Civil Engineering Reference
In-Depth Information
AE
L
AE
L
x
0
0
−
x
0
0
12
EI
L
6
EI
L
12
EI
L
6
EI
L
y
y
y
y
0
−
0
−
−
∆
∆
P
P
M
P
P
M
3
2
3
2
ix
ix
6
EI
L
4
EI
L
6
EI
L
2
EI
L
iz
y
y
y
y
iz
0
−
0
q
2
2
iy
iy
=
AE
L
AE
L
∆
∆
jx
−
x
0
0
x
0
0
jx
jz
jz
12
EI
L
6
EI
L
12
EI
L
6
EI
L
q
y
y
y
y
0
−
0
jy
jy
3
2
3
2
6
E
I
2
EI
L
6
EI
L
4
EI
L
y
y
y
y
0
−
0
2
2
L
(4.23)
4.11
ELAStic MEMbER StiffnESS, X-Y SYStEM
The X-Y coplanar coordinate system differs from the X-Z system in a few
ways. The derivation of elastic member stiffness in the X-Y system will
be derived in the two following examples using the area moment method
and conjugate beam method.
Example 4.17
q
iz
stiffness
Derive the
q
iz
stiffness using the area moment method for a linear member.
A free-body diagram is shown in Figure 4.30 with an imposed rota-
tion of 1 unit on the
i
i-end of the member. The moments are assumed in the
positive
z
direction using the right-hand rule and the Cartesian right-hand
coordinate system. The forces are shown consistent with the deformation.
The moment diagram divided by EI is shown for the reaction forces on the
i
i-end of the member.
P
iy
P
jy
M
jz
M
iz
θ
iz
EI
z
EI
z
Figure 4.30.
Example 4.17
q
iz
stiffness.
