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6SC )B:m&Bs&afYDe@p:,Ғ Ȝ>{o˸rꦄHrz_CӃRHRUyϹ ÕL G0S%A> ).;dG9rB ]tDuJRRjcp)>yscA*}RH.c0F3 ߍz^H͘Olly}tx{Hts5;W/~j ;)Q[]E33*VCUZy[lhJjjJ lXHyBo^9 `lWu9 %-R߰9l7)"Ejf|69՘$\.V6hQ6\ExC 4˥uYܩå" _C_@k3kLmXrPI Ҙ& 3v7/0*~w7^0FT/v Jڑig/J_pHJܻZzD:JABPK!sPdrs/downrev.xmlDN1&CsLIG)$^ ܰ;L\tl0?-t>Vb(E\xpests*&dg2psO~*%#s4PZǢ&q;bJ&6I]Y6;z@˷բ߯˪]Ï1WT!گVzTr>n0& H PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!7#*drs/shapexml.xmlPK-!sP{drs/downrev.xmlPKzpB   PJ?Line 9"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!3$drs/shapexml.xmlUn1}Z%,4B~ .l/|}H*ev=99s|}Uluv3Jfz_dz+J]4tfuNWNDZWM5U*jrǵlQ 4Pz3g6Xi3D9ehPu*4'#i=3 ak€QZxC@@/|5_j :FdCؤxiJ7xo+N☪"ۜbOv9t/(' {J.%)op߹9 r*\:JpHi(FakWݎepͥhoG4-G<3b0TPܑ'BMI-<[=2H;[u<7ңhP[iaNpr]~&F e~7,kNky6- pHMN5& k u(OkUvs ~>|K D/PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!3$*drs/shapexml.xmlPK-!96|drs/downrev.xmlPK|>p?B   R$?Line 10"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!yU"drs/shapexml.xmlUn0 }:tq.ڠnA}-ˉ] IΥ_?Jr,mQyxtsSluvfpL)2+J]4tfFuNWףNDZWྚk\U.;k)eٰ@hzf^lff(s.1YBshuIހ]Gia{h3^^ jטu"6#^IiJ7x*☪"bS9 F| ހK ljwn@BJ,<l·StL8MsÐmN,vF2 0]cidNd\9uSsKP9/}š|eDAdc%=*<\a<݃)!Aȵ4rz*s;!Å)R`hn)qJ$n1pzYFH=YKf'\_<>:0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!%RWT%drs/shapexml.xmlUn0 }:tq.͚nA}-ˉ] IΥ_?Jr,mQ!yxt}WluvfpL)*Ӌ+J]4fuNNDZ5Wྚk\Uծ:k)eٰ@hzG`Q洗](Ѡ0LhN]i]7 aQZaro m&k+XC͗C`M#4DOx&)q4`>0WV *csM9?B`0> (al $Syv|()DTHyn15=TWa;#Eήd 2[((fM -B* T5A!M*<0]i ޔ$[s=U?H_;!)Hv4!(d%jG4AP$%%P,s:z>%'o3}o7a,He[ pe<8L= FHYK'.[~y5%pfջů]-dgt{M,Ö/.Ex~A}EӢbģUPBqG<Zx2H <[u<7ңhQ[iaJpr]~&F e~7,kNky1- pHMN5 "Zl w^ vikwpH PSE)?$iLV;ُ7 ?J/~Xfdiȴ/J_pHJܻZzD:JQBPK! 4drs/downrev.xmlD]k0p†vFݘu۳ 6IIl._/nS;E\q Ddt;HDv w{ɍH#.e )cՒŸ=>X$C#u!NY"-ZRu+cLQ\w\kI+ȳ5cF_02(%6_PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!%RWT%*drs/shapexml.xmlPK-! 4}drs/downrev.xmlPKz7p8B   RJ?Line 12"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! 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Ԕ?"6@`NNN?N"f I 2 AhB T Ԕ?"6@`NNN?NX ; 2 BhB Tm Ԕ?"6@`NNN?N dz 1& Ch  1,$D' 0B Dh HZDԔ?"0@NNN?N&B Eh BDԔ?"0@NNN?N1SS Xf d Fh3  Z ,$D( 02 Gh T% Ԕ?"6@`NNN?N"f I 2 HhB T Ԕ?"6@`NNN?NX ; 2 IhB Zm Ԕ?"6@`NNN?N dI Jh <[  ?"6@ NNN?Nw ,$ 0 k%Joints have to be in the equilibrium!