laboratory testing and numerical simulation of a strip

Laboratory Testing and Numerical Simulation of a Strip Footing on Geosynthetically

Reinforced Loose Sand

Jiunnren Lai Associate Professor, Department of Construction Engineering, Chaoyang University of Technology, 168 Jifong E. Rd., Wufong District, Taichung City, Taiwan 41349. Bo-Huan Yang Ph.D. Candidate, Department of Construction Engineering, Chaoyang University of Technology, 168 Jifong E. Rd., Wufong District, Taichung City, Taiwan 41349.

Abstract

The objective of this paper is to study the bearing capacity and loading behavior of strip

footings on geosynthetically reinforced loose sand. Laboratory loading tests were conducted on

a rigid steel strip footing in a 0.9m x 0.9m x 1.0m sand box. Woven geotextile was used as

reinforcement material. A finite difference program was used to simulate the load-settlement

behavior of this footing. Two constitutive (Mohr-Coulomb and Double-Yield) models were

utilized for the loose sand. Direct shear tests were performed to obtain the parameters needed

for numerical simulation. The unreinforced ultimate bearing capacity obtained from this study

ranges between theoretical values calculated by assuming general and local shear failure

conditions. The bearing capacity predicted by these two soil models match with values

obtained from sand box tests. However, the Double-Yield model outperforms the traditional

Mohr-Coulomb model in simulating the loading behavior of this strip footing.

Keywords: bearing capacity, geotextile, Double-Yield model

Introduction

The bearing capacity of shallow foundation on weak soil reinforced with geosynthetics is one

of the most important subjects in geotechnical engineering. A great number of investigators

have been devoted their research efforts into this issue. In 1975, Binguet and Lee [1] reported

results of bearing capacity tests of footings on soil bed reinforced with galvanized steel strips

and defined qu,reinforced/qu,unreinforced as Bearing Capacity Ratio (BCR) to quantify the benefit of

reinforcement. For a better understanding of how the reinforced soil performs, several

researchers have been investigated the failure mechanism between soil and reinforcement

material [2-4]. The failure mechanism and bearing capacity of reinforced soil subjected to

footing load were studied by Binguet and Lee [5], Schlosser and Elias [6], Huang and Tatsuoka

[7], Huang and Menq [8], Yamamoto and Kusuda [9], and many others. For designing purpose,

some investigators have performed numerical simulations using finite element (FEM) method

[10-13] or finite difference program [14, 15] to predict the ultimate bearing capacity of

reinforced soil and to optimize the arrangement of reinforcement elements beneath the footing.

On the other hand, researchers such as: Fragaszy and Lawton[16], Verma and Char [17], Khing

et al. [18], Omar et al. [19], Dash et al. [20], Ghosh et al. [21], and Abu-Farsakh et al. [22,23]

have conducted laboratory tests or field observations to obtain the optimum arrangements for

various reinforcement materials. For single layer of reinforcement, they found the optimum

reinforcement length is about 4~7 times footing width (B), and the optimum depth ratio ranges

between 0.25B and 0.4B. Sharma [24] and Chen and Abu-Farsakh [25] have developed

analytical solutions to estimate the ultimate bearing capacity of strip footings on reinforced soil.

However, most of these studies have been focused on the ultimate bearing capacity of the

reinforced soil under general shear failure condition, which is reasonable for dense sand. For

cases when heavy construction equipment is not available or the compaction quality is not as

good as expected. The sand will be in loose condition and general shear failure may not occur.

Therefore, in this paper, a series of laboratory tests were conducted to study the loading

behavior of a reinforced strip footing in loose sand. The load-settlement curves of these tests

were simulated using a commercial finite difference program. Bearing capacity and loading

behavior of the strip footing from sandbox test and numerical simulation were compared. The

effects of various designing parameters, such as depth, length, and layer arrangement of the

reinforcement on the bearing capacity were also investigated.

