Civil Engineering Reference
In-Depth Information
(+)
T
0
()
-
T
-
T
0
Twist rotation prevented
(b) Torque
T
x
T
-
T
0
T
0
0
=0
L
= 0
-
Ta
(
L
a
)
GI
L
t
x
T
a
L
-
a
(+)
(a) Beam
(c) Twist rotation
Figure 10.17
Uniform torsion of a statically indeterminate beam.
members for which the use of the uniform torsion theory is appropriate, such as
thosewithhightorsionalrigidityandlowwarpingrigidity,theseadditionalstresses
decreaserapidlyasthedistancefromtheloadedpointincreases.Becauseofthis,the
anglesoftwistpredictedbytheuniformtorsionanalysisareofsufficientaccuracy.
10.2.3 Plastic collapse analysis
10.2.3.1 Fully plastic stress distribution
Theelasticshearstressdistributionsdescribedintheprevioussub-sectionsremain
validwhiletheshearyieldstress
τ
y
isnotexceeded.Theuniformtorqueatnominal
first yield
T
ty
may be obtained from equations 10.11, 10.16, 10.18, or 10.22 by
using
τ
t
,
max
=
τ
y
.Yielding commences at
T
ty
at the most highly stressed regions,
and then generally spreads until the section is fully plastic.
The fully plastic shear stress distribution can be visualised by using the 'sand
heap' modification of the Prandtl membrane analogy. Because the fully plastic
shear stress distribution is constant, the slope of the Prandtl membrane is also
constant, and its contours are equally spaced in the same way as are those of a
heap formed by pouring sand on to a base area of the same shape as the member
cross-section until the heap is fully formed.
ThisisdemonstratedinFigure10.18aforacircularcross-section,forwhichthe
sand heap is conical. Its fully plastic uniform torque may be obtained from
R
T
tp
=
τ
y
(
2
π
r
)
r
d
r
(10.24)
0
whence
T
tp
=
2
π
R
3
/
3
τ
y
(10.25)
Search WWH ::
Custom Search