Spectral Analysis of Offshore Structures Under Wave and Earthquake Loadings

Abstract

This chapter presents spectral analysis of offshore structures under wave and earthquake actions. It contains six sections. The first section explains problem description and gives some general information. In the second section, formulation of dynamic analysis of offshore structures in the frequency domain, transfer functions of wave, and earthquake forces are presented. The third section presents response transfer functions of wave and earthquake forces individually. Then, having presented hydrodynamic and inertia forces produced by earthquakes, response transfer functions of their combined effect are formulated. The fourth section explains calculation of response spectra of offshore structures under stochastic wave and earthquake forces. Then, the stochastic ground motion under earthquakes and its spectral representation are presented. It follows the calculation of transfer functions and response spectra of offshore structures under nonuniform earthquake ground motions. In section five, calculation of response statistical quantities is presented and. In section six, numerical demonstration of an example offshore structure is given.

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4.1 Introduction

Structural response analyses under dynamic loading are well documented in general in many text books, see e.g. [1–10], especially for offshore applications, see e.g. [11–14]. The response analysis can be carried out either in the time domain or in the frequency domain as depending on the analysis and the loading type [15]. For a nonlinear analysis, the time domain approach is traditionally used [16] and, for the linear analysis, both time domain and frequency domain analyses may be used. For the spectral analysis, a frequency domain approach is more adequate and therefore it is used generally. Offshore structures are continuously subjected to random water waves, and therefore, a spectral analysis method is essentially used to determine response statistical quantities that needed in the calculation of fatigue damages and averages of extreme-value responses [17–20]. In order to apply a spectral analysis procedure to offshore jacket structures, the calculation of wave forces on structural members and their linearization techniques has been presented in details in Chap. 3. These forces have been formulated in terms of the random water elevation so that the corresponding responses will also depend on this random variable. Since the stochastic description of the random water elevation is determined as presented in Sect. 3.4, stochastic structural responses can be calculated in terms of the spectral values of the random water elevation through a spectral analysis procedure that will be presented in this chapter.

One other important loading category of offshore structures is introduced by seismic ground motions [21–27] which are commonly known as earthquakes that may cause catastrophic consequences in both economical and social terms. In order to avoid these unpleasant consequences of earthquakes, structures to be built in seismically active offshore areas must be designed against probable structural failures. Earthquakes occurring in ocean environments cause also tsunamis [23], which are huge long water waves traveling long distances without dissipating sufficient energy to become harmless. Since tsunamis are water waves created by earthquakes their impacts on structural systems can be considered in the wave loading category. In this chapter, the earthquake loading due to random ground motion and related structural responses will be presented. Earthquakes are complex random phenomena and their sizes are used in practice to indicate their significances. The size of an earthquake can be measured in terms of magnitude and intensity scales [26, 27]. The magnitude of an earthquake is the amount of energy that released from its source, and the intensity is a measure of an earthquake hazard at a specific location, which differs from location to location. Earthquakes are recorded at different locations by seismometers as being accelerations of the ground motion in time domain, which are termed as time histories [28]. Time histories at different sites may differ considerably from each other in duration, frequency content, and amplitude. The average duration of strong-motion acceleration is about 45 s and, with increasing intensity, the duration decreases, and becomes on the average of 20–25 s [29]. The spatial variation of seismic ground motions has an important effect on the response of long structures [30]. Since they extend over long distances parallel to the ground, their supports undergo different motions during an earthquake. The spatial variation of the ground motions is described by a deterministic time delay function, which is required for the waveforms to reach faraway supports of the structures, and a stationary Gaussian process representing the random ground motion. The time varying property of the ground motion makes the process non-stationary and, in the spectral form, it is represented by a coherency function [31–37]. The detail of this subject will be presented in Sect. 4.5.

4.2 Dynamic Analysis of Structures in the Frequency Domain, the Transfer Function Approach

The calculation of response displacements of a linear structural system in the frequency domain is outlined in Sect. 1.7.2. The formulation of the global displacements in the frequency domain has been given by Eq. (1.229), which is rewritten below, in general for the convenience as,

$$ \{ D(\omega )\} = [H(\omega )]_{{\user1{DP}}} \{ P(\omega )\} \,\,\,\, \to \,\,\,\,[H(\omega )]_{{\user1{DP}}} = \left( {[K] + i\omega [C] - \omega^{2} [M]} \right)^{ - 1} $$

(4.1)

where {P(ω)} is the system load vector and [H(ω)]_DP is defined as the structural transfer function matrix between the displacement and load vectors. For each frequency, Eq. (4.1) must be solved to find the frequency content of displacements. Since this procedure requires multiple matrix inversion, the solution of Eq. (4.1) is practically time-consuming, and therefore, the modal analysis procedure is usually applied, in practice, to obtain the displacements. By using the modified modal analysis, the displacements vector is written from Eq. (1.265) as

$$ \{ D(\omega )\} = [K]^{ - 1} \{ P(\omega )\} + \left( {\sum\limits_{j = 1}^{q} {\alpha_{j} (\omega )\,f_{j} (\omega )\{ \phi \}_{j} } } \right) $$

(4.2)

in which the first term is the contribution of the quasi-static response and the second term in brackets is the contribution of the dynamic response. The vector {ϕ} _j is the jth^. natural mode vector, α _j (ω) and f _j (ω) are respectively the eigenmode participation factor and the generalized force, which are defined by Eq. (1.264a) and the generalized force is

$$ f_{j} (\omega ) = \{ \phi \}_{j}^{T} \{ P(\omega )\} $$

(4.3)

Once the load vector $\{ P(\omega )\}$ and structural eigenmodes information are known, the response displacements in the global coordinates can be easily calculated from Eq. (4.2). The stiffness matrix [K] needs to be inverted once for all frequency variations, and therefore, the calculation will be relatively very fast. In the case of wave loading, the structural load vector $\{ P(\omega )\}$ is stated in terms of the random water elevation η(ω) and a transfer function vector $\{ H(\omega )\}_{P\eta }$ in the frequency domain as explained in Sect. 3.7.3 for the consistent member forces. For the global system, it can be written as,

$$ \{ P(\omega )\}_{{\text{wave}}} = \{ H(\omega )\}_{P\eta } \eta (\omega ) $$

(4.4a)

where the subscript (wave) denotes loading due to waves. $\{ H(\omega )\}_{P\eta }$ denotes the system transfer function vector between the wave load vector $\{ P(\omega )\}$ and the random water elevation η(ω), which is obtained from the assembly process of member transfer functions. Having used Eq. (4.4a) in Eqs. (4.2) and (4.3), the global displacements can be stated as written by,

$$ \{ D(\omega )\}_{{\text{wave}}} = \{ H(\omega )\}_{D\eta } \eta (\omega )\,\,\,\, \to \left\{ \begin{gathered} \{ H(\omega )\}_{D\eta } = [K]^{ - 1} \{ H(\omega )\}_{P\eta } + \sum\limits_{j = 1}^{q} {\left\{ {h(\omega )} \right\}_{j\eta } } \hfill \\ \left\{ {h(\omega )} \right\}_{j\eta } = \alpha_{j} (\omega )\,\{ \phi \}_{j}^{T} \{ H(\omega )\}_{P\eta } \{ \phi \}_{j} \hfill \\ \end{gathered} \right. $$

(4.4b)

in which $\{ H(\omega )\}_{D\eta }$ is the response transfer function vector for the global displacements under the wave loading. When the structure is subjected to the ground motion with a random base acceleration, $\ddot{u}_{g}$, like in the case of an earthquake, the corresponding system load vector can be stated in a similar way to Eq. (4.4a) as written in the frequency domain by,

$$ \{ P(\omega )\}_{g} = \{ H(\omega )\}_{{P\ddot{u}_{g} }} \ddot{u}_{g} (\omega ) $$

(4.5a)

where the subscript (g) denotes loading due to a ground motion, $\{ H(\omega )\}_{{P\ddot{u}_{g} }}$ denotes the system transfer function vector between $\{ P(\omega )\}_{g}$ and the random base acceleration $\ddot{u}_{g} (\omega )$. Having used Eqs. (4.5a) in (4.2) and (4.3), the global displacements under an earthquake loading can be stated as written by,

$$ \{ D(\omega )\}_{g} = \{ H(\omega )\}_{{D\ddot{u}_{g} }} \ddot{u}_{g} (\omega )\,\,\,\, \to \left\{ \begin{gathered} \{ H(\omega )\}_{{D\ddot{u}_{g} }} = [K]^{ - 1} \{ H(\omega )\}_{{P\ddot{u}_{g} }} + \sum\limits_{j = 1}^{q} {\left\{ {h(\omega )} \right\}_{{j\ddot{u}_{g} }} } \hfill \\ \left\{ {h(\omega )} \right\}_{{j\ddot{u}_{g} }} = \alpha_{j} (\omega )\,\{ \phi \}_{j}^{T} \{ H(\omega )\}_{{P\ddot{u}_{g} }} \{ \phi \}_{j} \hfill \\ \end{gathered} \right. $$

(4.5b)

in which $\{ H(\omega )\}_{{D\ddot{u}_{g} }}$ is the response transfer function vector for the global displacements under an earthquake loading. Details of the calculations are presented in the following section.

4.3 Calculation of Response Transfer Functions of Offshore Structures

In the previous section, definitions of response transfer functions for the global displacements under wave and earthquake forces, $\{ H(\omega )\}_{D\eta }$ and $\{ H(\omega )\}_{{D\ddot{u}_{g} }}$, are presented. In this section, their calculations are explained in detail.

