Civil Engineering Reference
In-Depth Information
5.12 tORSiOn
The torsional stiffness of slender linear members is composed of two parts.
The first is known as St. Venant's torsion, which is uniform on a member
at any distance, r, from the longitudinal axis. This is the torsional stiffness
that is included in the elastic member stiffness Equation 4.33. This is the
primary resistance to torsion for circular crosses-sections that have area
distributed uniformly about the longitudinal axis. The second type of tor-
sional stiffness is known as warping torsion. Warping torsion is the primary
stiffness in thin-walled open cross-sections such as angles shapes and wide
flange shapes. The warping torsion will cause longitudinal deformations in
the cross-section that will cause certain portions to elongate and other por-
tions to shorten. This warping effect can be included in the derivation of the
stiffness. “Structural Analysis and Design,” by Ketter, Lee, and Prawel, Jr.,
covers this derivation (Ketter, Lee, and Prawel 1979). The torsional stiffness
at each degree of freedom is represented as two components instead of the
single values used in the normal elastic stiffness. Equation 5.35 shows the
elastic torsional member stiffness in the 3-D Cartesian coordinate system.
AE
L
AE
L
x
0
0 00 0
0
−
x
0
0 00 0
0
12
EI
L
6
EI
L
12
EI
L
6
EI
L
0
z
0 00 0
z
0
−
z
0
0
0
0
z
3
2
3
2
12
EI
L
6
EI
L
12
EI
L
6
EI
L
y
y
y
y
0
0
00
−
0
0
0
−
0
0
−
0
3
2
3
2
0
0
0
−−
TT
0
0
0
0
0
TT
−
2
0
0
1
2
1
0
0
0
−−
TT
0
0
0
0
0
TT
0
0
2
3
2
4
6
EI
L
4
EI
L
6
EI
L
2
EI
L
y
y
y
y
0
0
−
00
0
0
0
0
0
0
2
2
6
EI
L
4
EI
L
6
EI
L
2
EI
L
0
z
0
0
0
0
z
0
−
z
0 00 0
z
2
2
AE
L
AE
L
−
x
0
0 00 0
0
x
0
0 00 0
0
12
EI
L
6
EI
L
12
EI
L
6
EI
L
z
z
z
z
0
−
0
0
0
0
−
0
0
0
0
0
−
3
2
3
2
12
EI
L
6
EI
L
6
EI
L
1
2
EI
L
y
y
y
0
0
−
00
0
0
0
z
00
0
3
2
3
2
0
0
0
TT
0
0
0
0
0
−−
TT
0
0
1
2
1
2
0
0
0
−
TT
0
0
0
0
0
TT
0
0
2
4
2
3
−
6
EI
L
2
EI
L
6
EI
L
4
EI
L
y
y
y
y
0
0
00
0
0
0
0
0
0
2
2
2
EI
L
6
EI
L
6
EI
L
4
EI
L
y
0
z
0 00 0
0
−
z
0
0
0
0
z
2
2
(5.35)
