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Free flexural vibration analysis of one-way stiffened plates by the free interface modal synthesis method
IAN SMITH Department of Forest Engineering and Wood Science and Technology Centre, University of New Brunswick,
Box 4400, Fredericton, NB E3B 5A3, Canada
AND
LIN J. HU AND ALLISON B. SCHRIVER Department of Civil Engineering, University of New Brunswick, Box 4400, Fredericton, NB E3B 5A3, Canada
Received August 18, 1992
Revised manuscript accepted March 1, 1993
A numerical model is presented for predicting the natural frequencies of one-way stiffened plates with ribs having high ratios of flexural to shear rigidity. The model is based on the free interface modal synthesis method. Experimental validation using floors with wood I-joists and wood-based sheathing showed that the model has good numerical accuracy in the predictions of natural frequencies and mode shapes if analyses include shear deformation and rotatory inertia effects in ribs. Neglect of these effects can lead to large errors in the predicted natural frequencies for plates with ribs having high ratios of flexural to shear rigidity. Large errors can also be encountered in natural frequency prediction for plates with fairly low ratios of flexural to shear rigidity. This occurs with mode shapes that have multiple curvature along ribs if shear deformation and rotatory inertia effects are neglected.
Key words: free flexural vibration, natural frequencies, ribbed plates, flexural rigidity, shear rigidity, modal synthesis.
Cet article presente un modele numerique pour prevoir les frequences naturelles de plaques renforcees avec nervures dont les rapports de la rigidite en flexion la rigidite en cisaillement sont eleves. Le modttle est base sur la mtthode de synthttse modale a interface libre. Des experiences de validation a l'aide de planchers constitues de solives de bois en I et d'un revCtement a base de bois ont demontre que le modele avait une bonne precision numkrique lorsqu'il s'agit de prevoir les frequences naturelles et les formes de mode, dans la mesure ou les analyses integrent la deformation en cisaillement et les effets de l'inertie rotatoire dans les nervures. L'omission de ces effets peut entrainer de graves erreurs dans la prevision des frequences naturelles de plaques avec nervures dont les rapports de la rigidite en flexion a la rigidite en cisaillement sont Cleves. De graves erreurs de prevision peuvent egalement 2tre commises dans le cas des plaques dont les rapports de la rigidite en flexion a la rigidite en cisaillement sont relativement faibles. Cela se produit pour les formes de mode ayant des courbures multiples le long des nervures si la deformation en cisaillement et les effets d'inertie rotatoire ne sont pas pris en consideration.
Mots clPs : vibrations propres en flexion, frequences naturelles, plaques nervurees, rigidite en flexion, rigidit6 en cisaillement, synthltse modale.
[Traduit par la redaction]
Can. J . Civ. Eng. 20, 885-894 (1993)
Introduction The free flexural vibration behaviour of plates with ribs
having a high ratio of flexural to shear rigidity is not modelled well by current methods found in the literature. According to the classical one-dimensional theory of elastic beams, the governing equation for free vibration of an Euler beam is
The wave velocity of straight-crested waves is inversely proportional to the wavelength. The exact solution of [ l ] confirms its validity only for waves that are long in com- parison with the depth of the beam. Hence the Euler beam equation cannot be expected to give good results for deep beams, or for the natural frequencies of high-order modes of vibration of shallow beams (Timoshenko and Young 1964). Equation [I] is only valid for the natural frequencies of low-order modes of vibration of beams with low ratios
NOTE: Written discussion of this paper is welcomed and will be received by the Editor until April 30, 1994 (address inside front cover). Printed in Canada / In i~r in~c nu Canada
of flexural to shear rigidity ( E I , / d G,). To remove this limitation Timoshenko and Young (1964) included the effect of transverse shear deformation and rotatory inertia in the one-dimensional vibration theory. Chui and Smith (1990) provided solutions for free vibration response of deep beams with generalized end conditions, i.e., semirigid translational and rotational end springs, as an extension of Timoshenko and Young's work. It is logical to deduce that the transverse shear deformation and rotatory inertia in ribs should be included in the flexural vibration equations of plates stiffened with ribs having high ratios of flexural to shear rigidity. This should lead to reliable predictions of natural frequencies and mode shapes for a broad range of plates.
