Çift duvarlı tüplerde burkulmanın başlangıç değerleri yöntemiyle incelenmesi

Abstract

Bu çalışma da tek ve çift duvarlı çubukların burkulma yükleri başlangıç değerleri yöntemiyle hesaplanmıştır. Nanoteknoloji de kullanılan karbon nanotüpler genellikle tek duvarlı veya çift duvarlı çubuklardır. Özellikle çift duvarlı tüplerin mekanik davranışlarının incelenmesi çok önemlidir. Çift duvarlı tüplerde iki tüp arasındaki etkileşim van der waals kuvveti ile göz önüne alınır. Başlangıç değerleri yöntemi, probleme ait bilinmiyenlerin başlangıçtaki değerlerinin bilinmesi halinde, problemin çözümünün sistematik olarak veren bir yöntemdir. Bu yöntem ile tek duvarlı bir çubuğun burkulma yükü 2x2 lik bir determinant yardımıyla olmaktadır. Aynı problemin klasik yöntem ile çözümünde karşımıza çıkan determinant 4x4 mertebesindedir. Benzer şekilde çift duvarlı çubuğun burkulma yükleri başlangıç değerleri yöntemiyle 4x4 mertebesindeki determinantlarla klasik yöntemde ise 8x8 mertebesindeki determinantlarla yapılmaktadır. Tek ve çift duvarlı çubuklara ait burkulma yükleri çizelgeler halinde verilerek, elde edilen sonuçlar karşılaştırmıştır.Çift duvarlı çubukların burkulma yükleri, tek duvarlı çubukların burkulma yüklerinden daha büyüktür.

In this study, the buckling load of single and double walled beams are calculated by the method of initial values.In generally, Single and Double walled nanotubes are widely used in nanotechnology.The study of mechanical behavior of double walled carbon nanotubes has great importance.The interaction between the inner and outer tubes are denoted by van der waals force.The method of initial values gives the values of the displacements and stress resultants throughout the rod once the initial displacments and initial stress resultants are known.The buckling loads of a single-walled tube can be obtained by the determinant of 2x2 matrix in this method. 4x4 determinant appears in the solution of same problem with classical method.Similarly the buckling loads of a double-walled tube are calculated by the determinant of 4x4 matrix. 8x8 determinant appears in the solution of same problem.The buckling loads of single and double walled tubes are presented in tables. The buckling loads of double walled tubes are greater than the buckling loads of single walled tubes.The equations of equilibrium,the constitutive relations and geometrical compatibility. Conditions of a rod in the plane areWhere v is deflection, is rotation around the binomial, M is bending moment and T is shear force. Aboue system can be written in matrix form as belowOrWhere,;The solution of about system of differential equations is,Where,Example: The buckling load of a simple beam (figure 3.2),Figure 3.2: Single-walled beamIn this case, two of initial values are known,v(0)=M(0)=0The other initial values can be abtained by using the boundary conditions,v(L)=0, M(L)=0The boundary conditions reach us to the following system,The bucking determinant be comes,detThe elements of Carry- Over matrix can be obtained analytically. In this thesis Carry-Over matrix will be calculated opproximately. The series expansion of is (equation 3.10).(3.10)The elements of Carry- Over matrix is calculated by using about relation.The governing equations for a double-walled beam are (figure 3.3),Figure 3.3: Double-walled beamWhere v1 is the deflection in outer tube, v2 is the deflecton in inter tube, is the rotation about the binormail in outer tube, is the rotation about the binormail in inner tube, M1 is the bending moment in outer tube, M2 is the bending moment in inner tube,T1 is the shear force in outer tube, T2 is the shear force in inner tube. The tream c(v1-v2) displays the in teraction between the inner and outer tubes and called ad Van der waal force. This system can be written as,Example: The buckling loads of a double-walled simple beam (figure 3.4).Figure 3.4: Two-hinged, double-walled beamThe coefficient matrix A for a double walled tube is,The Carry-Over matrix can be found by using ,(equation 3.10).(3.10)In this case four of initial values are known,v1(0)=M1(0)=v2(0)=M2(0)=0The other initial values can be obtained by the help of end conditions,v1(L)=M1(L)=v2(L)=M2(L)=0About conditions give the following relation,Then the buckling determinan becomes,The buckling loads of single and double walled simple bars are presented in the following table (table 3.1).Table 3.1: Two-hinged single and double walled tip of the rod bucklingresultsNumber of TermsBuckling loadsingle-walled beamBuckling loaddouble-walled beam69.00009.980489.47809.6669109.914210.1029129.866810.0553149.869710.0582169.869610.0580189.869610.0580209.869610.0580