# The Ovalisation of Thin-walled Circular Tubes Subjected to Bending

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**ABSTRACT:**The bending collapse behaviour of rectangular and square section tubes is studied theoretically and experimentally. A limit analysis technique was employed and a set of formulae relating the hinge moment and associated angle of rotation was derived. The theoretical predictions were verified by comparison with 56 quasi-static bending tests on 27 different sections having aspect ratios () varying between 3.0 and 0.33 and with width to wall thickness ratios () between 128 and 9.14. Very good agreement was found between the theoretical predictions and experimental results for the whole range of sections.International Journal of Mechanical Sciences 01/1983; 25:623-636. · 2.06 Impact Factor -
##### Article: An investigation of the brazier effect of a cylindrical tube under pure elastic-plastic bending

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**ABSTRACT:**The Brazier effect of originally straight cylindrical tubes under pure elastic-plastic bending is investigated by the deformation theory based on some deformation assumptions agreeing with numerous experiments. The expressions for bending moment and flattening ratio in terms of curvature are finally obtained.International Journal of Pressure Vessels and Piping 01/1987; 30(2):77-86. · 1.08 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The moment-rotation properties of square thin-walled tubes were measured up to and beyond bending collapse with a unique bending machine that was capable of applying a pure bending moment to large deflections without significant shear or tensile forces. This allows an accurate determination of a tube's maximum moment and plastic behaviour prior to collapse, the rotation at which collapse is initiated, as well as the resulting collapse curve. The results were compared with existing analytical predictions. In addition, the finite length of the local plastic collapse mechanism was determined by adjusting the deformable length of the test specimens. The concept of a finite mechanism length is important for the purposes of modelling the behaviour of structures such as bus frames that are commonly made from thin-walled tubes.International Journal of Mechanical Sciences. 01/1993;

Page 1

Abstract— 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

tubes, one could rely on the use of finite element codes or

experiments. These tools provide results with high accuracy but

costly and require extensive running time. Therefore, an

approximated model of tubes collapse mechanism is an

alternative especially for the early step of design. This paper is

also aimed to develop a closed-form solution to predict the

response of circular tube subjected to pure bending, focusing on

the ovalisation regime. New ovalisation model was developed to

include the effect of curvature into account. In order to

compare, the experiment was conducted with a number of tubes

having various D/t ratios. In addition, the available predictions

from other investigators were also presented and compared.

Good agreement was found between the theoretical predictions

and experimental results. In addition, the present model

provides more accurate result compared to some available

theories.

Index Terms—Bending, Circular tube, Plasticity, Ovalisation

I. INTRODUCTION

Many researchers have been investigating the collapse

mechanism and energy absorption capacity of many

structures, majority focusing on thin-walled 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 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, it is costly and requires 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

hinge line method. When thin-walled members are crushed

C. Thinvongpituk1 and S. Poonaya are with Department of Mechanical

Engineering, Faculty of Engineering, Ubonratchathani University,

Warinchamrap Ubonratchathani Thailand 34190

(1corresponding author; Tel: 0066 81 5935002, Fax: 0066 45 353380, e-mail:

chawalit@rocketmail.com)

S. Choksawadee and M. Lee are with Department of Industrial

Engineering, Faculty of Engineering, Ubonratchathani University,

Warinchamrap Ubonratchathani Thailand 34190

by any load, the collapse strength is reached. Then, plastic

deformations are occurred over some folding lines and are

called “hinge lines”. When hinge lines are completed around

the structure, global or local collapse will progress. Some

examples of studies on collapse mechanism are as followed;

D. Kecman [1] studied the deep bending collapse of

thin-walled 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. L.C. Zhang and T.X. 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 indicated that the flattening of tube increases

nonlinearly as the longitudinal curvature increases. However,

this phenomenon is to limited at some maximum values. T.

Wierzbicki and S.U. Bhat., [3] derived a closed-form

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 hinge line. The

deformation of a ring was modeled into a “dumbbell” shape.

The analytical results agreed well with the experiments. T.

Wierzbicki and M.S. Suh., [4] conducted a theoretical

analysis of the large plastic deformations of tubes subjected

to combined load in the form of lateral indentation, bending

moment and axial force. The model is effectively decoupling

the 2-D problem into a set of 1- D problems. The theoretical

results gave good correlation with existing experimental data.

