Preprint SME Annual Meeting Feb. 28-Mar. 03, 2010, Phoenix, AZ ANALYTICAL DESIGN PROCEDURE USING THE WILSON EQUATION

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1 Feb. 8-Ma. 0, 00, Phoenix, AZ Pepint 0-05 ANALYTICAL DESIGN PROCEDURE USING TE WILSON EQUATION V. A. Soazzo, John T. Boyd Co., Rihland, WA INTRODUCTION The analytial pilla design poedue employing the Wilson equation is based on the site-speii mateial stength o the loo, pilla, and oo ok. Stable pillas so designed will sometimes be lage than those designed using empiial methods thus poiding moe stability o at othe times smalle than empiially designed pillas thus onseing natual esoues. Pesented hee is a detailed poedue o the analytial design o sot ok pillas using the Wilson Equation. An example o this poedue is pesented that uses data om a mine in the Owl Seam in Southen Appalahia. These Owl Seam pillas ae now in use and ae peoming as expeted. Also addessed is an assessment o the D. Soazzo s analytially designed pillas based on pilla peomane data gatheed by D. Mak in the Southen Appalahian Coal Field. PILLAR DESIGN PROCEDURE Beause o the eliane o analytial designs on auate mehanial popeties and the potential aiation o ok and oal mateial popeties that an be intodued at seeal stages o dilling, sample shipping, and testing, the highest standads must be adheed to duing these steps. The design poedue inludes the ollowing steps:. Dilling, logging, seleting, peseation, and shipping o oe samples.. Laboatoy mehanial testing o oo, pilla, and loo ok.. Analysis o test esults and deelopment o ailue enelopes. 4. Deelopment o a site speii pilla stength equation based on the Wilson Equation. 5. Load assessment and pilla design. The impotane o auate detemination o ok and oal mehanial popeties in analytial design annot be emphasized enough. To that end, dilling, sampling, and testing will be oeed hee in detail. GEOTECNICAL SAMPLING Geotehnial dilling, sampling, and testing must adhee to the Ameian Soiety o Testing and Mateials (ASTM) and Intenational Soiety o Rok Mehanis (ISRM) standads. These poedues help maintain the natual moistue and minealogy o the samples, edue beakage duing dilling, sampling, and shipping, and minimize onusion in the laboatoy. The sample diamete must be NX o Q size oe eoeed by wie-line split bael. The objetie is to minimize oe beakage duing dilling and oe extation. The oe bael ontaining the oe should not be emoed om the hole until the geologist is pesent and eady to log the oe. Logging o the oe should take plae immediately and quikly and doumented with photogaphs. The suae o the samples must not be allowed to dy, thus onstant wetting o the samples is equied. A qualiied geologist should geologially and geotehnially log the oe and selet samples o testing. Besides ok type and geologi desiption, the log must inlude suh data as hadness, joint oughness oeiient (JRC), ok quality designation (RQD), peent eoey, gouge and illing type and hadness, degee o weatheing, disontinuity types and spaing, all to the ISRM standads. The mehanial popeties o a ok type an ay aoss a mine blok and etially though the sequene beause o the dieent sediment soues and depositional enionments that existed o ok aboe and below the seam. As a esult, sandstone aboe and below the seam will not hae the same mehanial popeties. This aiation o mehanial popeties om aboe and below the seam is tue o all othe ok types. Thus, ok samples o testing should be seleted om holes (a minimum o thee holes) loated thoughout the esee and thoughout the dill hole. Geneally, 5 oe samples, thee times longe than the oe diamete, should be obtained o eah ok type ound in the oo, pilla, and loo. Coe samples should be obtained om the oal and patings, as well as om 5 metes (50 eet) aboe the top o oal to 6 metes (0 eet) below the base o the oal. In thik seams, samples should be obtained om eah oal ply o benh. The samples should be peseed and shipped to a laboatoy in aodane with the highest standads. Samples om the oe seleted o testing should be immediately wapped in plasti o othewise sealed to peent moistue loss and to stengthen the oe o shipping. Plasti oe sleees an be used but these sleees hae the disadantage o not stengthening the sample. The boehole numbe, sample numbe, and depths (to and om) maked lealy