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Nonlinear Dynamic Behavior of Granular Materials inBase Excited SilosPravin Jagtapa, Tanusree Chakrabortya & Vasant Matsagara

a Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, NewDelhi, IndiaPublished online: 25 Nov 2014.

To cite this article: Pravin Jagtap, Tanusree Chakraborty & Vasant Matsagar (2015) Nonlinear Dynamic Behaviorof Granular Materials in Base Excited Silos, Mechanics of Advanced Materials and Structures, 22:4, 313-323, DOI:10.1080/15376494.2014.947821

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Mechanics of Advanced Materials and Structures (2015) 22, 313–323Copyright C© Taylor & Francis Group, LLCISSN: 1537-6494 print / 1537-6532 onlineDOI: 10.1080/15376494.2014.947821

Nonlinear Dynamic Behavior of Granular Materialsin Base Excited Silos

PRAVIN JAGTAP, TANUSREE CHAKRABORTY, and VASANT MATSAGAR

Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India

Received 2 February 2012; accepted 18 July 2014.

Nonlinear behavior of granular materials stored in steel silos subjected to dynamic base excitation due to earthquake is presented inthe current article. Three-dimensional finite element (FE) modeling of the granular material silo is carried out under three-directionalearthquake ground acceleration time histories. Granular material is modeled by adopting a continuum approach. The nonlinearityof the granular materials is represented by a hypoplastic material law in the FE approximation. The interface between the granularmaterial and the silo wall is modeled by using surface-to-surface based contact formulation. The horizontal and vertical displacementsof the granular material under earthquake ground acceleration at various depths of the silo are studied. Moreover, the stresses inducedin the steel silo are also investigated. The static FE simulation and the analytical solution obtained by using Janssen’s theory areobserved to be in close agreement. Also, the dynamic FE simulations compare with the calculated results using Eurocode 8 part 4with reasonable accuracy. The stresses in the steel silo wall are higher for loose packing of the granular material as compared to thatfor the dense packing.

Keywords: dynamic, earthquake, granular material, hypoplastic, nonlinear, silo

1. Introduction

Silos are storage structures usually called bins. Silos have along history of use in industrial and commercial sectors forstorage of granular materials. Failure of silos during naturalcalamities, such as earthquakes, may lead to catastrophic situ-ations if hazardous chemicals are stored or lead to shortage offood supply if food grains are stored. Hence, it is very impor-tant to maintain the structural integrity of silos during strongearthquakes. Moreover, as granular materials are widely usedin many kinds of industrial and agricultural processes, thecomprehensive study and understanding of the static and dy-namic responses of granular materials under earthquake load-ing is crucial.

Dogangun et al. [1] reviewed various cases of failure of silosaround the world due to earthquake. During the El Salvador,2001 earthquake, three people lost their lives as a result ofa silo failure. During the Edgecumbe, 1987 earthquake, hugestainless steel milk silos collapsed spilling thousands of litersof milk. It was observed that the effect of the vertical com-ponent of earthquake load has less effect on relatively high

Address correspondence to Tanusree Chakraborty, Departmentof Civil Engineering, Indian Institute of Technology Delhi, HauzKhas, New Delhi 110016, India. E-mail: tanusree@civil.litd.ac.inColor versions of one or more of the figures in the article can befound online at www.tandfonline.com/umcm.

and heavy silo structures. However, the horizontal compo-nents of the earthquake increase the bending moment at thebase of the silo. Nonuniform pressure distribution appears atthe base of the silo due to the increased bending moment atthe base, which is significantly more than the pressure causedby gravity loads. The material in the silo oscillates during anearthquake and causes damage to the upper portion of thesilo.

Granular materials demonstrate highly nonlinear behaviorunder dynamic loading from the very initial stage of the load-ing. Hence, a suitable material constitutive law is necessaryto capture the nonlinear response of the granular materials.In the literature, the theory of hypoplasticity has been usedextensively to model the stress-strain behavior of the granularmaterials [2–4]. Hypoplasticity is a generalization of hypoe-lasticity (minimum elastic) in which the stress rate is expressedas isotropic tensorial functions of the invariants of stressesand rate of deformation tensors. A hypoplastic model pro-vides an improved approach to the conventional elasto-plasticconstitutive model for granular materials. As distinct fromelasto-plasticity theories, hypoplasticity was devised withoutthe introduction of yield surface, flow rule, and without the de-composition of the deformation into elastic and a plastic part.Kolymbas [5] developed a hypoplasticity approach in whichthe evolution equation for the stress tensor consists of the sumof a linear and nonlinear rate of deformation tensor expressedas tensorial functions. Niemunis and Herle [3] and von Wolf-fersdorff [4] considered the effect of intergranular strain in thehypoplastic constitutive model.

