The Spatial Analytical Method for the Dynamic Response of Submerged Floating Tunnel with Tension Leg Under Vehicle Load

Abstract

A spatial fully analytical model of the submerged floating tunnel (SFT) with tension leg under moving vehicle load is presented and its dynamic response is studied. Unlike the existing analytical model of SFT, the global spatial structure is divided into a series of tube and cable components along the span direction. Hamilton principle is used to build the model of components, where the cables and tubes are defined in local and global Cartesians, respectively. The drag and lift forces of the components of SFT from the wave are assumed by the Morison equation. The vehicle is assumed as a moving force with constant velocity. By assembling the equations of all components through the match conditions and boundary conditions, the global equation is established. The frequencies and the modes of each component are achieved from the characteristic equation corresponding to the linear undamped free vibratory case. The results show that the presented method can effectively obtain the natural out-of-plane vibration of SFT. The effects of stiffness ratios and mass ratios on the out-of-plane frequencies are investigated. The dynamic instability regions of some key parameters are obtained.

Appendices

Appendix 1. $\mathbb {M}_{i}$ for Out-of-Plane Vibration

When $i=1\sim n$, the elements for the modal connection matrix $\mathbb {M}_{i}$ between ith cable and the corresponding arch segment in Eq. (22) are

$$\begin{aligned} \begin{aligned} \mathbb {M}_{i}(1,1)=&\,0,~~~\mathbb {M}_{i}(1,2)=0,&\mathbb {M}_{i}(1,3)=&\,0,~~~\mathbb {M}_{i}(1,4)=0, \\ \mathbb {M}_{i}(2,1)=&\cos {l_{ai}}\eta _{i1}\csc {l_{ci}\zeta _i},&\mathbb {M}_{i}(2,2)=&\sin {l_{ai}}\eta _{i1}\csc {l_{ci}\zeta _i}, \\ \mathbb {M}_{i}(2,3)=&\cosh {l_{ai}}\eta _{i2}\csc {l_{ci}\zeta _i},&\mathbb {M}_{i}(2,4)=&\sinh {l_{ai}}\eta _{i2}\csc {l_{ci}\zeta _i}, \\ \end{aligned} \end{aligned}$$

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and when $i=n+1\sim n+m$ the elements are

$$\begin{aligned} \begin{aligned} \mathbb {M}_{i}(1,1)=&\,0,~~~\mathbb {M}_{i}(1,2)=0,&\mathbb {M}_{i}(1,3)=&\,0,~~~\mathbb {M}_{i}(1,4)=0, \\ \mathbb {M}_{i}(2,1)=&\cos {l_{ai}}\eta _{i1}\csc {l_{ci}\zeta _i},&\mathbb {M}_{i}(2,2)=&-\sin {l_{ai}}\eta _{i1}\csc {l_{ci}\zeta _i}, \\ \mathbb {M}_{i}(2,3)=&\cosh {l_{ai}}\eta _{i2}\csc {l_{ci}\zeta _i},&\mathbb {M}_{i}(2,4)=&-\sinh {l_{ai}}\eta _{i2}\csc {l_{ci}\zeta _i}, \\ \end{aligned} \end{aligned}$$

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Appendiix 2. $\mathbb {T}_{i}$ and $\mathbb {R}_{i}$ for Out-of-Plane Vibration

By substituting the shape functions of arch segments shown in Eq. (18) and the shape functions of cables shown in Eq. (17) into the connected matching conditions in Eq. (21), the relationships between two adjacent arch segments can be obtained as follow:

$$\begin{aligned} \cos \eta _{i1}l_{ai}A_{i1}&+\sin \eta _{i1}l_{ai}A_{i2}+\cosh \eta _{i2}l_{ai}A_{i3}+\sinh \eta _{i2}l_{ai}A_{i4}=A_{(i+1)1}+A_{(i+1)3}\end{aligned}$$

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$$\begin{aligned} -\eta _{i1}\sin \eta _{i1}l_{ai}A_{i1}&+\eta _{i1}\cos \eta _{i1}l_{ai}A_{i2}+\eta _{i2}\sinh \eta _{i2}l_{ai}A_{i3}+\eta _{i2}\cosh \eta _{i2}l_{ai}A_{i4}\nonumber \\&=\eta _{(i+1)1}A_{(i+1)2}+\eta _{(i+1)2}A_{(i+1)4}\end{aligned}$$

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$$\begin{aligned} -\eta _{i1}^2\cos \eta _{i1}l_{ai}A_{i1}&-\eta _{i1}^2\sin \eta _{i1}l_{ai}A_{i2}+\eta _{i2}^2\cosh \eta _{i2}l_{ai}A_{i3}+\eta _{i2}^2\sinh \eta _{i2}l_{ai}A_{i4}\nonumber \\&=-\eta _{(i+1)1}^2A_{(i+1)1}+\eta _{(i+1)2}^2A_{(i+1)3}\end{aligned}$$

