theory of structures problems solved by dk mamonai
TRANSCRIPT
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< > x U n l c
N' Zsuufp}z ? Zsuufp}z npc upfykmcm }f pczkz} }bc hfnmz']bcpc npc }bpcc }{ucz fi Zsuufp}z
>' Bkglc Zsuufp}? K} kz nh|n{z kg jhfja|kzc fp Ng}k$jhfja|kzc gf} bfpkfg}nh fp ycp}kjnh
0' Ikrcm Zsuufp}? Bfpkfg}nh fp Ycp}kjnh fp Jhfja|kzc
:' Jkpjshnp fp Pfhhcp Zsuufp}? Nh|n{z gfpdnh }f }bc zspinjc'
@' Hfnmz ?
Mcnm Hfnmz Hkyc Hfnmz
Sgkifpdk}{ Mkz}sp`cm Hfnm Dfykgl Hfnm
Sgkifpdh{ Ynp{kgl Hfnm \kgm Hfnm
Ufkg} Hfnm Zgf| Hfnm
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 0 x U n l c
@cndz ?
"k+ Mc}cpdkgn}c @cnd fp Zkduh{ Zsuufp}cm @cnd?
"kk+ Jfg}kgsfsz @cnd ? Fgc fp dfpc zung kz znkm nz Jfg}kgsfsz @cnd'
"kkk+Jng}khcycp @cnd ? Kg Zkgmbk k} kz znkm nz JBNON.
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< : x U n l c
Upf`hcd?> ? Mc}cpdkgn}c Ifpdz
U
H
Ipcc @fm{ Mknlpnd U
Dn H
Pn Bn
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn U 4 5
Pn4 U
Dfdcg}(
!yc Jhfja|kzc D 4 5
Dn! Pn / H 45
Dn ! UH 4 5
Dn 4 $ UH
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< = x U n l c
Upf`hcd?0 ? Mc}cpdkgn}c Ifpdz
U4 25h`
H 4 >5
Ipcc @fm{ Mknlpnd U 4 25 h`
Dn H 4 >5
Pn Bn
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn U 4 5
Pn4 25 h`
Dfdcg}(
!yc Jhfja|kzc D 4 5
Dn! Pn / H 45
Dn ! UH 4 5
Dn 4 $ 255 h`$i} " Ng}k$jhfja|kzc +
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 3 x U n l c
Upf`hcd?: ? Sgkifpdh{ mkz}sp`cm Hfnm
\ y&y
H
\H
Ipcc @fm{ Mknlpnd H&0 H&0
N J @
Dn H
Pn Bn
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn \H 4 5
Pn4 \H
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} J
Dn! Pn / H&0 45
Dn 4 $ " Pn / H & 0 +
Dn 4 $ " \H / H & 0+
Dn4 $ \ H0
& 0
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 9 x U n l c
Upf`hcd?= ? Sgkifpdh{ mkz}sp`cm Hfnm
: aku&d
0d
9 aku
Ipcc @fm{ Mknlpnd >'3 >'3
N J @
Dn 0d
Pn Bn
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn \H 4 5
Pn4 9 Aku
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} J
Dn! Pn / H&0 45
Dn4 $ \ H 0 & 0
Dn4 $ : "0+0
& 0
Dn4 $ 9 aku' d
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< < x U n l c
Upf`hcd?3? 95a : 95a
=
9 9 9
Ipcc @fm{ Mknlpnd 95a Ir 95a
I{
N J M 3:':5 @
Dn Bn P`
Pn 9 9 9
Ikpz} fi nhh jbnglkgl ]nglcg} kg}f Mclpcc
4 ]ng$> =&: 4 3:':5
Gf|( ikgmkgl R$[ jf$fpmkgn}cz fi }bc Kgjhkgcm Ifpjc
Ir4 I Jfz 4 95 Jfz 3:': 4 95 "5'3;+4 :9A 1 I{ 4 I Zkg 4 95 Zkg 3:': 4 95 "5'25>+4=2'>5A
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn Ir 4 5( Pn ! P` U $ I{ 4 5
Bn4Ir ( Pn ! P` $95 $=2 45
Bn4 :9 A( Pn ! P` 4 >52 $$$$$$$$$$$$$$$$$$$$$$$$$"k+
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 2 x U n l c
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
95 / 9 ! I{ / >0 P` />2 4 5
95 / 9 ! =2 / >0 4 >2 P`
P` 4 30 A
Us}}kgl Ynhsc fi P` kg C~sn}kfg "k+
Pn ! P` 4 >52
Pn ! 03 4 >52
Pn 4 39 A
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< ; x U n l c
Upf`hcd?9?
>0 a >9a >9a : a & i}
0 A & i}
2 = = = = 9
Ipcc @fm{ Mknlpnd
>9 a >0 a >9 a >9 a ; a
N J M C I L @
Pn Bn P`
= = = = = = 0 0 0
Nz }bcpc kz Y'M'H 4 >&0 / 9 / : 4 ; a 1 R 4 H & : 4 9 & : 4 0
Ifp S'M'H 4 0/ 2 4 >9 A 1 R4 H & 0 4 2 & 0 4 =
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn >9a >0a >9a >9a ! P` ;a 4 5
Pn ! P` 4 9; $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
">9 / =+ ! ">0 / >0+ ! ">9 />9+ ! ">9 / 05+ $ "P` / 0=+ ! "; / 09+4 5
>5>2 0=P` 4 5
P` 4 =0'=> A
Us}}kgl Ynhsc fi P` kg C~sn}kfg "k+
Pn ! P` 4 9; Pn ! =0'=> 4 9; Pn 4 09'9 A
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >5 x U n l c
Upf`hcd * < Ufkg} Hfnm
=5 A 35A :5 A
:&=
N C J I M L B @
= 3 : : 2 : :
"UNP] > + \c Mkz}sp` }bkz Mknlpnd kg}f }|f N Ufkg} }f J Ufkg}
Ipcc @fm{ Mknlpnd =5 a
N C J
Bn Bj
= 3
Pn Pj
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn $ Bj 4 5( Pn =5 ! Pj 4 5
Bn4 Bj Pn ! Pj 4 =5 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
$ Pj / ;5 ! =5 /= 4 55 4 $; Pj ! >95
Pj 4 >2 A
Us}}kgl Ynhsc fi Pj kg C~sn}kfg "k+
Pn ! Pj 4 =5 Pn ! >2 =5 4 5 Pn 4 00 A
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >> x U n l c
"Unp} 0+ 35a :0 A :5 A
J Bj I M L B @
: : = = : :
4 ]ng$>:&= :9'29
Ir 4 I Jfz 35 / Jfz :9'29 =5A
I{4 I Zkg 35 / Zkg :9'29 :5 A
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bj ! Ir 4 5( $ Pj I{ ! Pm :0 $:5 ! P` 4 5
Bj 4 $ =5 a P` ! Pm 4 >>5 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
$ Pj / >2 $ I{ / >3 ! Pm />0 :0/2 :5/0 4 5>0 Pm 4 >5;5
Pm 4 ;> A
Us}}kgl Ynhsc fi Pm kg C~sn}kfg "k+
P` ! Pm 4 >>5 P` ! ;> >>5 4 5 P` 4 >; A
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >0 x U n l c
Upf`hcd * 2 ?
