Cienkościan by Tommy www.PROJEKTOMANIA.com kom: 693-667-598
Dane: Nr tematu: 45 Grupa: 3 Nr tematu: 45
Typ przekroju
L1 L2 L3 L4 Po (i)obc
dB odc �
- cm cm cm cm - - - - - -
2 8.2 20 10 17 10 2 2 1 2 0.29
Zad 5 Cienkościan
by Tommy
kom 693 667 598
www.PROJEKTOMANIA.com
Schem belki
Py1 Pz1 qy1 qz1
L �ł
- m - - kN kN kN/m kN/m
2 5.85 0.74 0.35 0 0 0 6.7
1. Charakterystyki geometryczne
1.1 Środek ciężkości
Li
�i = Ai = �i �" Li A = Ai
"
d
i
A1 =
�1 = 0.820 cm 6.72
cm2
A2 =
�2 = 2.000 cm 40.00
cm2
A3 =
�3 = 1.000 cm 10.00
cm2
A4 =
�4 = 1.700 cm 28.90
cm2
A = Ł Ai = 85.62
cm2
Współrzędne środków ciężkości prętów w osiach y1,z1
y1 = -20.00 cm
z1 = -4.10 cm
y2 = -10.00 cm 0.00 cm
z2 =
5.00 cm 0.00 cm
y3 = z3 =
0.00 cm 8.50 cm
y4 = z4 =
Momenty statyczne
Sy1 =
6.72 ( -4.10 )+ 40.00 ( 0.00 )+ 10.00 ( 0.00 )+ 28.90 ( 8.50 )
Sz1 =
6.72 ( -20 )+ 40.00 ( -10 )+ 10.00 ( 5 )+ 28.90 ( 0 )
Sy1 =
218.08
cm3
Sz1 = -484.48
cm3
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Cienkościan by Tommy www.PROJEKTOMANIA.com kom: 693-667-598
Środek ciężkości
S
Sz1
y1
yc = zc =
A A
yc = -5.66 cm
zc =
2.55 cm
Współrzędne środków ciężkości prętów w osiach y0,z0
y1 = -14.34 cm
z1 = -6.65 cm
y2 = -4.34 cm
z2 = -2.55 cm
y3 = z3 = -2.55 cm
10.66 cm
y4 = z4 =
5.66 cm 5.95 cm
1.2 Centralne momenty bezwładności
Iy0 =
[1/12( 0.82 ( 8.20 6.72 ( -6.65 40.00 ( -2.55
)3)+ )2]+[ )2]+
+[ 10.00 ( -2.55 1.70 ( 17.00 28.90 ( 5.95
)2]+[1/12( )3)+ )2]
Iz0 =
[ 6.72 ( -14.34 2.00 ( 20.00 40.00 ( -4.34
)2]+[1/12( )3)+ )2]+
[1/12( 1.00 ( 10.00 10.00 ( 10.66 28.90 ( 5.66
)3)+ )2]+[ )2]
Iy0z0 =
6.72 ( -6.65 )( -14.34 )+ 40.00 ( -2.55 )( -4.34 )+
10.00 ( -2.55 )( 10.66 )+ 28.90 ( 5.95 )( 5.66 )
Iy0 =
2379.29
cm4
Iz0 =
5614.97
cm4
Iy0z0 =
1785.32
cm4
1.3 Osie główne
- 2J
y0z0
tg(2Ćo ) =
J - Jz0
y0
tg(2Ć0) = 1.1035
o
2Ć0 = 47.82
o
Ć0 = 23.91
1.4 Wykresy współrzędnych
i i
yi = (y1 - yc ) cosĆ0 + (z1 - zc ) sinĆ0
i i
zi = -(y1 - yc ) sin Ć0 + (z1 - zc ) cosĆ0
y1 =( -20 -( -5.66 )cosĆ0+( -8.2 -( 2.55 )sinĆ0= -17.47 cm
y2 =( -20 -( -5.66 )cosĆ0+( 0 -( 2.55 )sinĆ0= -14.14 cm
0 -( -5.66 )cosĆ0+( 0 -( 2.55 )sinĆ0= 4.14 cm
y3 =(
10 -( -5.66 )cosĆ0+( 0 -( 2.55 )sinĆ0= 13.28 cm
y4 =(
0 -( -5.66 )cosĆ0+( 17 -( 2.55 )sinĆ0= 11.03 cm
y5 =(
z1 =-( -20 -( -5.66 )sinĆ0+( -8.2 -( 2.55 )cosĆ0= -4.01 cm
