Dynamic Pressure for Circular Silos Under Seismic Force

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14th Symposium on Earthquake Engineering

Indian Institute of Technology, RoorkeeDecember 17-19, 2010

Paper No. A009

DYNAMIC PRESSURE ON CIRCULAR SILOS UNDER

SEISMIC FORCE

Indrajit Chowdhury 1 and Raj Tilak21Petrofac International Limited, Sharjah, UAE, [email protected]

2Petrofac International Limited, Sharjah, UAE, [email protected]

ABSTRACT

Circular silos (both steel and reinforced concrete) are often deployed to store material in various

industries like cement plants (clinkers), power plants( raw coal/coke), oil and gas industry( sulfur

pellets) etc. Technology that is in vogue for earthquake analysis of such structures is to consider the

silo and its content as a lumped mass and seismic effect of this mass is considered in design of thesupporting frame only. No effect of this seismic force is considered on silo wall when the content is

subjected to seismic vibration.

In the present paper a procedure has been suggested wherein the additional dynamic pressure due toearthquake can be incorporated in analysis of such circular silos. While carrying out this analysis,

conventional Jansens method has been modified to develop the additional dynamic pressure due to

seismic force and a parametric study has been done to study the effect of this dynamic pressure on thewall of silo for different structural configuration.

Keywords: Dynamic pressure, Jansens Method, IS code, Silo, Time period

INTRODUCTION

In conventional design office practice, static pressure on wall of rectangular, circular bunkers and silos

due to the fill material are usually estimated based on Airys or Jansens theory (Gray and Manning

1973). Though it is a well established fact that when seismic waves due to a major earthquake hits a

site, the whole system (i.e. the frame and the container together with its content) is subjected to

vibration and would induce additional dynamic pressure over and above the static pressure that isestimated based on the theories as cited above.

No rational theory exists till date in practice to estimate this additional pressure on the wall. Though

analysis based on Finite Element Method (FEM) could be one of the plausible method of analysis, yet

suffers from a major deficiency that - as the fill material is at different stages of compaction during its

operation, hence it is extremely difficult (if not impossible) to asses the in-situ elastic property of thefill material to carry out a comprehensive dynamic analysis of the problem. Thus it is completely leftto the judgment of the design engineer to provide additional reinforcement to cater to this unknown

factor. In recent past, techniques have been proposed to estimate the dynamic pressure due to seismic

force on rectangular bunkers (Chowdhury 2009) based on Airys method. A theory on similar line is

proposed herein and is extended to Jansens theory which remains the most popular technique inindustry for estimation of pressure on walls of circular silos.


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PROPOSED METHOD

As a first step the circular silo can be modeled as shown in the Fig. 1 below. The silo being supported

on columns and bracings, have been replaced by springs in vertical and horizontal direction. The mass

of the silo including the stored material is considered as M acting at the centre of gravity Z c from the

ring beam level from which the super structure is suspended on the frame as shown in Fig. 1.

Zc Kx

R R

Kz Kz

Fig. 1:- Structural arrangement of circular silo and its equivalent mathematical model

Based on the mathematical model as proposed in Fig. 1 the free vibration equation for the above

system can be expressed as (Meirovitch 1975)

0.

0

022

=

+

+

x

RKZcKZK

ZKKx

J

M

zxcx

cxx

&&

&&(1)

Here

M= Mass of the silo including its content

J= Mass moment of inertia of the silo and its content

Kx, Kz = Lateral and vertical stiffness of the supporting frame (magnitude elaborated later for different

structural configuration)

Zc= Center of gravity of the silo including its content above the collar or ring beam.R= Radius of the silo.

The Eigen solution of the problem is given by


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0.

22=

+

JRKZKZK

ZKMK

zcxcx

cxx(2)

Here = Eigen value of the of the problem

Considering 2= and T= /2 we can find out the first two fundamental time period of thesystem.

Considering time periods for the given two modes as T1 and T2, we can find out the value of Sa1/g and

Sa2/g for the first two modes for 5% damping say for a reinforced concrete structural system and 2-3%damping for a steel structure system.

Having estimated the acceleration the silo is subjected to based on its first two fundamental time

period the same can be fitted into Jansens theory based on figure -2 as shown below.

We take here a strip of the material stored in silo of depth dz and apply the applicable forces as shownin the figure below. The difference here is the additional force exerted on the wall due to the seismic

force generated by the stored material is also included.

These forces when applied on the wall using coefficient of friction ' we have the free body diagram

as shown in Fig. 2.

