ht. J. Pres. Vex & Piping 71 (1997) 19-28

ELSEVIER

0 1997 Elsevier Science Limited. All rights reserved Printed in Northern Ireland

PII:SO308-0161(96)00018-X 030%0161/97/$17.00

A simplified model of Brazier effect in plastic bending of cylindrical tubes

Tomasz Wierzbicki & Monique V. Sinmao Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

(Accepted 16 April 1996)

The Brazier effect in cylindrical tubes under pure plastic bending is investi- gated. A simple theoretical model is proposed to predict the bending moment and section deformation in terms of curvature, material properties and geometric parameters of a tube made of a perfectly plastic and strain-hardened material. A postulate of a minimum instantaneous bending moment is used to determine the degree of ovalization. The analytical solution compares well with the results of numerical calculations taken from ABAQUS 4-6 Example Manual. 0 1997 Elsevier Science Ltd.

NOTATION

initial (undeformed) tube radius current arc radius tube thickness length of tube section length of flat segment work of external forces internal (plastic) work of the tube internal work in bending of the tube internal work in crushing of the tube components of the stress tensor--a, /3 = r, 1‘3

flow stress yield stress distance to neutral axis external applied bending moment fully plastic bending moment of an undeformed (circular) section fully plastic bending moment of a deformed (ovalized) section fully plastic bending moment of a ring section bending moment tensor membrane force tensor flattening of tube minor axis, 6 = 2R - 2r flattening of tube minor axis, normalized with undeformed tube diameter angle of rotation of tube ends curvature of the tube middle axis curvature tensor strain in the axial direction strain tensor stress-strain curve constant strain hardening coefficient

correction factor for bending contribution correction factor for crushing contribution bending stress crushing stress crushing stress in front of moving hinge line, continuous stress contribution crushing stress behind moving hinge line crushing stress at moving hinge line, discon- tinuous stress contribution bending strain crushing strain crushing strain in front of the moving hinge line, continuous strain contribution crushing strain behind the moving hinge line dimensionless strain rate ratio Cartesian coordinates, see Fig. 2(a)

1 INTRODUCTION

Cylindrical tubes have wide application in pipe networks, support structures and energy-absorbing devices. During service, cylindrical tubes may be subjected to large bending due to accidental loading such as explosion, collision, or mass impact. These extreme loading conditions can result in structural collapse caused by large plastic deformation of the member tubes. Large plastic deformation causing ovalization of the initially circular cross-section of tubes subjected to pure bending is called the Brazier effect.’ The Brazier effect produces a gradual loss of bending resistance.

19

In relatively thick tubes, the growth of ovalization is

20 T. Wierzbicki, M. V. Sinmao

characterized by a hardening-softening behavior of the tube. This behavior derives from the opposing effects of material strain hardening, which increases the load-carrying capacity, and ovalization, which reduces the bending rigidity. By contrast, in thinner tubes, there is an abrupt transition from a stable to an unstable behavior as the tube suddenly loses most of its bending strength as the depth of the local dent increases. The response curves for thick and thin tubes are compared in Fig. 1. Experimental evidence has established a strong association between relatively thick tubes and the occurrence of ovalization in tubes subjected to pure bending.‘-’ Theoretical predictions for the ovalization of tubes corroborate these experimental observations.

A method of analysis of tube ovalization in the plastic range was first presented by Ades.236 The theoretical moment-curvature and ovalization- curvature relationships are determined using the principle of least work in an iterative numerical method, well-suited for computer applications but not suitable for development into an analytical model. Munz and Mattheck7 approached the analytical solution of this problem by considering the vertical component of the axial bending stresses as the primary cause of section ovalization. Following Munz and Mattheck, Prinja and Chitkara’ derived theoreti- cal solutions for the post-collapse bending moment and section flattening of a tube due to plastic bending, by analysing the vertical stresses developed on a section that is modelled after the static compression of an initially circular ring with four stationary hinges. Alternatively, Gerber4 and Ueda3 derived moment- curvature relationships by considering the strains developed at the surface of tubes in pure bending. In particular, a relationship between pipe surface strain and flattening of a pipe, obtained from experiments, is used to solve for the bending moment of a pipe. Zhang and Yu9 developed formulae for the ovaliza- tion of tubes with arbitrary cylindrical cross-sections and one symmetric plane. More recently, Yu, Reid, and Wang’ proposed a theoretical model to predict the hardening-softening behavior of tubular can-

