MATHEMATICAL MODELLING OF ARCH FORMATION IN GRANULAR MATERIALS

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1 6 th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE MATHEMATICAL MODELLING OF ARCH FORMATION IN GRANULAR MATERIALS Istva eppler SZIE Faculty of Mechaics, H-2103 Gödöllő Páter. 1., Hugary Abstract: The mathematical modellig of formatio, stability ad damage of self-supportig stagat arch like structures iside bulk material discussed i this paper. Some experimetal methods will also be preseted, for the measuremet of material properties used i the mathematical model. We preset a algorithm for the determiatio of critical outlet size for a simplified model silo. eywords: graular material, archig, fiite elemet method, triaxial apparatus 1. INTRODUCTION Arcig occurs mostly durig the discharge of silos, but pressure actig o udergroud structures is also iflueced by archig, ad smaller tha the weight of the overburde. Earth pressure applied o retaiig wall sometimes also smaller tha the theoretical value, ad this pheomea are also caused by the archig actio. The arch formatio iside silos ca chage the wall pressure distributio i a large scale, ad sometimes the whole silo collapses because of this chage i pressure distributio. The outflow of graular materials from cotaiers is also cosidered to a sequece of formatio ad collapse of arches. The evaluatio of stresses i graular assemblies without takig the archig actio ito cosideratio, always gives false, iaccurate results. 2. MATHEMATICAL MODEL The classical method, used for the evaluatio of stresses withi the arch uses two theoretical solutios [1]. The first oe evaluates the stresses that cosolidate the graular material ad give rise to its stregth, the secod oe aalyzes the stresses that act i a arch regarded as a structural member. The evaluatio of stresses withi these speculative arches is based o

2 assumptios o the shape of these arches. The classical defiitio of archig follows from these aspects: archig refers to spotaeous formatio of a arch like supported stagat mass of bulk material [1], which is capable of bearig the pressure origiatig from the mass above it. For the mathematical model, first we eed to determie the stresses iside our model silo. The model silo was a simple rectagle shaped cotaier, with variable outlet size o its bottom. For the determiatio of stresses actig iside the graular mass, we used a used liear elastic, isotropic, homogeous cotiuum material model for the graular assembly. Our assumptios were: 1. The graular material fills out cotiuously the cotaier, ad its material properties are idepedet of space coordiates, time ad orietatio. 2. We assumed, that the coectio betwee the stress ad the deformatio tesor is liear i every poit of our model silo. 3. The load origiated oly from material self weight. 4. The wall-material frictio ca be eglected. 5. The rigidity of the silo side wall supposed to be ifiite. 6. We assumed plai strai state. Our aim was to determie the critical outlet size of this model silo. Critical outlet size meas a border-lie case, amely if the outlet size is bigger tha this critical values, stable arches ca ot be take form. Fig. 1: Triaxial apparatus Fig. 2: Triaxial compressio

3 2.1. Material properties ad failure criteria For the determiatio of stresses, the specime s Youg modulus (E) ad the Poisso s ratio ( ν ) had to be determie. The measuremet process of these material properties described i [2]. We eeded a other material property for the descriptio of the collapse of the arch. This is the critical stress belogig to biaxial stress state. The arch collapses, whe o its free boudary where the material is i biaxial stress state the compressive stress exceeds the critical value σ. We used a special triaxial apparatus (fig. 2) for the determiatio of material properties. The descriptio of this apparatus ca be foud i [3]. To measure the critical stress belogig to biaxial stress state, we applied two differet kid of load o the graular material. I the first step we applied a triaxial pre-compressio o the specime. The we removed oe of the lateral sprigs (fig. 3), ad icreased the vertical load util the collapse of the specime. With the removal of oe of the lateral sprigs, we realized the biaxial stress state, eeded for the measuremet of the critical stress. owig the material properties, the stresses arisig iside the model silo ca be determied usig fiite elemet method. The fiite elemet model of silo is i fig 3. After the determiatio of stresses iside the model silo, we have to aalyze the process of arch formatio. For this, we have to determie the failure criterios cotrollig the arch formatio ad collapse process. The failure criterio for arch formatio is simple for this rectagle shaped model silo: graular assemblies are uable to resist tesio. It is also simple to formulate this coditios i mathematical form. After the evaluatio of stresses, we have to compute the Fig 3: The fiite elemet model eigevalues of the stress tesor i every poit of the graular material. These eigevalues are the so called pricipal stresses. If the biggest pricipal stress is positive, tha i that poit of the material tesio occurs, ad the