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314 P. Jagtap et al.

There have been very few studies in the literature whereinhypoplastic material constitutive models are used for granularmaterials under dynamic excitations due to the challengingnature of the problem. Holler and Meskouris [6] analyzeda granular material silo under a synthetic, two-dimensional(2D) earthquake base excitation using a hypoplastic granu-lar material model proposed by Niemunis and Herle [3]. Theyused a finite element (FE) approach. Nateghi and Yakhchalian[7] studied the seismic behavior of granular material silos ofdifferent height-to-diameter ratios subjected to unidirectionalearthquake excitation. However, as per the authors’ knowl-edge, analysis of a silo under real three-directional earthquakebase excitations considering a hypoplastic granular materialmodel and proper silo-granular material interaction has notbeen attempted yet.

The specific objectives of the present study are: (i) to de-velop a comprehensive three-dimensional FE model of granu-lar material silo including all possible sources of nonlinearitiesfor static and dynamic analyses; (ii) to study the behavior ofgranular material in a steel silo under three-directional baseexcitation due to real earthquake; and (iii) to compare the ef-fect of loose and dense packing of the granular material onthe seismic response of granular material silo.

In the present study, a three-dimensional (3D) FE analysisof a cylindrical steel silo has been performed considering: (i)three-directional earthquake ground acceleration time histo-ries, (ii) an advanced hypoplastic constitutive model [4] forthe granular material, and (iii) friction contact interactionbetween the steel silo and the granular material. The FE soft-ware, Abaqus (version 6.11) has been used for the purpose ofnumerical simulation. Abaqus/Standard [8] implicit dynamicprocedure has been chosen for the analysis over an explicitdynamic procedure in order to ensure convergence and accu-racy of the solution. The analyses have been performed forloose and dense packing for granular material in the steelsilo considering the Imperial Valley (May 18, 1940) earth-quake ground acceleration recorded at El Centro instrumentstation.

2. Hypoplastic Constitutive Model for GranularMaterials

The hypoplastic constitutive model for granular materials usedin the present study was developed by von Wolffersdorff [4].This material law describes the incremental stress (��′

ij) asa function of stress (�′

ij), incremental strain (�εij), and voidratio (e) and is well suited for cohesionless granular materials.The index notations used herein (viz. i, j, k, . . .) can take valuesfrom 1 to 3 depending on the number of Cartesian coordinates.The stress-strain relation of the hypoplastic constitutive modelproposed by von Wolffersdorff [4] is:

��′ij = Lijkl�εkl + Nij

√�εkl�εkl, (2.1)

where Lijkl is the fourth-order linear stiffness tensor and N ij isthe second-order nonlinear stiffness tensor. Both L and N arefunctions of stress (�′

ij), void ratio (e), Lode’s angle (�), and

critical state friction angle (�c) given by:

Lijkl = fs1

�mn · �mn

⎡⎣F2�ik�jl +

(√3 (3 − sin�c)

2√

2sin�c

)2

�ij · �kl

⎤⎦

(2.2)

and

Nij = fs fd F

(√3 (3 − sin�c)

2√

2sin�c

)(�ij + �∗

ij

)/(�kl · �kl),

(2.3)where �ij = �′

ij/(�mn�mn) is the normalized stress and �∗ij =

�′ij − (1/3)�ij is the normalized deviatoric stress. �ij is the Kro-necker’s delta. The parameter F in Eq. (2.3) is given by:

F =√

18

tan2� + 2 − tan2�

2+√2tan� cos3�

− 1

2√

2tan� , (2.4)

where tan� = √3√

�∗kl�

∗kl and cos3� =

−√6�∗

ij �∗jk�∗

ki/(�∗mn�∗

mn)3/2; note that � is the Lode’s an-gle. Multi-axial stress-strain response of granular materials iscaptured in this model through the dependence of L and Non the Lode’s angle (�). The factor f s in Eq. (2.2) is given by:

fs = hs

n

(ei

e

)� 1 + ei

ei

(−�ij�ij

hs

)1−n[

3 +(√

3(3 − sin �c)

2√

2 sin �c

)2

−(√

3 (3 − sin �c)

