Numerical investigation of earth pressure reduction on buried pipes ...

Geosynthetics International

Numerical investigation of earth pressure reduction on buried pipes using EPS geofoam compressible inclusions A. F. Witthoeft1 and H. Kim2 1

Project Engineer, Ninyo & Moore Geotechnical and Environmental Sciences Consultants, 475 Goddard, Suite 200, Irvine, CA 92618, USA, Telephone: +1 949 753 7070; Telefax: +1 949 753 7071; E-mail: [email protected] 2 Project Manager, Fugro Consultants, Inc., 6100 Hillcroft, Houston, TX 77081, USA, Telephone: +1 713 369 5454; Telefax: +1 713 369 5518; E-mail: [email protected] (Corresponding author) Received 07 August 2015, revised 20 October 2015, accepted 28 October 2015 ABSTRACT: This paper presents a numerical study performed to investigate the effect of expanded polystyrene (EPS) geofoam panels placed over a buried pipe. It is recognized that EPS geofoam panels as compressible inclusions over a buried pipe are effective in reducing the earth pressure acting on the pipe due to positive arching action. To date, however, there is no systematic methodology that links the earth pressure on a buried pipe with the geometry of EPS panels. To investigate the ‘optimal’ geometry of EPS panels, a two-step numerical modeling approach was employed and calibrated against results of a model-scale experimental study. First, material properties were estimated for each component used in the model-scale tests (i.e., soil, EPS geofoam and steel pipe). Second, the model-scale experiments were simulated using the selected material properties. These simulations resulted in reasonable agreement between model-predicted and measured vertical and lateral earth pressures. Using the calibrated model, additional cases that were not covered in the experimental study were investigated to examine different widths and thicknesses of EPS panels. The numerical analysis provided quantification of the effect of EPS compressible inclusion and a systematic approach to optimizing the design of buried pipes using EPS geofoam panels. KEYWORDS: Geosynthetics, Buried pipes, Compressible inclusion, EPS (expanded polystyrene) geofoam, Imperfect ditch condition, Numerical study REFERENCE: Witthoeft, A. F. and Kim, H. (2015). Numerical investigation of earth pressure reduction on buried pipes using EPS geofoam compressible inclusions. Geosynthetics International. [http://dx.doi.org/10.1680/jgein.15.00054]

1. INTRODUCTION The design of buried pipes typically involves computation to evaluate whether the specified pipe wall thickness and pipe stiffness are sufficient to support both the internal pressure and the external load. The vertical and horizontal earth pressures imposed on a buried pipe are controlling parameters for selection of both the pipe material and cross-section. For a buried pipe below a high fill, large earth loads acting on the pipe require a substantial pipe section to maintain its structural integrity. In such a case, the imperfect ditch construction method can be used to reduce earth loads on the pipe, resulting in a more economical pipe design. The imperfect ditch method generally employs one or multiple compressible inclusions over the pipe crown. Deformation of the compressible inclusion induces so-called ‘positive arching’ within the fill over the pipe. This positive arching action transfers a portion of the self-weight loading from the soil prism over

the pipe into adjacent soils; as a consequence, the pipe experiences vertical earth pressure less than that corresponding to the soil prism self-weight. Since Marston (1930) proposed the concept of the imperfect ditch method, several different types of materials (e.g., leaves, baled straw, sawdust, woodchips) have been suggested as compressible inclusions (Spangler 1958; Larson and Hendrickson 1962; Vaslestad et al. 1994a; McAfee and Valsangkar 2004, 2005). However, the engineering properties of these organic materials are difficult to control. Thus, several researchers have focused on the use of expanded polystyrene (EPS) geofoam as a compressible inclusion or a seismic buffer (Horvath 1995; Frydenlund and Aabøe 1996; Hazarika 2006; Zarnani and Bathurst 2007; Kim et al. 2010). EPS geofoam is a manufactured material; its engineering properties are fairly reliable and uniform, and its deformation behavior is predictable and controllable. Several researchers have investigated the use of EPS geofoam as a compressible material for use with buried

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Witthoeft and Kim steel pipe and Jumunjin sand). ‘In-isolation’ (i.e., individual component) tests were simulated for each of these three materials. 2.1. EPS geofoam The model-scale tests by Kim et al. (2010) used a single type of EPS geofoam, which had a density of approximately 15 kg/m3. Kim et al. (2010) performed two unconfined compression tests on specimens of this geofoam material. Results of these unconfined compression tests are reproduced in Figure 1. For reference, test results reported by Duskov (1997) for EPS with a density of approximately 15 kg/m3 and by Hazarika (2006) for EPS with a density of approximately 16 kg/m3 are also reproduced in Figure 1. The EPS geofoam used in the Kim et al. (2010) model tests was simulated as a linear-elastic/Mohr–Coulomb material with post-yield strain-hardening using FLAC’s built-in strain-hardening/softening model (Itasca 2011). Although a bilinear elastic model might have been used to produce similar behavior for monotonic loading conditions, a linear-elastic/strain-hardening model was adopted to allow for the possibility of non-recoverable plastic strain in the event of unloading. In light of the variation of test results on EPS materials having similar densities illustrated in Figure 1, the assumption of linear-elastic pre-yield behavior per se was considered unlikely to introduce significant error into the model. Post-yield strain-hardening behavior was assumed to be linear (i.e., linearly increasing strength with increasing strain level). This assumption of linear strain-hardening is considered to be reasonably accurate up to strain levels in the vicinity of 30% based on the test results shown in Figure 1 and those reported by other researchers (e.g., Horvath 1995). For comparison with the unconfined compression test results presented by Kim et al. (2010), a numerical model was developed to simulate unconfined compression of an EPS specimen. As shown in Figure 2, the simulation results are in reasonable agreement with the test results reported by Kim et al. (2010).

