Plastic collapse analysis of thinwalled circular tubes subjected to bending
ABSTRACT a b s t r a c t Circular tubes have been widely used as structural members in many engineering applications. Therefore, its collapse behavior has been studied for many decades, focusing on its energy absorption characteristics and collapse mechanism. In order to predict the collapse behavior of members, one could rely on the use of finite element codes or experiments. These tools are helpful and have high accuracy but are costly and require extensive running time. Therefore, an approximate model of tubes collapse mechanism is an alternative especially for the early step of design. This paper is also aimed to develop a closedform solution to predict the moment–rotation response of circular tube subjected to pure bending. The model was derived based on the principle of energy rate conservation. The collapse mechanism was divided into three phases. New analytical model of ovalisation plateau in phase 2 was derived to determine the ultimate moment. In phase 3, the Elchalakani et al. model [Int. J. Mech. Sci. 2002; 44:1117–1143] was developed to include the rate of energy dissipation on rolling hinge in the circumferential direction. The 3D geometrical collapse mechanism was analyzed by adding the oblique hinge lines along the longitudinal tube within the length of the plastically deformed zone. Then, the rates of internal energy dissipation were calculated for each of the hinge lines which were defined in terms of velocity field. Inextensional deformation and perfect plastic material behavior were assumed in the derivation of deformation energy rate. In order to compare, the experiment was conducted with a number of tubes having various D/t ratios. Good agreement was found between the theoretical prediction and experimental results.

Article: Buckling and postbuckling of gradient and nonlocal plasticity columns experiencing softening
[Show abstract] [Hide abstract]
ABSTRACT: The buckling and the postbuckling behaviors of a perfect axially loaded column are analytically investigated through a global bilinear moment–curvature elastoplastic constitutive law. Three plasticity cases are studied, namely the linear hardening plasticity law, the perfect elastoplastic case and the softening case. The applications of such a study can be found in various structural engineering problems, including reinforced concrete, steel, timber or composite structures. It is analytically shown that for all kinds of elastoplastic behaviors, the plasticity phenomena lead to a global softening branch in the load–deflection diagram. The propagation of the plasticity zone during the postbuckling process is analytically characterized in case of linear hardening or softening plasticity laws. However, it is shown that the unphysical elastic unloading solution necessarily occurs in presence of local softening moment–curvature constitutive law. A nonlocal plasticity moment–curvature softening law is then used to control the localization branch in the postbuckling stage. This nonlocal plasticity law includes the explicit and the implicit gradient plasticity law. Higherorder plasticity boundary conditions are derived from an extended variational principle. Some parametric studies finally illustrate the main findings of this paper, including the plasticity modulus effect on the postbuckling behavior of these plasticity structural systems.International Journal of Solids and Structures 08/2014; 51(2324):40524067. · 2.04 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: To improve the accuracy of the vehicle crashworthiness simulation, it is necessary as well as important to integrate the valid forming effects of key parts. It has been agreed by many that onestep simulation results should be used only as a qualitative trend of the part but not as an engineering result for further structural analysis, especially for a relatively complex part. The study shows that it is inaccurate to analyze the forming effects with onestep simulation based on the geometry of the final part through comparison with the incremental simulation and verification with the actual part, whether in thickness or in plastic strain. However, incremental simulation is very time consuming and infeasible in the early stage of vehicle design due to lacking of forming tools and process parameters. An engineering approach is proposed to meet the requirement of accuracy as well as the time efficiency, where onestep simulation is conducted based on the geometry of the transformed part instead of the final part. The geometry of the transformed part is generated by simple die design engineering and proves to offer much more accuracy than the onestep simulation based on the final part geometry.Science China Technological Sciences 55(6). · 1.19 Impact Factor  SourceAvailable from: Matteo Strano[Show abstract] [Hide abstract]
ABSTRACT: The role of an antiintrusion bar for automotive use is to absorb the kinetic energy of the colliding vehicles that is partially converted into internal work of the members involved in the crash. The aim of this paper is to investigate the performances of antiintrusion bars, made by tubes filled with aluminium foams. The reason for using a cellular material as a filler deals with its capacity to absorb energy during plastic deformation, while being lightweight. The study is mainly conducted by evaluating some key technical issues of the manufacturing problem and by running experimental and numerical analyses. The evaluation of materials and shapes of the closed sections to be filled is made in the perspective of a car manufacturer (production costs, weight reduction, space availability in a car door, etc.). Experimentally, foams are produced starting from an industrial aluminium precursor with a TiH2 blowing agent. Empty and foam filled tubes are tested in three point bending, in order to evaluate their performances in terms of several performance parameters. Different manufacturing conditions, geometries and tube materials are investigated. The option of using hydroformed tubes, with non constant cross section, for the production of foam filled side structures id also discussed.International Journal of Material Forming 01/2013; 6(1). · 1.42 Impact Factor
Page 1
Plastic collapse analysis of thinwalled circular tubes subjected to bending
S. Poonaya?, U. Teeboonma, C. Thinvongpituk
Department of Mechanical Engineering, Faculty of Engineering, Ubonratchathani University, Warinchamrap Ubonratchathani 34190, Thailand
a r t i c l e i n f o
Article history:
Received 15 May 2008
Received in revised form
7 November 2008
Accepted 7 November 2008
Available online 18 January 2009
Keywords:
Bending
Circular tube
Collapse mechanism
Energy absorber
a b s t r a c t
Circular tubes have been widely used as structural members in many engineering applications.
