1 Introduction

Debris flows endanger humans living in mountainous regions all over the world (Hübl et al. 2009). The design of structural mitigation measures, such as check dams and drainage channels, and the maintenance of infrastructures such as bridges require estimating the potential impact forces caused by debris flows (Scheidl et al. 2013). However, no entirely mechanistically and theoretically based impact model is currently sufficiently accurate and computable within periods common for design offices (Scheidl et al. 2013). The debris flow impact models used for engineering are mostly empirical or semi-empirical formulas. The impact force of debris flows refers to a dynamic load produced by a moving debris flow process that contacts other objects; this major external force causes engineering damage to the debris flow areas of roads, bridges, and housing projects (Liu and Wei 1997; Chen et al. 2007). A debris flow is a typical solid–liquid two-phase flow. In the process of debris flow movement, the flow carries a wide particle size distribution of solid particles ranging from a few millimeters to tens of dollars. For this reason, it is difficult to use density, velocity, or deep mud to explain the differences in conditions of debris flow impacts (Hu et al. 2006). Moreover, because of the randomness of the impacting solid particles, the impact force of the debris flow is not the same under the same conditions at different locations (Iverson et al. 2010), further increasing the difficulty of studying the debris flow impact force. Over the years, determining the debris flow characteristics and impact load has been scientifically difficult, and this is still the weakness of debris flow dynamics (Wu et al. 1990). The main methods of obtaining debris flow impact force data include field measurements, model tests and numerical simulation. Measuring the impact of prototypical debris flow fields is an important means of obtaining reliable data. Systematic observation of debris flow was conducted at Kamikamihori valley on the eastern slope of Mt. Yakedake in 1980, using a synthetic observation system equipped with many instruments and introducing various topographical surveys (Okuda and Okunishi 1980). Long-term observations were conducted using the natural advantages of Jiangjia Gully (Yunnan Province, China), where debris flows frequently occur. The time-history curves of the debris flow impact force were measured, and the eigenvalues of the debris flow impact force were analyzed (Zhang and Yuan 1985; Zhang 1993). As another example, debris flow impact piers were built in Jiangjia Gully (Yunnan Province, China). The time-history curves of the debris flow impact force were acquired using sensors and a microcomputer, and the debris flow impact load was taken into a sawtooth pulse, rectangular pulse and spike pulse (Wu et al. 1990). Field test devices were also established in Jiangjia Gully. Through the newly developed force sensors and data acquisition devices, the original impact signals at different positions of deep mud, long duration, and full waveform were measured and the real debris flow impact data were obtained by filtering the original signal (Hu et al. 2006, 2011). In situ testing of debris flow was also conducted (König 2006; Wendeler et al. 2007). In addition, the impact force test of large-scale debris flow has been studied (DeNatale et al. 1999; Bugnion et al. 2012; Wendeler and Bugnion 2012). Using the large-scale debris flow test or monitoring stations to test the in situ observations of debris flow impact has the advantage of not needing to consider the scaling problem, and some important data can be obtained. However, the ability to control the boundary condition to acquire data is limited. For example, the occurrence frequency of a debris flow time is unknown, and the installation and operation of measuring equipment is the main drawback of natural debris flow observations. For large-scale debris flow tests, the expected volume of the material can lead to a too-costly measurement design; furthermore, the fluid and material parameters are often unpredictable (Scheidl et al. 2013).

Because of the above reasons, a large number of scholars have designed different physical models of debris flow impact tests based on the scaling consideration to study the impact of debris flow. Experimental research on the debris flow head impacting dams was conducted using flume tests: the composition of the impact force peak was analyzed, the calculation formula of the impact force of the debris flow head was established, and the vertical distribution of the debris flow impact force on the dam was determined (Wei 1996). Based on dynamic equilibrium analysis, the debris flow hydrodynamic pressure was analyzed using flume tests (Armanini and Scotton 1993). A larger-scale debris flow ditch test model was constructed, and a series of debris flow impact characteristics tests were conducted. The influences of the debris flow slurry viscosity, solid particle size and solid phase on the debris flow impact characteristics were studied, and some useful conclusions were obtained (Chen et al. 2010; He et al. 2013, 2014a, b). The debris flow velocity and impact characteristics were explored by flume tests and the relationship between the surface velocity of debris flow and the surface layer impact force was revealed (Yang et al. 2011). Model tests of the debris flow impact properties were performed, and the maximum impact of the debris flow was obtained by analyzing the experimental data (Scheidl et al. 2013). Debris flow impact tests were conducted using large flume tests. The original impact force signal obtained was subjected to wavelet denoising, and the calculation formula of the debris flow liquid slurry impact, \(p = 0.5\rho_{f} v^{2}\), was studied (Tang et al. 2013). The debris flow impact force was measured by the debris flow impact model test, and the debris flow impact speed ranged from 2 to 13 m/s (Bugnion et al. 2012). Impact force of viscous debris flow was measured by a miniaturized flume test and the impact force could be divided into three phases, i.e. the sudden strong impact of the debris flow, continuous dynamic pressure of the body and slight static pressure of the tail (Zeng 2014; Cui et al. 2015). The particle flow of the flume test was conducted first, and then the experiment was simulated by the discrete element software PFC 2D. Finally, the run-out distance and impact spectral characteristics of the particle flow were obtained (Valentino et al. 2008).

