nosaci greda konzola, projektanti, statika
DESCRIPTION
grede nosioci konzola, statika, proračun, dimenzioniranjeTRANSCRIPT
![Page 1: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/1.jpg)
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7. dio
Nosa�i:
Greda i konzola
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Dijagrami:
1. Uzdužnih sila N2. Popre�nih sila TZ
3. Momenata savijanja My
![Page 3: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/3.jpg)
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Prosta greda
1. Optere�ena koncentriranom silom F
x
z
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azadovoljen 0F .3
R 0R 0M .2
R 0R 0M .1
x
AAB
BBA
=�
⋅=�=⋅−⋅=�
⋅=�=⋅−⋅=�
lbF
bFl
laF
aFl
I. Reaktivne sile:
x
z
![Page 5: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/5.jpg)
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( )
00
0
0
0R-R- 0F:Kontrola
BAz
=
=++⋅−
=+⋅−⋅−
=+=�
Fl
baF
Fl
aFl
bF
F
x
z
![Page 6: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/6.jpg)
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II. Dijagrami unutarnjih sila
Uzdužna sila N u nekom presjeku nosa�a jednaka je sumi projekcija svih sila koje djeluju s jedne strane presjeka u os nosa�a) u promatranom presjeku.
Dijagram uzdužnih sila N
N – dijagram nema
x
z
![Page 7: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/7.jpg)
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Dijagram popre�nih sila Tz – dijagram
Za karakteristi�ne to�ke: A, 1 i BTA = RA T1
d = RA – F= - RB
(T1l = RA) TB = - RB
x
z
![Page 8: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/8.jpg)
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Tz – dijagram
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Dijagram momenata savijanja My
Izme�u to�aka A i 1 Izme�u to�aka 1 i BMy= RA
. x My= RA. x – F. (x-a)
- za x = 0; MA = 0 - za x = a; M1 = RA. a
- za x= a; M1 = RA. a - za x = l; MB = 0
M1l = M1
d
x
z
![Page 10: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/10.jpg)
10My – dijagram
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A - 1 1 – BxRM Ay ⋅= ( )axFxRM Ay −⋅−⋅=
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To�ka A – 1 To�ka 1 – B
Diferencijalna veza:
)A(Ay
Ay
TRdx
dM
xRM
1−==
⋅= ( )( )
)B(Ay
Ay
Ay
TFRdx
dM
aFxFRM
axFxRM
−=−=
⋅+⋅−=
−⋅−⋅=
1
zy T
dx
dM=
Veza izme�u dijagrama My i TZ
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Derivacija momenta savijanja po nosa�ujednaka je popre�noj sili.
zy T
dx
dM=
Veza izme�u dijagrama momenata savijanja My i
dijagrama popre�nih silaTZ
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Primjer:
• F=8,66 kN RA= 6,50 kN• l = 4 m RB= 2,16 kN• a = 1 m
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Prosta greda
1 b) Kosa sila F !• F=10 kN α = 60°• l = 4 m• a = 1 m
Reakcije: RAH= 5,0 kN ; RAV= 6,50 kNRB= 2,16 kN
Dijagrami: N, Tz i My
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Prosta greda
2. Optere�ena kontinuiranim optere�enjem q
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18azadovoljen 0F .32
lqR 0
2l
lqlR 0M .2
2lq
R 02l
lqlR 0M .1
x
AAB
BBA
=�
⋅=�=⋅⋅−⋅=�
⋅=�=⋅⋅−⋅=�
I. Reaktivne sile
x
z
![Page 19: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/19.jpg)
1900 0
022
0R-R- 0F:Kontrola
BAz
==⋅+⋅−
=⋅+⋅−⋅−
=⋅+=�
lqlq
lqlqlq
lq
x
z
![Page 20: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/20.jpg)
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II. Dijagrami unutarnjih sila
N – dijagram
• Uzdužnih (normalnih) sila nema.
Tz – dijagram• Popre�na ili transverzalna sila Tz u nekom
presjeku nosa�a jednaka je sumi projekcija svih sila koje djeluju s jedne strane presjeka u normalu na os nosa�a u promatranom presjeku.
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Dijagram popre�nih sila Tz
Od to�ke A do to�ke B jedno podru�je:TZ = RA – q . x
(jednadžba pravca)
x=0; TA = RA
x=l; TB = RA – q l = - RB
x
z
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Tz - dijagram
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Dijagram momenata savijanja My
- za x = 0; MA = 0 - za x = l; MB = 0
2
22x
qxR
xxqxRM AAy ⋅−⋅=⋅⋅−⋅=
z
x
![Page 24: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/24.jpg)
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22
022
lq
ql
qR
x
xq
Rdx
dM
A
Ay
===
=⋅−=
2
2x
qxRM Ay ⋅−⋅=
Traženje mjesta ekstrema:
Prvu derivaciju izjedna�imo s nulom i dobivamo vrijednost x za koji je
moment savijanja ekstreman (maksimalan).
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2
2x
qxRM Ay ⋅−⋅=
2
lx =
8
84222222
2222
qlM
qlqllqlqlx
qxRM
ekst
Aekst
=
−=��
���
�⋅−⋅=⋅−⋅=
Vrijednost maksimalnog momenta savijanja:
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Dijagrami:
![Page 27: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/27.jpg)
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My - funkcija:
Tz - prva derivacija:
Optere�enje q - druga derivacija:
2
2x
qxRM Ay ⋅−⋅=
zAy TxqR
dx
dM=⋅−=
qdx
Md y −=2
2
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Diferencijalne veze izme�u unutarnjih sila i optere�enja:
zy T
dx
dM=
qdxdT
dx
Mdzy −=2
2
Druga derivacija momenta savijanja po nosa�u jednaka je – q (optere�enju nosa�a).
