ZURAIDAH BINTI AHMAD PEJABAT B, JKA, PUO http://mindasarjana!"#$sp#t %#m
&&'()* +TRU&TURA ANA-+I+ . &HAPTER ): ANA-+I+ O +TATI&A- DETERMINATE .D PIN JOINTED TRU++E+ E&TURE: ' HOUR+ TUTORIA: 0 HOUR+
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1.0 ANALYSIS OF STATICALLY DETERMINATE 2D PIN-JOINTED TRUSSES. )) Und1rstand th1 app"i%ati#n #2 m1th#d #2 j#ints and m1th#d #2 s1%ti#ns in d1t1rminin$ th1 2#r%1s in tr3ss1s ))) E4p"ain t5p1s #2 tr3ss1s )). Di61r1ntiat1 !1t711n d1t1rminat1 and
ind1t1rminat1 tr3ss1s ))0 E4p"ain t1nsi#n m1m!1rs and %#mpr1ssi#n m1m!1rs ))8 +tat1 th1 s5m!#" 2#r t1nsi#n and %#mpr1ssi#n m1m!1rs ))9 E4p"ain th1 m1th#d #2 j#ints ))' E4p"ain 1r#*2#r%1 m1m!1rs )); E
D1?niti#n: A tr3ss is a str3%t3r1 that %#nsists #2 ) A"" strai$ht m1m!1rs . &#nn1%t1d t#$1th1r 7ith pin j#ints 0 &#nn1%t1d #n"5 at 1nds #2 th1 m1m!1rs 8 And a"" 14t1rna" 2#r%1s @"#ads r1a%ti#ns m3st !1 app"i1d #n"5 at th1 j#ints N#t1: E<1r5 m1m!1r #2 a tr3ss is a . 2#r%1 m1m!1r Tr3ss1s ar1 ass3m1d t# !1 #2 n1$"i$i!"1 71i$ht @%#mpar1d t# th1 "#ads th15 %arr5 •
))) T5p1s #2 tr3ss1s a truss is a str3%t3r1 %#mprisin$ #n1 #r m#r1 trian$3"ar 3nits %#nstr3%t1d 7ith strai$ht m1m!1rs 7h#s1 1nds ar1 %#nn1%t1d at j#ints r121rr1d t# as n#d1s E4t1rna" 2#r%1s and r1a%ti#ns t# th#s1 2#r%1s ar1 %#nsid1r1d t# a%t #n"5 at th1 n#d1s and r1s3"t in 2#r%1s in th1 m1m!1rs 7hi%h ar1 1ith1r t1nsi"1 #r %#mpr1ssi<1 2#r%1s M#m1nts @t#rC31s ar1 14p"i%it"5 14%"3d1d !1%a3s1, and #n"5 !1%a3s1, a"" th1 j#ints in a tr3ss ar1 tr1at1d as r1<#"3t1s
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A p"anar tr3ss "i1s in a sin$"1 p"an1 P"anar tr3ss1s ar1 t5pi%a""5 3s1d in para""1" t# 2#rm r##2s and !rid$1s
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A p"anar tr3ss "i1s in a sin$"1 p"an1 P"anar tr3ss1s ar1 t5pi%a""5 3s1d in para""1" t# 2#rm r##2s and !rid$1s A spa%1 2ram1 tr3ss is a thr11* dim1nsi#na" 2ram17#r #2 m1m!1rs pinn1d at th1ir 1nds A t1trah1dr#n shap1 is th1 simp"1st spa%1 tr3ss, %#nsistin$ #2 si4 m1m!1rs 7hi%h m11t at 2#3r j#ints
Di61r1ntiat1 !1t711n d1t1rminat1 and ind1t1rminat1 tr3ss1s
This is a n1%1ssar5 %#nditi#ns 2#r stati%a""5 d1t1rmina%5 This is n#t s3%i1nt %#nditi#n +# 1<1n i2 a tr3ss satis?1s th1 a!#<1 r1"ati#n it ma5 n#t !1 d1t1rminat1 B3t i2 it is d1t1rminat1 than it satis?1s th1 a!#<1 r1"ati#n MF.n*r : stati%a""5 d1t1rminat1
14amp"1 • • • • • •
.n F mr mF9 rF0 nF8 .n F.48 F = mr F9 0 F=
14amp"1
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.n F mr m F)9 rF0 nF> .n F.4 > F )= mr F)9 0 F )=
mF m1m!1r rF r1a%ti#ns n F j#ints
Externa !"r#es$ #ads a%tin$ "n 5#3r str3%t3r1 N#t1: this in%"3d1s r1a%ti#n 2#r%1s 2r#m th1 s3pp#rts as 71"" Interna !"r#es: #r%1s that d1<1"#p 7ithin 1<1r5 str3%t3r1 that 11p th1 di61r1nt parts #2 th1 str3%t3r1 t#$1th1r
E4p"ain t1nsi#n m1m!1rs and %#mpr1ssi#n m1m!1rs
+tat1 th1 s5m!#" 2#r t1nsi#n and %#mpr1ssi#n m1m!1rs
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T5p1 #2 int1rna" 2#r%1s +5m!#"s )T1nsi#n @T .&#mpr1ssi#n @&
<1
*<1
Truss Ana%s&s ' Met(") "! J"&nts $ •
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M1th#d #2 j#int is a str3%t3ra" ana"5sis m1th#d t# d1t1rmin1 th1 int1rna" 2#r%1s #2 m1m!1rs in a tr3ss This m1th#d is d1ri<1d !as1d #n th1 1C3i"i!ri3m %#nditi#ns at j#ints It is !asi%a""5, a sp1%ia" %as1 #2 1C3i"i!ri3m #2 %#n%3rr1nt @int1rs1%tin$ 2#r%1s In this %as1 r#tati#na" #r m#m1nt 1C3i"i!ri3m is r1adi"5 satis?1d at th1 j#int Th1r12#r1 in th1 m1th#d #2 j#int 2#r p"an1 tr3ss th1r1 ar1 #n"5 t7# 1C3i"i!ri3m 1C3ati#ns r1C3ir1d: Pr"#e)ure !"r *et(") "! +"&nts$ +1"1%t a j#int ha
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Dra7 th1 2r11*!#d5 dia$ram @BD 2#r th1 Ma1 s3r1 thatt# a""!1 n#7n and 3nn#7n 2#r%1s ar1 a%%#3nt1d 2#r #r th1 3nn#7n 2#r%1s, th1s1"1%t1d dir1%ti#nj#int #2 th1 2#r%1s n11d ass3m1d M5 r1%#mm1ndati#n is t# a"7a5s ass3m1 that th1 3nn#7n 2#r%1s ar1 in t1nsi#n N1$ati<1
E4p"ain th1 m1th#d #2 j#ints I2 a tr3ss is in 1C3i"i!ri3m, th1n 1a%h #2 its j#ints m3st !1 in 1C3i"i!ri3m • Th1 m1th#d #2 j#ints %#nsists #2 satis25in$ th1 1C3i"i!ri3m 1C3ati#ns 2#r 2#r%1s a%tin$ #n 1a%h j#int •
L 4 F( L5F(
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&#nsid1r th1 tr3ss sh#7n Tr3ss ana"5sis in<#"<1s: @i D1t1rminin$ th1 ETERNA r1a%ti#ns @ii D1t1rminin$ th1 INTERNA 2#r%1s in 1a%h #2 th1 m1m!1rs @t1nsi#n #r %#mpr1ssi#n
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R1%a"": Externa !"r#es$ #ads a%tin$ "n 5#3r str3%t3r1 N#t1: this in%"3d1s r1a%ti#n 2#r%1s 2r#m th1 s3pp#rts as 71"" Int1rna" 2#r%1s: #r%1s that d1<1"#p 7ithin 1<1r5 str3%t3r1 that 11p th1 di61r1nt parts #2
M1th#d #2 j#ints
@i E4t1rna" r1a%ti#n
@ii Int1rna" #r%1s
J#int B
,!%0 CD / 20 0 CD -20N C"*ress&"n3
J#int &
L 4 F ( *B& A& &#s 89( )( F ( A& F )(/&#s 89( F )8)8N
4 De#e*5er 2016$ Met(") "! J"&nt Re#a
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h1r1: m F n# #2 m1m!1r r F n# #2 r1a%ti#ns j F n# #2 j#int mF.j*r : stati%a""5 d1t1rminat1 mG.j*r: 3nsta!"1 m .j*r: stati%a""5 ind1t1rminat1 N# #2 r1d3ndan%5 F * / r -2+ De7ree "! &n)eter*&nate De7ree "! Free)"*
E4amp"1 . D1t1rmin1 th1 d1$r11 #2 2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss .9 9( N
N
D
&
• 8m A
B
8m
• 8m
@) D1$r11 #2 r11d#mF m* .j r • mF9 •
rF0
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D#Fm*.j r D# F 9*@.480 D#F ( * N# r1d3ndan%5 * +tati%a""5 d1t1rminat1
E4amp"1 0 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss .9 9( @. E4t1rna" N N R1a%ti#ns D & L4F( • 8m A
B B4 8m
A5
8m B5
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B4 F ( LM!F( A5@89(@8F( A5 F *9(N L5F( A5 B5*.9*9( F( *9( B5 *;9 F(
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B5 F).9N
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@. Int1rna" 2#r%1s
J#int A
J#int B
r11 !#d5 Dia$ram AD
r11 !#d5 Dia$ram BD B&
AB
AB
9(N • • • • • •
L5F( AD &#s89( 9( F( AD F ;(;)N @t1nsi#n L4F( AB AD &#s89( F ( AB F *9(N @%#mpr1ssi#n
).9 N
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L4F( B& &#s89( * AB F ( B& &#s89( @*9( F ( B& F *;(;)N @%#mpr1ssi#n L5F( BD B& &#s89( ).9 F( BD @*;(;) &#s89 ( ).9F (
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BD F *;9N @%#mpr1ssi#n
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@. Int1rna" 2#r%1s
J#int & r11 !#d5 Dia$ram 9(N &D &B
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L4F( &D &B &#s89 ( F ( &D @*;(;) &#s89( F ( &D F 9(N @t1nsi#n
Int1rna" 2#r%1s
E4amp"1 0 D1t1rmin1 th1 d1$r11 #2 2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss
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@) D1$r11 #2 r11d#mF m* .j r • mF)0 •
rF0
D#Fm*.j r D# F )0*@.4=0 D#F ( * N# r1d3ndan%5 * +tati%a""5 d1t1rminat1
E4amp"1 0 R1a%ti#n 2#r%1s at s3pp#rts • • • • •
L4F( Q4 9@0 )9@0 F ( Q4 '( F( Q4F *'(N LM$F(
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8x
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8%
9%
H5@8*)9@0'>* 9@0'>F( H5F >(N L5F( Q5 H5 F( Q5 >( F( Q5 F*>(N
E4amp"1 0 Int1rna" 2#r%1s
8x
8%
9%
4amp 1 1 1rm n1 1 1$r11 # 2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in m1m!1r AB, AE, &D, DQ
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D1$r11 #2 r11d#mF m* • .j r mF)) •
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rF0
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D#Fm*.j r F ))* .4;0 D#F ( * N# r1d3ndan%5
@) E4t1rna" r1a%ti#n
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L5F( A5* 8 @).9 F( A5 F9(N LMaF( E4@.9* ).9@.8'F( E4 F '(N L4F( A4 E4 F ( A4 '( F(
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A4F *'(N
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A4
• • •
A5
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E4 • • •
@. Int1rna" 2#r%1s ).9 J#int D
J#int A
N
r11 !#d5 Dia$ram ).9 N '(N 9( N
D & AB
AE
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L5F( 9( *).9 AE F( AE F 0;9N @t1nsi#n L4F( AB *'( F (
•
AB F'(N @t1nsi#n
• • •
•
Ө
DQ • • • •
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r11 !#d5 Dia$ram DEF@'..9.)/ .
DE F '9m
L5F( *DQ+inӨ *).9 F( *DQ@.9/'9*).9 F ( DQF *0.9N @&#mpr1ssi#n L4F( D& DQ&#sӨ F ( D& @*0.9@'/'9 F( D&F 0(N @T1nsi#n
2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss )(N
A
B &
8m
D E .(N .m
Q 0(N
.m
.m
2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss
2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss
2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss
2r11d#m #2 th1 str3%t3r1 B5 3sin$ th1 m1th#d #2 j#ints, d1t1rmin1 th1 2#r%1 in 1a%h m1m!1r #2 th1 tr3ss
.m
9N 8m
8m
)(N
E4amp"1 >
.m 9N 8m
8m
)( • D#Fm*.j r N • D# F 9*@.480 •
@) D1$r11 #2 r11d#mF m* .j r • mF9 • rF0 • jF8
• •
D#F ( * N# r1d3ndan%5 * +tati%a""5 d1t1rminat1
E4amp"1 >
A4 .m
A5 S
B4 J#int D DA
D&
• •
)( • N • • • • •
@. E4t1rna" R1a%ti#ns • L5F( 9N )( • A5 *9*)( F( N 8m 8m • A5 F )9N ADF@..=.)/.F '= • LMaF( L5F( • B4@.F 9@8 )(@= DA +inӨ )( F( • B4 F 9(N DA@./'=*)( F( • L4F( DA F 8).0N @t1nsi#n L4F( • A4 B4 F ( D& DA &#s Ө F ( • A4 9( F( D&DA@=/'=F( • A4 F *9(N Ө
D& F *8(N @%#mpr1ssi#n
@. Int1rna" 2#r%1s
J#int B
J#int &
r11 !#d5 Dia$ram AB 9(N
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A&F@..8.)/.F .( B&
L5F( AB F( L4F( B& 9( F ( B& F *9(N @%#mpr1ssi#n
r11 !#d5 Dia$ram &A
&B
&D
9 N • • • • • • • •
L5F( &A +inS F 9 &A@ ./.( F9 &A F )))= N @t1nsi#n L4F( &B&A &#s S F&D &B)))= @8/.( F *8( B& F *9(N @%#mpr1ssi#n * pr#<1n
E4p"ain 1r#*2#r%1 m1m!1rs AB 9(N
• •
• • •
B&
L5F( AB F( 1r# 2#r%1 m1m!1r L4F( B& 9( F ( B& F *9(N @%#mpr1ssi#n
L4F( B& F ( B& is a 1r# 2#r%1 m1m!1r
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In th1 ?1"d #2 1n$in11rin$ m1%hani%s, a :er" !"r#e *e*5er r121rs t# a m1m!1r @a sin$"1 tr3ss s1$m1nt in a tr3ss 7hi%h, $i<1n a sp1%i?% "#ad, is at r1st: n1ith1r in t1nsi#n, n#r in %#mpr1ssi#n In a tr3ss a 1r# 2#r%1 m1m!1r is #2t1n 2#3nd at pins @an5 %#nn1%ti#ns 7ithin th1 tr3ss 7h1r1 n# 14t1rna" "#ad is app"i1d and thr11 #r 2171r tr3ss m1m!1rs m11t R1%#$niin$ !asi% 1r# 2#r%1 m1m!1rs %an !1 a%%#mp"ish1d !5 ana"5in$ th1 2#r%1s a%tin$ #n an indi
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NOTE: If the pin has an external force or moment applied to it, then all of the members attached to that pin are not zero force members UNLESS the external force acts in a manner that fullls one of the rules belo:
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I2 #n"5 t7# m1m!1rs m11t in an 3n"#ad1d j#int, !#th ar1 1r#*2#r%1 m1m!1rs I2 thr11 m1m!1rs m11t in an 3n"#ad1d j#int #2 7hi%h t7# ar1 in a dir1%t "in1 7ith #n1 an#th1r, th1n th1 third m1m!1r is a 1r#*2#r%1 m1m!1r I2 t7# m1m!1rs m11t in a "#ad1d j#int and th1 "in1 #2 a%ti#n #2 th1 "#ad %#in%id1s 7ith #n1 #2 th1 m1m!1rs, th1 #th1r m1m!1r is a 1r#*2#r%1 m1m!1r Reas"ns !"r ;er"-!"r#e *e*5ers &n a truss s%ste* Th1s1 m1m!1rs %#ntri!3t1 t# th1 sta!i"it5 #2 th1 str3%t3r1, !5 pr#
Id1nti25 th1 1r# 2#r%1 m1m!1r 2r#m th1 tr3ss !1"#7 • • •
8x
8%
9%
M1m!1r A& M1m!1r QH Th1 1r# 2#r%1 m1m!1r %arr5 n# int1rna" 2#r%1
Id1nti25 th1 1r# 2#r%1 m1m!1r 2r#m th1 tr3ss !1"#7 • •
M1m!1r AD M1m!1r B&
Id1nti25 th1 1r# 2#r%1 m1m!1r 2r#m th1 tr3ss !1"#7 M1m!1r AB
.m
9N 8m
8m
)(N
Id1nti25 th1 1r# 2#r%1 m1m!1r 2r#m th1 tr3ss !1"#7
M1m!1r AB M1m!1r BQ M1m!1r DI M1m!1r DE