Based on previous studies, the three main factors of debris flow bulk density, drainage channel slope and upstream surface gradient of the check dam were selected as the main factors to carry out the orthogonal experiment design. The influence degree of the above three factors for debris flow impact force was obtained by range analysis and variance analysis. Research results can provide technical data and theoretical support for optimizing the design of check dams.

2 Debris Flow Impact Model

A corresponding adjustment coefficient of λ can be multiplied to consider the effect of stones in the debris flow. Taking the empty check dam as an example the impact pressure of the debris flow fluid can be expressed as:

$$\delta = \lambda \frac{{\gamma_{c} }}{g}v_{c}^{2} \sin \alpha$$
(1)

where δ is the overall impact pressure of the debris flow fluid (kPa); v c is the section-averaged debris flow velocity (m/s); γ c is the debris flow fluid density (kN/m3); g is the gravitational acceleration; α is the angle between the buildings and the stress surface of the debris flow impact pressure direction (°), where sinα is equal to 1 with vertical impact; and λ is the building shape coefficient, where λ is equal to 1.0 for circular buildings, 1.33 for rectangular buildings, and 1.47 for square buildings. For the check dam, λ of 1.47 is suitable for the water impact, and the value should increase when considering the impact of water and stones. When D ≤ 0.5 m, λ is 1.47; when D = 3 m, λ is 4; and when D > 3 m, λ is no greater than 8; the values for other cases can be obtained by linear interpolation (Jiang 2014).

3 Orthogonal Experimental Design

3.1 Orthogonal Testing Method

The orthogonal experimental design method is a highly efficient way capable of dealing with multifactor experiments and screening optimum levels by using the orthogonal design table, experimental results of which are studied by statistical method(Jiang 1985). Before making an orthogonal design table, reasonable and representative levels of all factors are determined at first according to theories or a few experiments. And then experiments represent all the level groups of the experimental factors are performed. Positive and negative factors and their impact degrees (ID) to the objective of production are revealed by calculating the experimental results, e.g. conversion and yield. The possible optimum level can be concluded according to the impact of the factors. At last, a confirmatory experiment is performed following the concluded optimum level (Tang and Feng 2002). Being compared with the general test method, orthogonal test method has two outstanding advantages. Firstly, in the arrangement, the number of experiments is reduced as far as possible. Secondly, on the basis of test data obtained by fewer times of experiments, the correct conclusion of practical guidance can be given, which is always a better result (Jia et al. 2014). For example, for an experiment with four factors and four levels of each factor, an orthogonal design table L16(44) could be used, and the experiment program only contains 16 level groups, reflecting the overall situation of the comprehensive experiment containing 256 level groups in all. Thus it is much easier to find out the optimum level group. (Jia et al. 2014, Ji et al. 2014).

3.2 Optimal Factors and Orthogonal Table

The first aim of the orthogonal experiments was to find the most important experimental factor on the performance of debris flow impact force. The second aim was to detect the optimum operational parameters to obtain the minimum mean value of impact force. Before arranging the orthogonal experimental matrix, the reasonable and condensed levels of each experimental factor should be chosen.

According to the above impact pressure expression 1 and preliminary experimental experiences, these three factors shown as follows: L (drainage channel slope), M (upstream surface gradient of the check dam), N (debris flow bulk density) are chosen as the experimental factors, which are shown in Table 1. Each factor includes three levels as: L: 9°, 12° and 15°; M: 1:0, 1:0.15, and 1:0.3; N: 14.2, 18.1, and 19.5 kN/m3.