150

Axial stress (kPa)

pipes and culverts (e.g., Vaslestad et al. 1994b; Kim and Yoo 2005; Sun et al. 2005, 2009). For example, based on numerical analyses for box-type concrete culverts, Sun et al. (2005) found that the use of EPS panels reduces bending moments along both the top and sides of the culvert. They also generally found that load reduction due to placement of EPS panels is more pronounced for gravelly, sandy fill materials than for silty, clayey fill materials. Sun et al. (2009) performed both instrumented field tests and numerical analyses to investigate earth pressure reduction on concrete culverts with EPS panels in various configurations. Based on these results, Sun et al. (2009) indicated that EPS geofoam can be used effectively to reduce vertical load on rigid culverts resting on rigid foundations. However, few studies exist to ‘optimize’ the dimensions of an EPS geofoam compressible inclusion placed over a buried pipe via a systematic approach. The effect of an EPS geofoam compressible inclusion over a buried pipe depends on several factors, such as: (1) EPS geofoam density and panel thickness, which determine stress–strain behavior; (2) EPS geofoam width relative to pipe diameter; and (3) number and spacing of EPS geofoam panels when multiple EPS panels are placed over the pipe. Although Kim et al. (2010) researched these factors and associated changes in the earth pressure acting on a buried pipe, this experimental investigation was subject to limitations on the measurement of stresses and strains. Some of these limitations can be addressed through a numerical modeling approach. For example, model-scale tests provide stress and strain measurements at discrete points; numerical analyses can provide continuous stress and strain distributions over a cross-section. Moreover, a limited set of EPS panel configurations was evaluated in the model-scale tests; numerical analysis provides an efficient means of evaluating a significantly larger set of configurations. Consequently, additional study is warranted to develop design guidelines for ‘optimal’ compressible inclusion geometry. The purpose of this study was to evaluate earth pressure reduction on a buried pipe for various EPS geofoam compressible inclusion configurations and to illustrate ‘optimization’ of EPS panel geometry using a numerical approach. A numerical model was developed using FLAC. Calibration was performed for each material used in the Kim et al. (2010) model-scale tests (i.e., EPS geofoam, Jumunjin sand and the pipe) using laboratory test data reported by Kim et al. (2004, 2010) and Park et al. (2008). Numerical model results were compared against test results presented by Kim et al. (2010). After calibration of the FLAC model, more cases not covered in the Kim et al. (2010) model-scale test program were examined to provide a basis for a sample ‘optimization’ of EPS geofoam panel dimensions.

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50 Test results from Kim et al. (2010) Test results from Hazarika (2006) Test results from Duskov (1997)

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2. EVALUATION OF MATERIAL PROPERTIES

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10

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30

Axial strain (%)

A material model was calibrated for each of the three materials used in the model-scale test (i.e., EPS geofoam,

Figure 1. Unconfined compression test results on EPS geofoam specimens from the literature



Earth pressure reduction on buried pipes using EPS geofoam Table 1. Material properties assumed for EPS geofoam

150

Axial stress (kPa)

3

100

Value

Young’s modulus, Ef Poisson’s ratio, νf Cohesion, cf

2800 kPaa 0.09b 41 kPa at zero plastic axial straina 64 kPa at 30% axial straina 15 kg/m3a

Density, ρf

50 Test results from Kim et al. (2010) FLAC model results

0

Parameter

0

10

20

a

Estimated based on results of laboratory testing EPS geofoam reported by Kim et al. (2010). b Estimated based on regression equation proposed by Horvath (1995). 30

Axial strain (%)

Figure 2. Comparison of unconfined compression test results for EPS reported by Kim et al. (2010) with FLAC model results

The Young’s modulus of the EPS was estimated based on the stress–strain response of the Kim et al. (2010) EPS specimens up to an axial strain level of approximately 2.4%. Following this approach, the Young’s modulus for EPS was estimated to be approximately (Ef )avg = 2800 kPa (two test results showed 3190 and 2500 kPa, respectively). This value is in reasonable agreement with the density– stiffness relationships proposed by other researchers. For example, Hazarika (2006) and Horvath (1995) proposed the following equations: Ef ¼ 041ρf 28

ð1Þ

Ef ¼ 045ρf 3

ð2Þ

where Ef is the EPS Young’s modulus in MPa and ρf is the EPS density in kg/m3. Based on these equations, suggested Young’s modulus values are Ef = 3350 kPa and Ef = 3750 kPa, respectively. These values are approximately 20–34% higher than the selected value. According to Horvath (1995), the Poisson’s ratio of the EPS geofoam was expressed in terms of EPS density as: νf ¼ 00056ρf þ 00024

ð3Þ

where νf is the EPS Poisson’s ratio and ρf is the EPS density in kg/m3. A Poisson’s ratio value of approximately νf = 0.09 was selected using this equation. Strain-hardening parameters were estimated based on the unconfined compression test data reported by Kim et al. (2010). Strain-hardening was incorporated in the model by setting EPS cohesion to cf = 41 kPa at a plastic strain level of zero (i.e., at yield) and linearly increasing EPS cohesion to cf = 64 kPa at a strain level equivalent to 30% axial strain. The material properties assumed for the EPS geofoam model are summarized in Table 1. 2.2. Soil (air-pluviated Jumunjin sand) Kim et al. (2010) backfilled the test box using Jumunjin sand air-pluviated from a drop height of approximately 50 cm. This backfilling technique resulted in a sand dry unit weight of approximately γd = 14.8 kN/m3, corresponding to a void ratio of approximately e = 0.74 based on a specific gravity of 2.63.

The Jumunjin sand was modeled as a nonlinear-elastic/ Mohr–Coulomb material. Material properties for the Jumunjin sand were estimated based on test results reported by Kim et al. (2004, 2010) and Park et al. (2008) and on recommendations by Duncan et al. (1980). The elastic behavior of the sand was controlled by a stress-dependent tangent Young’s modulus, Et, and a fixed value of Poisson’s ratio, ν. The tangent Young’s modulus value was calculated as the product of the initial Young’s modulus value, Ei, and a hyperbolic degradation function. The initial Young’s modulus value was calculated as a function of mean effective stress, void ratio and Poisson’s ratio. Following the general form adopted by Papadimitriou et al. (2001), which is based on the equation proposed by Hardin (1978), the initial shear modulus (Gi) is calculated as: 0 0:5 Bpatm p Gi ¼ ð4Þ 03 þ 07e2 patm where B is the shear modulus number, patm is atmospheric pressure (assumed to be 101.3 kPa), e is the void ratio and p′ is the mean effective stress. The initial Young’s modulus, Ei, is related to the initial shear modulus according to Equation (5): Ei ¼ 2Gi ð1 þ νÞ

ð5Þ

where ν is Poisson’s ratio. Young’s modulus degradation with increasing shear stress was incorporated following the hyperbolic model proposed by Duncan and Chang (1970): Rf ð1 sin ϕÞðσ 01 σ 03 Þ 2 Ei ð6Þ Et ¼ 1 2c cos ϕ þ 2σ 03 sin ϕ where Et is the tangent Young’s modulus, Rf is the failure ratio, ϕ is the soil friction angle, σ1′ and σ3′ are the major and minor principal effective stresses, respectively, and c is the soil cohesion. The Poisson’s ratio of the Jumunjin sand was estimated based on the ratio of horizontal to vertical stresses for Test 1 (i.e., test box filled with sand only) reported by Kim et al. (2010) using the equation: σh ν ð7Þ ¼ σv 1 ν where σh and σv are the measured horizontal and vertical stresses, respectively, and ν is Poisson’s ratio. A value of



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According to the Park et al. (2008) regression equation, the direct shear friction angle for a void ratio of e = 0.74 is ϕds = 34°, which agrees reasonably well with the value of ϕds = 33° reported by Kim et al. (2010). The material properties assumed for Jumunjin sand are summarized in Table 2. For comparison with the Park et al. (2008) drained triaxial compression tests, a model was developed to simulate drained triaxial compression of a Jumunjin sand specimen. As shown in Figure 3, the simulation results are in reasonable agreement with the test results reported by Park et al. (2008).

