Back-calculation of elastic modulus of soil and subgrade from portable falling weight deflectometer measurements

Engineering Structures 34 (2012) 1–7

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Back-calculation of elastic modulus of soil and subgrade from portable falling weight deflectometer measurements C. Asli a,b, Z.-Q. Feng c,a,⇑, G. Porcher a, J.-J. Rincent b a

Université d’Evry-Val d’Essonne, LMEE, Evry, France Rincent BTP Services, 39 Rue Michel Ange, 91026 Evry, France c Southwest Jiaotong University, School of Mechanics and Engineering, Chengdu, China b

a r t i c l e

i n f o

Article history: Received 17 July 2011 Revised 28 September 2011 Accepted 3 October 2011 Available online 2 November 2011 Keywords: PFWD Pavement Back-calculation Elastic modulus Boussinesq theory Identification

a b s t r a c t In the field of nondestructive testing of pavement, the portable deflectometer devices have gained, in the recent years, a wide use for in situ assessment of elastic properties of soils, subgrade and pavement foundations. However, if the use of the elasto-static model based on the Boussinesq’s theory does not constitute a shortcoming in the back-calculation procedure of homogenous elastic modulus, questions have been arose about the reliability and accuracy of the peak value method commonly used to extract the static stiffness of soils and subgrade from the dynamic transient data. This paper deals with the use of minimization technique, based on least square algorithm as an alternative method for data analysis and soil elastic stiffness identification. Details of the mathematical basis, the implementation and different steps for elastic modulus back calculation, are presented. In this method, an equivalent spring-mass-dashpot system, where the complex dynamic soil behavior is reduced to a viscoelastic one-dimensional wave propagation problem, is used for modeling the loading plate/soil system. The comparative study shows the suitability and the accuracy of the minimization method for soil elastic stiffness identification and elastic modulus back calculation. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Since the mechanistic-empirical methods have been widely adopted in pavement design procedures, the nondestructive testing techniques (NDT) have seen an increasing interest and acceptance from pavement managers and engineers. Because they provide parameters that are useful for material optimization and pavement deterioration assessment. Nowadays, the most promising devices are based on the application of dynamic loads using vibratory or impulse sources, and the measurement of the resulting surface deflection or the phase difference between the motions recorded at various receivers. Among the deflection based devices, the falling weight deflectometer (FWD) is the most widely used and, actually, is considered as a standard test for pavement evaluation [1,2]. However, despite of many attempts and research works, it is still not preconized to be directly used on soil and subgrade for in situ assessment of the material properties [3]. This is due to many reasons related to the accessibility for pavement under construction, to the unevenness of the surface which leads to inaccurate deflection measurement during the test [4,5] and to the weak cohesive ⇑ Corresponding author at: Université d’Evry-Val d’Essonne, LMEE, Evry, France. Tel.: +33 0 169477501. E-mail address: [email protected] (Z.-Q. Feng). 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.10.011

property of soils and subgrade that makes the FWD back-calculation procedure not appropriate. To overcome this lack, the portable deflectometer devices is found to be a best alternative. Different types of devices have been developed in the world [3] (e.g., LWD PRIMA 100 of Carl Bro, PFWD of Dynatest, GDP of Zorn) and, over the years, they have experienced increasing popularity due to their light weight, quick measurements and high performance, compared to the conventional static tests (e.g., static plate test, CBR test and Benkalman beam). Publications dealing with their performance show that they can constitute a useful tool for the quality control and quality assurance (QC/QA) of newly constructed pavement foundations [6–9]. However, due to their simple concept (portable deflectometer devices use only central sensors such as geophone and accelerometer to measure deflection under the loading plate), they can only provide an evaluation of the homogeneous elastic modulus of layered media. Therefore, the half-space theory, where the soil is assumed to be homogenous, isotopic and linear elastic half-space, is often used in back calculation procedure. Furthermore, the estimation of the static stiffness is obtained by considering only the maximum values of the load and deflection. This is the so called peak value method. Recently, many authors shown that this method leads to inaccurate estimation of the static stiffness. Hoffman et al. [10] demonstrated theoretically that the peak values method leads to significant systematic errors. To overcome this problem, they

