Keywords

1 Introduction

The manuscript theme is soil retaining structures, with a special emphasis on in cantilever secant pile walls. The wall tip displacement as well the comparison between the methods of analysis that used to handle this problem presents the target of this study. The paper skeleton starts by hints on the literature leads to research questions. Then the research strategy presented by figures, tables, flowchart, the flowchart is the key figure. Then, the results presented, analyzed, discussed, and the conclusions offered.

Worku (2007) studied the use of springs to account for subgrade deformation. Bekdas et al. (2020) investigated the analysis of cantilever soldier pile walls in sandy soil based on the finite element and provides an optimal design. Ramadan et al. (2018) evaluated the design parameters needed to design contiguous pile supporting shallow excavations in clay. 3D- finite element model is used. Mahesh et al. (2019) examined diameter effects on pile capacity, bending moment and wall displacement. The case study involves deep excavation supported by secant piles of reinforced and plain concrete piles. Liao et al. (2014) performed comprehensive model tests to study the failure modes of secant faces. Khater (1991) published a case of sheet pile failures in clay, reported: “safety factor penetration ratio 20% is insufficient in clayey soil”.

The research statement could be presented by four research questions. Does wall penetration depth have a peak value Q1? Which analysis method is reliable: conventional method, Winkler or finite element, Q2? Is it reasonable to suggest empirical equations to predict the wall tip displacement ‘δ’, Q3? What is a ‘δ’ limitation for safe design? Q4. Satisfactory answer of those questions is the aim and importance of this work.

2 Methods Employed

Methodology is the strategy that outlines the way in which the research is to be carried out. Here, it is numerical analysis. Methods are the tools chosen to create quality data. Herein, they are the conventional method, Winkler analogy and 2D-FEA.

2.1 Parametric Study

Figure 1 is the variable interactions flowchart. Its basis is simple, 1st level ‘1-2-3’ is the analysis methods. The three doted rectangles present excavation depths. The numbers on the box show ‘routes’, e.g. ‘4-7-12-18’, is h = 5.0-m, loose soil S1, five wall rigidities I6: I12. This repeated three times, i.e. (1-4), (2-4) and (3-4). Box 20; explains that every single case has been investigated 10 times by increasing DO in steps of 10% each, DO is the minimum penetration depth calculated by the conventional method.

Fig. 1.
figure 1

Flow chart and research algorithm.

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Quality data is defined as a set of ‘focused-on’ data induced by analyzing suggested variables intentionally oriented to achieve the research goal. So, the parametric study technique is appropriate for that achievement. Briefly, two soils have been considered, loose and dense well graded dry sand, S1 and S2 respectively. Three excavation depths “h” have been studied, h = 3.0-m, 4.0-m and 5.0-m. This cover the practical range of cantilever secant pile. Ten wall rigidities “In”, integer “n” increases from 1 to 10, cover flexibility up to very rigid walls. The above have been analyzed three times, i.e. conventional method, then Winkler analogy and finite element analysis. Figure 1, Table 1 and Table 2 give a detailed description of every single variable used.

Table 1, summarizes the mechanical and physical properties. Even if the soil is dry sand, some parameters such as, γsat, ψo and c are needed for Plaxis-2D, by default. They have been used but never affect the physical meaning or behavior of dry sand.

Table 1. Physical and mechanical properties of the used soil.
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Table 2, summarizes the mechanical properties of the wall per meter run. The overlap of RC pile on bentonite pile is kept constant as 0.10-m. The Deqv stands for the equivalent pile diameter induced from neglecting bentonite piles mechanical properties.

Table 2. Physical and mechanical properties of the secant piles.
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2.2 Methods and Tools

The conventional method formulations described by Das (1987) and incorporated into an Excel sheet is developed to fulfill research needs. Spring analogy used here simulates Elasto-plastic soil response. Beam element models the sheet piles. This behavior has been formulated and runs with 2D noncommercial software ‘SPWALL20006’. It has been coded by Prof. Arnold Verruijt for Delft University of Technology (2006). The finite element analysis used is Plaxis-2D. Triangular 15-node elements were used to model the soil with Mohr-Coulomb material model. Plate and interface element were used for the sheet pile modeling. Staged construction procedure was used.