Therefore, its collapse behavior has been studied for many decades, focusing on its energy absorption
characteristics and collapse mechanism. In order to predict the collapse behavior of members, one could
rely on the use of finite element codes or experiments. These tools are helpful and have high accuracy
but are costly and require extensive running time. Therefore, an approximate model of tubes collapse
mechanism is an alternative especially for the early step of design. This paper is also aimed to develop a
closedform solution to predict the moment–rotation response of circular tube subjected to pure
bending. The model was derived based on the principle of energy rate conservation. The collapse
mechanism was divided into three phases. New analytical model of ovalisation plateau in phase 2 was
derived to determine the ultimate moment. In phase 3, the Elchalakani et al. model [Int. J. Mech. Sci.
2002; 44:1117–1143] was developed to include the rate of energy dissipation on rolling hinge in the
circumferential direction. The 3D geometrical collapse mechanism was analyzed by adding the oblique
hinge lines along the longitudinal tube within the length of the plastically deformed zone. Then, the
rates of internal energy dissipation were calculated for each of the hinge lines which were defined in
terms of velocity field. Inextensional deformation and perfect plastic material behavior were assumed in
the derivation of deformation energy rate. In order to compare, the experiment was conducted with a
number of tubes having various D/t ratios. Good agreement was found between the theoretical
prediction and experimental results.
& 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Many researchers have been investigating the collapse me
chanism and energy absorption capacity of many structures,
majority focusing on thinwalled structures such as shell, tubes,
stiffeners and stiffened sandwich panels. These structures have
been identified as a very efficient impact energy absorbing system
and called ‘‘energy absorber’’. The study of deformation in the
energy absorber accounts for various parameters such as
geometrical shape, mode of collapse, strain hardening and strain
rate effect. In general, there are several approaches to determine
the energy absorption of structural members, by using finite
element analysis, experiments and theoretical analysis. Although
finite element analysis and experimental approaches can provide
accurate results, they are costly and require extensive running
time. Therefore, the theoretical analysis is an alternative for the
early step of design.
Theoretical analysis of the collapse can be made by using
the hinge line method. When thinwalled members are crushed
by any load, the collapse strength is reached. Then, plastic
deformations occur over some folding lines and these are called
‘‘hinge lines’’. When hinge lines are completed around the
structure, global or local collapse will progress. The internal
energy in deformed structure is determined by the summation of
plastic energy dissipated in each hinge line.
Many researchers have studied the plastic collapse behavior
and energy absorption characteristics of thinwalled tubes
subjected to bending. Kecman [1], studied the deep bending
collapse of thinwalled rectangular columns and proposed a
simple failure mechanism consisting of stationary and rolling
plastic hinge line. The analytical solution was achieved using limit
analysis techniques. Zhang and Yu [2] studied the ovalisation of a
tube with an arbitrary cross section and one symmetric plane to
obtain a full moment–curvature response. Their analysis showed
that the flattening increases nonlinearly as the longitudinal
curvature increases up to a limiting maximum value. Wierzbicki
and Bhat [3] derived a closedform solution to predict the
pressure necessary to initiate and propagate a moving hinge on
the tube. The calculations were performed using a rigid–plastic
material, and a simple moving hinge model was assumed to occur
along the hinge line. The deformation of a ring was modeled into a
‘‘dumbbell’’ shape. The analytical results agreed well with the
experiments. Wierzbicki and Suh [4], conducted a theoretical
analysis of the large plastic deformations of tubes subjected to
ARTICLE IN PRESS
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/tws
ThinWalled Structures
02638231/$see front matter & 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tws.2008.11.005
?Corresponding author.
Email address: s_poonaya@yahoo.co.th (S. Poonaya).
ThinWalled Structures 47 (2009) 637–645
Page 2
combined load in the form of lateral indentation, bending
moment and axial force. The model was effectively decoupling
the 2D problem into a set of 1D problems. The theoretical results
gave good correlation with existing experimental data. Cimpoeru
and Murray [5] presented empirical equations of the momen
t–rotation relation of a square thinwalled tube subjected to pure
bending where the widthtothickness ratio was less than 26.
