Keywords

1 Introduction

Mountainous regions are susceptible to landslides, such as rock fall, avalanches, and debris flows (Froude and Petley 2018). To mitigate the effects of landslides, installing physical barriers on hillsides in densely populated mountainous areas, including Korea, Japan, Hong Kong, and Taiwan, will be increasingly important for protecting human life and infrastructure. For example, in Hong Kong, hundreds of barriers have been installed on hillsides in recent years. Under the current Landslip Prevention and Mitigation Program that was commissioned in 2010 by the Hong Kong Government, several new barriers are scheduled to be installed each year. Barriers not only require robust designs, but they also need to be sustainable. For instance, installing single large reinforced concrete barriers at the base of a slope is decreasingly viable in densely populated areas because of the challenges of land scarcity and the need to preserve the natural environment. Instead, the flexible barriers that blend in well with their natural surroundings are becoming the more sustainable and favorable option (Song et al. 2019a; Ng et al. 2017, 2020). Figure 1a shows the schematic of a typical flexible barrier with its superstructure and foundation. The superstructure, including the net, post and cables, directly arrests a dynamic landslide mass and transfers the load to the foundation. Figure 1b shows a flexible barrier installed in Kwun Yam Shan, Hong Kong.

Fig. 1
2 parts. A. A diagram of the typical flexible barrier labels the landslide mass, foundation anchor, cable, post, and net. B. A photograph presents a human looking at the flexible barrier installed in Kwun Yam Shan.

Landslide-resisting flexible barriers. (a) Schematic of a typical flexible barrier, and (b) the flexible barrier installed in Kwun Yam Shan, Hong Kong (photo from HKIE 2020)

Full size image

Over recent years, extensive research efforts have been placed on the optimization of the superstructure of landslide-resisting flexible barriers (Song et al. 2019b; Choi and Goodwin 2020; Vicari et al. 2022; Wang et al. 2022; Ng et al. 2016; Song et al. 2018; Kong et al. 2021). However, the design and optimization of the foundation of a flexible barrier has received little to no attention despite field evidence of anchor failure under dynamic loading (Yu et al. 2019; Margreth and Roth 2008). Flexible barrier foundations are generally constructed as grouted anchors, which mobilize shear resistance against the pullout force from cables attached to the net (Cerro et al. 2016). Existing guidelines suggest that foundation anchors should be designed and tested based on quasi-static theories (Stelzer and Bichler 2013; Cerro et al. 2016). However, the dynamic impact load imposed on a foundation anchor abruptly increases to a peak load before diminishing in less than a second (Bertrand et al. 2012; Koo et al. 2017; Yu et al. 2019, 2021; Zhao et al. 2020; Song et al. 2019a). More importantly, field evidence shows that the load distribution along anchors subjected to dynamic loading differs drastically compared to that of quasi-static loading (Platzer et al. 2020). Evidently, there is a need to assess whether the existing design and test methods are adequate for dynamic loading conditions.

Researchers have attempted to extend the quasi-static pullout theories to consider the effects of loading rate on the dynamic penetration of piles and pullout of anchors in dry sand. The effects of loading rate have been reported to have minimal to significant influence on shaft resistance (Dayal and Allen 1975; Heerema 1979; Leprt et al. 1988; Tan et al. 2008; Huy et al. 2005; Aprialdi et al. 2014) depending on the surface roughness and pullout velocity. It is worthwhile to mention that due to practical reasons, these studies were commonly conducted in idealized lab conditions to obtain a controlled rate of loading. More importantly, in the above-mentioned studies, dry sand was used. Therefore, the effects of pore pressure change under dynamic loading were ignored.

The importance of pore pressure change can be shown by comparing dynamic element test results in dry and saturated sand. For dry sand, triaxial tests suggest only a 10% increase in strength from dynamic loading in timescales of less than a second (Casagrande and Shannon 1949; Whitman and Healy 1962; Lee et al. 1969; Schimming et al. 1966). Such a small increase may be considered negligible for engineering applications (Tan et al. 2008). In contrast, triaxial tests and direct shear tests conducted using saturated sand showed that the pore pressure change during tests can significantly influence soil strength, even in drained tests (Schimming et al. 1966; Seed and Lundgren 1954). More specifically, dynamic loading does not allow enough time for the generated excess pore pressure to fully dissipate, resulting in a partially drained condition (Shen et al. 2017). Evidently, the drainage condition and pore pressure change need to be carefully considered in the assessment of the dynamic pullout resistance as well.

