GEOMETRICALLY FEATURED SHELLS

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME
TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 3, Issue 2, July- December (2012), pp. 251-264
IJCIET
© IAEME: www.iaeme.com/ijciet.html
Journal Impact Factor (2012): 3.1861 (Calculated by GISI) IAEME
www.jifactor.com

STUDIES ON GEOMETRICAL FEATURED METALLIC SHELL
STRUCTURES FOR INWARD INVERSION

Ram Ranjan Sahua, Dr. Pramod Kumar Guptab

a PhD scholar, Civil Engineering Department, IIT Roorkee, India
(Working as Assistant General Manager in Engineering Research Centre of TATA Motors-
Pune, through Tata Technologies, Pune, India)
A1-404, Kumar Prerana, Aundh, Pune (MH), Pin 411007 India
Email: RamRanjan.Sahu@tatatechnologies.com

b Associate Professor,
Structural Engineering Department of Civil Engineering,
IIT Roorkee, (Uttarakhand), Pin 247667 India
Email: pkgupfce@iitr.ernet.in

ABSTRACT

The geometrical inward inversion studies were planned on the metallic shell geometries.
These geometries are circular in shapes. They are closed at the top and open at the bottom. In
between top and bottom faces, the geometrical features are changed from sample to sample.
Feature changes are in shape, apical angle, steps, thickness etc. Studies were made to see how
these features play role in inversion process of large deformation. Force stroke graphs were
plotted for featured samples and discussed in details for deformation characteristics.
Comparative analysis is done for samples in context to energy absorption. The analytical
simulations were also done for experiments. Good correlations were found with experimental
results. The parameters which could not be obtained physically could be simulated
analytically for parametric studies. This paper also gives a guide line on parameters to be
taken for good energy absorption, in inward inversion process.

Keywords: Large Deformations, Shell structures, Energy absorption, load-deflection, Finite
Element analysis

I. INTRODUCTION
The geometrical inward inversion is of great interest for researcher in the field of energy
absorption phenomenon. Alghamadi [1] paper reviews the common shapes of collapsible
energy absorbers and their different modes of deformations. Also Alghamdi [2] introduced
direct inversion method on frusta. Aljawi et al. [3] simulated the inversion collapse process

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of frusta, through finite element analysis using ABAQUS software. Good agreement was
obtained between experimental results and theoretical predictions. Further inversion studies
were done on frusta during axial crush, by Aljawi and Alghamdi [4]. Reid [5] reported one of
the interesting energy absorber by tube inversion which basically involves the turning inside
out of a thin circular tube made of ductile material. This deformation process gives constant
inversion load for a uniform tube. Collapse studies on varying wall thickness of metallic
frusta subjected to axial compression was done by P. K. Gupta [6]. The mode of collapse
forms by the development of one concertina fold followed by the plastic zone. During the
development of mode of collapse some portions of the frusta move radially inward and some
radially outward. Nia and Hamedani [7] studied the axial crushing on various section shapes
(circular, square, rectangular, hexagonal, triangular, pyramidal and conical etc) made of
aluminium sheets of 1 and 1.5mm thickness. They investigated that in axial quasi-static tests,
the larger the number of section edges, the greater the energy absorption capacity. This is due
to an increase of the number of folds and plastic hinges in sections with larger number of
edges. For their test the absorbed energy per unit mass was maximum for cylindrical tubes. N
K Gupta et al. [8] Studied on collapse behavior of thin spherical shells under quasi-static
loading. Also three-dimensional numerical simulations were carried out for all the specimens
tested under quasi-static loading using ANSYS. They found that the relatively thick shells
deform axi symmetrically and major load is absorbed by the rolling plastic hinges. When the
thickness is reduced considerably, the inward dimpling is followed by non symmetric
multiple number of lobes which are caused by the formation of stationary hinges.

The shape, size, material, geometrical features could affect the inversion phenomenon a
lot. Few samples which were studied are shown in Figure 1, having different shapes and
geometrical features.

Figure 1: Test samples shape

II. EXPERIMENTS
2.1. Experimental setup

The experiment setup consist of
1) Machine to apply load
Samples were inverted at quasi-static condition by the use of a 4-ton Instron universal
testing machine (UTM), at a constant plunger crosshead speed of 15 mm/min. The plunger
can go up down by 125mm from its mean position. Hence it may happen that the start of

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graph can be from any +/- position depends on position of plunger, for that sample. The
schematic of test setup is shown in Figure 2.