z2 =-( -20 -( -5.66 )sinĆ0+( 0 -( 2.55 )cosĆ0= 3.48 cm
0 -( -5.66 )sinĆ0+( 0 -( 2.55 )cosĆ0= -4.62 cm
z3 =-(
10 -( -5.66 )sinĆ0+( 0 -( 2.55 )cosĆ0= -8.67 cm
z4 =-(
0 -( -5.66 )sinĆ0+( 17 -( 2.55 )cosĆ0= 10.92 cm
z5 =-(
wB(1) =
0.00 0.00 0.00 0.00 340.00
cm2 wB(2) = cm2 wB(3) = cm2 wB(4) = cm2 wB(5) = cm2
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1.5 Momenty bezwładności
Li
J = [2�"( f 1�" g1+ f 2�" g2) + f 1�" g2 + f 2�" g1]�"�i
6
Iy
Li
Pręt �i f1 f2 g1 g2 Wynik
1 8.20 0.82 -4.01 3.48 -4.01 3.48 31.96
2 20.00 2.00 3.48 -4.62 3.48 -4.62 231.94
3 10.00 1.00 -4.62 -8.67 -4.62 -8.67 455.65
4 17.00 1.70 -4.62 10.92 -4.62 10.92 868.27
1587.82
cm4
Iz
Li
Pręt �i f1 f2 g1 g2 Wynik
1 8.20 0.82 -17.47 -14.14 -17.47 -14.14 1685.84
2 20.00 2.00 -14.14 4.14 -14.14 4.14 2114.91
3 10.00 1.00 4.14 13.28 4.14 13.28 828.54
4 17.00 1.70 4.14 11.03 4.14 11.03 1777.16
6406.44
cm4
IwBz
Li
Pręt �i f1 f2 g1 g2 Wynik
1 8.20 0.82 -4.01 3.48 0.00 0.00 0.00
2 20.00 2.00 3.48 -4.62 0.00 0.00 0.00
3 10.00 1.00 -4.62 -8.67 0.00 0.00 0.00
4 17.00 1.70 -4.62 10.92 0.00 340.00 28196.94
28196.94
cm5
IwBy
Li
Pręt �i f1 f2 g1 g2 Wynik
1 8.20 0.82 -17.47 -14.14 0.00 0.00 0.00
2 20.00 2.00 -14.14 4.14 0.00 0.00 0.00
3 10.00 1.00 4.14 13.28 0.00 0.00 0.00
4 17.00 1.70 4.14 11.03 0.00 340.00 42908.45
42908.45
cm5
Ks =1/3[ 8.20 ( 0.82 20.00 ( 2.00 10.00 ( 1.00 17.00 ( 1.70 86.014
)3+ )3+ )3+ )3]= cm4
1.6 Główny biegun wycinkowy
Jwby
Jwbz
yA = yB + zA = zB -
J Jz
y
yA = -14.14 + ( 17.76 ) = 3.61 cm
zA =
3.48 - ( 6.70 ) = -3.21 cm
1.7 Główna współrzędna wycinkowa
wA( 2 )= 0
wA( 1 )= 0 + [( -14.14 -( 3.61 ))( -4.01 -( -3.21 ))-( 3.48 -( -3.21 ))( -17.47 -( 3.61 )]= 155.38 + ( 0 ) = 155.38
cm2
wA( 3 )= 0 + [( -14.14 -( 3.61 ))( -4.62 -( -3.21 ))-( 3.48 -( -3.21 ))( 4.14 -( 3.61 )]= 21.48 + ( 0 ) = 21.48
cm2
wA( 4 )= 21.48 + [( 4.14 -( 3.61 ))( -8.67 -( -3.21 ))-( -4.62 -( -3.21 ))( 13.28 -( 3.61 )]= 10.74 + ( 21.48 ) = 32.22
cm2
wA( 5 )= 21.48 + [( 4.14 -( 3.61 ))( 10.92 -( -3.21 ))-( -4.62 -( -3.21 ))( 11.03 -( 3.61 )]= 17.87 + ( 21.48 ) = 39.35
cm2
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SwA=0,5[(
155.38 +( 0.00 ))* 8.20 * 0.820 +
+( 0.00 +( 21.48 ))* 20.00 * 2.000 +
+( 21.48 +( 32.22 ))* 10.00 * 1.000 +
