Civil Engineering Reference
In-Depth Information
Table 5.14.
Example 5.3 Member stiffness
P
L
Final P
D
G
0.00
−
51.40
−
51.40
kips
∆
x4
0.00
16.87
31.87
kips
∆
y4
0.00
2132.90
2732.90
kip-in
q
z4
0.06
51.40
51.40
kips
∆
x2
−
2.26
−
16.87
−
1.87
kips
∆
y2
0.00
1915.62
1315.62
kip-in
q
z2
Table 5.15.
Example 5.3 Member stiffness
D
G
P
L
Final P
0.00
51.40
51.40
kips
∆
x5
0.00
16.13
16.13
kips
∆
y5
−
0.02
0.00
0.00
kip-in
q
z5
−
0.04
−
51.40
−
51.40
kips
∆
x3
−
2.25
−
16.13
−
16.13
kips
∆
y3
0.00
2322.89
2322.89
kip-in
q
z3
member are released, then the member is unstable. The axial components
of the member stiffness matrix are
k
1,1
,
k
1,7
,
k
7,1
, and
k
7,7
. The torsional
components of the member stiffness matrix are
k
4,4
,
k
4,10
,
k
10,4
, and
k
10,10
.
5.4
nOn-PRiSMAtic MEMbERS
Non-prismatic members have cross-sectional properties that vary along
the length of the member. The stiffness of these members can be handled
in two ways. First, the member could be divided into prismatic sections
and modeled with several different members of constant cross-section.
Second, the member can be modeled with stiffness derived from the math-
ematical model of the cross-sectional variation. The following derivation
is for the stiffness of a non-prismatic member in the X-Z system.
Example 5.4
Non-prismatic member stiffness
Derive the local member stiffness in the X-Z system for a non-prismatic
cross-section using Castigliano's theorems.
Figure 5.6 shows the beam with an applied deflection and an applied
rotation at the
i-
i-end. These can be used simultaneously to derive the stiff-
ness of the member.
