Civil Engineering Reference
In-Depth Information
k
0
0
0
0
0
k
0
0 000
11
,
17
,
0
k
0
0
0
k
0
k
0 00
k
22
,
26
,
28
,
212
,
∆
∆
∆
P
P
P
M
M
M
P
P
P
M
M
M
ix
ix
00
3
EI
L
3
EI
L
3
EI
L
y
y
y
0
−
000
−
000
iy
iy
3
2
3
00 0
k
0 000
0
k
00
iz
iz
44
,
410
,
q
q
q
00
3
EI
L
3
EI
L
3
EI
L
ix
iy
ix
iy
y
y
y
−
0
000
0000
2
2
0
k
0
0
0
k
0
k
0 00
k
iz
iz
=
62
,
66
,
68
,
612
,
∆
∆
∆
k
0
0
0
0
0
k
0
0 000
jx
jy
jx
jy
71
,
77
,
0
k
0
0
0
k
0
k
0 00
k
82
,
8
,6
8 8
,
8 12
,
00
3
EI
L
3
EI
L
3
EI
L
jz
jz
y
y
y
−
0
000
000
q
q
q
3
2
3
jx
jx
00 0
k
0 000
0
k
00
jy
jz
jy
jz
10 4
,
1
010
,
0
0
0
0
0 000
0 000
0
k
0
0
0
k
0
k
0 00
k
12 2
,
126
,
12 8
,
1212
,
(5.5)
When more than one of the four degrees of freedom is released on a flex-
ural member, two conditions may exist. The first is that the member will
provide no joint stiffness. This occurs when both deflection and rotation
at either end are released, when rotation is released at both ends, or when
deflection is released at one end and rotation is released at the other end.
The stiffness matrix for this condition is shown in Equation 5.6. The
resulting beam is either a cantilever beam, a pinned-pinned beam, or a
pinned-slotted beam.
k
00000 0 0000
0 000 0 0 00
0 00000000000
000
k
∆
∆
∆
P
P
P
M
M
M
P
P
P
M
M
M
11
,
17
,
ix
ix
k
k
k
k
22
,
26
,
28
,
212
,
iy
iy
iz
iz
q
q
q
00000 0 0
0 00000000000
0
k
k
44
,
410
,
ix
ix
iy
iy
k
000 0
k
k
0 00
k
, , , ,
,
000000 0 0000
0 000 0 0 00
0 00000000000
000
62
66
68
612
iz
iz
=
k
k
∆
71
77
,
j
x
jx
k
k
k
k
∆
∆
q
q
q
82
,
86
,
88
,
812
,
jy
jy
jz
jz
00000 0 0
0 00000000000
0
k
k
10
4
1010
,
jx
jx
jy
jy
k
000 0
k
k
0 00
k
12 2
,
126
,
12 8
,
1212
,
jz
jz
(5.6)
