https://orcid.org/0000-0002-0881-8136
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ABSTRACT
This paper presents the reliability analysis on the basis of the foundation failure against bearing capacity using the concept of fuzzy set theory. A surface strip footing is considered for the analysis and the bearing capacity is estimated using the conventional Finite Element Method (FEM). The spatial variability of the variables is taken into consideration to capture the physical randomness of the soil parameters for an isotropic field. A variation of the probability of failure (Pf) against a varying limiting applied pressure (q) is presented for different Coefficient of Variation (COV) of the variables and different scale of fluctuation (θ). The results reveal that the friction angle of soil (ϕ) is the most influencing parameter among the other variables. Further, the influence of the scale of fluctuation (θ) on the probability of failure (Pf) is also examined. It is observed that for a particular COV of ϕ, higher value of θ predicts higher Pf whereas, Pf increases as COV of ϕ increases for a particular θ value. Later, a comparison study is accomplished to verify the viability of the present method and it can be noticed that the present method compares well with the other reliability method (First Order Reliability Method) to a reasonably good extent.
Notations
| A | = | area of a small domain containing soil properties |
| A1 | = | dimension of the specified domain A in x direction |
| A2 | = | dimension of the specified domain A in z direction |
| AE | = | the area exceeding the applied pressure q of the resulting fuzzy number |
| AF | = | the area not exceeding the applied pressure q of the resulting fuzzy number |
| AT | = | the total area under the membership function of the resulting fuzzy number |
| α | = | value that indicates the α – level cut |
| β | = | reliability index |
| Bf | = | width of the footing |
| c | = | cohesion of soil |
| COV | = | coefficient of variation |
| E | = | modulus of elasticity of soil |
| F | = | fuzzy set |
| Fα | = | fuzzy set at α – cut level |
| γ | = | unit weight of soil |
| γvr(A) | = | variance reduction function over domain A |
| k | = | dispersion coefficient for triangular membership function |
| k1, k2 | = | dispersion coefficient for trapezoidal membership function |
| mz | = | the mode (most probable value) of the input variable |
| μF(z) | = | membership function of fuzzy set F |
| υ | = | Poisson’s ratio of soil |
| Pf | = | probability of failure |
| Φ | = | standard normal cumulative distribution function |
| ϕ | = | angle of internal friction of soil |
| ψ | = | dilatancy angle |
| q | = | applied pressure |
| qs | = | surcharge pressure |
| qult | = | ultimate bearing capacity of soil |
| R2 | = | coefficient of determination |
| ρ | = | correlation coefficient |
| s | = | inherent soil variability |
| σX2 | = | variance of the fuzzy number (soil parameter) |
| σXR2 | = | reduced variance of the soil parameter |
| τx | = | absolute distance between two points in the x direction |
| τz | = | absolute distance between two points in the z direction |
| θx | = | scale of fluctuation in the x direction |
| θz | = | scale of fluctuation in the z direction |
| u | = | trend function |
| Vx | = | COV of a parameter |
| Z | = | universal set |
| z1, z4 | = | the lower and upper values of the fuzzy number (z) at μF(z) = 0 |
| z2, z3 | = | the z values that correspond to μF(z) = 1 |
| ζ | = | in situ soil property |
Disclosure statement
No potential conflict of interest was reported by the authors.