14th Symposium on Earthquake Engineering Indian Institute of Technology, Roorkee

December 17-19, 2010

Paper No. A009

DYNAMIC PRESSURE ON CIRCULAR SILOS UNDER SEISMIC FORCE

Indrajit Chowdhury 1 and Raj Tilak2

1 Petrofac International Limited, Sharjah, UAE, Indrajit.Chowdary@petrofac.com 2Petrofac International Limited, Sharjah, UAE, raj.tilak@petrofac.com

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 the supporting 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 to earthquake can be incorporated in analysis of such circular silos. While carrying out this analysis, conventional Jansen’s 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 the wall of silo for different structural configuration. Keywords: Dynamic pressure, Jansen’s 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 Airy’s or Jansen’s 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 is estimated 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 the fill material to carry out a comprehensive dynamic analysis of the problem. Thus it is completely left to 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 Airy’s method. A theory on similar line is proposed herein and is extended to Jansen’s theory which remains the most popular technique in industry for estimation of pressure on walls of circular silos.

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 Zc 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.

00

22 =⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+−

−+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡θθθ

xRKZcKZK

ZKKxJ

M

zxcx

cxx

&&

&& (1)

Here M= Mass of the silo including its content Jθ= 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

0.

22 =⎥⎦

⎤⎢⎣

⎡Λ−+−

−Λ−

θJRKZKZKZKMK

zcxcx

cxx (2)

Here Λ = Eigen value of the of the problem Considering 2ω=Λ and T= ωπ /2 we can find out the first two fundamental time period of the system. 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 Jansen’s 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 shown in 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 'µ 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)(... =++′+′+−− AdzdzdPPUdzPAdz

gSdzAAP v

vhai

v µγβµγ (3)

0.. =⎟⎠⎞

⎜⎝⎛+′+′+−⇒ dz

dzdP

dzAUPdz

gS

dz vh

ai µγβµγ (4)

Considering Rh=A/U and v

h

PP

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

dzPRg

SdP vh

aiv

⎭⎬⎫

⎩⎨⎧ ′

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ′−=λµβµγ 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

vheg

SRP

λµ

λµ

βµγ1

1 (6)

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

⎟⎟⎠

⎞⎜⎜⎝

⎛−

′

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′±=

′− z

R

aih

heheg

SRP

λµ

µ

βµγ1

1 (7)

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

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

′

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′+=∴

′− z

R

aih

hehe

gS

RP

λµ

µ

βµγ1

1 (8)

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

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+′+

′=

′− z

Raiaihhev

heg

Sg

SRPλµ

ββµµγ 1

321 (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 gravity I= Importance factor as per IS code Pv= Vertical pressure on silo wall Phe= Horizontal pressure on silo wall Phev= Horizontal pressure on silo wall considering vertical acceleration also R= 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

Z=Zone factor as per IS code z =Any depth from top of silo

RZI2

=β a code factor

=γ Weight density of the fill material =′µ Coefficient of friction between wall and the fill material

φφλ

sin1sin1

+−

==v

h

PP

=φ Angle of repose of the fill material The final SRSS design pressure is given by

∑=

=2

1

2

ihihd 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 Janssen’s 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 and ii) 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 : 10m Number of Columns : 8 Number of Panels : 2 Number of Bracings : 16@8 each panel Column Height : 8m@ 4m each panel Column Cross-section : 0.5m×0.5m Beam Length : 3.83m Beam Cross-section : 0.5m×0.5m Bracing Length : 4.57m Bracing Cross-section : 0.5m×0.5m 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.

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= Young’s modulus of the column material Ic = Moment of inertia of the column cross section Nc= Number of columns Np= Number of panels hp= 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

vp

c

rppp

ccx KCKC

LNEAN

KNNhIEK

21

2

0

00

3 1

cos2

12112

++

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

θ (13)

Here =0A Area of cross section of diagonal braces =0E Modulus of elasticity of diagonal braces =0L Length of diagonal braces

=vθ 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

LEAhEA

cc

p

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

00

LEALEA

bb

b

( )v

c

p

NN

C θπ 222

1 sin1sin3

14

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

vp

p

NN

C θ22 cos

1−=

Axial stiffness of columns may be represented as below

pp

cccz Nh

NAEK

..= (14)

Alternatively the vertical and lateral stiffness of the staging frame can be determined as mentioned hereafter M M Rigid 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( 10≈K . We apply a unit load successively in the vertical and horizontal direction and find out the displacement at the ring beam level as δz and δx say. Once the displacement are known the corresponding stiffness can be obtained from the expression P=K.δ. 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 : 0.7 Density of fill material : 14.1 kN/m3 Horizontal to vertical pressure ratio (λ) : 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) : 1.5 Response reduction factor (R) : 3 (ordinary moment resisting frame) Modal damping ratio : 5% Calculated time period of silo for two types of staging :

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

(N/m) Kz

(N/m) m

(kg) Zc

(m) Jθ

(kg-m2) Modal Periods

1st mode (sec)

2nd mode (sec)

Case 1 8.48×107 5.43×109 1.38×106 2.85 1.62×108 0.804 0.217 Case 2 5.74×108 5.43×109 1.38×106 2.85 1.62×108 0.318 0.210 RESULTS AND DISCUSSION

Pressure profile along the depth has been calculated using the proposed method and compared with Janssen’s pressure profile. Pressure profiles in Case 1, Case 2 have been compared with Janssen’s 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 Janssen’s formula

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

vertical seismic force (phe), with vertical seismic force (phev) and Janssen’s formula

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

seismic force (phe)

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 Jansen’s 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 design significantly 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 design office 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 design office practices in terms of IS code.

REFERENCES

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

2. Dutta, Somnath, Mandal Aparna Dutta C Sekhar-2003 “Soil Structure interaction in dynamic behavior 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-1893-2002 Indian Standard Code of practice for earthquake resistant design of structures Bureau of Indian Standards New Delhi.

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