SHEAR-BEAM ANALYSIS FOR SEISMIC RESPONSE OF METAL WHEAT BINS

B. O. Hardin, R. A. Bucklin, I. J. Ross

ABSTRACT. Grain bins are often located in areas where the risk of earthquakes requires they be designed to resist seismic loading. A description of the shear-beam method of seismic analysis, including discussion of an appropriate constitutive model for particulate materials, is presented. This is followed by formulation of the constitutive properties database for two types of wheat at two packing densities and three moisture contents, and the procedure for obtaining composite shear-beam properties from the individual wheat and bin wall properties. The results of seismic response computations for a large steel bin filled with wheat are given to illustrate the shear-beam analysis method. Keywords. Grain bins, Wheat, Earthquakes, Seismic analysis. Constitutive properties.

Grain bins are often located in areas where the risk of earthquakes requires that they be designed to resist seismic loading. The accelerations produced by earthquakes cause inertial loads on

bins that can lead to structural failure (ASAE, 1994); however, little guidance for estimating the magnitude and nature of the effects of seismic loads on grain bins is available.

Literature has been developed dealing with the problem of seismic loads on tanks storing liquids. Rammerstorfer et al. (1990) surveyed this literature and compared analytical and numerical models to experimental data for cylindrical above-ground liquid storage tanks with a vertical axis. While some of the failure modes of liquid storage tanks and grain bins under seismic loading are similar, most of the design principles for liquid storage tanks do not apply to grain bins. This is because of the fundamental difference in the response of liquids which behave as true fluids and grain which behaves as a visco-elastic material with frequency dependent viscosity (hysteretic). The purposes of this article are to discuss the stress-strain behavior of particulate materials under cyclic loading and to present a constitutive model and constitutive properties database for wheat that can be used for seismic analysis of grain bins.

Article was submitted for publication in May 1995; reviewed and approved for publication by the Structures and Environment Div. of ASAE in October 1995.

This article is published with the approval of the Director of the Kentucky Agricultural Experiment Station and designated Paper No. 96-05-060.

The authors are Bobby O. Hardin, Professor, Civil Engineering Dept., University of Kentucky, Lexington; Ray A. Bucklin, ASAE Member Engineer, Professor, Agricultural Engineering Dept., University of Florida, Gainesville; and I. Joe Ross, ASAE Fellow Engineer, Professor, Biosystems and Agricultural Engineering Dept., University of Kentucky, Lexington. Corresponding author: L Joe Ross, Biosystems and Agricultural Engineering, 128 Agricultural Engineering Building, University of Kentucky, Lexington, KY 40506-0276; telephone: 606-257-3000; e-mail: <ijross@bae.uky.edu>.

Most design codes offer little guidance in dealing directly with the seismic loading of grain bins. However, ACI 313 (American Concrete Institute, 1991) does offer guidance specifically dealing with grain bins. ACI 313 contains the provision that not less than 80% of the weight of stored material shall be included in the mass used for seismic analysis, with the stored material modeled as a lumped mass located at its volumetric centroid. The Australian bulk storage standard (Standards Association of Australia, 1990) contains a similar provision allowing use of 80% of the weight of stored material.

Harris and von Nad (1985) tested two model silos constructed from steel pipe to determine if the use of 80% of the weight of stored material is appropriate. The pipes (0.457 m diameter x 3.05 m height; and 0.203 x 1.52 m) were welded to a steel base and a hydraulic actuator was used to shake the models horizontally at frequencies of 1, 3, 5, 7, and 9 Hz. The models were tested empty, filled with sand, and filled with wheat. They measured the maximum deflection at the tops of the models, and concluded that the assumption of an effective weight equal to 80% of the weight of stored material is conservative. They observed that the models vibrated in the first mode at the lower test frequencies, but did not analyze the vibration modes or system damping in detail.

Ziolkowski et al. (1985), Hardin (1987), and Hardin et al. (1990) investigated the stress-strain and strength properties of wheat. Their test results and analyses have been used as part of the methodology presented herein for estimating the seismic response of grain bins filled with wheat.

This article begins with a description of the shear-beam method of seismic analysis, including discussion of an appropriate constitutive model for particulate materials like wheat. This is followed by formulation of the constitutive properties database for soft and hard red winter wheat at two packing densities and three moisture contents, and presentation of the procedure for obtaining composite shear-beam properties from the individual wheat properties and bin wall properties. Finally, the results of seismic response computations for a large steel bin filled with

VOL. 39(2):677-687

Transactions of the ASAE

© 1996 American Society of Agricultural Engineers 0001-2351 / 96 / 3902-0677 677

wheat are given to illustrate the shear-beam analysis method.

