Konsol kirişlerde büyük yer değişimleri için sonlu elemanlar modeli geliştirilmesi
Finite element modelling for cantilever beams with large deflections
Yazar: DENİZ YILMAZER
Danışman: DOÇ. DR. CEMAL BAYKARA
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Konstrüksiyon ve İmalat Bilim Dalı
Derece: Yüksek Lisans, 2014
Konu: Makine Mühendisliği = Mechanical Engineering

Özet:
İnce konsol kirişlerin farklı yükler altında incelenmesi ve mümkün olan davranışlarına nümerik yaklaşımlar oluşturulması uzun süredir incelenmekte olan bir konudur. İnce konsol kirişlerin minimal kesit alanları yük altında yüksek yer değişimleri yaratırken, çok küçük şekil değiştirmelerde dahi yüksek geometrik lineer olmama durumu oluşturmaktadır. 
Bu çalışma, lineer olmayan malzemeler ile oluşturulan konol kirişlerde meydana gelen yüksek yer değişimleri, Ludwick tipi gerilme – şekil değiştirme bağıntısının nümerik sonuçlarını kullanarak, Ansys 13'te oluşturulan model ile karşılaştırmaktadır. Sunulan analizde, serbest ucundan noktasal yüke maruz bırakılmış bir konsol kiriş göz önünde bulundurulmuştur. Alüminyum alaşımı ve tavlanmış bakır için oluşturulan nümerik sonuçlar, Gilbert Lewis ve Frank Monasa'nın gerçekleştirdikleri çalışmadan elde edilmiş ve sonlu elemanlar yöntemiyle geliştirilmiş model sonuçları ile karşılaştırılmıştır. Öte yandan tavlanmış bakır malzemeli konsl kirişte de farklı geometrik modellerin etkileri incelenmiştir. 
Çalışmada, modelleme ve meshleme kodlarının oluşturulması ve çalışmaya bu kodların aktarılarak bilgi devamlılığının sağlanabilmesi için Ansys Workbench yerine Ansys APDL modülü kullanılmıştır. Her iki malzemede, Ludwick tipi malzeme bağıntılarının Ansys'e tanıtılması için herhangi bir modül ya da kod olmadığı için, malzeme özellikleri sisteme gerilme – şekil değiştirme grafiklerinin tanıtılması ile gerçekleştirilmiştir. Gerilme – şekil değiştirme grafikleri, Microsoft Excel kullanılarak ve uygun şekil değiştirme verileri girilerek elde edilmiştir. Öte yandan, Lewis ve Monasa'nın çalışmalarında elde ettiği bağıntılardan yola çıkarak çalışmada kullanılacak kuvvet değerleri Matcad programı ile elde edilmiştir. 
Hem alüminyum alaşım ve hem de tavlanmış bakır için, Ludwick bağıntısı ile nümerik olarak incelenen sonuçlara yakın sonuçlar elde edilmiştir. Alüminyum alaşımı ve tavlanmış bakırı karşılaştırdığımızda, sadece ilk hareket esnasında tavlanmış bakır sonuçlarının Ludwick bağıntısından elde edilen sonuçlara daha yakın olduğu görülmektedir. Bunun nedeni olarak da, Ludwick bağıntısında, elastisite modülünden bağımsız olarak elde edilen alüminyum alaşımının başlangıç değeri, Ansys simülasyonunda elastisite modülü girilerek olduşturulan başlangıç değerinden farklı olmasıdır. Fakat 3. kuvvet değeri uygulandığında farkların %10 mertebesine kadar gerilemekte ve geri kalan kuvvet analizlerinde farklar %1 civarında oluşmaktadır. Tavlanmış bakır içinse simülasyon ardından elde edilen sonuçlar bütün analiz boyunca Lewis ve Monasa sonuçlarına çok yakındır. 
Lineer değişken kesitli konsol kiriş ile dikdörtgen kesitli konsol kirişin karşılaştırılmasında, lineer değişken kesitteki v ve h yönlerindeki yer değiştirmelerin dikdörtgen kesitten daha yüksek olduğu görülebilir. Fakat bu her iki grafiğin birbirinden tamamen farklı olduğu anlamına gelmemektedir. Iki kesit tipinde de 1. tip hareketten 2. tip harekete geçiş noktalarının ve grafiklerin genel davranışlarının benzer olduğu söylenebilir. 
Parabolik değişken kesit ise lineer değişken kesit ile benzer davranışları gösterirken, yer değiştirme miktarları az da olsa birbirinden farklıdır. Parabolik kesit yer değiştirmeleri lineer kesit ile dikdörtgen kesit arasında kalmaktadır.

Abstract:
The investigation of thin beams under different loadings and numerical approaches to possible behaviors has been a long standing subject of practical interest. Minimal cross sectional area of thin beams creates large deflections under certain loadings by arising geometrical nonlinearities in small strains.
Earlier of these studies, solutions for large deflections of cantilever beams of linear elastic materials subjected to a point load were conducted by Frisch – Fay, Barten and Bishop and Drucker. Inelastic large deformation of cantilever beam with Ramberg – Osgood strain stress type material was studied by Prathap and Varadan. Lo and Gupta examined the bending problem considering elastic behavior on the sections deforming elastically and logarithmic function of strain for the sections stressed above elastic limit. However, this approach could not go further than special cases while the logarithmic function was approximated by a semi logarithmic relation.