4.3.1 Response Transfer Functions for Wave Loads

Response transfer functions of global displacements under the wave loading, which are defined in the vectorial form in Eq. (3.42b), are calculated by using transfer functions of global wave forces $\{ H(\omega )\}_{P\eta }$ and the eigenvalue information of the structural system. The transfer functions of global wave forces are obtained from the assembly process of the transfer functions of member wave forces. For a member, the transfer functions of wave forces in the global coordinates can be deducted from Eq. (3.60e) as written by,

$$ \left. \begin{gathered} \text{transfer\;functions\;of\;the\;wave\;} \hfill \\ \text{loading\;for\;a\;member} \hfill \\ \end{gathered} \right\} \, \, \to \,\,\,\,\left\{ {h_{w} (\omega )} \right\}_{G} = \omega \left\{ {\begin{array}{*{20}c} {q_{1} [T_{n} ]} \\ {q_{3} [T_{\theta } ]} \\ {q_{2} [T_{n} ]} \\ {q_{4} [T_{\theta } ]} \\ \end{array} } \right\}\{ \phi \} \, $$

(4.6a)

in which q _j (j = 1,2,3,4), $[T_{n} ]$, $[T_{\theta } ]$ and $\{ \phi \}$ are explained in Sect. 3.7.3. For each loaded member, the vector of transfer functions $\,\left\{ {h_{w} (\omega )} \right\}_{G}$ will be calculated from Eq. (4.6a) and then assembled to form the vector of transfer functions of the wave forces, $\{ H(\omega )\}_{P\eta }$, in the global coordinates. Once the vector $\{ H(\omega )\}_{P\eta }$ is calculated the transfer function vector $\{ H(\omega )\}_{D\eta }$ will be calculated by using Eq. (4.4b) in which the quasi-static term (the first term) can be calculated easily since the system stiffness matrix [K] is inverted only once. Calculation of the dynamic term (the second term) can also be carried out easily as stated below for a vibration mode (eigenmode).

$$ \left. \begin{gathered} \text{dynamic\;contribution\;of\;response\;}\hfill \\ \text{transfer\;functions\;for\;an\;eigenmode} \hfill \\ ( j){\text{\;under\;wave\;loading}} \hfill \\ \end{gathered} \right\} \, \, \to \,\,\,\left\{ \begin{gathered} \left\{ {h(\omega )} \right\}_{j\eta } = \beta_{j\eta } (\omega )\{ \phi \}_{j} \hfill \\ \beta_{j\eta } (\omega ) = \alpha_{j} (\omega )\,\{ \phi \}_{j}^{T} \{ H(\omega )\}_{P\eta } \hfill \\ \end{gathered} \right. $$

(4.6b)

where $\beta_{j\eta } (\omega )$ is a frequency dependent scalar function which defines the participation of the eigenmode vector j to the transfer functions of the global displacements. α _j (ω) is the natural frequency dependent eigenmode participation factor given by Eq. (1.264a), and $\{ \phi \}_{j}$ is the eigenmode vector for the jth^. vibration mode. As stated in Eq. (4.4b), the total dynamic contribution to the transfer functions of the global displacements is obtained by superimposing the contributions of eigenmodes considered.

4.3.2 Response Transfer Functions for Earthquake Loading

For offshore structures, the earthquake loading is somewhat different than that for onshore structures. The existence of water surrounding offshore structures produces additional hydrodynamic forces on structures during a ground motion like in the case of an earthquake [38]. Besides this hydrodynamic force, earthquakes produce inertia forces on offshore structures due to the ground acceleration as they occur for onshore structures. In this section, these forces are explained separately in the following sub-sections.

4.3.2.1 Hydrodynamic Forces Produced by Earthquakes

The additional hydrodynamic force produced by earthquakes can be calculated using the Morison’s equations which have been explained in detail in Chap. 3 for wave-current-structure interactions. In order to calculate this additional hydrodynamic force, the earthquake ground velocity and acceleration will be added to the relative water particle velocity and acceleration. For this purpose, it is assumed that the sea water is inviscid and do not move with the ground during the earthquake. It is also assumed that the structure is fixed at the bottom to make the same motion with the ground at the bottom. Under these conditions, the motion of the structure is like as shown in Fig. 4.1. The assumption of water at the sea bottom may be crude due to roughness of the bottom. In the reality, the water at the bottom moves with the ground. But, this motion decreases gradually toward the sea surface, and therefore, this assumption may be considered to be reasonable for deep water environments. Relative velocity and acceleration of water particles are used in the Morison’s equation to calculate the wave force as written by Eq. (3.88b) in the absence of earthquakes. In the case of earthquakes, the relative velocity and acceleration of water particles that used in Eq. (3.88b) will be modified to include the ground velocity and acceleration. These modified relative velocity and acceleration of water particles can be stated from Fig. 4.1 as written by, in the normal direction to the member axis,

$$ \left. \begin{gathered} {\text{relative}}\,{\text{water}}\,{\text{particle velocity and}}\, \hfill \\ {\text{acceleration during an earthquake }} \hfill \\ \end{gathered} \right\}\, \to \left\{ \begin{gathered} u_{r} = u + U_{c} - (\dot{d} + \dot{u}_{g} ) \hfill \\ \dot{u}_{r} = \dot{u} - (\ddot{d} + \ddot{u}_{g} ) \hfill \\ \end{gathered} \right.\,\, $$

(4.7a)

where u and U _c are respectively wave and current velocities, d and u _g are respectively structural and ground deformations as shown in Fig. 4.1. With these relative velocity and acceleration the Morison’s equation can be written similarly to Eq. (3.88b) as,

$$ f = C_{D} \left| {u + U_{c} - (\dot{d} + \dot{u}_{g} )} \right|\left( {u + U_{c} - (\dot{d} + \dot{u}_{g} )} \right) + C_{M} \dot{u} - C_{A} (\ddot{d} + \ddot{u}_{g} ) $$

(4.7b)

In the vectorial form, it can be stated similarly to Eq.(3.88d) as written by,

$$ \{ f_{n} \}_{G} = \{ f_{\,n} \}_{G,\text{wave}} + \{ f_{n} \}_{G,\text{current}} - \{ f_{dn} \}_{G,\text{struc.}} - \{ f_{{u_{g} \,n}} \}_{G,\text{ground}} $$

(4.7c)

where the force vectors $\{ f_{\,n} \}_{G,\text{wave}}$, $\{ f_{n} \}_{G,\text{current}}$ and $\{ f_{dn} \}_{G,\text{struc.}}$have been defined in Eq. (3.88e). The vector $\{ f_{{u_{g}\,n}} \}_{G,{\text{ground}}}$ denotes the hydrodynamic force vector in the global coordinates due to the ground motion. It is calculated from,

$$ \{ f_{{u_{g} \,n}} \}_{G,\text{ground}} = [T_{n} ]\left( {C_{D} A\,\{ \dot{u}_{g} \} + \,C_{A} \{ \ddot{u}_{g} \} } \right) $$

(4.7d)

in which $\{ \dot{u}_{g} \}$ and $\{ \ddot{u}_{g} \}$ are the velocity and acceleration vectors of the ground in the global coordinates, C _D and C _A are respectively the drag and added mass constants defined in Eqs. (3.45b) and (3.88c) respectively, [T _n ] is the transformation matrix defined in Eq. (3.55b), and A is the linearization coefficient of the Morison’s equation which has been explained in Sects. 3.8 and 3.9 in the absence of earthquakes. In the presence of earthquakes, it can be calculated as explained in Sect. 3.9 provided that the relative water velocity vector $\{ u_{rn} \}$ in Eq.(3.84a) is taken to be $\{ u_{rn} \}_{g}$ which is defined as,

$$ \{ u_{rn} \}_{g} \, = \{ u_{n} \} - \left( {\{ \dot{d}_{n} \} + \{ \dot{u}_{gn} \} } \right) \to \left\{ \begin{gathered} \{ u_{rn} \}_{g} \, = \{ u_{rn} \} - \{ \dot{u}_{gn} \} \hfill \\ \{ u_{rn} \} \, = \{ u_{n} \} - \{ \dot{d}_{n} \} \hfill \\ \end{gathered} \right. $$

(4.7e)

This relative velocity vector is a function of two independent random variables, one is the water elevation η and the other one is the earthquake ground acceleration $\ddot{u}_{g}$. Since the structural velocity vector $\{ \dot{d}_{n} \}$ is dependent on both the water elevation η and the earthquake ground acceleration $\ddot{u}_{g}$, the components, $\{ u_{rn} \}$ and $\{ \dot{u}_{gn} \}$, of the relative velocity vector $\{ u_{rn} \}_{g}$ are not independent. Since the linearization coefficient of the Morison’s equation A is dependent on the standard deviation of this relative normal water velocity, a similar procedure presented in Sect. 3.9.1 can be applied to calculate it. If the structural velocity vector $\{ \dot{d}_{n} \}$ is not taken into account in the calculation of hydrodynamic forces, the loading will be independent of the structural response velocities. In this case, calculation of the standard deviation of the relative normal water velocity can be simplified considerably. In the formulation presented here, it is assumed that the linearization coefficient A of the Morison’s equation is available. By introducing Eq. (4.7d) into the general statement of the consistent member force vector given by Eq. (3.60d) the hydrodynamic consistent force vector of a submerged member, which is produced by an earthquake, can be obtained in the global coordinates as stated by,

$$ \left\{ {p_{{\dot{u}_{g} }} } \right\}_{G,\text{hyro}} = \,\left\{ \begin{gathered} \left\{ {h_{{\dot{u}_{g} }} (\omega )} \right\}_{G,\text{hyro}} \dot{u}_{g} (\omega ) \hfill \\ \left\{ {h_{{\dot{u}_{g} }} (\omega )} \right\}_{G,\text{hyro}} = \left( {\int\limits_{0}^{\ell } {\left( {C_{D} A\, + \,i\omega C_{A} } \right)\left\{ {\begin{array}{*{20}c} {N_{1} [T_{n} ]} \\ {N_{3} [T_{\theta } ]} \\ {N_{2} [T_{n} ]} \\ {N_{4} [T_{\theta } ]} \\ \end{array} } \right\}ds} } \right)\,\{ \delta \} \hfill \\ \end{gathered} \right. $$

(4.7f)

where $\left\{ {h_{{\dot{u}_{g} }} (\omega )} \right\}_{G,\text{hyro}}$ is the transfer function vector between $\left\{ {p_{{\dot{u}_{g} }} } \right\}_{G,hyro}$ and the absolute ground velocity $\dot{u}_{g} (\omega )$ and {δ} is a constant vector containing cosine directions (translational components) of the ground motion in the global coordinates. The integration in Eq. (4.7f) can be calculated in a similar way explained in Sect. 3.10.1. The drag term (the term containing C _D A) in Eq. (4.7f) is similar to the statement of the consistent force vector due to a constant current which is given by Eq. (3.89a). From this similarity, the integration of the drag term is readily available from Eq. (3.89c). The integration of the inertia term (the term containing iω C _A ) in Eq. (4.7f) can be calculated easily using the shape functions N _j (j = 1 to 4) given by Eq. (3.59c). Having carried out these operations, the transfer function vector of consistent hydrodynamic forces due to a ground deformation can be obtained as written by,

$$ \left\{ {h_{{\dot{u}_{g} }} (\omega )} \right\}_{G,hyro} = \left( {C_{D} A_{1} \left\{ {\begin{array}{*{20}c} {q_{c1} [T_{n} ]} \\ {q_{c3} [T_{\theta } ]} \\ {q_{c2} [T_{n} ]} \\ {q_{c4} [T_{\theta } ]} \\ \end{array} } \right\} + \frac{\ell }{12}i\omega C_{A} \left\{ {\begin{array}{*{20}c} {6\,[T_{n} ]} \\ {\ell \,[T_{\theta } ]} \\ {6\,[T_{n} ]} \\ { - \ell \,[T_{\theta } ]} \\ \end{array} } \right\}} \right)\{ \delta \} $$

(4.7g)

where q _cj (j = 1 to 4) are presented in Table 3.13 and A ₁ is the value of A at the member end (1).

