Propagation characteristics of transient waves in low-strain integrity testing on cast-in-situ concrete thin-wall pipe piles

Abstract

The three-dimensional effects of pile head and the applicability of plane-section assumption are main problems in low-strain dynamic tests on cast-in-situ concrete thin-wall pipe piles. The velocity and displacement responses were calculated by a theoretical formula deduced by the authors. The frequency and influencing factor of high-frequency interference were analyzed. A numerical method was established to calculate the peak value and arrival time of incoming waves on top of the piles. The regularity along circumferential and the influence of radius or impulse width were studied. The applicability of plane-section assumption was investigated by comparison of velocity responses at different points in the sections at different depths. The waveform of velocity response at different points forked after the first peak, indicating that the propagation of stress waves did not well meet the plane- section assumption.

Keywords

pipe pile / low strain integrity testing / dynamic response / transient wave / three-dimensional effect

Introduction

The cast-in-situ concrete thin-wall pipe pile has been widely applied in reinforcing soft soil for its larger side frictional resistance and use of less concrete compared with common piles [1,2]. The pile casting quality is of great importance for such kind of cast-in-situ pipe piles. Low-strain testing has proved to be an effective method for testing pile shaft integrity. The foundation of this methodology is based on one-dimensional wave theory, which requires that the wave propagation satisfies plane-section assumption. However, the stress is nonaxisymmetrical when the pile head is subjected to load transient point load during the low-strain integrity testing. In fact, the wave propagation in pile shaft is a three-dimensional problem.

Previous studies have demonstrated that there is a serious size effect when the pipe pile is subjected to low-strain testing. Hence, the propagation of stress waves in the pipe pile does not satisfy plane-section assumption, and one- dimensional wave function cannot be applied in such a case. Because it is very difficult to solve three-dimensional wave equations for initial boundary conditions, the finite element method (FEM) is often employed to study three- dimensional boundary effect problems of propagation of stress waves [3-5]. The FEM needs complicated prior and post process and a lot of computing space, and is even difficult to apply to practical problems. Hence, to study a simplified analytical method is of great importance. Studies on infinite hollow cylinder are carried out, and the expressions of displacement components are obtained [6].

Considering the trivial variation of the dynamic response of thin-wall pipe pile along the radial direction, an analytical solution to the pile head dynamic response by the separation of variables and the method of variation of the parameters is proposed, in which it is assumed that all the variables are uniform along the pile radial direction. From this study, the stress wave propagation behaviors in pile head, peak value of incident waves, arrival time, the validity of plane-section assumption, and so on are further investigated.

Dynamic response characteristics on pile head

Velocity responses and displacement responses

The expressions of displacement and velocity of cast-in-situ concrete thin-wall pipe pile under transient concentrated load with low strain can be expressed as follows [7]:

(1)u_z(z,\theta ,t)={\displaystyle \sum _{n=\mathrm{0}}^{\infty }{\displaystyle \sum _{m=\mathrm{0}}^{\infty }\cos \left({\displaystyle \frac{n\pi }{H}}z\right)\cos (m\theta )T_{\mathrm{1}mn}(t),}}

(2)v_z(z,\theta ,t)={\displaystyle \sum _{n=\mathrm{0}}^{\infty }{\displaystyle \sum _{m=\mathrm{0}}^{\infty }\cos \left({\displaystyle \frac{n\pi }{H}}z\right)\cos (m\theta )T_{\mathrm{1}mn}^{\prime }}(t),}

where T_1mn(t) is a function of time t. The detailed expression can be found in Ref. [7].

Equations (1) and (2) indicate that the dynamic responses consist of infinite vibratory modal, in which the m=0 modal is axisymmetric because its dynamic responses are independent of coordinate q, whereas the m≠0 modal is nonaxisymmetrical because its dynamic responses are dependent of coordinate q.

Figures 1 and 2 show the velocity and displacement responses at five different points on pile head whose radius angles between exciting point and receiving point are 0°, 45°, 90°, 135°, and 180°, respectively. The parameters of this study are listed as follows: pile length is 15 m, inner radius is 0.38 m, outer radius is 0.5 m, Young’s modulus is 20 GPa, Poisson’s ratio is 0.17, density is 2400 kg/m³, peak value of the semisinusoidal exciting force impulse is 4 kN, and width of it is 1.5 ms. The damp of the concrete and the effect of the soil are ignored, and the pile bottom is assumed to be complete freedom.