4 cotiuity of the material fails. The material is fallig out from these poit through the ope outlet. The stability ad collapse of the arches depeds o two differet failure coditios. The firs failure coditio occurs, whe o the materials free boudary where the material is i biaxial stress state the compressive stress exceeds the critical value σ. The mathematical formulatio of this failure criterio is also i coectio with the pricipal stress values. If the third (smallest, ad i our case egative) pricipal stress value is smaller the the critical σ, i ay poits of the graular assembly s boudaries, tha the failure of the arch is due to happe. The value of this critical stress depeds o the magitude of pre-compressig stresses. The secod failure criterio is about the shear stresses actig iside the graular material. The value of the shear stresses iside graular materials caot be higher tha a critical value. The critical shear stress value is usually cosidered to be liear fuctio of the frictio agle ad cohesio of the graular material. Usig this failure coditios, it is possible to create a algorithm for the umerical simulatio of the archig process (fig 4) The algorithm 1. First we ope the outlet to a iitial size. 2. The determiatio of stresses iside the graular material comes ext. Usig fiite elemet method, this is possible; umerically. 3. owig the stresses, the eigevalues of stress tesors must be computed. 4. Usig the biggest eigevalues, the failure criterio for arch formatio must be applied. 5. After the removal of material elemets, where the first failure criterio prevailed, the domai, where the stresses were evaluated chaged, so the stresses must be evaluated agai. 6. owig the ew stresses, the eigevalues must be computed agai, ad the failure coditio for arch formatio must be applied. This goes util there are o more poits iside the material, where tesio occurs. 7. Whe there is o more tesio, the two failure criterio for arch collapse must be take ito accout. If oe of them comes to be true, the a stable arch formed. This arch belogs to the outlet size adjusted i step The outlet size ca be elarged, ad the the whole process starts agai from step The algorithm rus util oe of the arch failure criterios comes to be true. 10. Whe the arch collapse criterio occurs, the critical outlet size is determied.

5 ( ) Defie the domai T ad the boudary coditios. Evaluate the stresses: F + q = 0, 1 A = t + t, 2 [ ] 1 ν A = F FI E 2G 1+ ν. Solve the ( F σ E) = 0 eigevalue problem i T. σ 1 is the biggest, σ 3 is the smallest eigevalue of F. p T where σ 0? 1 > y Remove the elemets, where σ > 0 1 is true. The arch is stable, outlet size ca be elarged. p T, where σ 3 < σ < 0? (T is T-s boudary) y p T, where τ τ y The arch collapses Critical outlet size determied Fig 4.: The algorithm for determiatio of critical outlet size 3. RESULTS AND FURHTER RESEARCH A iterative method for modellig the archig actio i graular assemblies was developed. This method is differet, ad more efficiet tha ay methods existig i the literature for determiatio of critical outlet size i silos. Our further research will iclude the coic shaped cotaiers, where the possible slidig of the material at the cotaier wall must be also take ito accout.

6 4. REFERENCES 1. A. Drescher, A. J. Waters, C. A. Rhoades: Archig i Hoppers I-II., Powder Techology, 84, (1995), pp Csorba L., Balássy Z.Huszár I., Csizmadia B.: Determiatio of Poisso's ratio i elastic oedometer, 4th ICCPAM, Rostock, 1989, Proceedig, Volume 1, pp Csizmadia B. Oldal I. eppler I.: Quasi-Triaxial apparatus for the determiatio of mechaical properties of graular materials. 20th. Daubia Adria Symposium o Experimetal Methods i Solid Mechaics, September 24 27, Győr, Hugary. 2003