2C9 Design for seismic and climate changes. Jiří Máca
|
|
- Job Tate
- 5 years ago
- Views:
Transcription
1 2C9 Design for seismic and climate changes Jiří Máca
2 List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom systems I 4. Dynamic analysis of single-degree-of-freedom systems II 5. Dynamic analysis of single-degree-of-freedom systems III 6. Dynamic analysis of multi-degree-of-freedom systems 7. Finite element method in structural dynamics I 8. Finite element method in structural dynamics II 9. Earthquake analysis I 2
3 Finite element method in structural dynamics I Free vibration analysis 1. Finite element method in structural dynamics 2. Free vibration 3. Rayleigh Ritz method 4. Vector iteration techniques 5. Inverse vector iteration method + examples 6. Subspace iteration method 7. Examples
4 1. FINIE ELEMEN MEHOD IN SRUCURAL DYNAMICS Statics Kr f K A Ke N e e1 V K e B DB Dynamics dv Mr ( t) Kr( t) 0 Mr ( t) Cr ( t) Kr( t) f( t) Forces do not vary with time Structural stiffness K considered only N A e Operator e 1 denotes the direct assembly procedure for assembling the stiffness matrix for each element, e = 1 to N e, where N e is the number of elements Free vibration No external forces present Structural stiffness K and mass M considered Forced vibration External forces (usually time dependent) present Structural stiffness K, mass M and possibly damping C considered
5 Finite Element Method in structural dynamics Inertial force (per unit volume) ρ mass density Damping force (per unit volume) ξ damping coefficient f u f udv ˆi f u f udv ˆd i d V V Principle of virtual displacements WI W E internal virtual work = external virtual work W I ε σ dv σ stress vector ε strain vector V ˆ W u f dv u fˆ dv E i d V V
6 Finite Element Method in structural dynamics Finite Element context u Nr u Nr unr u r N u r N r nodal displacements N interpolation (shape) functions Strain displacement relations ε u Nr Br ε r B Constitutive relations (linear elastic material) σ D ε DBr D constitutive matrix
7 Finite Element Method in structural dynamics Internal virtual work ε σ r B DB r W I dv dv V V External virtual work W W E I 0 ˆ W u f dv u fˆ dv E i d V V dv V dv V u u u u V V dv r N N r r N N dvr r N N dv r N N dvr B DB dv r V V V Dynamic equations of equilibrium Mr Cr Kr 0 0 Mr Cr Kr f f nodal forces (external)
8 Finite Element Method in structural dynamics Mass matrix (consistent formulation) M N N V dv Damping matrix C N N V dv Stiffness matrix K B DB V dv Mass matrix - other possible formulations Diagonal mass matrix equal mass lumping lumped-mass idealization with equal diagonal terms special mass lumping - diagonal mass matrix, terms proportional to the consistent mass matrix
9 Finite Element Method in structural dynamics Beam element length L, modulus of elasticity E, cross-section area A moment of inertia (2nd moment of area) I z, uniform mass m Starting point approximation of displacements u uv, Exact solution (based on the assumptions adopted) - Approximation of axial displacements x x ux ( ) 1 u u huhu L L u 1 left-end axial displacement u 2 right-end axial displacement
10 Beam element - Approximation of deflections x x x x x vx ( ) 13 2 v1 2 L 1 L L L L L x x x x 3 2 v2 L 2 L L L L vx ( ) hvh hv h v1 1 v 1 h 3 v 2 2 v 1 left-end deflection v 2 right-end deflection φ 1 left-end rotation φ 2 right-end rotation 1 1 h 4 h 5 v h 6
11 Beam element Vector of nodal displacements r u, v,, u, v, Matrix of interpolation functions N h1 0 0 h h3 h4 0 h5 h 6 u Nr Constitutive relations Strain displacement relations du Nx EA 0 dx 2 M z 0 EI z dv 2 dx σ D ε ε u Nr Br B dh1 dh dx dx dh dh dh dh 0 0 dx dx dx dx
12 Beam element Element stiffness matrix K L B DB 0 dx K EA EA L L 12EI z 6EI z 12EI z 6EI z L L L L 6EI z 4EI z 6EI z 2EI z L L L L EA EA L L 12EI 6EI 12EI 6EI 0 0 L L L L 6EI z 2EI z 6EI z 4EI z L L L L z z z z