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C. Asli et al. / Engineering Structures 34 (2012) 1–7

adopted a spectral analysis approach. By assuming that the soil structure responds as a linear single degree of freedom (SDOF) system, they extrapolated the dynamic stiffness at zero frequency that refers to the static stiffness, by fitting the theoretical frequency response function (FRF) (mobility) curve of the SDOF system to the experimental mobility data obtained by performing the Fourier transform on the time dependent data. Ruta et al. [11], showed also, on the basis of the analytical solution of a half-space under impulse load, that erroneous estimation of static stiffness can be made when using the peak value method. In order to extract the static stiffness from the dynamic transient data, they used the static flexibility which is defined as the ratio between the total displacement and the total load (impulsion) over a certain range of time. More recently, Rhayma et al. [12] developed specific diagnostic devices to analyze the behavior of railway tracks. The aim of the present work is to assess the limitation and the reliability of the peak value method for field data interpretation and to develop an accurate, efficient and easily implementable method for soil stiffness identification. A minimization technique, using least square algorithm is used. In this method the loading plate/soil system is represented by a SDOF system, where the soil is assumed to be linearly visco-elastic [13–15].

The operation mode of the device is simple. As the sliding drop weight is released, it strikes the cylindrical shock absorbers so that an impulse load of 5–25 ms duration with a maximum force ranging up to 35 kN, is transferred through the loading plate into the ground. The load and the motion at the center point on the surface of pavement are simultaneously monitored using a load and a velocity transducer, respectively. During the tests, two filtered digitalization channels connected to the sensors are used for data acquisition and a 10 kHz sampling frequency is chosen. This allows acquiring 4096 samples during the entire time acquisition of 410 ms. A typical time history of a recorded signals that we can visualize on the lap-top, is illustrated in Fig. 2. 3. Data interpretation method and back calculation procedure As for many portable deflectometer devices, although the test is dynamic in nature, the elastostatic model, based on the Boussinesq’s theory [16], is used in back-calculation procedure. Therefore, for a distributed load on a circular area of the free surface of a homogenous, isotropic and linear elastic half-space, the elastic modulus can be obtained as follows [17,18]:

E¼ 2. Light dynaplaque The portable deflectometer device used in this study is developed and manufactured by Rincent BTP Company. It is a handhold device that can be used easily by only one operator. It includes two parts as shown in Fig. 1. The electronic part consists of a National Instrument USB NIDaq acquisition card, a load cell, a velocity transducer (geophone) and a lap-top. The mechanical part includes, mainly, a 30 cm diameter loading plate, a sliding drop weight (10 or 16 kg), a guide rod and cylindrical shock absorbers made of rubber.

ð1  m2 Þ k ba

ð1Þ

where m is Poisson’s ratio, a is the radius of the loading plate, b is the shape factor depending on the stress distribution under the loading plate (b = p/2, 2 and 3p/2 for uniform, inverse parabolic and parabolic stresses distribution, respectively), and k is the elastic stiffness of the loading plate/half-space system, often referred to as the soil stiffness. In practice, as the loading plate is considered stiffer compared to the soil, an inverse parabolic pressure distribution is commonly used for portable falling weight deflectometer data interpretation and elastic properties evaluation. It is of the form:

p0 pðrÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi 2 1  ar 2

ð2Þ

where r 2 [0, a] is the distance from the axis to any point in the circular loading area and p0 is the contact pressure at the center point. This form of distribution gives rise of a uniform normal displacement under the loading plate and therefore a stress concentration at the edge Fig. 3). However, it is worth noting that it remains a simplification of the real contact pressure distribution that would appear in the field, which is highly complex and depends on device parameters, load magnitude and soil profile and properties [19,20]. Besides, we can notice easily, that the main problem in the back-calculation procedure is how to extract the elastic stiffness from the time dependent data. From the static plate tests, the elastic stiffness k is simply defined as:

k¼

P d

ð3Þ Ra

where P is the applied load given by P ¼ 2p 0 pðrÞdr and d is the induced displacement at the center point or the average of the measured displacements at three points under the loading plate. However, in the case of a dynamic plate tests (e.g., PFWD), the dynamic response of the ground to the impact loading is affected by the inertia and the damping properties of the media, that must be taken into account in order to identify, accurately, the elastic properties, namely, the elastic stiffness k, from transient data. The method proposed here consists of two steps. The first step aims to identify the elastic stiffness k. By assuming that the behavior of the rigid loading plate/soil system is comparable to that of a mass – spring – damper system Fig. 3), its dynamic equilibrium is governed by the equation: Fig. 1. Portable deflectometer device.