Experimental Methods and Material Properties

Testing Equipment

A series of sandbox tests were conducted to study the loading behavior of strip footings on

reinforced loose sand. A schematic drawing of the testing equipment is shown in Fig. 1. It

consists of a test cell, a reaction frame with hydraulic ram, a hydraulic pump, load and

displacement sensors, and a data acquisition system. The dimensions of the test cell are 0.9m x

0.9m x 1.0m, respectively. The test cell was prepared by pouring dry sand from a #10 sieve at a

constant falling height of 0.3m. This sample preparation method can yield uniform sand with a

density of about 1410 kg/m3 (relative density, Dr = 12.4%) and a friction angle, , of about 34

degrees. A hydraulic loading system was used to apply normal force to a 0.85m x 0.10m x

0.05m rectangular steel plate. A load cell was used to measure the applied force while two

linear variable differential transformer (LVDT) were used to measure the settlement of the

footing.

Material Properties

The sand used in this study is an alluvium sand commonly used as fine aggregate for concrete.

Table 1 shows the properties of this sand. Its fine content (% finer than 0.075mm) is less than

5%. The Uniformity coefficient (Cu) and Coefficient of gradation (Cc) are 2.97 and 0.94,

respectively. It can be classified as poorly graded sand (SP) according to the Unified Soil

Classification System (USCS). A woven geotextile (model PP 70/70) was used as

reinforcement material. The properties of this geotextile are shown in Table 2.

Numerical Simulations

Soil Model

A commercial finite difference program - Fast Lagrangian Analysis of Continua (FLAC) was

used to perform the numerical simulation. The software provides a variety of soil model such

as the traditional elastic-plastic Mohr-Coulomb model as well as the Double-Yield model. In

addition to shear (line A-B in Fig. 2) and tensile yield function (line B-C in Fig. 2) of the

Mohr-Coulomb model, a volumetric yield function (Fig. 3) is also considered in the

Double-Yield model. In this model, the permanent volume change (plastic strain) induced by

change in normal stresses is taken into account [26]. The volumetric yield surface, also known

as “cap”, is defined by the cap stress (pc), and is related to the plastic volumetric strain. When pc

increases, the soil becomes denser, and its stiffness also increases. Therefore, it is suitable to

simulate the volume change behavior of reinforced soil. As shown in Fig. 3, the stiffness of soil

(i.e., the current bulk modules, Kc) is a function of cap stress and plastic volumetric strain, and

can be calculated by the following equation:

cc p

V

dpK R

d (1)

where R is the stiffness ratio and is equal to p eV V , while e

V and pV are elastic and plastic

volumetric strain, respectively. In FLAC, R is a parameter that should be input by the user. In

addition, FLAC calculates the current shear modules, Gc, by the following equation:

cc

KG G

K (2)

Where K and G are the initial (maximum) elastic bulk modulus and shear modulus,

respectively.

In the Double-Yield model, the shear yield function and plastic potential function are

controlled by mobilized friction angle (mob) and dilation angle () respectively. The shear

yield is a function of mobilized friction angle (mob) and plastic shear strain (p) and can be

obtained from plane strain tests or direct shear tests using the following equations:

1 yxmob

yy

Tan

(3)

max

yx yxp d d

G G

(4)

Where yx and yy are shear stress and normal stress, G and Gmax are current and maximum

(initial) shear modulus, dyx is the incremental shear stress, respectively.

The dilatancy angle controls the amount of plastic volumetric strain developed during

plastic shearing and can be obtained from a plane strain test using the following equation [27]:

1 1 3

1 3

p p

p pSin

(5)

where 1p and 3

p are the major and minor principle plastic strain increment, respectively. It

can also be obtained from the vertical displacement-shear displacement curve of a direct shear

test using the following equation [28]:

1 1yy

yx

d dyTan Tan

d dx

(6)

Wheredy and dx are the incremental vertical and shear displacement, respectively.

When using the Double-Yield model, in addition to the density, peak friction angle, shear

and bulk moduli needed for the Mohr-Coulomb model, a stiffness ratio, a table defining

volumetric strain (v) vs. normal stress (n, a second table defining mobilized friction angle

(mob) vs. plastic shear strain (p), and a third table defining dilation angle () vs. plastic shear

strain (p) are also needed.