Studies that investigate the effects of the loading rate on the pullout resistance in saturated sand are limited in the literature. Kyparissis (2016) pioneered pullout tests for a rough pipe buried in saturated sand subjected to different pullout velocities. In each experiment, the pore pressure at the soil-pipe interface and pullout force were measured. It was found that the peak negative pore pressure changed, and peak pullout force increased with pullout velocity. However, no displacement measurements at the pipe head were reported. Also, it is practically difficult to achieve a constant pullout velocity at such small displacements before the mobilization of the peak pullout force. Therefore, the pullout velocity, and thus, the loading rate could not be easily characterized. In addition, no quasi-static pullout resistances were reported, which limits the understanding of how dynamic effects contribute to the total pullout resistance. A rational analytical expression that can describe the dynamic pullout resistance in saturated sand does not yet exist in the literature.

This study aims to evaluate the dynamic pullout resistance of anchors in saturated sand to enhance the understanding of the effects of the loading rate on the pullout resistance of anchors and to improve existing design methods for the foundations of landslide-resisting flexible barriers. To achieve this, a new analytical model is proposed to estimate the peak dynamic pullout resistance by considering the effects of pore pressure change. Experimental investigations are performed by using a newly-developed experimental apparatus which can apply pullout force at designated velocities. A unique pullout test dataset is reported which explicitly includes quasi-static tests and tests in saturated sand. The proposed analytical model is evaluated by the produced dataset in this study and datasets in the literature.

2 New Dynamic Pullout Model

For quasi-static pullout, the effects of the loading rate are negligible so that the pullout resistance, F, is equal to the resistance induced by overburden, F0. For dynamic pullout in saturated sand, the effects of pore pressure change should be included:

$$ F={F}_{\mathrm{o}}+{F}_{\mathrm{p}} $$
(1)

where Fpis the pore pressure change induced resistance. Details of quasi-static and dynamic pullout are discussed below.

2.1 Overburden

Since the burial depth, as a geometric length scale, is not directly related to the pullout velocity, overburden induced resistance is considered rate independent. A common assumption adopted in quasi-static pullout resistance theories is that the shear stress distribution is uniform along the length of a horizontal anchor. Although this assumption is an idealization compared to real shear stress distributions, pullout theories developed based on this assumption have been widely accepted by engineering practitioners and produced reasonable predictions (Powell and Watkins 1991; Lazarte et al. 2015; Ostermayer and Scheele 1978). One of the most commonly used theories was firstly proposed by Schlosser and Guilloux (1981) and then further developed by GEO (2008). If cohesion is ignored, the equation for quasi-static pullout resistance only considers the soil properties and anchor geometry as follows:

$$ {F}_{\mathrm{o},\mathrm{peak}}=2 DL{\sigma}_{\mathrm{v}}^{\prime }{\mu}^{\ast } $$
(2)

where Fo, peak is the peak overburden induced resistance, D is the outside diameter of the anchor, L is the length of the anchor, \( {\sigma}_{\mathrm{v}}^{\prime } \) is the vertical stress acting on the anchor, and μ∗ is the apparent coefficient of friction. Normally, \( {\sigma}_{\mathrm{v}}^{\prime } \) is taken as γ′H, where γ′ is the effective unit weight of the soil and H is the burial depth of the anchor. Equation 2 is adopted in this study to characterize the resistance from overburden because of its practicality.

The apparent coefficient of friction, μ∗, is a key parameter that can be back-calculated from quasi-static pullout test results. In engineering design practice, μ∗is taken as tanϕ′, where ϕ′ is the friction angle of soil (GEO 2008). However, the μ∗ obtained from field measurements is usually four to 14 times larger than (Powell and Watkins 1991). This is because the soil at the rupture zone, which is a layer of soil that undergoes intensive shearing at the interface between the soil and anchor, dilates as the anchor is pulled. Dilation in turn increases the normal stress at the surface of an anchor (Luo et al. 2000).

