Pasternak zemini üzerinde ani ve hareketli yüke maruz dikdörtgen kalın kompozit bir plakanın dinamik analizi
Dynamic analaysis of a rectangular thick composite plate resting on a pasternak foundation subjected to sudden and moving load
Yazar: REHA GÜRSOY
Danışman: PROF. DR. VEDAT ZİYA DOĞAN
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Uçak ve Uzay Mühendisliği Ana Bilim Dalı / Uçak ve Uzay Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2016
Konu: Uçak Mühendisliği = Aircraft Engineering ; İnşaat Mühendisliği = Civil Engineering

Özet:
Bu çalışmada, bir ticari uçağın inişi sırasında pist üzerinde oluşabilecek çökmelerin hesaplanması amaçlanmıştır. Çalışma kapsamında pist modellemesi yapılırken dikdörtgen kalın kompozit bir plaka kullanılmıştır. Pistin altında bulunan toprak zemin tasarımı için ise Pasternak Modeli seçilmiştir. Analitik çözümde "Birinci Mertebeden Kayma Deformasyon Teorisi" kullanılmıştır. Formülasyon oluşturulurken kompozit plakanın dikdörtgen, kalın, asimetrik ve çapraz tabakalamaya sahip bir yapıda olduğu göz önüne alınmıştır.  Problemin çözümü için hem MATLAB hem de ABAQUS programlarından yararlanılmıştır. Analitik çözüm için gerekli olan basınç değeri, ABAQUS'te yapılan simülasyon sonucu elde edilmiştir. Çalışmanın sonucunda analitik ve nümerik çözümlerin sonuçları birbirleri ile kıyaslanmıştır. 

Toplam altı bölümden oluşan çalışmanın birinci bölümünde, uçaklardan ve havaalanı pistlerinden genel olarak bahsedilmiştir. Pist zemininde oluşabilecek sorunlar hakkında hem sözel hem de görsel bilgiler verilmiştir. Ayrıca uçakların iniş takımları ve çeşitleri kısaca anlatılmıştır. Daha sonra tezin amacı belirtilmiş ve literatürde yer alan çalışmalar hakkında kısa bilgiler verilmiştir.

İkinci bölümde, kompozit plakaların genel özelliklerinden bahsedilmiştir. Simetrik ve çapraz tabakalı plakaların nasıl oluşturuldukları anlatılmıştır. Ortotropik plakalar hakkında bilgiler verilmiş ve bu plakalar için elde edilen bağıntılar gösterilmiştir. Ayrıca pistin altında bulunan elastik zemin modellemesinin nasıl olması gerektiği belirtilmiştir.

Üçüncü bölümde, inişin nasıl yapıldığı anlatılmıştır. İniş tekerleği ve pist arasında meydana gelen temas belirtilmiştir. Uçak kütlesinin pist üzerinde oluşturduğu etkinin sayısal ve nümerik analize ne şekilde dahil olacağı incelenmiştir. Ayrıca inişi gerçekleştirecek uçağın tipi ve analiz için gereken karakteristik özellikleri verilmiştir. Pist bir kompozit plaka olarak dizayn edilmiş ve bu plakayı oluşturan tabakalar ayrıntılı olarak incelenmiştir. Her bir tabakaya ait önemli sayısal değerler belirtilmiştir.

Dördüncü bölümde, analitik çözüm için gereken denklemler, "Virtual Work" prensibine dayalı olan "Birinci Mertebeden Kayma Deformasyon Teorisi" kullanılarak elde edilmiştir. Daha sonra çökme analizinin önemli adımları anlatılmıştır. Pist üzerinde tekerleğin yaptığı temas basıncının bulunması gösterilmiştir. Çökme analizi için hangi bilgisayar programlarının kullanılacağı bahsedilmiştir. Ayrıca nümerik çözüm için gereken tekerlek  ve pist modellerinin, ABAQUS ve CATIA programlarında nasıl yapıldıkları gösterilmiştir.

Beşinci bölümde, MATLAB ve ABAQUS programları yardımıyla elde edilen sayısal sonuçlar grafikler halinde gösterilmiş ve birbirleriyle karşılaştırılmıştır. Plaka üzerinde oluşan maksimum basınç ve çökme miktarı belirlenmiştir.

Altıncı bölümde, elde edilen sonuçlar değerlendirilmiştir. Gelecek benzer çalışmalarda dikkat edilmesi gerekilen hususlardan bahsedilmiştir.

Abstract:
In this study,  during a commercial airplane's landing, deflections which occur on the runway pavement is calculated. In calculations, runway pavement is modeled as a rectangular thick composite plate which is resting on Pasternak foundation. In analytical solution, "First Order Shear Theory" is used. During the formulation process of the problem, plate is considered as rectangular, thick, aysmmetric and cross-ply laminated. MATLAB and ABAQUS programs are used for calculations. The pressure value, which is necessary in finding analytical solution, is obtained from ABAQUS simulation of the problem. In conclusion, analytical and numerical solutions are compared to each other.

