Gemi levhalarının elasto-plastik olarak incelenmesi

Abstract

ÖZET Bu çalışmada, bir geminin düzenli denizlerde seyretmesi sıra sında, bünyesinde oluşan iç kuvvetlerin alt ve üst sınır de ğerleri incelenmiş, bu değerlerin elastik, elasto-plastik ya da tam plastik sahada kalıp kalmayacağı araştırılacak, boyut- lamaya ya da yapısal gemi dizaynına esas olacak orta kesit mukavemeti ve dizayner açısından önemli olabilecek boyutlama yöntemi geliştirilmiştir. Ayrıca, boyuna simetri düzlemindeki boyuna eğilmeden doğan düşey eğilme momentinin gemi boyunca değişimi frekans domeninde, frekansın bağlısı olarak incelen miş ve elde edilen iç kuvvet değerleri esas alınarak, orta kesit boyutlaması yapılmıştır. Çalışma dört bölüm ve iki ek bölümden meydana gelmiştir. Birinci bölümde, konunun önemi belirtilerek, böyle bir çalış maya neden gerek duyulduğu açıklanmış, uygulamaya dönük nokta lar incelenerek, Gemilerin orta kesitlerinin boyutlaması üze rine yapılan elastik ya da plastik çalışmalar gruplara ayrıla rak, kısa özetler verilmiş, boyutlamada bu günkü durum anla - tılmıştır. İkinci bölümde, gemi ve çevre akışkan arasındaki, karşılıklı etkileşimleri de gözönünde bulundurarak sistemin hareket denk lemleri elde edilmiştir. Bu denklemlerin elde edilmesinde, ana litik mekaniğin varyasyon ilkelerinden Hamilton-Ostrogradskii ilkesi kullanılmıştır.- II - Gemi ve çevre akışkan ortamının kinetik ve potansiyel enerji leri ile birlikte sönüm kuvvetlerini içeren enerji ifadeleri yazılıp neticede Lagrange enerji fonksiyonluları elde edilmiş, ve bu fonksiyonun belli zaman aralığında varyasyonunun denge konumunda sıfır olduğu düşünülerek hareketi karakterize eden denklemler elde edilmiştir. Geminin boy istikametindeki si metri düzleminde yazılan kısmî-integro-diferansiyel denklem ler, seperasyon yöntemi kullanılarak, adi diferansiyel denk lemlere dönüştürülmüş ve modal form yazılarak öz frekans ve bunlara karşı gelen öz formlar ya da Mod' lar elde edilmiştir. Her Mod'a tekabül eden orta kesit eğilme momenti elde edilmiş, elastik sınırdaki akma başlangıcına ait moment ya da tam plas- tikleşmeye tekabül eden momentle karşılaştırmakla modun elas tik ya da plastik mod olup olmadığı saptanmıştır. Daha sonra, gemi orta kesiti, iki borda levhası, dip levhası, boyuna per de 'levhaları ve güverte levhalarının birleşiminden meydana gelmiş basit bir kutu kesit olarak ele alınmış, böylece `eş değer kesit yöntemi` kullanılarak frekansa bağlı olarak elde edilen, eğilme momenti, bu eşdeğer kesite uygulanarak ve elas- to-plastik inceleme yapılmıştır. Üçüncü bölümde, Ek II de ara işlemleri verilen denklemlere sayısal uygulama yöntemi belirtilmiş uygulanan bilgisayar programının içeriği belirtilmiştir. Sonuçlarda, sayısal uygulamada çıkan nümerik değerler ve ileriye dönük çalış mayı destekleyici önerilerde bulunulmuştur. Her iki ek bölümde de diğer bölümlerde sonuç olarak belir tilen denklemlerin ara işlemleri verilmiştir.