2.2 Pore Pressure Change

The soil in the rupture zone dilates as an anchor is pulled. In saturated sand, the volume increase results in a negative pore pressure change. When the pullout is quasi-static, a drained condition applies, so that the pore pressure change may be ignored. However, when the pullout is dynamic, a partially drained condition applies so that pore pressure change should be considered. Based on the principle of effective stress, the normal effective stress acting on the surface of the anchor will increase by the same magnitude of negative pore pressure change (Terzaghi et al. 1996; Yin 2009). Assuming that the pore pressure change is uniform at the anchor surface and Coulomb’s friction law applies, the change in pullout resistance induced by pore pressure change can be calculated as follows:

$$ {F}_{\mathrm{p}}=\pi DL\varDelta {\sigma}_{\mathrm{n}}^{\prime } tan\delta =\pi DL\left(-p\right) tan\delta $$
(3)

where \( \varDelta {\sigma}_{\mathrm{n}}^{\prime } \) is the change in effective normal stress at the anchor surface, p is the pore pressure change, and δ is the friction angle of the soil-anchor interface. For a rough soil-anchor interface, δ can be taken as ϕ′ (Uesugi and Kishida 1986; Zhang et al. 2009). It is evident from Eq. 3 that to calculate Fp, the negative pore pressure change needs to be determined. However, the negative pore pressure change is rate dependent. For instance, when the pullout velocity increases, there is less time available for the excess pore pressure to dissipate. Therefore, the change in negative pore pressure increases. Consequently, the relationship between the pullout velocity and the pore pressure change needs to be determined to quantify the effect of loading rate on pullout resistance.

The generation of excess pore pressure is governed by volume change. Based on the definition of the dilation angle (Bolton 1986; Nemat-Nasser 1980), the rate of plastic volume change of the soil skeleton can be calculated as \( \dot{V^{\mathrm{p}}}=\dot{u_{\mathrm{y}}}= tan\psi \dot{u_{\mathrm{x}}}= tan\psi v \), where uy is the thickness change of the rupture zone, uxis the horizontal displacement of the anchor, ψ is the dilation angle of the soil, and v is the pullout velocity. Derived from elasticity theories by Taylor (1948), the elastic volume change rate of the soil skeleton can be calculated as \( \dot{V^{\mathrm{e}}}=2\left(1+\mu \right)\left(1-2\mu \right)\varDelta {\dot{\sigma}}_m^{\prime }h/E=2\left(1+\mu \right)\left(1-2\mu \right)\left(-\dot{p}\right)h/E \), where μ is the Poisson’s ratio of the soil, E is the elastic modulus of the soil, \( \varDelta {\sigma}_m^{\prime } \) is the change in the mean effective stress, and h is the thickness of the rupture zone. The dissipation of excess pore pressure is governed by seepage. Based on Darcy’s law (Harr 1991), the pore fluid inflow rate can be calculated as q = ki, where k is the hydraulic conductivity of the soil and i is the hydraulic gradient at the outer boundary of the rupture zone. i can be expressed as \( \frac{1}{\gamma_w}\nabla p \), where γw is the unit weight of the pore fluid. The volume of the soil particles can be assumed to be constant. More specifically, the voids created from volume change must be filled with inflowing pore fluid, which is expressed as follows:

$$ \dot{V^p}+\dot{V^e}=q $$
(4)

When the negative pore pressure change is at its peak value, −pp, the rate of change is zero, i.e., \( -\dot{p}=0 \).

Although ψ is not constant during the pullout process, ψ can be assumed to be the peak angle of dilation, ψp, at the peak pullout resistance because the peak shear strength of soil is generally correlated with ψp (Luo et al. 2000). Therefore, Eq. 4 can be rewritten as:

$$ \tan {\psi}_{\mathrm{p}}v-k{i}_{\mathrm{p}}=0 $$
(5)

From Eq. 5, the peak hydraulic gradient can be determined. The hydraulic gradient at the rupture zone is assumed to be representative of the hydraulic gradient at a short distance from the rupture zone (i.e., less than 10% of the distance to the free surface). As such, the negative pore pressure change in the rupture zone can be estimated as follows:

$$ -p=-{p}_m+ is{\gamma}_w $$
(6)

where −pm is the measured negative pore pressure change and is the distance from the point of measurement to the rupture zone. However, it may not be feasible to obtain measurements of pore pressure changes in routine engineering practice. Therefore, the negative pore pressure change needs to be estimated without measurements.

