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:
St_V
Mx = Mx
St_V
Mx
w
+ Mx
is St. Venant's torsion = G⋅ KT⋅φ'
w
= −E⋅Iωω⋅φ'''
M x is warping torsion
M x is internal or external concentrated torque Tω
(1)
M x = G⋅ KT⋅φ' − E⋅ Iωω⋅φ'''
uniform (distributed) torque mx (torque per unit length) is related to Mx
equilibrium element => −mx =
d
M x =>
dx
differentiating (1) =>
mx = E⋅ Iωω⋅φ''' − G⋅ KT⋅φ''
(2)
solution of (1) has homogeneous and particular solution. rewriting:
φ''' −
G⋅ KT
E⋅ Iωω
homogeneous =>
⋅φ' =
−M x
2
let λ =
E⋅ Iωω
2
φ''' − λ ⋅φ' = 0
G⋅ KT
E⋅ Iωω
assume solution φH := e
m⋅xx
(
3
)
2
2
2d
2
3
2
3
φ − λ ⋅ φH → m ⋅exp(m⋅x) − λ ⋅m⋅exp(m⋅x) => m − λ ⋅m = m⋅ m − λ = 0
3 H
dx
dx
d
1
notes_16_pure_&_warping_torsion.mcd
roots
m := 0
m := λ
0
homogeneous solution =>
φH := c ⋅e + c ⋅e
1
3
2d
3
φP − λ ⋅
dx
dx
λ ⋅x
x
2
particular solution assume φP := A ⋅x
x
d
m := −λ
λ
+ c ⋅e
− λ ⋅x
2
G⋅ KT
φ''' −
E⋅ Iωω
⋅φ' =
−M x
E⋅ Iωω
Mx
2
φP → −λ ⋅A is a solution <=> A :=
2
2
−λ ⋅A →
λ ⋅E⋅IIωω
therefore:
φ( x) := c + c ⋅e
1
λ ⋅x
2
+ c ⋅e
− λ ⋅x
2
Mx
+
E⋅ Iωω
which can be rewritten as
⋅x
2
−M x
λ ⋅E⋅IIωω
φ(x) := A + B⋅cosh( λ⋅x) + C⋅sinh( λ⋅x) +
Mx
⋅x
2
λ ⋅E⋅Iωω
similarly equation (2) => φ
homogeneous =>
4
IV
2
− λ ⋅φ'' =
mx
E⋅ Iωω
2
assume solution φH := e
− λ ⋅φ'' = 0
2 d2
2
4
2
φ → m2 ⋅exp(m2⋅x) − λ ⋅m2 ⋅exp(m2⋅x)
2 H
d
φH − λ ⋅
4
dx
φ
IV
m2⋅xx
4
2
2
=> m2 − λ ⋅m2 = 0
dx
roots
m2 := 0
m2 := 0
m2 := λ
0
homogeneous solution =>
(double root)
m2 := −λ
λ
0
φH := c ⋅e + c ⋅x⋅e + c ⋅e
1
2
2
3
λ ⋅x
+ c ⋅e
− λ ⋅x
4
3
particular solution assume φP := A1⋅x + B⋅
Bx
4
d
4
dx
2 d2
2
φ → −λ ⋅(2⋅ A1 + 6⋅ B⋅x)
2 P
φP − λ ⋅
2
dx
−λ ⋅(2⋅ A1 + 6⋅ B⋅x) =
mx
is a solution <=> B := 0 and A1 :=
E⋅ Iωω
−mx
2
2⋅λ ⋅E⋅IIωω
2
notes_16_pure_&_warping_torsion.mcd
φP :=
−mx
2
2
4
⋅x
x
d
4
2⋅λ ⋅E⋅Iωω
dx
mx
2 d2
φ →
2 P
E⋅ Iωω
dx
check
φP − λ ⋅
therefore:
φH := c + c ⋅x + c ⋅e
1
2
λ ⋅x
3
+ c ⋅e
− λ ⋅x
4
−
mx
2
2
⋅x
which can be rewritten as
2⋅λ ⋅E⋅IIωω
φ( x) := A + B⋅x + C⋅cosh( λ⋅x) + D⋅sinh( λ⋅x) −
mx
2
2
⋅x
2⋅λ ⋅E⋅Iωω
boundary conditions for various situations:
fixed end
φ=0
no twist
φ' = 0
no slope
pinned end
φ=0
no twist
Bi = 0
free warping
free end
Bi = 0
free warping
φ''' = 0
no warping shear
no twist
φl' = φr'
Bil = Bir
continuous
φl' = φr'
Bil = Bir
continuous
continuous supports φ = 0
transition point
within span
general
φ l = φr
φ'' = 0
free end from bending
for a visual of Bi the bimoment see figure 6.13 in text
3
notes_16_pure_&_warping_torsion.mcd
Problem - Torsional response of Cantilever Girder
general solution for end
moment Mx and fixed at x =
0 (cantilever)
Vy=5000 lb
φ = φ' = 0
x=0
φ( 0) →
A+ B=0
2
2
C⋅λ +
2
λ ⋅E⋅ Iωω
2
or ... substituting λ =
⋅x
λ ⋅E⋅ Iωω
Mx
simplify
2
Mx
φ' = 0
→ C⋅λ +
collect , C
Mx
λ ⋅E⋅Iωω
λ ⋅E⋅ Iωω
d
φ( x) φ_pr(x) → C⋅cosh( λ⋅x) ⋅λ +
φ_pr( x) := φ
dx
φ_pr( 0)
x
φ(x) := A + B⋅cosh( λ⋅x) + C⋅sinh( λ⋅x) +
from restrained_torsion.mcd
Mx
Mx
L
φ'' := 0
x=L
1/2 in
Vy=5000 lb
Mx
=0
2
λ ⋅E⋅Iωω
G⋅ KT
C⋅λ +
E⋅ Iωω
Mx
G⋅ KT
=0
free end => φ'' = 0
φ_db_pr( x) :=
2
d
2
φ x)
φ(
dx
φ_db_pr(x) → C⋅sinh( λ⋅x) ⋅λ
φ_db_pr(L) → C⋅sinh( λ⋅L) ⋅λ
or .....
Given
A+ B=0
2
B⋅cosh( λ⋅L) ⋅λ + C⋅sinh( λ⋅L) ⋅λ = 0
2
2
2
B + C⋅tanh( λ⋅L) = 0
C⋅λ +