Figure 2: Schematic of Test set-up

The alignment of sample and its fixture is assured with machine axis.
2) Fixture to hold and position the sample
Top fixture is inversion rod whose one end is fixed to load cell of UTM and other end fixed
to the top of model, through washers. The bottom fixture consists of cylindrical vessel and
locating ring. The locating ring rests on the top of the vessel through stoppers. Vessel is kept
on moving ram of machine. The Photograph of test setup along with bottom fixture and
locating ring is shown in Figure 3.

a) b)

c)

Figure 3: Photograph of a) test set-up b) bottom fixture and c) locating ring

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3) Test Samples
The aluminum test samples of different shapes, having different geometrical features in it,
were picked up from market, where these were readily available. They have 11 categories
based on shape and features, as shown row wise and in increasing order towards right side, in
Figure 4.

Figure 4: Test Samples

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2.2. Material properties

The tensile test pieces were cut directly from the sample and shaped with dimensions as
per ASTM E8 [9] specifications. These test pieces were tensile tested with standard tensile
testing machine at room temperature. The typical stress strain graph is shown in Figure 5.

Figure 5: Stress-strain graphs for the sample

From the graph, the material tensile strength is 55 N/mm2 @ 4% elongation and 0.2% proof
stress is 45 N/mm2.
Since top and bottom fixture were made of steel, these were supposed to very rigid as
compared to model.

2.3. Experimental results

The typical force-displacement (F-H) graph is shown in Figure 6 for the sample category
7. Forces are measured in kN and displacement (machine stroke) in mm, hence here after
these units only are referred.

Figure 6: Typical F-H graph

The shapes of the graphs are similar for other category of samples. The deformation stages of
sample are also depicted in figure for easy understanding. The load quasi statically rises from
point 1 to 2 as shown in Figure 6, to a maximum value to point 2. This is the force required to

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generate the plastic zone at the top cap end. Afterwards it decreases, which is the indication
of start of the inversion process. This is shown from point 2 to 3. On further load application,
the graph nature depends on the geometrical features. It may go up and down depending on
rolling hinges volume, as well can be shaky in shape, based on wavy nature of geometry. The
representation of each samples category while testing are shown in Figure 7 a to c. In this
figure, the photographs of the samples are shown on its initial, mid and final stages of
deformations. Also corresponding load displacement (F-H) graph are shown in extreme right
side.

Prefix S suffix, prefix=sample category, S=sample, suffix=sample no

Figure 7a: Sample category 1 to 3

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7s1


Figure 7b: Sample category 4 to 9

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Figure 7c: Sample category 10 to 11

The samples test results are enlisted in Table 1. In this table the samples stroke, their
inversion length, specific energy and average force of the process is tabulated.

Table 1: Test results

The sample category 1 has shape of frusta with apical angle of 6 to 7 degree. On load
application it gives typical and smooth F-H graph (refer Figure 7a). Also from Table 1, this

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geometry gave highest energy absorption as compared to other samples. The samples 9s1,
9s2, 8s4 having step features (figure 4), gave very less specific energy as compared to other
samples. Refereeing sample category 3 of figure 7a, the local crushing took place. Since no
inversion process could be started, rather local crushing yielded a quite shaky graph. Though
its specific energy content is high, it cannot be used as energy absorber, whose basic
requirement is smoothness in F-H graph.

III. FINITE ELEMENT SIMULATION

Finite element simulation was used to simulate the models inward inversion process. It
was also used for weight calculation up to deformed portion of the samples. FE simulations
could be used to study in details, about the model deformation phenomenon, to extract salient
features during experiments like deformation, its effort, energy associated, correlation to test
data like F-H graph etc. HyperMesh [10] is used for FE model building. Aluminum models
were represented with 4 node shell elements at mid plane surface. The locating ring and
fixtures were represented with rigid elements representation. The LsDyna [11] explicit solver
was used to solve the problem. The material models *MAT_RIGID was used for fixtures and
*MAT_PIECEWISE_LINEAR_PLASTICITY was used for samples. The contact type
*CONTACT_AUTOMATIC_SURFACE_TO_SURFACE was used to define contacts
between disjoint parts. Also for self contact
*CONTACT_AUTOMATIC_SINGLE_SURFACE was used. Coulomb friction type was
used to define the coefficient of friction between samples and fixtures. Fig. 8 shows the FE
shell model and its deformed cut section is compared with actual sample cut section. Shell
element formulation choosen was Belytschko-Tsay because of its less computation cost with
good accuracy. FE model fully represent the sample and simulate the experimental process.
Result interpretation was done through LS-PREPOST.

Figure 8: FE model and cut section comparison

3.1. Simulation results
The energy balance graph of FE simulation for sample 7s1 is shown in Figure 9. From the
graph it is evident that the unwanted energies like kinetic energy, Hourglass energy, sliding
energy are minimum in the simulation. Hence the total energy is contributed only through
sample internal energy, which is caused by deformation. This graph represent perfect energy
balace of FE simulation and hence ensure the correctness of simulation

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Figure 9: Energy balance of FE simulation
The FE simulated F-H graph and vonMises stress contour at different deformation stages is
shown in Figure 10 for sample 7s1. The F-H graph resembles to experimental graph. The red
colored stress indicates the formation of plastic zone. The drift of hinges which rolls from top
to bottom can be clearly seen in this figure.

