Maxwell model for geosynthetic encased column (GEC) in soft ground improvement for construction works in Viet Nam

f stress response s (t)
under strain input e0 applied at t = 0 and kept con-
stant2. Thus, the Maxwell model is characterized by
the following simple relaxation function.
E (t) =
s (t)
e0
(6)
Figure 2: Stress relaxation
By adding multiple Maxwell elements, a generalized
Maxwell model, as shown in Figure 32, is built to ap-
proximate viscoelastic behavior of soft soil possible
better.
Figure 3: Schematic of a generalized Maxwell
model
With n the total number of Maxwell elements in the
generalized Maxwell model, the mechanical proper-
ties are described by summation of the stress in each
Maxwell element.
st = åni=1si (7)
Model Description
The behavior of geotextile encased stone column
(GEC) on soft foundation soil has been idealized by
the proposed foundation model (Figure 4) and as-
suming soft ground was supposed as a viscoelastic
material. The paper focused on finding the suitabil-
ity of the Maxwell model for soft soil with the sim-
ple analytical calculation of Raithel, M., Kempfert, H.-
G (1999), Alexiew, D., Brokemper, D. and Lothspe-
ich, S. (2005), Alexiew, D., Raithel, M. (2015) 3–5 for
GEC, which is formed on the unit cell concept where
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Science & Technology Development Journal – Engineering and Technology, 4(1):747-757
Figure 4: Proposed foundation model
the column is considered as rigid plastic, with infinite
modulus of elasticity.
A geotextile encased stone column (GEC) in soft soil
under a maintained load is usually subject to addi-
tional settlement, the magnitude of which differs be-
cause of changes in the stress-strain behavior over
time. Such stress relaxation behavior occurs in the soil
surrounding theGEC aswell as on theGEC-soil inter-
face itself. This article focuses only on the generalized
Maxwell model of viscoelastic soft soil and Relaxation
Modulus E (t).
The Maxwell model, which consists of spring and
dashpot element in series, can effectively reflect the
stress relaxation behavior of soft soil.
Estimating the viscosity and strain rate
The viscosity and strain rate of the soft clay will be
taken from the studies of Anders Augustesen, Morten
Liingaard and Poul V. Lade (2004)6, G. Qu, S. D.
Hinchberger and K. Y. Lo (2010) 7, Arindam Dey and
Prabir Kr. Basudhar (2012) 8, Hong-Hu Zhu, Lin-
Chao Liu, Hua-Fu (2012)9.
Varying the relaxation times and elasticity
modulus
The relaxation time is dependent on the value of
young’s modulus E and the viscosities h . In this paper
a = 0.5 will assumed for all the analysis2.
t =
a :h
(1a) :E (7)
Settlement calculationofGECand the time-
dependent behavior of soft clay soils
The ground improvement works are to provide a
steady platform to support the operation of SL6000
(Kobelco) and the CC2800 (Terex Demag) cranes on
the designated routes, which can support up to 500
kPa of transient loads and long-term primary set-
tlement should be less than 250 mm. The soft clay
is reinforced by the GEC with a diameter of 0.8 m,
depth of treatment from 12 m and the tensile stiff-
ness of the geotextile encasement J = 3000 kN/m. The
GEC are arranged in square grids with spacing 2.3 m.
The settlement was estimated based on Raithel and
Kempfert’s analytical calculation10. This calculation
is conducted using data obtained from soil parame-
ters in Table 1.
ANALYSIS, RESULTS AND
DISCUSSION
AMatlab code was written to simulate viscoelastic be-
havior of soft clay. The model was built first only with
the spring, then a dashpot was added to the model to
simulate the mechanics of a Maxwell element. The
generalized Maxwell model was simulated by com-
bining a spring and five Maxwell elements in this pa-
per.
For the present study, the design variables governing
the constitutive behavior of the generalized Maxwell
model for soft soil are as follows and listed in Table 2:
(1) Elastic coefficient of the Maxwell element Espring,
(2) Viscous coefficient of the Maxwell element h i,
(3) Strain rate :e
To study the suitability of the generalized Maxwell
model, the viscosity h is set to 0.1, 0.5, 1, 5 and 10
MPa
d and strain rate :e is checked with 1%, 5%, 10%
and 20%.
How to determine the Maxwell model parameters is a
major concern, in this section some issues are con-
sidered to evaluate the response of the generalized
Maxwell model described above. Several studies are
performed and relaxation time is considered, which
is usually used to assess the time-dependent behavior
of viscoelastic soft soil.
Constant strain rate with different viscosi-
ties
In this case the influence of viscosity on the general-
izedMaxwellmodel is studied. Thenumerical simula-
tions are carried outwith different number ofMaxwell
elements.