Reinforcement Model

A cable element was used to simulate the behavior of reinforcement (geotextile). In FLAC, the

cable is assumed to be divided into a number of segments of length, L, with nodal points

located at each segment end. The axial behavior of conventional reinforcement systems is

assumed to be governed entirely by the reinforcing element itself. In formulation, the axial

stiffness is described in terms of the reinforcement cross-sectional area (A) and Young’s

modulus (E). The shear behavior of the reinforcement/medium interface is represented as a

spring-slider system located at the nodal points, and is described numerically by the interface

shear stiffness (Kbond) using the following equation:

( )Sbond c m

FK u u

L (7)

where:

Fs = shear force that develops in the grout

(i.e., along the interface between the cable element and the grid);

Kbond = grout shear stiffness;

uc = axial displacement of the cable;

um = axial displacement of the medium (soil or rock); and

L = contributing element length.

The maximum shear force developed in the interface, per length of element, is a function of the

cohesive strength and the stress-dependent frictional resistance of the interface. The following

relation is used to determine the maximum shear force:

max

tan( )sbond c friction

FS S perimeter

L (8)

where:

Sbond = intrinsic shear strength or cohesion;

’c = mean effective confining stress normal to the element;

Sfriction = friction angle; and

perimeter = exposed perimeter of the element.

The mesh used in numerical simulation is shown schematically in Fig. 4. The element size

is 1cm x 1cm. For a strip footing, it is reasonable to assume the soil is subjected to plane strain

condition. Because of symmetry, only the right-half of the test cell was simulated. A roller

boundary was used for the left boundary (axis of symmetry), while a pinned boundary was used

for the right and bottom boundaries. Since a thick steel plate was used as strip footing in the

sandbox test, the settlement of soil underneath the footing should be uniform. Therefore, a

constant rate of deformation was assigned to the grids right beneath the footing to simulate the

application of footing load. Unbalance forces of all grids were calculated and redistributed for a

small time step until it reaches equilibrium. The stress applied on the footing was obtained by

summing the nodal force in the vertical direction of grid points beneath the footing, than

divided by half the width of the footing.

Results and Discussions

Parameters for Sand

Table 3 shows properties of the loose sand used in numerical simulation. Oedometer test and

direct shear test were performed to obtain the parameters (properties) needed for numerical

simulation. Stress-volumetric strain curve from the oedometer test is shown in Fig.5. The shear

modulus (G) and bulk modulus (K) are calculated using constrained modulus (M) obtained

from the first load increment and assuming a Poisson’s ratio () of 0.35 by the flowing two

equations:

(1 2 )

2(1 )

MG

(9)

(1 )

3(1 )

MK

(10)

The stiffness ratio (R) obtained from the 1st and 2nd loading-unloading cycles are 5.3 and 5.7,

with an average value of 5.5. The stress-strain data in Fig. 5 were also used as the input table of

volumetric strain (v) vs. normal stress (n) relationship in the Double Yield model.

The shear stress (yx) vs. shear displacement (dx) curves from three direct shear tests are

shown in Fig. 6. The cohesion intercept, c, and angle of internal friction, peak, obtained from

these tests are 0 kPa and 34.4o, respectively. For each data point in Fig. 7a, the yx/y value was

used to calculate the ordinate (mob) of shear yield function using Eq. (3), and the shearing

displacement (dx) value was used to calculate the abscissa (p) of shear yield function using Eq.

(4). Similarly, the vertical displacement (dy) and the shearing displacement (dx) in Fig. 7b were

used to calculate the dilation angle () of the plastic potential function using Eq. (6). These two

functions are shown in Fig. 8 and Fig. 7b and were used as input table for the Double Yield

model in numerical simulation. Negative values of dilation angle were obtained (Fig. 7b)

because the sand was prepared in loose condition and is expected to have contractive behavior

during shear.

Parameters for Reinforcement

Table 4 shows the properties of geotextile used in numerical simulation. The elastic (Young’s)

modulus was calculated by using the tensile strength at 2% elongation in the cross direction.

The interface parameters (Kbond and Sbond) were obtained from previous pullout tests. The

interface friction angle (interface) was assumed to be equal to 1/3 of the peak of the loose sand, a

typical reduction factor that is commonly used for the interface friction angle between soil and

retaining structures. Parameter study indicated that the bearing capacity is influenced mostly

by Kbond. The interface friction angle (1/3~1 of soil-soil) has minimum effect on the bearing

capacity.

