Sfeliouras Xristou [Main]
105
1 1.ΠΡΟΛΟΓΟΣ Οι σύγχρονες οικονομικές κατασκευές προϋποθέτουν την επίλυση πολύπλοκων στατικών συστημάτων. Οι σχετικές με το αντικείμενο γνώσεις πηγάζουν από τις αρχές και τις μεθόδους της στατικής. Η στατική είναι ο κλάδος της επιστήμης της μηχανικής που ασχολείται, με τη σύνθεση και την ανάλυση δυνάμεων, τον προσδιορισμό του κέντρου βάρους, τον υπολογισμό των εσωτερικών εντατικών μεγεθών ολόσωμων και δικτυωτών φορέων στο επίπεδο και στο χώρο. Η χρησιμοποίηση του ηλεκτρονικού υπολογιστή στο στατικό υπολογισμό άλλαξε εντελώς τον τρόπο παρουσίασης των μεθόδων της στατικής. Ο πολιτικός μηχανικός που σήμερα επιβάλλεται να χρησιμοποιεί ένα πρόγραμμα για το στατικό υπολογισμό, πρέπει να έχει την ικανότητα αφενός μεν να δώσει ένα σωστό για τη λειτουργικότητα της κατασκευής στατικό μοντέλο βάσει του οποίου θα εισάγει τα δεδομένα στον υπολογιστή και αφετέρου να ελέγξει τα αποτελέσματα που θα πάρει. Η ικανότητα αυτή αποκτάται μόνο με την πλήρη και σε βάθος γνώση των αρχών της κλασικής στατικής. Σκοπός της πτυχιακής αυτής εργασίας είναι να αποκτήσουμε τις γνώσεις και την οικειότητα της στατικής και συγκεκριμένα των επίπεδων ισοστατικών φορέων ώστε στη συνέχεια της πορείας μας ως μηχανικοί να μπορούμε να αξιολογούμε την ορθότητα των αποτελεσμάτων που θα μας δώσει ο ηλεκτρονικός υπολογιστής. Οι γνώσεις που αποκτήθηκαν κατά την επίλυση των ασκήσεων που υπάρχουν στην πτυχιακή εργασίας , αν και είναι δύσκολο να καταμετρηθούν, είναι: Oρισμός και αναγνώριση του φορέα μιας κατασκευής, καθώς και τα στοιχεία του. Aναγνώριση και σχεδίαση των στοιχείων του φορέα ανάλογα με το σχήμα και τον τρόπο καταπόνησής τους.
Transcript of Sfeliouras Xristou [Main]
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AutoCad 2007 AutoDesk.
00 == G
G
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12
BP=25KN
HA
VA
VB
1
..
KNVKNVVVVFy ABABA 5.12250550 ==+=+=
KNVVVM BBBA 5.1255.62055.2250 ===+=
B
q=5KN/m
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13
HA
VA
M(x)N(x)
Q(x)
x
q=5KN/m
-
=0m =0, Q=-12.5KN, M=0KNm =1m =0, Q=-7.5KN, M=10KNm
I-I
=0m =5.30, Q=5.30KN, M=10KNm =3m =-5.30, Q=-5.30KN, M=10KNm
00 == NFx
05.122
50 =+= XXXM
055.120 =+= XQFy
XRXR == 55.7555.12
HA
VA
M(x)
N(x) Q(x)
x
q=5KN/m
XRQ == 54.330.5cos
XRN == 54.330.5sin
02
5)5.0(5)1(5.120 =++++= XXXXMM
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14
B
10KNm10KNm
10KNm
10KNm
+
+
+
I-I
=0m =0, Q=7.5KN, M=10KNm =1m =0, Q=12.5KN, M=0KNm
M, N, Q
(M)
HA
q=5KN/m
M(x)N(x)Q(x)
x
VA
KNNFx 00 ==
051555.120 =+= XQFy
02
5)5.1(15)5.3(5)4(5.120 =++++++= XXXXXMM
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15
(Q)
(N)
5
B++
-
-
-12.5KN -7.5KN-5.3KN
5.3KN7.5KN
12.5KN
B-
+
-5.3KN
5.3KN
B
P1=5KN P2=5KN
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16
...
KNHFx A 00 ==
KNVVVFy ABA 50550 ==+=
KNVVM BBA 50103080 ===
-
B
P1=5KN P2=5KN
HA
VA VB
B
VB
x
N(x)Q(x)
M(x)
050 == XMM
KNQQFy 5050 ==+=
KNNFx 00 ==
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17
=0m =0, Q=-5KN, M=0KNm =2m =0, Q=-5KN, M=10KNm
I-I
=0m =0, Q=0KN, M=10KNm =4m =0, Q=0KN, M=10KNm
I-II
B
P2=5KN
VB
N(x)
Q(x)
M(x)
x
05)2(50 =++= XXMM
KNQQFy 00550 ==+=
KNNFx 00 ==
B
P1=5KN P2=5KN
VB
N(x)Q(x)
M(x)
x 05)4(5)6(50 =++++= XXXMM
KNQQFy 505550 ==+=
KNNFx 00 ==
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18
=0m =0, Q=5KN, M=10KNm =2m =0, Q=5KN, M=0KNm
M, N, Q
(M)
(Q)
(N)
B10KNm 10KNm
+
B
5KN
-5KN-5KN
5KN+
-
B0
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19
B
VB
HA
VA
P=280KN
3
..
