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Yashasvi Giridhar, IJPRET, 2015; Volume 3 (9): 104-111
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INTERNATIONAL JOURNAL OF PURE AND
APPLIED RESEARCH IN ENGINEERING AND
TECHNOLOGY
A PATH FOR HORIZING YOUR INNOVATIVE WORK
A QUALITATIVE ANALYSIS OF DYNAMIC AXIAL CRUSHING OF THIN WALLED
SQUARE COLUMNS
YASHASVI GIRIDHAR, MOHD. ZAHID ANSARI
PDPM-Indian Institute of Information Technology, Design and Manufacturing Jabalpur, Khamaria, Jabalpur, MP, India 482005
Accepted Date: 05/03/2015; Published Date: 01/05/2015
Abstract: The axial crushing phenomenon of dynamically loaded thin walled columns
provides an efficient means to improve the impact energy absorption and crashworthiness
of automotives. This work uses a finite element analysis to study the crushing of square
aluminium alloy tubular columns under dynamic load conditions. A mass of 400 kg moving at
two different velocities was used to apply the dynamic loads. The side length and wall
thickness of columns were changed to study their effect of crushed length and plastic strain
energy stored during the crushing. Finally, a qualitative analysis was performed over the
obtained results in comparison with super folding element theory.
Keywords: Column; nonlinear dynamics; crash worthiness; super folding element theory;
progressive dynamic collapse.
\
Corresponding Author: MR. YASHASVI GIRIDHAR
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Yashasvi Giridhar, IJPRET, 2015; Volume 3 (9): 104-111
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Research Article
Impact Factor: 4.226
Yashasvi Giridhar, IJPRET, 2015; Volume 3 (9): 104-111
ISSN: 2319-507X
IJPRET
INTRODUCTION
Thin walled metal columns have been a source of interest for a long time owing to their high
impact energy absorbing characteristics as well as their well defined collapse patterns. Due to
their high energy absorption capacity coupled with lighter weight they have been widely used
in automotive industry such as in cars and trains to not only provide high structural integrity but
also high crash worthiness. To improve their energy absorption ability, grooves [1], dents [2]
and pre-folds [3, 4] can be introduced to reduce the initial buckling load. The columns normally
have tubular profile with square, rectangular or circular section. Square and rectangular forms
are normally used in trains whereas the circular ones are more commonly used in landing gears
in airplanes and helicopters where the telescopic tubular arrangement can also impart good
vibration absorption feature to it.
The buckling and collapse of a circular tube column under axial compressive load shows various
collapse modes depending on tube width-to-thickness ratio. Short and thin square tubes are
commonly used in energy absorption mechanisms for their easy collapsing features, usually
collapse in the symmetric mode [4]. Many efforts have been made in understanding the
collapse pattern of thin walled tubes with different cross sections with emphasis on higher
energy absorption capacity. Wierzbicki and Abramowicz [5] presented a simple theoretical
model called basic or super folding element (SFE) to predict the collapse behaviour of tubular
structures loaded axially.
This element consisted of four folding surfaces, i.e. conical, cylindrical, trapezoidal and toroidal
surfaces, and the element is circumferentially in-extensional. The energy dissipation in SFE in
affected by the stationary plastic hinge lines, the travelling plastic hinge line and the associated
localized in-plane stretching of the toroidal surface. It was found that travelling plastic hinge
lines were most effective in energy dissipation [5]. The material properties are considered here
to be rigid-perfectly plastic with a constant value of the flow stress.
This paper aims to investigate the crushing behaviour of thin walled tubular square aluminium
alloy columns under axial compressive load. The height of the columns was fixed and its side
length and wall thickness were varied. The columns were subjected to a dynamic load of 400 kg
mass moving at velocities 7.5 m/s and 15 m/s, respectively. A commercial finite element
analysis software LS-DYNA Explicit was used to simulate the collapse deformation behaviour of
the columns. The collapse mechanism and the energy absorption characteristics of the columns
are studied.
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2. Theory and Modelling
The buckling and collapse behaviour of tubular columns can be into classified into three types,
i.e. static plastic buckling, dynamic plastic buckling and dynamic progressive buckling. In crash
scenarios of automotive chassis, generally this behaviour lies in the region of dynamic plastic
buckling and dynamic progressive buckling. In addition, it also depends on factors such as initial
impact velocity, mass ratio between the impact mass and the tube. The deformation mechanics
involved in all these is similar. The large plastic deformations are accommodated by the
formation of complicated pattern of folds and wrinkles in the thin wall of the columns. During
this collapse process, fold lines are formed continuously and bending occurs, leading to
formation of double curvature surfaces.
The basic plastic folding mechanism consists of five different deformation steps: (1)
deformation of a floating toroidal surface (2) bending along stationary hinge lines (3) rolling
deformations (4) opening of a conical surface and (5) bending of deformations along inclined
stationary hinge lines which follow the locking of travelling hinge line as mentioned in step 3.
SFE was proposed taking into consideration the material continuity and kinematic admissibility
criteria. The deformation mode consisting of four trapezoidal elements, a section of two
horizontal cylindrical surfaces, two inclined conical surfaces and a section of a toroidal surface is
called the basic folding mechanism. The mean crushing force (Pm) for a thin walled square
column can be given as [6,7]:
Pm  13.06  0 b 1/3 t 5/3
(1)
where, σ0 is flow stress of the column material and b and t are the side length and the wall
thickness of the square column. For elastic-strain hardening materials with power law
hardening behaviour, flow stress can be given as [8]:
0 
 y u
1 n
Where, σy is yield stress and σu is tensile strength, and n is power law exponent.
