Environmental Engineering Reference
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z
y
Undeformed
geo
metry
Deformed geometry
dy
Deformed elastic axis
w+dw
r
w
A
u+du
u
x
(a)
M
z
+dM
z
z
δP
F
z
+dF
z
y
M
y
+dM
y
M
x
F
y
+dF
y
F
x
r
a
F
y
(b)
dr
(a)
A
F
x
+dF
x
F
z
M
y
M
x
+dM
x
M
z
x
(b)
Figure 2: Loads and displacements of a beam element.
The above system is completed with appropriate boundary and initial conditions.
Boundary conditions at the two ends of the beam will either specify the load
(Neumann or static condition) or constrain the corresponding displacement or rotation
(Dirichlet or kinematic condition). Unconstrained or free ends will have zero loading.
3.2 Principle of virtual work and FE approximations
Considering the dynamic equations in the form
F
(
u
) = 0, then for any virtual dis-
placement
d
u
the work done by the loads
F
(
u
) must be zero. Work is a projection
operation defi ned by the inner product of integrable functions: (
f
,
g
)
f
(
x
)
g
(
x
) d
x
with the integral defi ned on the domain of defi nition which in the present case is
the length of the beam
L
. So,
≡
∫
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