SUBJECTED TO COMBINED LOADING

1

PLASTIC ANALYSIS OF DENTED TUBESSUBJECTED TO COMBINED LOADING

by

MYUNG SUNG SUH

B.S., Seoul National University, Korea

(1977)S.M., Massachusetts Institute of Technology

(1984)

SUBMITTED TO THE DEPARTMENT OF OCEAN ENGINEERINGIN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

MAY 1987

Massachusetts Institute of Technology, 1987

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Accepted by......

/ f Department

Signature redacted...........Prof ess

.... Signature redacted

of Ocean Engineering

May 20, 11987..............M ..... ,.......

or Tomasz Wierzbicki

Thesis Supervisor

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2

Plastic Analysis of Dented TubesSubjected to Combined Loading

byMyung Sung Suh

Submitted to the Department of Ocean Engineering on May 20, 1987 inpartial fulfillment of the requirements for the degree of Doctor of Philosophy.

Abstract

A theoretical analysis is presented of large plastic deformations of tubes subjected tocombined loading in the form of lateral indentation, bending moment and axial force.A considerable effort is made to develop and justify an accurate and yetmathematically simplified model of the shell. The model is capable of describing withsome realism local damage of tubes undergoing large strain, rotation and shapedistortion. The load-indentation characteristics of tubes are shown to dependstrongly on the magnitude of the bending moment and axial force applied to the tubeends. The calculations reveal that the resistance of the tube to lateral indentationand thereby the energy that the tube can absorb are sharply diminished withchanging the direction of axial force (or bending moment) from positive to negative.With increasing negative axial force (or bending moment) the tubes are found to losestability and fail by local plastic sectional collapse well above the negative fullyplastic axial force (or bending moment). The residual strength of dented tubes is alsodetermined based on the results of plastic instability load. The three-dimensional fullinteraction surfaces are constructed by combining each interaction curve forprescribed axial force and bending moment. The effect of shear deformation isdiscussed in an approximate way. Finally, the pinching of a tube as a symmetricdenting problem is discussed. The present results are shown to give good correlationwith existing documented experiments reported in the literature.

Thesis Supervisor : Tomasz Wierzbicki, Sc.D.

Title : Professor of Applied MechanicsDepartment of Ocean Engineering

3

Acknowledgements

I sincerely thank Professor Wierzbicki for his guidance and support during the course

of this study and for the opportunity to contribute to an interesting field of research.

Our frequent interaction and free exchange of idea assured a lively pace and

meaningful direction for this research. It has been a most rewarding experience.

The suggestions and perspective provided by Professor Karr, Professor Moshaiov and

Professor Gibson are also gratefully acknowledged, as are the frequent

communications with Dr. Abramowicz.

Special thanks are due to Professor Ogilvie and Patti LeBlanc-Gedney for getting me

stretched at MIT and for their continuous support.

I am grateful to Korean Presbyterian Church in Boston for giving me hope and

spiritual conviction.

I also thank my parents, parents-in-law, relatives and friends for their help and

encouragement.

Finally, for their inspiration to me, and for her loving patience and understanding

during these years of study, this thesis is dedicated to my children, Jun Su and

Esther, and to my wife, Kyung Hee.

The Lord is near to all who call on him,to all who call on him in truth.

He fulfills the desires of those who fear him,he hears their cry and saves them.

( PSALM 145 : 18-19 )

4

Table of Contents

Abstract 2Acknowledgements 3Table of Contents 4List of Figures 6Notations 101. Introduction 142. Formulation of the Problem 193. Simplified Shell Model 24

3.1. Crushing of Rings 293.2. Extension of Generators 44

4. Indentation of Tubes Subjected to Lateral Load 544.1. Indentation problems for different boundary conditions 54

4.1.1. Tube with full end fixity 544.1.2. Tube free to move axially but restricted from rotation 594.1.3. Tube with rotational and tranlational freedom 71

4.2. Discussion and comparison with Experiments 815. Indentation of Tubes Subjected to Combined Loading 93

5.1. Plastic Instability of Compressed Tubes 955.1.1. Lateral Load/Axial Force Interaction Based on Approximate 95

Global strain rate5.1.2. Lateral Load/Axial Force Interaction Based on Exact Global 102

Strain Rate5.2. Plastic Instability of Tubes Subjected to Bending 112

5.2.1. Lateral Load/Bending Moment Interaction Based on Approximate 112Global Strain Rate

5.2.2. Lateral Load/Bending Moment Interaction Based on Exact 121Global Strain Rate

5.3. Tubes Subjected to General Combined Loading 1255.3.1. P-M-N Interaction Based on Approximate Global Strain Rate 1255.3.2. P-M-N Interaction Based on Exact Global Strain Rate 136

6. Residual Strength of Dented Tubes 1517. Estimation of Shear Effect 155

5

8. Pinching of Tubes8.1. Introduction8.2. Theoretical Predictions

8.2.1. Model I8.2.2. Model II

8.3. Comparison between Experimental Results and Theoretical Predictions9. Conclusions and RecommendationsReferencesAppendix A. Crushing Force for an Unsymmetrically Deformed Ring

ModelAppendix B. Rate of Extensional Energy for an Unsymmetrically

Deformed Ring Model

B.1. Calculation of wO/R and wO*O/R2

B.2. Calculation of rate of extensional energyAppendix C. Rate of Extensional Energy

Deformed Ring Model I

C.1. Calculation of wO/R and wO*O/R2

C.2. Calculation of rate of extensional energyAppendix D. Rate of Extensional Energy

Deformed Ring Model H

for a Symmetrically

for a Symmetrically

D.1. Calculation of wO/R and wO*O/R2

D.2. Calculation of rate of extensional energyAppendix E. Lateral Load/Axial Force Interaction Formula by

Normality Requirements

161161163163172182191194199

201

201

204205

205

207208

208211213

List of Figures

Figure 1-1:

Figure 2-1:

Figure 3-1:Figure 3-2:

Figure 3-3:

Figure 3-4:

Figure 3-5:

Figure 3-6:

Figure 3-7:

Figure 3-8:

Figure 3-9:

Figure 3-10:

Figure 3-11:Figure 3-12:

Figure 3-13:Figure 3-14:Figure 3-15:Figure 3-16:Figure 4-1:

Photographs of a local plastic damage of a tube caused byunsymmetric indentation and symmetric pinching (afterSmith [37] and Montgomery [18]), respectively).Three components of external loading of the tube andcorresponding generalized velocities.Geometry of the plastically deforming zone.Present computational model of the shell consisting of asystem of rings and generators.Non-dimensional crushing strength of a ring versus denteddepth for n = 1.Non-dimensional crushing strength of a ring versus denteddepth for n = 2.Non-dimensional crushing strength of a ring versus denteddepth for n = 0.5.Non-dimensional crushing strength of a ring versus denteddepth for #0 = r/2.Initial and intermediate shapes of deformed rings with -0 =

?r/2 and n = 1.Intermediate shapes of deformed rings with 00= r/2 and n

= 4.Initial and intermediate shapes of deformed rings with #=? and n = 1.Initial and intermediate shapes of deformed rings with $=0 and n = 1.Generalized velocity components.Displacements of material points at the symmetry plane x= 0 as a function of the circumferential coordinate a atthree values of dented depth.Exact global strain rates for different $,.Exact global strain rates for different n.Exact global strain rate and its linear approximation.Exact global strain rate and its quadratic approximation.

Dependence of the indentation load with different values of nfor a fixed position #0.

20

2527

34

35

36

37

38

39

40

41

4548

49

50515260

7

Figure 4-2:

Figure 4-3:

Figure 4-4:

Figure 4-5:

Figure 4-6:

FigureFigure

4-7:4-8:

Figure 4-9:

Figure 4-10:

Figure 4-11:

Figure 4-12:

Figure 4-13:

Figure 4-14:

Figure 4-15:

Figure 4-16:

Figure 4-17:

Figure 5-1:

Figure 5-2:

Figure 5-3:

Figure 5-4:

Figure 5-5:

Dependence of the indentation load with different values of

00 for a fixed power n.Dependence of the extent of dented region with differentvalues of 40 for a fixed power n.

Dependence of the extent of dented region with differentvalues of n for a fixed position #0.Reduction of the rate of energy dissipated by generators forthe tube with freely-sliding boundaries, N = 0.Distribution of global strain rate for freely-sliding boundarycondition.Residual deformations and stresses after denting.Distribution of global strain for a tube with rotational andtranslational freedom.Distribution of exact global strain for a tube with twointersection points for freely-rotating boundary condition.

Distribution of exact global strain for a tube with fourintersection points for freely-rotating boundary condition.Comparison of exact and approximate load-indentationcharacteristics for three types of boundary conditions.Comparison of exact and approximate extents of dentedregion for three types of boundary conditions.Theoretical and experimental profile of the leadinggenerator in the damaged zone.Calculated and experimentally observed shape of the locallydamaged zone in a dented tube.Load-indentation characteristics for a freely-sliding tubeexperimental curve due to Smith [34] shows unloading andreloading.Actual stress-strain curve of the material and rigid-perfectlyplastic idealization at the level of average flow stress.Correlation of the present theoretical solutions with theresults of full scale test reported by Smith [34].

A construction illustrating changing sign of global strain rateand stresses to ensure development of prescribed axial forcein the tube with bo = 0Normalized lateral load/axial force interaction with bo = 0based on quadratic global strain rate.Variation of intersection points for different axial forcesbased on exact global strain rate with bo = 0Lateral load/axial force interaction with b0 = 0 for differentdented depths.Effect of radius to thickness ratio and imperfections on thebuckling strength of cylindrical shell (after Almroth andBrush).

61

62

63

65

67

6874

76

77

80

82

85

86

88

89

92

98

101

104

107

108

8

Figure 5-6:

Figure 5-7:

Figure 5-8:

Figure 5-9:

Figure 5-10:

Figure 5-11:

Figure 5-12:

Figure 5-13:

Figure 5-14:

Figure 5-15:

Figure 5-16:

Figure 5-17:

Figure 5-18:

Figure 5-19:

Figure 5-20:

Figure 5-21:

Figure 5-22:

Figure 5-23:

Figure 5-24:

FigureFigure

6-1:7-1:

Load-indentation characteristics of the dented tube forvarious values of axial force.Quadratic global strain rate profile with 6, = 0 depending

on the size of rotation rate and the magnitude of 66/ andRb.

0

Normalized lateral load/bending moment interaction with nO= 0 based on quadratic global strain rate.Lateral load/bending moment interaction with n0 = 0 for

different dented depths.Load-indentation characteristics of the dented tube forvarious values of bending moments.Quadratic global strain rate profile depending on the size ofrotation rate and the magnitude of bi/ and Rbo.

Bending moment/axial force interaction based on quadraticglobal strain rate influenced by lateral load.Normalized three-dimensional interaction surfaces subjectedto combined loading based on quadratic global strain rate.Normalized lateral load/bending moment interaction withN = 0 based on quadratic global strain rate.Normalized lateral load/axial force interaction with M = 0based on quadratic global strain rate.Exact global strain rate profile subjected to generalcombined loading (M/M = 0.51 and N/N = 0.66).

Exact global strain rate profile subjected to generalcombined loading (M/M, = -0.39 and N/N, = 0.74).

Exact global strain rate profile subjected to generalcombined loading (M/M, = -0.669 and N/Np = 0.3).

Exact global strain rate profile subjected to generalcombined loading (M/M = 0.852 and N/N, = 0).

Exact global strain rate profile subjected to generalcombined loading (M/M = 0 and N/N = 0.3).

Bending moment/axial force interaction based on exactglobal strain rate for different dented depths.Lateral load/bending moment interaction with N = 0 basedon exact global strain rate.Lateral load/axial force interaction with M = 0 based onexact global strain rate.Three-dimensional interaction surfaces subjected to generalcombined loading based on exact global strain rate.

Residual strength of dented tube versus dented depth.Conceptual model of a tube showing symmetric sectionalcollapse and unsymmetric collapse.

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153156

9

Figure 7-2:

FigureFigureFigureFigureFigureFigure

FigureFigureFigureFigureFigureFigureFigure

FigureFigure

8-1:8-2:8-3:8-4:8-5:8-6:

8-7:8-8:8-9:8-10:8-11:8-12:8-13:

8-14:8-15:

Figure 8-16:

Figure 8-17:

Figure 8-18:

Figure 9-1:

Transition from symmetric to unsymmetric sectional collapsethrough simple shear.Square ring model and diamond shape ring model.Geometry of deformed ring model I.Displacement of material points at midspan of ring model I.Global strain rate for different dented depths of ring model I.Intermediate deformed shapes of ring model I.Variation of intersection points for different dented depthswith N = 0 for ring model I.Geometry of deformed ring model H.Crushing strength of ring model 11.Intermediate deformed shapes of ring model H (n = 0.01).

Intermediate deformed shapes of ring model H (n = 0.5).Displacements of material points for ring model H.Global strain rate for ring model II.Load-indentation characteristics of a clamped tube for ringmodel II.Extent of dented region of a clamped tube for ring model II.Comparison between theoretical predictions andexperimental measurements extent of dented region of aclamped tube.Comparison between theoretical predictions andexperimental measurements extent of dented region of afreely-sliding tube.Comparison between theoretical predictions andexperimental measurements load-indentation characteristicsof a clamped tube.Comparison between theoretical predictions andexperimental measurements load-indentation characteristicsof a freely-sliding tube.

Localized shear-affected zone of dented tube.

158

162164166167168171

173176177178180181183

184187

188

189

190

193

g

rate of internal energy dissipation

r1 , r2

10

Notations

gravitational acceleration

nondimensional radii defined in Figure 3-1

coordinate in circumferential direction

length of each are

thickness of circular tube

axial displacement of a generic point

u at x =

vertical displacement of a generic point

indentation depth at midspan

coordinate in axial direction

vertical distance between material point and centroidal axis

for a ring

diameter of undeformed tube

rate of total crushing energy

rate of external work

rate of total extensional energy

s

sp, s 2, s 3

t

U

U 0

w

w

x

z

D

Erush

gen

11

k shear

L

instantaneous crushing force

X

M

rate of shear energy

total span of tube

total ring circumference

external bending moment

critical bending moment causing plastic instability

stress couple tensor

fully plastic bending moment of the wall

plastic moment capacity of undeformed cross section of the tube

external axial force

critical axial force causing plastic instability

threshold axial force separating strengthening and

softening range

fully plastic axial force of the wall

plastic force capacity of cross section of the tube

stress resultant tensor

shear force

lateral concentrated load

M

N

NC

Nth

N0

Np

NxO

P

P C(we)

R

S

v

12

radius of undeformed tube

radii of deformed tubes (defined in Figure 3-1)

surface area of continuously deforming region

velocity of moving plastic hinge

tangential velocity of each moving plastic hinge

angles (defined in Figure 3-1)

length of hinge line

indentation depth

rate of indentation depth

average strain

average through-thickness bending strain

axial strain rate

shear strain rate

circumferential strain rate

strain rate due to local denting

strain rate due to uniform compression or extension

strain rate due to overall bending and rotation of cross section

F

6

av

Eb

EX

X0

08

xOg

v 11v 2

13

b

K0

Zs

K

K 0

K6 0

rate of rotation

rate of curvature

initial curvature of undeformed tube

change in curvature

rate of longitudinal curvature

rate of twist

rate of circumferential curvature

half-length of dented region

mass density of tube material

average flow stress

yield stress

current position of lower plastic hinge

initial position of lower plastic hinge

rate of relative rotation on both sides of the hinge

jump in the enclosed quantity across stationary or moving hinges

0/a t ( ), differentiation with respect to time

(or i/0 # ( ) in Appendix A and Appendix B)

p

0

0

y

[1]

( )

14

Chapter 1

Introduction

Moderately thick fabricated or cold drawn metal tubes encountered in various

industrial applications are often subjected to accidental loading in addition to normal

service loading. The service loads consist of axial tension or compression, bending

moment, shear loading and sometimes twisting moment. The accidental loading may

be of various origin and typically applied in the transverse direction to the

longitudinal axis of a tube. Examples of such loading are minor collisions of supply

boats with multi-story offshore oil platforms, impact caused by dropped objects,

mishandling during launching or installation of marine structures, ice scouring of

Arctic pipelines, collisions of offshore installations with moving ice features and

hydrodynamic wave impact. The combination of service and accidental loads may

lead to a severe local shape distortion of tubes, loss of axial and bending strength and

stiffness and catastropic collapse of a given member.

From the point of view of economy of design and safety of operation it is

important to be able to predict the response of tubes under all possible combination

of external loads. With the exception of Ref.f18] and 1201, in all previous analyses of

similar problems only one component of external loading was applied to the tube at a

time. For example, tubes were subjected to either indentation or axial compression.

The problem of a combined loading, which often occurs in many practical situations,

has not been studied theoretically or experimentally in the literature. Also the effect

of different boundary conditions on a local crushing strength of tubulars appears not

to be fully understood.

The objective of the present thesis is to get an insight into the mechanisms of

15

plastic deformation of tubes undergoing large shape distortion and sectional collapse

and to derive the load-indentation characteristics of tubes subjected to lateral

concentrated loading under a variety of boundary conditions and combined loading,

as well as local imperfection of a given member. The present solutions can be useful

in various industrial applications, for example, in the assessment of local damage and

residual strength of offshore platforms subjected to accidental loads. Typical shapes

of a locally damaged zone in tubulars are shown in Fig. 1-1.

In contrast to the existing analyses of similar problems, all predictions in the

present thesis are made on a purely theoretical basis. Taby and Moan [41]

introduced an empirical factor in their formula for the axial stresses in the dented

zone. Smith in a series of publications [33-37] calculated the residual strength of

dented tubes using the concept of a reduced strength and stiffness of shell elements in

the damage affected zone. The reduction coefficient was then determined from the

best fit of the experimental data. In the computational model adopted by the above

mentioned authors, the radius of the tube was considered as constant with increasing

dent depth. This is a reasonable assumption for relatively shallow dents [45], but

limits the applicability and accuracy of the respective solutions for deeper dents.

Ueda and Rashed [43] reported a good agreement of the theoretically calculated

ultimate bending moments of tubes having dented zones with the independently

performed tests. In all cases, the maximum bending moment occurred early in the

loading process when the sectional collapse was still very small.

Experiences with the problem of propagating buckles in pipelines indicate that

much of the strength of a tube comes from the ring mode of deformations [49].

Plastic collapse of rings was studied among others by Reid and Reddy [28]. Also

experiments performed on short cylinders at the University of Manchester Institute of

Science and Technology (UMIST) proved that such tubes resisted transverse loads by

predominantly circumferential bending [42, 46, 47]. In addition to the already

mentioned experimental studies on model and full scale tubes with dents, performed

in Norway [38], [40], England [33-37] and Japan [43], tests on fabricated tubes with

16

a)

b)

Figure 1-1: Photographs of a local plastic damage of a tube caused byunsymmetric indentation and symmetric pinching

(after Smith [37] and Montgomery [18]), respectively).

17

large diameter were carried out in Canada [2, 31] and the United States [24, 32].

Birkemoe and Sato [2, 31] tested large diameter tubes in compression (D/t>70) and

observed a Odiamond" local failure mode on the compression side of the tube.

Sherman reported on a very thorough experimental study in which tubes with D/t

ranging from 30 - 70 were subjected to pure bending. In all cases the shape of the

locally damaged zone was similar and resembled much that obtained in the

indentation tests by a rigid punch. Ostapenko in a series of papers [24, 25]

determined experimentally design curves for tubes subjected to compression. He

found that tubes with D/t<50 developed their full compressive strength N . Thep

effect of residual strength of dented tubes was not studied by Ostapenko.

The intention in the introductory chapter is to discuss those references which

are directly related to the development of the present computational model of the

tube rather than to survey the vast literatures concerned with the behavior of

damaged tubes. For a comprehensive review of the state-of-the-art in this field, the

reader is referred to the paper by Ellinas and Valsgard [9]. A great deal of good

work has been published on the plastic response of thinner shells [21, 45]. As thin

tubes are outside the scope of the present study, the relevant publications are not

commented upon.