2√

2 sin �c

)√3(

ei0 − ed0

ec0 − ed0

)�]−1

, (2.5)

where hs is granular hardness; n is exponent of granular hard-ness; � and � are the material parameters; ed0, ec0, and ei0 re-spectively describe the conventional minimum, conventionalmaximum, and maximum possible void ratios, where ed0 < ec0< ei0; e is the current void ratio; and ei is the maximum possi-ble void ratio depending on the current stress state as definedlater. The factor f d in Eq. (2.3) is given by:

fd = ei − ed

ec − ed. (2.6)

The void ratio values mentioned above are not constantsin the present model, but they depend on the current stress inthe material as given below:

ei

ei0= ec

ec0= ed

ed0= e

e0= exp

[−(

−�′

ij�ij

hs

)n], (2.7)

where depending on the current stress state ec and ed arethe conventional maximum and minimum void ratios, respec-tively; and e0 is the initial void ratio.

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Nonlinear Dynamic Behavior of Granular Materials in Base Excited Silos 315

Fig. 1. Silo model.

The limit surface, � (�) of the present model follows theirreversible surface given by Chambon et al. [9]:

� (�) = (L−1 N) · (L−1 N) − 1 = 0. (2.8)

Fig. 2. 3D silo model FE meshing.

Table 1. Silo dimensions

Sr.no. Parameter Value

1 Height h = 1.1 m2 Diameter d = 1.0 m3 Thickness of steel silo t = 3 mm4 Foundation length L = 1.2 m5 Foundation breadth B = 1.2 m6 Foundation thickness T = 0.3 m

The flow rule considered in the present model is asso-ciative. The inter-granular strain effect has not been takeninto account. The hypoplastic model has been validated byvon Wolffersdorff [4]. The model predicts the nonlinear andinelastic loading-unloading behavior of granular material withreasonable accuracy. In the present study, the hypoplastic con-stitutive model has been incorporated in the FE softwareAbaqus through user material (UMAT) subroutine to simu-late the behavior of granular materials under earthquake baseexcitation.

3. Three-Dimensional Finite Element Modelingof Granular Material Silo

The 3D FE model of the granular material silo is developedin Abaqus (version 6.11) software. The schematic geometryand the FE model of the silo are shown in Figures 1 and 2,respectively. The model consists of a cylindrical steel silo, thegranular material inside the silo, and the concrete foundationfor the silo. The steel silo has an aspect ratio of 1.1 and isconsidered to be completely filled with the granular material.Table 1 shows the dimensions of the granular material siloconsidered in the present study, which has been chosen perHoller and Meskouris [6].

The steel silo is modeled using four node shell elements withreduced integration (S4R). The granular material is modeledusing a continuum approach. The FE modeling of the granularmaterial and the concrete foundation is done using 3D eightnode elements with full integration (C3D8) to avoid hourglass-ing of the elements. Appropriate material models are used foranalysis purposes as discussed later. Table 2 presents the ele-ment types and total number of elements used to model thesteel silo, granular material, and concrete foundation.

Table 2. Finite element information

Type ofelement Name of element

Number ofelements

Element usedto simulate

Shell element S4R (LinearQuadrilateral)

1260 Steel silo

Solid element C3D8 (LinearHexahedron)

8211 Granularmaterial

Solid element C3D8 (LinearHexahedron)

4500 Concretefoundation

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Table 3. Material properties of steel and concrete

Sr.no. Material

Density (kg/m3)

Modulus ofelasticity E (MPa)

Poisson’sratio

1 Steel 7850 2.1 × 105 0.32 Concrete

(M25)2500 25 × 104 0.2

Fixed boundary conditions are applied at the bottom sur-face of the foundation. The interface between the steel silo andthe granular material has been modeled using the surface-to-surface friction contact formulation available in Abaqus, withassumed constant coefficient of friction, � = 0.5. The hardcontact behavior has been considered in the radial direction;thereby, the granular material cannot penetrate the steel silo.The inner surface of the steel silo has been considered as themaster surface and the outer surface of the granular mate-rial as the slave surface. The granular material can move inthe circumferential direction. The total number of degrees offreedom in the model is 55107. Subsequently, static analysis isperformed and results are compared with Janssen’s analyticalsolution for silos in order to validate the FE modeling. Fur-ther, nonlinear dynamic analysis of the granular material silois performed for base excitation due to earthquake.

