Research ArticleThe Shear Strength and Dilatancy Behavior ofWheat Stored in Silos

Changnv Zeng1 and Yuke Wang 2

1College of Civil Engineering and Architecture Henan University of Technology Zhengzhou 450001 China2College of Water Conservancy Science and Engineering Zhengzhou University Zhengzhou 450001 China

Correspondence should be addressed to Yuke Wang ykewang163com

Received 25 May 2019 Accepted 9 July 2019 Published 22 July 2019

Academic Editor Marcin Mrugalski

Copyright copy 2019 Changnv Zeng and Yuke Wang This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper focuses on the shear and dilatancy behavior of wheat stored in silos with various densities and normal stressesThe goalis to find a quantitative relationship modeling the peak friction angle and maximum dilatancy angle of wheat stored in silos Atotal of 48 direct shear tests were carried out to research the evolution of shear and dilatancy of stored wheat in silos It is revealedthat strength of wheat in bulk attributes to the combination of frictional and dilatant during shearing in particular attributing toits elliptic shape An increase in relative density enhances the peak friction angle as well as the dilation The relationships betweenrelative density peak friction angle anddilatancy anglewere presentedbasedon the tests data andBoltonrsquos theoryThenan advancedmodel is developed to evaluate the peak shear behavior of wheat stored in silos considering the dilatancy of the stored wheat It isa practical method to predict the strength and dilatancy behavior of wheat stored in silos

1 Introduction

Grain silos are special structures in industry for storing agri-culture materials which are widely used in the whole world[1] The failure of silos may not only cause the loss of storedmaterial itself but also sometimes cause an explosion Wheatis one of the important rawmaterials which provides approx-imately twenty-one percent of the worldrsquos food [2] Becauseof large amount of wheat production and large populationin China to serve most of the wheat in China is stored insilos for three to five years The long-term storage system isvery different from those in other countries [3]Therefore thestorage of wheat in silos is of great concern to the governmentengineers and scientists The mechanical properties of thestoredmaterials are primarily important to design in the silos[4]

Janssenrsquo theory found in 1895 is still the basis of mostinternal standards to gain the loads applied on silos [5ndash10]It is simple to obtain the loads with some common materialproperties of the internal friction of stored materials frictioncoefficient of agriculture grain to wall etc These commonproperties of grain such as frictional angle bulk density

and friction angle between concrete and steel have beenwidely investigated by many researchers [11ndash16] Howeverit is found that the ldquoreal silo loadsrdquo measuring from silosare more complex than those obtained by Janssenrsquos theorywhich still provides new uncertainties in the design [17]The stored materials are undergoing various forces such asgravity loading friction with the silo walls which changewith depth during silo filling and discharging [18ndash25] Forexample the dilatancy phenomenon usually occurred duringsilo discharging As a result the numerical methods areusually adopted to accurately simulate silo loads during thestorage conditions It is necessary to investigate the stress-strain model including friction and dilatancy behavior foragricultural materials at different conditions in silos

Direct shear tests are widely used to investigate themechanical properties of materials especially for soils [26ndash30] However for agriculture material the mechanical prop-erties are still poorly understood [5] especially for the largeamount and long-term stored wheat in China In this workthe strength and dilatancy behavior of wheat is investigatedusing direct shear tests under different normal stresses andrelative densities The stress-dilatancy model was built based

HindawiComplexityVolume 2019 Article ID 1547616 9 pageshttpsdoiorg10115520191547616

2 Complexity

Table 1 Physical properties of wheat

Sample Maximum grainsize (mm)

D60(mm)

D10(mm)

Coefficient ofuniformity Cu

Specific gravity Gs

Wheat 5 412 336 123 134

Table 2 Properties for shear tests of wheat

Sample 119890max 119890min Initial void ratio values used Normal stress(kPa)

Wheat 0668 0450 0487049104960506051005150523053205500569058706020613062306410668 50100150

on the relationship of friction and dilatancy angleThe effectsof relative densities and normal stresses were also examined

2 Experimental Procedure and Materials

21 Materials The wheat samples used in this project wereuniformly graded winter wheat grown in Henan Province ofChina Different parameters were used to describe the wheatincluding

(1) Particle Shape and Particle Size Distribution The particleshapes were quantified by Zingg index tomeasure the particleshape The length of major axis (L) minor axis (B) and itsthickness (T) are used to describe the particle shape shownin Figure 2 The average Zingg index is 162 which indicatesthe rod-shaped particle for wheat The grain size distributioncurvewas shown inFigure 3 and the coefficient of uniformityforwheatwas obtained It showed thewheatwas very uniformand poorly grade The main properties of wheat used weresummarized in Table 1

(2) Void RatioThe granular mass is commonly considered tobe consisted of a network of solid particles enclosing voids ofvarying sizes The void ratio of samples is described as

119890 = 120588119904 minus 120588120588 =120588119904120588 minus 1 (1)

where 120588119904 is density of wheat particle with the same quantita-tive value of the specific gravity 119866119904=134 120588 is the density ofwheat mass and 119890 is the void ratio In this paper the valuesof 119890119898119886119909 and 119890119898119894119899 are 0668 and 0450 for wheat respectivelyaccording to SL237-1999 [31] The detailed properties ofvoid ratio were listed in Table 2 The density of samplecan be evaluated by comparing with the value of 119890119898119886119909 and119890119898119894119899 Here 119890119898119886119909 represents the state of void ratio at loosestcondition which is achieved by quickly inverting the wheatin a container And 119890119898119894119899 is measured by vibration withoutcausing wheat particle crushing

Specimens with different void ratio were prepared bypouring a specified mass of dry wheat into the shear box bydeposition with a spoon for loose samples in five lifts eachtamped by using a steel rammer for dense samples withoutdestroying grain particle until the required void ratio wasachieved for each layer

(3) Moisture Content The moisture content was obtainedfromwheat flour with oven dryingmethod Some representa-tive crushing wheat flour sample of 10 g was placed in a dishwhich was dried at 105∘C in an oven for at least 3 hours Themoisture content of wheat particle when crushing into flourwas 106 by dry basis

22 Testing Apparatus and Procedure The developed shearbox with circular cross section was 100 mm in diameter and100mm in depth as shown in Figure 1The shear cell diametershould be at least 20 times the maximum particle size and notless than 40 times the mean particle size as recommendedby Eurocode 1 Part 4 In the study the values of the meanparticle size and the mean particle size are 5 mm and 402mm according to the particle distribution curve respectivelywhich are partly satisfying the size of cell box in Eurocode1 However the size of shear box is not very accurate withthe requirement when it exceeds 40 times mean particle sizeMost of the shear boxes in the literatures are less than 100mm because of the limitation of the shear device used Inaddition the results in a large box of 143 mm considering theratio of 5 between sample size and grain size are similar tothose by the authors [21] Regardless the test results providedby these shear devices show considerable parameters for theagricultures materials

As shown in Figure 1 the upper part of the shear box isrestrained while the lower part is controlled by a motor toapply the horizontal shear load with displacement-controlledmode The vertical normal load is imposed by the upperplaten which is fixed with the upper part of shear box Theopening size of the two halves of shear box is 2 mm whichis corresponding to the requirement according to Shibuya etal [32] During the tests the loads and displacements at axialand horizontal directions were recorded automatically by acomputer-controlled data collection system

Test samples were prepared by dry pouring method with10 layers to obtain specified densities For dense sample itmay be vibrated or compaction without grain crushing Inthis study the influences of void ratio and normal stress onstress-dilatancy behavior were examined The values of voidratio ranged from0487 to 0668with normal stress of 50 kPa100 kPa and 150 kPa which usually existed for stored wheatin situ silos The shearing velocities are set as 24 mmminuntil the horizontal displacement reaches more than 20 mm

Complexity 3

transducer

transducer

load cell

shear box

lever

shear load

Figure 1 The direct shear box apparatus

T

BL

Figure 2 Size of wheat grain

Gradation data for test

Particle size (mm)

Perc

enta

ge fi

ner (

)

