Dynamic analysis of sandwich beams

Free Vibration Analysis of Curved Sandwich Beams: A Dynamic Finite Element

“…For each of the displacements, a Hermite polynomial shape function similar to expressions (10) and (11) was used to define the approximation space for both the field variables and weighting -or test -functions. FEM DFE 10 Elements Ahmed, 1971Ahmed, , 1972 3 Table 2 above, shows that for the first two natural frequencies, the DFE results are slightly larger than those obtained from both FEM formulations, but for the 3 rd -5 th frequencies, the DFE values are smaller than those found by the 3-DOF FEM formulation but larger than the 4-DOF FEM formulation. For 20-element FEM models, the largest difference is 1.4% seen between the 3-DOF and 4-DOF FEM formulations (in the 5 th natural frequencies), but when the number of elements is increased to 40, the difference reduces to 0.36%, which is still the largest when comparing all three models.…”

“…The results of the DFE, and 3-and 4-DOF/node FEM formulations along with those reported by Ahmed (1971Ahmed ( ,1972 are listed in Table 2 below. For the first set of results from Ahmed (1971), shown in the second column of Table 2 below, each node has 4-DOFs.…”

“…The 10-element FEM model developed employs similar polynomial Hermite shape functions such as those found in equations (10) and (11) for the approximation space of the field variables v, v', w and w', respectively. The results from Ahmed (1972), shown in the third column of Table 2, are from a 10-element FEM model where each node has 6-DOFs. The DOFs, in this case, are associated with circumferential displacement (v and v'), radial displacement (w and w') and transverse shear in the x-y plane (φ and φ', which none of the derived models takes into account).…”

“…The simplest sandwich beam model utilizes Euler-Bernoulli theory for the face layers and only allows the core to deform only in shear. This assumption has been widely used in several DSM and FEM studies such as those by Banerjee (2003), Ahmed (1971Ahmed ( ,1972, Mead and Markus (1968), Fasana and Marchesiello (2001), Baber et al (1998), and in earlier papers by the authors; see e.g., Adique & Hashemi (2007), and . In more recent publications, Banerjee derived two new DSM models which exploit more complex displacement fields.…”

“…In the this Chapter, we derive a DFE formulation for the free vibration analysis of curved sandwich beams and test it against FEM and DSM to show that DFE is another viable tool for structural vibration analysis. The face layers are assumed to behave according to EulerBernoulli theory and the core deforms in shear only, as was also studied by Ahmed (1971Ahmed ( ,1972. The authors have previously developed DFE models for two straight, 3-layered, sandwich beam configurations; a symmetric sandwich beam, where the face layers are assumed to follow Euler-Bernoulli theory and core is allowed to deform in shear only Hashemi, 2007, andHashemi &, and a more general nonsymmetric model, where the core layer of the beam behaves according to Timoshenko theory while the faces adhere to Rayleigh beam theory (Adique & Hashemi, 2008).…”