Scour effect on a single pile and development of corresponding scour monitoring methods

Beams and columns supported along their length are very common structure configurations, and the most routine method to treat the elastic foundation is the Winkler model (Ding 1993 and Coskun 2003). A pile embedded in soil is similar to a beam resting on a Winkler elastic foundation. Herein, assume the pile is fully buried in soil at the beginning. After the soil is eroded, the top part of the pile is exposed to the water flow, as shown in figure 1. The pile length is L with the origin at the top of the pile, and the unsupported depth is l, which is also the loading length with a distributed load q(x). The unsupported length l, due to initial scour and/or initial construction, is generically called the initial scour depth hereafter. The rest of the pile is embedded in the soil with an elastic spring coefficient k(x). The governing equation of the pile with a uniform cross-section and flexural rigidity and partially embedded in soil as shown in figure 1 can be expressed as

$$\begin{eqnarray} &&E I\frac{{\mathrm{d}}^{4}w}{\mathrm{d}{x}^{4}}=q\tqs 0\leq x\leq l \end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray} &&E I\frac{{\mathrm{d}}^{4}w}{\mathrm{d}{x}^{4}}+k(x)w=0\tqs l\leq x\leq L \end{eqnarray} \tag{ 1b }$$

where w(x) is the lateral displacement of the pile. Herein, k(x) is considered as a constant. The general solution of equation (1) consists of two parts, namely, the deflection of the pile is divided piecewise as

$$\begin{eqnarray} &&\fl w(x)\nonumber\\ &&\fl =\left \{ \begin{array}{@{}l@{}} \displaystyle {{w}_{1}(x)=C}_{1}\frac{{x}^{3}}{6}+{C}_{2}\frac{{x}^{2}}{2}+{C}_{3}x+{C}_{4}+\frac{{q x}^{4}}{2 4 E I},\\\displaystyle \quad \tqs 0\leq x\leq l \\ \displaystyle {w}_{2}(x)={\mathrm{e}}^{\beta x}({A}_{1}\sin \beta x+{A}_{2}\cos \beta x)\\\tqs \displaystyle +~{\mathrm{e}}^{-\beta x}({A}_{3}\sin \beta x+{A}_{4}\cos \beta x),\\\displaystyle \quad \tqs l\leq x\leq L \end{array}\right . \end{eqnarray} \tag{ 2 }$$

where β = (k/4EI)^1/4, and C_i and A_i (i = 1–4) are unknown constants. The boundary condition at x = l requires the geometric continuity of displacement and slope, and the continuity of bending moment and shear force, which are expressed as

$$\begin{eqnarray*} &&{w}_{1}(l)={w}_{2}(l),\tqs {w}_{1}^{\prime}(l)={w}_{2}^{\prime}(l),\\ &&{w}_{1}^{\prime\prime}(l)={w}_{2}^{\prime\prime}(l)\tqs \text{and}\tqs {w}_{1}^{\prime\prime\prime}(l)={w}_{2}^{\prime\prime\prime}(l). \end{eqnarray*}$$

Four additional conditions can be derived from boundaries at x = 0 and x = L. For a free-fixed pile, we have:

$$\begin{eqnarray*} &&{w}_{1}^{\prime\prime}(0)=0,\tqs {w}_{1}^{\prime\prime\prime}(0)=0,\tqs \nonumber\\ &&{w}_{2}(L)=0\tqs \text{and}\tqs {w}_{2}^{\prime}(L)=0. \end{eqnarray*}$$

From ${w}_{1}^{\prime\prime}(0)=0$ and ${w}_{1}^{\prime\prime\prime}(0)=0$, we have C₁ = C₂ = 0. This leads to six condition equations to solve six constants, C₃,C₄ and A_i (i = 1–4). The six equations can be rewritten in the matrix form as