100

90

80

70

60

50

40

30

20

10

0

1 2 3 4 5 6 7 8 9 10

Figure 3 Grain size distribution of wheat used

3 Experimental Results

31 Stress-Strain Behavior and Volumetric Response A seriesof direct shear tests were carried out with a consideration

of different levels of void ratios and normal stresses Theinitial void ratio of specimen ranged from 0487 to 0668including loose and dense specimens The typical curvesof three void ratios are shown in Figures 4ndash6 with normalstresses ranged from 50 kPa to 150 kPa The stress-strainevolutionwas plotted in terms of ratio between shear strengthand normal stress R=1205911205901015840V and horizontal displacement 119906The volumetric response was characterized as vertical normaldisplacement and horizontal shear displacement For thedense wheat sample with relative density119863119903 = 83at normalstress of 50 kPa 1205911205901015840V increases with the increase of horizontaldisplacement u while it reached the peak value at aboutu=5 mm After that the stress ratio 1205911205901015840V keeps constant ata relatively small range until it decreases to a stable state atlarge horizontal displacement which is called a critical stateDuring the shearing process significant dilative deformationoccurs then a small contraction occurs in the initial stageThe maximum amount of dilation reaches the peak strengthstate After the plateau stage of curve in Figure 4 the rateof dilation decreases with continuous shearing and then thecritical stage is achieved With the void ratio increasingboth the peak value of 1205911205901015840V and dilative deformation reducewith the generation of horizontal displacement Furthermoredilation nearly vanishes for the loose sample with 1198900 = 0641at 150 kPa

Comparing the responses of wheat at different initial voidratios the loose sample requires larger horizontal displace-ment to obtain the peak horizontal shear stress than that ofdense sample at the same normal stress However the post-peak strain softening of loose sample is gentler than thatof dense sample As expected the critical state is confirmedusing the results as shown in Figures 4ndash6 for all the testsamples The effect of normal stress on stress-strain responseis also examined As shown in Figures 4ndash6 at the same initial

4 Complexity

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

50kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

50kPae0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)

06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

(b)

Figure 4 Experimental results of stress-strain response (50 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

100kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10

Shea

r stre

ss ra

tio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

100kPa

(b)

Figure 5 Experimental results of stress-strain response (100 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

void ratio the final vertical displacement decreases with theincreasing normal stress and this suppresses the dilationThesamples with different void ratio show the same trend

32 Peak Strength Friction Angle The peak state representsthe maximum shear strength of the wheat sample whichwas determined when the horizontal shear stress meets itsmaximum value The peak direct shear friction angle 1206011015840119889119904 isdefined with

1206011015840119889119904 = tanminus1 ( 1205911205901015840V) (2)

where 120591 and 1205901015840V denote horizontal shear stress and normalstress

Two methods can be used to obtain the strength frictionangle at the peak state which are interpreted of a single

test (first method) or multiple tests (second method) Thefirst method is the direct measurement from shear stressratio of mobilized shear stress to normal stress at peak stateThe results obtained from the first method are shown inFigure 7 It shows that an increase of peak friction angleperforms with the increase of relative density However thepeak friction angle of wheat decreases with the increaseof normal stress Figure 8 shows an example of the secondmethod to obtain the friction angle with multiple tests curvesof shear stress versus normal stress The intercept equalstan 120593 in which 120593 is also called internal friction angle It isone of the useful mechanical properties in silo design asrecommended bymost of silo codes which is greatly affectedby the density of sample Figure 9 showed the effect of relativedensity on internal friction angle at peak state strength Agradual increase occurs with the increase of relative density

Complexity 5

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

150kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

150kPa

(b)

Figure 6 Experimental results of stress-strain response (150 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

45

40

35

30

25

2000 02 04 06 08 10

Relative density ＄Ｌ

Ｊ

Ｅ(∘)

50kPa100kPa150kPa

Figure 7 Relationships between peak friction and relative density

of samples which indicates denser sample exhibits higherinternal friction angle and thus higher shear strength

The shear behavior of wheat was similar to the angularsand which was also influenced by its angularity particleshape [33] As is well-known to us all wheat grain is ellipticshape which may cause large interparticle locking betweenthe particles when they slide or rotate during shearingHigh shear stress occurs to overcome the locking betweenparticles just before the dilation takes place Thus it alsoimplies that the shear resistance is the combined action ofparticle interlock and dilation In the initial stage of shearingthe interparticle locking dominates while dilation tends toplay prevailing effect after a coupled effect of dilation andinterlocking The plateau curve in Figure 6 describes theconsequence of couple effect More specially the interlocking

e0 = 0569

= 3019∘

Nor

mal

stre

ss (k

Pa)

Shear stress (kPa)

140

120

100

80

60

40

20

00 50 100 150 200 250

Figure 8 Shear stress against normal stress at peak state (1198900 =0569)

action decreaseswith the development of horizontal displace-ment It is noted that these observations are very similar tothose of sands [33ndash35]

33 Critical State Friction Angle 1206011015840119888V In this paper the frictionangle at critical state is used to represent the minimumshear strength condition during shearing It is fundamentalto describe the shear behavior at constant volume shearingcondition with the theory of dilatancy In this paper thecritical state friction angle is obtained based on the fitted lineof 1206011015840119901119890119886119896 against 120595 according to Simoni and Houlsby [26]Here 1206011015840119901119890119886119896 is the peak friction while 120595 is measured at thepoint corresponding to peak strength The intercept of thefitted line is the derived critical friction angle at the point ofdilatancy angle 120595 = 0 as shown in Figure 10 At least two testsare needed at various densities for this method which showsmore reliable than that obtained from single test results [26]

6 Complexity

40

38

36

34

32

30

28

26

24

22

200 10 20 30 40 50 60 70 80 90

Relative density ＄Ｌ

Inte

rnal

fric

tion

angl

e(∘)

Figure 9 Internal friction angles of different relative density at peakstate

The result is also compared with Boltonrsquos dilatancy equation[36 37] In this study 1206011015840119888V is derived from a wide range ofdensities of wheat direct shear tests The test results of peakfriction 1206011015840119901119890119886119896 and dilatancy angle 120595 are regressed as shownin Figure 10 A linear fitted line with the direct shear tests ofwheat in bulk is observed The derived 1206011015840119888V value for wheat atdifferent normal stresses shows substantially different valuesThe 120601119888V values are 3146∘ 2819∘ and 2571∘ for normalstresses 50 kPa 100 kPa and 150 kPa respectively It can beseen that the critical state friction angle decreases with theincrease of imposed normal stressTheobservation of derivedcritical state friction related to the normal stress is probablycaused by the effect of deformable wheat particle at largenormal stress which is somewhat similar to that of rockfillmaterials [38]

4 Dilatancy Behavior of Wheat

As showed in Figures 4ndash6 the vertical displacement andhorizontal displacement are also measured as well as theshear strength In the direct shear stress condition the rateof dilation of wheat 119889120592119889119906 is directly calculated by the ratioof vertical displacement V to the horizontal displacement 119906which is shown in

tan120595 = 119889120592119889119906 (3)

The differential value of 119889120592119889119906 should be smoothing asthe measured displacement is obtained at very short intervalof time The successive readings are tested for several timeuntil the appropriate moving average value of three readings119889u asymp 070 mm is obtained which shows a limit scatterand sufficient smooth relation between dilation ratio andshear displacement It is expected that value of (119889120592119889119906)119898119886119909is coincident with the value of (1205911205901015840V)119898119886119909 It is also used tocalculate the maximum dilation angle 120595119898119886119909

A group of test results of maximum dilatancy angles varywith densities and normal stresses as shown in Figure 11 Itis revealed that dilatancy angle increases with an increase

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

＝Ｐ = 3146∘

50kPa

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

(a)

＝Ｐ = 2819∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

100kPa

(b)

＝Ｐ = 2571∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

150kPa

(c)

Figure 10 The method used for determination of the critical statefriction angle (a) 50 kPa (b) 100 kPa (c) 150 kPa

of relative density while it decreases with increasing normalstress As an example for specimen with 1198900 = 0487 the

Complexity 7

00 02 04 06 08 10

Relative density ＄Ｌ

50kPa100kPa150kPa

ＧＲ

(∘)

12

10

8

6

4

2

0

Figure 11 Maximum dilatancy angle versus relative density atdifferent normal stress

average dilatancy angle of 113∘ and for 1198900 = 0668 thatis 286∘ under imposed normal stress of 50 kPa For thespecimen with 1198900 = 0487 the dilatancy angles were 113∘103∘ and 83∘ under normal stress of 50 kPa 100 kPa and150 kPa respectively It can be indicated that the dilation maybe suppressedwith the increase of imposed normal stressThestress and dilation behavior of wheat in bulk is consistent withthose reported results of various sands [26ndash28]

5 Discussion

Bolton [37 38] presented a new empirical equation based onRowersquos theory which is widely used for various sands shownin

1206011015840119901119890119886119896119901119904 minus 1206011015840119888V119901119904 = 08120595119898119886119909 (4)

In this paper the framework of Boltonrsquos equation isadopted to interpret the experimental result of direct sheartests of wheat in bulk

