Introduction to the Finite Element Method (3) [PDF]

Nov 28, 2012 - Practical aspects of finite element analysis. ❑ Examples of FE ... FEM programs. ➢general purpose. ○ simu

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Introduction to the Finite Element Method (3)

Petr Kabele Czech Technical University in Prague Faculty of Civil Engineering Czech Republic [email protected] ∗ people.fsv.cvut.cz/~pkabele

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

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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

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

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

Selection of analysis type Consider what physical phenomena should be analyzed. stress analysis stability static mechanical

dynamic

heat transport

... ...

mass transport fluid

... ...

linear nonlinear

modal analysis transient analysis ... ...

linear nonlinear

magnetism coupled, interaction ... ...

6

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,...)

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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, 2D 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.

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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

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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”.

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Example:

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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.

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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)

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Example 1

Perform analysis of a slab. Uniform distributed load 8 kN/m2 (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)

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

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Model 1 : plate elements Boundary conditions

u, v, ϕx, ϕy, ϕz ... free w ... fixed

ϕx, ϕy, ϕz ... free u, v, w ... fixed u, v, ϕx, ϕy, ϕz ... free w ... fixed v, ϕx, ϕy, ϕz ... free u, w ... fixed u, v, ϕx, ϕy, ϕz ... free w ... fixed

Note: these point BC are imposed to prevent rigid body movement in slab plane.

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Model 1 : plate elements - results Deflection

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Model 1 : plate elements - results Bending moment intensities Mesh 1: Element 59

mx Int point 1 Int point 2 Int point 3 Average:

-9.24586E-04 -4.27317E-04 -7.25490E-04 -6.92464E-4

my -5.78145E-03 -5.93220E-03 -5.66659E-03 -0.00579341

Mesh 2: Element 431

mx Int point 1 Int point 2 Int point 3 Average

-5.05462E-05 -5.62267E-04 -3.10636E-04 -9.23449e-4

my -5.63296E-03 -5.64567E-03 -5.64596E-03 -0.00564153

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Model 1 : plate elements - results Stress ... may be not directly accessible, calculated from σ y , ext = ±

6 my 2 h

σy,ext = ±1.54491 MPa

σy,ext = ±1.50441 MPa

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

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

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Model 2 : solid elements

mesh 1

20-node isoparametric solid elements 3 DOF/node (3 translations) mesh 2

mesh 3

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Model 2 : solid elements Boundary conditions

u, v ... free w ... fixed

z, w

x, u

u, v, w ... fixed u, v ... free w ... fixed

u, w ... fixed

y, v

u, v ... free w ... fixed

Note: these point BC are imposed to prevent rigid body movement in slab plane.

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Model 2 : solid elements - results Deflection

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

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Model 2 : solid elements - results Bending stress σy

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Models 1, 2, 3: comparison

Deflection

y-axis

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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

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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.

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

30

Deformed mesh

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Principal stresses

32

Maximum principal stress

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Maximum principal stress – smoothed plot

34

Convergence study – meshes Q4 elements

Q9 elements

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Convergence of extreme displacement

Convergence of max. princ. stress 3.0

-7.008E-03 -7.010E-03

2.5

-7.014E-03

Q4

-7.016E-03

Q9

-7.018E-03

Q9a

sig_max

u_ext

-7.012E-03 Q4

2.0

Q9

1.5

Q9a

1.0

-7.020E-03 -7.022E-03

0.5

-7.024E-03

0.0

-7.026E-03 100

1000

10000

100000

100

1000

DOF

Mesh 1 2 3 4 5 6 4r

El. type Q4 Q4 Q4 Q9 Q9 Q9 Q9

10000

100000

DOF

# of elem 106 408 1616 106 408 1616 378

# of DOF 262 914 3426 946 3458 13314 3114

u_ext -7.0239E-03 -7.0148E-03 -7.0107E-03 -7.0128E-03 -7.0095E-03 -7.0090E-03 -7.0097E-03

sig_max 1.671 2.328 2.631 2.346 2.606 2.760 2.782

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Maximum principal stress

Q4 elements

Q9 elements

37

Maximum principal stress

Q4 elements

Q9 elements

38

Maximum principal stress

Q4 elements

Q9 elements

39

Local refinement

40

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Convergence of extreme displacement

Convergence of max. princ. stress 3.0

-7.008E-03 -7.010E-03

2.5

-7.014E-03

Q4

-7.016E-03

Q9

-7.018E-03

Q9a

sig_max

u_ext

-7.012E-03 Q4

2.0

Q9

1.5

Q9a

1.0

-7.020E-03 -7.022E-03

0.5

-7.024E-03

0.0

-7.026E-03 100

1000

10000 DOF

100000

100

1000

10000

100000

DOF

42

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 9.3 -- Teacher Edition, 2008,

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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, 2007 – 2012

Last modified: 28.11.2012

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