$$\begin{eqnarray} &&{M}^{\ast }\left [\begin{array}{@{}c@{}} \displaystyle {C}_{3}\\ \displaystyle {C}_{4}\\ \displaystyle {A}_{1}\\ \displaystyle {A}_{2}\\ \displaystyle {A}_{3}\\ \displaystyle {A}_{4} \end{array}\right ]=\left [\begin{array}{@{}c@{}} \displaystyle 0\\ \displaystyle 0\\ \displaystyle q{l}^{4}/2 4 E I\\ \displaystyle q{l}^{3}/6\\ \displaystyle q{l}^{2}/2\\ \displaystyle q l/E I \end{array}\right ] \end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray*} \fl M=\left [\begin{array}{@{}cccc@{}} \displaystyle 0&\displaystyle 0&\displaystyle {\mathrm{e}}^{L \beta }\sin L \beta &\displaystyle {\mathrm{e}}^{L \beta }\cos L \beta \\ \displaystyle 0&\displaystyle 0&\displaystyle \beta {\mathrm{e}}^{L \beta }(\cos L \beta +\sin L \beta )&\displaystyle \beta {\mathrm{e}}^{L \beta }(\cos L \beta -\sin L \beta )\\ \displaystyle -l&\displaystyle -1&\displaystyle {\mathrm{e}}^{L \beta }\sin \beta l&\displaystyle {\mathrm{e}}^{L \beta }\cos \beta l\\ \displaystyle -1&\displaystyle 0&\displaystyle \beta {\mathrm{e}}^{L \beta }(\cos \beta l+\sin \beta l)&\displaystyle \beta {\mathrm{e}}^{L \beta }(\cos \beta l-\sin \beta l)\\ \displaystyle 0&\displaystyle 0&\displaystyle 2{\beta }^{2}{\mathrm{e}}^{L \beta }\cos \beta l&\displaystyle -2{\beta }^{2}{\mathrm{e}}^{L \beta }\sin \beta l\\ \displaystyle 0&\displaystyle 0&\displaystyle 2{\beta }^{3}{\mathrm{e}}^{L \beta }(\cos \beta l-\sin \beta l)&\displaystyle -2{\beta }^{3}{\mathrm{e}}^{L \beta }(\cos \beta l+\sin \beta l) \end{array}\right .\nonumber\\ \tqs \qquad \left .\begin{array}{@{}cc@{}} \displaystyle {\mathrm{e}}^{-L \beta }\sin L \beta &\displaystyle {\mathrm{e}}^{-L \beta }\cos L \beta \\ \displaystyle \beta {\mathrm{e}}^{-L \beta }(\cos L \beta -\sin L \beta )&\displaystyle -\beta {\mathrm{e}}^{L \beta }(\cos L \beta +\sin L \beta )\\ \displaystyle {\mathrm{e}}^{-L \beta }\sin (\beta \ast l)&\displaystyle {\mathrm{e}}^{-L \beta }\cos (\beta \ast l)\\ \displaystyle \beta {\mathrm{e}}^{-L \beta }(\cos \beta l-\sin \beta l)&\displaystyle -\beta {\mathrm{e}}^{-L \beta }(\cos \beta l+\sin \beta l)\\ \displaystyle -2{\beta }^{2}{\mathrm{e}}^{-L \beta }\cos \beta l&\displaystyle 2{\beta }^{2}{\mathrm{e}}^{-L \beta }\sin \beta l\\ \displaystyle 2{\beta }^{3}{\mathrm{e}}^{-L \beta }(\cos \beta l+\sin \beta l)&\displaystyle 2{\beta }^{3}{\mathrm{e}}^{-L \beta }(\cos \beta l-\sin \beta l) \end{array}\right ]. \end{eqnarray*}$$

This equation can be solved very conveniently by mathematical tools, such as Mathcad or MATLAB. The six constants, expressed as a function of the scour depth and other variables, are not presented here because they are very tedious. For a pinned–pinned pile, the boundary conditions are expressed as

$$\begin{eqnarray*} &&{w}_{1}(0)=0,\tqs {w}_{1}^{\prime\prime}(0)=0,\tqs \nonumber\\ &&{w}_{2}(L)=0\tqs \text{and}\tqs {w}_{2}^{\prime\prime}(L)=0 \end{eqnarray*}$$

and equations and solutions can be similarly obtained.