A number of numerical models for one-way stiffened plates have been developed to simulate their free flexural vibration behaviour (Hoppmann and Magness 1957; Huffington and Hoppmann 1958; Lenzen and Murray 1969; Long 1968, 197 1 ; Kirk 1970; Dowel1 1972; Aksu 1974; Nair and Rao 1984; Mukherjee and Mukhopadhyay 1986; Chui 1987; Smith and Chui 1988; Filiatrault et al. 1990; Kuravsky and Arnautov 1991). Nevertheless, none of the models include the effect of the transverse shear deformation and rotatory inertia in ribs. Their applications are limited to plates with ribs having low ratios of flexural to shear rigidity.
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FIG. 1. Geometric notation of the ribbed plate model.
A relatively sophisticated model is required to study the free flexural vibration behaviour for a broad class of stiffened plates. It seems desirable that any model is com- putationally efficient and easy to use. The work reported here is intended to fulfill these requirements by including the effect of transverse shear deformation and rotatory inertia of ribs in a numerical ribbed plate model. The free interface modal synthesis method proposed by Hou (1969) is used, because of the computational efficiency attainable with that method (Dowel1 1972).
Theory For a one-way stiffened plate such as that shown in Fig. 1,
it is assumed that transverse displacement, W, governs the displacement of the system; small deflection theory is valid; and the ribs may be rigidly or semirigidly attached to the plate.
The system is divided into disconnected plate and rib sub- structures. The transverse motions of the plate and rib sub- structures may be approximately expanded in terms of beam modes, which must satisfy the boundary conditions for the ribbed plate as a whole. Functions suitable for common combinations of boundary conditions can be obtained from references such as Cheung and Cheung (1971). Let
( j th rib; j = 1, 2, ..., J )
Equations [2] and [3] establish transformations from the physical (x-y-z, in Fig. 1) to the modal coordinate system (qk,n(t), qjk(t)).
If an orthogonal set of beam functions are selected in [2] and [3], the cross products vanish. Then the kinetic and potential energies for a ribbed plate can be written explicitly in terms of modal coordinates as follows: Kinetic energy:
Potential energy:
The modal characteristics, w ,,,,, Mu,,, 4, and M$, are obtained by modal analyses of the separated plate and rib substructures. Two-dimensional theory of flexural vibration of an orthotropic elastic plate is used to determine a plate's angular natural frequencies, w,,, values. One-dimensional theory of flexural vibration of a Timoshenko beam, equa- tion [6] (Timoshenko and Young 1964), is used to determine a rib's angular natural frequencies, dk values. This allows the effects of transverse shear deformation and rotatory inertia of ribs to be included explicitly.
Constraint equations are applied to synthesize the plate and rib substructures into the ribbed plate, and to form system stiffness and mass matrices. The constraint equations express the requirements of displacement compatibility and force equilibrium at the plate-to-rib connections. The trans- verse displacement compatibility requirement in the physical coordinate system is
As the equation is a continuous function, it also implicitly expresses the compatibilities in slopes and curvatures of the plate and rib substructures. Equation [7] is transformed into the modal coordinate system by substituting [2] and [3] in that equation to yield
To satisfy the requirements of horizontal displacement compatibility and force equilibrium, horizontal movements would normally be introduced into the system vibration equations. This would account for semirigid plate-to-rib connections. Here, however, an alternative approach is adopted. Effective flexural rigidity of a rib is used to satisfy horizontal displacement compatibility and force equilibrium requirements between the plate and rib components. The effective rigidity, ElT, of a rib semirigidly connected to a covering strip of the plate can be obtained, in an approxi- mate manner, from the following equation.
