Transactions of the VŠB Technical University of Ostrava No.2, 2010, Vol.X, Civil Engineering Series paper #20

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1 Trasactios of the VŠB Techical Uiversity of Ostrava No.2, 2010, Vol.X, Civil Egieerig Series paper #20 Petr JANAS 1, Richard ŠŇUPÁREK 2, Marti KREJSA 3 ad Vlastimil KREJSA DESIGNING OF ANCHORING REINFORCEMENT IN UNDERGROUND WORKINGS USING DOPROC METHOD VYUŽITÍ METODY POPV K NAVRHOVÁNÍ KOTEVNÍ VÝZTUŽE DŮLNÍCH DĚL Abstract The achorig reiforcemet (roof boltig) raks curretly amog importat reiforcig solutios available i the miig idustry ad udergroud egieerig. The reiforcemet is sized o the basis of various theoretical assumptios ad empirical pieces of kowledge. Geerally it is assumed that the iput values are clearly determiistic. This is valid for both the geotechical coditios uder which the achors will be applied as well as for properties of the achors that are iflueced also by istallatio procedures. But most data used as iputs i various achor sizig methods are radom. Recetly, applied Direct Optimized Probabilistic Calculatio (DOProC) has started developig for the assessmet of the reliability of buildig costructios. DOProC is also applicable for the desig of achors. A assumptio for the applicatio is a sufficiet database of radom quatities relatig to the eviromet where the achors will be istalled, properties of the achors ad achorig techologies, achor desig computatioal models ad suitable ad efficiet tools for the probabilistic desig of the achorig reiforcemet. Keywords Direct Optimized Probabilistic Calculatio, DOProC, achorig reiforcemet, udergroud workig, probabilistic calculatio, reliability assessmet, probability of failure, reliability fuctio, radom variable. Abstrakt Při avrhováí koteví výztuže se vychází z růzých teoretických předpokladů a empirických pozatků. Zpravidla se přitom předpokládá, že vstupí hodoty jsou jedozačě dáy determiisticky. Předložeý příspěvek ukazuje postup pravděpodobostího avrhováí a posuzováí spolehlivosti koteví výztuže při aplikaci metody POPV (Přímého Optimalizovaého Pravděpodobostího Výpočtu). Klíčová slova Přímý Optimalizovaý Pravděpodobostí Výpočet, POPV, koteví výztuž, důlí dílo, pravděpodobostí výpočet, posudek spolehlivosti, pravděpodobost poruchy, fukce spolehlivosti, áhodé proměé. 1 Doc. Ig. Petr Jaas, CSc., Katedra stavebí mechaiky, Fakulta stavebí, VŠB-Techická uiverzita Ostrava, Ludvíka Podéště 1875/17, Ostrava-Poruba, tel.: (+420) , petr.jaas@vsb.cz. 2 Doc. Ig. Richard Šňupárek, CSc., tel.: (+420) , Ústav geoiky AV ČR Ostrava, Studetská 1768, Ostrava-Poruba, richard.suparek@ug.cas.cz. 3 Ig. Marti Krejsa, Ph.D., Katedra stavebí mechaiky, Fakulta stavebí, VŠB-Techická uiverzita Ostrava, Ludvíka Podéště 1875/17, Ostrava-Poruba, tel.: (+420) , marti.krejsa@vsb.cz. 1