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C. Asli et al. / Engineering Structures 34 (2012) 1–7

(a)

400

(b)

0.2

(c)

100

Deflection Load

80

0.15

60 0.1 40

−200

Acceleration [m/s²]

0 Velocity [m/s]

Load [daN] and deflection [µm]

200

0.05 0 −0.05

−400

20 0 −20 −40

−0.1

−60

−600 −0.15

−800

0

10

20 Time(ms)

30

40

−0.2

−80

0

10

20 Time(ms)

30

40

−100

0

10

20 Time(ms)

30

40

Fig. 2. Typical field data: (a) load and deflection, (b) velocity and (c) acceleration time histories.

Fig. 3. Inverse parabolic contact pressure distribution.

€ þ cu_ þ ku ¼ f mu

ð4Þ

€ ; u_ and u are, respectively, the accelwhere f is the applied load and u eration, the velocity and the deflection. k, c and m are, respectively, the elastic stiffness, the damping coefficient and the equivalent mass. In order to simplify the writing, the time parameter is omitted all over this study (Fig. 4). The acceleration and the deflection are derived from the known velocity by using the central finite difference integration scheme. Then, by seeking to make the measured velocity u_ solution of Eq. (4) over a certain range of measure s, the parameters of the model m, c and k are identified, using the minimization technique based on the least square method. The measuring range s is defined by time interval including only the load values that are greater than 5% of the maximum value of the applied load. Over this range, the objective function is expressed by:

JðxÞ ¼

1 2

Z s

2

€ þ cu_ þ ku  f k dt kmu

ð5Þ

0

where (x) = (m, c, k) and k.k is the norm associated to the scalar product h.,.i between scalar functions of time. The development of Eq. (5) leads to:

JðxÞ ¼

 Z s  Z s Z s 1 2 €; u €idt þ c2 _ uidt _ hu hu; þk hu; uidt þ mc m2 2 0 0 0 Z s Z s Z s € ; uidt _ € ; uidt þ ck _ uidt  m hu þ mk hu hu;  0 0 0 Z Z s Z s Z s 1 s € ; f idt  c _ f idt  k hu hu; hu; f idt þ hf ; f idt  2 0 0 0 0

that can be written in a matrix form as follows: 2 3 2 3 €; u € ihu € ; u_ ihu € ;ui € ;f i Z Z s hu Z s hu 1 1 s 6 € _ _ _ _ 7 6 _ 7 T JðxÞ ¼ fxg hf ;f idt 4 hu; uihu; uihu;ui 5dtfxg  4 hu;f i 5dt þ 2 2 0 0 0 € _ hu;uihu;uihu;ui hu;f i

ð6Þ T

where {x} is the vector transpose of {x}. The minimization of the objective function J(x) implies that the gradient of J(x) is zero:

rJðxÞ ¼

Z s 0

Fig. 4. SDOF model of the loading plate/soil system.

2

3 2 3 €; u € ihu € ; u_ ihu € ; ui €; f i hu Z s hu 6 € _ _ _ _ 7 6 _ 7 4 hu; uihu; uihu; ui 5dt fxg  4 hu; f i 5dt ¼ 0 0 € ; uihu; _ uihu; ui hu hu; f i

ð7Þ

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C. Asli et al. / Engineering Structures 34 (2012) 1–7

Fig. 5. Soil profiles of experimental sites: (a), (b) and (c) are the three test beds; (d) the finished subgrade and (e) the embankment.

Finally, by solving Eq. (7), the elastic stiffness k, the damping coefficient c and the equivalent mass m can be estimated. Only the elastic stiffness is used in Eq. (4) to evaluate the homogenous elastic modulus. The second step aims to assess the reliability and the accuracy of the estimated parameters. For this, a forward problem is first solved using the estimated parameters k,c and m. Then, the calculated deflection is compared to the measured deflection using a quality index defined by

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Rs 2 kum ðtÞ  uc ðt; m; c; kÞk dt 0 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qi ¼ 1  Rs 2 kuc ðt; m; c; kÞk dt 0