EI, + EI, [9] EIT = - EI,
1 - z2/[(a + /3)(EIr + El,)]
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This equation is a special case of the Fourier series solution given by Smith (1980), and is exact when the mode shape of the rib is sinusoidal. For other mode shapes the expression is approximate. The EIp term avoids a double accounting of local flexural rigidity of the covering strip of the plate in the system stiffness.
Connection flexibility is introduced in the term a, which is inversely proportional to kc, the force per unit displace- ment per unit length of the plate-to-rib connection. For a rigid connection, kc tends to infinity, and [9] reduces to the standard solution for full composite action. EIT is sub- stituted for EI, in [6] if there is a plate-to-rib connection capable of transferring shear forces.
Applying Lagrange's equations in conjunction with the constraint equations, the undamped free vibration equations can be expressed in matrix form in terms of the independent modal coordinates q;,,,:
Appendix 2 gives more details on the derivation of [lo]. A computer program has been developed to perform the analysis, under selected combinations of boundary condi- tions at the edges of a ribbed plate.
The above derivation is not restricted to a particular ribbed plate geometry, i.e., not just to a system such as that shown in Fig. 1. The theory is applicable to plates with non- rectangular plans and plates with openings. This requires appropriate substructuring of the system and selecting suitable mode shape functions for the substructures. An example is a ribbed plate with the plan defined in Fig. 2. A relatively simple solution can be obtained for such a plate by dividing the system into substructures having rectangular plans. The complete system's kinetic and potential energies may be generated from the modal properties of the sub- plates. By requiring continuity of displacements, slopes, and curvatures at boundaries between the subplates, the com- patibility equations may be written as follows:
For x2 I x I a,
[I la] Wi(x, Y I , 1) = Wi(x, YI, 1)
For 0 I x I x l ,
For x2 5 x I a,
For x = xl and y = y2,
[ I ld ] W3(~1, y2, 1) = Wdxi, Y2, 1)
In the above a single prime applied to W signifies slope,
FIG. 2. An example of a ribbed plate with a nonrectangular plan and an opening.
and a double prime signifies curvature. Compatibility requirements for slopes and curvatures would be enforced relative to both the x- and y-axis directions.
Other arrangements may be dealt with in a similar fashion. The analysis can be extended to two-way stiffened plates and three-dimensional assemblages of plates, provided appropriate degrees of freedom are modeled within the sub- structures. For a ribbed plate with rigid intermediate sup- ports, extra constraint equations are required. For a plate with semirigid supports, extra energy terms are needed. An analysis of multi-span ribbed plates will be reported separately.
Numerical convergence Convergence is an important mathematical feature for
assessing the computational efficiency of a numerical pro- cedure. The efficiency of a model for free vibration analysis depends on the size of the eigenvalue problem, i.e., the order of the characteristic matrix, which is equal to the minimum number of degrees of freedom required for convergent solutions. The number of degrees of freedom of a modal synthesis model depends on the minimum number of modes required in the modal expansion to obtain a convergent solu- tion. The lower the number of modes required, the more efficient the model will be.
Table 1 shows the natural frequencies that were obtained for different numbers of modes in the modal expansions for a plate with the two edges containing ends of ribs simply supported and the other two edges free to vibrate (s-f-s-f plate). The plate considered has a 5.9 m span, a width of 3.46 m, and eight equally spaced ribs. The plate is semi- rigidly attached to ribs. The plate, rib, and plate-to-rib connection properties are taken to be the mean values given in Table 3. The results in Table 1 show that rapidly conver- gent solutions can be obtained with only a small number of terms in the modal expansions. For the demonstration problem, only five modes are necessary in the across-rib direction and one in the along-rib direction to obtain three- digit-precision solutions in the first five natural frequencies. Only one term is required in the along-rib direction, as the first five modes develop single curvature in that direction.
