DIFFERENTIAL EQUATIONS OF CONTINUUM EQUILIBRIUM FOR A PLANE STRAIN IN CYLINDRICAL COORDINATES AT BILINEAR APPROXIMATION OF CLOSING EQUATIONS

Problems of the formulation of differential equations of equilibrium in terms of displacements for a plane strain of continuous media at bilinear approximation of closing equations are considered leaving out of account geometric nonlinearity in the cylindrical coordinate system. Based on the assumption that the curves of volumetric and shear strain are independent from each other, six main cases of physical dependencies are considered, which are the functions of the relative position of break points on the bilinear curves of the volumetric and shear strain. Obtaining of bilinear physical dependencies is based on the calculation of secant moduli of the volumetric and shear strain. On the first line of the curves, secant moduli are constant for both volumetric and shear strain, while on the second line, the secant modulus of the volumetric strain is a function of the volumetric strain, and the secant modulus of the shear strain is a function of the shear strain intensity. Putting the corresponding bilinear physical equations into differential equations of continuum equilibrium, which disregard geometrical nonlinearity, the resulting differential equations of equilibrium are obtained in terms of displacements for a one-dimensional plane strain of continuum in the cylindrical coordinate system. These equations can be used when determining stress-strain state of continuous media under one-dimensional plane strains with no regard for geometrical nonlinearity, and whose physical relations are approximated by bilinear functions.

Stress-Strain State of Steam Turbine Lock Joint Under Plastic Deformation

The stress-strain state problem for the lock joint of the rotor blades of the first stage of the medium-pressure cylinder under plastic deformation is solved. When solving the problem, the theory of elastic-plastic deformations is used. The problem is solved using two different approaches to specifying plastic deformation curves. The applicability of using a simpler bilinear approximation instead of the classical multilinear one is estimated. Based on the example of solving this problem, the time required to perform the calculation with the use of the bilinear and multilinear approximations is shown. Comparison of the results obtained in the form of the distribution of plastic deformations, equivalent stresses, and contact stresses over support pads made it possible to assess the difference when the two types of approximation are used. The obtained result error value when using the bilinear approximation made it possible to draw conclusions about the applicability of this approach to the processing of plastic deformation curves for solving problems of this kind. The problem is solved using the finite element method. To objectively assess the effect of plastic deformation on the redistribution of loads in the lock joint, a finite element model is used, obtained when solving the problem of the thermally stressed state of the rotor blade lock joint. The distribution of contact stresses in the lock joint is shown. The results are compared with those obtained earlier when solving the problem of thermoelasticity. Significant differences in the level of contact stresses are noted. Results of the computational assessment of the stress-strain state of the lock joint of the rotor blades of the first stage of the medium-pressure cylinder of a steam turbine are presented, which allow characterizing the degree of relaxation and redistribution of stresses in the structure in comparison with the results obtained earlier when solving the problem of thermoelasticity. Conclusions are made about the economic viability of using the calculation methods presented.