Table 1 Factors of the orthogonal table and their levels
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There are three experimental factors and three levels for each factor, choosing an orthogonal table is the other important work. Orthogonal tables play an important role in the arrangement of the whole test process and results, which are considered to be the most important part of the whole orthogonal experiment design. As the factors and their levels are shown in the Table 1, an orthogonal table noted as L9(34) (shown in Table 2) can be chosen as the experimental scheme to arrange the experiments. “9” stands for the orthogonal table rows, namely the maximum number of tests and each row of orthogonal table represents a run with a specific set of levels to be tested. “4” means the orthogonal array that is the maximum number of factors can be arranged. Because this study has only three factors, more than one column can be used to arrange the error term (e). “3” represents the level of the factors. During the test, the experimental order was random to avoid possible subjective bias.

Table 2 Schemes of orthogonal test design of impact force
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3.3 Analytical Methods of Experiment Results

Impact force mean values of debris flow on the upstream surface of the check dam are taken as the test index. The points, which are on the dam abutment or overflow section of the check dam, are selected as the representatives to be studied, rather than only for a point or some points at a position. Only in this way, can impact force of debris flow of the whole dam upstream surface be understood comprehensively. A relatively more reasonable conclusion can be obtained by the comprehensive analysis of impact force of the multiple representative points. Figures 1 and 2 show the location diagram of representative points. As the experimental times have been reduced greatly, the method to process the orthogonal experimental data was very important. The range analysis and variance analysis are usually used to detect the most sensitive factor influencing the target index, and verify the distinctiveness in the researches. In this study, the range analysis and variance analysis are selected to deal with the test results, and the sensitivity of different factors is studied.

Fig. 1
figure 1

Plane arrangement of investigated points of the check dam

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Fig. 2
figure 2

The pressure sensor layout

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3.3.1 Range Analysis

The range analysis is an image visual analysis method, which is simple, concise, and easy to understand and operate. Range analysis, R method for short, normally involves two steps of calculation and judgment. Flow diagram of R method procedures is shown in the Fig. 3.

Fig. 3
figure 3

Flow diagram of R method procedures

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In this study, the range analysis is employed to discriminate the comparative significance of each factor, which was defined as the difference between the maximum and minimum value of \(\overline{{K_{ij} }}\) the j-th factor, noted as \(\overline{{R_{j} }}\)

$$\overline{{R_{j} }} = \hbox{max} \left( {\overline{{K_{1j} }} ,\overline{{K_{2j} }} , \cdot \cdot \cdot \overline{{K_{ij} }} } \right) - \hbox{min} \left( {\overline{{K_{1j} }} ,\overline{{K_{2j} }} , \cdot \cdot \cdot \overline{{K_{ij} }} } \right)$$
(2)

where \(\overline{{K_{ij} }}\) is the average targeting value of each experimental factor at the same level in the orthogonal experiments, which was used to determine the optimal level and the optimal combination of factors. \(\overline{{K_{ij} }}\) can be expressed as:

$$\overline{{K_{ij} }} = \frac{{K_{ij} }}{n}$$
(3)
$$K_{ij} = \sum\limits_{k = 1}^{{p_{1} }} {Y_{ijk} }$$
(4)

where the Arabic numerals i (i = 1, 2, 3) are the level number and j (j = L, M, N) donated a certain factor. K ij is the sum of the targeting indexes of all levels in each factor j; n is the total levels of the corresponding factor. Y ijk is the k-th test result index value, with the j-th factor and the i-th level. A larger \(\overline{{R_{j} }}\) means a greater importance of the factor.

3.3.2 Analysis of Variance

In the test process,because of the existence of the subjective and objective factors, there is a certain error between the experimental results and the real results. Range analysis shows the order of the variables influencing on the target by intuition. However, it cannot distinguish whether the undulation of each factor level is caused by the varying levels or by the experimental errors. Moreover, it cannot provide a standard to estimate whether the influence of variable is noticeable. In order to overcome this limitation, a variance analysis is necessary to obtain the magnitudes of the factor affecting the targeting index.