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Jo i s t
0
I - - I FIG. 3. Plan of test floor 3, showing the joist reference system. K1, K4, E l , and E4: locations of masses; A to 0: locations of I-joists.
TABLE 1 . Convergence of the natural frequencies in the modal synthesis model
Natural frequency* (Hz) Mode NO. N = 5 N = 7 N - 1 0 N = 2 0 N = 2 9
(a) M = 1 1 14.213 14.210 14.208 14.198 14.201 2 14.825 14.819 14.811 14.803 14.803 3 16.076 16.069 16.067 16.063 16.064 4 20.327 20.327 20.327 20.328 20.324 5 28.891 28.874 28.858 28.840 28.839
(b) M = 2 1 14.213 14.210 14.208 14.198 14.201 2 14.825 14.819 14.811 14.803 14.803 3 16.076 16.069 16.067 16.063 16.064 4 20.327 20.327 20.327 20.328 20.324 5 28.891 28.874 28.858 28.840 28.839
NOTES: M indicates the number of selected modes in the along-rib direction, and N indicates the number of selected modes in the across- rib direction.
*Estimated natural frequency for a model with M x N terms.
Therefore five degrees of freedom were required by the modal synthesis model to obtain the solutions for the plate in Table 1. By comparison, at least 71 degrees of freedom should be used to solve the same problem by the finite strip model of Filiatrault et al. (1990). An even greater number of degrees of freedom would be required using shell plus beam finite elements in an analysis. Other studies not reported also uphold the conclusion that a relatively small number of terms are required in the modal expansion to obtain convergent results for the natural frequencies of plates. This points to an advantage of the proposed model in computational efficiency, compared with the finite element types of models in the literature.
Verification Modal tests were carried out on a series of full-scale,
laboratory-built wood I-joist floors to provide data for the validation of the proposed model. Wood I-joist floor spec- imens were chosen because of their high degree of orthotro- picity in "plate" properties, which facilitates the evaluation of model capabilities.
The test floors consisted of eight 241 mm (9.5 in.) deep wood I-joists and 1.2 x 2.4 m sheets of 15.5 mm thick Type 0-1 oriented strand board (OSB) with tongue and groove edges (CSA 1985). The flooring was attached to the joists using No. 8 (4.37 mm diameter) wood screws spaced at 160 mm centers. The flooring panels were laid with their long axes normal to the joists, and were butt jointed over the joists, with staggering of joints between rows of panels. The I-joists had flanges of 38 x 89 mm machine stress rated Douglas-fir lumber and an 11.0 mm thick OSB web. Both ends of each I-joist were skew screwed to a double layer of 38 x 89 mm spruce-pine-fir lumber bearing plate, using two No. 8 wood screws per end through the bottom flange of the joist. The bearing plates were in turn secured to 292 x 292 mm heavy timbers which rested on the ground. Mea- surements were made to check that there was no detectable motion at the supports. Offcuts of the joists were inserted between joist ends as blockings to prevent rotations at the ends of the joists. It is assumed that the span of a floor was the center-to-center distance between end bearing plates. The widths of the s-s-s-s plates are assumed to be the center-to- center distances between the outer (supported) joists.
Figure 3 shows the plan of a typical floor, floor 3, and the arrangement of the joists. Ten series of tests were con- ducted on the floors having various spans, support condi- tions, and imposed masses (Table 2).
