Pseudo-analytical model for sliding immersed structures under time-history earthquake loadings

Abstract

Some specific projects require that the structures being constructed are not anchored while being immersed. It is the case for recent bridges designed for large earthquake loadings, masonry structures, unanchored immersed storage structures. This solution gives more flexibility in the localization and in the use of such structures and a decrease in the internal stress. However, one has to evaluate precisely the behavior of this sliding structure, and in particular, the cumulative sliding displacement during a seismic event in order to prevent any impact or loss of stability. Given the highly non-linear behavior of such immersed sliding structures, the computational time and capacity to estimate the cumulative sliding displacement is important. It is then almost impossible to do easily a predimensioning of the structure or to apply a stochastic method in order to cover the possible time-history earthquakes corresponding to the dimensioning spectrum. The aim of this paper is to present a pseudo-analytical model to estimate the sliding amplitudes of different simplified systems which represent a given dynamic behavior: a single immersed sliding mass and an immersed sliding beam system. In each model, the fluid-structure interaction between the immersed body and the pool is modeled as hydrodynamic masses. The sliding is represented by a solid Coulomb friction. The seismic loading can be any 3D time-history accelerograms. The pseudo-analytical solutions are obtained considering the different phases of the movement and the continuity between each phase. The results are then compared to the values computed using $ANSYS^{TM}$. The analytical curves show a good fit of the computational results. A comparison between 3 models is shown (single immersed sliding mass, immersed sliding mass-spring system, immersed sliding beam) stressing the importance of the model on the estimated sliding displacement.

Appendices

Appendix 1: Notation

1.1 Tensor notation

In this article, the tensor notation has been used. The dimension of each value is given by the number of underlines.

For example:

• $\underline{u} = (u_i)_{i \in [1, N]}$ is a 1-dimension vector where the basis is composed of N elements.

• $\underline{\underline{M}} = (m_{ij})_{i \in [1,N], j \in [1,M]}$ is a 2-dimension matrix with N lines and M columns.

1.2 Time and spatial derivatives

The time derivatives are noted with upper dots. The number of dots gives the order of the derivatives.

For example:

• $\dot{U} = \frac{\partial U}{\partial t}$: is a velocity,

• $\ddot{U} = \frac{\partial ^2 U}{\partial t^2}$: is an acceleration.

The Laplace operator is used in this article. It is used to simplify the notation. It depends on the coordinate system:

• For a Cartesian system: $\varDelta U = \frac{\partial ^2 U}{\partial x^2} + \frac{\partial ^2 U}{\partial y^2} + \frac{\partial ^2 U}{\partial z^2}$

• For a cylindrical system: $\varDelta U = \frac{\partial ^2 U}{\partial r^2} + \frac{1}{r} \frac{\partial U}{\partial r} + \frac{1}{r^2} \frac{\partial ^2 U}{\partial \theta ^2} + \frac{\partial ^2 U}{\partial z^2}$

Appendix 2: Fluid-structure coupling: Added mass

In this appendix, we will do a short summary of the governing equations and assumptions used for the added mass theory that models the fluid-structure coupling in the article. It will be illustrated using the example documented by Fritz (1972) of two concentric infinite cylinders with a perfect incompressible fluid annulus. Then, the generalized buoyancy theory will be developed.

2.1 Fluid-structure interaction: added mass theory

The fluid-structure coupling model presented in Sect. 2.4 is based on the added mass theory. This theory applies when the Froude and Stokes coefficients are important. It corresponds to a ‘perfect fluid’. Then, the hydrodynamic pressure in the fluid is given by:

$$\begin{aligned} c_f^2 \varDelta p_f = \frac{\partial ^2 p_f}{\partial t^2} \end{aligned}$$

