STUDY ON

THE SHEAR BENDING PROCESS

OF CIRCULAR TUBES

Doctoral Dissertation

MOHAMMAD GOODARZI

****************************************************************************************************************************

Department of Mechanical Engineering and Intelligent Systems

The University of Electro-Communications

2007-March

ACKNOWLEDGEMENTS I wish to express my sincere thank to my supervisors, Prof. Makoto Murata and Associate

Prof. Takashi Kuboki for taking the time to mentor and tutor me throughout the years of my

graduate study program. Their insight, wisdom, support, valuable advises and trust were

indispensable.

I also would like to give special thank all of my friends at Murata-Kuboki Lab. especially

Mr. K. Takahashi, Dr. J. Yao, Dr. T. Makiyama and Dr. Y. Ying.

Further, I would like to give my deep gratitude to the staffs of Technical Division especially

Mr. Nakazawa, Mr. Murakami, Mr. Saito, Mr. Arakawa and Mr. Tabata for their invaluable

assistance in technical areas.

Finally, I would like to thank the SANGO Co. Ltd. for technical supports.

ABSTRACT

Cold bending of metal tube products is one of the oldest metal forming processes and

the bent tubing parts are widely used in industries. Applying conventional tube bending

methods, the minimum bending radius is almost more than 1~2 times of the tube diameter

even using mandrels. Tube shear bending is a beneficial technique to realize the production

of unified and compact bent tubular parts through cold metal forming. It is an appropriate

technology to realize a considerable small bending radius, which is very difficult to be

achieved by conventional cold-bending methods.

In this research, the process of shear bending was studied both by experiments and

numerical simulations. In this work, mandrels were used inside the circular tube. Moreover,

an axial pushing pressure was applied on the tube. The main experiments were carried out

using A1050 circular extruded aluminum tubes. A 3D explicit analysis was conducted

using a commercial finite element code ELFEN.

The deformation behavior of a tube subjected to the shear bending process was studied.

In was found that during the process a combination of shearing and bending deformations

occurs. In this manner, the lateral side of the tube undergoes shearing deformation whereas

the deformation mode around the top and bottom sides of the tube is bending.

The effects of the axial pushing pressure on the process were examined. It was found

that a limited range of appropriate pressures to perform a successful forming process exists.

If the value of the applied pushing pressure is not selected within the appropriate range,

rupture or wrinkle occurs. Therefore, in order to perform the forming process successfully

and obtain a sound product without any failure, the amount of pushing force should be

appropriate.

The effects of the die corner radius on the process were investigated. It was found that

the appropriate pushing force is almost constant regardless the value of the die radius.

There is a limit range of die radii suitable for performing the process. Forming on dies with

radii larger than a critical value results in only rupture or wrinkle. The effect of the die

radius on thickness of the deformed tube is low. However, larger die radius decreases the

cross section ovality. Whilst a small bending radius results in high cross section

deformation, increasing the die corner radius the wrinkling tendency of the tube increases.

The effects of the initial thickness on the process were investigated. Increasing the initial

thickness, the forming limit of the tube expands. Employing higher pushing pressure

within the forming limit, the amount of thickness reduction decreases.

The effects of the material properties on the process were investigated. The experiments

were performed using copper and two kinds of aluminum tubes. Forming limits of tubes

with different materials were obtained. The experimental results show that implementation

of a successful shear bending process is feasible by providing sufficient elongation. The

simulation results indicate that the smaller the hardening exponent, the larger the shearing

deformation is. Consequently, more uniform thickness distribution can be obtained.

The effects of applying an eccentric axial pushing force as a way to prevent the tube

from extreme thinning were examined. Finite element simulation proves that exerting an

eccentric load is effective only when the tube is short enough.

