SOME BASIC CONCEPTS OF ENGINEERING ANALYSIS
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1 SOME BASIC CONCEPTS OF ENGINEERING ANALYSIS LECTURE 1 46 MINUTES I-I
2 SolIe basic ccnacepls of eugideeridg ualysis LECTURE 1 Introduction to the course. objective of lectures Some basic concepts of engineering analysis. discrete and continuous systems. problem types: steady-state. propagation and eigenvalue problems Analysis of discrete systems: example analysis of a spring system Basic solution requirements Use and explanation of the modern direct stiffness method Variational formulation TEXTBOOK: Sections: 3.1 and Examples:
3 Some basic concepts 01 engineering aulysis INTRODUCTION TO LINEAR ANALYSIS OF SOLIDS AND STRUCTURES The finite element method is now widely used for analysis of structural engineering problems. 'n civil, aeronautical, mechanical, ocean, mining, nuclear, biomechanical,... engineering Since the first applications two decades ago, - we now see applications in linear, nonlinear, static and dynamic analysis. - various computer programs are available and in significant use My objective in this set of lectures is: to introduce to you finite element methods for the linear analysis of solids and structures. ["Iinear" meaning infinitesimally small displacements and linear elastic material proeerties (Hooke's law applies)j to consider - the formulation of the finite element equilibrium equations - the calculation of finite element matrices - methods for solution of the governing equations - computer implementations.to discuss modern and effective techniques, and their practical usage. 1 3
4 Some basic concepts of engineering analysis REMARKS Emphasis is given to physical explanations rather than mathematical derivations Techniques discussed are those employed in the computer programs SAP and ADINA SAP== Structural Analysis Program ADINA=Automatic Dynamic Incremental Nonlinear Analysis These few lectures represent a very brief and compact introduction to the field of finite element analysis We shall follow quite closely certain sections in the book Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc. (by K.J. Bathe). Finite Element Solution Process Physical problem Establish finite element ~ model of physical problem Revise (refine) the model? I 1 : I,- S_ol_v_e_th_e_m_o_d_el I ~ ~ il-_i_n_te_r.;..p_re_t_t_h_e_re_s_u_lt_s_...j 1-4
5 SolIe basic concepts of engideering analysis 10 ft 15 ft I 12 at 15,. Analysis of cooling tower. K~~~~~-~,-Fault \\(no restraint assumed) Altered' grit E= toe c., Analysis of dam. 1 5
6 Some basic concepts of engineering analysis B. t W o E~~;;C= _... F Finite element mesh for tire inflation analysis. 1 6
7 SolDe basic concepts of engineering analysis Segment of a spherical cover of a laser vacuum target chamber. p l,w -0.2 Mil ""= x 16 MESH ~ C BENDING MOMENT DISTRIBUTION ALONG DC OF PINCHED CYLINDRICAL SHELL p PINCHED CYLINDRICAL SHELL EtW -50 P- 100 OD;:..., ~~~~~~C x 16 MESH DISPLACEMENT DISTRIBUTION ALONG DC OF PINCHED CYLINDRICAL SHELL 1-7
8 SoBle basic concepts 01 engineering analysis I Finite element idealization of wind tunnel for dynamic analysis SOME BASIC CONCEPTS OF ENGINEERING ANALYSIS The analysis of an engineering system requires: - idealization of system - formulation of equilibrium equations - solution of equations - interpretation of results 1 8
9 Some basic concepts of engineering analysis SYSTEMS DISCRETE response is described by variables at a finite number of points set of algebraic -- equations CONTINUOUS response is described by variables at an infinite number of points set of differential equations PROBLEM TYPES ARE STEADY -STATE (statics) PROPAGATION (dynamics) EIGENVALUE For discrete and continuous systems Analysis of complex continuous system requires solution of differential equations using numerical procedures reduction of continuous system to discrete form powerful mechanism: the finite element methods, implemented on digital computers ANALYSIS OF DISCRETE SYSTEMS Steps involved: - system idealization into elements - evaluation of element equilibrium requirements - element assemblage - solution of response 1 9
10 Some basic concepts of engineering analysis Example: steady - state analysis of system of rigid carts interconnected by springs Physical layout ELEMENTS I ~ : ~\l) k, u1 - F(2 ) F(2) --- 2, --- F(3) --, F(4) - F(', ) 1 U1 U 3.. F(4).. 3 '4 [1-1]["1]. [F1 4' ] u2-1 1 U F(4) 3 3 '2 [ 1-1]["I]fF}] u F( 2 ) 2 2 F(S) F(S) 2 3 u, u 2 -t] 2 '5 [1 [F(5l] k 3 F(3) -1 1 u = F1S ) '3 [ ] -r 1 ]f P l] -1 1 u F(3) 2 2
11 SolIe basic coicepls of engineering analysis Element interconnection requirements : F(4) + F(S) = R These equations can be written in the form K U = E. Equilibrium equations K U = R (a) U T = [u - 1 R T = [R - 1 +k 4 k 1 + k 2 + k 3 ~ -k 2 - k 3 : -k4....'" K = -k 2 - k 3 ~ k 2 + k 3 + k S ~ -k S 1 11
12 Some basic concepts of engineering analysis and we note that ~= t ~(i) i =1 where ::] o 0 etc... This assemblage process is called the direct stiffness method The steady- state analysis is completed by solving the equations in (a) 1 12
13 Some basic concepls 01 engineering analysis u,...: : ~ u, K= K1.... : : : U1...:. K ~ ~~.~ ~~ :... ~1---.JI.l\fl--r/A~,1\1\1\~~r/A 1 13
14 SOlDe basic concepts of engineering analysis u, K =... : :. 'O u, K= + K 4 ; K 1 +K 2 + K 3 ;-K 2 -K 3 'O'O'O 'O'O'O'O'O'O'O'O'O'O'O'O'O'O:'O'O'O'O'O'O'O'O'O'O'O'O'O'O:'O'O'O'O'O'O'O'O'O'O'O'O'O'O 'O'O + K 4 ;. u, K 1 +K 2 + K 3 ~-K2 -K 3 -K 4 'O'O K = -K 2 -K 3 ~K2+ K 3 + K 5 -K 5 'O'O : I K 1 14
15 Some basic concepts of engineering analysis In this example we used the direct approach; alternatively we could have used a variational approach. In the variational approach we operate on an extremum formu lation: u = strain energy of system W = total potential of the loads Equilibrium equations are obtained from an - 0 (b) ~- 1 In the above analysis we have U=~UT!!! W = U T R Invoking (b) we obtain K U = R Note: to obtain U and W we again add the contributions from all elements 1 15
16 SOlDe basic concepts of engineering analysis PROPAGATION PROBLEMS main characteristic: the response changes with time ~ need to include the d'alembert forces: For the example: m, a a M = a m 2 a a a m 3 EIGENVALUE PROBLEMS we are concerned with the general ized eigenvalue problem (EVP) A v = A B v!l,.!l are symmetric matrices of order n v is a vector of order n A is a scalar EVPs arise in dynamic and buckling analysis 1 16
17 Some basic concepts of engineering analysis Example: system of rigid carts ~lu+ku=o Let U = <p sin W(t-T) Then we obtain _w 2 ~~ sin W(t-T) + K <p sin W(t-T)= Hence we obtain the equation There are 3 solutions w,,~, (l)2 ' ~2 eigenpairs w3 ' ~3 In general we have n solutions 1 17
18 MIT OpenCourseWare Resource: Finite Element Procedures for Solids and Structures Klaus-Jürgen Bathe The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit:
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