BCURA PROJECT B54 on Arching propensity in coal … Projects/b54_final_report.pdf · to conduct a survey of existing power station coal bunker geometries; 2. to develop computational

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University of Edinburgh Institute of Infrastructure and Environment Silos and Granular Solids Research Group

BCURA PROJECT B54

on

Arching propensity in coal bunkers with non-symmetric geometries

Final report

Project start date: 01.09.01 end date: 31.08.05

Project Officer: Mr M. Jones

UK COAL PLC Harworth Park, Blyth Road,

Harworth, Doncaster DN11 8DB, UK Tel:01302 755137 Fax: 01302 755252

Email: [email protected]

Investigators: Dr J. Y. Ooi and Prof. J. M. Rotter Researcher: Dr Shiwen Wang

Project Manager: Dr J.Y. Ooi

SCHOOL OF CIVIL & ENVIRONMENTAL ENGINEERING The University of Edinburgh

Edinburgh EH9 3JN, UK Tel: 0131 6505725 Fax: 0131 6506781

Email: [email protected]

CONFIDENTIAL

September 2005

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Contents EEXXEECCUUTTIIVVEE SSUUMMMMAARRYY ........................................................................................................................................................................................................................................ 44

11.. IINNTTRROODDUUCCTTIIOONN ............................................................................................................................................................................................................................................................ 66 1.1 PROBLEM STATEMENT ................................................................................................................... 6 1.2 AIMS AND OBJECTIVES .................................................................................................................. 7 22.. RREEVVIIEEWW OOFF JJEENNIIKKEE AARRCCHHIINNGG TTHHEEOORRYY.................................................................................................................................................................. 77 2.1 EXAMINATION OF THE JENIKE FLOW FUNCTION TEST..................................................................... 7 2.2 MECHANICAL MODELS FOR THE MECHANICAL BEHAVIOUR OF WET COAL ..................................... 8 2.3 MULTI-STRAND COMPUTATIONAL STRATEGY ................................................................................ 8 33.. IINNDDUUSSTTRRIIAALL SSIITTEE SSUURRVVEEYY.................................................................................................................................................................................................................... 88 3.1 COAL PROPERTIES ......................................................................................................................... 9 3.2 BUNKER GEOMETRY...................................................................................................................... 9 3.3 LINING STRUCTURE AND WALL FRICTION .................................................................................... 10 3.4 FILLING SEQUENCE AND DISCHARGE AID ..................................................................................... 11 3.5 SOME INNOVATIONS .................................................................................................................... 11 44.. CCOOMMPPUUTTAATTIIOONNAALL MMOODDEELLLLIINNGG OOFF AA SSYYMMMMEETTRRIICC CCOONNIICCAALL HHOOPPPPEERR ...................................... 1111 4.1 EXPLORATION OF CAM CLAY MODEL AND STRESS HISTORY IN COAL ........................................... 11 4.2 GEOMETRY OF THE HOPPER ......................................................................................................... 12 4.3 FE MESH AND MATERIAL PARAMETERS....................................................................................... 12 4.4 BOUNDARY CONDITIONS AND LOADING....................................................................................... 12 4.5 RESULTS AND DISCUSSIONS ......................................................................................................... 12 4.6 COMPARISONS WITH ANALYTICAL MODELS ................................................................................. 13 4.6.1 Summary of analytical models....................................................................................................... 13 4.6.2 Comparisons.................................................................................................................................. 13 55.. SSTTRREESSSSEESS IINN TTHHEE CCOOAALL AABBOOVVEE AARRCCHHEEDD OOUUTTLLEETT OOFF SSYYMMMMEETTRRIICCAALL BBUUNNKKEERRSS 1144 5.1 INTRODUCTION............................................................................................................................ 14 5.2 THE DETAILS OF THE MODEL....................................................................................................... 15 5.3 RESULTS...................................................................................................................................... 15 5.3.1 Wall Pressure studies .................................................................................................................... 15 5.3.2 Comparisons for wall pressures predicted by theoretical and numerical models......................... 17 5.3.3 Stress ratio affected by arch and hopper half angles .................................................................... 19 5.4 STRESSES ABOVE OUTLET USING CRITICAL STATE THEORY ........................................................ 19 5.4.1 Case 1: Hopper with selfweight..................................................................................................... 19 5.4.2 Case 2: Hopper with selfweight and surcharge ............................................................................ 19 66.. SSTTRREESSSSEESS AABBOOVVEE AANN AARRCCHHEEDD OOUUTTLLEETT IINN UUNNSSYYMMMMEETTRRIICCAALL BBUUNNKKEERRSS .............................. 2200 6.1 THE DETAILS OF THE MODEL....................................................................................................... 20 6.2 THE EFFECTS OF WALL FRICTION ................................................................................................. 20 6.3 THE EFFECTS OF UNSYMMETRICAL ARRANGEMENTS................................................................... 21 6.4 PADDLE POSITION ....................................................................................................................... 21 6.5 THE EFFECTS OF OUTLET HEIGHT ................................................................................................ 22 6.6 THE EFFECTS OF HOPPER HEIGHT................................................................................................ 22 6.7 REMARKS .................................................................................................................................... 22 77.. SSTTUUDDYY OOFF AA TTYYPPIICCAALL UUKK CCOOAALL BBUUNNKKEERR........................................................................................................................................................ 2233 7.1 INTRODUCTION............................................................................................................................ 23 7.1.1 Geometry ....................................................................................................................................... 23 7.1.2 Mesh .............................................................................................................................................. 23 7.1.3 Material Parameters ..................................................................................................................... 23 7.1.4 Loading and boundary condition .................................................................................................. 24

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7.2 RESULTS...................................................................................................................................... 24 7.2.1 Flow propensity analysis ............................................................................................................... 24 7.2.2 Wall Friction Effect ....................................................................................................................... 24 7.2.3 Paddle Feeder Position ................................................................................................................. 25 7.2.4 Progressive filling ......................................................................................................................... 26 7.3 DISCUSSION ................................................................................................................................ 26 88.. CCOONNCCLLUUSSIIOONNSS.............................................................................................................................................................................................................................................................. 2277

RREEFFEERREENNCCEESS .......................................................................................................................................................................................................................................................................... 2299

AAPPPPEENNDDIIXX AA:: CCAAMM--CCLLAAYY MMOODDEELL ............................................................................................................................................................................................ 3322 A.1 GENERAL FORMULATION ............................................................................................................ 32 A.1.1 Cam clay model simplified into three dimensional model ............................................................ 33 A.1.2 Cam clay model simplified into tri-axial compression model....................................................... 33 A.2 HARDENING LAW ....................................................................................................................... 34 A.2.1 Exponential form........................................................................................................................... 34 A.2.2 Piecewise linear form.................................................................................................................... 34 A.3 CRITICAL STATE LINE................................................................................................................. 34 AAPPPPEENNDDIIXX BB SSTTUUDDYY OOFF SSTTRREESSSS HHIISSTTOORRYY BBYY TTRRIIAAXXIIAALL CCOOMMPPRREESSSSIIOONN ........................................ 3366 B.1 GEOMETRY OF THE MODEL ......................................................................................................... 36 B.2 MESH AND MATERIAL PARAMETERS........................................................................................... 36 B.3 BOUNDARY CONDITIONS AND LOADING...................................................................................... 36 B.4 HARDENING EFFECTS .................................................................................................................. 36 B.5 SOFTENING EFFECTS.................................................................................................................... 38 B.6 LOADING METHOD EFFECTS........................................................................................................ 38 B.7 ELEMENT EFFECTS ...................................................................................................................... 38 B.8 LOADING PATH EFFECTS ............................................................................................................. 39 B.9 DISCUSSION AND REMARKS......................................................................................................... 40 AAPPPPEENNDDIIXX CC SSUUMMMMAARRYY OOFF SSEEVVEERRAALL AANNAALLYYTTIICCAALL SSOOLLUUTTIIOONN .......................................................................... 4422

NNOOTTAATTIIOONN .................................................................................................................................................................................................................................................................................... 4433

FFIIGGUURREESS ............................................................................................................................................................................................................................................................................................ 4444

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Executive summary Coal handling problems can cause serious unanticipated economic losses when the coal does not flow as expected from a bunker. A comprehensive survey (EPRI, 1995) on coal handling problems concluded that plugged bunkers and feeders are the biggest handling problems in this industry. In the UK, problems of coal arching and bridging in bunkers frequently occur at some power stations and human intervention is needed to break up the arches. These blockages can be very costly. Arching or bridging at a bunker outlet occurs when the stresses in the coal near the outlet are not sufficiently large to overcome the cohesive strength that permits an arch to form over the outlet. The mechanics of the problem depend on two major considerations: a) coal flow properties (characterised by the coal’s Flow Function) ; b) the bunker design (geometry, hopper angles, wall friction, feeder arrangement etc.),

traditionally characterised by the Hopper Flow Factor. The aim of this project is to examine the practical coal bunker geometries and to investigate and quantify their propensity for arching. This was achieved through the use of modern computational modelling techniques that can predict the performance of a coal of given properties when placed in a bunker of given geometry and surface wall friction. The objectives of the whole project were: 1. to conduct a survey of existing power station coal bunker geometries; 2. to develop computational models of typical existing coal bunker geometries and feeder

arrangements using a finite element method with appropriate constitutive models for the coal and the coal-bunker interface;

3. to determine the stress history of the coal as it passes from the bunker to the outlet (so evaluating the major consolidating stress applied to the coal);

4. to model the coal in various arched geometries (so evaluating the stress at incipient arch collapse);

5. using this information, to extend the Jenike flow factor design approach to cover these cases of non-standard bunkers;

The influence of coal flow properties on handlability has been extensively investigated by the Edinburgh University Group recently through BCURA funding, leading to the successful design and development of two industrial testers. However, it is not sufficient only to measure the coal properties. Poor handling coal can pass easily through some bunkers, but even good handling coal may block a bad bunker geometry. The key scientific information on bunker outlets for arching prediction is only available for very simple conical and wedge shaped hoppers, with a horizontal hole outlet. Such geometries are rarely found in UK power stations, and the interpretation of current design methods to give appropriate predictions of arching requires much speculative engineering judgment. The most significant outcome of this project is the new design information that can be used to eliminate arching problems in typical coal bunkers with the typical geometries and outlet arrangements used in UK coal bunkers, which differ considerably from those considered in Jenike’s classic arching theory. The new calculations permit the Jenike design method to be extended to the hopper outlet forms used in power stations in the UK.

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This main work commenced with careful measurements of power station bunker geometries. Then a computational finite element model was created and verified for the modelling of coal in such bunkers. The model was first used to predict consolidation stress states when the bunker was filled, considering a range of different material properties for the coal and different wall friction coefficients for the hopper walls i.e. different hopper linings. The computational model was then used in more difficult and extensive calculations to examine the stresses that develop in different shapes of coal bridges or arches across the outlet, to determine the most critical arch form and to deduce the conditions under which cohesive strength in the coal would just cause a stable arch to form. These calculations led to different outcomes depending on the geometry of the hopper sides, the hopper wall friction (with or without low friction liners), and the size of the outlet. The calculations addressed all these items in a huge parametric study, where all were systematically varied. Not only are these calculations very extensive, yielding a huge mass of information, but they are complicated to interpret, and much effort has been put into transforming them into design-relevant information that can be used in the Jenike design method for these bunkers. The main achievement of this project is the development of a method for assessing typical UK coal bunker geometries according to a modified version of the Jenike method for predicting arching across the outlet of a classic symmetrical bunker. This gives a rigorous basis for addressing arching problems and evaluating arching propensity in coal bunkers. The results will be beneficial to the owners and manufacturers of coal bunkers.

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1. Introduction

1.1 Problem statement The bulk handling of coal is important in many industries, such as mining, power generation and steel making. Problems in coal handling can seriously affect the reliable supply of coal and cause unanticipated hidden costs. A comprehensive survey on coal handling problems concluded that plugged bunkers and feeders are the biggest handling problems facing the industry. Another survey from the North American Electric Reliability Council (NERC) showed that these problems are not rare events (nearly 1000 events were reported on bunker flow problems in the period 1982-87 alone) resulting in major costs to some plants (EPRI, 1995). In the UK, problems of coal arching in bunkers occur frequently in some power stations, and regular human intervention (e.g. the use of air lances) is needed to break up the arches. Arching at a bunker outlet occurs when the stress field in the coal near the outlet is not sufficient to break down an arch which is held together by the cohesive strength developed in the coal (Jenike, 1964). The mechanics of the problem depend on the coal flow properties (characterised by the coal’s Flow Function) and the bunker design (geometry, hopper angles, wall friction, feeder arrangement etc.), traditionally characterised by the Hopper Flow Factor. Bunker design is normally performed using the Jenike method (Jenike, 1964; Rotter, 2001), which includes some simplifying approximations, and which is difficult to apply to geometries where the flow factor has not been rigorously evaluated. Until this project, flow factors only existed for symmetrical conical or planar (wedge) bunkers (Fig. 1.1). The concept of arching used in the existing theory is illustrated in Fig. 1.2. This project was not concerned with mechanical arching, since this is easily prevented. Instead, it was exclusively concerned with cohesive arching. The standard method of assessing arching potential (Rotter, 2001) relies on an analysis of stresses in the hopper that relate only to these symmetrical geometries (Figs 1.3 & 1.4). British coal bunkers often have more complicated geometries and outlet arrangements (Fig. 1.5), and the standard method cannot be applied directly. As each coal bunker is different, each currently needs an individual evaluation, and it is far from a simple matter to determine which of several alternative remedies should be chosen to be the cheapest reliable solution for handling problems. There was thus a need to develop new more accurate predictions with a wider range of applicability. This has been achieved using modern nonlinear finite element computer modelling. The finite element method has been used with a variety of different constitutive material models to predict pressures and flow in silos (e.g. Haussler and Eibl, 1984; Rombach and Eibl 1989; Ooi and Rotter, 1990; Schwedes and Feise, 1993; Kolymbas, 1993; Ragneau et al, 1994; Ooi and She, 1997). However almost all these studies focused exclusively on improving constitutive models and modelling techniques, which were then applied to relatively simple geometries in symmetrical bottom-discharging gravity flow silos. A numerical model of arching in a typical UK coal bunker with a twin horizontal outlet and paddle feeder arrangement (e.g. Fig. 1.5) does not appear to have been attempted before the present study. The results of this study apply to most British coal bunkers. They have been calibrated against the analytical studies of Jenike which were undertaken in the 1960s, which are still used throughout the world as the best available information. The outcome of the present study does not depend on the method by which the flow properties of the coal are evaluated, since the calculations have adopted the Jenike philosophy. Thus, they are usable with any reliable method for measuring the flow properties of the coals, including the Jenike Shear

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Cell, Schulze Annular Shear Cell, Edinburgh Cohesion Tester etc. One of the most direct benefits of this project is that it should now be possible to re-evaluate each existing bunker and assign a maximum unconfined strength for reliable flow in it. The project has exploited recent research on coal handling and bunker design conducted at Edinburgh and funded by BCURA, in which the interactions between stress history, moisture content and particle size distribution as affected by segregation were found to have a major influence on coal handling performance (Zhong et al, 2000; Rotter and Ooi, 2000; Zhong et al, 2005).

1.2 Aims and objectives In this project, different practical bunker geometries were examined to investigate and quantify their propensity to stop flowing. This was achieved through the use of modern computational modelling techniques that can predict the performance of a coal of given properties when placed in a bunker of given geometry and surface wall friction. The objectives of the whole project were: 6. to conduct a survey of existing power station coal bunker geometries; 7. to develop computational models of typical existing coal bunker geometries and feeder

arrangements using a finite element method with appropriate constitutive models for the coal and the coal-bunker interface;

8. to determine the stress history of the coal as it passes from the bunker to the outlet (so evaluating the major consolidating stress applied to the coal);

9. to model the coal in various arched geometries (so evaluating the stress at incipient arch collapse); 10. using this information, to extend the Jenike flow factor design approach to cover these cases of

non-standard bunkers; These objectives have all been satisfactorily achieved.

2. Review of Jenike arching theory A review has been undertaken of the Jenike arching theory and its application in practice, including the determination of the flow function by testing and the flow factor evaluation using classical mechanics theory. This work has led to a number of interim conclusions a) The Jenike arching theory is relatively simple, so some modifications to it can be made even for

standard outlet geometries; b) The application of the Jenike method depends heavily on the analysis that leads to the flow

factor for the hopper. It has been found that the method is rather variable in its accuracy when different properties are assumed for the coal. New proposals for modifications to this classic theory are still being developed and verified;

c) The application of the Jenike method leaves quite a lot to be desired. In particular, the stress history of the solid that arches in the bunker is seriously over-simplfied because that was all that could be done in the 1960s. The computer programs that have been used in this project overcame this stress history difficulty and produced a much clearer image of true arching situations. However, the full stress history was found to be very complicated indeed, and many numerical problems had to be overcome in the search for sound solutions.

d) The Jenike method could not be applied to bunkers with the outlet geometries commonly used in Britain (Fig. 1.5). This project has focussed on these other geometries.

2.1 Examination of the Jenike flow function test The entire world bulk solids handling community depends on the Jenike flow function test for its predictions of arching in all bunkers. This test is conducted in a shear cell, in which the stress state is not well defined. Under this project, some work was undertaken to assess the real behaviour of coal placed in this tester, so that this behaviour can be related to the conditions under which arching can occur at a hopper outlet. However, the relationship proved to be more difficult to establish than was

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originally thought, and it was found that this part of the project could not be completed within the timescale, so it was abandoned before it was complete. Nevertheless the specific conclusions arising from this study include: a) The Jenike tester is difficult to use and slow, and it is not easy to produce identical results with

it every time. For this reason, some doubt must be cast on many of the precise values of properties that come from the test.

b) This tester is completely unsuitable for coal testing, because it cannot accommodate particles larger than a few millimetres in size.

c) Several research groups world-wide have attempted to devise new testers that can reveal the coal behaviour more accurately than the Jenike tester.

d) The task of relating measurements made in other testers, such as the Edinburgh Cohesion Tester, to those of the Jenike cell is quite complicated, and requires a relatively complicated numerical analysis to determine the relationships.

2.2 Mechanical models for the mechanical behaviour of wet coal The numerical modelling by computer used in this project required that the behaviour of the coal should be represented in an appropriate manner. The Jenike theory assumes that coal is a rigid-plastic material obeying a simple frictional failure criterion at all points. Whilst the computer model could also represent it approximately in this manner, it is a considerable oversimplification, and this idealisation omits key aspects of the stress history arising from placement and flow of the coal. Interim conclusions that have been reached in this project include: a) A simple elastic model should be used for parametric explorations of a wide range of

geometries of both bunkers and arch shapes; b) A simple elastic-plastic model using a Mohr-Coulomb or Drucker-Prager failure criterion

should be used to assess Jenike theory against more modern calculations. c) It was thought that it would be very useful to conduct a few analyses using a full stress history

for the coal. However, this was found to be a very complicated task. The Modified Cam Clay mathematical model was chosen as probably the most appropriate for this purpose, but it was known to have difficulties when the stress level becomes very low, as in the bunker arching problem. Although this full stress history study was attempted in this project, it ran into many difficulties associated with the highly overconsolidated material and the prediction of very high strength with very brittle behaviour.

d) The main predictive outcomes from this project were achieved with an elastic-plastic model using a Drucker-Prager failure criterion. This was calibrated onto the unconfined compressive strength of the coal.

2.3 Multi-strand computational strategy The plan for computational work was set out as follows: a) The different constitutive models were each used for the greatest applicable information. b) Elastic stress calculations were used to determine stress states in a wide range of arch

geometries to ensure that the worst conditions were identified. c) The classical bunker outlet geometry was used to calibrate the predictions. d) The geometries of existing British bunker outlets were carefully measured and documented and

adopted into the calculation plan.