Results from the empirical model were compared with the
analytical model of Kecman [1]. Wierzbicki et al. [6] studied the
bending collapse mechanism of thinwalled prismatic columns by
the concept of basic folding mode. They developed the collapse
mechanism by adding the toroidal and rolling deformation in the
compressive model. Closeform solutions were derived for the
moment–rotation characteristic of square column in the post
failure range. The stress profiles in the most general case of a
floating neutral axis were also shown. The simplified analytical
solution was used to predict the moment–rotation relationship
with an absolute error not greater than 7%. Wiezbriki and Sinmao
[7] studied the simplified model of circular tube in pure bending,
which was valid for large and very large sectional distortion. Good
agreement with numerical solution (ABAQUS) was obtained. Kim
and Reid [8] modified the mechanism model of Wierzbicki et al.
[6], and suggested that the toroidal deformation and conical
rolling should be defined differently from the case of axial
compression to satisfy the bending kinematics condition. Good
agreement was found between the model and the experiment.
Elchalakani et al. [9] predicted the response of a circular steel tube
under pure bending. They included the effect of ovalisation along
the length of the tube into the model. Work dissipated through
the toroidal and the rolling hinges was ignored. The hinge
mechanism was assumed straight and inextensible. Good agree
ment between analytical result and experiment was achieved. In
another report, Elchalakani et al. [10] presented a closedform
solution of the postbucking collapse of the slender circular
hollow section with D/t485 subjected to pure bending. The
theoretical analysis closely matched with the experimental
results.
The main objective of this paper is to develop a closeform
solution of thinwalled circular tube subjected to bending using a
rigid plastic mechanism analysis. The model of ovalisation phase
is derived to determine the ultimate moment. In addition to the
structural collapse phase, Elchalakani’s model [9] is developed to
include the rate of energy dissipation of rolling hinge in the
circumferential direction. The model is based on the principle of
energy rate conservation and is analyzed in the 3D problem by
adding the longitudinal hinges in plastic zone. The theoretical
model is presented and then resolved in the form of momen
t–rotation characteristics.
2. Theoretical predictions
In general, the collapse mechanism of tube subjected to
bending can be divided into three phases, elastic behavior,
ovalisation plateau and structural collapse. Each phase behaves
in a different deformation mode. The present study attempted to
develop the collapse model of each phase by focusing on their
moment–rotation relationship individually. The analytical model
of each phase is derived as explained.
2.1. Elastic behavior
In this phase, the moment increases linearly with constant
slope up to a yield moment–rotation. The elementary theory
of elasticity is generally used to predict the linear moment–rota
tion characteristic of a circular tube. Yield moment and the
corresponding angle are defined by
My¼2syI
Do
(1)
yy¼MyL0
EI
(2)
where Myis the yield moment, L0is the moment length, E is the
elastic modulus, I is the second moment of area, sy is the
measured yield stress, yyis the yield rotation angle and Dois
the outside diameter of the tube.
2.2. Ovalisation plateau
In phase 2, the circular cross section of tube subjected to
bending started to deform in an oval shape. In general, the
bending moment in this phase is assumed constant and ultimate.
Ueda [11] derived the moment–rotation relationship by consider
ing the strains developed at the surface of the tube under constant
moment. He assumed that an initially circular cross section is
deformed to an elliptical cross section. The ultimate bending
moment was obtained by integrating stress over the cross section.
His ultimate moment is expressed as
Mu¼ syZpþ ðsu?syÞZe
where sy is the yield stress, su is the ultimate tensile stress,
Zp¼ (4/3)(Ro3?Ri3) is the plastic bending section modulus,
Ze¼ (p/4Ro)(Ro4?Ri4) is the elastic bending section modulus, Ro
is the outer radius of tube, and Riis the inner radius of tube.
Recently, Elchalakani et al. [12,13] also determined the
ultimate moment of circular hollow section by approximating
the ovalised section as an elliptical shape. Their experiments
suggested that the ovalisation starts when the major axis reaches
1.10d and the minor axis reaches 0.9d. The solution for their
ultimate moment is shown in the following equation:
(3)
Mu¼ Sovalisedsy¼4
3ðR2
vRh? R2
viRhiÞsy
(4)
where Sovalisedis the plastic section modulus of an ovalised tube,
sy is the measured yield stress of an ovalised tube. Rh¼ Dh/
2 ¼ 0.55Doand Rv¼ Dv/2 ¼ 0.45Doare the external horizontal and
vertical radii of an ovalised tube, respectively. The internal
horizontal and vertical radii are Rhi¼ (Rh?t) and Rvi¼ (Rv?t),
respectively, and t is the thickness of tube.
The present paper aims to propose a new model for sectional
ovalisation by developing the model of those two works [11,12].
The new ovalisation model proposed here is shown in Fig. 1. The
radius of curvature R1which is formed at both ends of flattening
ARTICLE IN PRESS
Fig. 1. The proposed model of ovalisation of tube due to bending.
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
638
Page 3
sides is taken into account. Although the behavior of material
exhibits as slight hardening, the bending moment is assumed
constant during the increment of bending rotation. The rolling
hinge of the circumferential cross section is ignored. Then, the
ultimate moment of an ovalised tube and the corresponding angle
of rotation are determined by integrating the stress over the cross
section.