The velocity curves of Fig. 1 show that the peak value and arrival time of incident wave differ from the radius angles. The peak value at the exciting point is the largest and the arrival time is the earliest. The peak values of the remaining points are smaller and the arrival time is more or less delayed. Because of the interference of high-frequency wave, the velocity response curve is not steady between incident wave crest and reflected wave crest. As a result, many high-frequency oscillatory wave crests form before the reflected wave from pile bottom arrives. The peak values of high-frequency interference are not the same among different points. The 90° point suffers the least high-frequency interference, so its velocity response curve is the steadiest. The high-frequency interference waves form the wave crest and wave hollow at the same time. For instance, the time of wave crest at the 0° and 45° points is the same as that of the wave hollow at the 135° and 180° points. Whereas, the time of wave crest at the 135° and 180° points is the same as that of the wave hollow at the 0° and 45° points. Moreover, the intersecting points of these velocity response curves and time axis (zero point) are almost the same. In Fig. 1, the velocity response curves intersect at the time t=0.0017, 0.0024, 0.0031,..., which is just like the property of “stationary point”.

The above findings indicate that the frequency of high-frequency interference at different points is the same or presents diploid relation, which will be verified in the following study. The curves in Fig. 1 also show that the peak value and arrival time of reflected wave crest from pile bottom have little difference.

The displacement response curves are shown in Fig. 2. The displacement curves rise step by step and the displacement increases progressively, because the pile bottom is freedom and the downward rigid body displacement occurred when the pipe pile applied the exciting force. The time to form incident waves is 0-0.0015 s and the slope coefficients are different from different points; correspondingly, the peak values of incident wave on velocity curves are different. The curve between 0.0015 s and 0.0102 s approximately parallels the time axis, which indicates that the displacement is invariable during the time. The stress waves transmit downward the pile until arriving at pile bottom, and then reflect and continue to transmit upward the pile until arriving at pile top at the time about 0.0102 s. From 0.0102 s to 0.0117 s the displacement curves rise quickly, indicating that the reflected waves from pile bottom have arrived. After 0.0117 s, the curves turn out to be a steady platform, at which time the stress waves transmit downward the pile again and then repeat the former process. From 0.0015 s to 0.0102 s, the curves except that at the 90° point are not steady. There are also many high-frequency interference wave crests. The time of wave crest, wave hollow, and “stationary point” are similar with that of velocity response curve.

Frequency of high-frequency interference

The previous study has pointed out that the different points on top of pile suffered different high-frequency interference. The following will study the peak value and influencing factor of high-frequency interference in frequency domain by fast Fourier transformation (FFT).

Figure 3 is the field measurement waveform of velocity response of a cast-in-situ concrete thin-wall pipe pile in Yantong expressway soft soil improvement practice. Figure 4 is its Fourier spectrum. Figure 4 also shows the calculational result using the theory in this paper.

In the field low-strain testing, the sensor is a speed sensor, the radius angle between exciting point and receiving point is 90°, and the width of exciting impulse is about 1 ms. The waves suffer distinctive high-frequency interference. Although the measured and calculated values vary, the trends of the two can well match each other. The frequency difference between two adjacent peaks among the former six peaks is about 183 Hz, which is the harmonic peak of the reflected wave from pile bottom. The seventh peak differs from the others. It is a high-frequency wave crest. The interference frequency of the field measurement waveform is 1365 Hz and that of calculated waveform is 1390 Hz. The latter is a little larger than the former. The difference may be caused by ignoring damp and the radial wave propagation.

The frequency of high-frequency interference is dependent on the position of receiving point. Figure 5 shows the velocity spectrums at the 45°, 90°, and 135° angles, respectively. The frequency of high-frequency interference at the 45° and 135° angles is 685 Hz, whereas that at the 90° angle is 1370 Hz, the former being just half of the latter.

The peak value of the high-frequency interference wave crest at the 45° point is the largest, and that at the 135° point takes the second largest. That at the 90° point is the least, thus the radius angles between exciting point and receiving point should be 90°, which has been recommended in the Technical Code for Testing of Building Foundation Piles (JGJ 106-2003).

