Yarı düzlemde gömülü yapıların zorlanmış titreşimleri
Başlık çevirisi mevcut değil.
- Tez No: 46564
- Danışmanlar: PROF.DR. HASAN ENGİN
- Tez Türü: Doktora
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Konstrüksiyon, Titreşim, Construction, Vibration
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Elastik bir ortam içindeki yapıların dinamik etkiler altındaki davranışı geçmişte pek çok araştırmaya konu olmuştur. Yeraltı tünelleri ve boru Katları, yeraltı santralları, su altı boru Katlan bu tip araştırmaların temel problemleri olmuştur. Somuz ortam ile ilgili çalışma çokluğuna karşılık yan sonsuz ortamdaki yapıların dinamik davranışı ile ilgili çalışmalara yakın zamanlarda rastlanmaktadır. Yan sonsuz ortam kabulünün yeraltı yapılan için sonsuz ortam kabulüne göre daha uygun olmasına karşılık matematik yapısı bilhassa sınır koşullarının sağlanmasının güçlüğünden dolayı daha ağırdır. Bu çalışmada ise elastik bir yanını uzay içinde bulunan farklı malzemeden yapılmış ve harmonik iç zorlama etkisindeki silindirik kabuğun ve yarım uzayın davranışı analitik ve sonlu eleman olmak üzere iki ayrı yöntemle incelenmiştir. Birinci bölümde,problem tanıtılarak literatürde bulunan konu ile ilgili önceki çalışmatotriffî ve bu çalışmada kullanılan yöntemlerden bahsedilmiştir ikinci bölümde çalışmada kullanılan her iki yöntem için de geçerli olan temel denklemlerin elde edilişi açıklanmıştır. Üçüncü bölümde anlatılan analitik çözümde <p ve H» Helmholtz yerdeğiştirme potansiyellerinin bilinen tanımlan kullanılarak silindirik koordinatlarda iki adet dalga denklemi elde edildi.Bu denklemler Bessel fonksiyonları- trigonometrik seriler yardımı ile çözülmüştür. Integrasyon sonunda ortaya çıkan bilinmiyenler, silindirik kabuk iç yüzeyi, kabuk- dış ortam ara yüzeyi ve serbest yüzey üzerinde yazılan sınır koşullan yardımıyla hesaplanmıştır. Dördüncü bölümde Problemin Sonlu eleman yöntemi ile çözümü verildi. Bunun için bölge sonlu ve sonsuz kısımlara ayrıldı. Sonlu bölgelerde sonlu elemanların sonsuz bölgelerde sonsuz elemanların uygulanması anlatıldı. Beşinci Bölümde verilen sayısal işlemlerde kullanılan sonlu eleman ve analitik yöntem sonuçlan ayn ayrı elde edilip birbirleri ile karşılaştırılmıştır. Sayısal işlemler sonunda o> zorlama frekansı, silindirik kabuk yarıçapı ve ekseninin serbest yüzeyden olan derinliğine göre silindirik kabuk ve serbest yüzey üzerindeki çeşitli noktaların düşey doğrultudaki yerdeğişimlerine ait sonlu eleman ve analitik çözüm değerleri aynı grafikler üzerinde karşılaştırılacak şekilde verilmiştir. Küçük açısal hızlarda ve küçük yarıçaplarda sonlu eleman yönteminin, büyük açısal hızlar ve büyük yarıçaplarda ise seri yönteminin daha iyi sonuçlar verdiği anlaşılmıştır.
