Static and Dynamic Analyses of Rectangular Floating Plates Subjected to Moving Loads

Abstract

This chapter presents a general and simple analytical methodology for the static and dynamic problems of a rectangular floating plate, such as an international airport, in which the floating plate has an arbitrarily variable structural stiffness and locates on the elastic foundation with variable spring stiffness. The discontinuous variation in the rigidity and mass of the plates due to the voids is also expressed as a continuous function by means of the extended Dirac function. The effect of the variation of the moving mass due to a jumbo jet taking off and landing on the very large floating plate is considered. The closed-form approximate solutions for free and forced vibrations are also considered to be powerful in the preliminary design stage.

Appendix: Shape Function of Beam on Elastic Foundation, Supported with Translational and Rotational Stiffnesses

In the theory the shape function of the current rectangular floating plates is substituted with one of a uniform beam on the Winkler-type elastic foundation. So, we present the dynamic problem of uniform beams on a uniform elastic foundation supported with elastic translational stiffness and rotational stiffness at the both ends, as shown in Fig. 10.12.

Fig. 10.12

Beam supported with translational stiffness and rotational stiffness on elastic foundations [17]

Where $k_{c}$ is the translational stiffness of the elastic foundation, and $k_{a}$, $k_{b}$, $R_{a}$, and $R_{b}$ are the translational stiffnesses and rotational stiffnesses of the supports at both ends, respectively. $\hat{m}_{0}$ is the mass per unit length of the uniform beam.

The equation of motion for the uniform beam may be given by

$$ \hat{m}_{{0{\kern 1pt} }} \ddot{w} + [EI\,w^{\prime\prime}]^{\prime\prime} = p \, (x,y) - k_{c} w $$

(10.67)

where EI is the bending stiffness of the uniform beam.

The equation for the transverse free vibration becomes from Eq. (10.67) as

$$ \hat{m}_{0} \ddot{w} + [EI\,w^{\prime\prime}]^{\prime\prime} + k_{c} w = 0 $$

(10.68)

The boundary conditions are also obtained as

Elastic support

$$ k_{a} w_{a} = - EI\,w^{\prime\prime\prime}\quad {\text{at}}\,x = \, 0 $$

(10.69)

$$ k_{b} w_{b} = EI\,w^{\prime\prime\prime}\quad {\text{at}}\,x = l $$

(10.70)

Elastic restraint

$$ R_{a} w^{\prime}_{a} = \, EI\,w^{\prime\prime}\quad {\text{at}}\,x = \, 0 $$

(10.71)

$$ R_{b} w^{\prime}_{b} = - EI\,w^{\prime\prime}\quad {\text{at}}\;x = l $$

(10.72)

The various boundary conditions can be made up easily by changing the coefficients of the translational stiffness and rotational stiffness. For example,

• When $k_{a} \to \infty$, the unmoving support condition is $w_{a} = 0.$

• When $k_{a} \to 0$, the free support condition is $EI \, w^{{\prime \prime}{\prime}} = 0.$

• When $R_{a} \to \infty$, the constrained condition for rotation is $w^{\prime}_{a} = 0.$

• When $R_{a} \to 0$, the free condition for rotation is $EI\,{\kern 1pt} w^{\prime\prime} = 0.$

Expressing the deflection $w(x,t)$ in the form

$$ w(x, t) = \bar{w} ( {\text{x) }} {\text{e}}^{{i\upomega t}} $$

(10.73)

the frequency equation for the current beam may be expressed as

$$ \bar{w}^{{\prime \prime}{\prime \prime}}- k^{4} \bar{w} = 0 $$

(10.74)

in which $k^{2}$ is defined as

$$ k^{2} = \sqrt {\frac{{\omega^{2} \hat{m}_{0} - \bar{k}_{c} }}{EI}} $$

(10.75)

The solution of Eq. (10.74) is given as

$$ \begin{aligned} \bar{w}(x) & = C_{1} \left[ {\cos (kx) + \cosh (kx)} \right] + C_{2} \left[ {\cos (kx) - \cosh (kx)} \right] \\ & \quad + C_{3} \left[ {\sin (kx) + \sinh (kx)} \right] + C_{4} \left[ {\sin (kx) - \sinh (kx)} \right] \\ \end{aligned} $$

(10.76)

in which $C_{1} - C_{4}$ are arbitrary constants.

