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"CU= xOt; CUOx@ p=Owt p}rLD R= TB u; |m}t=v}O C=YNWt w xOW pOt

CQ=UN |@=}u=mt QO xDiQQ=mx@ |=yXN=W |iQat "2X=YDN= CQ=UN |@=}u=mt QO xOWxO=iDU= |=yXN=W |iQat x@ VN@ u}=C=kDWt w |Owt p=mW= 'xO=iDU=OQwt |m}t=v}O QDt=Q=B xm CU= QmP u=}=W "OQ=O

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|xQo QO s)=i Owt |Owt u=mtQ}}eD Qov=}@ �i(j�1) w �i(j+1) '�j '3 |x]@=Q QOhQat kij w �00ij w xOW|Q}oxR=Ov= \=kv |xrY=i L w u; Qw=Ht |xQo wO w s)=j|=Q@ |Owt pmW |=vLv= |x@U=Lt =@ "OvDUy s)=j |xQo QO i |Owt pmW |=vLv=|=vLv= Q}}eD XN=W 'sy R= =yu; p[=iD w xO}OCQ=UN w sr=U |xR=U Cr=L wO

%OwW|t h} QaD 4 |x]@=Q R= xO=iDU= =@ |Owt

��00ij = �00dij � �00ij (4)

w sr=U |xR=U |Owt pmW |=vLv= Qov=}@ ?}DQDx@ �00dij w �00ij '4 |x]@=Q QO xm"CU= xO}OCQ=UN

|=Q@ w Ov=xDi=} VQDUo R}v |Owt p=mW= w |a}@] |=yOt=U@ |=y|Q}oxR=Ov= |=v@txm |D=ar=]t u=}t u}= QO [9]"Ov=xOW xO=iDU= CQ=UN COW u}tND w |@=}u=mtxO=iDU= Ovv=t 'Ov=xOw@ p=Owt |WvQm |SQv= w |Owt p=mW= R= xO=iDU= |=v@t Q@|WvQm MU=B `@=D XN=W =} |WvQm |SQv= XN=W =} |Owt |=vLv= XN=W R="Ov=xDW=O CQ=UN X}NWD QO |rw@kp@=k C}kiwt w OvQ=O |}q=@ sHL '|Ot=U@[10]"CU= CQ=UN COW w u=mt x@XN=W u}= O}OWC}U=UL 'C}kiwt u}= p}rO=yu; u}@ xU}=kt w CQ=UN R= Oa@ w p@k xR=U Q=DiQ uDiQoQ_vQO =@ =yXN=W u}=

"OvwQ|t Q=mx@ CQ=UN |}=U=vW |=Q@|UQQ@ 'xrLQt 4 pt=W |}xR=U CQ=UN X}NWD '|twta |Ov@xk@] l} QO|v}@V}B w CQ=UN COW u}tND 'CQ=UN |@=}u=mt 'CQ=UN OwHw sOa =} OwHw