CONTENTS

CHAPTER 1 INTRODUCTION

1.1 Introduction to metal forming I-1

1.1.1 Forming methods I-3

1.2 Bending deformation I-4

1.2.1 Mechanism of bending deformation I-6

1.2.2. Bending factors I-7

1.2.2.1 Tube wall factor I-7

1.2.2.2 Bending factor I-8

1.2.2.3 Minimum bending radius I-8

1.2.3 Difficulty in tubing process design I-8

1.2.4 Bending equipment I-9

1.2.4.1 Mandrel I-10

1.3 Tube bending methods I-10

1.3.1 Rotary draw bending I-13

1.3.2 Compression bending I-14

1.3.3 Ram bending I-15

1.3.4 Press bending I-16

1.3.5 Stretch bending I-17

1.3.6 Roll bending I-18

1.3.7 Push bending I-19

1.3.8 Laser bending I-19

1.3.9 Air bending I-20

1.3.10 Bending using a polyurethane pad I-21

References I-23

CHAPTER 2 FUNDAMENTALS OF THE SHEAR BENDING PROCESS

2.1 Introduction II-1

2.2 Experiments II-4

2.2.1 Experimental set up II-4

2.2.2 Experimental conditions II-8

2.2.2.1 Material properties II-8

2.2.2.2 Dies II-9

2.3 Finite element simulation II-9

2.3.1 Simulation model II-11

3.3.2 Failure criteria II-14

References II-15

CHAPTER 3 DEFORMATION BEHAVIOR OF A TUBE SUBJECTED TO THE

SHEAR BENDING PROCESS

3.1 Introduction III-1

3.2 Deformation behavior and strains distributions III-1

3.2.1 pure bending process III-1

3.2.2 Pure shearing III-2

3.2.3 Actual shear bending III-4

3.3 Conclusion III-12

CHAPTER 4 EFFECT OF AXIAL PUSHING FORCE ON THE SHEAR

BENDING PROCESS

4.1 Introduction IV-1

4.2 Preliminary experiments IV-1

4.3 The effect of the pushing force on working loads IV-5

4.4 The effect of the pushing force on the distribution of thickness strain IV-8

4.5 The effect of the pushing force on the cross section deformation

of the deformed tube IV-11

4.6 Conclusions IV-12

CHAPTER 5 EFFECT OF DIE CORNER RADIUS ON THE SHEAR BENDING

PROCESS

5.1 Introduction V-1

5.2 Experimental conditions V-2

5.3 Simulation parameters V-2

5.4 Formability of the tube V-3

5.4.1 Preliminary experiments V-1

5.4.2 The effects of the die radius on the deformation V-5

5.4.3 FEM results of the tube formability V-6

5.4.4 Results of the experiments V-9

5.5 Dimensional accuracy V-12

5.5.1 Cross section deformation V-12

5.5.2 Thickness change V-14

5.6 Conclusions V-16

Reference V-17

CHAPTER 6 EFFECT OF TUBE INITIAL THICKNESS ON THE SHEAR

BENDING PROCESS

6.1 Introduction VI-1

6.2 Experiments VI-1

6.2.1 Preliminary experiment VI-1

6.3 Results of simulation VI-3

6.4 Deformation behavior VI-5

6.5 Forming velocity VI-7

6.6 Forming energy VI-8

6.7 Forming limit VI-11

6.8 Forming accuracy VI-12

6.8.1 Cross section deformation VI-12

6.8.2 Distribution of thickness strain VI-13

6.9 Conclusions VI-17

Reference VI-19

CHAPTER 7 EFFECT OF MATERIAL PROPERTIES ON THE SHEAR

BENDING PROCESS

7.1 Introduction VII-1

7.2 Experiments VII-2

7.2.1 Forming limit VII-4

7.2.2 Thickness strain VII-6

7.3 Effect of the work hardening exponent VII-9

7.3.1 Simulation Parameters VII-9

7.4 Conclusions VII-17

References VII-18

CHAPTER 8 THE SHEAR BENDING OF A CIRCULAR TUBE SUBJECTED

TO AN ECCENTRIC AXIAL PUSHING FORCE

8.1 Introduction VIII-1

8.2 Simulation parameters VIII-1

8.3 Stress distribution VIII-2

8.4 Thickness distribution VIII-4

8.5 Conclusions VIII-5

References VIII-6

CHAPTER 9 SUMMARY IX-1

Nomenclature

(X,Y,Z) Cartesian Coordinate System

P Axial Pushing Pressure

FP Axial Pushing Force

FS Shearing Force

SP Pushing Stroke

SS Shearing Stroke

Y Yield Stress

TS Tensile Strength

E Young’s Modulus

v Poisson’s Ratio

n Work Hardening Exponent

K Strength Coefficient

A0 Initial Cross Section Area of Tube

D0 Initial Diameter of Tube

L0 Initial Length of Tube

t Thickness of Tube

t0 Initial Thickness of Tube

rc Die Corner Radius

c Radial Clearance Between Tube and Tooling

ro Outside Bending Radius

R Bending (Centerline) Radius

Rmin Minimum Bending Radius

WF Tube’s Wall Factor

BF Bending Factor

η1 ,η2 ,η Flattening Factors of Cross Section

Dv , Dh Tube Diameters in Two Perpendicular Directions

α Shear Angle

εsh Shear Strain

γ Engineering Shear Strain

ε Effective Strain

pε Effective Plastic (Accumulative) Strain

σ Effective Stress

εI,II Principal Strains

εt Thickness Strain

µ Friction Coefficient

σx Normal Stress in X Direction

σy Normal Stress in Y Direction

τxy Shear Stress in XY Plane

σΕ Normal Stress Component

τΕ Shear Stress Component

θ Angular Position

e Elongation

ε True Strain

σ True Stress

h Height of Wrinkle

θ Distance Between each Layer to Neutral Layer

V Velocity of Metal Flow

Vy Velocity of Moving Die

δ Feed Material

U1 Thickening Energy

U2 Bending Energy

w Width of a Strip

l Length of a Unit Element

I-1

CHAPTER 1

INTRODUCTION

1.1 Introduction to metal forming

Taking a few moments to inspect the different objects used in the daily life, it is realized

that almost all of them have been transformed from various raw materials and assembled

into those objects through various processes that are called manufacturing processes.

Generally, the higher the level of manufacturing, the higher the life standard is.

According to DIN (Deutsches Institut für Normung) 8580, the manufacturing processes

are divided into six main groups:

I. Primary forming

II. Deforming

III. Separating

IV. Joining

V. Coating

VI. Changing the material properties.

Metal forming is used synonymously with deformation or deforming and comprises the

methods in group II of the manufacturing process classifications. The term “metal

forming” refers to a group of manufacturing processes by which the given shape of a

workpiece is converted to another shape without change in the mass or composition of the

material of the workpiece [1].

All metal objects, except castings, have at some time in their manufacture been

I-2

subjected to at least one metalworking operation. Several different operations may often be

necessary [2].

Both ferrous and nonferrous metals, unless cast directly into their final shape, pass

through either rolling mills or extrusion process. If one accept that from 20 to over 40% of

all rolled steel production is in the form of sheet and coils it is clear that many millions of

tons of steel go on to be worked by metal forming processes [1].