13 Beam element Element mass matrix consistent formulation L M N mn 0 dx M L L 2 2 ml 4L 0 13L 3L L 2 sym 4 L
14 Beam element Element mass matrix other possible approximation of deflections x x vx ( ) 1 v v hvhv L L Element mass matrix diagonal mass matrix vx ( ) hv hv where h 1 =1 for x l/2 h 1 =0 for x>l/2 h 2 =0 for x < l/2 h 2 =1 for x l/ ml M sym ml M sym 0
15 Comparison of Finite Element and exact solution
16 Comparison of Finite Element and exact solution Accuracy of FE analysis is improved by increasing number of DOFs
17 2. FREE VIBRAION natural vibration frequencies and modes Mr ( t) Kr( t) 0 equation of motion for undamped free vibration of MDOF system r ( t) ( A cos t B sin t) r n n n n n 2 ( t) nr( t) motions of a system in free vibrations are simple harmonic K M 0 2 n n 2 det n 0 free vibration equation, eigenvalue equation - natural mode ω n - natural frequency K M nontrivial solution condition n frequency equation (polynomial of order N) not a practical method for larger systems
18 3. RAYLEIGH RIZ MEHOD general technique for reducing the number of degrees of freedom K M 0 2 n n K M 0 K M K M Rayleigh s quotient Properties 1. When is an eigenvector n, Rayleigh s quotient is equal to the 2 corresponding eigenvalue n 2. Rayleigh s quotient is bounded between the smallest and largest eigenvalues n n
19 Rayleigh Ritz method u ( t) Ψ z ( t) Ψ z displacements or modal shapes are expressed as a linear combination of shape vectors j, j = 1,2 J <N Ritz vectors make up the columns of N x J matrix Ψ z is a vector of J generalized coordinates substituting the Ritz transformation in a system of N equations of motion Ku Cu Ku sp() t we obtain a system of J equations in generalized coordinates mz cz kz sp () t m Ψ M Ψ c Ψ C Ψ k Ψ K Ψ s Ψ s Ritz transformation has made it possible to reduce original set of N equations in the nodal displacement u to a smaller set of J equations in the generalized coordinates z
20 Rayleigh Ritz method substituting the Ritz transformation in a Rayleigh s quotient K z Ψ K Ψ z M z Ψ M Ψ z ( z) z i 0 i1,2,... J ( ) z and using Rayleigh s stationary condition we obtain the reduced eigenvalue problem kz mz m Ψ M Ψ k Ψ K Ψ, z solution of all eigenvalues and eigenmodes e.g. Jacobi s method of rotations J J i izi
21 Rayleigh Ritz method Selection of Ritz vectors Selection of Ritz vectors
22 4. VECOR IERAION ECHNIQUES Inverse iteration (Stodola-Vianello) Algorithm to determine the lowest eigenvalue Kx Mx x K Mx x 1 k1 k k1 k x k 1 k 1 1/2 ( xk 1Mxk1) k 1 1 as k x 1
23 Vector iteration techniques Forward iteration (Stodola-Vianello) Algorithm to determine the highest eigenvalue Mx Kx x M Kx x 1 k1 k k1 k k 1 k 1 1/2 ( xk 1Mxk1) n xk n as k 1 x
24 5. INVERSE VECOR IERAION MEHOD 1. Starting vector x! arbitrary choice R Mx 2. ( k 1) k1 k1 3. x Kx (Rayleigh s quotient) x Mx 4. Kx Kx 1 1 R 2 1 k1 Mx ( k1) ( k) ( k 1) k k1 k1 tol 5. If convergence criterion is not satisfied, normalize xk 1 x( k 1) 1/2 and go back to 2. and set k = k+1 x Mx k1 k1 x k1 1 K Mxk
25 Inverse vector iteration method 6. For the last iteration (k+1), when convergence is satisfied 2 ( k 1) 1 1 x k 1 1 ( k 1) 1/2 xk1mxk1 x Gramm-Schmidt orthogonalisation to progress to other than limiting eigenvalues (highest and lowest) evaluation of n1 correction of trial vector x x x Mx n j j1 jmj j
26 Inverse vector iteration method Vector iteration with shift μ the shifting concept enables computation of any eigenpair KM KKM eigenvectors of the two eigenvalue problems are the same inverse vector iteration converge to the eigenvalue closer to the shifted origin, e.g. to, see (c) 3
27 Inverse vector iteration method Example 3DOF frame, 1 st natural frequency k k 1,5 k k k 1,5 k K 1 flexibility matrix δ k m 0 0 M 0 m m m δm k Kx Mx x K Mx Mx 1 k1 k k1 k k x starting vector
