Page 331 - Marine Structural Design

Chapter15 offshore Structures Under Earthquake Loads 307

Geometrical and ultimate strength requirements for primary members and their
connections as given in API are satisfied. These requirements concern number of legs,
jacket foundation system, diagonal bracing configuration in vertical frames, horizontal
members, slenderness and diameter/thickness ratio of diagonal bracing, and tubular joint
capacities.

15.3 Equations and Motion

15.3.1 Equation of Motion
The equations of motion for a nonlinear offshore structure subjected to a earthquake loading
can be expressed as
[M]{diij+ [C]{dii}+ [K']{dU} = -[M]{diig}+ (&Ye} (15.1)
where {dU}, {dU} and {dd} are the increments of nodal displacement, velocity and
acceleration relative to the ground respectively. [MI is the structural mass matrix, while [C] is
the structural damping matrix. [KT] denotes the structural tangent stiffness matrix. (a} are
the increments of the hydrodynamic load. The ground acceleration vector {oz} is formed as
an assembly of three-dimensional ground motions.
We shall here assume that at the time of the earthquake there is no wind, wave or current
loading on the structure. According to the Morison equation (Sarpkaya and Isaacson, 1981),
the hydrodynamic load per unit length along a tubular beam member can be evaluated as

where p is the mass density of the surrounding water, D is the beam diameter, CA is an added
mass coefficient, CD the drag coefficient, A=xD2/4, and {ti"} denotes the normal components
of the absolute velocity vector. The absolute velocity vector is
{%J={4+bg) (15.3)

Using a standard lumping technique, Eq. (15.1) can be rewritten as
([MI+ [M, ){do}+ [c]{&}+ [K']{~u} = -([MI+ [M, D{diig}+ @F,} (15.4)

where [Ma] is an added mass matrix containing the added mass terms of Eq. (15.2). The
increments of drag force terms from time (t) to (t+dt) are evaluated as
(@D 1 = c [T+dt I' VD - c [T 1' {fD >o (15.5)
where denotes summation along all members in the water, while {fD} are results of
integration of the drag force terms of Eq. (15.2) along the member. [TJ is the transformation
matrix. the equations of motion Eq. (1 5.4) are solved by the Newmark-P method (Newmark,
1959).