It can be seen that when the viscosity or the number of
Maxwell elements increases greater values of stress are
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Science & Technology Development Journal – Engineering and Technology, 4(1):747-757
Table 1: Typical Subsoil Profile
Depth (m) Soil Type jorig c’orig (kPa) Elastic Modulus E
(MPa)
0 to 0.6 Gravel 40 0 -
0.6 to 1.5 Sand 30 0 -
1.5 to 2.5 Crust 0 35 -
2.5 to 13.5 Soft Clay 0 20 1.0
Table 2: Parameters of the generalizedMaxwell model
Item Value Unit
Young’s modulus E 1.0 MPa
Viscosity h varies MPa
d
Strain rate :e varies -
Linear elastic Modulus Espring 0.2 MPa
obtained. This is highlighted in Figure 5, in which the
effect of numbers of Maxwell elements on the maxi-
mum stress s reached it can be observed.
The general observation is that the higher the viscos-
ity, the higher the stresses for a certain strain rate.
For example, Figure 5 (a and b) show the comparison
of the stress versus time by viscosity 5 Mpa:d and 10
MPa:d with same constant strain rate 5% per day for
a period of 500 days, and it is observed that the stress
reached a maximum level 1,428 MPa and 2,365 MPa
respectively after a period of time of 20 days for case
5 Maxwell elements.
The evolution of stress with time for different num-
ber of Maxwell elements is shown in Figure 5. As can
be seen the stress response is nonlinear although the
strain rate has a linear variation with time.
Constant Maxwell elements with different
strain rates and viscosities
In this case two different viscosities with five different
strain rates are considered. From a point of view of
the sensitivity to constitutive parameters it can be said
that when viscosity decreases, the stress is lower for
the same time.
It is seen that the maximum stress moves to the right
faster for higher strain rate. In Figure 6, the results
of the generalized Maxwell model are shown, and it
seems that the bigger the strain rate, the stiffer the soil.
Figure 6 also shows the results obtained by changing
strain rate from1% to 20%while keeping the other pa-
rameters constant. It can be seen that the smaller the
value of strain rate, the longer the time to reach the
maximum stress. An example, Figure 6a shows the
result of strain rate 1% per day, during the first 100
days an increasing strain is applied, the strain rate is
constant and equals 1% per day, during these days the
stress appears to increase non-linearly with the strain.
After 100 days the strain remains constant at maxi-
mum strain, and the stress decreases exponentially to
a stress-relaxation limit.
Figure 7 presents the effects of strain rate on the stress
of the generalizedMaxwell model for soft soil. For the
short period of time (1 to 5 days), the stress is found to
increase linearly with strain rate before the required
strain value is reached. The stress developed almost
linearly with low strain rate (1% to 10% per day) un-
til the maximum strain is reached and then decreases
with high strain rate for a period of 10 to 20 days. Fi-
nally, for the larger period of more than 50 days, the
stress is gradually reduced from very low strain rate to
very high strain rate.
Settlement calculation of GEC
The settlement of a GEC is estimated using the elastic
modulus E and relaxation modulus E (t).
It is shown that the viscosity mainly affects the overall
relaxing rate of the foundation soil. For the same time
and same strain rate, the settlements increase with the
decrease of viscosity.
The increase of Maxwell elements causes smaller val-
ues of settlement, when the viscosity and strain rate
are same. Fig. 8 shows the results at different mod-
uli of relaxation modulus E (t) under constant load-
ing with the viscosity h = 5 Mpa:d and 10 Mpa:d.
It is easy to make the conclusion that the smaller the
modulus of relaxation E (t) is, the bigger the settle-
ment is, and the smaller the strain rate to reach the
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Figure 5: Stress versus time for constant strain rate
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Figure 6: Stress versus time for constant Maxwell elements
final settlement spends.
LIMITATIONOF THE PROPOSED
APPROACH
The proposed analysis of using relaxation modulus
for estimating the settlement of GEC is primarily in-
tended for understanding the behavior of viscoelas-
tic soils. It has been mentioned in the study that the
generalized Maxwell model is capable of representing
the behavior of a viscoelastic soil when considered in
terms of the short-term or long-term settlement un-
der loading followed by a relaxation phase under sus-
tained loading. The used data surely restricts the ap-
proach of estimating the generalized Maxwell model
parameters being valid only for the loaded viscoelastic
soft soils.
The verifiable relationship between the generalized
Maxwell model parameters and soft soil properties is
an important problem to be solved, which needs the
support of sufficient measurement data from labora-
tory and field experiments.
CONCLUSIONS
Based on the above studies, the following conclusions
can be made:
• Based on the theory of linear viscoelasticity, a
generalized Maxwell viscoelastic model is de-
veloped to account for the time-dependent be-
haviour of soil foundations improvement with
GEC under concentrated line load.