SHEAR-BEAM METHOD OF SEISMIC ANALYSIS

The seismic response of metal bins filled with wheat can be estimated by assuming that horizontal ground motion from an earthquake produces simple shear deformation within the bulk wheat and bin walls (fig. la). A structure deforming in this configuration is called a shear-beam. Seismic loading, which is applied to the shear-beam base at z « H, results from horizontal ground acceleration for a design earthquake (fig. lb). Shear stress-strain relations for bulk wheat and bin wall materials are required for the analysis. The bin walls are nearly elastic, but the shear stress-strain relations for particulate materials like wheat are very nonlinear (fig. Ic).

EQUIVALENT LINEAR ANALYSIS An iterative equivalent linear analysis in the frequency

domain can be used to estimate the response of this

nonlinear system (Schnabel et al., 1972). Considering vertical propagation of shear waves in a linear visco-elastic system, horizontal displacements u(z,t) within the shear-beam must satisfy the wave equation:

at^ az2 az^at (1)

where p = mass density G = shear modulus Ti = shear coefficient of viscosity t =time

The seismic loading is decomposed into harmonic components via Fourier transform, and the response of a linear system to the earthquake is obtained by superposition of displacements resulting from the set of harmonic loading components. The harmonic displacements for frequency (O have the form:

u(z,t)=(Ae^*^^ + Be- * ")ei«t (2a)

T ~ ^ T{Z)

r(z+dz)

wheat

wall—H

-r(2)

ground acceleration A

(c)

Figure 1-Shear-beam model for wheat and bin. (a) Cross-section showing shear deformation, (b) ground surface acceleration from design earthquake, and (c) shear stress-strain relationship for wheat.

where

e = base of natural logarithms A and B are constants determined by the loading, and

k = p(0^

W-'f)J 1/2

(2b)

Shear strains computed by the linear analysis are used to adjust material stress-strain properties for succeeding iterations, until assumed properties are compatible with computed strains.

CYCLIC SHEAR STRESS-STRAIN RELATION FOR

PARTICULATE MATERIALS The shear stress-strain relation for constant amplitude

cyclic loading of a particulate material is a hysteresis loop (fig. 2a). The slope of the loop reflects material stiffness (secant shear modulus G), and the area circumscribed by the loop is a measure of the energy dissipated during a cycle of loading. As shown in figure 2b, the ratio of energy dissipated in one loading cycle to elastic strain energy stored at maximum strain defines the visco-elastic damping ratio r|co/G = (AL/AJ)/(27I;). By definition, the hysteretic damping ratio D = (AL/AT)/(47C), giving:

215! = 2D G

(3)

for visco-elastic modeling of hysteretic materials. Damping in particulate materials is approximately

hysteretic, i.e., the damping ratio D is independent of frequency (Richart et al., 1970; Schnabel et al., 1972; Das, 1993). The hysteretic nature of damping in sands was observed by Hardin (1965). Many of the constitutive formulations used for sands have been shown to represent

678 TRANSACTIONS OF THE ASAE

backbone curve-

AL = area circumscribed by hysteresis loop

A T = area of cross-hatched triangle

strain amplitude

(a)

r^l pa r = dr/dt

rdT ^•-^Ilcyck,

T = Asin(jJt; t=Aojcoscot; dT= Acocoscotdt

AL = f^'^^(GAslncot + r) A co cos cot) Acocoscotdt = A^frrrjco)

1 2 A T = elastic strain energy stored at maximum strain = - GA

Compute A L / A J for Kelvin-Voigt visco-elastic model

(b)

Figure 2-(a) Hysteresis loop stress-strain relationship, and (b) visco-elastic damping ratio.

wheat behavior (Ziolkowski et al., 1985; Hardin et al., 1990). Thus, damping in wheat is assumed to be hysteretic, but this is a topic for future research. Equation 3 with D independent of co shows that x] must be taken inversely proportional to (O to represent hysteretic material behavior with a linear visco-elastic model.

Figure 3a illustrates a schematic representation of hysteresis loops for three different strain amplitudes 7, where the apex of each loop is located on the dashed backbone curve (points a, b, and c). The effect of increasing strain amplitude 7 is to decrease G and increase D. The reduction of G with 7 is specified by defining the backbone curve in terms of G^^ and x ax ( S- ^t)); G ax is the initial tangent modulus for the backbone curve (G approaches Gj ax as 7 approaches 0), and x ax is the simple shear strength of the particulate material. The reference strain:

(a)

(b)

Figure 3-Efifect of strain amplitude on cyclic stress-strain relation (Hardin and Drnevich, 1972).

that is nearly independent of state of stress in the particulate material (Hardin and Drnevich, 1972).

