improving performance of structures using active instantaneous optimal control
TRANSCRIPT
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�Q=}O=DU=� |QDWwW OtLtu=Oty =v}U |raw@ x=oWv=O '|UOvyt w |vi xOmWv=O
|L=Q] |=yxt=vu}}; C=t=Rr= j@=]t u=wD|t =Q |rQDvm |wQ}v p=ta= =@ xR=U MU=B Vy=m=@ Q=DWwv u}= QO "CN=U xv}y@ |]NQ}e p}rLD |=yVwQ R= xO=iDU= =@ OQmrta T=U=Q@Ow@y@ Q_v OQwt OQmrta x@ uO}UQ CyH QO xR=U Q=DiQ 'p=ai pQDvm sDU}U R= xO=iDU=
=@ xk@] 12 w 8 '4 |=y?=k x@ |}x_Lr |xv}y@ sD} Qwor= =@ pQDvm |wQ}v "OwW|t xO}WN@|xOwOLt QO =Q xR=U xm |}=Hx@=H =D 'OwW|t p=ta= xR=U |]NQ}e Q=DiQ ZQi
"OQ=O xov hOy |OQmrta K]U QO |]NQ}e |m}D=DU= p}rLD R= pY=L |=y|}=Hx@=HOvyO|t u=Wv G}=Dv "CU= |r=ta= |rQDvm |wQ}v Q=Okt Q=}at xR=U OQmrta |x]kv '`k=wQO
c/r=1E+9 C@Uv QO xR=U |}=Hx@=H O=} R Vy=m x@ QO=k |}x_Lr |xv}y@ sD} Qwor=xar=]t OQwt |=yxR=U |=Q@ "OQ=O R=}v |}x_Lqt p@=k pQDvm |wQ}v x@ C@Uv u}= xm &CU=u}= "Ovm|t ^iL IO |OQmrta |xOwOLt QO =Q xR=U '5E9 =D 1E9 u}@ c/r C@Uv
"CU= xO=iDU= p@=k R}v x@=Wt |=yxR=U |=Q@ C@Uv
[email protected]@[email protected]
sD} Qwor= 'hOy |}=Hx@=H 'OQmrta T=U= Q@ |L=Q] 'p=ai pQDvm %|O}rm u=oS=w"|OQmrta K]U '|}x_Lr |xv}y@ pQDvm
xtOkt "1Q=Qk =yxR=U |R=Uy@ w |L=Q] |=v@t xm 'CU= xDiQo CQwY FEMA w ATC
|xrRvtx@ �360 xQ=tW |x} QWv� u=Q}= |}xRQr |R=Uy@ ptar=QwDUO |x}yD "Ov=xDiQoOQmrta T=U=Q@ |L=Q] |=yVwQ OQ@ Q=m QO QwWm |UOvyt |xat=H s=o u} QDsytQO xOv}=Ri Q=@ p}rLD VwQ OQ@ Q=m '1380 p=U QO "CU= KQ]t Q}N= |=yp=U QO|=ysDU}U '2005 p=U QO u}vJsy [3]"CU= xDiQo Q=Qk |UQQ@ OQwt p=ai pQDvm[2]"OvDiQo Q=Qk|UQQ@ OQwt|vD@ |=yxR=U u=UWmQ}e Q=DiQ uDiQoQ_vQO =@ p=ai pQDvmp=U QO OQmrta T=U=Q@ |L=Q] w =[=kD� C}iQ_ |vLvt R= xO=iDU= EL@ QO
[4]"CiQo Q=Qk |UQQ@ OQwt p=ai xt}v pQDvm sDU}U R}v '2009|L=Q] QO |} wQ}v |=yVwQ |=y|DU=m x@ xHwD =@ 'OW QmP xm xvwou=tyuDiQoQ_vQO w |]NQ}e |=yp}rLD |=v@t Q@ O}OH |=yVwQ |x�=Q= x@ R=}v '|}xR=U|UOvyt VwQ 'x}=B u}= Q@ w OW Q=mW; xrRrR s=ovy x@ xR=U |=RH= |ak=w Q=DiQ'|]NQ}e |m}t=v}O p}rLD |=Hx@ VwQ u}= QO "OW x�=Q= OQmrta T=U=Q@ xrRrREa=@ w[wt u}= "OwQ|t Q=mx@ C}iQ_ h}] Q}_v '|]NQ}e |m}D=DU= |=yp}rLDu}= uOQmOQ=w |=Q@ VqD w ?rH C}iQ_ h}] VwQ x@ pQDvm u=UOvyt xHwD xm OW|L=Q] |=Q@ |@ wJQ=yJ '|}xR=U pQDvm "OW R=e; =yxR=U pQDvm |xv}tR QO EL@OwHw x@ 'CU= sm =L |L=Q] Q@ xR=U CmQL xm |r=L QO |}xR=U |=ysDU}UR= |m} |xrRvtx@ p=ai pQDvm |=ysDU}U |=y|oS} w u} QDsyt R= |NQ@ "OQw;|t
w[wt '|m}t=v}O |=yQ=@ QF= CLD =yxR=U MU=B Vy=m 'Q}N= |xyO xU |]T=U=Q@|L=Q] |=yVwQ QOXkv p}rOx@|} wU R= "CU= xOw@ |Q=}U@ |=yVywSB|D=Q}}eDVwNDUO =yxR=U|L=Q]hrDNt|=yxt=vu}}; QO xOWxDiQo Q_v QO Ow}k 'wQ}vOvv=t |DmQL |=yQDt=Q=B 'xR=U �|a=HDQ= Q}e� |ak=w Q=DiQ uDiQoQ_vQO =@ w Ov=xOWu}= |L=Q] QO |}xOvvmpQDvm pt=wa |xrRvtx@ |}xR=U |=y?=DW w =y|}=Hx@=Hx�=Q= OQmrta T=U=Q@ xrRrR |UOvyt VwQ T=U= u}= Q@ "Ov=xOW xDN=vW =yxR=U|m}t=v}O p}rLD |=H x@ xR=U u=UOvyt xm Ot; OwHwx@ hOy u}= =@ VwQ u}= "OW|m}D=DU= |=yp}rLD "Ovvm xO=iDU= |]NQ}e |m}D=DU= |=yp}rLD R= '|]NQ}e|]NQ}e|m}t=v}O p}rLD R= pY=L G}=Dv Ovv=wD|t|@ wN Q=}U@?} QkD =@|]NQ}e|=[Q= |=Q@ xOWxDiQo Q=mx@ u} wv |=yVwQ R= |m} |iQ] R= "Ovvm |v}@V}B =Q'1988 p=U QO "CU=yxR=U QO p=ai pQDvm |=ysDU}U R= xO=iDU= '|DmQL C=t=Rr=|=Q@ |]NQ}e Q=DiQ =@ |}=yxR=U QO p=ai pQDvm sDU}U R= xO=iDU= OQwt QO |D=k}kLDp=U R= R}v OQmrta T=U=Q@ u} wv|L=Q] |=yVwQ |=DU=Q QO [2w1]"OW s=Hv= Kr=YthrDNt |=yptar=QwDUO Q=WDv= ?r=k QO |Q=}U@ uwOt |=yVqD p=L x@ =D 1996
