Katmanlı kompozit panellerin anlık basınç yüküne dinamik cevabı

Abstract

Bu çalışmada, kenarlarından ankastre olarak mesnetli katmanlı kompozit panellerin anlık basınç yükü altındaki dinamik davranışları deneysel ve teorik olarak araştırılmıştır. Anlık basınç yükü, genellikle, patlama olaylarının sonucu olarak meydana gelen çok kısa bir zaman sürecinde etkin olan bir kuvvettir. Bu basınç yükü çok kısa sürede bir pik değerine ulaşıp exponansiyel olarak azalan bir nitelik taşımaktadır. Bu tür yükler, yüksek dinamik karakterlerinden dolayı, yapılar üzerinde etkili olmakta ve hasarlara neden olabilmektedirler. Dolayısıyla, son yıllarda uçak yapılarında kullanımı giderek artan katmanlı kompozit panellerin anlık basınç yüklerine cevaplarının incelenmesi oldukça önemlidir. Teorik çalışmada, katmanlı kompozit kabuğun hareket denklemleri Love'un ince elastik kabuk teorisi çerçevesinde türetilmiştir. Geometrik nonlineerlik etkileri von Karman varsayımları ile hesaba katılmıştır. Hareket denklemleri virtüel iş prensibi kullanılarak çıkarılmıştır. Anlık basınç yükünün ifade edilmesi için Friedlander sönüm fonksiyonu, uygun değişiklikler yapılmak suretiyle, kullanılmıştır. Anlık basınç yükünü ifade eden bu fonksiyondaki parametrelerin değerleri deney bulgularından elde edilmiştir. Hareket denklemleri bir seri çözüm fonksiyonu seçilerek ve Galerkin yöntemi uygulanarak zamana bağlı nonlineer diferansiyel denklemler şeklinde elde edilmiştir. Bu denklemlerin yaklaşık çözümü için Runge- Kutta yönteminden faydalanılmıştır. Ayrıca problem sayısal bir çözüm tekniği olan sonlu elemanlar yöntemi ile çözülmüştür. Bu amaçla ANSYS sonlu elemanlar yazılımı kullanılmıştır. Deneysel çalışmalarda anlık bir basınç yükü sağlayacak detonasyon tüpü ve gerekli donanım kurulmuştur. Detonasyon olayı LPG ve oksijen karışımının bir ucu kapalı bir tüp içine gönderilerek ateşlenmesi ile gerçekleştirilmiştir. Tüpün yakıt karışımı ile doldurulması ve ateşleme olayları bilgisayar kontrollü olarak yapılmıştır. Ateşleme sonucunda tüp içinde ses üstü hızlarda ilerleyen bir basmç dalgası geliştirilmiş ve bu tüpün açık olan ağzından atmosfere yayılmıştır. Bu basınç dalgası tüpün açık ucundan belirli uzaklığa yerleştirilen kompozit panellere çarptırılmıştır. Böylece paneller üzerinde bir anlık basmç yükü etkisi oluşturulmuştur. Panellerin orta noktalarındaki birim uzama değerlerinin zaman ile değişimleri deneysel olarak elde edilmiştir. Deneysel olarak elde edilen bu değerler birim uzama-yer değiştirme denklemlerinde kullanılarak yer değiştirme-zaman grafikleri oluşturulmuştur. Yaklaşık teorik analizler için seçilen çözüm fonksiyonlarının terim sayısı arttırılarak analiz sonuçlarının belli bir değere yakınsadığı görülmüştür. Analizler düz paneller için çözüm fonksiyonlarının ilk teriminin alınmasının yeterli olmasma karşın eğrilikti paneller için daha fazla terimin alınmasının gerekli olduğunu göstermiştir. Yaklaşık teorik ve sayısal analiz sonuçlan kullanılarak panellerin orta noktalarındaki birim uzama ve yer değiştirmelerin zaman ile değişimlerini gösteren grafikler oluşturulmuştur. Deney sonuçlan ile analiz sonuçları karşılaştınlarak aralannda iyi bir uyum olduğu görülmüştür.

