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10.3.1
Dynamic Stiffness Matrices and Exact Mass Matrices
The stiffness matrix is normally obtained by substituting the shape functions
u
=
Nv
in
the internal virtual work relationship (Eq. 10.2)
u
T
u
D
T
ED
u
u
dV
V
δ
giving
k
i
N
T
u
D
T
ED
u
N
dV
=
(10.40)
V
The potential accuracy of a structural analysis depends in part on the potential accuracy
of the element stiffness matrix
k
i
, which depends on the shape function
N
. Normally, the
more representative
N
is of the exact solution, the better the
k
i
. For example, if
N
is the
exact solution to the local (differential) governing equations for an element, the element
stiffness matrix
k
i
will be exact. For a beam, the usual stiffness matrix [Chapter 4, Eq.
(4.12)] is exact only for static responses since it is formed using
N
composed of static
response polynomials (Eq. 10.8), which are the exact solution of the static differential
equations for a beam.
Beams and bars are structural members for which exact solutions to the governing
differential equations are readily available. If dynamic effects are included, the beam
equations [Chapter 1, Eqs. (1.133)] become
∂w
∂
V
k
s
GA
x
=−
θ
+
∂θ
∂
M
EI
+
M
T
EI
x
=
(10.41)
∂
V
w
+
ρ
∂
2
w
x
=
k
t
2
−
p
z
(
x, t
)
∂
∂
∂
M
∂
r
y
∂
2
θ
k
∗
−
x
=
V
+
(
P
)θ
+
ρ
t
2
−
c
(
x, t
)
∂
where, in addition to the inclusion of inertia, several other terms such as the effects of
elastic foundations, applied moments, and thermal loading are displayed. The inertia
force per unit length of beam in the
x
direction is
2
t
2
−
ρ
w
=−
ρ(∂
¨
w/∂
)
,
while the i
ne
rt
ia
r
y
(∂
2
t
2
mom
en
t is
,
which is derived in Section 10.1.1. These, together with
p
z
, c,
and
M
T
act on the beam as the applied loading. In these equations,
−
ρ
θ/∂
)
w
,
θ
,V,
and
M
are
the deflection, rotation, shear force, and moment at a cross-section. Also,
k
s
=
Shear form factor
A
=
Area of the cross-section
G
=
Modulus of elasticity in shear
E
=
Young's modulus of elasticity
Moment of inertia taken about the neutral axis
M
T
=
A
E
I
=
α
TzdA
=
Magnitude of distributed thermal moment
T
=
Temperature change
α
=
Coefficient of thermal expansion
Elastic foundation modulus (force/length
2
)
k
=
ρ
=
Mass density (mass per unit length)
p
z
=
Distributed load (force/length)
k
∗
=
Rotary spring constant (force-length/radian)
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