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 = -FL�(1- � ) M = FL� (1- � )
ij �L
ji
2
Vij = F (1 + 2� )(1 - � )2
V = -F� (3 - 2� )
ji
F
M = -FL / 8
Mji = FL/8
ij
L/2 L/2
Vji = -F/2
Vij = F / 2
M = -FL�(1- � )
M = FL�(1- � )
ij
F F ji
�L
�L
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 = FL�(1-�2)/4 M = FL�(1-�2)/4
F F ji
ł L �L ł L
Vji = -F�(3-�2)/2
Vij = -F�(3-�2)/2
M = M�(2 - 3� )
M = M (1 - � )(3� -1)
ji
ij
M
�L
Vij = -6M�(1- � )/ L
Vji = -6M�(1-�)/ 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 = qL2� (4 -3�)/12
Mij = -qL2� [6 -�(8-3�)]/12
�L ji
q
Vji = -qL�3(1-� / 2)
Vij = qL�[1-�2(1-� /2)]
Mij = -11qL2 /192
M = 5qL2 /192
ji
L/2
q
Vij = 13qL / 32 V = -3qL / 32
ji
Mij = -qL2�(3-�2)/24
ł L �L
ł L Mji = qL2�(3-�2)/24
q
V = -qL� / 2
Vij = qL� / 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 = qL2� (5 - 3� )/ 60
M = -qL2� [10 - �(10 - 3� )]/ 60
�L ji
q
ij
2
3
Vij = qL�[1/ 2 -� (5 - 2�)/ 20]
V = -qL� (5 - 2� )/ 20
ji
M = -23qL2 / 960
M = 7qL2 / 960
ij
ji
L/2
q
Vij = 9qL / 40 V = -qL / 40
ji
3
Mij = -qL2�2[10-�(15-6�)]/30
M = qL2� (5 - 4� )/ 20
�L ji
q
2
Vij = qL�[1/ 2 - � (15 - 8� )/ 20]
Vji = -qL�3(15-8�)/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 = -qL2�2(4-3�)/12
M = qL2�2(4-3�)/12
ji
�L �L
q q
Vij = qL� / 2
V = -qL� / 2
ji
M = qL2�2(2-�)/12
Mij = -qL2�2(2-�)/12
q ji
�L �L
q
Vij = qL� / 2 V = -qL� / 2
ji
Mij = -qL2[1-�2(2-�)]/12 q Mji = qL2[1-�2(2-�)]/12
�L �L
Vij = qL(1- � )/ 2 V = -qL(1 - � )/ 2
ji
�i
M = 2EI�i / L
M = 4EI�i / L
ji
ij
V = -6EI�i / L2
Vij = -6EI�i / L2
ji
�i
M = 2EI� / L �j
M = 4EI� / L
ij j
ji j
Vij = -6EI� / L2 V = -6EI� / L2
�j
j ji j
M = -6EI� / L M = -6EI� / L
ij ji
� ="/L
V = 12EI� / L2
Vij = 12EI� / L2
ji
M = -EIąT ("td - "tg )/ h
M = EIąT ("td - "tg )/ h
ij
ji
"tg
Vij = 0 "td V = 0
ji
M = 0 M = EIąT ("td - "tg )/ h
ij ji
"tg
"td
Vij = -EIąT ("td - "tg )/(Lh) Vji = -EIąT("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 �" (3�"� - 2) �" �" "�
M = 2 �" (3 �"� -1) �" �" "�
ij
�L ji
L
L
EI
EI
"�
Vij = 6 �" (1- 2 �"� ) �" �" "�
V = 6 �" (1- 2 �"� ) �" �" "�
ji
L2
L2
EI
EI
M = 6 �" �" "h
M = 6 �" �" "h
ij
�L ij
L2
L2
"h
EI EI
Vij = -12 �" �" "h Vji = -12 �" �" "h
L3 L3
http://www.iil.pwr.wroc.pl/zukowski 3