In the early studies on local tube indentation, Morris and Calladine [19] and

Soreide and Amdahl [38] emphasized the importance of bending and extensional

deformation of the affected shell elements. Based on those observations, a simple

computational model was introduced in Ref. [20]. The model consisted of a system of

mutually interacting rings and generators. In this work, the importance of boundary

conditions on the resistance of tubes to denting was pointed out and the existence of

plastic instability of compressed tubes with an unsymmetric collapse mode was

predicted. However, along with other simplifications, a circular section was replaced

in reference [20] by an equivalent square section. The present thesis considerably

improves and extends the previous findings still preserving the appealing simplicity of

closed-form solutions.

18

The thesis is organized in the following way. First all assumptions are carefully

spelled out in Chapters 2 and 3. The finite strains and rotations of a cylindrical shell

is described by an accurate and yet mathematically tractable model. Then each of

the basic mechanisms of the tube resistance, i.e., crushing of rings and extension of

generators are analyzed in great detail. Further, the problems of dented tubes

subjected to lateral load are worked out in Chapter 4 for three typical boundary

conditions and compared with the existing experimental data. The task of predicting

the crushing resistance of tubes subjected to an arbitrary axial force or bending

moment combined with lateral load is performed in Chapter 5. The analysis has led

to the discovery of a new phenomenon not previously reported in the literatures. It is

found that the resistance of tubes to lateral load is reduced dramatically with the

increasing amount of axial compression or negative bending moment up to the point

of instability. The instability may occur well before the fully plastic axial force or

bending moment is obtained. The interaction between lateral load, axial force and

bending moment is studied and the three-dimensional interaction surfaces under

combined loading are constructed. The solutions for plastic instability contain special

cases, the prediction of the residual strength of tubes with initial dents (Chapter 6).

The influence of shear deformations on the plastic response is discussed in an

approximate way in Chapter 7. In Ref. [18] Montgomery reported on a

comprehensive experimental study on pinch loaded tubes in which the influence of

boundary conditions on the crushing strength is studied. A discussion of pinched

tubes as a symmetric denting problem is contained in Chapter 8. Finally, conclusions

and recommendations are presented in Chapter 9.

10

Chapter 2

Formulation of the Problem

The present concern is a class of problems for thick and medium thick tubes in

which large sectional collapse takes place under the action of a rigid indentor,

bending moment, axial compression or combination of the external loading, Fig. 2-1.

The magnitude of the local dent is not restricted and theoretically can be as large as

the diameter of the tube. For shells with the diameter to thickness ratio D/t < 50,

which are of interest in offshore applications, the maximum lateral displacement can

thus be fifty times larger than the wall thickness of the tube. Clearly, neither the

infinitesimal nor so-called 'moderately large deflection" theory of cylindrical shells

are applicable in this case. The deformation process of the shell will be described

here using the updated Lagrangian formulation by keeping track of the current

geometry of the shell. In contrast to the advanced numerical codes with capabilities

of handling large strain and displacement, the variable shell geometry will be updated

analytically, using continuously varying functions with a few free parameters. Such

an approach not only leads to a highly desirable closed-form solution to the problem

of tube indentation, but also provides an insight into the mechanism of shell

deformations with severe unsymmetric shape distortion.

Constitutive Behavior

Rigid-perfectly plastic material idealization will be adopted in conjunction with the

associated flow rule. The uniqueness of the rate of energy dissipation E = is then

preserved (a and i denote the work conjugate stress tensor and strain rate tensor,

respectively). A concept of a rigid-plastic solid provides a good representation of the

real physical behavior of the material in the range of large strains. The average flow

20

I

M2 Oa

Figure 2-1: Three components of external loading of the tube andcorresponding generalized velocities.

U0

IN

21

stress ao, which lies somewhere between the yield stress and the material's ultimate

strength, reflects the strain-hardening effect in an approximate way.

Indeed, the strains in the dented regions of the tube can reach considerable

values. Experiments show that in the course of the loading process the shell

curvature in the circumferential direction changes from the initial uniform KO= 1/R

to the highly localized which can be equal to tc= 6/R or more. The change in

curvature AZc = K - tc is then of the order of Ac ~ 5/R or Aic ~ 1/5t for a tube

with D/t = 50. Recalling that the average through-thickness bending strain is

(b = t/2 Ai, Eb can easily attain 10%. Similarly high strains may be developed in

the axial direction of the affected shell element. This strain can be estimated from the

approximate nonlinear formula EX= 1/2 (dw/dx) 2 where w is the lateral deflection of

shell element and x is axial coordinate. Taking in the first approximation the shape

of the dent to be linear, the formula for the axial strain becomes EX - 1/2 (w/ )2

where is the half-length of the dented zone. For severely distorted shells the central

deflection may be set equal to the shell radius w = R. Experiments show that at this

stage of the deformation, the extent of dented region equals = 3 - 5R: so that E,

3%. A simple calculation indicates that the strains in the dented region of the tube

can be one to two orders of magnitude higher than the maximum elastic strains that

the metal tube can tolerate. This observation justifies the neglect of elastic strains

and the use of the rigid-plastic material idealization in the present problem.

Equilibrium

The statement of global equilibrium is expressed via the principle of virtual work or

virtual velocity

ezt int (2.1)

The left hand side of the above equation represents the rate of work of external

forces on the corresponding velocities. In the absence of a twisting moment, the

expression for the rate of change of external energy is given by

22

Eex =P + N6+ 2N +2 i (2.2)

The first term is due to the lateral crushing of the member. The second term results

from the product of an axial force and the corresponding velocity of a neutral axis of

the shell. The third term is due to the global bending. In the case of plastic shells the

rate of internal work Eit is given by the sum of contributions due to continuous

deformation field and discontinuous velocity field along the stationary or moving

plastic hinge lines

E1 ~ ~ (M + Na a) dS + fi ) M a]1)dF (2.3)

where dS and dr denote respectively the current deformed surface element and hinge

line element and the symbol [ I denotes a jump in the corresponding quantity across

stationary or moving hinge line. The surface integration should be extended over the

plastically deforming part of the shell. The symmetric components of the stress

couple tensor (bending moments) and stress resultant tensor (membrane forces) are

denoted respectively by MO and N'O. The corresponding components of the

generalized strain rate tensors are curvature rates k and extension rates ;,,. The

rate of work at the "i-tho hinge line of the length IP is equal to the fully plastic

bending moment MO) - at 2/4 times the relative rotation rate 90) of a shell element

on both sides of the hinge. The yield condition, binding the internal forces in Eq.

(2.3), may be simplified in order to get either bounds or approximate solutions. A

rigorous analysis of the full-plastic behavior of a shell element subjected to stress

couple and stress resultant gives a complicated interaction formula; but in this type

of large shape distortion it will be satisfactory to use the approximate yield locus as

shown in Ref. [1].

In Ref. [20] the global equilibrium equation Eq. (2.1) and the normality

23

requirements served to eliminate unknown generalized velocity components in Eqs.

(2.2) and (2.3). In this thesis a different approach is used, which is more

straightforward and simpler to derive the interaction formula F(P,M,N,b) = 0. The

force equilibrium and the moment equilibrium in the axial direction of the tube in

addition to the global equilibrium equation (2.1) are considered. The present

solutions are derived by introducing a suitably chosen sub-class of the velocity fields

with only few degrees of freedom into the deformed geometry of the locally collapsing

tube. Specifically, there are three degrees of freedom, two in cross-sectional plane

(RI, R'2 ) and one in the axial direction (i).

Geometrical Relations

With reference to the current deformed configuration, the relations between velocities

and strain rates and curvature rates are linear and formally have the same form for

displacements and strains as those of the classical linear shell theory. In the present

description the difficulty in dealing with geometrical nonlinearities has been shifted to

the procedure of continuously updating the current shape of the shell in the dented

region and the necessity of calculating first and second quadratic forms at each point

of the shell. For the sake of obtaining closed-form solutions, the formulas for the

relevant components of the strain rate tensor will be derived using a simplified model

of the shell to be developed in the next section. This model effectively decouples the

problem of two-dimensional shell geometry into a set of one-dimensional problems,

each much simpler to deal with analytically.

24

Chapter 3

Simplified Shell Model

The present computational model is based on the following premises:

" A careful inspection of actually damaged tubes reveals that the plasticallydeforming zone undergoing severe shape distortion is restricted to fewdiameters of the shell on both sides of the dent center. It is assumed herethat the extent of the locally damaged zone is finite and is denoted by 2 ,Fig. 3-1. The length is unknown and is considered variable during theloading process. This assumption is fully compatible with the presentrigid-plastic material idealization.

" The cross-section at which the deformed part of the shell joins theundeformed part is taken to be plane and circular. Therefore, noovalization and warping of the tube exist beyond the dent-affected zone.Thomas, et. al. [42] reported severe ovalization and warping in shortdented tubes. The observed mode clearly reduces the amount of shearand extensions in the shell walls. For longer tubes the process ofovalization and warping is suppressed by the presence of the continuingtube, but this tendency can never be eliminated. This model tube will beslightly stiffer than the actual tube because certain deformation modeshave been eliminated.

* Inside the plastically deformed zone, the ovalization and its extreme form,the unsymmetrical shape distortion are permitted. However, at the actualplane of symmetry the axial displacements are identically zero. Sincethere are no warpings at both ends of a relatively short tube section0 < x < , it is reasonable to assume that in addition to crushing allcross-sections undergo rigid-body translations and rotations.

The above observations lead us to the present computational model of the shell.

The model consists of a series of unconnected rings or slices and a bundle of

unconnected generators, Fig. 3-2-a and 3-2-b. The rings and generators are loosely

connected, as shown in Fig. 3-2-c. This means that lateral deformations in both

W(X,a)

W (x,0) = WCta)

W(0,a) = WO(a)

W (0,0) = a

Geometry of the plastically deforming zone.

25

x

R2 a

R, 0 <

R ds

WC

Figure 3-1:

26

directions should be compatible. The resulting deformation shown in Fig. 3-2-d

resembles well the locally collapsed sections of actual tubes.

The rings are inextensional in circumferential direction, 26 0= 0, and the energy is

absorbed predominatly by circumferential bending in the continuous deformation

field and at the stationary or moving plastic hinges. Thus, the so-called crushing

energy per unit width of the ring consists of the terms

crush = tl 'c ds +0 ] (3.1)

where ,, is a circumferential curvature rate and s is a coordinate in circumferential

direction of the ring, t = 27rR is the total ring circumference and the summation is

extended over the number of active plastic hinges. The total crushing energy in the

dented zone is obtained by integrating Ecrush over the length of the dented zone

Zcrush = 2 f E h dx - (3.2)

Here, the circumscribed yield locus on the exact one will be used

INaI = No = t (3.3)

a t 2

|MA = M = - (3.4)0 4

where t is the thickness of the shell; at yield one or both of these relation is satisfied.

Therefore, the bending resistance of the ring described by M0 has been taken to be

independent of axial resistance of generators No. The unsymmetric crushing of a

plstic ring is a one dimensional problem and can be solved without calculating

quadratic forms of shell element in the deformed configuration.

27

a)

C)

b)

d )

Figure 3-2: Present computational model of the shell consisting of a systemof rings and generators.

28

The generators are treated in the present model as rigid-plastic beams which can

bend and stretch or compress as the depth of the dent increases. However, the

change in the longitudinal curvature of generators k, is much smaller than the

change in the circumferential curvature of rings k,,. In this calculation, the term

MX.,k will be neglected. The contributions of generators to the overall energy

dissipation reduce then to

E, = 2j 0 x , I dx (3.5)

where No = o t is the fully plastic membrane force in the generator of unit width.

Since the axial strain rate ;, can be tensile in a part of the circumference and

compressive in the remaining part, the absolute sign is introduced to ensure the non-

negativeness of E n. By assembling the dissipation of all generators an expression is

obtained for the total rate of work of the deformed part of the shell in the

longitudinal direction.

S= J 27RZ ds (3.6)gen fo gen

The compatibility of deformation of rings and generators is ensured by requiring that

the lateral displacement of the two types of one-dimensional structures be the same,

Fig. 3-2. The only components of the general expression for the internal energy

dissipation not accounted for by the present model are twisting and shear energy

MXkX0 and Nxexo' The calculations of those components require consideration of full

shell geometry. In the present calculations the energy due to the twisting deformation

is neglected. The twist rate kX8 is believed to be small compared to k,,. On the other

hand, the shear strain rate Xo does always exist in a cylindrical shell undergoing large

unsymmetric deformations. The dissipation due to shear is given by

29

shear ~ Is NX8 ;zO dS (3.7)

In view of difficulties in evaluating the shear strain rate even in the present simple

model, the shear energy will be calculated using a still more crude model of the

square tube with moving plastic hinges. The model will be described in Chapter 7.

3.1. Crushing of Rings

The experience with locally dented tubes and also observation of experiment

indicate that the cross-sectional shape of the tube in the damaged zone has a flat

upper part, as shown in Fig. 1-1. Within the present rigid-plastic material

idealization, such a flat segment can only be produced by moving plastic hinges.

Suppose a plastic hinge sweeps through the material points of the ring. From the

continuity of displacements it follows that the rate of rotation at the hinge [0] is

linearly related to the change in curvature on both sides of the hinge [K] = K1 -

(see for example Hopkins [14])

[9] = V[K] (3.8)

The moving hinge can thus impose, remove or change the curvature of the ring as it

passes through the material points. The concept of moving plastic hinges has been

proved useful in the past in the dynamic analysis of rigid-plastic straight beams [26],

[39]. More recently it has led to the improved solutions for axi-symmetric of quasi-

static crushing of cylinders [48] and propagating buckles in pipelines [49].

The computational model of the ring is shown in Fig. 3-1. The ring is initially

of radius R and thickness t. During the deformation process, the ring flattens out

and the amount of central displacement is denoted by w.. At any stage of the

crushing process, the ring consists of the top flat part and three arcs. The larger

radius R, increases from the initial value R as the deformation progresses while the

30

radii of two remaining arcs shrink. The flats and arcs are separated by four moving

plastic hinges. Initially, the length of the flat segment is zero and the top central

hinge splits into two hinges traveling in the opposite directions. Two other hinges are

also symmetric with respect to the vertical axis. They must be formed to ensure that

a sufficient degree of freedom exist in the ring. The position of the lower hinge at any

stage of the deformation process is denoted by $, the initial value of it being 00. The

tangential velocities in the current, deformed configuration of the ring are deformed

by

V - d(3.9)1 dt

d(sl+s2)

V d (3.10)2 dt

where the lengths of the are segments s, and s2 with the radii Ri and R2 respectively

are

s= R, (3.11)

s2 = R 2 (r-+) (3.12)

At the same time the length of the flat segment is

S3 = (RI-R2 ) sino (3.13)

According to the previous assumption, the ring is taken to be inextensible so that the

sum of the length of the flat segment and two arcs should be constant and equal to

the one half of the original circumference of the ring

31

81 + S2 + 83 = 7rR (3.14)

From Eq. (3.8) the rates of rotation in both type of hinges are

1 1V, - -R2i - R 1(3.15)

19 = V2R (3.16)

As the hinges travel down the ring, the curvatures of the two arcs change

continuously with reference to the deformed configuration. The rates of curvature

K6 6 are defined by

) =- (3.17)

-kO) (3.18)R2R2

in arcs of the lengths si and s2, respectively. By substituting Eqs. (3.9) to (3.18) into

the Eq. (3.1) the rate of energy per unit width in the crushing mode is obtained by

2 2 1 1 2Ecuh 2[IM |+JM ,(RT 1)I+IM ,O(--)|+IM ,(7-#0)(-- )11

(3.19)

In the present derivation the position of the lower hinge, 0, is taken as a

monotonically changing time-like parameter, which describes the process of

32

deformation. By the chain rule the derivative with respect to time is replaced by the

derivative with respect to 0, d/dt = d$/dt d/d$. This parameter is related to the

variable radii R , R2 and central deflection wc by

wC = 2R - [ Rj(1-cos$) + R2 (l+cos$) 1 (3.20)

In the present calculations, the following four non-dimensional parameters are used

Ri R2 WCr 1 =---, r2 F=-, = (3.21)

Using the condition of inextensibility of Eq. (3.14) and the relation of Eq. (3.20) two

out of four parameters can be eliminated. The same applies to the time rates of

those parameters. In order to describe the crushing problem in terms of a single

variable (w0 or 0) it remains to establish one more relation between the parameters

involved. Noting that the radius r2 decreases during the crushing process from the

initial value r = 1 at 0 = O, the power dependence between r 2 and 4 is suggested in

the form

r ( )n (3.22)

The rate form of the above equation is

r2 = n ( n)1 (3.23)0 0

The coordinate of the initiation point of the hinge 0 and the exponent n are

considered as constant during the crushing process and should be determined as a

part of the solution. It is noted that the parameters $ and n cannot be uniquely

determined from the analysis of rigid plastic rings alone. It is possible though that

33

certain deformation modes of tubes which dissipate less energy in crushing may

actually dissipate more energy in extension of generators (See next section) or in

shear. There may then be a unique mode which yields an optimum solution. Such an

optimization procedure will not be explored in the present thesis. The parametric

study is performed to find the dependence of the rate of energy due to crushing on

the magnitude of $ and n. This energy is a linear homogeneous function of the

displacement rate w. and, thus, can be represented in the form

E Pj(iff ) ;V (3.24)crush C cc(.4

where the function Pfw ) depends on the displacement. The function PCw)

represents an instantaneous crushing force necessary to deform the ring further from

a given deformed shape. A full set of dimensionless equations are derived for the4M

0calculation of crushing force in Appendix A. The non-dimensional function P. /()

is plotted in Figs. 3-3 to 3-6 as a function of W . The corresponding shapes of the

deforming rings are shown in Figs. 3-7 to 3-10. It is seen that the deformation mode

in which all hinges are initiated at one point $ = 7r gives initially an infinite

crushing force and thus should not be considered as a valid mode. The lowest value

of the crushing force corresponds initially to = 0, but the corresponding

deformation mode is symmetric and, thus, unrealistic for the present purposes. For a

fixed initial position of #, higher power of n tends to yield larger crushing force.

Approximate crushing energy

It appears that the realistic description of the unsymmetric deformation mode of the

ring is obtained for the set of parameters 0 = 7r/2 and n=1. It is desirable for the

present analysis to have a closed form solution for the crushing strength of the ring.

A simplified constant crushing force solution is suggested by the formula

34

C

(*\J

0

07

cc TC)H

Y-H-

C wd0W -70-

Fiur 33e ondiesina cuhig trnthofa ig eru

detddphfrnk1

35

CO

0J

Q

-D

Q

cycc

411CDI

Z

-A-

C 0\N

Figure 3-4: Non-dimensional crushing strength of a ring versusdented depth for in = 2.

36

(0

COco

I

Li

(C* Li

110 Li

CYCo0

o~ ~ if i) 0 0 f 0

Figure 3-5: Non-dimensional crushing strength of a ring versusdented depth for n = 0.5.

37

(0

|-4

,-4 0

(/

CO [

Q CL

0

Figure 3-6: Non-dimensional crushing strength of a ring versusdented depth for 00 = ?r/2.

38

0

= =1 .0

R =1.5

Figure 3-7: Initial and intermediate shapes of deformed rings withS0= 7/2 and n = 1.

39

IT

n =4 N =0.5

e j=1.o

A0=1.5

Figure 3-8: Intermediate shapes of deformed rings with

# = 7r/2 and n = 4.

40

=o

6S=0.5

* 6=1.0

=1.5

Figure 3-9: Initial and intermediate shapes of deformed rings with0 = 7 and n = 1.

41

.=0 =0

n=1 0

0.5

6 =1.5

Figure 3-10: Initial and intermediate shapes of deformed rings with#0= 0 and n = 1.

