Transverse seismic response of end‐bearing pipe piles to S‐waves

Abstract

This paper presents a method for the seismic analysis of open-ended pipe piles subjected to vertically propagating S-waves, that considers kinematic interaction between the pipe pile and its external and internal soil. Following the presentation of the elastodynamic continuum model, which is based on the assumptions of linear elastic soil response and uniform soil conditions, we employ the derived solution to investigate the sensitivity of the seismic response of pipe piles to certain key problem parameters, such as pile slenderness or the relative stiffness of the pipe pile compared to its surrounding soil. We demonstrate that the bending stiffness of pipe piles is governing their seismic response, and that thin-walled pipe piles offer the best material usage versus seismic performance ratio. The presented solution offers a low-cost alternative to complex numerical simulations for preliminary seismic design purposes, such as the selection of optimal pipe pile section geometry.

Lateral kinematic response of offshore pipe piles to S‐wave seismic excitation

“…The solution proposed in this study can be degenerated to the special case of onshore pipe piles fully embedded in soil by setting $${H}_{\rm{2}} \to 0$$. The kinematic response factors I u obtained by the degenerated solution of this study are compared against those obtained by the solution of Zheng et al 36 . in Figure 2 for different values of relative pile material moduli E p / E s .…”

“…Based on the confined elastodynamics, 36 governing equations of the outer and inner soil are formulated in the cylindrical coordinate system as: $$$$\begin{equation}{G}^*{\nabla }^2{u}_{rj} + ({\lambda }^* + {G}^*)\frac{{\partial {e}_j}}{{\partial r}} - \frac{{{G}^*}}{{{r}^2}}\left( {2\frac{{\partial {u}_{\theta j}}}{{\partial \theta }} + {u}_{rj}} \right) + {G}^*\frac{{{\partial }^2{u}_{rj}}}{{\partial {z}^2}} = - {\rho }_s{\omega }^2{u}_{rj}\end{equation}$$$$ $$$$\begin{equation}{G}^*{\nabla }^2{u}_{\theta j} + ({\lambda }^* + {G}^*)\frac{{\partial {e}_j}}{{r\partial \theta }} - \frac{{{G}^*}}{{{r}^2}}({u}_{\theta j} - 2\frac{{\partial {u}_{rj}}}{{\partial \theta }}) + {G}^*\frac{{{\partial }^2{u}_{\theta j}}}{{\partial {z}^2}} = - {\rho }_s{\omega }^2{u}_{\theta j}\end{equation}$$$$where …”

“…The pipe pile is modeled as a one‐dimensional Euler–Bernoulli beam 36 : $$$$\begin{equation}E_p^ * {I}_p\frac{{{\partial }^4{u}_{pw}}}{{\partial {z}^4}} - {\rho }_p{A}_p{\omega }^2{u}_{pw} + {f}_{w1} + {f}_{w2} = 0\;\;\;\;( - {H}_2 \le z \le 0)\end{equation}$$$$ $$$$\begin{equation}E_p^*{I}_p\frac{{{\partial }^4{u}_{ps}}}{{\partial {z}^4}} - {\rho }_p{A}_p{\omega }^2{u}_{ps} + {f}_{s1}{\rm{ + }}{f}_{s2} = 0\;\;\;\;\;(0 \le z \le {H}_1)\end{equation}$$$$where $${u}_{pw}$$ and $${u}_{ps}$$ are the lateral displacements of the pile segments embedded in seawater and in soil, respectively. The pipe pile features the cross‐sectional area $${A}_p$$, inertial moment $${I}_p$$, Young's modulus $${E}_p$$, mass density …”

“…The existence of pile will generate scattered waves in the soil medium. Hence, the resultant displacements of the outer soil $${u}_{r1}$$ and $${u}_{\theta 1}$$ can be assumed as the sum of the free‐field motion u ff and the scattered wavefield $$u_{r1}^s$$ and $$u_{\theta 1}^s$$ as 36 : $$$$\begin{equation}{u}_{r1}(r,\theta ,z) = u_{r1}^s(r,\theta ,z) + {u}_{ff}\cos \theta \end{equation}$$$$ $$$$\begin{equation}{u}_{\theta 1}(r,\theta ,z) = u_{\theta 1}^s(r,\theta ,z) - {u}_{ff}\sin \theta \end{equation}$$$$where $${u}_{ff}(z) = {u}_{g} \frac{{\cos (qz)}}{{\cos (q{H}_{1})}}$$ is the free‐field soil motion 39 ; $$ q = \sqrt {{{{\rho }_{s}{\omega }^{2}} / {{G}^{*}}}} $$ is the complex wavenumber of S‐waves. The governing equations of the scattered motions of the outer soil $$…”

“…The abovementioned studies are all about solid piles and only considered the pile‐soil interaction. Nevertheless, piles used in offshore engineering are generally pipe piles and part of the pile segments is embedded in seawater 30–36 . In this paper, we present an analytical formulation to describe the kinematic response of a single offshore pipe pile subjected to S‐wave excitation, in which kinematic soil‐pile‐seawater interactions are quantified by means of closed‐form expressions that provides the soil resistances and hydrodynamic pressures of seawater to pile deformation.…”