It is difficult to explicitly capture the pore pressure distribution. This is because the generation and dissipation of excess pore pressure are highly transient since the time scale of a dynamic load may be less than a second (Bertrand et al. 2012; Koo et al. 2017; Yu et al. 2019, 2021; Zhao et al. 2020). Moreover, the pullout velocity and angle of dilation of the soil in the rupture zone, which drives the change in pore pressure, varies non-linearly from zero to a maximum value during the loading process (Hossain and Yin 2015). Therefore, analytically calculating the pore pressure change under such transient seepage condition may not be feasible. Instead, a conservative but semi-empirical method, which is modified from steady state seepage solutions (Fernández and Alvarez Jr. 1994) to account for transient seepage, is proposed to estimate negative pore pressure change:

$$ -p=\eta i\frac{r^2}{4H}\frac{\left(1+\frac{4{H}_{\mathrm{w}}^2}{r^2}\right)\ln \left(1+\frac{4{H}_{\mathrm{w}}^2}{r^2}\right)}{\frac{2H}{r}}{\gamma}_w $$
(7)

where η is an empirical parameter accounting for the transient seepage, Hw is the distance from the anchor to the free surface of pore fluid, and r is the radius of the anchor. The value of η depends on the soil properties and the timescale of the problem. To develop the same hydraulic gradient, transient seepage requires lower pore pressure in the rupture zone compared to steady state seepage. Thus, the upper limit of η is unity, which represents a steady-state seepage condition. This value of η should provide an upper bound for the negative pore pressure change.

3 Physical Experiments

3.1 Dynamic Pullout Experimental Apparatus

A new experimental setup (Fig. 2) was developed to investigate the dynamic pullout resistance of a soil anchor and evaluate the proposed analytical model. This setup consists of two parts. The first part of the setup (Fig. 3a) is a box with dimensions of 1.2 m, 0.2 m, and 0.3 m in length, width, and height, respectively. Two plates, each with a thickness of 10 mm, were installed 0.7 m apart orthogonally to the pullout direction. The space in between the dividing plates forms a soil container with dimensions of 0.7 m, 0.2 m, and 0.3 m in length, width, and height, respectively.

Fig. 2
An isometric view of the pullout experimental apparatus. It labels the electric motor, model anchor with soil coating, pulley, and 0.7 L times 0.2 W times 0.3 H soil container.

Isometric view of the pullout experimental apparatus

Full size image
Fig. 3
3 grayscale photographs. A. It lables the rare end plate with opening, soil container, rare dividing plate with an opening, and others, along with geometric dimensions. B. It indicates the electric motor, gear box, and aluminum frame. C. It labels the anchor head, load cell, pulley, and others.

Details of the pullout experimental apparatus (all dimensions in m): (a) soil container, (b) electric motor, and (c) connection and instrumentation at the inclusion head

Full size image

The soil container is filled with sand before each test. At 10 mm beneath the centroid of each dividing plate, there is a circular opening with a diameter of 16 mm. An anchor, which is a steel bar with a length of 0.75 m and diameter of 16 mm was threaded through the circular openings. A layer of Toyoura sand was glued to the anchor surface to achieve a rough interface condition (Nazir and Nasr 2013; Milligan and Tei 1998). An eyebolt was connected to the head of the anchor. The second part of the setup consists of a 40-watt power electric motor mounted to the vertical aluminum frame (Fig. 3b). A digital speed governor and gearboxes with different reduction ratios were used to control the rotational speed of the motor. A steel wire is connected to the electric motor and the eyebolt using a pulley. When the motor is powered on, the steel wire pulls the anchor at different preset velocities.

3.2 Instrumentation

Figure 3c shows a laser distance sensor mounted near the pulley to measure the distance from it to a target plate, which was attached to the anchor head. Therefore, the head displacement and pullout velocity can be measured. To measure the pullout resistance, a load cell with a capacity of 500N was attached to the anchor head. The mobilized resistance is considered equal to the pullout force because the inertial force was estimated to be less than 2% of the pullout force, which can be considered negligible. To obtain pore pressure measurements, two pore pressure transducers (PPTs) with a capacity of 100 kPa and a precision of 1 kPa were placed in the sand surrounding the anchor. One of the PPTs was placed on the top, and another PPT was on the side of the anchor. The PPTs were positioned at a radial distance of 12 mm from the anchor.

3.3 Modelling Procedures and Test Program

To obtain pore pressure measurements, two pore pressure transducers (PPTs) with a capacity of 101 kPa were placed in the sand surrounding the anchor. One of the PPTs was placed on the top, and another PPT was on the side of the anchor (Fig. 3d). The PPTs were positioned at a radial distance of 12 mm from the anchor.