The study is composed of six chapters. In the first chapter, first of all, informations about airplanes and airport runways are given. The problems which can be occured on the runway's surface are indicated verbally and visually. Also, landing gear types of airplanes are expressed. And then, the purpose of thesis is emphasized, the studies in the literature are informed.

In the second chapter, the general properties of composite materials are expressed. The forms of symmetric and cross-ply plates are indicated. The general informations about orthotropic materials and the fundamental equations of orthotropic plates are written. In addition, the model of elastic foundation, which is under the composite plate, is described.

In the third chapter, the landing phase of analysis is indicated. The contact between landing gears and runway pavement is explained. The effect which is occured on the runway pavement by airplane's mass is described. Also, the airplane's type and  characteristic properties, which is used in analysis, is detailed. The runway is designed as a composite plate. The layers of composite plates are indicated. The numerical values of each layer are expressed.

In the fourth chapter, the equations, which are required for analytical solution, are obtained by using "First Order Shear Theory" and this theory is based on "Virtual Work Principle". After that, the crucial steps of deflection analysis are explained. The contact pressure is detailed which is applied by landing gear wheel onto runway pavement. The computer programs are expressed which are used for both analytical and numerical solutions. Also, the modelings of runway pavement and wheel are detailed.

In the fifth chapter, the results of both analytical and numerical solutions are shown in the graphics and compared to each other.
In the sixth chapter, the results are commented. The necessary points, which must be considered for the future similar studies, are expressed.

According to the International Civil Aviation Organization (ICAO), a runway is a defined rectangular area on a land aerodrome prepared for the landing and takeoff of aircraft. Runways may be a man-made surface (often asphalt, concrete, or a mixture of both) or a natural surface (grass, dirt, gravel, ice, or salt). In this thesis, runway is modelled, where the landing gear wheel touches, as a rectangular  composite plate.

Composite materials are made from two or more constituent materials with significantly different physical and chemical properties that, when combined, produce a material with characteristics different from the individual components. Orthotropic composite plates are basically a plate made out of composite materials. It has different mechanical properties in three mutually perpendicular planes. Most laminated composite plates fall into this category. 

In this study, composite plate consists of four layer. Each layer has its own mechanical properties. Composite plate is designed as symmetrical and cross-ply laminate. Also, the composite plate is rectangular and thick. 

One of the layers of the composite plate is reinforced concrete. Reinforced concrete is a composite material in which concrete's relatively low tensile strength and ductility are counteracted by the inclusion of reinforcement having higher tensile and ductility. The reinforcement is usually, though not necessarily, steel reinforcing bars (rebar) and is usually embedded passively in the concrete before the concrete sets. Reinforcing schemes are generally designed to resist tensile stresses in particular regions of the concrete that might cause unacceptable cracking and structural failure. Modern reinforced concrete can contain varied reinforcing materials made of steel, polymers or alternate composite material in conjuction with rebar or not.

Composite plates supported directly by the soil continuum is a very common construction form. It is used in residential, commercial, industrial and institutional structures. The behavior of the plate when it carries external loads is influenced by the soil and the behavior of the soil is, in turn, influenced by the action of plate under load. Developing a realistic mathemathical model for this complex soil-structure interaction problem is essential in order to provide safe and economical designs. Ultimately, all structure loads must be transferred to the soil continuum and bothe the soil and the structure act together to resist and support loads. In many practical design problem of this type, the soil continuum is layered and may be resting over rigid rock or a relatively stronger soil. Soil is truly a nonhomogeneous and an anisotropic medium that behaves in a non-linear manner, while structures can be adequately modeled and analyzed assuming isotropic and linear behavior.

The elastic foundation (soil support) model of this study is Pasternak Model. This model contains springs and damper. Thus, it is very realistic model and is capable of absorbing a lot of energy. Therefore, the impact load is less detrimental to the composite plate (runway pavement) during landing. 



The airplane type is Boeing 747-400 which performs landing. This airplane is major development and the best-selling model of the Boeing 747 family of jet arliners. While retaining the four-engine wide-body layout of its predecessors, the 747-400 embodies numerous technological and structural changes to produce a more efficient airframe. Boeing 747-400's main landing gear consists of sixteen wheel.

Analytical calculation in this thesis is done by using "Virtual Work Principle". This principle arises in the application of the principle of least action to the study of forces and mmovement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies. 

In this study, there is two type of loads. First is impact (sudden) load. Impact load occurs when the wheel of landing gear touches onto runway pavement (composite plate). Second load is moving load. Moving load occurs while the wheel of landing gear advances on the runway pavement. It changes the place in time which is applied. 

Contact mechanics is the study of the deformation of solids that touch each other at one or more points. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact. Central aspects in contact mechanics are the pressures and adhesion acting perpendicular to the surfaces of contacting bodies (known as the normal direction) and the frictional stresses acting tangentially between the surfaces.

During loading process, contact pressure occurs between the wheel and the composite plate. Contact pressure is obtained by using ABAQUS in finite element solution. After that, this value is used in finding analytical solution.