SUMMARY ELASTO-PLASTIC ANALYSIS OF THE SHIP PLATES In this study bending moment in the midship section of a ship in a seaway is evaluated in the frequency domain and, elastic and plastic boundaries of the system are investigated. The system of a ship and fluid is taken as a dynamic system and via the use of Hamilton-Ostrogradskii principle, which is one of the variational principles of analytical mechanics, the equations of motion of the body-fluid interaction are obtained. As it is well known, if a physical system is perturbed from it's equilibrium position it will perform a certain motion. Furthermore, if this system is considered as an elastic one there will be some unbalanced internal forces and couples in the cross-section and hence a distortion of this section will occur. Obviously, the amplitude of internal forces depends on the characteristics of the motion, and the types of the perturbation acting on the system. Ship motions, by nature, are quite complex to analyse and there is always an interaction effects between different modes. The exact formulation of this problem is very difficult and it is a know fact that this interaction problem has not been defined in a complete and rigorous manner. The aim of this study is to find out the characteristics of the variations of internal forces in the cross-sections of a ship and to relate the magnitude variations to the actual calculation. In order to formulate the equations of motion the usual simplyfiying assumptions related to the fluid and its motion are made together with the boundary conditions an concerning the body and the fluid domain. In order to obtain the equations of motion for the aforementioned body-fluid interaction phenomenon, the variational principles of analytical mechanics have been used. In fact, the use of the variational technique in?-İV- the formulation of dynamic systems has some advantages. For instance, once a scalar energy function is defined, it is possible to derive the equations of motion of this dynamic system under the linearity assumption, the energy function, can be formulated for a ship and it's surrounding fluid and by the use of superposition principle it becomes possible to derive the general equations of motion for the whole system. By employing the Hamilton-Ostrogradskii principle the equation of motion can be written in a general from as follows : ti n -* ^ -? / (<5T +.Z F.. 6r +. fff X div ör dv) dt = 0 (1) t- 1=1 1 v uo One can assume that some of the forces acting on the system are conservative and they are derived from a potential function. In addition to these in the variable calculations, one can consider that all the constraints are to be taken in to account. Consequently, equations (1) may be written as: t, n -> i / (<5T - 6n + iE Fi# 6r)dt = 0 (2) to ` Here: 5 stands for variation, T is kinetic energy of the system, Fj represents the forces on the system;X is a Lagrange multiplier, r is the position vector, v is volume element, t is time and ir is the potential energy of the system. All of the parameters appearing in the equation (2) should be defined with respect to the fixed coordinate axes. For that reason a cartesian coordinate system is defined according to right hand rule in such a way that x-axis will be in the longitudinal direction of ship, positive to the bow, the origin is located at the center of gravity of the section at aft perpendicular. The kinetic energy of the system is considered to be the sum of two components: T-, and T2, for ship and for the surrounding fluid, respectively T^ has the characteristics of the translation and rotation and may be defined as:?V- Tl ^`~ WT MW +[h]T[i]{w} (3a) 1 2 2 where [m] is a mass matrix, [l] is a mass of inertia, in the analysis, it is assumed that the cross-product of mass and mass of inertia terms are constant during the motion. The variation T^ may be written in terms of the variation of the vector elements of (v) and {w}. Hence we have, t - T- 6 / T. dt - - / / ([ m. v]<5v 4 [mnr> w] ow + t0 X t0 o X1 + [lxlV + I21$+ I310]«^ + [I12^+ x22* + I32^6lJ; + + [l31<P + I23^I33^] 66 )dx dt (3b) At the outset of the formulation, it is assumed that fluid motion is characterized by the velocity potential $ =.