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"""pQDvm
R=xO=iD
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[6]"Qw;VwB p}rLD s=ovy QO |Q}tN |=ypYit p}mWD OvwQ "1 pmW
[7]"Qw;VwB p}rLD R= pY=L |vLvt |wQ |OQmrta Kw]U "2 pmW
"R=}v w C}iQ_ h}] |@Uv C}akwt hrDNt |=yCr=L "3 pmW
|OQmrta |x]kv I}y xm |DQwY QO 'R=}v w C}iQ_ |=y|vLvt s}UQD R= TBxm OwQ|t Q=_Dv= '�OvW=@ xDW=Ov sy =@ |OQwNQ@ C}iQ_ w R=}v h}]� OwWv Ci=},qt=m =Q OwNCtw=kt w OwW|oDN}Uo Q=JO R=}vh}] `HQt |xrRrR uO=OMQ =@ xR=UOQmrta K]U R= Oa@ OQmrta |x]kv xm|DQwYQO "�hr=3 pmW� OyO@ CUO R=MU=B |xv}W}@ 'KQ] |xrRrR CLD xm CU= |vat u}O@ 'OQ}o Q=Qk xOW|v}@V}B"�?3 pmW� CiQ Oy=wN QD=Qi R=Ht OL R= =ypmWQ}}eD w =y|}=Hx@=H pt=W xR=Ux@ R=}v 'OW=@ Q_vOQwt OL =} w |v=H |vt}= OL R= p@k xR=U OQmrta |x]kv Qo =Ow@ Oy=wN R}v |QDnQR@ |=yxrRrR ptLD x@ QO=k xR=U |DL w CU}v |R=Usw=kt
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xR=U p=ai pQDvm sDU}U "2"2%OwW|t p}mWD =yVN@ u}= R= |rm Qw]x@ xR=U pQDvm sDU}U l}
%R= Ov=CQ=@a =yxR=U OQmrta K]U <=kDQ= |=yxv} RoQDu=RQ= w QDu=U; |xO=iDU= VQDUo w Kr=Yt w O=wt |Qw=:vi w Vv=O CiQW}B "1
"|rQDvm |=ysDU}U R="Q}otWJ C=Q}}eD p=ta= uwO@ 'OwHwt |=yxR=U |wQ Q@ u=U; ?Yv C}r@=k "2
"=yxR=U Q=DiQ Ow@y@ QO p=ai pQDvm sDU}U |q=@ C}r@=k "3R=i C=ar=]t p=@vOx@ =yu; C}rw@kt w pQDvm |=ysDU}U x@ O=tDa= C}r@=k "4OQmrta |x}=B Q@ |UOvyt O}OH VwQ |wQ NEHRP |xt=vu}}; swOp=aixt}v w p=ai pQDvm|=ysDU}U|xv} Ro|x�=Q= w �NEHRP-FEMA-273�
[5]"O}OH |=yu=tDN=U |}xRQr |L=Q] |=Q@sDU}U R= O}OH |=yxt=vu}}; T=U=Q@ u=wD|t =}; xm CU= KQ]t p=-wU u}= p=L|R=Uxv}y@ |=v@t xmv}= x@ xHwD =@u}vJsy #OQm xO=iDU= xR=U|R=Uy@ |=Q@ p=ai pQDvm|rQDvm |wQ}v |xv}y@ Q=Okt uDi=} 'pQDvm |wQ}v |R=Uxv}y@ |=yVwQ R= |Q=}U@ QO,=D}=yv w OyO Vy=m pwkat |rQDvm |wQ}v p=ta= OL QO =Q xR=U |}=Hx@=H xm CU='Ovm u}t-=D 'CU= xOW Qw_vt xR=U |}=yv |}=Hx@=H |xOwOLt |=Q@ xm =Q |}=y\QW|=Q@ �u} wv |L=Q] |=yVwQ x@ xHwD =@� |}xv}y@ w XNWt |xOwOLt u=wD|t =};
#OQw; CUOx@ |rQDvm |=ywQ}v j} Q] R= xR=U MU=B Vy=m|wQ}v 'xR=U OQmrta K]U Ow@y@ ut[ xm CU= u; Q@ |aU Q=DWwv u}= QOx@ "OwW p=ta= Q_vOQwt |OQmrta K]U |}=Hx@=H Vy=m |xOwOLt QO |rQDvm|x]kv '`k=wQO "Ci=} Oy=wN Vy=m umtt OL =D |r=ta= |rQDvm |wQ}v '?}DQD u}="OwW|t xDiQo Q_v QO p=ai pQDvm sDU}U QO wQ}v p=ta= Q=}at |xrRvtx@ xR=U OQmrtau}}aD� |}xRQr |@=} RQ= 'xar=]t OQwt |=yxR=U =OD@= 'VywSB hOy x@ p}v |=Q@OQmrta K]U =@ =yu; |=Q@ xOt;CUOx@ OQmrta K]U w OvwW|t �|OQmrta K]UpQDvm sDU}U R= xO=iDU= =@ xR=U '|R=Uy@ x@ R=}v CQwY QO "OwW|t xU}=kt hOyQO |r=ta= |xrRrR QF= Q@ xR=U |xv}W}@ |}=Hx@=H xm |}xvwox@ &OwW|t C} wkD p=ai|wQ}v Q=Okt |xOvQ}oQ@ QO xm pY=L G}=Dv "Ov=t@ |k=@ hOy |OQmrta |xOwOLt|wQ}v 'CU= |r=ta= |xrRrR R= pY=L |}=Hx@=H CLD |xR=U Qy |=Q@ R=}vOQwtQO xR=U |UQQ@ Q=}at xm =Q hOy |}=Hx@=H QO xR=U OQmrta ^iL |=Q@ R=}vOQwt