Toyoura sand with D50 = 0.18 mm and ϕ′ = 34° was used in the experiments. In each test, the sand was prepared in three layers in the container. Dry (Schiavon et al. 2016; Hao et al. 2019) and wet pluviation (Levesque 2003; Kim and Kim 2019) methods were used for the dry and saturated tests, respectively, to achieve a target unit weight of γ = 14.65 kN/m3. Under this state, the soil exhibits dilative behavior under shearing, and its peak dilation angle, is estimated to be 12° (Yang and Li 2004), and the hydraulic conductivity is estimated to be 0.06 cm/s (Yoshimi et al. 1975; Amer and Awad 1974). The anchor and PPTs were installed at the target depths. After the sand is prepared, the instrumentation and data logger is initiated. The speed governor is preset to a designated speed, and the motor is powered on. The anchor is then pulled by the steel wire.

In this study, pullout tests were conducted at velocities from 4 mm/s to 60 mm/s in both dry and saturated sand. The pullout velocities were back-calculated by considering the typical dynamic loading duration as reported in the literature (Yu et al. 2019; Platzer et al. 2020; Pradhan 2003; Lum 2007). A typical duration of 0.1 s and displacement of 2 mm gives an approximate pullout velocity of 20 mm/s. Table 1 gives a summary of the experimental program.

Table 1 Test program
Full size table

4 Result Interpretations and Discussions

4.1 Rate Effects from Dynamic Pullout

Figure 4(a, b) show comparisons of the normalized pullout resistance with head displacement at different pullout velocities in dry and saturated sand, respectively.

Fig. 4
2 multiline graphs of normalized pullout resistance versus head displacement. A plots 4 lines for 60, 25, and 4 millimeters per second and quasi-static, with ascending to nearly horizontal trends. B has 6 lines for 50, 35, 26, 16, and 4 millimeters per second and quasi-static, with ascending trends.

The normalized pullout resistance against the head displacement for tests conducted at different velocities in: (a) dry sand, and (b) saturated sand

Full size image

The normalized pullout resistance is defined as the measured pullout resistance divided by the peak quasi-static pullout resistance. It is noted that the magnitude of the pullout resistance of an anchor in saturated and dry sand cannot be directly compared with each other since they have different effective overburden stresses.

For each test, it can be observed that the pullout resistance increases to a peak value before decreasing. This is related to the initial increase and subsequent decrease of the mobilized friction and dilation angles of the soil in the rupture zone as the pullout displacement increases (Hossain and Yin 2010; Borana et al. 2016). The displacement required to mobilize the peak pullout resistance is referred to the critical shear displacement (Luo et al. 2000), which is governed by the shape and size of the granular material adopted (Billam 1972). Based on all tests, the critical shear displacement is from 2.0 to 2.5 mm. This range agrees with other studies that investigate the quasi-static pullout resistance of anchors (Chang et al. 1977; Luo et al. 2000). This finding, together with an obvious peak resistance, implies that the formation of the rupture zone is consistent for both dynamic and quasi-static loading in both dry and saturated sand. When comparing tests conducted at different pullout velocities, the pullout resistance increases with pullout velocity for both dry and saturated tests. This finding agrees with previous dynamic pullout tests (Kyparissis 2016; Tan et al. 2008; Aprialdi et al. 2014; Jenck et al. 2014), which attributed the increase in pull out resistance to rate effects. A comparison between Fig. 4(a, b) reveals that rate effects in saturated sand are more profound than those in dry sand. This phenomenon is because of the effect of negative pore pressure change from dilation. Evidently, the effects of pore pressure change must be considered when assessing dynamic pullout resistance.

Figure 5 shows the measured pore pressure changes with head displacement for the saturated sand tests. For each test, the negative pore pressure change increases abruptly before gradually decreasing with displacement. Additionally, the displacement that corresponds to the peak negative pore pressure change approximately coincides with the aforementioned critical shear displacement. This is because the change in pore pressure is driven by the initial increase and subsequent decrease of the dilation angle of the soil in the rupture zone. Before the critical shear displacement is reached, excess pore water pressures are generated because of the mobilization of the dilation angle. After the critical shear displacement is reached, the effect of soil dilatancy diminishes, and the effects of the dissipation of excess pore pressure becomes more apparent. When comparing tests conducted at different pullout velocities, the measured negative pore pressure change generally increases with pullout velocity. This is because a higher pullout velocity generates excess pore pressures at shorter timescales compared to that required for dissipation (i.e. towards undrained conditions), leading to more significant changes in pore pressure. The similarities of the above-mentioned trends observed between the pullout resistance and pore pressure changes show that the negative pore pressure change increases normal effective stress and thus pullout resistance, which is supported by Eq. 3. This implies that anchors installed in ground with high ground water tables will be more susceptible to rate effects.

