The aim of this paper is to present a solution algorithm for determining the frame element crosssection carrying capacity, defined by combined effect of bending moment and axial force. The distributions of stresses and strains inside a cross-section made of linearly hardening material are analysed. General nonlinear stress-strain dependencies are composed. All relations are formed for rectangular cross-section for all possible cases of combinations of axial force and bending moment. To this end, five different stress-strain states are investigated and four limit axial force values are defined in the present research. The nonlinear problem is solved in MATLAB mathematical software environment. Stress-strain states in the cross-sections are investigated in detail and graphically analysed for two numerical experiments.
The problems of optimal design of truss-type structures, aimed at determining the minimal volume (weight) of the structure, while optimizing the bar cross-sections and the truss height, are considered. The considered problem is treated as a nonlinear problem of discrete optimization. In addition to the internal forces of tension or compression, the elements of the truss can have the bending moments. The cross-sections of the bars are designed of the rolled steel profiles. The mathematical models of the problem are developed, taking into account stiffness and stability requirements to structures. Nonlinear discrete optimization problems, formulated in this paper, are solved by the iterative method using the mathematical programming environment MATLAB. The buckling ratios of the bars under compression are adjusted in each iteration. The requirements of cross-section assortment (discretion) are secured using the method of branch and bound.
This paper focuses on the creation and numerical application of physically nonlinear plane steel frames analysis problems. The frames are analysed using finite elements with axial and bending deformations taken into account. Two nonlinear physical models are used and compared -linear hardening and ideal elastic-plastic. In the first model, distributions of plastic deformations along the elements and across the sections are taken into account. The proposed method allows for an exact determination of the stress-strain state of a rectangular section subjected to an arbitrary combination of bending moment and axial force. Development of plastic deformations in time and distribution along the length of elements are determined by dividing the structure (and loading) into the parts (increments) and determining the reduced modulus of elasticity for every part. The plastic hinge concept is used for the analysis based on the ideal elastic-plastic model. The created calculation algorithms have been fully implemented in a computer program. The numerical results of the two problems are presented in detail. Besides the stress-strain analysis, the described examples demonstrate how the accuracy of the results depends on the number of finite elements, on the number of load increments and on the physical material model. COMSOL finite element analysis software was used to compare the presented 1D FEM methodology to the 3D FEM mesh model analysis.
Keywordsreduced modulus of elasticity, distributed plasticity, incremental method, linear hardening, plastic deformations 402| Grigusevičius and Blaževičius Period.
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