International Journal of Scientific and Research Publications, Volume 2, Issue 1, January 2012
ISSN 2250-3153
1
Mathematical Analysis of Section Properties of a
Platform Integrated with Vehicle Chassis
Prof. Deulgaonkar V.R, Prof. Dr. Kallurkar S.P, Prof. Dr. Matani A.G
Abstract- The present work depicts mathematical behavior of a
vehicle mounted platform/frame integrated with chassis structure
in terms of plane stresses and plane strains for non-uniform
loads. The load type considered in present work is concentrated
load for which the mathematical model is formulated. A different
type of combination of longitudinal and cross members in
platform/frame design is formulated. The dimensions of platform
members are determined using IS standards. After analysis of all
possible combinations of longitudinal and cross members present
design is anticipated. Section properties of longitudinal and cross
members of the platform are determined & deduction of bending
stress and shear force based on the load pattern are the
fundamental steps in design and analysis of platform structure.
The peculiarity of this analysis is the calculation of combined
section modulus of three members. These are evaluated by excel
programs developed indigenously.
Index Terms- concentrated load, section properties, shear
forces, plane stresses and strains, platform.
I. INTRODUCTION
T
ransportation system is a prominent factor which has great
impact on country’s economic augmentation. Transportation
of sophisticated equipment/cargo needs extreme concern for long
transportation distance and type of road also has an impact on
proper transportation of the cargo. To deal with the above and
many other related factors a significant research is carried on
vehicle chassis. Platforms are required to provide a leveled base
for accurate leveling vehicle mounted devices. The platforms are
generally of welded structure type, designed to accommodate
one, two or more ISO type shelters according to the application.
The design of the platform mainly depends upon load, its type
and pattern; these parameters are determined by the shelter
length, distance between the ISO corner blocks, the weight of
shelter and the load distribution inside the shelter. Selection of
type of cross members, determination of their dimensions and
locations with respect to expected load patterns are important
steps in the process of platform design.[1-5] Factors such as selfweight of platform, mounting of platform on vehicle chassis, axle
load distribution, departure angle and ramp angle of the vehicle
bear a significant effect on the platform design. Various types of
chassis frames are into use viz. conventional, integral, semiintegral frames, of which conventional being extensively used.
The platform members are made of three types of sections viz.
channel, tabular & box sections. Each section has its own
characteristic for the type of load it is subjected to. The loads
acting on the platforms are categorized as short duration load,
momentary load, impact, inertia, static and overloads.
Elementary analysis of platform comprises static and dynamic
loads. The structural configuration of a chassis is often very
complex due to normative and functional constraints. A vehicle
chassis is characterized by a high level of static indetermination
because of the complex interconnection of beams (longitudinal
elements, cross elements and pillars) and panels. Therefore it is
relatively difficult to perform an analytical calculation of stresses
and strains unless dire approximations are introduced. The
structural analysis is normally performed numerically using the
finite element method. However, in order to provide some design
criteria, it is necessary to understand the structural functionality
of the principal chassis components. For this reason it is
convenient to consider some basic layouts: Though these layouts
cannot provide precise quantitative information, they can prove
useful to explain the structural function of the chassis
components.[6-9] The conception of the parameters which
influence chassis behavior is useful both in the outlining process,
when the main configuration is selected and during results
analysis when the final design is refined. For precise
mathematical analysis boundary conditions needs to needs to be
thoroughly understood. A monocoque structure is one whose
thickness is small if compared to the section dimensions, e.g. a
folded metal sheet so that it forms a cylinder welded on a
generatrix. This structure is unable to support concentrated loads
causing local collapse of monocoque. To withstand concentrated
loads the structure is stiffened with longitudinal stiffeners and
ribs, such structures are termed as semi-monocoques.e.g.
Aeronautical structures as wings. The connections for the
application of concentrated loads are grasped by addition of
thicker sheet metal elements.
II. THEORY
The longitudinal members are presumed as long cylindrical or
prismatical bodies. These members are subjected to concentrated
loads/forces that are perpendicular to longitudinal elements and
invariable along the length. Dimension along z-direction is
extremely dominant as compared with the dimensions in x & y
directions. Microanalysis of forces acting on the body shows that
surface and body forces are into existence. Fundamental
assumption of the body being rigid reflects the fact that relative
distance between any two points on it is always constant. The
components of small displacements parallel to x, y & z axis are
u, v & w respectively. Then the components of normal and
shearing strain along x, y & z axes are given as
ex = du/dx; ey = dv/dy and ez = dw/dz
(1)
γxy = (du/dy) + (dv/dx); γxz = (du/dz) +(dw/dx);
γyz = (dv/dz)+(dw/dy)
(2)
The aim of considering complete differentials rather that partial
ones is that it gives complete strain in the presumed plane, as
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International Journal of Scientific and Research Publications, Volume 2, Issue 1, January 2012
ISSN 2250-3153
already known that length is the dominant dimension in present
analysis.[11-12].