Figure 12 presented the relationship of peak friction angle1206011015840119901119890119886119896119889119904 and the maximum dilation angle 120595119898119886119909 in direct sheartests As seen from Figure 12 the test data of wheat in bulk isfitted using (5) for wheat based on Boltonrsquos equation

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 08120595119898119886119909 (5)

From the fitted curve b=085 for normal stress of 50 kPais matched

A linear relationship between 1206011015840119901119890119886119896119889119904 and 120595119898119886119909 isobserved in Figure 12 and (5) fromexperiment investigationsThe presentation in (5) is deduced on the framework ofBolton [36 37] which agrees with (4) by Bolton for the planestrain conditions As above descriptions the theory for planestrain conditions proposed by Bolton [36 37] is also suit forthe case of direct shear tests for example of wheat direct

0 2 4 6 8 10 14120

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

９ = 085Ｒ

２2 = 09639

50kPa

Dilatancy angle ＧＲ (∘)

Figure 12 Peak friction angle 1206011015840119901119890119886119896119889119904 versus maximum dilationangle 120595119898119886119909(50 kPa)

shear tests Thus the equation can be deduced in terms of thefriction angle obtained from direct shear tests The series ofscattered test data of wheat is fittedwith the deduced equationaccording to Boltonrsquos theory

The influence of density confining pressure on peakfriction angle was also examined by Bolton [36 37] Herethe widely accepted definition of compaction is used in (6)which is named as relative density 119863r In this theory a newindex called relative dilatancy index 119868119877 is developed which inturn combines 119863r and normal stress with its shear resistanceand corresponding peak friction angle

119863119903 = 119890119898119886119909 minus 119890119890119898119886119909 minus 119890119898119894119899 (6)

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 5119868119877 (7)

119868119877 = 119863119903 (119876 minus ln (1205901015840V)) minus 1 (8)

In this work the effects of relative density and normalstresses are also examined for wheat in bulk The typical testresults are plotted in Figure 13 The quantitative correlationwas showed in (7) and (8) where the parameter Q value is745 to 764 with an average value of 754 for wheat

6 Conclusions

The shear and dilatancy behavior of wheat stored in silos arepresented in this paper The direct shear tests of wheat arecarried out at various levels of densities and normal stressesin order to investigate the influence of relative density andnormal stress on shear and dilation behavior of wheat Boththe peak friction angle and internal friction angle increasewith the increasing relative density of wheat sample It isobserved that denser sample with higher relative densityenhances the dilation by higher dilatancy angle whereas thedilation is suppressed with increasing normal stress Basedon the framework of Boltonrsquos theory the relationship between

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 3

transducer

transducer

load cell

shear box

lever

shear load

Figure 1 The direct shear box apparatus

T

BL

Figure 2 Size of wheat grain

Gradation data for test

Particle size (mm)

Perc

enta

ge fi

ner (

)

100

90

80

70

60

50

40

30

20

10

0

1 2 3 4 5 6 7 8 9 10

Figure 3 Grain size distribution of wheat used

3 Experimental Results

31 Stress-Strain Behavior and Volumetric Response A seriesof direct shear tests were carried out with a consideration

of different levels of void ratios and normal stresses Theinitial void ratio of specimen ranged from 0487 to 0668including loose and dense specimens The typical curvesof three void ratios are shown in Figures 4ndash6 with normalstresses ranged from 50 kPa to 150 kPa The stress-strainevolutionwas plotted in terms of ratio between shear strengthand normal stress R=1205911205901015840V and horizontal displacement 119906The volumetric response was characterized as vertical normaldisplacement and horizontal shear displacement For thedense wheat sample with relative density119863119903 = 83at normalstress of 50 kPa 1205911205901015840V increases with the increase of horizontaldisplacement u while it reached the peak value at aboutu=5 mm After that the stress ratio 1205911205901015840V keeps constant ata relatively small range until it decreases to a stable state atlarge horizontal displacement which is called a critical stateDuring the shearing process significant dilative deformationoccurs then a small contraction occurs in the initial stageThe maximum amount of dilation reaches the peak strengthstate After the plateau stage of curve in Figure 4 the rateof dilation decreases with continuous shearing and then thecritical stage is achieved With the void ratio increasingboth the peak value of 1205911205901015840V and dilative deformation reducewith the generation of horizontal displacement Furthermoredilation nearly vanishes for the loose sample with 1198900 = 0641at 150 kPa

Comparing the responses of wheat at different initial voidratios the loose sample requires larger horizontal displace-ment to obtain the peak horizontal shear stress than that ofdense sample at the same normal stress However the post-peak strain softening of loose sample is gentler than thatof dense sample As expected the critical state is confirmedusing the results as shown in Figures 4ndash6 for all the testsamples The effect of normal stress on stress-strain responseis also examined As shown in Figures 4ndash6 at the same initial

4 Complexity

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

50kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

50kPae0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)

06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

(b)

Figure 4 Experimental results of stress-strain response (50 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

100kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10

Shea

r stre

ss ra

tio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

100kPa

(b)

Figure 5 Experimental results of stress-strain response (100 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

void ratio the final vertical displacement decreases with theincreasing normal stress and this suppresses the dilationThesamples with different void ratio show the same trend

32 Peak Strength Friction Angle The peak state representsthe maximum shear strength of the wheat sample whichwas determined when the horizontal shear stress meets itsmaximum value The peak direct shear friction angle 1206011015840119889119904 isdefined with

1206011015840119889119904 = tanminus1 ( 1205911205901015840V) (2)

where 120591 and 1205901015840V denote horizontal shear stress and normalstress

Two methods can be used to obtain the strength frictionangle at the peak state which are interpreted of a single

test (first method) or multiple tests (second method) Thefirst method is the direct measurement from shear stressratio of mobilized shear stress to normal stress at peak stateThe results obtained from the first method are shown inFigure 7 It shows that an increase of peak friction angleperforms with the increase of relative density However thepeak friction angle of wheat decreases with the increaseof normal stress Figure 8 shows an example of the secondmethod to obtain the friction angle with multiple tests curvesof shear stress versus normal stress The intercept equalstan 120593 in which 120593 is also called internal friction angle It isone of the useful mechanical properties in silo design asrecommended bymost of silo codes which is greatly affectedby the density of sample Figure 9 showed the effect of relativedensity on internal friction angle at peak state strength Agradual increase occurs with the increase of relative density

Complexity 5

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

150kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

150kPa

(b)

Figure 6 Experimental results of stress-strain response (150 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

45

40

35

30

25

2000 02 04 06 08 10

Relative density ＄Ｌ

Ｊ

Ｅ(∘)

50kPa100kPa150kPa

Figure 7 Relationships between peak friction and relative density

of samples which indicates denser sample exhibits higherinternal friction angle and thus higher shear strength

The shear behavior of wheat was similar to the angularsand which was also influenced by its angularity particleshape [33] As is well-known to us all wheat grain is ellipticshape which may cause large interparticle locking betweenthe particles when they slide or rotate during shearingHigh shear stress occurs to overcome the locking betweenparticles just before the dilation takes place Thus it alsoimplies that the shear resistance is the combined action ofparticle interlock and dilation In the initial stage of shearingthe interparticle locking dominates while dilation tends toplay prevailing effect after a coupled effect of dilation andinterlocking The plateau curve in Figure 6 describes theconsequence of couple effect More specially the interlocking

e0 = 0569

= 3019∘

Nor

mal

stre

ss (k

Pa)

Shear stress (kPa)

140

120

100

80

60

40

20

00 50 100 150 200 250

Figure 8 Shear stress against normal stress at peak state (1198900 =0569)

action decreaseswith the development of horizontal displace-ment It is noted that these observations are very similar tothose of sands [33ndash35]

33 Critical State Friction Angle 1206011015840119888V In this paper the frictionangle at critical state is used to represent the minimumshear strength condition during shearing It is fundamentalto describe the shear behavior at constant volume shearingcondition with the theory of dilatancy In this paper thecritical state friction angle is obtained based on the fitted lineof 1206011015840119901119890119886119896 against 120595 according to Simoni and Houlsby [26]Here 1206011015840119901119890119886119896 is the peak friction while 120595 is measured at thepoint corresponding to peak strength The intercept of thefitted line is the derived critical friction angle at the point ofdilatancy angle 120595 = 0 as shown in Figure 10 At least two testsare needed at various densities for this method which showsmore reliable than that obtained from single test results [26]

6 Complexity

40

38

36

34

32

30

28

26

24

22

200 10 20 30 40 50 60 70 80 90

Relative density ＄Ｌ

Inte

rnal

fric

tion

angl

e(∘)