28 Inverse vector iteration method 1 st iteration 6 1 m (1) xmx 1 0 k x1 δmx0 12 0, 489 x1 2 6k 1 1 m 15 xmx 2,5 2 nd iteration 11 1 m (2) xmx 2 1 k x2 δmx1 24,5 0, 481 x2 2, 23 6k 2 2 m 32 xmx 2,91 12, 28 1 m (3) xmx 3 2 k x3 δmx2 27,70 0,481 x3 2,26 6k 3 3 m 36, 43 xmx 2,97 (3) 1 0,694 k 1 x3 m 3 rd iteration normalization u1 1
29 Appendix flexibility matrix k k u u 3k 2k 2k m 1 m 2 m u 3 k 1,5 k δ k 1,5 k k F k 3k 3k F k 3k 2k 3k 2k F k 3k 2k 3k 2k 2k
30 6. SUBSPACE IERAION MEHOD efficient method for eigensolution of large systems when only the lower modes are of interest similar to inverse iteration method - iteration is performed simultaneously on a number of trial vectors m m smaller of 2p and p+8, p is the number of modes to be determined (p is usually much less then N, number of DOF) 1. Staring vectors X 1 R MX 1 1 KX R Subspace iteration a) KX k1 MX k b) Ritz transformation K k1 Xk 1KXk1 M k1xk1mxk1
31 Subspace iteration method 2. Subspace iteration cont. c) Reduced eigenvalue problem (m eigenvalues) K Q Ω M Q 2 k1 k1 k1 k1 k1 d) New vectors X X Q k1 k1 k1 e) Go back to 2 a) 3. Sturm sequence check verification that the required eigenvalues and vectors have been calculated - i.e. first p eigenpairs Subspace iteration method - combines vector iteration method with the transformation method
32 7. EXAMPLES Simple frame exact solution elem elem elem f 1 [Hz] 7,270 7,282 7,270 7,270 f 2 [Hz] 28,693 34,285 28,845 28,711 EI = knm 2 μ = 252 kgm -1 f 3 [Hz] 46,799 74,084 47,084 46, shape of vibration 2. shape of vibration 3. shape of vibration
33 Examples Simple frame exact 3 6 solution elem elem f 1 [Hz] 28,662 34,285 28,845 f 2 [Hz] 41,863 65,623 42,393 EI = knm 2 μ = 252 kgm -1 f 3 [Hz] 50,653-51, shape of vibration Recommendation: it is good to divide beams into (at least) 2 elements in the dynamic FE analysis
34 Examples 3-D frame (turbomachinery frame foundation)
35 Examples 3-D frame (turbo-machinery frame foundation)
36 Examples 3-D frame (turbo-machinery frame foundation)
37 Examples 3-D frame + plate + elastic foundation
38 Examples cable-stayed bridge
39 Examples cable-stayed bridge
40 Examples cable-stayed bridge
Lecture 27: Structural Dynamics - Beams.
Chapter #16: Structural Dynamics and Time Dependent Heat Transfer. Lectures #1-6 have discussed only steady systems. There has been no time dependence in any problems. We will investigate beam dynamics
More informationCodal Provisions IS 1893 (Part 1) 2002
Abstract Codal Provisions IS 1893 (Part 1) 00 Paresh V. Patel Assistant Professor, Civil Engineering Department, Nirma Institute of Technology, Ahmedabad 38481 In this article codal provisions of IS 1893
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More informationLecture 7: The Beam Element Equations.
4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite
More informationJEPPIAAR ENGINEERING COLLEGE
JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationChapter 4 Analysis of a cantilever
Chapter 4 Analysis of a cantilever Before a complex structure is studied performing a seismic analysis, the behaviour of simpler ones should be fully understood. To achieve this knowledge we will start
More informationANALYSIS OF HIGHRISE BUILDING STRUCTURE WITH SETBACK SUBJECT TO EARTHQUAKE GROUND MOTIONS
ANALYSIS OF HIGHRISE BUILDING SRUCURE WIH SEBACK SUBJEC O EARHQUAKE GROUND MOIONS 157 Xiaojun ZHANG 1 And John L MEEK SUMMARY he earthquake response behaviour of unframed highrise buildings with setbacks
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationCIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass
CIV 8/77 Chapter - /75 Introduction To discuss the dynamics of a single-degree-of freedom springmass system. To derive the finite element equations for the time-dependent stress analysis of the one-dimensional
More informationMatrix Iteration. Giacomo Boffi.
