IlI lillill JAI II II I III

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 471h St., New York, N.Y. 10017.. 974A-96 The Society shall not be responsible for statements or opinions advancedln papers or deicussion at meetings of the Society or of Its Divisions or Sections, or printed In its publications. Discussion is printed only if the paper is published in an ASME Journal. Authorization to photocopy material for Internal or personal use under circumstance not fairing within the fair use -provisionsof the Copyright Act b granted by ASME to libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service provided that the base fee of $0.30 per page is paid directly to the CCC, 27 Congress Street Salem MA 01970. Requests for special pernessien or bulk reproduction should be addressed to the ASMETechricel Pubishing Department Copyright 0 1997 by ASME All Rights Reserved Printed in U.S.A VIBRATION AND BUCKLING OF SQUARE RLATES CONTAINING CENTRAL HOLES A.B.Sabir. Division to Structural Engineering School of Engineering. University of Wales, Cardiff PO Box 917 Cardiff, UK CF2 1XH. IlI lillill JAI II II I III G.T.Davies. Division Of Structural Engineering School of Engineering, University of Wales, Cardiff PO Box 917 Cardiff, UK CF2 1XH. ABSTRACT. The finite element method is used to determine the natural frequencies of flat square plates containing centrally located circular or square holes. The plates are subjected to either inplane uniaxial, biaxial or uniformly distributed shear along the four outer edges. These edges are either simply supported or clamped. To determine the stiffness and mass matrices, non conforming rectangular and triangular displacement elements are used to model the out of plane behaviour of the plate. The inplane stress distribution within the plates, which are required in the analysis are determined by using inplane finite elements having displacement fields based on assumed strains. These satisfy the exact requirements of strain free rigid body modes of displacements. The natural frequencies of simply supported and clamped plates are initially determined when no inplane loads are applied. This showed the influence of the size of the hole on the natural circular frequency. These plates were then subjected to inplane loads and the effect of these forces on the natural frequencies are given. The results show the natural frequencies of square plates with central circular holes decrease with increasing compressive forces, and that the frequencies become zero when the compressive forces are equal to the elastic buckling loads of the plates. By repeating this process for all boundary conditions and applied loads a comprehensive set of results is obtained for the buckling and vibrational properties of square plates containing centrally located holes. INTRODUCTION. In the present paper the finite element method is used to determine the natural circular frequencies and elastic buckling loads of square plates which contain centrally located circular or square holes. Kapur and Hartz(1966) were the first authors to use the finite element method in the buckling analysis of thin plates. They used the non conforming rectangular bending element derived by Melosh(1963), which has 3 degrees of freedom at each corner node. Further studies into the buckling loads of thin plates were undertaken by Piflco and Ikakson(1969), who used the conforming rectangular element. This 16 degree of freedom bending element was developed by Bogner et at (1966) and improves the convergence properties, since this element satisfies the compatibility criteria of the normal slope across inter element boundaries. Sabir(1973) used both bending elements to determine the buckling loads, and gave both stiffness and geometric matrices explicitly. This investigation also included the buckling loads of plates on elastic foundations. In all the above work the buckling loads were determined by assuming that the enplane stress distribution within the plate was known. Therefore these results were only dependant on the performance of the rectangular bending elements. However if the plates contain openings, inplane stress concentrations will occur in the vicinity of the hole. Therefore in any buckling analysis of a thin plate containing a hole, these stress distribution have to be accurately calculated. This can by done by using inplane finite elements. The elastic buckling loads of plates containing holes has been carried. out by many authors using different numerical methods. Presented at the ASME ASIA '97 Congress & Exhibition Singapore September 30-October 2,1997 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/ab