3. Industrial site survey The first industrial surveys at West Burton and Drax Power Stations were carried out on 26th and 27th July 2002. West Burton Power Station then belonged to the London Power Company (LPC) (now owned by EDF Energy Ltd). Besides West Burton Power station, LPC also owned Cottam Power station, Sutton Bridge

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Power station and several others. West Burton Power Station was opened on 25th April 1969. It is a 2000MW power station. The station comprises four coal-fired generating sets and two 20MW gas turbines. The station can burn up to 19,000 tonnes of coal a day, so the bunkers must be capable of handling at least 19,000 tonnes of coal within one day. Drax is now the largest coal-fired power station in Western Europe. It stands in 1854 acres and is situated in North Yorkshire on the south bank of the River Ouse, midway between Selby and Goole. It can produce enough electricity - about 4000 MW- to meet the needs of approximately four million people. The station was built in two 2000 MW stages. It has six 660 megawatt, coal-fired generating units. The first stage of construction began in 1965 and was completed in 1974. The second stage began in 1978 and was completed in 1986. Both power stations are coal-fired plant and have Flue Gas Desulphurisation (FGD) equipment associated with their generating units. The coal receiving bunker is typical and its filling and discharging operation is important to avoid disruption of power generation.

3.1 Coal properties The principal coal properties that cause the greatest concern to power stations include ash, sulphur, moisture and volatile matter contents, heating value and grindability. A high moisture content will lower the boiler efficiency, whilst ash, sulphur and nitrogen may contribute to air pollution, acid rain and global warming. As Power Station Fuel, coal should be easy to handle. This further requires that the coal should develop only a small amount of cohesion when compressed in a bunker. A well designed bunker can handle coals with very poor handling performance, but the challenge in this project is to recommend limits on the cohesion potential of coals so that they can be guaranteed to flow through the existing bunkers without modification. Table 3.1 shows some typical properties of UK coal. These data come from one coal mine belonging to RJB Mining (UK) Ltd (Zhong, 2001).

Table 3.1 Properties of typical UK coal

Coals Moisture range Ash content range Particle size (mm) CV§

Washed coal (46%) 8% ~ 12% 7% ~ 13% <10 Very high Singles (15%) 6% ~ 9% 5% ~ 8% 10~50 29270 Filter cake (6.5%) 25% ~ 34% 8% ~ 15% <1 19450 Foreign fines (7.5%) 8% ~ 12% 4% ~ 18% <6 2460 U/T# fines (25%) 7% ~ 9% 28% ~ 50% 10~50 Varying Final PSF* 8% ~ 12% 14.0% ~ 19.2% <50 24260

§CV --- Calorific Value (kcal/kg) #U/T --- Untreated fines *PSF --- Power Station Fuel

3.2 Bunker Geometry Four types of coal receiving bunker were found in the power stations visited during this survey. Their geometries are explained separately in the following. Type A (Fig. 3.1a): This is a concrete planar hopper, 55.47m long and 6.7m high with hopper wall slopes of 30° to vertical(Fig.3.1a). This bunker is very similar to bunkers at Fiddler’s Ferry (Rotter and Ooi, 2000). The bunker is partitioned into eight separate compartments (Fig.3.1b). At the top of the bunker, there are two railway tracks along the length of the bunker. At the level of the two outlets, there is an inverted cone making an angle of 60° with the vertical on a concrete platform. This structure is different from that of Fiddler’s Ferry. In Fiddler’s Ferry, a structure

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termed a “Coal Bunker Centre Extension Cone” (colloquially known as the “dog kennel”) was constructed at a date after its initial bunker construction. The two outlets are horizontal slots above a concrete table on which the coal was intended to lie at its angle of repose. For the coal to lie on this table without spilling onto the conveyor, it must have an angle of repose greater than the angle defined by the slope from lip of the concrete bunker wall to the edge of the table. From the original design, it measured an angle of 50°. Thus, unless the coal had an angle of repose in excess of 50°, it would continuously fall from the table onto the conveyor. Coals have a wide range of repose angles, varying from about 35° for very free-flowing coals to as high as 65° for highly cohesive coals. Additional plates (Fig.3.1c) are bolted onto the external wall of bunker on the outside to reduce the repose angle required of the coal and help to control the discharge of free flowing granular coals. This is similar to that of Fiddler’s Ferry. The outlet is served by two travelling feeding machines, each of which carries a pair of counter-rotating paddle feeders which withdraw coal from the feeding platform onto the conveyor. Other dimensions and comparisons for the bunker are listed in Table 3.2:

Table 3.2. Dimension comparisons between two receiving bunkers Fiddler’s Ferry West Burton Bunker width at the top 9.6 m 9.8 m Bunker width at the bottom 2.5 m 2.8 m Width of discharge opening 0.63 m 0.67 m Bunker half angle 30° 30°

Bunker lining material UHMWP sheets, good condition

Stainless steel sheets, good condition

Dog kennel (Inverted cone) 60° slope on lower part and 45° slope on the upper part 60° slope for the whole part

Type B: The second kind of bunker shown in Fig. 3.2 is the “so called” milling bunker. It is comprised by two arrays of rectangular compartments. Each compartment comprises six rectangular cells with a length of 45.72m and a width of 22.86m. Bunker structures, one typical cell and their dimensions are shown in Fig.3.2. Each cell is consisted by a vertical rectangular silo and an unsymmetrical trapezoidal hopper. The side walls of the trapezoidal hopper are inclined in different angles with vertical in two directions. The hopper walls are very steep, with angles of 67° to 75° degrees to the horizontal. The total height of each cell is 15.36m. Coals are discharged automatically when trains go through the tracks. Type C: Fig. 3.3 shows one type of concrete bunker with twin horizontal outlets. The whole structure of the bunker is in the shape of a letter W. Paddle feeders and belt conveyors are used to transfer coal from the hopper to the next process. Type D: Fig.3.4 shows an all metal hopper found in Drax power station. This hopper is completed encapsulated in the coal operation (Fig.3.4a). Vibration is applied at the outlet to assist the withdrawal of coal from the bunker (Fig.3.4b)

3.3 Lining structure and wall friction Three kinds of lining have been observed in coal bunkers in power industry. The first type is found at West Burton power station, in an un-used bunker. There are no linings at all. Heavy wear scratches can be observed on the inside of the bunker (Fig.3.5a). The second type is UHMWP linings bolted onto the inside of bunkers. This kind of lining has been observed in our previous survey at Fiddler’s Ferry power station (Rotter and Ooi, 2000). A layer of 10mm thick UHMWP sheets is bolted into the concrete bunker as the lining. The surface of the UHMWP sheets is very smooth with low wall frictions. This kind of lining is also cheaper and easier to install and replace. The main disadvantage for this lining is that it is susceptible to wear and tear. There is also the possibility of ageing in the

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UHMWP plates with varying temperatures. The third type observed is the all metal bunkers found at AES Drax power station (Fig.3.5b). No lining is needed as the inside of the bunker is already very smooth. The disadvantage for this bunker is that it is expensive and cannot be easily replaced when the inside is heavily worn.

3.4 Filling sequence and discharge aid Coal is delivered to West Burton Power station by the ‘merry-go-round’ rail system or by lorry. In Drax power station, large proportion of the coal is supplied by rail. The coal is unloaded automatically from the rail wagons to the receiving bunkers and transported by conveyor either to the other bunkers or to the stockpile. Paddle feeders are used to assist the discharge of coal from bunkers. In Drax power station, vibration is used to assist the coal discharge. This kind of discharge aid was observed only for completely encapsulated bunkers.

3.5 Some innovations To improve the flowability of coals within bunkers, ‘end wall additions’ were built to aid the coal to pass through the frame of ‘Tulley Gate’ in the bunker mouth. One method is to add new profiling at slot ends (Fig.3.6). The sloped end plates are fabricated in stiffened mild steel lined with 3mm thick stainless steel linings. The maximum downloading capacity for the profiling at slot ends was calculated to be 16.6t/m3, whereas in practice, it is not likely to exceed 10t/m3. The maximum design sideload was 10.63t/m3, which is significantly greater than 7.03t/m3 expected in practice. Bathtub profiling is another innovation used to improve the flowability of coal within the discharge system of a bunker. Two layers of beams are installed at the bottom of the bunker (Fig.3.7). The two layers of beams are installed so that coal flowing from the upper profiling beams is matched up with the lower beams. The slope plates and lower profiling beams in the feeder were fabricated in a stiffened mild steel and lined with 3mm thick 2B stainless steel with surface roughness less than 0.5 µm. Paddle feeders are used to withdraw coal from the bunker.

4. Computational modelling of a symmetric conical hopper The first modelling study is a simple axisymmetric hopper under filling conditions. The purpose of this study is to check the existing analytical solution (Jenike and others) to verify that the computational procedure gives good practical results. The coal will be analysed using three different constitutive relations: Elastic (EL), Mohr-Coulomb (MC) and Cam-Clay (CC). The Finite Element Method (FEM) has been proved a successful method to predict stress histories within bunkers. This has been verified elsewhere in the literature (e.g. Haussler and Eibl, 1984; Rombach and Eibl, 1989; Ooi and Rotter, 1990; Schwedes and Feise, 1993, Ooi, Chen et al. 1996; Ooi and She, 1997; Rotter et al, 1998). ABAQUS (Hibbit, Karlsson and Sorenson, 2001) software was used in our study.

4.1 Exploration of cam clay model and stress history in coal The Cam Clay models are based on the framework of volumetric hardening, pressure sensitivity and the critical state soil mechanics. The original and modified Cam clay models were developed by Roscoe et al (1963), and Roscoe and Burland (1968), respectively. The Cam Clay model is a classical plasticity model with a unique feature of a "critical state" surface allowing plastic hardening/softening. The details about the Cam clay model and its implementation in Abaqus finite element program can be found in Appendix A. The capability of the Cam clay model to predict the complex stress history in a coal bunker was investigated extensively using a triaxial test situation. Triaxial test is a widely used procedure to determine strength and stress-strain properties of granular solids such as coal and soils. In such a test, a cylindrical specimen is subjected to an axial compression stress aσ and a radial pressure stress rσ . The

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specimen is assumed to deform uniformly during the test so that the information obtained from the test represents the true material behaviour of a single granular solid element within bunker. In this modelling study, coal mass is modelled using the modified Cam clay model with exponential hardening rules. Two kinds of failure mechanism, hardening failure and softening failure, are investigated. Relationships between coal strength and different loading paths are explored. The results of this study can be found in Appendix B.

4.2 Geometry of the hopper A symmetric hopper with coal inside is shown in Fig.4.1. The outlet dimension is 300mm, the diameter of the hopper at the top is 3754mm, and the total height is 2992mm. The hopper half angle is 30° from the vertical. The thickness of hopper is 8.66mm. Hopper is assumed to be made of steel.

4.3 FE Mesh and material parameters For a symmetric bunker under symmetric loading, all the normal stress and strain should be symmetric about the central axis. The computational model can therefore be reduced to half of the complete structure. The mesh is shown in Fig. 4.2. The coal inside was modelled by 15×40=600 axisymmetric quadratic elements (CAX8). While hopper was modelled by 2×40=80 axisymmetric quadratic element (CAX8). The interface between the coal and hopper was modelled using contact pairs elements. The friction coefficient µ was assumed to be µ=0.3 in this study. The total number of elements was 761, and the total number of nodes was 2398. Table 4.1 and Table 4.2 show the material parameters for the hopper structure and the solid contained in the hopper (coal).

Table 4.1 Material parameters for hopper Young’s Modulus (MPa) Poisson’s ratio

Hopper walls 100 0.3

Table 4.2 Material parameter for coal Elastic Model Mohr-Coulomb Cam-Clay Model

γ (N/m3) E (MPa) ν c (kPa) φ (°) κ λ M e0 G (kPa) 7601.8 100 0.32 10 30 0.0092 0.0367 1.68 0.87 700

γ--- Bulk density E---Young’s modulus ν--- Poisson’s ratio c--- Cohesion φ--- Internal friction angle

κ--- Logarithmic bulk modulus λ--- Logarithmic hardening modulus M---Stress ratio at critical state e0--- Initial void ratio G--- Shear modulus

4.4 Boundary conditions and loading In this finite element analysis, the outlet and the external surface of the hopper were fixed in both directions, and the central axis was a line of symmetry for the model. Only gravitational load was applied to simulate end of filling of the bunker.

4.5 Results and discussions The target of this analysis is to determine the stress state after the coal has been placed in the bunker, with special attention on the stress states that may induce high cohesion. To this end, the full set of stresses throughout the stored coal must be examined. Contours for the most compressive in-plane principal stress are shown in Figs 4.3-4.5. Because ABAQUS is a general finite element package, it follows a convention of tension positive. Thus a negative value of stress here represents a compressive stress within the coal. From these figures, it can be observed that no major difference exists between the results obtained using different constitutive models for the coal: the EL model and the MC model are almost indistinguishable.

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Much larger deformations can be observed in Fig. 4.5 for the CC model. An arched compressive zone appeared near the outlet. The value of principal compressive stress for the arched layer of coal near the outlet increased from 6kPa at some distance above the outlet to approximately 9kPa vertically near the outlet. This indicates that arching may occur near this zone. The contours of most compressive in-plane principal strain are shown in Figs 4.6-4.8. Again, negative values represent compression. There are no major differences between the strain contours for the EL and MC models, but the contours for the CC model indicates larger strains than the other models. The arching zone is clear in all three analyses. For most engineering problems, the wall pressure at the interface between coal and hopper is the most interesting for design engineers. The wall pressures from these calculations are presented in Fig. 4.9. It can be observed from this figure that the maximum wall pressure occurs at about one third the total height above the outlet along the contact surface between coal and hopper. There are no major differences between the maximum wall pressures predicted by the three constitutive models (8.9kPa for CC model, 8.15kPa for EL and MC model). These finite element predictions indicate that the wall pressure along the coal/hopper interface varies in a smooth nonlinear parabolic way. Fig. 4.10 shows the slip distance along coal/hopper interface for each of the three models. It can be observed that a major difference exists between the CC model and the EL/MC models. The maximum slip occurs at the top interface in all three models, but the CC model has the largest slip distance (0.18m), while the slip distance at the top interface for the EL and MC models is very small (4.66×10-5m). This suggests that the parameters in the Cam-Clay model have been chosen to give a very compressible response for the coal. The maximum slip occurring at the top is also a result of modelling the whole stored solid in one go (without considering progressive filling).

4.6 Comparisons with analytical models

4.6.1 Summary of analytical models Most of the analytical theories for the distribution of wall pressures in hoppers are based on the assumption that the mass of solid within the hopper is in a plastic state of stress. This can be easily seen from the fact that failure properties, such as the angle of internal friction, are widely used in these theories. There are several classical theories for the pressures in hoppers, including those of Jenike, Walker, Walters, McLean, and Rotter’s design equations. These are used in this study to compare with the ABAQUS results. A complete review of the analytical models for bunkers can be found elsewhere (Ooi and Rotter, 1991). Appendix C lists the major features of these theories including the notation.

4.6.2 Comparisons The predicted pressures obtained with the EL, MC and CC models are compared with the analytical solutions in Figs 4.11-4.13. For the EL and MC models, the FEM simulations are quite close to McLean’s simple rule (F=1). The peak wall pressure values lie between the Rotter design theory (Rotter, 2001) and Walker’s flow theory (Walker, 1965). The Rotter design prediction indicates that the peak wall pressure should occur at one third of the total height between the vertex and transition, measuring from the vertex. The FEM calculations and McLean predict this point to be about 40% of the total height. For Walker flow theory, the position moves higher to 51% of the total height. So, it can be concluded that the Rotter design equation and Walker flow provide quite good bounds on the position of this peak found in the numerical calculations with the EL and MC models, whilst the simple McLean assumption provides a good match. For Walters’ flow theory, the peak pressure is even lower than the lower bound provided by Walker’s flow theory. The predicted peak wall pressure is located significantly higher above the vertex, at 71% of the total height. Jenike and Walker’s filling theory provide good estimates of the wall pressures before it reaches peak from the hopper transition. It can be treated as lower bound to the numerical simulations before peak is reached from hopper transitions. From these two figures, we can also see that Jenike peak theory provides a much larger upper bound for the peak wall pressure. This may indicates

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that traditional Jenike theory is too conservative in the hopper design and some improvements can be made to the Jenike theory. The corresponding comparisons between the CC constitutive model and the analytical equations are shown in Fig.4.13. Here, the Rotter design theory provides a close match with the numerical calculations, so it can be treated as a close upper bound, whilst McLean’s proposal provides an useful lower bound for the numerical estimation. Walker’s flow and Walters’ flow theories provide further lower bounds for the numerical computations. It is remarkable that the Jenike and Walker fill theories provide close estimations of wall pressure to the numerical results before it reaches peak values near the hopper transition. The Jenike peak pressure gives a very high estimate of the peak wall pressure and it is clear that an improvement of this theory is needed. Such an improvement is the purpose of this project.

5. Stresses in the coal above arched outlet of symmetrical bunkers

5.1 Introduction The occurrence of arching across a hopper outlet depends on many factors: the material properties of the coal, the stress history to which it has been subjected, the bin or hopper geometry, and the outlet size. Theoretical analyses of arching in hoppers originate from various studies by Jenike and co-workers (1961, 1964). Because the simple analyses they used ignored several features of the problem, the Jenike theory gives over-safe results, even for the symmetrical geometry examined. Modifications of Jenike’s theory have been made by others e.g. Walter et al (1966, 1967), Arnold et al (1976) and a description of Mroz and Szymanski’s theory can be found in Drescher (1991). A full description of current design procedures is given by Rotter (2001). It should be noted that all these authors assumed that the bulk material in a hopper can be regarded as a non-interacting stack of structural members: arches or domes. Arching may occur if the strength of each member is greater than the weight-induced stresses. The arch is static, and it is failure under static conditions that must be examined, even though the literature commonly refers to this as a problem of “flow”. The fundamentals for determining “flow properties” in a laboratory environment were developed by Jenike and others and a great number of authors presented results discussing the testing methodology, as well as describing modifications to the equipment or new designs and specimen preparation techniques (Stainforth et al 1973, Williams, et al 1967, Samath et al 1993). Further descriptions of relevant tests and their interpretations are given by Rotter (2001). In spite of extensive theoretical and experimental work, the prediction of the hopper critical outlet size preventing arching often is unsatisfactory. Wright (1972) claimed a high degree of accuracy in the estimation of the critical outlet width by the Jenike method, but experiments later showed that the method commonly led to over-design by more than a hundred percent (Eckhoff et al 1974). In several cases, the outlet size found in model or full-scale hopper tests was much smaller than the theories predict (Walker 1966 and 1967, Enstad 1981). Coal and other solids stored in silos and bunkers behave in very complicated ways that are very difficult to predict mathematically. As a consequence, it is natural to use computer methods to try to capture the complicated behaviour of the solid, and achieve more assured predictions of what will happen when a given solid is placed in a bunker. However, this has also turned out to be a very difficult problem, and the huge research efforts of the 1980s and 1990s have led to a realisation that the problems are particularly difficult even for the best formulations. Most of the effort on this problem has been directed towards the filling and discharge of silos, to determine flow patterns and wall pressures. These studies have explored the effects of silo geometry (Guines et al 2001), bulk material behaviour (Rombach et al 1995, and Tejchman 1998), interaction between bulk solids and silo walls (Ooi, Rotter 1990), and the filling and emptying processes (Chen et al. 2001, Ragneau et al. 1995,

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Keiter et al. 2001). However, by contrast, no numerical modelling studies appear to have explored the silo arching problem, so current understandings of this problem are still based on very simplified classical analyses developed in the 1960s and 1970s (Rotter 2001). The purpose of this section is to describe the results of models for the coal behaviour in various arch geometries, which show the stress state at incipient arch collapse. The effects of different arch geometries on the silo wall pressure have also been explored. Numerical model details and materials properties for coals are given in detail. Comparisons for wall pressures between theoretical and numerical computations are made. The relationships between the isotropic stress component p and the deviatoric stress component q, which controls when the arch will collapse, are explored for different arch geometries using the concepts of Critical State Theory in soil mechanics.

