The study presents the results of theoretical investigations into lateral torsional buckling (LTB) of bi-symmetric I-beams, elastically restrained against warping at supports. Beam loading schemes commonly used in practice are taken into account. The whole range of stiffness of the support joints, from free warping to warping fully restrained, is considered. To determine the critical moment, the energy method is used. The function of the beam twist angle is described with power polynomials that have simple physical interpretation. Computer programs written in symbolic language for numerical analysis are developed. General approximation formulas are devised. Detailed calculations are performed for beams with end-plate joints. Critical moments determined with programs and approximation formulas are compared with the results obtained by other researchers and with those produced by FEM. Very good accuracy of results is obtained.
This paper presents the resolution of the optimal reactive power dispatch (ORPD) problem and the control of voltages in an electrical energy system by using a hybrid algorithm based on the particle swarmoptimization (PSO) method and interior point method (IPM). The IPM is based on the logarithmic barrier (LB-IPM) technique while respecting the non-linear equality and inequality constraints. The particle swarmoptimization-logarithmic barrier-interior point method (PSO-LB-IPM) is used to adjust the control variables, namely the reactive powers, the generator voltages and the load controllers of the transformers, in order to ensure convergence towards a better solution with the probability of reaching the global optimum. The proposed method was first tested and validated on a two-variable mathematical function using MATLAB as a calculation and execution tool, and then it is applied to the ORPD problem to minimize the total active losses in an electrical energy network. To validate the method a testwas carried out on the IEEE electrical energy network of 57 buses.