Özet (Çeviri)
SUMMARY l.INTRODUCTION Dynamical analysis of the underground structures has been subject of a lot of studies. These probleme are of interest in earthquake engineering, and in other areas of engineering where it is important to know the dynamic resporae of the shell structures. Examples are buried pipelines tunnels,etc. Previous works like these were done by Datta and his colleagues for the case of haımonic waves incoming from infinity. in these works, some soiution technigues are used as like the method of matched asymptotic expansions, the method of successive leflections, hybrid finite element and eigenfunction expansion technique, series of Bessel-trigonometric functions, and ete. Balendra and his colleagues dealt with dynamic response of buildings due to trains in underground tunnels. Ih these woıks, the di&placemenis fields are foımulated by the method of wave function expansion and the boundary conditions are satisfied onty at a finite number of points along the traction fiee surface, the tunnel-soil interface and the soil-foundation interfece. The tunnel and the foundalion have been assumed as being rigid. This work discusses the dynamic response of a homogeneous, isotropic and perfectly elastic half-space \vhich contains an elastic circular cylindrical hollow shell of infinite length at a finite depth below the plane boundaıy and parallel to it. Loading is an inner hannonic pressure. 2. STATEMENT OF THE PROBLEM Since the geometry, the material properu'es and inner pressure are îndependent of the thiıd coordinate,the plane strain case is considered. A section of the medium by a plane normal to the axis of the circular cylindrical shell is represented in Fiğ. 1. x /' ı II/ rH / Fiğ. 1. Geometry of the problemThe y-axis is isken parallel to the free surface and the x-axis perpendicular to it. The z- axis are assumed to be drawn perpendicular to this plane section. Assuming that the materials of the shell(denoted by I Fig.l) and the half space (denoted by H) are linear, elastic,and homogeneous. 3. EQUATIONS OF MOTION In the case of loading of the inner surface of the shell by a harmonic pressure p = p0c«(ot (1) the Navier equations are written as equations of motion (A.+ p)VVfj+p.VaÛ+p<o2Ü = Ö (2) when the strains are asumed to be small. Here Ü stands for the displacement vector, p the density and X, p. the Lame constants of the medium. 4. ANALYTICAL SOLUTION We note the highly complex nature of the displacement equations (1) of motion. It is possible to obtain a simpler set of equations by introducing the scalar and vector potentials q> and^'^Ve, such that Ûö>=V«p(i)+Vx(M({i)^),V-(i|>(0^) = 0 (3) Here i=l region I (Shell), i = 2 region II (half space). We obtain V2<?m+(k?)2*(i> = 0 (4) Where kf ' = <d / Ş? * dilatational wave number kf = ©/p^ shear wave number pt(i) = ^(x(i)+2|i0))/Pö) dilatational wave velocity P2(i> = i/u.(i)/P(i) share wave velocity The polar components of ü and stress are given by ör r 56 u(0_iaq>w a»< 8 r a© Ör 0» = (X»+2lx»)ŞÜ+^(ü»+^, (6) ör r öv oe<i> = X(i)^-+l(Xe>+2|i(i))(Ur0)+^-) dr r 56 11 <*> flTT® Ml® a,=ii_(r£^_-U«+^L_) r ör 00 VIThese Equations (4) are solved by using the serieg of Bessel-trigonometric fiınctions. <P (° = İK“ Ja(k «r)+B« Y« (k«r)] Coş n6(7) a-O V« = 2[c.wJE(k2«r)+D«Y®(k«r)]8m0e n-O here A^, B^°, C(B°S D^° aretxxnstants and vwmedİhe syn^netry conditionfor 6. Equations (5) and (6) can be expressed in temıs Begsel-trigonometric series by substituting Eqs.(7) into Eqs. (5), and Eqs. (7) into (6). Ur(İ) = Zk(İ>(7J.<k«(İ)^^^^(8> +C,f)-JB(k2(i)r)+Df-Yll(k2ö)r)] C«n8 rrJ Ue(i) = î[-A,ö)-J.(kfr)-B«°YE(k<i>r)-C«f°JB(kfr)-kfJl+1(kfr)).-oL rTv*/ -D«)[|Y.(kfr)-k«>Yll+1(k«>r)j j SinnS c<» = ±2^ [f£!ç£-(l+0.5^-)(kf)2 V.(kfr)^^ J.+,<kı(i)r>l(9) «-o [ LV rI1yrJ +^>|”^-(1+0.5^.)(kf)2JYll(kf>r)+^Yll+1(k1<i)r)j +C«^JB(k«r)-k«^(kfr)j-HDf^Y.(kfr)-kf ^YB+1(kfr)j }C««16 c« =£2 J^^-^-^-Ckf )2 jJ.Ck^D-^Ck^)] +^{(-^~^(k<(i))a)Y'(k“>r)”JTY^(kt(ör)]+ +P»(-İJ£J.(k^«r)+kf f J^(k2<»r)j+D«^-^£Y.(kfr)+kf ^(tfV)j J Co.n6 viiT® VÎIIİA'''! n*~n T /ı.