RESEARCH PAPER

Behavior of Sand After a High Number of Cycles Applicationto Shallow Foundation

Hanane Dob1 • Salah Messast1 • Abdelhamid Mendjel2 • Marc Boulon3 •

Etienne Flavigny3

Received: 21 March 2016 / Revised: 3 June 2016 / Accepted: 3 June 2016 / Published online: 17 June 2016

� Iran University of Science and Technology 2016

Abstract Considerable strains appear in the structures

during accumulation of the irreversible strains of the sub-

grade under the effect of the cyclic loads. If the number of

cycles is very large, even a small strain after accumulation

becomes significant and sometimes harmful. In this study,

a simple numerical modeling of the behavior of sand under

cyclic loading is proposed. The suggested approach con-

sists, in drained condition, in determining the parameters

characterizing the average cyclic path of the soil under the

effect of the number of cycles duly characterized and

translating the cyclic effect by a volumetric strain cumu-

lated by a variation of the module of the soil. In this study,

we are interested in cyclic triaxial compression tests sim-

ulated by a finite element calculation. While proposing an

analogy between the cyclic pseudo creep and the soft soil

creep model (SSCM), on the first hand we propose an

equivalence between the cyclic parameters and the

parameters of SSCM, and on the other an equivalence time

number of cycles will be established. The application of the

formulation suggested on a shallow foundation under

cyclic loading confirms the good adaptation of the model

suggested to this type of problem.

Keywords Cyclic loading � Plaxis � SSCM � Modeling �Volumetric strain � Sand � Shallow foundation

1 Introduction

Many structures are likely to be subjected to cyclic loading

in either normal or accidental situations such as roads,

bridges, railroads, silos, tanks, foundations for vibrating

machines, etc. However, the accumulation of the soil strain

causes disorders and even adverse effects in these structures

or the neighboring structures. Thus, the importance of pre-

dicting the accumulated irreversible cyclic strains appears.

There are two categories of approaches for the prediction of

the cyclic behavior of materials: the implicit and the explicit

approaches. The implicit or the incremental models require

several (hundreds) increments per cycle, which causes an

accumulation of the numerical error and an imprecision of

the results for a high number of cycles. Generally, this type

of model is desirable for N\ 50 [1]; therefore, the impor-

tance of the explicit models especially for a high number of

cycles (N[ 1000) appears, and in such models only one or a

few initial cycles are calculated by means of an incremental

calculation and the rest of the cycles will be treated as a

pseudo creep.

Several studies have been conducted in this theme and

their application on the response of the shallow founda-

tion under cyclic loading. Some researchers have focused

their work on the triaxial and shear cyclic tests with low

and high number of cycles [2–6]. Others are interested in

the constitutive modeling of the behavior of sand under

cyclic loading [7–11]. Several researchers are interested in

the numerical modeling of the problem [12–14] and others

have studied the response of the foundation under large

number of cycles [15–20].

& Salah Messast

smessast@yahoo.fr

Hanane Dob

dobh88@yahoo.fr

1 Department of Civil Engineering, Laboratory LMGHU,

University of Skikda, Skikda, Algeria

2 Department of Civil Engineering, University of Annaba,

Annaba, Algeria

3 Laboratory 3s-r, University Joseph Fourier,

Saint-Martin-d’Heres, France

123

Int J Civ Eng (2016) 14:459–465

DOI 10.1007/s40999-016-0050-1

The objective of this paper is to study the behavior

of the pulverulent soils under cyclic loading and the

estimation of the accumulated volumetric strains, based

on an analogy between the soft soil creep model

(SSCM) and the cyclic pseudo creep, which assimilates

the latter as a deferred behavior of a fictitious material,

handled by a finite elements calculation using the Plaxis

program and taking the SSCM as a model of behavior

of the soil.

Moreover, this technique will be applied on a shallow

foundation under cyclic loading (centrifuge test of Helm

et al. [21]) to highlight the reliability of the proposed

formulation.

2 Cyclic Behavior

A cyclic triaxial test can be described by the cyclic stress

path in the p–q plane, where p is the mean pressure and q is

the deviatoric stress. Figure 1 shows the definition of cyclic

parameters in the p–q plane.

3 Presentation of Soft Soil Creep Model

The SSCM is a relatively new model that has been

developed for application to settlement problems of foun-

dations, embankments, and unloading.

The latter is most dominant in soft soils, and the

advantage of this model is that it takes into account the

viscous effect, creep and stress relaxation. Where all soils

exhibit some creep and primary compression, it is followed

by a certain amount of secondary compression.

In this approach, we consider that the pseudo creep (the

evolution of the strain according to time; see Fig. 2) is

controlled mainly by the effect of the stiffness parameters

of the SSCM.