Fig. 5
A multiline graph of measured negative pressure change versus head displacement. It plots 5 lines for 50, 35, 26, 16, and 4 millimeters per second. The lines have ascending to descending trends. 50 millimeters per second has the highest peak among others.

Measured negative pore pressure change against the head displacement for tests conducted at different velocities in saturated sand

Full size image

4.2 Model Evaluation

Figure 6 shows a comparison of measured and calculated normalized peak pullout resistance (Fpeak/Fo, peak) in saturated sand. Furthermore, the proposed model is evaluated against existing quasi-static ones. The peak overburden induced resistance, Fo, peak, is determined from the quasi-static tests conducted in this study. The peak overburden induced resistance is found to be 71N for experiments carried out in saturated sand. Therefore, the apparent friction coefficient, μ∗, can be back-calculated as 4 for saturated sand. This value agrees with the range reported from field quasi-static pullout experiments (Powell and Watkins 1991; Schlosser and Guilloux 1981). Experimental measurements from Kyparissis (2016), who conducted dynamic pullout tests in saturated sand, are shown for comparison. It is worthwhile to note that no quasi-static test data was reported in their study. Thus, Fo, peak is estimated by linear extrapolation of the pullout resistance with the pullout velocity. At each burial depth, the intercept of the linear extrapolation is considered to be Fo, peak. The peak quasi-static pullout resistance based on overburden (i.e., Eq. 2) is the same regardless of the pullout velocity. In contrast, the measured data, at different pullout velocities, exhibits an increase in resistance with pullout velocity. At a pullout velocity of 70 mm/s, the effects of the loading rate increase the pullout resistance by 200% compared to the overburden resistance.

Fig. 6
A scatter plot has 4 data points for measured of the proposed study and Kyparissis 2016, and F o + F p + F d, peak of the proposed study and Kyparissis 2016. There is a horizontal line for F o, peak at 1.0 on the y-axis. Measured of the proposed study has the highest peak.

Comparison of measured and calculated normalized peak pullout resistance for tests in saturated sand

Full size image

Existing quasi-static pullout theories (i.e., Eq. 2) underestimate the dynamic pullout resistance. For a soil anchor, such a significant underestimation may cause engineers to overlook the fact that the anchor can rupture before the failure of the interface between soil and anchor. For example, for a typical 13-m long anchor foundation of a debris flow resisting barrier, a soil-anchor interface strength was calculated to be about 313 kN (i.e., D= 125 mm, H = 8 m, tanϕ′ = 34°, and γ′ = 18 kN/m3). To prevent tendon failure, engineers designed a tendon with a capacity of 519 kN. However, under dynamic loading, the soil-anchor interface strength can increase to 1000 kN, which largely exceeds the designed strength of the tendon. Thus, a larger diameter should be assigned to the tendon. Evidently, there is a pressing need to progress towards a more rational estimation of the dynamic pullout resistance of anchors in saturated sand.

In saturated sand, the effects of pore pressure change should be considered in dynamic pullout. Equation 3 shows that the normal effective stress in the rupture zone is affected by changes in pore pressure. In this study, PPTs were not installed in the rupture zone to avoid disturbance to it. However, the pore pressure change measured by the PPT can be used to estimate the pore pressure change in the rupture zone by using Eq. 6. For the data reported by Kyparissis (2016), the PPT was placed at the anchor surface. Thus, the pore pressure change measured by the PPT can be regarded as that in the rupture zone. Correspondingly, the pore pressure induced resistance can be calculated by using Eq. 3. The normalized dynamic pullout resistance calculated by using Eq. 1 is shown in Fig. 6. The calculated values show the same trend and order of magnitude as the measured values. It is shown that Eq. 1 can be used to estimate the dynamic pullout resistance and Eq. 3 can be used to explain the effects of negative pore pressure change.