From Fig. 1, the geometry of circumferential cross section of
tube is assumed inextensible and can be expressed as
x
2þ R1ðp ? fÞ ¼ pR
where x/2 ¼ R1sin(f) and R is the initial radius of the tube.
The bending moment in a tube can be obtained by integrating
the stress over the circumferential cross section which is
expressed as
Z
where dA ¼ tds is the crosssectional area of an element of the
tube, t is the thickness of the tube, z is the distance from the
neutral axis of a sectional ovalisation to the circumferential area
and ds is the length of the circumferential cross section.
By integrating Eq. (6), the expression for moment can be
obtained as
ZpR
M ¼ s0tR2
where R1¼ pR/(p?f+sin(f)), for large deformation the R1is equal
to the outside radius of tube, R.
The ultimate bending moment can be determined by mini
mizing the bending moment in Eq. (7) with respect to the
deformation angle f. The ultimate bending moment is finally
obtained as
(5)
M ¼
AszdA
(6)
M ¼ 2s0t
0
zds
1ðsinð2fÞ þ 2 sinðfÞÞ
(7)
Mu¼ 3s0tR2
where s0is the ultimate stress of material, t is the thickness of the
tube, and R is the outside radius of the tube.
(8)
The changing angle of rotation between phase 1 (elastic
regime) and phase 2 (ovalisation regime) is determined by
yoval¼Mu
EI
(9)
where Muis the ultimate moment obtained from Eq. (8), E is the
elastic modulus, and I is the second moment of area.
The rotation angle at the end of phase 2 is, still, cannot be
determined theoretically. However, this study assumes the
intersection between the moment–rotation angle curves of phase
2 and phase 3 to be the end angle of phase 2 and, hence, to be the
onset angle of phase 3.
2.3. Structural collapse
In phase 3, the structure starts to collapse resulting in load
carrying capacity to decrease rapidly. The present paper proposes
a modified collapse model as shown in Fig. 2. The model involves
the flattened hinge (AB) in the circumferential cross section
and the oblique hinge lines along the longitudinal direction of
tube (AS and BS), as shown in Figs. 2(a) and (c), respectively.
The traveling hinges of the flattened region in the circumfer
ential cross section in Fig. 2(a) are modeled as a line connecting
points A and B. The rolling hinge has radius r and circular arc of
current radius R1. Figs. 2(b) and (c) show the deformation of four
oblique hinge lines along the longitudinal tube within the length
of plastic zone H.
2.3.1. Assumptions
In order to analyze the proposed collapse model in Fig. 2, the
following assumptions were made:
1. The tube material is ductile, rigid–perfectly plastic, isentropic,
homogeneous and material compatibility condition is main
tained.
2. The tube circumference is inextensible.
3. Shear deformation and twist of the deformed tube are
neglected.
ARTICLE IN PRESS
Fig. 2. Plastic collapse model of thinwalled circular tube subjected to pure bending.
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
639
Page 4
4. The collapse mechanism formed in the tube is kinematically
admissible and the cross section of the tube deforms in a
simplified manner as shown in Fig. 2.
5. All hinges are assumed straight. Work dissipation through the
toroidal region is ignored, but the rolling hinges in the
deformed cross section are considered.
6. The initial mean radius of tube (R) is the tube’s crosssection
radius at the beginning of the plastic hinge formation.
7. The tube does not elongate or contract in the axial direction.
8. The radius R1is constant and equal to the outside radius of
tube (R) during large deformation of the cross section.
9. The parameter H and r are constant during the collapse of
section.
2.3.2. Geometrical analysis
From the kinematics of the collapse mechanism shown in Fig.
2, the global geometry of collapse can be expressed as
x þ rf þ R1¼ ðpRÞ
where R is the mean radius of circular tube, R1is the radius of
deformed cross section, x is the length of hinge line ((AB) in Fig.
2(a)), and f is the mechanism angle defined in Fig. 2(a).
The vertical displacement D1of circular cross section, in Fig.
2(a) is defined by
(10)
D1¼ R1ð1 þ cos fÞ þ rð1 ? cos fÞ
and in Fig. 2(b) by
(11)
D1¼ 2R cos r ? H sin a
where a ¼ (p/2)?arcsin(1?(2R/H)sinr), r ¼ y/2, and y is the
bending rotation at the end of the tube.
Thus, the relationship between the mechanism angle f and the
bending angle r can be determined from Eqs. (11) and (12) using
the numerical techniques and can be expressed as
(12)
f ¼½4ðRHÞ1=2r1=2?1=2
ðR ? rÞ1=2
The velocity of hinge propagation, in Fig. 2(a) is expressed as
(13)
VA¼dx
dt¼_x
and
VB¼dR1ðp ? fÞ
dt
¼ ?R1_f
(14)
2.3.3. Plastic energy dissipation
The rate of internal energy dissipation resulting from contin
uous and discontinuous deformation rate fields is defined by
Z
i?1
where S denotes the current shell midsurface, n is the total
number of plastic hinge lines, liis the length of the ith hinge, and
½_ci? denotes a jump of the rate of rotation across the moving hinge
line. The components of the rate of rotation and the rate of
extension tensors are denoted by K˙aband _ ?ab, respectively. The
corresponding conjugate generalized stresses are the bending
moment Maband the membrane forces Nab. Mn¼ ð2=
is the plane strain plastic bending moment (per unit length)
normal to the hinge line. All components are expressed in the
cylindrical coordinate system on the surface of the shell.
