ࡱ> 02/` bjbjss .Lt%@@@@@@@T8P$To"$%h @@@)eeed@@eee@@. QOu?0o,7!E7!.7!@.> ,eL$pdoTTT TTTTTT@@@@@@  MINE SUPPORT MECHANISMS USING A LIMIT EQUILIBRIUM ANALYSIS LIMIT EQUILIBRIUM RELATIONSHIPS FOR MATERIAL INTERFACE PROBLEMS Figure 1 shows a representative section diagram of a slot-shaped excavation. The excavation is assumed to be indefinitely extended to the left ( EMBED Equation.3 in the region designated as mined in Figure 1) and is bounded by two planes A and B representing the floor and the roof of the excavation respectively. Assuming that the y-axis points into the plane of the diagram in Figure 1, the floor and roof planes are defined to be z = 0 and z = H respectively. The plane region defining the vertical edge of the excavation, separating the mined from solid material (designated as the face position in Figure 1), is bounded by the constraints EMBED Equation.3 , EMBED Equation.3 and EMBED Equation.3 . All material in the regions EMBED Equation.3 and EMBED Equation.3 is assumed to be elastic. The material to the right of the face position in Figure 1 (i.e. the region EMBED Equation.3 ,  EMBED Equation.3 and EMBED Equation.3 , designated as failed) is assumed to be in a state of so-called limit equilibrium that is defined by a specified constraint relationship between the stress components EMBED Equation.3 ,  EMBED Equation.3 and EMBED Equation.3 . Stress components EMBED Equation.3 ,  EMBED Equation.3 and EMBED Equation.3 are assumed to be constant or zero and are ignored. It is further assumed that boundary conditions are specified on the face that assume the forms EMBED Equation.3 and EMBED Equation.3 .  EMBED Equation.3 and EMBED Equation.3 are specified constants or functions derived from the deformations induced in a thin interface material layer such as a spray-on liner or a membrane material that is used to contain the movement of the failed material. The problem is to determine the stress state in the region EMBED Equation.3 and EMBED Equation.3 . It is also necessary to impose further assumptions to determine deformations in the failed material. These may include a specification of the roof and floor movements and suitable plastic flow rules to connect the stress and strain states in the failed region. Initially, it would be of interest to solve the special case where EMBED Equation.3 and EMBED Equation.3 .  