To evaluate Eq. 7, the calculated peak pullout resistance by the different proposed methods (i.e., with and without in-situ pore pressure measurement) and measured peak pullout resistances are compared in Fig. 7. The calculated normalized peak pullout resistance (Fpeak/Fo, peak), with reference to pore pressure measurements, shows close agreement with the measured data.

Fig. 7
A scatter plot of the calculated versus measured normalized peak pullout resistance. It has 4 data points for this study and Kyparissis 2016, plotted along an ascending reference line. There is an inset scatter plot that plots data points for this study and Kyparissis 2016, with ascending trends.

Comparison of calculated normalized peak pullout resistance

Full size image

As discussed, compared to the scatter of the data points in this study, a larger scatter for the external dataset (Kyparissis 2016) may be related to its lack of clear displacement measurements, and thus characterizable pullout velocities. If Eq. 7 is used without reference to pore pressure measurements, the empirical parameter, η, is determined by regression analysis to be 0.6 for this study and 0.4 for the external dataset. It is shown that the normalized calculated peak pullout resistance provided by Eq. 7 also produce close agreement with the measured values. Thus, Eq. 7 can be used to estimate pore pressure change reasonably without the need for additional measurements. If a value of unity is adopted for η, then Eq. 7 may be used as an upper bound for estimating the dynamic pullout resistance in all of the test cases in this study. Nevertheless, more experiment data can help to optimize the value of η.

5 Discussion

5.1 Radiation Damping

The measured and calculated normalized peak pullout resistance for dry sand is shown in Fig. 8.

Fig. 8
A scatter plot of normalized peak pullout resistance versus pullout velocity. It has 5 data points for the proposed study, Jenck et al 2014, Aprialdi et al 2014, Tan et al 2008, and F o + F d peak of proposed study. There is a horizontal line at 1.0 for F o peak. It indicates no failure in 2 tests.

Comparison of measured and calculated normalized peak pullout resistance for tests in dry sand

Full size image

As discussed, the peak overburden induced resistance, Fo,peak, is determined from the quasi-static tests conducted in this study. The peak overburden induced resistance is 157N for tests carried out in dry sand. Therefore, the apparent friction coefficient, μ∗, can be back calculated as 3 for dry sand. This value is in the range reported from quasi-static pullout experiments in the field (Powell and Watkins 1991; Schlosser and Guilloux 1981). In addition, dynamic pullout studies (Tan et al. 2008; Jenck et al. 2014; Aprialdi et al. 2014) in dry sand are also shown for comparison. Linear extrapolation of the pullout resistance was used to obtain Fo,peak for data where quasi-static pullout data was not reported (Aprialdi et al. 2014; Jenck et al. 2014). In this study, at a pullout velocity of 40 mm/s, the pullout resistance increased 40% compared to the quasi-static condition. In the study of Jenck et al. (2014), at a pullout velocity of 550 mm/s, the pullout resistance increased 100% compared to the quasi-static condition. Compared to tests in saturated sand (Fig. 6), the importance of pore pressure change is once again highlighted. Moreover, considering that the effects of pore pressure change are negligible in dry sand, there may also be other rate dependent resistance in dry sand.

It is proposed in the literature that radiation damping is a possible effect of dynamic loading on soil-anchor interaction (Tan et al. 2008). When a structure interacts with soil, elastic waves are generated. If a soil remains elastic, the energy carried by waves transmitted through an open boundary will dissipate. This energy dissipation phenomenon is called radiation damping and can be mathematically approximated by using dashpots (Dobry 2014; Novak 1974). Analytical solutions for a vibrating anchor in an infinite elastic soil medium have been reported (Nogami and Novak 1976; Novak et al. 1978; Gazetas and Makris 1991). During pullout, it is assumed that elastic shear waves are generated. However, the presence of a plastic rupture zone violates the assumption of an elastic soil. Therefore, the above-mentioned analytical solutions cannot be directly applied to a pullout problem. To address the above-mentioned shortcoming and quantify the damping effect on the pullout resistance of anchors, Tan et al. (2008) proposed an apparent damping coefficient:

$$ {F}_{\mathrm{d}}=L{c}^{\ast }v $$
(8)

where Fd is damping induced resistance and c∗ is the apparent damping coefficient.

