RESPONSE OF NONLINEAR STRUCTURES
123
limits for maximum displacement, horizontal shear load, and overturning moment are thus
^max — 8.82 in. and 5.88 in
Zmax — 6.02 x 105 1b and 5.56xl05 1b
Mmax = 119 x 109 in.-lb and 1.74 x 109 in.-lb
It is noted that these numerical results are quite sensitive to small changes in
the structural period. That is, since 7q lies in a trough, a 10 percent increase
or decrease in To would increase each of these calculated responses by about 50
percent. To avoid the many peaks and troughs that typify the Sv vs. To curves
calculated directly from earthquake data, such curves are often smoothed in a
statistical sense before being used for design purposes.
5.5
RESPONSE CHARACTERISTICS OF NONLINEAR
STRUCTURES
It was shown in Chapter 2 that offshore cable-stayed installations such as spread
moored ships, floating platforms, and compilant towers ail hâve nonlinear restraint forces. Under some spécial conditions these structures may be subjected
to disturbing forces that are approximately harmonie. For instance, Wilson
(1951) observed erratic surge oscillations of a ship moored in a harbor. The
ship’s excitation force was caused by harmonie harbor waves at a frequency
close to the natural frequency of oscillation of the water within the harbor
basin. The oscillation of the harbor waters was due to the waves in the adjacent
sea.
For a cable-stayed offshore structure, neither the wave-induced exciting force
amplitude p0 nor the excitation frequency eu ever remains constant. However, to
gain some physical insight about the response of the inherently nonlinear cablestayed structures, the following undamped structural model is investigated in
which po and uj are constant.
mi + k}v + k3v3 = po cos ait
(5.81)
Responses to Harmonie Excitation
Solutions for this nonlinear structural model are now derived using the perturbation technique discussed in Section 5.2. Cast équation (5.81) as
v + u)qV + ek3v3 = ep0 cos ut
(5.82)
where e = 1/m is arbitrarily small. A steady-state solution to équation (5.82)
is sought where this solution has the same frequency of oscillation as the wave
excitation frequency eu. It is required that both this solution v(t) and the frequency w not differ greatly from their corresponding values values fo(t) and iUq
of the linear System (k3 = 0). That is
v = v0(t) +evi(t)
(5.83)
123
limits for maximum displacement, horizontal shear load, and overturning moment are thus
^max — 8.82 in. and 5.88 in
Zmax — 6.02 x 105 1b and 5.56xl05 1b
Mmax = 119 x 109 in.-lb and 1.74 x 109 in.-lb
It is noted that these numerical results are quite sensitive to small changes in
the structural period. That is, since 7q lies in a trough, a 10 percent increase
or decrease in To would increase each of these calculated responses by about 50
percent. To avoid the many peaks and troughs that typify the Sv vs. To curves
calculated directly from earthquake data, such curves are often smoothed in a
statistical sense before being used for design purposes.
5.5
RESPONSE CHARACTERISTICS OF NONLINEAR
STRUCTURES
It was shown in Chapter 2 that offshore cable-stayed installations such as spread
moored ships, floating platforms, and compilant towers ail hâve nonlinear restraint forces. Under some spécial conditions these structures may be subjected
to disturbing forces that are approximately harmonie. For instance, Wilson
(1951) observed erratic surge oscillations of a ship moored in a harbor. The
ship’s excitation force was caused by harmonie harbor waves at a frequency
close to the natural frequency of oscillation of the water within the harbor
basin. The oscillation of the harbor waters was due to the waves in the adjacent
sea.
For a cable-stayed offshore structure, neither the wave-induced exciting force
amplitude p0 nor the excitation frequency eu ever remains constant. However, to
gain some physical insight about the response of the inherently nonlinear cablestayed structures, the following undamped structural model is investigated in
which po and uj are constant.
mi + k}v + k3v3 = po cos ait
(5.81)
Responses to Harmonie Excitation
Solutions for this nonlinear structural model are now derived using the perturbation technique discussed in Section 5.2. Cast équation (5.81) as
v + u)qV + ek3v3 = ep0 cos ut
(5.82)
where e = 1/m is arbitrarily small. A steady-state solution to équation (5.82)
is sought where this solution has the same frequency of oscillation as the wave
excitation frequency eu. It is required that both this solution v(t) and the frequency w not differ greatly from their corresponding values values fo(t) and iUq
of the linear System (k3 = 0). That is
v = v0(t) +evi(t)
(5.83)