Figure 10: FE simulated F-H graph and Stress contour

IV. DISCUSION

Referring Figure 7b, the sample category 4 wherein at the top, a wavy geometrical
features (curvature radius=3 mm, height=5 mm, numbers=3) exists, its F-H graph also have
wave, having 3 peaks. This waviness is attributed to the change in hinges volume due to
wavy geometry.
Sample category 9 have step of 2 mm at the height of 55 mm from base. On load
application it buckles at this step location. Hence the corresponding graph also have sudden
deep gradient from point 2 to 3. Similar behavior is also noticed in sample 8s5.
The sample 9s4 buckled at mid featured location, yielding very wavy graphs due to local
crushing as shown in Figure 11.

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Figure 11: Buckling at feature

The effect of apical angle change, on F-H graph is shown in Figure 12. The up gradient in
noticed in graph for 5 degree (sample 2s2) while it is down gradient for sample 11s1, whose
apical angle is 8 degree.

Figure 12: Effect of apical angle change on F-H graph

A plateau is noticed on F-H graph at point 2, for samples (8s1, 8s2) having straight
portion at the top end as shown in figure 13. This plateau is due to good resistance offered by
straight portion after onset of plastic zone (point 2).

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Figure 13: Effect of top straight portion on F-H graph

Thickness of sample play important role in energy absorption. Its effect on F-H graph can
be seen in Figure 14. More the thickness more is the energy absorption. The energy absorbed
by sample 2s2 (thickness=0.95 mm) is 4.22kJ/kg, while it has increased to 6.23kJ/kg for
sample 2s3 whose average thickness is 1.3 mm.

Figure 14: Thickness effect on F-H graph

The energy absorption in different stages of deformations (sample 7s1) was compared for
experimental and numerical simulation. Results matched well as shown in Fig. 15. This
shows that the numerical simulations where material properties are taken from the test,
geometrical representations and boundary conditions assumptions are perfectly all right.

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Figure 15: Energy absorption comparison

CONCLUSION

Experimental and computational studies of inward inversion on featured geometries were
done. It was found that all geometry has undergone the inward inversion process properly,
except those who had features like step and wave patterns. Step feature invited a local
buckling while wave feature yielded waviness in F-H graph or started local crushing. Also it
is noticed that the less the apical angle of geometry, more is the energy absorption capacity.
A Finite Element computational model of the development of inward inversion mode is also
presented. The FE deformation and actual deformation shape matched well. Also FE could be
for weight calculation of deformed portion of samples.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the support of engineering research centre of TATA
Motors at Pune, for the supporting experimental works. Also thanks to the proto shop for
fabrication of fixtures and material testing group for getting the material non linear properties
through test.

REFERENCES

[1] Alghamdi AAA (2001), “Collapsible impact energy absorbers: an overview”. Thin-
Walled Structures, Vol.3, No.2, pp. 189–213.
[2] Alghamdi AAA (1991), “Design of simple collapsible energy absorber, Master of Science
Thesis", Jeddah, Saudi Arabia: College of Engineering, King Abdulaziz University
[3] Aljawi AAN, Alghamdi AAA (1999), “Investigation of axially compressed frusta as
impact energy absorbers, In: Gaul L, Brebbia AA, editors. Computational methods in contact
mechanics IV. Southampton: WIT Press, pp. 431–43.
[4] Aljawi AAN, Alghamdi AAA (2000), “Inversion of frusta as impact energy absorbers”,
In: Hassan MF, Megahed SM, editors. Current advances in mechanical design and production
VII. New York: Pergamon Press, pp.511–9.
[5] Reid SR (1993), “Plastic deformation mechanisms in axially compressed metal tubes used
as impact energy absorbers”, Int J Mech Science, Vol. 35. No. 12, pp. 1035–52.
[6] P.K. Gupta (2008), “A study on mode of collapse of varying wall thickness metallic frusta
subjected to axial compression”. Thin-Walled Structures, Vol.46, pp. 561–571.

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[7] Nia and Hamedani (2010), “Comparative analysis of energy absorption and deformations
of thin walled tubes with various section geometries”, Thin-Walled Structures, Vol.48,
No.12, pp. 946-954.
[8] N.K. Gupta, N. Mohamed Sheriffb, R. Velmurugan (2008), “Experimental and theoretical
studies on buckling of thin spherical shells under axial loads”. International Journal of
Mechanical Sciences, Vol.50, pp.422–432.
[9] ASTM International: ASTM E8 / E8M - 09 Standard Test Methods for Tension Testing of
Metallic Materials
[10] HyperMesh11. A product of Altair Engineering HyperWorks
[11] LsDyna software and its user manuals. Livermore Software Technology Corporation,
Livermore, California 94550-1740.

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