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As for the dynamic free vibration, the equation of motion of the same system is derived as

$$\begin{eqnarray} &&\fl E I\frac{{\partial }^{4}w(x,t)}{\partial {x}^{4}}+\rho A\frac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}=0,\tqs 0\leq x\leq l \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} &&\fl E I\frac{{\partial }^{4}w(x,t)}{\partial {x}^{4}}+\rho A\frac{{\partial }^{2}w(x,t)}{\partial {t}^{2}}+k(x)w(x,t)=0,\nonumber\\ &&l\leq x\leq L \end{eqnarray} \tag{ 4b }$$

where w(x,t) is the time-dependent displacement of the pile, ρ is the mass density, and A is the cross-section area. The coefficient k(x) is also considered as a constant, K. A general solution can be obtained easily by separating the variables into time and space domains using

$$\begin{eqnarray} &&w(x,t)=W(x){\mathrm{e}}^{\mathrm{i}\omega t}. \end{eqnarray} \tag{ 5 }$$

The solution of the governing equation thus reduces to that of

$$\begin{eqnarray} &&\fl \frac{{\mathrm{d}}^{4}W(x)}{\mathrm{d}{x}^{4}}-{\lambda }^{4}W(x)=0,\tqs 0\leq x\leq l \end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray} &&\fl \frac{{\mathrm{d}}^{4}W(x)}{\mathrm{d}{x}^{4}}-({\lambda }^{4}-K/E I)W(x)=0,\tqs l\leq x\leq L \end{eqnarray} \tag{ 6b }$$

in which $\lambda =\sqrt[4]{{\omega }^{2}\rho A/E I}$. The characteristic roots of equation (6b) are derived as

$$\begin{eqnarray*} &&{m}_{1,2,3,4}=\pm \sqrt{\pm \sqrt{{\lambda }^{4}-K/E I}}=\pm \sqrt{\pm \sqrt{({\omega }^{2}\rho A-K)/E I}}. \end{eqnarray*}$$

Depending on the relationship between λ⁴ and K/EI, different solutions can be obtained next.

If λ⁴ > K/EI, the general solution is

$$\begin{eqnarray} &&W(x)=\left \{ \begin{array}{@{}l@{}} \displaystyle {W}_{1}={C}_{1}\sin \lambda x+{C}_{2}\cos \lambda x+{C}_{3}\sinh \lambda x\\\tqs \displaystyle +~{C}_{4}\cosh \lambda x,\tqs 0\leq x\leq l\\ \displaystyle {W}_{2}={A}_{1}\sin {\lambda }_{2}x+{A}_{2}\cos {\lambda }_{2}x+{A}_{3}\sinh {\lambda }_{2}x\\\tqs \displaystyle +~{A}_{4}\cosh {\lambda }_{2}x,\tqs l\leq x\leq L \end{array}\right . \end{eqnarray} \tag{ 7 }$$

where ${\lambda }_{2}=\sqrt[4]{({\omega }^{2}\rho A-K)/E I}$.

If λ⁴ < K/EI,

$$\begin{eqnarray} &&\fl W(x)\nonumber\\ &&=\left \{ \begin{array}{@{}l@{}} \displaystyle {W}_{1}={C}_{1}\sin \lambda x+{C}_{2}\cos \lambda x+{C}_{3}\sinh \lambda x\\\tqs \displaystyle +~{C}_{4}\cosh \lambda x,\tqs 0\leq x\leq l\\ \displaystyle {W}_{2}={\mathrm{e}}^{{\lambda }_{2}x}({A}_{1}\sin {\lambda }_{2}x+{A}_{2}\cos {\lambda }_{2}x)\\\tqs \displaystyle +~{\mathrm{e}}^{{-\lambda }_{2}x}({A}_{3}\sin {\lambda }_{2}x+{A}_{4}\cos {\lambda }_{2}x),\tqs \\\tqs \displaystyle l\leq x\leq L \end{array}\right . \end{eqnarray} \tag{ 8 }$$

where ${\lambda }_{2}=\sqrt[4]{(K-{\omega }^{2}\rho A)/4 E I}$.

If λ⁴ = K/EI, namely, ω² = K/ρA,

$$\begin{eqnarray} &&\fl W(x)\nonumber\\ &&=\left \{ \begin{array}{@{}l@{}} \displaystyle {W}_{1}={C}_{1}\sin \lambda x+{C}_{2}\cos \lambda x\\\displaystyle \quad +~{C}_{3}\sinh \lambda x+{C}_{4}\cosh \lambda x,\\\displaystyle \quad \tqs 0\leq x\leq l \\ \displaystyle {W}_{2}={A}_{1}\frac{{x}^{3}}{6}+{A}_{2}\frac{{x}^{2}}{2}{+A}_{3}x{+A}_{4},\\\displaystyle \quad \tqs l\leq x\leq L. \end{array}\right . \end{eqnarray} \tag{ 9 }$$