Derivation of a relationship between shear modulus and damping by Hardin and Drnevich (1972) is given in figure 4. Assuming loop area AL is a constant proportion of area abc [2kj (area abc)] in figure 4, for all strain amplitudes, the relationship:

Xmax (4) - ^ = 1 D„ G„

(5b)

can be used to normalize the backbone curve and define a modulus reduction function

Gmax Wr^ (5a)

is obtained, where D ^ x i the damping ratio at a large strain amplitude (maximum value of D).

Equations 5 provide a formulation of constitutive properties that can be used for equivalent linear seismic analysis of systems that include particulate materials such as wheat.

VOL. 39(2):677-687 679

1 A L Damping Ratio = D = '^'fi^

AL/2 = cross-hatched area = ki(areaabc)

:)

Damping measurements for wheat are a topic for future research; consequently, the value of D ax = ^ ^ ^ ^ ^ ^ wheat is based on damping measurements for clean sand reported by Hardin and Dmevich (1972). The damping function in figure 5b was obtained by substituting G/G^ax from figure 5a into equation 5b (except when G/G^ax = !» see table 2). The damping function in figure 5b has been used to model all wheat materials analyzed herein.

Values of y^ must be determined from equation 4 to use the modulus reduction and damping functions in figure 5 for a specific seismic analysis. This requires wheat models for Gn ax an^ ' max ^hich are presented in the following sections.

-n-^) assuming ki Is constant,

asG-^0 ;D-^2k i /7T = Dmax Dmax = 1 -

Figure 4-Relationship between modulus and damping for particulate materials (Hardin and Drnevich, 1972).

WHEAT PROPERTIES Test results have been presented by Hardin et al. (1990)

for soft and hard red winter wheat in dense and loose density states, and 8,13, and 18% (w.b.) moisture contents, for a total of 12 material states. The specific gravities of grains reported by Hardin (1987) were used to compute unit weights shown in table 1, assuming void ratios (volume of voids/volume of solids), e = 0.60 and 0.68 for dense and loose packings, respectively.

SHEAR MODULUS AND DAMPING VERSUS S T R A I N A M P L I T U D E

Hardin (1987) used the torsional resonant column test to measure the effect of strain amplitude on shear modulus for the 12 material states listed in table 1. He tested specimens subjected to five different isotropic states of stress, 03 = 17, 34, 69, 138, and 276 kPa, a total of 59 G versus 7 curves. All of these curves were approximated by a single modulus reduction relationship (eq. 5a) after normalization, using Gniax ^^^ ^ reference strain Yr (eq. 4). Based on these normalized test results, and considering the fact that resonant column vibration applies many more cycles than an earthquake, the shear modulus reduction function in figure 5a has been used to represent the hysteretic behavior of all wheat materials analyzed herein. Equivalent linear analysis involves log-linear interpolation between values in table 2.

Table 1. Bulk unit weights and specific gravities of grains

Wheat Type

Moisture Content % (wb)

Grain Specific Gravity

7wheat(kN/m3)

Lx)ose Dense

Soft red winter wheat

Hard red winter wheat

8 13 18

8 13 18

1.37 1.38 1.34

1.34 1.35 1.36

8.00 8.06 7.82

7.82 7.88 7.94

8.40 8.46 8.22

8.22 8.28 8.34

BULK WHEAT ELASTICITY, DETERMINATION OF G^ax Three-dimensional stress-strain relations for elasticity of

particulate materials were presented by Hardin and Blandford (1989). Hardin et al. (1990) have used these equations to model the elasticity of wheat. They show that when the principal axes of fabric and stress coincide, the value of Gjnax for simple shear deformation in the plane containing tiie major and minor principal effective stresses, o\ and 0 3 is equal to the elastic shear modulus G31 as defined by Hardin and Blandford:

0.5 H

o

100

(a)

100

(b)

Figure 5-Normalized hysteretic behavior of wheat, (a) Modulus reduction function, and (b) damping function.