2 1 INTRODUCTION Recetly, stochastic (probabilistic) methods have become more ad more popular for the appraisal ad desigig of structures. These methods have provided more iformatio about reliability ad resistace of carryig structures [8], [9], [10]. The reaso is that the iput data used whe desigig ad evaluatig the structures are of a rather radom ature. They are, however, frequetly processed as data that are clearly defied by their determiistic ature. Use of the probabilistic method i the regular appraisal ad desigig procedures has started recetly oly. Achorig supports raks curretly amog importat bracig solutios that are used i the miig idustry ad udergroud egieerig. The support is sized o the basis of various theoretical assumptios ad empirical pieces of kowledge. Geerally it is assumed that the iput values are clearly determiistic. This is valid for both the geo-techical coditios uder which the achors will be applied as well as for properties of the achors that are iflueced also by istallatio procedures. But most data used as iputs i various achor sizig methods are radom. I some places, the empirical ad aalytical methods have bee used successfully to desig the achor supports based o theoretical aalysis ad processig of data gaied from extesive empirical measuremets. I case of those methods, the probabilistic approach ca be used to desig a achor support oly if it assumed that the iitial basic data that describe dimesios of uderworkig, features of rock pillar, achors or achor types are ot of a determiistic ature, but are radom data i lie with respective iput parameter histograms. 2 PROBABILISTIC CALCULATIONS There are may probabilistic calculatio methods [Novák 2005]. They iclude simulatio methods (based o Mote Carlo, LHS, or Importace Samplig), approximatio methods (such as FORM ad SORM) as well as umerical methods, icl. Direct Determied Fully Probabilistic Calculatio - DDFPC. The method was published first i 2002 (Jaas Krejsa) - it was origially developed as a Mote Carlo alterative to SBRA (Marek, Guštar, Aagos 1995 [9]). The ame of the method - Direct Determied Fully Probabilistic Calculatio (DDFPC) meas that the calculatio procedure for a certai task is clearly determied by its algorithm, while Mote Carlo geerates calculatio data for simulatio o a radom basis. It, however, followed from a umber of cosultatios ad discussios that the word "determied is somewhat misleadig. The method requires high-performig iformatio systems for complex tasks. Therefore, efforts have bee made to optimize calculatios i order to reduce the umber of operatios, keepig, at the same time, reliable calculatio results. Chaces of optimisig the calculatio steps seem to be extesive. Havig cosulted the issue with experts i costructio reliability (Šejoha, Novák, Keršer, Teplý 2009), the ame of the method was made more precise ad reads ow Direct Optimized Probabilistic Calculatio DOProC. The radom ature of quatities eterig the probabilistic calculatio is ofte expressed by the histogram created o the basis of observatios ad, frequetly log-lastig measuremets. Those histograms are used the to assess the reliability of structures. I probabilistic calculatios, the idividual radom variables might be multiplied, divided, added, or subtracted. Sometimes, more complicated operatios are eeded. The eeded radom variable operatios are expressed i histograms. The operatios ca be carried out usig geeral priciples of the probabilistic theory. It is possible to use ay histogram expressig ay radom variable that eters the calculatio. Let the histogram B be a arbitrary fuctio f of histograms A j, where j rages from 1 to. The: B = f A, A, A,... A j... A ), (1) ( Each histogram A j cosists of i j iterval where each iterval is limited with a j,i from below ad a j,i+1 from above. 2

3 where where This meas, that for the iterval i j = 1, the values will be as follows: I i j, the followig formula is valid: a a a a (2) j, 1 j j,2 = a + a a (3) j, 2 j,1 j j,2 a j, max a j,mi a j = (4) i j a a a (5) j, i j j, i+ 1 Let us express a j i the iterval as a j (ij). Similar relatios are valid for the B histogram. if there are i itervals, the values of the histogram i the i th iterval rage from b i to b i+1, this meas b (i). The values ca be expressed as follows: ( i) ( i1) ( i2) ( i3) ( ij) ( i) b = f ( a, a, a,..., a,..., a ) (6) 1 2 for the specific combiatio of argumets: a 1 (i1), a 2 (i2),, a j (ij), a (i). The same value - b (i) ca be derived for other values too (or at least for some values too) - a j (ij). If the potetial combiatio of values a j (ij) is marked as l, the followig geeral formula ca be derived: ( i1) 1 ( i2) 2 3 ( i3) 3 j ( i) b = f ( a, a, a,..., a,..., a ) (7) The probability p bl i of occurrece of b (i) is the product of p aj (ij) (probabilities of occurrece of a j ij values). The: p i bl ( i1) aj ( i2) aj ( i3) 3 ( ij) j ( ij) aj j ( i) l ( i) aj = p a a... a... a (8) Fig.1: Priciples of umerical operatios with two trucated histograms 3