ð8Þ

where um ðtÞ is the measured deflection and uc ðt; m; c; kÞ is the calculated deflection using the previous back calculated parameters m,c and k. € c ðt; m; c; kÞ In order to calculate uc ðt; m; c; kÞ, u_ c ðt; m; c; kÞ and u that refers to a forward problem, the equation of motion Eq. (4) is solved by using the central finite difference integration scheme in which the velocity and the acceleration are approximated by:

u_ t ¼

utþDt  utDt 2 Dt

€t ¼ u

utþDt  2ut þ utDt Dt 2

ð9Þ

where Dt is the time step. The substitution in the equation of motion Eq. (4) leads to:

     m c 2m m c u þ ¼ f  k    u utDt tþDt t t Dt Dt 2 Dt 2 Dt 2 2 Dt



ð10Þ

By combining Eqs. (9) and (10), the deflection uc ðt; m; c; kÞ, the € c ðt; m; c; kÞ are calculated velocity u_ c ðt; m; c; kÞ and the acceleration u in an iterative manner. The calculated deflection is then compared to the derived deflection in order to assess the accuracy of the identified parameters.

consists of more than 3 m depth clayey soil layer whereas the finished subgrade consists of 50 cm of a lime-treated silt (LTS) overlying a marly limestone in situ soil. The underlying soil in the three test beds consists of marly clay with a high moisture content. Fig. 5 shows the soil profiles of the different experimental sites with the different compaction ratio cm and moisture content w. The test protocol consists of carrying out tests, using the 16 kg sliding drop weight, on three points on each bed and on five points on the embankment and finished subgrade. During the tests, the targeted load is obtained by varying the falling height of the sliding drop weight. Besides, in order to obtain a correct and useable measurements, two drops are performed to ensure a good contact between the loading plate and the ground. Then, three other drops are performed and averaged so as to identify the static stiffness and the homogenous elastic modulus E.

5. Experimental results and analysis To examine the effectiveness of the proposed method, a comparative study is performed by applying both the peak value method (PV) and the minimization method (MM) for the static stiffness identification and the elastic modulus back-calculation. Knowing that the contact pressure distribution under the loading plate is highly complex, in this study, the inverse parabolic pressure distribution presented in Section 3 is adopted. Poisson’s ratio of 0.499 is assumed for marly clay and clayey soils which are considered as incompressible materials whereas 0.4 is used for the concrete crushed aggregates and lime-treated silt. Table 1 shows the main

Table 1 Experimental and back calculated results (fmax are the maximum values of the measured loads, CCA: concrete crushed aggregates and LTS: lime treated silt). Subgrade

Test point

fmax (kN)

Marly clay (70%) Marly clay (98%) Clayey soil Clayey soil CCA CCA LTS LTS

1

4.80

4.86

4.86

1

7.70

9.41

1 2 1 2 1 2

9.78 6.78 6.13 17.1 20.5 19.7

13.0 19.4 22.0 29.7 72.2 110

4. Experimental sites and field tests The experimental data used in this study are obtained by using the portable deflectometer device applied on three test beds as well as on an embankment and a finished subgrade. The test beds are constructed by excavating the in situ marly clay soil to a depth of 35 cm and then filled with the same soil compacted up to 70% for the first bed and up to 98% for the second bed. For the third test bed, crushed concrete aggregates (CCA) compacted up to 98% are used. Each bed is 1.5 m wide and 6 m long. The embankment

kMM kPV EMM (MN/m) (MN/m) (MPa)

EPV (MPa)

timp (ms)

Qi (%)

12

12

20

98.3

8.80

23

22

18

97.4

13.4 20.0 33.1 41.2 104 146

32 48 61 83 202 308

33 50 92 115 290 409

16 15 13 12 10 08

97.2 95.5 97.5 97.2 95.9 96.2

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C. Asli et al. / Engineering Structures 34 (2012) 1–7

Measured deflection Calculated deflection Load

600

120

80

0.2

200

−400 −600

Acceleration [m/s²]

0.15

0 −200

Measured calculated

0.25

Velocity [m/s]

Load [daN] and Deflection [µm]

400

Measured Calculated

0.3

0.1 0.05 0

−0.1

−1000

0

−40

−0.05

−800

40

−80

−0.15

−1200

−0.2 0

10

20 Time(ms)

30

40

−120 0

10

20 Time(ms)

30

40

0

10

20 Time(ms)

30

40

Fig. 6. Typical field data: marly clay (70%).