680 TRANSACTIONS OF THE ASAE

liable 2. Normalized strain, modulus, and damping values for wheat (fig. 5)

yhr 0.01 0.03162 0.06 0.1 0.2 0.3162 0.6 1 2 3.162 6

10 31.62

100

G/Gmax

1 1 0.98 0.95 0.84 0.71 0.50 0.33 0.18 0.13 0.09 0.07 0.03 0.01

Dmax[l-(G/G„,ax)]

0 0 0.6 1.5 4.8 8.7

15.0 20.1 24.6 26.1 27.3 27.9 29.1 29.7

D Used (%)

0.1 0.2 0.6 1.5 4.8 8.7

15.0 20.1 24.6 26.1 27.3 27.9 29.1 29.7

>31

Pa 2 ( 1 + v) F(e)

(aso'i) 1/2

Pa (6a)

Equations 6a, 6b, and 6c have been used to compute the variations of Gj ax with depth shown in figure 6 for 12 wheat material states, where values of S31 and n in table 3 were used; with constant K(z) = 0.45 and e = 0.60 for dense wheat; and constant K(z) = 0.5 and e = 0.68 for loose wheat.

STRENGTH OF BULK WHEAT, DETERMINATION OF x^ax The Mohr-Coulomb strength envelopes for particulate

materials like wheat and sand are nonlinear, but the envelope for wheat exhibits significant curvature at relatively low stresses. This is illustrated in figure 7a which shows the strength envelope for three triaxial compression tests performed by Hardin (1987) on hard red winter wheat in a loose packing state at 13% moisture content (w.b.).

The peak strength of wheat can be expressed in terms of the principal effective stress ratio at the peak, R ^ x = (aj/a 3)peai . Hardin, et al. (1990) used the strength model by Hardm (1985) to model the nonlinear strength envelope for bulk wheat. The value of R ^ x defined by the model is:

where the elastic stiffness coefficients S31 and powers n, determined by Hardin et al. (1990) from resonant column tests for 12 wheat material states, are listed in table 3. Each pair of S31 and n values for a given material state define the "regression" relationship from five tests under isotropic states of stress ranging from 17 to 276 kPa. The symbol v is the elastic Poisson's ratio, F(e) = 0.3 + 0.7e2 is a function of void ratio e, and p^ = atmospheric pressure. As shown by Hardin et al (1990), elastic Poisson's ratio v should be distinguished from plastic and total Poisson's ratios; the elastic value of v = 0.1, whereas, the plastic and total values increase continuously as loading proceeds to the peak strength.

Referring to figure 1, the vertical stress a^ and horizontal stress a^ prior to the earthquake are assumed to be principal stresses. Thus,

R„

Oi = Ov = Twheatdz (6b)

and

03 = Oh = K Gv (6c)

where in general K = K(z) which may be a function of depth in the wheat, and Y^heat = Ywheat-

Table 3. Elastic parameters for wheat

Wheat Type

Soft red winter wheat

Hard red winter wheat

Moisture Content % (w.b.)

8 13 18

8 13 18

VOL. 39(2):677-687

Loose

S31

520 529 405

531 516 428

n

0.52 0.45 0.40

0.52 0.50 0.41

Dense

S31

530 520 411

536 501 428

n

0.51 0.45 0.39

0.53 0.48 0.46

Rev + [2K„i„ + R „ + 2(R,, - K^iJ F(b)] d,^, (7a)

where

l min — 1 + sin<|)n 1 + sin^cv

= , Kcv = • (7b) 1 - sin(t)n 1 - sin<t)c

sin(t)cv = kf(7c/2 - ^^ tan<|)n + (l - kf) sinct) (7c)

Figure 6-Variation of G„,3x ^^^ depth in wheat.

681

hard wheat loose

1 3 % (wb)

(a)

strength envelope

2 3 4

Normal stress/pa

(a)

equations 7 to each set of three tests, leading to the values of ^iioo ^ d ^oa i^ ^t)le 4, and the following equations relating of, r , and kf to moisture content:

5f = '6 hard wheat pa 1 + %(wb)/2

1.1 2f^ Pa 1 + %(wb)/2

soft wheat

l + %(wb)/112 l+%(wb)/2

(8a)

(8b)

Shear strength

Initial s

03 Ko^ r,

^woif^i

0 = center of circles = ( 1 + K ) ^ « (Rmax+I)"^

rj - radius of Initial stress circle - (1-1^ (K

•• radius of simple shear strength circle

- ( R m a x - D - :(1+I0 (Rmax+1) 2

max = (r | -r |2) l^

(b)

Figure 7-Strength of wheat (a) Nonlinear strength envelope, and (b) computation of x„„„ .