(35)

where $c_f$ is the velocity of sound in the water (1500 m/s), $\varDelta$ is the Laplace operator and $p_f$ is the hydrodynamic pressure in the fluid. At the fluid-structure interface, the structure is considered waterproof and since the fluid is modeled as a perfect fluid, there is no transverse friction. The interface conditions are:

$$\begin{aligned} \underline{\underline{\sigma _s}}.\underline{n} = -p_f \underline{n} \end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial \underline{u_s}}{\partial t}.\underline{n} = \frac{\partial \underline{u_f}}{\partial t}.\underline{n} \end{aligned}$$

(37)

where $\underline{\underline{\sigma _s}}$ is the Cauchy internal stress of the structure, $\underline{u_s}$ is the structure displacement, $\underline{u_f}$ is the fluid displacement and $\underline{n}$ is the fluid-structure interface surface vector.

These boundary conditions are usually expressed as:

$$\begin{aligned} - \frac{1}{\rho _f} \frac{\partial p_f}{\partial n} = \underline{n}.\frac{\partial ^2 \underline{u_s}}{\partial t^2} \end{aligned}$$

(38)

where $\rho _f$ is the fluid density.

Considering a virtual velocity $\underline{\hat{V}_s} = \frac{\partial \underline{u_s}}{\partial t} \hat{Q}$ and using a frequency notation $\underline{A}(t) = \underline{A_0} e^{i \omega t}$, one can express Newton’s second law for the structure as:

$$\begin{aligned} \left( - \omega ^2 \underline{\underline{M}} + i \omega \underline{\underline{C}} + \underline{\underline{K}} \right) \underline{q} = \underline{F_e} + \int _{\delta \Omega } \rho _f \omega ^2 S(q) \underline{n}.\frac{\partial \underline{u_s}}{\partial \underline{q}} da \end{aligned}$$

(39)

where $\underline{\underline{M}}$, $\underline{\underline{C}}$ and $\underline{\underline{K}}$ are respectively the mass, damping and stiffness generalized matrix, $\underline{q}$ is the generalized coordinate vector, $\rho _f$ is the fluid density, $\delta \Omega$ is the structure surface, $S(q) = \frac{p}{\rho _f \omega ^2}$ is the hydrodynamic coupling surface and $\underline{F_e}$ is the generalized external force.

It can then be demonstrated that the right element of equation (39), which is the fluid-structure coupling forces, is composed of:

• a part proportional to the acceleration of the structure for the low frequencies,

• a part proportional to the velocity of the structure for the middle frequencies,

• a part proportional to the displacement of the structure for the high frequencies.

Since we are dealing with seismic loadings, the range of exciting frequencies is the low frequency. Then, the fluid-structure coupling is proportional to the acceleration of the structure. It can then be modeled as an added mass that is applied on the structure degree of freedom.

2.2 Example of analytical calculation of hydrodynamic added mass: 2 concentric infinite cylinder with a fluid annulus

This example is thoroughly described in Fritz (1972). For further explanation, the reader is advised to refer to this article. The studied configuration is presented in Fig. 7.

Fig. 7

Configuration of the two concentric cylinder separated by a fluid annulus

Where $R_1$ and $R_2$ are respectively the radius of the internal and external cylinders. We will consider only a displacement of the cylinders along the X-axis. The same demonstration is applicable for the Y-axis. $U_{x,1}$ and $U_{x,2}$ are respectively the displacements of the internal and external cylinders.

Using Eq. (35) and considering an incompressible fluid $(c \rightarrow \infty )$, we deduce the pressure equation in cylindrical coordinates:

$$\begin{aligned} \frac{\partial ^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} + \frac{1}{r^2} \frac{\partial ^2 p}{\partial \theta ^2} = 0 \end{aligned}$$

(40)

The boundary conditions at the surface of each cylinder are:

$$\begin{aligned} - \frac{\partial p}{\partial r} \left( r = R_{1,2} \right) = \rho _f \frac{\partial ^2 U_r}{\partial t^2} \left( r = R_{1,2} \right) \end{aligned}$$

(41)

where $U_r (r=R_i) = U_{x,i} \cos \theta$ is the radial displacement at the surface of the cylinder i.