Table 1.1 Classification of manufacturing processes

Creation of

cohesion

Maintenance of

cohesion

Destruction of cohesion Increase of

cohesion

I Primary

forming

II Deforming Shape modification

III Separating

VI Changing the material properties

a-Addition of particles

b-Removal of particles

c-Rearrangement of particles

IV Joining

V Coating

Metal forming is an ancient art and was the subject of closely-guarded secrets in

antiquity. In many respects the old craft traditions have been retained until the present time,

even incorporating empirical rules and practices in automated production lines. Such

techniques have been successful when applied with skill, and when finely adjusted for

specific purposes. Unfortunately, serious problems arise in commissioning a new

production line or when a change is made from one well-known material to another whose

characteristics are less familiar. The current trend towards adaptive computer control and

flexible manufacturing systems calls for more precise definition and understanding of the

processes, while at the same time offering the possibility of much better control of product

I-3

dimensions and quality [3].

The following list outlines the most important areas of applications of workpiece

produces by deformation, underlying their technical significance [1]:

- Components for automobiles and machine tools as well as for industrial plants and

equipment.

- Hand tools such as hammers, Screwdrivers and surgical instruments.

- Fasteners, such as screws, nuts, bolts and rivets.

- Construction elements used in tunneling, mining, and quarrying

- Containers such as metal boxes , cans and canisters.

- Fittings used in the building industry such as for doors and windows.

1.1.1 Forming methods

The following classification of the deformation methods into 5 groups is based mainly

on the important differences in effective stresses [1].

1. Compressive forming

2. Combined tensile and compressive forming

3. Tensile forming

4. Forming by bending

5. Forming by shearing

Plastic processing technology can shape a material and improve its properties. With the

development of aerospace, automobile, and high-technology industries, and with the rise of

Economic Global Competition, Knowledge Economy, and Green Manufacturing, plastic

processing technology has been facing a challenge and an opportunity. Therefore, it is

required to develop advanced plastic processing technologies in order to manufacture parts

with light weight, high strength, high precision, high efficiency and at low cost, within a

I-4

short period, and a friendly environment, and with intellectualization and digitization. This

needs to combine plastic processing technologies with materials, mechanics, the

application of computer, etc. Thus, focused on precision plastic forming processes and

characterized with complex technologies, high-added value, Hi-Tech, and even complex

knowledge, advanced plastic processing technologies play a more and more important role

in the development of advanced manufacturing technologies [4].

Metal forming

Compressiveforming

Combined tensileand compressive

forming

Temsileforming

Formingby bending

Formingby

shearing

Inde

ntin

g

Clo

sed-

die

form

ing

Ope

n-di

e fo

rmin

g

Rol

ling

Pulli

ng th

roug

h a

die

Push

ing

thro

ugh

a di

e

Ups

et b

ulgi

ng

Spin

ning

Flan

ging

form

ing

deep

dra

win

g

Exp

andi

ng

Stre

tchi

ng

Jogg

ling

Ben

ding

wit

hrot

ary

tool

mot

ion

Ben

ding

with

line

arto

ol m

otio

n

Rec

essi

ng

Tw

istin

g

Figure 1.1 Classification of metal forming methods

1.2 Bending deformation

Bending is one of the most common metalworking operations. Bending is the plastic

deformation of metals about a linear axis called the bending axis with little or no change in

the surface area.

The potential advantages of using bending as a forming process are low tooling costs

and a flexible production route [5].

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Extrusions are used widely for the design of lightweight assemblies, especially if a high

specific stiffness is needed. Whilst some technical buildings such as bridges and

skyscrapers are often made from straight elements, other applications demand bent parts.

In modern production engineering, the elastic-plastic bending of strips, various beam

sections and sheets, has been extensively employed in forming of large metal members of

structures as well as various items of use in daily life [6].

A tube has high flexural and torsional rigidities respect to its weight. Utilization of tubes

in order to meet the demands of lightweight and low cost products has been increasing.

Thin-walled tube parts are playing an important role in automobile, aerospace, oil and

other various industries for their high strength/weight ratio [7].

Cold bending of metal tube products is probably one of the oldest metal forming

processes and the bent tubing parts are widely used in industry.

The principle for bending the tubes is much the same as for bending of sheets and bars.

Cold bending of metal tubes is a very important production method considering that metal

tubes are widely used in a great variety of engineering products, such as automobile,

aircraft, air conditioner, air compressor, exhaust systems, fluid lines. Although cold

bending of metal tubes is an old metal forming process, it is becoming a precision

metalworking process and requires high quality assurance. There is a variety of methods

for cold bending including rotary drawing bending, compression bending, empty-bending,

ram bending, rolling bending, etc. Bending machines range from hand benders, hydraulic

bending, to fully computerized CNC benders.

The problem that is facing tubing production industry is that with the customer's demand

on complex tubing parts and tight tolerances, there often exist defects and failures of

tubing parts, such as undesired deformation, inaccuracy of bend angles and geometry,

wall-thinning, flattening, wrinkling, cracks, etc. All of these are in close relationship with

the selection of bending methods, tool/die design, die set conditions, machine setup,

I-6

material effects, a number of bending process parameters such as minimum bending radius,

springback, wall factor, empty-bending factor, etc [8]. In today’s applications of formed

thin-walled parts, however, new challenges have arisen, including the prediction of

dimensional tolerances [9].

