Vibration and buckling of plates and shells using dynamic stiffness method

Метод динамичке крутости (МДК) представља алтернативу методу коначних елемената (МКЕ) у анализи вибрација и избочавања конструкција. Основни елемент у МДК је континуални елемент, односно његова матрица крутости, која је формулисана на основу тачног решења диференцијалне једначине проблема, па је самим тим избегнута потреба за дискретизацијом домена. Да би МДК могао да нађе ширу примену, потребна је одговарајућа база континуалних елемената. У оквиру дисертације су по први пут формулисани континуални елементи за анализу вибрација Mindlin-ове правоугаоне плоче и сегмента кружне цилиндричне љуске по Donnell-Mushtari-евој и Flügge-овој теорији. За решење проблема слободних вибрација коришћен је Gorman-ов метод суперпозиције, док је динамика матрица крутости формулисана помоћу метода пројекције. Такође, на основу решења у затвореном облику формулисани су следећи континуални елементи, односно одговарајуће матрице крутости, за анализу вибрација и избочавања: Maurice Lévy-еве плоче по Mindlin-овој теорији, кружне цилиндричне љуске и сегмента кружне цилиндричне љуске са специјалним граничним условима по Donnell-Mushtari-евој и Flügge-овој теорији. Изведене матрице крутости су имплементиране у за ту сврху написани Matlab програм за анализу вибрација и избочавања система плоча и љуски. Резултати многобројних нумеричких примера су упоређени са доступним резултатима из литературе, као и резултатима МКЕ, чиме је извршена верификација у раду формулисаних континуалних елемената.Dynamic stiffness method (DSM) is an alternative to the Finite element method (FEM) in the vibration and buckling analysis. The essential element in the DSM is a continuous element and the corresponding stiffness matrix. The stiffness matrix is formulated based on the exact solution of the governing equations. Consequently, the discretization of the domain is minimized. For a wider application of the DSM, a suitable base of the continuous elements is necessary. Within this thesis, the continuous elements and the corresponding dynamic stiffness matrices for vibration analysis of the Mindlin plate and segment of circular cylindrical shells based on the Donnell-Mushtari and Flügge theory are formulated for the first time. Gorman's method of superposition has been used for solution of the free vibrations problem, while the dynamic stiffness matrix is formulated by using the Projection method. In addition, based on the closed-form solutions of the of free vibration and buckling problem, the following continuous elements are formulated: Maurice Lévy plate element based on the Mindlin theory, circular cylindrical shell and segmented circular cylindrical shell with special boundary conditions element based on the Donnell-Mushtari and Flügge theory. The developed stiffness matrices are implemented in the Matlab program for the vibration and buckling analysis of plates and shells assemblies. The results of numerous numerical examples are compared with the available results in the literature, as well as with the results obtained using the FEM , and, in such way, the formulated continuous elements are verified

Dynamic stiffness method in the vibration analysis of circular cylindrical shell

In this paper the dynamic stiffness method is used for free vibration analysis of a circular cylindrical shell. The dynamic stiffness matrix is formulated on the base of the exact solution for free vibration of a circular cylindrical shell according to the Flügge thin shell theory. The matrix is frequency dependent and, besides the stiffness, includes inertia and damping effects. The derived dynamic stiffness matrix is implemented in the code developed in a Matlab program for computing natural frequencies and mode shapes of a circular cylindrical shell. Several numerical examples are carried out. The obtained results are validated against the results obtained by using the commercial finite element program Abaqus as well as the available analytical solutions from the literature.U ovom radu korišćena je metoda dinamičke krutosti za analizu slobodnih vibracija kružne cilindrične ljuske. Dinamička matrica krutosti formulisana je na osnovu tačnog rešenja sistema diferencijalnih jednačina problema slobodnih vibracija po Flügge-ovoj teoriji ljuski. To je frekventno zavisna matrica koja u sebi, pored krutosti, sadrži uticaj inercije i prigušenja. Izvedena dinamička matica krutosti implementirana je u za tu svrhu napisani Matlab program za određivanje sopstvenih frekvencija i oblika oscilovanja kružne cilindrične ljuske. Urađen je niz primera. Rezultati dobijeni primenom dinamičke matrice krutosti upoređeni su s rezultatima dobijenim pomoću komercijalnog programa zasnovanog na metodi konačnih elemenata Abaqus, kao i sa dostupnim analitičkim rezultatima iz literature