To adopt the apparent damping coefficient into this study, Fig. 9 shows the measured damping induced resistance against the pullout velocity for dry sand. The damping induced resistance can be calculated by subtracting the overburden induced resistance from the dynamic pullout resistance. A linear fit, corresponding to c∗ = 1500 Ns/m/m, can describe the relationship between Fd per unit length of anchor and v. The deduced damping value falls within the range reported by Tan et al. (2008). In addition, the peak frictional resistance calculated by (Fo + Fd)peak (i.e., c∗ = 1500 Ns/m/m) is shown in Fig. 8. It can be found that damping induced resistance from dynamic loading can be well captured in dry sand.

Fig. 9
A scatter plot of F peak over F o, peak versus v. It has 4 data points for the measured of proposed study and Kyparissis 2016, and F o + F p + F d peak of the proposed study and Kyparissis 2016. There is a horizontal line at 1.0 for F o, peak. Measured of the proposed study has the highest peak.

Measured damping induced resistance for tests in dry sand and determination of actual damping coefficient

Full size image

Considering the physical origin of radiation damping (i.e., propagation of shear waves during pullout), the actual damping coefficient calculated from the dry sand tests is assumed for simplicity to be applicable to saturated tests since the pore fluid cannot transmit shear waves. It is worthwhile to point out that the reported data from Kyparissis (2016) does not include dry sand measurements, so their dataset cannot be directly used to separately evaluate the effects of pore pressure change from soil damping. Consequently, damping induced resistance needs to be back-calculated by using linear regression between the measured peak damping induced resistance and pullout velocity.

If the damping effect (i.e., c∗ = 1500 Ns/m/m) is included, the calculated normalized peak pullout resistance is shown in Fig. 10 and the calculated peak pullout resistance by the different proposed methods is shown in Fig. 11. The close agreement between calculated and measured values suggests the theory proposed by Tan et al. (2008) may help to estimate the damping induced resistance in dynamic pullout. Nevertheless, there has been no direct measurement of elastic waves generated in dynamic pullout experiments. Therefore, more experimental data may be needed to support this theory and evaluate a rational range of the actual damping coefficient, which can hardly be calculated by closed-form equations due to the existence of the rupture zone.

Fig. 10
A scatter plot of damping induced resistance versus pullout velocity. It depicts the data points scattered along an ascending line, representing the linear regression. It presents c asterisk = 1500 Newton second per meter per m, and R squared = 0.97.

Comparison of measured and calculated normalized peak pullout resistance, considering damping effect

Full size image
Fig. 11
A scatter plot of the calculated versus measured normalized peak pullout resistance. It has 4 data points for this study and Kyparissis 2016, plotted along an ascending reference line. There is an inset scatter plot which plots data points for this study and Kyparissis 2016, with ascending trends.

Comparison of calculated normalized peak pullout resistance, considering damping effect

Full size image

6 Conclusions

The effects of dynamic loading on the pullout resistance of a foundation anchor in saturated sand were investigated by developing a new analytical model, which was then evaluated by data produced by a novel experimental setup and external datasets. Key findings may be concluded as follows:

  1. 1.

    The dynamic loading exerted on an anchor causes rate effects on the pullout resistance. The rate effects increase the pullout resistance to up to three times compared to quasi-static loading. However, existing design guidelines for foundation anchors of landslide-resisting flexible barriers adopt quasi-static pullout theories that overlooked rate effects. This can cause the unexpected tendon failure of the anchors because quasi-static design overlooks rate strengthening and assumes that the soil-grout interface is the weakest component of an anchor design. The need for the dynamic analysis of foundation anchors of landslide-resisting flexible barriers is highlighted in this study.

  2. 2.

    Dynamic loading effects are mainly attributed to the effects of pore pressure change. The pore pressure change effect is caused by a partial drainage condition, which is governed by the limited timescale for pore pressure dissipation. Because of the absence of pore pressure change, the rate effects in dry sand are less significant than those in saturated sand. Thus, for sites with anchors installed in the ground with high water tables, the rate effects are expected to be more significant and should be more carefully evaluated.

  3. 3.

    The proposed analytical model can well describe the effects of pore pressure change and provide estimations of the dynamic pullout resistance of anchors. Compared to existing quasi-static approaches, the proposed model can be used to provide a more rational estimation of the interface strength between the soil and anchor, which can contribute to scientifically designing the anchor foundation of landslide-resisting flexible barriers.

  4. 4.

    A new method is proposed to estimate the pore pressure change during dynamic pullout without the need for in-situ pore pressure measurements. This method serves as a quick and conservative guiding tool to assess the possible magnitude of pore pressure change caused by dynamic loading.