Abstract
In this paper, a quantitative uncertainty estimation of the random distribution of the soil material properties to the probability density functions of the failure load and failure displacements of a shallow foundation is portrayed. A modified Cam Clay yield stress model is implemented alongside with a stochastic finite element model. The random distribution of the reload path inclination \(\kappa\), the critical state line inclination c of the soil and the permeability k of the water flow relation, has been tested with Monte Carlo simulations accelerated implementing the Latin hypercube sampling. It is proven that both failure load and failure displacements follow Gaussian normal distribution despite the excessive nonlinear behaviour of the soil. In addition, as the eccentricity increases the mean value of failure load decreases in small values of the eccentricity and then increases and the failure displacement always increases. The uncertainty of the output failure stress with the increase in the eccentricity of the load remains the same. The failure spline of clays can be defined within an acceptable accuracy with the aforementioned numerical scheme in all possible geometry and load conditions, considering the eccentricity of the load in combination with nonlinear constitutive relations.
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Abbreviations
- \(N_f\) :
-
Friction variables
- \(N_q\) :
-
Friction variable indicating the influence of possible vertical load in the lateral of the foundation
- \(N_\mathrm{c}\) :
-
Friction variable indicating the influence of the cohesion of the soil
- \(N_{\gamma }\) :
-
Friction variable indicating the influence of the settlement dimensions alongside with the total weight of the soil
- \(S_f\) :
-
Shape variables
- \(S_q\) :
-
Shape variable indicating the influence of possible vertical load in the lateral of the foundation
- \(S_\mathrm{c}\) :
-
Shape variable indicating the influence of the cohesion of the soil
- \(S_{\gamma }\) :
-
Shape variable indicating the influence of the settlement dimensions alongside with the total weight of the soil
- \(\kappa\) :
-
Compressibility factor
- c :
-
Critical state line inclination
- k :
-
Permeability in units, \(\frac{\mathrm{m}^3 \mathrm{s}}{\mathrm{Mgr}}\)
- \(\phi _0\) :
-
Friction angle
- \({\mathbf {M}}\) :
-
Total mass matrix
- \({\mathbf {C}}\) :
-
Total damping matrix
- \({\mathbf {K}}\) :
-
Total stiffness matrix
- \(\mathbf {M_\mathrm{s}}\) :
-
Solid skeleton mass matrix
- \(\rho _\mathrm{d}\) :
-
Density of the soil
- \({\mathbf {B}}\) :
-
Deformation matrix
- \({\mathbf {E}}\) :
-
Elasticity matrix
- \(\mathbf {C_\mathrm{s}}\) :
-
Solid skeleton damping matrix
- \(\mathbf {K_\mathrm{s}}\) :
-
Solid skeleton stiffness matrix
- \({\mathbf {m}}\) :
-
Unity matrix
- \({\mathbf {b}}\) :
-
Loading vector
- \({\mathbf {k}}\) :
-
Matrix of permeability in units, \(\frac{\mathrm{m}^3 \mathrm{s}}{\mathrm{Mgr}}\)
- \(\mathbf {N^P}\) :
-
Shape functions for pore pressure
- \(\mathbf {N^u}\) :
-
Shape functions for displacements
- \({\mathbf {S}}\) :
-
Saturation matrix
- \(\mathbf {Q_\mathrm{c}}\) :
-
Coupling matrix
- \({\mathbf {H}}\) :
-
Permeability matrix
- \(\mathbf {f_S}\) :
-
Equivalent forces due to external loading
- Q :
-
Variable for combining the influence of bulk moduli of fluid and solid skeleton in porous problems
- \(\varvec{\sigma }\) :
-
Total stress tensor
- \({\mathbf {s}}\) :
-
Deviatoric component of the stress tensor
- \(p_\mathrm{h}\) :
-
Hydrostatic component of the stress tensor
- a :
-
Halfsize of the bond strength envelope
- \(\mathbf {s_L}\) :
-
Deviatoric component of the stress point of the centre of the plastic yield envelope
- \(p_L\) :
-
Hydrostatic component of the stress point of the centre of the plastic yield envelope
- \(\xi\) :
-
Similarity factor between the plastic yield envelope and bond strength envelope
- \(f_\mathrm{g}\) :
-
Generalized elliptic envelope
- \(f_\mathrm{p}\) :
-
Plastic yield envelope (PYE)
- F :
-
Bond strength envelope (BSE)
- \(\nu\) :
-
Specific volume of the soil
- q :
-
Von Mises stress
- e :
-
Deviatoric strain measure
- \(\varvec{\epsilon _{\mathrm{dev}}}\) :
-