Metoda dinamičke krutosti u analizi vibracija kružne cilindrične ljuske

In this paper the dynamic stiffness method is used for free vibration analysis of a circular cylindrical shell. The dynamic stiffness matrix is formulated on the base of the exact solution for free vibration of a circular cylindrical shell according to the Flügge thin shell theory. The matrix is frequency dependent and, besides the stiffness, includes inertia and damping effects. The derived dynamic stiffness matrix is implemented in the code developed in a Matlab program for computing natural frequencies and mode shapes of a circular cylindrical shell. Several numerical examples are carried out. The obtained results are validated against the results obtained by using the commercial finite element program Abaqus as well as the available analytical solutions from the literature.U ovom radu korišćena je metoda dinamičke krutosti za analizu slobodnih vibracija kružne cilindrične ljuske. Dinamička matrica krutosti formulisana je na osnovu tačnog rešenja sistema diferencijalnih jednačina problema slobodnih vibracija po Flügge-ovoj teoriji ljuski. To je frekventno zavisna matrica koja u sebi, pored krutosti, sadrži uticaj inercije i prigušenja. Izvedena dinamička matica krutosti implementirana je u za tu svrhu napisani Matlab program za određivanje sopstvenih frekvencija i oblika oscilovanja kružne cilindrične ljuske. Urađen je niz primera. Rezultati dobijeni primenom dinamičke matrice krutosti upoređeni su s rezultatima dobijenim pomoću komercijalnog programa zasnovanog na metodi konačnih elemenata Abaqus, kao i sa dostupnim analitičkim rezultatima iz literature

Granična čvrstoća pritisnutih podužno ukrućenih pločastih nosača

In this paper, two shear deformable dynamic stiffness elements for the free vibration analysis of rectangular, transversely isotropic, single- and multi-layer plates having arbitrary boundary conditions are presented. Dynamic stifness matrices are developed for the Reddy’s higher-order shear deformation theory (HSDT) and the Mindlin-Reissner’s first-order shear deformation theory (FSDT). The dynamic stiffness matrices contain both the stiffness and mass properties of the plate and can be assembled similarly as in the conventional finite element method. The influence of faceto-core thickness ratio and face-to-core module ratio of sandwich plate, as well as the influence of the shear deformation on the free vibration characteristics of sandwich plates have been analysed. The results obtained by proposed HSDT and FSDT dynamic stiffness element are validated against the results obtained using the conventional finite element analysis (ABAQUS), as well as the results obtained by 4-node layered rectangular finite element. The proposed model allows accurate prediction of free vibration response of rectangular layered plate assemblies with arbitrary boundary conditions.Zbornik radova Građevinskog fakultet

Slobodne vibracije ploča sa ukrućenjima primenom metode dinamičke krutosti

The free vibration analysis of stiffened plate assemblies is presented in this paper by using the dynamic stiffness method. The transformed dynamic stiffness matrix of completely free rectangular Mindlin plate is derived by using the transformation matrix. In addition, the global dynamic stiffness matrix of plate assembly is derived by using similar assembly procedure as in the finite element method. The natural frequencies of stiffened plate assemblies with arbitrary boundary conditions are computed and validated against the results obtained by using the finite element software Abaqus. High accuracy of the results is demonstrated.U okviru ovog rada analizirane su slobodne vibracije ploča sa ukrućenjima primenom metode dinamičke krutosti. Razvijena je transformisana dinamička matrica krutosti potpuno slobodne pravougaone Mindlin-ove ploče korišćenjem matrice transformacije. Takođe, izvedena je globalna dinamička matrica krutosti sistema ploča koristeći sličan postupak kao u metodi konačnih elemenata. Određene su sopstvene frekvencije ploča sa ukrućenjima za različite konturne uslove i upoređene sa vrednostima dobijenim po metodi konačnih elemenata primenom programskog paketa Abaqus. Dobijeni su rezultati visoke tačnosti

Primena metode dinamičke krutosti u numeričkoj analizi slobodnih vibracija ploča sa ukrućenjima

The free vibration analysis of stiffened plate assemblies has been performed in this paper by using the dynamic stiffness method. Rectangular Mindlin plate dynamic stiffness element has been formulated. Using the rotation matrices, dynamic stiffness matrices of single plates have been derived in global coordinate system. The global dynamic stiffness matrix of plate assembly has been derived by using similar assembly procedure as in the finite element method. The natural frequencies of stiffened plate assemblies with different boundary conditions have been computed and validated against the results obtained by using the commercial software package Abaqus. High accuracy of the results has been demonstrated.U okviru ovog rada, analizirane su slobodne vibracije ploča sa ukrućenjima, primenom metode dinamičke krutosti. Formulisan je pravougaoni element Mindlin-ove ploče, zasnovan na metodi dinamičke krutosti. Primenom matrica rotacije, formirane su dinamičke matrice krutosti pojedinačnih ploča u globalnom koordinatnom sistemu. Korišćenjem sličnog postupka kao u metodi konačnih elemenata, izvedena je globalna dinamička matrica krutosti sistema ploča. Određene su sopstvene frekvencije ploča sa ukrućenjima za različite konturne uslove i upoređene sa vrednostima dobijenim u komercijalnom programskom paketu Abaqus. Dobijeni su rezultati visoke tačnosti