on the plasti. The sample loation should be noted on the geologi/geotehnial log and also logged into a sample inentoy sheet whih inludes the boehole numbe, tube numbe, sample numbe, depths, and lithologi desiption. The sealed oe should be plaed in a PVC o simila pipe ut to. metes (4 eet) lengths o less (0.85 metes o.8 eet i 55 gallon dum is used o ating) with pakaging mateial plaed aound the sample to ushion and stabilize the sample within the pipe. Fille mateial between samples and at the ends o the pipe is needed to peent the samples om shiting and to allow the laboatoy to saw o the end aps without damaging the sample. PVC end aps ae then glued to both sides o the pipe. The boehole numbe, tube numbe, sample numbes, and depths o all the samples ontained in the pipe should be maked lealy on the pipe. Fou to six o these pipes ontaining samples should be plaed in a ate o shipping. This ate may need to be onstuted i ommeial ates annot be obtained. A ommon substitute is a 55 gallon steel dum. One a ate is ull and the pipe seued om shiting, it is to be shipped immediately to the lab without waiting o additional ates to be illed. Stoage o the pipes and ates that ontain oe should be out o the weathe in a ool tempeatue ontolled enionment. TESTING PROGRAM A typial minimum testing pogam is illustated by the ollowing: Numbe o Tests Uniaxial Intat and Residual Tiaxial Compessie @ Type & Density Maximum/4 Maximum/ Maximum Roo Rok Type Copyight 00 by SME

2 Feb. 8-Ma. 0, 00, Phoenix, AZ Rok Type Rok Type Pilla Coal Ply Pating Type Coal Ply Coal Ply Floo Rok Type Rok Type Rok Type The onining stess,, inally seleted o testing will depend on the imum depth o the esee o mine blok. Typially the imum onining stess seleted is.5 to 4 times the oebuden load to aount o the onining stess to be expeiened at the pilla oe. Test pogams nomally hange one dilling begins and ontinues hanging thoughout the testing phase. These hanges ae due to dieent ok types being enounteed than antiipated, inability to eoe oet oe lengths, satte in the test data, et. Emphasis should be plaed on ompleting the tiaxial tests ist, as uniaxial tests ae not needed to detemine the ailue enelope. Also, esults o the uniaxial ompessie stength test on ok ae ommonly satteed. The satte, in lage pat, is due to the laws within the ok sample and end eets, espeially o ube samples. Fotunately, the aeage stength o tiaxial onined ok speimens is muh geate than the unonined speimen stength, whih an be moe impotant to pilla stength. and tiaxial tests ae moe epeatable and auate beause end eets and laws within the sample hae less o an eet on the inal esults. Beause o the dieene in mateial popeties between pilla, oo, and loo mateials, the oal-loo and oal-oo ontats shea one mining ous and stess hanges take plae aoss these ontats. Fo these easons, esidual tiaxial stength is used in the Wilson Equation. Fotunately, esidual tiaxial stength test esults ae highly epeatable ompaed to intat tiaxial and, o ouse, uniaxial test esults, as illustated in Figue. DEVELOPMENT OF FAILURE ENVELOPE Thee ae only a ew ailue enelopes that math ok and oal behaio and these ae well deined in the liteatue. Rok and oal at the oninements needed o pilla design, on ae oasions, will math the Moh-Coulomb ailue enelope 4. τ C + n tanφ tan β + Whee τ is the shea stength at a nomal stess o n, C is ohesion and φ is the intenal ition angle. owee, at high oninement, oal and most ok stengths ae not linea. Almost all ok, and sometimes oal, math the oek and Bown ailue enelopes: + m + s n τ A T B ompession equation shea equation whee is the uniaxial ompessie stength, t is the tensile stength, and m, s, A, T, and B ae ue-shaping paametes. Coal aely mathes the oek and Bown ailue enelope o easons disussed by Baon 5. owee, seeal ailue enelopes hae been deeloped o the ompessie stength o oal, inluding; t a a B + A A K + R Cate ailue enelope Ramamuthy ailue enelope Bieniawski ailue enelope all as pesented by Bieniawski 6, and whee R, B, a, K, and A ae ue-shaping paametes. BOYD's expeiene has shown that the Cate ailue enelope moe equently mathes the oal test data. Cue itting test data to ailue enelopes is time onsuming but the ollowing should assist in this matte: Cue itting o the esidual ailue enelope should math US Amy Cop o Enginees