Curvature

Fig. 1. Comparison of bending characteristics of thick and thin tubes.

tilever beams using stress-strain relationships derived from experiments. The predictions for the Brazier effect in the plastic range obtained by the above methods compare well with experimental data. However, most of the existing analyses are restricted to relatively small degrees of ovalization.

The objective of the present paper is to develop a simplified analysis which is valid for large and very large sectional distortion. A second objective is to show the applicability of the minimum postulate for deriving a relationship between the increasing curvature of the tube axis and the amount of section ovalization. A simple theoretical model of a cylindrical tube of strain-hardened material in pure plastic bending is constructed, guided by the experimentally observed shape of section coIlapse of an initially circular tube subjected to bending.” Analytical predictions for the bending moment and section deformation in terms of longitudinal curva- ture, material properties, and geometric parameters of the tube are derived and compared with an ‘exact’ numerical solution of the problem obtained by means of the computer code ABAQUS.

2 ASSUMPTIONS

Consider a cylindrical tube of rigid-perfectly plastic, strain-hardened material, subjected to pure plastic bending (Fig. 2) so that there is no shear on transverse planes or twisting of the tube. As a reference value, the fully plastic bending moment of the undeformed cross-section of the rigid-perfectly plastic circular tube is defined as:

MO = i

aoz, dA = 4aoR2t (1) A

where go is the material yield stress, R is the tube mean raidus and t is the tube thickness.

The deformation process is controlled by the increasing longitudinal curvature, K, or increasing relative rotation, 0, of the middle axis of the considered segment of the tube of the length 1:

Kl = 8 (2)

L.. >‘r \

\

M k-l M

Fig. 2. Cylindrical tube under pure bending.

A simplified model of Brazier ejfect in plastic bending of cylindrical tubes 21

Global equilibrium of the loaded structure can be established by equating the external rate of work we,, with the internal energy dissipation W&f

R.?xt = %t (3)

In the case of pure bending, the rate of work of external forces, I@##,,, can be expressed as:

wext = Mb (4)

where M is the external applied bending moment and 4 is the relative rate of rotation of the tube ends. In the present model, the rate of internal plastic work of the tube, ~i~,, is comprised of two parts. wbend is the rate of energy to bend the tube, and l@crUsh is the rate of energy to crush the tube laterally, so that:

wint = wlxnd + wcrush (5)

Each of these contributions will be treated separately. Then they will be related through a yield condition.

The following assumptions are made throughout the paper:

(1)

(2)

(3)

(4)

the tube material is rigid-perfectly plastic. Generalization to a work-hardening material is performed later in an iterative way; the Love-Kirchhoff hypothesis is used, so that all cross-sections of the tube remain plane and normal to the tube axis. Thus, warping is not permitted but ovalization is; the tube is relatively thick (R/t 5 30) so that it deforms in the plastic range, and there is uniform ovalization of the cross-sections over the considered length 1; plastic compressive-tensile deformations in the axial direction resulting from bending are coupled with bending-induced hoop deforma- tions through an inscribed yield condition;

Fig. 3. Two models for ovalization.

(5) the tube circumference is inextensible; and (6) shear deformation and twist of the deforming

shell element are neglected.