 −Mx

⋅tanh( λ⋅L)

 λ⋅G⋅ KT

 M
x

⋅tanh( λ⋅L) 
Find(A, B , C) →

 λ⋅G⋅ KT


−M x



λ⋅G⋅ KT


4
Mx
G⋅ KT
=0
B + C⋅tanh( λ⋅L) = 0
notes_16_pure_&_warping_torsion.mcd
Mx
B :=
λ⋅G⋅K
KT
⋅tanh( λ⋅L)
A := −B
B
Mx
(λ⋅x) +
φ(x) := A + B⋅cosh( λ⋅x) + C⋅sinh
C
−M x
C :=
λ⋅G⋅K
KT
⋅x
2
λ ⋅E⋅Iωω
−M x
φ( x) →
λ⋅G⋅ KT
⋅tanh( λ⋅L) +
Mx
λ⋅G⋅ KT
⋅tanh( λ⋅L) ⋅cosh( λ⋅x) −
Mx
λ⋅G⋅ KT
⋅sinh( λ⋅x) +
Mx
2
⋅x
λ ⋅E⋅ Iωω
substituting for A, B, C, and λ^2
Mx
φ( x) :=
λ⋅G⋅K
KT
⋅tanh( λ⋅L) ⋅ ( cosh( λ⋅x) − 1) −
φ( x) :=
Mx
λ⋅G⋅KT
Mx
λ⋅G⋅K
KT
⋅sinh( λ⋅x) +
Mx
⋅x
G⋅ KT
⋅ tanh( λ⋅L) ⋅ ( cosh( λ⋅x) − 1) − sinh( λ⋅x) + λ⋅x
Mx
d
φ( x) collect , λ →
⋅ ( tanh( λ⋅L) ⋅sinh( λ⋅x) − cosh( λ⋅x) + 1)
G⋅ KT
dx
φ_pr( x) :=
2
d
2
φ( x) collect , λ →
dx
Mx
G⋅ KT
Mx
G⋅ KT
⋅ ( tanh( λ⋅L) ⋅cosh( λ⋅x) − sinh( λ⋅x)) ⋅λ
φ_db_pr( x) :=
3
d
3
dx
φ( x) collect , λ →
⋅ ( tanh( λ
λ⋅L) ⋅sinh( λ⋅x) − cosh( λ⋅x) + 1)
Mx
G⋅ KT
M x⋅λ
G⋅K
KT
⋅ ( tanh( λ⋅L) ⋅cosh( λ⋅x) − sinh( λ⋅x))
⋅ ( tanh( λ⋅L) ⋅sinh( λ⋅x) − cosh( λ⋅x)) ⋅λ
φ_tr_pr( x) :=
let's look at these in general
2
λ =
5
M x⋅λ
2
2
G⋅K
KT
⋅ ( tanh( λ⋅L) ⋅sinh( λ⋅x) − cosh( λ⋅x))
G⋅ KT
E⋅ Iωω
notes_16_pure_&_warping_torsion.mcd
2 < λ⋅L < 5
L := 1
λ := 5
λ⋅L = 5
F0(x) , F1( x) , F2( x) , F3( x)
x := 0, 0.1 .. L
above equations factoring out
Mx
, φ(x) ~ total twist
(
)
(
(
)
)
(
)
F0(x) := tanh λ⋅L ⋅ cosh λ⋅x − 1 − sinh λ⋅x + λ⋅x
G⋅ KT⋅λ
Mx
, φ'(x) ~ St V torsion
(
)
(
)
(
)
F1(x) := tanh λ⋅L ⋅sinh λ⋅x − cosh λ⋅x + 1
G⋅ KT
M x⋅λ
, φ''(x) ~ axial stress warping torsion
(
)
(
)
(
)
F2(x) := tanh λ⋅L ⋅cosh λ⋅x − sinh λ⋅x
G⋅ KT
M x⋅λ
F3(x) := tanh( λ⋅L) ⋅sinh( λ⋅x) − cosh( λ⋅x)
2
G⋅ KT
, φ'''(x) ~ shear stress warping torsion
respectively
5
4
F0( x)
3
F1( x)
F2( x)
F3( x)
2
1
0
1
0
0.2
0.4
0.6
0.8
x
total twist
St. V torsion
axial stress - warping torsion
shear stress - warping torsion
6
L
notes_16_pure_&_warping_torsion.mcd