Ritchie and Rhodes(l 975) used the finite element method to determine the inplane stress distribution and the Rayleigh-Ritz method to determine the buckling load; while authors such as Rockcy et a/ (1967), and Shanmugam and Narayanan(1982) used the finite element method to evaluate both the inplane stresses and the buckling loads. The inplane finite elements used by these authors were based on linear displacement fields, and the convergence properties of these elements were examined by Sabir(1983). In the same paper a new class of inplane elements were developed. These elements were based on assumed strain functions rather than assumed displacements and improved the convergence properties of plane stress problems. Sabir and Chow(1983) used these elements in conjunction with the non conforming bending element to determine the buckling loads of thin plates containing central holes; and plates containing eccentric holes Sabir and Chow(1986). The finite element method has also been used to determine the natural frequencies of thin plate. Dawe(1965) used the 12 degree of freedom non conforming bending element and Mason(1968) used the 16 degree of freedom conforming bending element. The natural circular frequencies of thin plates containing holes has been investigated by various authors. Paramsaivam(1973) used the finite difference method, while Gutierrez, et al (1987) and Grossi, et a/.(1997) used the optimised Rayleigh-Ritz method. The Rayleigh- Ritz method was also used by Lee and Lim (1992) to examine the effect of inplane loads on the natural frequency. Prabhalcara and Datta(1997) used the finite element method to examine the vibration and buckling behaviour of plates with holes when subjected to axial loadings. The buckling and vibration behaviour of plates containing central square holes. Sabir, et al. (1995) and Sabir and Davies(1995); eccentric square holes. Sabir and Davies(1997a) Sabir and Davies(1997b); and reinforced square holes Sabir and Davies(1997c) has also been determined using the finite element method. In the present paper the natural circular frequencies and the buckling loads of square plates containing circular and square holes are examined. These plates will be subjected to various inplane loads and be either simply supported or clamped on all four sides. THEORETICAL CONSIDERATIONS. To determine the natural frequency of a plate which is subjected to an inplane load then the total potential energy of the structural system has to be considered. The total potential energy up, is the sum of the strain energy of the plate due to bending Up, the stretching strain energy arising from membrane stresses Upg, and the kinetic energy of the vibrating system Vpv. lip = Up + Upg + Vpv (1) For a plate with Cartesian co-ordinates x and y, the expressions for the energy equations are given in terms of the out-of-plane displacement w. These are given by Timoshenko and Goodier(1970). d w zw ow 2 Upg=-Tific4-7,7 0 ) 2 +2.rxyf i+ayity-) idx.dy (2) Vpv p tro 2liw 2eix.dy where I) is the flexural rigidity, v is Poisson's ratio, a 1 a, and Try are the inplane stresses, p is the mass density, t is the thickness and to is the natural circular frequency. In order to calculate the natural frequencies the principle of minimum potential energy is used. The first variation of the total potential energy gives the necessary equilibrium equation, and from the second variation the following transcendental equation is obtained, which described the vibration of a structure subjected to inplane loads. d 1 np to O r Eic P Kg Kulp = 0 (3) In the finite element method this equation is applied to each individual element in the structure, and then the overall stiffness, geometric stiffness and consistent mass matrices are assembled. This then yields the following transcendental equation. RN PliCgl) e 2 1 1`49} (4) where 8 is the structural nodal displacements. For any element in the plate, the stiffness matrix K, geometric stiffness matrix Kg, and consistent mass matrix Km are given by, Ke =1/1-1 I T B T Kg. =1,4-1 1 7111G T.N.G.dx.d4A -1 1 K me = p fru,.w.dx.d44-1 1 where A is the transformation matrix, D the rigidity matrix, B the strain matrix and N is the function of the internal inplane stresses in the element. The N and G matrices are shown below N = x Try / rny cy and 6= Ow O x Ow y (6) 2 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/abou

