only a function of geometric variables but also the Poisson, s ratio of the plate material. the influence of u, however, is very limited Higher order natural frequencies ratios omm can also be calculated by using Eqs. (8),( 12)and (13). But from the Eqs(18)and(19), and Tables 1 and 2, it can be found that the contribution of the second terms for Timn and T2mu are very limited compared to the first terms. Besides, apmn in Eq (12) will become greater with the increase of m and n, and so, the influence of variation of Smnversus m and n can be neglected. In other words, the higher-order natural frequencies of a plate with voids can be approximated mu 1@0mm. This relation implies that the cylindrical voids influence all the natural frequencies of rectangular plates almost in the same way. Moreover, the following important conclusions, which is the purpose of this paper, can be obtained 1. Omm the natural frequencies of the plate with voids, can be obtained from &i and amm. That is to say, if we know for a voided plate, an Lamm can be used to obtain the natural frequencies for all orders, which provides a convenient approach to solve free vibrations of the plate with voids 2. From Figures 3-5, it can be found that &u is not always greater than l. When Ti1>1211 in equation(14), &1 <1.0, if we do not consider the second term in the expressions of Tll and T2ul, then we have h<2.45r. So for a plate with cylindrical voids of same radii, h=2.45r will approximately be the separating point between Wmn>@omn and @mn<@bmn. In fact, Figure 8 shows that the exact point is h=2,22r OPTIMAL DESIGN As we have mentioned above, the value of &n changes with the locations and radii of cylindrical voids. In this way, people can get the natural frequencies required by use of designing the void radii and location Figure 3 illustrates the rough influence method of voids locations on natural frequencies for xm0205,0,) 2. Distributed disposed voids with x, =0. 25, <2=0.5, x,=0.75 3.Side disposed voids with x, =7+50, x2=3K,I,=5X, It shows that twoside symmetrically disposed voids create larger &1 than one-side dis voids under most circumstances, especially with the increase of r/a Is that a common conclusion? In order to observe the detail influence of the voids dispositions, we obtain the maximum an1 and minimum &( Figure 5)versus N, respectively when h=0. 2, b=a and osF s0.09 as well as the according voids dispositions for equal radii Figures 6 and 7 ). It shows that optimization effectiveness will be greater if the cylindrical voids are designed in different radii than in equal radi
In other words, by means of Equations(21)(24), the plate can be optimized, For example, the voided plate can be designed to be of the same natural frequency (i. e, &1=@1/abn )as the solid plat but the plate will be much lighter. Figure 8 shows the void radius and locations with N when &1=l COMPOSITE MODEL Plate with voids can also be considered to be orthotropic, so the composite model could be another approximate analysis method although it neglects the locations of voids. It must be noted that orthotropic model is only suitable for much more densely located and small radii voids The natural frequencies of a simply supported orthotropic plate are (ones R M, 1998) D +2(D2+2D60 +D2 (25) where bending stiffnesses of the plate are Table 1 Contribution of the 2nd terms of Timm, T2mn to Timn, T2mn for different m and n( N3 and w/a0. 20 and r/a0.05) 2nd term of nd term of 10.04413903000003962015659720.00969861064584 0.04415548000023160.15659720009696861.064575 12212333 0.044171300.00000734015517340009340911.063701 0.044174300000004340.15517340.009340911.063686 0.044158540000020100.15659720.009696861.064573 0.044175600000003050.15517340.00934091t.063679 1004417761000000103015287780008767011.062263 0.04417791000000073015287780008767011.062251 004417812000000052015287780008767011.062243 Table 2 Contribution of the 2nd tems of timm T2mn to Timm, T2mn for ditterent m and n(Ne4 and h/a=0.20 and r/a0.06) 2nd term of 2nd term of T T1 0058865240.000039620.19586700.009696861.08183 0058881700.000023160.1958670 0058839520000007340.19444330.09340911.080896 2 2 0.05886621000000434019444330009340911.080880 0058884750.00002010019586700009696 1081826 005887773000000305019444330.009340911.080874 005883]39000000103019214770.008767011079363 058852550.000000730.19214770.008767011.079351 0058867790000000520.19214770008767011.079343 BDo, D12=ADo, d22 Do =阳,B= 2B