800

150 Measured deflection Calculated deflection Load

400

0 −200

Acceleration [m/s²]

0.15

200

0.1 0.05 0 −0.05

−400

Measured Calculated

100

0.2 Velocity [m/s]

Load [daN] and Deflection [µm]

600

Measured Calculated

0.25

50 0 −50

−0.1 −100

−0.15

−600

−0.2 −800

−150

−0.25

−1000 0

10

20 Time(ms)

30

0

40

10

20 Time(ms)

30

−200

40

0

10

20 Time(ms)

30

40

Fig. 7. Typical field data: marly clay (98%).

1000

0.3 Measured deflection Calculated deflection Load

600

0 −200

Acceleration [m/s²]

200

0.1 0.05 0 −0.05

−400

Measured Calculated

100

0.2 0.15

400

50 0 −50 −100

−0.1

−600 −800

0.25

Velocity [m/s]

Load [daN] and deflection [µm]

800

150 Measured Calculated

−150

−0.15 0

10

20 Time(ms)

30

40

−0.2

0

10

20 Time(ms)

30

Fig. 8. Typical field data: clayey soil.

40

−200

0

10

20 Time(ms)

30

40

6

C. Asli et al. / Engineering Structures 34 (2012) 1–7

700

0.06 Measured deflection Calculated deflection Load

Measured calculated

20

0.04

500

10

300 200 100

0.02

Acceleration [m/s²]

400 Velocity [m/s]

Load [daN] and deflection [µm]

600

30 Measured Calculated

0

−0.02

0

0 −10 −20 −30

−100

−0.04

−40

−200 −300

0

10

20 Time(ms)

30

−0.06 0

40

10

20 Time(ms)

30

40

−50

0

10

20 Time(ms)

30

40

Fig. 9. Typical field data: concrete crushed aggregates.

results obtained from different methods. When the peak value method is applied, the soil static stiffness (kPV) is identified by a simple reading of the maximum value of the load and deflection from the temporal records. Then, the elastic modulus (EPV) is calculated using Eq. (1). Although a slight decrease is noticed in the case of the limetreated silt (LTS) because of wave reflection, a high quality index Qi is obtained for all the tests. This demonstrates the reliability and the accuracy of the identified parameters using the minimization method. Comparing the two methods PV and MM in view of k or E, it is observed that the difference increases as the impulse duration timp decreases. This can be explained by the fact that, when the impulse duration is longer, the soil or the subgrade is loaded at low frequencies, therefore, the behavior of the soil is quasi-static, governed primarily by its stiffness. However, when the impulse duration is shorter, the frequency range of the excitation increases. Therefore, the media under test is excited at high frequency and

200

0.08

80 Measured Calculated

Measured deflection Calculated deflection Load

150

40

100

0

Acceleration [m/s²]

0.04

50

0.02

0

20 0 −20 −40 −60

−50 −0.02

−80

−100

−100

−0.04 −150 0

Measured Calculated

60

0.06

Velocity [m/s]

Load [100N] and deflection [µm]

the behavior is now mainly governed by the inertia that attenuates significantly the amplitude of dynamic response of the soil or the subgrade. Consequently, it leads to over estimate the static stiffness and hence, the elastic modulus, because only the peak values of the load and deflection are considered. In addition, as shown in Table 1, the impulse duration depends on the soil and subgrade stiffness. The stiffer is the soil or the subgrade, the shorter is the impulse duration. From experiments, marly clay test beds and the clayey soil are of relatively small stiffness, so there is no significant difference between the two methods. Whereas, the difference becomes significant for the crushed concrete aggregates and the lime-treated silt which are relatively stiffer. Figs. 6–10 represent the typical field data recorded during the tests and those calculated using the minimization method. In general, good agreement over the range of measure s is found. This demonstrates the high quality index obtained for all the tests as well as the reliability and the accuracy of the identified parameters using the minimization method.

10

20 Time (ms)

30

40

0

10

20 Time (ms)

30

Fig. 10. Typical field data: lime-treated silt.