S.J. Cimpoeru and N.W. Murray., [5] presented empirical

equations of the moment-rotation relation of a square

thin-walled tube subject to pure bending where the

width-to-thickness ratio less than 26. Results from the

empirical model were compared with the analytical model of

Kecman [1]. T. Wierzbicki et. al., [6] studied the collapse

mechanism of thin-walled prismatic columns subjected to

bending by using the concept of basic folding mode. They

developed the collapse mechanism by adding the toroidal and

rolling deformation in the compressive model. Close-form

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 shown to

predict the moment-rotation relationship with an absolute

error not greater than 7%. T. 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. T.H.Kim and S.R.Reid., [8] modified the

mechanism model of Wierzbicki el 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

The Ovalisation of Thin-walled Circular

Tubes Subjected to Bending

C. Thinvongpituk, S. Poonaya, S. Choksawadee and M. Lee

Proceedings of the World Congress on Engineering 2008 Vol II

WCE 2008, July 2 - 4, 2008, London, U.K.

ISBN:978-988-17012-3-7WCE 2008

Page 2

was found between the model and the experiment. M.

Elchalakani el 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 agreement between analytical result and

experiment was achieved. In another report, M. Elchalakani

el al., [10] presented a closed-form solution of the

post-bucking collapse of the slender circular hollow section

with D/t >85 subjected to pure bending. Their theoretical

analysis closely matched with the experimental results. The

main objective of this paper was to develop a close-form

solution for the ovalisation of thin-walled circular tube

subjected to bending using a rigid plastic mechanism

analysis. The model was derived to determine the ultimate

moment. The experiment was also conducted with a number

of circular tubes. The theoretical result was compared with

experimental and with some available predictions. Good

agreement between them was achieved.

II. THEORETICAL PREDICTIONS

In general, the collapse mechanism of tube can be divided

into three phases which are elastic behavior, ovalisation

plateau, and structural collapse. Each phase behaves in

different deformation modes. Many investigators have been

attempting to develop the collapse models of them by

focusing on their moment-rotation relationship individually.

In the ovalisation phase, the circular cross-section of tube

subjected to bending is starting to deform in oval shape. In

general the bending moment in this phase is assumed

constant and ultimate. S. Ueda [11] proposed an interesting

report which performed the analytical method of

moment-curvature relationship by considering the strains

developed at the surface of tube under a constant-moment. In

the ovalisation regime, 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:

(

uypu

MZ

σσ=+−

σ is the yield stress,

σ is the ultimate tensile stress,

(

3

(

4

o

R

⎝⎠

modulus,

radius of tube.

Recently, M. Elchalakani et al. [13] also determined the

ultimate moment of circular hollow section by approximating

the ovalised section as an elliptical shape. Their experimental

observation suggested that the ovalisation starts when major

axis reaches 1.10d and the minor axis reaches 0.9d . The

solution for their ultimate moment is shown in (2).

)

ye

Z

σ

(1)

where

y

u

)

3

o

3

i

4

p

ZRR

=−

is the plastic bending section modulus ,

)

4

o

4

ie

ZRR

π

⎛

⎜

⎞

⎟

=−

is the elastic bending section

o R is the outer radius of tube, and

iR is the inner

()

2

v

2

vi

4

3

uovalisedyh hiy

MS R RR R

σσ==−

(2)

where

ovalised

σ is the measured yield stress of an ovalised

D

55 . 0

2

the external horizontal and vertical radii of an ovalised tube,

respectively. The internal horizontal and vertical radii are

() tRR

h hi

−=

and

RR

vi

=

the thickness of tube.

The present paper aims to propose a new model for

sectional ovalisation by developing the model of those two

literatures [11, 13]. In order to simplify the problem, the

following assumptions are made;

S

is the plastic section modulus of an

ovalised tube,

y

tube.

o

h

h

DR

==

and

o

v

v

D

D

2

R

45. 0

==

are

() t

v

−

, respectively, and t is

1. The material is ductile, rigid-perfectly plastic, isentropic,

homogeneous and material compatibility condition is

maintained.

2. The tube circumference is inextensible.

3. Shear deformation and twist of the deformed tube are

neglected.

4. The collapse mechanism formed in the ovalisation of tube

is shown in Fig. 1.

5. The initial mean radius of tube (R ) is the tube’s cross

section radius at the beginning of the plastic hinge formation.

6. The tube does not elongate or contract in axial direction.

7. The radius

outside radius of tube (R ) during large deformation of the

cross-section.

1 R (see Fig. 1) is constant and equal to the

Fig.1 The model of ovalisation of tube due to bending

New ovalisation model proposed here is shown in Fig. 1.

The curvature of radius

flattening sides were 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 (3),

()

1

2

1 R which is formed at both ends of

RR

ξ

πφπ

+−=

(3)

Proceedings of the World Congress on Engineering 2008 Vol II

WCE 2008, July 2 - 4, 2008, London, U.K.