There are six important components in the first integration
term of Eq. (15); i.e., _ ?xx, _ ?xy, _ ?yy, K˙xx, K˙xy, and K˙yy. Each of these
components will be considered individually.
Regarding the assumption that shear deformation and twist of
the deformed shell are neglected, we have_ ?ay¼_Kab¼ 0. The tube
circumference is inextensible, hence _ ?yy¼ 0. Later, the local
change in the axial curvature is assumed to be small compared
to the change in the circumferential curvature, resulting in
_E ¼
SðMab_ kabþ Nab_ ?abÞdS þ
X
n
Z
liMi
n½_ci?dli
(15)
ffiffiffi
3
p
Þðsot2=4Þ
K˙xxoK˙yy, and thus K˙xxwill be neglected. Finally, the axial strain
rate is zero, _ ?xx¼ 0, due to Assumption 7 as explained in Section
2.3.1. Hence, the expression for the rate of internal energy
dissipation expressed in Eq. (15) can be reduced to
Z
i¼1
_E ¼
s
ðNxx_ ?xxþ Myy_ kyyÞds þ
X
n
Z
LiMi
n½_ci?dli
(16)
Eq. (16) is the governing equation that will be used to calculate
energy dissipation in plastic hinges. The individual plastic hinge is
analyzed as follows.
2.3.3.1. Rate of lateral crushing energy of the tube. The expression
for the rate of energy of shape distortion or lateral crushing of
tube can be derived from Eq. (16) and is obtained as
Z
i¼1
where the plastic bending moment Myy¼ syyt2/4, syyis the cir
cumferential stress and t is the thickness of the tube. The rates of
curvature, ð_ kyyÞ along the straight flattened length ð_ kyyÞAand
around the semicircular curve are defined by
_Ecrush¼
s
ðMyy_ kyyÞds þ
X
n
Z
LiMi
n½_ci?dli
(17)
ð_ kyyÞA¼ ?
_R1
R2
1
and
ð_ kyyÞB¼ ?
_ r
r2
(18)
Since the rolling hinge, r and R1are assumed constant along the
rotational angle y, the rate of curvature is equal to zero, ð_ kyyÞ ¼ 0.
By using the assumptions and geometrical analysis explained
earlier, the rate of lateral crushing energy dissipated in the tube
can be derived as follows:
Z
where Mnis the bending moment normal to the hinge line, and
the rate of rotation_c at the moving hinge line is calculated from
the condition of kinemetic continuity
_Ecrush¼
X
n
i¼1
LiMi
n½_ci?dli
(19)
_c ¼ V½K?
where V is the velocity of the traveling hinge in the tangential
direction and K is the circumferential curvature. [K] ¼ K+?K?is
the jump in curvature from either side of the hinge line. Ahead of
the hinge K+¼ 1/r and behind the hinge K?¼ 0 (flat sections), the
rate of rotation at the hinge is
? ?
where VAand VBare tangential velocities in the current deformed
configuration of the ring in Fig. 2(a) and are given by
(20)
_cA¼ VA
1
r
and
_cB¼ VB
1
r?1
R1
??
(21)
VA¼dx
The bending moment normal to the hinge line (Mn) is assumed
to be equal to the circumferential bending moment (Myy) on the
continuous deformation region, Mn¼ Myy¼ (syyt2/4).
An approximate yield condition for the present problem is
given by
?
The plastic compressivetensile stress (sxx) in axial direction
resulting from bending is coupled with bendinginduced hoop
stress (syy) through an inscribed yield condition, to one point with
the coordinate,
dt¼_x;
VB¼?dR1ðp ? fÞ
dt
(22)
Myy
M0
?2
þ
Nxx
N0
??2
¼ 1 (23)
syy
s0
¼Myy
M0
¼
1ffiffiffi
2
p ;
sxx
s0
¼Nxx
N0
¼
1ffiffiffi
2
p
(24)
ARTICLE IN PRESS
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
640
Page 5
The rate of energy dissipation in the crushing mode can be
rewritten by substituting Eqs. (18), (21), (22) and (24) in Eq. (19):
for full cross section of tube,
_Ecrush¼
M0t
ffiffiffi
82
p
R2H ðR1? rÞcos f1
r
? ?
þ R1
1
r?
1
R1
????
_f
(25)
where M0¼R
As0zdA ¼ 4s0R2t is the fully plastic bending mo
ment of the undeformed cross section [7].