*&>8 d2&C"z  K  Kh  [ ,$D 0" Lh <c  ?"6@ NNN?N  p SX = 04>0 d2 " Mh <x  ?"6@ NNN?N  p SY = 04>0 d2 D Nh <  ?"6@ NNN?N K  SMK = 0T>0 d2  Xf d Oh3 M̘q Tr ,$D 02 Ph T% Ԕ?"6@`NNN?N"f I 2 QhB T Ԕ?"6@`NNN?NX ; 2 RhB Zm Ԕ?"6@`NNN?N dB Sh@ BD>?"0@NNN?N/ d/ ,$D/ 02 Th NZ ?"6@ NNN?NA d'h ,$D 02 Uh NZ ?"6@ NNN?N txn,$D 02 Vh NZ ?"6@ NNN?N ,$D 07 Wh <~  ?"6@ NNN?N; ,$ 0 Y?(>0 d2C  2 Xh NZ ?"6@ NNN?N& M ,$D- 02 Yh NZ ?"6@ NNN?N %),$D, 07 Zh <|  ?"6@ NNN?N  ,$. 0 Y?(>0 d2C  9 [h <t  ?"6@ NNN?N(  ,$$ 0 [!?!(>0 d2C  9 \h <y  ?"6@ NNN?N/,$) 0 [!?!(>0 d2C   ^h <߫  ?#" `T BFrames  2G `h <8h  ?"xKV,$ 0 E STIFF JOINT 2 C ah <D  ?"l(J,$  0 G HINGED JOINT 2CH h 0޽h ?O@5h6h7h9h:h;h@hAh Bh Gh Hh Ih PhQhRhThUhVhXhYh 33.w&w___PPT10w.[R+j9Drt' [ = 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=1:Bvisible*o3>+B#style.visibility<*\h%(D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*FhD' =1:Bhidden*o3>+B#style.visibility<*Fh%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*\hD' =1:Bhidden*o3>+B#style.visibility<*\h%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Yh%(D' =-o6Bdissolve*<3<*YhD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Xh%(D' =-o6Bdissolve*<3<*XhDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Zh%(Dy' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Sh%(D' =-o6Bdissolve*<3<*ShD' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*ZhD' =1:Bhidden*o3>+B#style.visibility<*Zh%(+h+0+Jh0 ++0+Wh0 ++0+Wh0 ++0+Zh0 ++0+Zh0 ++0+[h0 ++0+[h0 ++0+\h0 ++0+\h0 ++0+`h0 ++0+ah0 +j  0 !!,/l=!(  lB l <DԔ?"0@NNN?N| ,$D 0^ l 6u fu ^ l 65^ l 61 f 1  l B ?"6@ NNN?Ne -e  l B ?"6@ NNN?N o  l <D&N  ?"6@ NNN?N7>-",$D 0 OWhat is the difference between beams and frames? 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NNN?NT F ,$D 0 vt H o?"6@ NNN?N  ,$D 0 xt <Ml  ?#" `T BFrames  2Gl ib  tj m,$D  0B t NЂDffo?"0@NNN?Nb g B t NЂDffo?"0@NNN?N B t B%TRl  ?"6@ NNN?Ni nsJ>0 d2CKCB t NЂDffo?"0@NNN?N|H t 0޽h ?_P t t>tktut 33qKiK___PPT10IK.