks can
  be obtained from graphs or expressions
• These are described in Lecture 1
• Refer Figures 2 and 3 of the Guidelines

65
Time period of convective mode

• For further simplification, expressions for mc
  and Kc are substituted in the formula for Tc
• Then one gets,

For circular tanks
For rectangular tanks
66
Time period of convective mode

• Graphs for obtaining Cc are given in Figures 5
  and 7 of the Guidelines
• These are reproduced in next two slides
• Convective mass and stiffness are not affected by
  flexibility of base or staging
• Hence, convective time period expressions are
  common for ground supported as well as elevated
  tanks
• Convective mode time periods are usually very
  large
• Their values can be as high as 10 seconds

67
Time period of convective mode
C
h/D
Fig. 5 For circular tanks
68
Time period of convective mode
Cc
h/L
Fig. 7 For rectangular tanks
69
Time period of convective mode

• Example For a circular tank of internal
  diameter, 12 m and liquid height of 4 m.
  Calculate time period of convective mode.
• Solution h 4 m, D 12 m,
• ? h/D 4/12 0.33
• From Figure 5 of the Guidelines, Cc 3.6

Time period of convective mode,
3.98 sec
70
At the end of Lecture 3

• Based on mechanical models, time period for
  impulsive and convective modes can be obtained
  for ground supported and elevated tanks
• For ground supported tanks, impulsive mode time
  period is likely to be very less
• Convective mode time period can be very large