Applying the above equations to the longitudinal member of the
platform we get components u and v of the displacement
functions are x and y & being independent of the z co-ordinate.
Hence the longitudinal displacement of the outer member is zero.
We get the following equations as
γxy = γxz = ez = 0
(3)
Using Hooke’s law the normal stress can be determined.
III. COMPUTION OF SECTION PROPERTIES
Section properties of individual members of the platform are
of paramount importance in design and formulation of the
structure of the platform. Longitudinal and cross member of
dimensions are selected by taking into account the concentrated
loading to which the platform will be subjected.
A. Section properties of main longitudinal member of the
platform.
The main longitudinal member of the platform is intended to
support the cross-members at the central location and is mounted
on chassis. The main longitudinal member is welded to cross
members and is integrated with vehicle chassis. The dimensions
as per IS808 1967 are chosen, neglecting the fillet radii are
125x75x5 in mm, shown in figure 1 below.
2
Knowing the exact centroid location (Refer Table 1), further
assists in determining the orientation of main longitudinal
member along axis of loading.
B. Calculation of Moment of Inertia for main longitudinal
member about X & Y axes
Table 2 Moment of Inertia X axis Calculations
Shape
1
2
3
IC
d = xi - x(left)
d²A
781.25
60
1350000
633698
0
0
781.25
-60
1350000
Moment of Inertia about X axis =
Table 3 Moment of Inertia Y axis Calculations
Shape
1
2
3
IC
1757
81.2
5
1197
.916
7
1757
81.2
5
d = xi - x(left)
15.188
-19.811
15.188
d²A
86510
.9914
22568
0.847
3
86510
.9914
6
Moment of Inertia about Y axis =
Table1. Location of centroid for main longitudinal member
Shape
Area (mm2)
01
02
03
Total
375
575
375
1325
yi (from bottom)
mm
122.5
62.5
2.5
xi (from left)
mm
37.5
2.5
37.5
Table1. Continued
xiAi (mm3)
yiAi
(mm3)
Centroid Xleft
mm
Centroid
Ybottom mm
14062.5
1437.5
14062.5
29562.5
45937.5
35937.5
937.5
82812.5
22.3113
62.5
IC + d²A
mm4
1350781
633698
1350781
3335260
IC + d²A
mm4
26229
2
22687
9
26229
2
75146
3
The numeric values of moment of inertia determined above
gives moment of inertia (Refer Tables 2 &3) of shape also
referred as second moment of area. These values directly predict
area of material in the cross section and displacement of that area
from the centroid. The efficiency/strength of the cross section to
resist bending due to concentrated loading is envisaged with the
aid of second moment of inertia. The numeric values of other
properties of main longitudinal member along x and y axes are
enlisted directly below. To avoid complexities in calculations as
well as in manufacturing the section selected is symmetric about
both the axes.
Table 4 below reflects various section
properties of main longitudinal member about X axis.
Table4. Section Properties of Main Longitudinal Member about
X axis
Parameter
Symbol
Numeric Value
Unit
Section
Sx
53364.16667
mm3
Modulus
Section
Modulus
S (bot)
53364.16667
mm3
(bottom)
Section
Modulus
S (top)
53364.16667
mm3
(top)
Radius of
rx
50.17148267
mm
Gyration
Plastic
Zx
61531.25
mm3
Modulus
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International Journal of Scientific and Research Publications, Volume 2, Issue 1, January 2012
ISSN 2250-3153
Shape Factor
3
1.153044334
Table5. Section Properties of Main Longitudinal Member about
Y axis
Parameter
Section
Modulus
Section
Modulus (left)
Section
Modulus
(right)
Radius of
Gyration
Plastic
Modulus
Shape Factor
Numeric
Value
Unit
Sy
14262.32841
mm3
S (left)
33680.80514
mm3
S (right)
14262.32841
mm3
Symbol
ry
23.81474487
mm
Zy
25921.875
mm3
1.817506529
Determination of section/elastic modulus is vital step in design
of main longitudinal member as it is undeviating measure of
strength. There is direct proportion between the load carrying
capacity and value of section modulus. Section modulus
comprises the fact that extreme fibres are subjected to load/stress
during bending; hence its values from both ends .i.e. top &
bottom are established. Value of radius of gyration determined
helps in envisaging the distribution of mass about central axis.