(a)

(b)

Figure 1.2 Bent tubular parts; (a) various bending radii, (b) 3D bending

1.2.1 Mechanism of bending deformation

Loaded with a pure bending moment, a beam will first be elastically stretched and then

upset in the outer zones. When the yield stress is reached first in the outermost layers,

zones of plastic deformation will increase and grow towards the neutral layer. Due to work

hardening of the plastically stretched and upset areas, the bending moment has to increase

to affect further bending. When the bending moment is released, the elastic shares of the

moment will be set free and cause elastic recovery of the respective layers, i.e. the

stretched layers will contract and the compressed layers will expand. Due to the elementary

bending method, this can be understood as the superposition of a fictitious moment

I-7

directed opposite to the bending moment: springback of the beam will appear [10].

As bending occurs, the outside diameter of the tube stretches while the material along

the inside diameter tends to crimp and wrinkle. The walls along the outside radius of the

bend tend to thin, while the walls along the inside radius thicken. Basic bending methods

are used when these conditions are acceptable, while more advanced methods counteract

the forces at work in bending.

MB

σx(y)εx(y)

Plastic

Elastic

Plastic

εx,σx

MR= -MB

σ'(y)ε'(y)

y y

+ =

MR=0

σR(y)

εR(y)y

Loading Unloading

Figure 1.3 Distributions of stress and strain during bending deformation

1.2.2 Bending factors

There are many factors to be considered for a tube bending process. Among them, the

wall factor and the bending factor are used to determine the severity of a bend.

1.2.2.1 Tube wall factor

A common objective in tube bending is to form a smooth round bend. This is simple

when a tube has a heavy wall thickness and it is bent on a large radius. To determine if a

tube has a thin or thick wall, its wall thickness to its outside diameter is compared. The

result is called the tube’s wall factor (WF):

WF=t/D. (1-1)

I-8

1.2.2.2 Bending factor

The same type of comparison is made to determine if a bend radius is tight or large.

Bending factor (BF) is described as the ratio of bending centerline radius (R) over the

outside diameter of the tube (D):

BF=R/D (1-2)

1.2.2.3 Minimum bending radius

In practice, an empirical formula for determining the minimum bending radius, Rmin, is

in wide use:

Rmin=D/2e (1-3)

where D is the outside diameter of the tube, and e is the elongation of the tube material

[11].

1.2.3 Difficulty in tubing process design

Generally, tube geometry and bending radius determine whether a mandrel is needed

and if so the type necessary.

Common failures and defects in metal tube bending parts can be classified as:

Deformation (wall thinning, flattening, wrinkling)

Inaccuracies (overbending, underbending, twisting, beyond the linear dimension

tolerance)

Breakage/crack.

Dents/marks.

Cross section deformation: The main characteristic of a tube under bending is the

distortion (ovalization) of its cross-section because of the inward stress components. The

ovalizing mechanism results in loss of stiffness in the form of limit point instability,

referred to as ovalization instability [12]. This distortion arises because the bending

I-9

moment is resisted by the cross-section of the tube. In the case of a tube with square cross

section, the tension and compression flanges undergo concave distortions, while the web

suffers a convex distortion. Using mandrel and employing a thick tube are effective means

to avoid ovalization.

Wrinkling: This deformation arises on compressive regions where the compressive stress

exceeds the buckling stress. To restrain wrinkling, one must use a mandrel, axial tension

and materials of high n value, a thick tube, and so on.

Folding: This deformation arises when the flattening distortion or wrinkling reduces the

bending rigidity of the section. The countermeasures against folding are essentially the

same as those against wrinkling.

Necking and splitting: This fracture arises on the tension flange and on tensile regions of

webs where the stress exceeds the deformation capacity of the materials. To restrain

necking and splitting, we must select a large bending radius, materials of high ductility,

additional axial compression and so on [13].

Wrinkling research has interested many scholars for a long period. The energy method

has always been a widely used approach to obtain the critical condition of wrinkling.

However, up to now, literature on the studies of wrinkling in the tube bending process has

been scant [14].

Since tube bending is influenced by many technical factors related to bending structure,

bending radius, material, wall thickness, diameter, tooling/die selection and condition,

bending methods, lubrication, and operating parameters, etc., it is often difficult to achieve

an optimal design of the tube bending process, in particular for bending parts with complex

configuration and geometry requirements.

1.2.4 Bending equipment

Tooling and die play an extremely important role in cold bending of metal tube products,

I-10

and are directly related to most failures in tube production. Basically, cold bending requires

at least three items: a center forming die, either fixed or movable (for rotary draw bending),

a pressure die and a clamping or following die. In the draw bending, a mandrel and a wiper

die are often equipped.

1.2.4.1 Mandrel

The mandrel is a tool inserted inside a tube, pipe, or other hollow section in the region of

the bend tangent. Its purpose is to support the outside wall of the workpiece as it is pulled

around the bending form and reduce the amount of flattening. In addition, the mandrel

helps prevent wrinkles from forming on the inner wall of the bend. The mandrel sometimes

has a secondary function as a sizing tool on extremely close tolerances in thin wall tubing,

as commonly used in the aircraft and aerospace industries.

It is cheaper to bend tubing without a mandrel. Trial bending is generally necessary to

find what bends can be made. Tubing with thick walls is more likely to be bendable

without a mandrel than thin-wall tubing. Bends with large radii are more likely to be

formable without a mandrel than those with small radii. Slight bends are more feasible than

acute bends. Wide tolerances on permissible flattening make a bend easier to form without

a mandrel. Springback is greater without a mandrel, but it can be compensated for by

overbending, or lessened by increasing force on the pressure die.