('),\. ı.O)flT /vö)r\ ]. «(öf n2~py M-(İ),N, fc(')By n-c>ril *re = JLZ(11A»r~Jn(k1 U+'l - Vl(*1 r> +B»r~Yn(k1 r)+*1 -Y»+l(k1 T) »-Q l V r')\ Tr) '( -2“At W 'l +C«> [-^+0.5(k«)2JjR(kfr)-^JB+1<k^)j +D«[f-^+0.5(kf)2JYR(k«>r)-^Yll+1(kfr)] j Smn6 4.a BOUNDARY CONDITIONS in the case of loading of the inner surface of the shell by a harmonic pressure p = p0Coe<ût, The boundary conditions at r = aare cf=-P.,t<« = 0, r=a(10) it will be assumed thai the contact is perfect, so that the displacements and tractions are continuous across the interface between regions I and ü. Thus Uf'-^.uj0-^ r = b(11) o«-0?»ft8-t« r = b(12) Furtiıermore, on the fiee gurface, x = H, we have of = 0,^ = 0 r = H/Co66(13) Because, difEtculty of using infinite series, we had to truncate the series in a ünite number N. The boundary conditions at the inner surfece of the ehell and continuity conditions at the interface are satisfied exactfy. But, the conditions at the free surface are too more much than we need. The boundaıy conditions at fiee surface are satisfied approximately as follows: J= î f(0l(2) )2+(t£> )2 j dy = minimum(14) ^f^il0T^”“^ ]*=o' ”u. n=°-u-- os) After this, the unknown integration constants are calculated. Choosing definite values for the shell geomeüy, the material propertieg, the depth of the shell axes fiom surface, the displacements of some special points have been investigated \vith respect to forcing frequency «. 5. FINITE-ELEMENT SOLUTION The finite element method with the use of a computer has lapidty become öne of the most powerful tools in solving the complex problems of continuous media in general, in this method, a continuum is divided into many small elements of convenient shapes-triangular,quadrilateral, ete. Choosing suitable points called“nodes”\vithin the elements, the variable in the differential equation is written as a linear combination of appropriately selected interpolation functions and the values of the variable ör its viiivarious derivatives specified at the nodes. Using variational principles or weighted residual methods, the governing differential equations are transformed into“finite element equations”governing all isolated elements. These local elements are finally collected together to form a global system of differential or algebraic equations with proper boundary and/or initial conditions imposed. The nodal values of the variable are determined from this system equations. 5.1 IZOPARAMETRIC ELEMENT Among the two-dimensional elements, the triangular element was the first investigated in the early days of development, although in recent years the four-sided isoparametric element has become more popular. The name“isoparametric”derives from the fact that the“same”parametric function which describes the geometry may be used for interpolating spatial variations of a variable within an element. The izoparametric element utilizes a nondimensionalized coordinate and therefore is one of the natural coordinate elements. Izoparametric element (quadratic variation) is described below. Consider an arbitrarily shaped quadrilateral element as shown in Fig.2. (-1,-1) (1,-D Fig.2. Izoparametric element (quadratic variation) The izoparametric coordinates ^ whose values range from 0 to ± 1 are established at the centroid of the element. The reference cartesian coordinates ^ are related to ^ as xi = ai+aij^+Op4j|k+ap^4k|1 (16) for the two-dimensional quadratic element in Fig.2. Here, i,j,k,l=l,2 and Op=0 for j = k = I. Writing (16) in terms of nodal values yields Here the subscript (N) denotes the node number and is not to be summed. In a matrix farm, we may rewrite (17) as IXHere the coefficient matrix C is given by (18) C = -1-11 1 1-1 -1 1-11-11-11 1111111 -111-111-1 0-10010 0 10 10 0 0 0 0 10 0 10 0 -10 10 0 0 0 Thus, 0,-e1^ (19) Substituting (19) into (18) yields Xi = »ft,a)XKi (20) Here H^C^i) is called the isoparametric function and has the form at coiner nodes: V(4Ni)=0.25(1+^1|,)(1+^2^)(^t|t + |K2^-1) (21) at midside nodes: V(^i)= 0.5(1-^)0+^ 4*) for ÇM, = 0 (22) M»(^i)=0.5(1+4Hilr)(1-^) for ÇM2 = 0 (23) 5.2 FINITE-ELEMENT-EQUATIONS For the finite-element-equations the Navier equations (2) can be expressed in the form dx (C,i au dx +c. 12 8V dy )-c a,au avv 2tt _ ay dy dx' (24)“ a,au av, a,”au“ av, 2”_ -C33- (-T-+-T-)-- (C12--+C22- -)-p<D2V = 0 ax ay dx dy ox. dy Where C^Cy = C^) are the elasticity(material) constants. The variational form of Eqs.