Similar to the static solution, the boundary condition at x = l requires the geometric continuity of displacement and slope, and the continuity of bending moment and shear force. They are expressed as:

$$\begin{eqnarray*} &&{W}_{1}(l)={W}_{2}(l),\tqs {W}_{1}^{\prime}(l)={W}_{2}^{\prime}(l),\tqs \nonumber\\ &&{W}_{1}^{\prime\prime}(l)={W}_{2}^{\prime\prime}(l)\tqs \text{and}\tqs {W}_{1}^{\prime\prime\prime}(l)={W}_{2}^{\prime\prime\prime}(l). \end{eqnarray*}$$

Four additional conditions from the boundaries at x = 0 and x = L depend on the particular geometry under consideration. For a free-fixed pile, we have:

$$\begin{eqnarray*} &&{W}_{1}^{\prime\prime}(0)=0,\tqs {W}_{1}^{\prime\prime\prime}(0)=0,\tqs \nonumber\\ &&{W}_{2}(L)=0\tqs \text{and}\tqs {W}_{2}^{\prime}(L)=0. \end{eqnarray*}$$

From ${W}_{1}^{\prime\prime}(0)=0$ and ${W}_{1}^{\prime\prime\prime}(0)=0$, we have C₁ = C₃ and C₂ = C₄. The remaining six equations can be rewritten in the matrix form as:

$$\begin{eqnarray} {M}^{\ast }\left [\begin{array}{@{}cccccc@{}} \displaystyle {C}_{1}&\displaystyle {C}_{2}&\displaystyle {A}_{1}&\displaystyle {A}_{2}&\displaystyle {A}_{3}&\displaystyle {A}_{4} \end{array}\right ]^{\mathrm{T}}=0. \end{eqnarray} \tag{ 10 }$$

The frequency equation is given by setting the determinant of the coefficient matrix M to be zero, i.e., |M| = 0. Different coefficient matrices can be derived for the three cases discussed above. Therefore, the natural frequency in different ranges should be solved by different frequency equations derived from the corresponding coefficient matrix, as described below.

If $\omega \gt \sqrt{K/\rho A}$, i.e. λ⁴ > K/EI, the coefficient matrix is derived from equation (7) and denoted as M₁. The natural frequencies are obtained by solving |M₁| = 0.

If $\omega \lt \sqrt{K/\rho A}$, i.e. λ⁴ < K/EI, the coefficient matrix is derived from equation (8) and denoted as M₂. The natural frequencies are obtained by solving |M₂| = 0.

If $\omega =\sqrt{K/\rho A}$, the natural frequencies are automatically obtained.

Example 1 On the basis of the above analytical solution, an example was used for a demonstration. A square concrete pile with properties: L = 24.38 m (40 ft), A = 0.37 m² (24 × 24 in²),EI = 2.48 × 10⁸ N m⁻² (599 424 kips ft²), and q = 146 kN m⁻¹ (10 kips ft⁻¹). The elastic coefficient of soil per unit is k = 2.76 × 10⁵ kN m⁻² (5760 kips ft⁻²). The pile head is free and the pile bottom is assumed to be fixed. A program was developed in MATLAB to solve the analytical equations. In the case of l = L, the pile without soil supporting is identical to a cantilever beam with a distributed loading, which is a special case and can be used for validation. Figure 2 shows that the pile deflections of both cases (cantilever beam and l/L = 1) are exactly the same as they are supposed to be, which validates the numerical solution procedure of the present study.

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The scour ratio is defined as the ratio between the length without soil supporting and the pile length, l/L, and ranges from 0 to 1. The pile deflection and moment at various scour ratios are shown in figures 2 and 3. It can be easily seen that the pile displacement, moment and the location of the maximum moment increase with the increase of the scour ratio. There exists a turning point in the moment curve at a location close to the interface between the water and soil. More details will be described in the next section.

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Figure 4 shows the pile displacement at different positions versus the scour ratio and all the curves have the same trend. Figure 5 shows the pile moment (curvature) at different positions versus the scour ratio. It is found that for any position of the pile, if it is buried deeply in the soil, its moment is not significant. Only if the scour depth approaches that position does the moment become significant. Take the x = 0.8L section as an example. As the scour ratio increases from 0 to about 0.6, the moment of this section is small and hardly changes because the section is buried in the soil. However, when the scour ratio is larger than 0.6, the moment increases quickly from a small negative value to a positive value. It reaches a maximum at a scour ratio of 0.8 and remains constant since the section is totally exposed to the water at and after this scour ratio. Figures 6 and 7 demonstrate the influence of pile stiffness, represented by beta = EI/k, and soil stiffness, represented by k.