5.2 The details of the Model The geometry of a hopper with an arch over the outlet is shown in Fig. 5.1a. To make the results dimensionless, and so applicable to all hoppers of the same shape, the outlet radius of the hopper was used as a unit dimension (b=1). The hopper height should not influence the stress state near the outlet, so a large hopper was used with a height of H=50 units. Hopper half angle, β, varied from 15°, 30°, 45°, 60°, while arch angle, θ, varied as 30°, 45°, 60°, 75° to 90°. In this study, the total height of hopper from vertex was represented by H. Coals were modelled as pseudo-elastic solids since this model satisfies equilibrium everywhere and the properties of a consolidated coal that is not at failure are quite well represented by an elastic model. The hopper was modelled using an analytical rigid surface. The finite element mesh used in this study is shown in Fig. 5.1b. Coal was modelled with 8-noded axi-symmetric elements (CAX8). There were 3,361 elements for coal and 10,164 nodes for the entire stored coal. The coal to hopper interface was modelled by 161 interface elements generated by ABAQUS and with Coulomb friction. The wall friction coefficient was varied from 0.1 to 0.6 in 0.1 increments. Geometric non-linearity was considered in the study. For the elastic properties for coals, the elastic modulus was taken as 10MPa and Poisson’s ratio as 0.3, though these values do not have very much impact on the predicted stresses. For the boundary conditions and loads in this analysis, the outside hopper of the model was assumed to be fixed in all directions (Fig. 5.1b). Arched outlet was stress free. The axis of the hopper was assumed symmetric boundary. Stress distributions for two load cases were investigated. Namely, self-weight (1kN/m3) only and self weight (1kN/m3) + surcharge along the transition (100kPa).

5.3 Results

5.3.1 Wall Pressure studies

5.3.1.1 Friction effects Wall pressures and vertical stresses are shown in Fig. 5.2 near the coal/bunker interface and p-q stress around arched outlet for different wall friction conditions. In this computation, both the hopper half angle and arch angle were in 30 degree and only self-weight was considered. It can be observed that both wall pressure and vertical stresses reduce with the increasing of wall friction. For the completely smooth wall friction, both wall pressure and vertical stresses exhibit the largest for all the friction situations. There are no obvious reductions for wall pressure and vertical stresses along wall if wall friction is greater than 0.4. Fig. 5.3(a) is p—q stress distribution along arched outlet. It can be seen from this figure that large wall friction will cause small p—q stresses at outlet. The stress ratio varies roughly between 1.4~1.5. The detailed relation is shown in Fig. 5.3(b). It can be observed from this figure there exists a transition point A at 60% of total height, H, of hopper. For portions between 60~100% of total height, H, of hopper, little difference exists for vertical stresses with maximum vertical stress arrived with coarse wall (µ=0.6) and minimum arrived with smooth hopper (µ=0.0). For portions up to 60% of

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total height, there is large difference for vertical stress at wall with the maximum vertical stress caused by smooth wall and minimum vertical stresses at wall are caused by coarse wall. Figures 5.4-5.6 show wall pressures, vertical stresses at the interface and p-q stresses around an arched outlet for hoppers with different wall friction and under a surcharge at the transition. It can be seen from Fig. 5.4 that wall pressures are translated along horizontal axis for different wall frictions. The peak of the wall pressure moves to the upper end of hopper in vertical directions and there is large overstress at the hopper outlet. For the vertical stresses with surcharge (Fig. 5.5). The transition point A moves up to 80% of total height, H. There are little differences for vertical stresses near top portions (80%~100% of total height). Below this portion (<80% of total height, H), large differences exist for vertical stresses with the maximum vertical stresses at wall are caused by smooth wall and minimum stresses are caused by coarse walls. The relationship between the stress invariants p and q is shown in Fig. 5.6a at an arched outlet. A large wall friction coefficient causes small p—q stresses near the outlet. The stress ratio, M=q/p, varies from 1.1~1.6. There are variations of stress ratio for wall friction between 0.0~0.2 (Fig. 5.6b), but after the wall friction reached over 0.2, stress ratios tend to constant. Figs. 5.7-5.9 are results for wall pressure, vertical stresses and p—q at arched outlet for hoppers with self-weight. In these figures, the hopper half angle is 30 degree, while the arch angle at outlet is 45 degree. It could be observed from Fig. 5.7 that wall pressure reduced in some extent compared with the Fig. 5.1 for hoppers with same angles for arch and half hoppers. The wall pressures vary with the same trends but the maximum wall pressures reduced in some extent. Vertical stresses at wall (Fig. 5.8) have very similar distributions compared with Fig. 5.2 besides some difference at the outlet. Compared for vertical stresses between Fig. 5.2 and Fig. 5.8, it can be observed that vertical stresses at wall will decrease with the increase of arch angles. Fig. 5.9a is p—q stress distributions at arched outlet. It can be seen that large wall friction will cause the p—q stress reduced to lower level. The stress ratio, M, varies from 1.05~1.45 (Fig. 5.9b), with the large stress ratio corresponding to large wall frictions. Fig. 5.10-5.12 are results for wall pressure, vertical stresses and p—q at arched outlet for hoppers with self-weight+surcharge. The wall pressure distributions for different wall frictions in Fig. 5.10 showed that there is translation of wall pressures. The maximum wall pressure was induced by smooth wall while, for coarse wall, the wall pressure will reduce. The vertical stress in Fig. 5.11 (β=30°, θ=45°) is similar with Fig. 5.5 (β=30°, θ=30°) except vertical stress near arch for the former is smaller than the other. For the p—q stress at arched outlet, lower wall friction result high p—q stresses (Fig. 5.12a). The stress ratio, M, varies from 1.0~1.6 depending on wall friction. Large wall friction causes large stress ratio.

5.3.1.2 Hopper stresses with different arch angles Wall pressures for hoppers with different arched outlets under self-weight are shown in Fig. 5.13. It can be observed from the figure that there is no major difference for wall pressures for most part of the hopper except near the outlet where there is major difference for wall pressure. There is large overstress at outlet if the hopper half angle, β, equals arch angle, θ ( In this case, β =θ=30°). For other situations where the arch angle is greater than hopper half angle (θ>β), wall pressures at outlet approaching towards zero with the increasing of arch angle. Fig. 5.14 shows the vertical stresses at walls for hoppers with different arch angle. It can be seen major differences only exist at the outlet, where some overstress exist if β =θ, while vertical stresses approaching towards zero with the increasing of arch angle from 45° to 90° with 15° increments. Fig. 5.15 is the wall pressures for hoppers with different arch angles under the self-weight and surcharge. For portions immediately away from outlet (8%-100% h/H), wall pressures are same for hoppers with different arch outlets. Near the outlets (from outlet~8% h/H), wall pressures gradually tends to zero with the increasing of arch angle except the case for which arch angle and hopper half angle are equal (β =θ=30°).

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Fig. 5.16 is the vertical stress distributions along the walls for hoppers with different arch angles under self-weight+surcharge. It can be found from this figure that vertical stress at the transition is just exact the value of surcharge for all the cases. For portions away from the arched outlet (8%~100% h/H), vertical stresses are same. At the outlet, overstress exist only for cases of arch angle equals to hopper half angle. For other situations, vertical stresses decrease with the increasing of arch angles.

5.3.1.3 Hopper stresses with different hopper half angles In this investigation, the arch angle was kept 30 degrees, while the hopper half angles vary from 15° to 60° with 15° increments. The wall frictions were kept 0.2. The inside coals were modelled as elastic media under self-weight and self-weight and surcharge at transition. Fig. 5.17 shows wall pressures distributions for hoppers with different hopper half angles. It could be observed from this figure that wall pressures decreasing with the reduction of hopper half angles. That means the steeper of the hopper, the smaller the wall pressures. The other obvious phenomenon is exhibited for wall pressure at outlet. There are overstresses if the hopper half angle is equal or greater the arch angle (β≥ θ), otherwise, no overstresses exist (β< θ). Larger hopper half angle normally cause large over-stresses at outlet. The variation of vertical stresses with hopper half angles is shown in Fig. 5.18. Small hopper half angle will cause lower vertical stresses at the wall. Meanwhile, there is overstresses if the hopper half angle is equal or greater than arch angle (β≥ θ). Figs. 5.19-5.20 show wall pressure and vertical stresses at wall for hoppers with self-weight and surcharge. It can be seen from Fig. 5.19 that there are some shifts for wall pressures with the increasing of hopper half angle. Overstresses exist at outlet if hopper half angle is equal or greater than arch angle (β≥ θ). For vertical stresses, small hopper half angle will cause lower vertical stresses. At the top transition zone, vertical stresses are equal to surcharge pressures exactly. There are overstresses for vertical stresses at the outlet if β≥ θ holds.

5.3.2 Comparisons for wall pressures predicted by theoretical and numerical models

5.3.2.1 Theoretical formulations Following the description given by Rotter [32], wall pressures for a conical hopper after filling or storing under symmetrical filling conditions can be represented by Eq. 1, after generalisation of what is commonly known as Walker’s theory [4], which was probably first devised by Dabrowski [29]:

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n

hvft

n

hh

hfvffnf h

xphx

hx

nhF

pFp

+

−

−

==1

γ (1)

in which: )1cot(2 −+= fhf FFn βµ (2)

where: γ is the upper characteristic value uγ of the solid unit weight; hh is the vertical height between the hopper apex and the transition; x is the vertical coordinate upwards from apex; hµ is the wall friction coefficient for the hopper; β is the hopper apex half angle; vftp is the mean vertical stress in the solid at the transition after filling. Rotter [32] suggests that the stress ratio, fF , for filling conditions can be represented by

βµβµ

cot1cot1

++

=aFf (3)

where, a is an empirical constant with a value between 0 and 1, for filling pressures, Rotter suggests 8.0=a . In this study, stress ratio, fF , is correlated between numerical computed wall pressure and

theoretical one.

5.3.2.2 Numerical predictions Fig. 5.21 shows comparisons for wall pressures between theoretical predictions obtained by Eq. 1 and numerical computations. In this case, the hopper half angle is 30°, while arch angles vary from 30°~90° with 15° increment. The wall friction between coal and hopper is 0.2. Wall pressures at the coal/hopper interface can be divided into three different portions. At the upper part (60%~100% h/H), small differences exist for wall pressures compared between theoretical and numerical ones. Numerical predictions are slightly larger than theoretical predictions. Each of them are very close with different arch angles, this indicates wall pressures are less influenced by different arch angles for portions far away from arch positions. Wall pressure comparisons in the second portion (6%~60% h/H) exhibit a very different results. In this portion of the hopper, theoretical predictions are larger then numerical computations. Wall pressure comparisons exhibit the largest if arch angle, θ=30°. And the theoretical prediction is 18.5% higher than the corresponding numerical prediction. The highest wall pressure occurs at a position of about 27.6% h/H. At the lower portion of the hopper (from outlet~6% h/H), numerical predictions are generally high than theoretical ones besides the case of θ=30° and besides outlet. At the outlet, theoretical predictions are high, while numerical predictions for wall pressures tend to zero. Fig. 5.22 shows stress ratio (ratio between wall pressure and vertical stress at wall, Ff=pnf/pvf) versus arch angle (θ). The hopper is with half angle:30°, wall friction: 0.2 and under self-weight. It can be observed from this figure that stress ratio varies almost linearly with the increasing of arch angle from 0.96 to 1.02. Fig. 5.23 illustrate comparisons for wall pressures predicted by theoretical formulation and by numerical calculations. Wall pressure distribution comparisons are a little bit different for hoppers with surcharge compared with that of without surcharge (Fig. 5.21). In general, they are more close if surcharge are considered. At upper part of the hopper (85.9%~100% h/H), theoretical predictions for wall pressure are greater than the numerical counterpart. The maximum difference is 28% over the numerical estimations if arch angle is 90°. For the middle part (4%~85.9% h/H), the maximum differences for wall pressures predicted by two methods are only 5.8% at a position with h/H=70%. For lower part of this portion (4%~40% h/H), numerical predictions are quite close to theoretical estimations. At the outlet, theoretical predictions are higher than numerical predictions except a special case when arch angle equals to hopper half angle.

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Stress ratio versus arch angle for hopper with different arch outlet with self-weight and surcharge (Fig. 5.24) showed similar variation with Fig. 5.22. Stress ratio varies from 0.916~0.941 with the increasing of arch angle. This range of stress ratio is smaller than the ones for hopper with only self-weight.

5.3.3 Stress ratio affected by arch and hopper half angles The variations of stress ratio vs. hopper half angle under different arch angles are studied. Coals within hopper are considered in two situations: self-weight only and selfweight+surcharge. The wall friction is 0.2. Under the situations of self-weight, Fig. 5.25 indicates that stress ratio becomes larger with the increasing of arch angles, θ. If the arch angles are kept constant, then stress ratio will increase with the increasing of hopper half angle, β. Stress ratio drops down if hopper half angle β>45°. The stress ratio varies between 0.91~1.03. Fig. 5.26 shows stress ratio variations for coals under self-weight and surcharge. The stress ratio increases with the increasing of arch angle, θ. If the arch angle is kept constant, then the stress ratio will increase monotonically with the increase of hopper half angle, β. The range of stress ratios varies between 0.84 and 0.99, lower than the stress ratios for hoppers with only self-weight.

5.4 Stresses above outlet using Critical State Theory To explore the p—q stress distributions at outlet with different hopper half angle and arch geometries (Fig. 5.27a), Concept of Critical State Soil mechanics are used in this study. The p—q stress at outlet are fitted to a straight line, q=Ap+B, in the p—q spaces (Fig. 5.27b), where A is gradients or stress ratio and B is interception. Fig. 5.27b also shows the possible failure zones at low stress regions with the concept of critical state soil mechanics. It can be seen from this figure that large interception, B, normally will cause the failure of coals within low stress regions. We assume that A and B are functions of hopper half angle, β, and arch angle, θ. In this study, the hopper half angles varies from 15° to 60° with 15° increments, while arch angles varies from 30° to 90° with 15° increments. The following sections will examine these relations carefully.

5.4.1 Case 1: Hopper with selfweight Fig. 5.28 shows the variations of A and B with different hopper geometry and arch angles. Generally, the stress ratio, A, increase with the increase of hopper half angle, β, from 15° to 60° (Fig. 5.28a). Stress ratio increase with the decreasing of arched angles from 90° to 60° at outlet. For arch angles of 30° and 45°, the stress ratio will first increase with the increasing of hopper angles, β. The stress ratio will reach maximum when hopper half angle equals the corresponding arch angle (i.e., β=θ). If the hopper half angles are increased further, then stress ratio will reduce. The overall value for stress ratios of the two arch angles are greater than the rest. Fig. 5.28b is the interception of fitted stress equation at q axis. It can be observed from this figure that interceptions are large with the increasing of arching angle from 60° to 90° for all the hopper geometries if hopper half angles are smaller than 45°. So, it can be concluded that coal arches are easily disrupted with the increasing of arch angles in this range. If hopper angles are increased further greater 45°. The interception will decrease. If arch angles are smaller than 45°, the variations of interceptions are moderate between 0~5 kPa.

5.4.2 Case 2: Hopper with selfweight and surcharge Fig. 5.29a shows the variation of stress ratio, A, for different hopper geometry and arch angles. It can be observed from this figure that stress ratio increases with the increasing of hopper half angle, β, for outlet with arch angle 60° and 75°. The stress ratio varies between 0.89~1.65. For outlet with 90° of arch angle, the stress ratio generally increase from 0.83~1.4 with the increasing of hopper half angle, except only a slightly decrease of stress ratio to 0.79 when hopper half angle is 30°. Similar with case 1,

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stress ratios reach peaks when hopper half angle and arch angle are equal with the arch angle being 30° and 45°, respective. The variations of stress ratio for these two situations is between 1.09~1.6. Fig. 5.29b is the variations of interceptions, B, for different hopper geometry and arch angles. The interceptions reach peaks at β=30° if arch angles are 60°,75° and 90°. Further increasing of hopper half angle cause the interception decrease rapidly to even lower values. The variations of the interceptions for the three situations between –4~21.9kPa. If the arch angle is at 30° and 45°, the interception will reach minimum when β=θ. The interception varies between –2~11.9kPa.

6. Stresses above an arched outlet in unsymmetrical bunkers

6.1 The details of the Model A hopper with an unsymmetrical geometry and with an arch across the outlet is shown in Fig. 6.1a. The height measured from the top of the arch is H, the slope of the left hand side of the hopper has a half angle, βL, and the right hand side a half angle, βR; the height of the arch is h, the width of the arch is b. The coal is initially modelled here as an elastic solid, and the hopper structure is modelled by an analytic rigid surface. The coal/hopper wall, coal/hopper bottom interfaces are modelled by frictional contact elements. The mesh of the finite element model is shown in Fig.6.1b, with the outlet zoomed so that the detail of the simulating mesh can be seen clearly. The model used 8-noded plane strain elements (CPE8) in this study. The coal inside is simulated by elastic material, with a Young’s modulus of 10MPa, a Poisson’s ratio of 0.3, and a gravity density of 1KN/m3. It should be noted that the gravity density is taken as unity so that the results of the calculation can be multiplied by the real density of a coal to give the results for all coal densities directly. It should also be noted that the results do not depend significantly on the value chosen for the Young’s modulus, so any errors in assessing this rather difficult parameter are not important. In terms of the boundary conditions: the top surface of the coal and the outline of the arch over the outlet are both treated as stress free. All other interfaces between the coal and the hopper wall and the coal and the hopper bottom are treated as contact surfaces between the coal as a continuum and a rigid hopper structure. The only source of stress in this study is the self-weight of the coal.

6.2 The effects of wall friction The wall pressure distributions for five different wall frictions are shown in Fig. 6.2. A completely smooth surface (µ=0), and rough wall surfaces with the friction coefficient varying from 0.1 to 0.4 with 0.1 increments are shown. With the horizontal outlet open at the base of the left wall (Fig.6.2a), the numerical model shows that a high local value of wall pressure exists near the outlet for all the wall friction conditions. The smooth wall surface causes the largest wall pressure, but the local wall pressure decreases as the wall friction coefficient increases. On the right hand wall (opposite the outlet), no increase in wall pressure was found for any of the wall conditions on the right hand hopper wall. This is because the zone near this wall remains completely filled with coal during the whole process of discharging. So, stresses are distributed homogeneously, hence, no rise in stresses can be produced. Also, at the right side wall, smooth wall surface induced the largest wall pressure, while wall pressures decrease with the increase of wall friction. Comparisons for wall pressures between theoretical predictions and numerical computations are made in Fig. 6.3 for both walls. It can be observed from both figures that fairly reasonable correlations exist for the two estimations. For pressures near outlet at the left wall side, the two predictions are accurate, within 5% of errors. The variations of wall stress ratio for different wall friction conditions are shown in Fig. 6.4. For the two walls, the wall stress ratio decrease with the increase of wall friction. It varies from 0.7 to 1.1. Wall

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stress ratio for the left hopper wall, with outlet on its side, is smaller than that of the counterpart. This shows strong evidence that the stress field at two sides are unsymmetrical due to the forming of a horizontal outlet at left. The stress field along outlet are discussed in the framework of Critical State Soil Mechanics (Fig. 6.5), where p, mean effective stress, q, equivalent shear stress. The Critical State Line (CSL) is also shown in this figure. It can be seen for all the wall friction conditions, the stress ratio q/p>1.68, which is the Critical State Stress Ratio. That means that the stress state in the coal can only be sustained if its stress history has caused it to develop cohesion. The extent to which the stress ratio exceeds this critical ratio is a direct measure of the cohesion required in the coal for stable arch formation.

6.3 The effects of Unsymmetrical arrangements The unsymmetrical properties are investigated by varying the half angles on right side of hopper from 15 degree to 45 degree with 15 degree of increment, while keeping the left half angle of the hopper at 30 degrees (Fig. 6.6). Numerical results indicate that there is large increase of wall pressure at outlet for left wall pressure. No substantial variations of wall pressures can be found if the right half of the hopper angles varies from 15 to 45 degrees (Fig. 6.6a). For the right half of the hopper, no increase of wall pressures exhibit, while substantial variations of wall pressures can be found for the three half angles (Fig. 6.6b). For the two sides, large half hopper angle result in large wall pressure, while smaller hopper half angle will result small wall pressure. The stress ratio near the outlet is shown in Fig. 6.7, under different unsymmetrical conditions. It can be seen that for all the three situations, the stress ratio near the outlet is locally greater than the Critical State Stress Ratio, indicating that the coal here must have cohesion to sustain this arch geometry. The wall stress ratios for the three situations are shown in Fig. 6.8. It should be noted that for the 30/30 situations, the wall stress ratio should be the same if no outlet is present. Due to the presence of the horizontal outlet at left, the wall stress ratio at left will reduce 5.7% of original value at right. If the right half angle increases, the wall stress ratio at left will reduce further. On the contrary, if the right half angle is smaller than the left half angle, the wall stress ratio at left will increase dramatically. For the present case of 30/15, the wall stress increase will increase 24.2% compared to the right half angle.

