Environmental Engineering Reference
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and surface loading. The balance of forces and moments leads to the following
equations[ 1 ]:
r
gV
d
+
t S
d
=
r
r V
d
,
r
g RV
×
d
+
t RS
×
d
=
r
r R V
×
d
(1 )
V
V
V
V
V
V
r , includes
In eqn (1), the current position
rr t
(,)
of a material point originally at
the elastic displacement and any rigid body motion while R denotes the distance
from the point with respect to which moments are taken. (An overhead arrow
will denote geometric vectors while matrices will be in bold.) Usually the volume
loading term is due to gravity. Then as regards surface loading, when V is part of a
solid body there will be two terms: the aerodynamic loading defi ned on the part of
∂ V in contact with air, and the internal loading defi ned on the rest of ∂ V in contact
with the remaining of the solid body. Internal surface loading is directly connected
to the stress tensor s :
= with n denoting the outward unit normal, which
together with Green's theorem allows transforming the surface integrals into vol-
ume integrals and thus deriving the dynamic equations in differential form. Elastic
models are next introduced which relate stresses with strains so that at the end the
equations are formulated with respect to the displacements and rotations contained
in the defi nition of r (for further reading the reader can consult [14] out of the
long list of modern textbooks on structural mechanics).
In structural analysis the differential form of the equations is seldom used.
Instead the equations are reformulated in weak form based on the principle of
virtual work. Weak formulations are the starting point of fi nite element discrete
models of the Galerkin type ([2] and Section 3.2). An alternative and defi nitely
more powerful way to formulate dynamic equations is to use Hamilton's principle
[ 3 ]. Let T and U denote the kinetic and strain energies, respectively, defi ned by a
set of displacements and rotations collectively denoted as u ( t ) = ( u 1 , u 2 ,…) T .
Assuming the presence of non-conservative loads F i connected to u i then
T
t
s
n
d(T )
∂−
U
∂−
(T )
U
+
+ =
F
0,
i
(2)
i
dt
u
u
i
i
This is also known as Lagrange equations. Each equation corresponds to balance
of either forces or moments depending on whether the associated u i is a displace-
ment or a rotation. Among the various advantages Hamilton's principle has, of par-
ticular importance is that the fi nal form of the dynamic equations is easily obtained
using symbolic mathematical software [ 4 , 5 ].
3 Beam theory and FEM approximations
3.1 Basic assumptions and equation derivation
Beam theory applies when one of the dimensions of the structure is much larger than
the others [6, 7]. Using asymptotic analysis, it is possible to eliminate the depen-
dence on the two shortest dimensions by appropriate averaging of the distributed
 
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