3.1. Material Properties of Steel Silo and ConcreteFoundation

The steel silo and the concrete foundation are considered tofollow linear-elastic material behavior. The density ( ), mod-ulus of elasticity (E), and Poisson’s ratio () values for steeland concrete, considered in the present study, are given inTable 3.

3.2. Hypoplastic Material Properties for Granular Material

The stress-strain response of the granular material is simulatedusing an advanced hypoplastic constitutive model proposedby von Wolffersdorff [4]. The parameters considered for thegranular material are summarized in Table 4. The void ratioconsidered for loose packing is eL = 0.8 and that for densepacking is eD = 0.5. The density, = 1610 kg/m3 is for thegranular material.

Table 4. Hypoplastic material properties [6]

Sr.no. Parameter Value

1 Critical state friction angle �c = 37◦2 Granular hardness hs = 11,700 MPa3 Conventional minimum void ratio ed0 = 0.5324 Conventional maximum void ratio ec0 = 0.6905 Maximum possible void ratio ei0 = 0.8006 Exponent of granular hardness n = 0.437 Material parameter � = 0.1058 Material parameter � = 1.0

4. 3D Finite Element Analysis

4.1. Static Analysis and Validation of FE Model

The validity of the numerical FE analysis is ensured by com-paring results calculated analytically using Janssen’s theoryfor design of silo under gravity loading. For this analysis, thebehavior of the granular material is considered to be linear-elastic with assumed density, = 1800 kg/m3; modulus ofelasticity, E = 50 MPa; and Poisson’s ratio, = 0.2. It is to benoted that in the FE simulation the steel silo wall is consid-ered to be rigid in order to compare the results with Janssen’stheory. Figure 3a shows the comparison of the vertical pres-sure [Pv (kPa)] distribution along the depth (m) of the steelsilo obtained from the FE simulation with the analytical re-sults calculated using Janssen’s theory as given by Eq. (4.1)below:

Pv = wR�

1 + sin �c

1 − sin �c

[1 − exp

(−�h(1 − sin �c)

R(1 + sin �c)

)], (4.1)

where w is weight density (= g); R is hydraulic mean radius(= d/4); � is coefficient of friction between wall and granularmaterial considered here as 0.577; and g is the gravitationalacceleration.

Figure 3b shows the comparison of the horizontal pres-sure [Ph (kPa)] distribution along depth (m) of the steel siloobtained from the FE simulation with the analytical resultscalculated using Janssen’s theory as given by Eq. (4.2) below:

Ph = 1 − sin �c

1 + sin �cPv. (4.2)

It is observed from Figures 3a and 3b that the vertical andhorizontal pressure distribution along the depth of the steelsilo predicted by the FE simulation and the analytical solutionobtained by using Janssen’s theory are in close agreement.The minor difference is attributed towards the dependence ofFE analysis on the linear-elastic constitutive behavior of thegranular material, whereas Janssen’s theory is independent ofany constitutive law while it takes into account the criticalstate friction angle.

4.2. Dynamic Analysis and Validation of FE Model

The earthquake analysis of a granular material silo is car-ried out with hypoplastic granular material properties as re-ported in Table 4. The silo dimensions are the same as thosegiven in Table 1. In the current analysis, the ground acceler-ation time histories of the Imperial Valley earthquake (May18, 1940) recorded at El Centro instrument station have beenconsidered. Figure 4a presents the ground acceleration timehistories for the Imperial Valley, 1940 earthquake in three di-rections along with corresponding peak ground acceleration(PGA) values. In Figure 4b, the earthquake excitation com-ponents applied along the three Cartesian coordinates, x, y,and z are shown. It is to be noted that S00E component isapplied in the x direction as xg, S90W component is appliedin the y direction as yg, and VERT component is applied in

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Nonlinear Dynamic Behavior of Granular Materials in Base Excited Silos 317

1.0

0.8

0.6

0.4

0.2

0.0

0 3 6 9 12 151.0

0.8

0.6

0.4

0.2

0.0

0 2 4 6

Janssen's Theory Results Abaqus Results

Vertical Pressure, Pv (kPa)

Dep

th (

m)

Janssen's Theory Results Abaqus Results

Horizontal Pressure, Ph (kPa)

Dep

th (

m)

(a) (b)

Fig. 3. (a) Silo depth (m) versus vertical pressure (Pv) (kPa). (b) Silo depth (m) versus horizontal pressure (Ph) (kPa).