(24) over an element Q* is given by multiplying the first equation with a test function w, and the second one with a test function w2 and integrating the results by parts We have aw, aw,,au av J [ -r-1(Ci,-r-+C12- -)+C33--!-(-- +-- )-WlP©2U]dxdy- 4w,t,<b=o (25) J dx dx dy dy dy dx J tr“ aw2/au avv aw2,”au“ avv ”, 2“,A. r”,.“ J[ C33-^(-T-+-r-)+-r-1(C12-- +CJJ-- )-W2p«2V]<kdK--4w2ty & = 0.J, 3x 3y dx dy dx dy L Where tx andt, denote the boundary forces defined as follows.tI = (C11--+C12--)nI4C33(- -+-- )ny=tI (26) ox oy oy ox ”,au avN,“ au ”av, V SC33(--+- -)nI+(C12-- + Ca-- )ny = t^ ' öy dx 8x dy ' y Let U(,) and V(,) (displacements over element e) be approximated by finite element interpolation of the form u« =£“«!#>, v^f^vf* (27) j-i j-t where i^e) are appropriate interpolation function for the variational form in Eqs. (24) Using the finite-element interpolation of the form (27) for Uand V, and substituting W, = vfi and W2 = q»j into Eqs.(24), we obtain {[^l-P-^MJJW+MW-M (28) [K»]{«} + {[K»]-p.»[M]}{v}«M Where «f.J,C,M+çD^l>** (29) i öx öx dy öy Mjj = j Vi M»j <k dy, Ff = ^Vi tx &, If 2)= JVi ty ds Note that coefficients matrix JK12],for example, corresponds of V in the first equation (i.e., the first superscript corresponds to the equation number and second to the variable number). 5.3. INFINITE ELEMENTS In the physical world, problems are frequently encountered in which the region to be analysed is of very small dimensions compared with those of the surrounding medium. Examples occur in underground openings, in either tunnelling or mining operations, and in wave propagation and fracture mechanics. It is natural to idealize these problems by considering the boundaries remote from the area of interest to extend to infinity, and hence their description as ”infinite domain“ problems, traditionally, in the finite element method, the unbounded domain is idealized by extending the conventional finite element mesh outward to a point where the influence of the xistructure is small enough to be neglected. This approach generally requires an experimentation with several grid sizes and assumed boundary conditions. Results obtained are generally good for static problems but the method is not suitable for many dynamic analyses. A recent development is the mapped infinite element [28] which is generally applicable to such unbounded problems, particularly when used in conjunction with conventional elements. In this paper the mapped infinite element is coupled with conventional isoparametric elements Mapped infinite elements are based on a simple mappimg technique that applies to both the modelling of the geometry and the field variables. The reference cartesian coordinates x,y are related to ^.n as x(4,Tl)=£xiMi, yfoitf^yiM, (30) i-1 i-1 where M( are the mapping functions. 5.3.a. SINGLE INFINITE ELEMENTS This term is used to denote elements which extend to infinity in one direction only (chosen to be then direction). Fig.3. 2 node superparametric element Mapping and shape functions for 2 node superparametric element are below. Mapping functions: Mf = <1-!X-T0/(1-n> (31) M2=(1+Ç)(-nV(1-Ti) M3 = (1+«0+i|)/[2(l-Tl)] M4=(1-Ç)(1+n)/[2(l-n)] Shape functions: ift = (1-S)0-T|)/4 (32) V2 = (1+4)(1-n)/4 xu5.3.b. DOUBLE INFINITE ELEMENTS This tram is used to denote elements which extend to infinite in two directions. They can be used as ”comer" elements in a mesh where transition is needed between zones extending to infinity along different directions. 4 Fig.4. 1 node superparametric element Mapping and shape functions for 1 node superparametric element are below. Mapping functions: M,= 4Çn/[(1-|)(1-Ti)] (33) M2 = (1+4)(-2n)/[(1-^(1-T,)] M3 = (1+i;)(1+T,)/[(1-Ç)(1-Ti)] M4=(-2Ç)(1+t,)/[(1-|)(1-tO] Shape functions: M», = (1-^)(1-h)/4 (34) XUl
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