6.4 Paddle Position The variation of the wall pressures down hopper height are shown in Fig. 6.9 for three paddle positions: w=0.7m, 1.0m and 1.3m. These have increased wall pressure at the bottom of left wall due to the existence of horizontal outlet at this side (Fig. 6.9a). For portions away from the bottom of the arch zone, less effect can be found for the wall pressures for the three paddle positions. Wall pressures are only affected near the outlet, where large paddle position (or distance deeper into the hopper) cause larger wall pressure atop the hopper wall. There is no increase of wall pressures at the right side of the hopper wall (Fig. 6.9b). Only at the very bottom of the right wall, some difference of wall pressures can be found, where larger paddle position cause bigger wall pressures. The p—q stress distribution along the outline of the outlet is shown in Fig. 6.10. For the greater part of the outlet, this stress ratio is larger than the Critical State Stress Ratio, so that it must have cohesion to be stable. The extent to which the stress ratio exceeds this critical ratio is a direct measure of the cohesion required in the coal for stable arch formation. Wall stress ratio for the wall pressures of the three paddle positions are fitted from numerical computations (Fig. 6.11). It can be seen from this figure wall stress ratio decrease when the paddle position increase from 0.7m to 1.0m. If paddle positions are further extended into the coal from 1.0 to 1.3m, wall stress ratio keeps the constant. Also, wall stress ratio for the left wall is smaller than that of the right side wall.

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6.5 The effects of outlet Height The effect of arch height to the wall pressure is shown in Fig. 6.12 with three outlet height: 0.4, 0.7, 0.1m. It can be observed from this figure that for portions away from the top of outlet, no significant reduction of wall pressures can be found. It is just at where near the top of the outlet, some effects on the wall pressure can be found, where higher outlet will induce larger wall pressure. Also, there exist increase of wall pressure for left wall with horizontal outlet (Fig. 6.12a). No increase can be found for the wall pressure at right wall (Fig. 6.12b). But at the very bottom of the right wall, the pressure reached zero. This is caused by the separation between coal and right corner bottom of the hopper. For real bunker, the bulk material should completely fill in every corner near the right wall. So, the wall pressure should reach a minimum. The stress relationship p—q along the outlet is shown in Fig. 6.13, together with the Critical State Stress Ratio. For all the three outlet heights, the q/p ratio is above the CSL, but by a rather simple ratio, which means that the required cohesion for stable arching is rather invariant with position in the arch. Fig. 6.14 is the wall stress ratio for the left wall and right wall. There is remarkable difference for the wall stress ratio between the two walls. The wall stress ratio for the left wall is still smaller than that of the right wall, but their trend is different. For the right wall, the wall stress ratio tends to increase again after drop to minimum at h=0.7m, while the stress ratio for the left wall tend to decrease further.

6.6 The effects of Hopper Height Fig. 6.15 shows the results for hoppers with three heights: 25m, 50m and 75m. For the left hand wall with horizontal outlet (Fig. 6.15a), there exist surcharges of wall pressures for three heights. The higher the height of hopper, the larger the wall pressure will be. They showed some self-similarities for the wall pressures of the three heights. Wall pressure at the right wall shows more similarities (Fig. 6.15b) except no increase are observed at the bottom of the hopper. Fig. 6.16 gives the results of p—q stress distributions at outlet. Generally speaking, higher hopper height will induce higher p—q stresses. For all the three heights, the stress ratio, q/p, is only slightly greater than the Critical State Stress Ratio. The peak ratio is very well defined, indicating that a rather well defined cohesion, after consolidation, is required for the arch to be sustained. The fact that this ratio is only slightly above the Critical State Stress Ratio means that relatively small cohesion in the coal will cause a stable arch to form. The stress ratios for each wall at three heights were fitted from numerical solutions of wall pressures with the theoretical formula given by Rotter [2001]. The results are shown in (Fig. 6.17), where the values are seen to be slightly less than unity, as proposed in the theory. Due to the presence of the horizontal outlet, the stress ratio at left wall is smaller than that of the right wall. The difference of the stress ratio between the two walls is generally large, while this difference will reduce with the increase of wall height.

6.7 Remarks The discharge of cohesive materials, such as coal, from silo and bunker is notoriously unreliable [Rotter 2001]. Often a material will discharge freely, but if the flow is stopped for a while it maybe very difficult to restart the discharge. This is because when coal is stationary inside bunker, consolidation takes place under the principal stress and an unconfined yield stress develops. A stable arch of coal will be formed across orifice and it has to be broken before the coal flow again. Jenike’s radial stress field theory indicates that radial stress during flow is a factor of 1-(r/r0)5 compared with radial stresses during arching, where r is radial distance from vertex of hopper [Nedderman 1992]. In power stations, the time-dependent consolidation of coal within bunker is more difficult to estimate. To avoid of stoppage of coals, it is a good practice to circulate small proportions of coal from time to time so that none of the coal is held stationary for long periods of time.

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To get reliable flow [Rotter, 2001], the silo or bunker has to be designed in such a way that the strength of the stored bulk solid at no time and in no place is sufficient to form stable arches (bridging), stable ratholes or other dead zones [Schwedes 2003]. The most critical area in a silo or bunker is the area directly above the outlet. Bulk solid flowing into the silo from top to bottom is subjected to different stresses. According to these stresses it is consolidated and gets its strength. Above the outlet the flow is converging with decreasing stresses towards the outlet. Due to these decreasing stresses the strength of the bulk solid also decreases towards the outlet. Whilst the solid is flowing, a passive plastic state of stress prevails in this converging flow. At each point within this region, it may be supposed that steady state flow exists. It should be noted that only at steady state flow a bulk solid sample loses its memory of the stress history, i.e. that any bulk solid element at a certain location in the convergent geometry is always exposed to the identical state of stress (in steady state flow) and hence has an identical strength independent of the stress history. This is independent of all the possible ways to reach this point.

7. Study of a typical UK coal bunker

7.1 Introduction

7.1.1 Geometry Fig. 7.1 is a typical cross section drawing of a coal bunker in power station, it is hopper in shape with hopper half angle β=30°, two horizontal outlet are formed between hopper and bottom. Arching occurs near the two outlets when poorly blended coals are filled into bunker. Sometimes, a “Dog-kennel” is designed near the outlets to improve the flowing properties of coals within bunkers. Paddle feeders are used near the two outlets to help the withdrawing of coals from bunkers into next working site. In this study, the total height of coal after filling is H=8.21m, Coal repose angle is φr=46°. When arching occurs, the contacting distance between coal and hopper is HL=HR=6.7m. To adjust the position of the paddle feeders, three paddle positions are investigated in this study, PL=PR=0.152m, 0.187m and 0.222m.

7.1.2 Mesh Fig. 7.2 shows the finite element mesh in this study, coals are modeled by 4-node plane strain element with reduced integration scheme (CPE4R). Finite element meshes are biased for coals near the hopper walls. There are 2743 elements and 3156 nodes for coal materials. To model the normal flowing and arching processes, coals are divided into three parts, they are main part of COAL, arched toe parts: TOE-L and TOE-R. The three coal parts are tied together along the arched boundary shown in Fig.7.2. The rests of the model: left hopper, right hopper, bottom and dog-konnel, are modeled as analytical rigid surface. These rigid surfaces are represented by reference points and fixed in all analysis. The interface between coal/hopper, bottom and dog-konnel are modeled by Coulomb frictional elements. The number of contact element is 205. Wall frictions are varied between 0.1 ~ 0.3 in this study.

7.1.3 Material Parameters In this study, coal materials are modeled by Cam Clay constitutive relations within the framework of Critical Sate soil mechanics. This is because Cam Clay model is comparable simple with few parameters. Furthermore, it is loading path dependent and can study the compressive properties for coals within bunkers. In ABAQUS, it uses Modified Cam Clay model. The Cam Clay parameters in this study are listed in Table 1.

Table 1 Parameters for Cam Clay model

Log. bulk modulus

(κ)

Poisson’s ratio (ν)

Log. harden

modulus (λ)

Critical stress ratio (M)

Initial void ratio (e0)

Intercept of virgin

consolidation line(e1)

Wet cap parameter

(β)

Third stress

invariant parameter

(K) 0.0092 0.32 0.0367 1.68 0.79 0.87 0.5 1

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Coals within bunkers are assumed fully saturated and they are completely drained when discharge. The specific gravitational weight γ for coal is 7.64kN/m3. An initial effective stress of 10Pa is applied on coals when they are dumped into bunkers.

7.1.4 Loading and boundary condition Only gravitational loads are applied in the analysis. Hopper, bottom and dog-konnel are fixed according to fixed reference point. The outside of the two horizontal outlet are free boundaries. There are two load steps, In loading step 1, Normal flowing of coals are assumed in the analysis; while in loading step2, Arching is assumed by excavating coals near the outlet along the arched boundaries. Excavation calculations are carried out by removal of elements at regions of TOE-L and TOE-R. In this step, ABAQUS stores forces that the regions to be removed is exerting on the remaining part of the model at the nodes on the boundary between removal and remaining elements. These forces are ramped down to zero during the removal step; therefore, the effect of the removed region on the rest of the model is completely absent only at the end of the removal step. The forces are ramped down gradually to ensure that element removal has a smooth effect on the model. For progressive filling studies, coals are divided into several layers, they are reactivated in a strain free state, these elements become fully active immediately at the moment of reactivation. Before reactivation, they are reset to an “annealed” state(zero stress, strain, plastic strain,etc) in the configuration in which they lie at the start of the reactivation step. It should be noted that this kind of reactivation scheme is usually used to model the creation of an undeformed and unstrained region of the model that is sharing a boundary with another, possibly stressed, deformed region. Besides material nonlinearity, geometric nonlinearity is considered in the study.

7.2 Results

7.2.1 Flow propensity analysis Fig.7.3 is the deformation propensity for coals from normal flowing to arching occurs. The wall friction is 0.2 for all the interfaces between coal/hopper, bottom and dog-konnel. Fig.7.3a shows the flow propensities at the start of normal flow. It can be seen from this figure that there is a sudden spillage of coals from the two horizontal outlets. Meanwhile, coals at the two top sides are dumped towards the center of the hopper. There are large deformations of coals along the center of bunker. That means the highest flow velocities occur on the center-line with the velocity decreasing towards the wall. Since the two top are dumped in coal repose angle, coal powders cascade down the surface so that the faster central core is fed from the slow moving outer parts of the top surface. It should be noted that numerical simulation indicates that this kind of phenomenon is more obvious if the wall friction becomes larger. This also indicates that bending effects exist for coals at both sides. When normal flowing achieved(Fig.7.3b), coals are flowing down fast at the two top side, and flowing steadily at the two outlet. When arching is formed(Fig.7.3c), only flow propensities at regions near dog-kennel are affected, there are no obvious effect to the flowing propensities for the vast part of the coals within bunkers.

7.2.2 Wall Friction Effect By changing wall frictions at interface between coal/bunker, the wall friction effects can be studied. The wall friction are changed by 0.1, 0.2 and 0.3, corresponding to wall friction angle 5.7°~16.7°. Fig.7.4 is the wall pressure distributions along walls. Since the hopper walls are symmetric, only results at left wall are presented in this study. For normal flowing(Fig.7.4a), wall pressures decrease with the increase of wall friction. This decrease is more obvious at the tied interface between coal and toe-l. It should be noted that due to the tied model in ABAQUS, only translation displacements along the tied arched boundaries are continuous, strain continuous can’t be guaranteed along these boundaries, hence discontinuous stresses are found along the boundaries. The present pressure values at interfaces are averages along the boundaries. When arching occurs, coals at the two outlets are excavated along the arched boundaries. Wall pressures (Fig.7.4b) are in similar distribution with that of normal flowing for

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most part of the hopper wall except near the arched boundaries, where large overstresses exhibit. This clear indicates that large radial stresses exist due to the forming of arch. Fig.7.5 show the p-q stress at the arched boundaries under different wall friction conditions. The non symbolized end in Fig. 7.5 stands for the starting point of arch from hopper wall. it can be seen from Fig. 7.5a that for normal flowing situations, the bottom of arch lies in the low stress region. The stresses magnitude at the bottom of arch is just below Critical State Line(CSL) or a little above it when wall friction is small. Compared with Fig. 7.5a, stress magnitudes at arched boundaries are much larger when arched occurred(Fig.7.5b). Large shearing stresses exist along the arched boundaries near the hopper walls. While at the bottom of the arch, stress magnitude are large than the corresponding CSL value. This indicates a possible failure near the bottom of the arch. Fig.7.6 shows the whole field of distribution of stress ratio q/p when wall friction is 0.2. It can be seen from the figures that for most volume part of the coal, the distributions are more or less the same except near the arched zones at the two legs(Fig.7.6a for normal flowing, Fig.7.6b for arching). On normal flowing, the possible failure zone lies at the top of the toe near the hopper. While for arching situations, the most dangerous failure zone occurs near the mid part of dog-konnel. For arched boundaries, the stress ratio are near to Critical State value, M=1.68. Fig.7.7 shows stress histories for point A(shown in Fig.7.2) at arching boundary from normal flowing to arching. It can be observed from the figure that the slope of the stress path are smaller than Critical State value, M=1.68 for all the three wall friction situations. For points just at the left of point A(node 2) and belong to part TOE-L, stress path going directly back to origin when arching occurs. This is caused by the unloading with the excavation of coals at TOE-L. For point just at the right of point A(node 25) and belong to COAL, the stress path goes differently when arching occurs. They are caused by continuous loading when arching occurs. It can be noted that for all the wall friction situations, these stress path finally goes somewhere higher than CSL indicating a possible failure of this point. It should also be noted that for normal flowing, the stress path should be same for either node 2 and node 25. Since tied model along the arching boundary can only guarantee the continuous of translational degree of freedom, strains are discontinuous on both sides of point A, hence stresses are not continuous at point A. Although such numerical errors exist, the stresses at two sides of arching boundary are close enough to trace the stress histories of coals from normal flowing to arching.

7.2.3 Paddle Feeder Position The effects of paddle feeder positions are studied by adjusting PL and PR in Fig.7.1. Three positions are studied with PL=0.152m, 0.187m and 0.222m. Wall pressures are shown in Fig.7.8. It can be shown in Fig.7.8a that wall pressure decrease with the increasing of paddle positions for normal flowing situation, but this decrease is not obvious. It should be noted that wall pressure at the interface between COAL and TOE-L is not accurate because of the tied model. When arching occurs(Fig.7.8b), wall pressures almost doubled the value of normal flowing at the arched boundary. This indicates that large radial stress field exists when arching occurs. Increasing the paddle position will lower the wall pressure at arched boundary, but not obvious. Fig.7.9 shows the p~q distributions along the arched boundaries for different paddle position with wall friction, µ=0.1. The sharp non symbolized end of curves corresponds to arch boundaries starting from hopper walls. It can be seen from Fig.7.9a that for normal flowing, stress levels around top part of the arch are well below the Critical State Line, indicating the stress are within elastic regions. For the lower portion of the arch, stress levels are very low but above the Critical State Line, this indicates a possible tensile failure zones exists. Increasing the paddle position will increase stress levels at the bottom of the arch. When arching is formed(Fig.7.9b), the stress levels for the whole arch boundaries are quite near to Critical State Line. Stress levels at both the top and bottom of the arch are quite near to the Critical State Line. Stress levels will increase with the increase of paddle position at bottom of arch, but this increasing is not obvious.

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The whole field distribution of stress ratio, q/p for paddle PL=0.222m, are shown in Fig.7.10. In normal flowing (Fig.7.10a), stress ratio at zones between toe and hopper shows great value than the Critical State Ratio, M=1.68. This indicates these zones are experiencing dilation, tensile failure will result. When arching is formed (Fig.7.10b), zones near the dog-konnel and at the bottom of the two legs experience dilation deformation. Tensile failure will first occurs in this zone. Compared between the two figures, arching will only cause stress redistributions near dog-konnel and outlet. For the rest part of the coal, the effects are small.

7.2.4 Progressive filling Fig.7.11 is results of wall pressure upon progressive filling on the left walls under normal flowing of coals at outlet (Fig.7.11a) and when arching had formed when filling (Fig.7.11b). Coals are filled three times. At the first time, coals are filled half of the original height shown in Fig.7.1, then, at the second filling, half of the remaining height is filled, last, the bunker is fully filled. It can be observed from the both figures that wall pressure increase with the progressive filling of coals into bunkers with or without arching formed. It should be noted that there is a sharp variations of wall pressures for the second and third filling operations. This is caused by numerical errors at interface between first filling and second filling. There are large deformations at this interface, layers are tied at interface between first filling and second filling. Hence, large mismatch pressures are caused numerically. Compared with second filling, there is no large difference in wall pressure at interface between second and third filling because there are relatively small deformations between layers. Studied for wall pressure between normal flow and arching indicates that wall pressure will increase when arching occurs. This increasing will be more obvious with the increasing of filling height. Fig. 7.12 illustrates p~q stress at the possible arching boundary for the situation of normal flowing and arching. It can be seen from Fig.7.12a that stress level is far lower below the Critical State Line at the top of the arching boundaries for normal flowing case. Only at the zones that is quite near to the bottom of arch, the stress level near Critical State Line. When arching occurs(Fig.7.12b), stresses at both the top and bottom of the arching boundaries are quite near to Critical State Line. So, the possible failure zone should start from these zones. Fig. 7.13 shows the stress ratio variations of the whole field of coal under progressive filling when arching occurred, it can be found from this figures that the possible failure zones around the two legs of coals decrease with the increasing of filling height. This implies that upon the forming of arching, further filling of coal will release stresses around the bottom zone, this is very harmful for the destroying of arch. In other words, arching should be demolished before further filling operations. It should be noted that there is clear discontinuous of stress ratios between different filling operations. The reason for this is that the interfaces between different filling are tied together, only translational displacement degree of freedom are guaranteed, hence there exists a small discontinuous of strain and stress among different layers. Fig. 7.14 illustrates the whole field of stress ratio distributions for coals in progressive filling when coals are free flowing from outlet. These figures show that the two top part of the layer is easy to be failed with the progressive filling. At the two outlets, stress ratio increases in a zone across the arched profiles.

7.3 Discussion Results by different wall friction, and progressive filling indicate that stresses and strains around outlet are relatively independent from the vast part of the coals within bunker when arching formed. This numerical results further verified the fact that the stress distribution in the lower part of a hopper is effectively independent of the height of the material or the presence of a surcharge [p304, RM Nedderman]. Thus, it can be concluded that it is pointless to try to extrude a granular material through a converging passage by the application of a stress or surcharging. This merely has the effect of increasing the wall stresses and hence the retarding frictional forces.

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Drescher, Cousens and Bransby (1978) analysised the velocity distribution of granular materials within a wedge shaped hopper. In their studies, materials in regions ABC and A’BC’ was assumed to flow with a constant velocity parallel to the wall while below a velocity discontinuous zone, velocity distributions are close to that predicted by the radial stress field. Although velocity or displacement discontinuous zones can not be found in the present studies, the incipient flow propensities really shows perfect agreement such a tendency in some extent. It should be noted that a detailed dynamic studies for the flow and velocities are needed so that a complete comparisons are made between numerical results and analytical one.

8. Conclusions In this project design information has been produced that can be used to eliminate arching problems in typical coal bunkers with the typical geometries and outlet arrangements used in UK coal bunkers, which differ considerably from those considered in Jenike’s classic arching theory. The new calculations permit the Jenike design method to be extended to the hopper outlet forms used in power stations in the UK. This project has included careful measurements of power station bunker geometries, careful development and verification of a computational finite element model using ABAQUS software for the coal in such a bunker, which was used first to predict consolidation stress states when the bunker was filled, considering a range of different material properties for the coal and different wall friction coefficients for the hopper walls. This computational model was then used in more difficult and extensive calculations to examine the stresses that develop in different shapes of coal bridges or arches across the outlet, to determine the most critical arch form and to deduce the conditions under which cohesive strength in the coal would just case a stable arch to form. These calculations led to different outcomes depending on the geometry of the hopper sides, the hopper wall friction (with low friction liners or not), and the size of the outlet. The calculations addressed all these items in a huge parametric study, where all were systematically varied. The following are the main conclusions from the project.