where Nx, Ny, Nxy are the direct inplane and shearing stresses within an element of the plate. The degrees of freedom are w the lateral deflection, ex and ey the rotation about the x and y axis respectively. Inclane Displacement Fields. To calculate the inplane stress matrix N for each element, strain based inplane elements, based on the work by Sabir(1983) are used. Two inplane elements are used to calculate the stresses prior to buckling. The first is the strain based rectangular inplane element (SBRIE) which has two degrees of freedom at each of its four corner nodes. The displacement field for this element contains the two inplane displacements u and v in the x and y direction respectively. The element and its displacement field are shown in figure I. Z, W Figure 3 Non conforming bending element v = a2 +03x f y, v a7 y 2 agy u = al - a3y+ a4x + asxy + 2 2 09? oar a6y + 0 7 xy + 2 2 7- (7) The rectangular non conforming bending element is shown in figure 3 and the displacement field for the out-of-plane displacement w is given by the following twelve term polynomial equation, w =0 + 02x + asy + a4x2 + op)/ +a6y2 +07? + 0 l3r 2 Y + 092Y 2 + WY 3 + al Ix 3 Y ± <7 12x31 3 (9) The displacement field for the lateral deflection of the triangular element is given by Figure 1 Inplane degrees of freedom. The second inplane element is the strain based triangular inplane element (SBTIE). This element also has two degrees of freedom at each corner node and the displacement field is shown below. This element is used to model the circular edge of the hole. y, v u = a 1 a3y + + at Y 0 2 to = a2 + a3x + 05y + 06-2- Figure 2 Inplane triangular element SBTIE. The displacement fields for SBRIE and SBT1E satisfy the exact requirements for rigid body modes of displacement, and are based on independent linear variations of the direct strains and constant shearing strain. Out-of-plane displacement fields. To calculate the out of plane stiffness matrix, the non conforming bending element and the triangular bending elements are used. Both these elements have three degrees of freedom at each corner nodes. ( 8 ) W = a + a2x + 07y + a 4x2 + asxy + 00 2 1- cyx 3 + a,(x 2 y xy) + a9y 3 (10) PROBLEM CONSIDERED. Square plate which are either simply supported or clamped on all four sides are examined, and the elastic properties and dimensions are taken to be width of the plate d = 600 mm, thickness t = 1mm, Young's modulus E = 20x104N/nun2, Poisson's ratio v = 0.3, and Mass density p 7850Kg/m 3. The plate will contain a circular or square hole, where the dimension of the hole diameter or width is shown in figure 4 The size of the square or circular hole will be referred to in terms of the hole size ratio b/d, where this ratio will vary from an unperforated plate (b/d) to b/d=0.5. To determine the natural frequency of a plate containing a centrally located hole, equation (4) is solved. The inplane stress distribution within the plate is first determined using the strain based elements. The values of these stresses at the Gaussian point of each element is then used in the calculation of matrix N for each element. The out of plane stiffness and mass matrices are then determined using the non conforming bending element, and equation (4) is assembled for the entire plate. The eigenvalues and vectors of equation (4) are then obtained using Jacob's numerical method. The out of plane stiffness and mass matrices are then determined using the non conforming bending element, and equation (4) is 3 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/ab