3 Presentation and Analysis of Results

Figure 2 and Fig. 3 demonstrates the case of an excavation depth h = 3.0-m, the other two cases, i.e. h = 4.0-m and 5.0-m having typical trend with different values. Figure 2 presents the FEA results, while Fig. 3 reports the spring analogy, comparison wise. Loose sand S1 and dense S2 have been investigated, with each, five wall rigidities, In have been inspected. DO is the minimum possible penetration depth (no safety factor). Di is a penetration depth with increasing factor of safety by steps 10% of DO. Hence, x-axis is the dimensionless ratio Di/DO. δO is the horizontal displacement of the wall tip corresponding to penetration depth DO. The δi is variable displacement of the wall tip at the penetration depth Di. The y-axis is the dimensionless ratio δi/δO, too.

3.1 Effective Penetration Depth and Design Hints

Figure 2 and Fig. 3, as In or soil density increases, δi decreases for the same Di/DO ratio. The effect of “In” on δi is more clear for S1 than for S2. ‘δi’ decreases as Di/DO increases, dramatically up to Di/DO is 1.50. Then curves get flatter, almost horizontal. It is concluded, for Di/DO beyond 1.5, no significant reduction of δi. Increasing DO by “design factor to satisfy safety”, of 20% appears not to be that generous! The later “point of view” is previously reported in a published case history, Khater (1991).

Fig. 2.
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Effect of penetration depth on wall tip displacement, [Plaxis cases]

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Figure 3 shows the results of the spring analogy for the same cases previously studied based on the Plaxis. The curve behaviors are typical in both techniques, but the Winkler approach has numerical values are higher. The numerical deviation is expected due to the lack of soil shear that leads to high values of δO, thus forcing all curves to be compacted. The two comments about Di/DO = 1.5 and Di/DO = 1.2 are valid herein and through the other two excavation depths. Again, Winkler overestimated ‘δ’.

Fig. 3.
figure 3

Effect of penetration depth on wall tip displacement, Winkler cases

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Table 3 displays the result variations among the three used methods. Conventional method results are the base of comparison. The ratio (%DO) definition is DO computed by spring analogy (or Plaxis) to DO calculated by the conventional method. The ratio (% BM) definition follows the same (%DO) concept, and both are decimals. Columns (2) and (6) show the values of (%DO) for the spring analogy and FEA- Plaxis 2D. The values range from 1.10 to 1.17. Columns (4) and (8) show the values of (%BM), for the spring analogy is 1.0; and for FEA- Plaxis 2D it ranges from 0.51 to 0.68. It is concluded that, the conventional method and the spring analogy underestimate the minimum possible penetration depth DO-min and overestimate the maximum expecting bending moment BM max, in general. This leads to higher values of δO predicted by spring analogy, and in comparison reasonable ones for FEA- Plaxis 2D.

Table 3. Comparison between the three analysis methods
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Figure 4-a, curves (1) and (2), for all soil densities or wall rigidities, increasing pile penetration depth does not affect maximum bending moment. Figure 4-b, curve (3), (4) and (5), for all soil densities or wall rigidities increases the pile penetration depth reduces the wall tip horizontal displacement. Most of the reduction induces from the increase of wall rigidity. To reduce excessive δi it is advised to increase the penetration depth up to 1.5 DO then increase the rigidity In, then change the structural system.

Fig. 4.
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Hints may enhance the secant pile design.

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3.2 Rational Empirical Equations

The variables that affect the displacement of the wall tip “δi” could be grouped in three categories. The first one will be ignored, here-in, because it is believed its effect is small, as well as its range of variation, i.e. soil and pile Poisson’s ratio.

Second category variables are; soil modulus of elasticity ES, pile flexural rigidity EI and Rankin’s earth pressure coefficient difference (kp – ka). Increases in the value of these variables reduce the expected δi i.e. inverse proportionally. Last category consists of only one parameter, the maximum bending moment Mmax. Increases in the bending moment increases δi, i.e. direct proportionally. The effect of soil density ‘γ’ is indirectly covered by the existence of ES, kp, ka and Mmax.