Figure 9 Internal friction angles of different relative density at peakstate

The result is also compared with Boltonrsquos dilatancy equation[36 37] In this study 1206011015840119888V is derived from a wide range ofdensities of wheat direct shear tests The test results of peakfriction 1206011015840119901119890119886119896 and dilatancy angle 120595 are regressed as shownin Figure 10 A linear fitted line with the direct shear tests ofwheat in bulk is observed The derived 1206011015840119888V value for wheat atdifferent normal stresses shows substantially different valuesThe 120601119888V values are 3146∘ 2819∘ and 2571∘ for normalstresses 50 kPa 100 kPa and 150 kPa respectively It can beseen that the critical state friction angle decreases with theincrease of imposed normal stressTheobservation of derivedcritical state friction related to the normal stress is probablycaused by the effect of deformable wheat particle at largenormal stress which is somewhat similar to that of rockfillmaterials [38]

4 Dilatancy Behavior of Wheat

As showed in Figures 4ndash6 the vertical displacement andhorizontal displacement are also measured as well as theshear strength In the direct shear stress condition the rateof dilation of wheat 119889120592119889119906 is directly calculated by the ratioof vertical displacement V to the horizontal displacement 119906which is shown in

tan120595 = 119889120592119889119906 (3)

The differential value of 119889120592119889119906 should be smoothing asthe measured displacement is obtained at very short intervalof time The successive readings are tested for several timeuntil the appropriate moving average value of three readings119889u asymp 070 mm is obtained which shows a limit scatterand sufficient smooth relation between dilation ratio andshear displacement It is expected that value of (119889120592119889119906)119898119886119909is coincident with the value of (1205911205901015840V)119898119886119909 It is also used tocalculate the maximum dilation angle 120595119898119886119909

A group of test results of maximum dilatancy angles varywith densities and normal stresses as shown in Figure 11 Itis revealed that dilatancy angle increases with an increase

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

＝Ｐ = 3146∘

50kPa

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

(a)

＝Ｐ = 2819∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

100kPa

(b)

＝Ｐ = 2571∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

150kPa

(c)

Figure 10 The method used for determination of the critical statefriction angle (a) 50 kPa (b) 100 kPa (c) 150 kPa

of relative density while it decreases with increasing normalstress As an example for specimen with 1198900 = 0487 the

Complexity 7

00 02 04 06 08 10

Relative density ＄Ｌ

50kPa100kPa150kPa

ＧＲ

(∘)

12

10

8

6

4

2

0

Figure 11 Maximum dilatancy angle versus relative density atdifferent normal stress

average dilatancy angle of 113∘ and for 1198900 = 0668 thatis 286∘ under imposed normal stress of 50 kPa For thespecimen with 1198900 = 0487 the dilatancy angles were 113∘103∘ and 83∘ under normal stress of 50 kPa 100 kPa and150 kPa respectively It can be indicated that the dilation maybe suppressedwith the increase of imposed normal stressThestress and dilation behavior of wheat in bulk is consistent withthose reported results of various sands [26ndash28]

5 Discussion

Bolton [37 38] presented a new empirical equation based onRowersquos theory which is widely used for various sands shownin

1206011015840119901119890119886119896119901119904 minus 1206011015840119888V119901119904 = 08120595119898119886119909 (4)

In this paper the framework of Boltonrsquos equation isadopted to interpret the experimental result of direct sheartests of wheat in bulk

Figure 12 presented the relationship of peak friction angle1206011015840119901119890119886119896119889119904 and the maximum dilation angle 120595119898119886119909 in direct sheartests As seen from Figure 12 the test data of wheat in bulk isfitted using (5) for wheat based on Boltonrsquos equation

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 08120595119898119886119909 (5)

From the fitted curve b=085 for normal stress of 50 kPais matched

A linear relationship between 1206011015840119901119890119886119896119889119904 and 120595119898119886119909 isobserved in Figure 12 and (5) fromexperiment investigationsThe presentation in (5) is deduced on the framework ofBolton [36 37] which agrees with (4) by Bolton for the planestrain conditions As above descriptions the theory for planestrain conditions proposed by Bolton [36 37] is also suit forthe case of direct shear tests for example of wheat direct

0 2 4 6 8 10 14120

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

９ = 085Ｒ

２2 = 09639

50kPa

Dilatancy angle ＧＲ (∘)

Figure 12 Peak friction angle 1206011015840119901119890119886119896119889119904 versus maximum dilationangle 120595119898119886119909(50 kPa)

shear tests Thus the equation can be deduced in terms of thefriction angle obtained from direct shear tests The series ofscattered test data of wheat is fittedwith the deduced equationaccording to Boltonrsquos theory

The influence of density confining pressure on peakfriction angle was also examined by Bolton [36 37] Herethe widely accepted definition of compaction is used in (6)which is named as relative density 119863r In this theory a newindex called relative dilatancy index 119868119877 is developed which inturn combines 119863r and normal stress with its shear resistanceand corresponding peak friction angle

119863119903 = 119890119898119886119909 minus 119890119890119898119886119909 minus 119890119898119894119899 (6)

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 5119868119877 (7)

119868119877 = 119863119903 (119876 minus ln (1205901015840V)) minus 1 (8)

In this work the effects of relative density and normalstresses are also examined for wheat in bulk The typical testresults are plotted in Figure 13 The quantitative correlationwas showed in (7) and (8) where the parameter Q value is745 to 764 with an average value of 754 for wheat

6 Conclusions

The shear and dilatancy behavior of wheat stored in silos arepresented in this paper The direct shear tests of wheat arecarried out at various levels of densities and normal stressesin order to investigate the influence of relative density andnormal stress on shear and dilation behavior of wheat Boththe peak friction angle and internal friction angle increasewith the increasing relative density of wheat sample It isobserved that denser sample with higher relative densityenhances the dilation by higher dilatancy angle whereas thedilation is suppressed with increasing normal stress Basedon the framework of Boltonrsquos theory the relationship between

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Complexity

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

50kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

50kPae0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)

06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

(b)

Figure 4 Experimental results of stress-strain response (50 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

100kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10

Shea

r stre

ss ra

tio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

100kPa

(b)

Figure 5 Experimental results of stress-strain response (100 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

void ratio the final vertical displacement decreases with theincreasing normal stress and this suppresses the dilationThesamples with different void ratio show the same trend

32 Peak Strength Friction Angle The peak state representsthe maximum shear strength of the wheat sample whichwas determined when the horizontal shear stress meets itsmaximum value The peak direct shear friction angle 1206011015840119889119904 isdefined with

1206011015840119889119904 = tanminus1 ( 1205911205901015840V) (2)

where 120591 and 1205901015840V denote horizontal shear stress and normalstress

Two methods can be used to obtain the strength frictionangle at the peak state which are interpreted of a single

test (first method) or multiple tests (second method) Thefirst method is the direct measurement from shear stressratio of mobilized shear stress to normal stress at peak stateThe results obtained from the first method are shown inFigure 7 It shows that an increase of peak friction angleperforms with the increase of relative density However thepeak friction angle of wheat decreases with the increaseof normal stress Figure 8 shows an example of the secondmethod to obtain the friction angle with multiple tests curvesof shear stress versus normal stress The intercept equalstan 120593 in which 120593 is also called internal friction angle It isone of the useful mechanical properties in silo design asrecommended bymost of silo codes which is greatly affectedby the density of sample Figure 9 showed the effect of relativedensity on internal friction angle at peak state strength Agradual increase occurs with the increase of relative density

Complexity 5

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

150kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

150kPa

(b)

Figure 6 Experimental results of stress-strain response (150 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

45

40

35

30

25

2000 02 04 06 08 10

Relative density ＄Ｌ

Ｊ

Ｅ(∘)

50kPa100kPa150kPa

Figure 7 Relationships between peak friction and relative density

of samples which indicates denser sample exhibits higherinternal friction angle and thus higher shear strength

The shear behavior of wheat was similar to the angularsand which was also influenced by its angularity particleshape [33] As is well-known to us all wheat grain is ellipticshape which may cause large interparticle locking betweenthe particles when they slide or rotate during shearingHigh shear stress occurs to overcome the locking betweenparticles just before the dilation takes place Thus it alsoimplies that the shear resistance is the combined action ofparticle interlock and dilation In the initial stage of shearingthe interparticle locking dominates while dilation tends toplay prevailing effect after a coupled effect of dilation andinterlocking The plateau curve in Figure 6 describes theconsequence of couple effect More specially the interlocking

e0 = 0569

= 3019∘

Nor

mal

stre

ss (k

Pa)

