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CALCULATION OF CROSS SECTIONAL PROPERTIES

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Gabor Vörös István Kirchner

Budapest University of Technology and Economics Budapest University of Technology and Economics
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 GÉPÉSZET 2004, Budapest, 2004.May.27-28. 421-425

CALCULATION OF CROSS SECTIONAL PROPERTIES

Gábor M. Vörös1, István Kirchner2
1
Budapest University of Technology and Economics, Department of Applied Mechanics, H-1111. Budapest,
Muegyetem rkp. 5. e-mail: voros@mm.bme.hu
2
Dunaújváros Polytechnic, Department of Applied Mathematics,
STRUSOFT Budapest, H-1111. Bem rkp. 32. e-mail: istvan.kirchner@strusoft.hu

Summary

Beam elements of arbitrary cross sections are important members of finite element models for
a variety of practical applications. In the effective use of such elements, at the modelling
stage, the user is faced with the time consuming problem of determination of beam cross
sectional properties. In this paper a computationally effective finite element method for
accurate calculation of torsion and shear related properties of solid or thin walled sections is
described.

The cross sectional properties usually are the input data of finite element programs.
Their accuracy – among others – basically influences the accuracy of final results. Calculation
of geometric properties such as area, centre of gravity, moments of inertias, principal
directions, is a simple problem. However, shear and z
torsion related properties – shear centre, torsion moment
of inertia, shear areas, etc. – are more difficult to zT T
determine accurately, especially for arbitrary sections.
Commonly accepted method to define these data is based y
on the Vlasov’s theory of thin walled beams, even if the
thin walled property is questionable. Usually, the low or C yT
uncontrollable accuracy of the input section properties
are in contradiction with the required high precision of
the finite element program systems. After performing a Fig. 1.
finite element analysis for a beam structure, the user
obtains the internal forces and moments of the beam elements. An accurate determination of
stress distribution – including the direct shear stress or the normal stresses due to constrained
torsion – for each element is necessary for the effective use of the finite element analysis in
the design process.
The first finite element formulation for the torsion of arbitrary shaped section seems to
have appeared in Reference [2] where the author used a three node linear triangular element.
In this paper a displacement based finite element method is described for the torsion and shear
problem of arbitrary cross sections. Although this work is limited to homogeneous isotropic
sections, the process can be extended to multi material composite bars as it was outlined in
References [4] and [6]. The general solution of elastic beam problems can be found in many
textbooks, for example in References [1] and [3], only the basic definitions and final results
will be presented here. Figure 1 shows the coordinate system and notations. The local axes y
and z are the principal axes of the section and Ix, Iy denote the principal moments of inertia.
The coordinates of the shear centre T relative to the C centroid in the plane of the section are
given by the yT, and zT.
1 TORSION

Consider a general section of a straight, uniform beam as shown in Fig 1. Due to a

single and constant torque Mx each section undergoes a ψ(x) angle rotation about the point T,
called of shear (or torsion) centre. The u(x,y,z) axial displacement is the out of plane torsion
called warping. In case of free torsion - when the warping is not constrained and its measure
is the same in each sections – the rate of twist is constant, thus the ψ(x) angle of twist is a
linear function. Accordingly, the displacement vector is

 u 
dψ
u = -ψ(z − z T )  ,
 u = ϑ(ϕ − yz T + zyT ) , = ϑ = const . , (1)
dx
 ψ(y − y T ) 
where the ϕ(y,z) is the so-called St’Venant warping function. Assuming a homogeneous,
linearly elastic beam material and small deformations, the shear stress distributions are given
as:
M ∂ϕ M ∂ϕ
τ xy = x ( − z) , τ xz = x ( + y) . (2)
J ∂y J ∂z
The cross section property is the torsion moment of inertia
 ∂ϕ ∂ϕ 
J = I y + I z − ∫  z − y  dA , (3)
A
∂y ∂z 
and the co-ordinates of the T shear centre are
1 1
y T = − ∫ zϕ dA , z T = ∫ yϕ dA . (4)
Iy A Iz A
It follows from the internal equilibrium of linear elasticity and the boundary conditions that
the ϕ(y,z) warping function is the solution of the following differential equation:

