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Influence of Surcharge and Foundation Soil on the Seismic Stability of Helical Soil-Nail Walls

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Abstract

This study proposes a closed-form solution to estimate the seismic factor of safety of helical soil–nail walls subjected to uniform surcharge. Unlike the existing solutions, the proposed method considers an underlying layer of foundation soil. The backfill and the foundation soil are modelled as a viscoelastic material which enables to incorporate effect of material dam**. The study investigates how the amplification of seismic waves in both the foundation and backfill soil influences the stability of the wall. The study also explores the impact of impedance ratio on wall stability as seismic waves propagate through the interface between the foundation soil and backfill. The impact of surcharge magnitude, foundation soil properties, base excitation frequency, and the diameter and spacing of helix in soil nail on the seismic factor of safety are also discussed. It is observed that the helical soil nails provide an enhanced stability to the vertical cuts compared to the conventional soil nails.

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Data Availability

The authors confirm that the data supporting the findings of this study are available within the article. Also, some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

a bh, a bv :

Horizontal and vertical acceleration in the backfill layer, respectively

a fh, a fv :

Horizontal and vertical acceleration in the foundation layer, respectively

a h0, a v0 :

Horizontal and vertical acceleration at the bedrock, respectively

A j :

Bearing surface area of the circular disc Aj = π(Dh – d)2/4

B :

Width of failure wedge at the backfill surface

d :

Diameter of the nail shaft

D, D f :

Dam** ratio of backfill and foundation soil, respectively

D h :

Diameter of the helix

f :

Frequency of base excitation

F D :

Total driving force

f n :

Fundamental frequency

Fo S :

Factor of safety of soil nailed wall

F R :

Total resistive force

g :

Acceleration due to gravity

H :

Height of the helical soil nailed wall

H f :

Depth of the foundation layer

k a :

Coefficient of active earth pressure

k h :

Horizontal seismic acceleration coefficient

k o :

Coefficient of earth pressure at rest

k v :

Vertical seismic acceleration coefficient

L :

Total length of soil nail

L e :

Effective length of the soil nail

N q :

Bearing capacity factor

P eq :

Total tensile force per unit horizontal spacing

\(P_{y}^{i}\) :

Pull-out capacity of ith helical soil nail

q :

Surcharge magnitude

Q h, Q v :

Total horizontal and vertical inertia forces acting on the failure wedge

Q hq, Q vq :

Horizontal and vertical seismic inertia forces due to surcharge, respectively

S hx :

Spacing between the helixes

S v :

Vertical spacing of soil nails

t :

Time

u bh, u bv :

Horizontal and vertical displacements in the backfill layer, respectively

u fh, u fh :

Horizontal and vertical displacements in the foundation layer, respectively

u ho :

Horizontal displacement at the bedrock

V s, V p :

Shear and primary wave velocities of backfill soil, respectively

V sf, V pf :

Shear and primary wave velocities of the foundation soil, respectively

W :

Weight of the failure wedge

β :

Wall slope

γ b :

Unit weight of the backfill layer

δ :

Peak interface friction angle between soil–nail interface

θ :

Angle of failure plane

λ, λ v :

Impedance ratio of shear and primary waves, respectively

ρ, ρ f :

Mass density of backfill and foundation soils, respectively

\(\sigma_{n}^{i}\) :

Overburden pressure acting on the nail-soil interface

φ :

Soil internal friction angle

ω :

Angular frequency

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Funding

This work was not supported by any external funding.

Author information

Authors and Affiliations

  1. FSG Geotechnics & Foundations, Level 2N/527 Gregory Terrace, Fortitude Valley, QLD, 4006, Australia

    V. S. Ramakrishna Annapareddy

  2. Department of Civil Engineering, University Institute of Technology, Himachal Pradesh University (HPU), Shimla,, 171005, India

    M. Sharma

  3. Sarathy Geotech and Engineering Services Pvt. Ltd, Gandhi Bazaar, Basavanagudi, Bengaluru, Karnataka,, 560004, India

    S. Godas

  4. Department of Engineering, College of Engineering and Computer Science, Texas A&M University-Corpus Christi, Corpus Christi, TX, 78412, USA

    E. Agarwal

Authors
  1. V. S. Ramakrishna Annapareddy
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  2. M. Sharma
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  3. S. Godas
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  4. E. Agarwal
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Contributions

V. S. Ramakrishna Annapareddy: Methodology, Writing—Original draft preparation, Investigation, Software, Data curation, Validation. Mahesh Sharma: Conceptualization, Writing—review & editing, Resources, Supervision. Srikar Godas: Conceptualization, Writing—review & editing, Resources, Supervision. Ekansh Agarwal: Writing—review & editing, Investigation, Resources, Supervision.