Figure 1. Force equilibrium of elementary material slice between two bounding surfaces. A) ILLUSTRATION OF A SIMPLE ONE-DIMENSIONAL LIMIT EQUILIBRIUM RELATIONSHIP The force equilibrium of a material slice is shown in Figure 1. It is assumed that at x = 0 the material is unconfined and that the surface-parallel stress component EMBED Equation.3 increases as x increases. From Figure 1 it can be inferred that the equilibrium force balance acting on the slice of height H and unit out of plane width requires that  EMBED Equation.3 . (1) Taking the limit  EMBED Equation.3 , equation (1) can be written in the form of the differential equation  EMBED Equation.3 . (2) Equation (2) can be solved for EMBED Equation.3 if a relationship exists between  EMBED Equation.3 and EMBED Equation.3 . This can be established by making the following assumptions. Assume that EMBED Equation.3 is related to the surface-normal stress  EMBED Equation.3 by a frictional slip condition of the form:  EMBED Equation.3  (3) where  EMBED Equation.3 is the friction coefficient. Assume that  EMBED Equation.3  is related to the average stress EMBED Equation.3 by a failure relationship of the form:  EMBED Equation.3 , (4) where C and m are specified constants. Substituting equations (4) and (3) into equation (2) yields the required differential equation as follows.  EMBED Equation.3 . (5) Equation (5) can be integrated directly to yield  EMBED Equation.3 , (6) The integration constant A can be evaluated by applying the boundary condition  EMBED Equation.3 when x = 0. This yields the value of A as  EMBED Equation.3 . (7) Substituting equation (7) into equation (6) and solving for EMBED Equation.3 gives the result  EMBED Equation.3 . (8) Letting  EMBED Equation.3 , equation (8) can be written as  EMBED Equation.3 . (9) Substituting equation (9) into equation (4) gives an expression for EMBED Equation.3 :  EMBED Equation.3 . (10) B) AN EXPANDED ANALYSIS The formulation presented in section A ignores the detailed stress variation in the z direction. Specifically, the following two equilibrium equations should be satisfied  EMBED Equation.3 , (11)  EMBED Equation.3 . (12) In addition, constraint relationships such as those