Table 2. Material properties assumed for Jumunjin sand Parameter

Value

Shear modulus number, B

150 for initial loadinga 450 for unloading/reloadingb 0.39c 0.85d 0.74e 1.54 Mg/m3e 34°e 0.5 kPaf 0g

Poisson’s ratio, ν Failure ratio, Rf Void ratio, e Total density, ρt Friction angle, ϕ Cohesion, c Dilation angle, ψ

a Assumed to be reduced by a factor of 3 from the value for unloading/ reloading based on recommendation by Duncan et al. (1980) for dense sands. b Estimated based on results of laboratory testing on Jumunjin sand reported by Kim et al. (2004). c Estimated based on results of laboratory testing on Jumunjin sand reported by Kim et al. (2004) and on results of model-scale testing reported by Kim et al. (2010). d Estimated based on results of laboratory testing on poorly-graded sands (SP) reported by Duncan et al. (1980). e Estimated based on results of laboratory testing on Jumunjin sand reported by Park et al. (2008). f Assumed to account for moisture effects, following the convention described by Hatami and Witthoeft (2008). g Negligible effect on model results based on preliminary simulations varying from 0 to 10°.

Deviatoric stress (kPa)

400

300

200

Test results from Park et al. (2008): loose state Test results from Park et al. (2008): medium state FLAC model results

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0

0

4

8

12

Axial strain (%)

Figure 3. Results of drained triaxial compression test simulation using Jumunjin sand constitutive model

2.3. Steel pipe The pipe used in the Kim et al. (2010) model-scale tests was made of steel and had an outside diameter of approximately 10 cm. According to Kim et al. (2010), the bending stiffness of the model-scale pipe was evaluated based on the results of two parallel plate loading tests. The estimated pipe bending stiffness values measured in these tests at displacement levels of approximately 5% of the initial pipe diameter were approximately 433 kN/m/m and approximately 506 kN/m/m. It is noted that laboratory data recorded during these tests (unpublished) show an approximately linear load-displacement response within this strain level (i.e., up to approximately 5% of pipe diameter). A model was developed to simulate parallel plate testing of the model-scale pipe for calibration with the Kim et al. (2010) parallel plate test data. The pipe was simulated using beam elements assigned a linear-elastic material model. Fixed values were assigned for the pipe Young’s modulus, cross-sectional area and density, and the moment of inertia was varied until agreement with the Kim et al. (2010) test data was achieved. The simulation results and the pipe parameters are shown in Figure 4 and Table 3.

4 Test results from Kim et al. (2010) FLAC model results Applied axial load (kN/m)

ν = 0.39 was estimated using this method. This value agrees well with the measured Poisson’s ratio value (i.e., ν = 0.38) reported by Kim et al. (2004). The shear modulus number was estimated based on stiffness values during unloading as reported by Kim et al. (2004). Kim et al. (2004) reported Young’s modulus values for two tests performed at different void ratios and confining stresses (i.e., with e = 0.79 at σ3′ = 150 kPa; and with e = 0.74 at σ3′ = 100 kPa given in the paper). The shear strength of the sand was estimated based on a regression equation for direct shear friction angle ϕds as a function of relative density proposed by Park et al. (2008). Substituting the maximum and minimum void ratios reported by Park et al. (2008) in place of the relative density term, this equation is: 0843 e 2 ϕds ¼00009 100 0226 ð8Þ 0843 e þ 00216 100 þ 31233 0226

3

2

1

0

0

2

4

6

Vertical displacement (mm)

Figure 4. Comparison of parallel plate test results for steel pipe reported by Kim et al. (2010) with FLAC model results



Earth pressure reduction on buried pipes using EPS geofoam

pipe axis) by 140 cm wide by 90 cm high. The base and three of the four walls of the box were made of steel, while one face of the four walls was made of transparent rigid acrylic plate. The model-scale test procedure consisted of three steps: (1) pipe placement and side fill; (2) EPS panel placement and backfilling; and (3) surcharging. During the first step, the pipe was placed along the base of the test box, and Jumunjin sand was pluviated into the box until the sand fill level reached the pipe crown. During the second step, an EPS geofoam compressible inclusion was placed at the level of the pipe crown (as applicable), and Jumunjin sand was pluviated into the box from a drop height of approximately 50 cm until the sand level reached the top of the box. During the final step, a surcharge load was applied in three stages (i.e., 49, 98 and 147 kPa) to the top of the Jumunjin sand fill using a steel plate advanced by a hydraulic system. Two pressure transducers approximately 5 cm in diameter were mounted on the pipe. One transducer was fixed at the pipe crown and was oriented to measure vertical

3. SIMULATION OF MODEL-SCALE TESTS The model-scale tests performed by Kim et al. (2010) were simulated using the calibrated material models for EPS geofoam, Jumunjin sand and the steel pipe. Photographs of the test box illustrating the testing procedure are shown in Figure 5. The details of the test chamber and test procedure were presented by Kim et al. (2010). The box had interior dimensions of 100 cm long (i.e., along the Table 3. Material properties assumed for model-scale pipe

a

Parameter

Value

Young’s modulus, E Moment of inertia, I Cross-sectional area, A Density, ρ

200 GPaa 47 × 10−12 m4/ma 1.5 × 10−3 m2/mb 8 Mg/m3a

5

Assumed as typical steel material properties. Estimated based on measurements reported by Kim et al. (2010).

b

(a)

(b)

(c)

Figure 5. Model test procedure by Kim et al. (2010): (a) cross-section of soil box, (b) air-pluviation Jumunjin sand, and (c) the placement of EPS geofoam Geosynthetics International


