3,0
3,0
2,0
2,0
4,0
P=20 kN
H = 29
D
5
11
H = 29
E
R
R = 49
A
B
R = 25
A
6
11
H = 29
R = 55
E
D
R = 5
C
C
R = 9
B
6
11
6
11
q = 10 kN/m
K=S
6
11
6
11
OBLICZANIE PRZEMIESZCZEŃ UKŁADÓW STATYCZNIE
WYZNACZALNYCH Z ZASTOSOWANIEM RÓWNANIA PRACY
WIRTUALNEJ
4,36
23
11
21
11
60
41
9
11
21
22
118
2
11
M
T
-20
35
-8
4
11
30
5
11
-9
6
11
6
11
6
11
29
-49
-29
N
-38
1
11
6
11
6
11
-49
41
2
11
118
5
9
11
5
20
20
20
10
11
41
20
10
11
20
9
11
11
2
max
118
=
M
[kNm]
200
=
dop
σ
[MPa]
11
6
max
49
=
N
[kN]
Po uwzględnieniu współczynnika bezpieczeństwa równego 1,2 przyjęto następujące wartości sił
przekrojowych:
82
,
141
max
=
M
[kNm]
45
,
59
max
=
N
[kN]
3
max
709cm
W
W
M
dop
≥
≥
σ
Przyjęto przekrój dwuteowy I 340:
W = 923 cm
3
I = 15 700 cm
4
A = 86,8 cm
2
EI = 32 185 kNm
2
EA = 1 779 400 Kn
GA = 684 383 [kN]
G = 78,846 GPa
χ = 2,345
środnika
A
A
E
G
=
+
=
χ
ν )
1
(
2
M
0,18
0,91
1,82
P = 1
R = 0,55
H = 0,45
A
R = 0
E
H = 0,45
R = 0,45
B
T
-0,04
0,55
-0,45
0,45
N
-0,71
-0,45
-0,45
Przemieszczenie pkt. K (składowa pionowa)
∑
∫
∫
∫
+
+
+
=
⋅
i
i
I
K
R
R
k
dS
EA
N
N
dS
GA
T
T
dS
EI
M
M
__
__
__
__
_
1
1
χ
δ
]
[
66
,
1
]
[
0166
,
0
58
,
536
185
32
1
]
82
,
1
4
118
82
,
1
4
20
82
,
1
4
118
91
,
0
2
5
91
,
0
2
)
18
,
0
91
,
0
(
2
5
18
,
0
91
,
0
(
2
)
91
,
0
18
,
0
(
2
41
18
,
0
41
5
[
1
)
(
3
2
11
2
2
1
2
1
3
2
3
2
11
2
2
1
2
1
3
2
3
2
11
10
2
1
2
1
2
1
3
2
2
1
3
2
11
10
2
1
3
1
3
2
11
9
2
1
3
2
11
9
2
1
__
cm
m
EI
dS
EI
M
M
M
V
K
=
=
⋅
=
=
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
+
⋅
−
⋅
⋅
⋅
+
⋅
−
⋅
⋅
⋅
+
⋅
−
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
=
=
∫
δ
]
[
04
,
0
88
,
105
684383
345
,
2
]
45
,
0
4
29
45
,
0
)
2
9
2
49
(
45
,
0
2
9
45
,
0
10
55
,
0
2
10
55
,
0
2
30
04
,
0
5
8
[
)
(
11
6
11
6
2
1
11
6
2
1
11
6
2
1
11
5
2
1
11
5
2
1
11
5
2
1
11
4
__
cm
EA
dS
GA
T
T
T
V
K
=
⋅
=
=
⋅
⋅
+
⋅
⋅
⋅
+
⋅
⋅
+
⋅
⋅
⋅
+
+
⋅
⋅
−
⋅
⋅
⋅
+
⋅
⋅
⋅
+
⋅
⋅
=
=
∫
χ
χ
δ
]
[
02
,
0
75
,
330
1779400
1
]
45
,
0
4
49
45
,
0
8
29
71
,
0
5
38
[
1
)
(
11
6
11
6
11
1
__
cm
EA
dS
EA
N
N
N
V
K
=
⋅
=
=
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
=
∫
δ
]
[
13
,
4
29
45
,
0
01
,
0
1
1
11
6
__
cm
EI
R
R
k
i
i
=
⋅
⋅
=
∑
Przemieszczenie pionowe punktu K wynosi:
]
[
86
,
5
13
,
4
02
,
0
04
,
0
66
,
1
)
,
,
(
cm
M
T
N
V
K
=
+
+
+
=
δ
Przemieszczenie pionowe punktu K wynosi (z pominięciem siły normalnej N i tnącej T):
]
[
79
,
5
13
,
4
66
,
1
)
(
cm
M
V
K
=
+
=
δ
Przemieszczenie pkt. K (składowa pozioma)
M
1,45
0,67
1,45
0,64
-0,36
-0,36
A
R = 0
E
