Civil Engineering Reference
In-Depth Information
Inserting
θ
from Equation (7.50) into [9]:
p
o
A
o
(
ε
r
−
ε
d
)2 sin
2
α
r
cos
2
ψ
=
α
r
(7.51)
ψ
Inserting
from Equation (7.51) into [10]:
−
ε
ds
)
A
o
2
p
o
t
d
(
sin
2
α
r
cos
2
α
r
=
(7.52)
ε
r
−
ε
d
)
(
Equation (7.52) is the basic compatibility equation describing the warping of the shear flow
zone in a member subjected to torsion. To eliminate
α
r
in Equation (7.52) we utilize the first
type of compatibility equations, (5.46) and (5.47), in Section 5.2.2, to express
α
r
in terms
of strains:
α
r
=
ε
−
ε
d
ε
r
−
ε
d
cos
2
(7.53)
ε
t
−
ε
d
ε
r
−
ε
d
sin
2
α
r
=
(7.54)
Substituting cos
2
α
r
and sin
2
α
r
from Equations (7.53) and (7.54) into Equation (7.52) and
taking
ε
ds
/
2
=
ε
d
from Equation [11] result in:
(
A
o
p
o
−
ε
d
)(
ε
r
−
ε
d
)
t
d
=
(7.55) or [19]
(
ε
−
ε
d
)(
ε
t
−
ε
d
)
It should be noted that the variable
t
d
is expressed in terms of strains in all the
-,
t
-,
r
- and
d
directions. The variable
t
d
is also involved in Equations [1], [2], [8] and [10] through the
terms
A
o
,
p
o
,
−
ρ
t
. Hence, in the solution procedure the variable
t
d
must first be assumed
and then checked by Equation [19].
ρ
and
7.1.5.3
ε
as a Function of
f
,
f
p
The strain
α
r
from the
equilibrium equation [1] and the compatibility equation (7.52). To do this, we first substitute
cos
2
ε
can be related to the stresses,
f
and
f
p
, by eliminating the angle
α
r
from Equation (7.53) into (7.52) to obtain the compatibility equation:
A
o
2
p
o
t
d
(
−
ε
ds
)
sin
2
α
r
=
(7.56)
(
ε
−
ε
d
)
Then substituting sin
2
α
r
from compatibility equation (7.56) into the equilibrium equation
[1], and utilizing the definitions of
ρ
=
A
/
p
o
t
d
and
ρ
p
=
A
p
/
p
o
t
d
result in:
A
o
(
−
ε
ds
)(
−
σ
d
)
ε
=
ε
d
+
(7.57)
2(
−
p
o
t
d
σ
+
A
f
+
A
p
f
p
)
For pure torsion,
σ
=
0. Also
ε
ds
/
2
=
ε
d
from Equation [11]. Equation (7.57) becomes:
A
o
(
−
ε
d
)(
−
σ
d
)
ε
=
ε
d
+
(7.58) or [20]
(
A
f
+
A
p
f
p
)
