PSI - Issue 73

Lenka Koubova et al. / Procedia Structural Integrity 73 (2025) 66–72 Lenka Koubova / Structural Integrity Procedia 00 (2025) 000–000

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Like the stiffness matrix, the mass matrix [ m ] is determined by localizing the global mass matrices of the elements [ m i ] into which the structure is divided. The global mass matrix [ m i ] of the i -th element is obtained by the transformation of the local mass matrix [ ∗ ] in Eq. (3). [ ∗ ] = ∙ ∙ ∆ ∙ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 1 3 0 0 1 6 0 0 0 1 3 3 5 − 11∙∆ 210 0 7 9 0 13∙∆ 420 0 − 11∙∆ 210 ∆ 2 105 0 − 13∙∆ 420 − ∆ 2 140 1 6 0 0 1 3 0 0 0 7 9 0 − 13∙∆ 420 0 1 3 3 5 11∙∆ 210 0 13∙∆ 420 − ∆ 2 140 0 11∙∆ 210 ∆ 2 105 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ (3) In Eqs. (2) and (3), A is the cross-sectional area, I is the moment of inertia, ∆ s i is the length of the element i , E is the elastic modulus of the material, and ρ is the density of the material. It is assumed that the construction vibrates in harmonic motion. Then Eq. (1) can be modified into Eq. (4). � [ ] − 2 ∙ [ ] �� ( ) � = {0} (4) We are looking for the zero determinant of the matrix � [ ] − 2 ∙ [ ] � ; this gives us the natural frequencies ω n (Koubova (2024)). 3. Natural frequencies and mode shapes of MRF structures The procedure presented in the previous chapter was used to determine the natural frequencies and mode shapes of the MRF structures. 3.1. Parameters of the solved MRF structures This paper studies the dependence of the natural frequencies of MRF structures that have from one to five floors. Furthermore, there can be from two to five columns on the floor. The column span is w = 4 m, and the height of one floor is h = 3 m. The cross-section is chosen to be square with a side of 0.4 m. The material is concrete with a modulus of elasticity E = 33 GPa and a density of ρ = 2400 kg/m 3 . In the solution, each member (column or beam) is divided into two parts. There are three degrees of freedom (DoF) at each joint and in the middle of each member. The columns are fixed at the bottom. So, for example, the frame structure in Fig. 1 is designed as a system with 210 degrees of freedom.

Fig. 1. Example of a solved MRF structure (5 floors, 5 columns per floor).