ISBN:978-988-17012-3-7 WCE 2008

Page 3

where

( )

φ

ξ

2

sin

1 R

=

and R is the initial radius of

tube.

The bending moment in a tube can be obtained by

integrating the stress over the circumferential cross section

which is expressed in (4),

∫

A

where

tdsdA =

is the cross- sectional area of an element

of the tube, t is the thickness of 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 (4), the expression for moment can be

obtained as shown in (5).

∫

0

((

φσ

2sin

10

=

tRM

π

sin

+−

=

zdAM

σ

(4)

=σ

2

π

zdstM

R

0.

)( )

φ

)

sin2

2

+

(5)

where

( )

φφπ

1

=

R

R

, for large deformation the

1 R is equal to the outside radius of tube, R.

The ultimate bending moment can be determined by

minimizing the bending moment in (5) with respect to the

deformation angleφ . The ultimate bending moment is finally

obtained as expressed in (6)

0

3

tRMu

σ=

where

0

σ

thickness of tube, and R is the outside radius of tube.

When you submit your final version, after your paper has

been accepted, prepare it in two-column format, including

figures and tables.

2

(6)

is the ultimate stress of material, t is the

III. EXPERIMENTS

A. Specimen preparation

In order to verify the proposed model, the experiment is

conducted with 18 tubes (UB1 to UB6) of mild steel with

different diameter to thickness ratios. The nominal diameter

to thickness ratios ranges from 21.16 to 42.57 and the length

of each specimen is 1,500 mm. The material properties are

determined by using the tensile coupons tested according to

the British Standard BSEN 10 002-1:1990 [14]. Results from

tensile tests are shown in Table 1.

TABLE I

DIMENSIONS AND MATERIAL PROPERTIES OF SPECIMENS

Specimen

No.

Diameter

(mm)

Thickness

(mm)

D/t

Modulus

of

Elasticity

E

(GPa)

Yield Stress

σ

(MPa)

y

Ultimate

Stress

σ

(MPa)

u

Yield

Angle

y

θ

(deg)

2.99

Yield

Moment

M

(kN)

y

UB1 59.25 2.80 21.16 128 330 383 2.21

UB2 59.00 2.30 25.65 160 270 314

1.96

1.51

UB3 46.85 1.80 26.03 173 320 355

3.17

1.01

UB4 59.35 1.80 32.97 178 354 370

2.3

1.61

UB5 58.55 1.60 36.59 128 257 295

2.44

1.02

UB6 74.50 1.75 42.57 133 306 380

2.41

2.45

B. Test setup and procedure

The experimental setup is 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,

S.J. 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.2.

As can be seen from the diagram in Fig.2, 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 tested specimen. Fig. 3 shows an experimental set

up and various views of deformed specimen. The

experimental ovalisation shape is found similar to the

proposed model shown in Fig.1.

Proceedings of the World Congress on Engineering 2008 Vol II

WCE 2008, July 2 - 4, 2008, London, U.K.

ISBN:978-988-17012-3-7WCE 2008

Page 4

Fig. 2 The diagram of the pure bending machine used in this study

(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) (f) Final deform (Plain view)

Fig. 3 The experimental setup, undeformed and deformed specimen in various views

Proceedings of the World Congress on Engineering 2008 Vol II

WCE 2008, July 2 - 4, 2008, London, U.K.

ISBN:978-988-17012-3-7WCE 2008

Page 5

IV. RESULTS AND DISCUSSIONS

The ultimate moment analysed in this paper are compared

with experimental results as well as with available models

such as Ueda’s [11] and Elchalakani’s models [13]. Table 2

shows a summary of the ultimate bending moment predicted

by those two models and newly derived model, compared

with experimental results.

From Table 2, it can be observed that the present prediction

(6), Elchalakani’s and Ueda’s formulae overestimate the

ultimate bending moments by 1.67%, 8.7% and 11.8%, in

average, respectively. The present study, which includes the

curvature into account, seems to give more accurate results

compared to experiments and other two predictions.

However, it still overestimates the ultimate moment,

especially for high D/t tubes. This can be explained that, for

tubes with high D/t ratios, the plasticity does not spread

linearly along the whole length as it is assumed in the

analysis. In contrast, the plasticity tends to concentrate at the

plastic hinge region and causes premature failure.

TABLE I I

COMPARISON OF ULTIMATE MOMENTS PREDICTED FROM SIMPLIFIED MODEL AND TEST RESULTS.