42

8M

P = R(3.25)

The above formulae will be used throughout the remainder of this thesis. It should

be noted that the crushing force corresponding to the simplest symmetric collapse

mode consisting of four stationary plastic hinges [6] is given by

4M -i

PC R -1/2 (3.26)

which is less than half of Eq. (3.25). This is consistent with the findings by Reid [28],who showed that initial collapse load and the entire load-deflection characteristics of

rings increases when the deformation changes from a symmetric to an unsymmetric

mode. The total crushing energy in the dented zone is obtained by substituting Eqs.

(3.20) and (3.25) into Eq. (3.2). This results in

8Mu = 2j - w(x)dx (3.27)

In order to perform the integration, the velocity profile of the leading generator

should be known. Following the analysis of large dynamic deformations of rigid-

plastic beam in Refs. [14] and [26], the velocity field of the leading generator can be

assumed to vary linearly with x, according to

;VC = i( l- -) (3.28)

where x = is the point of an instantaneous rotation. With the above expression,the integration of Eq. (3.27) can be performed to give

43

crush R8M 0 (3.29)

Exact crushing energy

The crushing energy of Eq. (3.29) is based on the constant crushing force. In order to

justify the above approximation the exact rate of crushing energy is considered and

compared. The deflection profile of the leading generator should be known to

calculate the exact Ecrush* It means that the iteration process is needed to reach the

converged values. It will be shown in Chapter 4 that the solution of the indentation

problem of a tube with full end fixity, based on Eq. (3.29) leads to the simple

quadratic displacement profile of the leading generator, corresponding to a = 0 in

Fig. 3-1, in the dented zone.

e= (3.30)

From Eqs. (3.2) and (3.19) the exact expression of the crushing energy has the form

8MO fwcXEcrush R 3jl ) (1-) dx (3.31)

where f(w,/R) is the crushing force given by Eq. (A.6) in Appendix A. From Eq.

(3.30) w,/R is a quadratic function of x/ . By letting x/ = q, Ecrush has the form

8M 1 8M

crush - R flq) (1-q) dq = R Y (3.32)

The value of Y is slightly above 1 for even larger deflection. Detailed explanation

will be provided in Chapter 4.

44

3.2. Extension of Generators

The global strain rate is decomposed into three parts:

j;,dx = | + 21I dx + z 60 (3.33)

where

* The strain rate ;1 due to local denting. This component is different forevery generator.

e The strain rate e2 due to the uniform compression or extension of thetube. It is the same for all generators.

* The displacement rate z00 due to the global rotation. This termcorresponds to the concept of plastic hinge in a beam subjected tobending. It changes from generator to generator.

The conceptual drawing of the generalized velocities, i, u and 6,, are shown in Fig.

3-11. Since the integration Eq. (3.33) can be carried out term by term, consider first

the contribution of ;1. From the definition of the strain rate in the Lagrangian

description

. dw dt; - (-) (3.34). dx dx

Assuming a linear velocity field described by Eq. (3.28), and integrating the first term

in Equation (3.33), the contribution of the rate of energy dissipation by a generator in

the denting mode becomes,

2 No 0 dx = 2 No (3.35)

The pure axial strain rate 2 results from the beam-like theory

45

20

o -- - -- -/

i a

U0

Conceptual drawing of generalized velocity components.Figure 3-11:

46

2 = (3.36)

The function uo describes a uniform elongation or shortening of the tube and thus is

independent of the coordinate s or a. The z-coordinate appearing in Eq. (3.33) can

be uniquely related to the angle a by identifying each generator on the undeformed

section of the tube.

z 0 = R cosa b (3.37)

Integrating with respect to the undeformed axis of the tube

Io2;2 = -- jx (3.38)

Observing that uK0 = 0 and denoting n| = U0, the global strain rate becomes

Wotbo0d + + R coga bo (3.39)

The energy dissipated by all the deforming generators is equal to

2 7rR WWEg 2 N ] | + 1 + R cosa 0 ,1 ds (3.40)

The magnitudes of the displacement, wa, andvelocity, v, of the generators are a

function of the are length s or the central angle a = s/R measured from the leading

generator. Those functions are calculated in the Appendix B. Again the parametric

study is performed to find the dependence of the rate of energy due to extension of

generator Egen on the magnitude of 00 and n. The nondimensional displacement

w./R and the global strain rate due to crushing woNw/R2 of generators are plotted in

47

Figs. 3-12 to 3-14 as a function of a for different $0 and n. For a fixed power n,

higher position of #0 tends to yield smaller extension in generators, which is the

opposite behavior with the case of crushing in rings. For a fixed position of 00, the

behavior changes depending on dented depth. The change of power n is less sensitive

to the extension than the change of initial position of moving hinge 00. Which

combination of 00 and n yields the lowest indentation force will be discussed by

considering both crushing energy and extensional energy in Chapter 4.

Approximate extensional energy

The areas under the respective curves are always positive and according to Eq. (3.6)

represent the total rate of energy absorbed in the dented zone by the generators in

the absence of any overall rotation 80 and translation uO. This extensional energy

was calculated in Ref. [20] in an approximate way using a straight line approximation

of the function

( a -)(3.41)R2 R2

Fig. 3-15 shows that the linear approximation always overestimate the actual

extensional energy by a considerable amount. The resulting dissipation rate is given

by the equation

E = 2 r NR - (3.42)

In the present approach the quadratic distribution of the extension rate due to local

denting which yields better results particularly for freely-sliding and freely-rotating

boundary conditions than the linear distribution in Ref. [201 is proposed (Fig. 3-16).

48

LO

0C\J

z7I4 .

0O

Ie I I

0

0)

0

a;

0

0

(~0

0

0

0

0

0

(\J

0

0

0

0

j / On~

Figure 3-12: Displacements of material points at the symmetry planex = 0 as a function of the circumferential coordinate a

at three values of dented depth.

CI_J

4g

II II ii

~SO 6~o

II I

/7

/

//

I,

/

if)Cd

'1I)

(f

/7/

-, )

-, / -~

6 jl/

0 / 'If/ (I(

II ' I 1

Ill ~i.lit

/ 4/ I

///

7

7 1'/1 ,1

0 if)

0

Figure 3-13: Exact global strain rates for different e0.

C

0

IF

-J

C,)

0D

Cij0D

0D

0D

0 0D L

Ni

~E* om

00

50

0D

CNJ 0

[FO 0U?'

Figure 3-14: Exact global strain rates for different n.

'- O

0 -

ii II /

cc Is

~JC44 /

II /I// /1

/ // (I

0*

II /

0/ II

/ /

/

/

/ /

/

,1 / ~I

I I

L h

0

0

CD

0

0-\

* * a / on on

4)

}C

\\ nil 0

0 x mx0'

51

Exact global strain rate and its linear approximation.Figure 3-15:

52

C0

I I IG)

C0

cv) 0

(V)

G)) W

onC

Exact global strain rate and its quadratic approximation.Figure 3-16:

53

0 ( 2(3.43)

R2 p2

Substituting Eq. (3.43) along with ;= u0 = 0 into Eqs. (3.40) and (3.6), a lower

value of the dissipation is obtained

4 5.E = - r N R- (3.44)

R 3 0 E

which is lower than Eq. (3.42).

Exact extensional energy

The rate of extensional energy is obtained by the calculation of the area under the

curve of w0i 0 /R 2. Detailed calculations are included in Appendix B. Therefore, the

exact rate of extensional energy with fully clamped boundary condition becomes

4N R3Zd

E - (3.45)gen dt

where Z is a nondimensional parameter given in Appendix B.

For a fixed depth of the dent 6, the extensional energy becomes a function of

four parameters f, , 6' and io. Those parameters will be determined from the

boundary conditions and the energy minimization procedure, as described in the next

chapter. )

54

Chapter 4

Indentation of Tubes Subjected toLateral Load

The boundary conditions for the present problem can be specified with the help

of the expression of Eq. (2.2) for the rate of external work. In this Chapter we shall

consider a finite length tube subjected to lateral load for different boundary

conditions. In all cases the deformation are induced by pressing into the tube a rigid

indentor at a constant rate 6. The indentation depth 6 is then an increasing function

of time. The reaction force under the punch P(3) is unknown and will be found as a

part of the solution. Out of four remaining parameters in the Eq. (2.2), two should be

prescribed to uniquely define E . There are four possible combinations depending

on which static or kinematic quantities are prescribed. This gives rise to four general

types of boundary conditions. In addition, a few subcases will be considered when one

or two quantities of interest vanish. This brings the total number of possible

boundary conditions to eight. All of them are described and illustrated in Table 4-1.

In order to illustrate the present methodology, the solutions for the cases 1, 2a and 4c

will be worked out in detail.

4.1. Indentation Problems for Different Boundary Conditions

4.1.1. Tube with full end fixity

The ends of the tube are constrained against rotation and axial displacement.

Since the sections of the tube outside the dented zone x > are rigid, the boundary

conditions imply that 6, = 0 and 60 = 0. The rate of external work is reduced to

eBt = Pi. The rate of internal work is a sum of the dissipation of rings and

generators E = Ecrush + E

No Prescribed

1 6 u0 0o aO~0

2 N &o=o --

2a No e=o

3 u= 00 M

4 N M

4a N =0 M

4b N M=0

4c N=o M=0

Eight possible cases of boundary conditions.

55

Table 4-1.

56


As a first approximation consider expressions of Eqs. (3.29) and (3.42) for crushing

and external energy respectively. The energy balance postulate of Eq. (2.1) yields

8M4 $ 4rN0 R6 (P65= R + 6 (4.1)

R 3

The term 6 can be dropped out from both sides of the above equation. The Eq. (4.1)

provides the solution to the indentation problem in terms of a single unknown

parameter . It is plausible that the length of the locally dented zone adjusts itself in

such a way as to minimize the required force at any stage of the indentation process.

Instead the minimum force exists for each value of the indentation depth and the

relation between and 6, found from OP/( = 0 is given by

. . (4.2)R 3t

Substituting Eq. (4.2) back into Eq. (4.1), we obtain the desired force-deflection

characteristics of the indentation process

= 16 7r(4 .3 )M -6 3t R

0

The present solution can be compared with a still more crude approximation derived

in Ref. [20], using a linear rather than quadratic variation of the rate of energy

dissipation by generators of Eqs. (3.29) and (3.42). The difference is only in the

coefficient

57

P 7D- 16 - (4.4)M, 2 t R

Despite their simplicity, the solutions of Eqs. (4.2) and (4.3) describe all important

features of the plastic behavior of tubes under lateral load. The crushing force is seen

to depend linearly on the average flow stress of the material and there is a square

root dependence on both the diameter-to-thickness ratio and the normalized dent

depth. Having found the dependence of on 6, it is possible to derive an expression

for the deflection profile the leading generator. The velocity field is linear '(x,t) =

i[1-x/ (t)]. However, because the extent of the damaged zone also changes in time,

the resulting permanent deflection shape is nonlinear. The deflection is a time-

integral of velocities

w(xi) = ti(xj) dt = J ( 1- ) dif)) (4.5)

where T(x) is the time at which x = . Changing the integration variable from t to 6,

Eq. (4.5) can be written in an alternative form

w(x,S) = ( I - ) d6 (4.6)RJ7-

j3t

Integrating the above expression and noting that at x = , (T) - 3tx2 /2,rR 2 , the

deflection profile becomes

w(x,3) = 6(1--)- (4.7)

It is worth noting that an identical expression for the deflection profile was derived in

Ref. [20] using a crude model of the shell. This rather surprising result follows from

58

the detailed calculation of the bending and extensional deformation mode. What

matters here is the functional dependence of Ecrush on and Egen on 1/ .


Combination of Eqs. (3.32) and (3.45) gives the energy balance equation

8M, 4NR3ZP R (4.8)

By elimination of d4/dt Eq. (4.8) simplifies to

8Mc$ 4N0 R3ZP = -j-Y+ (4.9)P R f dSldo

The minimum energy postulate gives the optimum extent of dented region

._ DRZ (4.10)R tdb/djo Y

The load-indentation relationship is obtained by combining Eqs. (4.9) and (4.10)

P D R- =16 - YZ (4.11)

M it d6/doo

Again, the parametric study is performed to find the dependence of the indentation

force P and the extent of dented region on the magnitude of and n. The solid

lines in Figs. 4-1 to 4-2 correspond to constant crushing force which acts as a lower

bound. For a fixed power n, lower position of 00 tends to yield smaller indentation

force. For a fixed position of 0,, lower power n tends to yield smaller indentation

force. Figs. 4-3 and 4-4 show that the behavior of the extent of dented region is

opposite to that of the indentation force. Whether we take the exact crushing force

59

or the approximate one does not make much difference for even larger deflections.

The deviation between them are about 8% for 6/R = 1.0 and 20% for 6/R = 1.5. In

actual tube the deflection may not go well beyond 6/R = 1.5 due to locking or global

deformation. It also shows that the force-deflection relation is not much influenced by

the change of the starting point 00 and the power n. The tendency supports the

validity of the use of constant crushing force. For simplicity, Y = 1 will be used

throughout the remainder of the thesis.

4.1.2. Tube free to move axially but restricted from rotation

This is the next step in complexity in the series of eight different boundary

conditions. With N = 0 and ie = 0, the rate of extensional energy is the same as in

the case of a fully clamped tube, i.e., Eext = Pi. The rate of internal work, and more

specifically the rate of energy dissipation by the generators will differ because,

according to Eq. (3.40), there are two independent components of the velocity 6 and

u0 . The missing relation between 6 and uO is obtained from the condition of zero axial

force N = 0. The total axial force in the cross-section is defined by

N =2uotR "sign(j ' dx) do, (4.12)

It is worth noting that the axial force calculated from the above definition is the same

for any cross-section 0 < z < f.


The condition N = 0 is met if the change of the sign of the extensional strain rate .occurs at the point a = ?r/2. The strain rate due to the denting alone is purely

extensional (positive). Therefore, in order to change the sign of ', a uniform

compressive velocity ui must be superimposed so that the point along the

circumference of the tube with zero strain rate corresponds to a = 7r/2, Fig. 4-5. The

60

25

mcI4.,

20

F 15

0C')

0z

to

3

0.0 0.2 0.{f 0.6 0.8 1.0 1.2 14. 1.6

DENTED DEPTH / RADIUS

Figure 4-1: Dependence of the indentation load with differentvalues of n for a fixed position #..

//

--- n = 0 .

o - 2- n =

---- n =

flu.| /

61

25

20

15

1to

Li

00.0 0.2 0. 0.6 0.8 1.0 1.2 1.A 1.6


Figure 4-2: Dependence of the indentation load with differentvalues of 00 for a fixed power n.

F.LE0(jJ

0

=1O

0 4

62

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1H 1.6


Figure 4-3: Dependence of the extent of dented region with differentvalues of 00 for a fixed power n.

1.2

Li

H

a(I)

H

SI I I |

4o4 317

O2 4

7------

1.0

0.8

0.6

0.2

0.0

I

I II I

63

1.2,91=

1 ---n=0.5

n/

/--- 2 /=2

0 8

0 1

H

'0.6

0.0 0.2 {. 0.6 0.8 1.0 1.2 . 1.6

DENTED DEPTH /RADIUS

Figure 4-4:6 Dependence of the extent of dented region with differentvalues of n for a fixed position # .

64

circumference of the tube with zero strain rate corresponds to a = 7r/2, Fig. 4-5. The

magnitude of nO relative to i would depend on the particular shape of the function

ww n/. It follows from simple calculation that the condition N = 0 is satisfied if

o = - - (4.13)S 4

Introducing the above result into Eq. (3.40) and taking proper care of the integration

with an absolute value sign we obtain an expression for the total work of internal

work of internal work in the generators

E = (4.14)gen

The sliding freedom of the tube has thus reduced the rate of dissipation in the

generators by the factor 3/4 compared to the tube with the axial freedom restricted.

In the present illustrative example the ring solution with the constant crushing force

is used.

The energy balance equation becomes

. 8ME ,rN0 R6P8 = R 8+ 8 (4.15)

After minimization with respect to we obtain

- (4.16)

and the load-indentation relationship takes the same form as before with a magnitude

reduced by the factor V3/4

65

LI

wow.

1

1/4 i

+ cr01

z

M MpR

N0 y

Figure 4-5: Reduction of the rate of energy dissipated bygenerators for the tube with freely-sliding boundaries, N = 0.

66

P r D-- 16 (4.17)

MO 4t R

Exact extensional energyW W0 0

The global equilibrium equation based on the exact with N = 0 and 00 = 0

becomes

8Mo$ wOTPR= R 6+ 4NR f| +t |O da (4.18)

The axial force N is expressed in terms of "intersection points", at which the

argument of the sign function in the integrand of Eq. (4.12) become zero. The

number of intersection points is three due to the characteristics of exact w /

N 2- --+-(a-a 2 + a3) = 0 (4.19)

N IrP

where a1 and a2 are between 7r/4 and 7r/2, and a3 is slightly beyond 7r/2. The stress

distributions along the circumference for b/R = 0.5 and b/R = 1.0 are plotted in Fig.

4-6. There exist two compressive regions when N = 0. It is good to compare with

residual stress distribution in Ref. [7]. Fig. 4-7-a shows the deflection profile around

the circumferential center line of the dent, to an exaggerated scale, and Fig. 4-7-b

shows the average axial residual stresses at midspan. Instead of Eq. (4.13) under the

quadratic approximation of w~x./ we have three conditions

+ )W = 0 (4.20)

67

ci 0 O (D

_ -- 4 0 n

Figure 4-6: Distribution of global strain rate for freely-slidingboundary condition.

00

II Iia I

0

CD

0

CD

C1)

CL

(~0

-- 4 0CDJ 0D

68

compressson0n30,-

0

0 3o,Uenson

a)

Residual deformations and stresses after denting.

I

Figure 4-7:

69

w0 (a)i(.)

+ U = 0 (4.21)

W (C'P a3 )+V UO= 0 (4.22)

Four unknowns al, a2, a3 and u0 are uniquely determined by solving four equations

(4.19) to (4.22). Since nO is independent of the circumferential coordinates a, the

integration of Eq. (4.18) can be performed easily

8MO . 4NR3 Z1 1P6= 8b+ (4.23)

R

where

Z = j G()d _ 2G(a)da +fac J 3G(a)da - G(a)da

w,(a)w,(a)G(a) =2

Eq. (4.23) reduces to

8M0 E 4N0R3 Z

R I dbldo

The expressions of the extent of dented zone should be

R

D R

- dT/do ZIII

and the load-indentation relation becomes

(4.24)

(4.25)

(4.26)

(4.27)

70

P D R-16 - Z (4.28)

MO t d/d d

Discussion

We can conclude that by relaxing the boundary conditions in the axial direction from

full fixity ni = 0 (N = N ) to freely-sliding (N = 0), the crushing strength reduces

by approximately 10%. In the previous analysis of the same problem the reduction

was found to be unrealistically high, equal to 1/v'2 which is about 30% [201. The

above types of boundary conditions provide bounds on the general Case 2, defined in

Table 1, where an arbitrary tensile force 0 < N < N was applied to the axis of the

tube during the indentation process. If the axial force is compressive rather than

tensile, the situation would change dramatically. The analysis of this problem will be

presented in Chapter 5.

We are now in a position to find the relation between the axial shortening of

the tube uO and the depth of the dent 6. Introducing Eq. (4.16) into (4.13) and

integrating with respect to time, the following expression is obtained for the

shortening of one side of the tube

uO = 0.133 (4.29)R

For example, a tube with R/t = 50 undergoing indentation equal to the tube radius 6

- R should suffer a total shortening equal to twice the wall thicknesses. Such a

shortening could be easy to detect experimentally although no such measurements are

known to the author.