"OR Oy=wN u}tND R}v 'CU= OQmrta T=U=Q@ |L=Q] |=yVwQ
|Q_v |v=@t "2OQmrta T=U=Q@ |R=Uy@ "1"2
�Qw;VwB� xOv}=Ri |m}D=DU= p}rLD R= 'OQmrta T=U=Q@ =yxR=U |R=Uy@ w |L=Q] QO|wor= =@ |@v=H Q=@ 'VwQ u}= QO "OwW|t xO=iDU= xR=U OQmrta |@=} RQ= |=Q@'O@=}|t V}=Ri= ,=H} QOD KQ] |xrRrR QO Q=_Dv= OQwt pmWQ}}eD x@ uO}UQ =D XNWt"OvwW|t xR=U QO |Q}tN |=ypYit p}mWD?@U|@v=H|}=Hx@=H w wQ}vV}=Ri= w|OQmrta Kw]U w |Q}tN |=ypYit p}mWD OvwQ ?}DQD x@ 2 w 1 |=ypmW
"OvyO|t u=Wv �Qw;VwB� C}iQ_ |vLvt |wQ Q@ =Q hrDNtOv=wD@ O}=@ xR=U 'CU= xOW ?=NDv= u=tDN=U |=Q@ xm |OQmrta K]U x@ xHwD =@1pmWQ}}eD� wQ}v|vLvt QO =ypmWQ}}eD xmv}= uwO@ =Q|@v=H|}=Hx@=H R=|v}at OLw FEMA-356 QO |}=Hx@=H u}= "Ovm ptLD 'OwQ QD=Qi R=Ht OL l} R= =[a=xO}t=v 3=[=kD |}=Hx@=H ATC-40 QO w 2hOy |}=Hx@=H '|R=Uy@ ptar=QwDUO
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78
"""w|tw
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'|QwYv
tx}Oyt |=yT} QD=t "OwW|t xv}y@ �t |v=tR |xR=@ Qy QO '(J(t)) xt} QH `@=D 'l}Uqm
x@ C@Uv Q T} QD=t QO nQR@ |=yx}=QO xm OvDUy |vRw |=yT} QD=t R w Q|wQ}v |oQR@ x@ C@Uv |}=Hx@=H Q}O=kt Vy=m C}ty= |xOvyOu=Wv 'R T} QD=t?} Q[ �(t) 'xDU@� R=@ |rQDvm sDU}U uDiQoQ_vQO =@ "TmaQ@ w CU= |rQDvm
[10�8]%�7 |x]@=Q� Ow@ Oy=wN Cr=L Q=OQ@ w xrRrR |=yl} QLD R= |a@=D 'Sv=Qo q�(t) = GZ(t) + h(t) (7)
%�11 |r= 8 |=yx]@=Q� R= Ov=CQ=@a d(t��t) w 'h(t) 'G x]@=Q u}= QO xmG = �(I + �t2
8 QBR�1BT )�1Q (8)
h(t) = G(Td(t��t) + �t2 W1�xg(t)) (9)
= T�1AT (10)
d(t��t) =e�tT�1fZ(t��t) + �t2 [Bu(t��t)
+W1�xg(t��t)]g (11)
|xS} w |=yQ=OQ@ u; |=yuwDU xmCU=|U} QD=t T w xQy@T} QD=tG '=yx]@=Q u}= QO%R= Ov=CQ=@a 13 w 12 |=yx]@=Q j@=]t Z(t) w u(t) u}vJty "CU= AT} QD=tu(t) = �t
4 R�1BT (GZ(t) + h(t)) (12)
Z(t) =(I � �t2
8 BR�1BTG)�1 � (I + �t2
8 BR�1BTG)
(Td(t��t) + �t2 W1�xg(t)) (13)
sDU}U u}= u=wD|t [11]'MATLAB Q=Ri=sQv |U} wvxt=vQ@ \}Lt R= xO=iDU= =@|xOwOLt QO MU=B u; pQDvm |=Q@ R=}v OQwt |wQ}v w xR=U MU=B w OQm pOt =Q CqO=at
"OQw; CUOx@ =Q Q_vOQwt
xar=]t OQwt |=yxR=U |iQat "3uD@ |xk@] 12 w 8 '4 |xv=yO 3 ?=k 3 'Q=DWwv u}= QO |UQQ@ OQwt |=yxR=UR= R}yQB w |oO=U |=Q@ xOW?=NDv= |=y?=k "OvDUy �|a=HDQ=Q}e Q=DiQ =@� KrUt|v=}t |=y?=k R= =y?=k u}= "Ov=xOW ?=NDv= |Oa@ wO 'xR=U QO VJ}B R= |W=v Q=F;QDt 5 pO=at |Q}oQ=@ ZQa 'CtU 2 R= ,=atH w Ov=xOW ?=NDv= |[Qi |xR=U|Q}PBpmW "CU= QDt 3 2 xk@] Qy `=iDQ= w 5 R}v =y?=k |xv=yO ZQa "OvQ=O|R}NxRQr Q]N =@ |xvyB QO =yu; |Q}oQ=Qk pLt w \UwDt |WtN |=y?=k u}=
"�u=Q}= 2800 OQ=Ov=DU= j@=]t� CU= II wv l =N |wQ Q@ w O=} RxrRrR Q@=Q@ QO =yu=tDN=U |L=Q] |xt=vu}}; j@=]t =y?=k |}xRQr |L=Q]xDiQo CQwY �=@;� u=Q}= uD@ |xt=vu}}; w [12]'pw= V}=Q} w �u=Q}= 2800 OQ=Ov=DU=�xOw@ ZQi u}= u}t-=D |=Q@ u=Q}= 2800 OQ=Ov=DU= pw= V}=Q} w R= xO=iDU= p}rO "CU=|Q@ Q=m w OwW|t xO=iDU= u; R= u=vJsy xm '|t}Ok u=tDN=U l} R= ?=k xm CU=u=tR QO |vwmUt |Q@ Q=m ZQi =@ "OW=@ xOW ?=NDv= 'CU= xOW Zwa R}v u;V}=Q} w T=U=Q@ x}rw= |L=Q] w CN=U u=tR QO =y?=k C}ty= ?} Q[ CN=UQ}}eD ZQi =@ xm |r=L QO [12]'CU= �I=1� l} Q@=Q@ u=Q}= 2800 OQ=Ov=DU= pw=
[13]"CU= xOW ^=Lr 1 2 =yu; |rai C}ty= ?} Q[ '|Q@ Q=m