The plastic section modulus is used for materials where plastic
behavior is dominant. The value predicted for this section is for
reference as component is designed by taking into account only
the elastic modulus. (Refer Tables 4 &5) Ratio of plastic to
elastic moment in a component subjected to bending is devised
by shape factor. This is of paramount importance to predict the
material behavior in under loading in elastic and plastic regions.
From the above analysis we get all the physical and geometrical
properties of main longitudinal member. These properties assist
in freezing the platform configurations. Similar analysis of other
members as outer longitudinal member (125x75x6), longitudinal
walkway support (50x30x5), and chassis of the vehicle
(250x100x6) leads us to all the individual properties of the
platform members. Now the calculation of combined section
modulus of chassis, main longitudinal member and cross member
is attempted. This combined section modulus as shown in figure
2 below is further used in evaluation of stress of the platform for
various types of load cases on the platform as load in stationary
condition, load during braking, load during vehicle travel on a
gradient etc. hence calculation of combined section modulus is
important at this design stage before further processing for other
higher analysis. This calculation is depicted in Table 6 below.
Figure2. Sections combined for section modulus calculation
Sr.
No.
Area (a)
mm2
01
02
03
04
05
06
07
08
09
10
800
575
1428
1072
375
600
1072
375
600
800
Total
7967
Cent.
Dist.
from
bottom
edge mm
(Yi)
521
312.5
125
450
372.5
247
450
252.5
3
379
Position of
centroid
from bottom
edge
ayi (mm3)
416.8 x 103
179.687 x 103
178.5 x 103
482.4 x 103
139.687 x 103
148.2 x 103
482.4 x 103
94.6875 x 103
1800
303.2 x 103
2.42736
x
106mm3
315.04mm
Knowing the centroid location for the combined section,
further the analysis is extended to calculate the moment of inertia
of the combined sections. This is shown in Table 7 below
Ic M.I.
axis
(mm4)
Yi-ӯ=d
(mm)
4266.66
205.6
633.69 x
106
6.740 x
106
1.604069
33 x 106
-2.9
-190.4
134.6
781.25
57.1
1800
-68.4
1.604069
33 x 106
134.6
781.25
-62.9
1800
-312.4
4266.666
-63.4
d2A (mm4)
I c + d 2A
(mm4)
33.8170 x
106
33.82135
106
638.5336
103
58.5087
106
21.0256
106
1.2234
106
2.8089
106
21.0256
106
4835.75
51.7680
106
19.4215
106
1.2226
106
2.8071
106
19.4215
106
1.4836
106
58.5562
106
3.2156
x
x
x
x
x
Iz = Σ(Ic + d2A)
(mm4)
x
x
x
x
x
x
x
x
1.484 x 106
x
58.5586 x
106
3.2199146
x
202.314715
2 x 106
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International Journal of Scientific and Research Publications, Volume 2, Issue 1, January 2012
ISSN 2250-3153
7
106
67 x 106
Section modulus values are Sbottom = 641.4543918 x 103 mm3
& Stop = 965.2419618 x 103 mm3. This combined section
modulus value determined gives us the combined strength of the
vehicle chassis, main longitudinal member and cross member
combination.
REFERENCES
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[4]
IV. CONCLUSION
For the design of platform component, the elementary concepts
of design as moment of inertia, location of centroid, computation
of section modulus through numeric technique is accomplished.
Plane stress and plane strain concepts are utilized to have
thorough insight of the stress and strains along the dominant
parameter i.e. length of the main longitudinal member. Applying
the identical modus operandi for working out of the combined
section modulus of vehicle chassis, main longitudinal member
and cross member as employed for estimating the section
modulus and other properties of main longitudinal member
provides us the novel way of determining the section modulus
and hence the strength/efficiency of combined sections. Thus a
numerical technique to calculate the section properties of
combined sections is formulated. This technique of calculating
section properties can be applied earlier than using advanced
design techniques as computer aided design and analysis. The
advanced concepts of plane strain and plane stress are employed
further in the analysis of normal components of stress and strain.
Calculation of stress at various locations and for different load
cases using this technique will be attempted in future.
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First Author – Prof. Deulgaonkar, M.E (Design) Mechanical Engineering, Sr.
Lecturer, MMCOE, Pune. [email protected]
Second Author – Prof. Dr. Kallurkar S.P, Ph.D. Principal A.G. Patil Institute of
Technology Solapur, [email protected]
Third Author – Prof. Dr. Matani A.G, Ph.D Mechancial Engg., Assistant
Professor Govt. College of Engineering. Amravati, [email protected]
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