Three-Dimensional Finite Element Stress Analysis Applied to Two Problems in Rock Mechanics

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Three-dimensional finite elem ent stressanalysis appl ied to two problems inrock mechanics

SYNOPSISBy T R Staeey M .Se. (Eng)*

T he fin ite e leme nt m eth od o f stre ss a na ly sis is n ow ad ays w id ely u se d in th e fie ld s o f m in in g, m ec ha nic al a ndc iv il e ng in ee rin g. T his p ap er p re se nts a s ho rt d esc rip tio n o f th e th re e-d im en sio na l m eth od , in w hic h th e e le -m en ts u se d a re a rb itra ry h exa he dro ns. T wo e xamp le s o f th e a pp lic atio n o f th e m eth od a re p re se nte d, n am ely,ca lc ula tio n o f th e e la stic s tre ss d is trib utio ns in a c oa l p illa r a nd in a n o pe nc as t m in e.

SINOPSISD ie e in dig ee leme ntm eto de v an d ru ks pa nn in g w ord d ee sd ae a lg em ee n g eb ru ik o p d ie g eb ie d v an m ynwe se ,megan ie se e n s iv ie le in ge nie urswe se . H ie rd ie v erh an de lin g g ee 'n kort beskr yw ing van d ie d rie -d imens iona lemeto de waa rin d ie e lemen te v an a rb itre re s es vla kk e g eb ru ik g emaa k word . Twee v oo rb ee ld e v ir d ie a anwend in gv an d ie me to de word v oo rg ele , n aam lik d ie b ere ke nin g v an d ie e la stie se d ru ks pa nn in g v ers pre id in g in 'n s teen -koo l pila ar en in 'n o op gro ef m yn .

INTRODUCTIONThe finite element method is becoming increasingly

well known as an extremely powerful method of stressanalysis of two-dim ensional structures. This is due to thefact that it allows arbitrary structural properties to beincluded in the analysis. This capability is inter aliavery useful in the study of the stress distributions aroundmining excavations in which the structural material isrock, inherently an inhomogeneous and anisotropicmedium. It is evident, therefore, that the extension ofthe finite element method to three dimensions wouldprovide an analytic technique which could treat com plexproblems in which no two-dimensional axes of sym-m etry exist. The developm ent of the three-dimensionalmethod has unfortunately been hindered by the factthat extremely large digital computers are required andthe com puting tim e becom es considerable. However, theanalysis has been applied to the calculations of thestresses in arch dams! with good success, and recentlyalso to the stability of rock slopes2.T his paper presents an outline of the three-dim ensional

finite element method and its application to two prob-lems in rock mechanics, namely the calculation of thethree-dim ensional stress distribution in a coal pillar andin an entire proposed open-cast m ine.

THE FINITE ELEMENT METHODA general ou tlin eThe finite element method has been described else-

where3 and a detailed explanation will therefore not be.Senior R esearch O fficer, R ock M echanics D ivision, N .M .E.R .I.,C .S .I.R ., P .O . B ox 39 5, P reto ria.

presented here. In essence, however, the method may beoutlined as follow s:The structure to be analysed is divided suitably into

'finite elements', assumed to be connected only at theircorners which are termed nodal points. From the ele-ment dimensions and material properties, the stiffnessof each element m ay be calculated. The elem ent stiffnessrelates the nodal point forces to the corresponding nodalpoint displacements. The stiffnesses of appropriate ele-ments are superimposed to evaluate the nodal pointstiffnesses. If the nodal point forces are known, e.g. fromboundary conditions and body forces, the nodal pointdisplacements, and hence element stresses, may easilybe calculated. This then provides the stress distributionthroughout the com plete structure.Develo pm en t of th e three-d im ensio nal an alysisIn the two-dimensional finite element analysis, the

simplest procedure is attained using the three nodetriangular element. Irregular boundaries may be easilyfitted with these elements and mesh size may be in-creased and decreased at will. The tetrahedron, whichis the three-dimensional version of the triangle, offersthe sam e advantages, but suffers from the disadvantagethat it is extremely difficult to form a mental three.dimensional picture of adjoining tetrahedrons. It is fa~easier to consider hexahedrons which can be moreeasily pictured in space.The element chosen for three-dimensional analys~s,therefore, is the arbitrary hexahedron isoparametric

element described by Zienkiewicz3. This element haseight arbitrary nodes, and faces of the element may bewarped. It satisfies the requirements of easy matchingof irregular boundaries and easy increase and decrease ofm esh size.