• A symmetric hopper simulation was carried out using three alternative constitutive models: Elastic, Mohr-Coulomb and Cam-Clay. The stress history of the coal within the hopper, the wall pressures and the slip distance of the coal down the coal/hopper interface were studied.

• Comparisons were made between the results of the numerical simulations and existing analytical predictive equations. Rotter’s (2001) design equation for hopper filling pressures provides a good upper bound for the numerical simulations.

• The peak wall pressures predicted by Jenike’s theory are very high and so unnecessarily conservative.

• The Cam Clay model for the constitutive behaviour of coal was reviewed with two hardening laws. Key governing equations for Critical State for Cam Clay model were examined. Two basic failure mechanisms: hardening failure and softening failure for coals within bunkers were found.

• Cohesion tests on coal were explored by computational modelling to find the relationship between the assumed properties of the coal and the measurements made using the Edinburgh Cohesion Tester.

• Six loading paths were designed to calibrate the Cam clay model. These paths are successful in tracing various loading history and different failure mechanisms for any granular solids. This study produced convincing evidence that the Cam Clay model is the best current model for coal studies within a bunker.

• A general review was made on the stress history and failure mechanism of coal in bunkers leading to arching. This review was made within the conceptual framework of soil mechanics with the critical state concept. The stress state of coals near the bunker outlet indicated that the failure of a coal arch is likely to be in the form of dilatant rupture, or a wedge of coal against the wall may slip down.

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• The bunker wall pressures, hydrostatic and deviatoric stresses and the stress ratios in the coal near the outlet were all found for many different values of wall friction for two hopper half angles (30°, 45°) and two loading situations (self-weight and self-weight+surcharge). For these two loading cases, the pressures at the bunker wall, and the stresses at the outlet all decreased with increasing wall friction. The stress ratios (i.e. ratio of equivalent shear stress, q, to mean effective stress, p) approached a maximum at high wall friction.

• Many different geometries for the arch formation across the outlet were explored. By varying the arch angle, the sensitivity of the arch geometry to wall pressures was analysed for hoppers with a hopper half angle of β=30° under two loading situations: self weight and self weight+surcharge. The numerical results indicated that for portions away from the outlet, the wall stresses and coal vertical stresses are less influenced by the arch angle. At the outlet, there is a residual stress if the hopper half angle, β, is equal to the arch angle, θ. If the arch angle, θ, is greater than hopper half angle, β, the wall pressures approach zero with an increasing arch angle, β.

• Excellent agreement was found for wall pressures predicted by theoretical and numerical methods. The stress ratios increase as the arch angle increases, making failure of the arch more probable.

• Wall pressures increase with the increasing of hopper half angles, while the stress ratio (i.e. the ratio of equivalent shear stresses, q, over mean effective stress, p) increase with the increasing of hopper half angle, β. For hoppers with same hopper half angle, β, stress ratio increase with the increasing of arch angle, θ.

• A further study of the stress distributions at outlet in the frame of Critical State Soil Mechanics was made. The variations of the stress ratio, A and intercept, B for arch angles from 60°~90° were found to be quite different from those for arch angles between 30°~45°.

• A large local increase in the wall pressures occur on the wall above the outlet when there is a vertical opening for an outlet. By contrast, the wall pressures on the opposite wall are less affected.

• For all the parametric studies, the stress ratio q/p at the hopper outlet is slightly greater than the Critical State Stress Ratio. In the framework of Critical State Soil mechanics, this means that the material above the outlet must have developed cohesion as a result of its stress history to permit the stable arch to form.

• The extent to which the stress ratio q/p exceeds the Critical State Stress Ratio is a direct measure of the degree of cohesion required for the arch to be stable.

• These calculations have shown that the level of cohesion required for a stable arch does not vary very much from one point to another in the proposed arch geometries: this indicates that the chosen arch geometries are very realistic and offer a good basis for the development of a new design rule.

• The Critical State Stress Ratio was only slightly exceeded in most of the calculations. This showed that a relatively small cohesion within the coal is sufficient to support a stable arch.

• The relationship between a measured cohesion and the Critical State Stress Ratios derived from these calculations has been established.

• The full set of stress states in the arches calculated within this project have been analysed to develop a modification of the Jenike procedure so that it can be applied to hoppers with the geometries found in typical UK power station bunkers.

• The calculations permit the relationship between the unconfined strength measured in an Edinburgh Cohesion Tester to be related to the geometry and wall friction in a bunker of a given geometry and the limiting value of unconfined strength can be found at which that bunker is just ready to form an arch.

• This is the first time that such calculations have ever been performed on arching in bunker outlets using a modern sophisticated computational tool which can model the true behaviour of coal without excessive assumptions.

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The project has achieved its intended goals, and represents a substantial step forward in understanding the conditions under which arching will occur in bunkers of this geometry (plough feeder table). Many difficulties of modelling were encountered during the project, and these led to additional explorations of coal behaviour that were not part of the original planned project. Nevertheless they were very necessary as part of the task to model the behaviour of sticky coal well. Much further useful work could be undertaken that would take advantage of what has been learned in this project to model sticky coal in chutes and in other handling operations.

References 1. Alawaji, H., Runesson, K. etal.(1992) “Implicit integration in soil plasticity under mixed control for

drained and undrained response”. Int. J. Numer. Anal. Meth. 16, 737-756. 2. Arnold, B.J., Lohnes, R.A. and O'Connor, D.C. (1993) “EPRI coal handleability project:

handleability index validation”, Iowa State University Report, 19pp. 3. Arnold, P.C and McLean, A.G.(1976), Powder Technology, 13, 255 4. Arnold, P.C and McLean, A.G.(1976), Powder Technology, 15, 279 5. Arnold, P.C., McLean, A.G. and Roberts, A.W. (1980) Bulk solids: storage, flow and handling,

Tunra, University of Newcastle, Australia, Sept. 6. Atkinson J.(1993) “An introduction to the mechanics of soils and foundations”. McGraw-Hill Book

Company 7. Bolton M.(1979) “A Guide to Soil Mechanics”. The MacMillan Presss Ltd 8. Borja, R.I.(1991) “Cam-clay plasticity, part II: implicit integration of constitutive equation based

on nonlinear elastic stress predictor”. Comp. Meth. Appl. Mech. Eng. 88, 225-240. 9. Borja, R.I., Lee, S.R.(1990) “Cam-clay plasticity, part I: implicit Integration of elasto-plastic

constitutive relations”. Comp. Meth. Appl. Meth. Eng., 78, 49-72. 10. Carter, J.P.(1982) “Prediction of the non-homogeneous behaviour of clay in the tri-axial test”.

Geotechnique, 32(1), 55-58 11. Chen, J.F., Yu, S.K., Ooi, J.Y. and Rotter, J.M.(2001), Finite element modelling of filling processes

in a full-scale silo, Journal of Engineering Mechanics, Oct., 1058-1066 12. Dabrowski, A.(1975) “Parcie Materialow Sypkich w Leju” (Pressures from bulk solids in hoppers),

Archiwum Inzynierii Ladowej, Warsawa, p325-328 13. Drescher, A., Waters A.J. and Rhoades, C.A. (1995) “Arching in hoppers: I. Arching theories and

bulk material flow properties” Powder Technology, 84, No.2, 165-176. 14. Drescher, A., Waters A.J. and Rhoades, C.A. (1995) “Arching in hoppers: II. Arching theories and

critical outlet size” Powder Technology, 84, No.2, 177-183. 15. Drescher, Analytical Methods in Bin-Load Analysis, Elsevier, Amsterdam, 1991 16. Eckhoff, R.K. and Leversen, P.G.(1974), Powder Technology, 10, 51 17. Eckhoff, R.K. Enstad, G.G and Leversen, P.-G(1974), A further contribution to the design of mass

flow hoppers for powder silos, International Scandinavian Congress on Chemical Engineering, Bella Centret, Copenhagen, 28-30, Jan.

18. Endo, Y and Alonso, M.(2002), An estimation of hopper outlet size and slope for mass flow from the flowability index, Trans, IchemE, 80A, 625-630.

19. Enstad, G.(1975) “On the theory of arching in mass flow hoppers”. Chemical Engineering Science, 30, 1273-1283

20. Enstad, G.G.(1981), A novel theory on the arching and doming in mass flow hoppers, The Chr. Michelsen Institute, Bergen.

21. Feise, H and Schwedes, J.(1995), Calculation of silo stresses and feeder loads by the hypoplastic material model, Bulk Solids Handling, 16:4, 537-540

22. Guines, D., Ragneau, E and Kerour, B.(2001), 3D Finite Element simulations of a square silo with flexible walls, Journal of Engineering Mechanics, ASCE, Oct., 1051-1057

23. Harder, J. and Schwedes, J.(1985), Particle Charact., 2, 149 24. Haussler, U. and Eibl, J. (1984) “Numerical investigations on discharging silos” J. Eng. Mech,

ASCE, 110, 957-971.

30

25. Hibbit, Karlsson and Sorensen (199x) “ABAQUS User’s Manual” Ver 5.x, Hibbit, Karlsson and Sorensen Inc., USA.

26. Hibbit, Karlsson and Sorensen (2001) “ABAQUS User’s Manual” Ver 6.2, Hibbit, Karlsson and Sorensen Inc., USA.

27. Hibbit, Karlsson and Sorensen (2002) “ABAQUS Theory Manual” Ver 6.3-1, Hibbit, Karlsson and Sorensen Inc., USA.

28. IChE (1989) “Standard shear testing technique for particulate solids using the Jenike shear cell” IChE/EFChE joint publication, IChE, England.

29. Jenike, A.W. (1961), Utah Univ. Eng. Exp. Stn. Bull. 108 30. Jenike, A.W. (1964) Storage and flow of solids, Bulletin No. 123, Utah Engineering Experiment

Station, University of Utah, Salt Lake City, Utah, 197pp. 31. Jenike, A.W. (1964), J. Appl. Mech., Trans. ASME, 5, 31- 32. Jenike, A.W. (1964), Utah Univ. Eng. Exp. Stn. Bull. 123 33. Jenike, A.W., Elsey, P.J. and Wooley, R.H.(1960), ASTM Proc., 60, 1 34. Jenike, A.W., Johanson, J.R. and Carson, J.W. (1973) “Bin loads - Part 2, 3 and 4” J. Eng. for

Industry Trans. ASME, 95 Series B(1), 6-16. 35. Keiter T. W.R. and Rombach, G.A.(2001), Numerical aspects of FE simulations of granular flow in

silos, Journal of Engineering Mechanics, ASCE, Oct., 1044-1050. 36. Kolymbas, D. (1993) Modern Approaches to Plasticity, Elsevier, Amsterdam. 37. Lahlouh, E.H., Ooi, J.Y. and Rotter, J.M. (1998) “Assessment of coal handlability” Proc. World

Congress on Particle Technology 3, 7-9 July, Brighton, UK. 38. McLean, A.G. (1985) “Initial stress fields in converging channels”, Bulk Solids Handling, 5, No.2,

49-54. 39. Motzkus, U.(1974) “Belastung von Siloboden und Auslauftrichtern durch kornige Schuttguter”, Dr.

–Ing Dissertation, technical University of Braunschweig, Braunschweig, Germany. 40. Nedderman, R.M. (1992) “Statics and Kinematics of Granular Materials”, Cambridge University

Press. 41. Ooi, J.Y. and Rotter, J.M. (1990) "Wall pressures in squat steel silos from finite element analysis",

Computers and Structures, 37, No. 4, 361-374 42. Ooi, J.Y. and Rotter, J.M. (1991) “Elastic predictions of pressures in conical silo hoppers”,

Engineering Structures, 13, 2-12. 43. Ooi, J.Y. and Rotter, J.M. (1994) “Flow observations and predictions in full scale silos”, Proc.,

Flows of Granular Materials in Complex Geometries, eds S.L. Passman, E. Fukushima, and R.E. Evans, Sandia Nat. Labs, Int. Energy Ag., Albuquerque, 32-44.

44. Ooi, J.Y. and Rotter, J.M.(1990), Elastic predictions of pressures in a conical silo hoppers, Engineering Structures, 13:2, 2-12

45. Ooi, J.Y. and Rotter, J.M.(1990), Wall pressures in a squat steel silos from simple finite element analysis, Computers & Structures, 37:4, 361-374

46. Ooi, J.Y. and She, K.M. (1997) "Finite element analysis of wall pressure in imperfect silos", Int. J. Solids and Structures, 34, No. 16, 2061-2072.

47. Ooi, J.Y., Chen, J.F., Lohnes, R.A. and Rotter, J.M. (1996) "Prediction of static wall pressures in coal silos" Construction and Building Materials, 10, No. 2, 109-116.

48. Ooi, J.Y., Rotter, J.M. and Lohnes, R.A. (1997) "Constitutive equations for crushed coal", Trans., Soc. for Mining, Metallurgy & Exploration Inc, 302, 135-140.

49. Ooi, J.Y., Rotter, J.M. and Lohnes, R.A. (1997b) "Constitutive equations for crushed coal", Trans., Soc. for Mining, Metallurgy & Exploration Inc, 302, 135-140.

50. Ooi, J.Y., Rotter, J.M., Lahlouh, E.H. and Zhong, Z. (1998) “Blind trial on coals for rapid handling assessment” , Proc., 6th Int. Conf. on Bulk Materials Storage Handling and Transportation, I. E. Australia, September, Wollongong, Australia, pp 73-78.

51. Peric, D., Ayari, M.A.(2002) “Influence of Lode’s angle on the pore pressure in generation soils”. Int. J. Plasticity. 18, 1039-1059.

52. Peric, D., Ayari, M.A.(2002) “On the analytical solutions for the three-invariant Cam clay model”. Int. J. Plasticity. 18, 1061-1082.

31

53. Ragneau, E. and Aribert, J.M.(1995), Silo pressure calculation: From a Finite Element approach to simplified analytical solutions, Bulk Solids Handling, 15:1, 71-84

54. Rombach, G. and Eibl, J. (1989) “Numerical simulation of filling and discharging processes in silos” Proc. 3rd Int. Conf. on Bulk Materials, Newcastle, 48-52.

55. Rombach, G. and Eibl, J.(1995), Granular flow of materials in silos: Numerical results, Bulk Solids Handling, 15:1, 65-70

56. Roscoe, K.H., Burland, J.B.(1968) “On the generalized stress-strain behaviour of ‘wet’ clay”. In: Heyman, J., Leckie, F.A.(Eds.), Engineering Plasticity. Cambridge University Press, 535-609.

57. Roscoe, K.H., Schofiled, A.N., Thurairajah, A.(1963) “Yielding of clays in states wetter than critical”. Geotechnique. 13, 211-240.

58. Rotter, J.M. (2001) “Guide for the Economic Design of Circular Metal Silos”, Spon, London. 59. Rotter, J.M. and Ooi, J.Y. (2000) “Report on Visit to Fiddler’s Ferry Power Station on 26th January

2000”, Technical Investigation Report C00-01, School of Civil and Environmental Engineering, February 2000, 10pp

60. Rotter, J.M., Holst, J.M.F.G., Ooi, J.Y. and Sanad, A.M. (1998) “Silo pressure predictions using discrete element and finite element analyses", Phil. Trans. Royal Society of London: Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Series A, Vol. 356, No. 1747, pp 2685-2712, 15 Nov 1998.

61. Samath, S., Puri, V.M., Manbeck, H.B. and Hogg, R.(1993), Powder Technology, 76, 277 62. Schwedes J.(2003) “Review of testers for measuring flow properties of bulk solids”, Granular

Matter, 5, 1-43 63. Schwedes, J. and Feise, H. (1993) “Modelling of pressures and flow in silos” Int. Symp. on Reliable

Flow of Particulate Solids II, Oslo, EFChE Pub No. 96, Aug., 193-216. 64. Schwedes, J.(1975), Powder Technology, 11, 59 65. Scwedes, J and Schultze, D(1990), Powder Technology, 61, 59 66. Stainforth, P.T. and Ashley, R.C.(1973), Powder Technology, 7, 215 67. Tejchman, J.(1998), Numerical simulation of filling in silos with a polar hypoplastic constitutive

model, Powder Technology, 96, 227-239 68. Walker, D.M. (1966) “An approximate theory for pressures and arching in hoppers” Chemical

Engineering Science, 21, 975-997. 69. Walker, D.M. (1967) “A basis for bunker design”, Powder Technology, 1, 228-236. 70. Walters, J.K. (1973) “A theoretical analysis of stresses in axially-symmetric hoppers and bunkers”,

Chemical Engineering Science, 28, 779-789. 71. Williams, J.C. and Birks, A.H.(1967), Powder Technology, 4, 199 72. Wood, D.M. (1990) “Soil Behaviour and Critical State Soil Mechanics”. Cambridge University

Press. 73. Wood, D.M.(1990) “Soil Behaviour and Critical State Soil Mechanics”. Cambridge University

Press. 74. Wright, H.(1972), An evaluation of the Jenike bunker design method. Trans. Of the ASME, Paper

No. 72-MH-7 75. Zhong, Z. (2001) Facsimile correspondence information with Welbeck Colliery. 76. Zhong, Z., Ooi, J.Y. and Rotter, J.M. (2000) “Nonlinear effects of blending on coal handling

behaviour”, Proc. I.Mech.E. Conference, June 2000. 77. Zhong, Z., Ooi, J.Y. and Rotter, J.M. (2005) “Predicting the handlability of a coal blend from

measurements on the source coals”, Fuel, Vol. 84, No.17 pp 2267-2274.

32

Appendix A: Cam-Clay Model The Cam Clay models are based on the framework of volumetric hardening, pressure sensitivity and the critical state soil mechanics. The original and modified Cam clay models were developed by Roscoe et al (1963), and Roscoe and Burland (1968), respectively. The basic difference between these two models is in the assumption pertaining to the plastic energy dissipation. According to the original Cam clay model the entire plastic energy is dissipated in friction, while the modified Cam clay model uses more complex energy dissipation mode. In most applications, the original Cam clay model was gradually replaced by the modified Cam clay model, due to the non-smooth shape of the yield surface that is exhibited by the former model. ABAQUS use an extension of the "modified Cam-clay" theory. In this model, the soil is assumed to be saturated with a permeating fluid that carries a pressure stress and flow according to Darcy's law. The mechanical deformation of the soil is decomposed into elastic and a plastic part; an elasticity theory; a yield surface; a flow rule; and a hardening rule. In ABAQUS, Cam clay model is implemented numerically using backward Euler integration of the flow rule and hardening rule. The main features of Cam Clay model include the use of an elastic model (either linear elasticity or the porous elasticity model, which exhibits an increasing bulk elastic stiffness as the material undergoes compression) and for the inelastic part of the deformation a particular form of yield surface with associated flow and a hardening rule that allows the yield surface to grow or shrink. A key feature of the model is the hardening/softening concept, which is developed around the introduction of a "critical state" surface: the locus of effective stress states where unrestricted, purely deviatoric, plastic flow of the soil skeleton occurs under constant effective stress. This critical state surface is assumed to be a cone in the space of principal effective stress (Fig. A.1), whose vertex is the origin (zero effective stress) and whose axis is the Mean Effective Stress (MES). The advantage of the Cam Clay model lies in their apparent simplicity and their capability to represent the stress-strain behaviour of soils realistically. The particular advantage includes description of the soil behaviour in terms of the effective stress, thus providing for the coupled effects of both: the pore pressure generation and shear stresses. The important influence of a stress history is addressed by means of simultaneous incorporation of the compression and shear behaviours. The simplicity of the model is also reflected in a small number of the material parameters. The model needs only five parameters, three critical state parameters: M (slope of critical state line); λ (slope of plastic loading path, or λ-line) and κ (slope of elastic loading-unloading path, κ-line); The other two parameters are initial void ratio, 0e , and hardening parameter, cp . Consolidation parameter, N , is also used in the elasto-plastic constitutive equations. Because the model is based on the physical notion of the way that soil deforms and yields plastically, so, all these material parameters have a readily understood physical meaning, they will be explained further in next section. Thus far, the Cam Clay model remains the most widely used plasticity model for characterization of the stress-strain behaviour of the cohesive soil subjected to a static loading(Roscoe and Burland, 1968; Carter, 1982; Wood, 1990; Borja and Lee, 1990; Borja, 1991; Alawaji et al., 1992; Peric and Ayari, 2002).