the z direction of the silo model as zg. Time history analysisof the granular material silo subjected to three components ofthe earthquake excitation has been carried out using the im-plicit/dynamic procedure in Abaqus. The dynamic pressureenvelopes on the left and right side of the silo wall along both

the S00E and S90W component of the earthquake excitationsare compared with the dynamic pressure distribution calcu-lated using Eurocode 8 part 4 [10]. Following the Eurocode,the reference dynamic pressure �ph,so on the silo wall at a ver-tical distance x from flat bottom of the circular silo are given

-4

-2

0

2

4

-2

-1

0

1

2

0 5 10 15 20 25 30 35 40 45 50 55

-2

-1

0

1

2

Acc

eler

atio

n m

/sec

2

S00E, PGA = 3.42 m/sec2

S90W, PGA = 2.10 m/sec2

Acc

eler

atio

n m

/sec

2

Imperial Valley, 1940

VERT, PGA = 2.06 m/sec2

Acc

eler

atio

n m

/sec

2

Time (sec)

(U1)

(U2)

(U3,

S22

)

y

x

x

z

(U1)

( 11 )S

( 11 )S

gy

gx

gz

(a) (b)

Fig. 4. (a) Horizontal component (S00E), (S90W) and vertical component (VERT) of Imperial Valley, 1940 earthquake. (b) Key figureof steel silo under Imperial Valley, 1940 earthquake.

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318 P. Jagtap et al.

0 1 2 30

3

6

9

12

Spe

ctra

l Acc

eler

atio

n ( m

/sec

)

Time Period (sec)

Damping = 0.02

0 1 2 30.00

0.15

0.30

0.45

0.60VERTS90WS00E

Time Period (sec)

Spe

ctra

l Dis

plac

emen

t (m

) Imperial Valley, 1940

(a) (b)

Fig. 5. (a) Acceleration response spectrum for Imperial Valley, 1940 earthquake. (b) Displacement response spectrum for ImperialValley, 1940 earthquake.

as per Eq. (4.3) below:

�ph,so = �(z)� min(r∗s ; 3x), (4.3)

where �(z) is the ratio of response acceleration of the silo ata vertical distance z from the equivalent surface of the storedcontents to the acceleration of gravity, � is the bulk unit weightof the particulate material in a seismic design situation. Also,rs

∗ is defined in Eq. (4.4) below:

r∗s = min(hb, dc/2), (4.4)

where hb is the overall height of the silo from flat bottom to theequivalent surface of the stored content, and dc is the inside

diameter of the silo parallel to the horizontal component ofthe seismic action.

For calculation of Eurocode pressure, �(z) is consideredconstant and it is obtained from 2% damped accelerationresponse spectrum of the earthquake excitation as shownin Figures 5a and 5b. Eigenvalue analysis of a silo is car-ried out considering elastic behavior of the granular material.The first translational mode frequency of the silo is observedas 77.650 Hz. Distribution of the dynamic pressure (�ph,so)on the silo wall is plotted in Figures 6a and 6b in accor-dance with the Eurocode along with the pressure distribu-tion on the left and right side of the silo wall along S00Eand S90W components of the earthquake excitation as ob-tained from the numerical simulation. It may be observedfrom Figures 6a and 6b that the dynamic pressure distribu-tion obtained from Eurocode is conservative under the three

Fig. 6. (a) Eurocode and Abaqus dynamic pressure distribution on left and right side of the silo wall vs. silo depth in the direction ofearthquake excitation (S00E). (b) Eurocode and Abaqus dynamic pressure distribution on the left and right side of the silo wall vs.silo depth in the direction of earthquake excitation (S90W).

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Nonlinear Dynamic Behavior of Granular Materials in Base Excited Silos 319

Fig. 7. (a) Vertical displacement (U3) along circumferential path for loose and dense packing of granular material under earthquakebase excitation. (b) Vertical displacement (U3) along diametric path for loose and dense packing of granular material under earthquakebase excitation.

directional earthquake excitation of the granular materialsilo.

4.3. Nonlinear Dynamic Analysis for Earthquake

The dynamic analysis of the granular material silo is carriedout using three-directional (two horizontal and one vertical)ground acceleration time histories of an earthquake, adoptingthe implicit dynamic procedure in Abaqus. A numerical �-damping is assumed to be in the slight range (� = −0.05)in the Hilber-Hughes-Taylor operator [11]. It may be notedthat in order to perform an earthquake time history analysisin Abaqus, gravity loading has been applied on the FE modelin the dynamic loading step.

