Transfer matrisleri kullanarak kirişlerde yerdeğişimleri metodunun uygulanması

Abstract

Çalışmada, kirişlerde kuvvet ve yerdeğişimleri kavramları incelenmiş olup, san yıllarda geliştirilmiş alan iki yöntemden Matris Yerdeğişimleri Metodu incelenmiştir, Matris Kuvvet Metodu ve Matris Yerdeğişimleri Metodu analitik yöntemler olup, bilgisayarın kullanım alanlarının genişlemesi ve uygun programların yazılması ile kirişlerin incelenmeleri, belirli modellerin çözümleri kolaylaşmaktadır. El ile elde edilmeleri zor olan sonuçlar, konuya vakıf olan kişiler tarafından programa girilen datalar ile çabuk bir şekilde elde edilebilmektedir. Çalışmada önce genel olarak Matris Kuvvet Metoduna:, değinilmiş, her iki metodun ortak kullanıldığı tablolar verilmiştir. Daha sonra Matris Yerdeğişimleri Metodu incelenmiş, bu iki metodun karşılaştırılmaları ve kulla nımlarının sağladığı kolaylıklar açıklanmıştır. Çalışmanın son aşamasında, Matris Yerdeğişimleri Metodu için GU-Basıc dilinde hazırlanmış bir bilgisayar programı verilmiştir. Bu programın çalışmasıyla elde edilen sonuçlar el ile yapılan çözümle karşılaştırılmış, beklenen doğru değerlere ulaşılmıştır. Ancak, bu proğramın etkin kullanımı için, yani değişik sistemlerde sonuç arandığı takdirde, konuya yabancı olmayan kullanıcı ta rafından dataların programa verilmesi, fakat programın geri kalan prosedürüne müdahalede bulunulmaması gerekmektedir.

In this work, we shall consider the effect of external loading (either static or dynamic) on an elas tic system which consists of an assembly of elastic elements counected at a finite number of jiints (or nodal points). The system is said to be statically determinate when the forces acting on each of the elements can be determined by means of the equations of equilibrium. Should the system possess more ele ments or external supports than are necessary far stability, then it is described as being statically indeterminate. The excess forces carried by these elements and supports are described as redundants. The correct problem has been found if the follow ing requirements are satisfied: 1- Exuilibrium; The external forces and the inter nal forces they induce are in equilibrium at each joint. 2- Compatibility ; The elements are so deformed that they can all fit together. 3- Force-deflection relationship; The external far ces and deformations satisfy the stress-strain relationship of the element. Two basic methods of analysis are available for salving the problem, and are referred to as the force method and the displacement method. In the farce method the redundant farces are taken as the unknowns of the problem, so that all the internal forces can be expressed (in terms of) the external and the redundant forces. Then by using the stress-strain relationship the deformations of the elements can also be expressed in terms of the external and redundant forces Finally, by applying the compatibilty criterion that the deformed elements must fit together, it is possible to formulate a set of linear simultaneous equations which yield the values of the rudundant for ces. We may then calculate the stresses in the elements of the structure as well as the displacements of its joints in the directions of the external forces which are related by the flexibilty matrix F. In the displacement method the displacements of the joints necessary to describe fully the deformed state of the structure are imposed on teh structure. The de formations of the elements can then be calculated in terms of these displacements, and by use of thi stress- strain relationship, so can the internal forces. Finally, by applying the equilibrium criterion at each joint, we obtain a set of linear simultaneous equations from which we can solve for the unknown forces necessary to cause the imposed displacements, thereby obtaining the stiffness matrix K. In choosing the method most suited to o particular problem one would certainly be influenced by the number of unkown quantities. Referring to the shorf cataloque of flexibility and stiffness matrices, it is advantageous to use for the structural element A. The relation between the deformations v of the gth structural element and its associated internal for ces is described by; P = H v g g g p = «p v stress-strain relation ship VI Where ; cambines the stiffuess matrices of the s unassembled sturctural elements. Since in the problem the displacements d are deficient by the displacements y, in order tc describe completely all possible displacements of the joints, the displacements y are the kinematic deficiencise of the problem. The kinematic deliciencieis y are exactly analogous to the force redundants. Hence in order to compute the stiffness matrix H, ve have to form both teh matrix. A for teh "given" displacements d an the matrix A. for the kinematic deficiencies y. The same tabular from of the calculations for the force method can be used in the displacement method. VI 1 In the suffix, a camputerpragramm of this calcula tion procedure is prepared in Basic language. Bo, uith the aid of this programm it could be calcula ted in second final form of matrix A, and final form of matrix Hf, when matrix A, A and K are given. And if teh displacements are known, the computer prog ramm compute the deformations v and the external forces f of the system. The force method has been traditionally favored by engineers because the systems formerly occurring in practice could mostly be idealized to possess only a few force redundancies. With teh avilabilitly of digi tal computers, much more complicated systems can now be analyzed and a comparison of the number of unknown redundancies with the number of kinematic deficiencies is seldom of overriding importance in deciding wihceh of the two methods is to be used. The various factors affecting this decision are now discussed. A mentioned earlier, the force method is to be pre ferred when the number of unknown redundancies is much smaller than the number of kinematic deficiencies, and vice versa. Gne clear advantage of the force method., when applied to the static problems, can be see from the following discussion. The expressions obtained from the force method are d = F,f d and p= Bf That is, '.the displacements and internal forces are expressed in terms of the applied forces. This is an advantage of the force method because usully the applied forces are given. In the displacement method, on the other hand, the corresponding expressions obtained are f = Kxd and V = Ad VI 11 That ia, the applied farces and internal deforma tions are given in terms of the displacements d, which are normally unknown, In order to determine the dis-t placements when the forces are given, it is necessary to inveryt the stiffness matrix Kj. A severe drawback of the force me in the fact that considerable skill is ing the best redundant farce gruops, t being those which cause only local eff diffuse into the system. This can be tage when the system is one of great c displacement method does not suffer fr since the selection of displacemenets Furthermore the effects of impressing on the structure at a tlme-with the ot are purely local. Because of this adv appears to be a definite swing in prac use of the displacement method. thod must be seen needed in choosi he best groups ects and do not a real disadvan- omplexity. The om this defect is usually obvious, one displacement hers kept zero- antage there tice toward the