A.1 General Formulation The yield surface for the modified Cam clay model in ABAQUS takes the shape of an ellipse in the 'p , t space, and it is given by

2 2

2

1 ' 1 1 0p ta Maβ

− + − =

(1)

where, '1

3' ( )ijp trace σ= − Mean Effective Stress or equivalent pressure;

33

31 1 11 12

rt qK K q

= + − −

Deviatoric stress measure;

32 ij ijq s s= Equivalent shear stress; ' 'ij ij ijs pσ δ= + Deviatoric stress;

( )139

2 ij jk kir s s s= Third stress invariant;

M constant that defines the slope of the critical state lines; β constant that is equal to 1.0 on the “dry” side of critical state line( 't Mp> ) but may be different from 1.0 on the “wet” side of the critical state line( 1.0β ≠ introduces a different ellipse on the wet side of the critical state line; i.e., a tighter “cap” is obtained if 1.0β < as shown in Fig. A.2;

0a hardening parameter that defines the initial size of the yield surface (Fig.A.2); K ratio of the flow stress in tri-axial tension to the flow stress in tri-axial compression and determines the shape of the yield surface in the plane of principal deviatoric stress(the “Π-plane”, Fig.A.3); ABAQUS requires that 0.778 1.0K≤ ≤ to ensure that the yield surface remains convex; It should be noted that parameters ,M β and K are defined by users and it can also depend on temperature as well as other predefined field variables.

A.1.1 Cam clay model simplified into three dimensional model For Cam clay model in tri-axial space with 1K β= = , equation (1) can be simplified as

2 2' 1 1 0p qa Ma

− + − =

(2)

where ' ' ' '1 1

3 3' ( ) ( )ij xx yy zzp trace σ σ σ σ= − = − + + (3)

( ) ( ) ( ) ( )

2 2 2 2 2 23 32 2

2 2 2 2 2 212

( 2 2 2 )

3

ij ij xx yy zz xy yz xz

xx yy yy zz zz xx xy yz xz

q s s s s s s s s

σ σ σ σ σ σ τ τ τ

= = + + + + +

= − + − + − + + +

(4)

A.1.2 Cam clay model simplified into tri-axial compression model Tri-axial compression model is a special case for axis-symmetric model. Besides the uniform deformation on the radial direction, if uniform deformation also exist along axial direction, then all the shear stress components within axi-symmetric model are zero, and rr rθθσ σ σ= , zz aσ σ= ; If solid is modelled with Cam clay constitutive relation and with 1K β= = , Then, the yielding surface again has the following form

2 2' 1 1 0p qa Ma

− + − =

(2.a)

but with ' ' '1 1

3 3' ( ) ( 2 )ij a rp trace σ σ σ= − − + (5)

( ) ( ) ( )

2 2 23 32 2

2 2 212

( )ij ij rr zz

rr zz zz rr

a r

q s s s s sθθ

θθ θθσ σ σ σ σ σ

σ σ

= + +

= − + − + −

= −

(6)

34

A.2 Hardening Law Two hardening laws are employed in ABAQUS for Cam clay models: exponential form and piecewise linear form.

A.2.1 Exponential form The exponential form of the hardening law is written in terms of some of the porous elasticity parameters. The size of the yield surface at any time is determined by the initial value of the hardening parameter, 0a , and the amount of inelastic volume change occurs according to the equation

0 01exp (1 )

pl

pl

Ja a eJλ κ

−= + −

(7)

where plJ inelastic volume change(the ratio of current volume to initial volume, attributable to inelastic

deformation); κ logarithmic bulk modulus of the material; λ logarithmic hardening constant;

0e initial void ratio; The initial size of the yield surface, 0a , can be defined directly or indirectly by specifying 1e , which is the intercept of the virgin consolidation line with the void ratio axis in the plot of void ratio, e , versus the logarithm of the effective pressure stress, ln p , In this option, 0a is defined by

1 0 00

ln1 exp2

e e pa κλ κ

− − = − (8)

where 0p is the initial value of the equivalent hydrostatic pressure stress.

A.2.2 Piecewise linear form In this hardening law, relationship between the yield stress in hydraulic compression, cp , to the

corresponding volumetric plastic strain, plvolε is defined by user in a piecewise linear way shown in

Fig.A.4 in the form of ( )pl

c c volp p ε= (9) The evolution parameter, a , is then given by

1cpaβ

=+

(10)

The volumetric plastic strain axis has an arbitrary origin: 0pl

volε is the position on this axis

corresponding to the initial state of the material, thus defining the initial hydrostatic pressure, 0cp ,

and hence the initial yield surface size, 0a .

A.3 Critical State Line When soil samples are sheared they approach the Critical State Line (CSL). The equations for CSL are (Fig. A.2, Fig. A.5)

1

'ln '

q Mpe e pλ

== −

(11)

where M represents slope of CSL in ( ',p q ) space, and 1e represents the location of CSL in ( ',p e ) plot. The critical state line represents the final state of soil samples in tri-axial tests when it is possible to continue to shear the sample with no change of imposed stresses or volume of the soil. The equation of swelling or recompression line is given by

ln 'e e pκ κ= − (12)

35

where eκ depends upon which κ -lines the soil is on, it stays constant while the soil is moving up or down the same line. Thus, the particular unloading-reloading line which corresponding to the yield locus with size )1( β+a is

( ) ( )1 'ln (1 ) ln (1 )ape e aλ β κ β= − + + + (13)

At a mean stress, '' csp p a= = , the void ratio at critical state is then

( )1 ( ) ln 1 ln 'ln '

cs cs

cs

e e pe p

λ κ β λλΓ

= − − + −

= − (14)

This is the CSL in the void ratio e - logarithmic pressure ln p plot.

36

Appendix B Study of stress history by Triaxial Compression Tri-axial compression test is presently the most widely used procedure to determine strength and stress-strain properties of granular solids such as coal and soils. In such a test, a cylindrical specimen is subjected to an axial compression stress aσ and a radial pressure stress rσ . The specimen is assumed to deform uniformly during the test so that the information obtained from the test represents the true material behaviour of a single granular solid element within bunker. Thus, the uniformity of stress and strain in the specimen is an essential part for the tri-axial test concept. In this modelling study, coal mass is modelled by the modified Cam clay model with exponential hardening rules. Two kinds of failure mechanism, hardening failure and softening failure, are investigated. Relationships between coal strength and different loading paths are explored. Perfect drainage is assumed because it is assumed that there is enough time for the coals to drain out from bunker before they are discharged.

B.1 Geometry of the Model The original arched geometry is shown in Fig. B.1a. To study the stress history of coals within bunker, samples with the shape of a cylinder are taken from different parts within bunker to begin tri-axial compression tests. Due to the inherent axi-symmetric nature of the problem, only one half section was used in the numerical model. The radius and height of the cylinder is 50mm and 200mm, respectively (Fig.B.1).

B.2 Mesh and Material Parameters Two sets of meshes are established for the tri-axial compression study. Each model has 200 elements, 10 elements along radial and 20 elements along height. The finite element meshes are also biased towards the all-round outside edge as wee as towards the top and bottom. Four kinds of axi-symmetric elements are used: 8-node axi-symmetric bi-quadratic element (CAX8), 8-node axi-symmetric element with reduced integration scheme(CAX8R), 4-node axi-symmetric linear element (CAX4) and 4-node axi-symmetric linear with reduced integration scheme(CAX4R). The two sets of meshes are shown in Fig. B.2. Table B.1 lists the parameters used in Cam clay model.

Table B.1 Parameters for Cam Clay model

Log. bulk modulus

(κ)

Poisson’s ratio (ν)

Log. harden

modulus (λ)

Critical stress ratio (M)

Initial void ratio (e0)

Intercept of virgin

consolidation line(e1)

Wet cap parameter

(β)

Third stress

invariant parameter

(K) 0.0092 0.3 0.0367 1.68 1.02 0.87 1 1

B.3 Boundary Conditions and Loading All the boundary condition and loading sides are shown in Fig. B.2. As shown in this figure, the bottom and axis of the model are constrained from moving along vertical and horizontal directions, respectively. Cell pressures are applied at the all-round outside edge. Force or displacement is applied at the top of the model. Four loading steps are used for all the model studies. In step 1, only self-weight of the coal was applied to the model, while in step 2, the all-round cell pressure is 50kPa, top pressure is 100kPa. It should be noted that step 1 and step 2 are all the same for all models in this study. The purpose of these two steps is to produce high initial stresses within coal samples, so that different overstress history within coal can be modelled in the subsequent two load steps. Table B2 lists some of the typical loading steps in this study.

B.4 Hardening Effects To explore the hardening failure, four specially designed load steps are used to achieve this. Typical results for one element (element 92) within specimen were selected. The results are shown in Fig. B.3 (a-d). After loading steps 1 and 2, the loading vector reached a point at reference yield surface. Then,

37

tri-axial specimen was unloaded elastically back in step 3 from the reference yielding surface to point (53.3, 40) in ( ',p q ) plot. The Over-Consolidation Ratio (OCR) is 1.5(80/53.3=1.5). Then, specimen was load vertically in the ( ',p q ) plot at step 4. The last loading step is achieved by adjusting the cell

Table B2 Loading steps for tri-axial compression model

Step 1 Step 2 Step 3 Step 4

Load position Self weight (kN/m3)

Top (kPa)

Cell (kPa)

Top (kPa)

Cell (kPa)

Top (kPa)

Cell (kPa)

Hardening 7.8 100 50 80 40 160 0 Softening 7.8 100 50 20 10 40 0 Path 1 7.8 100 50 40 20 80 0 Path 2 7.8 100 50 40 20 120 20 Path 3 7.8 100 50 40 20 240 220 Path 4 7.8 100 50 40 20 40 0 Path 5 7.8 100 50 40 20 40 200 Path 6 7.8 100 50 40 20 0 20

pressure and top pressure simultaneously so that the Mean Effective Stress kept constant. With Cam clay constitutive relation and tri-axial compression model, ABAQUS shown specimen fail when shear stress reached the critical state, a interception point between loading path and critical state line (Fig. B.3a). When failure occurs, the final yielding surface is greater than the previous reference yield surface. This indicates that the coal sample hardens when failure. Void ratio and logarithmic of Mean Effective Stress (MES) curve( , lne p ) (Fig.B.3b) shows that void ratio starts with initial value(1.02), it first decrease along elastic unloading and reloading line(κ-line). After specimen failed under self weight, it changed its slope and further decrease along a line almost parallel to the normal compression line(NCL, λ-line). Step 1 complete somewhere quite near NCL. From Cam-Clay theory, it is known that void ratio should follow the λ-line when plastic deformation becomes available. So, this small discrepancy may be caused by the numerical errors. Upon further increase loading at step 2, plastic deformation increased, void ratio decreased further along the NCL, it stopped decrease when load step 2 ends. This continuous decrease of void ratio means specimen is contracting when loading at step 1 and step 2. When unloading occurs at step 3, void ratio increased along κ-line and specimen dilates elastically, this process ends with the stoppage of unloading at step 3. If we further increase the equivalent shear load q while kept MES constant in step 4, void ratio decrease rapidly along vertical line in void ratio vs logarithmic MES plot (Fig.B.3b), specimen fail when void ratio reached its critical state value. The equivalent shear stress q vs logarithmic axial strain aε plot (Fig.B.3c) shows that during the first step of loading, little equivalent shear stress was induced within specimen although some plastic deformation had occurred, when loading is further increased in load step 2, equivalent shear stress increases unsteadily with the increase of mean effective stress. When unloading happens in step 3, equivalent shear stress decreases instantly with small axial strain. When specimen was loading again in step 4, equivalent shear stress increase sharply with small increase of axial strain, it then increases gradually. Finally it turns to horizontal when reaching critical state. It can also be observed from this figure that when critical stage arrive, equivalent shear stresses will not increase anymore within specimen while axial strain increase to infinite. This is a very clear sign for the failure of specimen. Fig. B.3d also indicates that when specimen fails, volumetric strain has very little change. Fig.B.4 (a-d) shows contours of the specimen when failure occurs. The equivalent shear stress and mean effective stress are homogeneous along the height of the specimen (Fig.B.4a, b). The maximum stress is at the bottom of the specimen. This is caused by the consideration of self-weight within specimen. Homogeneous axial strain are resulted from homogeneous stress (Fig.B.4c) and larger axial strain at bottom of specimen is believed to be caused by self-weight. Shear strain showed a interesting variation. Although very small, the biggest shear strain near outside is almost ten times shear strain at somewhere near top and bottom of specimen.

38

B.5 Softening Effects Fig.B.5 (a-d) shows results for the soften failure modelling. In this study, specimen is unloaded to lower stress level somewhere within “dry side” in ( ',p q ) plot at load step 3. The over-consolidation ratio(OCR) is 6 (80/13.3=6.0). Cell pressure and top pressure are adjusted simultaneously so that the Mean Effective Stress kept constant at load step 4. So, the loading path moves upward until the specimen fail when stresses reach critical state. It can be seen from Fig.B.5b that the loading path had reached to the left of CSL in ( pe ln, ) plot after the third load step. Upon unloading, more dilation can be observed compared with the hardening case. When further apply the MES at step 4, void ratio going vertically until it meets critical state line. This means that specimen dilate rapidly at step 4 until it failed at critical state. Because the loading path at step 4 in ( ',p q ) plot goes vertically and it intercepts with both critical state line and reference yielding surface, it is difficult to judge how the stress path goes with this two interception points. However, the loading path in shear stress vs axial strain( aq ε, ) plot indicates that loading vector at step 4 will first arrive peak(50kPa), then it drops to critical state(22kPa). A three dimensional check of the yielding surface also verified that stress path will first intercept with reference yield surface, then with rapid increase of the void ratio, loading path moves to critical state line. Hence, the final yield surface shrink. The volumetric strain and axial strain plot( av εε , ) in Fig.B.5d shows that during step 1 to 2 volume strain increase linearly, this indicates that the compression volume increase proportional to external load. At step 3, with the decrease of radial and axial load, volumetric strain decrease linearly due to elastic unloading. Further load increase at step 4 caused the specimen keep volume strain constant for a little while with the rapid increase of shear stress q, then volumetric strain drops to constant when critical state reached. At last, specimen has very little change of shear stress, very little change of volumetric strain, but very large axial strain. That means a kind of uncontrollable failure of the whole specimen. Fig. B.6 (a-d) is stress and strain distribution within specimen correspondence to softening failure. Compared with contours for hardening failure, it can be seen that the stress level is much lower than hardening failure (smaller than one third of the hardening value, Fig. B.6a,b), although it showed the same trend of homogenous distribution along vertical height of the specimen. The axial strain is greater than values for hardening failure (Fig.B.6c). Larger shear strain exists compared with hardening failure (Fig.B.6d). This stress and strain distribution may indicate that softening failure is more severe than hardening failure.

B.6 Loading Method Effects Two kinds of loading methods are used in real tri-axial compression test: force controlled loading and displacement controlled loading. To investigate the feasibility of these two methods in the application of numerical model, two models are set to carry out this study. The results are shown in Fig.B.7 (a-d). In this model, specimen is loaded on the top edge either with force loading or displacement loading. Load steps are carefully selected to make the specimen failure in the form of softening rupture. The reason for this kind of loading selection is because softening failure study is more difficult than hardening failure study. The over-consolidated ratio (OCR) for this investigation is 6. The problem is difficult with both geometric and material non-linearity present. To overcome these difficulties, different types of finite elements are tried. Numerical study indicates linear axi-symmetric element with reduced integration option scheme is very effective for such problem. With this knowledge, stable solutions are obtained with ABAQUS program. To simplify the comparisons, only results at load step 4 are compared between the two kinds of loading method at Fig. B.7. It can be observed from these figures that all the curves show good correspondence. Numerical studies for loading method effects also indicates that when over-consolidation ratio is further increased, then the displacement controlled loading is more stable than force loading. In most cased, displacement controlled loading will yield a stable solution while force loading interrupted by numerical difficulties. It should be noted that these numerical practices are also in agreement with real tri-axial experimental tests.

B.7 Element Effects Four kinds of finite elements (CAX8, CAX8R, CAX4, CAX4R) are used to investigate the suitability of these elements for the present problem. For the hardening failure, all the four elements showed

39

excellent behaviour. Typical results are shown in Fig.B.8(a-d). To simplify the comparisons, only results at step 4 are compared. Excellent correspondence can be observed for all the four elements. However, when the specimen is unloaded into “dry side” with high Over-Consolidation Ratio (OCR) and then loading again to critical state in the form of soft failure, Quadratic elements (CAX8, CAX8R) show poor behaviour even for small loading increments, the computation will stop due to numerical difficulties. Although some improvements can be made for elements CAX8R, i.e., quadratic elements with reduced integration scheme, in the softening failure modelling, linear elements (CAX4, CAX4R) show excellent behaviour than quadratic elements. It should be noted that element CAX4R is very successful in modelling all the failure behaviour and loading paths in the present investigation.

B.8 Loading Path Effects Cam clay failure theory is very much path dependent. This made the model an ideal tool for coal simulation within bunker. Several selected loading paths are investigated in this work. All the loading paths are designed with four load steps. Load steps 1 through 3 are used to make the coal specimen reach certain point within ( ',p q ) plot with some extent of over consolidated condition. In this study, the Over-Consolidation Ratio is 2.97(80/26.9=2.97). The first three loading steps are all the same for all the models in this study. After the first three loading steps, it can be seen from Fig. B.9(a-d) that loading path ends at somewhere at “dry side” in ( ',p q ) plot and left of critical state line in ( pe ln, ) plot. Different loading paths are designed at load step 4. They are explained below. Path 1: Mean Effective Stress kept constant

To make the Mean Effective Stress(MES) constant, the loading path will start from step 3 directly vertical along path1. The ( pe ln, ) plot shows that the start point lies at the left of critical state line. When equivalent shear loading increase, void ratio increase monotonically until the failure point intercepted with critical state line. The failure of the specimen was caused by swelling of the void as well as the specimen, The ( aq ε, ) plot indicates that equivalent shear stress first increased to peak then dropped to a constant value at critical state. When the final stage reached, there are large variations of shear stress within specimen, axial strain become infinite while volumetric strain remains constant. These deformation characters indicate specimen is failing uncontrollably.

Path 2: Cell pressure constant, loading axial

The specification for stress path 2 at step 4 is to keep the cell pressure constant while the equivalent shear stress increase ( 0,r aqδσ δ δσ= = ). Therefore (2 )

3 3r a qp δσ δσ δδ += = . This means that the mean

effective stress p increase at one-third the rate of the equivalent shear stress q . So, in ( ',p q ) plot at step 4, specimen will keep this stress path until failure when it intercepted with critical state line. The plastic compression line plot( pe ln, ) indicates that void ratio decrease with the increase of effective pressure along κ-line, after reach critical state, the void ratio rapidly decrease to failure along the critical failure line. The ( aq ε, ) and ( av εε , ) plot showed that large shear stress induced when failure occurs and volume strain arrive a constant value while axial strain tends infinite. Large shear stress at failure also means large shear strains. Path 3: Equivalent shear Stress constant

To keep in the same increment for both cell pressure and axial pressure, the equivalent shear stress q will kept constant in ( ',p q ) plot, this cause the effective stress path 3 move horizontally. The ( pe ln, ) plot indicates that void ratio will first decrease along κ-line because of elastic loading. After loading path pass through the reference yield surface, plastic yielding takes place, hence, void ratio continued dropping even more rapidly along λ-line. So, it can be concluded that specimen contract all the way down to failure for this kind of loading. Different from the above two loading paths, when failure occur, significant volumetric strain increase can be observed for this loading path from ( av εε , ) plot. It should also be noted that very large effective stress is needed for the specimen to achieve failure in this loading path. For the present loading case, an unimaginable 4,000kPa of effective stress is needed to achieve failure. So, specimen is thought to be stable under this loading method.