The variance analysis first divided the total fluctuation into two parts: the fluctuation coming from the experimental conditions and generated from the experimental error; and then the two part fluctuation are compared with each other by statistics analysis; finally, the conclusion can be obtained according to the comparative results. As the F value of each factor can be used to indicate the ratio of the sum of the square of each factor’s mean deviation to that of the experimental error. Therefore, the F-test is employed to analyze the experimental data, and the F value is used to indicate the degree of the factor’s influencing, which can be calculated by the following formulas (Jia et al. 2014).

$$S_{sum} = \sum\limits_{i = 1}^{n} {\sum\limits_{k = 1}^{{p_{1} }} {\left( {Y_{ijk} } \right)^{2} } } - \frac{{T^{2} }}{{p_{0} }}$$
(5)
$$S_{j} = \frac{{\sum\nolimits_{i = 1}^{n} {\left( {K_{ij} } \right)} }}{{p_{1} }}^{2} - \frac{{T^{2} }}{{p_{0} }}$$
(6)
$$S_{e} = S_{sum} - \sum\limits_{{{\text{j}} = 1}}^{m} {S_{j} }$$
(7)
$$T = \sum\limits_{i = 1}^{n} {\sum\limits_{k = 1}^{{p_{1} }} {Y_{ijk} } }$$
(8)

where S sum stands for total differences; S j reflects the differences of experimental results caused by the change in every level of factor j, which shows the influence of factor j on the experimental results; S e reflects the differences of experimental results caused the error; m is the number of the factors; p 1 stand for test times with the j-th factor and the i-th level; p 0 stand for the total test times; T stands for the sum of test indexes; Other symbols are the same as the above.

Given a specific test level, ζ (degree of confidence), the critical value, F ζ (f j , f e ) can be found in the F distribution table. The F values of each factor are compared with F ζ (f j , f e ). A larger F value of the factor means a more significant effect for test index. F can be expressed as:

$$F = \frac{{{\raise0.7ex\hbox{${S_{j} }$} \!\mathord{\left/ {\vphantom {{S_{j} } {f_{j} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${f_{j} }$}}}}{{{\raise0.7ex\hbox{${S_{e} }$} \!\mathord{\left/ {\vphantom {{S_{e} } {f_{e} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${f_{e} }$}}}}$$
(9)

where f j represents degree of freedom of each factor; f e stands for degree of freedom of test error; and other symbols are the same as the above. f j and f e can be expressed as:

$$\left\{ \begin{aligned} f_{e} = f_{T} - \sum {f_{j} } \hfill \\ f_{j} = n - 1 \hfill \\ f_{T} = p_{0} - 1 \hfill \\ \end{aligned} \right.$$
(10)

4 Debris Flow Impact Force on the Surface of Check Dams

4.1 Experimental Setup

4.1.1 Experimental Materials

The experiments were conducted at the Dongchuan Debris Flow Observation and Research Station at the Chinese Academy of Sciences. An original sample of the debris flow materials from the Jiangjia Gully was used in the experiment because debris flows from this area are typical of the flows that were found in most of the mountainous regions in China. The particle diameter of the test model materials is limited by the use of a 20 × 20 mm steel mesh, which rejects gravel soil particles with diameters >20 mm. The particle size distribution of the soil sample is shown in Fig. 4. The median particle size of the soil sample is 2.5 mm, and the contents of particles <1 mm and <0.1 mm are approximately 24.3 and 5.6%, respectively.

Fig. 4
figure 4

Gradation/particle size distribution of the debris flow in the test model

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4.1.2 Experimental Model

The test model mainly consists of the following four components: a drainage channel, check dam, tailing pool, and hopper. The experiments were conducted in an 8-m-long open-water channel with a rectangular cross section of 0.4 × 0.5 m (width × height). All of the sidewalls in the test section were reinforced glass, the bottom slab was steel plate, and the check dam was steel plate. The upper portion of the dam section was 0.2 m, the underside was 0.34 m, the dam height was 0.35 m, and the dam length was 0.4 m. The slope of the front of the dam was 1:0.3, while the slope behind the dam was 1:0; the width and height of the overflow mouth were 0.2 and 0.05 m, respectively. Figure 5 shows the three-dimensional diagram and panorama of the test model.

Fig. 5
figure 5

Three-dimensional diagram and panorama of the test model

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4.1.3 Measurement System

The measuring device consisted of the following three components: eight modified pressure sensors, a high-speed continuous data acquisition instrument, and data collection procedures.

4.1.3.1 Pressure Sensors

Eight pressure sensors, which were used to measure the debris flow fluid impact, were arranged vertically and horizontally along the check dam upstream face. The two-dimensional layout schematic of the pressure sensors and the numbers of the eight sensors are shown in Fig. 1. Figure 2 shows the arrangement of the pressure sensors along the check dam upstream face. The pressure sensors are piezoresistive sensors, and the relevant parameters of the pressure sensors are shown in Table 3.

Table 3 Technical parameters of the pressure sensors
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4.1.3.2 Data Acquisition Instrument

The data acquisition instrument is the high-speed continuous collection instrument. The relevant model and characteristic parameters are shown in Table 4.