3 COMPUTATIONAL MODEL

Two simple models of the deformed cross-section were considered, each consisting of four travelling hinge lines and segments of continuously changing curvature. In the first model, four plastic hinges are formed on the vertical axis of symmetry, moving outward with velocity V with respect to this axis, Fig. 3(a). The hinges separate flat regions each of length b, from uniform but continuously shrinking semi-circular arcs of current radii Y. This model is strongly supported by transient cross-sectional shapes meas- ured by NNC.‘,” The inextensibility condition [assumption (5)],

2b f 27~~ = 27cR (6)

provides a relationship between variables b and Y. Defining the relative displacement (flattening of the rube minor axis) by:

S=2R-2r (7)

the parameters b and Y can be expressed in terms of 6:

Finally, differentiating the inextensibility condition wtih respect to time provides a relationship between the velocity of hinge propagation and the other geometrical parameters involved:

In the second model, the two pairs of plastic hinges that are formed on the horizontal axis of symmetry, propagate inward and separate arcs of constant radius RI from arcs of increasing radius Rz, Fig. 3(b). In contrast to the first model, which was a one-degree-of- freedom system, the second model involves two parameters. An additional assumption is required to derive a simplified solution. To proceed further with this analysis, guided by the so-called ‘dog bone’ shape of the deformed section in the related problem of buckle propagation in submerged pieplines,” it was assumed that R, = 0.2R,. However, the resulting ‘oval’ shape does not resemble reality well, and the accuracy of the second model in terms of cross- sectional resistance was poor. It was decided not to pursue this model any further.

22 T. Wierzbicki, M. V. Sinmao

4 PLASTIC WORK DISSIPATION

In rigid-perfectly plastic shells, the rate of plastic work results from the continuous and discontinuous velocity fields:

(10)

where M, = (2/fi)g0(t2/4) is the plane strain plastic bending moment (per unit length) normal to the hinge line, L’ is the length of the ith hinge line, r,&’ is the rotation of the ‘ith hinge, and m is the total number of stationary or moving hinge lines. Strain rate tensors conjugate to the bending moment MLvp, and membrane forces Nap are respectively the curvature rate IC=, and extension rate Cap. All components are expressed in the cylindrical coordinate system on the surface of the shell.

The integrand in the first term of eqn (10) has, in general, six components. In view of assumption (6), kxe = &x0 = 0. Plastic inextensibility [assumption (5)] implies that tee = 0. Finally, the local change in the axial curvature is assumed to be small compared with the change in the circumferential curvature, L << Gm and thus kXX will be neglected. The effect of the overall curvature of the tube axis K is retained, in accordance with assumption (2):

&x =I?2 (11)

where z is the distance from the neutral axis of the tube cross-section.

With the above simplifications, the expression for the rate of plastic work reduces to:

hi,, = N,,~z dS + I

MeeK,, dS + 4M,llC, (12) S

where the plastic membrane force NXX = a,,t and the plastic bending moment Mee = aee(t2/4). The first term in eqn (12) represents the rate of plastic work to bend a tube section of the length I:

where

M,,= I

u.&z ds = I @xxZ d‘4 (14)

21rR A

is the plastic bending capacity of the deformed section, ds is the hoop element length, and dA = t ds is the cross-sectional area element of the tube. The remaining two terms in eqn (12) correspond to the rate of energy of shape distortion or lateral crushing of the tube:

~CNSh = k,&2xrlMBe + 4M,l$ (15)

Each of the two above components of internal work dissipation will be analysed separately.

5 BENDING ENERGY

From the definition [eqn (14)] and the assumed kinematics, the fully plastic bending moment, Mp, of a deformed section is:

Mp = 4axxr2t + 2btra,, (16)

The inextensibility condition [eqn (6)] and the section deformation parameter 6 [eqn (7)] can be used to rewrite Mp in terms of 6:

Mp = 4a,,t[(R - S/2)2 + n(R - a/2)6/4] (17)

or, using eqn (l),

Mp = M,(l - 6)(1 + 0.576)(r,,/(~,) (18)

where 8 = 6/(2R) is the dimensionless section deformation parameter. Finally, using eqns (2), (13) and (18), the rate of energy to bend the tube is found to be:

l@,,end = M,,(l - 6)(1 + 0~576)8(~,,/a,,). (19)