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$$\begin{aligned} \chi _{wi}[\eta _{i1}^3\sin \eta _{i1}l_{ai}A_{i1}&-\eta _{i1}^3\cos \eta _{i1}l_{ai}A_{i2}+\eta _{i2}^3\sinh \eta _{i2}l_{ai}A_{i3}+\eta _{i2}^3\cosh \eta _{i2}l_{ai}A_{i4}]\nonumber \\&=\chi _{wi}[-\eta _{(i+1)1}^3A_{(i+1)2}+\eta _{(i+1)2}^3A_{(i+1)4}]-\zeta _{i}\sin \zeta _{i}l_{ci}C_{i1}+\zeta _{i}\cos \zeta _{i}l_{ci}C_{i2} \end{aligned}$$

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Therefore, Eq. (23) is built, when $i=1\sim n$ the elements of $\mathop {\mathbb {T}_{i}}$ and $\mathop {\mathbb {R}_{i}}$ are

$$\begin{aligned} \begin{aligned} \mathbb {T}_{i}(1,1)=&\,\frac{r_{i1}\eta _{(i+1)1}}{\eta _{i1}}\cos \eta _{i1}l_{ai}; \mathbb {T}_{i}(1,2)= \frac{r_{i1}\eta _{(i+1)1}}{\eta _{i1}}\sin \eta _{i1}l_{ai};\\ \mathbb {T}_{i}(1,3)=&\,\frac{r_{i2}\eta _{(i+1)1}}{\eta _{i2}}\cosh \eta _{i2}l_{ai}; \mathbb {T}_{i}(1,4)= \frac{r_{i2}\eta _{(i+1)1}}{\eta _{i2}}\sinh \eta _{i2}l_{ai};\\ \mathbb {T}_{i}(2,1)=&\,-r_{i1}\sin \eta _{i1}l_{ai}; \mathbb {T}_{i}(2,2)= r_{i1}\cos \eta _{i1}l_{ai};\\ \mathbb {T}_{i}(2,3)=&\,r_{i2}\sinh \eta _{i2}l_{ai}; \mathbb {T}_{i}(2,4)= r_{i2}\cosh \eta _{i2}l_{ai};\\ \mathbb {T}_{i}(3,1)=&\,\frac{r_{i3}\eta _{(i+1)2}}{\eta _{i1}}\cos \eta _{i1}l_{ai}; \mathbb {T}_{i}(3,2)= \frac{r_{i3}\eta _{(i+1)2}}{\eta _{i1}}\sin \eta _{i1}l_{ai};\\ \mathbb {T}_{i}(3,3)=&\,\frac{r_{i4}\eta _{(i+1)2}}{\eta _{i2}}\cosh \eta _{i2}l_{ai}; \mathbb {T}_{i}(3,4)= \frac{r_{i4}\eta _{(i+1)2}}{\eta _{i2}}\sinh \eta _{i2}l_{ai};\\ \mathbb {T}_{i}(4,1)=&\,-r_{i3}\sin \eta _{i1}l_{ai}; \mathbb {T}_{i}(4,2)= r_{i3}\cos \eta _{i1}l_{ai};\\ \mathbb {T}_{i}(4,3)=&\,r_{i4}\sinh \eta _{i2}l_{ai}; \mathbb {T}_{i}(4,4)= r_{i4}\cosh \eta _{i2}l_{ai};\\ \end{aligned} \end{aligned}$$

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and

$$\begin{aligned} \begin{aligned} \mathbb {R}_{i}(1,1)=&\,0;~ \mathbb {R}_{i}(1,2)=0;~ \mathbb {R}_{i}(2,1)=\frac{r_{i5}\zeta _i\sin \zeta _il_{ci}}{\eta _{(i+1)1}\chi _{wi}};~ \mathbb {R}_{i}(2,2)=-\frac{r_{i5}\zeta _i\cos \zeta _il_{ci}}{\eta _{(i+1)1}\chi _{wi}};\\ \mathbb {R}_{i}(3,1)=&\,0;~\mathbb {R}_{i}(3,2)=0;~ \mathbb {R}_{i}(4,1)=-\frac{r_{i5}\zeta _i\sin \zeta _il_{ci}}{\eta _{(i+1)2}\chi _{wi}};~ \mathbb {R}_{i}(4,2)=\frac{r_{i5}\zeta _i\cos \zeta _il_{ci}}{\eta _{(i+1)2}\chi _{wi}};\\ \end{aligned} \end{aligned}$$

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where $r_{i1}\sim r_{i5}$ are simplifying coefficients.

Appendix 3. Boundary Matrixes for Out-of-Plane Vibration

$$\begin{aligned} \mathop {\mathbb {B}_{\text {L}}}= \left[ \begin{array}{cccc} 1 &{}0 &{}1 &{}0 \\ 0 &{}\eta _{11} &{}0 &{}\eta _{12} \\ \end{array} \right] , \end{aligned}$$

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$$\begin{aligned} \mathop {\mathbb {B}_{\text {R0}}}= \left[ \begin{array}{cccc} 1 &{}0 &{}1 &{}0 \\ 0 &{}\eta _{(n+m)1} &{}0 &{}\eta _{(n+m)2} \\ \end{array} \right] , \end{aligned}$$

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