Calculation of penetration depth of a support retaining system anchored on the top, for underground network and buried energy pipelines

Abstract

The object of the present paper is to determine the minimum support requirements provided for the safe execution of trench excavation works for underground network receptacles. This term is used to define the open excavations, which are carried out for underground networks to be installed or constructed, and which are classified in terms of place of execution and soil categories. This paper focuses on the study of the free earth support method in excavations carried out in earthy soils and in places where the passage of the pipeline requires deep excavations of vertical front. In these cases, for reasons of safety of both the personnel and the equipment, it is necessary to support the vertical excavation fronts, even when the analyses indicate their stability. In this context, and to avoid the consequences, before the start of the excavation and at its limit, the formation of vertical support structures is required, which are commonly known as "retaining bulkheads". The basic criteria for selecting the type of the support system are soil characteristics, local conditions (surface load, groundwater table) and finally, the lifetime of the trench. Based on the above, the objective of the study is to investigate the temporary support of vertical fronts of depth trenches about 10 m, the excavation of which is carried out on earthy soil. The study is based on the implementation of the limit equilibrium method, which is applied according to the provisions—guidelines set out in Eurocode 7 (EN 1997). The study investigates the distribution of earth pressures in an anchored retaining structure of a deep excavation by extracting the main design parameters of similar systems. The goal of this work is to select the type of the support system depending on the soil properties (c, γ, φ and Ε) and local conditions, as a prerequisite for safe and rational design, while in the end of the project useful conclusions are drawn.

1 Introduction

Partition retaining walls are an integral part of the design of geotechnical infrastructure projects, as they are used for the permanent or temporary retention of soil material, in which it is not possible to achieve stability conditions through its shear strength.

Such conditions may occur during the digging of vertical excavation fronts, such as power lines, where there is a sharp change in the soil surface after the removal of large volumes of soil [1]. In this case it becomes necessary to apply support retaining systems, which will stabilize the vertical slopes and ensure the safe and successful implementation of the project.

The characteristic depth of the retaining bulkheads is the depth of penetration, so that the active ground thrusts are balanced by the downstream ground. However, most of the time and especially in deep excavations, soil support is achieved by applying additional reinforcing agents, which are applied to develop the necessary lateral reactions for stability [2]. The main lateral retainer systems are struts and anchors (Fig. 1), with the latter being the subject of this study.

Fig. 1

Application of bulkhead retaining walls with an anchor level on the head

The necessary use of these additional elements is due to the fact that deep excavations require the engineer to be open to implementing a safe design, as the usefulness of lateral support is justified by the following reasons [3]:

• Their absence in deep excavations would lead to unacceptable movements of the retainer, and therefore of the retained front.

• To achieve acceptable movements in a retainer without lateral supports, the cost would be prohibitive.

Despite their structural simplicity, the stability analysis of these systems is a very difficult problem, as it depends on the interaction wall-ground soil. For this reason, it is imperative that during their design, forms of failure be investigated, such as that of limited anchoring and rotation around the anchorage point or around the base of the wall, failure of the anchoring system itself and finally failure of the device bodies due to bending or shear [4].

2 Literature review

The free earth support method is provided and analyzed in this study in order to determine the required penetration depth (D). Marinos et al. [1] looked into the circumstances in which some areas experience an upsurge in geo-risks. Alamanis and Dakoulas [2] investigate the impact of random fields of soil properties on the seismic behavior of pipelines travelling through natural slopes in two dimensions. The retaining wall design in supported excavations and foundations was provided by the Atkinson book [3], along with Kempfert and Gebreselassie [4]. Budhu [5] in his book discussed soil mechanics principles with their application to foundation studies, while Sabatini et al. [6] and Das [7] provide essential articles on ground anchors, earth anchors, and anchored systems. The book by Komodromos [9] covers the analysis and design of foundations and supports using simple to highly complex methods, depending on the specific requirements of each problem. Gaba et al. [8] provide good practice guidance on the selection and design of vertical embedded retaining walls to satisfy the requirements of Eurocodes. Furthermore, Eurocode 7 [10], established by the European Committee for Standardization (CEN, 2004), is intended to be used in conjunction with EN 1990:2002, which establishes the principles and requirements for safety and serviceability, describes the basics of design and verification, and provides guidelines for related aspects of structural reliability. Professor Anagnostopoulos et al. [11] give foundation and retaining system calculations for geotechnical constructions in their book, which were particularly useful in compiling the theoretical framework for this study. Most engineers involved in the design of sheet pile retaining structures use US Army Engineer Manual [12], (Design of Sheet Pile Walls). These techniques have regularly resulted in successful retaining structures that have operated well in the field. An intriguing Analysis of Anchored Sheet Pile Wall Deformations and Geo-Support is presented by Bilgin et al. [13]. The example documentation for pywall, a tool for the analysis of flexible retaining structures, was written by Reese et al. [14]. Zachos et al. [15] conducted research to investigate the distribution of soil pressures as a function of soil parameters (c, E, and F). They also proposed a method for determining the cross section of a support system that is necessary. The guidelines for choosing the diameter, length, and strength of the anchor can be found in the CFMS manual [16]. Many analyses were carried out in which, on the one hand, the values of soil attributes and excavation depth were varied in order to examine a wide variety of soils, and on the other hand, the required penetration depth and intensive quantities created in the structure were determined. The analysis' findings are represented in a series of diagrams that indicate the relationship between the soil characteristics' typical values and the retaining system's qualities. Other useful results that set the current work apart from others in the literature are also drawn.

3 Bulkhead wall analysis methodology with an anchor level

Bulkhead retaining walls, in accordance with the structural constructions, should be secured against the limiting states of failure and functionality. When checking their adequacy, in the context of considering a limit state of equilibrium, the areas of development of active and passive pressures are identified (Fig. 2), depending on the type and operating characteristics of the device [5].

Fig. 2

Graphic illustration of the development of active and passive thrusts on the bulkhead

The design of bulkhead retaining walls by applying a simple anchor to the head of the wall can be done by adopting one of the following assumptions [6]:

• Free earth support, which corresponds to the free lower extremity method.

• Fixed earth support at the base of the wall, which corresponds to the method of flexible curtain or three joint arch.

The difference between the above two assumptions lies in the effect that the penetration depth of the pile has in the form of the deviation of the wall from the vertical position and the different distribution of stresses [7]. The target of both methods is to calculate the required penetration depth (D) and the design bending moment (M_d).

Assuming the Free Earth Support Method, which is the method of calculating the above-mentioned unknowns in the present work, the designed wall can be considered as a simple vertical beam, supported by the ground stresses acting in the part of the wall that is "sunk" in the ground and the tension created by the anchor at the top of the wall. In this case the depth of penetration is such to prevent the wall from tipping over, but not necessarily preventing it from rotating in relation to its base [8].

In the present work, the case of applying a bulkhead retaining wall with an anchorage level on the head is considered, as a stability device of a vertical front excavation of loose soil, considering a horizontal free surface without load and development of a type of soil material along its entire height. The methodology which describe the calculation of the active and passive stresses is given as follow (Fig. 3). The recommended forces and their positions of application, as well the bending moments, the shear forces and the tensile force of the anchor, are also calculated in order to determine the minimum required depth of penetration of the bulkhead. Finally, the zero position of the shear force is determined while the maximum value of the bending moment, necessary to select the required cross-section of the bulkhead carrier, is calculated [9].

Fig. 3

Diagram of distribution of soil pressures of a bulkhead wall with anchorage in the top

Step 1: Calculation of pressures (kPa) (Fig. 4).

Fig. 4

Active and passive soil pressures bulkhead surfaces

Typical length sizes

$$\mathrm{e}=(0.2 \sim 0.4)\cdot \mathrm{H}$$

(1)

$$\mathrm{a}=\frac{{\upsigma }_{\mathrm{a}}}{\upgamma \cdot \left({\mathrm{K}}_{\mathrm{p}}-{\mathrm{K}}_{\mathrm{a}}\right)}$$

(2)

Soil thrusts

$${\upsigma }_{\mathrm{a}}=\upgamma \cdot \mathrm{H}\cdot {\mathrm{K}}_{\mathrm{a}}$$

(3)

$${\upsigma }_{\mathrm{p}}=\upgamma \cdot \mathrm{d}\cdot \left({\mathrm{K}}_{\mathrm{p}}-{\mathrm{K}}_{\mathrm{a}}\right)$$

(4)

Step 2: Calculation of recommended active and passive forces (kN/m) (Fig. 5)

Fig. 5

Recommended active and passive forces and their application distance from the anchorage position

$${\mathrm{P}}_{\mathrm{a}1}=\frac{1}{2}\cdot {\mathrm{K}}_{\mathrm{a}}\cdot\upgamma \cdot {\mathrm{H}}^{2}$$

(5)

$${\mathrm{P}}_{\mathrm{a}2}=\frac{1}{2}\cdot {\mathrm{K}}_{\mathrm{a}}\cdot\upgamma \cdot \mathrm{H}\cdot a$$

(6)

$${\mathrm{P}}_{\mathrm{p}}=\frac{1}{2}\cdot \left({\mathrm{K}}_{\mathrm{p}}-{\mathrm{K}}_{\mathrm{a}}\right)\cdot\upgamma \cdot {\mathrm{d}}^{2}$$

(7)

Step 3: Calculate moment level relative to the anchorage position (m) (Fig. 5)

$${\mathrm{L}}_{1}=\frac{2}{3}\cdot \mathrm{H}-\mathrm{e}$$

(8)

$${\mathrm{L}}_{2}=\frac{1}{3}\cdot \mathrm{a}+\mathrm{H}-\mathrm{e}$$

(9)

$${\mathrm{L}}_{3}=\frac{2\cdot \mathrm{d}}{3}+\mathrm{H}+\mathrm{a}-\mathrm{e}$$

(10)

Step 4: Equation of moment equilibrium with respect to the anchorage point, according to DA2*/EN 1997.

In the case of the anchored bulkhead with an anchor level on the top, the equilibrium of bending moments is taken with respect to the anchorage point (O) (Fig. 3). From the system of the resulting equations, the depth (d) will be calculated (Fig. 3), the value of which determines the required penetration depth of the diaphragm (D), in order for the passive thrusts to offer the required support.

In the context of the implementation of EN 1997, the control against failure is done by comparing the intensities E with the corresponding resistance R and the requirement that the resistance exceeds the intensities R > E [10]. In the present work, the design value of the intensity E_d and the resistance R_d, will follow the data required by the application of the analysis method DA-2*, which becomes from the national regulations.

$${\mathrm{E}}_{\mathrm{d}}=\mathrm{E}\left({\upgamma }_{\mathrm{F}}\cdot {\mathrm{F}}_{\mathrm{K}}{\mathrm{X}}_{\mathrm{K}}\right)\le {\mathrm{R}}_{\mathrm{d}}=\frac{1}{{\upgamma }_{\mathrm{R}}}\cdot \mathrm{R}\left({\mathrm{F}}_{\mathrm{K}}{\mathrm{X}}_{\mathrm{K}}\right)$$

(11)

In this case, the stresses resulting from the active thrusts on the right side of the wall are considered as tensions. To calculate the intensity design value, the characteristic values of the moments are multiplied by the individual coefficient γ_Ε = 1.35 if they come from permanent loads (earth loads) (γ_E from table Α.3 of EN 1997–1) [11].

$${\mathrm{M}}_{\mathrm{a}1}={{\mathrm{P}}_{\mathrm{a}1}\cdot \mathrm{L}}_{1}=\left(\frac{1}{2}\cdot {\mathrm{K}}_{\mathrm{a}}\cdot\upgamma \cdot {\mathrm{H}}^{2}\right)\cdot \left(\frac{2}{3}\cdot \mathrm{H}-\mathrm{e}\right)$$

(12)

$${\mathrm{M}}_{\mathrm{a}2}={{\mathrm{P}}_{\mathrm{a}2}\cdot \mathrm{L}}_{2}=\left(\frac{1}{2}\cdot {\mathrm{K}}_{\mathrm{a}}\cdot\upgamma \cdot \mathrm{H}\cdot \mathrm{a}\right)\cdot \left(\frac{1}{3}\cdot \mathrm{a}+\mathrm{H}-\mathrm{e}\right)$$

(13)

$${\mathrm{E}}_{\mathrm{d}}={\mathrm{M}}_{\mathrm{Ed}}={\gamma }_{\mathrm{E}}\cdot {\mathrm{M}}_{\mathrm{a}}=1.35\cdot {\mathrm{M}}_{\mathrm{a}} \Rightarrow {\mathrm{E}}_{\mathrm{d}}=1.35\cdot \left({\mathrm{M}}_{\mathrm{a}1}+{\mathrm{M}}_{\mathrm{a}2}\right)$$

(14)

The resistance is the moment resulting from the passive thrust on the left side of the wall. The determination of the design value of the resistance is obtained by dividing the resistance moment by the individual coefficient γ_R = 1.40, which is considered a favorable stability factor and is therefore subject to reduction (γ_R from table A.13 of EN 1997–1) [10].

$${\mathrm{M}}_{\mathrm{p}}={{\mathrm{P}}_{\mathrm{p}}\cdot \mathrm{L}}_{3}=\left(\frac{1}{2}\cdot \left({\mathrm{K}}_{\mathrm{p}}-{\mathrm{K}}_{\mathrm{a}}\right)\cdot \gamma \cdot {\mathrm{d}}^{2}\right)\cdot \left(\frac{2\cdot \mathrm{d}}{3}+\mathrm{H}+\mathrm{a}-\mathrm{e}\right)$$

(15)

$${\mathrm{R}}_{\mathrm{d}}=\frac{1}{{\gamma }_{\mathrm{R}}}\cdot {\mathrm{M}}_{\mathrm{p}} \Rightarrow {\mathrm{R}}_{\mathrm{d}}=\frac{1}{1.4}\cdot {\mathrm{M}}_{\mathrm{p}}$$

(16)

The inequality (11) is transformed into the inequality (17), through which the depth (d) is calculated, which together with the depth (a), which is respectively determined by Eq. (2), give the minimum required depth of penetration (D) of the diaphragm [12].

$${\mathrm{E}}_{\mathrm{d}}\le {\mathrm{R}}_{\mathrm{d}} \Rightarrow 1.35\cdot \left({\mathrm{M}}_{\mathrm{a}1}+{\mathrm{M}}_{\mathrm{a}2}\right)\le \frac{1}{1.4}\cdot {\mathrm{M}}_{\mathrm{p}} \Rightarrow \mathrm{d}\ge \dots (\mathrm{m})$$

(17)

$${\mathrm{D}}_{\mathrm{min}}=\mathrm{d}+\mathrm{a} \Rightarrow {\mathrm{D}}_{\mathrm{min}}=\dots (\mathrm{m})$$

(18)

Step 5: Calculate anchor force (N).

In designing an anchorage system, each anchor is considered to cover a wide band, which is determined by the vertical and horizontal distance between adjacent anchors. The choice of diameter, length and strength of the anchor, play a crucial role in ensuring the assumption of the required load, throughout the life of the construction [16].

$${\mathrm{\Sigma F}}_{\mathrm{H}}=0 \Rightarrow {\upgamma }_{\mathrm{E}}\cdot {\mathrm{P}}_{\mathrm{a}}-{\upgamma }_{\mathrm{F}}\cdot \mathrm{T}=\frac{1}{{\upgamma }_{\mathrm{R}}}\cdot {\mathrm{P}}_{\mathrm{p}} \Rightarrow$$

$$\mathrm{T}={\mathrm{F}}_{\mathrm{k}}=1.35\cdot \left({\mathrm{P}}_{\mathrm{a}1}+{\mathrm{P}}_{\mathrm{a}2}\right)-\frac{1}{1.4}\cdot {\mathrm{P}}_{\mathrm{p}} \Rightarrow$$

(19)

From the equilibrium of the horizontal forces (design values of both thrusts and the reaction force) the design value of the anchorage force T = F_k (19) is calculated. Because the anchor force is considered to be a favorable permanent action, the individual coefficient of γ_F is 1.00.

Step 6: Check against anchor extraction.

Having calculated the required anchorage force, the anti-extrusion test follows, which is carried out according to EN 1997 and application of the DA-2 method, which states that adequacy is achieved if the inequality R_a,d ≥ F_d is valid.

The design value of the draw resistance (R_a,d) is derived from the quotient of the characteristic value of the draw resistance (R_{a, k}) to the factor (γ_a), the values of which are defined in Table 6.4 (Table A.12/Annex A of EN 1997–1) [10].

$${\mathrm{R}}_{\mathrm{a},\mathrm{d}}=\frac{1}{{\upgamma }_{\mathrm{a}}}\cdot {\mathrm{R}}_{\mathrm{a},\mathrm{k}}$$

(20)

The characteristic value of the extrusion resistance (R_a,k) can either be theoretically calculated by applying Eq. (21) or determined from the results of extrusion test loads [11].

$${\mathrm{R}}_{\mathrm{a},\mathrm{k}}=\mathrm{min }\left\{\left(\uppi \cdot {\mathrm{D}}_{\mathrm{a}}\cdot {\mathrm{L}}_{\mathrm{f}}\cdot {\mathrm{f}}_{\mathrm{su},\mathrm{k}}\right),{(\mathrm{A}}_{\mathrm{s}}\cdot {\mathrm{f}}_{\mathrm{y},\mathrm{k}})\right\}$$

(21)

where: D_a is the anchor grout hole diameter, L_f is the anchoring length, f_su,k is the characteristic value of grout lateral friction grout—soil, Α_s is the area of steel tendon, f_y,k is the characteristic value of tendon steel leakage limit.

The design value of the anchor force (F_d) is calculated from the product of the characteristic value (F_k), which resulted from the previous calculation procedure, and the safety factor γ_F.

$${\mathrm{F}}_{\mathrm{d}}={\upgamma }_{\mathrm{F}}\cdot {\mathrm{F}}_{\mathrm{k}}=1.35\cdot {\mathrm{F}}_{\mathrm{k}}$$

(22)

Step 7: Determine the zeroing position of the shear force.

According to the free-edge theory, the maximum value of the active thrusts (σ_a) develops at the level of the excavation surface, as shown graphically in Fig. 4. To find the zero point of the shear forces, which is located between the anchorage point and the final excavation surface, a depth u is taken from the initial ground surface. At this depth, the value of the distributed triangular load is defined as σ_u and determined according to the theory of similar triangles (23) [13] (Fig. 6).

Fig. 6

a Active soil thrusts at depth u, b Diagram of shear forces where V = 0 at depth u

$$\frac{{\upsigma }_{\mathrm{u}}}{\mathrm{u}}=\frac{{\upsigma }_{\mathrm{a}}}{\mathrm{H}} \Rightarrow \frac{{\upsigma }_{\mathrm{u}}}{\mathrm{u}}=\frac{\upgamma \cdot \mathrm{H}\cdot {\mathrm{K}}_{\mathrm{a}}}{\mathrm{H}} \Rightarrow$$

$${\upsigma }_{\mathrm{u}}=\upgamma \cdot \mathrm{u}\cdot {\mathrm{K}}_{\mathrm{a}}$$

(23)

An equilibrium equation of the horizontal forces that develop from the initial surface of the ground to the depth u is applied, in order to determine this depth and to determine the zero position of the shear forces.

$${\mathrm{\Sigma F}}_{\mathrm{H}(\mathrm{u})}=0 \Rightarrow \mathrm{T}-\frac{1}{2}\cdot \mathrm{u}\cdot {\upsigma }_{\mathrm{u}}=0 \Rightarrow$$

$$\mathrm{u}= \sqrt{\frac{2\cdot \mathrm{T}}{\upgamma \cdot {\mathrm{K}}_{\mathrm{a}}}}$$

(24)

Step 8: Calculation of maximum bending moment (kNm/m).

At the point of zero of the shear forces, which is located at a depth u from the initial surface of the ground, is also a point of development of the maximum value of the bending moment (Fig. 7), which is calculated according to (28) [14].

Fig. 7

a Moment lever relative to the anchorage position, b Diagram of bending moments

The calculation of the recommended active thrust at depth u and the levers that determine the distance between the positions of application of the forces and the position of maximization of the bending moment, is done through Eqs. (25), (26) and (27) respectively.

$${\mathrm{P}}_{\mathrm{u}}=\frac{1}{2}\cdot {\mathrm{K}}_{\mathrm{a}}\cdot\upgamma \cdot {\mathrm{u}}^{2}$$

(25)

$${\mathrm{h}}_{1}=\frac{1}{3}\cdot \mathrm{u}$$

(26)

$${\mathrm{h}}_{2}=\mathrm{u}-\mathrm{e}$$

(27)

Based on the above, the calculation of the maximum bending moment is achieved through the difference of the product of the forces of action and reaction, on their respective distance to the depth of zeroing of the shear force.

$${\mathrm{M}}_{\mathrm{max}}=\mathrm{T}\cdot {\mathrm{h}}_{2}-{\mathrm{P}}_{\mathrm{u}}\cdot {\mathrm{h}}_{1}$$

$${\mathrm{M}}_{\mathrm{max}}=\mathrm{T}\cdot \left(\mathrm{u}-\mathrm{e}\right)-\frac{{\mathrm{K}}_{\mathrm{a}}\cdot\upgamma \cdot {\mathrm{u}}^{3}}{6}$$

(28)

Step 9: Select the minimum required cross section.

The selection of the minimum required cross-section for the stability of the system is achieved by calculating the value of the cross-section's moment of resistance W_el(min) [15].

$${\mathrm{W}}_{\mathrm{el}(\mathrm{min})}=\frac{{\mathrm{M}}_{\mathrm{max}}}{{\upsigma }_{\mathrm{B}}}\cdot \mathrm{FS}$$

(29)

σ_Β is the leakage limit of the construction material of the support beam, FS is the safety factor.

4 Parametric solution data

The calculations prepared in the context of the study, concern an extensive parametric investigation of soils with different mechanical properties, which obtained from available published data. Specifically, the illustrative example concerns a trench dug at a depth of 10 m with vertical side fronts and a horizontal unloaded surface, which has also practical interest [15].

The parameters used in the study of the support system are the following:

• Weight density: γ = 20 kΝ/m³.

• Friction angle: φ = 20°, 30°, 40°.

• Cohesion: c = 0, 5, 10, 15 kPa.

• Wall friction angle: δ = 0, 2/3φ, φ.

• Excavation depth: H = 10 m.

The solutions performed concern all the possible combinations of soil parameters and excavation depth, resulting in 108 different control cases. The investigation of the effect of the soil parameters on the value of the penetration depth of the retaining system and on the calculation of the maximum bending moment and shear force developed in it, was prepared using the software GEO5. The application of EN 1997 for Greek data (DA-2*), which has been already defined, requires:

• Calculation of penetration depth (D), through GEO analysis with DA-2*.

• Static adequacy control of the wall, through STR analysis with DA-2*.

• Calculation of active stresses according to the Coulomb method.

• Calculation of passive stresses according to the Coulomb method.

• Seismic analysis according to the Mononobe–Okabe method.

5 Results

The following figures show the effect of soil characteristics on the penetration depth of the bulkhead, the bending moments and the shear forces developed on it, as well as the force required to be applied by the anchor, in the case of a 10 m vertical excavation of homogeneous soil (Figs. 8, 9, 10).

Fig. 8

Variation of penetration depth D (m), bending moment M (kNm/m), shear force V (kN/m) and anchorage force F_k (kN), as a function of cohesion c (kPa) and wall friction angle δ, for soil with friction angle φ = 20°

Fig. 9

Variation of penetration depth D (m), bending moment M (kNm/m), shear force V (kN/m) and anchorage force F_k (kN), as a function of cohesion c (kPa) and wall friction angle δ, for soil with friction angle φ = 30°

Fig. 10

Variation of penetration depth D (m), bending moment M (kNm/m), shear force V (kN/m) and anchorage force F_k (kN), as a function of cohesion c (kPa) and wall friction angle δ, for soil with friction angle φ = 40°

6 Conclusions and suggestions for future research

A thorough understanding of the behavior of a support system used during trench digging leads to a proper analysis and design of the system, resulting in the avoidance of collapse occurrences, which, apart from financial implications, can endanger human lives. Due to the interaction between wall and earth, the response of a retaining wall that supports even a single layer of soil is a difficult problem [15].

The insertion of the beam before the trench excavation and the positioning of the anchors in the first stage of excavation, which does not exceed 3 m, are two significant advantages of using a bulkhead wall with an anchor level on the head. The disturbance of the supported soil's prevailing soil conditions is reduced to a minimum, if not entirely prevented, and the release of soil stresses is limited. As a result, a bulkhead wall anchored with one or more anchor levels is an ideal solution for the passage of energy pipelines in ditches where the topography and the pipeline's necessary slope demand an excavation depth more than 10 m.

The outcomes of the solutions are shown in the form of graphs in the preceding paragraph. The following are the findings of the investigation.

The maximum values of the penetration depth are observed in soils with reduced shear strength τ and reduced value of the ratio of friction d/f. In brief, while the angle of internal friction decreases φ↓ (20 < 30 < 40) and simultaneously the same applies to coherence c↓ (0 < 5 < 10 < 15) and ratio δ/φ↓ (0 < 2/3φ < φ), the required penetration depth of the diaphragm wall D↑ increases. The numerical results of the present study show that the maximum value of the penetration depth occurs in the case of the following soil characteristic values: φ = 20°, c = 0 kPa and friction ratio δ/φ = 0, which implies that the penetration depth is equal to excavation depth (D = H).

Depending on the depth of penetration, the maximum value of bending moment, shear force and anchorage force are observed in soils with both shear strength τ and value of friction ratio d/f reduced. Therefore, as the angle of internal friction decreases φ↓ (20 < 30 < 40) in parallel with the cohesion c↓ (0 < 5 < 10 < 15) and the ratio δ/φ↓ (0 < 2/3φ < φ), the values of: the bending moment Μ_max↑, the shear force V_max↑ and the value of the anchor's tensile force Fk_max↑, are corres-pondingly increased.

In all the diagrams there is a convergence of the values of the curves that reflect the ratio d/φ (0 < 2 / 3φ < φ), with the values coinciding with the absolute, in cases where the soil cohesion receives the maximum value (c = 15 kPa). Finally, in the soils with the highest value of internal friction angle, there is a smaller deviation between the maximum and the minimum value of the curves, which in combination with the cohesion demonstrates the effect that the soils with improved mechanical characteristics have on the support systems.

Finally, it should be mentioned that in the context of the parametric numerical simulation carried out in the present work, it was not possible to adequately cover all the issues related to the phenomenon of wall-ground interaction. The problem is extremely complex. For a further investigation it is proposed to extend the parametric analysis with more simulations, in order to determine the required penetration depth, considering the effect of:

• The multilayer soil.

• The existence of underground water.

• The action of surface loading.

• The load at the excavation depth.

• The presence of a slope on the free surface behind the support system.