In this thesis, laminated composite panels subjected to air blast loading are studied using experimental, approximate theoretical and numerical methods. Flat and cylindrically curved laminated composite panels are examined in the study. The panel is clamped at its all edges. In the first chapter, the importance of the effects of air blast loading on the panel structures is discussed. In this chapter, composite panel applications are introduced. Also several studies related to the effects of air blast loading on the panel structures are reviewed. Isotropic plate structures subjected to air blast loading are investigated by Houlston et al. [3,4]. Numerical results obtained by using ADINA finite element code are compared with experimental results. A single degree of freedom elastodynamic analysis of the response of a rectangular plate subjected to an explosive blast has been conducted assuming a Navier form of displacement function and a modified Friedlander reflected blast overpressure which exponentially decays with time [5]. Analytical results are compared with ADINA results, and a good agreement is found for the linear elastic dynamic case. In the investigation of the dynamic response of steel cylindrical shell panels subjected to air blast loading, approximate theoretical solutions are presented for linear and nonlinear geometric behavior [16]. Theoretical results are compared with ADINA results for several panel rise cases. A numerical model for large deflection, elastoplastic analysis of cylindrical steel structures under air blast loading conditions is presented by Jiang and Olson [18,19]. The model is based on a transversely-curved finite strip formulation. The dynamic response of a steel toroidal shell panel subjected to blast loading is investigated by Redekop [17]. Numerical results are computed using the theory based on the Mushtari-Vlasov-Donnell assumptions. Response of laminated composite flat panels to blast loading is studied by Librescu and Nosier [6]. The results obtained within a higher order plate theory are compared with their first-order transverse shear deformation and classical counterparts. For thin plates, the results reveal that the response characteristics obtained within classical and shear deformable theories are in perfect agreement. In the present study, experimental results are obtained for thin laminated panels. The analytical expressions that can be used in preliminary design are obtained. These analytical expressions are correlated with experimental results. In the second chapter, nonlinear dynamic equations of the cylindrically curved laminated composite panels are derived. The governing equations are classified in three sets: (i) Strain displacement relations, (ii) equations of equilibrium, and (iii) constitutive laws. The effects of geometrical nonlinearities are found in the first and xii second sets of governing equations, and the effects of material nonlinearities are found in the third set of equations. The geometrical nonlinearities can be a result of large strains, rotations, and displacements of the fibers of a differential volume element that has undergone a transformation from some original configuration. Material nonlinearities are usually the result of straining beyond the limit of proportionality caused by large strains. Beyond this limit, the stress strain relationship is nonlinear; a special case is the material that has been strained beyond its yield point and plastic flow results. Therefore, it appears that nonlinearity is of two general types, geometrical and material, and each is treated independently. In this regard, Novozhilov categorizes four basic problems in the theory of elasticity:. Geometrically/materially linear. Materially nonlinear. Geometrically nonlinear; and. Geometrically/materially nonlinear. The basic equations which describe the behavior of a thin elastic shell were originally derived by Love in 1888. These equations, together with the assumptions upon which they are based, form a theory of thin elastic shells which is commonly referred to as Love's first approximation. In this chapter, the nonlinear strain- displacement relations based on the von Karman assumption and Love shell theory are used. In the three dimensional theory of elasticity, the fundamental equations occur in three broad categories. Thus, it is recalled that in elasticity we have equations of motion which are obtained from a balance of the forces acting on some fundamental element of the medium considered, that we have strain-displacement relations which are obtained strictly geometrical consideration of the process of deformation, and that we have the constitutive law of elasticity which is introduced in order to provide a relationship between the stresses and the strains in the elastic medium. However, the solution of problems in the three-dimensional theory of elasticity involves vast complications. Thus, a group of simplifying assumptions that provide a reasonable description of the behavior of thin elastic shells is proposed by Love and is led to the development of a subclass of the theory of elasticity known as the theory of thin elastic shells. Love's first approximation to the theory of thin elastic shells is based upon the following postulates:. The