Unreinforced Footing

In order to validate the soil models used in this study, numerical simulations were performed

using the Mohr-Coulomb model and the Double Yield model. Results from the numerical

simulation are then compared with results from laboratory testing. Figure 9 shows the tested

and simulated load-settlement curves, together with the theoretical ultimate bearing capacity

calculated using Terzaghi’s equation. It can be seen that the ultimate bearing capacity from the

two sandbox tests are between the theoretical values by assuming general and local shear

failure. This seems reasonable because the sand was prepared under loose condition and

general shear failure might not occur thus resulted a lower value of ultimate bearing capacity.

Although the two soil models yield about the same ultimate bearing capacity and match the test

result, the Double Yield model outperforms the Mohr-Coulomb model in simulating the

loading behavior of the unreinforced strip footing. Therefore, only the Double Yield model was

used in simulating the reinforced footing.

Effects of Design Parameters on the Bearing Capacity

Both numerical simulation and laboratory test were performed to study the effects of design

parameters such as buried depth and reinforcement length on the bearing capacity. Comparison

between simulated and tested load-settlement curves of footing with single layer of

reinforcement are shown in Fig.10. It can be seen that the simulated curve matches the tested

curve reasonably well. The effects of buried depth (for a constant length of 6B) and

reinforcement length (for a constant depth of 0.4B) on the bearing capacity using single layer of

reinforcement from numerical simulation are summarized in Figs. 11(a) and 11(b). The

optimum buried depth and reinforcement length are 0.5B and 5B, respectively. For a given

buried depth or reinforcement length, the BCR increases as the normalized settlement (s/B)

increases. Because the stiffness of geotextile is relatively lower than the other geosynthetics

such as geogrids or geonets, higher deformations are required for the geotextile to mobilize its

tensile strength.

The simulated and tested BCR at various buried depths and reinforcement length are

compared in Fig. 12 and Fig.13 at various s/B. At 20% of s/B, the maximum BCR are 1.46 (test)

and 1.32 (simulation), the optimum buried depth are 0.4B (test) and 0.5B (simulation), and the

optimum reinforcement length are 4B (test) and 5B (simulation), respectively. The curves

obtained from numerical simulation are smoother than those of sandbox test. Since the width of

the footing is only 10cm, the variations in buried depth are very small. It is very difficult to

place the reinforcement at the right location. Furthermore, the sand was test under very loose

condition (Dr = 12.4%), the differences in bearing capacity are only a few kPa. Therefore, more

scatter within the test data can be expected. Although the simulated curves do not match with

the tested curves very well, the trends are very similar. And, the difference in the optimum

buried depth and reinforcement length obtained from numerical simulation and sand box test is

small.

The effectiveness of enhancing the bearing capacity of the footing with additional layer of

reinforcement and its optimum arrangement were also studied. Two different configurations

were compared. The first configuration is arranging the first layer of reinforcement at a buried

depth of 0.4B, while allocating the second layer 0.2B above it (i.e., d = -0.2B). In contrast, the

second configuration is arranging the second layer 0.2B beneath the first layer (d = 0.2B). The

reinforcement lengths are all kept at 3B. Comparison of the effectiveness of these two

configurations is shown in Table 5. Under the same reinforcement arrangement, the BCR

obtained from numerical simulation are slightly lower than values obtained from sand box test.

Nevertheless, the BCR of configuration 1 (d = -0.2B) are all higher than configuration 2 (d =

0.2B), from both numerical simulation and sandbox test. At 20% of s/B, the BCR only increase

slightly (1.46 to 1.55) from sandbox test, but more significant (1.23 to 1.39) from numerical

simulation.

Conclusions

This paper investigates the behavior and effectiveness of a strip footing reinforced with woven

geotextile through sandbox testing and numerical simulation. Based on the results of this study,

the following conclusions can be drawn:

(1) The ultimate bearing capacity obtained from both sandbox test and numerical simulation is

lower than the value calculated using Terzaghi’s equation, indicates that general shear

failure of the footing does not occur.

(2) The Double Yield model is very suitable in simulating the behavior of loose sand. The

procedures used in study to obtain the parameters for the Double Yield model are also

appropriate.