Fig. 4.1

Deformation of an offshore structure under wave and earthquake actions

4.3.2.2 Inertia Forces Produced by Earthquakes

In addition to hydrodynamic forces, earthquakes produce also inertia forces on structural elements due to structural masses vibrating with the earthquake ground acceleration. These forces are calculated at member (element) levels from the dynamic equilibrium equation of a member in which the inertia term (term with the structural mass) is calculated by using the total acceleration vector of the member. For offshore structures subjected to wave, current, and earthquakes, the dynamic equilibrium equation of a member can be stated as similar to Eq. (1.84). It is written by,

$$ [k]\{ d\} + [c_{s} ]\{ \dot{d}\} + [m_{s} ]\left( {\{ \ddot{d}\} + \{ \Updelta \} \ddot{u}_{g} } \right) - \left\{ p \right\}_{G} = \{ f_{{\text{int} .}} \}_{G} $$

(4.8a)

in which the vectors $\{ \Updelta \}$, $\left\{ p \right\}_{G}$ and $\{ f_{{\text{int} .}} \}_{G}$ are respectively the earthquake direction vector, consistent applied load vector and the vector of internal forces at the ends of the member, which are defined in the global coordinates. The earthquake direction and consistent applied load vectors are defined as written by,

$$ \begin{gathered} \{ \Updelta \}^{T} = \left\{ {\,\{ \delta \}^{T} \,\{ 0\}^{T} \,\{ \delta \}^{T} \,\{ 0\}^{T} } \right\} \, \hfill \\ \left\{ p \right\}_{G} = \left\{ {p_{w} } \right\}_{G} + \left\{ {p_{c} } \right\}_{G} - \left\{ {p_{{\dot{u}_{g} }} } \right\}_{G,hyro} - \left\{ {p_{{\dot{d}}} } \right\}_{G} - \left\{ {p_{{\ddot{d}}} } \right\}_{G} \hfill \\ \end{gathered} $$

(4.8b)

where the vector $\{ \delta \}$ is the same as defined in Eq. (4.7f) and the calculation of the vector $\left\{ p \right\}_{G}$ is carried out by using Eq. (4.7c). The terms, $\left\{ {p_{w} } \right\}_{G}$ and $\left\{ {p_{c} } \right\}_{G}$, of this vector are calculated respectively from Eqs. (3.60e) and (3.89c). The term $\left\{ {p_{{\dot{u}_{g} }} } \right\}_{G,\text{hyro}}$ is calculated from Eq. (4.7f) and the terms, $\left\{ {p_{{\dot{d}}} } \right\}_{G}$ and $\left\{ {p_{{\ddot{d}}} } \right\}_{G}$ are calculated respectively from Eqs. (3.90c) and (3.92b). Having substituted the statements of these force vectors in Eq. (4.8b) and rearranged Eq. (4.8a) it can be written that,

$$ [k]\{ d_{G} \} + [c]\{ \dot{d}_{G} \} + [m]\{ \ddot{d}_{G} \} - \left( {\left\{ {p_{w} } \right\}_{G} + \left\{ {p_{c} } \right\}_{G} - \left\{ {p_{{\ddot{u}_{g} }} } \right\}_{G} } \right)\, = \{ f_{{\text{int} .}} \}_{G} $$

(4.8c)

in which [c] and [m] are the total damping and mass matrices of the member in the global coordinates defined as,

$$ [c] = [c_{s} ] + \left[ {c_{h} } \right]\quad \text{and}\quad [m] = [m_{s} ] + \left[ {m_{a} } \right] $$

(4.8d)

The terms [c _s ] and [m _s ] in Eq. (4.8d) are the structural damping and mass matrices, [c _h ] and [m _a ] are the hydrodynamic damping and added mass matrices which are calculated respectively from Eqs. (3.90d) and (3.92c). The vector $\left\{ {p_{{\ddot{u}_{g} }} } \right\}_{G}$ in Eq. (4.8c) is the total force vector of the member in the global coordinates produced by the earthquake. This force vector is defined as,

$$ \left\{ {p_{{\ddot{u}_{g} }} } \right\}_{G} = \left\{ {h_{{\dot{u}_{g} }} (\omega )} \right\}_{G,\text{hyro}} \dot{u}_{g} + [m_{s} ]\{ \Updelta \} \ddot{u}_{g} $$

(4.8e)

in which the first term is the hydrodynamic force vector as defined in Eq. (4.7f) and the second term is the inertia force vector produced by the structural mass vibrating with the ground acceleration. The response transfer functions for the total earthquake forces are presented in the following section.

4.3.2.3 Response Transfer Functions for Combined Earthquake Hydrodynamic and Inertia Forces

The combination of the earthquake hydrodynamic and inertia forces of a member is calculated from Eq. (4.8e). In order to find the transfer function of this combined earthquake force, the ground velocity $\dot{u}_{g}$ is stated in terms of the ground acceleration $\ddot{u}_{g}$ in the frequency domain. Having carried out this operation, the earthquake force vector can be written as,

$$ \left\{ {p_{{\ddot{u}_{g} }} } \right\}_{G} = \left\{ \begin{gathered} \left\{ {h_{{\ddot{u}_{g} }} } \right\}_{G} \ddot{u}_{g} (\omega ) \hfill \\ \left\{ {h_{{\ddot{u}_{g} }} } \right\}_{G} = [m_{s} ]\{ \Updelta \} - \frac{i}{\omega }\left\{ {h_{{\dot{u}_{g} }} (\omega )} \right\}_{G,\text{hyro}} \hfill \\ \end{gathered} \right. $$

(4.8f)

in which $\left\{ {h_{{\ddot{u}_{g} }} } \right\}_{G}$ is the transfer function vector of the total earthquake forces of a member. For the whole system, the transfer function vector $\{ H(\omega )\}_{{P\ddot{u}_{g} }}$ defined in Eq. (4.5a) will be obtained by using the assembly process of member transfer functions $\left\{ {h_{{\ddot{u}_{g} }} } \right\}_{G}$. The contribution of concentrated masses at the deck of the platform will be added to this assembled transfer function, $\{ H(\omega )\}_{{P\ddot{u}_{g} }}$. Having calculated the vector $\{ H(\omega )\}_{{P\ddot{u}_{g} }}$for the system, the associated response transfer function vector $\{ H(\omega )\}_{{D\ddot{u}_{g} }}$ will be calculated by using Eq. (4.5b) in a similar way presented in Sect. 4.3.1 for the wave loading. Calculation of the quasi-static contribution from Eq. (4.5b) is straightforward and calculation of the dynamic contribution for an eigenmode is similar to Eq. (4.6b). It is written as,

$$ \left. \begin{gathered} {\text{dynamic contribution of response }} \hfill \\ {\text{transfer functions for an eigenmode}} \hfill \\ ( { }j ) {\text{ under earthquake loading}} \hfill \\ \end{gathered} \right\} \, \to \left\{ \begin{gathered} \left\{ {h(\omega )} \right\}_{{j\ddot{u}_{g} }} = \beta_{{j\ddot{u}_{g} }} (\omega )\{ \phi \}_{j} \hfill \\ \beta_{{j\ddot{u}_{g} }} (\omega ) = \alpha_{j} (\omega )\,\{ \phi \}_{j}^{T} \{ H(\omega )\}_{{P\ddot{u}_{g} }} \hfill \\ \end{gathered} \right. $$

(4.8g)

where $\beta_{{j\ddot{u}_{g} }} (\omega )$ is a frequency dependent scalar function which is similar to $\beta_{j\eta } (\omega )$ defined in Eq.(4.6b). The only difference is, in $\beta_{{j\ddot{u}_{g} }} (\omega )$, the earthquake loading is used instead of wave loading.

4.4 Calculation of Response Spectra of Offshore Structures

In the preceding sections, response transfer functions for the global displacements have been presented for both wave and earthquake loadings. By using these transfer functions, response spectra of the structure that required, which may be for displacements, forces or stresses, will be presented in this section. The calculation is explained firstly for the stochastic wave and earthquake loadings separately, and then for the combination of these loading cases. In these calculations, current loads do not produce any dynamic response. Since a constant current profile is assumed, they are purely static as given by Eq. (3.89c).