1.3 Tube bending methods

Unlike conventionally straight structural parts, bending is required in producing

automobile body frames with various profiles. Thus, the development of a flexible bending

technology that can be used for the generation of various profiles of structural frames is

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required to improve the design and production of bending. Moreover, the increasing

market requirement for small lot production rather than mass production has increased the

development of such flexible bending methods.

There are several ways to bend a tube, and the type of equipment selected depends on

the desired quality of parts. Each type characteristically has certain applications and

limitations with regards to the kinds of bends it produces and the maximum angle of bend

it achieves as indicated in the Table 1.2 [15].

Selection of a bending process for tubing depends on: Quality of bend and production

rate required, diameter, wall thickness and minimum bend radius desired. It is essential to

select a suitable bend method according to the tube material, the relative bend radius, R/D,

the relative thickness, t/D and the desired precision, where D is the outside diameter, R the

centerline radius and t the wall thickness.

There are two criteria for the ordering, which are: the distinction between kinematic

bending and bending with shape-defining rigid tools; and the distinction between the kinds

of forces and moments generating the local deformation.

Forming with shape-defining dies is here defined as forming with rigid tools that contain

the desired workpiece geometry, especially with respect to the curvature, corrected by the

springback of the tube during unloading. Due to the fixed geometry of the tool, the

geometry of the workpiece is also fixed, yielding a high reproducibility and a shorter

processing time in many cases. As examples: press bending, stretch bending, rotary draw

bending.

The kinematics bending processes are more flexible. The final shape of the part is not

determined by the shape of the tool, but rather by the relative movement of the tool and the

workpiece. Normally, the final shape is produced by a number of successive steps that can

be changed easily from workpiece to workpiece, yielding the high flexibility of these

processes [16].

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Generally, tube bending methods can be categorized into two main groups: one which a

bending moment is applied on the tube. Pure bending is a bending method with constant

bending moment and air bending is a method of bending with a non-constant moment

along the bending axis. On the other hands, there are methods of bending which use no

moment for the bending process itself. Thermally induced bending or laser bending,

partially change in wall thickness are some examples.

Table1.2 characteristics of the basic bending methods

Bending process Types of bends

Usually accomplishes

Min

bending

radius

Maximum

angle

of bending

Mass

production

Draw Single, multiple, compound 1D0 Up to 180º ✔

Compression

Ram and press

Manual

Single

Series of different bend angles

Single, compound, spiral

2D0

2D0

Up to 180º

Up to 165º

360º

✔

✔

Roll Circular, spiral, helical 5D0 360º -

Stretch

linear

radial

Variable curvature

Circles, ovals, rectangles, spirals

-

180º

360º

-

It might be surprising that there are bending methods working without a moment for the

generation of the plastic strains necessary for bending. Usually bending is realised by

compressive strains in the x-direction above the neutral axis and tensile strains below it,

which are introduced simultaneously by a bending moment. If the position of the neutral

axis is at the top of the workpiece, the zone with compressive strains in x-direction

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diminishes to zero, only tensile strains act during bending. These can be introduced by a

local compression in y-direction, e.g. by local hammering without applying any moment. If

the neutral axis is shifted to the bottom of the workpiece, another principle, i.e. laser

forming, may be applied. This uses thermal stresses that produce local compression of the

material, also resulting in the curvature of the tube.

Basic bending methods, which are widely used, are as follows:

Rotary draw bending

Compression bending

Roll bending

Stretch forming

Here, important tube bending methods are introduced and their main characteristics are

investigated:

1.3.1 Rotary draw bending

Rotary draw bending is a widely used method for bending the tubes, particularly for

tight bending radii and thin wall tubes [17-19]. A die set-up for a rotary draw bending is

shown in Figure 1.4. Rotary draw bending is called because the tube is being drawn into

the bending area past the tangent point. At one end, the tube is tightly pressed between a

bending die and a clamp die at just beyond the front tangent point against the clamp die. At

the other end, the workpiece is held by a pressure die and/or a wiper die and a mandrel

when they are necessary. The mandrel is inserted inside the tube to minimize the stretching

that occurs along the outside radius of the tube, while a wiper die reduces wrinkling along

the inside radius. The pressure die restrains the free end of the workpiece and allows it to

move in a straight line. As the workpiece is being drawn and rotating around the center die,

the pressure die, which is either static or boosted, transfers the workpiece to the center die

at the tangent point, so as to get the desired angle and radius.

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The draw bending machine can be powered (hydraulic, pneumatic, electric/mechanival),

manual, or numerically controlled. These machines handle about 95% of the tube-bending

operations. Rotary draw bending is the most versatile and flexible bending method. When

deformities are unacceptable, say for pipes that will carry liquids or gases; this method is

suitable. Rotary draw bending produces the highest quality products, although it is too

time-consuming and expensive. Regarding mass production it is an excellent method.

TubeWiper die

Bending die

Pressure dieMandrelClamp die

Boost pressure

Figure 1.4 Schematic sketch of rotary draw bending equipments

1.3.2 Compression bending

Compression bending is another common method for cold bending. This is also the

simplest and most economic operation for bending the metal tube parts.

Figure 1.5 shows a set-up of compression bending. The difference from the rotary draw

bending is that the center-forming die is fixed rather than rotatable, and the clamping die is

replaced by a movable following die. The following die by means of a rotary arm presses

the workpiece around the center-forming die to form the desired shape.

