ANALYSIS OF NON-CIRCULAR MEMBERS SUBJECTED TO
TWISTING LOADS: A FINITE DIFFERENCE APPROACH
Goteti Chaitanya
R.V.R&J.C College of Engineering, India
E-mail: chaitanyagoteti16@gmail.com
Reddy Sreenivasulu
R.V.R&J.C College of Engineering, India
E-mail: rslu1431@gmail.com
Submission: 03/04/2015
Revision: 19/04/2015
Accept: 24/04/2015
ABSTRACT
Many
torque carrying members have circular sections such as shafts. However, there
are certain structural members like automotive chasis frames, cross members and
machine frames which are often subjected to twisting loads and their cross
sections are non-circular. Several methods were developed to analyze such
sections such as Saint Venant’s semi inverse method, Prandtl’s elastic membrane
analogy...etc. In this paper, the second order partial differential stress
function equation for non-circular torsional members is applied on a
rectangular section for different b/h (height /width of section) values and the
solutions for maximum torsional shear stress are found by employing second
order finite difference method. The results are compared to the results
obtained from commercial finite element software (ANSYS version 10 of ANSYS
Inc. which is the acronym for Analysis system) and by direct solution of the
stress function equation using analytical correlations available for
rectangular sections. The results obtained by different approaches are in close
congruence with a percentage deviation of only 3.22. It is observed that, in
implementing second order finite difference scheme, the error in estimating
stress is proportional to S2. Where “S” is the grid size.
Keywords: Non-Circular
Section, Prandtl’s stress function, Finite difference scheme, Grid size
1. INTRODUCTION
Simple torsion formulae applicable
for circular sections cannot be applied directly to irregular sections as
warping of sections should be considered due to uneven distribution of area
around their axes of rotation. Saint-Venant (SINGH, 2013) presented stress
correlations for irregular shaped members subjected to torsion by considering
warping effect.
Prandtl (BORESI; SCHMIDT, 2014)
developed membrane analogy for non-regular sections subjected to torsion. In
this, a homogenous membrane is considered which is supported at the edges
having same outline as the cross section of twisted bar and is subjected to
uniform tension at the edges along with uniform lateral pressure per unit area
of the membrane.
Lord Kelvin (UGURAL; FENSTER, 2003)
presented hydrodynamic analogy based on similarities between torsional stress
function and stream function governing motion of ideal irrotational fluids
contained in a vessel having same cross section as that of the twisted bar.
This analogy is especially useful for determining stress concentrations
developed at sharp corners and notches in irregular shaped sections.
NewChen (1999) presented
differential quadrature method. In this method a new discretization scheme is
proposed for any generic problem with arbitrary domain. According to this
method, a global algebraic system of equations can be developed by assembling
all the discretized elements.
Turken, Kadıoğlu and Ataoğlu (2002)
proposed a solution to torsion of irregular sectioned bars using boundary
element method. In this method the boundary is divided into linear elements and
the integral equation for torsion is reduced to a system of linear algebraic
equations. Unlike in case of regular
circular sections, exact numerical solutions to torsional loads are not
available for irregular shaped sections.
However, Boresi and Chong (2010)
presented approximate solutions to non-circular boundaries by solving the two
dimensional harmonic and biharmonic stress function equations using Finite
difference schemes. Strikwerda (2004) provided detailed introduction to various
finite difference schemes such as Newton’s forward difference formula, central
difference formula…etc, for various univariate, bivariate and multi variate
functions to obtain numerical solutions to partial differential equations.
Ward (1998) provided solutions to linear
and second order finite difference schemes with variable coefficients.
Chatopadhyaya (2013) presented and discussed the warping effects in various non
circular geometries subjected to twisting loads in an elaborative manner from a
series of experimental results.
In this work, a structural member of
rectangular section is analyzed for different (b/h) (Section height / width)
values using second order finite difference scheme and the results obtained are
compared to the results obtained from (ANSYS version 10 of ANSYS Inc which is
the acronym for Analysis system) finite element software and from the
analytical correlations developed for non-circular sections having regular
geometry. Matweb provided necessary mechanical and physical properties of
chosen structural steel for modeling the member.
2. PROBLEM DESCRIPTION AND ANALYSIS
In the present work, a rectangular member with b/h ratios 1.0, 1.5, 2.0,
2.5, 3.0 is considered. The member is made of EN 24 T structural steel with
modulus of elasticity 200 Gpa, shear modulus 76.92 Gpa and Poisson’s ratio 0.3.
A torque of 2500 N-mm is applied by constraining one end and the analysis is
carried out by using Prandtl’s Analytical equations, Finite difference method
and Ansys.
3. ANALYTICAL CORRELATIONS
Recalling the Prandtl’s stress function
(1)
Where
and is the stress function which is zero around
the periphery of the section for equilibrium. Also, from
Prandtls membrane analogy for rectangular section (2)
For the rectangular
section shown in Figure 1, the stress function takes the following form
(3)
Figure 1: Rectangular Section
where
(4)
Where V is a shear stress function in x and y. Employing method of
separation of variables, the shear function V is expressed as:
(5)
First
of equations 4 and equation 5 yields:
(6)
Prandtl found the following closed form solutions for rectangular
sections by solving equation no 6. The solution methodology was also extended
to other regular sections like equilateral triangle and ellipse by selecting
appropriate equation form for Ø.
