Introduction to the Finite Element Method (3)

Size: px

Start display at page:

Download "Introduction to the Finite Element Method (3)"

Rosaline Lloyd
6 years ago
Views:

1 Introduction to the Finite Element Method (3) Petr Kabele Czech Technical University in Prague Faculty of Civil Engineering Czech Republic people.fsv.cvut.cz/~pkabele 1

2 Outline Types of finite element programs Practical aspects of finite element analysis Examples of FE modeling 2

3 Finite element programs classification and structure FEM programs general purpose simulation of general physical problems (statics, dynamics, heat/mass transport, magnetism,..., coupled problems) more complex problem definition/input (choice from many options) user must perfectly understand the mathematical and physical essence of analyzed problem e.g DIANA, ADINA, ABAQUS specialized, engineering simulation of specific engineering problems (e.g. elastic truss structure) user-friendly input (mouse-click, predefined material models, structural members, cross-sections etc., close linkage to design codes) use in engineering practice (structural design) e.g. SAP 3

4 Structure of finite element programs Preprocesor graphical interface for data input Computational core FE program itself Postprocesor graphical interface for processing and visualization of results 4

behavior of real structures by performing an algebraic solution of a set of

5 Practical aspects of finite element analysis General consideration: Finite element analysis is essentially an approximate method for calculating the behavior of real structures by performing an algebraic solution of a set of equations describing idealized structures Physical reality Finite element model 5

6 Selection of analysis type Consider what physical phenomena should be analyzed. mechanical static dynamic stress analysis stability linear nonlinear heat transport mass transport fluid magnetism modal analysis transient analysis linear nonlinear coupled, interaction

7 Selection of modeling hypotheses The most difficult part Geometry and morphology (model scope and detail, structural form, internal composition, connections between the structural elements, ) Material models and properties Actions (mechanical, physical, chemical ) Existing alterations and damage (cracks, constructional mistakes, disconnections, crushing, leanings, ) The interaction of the structure with its surroundings (soil, fluids, other structural parts,...) 7

8 To this end: Clarify what result is anticipated (e.g. overall deformation of a large structure vs. crack propagation at a detail). Consider, what information about the analyzed structure is available (geometry, material, surroundings/supports, loading). Think of suitable simplification, reduction of dimension, substructuring, decomposition, use of symmetry. Select suitable kinematic assumptions and dimension (truss, beam, 2- D solid, plate, shell, 3-D solid). Bear in mind the complexity of model, solution time, postprocessing time and visualization of results. In complex problems, combining various kinematic assumptions may be efficient (e.g. beam + plate). However, proper linkage of all DOF s must be ensured. 8

9 Pre-analysis Make a rough estimation of the expected result (e.g. simplified calculation by hand). Estimate locations of strain concentration and locations of uniform strain use denser mesh in locations with steeper gradients. Run a pilot analysis with coarser mesh compare results with the rough estimate use the results to identify further locations of strain concentration Refinement and analysis Refine the hypotheses and FE mesh as necessary based on the previous step and run the analysis 9

10 Preliminary results check Always check after analysis plot magnified displacement of the model, display the stresses (generalized stresses), reactions Compare results with the rough estimate. Check that loading and kinematic boundary conditions act as expected (stress under loading must correspond to imposed distributed load, outer reactions must be in equilibrium with imposed loading). Check for possible discontinuities due to improper meshing (overlaps of mesh, unexpected stress concentrations) If check fails, find and correct mistakes in input and return to Refinement and analysis. 10

11 Example: 11

12 Rigorous results check Analysis verification: Is the mathematical formulation solved correctly? Check error/accuracy/convergence messages. Check mesh quality criteria Analysis validation Does the mathematical model correctly represent the physical reality? Validation of modeling hypotheses... see SA2 Lecture 1. 12

13 Results processing and presentation FE analysis usually produces huge amount of data. These must be sorted out and presented in an easy-to-understand way. Some examples: plot of deformed configuration contour plots of field variables (displacement, stress, strain, components or principal values,...) vector plots (displacements, principal stress, strain,...) line plots of field variables along line, section time history plots/tables of values in given points extreme values of field variables (see idiana intro for examples) 13

14 Example 1 Perform analysis of a slab. Uniform distributed load 8 kn/m 2 (incl. self weight) Thickness: 0.15 m Plan: 2 x 3 m Material (R/C): E= 30 GPa ν= 0.2 Supports allow free sliding and rotation but no vertical movement (up or down) 14