The mechanical properties for the ribs and the plates were determined from vibration tests, following the techniques described by Chui (1991) and Nakao and Okano (1987) respectively. The load-slip response of the floor sheathing- to-joist connections was estimated by a static test, after the
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TABLE 2. Summary of verification test floors
Imposed mass
Reference Span Per point Position No. (m) Supports (kg) (Fig. 3)
1 6.8 S-f-S-f 2 6.8 s-s-s-s 3 5.9 s-f-s-f
3-1 5.9 S-f-S-f 79.4 E l , E4, K l , K4 3-2 5.9 S-f-S-f 79.4 K1, E4 3-3 5.9 S-f-S-f 79.4 K 1 3 -4 5.9 s-f-s-f 79.4 Center 4 5.9 s-s-s-s 5 4.45 s-f-s-f 6 4.45 S-S-S-s
NOTES: (i) The width of all floors is 3.46 m. (ii) The coordinate system for positions K1, K2, E l , and E4 is illustrated in Fig. 3. (iii) s-s-s-s signifies plates with the four edges simply supported. (iv) s-f-s-f signifies plates with the two edges containing ends of ribs simply supported and the other two edges free to vibrate.
TABLE 3. Measured mean values of mechanical properties of test floors
Properties Mean value
Mass per unit length of ribs 4.42 (kg/m) (range 4.25-4.7)
Flexural rigidity of ribs (EZ,) 9.53 ( x lo5 ~ . m ' ) (range 8.64-1 1.45)
Shear rigidity of ribs (jAG,) 1.32 ( x lo6 N) (range 1.21-1.47)
Mass density of sheathing 644 (kg/m3) E, of sheathing 2.34 (GPa)
TABLE 4. Predicted vs. test natural frequencies (Hz)
Mode
Floor 1 2 3 4 5
Test Model Error (070)
Test Model Error (070)
Test Model Error (070)
Test Model Error (070)
Test Model Error (070)
Test Model Error (070)
Test Model Error (070)
Test Model Error (070)
Test Model
E; of sheathing 7.32 ( G P ~ ) Error (070) 27.5 14.8 -2.8 -6.6 5.3 Shear modulus of sheathing 1.38 (GPa) Secant slip modulus at 0.1 mm slip 6 Test 17.1 23.7 28.4 34.0 43.0
of connection 2.82 ( x lo7 N/mZ) Model 22.1 22.8 25.4 32.1 43.0 Error (070) 29.5 - 3.8 - 10.7 - 5.7 0.0
floors were tested and dismembered. The mean values of the measured mechanical properties of the components are summarized in Table 3 .
The impact excitation approach (e.g., Ohlsson 1982; Chui 1986) was used to characterize the vibration responses of the test floors. Typically, an impact was applied at one position on the plate surface and the acceleration response was measured at various positions. The impact positions were selected so that all of the low-order modes of vibration were excited. The response was measured at mid and quarter span positions of each rib. This gave sufficient information for the definition of natural frequencies, mode shapes, and damping for the first five or six vibration modes of each floor. Impacts were applied by an instrumented hammer with a force transducer attached to its tip. The acceleration was measured by an accelerometer that could be fixed firmly to the upper plate surface at any desired location. The specimens were excited into motion by gently tapping on their upper surfaces. A spectrum analyzer was used to record and analyze the impact and response signals. Both signals were analyzed using the Fast Fourier Transform technique (Newland 1984) to calculate their spectra, which were then curve-fitted to output the natural frequencies, mode shapes,
and viscous damping ratios (Chui 1986). Hammer impact testing is recognized to be a quick and reliable means of estimating the response characteristics of specimens (Ewins 1986).
able 4 compares the measured natural frequencies with the model predictions. The flexural and shear rigidities of individual ribs and the mean plate and connection properties were used to obtain "model" estimates of natural frequen- cies. Thus, although the predictions in Table 1 and those in Table 4 for floor 3 nominally refer to the same system, the results differ, as rib properties were taken to be uniform in the former case. Figure 4 shows typical comparisons between mode shapes measured in the across-joist direction and those predicted by the model for floors 1, 2, and 3-3. Both the measured and predicted mode shapes were nor- malized by converting the absolute maximum amplitude to one unit. The mode shapes shown for floor 3-3 represent a situation where a mass is im~osed on the surface of the floor. Lack of symmetry in bo t i the measured and predicted mode shapes for floors 1 and 2 reflects nonuniformity in masses and mechanical properties of the ribs.