6 CRUSHING ENERGY

The expression for the rate of crushing energy is given by eqn (15). The rate of rotation ~,J!J at the moving hinge line is calculated from the condition of kinetmatic continuity:‘”

li, = V[K] (20)

where V is the velocity of hinge propagation and [K] = K+ + K- is the jump in curvature from either side of the hinge line. Ahead of the hinge Kt = l/r and behind the hinge K- = 0 (flat sections). Using eqn (9), the rate of rotation at the hinge is:

li,,-E 2r

(21)

The curvature rate in the first term of eqn (15) is:

(22)

The bending moment normal to the hinge line M, is assumed to be equal to the circumferential bending moment in the continuous deformation region. M, = Mee. Substituting eqns (21) and (2?) into eqn (15) and expressing i/r in terms of 8 and 6 gives

(23

A simplified model of Brazier effect in plastic bending of cylindrical tubes 23

It should be noted that the continuous and discontinuous energy dissipation mechanisms contrib- ute equally to the rate of internal dissipation in the crushing mode. The above equations can be rewritten in terms of the fully plastic bending moment of the undeformed section given by eqn (1):

8 BENDING RESISTANCE AND MINIMIZATION

Substituting eqns (19) (24) and (27) into eqn (12) the expression

M 1

MO = ti [(l - S)(l + O-576)]

7 YIELD CONDITION

An approximate yield condition selected for the present problem is given by:i3

(25)

The strain rate vector with the components (kes, &) is assumed to be normal to the yield curve. The position of the stress point on the yield locus depends on the strain rate ratio:

is obtained for the external applied bending moment, with:

a! = [(?TTl)/(4R2)][&B] (29)

A minimization procedure is invoked to obtain the final expression for the normalized external applied bending moment. The normalized bending moment depends on the parameter 8 and the unknown ratio, LY, of crushing and bending velocities. It is postulated that for a given 01, the amount o,f ovalitation (measured by 8) adjusts itself so as to minimize the external applied bending moment:

Rkss 1S -= %X3 &Z(l - S) (26)

This ratio, in turn, changes from point-to-point along the cross-section and varies with the amount of crush, 8. This dependence is difficult to capture in the present approximate theory. Therefore, the stress profile is confined, in accordance with assumption (4), to one point with the coordinates:

*ee -M.s- 1 a;, Xx 1

ro M0 ti a0 No ti (27)

The above assumption is equivalent to considering a limited interaction curve inscribed to the exact yield condition, Fig. 4.

Fig. 4. The nonlinear yield condition and the inscribed square.

dM z- =0 (30)

The solution of the above equation is:

a = (1 - 6)2[0.43 + 1*146] (31)

Substituting eqn (31) for LY in eqn (28), the minimized expression for the normalized external applied bending moment in terms of 8 is obtained:

+ = & (1 - 6)[1*43 + 1.7161 0

The relationship between 8 and 0 can be used to express the normalized external bending moment in terms of curvature K, which is the controlling parameter for the process. Now, eqn (29) with eqn (31) can be integrated in time:

s (1 - 8)” + (0.43 + 1.146) dt = (33)

with the initial conditions 8 = 8 = 0, for t = 0. The result is:

15 = (tl/R2)[0*363 ln(0.43 + 1.146) - 0.363 ln(1 - 8)

- 0.194 + 0~500/(1 - S)] (34)

A plot of the function e(s) is shown in Fig. 5. It is seen that for 8 < 0.5, 8 is almost a linear function. Therefore, with eqn (2), 8 can be approximated by:

8 = 0*533R’K/t (35) Substituting eqn (35) into eqn (32), the minimized expression for the normalized external applied bending moment can be expressed in terms of curvature as:

; = (1 - 0533R2K/t)(1~011 + 0.644R’Klt) (36) 0

24

0 - 1.00 H/R2

T. Wierzbicki, M. V. Sinmao

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

F

Fig. 5. Approximating 8 as a function of 8.

The graph of eqn (36) (Fig. 6), shows the reduction in bending rigidity of the tube due to ovalization. The bending moment starts from the fully plastic value of the undeformed cross-section, given by eqn (1) and diminishes with K. The current solution shows the softening effect due to ovalization, and needs to be modified to include the stiffening effect due to strain-hardening.

