Differential equation and solution
aka Pure and Warping Torsion
aka Free and Restrained Warping
ref: Hughes 6.1 (eqn 6.1.18)
the development of warping torsion up to this point was assumed to be "pure" or "free" i.e. it was the
only effect on a beam and it's behavior was unrestrained. this led us to state
φ
'' and
φ
''' were constant.
the development of St. Venant's torsion in 13.10 was developed the same way,
φ
' constant. we will now
address the situation where boundary conditions may affect one or both of these effects.
combined torsional resistance determined by:
M
x
=
M
x
St_V
+
M
x
w
M
x
St_V
is St. Venant's torsion =
GK
T
⋅φ
'
⋅
M
x
w
is warping torsion
=
−
E
⋅
I
ωω
'''
M
x
is internal or external concentrated torque
T
ω
M
x
=
T
'
−
EI
ωω
'''
(1)
⋅
⋅
uniform (distributed) torque
m
x
(torque per unit length) is related to Mx
equilibrium element =>
−
m
x
=
d
M
x
=>
dx
differentiating (1) =>
m
x
=
ωω
'''
−
T
''
(2)
⋅
⋅
solution of (1) has homogeneous and particular solution. rewriting:
⋅
⋅
T
−
M
x
let
λ
2
=
T
φ
'''
−
'
=
⋅
⋅
⋅
ωω
ωω
ωω
mx
homogeneous =>
φ
'''
− λ
2
'
=
0
assume solution
φ
H
:=
e
⋅
x
⋅
(
d
d
x
3
3
φ
H
− λ
2
⋅
d
d
x
φ
H
→
m
3
⋅
exp(m
⋅
x)
− λ
2
⋅
m
⋅
exp(m
⋅
x)
=>
m
3
− λ
2
⋅
m
=
mm
2
− λ
2
)
=
0
1
notes_16
_pure_&_warping_torsion.mcd
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ωω
λ
λ
λ
x
λ
λ
x
I
ωω
I
ωω
A
roots
m
:=
0
m
:=
m
:=
−
0
λ⋅
x
−
x
homogeneous solution =>
φ
H
:=
c
1
⋅
e
+
c
2
⋅
e
+
c
2
⋅
e
particular solution assume
φ
P
:=
Ax
φ
'''
−
GK
T
⋅φ
'
=
−
M
x
⋅
x
⋅
EI
ωω
ωω
⋅
⋅
d
3
3
φ
P
− λ
2
⋅
d
φ
P
→
−λ
2
⋅
A
is a solution <=>
A
:=
M
x
dx
dx
λ
2
⋅
E
⋅
−λ
2
⋅
A
→
−
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 Spring '03
 DavidBurke
 Strain, Torsion

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