A real polynomial P in one real variable is hyperbolic if its roots are all real. The composition of Schur-Szegö of the polynomials P = n j=0 C j n ajx j andIn the present paper we show how for n = 2 and when P and Q are real or hyperbolic the roots of P * Q depend on the roots or the coefficients of P and Q. We consider also the case when n ≥ 2 is arbitrary and P and Q are of the form (x−1) n−1 (x+b). This case is interesting in the context of the possibility to present every polynomial having one of its roots at (−1) as a composition of n − 1 polynomials of the form (x + 1) n−1 (x + b).

The recent failure of buildings because of punching shear has alerted researchers to assess the reliability of the punching shear design models. However, most of the current research studies focus on model uncertainty compared to experimentally measured strength, while very limited studies consider the variability of the basic variables included in the model and the experimental measurements. This paper discusses the reliability of FRP-reinforced concrete slabs’ existing punching shear models. First, more than 180 specimens were gathered. Second, available design codes and simplified models were selected and used in the calculation. Third, several reliability methods were conducted; therefore, three methods were implemented, including the mean-value first-order second moment (MVFOSM) method, the first-order second moment (FOSM) method, and the second-order reliability method (SORM). A comparison between the three methods showed that the reliability index calculated using the FOSM is quite similar to that using SORM. However, FOSM is simpler than SORM. Finally, the reliability and sensitivity of the existing strength models were assessed. At the same design point, the reliability index varied significantly. For example, the most reliable was the JSCE, with a reliability index value of 4.78, while the Elgendy-a was the least reliable, with a reliability index of 1.03. The model accuracy is the most significant parameter compared to other parameters, where the sensitivity factor varied between 67% and 80%. On the other hand, the column dimension and flexure reinforcement are the least significant parameters compared to other parameters where the sensitivity factor was 0.4% and 0.3%, respectively.

Turbulent flow in fluid dynamics is used to describe fluid motion characterized by unpredictable fluctuations in pressure and flow velocity. Turbulence is generated when an area of fluid flow has an excessive amount of kinetic energy, which exceeds the damping impact of the viscosity of a fluid. The primary goal of turbulence modeling is to establish a mathematical model to predict time-averaged velocity, turbulence kinetic energy, and pressure rather than compute the fully turbulent flow pattern as a function of time, as is done in large eddy simulation (LES) and Reynolds-averaged Navier–Stokes simulations. Computationally solving the Navier–Stokes equation of motion to simulate turbulent flows necessitates resolving a wide range of length scales and times, all of which impact the flow field. The current study is concerned with the investigation of turbulence kinetic energy through the use of an LES model. The kinetic energy caused by turbulence is analyzed at the outlet and inlet. Along with the pressure, the fluctuations, as well as the mean velocity at the outlet and inlet, are examined. The C++-based programming is done to compute the turbulent flow in OpenFOAM. The computations made in OpenFOAM and Python show great agreement. For a better understanding of readers, graphs and animation are given.

Machine foundations are subjected to significant dynamic impacts. These impacts could spread to the surrounding regions, affecting workers, sensitive equipment in the same institution, or nearby areas. This study analyzes the response of machine-supporting foundations to harmonic and explosive loads under operational conditions and provides closed-form solutions for predicting responses in terms of displacement, velocity, and acceleration time-histories to two common types of blast loads: a more accurate typical profile and simplified triangular profile. The soil-machine foundation system is regarded as a single-degree-of-freedom (SDOF) system that exhibits elastic–perfectly flexible resistance behavior. For the analysis of the SDOF system, two cases are considered: one assumes that the supporting soil keeps elastic during the explosion, and the peak displacement is less than the elastic one, while the other assumes that the blast occurs in an elastic state, and the peak displacement occurs in a plastic state. By using the closed-form analytical solutions, a detailed parametric analysis is carried out to evaluate the impacts of significant soil-foundation system characteristics such as mass, stiffness, and damping ratio on the response-time history of machine foundations. The findings are compared to those reported in the literature, and relevant conclusions are derived. Obtained results demonstrated that, despite its simplicity and usage of only positive phase to simulate blast loads, the simplified model’s response behavior differs significantly from the typical one. Furthermore, the derived solutions are utilized to design the foundations supporting vibrating machines for both harmonic and blast loads in a variety of conceivable scenarios depending on the blast magnitude.

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