shell is thin.. The transverse normal stress is negligible.. Normals to the reference surface of the shell remain normal to it and undergo no change in length during deformation. In addition, some assumptions are used for laminated composite panels. It is assumed that the laminate thickness is small compared to its lateral dimensions. Therefore, stresses acting on the interlaminar planes in the interior of the laminate, that is, away from the free edges, are negligibly small. It is also assumed that there exists a perfect bond between any two laminae. That being so, the laminae are not capable of sliding over each other and there are continuous displacements across the bond. Equations of motion are derived from the virtual work principle. Transverse shear stresses and in-plane inertias are ignored. Thus the virtual work principle is applied to the shell and the following equation is obtained: xiii 5Je = Idt jj[ax5sx +as8ss +axs8sxs]dA A -JJköu + qs8v + qz8w]dA (1) A - J J[m(û8û + v8v + w8w)]dA = 0 Substituting the constitutive equations in equation (1), we obtain the equilibrium equations as follow dN^ 8N^ 15MS 15MXS " as " ax ~7-^s~-7^T-qs~mv =0 (3) a2Mv a2Mc a2Mxs ns faNx a*Oaw°.2.. +- -- - rJL + - dx ds dxds r \ 8x ds J dx aNs ao aw ?+,0 as dx J 8s.qz -mw° =0 r _ajV aV a2w0> lN* ax2 +Nxs axas +Ns as2, (4) Using the constitutive equations and the strain-displacement relations, equations (4) can be written in terms of displacements as follow Lnu° +LI2v° +L13w° +N1(w°) + mü° -qx = 0 L21u° +L22v° +L23w° +N2(w°) + mv° -q, = 0 (5) L31u° + L32v° + L33w° +N3(u°,v°,w°) + mw° ~qz = 0 with the following boundary conditions u°(0,s,t) = u°(4s,t) = u°(x,0,t) = u°(x,s0,t) = 0 au° / s au° / x au° / \ au° / \ - (0,s,t) = -^(tAt) = ^-M,t) = ^-(x,s0,t) = 0 v°(0,s,t) = v°(£,s,t) = v°(x,0,t) = v°(x,s0,t) = 0 (6) 8v°, N dv° { \ dv° / s dv° i \ ^(0,s,t) = ^fes,t) = -^-(x.0, t) = ^(x,s0,t) = 0 XIV w°(0,s,t) = w°(i,s,t) = w°(x,0,t) = w°(x,s0,t) = 0 5w° / ^ Sw° / x 9w° / >. dw° / \ ^(0,s,t) = -fo(tAt) = ~^-(*At) = -X-(x,şD>t) = 0 as and initial conditions u°(x,s,0) = 0 v°(x,s,0) = 0 w°(x,s,0) = 0 ü°(x,s,0) = 0 v°(x,s,0) = 0 w°(x,s,0) = 0 (7) Also the air blast loading approximation is presented in this chapter. Friedlander decay function is used to approximate the air blast loading with appropriate modifications [5]. p(x,s,t) = [(pm - pc)sin(7tx / 1)2 sin(7ts/ s0)2 + pc](l - 1 / tp)e -at/t" (8) In the third chapter, approximate theoretical analysis and numerical analysis are presented. The solution methods for the nonlinear ordinary differential equations are presented. Some examples related to the solution of this type differential equations are reviewed. Displacement functions for the clamped boundary conditions are assumed in the following form: M N u° = XZUmn(t)l-cos in=l n=l M N 2m7ix v° = £5X(t) l-cos m=l n=l M N £ 2m7ix 2mts 1 - cos v s0 J ( 1 - cos- w° = Z2>mn(t) i_cos m=l n=l 2m7tx £ 1-cos- 2n7rs s0 J 2n7ts (9) V The Galerkin method is used to obtain the nonlinear differential equations from the equations of motion which are derived in the second chapter. The three equations of motion are obtained by accounting the first terms of displacement functions. The panel is affected by normal blast pressure wave. In-plane inertias are ignored. Therefore U and V are calculated from the first two equations as zero. Thus the following equation is obtained: W + c,W + c2W2 + c3W3 = c4Pt (10) with the initial conditions xv U(0) = 0 V(0) = 0 w(o) = o (11) u(o) = o v(o) = o w(o) = o In this chapter, approximate solution techniques are discussed. The approximate approaches divide roughly into two categories. In the first category, a minimization of energy approach is used. The variational integral method, the Galerkin method, and the Rayleigh-Ritz method are of this type. The finite element and finite difference methods based discretizing techniques are in the second category. For the approximate theoretical analysis of the laminated composite panels subjected to air blast loading, two different computer programs are utilized. In the first program, the nonlinear analysis is achieved. For this aim, a PASCAL program is written. In this program, the first term of displacement functions is used. The fourth order Runge-Kutta method is used to solve the nonlinear equation of motion. In the other program, linear analysis is realized using the 1, 4 and 9 terms of the displacement functions. The FORTRAN program which is used the fifth or six order Runge-Kutta- Verner method is utilized. The fourth order Runge-Kutta method is presented for the nonlinear differential equation of motion. Moreover, finite element method is utilized for obtaining the dynamic response of the panel. The ANSYS finite element software is used for this purpose. Panel structure is discretized to 64 eight nodded layered shell elements. There are six degrees of freedom in each nodes. Application of the finite element method to the problem gives the equations of equilibrium in the matrix form as: MU+KU=F (12) The Newmark integration technique is used in the finite element analysis. The Newmark integration scheme can also be understood to be an extension of the linear acceleration method. The following assumptions are used: Ût+At=Ût+[(l-8)Üt+5Üt+At]At (13) U.