Then, the above parameters have been re-arranged in a common fraction denoted as ‘β’, Eq. (1). Its denominator is second group variables, while the numerator is Mmax. It has been assumed, with some confidence, the powers of all variables equal to “1”, which is rational and matches the engineering sense.

$$\beta = \frac{{M}_{max}}{{E}_{S} EI \left({k}_{p}- {k}_{a}\right) }$$
(1)

Consider β is x-axis of and δ/h is its y-axis, Fig. 5. For each excavation depths the input data is fed into Plaxis, run by run, and have been used to calculate βi, and the corresponding output per each run used to calculate δi/h. Then, Fig. 5 graphs have been plotted containing three curves presenting three excavation depths; linear curve fitting is the best fit. Equations (2), (3) and (4) present each curve trend.

Fig. 5.
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Prediction of wall tip horizontal displacement, first approximation.

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Case of h = 5.m, R-sq = 0.76

$$\updelta{/}{\rm h} = 107202.4\,\upbeta + 0.010815 $$
(2)

Case of h = 4.m, R-sq = 0.97

$$\updelta{/}{\rm h} = 70791.446\,\upbeta + 0.00889 $$
(3)

Case of h = 3.m, R-sq = 0.95

$$\updelta{/}{\rm h} = 38582.394\,\upbeta + 0.0046 $$
(4)

Unit’s Key.

The above equations units are: Mmax (kN.m/m), Es (kN/m2), EI (kN.m2/m), β (m/kN), δ (m), h (m), the constants “107202.4 - 70791.446 - 38582.394” (kN/m) while “0.010815 - 0.00889 - 0.0046” (dimensionless).

Those equations present an improvement to the conventional method of design cantilever secant pile walls, as first approximation. The procedures are: (a) calculate penetration depth “DO” and Mmax based on the conventional method. (b) Increase DO by 50% and reduce Mmax by 35%. (c) Calculate “β”, then pick up the equation matches the excavation depth to get δ. At this point, one can acquire reasonable bending moment, safe penetration depth and approximate wall-tip displacement. It could be used for dry sand densities, pile rigidities and any type of loading because the above equations use Mmax, regardless of loads or soil conditions produced it. This upgrade is limited to homogeneous sand. A look back at Eqs. 2, 3 and 4 says minimum ‘δ’ occurs if ‘β’ is “zero”. This will never happen unless Mmax = 0.0, not logical, or ‘β’ denominator is “infinity”. The only reason to accept this infinity is the existence of rigid capping beam and corners struts. The capping beam effect is the running research by me and research assistant. Until then, ‘δ’ is acceptable if it is ≤ 5.0-cm, 3.5-cm or 1.5-cm for wall heights 5.0-m, 4.0-m, and 3.0-m respectively. That is to say acceptable limit is δi ≤ 0.8% excavation depth, ‘h’.

4 Conclusions

Within the limits and assumptions of this study the following facts are concluded:

  1. 1.

    As the wall penetration depth increases, the wall tip displacement ‘δ’ decreases. Beyond 50DO% increase, no weighty ‘δ’ improvement is noticed.

  2. 2.

    It is suggested to avoid the 20% DO depth increases and to replace it by 50%.

  3. 3.

    High flexural rigidity and all cases of dense soils decreases δ.

  4. 4.

    Conventional method and spring analogy underestimate penetration depth and overestimate bending moment. Also, spring analogy gives higher values of ‘δ’ if all the above compared to finite element analysis.

  5. 5.

    Generally, increases penetration depth, does not change the values of maximum bending moment. But, it reduces wall displacement, effectively.

  6. 6.

    To reduce excessive δi first increase penetration depth up to 50%, and then increase wall rigidity, if neither insufficient, introduce anchors or struts.

  7. 7.

    The acceptable limit of ‘δi’ is ≤ 0.8% ‘h’, unless stringent values are imposed by adjacent structure. This ‘δi’ is the one that calculated by 2D-FEA model.

  8. 8.

    A rational technique has been suggested using empirical equations to upgrade the capability of conventional method to predict wall tip displacement.