The boundary conditions (10.69)–(10.72) can be rewritten in the form

$$ \alpha_{a} \bar{k}_{a1} \bar{w}_{(0)} + \bar{k}_{a2} \bar{w}^{\prime\prime\prime}_{(0)} = 0;\quad \alpha_{b} \bar{k}_{b1} \bar{w}_{(l)} - \bar{k}_{b2} \bar{w}^{\prime\prime\prime}_{(l)} = 0 $$

(10.77)

$$ \beta_{a} \bar{R}_{a1} \bar{w}^{\prime}_{(0)} - \bar{R}_{a2} \bar{w}^{\prime\prime}_{(0)} = 0;\quad \beta_{b} \bar{R}_{b1} \bar{w}^{\prime}_{(l)} + \bar{R}_{b2} \bar{w}^{\prime\prime}_{(l)} = 0 $$

(10.78)

where $\alpha_{a}$, $\alpha_{b}$, $\beta_{a}$, and $\beta_{b}$ are defined to be

$$ \alpha_{a} = \frac{{k_{a} }}{EI};\quad \alpha_{b} = \frac{{k_{b} }}{EI} $$

(10.79)

$$ \beta_{a} = \frac{{R_{a} }}{EI};\quad \beta_{b} = \frac{{R_{b} }}{EI} $$

(10.80)

The constants $\bar{k}_{a1} ,\bar{k}_{a2} ,{ \ldots }$, $\bar{R}_{b1}$, and $\bar{R}_{b2}$ are constants and take the value 1 for the general boundary condition including the translational stiffness and rotational stiffness. When the translation or rotation degree is constrained, the constants $\bar{k}_{a1}$, $\bar{k}_{b1}$, $\bar{R}_{a1}$, and $\bar{R}_{b1}$ takes 1, and $\bar{k}_{a2}$, $\bar{k}_{b2}$, $\bar{R}_{a2}$, and $\bar{R}_{b2}$ are 0.

Conversely, when the translational or rotational is free, the constants $\bar{k}_{a1} { \ldots }$ and $\bar{R}_{b1}$ take 0, and $\bar{k}_{a2} ,{ \ldots }$ and $\bar{R}_{b2}$ are 1. These constants are the general boundary conditions and include extreme (classic) boundary conditions such as perfectly constrained or free for translation and rotation. Substituting Eq. (10.76) into Eqs. (10.77) and (10.78) and using the condition that the determinant $\left| {A_{ij} } \right|$ of the reduced equations is zero, the following algebraic equation for $kl$ is obtained

$$ \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } & {A_{13} } & {A_{14} } \\ {A_{21} } & {A_{22} } & {A_{23} } & {A_{24} } \\ {A_{31} } & {A_{32} } & {A_{33} } & {A_{34} } \\ {A_{41} } & {A_{42} } & {A_{43} } & {A_{44} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {C_{1} } \\ {C_{2} } \\ {C_{3} } \\ {C_{4} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right] $$

(10.81)