"CU= xR=U |xOv=t}k=@ Qtal}miDVN@ wO QOu=wD|t =Qp=Owt|=yQDt=Q=B|=v@t Q@CQ=UN|@=}u=mtVwQ"2 &|m}t=v}O C=YNWt u}}aD |=Q@ p=Owt |=yQDt=Q=B |}=U=vW sD} Qwor= "1 %OQm[11]"|r@k |xrLQt QO xOWxORu}tND |=yQDt=Q=B \UwD CQ=UN |@=}u=mt sD} Qwor=|Ov@xk@] |]NQ}e w |]N |xDUO wO QO Ov=wD|t xR=U Q@ CQ=UN Q}F-=D|v=UWm |x}rw= |xR=U xm OwW|t jq]= |OQ=wt x@ |]N CQ=UN C}a[w "OwWu}O@ &OyO u=Wv OwN R= |v=UWm w |]N |Q=DiQ R}v CQ=UN wkw R= Oa@ |]NsQH |va} xOvyOp}mWD O=wt C=YNWt =} w |UOvy C=YNWt OvJ Qy xm |vatCqO=at R= xO=iDU= =@ u=wD@ =Q xR=U |=yMU=B |rw 'Ovm Q}}eD |v=UWm pwOt =}xm OwW|t jq]= |OQ=wt x@ |]NQ}e CQ=UN C}a[w "OQm u=}@ |]N CmQLu=Wv OwN R= |]NQ}e Q=DiQ CQ=UN wkw R= Oa@ |]N |v=UWm |x}rw= |xR=U|xDUO %OvwW|t |Ov@xk@] xwQo wO QO R}v CQ=UN |=ypOt 'u}= Q@xwqa [5]"OyOQO w �xvwoQ}D� pmW|]N QY=va R= pmWDt |=yxR=U |=Q@ ,=DOta xm |}=ypOt 'pw=xDU@=w |=ywQ}v Q}_v |}=yQDt=Q=B R= CQ=UNXN=W xm OvwQ|t Q=mx@ xrRrR |UOvyt|@=} RQ= xR=U |=[a= pmW Q}}eD x@ \w@ Qt |SQv= =} |Q}PBpmW '`]kt K]U x@Kr=Yt pL=Qt |=yCr=L xm OvDUy |DQ=UN |=ypOt 'swO |xDUO w &OvwW|t[12]"OvQ=wDU= l}t=v}OwtQD pwY= |=v@t Q@ w Ovvm|t |UQQ@ =Q xR=U \=kv x@ \w@ Qt=yQ}D QO CQ=UN |@=}u=mt |=Q@ =yVwQ R= |Q=}U@ 'OW xQ=W= QDV}B xJv=vJxU |=y?=k XwYNQO "CU= xDiQ Q=mx@ |Oa@ wO |=y?=k =} �|Oa@ l} |=[i�=} Q}D QO CQ=UN X}NWD x@ xm CU= xDiQo CQwY xOvm =QB |D=k}kLD R}v |Oa@T} QD=t |x} RHD R= xO=iDU= =@ Q}D =} uwDU QO u; j}kO pLt R= Q_vhQY 'uwDUC=ar=]t R= |O=OaD QO [13]"OQ=O X=YDN= |[Qa w |QwLt |=yxir-wt x@ |DNUQDW}@ |m}t=v}O |=yQDt=Q=B QO pqDN= OwHw w CQ=UN COW u}tND |xr�Ut R}vwO |=y?=k QO CQ=UN X}NWD |Oa@ C=k}kLD QO [15w14]"CU= xOw@ Q_vOQwtxm CU= QmP u=}=W "CU= xOW |UQQ@ �xk@] 2� |Oa@ xU w �xk@] 10� |Oa@|}=yDv= =} |}=OD@= |xt}v QO Qo}O Q=@ w =[a= pw] s=tD QO Q=@l} CQ=UN OwHwxHwD OQwt QDtm CQ=UN QDj}kO pLt u}}aD p=Lu}= =@ "CU= xOW |}=U=vW =[a=QO u; Q}F-=D w |DNU T} QD=t QO pqDN= |R=Ux}@W XwYNQO xD@r= xm 'CU= xOw@QO [16]"CU= xOt; CUOx@ |}=yC}kiwt 'xr�Ut pL |}=Qosy w CQ=UN |}=U=vWx@ l}ORv =} |k}kL T=}kt QO w |ak=w |=ypOt R= xO=iDU= C=ar=]t R= |Q=}U@XwYNQO QDj}kO G}=Dv ?Um |=Q@ u}= OwHw =@ [18w17]'CU= xOw@ xHwD OQwt u;

"CU= R=}vOQwt |QDW}@ |=y|UQQ@ 'CQ=UN |@=}u=mtR= xO=iDU= =@ xtQ;uD@ |Oa@ xU |=y?=k QO CQ=UN |@=}u=mt x@ j}kLD u}=|=Q@ "OQ=O X=YDN= p=Owt |WvQm |SQv= w |Owt |=vLv= Q}}eD R= xO=iDU= |=yVwQ|Owt p=mW= '|Owt C}U=UL w |Owt |=vLv= Q}}eD VwQ QO QDj}kO |@=}u=mt

70

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[Q h} QaD 17 w 16 |=yx]@=Q R= xO=iDU= =@ CQ=UN R= Oa@ w p@k Cr=L wO QO[19]%OwW|t

MSEij = �Ti �Kj � �i (16)

MSEdij = �dT

i �Kj � �di (17)

R= Oa@ |v=tr= |DNU Q=Okt uOw@vswrat x@ xHwD =@ xm CU= u; xHwDp@=k |xDmv|WvQm |SQv= |x@U=Lt |=Q@ x}rw= Kj R= u=wD|t ?} QkD R= xO=iDU= =@ 'CQ=UN

"OQm xO=iDU= xO}O?}U; w[a |v=tr=xO}O?}U; w sr=U u=tr= |Owt p=mW= hQat �di w �i' 17 w 16 |=yx]@=Q QOp=Owt |WvQm |SQv= hQat MSEdij w MSEij w |v=tr= |DNU hQat Kj w18 |x]@=Q R= xO=iDU= =@ p=Owt |WvQm |SQv= Q}}eD 'Ov=xO}O?}U; w sr=U u=tr=