CaQU '|}=Hx@=H� xR=U |=yMU=B w|HQ=N |wQ}v |Q}oxR=Ov= |=Q@|}=yQoTL �'�?=DW w
sRq |rQDvm |wQ}v u}}aD w =yQoTL R= xOt;CUOx@ C=aq]= VR=OQBsDU}U �'ZwQit |rQDvm sD} Qwor= l} T=U=Q@
"xOWx@U=Lt |rQDvm |wQ}v p=ta= |=ysR}v=mt �?=NDv= xm OQ=O OwHw |irDNt |=yVwQ |r=ta= |rQDvm |wQ}v |x@U=Lt |=Q@sD} Qwor= R= VywSB u}= QO "CU= xDU@=w xR=U \}=QW uwJ |OOaDt pt=wa x@ u;|r=ta= |wQ}v |@Uv pQDvm CyH |QDt=Q=B uDW=Op}rOx@ |}x_Lr |xv}y@ pQDvmQ@ sm =L p}Uv=Qi}O |xrO=at "CU= xOW xO=iDU= hOy |}=Hx@=H x@ uO}UQ |=Q@%CU= 1 |x]@=Q j@=]t 'CU= xOW RyHt p=ai pQDvm sDU}U x@ xm |}xR=U V=aDQ=M �x+ C _x+Kx = E�xg(t) +DU(t) (1)
"OvDUy xR=U |DNU w '|}=Q}t 'sQH |=yT} QD=t ?}DQD x@ K w C 'M 'u; QO xm|=yT} QD=t E w D w 'u}tR ?=DW �xg(t) 'pQDvm |wQ}v Q=OQ@ U(t) u}vJsy"Ovvm|t XNWt =Q xrRrR |wQ}v w pQDvm |wQ}v C}akwt ?}DQD x@ xm OvDUy |v=mt
%OwW|t pY=L 2 |x]@=Q 'Cr=L |=[i QO 1 |x]@=Q uDWwv =@_Z(t) = AZ(t) +W1�xg(t) +Bu(t) (2)
%R= Ov=CQ=@a 4 w 3 |=yx]@=Q j@=]t B w W1 'A 'Z(t) ' _Z(t) 'u; QO xm_Z(t) =
"_x
�x
#; Z(t) =
"x
_x
#(3)
A =
" 0 I
�M�1K �M�1C
#; W1 =
" 0M�1E
#;
B =
" 0M�1D
#(4)
VwQ x@ pQDvm 'OwW xDiQo Q_v QO u}tR |=yl} QLD R= |a@=D \ki u(t) xm|t=ovy|}=Hx@=H� |rQDvm sDU}U MU=B R= |a@=D u(t) xm |DQwY QO "Ow@ Oy=wN R=@ |xkrLsDU}U MU=B R= |a@=D u(t) Qo = w xDU@ |xkrL VwQ x@ pQDvm 'OW=@ �CaQU wuDiQoQ_vQO =@ "CU= xDU@� R=@ |xkrL VwQ x@ pQDvm 'OW=@ u}tR |=yl} QLD wxt} QH `@=D '|}x_Lr |xv}y@ |rQDvm sD} Qwor= w xDU@� R=@ |xkrL VwQ x@ pQDvm
%O};|t CUOx@ 5 |x]@=Q j@=]t 'J(t)
J(t) = ZT (t)QZ(t) + uT (t)R u(t) (5)
%R= Ov=CQ=@a 6 |x]@=Q j@=]t R w Q x]@=Q u}= QO xm
Q =
"c�K 0
0 c�K#
2n � 2nR =
2666664r 0 : 00 r : 0: : : :
0 0 : r
3777775m�m
(6)
sDU}U |O=R; C=HQO ?}DQD x@ m w n w xR=U |DNU T} QD=t K 'x]@=Q u}= QO|xv}y@ pQDvm sD} Qwor= hqNQ@ 'sD} Qwor= u}= QO "OvDUy pQDvm |=ywQ}v O=OaD w
79
"""pQDvm
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U<=kDQ="xk@] 8 ?=k |=yQ}D `]=kt C=YNWt "4 pwOH
|=yOQor}t |=yOQor}t `=iDQ= ZQa xk@]u}}=B q=@ �mm� �mm�3�20 6�20 300 300 83�20 5�20 400 350 73�20 6�20 400 400 63�20 6�20 500 400 53�20 6�20 500 400 43�20 6�20 500 400 33�20 6�20 500 400 23�20 5�20 500 400 1
"xk@] 8 ?=k |=yuwDU `]=kt C=YNWt "5 pwOH|Q=vm |=yuwDU |v=}t |=yuwDU
xk@]|=yOQor}t Oa@ |=yOQor}t Oa@|rw] �mm� |rw] �mm�8�20 350 8�20 350 812�20 400 12�20 400 712�20 400 16�20 400 616�20 450 16�20 450 516�20 450 20�20 600 420�20 600 24�20 600 324�20 600 24�25 600 224�20 600 24�25 600 1
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uD@ |xt=vu}}; T=U=Q@ |L=Q] pwY= C}=aQ =@ 'xR=U |L=Q] w p}rLD |=Q@R= 'OQmrta |UQQ@ Qw_vt x@ '=yu; |]NQ}e |m}D=DU= p}rLD u}vJsy w �=@;� u=Q}=
"CU= xOW xO=iDU= [7]'SAP2000 Q=Ri=sQv|=ypm}U QO Kr=Yt |]NQ}e Q=DiQ VywSB u}= QO 'OW u=}@ xm xvwou=tyuD@ T} RQDU}y Q=DiQ |R=UpOt |=Q@ "CU= xOW xDiQo Q_v QO R}v |Q=PoQ=@ |r=wDtSuneja |O=yvW}B pOt '=yu; u=}t R= xm OQ=O OwHw |irDNt |=ypOt 'KrUtOQwt 'SAP2000 Q=Ri=sQv QO xS} wx@ '|R=UpOt QO CkO p}rOx@ '1999 p=U QO'|vD@ |=[a= QO |oOQwNlQD C=QF= uOQmQw_vt |=Q@ [14]"CU= xDiQo Q=Qk xO=iDU=|xt=vu}}; 'ACI 318-02 xrtH R=� |R=Uy@ w |L=Q] |=yptar=QwDUO =@ j@=]t|=yptar=QwDUO ?r=e w [13]'�swU V}=Q} w� u=Q}= 2800 OQ=Ov=DU= '�=@;� u=Q}= uD@Q=Okt 0 35 =yQ}D |=Q@ w 0 7 =yuwDU |=Q@ xOQwNlQD `]kt |UQv}= u=tt �|R=Uy@1 pwOH j@=]t |R=UpOt QO xDiQQ=mx@ |=yZQi Q}=U "CU= xOW Qw_vt x}rw=
