2D Finite Element Method on the Scaling

Law for Strip Footing

Myungjae Lee1 and Heejung Youn1

1 Department of Civil Engineering, Hongik University, Sangsu-dong, Mapo-gu, Seoul,

Republic of Korea.

Abstract. This paper investigates the similitude law for strip footing resting on

cohesionless soils using a 2D finite element analyses. The 2D finite element

analyses were conducted considering three different conditions: laboratory

model test under 1-g, centrifuge test under n-g, and full scale test under 1-g.

The scaling relations in stress and displacement among the tests were examined

for varying internal friction angle. The friction angles used were 32, 35, 38,

41 and 44°. The cohesionless soils were simulated using the Hardening Soil

model, which enables the increase in shear strength and stiffness with depth.

Based on series of numerical results, the scaling relations were suggested for

bearing pressure and settlement of strip footing

Keywords: friction angle, strip footing, similitude law, scaling relation, unit

bearing capacity, 2D finite element

1 Introduction

It is possible to easily predict the bearing capacity of shallow foundation in sandy soil

through an indoor experiment. However, the bearing capacity as to the foundation in

sandy soil is largely affected by confining pressure. Moreover, an indoor experiment

can hardly simulate an increase in confining pressure in relation to depth. On that

account, it is hard to make an accurate prediction in regard to the full scale. To solve

the aforementioned problem, Ko (1988) presented the scaling relation of each of the

following tests in Table 1: full scale test, centrifuge test and laboratory model test. Ko

(1988) leveraged geometric scale ratio as n and stress scale ratio as N.

This paper calculated the scaling relations between the 1g laboratory model test,

centrifuge test and full scale test by means of the 2D finite element program called

‘PLAXIS’. Moreover, this paper analyzed which impact this scaling relations would

create with the changes in friction angle, elastic coefficient and fitting parameter ‘m’.

This paper leveraged gravitational acceleration of 1g (9.81 m/s2) and 20-g (196.2

m/s2).

1 Heejung Youn, Assistant Professor, School of Urban and Civil Engineering, Hongik

University E-mail: geotech@hongik.ac.kr

Advanced Science and Technology Letters Vol.120 (GST 2015), pp.688-691

http://dx.doi.org/10.14257/astl.2015.120.137

ISSN: 2287-1233 ASTL Copyright © 2015 SERSC

Table 1. Scaling relations (modified from Ko(1988))

Full scale model Centrifuge model at

equal stress level Laboratory model

Length 1 n n

Stress 1 1 N

Strain 1 1 1

Displacement 1 n n

Force 1 n2 Nn2

Void ratio 1 N/A em=ep+λln(N)

n: geometric scale ratio, N : stress scale ratio, N/A : not available

2 Numerical model

This paper conducted the numerical analysis by utilizing PLAXIS. This paper

calibrated the load-settlement curve and numerical analysis result in relation to the

strip footing obtained through the previous indoor experiment (Mandal and

Manjunath, 1995). Figure 1(a) is the lateral view of the soil bin used in the indoor

experiment. The width of strip footing was 100mm. This soil bin along with the strip

footing were modeled by using PLAXIS as shown in Figure 1(b). The configuration

model simulated an increase in the confining pressure depending on the depth by

utilizing Hardening Soil model.

Fig. 1. (a) Side view of laboratory scale experiment on strip footing (Mandal and Manjunath,

1995), (b) Mesh generation and boundary condition in the PLAXIS model

Table 2. Material parameters of 1-g laboratory test and input parameters for the Hardening

Soil model

Laboratory test Numerical analysis

Dry unit weight (kN/m3) 18.1 18.1

Relative density (%) 73 -

Friction angle (°) 38 32 35 38 41 44

Dilation angle (°) - 2 5 8 11 14

Specific gravity, Gs 2.65 -

𝐸50𝑟𝑒𝑓

(kPa) - 7,500

m - 0.7

(a) (b)

Advanced Science and Technology Letters Vol.120 (GST 2015)

Copyright © 2015 SERSC 689

Table 2 shows the material properties used in the indoor experiment along with the

input values used in the numerical analysis program called ‘PLAXIS’. This paper then

calibrated the results of indoor experiment on the basis of the aforementioned value.