40

Path 4: Active failure

To simulate the so-called active failure in geomechanics, axial pressure should be kept constant while cell pressure be reduced slowly to zero( δσδσδσ == ra ,0 ). This causes loading path moves along path 4 in the slope of -1.5( 2

3'/ −=pq δδ ). The ( ',p q ) plot indicates specimen will first unloading, this elastic unloading will ends when loading path hit the reference yield surface. After this point, plastic deformation mobilise, this mobilised plastic deformation cause the mean effective stress within specimen even increase, specimen fail when it reached critical state. This process can also be verified from ( aq ε, ) plot. The void ratio variation is also very interesting for this kind of loading case, void ratio first increase along κ-line away from critical state line due to elastic unloading, upon the mobilisation of plastic deformation, void ratio increased again but approaching critical state in a very different line. It can be imagined that specimen is kept dilation during the whole process of unloading. The final failure should be in the form of dilate rupture. Numerical simulations also indicate the volumetric strain remains constant when failure occur.

Path 5: Passive failure

Opposite to path4, in this loading step, the axial effective stress is kept constant and cell pressures are increased gradually. Thus, specimen experiences some form of passive failure. Loading path will go downward along path 5 in ( ',p q ) plot in the slope of –1.5. In this study, the loading path ends when cell pressure and axial pressure becomes equal. In this loading step, void ratio decreases along κ-line because effective stress is constrained inside the reference yield surface. Axial compression strain decrease during the loading increase. Volumetric strain increases with the decrease of axial strain. All these deformation behaviour suggests that although the void within the coal sample is contracting from micro-mechanics point of view, its whole volume is increasing from the macro-mechanics measurement. This dilation deformation maybe attributed to the dilatant deformation of skeleton materials within coal samples. If cell pressures are increased further, loading path5 will going downward beyond the ( ',p q ) plot and specimen will experience axial compression deformation before specimen reach critical deformation state. Large effective stress is needed to cause the specimen failure.

Path 6: Cell Pressure constant, unloading axial

This loading path is achieved by keeping the cell pressure constant, then reducing the top edge pressure to zero. The mean effective stress p decreases at one-third the rate of the equivalent shear stress q . Void ratio increase along the κ-line due to the elastic unloading property of this loading step. The equivalent shear stress vs axial strain plot ( aq ε, ) indicates that axial compression strain reduce with the decrease of axial load. That means specimen dilate during this loading step. The volumetric strain vs. axial strain plot ( av εε , ) indicates that the volumetric strain decreasing with the decrease of axial compression strain. Our simulation for this study ends when axial compression pressure equals to cell pressure. If axial compression is further unloaded, the loading path will going downward ( ',p q ) plot along path6 until it meets the critical state.

B.9 Discussion and Remarks Generally speaking, when a particulate material is loaded, particles rearrange, agglomerate and fracture, causing the volume to either decrease or decrease. The material then reaches some point at which additional load may change the shape of a given mass, but not its volume. This state point is termed critical state. Before reaching the critical state, the material undergoes successive yield states, and hardens(or softens) after each yield state, depending upon the material’s response to loading conditions and stress history of consolidation. For soils with heavily Over-Consolidation Ratio, it will become ‘dense’ with respect to critical states; they need a very large increase in Means Effective Stress(or spherical pressure) if they are to undergo plastic compression. Without this extra MES (or spherical pressure) they have a strong tendency to dilate and soften, by the agency of transient pore suction.

41

For soils with lightly over-consolidated ratio or virgin, it will become ‘loose’ with respect to critical states; they need only a little extra pressure to undergo plastic compression. They not only compress readily under MES (or spherical pressures), but also contract strongly when sheared, generating transient positive pore pressure. Once contracted, they become stiffer and stronger. Different loading paths can result in different failure mechanisms. Fig.B.10 shows a general descriptions of possible stress paths in tri-axial compression test with a full Cam-Clay yield surface around a stress point in arbitrary position. By changing different loading increments for cell pressures and axial load, several loading paths can be achieved. These loading paths can be defined in the equation (15):

1 3'1 3 1

1 3 3'3 1 3

1 3'1 3 1

1 3 3

' ; 0, ', 0, , / ' 3

, 2 0, ' 0, 3, 0, , / ' 3 / 2

' ; 0, ', 0, , / ' 3

, 2 0, ' 0

p q pq p

q p qq p

p q pq p

q p

δσ δσ δσ δ δ δσσ δσ δσ δσ δ δ

δσ δσ δσ δσ δ δ δσσ δσ δσ δσ δ δ

δσ δσ δσ δ δ δσσ δσ δσ δσ δ δ

δσ δσ δσ δσ δ

+ = = ∴ = =+ = = ∴ =

+ + = = − ∴ = =− = = − ∴ = −− = = − ∴ = = −− = = − ∴ =

− + = = ∴ ='3 1 3

, 3, 0, , / ' 3 / 2

qq p

δ δσσ δσ δσ δσ δ δ

= − − = = ∴ = −

(15)

where, 3

)2( 31' σσ +=p , is Mean Effective Stress(MES); q, 31 σσ −=q , is equivalent shear stress. From this figure, it can be concluded that loading vectors tend to generate contractions while unloading vectors tend to generate dilatant rupture, failure mode also depend on the starting point. Likewise a reduction of pore pressure inevitably leads to contraction of void while an increase in pore pressure inevitably leads to dilatant rupture. For the coal bunker problem with the formation of arch at outlet shown in Fig. B.1(a), the left of the arch is exposed at the bottom, classical theory of Jenike (1964, 1973) and Enstad (1975) indicates that the most compressive stress is along a arched zone, so stress path at here should follow −'

sσ vector: failure could be sudden and involve a wedge of coal wall slip down a rupture plane caused by dilatant compression. At the right side of the exposed arch, the lateral stress is zero, the arch theory suggests that this direction is just in the direction of most compressive stress, so stress paths here should follow the −'

1σ vector: failure could be dilatant extension with rupture softening. Hence, this part of arch failure should be caused by dilation of void within coal. If water table increases within bunker coal, rupture failure propensity near outlet will increase.

42

Appendix C Summary of several analytical solution

Wall pressure Normal stress Memo

Walker Fill zii Fp σ= ( ) ztz zH σγσ +−= βµ cot1

1+

=iF

Walker Flow

zff DFp σ= ;

( )f

ffF

εβφεφ

22cossin12cossin1+−

+=

n

zt

n

z Hz

Hz

Hz

nH

+

−

−= σγσ

1;

( )( )f

fBεβφ

εβφ22cossin1

22sinsin+−

+=

βtan2BDn = ; D=1 for Walker flow;

+−+= −−

2

11

1sincostan

22

µφµµπε f

Walters Fill (+) Walters Flow (-1) Same as Walker Flow

Same as Walker Flow;

( )( )[ ]φφη

ηφφηsin2sin1cos

sinsin2sin1cos2

222

yD

±+−±+

= ;

( )

+±+= −−

2

11

1sincostan

22

µφµµπε

Besides same as Walker Flow [ ]1cot −+= FDFDn βµ ;

( )( )231132 cc

y −−= ;

2

tantan

=

φηc ;

( )( )

++

+= −

φβεφβεη

sin22cos1sin22sintan 1 ;

McLean Same as Walker Flow Same as Walker Flow Same with Walker flow; β

µtan2

=n

Jenike Radial ( )p K H zγ= − tan

tanK β

µ β=

+

Jenike Peak (1 / )(tan )p

Rpz H

γβ µ

=+ +

JMR Design nf f vfp F p=

1

n n

vf ztH z z zp

n H H Hγ σ

= − + −

1 cot1 cotf

aF µ βµ β

+=

+; 2 coteffn aµ β=

43

Notation H vertical height between the hopper apex and the transition; h outlet height; B ratio of vertical shear stress to vertical direct stress at the wall; D distribution factor, it is the ratio of vertical stress at the wall to the mean vertical stress at

any given level. D is assumed unity; σz mean vertical stress at height z above the hopper apex; σzt mean vertical stress in the solid at the transition; β hopper half-angle (the angle between the vertical and the transition; γ solid unit weight; φ internal friction angle; µ wall friction coefficient; µeff effective wall friction coefficient for the hopper; z vertical coordinate upwards from hopper apex; a empirical constant, taken here as a=0 MES Mean Effective Stress NCL Normal Compression Line or Virgin Compression Line CSL Critical State Line OCR Over-Consolidation Ratio RYS Reference Yield Surface βL left hopper half-angle (the angle between the vertical and the left wall); βR right hopper half-angle (the angle between the vertical and the right wall); w outlet width; b radius of hopper, b=1m in this study; θ arch angle A ratio of q over p; q equivalent shear stress, raijij ssq σσ −== 2

3

p mean effective stress, ( )rakkp σσσ 231

31 +==

aσ Axial stress;

rσ radial stress;

fF vfnff ppF = ;

nfp hopper pressure normal to the wall after filling or during storing;

vfp mean vertical stress in the solid after filling or during storing; Symbols

'13' ( )ijp trace σ= − Mean Effective Stress or equivalent pressure;

32 ij ijq s s= Equivalent shear stress;

β constant that is equal to 1.0 on the “dry” side of critical state line( 't Mp> ) but may be different from 1.0 on the “wet” side of the critical state line( 1.0β ≠ introduces a different ellipse on the wet side of the critical state line; i.e., a tighter “cap” is obtained if

1.0β < ;

0a hardening parameter that defines the initial size of the yield surface; K ratio of the flow stress in tri-axial tension to the flow stress in tri-axial compression and

determines the shape of the yield surface in the plane of principal deviatoric stress(the “Π-plane”, ABAQUS requires that 0.778 1.0K≤ ≤ to ensure that the yield surface remains convex;

κ logarithmic bulk modulus of the material; λ logarithmic hardening constant; 0e initial void ratio

44

Figures

Figure 1.1 Jenike concept of a bunker outlet with arching

σ1 σ1

Stress actingin arch

Cohesive arch Mechanical arch

Figure 1.2 Classical arching concepts

45

β

σ1 σ1 B

W

T

Figure 1.3 Jenike analysis of stresses in standard bunker

σa

σc σm

σa

Critical size when σa = σc

σc, σa, σmB

Figure 1.4 Jenike analysis of consolidation and cohesion in standard bunker

46

Coal

Compressedcoal

Backwall

Naturalrepose angle

of coal

Paddle offeeder

Feedertable

Figure 1.5 Arching at outlet of a typical UK coal bunker

47

Figure 3.1 (a) West Burton receiving bunkers; (b) Top view; (c) Steel plate bolted at

outlet of bunker

(a)

(b) (c)

48

Figure 3.2 Filling bunker with rectangle compartments (West Burton).

Figure 3.3 Concrete bunker in “W” shape with horizontal outlet(West Burton)

49

(a)

(b)

Figure 3.4 All metal encapsulated bunker: (a) side view; (b) discharge aid(Drax)

50

Figure 3.5 Bunker wall condition. (a) concrete with imposed reinforcing bars(West

Burton), (b)smooth metal finish (Drax)

3,048

Existing profiled beams70

75

Gate area

762

67

70

480

90

New profiling at slot ends

Figure 3.6 End wall additions to cross beam profiling(West Burton).

36023

5

50 60

285

3,048

Direction of coal flow

Chain area

Gate area

top of chain

existing profiled beams762

280

60

Figure 3.7 Bath type profiling in feeder feedbox(West Burton)

(a) (b)

51

Figure 4.1 Geometry of symmetric bunker Figure 4.2 Finite element mesh

Figure 4.3 Maximum in-plane compressive principal stress(Elastic model)

Figure 4.4 Maximum in-plane compressive principal stress(Mohr-Coulomb model)

52

Figure 4.5 Maximum in-plane compressive

principal stress(Cam-Clay model) Figure 4.6 Maximum in-plane compressive

principal strain for elastic model

Figure 4.7 Maximum in-plane compressive principal strain for Mohr-Coulomb model

Figure 4.8 Maximum in-plane compressive principal strain for Cam-Clay model

53

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10Wall pressure, kPa

Vert

ical

pos

ition

, m

Elastic Mohr-CoulombCam-Clay

Figure 4.9 Wall pressure along contact interface for three model

Figure 4.10 slip distance between contact surface for three model

Hopper with Elastic Model

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16

wall pressure(kPa)

vert

ical

pos

tion(

m)

walker fillwalker flowwalters flowMcLeanJenike peakJenikeJMR DesignABAQUS

Figure 4.11 Comparisons between elastic model and analytical solution

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20Slip along contact surface, m

Vert

ical

pos

ition

, m

ElasticMohr-CoulombCam-Clay

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05Slip along contact surface, m

Vert

ical

pos

ition

, m

Elastic

Mohr-Coulomb

54

Hopper with Mohr-Coulomb Model

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16

wall pressure(kPa)

vert

ical

pos

tion(

m)


Figure 4.12 Comparisons between Mohr-Coulomb model and analytical solution

Hopper with Cam-Clay Model

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6 8 10 12 14 16

wall pressure(kPa)

vert

ical

pos

tion(

m)


Figure 4.13 Comparisons between Cam-Clay model and analytical solution

55

θ

β

Figure 5.1a The geometry of a hopper with an arch

Figure 5.1b The finite element mesh

56

Figure 5.2a Wall pressure with different wall friction (CAX8, selfweight, β=30°,

θ=30°)

Figure 5.2b Vertical stress at coal/wall interface under different wall friction (CAX8, selfweight, β=30°, θ=30°)

Wall friction effect(CAX8, selfweight, β=30 degree, θ=30 degree)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60

Wall pressure, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50 60


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

A

57


0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40

p, (kPa)

q, (k

Pa)

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

(a)

Stress Ratio vs. Wall friction(CAX8, selfweight, β=30 degree, θ=30 degree)

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Wall friction, µ

Stre

ss R

atio

, M=q

/p'

Stress Ratio, M

(b)

Figure 5.3 (a) p—q distribution along arched outlet for different wall friction(CAX8,

selfweight, β=30°, θ=30°); (b) stress ratio at arched outlet.

58

Figure 5.4 Wall pressure with different wall friction(CAX8, selfweight+surcharge,

β=30°, θ=30°)

Figure 5.5 p—q distribution along arched outlet for different wall friction(CAX8, selfweight+surcharge, β=30°, θ=30°)

Wall friction effect(CAX8, selfweight+surcharge, β=30 degree, θ=30 degree)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250

Wall pressure, kPa

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 50 100 150 200 250

Wall pressure, kPa

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120 140σ22, kPa

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100 120 140σ22, kPa

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

A

59


0

50

100

150

200

250

0 20 40 60 80 100 120 140 160

p, (kPa)

q, (k

Pa)

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

(a)

Stress Ratio vs. Wall friction(CAX8, selfweight+surcharge, β=30 degree, θ=30 degree)

1

1.1

1.2

1.3

1.4

1.5

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6

Wall friction, µ

Stre

ss R

atio

, M=q

/p'

Stress Ratio, M

(b)

Figure 5.6 (a)p—q distribution along arched outlet for different wall friction(CAX8, selfweight+surcharge, β=30°, θ=30°); (b) Stress ratio at arched outlet.

60

Figure 5.7 Wall pressure for different wall friction(CAX8, selfweight, β=30°, θ=45°)

Figure 5.8 Vertical stress along wall for different wall friction(CAX8, selfweight,

β=30°, θ=45°)


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35 40 45 50


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35 40 45 50


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

61


0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45

p, (kPa)

q, (k

Pa)

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

(a)

stress ratio, M

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

0 0.1 0.2 0.3 0.4 0.5 0.6

Wall friction, µ

Stre

ss r

atio

, M=q

/p

stress ratio, M

(b)

Figure 5.9 p—q along arched outlet(CAX8, selfweight, β=30°, θ=45°)

62

Figure 5.10 Wall pressure for different wall friction(CAX8, selfweight+surcharge, β=30°, θ=45°))

Figure 5.11 Vertical stress along wall for different wall ction(CAX8, selfweight+surcharge, β=30°, θ=45°)


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120 140σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100 120 140σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120 140 160 180 200


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6


0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100 120 140 160 180 200


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

63


0

50

100

150

200

250

0 20 40 60 80 100 120 140 160 180

p, (kPa)

q, (k

Pa)

WF: 0.0WF: 0.1WF: 0.2WF: 0.3WF: 0.4WF: 0.5WF: 0.6

(a)

Stress ratio, M

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6

Wall friction, m

Stre

ss r

atio

, M=q

/p

Stress ratio, M

(b)

Figure 5.12 p—q along arched outlet(CAX8, selfweight+weight, β=30°, θ=45°)

64

Figure 5.13 Wall pressure for hoppers with different arched outlet;(selfweight, CAX8, wall friction: 0.2)

Figure 5.14 Vertical stress at wall for different arched outlet; (selfweight, CAX8, wall friction: 0.2)

Wall pressure(β=30 degree, selfweight only, CAX8, WF: 0.2)

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30 35


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30θ=45θ=60θ=75θ=90

Wall pressure(β=30 degree)

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H θ=30

θ=45θ=60θ=75θ=90

Vertical stress(β=30 degree, selfweight only, CAX8, WF: 0.2)

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25

σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30θ=45θ=60θ=75θ=90

Vertical stress(β=30 degree, selfweight only, CAX8)

0.02

0.04

0.06

0.08

0.10

0.12

0 2 4 6 8 10 12 14 16 18

σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H θ=30

θ=45θ=60θ=75θ=90

65

Figure 5.15 Wall pressure for hoppers with different arched outlet; (selfweight+surcharge, CAX8, wall friction: 0.2)

Figure 5.16 Vertical stress for hoppers with varied arched outlet; (seflweight+surcharge, CAX8, wall friction: 0.2)

Wall pressure(β=30 degree, selfweight+surcharge, CAX8, WF: 0.2)

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30θ=45θ=60θ=75θ=90

Wall pressure(β=30 degree)

0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100 120


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H θ=30

θ=45θ=60θ=75θ=90

Vertical stress(β=30 degree, selfweight+surcharge, CAX8, WF: 0.2)

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120

σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30θ=45θ=60θ=75θ=90

Vertical stress(β=30 degree, selfweight+surcharge, CAX8)

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50 60

σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30θ=45θ=60θ=75θ=90

66

Wall pressure for different hopper angle(θ= 30 degree, CAX8, selfweight, WF: 0.2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120 140 160


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

β=15β=30β=45β=60

Figure 5.17 Wall pressure for different hopper half angle;(Arched angle: 30degree, selfweight, wall friction: 0.2)

σ22 for different hopper angle(θ=30 degree, CAX8, selfweight, WF: 0.2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120

σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

β=15β=30β=45β=60

Figure 5.18 Vertical stress under different hopper half angle; (selfweight, wall friction: 0.2)

67

Wall pressure varies for different β (θ=30 degree, selfweight+surcharge, WF: 0.2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300 350 400 450


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

β=15β=30β=45β=60

Figure 5.19 Wall pressure varies with different hopper half angle;(selfweight+surcharge, arched angle: 30 degree, wall friction: 0.2)

σ22 for different hopper angle(θ=30 degree, CAX8, selfweight+surcharge, WF: 0.2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300 350

σ22, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

β=15β=30β=45β=60

Figure 5.20 Vertical stress with different hopper half angle; (selfweight+surcharge, arch angle: 30 degree, wall friction: 0.2)

68

Figure 5.21 Comparison of wall pressures between theoretical and numerical;(selfweight, wall friction: 0.2, CAX8)

Ff vs. θ(Selfweight, CAX8, β=30 degree, WF: 0.2)

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

0 20 40 60 80 100

θ

Ff

β=30

Figure 5.22 Stress ratio, Ff, for hoppers with different arched angle;(selfweight, wall friction: 0.2, CAX8)

Wall pressure(Selweight, β=30 degree;(1)--theoretical, (2)--numerical, WF: 0.2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30(1)θ=30(2)θ=45(1)θ=45(2)θ=60(1)θ=60(2)θ=75(1)θ=75(2)θ=90(1)θ=90(2)

Wall pressure(Selweight, β=30 degree;(1)--theoretical, (2)--numerical, WF: 0.2)