Shear stress (kPa)

140

120

100

80

60

40

20

00 50 100 150 200 250

Figure 8 Shear stress against normal stress at peak state (1198900 =0569)

action decreaseswith the development of horizontal displace-ment It is noted that these observations are very similar tothose of sands [33ndash35]

33 Critical State Friction Angle 1206011015840119888V In this paper the frictionangle at critical state is used to represent the minimumshear strength condition during shearing It is fundamentalto describe the shear behavior at constant volume shearingcondition with the theory of dilatancy In this paper thecritical state friction angle is obtained based on the fitted lineof 1206011015840119901119890119886119896 against 120595 according to Simoni and Houlsby [26]Here 1206011015840119901119890119886119896 is the peak friction while 120595 is measured at thepoint corresponding to peak strength The intercept of thefitted line is the derived critical friction angle at the point ofdilatancy angle 120595 = 0 as shown in Figure 10 At least two testsare needed at various densities for this method which showsmore reliable than that obtained from single test results [26]

6 Complexity

40

38

36

34

32

30

28

26

24

22

200 10 20 30 40 50 60 70 80 90

Relative density ＄Ｌ

Inte

rnal

fric

tion

angl

e(∘)

Figure 9 Internal friction angles of different relative density at peakstate

The result is also compared with Boltonrsquos dilatancy equation[36 37] In this study 1206011015840119888V is derived from a wide range ofdensities of wheat direct shear tests The test results of peakfriction 1206011015840119901119890119886119896 and dilatancy angle 120595 are regressed as shownin Figure 10 A linear fitted line with the direct shear tests ofwheat in bulk is observed The derived 1206011015840119888V value for wheat atdifferent normal stresses shows substantially different valuesThe 120601119888V values are 3146∘ 2819∘ and 2571∘ for normalstresses 50 kPa 100 kPa and 150 kPa respectively It can beseen that the critical state friction angle decreases with theincrease of imposed normal stressTheobservation of derivedcritical state friction related to the normal stress is probablycaused by the effect of deformable wheat particle at largenormal stress which is somewhat similar to that of rockfillmaterials [38]

4 Dilatancy Behavior of Wheat

As showed in Figures 4ndash6 the vertical displacement andhorizontal displacement are also measured as well as theshear strength In the direct shear stress condition the rateof dilation of wheat 119889120592119889119906 is directly calculated by the ratioof vertical displacement V to the horizontal displacement 119906which is shown in

tan120595 = 119889120592119889119906 (3)

The differential value of 119889120592119889119906 should be smoothing asthe measured displacement is obtained at very short intervalof time The successive readings are tested for several timeuntil the appropriate moving average value of three readings119889u asymp 070 mm is obtained which shows a limit scatterand sufficient smooth relation between dilation ratio andshear displacement It is expected that value of (119889120592119889119906)119898119886119909is coincident with the value of (1205911205901015840V)119898119886119909 It is also used tocalculate the maximum dilation angle 120595119898119886119909

A group of test results of maximum dilatancy angles varywith densities and normal stresses as shown in Figure 11 Itis revealed that dilatancy angle increases with an increase

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

＝Ｐ = 3146∘

50kPa

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

(a)

＝Ｐ = 2819∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

100kPa

(b)

＝Ｐ = 2571∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

150kPa

(c)

Figure 10 The method used for determination of the critical statefriction angle (a) 50 kPa (b) 100 kPa (c) 150 kPa

of relative density while it decreases with increasing normalstress As an example for specimen with 1198900 = 0487 the

Complexity 7

00 02 04 06 08 10

Relative density ＄Ｌ

50kPa100kPa150kPa

ＧＲ

(∘)

12

10

8

6

4

2

0

Figure 11 Maximum dilatancy angle versus relative density atdifferent normal stress

average dilatancy angle of 113∘ and for 1198900 = 0668 thatis 286∘ under imposed normal stress of 50 kPa For thespecimen with 1198900 = 0487 the dilatancy angles were 113∘103∘ and 83∘ under normal stress of 50 kPa 100 kPa and150 kPa respectively It can be indicated that the dilation maybe suppressedwith the increase of imposed normal stressThestress and dilation behavior of wheat in bulk is consistent withthose reported results of various sands [26ndash28]

5 Discussion

Bolton [37 38] presented a new empirical equation based onRowersquos theory which is widely used for various sands shownin

1206011015840119901119890119886119896119901119904 minus 1206011015840119888V119901119904 = 08120595119898119886119909 (4)

In this paper the framework of Boltonrsquos equation isadopted to interpret the experimental result of direct sheartests of wheat in bulk

Figure 12 presented the relationship of peak friction angle1206011015840119901119890119886119896119889119904 and the maximum dilation angle 120595119898119886119909 in direct sheartests As seen from Figure 12 the test data of wheat in bulk isfitted using (5) for wheat based on Boltonrsquos equation

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 08120595119898119886119909 (5)

From the fitted curve b=085 for normal stress of 50 kPais matched

A linear relationship between 1206011015840119901119890119886119896119889119904 and 120595119898119886119909 isobserved in Figure 12 and (5) fromexperiment investigationsThe presentation in (5) is deduced on the framework ofBolton [36 37] which agrees with (4) by Bolton for the planestrain conditions As above descriptions the theory for planestrain conditions proposed by Bolton [36 37] is also suit forthe case of direct shear tests for example of wheat direct

0 2 4 6 8 10 14120

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

９ = 085Ｒ

２2 = 09639

50kPa

Dilatancy angle ＧＲ (∘)

Figure 12 Peak friction angle 1206011015840119901119890119886119896119889119904 versus maximum dilationangle 120595119898119886119909(50 kPa)

shear tests Thus the equation can be deduced in terms of thefriction angle obtained from direct shear tests The series ofscattered test data of wheat is fittedwith the deduced equationaccording to Boltonrsquos theory

The influence of density confining pressure on peakfriction angle was also examined by Bolton [36 37] Herethe widely accepted definition of compaction is used in (6)which is named as relative density 119863r In this theory a newindex called relative dilatancy index 119868119877 is developed which inturn combines 119863r and normal stress with its shear resistanceand corresponding peak friction angle

119863119903 = 119890119898119886119909 minus 119890119890119898119886119909 minus 119890119898119894119899 (6)

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 5119868119877 (7)

119868119877 = 119863119903 (119876 minus ln (1205901015840V)) minus 1 (8)

In this work the effects of relative density and normalstresses are also examined for wheat in bulk The typical testresults are plotted in Figure 13 The quantitative correlationwas showed in (7) and (8) where the parameter Q value is745 to 764 with an average value of 754 for wheat

6 Conclusions

The shear and dilatancy behavior of wheat stored in silos arepresented in this paper The direct shear tests of wheat arecarried out at various levels of densities and normal stressesin order to investigate the influence of relative density andnormal stress on shear and dilation behavior of wheat Boththe peak friction angle and internal friction angle increasewith the increasing relative density of wheat sample It isobserved that denser sample with higher relative densityenhances the dilation by higher dilatancy angle whereas thedilation is suppressed with increasing normal stress Basedon the framework of Boltonrsquos theory the relationship between

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

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Mathematical Problems in Engineering

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Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 5

e0 = 0487 ＄Ｌ = 83

e0 = 0569 ＄Ｌ = 45

e0 = 0641 ＄Ｌ = 12

Onset of dilation

Critical state

150kPa

5 10 15 20 250

Horizontal displacement (mm)

00

01

02

03

04

05

06

07

08

09

10Sh

ear s

tress

ratio

Ｐ

(a)

0 5 10 15 20 25

e0 = 0487

e0 = 0569

e0 = 0641

Horizontal displacement (mm)06

04

02

00

minus02

minus04

minus06

minus08

minus10

Vert

ical

disp

lace

men

t (m

m)

150kPa

(b)

Figure 6 Experimental results of stress-strain response (150 kPa) (a) Shear stress ratio-horizontal displacement curve (b) Verticaldisplacement-horizontal displacement

45

40

35

30

25

2000 02 04 06 08 10

Relative density ＄Ｌ

Ｊ

Ｅ(∘)

50kPa100kPa150kPa

Figure 7 Relationships between peak friction and relative density

of samples which indicates denser sample exhibits higherinternal friction angle and thus higher shear strength

