<s>
The	O
torsion	B-Algorithm
constant	I-Algorithm
is	O
a	O
geometrical	O
property	O
of	O
a	O
bar	O
's	O
cross-section	O
which	O
is	O
involved	O
in	O
the	O
relationship	O
between	O
angle	O
of	O
twist	O
and	O
applied	O
torque	O
along	O
the	O
axis	O
of	O
the	O
bar	O
,	O
for	O
a	O
homogeneous	O
linear-elastic	O
bar	O
.	O
</s>
<s>
The	O
torsion	B-Algorithm
constant	I-Algorithm
,	O
together	O
with	O
material	O
properties	O
and	O
length	O
,	O
describes	O
a	O
bar	O
's	O
torsional	O
stiffness	B-Algorithm
.	O
</s>
<s>
The	O
SI	O
unit	O
for	O
torsion	B-Algorithm
constant	I-Algorithm
is	O
m4	O
.	O
</s>
<s>
In	O
1820	O
,	O
the	O
French	O
engineer	O
A	O
.	O
Duleau	O
derived	O
analytically	O
that	O
the	O
torsion	B-Algorithm
constant	I-Algorithm
of	O
a	O
beam	O
is	O
identical	O
to	O
the	O
second	O
moment	O
of	O
area	O
normal	O
to	O
the	O
section	O
Jzz	O
,	O
which	O
has	O
an	O
exact	O
analytic	O
equation	O
,	O
by	O
assuming	O
that	O
a	O
plane	O
section	O
before	O
twisting	O
remains	O
planar	O
after	O
twisting	O
,	O
and	O
a	O
diameter	O
remains	O
a	O
straight	O
line	O
.	O
</s>
<s>
For	O
non-circular	O
cross-sections	O
,	O
there	O
are	O
no	O
exact	O
analytical	O
equations	O
for	O
finding	O
the	O
torsion	B-Algorithm
constant	I-Algorithm
.	O
</s>
<s>
Non-circular	O
cross-sections	O
always	O
have	O
warping	O
deformations	O
that	O
require	O
numerical	O
methods	O
to	O
allow	O
for	O
the	O
exact	O
calculation	O
of	O
the	O
torsion	B-Algorithm
constant	I-Algorithm
.	O
</s>
<s>
The	O
torsional	O
stiffness	B-Algorithm
of	O
beams	O
with	O
non-circular	O
cross	O
sections	O
is	O
significantly	O
increased	O
if	O
the	O
warping	O
of	O
the	O
end	O
sections	O
is	O
restrained	O
by	O
,	O
for	O
example	O
,	O
stiff	O
end	O
blocks	O
.	O
</s>
<s>
And	O
the	O
torsional	O
stiffness	B-Algorithm
,	O
</s>