40

−120

0

10

20 Time (ms)

30

40

C. Asli et al. / Engineering Structures 34 (2012) 1–7

7

6. Conclusion

References

In this work, an accurate and easily implementable technique is proposed as an alternative method to the peak value method, commonly used for the static stiffness identification from portable deflectometer measurements. Based on a single degree of freedom modeling of the loading plate system and the linear visco-elastic behavior of soil and subgrade assumption, the proposed method makes use of a minimization technique combined with the least square method for identification of the soil and subgrade static stiffness. The comparative study conducted here allows for the following conclusions:

[1] Roesset JM. Nondestructive dynamic testing of soils and pavements. Tamkang J Sci Eng 1998;1:61–81. [2] Lytton RL. Backcalculation of layer moduli – state of the art. In: Bush AJ, Baladi GY, editors. Nondestructive testing of pavements and backcalculation of moduli. Philadelphia: ASTM; 1994. [3] Fleming PR, Thom NH. Experimental and theoretical comparison of dynamic plate testing methods. In: Correia AG, Branco FEF. (Eds.), Bearing capacity of roads, railways and airfields: 6th international conference. CRC Press; 2002. p. 731–40. [4] Nazzal MD. Dynamic soil properties out of SCPT and bender element tests with emphasis on material damping. Ph.D. thesis, Louisiana State University; 2003. [5] Nazzal MD, Abu-Farsakh MY, Alshibli K, Louay M. Evaluating the light falling weight deflectometer device for in situ measurement of elastic modulus of pavement layers. J Transport Res Board 2007;2016:13–22. [6] Steinert BC, Humphrey DN, Kestler MA. Portable falling weight deflectometer study. Technical report. Department of Civil and Environmental Engineering. University of Maine; 2005. [7] Frost MW, Fleming PR, Lambert JP. Lightweight deflectometers for quality assurance in road construction. In: Tutumluer E, Al-Qadi I. (Eds.), Bearing capacity of roads, railways and airfields: 8th international conference. CRC Press; 2002. p. 809–18. [8] Sargand SM, Edwards WF, Salimath S. Evaluation of soil stiffness via nondestructive testing. Technical report. Ohio Research Iinstitute for Transportation and the Environment; 2001. [9] Hossain MS, Apeagyei AK. Evaluation of the lightweight deflectometer for in situ determination of pavement layer moduli. Technical report. Virginia Transportation Research Council; 2010. [10] Hoffmann OJ-M, Guzina BB, Drescher A. Stiffness estimates using portable deflectometers. Transport Res Rec 2004;1869:59–66. [11] Ruta P, Szydlo A. Drop-weight test based identification of elastic half-space model parameters. J Sound Vib 2008;282:411–27. [12] Rhayma N, Bressolette Ph, Breul P, Fogli M, Saussine G. A probabilistic approach for estimating the behavior of railway tracks. Eng Struct 2011;33:2120–233. [13] Adam C, Adam D. Modelling of the dynamic load plate test with the light falling weight device. Asian J Civil Eng Build Housing 2003;4:73–89. [14] Kim S-M, McCullough BF. Dynamic response of plate on viscous winkler foundation to moving loads of varying amplitude. Eng Struct 2003;25:1179–88. [15] Cornejo Cordova C-J. Elastodynamics with hysteretic damping. DUP ScienceDelft University Press; 2002. [16] Boussinesq J. Application des potentiels a l’Ttude de l’Tquilibre et du mouvement des solides Tlastiques. Paris, France: Gauthier-Villard; 1885. [17] Johnson KL. Contact mechanics. Cambridge University Press; 1985. [18] Verruijt A. An introduction to soil dynamics. New York: Springer; 2010. [19] Issa SS. On the contact pressure of circular plates on elastic subgrade. Eng Struct 1985;7:64–8. [20] Mooney MA, Miller PK. Analysis of lightweight deflectometer test based on in situ stress and strain response. J Geotech Geoenviron Eng 2009;135:199–208.

(1) For all tests carried out in this work, the proposed method allows for a good correlation between the numerical modeling and the experimental data which proves the accuracy and the reliability of the identified parameters. (2) It is found that the ratio between the peak value of the load and the deflection does not, often, correspond to the static stiffness of the media under test. The analysis shows that the impulse duration, which is highly dependent on the soil and subgrade stiffness, has a significant influence on the results, since the dynamic response of the soil and subgrade depends rigourously on the frequency range of the excitation. (3) Unlike the peak value method, the results obtained from the proposed method depends only on the actual elastic properties (elastic modulus) of the soil or subgrade under test (for an assumed Poisson’s ratio and a pressure distribution shape). (4) The minimization method allows more appropriately for the elasto-static back-analysis in spite of the dynamic nature of the test. This method enables to extract, from the dynamic response of the soil or subgrade, the real static part.

Acknowledgments The authors acknowledge ‘‘Rincent BTP Matriaux’’ company for the construction of test beds and for the help in providing data and performing field tests of this research.