The SSCM requires the following material constants:

• Failure parameters as in the Mohr–Coulomb model:

• u: friction angle (�).• c: cohesion (kN/m2).

• w: dilatancy angle (�).

• Basic stiffness parameters

• k*: the modified swelling index.

• j*: the modified compression index.

• l*: secondary compression index.

4 Analogy Between the Cyclic Pseudo Creepand SSCM

This section consists of reproducing the curves of the

volumetric strain as a function of the number of cycles by a

pseudo creep treated by analogy using SSCM.

Fig. 1 The cyclic stresses path in the p–q plane

Fig. 2 Response of the materials by the SSCM

Fig. 3 Evolution of strain in a cyclic triaxial test

460 Int J Civ Eng (2016) 14:459–465

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In this approach, the pseudo creep is treated by analogy

as a behavior of a fictitious material by the SSCM (see

Fig. 3), the parameters k*, j* and l* were estimated

according to the cyclic parameters, and then we looked for

an equivalence between the time and the number of cycles.

Figure 3 shows the evolution of strain according to:

– time (t) in the case of the response of the fictitious

material by the SSCM;

– number of cycles (N) in the case of the cyclic pseudo

creep.

5 Numerical Application and Results

This work is based on the drained cyclic triaxial tests

realized by Thanopoulos [22] on the fine sand of Plancoet

characterized by its dry volume weight of 12.5 kN/m3,

u = 37.6�, c = 0 kN/m2, W = 6�, m = 0.2.

Ten (10) cyclic tests have been achieved to impose

stresses for three values of lateral stress (40, 80 and

160 kPa). The number of cycles varied from 200 to 2760

cycles, except for test 9, which contains only 23 cycles,

tests 4 and 14 contain two series of cycles (indications a

and b), and test 8 is a monotonic test at a lateral stress equal

to 80 kPa. The characteristics of the tests are shown in

Table 1.

c: Cohesion, W: dilatancy angle, u: friction angle, m:Poisson’s ratio.

Figure 4 shows in the p–q plane the band in which the

tests of Thanopoulos [22] was achieved.

After several numerical tests, the simulated and experi-

mental curves are shown in Fig. 5 with the parameter sets for

each test. Table 2 summarizes the parameters of the cyclic

tests of Thanopoulos [22] and the parameters of the SSCM

for the tests which present a conformity with the simulated

tests that is to say tests: 2, 4a, 12, 14a and 14b [12].

According to the curves in Fig. 5, there is a good con-

vergence between the simulated and experimental curves,

except that tests 2 and 4a show a slight gap between the

simulated and experimental curves, this gap is acceptable.

These two tests are located closer to the characteristic line

than other tests (see Fig. 4), which supports the basic idea

of this work. The equivalence between time number of

cycles is established as 1 day for one cycle.

Tables 1 and 2 show the parameters of fictitious mate-

rial. The procedure adopted is as follows: for constant

values of gav, the curves of k*, j* and l* according to Dg/gl

were traced, and from these curves Eq. (1) is written:

k� ¼ A1 �Dggl

þ B1

j� ¼ A2 �Dggl

þ B2

l� ¼ A3 �Dggl

þ B3:

ð1Þ

The resolution of the equations system (1) allows the

determination of Ai and Bi. The Ai and Bi dependence curves

according to gav allow to write the equations system (2):

A1 ¼ �55:531 � gav þ 50:001ð Þ � 10�5;

B1 ¼ �55:53 � gav � 50:0013ð Þ � 10�5;

A2 ¼ �55:53 � gav þ 50:0013ð Þ � 10�5;

B2 ¼ �20:795 � gav � 16:7622ð Þ � 10�5;

A3 ¼ �452:309 � gav þ 413:566ð Þ � 10�5;

B3 ¼ �172:553 � gav � 125:28ð Þ � 10�5 :

ð2Þ

After the substitution of Eq. (2) into Eq. (1), we can

express k*, j* and l*, respectively, by Eqs. (3)–(5):

Table 1 Parameters of the cyclic tests in compression achieved on

the fine sand of Plancoet, according to Thanopoulos [22]