71

4.1.3. Tube with rotational and tranlational freedom

The bending moment in the dent-affected zone is defined by

M = 2ortR] sign(] C ;dx) z da (4.30)

where the dependence of z on s is different for every cross-section. The bending

moment at the end of the plastically deforming zone x = C can be calculated taking z

= R cosa. Simple integration shows that the fully plastic bending moment of the

cross-section becomes

M = 4,R 2 t (4.31)p 0

We shall consider now a tube totally unrestrained from rotational and axial

motion. Because both the bending moment and axial force vanish at the tube ends,M = N = 0, the only contribution to the rate of work of external forces comes from

the lateral force P. Like in the previous cases, two solutions will be presented. First,the approximate closed-form solution will be derived to illustrate the method. Then,an exact load-indentation characteristics will be determined numerically.


In order to satisfy both N=0 and M=0, the axial strain rate and stress should

change the sign twice between the end points a = 0 and a = r. From the definitions

of Eqs. (4.12) and (4.30), the coordinates of points at which the global extensional

strain rate f idx vanishes are found to be al = 7r/4, and a, = 37/4. Introducing

the quadratic function in Eq. (3.43) into Eq. (3.39), the global extensional strain rate

in any generator can be expressed as the function of a

72

dx = -(1--)" ++ Rcosa (4.32)

Since the global strain rates at intersection points al and a2 are zero, we have two

algebraic equations for ii and eO

Q 66 .-+ U

16 + 0

1 3 .-+ U

1 ( 0

1.+-Re+ - RV2

1

= 0 (4.33)

Re6 = 0a (4.34)

whose solutions are

. 533iS= (4.35)S 16

. 1 3360 = -- (4.36)

2 V-R

Because the integrand of Eq. (3.5) involves an absolute sign and the function

f x(a) dx changes sign twice, the integration in Eq. (3.5) should be split into three

parts according to

Egen = 2 [al EgenRda fEgen R da + J' Egen R da

where

(4.37)

73

E 2 N + Z (4.38)

After straightforward calculations, we obtain

7rE = - N R- (4.39)

It is seen that the relative rotation and translation of tube end cross-sections release

the tensile stresses in the dented zone and reduce more than twice the energy

dissipated by the generators. This is illustrated in Fig. 4-8 where the shaded area

represents the actual dissipation. The rest of calculations are straightforward. The

reaction force P(6, ) is defined by the rate of energy balance equation

8M 7rN RPi= 0 + 3 (4.40)

R Aq

The optimal condition oP/o9 = 0 leads to

(4.41)R 8 lt

Combining Eqs. (4.40) and (4.41), the strength characteristics of the tube under

lateral concentrated force is given by the formula with a magnitude reduced by the

factor

P i D6M - 16 - - (4.42)M, 16 t R

0


The global equilibrium equation with M = N = 0 based on the constant crushing

energy becomes

74

I

n/2

n/2

M =0

N =0

Figure 4-8:

Az

. . y

Distribution of global strain for a tube with rotationaland translational freedom.

wO wO

K-R00

V I

00

0-0

a

I I 1

"a i

I

8MO$ + NR wP = +I4N-R | - +u +Rcosao 0 Ida (4.43)

The number of intersection points changes depending on the dented depth as shown

in Figs. 4-9 and 4-10 we can consider two cases.

i) Case I (two intersection points)

For large deflection the number of intersection points becomes two as shown in Fig.

4-9. The points should be the same as the quadratic approximation case to be

a = 7r/4, and a2 = 31/4

wO(7r/4)tb( 7r/4) ++ UO+

WO(37r/4);'vO(37r/4) .

A -+ u,

R.-- = 0

R.-- # = 0

The axial velocity O and the rate of rotation O are expressed in terms of intersection

points

. 1 wO(r/4)ti (/4) Wo(37r/4) t i 0(37f/4)U+

. 1 wo(r/4) ti0(7r/4) WO(3 7r/4) t i,(3r/4)

V2R

Substitution of Eqs. (4.46) and (4.47) back into Eq. (4.43) gives

(4.46)

(4.47)

(4.44)

(4.45)

76

0~0

ccc0 0

C.)C/

I 0

0 T

00

N C/

on 0 n

Figure~~~/ 4-9 Ditiuinocxatgoa tan o uewttwo ntesecionpoits fr feel-roatig bonday cndiion

77

CD Q

on on'

Figure 4-10: Distribution of exact global strain for a tube withfour intersection points for freely-rotating boundary condition.

I +

W I -

0

C.)

cc

0D

0

(0

0D

CL

0o0D

CD

00n

0D 0

0

0

0-4

CD0

78

. 8Mo 4NOR3Zy IP 8M=+ 4RZ (4.48)

R

where

Z1= f iG(a)da _ 2G(a)da + G(a)da (4.49)

The extent of dented region and the indentation force have the similar froms with

Eqs. (4.25) and (4.26), respectively

SD R.-- Z(4.50)

R t ds/dp (I5

P D RM .- 16 - Z (4.51)

ii) Case II (Four intersection points)

For 6/R < 0.8 the number of intersection points becomes four (Fig. 4-10). The

bending moment and the axial force are functions of ai (i = 1, 2, 3, 4)

M

M sinal - sina2 + sina3 sina4 (4.52)P

N 2N 1+-(a,-- a2 + a3-c 4) (4.53)

P

M = 0 and N = 0 give two conditions, and the zero global strain rate requirement at

a = a (i = 1, 2, 3, 4) gives four more equations

79

+ + Rcosa,. O = 0 (4.54)

Therefore, we have six equations for six unknowns al, a2, , cr , a40 and 0 . The

same procedure with the previous case leads to the expressions of the extent of

dented region and the indentation force

_ D RZyD ts/d s (4.55)

R t d6ZdV

P D RZW- 16 - /(4.56)

M 0 t dbldo

where

Z = j 1G(a)da - 2G(a)da + f 3 G(a)da - / 4G(a)da +at 1a2 fa3

JG(a)da (4.57)

Discussion

The resulting load-indentation relation is compared with the formula of Eq. (4.42) in

Fig. 4-11 showing a good agreement of both solutions. Figs. 4-9 and 4-10 illustrates

how the free translation and rotation of the tube ends reduce the energy dissipation

of generators. The last step in the present analysis is the calculation of the rotation

angle 00 as a function of the indentation depth 6. The relation of Eq. (4.36) between

the rates or increments of these quantities is linear. Integrating this relation with the

help of the expression of Eq. (4.41) gives the sought formula

c ot0 = 0.3761 6 - in radians (4.58)

R R 2

80

exact

-- -- approximate20

15

10

0

/

/

//

I,,

///

/

ii ~/ A

'I

PA

fully-clamped

A

A A

AZ' ~/

A A/ A

/

// A// A

/ A

/ / ~ A*~ freely-slidingA

/

/

freely-rotating

0.0 0.2 0.4 0.6 0.8 1.0 [.2


Figure 4-11: Comparison of exact and approximate load-indentationcharacteristics for three types of boundary conditions.

25

4-

F-

0(h

/

/

0z

0~

1.1

-XI

81

6 3t= 21.55 - in degrees (4.59)

R R 2

For example, the rotation angle of one side of the tube having the aspect ratio D/t =

50 is predicted by Eq. (4.59) to be equal to 4.31 degrees when the indentation reaches

the tube diameter 6 = R. Rotations of free-free tubes were observed by Smith, et. al.

[34] in their model and full scale experiments with much smaller indentation depths.

No attempts were made in the open literature to quantify theoretically this effect.

The formulas derived appear to be the first solution of this important and practical

problem.

4.2. Discussion and comparison with Experiments

The present analysis draws attention to the fact largely overlooked in the

literature which is the influence of boundary conditions on the strength

characteristics of the tube subjected to transverse concentrated loads. In order to

focus on the local process of plastic indentation and to eliminate the interaction

between the local and global modes of tube failure, the analysis is restricted to those

types of boundary conditions in which the external work is done only by the

transverse force P on the corresponding velocity i. With this limitation four

conditions need to be considered out of which the one with 6i = 0 and M = 0 is0

insignificant in practical application. The remaining three conditions is studied in

great detail and the corresponding solutions were derived in the previous section. In

all cases, predictions were made on the strength resistance of the tube as well as on

the magnitude of kinematic quantities E(6), w,(x,b), u0 (6) and 00(6) describing the

geometry of the locally damaged zone.

Taking the solution for a fully fixed tube as a reference, the reduction of the

tube strength was found to be \//~4 and \/3/16 for the freely sliding and freely

rotating case respectively. The extents of the plastically deforming zone as a function

82

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 14-


Figure 4-12: Comparison of exact and approximate extents of dentedregion for three types of boundary conditions.

exact

- - -approximate

fully-clamped

- - a ,-.

//

Sfreely -sliding

4-

H-~ri0(p

(p2H

Li

C~)

83

of the indentation depth are compared in Fig. 4-12 for fully fixed, freely-sliding and

freely-rotating case, respectively. From Fig. 4-12 we can see that there is a tendency

for the locally dented zone to shrink by relaxing the kinematical constraints at the

tube ends.

It is important to note that all the derived relations between the geometrical

quantities involved are independent of the parameters characterizing the material of

the tube. This property of the solution can be explained by the fact that local

geometry of the damage zone depends on relative values of stresses and deformations

in the shell rather than on the absolute magnitudes of those quantities. A limited

amount of reliable experimental data exist in the literature to verify the present

theory. In some cases incomplete data was published for confidential reasons (see

Ref. [10]) or precise description of the test procedure and the end conditions were

missing (for example [41]). Smith, et. al. [34] reported on a carefully executed

indentation tests on full scale and model tubes with almost identical length to

diameter and diameter to thickness ratios. One full scale test (F2) and one small

scale test (F2S) were fully documented as far as the measurements of the force-

deflection characteristics P-6 and total rotation angle 00 were concerned. In addition,

the profile of the dented zone w,(x) in the F2 test was measured.

The circles in Fig. 4-13 represent the measured shape of the leading generator,

normalized with respect to the central deflection wc(x)/&. The actual profile results

from the local denting and overall bending. In order to compare the shape of the

generator produced by a pure indentation, with the prediction of Eq. (4.7), the

experimental profile should be corrected for the global rotation. The rotation angle

(Oo)exp measured in the test was equal to 60 = 26 /L = 0.01[rad]. The points

corresponding to the corrected normalized profile of the dent are denoted in Fig. 4-13

by triangles while the solid line represents the present theoretical solution. The

agreement is good which supports the presently assumed computational model of the

tube.

84

A three-dimensional sketch of the dented zone, showing theoretically predicted

shapes of rings and generators is shown in Fig. 4-14. Again, the picture closely

resembles a photograph of the dent-affected zone of the tube, shown in Fig. 1-1. The

shaded area in Fig. 4-14 represents a portion of the tube with zero Gaussian

curvature. This portion is composed of flat segments of rings, as explained in

Chapter 3.

The overall lateral deflection 6, measured in Ref. [34] in a test with the small

scale tube F2S was equal to 6= 0.005L, where L is the length of the tube. This is

equivalent to the experimental rotation

26

(o)experiment = 26= 0.010 (4.60)

attained when the depth of the dent was equal to 8 = 5.2t. The diameter to

thickness ratio of the tube in question was D/t = 40.9. Substituting those values

into the formula of Eq. (4.58), the rotation predicted by the present model becomes

(O)theory = 0.0107 (4.61)

The agreement with experiments is excellent considering the complexity of the

problem.

The present analysis also indicates that tubes with the same D/t and b/R ratios

should suffer the same rotations. However, the rotation in the full scale tube was

reported in Ref. [34] to be more than two times smaller than that for the small scale

tube with almost identical D/t. It indicates that the dead weight of the tube could

provide for a sufficient restraint bending moment, which prevents large rotations

from developing. The bending moment acts as soon as the end of the tube is lifted. Its

magnitude equals to

85

0

0ow

wn

o <

<)

4:)

600

40

D

ID

Figure 4-13: Theoretical and experimental profile of the leading generatorin the damaged zone.

86

YR 1.0n 1

Figure 4-14: Calculated and experimentally observed shape of the locally

damaged zone in a dented tube.

lot-.

87

IM = - R t Lp g (4.62)

4

where p and g denote respectively mass density of the tube material and the

gravitational acceleration. The ratio of this moment to the fully plastic bending

moment of the cross-section given by Eq. (4.31) is

M - (pgL) L (4.63)

M 8 %o D

The scale effect is clearly visible, since in tubes with the same L/D but different L

the restoring bending moment will not be the same. The indentation problem in

tubes with the prescribed bending moment and axial force is more difficult to treat

analytically and its solution will be reported in Chapter 5.

The present theoretical solution for the strength characteristics of the freely-

rotating tube, Eq. (4.42), will be correlated with two sets of data reported by Smith,

et al [34]. Fig. 4-15 shows the force-deflection characteristics based on the exact

global strain rate measured in the indentation test of the model tube with the

dimensions D = 65.1 mm, t = 1.59 mm and L = 1325 mm and the yield stress a =

280 N/mm 2. Transforming Eq. (4.42) based on quadratic global strain rate to the

coordinates as shown in Fig. 4-15, P is obtained

PL 0' Lt b a 0P 0.627 - - = 0.312 (4.64)

4tD y D2 t t

From Eqs. (4.51) or (4.56) based on exact global strain rate P has the expression

88

PL4D 2 t o

1.5.EXPERIMENT 1.3

1.21.1

1.01.0-THEORY

0.5-

0 2 4 6 u

Figure 4-15: Load-indentation characteristics for a freely-sliding tubeexperimental curve due to Smith [34] shows unloading and reloading.

89

Oly

1 Ca 2

Figure 4-16: Actual stress-strain curve of the material and rigid-perfectlyplastic idealization at the level of average flow stress.

ao Lt 0! 0 iIP = 0.627 - - - = 0.312 - (4.65)

yD2 t ay d F6d 0

The formula of Eqs. (4.64) or (4.65) has been nondimensionalized with respect to the

initial yield stress of the material a whereas in the present analysis the average Oflow

stress", aO, is used. The average flow stress is understood as the best fit of the actual

stress-strain curve in the expected range of strains with the present rigid-plastic

material idealization. As pointed out in Section 2, the bending and axial strains

reach 3% for the depth of the dent equal to the tube radius 6 = R. The average

strain is e = 1.5% and the average flow stress can be defined as ao= oro(Eav), Fig.

4-16. According to the above definition, u, is a function of the work-hardening

property of the material as well as the magnitude of the average strain attained in

the deformation precess. While the latter quantity can always be calculated by the

present shell model, the information on the work-hardening parameter of the steel

tubes used by Smith, et. al. is not available. Therefore, theoretical curves based on

exact global strain rate were drawn in Fig. 4-15 for a few constant values of the ratio

a"a "y= 1.0; 1.1; 1.2; 1.3. The present theory is seen to predict correctly the

functional dependence of the solution on all geometrical parameters of the problem.

However, in the absence of the data on the strain-hardening parameters, no definite

conclusion can be drawn regarding the quantitative agreement between the theory

and experiment.

The experimental results for the full scale tubes were presented in Ref. [34] in

physical quantities. Introducing the definition of M into Eqs. (4.42) or (4.51) (or

(4.56)), one gets

P[KN| = 4 x 10-3 e t2 (4.66)0 Fit

or

91

FKNI = 4 x 10- 3 , t2 D R_ ZR (4.67)

Taking the tube D = 396 mm, t = 9.9 mm, L = 7754 mm and the same yield stress

as before, the formula of Eqs. (4.42) or (4.51) reduces to

FPKN = 21.86 -Vm (4.68)

or

P[KN = 694.25 - (4.69)oe ds/d4

Again, plots of the above function were made for several different values of the ratio

ao/ay and those are shown by full lines in Fig. 4-17. The broken line is the

experimental curve. The correlation is seen to be good considering the fact that, no

"forge factor" was introduced in the present analysis. The shear effects in the

dented zone would bring the present prediction right on the experimental curve.

This problem will be addressed in the approximate way in Chapter 7.

92

p(kN)

400-

EXPERIMENT 0eo/cy =1.3

300- 1.2

1.0

200 - THEORY

100 -

p (MM

0 25 50 75

Figure 4-17: Correlation of the present theoretical solutions with the resultsof full scale test reported by Smith [34].

03

Chapter 5

Indentation of Tubes Subjected toCombined Loading

The global energy equilibrium should satisfy Eq. (2.1). This equation can be

generally expressed in the following form

P6+ 2Mb + 2Nu* = A(,6)6 (5.1)

where is related to b. The interaction function g(P, M, N, 6) = 0 determines all the

possible combination of the generalized force which produce plastic flow in the

element for a given level of indentation 6. In order to obtain the interaction surface

under combined loading two approaches are considered. In DnV report [20], de

Oliveira et al. derived the interaction formula using the global equilibrium and the

normality postulate. In the present approach the force and the moment equilibrium

satisfying Eqs. (4.12) and (4.30) will be used instead of the normality postulate. In

Appendix E the approach in Ref. [20] is applied to the lateral load/axial force

interaction with io = 0 by using the quadratic approximation for the extension rate,

resulting in the identical results with the present one.

Normality requirements approach

If the interaction curve is a valid yield criterion in the space of generalized stress

according to the associated flow rule, plastic flow follows the direction normal to the

yield surface.

94

. F= - (5.2)

0P

i= (2M)(.)

S= a (5.4)i9(2N)

where p is a positive scalar multiplier, which is nonzero only when plastic

deformation occur. The global equilibrium Eq. (5.1) and the normality postulate (5.2)

to (5.4) furnish four equations for four unknown parameters (5, U, 00 and .

Elimination of these parameters leads to the desired interaction g(P,M,N,6) = 0.

Force and moment equilibrium approach

The global strain rate in generators is defined by Eq. (4.32). The argument of the

sign function in the integrand of Eqs. (4.12) or (4.28) changes sign at a = ai. For a

prescribing bending moment and/or axial force, intersection points a, are uniquely

determined. At a = ai, uO and bo can be expressed in terms of the remaining

parameters.

= f(P, M, N, 6) (5.5)

b= f 2(P, M, N, 6) (5.6)

Substituting Eqs. (5.5) and (5.6) into (5.1) and dropping the homogeneous term 6, the

interaction formula are obtained. This approach is more straightforward and simpler

to derive the interaction formula than the previous approach.

We shall consider two-dimensional interaction curves (lateral load/axial force,

95

lateral load/bending moment and bending moment/axial force) using the force and

the moment equilibrium approach in order to construct the three-dimensional

interaction surfaces. It should be noted that the interaction curves based on lateral

load/axial force interaction with = 0 are projections of the three-dimensional

interaction surface onto the M = 0 plane, rather than intersections of the surface

with that plane. Similarly, the intersection curves based on lateral load/bending

moment interaction with i, = 0 are the projections of the three-dimensional

intersection surface onto the N = 0 plane.

5.1. Plastic Instability of Compressed Tubes

So far the axial force N applied at the tube ends was either N = N or N = 0

The simplicity and relative accuracy of the present method encourages to look at the

more general case of loading in which the axial force (tensile or compressive) of a

specified magnitude is applied to the tube prior to the local indentation. Such a pre-

stressed tube is then subjected to local lateral indentation. The task is to determine

the load-indentation characteristics of the tube as a function of axial force. The

calculations to be shown reveal the existence of a critical magnitude of the

compressive force N. below which a spontaneous sectional collapse of the tube takes

place under an arbitrarily small lateral disturbance. The tensile force is taken to be

positive while the compressive force is negative.

5.1.1. Lateral Load/Axial Force Interaction Based on Approximate Global

Strain Rate

The rate of energy equation for the tube subjected to combined denting and

compression is

8M~q.x5Pi+2Nit0 = M+AN R |-( -- )+ |da (5.7)

Since two components of the external loading P and N are prescribed independently,

96

Since two components of the external loading P and N are prescribed independently,

there are two terms in the rate of external energy. The second term on the left hand

side of Eq. (5.7) vanishes when N = 0 or O= 0. Those two special cases were

already considered earlier in Section 4. In the absence of denting deformation s= 0,


2Nti = 4NOR j tI 0 I da (5.8)

which yields after integration

N = N, I' (5.9)

The above equations have two solutions

N =N P

N =-N (5.10)P

which mean that the tube developes its full axial strength by reaching yield stress

respectively in tension or compression. The tube remains rigid, if the axial force

stays inside those limiting values - Ne < N < N . The axial force acting alone may

not lead to the tube failure but may dramatically change the denting strength of the

pre-stressed tube.