F(b)«4b"d-b"^')

1

1 +10gio[Rcv(l +(Wx)]

Oi - G 3

m'

tan(t)ji« tan(t)jioa ro + l - r g

1 + 03/Of

(7d)

(7e)

(7f)

In the interest of brevity, the references should be consulted for the details of model development.

Referring to figure 7b, the relationship between x j x and R ^ is:

Tmax — O1 .^ . ( i -K) ( l+K)(Rmax~l)

( l -K)(Rmax+l)J

12 - 1

1/2

(9)

The model parameters for wheat in table 4 and those defined by equations 8 have been used with equations 6, 7, and 9 to compute the variations of Xj^^ with depth shown in figure 8 for 12 wheat material states, where K =* constant K(z) « 0.45 and 0.5 for dense and loose wheat, respectively.

BULK WHEAT REFERENCE STRAIN

Values of G ^ x ^ ^ ' max ^^^^ figures 6 and 8 were substituted into equation 4 to obtain the variations of y^ with depth shown in figure 9 for 12 wheat materials.

EFFECT OF DEPTH ON MODULUS AND DAMPING FUNCTIONS The shear modulus and damping in bulk wheat vary

continuously with depth. This variation is accommodated in the analysis used herein by dividing the shear-beam in figure la into several horizontal sublayers, where the material properties are assumed constant within a sublayer. For a given sublayer in a specific material, the value of 7r in figure 9 corresponding to sublayer mid-depth is used to convert normalized strain in figure 5 (or table 2) to 7. To illustrate, modulus reduction and damping functions for 1,5, and 30 m depths (7 - 0.000282, 0.000574, and

and

dmax ' doo 26^ 1 + lo^/o^ 03/Od + Od/03

(7g)

in which 02 is the intermediate principal effective stress, and d = 0.2 will approximate the behavior of many particulate materials. For simple shear strength (fig. 7b) o^ - o;(l + K)/(Rn,ax + 1); 02 = Ko;; and o{ = Rmax<>3-

Hardin (1987) conducted three triaxial compression tests with 03 = 17, 69, and 276 kPa on each of 12 material states, a total of 36 tests. Hardin et al. (1990) fitted

Table 4. Strength model coefficients

Moisture Content (%) w.b. %oa

Loose Dense

doo ^lioa doo

Soft Red Winter Wheat

8 13 18

26.3 26.3 32.0

0.150 25.2 0.150 27.1 0.100 32.5

0.513 0.412 0.400

Hard Red Winter Wheat

8 13 18

24.4 23.1 27.0

0.071 22.7 0.150 23.1 0.150 28.6

0.400 0.445 0.400

682 TRANSAcnoNS OF THE A S A E

Shear Strength, "U^ /Pa

Figure 8-Variation of x„u„ with depth in wheat.

0.001088, respectively) in loose hard red winter wheat at 13% (w.b.) moisture content are shown in figure 10.

COMPOSITE PROPERTIES FOR WHEAT AND BIN WALLS

Considering a horizontal slice of the shear-beam at depth z in figure la, with thickness dz, the shear strain in

0.002 0 0.001 Reference Strain, ^^

Figure 9-Variatioii of 7^ with depth in wheat

1

0.5-

0 -

Depth = 1 m.

I 1

y sV \ 3 0

— 1

hard wheat loose e s 0.68 13%(wb)

. • =a*=a p 1 0.00001 0.0001 0.001

Shear Strain, r

(a)

0.01 0.1

0.00001 0.0001 0.001

Shear strain, r

(b)

Figure 10-Modulus and damping functions for loose, hard wheat, 13% (w.b.).

both wheat and bin walls is equal to 7. Letting A^ eat ^ ^ Avails denote cross-sectional areas of wheat and bin walls, respectively, with G heat ^^^ ^walls denoting the shear moduli of corresponding materials, the shear stresses in wheat and bin walls are x heat ' G heatY and x lls -^wallsV- The average shear stress Xcomp ^ at satisfies equilibrium of the composite slice is defined by:

'^compv^wheat "*" ^walls/ ~ '^wheat^wheat "*" '^walls^walls

which leads to the composite shear stiffness:

•'comp "^comp __ ^ w h e a t ^ h e a t "*" W a l l s ^ a l l s

^ ^ h e a t "^ ^ a l l s

(10a)

Turning to energy dissipation, with the symbol E denoting energy, the hysteretic damping ratios in wheat and bin walls, respectively, are:

D, AE,

wheat • *eaL and D, AE,

4nE, walls "

walls

wheat 471 E walls

where AE^ eat and AE^ ug are quantities of energy dissipated in wheat walls and bin walls in one strain cycle.