4 Probability p b (i), beig the probability of occurrece of all potetial combiatios (a 1 i1, a 2 i2,, a j ij, a i ) l of f, with the result of b (i) is the sum of p pl (i) probabilities pursuat to (9): l ( i) ( i) p b = p (9) l = 1 The umber of itervals (i j ) i each histogram (A j ) ca vary similarly as the umber of i itervals i the histogram (B). The umber of itervals is extremely importat for the umber of eeded umerical operatios ad required computig time. O top of this, the accuracy of the calculatio depeds cosiderably o the umber of itervals. DOProC optimisig methods as metioed, for istace, i [6]. However, ot more details are available there. 3 PROBABILISTIC CALCULATION OF ANCHOR SUPPORTS For purposes of probabilistic calculatio of the achor support, it is ecessary to draft: guidelies to be followed durig the sizig of the support, database of parameters eeded for probabilistic calculatios (radom quatity sets for proposals of the achors), software tools for probabilistic calculatios. May guidelies exist for the desigig process of the achor supports (Pruška 2002 [11]). There are may procedures based o empirical ad/or aalytical ad experimetal methods that ca be used for the sizig of the achor supports. The empirical-aalytical methods are based, typically, o simplified aalytical solutios. They, however, use i the calculatio coefficiets that deped o relatively easy-to-determie parameters. They iclude properties of materials (i this case, the properties of rock are determied typically i laboratories) as well as parameters that ca be measured o site or determied by observatios. A valuable source of iformatio is the results of covergece measuremets, determiatio of o-elastic deformatio rage i miig workigs ad uderworkig that might load, by its weight, the support which should be, if ecessary, stabilised by meas of achors. Though the empirical ad aalytical methods apply oly to the area where the ecessary kowledge was acquired, they ca be used i other situatios, oce the ecessary data are verified ad specified. For dozes of years, they have bee used i coal mies i the Ostrava - Karviá Colliery for purpose of desigig of the bracig supports i log miig workigs as well as whe for desigig of bolt supports. I order to desig sigle ad combied roof boltig of uderworkigs i the Ostrava - Karviá Colliery, a calculatio programme ANKER was developed (Šňupárek, Jaas, Slavík [13]). Whe desigig the bolt support for certai coditios, followig parameters eed to be defied: the legth of bolts, the umber ad locatio of the bolts ear the miig workig or uderworkig, parameters of the bolts (the type, diameters, material, achorig method...) Extesive measuremets were carried out i the miig workigs i the Ostrava-Karviá Colliery (Škrabiš 1977 [12]). It follows from the measuremets that the covergece, this meas dislocatio of rock ito the miig workig, ca be calculated from the followig formula: u = 0,1B pl 0,015t ( ) r e e 1,2 H q σ (10) 4

5 Followig parameters characterize the coditios ad are used i the formula (10): H... the efficiet depth uder the surface [m], B... the dimesio (typically, the width) of the miig workig [m], t... the time i days, q... the load-carryig capacity of the support [knm -2 ], σ r... the reduced stregth of the rock [MPa]. The reduced stregth of the hagig rock, σ r, is determied as follows: σ dimi 1 σ r = β 2B (11) where β is the stratificatio coefficiet pursuat to Table 1, σ di is the stregth i oe-axis compressio of the i th strata ad m i is the thickess of the i th strata. Tab.1: Stratificatio coefficiet - β Number of strata β 1,0 0,95 0,90 0,86 0,82 0,79 0,76 0,73 0,71 0,70 I past, geo-physical ad exteso-metric measuremets were used to determie the o-elastic deformatio rage, B, that is the basis for specificatio of loadig ad legth of the bolt. Evaluatios have, however, idicated that the o-elastic deformatio rage ca be defied o the basis of the covergece measuremets usig the formula (12) (Škrabiš 1977 [12]): Oce (10) is used i (12), for t the formula is followig B B = 0, B K 0,4 0,6 = K B u (12) e 1,2 H q 45 r 0,6 σ 1 (13) K characterizes the relatio betwee the o-elastic deformatio i the miig workig or uderworkig with the B dimesio ad covergece i (12). I past, a sigle oe determiistic value, K = 8.3, was used ([2], [3] a [12]). The reaso was that eve with a variable value, or, better to say, with a set of K variable values, it was difficult or rather impossible to make flexible calculatios i spite of the fact that the data were available. The calculatios of B (the dimesio) or σ r (the reduced stregth) were similar determiistic values were used eve if that was ot the case i reality. The load to be trasferred by the bolted support should be suitable for the o-elastic deformatio rage (B ), rock weight (γ) as well as for a certai level of self-bearig capacity of rock strata that does ot exist i the o-elastic deformatio rage. Usig RMR (a geo-mechaical classificatio parameter) proved to be a good solutio (Bieawski (1989) [1]). The, the load of the bolted support was determied by the followig formula: 5

