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In this development of the consistent mass matrix, only the inertia associated with the
transverse deflection
of the beam centroidal axis is considered. Normally, this approx-
imation leads to adequate precision for slender beams. For deeper beams, neglecting the
inertia due to the rotation of the beam cross-section may cause some error. To take the
rotary inertia of the cross-section into account, begin with the beam segment in Chapter 1,
Fig. 1.15. If
u
0
in Chapter 1, Eq. (1.98) is ignored, the displacement of the area
dA
in the
x
direction is
u
w
z
¨
=
z
θ
,
where
θ
=−
∂w/∂
x
.
Then the acceleration of the area is
u
=
θ.
z
¨
I
¨
The integral of the moment of the force
−
γ
u
(
=−
γ
θ)
over the cross section is
−
γ
θ
,
=
A
z
2
dA,
where
A
is the area of the cross-section. If
r
y
is the radius of gyration
with
I
¨
and
ρ
=
γ
A
is the mass per unit length, the inertia moment can be written as
−
ρ
r
y
θ
,
since
r
y
=
I
/
A
.
The inertia moment can be taken as a d'Alembert distributed moment
applied on the beam element, so that an external virtual work term
x
δθρ
¨
dx
should be
added in the expression of the principle of virtual work of Eq. (10.7). Use the trial function
θ
=−
∂w/∂
r
y
θ
x
=−
∂
N
/∂
x
v
to obtain the consistent matrix due to the rotary inertia of the
cross-section
36
−
3
−
36
−
3
=
ρ
r
y
30
−
3
4
2
3
−
2
∂
∂
r
y
∂
∂
m
r
=
x
N
T
ρ
x
N
dx
(10.10c)
−
36
3
36
3
−
3
−
2
3
4
2
Often this is expressed with
.
Since rotary inertia does not affect axial motion,
m
r
expanded to include axial motion
would appear as
ρ
r
y
=
γ
I
0
0
0
0
0
0
0 6
−
3
0
−
36
−
3
r
y
30
m
r
=
ρ
0
−
3
4
2
03
−
2
(10.10d)
0
0
0
0
0
0
0
−
36
3
0 63
2
2
0
−
3
−
03
4
Although the consistent mass matrix, which is based on “static” shape functions, is the
most frequently employed mass matrix, more accurate mass matrices can be computed by
using shape functions that more closely represent the dynamic response. For example, use
of the exact dynamic shape functions in
w
=
Nv
leads to an exact mass matrix. This topic
is considered in some detail in Section 10.3.1.
After the consistent mass or other mass matrices for the elements are established, the
global mass matrix can be assembled in the same fashion as the global stiffness matrix
(Chapter 5).
EXAMPLE 10.1 Consistent Mass Matrices for a Frame
Find the element and global consistent mass matrices of the frame shown in Fig. 10.2. This
same frame has been treated earlier in Chapter 5, Example 5.5. The element properties are
E
=
200 GN/m
2
,
γ
=
7800 kg/m
3
, and
2356 cm
4
10
−
5
m
4
,
Elements 1 and 2:
I
=
=
2
.
356
×
32 cm
2
10
−
4
m
2
A
=
=
32
×
(1)
5245 cm
4
10
−
5
m
4
,
Element 3:
I
=
=
5
.
245
×
66 cm
2
10
−
4
m
2
=
=
×
A
66
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