Lower-Bound Axisymmetric Formulation for Geomechanics Problems Using Nonlinear Optimization

A lower-bound limit analysis formulation, by using two-dimensional finite elements, the three-dimensional Mohr-Coulomb yield criterion, and nonlinear optimization, has been given to deal with an axisymmetric geomechanics stability problem. The optimization was performed using an interior point method based on the logarithmic barrier function. The yield surface was smoothened (1) by removing the tip singularity at the apex of the pyramid in the meridian plane and (2) by eliminating the stress discontinuities at the corners of the yield hexagon in the pi-plane. The circumferential stress (sigma(theta)) need not be assumed. With the proposed methodology, for a circular footing, the bearing-capacity factors N-c, N-q, and N-gamma for different values of phi have been computed. For phi = 0, the variation of N-c with changes in the factor m, which accounts for a linear increase of cohesion with depth, has been evaluated. Failure patterns for a few cases have also been drawn. The results from the formulation provide a good match with the solutions available from the literature. (C) 2014 American Society of Civil Engineers.

Bearing capacity of circular footings over rock mass by using axisymmetric quasi lower bound finite element limit analysis

The ultimate bearing capacity of a circular footing, placed over rock mass, is evaluated by using the lower bound theorem of the limit analysis in conjunction with finite elements and nonlinear optimization. The generalized Hoek-Brown (HB) failure criterion, but by keeping a constant value of the exponent, alpha = 0.5, was used. The failure criterion was smoothened both in the meridian and pi planes. The nonlinear optimization was carried out by employing an interior point method based on the logarithmic barrier function. The results for the obtained bearing capacity were presented in a non-dimensional form for different values of GSI, m(i), sigma(ci)/(gamma b) and q/sigma(ci). Failure patterns were also examined for a few cases. For validating the results, computations were also performed for a strip footing as well. The results obtained from the analysis compare well with the data reported in literature. Since the equilibrium conditions are precisely satisfied only at the centroids of the elements, not everywhere in the domain, the obtained lower bound solution will be approximate not true. (C) 2015 Elsevier Ltd. All rights reserved.

Lower Bound Axisymmetric Limit Analysis Using Drucker-Prager Yield Cone and Simulation with Mohr-Coulomb Pyramid

This paper presents a lower bound limit analysis approach for solving an axisymmetric stability problem by using the Drucker-Prager (D-P) yield cone in conjunction with finite elements and nonlinear optimization. In principal stress space, the tip of the yield cone has been smoothened by applying the hyperbolic approximation. The nonlinear optimization has been performed by employing an interior point method based on the logarithmic barrier function. A new proposal has also been given to simulate the D-P yield cone with the Mohr-Coulomb hexagonal yield pyramid. For the sake of illustration, bearing capacity factors N-c, N-q and N-gamma have been computed, as a function of phi, both for smooth and rough circular foundations. The results obtained from the analysis compare quite well with the solutions reported from literature.

Lower Bounds for Sums of Powers of Low Degree Univariates

We consider the problem of representing a univariate polynomial f(x) as a sum of powers of low degree polynomials. We prove a lower bound of Omega(root d/t) for writing an explicit univariate degree-d polynomial f(x) as a sum of powers of degree-t polynomials.

Bearing capacity factors for a conical footing using lower- and upper-bound finite elements limit analysis

Bearing capacity factors, N-c, N-q, and N-gamma, for a conical footing are determined by using the lower and upper bound axisymmetric formulation of the limit analysis in combination with finite elements and optimization. These factors are obtained in a bound form for a wide range of the values of cone apex angle (beta) and phi with delta = 0, 0.5 phi, and phi. The bearing capacity factors for a perfectly rough (delta = phi) conical footing generally increase with a decrease in beta. On the contrary, for delta = 0 degrees, the factors N-c and N-q reduce gradually with a decrease in beta. For delta = 0 degrees, the factor N-gamma for phi >= 35 degrees becomes a minimum for beta approximate to 90 degrees. For delta = 0 degrees, N-gamma for phi <= 30 degrees, as in the case of delta = phi, generally reduces with an increase in beta. The failure and nodal velocity patterns are also examined. The results compare well with different numerical solutions and centrifuge tests' data available from the literature.