standads 7 that / o the test esults lay aboe the ailue enelope. The intat ailue enelope should be best it as the intat enelope will geneate paametes needed to deelop the esidual ailue enelope. Fo ok, ist ue it the intat esults to the oek and Bown ompession equation as the deeloped uniaxial ompessie stength,, will be needed o subsequent ailue enelopes. Stat by using the suggested ue itting paametes o the ok type being analyzed as pesented by oek and Bay. Figue. Intat and Residual Conined Compessie Stength o Owl Seam Roo Shale Test Results Showing Satte. The seletion o a laboatoy is paamount as ew laboatoies ae apable o both high oninement and the ability to omplete a esidual test. In the United States, the laboatoy should be Amy Cops o Enginees etiied. Use the deeloped intat uniaxial ompessie stength om the intat ue it to deelop the esidual oek and Bown ompession equation. The ue itting paametes o m and s deeloped duing this step ae used to alulated ue itting paamete T. ( m ( m 4 ) ) 0.5 T s Copyight 00 by SME

3 Using the esidual T and the deeloped uniaxial ompessie stength,, deelop the esidual oek and Bown shea stength equation. One all esidual shea ues ae deeloped o all the ok and oal, eah shea ue in the loo is ompaed to the othes and to that o the oal so that the weakest ok in esidual shea is seleted to epesent the loo shea stength. A ompaison is also made with the oo ok shea stengths. Coal esidual stength test esults ae used to deelop the esidual ompession ailue enelope equation. These ue itting paametes ae enteed in the Wilson Pilla Equation and pilla stength is alulated. The esulting equation will need to be integated. owee, suh integations ae time onsuming and, depending on the ailue enelope equations used, may not be possible to integate to a losed om. The autho uses MathCAD sotwae (MathSot, In). o this integation. Integating this equation podues a stess poile whih ous within the pilla om the pilla edge to its oe. Fo this equation to geneate the load aying apaity o the pilla, the Wilson equation will need to be double integated. Double integations o this equation hae poen to be poblemati, and until sotwae is deeloped o this type o double integation, the autho imbeds MathCAD into an Exel (Miosot, In.) speadsheet poduing a matix o the aying stess whih is used to deeloped pilla stengths. TE WILSON PILLAR EQUATION The Wilson Equation is ound as equation () in Wilson and Ashwin 8 : ( + d ) ml ml d ml whee l and dy ae the unit length and width o the element, m is extation height, and is the hoizontal stess eated by Poisson s and othe eets within the pilla, as illustated in Figue. SME Annual Meeting Feb. 8-Ma. 0, 00, Phoenix, AZ ( τ + τ ) ( τ + τ ) dy m Setting the integation: ldy d ml d ( τ + τ ) dy d m d τ τ y m + d whee y is the distane om the ib o the pilla and is the etial stess in the pilla at. Note that the pilla stength is a deiatie. This equation geneates a stess poile om pilla edge, whih has the stength o, the esidual ompessie stength o oal to the oe o the pilla whih suppots a etial stess o. The etial stess,, aies as the loation, y, aies, see Figue. Figue. Stess Poile in a Sot Pilla. The deiatie o the Cate and Moh-Coulomb equations ae: d d t R tan β R R d o Cate d o Moh-Coulomb Assuming the oal s esidual ailue enelope mathes the Cate equation, that the oo ok esidual ailue enelope mathes the oek and Bown equation, and that the loo ok esidual ailue enelope mathes the Moh-Coulomb equation, the esulting Wilson equation om would be: Figue. Fee Body Diagam o a Pilla Slie, Ate Wilson and Ashwin. Oe distane dy the hoizontal stess inease is deined by: dy d ml. This hoizontal stess is esisted by the shea stength o the top and bottom o the pilla, τ and τ, whee and ae subsipts epesenting the mateial popety at the oo and loo o the pilla (ompae to equation (4) in Wilson and Ashwin). Thus, y m t R B A T + C R R + tan φ d On oasion, and moe ommonly in the loo, the esidual ailue enelopes o one ok type will be weake than the othe ok types at low nomal stess but will swith at highe nomal stess. I these two ok types ae thin layes in the loo, the pilla may spead in one ok type then swith to anothe. This an be handled by the Wilson equation. The nomal stess at the intesetion o the two ailue Copyight 00 by SME

