MOMENTY I SIA
Y BRZEGOWE
PR
T OBUSTRONNIE UTWIERDZONY
EI =const
M
M
M ji
ij
ij
M
Vij
Vij L Vji ji V
ji
2
2
F
M = -FL¾(1- ¾ ) M = FL¾ (1- ¾ )
ij ¾L
ji
2
Vij = F (1 + 2¾ )(1 - ¾ )2
V = -F¾ (3 - 2¾ )
ji
F
M = -FL / 8
Mji = FL/8
ij
L/2 L/2
Vji = -F/2
Vij = F / 2
M = -FL¾(1- ¾ )
M = FL¾(1- ¾ )
ij
F F ji
¾L
¾L
V = -F
Vij = F
ji
M = -FL(n -1/ n)/12 M = FL(n -1/ n)/12
ij ji
F F F
L/n L/n L/n L/n
Vij = F(n -1)/ 2
V = -F(n -1)/ 2
ji
M = -FL(n + 0.5 / n)/12 M = FL(n + 0.5/ n)/12
L F F F F L
ij ji
L/n L/n L/n
2n 2n
Vij = Fn / 2
V = -Fn / 2
ji
Mij = FL¾(1-¾2)/4 M = FL¾(1-¾2)/4
F F ji
Å‚ L ¾L Å‚ L
Vji = -F¾(3-¾2)/2
Vij = -F¾(3-¾2)/2
M = M¾(2 - 3¾ )
M = M (1 - ¾ )(3¾ -1)
ji
ij
M
¾L
Vij = -6M¾(1- ¾ )/ L
Vji = -6M¾(1-¾)/ L
M = M / 4 M = M / 4
ij M ji
L/2 L/2
Vij = -1.5M / L V = -1.5M / L
ji
Mji =(2q1 +3q2)L2 /60
q1
Mij = -(3q1 +2q2)L2 /60 q2
V = -(3q1 + 7q2 )L / 20
Vij = (7q1 + 3q2 )L / 20
ji
Mij = -qL2 /12 Mji = qL2 /12
q
Vij =qL/2
V = -qL / 2
ji
Mij = -qL2 /20 Mji = qL2 /30
q
V = -3qL / 20
Vij = 7qL / 20
ji
3
2
M = qL2¾ (4 -3¾)/12
Mij = -qL2¾ [6 -¾(8-3¾)]/12
¾L ji
q
Vji = -qL¾3(1-¾ / 2)
Vij = qL¾[1-¾2(1-¾ /2)]
Mij = -11qL2 /192
M = 5qL2 /192
ji
L/2
q
Vij = 13qL / 32 V = -3qL / 32
ji
Mij = -qL2¾(3-¾2)/24
Å‚ L ¾L
Å‚ L Mji = qL2¾(3-¾2)/24
q
V = -qL¾ / 2
Vij = qL¾ / 2
ji
http://www.iil.pwr.wroc.pl/zukowski 1
EI =const
M
M ji
M
ij
ij
M
V
Vij Vij L Vji ji
ji
3
2
M = qL2¾ (5 - 3¾ )/ 60
M = -qL2¾ [10 - ¾(10 - 3¾ )]/ 60
¾L ji
q
ij
2
3
Vij = qL¾[1/ 2 -¾ (5 - 2¾)/ 20]
V = -qL¾ (5 - 2¾ )/ 20
ji
M = -23qL2 / 960
M = 7qL2 / 960
ij
ji
L/2
q
Vij = 9qL / 40 V = -qL / 40
ji
3
Mij = -qL2¾2[10-¾(15-6¾)]/30
M = qL2¾ (5 - 4¾ )/ 20
¾L ji
q
2
Vij = qL¾[1/ 2 - ¾ (15 - 8¾ )/ 20]
Vji = -qL¾3(15-8¾)/20
Mji =3qL2 /160
Mij = -qL2 /30
L/2
q
Vij = 29qL /160
V = -11qL /160
ji
Mij = -5qL2 /96
q M = 5qL2 /96
ji
L/2
V = -qL / 4
Vij = qL / 4
ji
M = qL2 /32
Mij = -qL2 /32
ji
q
L/2 q
V = -qL / 4
Vij = qL / 4
ji
Mij = -qL2¾2(4-3¾)/12
M = qL2¾2(4-3¾)/12
ji
¾L ¾L
q q
Vij = qL¾ / 2
V = -qL¾ / 2
ji
M = qL2¾2(2-¾)/12
Mij = -qL2¾2(2-¾)/12
q ji
¾L ¾L
q
Vij = qL¾ / 2 V = -qL¾ / 2
ji
Mij = -qL2[1-¾2(2-¾)]/12 q Mji = qL2[1-¾2(2-¾)]/12
¾L ¾L
Vij = qL(1- ¾ )/ 2 V = -qL(1 - ¾ )/ 2
ji
Õi
M = 2EIÕi / L
M = 4EIÕi / L
ji
ij
V = -6EIÕi / L2
Vij = -6EIÕi / L2
ji
Õi
M = 2EIÕ / L Õj
M = 4EIÕ / L
ij j
ji j
Vij = -6EIÕ / L2 V = -6EIÕ / L2
Õj
j ji j
M = -6EIÈ / L M = -6EIÈ / L
ij ji
È ="/L
V = 12EIÈ / L2
Vij = 12EIÈ / L2
ji
M = -EIÄ…T ("td - "tg )/ h
M = EIÄ…T ("td - "tg )/ h
ij
ji
"tg
Vij = 0 "td V = 0
ji
M = 0 M = EIÄ…T ("td - "tg )/ h
ij ji
"tg
"td
Vij = -EIÄ…T ("td - "tg )/(Lh) Vji = -EIÄ…T("td - "tg)/(Lh)
2 http://www.iil.pwr.wroc.pl/zukowski
"
h
h
EI =const
M
M ji
M
ij
ij
M
V
Vij L
Vij Vji ji
ji
EI
EI
M = 2 Å" (3Å"¾ - 2) Å" Å" "Õ
M = 2 Å" (3 Å"¾ -1) Å" Å" "Õ
ij
¾L ji
L
L
EI
EI
"Õ
Vij = 6 Å" (1- 2 Å"¾ ) Å" Å" "Õ
V = 6 Å" (1- 2 Å"¾ ) Å" Å" "Õ
ji
L2
L2
EI
EI
M = 6 Å" Å" "h
M = 6 Å" Å" "h
ij
¾L ij
L2
L2
"h
EI EI
Vij = -12 Å" Å" "h Vji = -12 Å" Å" "h
L3 L3
http://www.iil.pwr.wroc.pl/zukowski 3