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pY=L G}=Dv w |R=UpOt "4xOQ@s=v |=yptar=QwDUO QO xOWK} QWD VwQ x@ OQmrta |x]kv uOQw;CUOx@ |=Q@|R=UpOt =@ \ki w |DL=Qx@ SAP2000 Q=Ri=sQv R= xO=iDU= =@ 'p@k |=yOv@ QO|=y|}=Hx@=H u=wD|t ?}DQD u}ty x@ "O}UQ xH}Dv x@ u=wD|t Q=Ri=sQv u}= QO =y?=kx@ 'xR=U C}a[w Ow@y@ w xU}=kt |=Q@ =Q hrDNt |OQmrta Kw]U R= GwQN =@ pO=at'pwtat |=y|R=UpOt Q@xwqa Qw_vt u}=x@ "OQw; CUOx@ |rQDvm |=ysDU}U ltm
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OQmrta |x]kv |x@U=Lt w |]NQ}e |m}D=DU= p}rLD G}=Dv "1"4�|}xRQr |@=} RQ=�
C=L}[wD x@ xHwD =@|UQQ@ OQwt |=y?=k|}xRQr|@=} RQ= R= pY=L G}=Dv |xYqN"CU= xOt; 12 |r= 9 |=ypwOH QO 'p@k CtUk QO xOWQmP |=yZQi w
OQmrta |x]kv xm OwW|t x_Lqt =ypwOH QO xOWx�=Q= C=aq]= x@ xHwD =@CU= u=Um} ,=@} QkD FEMA-356 w FEMA-440 |=yxt=vu}}; QO xOWx@U=Ltu}= "CU= QDW}@ ATC-40 |xt=vu}}; R= pY=L G}=Dv 'xOWx@U=Lt Q}O=kt u}@ R= wO}OH |=yptar=QwDUO QO QDu=v}t]= p@=k VQ}PB |xOwOLt |xOvyOu=Wv hqDN=|xOwOLt QO xR=U |=[a= R=}v Ctw=kt 'Qo}O u=}@ x@ "CU= ATC-40 x@ C@Uv
"xk@] 12 ?=k |=yuwDU `]=kt C=YNWt "7 pwOH|Q=vm |=yuwDU |v=}t |=yuwDU
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QO OQmrta |x]kv |x@U=Lt |=Q@ Q=Ri=sQv x@ xOW|iQat |=yQDt=Q=B "8 pwOH"FEMA-356
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C=YNWt =@ '�u=Q}= 2800 OQ=Ov=DU=� xrRrR Q@=Q@ QO =yu=tDN=U |L=Q] |xt=vu}};|=yxO=O Q}=U "CU= xOW xO=iDU= O=} R Q]N =@ |R}NxRQr |xk]vt w II wv l =N
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"Ov=xOW QmP 13 pwOH QO Q}O=kt u}= |t=tD "CU=xOW xO=iDU= R}v �O}OHsRq |xv}W}@ pQDvm |wQ}v w C=k@] |}=Hx@=H |xv}W}@ CQwYx@ p}rLD G}=Dv'6 w 5 |=ypmW |=yQ=Owtv "CU= xOW G=QNDU= =y?=k |}=Hx@=H Vy=m |=Q@C=k@] |}=Hx@=H =yQ=Owtv u}= QO "CU=y?=k pQDvm G}=Dv |xYqN R= |}xvwtv
"CU= xOW sUQ T@] |xrRrR CLD R=}vOQwt |rQDvm |wQ}v w xk@] 4 ?=k|wQ}v =yCr=L |t=tD QO '6 w 5 |=ypmW QO xOWsUQ |=yQ=Owtv x@=Wtu}vJsy "CU= xOW |v=kwi C=k@] QO XwYNx@ |}=Hx@=H Vy=m ?@U pQDvm|QDW}@ Vy=m |}=Hx@=H '|rQDvm |wQ}v V}=Ri= xH}Dv QO w c/r C@Uv V}=Ri= =@
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"FEMA-440 T=U= Q@ OQmrta |x]kv C=YNWt "10 pwOHV (kN) D (m) Sa (g) Sd (m) C=k@] O=OaD479 08 0 25 0 25 0 19 4906 62 0 32 0 23 0 24 81232 3 0 48 0 21 0 36 12
xO=iDU= C}r@=k K]U |=Q@ hrDNt |=y xt=v u}}; QO hOy |}=Hx@=H "11 pwOH"xikw |@
ATC-40 FEMA-356 FEMA-440 C=k@] O=OaD0 21 0 25 0 25 40 3 0 31 0 32 80 42 0 49 0 48 12
"s=@ hrDNt |=y|}=Hx@=H QO pY=it p}mWD OvwQ "12 pwOH�m� hrDNt |OQmrta Kw]U pO=at |}=Hx@=H
CP LS IOQO xR=U |}=Hx@=H O=OaDpYit u}rw= p}mWD C=k@]
0 20 0 13 0 12 0 09 40 34 0 21 0 19 0 13 80 51 0 35 0 32 0 24 12