http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 12, 2016 Outline Second -Ritz Method Dynamic analysis of MDOF
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationEffect of Mass Matrix Formulation Schemes on Dynamics of Structures
Effect of Mass Matrix Formulation Schemes on Dynamics of Structures Swapan Kumar Nandi Tata Consultancy Services GEDC, 185 LR, Chennai 600086, India Sudeep Bosu Tata Consultancy Services GEDC, 185 LR,
More informationLecture 15: Revisiting bars and beams.
3.10 Potential Energy Approach to Derive Bar Element Equations. We place assumptions on the stresses developed inside the bar. The spatial stress and strain matrices become very sparse. We add (ad-hoc)
More informationInstitute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationDr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum
STRUCTURAL DYNAMICS Dr.Vinod Hosur, Professor, Civil Engg.Dept., Gogte Institute of Technology, Belgaum Overview of Structural Dynamics Structure Members, joints, strength, stiffness, ductility Structure
More informationReduction in number of dofs
Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole
More informationFREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS
FREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS A Thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Technology in Civil Engineering By JYOTI PRAKASH SAMAL
More informationAdvanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One
Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationAdvanced Vibrations. Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian
Advanced Vibrations Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian ahmadian@iust.ac.ir Distributed-Parameter Systems: Exact Solutions Relation between Discrete and Distributed
More informationVIBRATION PROBLEMS IN ENGINEERING
VIBRATION PROBLEMS IN ENGINEERING FIFTH EDITION W. WEAVER, JR. Professor Emeritus of Structural Engineering The Late S. P. TIMOSHENKO Professor Emeritus of Engineering Mechanics The Late D. H. YOUNG Professor
More informationInstitute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationStructural Matrices in MDOF Systems
in MDOF Systems http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 9, 2016 Outline Additional Static Condensation
More informationIntroduction to structural dynamics
Introduction to structural dynamics p n m n u n p n-1 p 3... m n-1 m 3... u n-1 u 3 k 1 c 1 u 1 u 2 k 2 m p 1 1 c 2 m2 p 2 k n c n m n u n p n m 2 p 2 u 2 m 1 p 1 u 1 Static vs dynamic analysis Static
More informationStochastic Dynamics of SDOF Systems (cont.).
Outline of Stochastic Dynamics of SDOF Systems (cont.). Weakly Stationary Response Processes. Equivalent White Noise Approximations. Gaussian Response Processes as Conditional Normal Distributions. Stochastic
More informationME FINITE ELEMENT ANALYSIS FORMULAS
ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness
More informationOutline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices
Outline in MDOF Systems Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano May 8, 014 Additional Today we will study the properties of structural matrices, that is the operators that
More informationMechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem
Mechanical Vibrations Chapter 6 Solution Methods for the Eigenvalue Problem Introduction Equations of dynamic equilibrium eigenvalue problem K x = ω M x The eigensolutions of this problem are written in
More informationDesign of Structures for Earthquake Resistance
NATIONAL TECHNICAL UNIVERSITY OF ATHENS Design of Structures for Earthquake Resistance Basic principles Ioannis N. Psycharis Lecture 3 MDOF systems Equation of motion M u + C u + K u = M r x g(t) where:
More informationAdvanced Vibrations. Distributed-Parameter Systems: Approximate Methods Lecture 20. By: H. Ahmadian
Advanced Vibrations Distributed-Parameter Systems: Approximate Methods Lecture 20 By: H. Ahmadian ahmadian@iust.ac.ir Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz
More informationME 475 Modal Analysis of a Tapered Beam
ME 475 Modal Analysis of a Tapered Beam Objectives: 1. To find the natural frequencies and mode shapes of a tapered beam using FEA.. To compare the FE solution to analytical solutions of the vibratory
More informationGiacomo Boffi. April 7, 2016
http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 7, 2016 Outline Analysis The process of estimating the vibration
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationChapter 23: Principles of Passive Vibration Control: Design of absorber
Chapter 23: Principles of Passive Vibration Control: Design of absorber INTRODUCTION The term 'vibration absorber' is used for passive devices attached to the vibrating structure. Such devices are made
More informationFREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS
Lecture Notes: STRUCTURAL DYNAMICS / FALL 2011 / Page: 1 FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS : : 0, 0 As demonstrated previously, the above Equation of Motion (free-vibration equation) has a solution