represented by equations (3) and (4), must be satisfied. The limit equilibrium constraint may be expressed in the form of a function between the principal stress components, determined from the stress components EMBED Equation.3 ,  EMBED Equation.3 and EMBED Equation.3 . Further assumptions regarding the plastic deformation field are required as suggested in Jaeger and Cook, Chapter 9. References Jaeger, J.C. and Cook, N.G.W. 1979. Fundamentals of Rock Mechanics, 3rd Edition, Chapman and Hall, London. Hill, R. 1950. The mathematical theory of plasticity, Oxford, Clarendon Press.     PAGE  PAGE 1 =[|}~  " # $ % H T U : ; D E H I W   & ' ( ) * + > Եԛԛԏԇ{hW{{!j$hShh~CJEHUaJ%jqH h~CJUVaJmH sH jhShCJUaJhShCJaJhJ h 6CJaJh 6CJaJ!jh h CJEHUaJ%jNH h CJUVaJmH sH jh CJUaJh CJaJh/CJaJh/5CJaJh35CJaJh t5CJaJ <=}~fgiuv  12$ & Fdha$gd1[ $dha$gdpA $dha$gd t $dha$gd3t> ? @ A D E X Y Z [ x y ǻ|kXG?h tCJaJ!j h~h~CJEHUaJ%jH h~CJUVaJmH sH !jh~h~CJEHUaJ%jH h~CJUVaJmH sH !jwh~h~CJEHUaJ%jH h~CJUVaJmH sH h~CJaJjh~CJUaJhShCJaJjhShCJUaJ!jHhShhShCJEHUaJ%j2H hShCJUVaJmH sH    " # $ % ' ( ; < = > A B U V W X p         0 xeT!johJ hJ CJEHUaJ%jNH hJ CJUVaJmH sH hc/BCJaJ!j>h~hJ CJEHUaJ%jH hJ CJUVaJmH sH !jhShhJ CJEHUaJ%j2H hJ CJUVaJmH sH !j hShhJ CJEHUaJ%jH hJ CJUVaJmH sH jhJ CJUaJhJ CJaJ0 1 2 3 6 7 J K L M ` a t u v w y z Ǵǐl[H%j H hc/BCJUVaJmH sH !jhc/Bhc/BCJEHUaJ%jH hc/BCJUVaJmH sH !jhc/Bhc/BCJEHUaJ%jH hc/BCJUVaJmH sH !jhJ hJ CJEHUaJ%jH hJ CJUVaJmH sH hJ CJaJjhJ CJUaJ!jhJ hJ CJEHUaJ%jH hJ CJUVaJmH sH  =>?RSTUXYlmnoqrʾҫʎ{jbVNh,tCJaJjh,tCJUaJh tCJaJ!j"hc/Bh,tCJEHUaJ%j"H h,tCJUVaJmH sH jhfpCJUaJ!j hc/Bhc/BCJEHUaJ%jQ!H hc/BCJUVaJmH sH jhc/BCJUaJhfpCJaJhc/BCJaJhJ CJaJjhJ CJUaJ!jhc/Bhc/BCJEHUaJ }ǴǛǛǛǛ|kXG!j`+h~hjCJEHUaJ%jH hjCJUVaJmH sH !j<)hShhjCJEHUaJ%jH hjCJUVaJmH sH jhjCJUaJhjCJaJ!j*'h,th,tCJEHUaJ%j+#H h,tCJUVaJmH sH h,tCJaJjh,tCJUaJ!j%h,th,tCJEHUaJ%j#H h,tCJUVaJmH sH  )*UX34GHIJMNabcdefghiۼۘti`Wh/5CJaJhpA5CJaJhthpACJaJj2hthtUh/CJaJ!j/hpAhpACJEHUaJ%jH hpACJUVaJmH sH !j-hpAhpACJEHUaJ%jH hpACJUVaJmH sH jhpACJUaJhpACJaJh tCJaJhjCJaJjhjCJUaJ PQghIJKgwxݸ݈ud\\P\jh1[CJUaJh1[CJaJ!js=h3h &CJEHUaJ%jSH h &CJUVaJmH sH jh3CJUaJ!jY;h1[h6CJEHUaJ%jH h6CJUVaJmH sH h6CJaJjh6CJUaJh36CJaJhGCJaJh3CJaJh/CJaJh/5CJaJhG5CJaJ   !"