Civil Engineering Reference
In-Depth Information
4.4
Effective density and bulk modulus for cylindrical
tubes having triangular, rectangular and hexagonal
cross-sections
The effective density has been calculated by Craggs and Hildebrandt (1984, 1986) using
the finite element method (Hubner 1974, Zienkiewicz 1971). The calculation has been
performed for slits, for cylindrical tubes having a circular cross-section, and also for
cylindrical tubes having triangular, rectangular and hexagonal cross-sections. With the
notation of Craggs and Hildebrandt, the effective density
ρ
can be written
ρ
=
ρ
e
+
e
/j ω
(4.50)
where
ρ
e
and
e
are real parameters. Let ¯
r
be the hydraulic radius, given by
cross - sectional area
cross - sectional perimeter
¯
r
=
2
×
(4.51)
e
¯
r
2
/η
and
ρ
e
/ρ
o
are calculated by Craggs and Hildebrandt as func-
tions of the variable
β
given by
The quantities
(ρ
o
ω/η)
1
/
2
β
=
¯
r
(4.52)
In the case of a circular cross-section, ¯
r
is the radius
R
,and
β
is equal to
s
.Inthe
case of a slit, ¯
r
is equal to the width 2
a
,and
β
is equal to 2
s
.
The quantities
e
¯
r
2
/η
and
ρ
e
/ρ
o
have been fitted by Craggs and Hildebrandt (1986)
to polynomials in the case of slits, circular, triangular and square cross-sections,
e
¯
r
2
/η
=
a
1
+
a
2
β
+
a
3
β
2
+
a
4
β
3
(4.53)
b
3
β
2
b
4
β
3
ρ
e
/ρ
o
=
b
1
+
b
2
β
+
+
(4.54)
This representation is valid in the range 0
<β<
10. The coefficients
a
i
and
b
i
are given by Craggs and Hildebrandt (1986). The quantity
F(ω)
in Equation (4.45) is
equal to
F(ω)
=
ρ
o
/(ρ
e
+
e
/j ω)
(4.55)
and the bulk modulus can be evaluated with the use of Equation (4.46), where
F(B
2
ω)
is given by
=
1
(b
1
+
F(B
2
ω)
b
3
B
2
β
2
b
4
B
3
β
3
)
b
2
Bβ
+
+
a
4
B
3
β
3
)
1
jωB
2
η
ρ
o
¯
r
2
(a
1
a
3
B
2
β
2
+
+
a
2
Bβ
+
+
(4.56)
A more general method for predicting the effective density and bulk modulus is given
in Section 4.7.