Vibration and buckling of plates and shells using dynamic stiffness method

Метод динамичке крутости (МДК) представља алтернативу методу коначних елемената (МКЕ) у анализи вибрација и избочавања конструкција. Основни елемент у МДК је континуални елемент, односно његова матрица крутости, која је формулисана на основу тачног решења диференцијалне једначине проблема, па је самим тим избегнута потреба за дискретизацијом домена. Да би МДК могао да нађе ширу примену, потребна је одговарајућа база континуалних елемената. У оквиру дисертације су по први пут формулисани континуални елементи за анализу вибрација Mindlin-ове правоугаоне плоче и сегмента кружне цилиндричне љуске по Donnell-Mushtari-евој и Flügge-овој теорији. За решење проблема слободних вибрација коришћен је Gorman-ов метод суперпозиције, док је динамика матрица крутости формулисана помоћу метода пројекције. Такође, на основу решења у затвореном облику формулисани су следећи континуални елементи, односно одговарајуће матрице крутости, за анализу вибрација и избочавања: Maurice Lévy-еве плоче по Mindlin-овој теорији, кружне цилиндричне љуске и сегмента кружне цилиндричне љуске са специјалним граничним условима по Donnell-Mushtari-евој и Flügge-овој теорији. Изведене матрице крутости су имплементиране у за ту сврху написани Matlab програм за анализу вибрација и избочавања система плоча и љуски. Резултати многобројних нумеричких примера су упоређени са доступним резултатима из литературе, као и резултатима МКЕ, чиме је извршена верификација у раду формулисаних континуалних елемената.Dynamic stiffness method (DSM) is an alternative to the Finite element method (FEM) in the vibration and buckling analysis. The essential element in the DSM is a continuous element and the corresponding stiffness matrix. The stiffness matrix is formulated based on the exact solution of the governing equations. Consequently, the discretization of the domain is minimized. For a wider application of the DSM, a suitable base of the continuous elements is necessary. Within this thesis, the continuous elements and the corresponding dynamic stiffness matrices for vibration analysis of the Mindlin plate and segment of circular cylindrical shells based on the Donnell-Mushtari and Flügge theory are formulated for the first time. Gorman's method of superposition has been used for solution of the free vibrations problem, while the dynamic stiffness matrix is formulated by using the Projection method. In addition, based on the closed-form solutions of the of free vibration and buckling problem, the following continuous elements are formulated: Maurice Lévy plate element based on the Mindlin theory, circular cylindrical shell and segmented circular cylindrical shell with special boundary conditions element based on the Donnell-Mushtari and Flügge theory. The developed stiffness matrices are implemented in the Matlab program for the vibration and buckling analysis of plates and shells assemblies. The results of numerous numerical examples are compared with the available results in the literature, as well as with the results obtained using the FEM , and, in such way, the formulated continuous elements are verified

Dynamic stiffness – based free vibration study of open circular cylindrical shells

In this paper, the dynamic stiffness method has been applied in the free vibration analysis of open circular cylindrical shells based on Flügge thin shell theory. The dynamic stiffness matrix for a completely free open cylindrical shell element has been derived and coded in MATLAB to compute the natural frequencies and mode shapes. The results of the numerical study have been compared with the results from the finite element analysis, as well as with the results from the literature. After a detailed convergence study, considering the number of terms in the proposed solution, applied boundary conditions and geometry, limitations and recommendations for application of the proposed method have been given

Free vibration study of axisymmetric assemblies using dynamic stiffness method

U ovom radu primenjena je metoda dinamičke krutosti za analizu slobodnih vibracija sistema koji se sastoji od kružne cilindrične ljuske i kružnih i/ili prstenastih ploča. Pri formulaciji dinamičkih matrica krutosti korišćena je Flügge-ove teorija ljuski, kao i Kirchoff-ova teorija tankih ploča. Na osnovu razvijenog programa u Matlab okruženju, urađeno je nekoliko numeričkih primera. Rezultati za sopstvene frekvencije dobijeni primenom metode dinamičke krutosti upoređeni su sa rezultatima dobijenim pomoću komercijalnog programa Abaqus zasnovanog na metodi konačnih elemenata.In this paper, the dynamic stiffness method is used for the free vibration analysis of assemblies built up of circular cylindrical shells and circular and/or annular plates. Flügge's shell theory as well as Kirchoff's theory of thin plates were used in the formulation of dynamic stiffness matrices. Based on the Matlab code, developed for this particular purpose, several numerical examples were presented. The results for natural frequencies obtained using the dynamic stiffness method are compared with the results obtained using the commercial finite element method - based software Abaqus

Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies

The dynamic stiffness matrix of completely free rectangular Mindlin plate element is presented in this paper. The system of three coupled equations of motion is transformed into two uncoupled equations introducing a boundary layer function. The dynamic stiffness matrix is derived by use of the superposition method and the projection method. Using the proposed method natural frequencies of individual plates and plate assemblies with arbitrary boundary conditions are computed and validated against the results available in the literature and the finite element analysis. High efficiency and accuracy of the results are demonstrated