4.4.1 Response Spectra Under Stochastic Wave Loads

Stochastic wave loads are formulated in terms of transfer functions and the water elevation η of random waves as stated vectorially in the frequency domain in Eq. (4.4a). The water elevation η is a random scalar quantity which characterizes the randomness of waves. It is taken to be the input variable of response spectra of offshore structures under stochastic wave loads. Its stochastic description and spectral representation are presented in Sect. 3.4. Knowing its spectral values the calculation of response spectra of structures, which are subjected to only wave loads for the time being, is presented in this section. The spectral analysis and input–output relations have been presented generally in Sects. 2.5 and 2.6. Their applications to random wave loads and corresponding structural responses are explained here. If we assume that the outputs are global displacements of the structure, the input–output relation has already been constructed as stated vectorially in Eq. (4.4b), which is rewritten below for the convenience.

$$ \{ D(\omega )\}_{\text{wave}} = \{ H(\omega )\}_{D\eta } \eta (\omega ) $$

(4.9a)

By using Eq. (2.136) and Eq. (4.9a), the spectral matrix of displacements can be readily obtained as written by,

$$ \left. \begin{array}{l} {\text{Spectrum of displacements}}\\ {\text{under random wave loads}}\\ \end{array} \right\} \to [S_{{D_{\text{wave}} }} (\omega )] = \{ H(\omega )\}_{D\eta }^{*} \{ H(\omega )\}_{D\eta }^{T} \,S_{\eta \eta } (\omega )\,\, $$

(4.9b)

in which $S_{\eta \eta } (\omega )$ is the spectral function of the random water elevation η, the transfer function vector $\{ H(\omega )\}_{D\eta }$ is calculated from Eq. (4.4b), the superscripts, (*) and (T), denote respectively a complex conjugate and transposition. Any structural response output which is derived from displacements can be calculated in a similar way step-by-step. Since every step requires a matrix operation, the step-by-step calculation procedure is not efficient in terms of calculation time. An alternative and most powerful algorithm is to calculate firstly the transfer function of the response which is required, and then to apply the spectral calculation. This can be carried out only if the transfer function vector $\{ H(\omega )\}_{D\eta }$ is evaluated at all frequency points considered. This evaluation can be carried out easily using Eq. (4.4b). Response transfer functions of member displacements are extracted from the transfer functions of the global displacement $\{ H(\omega )\}_{D\eta }$, and consequently transfer functions of member internal forces and stresses are calculated for the frequencies considered. Then, the spectral values of the response required are calculated by using the procedure presented above in Eq. (4.9b). For example, if the response is assumed to be a stress in a member, its transfer function is calculated by using transfer functions of displacements. It is denoted by h _sη (ω). Since this stress transfer function is a scalar function, the spectrum of the stress is calculated similarly to Eq. (4.9b) from,

$$ S_{ss} \left( \omega \right) = h_{s\eta }^{*} \left( \omega \right)h_{s\eta } \left( \omega \right)S_{\eta \eta } \left( \omega \right) \to S_{ss} \left( \omega \right) = \left| {h_{s\eta } \left( \omega \right)} \right|^{2} S_{\eta \eta } \left( \omega \right) $$

(4.9c)

The calculation algorithm of a stress spectrum is summarized below in Fig. 4.2.

Fig. 4.2

Calculation algorithm of a stress spectrum under stochastic wave loads

In the case of wave-current actions, the response will be nonzero-mean process due to static current loads which are given by Eq. (3.89c) for members. But, as explained in Sect. 3.5, the existence of current alters the spectral form of the water elevation η. In this case, the global displacement vector {D} can be stated in two terms in the frequency domain as,

$$ \{ D(\omega )\} = \{ D(\omega )\}_{\text{wave}} + \{ \mu_{D} \} $$

(4.10a)

in which $\{ D(\omega )\}_{wave}$ is given by Eq. (4.9a) and $\{ \mu_{D} \}$ is a mean value displacement vector. $\{ D(\omega )\}_{wave}$ is a stochastic process with zero mean and $\{ \mu_{D} \}$ is calculated from the stiffness equation as written by,

$$ [K]\{ \mu_{D} \} = \left\{ {P_{c} } \right\}_{G} $$

(4.10b)

where {P _c } _G is the vector of system forces in the global coordinates due to a constant current. Similar to the displacements, any response quantity of the structure comprises a stochastic term with zero mean and a constant term defining the mean value response.

4.4.2 Response Spectra under Stochastic Earthquake Loading

The response spectra of offshore structures under an earthquake loading are similar to those calculated from wave loads. They are calculated by using the response transfer functions defined in Eq. (4.5b). For the convenience, the system displacement vector $\{ D(\omega )\}_{g}$ due to an earthquake ground motion is rewritten in terms of transfer functions $\{ H(\omega )\}_{{D\ddot{u}_{g} }}$, from Eq. (4.5b), as,

$$ \{ D(\omega )\}_{g} = \{ H(\omega )\}_{{D\ddot{u}_{g} }} \ddot{u}_{g} (\omega ) $$

(4.11a)

in which $\ddot{u}_{g} (\omega )$ is the acceleration of the ground in the frequency domain due to the earthquake motion. This acceleration is a random process and constitutes the input of the response spectra under earthquake loadings. Once a spectral function of the ground acceleration, $S_{{\ddot{u}_{g} }} (\omega )$, is known, the calculation of any response spectrum of the structure will be carried out in the same way as explained in the preceding section for the wave loads. Thus, as similar to Eq. (4.9b), the spectral matrix of system displacements is written by,

$$ \left. \begin{gathered} \text{Spectrum\;of\;displacements} \hfill \\ \text{under\;earthquake\;loadings} \hfill \\ \end{gathered} \right\} \to [S_{{D_{g} }} (\omega )] = \{ H(\omega )\}_{{D\ddot{u}_{g} }}^{*} \{ H(\omega )\}_{{D\ddot{u}_{g} }}^{T} \,S_{{\ddot{u}_{g} }} (\omega ) $$

(4.11b)

Any response spectrum can also be calculated by using the calculation algorithm presented in Fig. 4.2. As it can be realized from these spectral calculations, the determination of the earthquake spectrum $S_{{\ddot{u}_{g} }} (\omega )$ is the central issue in the spectral earthquake analysis of structures. In the following sections, the stochastic ground motion and its spectral representation under earthquakes are presented briefly.

4.4.2.1 Stochastic Earthquake Ground Motion

Earthquakes are random ground motions displaying a broadband character. They may result from various sources among which the tectonic-related earthquake motions are the largest and most important [23–28] for engineering, economical, and social points of view since their consequences are mostly catastrophic, especially in the near fault region. Earthquakes are measured in terms of magnitudes and intensities. The magnitude is the amount of energy that released during the earthquake. It is a unique measure for all locations, i.e. it is site independent. The best known measure of the earthquake magnitude is the Richter scale [39]. The intensity is a scale of the effect of an earthquake hazard at a specific location. It is based on observed human behavior and structural damages at a specific location so that the earthquake intensity scale is site dependent. There are numerous intensity scales that are in use currently in different parts of the world [26] such as the modified Mercalli (MMI) scale in the United States, the European Macroseismic scale (EMS-98) in Europe, the Shindo intensity scale in Japan, etc. The earthquake ground motion decreases with distance from the source of the earthquake and gradually dies away. This feature of the earthquake motion is termed as the attenuation. It is a function of not only the distance but also earthquake magnitude and geological site conditions [40–43]. The attenuation relations can be obtained by a statistical process of earthquake data (a regression analysis) measured at different locations. The basic data of earthquakes are recordings of ground accelerations at different sites during different earthquakes in time series, which contain valuable, and complete information that used in seismic analysis [23, 25]. These data vary significantly from site-to-site due to various factors. Depending on these factors, especially the magnitude and local site response, the recorded strong ground motions can display long duration that cause potential damages. The maximum amplitude of a recorded acceleration is defined as the peak ground acceleration (PGA), and similarly, the peak ground velocity (PGV) and peak ground displacement (PGD) are defined to indicate respectively the maximum amplitudes of the ground velocity and displacement [23]. The time histories of the ground velocity and displacement are obtained from integrations of the recorded time history of the ground acceleration. Based on the recorded time histories of the ground acceleration, which are mostly broadband random processes, spectral functions of the ground acceleration are determined. This spectral representation is presented briefly in the following section.

4.4.2.2 Spectral Representation of Stochastic Earthquake Ground Motion

The earthquake ground motion is provided by accelerograms in time domain that are recorded by accelerographs. For an earthquake occurrence, the earthquake accelerograms recorded at different locations are typically different from each other even the locations are within the range of dimensions of engineering structures and even there are similarities in accelerograms. These differences become larger for larger distances between recording stations (separations distance) due to different arrival time of seismic waves at different stations (wave passage effect), different soil conditions at different locations (site effect). These phenomena of the earthquake motion are fully described by so-called the coherency effect and being considerable for lifeline and large-scale engineering structures, which can be properly modeled by using multi-support excitation input [44–50]. In the literature, there are numerous publications [30–33, 37, 44, 51–63] to address the coherency of the earthquake motion. In this case, for linear structures, a response value of the structure (e.g. displacement, force, stress) is calculated by superimposing relevant responses under multiple support excitations due to a nonuniform earthquake ground motion. Multiple support excitations are shown schematically in Fig. 4.3 for an example offshore structure with three supports which are numbered as (1), (2), and (3). Each support is subjected to a random ground acceleration, namely $\ddot{u}_{g1}$, $\ddot{u}_{g2}$ and $\ddot{u}_{g3}$. A response value (displacement, force, stress) at a location in the structure, which is denoted by r, can be calculated generally in the frequency domain as written by,

$$ \left. \begin{gathered} \text{Frequency\;response\;under} \hfill \\ \text{multiple-support\;excitations} \hfill \\ \end{gathered} \right\} \to r(\omega ) = \sum\limits_{j = 1}^{n} {H_{{r\,\ddot{u}_{gj} }} (\omega )} \,\ddot{u}_{gj} (\omega ) $$

(4.12a)

where $H_{{r\,\ddot{u}_{gj} }} (\omega )$ is the transfer function between the response r and the ground acceleration at the support j, $\ddot{u}_{gj} (\omega )$, and n is the number of prescribed supports of the structure. The frequency response r(ω) given by Eq. (4.12a) can be conveniently written in the vector notation as,

$$ r(\omega ) = \{ H_{{r\,\ddot{u}_{g} }} (\omega )\}^{T} \{ \ddot{u}_{g} (\omega )\} $$

(4.12b)

Fig. 4.3

Multiple support excitation of an offshore structure under a nonuniform earthquake ground motion

in which the vector $\{ H_{{r\,\ddot{u}_{g} }} (\omega )\}$ contains all transfer functions $H_{{r\,\ddot{u}_{gj} }} (\omega )$ for (j = 1 to n), and the vector $\{ \ddot{u}_{g} (\omega )\}$ contains all support accelerations $\ddot{u}_{gj} (\omega )$ for (j = 1 to n). From Eq. (4.12b), the spectrum of the response r, S _rr (ω), can be easily stated by using the input–output relation of a linear systems as similar to Eq. (2.136). It is written as,

$$ \left. \begin{gathered} \text{Response\;spectrum\;under} \hfill \\ \text{multi-support\;random\;excitations} \hfill \\ \end{gathered} \right\} \to S_{rr} (\omega ) = \{ H_{{r\,\ddot{u}_{g} }} (\omega )\}^{*T} [S_{{\ddot{u}_{g} }} (\omega )]\{ H_{{r\,\ddot{u}_{g} }} (\omega )\} $$