The frequency and peak value of high-frequency interference vary with the pile radius and impulse width. Figure 6 shows the velocity response spectrums of pipe piles with different radii, where the radius is the average of inner and outer radius. The figure shows that the frequencies of high-frequency interference of pipe piles with 0.3 m , 0.4 m, and 0.5 m radius are 2010 Hz, 1500 Hz, and 1210 Hz, respectively. The larger the radius of the pile, the smaller the frequency of high-frequency interference, but the variation of peak values can be ignored.

The velocity response spectrums of the pipe piles with different impulse widths at 90° angle are shown in Fig. 7. The frequency of high-frequency interference of each curve is 1370 Hz, but the peak values have rather great differences. The larger the impulse width, the smaller the peak value. It indicates that the wide impulse is propitious to weaken high-frequency interference; thus, soft hammerhead should be chosen to obtain wider impulse.

Peak value and arrival time of incident wave on top of pile

Calculation method of peak values and arrival time of incident waves

The peak value and arrival time of incident waves can be calculated by Eq. (2). It is difficult to obtain the explicit expressions of peak value and arrival time for the complexity of this equation, so the numerical method and results are presented in this paper.

In Eq. (2), given z=0, then differentiates the equation by time, the acceleration expression is obtained:

(3){\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_z(\mathrm{0},\theta ,t)}{\partial t^{\mathrm{2}}}}={\displaystyle \sum _{n=\mathrm{0}}^{\infty }{\displaystyle \sum _{m=\mathrm{0}}^{\infty }\cos (m\theta )T_{\mathrm{1}mn}^{″}(t).}}

When coordinate q is known, given:

(4){\displaystyle \sum _{n=\mathrm{0}}^{\infty }{\displaystyle \sum _{m=\mathrm{0}}^{\infty }\cos (m\theta )T_{\mathrm{1}mn}^{″}(t)=\mathrm{0.}}}

The zero points of acceleration in Eq. (4) t₁, t₂,..., t_k can be obtained by numerical method, in which t₁ is the time of incident wave crest on the velocity response curve. The peak value of the incident wave crest can be calculated by Eq. (2), where t=t₁, t₂,..., t_k are the time of high-frequency wave crest and reflected wave crest from pile bottom.

Reference [8] also shows the method to calculate the arrival time of incident wave and the test results. The comparison of calculational results and test results [8] with the calculational results in this paper is listed in Table 1. In calculation, the outer diameter of pipe pile is 400 mm, the thickness of wall is 90 mm, the pile length is 8 m, C_R is 2529 m/s, and the width of exciting impulse is 0.3 ms. The results in Table 1 show that the calculational results in this paper can well match the results in Ref. [8], indicating that the method in this paper is reliable.

Effect of radius and exciting impulse width

The arrival time of incident wave is related to the pile radius and exciting impulse width. Figures 8 and 9 show the arrival time of incident wave peaks at different points on top of pile with different radii and different impulse widths. Figure 8 shows that the larger the radius, the farther the transmitting distance of stress wave, and as a result, the longer the arrival time. Figure 9 shows that the wider the exciting impulse, the longer the arrival time of incident wave peaks. The figure also presents that the arrival time of incident wave peaks does not vary with q linearly as the expression in Ref. [8]. Within the angle of 0° to 135°, the arrival time of incident wave peaks gets longer and longer as q increases, whereas within the angle of 135° to 180° the curves parallel the abscissa axis, indicating that during the period the arrival time is almost the same and not lagged.

The incident wave on top of pile is the superposition of waves with different wave velocities, which is formed at the exciting point and then transmitted towards the 180° point along pile wall clockwise or counter-clockwise. The wave crest within the angles of 0° to 135° is the superposition of waves from the same side, thus the arrival time of incident wave peaks is longer than the increase of q. However, within the angles of 135° to 180°, when the slow stress wave from the same side has not come, the speedy stress wave from another side has come; as a result, the hysteresis of the arrival time of incident wave peaks is no longer enhanced. Particularly, the superposition of waves at the 180° point is more complex [8], thus the arrival time of incident wave peaks may be earlier than the 135° point.