By comparing the natural frequencies observed in the tests with the predictions, it is found that the modal synthesis
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Mode Floor 1 Floor 2 Floor 3-3
FIG. 4. Predicted (solid lines) vs. test (broken lines) mode shapes for floors 1, 2, and 3-3.
model generally yields good results for the type of ribbed plates considered. Table 4 indicates that among the 50 predicted natural frequencies for the floors tested, the per- centage errors in 28 predictions are less than 10%; and three predictions have errors greater than 25%. The best agree- ment between the tests and the predictions is obtained in the estimates of the fundamental natural frequencies, except for floors 5 and 6. The predictions for the third and fourth natural frequencies and the fundamental natural frequencies of floors 5 and 6 are not so favorable. One large percentage error, 28.4'70, occurs in the predicted fourth natural fre- quency for floor 3-2, which has two imposed masses on the system.
The eigenvalue solution technique used in the model is a modified QR method (Moler and Stewart 1973). As the model approximates a real system, some errors in the numer- ical predictions are expected. These errors come from the discretization within a solution which approximates the solu- tion to a continuum problem. This is not, however, sus- pected as being the primary source of the errors between the test and predicted natural frequencies.
By examining Fig. 4, it is found that the predictions of the mode shapes match the test results well with respect to their shapes and number of nodes and peaks. The largest discrepancies occur in the mode shapes of three higher-order modes for floor 3-3 which has one imposed mass, and in the mode shape of the fifth mode for the s-s-s-s floor, floor 2.
There are several potential error sources associated with the physical tests. First, there may have been systematic errors from the method of instrumentation, but none could be identified. Second, it was impossible to establish com- pletely simple support conditions at the edges of full-size wood plate systems. There would have been some partial restraint of rotations. Chui and Smith (1990) demonstrated theoretically that partial end fixities can significantly affect the natural frequencies of beams. This second error source in the tests seems to be the main reasons for discrepancies,
as the predictions for the s-f-s-f floors are closer to the test results than those for the s-s-s-s floors. The s-f-s-f- case is thought to reflect a more accurate across-ribs physical rep- resentation. The influence of partial fixities a t plate edges is also thought to explain why the predictions for the floors with shorter spans (floors 5 and 6) do not match the test results well.
Overall, the modal synthesis model produces good predic- tions of natural frequencies and mode shapes for ribbed plates of the type studied. It is believed that the model devel- oped can be applied to other types of ribbed plates, e.g., steel girder - concrete slab construction, reinforced plastic composite ribbed plates. Extension of the model to forced vibration analysis is fairly straightforward and requires the addition of a damping matrix and velocity and force vectors to equation [lo]. Such an extension is achieved following the procedures outlined by Hu (1992).
Application of the model This section examines how rib properties can affect the
free vibration behaviour of ribbed plates, as an illustration of the benefits from using the new model.
An examination of the sensitivity of the natural frequen- cies to the ratio EI,/pAG, was carried out based on a ribbed plate with the same geometry and constructional arrangement as the test floor 1 (Table 2). The mechanical properties of the plate and plate-to-rib connections were the same as in the test floor 1. The ratio EIr/p.AGr of ribs is varied by changing the flexural rigidity (EI,) and keeping the shear rigidity W G , ) constant. The ribs in the ribbed plate are assumed to have uniform geometric and material properties. For the reference system (floor 1, Table 2), the ratio EIr/pAGr is 0.72. Figure 5 shows the relationship of the natural frequencies and the EIr/p.AG, ratios. The sen- sitivity of the natural frequencies to shear deformation and rotatory inertia is measured by the ratio f,/ f. The fre- quency f, includes the influences of shear deformation and rotatory inertia in ribs, while the frequency f excludes these
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Mode 5
----------------- M o d e 4
Mode 3
1 Mode 1
* 1
*"
EI,lpAGr (m2) FIG. 5. The relationship f,/ f vs. EIr/pAGr.