VOL. 39(2):677-687 683

roof and E^heat and Ev aiis are the corresponding energies stored i : r ^ o at maximum strain 7. For the horizontal sHce in figure la:

Ewheat = r '^wheatY ^ v h e a t ^ ^ a n d

and the corresponding energy stored at maximum strain is:

^wheat •*• ^walls '^ U^^) V^wheatAvheat "^ ^walls ^ a l l s ) ^

The composite hysteretic damping ratio:

D comp 47t (AE i ,, + AE^^jiJ

This leads to:

J ) ^ ^ w h e a t W h e a t ^ h e a t "*" ^ w a l l s W a l l s ^ a l l s comp

(10b) W h e a t ^ h e a t "^ W a l l s ^ a l l s

Finally, the total mass of the horizontal slice in figure la is:

P w h e a t ^ h e a t ^ "•" P w a l l s ^ a l l s ^ ^ " " Pcomp V ^ h e a t "*" ^ a l l s ) ^

where p^heat and pwaiis are mass densities of wheat and bin walls, respectively, and Pcomp is the composite mass density. This gives:

Q __ Pwhcat^vheat Pwalls^^valls ( l O c )

^ h e a t •*" Avails rcomp

EXAMPLE OF SEISMIC RESPONSE COMPUTATIONS

The shear-beam analysis described above has been used to compute the response of a cylindrical wheat storage bin with height = 24.4 m (80 ft) and diameter = 27.3 m (89.5 ft), denoted herein as "bin 5". The bin was filled with loose, hard wheat at 13% (w.b.) moisture content. Considering variations in thickness of steel bin walls from top to bottom, 11 composite sublayers plus a sublayer representing the mass of the bin roof were used to model the shear-beam. The variation of A^^ils/^wheat ^^^ depth is shown in figure 11a. The bottom wall section consists of double 10-gage steel plates with three 2-gage stiffeners per sheet. The least wall section (second from top) consists of

^walls "" ~ '^walls'V ^ a l l s ^ ^ 2 »

9

Thus, the energy dissipated by both wheat and bin walls is: 10

11

^ E ^ h e a t + ^ E ^ a l l s =

2 ^ 7 ^ ( W h e a t ^ h e a t W h e a t "*" W a l l s ^ a l l s W a l l s ) ^ ^ (a) (b)

0 1 2 3 Composite D (%)

(c)

Figure 11-Bin 5 sublayer properties, (a) Area ratio, (b) strain compatible shear modulus, and (c) strain compatible hysteretic damping ratio.

14-gage plate with three 16-gage stiffeners per sheet. The steel walls were assumed to have the properties G^^lis/Pa"" 884,000, and hysteretic damping ratio D^^lls"" 2^-

Variations of G^heat/^max ^ ^ ^ wheat w^^ normalized strain amplitude 7/7^ are defined in figure 5 and table 2. The variations of Gj ax ^ d 7 with depth of loose, hard wheat (13% w.b.) are defined in figures 6 and 9, respectively. These wheat and wall properties, and the area ratios in figure 11a were used with equations 10 to compute composite properties for each of the 11 sublayers. Variations of composite properties Gcomp and Dcomp with strain amplitude 7 are shown in figure 12 for each sublayer.

The shear-beam model for bin 5 was subjected to the N21°E horizontal component of the Castaice earthquake scaled to a maximum acceleration of 0.1 g (fig. 13c). Earthquake records are available from the National Center for Earthquake Engineering Research (Earthquake Strong Motion Database, NCEER, Lamont-Doherty Earth Observatory, Columbia University, Palisades, N.Y.). Response computations were made with the computer program SHAKE91 (Idriss and Sun, 1992).

Starting with values of G omp corresponding to small strain amplitudes in figure 12a (maximum Gcomp for each sublayer), and with D Q p = 5% for each sublayer, the strain history at mid-depth of each sublayer was computed. The cyclic strain amplitudes in a sublayer are irregular. The equivalent constant amplitude cyclic strain is usually assumed to be 0.4 to 0.75 times the maximum strain (Idriss and Sun, 1992). The value 0.5 was used for the bin 5 analysis. The relationships in figure 12 were used with the equivalent strains to modify Gcomp and Dcomp for each sublayer for the next iteration. The iteration process continued until the properties assumed were compatible with computed strains. Converged values of G . ™ and Dcomp for bin 5 after six iterations are shown in figures l ib and l ie .

