STRUCTURAL RESPONSE STATISTICS
239
That is, the displacements Ç, behave as
« ^sinu.t
(9.58)
However, if any value of
is négative or complex, the System will be dynamically unstable. For instance, if -vt — —oP where a is real and positive, then
wi = ±ja. With uq = —ja, équation (9.58) becomes
G~êieat
(9.59)
which is unbounded as time increases. If u>2 is complex, one of the roots produces
this same unboundedness. Computer codes that extract roots of a déterminant,
such as subroutine Eigenvalues of Mathematica® (1999), are generally available
to the engineer, so it is a relatively straightforward task to check the dynamic
stability of a structure.
With respect to a gravity platform on an elastic soil foundation, dynamic
instability leading to toppling would occur for critical combinations of its mass,
its center of buoyancy, its center of gravity, and its foundation stiffness. For
instance, for sufficiently high values of platform mass and hc, it is visualized that
a moderately weak soil foundation could offer an insufficient restoring moment to
resist both the angular momentum of the structure and the overturning moment
due to its deadweight. Then the structure would topple.
In conclusion, it is noted that if the criterion for dynamic stability is observed
(that ail values of
æ-g reaj anc] positive), then the static criteria discussed
in Chapter 1, applied to gravity platforms, will also be satisfied. It is now well
recognized that a dynamic stability analysis includes the results obtained from
a static stability analysis; but the converse is not true.
9.3 STRUCTURAL RESPONSE STATISTICS
FOR WAVE LOADING
As discussed in Chapters 6 and 7, the wave data available to the analyst and
designer of offshore structures are most often in the form of a surface wave
height spectrum, S^u). In Chapter 6 a method was presented for representing
this spectrum as harmonie waves forms. These forms, when converted to structural forces through suitable transfer functions G(u>), are then used to calculate
the time history of structural response. An alternative approach was discussed
in Chapter 7 where the spectral density of the response and its variance were
calculated directly from S^(w) and G(w) for the single degree of freedom case.
This latter approach is now employed for the case of a linear structure with
N degrees of freedom. Summarized in the following ten steps are the critical
assumptions and the methodology leading to an analytical form for the spectral
density defined as S(Çk, w) and the variance
1. Set up the mathematical model and dérivé the équations of motion for
the structure in the form of équation (9.1). Identify the coefficient matrices M
and K.
239
That is, the displacements Ç, behave as
« ^sinu.t
(9.58)
However, if any value of
is négative or complex, the System will be dynamically unstable. For instance, if -vt — —oP where a is real and positive, then
wi = ±ja. With uq = —ja, équation (9.58) becomes
G~êieat
(9.59)
which is unbounded as time increases. If u>2 is complex, one of the roots produces
this same unboundedness. Computer codes that extract roots of a déterminant,
such as subroutine Eigenvalues of Mathematica® (1999), are generally available
to the engineer, so it is a relatively straightforward task to check the dynamic
stability of a structure.
With respect to a gravity platform on an elastic soil foundation, dynamic
instability leading to toppling would occur for critical combinations of its mass,
its center of buoyancy, its center of gravity, and its foundation stiffness. For
instance, for sufficiently high values of platform mass and hc, it is visualized that
a moderately weak soil foundation could offer an insufficient restoring moment to
resist both the angular momentum of the structure and the overturning moment
due to its deadweight. Then the structure would topple.
In conclusion, it is noted that if the criterion for dynamic stability is observed
(that ail values of
æ-g reaj anc] positive), then the static criteria discussed
in Chapter 1, applied to gravity platforms, will also be satisfied. It is now well
recognized that a dynamic stability analysis includes the results obtained from
a static stability analysis; but the converse is not true.
9.3 STRUCTURAL RESPONSE STATISTICS
FOR WAVE LOADING
As discussed in Chapters 6 and 7, the wave data available to the analyst and
designer of offshore structures are most often in the form of a surface wave
height spectrum, S^u). In Chapter 6 a method was presented for representing
this spectrum as harmonie waves forms. These forms, when converted to structural forces through suitable transfer functions G(u>), are then used to calculate
the time history of structural response. An alternative approach was discussed
in Chapter 7 where the spectral density of the response and its variance were
calculated directly from S^(w) and G(w) for the single degree of freedom case.
This latter approach is now employed for the case of a linear structure with
N degrees of freedom. Summarized in the following ten steps are the critical
assumptions and the methodology leading to an analytical form for the spectral
density defined as S(Çk, w) and the variance
1. Set up the mathematical model and dérivé the équations of motion for
the structure in the form of équation (9.1). Identify the coefficient matrices M
and K.