0,36
-0,36
B
T
0,29
N
P = 1
H = 0,64
R = 0,36
R = 0,36
H = 0,36
]
[
6
,
0
71
,
197
32185
1
]
45
,
1
118
4
45
,
1
4
20
45
,
1
118
4
45
,
1
4
20
45
,
1
41
4
45
,
1
41
5
[
1
)
(
3
2
11
2
2
1
2
1
3
2
3
2
11
2
2
1
2
1
3
2
3
2
11
9
2
1
3
2
11
9
2
1
__
cm
EI
dS
EI
M
M
M
H
K
=
⋅
=
=
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
+
+
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
−
=
=
∫
δ
]
[
3
,
3
29
36
,
0
01
,
0
1
1
11
6
__
cm
EI
R
R
k
i
i
=
⋅
⋅
=
∑
Przemieszczenie poziome punktu K wynosi (z pominięciem siły normalnej N i tnącej T):
]
[
9
,
3
3
,
3
6
,
0
1
)
(
__
cm
R
R
k
M
i
i
H
K
H
K
=
+
=
+
=
∑
δ
δ
Obrót przekroju K
M
0,64
0,82
A
R = 0
E
B
0,18
0,36
M = 1
H = 0,09
R = 0,09
R = 0,09
H = 0,09
Wzajemne przemieszczenie punktów R i S
0,63
P = 1
1,26
0,32
M
R = 0
E
P = 1
0,9482
0,3162
0,9482
0,3162
H = 0,16
H = 0,16
]
[
97
,
0
92
,
312
32185
1
]
32
,
0
2
59
)
36
,
0
26
,
1
(
2
59
)
32
,
0
26
,
1
(
2
118
26
,
1
4
20
26
,
1
4
118
63
,
0
2
5
63
,
0
2
20
63
,
0
2
20
63
,
0
)
2
5
2
20
2
20
2
41
(
63
,
0
5
41
[
1
)
(
3
2
11
1
2
1
3
2
3
1
11
1
2
1
3
1
3
2
11
2
2
1
2
1
3
2
3
2
11
2
2
1
2
1
3
2
3
2
2
1
3
2
11
10
2
1
3
2
2
1
11
10
2
1
11
9
2
1
3
2
11
9
2
1
__
cm
EI
dS
EI
M
M
M
RS
=
⋅
=
=
⋅
⋅
⋅
⋅
−
⋅
−
⋅
⋅
⋅
+
⋅
−
⋅
⋅
⋅
+
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
−
⋅
⋅
⋅
−
⋅
⋅
−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
⋅
⋅
=
=
∫
δ
]
[
47
,
1
29
)
16
,
0
(
01
,
0
1
1
11
6
__
cm
EI
R
R
k
i
i
−
=
⋅
−
⋅
=
∑
Wzajemne przemieszczenie punktów R i S wynosi:
]
[
5
,
0
47
,
1
97
,
0
1
)
(
__
cm
R
R
k
M
i
i
RS
RS
−
=
−
=
+
=
∑
δ
δ
]
[
0046
,
0
97
,
147
32185
1
]
36
,
0
4
118
36
,
0
4
20
36
,
0
4
118
18
,
0
2
5
18
,
0
2
20
18
,
0
2
20
2
5
)
18
,
0
64
,
0
(
2
20
)
18
,
0
64
,
0
(
2
20
)
18
,
0
64
,
0
(
2
41
64
,
0
5
41
[
1
)
(
3
2
11
2
2
1
2
1
3
2
3
2
11
2
2
1
2
1
3
2
3
2
2
1
3
2
11
10
2
1
2
64
,
0
82
,
0
3
2
3
2
2
1
3
2
11
10
2
1
3
1
11
9
2
1
3
2
11
9
2
1
__
rad
EI
dS
EI
M
M
M
K
=
⋅
=
=
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
+
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
−
⋅
+
⋅
⋅
−
⋅
+
⋅
⋅
+
+
⋅
+
⋅
⋅
+
⋅
⋅
⋅
⋅
=
=
+
∫
ϕ
δ
]
[
0083
,
0
29
09
,
0
01
,
0
1
1
11
6
__
rad
EI
R
R
k
i
i
=
⋅
⋅
=
∑
Obrót przekroju K wynosi:
]
[
013
,
0
0083
,
0
0046
,
0
1
)
(
__
rad
R
R
k
M
i
i
K
K
=
+
=
+
=
∑
ϕ
ϕ
δ
δ
Obrót cięciwy RS
]
[
25
,
0
78
,
79
32185
1
]
17
,
0
2
59
)
17
,
0
24
,
0
(
2
59
)
17
,
0
24
,
0
(
2
118
24
,
0
4
20
24
,
0
4
118
12
,
0
2
5
12
,
0
2
)
20
20
(
)
12
,
0
06
,
0
(
2
5
)
06
,
0
12
,
0
(
2