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"CU= u=Um} FEMA-274 w ATC-40 |xt=vu}}; wO |=Q@C}=aQ =@ xm|}=y?=k R=l}I}y xmCi=} QO u=wD|t C=@U=Lt G}=Dv QOCkO =@w p}rLD �=@;� u=Q}= uD@ |xt=vu}}; w �pw= V}=Q} w� u=Q}= 2800 OQ=Ov=DU= \@=w[,qFt "OvDU}v CP |xOwOLt QO |DL |OQmrta OwOL u}t-=D x@ QO=k 'Ov=xOW |L=Q]|=Q@ =Q hOy |}=Hx@=H 'IO |xOwOLt QO Q=DiQ |=Q@ FEMA-356 |xt=vu}};|}=Hx@=H =D CU= sRq |va} 'OyO|t CUOx@ QDt|Dv=U 31 pO=at xk@] 8 ?=kR= V}@ pmWQ}}eD OwOL =@ |rYit '?=k u}= |=[a= R= l}I}y QO 'QDt|Dv=U 31K]U R= |DL |QDt|Dv=U 21 |}=Hx@=H QO ?=k u}= |rw &OwWv p}mWD IO K]U
"p}rLD QO xOW xO=iDU= c/r |=yC@Uv "13 pwOH1 00E + 09 1 00E + 08 7 75E + 07 5 50E + 07 3 25E + 07 1 00E + 07
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"""w|tw
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|xv=}t QO QDW}@ C=k@] |}=Hx@=H Vy=m QO OwHwt |=yVyH "CU= xDi=}|xU}=kt "OwW|t xO}O 1E9 w 1E8 u}vJsy w 1E8 w 1E7 u}@ c/r |=yC@UvQ_v R= =Q G}=Dv CLY [2]'2004 p=U QO xOWx�=Q= G}=Dv =@ xQ=W=OQwt |=yQ=Owtv"Ovvm|t O}}-=D |rQDvm |=ywQ}v |@} QkD |xOwOLt w |}=Hx@=H |=yVyH |ovwoJpQDvm |wQ}v 'hrDNt c/r |=yC@Uv QO xm OvyO|t u=Wv |rQDvm |wQ}v |=yQ=Owtv|=Q@ pQDvm |wQ}v Q}O=kt Cw=iD w Ovm|t =O}B V}=Ri= |}q=@ C=k@] QO G} QODx@x@ xHwD =@ "CU}v RQ=@ u=OvJ q=@ |=yC@Uv QO RHx@ c/r hrDNt |=yC@UvQ_vOQwt |OQmrta |}=Hx@=H x@ uO}UQ hOy =@ xR=U MU=B Vy=m xm w[wt u}="CU= xDi=} xt=O= ?wr]t OQmrta x@ uO}UQ OL =D MU=B Vy=m u=R}t 'CU=R= |W=v |}=Hx@=H Vy=m QO OwHwt |=yVyH 'OW KQ]t QDW}B xm Qw]u=tyOL R= QD=Qi =Q |}=Hx@=H Q}O=kt xm OvDUy |}xvwo x@ |rQDvm O=} R |=ywQ}v p=ta=Vy=m CyH R}v c/r C@Uv |v=}t Q}O=kt R= wQu}= R= "OvyO|t Vy=m ?wr]tx@ xHwD =@ ',qFt "CU= xOW xO=iDU= Q_vOQwt hOy x@ uO}UQ |=Q@ 'pQDvm |wQ}vj@=]t xm |r=L QO &CU= QDt|Dv=U 13 Q@=Q@ ,=@} QkD s=@ |}=Hx@=H '5 pmW Q=OwtvQDt|Dv=U 12 pO=at IO OQmrta K]U R= GwQN pO=at |}=Hx@=H '12 pwOHxOW GQ=N R}v LS w IO |OQmrta K]U R= ?=k u}= wQu}= R= "CU= xOt; CUOx@pQDvm |wQ}v p=ta= =@ Cr=L u}= QO "CU= xO}UQ CP |OQmrta K]U RQt x@ wQ_v OQwt xOwOLt QO ?=k s=@ |}=Hx@=H ^iL QO |aU 'hrDNt |=yxR=Ov= QO
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"xk@] 4 ?=k -IO |OQmrta xOwOLt QO xR=U ^iL |=Q@ sRq |=yQDt=Q=B "14 pwOH|rQDvm |wQ}v |}=yv |}=Hx@=H c/r C@Uv uwO@ |}=Hx@=H |xv}W}@ s=v(kN) xv}W}@ (m) xOW ?=NDv= (m) pQDvm |wQ}v CW=ov?=DW
31 0 12 5 50E + 07 0 13 Q@?;32 0 12 5 50E + 07 0 13 T@]31 0 12 5 50E + 07 0 13 Q=@OwQ
"xk@] 8 ?=k -IO |OQmrta |xOwOLt QO xR=U ^iL |=Q@ sRq |=yQDt=Q=B "15 pwOH|rQDvm |wQ}v |}=yv |}=Hx@=H c/r C@Uv uwO@ |}=Hx@=H |xv}W}@ s=v(kN) xv}W}@ (m) xOW ?=NDv= (m) pQDvm |wQ}v CW=ov?=DW
47 0 19 7 75E + 7 0 21 Q@?;55 0 19 7 75E + 7 0 21 T@]62 0 19 7 75E + 7 0 21 Q=@OwQ
"xk@] 12 ?=k -IO |OQmrta |xOwOLt QO xR=U ^iL |=Q@ sRq |=yQDt=Q=B "16 pwOH|rQDvm |wQ}v |}=yv |}=Hx@=H c/r C@Uv uwO@ |}=Hx@=H |xv}W}@ s=v(kN) xv}W}@ (m) xOW ?=NDv= (m) pQDvm |wQ}v CW=ov?=DW
72 0 32 1 00E + 08 0 35 Q@?;81 0 32 1 00E + 08 0 35 T@]91 0 32 1 00E + 08 0 35 Q=@OwQ
83
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"OQw; CUOx@ =Q pQDvm |wQ}v R= xO=iDU= =@ ?=k Qy |R=Uy@ |=Q@ R=}vOQwt c/r
