Pravougaona plo č a Za datu datu plo plo č u i opte opterre ć enje enje sra sra č unat unatii inacrt inacrtat atii dija dijagr gram amee:
1. Ugibaw; 2.
Prese č nih sila M x ,M y ,M xy, T x iT y ;
3.
ea!c ea!cij ijee oslon oslonaca aca " x i " y # $eometrijs!e $eometrijs!e !ara!teristi!e !ara!teristi!e ploč plo č e : % x =1.9 m & =1.25 ' =0.5 ( =0.2 h =0.20 m ) =0.20
Opterećenje ploče: 2
p=12 !* / m + =16 !* / / m =24 !*
& =
% y % x
=¿ 1.25=
% y 1.9
=¿ % y =1.9∗1.25=2.375 m
l x = 4∗% x =4∗1.9 =7.6 m l y = 4∗% y = 4∗2.375=9.5 m 2∗ ' ∗l x =2∗0.5∗7.6 =7.6 m 2∗ ( ∗l y =2∗0.2∗9.5 =3.8 m
5 7 3 . 2
N k 4 2 = F
F=24 kN 4
3
2 5 7 3 . 2
q=16 kN/m
0 8 . 3
5
6
0 5 . 9
1 5 7 3 . 2
p=12 kN/m²
F=24 kN 5 7 3 . 2
1.90
1.90
1.90
1.90
7.60
F=24 kN
q=16 kN/m p=12 kN/m²
X
² m / N k 2 1 = p N k 4 2 = F
18
17
16
9
10
11
4
3
2
Y
19
8
12
15
X 20
7
5
6
1
13
14
-onturni uslovi w 18= w 4 w 17= w3 w 16= w2 w 15= w2 w 14= w 1 w 19=−w4 w 20=−w5 w 7=w 8=w 9= w10=w 11= w12= w13= 0 Proračunelemenata al.a / eme 2
& =1.5625 1 2
&
=0.6400
4∗( 1 + & ) =10.2500 2
( )= ∗ + + = ( )
4∗ 1 +
6
2
&
2
1
6.5600
2
&
1
8
2
21.215
&
4
& ∗% x =20.3627
2 -10.2500 2
1.5625
0.6400 -6.5600 21.2150 -6.5600 0.6400
Proračun slobodnih članova 3
7
∗0.203 = = 20833,3333 ! = 2 2 12∗( 1−) ) 12∗( 1 −0.20 ) 0∗h
3.0∗10
´ i∗& ∗% x !w 0i =Z 2
4
2 -10.2500 2
1.5625
q=16 kN/m 1
m 5 7 3 . 2 = y S
p=12 kN/m² Sx=1.9 m
´ 1= Z
12∗1,9∗2,375 + 16∗1,9 1,9∗2,375
=18,7368 !* / m2
!w 01=18,7368∗20,3627 =381,5318 !w 05=!w 01= 381,5318
2 p=12 kN/m²
m 5 7 3 . 2 = y 5 S 2 1 7 . 0
Sx=1.9 m
´ 2= Z
12∗1,9∗0,7125 1,9∗2,375
=3,6000 !* / m2
!w 02=3,6000∗20,3627 = 73,3057 !w 04=!w 02=73,3057
m 5 7 3 . 2 = y 5 S 2 1 7 . 0
F=24 kN 3 p=12 kN/m² Sx=1.9 m
´ 3= Z
24 + 12∗1,9∗0,7125 1,9∗2,375
=8,9186 !* / m2
!w 03=8,9186 ∗20,3627 =181,6068
m 5 7 3 . 2 = y S
q=16 kN/m 6 p=12 kN/m² Sx=1.9 m
´ 6= Z
16∗1,9 + 12∗1,9∗2,375 1,9∗2,375
=18,7368 !* / m2
!w 06=18,7368∗20,3627 =381,5318
1 1
1 2
22,7775 -6,5600 2,0000 0 1,5625 10,2500
1 4
1 3
-13,1200 24,0575 -10,2500 1,5625 0 4,0000
4,0000 -10,2500 22,4950 -10,2500 4,0000 -13,1200
1 5
0 1,5625 -10,2500 20,9325 -13,1200 4,0000
-1 o
1 6
1,5625 0 2,0000 -6,5600 19,6525 -10,2500
-10,2500 2,0000 -6,5600 2,0000 -10,2500 21,2150
381,5318 73,3057 181,6068 73,3057 381,5318 381,5318
U$232 1 1
84,6938
1 2
44,4709
1 3
79,9967
1 4
60,5498
1 5
110,9860
1 6
142,1980
9
10
11
8
4
3
2
12
4
7
5
6
1
13
5
60.5498
79.9967 2
110.9860
4
4
3
2
44.4709 60.5498 79.9967 5
6
1
84.6938 110.9860 142.1980
!w i [ m ]
%tvarniugibi
( )[ !wi !