Fig. 2: Free body diagram of pressure equilibrium for an element of strip dz

For equilibrium, summation of forces in vertical direction, leads to the expression

0)(... =++++ Adzdz

dPPUdzPAdz

g

SdzAAP vvh

aiv (3)

0.. =

+++ dz

dz

dPdz

A

UPdz

g

Sdz vh

ai (4)


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Considering Rh=A/U andv

h

P

P= and substituting in Eq. (4) we have,

dzPRg

SdP v

h

aiv

=

1 (5)

Re-arranging and integrating over the entire depth of fill with appropriate boundary condition, we

finally get the total static plus dynamic pressure due to seismic force as

+

=

z

R

aih

vhe

g

SR

P

1

1

(6)

Horizontal intensity of pressure on the silo wall is then finally given by

=

zR

ai

h

heheg

SR

P

1

1

(7)

The sign components within the brackets will fluctuate depending upon the direction of the seismicforce. Therefore, to arrive at the maximum possible value of Phe, we must take positive sign.

+

=

z

R

aih

hehe

g

SR

P

1

1

(8)

Considering the vertical seismic component also, along with earlier defined horizontal component ofseismic force on the wall of the silo we can modify Eq. (8) to

++

=

z

Raiaihhev

heg

S

g

SRP

1

3

21 (9)

In Eq. (9) the third term in the first parenthesis represents the vertical component of seismic

acceleration as per Clause 6.4.5 of IS-1893 (2002).

In the above derivation the acronyms used are as defined here after

A= Area of cross section of the silo

g = Acceleration due to gravityI= Importance factor as per IS codePv= Vertical pressure on silo wall

Phe= Horizontal pressure on silo wall

Phev= Horizontal pressure on silo wall considering vertical acceleration alsoR= Ductility factor as per IS code

Rh= Hydraulic mean depth of the section @ A/U

Sai= Horizontal acceleration due to seismic force in the ith

mode where i=1,2.U= Perimeter of the section


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Z=Zone factor as per IS code

z =Any depth from top of silo

R

ZI

2= a code factor

= Weight density of the fill material

= Coefficient of friction between wall and the fill material

sin1

sin1

+==

v

h

P

P

= Angle of repose of the fill material

The final SRSS design pressure is given by

=

=2

1

2

i

hihd PP (10)

CASE STUDY

A silo of circular cross-section is being analyzed here for the pressure distribution along the wall of

the silo, using Janssens Formulation and is being compared with the increase or decrease in pressure

on the walls on introduction of seismic forces.

Seismic forces have been further split into two cases i.e.

i) With horizontal acceleration only andii) With both the horizontal and vertical accelerations.

The geometry of the silo considered has been taken as defined below:

Diameter of Silo : 10m

Height of Silo : 10mNumber of Columns : 8

Number of Panels : 2Number of Bracings : 16@8 each panel

Column Height : 8m@ 4m each panelColumn Cross-section : 0.5m0.5m

Beam Length : 3.83m

Beam Cross-section : 0.5m0.5m

Bracing Length : 4.57mBracing Cross-section : 0.5m0.5m

In order to study the behaviour of different structural arrangements, two types of structural

arrangements have been studied as given below:

Case 1: Staging configuration with circumferential beams (infinite stiffness)

Case 2: Staging configuration with circumferential beams and diagonal braces

The structural stiffness Kz and Kx as used in Eq. (1) are adapted from Dutta et. al. (2003) for above

defined configuration or cases and have been used here.


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hp(typ.)

(a) (b)

Fig. 3 Different structural configuration used as the staging frame

Case 1: For staging type as shown in Fig. 3a stiffness of the frame is given by

pp

cccx

Nh

NIEK

3

..12= (11)

Here Ec= Youngs modulus of the column material

Ic = Moment of inertia of the column cross sectionNc= Number of columns

Np= Number of panelshp= Height of panel

Ac= Area of cross section of column

Axial stiffness of columns may be represented as below

pp

cccz

Nh

NAEK

..= (12)

Case 2: For staging type as shown in Fig. 3b stiffness of the frame is given by

( ) arbarc

v

p

c

rppp

ccx

KCKC

LN

EAN

KNNh

IEK

21

2

0

00

31

cos2

12

112

++

+

+=

(13)

Here

=0A Area of cross section of diagonal braces

=0E Modulus of elasticity of diagonal braces

=0L Length of diagonal braces

=v Angle of inclination of diagonal brace with horizontal direction

=arcK Relative axial stiffness of diagonal braces, with respect to that of the column @ 0

00

LEA

hEA

cc

p

=arbK Relative axial stiffness of diagonal braces, with respect to that of the beams @0

00

LEA

LEA

bb

b

( )v

c

p

N

NC

222

1 sin1sin3

14

+

=


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v

p

p

N

NC 22 cos

1=

Axial stiffness of columns may be represented as below

pp

cccz

Nh

NAE

K

..