P a 9 a : a &i}
2 9 0
Ipcc @fm{ Mknlpnd =2a 9a
P a
Dn Pn Bn 2 2
2 9 0
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn $=2 :< $ 9 4 5
P` ! Pm 4 >=: $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
Dn ! Pn />9 " =2 ! :< + / 2 $ 9 / 04 5
5 4 Dn ! >=: />9 >5;9 $ >0
Dn 4 $>>25 A' i}
Bcgjc }bc mkpcj}kfg fi Dfdcg} N zbfshm `c }nacg nz pcycpzcm'
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >: x U n l c
Upf`hcd * ; ?
P A &i}
N @
05
Ipcc @fm{ Mknlpnd
P/05 a
Bn
9'99 9'99 9'99
Pn Pn
Ufkg} Hfnm 4 >&0 / :< / 05 2;5 a
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn 2;5 ! P` 4 5
Pn ! P` 4 2;5 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
2;5 / 9'99 $ P` / 05 4 5 05 P` 4 3;0
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >= x U n l c
Upf`hcd * >5? Ikgm Sgagf|g Pcnj}kfgz
P a 2a
0'3 a&i} = a&i}
N J M @
= 2 9
Ipcc @fm{ Mknlpnd >5 a P a :0 a 2a
Pn N C J M @
Dn Bn
0 0 = = 9
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn >5 $ ::; a
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
Dn ! Pn / >2 >5/ >9 :< / >=:0 / >5 4 5 5 4 Dn ! 0350 $>
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >3 x U n l c
Upf`hcd * >>?
2 A P A 05 a
"5'0!P + A
=
2 3 3 ; 3
Ipcc @fm{ Mknlpnd?
2a "P !5'0 +A P A 05a
J N @
Pj P`
Bj Pn B`
=
2
; 3 ; 3
:0
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
B` 4 5( 2 >95 :< ! Pn 05 ! P` 4 5
Pn ! P` 4 >
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >9 x U n l c
Upf`hcd * >0?
" >5! P&0+ A&i}
05
\c `pcna }bkz }pknglhc kg}f }|f unp}z
UNP] ? >
">5!P&0+ A Ipcc @fm{ Mknlpnd
">&0 / >5 / Q>5!P&0T +
N
Pn
Y'M'H 4 >& 0 / >5 / 3='3 Bn
4 0
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >< x U n l c
Unp} 0?
" >&0 / >5 / Q>5!P&0T +
@
P`
:':: :':: :'::
Y' M'H 4 >&0 / >5 / 3='3
4 0
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >2 x U n l c
Upf`hcd * >:?
>5 a 05 a P a
>&0
N @ C J I M
2 = 9 = 3 3 = 9
Mkz}sp`kgl kg}f : unp}z
Unp} > >5 a Ipcc @fm{ Mknlpnd >5 a
2 = 9
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn $ Bc 4 5( Pn $>5 ! P` ! Pc 4 5
Bn 4 0 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
>5 / 2 $ P` / >0 Pc / >2 4 5 $>0 P` 4 $ 0'>0 />2 P` 4 :'93 A
Us}}kgl Ynhsc fi Pn kg C~sn}kfg "k+
Pn ! P` 4 >0'>0 Pn ! :'93 >0'>0 4 5 P` 4 2'=< A
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< >; x U n l c
Unp} 0 ? Ipcc @fm{ Mklpnd?
05 a 05 a
= 3 3
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bc $ Bi 4 5( Pc ! $05 !Pj $Pi 4 5
Bc 4
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 0> x U n l c
Upf`hcd * >=?
5'9 a & i} 0'= a &i}
>9
K} kz mkz}sp`cm hkac }bn}
>9
Ipcc @fm{ Mknlpnd?
>='= a
>='=
N @
Bn 2 3'::
Pn P`
;'9 a
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 00 x U n l c
Ufkg} Hfnm fi S'M'H 4 >9 / 5'9 ;'9 a
Ufkg} Hfnm fi Y'M'H 4 >&0 / >9 / " 0'=5'9 + >='= a
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn $ ;'9 >='= ! P` 4 5
Pn ! P` 4 0= $$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
;'9 / 2 ! >='= / >5'99 $ P` / >9 4 5 5 4 3 P` P` 4 >=':; A
Us}}kgl Ynhsc fi P` kg C~sn}kfg "k+
Pn ! P` 4 0= Pn ! >=':; 0= 4 5 Pc 4 ;'9> A
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 0: x U n l c
Upf`hcd * >3 ?
P A 055 h` &i}
>55 h`&i}
N J M
= = = 2
Zfhykgl }bkz c~sn}kfg `{ unp}z '
Unp} >?
J M
9 0 =
Zfhs}kfg?
@{ zfhykgl }bc n`fyc iklspc?
Ikpz} jfgycp} }bc S'M'H ngm Y'M'H kg}f zkglhc ufkg} hfnmz'
Ifp S'M'H 4 >0 / >55 4 >055 a Ifp Y'M'H 4 >&0 / >0 / >55 4 955 a
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pj $>055 955 ! Pm 4 5
Pn ! P` 4 >255 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
955 a
>055 a
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 0= x U n l c
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} J
>055 / 9 ! 955 / 2 Pm / >0 4 5 5 4 0 Pm Pm 4 >555 A
Us}}kgl Ynhsc fi P` kg C~sn}kfg "k+
Pj ! Pm 4 >255 Pn ! >555 >255 4 5 Pc 4 255 A
Unp} 0? P a 255 a
= = =
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn $ 2; 255 ! P` 4 5
Pn ! P` 4 22; $$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
2; / = ! 255 / 2P` / >0 4 5 5 4 :39 ! 9=55 $>0 P` P` 4 39: A
Us}}kgl Ynhsc fi P` kg C~sn}kfg "k+
Pn ! P` 4 22; Pn ! 39: 22; 4 5 Pc 4 :09 A
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 03 x U n l c
Zbcnp Ifpjc ngm @cgmkgl Dfdcg}
Zbcnp Ifpjc ?
]bc Zbcnp ifpjc n} n jpfzz$ zcj}kfgnh fi n `cnd kz }bc sg`nhngjcm ycp}kjnh ifpjc }f }bc
}klb} fp hci} fi }bc zcj}kfg'
@cgmkgl Dfdcg}?
]bc `cgmkgl dfdcg} n} n jpfzz$zcj}kfg fi n `cnd kz }bc nhlc`pnkj zsd fi
}bc dfdcg}z fi }bc ifpjcz }f }bc pklb} fp hci} fi }bc zcj}kfg'
FP
Nhlc`pnkj zsd fi }bc jhfja|kzc ngm ng}k$jhfja|kzc dfdcg} kz jnhhcm @cgmkgl Dfdcg}'
Zbcnp Ifpjc ) @cgmkgl Dfdcg} Mknlpnd?
]bc zbcnp ifpjc ngm `cgmkgl dfdcg} |khh `c mkiicpcg} n} mkiicpcg} jpfzzzcj}kfgz mcucgmkgl sufg }bc
hfnmkgl fg }bc `cnd' ]bc ynpkn}kfg kg }bc }bcpc jng `c zbf|g lpnubkjnhh{ `{ uhf} }bc Z'I ngm @'D
mknlpnd'
Zbcnp Ifpjc Mknlpnd ifp Mkiicpcg} ]{ucz fi Hfnmkgl?
JNZC$K ?
Ifp ng{ jfgjcg}pn}cm & Ufkg} Hfnm zbcnp ifpjc kz z}pnklb} hkgc'
JNZC$KK ?
Ifp S'M'H }bc zbcnp ifpjc mknlpnd |khh `c kgjhkgcm hkgc' Fp ]pknglshnp Ifpd
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 09 x U n l c
JNZC $ KKK ?
Ifp S'Y'H sgkifpd ynp{kgl hfnm 05 Jspyc |khh `c }bcpc kg k}z mknlpnd
05
ZKLG JFGYCPZKFG ?
Ng su|npm ifpjc }f }bc pklb} fi n zcj}kfg jfg}pk`s}cz }f n Ufzk}kyc Zbcnpkgl
ifpjc kg su|npm
H'B'Z
!YC $YC
P'B'Z
!YC $YC
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 0< x U n l c
@cgmkgl Dfdcg} Mknlpnd ifp Mkiicpcg} Hfnmz
JNZC$KIfp Jfgjcg}pn}cm Ifpjc " Ufkg} Hfnm + }bc `cgmkgl dfdcg} mknlpnd |khh `c kgjhkgcm hkgc'
JNZC$KK
Ifp SMH }bc zbnuc fi @'D mknlpnd |khh `c unpn`fhkj n} 05
jspyc
05
JNZC$KKK
]bc zbnuc ifp Y'M'H fi @'D |khh `c n} :5 jspyc'
:5
ZKLG JFGYCPZKFG? N jhfja|kzc dfdcg} }f }bc pklb} fi }bcg zcj}kfg fp ng ng}kjhfja|kzcdfdcg} }f }bc hci} fi }bc zcj}kfg jfg}pk`s}cz }f ufzk}kyc `cgmkgl dfdcg}'
H'B'Z? P'B'Z ?
!YC $YC $YC !YC
Ufzk}kyc @cgmkgl Dfdcg} kz nhzf agf|g nz Bfllkgl @cgmkgl Dfdcg} ngm Gcln}kyc @'D kz nhzf
agf|g nz znllkgl @'D
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 02 x U n l c
Upf`hcd ? >
0'3 a >'3 a 0 a
5'9 d 5'9 d 5'9 d
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( $ 0'3 ! Pn >'3 $ 0 4 5
Pn 4 9 a $$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} @
Dn ! Pn / 5'>2 $ 0'3 / 5' >0>'3 / 5'9 4 5 5 4 Dn ! >5'2 :5'; Dn 4 $ 9'; A
Bcgjc }bc mkpcj}kfg fi dfdcg} |khh `c pcycpzcm }f |bkjb |c bnyc }nacg '
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 0; x U n l c
Zbcnp ifpjc > 0 :
0'3a >'3 a 0 a
N Pn J M @
Bn
5'9 d 5'9 d 5'9 d
R>
R0
R:
Zcj}kfg " >$>+
Y> 4 Pn ! 9 } $$$$$$$$$$$ "k+
Ki 4 5 E N ! 9 }
Ki 4 5'9 E " N$J+ ! 9 }
Zcj}kfg " 0$0+
Y0 4 Pn0'3 90'3 ! :'3 } $$$$$$$$$$$ "kk+
Ki 4 5'9 E J ! :'3 }
Ki 4 >'0 E " N $M + ! :'3 }
Zcj}kfg " :$:+
Y: 4 Pn 0'3>'3 9 0'3>'3 ! 0 } $$$$$$$$$$$ "kkk+
Ki 4 >'0 E M ! 0 }
Ki 4 >'2 E " N $ @ + ! :'3 }
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :5 x U n l c
@cgmkgl Dfdcg} ?
> 0 :
0'3 a >'3 a 0 a
Dn Pn
5'3 d 5' 9 d 5'9 d
R>
R0
R:
Zcj}kfg " >$>+
@'D> 4 Pn / >! Dn 9 > 9'; $$$$$$$$$$$ "k+
Ki > 4 5 E N 9 " 5+ 9'; $ 9'; }'d
Zcj}kfg " 0$0+
@'D0 4 Pn / 0 ! Dn0'3 / " 0 5'9+ 9 0 9;0'3" 0 5'9 + 9 0 9';0'3 0 >'3"kk+
Ki 0 4 5'9 E J 9 / 5'9 9';0'3 / 5'9 ! >'3 $ :': }d
Zcj}kfg " :$:+
@'D: 4 Pn / : ! Dn0'3 / " : 5'9+ >'3 " : >'0+ 9 : 9;0'3" : 5'9 +>'3 " : >'0 + $$$$$$$$$$$ "kkk+
Ki : 4 >'0 E M 9 / >'0 9; 0'3 " >'0 5'9 + >'3 " >'0 >'0 + $ >'0 }'d
Ki : 4 >'2 E @ 5 }'d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
31/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :> x U n l c
Zbcnp ngm @cgmkgl Dfdcg} Mknlpnd
0'3 a >'3 a 0 a
5'9 d 5'9 d 5'9 d
! 9 } ! :'3} ! 0'> }
$9'; }d $:': }d $>'0 }d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
32/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :0 x U n l c
Upf`hcd ? 0
\ h`&i}
N @
H
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn \H 4 5
Pn 4 \H $$$$$$$$$$$$$$$$$$$$$$$$$$$$$"k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} @
Dg ! \ / H&0 5 4 Dn ! \H0& 0 Dn 4 $ \H
0& 0 A
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
33/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :: x U n l c
Ipcc @fm{ Mknlpnd fi Zbcnp ifpjc
N Pn @
Bn
H
R>
Zcj}kfg " >$>+
Y> 4 Pn $ \ \H $ \ $$$$$$$$$$$ "k+
Ki 4 5 E N \H \ "5+ ! \H
Ki 4 H E " @ + \H \ "H+ 5
@cgmkgl Dfdcg}
N Pn @
Dn
H
R>
@'D> 4 Dn ! Pn / \ / &0 Dn ! Pn / \0 &0 $$$$$$$$$$$ "k+
Ki > 4 5 E N Dn ! 5 $ \H0
& 0 }'d
Ki > 4H E @ $ \H0 & 0 ! \H0 \H0&0 5 }'d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
34/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< := x U n l c
Zbcnp ngm @cgmkgl Dfdcg} Mknlpnd
\ h`&i}
N @
H
!\H
5
5
$\H0
&0 05
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
35/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :3 x U n l c
Upf`hcd ? :
N jng}khcycp `cnd N@ 0d hfgl jnpp{kgl n sgkifpd mkz}sp`cm hfnm fi 0'3 Ag&d fycp n hcgl}b fi >'9 d ipkg
ipcc bngm ' mpn| }bc Z'I ngm @'D mknlpnd ifp }bc `cnd'
0'3 Ag&d
N @
5'= d >'9 d
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn = 4 5
Pn 4 = AG
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
Dn 4 Dn ! = / >'9 & 0 5 4 Dn ! ='2 Dn 4 $ ='2 AG ' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
36/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :9 x U n l c
Ipcc @fm{ Mknlpnd fi Zbcnp ifpjc
> 0
= Ag
N
@
Pn Bn
5'= d 5'2 d 5'2 d
R>
R0
Zcj}kfg " >$>+
Y> 4 Pn = AG $$$$$$$$$$$ "k+
Ki 4 5 E N = AG
Zcj}kfg " 0$0+
Y0 4 Pn 0'3 /" 0 5'= + Pn 0'3 0 ! >
Ki 0 4 5'= E J = 0'3 " 5'=+ ! > = AG $$$$$$$$$$$ "kk+
Ki 0 4 0 E " @ + = 0'3 "0+ ! > 5 $$$$$$$$$$$ "kkk+
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
37/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :< x U n l c
@cgmkgl Dfdcg}
> 0
= Ag
N @
Pn
5'= d 5'2 d 5'2 d
Dn
R>
Zcj}kfg " >$>+
@'D> 4 Dn ! > / Pn $$$$$$$$$$$ "k+
Ki > 4 5 E N Dn ! 5 $ ='2 }'d
Zcj}kfg " 0$0+
@'D0 4 Pn / 0 ! Dn0'3 / " 05'=+ q " 0 5'=+&0 t Dn ! Pn 0 $ >'03 " 0 5'= +
0
$ ='2 ! = 0 $ >'03" 0 $ 5'= +0
Ki 0 4 5'= E J $ ='2 ! >'9 $ 5 $ :'0 AG' d
Ki 0 4 0 d E @ 5 AG' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
38/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :2 x U n l c
Zbcnp ngm @cgmkgl Dfdcg} Mknlpnd
0'3 Ag&d
N @
5'= d >'9 d
!= Ag ! = Ag 5
$='25 ag'd $:'0 Ag'd 5
0f
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
39/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< :; x U n l c
Upf`hcd ? =
N jng}khcycp `cnd fi >'2 d kz hfnmcm nz zbf|g kg iklspc ' Mpn| }bc zbcnp ifpjc ngm `cgmkgl dfdcg}
mknlpnd'
0 Ag >'3 Ag > AG
0'3 Ag&d
N @ J M
5': d 5'9 d 5':d 5'9 d
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn 0 >'3 " 5'2 / 5'9 + $> 4 5
Pn 4 =';2 AG
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
Dn 4 Dn ! 0 / 5': ! >'3 / 5'; ! 5'=2 / >'3
5 4 Dn ! 5'9 ! >':3 ! 5''2
Dn 4 $ ='=< AG ' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
40/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =5 x U n l c
Ipcc @fm{ mknlpnd ngm Zbcnp Ifpjc
> 0 : =
0 ag >'3 ag 5'=2 Ag > Ag
N J M @
Pn Bn
5': d 5'9 d 5': d 5': d 5': d
R>
R0
R:
R=
Zcj}kfg " >$>+
Y> 4 Pn ! =';2 AG $$$$$$$$$$$ "k+ Ki > 4 5 E N !=';2 AG
Zcj}kfg " 0$0+
Y0 4 Pn 0 =';2 0 !0';2 AG Ki 0 4 5': E @ ! 0';2 AG
Zcj}kfg " :$:+
Y: 4 Pn 0>'3 =';2 :'3 >'=2 AG Ki : 4 5'; E J ! >'=2 AG
Zcj}kfg " =$=+
Y= 4 Pn 0 >'35'2" 0 >'0 + >'=2 $ 5'2 " 0 >'0 +
Ki = 4 >'0 E M ! >'=2 AG
Ki = 4 >'2 E C >'=2 5'2 " >'2 >'0 + > AG
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
41/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< => x U n l c
@cgmkgl Dfdcg}
> 0 : =
0 ag >'3 ag 5'=2 Ag > Ag
Dn
N @
Pn
5': d 5'9 d 5': d 5': d 5': d
R>
R0
R:
R=
Zcj}kfg " >$>+
@'D> 4 Pn / > ! Dn $$$$$$$$$$$ "k+
Ki > 4 5 E N Dn ! 5 $='=< Ag'd
Zcj}kfg " 0$0+
@'D0 4 Pn / 0 ! Dn 0 / " 0 5': +
=';2 0 ='= $ 0 0 ! 5'9
Ki 0 4 5': E @ =';2 " 5':+ $ ='= 0 " 5':+ ! 5'9 $ 0';< AG' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
42/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =0 x U n l c
Zcj}kfg " :$:+
@'D: 4 Pn / : ! Dn 0 / " : 5': + >'3 ": 5'; +
$='=< ! =';2 : $ 0 : ! 5'9 >'3 : !>':3
>'=2 : 0'30
Ki : 4 5'; E J >'=2 " 5'; + $ 0'30 $ > '>22 AG' d
Zcj}kfg " =$=+
@'D= 4 Pn / = ! Dn 0 / " = 5': + >'3 "= 5'; + 5'2 "= >'0+ "= $>'0 & 0+
$='=< ! =';2 : $ 0 : ! 5'9 >'3 : !>':3 5'= "= >'0+0
$ 0'30 ! >'=2 = 5'= "= >'0+0
Ki = 4 >'0 E M $0'30 ! >'=2 ">'0 + 5'= " >'0 >'0 +0 $ 5''2 >'0 +0 5 AG' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
43/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =: x U n l c
Zbcnp ngm @cgmkgl Dfdcg} Mknlpnd
0 Ag >'3 Ag > AG
0'3 Ag&d
N @ J M
5': d 5'9 d 5':d 5'9 d
!=';2 ag !0';2 ag !>'=2 ag !>'=2 ag >ag
$ ='=< ag'd $0';< ag'd $>'>2 ag'd $ 5'
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
44/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< == x U n l c
Upf`hcd ? 3
N 3d hfgl jng}khcycp `pcnd jnppkcz n ufkg} hfnm fi P } n} }bc ipcc bngm ngm n SMH fi " >! 5'0 P + } &d
fycp n hcgl}b fi :d ipfd }bc ipcc cgm ' Mpn| }bc zbcnp ifpjc ngm `cgmkgl dfdcg} mknlpnd'
P }
>2'2 }
N @ J
0d : d
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn 39'= 2; 4 5
Pn 4 >=3'= }
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
5 4 Dn ! 39'= / :'3 !2;/ 3
5 4 Dn ! >;
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
45/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =3 x U n l c
Ipcc @fm{ mknlpnd ngm Zbcnp ifpjc
> 0
">!5'0 P +/: P }
N @ J
Pn Bn
0 d >'3 d >'3 d
R>
R0
ZBCNP IFPJC
Zcj}kfg " >$>+
Y> 4 Pn ! >=3'= } $$$$$$$$$$$ "k+ Ki > 4 5 E N !>=3'= }
Zcj}kfg " 0$0+
Y0 4 Pn >2'2 "0 $0 + >=3'= >2'2 "0 $0 +
Ki 0 4 0 E @ >=3'= >2'2 " 0 $ 0 + ! >=3'= }
Ki 0 4 3 E J >=3'= >2'2 " 3 $ 0 + ! 2; }
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
46/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =9 x U n l c
@cgmkgl Dfdcg} ?
> 0
">!5'0 P +/: P }
Dn
N @ J
Pn
0 d >'3 d >'3 d
R>
R0
Zcj}kfg " >$>+
@'D> 4 Pn / > ! Dn >=3 > $ 90='= $$$$$$$$$$$ "k+
Ki > 4 5 E N Dn ! 5 $ 9=0'= }'d
Zcj}kfg " 0$0+
@'D0 4 Pn / 0 ! Dn ! >2'2 / " 0 0 + " 0 0 & 0 +
>=3'= 0 90='= $ ;'= " 0 ! 0+0
Ki 0 4 0 E @ >=3'= / 0 9=0'= ;'= " 0$ 0 +0 $ :3>'9 }' d
Ki 0 4 3 E J >=3'= / 3 9=0'= ;'= " 3$ 0 +0 5 }' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
47/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =< x U n l c
Zbcnp ngm @cgmkgl Dfdcg} Mknlpnd?
P }
>2'2 }
N @ J
0d : d
!>=3'= } !>=3'=} 2;}
5
$9=0'= }d 05
$:3>'9 }d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
48/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =2 x U n l c
Upf`hcd ? 9
N jng}khcycp `cnd fi >35 jd zung jnppkcz n hfnm fi 0} n} 95 jd ipfd }bc ikrcm cgm ( n hfnm fi >} n} }bc
ipcc cgm ngm n hfnm fi = }fg SMH fycp 95 jd ipfd 0 }fg hfnm }f|npmz }bc ipcc cgm ' Mpn| Z'I ngm @(D
mknlpnd
0 } > }
9 } &d
N @ J
5'9 d 5'9 d 5': d
Hfnm kg}cgzk}{ 4 Hfnm & Hcgl}b 4 = & 5'99 4 9 }&d
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn 0 "=+ $ > 4 5
Pn 4 < }
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
5 4 Dn ! 0 / 5'99 ! = / 5';; ! > / >'3
5 4 Dn ! >':: ! :';; ! >'3
Dn 4 $ 9':: } ' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
49/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< =; x U n l c
Ipcc @fm{ mknlpnd ngm Zbcnp ifpjc
> 0
0 }
= } > }
N @ J
Pn Bn
5'9 d 5': d 5': d 5': d
R>
R0
Zcj}kfg " >$>+
Y> 4 Pn ! < } $$$$$$$$$$$ "k+ Ki > 4 5 E N ! < }
Zcj}kfg " 0$0+
Y0 4 Pn 9 " 0 5'99 + Ki 0 4 5'99 E @ ! < }
Zcj}kfg " :$:+
Y: 4 Pn 9 "0 5'99 + $ 0 }
Ki : 4 >':: E J ! > }
Ki : 4 >'3 E M ! > }
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
50/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 35 x U n l c
@cgmkgl Dfdcg} ?
> 0
0 }
Dn = } > }
N @ J M
Pn
5'9 d 5': d 5': d 5': d
R>
R0
Zcj}kfg " >$>+
@'D> 4 Pn / > ! Dn $$$$$$$$$$$ "k+
Ki > 4 5 E N Dn ! 5 $ 9':: }'d
Zcj}kfg " 0$0+
@'D0 4 Pn / 0 ! Dn0 / " 0 5'9 + 9 / " 0 5'9 + " 0 5'9 +&0 < 0 9':: 0 "0 5'9 + : "0 $ 5'9 +
0
Ki 0 4 5'99 E J < / 5'99 9':: $ 0'>9 }' d
Zcj}kfg " :$:+
@'D: 4 Pn / : ! Dn0 / " : 5'9 + = / " : 5';+ < : 9':: 0 ": $ 5'9 + = ": 5'; +
Ki : 4 5'; E M < / 5'; 9': 0 " 5'; 5'9 + 5 $ 5'9 } ' d
Ki : 4 >'3 E @ < / >'3 9': 0 " >'3 5'9 + = " >'3 $ 5+ 5 } ' d
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
51/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 3> x U n l c
@cgmkgl Dfdcg} ngm Zbcnp Mknlpnd
0 } > }
9 } &d
N @ J
5'9 d 5'9 d 5': d
! } >}
$9':> }'d $0>'9 }'d 05 $5'9 }'d 5
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
52/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 30 x U n l c
]PSZZCZ
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
53/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 3: x U n l c
Upf`hcd > ?
05 a >3 a
@ M
95f
955
95f
95f
N 3 3 J 3 3 C
>5 >5
Pn P`
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn 05 >3 ! Pc 4 5
Pn ! Pc 4 :3 $$$$$$$$$$$$$$$$$$$$$$ "k+
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
5 4 05 / 3 ! >3 / >3 05 Pc 4 5
5 4 :03 05 Pc
Pc 4 >9' 03 a Pn 4 >2'
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
54/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 3= x U n l c
@
N 95f J
Ofkg} N ? Pn 4 >2'5'20
C~sn}kfg "k+ |khh `c I nj 4 $ >5'20 " J +
05 a
@ M
N J
Ofkg} @ ?
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
I n` r I `m ! I j` r 4 5( I n` { 05 I j` { 4 5
>5'20 ! I j` r $ I `m $$$$$$$$$$$$$$$$$$$$$$$$"kk+ >2''03
I j` { 4 I j` Zkg 955 I j` 4 >'== " J + ( I j` r 4 I j` Jfz 95
5 I j`r 4 $ 5'
C~sn}kfg "kk + |khh `c >5'20 I `m5' I `m 4 $ >5'20 " J +
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
55/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 33 x U n l c
@ M
J
Ofkg} J ? N 95f 95
fC
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
I `j r I nj $ I jm r $ I jc 4 5( $ I `j { $ I jm { 4 5
Ijc ! I jm r 4 $>5'> $$$$$$$$$$$$$$$$$$$$$$$$"kkk+ $>'03 I jm { 4 5 I jm { 4 $ >'03
I jm { 4 I jm Zkg 955 I jm 4 >'=: " ] + ( I j` r 4 I j` Jfz 95
5 I j` r 4 $5'
C~sn}kfg "kkk + |khh `c I jc 5' 4 $ >5'> I jc 4 $ ;':3 A
@ M
J
C
Ofkg} M ?
C~skhk`pksd(
!yc Ir 4 5(
I `m I jm r $ I mc r 4 5(
Imc r 4 $ ;':2'
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
56/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 39 x U n l c
Upf`hcd * 0 ? 0 a
:a I : a
L C
J M
N :5f
95f
95f
95f
95f
:5f
@
: : :
Pn P`
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn 4 5( Pn : 0 : ! P` 4 5
Pn ! P` 4 2 $$$$$$$$$$$$$$$$$$$$$$ "k+
@{ nuuh{kgl Zkgc Hn|
Zkg 95 & NL 4 Zkg ;5 & : NL 4 : Zkg 95 NL 4 0'3;
Jfz :5 4 NJ & NL NJ 4 0'3;
Zkg 95 & : 4 Zkg 95 & JI JI 4 :
Jfz 95 4 JM & : JM 4 >'3
Dfdcg}(
!yc Jhfja|kzc D 4 5 n} ufkg} N
5 4 : / NJ ! 0 / " : ! JM + ! : / " 9 ! M@ + P` / ;
5 4 02'=9 ; P`
P` 4 ='0< a Pn 4 = a
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
57/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 3< x U n l c
L
N 95f J
Pn
Ofkg} N ?
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
$ I nl r ! I nj 4 5( Pn $ I nl { 4 5
I nl r 4 I nj $$$$$$$$$$$$$$$$$$$$$$$$"k+ I nl { 4 = a
I nl { 4 I nl Zkg 955 I nl 4 2 a " J + ( I nl r 4 I nl Jfz 95
5 I nlr 4 9';0 a
C~sn}kfg "k + |khh `c I nj 4 I nl r I nj 4 9';0 a " ] +
:a
I
L
N J
Bcpc ki |c zfhyc k} mkpcj}h{ k} |khh `c dfpc mkiicpcg} ifp sz }f lc} }bc nzzsdcz `cnjsnzc 0sgagf|g |khh jfdc kg cnjb c~sn}kfg fi c~skhk`pksd'
Zf |c pf}n}c k} nz
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
58/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 32 x U n l c
{
:5 a
:55
N I r
L
C
Ofkg} L ?
Ir4 I Jfz 95 : Jfz 95 >'3 ( I{ 4 I Zkg 95 : Zkg 95 0'9
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
I nl ! I li I r 4 5( I lj $ I{ 4 5
I li 4 9'3 G " J + I lj 4 0'9 " J +
Gf}c mcgf}cz }
ynhscz |bkjb npc
nhpcnm{ agf|g }f
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
59/72
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
60/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 95 x U n l c
Dc}bfm fi Zcj}kfgz
Upcykfszh{ |c bnyc z}smkcm dc}bfmz fi ofkg}z ifp ngnh{zkz fi }pszzcz
Kg }bkz dc}bfmz |c jng mkpcj}h{ ngnh{zkz n }pszz ifpjc |k}bfs} jnhjshn}kgl ipfd kgk}knh ufkg} '
Bcpc jfgmk}kfg fi dfdcg} kz szcm pn}bcp }bng jfgmk}kfg fi z}n}kz}kjz " crjcu} kg crjcu}kfgnh jnzcz +
Gccm }f js} n zcj}kfg |bcpc dnrkdsd sgagf|g dcd`cpz ifpjc `cjfdcz cpf fp jng `c gclhcj}cm'
UPFJCMSPC
Js} n zcj}kfg
Mpn| n ipcc `fm{ mknlpnd
Nzzsdc jfdupczzkfg ngm }cgzkfg
Nuuh{ jf$uhngcp nuuhkjn}kfgz fi ifpjcz ngm c~sn}kfg fi dfdcg} }f mc}cpdkgc dcd`cpz ifpjc
-
7/23/2019 Theory of Structures Problems solved by Dk Mamonai
61/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 9> x U n l c
Upf`hcd ? > M
d 4>&:
@ I
d4:&0
N >0 J >0 C >0 L >0 B
Pn >055 >055 >055 Pb
!yc I{ 4 5 Pn ! Pb >055 $>055>055 4 5 Pn ! Pb 4 :955$$$$ " k +
]nakgl dfdc} n} N'
Dn 4 5 >055 />0 ! >055 / 0= ! >055 / :9 $ Pb / =2 4 5 =2 Pb 4 29=55 Pb 4 >255 h`
M Pn 4 >255 h`
@
Pn4>255 J >055 C
]nac }bc ufkg} |bcpc dnrkdsd sgagf|gz jng `c gclhcj}cm'
]nakgl dfdcg} n} @' >255 / >0 I jc / 2 4 5 I jc 4 00Ijc / >0 ! >255 / 0= 4 5 I `c r4 0=55 h`z
I `c r 4 I`c / Jfz ::'9; I `c 4 022='=: " J +
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 90 x U n l c
Upf`hcd * 0?
Pc~skpcm gccmcm ifpjcz
Pjc 4 6 ( P`m4 6 ( Pjm4 6 ( P li4 6
Ifp Ijc ( |c }nac dfdcg} n} M'
I jc / 05 ! Pn / =5 05 / 05 4 5
05 Ijc 4$ $:5 / =5 ! =55
Ijc 4 =5 a " J + fp Ijc 4 =5 a " ] +
Ifp I`m }nakgl dfdcg} n} J'
I`m / 05 ! :5 / 05 4 5
I`m 4 $ :5 " ] + fp :5 " J +
Ifp Ijm |c }nac dfdcg} n} @'
Ijm { / 05 ! Ijc / 05 ! :5 / 05 4 5 05 Ijm { 4 055 Ijm { 4 >5 a " J+
Ijm { 4 I jm / Zkg =3 >5 4 Ijm / Zkg =3 Ijm 4 >='>= a
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 9= x U n l c
NPJBCZ
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
65/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 93 x U n l c
Upf`hcd > ? : bkglc unpn`fhkj npjb fi 05 i} ngm z}ngmnpmkc fi 3 jnpkgl n jfgjcg}pn}c hfnm fi 05 akuz n}
9 ipfd }bc hci} zkmc zsuufp}' Mpn| }bc `cgmkgl dfdcg} mknlpnd ifp }bc npjb( ufzk}kfg ngm dnlgk}smc
fi dnrkdsd `cgmkgl dfdcg} ifp }bc npjb nhzf ikgm }bc ynhsc fi }bc pcnj}kfg ifpjcz
J
05 akuz
{j43
N @
05 i}
MN]N?
[j 4 3
U 4 05 akuz
H 4 05
@'D 4 6
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn $ B` 4 5( Yn ! Y` 4 05 $$$$$$ " k +
Bn 4 B`
!yc Jhfja|kzc D 4 5 n} ufkg} N
05/ 9Y` / 05 4 5 Y` 4 9 akuz us}}kgl }bkz ynhsc kg c~"k+ Yn 4 >= akuz
P'B'Z
!yc Jhfja|kzc D 4 5 n} ufkg} J
Y` / >5 ! B` / 3 4 5 B`4 >0 akuz nz B`4Bn ( Zf ( Bn 4 >0 akuz
PCZSH]NG]
Pn 4 Bn0
! Yn0
>2'== akuz
P`4 B`0
! Y`0
>:'= akuz
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 99 x U n l c
@'D dnr " !yc +
Szkgl Ifpdshn
[ 4 = [j " H + & H0
[ 4 = / 3/ 9 " 05 9 + & "05+0
[ 4 = '0 z'i}
Yn / 9 Bn / [ 4 >= / 9>0 / ='0 ::' 9 a'i}
@'D dnr " $yc +
B` / [Y` / $$$$$$$$$$$$$$$ "kkk+
Szkgl Ifpdshn
[ 4 = [j " H + & H0
[ 4 = / 3 / " 05 + & H0
[ 4 $ 0
& 05$$$$"ky+ us}}kgl [ ynhsc kg c~sn}kfg "kkk+
>0 / " 0
& 05+9 /
>0 >0 0&059
9 >0 0&05
Mkiicpcg}kn}kgl |'p'}
4 3 u us}}kgl }bkz ynhsc kg c~ "ky+ [ 4 $ >3
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 9< x U n l c
Upf`hcd 0 ? : bkglc unpn`fhkj npjb fi 05 d ngm z}ngmnpmkc fi = d jnpkgl n jfgjcg}pn}c S'M'H fi
0 }fg&d ipfd }bc hci} zkmc zsuufp} fycp }bc hcgl}b fi 2 d ' Mpn| }bc `cgmkgl dfdcg} mknlpnd ifp }bc
npjb( ufzk}kfg ngm dnlgk}smc fi dnrkdsd `cgmkgl dfdcg} ifp }bc npjb nhzf ikgm }bc ynhsc fi }bc
pcnj}kfg ifpjcz
J
N 2d @
MN]N? 05 d
[j 4 = d
U 4 0 }f & d
H 4 05 d
@'D 4 6
C~skhk`pksd(
!yc Ir 4 5( !yc I{ 4 5
Bn $ B` 4 5( Yn ! Y` 4 >9 $$$$$$ " k +
Bn 4 B`
!yc Jhfja|kzc D 4 5 n} ufkg} N
>9 / =Y` / 05 4 5 Y` 4 :'0 } us}}kgl }bkz ynhsc kg c~"k+ Yn 4 >0'2 }
P'B'Z
!yc Jhfja|kzc D 4 5 n} ufkg} J
Yn / >9 ! Bn / =>9 / 9 4 5 B`4 2 } nz B`4Bn ( Zf ( Bn 4 2 }
PCZSH]NG]
Pn 4 Bn0
! Yn0
>3'5;= }
P`4 B`0
! Y`0
2'9> }
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
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]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:< 92 x U n l c
@'D dnr " !yc +
Yn / Bn / [0 / / &0 $$$$$$$$ "y+
Szkgl Ifpdshn
[ 4 = [j " H + & H0
[ 4 = / = / " 05 + & "05+0
[ 4 5'2 5'5= 0
Us}}kgl Ynhsc fi [ kg c~ " y+
@'D4 >0'2 / 2 / " 5'2 5'5= 0
+ $ 0
@'D 4 >0'2 9'= ! 5':0 0
0
@'D 4 9'= 5'92 0
$$$$ " yk+
Mkiicpcg}kn}kgl |'p'}'
5 4 9'=" 5'92 + " 0 +
4 ='< d ipfd hci} bngm zkmc zsuufp}
Us}}kgl ynhsc fi kg c~ " yk +
@'D 4 9'= / ='
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
69/72
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
70/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
5;JC:<
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7/23/2019 Theory of Structures Problems solved by Dk Mamonai
72/72
]bcfp{ fi Z}psj}spcz Zkp Mp'_s`nkp
Upf`hcd 0 ? N jnz} kpfg : d kg hcgl}b |bcg bkglcm n} `f}b cgmz bnz n jpk}kjnh ukjahkgl hfnm fiU akhflpnd
|bcg }bc jfhsdg kz ikrcm n} `f}b }bc cgmz k}z jpk}kjnh hfnm pkzcz }f U ! :5(555 akhflpnd ' ki pn}kf fi
cr}cpgnh mkndc}cp kz >'03 ngm C kz c~snh }f > / >5 9 ' Mc}cpdkgc }bc cr}cpgnh ngm kg}cpgnh mkndc}cp fi
jfhsdg'
H 4 :555 jd ( mf&mk 4 >'03 mf 4 >'03 mk
Hfnm |bcg `f}b cgmz npc bkglcm ? U al
Hfnm |bcg `f}b cgmz npc ikrcm ? U ! :5(555
C 4 > / >59
Szkgl Cshcpz }bcfpcd |bcg `f}b cgmz npc bkglcm H 4 h
U 4 0
C K & H0
$$$$$$$$$$$$$$$$$$$$$$$$$$ " k +
Ikgmkgl kg}cpkn " k+ 4 k 4 & 9= " mf0 mk0 + Us}}kgl ynhsc fi mf
k4 & 9= " >'03 mk +0
$ " mk +0
k4 & 9= " 5'03 mk + = $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ "kk+
us}}kgl ynhsc fi k kg c~sn}kfg "k+
U 4 0 C K & H 0 U 4 0C " & " 5'03 mk + = + & 9= / H 0 U 4 : "5'03 mk+ = $$$$$$$$$$$$$$$$$$$$$$$$$$ "kkk+
Szkgl Cshcpz }bcfpcd |bcg `f}b cgmz npc ikrcm' H 4 h & 0 :555 & 0 >355 jd
U ! :5( 555 4 = 0 C K & H 0
Us}}kgl ynhsc fi k
U ! :5 ( 555 4 = 0C q & " 5'03 mk + = t & 9= / H 0 : " mk + =
U ! :5(555 4 :
" mk +=
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ "ky+
Us}}kgl ynhsc fi U ipfd c~ "kkk+ kg}f c~ " ky+
:
"5 03 mk+=
! :5 555 4 :
" mk +=