The computed acceleration histories at the top of bin 5 and at the top of sublayer 7 are presented in figures 13a and 13b. The composite shear stress histories at the tops of sublayers 2 and 7 and at the base of the shear-beam are plotted in figures 14a, 14b, and 14c, respectively, and the shear strain history at the top of sublayer 7 is given in figure 14d. Comparison of the response histories in

684 TRANSACTIONS OF THE ASAE

2000

1500

i 1000

E

5 500 H

0.0001 0.001 0.01 0.1 strain (%)

(a)

10

0.0001 0.01 0.1 Strain (%)

(b)

Figure 12-Composite properties for the 11 layers of the Bin 5 model. (a) Shear modulus, and (b) hysteretic damping ratio.

figures 13 and 14 to the input acceleration history in figure 13c shows that the ground motion is amplified in the bin and the frequency contents of the response and input histories are different.

Figure 15b shows the displacement amplification spectrum for bin 5. The first mode natural frequency is approximately 4 Hz. The Fourier spectra for the acceleration histories at the top and base of bin 5 are given in figures 15a and 15c, respectively. The influence of the 4 Hz first mode frequency on response is evident. Inspection of response histories in figures 13 and 14 show that maximum response occurs between 3 and 4 s. Variations of maximum response values with depth are plotted in figure 16. The maximum, ground acceleration (0.1 g) is amplified by a factor of 2.9 at the top of bin 5 (fig. 16a).

Figures 16b and 16c show variations of maximum composite shear stress and maximum shear strain with depth. For a shear-beam (fig. la) the shear strain in walls and wheat are identical, and the strain distribution in figure 16c can be used with the respective shear moduli to

acceleration (g)

0.3

Bins

roof 1 2

3 4 5

6

7

8

9 10

11

10

1

20

•

6 8 10 12 14 16 time (sec)

0 W « j | ^ ^

-0 .1

(c) [^/^V<A.^'A<y^A^^^vV*^^'>»•*V^

0 2 4 6 8 10 12 14 16 time (sec)

Figure 13-Bin 5 acceleration response, (a) Top of bin, (b) top of sublayer 7, and (c) N21''E horizontal component of Castaice earthquake scaled to 0.1 g maximum acceleration.

Bins

roof

shear stress (kPa) 10n

Lm_

8

9 ~

10

-^ 0

-10

•• v . VJ\ /W\ /V»MV\/VW\M/SPV. (a)

0 2 4 6 8 10 12 14 16 time (sec)

6 8 10 12 14 16 time (sec)

6 8 10 12 14 16 time (sec)

shear strain (%)

0.02

2 4 6 8 10 12 14 16 time (sec)

Figure 14-Bin 5 composite shear stress and strain response histories.

VOL. 39(2):677-687 685

bins

roof

h10

8

10

(a) Top of Bin 5

Acceleration Spectrum

- /v ;^^"^. ,

10 20 Frequency (Hz)

30

:20

t i o u

(b) Bin 5 Top to Base

Displacement , Amplification 1 Spectrum

ft ULA_^

20 10 20

Frequency (Hz)

(c) Base of Bin 5

Input Acceleration Spectrum

10 20 Frequency (Hz)

Figure 15-Response spectra for bin 5. (a) Fourier spectrum of acceleration at the top, (b) top-to-base displacement amplification, and (c) Fourier spectrum of base input acceleration.

compute maximum shear stresses in walls and wheat. Using G^alls/Pa = 884,000 with values of A^ lls/Awheat and ^comp/Pa ^ figures 11a and lib, the variation of G^heat/Pa with depth was computed from equation 10a. At each depth, values of G heat ^ ^ ^walls were multiplied by the

Bins

roof 1 ! ^

2

3 _ 4 5

6

8

10

0 0.1 0.2 0.3 0 10 20 30 Max. Acceleration (g) Max. Composite

Shear Stress (l(Pa)

(a) (b)

0 0.01 0.02 0.03 Max Shear Strain (%)

(C)

corresponding maximum shear strain in figure 16c to determine the distributions of maximum shear stress in wheat and walls shown in figures 17a and 17b, respectively. The shear strength of loose hard wheat (13% w.b.) from figure 8 is plotted in figure 17a for comparison. The most important result is the distribution of maximum shear stress in bin walls (fig. 17b) which can be used to assess the safety of bin 5 when subjected to the earthquake ground motion in figure 13c.