• Analytical solution of settlement in the foun-
dation was estimated based on Raithel and
Kempfert’s analytical calculation model with
Relaxation Modulus.
• The viscoelastic theory shows its potential in
modeling the long-term foundation deforma-
tion.
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Figure 7: Stress versus strain rate for constant Maxwell elements
• The presented analytical calculation can be ex-
tended to solve other geotechnical problems.
CONFLICT OF INTEREST
The authors pledge that there are no conflicts of inter-
est in the publication of the paper.
AUTHORS’ CONTRIBUTION
Pham Tien Bach presented the idea of study and car-
ried out the collecting data, writing codes and writing
the paper manuscripts. Dr. Vo Dai Nhat, Assoc. Prof.
Dr. Nguyen Viet Ky participated in the scientific idea
of research, guided to writing the paper, reviewed the
results of study. Le Quan contributed to review the
input data, output data and reviewing the paper.
APPENDIX: MATLAB CODE
%Setting the time
dt=0.1;
t=0:dt:500;
%Boundary conditions
strain(1) = 0;
straindot(1) = 0;
stress1(1) = 0;
stressdot(1) = 0;
prompt={’Enter the maximum strain’,’Enter the strain
rate’,’Enter Viscosity Eta1’,’Enter Viscosity Eta 2’,’Enter
Viscosity Eta 3’, ’Enter Viscosity Eta 4’,’Enter Viscosity
Eta 5’,’Enter YoungsModulus’,’Enter LinearModulus’};
name=’Sample setup’;
numlines=1;
defaultvalue={’1’,’0.2’,’10’,’10’,’10’,’10’,’10’,’1’,’0.2’};
answer=inputdlg(prompt,name,numlines,defaultvalue);
maxstrain = str2num(answer{1});
strainrate = str2num(answer{2});
visco1 = str2num(answer{3});
visco2 = str2num(answer{4});
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Science & Technology Development Journal – Engineering and Technology, 4(1):747-757
Figure 8: Settlements calculated using elastic and generalized Maxwell model
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visco3 = str2num(answer{5});
visco4 = str2num(answer{6});
visco5 = str2num(answer{7});
E = str2num(answer{8});
Espring = str2num(answer{9});
strain = zeros(size(t));
stress = zeros(size(t));
stress1 = zeros(size(t));
stress2 = zeros(size(t));
stress3 = zeros(size(t));
stress4 = zeros(size(t));
stress5 = zeros(size(t));
stressspring = zeros(size(t));
stressdot1 = zeros(size(t));
stressdot2 = zeros(size(t));
stressdot3 = zeros(size(t));
stressdot4 = zeros(size(t));
stressdot5 = zeros(size(t));
straindot = zeros(size(t));
alpha = 0.5;
relaxationtime1 = (alpha*visco1)/((1-alpha)*E);
relaxationtime2 = (alpha*visco2)/((1-alpha)*E);
relaxationtime3 = (alpha*visco3)/((1-alpha)*E);
relaxationtime4 = (alpha*visco4)/((1-alpha)*E);
relaxationtime5 = (alpha*visco5)/((1-alpha)*E);
for i = 2:length(t)
strain(i) = strain(i-1)+ strainrate * dt;
if strain(i) > maxstrain
strain(i) = maxstrain;
end
straindot(i) = (strain(i)-strain(i-1))/dt;
if relaxationtime1 <= 0
stressdot1(i) = 0;
stress1(i) = 0;
else
stressdot1(i) = (- stress1(i-1) + visco1* straindot(i) ) /
relaxationtime1;
stress1(i) = stress1(i-1) + stressdot1(i) * dt;
end
if relaxationtime2 <= 0
stressdot2(i) = 0;
stress2(i) = 0;
else
stressdot2(i) = (- stress2(i-1) + visco2* straindot(i) ) /
relaxationtime2;
stress2(i) = stress2(i-1) + stressdot1(i) * dt;
end
if relaxationtime3 <= 0
stressdot3(i) = 0;
stress3(i) = 0;
else
stressdot3(i) = (- stress3(i-1) + visco3* straindot(i) ) /
relaxationtime3;
stress3(i) = stress3(i-1) + stressdot3(i) * dt;
end
if relaxationtime4 <= 0
stressdot4(i) = 0;
stress4(i) = 0;
else
stressdot4(i) = (- stress4(i-1) + visco4* straindot(i) ) /
relaxationtime4;
stress4(i) = stress4(i-1) + stressdot4(i) * dt;
end
if relaxationtime5 <= 0