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationEQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION
1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development
More informationMulti Degrees of Freedom Systems
Multi Degrees of Freedom Systems MDOF s http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System
More informationProgram System for Machine Dynamics. Abstract. Version 5.0 November 2017
Program System for Machine Dynamics Abstract Version 5.0 November 2017 Ingenieur-Büro Klement Lerchenweg 2 D 65428 Rüsselsheim Phone +49/6142/55951 hd.klement@t-online.de What is MADYN? The program system
More informationEML4507 Finite Element Analysis and Design EXAM 1
2-17-15 Name (underline last name): EML4507 Finite Element Analysis and Design EXAM 1 In this exam you may not use any materials except a pencil or a pen, an 8.5x11 formula sheet, and a calculator. Whenever
More informationPart 1: Discrete systems
Part 1: Discrete systems Introduction Single degree of freedom oscillator Convolution integral Beat phenomenon Multiple p degree of freedom discrete systems Eigenvalue problem Modal coordinates Damping
More informationon the figure. Someone has suggested that, in terms of the degrees of freedom x1 and M. Note that if you think the given 1.2
1) A two-story building frame is shown below. The mass of the frame is assumed to be lumped at the floor levels and the floor slabs are considered rigid. The floor masses and the story stiffnesses are
More informationStatic & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering
Static & Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward LWilson Professor Emeritus of Civil Engineering University of California, Berkeley Fourth Edition
More informationCIVL4332 L1 Introduction to Finite Element Method
CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such
More information18. FAST NONLINEAR ANALYSIS. The Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis
18. FAS NONLINEAR ANALYSIS he Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis 18.1 INRODUCION he response of real structures when subjected
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationGeneralized Single Degree of Freedom Systems
Single Degree of Freedom http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March, 8 Outline Until now our were described
More informationGeneral elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
More informationResponse Analysis for Multi Support Earthquake Excitation
Chapter 5 Response Analysis for Multi Support Earthquake Excitation 5.1 Introduction It is very important to perform the dynamic analysis for the structure subjected to random/dynamic loadings. The dynamic
More informationM.S Comprehensive Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive
More informationChapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14
Table of Contents Chapter 1: Research Objectives and Literature Review..1 1.1 Introduction...1 1.2 Literature Review......3 1.2.1 Describing Vibration......3 1.2.2 Vibration Isolation.....6 1.2.2.1 Overview.
More informationEFFECTIVE MODAL MASS & MODAL PARTICIPATION FACTORS Revision F
EFFECTIVE MODA MASS & MODA PARTICIPATION FACTORS Revision F By Tom Irvine Email: tomirvine@aol.com March 9, 1 Introduction The effective modal mass provides a method for judging the significance of a vibration
More informationHydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition
Fluid Structure Interaction and Moving Boundary Problems IV 63 Hydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition K.-H. Jeong, G.-M. Lee, T.-W. Kim & J.-I.
More informationLecture 8: Assembly of beam elements.
ecture 8: Assembly of beam elements. 4. Example of Assemblage of Beam Stiffness Matrices. Place nodes at the load application points. Assembling the two sets of element equations (note the common elemental
More informationStructural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma).
Structural Dynamics Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). We will now look at free vibrations. Considering the free
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,
More informationChapter 3 Variational Formulation & the Galerkin Method
Institute of Structural Engineering Page 1 Chapter 3 Variational Formulation & the Galerkin Method Institute of Structural Engineering Page 2 Today s Lecture Contents: Introduction Differential formulation
More informationIntroduction to Geotechnical Earthquake Engineering
Module 1 Introduction to Geotechnical Earthquake Engineering by Dr. Deepankar Choudhury Professor Department of Civil Engineering IIT Bombay, Powai, Mumbai 400 076, India. Email: dc@civil.iitb.ac.in URL:
More informationTWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A. By Tom Irvine February 25, 2008
TWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A By Tom Irvine Email: tomirvine@aol.com February 5, 008 Introduction Consider a base plate mass m and an avionics mass m modeled as two-degree-of-freedom.