4 Feb. 8-Ma. 0, 00, Phoenix, AZ enelopes an be taken as ni. I this ous in the loo o mateial types and, then the Wilson equation takes the om o: d τ ni y m + d + τ ni d τ + τ d O ouse, this an ou in both the loo and oo o any numbe o mateials,,, whih will equie adding omponents to the Wilson equation, ni, n(i+), n(i+),. Anothe example o this equation s lexibility is its ability to handle a squeezing loo, within eason, by pediting the edued pilla stength using the ollowing modiiation: y m d τ τ d SAFETY FACTOR, REVIEW, PILLAR LOADING, AND PILLAR FOUNDATION The autho eommends a saety ato o o the Wilson Equation i testing and loading is undestood, but this alue an be as high as.5 to o bump onditions and highe i loading onditions o mateial stengths ae not well known. Thoughout the design poess, the designe should eiew his esults and detemine, most impotantly, i the esulting equation epesents the obseed behaio o the pilla. Fo example, the eiewe would note i moement is ouing in the oal o in the ok o the loo; o i spalling is ouing and whethe this is inluded in the design, et. Impotant to the design o a pilla is knowing the load on the pilla. Pilla loading is beyond the sope o this pape but most o the time, the autho employs equal aea loading o uses LAMODEL 9 to estimate the load. Sine a geat deal o load pedited by equal aea loading would atually be tanseed to the oal blok o baies, using equal aea loading oes edundany. In high extation settings, a tanse load is applied, eithe as eommended by Mak 0 o by Able. Fo designs that ae moe ompliated, LAMODEL is used to alulate loads and o patiulaly ompliated onditions ANSYS is employed. The Wilson equation only addesses pilla stength; a pilla oundation stength analysis needs to be ompleted beoe inal pilla dimensions ae eommended. AN EXAMPLE The Owl Seam (aka Taggat, Nosben, Uppe Elkhon, 4 Inh, and Uppe Ceda Goe) is extensiely mined in Kentuky and Viginia. The test esults data base o this seam is to a point whee tends an be deined aoss the egion and bloks o simila behaing pillas hae been delineated and sepaate pilla eommendations o eah blok hae been ompleted. One suh blok is an aea o stong loo. The ollowing is the pilla design poedue used o this blok. Ate dilling and testing, the stength data was analyzed to deelop mateial popeties o all the ok types in the immediate iinity o the Owl Seam inluding sandstone, sandy shale, and shale in the oo and loo and the oal and binde (shale) in the seam. In the oo and loo, the shales wee the weakest mateials and the oal is the weakest in the seam. Figues 4 and 5 show the ue itting o the oek-bown Failue iteion o the esidual test esults o the shale in the oo and loo o the Owl Seam. Figue 4. Shea Stength Plot o Owl Seam Roo Shale Residual Test Results and oek-bown Failue Enelope. Figue 5. Shea Stength Plot o Owl Sea Floo Shale Residual Test Results and oek-bown Failue Enelope. Similaly, the ue itting o the Cate and oek-bown Failue iteion o the esidual test esults the Owl Seam ae shown in Figue 6 and Figue 7. Figue 7 shows that the shea stength o the Owl Seam oal is a less than the shea stengths o the suounding shales, theeoe ailue and moement at the oo-pilla and pilla-loo ontat will ou in the oal. Beause o this, only oal paametes will be used in the Wilson pilla equation. Figue 8 is a seen shot o a MathCAD woksheet used to integate the Wilson equation. Note that oo () and loo () paametes use the oek-bown alues o the Owl Seam. Figue 8 shows the plot o the stess poile o both the Wilson Pilla Equation using Owl Seam data (ed ontinuous line) and the Mak-Bieniawski Equation (blue dashed line). This is one o the ew oasions whee thee is a lose math between the esults o these two equations. Figue 8 shows the stength o a squae pilla 45 metes on a side with 90 ossut angles to be 6,4 MN. Employing the eommended saety ato o esults in a pilla stength o 68,06 MN. The oesponding Bieniawski alulated stength o the same pilla is 09,7 MN and using the eommended stability ato o.5 esults in a pilla stength o 7,54 MN. In this atypial ase, the Bieniawski equation esults in a smalle pilla design than the Wilson equation. 4 Copyight 00 by SME