assembled for the entire plate. The eigenvalues and vectors of equation (4) are then obtained using Jacob's numerical method. a (b) Figure 5 Finite element mesh (a) square hole; (b) circular hole. Figure 4 Dimensions of the perforated plates. In the present paper the square plate will be subjected to either inplane compression or uniformly distributed shear and the natural frequency will be determined. The natural frequencies and the inplane loads will be given in terms of the following non dimensional quantities as given by Vb=wd 2 Pa d 2 = ft. 2 D Pad 2 Kb- ir 2 D K. = d2 z 2 D fa' for the natural circular frequencies, for uniaxial loads, for biaxial loads, and for shear loads, where Vb is the frequency coefficient, Ku is uniaxial load factor, Kb is the biaxial load factor and Ks is the shear load factor. Pu, Pb and Ps are the inplane uniaxial, biaxial and shear loads respectively. NUMERICAL RESULTS. Preliminary calculations were initially undertaken to determine the finite element mesh which would produce acceptable converged results. Such a study lead to the conclusion of the use of the meshes shown in figure 5, for all loading and boundary conditions considered in the present paper. Natural frequencies of unloaded plates with holes. The variation of the frequency coefficient Vb with hole size ratio b/d in the absence of inplane loads for either simply supported or clamped plates is given in figure 6. The bottom set of curves in this figure represent the case of a simply supported plate while the upper set are for the clamped plate. Figure 6 Frequency coefficient for simply supported and clamped plates. The frequency coefficient Vb, for plates containing circular holes are given by the solid lines. For the case of the simply supported plate this line shows that for small openings up to that given by b/d1.3, Vb remains almost constant. Thereafter Vb starts to increase slightly. These results compare well with the results given by Prabhakara and Datta( 1997). The frequency coefficient for a plate containing a square hole is also given for comparison, and this curve is represented by the dotted line. The same variation is also observed when the plate is clamped on all four sides. For a plate containing a circular hole, the frequency coefficient gradually rises as b/d is increased. This variation in Vb is also observed in the results given by Grossi, es al.(1997). Similarly for a plate containing a square hole, the frequency coefficient also rises with hole size, but the increase is to a larger extent. Examination of the equation defining the frequency coefficient (Eq. II) shows that Vb is proportional to the square root of the ratio of the mass density to the flexural rigidity of the plate. Hence the results shown in figure 6 indicate that this ratio remains almost constant for simply supported plates, suggesting that the presence of holes effects equally the reduction in mass and stiffness. For the case of the clamped plate the results show that the presence of holes, while decreasing the mass of the plate will increase the stiffness by a greater 4 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/abo

extent. This may be because the hole is in a portion where the curvature is more pronounced. All the values of Vb given in figure 6 are for the lowest natural frequencies. Their modes of vibration were also determined and a typical example is given in figure 7 for a clamped plate containing a large square hole( b/d=0.4). hole sizes of b/d3.2 & b/d=0.3. Good agreement of results are indicated in this figure. The same variation of Vb with Ku is given in figure 9 for clamped plates containing different sized circular holes. All the curves show how the frequency coefficient for each plate reduces as the uniaxial load factor is increased and the results follow the same pattern as for he simply supported ce$es. 1-4-300010 -o- lotp11 -a-tru12 r-b13413 - -trarit Figure 7 Mode of vibration for a plate with a hole - bid=0.4. Natural frequencies of plates subjected to uniaxial compression. In this section the square plates are subjected to uniform uniaxial compression, and the effect of this load on the natural frequency is examined. Figure 8 shows the variation of Vb with the inplane load factor Ku for simply supported plates containing circular holes with varying hole sizes. MAIO -o- Droll -a- brde02 a trociira(1937) fl C3 o tokl3ftrow) -4-01:034 t03.35 2 ID tanneryku Figure 9 Variation of Vb with Ku for a clamped plate. Buckling loads of plates subjected to uniaxial compression. In figures 8 and 9, the critical buckling load for each plate is obtained from the point where the frequency curve intercepts the horizontal axis. These points are plotted in figure 10 for both simply supported and clamped plates, and shows the variation of the buckling coefficient Ku with the hole size ratio b/d. For a simply supported plate containing a circular hole, the buckling coefficient Ku will fall from Ku=4 to Ku=2.96 as the hole size ratio is increased from b/d) to b/d3.5. Similar results are also observed when the simply supported plates contain square holes. This is also given by the dotted line on figure 10. 