Failure modes and bearing capacity of strip footings on soft ground reinforced by floating stone columns

Abstract

This study evaluates the failure modes and the bearing capacity of soft ground reinforced by a group of floating stone columns. A finite difference method was adopted to analyze the performance of reinforced ground under strip footings subjected to a vertical load. The investigation was carried out by varying the aspect ratio of the reinforced zone, the area replacement ratio, and the surface surcharge. General shear failure of the reinforced ground was investigated numerically without the surcharge. The results show the existence of an effective length of the columns for the bearing capacity factors N _c and N _γ. When certain surcharge was applied, the failure mode of the reinforced ground changed from the general shear failure to the block failure. The aspect ratio of the reinforced zone and the area replacement ratio also contributed to this failure mode transition. A counterintuitive trend of the bearing capacity factor N _q can be justified with a shift in the critical failure mode. An upper-bound limit method based on the general shear failure mode was presented, and the results agree well with those of the previous studies of reinforced ground. Equivalent properties based on the area-weighted average of the stone columns and clay parameters were used to convert the individual column model to an equivalent area model. The numerical model produced reasonable equivalent properties. Finally, a theoretical method based on the comparison of the analytical equations for different failure modes was developed for engineering design. Good agreement was found between the theoretical and numerical results for the critical failure mode and its corresponding bearing capacity factors.

Appendix

The incremental external work and internal energy dissipation can be calculated as follows.

1.1 Velocity field

The velocity hodographs are shown in Fig. 10b: v _p is the velocity of the triangular wedge; the velocity v ₀ at the start point B of the log-spiral curve and the relative velocity v _0p of the triangular block and the log-spiral region are determined from the velocity hodograph as follows.

$$ v_{0} = \frac{{v_{\text{p}} \cos \left( {\alpha - \varphi_{\text{eq}} } \right)}}{{\cos \varphi_{\text{eq}} }} $$

(18)

$$ v_{{0{\text{p}}}} = \frac{{v_{\text{p}} \sin \alpha }}{{\cos \varphi_{\text{eq}} }} = \frac{{v_{0} \sin \alpha }}{{\cos \left( {\alpha - \varphi_{\text{eq}} } \right)}} $$

(19)

And the velocities v ₁ and v ₂ in the Tresca material are determined as follows

$$ v_{1} = v_{2} = v_{0} \times {\text{e}}^{{(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} $$

(20)

1.2 Geometry

The lengths of l _AB , l _AC , l _AD , and l _DE (Fig. 10a) are given as follows.

$$ l_{AB} = \frac{B}{2\cos \alpha } $$

(21)

$$ l_{AC} = l_{AD} = l_{AB} {\text{e}}^{{(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} $$

(22)

$$ l_{DE} = \frac{{l_{AD} }}{\sin \beta }\sin \left( {\frac{\pi }{2} - \beta } \right) = l_{AB} \times {\text{e}}^{{(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} \times \cot \beta $$

(23)

1.3 Incremental internal energy dissipation

The incremental internal energy dissipation along the different velocity discontinuities can be calculated as follows.

• 1. The energy dissipation along the discontinuity surface AB

  $$ D_{{{\text{c}}1}} = c_{\text{eq}} \times \cos \varphi_{\text{eq}} \times v_{{0{\text{p}}}} \times l_{AB} $$

  (24)

• 2. The energy dissipation in the log-spiral shear zone is the summation of the discontinuity surface BC and the spiral surface.

  $$ D_{{{\text{c}}2}} = 2c_{\text{eq}} \int_{0}^{\pi /2 - \alpha } {r \times v_{0} {\text{d}}\theta } = c_{\text{eq}} \times l_{AB} \times v_{0} \times \left( {{\text{e}}^{{2(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} - 1} \right) \times \cot \varphi_{\text{eq}} $$

  (25)

• 3. The energy dissipation in the radial shear zone is the summation of the discontinuity surface CD and the circular arc surface.

  $$ D_{{{\text{c}}3}} = 2c_{\text{s}} \int_{0}^{\beta } {v_{1} \times l_{AC} {\text{d}}\theta } = 2c_{\text{s}} \times v_{0} \times l_{AB} \times {\text{e}}^{{2(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} \times \beta $$

  (26)

• 4. The energy dissipation along the discontinuity surface DE.

  $$ D_{{{\text{c}}_{4} }} = c_{\text{s}} \times v_{2} \times l_{DE} = c_{\text{s}} \times v_{0} \times l_{AB} \times {\text{e}}^{{2(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} \times \cot \beta $$

  (27)

The total incremental energy dissipation is the summation of these four parts

$$ D_{\text{total}} = \sum\limits_{i = 1}^{4} {D_{{{\text{c}}i}} } $$

(28)

1.4 Incremental external work

The incremental external work is provided by the self-weight of soil, vertical load, and surcharge.

• 1. The incremental external work due to the self-weight of the triangular wedge OAB is

  $$ W_{\gamma 1} = \frac{1}{2} \times \frac{{\gamma_{\text{eq}} }}{{\gamma_{\text{s}} }} \times l_{AB}^{2} \times \sin \alpha \times \cos \alpha \times v_{0} \times \frac{{\cos \varphi_{\text{eq}} }}{{\cos \left( {\alpha - \varphi_{\text{eq}} } \right)}} $$

  (29)

• 2. The incremental external work due to the self-weight of the log-spiral shear zone ABC is

  $$ \begin{aligned} W_{\gamma 2} = & \int_{0}^{{\left( {\frac{\pi }{2} - \alpha } \right)}} {\frac{1}{2}} \times \frac{{\gamma_{\text{eq}} }}{{\gamma_{\text{s}} }} \times l_{AB}^{2} \times v_{0} \times \cos \left( {\alpha + \theta } \right) \times {\text{e}}^{{3 \times \theta \times \tan \varphi_{\text{eq}} }} {\text{d}}\theta \\ = & \frac{1}{2} \times \frac{{\gamma_{\text{eq}} }}{{\gamma_{\text{s}} }} \times l_{AB}^{2} \times v_{0} \frac{{{\text{e}}^{{3 \times \left( {\frac{\pi }{2} - \alpha } \right) \times \tan \varphi_{\text{eq}} }} - \sin \alpha - 3 \times \cos \alpha \times \tan \varphi_{\text{eq}} }}{{9 \times \tan^{2} \varphi_{\text{eq}} + 1}} \\ \end{aligned} $$

  (30)

• 3. The incremental external work due to the self-weight of the arc shear zone ACD is

  $$ W_{\gamma 3} = - \int_{0}^{\beta } {\frac{1}{2} \times \gamma_{\text{s}} \times l_{AC}^{2} \times v_{2} \times \sin \theta {\text{d}}\theta } = \frac{1}{2} \times \gamma_{\text{s}} \times l_{AB}^{2} \times v_{0} \times {\text{e}}^{{3 \times \left( {\frac{\pi }{2} - \alpha } \right) \times \tan \varphi_{\text{eq}} }} \times \left( {\cos \beta - 1} \right) $$

  (31)

• 4. The incremental external work due to the self-weight of the passive Rankine zone ADE is

  $$ W_{\gamma 4} = - \frac{1}{2} \times \gamma_{s} \times l_{AD} \times l_{DE} \times v_{2} \times \sin \beta = - \frac{1}{2} \times \gamma_{\text{s}} \times r_{0}^{2} \times v_{0} \times {\text{e}}^{{3 \times \left( {\frac{\pi }{2} - \alpha } \right) \times \tan \varphi_{\text{eq}} }} \times \cos \beta $$

  (32)

• 5. The incremental external work due to the vertical load is

  $$ W_{\text{f}} = q_{\text{u}} \times \frac{B}{2} \times v_{\text{p}} $$

  (33)

• 6. The incremental external work due to the surcharge is

  $$ W_{\text{q}} = q \times l_{AE} \times v_{2} \times \sin \beta = q \times l_{AB} \times v_{0} \times {\text{e}}^{{2(\pi /2 - \alpha )\tan \varphi_{\text{eq}} }} $$

  (34)

The total incremental external work is the summation of these six contributions from Eqs. (32)–(35).

$$ W_{\text{total}} = W_{\text{f}} - W_{\text{q}} + \sum\limits_{i = 1}^{4} {W_{\gamma i} } $$

(35)