#;<=>PQdefgǿǿǬǿǿLjwdS!jFh1[h1[CJEHUaJ%jH h1[CJUVaJmH sH !jDh1[h1[CJEHUaJ%jH h1[CJUVaJmH sH !jCBh1[h1[CJEHUaJ%jH h1[CJUVaJmH sH hFCJaJh1[CJaJjh1[CJUaJ!j@h1[h1[CJEHUaJ%j1H h1[CJUVaJmH sH  CDWXYZǴǐl[OGhFCJaJjhFCJUaJ!jNh1[hFCJEHUaJ%j"H hFCJUVaJmH sH !jLh1[h1[CJEHUaJ%jH h1[CJUVaJmH sH !jJh1[h1[CJEHUaJ%jH h1[CJUVaJmH sH h1[CJaJjh1[CJUaJ!jHh1[h1[CJEHUaJ%j,H h1[CJUVaJmH sH h !FGxy./UV $dha$gdF $dha$gd6$dh^a$gdF$ & Fdha$gdF $dha$gd1[$dh^a$gd1[ *+>?@Ahi|}~ǿǬLjwdSKKh6CJaJ!jaWhFhFCJEHUaJ%juH hFCJUVaJmH sH !jGUh1[hFCJEHUaJ%jH hFCJUVaJmH sH !j,Sh1[hFCJEHUaJ%jH hFCJUVaJmH sH h1[CJaJhFCJaJjhFCJUaJ!j6QhFhFCJEHUaJ%joH hFCJUVaJmH sH #$789:Znwxy{|ۼl[Rh6CJaJ!jh\hhCJEHUaJ%jH hCJUVaJmH sH jhCJUaJhCJaJhBVCJaJhLwWCJaJhGCJaJ!jYhFhLwWCJEHUaJ%jSH hLwWCJUVaJmH sH jhFCJUaJhFCJaJh6h6CJaJh6CJaJh66CJaJ   )*-.12EFGHRSmn룒~kZ~Rh57TCJaJ!jch1[hGCJEHUaJ%j)H hGCJUVaJmH sH jhGCJUaJhGCJaJ!j`ahkvhkvCJEHUaJ%jH hkvCJUVaJmH sH hCJaJhkv6CJaJ!j%_hkvhkvCJEHUaJ%j]H hkvCJUVaJmH sH hkvCJaJjhkvCJUaJ  !FG $dha$gdi $dha$gdG $dha$gd1[$dh^`a$gd6 $dha$gdF  #$789:CDF^_뿷믣l[SSh6CJaJ!jjh hx>CJEHUaJ%jH hx>CJUVaJmH sH !jhh h CJEHUaJ%j+H h CJUVaJmH sH jh CJUaJh CJaJhGCJaJhQLdCJaJ!jeh57Th CJEHUaJ%jRH h CJUVaJmH sH h57TCJaJjh57TCJUaJ89Wez봣됈vnnbnO%j\H hV),CJUVaJmH sH jh]CJUaJh]CJaJhG6CJaJhG5CJaJhFCJaJhkvh6CJaJhGCJaJ!joh6h[)CJEHUaJ%j; H h[)CJUVaJmH sH !j~mh1[h6CJEHUaJ%j H h6CJUVaJmH sH h6CJaJjh6CJUaJ8KLOmƳΚښڒښښښ{hW{{!j+whJ hKaCJEHUaJ%jNH hKaCJUVaJmH sH jhKaCJUaJhKahKaCJaJhiCJaJhKaCJaJ!jth thV),CJEHUaJ%jH hV),CJUVaJmH sH h tCJaJjh tCJUaJh]CJaJjh]CJUaJ!jqh thV),CJEHUaJ!./01 !stuwxz{}~ǴǛǓ~sh`\`\`\`\Rjh"0JUhV~jhV~Uh3hiCJaJh^S[hiCJaJh9VhiCJH*aJhi5CJaJhGCJaJhiCJaJ!jc{hJ hKaCJEHUaJ%jH hKaCJUVaJmH sH hKaCJaJjhKaCJUaJ!jGyhJ hKaCJEHUaJ%jH hKaCJUVaJmH sH !"stvwyz|}h]hgd" &`#$gd" $dha$gdG $dha$gdih3hiCJaJhV~h0JmHnHuhnh"jh"0JU h"0J,1h/ =!"#$% $Dd b  c $A? ?3"`?2n@糨8t5BJD^`!B@糨8t5B xcdd`` @c112BYL%bpu 1%_iDd Xb  c $A? ?3"`?2i{Z|RҦb{ lE ^`!={Z|RҦb{ lR xcdd`` @c112BYL%bpu4L=fdbR ,.IeHԡ"|b@3X?!a4/Dd `b  c $A? ?3"`?2yQ3QShqUS^`!MQ3QShq  !"#$%&()*+,-.14a65798:<;=?>@BACEDFHGIKJLNMOPQSRTVUWYXZ\[]_^`bcfdeghikjlnmopqsrtvuwxy{z|~}Root Entry FQO3=Data '}WordDocument.LObjectPool LOQO_1223563598F LO LOOle CompObjfObjInfo  %*/49>CFILOTY^chklotwz FMicrosoft Equation 3.0 DS Equation Equation.39q0^ x<0 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native 1_1223564401 F LO LOOle CompObj fObjInfo Equation Native  1_1223564338F EOO EOOOle  (~ x=0 FMicrosoft Equation 3.0 DS Equation Equation.39q!