fe analysis with beam elements e. tarallo, g. mastinu politecnico di milano, dipartimento di...
TRANSCRIPT
FE analysis with beam elements
E. Tarallo, G. Mastinu
POLITECNICO DI MILANO, Dipartimento di Meccanica
Es02Es-02
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 solutionsOther few exercises (to include in exercises-book)
Es02Es-02
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 problem is 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)
Es02Es-02
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)
Es02Es-02
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
Es02Es-02
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
EILpLx
Lp
Lxpxv
LxpM
8)(
624)(
)(
2
)(
434
2
Es02Es-02
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 resultsNote: sensitive variables areNumber of elementsLinear or quadratic order
Es02Es-02
Exercise 1 – Modeling geometry and property
8
1 2 3
4
5
Es02Es-02
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
Es02Es-02
Exercise 2 – Results (analytic vs numeric) 10
FEM Results
Comparison btw analytic solution and FEM resultsNote: sensitive variables areNumber of elementsLinear or quadratic order
Exact solution:θ2=-1.1904e-5 radF1=625NF2=375NM1 =1.25e5 Nmm
Es02Es-02
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
Es02Es-02
Exercise 3 - results 12
Es02Es-02
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 variable
Material: E=206 000 MPa, ν=0.3 (steel)
Load: P=10 kN
L
H
P