Test r3 (kPa) N qmin (kPa) qmax (kPa) gav Dg

1 40 1670 4 89 0.837 1.18

2 40 2300 38 56 0.844 0.233

4a 80 406 9 166 0.801 1.118

4b 80 201 115 237 1.269 0.518

7 80 1274 67 166 0.980 0.571

8 80 1 0 0

9 80 23 116 172 1.125 0.823

11 40 2758 37 125 1.209 0.669

12 160 708 6 148 0.414 0.669

13 160 352 3 323 0.760 1.188

14a 160 256 37 154 0.497 0.514

14b 160 1045 37 250 0.69 0.812

16 160 1932 143 182 1.211 0.173

Fig. 4 Tests of Thanopoulos [22] with different average cyclic level

gav in the p–q plane. gl: the limit cyclic level, gc: the characteristic

cyclic level

Int J Civ Eng (2016) 14:459–465 461

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k� ¼ Dggl

�555:31 � gavþ 500:01ð Þþ 207:96 � gav� 15:762

� �

� 10�5; ð3Þ

j�¼ Dggl

�555:3 �gavþ500:013ð Þþ207:951 �gav�167:622

� �

�10�5; ð4Þ

l� ¼ Dggl

�4523:09 � gav þ 4135:66ð Þ þ 1725:535 � gav�

� 1252:8� � 10�5: ð5Þ

The relationship between the parameters characterizing

the fictitious material and the cyclic parameters is defined

by Eqs. (3)–(5) [13].

The characterization of the fictitious material is defined

by the determination of the parameters: k*, j* and l*, andthe others parameters are supposed to be the same as those

of the real material.

6 FEM Calculation of Shallow Foundations UnderCyclic Loading

In the recalculation of the centrifuge model test of Helm

et al. [21] (strip foundation under cyclic loading) the same

set of parameter was used to calculate the settlement after a

large number of cycles [23]. The calculation of settlement

using the PLAXIS program is done after the calculation of

the parameters k*, j* and l* of SSCM with the proposed

formulations. With the aim of testing the reliability of the

proposed formulation, we proceed to the comparison

between results obtained by this method and those derived

from the test of Helm et al. [21] and those obtained by

Wichtmann [24].

The numerical simulations were performed by FEM

calculation, where the discretization was carried out by the

elements of 15 nodes. Our model has 7819 nodes and 955

elements.

In the centrifuge model test of Helm et al. [21], fine sand

was used (c = 27 kN/m3, u = 32.8�, W = 3�).The centrifuge model test was recalculated with the

following boundary conditions:

• Dimensions of the test container: width 18.1 m, height

7.3 m (prototype). Using the symmetry, only half of the

soil was discretized (9.05 m 9 7.3 m) Fig. 6.

• Foundation: width b = 1.0 m, height h = 0.6 m, depth

of embedding t = 0 m.

• Material of the foundation: aluminum with

weight = 27 kN/m3, E = 25 000 MPa and m = 0.3.

• The average load rav = 89 kPa, amplitude ramp =

75 kPa.

Determining the response of the foundation under cyclic

loading is effected by a model that replaces the cyclic

loading after a number of cycles by the application of the

cumulative volumetric strain of the surrounding soil after

the same number of cycles. The soil mass is discretized in

several regions. The stresses developed in each region after

a loading cycle defined the parameters of an equivalent

triaxial test. These parameters determine later cyclical

parameters of each region. The behavior of the global

model will be determined by the application cumulative

strains of each region.

Figure 7 shows the cyclic loading fields applied in dif-

ferent regions of the soil in the p–q plane in terms of

average cyclical level.

Figure 8 shows that the evolution of settlement of the

shallow foundation presented in this study is very similar to

Table 2 Parameters of the

triaxial cyclic tests in

compression carried out by the

fine sand of Plancoet [22], and

the parameters of the simulated

tests of SSCM

Test r3 (kPa) qav (kPa) gav Dg k* j* l*

2 40 47 0.844 0.233 2.00E-04 1.00E-04 2.30E-03

4a 80 87 0.801 1.118 4.50E-04 3.50E-04 4.70E-03

12 160 77 0.414 0.669 4.50E-04 3.500E-04 4.40E-03

14a 160 96 0.497 0.514 2.00E-04 1.00E-04 2.30E-03

14b 160 143 0.69 0.812 4.50E-04 3.50E-04 3.50E-03

462 Int J Civ Eng (2016) 14:459–465

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Fig. 5 Curves (ev–N) simulated and experimental according to Thanopoulos [22]

Int J Civ Eng (2016) 14:459–465 463

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experimental curves of Helm et al. [21] and that numerical

calculation by Wichtmann [25].

7 Conclusion

The estimation of the evolution of the accumulated volu-

metric strain is formulated by an analogy between the

cyclic pseudo creep and the behavior of a fictitious mate-

rial, according to the SSCM behavior.

In this approach, the mechanical characteristics of the

fictitious material (k*, j* and l*) are determined according

to the cyclic parameters (gl, gav and Dg).The equivalence between time (SSCM) and the number

of cycles (cyclic pseudo creep) is set as per 1 day for one

cycle; it means that the modeling of cyclic behavior after

105 cycles can be simulated by the response of a fictitious

material with SSCM for a fictitious time (105 days).

The comparison between the proposed method and those

derived from the centrifuge model test of Helm et al. [21]

and those obtained by Wichtmann [24] confirms a good

adaptation of the proposed model for this type of problem.

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