Transformational deformation models of continuous thin‐walled structural elements with support elements of finite sizes: Theoretical foundations, computational, and physical experiments

Abstract A method of experimental study of forced bending vibrations of unfixed spans of continuous thin‐walled structural element fixed on rigid support elements of finite dimensions has been developed using instrumental means for registration of vibration acceleration amplitudes. The technique is realized on beams with different types of their fixing under the loading of one of the beam section with an external force that varies in time according to a harmonic law. Experimental dependences of amplitude values of vibration accelerations in specific points of certain spans of the beam on the frequency of external load have been obtained. These results indicate the passage of vibrations through the fixed sections due to the transformation of bending vibration modes of the loaded section into longitudinal transverse‐shift vibrations of the fixed sections during the transition across the boundary from the unfixed section to the fixed one (from the fixed to the unfixed section). For the theoretical study of the described phenomenon a transformational model of the flat beam deformation is constructed on the basis of the refined Timoshenko model with additional account of the deformability of the section with fixation to a rigid support element of finite length on one of its face surfaces. The corresponding equations of motion of the unfixed and fixed sections are derived. The corresponding kinematic and force conditions of their conjugation are formulated. On their basis, analytical solutions of two specific problems are found in the linear approximation. One‐dimensional finite elements for modeling the dynamic response of continuous beam of the considered class are also constructed. Numerical experiments have been carried out for a flat beam made of aluminum alloys D16AT, AMg‐6 and unidirectional fiber reinforced plastic. The presence of a significant transformation of the stress‐strain state parameters of the considered beams at the transition across the boundary from unfixed to fixed sections and a significant increase in the level of transverse tangential stresses on the fixed section of the beam in the vicinity of the interface of the unfixed section with the fixed one have been revealed.