03 I 1.5 2 15 3 15 Lantana Ku Figure 8 Variation of Vb with Ku for a simply supported plate. For each plate the frequency coefficient steadily decreases as the applied inplane uniaxial compression is increased. This behaviour is exhibited up to a load factor of about 2.5 in the case of the large circular holes, and a load factor of about 3.5 for the smaller holes. These load factors represent about 85% of the inplane loads which cause the plates to buckle. For larger inplane loads Vb decreases sharply becoming zero when the inplane loads equal the buckling loads. This figure also shows the results obtained by Prabhakara and Dana(1997) for simply supported plates containing circular holes with --4 or Wan D. mu Wall K nor Flie(1917) nor SttrOSSA x nor atratrall o nor Srannyilt013 o ct or Mote Figure 10 Buckling coefficients for simply supported and clamped plates. Also plotted on this figure are the finite element results of Sabir and Chow(1983) and Shanmugam and Narayanan(1982); the 5 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/ab

Rayleigh-Ritz results of Prabhakara eta! (1997) and the experimental results of Chow(1983). All of these results are in good agreement with the present analysis; except those given by Shanmugam and Narayanan(1982) where higher values of Ku were observed. Figure 10 also gives the variation of the buckling load for clamped plates. It is seen that for the case of circular holes the mode of buckling giving the smallest buckling load remains the same, while for square holes a change of mode is observed at b/d).35. The mode shape for the lowest buckling load for a clamped plate containing a large hole (b/d4i.4) is shown in figure II. This antisymmetric mode contains 3 half sinusoidal waves in the direction of the applied compression, and 2 half waves in the other transverse direction. 1-0-6t1010-8-616631 - -t03.02-6-ticlma3-6-136114 2 3 0 5 6 7 8 9 10 laitador Ks Figure 13 Variation of Vb with Ks for a simply supported plate. Figure 11 Antisymmetric mode of vibration for a clamped plate having b/d=0.4. j- -tstlad 1-0- 61P31 1-*- 36:62 1-0-boa3 i-e-3/363.4, -0-u0015 Natural frequencies and buckling loads of plates subjected to biaxial compression. Similar results to those given in figures 8, 9 and 10 were obtained for plates subjected to uniform biaxial inplane loads, and a typical mode of vibration is shown on figure 12. For details of results the reader is directed to a thesis for the degree of Doctorate of Philosophy by G.T.Davies to be submitted to the University of Wales. Figure 12 Mode of vibration for a clamped plate containing a large hole (b/c1=0.5) and subjected to biaxial compression. Natural frequencies and buckling loads of plates subjected to uniformly distributed shear. Figures 13 and 14 give the variation of the frequency coefficient Vb with the shear load factor Ks for simply supported and clamped plates containing circular holes respectively. Figure 14 Variation of Vb with Ks for a clamped plate. All these curves show similar properties to earlier results where uniaxial and biaxial compression are applied, in that an applied load will reduces the frequency coefficient. However for a plate containing a large opening i.e. b/d0.4 or 0.5, Vb will fall sharply, as the applied shear load is increased. Hence the buckling coefficients of a plate containing a large holes is much less than for a plate without a hole. Results for the variation of the buckling loads are inferred from figures 13 and 14 and are shown on figure 15. For both boundary conditions, the shear buckling coefficient Ks steadily decreases as the size of the hole is increased. For the simply supported plate containing circular holes, the reduction in the shear buckling capacity when a large openings is present (b/d1.5), is 66% while for a clamped plate it is 60%. Also plotted on figure 15 are the buckling coefficients for the simply supported and clamped plates containing square holes. The dotted lines show a reduction in the buckling coefficient Ks as b/d is increased from b/d) to b/d9.5. When bid-0.5 the reduction in the shear buckling capacity when all edges are simply supported is 70% and when the edges are clamped is 67%. 5 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/abo