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Example 2 A pile similar to example 1 was adopted for a demonstration of the dynamic solution: L = 24.38 m (40 ft), A = 0.37 m² (24 × 24 in²), E = 2.15 × 10¹⁰ Pa (3122 ksi), I = 0.0115 m⁴ (1.333 ft⁴), and density = 2403 kg m⁻³ (150 pcf). In order to validate the relationship between ρA and K for different cases, the coefficient of soil here is 2760 kN m⁻² (57.6 kips ft⁻²). The pile head is free and the pile bottom is assumed to be fixed. A program was developed in MATLAB to obtain the frequency equation from the determinant and then to solve the equations. Again, the special case of l = L was used for verification. The first four natural frequencies from the present study are 0.4957 Hz, 3.1059 Hz, 8.6986 Hz and 17.038 Hz, respectively, while from the reference (Clough and Penzien 2003) they are 0.4957, 3.1065, 8.6992 and 17.0459 Hz, which are very close to each other.

Figures 8 and 9 show the first and fourth natural frequencies of the pile versus the scour ratio. The natural frequency decreases as the scour ratio increases, with the frequency of the first mode dropping much faster than that of the fourth mode. It means that the scour effect on the lower mode is more significant than on the higher mode, which will benefit the scour monitoring discussed later. Figure 10 shows the effect of the soil stiffness on the first natural frequency, mainly affecting the natural frequency when the scour ratio is small.

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The first step of damage detection is to determine the occurrence of damage (Rytter 1993). From the results of example 2 shown in figures 8–10, it is found that the reduction of soil around the pile will reduce the pile's natural frequency, especially the low-order frequencies. Since the low-order frequencies are easy to measure, the change of frequency is an alternative feature for detecting the occurrence of scour damage. Figure 11 presents the change ratio of the first natural frequency versus the initial scour depths (the initial position of the interface between soil and water) for three different additional scour ratios, namely 5% (scour depth = 4 ft), 10% (8 ft) and 15% (12 ft). It can been seen that for this 34.38 m (80 ft) pile, when the initial soil position is above the 15.24 m (50 ft) line (measured from the pile top), an additional scour ratio of 5% (4 ft) can result in more than 10% change in the first natural frequency. It implies that if an additional scour with a ratio of more than 5% occurs on a pile with an unsupported length of less than 62.5% (50/80 = 0.625), it can be detected through the change of the fundamental frequency. Figure 12 displays the same observation on the piles supported with soil of different stiffness.

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Figures 3 and 5 have shown that the maximum moment of the pile changes with the scour ratio and its location is slightly below but close to the scour depth. Therefore, a method based on the bending moment profile was proposed here for scour monitoring. As shown in figure 13, the turning points of the three curves are 2.68 m (8.8 ft), 5.12 m (16.8 ft) and 7.56 m (24.8 ft), compared to the total unsupported length (scour depth) of 2.44 m (8 ft), 4.88 m (16 ft), and 7.32 m (24 ft), respectively. As expected, the detected turning point of moments is typically lower than the true soil (scour) line because the soil provides an elastic (not rigid) support to the pile. The turning point of the pile moment profile can still be considered as the detected scour depth since it is very close to the true scour depth. From an engineering practice point of view, the top soil is typically weak after being disturbed and it does not provide much support to the pile. Therefore, the fact that the detected scour depth is slightly lower than the true soil position has a practical significance. For a single pile, it is easy to obtain the moment profile through strain sensors attached along the pile under applied testing loads or hydraulic loads.

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Modal information such as natural frequency and mode shape is easy to obtain from the time-history data recorded from dynamic tests, which makes it attractive for damage and scour detection. Similar to the moment profile method, a strategy based on the modal strain (curvature) profile was also proposed in the present study. As shown in figure 14, the modal strain extracted from the first modal shape is similar to the bending moment profile. The detected scour based on this strategy would be 8.78 m (28.8 ft), 15.36 m (50.4 ft) and 20.24 m (66.4 ft), compared to the true scour depth 7.32 m (24 ft), 14.63 m (48 ft), and 19.51 m (64 ft), respectively. The same as the bending moment case, the detected turning point of strains is typically lower than the true soil (scour) line because the soil provides an elastic (not rigid) support to the pile. Since the modal information is related to the physical property of the structure, it should be more applicable and practical than the bending moment from static loading.