5 Feb. 8-Ma. 0, 00, Phoenix, AZ Figue 6. Residual Conined Compessie Stength o Owl Seam Test Results with Cate Failue Enelope. Figue 8. Seen Shot o Mathad Analysis o Owl Seam Pilla Stength Using Wilson Pilla Equation, note plot o Wilson Stess Poile esus Mak Bieniawski Stess Poile. Figue 7. Shea Stength Plot o Owl Sea Residual Test Results and oek-bown Failue Enelopes o Coal and Shale. Residual Failue Citeion Cue-Shapping Paametes Rok Type MPa t Mpa A T B R Roo Shale Coal, Owl Seam -oek-bown Cate Floo Shale MathCAD soles o the etial stess,, at seeal point, y, om pilla edge to pilla ente, poduing an aay o y(k) eses s(k). Unotunately, MathCAD o othe pogams annot sole the double integation need to alulate the pilla stength om the Wilson stess poile so the aay geneated by MathCAD o the Wilson equation is expoted to an Exel woksheet as shown in Figue 9. This speadsheet an alulate the pilla stength o any width, length, o oss-ut angle. The inby and outby ossut angle an be dieent. DOES IT WORK D. Chistophe Mak o NIOS gatheed data on 6 pilla ases in Southen Appalahian Coal Fields. These ases wee taken om dieent mines. D. Soazzo, using the analytial design poedue pesented aboe, has designed pillas at 0 o these mines. D. Mak lassiied these pilla ases as satisatoy o ailed. O the 0 ases not designed by D. Soazzo, only 55% o the pillas wee lassiied as satisatoy ( ases). y(k), metes s(k), MPa Stength pe slie, MN (length) Stength, MPa (length) Stength pe slie, MN (width) Stength, MPa (width) Stength pe slie, MN (length) Stength, MPa (length) Stength pe slie, MN (width) Stength, MPa (width) Total Stength, MN Width to slie, metes Angle o ist ossut 90 seond 90 Pilla Length, m 45 Pilla Width, m 45 Figue 9. Seen Shot o Speadsheet Used to Calulate Pilla Stength Based on the Stess Poile. O the 4 ases that wee designed by D. Soazzo, suessul ases (9) made up 9%. The ollowing whee NIOS s eason o lassiying the 4 emaining ases as ailed: bumped squeezed beause the mine let pilla duing eteat. The two bump ases oued in the same mine. The onditions in whih these and othe bumps oued at this mine wee analyzed and new suessul pilla designs wee undetaken to be used in aeas whee these onditions eoued. Fo the squeeze ase, this pilla was designed using the poedue o squeezing loo as disussed aboe and the pilla is stable. Fo the let pilla ase, these pillas wee let in plae, at BOYD's equest, to absob the stess o pillas let in a mine aboe. In this eassessment o pilla suess, o the 4 ases that wee designed by D. Soazzo, suessul ases (4) made up 95%. 5 Copyight 00 by SME

6 Feb. 8-Ma. 0, 00, Phoenix, AZ The onlusion is that the analytial design poedue is highly suessul appoah to designing pillas and when the peomane poes to be unsatisatoy the onditions an be ealuated and the Wilson pilla equation adjusted to detemine a stable edesigned pilla. Intenational Soiety o Rok Mehanis, , Commission on Standadization o Laboatoy and Field Testing, Suggested Methods, Pegamon Pess Ltd, Geat Bitain. Able J., 988, Sot Rok Pillas, in Geotehnial and Geologial Engineeing, Spinge Nethelands, pp Ameian Soiety o Testing and Mateials, 966, Testing Tehniques o Rok Mehanis, STP 40, Philadelphia, PA, p oek, Eet, and Bay, John T., "Rok Slope Engineeing, Institution o Mining and Metallugy, London, Baon, K. and Tao Yang, 99, Inluene o Speimen Size and Shape on Stength o Coal, in Poeedings o the Wokshop on Coal Pilla Mehanis and Design, U.S. Bueau o Mines, IC 95, pp Bieniawski, Z.T. and G. S. Kalamaas, 99, A Rok Mass Stength Conept o Coal Seams, in th Intenational Coneene on Gound Contol in Mining, West Viginia Uniesity, Mogantown, pp U.S. Amy Cop o Enginees, 970, Engineeing and Design, Enginee Manual 0--90, Washington D.C., pp V888-, see page. Wilson, A.., and D.P. Ashwin, 97, Reseah into the Detemination o Pilla Size, in The Mining Enginee, Volume Pat 9, Numbe 4, Institution o Mining Enginees, London, June, pp easley, K.A., and G. Chekan, 998, Patial Stess Modeling o Mine Planning, 7 th Intenational Coneene on Gound Contol in Mining, West Viginia Uniesity, Mogantown, pp Mak, Chistophe, 99, Analysis o Longwall Pilla Stability (ALPS): An Update, in Poeedings o the Wokshop on Coal Pilla Mehanis and Design, U.S. Bueau o Mines, IC 95, pp Able, John F. J., 98, Rok Mehanis andouts o MN. ANSYS, In., 75 Tehnology Die, Canonsbug, PA P:\ENG_WP\00.999\009\VAS\Analytial Design Poedue Using the Wilson Equation.do 6 Copyight 00 by SME