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stress. The other transducer was attached to the pipe springline and was oriented to measure horizontal stress. Both vertical and horizontal stress measurements were recorded at the end of backfilling and after each surcharge loading stage. 3.1. Geometry and boundary conditions A plane-strain model was developed using FLAC (Itasca 2011) to simulate the Kim et al. (2010) model-scale test sequence. Due to symmetry, only half of the test box was included in the simulation. The geometry and boundary conditions of the numerical model are shown in Figure 6. Each simulation was separated into two phases: a pluviation phase and a surcharging phase. The pluviation phase consisted of a sequence in which one row of soil numerical zones at a time (for a lift thickness ranging from 1.0 cm to approximately 3.3 cm) was activated and the model was allowed to reach static equilibrium. This sequence was performed for the first (i.e., lowest) row of numerical zones and was repeated until all soil numerical zones in the model were active (i.e., rows of numerical zones were activated sequentially from the bottom to the top of the model). During the surcharging phase, a vertical stress boundary condition was applied along the top of the model. The magnitude of this vertical stress was increased gradually (i.e., in increments of approximately 2.5 kPa, allowing the model to equilibrate between increments) until reaching the maximum surcharge load applied in the model tests by Kim et al. (2010). Displacement constraints were applied as shown in Figure 6. A horizontal displacement constraint was used along the plane of symmetry. The lateral and rotational degrees of freedom were fixed for the structural nodes lying on this boundary at all stages of the analyses. During the pluviation phase, the lower boundary was fixed in the vertical direction, and the right boundary was fixed in the horizontal direction. However, these boundaries were fixed in both horizontal and vertical directions during the surcharging phase. It was assumed that friction developed along these boundaries could be significant (e.g., USACE 1994, recommendations suggest a sand– steel interface friction angle of approximately one-half of the sand internal friction angle, or approximately 17° to 18° for the sand considered here) and could be idealized using a no-slip condition. It is noted that boundary friction effects are evident in the test results by Kim et al. (2010) test: for Test 1 (i.e., soil only), the measured vertical earth pressure at the end of surcharging (i.e., 139.3 kPa) was approximately 13% less than the sum of the self-weight and surcharge loads (i.e., 12.1 kPa + 147.1 kPa = 159.2 kPa). Thus, the model allowed the development of friction between the sand fill and the right boundary. 3.2. Pipe–soil interaction behavior Pipe–soil interaction was modeled as a frictional interface using the FLAC software’s built-in interface logic. Based on recommendations by USACE (1994), the interface

shear strength was assumed to be one-half of the estimated soil shear strength. The interface normal and shear stiffnesses were estimated iteratively by fixing stiffness values and comparing model-predicted stresses against stresses in Test 1 (i.e., test box filled with sand only) reported by Kim et al. (2010). Assumed properties for the pipe–soil interface are listed in Table 4. 3.3. EPS–soil interaction behavior EPS–soil interaction was modeled assuming a no-slip condition. However, preliminary simulations were performed to evaluate the effect of this assumption on model-predicted stresses. The effect was evaluated as change from the ‘baseline’ (i.e., no-slip assumption) model-predicted stresses at the pipe crown and springline. For these preliminary simulations, an EPS–soil interface was included in the model, and interface shear and normal stiffness values as well as interface friction angle values were varied. Interface stiffness values were varied from 600 × 103 kN/m/m to 9600 × 103 kN/m/m, and interface friction angle values were varied from 1° (i.e., similar to a full-slip condition) to 34° (i.e., similar to a no-slip condition). For the parameter ranges evaluated, model-predicted stresses generally showed modest sensitivity to interface stiffness (i.e., approximately 5–15% variation from ‘baseline’ values) and interface friction angle (i.e., approximately 5–10% variation from ‘baseline’ values). In light of this modest sensitivity the no-slip assumption was considered to be adequate for the purposes of this study. 3.4. Effects of soil and EPS constitutive behavior Although the model results presented here were developed using nonlinear-elastic/Mohr–Coulomb soil constitutive behavior, preliminary simulations were performed to evaluate the effects of using linear-elastic/Mohr–Coulomb soil constitutive behavior on model-predicted stresses. For these simulations, constant values of Young’s modulus and Poisson’s ratio were assigned to all soil zones in the numerical grid. Poisson’s ratio was fixed at a value of ν = 0.39, while Young’s modulus was varied between a ‘lower-bound’ value of approximately 48 MPa and an ‘upper-bound’ value of approximately 183 MPa. These Young’s modulus values were taken as representative of model-calculated (using the nonlinear-elastic/ Mohr–Coulomb soil model) values at the end of backfilling and the end of surcharging, respectively. The effect of the soil constitutive model was evaluated as change from the ‘baseline’ (i.e., nonlinear-elastic/ Mohr–Coulomb) model-predicted stresses at the pipe crown and springline. While agreement of modelpredicted stresses with the ‘baseline’ was reasonably good for the ‘lower-bound’ Young’s modulus value (i.e., approximately 20–25% variation from ‘baseline’ values), the variation was significant for the ‘upper-bound’ Young’s modulus value (i.e., approximately 60–65% variation from ‘baseline’ values). In light of this sensitivity to the constant stiffness value selected, it is recommended to assume nonlinear-elastic soil behavior,




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Surcharge load (surcharging phase)

Jumunjin sand

Model boundary fixed in x-direction (pluviation phase) and x- and y-directions (surcharging phase)

Model boundary fixed in x -direction

+y +r +x

Structural nodes fixed x- and rdirections

EPS geofoam

90 cm

Model boundary fixed in y-direction (pluviation phase) and x- and y-directions (surcharging phase)

Steel pipe with frictional pipe soil interface

70 cm (a)

Surcharge load (surcharging phase)

Jumunjin sand

Model boundary fixed in x-direction

+y

Model boundary fixed in x-direction (pluviation phase) and x- and y-directions (surcharging phase)

+r +x

90 cm

EPS geofoam (two-panel configuration shown) Structural nodes fixed in x- and rdirections

Model boundary fixed in y-direction (pluviation phase) and x- and y-directions (surcharging phase)

Steel pipe with frictional pipe soil interface

70 cm (b)

Figure 6. Geometry and boundary conditions for the test box model, (a) single layer of EPS geofoam, and (b) double layer of EPS geofoam

particularly when representative soil stress–strain data are available. The effect of EPS constitutive behavior was not systematically evaluated. However, for the EPS material properties assumed and the stress levels examined in this study, the EPS material did not enter the plastic range. Therefore, practically, the EPS constitutive behavior used in the simulations was equivalent to linear-elastic constitutive behavior. However, this equivalency should not

Table 4. Properties assumed for pipe–soil interface Parameter

Value

Normal stiffness, kn Shear stiffness, ks Friction angle, δ

600 × 103 kN/m/ma 600 × 103 kN/m/ma 17°b

a

Estimated based on calibration model for Kim et al. (2010) Test 2. Estimated as one-half of soil friction angle based on USACE (1994) recommendations.

b



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Table 5. Model-scale test variables (D denotes diameter of pipe, Wgeofoam denotes the width of EPS, and SEPS panel denotes spacing between EPS panels The presence of EPS geofoam inclusion

Test factor

Variables

Test no.