In the current analysis, the ground acceleration time histo-ries of the Imperial Valley earthquake (May 18, 1940) recordedat El Centro instrument station have been considered along thethree Cartesian coordinates, x, y, and z, as discussed before.The stress and displacement response of the granular materialsilo have been studied through the analyses. In Figure 4b, thecomponents of displacements U1, U2, and U3 along the threeCartesian coordinates, x, y, and z and stress components, e.g.,(i) hoop stress (S11) and (ii) vertical stress (S22), are shown.

5. Results and Discussion

The results of the earthquake analysis of the granular materialsilo have been studied for (i) loose packing of the granular ma-terial having void ratio (eL) of 0.8 and (ii) dense packing of the

granular material having void ratio (eD) of 0.5. The displace-ment of the granular material is checked in both horizontaland vertical directions. Figures 7a and 7b show the verticaldisplacement (U3) results at top, middle, and bottom surfacesof the granular material for both loose and dense packing.The results in Figure 7a are obtained along a circumferentialpath at a radial distance of 0.105 m from the inner surface ofthe steel silo in the granular material for the maximum peakvertical displacement. Figure 7b shows the peak vertical dis-placement plot along a diametric path along global x axis. Thepeak vertical displacement values are summarized in Tables 5aand 5b, respectively, for circumferential and diametric pathswhen the steel silo is subjected to the Imperial Valley, 1940earthquake.

The densely packed granular material exhibits more pos-itive vertical (upward) displacement at the top surface as

Table 5a. Peak vertical displacement (U3) values along circum-ferential path for loose and dense packing of granular materialunder earthquake base excitation

Granularmaterial surface Void ratio

Displacement of granularmaterial (m) × 10−4

Top surface 0.8 (loose packing) 2.6110.5 (dense packing) 2.066

Middle surface 0.8 (loose packing) 10.1350.5 (dense packing) 2.399

Bottom surface 0.8 (loose packing) 0.1620.5 (dense packing) 0.023

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Fig. 8. (a) Horizontal displacement (U1) along x directional vertical path for loose and dense packing of granular material underearthquake base excitation. (b) Horizontal displacement (U1) along y directional vertical path for loose and dense packing of granularmaterial under earthquake base excitation.

compared to the same for the loosely packed granular ma-terial. This is attributed to the dilative behavior exhibited bythe densely packed granular materials under the base excita-tion. On the other hand, the loosely packed granular materialexhibits contractive behavior under the base excitation.

The peak vertical displacement at the center of the steel silois higher than the same along the periphery. This is expectedbecause the friction contact interface between the silo walland the granular material restrains the vertical displacementof the granular material more along the periphery of the steelsilo than at its center.

Figure 8 shows the distribution of horizontal displacements(U1) of densely and loosely packed granular materials alongthe depth of the steel silo. Along the vertical axes, the distri-bution of the horizontal displacements are shown at a chosentime instance, t. The vertical axes are chosen at three differ-ent radial distances, i.e., −0.4995 m, 0, +0.4995 m from the

Table 5b. Peak vertical displacement (U3) values along diametricpath for loose and dense packing of granular material underearthquake base excitation

Granularmaterial surface Void ratio

Displacement of granularmaterial (m) × 10−4

Top surface 0.8 (loose packing) 3.4780.5 (dense packing) 4.125

Middle surface 0.8 (loose packing) 5.5890.5 (dense packing) 9.587

Bottom surface 0.8 (loose packing) 0.0150.5 (dense packing) 0.022

origin along the x axis in Figure 8a and along the y axisin Figure 8b. The time instances chosen for the densely (tD)and loosely (tL) packed granular materials along the x axisare tD = 0.028 s and tL = 0.019 s, respectively. The time in-stances chosen for the densely and loosely packed granularmaterials along the y axis are tD = 0.024 s and tL = 0.029 s,respectively.

Along the depth of the steel silo, the displacement of thegranular material diminishes, becoming zero with respect toground. It is to be noted that at the bottom, the granularmaterial is directly in contact with a 0.3-m-thick concretefoundation without any friction contact interaction betweenthe two. The densely packed granular material shows smallerdisplacement as compared to the loosely packed granular ma-terial in the circumferential direction. The peak horizontaldisplacement values are given in Tables 6a and 6b when sub-jected to the Imperial Valley, 1940 earthquake.