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Correspondence to M. Sharma.

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Appendices

Appendix A: Coefficients to Estimate Seismic Horizontal Acceleration

$${\text{U}}_{{{\text{hbz}}}} {\text{ = cos}}\left( {{\text{y}}_{{{\text{hb1}}}} \frac{{\text{z}}}{{\text{H}}}} \right){\text{sin}}\left( {{\text{y}}_{{{\text{hb2}}}} \frac{{\text{z}}}{{\text{H}}}} \right)$$
(24)
$${\text{V}}_{{{\text{hbz}}}} {\text{ = sin}}\left( {{\text{y}}_{{{\text{hb1}}}} \frac{{\text{z}}}{H}} \right){\text{cos}}\left( {{\text{y}}_{{{\text{hb2}}}} \frac{{\text{z}}}{{\text{H}}}} \right)$$
(25)
$$U_{hfz} = A_{hzf} - C_{hzf} P\chi_{h} - D_{hzf} Q\chi_{h}$$
(26)
$$V_{hfz} = B_{hzf} + D_{hzf} P\chi_{h} - C_{hzf} Q\chi_{h}$$
(27)
$$U_{h} = A_{hf} - C_{hf} P\chi_{h} - D_{hf} Q\chi_{h}$$
(28)
$$V_{h} = B_{hf} + D_{hf} P\chi_{h} - C_{hf} Q\chi_{h}$$
(29)
$$P = 1 + D \cdot D_{f}$$
(30)
$$Q = D_{f} - D$$
(31)
$$\chi_{h} = \frac{\lambda }{{1 + D_{f} }}$$
(32)
$$\lambda = \frac{{\rho .V_{s} }}{{\rho_{f} .V_{sf} }}$$
(33)
$$\begin{aligned} {\text{A}}_{{{\text{hfz}}}}&= {\text{ cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right){\text{cos}}\left( {{\text{y}}_{{{\text{hf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ &\quad-{\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right){\text{sin}}\left( {{\text{y}}_{{{\text{hf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(34)
$$\begin{aligned} B_{hfz} &= \cos \left( {y_{hb1} } \right)\cosh \left( {y_{hb2} } \right)\sin \left( {y_{hf1} \frac{{z_{f} }}{{H_{f} }}} \right)\sinh \left( {y_{hf2} \frac{{z_{f} }}{{H_{f} }}} \right) \hfill \\ &\quad + \sin \left( {y_{hb1} } \right)\sinh \left( {y_{hb2} } \right)\cos \left( {y_{hf1} \frac{{z_{f} }}{{H_{f} }}} \right)\cosh \left( {y_{hf2} \frac{{z_{f} }}{{H_{f} }}} \right) \hfill \\ \end{aligned}$$
(35)
$$\begin{aligned} C_{{{\text{hfz}}}}&= {\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{hf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ &\quad-{\text{cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{hf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(36)
$$\begin{aligned} D_{{{\text{hfz}}}}&= {\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{hf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\&\quad+ {\text{cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{hf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(37)
$$\begin{aligned} {\text{A}}_{{{\text{hf}}}} &={\text{cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right){\text{cos}}\left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ &\quad-{\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right){\text{sin}}\left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ \end{aligned}$$
(38)
$$\begin{aligned} B_{{{\text{hf}}}}&= {\text{cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ &\quad+{\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ \end{aligned}$$
(39)
$$\begin{aligned} C_{{{\text{hf}}}}&= {\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ &\quad-{\text{cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ \end{aligned}$$
(40)
$$\begin{aligned} D_{{{\text{hf}}}} &={\text{sin}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\&\quad+ {\text{cos}}\left( {{\text{y}}_{{{\text{hb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{hb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{hf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{hf2}}}} } \right) \hfill \\ \end{aligned}$$
(41)
$$y_{hb1} = \frac{\omega H}{{V_{s} }}\sqrt {\frac{{\sqrt {1 + 4D^{2} } + 1}}{{2\left( {1 + 4D^{2} } \right)}}}$$
(42)
$$y_{hb2} = - \frac{\omega H}{{V_{s} }}\sqrt {\frac{{\sqrt {1 + 4D^{2} } - 1}}{{2\left( {1 + 4D^{2} } \right)}}}$$
(43)
$$y_{hf1} = \frac{{\omega H_{f} }}{{V_{sf} }}\sqrt {\frac{{\sqrt {1 + 4D_{f}^{2} } + 1}}{{2\left( {1 + 4D_{f}^{2} } \right)}}}$$
(44)
$$y_{hf2} = - \frac{{\omega H_{f} }}{{V_{sf} }}\sqrt {\frac{{\sqrt {1 + 4D_{f}^{2} } - 1}}{{2\left( {1 + 4D_{f}^{2} } \right)}}}$$
(45)

