106
SINGLE DEGREE OF FREEDOM STRUCTURES
for the deck. If this jackup platform were modified so that the deck was instead
hinged instead of clamped to each leg, then an approximate mode shape would
be
7TX
= 1 - cos —
(5.14)
which satisfies the géométrie constraints of full fixity at the sea floor: ^(0) =
^'(0) = 0; and of nonzero displacement and slope at the deck level: V>(f) = 1
and ÿ'(t) = ?r/(2£).
For the Rayleigh Method, the actual magnitudes of the nonzero end slopes
and deflections are unimportant, provided that the chosen mode shape V>(x)
satisfies the beam’s géométrie constraints. Then this method leads to an upper
bound value for ujq. A straightforward proof of this upper bound property was
provided by Den Hartog (1947).
The steps of the Rayleigh Method used to calculate cvq for beam-type structures are summarized as follows:
1. Choose a simple mode shape ip(x) that satisfies the géométrie boundary
conditions of deflection and slope. Express the latéral displacement of the beam
at location x along its length as
v = v(x, t) = vq V'(æ) sin LüQt
(5.15)
where tp is an arbitrary constant.
2. From the latter équation, calculate V = U + Vg, the total potential energy
for the beam. Here U is the elastic strain energy for bending of the beams (legs
and/or cross braces) and Vg is the loss in potential energy due to a downward
displacement of a mass, as for instance, the decrease in the gravitational energy
of a deck mass
due to the latéral displacement of its vertical supporting legs.
3. Based on équation (5.15), calculate the total kinetic energy K: that for
the beam plus that for its end mass, if any.
4. Calculate up by equating the expressions for the maximum kinetic energy
(when V = 0) to the maximum potential energy (when K = 0), or
A™ = Vmax
(5.16)
Before the Rayleigh method is illustrated, consider some general forms of
System energy. From elementary beam theory, the strain energy for bending of
a beam element of length dx at position x is
El fd2v\2
.....
dU -
I ^-5
dx
(5-1 ’)
2 \dx2)
where v = v(x,t) is given by équation (5.15). The total potential energy of
. identical beams each of length f, and each with the identical displacement
v(x,t) is thus
U=^ f
dx
(5.18)
2 Jo
\dx2 J
SINGLE DEGREE OF FREEDOM STRUCTURES
for the deck. If this jackup platform were modified so that the deck was instead
hinged instead of clamped to each leg, then an approximate mode shape would
be
7TX
= 1 - cos —
(5.14)
which satisfies the géométrie constraints of full fixity at the sea floor: ^(0) =
^'(0) = 0; and of nonzero displacement and slope at the deck level: V>(f) = 1
and ÿ'(t) = ?r/(2£).
For the Rayleigh Method, the actual magnitudes of the nonzero end slopes
and deflections are unimportant, provided that the chosen mode shape V>(x)
satisfies the beam’s géométrie constraints. Then this method leads to an upper
bound value for ujq. A straightforward proof of this upper bound property was
provided by Den Hartog (1947).
The steps of the Rayleigh Method used to calculate cvq for beam-type structures are summarized as follows:
1. Choose a simple mode shape ip(x) that satisfies the géométrie boundary
conditions of deflection and slope. Express the latéral displacement of the beam
at location x along its length as
v = v(x, t) = vq V'(æ) sin LüQt
(5.15)
where tp is an arbitrary constant.
2. From the latter équation, calculate V = U + Vg, the total potential energy
for the beam. Here U is the elastic strain energy for bending of the beams (legs
and/or cross braces) and Vg is the loss in potential energy due to a downward
displacement of a mass, as for instance, the decrease in the gravitational energy
of a deck mass
due to the latéral displacement of its vertical supporting legs.
3. Based on équation (5.15), calculate the total kinetic energy K: that for
the beam plus that for its end mass, if any.
4. Calculate up by equating the expressions for the maximum kinetic energy
(when V = 0) to the maximum potential energy (when K = 0), or
A™ = Vmax
(5.16)
Before the Rayleigh method is illustrated, consider some general forms of
System energy. From elementary beam theory, the strain energy for bending of
a beam element of length dx at position x is
El fd2v\2
.....
dU -
I ^-5
dx
(5-1 ’)
2 \dx2)
where v = v(x,t) is given by équation (5.15). The total potential energy of
. identical beams each of length f, and each with the identical displacement
v(x,t) is thus
U=^ f
dx
(5.18)
2 Jo
\dx2 J