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The GFRP pipe was first put on the ground without sand and clicked on the top, and was then buried in the sand with different heights, 0.2, 0.4 and 0.6 m. The time histories of strain response are transferred into the frequency domain using the fast Fourier transform (FFT), as shown in figure 18. The first frequencies of the four cases are 34.85 Hz, 35.25 Hz, 38.30 Hz and 42.00 Hz, respectively. Therefore, it confirms that the fundamental frequency of the structure decreases with the reduction of sand height, which indicates a scour activity.

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As observed earlier in the numerical simulation, the location of the maximum moment in the pile changes with the scour depth. In this section, tests of the GFRP pipe under different supporting conditions were conducted: without sand and with different heights of sand. A lateral force was applied intermittently on the top of the GFRP pipe to generate the bending moment in the pipe, and the time history of the strain response at different locations was recorded by the five sensors.

Without sand: The GFRP pipe was on the ground without any sand around and six runs were conducted. The strain responses of the five sensors of the first run are shown in figure 19. For each run, the strain value of the five sensors at a specific time was extracted to demonstrate the strain distribution along the pipe length. Figure 20 is the strain value along the pile length at the five peak times shown in figure 19. Those values are then normalized based on the ratio to the strain of the bottom sensor, namely, the strain value of the sensor at the position of 0.0 m is set to be 1, as shown in figure 21. The average of the normalized strain value at each run is then summarized in figure 22. It can be found that the stain distribution is nearly linear.

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With sand 0.2 m high and 0.4 m high: The GFRP pipe was buried in the sand with a height of 0.2 m and 0.4 m, respectively. With the same procedure, the normalized strain distributions of all the three runs are summarized in figures 23 and 24. The maximum strain in figure 23 is at the position of 0.2 m and that in figure 24 is at the position of 0.4 m, which is equal to the sand height of each case and indicates the scour depth.

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In this test, the specimen was put into the water tank on the shaking table to simulate the water flow action and hanging weights were used to simulate debris in the river water. The GFRP pipe with the FRP bar was buried in different heights of sand. In the first case, the sand height was 0.4 m and one hanging weight was at the position 0.6 m. In the second case, the sand height was 0.2 m and two hanging weights were arranged at the positions 0.2 m and 0.6 m, respectively. The water height was 0.8 m and the speed of the shaking table was set to be 0.1, 0.25 and 0.5 m s⁻¹.

The time histories of strains at a speed of 0.5 m s⁻¹ are shown in figures 25 and 26. Figure 25 is the strain response from sensors on the FRP bar and figure 26 is that from sensors attached on the GFRP pipe. It is observed that the response of the FRP bar is much stronger than that of the GFRP pipe, especially at the 0.6 m position where the hanging weight is located. Also, the curves of both cases look totally different. The one in figure 25 for height 0.6 m looks like an impulse signal and is very obviously distinguished from the others, due to the discontinuous impact of the hanging weight.

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The same observation can be seen from the results of the second case with 0.2 m sand, as shown in figures 27 and 28. Besides the 0.6 m position, figure 27 reveals an impulse-like signal appears at the position of 0.2 m as well. From case 1 to case 2, the sand height reduces from 0.4 to 0.2 m, which exposes the sensor and the hanging weight at the 0.2 m position. It indicates that as the soil is scoured, the sensor originally buried in the soil is exposed to the water and begins to respond recognizably differently due to the water current and debris impact. It is a good sign for scour detection.

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With the same procedure as described above, the strain distribution of the pipe at a given time was obtained by extracting from the time-history results of the five sensors, as shown in figure 29, with a sand height of 0.4 m and water speed of 0.25 m s⁻¹. The maximum strain is at the 0.2 m position, which is lower than the sand height. A possible reason could be that the elastic coefficient of sand is not constant, but rather is linearly distributed along the depth, which means that the stiffness of the top sand is very small. Moreover, the sand saturated in water weakens its supporting ability and the water flow drags the top sand and makes it loose. Therefore, the identified scour depth is not exactly the same as the actual scour depth.

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