Predicted Ultimate

Moment,

(6)

Elchalakani’s

Ultimate

Moment [13],

Ueda’s Ultimate

Moment [11],

Specimen

No.

/

D t

Experimental

Ultimate

Moment

M

kN.m

Exp

,

u

M ,

kN.m

xp

u

E

M

M

u

M ,

kN.m

xp

u

E

M

M

u

M ,

kN.m

xp

u

E

M

M

BC1 21.16 3.20 2.83 0.88 3.02 0.94 2.93 0.92

BC2 25.65 1.85 1.87 1.01 2.12 1.15 2.1 1.13

BC3 26.03 1.12 1.05 0.94 1.18 1.05 1.17 1.04

BC4 32.97 1.76 1.76 1.00 2.09 1.19 2.11 1.2

BC5 36.59 1.09 1.22 1.11 1.13 1.04 1.34 1.23

BC6 42.57 2.38 2.76 1.16 2.76 1.16 2.84 1.19

average 1.0167 1.087 1.12

V. CONCLUSION

This paper provides a theoretical model to predict the

ovalisation mechanism of thin-walled circular tube subjected

to pure bending. The effect of curvature is taken into account

for the ovalisation phase. This model predicts the ultimate

moment with higher accuracy than other predictions, but

seems to overestimate for high D/t tubes. The mechanism in

other collapse phases of tube are also being investigated and

the results will be presented in the near future.

ACKNOWLEDGMENT

Authors wish to thank, Asst. Prof. S. Choksawat, Mr. M.

Lee and Asst. Prof. S. Lee, Department of Industrial

Engineering, Ubonratchathani University for support of the

test rig.

REFERENCES

[1] D. Kecman, “Bending collapse of rectangular and square section

tubes,” Int. J. Mech. Sci., 1983, Vol.25, No. 9-10, pp 623-236

[2] L.C. Zhang and T.X. Yu, “An investigation of the brazier effect of a

cylindrical tube under pure elastic-plastic bending,” Int. J. Pres. Ves.

& Piping, 1987, Vol.30, pp 77-86

[3] T. Wierzbicki and S.U. Bhat, “Initiation and propagation of buckles

in pipelines,” Int. J. Solids Structures, 1986, Vol. 22, No. 9, pp

985-1005

[4] T.Wierzbicki and M.S. Suh, “Indentation of tubes under combined

loading,” Int. J. Mech. Sci., 1988, Vol.30 No. 3-4, pp 229-248

[5] S. J. Cimpoeru and N. W. Murray, “The large-deflection pure

bending properties of a square thin-walled tube,” Int. J. Mech. Sci.,

1993, Vol.35, No. 3-4 ,pp 247-256

[6] T. Wierzbicki et al, “Stress profile in thin-walled prismatic columns

subjected to crush loading-II,” Computer&structure, 1994, Vol. 51,

No.6, pp 625-641

[7] T. Wierzbicki and M. V. Sinmao, “A simplified model of brazier

effect in plastic bending of cylindrical tubes,” Int. J. Pres. Ves.&

Pipe, 1997, Vol.71, pp 19-28

[8] T.H. Kim, and S. R. Reid, “Bending collapse of thin-walled

rectangular section columns,” Computer & Structures, 2001, Vol.79,

pp 1897-1991

[9] M. Elchalakani, X. L. Zhao and R. H. Grzebieta, “Plastic mechanism

analysis of circular tubes under pure bending,” Int. Mech. Sci., 2002,

Vol. 44, pp 1117-1143

[10] M. Elchalakani, R. H. Grzebieta, and X. L. Zhao “Plastic collapse

analysis of slender circular tubes subjected to large deformation pure

bending,” Advances in structural engineering, 2002, Vol. 5, No. 4 ,

pp 241-257

[11] S. Ueda., “ Moment-rotation relationship considering flattening of

pipe due to pipe whip loading.,” Nuclear Engineering and Design ,

1985, Vol. 85, pp 251-259

[12] T. S. Gerber. Plastic deformation of piping due to pipe whip loading.

ASME paper 74-NE-1, 1974

[13] M. Elchalakani, X. L. Zhao, R. H. Grzebieta., “Plastic slenderness

limits for cold-formed circular hollow sections,” Australion Journal

of Structural Engineering, 2002, Vol. 3, No. 3, pp. 127-141

[14] British standard, “Tensile testing of metallic materials,” 1991

Proceedings of the World Congress on Engineering 2008 Vol II

WCE 2008, July 2 - 4, 2008, London, U.K.

ISBN:978-988-17012-3-7WCE 2008