^iL |=Q@ xm OQm|v}@V}B u=wD|t 10|r= 8 |=ypmW |=yQ=Owtv|@=}uwQ@ R=12 w 8 '4 |=y?=k |=Q@ ?}DQDx@ u=wD|t 'hOy |}=Hx@=H CLD xR=U OQmrtac/r=1 28E+9 w c/r=9 87E+8 'c/r=1 32E+9 |=yC@Uv R= xk@]?U=vt pQDvm |wQ}v p=ta= xm Cio u=wD|t G}=Dv u}= |xU}=kt R= "OQm xO=iDU=w xR=U |x}rw= OQmrta K]U x@ '?wr]t OQmrta x@ uO}UQ |=Q@ c/r ?=NDv= =@?U=vt C@Uv ?=NDv= wQu}= R= w CU= xDU@=w xR=U |x}rw= Ctw=kt x@ QDy@ CQ=@ax@xR=U C=YNWt x@ |rm Qw]x@ w xOw@ Cw=iDt hrDNt |=yxR=U w =yxrRrR |=Q@ c/r
wQu}= R= "O=O O=yvW}B =y?=k |t=tD |=Q@XNWt Q=Oktl} u=wD|tv wCU= xDU@=w|}xRQr Q]N p}rLD R= TB xR=U Qy |R=Usw=kt |=Q@ CU= QDxDU}=W OUQ|t Q_v x@|wQ}v Q=Okt =D 'OwW p}rLD Q_vOQwt |xR=U '�xk]vt ptDLt |xrRrR u}}aD� x=oDN=U"O}; CUOx@ p}rLD G}=Dv =@ ?U=vDt |rQDvm C=R}yHD xH}Dv QO w R=}v OQwt |rQDvmx}YwD MU=B Vy=m |=Q@ sDU}U u}= R= xO=iDU= uOw@v|O=YDk= x@ xHwD =@ u}vJsy�pQDvm |=yx=oDUO� =yQrQDvm u=mt|@=}xv}y@ 'xOWu=}@ |=yp}rLD Q@xwqa xm OwW|tOQwt OQmrta x@ uO}UQ ut[ '|rQDvm |wQ}v u} QDtm p=ta= =@ =D OQ}PB CQwY R}v
"OW=@ x}HwD p@=k R}v |O=YDk= Q_v R= KQ] 'Q_v
|Q}oxH}Dv "5u}= u=wD|t '=yu; R= xOt;CUO x@ G}=Dv w xar=]t OQwt |=yxR=U p}rLD T=U=Q@
%OQm u=}@ =Q OQ=wtxR=U |}=Hx@=H O=} R Q=}U@ Vy=m x@ QO=k |}x_Lr |xv}y@ p=ai pQDvm sD} Qwor= �p@=k pQDvm |wQ}v Q=Okt 'C@Uv u}= xm CU= c/r=1E+9 C@Uv R= xO=iDU= =@
"OQ=O R=}v R}v =Q |}x_Lqt=D 1E9 u}@ c/r C@Uv �xk@] 12 w 8 '4 |=y?=k� xar=]t OQwt |=yxR=U |=Q@ �xOwOLt u}= wQu}= R= "Ovm|t ^iL IO |OQmrta |xOwOLt QO =Q xR=U '5E9sRq C=R}yHD |v}@V}B |=Q@ "OUQ|t Q_vx@ ?U=vt 'x@=Wt |=yxR=U pQDvm |=Q@"OwW|t x}YwD Q_vOQwt |xR=U x@ XDNt p}rLD 'hOy u}= x@ uO}UQ CyHOQmrta K]U x@ uO}UQ |=Q@ c/r C@Uv |xv}y@ ?=NDv= xm OvyO|t u=Wv G}=Dv �w �|rai OQmrta K]U� xR=U |x}rw= Ctw=kt uwJ |irDNt pt=wa x@ hOyCU= sRq w CU= xDU@=w xR=U u; CN=U \}=QW x@ XwYNt ptDLt |xrRrRQ@xwqa xSwQB u=ty x@ XwYNt Q]N p}rLD 'xOWEL@ VwQx@ |R=Uy@ |=Q@
"OQ}PB CQwY u; |OQmrta p}rLD
"xk@] 4 ?=k QO s=@ |}=Hx@=H Vy=m u=R}t "8 pmW
"xk@] 8 ?=k QO s=@ |}=Hx@=H Vy=m u=R}t "9 pmW
"xk@] 12 ?=k QO s=@ |}=Hx@=H Vy=m u=R}t "10 pmW
'4 |=y?=k OQmrta K]U |UQQ@ |=Q@ xm CU= |r=L QO u}= "OyO pmWQ}}eD CP
?=k x@ =Q QDt|Dv=U 49 w 32 '25 Q@=Q@ |}=y|}=Hx@=H ?}DQD x@ O}=@ xk@] 12 w 8p=ta= =@ 'IO K]U QO |OQmrta uDW=O |=Q@ =y?=k xm CU= sRq w OQm p=ta=xU |=[=kD� C}iQ_ |=y|vLvt "Ovv=t@ Q_vOQwt |xOwOLt QO \w@ Qt |}=Hx@=H|=y|}=Hx@=H QO ?}DQDx@ =yxR=U u}= xm OvyO|t u=Wv xk@] 12 w 8 '4 ?=k|=y?=k xm |r=L QO &Ov=xOW GQ=N IO K]U R= s=@ |QDt|Dv=U 32 w 19 '12R= QDtm ?D=Qt x@ |}=y|}=Hx@=H 'xOWxO=iDU= |=yCW=ov?=DW CLD xar=]tOQwtwQ u}= R= "Ov=xOW ptLDt =Q OwN OQmrta K]U |=Q@ �hOy |}=Hx@=H� sRq Q}O=ktxR=U x@ |}xrRrR p=ta= CQwY QO =D OwW p=ta= |rQDvm |wQ}v xR=U x@ CU= sRq
84
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1. force- deformation2. target displacement3. demand displacement
�References� `@=vt1. Yang, J.N., Long, F.X. and Wong, D. \Optimal con-
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85
Abs