0 01 02 03 bid Figure 15 Buckling coefficients for simply supported and clamped plates. Finite element results by Rocky et 01.(1967) and Sabir and Chow (1983), are also plotted on figure 15 for plates containing circular holes, and good agreement is observed. REFERENCES. Bogner, F.K., Fox, Rt., and Schmidt, L.A. 1966, "The generation of inter element compatible stiffness and mass matrices by the use of interpolation formulae." Proc. Conf on matrix methods in structural mechanics. Dawe, Di. 1965, " A finite element approach to plate vibration problems" J.Mech.Engng. Sci. Vol 7, No. I. Grossi. R.O.. del V. Arenas, B.. and Laura, P.A.A. 1997" Free vibration of rectangular plates with circular openings" Ocean engineering, Vol 24, No. 1, pp 19-24. Gutierrez, R.H., Laura, P.A.A., and Pombo, J.L. 1987, "Higher frequencies of transverse vibration of rectangular plates elastically restrained against rotation at edges with central free hole" J. Sound & Vibration, Vol 1170), pp 202-206. Kapur, K.K., and Hartz, 13.1. 1966, "Stability of plates using finite element method." Proc. American Society of civil engineers, Journal of Eng. Mech. Div. vol.92, EM2, pp.177-195. Lee. H.P.. and Lim, S.P. 1992, "Free vibration of isotropic and onhotropic square plates with square cutouts subjected to inplane forces." Comp. and Struct Vol. 43, No. 3. 431-437 Mason, V. 1968, "Rectangular finite elements for analysis of plate vibration," J. Sound & Vibration, Vol 7(3), pp437-448. Melosh, I963,. "Basic derivation of matrices for the direct stiffness method." AIAA Journal, Von, No.7, pp.1631-1637. Paramasivam P. 1973, "Free vibration of square plates with square openings." Journal of Sound and Vibration. Vol 300), pp. 173-178. Pifko, A.B.. and Ikason, G. 1969. "A finite element method for the plastic buckling analysis of plates." AIAA Journal. 04 as Prabhalcara, DL., and Dana, RIC 1997, "Vibration, buckling and parametric instability behaviour of plates with centrally located cutouts subjected to inplane edge loading." Thin walled structures, Vol. 27, No. 4, pp287-310. Ritchie D, and Rhodes 1, 1975, "Buckling and post buckling behaviour of plates." Aeronautical Quarterly. pp28i-296. Rockey, K.C., Anderson, R.G., and Cheung, Y.K. 1967, " The behaviour of square shear webs having a circular hole," Proc. Thin Walled Structures, Crosby Lockwood, pp 148-169. Sabir, A.B. 1973, "The application of the finite element method to the buckling of rectangular plates and plates on elastic foundations." Stravebnicky Casopsis Say XXI,I0, Bratilava, pp.689-712. Sabir A.B. 1983, "A new class of finite elements for plane elasticity problems." Proc. 7th Int seminar on Computational aspect of the FEM. Sabir A.B., and Chow F.Y. 1983, "Elastic buckling of flat panels containing circular and square holes." Instability and plastic collapse of steel structures, Granada, London, pp.311-321. Sabir A.B. and Chow F.Y. 1986, "Elastic buckling of plates containing eccentrically located circular holes.", Int. 1. Thin Walled Structures, Vol. 4, pp. 135-149. Sabir A.B, Djoudi M.S., and Davies, G.T. 1996, "Vibration and buckling of uniaxially loaded square plates with square holes." Energy Week Conference. Engineering Technology, pp182-189. Sabir A.B and Davies. G.T. 1996 "Vibration and buckling of square plates with square holes subjected to biaxial and shear loads." Energy Week Conference. Engineering Technology, ppi90-199. Sabir, A.B. and Davies, G.T. I997a "Natural frequencies of square plates with eccentrically located square holes when loaded by inplane uniaxial or biaxial compression" ASME Energy Week, Book IV, Vol IV, pp 490497. Sabir, A.B. and Davies, G.T. 19976 "Natural frequencies of square plates with eccentrically located square holes when loaded by inplane shear stresses" ASME Energy Week Book IV, Vol IV, pp 425-431. Sabir, A.B. and Davies, G.T. 1997c "Natural frequencies of square plates with reinforced central holes subjected to inplane loads." ASME Energy Week Book IV. Vol IV, pp 383-390. Shanmugam N.E., and Narayartan R. 1982, "Elastic buckling of perforates square plates for various loading and edge conditions." Int. Conf on finite element method, Shanghai. Timoshenko, S.P., and Goodier, IN. 1970, "Theory of Elasticity". Second edition, McGraw-Hill Book Company. 7 Downloaded From: http://proceedings.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/conferences/asmep/82075/ on 11/21/2017 Terms of Use: http://www.asme.org/abo