Compression bending is a process whereby a tube is bent to a reasonable smaller radius,

usually without the use of mandrel, wiper and precision tooling. The equipment is very

cheap. However, thinning and wrinkling outbreak easily. Regarding the accuracy, this

method creates a consistent outside diameter, but the inside of the tube is deformed.

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Tube Clamp

MovablePressure die

Stationarybending die

Figure 1.5 Compression bending

1.3.3 Ram bending:

Ram bending is one of the oldest and simplest methods of bending pipe and tubing. As

shown in Figure 1.6, two supporting dies hold the tube and sufficient force is applied by

means of a hydraulic ram to the center of the workpiece.

This process bends the tube to the desired angle and bending radius. Ram bending is

often used to bend larger diameter tubes where deformities are acceptable.

Ram

Supportdie

Tube

Figure 1.6 Ram bending

The simplicity of ram bending limits the types of work handled. Tubing can be bent

through angles up to 120, however this method cannot provide bends with close tolerances.

Ram bending is best suited for bending thick wall tubing. Although it is not recommended

I-16

for bending stainless steel tubing with unsupported walls if the desired radius of bend is

less than 6 times the tube diameter.

1.3.4 Press bending

Press bending, employing a fully supporting wing-type tooling as shown in Figure 1.7, is

a combination of ram and compression bending [20]. It operates in a manner similar to the

ram bender but is considerably faster and more flexible. The tube or pipe is placed on top

of adjacent wing-type dies set at the same levels. The dies simultaneously separate and

rotate with the tube as it deflects and bends from pressure applied by the descending ram

die. Cushion cylinders maintain constant torque on the wing dies. This cushioning force

confines the workpiece in the dies under properly applied pressure, accurately controlling

metal flow. The nearly constant cushion force is key to preventing wrinkles and producing

accurate bends with minimum distortion of the cross section.

Bendingpress

Wingdie

Tube

Cushion

Figure 1.7 Press bending

The major advantage of press bending is its high production capabilities, which makes it

an appropriate method for mass production. Bends can usually be made three to four times

I-17

faster than by conventional equipments. Bending presses use simple and cheap tooling and

are quickly and easy set up. In press bending, it is not practical to use a mandrel. Because

of this limitation, a slight reduction of the work diameter on the inside of the bend occurs.

Therefore, it is not a precise method. Bend angles greater than approximately 165º are

impractical. Tubing, pipe, rod and some formed sections are easily bent but rolled shapes

or thin walled parts are usually processed on rotary bending machines. As the deformation

is concentrated on the central part, this region is break easily.

1.3.5 Stretch bending

One of the bending methods that is mostly used is stretch bending [21-26], which is

shown in Figure 1.8. The process is usually done in three steps. First, the tube is stretched

by pulling it with the jaws at its end. The magnitude of the pre-stretch force lies between

zero and a small fraction above the yield point. The tube does not touch the forming tool at

this moment. Then the jaws or the punch move, resulting in an increasing contact between

the punch and the tube. A forming zone occurs first in the center of the tube. It is then

divided into two forming zones shifting away from each other. The movement stops, if

forming is complete. Then an additional increase of the stretching force may be applied.

F r

F aFa

Die

Tube

Figure 1.8 Stretch bending

I-18

Due to applying a tension force on the tube, the neutral axis moves from its normal

position toward inside the bending. This can prevent from occurrence of wrinkles in

bending inside. However, the cross section undergoes high deformation due to axial

tension. Moreover, as the tube ends must be constraint, the waist material is large.

1.3.6 Roll bending

The principle of roll bending [27] is shown in Figure 1.9. The workpiece is laid on two

rolls and is bent by an additional roll between the two lower rolls. The resulting bending

moment, which varies from zero to a maximum from the outer to the inner roll, leads to a

curvature of the tube similar to that of the elementary air bending process. By turning the

rolls the material is moved in the axial direction and bent continuously. The local curvature

can be varied by the indentation depth of the center roll.

Adjustableroller

Fixed rolls(drivers)

Figure 1.9 Roll bending

There are some variations of this process, using four or six rolls. The advantage of the

four-roll bending compared with the three-roll bending is the enhanced accuracy of the

cross section of the bent part, as the fourth roll is used to support the lower wall of the tube,

reducing the deformations of the cross section. The six-roll bending is a kind of mirrored

I-19

four-roll bending with the capability of bending S-shapes. Roll bending provides a simple

means for bending a wide range of cross sections. Different radii are achieved by changing

the position of one or two rolls . However, this method cannot produce small bending radii.

Moreover, execution of bends on roll bender requires a skilled operator to run the machine.

1.3.7 Push bending

The outline of tube push-bending [28-30] is shown in Figure 1.10. The principle of this

method is that a male die pushes a tube into a bend female die to make it deform to the

desired bend shape.

The equipment is cheap. Regarding working loads, it is suitable for thin tubes; however

from the view of undesirable deformation, it is appropriate for thick tubes. This method is

suitable for production of short elbow bent tubular parts. However, undesirable

deformation in front part of tube and thinning in bent part are the disadvantaged of this

forming method.

Die

Plunger

Internal pressureTube

Figure 1.10 Push bending

1.3.8 Laser bending

Laser forming [31,32], which schematically is shown in Figure 1.11, is one possibility of

I-20

introducing a local plastic compressive strain into the workpiece without any external

forces. The laser is used for local heating of the tube. Thermal stresses develop and reach

the flow stress, resulting in local compression of the material.

