Civil Engineering Reference
In-Depth Information
y
M
j
,
u
j
M
i
,
u
i
i
j
x
F
IGURE
17.6
Forces and
moments on a beam element
F
y, i
,v
i
F
y, j
,v
j
17.2 S
TIFFNESS
M
ATRIX FOR A
U
NIFORM
B
EAM
Our discussion so far has been restricted to structures comprising members capable
of resisting axial loads only. Many structures, however, consist of beam assemblies in
which the individualmembers resist shear andbending forces, in addition to axial loads.
We shall now derive the stiffness matrix for a uniform beam and consider the solution
of rigid jointed frameworks formed by an assembly of beams, or beam elements as
they are sometimes called.
Figure 17.6 shows a uniform beam
ij
of flexural rigidity
EI
and length
L
subjected to
nodal forces
F
y
,
i
,
F
y
,
j
and nodal moments
M
i
,
M
j
in the
xy
plane. The beam suffers
nodal displacements and rotations
v
i
,
v
j
and
θ
i
,
θ
j
. We do not include axial forces here
since their effects have already been determined in our investigation of trusses.
The stiffness matrix [
K
ij
] may be written down directly from the beam slope-deflection
equations (16.27). Note that in Fig. 17.6
θ
i
and
θ
j
are opposite in sign to
θ
A
and
θ
B
in
Fig. 16.32. Then
6
EI
L
2
4
EI
L
θ
i
+
6
EI
L
2
2
EI
L
θ
j
M
i
=−
v
i
+
v
j
+
(17.28)
and
6
EI
L
2
2
EI
L
θ
i
+
6
EI
L
2
4
EI
L
θ
j
M
j
=−
v
i
+
v
j
+
(17.29)
Also
12
EI
L
3
6
EI
L
2
12
EI
L
3
6
EI
L
2
−
F
y
,
i
=
F
y
,
j
=−
v
i
+
θ
i
+
v
j
+
θ
j
(17.30)
Expressing Eqs (17.28), (17.29) and (17.30) in matrix form yields
.
/
!
.
/
!
12
/
L
3
6
/
L
2
12
/
L
3
6
/
L
2
F
y
,
i
M
i
F
y
,
j
M
j
−
−
−
v
i
θ
i
v
j
θ
j
6
/
L
2
6
/
L
2
−
4
/
L
2
/
L
=
EI
(17.31)
0
"
12
/
L
3
6
/
L
2
12
/
L
3
6
/
L
2
0
"
−
6
/
L
2
6
/
L
2
−
2
/
L
4
/
L