TABLE 5. Effects of shear deformation and rotatory inertia of wood I-joists on the fundamental natural frequencies of systems tested by WSTC
Frequency Percentage Floor construction details from error in
frequency Joist Number Floor Test Model 1 using model 2 Span Width depth o f No. (Hz) (Hz) (070) (m) (m) (m) joists
NOTES: Model I - with allowance for shear deformation and rotatory inertia in joists; Model 2 - without any allowance for shear deformation and rotatory inertia in joists.
influences. A low value off,/ f indicates a high sensitivity of the natural frequency of a floor to the shear deformation and rotatory inertia effects. It is found that the higher the ratio EIr/pAGr, the lower the f,/ f will be. For the modes containing only one half wave along the ribs, the lower the mode number the higher is the sensitivity to shear defor- mation and rotatory inertia. Mode 6, containing two half waves in the along-rib direction, has the most sensitive natural frequency among the six considered.
The reference system represents the most extreme influence on f,/ f illustrated in Fig. 5. Thus, it is seen that shear deformation and rotatory inertia in ribs can have a
significant influence on the first few natural frequencies of practical ribbed plate structures. For systems with ribs having a high ratio of EIJpAG,, predictions of the natural frequencies should include the effects of shear deformation and rotatory inertia in ribs. For the systems underlying Fig. 5, the errors in the predicted natural frequencies associated with neglecting shear deformation in ribs will be less than, for example, 5 % for all of the first six modes if EIr/pAGr is less than 0.14.
As a further demonstration of the influence of shear deformation and rotatory inertia in ribs, the fundamental natural frequencies of 17 single-span wood I-joist floor
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systems were studied. Experimental results were available from the Wood Science and Technology Centre (WSTC), University of New Brunswick, Fredericton, New Brunswick. Table 5 demonstrates the theoretical influence of shear deformation and rotatory inertia in ribs on the fundamental natural frequencies of WSTC floors. This table also illus- trates a comparison with the test results and summarizes the construction arrangement. The corrections to the first natural frequencies for shear deformation and rotatory inertia are in the range of 10% to 40%. Thus, substantive errors in the fundamental natural frequencies predicted by numerical models are possible if the influence of shear defor- mation and rotatory inertia in ribs is neglected.
Conclusions The primary potential of the free vibration model
described here is in the prediction of free vibration modal characteristics of one-way stiffened plates with ribs having high ratios of flexural to shear rigidity. The following specific conclusions are drawn based on the work described.
1. The numerical model developed can yield accurate results for free vibration flexural responses of plates with ribs having high ratios of flexural to shear rigidity.
2. Convergence studies show that this modal synthesis model has very good numerical convergence characteristics. Relatively few terms are required in the modal expansions for substructures to obtain accurate estimates of natural frequencies. As a result, computer programs based on the model can be implemented in computers with relatively small memory capacities.
Acknowledgements The authors acknowledge the financial support of the
Natural Sciences and Engineering Research Council of Canada, Operating Grant OGP0004417. Suggestions made by Dr. Y.H. Chui (University of New Brunswick) during execution of the work are much appreciated.
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Chui, Y.H. 1986. Evaluation of vibrational performance of light- weight wooden floors: determination of effects of changes in construction variables on vibration characteristics. Research Report 2/86, Timber Research and Development Association, High Wycombe, United Kingdom.
Chui, Y.H. 1987. Vibration performance of wooden floors in domestic dwellings. Ph.D. thesis, Brighton Polytechnic, Moulsecoomb, Brighton, United Kingdom.
