İki parametreli elastik zemine oturan sonlu kiriş eleman için genel bir rijitlik matrisi ve tesir çizgileri
İki parametreli elastik zemine oturan sonlu kiriş eleman için genel bir rijitlik matrisi ve tesir çizgileri
Dosyalar
Tarih
1994
Yazarlar
Marmara, Murat
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Özet
Bu çalışma iki bölümden oluşmaktadır. Birinci bölümde iki parametreli elastik zemine oturan sonlu kiriş elamanın statik, dinamik ve burkulma hesabında kullanılabilen genel bir rijitlik matrisi elde edilmiş, ikinci bölümde ise aynı kiriş elemanın tesir çizgilerini çizebilmek için gerekli olan tesir fonksiyonları bulunmuştur. Rijitlik matrisi çıkarılırken, problemin diferansiyel denkleminin homojen çözümüne ait karakteristik denklemin kökleri kompleks ve-veya gerçel değişkenler olarak alınmıştır. Bu kökler kullanılarak uç kuvvetlerinin uç yer değiştirmeleri cinsinden ifadeleri, rijitlik matrisini sayısal olarak bulabilecek şekilde elde edilmiştir. Tesir fonksiyonlarının bulunmasında, diferansiyel denklemin homojen çözümüne ait karakteristik denklemin kökleri trigonometrik ve hiperbolik fonksiyonlar olarak bulunmuştur. Bu kökler yardımıyla, başlangıç parametreleri yöntemi kullanılarak tesir fonksiyonları bulunmuştur. Başlangıç parametreleri sınır şartlarından elde edilmiştir. Bulunan bu ifadeler, bilgisayar programında kullanılarak tesir çizgisi tabloları elde edilmiştir.
This study has two parts. In the first part of study, a general stiffness matrix for static, dynamic and buckling analysis of the beam element resting on two parameter elastic foundation is obtained. And the second part of the study, influence functions for influence lines of beams resting on two parameter elastic foundations is obtained. A GENERAL STIFFNESS MATRIX FOR BEAMS RESTING ON TWO PARAMETER ELASTIC FOUNDATION A beam element in figure which has 1 length and six degrees of freedom at two ends. N- W&//&J ptx.t) mu d,,P, k,k,. Y1 «p J5'Ps d3'P3 First, (4*4) stiffness matrix which consider 2., 3., 5., and 6. freedoms is obtained-. Then, it widen to (6*6) stiffness matrix with 1. and 4. freedoms. The differantial equation of beam resting on two parameter elastic foundation: Ely(m- (JCi+W) y" + lcy--(m1+in0) J% VI Here, y EI k ki Vertical displacement of beam Stiffness of beam 1. foundation modül 2. foundation modül Respectively mass of beam and foundation The characteristic equation of this differantial The roots of equation is obtained with w»em. characteristic equation may be reel and- or imaginary. The roots are shown m. m 2' tu rn,, Elastic curve of beam: w- j^ e%*+ A, e****- a, e^+ AA e**x This equation is writen matrix form: w= [G] [A] Here; [A] otained from d2=w(0), dn=w' (0), dc=w(l), d6=w' (1) =w/ boundary conditions dependent d- 2' A.3 Oc, Shape functions are obtained with using matrix [A]" [d] = [BJ[A] [A]=inv[B] [d] w=[G]inv[B] [d] = [N] [d] [N] : Shape functions. The (4*4) stiffness matrix is obtained from differantial equations of shear force and bending moment and degrees of freedom matrix [P]. V--EI d3w _2z2 dw dx3 dx M--EI dzw dx2 [P]- [P] -EC] U] = [C] [B)-Hd] VII The (4*4) stiffness matrix: LSJ-EC] lB]~l This (4*4) stiffness matrix widen to (6*6) matrix with 1. and 4. freedoms. And (6*6) general stiffness matrix: IS] - AE J Sx(l,l) Sx(l,2) ^(2,2) Syme Iri c AE 1 0 0 AE 1 0 0 5^(1,3).5^(1,4) ^(2,3) ^(2,4) 5X(3,3) ^(3,4).5^(4,4) INFLUENCE LINES FOR BEAMS RESTING ON TWO PARAMETER ELASTIC FOUNDATIONS Differantial equation of beam resting on two parameter elastic foundation: rffX-2r*-£fZ + s4y»_P dxr4 dx2 EI EI EI Here ; k t EI P y 1. foundation modül 2. foundation modül Stiffness of beam External load Vertical displacement of beam The characteristic equation of this differantial equation is obtained with y=e' The roots of characteristic equation are written as trigonometric and hyperbolic functions.
This study has two parts. In the first part of study, a general stiffness matrix for static, dynamic and buckling analysis of the beam element resting on two parameter elastic foundation is obtained. And the second part of the study, influence functions for influence lines of beams resting on two parameter elastic foundations is obtained. A GENERAL STIFFNESS MATRIX FOR BEAMS RESTING ON TWO PARAMETER ELASTIC FOUNDATION A beam element in figure which has 1 length and six degrees of freedom at two ends. N- W&//&J ptx.t) mu d,,P, k,k,. Y1 «p J5'Ps d3'P3 First, (4*4) stiffness matrix which consider 2., 3., 5., and 6. freedoms is obtained-. Then, it widen to (6*6) stiffness matrix with 1. and 4. freedoms. The differantial equation of beam resting on two parameter elastic foundation: Ely(m- (JCi+W) y" + lcy--(m1+in0) J% VI Here, y EI k ki Vertical displacement of beam Stiffness of beam 1. foundation modül 2. foundation modül Respectively mass of beam and foundation The characteristic equation of this differantial The roots of equation is obtained with w»em. characteristic equation may be reel and- or imaginary. The roots are shown m. m 2' tu rn,, Elastic curve of beam: w- j^ e%*+ A, e****- a, e^+ AA e**x This equation is writen matrix form: w= [G] [A] Here; [A] otained from d2=w(0), dn=w' (0), dc=w(l), d6=w' (1) =w/ boundary conditions dependent d- 2' A.3 Oc, Shape functions are obtained with using matrix [A]" [d] = [BJ[A] [A]=inv[B] [d] w=[G]inv[B] [d] = [N] [d] [N] : Shape functions. The (4*4) stiffness matrix is obtained from differantial equations of shear force and bending moment and degrees of freedom matrix [P]. V--EI d3w _2z2 dw dx3 dx M--EI dzw dx2 [P]- [P] -EC] U] = [C] [B)-Hd] VII The (4*4) stiffness matrix: LSJ-EC] lB]~l This (4*4) stiffness matrix widen to (6*6) matrix with 1. and 4. freedoms. And (6*6) general stiffness matrix: IS] - AE J Sx(l,l) Sx(l,2) ^(2,2) Syme Iri c AE 1 0 0 AE 1 0 0 5^(1,3).5^(1,4) ^(2,3) ^(2,4) 5X(3,3) ^(3,4).5^(4,4) INFLUENCE LINES FOR BEAMS RESTING ON TWO PARAMETER ELASTIC FOUNDATIONS Differantial equation of beam resting on two parameter elastic foundation: rffX-2r*-£fZ + s4y»_P dxr4 dx2 EI EI EI Here ; k t EI P y 1. foundation modül 2. foundation modül Stiffness of beam External load Vertical displacement of beam The characteristic equation of this differantial equation is obtained with y=e' The roots of characteristic equation are written as trigonometric and hyperbolic functions.
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1994
Anahtar kelimeler
Elastik zemin,
Kirişler,
Rijitlik matrisi,
Elastic ground,
Beams,
Stiffness matrix