= (14)

Alternatively the vertical and lateral stiffness of the staging frame can be determined as mentioned

hereafter

M MRigid link (typ.)

Zc Zc

hp(typ.)

(a) (b)

Fig. 4 Mathematical Model to determine Kz and Kx of the staging frame

As shown in Fig. 4 is the three dimensional mathematical model of the staging frame that can be easily

modeled in any commercially available structural software like STAAD,SAP 2000, GTSTRUDL etc.The mass of the silo including its content can be lumped at the node M at a height Zc from the ring

beam and are connected to the staging frame by mass less rigid link of high stiffness ()10(

10K

. We

apply a unit load successively in the vertical and horizontal direction and find out the displacement at

the ring beam level as zand x say. Once the displacement are known the corresponding stiffness can

be obtained from the expression P=K.. Where, P is the applied unit load.

Material Properties:

Modulus of elasticity of columns, beams and bracings : 21.7 GPa

Coefficient of friction of fill material : 0.7Density of fill material : 14.1 kN/m3

Horizontal to vertical pressure ratio () : 0.27

Seismic Parameters:

Here IS 1893: 2002 is being used to calculate the seismic force, using the following parameters-Seismic zone factor for Zone V (Z) : 0.36

Type of spectra : III (soft soil)

Importance factor (I) : 1.5Response reduction factor (R) : 3 (ordinary moment resisting frame)

Modal damping ratio : 5%

Calculated time period of silo for two types of staging :


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Table 1: Time period of first two modes of vibrations of the silo

Kx

(N/m)

Kz(N/m)

m

(kg)

Zc(m)

J(kg-m2)

Modal Periods

1st

mode

(sec)

2nd

mode

(sec)

Case 1 8.48107 5.43109 1.38106 2.85 1.62108 0.804 0.217

Case 2 5.74108 5.43109 1.38106 2.85 1.62108 0.318 0.210

RESULTS AND DISCUSSION

Pressure profile along the depth has been calculated using the proposed method and compared with

Janssens pressure profile. Pressure profiles in Case 1, Case 2 have been compared with Janssens

pressure profile in Fig.1, 2 and 3 respectively.

Fig. 5: Comparison of effective pressure profile for Case 1, considering seismic analysis without

vertical seismic force (phe), with vertical seismic force (phev) and Janssens formula


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Fig. 6: Comparison of effective pressure profile for Case 2, considering seismic analysis without

vertical seismic force (phe), with vertical seismic force (phev) and Janssens formula

Fig. 7: Comparison of effective pressure profile for all cases, considering horizontal component of

seismic force (phe)


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Fig. 8: Comparison of effective pressure profile for all cases, considering horizontal and vertical

components of seismic force (phev)

Perusing the above data we can come to some very important and interesting conclusions mentioned

hereafter:

CONCLUSION

1. Ignoring the seismic effect we significantly under design the silo wall design pressure.2. The dynamic pressure can be as high as 37-40 % over Jansens static pressure depending on

the staging configuration and type of foundation on which it is resting like soft, medium stiff

soil or rock.

3. The vertical component of earthquake that is usually ignored in conventional structural designsignificantly enhances the lateral dynamic pressure on the silo wall and should not be ignored

especially when the silo is of large capacity.4. The mathematical model proposed herein while sound in logic is easy to apply within a designoffice framework and does not need an elaborate FEM analysis and can very well be adapted

in a spread sheet or a Mathcad shell.

5. Code committee could examine this phenomenon and consider incorporating this in designoffice practices in terms of IS code.

REFERENCES

1. Chowdhury I -2009 Dynamic response of reinforced concrete rectangular bunkers underearthquake force. The Indian Concrete Journal Vol-83 #2pp 7-18.

2. Dutta, Somnath, Mandal Aparna Dutta C Sekhar-2003 Soil Structure interaction in dynamicbehavior of elevated tanks with alternate frame staging configurations, Journal of Sound and

Vibration pp 1-29.3. Gray W.S. & Manning G.P- 1973 Reinforced Concrete Water tower Bunkers and Silos

Concrete Publications limited London.

4. IS-1893-2002Indian Standard Code of practice for earthquake resistant design of structuresBureau of Indian Standards New Delhi.

5. Meirovitch L -1975Elements of vibration analysis Allied Publishers Inc. New York USA.

Dynamic Pressure for Circular Silos Under Seismic Force

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