Quasi-static response and multi-objective crashworthiness
optimization of oblong tube under lateral loading
A.Baroutaji
a
, E.Morris
a
, A.G.Olabi
b
(a) School of Mechanical and Manufacturing Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland
(b) University of the West of Scotland, School of Engineering, High Street, Paisley, PA1 2BE, UK.
Abstract:
This paper addresses the energy absorption responses and crashworthiness optimisation of thin-walled
oblong tubes under quasi-static lateral loading. The oblong tubes were experimentally compressed
using three various forms of indenters named as the flat plate, cylindrical and a point load indenter.
The oblong tubes were subjected to inclined and vertical constraints to increase the energy absorption
capacity of these structures. The variation in responses due to these indenters and external constraints
were demonstrated. Various indicators which describe the effectiveness of energy absorbing systems
were used as a marker to compare the various systems. It was found that unconstrained oblong tube
(FIU) exhibited an almost ideal response when a flat plate indenter was used. The design information
for such oblong tubes as energy absorbers can be generated through performing parametric study. To
this end, the response surface methodology (RSM) for the design of experiments (DOE) was
employed along with finite element modelling (FEM) to explore the effects of geometrical parameters
on the responses of oblong tubes and to construct models for the specific energy absorption capacity
(SEA) and collapse load (F) as functions of geometrical parameters . The FE model of the oblong
tube was constructed and experimentally calibrated. In addition, based on the developed models of the
SEA and F, multi-objective optimization design (MOD) of the oblong tube system is carried out by
adopting a desirability approach to achieve maximum SEA capacity and minimum F. It is found that
the optimal design of FIU can be achieved if the tube diameter and tube width are set at their
minimum limits and the maximum tube thickness is chosen.
Keywords: Energy absorbing systems, Oblong tubes, Finite element method (FEM), Quasi-static,
Design of experiment (DOE), Response surfaces method (RSM), Multi-objective optimization design
(MOD)
Nomenclature
E: The energy absorption capacity
F: collapse load
Abbreviations
FIU: flat plate indenter-unconstrained system.
FISC: flat plate indenter-side constraints system.
FIIC: flat plate indenter-inclined constraints system.
CIU: cylindrical indenter-unconstrained system.
CISC: cylindrical indenter-side constraints system.
CIIC: cylindrical indenter-inclined constraints system.
PIU: point indenter-unconstrained system.
PISC: point indenter-side constraints system.
PIIC: point indenter-inclined constraints system.
1 Introduction
Tubular systems which consist of one or more circular or square sectioned tubes are
commonly used to absorb kinetic energy through plastic deformation.
These structural elements can absorb kinetic energy from many types of deformation leading to
various energy absorption responses. The principle deformation mechanisms of tube include lateral
compression, lateral indentation, axial crushing, tube splitting, and tube inversion. A significant
amount of research has been conducted on the energy dissipated by tubular systems over the last three
decades. The main findings were outlined and presented in a research article by Olabi et al. [1] and
Alghamdi [2].
The lateral compression of circular tube were analysed by DeRuntz and Hodge [3], flat plate indenter
was used to compress the tubes. The authors used rigid perfectly plastic material model to predict the
force-deflection response of the tube. They found that the collapse load is affected by the geometrical
factors and material properties of the tube. Reid and Reddy [4] performed further investigations on
strain hardening effects. They developed an accurate material model which considered both geometric
and material strain hardening effects. The authors reported that the energy absorbing capacity can be
maximised by choosing appropriate tube dimensions.
Gupta et al [5] examined numerically and experimentally the lateral crushing of circular metallic
tubes under quasi-static conditions. Aluminium and mild steel tubes with different diameter to
thickness ratios were used in this investigation. The authors found that the energy absorbing capacity
and mean collapse load increases with increase in thickness and decrease in diameter.
Increasing the energy absorption capacity of the tubular system by means of external constraints was
applied by Reddy and Reid [6]. The authors built a tubular system with side constraints in which the
horizontal diameter of the tube was prevented from translating laterally. It was found that the energy
absorbed by a system with side constraints was three times more than the system with no constraints.