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30(1)θ=30(2)θ=45(1)θ=45(2)θ=60(1)θ=60(2)θ=75(1)θ=75(2)θ=90(1)θ=90(2)

69

Figure 5.23 Comparisons of wall pressures between theory and numerical; (selfweight+surcharge, CAX8, wall friction: 0.2)

Ff vs. θ(selfweight+surcharge, CAX8, β=30, WF: 0.2)

0.915

0.920

0.925

0.930

0.935

0.940

0.945

0 20 40 60 80 100

θ

Ff

β=30

Figure 5.24 Stress ratio varies with different arched outlet;(selfweight+surcharge, CAX8, wall friction: 0.2)

Wall pressure(selfweight+surcharge, β=30 degree; (1)--theoretical, (2)--numerical)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 20 40 60 80 100 120


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30(1)θ=30(2)θ=45(1)θ=45(2)θ=60(1)θ=60(2)θ=75(1)θ=75(2)θ=90(1)θ=90(2)

Wall pressure(selfweight+surcharge, β=30 degree; (1)--theoretical, (2)--numerical)

0.02

0.04

0.06

0.08

0.10

0.12

0 20 40 60 80 100 120


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

θ=30(1)θ=30(2)θ=45(1)θ=45(2)θ=60(1)θ=60(2)θ=75(1)θ=75(2)θ=90(1)θ=90(2)

70

Ff vs. β for different arch angle(selfweight only, CAX8, WF: 0.2)

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

10 15 20 25 30 35 40 45 50 55 60Hopper half angle, β(degree)

Ff

θ=30θ=45θ=60θ=75θ=90

Figure 5.25 Stress values varies with hopper half angle,β and arch angle θ; (wall friction: 0.2)

Ff vs. β for different arch angle(selfweight+surcharge, CAX8, WF: 0.2)

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

10 15 20 25 30 35 40 45 50 55 60Hopper half angle, β(degree)

Ff

θ=30θ=45θ=60θ=75θ=90

Figure 5.26 Stress values varies with hopper half angle, β and arch angle, θ; (wall friction: 0.2)

71

(a)

(b)

Figure 5.27. (a)Hopper with different arch configuration; (b) illustration of pq stress

B

A 1

p

q

Possible failure zone

Fitted line

Critical State Line

θ

β

72

A vs. β

0.6

0.8

1.0

1.2

1.4

1.6

1.8

15 25 35 45 55 65

β, (degree)

A

θ=30θ=45θ=60θ=75θ=90

(a)

B vs. β

-2

0

2

4

6

8

10

12

14

15 20 25 30 35 40 45 50 55 60 65

β, (degree)

B, (

kPa)

θ=30θ=45θ=60θ=75θ=90

(b)

Figure 5.28. p—q stress definitions along outlet with different geometries(self weight); (a) Interception; (b) Gradient

73

A vs. beta

0.6

0.8

1

1.2

1.4

1.6

1.8

15 25 35 45 55 65

β, (degree)

A

θ=30θ=45θ=60θ=75θ=90

(a)

B vs. beta

-5

0

5

10

15

20

25

15 20 25 30 35 40 45 50 55 60 65

β, (degree)

B, (

kPa)

θ=30θ=45θ=60θ=75θ=90

(b)

Figure 5.29. p—q stress definitions along outlet with different geometries(selfweight and surcharge); (a) Interception; (b) Gradient

74

(a)

(b)

Figure 6.1 Unsymmetrical hopper; (a) Geometry; (b) FEM model

h

w

H

βLβR

b b

75

Wall pressure at Left Wall(Hopper half angle, β=30 degree, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70Left wall pressure, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

µ= 0.0µ= 0.1µ= 0.2µ= 0.3µ= 0.4

(a)

Wall Pressure at Right Wall(Hopper half angle, β=30 degree, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35 40 45 50

Right wall pressure, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

µ= 0.0µ= 0.1µ= 0.2µ= 0.3µ= 0.4

(b)

Figure 6.2 Wall pressures vs. wall friction. (a) Left wall, (b) Right wall

76

Wall Friction effect(Hopper half angle, β=30 degree, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70

Left Wall pressure, (kPa)

Nor

mal

ized

hei

ght a

bove

ver

tex,

y/H

Analytical(0.0)Numerical(0.0)Analytical(0.2)Numerical(0.2)Analytical(0.4)Numerical(0.4)

(a)

Wall Friction effect(Hopper half angle, b=30 degree, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35 40 45 50


Nor

mal

ized

hei

ght a

bove

ver

tex,

y/H

Analytical(0.0)Numerical(0.0)Analytical(0.2)Numerical(0.2)Analytical(0.4)Numerical(0.4)

(b)

Figure 6.3 Wall pressure comparisons. (a) Left wall; (b) Right wall Hopper half angle, β=30 degree, CPE8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Friction Coefficient

Wal

l str

ess r

atio

, Ff

Left WallRight Wall

Figure 6.4 Stress ratio comparisons

77

p', q at arch boundary(Hopper half angle, β=30 degree, CPE8)

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120

p', (kPa)

q, (k

Pa)

µ= 0.0µ= 0.1µ= 0.2µ= 0.3µ= 0.4CSL(M=1.68)

Figure 6.5 p-q at outlet for different wall friction conditions

Wall Pressure comparisons at Right Wall(µ=0.2, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

β=30/15

β=30/30

b=30/45

(a)

78

Wall pressure comparisons at Left Wall(CPE8, µ=0.2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

β=30/15

β=30/30

β=30/45

(b)

Figure 6.6 Comparisons for wall pressures with unsymmetric hopper geometry. (a)

Left wall; (b) Right wall. (Hopper half angle at left kept 30 degree, right half angle varies from 15 to 45 with 15 degree increment)

p', q comparisons at arch boundary(µ=0.2, CPE8)

0

15

30

45

60

75

0 10 20 30 40

p', (kPa)

q, (k

Pa)

CSL(M=1.68)β=30/15β=30/30β=30/45

Figure 6.7 Stress distribution at outlet with different hopper geometry

79

µ=0.2, CPE8, selfweight

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

30/15 30/30 30/45

Hopper half angle in degrees, (left/right)

Wal

l str

ess r

atio

, Ff

Left WallRight Wall

Figure 6.8 stress ratio comparisons

Wall Pressure at Right Wall(β=30 degree, µ=0.2, CPE8)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

w=0.7

w=1.0

w=1.3

(a)

80

Wall pressure at Left Wall(β=30 degree, µ=0.2, CPE8)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

w=0.7

w=1.0

w=1.3

(b)

Figure 6.9 Wall pressure distributions affected by different paddle position. (a) Left

wall; (b) Right wall

p', q at arch boundary(β=30 degree, µ=0.2, CPE8)

10

20

30

40

50

60

70

80

5 10 15 20 25 30 35 40 45

p', (kPa)

q, (k

Pa)

w=0.7w=1.0w=1.3CSL

Figure 6.10 p-q stress at outlet

81

Hopper angle, β=30 degree, µ=0.2, CPE8

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

0.5 0.7 0.9 1.1 1.3 1.5

Paddle Position, w, (m)

Wal

l str

ess r

atio

, Ff

Left WallRight Wall

Figure 6.11 Stress ratio at outlet

Wall pressure at Left Wall(Hopper half angle, β=30 degree, µ=0.2, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

h=0.4

h=0.7

h=1.0

(a)

82

Wall Pressure at Right Wall(Hopper half angle, β=30 degree, µ=0.2, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35 40 45 50


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

h=0.4

h=0.7

h=1.0

(b)

Figure 6.12 Wall pressure distributions affected by outlet height. (a) Left wall; (b)

Right wall

p', q at arch boundary(Hopper half angle, β=30 degree, µ=0.2, CPE8)

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40

p', (kPa)

q, (k

Pa)

h=0.4h=0.7h=1.0CSL

Figure 6.13 p-q stress at hopper outlet affected by height of outlet

83

Hopper half angle, β=30 degree, µ=0.2, CPE8

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

0.89

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Outlet height, h, (m)

Wal

l str

ess r

atio

, Ff

Left WallRight Wall

Figure 6.14 stress ratio variations at outlet for different hopper height

Wall pressure at Left Wall(Hopper half angle, β=30 degree, µ=0.2, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

H=25Η=50H=75

(a)

84

Wall Pressure at Right Wall(Hopper half angle, β=30 degree, µ=0.2, CPE8)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30 35 40 45 50


Nor

mal

ized

hei

ght a

bove

ver

tex,

h/H

H=25Η=50H=75

(b)

Figure 6.15 Wall pressure distributions for different filling height. (a) Left wall; (b)

Right wall

p', q at arch boundary(Hopper half angle, β=30 degree, µ=0.2, CPE8)

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35 40 45 50

p', (kPa)

q, (k

Pa)

H=25Η=50H=75CSL

Figure 6.16 p-q distribution around arched outlet

85

Hopper half angle, β=30 degree, µ=0.2, CPE8

0.80

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

25 50 75

Hopper height, H, (m)

Wal

l str

ess r

atio

, Ff

Left WallRight Wall

Figure 6.17 Stress ratio affected by different filling height

β β

φ

φ

Figure 7.1 Geometry of the coal bunker and possible arching boundaries at outlet

86

Figure 7.2 Finite Element Mesh

25 2

87

Figure 7.3 Flowing and arching propensities(Wall friction: 0.2)

(b)

(c)

(a)

88

Figure 7.4 Wall pressures affected by different wall friction. (a) Normal flowing; (b) Arching

Wall pressure at Left wall(Normal Flow)

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80


Hei

ght,

(m)

WF=0.1WF=0.2WF=0.3

Wall pressure at Left wall(Arching)

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80


Hei

ght,

(m)

WF=0.1WF=0.2WF=0.3

(a)

(b)

89

Figure 7.5 p-q stress along the arched boundaries. (a) Normal flow; (b) Arching

pq at left Arch(Normal flowing)

0

5

10

15

20

25

0 5 10 15 20 25 30 35

p, (kPa)

q, (k

Pa)

WF: 0.1WF: 0.2WF: 0.3CSL

pq at left Arch(Arching formed)

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50

p, (kPa)

q, (k

Pa)

WF: 0.1WF: 0.2WF: 0.3CSL

(a)

(b)

90

Figure 7.6 stress ratio M=q/p distribution for the whole field of coal. (a) Normal flow; (b) Arching(Wall friction: 0.2)

stress history comparisons between flowing and arching

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10 12

p, (kPa)

q, (k

Pa)

WF: 0.1( 2-toel)WF: 0.1(25-coal)WF: 0.2( 2-toel)WF: 0.2(25-coal)WF: 0.3( 2-toel)WF: 0.3(25-coal)CSL

Figure 7.7 stress history from normal flowing to arching.

(a)

(b)

91

Figure 7.8 Wall pressures for three paddle positions. (a) Normal flowing; (b) Arching(Wall friction: 0.1)

Normal Flowing(Wall friction: 0.1)

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80


Hei

ght,

(m)

PL=0.152PL=0.187PL=0.222

Arching formed(Wall friction: 0.1)

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80


Hei

ght,

(m)

PL=0.152PL=0.187PL=0.222

(a)

(b)

92

Figure 7.9 p-q at arching boundaries. (a) Normal flowing; (b) Arching(Wall friction: 0.1)

Paddle position effect(pq at left arch, normal flowing)

0

5

10

15

20

25

0 5 10 15 20 25 30 35p, (kPa)

q, (k

Pa)

PL=0.152PL=0.187PL=0.222CSL

Paddle position effect(pq at left arch, Arching formed)

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45

p, (kPa)

q, (k

Pa)

PL=0.152PL=0.187PL=0.222CSL

(a)

(b)

93

Figure 7.10 Distribution of stress ratio, M=q/p. (a) Normal flowing; (b) Arching(Wall friction: 0.1)

(a)

(b)

94

Figure 7.11 Wall pressure during progressive filling. (a) Normal flowing; (b) Arching formed(Wall friction: 0.1)

(a)

(b)

Wall pressure at Left wall(Normal Flow)

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60


Hei

ght a

bove

arc

h, (m

)First fillingSecond fillingThird filling

Wall pressure at Left wall(Arching)

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60


Hei

ght a

bove

arc

h, (m

)

First fillingSecond fillingThird filling

95

Figure 7.12 p-q at arched outlet. (a) Normal flowing; (b) Arching formed(wall friction: 0.1)

(a)

(b)

pq at left arch(Normal flowing,no symbol end correspond to path start)

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

p, (kPa)

q, (k

Pa)

First fillingSecond fillingThird fillingCSL

pq at left arch(Arching formed,no symbol end correspond to path start)

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30 35 40

p, (kPa)

q, (k

Pa)

First fillingSecond fillingThird fillingCSL

96

Figure 7.13 stress ratio, q/p, upon progressive filling with the arching formed(wall friction: 0.1)

97

Figure 7.14 STRESS RATIO, Q/P, UPON PROGRESSIVE FILLING WITH NORMAL FLOWING(WALL FRICTION: 0.1)

98

Figure A.1 Cam-clay yield and critical state surfaces in principal stress space

Figure A.2 Clay yield surfaces in the p-t plane

Wet side Dry side

q=Mp

99

Figure A.3 Clay yield surface sections in the Π-plane

Figure A.4 piecewise linear hardening/softening curve

100

Figure A.5 assumed soil response in pure compression(exponential hardening/softening case)

101

Figure B.1 (a) coal within bunker with arch formation; (b)model geometry

(a)

(b)

102

Figure B.2 Mesh and elements for the computation model

Figure B.3 Hardening failure; (a) p-q; (b) e-lnp; (c) q- εa; (d) εv -εa

Element 92(CAX4r, const. mean effective stress, force loading)

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100 110p, (kPa)

q, (k

Pa)

step 1(self weight) step 2(ver. 100kPa,hor. 50kPa)step 3(ver. 80kPa, hor. 40kPa) step 4(ver. 160kPa, hor. 0kPa)Critical State Line Initial yield surfaceReference failure surface Cam-Clay(final)

(a) (b)

(c) (d)


0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

-8 -6 -4 -2 0 2 4 6Logarithmic pressure, ln(p)

Void

ratio

, e

Normal Consolidate LineCritical State Linestep 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 80kPa, hor. 40kPa)step 4(ver. 160kPa, hor. 0 kPa)


0

20

40

60

80

100

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28Logarithmic axial strain, (εa)

q, (k

Pa)

step 1(self weight)

step 2(ver.100kPa, hor. 50kPa)

step 3(ver. 80kPa, hor. 40kPa)


Element 92(CAX4r, const mean effective stress, force loading)

0.00

0.04

0.08

0.12

0.16

0.20

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28

Logarithmic axial strain, (εa)

volu

met

ric s

trai

n, (

ε v)

step 1(self weight)




103

Figure B.4 Harding failure; (a)equivalent shear stress; (b) Mean Effective Stress; (c)

axial strain; (d) shear strain

Figure B.5 Softening failure; (a) p-q; (b) e-lnp; (c) q- εa; (d) εv -εa

(a) (b)

(c) (d)

(a) (b)

(c) (d)


0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90p, (kPa)

q, (k

Pa)

step 1(self weight) step 2(ver. 100kPa,hor. 50kPa)step 3(ver. 20kPa, hor. 10kPa) step 4(ver. 40kPa, hor. 0kPa)Critical State Line Initial yield surfaceReference failure surface Cam-Clay(final)

Element 92(CAX4r, const. mean effective stress , force loading)

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05


Void

ratio

, e

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 20kPa, hor. 10kPa)step 4(ver. 40kPa, hor. 0 kPa)Critical State LineNormal Consolidate Line


0

10

20

30

40

50

60

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40


q, (k

Pa)

step 1(self weight)





0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22


volu

met

ric s

trai

n, (

ε v)

step 1(self weight)




104

Figure B.6 Softening failure; (a)equivalent shear stress; (b) Mean Effective Stress; (c) axial strain; (d) shear strain

Figure B.7 Loading method effect; (a) p-q; (b) e-lnp; (c) q- εa; (d) εv -εa

(a) (b)

(c) (d)

(a) (b)

(c) (d)

Element 92(CAX4r, constant radial stress)

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90p, (kPa)

q, (k

Pa)

step 1(self weight) step 2(ver. 100kPa,hor. 50kPa)step 3(ver. 20kPa, hor. 10kPa) Critical State LineInitial yield surface Reference failure surfacestep 4(displacement loading) step 4(force loading)Cam-Clay(final)


0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05


Void

ratio

, e

Normal Consolidation LineCritical State Linestep 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 20kPa, hor. 10kPa)step 4(displacement loading)step 4(force loading)


0

10

20

30

40

50

60

70

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40


q, (k

Pa)

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 20kPa, hor. 10kPa)step 4(displacement loading)step 4(force loading)


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.44 0.48


volu

met

ric s

trai

n, (

ε v)

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 20kPa, hor. 10kPa)step 4(displacement loading)step 4(force loading)

105

Figure B.8 Different Finite Element Method; (a) p-q; (b) e-lnp; (c) q- εa; (d) εv-εa

Figure B.9 Effect of different loading path; (a) p-q; (b) e-lnp; (c) q- εa; (d) εv-εa

(c) (d)

(a) (b)

Node 50001(Element comparison, force loading)

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100 110p, (kPa)

q, (k

Pa)

step 1(self weight) step 2(ver. 100kPa,hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa) Critical State LineInitial yield surface Reference failure surfacecax4r(ver.120kPa,hor.20kPa) cax4(ver.120kPa, hor.20kPa)cax8r(ver.120kPa,hor.20kPa) cax8(ver.120kPa,hor.20kPa)Cam-Clay(final failure)


0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

-10 -8 -6 -4 -2 0 2 4 6Logarithmic pressure, ln(p)

Void

ratio

, e Normal Consolidate LineCritical State Linestep 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa)cax4r(ver.120kPa, hor. 20kPa)cax4(ver.120kPa,hor.20kPa)cax8r(ver.120kPa,hor.20kPa)cax8(ver.120kPa,hor.20kPa)


0

20

40

60

80

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4Logarithmic axial strain, (εa)

q, (k

Pa)

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa)cax4r(ver.120kPa,hor. 20kPa)cax4(ver.120kPa,hor.20kPa)cax8r(ver.120kPa,hor.20kPa)cax8(ver.120kPa,hor.20kPa)


0.00

0.04

0.08

0.12

0.16

0.20

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32


volu

met

ric s

trai

n, (

ε v)

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa)cax4r(ver.120kPa, hor.20kPa)cax4(ver.120kPa,hor.20kPa)cax8r(ver.120kPa, hor.20kPa)cax8(ver.120kPa,hor.20kPa)

Element 92(CAX4r, force loading)

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100 110p, (kPa)

q, (k

Pa)

step 1(self weight) step 2(ver. 100kPa,hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa) Critical State LineReference failure surface path1(MES constant)path2(radial stress const) path3(q constant)path4(active failure) path5(passive failure)path6(unloding top) failure locus for path1failure locus for path2 failure locus for path4

(a) (b)

(c) (d)


0.50

0.60

0.70

0.80

0.90

1.00

1.10

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10Logarithmic pressure, ln(p)

Void

ratio

, e

Normal Consolidate LineCritical State Linestep 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa)path1(MES constant)path2(radial stress const)path3(q constant)path4(active failure)path5(passive failure)path6(unloding top)


0

20

40

60

80

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28Logarithmic axial strain, (εa)

q, (k

Pa)

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa)path1(MES constant)path2(radial stress const)path3(q constant)path4(active failure)path5(passive failure)path6(unloding top)


0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28


volu

met

ric s

trai

n, (

ε v)

step 1(self weight)step 2(ver.100kPa, hor. 50kPa)step 3(ver. 40kPa, hor. 20kPa)path1(MES constant)path2(radial stress const)path3(q constant)path4(active failure)path5(passive failure)path6(unloading top)

106

Figure B.10 Loading path and general failure mechanism

Contractile compression Ductile hardening

Dilatant compression Rupture softening

Contractile extension Ductile hardening

Dilatant extension Rupture softening

u- or p’+ u+ or p’-

σ’1+

σ’3+

σ’3-

σ’1-

q

p’

q+

q-

Documents

BCURA PROJECT B54 on Arching propensity in coal … Projects/b54_final_report.pdf · to conduct a survey of existing power station coal bunker geometries; 2. to develop computational