Lecture 5 January 31, 2006. Sudhir K. Jain, IIT Kanpur E-Course on Seismic Design of Tanks/ January...

Lecture 5

January 31, 2006

Sudhir K. Jain, IIT Kanpur E-Course on Seismic Design of Tanks/ January 2006 Lecture 5/ Slide 2

In this Lecture

Impulsive and convective base shear Critical direction of seismic loading


Base shear

Previous lectures have covered Procedure to find impulsive and convective

liquid masses This was done through a mechanical analog model

Procedure to obtain base shear coefficients in impulsive and convective modes

This requires time period, damping, zone factor, importance factor and response reduction factor

Now, we proceed with seismic force or base shear calculations


Base shear

Seismic force in impulsive mode (impulsive base shear)

Vi = (Ah)i x impulsive weight Seismic force in convective mode

(convective base shear) Vc = (Ah)c x convective weight (Ah)i = impulsive base shear coefficient (Ah)c = convective base shear coefficient

These are described in earlier lectures


Base shear

Now, we evaluate impulsive and convective weights Or, impulsive and convective masses Earlier we have obtained impulsive and

convective liquid mass Now, we consider structural mass also


Base shear : Ground supported tanks

Impulsive liquid mass is rigidly attached to container wall Hence, wall, roof and impulsive liquid vibrate

together In ground supported tanks, total impulsive

mass comprises of Mass of impulsive liquid Mass of wall Mass of roof



Hence, base shear in impulsive mode

gmmmAV twihi i

mi = mass of impulsive liquid mw = mass of container wall mt = mass of container roof g = acceleration due to gravity



This is base shear at the bottom of wall Base shear at the bottom of base slab is : Vi’ = Vi + (Ah)i x mb

mb is mass of base slab

Base shear at the bottom of base slab may be required to check safety against sliding



Base shear in convective mode

mc = mass of convective liquid

gmAV cchc



Total base shear, V is obtained as:

22ci VVV

Impulsive and convective base shear are combined using Square Root of Sum of Square (SRSS) rule

Except Eurocode 8, all international codes use SRSS rule Eurocode 8 uses absolute summation rule

i.e, V = Vi + Vc



In the latest NEHRP recommendations (FEMA 450), SRSS rule is suggested Earlier version of NEHRP recommendations

(FEMA 368) was using absolute summation rule

FEMA 450, 2003, “NEHRP recommended provisions for seismic regulations for new buildings and other structures”, Building Seismic Safety Council, National Institute of Building Sciences,, USA.

FEMA 368, 2000, “NEHRP recommended provisions for seismic regulations for new buildings and other structures”, Building Seismic Safety Council, National Institute of Building Sciences,, USA.


Bending moment:Ground supported tanks

Next, we evaluate bending or overturning effects due to base shear

Impulsive base shear comprises of three parts (Ah)i x mig (Ah)i x mwg (Ah)i x mtg



mw acts at CG of wall mt acts at CG of roof mi acts at height hi from bottom of wall

If base pressure effect is not included mi acts at hi

*

If base pressure effect is included Recall hi and hi

* from Lecture 1



Bending moment at the bottom of wall Due to impulsive base shear

ghmhmhmAM ttwwiiihi

hi = location of mi from bottom of wall hc = location of mc from bottom of wall hw = height of CG of wall ht = height of CG of roof

ghmAM ccchc )(

Due to convective base shear



For bending moment at the bottom of wall, effect of base pressure is not included Hence, hi and hc are used



Bending moment at the bottom of wall

ghmhmhmAM ttwwiiihi

ghmAM ccchc )(

Ground level

(Ah)imihi

(Ah)imw

hw

(Ah)imt

(Ah)cmc

hc

ht



Total bending moment at the bottom of wall

22ci MMM

SRSS rule used to combine impulsive and convective responses


Overturning moment:Ground supported tanks

Overturning moment This is at the bottom of base slab Hence, must include effect of base pressure

hi* and hc

* will be used

Ground level


Overturning moment:Ground supported tanks

Overturning moment in impulsive mode

tb = thickness of base slab

g/tmthm

thm)th(mAM

bbbtt

bwwb*ii

ih*i

2

Overturning moment in convective mode

gthmAM bccchc )()( **



Overturning moment is at the bottom of base slab Hence, lever arm is from bottom of base slab Hence, base slab thickness, tb is added to

heights measured from top of the base slab

Total overturning moment

2*2**ci MMM


Example

Example: A ground-supported circular tank is shown below along with some relevant data. Find base shear and bending moment at the bottom of wall. Also find base shear and overturning moment at the bottom of base slab.