Due to the asymmetry of heating and cooling, the compressive strains remain in the

workpiece after cooling. The shaping of the tube can be done by repeating the irradiation at

a different position along the bending axis or even at the same position. It is easily possible

to get three-dimensionally bent tubes by this process. There are restrictions on this process

for multi-chamber tubes and concern regarding the processing time, as the process is slow.

Laser forming as a springback-free and non-contact forming technique has been under

active investigation over the last decade. The extensive variety of possible applications

result from the reasonable high degree of control over the energy transfer, the high levels

of accuracy and reproducibility, the very high degrees of flexibility and the non-contact

nature of the technique.

Laserbeam

Tube

Clamp

Figure 1.11 Laser bending

1.3.9 Air bending

If some forces are applied on the tube in addition to the moment, a local control of the

bending moment and in turn the control of the local curvature, is possible. The forces and

moments applied lead to the formation of a local plastic zone where plastic bending takes

place. The zone can be shifted along the whole workpiece to be bent sequentially.

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One example of locally controlled air bending is shown in Figure 1.12, which is called

MOS bending [33]. The tube is guided in a shape-dependent guiding cylinder, and pushed

through it in the axial direction. At a certain distance from the exit there is a movable

bending die. A spherical bearing defines a supporting point for the tube. Depending on the

position of the bending die, the tube will be bent according to the moment introduced. This

method can be used for flexible bending of tubes with spatial curvatures.

Guiding cylinderSpherical bearingTube

Movable bendingdie

Axial force

Figure 1.12 MOS bending

1.3.10 Bending using a polyurethane pad

The length of the unbent ends after roll bending can be reduced significantly by a

method that is similar to roll bending but shows a much higher flexibility [34]. The method,

shown in Figure 1.13, is bending of tubes using one rigid roll and a flexible polyurethane

pad. The tube is laid on the flat polyurethane pad and pushed into it by the roll. The

moment due to the different pressure distributions between the roll and the tube and

between the tube and the pad results in bending of the tube. By shifting the roll along the

axis of the tube it can be bent along almost the whole of its length. The relative movement

between the roll and the pad can be described as a path curve

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F

P

Tube

padRoller

Figure 1.13 Pad bending

Many studies of traditional bending technologies have been made to date [35-43].

However, these processes have disadvantages concerning die set complexity,

manufacturing cost and production time. The draw bending has good reproducibility, but

requires a large bending die set. It is a very complex process requiring high production cost.

In the roll bending process, the overall cost of the die set is moderate and profiles of large

dimensions can be produced, but the difficulties of production and the constraints of profile

design make it unsuitable for the practical application in the automobile industry. Stretch

bending can be expanded to three-dimensional bending, but the process requires a die set

as large as the specimen and also high die set costs. Thus, a new applicable bending

technology is needed. This technology should allow for 2D/3D bending, with a high degree

of freedom in the process design, more flexibility and low production costs.

To meet the above-mentioned requirements, several techniques have been developed and

have begun to be applied in industry. Laser bending has been investigated intensively in

recent years due to its benefits of dieless and contactless forming. Moreover, the thermally

induced forming processes result in very little springback, leading to high product

accuracy.

I-23

References

[1] K. Lange, Handbook of metal forming, McGraw-Hill (1985), USA

[2] G.W. Rowe, Principles of industrial metalworking processes, Edward Arnold Ltd

(1977), London.

[3] G.W. Rowe, Finite element plasticity and metalforming analysis, Cambridge University

Press (1991).

[4] H. Yang, M. Zhan, Y.L. Liu, F.J. Xian, Z.C. Sun, Y. Lin, X.G. Zhang, Some advanced

plastic processing technologies and their numerical simulation, Journal of Materials

Processing Technology, Volume 151, Issues 1-3 (2004), pp. 63-69

[5] F. Paulsen and T. Welo, Application of numerical simulation in the bending of

aluminium-alloy profiles, Journal of Materials Processing Technology, Volume 58, Issues

2-3 (1996), pp. 274-285

[6] T.X. Yu and L. C. Zhang, Plastic bending, theory and applications, World scientific

(1996), Singapore

[7] H. Yang and Y. Lin, Wrinkling analysis for forming limit of tube bending processes

Journal of Materials Processing Technology, Volume 152, Issue 3 (2004), pp. 363-369

[8] Z. Jin, S. Luo and X. Daniel Fang, KBS-aided design of tube bending processes,

Engineering Applications of Artificial Intelligence, Volume 14, Issue 5 (2001), pp.

599-606

[9] F. Paulsen, T. Welo and O. P. Søvik, A design method for rectangular hollow sections in

bending, Journal of Materials Processing Technology, Volume 113, Issues 1-3 (2001),

pp.699-704

[10] M. Elchalakani, X.L. Zhao, R.H. Grzebieta, Plastic mechanism analysis of circular

tubes under pure bending, International Journal of Mechanical Sciences 44 (2002),

pp.1117-1143.

[11] Greg G Miller, Tube forming processes- A comprehensive guide, SME (2003), USA.

I-24

[12] S. A. Karamanos, Bending instabilities of elastic tubes, International Journal of Solids

and Structures, Volume 39 (2002), pp. 2059–2085

[13] N. Utsumi and S. Sakaki, Countermeasures against undesirable phenomena in the

draw-bending process for extruded square tubes, Journal of Materials Processing

Technology, Volume 123, Issue 2 (2002), pp.264-269

[14] H. Yang and Y. Lin, Wrinkling analysis for forming limit of tube bending processes,

Journal of Materials Processing Technology, Volume 152, Issue 3 (2004), pp.363-369

[15] Tube forming, Corona Publishing Co., Ltd. (1994), Japan (In Japanese).