+( 21.48 +( 39.35 ))* 17.00 * 1.700 ]= 2099.59
cm4
SwA
wi = wi -
A
A
w(1) = 155.38 -( 24.52 )= 130.86
cm2
w(2) = 0.00 -( 24.52 )= -24.52
cm2
w(3) = 21.48 -( 24.52 )= -3.04
cm2
w(4) = 32.22 -( 24.52 )= 7.70
cm2
w(5) = 39.35 -( 24.52 )= 14.83
cm2
1.8 Główny wycinkowy moment bezwładności
Iw
Li
Pręt �i f1 f2 g1 g2 Wynik
1 8.20 0.820 130.86 -24.52 130.86 -24.52 32537.00
2 20.00 2.000 -24.52 -3.04 -24.52 -3.04 9133.82
3 10.00 1.000 -3.04 7.70 -3.04 7.70 150.50
4 17.00 1.700 -3.04 14.83 -3.04 14.83 1773.33
43594.65
cm6
1.9 Współczynnik giętno-skrętny
G �" Ks
ą =
E �" Jw
ą = ( 0,4* 86.014 )/ 43594.65 = 0.028093 1/cm = 2.809 1/m
1.10 Momenty statyczne odciętej części przekroju
y(odc 2 0.29 )= -14.14 + 0.29 ( 4.14 -( -14.14 ))= -8.84 cm
;ś =
z(odc 2 0.29 )= 3.48 + 0.29 ( -4.62 -( 3.48 ))= 1.13 cm
;ś =
w(odc 2 0.29 )= -24.52 + 0.29 ( -3.04 -( -24.52 ))= -18.29
;ś =
cm2
Sy(C)=0,5[( -4.01 +( 3.48 ))* 8.20 * 0.820 +( 3.48 +( 1.13 ))* 20.00 * 2.000 * 0.29 ]= 25.00
cm3
Sz(C)=0,5[( -17.47 +( -14.14 ))* 8.20 * 0.820 +( -14.14 +( -8.84 ))* 20.00 * 2.000 * 0.29 ]= -239.58
cm3
Sw(C)=0,5[( 130.86 +( -24.52 ))* 8.20 * 0.820 +( -24.52 +( -18.29 ))* 20.00 * 2.000 * 0.29 ]= 109.20
cm4
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2. Siły wewnętrzne
2.1 Siły pochodzące od zginania
50
46.46
40
30
Mz
20
19.63
10
0.00
0
0 1 2 3 4 5 6 7
3.14
20
15.89
15 Ty
10.33
10
4.13
5
0
0 1 2 3 4 5 6 7
0.00
W płaszczyz
nie XY
qy = 2.72 kN/m
Ra = 15.89 kN
Ma = -46.46 kNm
Mz = 3.14 kNm
Ty = 4.13 kN
Schemat 2 - q - Płaszczyzna XY
qy = 6.7 * 0.405 = 2.72kN/m
Ra = 2.72 * 5.85 = 15.89kN
Ma = - 0.5 * 2.72 * 5.85^2 = -46.46kNm
Mz = 0.5 * 2.72 * (5.85 - 0.74 * 5.85)^2 = 3.14kNm
Ty = 2.72 * (5.85 - 0.74 * 5.85) = 4.13kN
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Cienkościan by Tommy www.PROJEKTOMANIA.com kom: 693-667-598
-120
-100
-104.81
-80
My
-60
-44.28
-40
-7.09
-20
0 1 2 3 4 5 6 7
0
0.00
20
40
35
35.83
30
25
Tz
20
23.29
15
10
5 0.00
9.32
0
0 1 2 3 4 5 6 7
W płaszczyz
nie XZ
qz = 6.13 kN/m
Ra = 35.83 kN
Ma = -104.81 kNm
My = -7.09 kNm
Tz = 9.32 kN
Schemat 2 - q - Płaszczyzna XZ
qz = 6.7 * 0.914 = 6.13kN/m
Ra = 6.13 * 5.85 = 35.83kN
Ma = - 0.5 * 6.13 * 5.85^2 = -104.81kNm
My = - 0.5 * 6.13 * (5.85 - 0.74 * 5.85)^2 = -7.09kNm