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how
More informationABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in s
ABSTRACT Modal parameters obtained from modal testing (such as modal vectors, natural frequencies, and damping ratios) have been used extensively in system identification, finite element model updating,
More informationChapter 12 Elastic Stability of Columns
Chapter 12 Elastic Stability of Columns Axial compressive loads can cause a sudden lateral deflection (Buckling) For columns made of elastic-perfectly plastic materials, P cr Depends primarily on E and
More informationD && 9.0 DYNAMIC ANALYSIS
9.0 DYNAMIC ANALYSIS Introduction When a structure has a loading which varies with time, it is reasonable to assume its response will also vary with time. In such cases, a dynamic analysis may have to
More informationA Guide to linear dynamic analysis with Damping
A Guide to linear dynamic analysis with Damping This guide starts from the applications of linear dynamic response and its role in FEA simulation. Fundamental concepts and principles will be introduced
More informationPART I. Basic Concepts
PART I. Basic Concepts. Introduction. Basic Terminology of Structural Vibration.. Common Vibration Sources.. Forms of Vibration.3 Structural Vibration . Basic Terminology of Structural Vibration The term
More informationFinite element analysis of rotating structures
Finite element analysis of rotating structures Dr. Louis Komzsik Chief Numerical Analyst Siemens PLM Software Why do rotor dynamics with FEM? Very complex structures with millions of degrees of freedom
More informationCHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
More informationStructural Dynamics Lecture 7. Outline of Lecture 7. Multi-Degree-of-Freedom Systems (cont.) System Reduction. Vibration due to Movable Supports.
Outline of Multi-Degree-of-Freedom Systems (cont.) System Reduction. Truncated Modal Expansion with Quasi-Static Correction. Guyan Reduction. Vibration due to Movable Supports. Earthquake Excitations.
More informationPrepared by M. GUNASHANKAR AP/MECH DEPARTMENT OF MECHANICAL ENGINEERING
CHETTINAD COLLEGE OF ENGINEERING AND TECHNOLOGY-KARUR FINITE ELEMENT ANALYSIS 2 MARKS QUESTIONS WITH ANSWER Prepared by M. GUNASHANKAR AP/MECH DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More informationmidas Civil Dynamic Analysis
Edgar De Los Santos Midas IT August 23 rd 2017 Contents: Introduction Eigen Value Analysis Response Spectrum Analysis Pushover Analysis Time History Analysis Seismic Analysis Seismic Analysis The seismic
More informationDynamics of Structures
Dynamics of Structures Elements of structural dynamics Roberto Tomasi 11.05.2017 Roberto Tomasi Dynamics of Structures 11.05.2017 1 / 22 Overview 1 SDOF system SDOF system Equation of motion Response spectrum
More informationFinite Element Method-Part II Isoparametric FE Formulation and some numerical examples Lecture 29 Smart and Micro Systems
Finite Element Method-Part II Isoparametric FE Formulation and some numerical examples Lecture 29 Smart and Micro Systems Introduction Till now we dealt only with finite elements having straight edges.
More informationSlender Structures Load carrying principles
Slender Structures Load carrying principles Basic cases: Extension, Shear, Torsion, Cable Bending (Euler) v017-1 Hans Welleman 1 Content (preliminary schedule) Basic cases Extension, shear, torsion, cable
More informationIntroduction to Mechanical Vibration
2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization Single-Degree-of-Freedom
More informationParametric Identification of a Cable-stayed Bridge using Substructure Approach
Parametric Identification of a Cable-stayed Bridge using Substructure Approach *Hongwei Huang 1), Yaohua Yang 2) and Limin Sun 3) 1),3) State Key Laboratory for Disaster Reduction in Civil Engineering,
More informationSome Aspects of Structural Dynamics
Appendix B Some Aspects of Structural Dynamics This Appendix deals with some aspects of the dynamic behavior of SDOF and MDOF. It starts with the formulation of the equation of motion of SDOF systems.