9 MODIFICATION FOR STRAIN-HARDENING

The effect of strain-hardening can be accounted for in an iterative way. The solution for a rigid-perfectly plastic material provides a unique solution for bending and crushing deformations as functions of the increasing curvature. From this solution, strains and

average strains can be calculated at any instant of the deformation process. For a given stress-strain relationship, the new stresses, corrected for strain- hardening, can be found.

The elastic-plastic strain-hardening material can be described by a power-type stress-strain law:

EE, ff=

E 5 Ey

fly + A(& - &$ E > Ey (37)

where A is a constant, try and &y are the yield stress and strain, respectively, and n is the strain-hardening coefficient. A rigid-plastic counterpart of eqn (37) is:

(T = uy + A(E)” (38)

The above equation applies to a monotonic loading. In the case of strain reversal. the stress level will be

0.8 -

0.4-

Bending

0.2 -

0.0 0.3 0.8 0.9 1.2 1.5 1.8

KR2 T

Fig. 6. Bending and crushing moment contributions, uncorrected for strain-hardening,

A simplified model of Brazier qffect in plastic bending of cylindrical tubes 25

different for isotropic and kinematic strain-hardening. It is observed through experiment that materials commonly exhibit the Bauschinger effect,14 or yield at a lower stress following a strain reversal than for continued forward deformation, due to changes in internal stresses and dislocation substructure.‘5 Kine- matic theory predicts the Bauschinger effect and is implemented in most FEM codes. For this reason, the hypothesis of a kinematic strain-hardening is specified. The solution for the isotropic hardening rate is also shown for comparison.

In general, the strain, E, in eqn (38) is different in the bending and crushing deformation modes. This is taken into account by defining two stress correction factors, one for bending, nb and one for crushing, q,, consistent with the reference moment defined in eqn (1):

77b=b qc=o” UY UY

where vi, and CT, are the stresses obtained by eqn (38) for bending and crushing, respectively. From eqns (11) and (39), the correction factors vb and vC can now be calculated for any set of material constants (A, uY, cy, n). Now, eqn (28) for a dimensionless bending moment can be modified to include the effect of strain-hardening:

; = 7J?b & [(l - 6)(1 + 0576)] 0

+ 71, $j [a/(1 - S)]

9.1 Bending strain and stress

(40)

The average bending strain defines a bending stress and corresponding bending stress correction factor, qb. From eqn (ll), the average bending strain is:

where: El, = Kz,, (41)

with & equal, as before, to the arc length of the tube circumference. For the present computational model, eqn (42) yields:

2 a” = : (1 - 6)(1 + 0.576)

where 8 is related to K by eqn (35). The expression for the average bending strain is:

E’, = 1.194; 6(1 - s)(l + 0376) (44)

or,

lb = 0+636KR[l - 0.533(KRZ/t)J

x [l-t 0*304(KR2/t)] (45)

Using eqn (39) with eqn (45) the expression for bending stress is obtained and the final expression for the bending stress correction factor 7]b is:

?Ibxl+*(Eb)n - = 1 + A(0.636KR)” (+Y

x {[l - 0*533(KR’/t)][l + 0&304(KR’/t)])”

BY

(46)

9.2 Crushing strain and stress

The two components of crushing strain define an average crushing stress and corresponding crushing stress correction factor, qC. Both continuous and discontinuous bending strains contribute equally to the energy dissipation. Therefore, the crushing stress is defined as an arithmetic average of the continuous and discontinuous bending stresses:

UC= (VI+ 0-H) (47)

The crushing stress contribution, (pi, due to continuous bending of the deforming semi-cirular arcs [Fig. 3(a)] is:

AT!= l +(! (El)”

uY UY

where the average strain, sI, is:

(48)

At the moving hinge line, there is a strain reversal as the hinge is straightening out the curved segments of the shell. The magnitude of the strain reversal is:

In Fig. 7, the kinematic hardening rule is specified for

Stress, 0 B

( Strain, c

Fig. 7. Kinematic vs isotropic strain-hardening.