∂ 2ϕ ∂ 2ϕ  ∂ϕ   ∂ϕ 
+ =0 ,  − z n z +  − y n y = 0 , (5)
∂y 2 ∂z 2  ∂y   ∂z 

where ny nz are the components of outward unit normal vector of section contour.
If the torsion warping – the axial displacement – of the cross section of the straight
beam is constrained, in addition to the (2) shear stresses a secondary normal stress distribution
appears:
d2ψ B
σ x = Eε x = E 2 (ϕ − yz T + zyT ) = ( ϕ − yzT + zyT ) . (6)
dx Iω
In this equation B(x) is the bimoment and the cross sectional property is the warping
parameter defined as:

Iω = ∫ ( ϕ − yz T + zyT ) dA .
2
(7)
A
For thin walled sections
ϕ − yz T + zyT ≈ −ω ,
where ω is the sector area function or sector area co-ordinate with pole T.
2 SHEAR

The beam is free of torsion if the Vy and Vz shear forces are passing through the T
point. The shear stress distributions can be calculated from the ψ1(y,z) and ψ2(y,z) shear stress
functions as:
Vy ∂ψ1 Vz ∂ψ 2 Vy ∂ψ1 Vz ∂ψ 2
τ xy = + , τ xz = + . (8)
A ∂y A ∂y A ∂z A ∂z

From the condition of internal equilibrium two boundary value problems can be
derived:
∂ 2 ψ1 ∂ 2 ψ1 A ∂ψ1 ∂ψ
- if Vy = 1 and Vz = 0, + = − y, ny + 1 nz = 0 , (9a)
∂y 2
∂z 2
Iz ∂y ∂z
∂ 2ψ2 ∂ 2ψ2 A ∂ψ 2 ∂ψ
- if Vy = 0 and Vz = 1, + = − z, ny + 2 nz = 0 . (9b)
∂y 2
∂z 2
Iy ∂y ∂z
where ny, nz are the components of outward unit normal vector of section contour.
Using the finite element method for elastic beam structures, the stiffness matrix is
derived from the U internal energy. The Us shear contribution of the beam internal energy per
unit length, with shear modulus G, is

1  Vy Vz2 Vy Vz 
2

Us =
1
2G A∫
2
( 2
)
τ xy + τ xz dA =
2GA  ρ y
+
ρz
+
ρ yz
.


The cross sectional properties are the shear factors:

ρ −y1 = ∫ ψ1 y/I z dA , ρ −z 1 = ∫ ψ 2 z/I y dA , [ ]

ρ −yz1 = ∫ ψ1 z / I y + ψ 2 y / I z dA . (10)
A A A

3 FINITE ELEMENT SOLUTIONS

The (5) and (9a-b) elliptic boundary value problems can be transformed into the
following energy principles or weak forms:

 1  ∂ϕ  2  ∂ϕ  2   ∂ϕ ∂ϕ 
Π 0 = ∫    +    −  z + y  dA = extr. (11a)
A
2  ∂y   ∂z    ∂y ∂z 
   
 1  ∂ψ  2  ∂ψ  2  A 
Π 1 = ∫   1  +  1   − ψ1 (I y y − I yz z )  dA = extr. (11b)
2 ∂y   ∂z   D
A    
 1  ∂ψ   ∂ψ 2  
2 2
A 
Π 2 = ∫   2  +    − ψ 2 (I z z − I yz y )  dA = extr. (11c)
2 ∂y   ∂z   D
A   

These problems can be solved by a 2D finite element method, where the “stiffness”
matrix, derived from the quadratic part of (11a.c) principles, are the same. The only
differences are in the linear parts, which are leading to three different right hand sides. Using
a quadratic (8 or 6 node) isoparametric finite element formulation with one degree of freedom
per node, the nodal values of φ(y,z), ψ1(y,z) and ψ2(y,z) functions and then – among others –
the (3), (4), (7), (10) cross sectional properties can be calculated.

4 NUMERICAL EXAMPLES

The first two examples serve to test and compare the usual thin walled and the
converged finite element (FEM) results.
To verify the accuracy and convergence of the presented finite element formulation the
semicircular section on Fig 2. was selected as the first example. Adopting the thin walled
assumption the section properties can be calculated as:

4 π
t/R =
π
= 1, 273 , ( )
J / v3 R =
3
= 1,0472
R
T (12)
 π3 8 
( )
Iω / vR 5 =  −  = 0 ,03738
 12 π 
v
t
The finite element results listed in the table bellow were
Fig. 2 obtained for sections R/v = 1 and 5 using decreasing size (e)
of quadratic elements. It is worth noting that the rate of
convergence is higher for thinner sections.