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY MAY1972 251

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.~~ ~!m r c:)~::j+ i !6JZ~6rl-- ~~~ ~1-6J 4m-4-6J4 m-I-- 6)4rn..j( a ) P L A N V IE W

~B o r dA r e a

~A R E A E N C L O S E D B Y D O T T E D L IN E S W A S C O N S ID E R E D F O R T H EF IN IT E E L E M E N T A N A L Y S ISU N IF O R M V E R T IC A L S T R E S S 0)69 M P a

( b ) S E C T IO N

E = 1 3 8 G P ay = 0 , 3'l f = 2 2 7 5 k g /m 'E = 34)5 G P a~ = 0 ,3? I = 2275 k g / m 2

~ P L A N E S O F

I A ~ ~S Y M M E T R Y

M ID - P L A N EL O A D E DS U R F A C E

H O R I Z O N T A L

( c ) P E R S P E C 1 1 V E V IE W O F A C T U A L S Y S T E M A N A L Y S E D .F ig . 1 -G E O M E T R Y A N D D IM E N S IO N S O F T H E R E G IO N C O N S ID E R E D F O R T H E F IN IT E E L E M E N T A N A L Y S IS .

( E = Y o u n g ' s mod ulu s, v= Po isso n's ra tio , y =D ensity )

252 MAY 1972 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF M INING AND M ETALLURGY

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The derivation of the elem ent stiffness m atrix, perhapsthe most critical part of the finite element analysis, hasb een described by Z ien kiew icz3 and presen ted elsew here4.The evaluation of the element stiffness is relativelycomplicated, requiring numerical integration, and adescription of this procedure is beyond the scope of thep resen t p ap er.The computer programme developed for the three-dimensional analysis is based on the two-dimensionalversion described by W ilson5. The analysis of evensimple problems using this programme cannot beattempted unless a digital computer with a large corestorage is available. For more complicated problemseven this is not sufficient, and, for this reason, the pro-gramme discussed in the present paper was designed tooperate on parts of the structure to be analysed. Thus,the structure is divided into appropriate parts, with acertain overlap which is required for continuity. Theimplementation of this necessitates the use of magneticdisc storage as well as computer core storage. For each

part of the structure, the element stiffnesses and hencenodal point stiffnesses are calculated, and the latterwritten onto m agnetic disc. Therefore, for the calculationof each part, the same core storage can be used. Solutionfor the nodal point displacements is also carried out inparts. W ith this method of operation, it is possible tohandle a problem involving a large number of elements.

CALCULATION OF TH E STRESS DISTRIBUTIONIN A COAL PILLARA recent investigation by van Heerden6 into the in-

situ stress distribution in a coal pillar prompted theanalysis of this problem by the three-dimensional finiteelement method. Fig. 1 shows the geometry and thedimensions of the pillar and the surrounding regionconsidered for the analysis. The coal seam was 1.8 mthick above the roof. It was overlain by shale right up tothe surface. However, only a thickness of 3 m of theshale overburden was taken into consideration and it wasassumed that a vertical overburden pressure of 0.69 M Paacted at the system boundary 4.8 m above the roof. Thepillars had a 6.4 m square horizontal cross-section, andadjacent pillars were separated by bords 6.4 m wide.The pillar height was 2.7 m. The material propertiesadopted for the coal and shale are noted on Fig. 1. Boththe coal and shale were assumed to be homogeneous,isotropic and to behave perfectly elastically.

The boundary conditions applied to the region con-sidered (see Fig. 1 (c)) were as follows: Horizontal dis-placement of all vertical in-situ boundary surfaces, i.e.excepting for the free faces of the pillar, was restricted.Vertical displacement of the horizontal m idplanethrough the pillar was restricted. The upper surface ofthe region was loaded as described above.The region was divided into a total of 141 elementsw ith 256 nodal points. For computer solution, theproblem was divided into two parts.

The distribution of the vertical stresses in the pillarhorizontal m idplane, along lines from the centre of thepillar normal to the free faces, and along diagonals, areplotted in Figs. 2 and 3 respectively. For comparison

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7,0 . --- Sa'amon and Oravccz 'heory- 3 -D F in ite elem en t an aly sis. . . . V an H urden$ in situ results"

Cons tr an ed h or iz on ta l m i d- pl an eperimeter

d..>: .6,0 .~~.~

:- . ..~,: C en t.. o f p ',l ra r

2,0 . .. . .,0 0,5 ',0D IS TA NC E F R OM F AC E ( Me tr es ) 3,0

Fig. 2-DISTRIBUTION OF VERTICAL STRESS ALONGNORMAL LINE IN PILLAR HORIZONTAL MID-PLANE

purposes the results of in situ stress measurementscarried out by van Heerden6 and also results, calculatedby van Heerden, using a theoretical method developedby Salamon and Oravecz7, are also shown in the twofigures. It can be seen that the agreement is close, exceptnear the faces of the pillar. The results of the finiteelement analysis indicate that the peak vertical stressesacting in the pillar horizontal m idplane do not occurat the face, but a little inside. For an elastic analysis atany rate, the form of these results is confirmed by thework of Logie8, who conducted three-dimensionalphotoelastic tests on models of bord and pillar exca-vations. The in situ stress measurements were based onthe assumption of elasticity at each measuring point.The elastic properties could, however, vary for differentmeasuring points. For the finite element analysis theelastic properties were assumed to be the same through-out the pillar. This may be the reason for the discrepancybetw een calculated and in situ stresses.

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