PSI - Issue 6

V.A. Meleshko et al. / Procedia Structural Integrity 6 (2017) 140–145 Meleshko V. А ., Rutman Y.L. / StructuralIntegrity Procedia 00 (2017) 000–000

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A difference of generalized methods from the finite elements method (FEM) is in exemption of finite elements during generation of algebraic equations system and their replacement with the sections where distributed stiffness parameters are integrated (Fig. 1). As a result, a number of equations in the system will correspond to the number of static (or kinematic) indetermination of the framed structure. Herewith, if the framed structure is statically determined then it is not required to solve the equations system. Here any distribution of fluid parameters by rod length is taken into account by means of the tangent stiffness matrix and the generalized Mohr formula.

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Fig.1. Replacement of finite elements with sections where there are tangent stiffness. To implement the proposed generalization, the generalized Morh formula and the tangent stiffness matrix were developed by Meleshko V. A. and Rutman Y. L. (2015); Meleshko V. A. and Rutman Y. L. (2017) . This matrix was obtained as an integral characteristic of intensely deformed state of all points of rod cross-section. The specified articles present that, when calculating the statically indeterminate systems, the generalized flexibility method consists of the following steps: -determination of stiffness matrix [R e ( T )] or rod element admittance matrix by means of the generalized Mohr formula; -creation of equilibrium equations or strains compatibility equations in the form of displacement method { P }=[ R ]{  } or the flexibility method {  }=[  ]{ P }; -determination of stresses in cross-section through the rod curvature in the cross-section {  }=[ A ][ S ]{  }; -determination of rod section tangent stiffness matrix for elestoplastic deformation [ T ]=  [ L ][ A ][ S ] dF , where ψ – column matrix of kinematic parameters (angular rates and sections curvatures); σ – column matrix of stresses in a section point; L – matrix depending on section point coordinates, and coupling between stresses and elementary internal forces in the cross-sections. The matrix [ S ] describes the coupling between section kinematic parameters and deformation rates. Components [ S ] depend on the coordinates of the considered rod section point. Matrix [ A ] components depend on stresses in a section point at each time step. In case of elastoplastic deformation the matrix [ A ] corresponds to the differential analogue of the Hooke’s law with pseudo elastic factors depending on stress mode in the point. When creating computational algorithms at each time step, the above formulas are written as incremental ratios. 2 Mathematical Scheme for Determination of Elastoplastic Deformations in the Plane Framed Structure For plane framed structures, where a bend of rods composing the structure is taken into account only, tangent stiffness determination can be significantly simplified. For simplification of the procedure to calculate tangent stiffness, the integral function of section state is used, which was derived by Kovaleva , N.V ., Skvortzov V.R., Rutman Y.L. (2007). The integral state function for the rectangular cross-section was obtained in Kovaleva N.V ., Skvortzov V.R., Rutman Y.L. (2007 ), and for the round one - in Ostrovskaya N.V. (2015). When the effect of transversal force is not taken into account, then bending methods are proportional to the integral state function. For example, for the rectangular cross-section, this function is as follows ( ), 2 2     bh M s (1) s – deformation corresponding to the yield strength,  s – yield strength,  – rod curvature in the cross-section under consideration, b , h – rectangular cross-section parameters. -determination of reactions in rods nodes { P e }=[ R e ]{  e }; -determination of internal forces in rods M (  ), Q (  ); where  (  ) – integral state function, , 2 s h     