stressdot5(i) = 0;
stress5(i) = 0;
else
stressdot5(i) = (- stress5(i-1) + visco5* straindot(i) ) /
relaxationtime5;
stress5(i) = stress5(i-1) + stressdot5(i) * dt;
end
stressspring(i) = Espring * strain(i);
stress(i) = stress1(i) + stress2(i) + stress3(i) +
stress4(i) + stress5(i)+ stressspring(i);
end
%Plotting
subplot(211); plot(t,stress,’black’,’LineWidth’,1); xla-
bel(’time (days)’,’FontSize’, 12); ylabel(’stress (MPa)’,
’FontSize’,12);
subplot(212); plot(t,strain, ’black’,’LineWidth’,1); xla-
bel(’time (days)’,’FontSize’,12); ylabel(’strain’, ’Font-
Size’,12);
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Tạp chí Phát triển Khoa học và Công nghệ – Kĩ thuật và Công nghệ, 4(1):747-757
Open Access Full Text Article Bài Nghiên cứu
Khoa Kỹ thuật Địa chất và Dầu khí,
Trường Đại học Bách khoa,
ĐHQG-HCM, Việt Nam
Liên hệ
Phạm Tiến Bách, Khoa Kỹ thuật Địa chất và
Dầu khí, Trường Đại học Bách khoa,
ĐHQG-HCM, Việt Nam
Email: tienbachpham@gmail.com
Lịch sử
 Ngày nhận: 24-9-2020
 Ngày chấp nhận: 23-3-2021 
 Ngày đăng: 31-3-2021
DOI : 10.32508/stdjet.v4i1.772 
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Mô hình Maxwell cho cọc bọc vải địa kỹ thuật trong xử lý nền 
đất yếu cho các công trình xây dựng ở Việt Nam
Phạm Tiến Bách*, Võ Đại Nhật, Lê Quân, Nguyễn Việt Kỳ
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TÓM TẮT
Trong lĩnh vực địa kỹ thuật – cải tạo nền đất yếu, các mô hình toán hay cơ là một trong những
thông số đầu vào rất quan trọng phục vụ thiết kế tính toán hay nghiên cứu. Việc xác định sự phù
hợp của các mô hình có ảnh hưởng rất lớn đến độ chính xác kết quả thiết kế và tính toán cũng
như tính ổn định bền vững của nền đất yếu sau khi được cải tạo. Ngược lại, việc lựa chọn các mô
hình tính toán không phù hợp sẽ dẫn đến chi phí cải tạo nền đất yếu tăng cao, thậm chí có thể
dẫn đến việc mất ổn định công trình và gây ra các thiệt hại to lớn về người về của.
Gần đây, rất nhiều dự án đường cao tốc lớn sau khi thiết kế thi công đưa vào sử dụng thì không
đáp ứng yêu cầu của tiêu chuẩn dẫn đến hao tổn kinh phí của các cá nhân, tổ chức và nhà nước để
xử lý hậu quả. Do đó việc nghiên cứu và ứng dụng sử dụng các mô hình toán hay cơ phù hợp với
phương pháp cải tạo nền đất yếu mới sẽ giúp ích rất nhiều cũng như bổ sung thêm các lựa chọn
cho công tác cải tạo đất yếu tại Việt Nam. Biến dạng của nền đất yếu không chỉ liên quan đến tải
trọng mà còn liên quan đến thời gian gia tải. Sự thay đổi ứng suất và biến dạng của nền đất yếu
theo thời gian được gọi là đặc tính lưu biến, và trong nghiên cứu này là ứng xử có tính đàn – nhớt.
Từ những lý do trên, chúng tôi cố gắng áp dụng mô hình tổng quát Maxwell để giải thích ứng xử
có tính đàn – nhớt của nền đất yếu. Đặc biệt, ứng xử phụ thuộc vào thời gian của nền đất yếu có
tính đàn – nhớt được thể hiện bằng cách sử dụng mô hình Maxwell. Mã lập trình Matlab giúp giải
quyết bằng số tất cả các phương trình toán học thể hiện các kết quả mô hình tổng quát Maxwell.
Chúng tôi thừa nhận rằng mô hình tổng quát Maxwell có thể ưu việt hơn trong việc thể hiện ứng
xử phụ thuộc theo thời gian của nền đất yếu. Kết quả cho thấy có lẽ đây là một trong những mô
hình hiệu quả để dự đoán ứng xử của nền đất yếu trong việc cải tạo nền đất yếu với cọc cát/đá
bọc vải địa kỹ thuật.
Từ khoá: Mô hình Maxwell, cọc cát/đá bọc vải địa kỹ thuật, cải tạo nền đất yếu, ứng xử đàn nhớt
Trích dẫn bài báo này: Bách P T, Nhật V D, Quân L, Kỳ N V. M ô hình Maxwell cho cọc bọc vải địa kỹ thuật 
trong xử lý nền đất yếu cho các công trình xây dựng ở Việt Nam. Sci. Tech. Dev. J. - Eng. Tech.; 
4(1):747-757.
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