Krzysztof Magnucki

The subject of this paper is a simply supported functionally graded beam in three variants: (A) without rigid partitions, (B) with two rigid partitions, and (C) with four rigid partitions. The mechanical properties, specifically Young’s modulus, vary symmetrically in the depth direction of these beams. An analytical model of the beam, considering the individual nonlinear shear deformation of a plane cross-section, has been developed. Two differential equations of equilibrium for the beam, based on the principle of stationary total potential energy, were derived. This system of two equations was approximately solved for each beam variant, and the critical forces were determined with the distinguished shear effect coefficient. Detailed calculations were performed for exemplary beams with selected dimensionless sizes and mechanical properties. The results of these calculations are presented in tables and figures. The main goal of the research is to develop an analytical model of this beam and investigate the impact of rigid partitions placed within the beam on the critical force.

The subject of the paper is the three-span rail (UIC 60) simply supported on four sleepers. The middle span is subject to the concentrated force at half of its length. Analytical study of the bending problem of the three parts of this rail is realized with consideration of the Timoshenko beam theory. Moreover, the bending problem is studied numerically using the finite element method (FEM). Based on the obtained analytical results it is seen that structures like rails should be analysed with shear effect taking into account. The result obtained this way is consistent with the results of numerical investigation. Ignoring the shear effect results in a significant underestimation of the deflection.

The subject of the paper is a homogeneous I-beam with an individual shape web. The beam, simply supported at one end and at the other simply supported with elastic limitation of rotation, is subjected to axial compression by a force F. The analytical model of the beam is developed with...

This study presents an analytical model of a clamped circular sandwich plate with a functionally graded core, developed within the framework of an individual nonlinear shear deformation theory. The mathematical formulation is an original contribution, providing a unified representation of various structural configurations -- including three-layer-like, five-layer-like, homogeneous single-layer, and intermediate forms -- through a single continuous function. The variation of Young's modulus across the core is controlled by three parameters, enabling flexible modeling of material gradation. The governing equations are derived using Hamilton's principle and are solved analytically through an approximate solution method. The proposed model allows investigation of the effects of material property variation and the core-to-plate thickness ratio on the fundamental natural frequency, as well as on the shear effect coefficient and plate mass, yielding general conclusions supported by a coherent theoretical framework.

This paper is devoted to the behavior of an axially compressed two-part beamr esting on elastic foundation. The beam was composed of two parts: rigid homogeneous part and sandwich three-layered part. The analytical model of the beam on the elastic foundation has been prepared. Winkler model has been assumed for the foundation description. Moreover, original function of deflection has been introduced and employed by the authors. The critical loads as a function of geometric and mechanical properties of the two-part beam and the parameters of Winkler's foundation were calculated. In addition, numerical analysis has been performed. Sample analytical and numerical calculations have been carried out, demonstrating good compatibility between the results attained with both models. The numerical analysis included the critical loads calculations for the beam with and without the elastic foundation. In both cases, the difference between analytical and numerical values of the critical loads did not exceed 8%.

The work concerns a sandwich beam with an individual core structure giving a shear effect exactly in accordance to the “broken line” theory. According to the general theoretical scheme of a planar cross section deformation, longitudinal displacements, strains and stresses are analytically formulated. Moreover, the unknown deformation function of the core, with consideration of the classical shear stress formula, is analytically derived. Based on the condition regarding the linear deformation function of the core, according to the “broken line” theory, the differential equation is obtained. The solution of this equation is the sought individual core structure. Then, the bending problem of a clamped sandwich beam under three-point bending is studied.

The subject of the paper is a simply supported standard wide-flange H-beam. Cross sections of this beam is analytically described as a three-layer structure. The shear effect in its succesive layers is taking into account with consideration of the classical shear stress formula called Zhuravsky shear stress. Based on Hamilton’s principle, two differential equations of motion are obtained. These equations are analytically solved and the fundamental natural frequency of flexural vibration for this beam is derived. Examplary calculations are carried out for selected five I-beams.

The subject of the work is a five-layered composite beam with clamped ends subjected to a uniformly distributed load along its length. Two analytical models of this beam are developed with consideration of the shear effect. The first model is formulated on the basis of the classical zig-zag theory. Whereas, the second model is developed using an individual nonlinear shear deformation theory with consideration of the classical shear stress formula (called Zhuravsky shear stress). The system of two differential equations of equilibrium for each beam model is obtained based on the principle of stationary total potential energy. These systems of equations are exactly analytically solved. The shear effect function and the maximum deflection are determined for each of these two beam models. Detailed calculations are carried out for exemplary beams of selected dimensionless sizes and material constants. The main goal of the research is to develop two analytical models of this beam, determine the shear effect function, the shear coefficient, and the maximum deflection in the elastic range for each model, as well as to perform a comparative analysis.

This paper is devoted to a thin-walled sandwich plate with an individual functionally graded core. The nonlinear shear deformation theory of a straight normal line is applied. A system of three differential equations of equilibrium of this plate is obtained, based on the principle of stationary...