R/v = 1 R/v = 5
v/e t/R J/(v3R) Iω/(vR5) t/R J/(v3R) Iω/(vR5)
1 1,050 0,8600 0,1332 1,261 1,0127 0,04536
2 1,055 0,8479 0,1334 1,261 1,0070 0,04536
3 1,056 0,8439 0,1332 1,261 1,0062 0,04537
4 1,057 0,8432 0,1333 1,261 1,0060 0,04537
10 1,057 0,8429 0,1334 1,261 1,0060 0,04537

The Fig 3. plots the ratio of converged FEM and (12) thin walled results with different R/v.
From these results we may observe that for R/v >10 the thin walled theory is acceptable with
5% - or less – error limit. Obviously, this is true for this specific shape only.

25

20 J
Ιω
15
t
%

10

0 R/v
0 10 20 30 40 50

Fig 3.
 As the second example, consider the U profile shown on Fig.4. The section properties
using the thin walled assumption can be calculated as:

3b 2 1 3
t = , J= v ( 2b + h ) ,
v 6b + h 3
. (13)
h=2b
T vh b 3b + 2h
2 3
Iω =
12 6b + h

t b The numerical results for specific dimensions h = 2b and

v = 1, are listed in the following table:

Thin section, Eq. (12) FEM solution Eq. (11) Thin/FEM

h/v t J Iω t J Iω J Iω
5 0,937 3,333 2,848 e1 0,5734 2,603 1,159 e1 1,28 2,46
10 1,875 6,667 9,114 e2 1,604 5,941 5,686 e2 1,12 1,60
50 9,375 33,33 2,848 e6 9,172 32,62 2,580 e6 1,02 1,10
100 18,75 66,67 9,114 e7 18,55 65,99 8,672 e7 1,01 1,05
500 93,75 333,3 2,848 e11 93,56 335,5 2,820 e11 0,99 1,01

The figures in the last (Thin/FEM) columns indicate the measure of error introduced with the
thin walled approximation. Comparing these results with the previous one on Fig 3, it is seen
that the reasonable accuracy of the thin walled solution is achieved for higher h/v aspect ratio.
This is due to the two corners in the U section.

The third example is taken from Reference [7], where the welded section - consisting of
U300 and L160x80x12 (DIN) - was replaced by a thin walled section. In the finite element
model the exact cold rolled geometry was analyzed as it is seen on Fig. 5.

Z z Ref [7] FEM

2
A (cm ) 86,76 86,30
T YC (cm) 1,21 1,44
ZC (cm) 19,20 19,22
y Iy (cm4) 11379,9 11431,1
C Iz (cm4) 4513,3 4372,8
Iyz (cm4) 3013,2 3054,7
J (cm4) 48,83 52,11
Y yT (cm) 1,39 0,72
zT (cm) 10,06 10,36
Iω (cm6) 203061,9
Fig. 5

The presented section property calculation method along with an effective mesh
generator [8] has been implemented to the finite element code FEM-Design [9], which is
developed and maintained by Structural Design Software (STRUSOFT Kft), Budapest.
 REFERENCES

[1] Vlasov, V.Z.: Thin-walled elastic beams, National Science Foundation, Washington 1961.
[2] Zienkiewicz, O.C.: The finite element method in engineering, McGraw-Hill, London,
1971.
[3] Surana, K.S.: Isoparametric elements for Cross Sectional Properties and Stress Analysis of
Beams, Int. J. for Num. Methods in Engng. Vol.14. 1979. pp. 475-497.
[4] Vörös, G.: A variational principle in torsion problem of composite rods, Periodica
Polytechnica, Vol 23. 1979. pp. 367-376.
[5] Wempner, G.: Mechanics of Solids with Application to thin Bodies, Sijthoff Noordhoff,
1981.
[6] Sapountzatkis, E.J.: Nonuniform torsion of multi-material composite bars by the boundary
element method, Computers and Structures, Vol. 79. 2001, pp. 2805-2816.
[7] Wagner, W., Gruttmann, F.: A displacement method for the analysis of flexural shear
stresses in thin walled isotropic composite beams, Computers and Structures, Vol. 80.
2002, pp. 1843-1851.
[8] Kirchner, I.: Interactive graphical input-output systems for FEM plate bending programs,
Microcomputers in Civil Engng. Vol 8. 1993. pp. 97-103.
[9] FEM-Design Felhasználói Kézikönyv, STRUSOFT Kft, Budapest, 2003.

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