Presented is a rigorous solution for the three-dimensional stability analysis of convex slopes with corners in plan view. The method is based on the UPPER-BOUND THEOREM of limit analysis approach. A rigid-block translational collapse mechanism is considered, with energy dissipation taking place along planar velocity discontinuities. This mechanism is optimized to obtain the minimum factor of safety for stability of the corners. The algorithm can also be used to determine the ultimate limit load of a foundation located on a corner. Based on comparisons with known solutions, the method was generally found to be accurate in predicting the stability of such slopes. The numerical results indicate that the unloaded corners are more stable than the straight slopes. Dimensionless diagrams for various corner angles are also presented.

Today, large dams are constructed with the different purposes of hydroelectric energy production, tourist attraction, increasing downstream life dams, etc. Concrete arch dams are one of the most affordable large-scale dam construction methods. Much of the reservoir pressure are supported by the abutments in concrete arch dams. This, in turn, highlights the significance of the stability of concrete arch dam abutment. Traditional method of abutment stability analysis calculates the safety factor of potentially slipping wedges using limit equilibrium. In this method rock wedges are considered rigid with no internal discontinuity. In this research, after a short review of geological structure of Bakhtiary dam site and recognition of potential slip wedge, the load distribution of abutments are estimated using finite element method. In numerical modeling, a least simplification in dam body and abutment geometry is purposed. Infinite elements are used to reduce effect of BOUNDary condition. Finally, stability of Bakhtiary arch dam abutments slip wedges has been approved by the application of UPPER BOUND THEOREM. Despite the limit equilibrium method, the proposed method is based on energy-work balances and is able to account for the effect of internal discontinuities yield.

In this paper, a mathematical model for symmetrical multi-layer sheet rolling, in which the layers are unBOUNDed before rolling, by using the UPPER BOUND method and stream function THEOREM is proposed. Using this model, we can investigate the plastic deformation behavior of sheets at the roll gap during rolling. Effect of various rolling conditions such as initial and final thickness and flow stress of sheets, friction factors, rolling velocity and etc. on the rolling power and force, the thickness reduction of each layer, the relative length of plastic region in each layer and etc. are discussed. The velocity field derived from the newly proposed stream function can automatically satisfy the volume constancy and velocity BOUNDary conditions within the roll gap. The optimized velocity fields are obtained through the minimization of total power, which is expressed by the function of five pseudo-independent parameters, during the plastic deformation. The analytical predictions from the proposed model were compared with the analytical and experimental results of other investigators and a good agreement is shown. Present model is applicable for simulating and online control applications of the rolling process of multilayer sheets.

In this paper, A mathematical model for cold rolling of symmetrical tri-layer strip is proposed using the UPPER BOUND THEOREM to investigate the plastic deformation behavior of the strip at the roll gap. With implementation steps to express in order to analysis of cold rolling process, can calculate and forecast the rolling power, rolling force, average pressure between layers and thickness reduction of each layer. In order to validity the theoretical model ,experiments on sandwich strip rolling are conducted by employing aluminum, mild steel (Al/St/AI)as the layers of the sandwich strips and results is compared together. It is found that all of theoretical predictions are in good agreement with the experimental measurements.

Bearing Capacity of concave slopes is such a three-dimensional problem that two-dimensional assumptions do not result in satisfactory and acceptable results. Presented in this article is an algorithm to calculate the bearing capacity of foundations located on such slopes based on the UPPER BOUND THEOREM of the limit analysis approach.Limit analysis method is based on the extensions of maximum work principle derived by Hill, and was given in the form of THEOREMs by Drucker, Prager and Greenberg. Using this approach, the true solution from a lower BOUND to an UPPER BOUND would be bracketed. Applicability of this THEOREM requires that the soil’s behavior be perfectly plastic and the deformation be governed by the normality rule. Considering the THEOREM of kinematic approach (UPPER-BOUND), the rate of work done by traction and body forces would be less than or equal to the energy dissipation rate in any assumed kinematically admissible failure mechanism. In this algorithm, initially the collapse mechanism is assumed as a set of five or six-face rigid blocks slipping on each other with enrgy dissipation taking place along planar velocity discontinuities. and then, during an optimizing process the minimum result would be considered as the ultimate load of the slope. This approach can be considered as an extension of the procedure proposed by Farzaneh and Askari in 1999. The numerical results indicate that the bearing capacity of a shallow foundation on concave slope is more than that of the same foundation on linear slope and also fewer is the ratio of slope radius to its height, more would be the bearing capacity of the foundation located on it. In addition, unloaded slopes are more stable when they are concave comparing with linear slopes. In bearing capacity, the curvature effect in concave slopes is more considerable for frictional soils than cohesive ones. The results are presented in the form of applicable charts to evaluate the bearing capacity of a foundation located on a concave slope.