[R+Z'DH' [ = @B DH' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0t%(D' =%(D' =%(DC' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/t%(D' =-6B#blinds(vertical)*<3<*/tD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ht%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Dt%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*et%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*ht%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Ft%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*st%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*tt%(D' =-o6Bdissolve*<3<*ttDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*it%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*ut%(D' =-o6Bdissolve*<3<*utDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Et%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*ft%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*vt%(D' =-o6Bdissolve*<3<*vtD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Vt%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-u6Bwipe(right)*<3<*t+0+0+t0 ++0+t0 ++0+t0 ++0+t0 ++0+Dt0 ++0+Et0 ++0+Ft0 ++0+et0 ++0+ft0 ++0+ht0 +>@  0  x$(  x^ x 65 x B ?"6@ NNN?Ne -e  x B ?"6@ NNN?N ^ x 6e me Q x < ?"6@ NNN?N. {,$ 0 s/Example: Diagrams of M, Q, N for a simple frame(0>0 d20Cz c6   x u,$D 0B  x <D>?"0@NNN?Nc6 B  x <D>?"0@NNN?NB  x <D>?"0@NNN?N6 z  f  x  f,$D 0B x <Do?"0@NNN?N B x <Do?"0@NNN?N f B x <Do?"0@NNN?NT Tz c6  x   f ,$D 0B x <Do?"0@NNN?Nc6 B x <Do?"0@NNN?NB x <Do?"0@NNN?N6 z c6  x h u,$D 0B x <Do?"0@NNN?Nc6 B x <Do?"0@NNN?NB x <Do?"0@NNN?N6  z  x r ,$@  0B x <Do?"0@NNN?NB x H30Do?"0@NNN?N B x H30Do?"0@NNN?NB x H30Do?"0@NNN?NjnB x H30Do?"0@NNN?NB x H30Do?"0@NNN?NB  x H30Do?"0@NNN?NBGB !x H30Do?"0@NNN?NB "x H30Do?"0@NNN?NB #x H30Do?"0@NNN?NB $x H30Do?"0@NNN?NafB %x H30Do?"0@NNN?NV &x BLX' ?"6@`NNN?N,$  0 r1,5 kN/m, >0 d2 G1z W   'x -/,$@ 0R (x H o?"6@ NNN?Ng  B )x <Do?"0@NNN?NW  R *x H o?"6@ NNN?Nu,$D 0B +x BDԔ?"0@NNN?N ,$@  0R ,x BL$2 ?"6@`NNN?N- ,$  0 n2 kN,>0 d2G1 z r/ e  -x r/ ,$D 0B .x <D1?"0@NNN?Nr re B /x <D1?"0@NNN?N e B 0x <D1?"0@NNN?N e B 1x HZD?"0@NNN?Nr  B 2x HZD?"0@NNN?N  ! 3x BL2 ?"6@`NNN?Nt   i2 m(>0 d2CN  / R  4x  / R B 5x <D1?"0@NNN?N / B 6x <D1?"0@NNN?N N / R B 7x BD?"0@NNN?N  N # 8x BL2 ?"6@`NNN?N   k1,5 m(>0 d2C! 9x BLУ  ?"6@`NNN?N   i1 m(>0 d2CR :x BL  ?"6@`NNN?N  ,$ 0 n2 kN,>0 d2GB ;x BD1Ԕ?"0@NNN?N ,$D  0B 0 d2GB >x HZDԔ?"0@NNN?N ,$@ 0R ?x BL   ?"6@`NNN?N* ,$ 0 n2 kN,>0 d2GB @x <D1o?"0@NNN?N],$@ 0B Ax BD1o?"0@NNN?NnWW,$@  0B Bx BD1o?"0@NNN?N},$D" 0z rG  Cx rGQ,$D 0B Dx <D1?"0@NNN?NrW GW L2 Ex 3 BLCQENG,˳HJQQ jJ? `TQLb.