No EPS

Stress distribution of fill soil

Single layer of EPS

The width of EPS (single layer of EPS geofoam inclusion)

Double layer of EPS

Spacing between EPS geofoam inclusions (for all cases, Wgeofoam = 1.0D)

Sand deposit without pipe Sand deposit with pipe Wgeofoam = 1.0D Wgeofoam = 1.5D Wgeofoam = 2.1D SEPS panel = 0.5D SEPS panel = 1.0D SEPS panel = 1.2D SEPS panel = 1.5D

Test Test Test Test Test Test Test Test Test

• • •

Test 1 – the test box contained sand only (i.e., no pipe or compressible inclusion). Test 2 – the test box contained sand and the pipe (i.e., no compressible inclusion). Test 3 – the test box contained sand, the pipe and a 100 cm long × 10 cm wide × 5 cm thick layer of EPS geofoam panel over the pipe.

Model-predicted values of vertical stress at the pipe crown and horizontal stress at the pipe springline were plotted against corresponding test results. The comparisons of numerical results with model-scale test results are presented in Figure 7. As shown in Figure 7, the numerical results agree reasonably well with the measurements reported by Kim et al. (2010). Based on the agreement between the simulation results and the model-scale test results, it appears that the numerical model described here provides a reasonably accurate means to assess the effects of an EPS geofoam compressible inclusion on the static earth pressures around a buried pipe. As was observed in model test results by Kim et al. (2010), the numerical results using FLAC also showed the benefit of using EPS geofoam over the pipe.

4. ‘OPTIMIZATION’ OF EPS PANEL DIMENSIONS Using the calibrated numerical model, a series of simulations was performed to demonstrate a possible ‘optimization’ technique to select dimensions for a single

Vertical stress at pipe crown (kPa)

3.5. Comparison of numerical results with model-scale test results Based on the calibrations of materials used in the tests, numerical analysis using FLAC was performed to simulate the model-scale tests presented by Kim et al. (2010). Table 5 summarizes the model-scale test variables presented in Kim et al. (2010). Calibration simulations were performed for three of the model-scale tests reported by Kim et al. (2010):

150 Test 1 results (Kim et al. 2010) Test 2 results (Kim et al. 2010) Test 3 results (Kim et al. 2010) FLAC simulation of Test 1 FLAC simulation of Test 2 FLAC simulation of Test 3

100

50

0 0

30

60

90

120

150

Applied surcharge pressure (kPa) (a)

Horizontal stress at pipe springline (kPa)

be expected to hold for all sets of EPS material properties and loading conditions.

1 2 3 4 5 6 7 8 9

150 Test 1 results (Kim et al. 2010) Test 2 results (Kim et al. 2010) Test 3 results (Kim et al. 2010) FLAC simulation of Test 1 FLAC simulation of Test 2 FLAC simulation of Test 3

100

50

0 0

30

60

90

120

150

Applied surcharge pressure (kPa) (b)

Figure 7. Simulation results versus model-scale test results reported by Kim et al. (2010) for (a) vertical earth pressure measured at the pipe crown; and (b) horizontal earth pressure measured at the pipe springline

EPS panel under the test conditions modeled (e.g., soil type, surcharge load, etc.). Use of two layers of EPS geofoam panel for further earth pressure reduction was also investigated. Knowing that reduction of overburden stress on the pipe crown is an important consideration, the placement of EPS geofoam over a buried pipe should aid in reducing the vertical earth pressure on the pipe. However, uniformity of stresses around the pipe should also be considered to reduce the potential for ovaling deformation, as illustrated




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σv, crown (a): σv,crown / σh,springline = 1 (a)

σv,crown

σh,springline

σv, crown (a): σv,crown /σh,springline < 1

(b)

σh,springline

(a): σv,crown / σh,springline > 1 (c)

σh,springline

Figure 8. Schematic illustration showing possible pipe deformation patterns for (a) uniform stress around the pipe; (b) vertical stresses larger than horizontal stresses; and (c) horizontal stresses larger than vertical stresses. Solid and dashed lines indicate original and deformed pipe cross-section

schematically in Figure 8. Therefore, ‘optimization’ of EPS panel dimensions should consider at least the following criteria: (1) the magnitude of vertical stress at the pipe crown, σv,crown, should be reduced to the extent practicable; and (2) an acceptable ratio of horizontal stress at the pipe springline to vertical stress at the pipe crown, σh,springline/σv,crown, should be maintained. In general, a lower overburden stress is preferred to a higher overburden stress. Additionally, based on the σh,springline/σv,crown ratio reported by Kim et al. (2010) for Test 2 (i.e., sand and pipe only), it appears that σh,springline/ σv,crown = 1.0 ± 0.1 is a reasonable target. Therefore, the goal of this example ‘optimization’ is to minimize σv,crown while maintaining 0.9 ≤ σh,springline/σv,crown ≤ 1.1.

4.1. Effects of EPS panel width The first variable investigated during the ‘optimization’ procedure was the ratio of EPS panel width to pipe diameter, wgeofoam/Dpipe, with a fixed EPS panel thickness of tgeofoam = 5 cm. Numerical results for the variation of wgeofoam/Dpipe are shown in Figure 9. For comparison, model test results by Kim et al. (2010) are also plotted in Figure 9. As shown in Figure 9a, the magnitude of σv,crown initially decreases with the addition of an EPS panel but increases gradually as the EPS panel is widened. Applying the first ‘optimization’ criterion (i.e., significant reduction of σv,crown), a narrow panel is preferable over a wide panel (i.e., the panel should be as narrow as possible). Meanwhile, as shown in Figure 9b, the σh,springline/ σv,crown ratio is relatively high for a narrow EPS panel but approaches a value of σh,springline/σv,crown that is practically close to 1.0 as the panel is widened. Applying the second ‘optimization’ criterion (i.e., 0.9 ≤ σh,springline/ σv,crown ≤ 1.1), the ratio of EPS panel width to pipe diameter should be at least wgeofoam/Dpipe = 1.5.