In the tube with constraint against rotation, ;3= 0, the axial force can be

expressed by the terms ; and ;2, determined in the preceding section.

N = 2iotR r { sign [(1- )2+ u }da (5.11)0Jo 7r

The argument of the sign function in the integrand of Eq. (5.11) changes sign at

a = al defined by Fig. 5-1.

97

U =(5.12)0X

so that after integrating, Eq. (5.11) becomes

2a1 2a1N = -2 7rRto0(1--) = -N (1-- ) (5.13)

7rP 7r

The above condition says that for any choice of N from the interval - N P < N < N Pthere is a unique point on the tube circumference a = al at which axial strain rate

vanishes and stresses change from tension to compression (Fig. 5-1). Having

determined this switching point, the integration of Eq. (5.7) with the absolute sign

can be performed to give

. . 8MA . 1 NP6+ 2 N u = R +2Nu + 6 [ I -- ( 1 - (5.14)

0 3 [1-4 1 N-J (.4P

where u0 is related to and N by Eqs. (5.12) and (5.13). The two identical terms on

both side of Eq. (5.14) can be dropped out and the expression for P takes the form

8M4 $ 4rNR 1 N3P + [1--(1- (5.15)

R 3( 4 NP

Eq. (5.15) provides the solution to the indentation problem in terms of a single

unknown parameter at any stage of the indentation process. Indeed the minimum

force exists for each value of the dented depth and the relation between and 6,fP

found from =0, is given by

98

a

L :a 1 n/2 TK

n Z

Figure 5-1: A construction illustrating changing sign of global strain rateand stresses to ensure development of prescribed axial force in

the tube with b = 0

0

0

+ j.

0

-

I - I

99

P

Substituting Eq. (5.16) back into Eq. (5.15) yields the desired force-deflection

characteristics of the indentation process

P t D-- 16 - - - Y (N) (5.17)M 3 t R

where

1 N7 (N) = _-( 1--)3 (5.18)

4 N

The crushing force is seen to depend linearly on the average flow stress of the

material and there is a square root dependence on both the diameter-to-thickness

ratio and the normalized dent depth. Finally, the denting strength of the tube is seen

to depend on the magnitude of the compressive/tensile force N. Several special cases

can now be recovered from the general solution, Eqs. (5.16) and (5.17).

Tube with full end fixity

Substituting N = N into Eq. (5.18), i(N) becomes unity.

Tube free to move axially

Substituting N = 0 into Eq. (5.18), n(N) becomes 1 .

Tube subjected to the ultimative compressive load (squash load) N = - Np

The term under the square root in Eq. (5.18) becomes negative and a real solution for

P does not exist.

Discussion

The dependence of the normalized lateral load on the magnitude of the

100

nondimensionalized axial force N/N expressed by Eq. (5.17), is shown in Fig. 5-2.

The resistance of the tube against lateral denting diminishes as N/N decreases from

full tension through zero towards compression. When the compressive force attains

the value

NC1/

N = 1 -4 = -0.5874 (5.19)NP

the tube resistance drops to zero and no lateral force can be equilibrated by the

system if the compressive force is further increased towards full squash load - N .

The existence of a critical value of the compressive force N. under which local

spontaneous sectional collapse of the tube takes place has not been reported

previously in the literature and thus requires a careful examination. However, a

similar problem for a rigid-plastic solid section beam was formulated and solved by

Jones [14]. Present results will be interpreted within the realm of the present

approximate theory. Consider the equilibrium statement of Eq. (5.14). In the general

case, the rate of external work goes on both plastic dissipation of rings (first term on

the right hand side of Eq. (5.14) and generators (two last terms). Each of those two

contributions are non-negative. However, the term 2N 0 on the right hand side of

Eq. (5.14) is seen to equilibrate the rate of external work due to axial compression. In

the resulting expression for P (Eq. (5.15)) the first term on the right hand side is

always positive while the second term can be positive or negative depending on

whether N is greater or smaller than N.. Thus, if N > Ne, the analytical minimum

for P with respect to exists as given by Eq. (5.15). If, on the other hand, N < NC,the second term in Eq. (5.15) changes sign into negative. Remembering that negative

values of P do not have physical sense, it is seen that a non-analytical minimum

exists at the same magnitude of as before, but the corresponding force P is zero.

From the above mathematical proof follows a physical interpretation of this

interesting phenomenon. First, observe that according to the present model any

101

P/'PC

1.0

6. 0

-1.0 -0.5874 0 1.0

N/Np

Figure 5-2: Normalized lateral load/axial force interaction withi = 0 based on quadratic global strain rate.

102

infinitesimal increment of 6, i.e., any positive 6 will bring all the generators to the

yield point. Furthermore, i uniquely determines the axial velocity ui, for a given set

of 6 and , as specified by Eq. (5.11). Now the increment of external work should be

equilibrated by the increment of the internal work. At N = N, the only available

component of the internal rate of work is equal to 2 N 6i which just suffices to

equilibrate the work increment in axial direction. Thus, no surplus of internal energy

on the tube strength exists to equilibrate the lateral force which therefore must

vanish. For any magnitude of the lateral force greater than zero, the increment of

external energy exceeds that of internal energy meaning unstable behavior of the

system.

Another way of interpreting the phenomenon of plastic instability is to observe

that the magnitude of al which corresponds to the critical force NC is chosen in such

a way that a net rate of energy of generators are equal to the rectangular area 7ru0 .

Therefore, the tube can bifurcate from pure symmetric compression into localized

unsymmetric denting with no energy added to the system. All that is needed is to

disturb the tube from one equilibrium state into the other by imposing an arbitrary

transverse velocity i at the lateral surface of the tube. From the above discussion, it

transpires that the present topic is a classical example of unstable structural

behavior.

5.1.2. Lateral Load/Axial Force Interaction Based on Exact Global Strain

Rate

All the above conclusions in the previous section are valid within the

assumptions of quadratic distribution of the rate of extension, given by Eq. (3.43) and

the rotational restraints at the tube ends. As shown in Fig. 3-16, the shape of actual

functions wowo/ differs from that of quadratic approximation. The exact global

equilibrium equation based on constant crushing force becomes

103

8M, 0 7, w tb

P6+ 2Nui = R 3+ 4NR I 0 0 + O I da (5.20)

The number of intersection points varies depending on the dented depth. Two cases

as shown in Fig. 5-3 will be considered. The axial force is determined by

2o, 1f 'T0 0 ]Ia(.1N = 20 R {sign[ + J} dc (5.21)

i) Case I (one intersection point)

With 6b = 0 the total axial strain rate Eq. (3.39) reduces to

f. 0f dx = + u0 (5.22)

The zero global strain rate requirement at an intersection point gives the expression

of the axial velocity

W (aj);VO(aj)U0 = - (5.23)

The intersection point al in Fig. 5-3-a and 5-3-c is obtained by Eq. (5.13) in terms of

prescribed axial force. After integration Eq. (5.20) becomes

8M( 4N R3 Z

0 I.P 0+ 2NR 6+ R + 2 Nu0 (5.24)

where

104

wow,

U 0

o a2 a3 a)a

o a 1 T2 3

b)

C)

Figure 5-3: Variation of intersection points for different axialforces based on exact global strain rate with 0 = 0

105

Z, = IG(a) da - J G(a) da (5.25)

In Eq. (5.24) the time derivative is replaced by the time-like parameter derivatived3 do dp

dt - db d. Eq. (5.24) reduces to

8M4 $ 4NR3 ZP = + (5.26)

R EdS/dP

The minimization postulate gives the extent of dented zone

_ D RZIR td3q (5.27)R ft dbld o

Substitution of Eq. (5.27) into Eq. (5.26) leads to the lateral load

D RZ,P = 16M - (5.28)

t db/do

ii) Case II (three intersection points)

There exist three intersection points mainly for negative axial force and the positions

of ai (i = 1, 2, 3, 4) change depending on dented depth. The axial force is determined

from the formula (5.21)

N 2-1-- (a--a2+a3) 1 (5.29)

P

The zero strain rate requirement at intersection points provides three additional

conditions given in Eqs. (4.20) and (4.22). Four unknowns al, a2, a 3 and 0 are

uniquely determined by solving four equations (4.20-22) and (5.29) under prescribed

106

axial force N. The axial velocity in Eq. (5.23) can be used for the present purpose.

The same procedure with Case I is applied to the present case. Eq. (5.24) reduces to

8MO$ 4NOR3 Zy;P + (5.30)

R d6ldo

where Zlil is given by Eq. (4.25). The expressions of the extent of dented zone and

the lateral load become, respectively

D RZIII(5.31)R It d6ldo

and

R RZiyP = 16 4 - (5.32)

Discussion

The resulting lateral load/axial force interaction is plotted in Fig. 5-4. It is seen that

the "exact" numerical solution differs from the analytical solution of Eq. (5.17). The

instability compressive force, at which the required indentation force becomes zero, is

different for given dented depth. The phenomenon could not be observed in Fig. 5-2,

where the instability force is constant irrespective of dented depth. The lateral

load/axial force interaction curves are now moved slightly to the left so that

the"exact3 critical value of the compressive force is N. = -0.764 N . The coefficient

in the approximate closed form solution was -0.5874.

The question arises how present findings relate to experimental results

performed on cylindrical shells in hundreds of laboratories around the world. The

fact that compressed tubes may loose stability at axial force equal to 0.764 of the

squash load of the tube suggests that a catastrophic sectional collapse should be a

107

1.5R

............. ............

0NNPp

Figure 5-4 Lateral load/axial force interactionwith 5 = 0 for different dented depths.

-1.0 1.0

-' =1.0

*15

A=0.5

6= 0.01

A

:

108

0E14

0

-A1-1

W

0

C

.H

M.

Figure 5-5: Effect of radius to thickness ratio and imperfections on thebuckling strength of cylindrical shell (after Almroth and Brush).

I .

0

0.

500 1.000 1.500 2,000 2,500 3.000 3,500

Shell Radius/Shell Thickness

Theory (Perfect Geometry)

0

0.8 -

6- 00

4 0 - (P 0

0 0

8 o w*

o e

.2 - Or 0 %

0 0~ 00

0 0 00

0 0 coS 6 00

A design recommendataon.I -0 0 0

0.

I.1

0

V( 0

109

common phenomenon easily observable in simple tests. Actually, general experience

with axially compressed cylinders confirms present conclusion. Fig. 5-5 shows the so-

called 'knock-down" factor, i.e., the ratio of experimental to theoretical ultimate

strength of cylindrical shells as a function of R/t. This analysis is applicable for

moderate to thick tubes 20<D/t<60 and predicts 25 - 30% reduction of the tube

strength with single dimple-like imperfections. This finding is generally in accord with

the trend of experimental points. However, there is lack of sufficient data on the

diagram with experimental points to fully confirm the present results. Thicker tubes

with or without local dents were tested in Britain [31] and Norway [34] under

different end conditions to the present one. If the simple-supported rather than

clamped boundary conditions are imposed, an additional bending moment resulting

from the load eccentricity will have a decremental effect on the measured knock-

down factor, especially for larger dent depth.

Another interesting result of the present solution is the existence of a threshold

axial force, NOb, of the lateral disturbance which brings the system into an unstable

pattern. This property of the solution is best illustrated in Fig. 5-6, where the results

of Fig. 5-4 are replotted in the coordinate system with the dent depth as an

independent variable and axial force as a parameter. It is seen that for N > Nth, all

curves are monotonically increasing functions of 6/R. For N < Nth, the situation

changes. Consider, for example, the curve corresponding to N/N = - 0.6. The

lateral resistance of the tube increases initially, and reaches a peak value at

6/R: 0.5. The magnitude of b corresponding to the maximum lateral load is called

the threshold dented depth for given axial force. If the axial load is further increased

towards the squash load, the threshold dented depth s/R becomes less than 0.1. This

means that the tube which brought in compression up to 70% of its load-carrying

capacity can tolerate in the stable way only small local indentation equal to 10% of

its radius. If the depth of the dent exceeds this threshold value, a spontaneous

collapse of the section takes place.

The present findings may have a profound implication in the offshore industry

110

P tM N D Ri -1 0

=020- Np

00 =0

15-

= -0.4Np

10-

= -0.5

5N= -0.55Np

== -0.65 = -0.6=-0.75 Np =p. N 6NP

0 0.5 1.0 1.5 R

Figure 5-6: Load-indentation characteristics of the dented tube forvarious values of axial force.

111

when designing tubular members against accidental and impact loads. If a short

tubular transmitting axial compression is designed with a factor of safety less than

1.5, a small lateral force caused by a collision may trigger a complete collapse of the

member. The amount of the impact energy dissipated by such a tube is only a small

fraction of the energy which otherwise could be absorbed by a similar tube without

compressive load. The present theory can also explain the mechanism of progressive

collapse of complex frameworks made of tubulars. It is plausible that the load

redistribution after one member has failed can bring some other members closer to

the squash load and make them vulnerable to the local collapse described above

should additional transverse loading occur, for example in the form of an extreme

hydrodynamic wave impact.

Finally, the present results bear important implications on the energy

absorption capabilities of tubular members subjected to lateral concentrated load and

axial force. The maximum energy that the tube can absorb is obtained by

integrating the P-6 function from 6= 0 to 6= 2 R, based on quadratic extension rate

E(N) = 2R P() d6 = 16 ]I t1.5D1.5n(N) (5.33)

where i(N) is defined by Eq. (5.18). The energy is seen to rapidly diminish with

increasing compression and reach zero at N = NC. In view of the above results a

word of caution should be given as to the applicability of similar analysis performed

in the past [10] in which the influence of the axial compression on energy absorption

capabilities of tubulars was not taken into account.

112

5.2. Plastic Instability of Tubes Subjected to Bending

Consider the tube free to rotate but restricted from axial movement, 0 = 0.

The tube prestressed in bending is then subjected to local indentation. The present

type combined loading condition corresponds to case 3 in Table 1. From the analysis

made in the previous chapter we anticipate the existence of the critical bending

moment. The bending moment is taken to be positive when the deflected tube is

convex upwards.

5.2.1. Lateral Load/Bending Moment Interaction Based on Approximate

Global Strain Rate

Since di = 0, the global energy equation using the quadratic approximation of

the rate of extension has terms related with indentation 3 and rotation 9 only.

. 2 M 8 M . I

P6+2Mb = R 6+4N R + Rcosa b,\ dda (5.34)

When 0b = 0, Eq. (5.34) corresponds to the case of tube with full end fixity in Eq.

(4.1). In the absence of denting deformation 3= 0, Eq. (5.34) reduces to

2M 6O = 4NORf0 I R cosa O0 1da (5.35)

which yields after integration

MO0 =Mjej (5.36)

or Eq. (5.36) has the solutions

113

M = MP

M=-M (5.37)P

which means that the tube developes its full bending moment capacity. In the tube

which has constraint against axial movement, the bending moment is given,

according to the definition of Eq. (4.30), by

M = 2e tR {sign 1 - )2+ R cosa 0 }R cosa da (5.38)

In addition to bending moment the axial force, induced by the axial constraints at

both ends of the tube, is coupled with the bending moment. Four different cases can

be distinguished depending on the sign of rotation rate bo and the magnitudes of

and 0o

i) Case I ( i0>0 )There is a unique intersection point at a = al, corresponding to positive bending

moment, at which the rate of axial strain vanishes and stress changes from tension to

compression (Fig. 5-7). By the definition of Eqs. (4.12) and (4.30) the bending

moment and the axial force become

MM sin a1 (5.39)

P

N 2a1- -(5.40)

N onP

Combination of Eq. (5.39) and Eq. (5.40) leads to an interaction formulae

114

wow.

0

O

w,0

W0

Figure 5-7:

Rb.cos a

-ann/2

-- ~~a)

n a

71/2

C)

n a

Quadratic global strain rate profile with == 0depending on the size of rotation rate and the magnitude of

6i/E and Rbo.

I

n/2\

b)

115

-- - cos ( 7 ) = 0 (5.41)M 2 NP P

The expression of Eq. (5.41) is exactly equal to bending moment/axial force

interaction for the thin-walled undeformed circular tube. From the distribution of

strain rate the switching point should be in 7r/2 < a < r and can be expressed in

terms of prescribed bending moment from Eq. (5.39)

Ma i r - sin ( T ) (5.42)

P

Under the condition n= 0, Eq. (3.39) reduces to

;xxdx 1 - )2 + R Cosa O (5.43)

The zero strain rate at a = a, gives the expression of the rate of rotation

( 1 - a )2

0 = -- (5.44)cosaI RE

The integration of Eq. (5.34) can be performed to give

8ME 41NR 1 MP4 2Mb = - + 2Mo + - { I - 2 sin- -

R 0 3 (1 MP

(5.45)

where 0 is related to i and M by Eq. (5.42) and (5.43). The expression for the lateral

force takes the form

116

8MOE 4NR& 1 Ml 3P = - 3 -{1-2[-sin-1 ( _)1} (5.46)R 3 7r

p

After minimizing with respect to , the extent of dented region and the load-

indentation relationship take the forms, respectively

S 2 n

R 3t nl(M (5.47)

and

- 16 -- D n(M) (5.48)M, Fit R

where

7 1 (Al) = 1 - 2 [ - sin-l( ) 13 (5.49)P

;I(M) can be interpreted as the lateral indentation force for a prescribed bending

moment normalized by the indentation force due to fully clamped tube. Two special

cases can be recovered from the above expression.


On substituting M = 0 into Eq. (5.49), the load factor 7 1(M) becomes unity and Eq.

(5.48) reduces to Eq. (4.3).

Tube free to move axially

On substituting M = M into Eq. (5.49), the load factor becomes and Eq. (5.48)

reduces to Eq. (4.17).

ii) Case II ( 0;<0 and I Re0 I <1I)

117

There are two intersection points corresponding to negative bending moment in Fig.

5-7-b. The bending moment and the axial force have the forms

(5.50)

N 2 2N- I +-a 1 - a2 (5.51)N 7r X2

P

From the above equations the intersection point al and a2 can be expressed in terms

of bending moment and axial force

1

2

M

p- + (

I M )2

2 MP

M 202(7 N()2-Cos(- -)

M 2 Np p

2 [1 - sin( 7)2 N

P

}

r 7r Na2 =a1+2 -2 N

P

(5.52)

(5.53)

The zero strain rate requirement at intersection

conditions

33

33

(1--)2

a2 2

7r

points provides two additional

+ Rcosa 1 0 = 0

+ R cosa 0 = 0

(5.54)

(5.55)

Four unknowns al, a' 2o and N are uniquely determined by solving equations Eqs.

(5.52) to (5.55). The rate of rotation can be obtained by Eqs. (5.54) or (5.55)

MM - sinal - sinaM pP

a sin {

118

cr1( 1 - )2.

60 = - - (5.56)

The global equilibrium equation becomes

.+ . 8M, . A7NO R3 i a 2 aIPi6 + 2Mi 0 = L+ {+2[ (1--)_ (1__)31 }+2M 0* R 3 ER2 7

(5.57)

The optimality condition leads to

2~ t2(M) (5.58)R 3"t2

where

= + 2 [ (1 - )3_(_i)3 (5.59)

and al and a2 represent positions of zero axial strain rate under negative bending

moment. The load-indentation relationship has the form

P 7rD - 16 - -- 2(M (5.60)

0

The tube cannot sustain the fully negative bending moment M= -M= , because the

resistance of the tube against lateral indentation diminishes as M/I decreases. The

dependence of the normalized lateral load on the magnitude of bending moment

expressed by Eq. (5.48) and Eq. (5.60) is plotted in Fig. 5-8. When

M = - 0.6383 MP, the tube resiatance drops to zero and no lateral force cannot be

119

P/ Pc

I1.0

-0.6383

Figure 5-8:

0

M/ Mp

Normalized lateral load/bending moment interactionwith ii = 0 based on quadratic global strain rate.

