192
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
Use this équation and employ numerical intégration, with reasonable limits for
the intégral, to compute the area A under the curve of Figure 7.5. Then verify
the resuit discussed in Chapter 6 that Hs = 4A1//2 = 15 m.
7.6 In Example Problem 74, three rms displacement responses av for the
jackup rig were computed, each based on an idealized model. band-limited white
noise, white noise with a cut-off frequency, and idéal white noise. Using the saine
parameters as for this example problem, solve équation (7.63) by numerical
intégration to obtain crv for each of these three idealized models. Compare your
results to the corresponding results in the text that were derived from closed
form solutions to the intégral. Explain possible différences in the results. Also
explain why these idealized models give results for crv that are about six times
that obtained from numerical intégration of équation (7.61).
7.7 Reconsider the jackup rig described in Example Problem 54, with the
design parameters given in Table 5.2. The statistical responses to this saine
structure were discussed in Example Problem 7.3. It is proposed to add more
equipment to the deck of this jackup rig so that the deck weight m^g would
increase from 1.02 x 10' 1b to 5 x 107 1b.
(a) What percentage of the Euler buckling load is this new deck load? Use
équation (5.36) to answer this and to explain whether the new deck weight will
increase the chance of structural buckling.
(b) With the new deck weight, the other parameters of Table 5.2, and
équation (5.36), calculate the following quantifies: the équivalent bending stiffness ki, the équivalent mass m, the équivalent damping constant c, based on
C = Ci/\/4kim = 0.05, and the undamped structural frequency cuo(c) Based on the same parameters of Example Problem 7.3, except for the
modified values of ki,m, and Cj, use équation (7.61) and numerical intégration
to compute the variance tr2. Would you expect av to be higher or lower when
compared to the original rig with the lower deck mass? Explain.
(d) Compute the response av of the modified structure to idéal white noise.
Compare this resuit to that obtained in part (c). What assumptions account
for the différences in these two results?
7.8 Consider the concrète monotower shown in Figures 2.2 and 5.2. This
structure is idealized as a rigid body consisting of a rectangular box caisson and
a uniform leg. The structure rotâtes in the plane with angle Ô about the base
point 0. The foundation is assumed to be linearly elastic with a stiffness kg and
damping cg, modeled respectively by équations (2.78) and (2.79), in which the
frequency u is identified as the fundamental frequency in free vibration, or wq.
Let Jq dénoté the Virtual mass moment of the structure’s inertia about point 0,
and let Af (t) dénoté the moment on the structure induced by wave action. The
governing équation (2.6) for rotational motion thus has the following form:
JoS + Cg6 + kgO = M(t)
I a | B> comparing the symbols of the above équation to those for translational
motion v, or équation (7.27), deduce for rotational motion the harmonie response
function H(w) in a form similar to équation (7.28).
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
Use this équation and employ numerical intégration, with reasonable limits for
the intégral, to compute the area A under the curve of Figure 7.5. Then verify
the resuit discussed in Chapter 6 that Hs = 4A1//2 = 15 m.
7.6 In Example Problem 74, three rms displacement responses av for the
jackup rig were computed, each based on an idealized model. band-limited white
noise, white noise with a cut-off frequency, and idéal white noise. Using the saine
parameters as for this example problem, solve équation (7.63) by numerical
intégration to obtain crv for each of these three idealized models. Compare your
results to the corresponding results in the text that were derived from closed
form solutions to the intégral. Explain possible différences in the results. Also
explain why these idealized models give results for crv that are about six times
that obtained from numerical intégration of équation (7.61).
7.7 Reconsider the jackup rig described in Example Problem 54, with the
design parameters given in Table 5.2. The statistical responses to this saine
structure were discussed in Example Problem 7.3. It is proposed to add more
equipment to the deck of this jackup rig so that the deck weight m^g would
increase from 1.02 x 10' 1b to 5 x 107 1b.
(a) What percentage of the Euler buckling load is this new deck load? Use
équation (5.36) to answer this and to explain whether the new deck weight will
increase the chance of structural buckling.
(b) With the new deck weight, the other parameters of Table 5.2, and
équation (5.36), calculate the following quantifies: the équivalent bending stiffness ki, the équivalent mass m, the équivalent damping constant c, based on
C = Ci/\/4kim = 0.05, and the undamped structural frequency cuo(c) Based on the same parameters of Example Problem 7.3, except for the
modified values of ki,m, and Cj, use équation (7.61) and numerical intégration
to compute the variance tr2. Would you expect av to be higher or lower when
compared to the original rig with the lower deck mass? Explain.
(d) Compute the response av of the modified structure to idéal white noise.
Compare this resuit to that obtained in part (c). What assumptions account
for the différences in these two results?
7.8 Consider the concrète monotower shown in Figures 2.2 and 5.2. This
structure is idealized as a rigid body consisting of a rectangular box caisson and
a uniform leg. The structure rotâtes in the plane with angle Ô about the base
point 0. The foundation is assumed to be linearly elastic with a stiffness kg and
damping cg, modeled respectively by équations (2.78) and (2.79), in which the
frequency u is identified as the fundamental frequency in free vibration, or wq.
Let Jq dénoté the Virtual mass moment of the structure’s inertia about point 0,
and let Af (t) dénoté the moment on the structure induced by wave action. The
governing équation (2.6) for rotational motion thus has the following form:
JoS + Cg6 + kgO = M(t)
I a | B> comparing the symbols of the above équation to those for translational
motion v, or équation (7.27), deduce for rotational motion the harmonie response
function H(w) in a form similar to équation (7.28).