Considering both ‘optimization’ criteria, it appears that an ‘optimal’ wgeofoam/Dpipe ratio should be as narrow as possible but at least wgeofoam/Dpipe = 1.5. Therefore, for the conditions considered here, the ‘optimal’ value of the wgeofoam/Dpipe = 1.5 ratio is approximately wgeofoam/ Dpipe = 1.5 (i.e., wgeofoam = 15 cm). Regression equations may be used to facilitate the ‘optimization’ process. For the configurations examined here, the effect of the normalized EPS panel width (wgeofoam/Dpipe) on the normalized vertical and horizontal stresses at the pipe crown and springline, respectively, may be approximated by: wgeofoam σ v;crown ¼ 008 ln þ 027; σ surcharge Dpipe ð9Þ wgeofoam 05 25 Dpipe and σ h;springline wgeofoam 0:39 ; ¼ 133 σ v;crown Dpipe wgeofoam 05 25 Dpipe

ð10Þ

Following this approach, the ‘optimization’ objective is to minimize σv,crown/Δσsurcharge, by changing wgeofoam/ Dpipe, subject to the constraint 0.9 ≤ σh,springline/ σv,crown ≤ 1.1. The constraint is satisfied on the interval of approximately 1.5 ≤ wgeofoam/Dpipe ≤ 2.5. For this interval, σv,crown/Δσsurcharge is minimized at a value of wgeofoam/ Dpipe = 1.5. 4.2. Effects of EPS panel thickness The second variable investigated during the ‘optimization’ procedure was the thickness of the EPS panel, tgeofoam, with a fixed ratio of EPS panel width to pipe diameter of



10

Witthoeft and Kim 1.2

1.2 Regression line FLAC parametric results Model test results (Kim et al. 2010)

σv,crown /σsurcharge (unitless)

Test 1

Test 1

0.8

σv,crown /σsurcharge = 0.08 ln(Wgeofoam /Dpipe ) + 0.27 (R2 = 0.99)

0.6

Test 2

0.4

0.2

0

σv,crown /σsurcharge = –0.09 ln(t geofoam ) + 0.45 (R2 = 0.98)

0.6

Test 2

0.4

Test 4 0

0

0.5

1

1.5

2

2.5

0

3

4

8

12

Wgeofoam /Dpipe (unitless)

t geofoam (cm)

(a)

(a)

16

20

2.1

2.1 Regression line FLAC parametric results Model test results (Kim et al. 2010)

Test 3 1.5 1.2

Test 2

0.9

Test 4 Test 5

0.6

Test 1

1.5 1.2 0.9

(R = 0.95)

Test 4

Test 2 Test 1

0.6

σh,springline /σv,crown = 0.94 (t geofoam )0.12

σh,springline /σv,crown = 1.33 (Wgeofoam /Dpipe )–0.39 2

0.3

Regression line Analytical results Model test results (Kim et al. 2010)

1.8 σh,springline /σv,crown (unitless)

1.8

σh,springline / σv,crown (unitless)

0.8

0.2

Test 5

Test 4

Test 3

Regression line Analytical results Model test results (Kim et al. 2010)

1.0 σv,crown / σsurcharge (unitless)

1.0

(R2 = 0.95)

0.3 0

0

0

0.5

1 1.5 2 Wgeofoam /Dpipe (unitless)

2.5

0

3

4

8

12

16

20

tgeofoam (cm) (b)

(b)

Figure 9. Effects of varying ratio of EPS panel width to pipe diameter while maintaining constant EPS panel thickness (tgeofoam = 5 cm) on (a) vertical earth pressure measured at the pipe crown; and (b) ratio of horizontal stress at the pipe springline to vertical stress at the pipe crown

wgeofoam/Dpipe = 1.5. It is noted that the effect of EPS panel thickness is also a function of EPS density because EPS density controls strain above a buried pipe when placed above it. In this study, the EPS density of 15 kg/m3 was investigated. Numerical results of stresses according to the variation of tgeofoam are shown in Figure 10. As shown in Figure 10a, the magnitude of σv,crown decreases with an increasing thickness of the EPS panel but tends toward a lower bound. Applying the first ‘optimization’ criterion (i.e., significant reduction of σv,crown), a thick panel is preferable over a thin panel (i.e., the panel should be at least 8 cm). Meanwhile, as shown in Figure 10b, the σh,springline/ σv,crown ratio is close to 1.0 for a small thickness of EPS but increases gradually as the panel thickness increases. Applying the second ‘optimization’ criterion (i.e., 0.9 ≤ σh,springline/σv,crown ≤ 1.1), the EPS panel thickness

Figure 10. Effects of varying EPS panel thickness while maintaining constant ratio of EPS panel width to pipe diameter (wgeofoam = 1.5Dpipe) on (a) vertical earth pressure measured at the pipe crown; and (b) ratio of horizontal stress at the pipe springline to vertical stress at the pipe crown

should not be larger than tgeofoam = 5 cm associated with the model test case. Considering both ‘optimization’ criteria, it appears that an ‘optimal’ EPS geofoam panel thickness should be as large as possible but not larger than tgeofoam = 5 cm. Therefore, for the conditions considered here, the ‘optimal’ thickness of the EPS panel is approximately tgeofoam = 5 cm. Following the regression equation ‘optimization’ approach, assuming an EPS panel width of wgeofoam/ Dpipe = 1.5, the effect of the EPS panel thickness (tgeofoam, cm) on the normalized vertical and horizontal stresses at the pipe crown and springline, respectively, may be approximated by: σ v;crown ¼ 009 ln tgeofoam þ 045; σ surcharge ð11Þ 1 cm tgeofoam 20 cm and



Earth pressure reduction on buried pipes using EPS geofoam 0:12 σ h;springline ¼0 94 tgeofoam ; σ v;crown 1 cm tgeofoam 20 cm

11 1.2

ð12Þ

5. SUMMARY AND CONCLUSIONS A series of numerical analyses was performed to study the use of EPS compressible inclusions for reducing earth pressures on a buried pipe. A numerical model

σv,crown / σsurcharge (unitless)

Model test results (Kim et al. 2010)

0.8

0.6

0.4 Test 8

Test 9

0.2 Test 6

Test 7

0 0

0.4

0.8

1.2

1.6

SEPS panel /Dpipe (unitless) (a) 2.1

(unitless)

1.8

v,crown

4.3. Effects of second EPS panel and spacing between panels Using two layers of EPS geofoam panels in a stacked configuration (see Figure 6) was also investigated. Numerical results, together with test results from Kim et al. (2010), for this dual-panel configuration are presented in Figure 11. As shown in the figure, the ‘optimized’ dimensions of wgeofoam/Dpipe = 1.5 and tgeofoam = 5 cm were used for both panels. Also, in order to compare the numerical results with model-scale test results by Kim et al. (2010), numerical analysis with dimensions of wgeofoam/Dpipe = 1.0 and tgeofoam = 5 cm were performed. Spacing between panels (spanels, i.e., distance from the top of the lower panel to the bottom of the upper panel) was varied between spanels/Dpipe = 0.0 (equivalent to a single EPS panel with tgeofoam = 10 cm) and a single EPS panel with spanels/Dpipe = 1.5. Based on the results shown in Figure 11a, it appears that vertical stress at the pipe crown is relatively insensitive to the spacing between panels. It is noted that the value of σh,springline/σv,crown is outside the interval 0.9 ≤ σh,springline/ σv,crown ≤ 1.1, as shown in Figure 11b. Therefore, the ‘optimization’ criteria are not satisfied for the doublepanel configurations examined here. A reason for these results is suggested by the patterns of stress and displacement presented in Figures 12a and 12b, respectively. In general, a second compressible inclusion generates additional stress reduction when the second compressible layer is placed above the plane of equal settlement (i.e., the level above which the vertical displacement in the soil prism overlying the pipe is approximately equal to that of the adjacent soil). As illustrated in Figure 12b, for the model geometry considered here, no plane of equal settlement is developed; consequently, the effectiveness of the second compressible layer is limited for this geometry. In addition, Figure 12b indicates that the displacement induced to the steel pipe is less than 0.15 mm. As shown in Figure 4, because the steel pipe should behave elastically for displacements less than 0.15 mm, the soil–pipe system in the model test should represent a rigid pipe condition.