mi = 141.4 t; mc = 163.4 t

mw = 65.3 t, mt =33.1 t,

mb = 55.2 t,

hi =1.5 m, hi* = 3.95 m,

hc = 2.3 m, hc* = 3.63 m

(Ah)i = 0.225, (Ah)c = 0.08

Roof slab 150 mm thick

Base slab 250 mm thick

4 m

10 m

Wall 200 mm thick


Example

Solution:Impulsive base shear at the bottom of wall is Vi = (Ah)i (mi + mw + mt) g

= 0.225 x (141.4 + 65.3 + 33.1) x 9.81 = 529.3 kN

Convective base shear at the bottom of wall is Vc = (Ah)c mc g

= 0.08 x 163.4 x 9.81 = 128.2 kN Total base shear at the bottom of wall is

544.6kN128.2529.3VVV 222c

2i


Example

For obtaining bending moment, we need height of CG of roof slab from bottom of wall, ht.

ht = 4.0 + 0.075 = 4.075 m

Impulsive bending moment at the bottom of wall is

Mi = (Ah)i (mihi + mwhw + mtht) g = 0.225 x (141.4 x 1.5 + 65.3 x 2.0 + 33.1 x 4.075) x

9.81 = 1054 kN-m

Convective bending moment at the bottom of wall is Mc = (Ah)c mc hc g = 0.08 x 163.4 x 2.3 x 9.81 = 295 kN-m

Total bending moment at bottom of wall is m-1095kN2951054MMM 222

c2i


Example

Now, we obtain base shear at the bottom of base slab

Impulsive base shear at the bottom of base slab is Vi = (Ah)i (mi + mw + mt + mb) g = 0.225 x (141.4 + 65.3 + 33.1 + 55.2) x 9.81 = 651.1 kN Convective base shear at the bottom of base slab is Vc = (Ah)c mc g = 0.08 x 163.4 x 9.81 = 128.2 kN Total base shear at the bottom of base slab is

663.6kN128.2651.1VVV 222c

2i


Example

Impulsive overturning moment at the bottom of base slab Mi

* = (Ah)i [mi (hi* + tb) + mw(hw + tb) + mt(ht +tb) + mb

tb/2]g = 0.225 x [141.4(3.95 + 0.25) + 65.3(2.0 + 0.25) + 33.1(4.075 + 0.25) + 55.2 x 0.25/2] x 9.81 = 1966 kN-mConvective overturning moment at the bottom of base slab Mc

* = (Ah)c mc (hc* + tb) g

= 0.08 x 163.4 x (3.63 + 0.25) x 9.81 = 498 kN-m

Total overturning moment at bottom of base slab

Notice that this value is substantially larger that the value at the

bottom of wall (85%)

m-kN 20284981966MMM 22*c

*i

* 22


Base shear : Elevated tanks

In elevated tanks, base shear at the bottom of staging is of interest

Ms is structural mass Base shear in impulsive mode

gmmAV siihi

Base shear in convective mode

gmAV cchc

Total base shear22ci VVV


Bending moment:Elevated tanks

Bending moment at the bottom of staging Bottom of staging refers to footing top

Impulsive base shear comprises of two parts (Ah)i x mig Ah)i x msg

Convective base shear has only one part (Ah)c x mcg



mi acts at hi*

mc acts at hc*

Bending moment at bottom of staging is being obtained

Hence, effect of base pressure included and hi*

and hc* are used



Structural mass, ms comprises of mass of empty container and 1/3rd mass of staging ms is assumed to act at CG of empty container CG of empty container shall be obtained by

considering roof, wall, floor slab and floor beams



Bending moment at the bottom of staging

ghmhhmAM cgss*iiih

*i

ghhmAM s*ccch

*c

hs = staging height Measured from top of footing to bottom of wall

hcg = distance of CG of empty container from bottom of staging



Bending moment at the bottom of staging

Top of footing

(Ah)i msg

hs

hcg

hi*

hs

hc*

ghhmAM s*ccch

*c ghmhhmAM cgss

*iiih

*i

(Ah)i mig

(Ah)c mcg



Total bending moment

22c*

i** MMM

For shaft supported tanks, M* will be the design moment for shaft



For analysis of frame staging, two approaches are possible

Approach 1: Perform analysis in two steps Step 1:

Analyze frame for (Ah)imig + (Ah)imsg Obtain forces in columns and braces

Step 2: Analyze the frame for (Ah)cmcg Obtain forces in columns and braces

Use SRSS rule to combine the member forces obtained in Step 1 and Step 2



Approach 2: Apply horizontal force V at height h1 such

that V x h1 = M* V and M* are obtained using SRSS rule as

described in slide nos. 26 and 32 In this approach, analysis is done in single step

Simpler and faster than Approach 1


Example

Example: An elevated tank on frame staging is shown below along with some relevant data. Find base shear and bending moment at the bottom of staging.

AA is CG of empty container

mi = 100t; mc = 180 t

Mass of container = 160 t

Mass of staging = 120 t

hi* = 3 m, hc

* = 4.2 m

(Ah)i = 0.08, (Ah)c = 0.04

GL

hs = 15 m

2.8 m


Example

Structural mass, ms = mass of container + 1/3rd mass of staging = 160 + 1/3 x 120 = 200 t

Base shear in impulsive mode

gmmAV sihi i

819x200100x080 ..

Base shear in convective mode gmAV cchc

819x180x040 .. kN6.70

= 78.5 + 157 = 235.5 kN


Example

Total base shear

22ci VVV

22 6705235 .. kN8.245

Now, we proceed to obtain bending moment at the bottom staging

Distance of CG of empty container from bottom of staging, hcg = 2.8 + 15 = 17.8 m


Example

Base moment in impulsive mode

ghmhhmAM cgss*iiih

*i

819x817x2001503x100080 ....

= 78.5 x 18 + 157 x 17.8

= 4207 kNm

Note: 78.5 kN of force will act at 18.0m and 157 kN of force will act at

17.8 m from top of footing.


Example

Base moment in convective mode

ghhmAM s*ccch

*c

819x1524x180x040 ...

Total base moment22c*

i** MMM

22 13564207 kNm4420

= 70.6 x 19.2= 1356 kNm

Note: 70.6 kN of force will act at 19.2 m from top of footing.


Example

Now, for staging analysis, seismic forces are to be applied at suitable heights

There are two approaches Refer slide no 33

Approach 1: Step 1: Apply force of 78.5 kN at 18 m and 157 kN at

17.8 m from top of footing and analyze the frame Step 2: Apply 70.6 kN at 19.2 m from top of footing and

analyze the frame Member forces (i.e., BM, SF etc. in columns and braces)

of Steps 1 and 2 shall be combined using SRSS


Example

Approach 2: Total base shear, V = 245.8 kN will be applied at height

h1, such that

V x h1 = M*

245.8 x h1 = 4420

h1 = 17.98 m Thus, apply force of 245.8 kN at 17.98 m from top of

footing and get member forces (i.e., BM, SF in columns and braces).


Elevated tanks:Empty condition

Elevated tanks shall be analysed for tank full as well as tank empty conditions Design shall be done for the critical condition

In empty condition, no convective liquid mass Hence, tank will be modeled using single

degree of freedom system Mass of empty container and 1/3rd staging

mass shall be considered


Elevated tanks:Empty condition

Lateral stiffness of staging, Ks will remain same in full and empty conditions

In full condition, mass is more In empty condition mass is less

Hence, time period of empty tank will be less Recall, T = Hence, Sa/g will be more

Usually, tank full condition is critical However, for tanks of low capacity, empty

condition may become critical

K

M2Π


Direction of seismic force

Let us a consider a vertical cantilever with rectangular cross section

Horizontal load P is applied First in X-Direction Then in Y-direction (see Figure below) More deflection, when force in Y-direction Hence, direction of lateral loading is important !!