The application of inclined constraints as an alternative to vertical ones also increases the energy
absorbing capacity of tubes/rings. Reid [7] studied the effect of varying the inclination angle for the
lateral compression of tubes. In general it was concluded that an externally constrained system is a
viable method to increase its energy absorbing capacity.
The above studies were concerned with the compression of tubes under rigid platens. However,
alternative ways of compressing these systems are possible by incorporating point-load indenters as
shown by Reid and Bell [8]. The load deflection for this type of compression tends to be unstable
once the collapse load has been reached. This behaviour is termed deformation softening as opposed
to deformation-hardening.
Instead of using flat plat or point-load indenter, Shim and Stronge [9] used cylindrical indenters to
compress ductile, thin walled tubes. The authors investigated the post-collapse response of these
tubes. The unstable response was also noticed for this kind of indenter.
In addition to circular tubes, various geometry shapes of tubes have been proposed by researchers to
use as energy absorbers under lateral loading such as elliptical tubes [10, 11] and oblong tubes [12].
Recently, researchers have utilized the finite element method (FEM) to predict the responses of
energy absorption systems under quasi-static [12, 13] and dynamic [14, 15] lateral loading. Morris et
al. [13] employed ANSYS to predict the responses of nested circular tubes compressed by two types
of indenters and subjected to external constraints. Close agreements between the computational and
experimental results were obtained.
Another numerical technique which is response surfaces method (RSM) was employed by researchers
to seek an optimal design and to perform the multi-objective optimization design (MOD) of energy
absorption system under pure axial [16, 17], lateral [18] and oblique loads [19].
Much of the research on thin-walled tube energy absorbers crushed laterally has focused on those of
circular cross section. However, the oblong tubes which are a modified form of circular tubes have
received less attention. In the present work, the oblong tubes are proposed as energy absorption
components. These tubes are expected to have high energy absorptions performance as they have a
greater lateral displacement stroke compared with circular tube systems. The crushing responses of
these tubes under quasi-static lateral loading have been investigated experimentally. The lateral
compressions were applied through three types of indenters named as flat plate, cylindrical, and point-
load indenter. Different variations of external constraints were incorporated into the oblong tube
energy absorption system to increase the energy absorption capacity of these systems. In addition,
with the aim of generating the design guidelines for such oblong tube as energy absorbing devices
under lateral loading, the response surface method (RSM) for design of experiments (DOE) was used
in conjunction with the finite element modelling (FEM). The FE model was developed using
commercial finite element code (ANSYS) and validated using experimental techniques. The specific
energy absorption capacity (SEA) and the collapse load (F) of the oblong tube were modelled as
functions of geometrical parameters such as thickness (t), diameter (D), and width (W). Factorial
study was performed to investigate the primary and interaction effects of geometric parameters on the
specific energy absorbed and collapse load. Furthermore, Based on the developed models of the SEA
and F, the approach of multi-objective optimization design was applied to find the optimal
configuration of the oblong tube.
2 Experimental work
2.1 Material and Geometrical properties
Mild Steel tubes were used in this work. The tubes were cold finished, manufactured according to the
DIN standards (DIN 2393 ST 37.2) and containing around 0.15% carbon. Tensile tests were carried
out in order to determine the mechanical properties of the tubes as shown in Figure 1. Figure 1
displays the true stress-strain curve of the tensile sample. Upon examination of this figure, it can be
seen that the stress-strain curve displays unusual behaviour in which strain softening occurred almost
immediately after yielding with no evidence of strain hardening. This phenomenon is due to sample
necking which takes place immediately after yielding. This behaviour is termed as tension instability
and the cold rolling process might be the reason for this. Table 1 shows the mechanical properties of
the mild steel material derived from the true stress-strain curve and used in the FE modelling. The
yield stress is validated according to DIN standards, which state that the yield stress of this material is
within the range of 450–525MPa [13, 14]. A non-zero value of 1500 MPa was employed to represent
the hardening modulus of this material. The same value of hardening modulus was used by [14, 15] to
define the softening stage of the same material.