[19]%O};|t CUOx@MSECij = MSEdij �MSEij (18)

|=Q@ |YN=W |xrRvtx@ `@=vt |NQ@ QO xm R}v p=Owt |WvQm |SQv= C=Q}}eD C@Uv%OwW|t h} QaD 19 |x]@=Q R= xO=iDU= =@ [19]'OwW|t |iQat CQ=UN |@=}u=mtMSECRij =

MSEdij �MSEijMSEij

(19)

|SQv= x@ |v=tr= |WvQm |SQv= C@Uv R= pY=L �XN=W "4"2xR=U pm |WvQm

|SQv=� s)=j u=tr= p=Owt |DNU u}t)=i C}U=UL 'OW xQ=W= QDV}B xm xvwou=tyOwW|t h} QaD 6 w 5 |=yx]@=Q R= xO=iDU= =@ xO}O?}U; w sr=U |xR=U QO �|v=tr=

[13]%CU= Q=QkQ@ 21 w 20 |=yx]@=Q 'u; VQDUo w \@=wQ u}= R= xO=iDU= =@ w1 + F dij1 + Fij

= 1 (20)

("dij � "di )"i("ij � "i)"di = 1 (21)

|xR=U |DNU T} QD=t uOw@vswrat Crax@ "ij w "dij |x@U=Lt QO xmv}= x@ xHwD =@QO |DNU T} QD=t u; |=H x@ w OQm xO=iDU= u; j}kO Q=Okt R= u=wD|tv xO}O?}U;Oy=wNv 1 OOa Q@=Q@ 21 |x]@=Q xm CiQo xH}Dv u=wD|t 'OwQ|t Q=mx@ sr=U Cr=L

"OwW|t |iQat �22 |x]@=Q� � ?} Q[ u; |=Hx@ w Ow@�J =

PNmi=1���Ti Kj��i + ��Ti K��i

��Ti K�iPNm

i=1 (�Ti Kj�i + �Ti K�i) ��Ti K��i(22)

"CU= xOWG=QNDU= |=yOt O=OaD Qov=}@ Nm '22 |x]@=Q QO xm

|Oa@ xU ?=k QO CQ=UN |@=}u=mt "3'xOW|iQat |=yXN=W R= xO=iDU= w |Oa@ xU |=y?=k QO CQ=UN |@=}u=mt |=Q@|R=UpOt Ansys |OwOLt u=tr= Q=Ri=sQv =@ xk@] l} |vD@ |Oa@ xU ?=k l}xOw@ |]N |xOwOLt QO xOWs=Hv= |=yp}rLD xm CU= QmP u=}=W [20]"CU= xOW%CU= Q} R pL=Qt pt=W 'j}kLD u}= QO CQ=UN |@=}u=mt xYqNQw]x@ w CU=

&sr=U Cr=L QO |Owt p=mW= G=QNDU= w ?=k |x}rw= |R=UpOt "1

Fij |Owt C}U=UL C@Uv "2"2w sr=U |xR=U |wQ Q@ �|v=tr= |SQv=� s)=j u=tr= p=Owt |DNU u}t)=i C}U=UL

[13]%OwW|t h} QaD 6 w 5 |=yx]@=Q R= xO=iDU= =@ xO}O?}U;

Fij = "ij"i

(5)

F dij ="dij"di

(6)

|DNU u}t)=i Qov=}@ "i w s)=j u=tr= p=Owt |DNU u}t)=i Qov=}@ "ij '5 |x]@=Q QO xm|DNU T} QD=t p}mWD x@ R=}vuwO@ "j w "ij |x@U=Lt |=Q@ "CU= xR=U pm p=Owt[9]%CU= xOW x}YwD 8 w 7 |@} QkD |=yx]@=Q xR=U |WtN Q=DiQ uDiQoQ_vQO =@ w("i)M = (EI) ( i)M (7)

("ij)M = (EI)j ( ij)M (8)

"OwW|t xO=iDU= 10 w 9 |=yx]@=Q R= |x@U=Lt Qw_vtx@ 8 w 7 |=yx]@=Q QO xm( ij)M =

Zj

��00i (x)

�2dx (9)

( i)M =X

(�00i (x))2 (10)

wtHt 10 w 7 |=yx]@=Q QO w j u=tr= |WtN p=Owt |DNU 9 w 8 |=yx]@=Q QOQ=DiQ uDiQoQ_vQO =@ |DNU Q=Okt x@=WtQw]x@ "OwW|t x@U=Lt =yu=tr= pm |DNU%OwW|t x@U=Lt |@} QkD CQwYx@ 12 w 11 |=yx]@=Q R= xO=iDU= =@ xR=U |QwLt

("i)A = (EA) ( i)A (11)

("ij)A = (EA)j ( ij)A (12)

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