The shear behavior of wheat was similar to the angularsand which was also influenced by its angularity particleshape [33] As is well-known to us all wheat grain is ellipticshape which may cause large interparticle locking betweenthe particles when they slide or rotate during shearingHigh shear stress occurs to overcome the locking betweenparticles just before the dilation takes place Thus it alsoimplies that the shear resistance is the combined action ofparticle interlock and dilation In the initial stage of shearingthe interparticle locking dominates while dilation tends toplay prevailing effect after a coupled effect of dilation andinterlocking The plateau curve in Figure 6 describes theconsequence of couple effect More specially the interlocking

e0 = 0569

= 3019∘

Nor

mal

stre

ss (k

Pa)

Shear stress (kPa)

140

120

100

80

60

40

20

00 50 100 150 200 250

Figure 8 Shear stress against normal stress at peak state (1198900 =0569)

action decreaseswith the development of horizontal displace-ment It is noted that these observations are very similar tothose of sands [33ndash35]

33 Critical State Friction Angle 1206011015840119888V In this paper the frictionangle at critical state is used to represent the minimumshear strength condition during shearing It is fundamentalto describe the shear behavior at constant volume shearingcondition with the theory of dilatancy In this paper thecritical state friction angle is obtained based on the fitted lineof 1206011015840119901119890119886119896 against 120595 according to Simoni and Houlsby [26]Here 1206011015840119901119890119886119896 is the peak friction while 120595 is measured at thepoint corresponding to peak strength The intercept of thefitted line is the derived critical friction angle at the point ofdilatancy angle 120595 = 0 as shown in Figure 10 At least two testsare needed at various densities for this method which showsmore reliable than that obtained from single test results [26]

6 Complexity

40

38

36

34

32

30

28

26

24

22

200 10 20 30 40 50 60 70 80 90

Relative density ＄Ｌ

Inte

rnal

fric

tion

angl

e(∘)

Figure 9 Internal friction angles of different relative density at peakstate

The result is also compared with Boltonrsquos dilatancy equation[36 37] In this study 1206011015840119888V is derived from a wide range ofdensities of wheat direct shear tests The test results of peakfriction 1206011015840119901119890119886119896 and dilatancy angle 120595 are regressed as shownin Figure 10 A linear fitted line with the direct shear tests ofwheat in bulk is observed The derived 1206011015840119888V value for wheat atdifferent normal stresses shows substantially different valuesThe 120601119888V values are 3146∘ 2819∘ and 2571∘ for normalstresses 50 kPa 100 kPa and 150 kPa respectively It can beseen that the critical state friction angle decreases with theincrease of imposed normal stressTheobservation of derivedcritical state friction related to the normal stress is probablycaused by the effect of deformable wheat particle at largenormal stress which is somewhat similar to that of rockfillmaterials [38]

4 Dilatancy Behavior of Wheat

As showed in Figures 4ndash6 the vertical displacement andhorizontal displacement are also measured as well as theshear strength In the direct shear stress condition the rateof dilation of wheat 119889120592119889119906 is directly calculated by the ratioof vertical displacement V to the horizontal displacement 119906which is shown in

tan120595 = 119889120592119889119906 (3)

The differential value of 119889120592119889119906 should be smoothing asthe measured displacement is obtained at very short intervalof time The successive readings are tested for several timeuntil the appropriate moving average value of three readings119889u asymp 070 mm is obtained which shows a limit scatterand sufficient smooth relation between dilation ratio andshear displacement It is expected that value of (119889120592119889119906)119898119886119909is coincident with the value of (1205911205901015840V)119898119886119909 It is also used tocalculate the maximum dilation angle 120595119898119886119909

A group of test results of maximum dilatancy angles varywith densities and normal stresses as shown in Figure 11 Itis revealed that dilatancy angle increases with an increase

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

＝Ｐ = 3146∘

50kPa

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

(a)

＝Ｐ = 2819∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

100kPa

(b)

＝Ｐ = 2571∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

150kPa

(c)

Figure 10 The method used for determination of the critical statefriction angle (a) 50 kPa (b) 100 kPa (c) 150 kPa

of relative density while it decreases with increasing normalstress As an example for specimen with 1198900 = 0487 the

Complexity 7

00 02 04 06 08 10

Relative density ＄Ｌ

50kPa100kPa150kPa

ＧＲ

(∘)

12

10

8

6

4

2

0

Figure 11 Maximum dilatancy angle versus relative density atdifferent normal stress

average dilatancy angle of 113∘ and for 1198900 = 0668 thatis 286∘ under imposed normal stress of 50 kPa For thespecimen with 1198900 = 0487 the dilatancy angles were 113∘103∘ and 83∘ under normal stress of 50 kPa 100 kPa and150 kPa respectively It can be indicated that the dilation maybe suppressedwith the increase of imposed normal stressThestress and dilation behavior of wheat in bulk is consistent withthose reported results of various sands [26ndash28]

5 Discussion

Bolton [37 38] presented a new empirical equation based onRowersquos theory which is widely used for various sands shownin

1206011015840119901119890119886119896119901119904 minus 1206011015840119888V119901119904 = 08120595119898119886119909 (4)

In this paper the framework of Boltonrsquos equation isadopted to interpret the experimental result of direct sheartests of wheat in bulk

Figure 12 presented the relationship of peak friction angle1206011015840119901119890119886119896119889119904 and the maximum dilation angle 120595119898119886119909 in direct sheartests As seen from Figure 12 the test data of wheat in bulk isfitted using (5) for wheat based on Boltonrsquos equation

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 08120595119898119886119909 (5)

From the fitted curve b=085 for normal stress of 50 kPais matched

A linear relationship between 1206011015840119901119890119886119896119889119904 and 120595119898119886119909 isobserved in Figure 12 and (5) fromexperiment investigationsThe presentation in (5) is deduced on the framework ofBolton [36 37] which agrees with (4) by Bolton for the planestrain conditions As above descriptions the theory for planestrain conditions proposed by Bolton [36 37] is also suit forthe case of direct shear tests for example of wheat direct

0 2 4 6 8 10 14120

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

９ = 085Ｒ

２2 = 09639

50kPa

Dilatancy angle ＧＲ (∘)

Figure 12 Peak friction angle 1206011015840119901119890119886119896119889119904 versus maximum dilationangle 120595119898119886119909(50 kPa)

shear tests Thus the equation can be deduced in terms of thefriction angle obtained from direct shear tests The series ofscattered test data of wheat is fittedwith the deduced equationaccording to Boltonrsquos theory

The influence of density confining pressure on peakfriction angle was also examined by Bolton [36 37] Herethe widely accepted definition of compaction is used in (6)which is named as relative density 119863r In this theory a newindex called relative dilatancy index 119868119877 is developed which inturn combines 119863r and normal stress with its shear resistanceand corresponding peak friction angle

119863119903 = 119890119898119886119909 minus 119890119890119898119886119909 minus 119890119898119894119899 (6)

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 5119868119877 (7)

119868119877 = 119863119903 (119876 minus ln (1205901015840V)) minus 1 (8)

In this work the effects of relative density and normalstresses are also examined for wheat in bulk The typical testresults are plotted in Figure 13 The quantitative correlationwas showed in (7) and (8) where the parameter Q value is745 to 764 with an average value of 754 for wheat

6 Conclusions

The shear and dilatancy behavior of wheat stored in silos arepresented in this paper The direct shear tests of wheat arecarried out at various levels of densities and normal stressesin order to investigate the influence of relative density andnormal stress on shear and dilation behavior of wheat Boththe peak friction angle and internal friction angle increasewith the increasing relative density of wheat sample It isobserved that denser sample with higher relative densityenhances the dilation by higher dilatancy angle whereas thedilation is suppressed with increasing normal stress Basedon the framework of Boltonrsquos theory the relationship between

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

40

38

36

34

32

30

28

26

24

22

200 10 20 30 40 50 60 70 80 90

Relative density ＄Ｌ

Inte

rnal

fric

tion

angl

e(∘)

Figure 9 Internal friction angles of different relative density at peakstate

The result is also compared with Boltonrsquos dilatancy equation[36 37] In this study 1206011015840119888V is derived from a wide range ofdensities of wheat direct shear tests The test results of peakfriction 1206011015840119901119890119886119896 and dilatancy angle 120595 are regressed as shownin Figure 10 A linear fitted line with the direct shear tests ofwheat in bulk is observed The derived 1206011015840119888V value for wheat atdifferent normal stresses shows substantially different valuesThe 120601119888V values are 3146∘ 2819∘ and 2571∘ for normalstresses 50 kPa 100 kPa and 150 kPa respectively It can beseen that the critical state friction angle decreases with theincrease of imposed normal stressTheobservation of derivedcritical state friction related to the normal stress is probablycaused by the effect of deformable wheat particle at largenormal stress which is somewhat similar to that of rockfillmaterials [38]