6 Q = B B 1,2H q 0,6 100 RMR RMR 45 = 2,51189 σ r γ B γ K 1 e (14) 100 I (14), γ is the specific gravity of rock [10 3 kg/m 3 ] ad Q is the total load of the bolted support per ruig meter i the workig [kn]. The appraisal of reliability of bolted supports i miig workigs is based o the reliability fuctio aalysis that is described with the followig formula: 100 FS = Q Q (15) where Q sv is the load-bearig capacity of the bolts ad Q is the bolt loadig per ruig meter i the workig. sv The load-carryig capacity of the bolts is based o the followig formula: Q sv. q 2 sv π ( d1 d2 ). σ sv = svqsv = = (16) d 4d s where sv is the total umber of achors per ruig meter i the workig, is the umber of achors i a row, typically, vertically to the workig s axis, q sv is the load-carryig capacity of oe bolt, d 1 is the bolt s outside diameter, d 2 is the bolt s iside diameter, d s is the spa betwee the achor rows ad σ sv is the ormal stress i oe bolt. I additio to the load or the required load-carryig capacity of the achor support, the legth of achors raks amog importat parameters that eed to be determied. The legth of the achors should correspod to the o-elastic deformatio rage, B, ext to the miig workig or uderworkig. It has followed from practical observatios ad measuremets i mies that, if the achor supports are istalled, the covergece ito the miig workig is less that that calculated from (10) where the covergece is determied for the workigs supported by bracig supports. The reaso is that the resistace agaist dislocatio of rock pillar appears oly after the rock-support cotact is established. This results i more extesive deformatio of the rock pillars, if compared with the achor supports. Data resultig from the compariso of deformatio i the workigs supported by the achor supports ad u i (10) ca be used to calculate the legth of achors, l, i the hagig wall as follows: s 0,6 1,5 H q l = 0, K... 45σ 1 B K e (17) I (17), K is a set of experimetal data. The workig ame of this quatity is a covergece coefficiet. Ulike the parameter resultig from the determiistic calculatio i (Šňupárek et al [13]), this parameter is of a radom ature. 6

Fig.2: Histogram of primary data for compressio stregth i carboiferous sadstoe [MPa] Specific databases of the radom iput variables were created o the basis of measuremets doe by maufacturers of the

The width of the mie workig, which depeds, i particular, o the headig techology, was measured i the Ostrava-Karviá Colliery mies i roadways supported by the bolt supports.

7 Fig.2: Histogram of primary data for compressio stregth i carboiferous sadstoe [MPa] Specific databases of the radom iput variables were created o the basis of measuremets doe by maufacturers of the achorig compoets ad i mies where the bolt supports were istalled. The width of the mie workig, which depeds, i particular, o the headig techology, was measured i the Ostrava-Karviá Colliery mies i roadways supported by the bolt supports. The database of the oe-axis stregth ad specific desity parameters of the rock was prepared o the basis of a extesive laboratory research of test drill cores from the carboiferous rock pillar (the data have bee collected i the past forty years by the Coal Research Istitute ad Istitute of Geoics established at the Czech Academy of Sciece). The both coefficiets, K ad K, are also based o extesive measuremets of covergeces i mie gate roads i the Ostrava-Karviá Colliery (carried out i past by the same istitutes). The database of tesile stregth of the achorig compoets was provided by the archive of the maufacturer, Akra Petřvald, ad by Miova Bohemia s.r.o. The measuremets were coducted i state-authorized testig laboratories ad i the Istitute of Geoics. Fig.3: Histogram of parametric distributio for compressio stregth i carboiferous sadstoe [MPa] 7