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a 2(Omega(root d.log N)) size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d(3) in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Sigma(i) Pi(j) Q(ij), where the Q(ij)'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that Sigma(i,j) (Number of monomials of Q(ij)) >= 2(Omega(root d.log N)). The above mentioned family, which we refer to as the Nisan-Wigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on the recent lower bound results 1], 2], 3], 4], 5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of 6] and the N-Omega(log log (N)) lower bound in the independent work of 7].

Synthesis of Hollow Nanotubes of Zn2SiO4 or SiO2: Mechanistic Understanding and Uranium Adsorption Behavior

We report a facile synthesis of Zn2SiO4 nanotubes using a two-step process consisting of a wet-chemical synthesis of core-shell ZnO@SiO2 nanorods followed by thermal annealing. While annealing in air leads to the formation of hollow Zn2SiO4, annealing under reducing atmosphere leads to the formation of SiO2 nanotubes. We rationalize the formation of the silicate phase at temperatures much lower than the temperatures reported in the literature based on the porous nature of the silica shell on the ZnO nanorods. We present results from in situ transmission electron microscopy experiments to clearly show void nucleation at the interface between ZnO and the silica shell and the growth of the silicate phase by the Kirkendall effect. The porous nature of the silica shell is also responsible for the etching of the ZnO leading to the formation of silica nanotubes under reducing conditions. Both the hollow silica and silicate nanotubes exhibit good uranium sorption at different ranges of pH making them possible candidates for nuclear waste management.

Meridional gradients in aerosol vertical distribution over Indian Mainland: Observations and model simulations

Multi-year observations from the network of ground-based observatories (ARFINET), established under the project `Aerosol Radiative Forcing over India' (ARFI) of Indian Space Research Organization and space-borne lidar `Cloud Aerosol Lidar with Orthogonal Polarization' (CALIOP) along with simulations from the chemical transport model `Goddard Chemistry Aerosol Radiation and Transport' (GOCART), are used to characterize the vertical distribution of atmospheric aerosols over the Indian landmass and its spatial structure. While the vertical distribution of aerosol extinction showed higher values close to the surface followed by a gradual decrease at increasing altitudes, a strong meridional increase is observed in the vertical spread of aerosols across the Indian region in all seasons. It emerges that the strong thermal convections cause deepening of the atmospheric boundary layer, which although reduces the aerosol concentration at lower altitudes, enhances the concentration at higher elevations by pumping up more aerosols from below and also helping the lofted particles to reach higher levels in the atmosphere. Aerosol depolarization ratios derived from CALIPSO as well as the GOCART simulations indicate the dominance of mineral dust aerosols during spring and summer and anthropogenic aerosols in winter. During summer monsoon, though heavy rainfall associated with the Indian monsoon removes large amounts of aerosols, the prevailing southwesterly winds advect more marine aerosols over to landmass (from the adjoining oceans) leading to increase in aerosol loading at lower altitudes than in spring. During spring and summer months, aerosol loading is found to be significant, even at altitudes as high as 4 km, and this is proposed to have significant impacts on the regional climate systems such as Indian monsoon. (C) 2015 Elsevier Ltd. All rights reserved.

Reply to Discussion on ``Bearing capacity of circular footings over rock mass by using axisymmetric quasi lower bound finite element limit analysis'' Comput. Geotech. 70 (2015) 138-149]

A discussion has been provided for the comments raised by the discusser (Clausen, 2015)1] on the article recently published by the authors (Chakraborty and Kumar, 2015). The effect of exponent alpha for values of GSI approximately smaller than 30 becomes more critical. On the other hand, for greater values of GSI, the results obtained by the authors earlier remain primarily independent of alpha and can be easily used. (C) 2015 Elsevier Ltd. All rights reserved.