PSI - Issue 44

Alberto Castellani et al. / Procedia Structural Integrity 44 (2023) 19–26 Author name / Structural Integrity Procedia 00 (2022) 000–000

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application dealt with in a previous study, Paolucci et Al (2009), has shown that the high frequency content of the rotation motion has little impact on civil engineering structures. On the other hand, excitation provided by rotation in the frequency range 1  5 cps, may be meaningful for tall buildings, chimneys, and bridges. Underground tubes are sensitive to excitations at quasi-static frequency. For this application, strains have been computed by Smerzini, and Paolucci (2008). 3. Representation of the ground motion In a Cartesian space, let x, y, and z be the orientation axes, the latter being vertical. Along these axes, the acceleration components during a ground motion are u x (t), u y (t), and u z (t). In the following, the attention is focused on the vertical u z (t) acceleration, but the elaboration to follow can be applied to each of the three space components. The set of records will be represented through an interpolation function, � � t, , � � � � � t � � � � � t � � � � � t � � � � � � t � � � � � t � � � � � � t � � � � � t � � � � � � t � � (1) where d 1 (t), d 2 (t) .. are unknown interpolation coefficients. Continuum mechanics defines rotations of the elementary volume around the Y-axis, ψ � � � � � � � �  � � � �  � (2) and the shear strain as γ �� � � � � �  � �  � (3) Instruments are at the soil surface, whereby definition,  xz = 0. Thus: ψ � � �  or �  (4) and, analogously ψ � � �  or �  (5) Rotation in x-z plane and rotation in the y-z plane are analyzed independently. On the basis of these evaluations one cannot evaluate the max rotation component. The above evaluations should be referred to as rotations in a two dimensions’ representation. See the previously mentioned, Flugge 1962 Handbook, chapter 32. 4. Mathematical procedure The vertical acceleration is represented through the interpolation function, eq. (1). In matrix arrangement, let A be the matrix of the coordinates of the recorders, and u z (t) be the vector of the recorded acceleration:

� 1 � � �� �� � � �� � � �� 1 � � �� �� � � �� � � �� … … … … … … … … 1 �� �� �� � �� � �� �� �� � �� �� �� � �

(6)