`TQLb.QLb.Q"6@ NNN?Nt  Fx <H  ?"6@ NNN?Nv  \a*>0 d2 v Gx <t  ?"6@ NNN?N%,$ 0 $sina=0,6 cos a=0,86>0 d2C B Hx BD1o?"0@NNN?Nmn,$@$ 0z  i  Ix  c ,$D/ 0B Jx <D1o?"0@NNN?N i B Kx <D1o?"0@NNN?N i B Lx <D1o?"0@NNN?N    i  Mx# Zk t ,$D* 0B Nx <D1o?"0@NNN?N i B Ox <D1o?"0@NNN?N i B Px <D1o?"0@NNN?N  z Q / Qx K ),$D9 0B Rx BN~D1o?"0@NNN?N<z</B Sx BCD1o?"0@NNN?N^Q ^B Tx <D1o?"0@NNN?NSz o 9  Ux | c ,$DK 0  Vx <G  ?"6@ NNN?No 9  Y+(>0 d2C 2 Wx H 1?"6@`NNN?Nz / Sz n U  Xx  ,$DJ 0  Yx <L  ?"6@ NNN?Nn U  Y-(>0 d2C 2 Zx H 1?"6@`NNN?N K Sz n U  [x  f1 ,$D3 0  \x <Q  ?"6@ NNN?Nn U  Y-(>0 d2C 2 ]x H 1?"6@`NNN?N K Sz o 9  ^x B ),$D. 0  _x <U  ?"6@ NNN?No 9  Y+(>0 d2C 2 `x H 1?"6@`NNN?Nz / B ax BD1?"0@NNN?N` ,$D' 0N bx <Z  ?"6@ NNN?Nw q ,$ 0 pQ [kN]:>0 d2F1C i cx <`  ?"6@ NNN?NL\OF ,$ 0 M [kNm]:>0 d2F1C&B dx HDo?"0@NNN?N]^,$D( 0= ex B:f  ?"6@ NNN?Not,$: 0 Y0,8&>0 d2C ! fx <j  ?"6@ NNN?N<,$ 0 C3&>0 d2C! gx <@o  ?"6@ NNN?NCn*,$# 0 C1&>0 d2C! hx <Ts  ?"6@ NNN?N\UC,$ 0 C3&>0 d2C> ix Bw  ?"6@ NNN?NH ,$@ 0 Z0,33&>0 d2C ' jx Bz  ?"6@ NNN?N,$& 0 C1&>0 d2C kx  0e0e    B#C DEpF0 @ 1 o 8c8c     ?A)BCD|E|| Kp%s;eCdcK2 (Jl[*i?#@         # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab% H,$D% 0B lx@ <D1o?"0@NNN?NnW},$@! 0B mx@ <D1o?"0@NNN?NT,$D 05 nx <~  ?"6@ NNN?N* ) ,$C 0 W2&>0 d2C = ox B;i  ?"6@ NNN?Ni (~P ,$; 0 Y1,6&>0 d2C 5 px <d  ?"6@ NNN?N G } ,$, 0 W2&>0 d2C = qx B{;d  ?"6@ NNN?N  ,$I 0 Y1,2&>0 d2C = rx BrL  ?"6@ NNN?N! % ,$DH 0 Y0,6&>0 d2C 5 sx <$  ?"6@ NNN?N U ,$2 0 W2&>0 d2C 5 tx <   ?"6@ NNN?N  ,$1 0 W2&>0 d2C   i  ux# Z ]" ,$DA 0B vx <D1o?"0@NNN?N i B wx <D1o?"0@NNN?N i B xx <D1o?"0@NNN?N  B yx BN~D1o?"0@NNN?N ,$@E 0B zx BCD1o?"0@NNN?N F ,$DG 0B {x <D1o?"0@NNN?N% j ,$@F 0z 4 |x .,$D< 0B }x 6D?"0@NNN?N4V5B ~x 6D?"0@NNN?N\b x B ?"6@`NNN?Nk5  x <T  ?"6@ NNN?N5 [0,5 m&>0 d2C Sz o 9  x X ?,$D= 0  x <  ?"6@ NNN?No 9  Y+(>0 d2C 2 x H 1?"6@`NNN?Nz / Sz n U  x  H ,$D> 0  x <X  ?"6@ NNN?Nn U  Y-(>0 d2C 2 x H 1?"6@`NNN?N K Sz n U  x   ,$DD 0  x <,   ?"6@ NNN?Nn U  Y-(>0 d2C 2 x H 1?"6@`NNN?N K 5 x <,%  ?"6@ NNN?N{ =b,$B 0 W2&>0 d2C  x H jJ?"6@ NNN?N'\b ,$@ 0jz V x P ,$D 0" x <l)  ?"6@ NNN?Nz pN [kN]:>0 d2F1C  x H jJ?"6@ NNN?NVj x H jJ?"6@ NNN?Nc G ,$@ 05 x <.  ?"6@ NNN?NL ,$+ 0 W2&>0 d2C  x <3  ?#" `T BFrames  2G z   x ^  ,$D) 0N [   x  ?B x <DjJ?"0@NNN?N[  B x <DjJ?"0@NNN?N  B x <DjJ?"0@NNN?N[  B x <D>?"0@NNN?N[  N cf  x B x BD>?"0@NNN?Nf  x <<9  ?"6@ NNN?NJFcf  UQ$>0 d2@ N HA  x B x BD>?"0@NNN?NH x <h>  ?"6@ NNN?N(A  UN$>0 d2@ #N ` @ x  62 x H jJ?"6@ NNN?N@ x <  ?"6@ NNN?N`  Uo$>0 d2@ N   x B x <Dg ?"0@NNN?NJ B xB HDg ?"0@NNN?N  x <  ?"6@ NNN?N1 Yn(>0 d2C 6l  x/li ,$D6 0zB xB <D?"B xB BZD?"B x HZDԔ?"0@NNN?Nll   j  x:,$@7 0B xB TLDjJ?"  j B xB TDjJ?" j B xB TDԔ?"0@NNN?N     x <  ?"a,$8 0 [3kN  2GH x 0޽h ?Ex 33.&___PPT10.[R+D' 4 = @B DU' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* x%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'x%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**x%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-x%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Cx%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Gx%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&x%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,x%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;x%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+x%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?x%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+B#style.visibility<*=x%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>x%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:x%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* x%(DY' =%(D' =%(D' 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' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(Dc' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D\' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D'  =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D ' =%(D. ' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D ' =%(DL ' =%(D' =4@BB*BB%()?)?D(' =.7 BBBBB-M 0 0 L 0 0.33302 E*3>*B ppt_xB ppt_y=0BB<* D' =4@BB*BB%()?)?D(' =.7 BBBBB-M 0 0 L 0 0.33333 E*3>*B ppt_xB ppt_y=0BB<* D' =4@BB*BB%()?)