m]
44.4764
3
142.1980 6
60.5498
3
84.6938 1
79.9967
2
44.4764
1 1
1 2
0.004065
0.002135
1 4
1 3
0.003840
1 6
1 5
0.002906
0.005327
0.006826
%ile u presecima( Mx )
v 2
&
=
0,2 1,5625
=0,128
( )=
2∗ 1 +
-0,128 -1
2,256
-0,128
M x , 1= M x , 2=
! 2
% x ! 2
% x
M x , 3 =
!
M x , 4=
!
2
% x 2
% x
M x , 5 =
!
M x , 6 =
!
M x , 7 = M x , 8 = M x , 9 =
2
% x 2
% x ! 2
% x ! 2
% x ! 2
% x
-1
¿
{
!
= M x , !
2
% x
∗ ( 2,256 ∗w1 + 2∗(−0,128 )∗w 2+(−1)∗w 6 )=10,3841 !*m /m ∗( 2,256∗w2 + (−0,128 )∗w1 +(−1 )∗w3 ) =2,6285 !*m / m ∗( 2,256 ∗w3 + (−1 )∗w4 + (−1 )∗w2 +(−0,128 )∗ w6 ) =15,8589 !*m / m ∗( 2,256∗w4 +(−1 )∗w3 + (−0,128 )∗w5 ) =11,7444 !*m/ m ∗( 2,256 ∗w5 + (−1 )∗w6 + 2∗(−0,128 )∗w 4 ) =25,6747 !*m / m ∗( 2.625∗w6 + (−1 )∗w5 + (−1 )∗w1+ 2∗(−0,128 )∗w3 ) =28,9861 !*m / m ∗( (−1 )∗w 5−(−1 )∗w5 )= 0 !*m / m ∗( (−1 )∗w 4− (−1 )∗w 4 )= 0 !*m / m ∗( 2∗(−0,128 )∗w 4 ) =−4,2938 !*m / m
v
2
&
2,256
M x , 10= M x , 11= M x , 12= M x , 13=
! 2
% x ! 2
% x !
2
% x ! 2
% x
∗( 2∗(−0,128 )∗w3 ) =−5,6729 !*m / m ∗( 2∗(−0,128 ) ¿ w2 )=−3,1536 !*m / m ∗( 2∗(−1 ) ¿ w2 )=−24,6376 !*m / m ∗( 2∗(−1 ) ¿ w1 )=−46,9218 !*m / m
9
10
11 4.2938
8
4
3
2
5.6729
9
11.7444
4
12
3.1536 10
5
6
1
25.6747
5
13
4
4.2938
11.7444
6
3
5.6729
24.6376
8
3
4 + 11.7444
2
-
12
2.6285 46.9218
15.8589
7
5
6
1
13
+ 10.3841 25.6747
28.9861
%ile u presecima ( My )
−v∗& 2=−0,3125 2∗( 1 + v∗&
2
)=2,625 -1
-0,3125
2,625 -1
-0,3125
+ 28.9861
1
15.8589
2
10
9
¿
{
! 2
= M y ,!
% y
2.6285
2
+
+ 7
15.8589
3
11
3.1536
11
10.3841
2.6285
M y , 4 =
M y , 5= M y , 6= M y , 7= M y , 8= M y , 9=
! 2
% y ! 2
% y ! 2
% y ! 2
% y ! 2
% y ! 2
% y
M y , 10= M y , 11= M y , 12= M y , 13=
∗( 2,625∗w4 +(−1)∗w5 + (−0,3125 )∗w3 ) = 4,0702 !*m / m ∗( 2∗(−1 )∗w 4 + 2,625 ∗w5 +(−0,3125 )∗w 6 )=22,3028 !*m / m ∗( 2,625∗w 6 +2∗(−1.00 )∗w3 + (−0,3125 )∗w5 +(−0,3125 )∗w1 ) =26,9698 !*m/ m ∗( (−0,3125 )∗w 5+ (−0,3125 )∗(−w5 ) )=0 !*m / m ∗( (−0,3125 )∗w 4 + (−0,3125 )∗( −w4 ) )=0 !*m / m ∗( (−1 )∗w4 + (−1 )∗w 4 )=−21,4692 !*m / m
! 2
% y ! 2
% y ! 2
% y ! 2
% y
∗( (−1 )∗ w3 + (−1 )∗w 3 )=−28,3645 !*m /m ∗( ( −1 )∗w2 +(−1 )∗ w2 )=−15,7681 !*m / m ∗( (−0,3125 )∗w2 + (−0,3125 )∗w2 )=− 4,9275 !*m / m ∗( (−0,3125 )∗w1 + (−0,3125 )∗w1 )=−9,3844 !*m / m 9
10
11 21.4692
8
3
4
2
28.3645
9
4.0702
4
12
15.7681 10
7
5
6
13
1
4
22.3028
4.0702
3
4.9275 8
3
4 + 4.0702
2
12
1.2487 6.1949
7
5
6
1
+ 15.7682 22.3028
26.9698
13
9.3844
+ 28.9861
6.1949
28.3645
1
2
10
9 21.4692
6
1.2487
2
+
+ 5
6.1949
3
11
11 15.7681
15.7682
1.2487
%ile u presecima( Mxy )
1
l
-1
k-1
k
k+1
-1
i
1
M xy , 1=
M xy , 2=
M xy , 3=
M xy , 4=
M xy , 5=
M xy , 6=
M xy , 7=
M xy , 8=
M xy , 9=
! ∗( 1 −) ) 2
( 1∗w +(−1 )∗w )=0 !*m/ m 3
4∗& ∗% x
3
! ∗( 1 −) ) 2
( (−1 )∗w )=−6,3024 !*m / m 6
4∗& ∗ % x
! ∗( 1 −) ) 2
( (−1 )∗w + 1∗w )=−1,1653 !*m / m 5
4∗& ∗ % x
1
! ∗( 1−) ) 2
4∗ & ∗% x
( 1∗w )=6,3022 !*m / m 6
! ∗(1 −) ) 2
( (−1 )∗w + 1∗w )= 0 !*m/ m 3
4∗& ∗% x
3
! ∗( 1−) ) 2
( 1∗w + (−1 )∗w + 1∗w + (−1 )∗w ) =0 !*m / m 4
4∗& ∗ % x
4
2
2
! ∗( 1 −) ) 2
( 1∗w + (−1 )∗w )= 0 !*m / m 4
4∗& ∗ % x
4
! ∗( 1 −) ) 2
4∗& ∗ % x
( 1∗w + 1∗w ) =4,9189 !*m / m 5
5
! ∗( 1−) ) 2
( w −w )=0 !*m/ m 3
4∗& ∗% x
M xy , 10=
M xy , 11=
M xy , 12=
M xy , 13=
3
! ∗( 1 −) )
( (−1 )∗w + 1∗w + (−1 )∗w + 1∗w ) =0 !*m / m 4
2
4∗& ∗ % x
4
2
2
! ∗( 1−) )
(1∗w + (−1 )∗w ) =0 !*m /m
2
4∗ & ∗% x
3
3
! ∗( 1− ) )
( 1∗w + (−1 )∗w ) =0 !*m / m
2
4∗& ∗% x
1
1
! ∗( 1− ) )
( 1∗w + (−1 )∗w + 1∗w + (−1 )∗w ) =0 !*m /m
2
4∗& ∗ % x
2
2
2
2
9
10
11 11
8
4
3
2
12
7
5
6
1
13
6.3024
1.1653 2
3
1
6
6.3024
2
5
6
1
3
4
5
1.1653 6.3022
4
10
11
7
9
10
9
13
6.3024 1.1653 8
3
4
2
12
4.9189 6.3022
%ileu presecima ( Tx ) 1 2
&
=0,6400
( )=
2∗ 1 +
1
2
3,280
&
0,640 1
-3,280 0,640
T x , 1=
-
-0,640 k
3,280
-1
¿
{
!
=T x ,!
3
2∗ % x
-0,640
∗( (−3,280 )∗1 6 +2∗0,640∗1 3 +1∗1 5 +(−1 )∗1 1 ) =−24,6189 !* / m 3 ∗ 2 % 4
1∗1 4 + 0,640∗1 6 + (−3,280 )∗¿
¿
T x, 2=
- 3
∗¿
2∗ % 4
6.3022
T x , 3=
T x , 4 =
T x , 5=
T x , 6=
-
∗( 3,280∗1 2 +(−0,640 )∗1 1+ (−3,280 )∗1 4+ 0,640∗1 5 )=−2,6179 !* /m
3
2∗ % 4
- 3
∗( 3,280∗1 3 + (−0,640 )∗1 6 + (−1 )∗1 2+ (−1 )∗1 4 ) = 4,8376 !* / m
2∗ % 4
-
∗( 2∗(−0,640 )∗1 3+ 3,280∗1 6+ (−1 )∗1 1+ (−1 )∗1 5 ) =12,2710 !* / m
3
2∗ % 4
-
∗( (−3,280 )∗1 5 +3,280∗1 1 + 2∗0,640∗1 4 + 2∗(−0,640)∗1 2 ) =−4,7862 !* / m
3
2∗% 4
p=12 kN/m² m 5 7 3 . 2 = y S
q=16 kN/m 7
0.95 m Sx=1.9 m
´ 7= Z
16∗0,95 + 12∗0,95∗ 2,375 0,95∗2,375
´ 7 =Z ´ 13=18,7368 !* /m Z
=18,7368 !* / m 2
2
m 5 7 3 . 2 = y 5 S 2 1 7 . 0
8 p=12 kN/m² 0.95 m Sx=1.9 m
´ 8= Z
12∗0,98∗ 0,7125 0,95∗2,375
=3,6000 !* / m2
´ 8 =Z ´ 12=3,6000 !* / m Z
2
Zamenjujuće T sile( ´ T ) 2 −v 2
&
=1,152 1,152 1
-4,304 1,152
-1,152 k
4,304 -1,152
-1
¿
{
! 3
2∗ % x
´ x ,! =T
25
26
27
18
17
16
9
10
11
4
3
2
Y
19
22
8
12
15
23
X 20
21
7
5
6
1
13
14
24
Ugibitača!a1 21 , 1 22− slobodnooslonjenaivica ( dvetač!e van !ontura) 4
1 ! + 2=−1 ! − 2 +
Z ! ∗ % x ! 4
1 21=−1 6 +
Z 7∗% x !
=
4
1 22=−1 3 +
Z 8∗% x !
=
−142,198 !
−79,9967 !
+ +
18,7368∗1,9
! 3,6000∗1,9
!
4
=101,819
4
=−33,0811
Ugibitača!a1 23 , 1 24 − u!lje/tenaivica( dvetač!e van!ontura) 4
1 ! + 2=
Z ! ∗ % x ! 4
1 23=
1 23=
Z 12∗% x !
8
3,6000∗1,9
! 4
1 24=
( ) + ∗ + ¿ ( ) + 8∗ 1 +
Z 13∗% x !
1
4
1
2
& 1
2
&
¿ 1 ! −1− 1 2−
4 2
&
4 2
&
∗( 1 i−1 +1 l−1 )− 1 ! −2
∗( 1 1 + 1 11 ) −1 3
+ 8∗( 1 +0,640 )∗44,4709 −
( )
+ 8∗ 1 +
1
2
&
¿ 1 1−
4 2
&
4 1,5625
∗( 1 2 + 1 2 ) −1 6
∗( 84,6938 + 0 ) −79,9967 =333,5610
1 23=
18,7386∗1,9
4
!
+ 8∗(1 + 0,640 )∗84,6938 −
4 1,5625
∗( 44,4709 + 44,4709 )−142,198 =333,5610
−1 (¿¿ 4 )+ 2∗(−1,152 )∗1 4 −1 6 + 1 21 4,304∗1 5+ 4,304∗1 5 + 2∗1,152∗¿=46,3725 !* / m ´ x , 7= ! ∗¿ T 3 2∗ % x −1 (¿¿ 5)+ 2∗(−1,152 )∗1 5−1 3+ 1 22 4,304∗1 4 + 4,304∗1 4 + 2∗1,152∗¿=11,1112 !* / m ´ x, 8= ! ∗¿ T 3 2∗ % x ´ x , 12= T ´ x , 13= T
! 3
∗(1 3 −1 23)=−18,4841 !* / m
3
∗( 1 6−1 24 )=−61,4728 !* /m
2∗ % x
! 2∗ % x
9
10
11 9
8
4
3
2
4.8376
12
11
10
2.6179
3
4
8.0794
2
+ 12.2710 7
5
6
1
13
12.2710
24.6189
4.7862 6
5
-
24.6189
1
4.7862 +
+
4
4.8376
2.6179
8.0794
10
9
2
3
11
61.4728 24.6189 -
4.7862 +
7
5 12.2710
6
13
1
46.3725
18.4841 8.0794 8
2.6179 3
4 +
2
-
12
4.8376
11.1112
ea!cije oslonaca ( " 7 , " 8 , "12 , "13 ) 2∗2− v 2
&
(
4∗ 1 +
=2,304 -2,304
2− v 2
&
)
l
+ Z ! ∗% x4
=8,608 -2
8,608
k
-2,304
i
!
{
∗
! = " x, ! 3 2∗% x
" x, 7=
" x, 8=
- 3
3
(
Z 8∗% x
∗
2∗ % 4
-
(
Z 7∗% x
∗
2∗ % 4
)
4
!
+ 8,608∗w 20+ (−2 )∗w 21+ 2∗(−2,304 )∗ w19 =−46.3725 !* / m
)
4
!
+ 8,608∗w 19+ (−2 )∗w 22+ (−2,304 )∗w 20 =−11.1112 !* / m
−u!lje/tena ivica "12 , " 13 4
-2,560
l
2
&
+ Z ! ∗% x4 -2
( )=
8∗ 1 +
1
2
13,120
k
-2,560
i
!
{
∗
!
=2.560
= " x, !
3
2∗% x
13,120
&
( (
4
Z 8∗% x - ∗ + 13,120∗w2 + (−2 )∗w3 +(−2,560 )∗w1 " x, 12= 3 ! 2∗% 4 4
)
=18,4841 !* / m
Z ∗% - ∗ 7 x + 13,120∗w1 + 2∗ (−2,560 )∗w2 +(−2 )∗w 6 " x, 13= 3 ! 2∗ % 4
)
=61,4728 !* / m
%ile u presecima( Ty ) 2
& =1.5625 2∗( 1 + & ) =5.125 2
-1 -1,5625
5,125
-1,5625
¿
k
1,5625
-5,125
{
!
=T y , !
3
2∗ % y
1,5625
1
T y, 1=
- 3
2∗ % y
∗( (−5,125 )∗1 2 + 1,5625∗1 3+ 5,125∗1 2+ (−1,5625 )∗1 3 )= 0 !* / m
T y, 2= T y, 3= T y, 4 =
T y, 5= T y, 6=
- 3
∗( (−1 )∗1 16 + (−5,125 )∗1 1 + 1,5625∗1 6 +1∗1 2) =−7,90776 !* / m
3
∗( (−1 )∗1 17 +(−5,125 )∗1 6 + 1,5625 ∗1 5 + 1,5625 ∗1 1 + 1∗1 3 ) =−15,7883 !* / m
3
∗(−1 18 + (−5,125 )∗1 5 + 1,5625 ∗1 6 + 1 4 )=−12.9369 !* / m
3
∗( 5.125∗1 4 +(−5,125 )∗1 4 +(−1,5625 )∗1 3 + 1,5625∗1 3 ) =0 !* /m
3
∗( 5.125∗1 3+ (−5,125 )∗1 3 +(−1,5625 )∗1 4 + 1,5625∗1 4 + (−1,5625 )∗1 2+ 1,5625 ∗1 2 ) =
2∗ % y
- 2∗ % y
- 2∗% y
- 2∗% y
- 2∗ % y
´ 9 =Z ´ 10= Z ´ 11 Z
Zamenjujuće T sile( ´ T ) & ∗( 2− v ) =2,8125 2
(
2∗ 1+ & ∗ ( 2− v )
-1 -1,5625
5,125
-1,5625
¿
k
1,5625
-5,125
{
! 3
=T y , !
2∗ % y
1,5625
1
Ugibitača!a1 25 , 1 26 , 1 27−u!lje/tena ivica( dvetač!e van !ontura) 4
1 m =
Z ! ∗% y
+ 8∗( 1 + & 2) ¿ 1 i −4∗ & 2∗( 1 i−1+ 1 i+1 ) −1 h
!
4
1 25=
Z 9∗% y !
+ 8∗( 1 + & 2 ) ¿ 1 4 −4∗ & 2∗( 1 8 + 1 3 ) −1 5=630,3055
4
1 26=
Z 10∗% y ! 4
1 27=
Z 11∗% y !
+8∗( 1 + & 2 ) ¿ 1 3− 4∗& 2∗( 1 4 + 1 2 )−1 6=841,3550 +8∗( 1+ & 2 ) ¿ 1 2− 4∗& 2∗( 1 3+1 12) −1 1= 326,9803
2
) =7,625
(−7,625 ) ¿ 1 4 +7,625 ¿ 1 4 + 2,8125 ¿ 1 3−2,8125 ¿ 1 3 + 1 5−1 25=−19,3827 !* / m ´ y ,9= ! 3 ∗¿ T 2∗ % y (−7,625 ) ¿ 1 3 + 7,625 ¿ 1 17 + 2,8125 ¿ 1 4 −2,8125 ¿ 1 18+2,8125 ¿ 1 2 −2,8125 ¿ 1 16+ 1 6−1 26 =−26,09 ´ x , 10= ! 3 ∗¿ T 2∗% y (−7,625 ) ¿ 1 2 + 7,625 ¿ 1 16 + 2,8125 ¿ 1 3−2,8125 ¿ 1 17 +1 1−1 27=−9,0429 !* / m ´ y , 11= ! 3 ∗¿ T 2∗ % y 9
10
11 19.3827
9
9.0429
26.0948
-
-
8
4
3
2
12
7
5
6
1
13
12.9369
11
10
15.7883
4
7.9077
3
5
6
1
3
12.9369
4
15.7883
+
10
11
19.3827
+
8
5
6
1
13
4
3
2
12
-
2
26.0948
7.9077
12.9369 15.7883
ea!cije oslonaca ( " 9 , " 10 , " 11)
−u!lje/tena ivica 2
4∗ & =6,250
k-1
k
k+1
2
8∗( 1 + & )= 20,500
+ Z ! ∗% y4 -6,250
20,500
-6,250
!
∗
{
! = " y, ! 3 2∗% y
-2,000
" y , 9=
- 3
2∗ % y
(
∗
4
Z 9∗% y !
7.9077
+
9
7
2
)
+ 20.500 ∗w 4 + (−6.250 )∗w3 + (−2.0 )∗w5 =19.3827 !* / m
9.0429
" y ,10 =
" y ,11=
- 3
2∗% y
- 3
2∗ % y
( (
∗
∗
)
4
Z 9∗ % y ! 4
Z 9∗ % y !
+ 20.500∗w3 + (−6.250 )∗w 4 + (−6.250 )∗w 2+ (−2.0 )∗w 6 = 26.0948 !* / m
)
+ 20.500 ∗w 2+ (−6.250 )∗ w3 +(−2.0 )∗w1 =9.049 !* /m