In the present paper, the deformation zone of the ECAE (Equal Channel Angular Extrusion) process was investigated using the UPPER BOUND THEOREM. For this purpose, the shape of the streamlines in the deformation zone was assumed to be cubic Bezier curves. Then, the force required for the ECAE process was optimized with regard to the parameters defining the shape of streamlines by the UPPER BOUND THEOREM. By using of these streamlines, equivalent stain induced through one pass of the ECAE was calculated and the shape of the dead metal zone was determined. The influence of different friction conditions on the deformation field was studied. The force applied through the process was determined by optimizing an UPPER BOUND solution and was compared with other works including experimental and theoretical data. Also using this comparison, it was seen that the difference between the theoretical and experimental value for the force was reduced in the present method.

For a graph G = (V, E), a triple Roman dominating function (3RD-function) is a function f: V →, {0,1,2,3,4} having the property that (i) if f(v) = 0 then v must have either one neighbor u with f(u) = 4, or two neighbors u, w with f(u) + f(w) ≥,5 or three neighbors u, w,z with f(u) = f(w) = f(z) = 2, (ii) if f(v) = 1 then v must have one neighbor u with f(u) ≥,3 or two neighbors u, w with f(u) = f(w) = 2, and (iii) if f(v) = 2 then v must have one neighbor u with f(u) ≥,2. The weight of a 3RDF f is the sum f(V ) = ∑,v, v f(v), and the minimum weight of a 3RD-function on G is the triple Roman domination number of G, denoted by ɤ,[3R](G). In this paper, we prove that for any connected graph G of order n with minimum degree at least two, ɤ,[3R](G) ≥,3n/2.

In this paper, new UPPER BOUNDs for the ultraspherical coefficients of differentiable functions are presented. Using partial sums of ultraspherical polynomials, error approximations are presented to estimate differentiable functions. Also, an error estimate of the Gauss-Jacobi quadrature is obtained and we state an UPPER BOUND for Legendre coefficients which is sharper than UPPER BOUNDs proposed so far. Numerical examples are given to assess the efficiency of the presented theoretical results.

Deformation of the material during cyclic expansion extrusion (CEE) is investigated using UPPER-BOUND THEOREM. The analytical approximation of forming loads agrees very well with the FEM results for different amounts of chamber diameter, friction factor and also for lower die angles. However, the difference between analytical and numerical solution increases at higher die angles, which is explained by the formation of dead-metal zones at these angles. The results show that the forming load increases at higher friction coefficients, higher chamber diameters, and lower amounts of corner fillet radius, but for the die angle there is a maximum value of load at about 60o. Forming load is enhanced by the increase of the die chamber diameter and friction factor. Increasing the die chamber diameter causes higher strains and, therefore, higher rate of homogenous work. The load slightly decreased by an increase of the die corner radius because of the lower and more homogeneous strain distribution in the material.

In this paper, an UPPER BOUND analysis of novel backward extrusion was presented. Initially, deformation zone was divided in four separate regions and an admissible velocity field for each was suggested. Then, total power in this process was calculated for each region, therefore, extrusion force was obtained. Moreover, investigation of relevance of extrusion force and process powers (friction, deformation, velocity discontinuity) with process parameters revealed better understanding in load estimation and process efficiency in this method. Finite element analysis by DEFORMTM3D was utilized for validation of UPPER BOUND results. UPPER BOUND analysis showed increasing initial billet diameter enhances extrusion force by nonlinear relation. In addition, big billet size remodels novel backward extrusion to conventional backward extrusion and proves lower requirement extrusion load in novel backward extrusion in comparison with conventional backward extrusion. Moreover, increasing the first region’s thickness in this process diminishes extrusion force by exponential relation and no considerable change in extrusion force can be seen in a particular thickness domain. Investigation of process parameters in power efficiency showed that increasing the extruded part’s diameter created a critical condition in process efficiency because of high friction power. But increasing the thickness enhances power efficiency. Finally, UPPER BOUND analysis results had good agreement with FEA.