P

P

X

Y



On the other hand, if cantilever is circular Direction is not of concern Same deflection for any direction of loading

Hence, it is important to ascertain the most critical direction of lateral seismic force Direction of force, which will produce maximum

response is the most critical direction In the rectangular cantilever problem, Y-direction is the

most critical direction for deflection



For frame stagings consisting of columns and braces, IS 11682:1985 suggests that horizontal seismic loads shall be applied in the critical direction

IS 11682:1985, “Criteria for Design of RCC Staging for Overhead Water Tanks”, Bureau of Indian Standards, New Delhi

Clause 7.1.1.2 Horizontal forces – Actual forces and moments resulting from horizontal forces may be calculated for critical direction and used in the design of the structures. Analysis may be done by any of the accepted methods including considering as space frame.

Clause 7.2.2 Bending moments in horizontal braces due to horizontal loads shall be calculated when horizontal forces act in a critical direction. The moments in braces shall be the sum of moments in the upper and lower columns at the joint resolved in the direction of horizontal braces.



Section 4.8 of IITK-GSDMA Guidelines contains provisions on critical direction of seismic force for tanks

Ground-supported circular tanks need to be analyzed for only one direction of seismic loads These are axisymmetric Hence, analysis in any one direction is sufficient



Ground-supported rectangular tanks shall be analyzed for two directions Parallel to length of the tank Parallel to width of the tank Stresses in a particular wall shall

be obtained for seismic loads perpendicular to that wall



RC circular shafts of elevated tanks are also axisymmetric Hence, analysis in one direction is sufficient

If circular shaft supports rectangular container Then, analysis in two directions will be

necessary



For elevated tanks on frame staging Critical direction of seismic loading for columns

and braces shall be properly ascertained Braces and columns may have different

critical directions of loading For example, in a 4 - column staging

Seismic loading along the length of the brace is critical for braces

Seismic loading in diagonal direction gives maximum axial force in columns

See next slide



Critical directions for 4 - column staging

Critical direction for shear force in brace

Critical direction for axial force in column

Bending Axis



For 6 – column and 8 – column staging, critical directions are given in Figure C-6 of the Guideline

See next two slides More information available in Sameer

and Jain (1994) 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. (http://www.nicee.org/ecourse/Tank_ASCE.pdf)




Critical direction for shear force and bending moment in columns

Critical direction for shear force and bending moment in braces and axial force in columns

Bending Axis




Critical direction for shear force and bending moment in braces

Critical direction for shear force, bending moment and axial force in columns

Bending Axis



As an alternative to analysis in the critical directions, following two load combinations can be used

100 % + 30% rule Also used in IS 1893(Part 1) for buildings

SRSS rule



100%+30% rule implies following combinations

ELx + 0.3 ELY

ELY + 0.3 ELx

ELx is response quantity when seismic loads are applied in X-direction

ELY is response quantity when seismic loads are applied in Y-direction



100%+30% rule requires Analyze tank with seismic force in X-direction;

obtain response quantity, ELX

Response quantity means BM in column, SF in brace, etc.

Analyze tank with seismic force in Y-direction; obtain response quantity, ELY

Combine response quantity as per 100%+30% rule

Combination is on response quantity and not on seismic loads



Important to note that the earthquake directions are reversible

Hence, in 100%+30% rule, there are total eight load combinations



SRSS rule implies following combination

22yx ELEL

Note: ELx is response quantity when seismic loads

are applied in X-direction ELY is response quantity when seismic loads

are applied in Y-direction Hence, analyze tank in two directions and

use SRSS combination of response quantity


At the end of Lecture 5

This completes seismic force evaluation on tanks

There are two main steps Evaluation of impulsive and convective masses Evaluation of base shear coefficients for

impulsive and convective modes SRSS rule is used to combine impulsive

and convective responses Critical direction of seismic loading shall

be properly ascertained Else, 100%+30% or SRSS rule be used

Lecture 5 January 31, 2006. Sudhir K. Jain, IIT Kanpur E-Course on Seismic Design of Tanks/ January...

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