In regard to the modeling calibrated based on the existing indoor experiment, this

paper conducted the three numerical analyses as shown in Table 3.

Table 3. Dimensions and gravity used in the numerical analyses

Test type Footing width(mm) Soil container(W×L, mm) Gravity (g)

Laboratory test 100 610×460 1

Centrifuge test 100 610×460 20

Full scale test 2,000 12,200×9,200 1

3 Numerical results

Figure 3 is the unit bearing pressure-settlement curve in relation to centrifuge test and

full scale test. The results of centrifuge test were compared by applying the geometric

scale ratio in the displacement and 20 in n as presented in Table 1. As shown in

Figure 3, there was no significant difference in the unit bearing pressure between the

centrifuge test and the full scale test. The results of these two condition states were

found to be similar to each other. Also, it was analyzed that the friction angle did not

have any significant impact.

Fig. 3. Unit bearing pressure-settlement curve (a) ∅ = 32°, (b) ∅ = 35° and (c) ∅ = 38° of

centrifuge test vs. full scale test with different friction angles

This paper applied the geometric scale ratio in the laboratory scale test and n in 20

by utilizing Table 1 in order to compare the laboratory scale test with the full scale

test. In addition, this paper obtained the values as shown in Table 4 by utilizing the

peak force value in regard to the stress scale ratio ‘N’. As a result, the unit bearing

pressure-settlement curve when the friction angle is 38°. There was a significant

difference in the unit bearing pressure depending on the depth. It was determined that

the difference in the geometric scale ratio ’n’ for displacement was the main factor

causing a significant difference in the unit bearing pressure. Moreover, it was found

that there was a difference of approximately 2n rather than n in the geometric scale

ratio for settlement with the same unit bearing pressure from the laboratory test and

full scale test. Thus, this paper utilized 2n rather than n in the geometric scale ratio as

shown in Figure 4.

Advanced Science and Technology Letters Vol.120 (GST 2015)

690 Copyright © 2015 SERSC

Table 4. Stress scale ratio with different friction angles obtained from peak force

Fig. 4. Unit bearing pressure-settlement curve (a) ∅ = 32°, (b) ∅ = 35° and (c) ∅ = 38° of

laboratory test vs. full scale test with different friction angles (Geometric scale ratio = 2n)

4 Conclusions

To examine the scaling relations with different friction angles in the strip footing, this

paper conducted the numerical analyses under the three conditions by utilizing the 2D

finite element method. The conclusions thereof are as follows:

1) This paper obtained the stress scale ratio ’N’ by utilizing the peak load in

order to compare the laboratory model test with the full scale test. N was

calculated at 15.6 to 16.9.

2) It was found that there was a difference of approximately 2n rather than n in

the geometric scale ratio for settlement from the laboratory test and full

scale test. However, it is believed that the aforementioned difference is

caused due to the fact that the elastic coefficient varying with different

depths is not taken into consideration rather than the improper consideration

of geometric scale ratio.

Acknowledgments. This work was supported by National Research Foundation of

Korea (NRF) funded by Ministry of Science, ICT & Future Planning (NRF-

2013R1A1A1011983)

References

1. Ko, H. (1988). "Summary of the state-of-the-art in centrifuge model testing." Centrifuges in

soil mechanics, 11-18.

2. Mandal, J., and Manjunath, V. (1995). "Bearing capacity of strip footing resting on

reinforced sand subgrades." Construction and Building Materials, 9(1), 35-38.

Friction angle ( ° ) 32 35 38 41 44

Stress scale ratio (N) 15.6 15.9 16.2 16.9 16.6

Advanced Science and Technology Letters Vol.120 (GST 2015)

Copyright © 2015 SERSC 691