-1.0 1.0a a

0. =0

120

tolerated for further increase of negative bending moment. The physical

interpretation of the instability of a tube subjected to critical bending moment will be

similar to that in the previous section.

iii) Case III ( bo <0 and | Rio | >

Case III in Fig. 5-8-c is similar to Case I except the sign of net axial strain rate

distribution and the range of intersection point, 0 < a1 < . The bending moment

and the axial force are respectively

M

M - sin a (5.61)P

N 2N - -- a1 (5.62)

P

The bending moment-axial force interaction becomes

M r N- +c os(--) = 0 (5.63)M 2 NP P

From the zero strain rate requirement the rate of rotation is equal to Eq. (5.56). The

load-indentation relation becomes

- 16 --- [2(1 _ )3 ] (5.64)MO 3 t R 7r

However, for any arbitrary value of switching point (0 < a1 < 0.377r) the

magnitude of I Ro I is always less than that of 1 | 1. It violates the original condition

I Roo I > - |. Also for 0.20637r < a1 < 7r, terms in square root in Eq. (5.64) give

121

negative values, which has no physical meaning. It is concluded that Case III is not a

valid condition for lateral load/bending moment interaction with bo = 0.

Discussion

The lateral load/axial force interaction in Fig. 5-2 shows the monotonically increasing

bahavior from N = N. to N = N . Under the lateral force-bending moment loading

condition the interaction curve has the maximum value at M = 0 (Fig. 5-8)

corresponding to N = N . At M = 0, rotation as well as axial displacement is

restrained, and generators are subjected to full tension, which lead to the same

condition as the tube with full end fixity. At M = MP, the tube has the maximum

moment capacity with zero axial force. Under the present behavior of axial strain

rate the switching point cannot reach a = 7r/2, which means that the tube cannot

reach the full positive moment capacity together with zero axial force. The rate of

rotation in Eq. (5.44) is very high near al = 7/2, and the resiatance against positive

rotation is very small. Part of generators suffer full tension, while part of generators

suffer full compression. The behavior of generators may be the part of explanation

why the indentation force corresponding to M = 0 is slightly smaller than that

corresponding to M = M .p

5.2.2. Lateral Load/Bending Moment Interaction Based on Exact Global

Strain Rate

From the results of previous section Case I and Case II will be considered. The

global equilibrium equation based on exact global strain rate becomes

8AI 7r W w0wb0P+ 2Me =- R + 4NRf + R cosa 90 da (5.65)

i) Case I ( 00>0 )

With U-0= 0 the total axial strain rate Eq. (3.39) reduces to

122

m (a)W,(a)f ( (dx= +Rcosa 90 (5.66)

The rate of rotation at a=a is obtained from the zero strain requirement

. 1 W'(aj)tb'(aj)= - 1w((5.67)

cosa1 R

The relation between al, bending moment and axial force are given in Eqs. (5.39) and

(5.40). Substitution of Eq. (5.67) into Eq. (5.65) gives

.r .

where Z, is given by Eq. (5.

8M 4N R3 Z,V)0

-+ 2 moo (5.68)

25). From Eq. (5.68) the time derivative is replaced bydb db d$

the time-like parameter derivative, -t do dt.


8ME 4N R3 Z= +

- P d|d

The minimization postulate yields the extent of dented zone

_ DRZIR t db/do

The indentation load becomes

P (5.69)

(5.70)

123

P D RZI- 16- __(5.71)

Al t d&/dk

ii) Case I ( 0<0 and | RO I < I 1 : 2 intersection points)

The bending moment and the axial force have the same expression with Eq. (5.50)

and Eq. (5.51), respectively. At the intersection point the rate of extension becomes

zero

+ R cosal o= 0 (5.72)

+ R cosa2O0 = 0 (5.73)

The rate of rotation is obtained by Eqs (5.72) and (5.73).

____________ W(aP)t Ja&)R 0 = COS (5.74)

The same procedure with Case I yields the extent of dented zone and the lateral

indentation force

S D RZII(5.75)R t dZId

and

124

C0I-

0

cc

,a\

I I

LO

0 0ccj46

011

0I,

0(NJ

U-) 0D Ci)

0ci

1*10

U|

F

0

0

CD~0

0D

0

( C

Figure 5-9: Lateral load/bending moment interaction with

A0 = 0 for different dented depths.

Lfl

0

0~

z

( I

I

125

P D RZ,- 16 - (5.76)

MO t ds/do

where Z1, is given by Eq. (4.49).

Discussion

The exact lateral load/bending moment interaction curves are plotted for different

dented depths as shown in Fig. 5-9. The trend of interaction curves is similar to

approximate case as shown in Fig. 5-8. The exact critical bending moment is

MC = -0.778M,, while the approximate one is MC= ~-0.6383M . Fig. 5-9 is

replotted to observe the behavior of instability as lateral force-indentation curves for

prescribed bending moments in Fig. 5-10. Again, such behavior is not observed in the

approximate case of Section 5.2.1. When the bending moment is larger than -0.5M,,

the lateral load-indentation curves have monotonically increasing behavior. When the

bending moment is less than -0.5M,, curves show the strengthening-softening

behavior. After reaching the peak value of indentation force for prescribed bending

moment, a spontaneous collapse of the tube takes place.

5.3. Tubes Subjected to General Combined Loading

Finally, three-dimensional interaction surfaces corresponding to general

combined loading (case 4 in Table 4-1) are described. The previous combined loading

conditions are special cases of the present loading condition.

5.3.1. P-M-N Interaction Based on Approximate Global Strain Rate

The global equilibrium equation based on the quadratic approximation of global

strain rate becomes

126

-M-0.77

0.0M.1 0 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 5-10: Load-indentation characteristics of the dented tube

for various values of bending moments.

20

4-

F-LI:0(I) 10

I I I

Mp

jg 0

~p~

Mp

- 0.5

Mp

-0.6

M=.7 p=-0.7 M-0.65M =-0 7 7 p

p/

0z

127

P6 + 2MO + 2Ni = _ -+ 4NR I (1 - )2 +i0 + R cosa oI| da

(5.77)

Depending on the prescribed external loading, the number of intersection points

varies.

i) Case I ( b0>0 )

The bending moment and the axial force are functions of intersection points a1 in

Eqs. (5.39) and (5.40), which lead to

Ma = sin-1 (-) (5.78)

P

SNa 12 (1 + - (5.79)

S2 NP

The zero strain rate requirement at intersection point a = a, in Fig. 5-11a provides

the expression of axial velocity

65a1= - (1--)2 - R cosal 0b (5.80)

The rate of rotation is not specified; but the terms with 00 are cancelled out after

integration. The integrand of Eq. (5.77) with the absolute sign can be performed to

give

. 8M 47rN RyaP + 2M O + 2N = -5+M + N0 [1 - 2(1-- )31+ 2M0 + 2Nu

P+9 u R 3 0 0

(5.81)

128

w.w, - - - .+R6 .cos a

o ->R/2 n

a)

wOw.

na0

n/2

b)

wow

0 a

n/2

C)

Figure 5-11: Quadratic global strain rate profile depending onthe size of rotation rate and the magnitude of

bi/ and Rbo.

129

The two sets of identical terms in both sides of Eq. (5.81) can be dropped out and the

expression for P takes the form

8MO 4NRb aYP 1 + [1 - 2(1 - -)] (5.82)

R 3E (8

After minimization with respect to the indentation force is expressed as a function

of either bending moment or axial force

P xDs M- 16 T {1 - 2[1 - -sin~ (-3} (5.83)

M, 3 t R 7r M,

P xD3 1 N16 { -{I -- )3} (5.84)

M, 3 t R 4 (584

Eq. (5.83) corresponds to lateral load/axial force interaction with ,= 0, which is

valid for 0 < M/M < 1. Eq. (5.84) corresponds to lateral load/axial force

interaction with bo = 0, which is valid for -0.5874 < N/N < 1.

0 p

ii) CaselII( 0>O and 1 R001 >1|-I)

The bending moment and the axial force are expressed by al from Fig. 5-11b.

M- -sina (5.85)

P

N 2- 1--a (5.86)N 7r1

P

Again the axial velocity has the same form by Eq. (5.80). Since 9O < 0 and

R|0 I > 0, 0 must be positive. After integration of Eq. (5.77) and dropping out

uO and b terms we obtain

130

8ME 47rN R5 cP = + [2(1 - 1]

R 3 7(5.87)

Substituting of Eqs. (5.85) and (5.86) into Eq. (5.82) and minimization with respect to

give the indentation force as a function of axial force

P IxD- 16 -- { 2 [1

MA 1 3 t R

1 -1- -sin-(--)- 1}

ir Mp

P [r D- 16 { - (

M, 3 tR 4' (5.89)N

N

Eqs. (5.88) and (5.89) are valid for -0.6037 < M/M < 0 and 0.5874 < N/N < 1.

iii) Case III ( b0<0 and I RI <l )

Fig. 5-11c corresponds to the present case. The bending moment and the axial force

are expressed in terms of two intersection points in Eqs. (5.50) and (5.51). The global

strain rates at al and a2

-( 1-)2 + +Rcosal B=-(-)rt 0 (5.90)

66 a 2-(1--)2 +u + R cosa2 0 = 0

7rI (5.91)

The rotation rates and the axial strain rates are obtained from the above equations

(5.88)

131

0= - -- 6i (5.92)Cosa I-cosao R

U0 = -- 1--) -R cosar 0 (5.93)

The integration of Eq. (5.76) yields to

8M& 47rN Rg 13Pi + 2M + 2No= R -4 3 [1- 2(1-- )3 + 2(1--- )3 +

2MbG + 2Nuo (5.94)

After minimization the final form of load - indentation relation becomes

P [7D 6 a a16 - 1 - 2[(1 - _)3 _ (1 _ 31} (5.95)

Mo 3 t R 7r 7r

For prescribed bending moment and axial force al and a2 can be expressed uniquely

by Eqs. (5.51) and (5.52).

Discussion

When 3 = 0 corresponding to an undeformed tube, the bending moment/axial force

interaction has the relations of Eqs. (5.41) and (5.63). Once the lateral load applies,the boundary AB shrinks to AE and the boundary CD shrinks to GD as shown in

Fig. 5-12. Eq. (5-95) determines the boundary EFG by connecting the instability

points for prescribed bending moments and axial forces. Plastic instability resulting

from the combined loading reduces the area enclosed by the interaction curve, which

means that the combination of bending moment and axial force with lateral

132

indentation precipitates the instability of tubes. The axial velocity tO becomes zero

along the dotted line AF, which was discussed in Section 5.2. The shaded region has

positive axial velocity, while the remaining region has negative axial velocity. The

rates of global rotation are always negative. That is why the upper interaction

surface in Fig. 5-13 is inclined toward the negative bending moment axis.

The three-dimensional full interaction surfaces are constructed by combining

each interaction curve for prescribed bending moment and axial force. The lateral

load/bending moment interaction curve with N = 0 in Fig. 5-14 represents the

intersection of the P - M - N interaction surface with N = 0 plane. The interaction

curve has the maximum value at M = M , which gives the identical results with a

tube free to slide in Case 4b of Table 4-1. At M = 0, the response corresponds to a

tube free to rotate in Case 4c of Table 4-1. The critical bending moment is

MC = -0.2607M . It represents that a tube subjected to lateral load/bending

moment combined loading with N = 0 is more susceptible to plastic instability than

one subjected to lateral load/bending moment combined loading with O = 0 in

Section 5.2.

The lateral load/axial force interaction curve with M = 0 in Fig. 5-15

represents the intersection of the P - M - N interaction surface with M = 0 plane.

As special cases, N = No corresponds to fully clamped tube and N = 0 corresponds

to freely-rotating tube. The critical axial force becomes NC = -0.1921N,, which

means that a tube subjected to lateral load/axial force combined loading with M = 0

is more suseptible to plastic instability than one subjected to lateral load/axial force

combined loading with bo = 0 in Section 5.1. Still, there is lack of sufficient data to

confirm the present findings.

133

N/Np

A11.0

N

NNC

G

1.0

0D M/Mp

Figure 5-12: Bending moment/axial force interaction based onquadratic global strain rate influenced by lateral load.

E

F

/B

-1.0

134

1 N Np

-.0

Figure 5-13: Normalized three-dimensional interaction surfacessubjected to combined loading based on quadratic global strain rate.

M/Mp0 1.0

'-1.000

135

P/P

1.0

N=0

-1.0 -0.2607 0 1.0

M/Mp

Figure 5-14: Normalized lateral load/bending moment interactionwith N = 0 based on quadratic global strain rate.

136

5.3.2. P-M-N Interaction Based on Exact Global Strain Rate

The general tendency of the plastic response based on the exact global strain

rate is the same as that based on the approximate global strain rate. However, the

former shows new phenomena which the latter cannot find, because the distribution

of global strain rate varies with dented depth. The new terminologies were

introduced to explain the phenomena

" threshold value of axial force (or bending moment)

" strengthening - softening behavior in load - Indentation

The global equilibrium equation based on the exact strain rate becomes

PF+2M0,+2N0 = 8M- R + 4NRfI + 0 + R cosa o0 I da (5.96)

i) Case I ( 0>0 )

Typical distribution of global strain rate with one intersection point is plotted in Fig.

5-16. The bending moment and the axial force are related to an intersection point al

in Eqs. (5.39) and (5.40). The zero strain rate at a = a, yields to

W tb0 0

+ O + Rcosao0 = 0 (5.97)

Since uO and are independent of circumferential coordinates, Eq. (5.95) can be

simplified after some manipulation.

8M 4NR3 Z1

P + 2M O + 2N s0 = + I "+2MO+ 2Nu0 (5.98)

where the intersection point al is determined from the prescribed bending moment or

axial force. Again 2M and 2NuO are cancelled out. Eq. (5.97) simplifies to

137

P/PC

10

-0.1921 0

Figure 5-15: Normalized lateral load/axial force interaction with M = 0based on quadratic global strain rate.

.

M=0

1.0

N/Np

1.0t

138

ciJ 0

0

d / 0CD ~ 0(T)

Figure 5-16: Exact global strain rate profile subjected togeneral combined loading

(M/M, = 0.51 and N/No = 0.66).

cD

0

(0

fl

0

(D

0D

Qy-CD 1O

od- z I

CO(D0

0D

-\J

0 0

C-

-J

139

8M 4NOR3 Z

R d6/d (5.99)

The minimization postulate gives the extent of dented zone

SD RZI_R it DRZIo (5.100)R t d/d

Substitution of Eq. (5.100) back into Eq. (5.99) leads to load - indentation relation

P D RZI-- 16 - (5.101)

M it d3/d

w(O)i 'O)ii) Case II (O<0, |Ro0I > )

The present case is plotted in Fig. 5-17. The only difference with Case I is the

expression of Z, which has the form

Z, - ] G(a)dx+ G(a)da (5.102)

iii) Case III (o <0 : two intersection points)

Fig. 5-18 corresponds to two intersection points case subjected to P - M - N combined

loading. The bending moment and axial force are functions of intersection points a

and a2. in Eqs. (5.49) and (5.50). The global strain rates become zero at al and a2

+ U0 + Rcosai, = 0 (5.103)

140

CW (0 - \J + J + (D 0)

rm r rz rm r rm rc rz Q

On on~


(M/MP = -0.39 and N/Ne = 0.74).

I I - I

//

0

z >Oi

//

/

- I

I I I I ~--

CD

(0

0

C')

CL

I I I I

141

Co0

(D0

Figure 5-18:

0

C\J 0 (\J

0D

Exact global strain rate profile subjected togeneral combined loading

(M/M = -0.669 and N/Np = 0.3).

0I-QD/

0/

d/

C) CL

z/.........

z/

0O

(D0

CV)

cr

142

Ja2)U,(a )+ LO + Rcosa2bo = 0 (5.104)

The rate of rotation is given Eq. (5.74) and the axial velocity is obtained from the

above equations

w (aj),(aj)0 = - - Rcosc 1 00 (5.105)

After integration Eq. (5.96) becomes

8M,, 4N R3Z11

P5+2M;6+ 2Nu = R 6+ 4 + 2MG+ 2Nu0 (5.106)

where ZI is given in Eq. (4.49). The same procedures with the previous case give the

expression of the extent of dented region and the load - indentation relation

S D RZII

R tdb/do (5.107)

and

P D RZn- 16 - (5.108)

M, t d3/dq+

iv) Case IV (00<0 : three intersection points)

Fig. 5-19 shows the case with three intersection points. The bending moment and

axial force are functions of intersection points al, a2 and a3

143

CO (D0

Figure 5-19:

0

C\J

0

0

0

(NJ

0

C\JIC)

odCLOL

z II

0

Exact global strain rate profile subjected togeneral combined loading

(M/MP = 0.852 and N/Np = 0).

cc0

(NJ

0

0

0

0

0O

0D

-

-j

144

M

M - sinal - sina2 + sina3 (5.109)

P

N 2

N - - 1+ -(a1 a 2 + 3 ) (5.110)P

The rate of rotation and the axial velocity are given by Eq. (5.105) and (5.106).

Similarly, we obtain the extent of dented zone and the load-indentation relation

S D RZ ,(5.111)

R t d6/dk

P D RZM - 16 - (5.112)M t d3/d4

where Z is given by Eq. (4.24).

v) Case V (bo > 0 ; four intersection points)

Fig. 5-20 shows the distribution of global strain rate for 6/R = 0.5. The whole

derivations are given in Eqs. (4.52) to (4.57) in Section (4.1.3).

Discussion

Fig. 5-21 shows the bending moment/axial force interaction curve based on exact

wow 0/ for different dented depths. As the dented depth goes deeper, the area

enclosed by the bending moment/axial force interaction curve decreases.

145

0D

(NJ

0

0D

on On


(M/M = 0 and N/Np = 0.3).

0)

CD0

(D

CD

CD

CO- d0

Q.Q-

Z

II II /

I I

CO Co

0 0D

I_

146

The lateral load/bending moment interaction with N = 0 is plotted for various

values of dented depth in Fig. 5-22. The critical bending moment under N = 0 is

M/M, = -0.618. The general behavior of interaction curves in Fig. 5-22 is similar

to Figs. 5-4 and 5-9, which means the tube become unstable after the threshold axial

force. The lateral load/axial force interaction with M = 0 is shown for different

dented depths in Fig. 5-23. The critical axial force under M = 0 is N,/N = -0.528.

The three-dimensional full interaction surfaces in Fig. 5-24 are constructed by

combining each interaction curve.

N/Np

1.0

M/Mp

0 1.0

= 0.01

Figure 5-21: Bending moment/axial force interaction based onexact global strain rate for different dented depths.

147

6= 1.0

S= 0.5

/R

/10

NN

NN

-1.0

148

-0.5

-15 =1_

.0.5

.5

6_R = 0.0 1

0.50M/Mp

Figure 5-22: Lateral load/bending moment interaction withN = 0 based on exact global strain rate.

N =0

-1.0 1.0

149

-0.5

M ON[D

-15 (=1.0.15

50.5

15

S=0.01

0.50N/Np

Figure 5-23: Lateral load/axial force interaction withM = 0 based on exact global strain rate.

M=0

-1.0 1.0a

150

20

/

1

1.0

6=0--1.0.

/.0

-.0

Figure 5-24: Three-dimensional interaction surfaces subjected togeneral combined loading based on exact global strain rate.

jpo't

N Np

100M

M Mp1.0

0000.000

151

Chapter 6

Residual Strength of Dented Tubes

The results presented in the preceding chapter offers an attractive possibility of

estimating the remaining axial strength (or bending moment capacity) of the tube

weakened by a local dent of the depth 6. A distinctive feature of the present method

is that the prediction is made on a purely theoretical basis. In the previous analysis of

this problem semi-emperical or emperical methods were used to assess the strength of

the tube in the damaged zone [36], [411. Gellin studied the effect of imperfections on

plastic buckling of short cylinders subjected to axial compression [131. He calculated

the decrease of axial strength as a function of the amplitude of imperfection. These

results are valid, however, for small amplitudes comparable to the wall thickness t.