` ""<y<"CompObj fObjInfo Equation Native =_1223564447F EOO EOOOle CompObjfObjInfoEquation Native 9 FMicrosoft Equation 3.0 DS Equation Equation.39q` 0d"zd"H FMicrosoft Equation 3.0 DS Equation Equation.39q_1223564528@F EOO EOOOle CompObjfObjInfoEquation Native 1_1223564551F EOO EOOOle CompObj fpD9 z<0 FMicrosoft Equation 3.0 DS Equation Equation.39q@ z>H FMicrosoft Equation 3.0 DS EqObjInfo!Equation Native 1_1223564728,$F EOO EOOOle CompObj#%fObjInfo&!Equation Native "1_1223564878)F QO QOuation Equation.39q x>0 FMicrosoft Equation 3.0 DS Equation Equation.39q8  xxOle #CompObj(*$fObjInfo+&Equation Native ':_1223564958'1.F QO QOOle (CompObj-/)fObjInfo0+ FMicrosoft Equation 3.0 DS Equation Equation.39qI$  xz FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native ,:_12235649723F QO QOOle -CompObj24.fObjInfo50Equation Native 1:_1223565278";8F EOO EOOOle 2J  zz FMicrosoft Equation 3.0 DS Equation Equation.39qx  yyCompObj793fObjInfo:5Equation Native 6:_1223565298=F EOO EOOOle 7CompObj<>8fObjInfo?:Equation Native ;: FMicrosoft Equation 3.0 DS Equation Equation.39qS  yz FMicrosoft Equation 3.0 DS Equation Equation.39q_12235653276TBF EOO EOOOle <CompObjAC=fObjInfoD?Equation Native @:_1223565649GF EOO EOOOle ACompObjFHBfs|T  xy FMicrosoft Equation 3.0 DS Equation Equation.39qM  xx | x=0 =f(z)ObjInfoIDEquation Native Ei_1223566055EOLF EOO EOOOle GCompObjKMHfObjInfoNJEquation Native Ki_1223566103QF EOO EOO FMicrosoft Equation 3.0 DS Equation Equation.39qM0  xz | x=0 =g(z) FMicrosoft Equation 3.0 DS Equation Equation.39qOle MCompObjPRNfObjInfoSPEquation Native Q5L f(z) FMicrosoft Equation 3.0 DS Equation Equation.39qpp4 g(z)_1223566123J^VF EOO EOOOle RCompObjUWSfObjInfoXUEquation Native V5_1223620814[F EOO EOOOle WCompObjZ\Xf FMicrosoft Equation 3.0 DS Equation Equation.39q!o f(z)=c FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo]ZEquation Native [=_1223620838Y`F EOO EOOOle \CompObj_a]fObjInfob_Equation Native `=_1222769652m|eF EOO EOO!p5 g(z)=0 FMicrosoft Equation 3.0 DS Equation Equation.39q  s FMicrosoft Equation 3.0 DS EqOle aCompObjdfbfObjInfogdEquation Native e6_1222769235jF EOO EOOOle fCompObjikgfObjInfoliuation Equation.39qq H s (x+x)=H s (x)+2x FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native