8 Usig SW applicatios, the measured data ca be processed to create histograms ad derive parameters. The best distributio is chose from amog of dozes of kow parametric distributios o the basis of a coefficiet that is referred to as a tightess coefficiet. Fig.2 shows the histogram of primary data prepared o the basis of 102 measuremet data whe the compressio stregth was measured i carboiferous sadstoe. The horizotal axis shows the compressio stregth i [MPa], while the vertical axis shows the probability of occurrece. The umber of classes was chose i the same way both for the primary data ad for the parametric distributio Gamma that is show i Fig.3. I this case, the solutio is the best the tightess is If aother distributio or more classes are chose, the tightess will decrease. 4 SOFTWARE FOR THE PROBABILISTIC CALCULATION OF RELIABILITIES OF THE ANCHOR SUPPORTS A software applicatio amed Kotveí ("Achorig ) was created for the probabilistic appraisal of reliabilities of the achor supports used i the miig workig. This software applicatio is based o the Director Optimized Probabilistic Calculatio ( DOProC ). Fig.4 shows the applicatio workplace i the Achorig applicatio. Values of a radom value are etered ito the iput form sheet. They iclude the width of the miig workig, B, (Fig.5), thickess ad stregth of idividual strata i the oe-axis compressio, σ d, K coefficiet (Fig.6), coefficiet of ifluece of the achor support o covergece decrease, K, specific gravity of rock γ, resistace of bracig support, q, if a combied support is used, ad the stregth of the bolt, σ H. The values ca be chose from a existig database that is available i the software. Fig.4: Applicatio workspace i the Achorig software 8

Fig.5: Histogram of the width of the miig workig B [m] Fig.

expressed by determiistic descriptio: stratificatio coefficiet, β,

i, outside ad iside diameters of the bolts, d 1 ad d 2, ad distace

9 Fig.5: Histogram of the width of the miig workig B [m] Fig.6: K coefficiet There are still some iput variables that are expressed by determiistic descriptio: stratificatio coefficiet, β, efficiet depth uder the surface, H, thickess of idividual strata, m i, outside ad iside diameters of the bolts, d 1 ad d 2, ad distace betwee the achor rows, d s. Fig.7: Histogram of the reduced stregth of hagig rock [MPa] 9

Fig.8: Workplace with a table for determiatio of RMR (the geomechaical classificatio coefficiet) by Bieawski s method (1989) I the first stage of the probabilistic calculatio, the histogram is

This histogram is eeded for determiatio of the legth ad the loadig of the achors ad for the geomechaical classificatio coefficiet, RMR, (to be defied by Bieawski s method (1989).

10 Fig.8: Workplace with a table for determiatio of RMR (the geomechaical classificatio coefficiet) by Bieawski s method (1989) I the first stage of the probabilistic calculatio, the histogram is determied for the reduced stregth of the hagig rock, σ r, pursuat to (2) (Fig.7). This histogram is eeded for determiatio of the legth ad the loadig of the achors ad for the geomechaical classificatio coefficiet, RMR, (to be defied by Bieawski s method (1989). For that purpose, a separate table i the applicatio is used (Fig.8). Values to be etered describe the coditios i the eviromet where the achor support is istalled. The result is the histogram for the geomechaical classificatio coefficiet, RMR (Fig.9). The, it is possible to determie the histogram for the l legth of the desiged bolt pursuat to (8) (Fig.10). It follows from the histogram that the bolt should be 2.54 m log to reach the reliability of Fig.9: Histogram of RMR (the geomechaical classificatio coefficiet) by Bieawski s method (1989) 10

Fig.10: Histogram of the bolt legth, l [m] The histograms for the Q loadig ad Q sv load-carryig capacity of the bolt are determied o the basis of (5) ad (7), respectively.

11: Histogram of the FS reliability fuctio where the failure probability is P f = 7.72.10-11,

11 Fig.10: Histogram of the bolt legth, l [m] The histograms for the Q loadig ad Q sv load-carryig capacity of the bolt are determied o the basis of (5) ad (7), respectively. The histograms, Q ad Q sv, ca be used i the reliability fuctio (6). The, the resultig probability appraisal is made o the basis of the failure probability P f (Fig.11). Fig.11: Histogram of the FS reliability fuctio where the failure probability is P f = , for 5 achors per oe meter of the workig. I this case the resultig failure probability is P f = 7, Accordig to strict criteria defied i ČSN Desigig of Steel Structures (1998), the achor support would be sufficiet because the desiged probability, P d, for the icreased reliability is ad the reliability coditio, P f P d, is fulfilled (Fig.11). It was chose to use 5 steel achors per each ruig meter of the workig. The diameter of each achor was 22 mm. If less achors were chose (for istace, 4 achors oly), the situatio would be differet: eve if the steel grade ad achor diameter were same, the failure probability would be P f = The reliability coditio, P f P d, would ot be fulfilled eve for a decreased probability pursuat to ČSN (Fig.12). It is possible to desig ad size the achor supports i the Achorig software very flexibly. The Achorig software 11