4 Dilatancy Behavior of Wheat

As showed in Figures 4ndash6 the vertical displacement andhorizontal displacement are also measured as well as theshear strength In the direct shear stress condition the rateof dilation of wheat 119889120592119889119906 is directly calculated by the ratioof vertical displacement V to the horizontal displacement 119906which is shown in

tan120595 = 119889120592119889119906 (3)

The differential value of 119889120592119889119906 should be smoothing asthe measured displacement is obtained at very short intervalof time The successive readings are tested for several timeuntil the appropriate moving average value of three readings119889u asymp 070 mm is obtained which shows a limit scatterand sufficient smooth relation between dilation ratio andshear displacement It is expected that value of (119889120592119889119906)119898119886119909is coincident with the value of (1205911205901015840V)119898119886119909 It is also used tocalculate the maximum dilation angle 120595119898119886119909

A group of test results of maximum dilatancy angles varywith densities and normal stresses as shown in Figure 11 Itis revealed that dilatancy angle increases with an increase

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

＝Ｐ = 3146∘

50kPa

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

(a)

＝Ｐ = 2819∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

100kPa

(b)

＝Ｐ = 2571∘

Peak

fric

tion

angl

e Ｇ

Ｒ(∘)

Dilatancy angle (∘)

45

40

35

30

25

20

15

100 1 2 3 4 5 6 7 8 9 10 11

150kPa

(c)

Figure 10 The method used for determination of the critical statefriction angle (a) 50 kPa (b) 100 kPa (c) 150 kPa

of relative density while it decreases with increasing normalstress As an example for specimen with 1198900 = 0487 the

Complexity 7

00 02 04 06 08 10

Relative density ＄Ｌ

50kPa100kPa150kPa

ＧＲ

(∘)

12

10

8

6

4

2

0

Figure 11 Maximum dilatancy angle versus relative density atdifferent normal stress

average dilatancy angle of 113∘ and for 1198900 = 0668 thatis 286∘ under imposed normal stress of 50 kPa For thespecimen with 1198900 = 0487 the dilatancy angles were 113∘103∘ and 83∘ under normal stress of 50 kPa 100 kPa and150 kPa respectively It can be indicated that the dilation maybe suppressedwith the increase of imposed normal stressThestress and dilation behavior of wheat in bulk is consistent withthose reported results of various sands [26ndash28]

5 Discussion

Bolton [37 38] presented a new empirical equation based onRowersquos theory which is widely used for various sands shownin

1206011015840119901119890119886119896119901119904 minus 1206011015840119888V119901119904 = 08120595119898119886119909 (4)

In this paper the framework of Boltonrsquos equation isadopted to interpret the experimental result of direct sheartests of wheat in bulk

Figure 12 presented the relationship of peak friction angle1206011015840119901119890119886119896119889119904 and the maximum dilation angle 120595119898119886119909 in direct sheartests As seen from Figure 12 the test data of wheat in bulk isfitted using (5) for wheat based on Boltonrsquos equation

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 08120595119898119886119909 (5)

From the fitted curve b=085 for normal stress of 50 kPais matched

A linear relationship between 1206011015840119901119890119886119896119889119904 and 120595119898119886119909 isobserved in Figure 12 and (5) fromexperiment investigationsThe presentation in (5) is deduced on the framework ofBolton [36 37] which agrees with (4) by Bolton for the planestrain conditions As above descriptions the theory for planestrain conditions proposed by Bolton [36 37] is also suit forthe case of direct shear tests for example of wheat direct

0 2 4 6 8 10 14120

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

９ = 085Ｒ

２2 = 09639

50kPa

Dilatancy angle ＧＲ (∘)

Figure 12 Peak friction angle 1206011015840119901119890119886119896119889119904 versus maximum dilationangle 120595119898119886119909(50 kPa)

shear tests Thus the equation can be deduced in terms of thefriction angle obtained from direct shear tests The series ofscattered test data of wheat is fittedwith the deduced equationaccording to Boltonrsquos theory

The influence of density confining pressure on peakfriction angle was also examined by Bolton [36 37] Herethe widely accepted definition of compaction is used in (6)which is named as relative density 119863r In this theory a newindex called relative dilatancy index 119868119877 is developed which inturn combines 119863r and normal stress with its shear resistanceand corresponding peak friction angle

119863119903 = 119890119898119886119909 minus 119890119890119898119886119909 minus 119890119898119894119899 (6)

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 5119868119877 (7)

119868119877 = 119863119903 (119876 minus ln (1205901015840V)) minus 1 (8)

In this work the effects of relative density and normalstresses are also examined for wheat in bulk The typical testresults are plotted in Figure 13 The quantitative correlationwas showed in (7) and (8) where the parameter Q value is745 to 764 with an average value of 754 for wheat

6 Conclusions

The shear and dilatancy behavior of wheat stored in silos arepresented in this paper The direct shear tests of wheat arecarried out at various levels of densities and normal stressesin order to investigate the influence of relative density andnormal stress on shear and dilation behavior of wheat Boththe peak friction angle and internal friction angle increasewith the increasing relative density of wheat sample It isobserved that denser sample with higher relative densityenhances the dilation by higher dilatancy angle whereas thedilation is suppressed with increasing normal stress Basedon the framework of Boltonrsquos theory the relationship between

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 7

00 02 04 06 08 10

Relative density ＄Ｌ

50kPa100kPa150kPa

ＧＲ

(∘)

12

10

8

6

4

2

0

Figure 11 Maximum dilatancy angle versus relative density atdifferent normal stress

average dilatancy angle of 113∘ and for 1198900 = 0668 thatis 286∘ under imposed normal stress of 50 kPa For thespecimen with 1198900 = 0487 the dilatancy angles were 113∘103∘ and 83∘ under normal stress of 50 kPa 100 kPa and150 kPa respectively It can be indicated that the dilation maybe suppressedwith the increase of imposed normal stressThestress and dilation behavior of wheat in bulk is consistent withthose reported results of various sands [26ndash28]

5 Discussion

Bolton [37 38] presented a new empirical equation based onRowersquos theory which is widely used for various sands shownin

1206011015840119901119890119886119896119901119904 minus 1206011015840119888V119901119904 = 08120595119898119886119909 (4)

In this paper the framework of Boltonrsquos equation isadopted to interpret the experimental result of direct sheartests of wheat in bulk

Figure 12 presented the relationship of peak friction angle1206011015840119901119890119886119896119889119904 and the maximum dilation angle 120595119898119886119909 in direct sheartests As seen from Figure 12 the test data of wheat in bulk isfitted using (5) for wheat based on Boltonrsquos equation

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 08120595119898119886119909 (5)

From the fitted curve b=085 for normal stress of 50 kPais matched

A linear relationship between 1206011015840119901119890119886119896119889119904 and 120595119898119886119909 isobserved in Figure 12 and (5) fromexperiment investigationsThe presentation in (5) is deduced on the framework ofBolton [36 37] which agrees with (4) by Bolton for the planestrain conditions As above descriptions the theory for planestrain conditions proposed by Bolton [36 37] is also suit forthe case of direct shear tests for example of wheat direct

0 2 4 6 8 10 14120

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

９ = 085Ｒ

２2 = 09639

50kPa

Dilatancy angle ＧＲ (∘)

Figure 12 Peak friction angle 1206011015840119901119890119886119896119889119904 versus maximum dilationangle 120595119898119886119909(50 kPa)

shear tests Thus the equation can be deduced in terms of thefriction angle obtained from direct shear tests The series ofscattered test data of wheat is fittedwith the deduced equationaccording to Boltonrsquos theory

The influence of density confining pressure on peakfriction angle was also examined by Bolton [36 37] Herethe widely accepted definition of compaction is used in (6)which is named as relative density 119863r In this theory a newindex called relative dilatancy index 119868119877 is developed which inturn combines 119863r and normal stress with its shear resistanceand corresponding peak friction angle

119863119903 = 119890119898119886119909 minus 119890119890119898119886119909 minus 119890119898119894119899 (6)

1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 = 5119868119877 (7)

119868119877 = 119863119903 (119876 minus ln (1205901015840V)) minus 1 (8)

In this work the effects of relative density and normalstresses are also examined for wheat in bulk The typical testresults are plotted in Figure 13 The quantitative correlationwas showed in (7) and (8) where the parameter Q value is745 to 764 with an average value of 754 for wheat