Appendix B: Coefficients to Estimate Seismic Vertical Acceleration

$$U_{vbz} = \cos \left( {y_{vb1} \frac{z}{H}} \right)\sin \left( {y_{vb2} \frac{z}{H}} \right)$$
(46)
$$V_{vbz} = \sin \left( {y_{vb1} \frac{z}{H}} \right)\cos \left( {y_{vb2} \frac{z}{H}} \right)$$
(47)
$$U_{vfz} = A_{vzf} - C_{vzf} P\chi_{v} - D_{vzf} Q\chi_{v}$$
(48)
$$V_{vfz} = B_{vzf} + D_{vzf} P\chi_{v} - C_{vzf} Q\chi_{v}$$
(49)
$$U_{v} = A_{vf} - C_{vf} P\chi_{v} - D_{vf} Q\chi_{v}$$
(50)
$$V_{v} = B_{vf} + D_{vf} P\chi_{v} - C_{vf} Q\chi_{v}$$
(51)
$$\chi_{v} = \frac{{\lambda_{v} }}{{1 + D_{f} }}$$
(52)
$$\lambda_{v} = \frac{{\rho .V_{p} }}{{\rho_{f} .V_{pf} }}$$
(53)
$$\begin{aligned} {\text{A}}_{{{\text{vfz}}}}&= {\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right){\text{cos}}\left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\&\quad- {\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right){\text{sin}}\left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(54)
$$\begin{aligned} B_{{{\text{vfz}}}}&= {\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\&\quad+ {\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(55)
$$\begin{aligned} C_{{{\text{vfz}}}}& ={\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\&\quad- {\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(56)
$$\begin{aligned} D_{{{\text{vfz}}}}&= {\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\&\quad+ {\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{vf1}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} \frac{{{\text{z}}_{{\text{f}}} }}{{{\text{H}}_{{\text{f}}} }}} \right) \hfill \\ \end{aligned}$$
(57)
$$\begin{aligned} {\text{A}}_{{{\text{vf}}}}&= {\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right){\text{cos}}\left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ &\quad-{\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right){\text{sin}}\left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ \end{aligned}$$
(58)
$$\begin{aligned} B_{{{\text{vf}}}} &={\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ &\quad+{\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ \end{aligned}$$
(59)
$$\begin{aligned} C_{{{\text{vf}}}} &={\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ &\quad-{\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ \end{aligned}$$
(60)
$$\begin{aligned} D_{{{\text{vf}}}}&= {\text{sin}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\cos \left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\&\quad+ {\text{cos}}\left( {{\text{y}}_{{{\text{vb1}}}} } \right){\text{sinh}}\left( {{\text{y}}_{{{\text{vb2}}}} } \right)\sin \left( {{\text{y}}_{{{\text{vf1}}}} } \right){\text{cosh}}\left( {{\text{y}}_{{{\text{vf2}}}} } \right) \hfill \\ \end{aligned}$$
(61)
$$y_{vb1} = \frac{\omega H}{{V_{p} }}\sqrt {\frac{{\sqrt {1 + 4D^{2} } + 1}}{{2\left( {1 + 4D^{2} } \right)}}}$$
(62)
$$y_{vb2} = - \frac{\omega H}{{V_{p} }}\sqrt {\frac{{\sqrt {1 + 4D^{2} } - 1}}{{2\left( {1 + 4D^{2} } \right)}}}$$
(63)
$$y_{vf1} = \frac{{\omega H_{f} }}{{V_{pf} }}\sqrt {\frac{{\sqrt {1 + 4D_{f}^{2} } + 1}}{{2\left( {1 + 4D_{f}^{2} } \right)}}}$$
(64)
$$y_{vf2} = - \frac{{\omega H_{f} }}{{V_{pf} }}\sqrt {\frac{{\sqrt {1 + 4D_{f}^{2} } - 1}}{{2\left( {1 + 4D_{f}^{2} } \right)}}}$$
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Annapareddy, V.S.R., Sharma, M., Godas, S. et al. Influence of Surcharge and Foundation Soil on the Seismic Stability of Helical Soil-Nail Walls. Indian Geotech J (2024). https://doi.org/10.1007/s40098-024-01013-5

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