Gilbert Lewis and Frank Monasa considered the problem of finite deflections of cantilever beams of nonlinear elastic materials and subjected to a concentrated load at the free end. Materials represented with Ludwick strain–stress relation was used in the study. Nonlinear bending theory was used to formulate the large deflections and geometrical nonlinearity. Moment – curvature relationship was built with the exact expression of the curvature. 4th order Runge Kutta Method was used to solve second order nonlinear differential equation, which is a result of moment curvature relationship. Numerical algorithm was solved over two sample materials obtaining the deflections and rotations along the central axis. Many different subjects for nonlinear mechanics were examined by Demeter G. Fertis.
For recent studies, a new methodology comparing to Ludwick method, was studied by Ghaffarzadeh and Nikkar to analyze the large deformation of a cantilever beam under point load at the free tip, known as variational method – II, which can be considered as explicit solution. Belendez and Niepp realized experimental and numerical analysis to introduce large deflection on cantilever beams subjected to a point load at the free end and uniformly distributed load along its length (its own weight). Also, He, Cao, Li, Hu and Sun was also investigated the uniformly distributed load effects in addition with the gradient of beams which can be studied under geometrical nonlinearity. Force directions at the free end of the cantilever beam were considered in several studies recently. Also, horizontal force, vertical force and bending torque's multiple effects on asymmetric Ludwick cantilever beam were investigated by Borboni and Santis. Vázquez-Leal et.al. conducted their researches on follower force on the free end of the cantilever beams and also nonlinear pendulum which is a common subject for shear deformable cantilever beam of Batista's investigations. M. Brojan is carrying out most of his studies on large deflections of nonlinearly elastic cantilever beams.
This paper deals with comparing Ludwick type strain-stress relationship numerical results for large deflections of cantilever beams of nonlinear elements using the models created at Ansys 13. In the present analysis, a cantilever beam subjected to a point load at the free end was taken into consideration. Numerical results were derived from the studies of Gilbert Lewis and Frank Monasa for aluminum and copper elements and compared with the generated finite element models' results. 
Additional geometries were also studied for copper material, which has much close values than aluminum material even for 1st type behaviour zone. The cross section shrinking model of bottom face was analyzed for linear varying and parabolic varying. Even the cross sections are similar; there is a distinctive deflection difference between the section types.
Ansys APDL module was used instead of Ansys Workbench, in order to create modelling and meshing codes and obtain information continuance by implying these codes in the study. For both materials, material properties were introduced to the system by defining their stress – strain charts instead of defining Ludwick type equation because any code or approach couldn't be found on Ansys codes for defining Ludwick type equation. Microsoft Excel was used to create the stress – strain charts by defining the proper strain values. Besides, Mathcad was used to define the proper force values by deriving from the equations of Lewis and Monasa's studies. 
Hence, Lewis and Monasa used non-dimensional results to define deflections, in this study; a sample problem with a thin beam was built targeting to compare numerical analysis results. A thin beam with 20 inch length (L), 0.25 inch height (h) and 1 inch width (b) was chosen for the analysis. Both material and geometrical nonlinearities were conducted to this cantilever beam. Numerical non-dimensional results derived for annealed copper and aluminum alloy. Experimental data of annealed commercially pure copper and NP8 aluminum alloy was integrated to their moment-curvature relationships and end deflections and end rotation vs. L^(n+1)⁄K_n   were derived by Lewis and Monasa. From these non-dimensional ratios, point loads at the end of the free end were calculated and implemented to Ansys models for simulation. 
For Ansys simulation, SOLSH190 element was selected, which gives ability to simulate shell structures with a wide range of thickness varying from thin to moderately thick. The element possesses the continuum solid element topology and features eight-node connectivity with three degrees of freedom at each node, translation in the nodal x, y, and z directions. The element has plasticity, hyperelasticity, stress stiffening, creep, large deflection and large strain capabilities. The element formulation is based on logarithmic strain and true stress measures. Elasticity modulus and Poisson's ratio of annealed copper is 1.7∙〖10〗^7psi and 0.343 as consecutive, while 1.03∙〖10〗^7psi and 0.33 for aluminum alloy. Multilinear kinematic hardening approach was used for nonlinear material. With KINH command, experimental stress – strain curve of Ludwick was defined to the simulation. Cantilever beam was meshed homogenously considering these details. Point loads, which were derived at previous calculations, were defined at the free end of cantilevers, and the analysis was conducted.
With both materials, close results to Ludwick type equation was established. Comparing aluminum alloy with annealed copper, it is found out that annealed copper results are more in accordance with Ludwick type equation. Especially on first movement zone, annealed copper performance is better than aluminum alloy, based on the reason of elasticity modulus definition in Ansys simulation for initial value determation exists but for numeric analysis Ludwick type equation was used by eliminating elasticity modulus of the material. On the other hand, for annealed copper, the appropriate results matching with elastic deformation zone of Ludwick type equation, follows the graph closely till the end of simulation.
Comparing of linear varying with original Ludwick type block section, it is seen that linear varying part deflection of both v and h directions, is higher than block section. However, it cannot be said that two graphs are completely different. At both section types, elastic to plastic deformation points are similar. On the other hand, two graph's characteristics are similar. 
Parabolic varying section generally shows the same behavior with linear varying one; only the deflection amount is slightly different. Parabolic varying deflections are between linear varying and original Ludwick type block section.