FE analysis with beam elements

E. Tarallo, G. Mastinu

POLITECNICO DI MILANO, Dipartimento di Meccanica

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Summary 2

Subjects covered in this tutorial

� An introduction to beam elements

� A guided example to evaluate a simple structure through

the use of FEM

� Comparison analytical vs numerical solutions

�Other few exercises (to include in exercises-book)

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Beam element – topic 3

� The element library in Abaqus contains several types of beam elements

� A “beam” is an element in which assumptions are made so that the problemis reduced to one dimension mathematically: the primary solution variables are functions of position along the beam axis only (as bar element)

� A beam must be a continuum in which we can define an axis such that the shortest distance from the axis to any point in the continuum is small compared to typical lengths along the axis

� The simplest approach to beam theory is the classical Euler-Bernoulli assumption, that plane cross-sections initially normal to the beam's axis remain plane, normal to the beam axis, and undistorted (called B23, B33 )

� The beam elements in AbaqusCAE allow “transverse shear strain”(Timoshenko beam theory); the cross-section may not necessarily remain normal to the beam axis. This extension is generally considered useful for thicker beams, whose shear flexibility may be important (called B21, B22, B31, B32 and PIPE )

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Beam element – shape function 4

Classic mechanical approach uses 3rd order interpolation function (elastic line theory)

To follow this theory use element B23, B33

Beam defined in Abaqus CAE has linear or quadratic interpolation function (element B21, B22, B31, B32)

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Beam element – topic (stiffness matrix) 5

Let us consider an Euler-Bernoulli beam:

−−−−

−−−

−

=

LEJLEJLEJLEJ

LEJLEJLEJLEJ

LEALAE

LEJLEJLEJLEJ

LEJLEJLEJLEJ

LEALEA

/4/60/2/60

/6/120/6/120

00/00/

/2/60/4/60

/6/120/6/120

00/00/

22

2323

22

2323

k

where the stiffness matrix is:

[ ] [ ]TT

yxyx yxyxkMFFMFF 222111222111 ϑϑϑϑ ×=

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Exercise 1 – data problem 6

Geometry: L=1 m; A=100x100 mm

Material: E=210 GPa; ν=0.3

Load: p=1 N/mm

Write the relation of internal load e solve the analytic problem of the deformed shape of the isostatic beam

EIL

pLxL

pLx

pxv

LxpM

+−+−−=

−−=

8)(

624)(

)(

2

)(

434

2

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Exercise 1 – Results (analytic vs numeric) 7

Exact solution:v2=-0.07142mmθ2=-9.52381 e-5 radM1 = 5e5NmmF1Y=1000N

FEM Results

Comparison btw analytic

solution and FEM results

Note: sensitive variables are

Number of elements

Linear or quadratic order

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Exercise 1 – Modeling geometry and property

8

1 2 3

45

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Exercise 2 – data problem 9

Geometry: L=1 m; A=100x100 mm

Material: E=210 GPa; ν=0.3

Load: p=1 N/mm

Write the relation of internal load e solve the analytic problem of the deformed shape of the iperstatic beam

)23(48

)()(

)(8

3

2

)(

2

2

xLEI

xLxpxv

LxpLLx

pM

−

−=

−+−−=

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Exercise 2 – Results (analytic vs numeric) 10

FEM Results

Comparison btw analytic

solution and FEM results

Note: sensitive variables are

Number of elements

Linear or quadratic order

Exact solution:θ2=-1.1904e-5 radF1=625NF2=375NM1 =1.25e5 Nmm

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11Excercise 3 - data

Material Property:E=210GPaForces:F1=-20kN (Z)F2=30kN(Y)P=80N/mm(X)Note: All the written dimensions are 500mmProblem:Solve the system and report max displacement and max stress

A1

A3

A2

A4

A1 A2 A3

A4

F1

F2

P

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Exercise 3 - results 12

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13Excercise 4

Compare max stress and displacement of the structures used in the previous lesson using beam elements

Geometry: L=1m, H=0.2 m, Section variableMaterial: E=206 000 MPa, ν=0.3 (steel)

Load: P=10 kN

L

H

P