(4.13)

in which the superscript (*) denotes the complex conjugate of a row vector, and $[S_{{\ddot{u}_{g} }} (\omega )]$ is the spectral matrix of the ground accelerations at the supports, $\ddot{u}_{gj} (\omega )$ for (j = 1 to n). This spectral matrix is formed as written by,

$$ [S_{{\ddot{u}_{g} }} (\omega )] = \left[ {\begin{array}{*{20}c} {S_{11} (\omega )} & . & . & {S_{1j} (\omega )} & . & . & {S_{1n} (\omega )} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ {S_{k1} (\omega )} & . & . & {S_{kj} (\omega )} & . & . & {S_{kn} (\omega )} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ {S_{n1} (\omega )} & . & . & {S_{nj} (\omega )} & . & . & {S_{nn} (\omega )} \\ \end{array} } \right] $$

(4.14)

in which the term, in general, $S_{kj} (\omega )$ denotes the cross spectrum of the ground accelerations at the supports k and j respectively, i.e. $\ddot{u}_{gk} (\omega )$ and $\ddot{u}_{gj} (\omega )$. Based on the recorded earthquake data at different sites, the spectral matrix of the earthquake ground motion can be calculated in the smoothed forms. By using the spectral information of two recording stations of the earthquake, say k and j, a coherency function of the seismic motion is defined [33, 51–57] as written by,

$$ \left. \begin{gathered} \text{Coherency\;function\;of} \hfill \\ \text{a\;seismic\;motion} \hfill \\ \end{gathered} \right\} \to \gamma_{kj} (\delta ,\omega ) = \frac{{S_{kj} (\omega )}}{{\sqrt {S_{kk} (\omega )S_{jj} (\omega )} }} $$

(4.15a)

in which δ denotes the separation distance between the locations k and j in the earthquake wave propagation direction. Its absolute value varies between zero and one, i.e. $0 \le \,|\gamma_{kj} (\delta ,\omega )|\, \le 1$. The coherency function is conveniently stated in an alternative form [56] written as,

$$ \left. \begin{gathered} \text{Alternative\;statement\;of} \hfill \\ \text{the\;coherency\;function} \hfill \\ \end{gathered} \right\} \to \gamma_{kj} (\delta ,\omega ) = |\gamma_{kj} (\delta ,\omega )|\exp [i\theta_{kj} (\delta ,\omega )] $$

(4.15b)

where $\left( {i = \sqrt { - 1} } \right)$, $|\gamma_{kj} (\delta ,\omega )|$ is the modulus, which is also called as loss of coherency or lagged coherency, and $\theta_{kj} (\delta ,\omega )$ is the phase of the coherency function describing the wave passage effect between the locations k and j. The real part of $\gamma_{kj} (\delta ,\omega )$, i.e. $\text{Re} \,\gamma_{kj} (\delta ,\omega )$, is called the unlagged coherency. The modulus $|\gamma_{kj} (\delta ,\omega )|$ is a measure of the similarity in the seismic motion which indicates the degree of linearity between recorded data at the stations k and j. The phase of the coherency function is defined [33, 54, 56] as written by,

$$ \left. \begin{gathered} \text{Definition\;of\;phase\;of} \hfill \\ \text{the\;coherency\;function} \hfill \\ \end{gathered} \right\} \to \theta_{kj} (\delta ,\omega ) = \arctan \left( {\frac{{\text{Im} S_{kj} (\omega )}}{{\text{Re} S_{kj} (\omega )}}} \right) $$

(4.16a)

For a seismic wave propagating with an approximately constant velocity c along the line between the stations k and j, the phase of the coherency function can be obtained [64] as written by,

$$ \theta_{kj} (\delta ,\omega ) = - \frac{\delta \,\omega }{c} $$

(4.16b)

which is a function of the separation distance δ, frequency ω, and the apparent velocity of the wave propagation c. For the lagged (loss of) coherency, or the modulus $|\gamma_{kj} (\delta ,\omega )|$, there have been numerous parametric expressions reported in the literature, see e.g. [30, 56] for a survey. The most popular one mentioned in the literature is introduced in [33]. It is written as,

$$ {\text{the lagged coherency}}\,\, \to \left\{ \begin{gathered} |\gamma_{kj} (\delta ,\omega )|\, = A\exp \left( { - \frac{B(\omega )}{a\,}} \right) + (1 - A)\exp \left( { - B(\omega )} \right) \hfill \\ B(\omega ) = \frac{2|\delta |}{\beta (\omega )}\,\,\left[ {1 + A\left( {a - 1} \right)} \right]\;\text{and} \hfill \\ \,\,\beta (\omega ) = k\left[ {1 + \left( {\frac{\omega }{{\omega_{0} }}} \right)^{b} } \right]^{\, - 1/2} \,\,\, \hfill \\ \end{gathered} \right. $$

(4.17a)

where A, a, k, b, and ω ₀ are five empirical parameters. These parameters can be obtained from the minimization of the error function defined [54] as,

$$ f(A,\,a,\,k,\,b,\,\omega_{0} ) = \sum\limits_{i = 1}^{{n_{s} }} {\sum\limits_{j=1}^{{n_{\omega } }} {\left[ {\arctan \left( {\gamma (\delta_{i} ,\omega_{j} )} \right) - \arctan \left( {|\gamma (\delta_{i} ,\omega_{j} )|} \right)} \right]^{2} } } $$

(4.17b)

in which n _s and n _ω are respectively the station pairs and discrete frequencies, and δ _i denotes the separation distance between the station pair i. It has been reported [54] that, for large separation distances and frequencies, Eq. (4.17a) produces erroneous values for a and k parameters. Therefore, for large δ and ω values, the coherency model given in Eq (4.17a) is degenerated to a simpler form [54] written as,

$$ |\gamma_{kj} (\delta ,\omega )|\, = A\,\exp \left( { - \frac{2|\delta |}{\chi }\left[ {1 + \left( {\frac{\omega }{{\omega_{0} }}} \right)^{b} } \right]^{1/2} } \right) + (1 - A) $$

(4.17c)

in which $\left( {\chi = a\,k/(1 - A)} \right)$. This model of $|\gamma_{kj} (\delta ,\omega )|$ contains four empirical parameters as A, χ, b, and ω ₀. In order to give an idea about the magnitudes of the empirical parameters, the values estimated from data [33] are written below.

$$ A{ = 0} . 7 3 6 ,\quad{ }a{ = 0} . 1 4 7 ,\quad { }k{ = 5210, }\quad\omega_{ 0} { = 6} . 8 5 ,\quad { }b{ = 2} . 7 8 $$

(4.17d)

These values may be varied according to soil conditions and stations of the data that they are estimated from. It is, usually, assumed that the supports of the structure have the same site conditions with different separation distances and frequencies (homogeneous ground condition), and therefore, the processes of ground accelerations at the supports will be identical, i.e. $\left( {S_{kk} (\omega ) = S_{{\ddot{u}_{g} }} (\omega )} \right)$ where (k = 1 to N _s ) with N _s is the number of supports, and $S_{{\ddot{u}_{g} }} (\omega )$ is the spectrum of the ground acceleration at a point (point earthquake spectrum). In this case, the spectral matrix of ground accelerations at the supports given by Eq. (4.14) will be stated in terms of the coherency functions and the point earthquake spectrum as written by,

$$ [S_{{\ddot{u}_{g} }} (\omega )] = \left[ {\begin{array}{*{20}c} {\gamma_{11} (\delta ,\,\omega )} & . & . & {\gamma_{1j} (\delta ,\omega )} & . & . & {\gamma_{1n} (\delta ,\omega )} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ {\gamma_{k1} (\delta ,\omega )} & . & . & {\gamma_{kj} (\delta ,\omega )} & . & . & {\gamma_{kn} (\delta ,\omega )} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ {\gamma_{n1} (\delta ,\omega )} & . & . & {\gamma_{nj} (\delta ,\omega )} & . & . & {\gamma_{nn} (\delta ,\omega )} \\ \end{array} } \right]\,\,S_{{\ddot{u}_{g} }} (\omega ) $$

(4.18)

The point earthquake spectrum $S_{{\ddot{u}_{g} }} (\omega )$ is also defined as the homogeneous earthquake spectrum for random ground motion which is assumed to be a zero-mean ergodic-Gaussian process of finite duration represented by a filtered white noise. This spectrum is represented by the modified Kanai–Tajimi spectrum [4] which is stated as written by,

$$ S_{{\ddot{u}_{g} }} (\omega ) = \frac{{1 + 4\xi_{g}^{2} \left( {\omega /\omega_{g} } \right)^{2} }}{{\left[ {1 - \left( {\omega /\omega_{g} } \right)^{2} } \right]^{2} + 4\xi_{g}^{2} \left( {\omega /\omega_{g} } \right)^{2} }}\,\,S_{f} (\omega )\,\,S_{0} $$

(4.19a)

where $\xi_{g}$ is the characteristic ground damping ratio, $\omega_{g}$ is the characteristic ground frequency, $S_{f} (\omega )$ is the filter spectrum for low frequency region to avoid difficulties and singularity when the frequency approaches zero, and S ₀ is an intensity factor. The filter spectrum is given [4] by

$$ S_{f} (\omega ) = \frac{{\left( {\omega /\omega_{f} } \right)^{4} }}{{\left[ {1 - \left( {\omega /\omega_{f} } \right)^{2} } \right]^{2} + 4\xi_{f}^{2} \left( {\omega /\omega_{f} } \right)^{2} }} $$

(4.19b)

in which $\omega_{f}$ and $\xi_{f}$ are respectively the characteristic frequency and damping ratio of the filter. The intensity factor S ₀ is calculated from the integration of Eq. (4.19a) as it is obtained to be,

$$ S_{0} = \,\,\frac{{4\xi_{g} \sigma_{{\ddot{u}_{g} }}^{2} }}{{\pi \left( {1 + 4\xi_{g}^{2} } \right)\,\omega_{g} }} $$

(4.19c)

in which $\sigma_{{\ddot{u}_{g} }}$ is the standard deviation (rms) of the ground acceleration. The original Kanai–Tajimi spectrum, which does not include the filter spectrum $S_{f} (\omega )$, has a constant value equal to S ₀ at the zero frequency, ($\omega = 0$). When this acceleration spectrum is transformed to the spectrum of ground displacement, at the zero frequency point, there will be a singularity and, in the small frequency region, high values of the displacement spectrum are obtained. This occurrence may result in exaggerated response spectral values in the small frequency region, and consequently, wrong response statistical values. Using the filter spectrum in the original Kanai–Tajimi spectrum, as stated in Eq. (4.19a), prevents this undesirable situation. In order to demonstrate the effect of the filter spectrum, the modified Kanai–Tajimi spectrum is plotted for ξ _f  = 0.6 and for different values of the characteristic filter frequency ω _f . In this demonstration, the standard deviation of the ground acceleration, the characteristic of ground frequency, and damping ratio are respectively assumed to be,

$$ \sigma_{{\ddot{u}_{g} }} = 0.25\,{\text{m/s}}^{ 2} ,\quad \omega_{\text{g}} { = 15}\,{\text{rad/s,}}\quad \xi_{\text{g}} { = 0} . 6 $$

(4.19d)

The shapes of these spectra are shown in Fig. 4.4 where it is seen that the filter spectrum dominates the small frequency region while, in the high frequency region, it has inconsiderable effect. If all supports undergo the same random ground motion, i.e. $\left( {\ddot{u}_{gk} (\omega ) = \ddot{u}_{g} (\omega )} \right)$ for (k = 1 to N _s ), then a unified ground motion is obtained with the homogeneous earthquake spectrum given by Eq. (4.19a). In this case, the response transfer functions will be calculated as explained in Sect. 4.3.2.3, and the response spectra will be calculated by using the response transfer functions explained in Sect. 4.4.2 and presented by Eq. (4.11b) for the displacements. Under multiple support excitations, as in the case of nonuniform ground motion, the calculation of response transfer functions is somewhat different than that explained in the Sect. 4.3.2.3. This calculation is explained briefly in the following section.