Table 6a. Peak horizontal displacement (U1) values along x di-rectional vertical path for loose and dense packing of granularmaterial under earthquake base excitation

Granularmaterial surface Void ratio

Displacement of granularmaterial (m) × 10−4

Left surface 0.8 (loose packing) 2.198(−0.4995 m) 0.5 (dense packing) 0.931

Middle surface 0.8 (loose packing) 1.056(0 m) 0.5 (dense packing) 1.294

Right surface 0.8 (loose packing) 0.889(+0.4995 m) 0.5 (dense packing) 1.838

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Nonlinear Dynamic Behavior of Granular Materials in Base Excited Silos 321

Table 6b. Peak horizontal displacement (U1) values along y di-rectional vertical path for loose and dense packing of granularmaterial under earthquake base excitation

Granularmaterial surface Void ratio

Displacement of granularmaterial (m) × 10−4

Left surface 0.8 (loose packing) 7.836(−0.4995 m) 0.5 (dense packing) 1.459

Middle surface 0.8 (loose packing) 5.745(0 m) 0.5 (dense packing) 0.430

Right surface 0.8 (loose packing) 5.984(+0.4995 m) 0.5 (dense packing) 1.489

Figures 9a and 9b show the U1 displacement of granularmaterial at radial distances of −0.4995 m and −0.3839 mfrom the steel silo along x and y directions. Note that thesilo-granular material interface is at -0.4995 m and higherdisplacement of granular material in a horizontal direction(U1) are observed at about −0.3839 m. Tables 7a and 7bshow the peak horizontal displacement values in the granularmaterial when the steel silo is subjected to the Imperial Valley,1940 earthquake.

Higher horizontal displacement is observed in the granularmaterial as compared to the silo-granular material interface.The lower displacement at the silo-granular material interfacecan be attributed to the restraint coming from the steel silo

Table 7a. Peak horizontal displacement (U1) values along x di-rectional adjacent vertical path for loose and dense packing ofgranular material under earthquake base excitation

Granularmaterial surface Void ratio

Displacement of granularmaterial (m) × 10−4

Left surface 0.8 (loose packing) 2.198(−0.4995 m) 0.5 (dense packing) 0.931

Left surface 0.8 (loose packing) 2.406(−0.3839 m) 0.5 (dense packing) 1.593

Table 7b. Peak horizontal displacement (U1) values along y di-rectional adjacent vertical path for loose and dense packing ofgranular material under earthquake base excitation

Granularmaterial surface Void ratio

Displacement of granularmaterial (m) × 10−4

Left surface 0.8 (loose packing) 7.836(−0.4995 m) 0.5 (dense packing) 1.459

Left surface 0.8 (loose packing) 6.150(−0.3839 m) 0.5 (dense packing) 1.662

wall. The higher displacement is observed in loosely packedgranular material as compared to that for the densely packed.Moreover, the displacement of the granular material towardsthe bottom of the steel silo diminishes, and peak horizontal

Fig. 9. (a) Horizontal displacement (U1) along x directional adjacent vertical path for loose and dense packing of granular materialunder earthquake base excitation. (b) Horizontal displacement (U1) along y directional adjacent vertical path for loose and densepacking of granular material under earthquake base excitation.

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322 P. Jagtap et al.

Fig. 10. (a) Hoop stress (S11) in silo wall along x direction for loose and dense packing of granular material under earthquake baseexcitation. (b) Hoop stress (S11) in silo wall along y direction for loose and dense packing of granular material under earthquakebase excitation.

Fig. 11. (a) Vertical stress (S22) in silo wall along x direction for loose and dense packing of granular material under earthquake baseexcitation. (b) Vertical stress (S22) in silo wall along y direction for loose and dense packing of granular material under earthquakebase excitation.

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Nonlinear Dynamic Behavior of Granular Materials in Base Excited Silos 323

displacement does not occur at the top free surface but at acertain depth.

Figures 10a and 10b show the distribution of hoop stresses(S11) in the steel silo, whereas Figures 11a and 11b showthe vertical stress distribution (S22) in the steel silo respec-tively along the depth of the steel silo at first (SP1) and fifth(SP5) section points of the shell elements. The response isobtained for loosely and densely packed granular materialswhen the steel silo is subjected to the Imperial Valley, 1940earthquake.

For both cases, the stresses in the steel silo wall are higherfor loose packing of the granular material as compared tothat for the dense packing. The higher stresses observed inthe case of the loose packing can be attributed to the higherhorizontal displacement observed for the loosely packed gran-ular material. The maximum hoop stress observed in the steelsilo is 120.854 kPa, which occurs at a certain distance fromthe bottom. The stresses at SP1 and SP5 are observed to haveapproximately the same magnitudes however opposite sign.