15 Model 1 : plate elements mesh 1 3-node plate elements 6 DOF/node (3 translations + 3 rotations) mesh 2 15

16 Model 1 : plate elements Boundary conditions u, v, ϕ x, ϕ y, ϕ z... free w... fixed ϕ x, ϕ y, ϕ z... free u, v, w... fixed u, v, ϕ x, ϕ y, ϕ z... free w... fixed v, ϕ x, ϕ y, ϕ z... free u, w... fixed u, v, ϕ x, ϕ y, ϕ z... free w... fixed Note: these point BC are imposed to prevent rigid body movement in slab plane. 16

17 Model 1 : plate elements - results Deflection 17

18 Model 1 : plate elements - results Bending moment intensities Mesh 1: Element 59 m x m y Int point E E-03 Int point E E-03 Int point E E-03 Average: E Mesh 2: Element 431 m x m y Int point E E-03 Int point E E-03 Int point E E-03 Average e

19 Model 1 : plate elements - results Stress... may be not directly accessible, calculated from σ = ± 6 y, ext 2 h my σ y,ext = ± MPa σ y,ext = ± MPa 19

20 Model 1 : plate elements - results Deformed shape and reactions (notice corner forces) 20

21 Model 1 : plate elements - results Deformed shape and reactions (notice corner forces) 21

22 Model 2 : solid elements mesh 1 20-node isoparametric solid elements 3 DOF/node (3 translations) mesh 2 mesh 3 22

23 Model 2 : solid elements Boundary conditions u, v... free w... fixed u, v, w... fixed u, v... free w... fixed z, w u, w... fixed x, u y, v u, v... free w... fixed Note: these point BC are imposed to prevent rigid body movement in slab plane. 23

24 Model 2 : solid elements - results Deflection 24

25 Model 2 : solid elements - results Deformed shape and reactions (notice corner forces) 25

26 Model 2 : solid elements - results Bending stress σ y 26

27 Models 1, 2, 3: comparison Deflection y-axis 27

28 Models 1, 2, 3: comparison Model Extreme stress (MPa) Plate 1 ±1.54 Plate 2 ±1.50 Solid 1 ±1.64 *) Solid 2 ±1.57 *) Solid 3 ±1.59 *) *) extrapolated values 28

29 Example 2 Perform a stress analysis of a wall exposed to uniform load, self-weight and foundation settlement. Identify the locations and magnitudes of maximum tension. 29

30 Initial calculation 4-node isoparematric quarilateral plane stress elements (Q4) 30

31 Deformed mesh 31

32 Principal stresses 32

33 Maximum principal stress 33

34 Maximum principal stress smoothed plot 34

35 Convergence study meshes Q4 elements Q9 elements 35

36 Convergence of extreme displacement Convergence of max. princ. stress E u_ext E E E E E E-03 Q4 Q9 Q9a sig_max Q4 Q9 Q9a E E E DOF DOF Mesh El. type # of elem # of DOF u_ext sig_max 1 Q E Q E Q E Q E Q E Q E r Q E

37 Maximum principal stress Q4 elements Q9 elements 37

38 Maximum principal stress Q4 elements Q9 elements 38

39 Maximum principal stress Q4 elements Q9 elements 39

40 Local refinement 40

41 41

42 Convergence of extreme displacement Convergence of max. princ. stress E u_ext E E E E E E-03 Q4 Q9 Q9a sig_max Q4 Q9 Q9a E E E DOF DOF 42

43 References K.J. Bathe: Finite Element Procedures, Prentice Hall, Inc., 1996 ADINA R&D, Inc.: Theory and modeling guide, Volume I: ADINA, November 2006 TNO DIANA BV.: DIANA User's Manual -- Release Teacher Edition, 2008, 43

44 Remark This document is designated solely as a teaching aid for students of CTU in Prague, Faculty of Civil Engineering, course Numerické metody v inženýrských úlohách. This document is being continuously updated and corrected by the author. Despite author s utmost effort, it may contain inaccuracies and errors. Limitation on Liability. Except to the extent required by applicable law, in no event will the author be liable to any user of this document on any legal theory for any special, incidental, consequential, punitive or exemplary damages arising out of the use of the work, even if author has been advised of the possibility of such damages. This is a copyrighted document Petr Kabele, Last modified:

Finite Element Method. Chapter 7. Practical considerations in FEM modeling

Finite Element Method. Chapter 7. Practical considerations in FEM modeling Finite Element Method Chapter 7 Practical considerations in FEM modeling Finite Element Modeling General Consideration The following are some of the difficult tasks (or decisions) that face the engineer