The pressure in the fluid is then calculated by solving Eq. (40) using the boundary conditions (41):

$$\begin{aligned} p_f (r, \theta ) = A r \cos \theta + \frac{B}{r} \cos \theta \end{aligned}$$

(42)

$$\begin{aligned} {\left\{ \begin{array}{ll} A = \frac{\rho _f}{R_1^2 - R_2^2} \left( R_2^2 \ddot{U}_{x,2} - R_1^2 \ddot{U}_{x,1} \right) , \\ B = \frac{\rho _f R_1^2 R_2^2}{R_1^2 - R_2^2} \left( \ddot{U}_{x,2} - \ddot{U}_{x,1} \right) . \end{array}\right. } \end{aligned}$$

(43)

The fluid-structure interaction forces are obtained by projecting the fluid pressure on the surface of each cylinder. Considering a unit-length of cylinder, the force applied on cylinder i is:

$$\begin{aligned} \underline{F_i} = \int _{\theta = 0}^{2 \pi }{p(R_i, \theta ) R_i (\cos \theta \underline{e_x} + \sin \theta \underline{e_y}) d\theta } \end{aligned}$$

(44)

The fluid-structure interaction forces applied on the internal and external cylinders are null along the Y-axis. The forces along the X-axis are expressed as:

$$\begin{aligned} F_1= & {} - M_H \ddot{U}_{x,1} + (M_1 + M_H) \ddot{U}_{x,2} \end{aligned}$$

(45)

$$\begin{aligned} F_2= & {} (M_1 + M_H) \ddot{U}_{x,1} - (M_1 + M_2 + M_H) \ddot{U}_{x,2} \end{aligned}$$

(46)

where:

• $M_1 = \rho _f \pi R_1^2$: is the equivalent mass of water occupied by the internal cylinder per unit length,

• $M_2 = \rho _f \pi R_2^2$: is the equivalent mass of water stored in the external cylinder per unit length,

• $M_H = M_1 \frac{R_2^2 + R_1^2}{R_2^2 - R_1^2}$: is the hydrodynamic mass per unit length.

$M_1$ and $M_2$ are directly linked to the generalized buoyancy effect presented in Appendix 2. $M_H$ is the mass involved in the fluid-structure interaction when the two cylinders have different accelerations and is called hydrodynamic mass.

The expressions obtained in Eqs. (45) and (46) can be generalized to more complex systems as the ones developed in this article.

2.3 Fluid-structure interaction: generalized buoyancy

When an immersed structure is subjected to an earthquake loading, the local coordinate system is no longer a Galilean system. In this local coordinate system, the ground motion creates an acceleration field that is applied to the structure. This horizontal acceleration field also creates an horizontal buoyancy: the ’generalized buoyancy’.

To illustrate this phenomenon, let us consider an unidirectional ground motion $U_{g} (t)$ that is applied on two structures A and B. The motion of the structures is expressed using the vector $\underline{U_{gal}} (t) = \left( U_{A,gal} (t), U_{B,gal} (t) \right)$ in the Galilean system and $\underline{U_{l}} (t) = \left( U_{A,l} (t), U_{B,l} (t) \right)$ in the local system. We obtain:

$$\begin{aligned} \underline{U_{gal}} (t) = \underline{U_{l}} (t) + \left( \begin{array}{l} U_g (t) \\ U_g (t)\\ \end{array} \right) \end{aligned}$$

(47)

