Değişken kalınlıklı halka plakların statik ve dinamik stabilitesi

Abstract

ÖZET Bu çalışmada değişken kalınlıklı halka plakların, radyal doğrultuda etkiyen uniform düzlem içi yükler altında, statik ve dinamik stabilitesi incelenmiştir. Bu 3maçla sonlu elemanlar metodu kullanılmıştır. Plağın sonlu eleman modeli kurulurken dairesel simetri özelliği dikkâte alınmış ve plağın sektör tipi eşparametrik elemanlardan oluşan, daire dilimi şeklindeki simetrik bir parçası modellenerek dalga yayılım tekniği ile plağın bütünü için analiz gerçekleştirilmiştir. Bu yaklaşım bilgisayar kullanımında gerek en az düzeyde hafıza gereksinimi göstermesi, gerekse hızlı sonuç alınması bakımından önemli üstünlüklere sahiptir. Eleman kalın ve ince plaklar için kullanıma uygundur. Halka plağın kalınlığı, yarıçap doğrultusunda h=hm3X(r/r0)+5 bağıntısına göre dış kenara doğru kalınlığı artarak veya h=hmax(r/ri)-*' bağıntısına göre kalınlığı azalarak değişmektedir. Yapılan çalışmada değişken kalınlıklı halka plakların statik stabilitesi incelenirken, kalınlık değişi miride ki bu iki farklı durumun, katsayısı ndaki artışın, sınır şartlarının ve delik büyüklüğünün, kritik burkulma yüküne ve burkulma modlarına etkisi araştırılmıştır. Dinamik stabilite analizinde bu faktörlerin ve sabit yükün dinamik kararsızlık bölgelerine etkisi incelenmiştir. Kullanılan modelin doğruluğunun kanıtlanabil mesi için elemanın geometrisi değiştirilerek kalınlığı lineer olarak değişen halka plakların serbest titreşimleri ve statik stabilitesi ile ilgili elde edilen sonuçlar, diğer araştı rmacıların çalışmalarıyla mukayese edilmiş ve bulunan değerlerin birbirine yakın olduğu görülmüştür. Geliştirilen bilgisayar programıyla karmaşık kalınlık geometrisine sahip dairesel plakların statik ve dinamik stabilite karakteristiklerini incelemek mümkündür. 3d

STATIC AND DYNAMIC STABILITY OF VARIABLE THICKNESS ANNULAR PLATES SUMMARY Recenty, lightweight stuctures have been extensively used as fundamental structural elements in various industrial fields. Therefore the buckling and parametric resonance analysis of plates have increased in practical importance. In this study the static and dynamic stability of annular plates subject to in- plane loads is investigated. It is a well known fact that when in- plane load reaches to a critical value, the plate looses its orijinal shape and out-of- plane deformations take place. This is called static stability. The load at which buckling occurs is thus a desing criterion for the plate. The other well known fact is that a plate subject to in- plane forces generally experience forced in- plane vibrations and when exciting frequencies overlap with the natural frequencies of the plate, resonance will occur. Apart from this phenomena, for certain values of exciting frequencies an entirely different type resonance will occur in transverse direction and annular plate is said to be dynamically unstable. This type of resonance is also called parametric resonance. In the dynamic stability problem the spectrum of values or the parameters causing this unstable motion is investigated and the regions of the dynamic instability is determined. Constructions of the light structures in the present day technology makes it essential to predict these dynamically unstable regions for different structural elements In the present study variable thickness sector type isoparametric annular plate element is employed to investigate the static and dynamic stability problem. All the applications presented in this study are analyzed using the wave propagation tecniques. The advantage of this method is that only a single sub-structure is represented by a finite element model. The thickness is either increased or decreased in outward radial direction by the equations h=hmax( r/r0)+x or h=nmax(r/n)-* respectively. The effect of thickness variation, boundary conditions, hole size on the critical buckling load and buckling mode is investigated. The instability regions of annular plates with parobolic thickness are determined for wide range of exciting frequencies using Bolotin Method's with effect of static forces taken into account. The choice of the element in sector form enables the structural analysis to solve dynamic stability of annular plates with sectoral cut-outs. The sector element developed İ3 based on the Mirdlin plate theory. The element, which is shown in Figure 1, has 8 nodes, 24 degress of freedom and has variable thickness. The degress of freedom per node are the transverse displacement, and the radial and angular slopes. Numerical integration is used to evaluate the kinetic and potential energies of the element. The element yields good results for both thin and thick plate. The sector element is referred to a set of cyli ndrical coordi nates, r, 8 and z, and a set of non-dimensional curvilinear coordinates