PSI - Issue 11

R. Capozucca et al. / Procedia Structural Integrity 11 (2018) 402–409 R. Capozucca, E. Magagnini, M.V. Vecchietti / Structural Integrity Procedia 00 (2018) 000–000

405

4

� � �� � � + � ∙

(3)

The differential Eq. (3) is of fourth order so the general solution is of the type: � � = � sinh � cosh � sinh � cosh (4) Eq. (4) has five unknown parameters � ( i =1,2,3,4), and an eigenvalue λ . To reduce the errors of experimental measures, the boundary conditions used in the vibration tests were free-free ends; so that in the case of free ends for beam, applying the following boundary conditions: �� | ��� =0 ; ��� | ��� =0 ; �� | ��� =0 ; ��� | ��� (5) an algebraic linear system in the unknown constants � may be obtained. A non-trivial solution exists when the determinant becomes: cos (6) The eigenvalue for a free end beam at the r mode � � may be correlated to the value � for a simply supported beam in the following way: � � = ξ ∙ � with ξ=coefficient that depends on the different r mode of vibration, equal to 1.506, 1.25, 1.167 and 1.125, respectively, for the first four modes. In the case of a simply supported, the expression of a circular natural frequency is the following: � � � � � � � ∙ � � � � � (7) By considering only the first three values of the eigenvalue � � and substituting the parameters of the experimental prototype (Tab. 1), the first three theoretical natural frequencies were calculated (Tab. 2). These frequencies were compared with the experimental frequencies recorded in the undamaged condition D 0 . 2.2. Dynamic analysis of undamaged beam by FEM The linear elastic FE modeling of beam B2, tested with free-free ends condition, was carried out with 3D ANSYS code (Fig. 3). Eight-node solid brick elements - Solid65 - were used to model the concrete. The solid element has eight nodes with three degrees of freedom at each node with capability of plastic deformation, cracking in three orthogonal directions and crushing. Generally, three techniques to model steel reinforcement in concrete by finite element are adopted: the discrete model, the embedded model, and the smeared model. The reinforcement in the discrete model uses beam elements that are connected to concrete mesh nodes. The embedded model overcomes the concrete mesh restrictions although increasing the number of nodes and degrees freedom. The smeared model assumes that reinforcement is spread uniformly throughout the concrete elements in a defined region. This approach is used for large-scale models where reinforcement does not significantly contribute to the overall response of the structure. In this study, the smeared model was used to model steel reinforcement. Furthermore, the Solid185 element was used for epoxy adhesive. Solid185 is used for the three-dimensional modeling of solid structures. The element is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. The Beam188 element was assumed to model the CFRP circular rods. Beam188 is a linear beam element in 3D with six degrees of freedom at each node. Element Combin 14 was inserted in the numerical modelling in addition to the aforementioned elements in order to model the beams’ suspension springs simulating the same free beam conditions in vibration. The element is defined by two nodes, a spring constant k and damping coefficients. The first three bending vibration modes for beam B2 are represented in Figure 3. The theoretical and experimental frequency values obtained for undamaged beam are shown and compared in Table 2. The frequency values obtained for damage degree D 0 for the hinge-hinge beam, as shown in Figure 4, are contained in the same table.