j_1222769457hroF EOO EOOOle mCompObjnpnfObjInfoqpEquation Native q5_1222769557tF EOO EOOOle rZ x!0 FMicrosoft Equation 3.0 DS Equation Equation.39qDZ d s dx=2/HCompObjsusfObjInfovuEquation Native v`_1222769693yF EOO EOOOle xCompObjxzyfObjInfo{{Equation Native |) FMicrosoft Equation 3.0 DS Equation Equation.39q   FMicrosoft Equation 3.0 DS Equation Equation.39q_1222769708w~F EOO EOOOle }CompObj}~fObjInfo  s FMicrosoft Equation 3.0 DS Equation Equation.39q  nEquation Native 6_1222769835F EOO EOOOle CompObjfObjInfoEquation Native 6_1222769954cF EOO EOOOle CompObjfObjInfoEquation Native B_1222770031F EOO EOO FMicrosoft Equation 3.0 DS Equation Equation.39q&( = n FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native )   FMicrosoft Equation 3.0 DS Equation Equation.39qx  s_1222770118F EOO EOOOle CompObjfObjInfoEquation Native 6_1222770293F EOO EOOOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q<   n =C+m s FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native X_1222770515F EOO EOOOle CompObjfObjInfoEquation Native _1222771600F EOO EOOo d s dx=2H(C+m s ) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native mH[ 1mln(C+m s )=2xH+A FMicrosoft Equation 3.0 DS Equation Equation.39q#  s =0_1222771805F EOO EOOOle CompObjfObjInfoEquation Native ?_1222771896F EOO EOOOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q6 A=1mln(C) FMicrosoft Equation 3.0 DS EqObjInfoEquation Native R_1223621673F QO QOOle CompObjfObjInfoEquation Native 6_1222772306F QO QOuation Equation.39qp  s FMicrosoft Equation 3.0 DS Equation Equation.39qm  s =Ole CompObjfObjInfoEquation Native Cme 2mx/H "1[] FMicrosoft Equation 3.0 DS Equation Equation.39q%[ =2m/H_1222772523F QO QOOle CompObjfObjInfoEquation Native A_1222772607F QO QOOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qQԦ  s =C(e x "1)/m FMicrosoft Equation 3.0 DS EqObjInfoEquation Native m_1222774048F QO QOOle CompObjfObjInfoEquation Native 6_1222774075F QO QOuation Equation.39qxt^  n FMicrosoft Equation 3.0 DS Equation Equation.39q8<d  n =CeOle CompObjfObjInfoEquation Native T x FMicrosoft Equation 3.0 DS Equation Equation.39qk[ " xx "x+" xz "z=0_1223874652F QO QOOle CompObjfObjInfoEquation Native _1223874757F QO QOOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qk " xz "x+" zz "z=0Oh+'0ObjInfoEquation Native 1TableG!SummaryInformation(X xuPJA}3wK*.Z*bN<#$p,5Vbiϰ!X @c112BYL%bpu @c112BYL%bpu @c112BYL%bpu @c112BYL%bpuO]2'P`D50@#^`!8>O]2'P`D50@@||xcdd``> @c112BYL%bpu @c112BYL%bpudKF&&\ s: @> 1,MMȂDd <@b  c $A? ?3"`?2PaW+h*Hl)X ^`!PaW+h*Hl)` nxcdd``> @c112BYL%bpu?xdA%?I\F(+~F .0%#(j+* 0+ssٹL|"?