Fig. 4.4

The modified Kanai–Tajimi earthquake spectrum for different filter frequencies, ω _f

4.4.2.3 Calculation of Structural Response Transfer Functions Under NonUniform Earthquake Ground Motion

When the structure is subjected to a random multiple ground motion the calculation of response transfer functions differs from that under a uniform ground motion which is presented in the Sect. 4.3.2.3. In this case, a matrix partitioning is used in the dynamic equilibrium equation, for which the displacements of the structure are separated into two parts as,

• 1. prescribed displacements at the supports (base displacements of the structure) which are denoted by the vector {u _g },

• 2. all other structural displacements excluding the supports (displacements of superstructure at the unconstrained degrees of freedom) which are denoted by the vector {D} in the global coordinate system.

It is now assume that the structure is subjected to ground motion only. Thus, the forces acting on the structure are the support reactions which are denoted by the vector {P _g }, and forces at the superstructure are zero. Under this condition, the dynamic equilibrium equation of the structure can be written as,

$$ \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{\text{dd}} } & {{\mathbf{K}}_{\text{du}} } \\ {{\mathbf{K}}_{\text{du}}^{T} } & {{\mathbf{K}}_{\text{uu}} } \\ \end{array} } \right]\left\{ \begin{gathered} {\mathbf{D}} \hfill \\ {\mathbf{u}}_{g} \hfill \\ \end{gathered} \right\} + \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{\text{dd}} } & {{\mathbf{C}}_{\text{du}} } \\ {{\mathbf{C}}_{\text{du}}^{T} } & {{\mathbf{C}}_{\text{uu}} } \\ \end{array} } \right]\left\{ \begin{gathered} {{\dot{\mathbf{D}}}} \hfill \\ {\dot{\mathbf{u}}}_{g} \hfill \\ \end{gathered} \right\} + \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{\text{dd}} } & {{\mathbf{M}}_{\text{du}} } \\ {{\mathbf{M}}_{\text{du}}^{T} } & {{\mathbf{M}}_{\text{uu}} } \\ \end{array} } \right]\left\{ \begin{gathered} {\ddot{\mathbf{{D}}}} \hfill \\ \ddot{\mathbf{{u}}}_{g} \hfill \\ \end{gathered} \right\} = \left\{ \begin{gathered} {\mathbf{0}} \hfill \\ {\mathbf{P}}_{g} \hfill \\ \end{gathered} \right\} $$

(4.20)

in which the bold characters denote matrices and vectors, i.e. ${\mathbf{D}} = \{ D\}$, ${\mathbf{u}}_{g} = \{ u_{g} \}$, ${\mathbf{P}}_{g} = \{ P_{g} \}$ and K _dd = [K_dd], K _du = [K_du], etc. Eq. (4.20) gives two sets of equations. The zero loading set can be stated explicitly in terms of the support motion as written by,

$$ {\mathbf{K}}_{\text{dd}} {\mathbf{D}} + {\mathbf{C}}_{\text{dd}} {\dot{\mathbf{D}}} + {\mathbf{M}}_{\text{dd}} \ddot{\mathbf{{D}}} = - \left( {{\mathbf{K}}_{\text{du}} {\mathbf{u}}_{g} + {\mathbf{C}}_{\text{du}} {\dot{\mathbf{u}}}_{g} + {\mathbf{M}}_{\text{du}} \ddot{\mathbf{{u}}}_{g} } \right) $$

(4.21a)

At this time, we assume that the displacement vector D is decomposed into a pseudo-static and a dynamic components as written by,

$$ {\mathbf{D}} = {\mathbf{D}}_{s} + {\mathbf{D}}_{d} \to \left\{ \begin{gathered} {\mathbf{D}}_{s} { : }\;\text{pseudo-static\;component} \, \hfill \\ {\mathbf{D}}_{d} { : }\;\text{dynamic\;component} \, \hfill \\ \end{gathered} \right. $$

(4.21b)

The pseudo-static component is calculated from Eq.(4.21a) by setting all dynamic terms to zero which leads to the following equation.

$$ {\mathbf{D}}_{s} = - {\mathbf{Ru}}_{g} \quad \text{where}\quad {\mathbf{R}} = {\mathbf{K}}_{dd}^{ - 1} {\mathbf{K}}_{du} $$

(4.21c)

Having substituted Eqs. (4.21c) and (4.21b) into (4.21a) it can be obtained that,

$$ {\mathbf{K}}_{\text{dd}} {\mathbf{D}}_{d} + {\mathbf{C}}_{\text{dd}} {\dot{\mathbf{D}}}_{d} + {\mathbf{M}}_{\text{dd}} \ddot{\mathbf{{D}}}_{d} = - \left( {{\mathbf{C}}_{\text{du}} - {\mathbf{C}}_{\text{dd}} {\mathbf{R}}} \right){\dot{\mathbf{u}}}_{g} - \left( {{\mathbf{M}}_{\text{du}} - {\mathbf{M}}_{\text{dd}} {\mathbf{R}}} \right)\ddot{\mathbf{{u}}}_{g} $$

(4.21d)

In Eq. (4.21d), the damping force term is usually small in comparison with the inertia force term [45], and therefore, it is neglected. Then, the differential equation of the dynamic component D _d of the displacement vector becomes as written by,

$$ {\mathbf{K}}_{\text{dd}} {\mathbf{D}}_{d} + {\mathbf{C}}_{\text{dd}} {\dot{\mathbf{D}}}_{d} + {\mathbf{M}}_{\text{dd}} \ddot{\mathbf{{D}}}_{d} = - \left( {{\mathbf{M}}_{\text{du}} - {\mathbf{M}}_{\text{dd}} {\mathbf{R}}} \right)\ddot{\mathbf{{u}}}_{g} $$

(4.22a)

which can be solved as explained in Sect. 4.2. The response transfer function matrix of this displacement vector, ${\mathbf{H}}_{{D_{d} \ddot{u}_{g} }} (\omega )$, can be stated implicitly from Eq. (4.1) as written by,

$$ \begin{gathered} {\mathbf{D}}_{d} (\omega ) = {\mathbf{H}}_{{D_{\text{d}} \ddot{u}_{g} }} (\omega )\,\ddot{\mathbf{{u}}}_{g} (\omega ) \hfill \\ {\mathbf{H}}_{{D_{\text{d}} \ddot{u}_{g} }} (\omega ) = \left( {{\mathbf{\rm K}}_{\text{dd}} + i\omega {\mathbf{C}}_{\text{dd}} - \omega^{2} {\mathbf{M}}_{\text{dd}} } \right)^{ - 1} \left( {{\mathbf{M}}_{\text{dd}} {\mathbf{R}} - {\mathbf{M}}_{\text{du}} } \right) \hfill \\ \end{gathered} $$

(4.22b)

By using the modal analysis method the response transfer function matrix ${\mathbf{H}}_{{D_{\text{d}}\ddot{u}_{g} }} (\omega )$ can be obtained explicitly from Eq.(4.2) as stated by,

$$ {\mathbf{H}}_{{D_{\text{d}} \ddot{u}_{g} }} (\omega ) = \left( {{\mathbf{K}}_{\text{dd}}^{ - 1} + \sum\limits_{j = 1}^{q} {\alpha_{j} (\omega )\,{\boldsymbol {\upvarphi}}_{j} {\boldsymbol {\upvarphi}}_{j}^{T} } } \right)\left( {{\mathbf{M}}_{\text{dd}} {\mathbf{R}} - {\mathbf{M}}_{\text{du}} } \right) $$

(4.22c)

in which ${\boldsymbol {\upvarphi}}_{j}$ denotes the eigenmode vector of the jth^. natural mode, i.e. ${\boldsymbol {\upvarphi}}_{j} = \{ \phi \}_{j}$, $\alpha_{j} (\omega )$ is the modal participation factor, see Eq. (4.2), and q is the number of eigenmodes included. The response transfer function matrix of the quasi-static displacements can be readily written from Eq. (4.21c) as,

$$ {\mathbf{D}}_{s} (\omega ) = - {\mathbf{Ru}}_{g} (\omega ) = {\mathbf{H}}_{{D_{s} \ddot{u}_{g} }} (\omega )\ddot{\mathbf{{u}}}_{g} (\omega ) \to {\mathbf{H}}_{{D_{s} \ddot{u}_{g} }} (\omega ) = \frac{1}{{\omega^{2} }}{\mathbf{R}} $$

(4.22d)

Since the total displacements, D, are obtained from the superimposition of the quasi-static and dynamic components of the displacements as written in Eq. (4.21b), the transfer function matrix of the total displacements can be stated as,

$$ {\mathbf{D}}(\omega ) = {\mathbf{H}}_{{D\ddot{u}_{g} }} (\omega )\,\ddot{\mathbf{{u}}}_{g} (\omega ) \to {\mathbf{H}}_{{D\ddot{u}_{g} }} (\omega ) = \left( {{\mathbf{H}}_{{D_{s} \ddot{u}_{g} }} (\omega ) + {\mathbf{H}}_{{D_{d} \ddot{u}_{g} }} (\omega )} \right) $$

(4.23a)