(3) With single layer of reinforcement, there is a 32% (from simulation) or 46% (from test)

increase in bearing capacity. The optimum buried depth is 0.4B from sandbox test, 0.5B

from numerical simulation. The optimum reinforcement length is 4B from sandbox test,

5B from numerical simulation. These two values are within the range reported by the

investigators in the reviewed literatures. Therefore, for design purpose, it is recommended

to use 0.4B for buried depth and 5B for reinforcement length.

Acknowledgments

The authors wish to thank Mr. Henry Sie of ACE Geosynthetics for providing the geotextile

used in this study. His support is deeply appreciated.

References

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TABLE 1. Properties of sand used in this study.

Properties Value

Specific gravity 2.70

Grain size (mm) D10 0.17

D30 0.28

D60 0.50

Dry density (kg/m3) Maximum 1743

Minimum 1375

Tested 1412

Relative density (%) 12.4

Cohesion (kPa) 0

Angle of friction (degrees) 34.4

TABLE 2. Properties of geotextile used in this study.

Properties Value

Tensile strength (kN/m) MD* CD*

2% elongation 9.6 14.3



Mass density (g/m2) 338

Thickness (mm) 0.86

* - MD: Machine Direction; CD: Cross Direction

TABLE 3. Properties of sand used in numerical simulation

Density (kg/m3)

Dilation Angle min. (

o) Friction Angle

peak (o)

Shear Modulus(Pa)

Bulk Modulus (Pa)

Stiffness Ratio(R )

1412 -3 34.4 3.84x105 1.15x106 5.5

TABLE 4. Properties of geotextile used in numerical simulation

Basic Property Axial Property Interface Property

Thickness

(mm)

Mass Density (g/m2)

Ultimate Strength (kN/m)

Elastic Modulus

(kPa)

Stiffness kbond

(kN/m/m)

Cohesive Sbond

(kN/m)

Friction Angleinterface

(o)

0.86 338 63.5 8.31x105 750 25 11.5

TABLE 5. Effectiveness of two reinforcement layers

BCRd

s/B d/B = -0.2 d/B = 0 (single layer) d/B = 0.2

Test Numerical Test Numerical Test Numerical

5% 1.01 1.12 1.20 1.09 1.01 1.13

10% 1.15 1.20 1.26 1.12 1.15 1.18

15% 1.34 1.29 1.35 1.18 1.29 1.27

20% 1.55 1.39 1.46 1.23 1.43 1.36

Figures Caption

Fig. 1. Schematic drawing of the experimental setup

Fig. 2. Mohr-Coulomb failure criterion in FLAC [26]

Fig. 3. Cap stress vs. volumetric strain of Double-Yield Model [26]

Fig. 4. Schematic drawing of finite difference mesh used in numerical simulation

Fig. 5. Stress- volumetric strain curve of the loose sand from Oedometer test

Fig. 6. Shear stress-shear displacement curves of the loose sand from direct shear tests

Fig. 7. Shear stress ratio and dilation angle of the loose sand from direct shear test

Fig. 8. Shear yield function of the loose sand obtained from direct shear test

Fig. 9. Comparison between simulated and tested load-settlement curves of strip footing on

unreinforced sand


unreinforced sand

Fig. 11. Effects of depth and length of single reinforcement on the bearing capacity from

numerical simulation

Fig. 12. Comparison between simulated and tested BCR at various buried depths

Fig. 13. Comparison between simulated and tested BCR at various reinforcement lengths

Fig. 1. Schematic drawing of the experimental setup

Fig. 2. Mohr-Coulomb failure criterion in FLAC [26]

Fig. 3. Cap stress vs. volumetric strain of Double-Yield Model [26]

Fig. 4. Schematic drawing of finite difference mesh used in numerical simulation

Fig. 5. Stress- volumetric strain curve of the loose sand from Oedometer test

Fig. 6. Shear stress-shear displacement curves of the loose sand from direct shear tests

Fig. 7. Shear stress ratio and dilation angle of the loose sand from direct shear test

Fig. 8. Shear yield function of the loose sand obtained from direct shear test


unreinforced sand

Fig. 11. Effects of depth and length of single reinforcement on the bearing capacity from

numerical simulation

Fig. 12. Comparison between simulated and tested BCR at various buried depths

Fig. 13. Comparison between simulated and tested BCR at various reinforcement lengths

laboratory testing and numerical simulation of a strip

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