tj, Ç, Ç with the following relationship between the two spaces [26]: Xu! = 0.5q0.5(-iî+ıj2)h1 +(l-r2)]i2 +0.5(r+r2)li3 r=0.5(r3-r1)+r1+0.5(r3-r1)n 8 = -6% 6T 5- T] O: Displacerr^nt rı«te o : Geometric node (D Figure 1. Isoparametric sector element. Let Ni(^ rj), i=l, 2,, 8 be the tvo dimensional parabolic stepe functions of the isoparametric element. Also let Vj, §/, arid % be the transverse displacement, radial slope and angular slope of the element nodes, respectively. With reference to the Mindlin's plate theory, the displacement field within the element is approximated by [26] 0 sNiCtn) 0 O O -zNi&ri) Ni(tn) O O. ^i (2) where u, v, v are the deformations in the r, e, z directions respectively and shape functions of the isoparametric element are [25] Hi(tn) = /v +&»)fl +no)«o +no - *) i= 1,3,5,7 Ni(tn)=i(ı-^)(i+îio) Ni(tn)=;(i+^o)(ı-n2) i =2,6 i =4,8 (3) XillThe basic assumptions of thick plate theory as presented by Mindlin are 1 25]: 1: The deflection of the mid-surface of the plate is small in comparion with the plate thickness. 2: The transverse normal stress is negligible. 3: Normals to the mid-surface of the plate before deformation remain straight, but not necessarily normal to the midsurface, after deformation. With assumption 1 and 2, strain energy of the plate is given by [26] U=^Jv(e]T[D][c]dv (4) where V is the volume and, in cylindrical coordinates. [c]T = [err e* 2t# 2e` 2^] (5) and [D] = (6) The strain-displacement relations are Hz 2C T2 2e, 9zJ i « i-i n O 0 3_ 3r 12. r 36 J Using assumption 3, the displacement can be expressed in the form (7) (8) where y, $ are the rotations about the r and 8 axes and w is the normal displacement of the middle plain. Equations (4) to (8) together give the strain energy. XIVThe kinetic energy is given by [24] lf f'.2.2.2^ T=- pi +y +v dv (9) where u, v, w, are the velocities. The additional strain energy in bending due to in- plane stress trr, Tre, Tee is 126] lf f (b&f 3dl3d fl3df dv (10) Substituting equations (1) arid (2) into equations (4), (9) and (10) give the stiffnes, mass and stability (geometric stiffness) matrices respectively. Numerical integration using (2x2), (3x3), arid (2x2) Gaussian meshes is used. Figure 2. The in- plane loads acting on the annular plate. The in-plane loads are assumed to be acting on the annular plate along the inner and outer edges as shown in Figure 2. Under the uniform loads per unit length of P^ snd Pout respectively, the force distribution in the radial and angular directions are given by [27] XVr2r2 1 Pr2-Pjî n r2-r2(W ^r2 rg-r? Pf«=0 When a flat plate is subjected to periodic in- plane forces the governing matrix equation of motion is [MJ{Y)+[Kj{t}-[Kj{q} = 0 (12) where [M] = Mass matrices of total sturucture, [K*] = Elastic stiffness matrix of the total stucture, [Kg] = Stability (geometric stiffness) matrix of the total stucture which is a function of the axial load P. It is customary to express the periodic radial force as follows PW = Po+ptCOSCDt <13) where P0 = In- plane static load Pt = In- plane amplitude of time dependent load, If the static buckling load P* is chosen as a measure of the magnitude of P0 and Pt then the expression can be written in the form, F(t) = c4P*+pP*cosort (14) in which a and p are percentages of P* and oj is the frequency of the exciting force. Substitution of equation ( 1 4) i nto equation ( 1 2) gives a more general equation [M]Ü} ¦ {M-«P*[Kgs]- PP*cos artpKj} {4} = O (15) The analysis of a given stuctural system for a dynamic stability implies the determination of boundaries between the stable and unstable regions. In Bolotin's method of analysis, the boundaries between stable and unstable solutions are formed by periodic solutions of periods T and 2T where T=2n-/to [9Î. For solutions with 2T, «a* K-* T foot but! U}= I KalkSin^`--*- Wkcos- (16) k=l,3,5L z 2 J and periodic solutions with a periodic T in the form XVI.OS» (4}=rO>}o+ I {a^sin- +{^003 - (17) where {a^} arid {ft^} are the vectors independent of time. Substitution of equations ( 1 6) and ( 1 7) into equation ( 1 5) leads to eigenvalue system for the dynamic stability boundary. For the solutions with period 2T ^]-^*[K«s]±pP*[K«t]-TtMl = 0 for the solutions period T [ig-c<P*[Kgî]-tû2[M] = 0 and (18) (19) = 0 (20) It has been shown by Bolotin that solutions with period 2T are the ones of greates practical importance and that as a first approximation the boundaries of the principal regions of dynamic instability can be determined from the equation [9]. M-^l^VfoJ-^M {*} = 0 (21) The two matrices [Kg^J and [Kğtl will be identical if the static arid time dependent components of the loads are applied in the same manner. If [K^]=[KgtMS], arid elastic stiffness matrix is repsesented by matrix [K], the equation (21 ) becomes W-(a*5p) P*[S]-T[M] fo} = 0 The solutions of equation (22) provides; (i)- static buckling load when oc= 1, p=0 arid ©=0 [m-p*ı«fl{ı)-o (22) (23) xvn(ii) -dynamic stability when all terms are presented When ot=1, p=0 and oj=0 then the solution of eigenvalue problem defined by equation (22) yields the non-dimensional critical buckling load parameter k v/hich İ3 defined as K = P*r2 (24) where D is the flexural rigidity of the plate given by E 4; D= - ^V (25) 12(1-D2) V ' Solution of equation (21 ) for the case of dynamic instability regions while &, and j3 varying yields the non-dimensional frequency parameter of the dynamic in- plane force which is given as * o to =ojî3 *mm (26) Consider the periodic structures shown schematically in Figure 3. Each periodic component could be described by an identical arrangement of element. These periodically occuring elements grids are linked by the unconstrained nodal displacement on the boundaries. In such a case it is more convenient to analyse the response of the sturueture using a wave approach rather than the usual modal tecniques. When a harmonic wave of frequency travels along the structure, the ratio of the amplitutes at corresponding points in adjacent sub- stuctures can be expressed in the form e*1, where u, is a complex propagation constant. Using this fact, it is sufficient to analyze a single periodic sud-stucture. {q>1 {q}` ^- T~- ^ (q>. Wn-i Figure 3. Rotationary periodic structure and an identical substructure XViilThe symmetry of the problem makes it possible to apply the wave propagation technique ifa repetitive section of the plate as shown in Figure 4. is considered then it is possible to classify three groups of degrees of freedom {qi_}, {qi} and {qR} respectively. If the static and dynamic components of the in- plane force P acts in the same manner, the stability boundaries can be written as follows for repetitive structure [M]{q'} +[[K]- (oP*+pP*cos wt)[S]]{q} = {F} (27) The first approximation of the dynamic unstabity regions for the repetitive structure can be determined from [[K]-^±ipjp*[Sl-^[M]j{ii}= {F} (28) 1 Element R/4 Element Figure 4. Repetitive structure of the annular plate with 4 along the radial and 1 element along the annular direction For a repetitive structure degress of freedom and the forces acting will be expressed in vector form as follows. {«} = Ql «E (F} = 1*. A. (29) I n a fi nite cyclic structure duri rig a conti nuous wave propagation { F}=0. Also the relation between the two repetitive substructures is such that [37] {fcl-^Cfc} (30) ow- mi (31) xixwhere >ı is complex wave propagation constant. If the equations (28) and (30) are used together with the {F}=0 constraint then the dynamic stability equation for a repetitive structure becomes. [K<i)]-(a± ipjplSûı)]- î^[Müı)]j*J = 0 (32) The complex wave propagation constant jn can be written in terms of real and imaginary parts; [38] M = Mr+)'Mi (33) similarly the related matrices can be expressed in terms of real and imaginary components; [K01)] = [Kr(M)+jKi(»)] (34) [SÛO]-[*00+İ*ÛO] (35) (36) (37) With these equations subsituted in equation (22) a real eigenvalue problem is formulated; (te t]-H*>r* 3-?g si}M m the submatriees [Kr] and [K1] in their explicit form are as follows. 1X00] = Kll +krr +<** M (KLr +Krl) kli +Kri cos J* Kn. +Km pc13 M n J (39) IVi/ xl [(kLK-kRl)süiM -KMsm^ (40) The other submatrices have similar forms. If there are N identical cyclic repetitive substructures then the complex wave propagation constant ji takes the following values 122]. For even H. 2ti 4ti 0, -, - N N GH N, * (41) and for odd N. XX2tı 4k ° ' N` ' N` '' 2ii (N-l) (42) The stiffness, mess, and geometric stiffness (stability) matrices are formed once and for each ^ value the eigenvalue problem is repeated for every nodal diameter configuration. To verify the theoritical model, a 1 0° sector of the plate is divided into four by one elements along the radial and angular directions respectively. The effect of Xthe thickness parameter, on the physical apperance of the plate can be seen in Figure 5. and Figure 6. Ia4I>Ia3 Figure 5. Thickness variation of the annular plate for positive A S'alues <h=hmaX(r/r0)-*) Figure 6. Thickness variation of the annular plate for negative A values (h=hrr,ax(r/ri)-^) Several examples were investigated for the static and dynamic stability of variable thickness annular plates with different geometry, lvalues, hole si2e arid boundary conditions. The results are presented in a series of tables figures plotted non-dimensionally in terms of parameters k, «x, p, and oo/ The results of peresent 3tudy is compared with the analytical and finite element solutions of other researchers results wherever possible and a good agreement is observed in all cases. It has been shown that the present method can be used effectively for variable thickness annular plates with complex geometry.