6 Conclusions

The shear and dilatancy behavior of wheat stored in silos arepresented in this paper The direct shear tests of wheat arecarried out at various levels of densities and normal stressesin order to investigate the influence of relative density andnormal stress on shear and dilation behavior of wheat Boththe peak friction angle and internal friction angle increasewith the increasing relative density of wheat sample It isobserved that denser sample with higher relative densityenhances the dilation by higher dilatancy angle whereas thedilation is suppressed with increasing normal stress Basedon the framework of Boltonrsquos theory the relationship between

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

0

2

4

6

8

10

12

Ｊ

Ｅ＞Ｍ- ＝Ｐ

＞Ｍ(∘)

50kPa

00 01 02 03 04 05 06 07 08 09 10

Relative density ＄Ｌ

ＪＥ＞Ｍ-

＝Ｐ＞Ｍ = 5 lowast [＄Ｌ lowast (745-ＦＨ Ｐ)-1]

Figure 13 Relationship between 1206011015840119901119890119886119896119889119904 minus 1206011015840119888V119889119904 and relative density(1205901015840V = 50 kPa)

peak friction angle and maximum dilatancy angle is deducedwith a consideration of relative density effect

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was substantially supported by Plan for YouthCore Teachers of Henan University of Technology (2015004)Plan for the Key Projects of High Schools of Henan province(19A560009) and Provincial Key Laboratory for Grain andOil Storage Facility amp Safety Henan University of Technology(2016KF-B03)

References

[1] H LiAnalysis of Steel Silo Structures onDiscrete Supports [PhDthesis] The University of Edinburgh Edinburgh Scotland UK1994

[2] R Ortiz K D Sayre B Govaerts et al ldquoClimate change canheat beat the heatrdquoAgriculture Ecosystemsamp Environment vol126 no 1-2 pp 46ndash58 2008

[3] X Li Z Cao Z Wei Q Feng and J Wang ldquoEquilibriummois-ture content and sorption isosteric heats of five wheat varietiesin Chinardquo Journal of Stored Products Research vol 47 no 1 pp39ndash47 2011

[4] A Dogangun Z Karaca A Durmus and H Sezen ldquoCause ofdamage and failures in silo structuresrdquo Journal of Performanceof Constructed Facilities vol 23 no 2 pp 65ndash71 2009

[5] F Ayuga P Aguado E Gallego and A Ramırez ldquoNew stepstowards the knowledge of silos behaviourrdquo International Agro-physics vol 19 no 1 pp 7ndash17 2005

[6] ldquoEN 1991-4 Eurocode 1 - Actions on structures ndash Part 4rdquo Silosand Tanks European Committee for Standardization Brussels2006

[7] ldquoACI 313-97 Standard practice for design and construction ofconcrete silos and stacking tubes for storing granularmaterialsrdquoAmerican Concrete Institute Farmington Hills MI 1997

[8] ldquoAS 3774-1996 Loads on bulk solids containersrdquo StandardsAustralia Homebush NSW 1996

[9] GB50077-2003 ldquoDesign Code for concrete silosrdquo (Chinese)2004

[10] GB50322-2011 ldquoDesign Code for steel silosrdquo (Chinese) 2011[11] Y Zhao Q-S Cao and L Su ldquoBuckling design of large circular

steel silos subject to wind pressurerdquoThin-Walled Structures vol73 pp 337ndash349 2013

[12] J Horabik andMMolenda ldquoMechanical properties of granularmaterials and their impact on load distribution in silo Areviewrdquo Scientia Agriculturae Bohemica vol 45 no 4 pp 203ndash211 2014

[13] C Zeng and Y Wang ldquoCompressive behavior of wheat fromconfined uniaxial compression testsrdquo International Agrophysics2019

[14] J M Boac R Bhadra M E Casada et al ldquoStored grainpack factors for wheat Comparison of three methods to fieldmeasurementsrdquo Transactions of the ASABE vol 58 no 4 pp1089ndash1101 2015

[15] A Ramırez J Nielsen and F Ayuga ldquoPressuremeasurements insteel silos with eccentric hoppersrdquo Powder Technology vol 201no 1 pp 7ndash20 2010

[16] M Gao X Cheng and X Du ldquoSimulation of bulk densitydistribution of wheat in silos by finite element analysisrdquo Journalof Stored Products Research vol 77 pp 1ndash8 2018

[17] J Durack and C Tranberg ldquoA challenge for designers of steelsilosrdquo Australian Bulk Handling Review pp 84ndash89 2010

[18] S A Thompson and I J Ross ldquoCompressibility and frictionalcoefficients of wheatrdquo Transaction of American Society Agricul-tural Engineering vol 26 no 4 pp 1171ndash1180 1983

[19] M F Moya M Guaita P J Aguado and F Ayuga ldquoMechanicalproperties of granular agricultural materials part 2rdquo Transac-tion of American Society Agricultural and Biological Engineeringvol 49 no 2 pp 479ndash489 2006

[20] M F Moya F Ayuga M Guaita and P J Aguado ldquoMechanicalproperties of granular agricultural materialsrdquo Transaction ofAmerican Society Agricultural Engineering vol 45 no 5 pp1569ndash1577 2002

[21] M F Moya P J Aguado and F Ayuga ldquoMechanical propertiesof some granular agricultural materials used in silo designrdquoInternational Agrophysics vol 27 no 2 pp 181ndash193 2013

[22] A Ramırez M F Moya and F Ayuga ldquoDetermination of themechanical properties of powdered agricultural products andsugarrdquo Particle amp Particle Systems Characterization vol 26 no4 pp 220ndash230 2009

[23] S D Liu Z Y Zhou R P Zou D Pinson and A B Yu ldquoFlowcharacteristics and discharge rate of ellipsoidal particles in a flatbottom hopperrdquo Powder Technology vol 253 pp 70ndash79 2014

[24] Q Zhang andM G Britton ldquoAmicromechanics model for pre-dicting dynamic loads during discharge in bulk solids storagestructuresrdquo Canadian Biosystems Engineering vol 45 pp 521-527 2003

[25] S Zhang P Lin CWang Y Tian JWan and L Yang ldquoInvesti-gating the influence of wall frictions on hopper flowsrdquoGranularMatter vol 16 no 6 pp 857ndash866 2014

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 9

[26] A Simoni and G T Houlsby ldquoThe direct shear strength anddilatancy of sandndashgravel mixturesrdquoGeotechnical and GeologicalEngineering vol 24 no 3 pp 523ndash549 2006

[27] J-J Wang J-J Guo J-P Bai and X Wu ldquoShear strength ofsandstonendashmudstone particle mixture from direct shear testrdquoEnvironmental Earth Sciences vol 77 no 12 article 442 2018

[28] S Nam M Gutierrez P Diplas and J Petrie ldquoDeterminationof the shear strength of unsaturated soils using the multistagedirect shear testrdquoEngineering Geology vol 122 no 3-4 pp 272ndash280 2011

[29] L E Vallejo and R Mawby ldquoPorosity influence on the shearstrength of granular material-clay mixturesrdquo Engineering Geol-ogy vol 58 no 2 pp 125ndash136 2000

[30] J Chen P C Hagan and S Saydam ldquoShear behaviour of acement grout tested in the direct shear testrdquo Construction andBuilding Materials vol 166 pp 271ndash279 2018

[31] Trade Standard of P R China ldquoStandard method for directshear test of coarse soilrdquo in Specification of Soil Test SL237ndash059TheMinistry ofWaterResources of China Beijing China 1999

[32] S Shibuya T Mitachi and S Tamate ldquoInterpretation of directshear box testing of sands as quasi-simple shearrdquo Geotechniquevol 47 no 4 pp 769ndash790 1997

[33] P Guo and X Su ldquoShear strength interparticle locking anddilatancy of granularmaterialsrdquoCanadian Geotechnical Journalvol 44 no 5 pp 579ndash591 2007

[34] A Cresswell and M Barton ldquoDirect shear tests on an unce-mented and a very slightly cemented locked sandrdquo QuarterlyJournal of Engineering Geology and Hydrogeology vol 36 no 2pp 119ndash132 2003

[35] A Cresswell and W Powrie ldquoTriaxial tests on an unbondedlocked sandrdquo Geotechnique vol 54 no 2 pp 107ndash115 2004

[36] M D Bolton ldquoStrength and dilatancy of sandsrdquo GΘotechniquevol 36 no 1 pp 65ndash78 1986

[37] MD Bolton ldquoDiscussionThe strength and dilatancy of sandsrdquoGeotechnique vol 37 no 2 pp 219ndash226 1987

[38] Y Xiao H Liu X Ding Y Chen J Jiang and W ZhangldquoInfluence of particle breakage on critical state line of rockfillmaterialrdquo International Journal of Geomechanics vol 16 no 1pp 04015031-1ndash04015031-18 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom