meh1-15_16-p11
TRANSCRIPT
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MEHANIKA I(STATIKA)
MEH1−15.16−P11
3/12/16 1
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rimer Gerberovog rama
0 1 2 3 4 5PODACI: q = 20 kN/m, F = F = F = F = F =10 kN,
a =1 m .
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1) Reakcije i reaktivni moment
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Statiki !"#$%i &a%'$t()(:
... *1+
3 4
01 2 5
04 5
1 3 2 4
05
0
30
23
4 5 2 0
23
3 3 5 6 0
2
A
A B
d
G B
A A
B
H H F F
q aV V F F V F
q a M V a F a F a a
M M F a F a F a F a
q aa V a F a
= − − + =
= − − − + + =
= × + × + × − × =
= + × + × − × − × −
− × + × + × =
∑∑
∑
∑
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I t&(-( !"#$%a ! *1+ $iam$ ,
I &%$ !"#$%a ! *1+ $iam$ HA,
... *2+
... *3+
... *4+
... *5+
( )
( )
4 5 0
15 3
41
10 5 10 3 20 1 04
B
V F F q a= − − + =
= − − × + × × =
0 BV =
3 4 10 10 0 A H F F = − + = − + =
0 A H =
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Sa = 0, i (t%&t$ !"#$%a ! *1+ $iam$ MA,
... *+
... *7+
1 3 2 4
05
33 3 5 6
2 10 1 10 1 10 1 10 3 1
3 20 1 3 1 0 5 1 10 6 1 50 kNm2
A
B
M F a F a F a F aq a
a V a F a
= − × − × + × + × +
+ × − × − × =
= − × − × + × + × × +
× ×+ × × − × × − × × =
50 kNm A M =
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2) Izrazi za presečne veličine u
proizvoljnim presecima delova
(polja) rama
2
4
10 kNm
10 kNm
M
M
=
=
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... *10+
... *11+
( )
( ) ( )
( )
,
,
,
50 kNm
0
40 kN
x I A
x I
y I
I A
M z M
dM z T z
dz
N z V
= − = −
= =
= − = −
( )( )
( )
,
,
50 kNm0
40 kN
x I
y I
I
M z T z
N z
= −=
= −
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Polje II: 1 ≤ ≤ 2
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... *13+
... *12+
( ) ( )
( )
( ) ( )
( )
, 3
,
,
50 10 1 10 60
10 kN
40 kN
x II A
x II
y II
II A
M z M F z a
z z
dM z T z
dz
N z V
= − + − =
= − + − = −
= =
= − = −
( )
( )
( )
,
,
10 60
10 kN
40 kN
x II
y II
II
M z z
T z
N z
= −
=
= −
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Polje III: 2 ≤ ≤ 4
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... *15+
... *14+
( ) ( )
( )
( ) ( )
( )
, 3 2
,,
2
50 10 1 10 10 50
10 kN
40 10 30 kN
x III A
x III
y III
III A
M z M F z a M
z z
dM z T z
dz
N z V F
= − + − + =
= − + − + = −
= =
= − + = − + = −
( )
( )
( )
,
,
10 50
10 kN
30 kN
x III
y III
III
M z z
T z
N z
= −
=
= −
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Polje IV: 0 ≤ ≤ 1
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... *1+
... *16+
( )
( ) ( )
( )
, 1
,
,
10
10 kN
0
x IV
x IV
y IV
IV
M z F z z
dM z
T z dz
N z
= − = −
= = −
=
( )( )
( )
,
,
1010 kN
0
x IV
y IV
IV
M z z
T z
N z
= −
= −
=
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Polje V: 1 ≤ ≤ 2
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... *17+
... *1+
( ) ( ) ( )
( ) ( )
( ) ( )
( )
, 5 4 0
,
,
4
3 4
2
3 10 10 20 4 20 402
20 kN
10 kN
x V
x V
y V
V
M z F a z M q a a z
z z z
dM z T z
dz
N z F
= − + − − =
= × − + − × × − = −
= =
= =
( )
( )( )
,
,
20 40
20 kN10 kN
x V
y V
V
M z z
T z
N z
= −
=
=
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Polje VI: 2 ≤ ≤ 3