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Appendix 1. List of symbols a plate span A, cross-sectional area of a rib b plate width d rib depth Ex, EY elastic moduli of the plate in the x- and
y-directions
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EA, EA s
EIP Err h kc
[K l , [MI
M'k Mm, n
{4p>9 (4,)
4 i 4 i 4k,n 4k,n S
T u w;, w,, w x'k XL, YE
Y' Z CY
P p p pAr PI Gr w i wtn,n
axial rigidity of a rib sum of axial rigidities of a rib and plate strip ~ , h 's / 1 2 flexural rigidity of a rib thickness of the plate force per unit displacement per unit length of plate-to-rib connection modal stiffness and mass matrices of the ribbed plate system kth modal mass of the jth rib modal mass of the plate independent modal coordinate and accelera- tion vectors of the ribbed plate system kth modal coordinate function of the jth rib kth modal velocity of the jth rib modal coordinate function of the plate modal velocity of the plate spacing of ribs kinetic energy of the ribbed plate system potential energy of the ribbed plate system transverse displacements of the jth rib, the plate, and the ribbed plate kth mode shape function for the jth rib mode shape functions for the plate in the x and y directions y-coordinate of jth rib in a ribbed plate (h + d ) / 2 k2r2/k,a2, where k is the number of half waves in the along-rib direction l/shEx + l /EAs + Z 2 / ( ~ I r + EI,) mass per unit volume of the plate mass per unit length of a rib shear rigidity of a rib kth angular natural frequency of the jth rib angular frequency of the plate
Superscripts and subscripts
j rib number ( j = 1 , ..., J ) k mode number for a rib (k = 1 , .. ., K ) P plate substructure r rib substructure m, n modal expansion numbers in x and y direc-
tions of the plate (m = 1, ..., M; n = 1 , ..., N)
Appendix 2. Derivation of equation [lo] The elements of the modal mass matrices of the plate and
rib substructures are
The m,nth angular natural frequency of an orthotropic plate is obtained from
where, from Hu (1992),
G is the through thickness shear modulus of the plate, and vx and vy are Poisson's ratios of the plate.
The angular natural frequencies of the modal expansion terms for the ribs are obtained from
[AS] (wjk12 = K'/M' k k
In the model discussed here, wjk is determined from [6] and Mi from [A2].
As the plate mode shapes in the x direction are well approximated by the beam modes of the ribs, it can be assumed in the constraint equation [8] that
m = k , M = K, and Xk(x) = ~jk (x )
which produces the modified constraint equation:
Since the functions Xk(x) are continuous and indepen- dent, and are nonzero over the plate span (0 I x I a), [A61 is equivalent to J x M equations of the form
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The matrix form
where { q ) is a vector of modal coordinates for the system, dimension M x N+ J x M; { q , ) is a vector of independent modal coordinates for the ribbed plate system, dimension M x N ; and [PI is the constraint or transformation matrix containing submatrices [ P , ] , [ P ! l , . . . , [ P i ] .
Submatrix [P , ] is an identity matrix. The jth submatrix [ P { ] is of order M by M x N and has the form
Y?(Y) -.. Y>(Y) = tP-',I{q,) = [A101
(zero)
The Lagrangian expression is written so that the constraint and equations [A71 are included in the form of undetermined multipliers, A,,,. The Lagrangian expression for the ribbed plate is
Applying Lagrange's equations (Clough and Penzien 1975), a set of motion equations for the ribbed plate system is obtained in terms of the Lagrange multipliers and { q ) . The motion equation is
[A121 [Mdl{9) + [Kd1{4) = [Al{A)
where
The elements of submatries [M,] and [M{] ( j = 1 , 2, 3 , ..., J ) are obtained from [All and [A2] . The elements of submatrix [K,] are the Km,,, terms defined in [A4] . The elements of submatrices [K-',] ( j = 1 , 2, 3, ..., J) are obtained from [A5] . The matrix [A] in [A121 consists of [PI and an identity and a null submatrices.
Combining [A121 with the constraint equations [AS] elim- inates the Lagrange multipliers and the dependent modal coordinates. This yields the equation ( [ l o ] ) for undamped free vibration of the ribbed plate system in terms of the inde- pendent modal coordinates q;,,,.
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