Birbirine eksenel normal kuvvet alan çubuklar ile bağlı düzlemsel iki çubuğun taşıma matrisi ile çözümü
Başlık çevirisi mevcut değil.
- Tez No: 75308
- Danışmanlar: PROF. DR. MEHMET BAKİOĞLU
- Tez Türü: Yüksek Lisans
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Taşıma matrisi, Çubuklar, Transfer matrix, Bars
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: İnşaat Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Yapı Mühendisliği Bilim Dalı
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada, taşıma matrisinden faydalanarak birbirine eksenel normal kuvvet alan çubuklar ile bağlı düzlemsel iki çubuk incelenmiştir. Birinci bölümde; konunun kısaca özeti yapılmıştır. İkinci bölümde; çubuk geometrisi hakkında bilgi verilmiş ve çubuğa ait genel denklemler elde edilmiştir. Üçüncü bölümde; taşıma matrisi ve tekil etkiler hakkında bilgi verilmiştir. Bu bölümde, çubuk probleminin başlangıç değerleri ile çözümünün genel formülasyonu yapılmıştır. Dördüncü bölümde; doğru eksenli çubuğun taşıma matrisi çıkartılmış ve kapalı şekilde elde edilemeyen taşıma matrisleri için bir sayısal yöntem formüle edilmiştir. Beşinci bölümde; birbirine eksenel normal kuvvet alan çubuk ile bağlı iki çubuğun formülasyonu, daha önceki bölümlerde elde edilen bilgiler yardımı ile yapılmıştır. Formülasyonda çubuklardan biri doğru, diğeri ise parabol alınmıştır. Altıncı bölümde; oluşturulan bilgisayar programı hakkında bilgi verilmiş ve çözülen örnek sistemlere ait bazı özellikler ve sonuç değerleri verilmiştir.
Özet (Çeviri)
SUMMARY THE ANALYSIS OF TWO RODS CONNECTED WITH RODS CARRYING ONLY NORMAL FORCES BY USING THE CARRY-OVER MATRIX In this work, two rods that are connected with rods only carrying normal forces have been solved by the carry-over matrix. As it is well-known, the 12*12 carry-over matrices of plane rods may be separated into two 6*6 matrices. One of these matrices is for the coplanar forces, the other one is for the forces perpendicular to the plane. We have used for parabolic arcs 6*6 carry-over matrix. For straight rods the order of this matrix is 4*4 The work is composed of seven parts. In the second part, the geometry of the plane rod has been introduced. The tangent to the elastic curve is t-direction, the direction of the normal in the plane to t- direction is n-direction and the direction normal to the plane is b-direction. The equilibrium equations are: -> dM -»-»-> - -+txT+m = 0 as ^ + P = ° Where T is the resultant of the stresses in the cross-section, while M is the resultant of the couples obtained from transformation of the distributed moments to the -* -> centroid. On the other hand p and m are distributed forces and couples -» respectively. Here, the independent variable s represents the arch length and t is the unit tangent vector. The compatibility equations are: dU - -> -* -- = y+Qxt dsda ds = co Where U is the displacement vector, Q is the rotation of the cross-section, y is -> correlative deformation of unit length of the rod and a is correlative rotation of unit -> -> length of the rod. Due to these equations show the relation between U and Q, it is named as compatibility equation. -> -» -> -> Constitution equations show relation between T, M and y,co which can be written in the follownig form: -> -» T=C.y -* where D is the rigidity of the rod with respect to couples M. C is the rigidity of the -> rod with respect to forces T. The four vectorial equations form a system of 12 equations which can be written as: ^ = AS+P For the non-homogenous case, S is composed of U, Q, T, M. The principal matrix is of course 12*12. But as mentioned above it is 6*6 for parabolic rods and 4*4 for straight rods, s can be written as curved coordinate by s = p.d<p where p is the radius of the curved rod. Figure 1. The coordinates of the parabolic rod xu?=.3 cos3 <p where p0 is the radius of the curvature of the crown and ç is the angle between the normal and the symmetry axis of the parabola. As a result of this acceptance, the equation given above can be written as: ^ = AS+P In the third part, firstly differential transition matrix has been obtained using the canonic form of the system. It can be written as: S«(s) = A.S(s) S(s) is the position matrix is composed of U, Q, T, M. S' (s) is derivation of S(.s) and A is the differantial passing matrix. There is also relation between S(.s) and S(s + ds), it is: S(s) = F(s).S(0) where S(0) and F(s) represent initial value and the carry-over matrix respectively. This equation is only for homogenous case. If any arbitrary action K(£) effects at a point ğ of the rod. The equation above can be written as: S(s) = F(s).S(0) + F(*-£.K(£ If there are more than one effect the equation above is: m S(s) = F(s).S(0) + £F(s-£).K(£) where m is the number of the effect and if there is also continued effect the equation can be written as: XIII* m S(s) = F(5).S(0) + J F(5- frv(frdğ+ZV(s- £).K(£) O <=1 for s=L (L is the length of the rod) the equation may be written in the following form: S(L) = F(L). S(0) + j F(Z - 0. p(Ç). dğ +£ F(Z - £ ). K(£ ) 0 1=1 If the system has unconditional effects, in the S(L) and S(0) there are 12 elements for the parabolical rods and 8 elements for the linear rods. Half of these elements are known because of the boundary conditions. The unknown elements can be found using the equation above. If the rod has conditional effects (support, roller support or hinged support) between s=0 and s=L because of effecting conditional effects, the system has additional unknowns for each effect These unknowns can be computed using the support conditions of the rod. The conditional and unconditional effects and boundary conditions mentioned above can be seen in the figure below. Figure 2. The conditional and unconditional effects and boundary conditions of the rod at the point A the support conditions are Ut (a) = U" (a) -Ub(a) = 0, at the point B the support conditions is Un(b) = 0 and at the point C the support conditions are Mt(c) = Mn(.c) = Mb(c) = 0 In the fourth part, firstly the carry-over matrix of the straight rod has been obtained. This matrix is: XIV¥(s) = e SJL A matrix is only related with s, so A matrix can be calculated easily. Therefore, this it is very easy to obtain the carry-over matrix of the straight rods. dS Solution of the -j- = A.S+ p differantial equation with numerical method: Sometimes the carry-over matrix cannot be calculated directly due to this equation. dS -r- = A.S+ p can be calculated only with numerical methods. dq> * m S(5) = F( s). S(0) + J F(s - Ç). p(£. dğ +£ F(s - 4 ). K(£ ) 0 '=1 For s = tp and F(s) = eSA equation written above may be shown in the following form: S(<p) = eA*. S(0) + J e A(p_r). p(r). dr +£ e A(p-r). K(^ ) o i=i In mis part, solution of this equation have been mentioned when A is variable or not. In the fifth part, some informations have been given about two rods that are connected to each other with rods carrying normal forces only. This can be seen from the figure given below. Figure 3. The two rods connected with rods carriying only normal forces XVAt general solution of this problem, the system has additional unknowns for each connected rod. The required equations for these unknowns can be written using the equality of the displacement both terms of the connected rod. For practical solution of the system, some basic assumptions are made to be considered as being fulfilled in the following. One of the rod in the system is straight and the other one is the parabolic. These two rods are connected with vertical rods, as shown in figure below. Figure 4. The system with parabolic and straight rod In the sixth part, some informations have been given about the computer program made for personal computer. Two examples are given. The seventh part is devoted to conclusions. In this part a comparative evaluation of the method used in this study with the other methods including finite elements have been made and it's advantages have been presented. Some information about the sensitivity of the method used in this study has also been given. XVI
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