where $A_{11} \sim A_{44}$ are defined as

$$ \begin{aligned} & \left[ {\begin{array}{*{20}l} {A_{11} = \, 2\alpha_{a} \bar{k}_{a1} } \hfill \\ {A_{12} = \, 0 \, } \hfill \\ {A_{13} = 0 \, } \hfill \\ {A_{14} = \, - 2\bar{k}_{a2} \, (kl)^{3} } \hfill \\ \end{array} } \right. \\ & \left[ {\begin{array}{*{20}l} {A_{21} = \alpha_{b} \bar{k}_{b1} \, \left[ {\cos kl + \cosh kl} \right] - \, \bar{k}_{b2} \, (kl)^{3} \left[ { \, \sin kl + \sinh kl} \right]} \hfill \\ {A_{22} = \alpha_{b} \bar{k}_{b1} \, \left[ {\cos kl - \cosh kl} \right] - \, \bar{k}_{b2} \, (kl)^{3} \left[ { \, \sin kl - \sinh kl} \right]} \hfill \\ {A_{23} = \alpha_{b} \bar{k}_{b1} \, \left[ {\sin kl + \sinh kl} \right] - \, \bar{k}_{b2} \, (kl)^{3} \left[ { - \cos kl + \cosh kl} \right]} \hfill \\ {A_{24} = \alpha_{b} \bar{k}_{b1} \, \left[ {\sin kl - \sinh kl} \right] + \, \bar{k}_{b2} \, (kl)^{3} \left[ { \, \cos kl + \cosh kl} \right]} \hfill \\ \end{array} } \right. \\ & \left[ {\begin{array}{*{20}l} {A_{31} = \, 0 \, } \hfill \\ {A_{32} = \, 2\bar{R}_{a2} \, (kl)^{2} \, } \hfill \\ {A_{33} = 2\beta_{a} \bar{R}_{a1} (kl) \, } \hfill \\ {A_{34} = 0 \, } \hfill \\ \end{array} } \right. \\ & \left[ {\begin{array}{*{20}l} {A_{41} = \, \beta_{B} \bar{R}_{b1} (kl) \, \left[ { - \sin (kl) + \sinh (kl)} \right] + \, \bar{R}_{b2} \, (kl)^{2} \left[ { - \cos (kl) + \cosh (kl)} \right]} \hfill \\ {A_{42} = - \beta_{B} \bar{R}_{b1} (kl) \, \left[ { \, \sin (kl) + \sinh (kl)} \right] - \, \bar{R}_{b2} \, (kl)^{2} \left[ { \, \cos (kl) + \cosh (kl)} \right] \, } \hfill \\ {A_{43} = \, \beta_{B} \bar{R}_{b1} (kl) \, \left[ { \, \cos (kl) + \cosh (kl)} \right] + \, \bar{R}_{b2} \, (kl)^{2} \left[ { - \sin (kl) + \sinh (kl)} \right]} \hfill \\ {A_{44} = \, \beta_{B} \bar{R}_{b1} (kl) \, \left[ { \, \cos (kl) - \cosh (kl)} \right] - \, \bar{R}_{b2} \, (kl)^{2} \left[ { \, \sin (kl) + \sinh (kl)} \right] \, } \hfill \\ \end{array} } \right. \\ \end{aligned} $$

(10.82)

The constants $k_{1} \sim k_{4}$ take the following values for the four cases:

for $A_{11} \ne 0 \, \,{\text{and }}A_{32} \ne 0$

$$ k_{1} = - \frac{{A_{14} }}{{A_{11} }};\quad k_{2} = - \frac{{A_{33} }}{{A_{32} }}k_{3} ;\quad k_{3} = \frac{{A_{32} }}{{A_{11} }}\left( {\frac{{A_{14} A_{21} - A_{24} A_{11} }}{{ - A_{33} A_{22} + A_{23} A_{32} }}} \right);\quad k_{4} = 1 $$

(10.83)

for $A_{11} \ne 0 \, ,A_{32} = 0 \, ,A_{33} \ne 0$

$$ k_{1} = - \frac{{A_{14} }}{{A_{11} }};\quad k_{2} = \frac{{A_{33} }}{{A_{11} }}\left( {\frac{{A_{14} A_{21} - A_{24} A_{11} }}{{A_{33} A_{22} - A_{23} A_{32} }}} \right);\quad k_{3} = - \frac{{A_{32} }}{{A_{33} }}k_{2} ;\quad k_{4} = 1 $$

(10.84)

for $A_{11} = 0 \, , \, A_{14} \ne 0 \, , \, A_{32} \ne 0$

$$ k_{1} = \frac{{A_{14} }}{{A_{32} }}\left( {\frac{{A_{33} A_{22} - A_{23} A_{32} }}{{A_{21} A_{14} - A_{11} A_{24} }}} \right);\quad k_{2} = - \frac{{A_{33} }}{{A_{32} }};\quad k_{3} = 1;\quad k_{4} = - \frac{{A_{11} }}{{A_{14} }}k_{1} $$

(10.85)

for $A_{11} = 0 \, ,\quad A_{14} \ne 0,\quad A_{32} = 0,\quad A_{33} \ne 0$

$$ k_{1} = A_{14} \left( {\frac{{A_{22} + k_{3} A_{32} }}{{A_{21} A_{14} - A_{11} A_{24} }}} \right);\quad k_{2} = 1;\quad k_{3} = - \frac{{A_{32} }}{{A_{33} }};\quad k_{4} = - \frac{{A_{11} }}{{A_{14} }}k_{1} $$

(10.86)