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- 4979 -

Analysis of the Mechanical and

Deformation Characteristics of a

Circular Diaphragm Wall

Chongfu WuProfessor, Department of Civil Engineering, Yanshan University

Qinhuangdao, China

e-mail:wcfysu@163.com

Huanchao AnMaster

School of Civil Engineering and Mechanics, Yanshan University

Qinhuangdao, China

e-mail: anhuanchao@yeah.net

Fengzhu LiEngineer, Yan Sheng Soil Engineering Limited company, Qinhuangdao, China

e-mail: ysyt@qhdysyt.com

ABSTRACTThe construction of a swirling pool circular diaphragm wall is monitored and the mechanical

and deformation characteristics are analyzed. After collecting the monitoring data andapplying the cylindrical shell theory with moment, formulas for the structural internal force

and deformation are developed. FLAC3D is utilized to model the foundation pit excavation;

the interaction of the structure and the soil is considered. The calculated results and themeasured results are compared and are relatively consistent. However, there are large

differences between the cylindrical shell theory with moment and the actual measured earth

pressures. The values are distinctly different, particularly under the excavation face.

Consequently, this theory can only be used for preliminary calculations. The FLAC3D results

are closer to the measured values and have higher reliability.

KEYWORDS: Circular diaphragm wall, field monitoring, theory with moment,Flac3D, swirling Pool

INTRODUCTIONThe circular diaphragm wall has been widely applied in the swirling pools of steel works

because of its high rigidity, improved safety and good impermeability. This particular support

form can also take full advantage of the compressive properties of the surrounding materials

because of the arch effect, thereby improving the integrity and stability of the structure and

reducing the project cost [1]. Scholars have analyzed the ground displacement and stress during

installation; Gunn and Clayton [2] and Kutmen [3] stressed the importance of the effects of

installing the retaining wall. de Moor[4]conducted a two-dimensional FEM analysis series on a

plan (horizontal) section using a series of wall panels for a given depth. Ng et al. [5]conducted

pseudo three-dimensional FEM analysis of the effects of diaphragm wall installation to examine

load transfer. Scholars and engineers worldwide have reported some studies of preliminary

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7 silty 18.0 10 35 36.1 14.8

THE SOLUTION OF INTERNAL FORCE AND

DISPLACEMENT OF CYLINDRICAL DIAPHRAGM WALLThe non-moment theory of cylindrical shells assumes there is no moment and torque on all

cross-sections of the entire sheet. The vertical beam on elastic foundation method ignores the arch

effect of the cylindrical diaphragm wall. Consequently, these two methods do not accurately

simulate the actual state of the structure. The three-dimensional elastic foundation plate method

and the finite element method more accurately simulate the structural deformation and force, but

involve high computational costs and a heavy workload. The cylindrical shell theory with

moment is the classical theory for the analysis of cylindrical shell deformation and internal force.

However, solving the basic differential equation is time-consuming, which limits the availabilityof engineering and technical personnel. This paper considers the cylindrical diaphragm wall as a

cylindrical shell and utilizes the cylindrical shell theory with moment to develop the deformation

and internal force formulas. These formulas may facilitate the analysis conducted by engineeringand technical personnel.

Model Analysis

The diaphragm wall of the swirl pool is primarily cylindrical, with a thickness generally

between 0.8 ~ 1.0 M and a radius in the range of 8 ~ 20 m. It can be simplified into an elastic

shell [15]. To simplify the calculation, the following assumptions are made for the cylindrical

diaphragm wall:

(1) The structure is considered an elastic axisymmetric thin cylindrical shell and the boundary

support is axisymmetric;

(2) The structure only withstands the normal load symmetrical to the central axis of the

cylinder;

(3) Because the normal deformation of the structure is small, the earth pressure at rest is

assumed to be the horizontal earth pressure at the excavation side and the trenchless side,

ignoring the role of groundwater;

(4) The valid calculation depth of the cylindrical diaphragm wall is determined according to

the Bloom filter simplified calculation method and is generally less than the actual height of the

wall.

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The basic differential equations

Figure 1:The infinitesimal middle plane of shell

The cylindrical diaphragm wall is divided using the excavation face to form two cylindrical

shells. Each shell is analyzed; the basic differential equations are listed and then solved with the

boundary conditions and continuity conditions. It is assumed that the elastic modulus of the

cylindrical shell is E, the thickness of the cylindrical shell is h, the effective calculation depth is

L, the middle surface radius is R, and the depth direction of the cylindrical diaphragm wall is .

The internal force and deformation of each excavation can be initially calculated by changing the

effective calculation depth. The infinitesimal middle plane of the shell is as shown in Figure 1,

and the cylindrical diaphragm wall diagram is shown in Figure 2.

For the axisymmetric bending of the cylindrical shell, the basic differential equations can be

reduced to ordinary differential equations [16]

8 4

8 2 4

d d

d d

F Eh F Z

R D RD (1)

In this formula, F refers to the displacement function ( )F F .

D is the cylindrical shell stiffness,

3

212(1 )

KEhD

, where is the Poisson's ratio of the

cylindrical shell, and is the integrity coefficient, which has a value from 0.7 to 1.1. takes into

account the integrity improvement for the stiffness loss due to errors in wall segment connections

and construction, uneven distribution in strata and ground loads, and the top beam and ring beam.

Its value is greater when the integrity of the cylindrical diaphragm wall is higher.

Ma

MadMada

da+N

N

Q

N

N1

1

2

2

dQada

daQa+

dN1da

da+

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Figure 2: Circular diaphragm wall

4

4

d

d

Fw R

(2)

4

4 2

d

d

w Eh Z w

R D D (3)

Lead in the following parameters:

1/ 4

21/ 4

2 2 2

3 1

4

Eh

R D R h

(4)

(5)

Equation (3) is converted into

4 2

4

d 44

d

w R Zw

Eh (6)

The solution of the basic differential equations

The upper half of the cylinder H

Assume when H , the severe weighted mean of the soil at various layers above the height

of on the trenchless side is 0 . The ground overload is q, and the normal pressure (earth

pressure) is

01 0 0 0k

Z k q q

(7)

horizontal

bracing

earth pressure

excavationsurface

L

H

the earth's surface

R

h

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In the above formula, k0is the coefficient of earth pressure at rest.

Formula (6) can be converted to

24

004

4d 4d

R kw w qEh

(8)

The solution of the equation is shown as follows

*

1 1 2 3 4 1sin sin cos cosw C sh C ch C sh C ch w (9)

*

1w is a particular solution of equation (7)

2

* 01 0

R kw q

Eh

(10)

The bottom half of the cylinder H

Assume that when H , the severe weighted mean of the soil at various layers above the

height of on the Trenchless side is 1 the severe weighted mean of the soil at various layers

above the height of on the excavation side is 2 .

The normal pressure (earth pressure) is

qkHkZ 10202

qHk 2120 (11)

Formula (6) can be converted into

qHEh

kRw

w 212

0

2

4

4 44

d

d

(12)

The solution of the formula is shown as follows

*

2 5 6 7 8 2sin sin cos cosw C sh C ch C sh C ch w (13)

*

2w is a particular solution of the equation (12),

qHEh

kRw 212

0

2*

2 (14)

The boundary conditions

iC 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8i can be determined by the boundary conditions and the

continuity conditions.

Boundary conditions:

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the upper end is unfixed1 0

0M

1 0 0Q (15a)

the bottom end is fixed2 0Lw

2d 0

d L

w

(15b)

Continuity condition H

Displacement continuityu dw w

d d

d d

u dw w

(15c)

Force continuity 1 1u dM M 1 1

u dQ Q (15d)

In the formula, the superscript u represents the upper half of the cylindrical shell, and the

superscript d represents the bottom half of the cylindrical shell.

Integral constant solution

In summary, linear equations in eight unknowns, using the boundary and continuity

conditions to solve the above equations, are established.

1 5 0C C (16a)

2 3 6 7C C C C

2

0

1 2 22 2 2 2 2 2

sin cos

sin cos 2cos

R KLch L Lsh L

L H qLch L Lsh L Lch L Eh

(16b)

2

04 8 1 2 22 2 2 2 2 2

2cos

sin cos 2cos

R KLch LC C L H q

Lch L Lsh L Lch L Eh

(16c)

In the formula (16b) and (16c),when H , H ; when H , L .

Displacement and internal force calculation

Displacement calculation

2

01 2 4 0sin cos cos

R kw C ch sh C ch q

Eh

H (17a)

2

02 2 4 2 1 2sin cos cos

R kw C ch sh C ch H q

Eh

H

(17b)

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Calculation of radial pressure

2

EhN w

R

(17c)

Calculation of moment

2

2 2

1 2 422 (sin cos ) sin

d wM D D C ch sh C sh

d

(17d)

Calculation of shear

3

3 3

1 2 434 sin 2 (sin cos )

d wQ D D C sh C ch sh

d

(17e)

FIELD MONITORING ANALYSIS

Site monitoring

Monitoring basis

Due to the complexity of the geological conditions, the results of this exploration are not

comprehensive enough to reflect the actual soil situation. Consequently, there will be some errorsin the soil sampling and testing process, which will inevitably be simplified in the design process.

In addition, the construction process may cause some deviation, resulting in inconsistencies

between the deformation and internal forces of the structure with the design values. Therefore, it

is necessary to understand the real-time changes in the soil and structure to ensure construction

safety.Monitoring content

The internal forces, displacement, and the earth pressure of the swirling pool wall are

monitored during the soil excavation and support structure application process. The specific

arrangement of the measuring points is shown in Figure 3.

MONITORING INSTRUMENTSA stress gauge is used to monitor the walls internal force; an inclinometer is used to monitor

the lateral displacement of the wall; and vibrating wire earth pressure cells are used to monitor the

earth pressure. Selected instruments are shown in Figures 4 and 5.

The frequency of monitoring

All of the instruments are monitored once a day during the excavation period and every other

day during the trenchless period.

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Figure 3: Layout of the measuring points

Figure 4: Earth pressure cells

Figure 5: Pictorial diagram of the stress gauge

40m

35m

30m

25m

20m

15m

10m

5m

measure line

Legendinclinometer tube

earth pressure cell

stress gauge

stress gauge

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The measured results

0.0 0.5 1.0 1.5 2.0 2.5 3.0

45

40

35

30

25

20

15

10

5

0

condition 1condition 2condition 3condition 4condition 5condition 6condition 7condition 8

de

pth/m

radial displacement/mm

Figure 6: Radial displacement

The measured results are organized and the radial displacement graph and circumferential

stress graph under the various conditions are shown in Figures 4 and 5, respectively. The

measuring points range from 5 m to 40 m, which does not account for the highest and lowest

points of the underground walls. In Figure 6, the radial displacement graph represents the inward

direction as the positive direction, whereas in Figure 7, the circumferential stress graph represents

tension as the positive direction. It can be observed from Figure 6 that the radial displacement is

larger in the middle and smaller on each end; the maximum lateral displacement occurs around

the excavation face. The development of circumferential stress is similar to that of the lateral

displacement. The maximum radial and lateral displacement of the wall is 2.13 mm, and themaximum compressive stress is 3.93 MP. There may be some error in the maximum value due to

the larger interval between the measuring points. The small radial displacement indicates the

significant arch effect of the cylindrical diaphragm wall, which has significantly limited the

lateral deformation. The space effect also leads to a large circumferential stress. Therefore,

prevention of brittle failure of the circular diaphragm wall resulting from circumferential stress

should be a priority.

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0 -1 -2 -3 -4

45

40

35

30

25

20

15

10

5

0

condition 1condition 2condition 3condition 4condition 5condition 6condition 7condition 8

dep

th/m

circumferential stress/MPa

Figure 7: Circumferential stress

RESULTS AND DISCUSSION

Numerical analyses

FLAC3D is used to establish the three-dimensional model considering the interaction of the

structure and the soil and simulating the pit excavation process. The soil constitutive model

adopts

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Table 2:Excavation conditions

construction steps conditions

1 excavate the pit to -4.0 m, construct the top beam

2 excavate the pit to -8.0 m, construct the first middle beam

3 excavate the pit to -12.0 m, construct the second middle beam

4 excavate the pit to -16.0 m, construct the third middle beam

5 excavate the pit to -20.0 m, construct the fourth middle beam

6 excavate the pit to -23.0 m, construct the fifth middle beam

7 excavate the pit to -26.0 m, construct the sixth middle beam

8 excavate the pit to -28.0 m, construct the cushion and floor

The Mohr-Coulomb model is used as the soil constitutive model and the lining element is

adopted for the wall. In addition, because the lining element contains a contact surface, there is no

need to add an additional one. Accounting for the disturbance of the surrounding soil caused by

excavation, the model size is 160 m 160 m 80 m. The top surface of the model is free; the

bottom surface of the model is completely constrained. Horizontal and normal restraints are

applied to the four sides. The excavation includes eight steps and the conditions of each step are

shown in Table 2. Because the boundary conditions of the symmetry plane may not accurately

reflect the actual situation, the entire model is selected to achieve accurate calculations The wall

elastic modulusc

E = 31.5 GPa, Poisson's ratio 0.2 , and the weight c = 26 KN / m3.

Numerical analysis results

Unbalanced force

When the ratio of the maximum unbalanced force and the internal force is small enough, the

operation stops. In FLAC3D, all of the grids are quadrilateral difference grids. For each grid,there are at least four surrounding grids that apply the internal action. The resultant force of these

forces is called the unbalanced force, which is the main control standard to calculate the

convergence using FLAC. The maximum value of the unbalanced force of the node is known as

the unbalanced force. When the system achieves balance, the maximum imbalance force

gradually tends to zero. Figure 8 shows a sampling of the maximum imbalance forces in the

calculation process.

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0 5000 10000 15000 20000 25000 30000

0

1000

2000

3000

4000

5000

6000

7000

8000

un

ba

lance

d

force

/kN

Unbalanced force

Figure 8: Unbalanced force

Radial displacement

Figure 9: Displacement contour in the Y-direction

XY

Z

SEL Y-Displ Magfac = 1.000e+000

-2.0758e-003 to -2.0000e-003-2.0000e-003 to -1.5000e-003-1.5000e-003 to -1.0000e-003-1.0000e-003 to -5.0000e-004-5.0000e-004 to 0.0000e+0000.0000e+000 to 5.0000e-0045.0000e-004 to 1.0000e-0031.0000e-003 to 1.5000e-0031.5000e-003 to 2.0000e-003

2.0000e-003 to 2.0705e-003 Interval = 5.0e-004

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-0.5 0.0 0.5 1.0 1.5 2.0 2.545

40

35

30

25

20

15

10

5

0

condition 1condition 2condition 3condition 4condition 5condition 6condition 7

condition 8

dep

th/m

radial displacement/mm

Figure 10: Simulation of radial displacement

The lining element is used to simulate the wall. Using the global coordinate system in the

extraction process of the lining element, the radial displacement could be obtained by extracting

the displacement in the y-direction at the point on the line (0,14,z), according to the symmetry.Figure 9 shows the deformation contour in the y-direction under condition 8. Figure 10 shows the

lateral displacement of the wall at depth under the various conditions. The numerical simulation

results are consistent with the measured trends. With advancement of the excavation, the

maximum displacement surface decreases slowly and the maximum displacement value

increases. The increase in the radical displacement of the walls above the excavation face is notsignificant. The maximum displacement value of 2.076 mm occurs at the excavation depth of 25

m.

Circumferential stress

Figure 11: Stress contour in the XXdirection

SEL stress-XX

Magfac = 1.000e+000-3.5836e+006 to -3.5000e+006-3.5000e+006 to -3.0000e+006-3.0000e+006 to -2.5000e+006-2.5000e+006 to -2.0000e+006-2.0000e+006 to -1.5000e+006-1.5000e+006 to -1.0000e+006-1.0000e+006 to -5.0000e+005-5.0000e+005 to -9.5745e+001

Interval = 5.0e+005depth factor = 1.00

XY

Z

XY

Z

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0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

dep

th/m

r a d i a l d i s p l a c e

a n a y c an u m e r i c a l s im e a s u r e d v a l

Figure 13: Comparison of radial displacement

0 -1 -2 -3 -4 -5 -6

45

40

35

30

25

20

15

10

5

0

analytical solutionnumerical simulate solutionmeasured value

dep

th/m

Figure 14: Comparison of circumferential stress

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CONCLUSIONSThe cylindrical diaphragm wall radial displacement and circumferential stress model is

obtained using the measured data. The overall trend is to first increase and then decrease along

the depth of the wall. The maximum value occurs around the excavation face. The arch effectlimits the radial displacement and may cause a large tangential stress.

An analytical solution model of the internal force deformation applied to a swirl pool circular

diaphragm wall using the cylindrical shell theory with moment is developed. The measured data

are used to verify the validity of the formula. The earth pressure value does not accurately reflect

the actual conditions because the calculation results beneath the excavation face are significantly

different from the measured value.

The FLAC3D simulation calculation involves a complex calculation process and a heavy

workload, but its calculated results are much closer to the measured value because it accounts for

the interaction between the structure and the soil.

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