More informationTruncation Errors Numerical Integration Multiple Support Excitation
Errors Numerical Integration Multiple Support Excitation http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 10,
More informationChapter 7 Vibration Measurement and Applications
Chapter 7 Vibration Measurement and Applications Dr. Tan Wei Hong School of Mechatronic Engineering Universiti Malaysia Perlis (UniMAP) Pauh Putra Campus ENT 346 Vibration Mechanics Chapter Outline 7.1
More informationFINAL EXAMINATION. (CE130-2 Mechanics of Materials)
UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,
More information10. Applications of 1-D Hermite elements
10. Applications of 1-D Hermite elements... 1 10.1 Introduction... 1 10.2 General case fourth-order beam equation... 3 10.3 Integral form... 5 10.4 Element Arrays... 7 10.5 C1 Element models... 8 10.6
More informationFinite Element Analysis Lecture 1. Dr./ Ahmed Nagib
Finite Element Analysis Lecture 1 Dr./ Ahmed Nagib April 30, 2016 Research and Development Mathematical Model Mathematical Model Mathematical Model Finite Element Analysis The linear equation of motion
More informationToward a novel approach for damage identification and health monitoring of bridge structures
Toward a novel approach for damage identification and health monitoring of bridge structures Paolo Martino Calvi 1, Paolo Venini 1 1 Department of Structural Mechanics, University of Pavia, Italy E-mail:
More informationChapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction
More informationSPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES
2010/2 PAGES 1 8 RECEIVED 21. 9. 2009 ACCEPTED 20. 1. 2010 Y. KOLEKOVÁ, M. PETRONIJEVIĆ, G. SCHMID SPECIAL DYNAMIC SOIL- STRUCTURE ANALYSIS PROCEDURES DEMONSTATED FOR TWO TOWER-LIKE STRUCTURES ABSTRACT
More informationCE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao
CE 6701 Structural Dynamics and Earthquake Engineering Dr. P. Venkateswara Rao Associate Professor Dept. of Civil Engineering SVCE, Sriperumbudur Difference between static loading and dynamic loading Degree
More informationMultiple Degree of Freedom Systems. The Millennium bridge required many degrees of freedom to model and design with.
Multiple Degree of Freedom Systems The Millennium bridge required many degrees of freedom to model and design with. The first step in analyzing multiple degrees of freedom (DOF) is to look at DOF DOF:
More informationMECh300H Introduction to Finite Element Methods. Finite Element Analysis (F.E.A.) of 1-D Problems
MECh300H Introduction to Finite Element Methods Finite Element Analysis (F.E.A.) of -D Problems Historical Background Hrenikoff, 94 frame work method Courant, 943 piecewise polynomial interpolation Turner,
More informationResponse Spectrum Analysis Shock and Seismic. FEMAP & NX Nastran
Response Spectrum Analysis Shock and Seismic FEMAP & NX Nastran Table of Contents 1. INTRODUCTION... 3 2. THE ACCELEROGRAM... 4 3. CREATING A RESPONSE SPECTRUM... 5 4. NX NASTRAN METHOD... 8 5. RESPONSE
More information3. Mathematical Properties of MDOF Systems
3. Mathematical Properties of MDOF Systems 3.1 The Generalized Eigenvalue Problem Recall that the natural frequencies ω and modes a are found from [ - ω 2 M + K ] a = 0 or K a = ω 2 M a Where M and K are
More information. D CR Nomenclature D 1
. D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the
More informationExperimental Modal Analysis of a Flat Plate Subjected To Vibration
American Journal of Engineering Research (AJER) 2016 American Journal of Engineering Research (AJER) e-issn: 2320-0847 p-issn : 2320-0936 Volume-5, Issue-6, pp-30-37 www.ajer.org Research Paper Open Access
More informationFinite Element Method
Finite Element Method Finite Element Method (ENGC 6321) Syllabus Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered in this course one dimensional
More informationCollocated versus non-collocated control [H04Q7]
Collocated versus non-collocated control [H04Q7] Jan Swevers September 2008 0-0 Contents Some concepts of structural dynamics Collocated versus non-collocated control Summary This lecture is based on parts
More informationSeismic Design of Tall and Slender Structures Including Rotational Components of the Ground Motion: EN Approach
Seismic Design of Tall and Slender Structures Including Rotational Components of the Ground Motion: EN 1998-6 6 Approach 1 Chimneys Masts Towers EN 1998-6: 005 TOWERS, CHIMNEYS and MASTS NUMERICAL MODELS
More informationDynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras
Dynamics of Ocean Structures Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Lecture 25 Continuous System In the last class, in this, we will
More informationIntroduction to Vibration. Professor Mike Brennan
Introduction to Vibration Professor Mie Brennan Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions Fundamentals For free vibration
More information