26 T. Wierzbicki, M. V. Sinmao

unloading from &I to .z[[, so that the absolute stress bending and for crushing, the corrected (Y after behind the hinge line, rII, corresponding to .sII, is: minimization is:

Within the width of the hinge, a continuous change of strain, and therefore of curvature, is assumed, and (TH at the moving hinge line is defined as an arithmetic average of the two absolute stresses corresponding to &I and Ebb:

Using eqn (47) with eqn (52), the expression for bending stress is:

u, = $(a1 + UH) = :a1 + $a,1 (53)

and with eqn (39), the final expression for the bending stress correction factor is obtained as:

1 n

(1 - 0*533(KP/t)) -1

qC=l+$A uY

= +$A (1 - 0533(KR’/t))

(54) uY

A plot of the suitably normalized bending and crushing stresses obtained from eqns (46) and (54), for IZ = 1, is shown in Fig. 8. It is seen that crushing strains are larger than bending strains for 8 > 0.6, with the maximum value of the bending strain at approximate- 1Y e b,max = Oa44t I R.

Repeating the same minimization process for solution with strain-hardening correction factors for

a = (~J~J(l- 6)2[0*43 + 1.1461 (55)

where the expression for &vc is obtained from eqns (46) and (54). The corrected relationship between 8 and 0 can be established by integrating eqn (29) with eqn (55):

so that the corrected relationship between 8 and K is:

(57)

Substituting eqn (57) for 8 and eqn (55) for LY into eqn (40), we obtain the expression for the normalized external applied bending moment corrected for strain hardening of the tube material:

2 = (l)b)[ 1 - ( ;)0.533R2K/t]

1 (58) The change in the minor axis of the tube due to ovalization can also be expressed in terms of curvature using 8 = 6/2R and eqn (52):

2r = 2R I- 2Lb [ 0

0.533R2Klt I (59)

?c

10 COMPARISON WITH FEM SOLUTION

The Example Manual of version 4-6 of ABAQUS (Section 4.2.1)16 gives a numerical solution for pure bending of a pipe section. The geometrical parameters

1.2 - /

,:

EI 2 0.6 -- x

-St

P

9 0.4 -~

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6

s

-Crushing, kinematic ~ Crushing, isotropic

Fig. 8. Comparison of bending and crushing contribution.

A simplified model of Brazier effect in plastic bending of cylindrical tubes 27

500

400

m 300

!ik

I F?!

g 200

100

0

0.0 1.0 2.0 3.0 4.0 5.0

Strain, %

Fig. 9. Material stress-strain diagram for ABAQUS example 4.2.1.

are R = 0.2 m and t = 0.01 m, giving R/t = 20. The stress-strain diagram is shown in Fig. 9. A power-law fit of this curve gives A = 69.82, cry = 25 X lO’N/m*, ey = 0.002 and n = 0.399. The above values were used in eqns (58) and (59) to calculate bending resistance and the section ovalization as a function of the increasing curvature.

A comparison of the present solution with the result of ABAQUS calculations is presented in Figs 10 and 11. The present solution with kinematic hardening predicts correctly the shape of the moment-curvature curve and with only 1% error of the maximum bending moment. For comparison, the solution corresponding to isotropic hardening is also shown in

1.6

1.4

Fig. 10 as a dotted line. The maximum bending moment occurs at KRIM,Mo=max = 0.021. Where the ovalization curve in Fig. 11 is shifted by this amount, the agreement between the present theory and FEM calculations is excellent.

11 CONCLUSIONS

As a segment of a thick tube bends to a uniform and diminishing radius, the initial circular cross-section of the tube deforms into an elliptical shape. The lateral or crushing deformation of the tube requires certain energy input, which increases its bending resistance. At the same time, the flattened cross-sectional shape offers smaller bending resistance. The combined effect of these two distinct mechanisms together with work-hardening properties of the material give a typical load vs curvature (rotation) characteristic with a rising part, smooth maximum, and a descending part.