%, #l+?\p~fdsQŸսWdnq4rS68@Ӭ;+KRsA2u(2t5= MDd (@b  c $A? ?3"`?2ԴE @c112BYL%bpu 1,RDd ,b  c $A? ?3"`?2\D[7DY8n'^`!0D[7DYxcdd`` @c112BYL%bpu4L=fdbR ,.IeHԡ"|b@3X?!a41Dd b  c $A? ?3"`?2{:B GטDW+o`!O:B GטD xcdd``fed``baV d,FYzP1n: X,56~) @ ,' ㆪaM,,He`0i?&meabM-VK-WMc1sC VPZ^T TY@:=*a|O&|׀P 27)?* B8gdQgn&#n)J.hqCl,4=`dbR ,.Ie`ԡ"|b@3X?`9Dd ,b  c $A? ?3"`?2|V Zf#``Lwe_-o`!W|V Zf#``Lwe8%xcdd``ed``baV d,FYzP1n: B@?b u00 UXRY7ӂ`'0L ZZpc@𵜛#'P]b mҙ\ T0i I9 \Y > ᰏD@Wd-6sI\FL #47G \[ Cv0o8n+KRs~aPdk{> 1e9Dd ,b  c $A? ?3"`?2b,l(ZS_0o`!Wb,l(ZS8%xcdd``ed``baV d,FYzP1n: B@?b u00 UXRY7ӂ`'0LY ZZpc@\# h(.P56r`ei5@|M8_J,1@penR~:׳p? 2\F1n2 Up{3\P48a( #RpeqIj.CE.ղ/0۵dٳ%-دᰨI;ڧG̙I <&ߗ낁UVUu埦ei;Ngn my3V?/#It=v8!|TB 2u]Yovѕe;{SM kuƪPʃj;cUT~3we}mUy(O}~w t^ē=gdpMLGIFV&k""l?E^L-qPc1b99'g9JZCW3={&9ã;Iޘzc~<&62}#Ff+i7:vRkK6pM}<dl03Akv"fiHlnjt,7&xǑ.8R߫雝q !Džz?'y;²刴'Y?aұMh8:O<)[՟bqۉ/މ7Jƚ%XR i[bkf`:|vŁN=p\qV#T6#I'D$JORBOV5HW N. _خ` x>\1N|Bț?دىwZȺ_Yx2V2lb&nxC4qMt}zo6Y4A6Q> M TBˌCA?&5Ex67hnoh@ D>+&&ecF롉F׀&hk},rbB46 @faKYH_! 4ӌbte!6* ) <=a÷ @c112BYL%bpu @c112BYL%bpu/> /%JBo`!IEg >/> /%J4 hxڝR=KP=&-XjVw(HňBM6`V7]%?EQ). qsy< $6C({DDZB4Ҿ"C,Lnce3J_  !f"n#8t 7reDUIkWj t~Su stՖ @c112BYL%bpu @c112BYL%bpuOA#Mo`!9 &dO X>O xcdd``> @c112BYL%bpuOo`!ZO^*@~VAm"? (xcdd`` @c112BYL%bpu\P$#t] `p321)W2ԡ"b> 1d'Dd b # c $A? ?3"`?"2@PS5 j1`zQo`!PS5 j1`:`!Rxcdd`` @c112BYL%bpu 1ܘ!+(|-/L[T T080ܛ[ F;=`321)W20cPdkYb#a18]Dd ,@b $ c $A? ?3"`?#2e &dO X>OApSo`!9 &dO X>O xcdd``> @c112BYL%bpu @c112BYL%bpu 0D(1 z.C``ÄI)$5a\E.B 0Cbd`M^WDd (@b & c $A? ?3"`?%2!;;r6ĬW}Wo`!u!;;r6ĬW@ Cxcdd``vdd``baV d,FYzP1n:&&! KA?H1Z q00 UXRY7&meabM-VK-WMc1sC VPZ˙&T T 4klvJ_lR^%3Dۓ 2!`'27)?a5'ׁpb#ܞ6d$03@ɞ\pۃRp{@|J.hlqclnv0o8 021)W2aPdh,ĀGf 8y'Dd Db ' c $A ? ?3"`?&2/p >b}e\)ܫ ԥx(Зe+w<\ԎYrA>*n23ɹ Uԍ[s4Xs,sqT3up}_Qb&mx[j&:NAQw2VXЖ< *& dB^dMDd ` Db ( c $A!? ?3"`?'2۹Cd@=#d-\o`!۹Cd@=#d-@xcdd``>$d@9`,&FF(`Ti3 A?d| >=5< %! 8 :@u!f0`t\F7L,L AtOA $37X/\!(?71XAk%δ{q90=tCsP~"rnυA|gh<} s8<>9i;ȄJt +ssvqr9/(s2Bo -_L,H^m/rc7NKdTc?JN(#02itԁc0 ni=Ĥ\Y\˰E.