Also an assumption was made in his analysis that the variation of amplitudes of

imperfections in both axial and circumferential directions could be described by

harmonic functions. By contrast we are concerned with a single dent-like

imperfection.

The present prediction of the residual strength of dented tubes is described by

the following argument. For each fixed axial force (or bending moment), the load-

indentation curves calculated represent equilibrium paths. A family of load-

indentation curves is shown in Fig. 5-6 (or Fig. 5-10). Consider one curve out of this

family exhibiting an unstable behavior. There are two equilibrium points at which the

lateral load is zero. One such point corresponds to 6 = 0 i.e., to the equilibrium state

in the undamaged tube. The resulting uniform compressive stresses are below yield

and the tube remains rigid. Another equilibrium point occurs at 6 = 6*. For any

0 < S < 6* a finite lateral load is needed to deform the tube further. As the dent

152

depth increases under constant axial force (or bending moment), the point 6 = 6*is

reached when the tube collapses at no lateral load. Conversely with a constant

prescribed dent depth one can increase the compressive axial force (or negative

bending moment) until the so-called residual strength of the tube is reached. Plotting

the coordinates of the intersection points of the load-indentation curve with the

horizontal axis for each constant axial force (or bending moment), subsequent points

of the residual strength curve are obtained. This solution is shown in Fig. 6-1 by the

full line. The present theory also predicts the existence of an asymptotic value of the

axial force (or bending moment) meaning that no matter how deep the dent may be,the tube will always support the axial force (or bending moment) smaller than N, (or

M). The accuracy of the present analysis depends to a large extent on the choice of

the ring deformation mode. Our choice is believed not to be good for very small dent

depths. For comparison also shown in Fig. 6-1 is the prediction of the approximate

theory, based on Eqs. (5-17). The tube strength suddenly drops to the critical value

NC (or Mc) and stays constant independent of 6. The actual residual strength of tubes

is clearly underestimated by this approximate theory. However, the asymptotic values

of the critical axial force (or bending moment) are almost the same in the exact and

approximte solutions.

A practical lesson learned from the above discussion in that short tubular

compressive members should be designed for axial load smaller than 0.587 N or thep

safety factor should be greater than 1/0.587 = 1.7. Such a design will be indifferent

to the presence of small or even large dents and therefore will be unconditionally safe

against accidental dents.

For more slender members the rotational deformations comes into play and the

above conclusion is no longer valid, as mentioned earlier in the experiments on the

residual strength of dented tubes performed by Smith [34] pin-pin end condition were

used. It is therefore not appropriate to correlate the present theory with his data. In

the absence of any other experimental data we have reproduced the test points due to

smith, recognizing the limitation of such a comparison. We took only the data

153

Np - 27rooRt

F

N

t

2R

-- exac

- - appr

x expeIS'

-4-

oximaterimentmithi

1.0 1.5 R

Figure 6-1: Residual strength of dented tube versus dented depth.

oWO

NNd

1.01

0.764

0.5871

0.5

0.5

--

a

t

m

154

corresponding to short tubes (small slenderness parameter x) and no or very little

initial overall bending amplitude. It is seen that for shallow dent depth 6 < 0.1R, the

experimental points lie above the present "exact" solution. The present prediction is

too conservative in this range because the details of the local dented zone is not

adequately described by the ring model with four moving hinges. On the other hand,the tested tubes were much weaker for 6 > O.1R than the theoretical curve mainly

because of the large imposed bending moment due to load eccentricity.

155

Chapter 7

Estimation of Shear Effect

An approximate expression for the rate of energy due to shear deformation will

be derived using a simple model of an equivalent tube with rectangular cross-section.

This model has previously been used in Ref. [20] to estimate the rate of extensional

energy in the tube indentation problem. A similar model has been proved useful in

explaining the importance of shear deformation in the problem of confined and

unconfined buckle propagation [50].

Consider a thin-walled square tube whose circumference is the same as that of

the original circular tube 4a = 27rR. The equivalent length of each side is thus equal

to

,rRa = (7.1)

Suppose the tube is subjected to a symmetric denting of the depth 6/2 on each side,

Fig. 7-1-a. It is possible to construct a paper model of such a deformed tube without

cutting the paper along the fold lines to release possible extensions or shear. This

construction provides the simplest proof that symmetric collapse of the square or

rectangular section tube does not produce any shear strains.

By contrast, the unsymmetric sectional collapse, which resembles the tube

indentation mode, cannot be reproduced by simply folding the tube walls along hinge

lines. This means that there must be considerable shear and/or extensions present in

this model. And the unequaled lengths of the generators in upper deformed plane and

156.

a)

b)

C)

Figure 7-1: Conceptual model of a tube showing symmetricsectional collapse and unsymmetric collapse.

157

lower deformed plane support the conclusion about the shape distortion during

deformation. The unsymmetric mode is shown in Fig. 7-1-b. It can be obtained from

the symmetric mode by shearing the side walls through the angle y as explained in

Fig. 7-2. the angle - is related to the geometry of the dented zone by

45= (7.2)

The rate of shear strain is defined as

Sd, 1 d(5/)- (7.3)dt 2 dt

The rate of energy dissipation due to shear is

Shear = t J k ,IdA (7.4)

where k is the yield stress in simple shear and the integration is performed over the

shear affected zone of the equivalent tube. For the Tresca yield condition k = a0/2.

The integral of Eq. (7.4) can easily be calculated assuming the strain rate ' to be1

uniform over the two lateral trapezoidal surfaces of the area - (2a-6) (

02

. JetEher = - 2 (2 a-6)& (7.5)

The above expression involves a time rate of the unknown quantity (. In order to

proceed with the solution further, an iterative procedure will be used.

The function appearing in the definition of * will first be estimated using a

closed form solution derived for the same problem without shear. This will provide a

158

I.

6

a

2+ 2

Figure 7-2: Transition from symmetric to unsymmetric sectionalcollapse through simple shear.

159

unique expression for Eshear which is linear in 6. The calculated shear dissipation will

be substituted back into the global rate of energy balance and the entire solution

process can be repeated. We shall apply this procedure to the case of a fully clamped

tube. Using Eq. (4.2), the strain rate is found to be

S 13t-4R =x (7.6)

=4R 27r v;

Substituting Eqs. (7.1) and (7.6) into Eq. (7.5), the rate of shear energy becomes

e= 1 N ( rR- ) 3t (7.7)hear 2 2 R 2r V

Now, adding the above term into the right hand side of the energy balance postulate

Eq. (4.1) the corrected or first iterative solution is obtained in the form

Pwith shear 0 without shear (7.8)

where Pwithout shear is given by Eq. (4.3) and the correction factor is defined by

R 6 3tR+

= [ 7+- 7(- ---(7.9)8t '27r R b

Consider a tube with the ration D/t = 40.9, as used in Smith's experiments. The

function I(s/R) for few values of the indentation depth 6/R is shown below

6/R 0.5 1.0 1.5 2.0

I 1.568 1.355 1.234 1.147

With increasing indentation depth, the correction factor is decreasing to an

asymptotic value asymp = 1.0. At very small central deflections 6/R, the formula of

160

Eq. (7.9) predicts an unrealistically high resistance of the tube. Clearly, there must be

an alternative mode of tube deformation which suppresses large shear strain rates

developed at the onset of the plastic indentation process. It is plausible that the tube

undergoes initially a symmetric mode even though the force is applied on the top

only, Fig. 7-1-c. The resulting initial symmetric ovalization of the tube section,

provides for a mechanism which eliminates the shear strains and minimizes initially

the denting resistance of the tube. The ovalization of the tube under bending, known

as the Brazier effect, has been observed and reported by many authors.

The infinite value of shear energy is a consequence of the singularity in the

slope of the -b function at b = 0. In reality, however, the length of the dented zone is

initially finite and so must be the contribution of shear. The finite width of the

plastic zone could be predicted by the present theory if the terms MXi, were

retained in the energy balance equation.

In conclusion of the present discussion of the shear effects we can state the

following :

" Shear strain may contribute significantly to the overall rate of energydissipation and thus may rise the resistance of the tube to denting.

* The effect of shear is most pronounced for small dented depth. For deepdents of the order of tube radius shear increases the tube resistance bynot more than 10%.

* The model of the equivalent square tube is too crude to make a realisticestimate on the shear effects for small 6. Thus, no attempt was made toquantify this effect in order to compare it with experimental datarecorded for 6/R < 0.1.

161

Chapter 8

Pinching of Tubes

8.1. Introduction

In the previous chapters the unsymmetric denting of a cylindrical tube has been

studied. In this chapter similar approaches are applied to analyze the symmetrical

denting of a tube, the so-called *pinching" of a tube. Montgomery [18] reported on

an extensive experimental study for different boundary and loading conditions. He

took the diamond shape square ring model (Fig. 8-1b) to analyze the pinching of a

tube, and found that as for deformed shapes the diamond shape model would be more

adequate than the square model (Fig. 8-1a) used by de Oliveira et al [20] . The same

conclusion was also made by the present author from an independent study.

However, the model due to Montgomery is not consistent in a sense that the crushing

force is calculated based on the behavior of a thin ring crushed between flat plates

and the rate of extensional energy is calculated based on the diamond-shaped ring

model.

Lateral line load is applied to the midspan of a tube in opposite directions and

normal to the tube axis. The extent of dented region is calculated as a part of the

solutions. The ovalization as a reversing mode is not considered. The simplified

square section model with symmetrical sectional collapse is shown in Fig. 7-1. The

shape distortion resulting from the unequal extension between generators seem to be

less significant than that in unsymmetrical denting of a tube.

Combination of Eqs. (2.2), (3.2) and (3.5) gives the expression of the global

energy equilibrium equation subjected to lateral indentation force and axial force.

162

a)

b)

Square ring model and diamond shape ring model.Figure 8-1:

163

+N/2, E + tO02 Pi+ 2 N = + 8N 0R +0+ R cosa 0 Ida (8.1)

Two different ring models are proposed and the new solutions are compared with

experiments and theoretical predictions by Montgomery in Ref.[18].

8.2. Theoretical Predictions

8.2.1. Model I

The simplest ring model for pinching analysis is first proposed. It is assumed

that Model I has four moving plastic hinges during deformation as shown in Fig. 8-2a.

Initially, hinges start at AT and AB. As the deformation proceeds, each of these

hinges splits into two hinges travelling in opposite directions. The deformation

behavior of each quadrant for a ring is identical due to symmetry. A hinge B in Fig.

8-2b leaves behind a flat region and goes into a region of curvature 1/r. In view of

the inextensibility postulate, the total perimeter of a ring remains constant. For one

quadrant

7rR 7 r- b + - (8.2)

2 2

where b is the length of flat region of one quadrant and r is the deformed radius of a

circular ring. The deflection 6 is obtained from geometry

6= R - r (8.3)

Differentiating Eqs. (8.2) and (8.3) with respect to time and combining them give

b= - r (8.4)2

AB

r\

Geometry of deformed ring model I.

164

a)

6I

b)

aR

r

A-r

Figure 8-2:,

165

The energy dissipation in a ring is composed of contribution due to continuous

deformations and discontinuous velocity fields in moving plastic hinges

E . 2=4 MO -+ 27rM. ( - )1 (8.5)r ~ r2

or using Eqs. (8.2) and (8.3)

6E . = 4M - (8.6)ring o R-6

Eq. (8.6) reveals that Erig is a function of dented depth only. The crushing force per

unit width is obtained from the rate of energy balance

M = /R (8.7)4 MO /R I- 6 R

The total rate of energy in crushing is obtained by integration of E ring over the

length of the dented zone. It is assumed that the velocity field of the leading

generator varies linearly along the tube axis and the deflection profile of the leading

generator is quadratic to the first approximation (see Eqs. (3.28) and (4.7))

cr.ush = 4 6 In (I- ) 6 (8.8)

The displacement w./R and the global strain rate due to crushing wowi/R 2 for ring

model I are derived in Appendix C. They are plotted in Fig. 8-3 and Fig. 8-4 and the

corresponding intermediate shapes of deformed rings are shown in Fig. 8-5. Curves

are symmetric with respect to a = 7r/2, which was different from the unsymmetric

denting of a tube. The rate of external energy becomes

166

u~) ~I- (V) Cu

0000

dj/O0flY

Figure 8-3: Displacement of material points at midspanof ring model I.

0

CD

S I

d I O

lC;*oIf,

0 0)

0

CO (D0

0

0

00

CL

C

(D

CD~

CD

167

00

0D

(D nO 1f - 00 (N - 0

I I00

-ic

- .olN

0

(0

CD~

Figure 8-4: Global strain rate for different dented depthsof ring model I.

TC--J

0 0)

0

6 = O.25

= 0.50

bY

6 = 0.75

Figure 8-5: Intermediate deformed shapes of ring model I.

168

+

- a - I - V

4/00007

/0000

169

8N R2Z

E = (8.9)

where the area Z under the extension curve for a = 0 to a = 7r / 2 is calculated in

Appendix C.


The ends of a tube are constrained against rotation and axial displacement. The rate

of external work is reduced to Eext = 2Pi from Eq. (8.1). Since u0 = 0, the global

equilibrium equation for a fully clamped tube should be

4;r M0 . 8N R2 Z0 0

2P = - n(1-- ) 6+ 6 (8.10)3 R

apThe optimality condition = 0 leads to

4 8 z - = - - Z -(8.11)R 7r t In ( 1 - 6 / R)

By combining Eqs. (8.10) and (8.11) the load-indentation relation has the form

2 7 Rln( I- 61R)P = 4Ao t2 - -- (8.12)

* 2 t 6 /R

Tube free to move axially but restrained from rotation

The number of intersection points for one quadrant will be one or two depending on

dented depth of a tube from the characteristics of global strain rate as shown in Fig.

8-6.

i) Case I (one intersection point)

170

For large dented depth (approximately 3/R > 0.5), the number of intersection point

becomes one and the position should be al = 7r/4 to satisfy zero axial force (Fig.

8-6a). The axial velocity is obtained by the condition at a = 7r/4

-+ = 0 (8.13)

Since 60 is independent of the circumferential coordinates a, the integration of Eq.

(8.1) can be performed easily

47r' M f . 8 N0 R2 Z,2Ps = - ln(1--)3+ 3 (8.14)

R

where

Z, = H(a) da - / H(a) da (8.15)

and

W 0(a)w 0 (a) 1H(a) R2 (8.16)

The extent of dented region and the load-indentation relation have the same form

with Eqs. (8.11) and (8.12) except Z1.

ii) Case II (two intersection points)

For small dented depth (approximately b/R < 0.5), the number of intersection points

become two as shown in Fig. 8-6b. The condition N = 0 is satisfied only if

a2 -1 = a(8.17)1 4

and

171

Wo\I~I

0 11/4 fl/2 a

a)

YYQ

Cio --

o a1 a 2 n/2 U

Figure 8-6: Variation of intersection points for differentdented depths with N = 0 for ring model I.

172

W (a J)zb (a ) + u = 0(8.18)

W (ce bo(a2) .O + u = 0(8.19)

After integration of Eq. (8.1), the rate of energy dissipation for genarators has the

form

8N0 RZJJFg = en (8.20)

where

Z = 'H(a) da - 2H(a) da + J'H(a) da (8.21)

8.2.2. Model II

We shall consider now a more realistic model based on the observation of the

deformed shape of a ring subjected to pinching load. It has eight moving hinges as

shown in Fig. 8-7a. Hinges (AT and AB) moving down on undeformed surface leave

behind a curvature -1/R1 , and the region of side hinge (BR and BL) leave behind a

region of curvature 1/R 2. Due to symmetry, one quadrant of a ring is considered.

Points A and B shown in Fig. 8-7b are originated at the material point AO and BO,

respectively. For one quadrant

R R + R2 +R(-+s- ) = -R (8.22)1 2 2

The deflection of the loading point should be

173

BL)

b)+

Geometry of deformed ring model II.

AT

4 BR

AB

R1

R B\ R2

Figure 8-7:

174

6 = (R-R 2 )sin " (8.23)

Two of five parameters R 1, R2, e, 6 and 3 can be eliminated from Eqs. (8.22) and

(8.23). Two more relations between parameters involved has to be established. It is

suggested that smaller radius R2 starts from the original radius R and decreases to

zero as the deformtion progresses up to 6 = R, while larger radius Ri increases to

infinitely from the initial value R2 = R. Thus, the radii Ri and R 2 change

continuously during deformation. The expression of Ri and R2 are suggested to be

r= 1 - (-)R

1r=

r 2

(8.24)

(8.25)

where

Ri

ri = -RR2

and r2 - R (8.26)

and the exponent n is considered as constant during deformation. Again n is not

determined uniquely from the present analysis alone. The rate forms of formulas

(8.24) and (8.25) are respectively

n= -12

(8.27)

(8.28)

175

The energy dissipation of a ring are composed of contribution due to continuous

deformations and discontinuous velocity fields in moving hinges.

11 1 R

fM0 R2a (----)|] (8.29)

R2i

where

VA - h, f + R, (8.30)

VB - h 2 p+1L 2 p (8.31)

The crushing force per unit width is

c 1. 1 . .1 r24 /R= - [(r-+rj ) (-+1)| + |(r2 +r 2) (--1)| + Ir + rf -1

4 M, / R 2 r r r2 rI r2= h(n, 6) (8.32)

A parametric study is performed to find the dependence of the crushing force on the

magnitude of n. Crushing forces are plotted in Fig. 8-8 as a function of dented depth,

and the corresponding intermediate deformed shapes of ring model II are plotted in

Fig. 8-9 and 8-10. Unlike the previous unsymmetrical deformation ring model in

Chapter 3, the ring model with larger n gives the more realistic deformed shape. The

present ring model is sensitive to the value of power n. The crushing force for each

power n has the same initial value

4 MP (0 ) = R (8.33)

10

hW

-

n=o.

99o

6n

=0.

75

aq

CIO

- n=

0.5

0oq

n-

-0.2

5s

2 --

0

n =o

.o1

0 0.0

0

.1

0.2

0

.3

0.4

- 0.5

0

.6

0.7

0.8

0

.9

1.0

DENTED DEPTH

/ RADIUS

177

S= o.o16R~- 0.25

= 0.75

Figure 8-9: Intermediate deformed shapes of ringmodel II (n = 0.01).

--L= 0.50

178

-4- =w= 0 .2 5

+{ 0.50

- =0.75

Figure 8-10: Intermediate deformed shapes of ringmodel II (n = 0.5).

f =0.5

179

and increases with different rates for different powers n. The smaller power n gives

the lower value of crushing force of a ring model. However, it does not indicate the

lower pinching force, since the contribution of energy from generators also has an

effect on the total indentation force. The total crushing energy based on the

quadratic approximation of a deflection profile in Chapter 3

16M YEcrush = R 5(8.34)

where

YIgj = fo h(n, 6) (1-- dx (8.35)

The derivation of w./R and w0 ,' V/R 2 based on ring model II are contained in

Appendix D. Plots of w./R and w.Nr0/R 2 are shown in Figs. 8-11 and 8-12 for

different values of n. The rate of extensional energy becomes

7r/2 1 0 bo .Egen =8N R f+ u da (8.36)


The ends of a tube are constrained against rotation and axial displacement. The rate

of external work is reduced to Eext = 2P from Eq. (8.1). Since 60= 0, the global

equilibrium equation for a fully clamped tube should be

16 M Yg . 8 N R2 7 C2P& = 6+ 3 (8.37)

where Z,,e is the area under the extension curves from a = 0 to a = 7r/2 and is

calculated in Appendix D.

The minimization postulate leads to

m1.

0 I

I

r..

0.9

-_

n =

0.0

1

0.8

--

n

= 0

.99

.T

-=0.7

5

0.'

7

-R 0.6

=0.

50

R

0 . -3

0.

25

0.2

0.1

0

00.