σh,springline /σ

Following this approach, the ‘optimization’ objective is to minimize σv,crown/Δσsurcharge, by changing tgeofoam, subject to the constraint 0.9 ≤ σh,springline/σv,crown ≤ 1.1. The constraint is satisfied on the interval of approximately 0 < tgeofoam ≤ 5 cm. For this interval, σv,crown/Δσsurcharge is minimized at a value of tgeofoam = 5 cm.

FLAC parametric results (w geofoam = 1.5D) FLAC parametric results (w geofoam = 1.0D)

1.0

Test 9

1.5 1.2 Test 7 Test 8 0.9 0.6 FLAC parametric results (w geofoam = 1.5D) FLAC parametric results (w geofoam = 1.0D) Model test results (Kim et al. 2010)

0.3 0.0 0

0.4

0.8

1.2

1.6

SEPS panel /Dpipe (unitless) (b)

Figure 11. Effects of varying spacing between EPS panels while maintaining constant ratio of EPS panel width to pipe diameter (wgeofoam = 1.5Dpipe) and constant EPS panel thickness (tgeofoam = 5 cm) on (a) vertical earth pressure measured at the pipe crown; and (b) ratio of horizontal stress at the pipe springline to vertical stress at the pipe crown

was developed and calibrated against measurements from an experimental study following a two-step process. First, a material model was developed and calibrated for each of the three materials used in the model-scale tests (i.e., EPS geofoam, steel pipe and Jumunjin sand). Second, a numerical model was developed to simulate the test box used in the model-scale tests. The numerical model was calibrated against the model-scale test results and was used to examine more cases that were not covered in the experimental program; results from these additional cases were presented in this paper. Notably, the use of numerical analysis methods provides for a systematic approach to evaluate the ‘optimal’ configuration of the EPS geofoam panel. ‘Optimization’ criteria were developed that considered both the magnitude of vertical stress at the crown of the pipe and the uniformity of stresses around the pipe. Based on the



12

Witthoeft and Kim equal settlement, which was not achieved for the geometry used in the analysis. More refined study should be performed in the future. While the ‘optimization’ presented here illustrates one possible design approach for compressible inclusions above buried pipes, additional research is anticipated to provide a more broadly applicable design approach for various pipe characteristics and site conditions. Key items to be considered in developing a design approach for EPS compressible inclusions include:

Jumunjin sand

+y

140

+r +x 120

•

100 80

EPS geofoam

60

• (a)

•

Jumunjin sand – 1.05

– 0.90

– 0.75

+y +r

– 0.60

+x

– 0.45

EPS geofoam

•

– 0.30

– 0.15

•

(b)

Figure 12. Numerical results for EPS panel width to pipe diameter ratio wgeofoam = 1.5Dpipe and EPS panel thickness tgeofoam = 5 cm: (a) vertical stress contours (kPa); and (b) vertical displacement contours (mm)

simulation results and the ‘optimization’ objectives of minimizing σv,crown while maintaining 0.9 ≤ σh,springline/ σv,crown ≤ 1.1, ‘optimal’ EPS panel dimensions geometry for the problem considered were estimated to be approximately wgeofoam/Dpipe = 1.5 (i.e., wgeofoam = 15 cm) and tgeofoam = 5 cm. The numerical model was also used to investigate the use of a double-panel configuration. The configurations examined in this paper did not satisfy the ‘optimization’ criteria. This is due to the fact that the second upper layer of the EPS panel should be placed above the plane of

Soil effects: the ‘optimal’ geofoam configuration depends on the amount of displacement required to mobilize shear stresses above the pipe. The amount of displacement required depends on factors including soil type and relative density. Pipe stiffness effects: the stresses and strains developed within a given fill material also depend on pipe stiffness. Pipe stiffness also depends upon the shape of the pipe (e.g., circular or different curvature) in addition to the pipe material. Geofoam effects: the amount of displacement achieved using a compressible inclusion depends on the properties of the inclusion, including stiffness, strength and creep effects. These properties are strongly correlated with the EPS density. Only one EPS density was evaluated in this study. Consequently, it is considered possible that ‘optimized’ geometry of the EPS panel might vary depending on the EPS density specified. Also, analysis should consider actual (delivered) EPS densities, not nominal densities, because the EPS density may be manufacturerdependent and different from its nominal density. Note that this variation of the EPS material quality should be carefully discussed and accounted for from the design stage. Economic factors: for practical purposes, an ‘optimized’ design should account for both expected savings due to a reduced pipe section and cost increases due to EPS material and installation costs. Practical considerations: while, in general, lower EPS density results in lower EPS stiffness and improved compressible inclusions performance, practical lowerbound limits on EPS density may apply. For example, in the authors’ experience, manufacturing quality control issues may arise for EPS densities at or below approximately 12 kg/m3. Additionally, regulatory agencies may impose lower-bound limits on EPS density.

ACKNOWLEDGEMENTS The authors thank S. R. Gudavalli of Fugro Consultants, Inc., Houston, Texas; P. Strandberg of Fugro (USA), Inc., Houston, Texas; and D. Chu of Ninyo & Moore Geotechnical and Environmental Sciences Consultants, Irvine, California, for their reviews of the original manuscript and valuable comments.