3. Finite Element Analysis
Figure 1 shows the schematic design of the thin square columns used in this study. The height
of the column was fixed at 300 mm but its side length was changed as 30 mm and 60 mm and
its wall thickness as 1.5 mm and 3 mm. The crash conditions were simulated as a mass of 400 kg
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striking the top, free-end of the square columns at two different velocities as 7.5 m/s and 15
m/s. Thus, in total 8 cases were studied. The bottom end of the columns was fixed to a rigid
base. The columns were made of aluminium alloy and its material properties are listed in Table
1.
Fig.1 Schematic of thin square column.
Table 1 Mechanical properties of aluminium alloy
Properties
Young’s modulus (GPa)
Density (kg/m3)
Poisson’s ratio
Yield stress (MPa)
Tangent modulus (MPa)
Al
71
2770
0.33
280
500
The crash simulations were conducted using a commercial finite element analysis software LSDYNA. The column material was considered as piecewise linear plasticity material with bilinear
isotropic hardening. The mechanical properties of the column material are listed in Table 1. The
finite element models were meshed using shell elements. This was done to reduce the
computational time required as well as for better predictions as compared to usual solid
elements in the case of thin walled structures. The shell elements chosen were the full
integration 4-node Belytschko-Tsay elements. The boundary conditions were chosen to
simulate the impact of a heavy mass over the top surface of the tubes that remained in contact
throughout the transient phase of deformation. This was achieved using mass elements of 400
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kg added to the topmost nodes of the columns and these nodes were initialised with the
deforming loads in form of kinetic energy velocities. The complete impact time was set to be 10
ms over which the entire crash analysis was made. This short duration over which heavy loads
were applied ensured that the dynamic effect of inertia was activated as opposed to the quasi
static conditions of collapse.
4. RESULTS AND DISCUSSION
Figure 2 shows the finite element results for a typical crushing sequence behaviour at different
time steps in the aluminium alloy square columns under the impact load. The entire crash time
was 10 ms. Both solid and wireframe results are shown to illustrate the buckling and collapse
behaviour by observing the folding mechanism and its propagation.
Fig.2 Solid (left) and wireframe (right) deformed shapes of the axially crushed square
aluminium alloy tube column at different time steps.
Figure 3 shows crushed length or the reduction in total height of the square columns under the
dynamic impact load of 400 kg mass moving at 7.5 m/s and 15 m/s, respectively. The initial
height of the columns was 300 mm. It can be seen in the figures that crushed length is
decreased with increase in column wall thickness and side lengths, which indicates the
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Yashasvi Giridhar, IJPRET, 2015; Volume 3 (9): 104-111
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increased resistance offered by the columns to buckling and crushing. This observation is
consistent with the relation given by Eq. 1. It can also be observed in the figures that the
crushed length is more than doubled when the impact velocity is doubled from 7.5 m/s to 15
m/s. It should be noted that the strain-hardening property of the aluminium alloy is already
included and implemented by the finite element software used in this study [9].
Fig.3 Crushed length in different (b, t) square aluminium alloy columns for impact velocity 7.5
m/s (left) and 15 m/s (right) at 10 ms.
Figure 4 shows the plastic energy stored in the square columns due to the plastic deformation
under the dynamic impact load of 400 kg mass moving at 7.5 m/s and 15 m/s, respectively. The
kinetic energy of the moving mass in converted into elastoplastic strain energy and the
resulting deformation in the columns. The deformation occurs in form of localized buckling and
fold formation which lead to collapse and crushing of the columns.
Fig.4 Plastic strain energy stored in different (b, t) square aluminium alloy columns for impact
velocity 7.5 m/s (left) and 15 m/s (right) at 10 ms.
Figure 5 show the effective plastic strain contour plot in the deformed square aluminium alloy
columns for different side lengths and wall thicknesses when crushed by a mass of 400 kg
striking axially at, respectively, 7.5 m/s and 15 m/s. Red indicates maximum plastic
deformation, whereas, blue represents the minimum or no plastic deformation. The plastic
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folding mechanism can easily be observed in the figures. The local buckling and first folds
generally started at the bottom of the columns triggering further buckling and collapse of the
column.
Fig.5 Effective plastic strain in different (b, t) deformed square aluminium alloy columns at
impact velocity 7.5 m/s (top row) and 15 m/s (bottom row) at 10 ms.
5. CONCLUSIONS
In this paper we studied the crash behaviour of square thin walled tubes under dynamic load.
The effect of different column designs and impact velocities were studied using finite element
software. Results showed good conformity to the deformation mechanisms predicted by super
folding element theory. The crushed length of the columns decreased with increase in wall
thickness and side length. In addition, doubling the impact velocity more than doubled the
crush length. Since the increase in column side length and wall thickness increased its
resistance to buckle, the plastic energy stored due to the permanent deformation of the
columns also increased. The real-time study of crushing sequence found that the crushing and
the first folding generally started at the bottom of the columns and the consequent fold then
propagated upwards and piled over it.
ACKNOWLEDGEMENT
This study was supported by PDPM-IIITDM-Jabalpur.
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