(7)
Where T=GJθ and
Where
(8)
(9)
and
(10)
Table 1 shows the values of torsional parameters k1 and k2
estimated for various b/h values.
Table1: Torsion
Parameters for Rectangular section
b/h |
1.0 |
1.5 |
2.0 |
2.5 |
3.0 |
K1 |
0.141 |
0.196 |
0.229 |
0.249 |
0.263 |
K2 |
0.208 |
0.231 |
0.246 |
0.256 |
0.267 |
4. FINITE DIFFERENCE METHOD
The section of the member under consideration for b/h=1 is discretized
as shown in Figure 2.
Taking b=10mm and h=10mm, the grid size S is chosen as 10/4 or 2.5mm.
Recalling the second order finite difference equations for bi variate
functions, (i,e) for functions of x and y variables we have:
(11)
Figure 2: Finite
grid for square section (b/h=1)
From Prandtl’s analogy, the stress function at boundary shown in figure
2 is zero and exists throughout the region R shown in figure 2 except at the
boundary.
Mathematically, on region
defined by R with in the boundary. =0 on the
boundary of the member defined by C. Due to symmetry of the section shown in
figure2, only the stresses at the three node points are considered. As Ø is
function of x and y, equations shown by 11 are applied to the present problem
as follows.
Where
h=k= S and i ,j denote nodal positions in x and y directions. Therefore,
(12)
Due to symmetry only one quarter of the region R shown in Figure 2 is
considered for analysis. Where i , j shown in equation 12 denote the node point
in region R of Figure 2. Due to symmetry of node position 2 with respect to
node position 1, we have F1,2 = F2,1. Here, it must be noted that the function Ø is
analogous to F shown in Figure 2. The following equations are deduced for nodal
positions (1,1), (1,2) and (2,2).
(13)
( Ø0,1 and Ø1,0 denote F2,1 and
Ø1,1 denotes F1,1)
(14)
(j+1 denotes boundary C, therefore 5th term is
zero)
(15)
(i+1, j+1 denote boundary C, therefore, 2nd
& 5th term are zero)
Equations
13 to 15 can be expressed as shown by equations 16, 17 and 18.
(16)
(17)
(18)
Solving
equations 16 to 18 we have the following solutions for stresses.
(19)
For
one quarter of region R, the maximum shear stress occurs at i=3 and j=1, That
is at x=b/2, y=0 position For the entire region R.
Using
fourth order backward difference formula
τmax may be approximated as:
(20)
For
i=3, j=1 equation 20 becomes:
(21)
From
equations 19 and 20 τmax = 2.583GθS =
0.646Gθb
5. FINITE ELEMENT METHOD
Ansys 10 version of ANSYS Inc. is used to model and analyze the present
problem. The section is modeled using SOLID 45 brick elements. The entire
section is meshed with uniform map mesh of initial element size 10. Torque of
2500 N-mm is applied to the model using a pilot node modeled outside the model.
Figure 3 shows the finite element model of the rectangular section for b/h=1.
Figure 3: FEM
model of Rectangular plate for b/h=1
6. RESULTS & DISCUSSION
Figures 4 and 5 show the maximum shear stress and angle of twist for the
rectangular section with b/h=1. The variations in maximum angle of twists and
maximum shear stress for different b/h values obtained from ANSYS, Finite
Difference Method and analytical equations are shown in tables 2 and 3
respectively.
The magnitudes of maximum shear stress and angle of twist reduced with
increase in the ratio b/h as seen in tables 2 and 3. From tables 2 and 3 it can be observed that
the percentage variations in the results obtained Finite difference and Ansys
are found to be minimal. This may be attributed to the fact that, both methods
employ the concept of discretization of the domain.
Figure 4:
xy-shear plot for the rectangular section with b/h=1
Figure 5: Angle of twist about Z-axis for section with
b/h=1
Table
2: Maximum Shear stress variation for different b/h values
Maximum Shear Stress in N/mm2 |
|||||
b/h |
1.0 |
1.5 |
2.0 |
2.5 |
3.0 |
ANSYS |
11.10 |
7.002 |
4.967 |
3.521 |
2.906 |
FINITE DIFFERENCE METHOD |
11.45 |
7.342 |
4.993 |
3.648 |
2.997 |
ANALYTICAL CORRELATIONS |
11.92 |
7.517 |
5.081 |
3.906 |
3.121 |
Table 3: Maximum
variation of angle of twist for different b/h values
Maximum Angle of twist in radians |
|||||
b/h |
1.0 |
1.5 |
2.0 |
2.5 |
3.0 |
ANSYS |
0.48E-04 |
0.12E-04 |
0.063E-04 |
0.0291E-04 |
0.0094E-04 |
FINITE DIFFERENCE METHOD |
0.427E-04 |
0.109E-04 |
0.060E-04 |
0.033E-04 |
0.0099E-04 |
ANALYTICAL CORRELATIONS |
0.402E-04 |
0.099E-04 |
0.058E-04 |
0.036E-04 |
0.0101E-04 |
7. CONCLUSIONS
A structural member of rectangular section with
different b/h values is analyzed using ANSYS, Finite difference method and
Prandtl’s closed form solutions for regular non-circular sections. The
following conclusions are drawn.
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