?D(' =.7 BBBBB-M 0 0 L 0 0.33333 E*3>*B ppt_xB ppt_y=0BB<*D' =4@BB*BB%()?)?D(' =.7 BBBBB-M 0 0 L 0 0.33333 E*3>*B ppt_xB ppt_y=0BB<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(Dd' =%(D ' =%(D' =4@BBBB%(D' =+4 8?`CB ppt_xBCB1+ppt_w/2B*Y3>B ppt_x<*D ' =+4 8?XCB ppt_yBCB ppt_yB*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(DP' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BB@BB%()?)?D*' =.7 BBBBB/M 0 0 L 0 -0.33302 E*3>*B ppt_xB ppt_y=0BB<*D' =4@BB@BB%()?)?D*' =.7 BBBBB/M 0 0 L 0 -0.33302 E*3>*B ppt_xB ppt_y=0BB<*D' =4@BB@BB%()?)?D*' =.7 BBBBB/M 0 0 L 0 -0.33302 E*3>*B ppt_xB ppt_y=0BB<*D' =4@BB@BB%()?)?D*' =.7 BBBBB/M 0 0 L 0 -0.33302 E*3>*B ppt_xB ppt_y=0BB<*!D ' =%(D. ' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*$D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*$D' =-g6B fade*<3<*$Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*%D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*%D' =-g6B fade*<3<*%D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*5D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*5DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(+p+0+40 ++0+60 +F  0 _SWS` UXgR(  0z V   ^ ,$D  0B  <D?"0@NNN?N B  <D?"0@NNN?NC C|  <%  ?"6@ NNN?N# w q Cx&>0 d2B  <%  ?"6@ NNN?NV! Cy&>0 d2BB  BD?"0@NNN?N;R R  <4%  ?"6@ NNN?N, D2l&>0 d2BB   <D1?"0@NNN?N  B  @ <D?"0@NNN?N #,$D  0z   $    |q ,$D 0T ^  #  t $ 42 B BLENH@Q Ԕ? `TL/`TL/`TL/`T"6@ NNN?N^42  BLENH@Q Ԕ? `TL/`TL/`TL/`T"6@ NNN?N T )r # HR  H o?"6@ NNN?N)r2  H1?"6@`NNN?NGt5T  &  # A  HR  H o?"6@ NNN?N ,& 2  H1?"6@`NNN?N  C4z E :H  f ,$D 0T )r # E  HR  H o?"6@ NNN?N)r2  H1?"6@`NNN?NGt5T  &  # :HR  H o?"6@ NNN?N ,& 2  H1?"6@`NNN?N  CZ '   3  2  N Ԕ?"6@`NNN?N'  2   N Ԕ?"6@`NNN?N'  B ! <D?"0@NNN?N,$@ 0B " <D?"0@NNN?N|,$@ 0B # <D?"0@NNN?Nq,$D 0x2 $ 3 >BmSC9ENGTH}eJ9Q ? .`T=mS,.`T=mS,9=mS,9"6@ NNN?N|V,$@ 02 % HjJ?"6@ NNN?N`D|,$D  02 & HjJ?"6@ NNN?N,$D  0$ ' <%  ?"6@ NNN?Nq],$ 0 Fj(>0 d2^ ( <,%  ?"6@ NNN?N%,$ 0 xCH>0 d2BKC! ) <X%  ?"6@ NNN?N-,$ 0 Cx&>0 d2B^ * <%  ?"6@ NNN?N,$ 0 yCH>0 d2BKCB + BD?"0@NNN?N,$@ 0B , BD?"0@NNN?N,$@  0# - <8s%  ?"6@ NNN?NT;,$  0 EC(>0 d2C# . < u%  ?"6@ NNN?NyT,$ 0 EC(>0 d2C1 / <%  ?"6@ NNN?N!F,$ 0 Sy6>0 d2BC. 0 <v%  ?"6@ NNN?NI,$ 0 PParabolic arch&>0 d2C4 1 <%  ?"6@ NNN?N,$ 0 VSemi-circular arch(>8 d2C 2 <V%  ?"6@ NNN?N): ,$  0 rTo determine reactions we only need to know position and magnitude of loads and position of the hinge and supports&s>0 d2sC 3 <-  ?"6@ NNN?Nx  ,$ 0 But to determine the cross-sectional forces we do need the equation describing shape of the arch: including coordinates of any point and its tangent. &>0 d2Cp 4 <d-  ?"6@ NNN?N x ,$ 0 y = a + bx + cx2D>0 d2BJB 5 <X-  ?"6@ NNN?NB 6< ,$$ 0 x = rcosjX >0 d2Bbb1 6 <-  ?"6@ NNN?Nxr,$ 0 Sr6>0 d2BCz | 7 |,$D 0  8 <-  ?"6@ NNN?Nr njCN>0 d2B 9 <D?"0@NNN?N_} F2 : # B@eENG9vIQ 1? @e@e`T@e@e`T`T@e`T`T"6@`NNN?N2 ; < -  ?"6@ NNN?N|v rCH>0 d2BKC < <-  ?"6@ NNN?N ,$ 0 xC , yCh >0 d2BKCBK$R = <P-  ?"6@ NNN?N ,$ 0 t r, jCN>0 d2B > BD?"0@NNN?N,$D! 0! ? <4 -  ?"6@ NNN?N,$( 0 Ch&>0 d2B @ <<%-  ?"6@ NNN?N  ,$ 0 1a,b,c from: for x = 0 y = 0 |2>0 d2BCCB CBC-.. A <)-  ?"6@ NNN?N3 C -,$  0 for x = l y = h>0 d2CBCBCBCB.. B <=-  ?"6@ NNN?Nx  r ,$ 0 for x = 2l y = 0z>0 d2CBCBCBC..B C 6D?"0@NNN?N8 T 8 Q D <L-  ?"6@ NNN?NH F,$" 0 s(Symetric arch)&>0 d2Cz ~ E ~,$@' 0 F <>-  ?"6@ NNN?NJ Cn&>0 d2BR GB Nu)G o?"6@ NNN?N"B HB B/;D?"0@NNN?NnB IB <D1?"0@NNN?N~5( J <X`-  ?"6@ NNN?N60 Fj(>0 d2L2 KB 3 |2BoaENGܓHIDQ 1? n1oa'n1oa'D`T1oa'D`T"6@`NNN?N(>z  K[ L K{s[,$D& 0R MB NZ&G o?"6@ NNN?N Wt[B NB BHD?"0@NNN?N=HI O <0e-  ?"6@ NNN?NK@E Cn&>0 d2BB P <D1?"0@NNN?Nd Q <\j-  ?"6@ NNN?N Fj(>0 d2L2 R 3 |2Bo[ENGܓHIDQ 1? n1o[n1o[D`T1o[D`T"6@`NNN?N( S <$n-  ?"6@ NNN?N t ,$# 0 j = arctg dy/dxh>0 d2 ,. T <u-  ?"6@ NNN?NI,$) 0 PParabolic arch&>0 d2C4 U <z-  ?"6@ NNN?N,$* 0 VSemi-circular arch(>8 d2C6 W <8-  ?"6@ NNN?Nq w _E ,$% 0 X(in polar coordinates)&>0 d2C X <t-  ?#" `T BArches  2GH  0޽h ? $:KR 33_____PPT10g_.=+ ԭWDY' - = @B D~Y' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*&D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*&DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*%DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D2' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(+x+0+'0 ++0+(0 ++0+)0 ++0+*0 ++0+-0 ++0+.0 ++0+/0 ++0+00 ++0+10 ++0+20 ++0+30 ++0+40 ++0+50 ++0+60 ++0+<0 ++0+=0 ++0+?0 ++0+@0 ++0+A0 ++0+B0 ++0+D0 ++0+S0 ++0+T0 ++0+U0 ++0+W0 +  0 BBp ?LA(  R  <` ?"6@ NNN?NV=,$ 0 t0Example: parabolic arch under concentrated force(1>0 d21Cz     Eb ,$D 042 B BLENH@Q Ԕ? `TL/`TL/`TL/`T"6@ NNN?NA 42  BLENH@Q Ԕ? `TL/`TL/`TL/`T"6@ NNN?N T )r #  StR   H o?"6@ NNN?N)r2   H1?"6@`NNN?NGt5T )r  #  [ R   H o?"6@ NNN?N)r2   H1?"6@`NNN?NGt5B  <D?"0@NNN?NQv,$D 0.z V   f= ,$@ 0   <p ?"6@ NNN?N# 7 1  Wx&>0 d2B    <4 ?"6@ NNN?NV! Wy&>0 d2B B  <D?"0@NNN?NCC< B  <D1?"0@NNN?NN    < ?"6@ NNN?NP3J  Wl&>0 d2B B  <D1?"0@NNN?N  S B  BD?"0@NNN?N;! ! B  BD?"0@NNN?N! !    < ?"6@ NNN?NeP J  Wl&>0 d2B Ez n  7 G ,$D  0B  BD?"0@NNN?Nn   <3 ?"6@ NNN?N Wh&>0 d2B B  BDԔ?"0@NNN?N:Y ,$@  0B  BDԔ?"0@NNN?NMY ,$D 0#  <p ?"6@ NNN?N+ ,$  0 EC(>0 d2C7  <& ?"6@ NNN?N,$ 0 YA(>0 d2C 7   <  ?"6@ NNN?N: ,$ 0 YB(>0 d2C B !@ HDԔ?"0@NNN?NORY ,$@ 0B " BDԔ?"0@NNN?NW,$@ 0B # HDԔ?"0@NNN?NX ,$@ 0B $@ BDԔ?"0@NNN?N,$D 0p % <P) ?"6@ NNN?N` }Z ,$$ 0 Mc = 0\>0 d2 Z & <x ?"6@ NNN?NE , ,$ 0 |RAJ>0 d2CKC Z ' < ?"6@ NNN?N< # ,$ 0 |RBJ>0 d2CKC Z ( < ?"6@ NNN?N3  ,$ 0 |VBJ>0 d2CKC Z ) <@ ?"6@ NNN?Np8W ,$ 0 |HBJ>0 d2CKC R * <\ ?"6@ NNN?Na [ ,$0 0 t SX = 08>0 d2 v + <L ?"6@ NNN?NV TP ,$ 0 SMA = 0Z>0 d2  , <$R ?"6@ NNN?NV 4 P ,$  0 )Pl - VB 2l =0>0 d2CcbbckCBCH   2 - HjJ?"6@ NNN?N ! X,$D  0B . BD>?"0@NNN?N  3,$@   0Z / <Hf ?"6@ NNN?NG L. ,$ 0 |VAJ>0 d2CKC Z 0 <V ?"6@ NNN?NgN ,$ 0 |HAJ>0 d2CKC q 1 <7 ?"6@ NNN?N_ qF ,$D! 0  VB = P/2N >0 d2ckC  B 2 <D?"0@NNN?Ngn 8 ,$D* 0B 3@ <D?"0@NNN?N  G ,$D% 0B 4@ BDԔ?"0@NNN?Ng ,$D& 0B 5 BDԔ?"0@NNN?NSgb ,$D+ 0B 6 BD1?"0@NNN?Ngg ,$D, 0B 7 BD1?"0@NNN?Nl ,$D' 0B 8 BDԔ?"0@NNN?N,$D)  0B 9@ BDԔ?"0@NNN?NXe,$D.  03 : <G ?"6@ NNN?Na  [ ,$/ 0 UHA/VA = HB/VB = l/h>0 d2CKCKCKCckcbbbC  ; B o?"6@ NNN?N ^,$D2  0 Higher the ratio l/h (i.e. lower the ratio h / l)  higher the value of horizontal reaction H`>8 d2CBC@C@CBCB,C.,1 < B o?"6@ NNN?Na m ,$D1 0 MHA = HB = VA l/h>0 d2CKCKkckcbbbC0   ' = < ?"6@ NNN?N; ",$   0 IP,>0 d2GT > < ?"6@ NNN?NV P ,$" 0 vSY = 0 8>0 d2 l ? <P ?"6@ NNN?NH / ,$D# 0 = VAN>0 d2Cck  @ < ?#" `T BArches  2GB L <D?"P 8\ ,$D3  0H  0޽h ?/  33___PPT10.=+5hD' = @B DC' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dn' =%(D' =%(D' 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