Deviatoric component of the strain tensor
- f :
-
Random function
- \(f_i\) :
-
Value of the random function at nodal points
- \(N_i\) :
-
Shape functions
- \(N_0\) :
-
Total number of shape functions
- \(h_1\) :
-
Truncated normal PDF
- \(\phi (x)\) :
-
Standard normal PDF
- \({\varPhi }(x)\) :
-
Standard normal CDF
- \(\sigma _d\) :
-
Standard deviation of the random variable before truncation
- \(A, B, X_0\) :
-
Normalized coordinates of the subspace of the truncated PDF limits and x, respectively
- \(H_1({\mathbf {x}},\omega )\) :
-
Karhunen–Loeve random field
- \(N_s\) :
-
Number of subintervals in the Latin Hypercube Sampling
- \(\mu ({\mathbf {x}})\) :
-
Mean value of the random field
- \(X(x_1 , x_2 , \ldots x_n)\) :
-
Random vector created by the Latin Hypercube Sampling
- \(M_\mathrm{e}\), \(\lambda _i\), \(\phi _i\) :
-
Total number of eigenvalues \(\lambda _i\) and eigenfunctions \(\phi _i\), respectively
- b :
-
Correlation length
- \(\mathrm{COV}(\xi _i ,\xi _j)\) :
-
Covariance function
- \(\lambda ^*\) :
-
Load factor causing failure of the body at exactly the time which ends the rampload function
- \(a_\mathrm{w}\), \(b_\mathrm{w}\) :
-
Weight coefficients of the proposed algorithm for failure load calculation
- T :
-
Time which the rampload function ends
- \(T_p\) :
-
Symmetrization factor of the stochastic process
- \(\lambda _n\) :
-
Trial load factor of step n causing failure
- \(t_n\) :
-
Time of failure at the generalized load factor \(\lambda _n\)
- \(\lambda _{\text{max-no-failure}}\) :
-
Maximum trial load factor which causes safety
- \(\lambda _{1,\mathrm{fail}}\) :
-
Initial trial load factor causing failure
- \(\lambda _{1,\text{no-failure}}\) :
-
Initial trial load factor which causes safety
- \(q_1 - q_4\) :
-
Equivalent forces of the shallow foundation
- \((e_x=\frac{M_x}{N}, e_y=\frac{M_y}{N})\) :
-
Eccentricities of the footing settlement
- h :
-
Corresponding dimension of the footing settlement for each eccentricity
- \(l_x - l_y - l_z\) :
-
Dimensions of the total finite element mesh
- \(\sigma _v , \sigma _x , \sigma _y\) :
-
Geostatic stresses in vertical direction and directions x and y, respectively
- \(\lambda\) :
-
Inclination of isotropic compression line for the respective normally consolidated clay
- \(a_{\mathrm{initial}}\) :
-
Initial halfsize of the ellipse
- \(a_{\mathrm{residual}}\) :
-
Residual halfsize of the ellipse
- OCR:
-
Overconsolidation ratio
- G :
-
Shear modulus
- \(K_{\mathrm{bulk}}\) :
-
Bulk modulus
- \(\gamma\) :
-
Specific weight
- \(\nu _0\) :
-
Initial specific volume of the soil
- \(\mathbf {u_x}\) :
-
Displacement vector in direction x
- \(\kappa _{z=0}\) :
-
Compressibility factor at depth = 0
- \(\kappa _{z=\mathrm{max}}\) :
-
Compressibility factor at maximum depth
- \(R=\frac{\kappa _{z=\mathrm{max}}}{\kappa _{z=0}}\) :
-
Ratio of the compressibility factors measured at depth = 0 and at maximum depth
- \(\mu _R\) :
-
Mean value of ratio R
- \(\kappa _{z=\mathrm{max,mean}}\) :
-
Compressibility factor at maximum depth when the ratio R has its mean value
- \(\kappa _\mathrm{L}\) :
-
Linear distribution over depth for the compressibility factor
- \(\kappa _\mathrm{C}\) :
-
Constant distribution over depth for the compressibility factor
- \(\kappa _{\mu }\) :
-
Mean value of \(\kappa\)
- \(c_\mathrm{R}\) :
-
Random variable case for the critical state line inclination
- \(c_\mathrm{D}\) :
-
Deterministic case for the critical state line inclination
- \(k_\mathrm{R}\) :
-
Random variable case for the permeability
- \(k_\mathrm{D}\) :
-
Deterministic case for the permeability
- \(\mu _{\phi }\) :
-
Mean value of the friction angle
- \(\sigma _{\phi }\) :
-
Standard deviation of the friction angle
- \(\mu _{k}\) :
-
Mean value of the permeability
- \(P\mathrm{CoV}_{k}\) :
-
Coefficient of variation of the friction angle
- \(\kappa _{\mathrm{mean}}\) :
-
Mean value of the compressibility factor in the random field representation
- \(c_{\mathrm{mean}}\) :
-
Mean value of the critical state line inclination in the random field representation
- \(k_{\mathrm{mean}}\) :
-
Mean value of the permeability in the random field representation
- \(\sigma _{\kappa }\) :
-