B is a non-dimensional volume factor. If A-1, Eq(25)is reduced to the isotropic solution (Cao, 1989). Then from equations(16)and(25), we can obtain 2m2n2(1-μ+AB) B6 (26) ubstituting b=a, m=l, n=l into equation(26), we can get 1+1+24 (27) 4B By using equation(27), a series of &u for simply supported plates with voids of equal radius r=r are calculated for different total void numbers N=3, 4, 5, 6, in which Poisson's ratio, A is 0.17; b=l, and the results are also shown in Figure 2 COMPARISON OF TWO METHODS From the above derivation and calculation, we draw the following conclusions: 1. Either of the two methods can provide a simple formulae to calculate the natural frequency coefficients Sl, which is of engineering significance. The Extended Dirac Function method will be more powerful 2. Both of the natural frequency coefficients a obtained by these two methods are related to the poisson s ratio 3. As compared with the Extended Dirac Function Method, the Composite Model gives a lower solution of &. A likely reason is the neglect of the influence of voids locations and the thickness of the plate. A CKNOWLEDGMENTS The authors are indebted to Dr. Cai for suggesting the use of Composite Method to simplify the proof and also to the csc for financial support REFERENCES Baidar Bakht, Mo S Cheung and Leslie G. Jaeger( 1981)Cellular and Voided Slab Bridges ASCE Journal of the Structural Division 107(9),1797-1813 2. Cao, Z.Y.(1989). vibration Theory of Plates and Shells. China Railway Publishing Houses of, Beijing.( in Chinese 19
3. Cope, R.J., Harris, G, Sawko, F. A New Approach to the analysis of Cellular Bridge Decks. Analysis of Structural systems for Torsion, ACL, Sp35-5, 185-210 4. Crisfield, M.A. and Twemlow, R. P( 1971) The Equivalent Plate Approach for the Analysis of Cellular Structures. Civil Engineering and Public Works Review, March 259~263 5. Elliotto, G and Clark, L A (1982)Circular Voided Concrete Slab Stiffness. ASCEJournal of the Structural Division 108(11), 2379-2393 6. Gorman, D. J (1995). Free Vibration of Orthotropic Cantilever Plates with Point Sup Journal of Engineering Mechanics, 121(8), 851-857 7. Gorman, D J.(1982). Free Vibration Analysis of Rectangular Plates. Elsevier-North Holland, Amsterdam 8. Jones, R. M.(1998)Mechanics of Composite Materials(second Edition). Taylor francis 9. Liew, K. M. and Xiang, Y,(1994). Vibration of Mindlin Plates on Point Supports Using Constraint Functions. Journal of Engineering Mechanics, 120(3), 499~513 10.Osama K. Bedair (1997). Stability, Free Vibration, and Bending Behavior of Multistiffened Plates Journal of Engineering Mechanics, 123(4), 328-337 Rajan Sen, Mohsen Issa, Xianfhong Sun and Antoine Gerges(1994) Finite Element Modeling of Continuous Posttensioned Voided Slab Bridges. ASCE Journal of the Structural Division 120(2), 651-667 12. Takabatake, H(1991). Static Analyses of Elastic Plates with Voids. International Journal of solid and structures, 28(2), 179-196 13. Takabatake, H. (1991). Dynamic Analyses of Elastic Plates with Voids, International Journal of solid and Structures, 28(7), 879-895 14. Wu, X. L and Zhang X. Z. (1991). Solution Method of the Lateral Bending Problem of Clamped Plate and It's Application. ACTA Aeronautica et Astronautica Sinica, 12 (4), 161~167 15. Wu, X. L and Zhang X. Z(1989). Analogy between Thermal Stress and Lateral Bending of Orthotropic Media. Proceedings of the International Conference on Applied Mechanics, 13~19 Unit 6 Collection of English Abstracts-Latest Development in Civil Engineering英文摘要集—土木工程最新进展 为了提高英语科技写作水平,结合英文摘要的写作,下面提供一些1999年以来部分国 外土木工程专业博士学位论文的英文摘要,使大家在提高科技英语写作能力的同时了解国际
最新土木工程的发展方向。 6.1 Towards 3D Plastic-zone Advanced Analysis of Steel Frames The latest Australian steel structures standard. AS4100-1990 is the first in the world to formally incorporate the provisions for a certain method of second-order inelastic analysis which can model the in-plane strength and stability of frames to such accuracy that the individual specification checks of member capacity are waived. This type of 2-D inelastic analysis is referred to as advanced analysis. Three-dimensional advanced analysis is not provided for in the standard, nor in any other existing standard around the world, apparently due to the_nercentign that some technical nrnhlems in practical 3-D advanced analvsis of steel