The governing equation is in the Galilean system:

$$\begin{aligned}&\underline{\underline{M}}.\left[ \underline{\ddot{U}_{l}} (t) + \left( \begin{array}{lll} \ddot{U}_g (t) \\ \ddot{U}_g (t) \end{array}\right) \right] + \underline{\underline{C}}.\left[ \underline{\dot{U}_{l}} (t) + \left( \begin{array}{lll} \dot{U}_g (t) \\ \dot{U}_g (t) \end{array}\right) \right] \nonumber \\&\quad + \underline{\underline{K}}.\left[ \underline{U_{l}} (t) + \left( \begin{array}{lll} U_g (t) \\ U_g (t) \end{array}\right) \right] = \underline{F_e} \nonumber \\&\quad + \left( \begin{array}{lll} M_A &{} M_{AB} \\ M_{AB} &{} M_B \end{array}\right) \cdot \left[ \underline{\ddot{U}_{l}} (t) + \left( \begin{array}{lll} \ddot{U}_g (t) \\ \ddot{U}_g (t) \end{array}\right) \right] \end{aligned}$$

(48)

Four terms are linked to the ground displacement:

• $\underline{\underline{M}}.\left( \begin{array}{l} \ddot{U}_g (t) \\ \ddot{U}_g (t) \end{array}\right)$: corresponds to the seismic inertial force. It is the usual seismic loading.

• $\underline{\underline{C}}.\left( \begin{array}{l} \dot{U}_g (t) \\ \dot{U}_g (t) \end{array}\right)$: corresponds to a seismic damping force. It is null when the damping is produced by actual dampers.

• $\underline{\underline{K}}.\left( \begin{array}{l} U_g (t) \\ U_g (t) \end{array}\right)$: corresponds to a seismic elastic force. It is null when the stiffness matrix is constituted from actual springs.

• $\left( \begin{array}{ll} M_A &{} M_{AB} \\ M_{AB} &{} M_B \end{array}\right) . \left( \begin{array}{l} \ddot{U}_g (t) \\ \ddot{U}_g (t) \end{array}\right) = \left( \begin{array}{l} M_A^1 \ddot{U}_g (t) \\ M_B^1 \ddot{U}_g (t) \end{array}\right)$: corresponds to the generalized buoyancy. It is applied on each individual structure considering an equivalent ’displaced water mass’ $M_A^1 = M_A + M_{AB}$ for structure A and $M_B^1 = M_B + M_{AB}$ for structure B. This displaced water mass is implemented in a generalized buoyancy matrix $\underline{\underline{M_1}}$.

For the configuration considered in 1, the fluid structure interaction matrix corresponds to $M_A = -M_H$, $M_{AB} = M_H + M_1$ and $M_B = - (M_H + M_1 + M_2)$. In this case, $M_A^1 = M_1$ which corresponds to the equivalent water mass displaced by the internal cylinder and $M_B^1 = M_2$ which corresponds to the equivalent water mass contained in the external cylinder. These two masses corresponds to the classical masses considered for the vertical buoyancy effect but applied to an horizontal acceleration field (the earthquake loading).

For classical structures, a reformulation of the governing equations for the fluid-structure interaction of structure submitted to earthquake loadings is:

$$\begin{aligned}&(\underline{\underline{M}} + \underline{\underline{M_H}}). \underline{\ddot{U}} (t) + \underline{\underline{C}}. \underline{\dot{U}} (t) + \underline{\underline{K}} \cdot \underline{U} (t) \nonumber \\&\quad = \underline{F_e} (t) - (\underline{\underline{M}} - \underline{\underline{M_1}}) \underline{\ddot{U}_e} (t) \end{aligned}$$

(49)

where $\underline{\underline{M_H}} = - \left( \begin{array}{ll} M_A &{} M_{AB} \\ M_{AB} &{} M_B \end{array}\right)$ is the hydrodynamic coupling mass matrix, $\underline{\underline{M_1}} = \left( \begin{array}{ll} M_A^1 &{} 0 \\ 0 &{} M_B^1 \end{array}\right)$ is the generalized buoyancy mass matrix, $\underline{U} (t) = \underline{U_{l}} (t)$ is the local structural displacement and $\underline{U_e} (t) = \underline{U_g} (t)$ is the ground displacement. Equation (49) is the equation that is used in this article in its generalized form.