Standard deviation of the compressibility factor in the random field representation
- \(\sigma _{\phi }\) :
-
Standard deviation of the critical state line inclination in the random field representation
- \(\sigma _{k}\) :
-
Standard deviation of the permeability in the random field representation
- \(\mu _{\nu }\) :
-
Mean values of the results
- \(\sigma _{\delta }\) :
-
Coefficient of variation of the results
- M :
-
Maximum values of the results
- \(\mu\) :
-
Minimum values of the results
- N :
-
Normal force
- \(u_{\mathrm{max}}\) :
-
Maximum displacement at failure
- \(u_{\mathrm{min}}\) :
-
Minimum displacement at failure
- \(\phi\) :
-
Rotation of the foundation at failure
- \(p_{\mathrm{vol}}\) :
-
Volumetric stress at failure
- \(q_{\mathrm{dev}}\) :
-
Von Mises stress at failure
- \(e_{\mathrm{vol}}\) :
-
Volumetric strain at failure
- \(e_{\mathrm{dev}}\) :
-
Deviatoric strain at failure
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Acknowledgements
This work has been supported by the European Research Council Advanced Grant MASTER Mastering the computational challenges in numerical modelling and optimum design of CNT-reinforced composites (ERC-2011-ADG-20110209). The author also acknowledges support from the Bodossaki Foundation.
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Appendix
Appendix
The following tables and figures are placed in this section
See Figs. Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22.
See Tables 5, 6, 7, 8, 9, 10 and 11.
PDFs of the maximum settlement stresses in kPa for non-porous medium
PDFs of the Normal force of the settlement in kN for non-porous medium
PDFs of the maximum displacement in m for non-porous medium
PDFs of the minimum displacement in m for non-porous medium
PDFs of the rotation of the settlement in rad for non-porous medium
PDFs of the maximum settlement stresses in kPa for porous analyses with deterministic shape functions for the stochastic material variables
PDFs of the Normal force of the settlement in kN for porous analyses with deterministic shape functions for the stochastic material variables
PDFs of the maximum displacement in m for porous analyses with deterministic shape functions for the stochastic material variables
PDFs of the minimum displacement in m for porous analyses with deterministic shape functions for the stochastic material variables
PDFs of the rotation of the settlement in rad for porous analyses with deterministic shape functions for the stochastic material variables
PDFs of the maximum settlement stresses in kPa for porous analyses with random field representation for the stochastic material variables
PDFs of the Normal force of the settlement in kN for porous analyses with random field representation for the stochastic material variables
PDFs of the maximum displacement in m for porous analyses with random field representation for the stochastic material variables
PDFs of the minimum displacement in m for porous analyses with random field representation for the stochastic material variables
PDFs of the rotation of the settlement in rad for porous analyses with random field representation for the stochastic material variables
Histograms of monitored output variables and the normal distribution fitting for 3 randomly selected analyses. a Porous analysis with constant distribution for \(\kappa\), random distribution for c and k with \(e=\,0\) and monitored output variable the Normal force. b Porous analysis of eccentricities \((e_x,e_y)=(\frac{h}{3},\frac{h}{6})\) with random field representations for all stochastic material variables and \(b=2\) m and monitored output variable the maximum failure displacement. c Porous analysis with linear distribution for \(\kappa\), random distribution for c and k with \((e_x,e_y)=(\frac{h}{3},\frac{h}{3})\) and monitored output variable the maximum settlement stress in kPa
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Savvides, A.A. Stochastic Failure of a Double Eccentricity Footing Settlement on Cohesive Soils with a Modified Cam Clay Yield Surface. Transp Porous Med 141, 499–560 (2022). https://doi.org/10.1007/s11242-021-01731-x
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DOI: https://doi.org/10.1007/s11242-021-01731-x