NOTATION Basic SI units are given in parentheses. A B c cf Dpipe E Ef (Ef )avg Ei Et e Gi I kn ks p′ patm Rf spanels tgeofoam wgeofoam Δσsurcharge δ ϕ

ϕds

γd ν νf ρ ρf ρt

σh, σh,springline σv, σv,crown σ1′ σ3′ ψ

pipe cross-sectional area (m2/m) shear modulus number for soil (dimensionless) soil cohesion (Pa) cohesion (shear strength) of EPS (Pa) diameter of pipe (m) pipe Young’s modulus (Pa) EPS Young’s modulus (Pa) average Young’s modulus of EPS test specimens (Pa) initial Young’s modulus of soil (Pa) tangent Young’s modulus of soil (Pa) void ratio of soil (dimensionless) initial shear modulus of soil (Pa) pipe moment of inertia (m4/m) interface normal stiffness (N/m/m) interface shear stiffness (N/m/m) mean effective stress (Pa) atmospheric pressure (Pa) failure ratio for soil (dimensionless) spacing between EPS panels (m) thickness of EPS panel (m) width of EPS panel (m) surcharge load applied at top of soil box (Pa) interface friction angle (degrees) soil friction angle (degrees) soil friction angle measured in direct shear test (degrees) dry unit weight of soil (N/m3) Poisson’s ratio of soil (dimensionless) Poisson’s ratio of EPS (dimensionless) pipe density (kg/m3) EPS density (kg/m3) soil total density (kg/m3) horizontal stress at pipe springline (Pa) vertical stress at pipe crown (Pa) major principal effective stress (Pa) minor principal effective stress (Pa) soil dilation angle (degrees)

REFERENCES Duncan, J. M. & Chang, C. Y. (1970). Nonlinear analysis of stress and strain in soils. Journal of the Soil Mechanics and Foundations Division, 96, No. SM5, 1629–1653. Duncan, J. M., Byrne, P., Wong, K. S. & Mabry, P. (1980). Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analysis of Stresses and Movements in Soil Masses, Report No. UCB/GT/80-01. Department of Civil Engineering, University of California Berkeley, Berkeley, CA, USA. Duskov, M. (1997). Materials research on EPS20 and EPS15 under representative conditions in pavement structures. Geotextiles and Geomembranes, 15, No. 1, 147–181. Frydenlund, T. E. & Aabøe, R. (1996). Expanded polystyrene – the light solution. In Proceedings of the International Symposium

13 on EPS Construction Method (EPS Tokyo ‘96), EPS Construction Method Development Organization, Tokyo, Japan, pp. 31–46. Hardin, B. O. (1978). The nature of stress-strain behavior for soils. In Proceedings of the ASCE Geotechnical Engineering Division Specialty Conference on Earthquake Engineering and Soil Dynamics, American Society of Civil Engineers, Pasadena, CA, USA, pp. 3–90. Hazarika, H. (2006). Stress-strain modeling of EPS geofoam for largestrain applications. Geotextiles and Geomembranes, 24, No. 2, 79–90. Horvath, J. S. (1995). Geofoam Geosynthetic, Horvath Engineering, PC, Scarsdale, NY, USA. Hatami, K. & Witthoeft, A. F. (2008). A numerical study on the use of geofoam to increase the external stability of reinforced soil walls. Geosynthetics International, 15, No. 6, 452–470. Itasca (2011). FLAC User’s Manual, Fifth edn. (FLAC Version 7.0), dated September 2011, Itasca Consulting Group, Inc., Minneapolis, MN, USA. Kim, K. & Yoo, C. H. (2005). Design loading for deeply buried box culverts. Journal of Geotechnical and Geoenvironmental Engineering, 131, No. 1, 20–27. Kim, S. I., Choi, J. S., Park, K. B., Seo, K. B. & Lee, J. H. (2004). Simulation of dynamic response of saturated sands using modified DSC model. In Proceedings of the 13th World Conference on Earthquake Engineering, WCEE Secretariat; Paper No. 1208, Vancouver, BC, Canada. Kim, H., Choi, B. & Kim, J. (2010). Reduction of earth pressure on buried pipes by EPS geofoam inclusions. Geotechnical Testing Journal, 33, No. 4, 1–10. Larson, N. G. & Hendrickson, J. G. (1962). A practical method for constructing rigid conduits under high fills. In Proceedings of the 41st Annual Meeting of the Highway Research Board (eds Burggraf, F., Orland, H. P. & Jackson, E. W.), vol. 41, pp. 273–280. Highway Research Board, Washington, DC, USA. Marston, A. (1930). The theory of external loads on closed conduits in the light of the latest experiments. Bulletin 96, Iowa Engineering Experiment Station, Iowa State College, Ames, IA, USA. McAfee, R. P. & Valsangkar, A. J. (2004). Geotechnical properties of compressible materials used for induced trench construction. Journal of Testing and Evaluation, 32, No. 2, 143–152. McAfee, R. P. & Valsangkar, A. J. (2005). Performance of an induced trench installation. Transportation Research Record, No. 1936, 230–237. Papadimitriou, A. G., Bouckovalas, G. D. & Dafalias, Y. F. (2001). Plasticity model for sand under small and large cyclic strains. Journal of Geotechnical and Geoenvironmental Engineering, 127, No. 11, 973–983. Park, L. K., Suneel, M. & Chul, I. J. (2008). Shear strength of Jumunjin sand according to relative density. Marine Georesources & Geotechnology, 26, No. 2, 101–110. Spangler, M. G. (1958). A practical application of the imperfect ditch method of construction. In Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board (eds Burggraf, F., Ward, E. M. & Orland, H. P.), vol. 37, pp. 271–277, Highway Research Board, Washington, DC, USA. Sun, L., Hopkins, T. C. & Beckham, T. L. (2005). Use of Ultralightweight Geofoam to Reduce Stresses in Highway Culvert Extensions, Publication KTC-05-34/SPR-297-05-1I. Kentucky Transportation Center, University of Kentucky, Frankfort, KN, USA. Sun, L., Hopkins, T. C. & Beckham, T. L. (2009). Reduction of Stresses on Buried Rigid Highway Structures Using the Imperfect Ditch Method and Expanded Polystyrene (Geofoam), Publication KTC-07-14/SPR-228-01-1F. Kentucky Transportation Center, University of Kentucky, Frankfort, KN, USA. USACE (1994). Design of Sheet Pile Walls. Engineer Manual EM 1110-2-2504, U.S. Army Corps of Engineers, Department of the Army, Washington, DC, USA.



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Vaslestad, J., Johansen, T. H. & Holm, W. (1994a). Load reduction on rigid culverts beneath high fills: long-term behavior. Transportation Research Record, No. 1415, 58–68. Vaslestad, J., Johansen, T. H. & Holm, W. (1994b). Load Reduction on Buried Rigid Pipes; Load Reduction on Rigid Culverts Beneath High Fills: Long Term Behaviour; Long-Term

Behaviour of Flexible Large-Span Culvert, Director of Public Roads, Norwegian Road Research Laboratory, Oslo, Norway. Zarnani, S. & Bathurst, R. J. (2007). Experimental investigation of EPS geofoam seismic buffers using shaking table tests. Geosynthetics International, 14, No. 3, 165–177.

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