Extreme scour effects on the buckling of bridge piles considering the stress history of soft clay

Abstract

Scour is a natural phenomenon caused by the erosion or removal of streambed or bank material from bridge foundations due to flowing water. As the buckling capacity of bridge piles varies inversely with the square of the unsupported pile height, an extreme scour at the pile caused by floodwater could result in a buckling failure of the pile and collapse of the bridge. The common way to analyze the scour-affected buckling stability of bridge piles is to remove the scoured soil layers while possible changes in stress history of the remaining soils are ignored. In reality, however, the remaining soils undergo an unloading process due to scour, and its overconsolidation ratios are increased. In this study, an analytical model with modified lateral subgrade modulus is presented to investigate the extreme scour effect on the buckling of bridge piles in soft clay considering the stress history of the remaining soils. A case study is used to compare the calculated results by considering and ignoring the stress history effect. The results show that ignoring the stress history of the soft clay will overestimate the buckling capacity of bridge pile foundation under scour.

Appendices

Appendix 1

For a pinned top-free tip boundary condition, the coefficient determinant is expressed as:

$$\Delta = \left| {\begin{array}{*{20}c} {b_{1,1} - {P \mathord{\left/ {\vphantom {P L}} \right. \kern-0pt} L}} & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{(n - 1)^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{(n - 1)^{2} \pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} \\ \end{array} } \right| = 0$$

(A1)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{1,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \frac{z}{L}} \right)}^{2} } {\text{d}}z$$

(A2)

$$b_{1,jj} = b_{jj,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \frac{z}{L}} \right)} } \sin \frac{(jj - 1)\pi z}{L}{\text{d}}z\quad (jj = 2, \ldots ,\, n)$$

(A3)

$$b_{ii,jj} = \frac{{(jj - 1)^{4} {\text{EI}}\pi^{4} }}{{2L^{3} }} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \sin \frac{(ii - 1)\pi z}{L}} } \, \sin \frac{(jj - 1)\pi z}{L}{\text{d}}z\quad (ii = jj = 2, \ldots ,\,n)$$

(A4)

$$b_{ii,jj} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \sin \frac{(ii - 1)\pi z}{L}} } \, \sin \frac{(jj - 1)\pi z}{L}{\text{d}}z\quad (ii = 2, \ldots ,\,n) \, (jj = 2, \ldots ,\,n)$$

(A5)

For a fixed top-free tip boundary condition, the coefficient determinant is expressed as:

$$\Delta = \left| {\begin{array}{*{20}c} {b_{1,1} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{9\pi^{2} P} \mathord{\left/ {\vphantom {{9\pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{(2n - 1)^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{(2n - 1)^{2} \pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} \\ \end{array} } \right| = 0$$

(A6)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{ii,jj} = \frac{{(2jj - 1)^{4} {\text{EI}}\pi^{4} }}{{32L^{3} }} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \cos \frac{(2ii - 1)\pi (L - z)}{2L}} \right)} } \left( {1 - \cos \frac{(2jj - 1)\pi (L - z)}{2L}} \right){\text{d}}z\quad (ii = jj = 1, \ldots ,n)$$

(A7)

$$b_{ii,jj} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \left( {1 - \cos \frac{(2ii - 1)\pi (L - z)}{2L}} \right)} } \left( {1 - \cos \frac{(2jj - 1)\pi (L - z)}{2L}} \right){\text{d}}z\quad (ii = 1, \ldots ,n) \, (jj = 1, \ldots ,n)$$

(A8)

For a fixed-sway top-free tip boundary condition, the coefficient determinant is expressed as:

$$\Delta { = }\left| {\begin{array}{*{20}c} {b_{1,1} } & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{(2n - 3)^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{(2n - 3)^{2} \pi^{2} P} {8L}}} \right. \kern-0pt} {8L}}} \\ \end{array} } \right| = 0$$

(A9)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{1,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} } } {\text{d}}z$$

(A10)

$$b_{1,jj} = b_{jj,1} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} } } \, \sin \frac{(2jj - 3)\pi z}{2L}{\text{d}}z\quad (jj = 2, \ldots ,n)$$

(A11)

$$b_{ii,jj} = \frac{{(2jj - 3)^{4} {\text{EI}}\pi^{4} }}{{32L^{3} }} + \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \, \sin \frac{(2ii - 3)\pi z}{2L}} } \sin \frac{(2jj - 3)\pi z}{2L}{\text{d}}z\quad (ii = jj = 2, \ldots , \, n)$$

(A12)

$$b_{ii,jj} = \sum\limits_{i = 1}^{m} {\int\limits_{{h_{i - 1} }}^{{h_{i} }} {k_{i} \sin \frac{(2ii - 3)\pi z}{2L}} } \sin \frac{(2jj - 3)\pi z}{2L}{\text{d}}z\quad (ii = 2, \ldots ,n) \, (jj = 2, \ldots , \, n)$$

(A13)

Appendix 2

The deflection function for the pinned top-pinned tip boundary condition is given as:

$$y = \sum\limits_{n = 1}^{\infty } {c_{n} } \sin \frac{n\pi }{L}z$$

(B1)

where c _n is the unknown constant.

Similarly, for the buckling load of a pile in a homogeneous soil with pinned top-pinned tip boundary condition, the coefficient determinant is expressed as follows:

$$\Delta { = }\left| {\begin{array}{*{20}c} {b_{1,1} - {{\pi^{2} P} \mathord{\left/ {\vphantom {{\pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} & {b_{1,2} } & {b_{1,3} } & \cdots & {b_{1,n} } \\ {b_{2,1} } & {b_{2,2} - {{4\pi^{2} P} \mathord{\left/ {\vphantom {{4\pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} & {b_{2,3} } & \cdots & {b_{2,n} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {b_{n,1} } & {b_{n,2} } & {b_{n,3} } & \cdots & {b_{n,n} - {{n^{2} \pi^{2} P} \mathord{\left/ {\vphantom {{n^{2} \pi^{2} P} {2L}}} \right. \kern-0pt} {2L}}} \\ \end{array} } \right| = 0$$

(B2)

where b _i,j are the intermediate parameters for calculation and can further be described as:

$$b_{ii,jj} = \frac{{jj^{4} \, {\text{EI}}\pi^{4} }}{{2L^{3} }} + \frac{Lk}{2}\quad \left( {ii = jj = 1, \ldots ,n} \right)$$

(B3)

$$b_{ii,jj} = 0\quad \left( {ii = 1, \ldots ,n} \right)\quad \left( {jj = 1, \ldots ,n} \right)$$

(B4)