=C;;Dd @b  c $A ? ?3"`?2?f!mM}eNai_^`!Y?f!mM}eNB@2 'xcdd``Vdd``baV d,FYzP1n:&,B@?b 030 UXRY7ӂ`'0LY ZZpc@<-NHq%0J3Tb1Bܤ\ > 0D@2W d-\0sIt'\F;&c!,Hu3@傆68F kv0o8+KRs@2u(2tA4T}b@3XsZaDd tDb  c $A ? ?3"`?2C -*B[IcxOa^`!C -*B[IcxOX  MxڕQJA};k. xD (}`&؟p(9VIo!ئ#,XFE%hs{o̍@_X>9 !Ai.治/^:ϗ-9ӘG8ct;5 >'!pI}#gcMڳ,IFRN˼O'Q¶ߌa_ok nQhJRPˡe4cC;9f,+l+eDj:_&%~Z6@uҟZ_5W5{2?Cd=YAXKZqe\&%(nDd ,@b  c $A? ?3"`?2cՂ%'EDOS2?d^`!7Ղ%'EDOS2 xcdd``> @c112BYL%bpu 0D(1 z.C``ÄI)$5Ad.P"CDHg!!v120e(9LDd 0Db  c $A? ?3"`?21=PU"^ds47f^`!1=PU"^ds47ˆ Hxcdd``vf 2 ĜL0##0KQ* Wô[RcgbR 73PT obIFHeA*/&`b]F"L/՘A,a +L}/UN9 +sspr/ pc@"gLF]F\vL=, ѕՀ<&[? dD,2 M-VK-WMNLPsBUO]xd~%?7~pQhrЭ@|[ 0R pLf% ؆ t1R?7)jO#3#`s2f_Q6=Xb.0._ pX.p'7] `p9Ĥ\Y\ 2C  KJDd ,b  c $A? ?3"`?2HV6&IH~"ph^`!hHV6&IH~"r`hn 6xcdd``dd``baV d,FYzP1n: B@?b u ㆪaM,,He`i?&meabM-VK-WMc1sC VPZfT T߰tzTL, ~n% $? cGnO#oTL {+ss8|={!@Dlab@g/'?n{0SqJ.hpcLи``㜑I)$5d.P"CDHg!t?1e4vDd hb . c $A'? ?3"`?-2_l# "` 7ko`!_l# "` b @(|wxڝRJA\r&BTES+1)m"(xFH ng^L#b#[ eDzngߛ;) @8u#B" t4cѿP"Lb& P0nstH<3Y:jS֧.NEw}y wWxok{;gq^D )uҗ]|j)<SxEzn?=]b8tޱk +A)(s /1,z/Ձ>Dd ,@b / c $A(? ?3"`?.2ec͟?,RV]Amo`!9c͟?,RV] xcdd``> @c112BYL%bpuA圄V um17+n+ge4܉s.LcqM˾ GjoBˬ[.j͓L^+ļԊÞ*.d".uz\傌cDd +Db 1 c $A*? ?3"`?02"枤`4ro`!"枤`Z MxڝR=K`7mX0T (.dP["T`Z&''quhWA?AqTHJ u7\{@4E!R[ı06P=ܛM"LB%kœ EG?HUq[@=J/hKo{s,(vRKU 0{S֜^uouO,QշsJaӳ.J|?[}no_H+Ev&yML^?7@xǹUdR04{ u:}\5#lUgLfto`!5>lUgLfZ xڝR=H`m+;"BGm.U" -& 8;uPI)tPA.ܻܽwwB@EOYB}l2V,S0>O>z.PQa5ANJ[@cOQ䞡W7ͼZj+x_W(8©cmaVRf8ЎRnѓK|^#:3>xSbÕ%81v8\x9ʟ?-$ttc31 $?ϓ7bRc{5}s̓b5/ܛRŹ6 %PO6nn;:*R 3*ݢDd h@b 3 c $A? ?3"`?22fq{cR1d«mBowo`!:q{cR1d«mΒ@| xcdd``> @c112BYL%bpu @c112BYL%bpu @c112BYL%bpu 0  !"#%$"$&(*>@DXZx"$';=AUW026JL`tvy>RTXlnq3 G I M a c w " P d f    C W Y *>@h|~#79{1EG#79.0::::::::::::::::::::::::::::::::::::::::::::::::::::  '!!   tS &S  aZZ``9*urn:schemas-microsoft-com:office:smarttagsplace8*urn:schemas-microsoft-com:office:smarttagsCity  ttvvwwyz|};i r  1rttvvwwyz|}333333333%)*AD[x%36M`wy>UXoq3 J M d i s w # - P g C Z *Ah#:B{1HQ#:BEsttvvwwyz|}ttvvwwyz|}Wfl^`o(() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.Wfz        *)J lc j6 &[)V),pAc/BG57TLwWwK]QLdShfpp s tkvV~pBVF intKa{2(3 ,tf~1[/x>"]@F>`@UnknownGz Times New Roman5Symbol3& z Arial"1hXX3A &3A &!4djj2QHP ?323DERIVATION OF SIMPLE LIMIT EQUILIBRIUM RELATIONSHIPNapierStudent