Tz = 6.13 * (5.85 - 0.74 * 5.85) = 9.32kN
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2.2 Siły pochodzące od skręcania pręta cienkościennego
m = -( 2.72 )*( -4.01 -( -3.21 6.13 )*( -17.47 -( 3.61
))*10-2 + ( ))*10-2 = -1.270 kNm/m
x x Ms B Mw
�L
4.33 -1.93729 -0.15861 0.00626
0 0.0 0.00000 2.48286 -7.42703
łL
2.05 -4.80399 -0.15246 -0.02358
łL
2.05 -4.80399 -0.15246 -0.02358
L 5.85 -0.45192 0.00000 0.45192
1
0.000
0
-0.452
01234567
-1
-2
-1.937
Ms
-3
-4
-4.804
-4.804
-5
-6
3
2.5 2.483
2
1.5
1
B
0.5
0 0.000
-0.152
-0.152
01234567
-0.5 -0.159
1
0.452
0.006
0 -0.024
-0.024
Mw
01234567
-1
-2
-3
-4
-5
-6
-7
-7.427
-8
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3. Naprężenia
3.1 Naprężenia normalne
M
M B
y
i
z
� = �" zi - �" yi + �" wi
x
J Jz Jw
y
-7.09 3.14 -0.159
*10-3 *10-3 *10-3
�x1 = ( -4.01 ( -17.47 ( 130.86
)*10-2 - )*10-2+ )*10-4
1587.82 6406.44 43594.65
*10-8 *10-8 *10-12
-7.09 3.14 -0.159
*10-3 *10-3 *10-3
�x2 = ( 3.48 ( -14.14 ( -24.52
)*10-2 - )*10-2+ )*10-4
1587.82 6406.44 43594.65
*10-8 *10-8 *10-12
-7.09 3.14 -0.159
*10-3 *10-3 *10-3
�x3 = ( -4.62 ( 4.14 ( -3.04
)*10-2 - )*10-2+ )*10-4
1587.82 6406.44 43594.65
*10-8 *10-8 *10-12
-7.09 3.14 -0.159
*10-3 *10-3 *10-3
�x4 = ( -8.67 ( 13.28 ( 7.70
)*10-2 - )*10-2+ )*10-4
1587.82 6406.44 43594.65
*10-8 *10-8 *10-12
-7.09 3.14 -0.159
*10-3 *10-3 *10-3
�x5 = ( 10.92 ( 11.03 ( 14.83
)*10-2 - )*10-2+ )*10-4
1587.82 6406.44 43594.65
*10-8 *10-8 *10-12
-7.09 3.14 -0.159
*10-3 *10-3 *10-3
�xC = ( 1.13 ( -8.84 ( -18.29
)*10-2 - )*10-2+ )*10-4
1587.82 6406.44 43594.65
*10-8 *10-8 *10-12
�x1 = 26.47 +( -47.61 )= -21.14 MPa
�x2 = -8.61 +( 8.92 )= 0.31 MPa
�x3 = 18.59 +( 1.11 )= 19.70 MPa
�x4 = 32.19 +( -2.80 )= 29.39 MPa
�x5 = -54.13 +( -5.40 )= -59.53 MPa
�xC = -0.72 +( 6.65 )= 5.93 MPa
3.2 Naprężenia styczne w punkcie C
M
s
� = �"�
s
KS
Ą#Tz S y Ty S z MW SW ń#
� = -ó# Ą#
+ +
M +W
J � J � JW�
ó# y z Ą#
Ł# Ś#
-1.94
�s = 2.00
* 10-2
86.014
* 10-8
�s = -45045.85 kPa = -45.05 MPa
9.32 * 25.00 0.01 * 109.20
* 10-6 4.13 * -239.58
* 10-6 * 10-8
�M+W = - [ ++
1587.82 2.00 2.00 2.00
* 10-8 * * 10-2 6406.44 * 10-8 * * 10-2 43594.65 * 10-12 * * 10-2
�M+W = - [ 733.56 + -772.27 + 7.84 ] = 30.87 kPa = 0.03 MPa
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