CONCLUSIONS A methodology for estimating the seismic response of

wheat bins has been presented where wheat and bin walls are modeled as a composite shear-beam. A constitutive

£ 1 0

20

(a)

shear strength

max. shear stress resulting from

earthquake

z —r 10 20 30 40 50 Wheat Shear Stress (kPa)

(a)

0 10 20 Max. Wall Shear Stress (MPa)

Figure l^Maximum response values for bin 5 subjected to N21°E horizontal component of Castaice earthquake scaled to 0.1 g maximum acceleration, (a) Acceleration, (b) composite shear stress, and (c) shear strain.

(b)

Figure 17-Maximum shear stress distributions in (a) wheat and (b) bin 5 walls.

686 TRANSACTIONS OF THE A S A E

model for cyclic loading of wheat is included. The variations with depth of low-strain shear modulus G^ax (fig. 6) and reference strain y,. (fig. 9) for hard and soft wheat in dense and loose density states, at moisture contents of 8, 13, and 18% (w.b.), along with the normalized modulus reduction and damping relationships in figure 5 and table 2 provide a constitutive model database for cyclic simple shear loading of wheat. This constitutive database can be used to analyze the seismic response of any bin configuration that deforms approximately as a shear-beam. Equations are given for computation of composite shear-beam properties from properties of the bin walls and wheat properties found in the constitutive database.

Seismic response of a wheat-filled bin is computed by equivalent linear analysis using the computer program SHAKE91. Shear strains computed by linear analysis are used to adjust material stress-strain properties until assumed properties are compatible with computed strains.

The procedure has been illustrated by analysis of a large bin (24.4 m high x 27.3 m diameter) subjected to a specified horizontal ground acceleration. Shear stress, shear strain, and acceleration histories and acceleration Fourier spectra at various depths, in addition to the displacement amplification spectrum for the bin have been computed. The profile of maximum shear strains was used to compute maximum shear stresses in wheat and bin walls. The latter can be used to assess the safety of bin walls when subjected to earthquake loading.

REFERENCES American Concrete Institute. 1991. Standard practice for design

and construction of concrete silos and stacking tubes for storing granular materials (ACI313-91) and commentary (ACT 313R-91). Detroit, Mich.: ACI.

ASAE Standards, 41st Ed. 1994. EP 433. Loads exerted by free-flowing grain on bins. St. Joseph, Mich.: ASAE.

Das, B. M. 1993. Principles of Soil Dynamics. Boston, Mass.: PWS-KENT Publishing.

Hardin, B. 0.1965. The nature of damping in sands. J. of the Soil Mechanics and Foundations Div., ASCE 91(l):63-97.

Hardin, B.O. and V. P. Dmevich. 1972. Shear modulus and damping in soils: Design equations and curves. / . of the Soil Mechanics and Foundations Division, ASCE 98(7): 667-692.

Hardin, B. O. 1985. Strength of soils in terms of effective stress. In Proc. of the Richart Commemorative Lectures, ASCE, 1-78.

Hardin, B. O. and G. E. Blandford. 1989. Elasticity of particulate materials. 7. ofGeotechnicalEng.ASCE 115(6):788-805.

Hardin, B. O., K. O. Hardin, I. J. Ross and C. V. Schwab. 1990. Triaxial compression, simple shear, and strength of wheat en mass. Transactions of the ASAE 33(3):933-943.

Hardin, K. 0.1987. The effect of strain amplitude on the shear modulus of wheat. M.S. thesis, Univ. of Kentucky, Lexington.

Harris, E. C. and J. D. von Nad. 1985. Experimental determination of effective weight of stored material for use in seismic design of silos. AC/7. 82(6):828-833.

Idriss, I. M. and J. I. Sun. 1992. User's manual for SHAKE91. Computer program distributed by NISEE, Univ. of Califomia, Berkeley.

Rammerstorfer, F. G., K. Scharf and F. D. Fisher. 1990. Storage tanks under earthquake loading. Applied Mechanics Review 43(ll):261-282.

Richart Jr., F. E., J. R. Hall Jr. and R. D. Woods. 1970. Vibrations of Soils and Foundations. Englewood Cliffs, N.J.: Prentice-Hall.

Standards Association of Australia. 1990. AS 3774-1990. Loads on bulk solids containers. North Sydney, Australia: SAA.

Schnabel, R B., J. Lysmer and H. B. Seed. 1972. SHAKE, A computer program for earthquake response analysis of horizontally layered sites. Computer program distributed by NISEE, Univ. of Califomia, Berkeley.

Ziolkowski, D. R, B. O. Hardin, C. V. Schwab and I. J. Ross. 1985. Shear modulus of wheat at low strain amplitude. Transactions of the ASAE 28(3): 884-888.

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