![Page 29: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/29.jpg)
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Primjer:
• q = 10 kN/m`• l = 4 m
Reakcije:RA = 20 kNRB = 20 kN
M1 = Mmaks.=ql2/8= 20 kNm
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Konzola
Optere�enje:1. Vektor sile F kolinearan sa osi štapa2. Vektor momenta M kolinearan sa osi štapa
3. Vektor momenta M okomit na os štapa4. Vektor sile F okomit na os štapa
5. Vektor sile F pod kutom u odnosu na os štapa
6. Kontinuiranim optere�enjem q
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1.a Vektor sile F
kolinearan s osi štapa
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1.b Vektor sile F
kolinearan sa osi štapa
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2. Vektor momenta M
kolinearan sa os štapa
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3. Vektor momenta M okomit na os štapa
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4. Optere�ena koncentriranom silom F –vektor sile okomit na os štapa
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lFlF
FF
⋅=�=⋅−=�
=�=−=�
=�
AAA
AAz
x
M 0M 0M .3
R 0R 0F .2
azadovoljen 0F .1
I. Reakcije:
z
x
![Page 37: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/37.jpg)
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00 0
0R- 0F:Kontrola
Az
==+−=+=�
FF
Fz
x
![Page 38: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/38.jpg)
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N - dijagram
• nema
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Tz - dijagram
TA = RA
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My - dijagram
- za x=0M = - MA
- za x=lMB = 0
A
xRMM AAy ⋅+−= (jednadžba pravca)
![Page 42: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/42.jpg)
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Dijagrami
![Page 43: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/43.jpg)
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4. Vertikalna sila F = 8,66 kNl = 0,5 m
Rješenje:RA = 8,66 kNMA = 4,33 kNm
Dijagrami: Tz i My
![Page 44: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/44.jpg)
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5. Kosa sila
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F = 10 kN α = 60°l = 0,5 m
Rješenje:FV = F sin α = 10 sin 60° = 8,66 kNFH = F cos α = 10 cos 60° = 5,00 kN
RAV = 8,66 kNRAH = 5,00 kNMA = 4,33 kNm Dijagrami: N, Tz i My
5. Kosa sila F
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“Desna” konzola !
• Desni kraj uklješten (upet) oslonac.
• Za optere�enje vertikalnom silom F nacrtajte dijagrame unutarnjih sila.
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6. Konzola optere�ena kontinuiranim optere�enjem q
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482M 0
2M 0M .3
R 0R 0F .2
azadovoljen 0F .1
2
AAA
AAz
x
lqllq
lqlq
⋅=�=⋅⋅−=�
⋅=�=⋅−=�
=�
I. Reakcije
x
z
![Page 49: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/49.jpg)
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00 0 -
0R- 0F:Kontrola
Az
==⋅+⋅=⋅+=�
lqlq
lq
x
z
![Page 50: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/50.jpg)
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Dijagrami:
• N – dijagrama – nema
• Tz – dijagramTz = RA – q x
- za x = 0; TA = RA
- za x = l; TB = 0
![Page 51: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/51.jpg)
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Tz - dijagram
x
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My - dijagram
M = - MA
2x
xqxRMM AAy ⋅⋅−⋅+−=
AA
- za x = 0
- za x = lM = 0
BB
![Page 53: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/53.jpg)
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Dijagrami:
![Page 54: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/54.jpg)
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Ponovo: Greda3. Trokutno optere�enje q0
lx
qqx 0=
x
qx
x
z
![Page 55: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/55.jpg)
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l6xq
x6
lqx
31
2xq
xRM
l2xq
6lq
2xq
RT
lx
qq 2
lq
32
R 2
lq
31
R
30x
Ay
20x
Az
0xBA
⋅⋅−−⋅⋅=⋅⋅⋅−⋅=
⋅⋅−⋅=⋅−=
=⋅⋅=⋅⋅=
Popre�na sila
Moment savijanja
Reaktivne sile
![Page 56: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/56.jpg)
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Tz dijagram
![Page 57: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/57.jpg)
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Dijagrami:
![Page 58: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/58.jpg)
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Greda s prepustom
1
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( ) ( )
azadovoljen 0F .3
R 0R 0M .2
R 0R 0M .1
x
AAB
BBA
=�
⋅=�=⋅−⋅=�
+⋅=�=+⋅−⋅=�
laF
aFl
lalF
alFlI. Reaktivne sile
x
z1
![Page 60: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/60.jpg)
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( )
00
0
0
0R-R 0F:Kontrola
BAz
=
=+⋅−−⋅
=++⋅−⋅=+=�
Fl
aFF
laF
Fl
alFl
aF
F
x
z1
![Page 61: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/61.jpg)
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F
RRT
RT
RT
BAd
B
Al
B
AA
==+−=
−=
−=
1T
F
1
![Page 62: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/62.jpg)
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Tz - dijagram
1z
x
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0
0
1 =⋅−=⋅−=
⋅−==
M
aFlRM
xRM
M
AB
Ay
A
x
z1
![Page 64: Nosaci Greda Konzola, projektanti, statika](https://reader030.vdocumento.com/reader030/viewer/2022033004/55cf9c50550346d033a964ec/html5/thumbnails/64.jpg)
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1