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... *21+
... *20+
( ) ( ) ( )
( ) ( )
( ) ( )
( )
, 5 4 0
,
,
4
3 4
2
3 10 10 20 4 20 402
20 kN
10 kN
x VI
x VI
y VI
VI
M z F a z M q a a z
z z z
dM z T z
dz
N z F
= − + − − =
= × − + − × × − = −
= =
= =
( )
( )( )
,
,
20 40
20 kN10 kN
x VI
y VI
VI
M z z
T z
N z
= −
=
=
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Polje VII: 3 ≤ ≤ 5
( )0 6
3
q a z q a
−
=
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... *23+
( ) ( )
( ) ( )
( )
3
,
2,
100 10 6
7
1010 63
10
x VII
y VII
VII
M z z z
T z z
N z kN
= − − −
= − + −
=
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Polje VIII: 5 ≤ ≤ 6
( )0 6
3
q a z
q a
−
=
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... *24+
( ) ( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
2
, 5
305
3
3
2,,
1 6
6
61
10 10 6
7
10 0 10 6
7
1010 6
30
x VIII
x VIII
y VIII
VIII
M z F a z q a z
q F a z a z
a
z z
z z
dM z T z z
dz N z
= − − − =
= − − −
= × − − − =
= − − −
= = − + −
=
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... *25+
( ) ( )
( ) ( )
( )
3
,
2
,
100 10 6
7
1010 63
0
x VIII
y VIII
VIII
M z z z
T z z
N z
= − − −
= − + −
=
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Polje IX: 6 ≤ ≤
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... *2+
... *26+
( ) ( ) ( )
( )
( )
, 5
,
10 10 0
10 kN
0
x IX
y IX
IX
M z F a z z z
T z
N z
= − = − = − +
= −
=
( )
( )
( )
,
,
10 0
10 kN
0
x IX
y IX
IX
M z z
T z
N z
= − +
= −
=
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3) resečne veličine u
karakterističnim presecima
P$#( 8m9 ; 8kN9 M
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P$#( 8m9 ; 8kN9 M
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P$#( 8m9 ; 8kN9 M
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... *2+
... *27+
... *30+
... *31+
... *32+
( ) ( )2
,
1010 6 0
3 y VII T z z = − + − =
4, 26 m z =
( ) ( )3
,
100 10 6
7 x VII M z z z = − − −
⇓
⇓
( ) ( )3
, 4,26
100 10 4, 26 6 4,26 31,54 kNm
7 x VII z
M z =
= − × − − =
( ), 4,26 31,54 kNm x VII z M z = =
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'tatički di$arami zadat# rama
) !tatički dijagrami
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Rešetke
>(?(tka ( k$'"t&!k@ia $&a$%a'a $ &a%i?ta$%a k$ima "! *m$)( "( "mat&ati+ k&a(%i#$'$ %(a'i.
Ak$ "%i ?ta$%i #()( ! ('$ &a%'i &( ( $&a%'$ &(?(tki. B &$ti%'$m ( &( $ &$"t$&'$&(?(tki.ta$%i &(?(tk( "( ! &ak"i %()! ak$%i@ama,
a%&t'(%ima i#i a%a&('im "$(%ima.I tak%( %(( "( "mat&a! #$'im %(ama *t.
%(ama k$( '( &('$"( m$m('t+
3/12/16 34
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>(?(tka ( $t(&(-('a "i#ama k$( (#!! !#$$%ima tak#zvanim čv#r#vima.()i'( ?ta$%a "( a'(ma&!!.&('( ! #$$%ima "( a'(ma&!(.B "#!a! a "( t()i'( ?ta$%a !m! ! $i&,
&$$&@i$'a#'$ i t&(a &a"$&(iti 'a %a%$&a 'a k&a(%ima "%ak$ $ ?ta$%a.Smat&a "( a "! ?ta$%i &(?(tk( $t(&(-('i
*'a&('!ti+ "am$ 'a at(a'( i#i &iti"ak.>(?(tk( "( $&a!! $ &a%i ?ta$%a k$i
"( #$'$ %()! ! t&$!#$%(.
3/12/16 35
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#$'im %(i%a'(m &a%i ?ta$%a !t&$!#$%( $iam$ "tai#'! &(?(tk!.Stai#'$"t ?ta$%a #$'$ %(a'i !
t&$!a$, &(ma 'a&('$ "#i@i ( $i#('a.
3/12/16 36
(z sta)iln#st *ta"#va zl#)n# vezani+ , tr#,a#
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Ak$ i "( &(?(tka $&a$%a#a $ '&. ?ta$%a#$'$ %(a'i ! (t%$&$!a$, tak%a
&(?(tka i i#a '("tai#'a. N("tai#'$"t ?ta$%a #$'$ %(a'i !
(t%$&$!a$ &ikaa'a ( 'a 'a&($ "#i@i.
3/12/16 3
(z nesta)iln#st *ta"#va zl#)n# vezani+ , četv#r#,a#
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P&i $&mi&a'! &(?(tk( 'a&( "( m(G!"$'$ $%()! t&i ?taa, a atim "( 'a$&aG!!
$"ta#i t&$!#$%i.a 'a$&a'! '$%$ ?ta'$ t&$!#a $t&('a "! %a *2+ '$%a ?taa i (a' *1+
'$%i %$&.>(?(tka "a %a t&$!#a ima#a i !k!'$ =4
%$&a.
'a$&a'! '$%$ t&$!#a $t&('$ (:◦ − 3 '$%i %$&$%a i
◦ ? = 2 − 3 '$%i ?ta$%a.
3/12/16 3
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Ak$ (
&(?(tka ( "tai#'a i "tatiki$&(G('a *m$)(m$ $ $&(iti "i#( !"%im ?ta$%ima+.>(?(tka ka$ k$'"t&!k@ia $"#a'a "( tak$ a
$"t$( t&i *3+ &(ak@i( %(a k$( "( m$!$&(iti i t&i *3+ "tatika !"#$%a &a%'$t()(.A'a#ia k$'k&(t'( &(?(tk( "%$i "( 'a
$&(Gi%a'( &(ak@ia, a atim i $&(Gi%a'("i#a ! ?ta$%ima.
3/12/16 37
2 3 *č = −
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>(?(tk( k$( -(m$ i!a%ati &iaa-( &!i&a%'i &(?(tki.Si#( ! ?ta$%ima &(?(tki $&(Gi%a-(m$ &im('$m:◦ M(t$a i"(@a'a *i&(i%a'a+ %$&$%a i
◦
M(t$a &("(ka *>it(&$%$ m(t$a+.Oa 'a%(('a m(t$a "! a'a#itika.B #it(&at!&i "( a $&(Gi%a'( "i#a !
?ta$%ima &(?(tki "&(-! i &aiki m(t$i ka$*&(m$'i' i !#ma'$%+ k$i "! "a &a%$(m&a!'a&"k( t('ik( $ti"'!ti.
3/12/16 40
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Primena metoda isecanja čvorova
O%a m(t$ "( "a"t$i ! "#((-(m:◦ I "tatiki !"#$%a &a%'$t()( $&(( "(
&(ak@i(.
◦ atim "( ami?#('im "(('(m ?ta$%a
i"(@a %$& $ %$& i ami?#('$ $%aa $$"tatka &(?(tk(.
◦ Bti@a &($"ta#$J (#a &(?(tk('a$m(?tam$ "i#ama ! &a%@! ?ta$%a.
◦ Sm(& "i#a ! &a%@! ?ta$%a "( &(t$"ta%i, a&a!' -( 'a k&a! $kaati a #i "m$ !
&(t$"ta%@i $&(?i#i.
3/12/16 41
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◦ S%aki $ %$&$%a $t(&(-(' ( "i"t(m"!(#'i "i#a.
◦ P&i $&(Gi%a'! "i#a ! ?ta$%ima k&(-(m$ $%$&a ! k$(m "! %( ?ta'( "i#( '($'at(*taka% %$& !%(k $"t$i+.
◦ Nak$' t$a "( &(#ai 'a 'a&('i %$& "a'a%i?( %( '($'at( ?ta'( "i#(.
◦ $ 'a&('$ %$&a &$m('i "( "m(& "i#( !?ta! k$i ( a('iki "a &(t$'im
%$&$m.
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(z "rimer -
rimer ravne re"etke
a &(?(tk! $&a$%a'! $ ?ta$%a #$'$%(a'i ! t&i ('ak$"t&a'i'a t&$!#a "t&a'i@(!)i'( l , m(t$$m i"(@a'a %$&$%a $&(iti "i#(! ?ta$%ima.
Podaci:F1 = 20 kNF2 = 40 kN
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1) Reakcije
K *1+
Statiki !"#$%i &a%'$t()(:
1
1 2
00
32 0
2 2
A
A B
B A
X X F . . . F
l M . l F l F
= − + == + − =
= × + × − =
∑∑
∑
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I &%$ !"#$%a ! *1+ $iam$ A,
K *2+
I t&(-( !"#$%a ! *1+ "#(i LA,
K *3+
K *4+
Sa *4+, i &!$ !"#$%a ! *1+ "#(i L,
K *5+
K *6+
1 10 kN A X F = =
( ) ( )2 11 1
3 40 20 3 1,344 4
A. F F kN = − = × − =
1,34 A. kN =
2 40 1,34 3,66 B A. F . kN = − = − =
3,66 B. kN =
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2) !ile u "tapovima rimena
metoda isecanja čvorova
Čvor 4
Statiki !"#$%i &a%'$t()(:K *+24 45
45
@$"60 0
"i' 60 0 B
X ' '
. . '
= − + =
= − =
∑
∑
o
o
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3/12/16 4
I *+ $iam$ a "!:
K *+
K *7+
24 45
45
1@$"60 44,64 22,32 kN23,66
44,64 kN"i'60 3 2
B
' '
. '
= = × =
= = =
o
o
24
45
22,32 kN
44,64 kN
'
'
=
=
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3/12/16 4
Čvor 5
Statiki !"#$%i &a%'$t()(:
K *10+
35 25 45
25 45 2
@$"60 @$" 60 0"i' 60 "i' 60 0
X ' ' ' . ' ' F
= + − == + − =
∑∑
o o
o o
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I *10+ "#((:
K *11+
K *12+
( ) ( )
35 45 25
225 45
1
@$"60 44,64 1,54 2 21,546 kN
4044,64 1,54 kN
"i'60 3 2
' ' '
F ' '
= − = − × =
=
= − = − =
o
o
35
25
21,546 kN
1,54 kN
'
'
=
=
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Čvor 3
Statiki !"#$%i &a%'$t()(:
K *13+
1 35 13 23
13 23
@$"60 @$"60 0"i' 60 "i' 60 0
X F ' ' ' . ' '
= − + + == − =
∑∑
o o
o o
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K *14+
K *15+
I *13+ "#((:
35 113 23
13 23
21,546 20 3,072 kN@$"60 1 2' F ' '
' '
− −+ = = =
=
o
13 23 1,546 kN' ' = =
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3/12/16 52
Statiki !"#$%i &a%'$t()(:
Čvor 2
K *16+
12 24 23 25
23 25
@$" 60 @$"60 0
"i' 60 "i' 60 0
X ' ' ' '
. ' '
= − + − − =
= − =
∑
∑
o o
o o
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K *1+
K *1+
I *16+ "#((:
( )( )
12 24 23 25
23 25
@$"60
1 22,32 1,546 1,546 20,4 kN
2
1,546 kN
' ' ' '
' '
= − + =
= − + × =
= =
o
12
23 25
20,4 kN
1,546 kN
'
' '
=
= =
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Statiki !"#$%i &a%'$t()(:
Čvor 1
K *17+
12 13
13
@$" 60 0
"i' 60 0
A
A
X / ' '
. . '
= − − − =
= − =
∑
∑
o
o
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K *20+
K *21+
I *17+ "#((:
12 13
13
1@$" 60 20 1,54 20,4 kN21,34
1,54 kN"i'60 3 2
A
A
' / '
. '
= + = + × =
= = =
o
o
12
13
20,4 kN
1,54 kN
'
'
=
=
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( )
( )
45
24
44,64
22,32
' kN
' kN
− =
+ =
( )
( )
( )
25
35
45
1,54
21,546
44,64
' kN
' kN
' kN
− =
− =
− =
( )
( )
( )
13
23
M
35
1,546
1,546
21,546
' kN
' kN
' kN
− =
+ =
− =
( )
( )( )
( )
M
23
M
25
12
M
24
1,546
1,54620,4
22,32
' kN
' kN ' kN
' kN
+ =
− =
+ =
+ =
( )
( )
M
13
M
12
1,54
20,4
' kN
' kN
− =
+ =
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Štap Sila u tapu !"#$
at('a*+
P&iti"'a*−+
20,4
22,320
1,54
1,546
1,54
44,640
21,546
24
12
13
23
25
4535