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In two recent decades, a new method has been inventedwhich is called Active Structural Control. Applying theexternal force, when it is needed; not only can reduce theresponse displacement but also is able to make a highreduction in response velocity and acceleration. Pre-dict the building behavior during an earthquake eventhas an important role in order to have a safe structure.The structure engineers usually use two methods to com-pute the structure response: i) Dynamic Analysis or ii)Nonlinear Static Analysis. Since the dynamic analysisdepends on several items such as frequency content ofearthquake record, the most structure specialists preferto employ the nonlinear static analysis (pushover anal-ysis) to compute the target displacement in IO, LS andCP levels.The essential goal of this research is to create a combina-tion between active structural control concept and non-linear static analysis. Three reinforced concrete build-ings with 4, 8 and 12 oors are selected. All buildingshave three bays and are analyzed using the pushovermethod and for each case a performance level is found.The main purpose of using active structural control equip-ment in the selected buildings is to keep the building per-
formance in that limit which is found at the analyticalpart.An important aspect of this investigation is the incorpo-ration of the possibility of inelastic deformation of themembers into the control algorithm. In analytic investi-gation, in order to solve the control formula, the Instan-taneous Optimal Control algorithm is employed whichis conducted to introduce \c/r" ration. This ratio showsthe importance of the amount of control force againstthe importance of reduction in structural response. Theresults show that the best c/r ratio for the selected struc-tures is between 1�109 and 5�109 which can keep thestructures in IO performance level. Generally, the opti-mal amount of c/r ratio to get the target performancelevel depends on di�erent items such as the primarystrength of structure (the initial performance level) andprobable seismic hazard, therefore; it seems that the haz-ard risk analyses is necessary as an additional analysis.
Key Words: Active control, performance based design,target displacement, instantaneous optimal control algo-rithm, performance level.
132
IMPROVING PERFORMANCE OFSTRUCTURES USING ACTIVEINSTANTANEOUS OPTIMALCONTROLM. [email protected]. Massumi(corresponding author)[email protected]. of Civil EngineeringKharazmi UniversityM. [email protected]. of Civil EngineeringBu-Ali Sina University
Sharif Civil Engineering JournalVolume 29, Issue 2, Page 77-85, Original Article
c Sharif University of Technology
� Received 18 June 2011; received in revised form 3 December2011; accepted 7 January 2012.
AbstractDuring recent earthquakes, the behavior of structureshas demonstrated that the main reason for building col-lapse is the big deformation during strong seismic events.Therefore, the structural designers would like to reducethe lateral displacement of structures under a speci�edlevel. Using the shear walls and bracing and increasingthe members size are the common ways which can pro-tect the buildings against the earthquake or wind load.