Having substituted the statements of ${\mathbf{H}}_{{D_{s} \ddot{u}_{g} }} (\omega )$ and ${\mathbf{H}}_{{D_{\text{d}} \ddot{u}_{g} }} (\omega )$ from Eqs. (4.22d) and (4.22c) into (4.23a) it can be obtained that,

$$ {\mathbf{H}}_{{D\ddot{u}_{g} }} (\omega ) = \frac{1}{{\omega^{2} }}{\mathbf{R}} + \left( {{\mathbf{K}}_{\text{dd}}^{ - 1} + \sum\limits_{j = 1}^{q} {\alpha_{j} (\omega )\,{\boldsymbol {\upvarphi}}_{j} {\boldsymbol {\upvarphi}}_{j}^{T} } } \right)\left( {{\mathbf{M}}_{\text{dd}} {\mathbf{R}} - {\mathbf{M}}_{\text{du}} } \right) $$

(4.23b)

When a lumped mass matrix is used, the off diagonal term will be zero, i.e. ${\mathbf{M}}_{du} = {\mathbf{0}}$. In this case, Eq. (4.23b) will be simplified as written by,

$$ {\mathbf{H}}_{{D\ddot{u}_{g} }} (\omega ) = \left[ {\frac{1}{{\omega^{2} }}{\mathbf{I}} + \left( {{\mathbf{K}}_{\text{dd}}^{ - 1} + \sum\limits_{j = 1}^{q} {\alpha_{j} (\omega )\,{\boldsymbol {\upvarphi}}_{j} {\boldsymbol {\upvarphi}}_{j}^{T} } } \right){\mathbf{M}}_{\text{dd}}} \right]{\mathbf{R}} $$

(4.23c)

in which I denotes a unit matrix. The calculation of the matrix R can be carried out easily from Eq. (4.21c) by applying unit displacements at supports and solving to the structural displacements D _s.

4.4.2.4 Calculation of Spectral Responses Under Non-Stationary Multi-Support Ground Excitation

In the spectral earthquake analysis, it is mostly assumed that the random earthquake ground acceleration is a filtered white noise stationary Gaussian process. Its spectral representation is given by the Kanai–Tajimi spectrum as shown in Fig. 4.4. This is an adequate simplifying assumption to ease the calculation of spectral responses of structures. However, it is realized from the study of recorded accelerograms that the earthquake ground excitation is generally a non-stationary random process [65], and therefore, the due response spectra depend on not only the frequency content but also a function of time. Non-stationary responses of structures have been studied in the past and reported in the literature, see i.e. [34–36, 58] and [65–89]. In this section, calculation of response spectra of structures under non-stationary multi-support earthquake excitation are explained briefly.

A non-stationary unified earthquake ground motion is represented by a uniformly modulated random process [71, 76] which is defined as,

$$ \ddot{x}_{g} (t) = \alpha (t)\,\ddot{u}_{g} (t) $$

(4.24a)

In Eq.(4.24), $\ddot{x}_{g} (t)$ is the non-stationary earthquake ground acceleration, α(t) is a deterministic envelope time function and $\ddot{u}_{g} (t)$ is the unified stationary earthquake ground acceleration for which the spectral function is defined by Eq. (4.19a). Similar to the stationary multiple support excitations presented in the previous sections, the non-stationary multiple support excitations can be written from Eq. (4.24a) as,

$$ \{ \ddot{x}_{g} (t)\} = [\alpha (t)]\{ \ddot{u}_{g} (t)\} $$

(4.24b)

where [α(t)] is a diagonal matrix allowing different deterministic time functions at different supports. The calculation procedure of non-stationary random vector processes is explained below in general. Then, the spectral formulation of multi-support earthquake random excitations is obtained from the general formulation. For this purpose, two non-stationary input processes and the corresponding response output processes are used. The assumed non-stationary input vector processes are denoted by {y ₁(t)} and {y ₂(t)}, and the corresponding response output vector processes are denoted by {z ₁(t)} and {z ₂(t)}, respectively. As similar to Eq. (4.24b), the non-stationary input processes are defined as,

$$ {\text{non-stationary input processes }} \to \left\{ \begin{gathered} \{ y_{1} (t)\} = [\alpha_{1} (t)]\{ x_{1} (t)\} \hfill \\ \{ y_{2} (t)\} = [\alpha_{2} (t)]\{ x_{2} (t)\} \hfill \\ \end{gathered} \right. $$

(4.25a)

whrere {x ₁(t)} and {x ₂(t)} are stationary input vector processes with zero mean, [α ₁(t)] and [α ₂(t)] are diagonal matrices containing deterministic time functions for the elements of the stationary {x ₁(t)} and {x ₂(t)} processes. The corresponding output processes for a linear system at the time stations, t ₁ and t ₂, are obtained from the convolution integral explained in Sect. 2.6 as written by,

$$ \begin{gathered} \{ z_{1} (t_{1} )\} = \int\limits_{ - \infty }^{{t_{1} }} {[h(t_{1} - \tau_{1} )]\{ y_{1} (\tau_{1} )\} d\tau_{1} } \hfill \\ \{ z_{2} (t_{2} )\} = \int\limits_{ - \infty }^{{t_{2} }} {[h(t_{2} - \tau_{2} )]\{ y_{2} (\tau_{2} )\} d\tau_{2} } \hfill \\ \end{gathered} $$

(4.25b)

in which [h(t)] is a matrix of system impulse response functions which are shown in Fig. 4.5 with non-stationary input processes. Having introduced Eqs. (4.25a) into (4.25b) the response output processes at time stations, t ₁ and t ₂, can be obtained as written by,

$$ {\text{non-stationary output processes }} \to \left\{ \begin{gathered} \{ z_{1} (t_{1} )\} = \int\limits_{ - \infty }^{{t_{1} }} {[h(t_{1} - \tau_{1} )][\alpha_{1} (\tau_{1} )]\{ x_{1} (\tau_{1} )\} d\tau_{1} } \hfill \\ \{ z_{2} (t_{2} )\} = \int\limits_{ - \infty }^{{t_{2} }} {[h(t_{2} - \tau_{2} )][\alpha_{2} (\tau_{2} )]\{ x_{2} (\tau_{2} )\} d\tau_{2} } \hfill \\ \end{gathered} \right. $$

(4.25c)

Fig. 4.5

An impulse response function and non-stationary input functions. a an impulse response function, b non-stationary input functions

The cross-correlation matrix of the non-stationary output processes, {z ₁(t ₁)} and {z ₂(t ₂)}, can be written from Eq. (2.131a) as,

$$ [R_{{z_{1} z_{2} }} (t_{1} ,t_{2} )] = E\left[ {\{ z_{1} (t_{1} )\} \{ z_{2} (t_{2} )\}^{T} } \right] $$

(4.26a)

Since {x ₁(t ₁)} and {x ₂(t ₂)} are stationary processes Eq. (4.26a) can be stated as written by,

$$ \begin{gathered} [R_{{z_{1} z_{2} }} (t_{1} ,t_{2} )] = \int\limits_{ - \infty }^{{t_{1} }} {\int\limits_{ - \infty }^{{t_{2} }} {[h(t_{1} - \tau_{1} )][\alpha_{1} (\tau_{1} )][R_{{x_{1} x_{2} }} (\tau )]} \,[\alpha_{2} (\tau_{2} )]^{T} [h(t_{2} - \tau_{2} )]^{T} d\tau_{1} d\tau_{2} } \hfill \\ \text{where}\quad \tau = \left( {\tau_{2} - \tau_{1} } \right) \hfill \\ \end{gathered} $$

(4.26b)

in which $[R_{{x_{1} x_{2} }} (\tau )]$ is the cross-correlation matrix of the stationary {x ₁(t)} and {x ₂(t)} processes. It is obtained from the Fourier transform of their cross-spectral matrix as written by,

$$ [R_{{x_{1} x_{2} }} (\tau )] = \int_{ - \infty }^{\infty } {\,[S_{{x_{1} x_{2} }} (\omega )]\,e^{i\omega \tau } d\omega } \quad \text {with} \quad (\tau = \tau_{2} - \tau_{1} ) $$

(4.26c)

Having introduced Eqs. (4.26c) into (4.26b), the cross-correlation matrix of the response outputs $[R_{{z_{1} z_{2} }} (t_{1} ,t_{2} )]$ can be expressed as written by,

$$ \left. \begin{gathered} \text{Cross-correlation\;of} \hfill \\ \text{non-stationary\;functions} \hfill \\ \end{gathered} \right\} \to [R_{{z_{1} z_{2} }} (t_{1} ,t_{2} )] = \int_{ - \infty }^{\infty } {\,[A_{1}^{*} (t_{1} ,\omega ]\,} [S_{{x_{1} x_{2} }} (\omega )]\,[A_{2} (t_{2} ,\omega ]^{T} \,d\omega $$

(4.27a)

in which [A _j (t,ω)], where (j = 1,2), is a matrix of modulating functions as depending on the time t and frequency ω, the superscript (*) denotes a complex conjugate. The matrix of modulating functions is defined as stated by,

$$ [A_{j} (t,\omega ] = \int\limits_{ - \infty }^{t} {[h(t - \tau )][\alpha_{j} (\tau )]e^{i\omega \tau } d\tau } $$

(4.27b)

When (t ₁ = t ₂ = t) the integrant of $[R_{{z_{1} z_{2} }} (t_{1} ,t_{2} )]$ in Eq. (4.27a) is defined as the evolutionary power cross-spectral matrix. It is written as,

$$ \left. \begin{gathered} {\text{Evolutionary cross-}} \hfill \\ {\text{specral matrix}} \hfill \\ \end{gathered} \right\} \to [S_{{z_{1} z_{2} }} (t,\omega )] = [A_{1}^{*} (t,\omega ][S_{{x_{1} x_{2} }} (\omega )]\,[A_{2} (t,\omega ]^{T} $$

(4.28a)

For this special case, since {z ₁(t)} and {z ₂(t)} are zero mean value non-stationary processes, the cross-correlation matrix $[R_{{z_{1} z_{2} }} (t_{1} ,t_{2} )]$ becomes as to be the cross-covariance matrix. It is stated from Eqs. (4.27a) and (4.28a) as written by,

$$ \left. \begin{gathered} {\text{Cross-covariances of}} \hfill \\ {\text{non-stationary processes}} \hfill \\ \end{gathered} \right\} \to [R_{{z_{1} z_{2} }} (t,t)] = [\sigma_{{z_{1} z_{2} }} (t)] = \int_{ - \infty }^{\infty } {\,[S_{{z_{1} z_{2} }} (t,\omega )]} \,d\omega $$

(4.28b)

When ({z ₁(t)} = {z ₂(t)} = {z(t)}), which is the case of multi-support non-stationary earthquake excitations, the cross-covariance matrix given by Eq. (4.28b) becomes as the auto-covariance matrix. When it is compared with the stationary processes, the statements given by Eqs. (4.28a) and (2.133) become similar, i.e.,

$$ \begin{gathered} {\text{stationary}}\,{\text{to}}\,{\text{non-stationary:}}\,\,[H_{z} (\omega )]\to [A_{z} (t,\omega ] \hfill \\ {\text{non-stationary}}\,{\text{to}}\,{\text{stationary:}}\,\,[A_{z} (t,\omega ] \to [H_{z} (\omega )] \hfill \\ \end{gathered} $$

(4.29)

The stationary case is obtained from the non-stationary case when the matrix of deterministic time functions in Eq. (4.27b) becomes a unit matrix, i.e. [α(t)] = I where I is a unit matrix. The calculation of the evolutionary power cross-spectral matrix from Eq. (4.28a) requires the calculation of the matrix of modulating functions, [A _j (t,ω)]. Under assumed deterministic time functions, i.e. [α _j (t)], it is calculated by using Eq. (4.27b). For different deterministic time functions, the corresponding modulating functions have been calculated and reported in the literature [90].