(j)(x,y,z;t). There fore, the kinetic energy of fluid is, 1 T. = - p / (V<>.Vcj>)dv (3c) 2 D and is variation is, ti t, 1 6 / T dt = / <$[-- p/(V<J>.V<>)dv]dt (3d) Co to 2 D where p is the density of the fluid, D is a total volume of the. fluid domain.-VI- It is, further assumed that the surface and volume variations caused are negligible by small waves. Eq.(3d)l is replaced by the equivalent relation, 16 T2 dt = / p / V<J> Vöf dv dt (3e) to t0 D By means of Green Theorem above equation is rewritten as follows 2 9<s* nn p / (V(j> V6<J> dv= -p / <f> V (6<J>)dv + p / <f>ds KJZ) D D I 3n where S is a fluid surface n* is an outward normal of surface. Ön the other hand we also have, V2<f> - 0 and A2(6(J>) =^SV2<f> =0 (3g) d<t> 94> p/4>ds=lD / <j>ds (3h) Z 9n S^+Sf an and hence, ti ti 86<J> / 6* T dt = / 6 / <j>ds dt (31) where 0 is the velocity potential and taken as the sum of two velocity potential namely <f>, and $2 an<* may be written as follows -f 4>(x,y,z;t) = ^1(x,y9z;t) + <>2(x,y,z;t) (3k)-VII- tx ti 6 f T2 dt = -p / / {[(HvY2)Y2+(Hiiry3)Y2+(HVY4)y2+ t0 t0 (H$Y5)Y2 + (HeY6)Y2]<Sv+[(HVY2)Y3+(HWY3)Y3 + (H<py4)Y3 +(Hİj;Y5)Y3+ (H6y6)Y3]<5w+[(HvY2)y4 + (H^Y3)Y4+(H^Y4)Y4+-(H^5)Y44- (Höy^yJ 6<P + [(hvy2)y5+(h^3)y5+(h«Jy4)y5->-(hİİ>»'5)y5 + (H8y6)y51^+[(Hvy2)Y6 +(Hwy3)y6-+(H(J;y4)Y6 + (H^5)Y6+(HeY6)Y6]^^2+HwT3+Hvr4+HÎnr5+HeYjST +Hxf Y26v + H^y3 <$w + H^Y4<5 -tH^ &f> +K{Çj6 59}ds d t (3m) Referring to the usual assumptions already made earlier in the analysis the elastic potential energy of ship-like body may be defined as nx = - / (Lw]TU]{w> + [flTtcQ {$}) Jx (4a) 1 o During the motion, [b] and [c] rigidity matrix elements are to be constant and hence we have,-VIII- t, fcı L <5 / nx dt - - / / <[b ^]' ^[b^^fr^e]' 66 t0 -*[cu(v-Y>)] 6v4-cu(v/ -&&p+[c22(w-}/>j}' 6w + c22(w-i;)6i;)dx dt (4b) At the same time, the variation of II may be written as follows, t i «-i 6 S 7T2 dt = pg / / -{TrY2öv+tY35w+ t?Y4«S^5ÖTp+nÇY66e trt t0 SF + CC +vy2-*wy3+V^4+ 4fY5^ÖY6)öîT} ds dt (4c) on the system, the potential forces and their variations -muv - p / [(HvY2)Y2+(HwY3)Y2+(Hn4)Y2+(HifY5)Y2+ x (H6y6)Y2] dlx -p/ (H1C+gç)Y2dlx*[c11<v'-«]' T(x) +[a4c11(v/-^J-/ [a^v + a*2w +a*3* +a*^ +*£.§] dlx = 0 *x (5a) ~m_0w-p / (HvYJYQ-(HwYJY-r^H^.)Y`-(Hi;Yr)Y.-(HeY.)Yjdl: J-X p / (H1C-gÇ)Y3 dlx-C22(w-^)-a5C22(w-i(;) lx ` ' I a21V ` 322W ` a23 ` a2h* ~ a256 I dlX = ° 1(X) (5b)-IX-.Iu<? -I214'-I3ie-P / [(HvY2)T4+(H^Y3)Y4+(Hn4)y4^(Hii;Y5)Y4+ (H9Y6)Y4] dlx - px/ (H1ç+gÇ)Y4 dl^C^fK^b^']' +Cn(v'- V)+a4 Cn(v'-Cf>)' -/ [a^v +*%2**i%$ +*%$ ^^I^-jT0 lx (5c) -I12lf-I22^-I32e-p / [(H^Y2)Y5+(HwY3)Y5^(H(fY4)Y5+WY5)Y5+ ^x (HeY6)Y5]dlx-p / (H1£+gÇ)Y5 dlx+tb22 l*Ca2 b22^' lx + C22(w-i^ )+a5 c22(w-i; )`+ / [ (a^+a^w+a^^ + lx a*ıj» +-a.,01 dl = 0 (5d) 44 45 -1 x.I13*-I23'i; `W ` p/ [(H5Y2)Y6+(HwY3)Y6-(HVY4)Y6* X (H^y5)Y6+(H0Y6)Y6]dlx-p / (H^+gOYg dlx + 0>336]'+ ?'?x + [a3 b339f- / [a^v +a*,,w ta*3« +a«4* +a^SJ dlx - 0 *x (5e) HvY2+HwY3+H(pY44-HİjJY5+HeY6 +H]LÇ+ g(C+vY2+wY3+y>Y4^ ^Y5+ÖY6) = 0 (5f)-X- The equations could be written as into two ordinary differential equations for the vertical plane, of the ship. From the solution of these equations we obtain m22(x)w(x,t)- [c22(x) {3(x,t)4a5(x,t)}j -?(x,t) (6a) I22(x)$ (x,t)-[b22(x){iJ>'(x,t)-KX2(x) $'(x,t)}]- -c22(x) [S(x,t)+a5(x) 3 (x,t)] - ~£#(x,t) (6b) w (x,t) = i/> (x,t) + $ (x,t) (6c) where 3 (x,t) represents shear strain. These equations may be solved in modal form w(x,t) = I qk(t) wk(x) (7a) *<*»*> = fcSo %(t) Vx) (7b) ^X't} = k^ oqk(t> 3k(x) (7c) The generalised coordinate q (t) is a measure of the deformation in the kth principal mode wk(x). The expressions for the deflections given by equations (7a to 7b) are substituted into the equations (6a to 6c), Following this, the equations of vertical and rotary are multiplied by ws(x) and ^s(x), respectively and these products are integrated along the ship. After the necessary orthogonality conditions have been employed, a set of equations is found to govern the variation of qs(t). They are-XI- ı^ıf>s]dx (s-0,1,2,3,...) (8) where y, is the generalised mass, <o, is the natural frequency, &,A. are the damping coefficients, ^ is the vertical fluid force per unit length, (J^ is the fluid moment per unit length.