2.3.3.2. Rate of energy dissipation over the central hinge AB. Energy
dissipated in the central hinge line (AB) can be expressed as
shown in Eq. (26). Its length depends on the angle a:
X
where i is the number of hinge lines, li¼ x is the length of hinge
line,_c ¼ 2_ a is the rate of rotation angle of hinge lines, and _ a can
be determined from Eq. (10).
_EAB¼
i
Mnli_ci¼
M0t
ffiffiffi
82
p
R2x
ð Þ_ a
(26)
2.3.3.3. Rate of energy dissipation over the oblique hinge lines SA, SB,
OA, and OB. The plastic energy dissipated over four oblique hinge
lines SA, SB, OA, and OB, as shown in Fig. 2(b) and (c) can be
calculated by using the following expression:
_EOB¼
X
i
Mnli_ci¼
M0t
ffiffiffi
42
p
R2_x
(27)
where i ¼ 4 is the number of hinge lines, lh¼
length of each hinge line, and_c ¼ VA=lh¼_x=lh is the rate of
rotation angle of each hinge line._c is assumed to vary along the
tangential velocity
VA
ofthe
2ðR1? rÞ cosðfÞ_f.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H2þ x2
q
is the
movinghinge ABand
_x ¼
2.3.3.4. Instantaneous moment. The global energy balance of a
single structural element can be established by equating the rate
of energy dissipation and the rate of external work.
_Eext¼_Eint
where_Eextis the rate of external work, and_Eintis the internal rate
of energy dissipation at the plastically deformed region.
The rate of total internal energy dissipation_Eintis defined by
(28)
_Eint¼_Ecrushþ_EABþ_EOB
External rate of energy dissipation is defined by
(29)
_Eext¼ M_y
The instantaneous moment can be determined by substituting
all expressions for plastic energy into Eq. (29) and substituting Eq.
(30) in Eq. (28), finally the corresponding moment is obtained as a
function of H, r, and y:
(30)
M ¼ MðH;r;yÞ
The mean crushing moment (Mm) can be determined by
integrating moment (M) in Eq. (31) with respect to the rotation
angle from r ¼ 0 to r ¼ rf(H):
MmðH;rÞ ¼1
rf
0
(31)
ZrfðHÞ
Mdr
(32)
where rf(H) is the contact angle where the upper and lower
sections gets in contact. This angle can be determined by defining
term D1in Eq. (12) equal to zero and hence
? ?
Similarlythemechanismangle
obtained from Eq. (13) by defining term D1¼ 0. Hence, ff is
rfðHÞ ¼ cos?1
R
H
(33)
of finalcollapse ff
is
obtained as
ff¼
2
ðR ? rÞ½ðR ? rÞR?1=2
The length of the plastic folding region (H) and the rolling
radius (r) can be determined from the postulate of minimum
mean crushing moment in Eq. (32). Then, the solution can be
expressed as
(34)
H ¼ 1:31R
The value of H and r are then substituted into Eq. (31) to give the
expression for the instantaneous moment in terms of the rotation
angle y:
and
r ¼ 0:6R
(35)
M ¼ MðyÞ
Details of each moment component are presented in the
Appendix.
(36)
3. Experiments
3.1. Specimen preparation
In order to verify the proposed model, the experiment was
conducted with 18 tubes. These tubes were divided into 6 groups,
each group consisted of 3 cold formed mild steel tubes with
similar geometries. The nominal diametertothickness ratios
range from 21.16 to 42.57 and the length of each specimen is
1500mm. The material properties were determined by using the
tensile coupons tested according to the British Standard BSEN 10
0021:1990 [14]. Results from tensile tests are shown in Table 1. It
is observed from Table 1 that the value of modulus of elasticity of
UB1, UB5, and UB6 are a bit low compared to other specimens.
This is due to the variation of material property in the local
market. However, this does not affect the study results because
the comparisons of experimental and analytical results were
based on each corresponding material.
3.2. Test setup and procedure
The experimental setup was designed to obtain a pure bending
moment over middle span of the specimen. The influence of shear
and axial forces should be avoided or minimized as much as
possible. To meet this requirement, Cimpoeru et al. [5] introduced
a machine that is able to apply a pure bending moment without
imposing shear or axial forces. A machine based on that concept
has been built at Ubonratchathani University to apply a pure
bending test on those 18 specimens. The diagram of this machine
is shown in Fig. 3.
As can be seen from the diagram in Fig. 3, the machine consists
of two load application wheels on its left and right ends. These
two wheels are connected to the tensile testing machine via two
connecting rods. The tested tube is placed on the load application
wheels and locked with two bolts on each side. As the tensile
machine pulls the connecting rods upward, the wheels start to
rotate and apply pure bending moment on the tested specimen.
Fig. 4 shows an experimental setup and various views of deformed
specimen. The experimental collapse mode was found similar to
the proposed model shown in Fig. 2.
4. Results and discussions
Two main findings of this paper are the ultimate moment
which was analyzed from the ovalisation regime and the model of
ARTICLE IN PRESS
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
641
Page 6
plastic collapse in the structural collapse regime. The results and
their discussions are presented below.