Table 4 Technical parameters of the data acquisition instrument
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4.1.4 Data Collection Procedures and Binary Data Export Tool

A denoising method based on wavelet transformation was used in the software components; this method has better filtering in the sharp white noise signal of the time domain signals acquired by the pressure sensors. The acquisition frequency, acquisition time and other physical quantities can be set up in the master interface of the data collection procedures. The acquisition frequency of the data collection procedures can range from 0.1 to 2200 Hz. The acquisition frequency in the experiment was set to 1000 Hz according to the research needs. Due to the large amount of data and to improve the collection efficiency of the data collection procedures, the data collected by the data acquisition program were automatically saved as binary data in a .bat file format; the binary data were then converted to decimal data with an .xlsx file format via an export tool.

4.2 Analysis of Experimental Results

According to orthogonal test design shown in the Table 2, nine sets of numerical simulation are carried out, and finally, the required data are extracted. The specific statistical data of mean values of debris flow impact force of investigated points are shown in Table 5.

Table 5 Mean values of debris flow impact force of investigated points (Unit: kPa)
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Table 5 shows that, among the nine test cases, mean values of impact force in the position of 8# point is the maximum value, while the position of 6# point has the minimum value. The impact force mean values of the overflow section are greater than that of the abutment position of the check dam. The mean values of impact force decrease as the check dam’s height increases.

4.3 Range Analysis

Using the range analysis method introduced in Sect. 3.3.1, the range analysis of the impact force mean values of the investigated points is carried out and the range analysis results are given in Table 6. The range value (R j ) expresses the significance of the influence of the factors. The factor with larger R j has a great effect on debris flow impact force. Trend chart of mean values of impact force and factor levels is shown in the Fig. 5.

Table 6 Extreme difference analysis of impact force mean values
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Table 6 indicates that, among the eight test index, number of times of the most significant factor is: eight times with L, zero time with M, zero time with N; number of times of the most secondary factor is: zero time with L, eight time with M, zero time with N. According to the R j value of each factor, among the eight test index, the order of the factors’ impact is as follows: the drainage channel slope > debris flow bulk density > the upstream surface gradient of the check dam. The R L is the largest among all these ranges, implying that the drainage channel slope is the most significant factor influencing on the debris flow impact force. The mean values of impact force debris flow of the eight key points are the largest under the working condition of L 3 M 1 N 3.

Figure 6 shows the average value of each factor. It should be mentioned that the graph is only used to show the trends of each factor, not to predict other levels untested experimentally in this study. The statistical parameter \(\overline{{K_{ij} }}\) increases with the increase in the level values of the drainage channel slope, decreases in a small scale with the increase in the level values of the upstream surface gradient of the check dam except for the key point of 7# and 8#, and increases and then decreases with the increase in the level value of debris flow bulk density.

Fig. 6
figure 6

Tendency of impact force mean values and influence level of the factor

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4.4 Analysis of Variance

Using the analysis of variance method introduced in Sect. 2.3.2, the range analysis of the impact force mean values of the investigated points is carried out and the variance analysis results are given in Table 7. F 0.05 (2, 2) = 19.0, F 0.1 (2, 2) = 9.0, and F 0.25 (2, 2) = 3.0 have been queried by the F distribution table and listed into the Table 7.

Table 7 Variance analysis of impact force mean values
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As shown in Table 7, eight times with L, zero time with M and N meet the expression of F > F 0.05 (2, 2) = 19.0; eight times with L, zero time with M and three times with N meet the expression of F > F 0.1 (2, 2) = 9.0 and eight times with L, one time with M and six times with N meet the expression of F > F 0.25 (2, 2) = 3.0. The above results indicate the order of the effect of the three factors is as follows: the drainage channel slope > debris flow bulk density > the upstream surface gradient of the check dam, which is in accord with the results of range analysis.

5 Conclusions

In this study, design of orthogonal test first carried out to determine the number of test groups and obtain the orthogonal table. Secondly, miniaturized flume experiments are carried out to measure impact force of debris flow according to the orthogonal table and the test date are obtained and dealt with. At last, the range and variance analysis of the test date are carried out. The main points could be concluded as the following:

  1. 1.

    The range analysis and variance analysis of the test results show that indicate the order of the effect of the three factors is as follows: the drainage channel slope > debris flow bulk density > the upstream surface gradient of the check dam, which indicates the drainage channel slope is the most significant factor.

  2. 2.

    As shown in statistical table of mean values of debris flow impact force, the mean values of impact force debris flow of the above eight key points have the largest values under the working condition of L 3 M 1 N 3.