+* = Ut + Ût At + [(1 / 2 - r,)Üt + r,Üt+At]At2 (14) In addition to above equations, for solution of the displacements, velocities, and accelerations at time t+At, the equilibrium equations at time t+At are also considered: MÜt+At+KUt+At=Ft+At (15) Finally, the equations of motion are obtained in the following form. (e0M + K)ut+At =F + M(e0Ut +e2Ût +e,Üt) (16) In the fourth chapter, the experimental studies are presented. First of all, the experimental setup is introduced. The experimental setup consists of two parts. These parts are external setup and internal setup. In the external setup, the detonation tube and the panel mounting frame exist. The internal setup consists of xvi the computer controlled combustion system together with the computer controlled measurement and data acquisition system. LPG and Oxygen tanks, solenoid valves, pressure manometers, flame arrester, air circulation pump, induction coil, a power supply, role driver card, PC 386 computer and PCL818 card are used in the combustion system. Piezo electric quartz crystal pressure transducer, strain-gauge, bridge circuit, charge amplifier, dynamic strain-meter, digital scope, RS232C serial interface, filter circuit are used in the measurement and data acquisition system. Experiments are carried out in the two parts. In the first part, the experiments are carried out to obtain the air blast pressure magnitude and the air blast pressure distribution on the panel. In the other part, the experiments are carried out to obtain the strain-time history at the center point of the laminated composite panels subjected to air blast loading. In the fifth chapter, the experimental and the analytical results are presented. Air blast peak pressure distribution on both flat and cylindrically curved panels which is obtained experimentally and its approximation are presented. Then the air blast pressure variation with time at the center point of the panel is presented for two different distances from the open end of the detonation tube. Furthermore, displacement-time history and strain-time history results are presented for the laminated flat panels subjected to air blast loading. It is found that there is a good agreement between the analysis and experiment results. In the analysis of cylindrically curved laminated panels subjected to air blast loading, displacement functions which include 1, 4 and 9 terms are selected. Strain-time history results are presented for the cylindrically curved laminated composite panels subjected to air blast loading. A good qualitative agreement between the analytical and experimental results are found. A theoretical analysis and the correlation with numerical and experimental results of the displacement-time and strain-time histories of the laminated composite panels exposed to normal blast shock waves are achieved in this study. The blast wave is assumed to be exponentially decaying with time and either uniformly distributed on the panel surface or sinusoidal varying with coordinates. The following conclusions apply to the case of laminated panels with fully fixed boundary conditions as considered herein: The blast pressure measurements on the panel show that the character of the pressure variation is strongly dependent on the distance between the open end of tube and the target panel. For example, if we decrease this distance about three times, the peak pressure on the panel increases approximately ten times. Furthermore, the ratio of the positive peak pressure to the negative peak pressure increases with the increasing distance. The pressure distribution on the panel has a sinusoidal variation for the cases of low distance. On the other hand, the spatial variation of the pressure becomes more uniform as the distance increases. From the time response curves for flat panels, one can conclude that experimental, theoretical and numerical results are in a good agreement during the first four msec. Afterwards, experimental results differ from the theoretical and numerical results. The discrepancy between experimental results and the others is more clear for the low distance case. Because, in this case, the plate moves rapidly due to the blast load with high velocity, and therefore the structural damping becomes a more significant factor restricting the plate response. Strain-time history curves obtained from the cylindrically curved laminated composite panels exposed to xvn blast wave show a qualitative agreement between the experimental and analytical results. On the other hand, if the frequencies are considered, a good agreement is found between experimental results and the others. It is important to note that the numerical solution procedure requires much more computer time than the approximate theoretical solution procedure. Thus the theoretical solution may be used for providing material in the preliminary design stage. The effect of fiber orientation and loading conditions on the dynamic response of the laminated composite panel can be examined by this method. The structural damping and hygro-thermal effects may be interesting in the aspect of the dynamic response of the panel. The stiffener and cutout effects on the dynamic behavior of the panel can be studied using this method. These will be the topics of future studies.