4.1. The comparison of ultimate moment with experimental results
and other simplified models
The ultimate moment analyzed in this paper are compared
with experimental results as well as with available models such as
Ueda’s [11] and Elchalakani’s [12]. Table 2 shows a summary of
the ultimate bending moment predicted by those two available
models and the newly derived model (Eq. (8)), compared with
experimental results.
From Table 2, it can be observed that the present prediction
(Eq. (8)), Elchalakani’s and Ueda’s formulae overestimate the
ultimate bending moments by 1.7%, 8.7%, and 11.8%, in average,
respectively. The present study, which has included the curvature
into account, seems to give more accurate results compared to
experiments and other two predictions. However, it still over
estimates the ultimate moment, especially for high D/t tubes. This
may be explained that, for tubes with high D/t ratios, the plasticity
does not spread linearly along the whole circumference as it is
assumed in the analysis. In contrast, the plasticity tends to
concentrate at the plastic hinge region and causes premature
failure.
Considering the coefficient of variation shown in the last row
of Table 2, it indicates that the present prediction provides the
results with slightly higher degree of variation than that of
Elchalakani’s but less than that of Ueda’s results.
4.2. Comparison of theoretical moment–rotation angle history with
experimental results
Figs. 5(a)–(f) show the moment–rotation angle relationship
analyzed in this paper compared to experimental results. In the
elastic regime, the theoretical moment–rotation curves lie above
the experimental results of every case. This may be because of the
effect of flow stress. Kim [8] suggested to use lower flow stress for
relatively thin sections and used a higher flow stress for relatively
thick sections. However, this study assumes the flow stress as sy
for every calculation. In phase 2, the ovalisation regime, the
theoretical curves are constant while the experimental curves
slightly increase as the ovalisation progresses. This is due to the
effect of strain hardening of material. Considering on phase 3, the
structural collapse, the theoretical curves lie slightly above
the experimental results, especially at large deflection of tubes.
This is because the model does not consider the softening due to
the formation of plastic hinges.
ARTICLE IN PRESS
Table 1
Dimensions and material properties of specimens..
Specimen
no.
Diameter
(mm)
Thickness
(mm)
D/t
Modulus of
elasticity, E (GPa)
Yield stress,
sy(MPa)
Ultimate stress,
su(MPa)
Yield angle,
yy(deg.)
Yield moment,
My(kNm)
UB1
UB2
UB3
UB4
UB5
UB6
59.25
59.00
46.85
59.35
58.55
74.50
2.80
2.30
1.80
1.80
1.60
1.75
21.16
25.65
26.03
32.97
36.59
42.57
128
160
173
178
128
133
330
270
320
354
257
306
383
314
355
370
295
380
2.99
1.96
3.17
2.30
2.44
2.41
2.21
1.51
1.01
1.61
1.02
2.45
Fig. 3. The diagram of the pure bending machine used in this study.
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
642
Page 7
5. Conclusion
This paper provides a theoretical model to predict the collapse
mechanism of thinwalled circular tube subjected to pure
bending. The collapse mechanism was divided into 3 phases;
elastic regime, ovalisation regime, and structural collapse regime.
The elementary theory of elasticity was adopted to explain the
elastic regime. The effect of curvature was taken into account for
the ovalisation phase. This model predicts the ultimate moment
accurately, but seems to overestimate for high D/t tubes.
This paper also developed the structural collapse regime by
considering the energy dissipation in rolling hinge in the
circumferential direction. The analytical moment–rotation curves
lie slightly above the experimental results. In general, it could be
concluded that the model developed here provides the results that
agree reasonably well with experimental results.
ARTICLE IN PRESS
Fig. 4. The experimental setup undeformed and deformed specimen in various views: (a) the experimental setup, (b) undeformed tube (UB1) in place, (c) starting in to
deform, (d) plastic zone in deformed tube, (e) final deform (side view) and (f) final deform (plain view).
Table 2
Comparison of ultimate moments predicted from simplified model and test results..
Specimen
no.
D/t
Experimental ultimate
moment
Predicted ultimate moment,
Eq. (8)
Elchalakani’s ultimate moment,
Eq. (4)
Ueda’s ultimate moment,
Eq. (3)
MExp(kNm)
Mu(kNm)
Mu/MExp
Mu(kNm)
Mu/MExp
Mu
(kNm)
Mu/MExp
UB1
UB2
UB3
UB4
UB5
UB6
21.16
25.65
26.03
32.97
36.59
42.57
3.20
1.85
1.12
1.76
1.09
2.38
2.83
1.87
1.05
1.76
1.22
2.76
0.88
1.01
0.94
1.00
1.11
1.16
3.02
2.12
1.18
2.09
1.13
2.76
0.94
1.15
1.05
1.19
1.04
1.16
2.93
2.10
1.17
2.11
1.34
2.84
0.92
1.13
1.04
1.20
1.23
1.19
Average
Coefficient of variation
1.017
0.093
1.088
0.080
1.118
0.096
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
643
Page 8
Acknowledgment
Authors wish to thank the Department of Industrial Engineer
ing, Ubonratchathani University for support of the test rig.