0.1

0.2

0.3

0

.4

0.5

0

.6

0.7

0.8

0

.9

1.0

ALP

HA

P

I

1.0

0.9

0.8

(\J0.7

0.6

0.5

0.+

o

0.3

0.2

0.

1

0.0

n =

0.0

1

---

n

=0.9

9 -

N- 0.

75R

6=0.

5

-/

-

=0.2

57

IR

0.0

0.1

0

..2

0.3

0.4

- 0.5

0.6

0.7

0.8

0

.9

1.0

ALP

HA

/

PI

00km 0 oq 0

00

182

_ _ZIICR

- 2 (8.38)R Y t

and

RP = 16 M0

2 YZcT_: (8.39)

The load-indentation curves of a clamped tube for ring model II are shown in Fig.

8-13 for different n. Small power n gives low force response, while large n gives more

realistic deformed shape. Except n = 0.99, the effect of n on the load-indentation

characteristics is not large, particular for small dented depth. According to Fig. 8-14

smaller n leads to larger extent of dented zone. It means that the dented zone for

smaller n may propagate easily, as the load increases. From now on the ring model

with n = 0.5 will be used to obtain the plastic response of a tube subjected to pinch

load.

Tube free to move axially but restricted from rotation

The same procedures with ring model I are applied to calculate the extent of dented

region and load-indentation reltion. The only difference is the expression of the rate

of extension H(a) resulting from the different deformation mode.

8.3. Comparison between experimental results and theoreticalpredictions

In this section experimental results and theoretical predictions by Montgomery

are compared with the predictions from model I and model II. All tubes used for

experiments in Ref. [18] have 2 inches of outer diameter and 1/16 inches of wall

thickness, which give R/t = 16. Two nondimensional parameters are introduced to

compare with Montgomery's results directly.

183

iF)Nd

0Cd

if)-D

if) 0D

U')d

d

C5C

( * * 4 t* 0D)

Figure 8-13: Load-indentation characteristics of a clampedtube for ring model II.

-)

(D0

0

0-Th-4

cm

H-t1~wcmcmwF-

cm

0C')

184

I I I I I

01)

C5C

U)U0

C5 CI I

C

CI)(-C U-) -f (V (\

sn i vai

--- 00C

./ I sA

Extent of dented region of a clamped tubefor ring model II.

0O

(VO

C')

0

6 i

-)

Figure 8-14:

185

" extent of dented zone : / R

* lateral load :P/ ( t2 )

Extent of dented zone

From the experimental observations for tubes free to move axially but restrained

from rotation, Montgomery identified four collapse modes depending on the ratio

L/D ; ring mode, transitional mode, reversing mode and localized mode. The present

analysis corresponds to localized mode, which belongs to 6a long tube* (L/D > 4.5).

Figs. 8-15 and 8-16 show the extent of dented region of a fully clamped tube and

freely-sliding tube, respectively. The present prediction for extent of dented region

corresponding to n = 0.5 locates between F = 0 and F = 1 curves by Montgomery,where F is the emperical constant reflecting the strain-hardening effect. In both cases

the theoretical predictions by Montgomery and the present author underestimate the

experimental measurements. Experiments in Ref. [18] show that the extents of

dented region does not change significantly for different boundary conditions.

Montgomery suggested from experimental results that instead of the dent length

increasing smoothly from zero, the dent will form at a set of distance from the

midspan and then begin to grow outwards. He added the constant values determined

approximately from experiments to shift dented length curves. Calladine commented

in a study of a buckle propagating in a confined, externally pressurized pipe line [5]

that the geometry of travelling hinge patterns cannot always be determined reliably

by minimization of the energy absorption equations with respect to the pattern

parameters.

Load-indentation

Load-indentation curves are plotted in Figs. 8-17 and 8-18. The slope of the

experimental curves for small dented depth is noticeably steeper than that of the

theoretical curves. Therefore, in Ref. [18] the idea that the dent length grows from an

initial value is incorporated in the theoretical load-deflection curves with the addition

of a constant term. Present calculations predict the load-indentation relation well.

186

In particular, the predictions for large dented depth are excellent due to the increase

of crushing resistance compared with Montgomery's calculations. As observed in

Chapter 7 the shear effect can be small resulting from the symmetric ovalization of a

tube. That may be the possible explanation why the correlation between present

theoretical predictions and experimental measurements is somewhat better than that

in unsymmetric indentation.

187

0 12"

o 16"o 20" t

--- Montgomery 08- Present (n = 0.5) 0

0

o 0

00

OF=O

F =1

4-model 1I

2-

6/R0 0.2 0.4 0.6 0.8

Figure 8-15: Comparison between theoretical predictionsand experimental measurements

extent of dented region of a clamped tube.

188

RtR8- 24"' tube R--- Montgomery

Present (n= 0.5)

6-

F 0

model 11

F=O

model 12-

I I I

0 0.2 0.4 0.6 0.8 6/R


extent of dented region of a freely-sliding tube.

189

Pu0ot2 0 12"

30 A 20" R =16--- Montgomery

Present (n = 0.5)

model IA

20model 1\

F

10- F 0----

6/R

0 Q2 0.4 0.6 0.8


load-indentation characteristics of a clamped tube.

190

P

* 8"/

30- 1 ""o 16" R =16

--- Montgomery tPresent (n = 0.5)

20model I

model 11

6/R

0 Q2 0.4 0.6 0.8


load-indentation characteristics of a freely-sliding tube.

191

Chapter 9

Conclusions and Recommendations

The most important and new conclusion of the present paper is that the

resistance of the tube to the local indentation by a rigid object strongly depends on

the type of boundary conditions. Several typical boundary conditions were examined

including axially and rotationally unrestrained and restrained tubes. The variation in

the axial force covered the whole possible range from fully plastic tension N to fully

compressive squash load -N . The same variations was also applied to a tube

subjected to bending moment. Simple closed-form solutions were derived for tubes

with different boundary conditions. All solutions considered in the thesis were chosen

to be of the same functional form with different numerical factors. Exact solutions

based on the constant crushing force were obtained and compared with approximate

solutions.

Present calculation revealed the existence of a critical magnitude of the

compressive axial force (or bending moment). The load which is at or above critical

load -N < N < N, (or -M < M < Me) causes a sectional collapse of an initially

undeformed (perfect) tube in the unsymmetric mode. By imposing initial "dimple-

like" imperfections or local dents, the strength of the tube is further decreased. A

practical implication of present findings is that short tubes with rotational restraint

should be designed to transmit not more than 1/2 of the fully plastic load carrying

capacity. Such a design would appear to be insensitive to the local dents, even of

quite large amplitudes. The members can take larger axial force (or bending moment)

up to approximately -0.764N (or -0.778M ) but the safety factor will gradually

diminish with increasing dent depth. Tubes loaded beyond this range will

unconditionally collapse.

192

The present solutions for the load-deflection characteristics of dented tubes

were shown to be within 10 - 20% of the experimentally measured values. This

relatively high degree of correlation together with the fact that the shape of the

dented zone is accurately predicted by the theory justifies the practical value of

present analysis.

Three-dimensional interaction surfaces were constructed by combining each

interaction curve corresponding to lateral load/axial force, lateral load/bending

moment, or lateral load/bending moment/axial force interaction. It is shown that

the lateral load/axial force interaction curves with 00 = 0 are projections of the

three-dimensional interaction surfaces onto M = 0 plane, rather than intersection of

the surfaces with M = 0 plane. Similarly, the lateral load/bending moment

interaction curves with uO = 0 are projections of the three dimensional interaction

surfaces onto N = 0 plane rather than intersection of the surfaces with N = 0 plane.

Future Research

The effect of shear energy dissipation on the plastic response of a dented tube was

studied approximately in Chapter 7. The dissipation in a thin-walled tube may be

concentrated on the localized zone (Fig. 9-1) rather than spreaded over the whole flat

part of the dented tube (Fig. 4-14). Therefore, the rigorous study on shear energy

dissipation in the localized zone should be done.

Structure is an assemblage of elements acting together, so that a systematic

study of models containing several elements, or even a complete three-dimensional

tubular structure is needed.

As an extension of the present study, work on stiffened tube subjected to

combined loading is also suggested.

193

Fig. 9-1. Localized shear-affected zone in the dented tube.

194

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44. Walker, A.C. and David, P., Effect of Impact Loading on Denting ofTubulars, Unpublished Report, Kenny, J.P. and Partners Ltd., London,Englnad.

45. Walker, A.C. and Kwok, M., Process of Damage in Thin-WalledCylindrical Shells, Proc. 5th OMAE Symposium, April, 1980, Tokyo,Japan.

46. Watson, A.R., Reid S.R. and Johnson, W., Large Deformations of Thin-Walled Circular Tubes Under Transverse Loading - III, Int. J. Mech.Sci., Vol. 18, 1976, pp. 501-509.

47. Watson, A.R., Reid, S.R., Johnson W. and Thomas, S.G., LargeDeformations of Thin-Walled Circular Tubes Under Transverse Loading -II, Int. J. Mech. Sci., Vol. 18, 1976, pp. 387-397.

198

48. Wierzbicki, T. and Bhat, S., A Moving Hinge Solution for AxisymmetricCrushing of Tubes, Int. J. Mech. Sci., Vol. 28, No. 3, 1986, pp. 135-151.

49. Wierzbicki, T. and Bhat, S., On the Initiation and Propagation of Bucklesin Pipelines, Int. J. Solids Structures, Vol.22, No.9, 1986, pp985-1005.

50. Wierzbicki, T. and Bhat, S., On the Transition Zone in UnconfinedBuckle Propagation, Proceedings Symp. on Current Conference andExhibition, Feb 23-28, 1986, New Orleans, in print, J. Energy ResourcesTechnology.

51. Wierzbicki, T. and Suh, M.S., Denting Analysis of Tubes under CombinedLoadings, MITSG 86-5, March, 1986.

199

Appendix A

Crushing Force for an UnsymmetricallyDeformed Ring Model

A complete set of dimensionless equations which describe the crushing behavior

of a unit width ring is:

The rate form of smaller radius r2 becomes

(A.1)2 = n ( )-0 00

The rate form of larger radius r1 becomes

[2(n--sin$)+r2(1+cos$)](0+sin$)-[7-r2( --- sin$)](I+cos$)

(O+sin#)2

(A.2)

The dented depth is expressed in terms of r, r2 and $

I -- ir(1-cos#) - rsine - 2(1+cos#) + r2 sino (A.3)

The velocity of lower moving plastic hinge becomes

V, = - 1# - r, (A.4)

200

The velocity of upper moving plastic hinge becomes

9 == ( i- 2 )sin# + (r1-r2)coss

The crushing force has the form

1 r 2- -)+ 1(7r-)-

1 r2+1

PC

4M0/R

(A.5)

1

~ 2i[

V2

r,

__1+| V(-

r2C 2z(A.6)

201

Appendix B

Rate of Extensional Energy for anUnsymmetrically Deformed Ring Model

B.1. Calculation of w 0/R and w,/R2

As shown in Fig. 3-1, any point on the original cross section suffers a

translation wo(a) as the section deforms. Each point on the original cross section

defined by the angle a. Thus, for a given value of the local coordinates s, the

distance along the circular are to the vertical axis is equal to Ra, where R is the

radius of a circle. Assuming that the section is inextensional in the circumferential

direction, then on the deformed section, the new location of this point is such that it

lies also at a distance along the deformed are from the vertical axis.

(i) 0 < a < (r1-r2)sin#

The original coordinates of any point are

X = R sina (B.1)

Y = R cosa (B.2)

The new coordinates of any point are

X = R a (B.3)

Y =R-8n (B.4)

202

The translation becomes

W(02 =

(a - sina)2 + (1 - - cosa)2

R?

Differentiation with respect to e gives the rate of global strain due to crushing

0 0

R2

(ii) (rl-r2 )sino < a < 7 - r1#


X = (RI - R2 ) sino + R2 sino

Y R-6-R 2 (1 ~cos5)

where

R

R2- (,-: - 1) sin#

R2

Similarly,

= [ (rI - r2)sin$ + r2sino - sina]2 + [- -r 2(1-cos6)-cosa]2

(B.10)

= [(rl-r2)sino+r2sino-sinal ( )sinO+(rj-r)cos sinO+

r2cosm]d + [I-R-r (-cos)-cosce [-R~2(1 coso)-r2 in# ]

(B.11)

(B.5)

(B.6)

(B.7)

(B.8)

(B.9)

w0)2

ww1100

(1 -6-Cosa) ( ~)

203

where

- a

r2 2

r Iir 2( I1r)2 r cI- sin$ - ( - - 1) COS$

r 22 r 2

(B.12)

(iii) 7r-ro$ < a < Ir


X = R sin(O-0)

Y= R, [1-cos($-) J - R

where

R

1

Similarly,

(')2R

0 0

(B.13)

(B.14)

(B.15)

= [rIsin($-tp)-sina]2 + {r1 [1-cos($S-tp)] - 1 - cosa}2 (B.16)

[rsin($-ip)-sina [r1sin($-P)+rjcos($-ip)(1-p)J+

{rl[1-cos(e-0)]-1-cosa}{r1[1-cos($-pO)]+rlsin($-p)(1-o')} (B.17)

where

S= (x-a) - +1 (.r22 (B. 18)

204

B.2. Calculation of rate of extensional energy

The rate of extensional energy is obtained by the calculation of the area under

the curve of wowe/R

(i) 0 < a < (r1 -r 2)sin$

A1 - a (1 -- cosa) (- ) dajo R (B.19)

where

af = (rl-r 2) sine (B.20)

(ii) (r,-r 2)sine < a < n-r 1

A 2 = Jf{ [(r1 -r)sin+r2 sin -sina[ (r -K2)sino+(rl-r2 )cos +rsino+

r2cos##J+[1 r2(1-cos #)-cosa] P r2(1-cos#)-r 2sinoj }da

(B.21)

where

f = 7r-r 4 (B.22)

(iii) r-r1 # < a < 7

A 3 = J t[risin(O-7)-sina] [b'sin(#-$)+rjcos(#-7)(1- )j +ff

{rj[1-cos($-p)]-1-cosa} {[1-cos(#-)j+rIs(-)(-)} da

(B. 23)

Z = Al +A 2 + A3 (B.24)

205

Appendix C

Rate of Extensional Energy for aSymmetrically Deformed Ring Model I

C.1. Calculation of w./R and w.',v/R 2

Similar procedures with Appendix B are performed based on the ring model I in Fig.

8-2.

(i) 0 < a < b/R

The original coordinates of any point are

X= Rcr

YT =Ra-


(C.1)

(C.2)

W( -)2 = ( -sinR

a )2 + (1 COS a)2R

The rate of global strain due to crushing becomes

0 0

R2

6 R= ( 1-- cos a)(--)

R R

(C.3)

(C.4)

206

(ii) b/R < a < rn/2


Xn = - 6+ R - 6) sin

Y= (R -5)cosfi

where

R

R a 2 ( R-)

Similarly,

r r3

2R

7r b

S2R

R

3- - )R

sin # - sin a ]2 + [ ( 1 -

(1 - )sinf - sinaiR

R) cos 0 - cos a

7r3

2 R

3.sin

R

cos 0 - ( 1

) cos 6 - cos a ]2

3+ (1 - -)cosiI] +

R

) sinR

__ R

R. R-.(C.10)

2(R -) 2

(C.5)

(C.6)

(C.7)

w

R

0 0

R2

(C.8)

where

(C.9)

207

C.2. Calculation of rate of extensional energy

(i) 0 < a < b/R

a WW to 1A1 =o ff

AhR 2 6/R

where

?r 6

(ii) b/R < a < 7r/2

7r/2 Wo 'b 1A 2 a 2 ( )da

fc f R? / R

Z, = Al + A 2

(.11)

(C.12)

(C.13)

(C.14)

208

Appendix D

Rate of Extensional Energy for aSymmetrically Deformed Ring Model II

Similar procedures with Appendix B are performed based on

8-7.

D.1. Calculation of wO/R and wo,/R2

the ring model in Fig.

(i) 0 < a < r, f


Xn = Ri sin f, (D.1)

Y = (R+R)cos g - (R- R2 ) sin # - R O S

where

r (D.3)


[ r, sin f, - sin a ] 2 + [(1 + r1 ) cos - ( 1 -r 2 ) n -

r Cos q - cosaJ 2

(D.2)

w0)2

(D.4)

209

The extension becomes

w w0

- (r sinc1 -sina)(rjcosq risinq)+[(I+rI)cos -(1-r2

r cosq1-cosa] [-(1+rl)sin

2 sin- 1co 1 + rIsinq

+ rcos - (1-r2)cos# # +li]

where

r I

1

(ii) r, a .< i/2-r 2


Xn = (R + R,) sin + R sin ( 2 -

Y R cos (R - R2 ) sinfi

where

2= a -r


= [(1+ r, ) sin c+sin ( -2

(1 - r2 ) sin f - COS a2

C) - sin a 12 + [cos ( C2-

(D.5)

(D.6)

(D.7)

(D.8)

w 0

(D.9)

0 -

(D.10)

210


o 0

R2((1 + r,)sin + sin( 2 - c) - sinal [(1 + rj)cos + isin

+ cos( 2 - f) (2 - c) I + [cos(c2- ) - (1-r2) sin a - cosa

[-sin( 2 -) ( 2- ) - (1-r2)cos# f + r2sino ]

where

(iii) 7/2-r2i< a < r


Xn= (R+R, )sin f+(R-R 2 )cos0+R2COS(0-3 )

Y = R 2 sin(,-~f3)

where

r 21+ r ) ;--

r2 r22


)2 = + r, ) sin S+ ( 1 - r2 ) cos 0 + r2 cos ( 0 - 3 ) - sin a 12+

[ r2 sin ( 0 - 3 ) - COS a ]2D

(D.11)

(D.12)

(D. 13)

(D.14)

(D. 15)

(D. 16)

211


0 00 0

R2= [(1+r,)sin +( 1-r 2 )cos 0 +r 2 cos(f-f 3 )-sinaI

[(1 + r, ) cos *+ rI sin - ( 1

r2 sin(

[r2 Cos

- 2

r2 2

- $(3)(i ~-

.3 ) (

S3) + 2cos( ~ c3) - + [r2sin(fi

~ 3) + 2 sin( - C

r2

D.2. Calculation of rate of extensional energy

(i) 0 < a < r1I

1(- )da

(ii) rf _< a < /2-r20

A = [' W(2 Jrf R2 %5/R

where

2 - ri

()

where

- r2 ) sin # # - r2 cos -

3) - cosa]

(D.17)

r2

2r2+( (D.18)

A 1 =/ o

I fo R2(D.19)

)da (D.20)

-

r2 2

(D.21)

212

(iii) r/2-r2# < a < 7r/2

7r/2wo WO IA R2 ( )da (D.22)

Z I A, R 2 + /R

Zy =A, A2 A3(D.23)

213

Appendix E

Lateral Load/Axial Force InteractionFormula by Normality Requirements

The global equilibrium equation based on quadratic extension rate becomes

Pi+ 2 Nti =

We discuss the case 1nej

8ME

R +4N R]

<I.~1.In nondimensional form Eq. (E.1) becomes

4D-L P + 2nn--

4t

rD2

25

34

85+ 21 + -i (-- )3/2

3 5

where

N N

= 21rDot0 0

U0

y1 = -17 6

P

0

I (,_')2 + n da (E.1)

(E.2)

(E.3)

(E.4)

(E.5)

214

4D2o tP =0 L

(E.6)

The normality requirements lead to

6

0

4L OF/op

2D OF/O9n

irL/2D

dp/dn

1= - (E.7)

Taking derivative of Eq. (E.2) with respect to p and using the result Eq. (E.7), ordp 7rL

dn - - ~ ,, we obtain

n = 1 - 2 (E.8)

Combining equations (E.2) and (E.8) we get

4D

L

4tQ 26 1- + -[1-- (1-n)3

irD2 34 4(E.9)

Eq. (5.15) is recovered from Eq. (E.9). The above approach was first suggested in Ref.

[20].

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SUBJECTED TO COMBINED LOADING