KNHFx A 00 ==
KNVVVFy ABA 8004700 ==+=
KNVVM BBA 2000547070 ===
q=70KN/m
B
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20
-
=0m =0, Q=-200KN, M=0KNm =4m =0, Q=80KN, M=240KNm
I-I
KNNFx 00 ==
=0m =0, Q=80KN, M=240KNm =3m =0, Q=80KN, M=0KNm
q=70KN/m
BN (x)
Q (x)M (x)
VBx
KNNFx 00 ==
02
702000 =+= XXXMM
0702000 =+= XQFy
q=70KN/m
B
VB
N(x)
Q(x)M(x)
x
KNQQFy 8004702000 ==+=
0)2(470)4(2000 =+++= XXMM
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21
M, N,Q
(M)
(Q)
(N)
B
240KNm
+
B
-200KN
80KN80KN+
-
B0
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22
B
VB
HA
VA
P=16KN P=100KN
B
q=8KN/m q=20KN/m
4
..
KNHFx A 00 ==
KNVVVFy ABA 2.690520280 ==+=
KNVVM BBA 8.460285.25205 ==+=
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23
B
q=20KN/m
VB
x
N(x)Q(x)M(x)
B
q=20KN/m
VB
HA
VA
q=8KN/m
N(x)Q(x)M(x)
x
-
=0m =0, Q=-46,8KN, M=0KNm =5m =0, Q=53,2KN, M=-16KNm
I-I
KNNFx 00 ==
082.695208.460 =++= XQFy0
282.69)5.2(520)5(8.460 =++++= XXXXXMM
=0m =0, Q=-16KN, M=-16KNm =2m =0, Q=0KN, M=0KNm
KNNFx 00 ==
0208.460 =+= XQFy
02
208.460 =+= XXXMM
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24
B00
M, N, Q
(M)
(Q)
()
B
0 0
-16KNm
-
+
B
-16KN
-46.8KN
53.2KN
0
+
-
-
-
25
q=20KN/m
B
B
P=240KN
HA
VA VB
N(x)Q(x)
M(x)
q=20KN/m
x
5
..
-
KNVVM BBA 1200424080 ===
KNVVVFy ABA 12002400 ==+=
00 == AHFx
02
200 =+= XXMM
0200 == XQFy
KNNFx 00 ==
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B
VB
q=20KN/m
N(x)Q(x)
M(x)
x
B
x
HA
VAVB
N(x)
Q(x)M(x)
q=20KN/m
=0m =0, Q=0KN, M=0KNm =2m =0, Q=40KN, M=-40KNm
I-I
=0m =0, Q=-80KN, M=-40KNm =8m =0, Q=80KN, M=-40KNm
I-I
KNNFx 00 ==
020160401201200 =++= XQFy
02
20)4(160)9(40)8(1201200 =++++++= XXXXXXMM
=0m =0, Q=-40KN, M=-40KNm =2m =0, Q=0KN, M=0KNm
02
20120)1(400 =+++= XXXXMM
XQXQFy +==+= 2080020401200
KNNFx 00 ==
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27
M, N, Q
()
(Q)
(N)
B+
-
--
-
-40KNm -40KNm
B++
-
-
-40KN-80KN
40KN80KN
B0 0 0
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28
A B
q=10KN /m
A B
P=120KN
HA
VA VB
6
...
KNHFx A 00 ==
KNVVVFy ABA 6001200 ==+=
KNVVM BBA 600412080 ===
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29
N(x)
Q(x)M(x)
x
q=10KN/m
-
=0m =0, Q=0KN, M=0KNm =2m =0, Q=20KN, M=-20KNm
-
=0m =0, Q=-40KN, M=-20KNm =8m =0, Q=40KN, M=-20KNm
KNNFx 00 ==
02
100 === XXMM
0100 == XQFy
B
q=10KN/m
N(x)
Q(x)M(x)
x
VB
02
1060)1(200 =+++= XXXXMM
01020600 =+= XQFy
KNNFx 00 ==
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30
A B
q=10KN/m
N(x)
Q(x)M(x) HA
VAVB
x
-
KNNFx 00 ==
010802060600 =++= XQFy
02
10)4(80)9(20)8(60600 =++++++= XXXXXXMM
=0m =0, Q=-20KN, M=-20KNm =2m =0, Q=0KN, M=0KNm
M, N, Q
(M)
(M)
A B
20KNm
+
20KNm
-- -
-
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31
A B
q1=5KN/m
q2=5KN/mq2=3KN/m
(Q)
(N)
7
A B
++
-
-
20KN
-40KN
40KN
-20KN
00
A B0 0 0
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32
A BHA
VA VB
P=60KNP=6KN P=6KN..
KNHFx A 00 ==
KNVVVFy ABA 36085)28(20 ==+= KNVVM BBA 36092592348512512380 ==++=
-
KNNFx 00 ==
0530 == XXQFy
280 XXMM ==
=0m =0, Q=0KN, M=0KNm =2m =0, Q=16KN, M=-16KNm
q 1=5KN /m
q 2=3KN/m
N (x)Q (x)M (x) x
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33
B
q1=5KN/m
q2=3KN/m
x
VB
N(x)Q(x)M(x)
A B
q1=5KN/m
q2=3KN/mq2=3KN/m
HA
VA VB
N(x)Q(x)M(x)
x
I-I
=0m =0, Q=-20KN, M=-16KNm =8m =0, Q=20KN, M=-16KNm
II-II
KNNFx 00 ==
02885836360 =++= XQFy
02
52
3)4(40)9(10)9(6)8(36360 =+++++++++= XXXXXXXXXMM
=0m =0, Q=-16KN, M=-16KNm =2m =0, Q=0KN, M=0KNm
KNNFx 00 ==
0)1(6)1(102
5360 =+++++= XXXXXMM
023525360 =+= XQFy
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34
M, N, Q
(M)
(Q)
(N)
A B+
-16KNm -16KNm
--
- -0 0
A B
0 0+ +--
20KN
-16KN
16KN
-20KN
A B0 0 0
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35
A
A
VA
HAMA
A
VA
HAMAM(x)N(x)
Q(x)
8
...
KNHFx A 00 ==
KNVFy A 00 ==
MMMMM AA === 00
TOMH I-I
-
36P
A
M, N, Q
(M)
(Q)
(N)
9
A
M ..KNm
+
A 0
A0
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37
P
A
VA
HAMA
A
VA
HAMAM(x)N(x)
Q(x)
...
KNHFx A 00 ==
PVPVFy AA ==+= 00
LPMLPMM AA === 00
-
M, N, Q
-
38
A
M ..KNm
+
A +P ..KN
A 0
q
A
()
(Q)
()
10
-
39
A P
VA
HAMA
A
VA
HAMAM(x)N(x)
Q(x)
...
KNHFx A 00 ==
AA VLqLqVFy ==+= 00
20
20
2LqMLLqMM AA ===
-
M, N, Q
-
40
A
M ..KNm
+
A+
P ..KN
A 0
P=15KN
q=6KN/m
A
()
(Q)
()
11
-
41
P=15KN
A
P=9KN
VA
HAMA
...
-
KNmMMM AAA 75.36021575.05.160 ==++=
KNVVFy AA 2405.16150 ===
KNHFx A 00 ==
-
42
=0m =0, Q=15KN, M=0KNm =0,5m =0, Q=15KN, M=-7,5KNm
-
=0m =0, Q=15KN, M=-7,5KNm =1,5m =0, Q=24KN, M=-36,75KNm
P=15KN
N(x)
Q(x)M(x)
x
0150 =+= XMM
KNQFy 150 ==
KNNFx 00 ==
P=15KN
x
N(x)
Q(x)M(x)
q=6KN/m
0)5.0(152
60 =+++= XXXMM
06150 == XQFy
KNNFx 00 ==
-
43
q2=3KN/m
P2=20KN
P1=15KN
M, N, Q
(M)
(Q)
(N)
A
-
-
-36.75KNm
-7.5KNm
A+ +
24KN15KN
A 0
-
44
P2=20KN
P1=15KN
A
B
C
P1=4.5KN
P2=4.5KN
M=3.375KNm
M=3.375KNm
VA
MA
HA
12
..
I-I
KNHHFx AA 150150 ===
KNVVFy AA 2905.45.4200 ===
KNmMMM 5.6705.4150 ===
-
45
A
BP1=4.5KN
M=3.375KNm
VA
MA
HA
N(x)Q(x)
M(x)
KNQQFx 150150 ===
KNNNFy 290290 ==+=
KNmMMM 5.6705.670 ==+=
II-II
=0m =-24.5, Q=15KN, M=48.375KNm =1.5m =-24.5, Q=15KN, M=25.875KNm
III-III
A
VA
MA
HA
N(x)Q(x)
M(x)
x
KNQQFx 150150 ===
KNNNFy 5.2405.4290 ==+=
0)5.1(15775.35.670 =++= XMM
-
46
=0m =-20, Q=15KN, M=22.5KNm =1.5m =-20, Q=15KN, M=0KNm
M, N, Q
(M)
x
A
B
C
P1=4.5KN
P2=4.5KN
M=3.375KNm
M=3.375KNm
VA
MA
HA
N(x)Q(x)
M(x)KNQQFx 150150 ===
KNNNFy 200295.45.40 ==+=
05.67375.3375.3)3(150 =+++= XMM
A
B
C-
-
-
-
-
-3.375KNm
0
-22.5KNm-25.875KNm
-48.375KNm
-67.5KNm
-3.375KNm
-67.5KNm
-
47
P2=5KN P2=5KNq=20KN/m
B C D
(Q)
(N)
13
A
B
C
+
+
+
15KN
15KN
15KN
15KN
-
-
-4.5KN
-4.5KN
A
B
C
-
-
-
-20KN
-20KN-24.5KN
-24.5KN-29KN
-29KN
-
48
P2=5KN P2=5KNP=1200KN
A
B C D
HA
VAMA
N(x)Q(x)
M(x)
...
KNHFx A 00 ==
KNVVFy AA 121001200550 ===
KNmMMHVM AAB 0060530602080300 ==+=
I-I
-
49
KNQFx 00 ==
KNNNFy 1210012100 ==+=
II-II III-III ( )
KNNFx 00 ==
0206050 =+= XQFy
02
206050 =+= XXXMM
=0m =0, Q=-605KN, M=0KNm =30m =0, Q=-5KN, M=9150KNm
M, N, Q
= 0 [ M =O KNm
A
C
HA
VAMA
q=20KN/m
N(x)Q(x)M(x)
x
-
50
A
B C D
- -
-9150KNm
A
B C D
- -
-605KN
-5KN -5KN
A
B C D
-1210KN
-1210KN
-
P=15KNq=30KN/m
AC
()
(Q)
()
14
-
51
P=15KN
A
B
C
P=3600KN
HA
VA
VB
...
KNHFx A 00 ==
KNVVVFy BBA 3630015120300 ==+=
KNVVM AAB 1506015600 ===
I-I
-
52
KNQFx 00 ==
KNNFy 36300 ==
KNmMM 00 ==
II-II
KNNFx 00 ==
030)15(0 =+= XQFy
02
30)15(0 =+= XXXMM
=0m =0, Q=-15KN, M=0KNm =60m =0, Q=-1815KN, M=54900KNm
III-III
B
HA
VB
N(x)Q(x)
M(x)
x
A
VA
q=30KN/m
M(x)N(x)
Q(x)
x
-
53
=0m =0, Q=1815KN, M=0KNm =60m =0, Q=15KN, M=-54900KNm
M, N, Q
(M)
KNNFx 00 ==
03018150 =+= XQFy
B
C
HA
VB
q=30KN/m
M(x)N(x)
Q(x)x
02
3018150 =+= XXXMM
A
B
C
-54900KNm
00
- -
-
54
(Q)
(N)
A
B
C
-15KN -15KN
-1815KN
- -
A
B
C
-3630KN
-
-3630KN
-
55
q=10KN/m q=5KN/m P1=15KN P2=10KN P3=20KN
B C D E
BC
CB
P=15KN
VB
HB
VC
HC
CE
P1=15KN P2=10KN
D E
P3=20KN
HC
VDVE
VC
P3X*COS45
P3*SIN45
HD
15
CBCB HHHHFx === 00
KNVVVFy BCB 5.70150 ==+=
KNVVM CCB 5.7035.1150 ==+=
-
56
014.140 == DC HHFx
KNVVVVFy DEDC 94.20014.1410150 ==++=
KNVVVM EECD 70.2505.914.1486102155.20 ==+=
TOMH I-I
=0m =-14,14, Q=14,14KN, M=0KNm =1.5m =-14,14, Q=14,14KN, M=-21,21KNm
I-I
KNNNFx 14.14014.140 ==+=
014.140 =+= XMM
E
P3=20KN
P3X*COS45N(x)
Q(x)M(x)
x
P3*SIN45
VE KNQQFy 56.11014.1470.250 ==+=
KNNNFx 14.14014.140 ==+=
070.25)5.1(14.140 =++= XXMM
P3=20KN
P3*COS45
P3*SIN45
N(x)
Q(x)M(x)
x
KNQQFy 14.14014.140 ===
-
57
=0m =-14,14, Q=-11,56KN, M=-21,21KNm =2m =-14,14, Q=-11,56KN, M=1,91KNm
I-I
KNNNFx 14.14014.140 ==+=
KNQQFy 56.1014.1470.25100 ==+= 010)2(70.25)5.3(14.140 =++++= XXXMM
=0m =-14,14, Q=-1,56KN, M=1,91KNm =4m =-14,14, Q=-1,56KN, M=8,15KNm
P2=10KN
E
P3=20KN
VE
P3X*COS45
P3*SIN45
N(x)
Q(x)M(x)x
-
58
P1=15KN P2=10KN
E
P3=20KN
VE
P3X*COS45
P3*SIN45
N(x)
Q(x)M(x)
x
P1=15KN P2=10KN
D E
P3=20KN
VDVE
P3X*COS45
P3*SIN45
HD
N(x)
Q(x)M(x)
x
V-IV
KNNNFx 14.14014.140 ==+=
KNQQFy 44.13014.1470.2510150 ==+=
015)4(10)6(70.25)5.7(14.140 =++++++= XXXXMM
=0m =-14,14, Q=13,44KN, M=8,15KNm =2m =-14,14, Q=13,44KN, M=-18,73KNm
V-V
KNNNFx 14.14014.140 ==+=
KNQQFy 5.7014.1470.25101594.200 ==++=
094.20)2(15)6(10)8(70.25)5..9(14.140 =+++++++= XXXXXMM
-
59
C
VC
HCN(x)
Q(x)M(x)
q=5KN/m
x
N(x)
Q(x)M(x)
VA
HA
MA
q=10KN/m
x
=0m =-14,14, Q=-7,5KN, M=-18,73KNm =2,5m =-14,14, Q=-7,5KN, M=0KNm
V-V
=0m =0, Q=-7,5KN, M=0KNm =3m =0, Q=-7,5KN, M=0KNm
V-V
=0m =0, Q=57,5KN, M=-162,5KNm =5m =0, Q=7,5KN, M=0KNm
KNNFx 00 ==
02
55.70 =+= XXXMM
055.70 =+= XQFy
05.57100 =+= XQFy
KNNFx 00 ==
02
105.575.1620 =++= XXXMM
-
60
P=10KNq=2KN/m q=4KN/m
A BC D
E FG
H
P=10KN
A BC D
E FG
H
P=35KNP=14KN
CD
P=10KN
VD
HD
VC
HC
16
...
DCDC HHHHFx === 00
KNVVVFy CDC 50100 ==+=
KNVVM DDC 5035.1100 ==+=
-
61
GH
GH
P=12KN
VHVG
HG
KNHFx G 00 ==
KNVVVFy GHG 60120 ==+=
KNVVM HHG 6035.1120 ==+=
GD HHFx == 0
KNVVVVVFy FGFED 8.2105.740 ==++=
KNVVVVM EGEDE 2.1965.51455.25475.05.145.10 ++=
DG
VE VF
HG
VD
HD
VG
D E F G
q=4KN/m
-
62
CA HHFx == 0
KNVVVFy ABA 3.805140 ==++=
KNVVVM BDB 7.1005.655.2521220 ==+=
-
KNNFx 00 ==
0460 =+= XQFy
062
40 =+= XXXMM
=0m =0, Q=-6KN, M=0KNm =3m =0, Q=6KN, M=0KNm
-D
q=2KN/m
A
VE VB
HDHG
VG
B C
q=4KN/m
HN(x)Q(x)M(x)
x
VH
-
63
F HG
VG
N(x)Q(x)M(x)
x
VF
q=4KN/m
I-I
KNNFx 00 ==
0460 == XQFy
062
40 =++= XXXMM
=0m =0, Q=6KN, M=0KNm =1m =0, Q=10KN, M=-8KNm
II-II
KNNFx 00 ==
06)1(48.210 =++= XQFy
0)1(68.21)5.0(42
40 =+++++= XXXXXMM
=0m =0, Q=0KN, M=-8KNm =5m =0, Q=0KN, M=1KNm
G
x
HG
VGq=4KN/m
N(x)Q(x)M(x)
-
64
D E F G
x
N(x)Q(x)M(x)
VFVE
HG
VGq=4KN/m
IV-IV
KNNFx 00 ==
0420468.212.190 =++= XQFy
0)6(6)5.5(4)5(8.21)5.2(542.192
40 =++++++++= XXXXXXXMM
=0m =0, Q=-11KN, M=-8KNm =1.5m =0, Q=-5KN, M=1KNm
V-V
KNNFx 00 ==
KNQQFy 5050 ==+=
050 == XMM
=0m =0, Q=-5KN, M=0KNm =1.5m =0, Q=-5KN, M=7.5KNm
VDHDD
N(x)Q(x)M(x)
x
-
65
VI-VI
KNNFx 00 ==
KNQQFy 501050 ==+=
0)5.1(5100 =++= XXMM
=0m =0, Q=5KN, M=7.5KNm =1.5m =0, Q=5KN, M=-7.5KNm
VII-VII
KNNFx 00 ==
KNQFy 50 ==
050 =+= XMM
=0m =0, Q=5KN, M=0KNm =1.5m =0, Q=5KN, M=-7.5KNm
VDHDD
N(x)Q(x)M(x)
x
P=10KN
VC
HCCN(x)
Q(x)M(x)
x
-
66
VIII-VIII
KNNFx 00 ==
027.1050 =+= XQFy
0)5.1(57.102
20 =+++= XXXXMM
=0m =0, Q=-5.7KN, M=-7.5KNm =5m =0, Q=4.3KN, M=-4KNm
IX-IX
KNNFx 00 ==
025257.103.80 =++= XQFy
0)5.6(5)5(7.10)5.2(5.23.82
20 =++++++= XXXXXXMM
=0m =0, Q=-4KN, M=-4KNm =2m =0, Q=0KN, M=0KNm
q=2KN/m
B C
x
VB
VC
HCN(x)
Q(x)M(x)
q=3KN/m
A B C
VC
HCN(x)
Q(x)M(x)
x
VA VB
-
67
A BC D
E FG
H
--
--
----
+ ++ + +
-8KNm
0-1KNm
7.5KNm
-7.5KNm-4KNm
0 00
A B C D E F G H+ + +
+ + +
-
-
-
-
--
-6KN
6KN10KN8.2KN
5KN
-5.7KN -5KN -5KN
-11KN
-4KN
4.3KN
11.8KN
5KN5KN
A BC D
E FG
H
0 0 0 0 0 00
q=5KN/m
C DBA
M, N, Q
17
-
68
C DBA
P=35KN
CE
C D
q=5KN/m
VC VD
VB
HBHC
..
1
A
q=5KN/m
VB
HB
VA
KNHFx B 00 ==
KNVVVFy ABA 100250 ==+=
KNVVM BBA 005.01515.0150 ==+=
-
69
q=5KN/m
D
N(x)Q(x)M(x)
x
VD
KNHFx B 00 ==
KNVVVFy CDC 84.20035250 ==+=
KNVVM DDC 16.4035.1351250 ==+=
I-I
=0m =0, Q=-4.16KN, M=0KNm =3m =0, Q=10.84KN, M=-10.02KNm
II-II
KNNFx 00 ==
KNNFx 00 ==
0516.40 =+= XQFy
02
516.40 =+= XXXMM
q=5KN/m
C D
x
N(x)Q(x)M(x)
VDVC
-
70
053516.484.200 =++= XQFy
02
584.20)5.1(15)3(16.40 =++++= XXXXXMM
=0m =0, Q=-10KN, M=-10.02KNm =2m =0, Q=0KN, M=0KNm
III-III
KNNFx 00 ==
050 == XQFy
02
50 =+= XXMM
=0m =0, Q=0KN, M=0KNm =1m =0, Q=5KN, M=-2.5KNm
IV-IV
VB
HB
q=5KN/m
N(x)Q(x)M(x)
x
-
71
=0m =0, Q=-5KN, M=-2.5KNm =1m =0, Q=0KN, M=0KNm
M, N, Q
()
A
VA
q=5KN/m
N(x)Q(x)M(x)
VB
HB
x
KNNFx 00 ==
0)5.0(52
5100 =+++= XXXXMM
0105150 =+= XQFy
C DBA
-2.5KNm
-10.02KNm
00 - -- -
+
-
72
q=20KN/m
C
P2=15KN
P3=20KN
P1=10KN
BA
(Q)
(N)
18
C DBA
10KN
-4.16KN-10KN
5KN
-5KN-
--
+ +
C DBA
0 0 0
-
73
C
P2=15KN
P3=20KN
P1=10KN
BA
P=100KN
...
KNHHFx BB 200200 ===
KNVVVFy CCB 35015100 ==+=
KNVVM BBC 10041521040 ==+=
2
C
P2=15KN
P3=20KN
P1=10KN
VC
HB
VB
-
74
1
A
q=20KN/m
VA
H AM A
VB
HB
KNHHHFx ABA 2000 ===
KNVVVFy ABA 9005200 ==+=
KNmMMM 20005105.25200 ==+=
I-I
KNNFx 200 ==
KNQFy 150 ==
0150 =+= XMM
=0m =-20, Q=15KN, M=0KNm =4m =-20, Q=15KN, M=-60KNm
P2=15KN
P3=20KNN(x)
Q(x)M(x)
x
-
75
C
P2=15KN
P3=20KN
P1=10KN
x
N(x)Q(x)M(x)
VC
II-II
KNNFx 200 ==
KNQQFy 20015350 ==+=
0)4(15350 =++= XXMM
=0m =-20, Q=20KN, M=-60KNm =2m =-20, Q=20KN, M=-20KNm
III-III
KNNFx 200 ==
KNQQFy 1001535100 ==+=
0)6(15)2(35100 =++++= XXXMM
=0m =-20, Q=-10KN, M=-20KNm =2m =-20, Q=-10KN, M=0KNm
C
P2=15KN
P3=20KN
x
VC
N(x)Q(x)M(x)
-
76
IV-IV
KNNFx 200 ==
020100 =+= XQFy
0102
200 =+= XXXMM
=0m =-20, Q=-10KN, M=0KNm =5m =-20, Q=90KN, M=200KNm
M, N, Q
(M)
(Q)
q=20KN/m
VB
HB
x
N(x)Q(x)M(x)
CBA
-
-
+
-
0
-60KNm
-20KNm
-200KNm
CBA-
--
+ +
-20KN-20KN-10KN
-10KN
90KN
15KN 15KN
-
77
(N)
CBA-
- - -
-20KN-20KN-20KN-20KN-20KN
-
78
4.
,
.
, .
.
, 3 .
[],[Q],[], (x), Q(x), M(x), . x, .
,
.
.
(x) , Q(x). (x): , .
Q(x): ,
.
(x): , .
.
l-x Q(x), M(x) -. ,
.
.
( , Q, M ), , .
-
79
,
, 3 , ,
. 3 .
-
,
.
.
, 3 4 .
G ,
:
3 4 .
.
[] , .
[, [] [Q] .
Beam Engilab.
AutoCad 2007 AutoDesk
00 == G
G
-
80
P1=15KN
P2=5KN
B
E
DC
P1=15KN
M=20KNm
M=30KNm
HA
VA VB
P=90KN
1
P1=15KN
P2=5KN
q=10KN/m q=15KN/m
B
E
DC
...
==
50F
KNVVVF ABAy 58.6401590100 ==+=
KNVVM BBA 42.6002036156155.153060 ==++=
-
81
B
VB
N(x)Q(x)
M(x)
x
P1=15KN
P2=5KN
B
E
M=30KNm
VB
x
N(x)Q(x)
M(x)
I-I
I-I
=0m =-45,42, Q=5KN, M=-30KNm =1,5m =-45,42, Q=5KN, M=-37,5KNm
I-I
KNQFx 00 ==
KNNFy 42.600 ==
KNmMM 00 ==
KNQFx 50 ==
KNNNFy 42.4501542.600 ==+=
015300 =+= XMM
-
82
P1=15KN
P2=5KN
B
E
DC
P1=15KN
M=20KNm
M=30KNm
VB
M(x)
Q(x)N(x)
x
KNNFx 50 ==
=0m =-5, Q=-45,42KN, M=-37,5KNm =6m =-5, Q=44,58KN, M=-34,98KNm
I-I
=0m =-64,58, Q=-5KN, M=-15KNm =6m =-64,58, Q=-5KN, M=0KNm
P1=15KN
P2=5KN
B
E
D
M=30KNm
q=15KN/m
N(x)Q(x)M(x)
x
VB
0151542.600 =+= XQFy0
215151553042.600 =++++= XXXXMM
KNQFx 50 ==
KNNXNFy 58.64042.601515200 ==+++=
05642.60306153615200 =+++= XMM
-
83
M, N, Q
()
(Q)
(N)
B
E
DC-37.5KNm
-30KNm
-30KNm
-37.5KNm-34.98KNm
-20KNm
-15KNm +
-
-
-
-
-
-
B
E
DC
44.58KN
-20KN
-5KN
15KN5KN
-45.42KN
-
+
++
-
-
15KN
B
E
DC
-45.42KN-60.42KN
-64.58KN
-5KN-5KN
-5KN
-5KN
-60.42KN
-64.58KN
-
-
-
-
-
-
84
B
C
P=200KN
2
...
KNHFx A 00 ==
KNVVVFy ABA 100010200 ==+=
KNVVM BB 100051020100 ===
-
85
-( )
KNQA 1001100cos100 === KNN A 1001100sin100 ===
KNmMMM CCC 25002552051000 ===
M, N, Q
(M)
q=20K N /m
N (x)
Q (x)M (x)
x
B
C
250KNm
+
250KNm
+
-
86
P1=15KN
P1=15KN
C B
(Q)
(N)
3
B
C
100KN+100KN +
B
C
-100KN
-
-
-100KN
-
87
P1=15KN
P1=15KN
C B
HA
VA
VB
...
KNHHFx AA 100100 ==+=
KNVVVFy AAB 56.00100 ==+=
KNVVM BBA 44.905.31051090 ===
-
KNNFx 00 ==
KNQFy 44.90 ==
XMM == 44.90
=0m =0, Q=-9,44KN, M=0KNm =4m =0, Q=-9,44KN, M=37,76KNm
B
VB
x
N(x)Q(x)M(x)
-
88
I-I
=0m =0, Q=0,56KN, M=37,76KNm =5m =0, Q=0,56KN, M=34,96KNm
I-I
KNQFx 00 ==
P1=15KN
B
VB
x
N(x)
Q(x)M(x)
KNNFx 00 ==
KNQQFy 56.001044.90 ==+=
010)4(44.90 =++= XXMM
P1=15KN
C B
VB
x
N(x)
Q(x)
M(x)
KNNNFy 56.0044.9100 ==+=
KNmMMM 96.340944.95100 ==+=
-
89
V-IV
=0m =-0.56, Q=0,56KN, M=34.96KNm =3.5m =-0.56, Q=0,56KN, M=0KNm
M, N, Q
(M)
P1=15KN
P1=15KN
C B
VB
N(x)
Q(x)
M(x)
x
KNQFx 100 ==
0944.9510100 =++= XMM
KNNNFy 56.0044.9100 ==+=
C B
34.96KNm
34.96KNm
0
37.76KNm
0
+
+
-
90
(Q)
(N)
C B
-9.44KN-9.44KN
0.56KN
10KN
10KN
+
+
-
0.56KN0
C B-0.56KN
-0.56KN
-
0
-
91
P=10KNq=10KN/m
A B
DC
P=10KN
A B
DC
P=40KN
HA
VA VB
4
...
KNHHFx AA 4004100 ==+=
KNVVVFy ABA 42.60100 ==+=
KNVVM BBA 42.16024105.31070 ===
-
92
-
KNQFx 00 ==
KNNFy 42.160 ==
KNmMM 00 ==
-
KNNFx 00 ==
KNQFy 42.160 ==
042.160 == XMM
=0m =0, Q=-16.42KN, M=0KNm =3.5m =0, Q=-16.42KN, M=57.5KNm
B
VB
N(x)Q(x)
M(x)
x
B
D
VB
N(x)
Q(x)M(x)
x
-
93
I-I
=0 =0, Q=-6.42KN, M=57.5KNm =3.5 =0, Q=-6.42KN, M=79.94KNm
V-V
P=10KN
B
D
VB
x
N(x)
Q(x)M(x)
0)5.3(42.16100 =++= XXMM
KNQQFy 42.601042.160 ==+=
KNNFx 00 ==
P=10KN
B
DC
VB
x
N(x)Q(x)
M(x)
q=10KN/m
-
94
0100 == XQFx
KNNNFy 42.6042.16100 ==+=
0742.165.3102
100 =++= XXMM
=0m =6.42, Q=0KN, M=79.94KNm =4m =6.42, Q=40KN, M=0KNm
M, N, Q
(M)
(Q)
A B
DC+
79.94KNm
79.94KNm 57.5KNm
0
00
+
+
A
B
D
C
40KN
0-6.42KN
-16.42KN
-6.42KN
-16.42KN-
0
-
+
-
95
C DG
P = 1 0 K N
q = 5 0 K N /m
(N)
5
AB
DC6.42KN
6.42KN
-16.42KN
-16.42KN
-
+
-
96
C DG
P = 1 0 K N
P = 3 0 0 K N
H A
V A
H B
V B
DG
V B
H B
q = 5 0 K N /mH G
H G
...
1) :
BGX HHF == 0
( ) KNVmm
KNVVF GBGy 145.4202500 ==
+=
1003201250230 ==+= BBBBG HVVH (1)
94565.105.410365065,1 =+=+= BBB VHV
(2)
(1) (2) : KNV B 145.142= = 43.61BH
-
97
-
KNNFY 145.1420 ==
KNQFX 43.610 ==
== 43,610
=m =-142,145, Q=61,43, =0 m =3m =-142,145, Q=61,43, =-184,29 m
I-I
=0m =-61,43, Q=-142.145KN, M=-184.29KNm =2m =-61,43, Q=-42.145KN, M=0KNm
D
VB
HB
M(x)
N(x)Q(x)
x
DG
VB
HB
q=50KN/m
N(x)Q(x)M(x)
x
050145.1420 =+= XQFY
KNNFX 43.610 ==
0343,61145,1422
500 =++=
-
98
C
P=10KN
q=50KN/m
VG
HG
M
QN
x
I-II
KNNFX 00 ==
0145.42500 =+= XQFY
0145.422
500 =+= XXXMM
=0m =0, Q=-42.145KN, M=0KNm =4m =0, Q=157,86KN, M=231,42KNm
V-IV
=0m =-157,85, Q=10KN, M=-231,42KNm =4m =-157,85, Q=10KN, M=0KNm
q=50KN/m
VG
HGN
QM
x
KNQQFX 100100 ===
KNNNFY 85.1570145.424500 ==+=
04145.422450100 =++= X
-
99
M, N, Q
()
(Q)
C DG
-184.29KNm
-184.29KNm
-231.42KNm
-231.42KNm
-
--
+
-
C DG
61.43KN
+
+
-
-
-142.145KN
-42.145KN
-10.0KN
157.86KN
-
100
B
EC
D
q=10KN/m
q=20KN/m
B
EC
D
P=240KN
P=40KN
VA
HA
VB
(N)
6
...
C DG
-
-
-
-142.145KN
-61.43KN
-157.85KN
-
101
VA
HA
N(x)Q(x)
M(x)
x
q=10KN/m
C
HA
q=10KN/m
N(x)
Q(x)
M(x)
q=20KN/m
x
VA
KNHHF AAx 1004100 ==+= KNVVVF ABAy 34.113012200 ==+=
KNVVM BBA 66.12601262402400 ==+=
-
010400 =+= XQFx
KNNFy 34.1130 ==
02
10400 =+= XXXMM
=0m =-113,34, Q=40KN, M=0KNm =4m =-113,34, Q=0KN, M=80KNm
-
-
102
C
D
VA
HA
q=10KN/m
q=20KN/m
N(x) Q(x)M(x)
x
Tan=2/6=0,333 R=113.34-20*X
=18,43 R1=35.81-6.32*X => R1=N
sin=0.316 R2=107.45-18.96*X => R2=Q
cos=0,948
=0m =-35,81, Q=107,45KN =6m =2,11, Q=-6,31KN
02
2034.1130 =+= XXXMM
=0m M= 0KNm =6m M= 320KNm
-
R=113.34-120-20*X = -6,66-20* R1=-2,10-6.32*X => R1=N
R2=-6,31-18.96*X => R2=Q
-
103
=0m =-2,10, Q=-6,31KN =6m =-40,02, Q=-120,07KN
02
2012034.1130 =++= MM
=0m M= 320,04 KNm =6m M= 0 KNm
V-V
KNQFx 00 ==
KNNNFy 66.126034.1132400 ==+=
04040136014400 =++=
=0m =-126.66, Q=0KN, M=0KNm =4m =-126.66, Q=0KN, M=0KNm
EC
D
VA
HA
q=10KN/m
q=20KN/m
M(x)
Q(x)N(x)
x
-
104
B
EC
D
+
++
80KNm
320.04KNm320.04KNm
80KNm 80KNm
0
0
M, N, Q
()
(Q)
(N)
B
EC
D
40KN
0
107.45KN
-6.31KN-120.07KN
-
+
+
-
0
0
B
EC
D
-113.34KN
-
-113.34KN
-35.81KN 2.10KN-
+
-2.10KN
-40.02KN-
-126.66KN
-126.66KN
-
-
105
5.
1. . 2003
2. 2002
3. 1997
4. . 1980
5. .
6. . () , 1983
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