4.5 Calculation of Response Statistical Quantities

In the previous sections, calculation of response spectra of offshore structures under wave and earthquake loadings has been presented. Since the input random wave elevation η(t) for the wave loading, and the earthquake ground acceleration $\ddot{u}_{g} (t)$ for the earthquake loading, are zero mean stationary processes, the corresponding responses become also zero mean random processes. If the non-stationary earthquake motion is considered, then the response becomes also non-stationary with zero mean. Statistical quantities of a response random variable are defined as the mean value, variance and probability distribution information such as skewness, kurtosis, and spectral bandwidth which are explained in Chap. 2. For stationary processes, these statistical quantities are calculated from spectral moments of the response process considered using Eq. (2.96) in Chap. 2. The spectrum of the response considered, i.e. a stress process due to wave loading, will be calculated from Eq. (4.9c). In the case of stationary multi-support earthquake excitations, a response spectrum will be calculated from Eq. (4.13). In the case of non-stationary earthquake excitations, variances, and covariances of the derived processes cannot be calculated directly from spectral moments of responses which are given by Eq. (2.100). Calculation of the variance $\sigma_{zz} (t)$ of a response non-stationary process z, which is a time function, is explained in the previous section. The variances and covariances of the derived processes of z, which may be $\sigma_{{z\dot{z}}} (t)$, $\sigma_{{\dot{z}\dot{z}}} (t)$, $\sigma_{{z\ddot{z}}} (t)$, $\sigma_{{\dot{z}\ddot{z}}} (t)$ and $\sigma_{{\ddot{z}\ddot{z}}} (t)$, can be calculated as similar to the calculation of $\sigma_{zz} (t)$ by using the evolutionary spectra of the derived processes. This subject has been studied and reported in the literature, see e.g. [91–98]. Statistical characteristics of a non-stationary response process are defined in terms of the standard deviations of z and its derivative processes. These characteristics are:

$$ \begin{gathered} {\text{Up-crossing frequency }} \ldots \ldots \ldots: \, \to \omega_{0} (t) = \sigma_{{\dot{z}}} (t)/\sigma_{z} (t) \hfill \\ {\text{Average frequency of maxima:}}\,\,\, \to \omega_{m} (t) = \sigma_{{\ddot{z}}} (t)/\sigma_{{\dot{z}}} (t) \hfill \\ {\text{Spectral Bandwidth}} \ldots \ldots \ldots \ldots: \,\to \,\varepsilon (t) = \sqrt {1 - \omega_{0}^{2} (t)/\omega_{m}^{2} (t)} \hfill \\ \end{gathered} $$

(4.30)

where $\sigma_{z} (t)$, $\sigma_{{\dot{z}}} (t)$ and $\sigma_{{\ddot{z}}} (t)$ are respectively standard deviations of z, $\dot{z}$ and $\ddot{z}$. It is worth noting that the process z can be a narrow banded at a time and a broad banded at another time instances.

4.6 Example

For the demonstration purposes, a jacket type offshore structure is analyzed by using the SAPOS program [99]. The structure, geometrical data and the 3D calculation model are shown in Fig. 4.6. It is supported on piles of 25 m depth in the soil as shown in Fig. 4.6. The soil properties are given in Table 4.1. Member dimensions and material properties of the structure are given in Table 4.2. It is assumed that the members in water are empty. Added masses of surrounding water of submerged members are taken into account by increasing the structural mass density according to the following statement:

$$ {\text{Total mass density of submerged members}}:\to \rho_{\text{tot}} = \left( {\rho_{s} + \frac{{\rho_{w} \gamma }}{4}\frac{D}{h}} \right) $$

(4.31)

where ρ _s is the mass density of structural material, ρ _w is the mass density of water, γ is the added mass coefficient, D and h are respectively diameter and wall thickness of the structural member, which are given in Table 4.2. It is assumed that the water depth is 75.0 m and the jacket is subjected to uni-directional random waves in the global X direction as shown in Fig. 4.6. The Pierson–Moskowitz (PM) sea spectrum, which is given by Eq. (3.31a) in Chap. 3, for an assumed sea state of H _s  = 9.0 m is used in the analysis. It is shown in Fig. 4.7 where ω _z is the zero crossings frequency and ω _p is the peak frequency of waves. It is also used a long-term Weibull probability distribution function of sea states given by Eq. (3.41) as shown in Fig. 4.8. In the analysis, the water structure interaction is taken into account with one iteration. The wave force data, marine growth, density of water and parameters of the Weibull probability distribution are given in Table 4.3 where c _d and c _m are respectively drag and inertia force coefficients, A, B and C are the parameters of the Weibull distribution. In the spectral analysis of the structure, both the quasi-static and dynamic contributions of the response are considered.

Fig. 4.6

An example jacket type offshore structure, geometrical data and 3D calculation model

Table 4.1 Properties of the soil under the example jacket structure

Table 4.2 Member dimensions, material properties and mass of the deck of the example jacket

Fig. 4.7

Pierson–Moskowitz sea spectrum

Fig. 4.8

Weibull probability distribution

Table 4.3 Wave force data and parameters of the Weibull probability function

Two natural mode shapes are used for the calculation of the dynamic response contribution. The mode shapes with corresponding natural frequencies are shown in Fig. 4.9. These two mode shapes are the same in orthogonal directions with the same natural frequencies. The spectrum and spectral moments of the hot-spot normal-stress at the bottom of a leg of the jacket are calculated for the assumed sea state of H _s  = 9.0 m. For the calculation of the hot-spot normal-stress, a stress concentration factor of (SCF = 2.0) is assumed. The calculated hot-spot stress transfer function is illustrated in Fig. 4.10 where the peak corresponds to the natural frequency of (ω ₁ = 2.98 rad/s). The stress spectrum is illustrated in Fig. 4.11 where the first peak corresponds to the peak frequency of waves at ω _p  = 0.42 rad/s and the second peak corresponds to the lowest natural frequency of the structure at ω ₁ = 2.98 rad/s. The stress spectral moments calculated from this spectrum are m ₀ = 0.345 × 10¹⁵, m ₂ = 1.626 × 10¹⁵ and m ₄ = 13.72 × 10¹⁵. By using Eqs. (2.116) and (2.117) in Chap. 2, the mean frequencies of zero crossings and maxima of the hot-spot stress are calculated to be ω ₀ = 2.172 rad/s and ω _m = 2.905 rad/s. From Eq. (2.102) the spectral bandwidth of the hot-spot stress process is calculated to be ε = 0.664 which indicates that, for the assumed sea state, the stress process is not narrow banded. It is a stochastic process between narrow and broadband.

Fig. 4.9

Natural mode shapes and frequencies of the example jacket structure. a First natural mode shape. b Second natural mode shape

Fig. 4.10

Transfer function of hot-spot normal-stress

Fig. 4.11

Spectrum of the hot-spot stress

Exercise 1

A monopod tower shown in Fig. 4.12 is subjected to uni-directional random waves. The tower is fixed at the bottom. It is made of steel and has a length of h _s  = 120.0 m, a diameter of D = 15.0 m with a wall thickness of t = 0.08 m. The mass of the deck is M _dec = 300.0 ton. The water depth is d = 100.0 m, the drag and inertia force coefficients are respectively c _d  = 1.3 and c _m  = 2.0. The structural damping ratio is assumed to be ξ = 0.01. The random waves are represented by the Pierson–Moskowitz sea spectrum given by Eq. (3.31a) in Chap. 3 with a sea state of H _s  = 9.0 m. The mass density of water is assumed to be ρ _w  = 1,024 kg/m³ and added mass coefficient is γ = 0.9. The deep water condition is used in the analysis. The stress concentration factor at the bottom of the tower is assumed to be SCF = 2.0. The following items are required:

Fig. 4.12

A monopod tower under random wave loading

• 1. Calculate natural frequency of the tower with and without containing water up to still water level (SWL)

• 2. Calculate the added damping ratio due to surrounding water of the tower

• 3. Plot transfer function of the normal hot-spot stress at the bottom of the tower

• 4. Calculate and plot the spectrum of the hot-spot stress

• 5. Calculate stress spectral moments, variance, frequencies of zero-crossings and maxima, and the bandwidth parameter of the hot-spot stress process at the bottom of the tower.

Exercise 2

The monopod tower shown in Fig. 4.12 is now subjected to a stationary earthquake random ground motion which is represented by the modified Kanai–Tajimi spectrum given by Eq. (4.19a). It is assumed that the characteristic ground damping ratio and frequency are respectively ξ _g  = 0.6 and ω _g  = 15.0 rad/s. The characteristic damping ratio and frequency of the filter are assumed respectively to be ω _f  = 1.5 rad/s, ξ _f  = 0.6. The standard deviation of the ground acceleration is assumed to be $\sigma_{{\ddot{u}_{g} }} = 0.25$ m/s². The followings are required to be calculated.

• 1. Calculate and plot the transfer function of the absolute horizontal displacement at the top of the tower

• 2. Calculate and plot the spectra of the displacement at the top and hot-spot normal stress at the bottom

• 3. Calculate stress spectral moments, variances, frequencies of zero-crossings and maxima, and the bandwidth parameters of the displacement at the top and the hot-spot stress at the bottom of the tower.