20
)
06
,
0
12
,
0
(
2
41
06
,
0
5
41
[
1
)
(
3
2
11
1
2
1
3
2
3
1
11
1
2
1
3
1
3
2
11
2
2
1
2
1
3
2
3
2
11
2
2
1
2
1
3
2
3
2
11
10
2
1
2
1
2
1
3
2
3
1
3
2
11
10
2
1
3
2
3
1
11
9
2
1
3
2
11
9
2
1
__
rad
EI
dS
EI
M
M
M
RS
−
=
⋅
−
=
⋅
⋅
⋅
⋅
−
⋅
+
⋅
⋅
⋅
−
−
⋅
+
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
−
−
⋅
⋅
⋅
−
+
⋅
−
⋅
⋅
⋅
+
⋅
−
⋅
⋅
⋅
+
+
⋅
−
⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
−
=
=
∫
ϕ
δ
]
[
73
,
0
29
)
8
,
0
(
01
,
0
1
1
11
6
__
rad
EI
R
R
k
i
i
−
=
⋅
−
⋅
=
∑
Obrót cięciwy RS wynosi:
]
[
98
,
0
73
,
0
25
,
0
1
)
(
__
rad
R
R
k
M
i
i
RS
RS
−
=
−
−
=
+
=
∑
ϕ
ϕ
δ
δ
M
R = 0
E
0,15
B
H = 0,08
0,06
P
0,12
0,24
0,17
P = 1/6,32 = 0,1581
0,15
0,05
0,05
H = 0,08
A
R = 0,09
R = 0,09
N
0,12
0,08
0,03
0,06
-0,09
Przemieszczenie pionowe pkt. K wywołane działaniem temperatury
)
(
)
(
0
t
t
V
K
V
K
V
K
∆
+
∆
=
δ
δ
δ
]
[
017
,
0
75
,
57
34
,
0
10
]
5
82
,
1
10
)
6
82
,
1
6
91
,
0
(
5
)
2
91
,
0
2
18
,
0
(
5
5
18
,
0
5
[
)
(
6
2
1
2
1
2
1
2
1
2
1
2
1
___
cm
h
dS
M
h
t
t
t
t
V
K
−
=
⋅
−
=
⋅
⋅
⋅
−
−
⋅
⋅
−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
−
+
⋅
⋅
⋅
−
=
∆
⋅
=
∆
−
∫
α
α
δ
[
015
,
0
925
,
14
10
]
4
45
,
0
8
8
45
,
0
5
,
10
5
71
,
0
5
,
10
[
)
(
6
__
0
cm
dS
N
t
t
t
t
V
K
−
=
⋅
−
=
⋅
⋅
−
⋅
⋅
−
⋅
⋅
−
=
⋅
∆
⋅
=
∆
−
∫
α
α
δ
]
[
32
,
0
015
,
0
017
,
0
)
(
)
(
0
cm
t
t
V
K
V
K
V
K
=
−
−
=
∆
+
∆
=
δ
δ
δ
Obrót przekroju K wywołany działaniem temperatury
)
(
)
(
0
t
t
K
K
K
∆
+
∆
=
ϕ
ϕ
ϕ
δ
δ
δ
]
[
10
3
39
,
9
10
89
,
6
34
,
0
10
]
2
09
,
0
8
2
06
,
0
8
)
6
03
,
0
2
08
,
0
(
5
,
10
5
12
,
0
5
,
10
[
]
2
17
,
0
8
)
2
17
,
0
2
24
,
0
(
8
)
6
24
,
0
6
12
,
0
(
5
)
2
12
,
0
2
06
,
0
(
5
5
06
,
0
5
[
5
6
6
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
__
0
__
rad
h
dS
N
t
dS
M
h
t
t
t
t
t
V
K
−
−
−
⋅
=
⋅
+
⋅
=
=
⋅
⋅
−
⋅
⋅
+
⋅
+
⋅
+
⋅
⋅
+
+
⋅
⋅
⋅
+
⋅
⋅
+
⋅
⋅
+
⋅
⋅
+
⋅
⋅
−
+
+
⋅
⋅
−
⋅
⋅
+
⋅
⋅
⋅
=
⋅
∆
⋅
+
∆
=
∫
∫
α
α
α
α
δ
Przemieszczenie poziome pkt. K wywołane osiadaniem podpór
∑
=
∆
+
⋅
0
1
__
i
i
H
K
B
δ
]
[
08
,
3
]
35
,
0
02
,
0
64
,
0
02
,
0
36
,
0
03
,
0
[
cm
H
K
=
⋅
−
⋅
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Obrót przekroju K wywołany osiadaniem podpór
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09
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0
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,
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0
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rad
K
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