12 has bee authorized uder File No. 001/ _SW ad is available o the web site: Fig.12: Histogram of the FS reliability fuctio where the failure probability is P f = , for 4 achors per oe meter of the workig 5 CONCLUSION Usig the proposed method, it is possible to apply probability calculatios i the desigig ad appraisal of reliability of the achor supports istalled i log miig workigs ad uderworkigs. Thus, it is possible to determie the legth, l, umber,, ad load-carryig capacity, Q sv. The pre-requisite is, however, a sufficiet database of iput quatities icludig the experiece from practical operatio because may iput quatities caot be based o models ad laboratory measuremets oly. The probabilistic approach which has bee described above i a example from the miig, as well as the available databases ca be used for other structures ad methods for calculatio of udergroud costructios. ACKNOWLEDGEMENT This project has bee completed thaks to the fiacial cotributio of state fuds provided by the Grat Agecy of the Czech Republic. The registratio umber of this project is 105/07/1265. REFERENCES [1] Bieiawski, Z.T., Egieerig rock mass classificatios, Wiley New York, [2] Jaas, P., Bláha, F., Dimesioig of supports i log miig workigs, crossigs ad braches i the Ostrava-Karviá Colliery, Uhlí č. 9, [3] Jaas, P., Dimesioig of roadway supports i coditios of the Ostrava-Karviá coal field, A.A.Balkema/Rotterdam/Brookfield,

13 [4] Jaas, P., Krejsa, M., Krejsa, V., Structural Reliability Assessmet Usig Direct Determied Fully Probabilistic Calculatio, Iteratioal ASRANet Colloquium, Glasgow, UK, ISBN / , [5] Jaas, P., Krejsa, M., Krejsa, V., Curret possibilities of applyig the Direct Determied Probabilistic Calculatio i Structural Reliability Assessmet, Trasactios of the VŠB Techical Uiversity of Ostrava, No.1, Vol.VI, Civil Egieerig Series, ISSN ; ISBN , [6] Jaas, P., Krejsa, M., Krejsa, V., Curret Possibilities of Direct Determied Fully Probabilistic Method (DDPFM), 4 th Iteratioal ASRANet Colloquium, Athes, Greece, ISBN , [7] Jaas, P., Krejsa, M., Krejsa, V., Curret possibilities of applyig the Direct Determied Probabilistic Calculatio, Iteratioal coferece 70 years of SvF STU, Bratislava, Slovakia, ISBN [8] Králik, J., Comparig the efficiecy of probabilistic methods i FEM costructio reliability. I IX. Iteratioal coferece Spolehlivost kostrukcí, 2008, Prague, ISBN [9] Marek, P., Guštar, M., Aagos, T., Simulatio-Based Reliability Assessmet for Structural Egieers, CRC Press Ic., Boca Rato, 1995, ISBN [10] Novák, D., Vořechovský, M., Rusia, R., Small-sample probabilistic assessmet software FREET. 9 th Iteratioal Coferece o Applicatios of Statistics ad Probability i Civil Egieerig - ICASP 9, Sa Fracisco, USA, 91-96, Rotterdam: Millpress, [11] Pruška, J., Ifluece of the bolt support o state of stress ad deformatio of discotiuous rock pillar, CTU i Prague, Fac. of Civil Egieerig, [12] Škrabiš, A., Forecast ad evaluatio of compressio ad deformatio pheomea i horizotal developmet ad preparatory workigs i the Czechoslovak part of the Upper Silesia Coalfield by a empirical ad aalytical method, thesis, Prague, [13] Šňupárek, R., Jaas, P., Slavík, J., Calculatio of the bolt support, [14] Šňupárek, R., Bolt supports i the miig ad udergroud costructio, FAST VŠB-TU Ostrava. Reviewer: Doc. Dr. Ig. Ja PRUŠKA, CTU i Prague, Fac. of Civil Egieerig, Thákurova 7, Praha 6. 13

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig