Bearing Capacity Factors for Isolated Surface Strip Footing Resting on Multi-layered Reinforced Soil Bed

Abstract

In this paper, the ultimate bearing capacity of an isolated surface strip footing resting on a reinforced c–ϕ soil bed is determined using the upper-bound limit analysis. The soil is assumed to obey the Mohr–Coulomb failure criteria along with associated flow rule. A Prandtl type failure mechanism along with kinematically admissible velocity field is considered in this analysis. The variation of bearing capacity is studied with respect to the angle of internal friction of soil (ϕ) and the number of reinforcement layers (m). The results obtained from the upper bound limit analysis are compared with the finite element solution performed separately using PLAXIS 2D. The present theoretical observations are generally found in good agreement with those results available in literature.

Appendices

Appendix 1: External Work Done

• (a) Considering the complete failure mechanism on both sides of the footing, the external work done by the self-weight of the whole triangular block below the footing (Fig. 2a) can be expressed as,

  $$ \Delta W_{block} = \frac{{\gamma B^{2} }}{2}\left[ {f_{1} \left( {\alpha_{i} ,\beta_{i} ,\theta } \right)} \right]V_{o} $$

  (10)

  where $f_{1} = \frac{\tan \theta }{2}$

• (b) The external work done by the self-weight of the 2n triangular rigid blocks on either side of the footing can be expresses as,

  $$ \sum\limits_{j = 1}^{2n} {[\Delta W]_{j} = \frac{{\gamma B^{2} }}{2}} [f_{2} (\alpha_{i} ,\beta_{i} ,\theta )]V_{0} $$

  (11)

  where,

  $$ \begin{aligned} f_{2} &= \frac{\cos (\theta - \phi )}{{2 \cdot \cos^{2} \theta \cdot \sin (\beta_{1} - 2\phi )}}\sum\limits_{i = 1}^{n} {\left[ {\frac{{\sin \alpha_{i} \sin \beta_{i} }}{{\sin (\alpha_{i} + \beta_{i} )}}\sin \left( {\beta_{i} - \theta - \phi - \sum\limits_{j = 1}^{i - 1} {\alpha_{j} } } \right)} \right.} \hfill \\ &\quad \left. {\prod\limits_{j = 1}^{i - 1} {\frac{{\sin^{2} \beta_{j} }}{{\sin^{2} (\alpha_{j} + \beta_{j} )}}} \frac{{\sin (\alpha_{j} + \beta_{j} - 2\phi )}}{{\sin (\beta_{j + 1} - 2\phi )}}} \right] \hfill \\ \end{aligned} $$

• (c) The external work done by the foundation load, P_urf can be expressed as,

  $$ \Delta W_{f} = P_{urf} V_{o} $$

  (12)

Appendix 2: Internal Dissipation of Energy

Considering the complete failure mechanism on both sides of the footing, the internal dissipation of energy can be derived as follows

• (a) The internal dissipation of energy along the line of discontinuity, DC₁ can be expressed as (Fig. 2a),

  $$ \Delta IWD_{{DC_{1} }} = cB[f_{3} (\alpha_{i} ,\beta_{i} ,\theta )]V_{o} $$

  (13)

  where $f_{3} = \frac{{\cos \phi \cos (\beta_{1} - \theta - \phi )}}{{\cos \theta \sin (\beta_{1} - 2\phi )}}$

• (b) The internal dissipation of energy along d_i (i = 1, 2, 3 … n) can be expressed as,

  $$ \Delta IWD_{d} = cB[f_{4} (\alpha_{i} ,\beta_{i} ,\theta )]V_{o} $$

  (14)

  where,

  $$ f_{4} = \frac{\cos \phi \cos (\theta - \phi )}{{\cos \theta \sin (\beta_{1} - 2\phi )}}\sum\limits_{i = 1}^{n} {\left[ {\frac{{\sin \alpha_{i} }}{{\sin (\alpha_{i} + \beta_{i} )}}\prod\limits_{j = 1}^{i - 1} {\frac{{\sin \beta_{j} }}{{\sin (\alpha_{j} + \beta_{j} )}}\frac{{\sin (\alpha_{j} + \beta_{j} - 2\phi )}}{{\sin (\beta_{j + 1} - 2\phi )}}} } \right]} $$

• (c) The internal dissipation of energy along l_i (i = 2, 3, 4 … n) can be expressed as,

  $$ \Delta IWD_{l} = cB\left[ {f_{5} \left( {\alpha_{i} ,\beta_{i} ,\theta } \right)} \right]V_{o} $$

  (15)

  where,

  $$ \begin{aligned} f_{5} & = \frac{\cos \phi \cos (\theta - \phi )}{{\cos \theta \sin (\beta_{1} - 2\phi )}}\sum\limits_{i = 2}^{n + 1} {\left[ {\frac{{\sin (\beta_{i - 1} - \beta_{i} + \alpha_{i - 1} )}}{{\sin (\beta_{i} - 2\phi )}}\prod\limits_{j = 1}^{i - 1} {\frac{{\sin \beta_{j} }}{{\sin (\alpha_{j} + \beta_{j} )}}} } \right.} \\ & \quad \left. {\prod\limits_{j = 1}^{i - 2} {\frac{{\sin (\alpha_{j} + \beta_{j} - 2\phi )}}{{\sin (\beta_{j + 1} - 2\phi )}}} } \right] \\ \end{aligned} $$

• (d) The internal dissipation of energy along the reinforcement embedded in the rigid blocks i = 1, 2, 3 … n can be expresses as,

  $$ \Delta IWD_{rf} = \Delta IWD_{rf}^{Cohesion} + \Delta IWD_{rf}^{Soil\,weight} + \Delta IWD_{rf}^{Footing\,load} $$

  (16)

  where $\Delta IWD_{rf}^{Cohesion}$ = internal dissipation of energy due to the adhesion between the soil and the reinforcement = $cf_{c} BM_{c} V_{0}$, $\Delta IWD_{rf}^{Soil\,weight}$ = internal dissipation of energy due to the frictional force developed along the reinforcement for the soil weight = $\frac{{s_{v1} }}{B}\gamma \mu B^{2} M_{\gamma } V_{0}$, $\Delta IWD_{rf}^{Footing\,load}$ = internal dissipation of energy dissipation due to the frictional force developed along the reinforcement for the footing load = $q_{urf} \mu B\frac{{s_{vi} }}{B}M_{P} V_{0}$.

M_c, M_γ and M_p are the factors which can be derived from the analysis. When the reinforcement layers pass above the points C₁ and C_n (Fig. 2a), the values of M_c and M_γ become equal and therefore, a common factor M may be used as given below,

$$ M_{c} = M_{\gamma } = M $$

(17)

It is worth noting here that a single value of M- or M_P-factor can be obtained if the whole optimization is performed for the failure mechanism considering both cohesion and unit weight of soil together as proposed by Michalowski [11]. However, if the method of superposition is used along with individual optimization, the values of M- and M_P-factor become different for cohesion as well as unit weight component. In this study, M₁ and M_P1 are used to define the M- and M_P-factor respectively for cohesion component, whereas M₂ and M_P2 are used to define the M- and M_P-factor respectively for unit weight component.