Free flexural vibration analysis of one-way stiffened plates by the free interface modal synthesis method:' Discussion MOHAMED ABDEL-MOOTY Institute for Research in Constr~iction, National Research Council Canada, Ottawa, ON KIA OR6, Canada Received April 19, 1993 Manuscript accepted July 7, 1993 Can. J. Civ. Eng. 21, 167-168 (1994) The authors have made a useful contribution in pointing out the first and second modes (M = 2) in the bottom half of the importance of including shear deformation and rotary Table 1 is misleading. That is because the vibrational modes inertia in the free vibration analysis of one-way stiffened for such uniform, symmetric floor would be either symmetric plates with ribs having high ratios of flexural to shear rigidity. or antisymmetric. Since the first five modes are symmetric This is of practical importance when designing wooden (with single curvature) in the along-rib direction, the second floors against uncomfortable vibrations. The authors used beam mode in that direction, which is antisymmetric, would an energy approach to the problem where they expanded have no contribution in those first five modes. That explained the dynamic response of the plate and ribs in terms of a complete set of functions that satisfy all boundary conditions. The theoretical proof of convergence for such method can be found in Leipholz (1977) which confirms the numerical convergence presented in the paper. The authors' method is easy to apply and can yield approximate closed-form mathematical expressions for the vibrational modes, which gives valuable insight into the system behaviour. Although not mentioned in the paper, variations and singularities in stiffness distribution, such as those due to partial fixities near edges, and in mass distribution, such as those due to concentrated masses, can be easily considered by adding appropriate terms to the kinetic and potential energy expres- sions. This is of practical importance for lightweight wooden floor vibrations, since the added occupancy loads are com- parable to the mass of the floor and this may substantially alter the natural frequencies of the floor. The writer would like to discuss some of the assumptions and approximation made in the paper and to point out sources of error that might be responsible for the discrepancies between the numerical and experimental results. In the paper, the transverse motions of the plate and rib substructures were expanded in terms of beam modes of vibrations. That is a valid approximation, since those modes constitute a complete set of functions that satisfy at least the geometric boundary conditions. Such expansion should yield upper bound approximation of the natural frequencies that reach the exact frequencies as the number of terms in the expansion approaches infinity. That convergence was evident from Table 1. The authors claimed, based on the result of the why the numbers in the top and bottom halves of the table were exactly the same. The inclusion of other odd modes would generally have improved the approximation The second terms in [4] and [5] representing the kinetic and potential energies of the rib substructures were derived based on the elementary Bernoulli-Euler beam theory of flexural vibrations. Then the authors replaced the rib natural fre- quencies, oi, by those of Timoshenko beam in the potential energy expression as an "explicit" inclusion of the effect of transverse shear deformation and rotary inertia of the ribs. That is an approximation that needs verification. Alternatively, in order to properly account for the effect of shear deformation and rotary inertia, the rib kinetic energy, T,, and potential energy, U,, should read (Langhaar 1962) Table 1, that only one mode in the along-rib direction and five modes in the across-rib direction are required to get where p is the shear distortion (Clough and Penzien 1975) the first five modes as opposed to 71 degrees of freedom ?Ind w!(x. '1. j = ?Ire given the l31. The using the finite strip model (Filiatrault et al. 1990) as an rotary inertia effect is accounted for in the second term of indication of the efficiency of their model over other numer- [Dl] while the shear deformation effect is considered in ical techniques. This may be true for the one-way uniform [D21. The composite action in bending of ribs semirigidly floor, simply supported on two sides, with no concentrated connected the plate was in a way mass analyzed in ~ ~ b l ~ 1. ~h~ authors did not mention the that avoids additional degrees of freedom through the use number of terms required to yield the vibrational frequencies the effective bending 19i. with reasonable accuracy for more general nonuniform floors was hken of shear deformation and with different boundary conditions and supporting concen- rotary inertia for the composite T-section of the rib and the trated masses. Furthermore, the writer sees that considering covering plate strip. The model used in the paper tends to underestimate the 'paper by Ian Smith, Lin J. Hu, and Allison B. Schriver. 1993. natural frequencies of vibration for the stiffened plate as Canadian Journal of Civil Engineering, 20(6): 885-894. shown in Table 4, although as stated earlier it is supposed to Prinled ~n C;!nndo / Imprimb ;hu Canada Can. J. Civ. 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