Various models were proposed in the past to describe the above effect and to predict the peak bending moment. Most of the analyses were based on considerations of local equilibrium of the shell element. In this paper an alternative approach was used, based on a minimum postulate. This postulate may, but does not have to, be interpreted as an application of the upper bound theorem in plasticity with a continuously updated configuration.

The effect of strain-hardening is included in an iterative way by calculating strain histories and adjusting stress levels in two load-resisting mechan- isms. A good correlation with a numerical solution obtained by means of a non-linear general purpose code, ABAQUS, proves the correctness of the new global approach. A closed-form solution is derived for the moment-curvature characteristic and a peak

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.06

KR

1 0 ABAQUS -“-Kinematic __ Isotropic

Fig. 10. Comparison of analytical results with ABAQUS calculations-moment.

28 T. Wierzbicki, M. V. Sinmao

2r 0.6 -~ 2R

0.4 .- 0

0.2 --

0.0 7

0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

KR

I 0 ABAQUS -2rlZR

I

Fig. 11. Comparison of analytical results with ABAQUS calculations-deforming radius.

bending moment, which includes all the geometrical and material parameters of the problem. The present result can be used for a rapid and accurate estimate of plastic bending response of thick metal tubes.

REFERENCES

1.

2.

3.

4.

5.

6.

I.

Brazier, L. G., On flexure of thin cylindrical shells and other thin sections. Proceedings of the Royal Society of London Series A, 1927,116,104-114. Jirsa, J. O., Lee, F. K., Wilhoit, Jr., J. C. and Merwin, J. E., Ovaling of pipelines under pure bending, OTC paper no. 1569, 1972. Ueda, S., Moment-rotation relationship considering flattening of pipe due to pipe whip loading. Nuclear Engineering Design, 1985, S&251-2.59. Gerber, T. L., Plastic deformation of piping due to pipe-whip loading. ASME paper 74-NE-l. Yu, T. X., Reid, S. R. and Wang, B., Hardening- softening behaviour of tubular cantilever beams. International Journal of Mechanical Sciences, 1993, 35, 1021-1033. Ades, C. S., Bending strength of tubing in the plastic range. Journal of Aeronautical Sciences, 19.57, 605-610. Munz, D. and Mattheck, C., Cross-sectional flattening of pipes subjected to bending. International Journal of Pressure Vessels and Piping, 1982, 10, 421-429.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Prinja, N. K. and Chitkara, N. R., Post-collapse cross-sectional flattening of thick pipes in plastic bending. Nuclear Engineering Design, 1984, 83, 113-121. Zhang, L. C. and Yu, T. X., An investigation of the Brazier effect of a cylindrical tube under pure elastic-plastic bending. Znternational Journal of Pressure Vessels and Piping, 1987, 30, 77-86. Post-collapse bending behaviour of steel pipes, National Nuclear Corporation Ltd. UK, Interim Report, 1983. Kiriakides, S., Yeh, M.-K. and Roach, D., On the determination of propagation pressure of long circular tubes. Journal of Pressure Vessel Technology Trans- actions of the ASME, May 1984,106,150-159. Hopkins, H. G., On the behaviour of infinitely long rigid plastic beams under transverse concentrated load. Journal of the Mechanics and Physics of Solids, 1955, 4, 38-52. Sawczuk, A., Yield surfaces. Solid Mechanics Division, University of Waterloo, Ontario Canada, Technical Note No. 1, March 1971. Bauschinger, J., Ueber die Veranderungen der Elas- tizitatsgrenze und der festigkeit des Eisens und Stahls durch Strecken, Quetschen, Erwarmen Abkuhlen und durch oftmals wiederholte Belastung, Mitt: Mech-Tech Lab, XIII Munchen, 1886. Suresh, S., Fatigue of Materials. Cambridge University Press, Cambridge, 1991. ABAQUS, Example Problems Manual, Version 4.8, Hibbitt, Karlsson, and Sorensen, Inc., California, 1989, example 4.2.1.