6. Conclusions

The present study investigates the nonlinear behavior of gran-ular materials stored in steel silos and the stresses induced inthe steel silo when subjected to dynamic base excitation dueto Imperial Valley (May 18, 1940) earthquake ground accel-eration recorded at El Centro instrument station. 3D finiteelement (FE) analysis is performed under three-directionalearthquake ground acceleration time histories using the FEsoftware Abaqus. The validity of the FE analysis is ensuredby comparing the static analysis results under gravitationalload considering linear-elastic material properties with theanalytical solution given by Janssen. In the dynamic anal-ysis, the stress-strain response of the granular materials issimulated by a hypoplastic material law. The interface be-tween the granular material and the steel silo is modeled us-ing surface-to-surface based contact formulation in Abaqus.The validity of the dynamic analysis results under earth-quake excitation applied along the three Cartesian coordi-nates is ensured by comparing the dynamic analysis resultswith that calculated using Eurocode 8 part 4. Effect of earth-quake loading on loosely and densely packed granular ma-terials is studied. The horizontal and vertical displacementsof the granular material at various depths and radial loca-tions and the stresses along the depth of the steel silo arestudied. The following conclusions are made from the presentstudy.

1. The densely packed granular material exhibits more up-ward displacement at the top surface as compared to thesame for the loosely packed granular material. The peak

vertical displacement at the center of the steel silo is higherthan the same along the periphery.

2. Along the depth of the steel silo, the horizontal displace-ment of the granular material diminishes, becoming zerowith respect to ground. The densely packed granular mate-rial shows smaller horizontal displacement as compared tothe loosely packed granular material in a circumferentialdirection.

3. Higher horizontal displacement is observed in the granularmaterial as compared to the silo-granular material inter-face. At the interface, the higher displacement is observedin loosely packed granular material as compared to thatfor the densely packed. Moreover, the displacement of thegranular material towards the bottom of the steel silo di-minishes, and peak horizontal displacement does not occurat the top free surface but at a certain depth.

4. The hoop and vertical stresses in the steel silo wall arehigher for loose packing of the granular material as com-pared to that for the dense packing.

References

[1] A. Dogangun, Z. Karaca, A. Durmus, and H. Sezen, Cause ofdamage and failures in silo structures, J. Perform. Const. Facil.,ASCE, vol. 23, no. 2, pp. 65–71, 2009.

[2] D. Kolymbas, An outline of hypoplasticity, Arch. Appl. Mech., vol.61, pp. 143–151, 1991.

[3] A. Niemunis and I. Herle, Hypoplastic model for cohesionless soilswith elastic strain range, Mech. Cohes.-Frict. Mater., vol. 2, no. 4,pp. 279–299, 1997.

[4] P.-A. von Wolffersdorff, A hypoplastic relation for granular materi-als with a predefined limit state surface, Mech. Cohes.-Frict. Mater.,vol. 1, no. 3, pp. 251–271, 1996.

[5] D. Kolymbas, Introduction to Hypoplasticity: Advances inGeotechnical Engineering and Tunnelling (Vol. 1), A. A. Balkema,Rotterdam, 1999.

[6] S. Holler and K. Meskouris, Granular material silos under dynamicexcitation: numerical simulation and experimental validation, Jour-nal of Structural Engineering, ASCE, Reston, VA, vol. 132 (10), pp.1573–1579, 2006.

[7] F. Nateghi and M. Yakhchalian, Seismic behaviour of silos withdifferent height to diameter ratios considering granular material-structure interaction, Int. J. Eng., vol. 25, no. 1, pp. 27–37, 2012.

[8] Abaqus/Standard User’s Manual, Version 6.11. Dassault SystemesSimulia Corporation, Providence, Rhode Island, USA.

[9] R. Chambon, J. Desrues, W. Hammad, and R. Charlier, CLoE, Anew rate-type constitutive model for geomaterials theoretical basisand implementation, Int. J. Numer. Anal. Methods Geomech., vol.18, pp. 253–278, 1994.

[10] European Committee for Standardization, Eurocode 8: Designof structures for earthquake resistance, Part 4: Silos, tanks andpipelines, European Committee for Standardization, Brussels, 2006.

[11] H.M. Hilber, T.J.R. Hughes, and R.L. Taylor, Improved numericaldissipation for time integration algorithms in structural dynamics,Earthq. Eng. Struct. Dyn., vol. 5, pp. 283–292, 1977.

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