Appendix
Following are the details of the moment components given in
(36). These formulae were calculated by using the numerical
techniques and Mathcad [15]:
M1¼
ð?1=4Þðð?R þ rÞ=rÞM0ðp=RÞt
ffiffiffi
2
p
f
r
!
is a moment component for the crushing of ring.
M2¼
5 ? 9 ? 10?2ðð?R þ rÞ2=ðR2HÞM0tf3
r
!
is a moment component for the central hinge.
M3¼
ð1=32Þð
ffiffiffi
2
p
=R2ÞðR ? rÞM0tf
r
!
is a moment component for the oblique hinge.
Here H and r can be determined from Eqs. (35) and (36),
respectively. M0¼ 4s0R2t, t is the thickness of tube, R is the
outside radius of tube, f is the mechanism angle of tube (see
Eq. (13)), and r is the bending rotation at the end of tube, as
shown in Fig. 2(b).
References
[1] Kecman D. Bending collapse of rectangular and square section tubes. Int J
Mech Sci 1983;25(9–10):623.
[2] Zhang LC, Yu TX. An investigation of the brazier effect of a cylindrical tube
under pure elastic–plastic bending. Int J Press Vessels Pip 1987;30:77–86.
[3] Wierzbicki T, Bhat SU. Initiation and propagation of buckles in pipelines. Int J
Solids Struct 1986;22(9):985–1005.
ARTICLE IN PRESS
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0510152025303540
Angle (deg)
0510152025303540
Angle (deg)
Moment (kN.m)
Moment (kN.m)
Experiment
Theory
Experiment
Theory
Experiment
Theory
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
05 10 152025 30 3540
Angle (deg)
Moment (kN.m)
Experiment
Theory
Experiment
Theory
Experiment
Theory
0
0.5
1
1.5
2
05 101520 253035 40
Angle (deg)
Moment (kN.m)
0
0.5
1
1.5
0
0.5
1
1.5
2
2.5
3
05 101520 2530 3540
Angle (deg)
05 10152025 30 3540
Angle (deg)
Moment (kN.m)
Moment (kN.m)
Fig. 5. (a–f) The relationship between rotation angle and bending moment achieved from experiment compared to theoretical analysis. (a) UB1—59.25?2.8 (D/t ¼ 21.16),
(b) UB2—59.0?2.3 (D/t ¼ 25.65), (c) UB3—46.85?1.8 (D/t ¼ 26.03), (d) UB4—59.35?1.8 (D/t ¼ 32.97), (e) UB5—58.55?1.6 (D/t ¼ 36.59) and (f) UB6—74.50?1.8 (D/
t ¼ 41.38).
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
644
Page 9
[4] Wierzbicki T, Suh MS. Indentation of tubes under combined loading. Int J
Mech Sci 1988;30(3–4):229–48.
[5] Cimpoeru SJ, Murray NW. The largedeflection pure bending properties of a
square thinwalled tube. Int J Mech Sci 1993;35(3–4):247–56.
[6] Wierzbicki T, et al. Stress profile in thinwalled prismatic columns subjected
to crush loading—II. Bending. Comput Struct 1994;51(6):625–41.
[7] Wierzbicki T, Sinmao MV. A simplified model of brazier effect in plastic
bending of cylindrical tubes. Int J Press Vessels Pip 1997;71:19–28.
[8] Kim TH, Reid SR. Bending collapse of thinwalled rectangular section
columns. Comput Struct 2001;79:1897–991.
[9] Elchalakani M, Zhao XL, Grzebieta RH. Plastic mechanism analysis of circular
tubes under pure bending. Int Mech Sci 2002;44:1117–43.
[10] Elchalakani M, Grzebieta RH, Zhao XL. Plastic collapse analysis of slender
circular tubes subjected to large deformation pure bending. Adv Struct Eng
2002;5(4):241–57.
[11] Ueda S. Moment–rotation relationship considering flattening of pipe due to
pipe whip loading. Nucl Eng Des 1985;85:251–9.
[12] Elchalakani M, Zhao XL, Grzebieta RH. Plastic slenderness limits for cold
formed circular hollow sections. Aust J Struct Eng 2002;3(3):127–41.
[13] Elchalakani M, Zhao XL, Grzebieta RH. Bending tests to determine slenderness
limitsforcoldformed circular
2002;58:1407–30.
[14] British standard. Tensile testing of metallic materials.1990.
[15] Mathcad 2000. User’s Guide, Math Soft, Inc.
hollowsections.J ConstrSteelRes
ARTICLE IN PRESS
S. Poonaya et al. / ThinWalled Structures 47 (2009) 637–645
645
View other sources
Hide other sources
 Available from Somya Poonaya · May 30, 2014
 Available from ubu.ac.th