Civil Engineering Reference
In-Depth Information
where
b
is the developed length of the mid-line and
t
the thickness of each thin-
walledelementofthecross-section,whilethesummationiscarriedoutforallsuch
elements.
The maximum shear stress away from the concentrations at re-entrant corners
can also be approximated as
τ
t
,
max
≈
T
t
t
max
I
t
,
(10.18)
where
t
max
is the maximum thickness.When more accurate values for the torsion
section constant
I
t
are required, the formulae developed in [3] can be used. The
values given in [4-6] for British hot-rolled steel sections have been calculated in
this way.
10.2.1.5 Thin-walled closed cross-sections
The uniform torsional behaviour of thin-walled closed-section members is quite
different from that of open-section members, and there are dramatic increases in
thetorsionalstiffnessandstrength.Theshearstressnolongervarieslinearlyacross
thethicknessofthewall,butisconstantasshowninFigure10.10b,andthereisa
constant shear flow around the closed section.
This shear flow is required to prevent any discontinuities in the longitudinal
warping displacements of the closed section. To show this, consider the slit rect-
angular tube shown in Figure 10.11a. The mid-thickness surface of this open
sectionisunstrained,andsothewarpingdisplacementsofthissurfaceareentirely
due to the twisting of the member.The distribution of the warping displacements
caused by twisting (see Section 10.3.1.2) is shown in Figure 10.11b. The relative
b
f
Relative warping
displacement
t
w
S
E
C
d
z
E
C
S
E
0
0
d
s
y
-
d
O
d
f
O
x
y
Slit
z
y
0
t
f
z
(a) Cross-section of a slit tube
(b) Warping displacements
Figure 10.11
Warping of a slit tube.
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