If the width of exciting impulse is narrow, the component of stress wave will be uniformly distributed, and the hysteresis within the angles of 135° to 180° no longer presents and the arrival time of incident wave peaks varies with q linearly (the 0.5 ms impulse curve in Fig. 9). If the pile radius is not very large, the transmitting distance of stress wave is not very long, thus the hysteresis also disappears, which can be explained by the results in Table 1.

Figures 10 and 11 show the peak values of incident wave peaks at different points on top of pile with different radii and different impulse widths, respectively. Generally speaking, because of the influence of three-dimensional effect and damp effect, the stress wave will decay as the propagation along circumference, and the further the propagation distance, the more the decay time. The curve within the angles of 0° to 90° can meet the above speciality, but within the angles of 90° to 180°, the peak values gradually increase but not decay, which can be interpreted by vibratory modals. Both the study of Gazis [6] and Eqs. (1) and (2) in this paper show that the displacement and velocity responses vary with cosmθ along circumference. At the 90° point,the values of all the odd modals are 0, therefore the dynamic responses at this point is the smallest. Whereas at the 180° point, the value of cosmθ is 1 or -1, so the dynamic responses of this vibratory modal are always the largest. As a result, the peak value of the incident wave at the 180° point is larger than that at the 90° and 135° points.

Applicability of plane-section assumption

The longitudinal bend modal is very easy to be excited in low-strain testing of pipe pile because the load is nonaxisymmetrical, so the plane-section assumption is difficult to match. As for solid pile, the plane-section assumption cannot be well matched near pile top, and it will be satisfied faraway. As for pipe pile, the plane-section assumption is not gradually satisfied as the depth increases. The criteria for plane cross section adopted in Refs. [3, 8] is: at the time the first peak at the point below the exciting point arrives, if the difference between the largest and the smallest velocity responses on the section is less than 10%, the assumption is valid. However, the results in this paper indicate that even though the first peaks at the same section are completely matched, the waveform will fork after it. Figures 12 and 13 show the velocity responses at different points on the section of 5 m in depth and pile bottom. The results indicate that the first peaks have a little difference if the depth is not very deep. It will match quite well if the depth is deeper, but after the first peak the dynamic responses is not the same. Moreover, the reflected wave peak from pile top or pile bottom has biggish distinction because the longitudinal bend modal has been excited. As the depth increases, the phenomenon of fork will be weakened but will not disappear (see Fig. 13). The velocity responses at the points of pile bottom also have difference. So long as the longitudinal bend modal is excited, the plane cross section assumption is difficult to be matched and the plane-section assumption is not suitable for pipe pile. On the other hand, the inadaptability of the plane-section assumption does not affect the application of low-strain testing on pipe piles, since only the dynamic responses on pile head are concerned and not the dynamic responses on the section below pile top. The former study shows that the arrival time of reflected wave from pile bottom at different points on pile top is the same, so the pile length or wave velocity can be correctly calculated by amending the arrival time of incident wave.

Conclusions

1) Because of the interference of high-frequency waves, on the velocity response curves many high-frequency oscillatory wave crests form before the reflected wave from pile bottom arrives. The high-frequency interference waves form the wave crest and wave hollow at the same time, which is just like the property of “stationary point”.

2) The frequency of high-frequency interference is dependent on the position of receiving point. The frequency of high-frequency interference at the 45° and 135° points is just half of that at the 90° point. The larger the radius, the smaller the frequency of high frequency interference, but the peak value has little difference. The larger the impulse width, the smaller the peak value. It indicates that the wide impulse is propitious to weaken high frequency interference; thus, the soft hammerhead should be chosen to obtain wider impulse.

3) The peak value and arrival time of incident wave on top of pile vary with the radius angles between exciting point and receiving point. The calculational results in this paper can well match the results in Ref. 8, indicating that the method in this paper is reliable.

4) Within the angles of 0° to 135°, the arrival time of incident wave peaks gets longer and longer as q inceases, but within the angles of 135° to 180° the curves parallel the abscissa axis, indicating that during the period the arrival time is almost the same and not lagged. The peak value of incident wave at the 180° point is larger than that at the 90° and 135° points.

5) Even though the first peaks at the same cross section can be completely matched, the waveform will fork after it. The longitudinal bend modal is very easy to be excited in low-strain testing of pipe piles, so the plane-section assumption is difficult to be satisfied.

References

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