Civil Engineering Reference
In-Depth Information
The area of the web below the centroid will be used to find
A
z
.
()
It
Az
884 02
295
95
2
.
yy
2
A
=
=
=
19 59
.
in
z
.
′
'
(
)
.
Table 5.16 shows the calculations in tabular form.
Table 5.16.
Example 5.13 Shear area
Element
b
h
z
A
Az
I
d
Ad
2
Chord
12
2
13.0
24
312
8
3.5
294
Web
2
12
6
24
144
288
-3.5
294
Σ
z = 9.5
48
456
296
588
Iy = 884.00
A'z' = 90.25
A
z
= 19.590
5.8
gEOMEtRic StiffnESS, X-Y SYStEM
An ordinary stiffness analysis, whether it includes shear deformations or
not, makes no adjustments for the changing geometry of a loaded struc-
ture. Forces and moments are calculated from the original positions of
the joints, not from their deformed positions. Elastic buckling, which is
a function of joint deformations, is therefore impossible to predict using
ordinary stiffness analysis. A procedure to include member and joint defor-
mations in force and moment calculations can be developed by assuming
a deformed shape and calculating the additional moment such as defor-
mation would cause. Figure 5.16 shows a member subjected to bending
and axial force in an un-deformed and deformed shape. An alternate, yet
similar, derivation is published by Ketter, Lee, and Prawel (1979).
The first is a beam in Figure 5.16 that was used to derive the ordinary
elastic stiffness matrix in Section 4.11 for the X-Y system. In that case, the
M
iz
P
ix
P
iy
∆
iy
P
ix
M
iz
L
L
P
iy
Figure 5.16.
Geometric stiffness.