[16] F. Vollertsen, A. Sprenger, J. Kraus and H. Arnet, Extrusion, channel, and profile

bending: a review, Journal of Materials Processing Technology, Volume 87, Issues 1-3

(1999), pp. 1-27

[17] J. b. Yang1, B. h. Jeonb and S. I. Oh, The tube bending technology of a hydroforming

process for an automotive part, Journal of Materials Processing Technology, Volume 111,

Issues 1-3 (2001), pp.175-181

[18] M. Sukimoto, Y. Taguchi, M. Sakaguchi, H. Akiyoshi and J. Endou, Deformation of a

cross section of 6063 alloy circular tube by rotary draw bending, Keikinzoku/Journal of

Japan Institute of Light Metals, Volume 44, Issue 9 (1994), pp. 475-479

[19] N. Utsumi and S. Sakaki, Countermeasures against undesirable phenomena in the

draw-bending process for extruded square tubes, Journal of Materials Processing

Technology,Volume 123, Issue 2 (2002), pp. 264-269.

[20] J. Gillanders, Pipe and Tube bending Manual, Second edition, International Rockford,

Illinois (1994), USA.

[21] F. Paulsen and T. Welo, A design method for prediction of dimensions of rectangular

hollow sections formed in stretch bending, Journal of Materials Processing Technology,

Volume 128, Issues 1-3 (2002), pp. 48-66.

[22] A. H. Clausen, O. S. Hopperstad and M. Langseth, Sensitivity of model parameters in

I-25

stretch bending of aluminium extrusions, International Journal of Mechanical Sciences,

Volume 43, Issue 2 (2001), pp.427-453.

[23] A. H. Clausen, O. S. Hopperstad and M. Langseth, Stretch bending of aluminium

extrusions for car bumpers, Journal of Materials Processing Technology, Volume 102,

Issues 1-3 (2000), pp.241-248.

[24] J.E. Miller, S. Kyriakides, Three-dimensional effects of the bend-stretch forming of

aluminum tubes, International Journal of Mechanical Sciences, Volume 45 (2003), pp.

115-140.

[25] E. Corona, A simple analysis for bend-stretch forming of aluminum extrusions,

International Journal of Mechanical Sciences, Volume 46, Issue 3 (2004), pp.433-448

[26] H. Zhu and K. A. Stelson, Modeling and Closed-Loop Control of Stretch Bending of

Aluminum Rectangular Tubes, Journal of Manufacturing Science and Engineering, Volume

125 (2003), p.113.

[27] W. L. Hu and Z. R. Wang, An experimental study of the roll-bending of

double-curvature workpieces, Journal of Materials Processing Technology, Volume 55,

Issue 1 (1995), pp.28-32.

[28] Y. Zeng and Z. Li, Experimental research on the tube push-bending process,

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[29] S. Baudin, P. Ray, B. J. Mac Donald and M. S. J. Hashmi, Development of a novel

method of tube bending using finite element simulation, Journal of Materials Processing

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[30] F. Stachowicz, Bending with upsetting of copper tube elbows, Journal of Materials

Processing Technology, Volume 100 (2000), pp. 236-240.

[31] N. Hao and L. Li, An analytical model for laser tube bending, Applied Surface

Science, Volumes 208–209 (2003), pp. 432–436.

[32] N. Hao and L. Li , Finite element analysis of laser tube bending process , Applied

I-26

Surface Science, Volumes 208-209 ( 2003), pp. 437-441.

[33] M. Murata and Y. Aoki, Analysis of circular tube bending by MOS bending method.

In: T. Altan Editor, Advanced Technology of Plasticity I (1996), pp. 505–508.

[34] J. W. Lee, H. C. Kwon, M. H. Rhee and Y. T. Im, Determination of forming limit of a

structural aluminum tube in rubber pad bending, Journal of Materials Processing

Technology, Volume 140, Issues 1-3 (2003), Pages 487-493.

[35] H.A. Al-Qureshi, Elastic-plastic analysis of tube bending, International Journal of

Machine Tools & Manufacture, Volume 39 (1999), pp.87–104

[36] S. Kyriakides, E. Corona, J.E. Miller, Effect of yield surface evolution on bending

induced cross sectional deformation of thin-walled sections, International Journal of

Plasticity, Volume 20 (2004), pp. 607-618

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formulations, experimental data and finite element analyses, Thin-Walled Structures,

Volume 41 (2003), pp. 127–147

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bending: Part II analytical models, International Journal of Mechanical Sciences, Volume

43 (2001), pp. 131-152

[39] T. H. Kim and S. R. Reid, Bending collapse of thin-walled rectangular section

columns, Computers & Structures, Volume 79, Issues 20-21, August 2001, Pages

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flexure, International Journal of Solids and Structures, Volume 40 (2003), pp. 6111–6123

[41] M. Elchalakani, X.L. Zhao, R.H. Grzebieta, Plastic mechanism analysis of circular

tubes under pure bending, International Journal of Mechanical Sciences, Volume 44 (2002),

pp. 1117-1143.

[42] H. Yang and Y. Lin, Wrinkling analysis for forming limit of tube bending processes,

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Journal of Materials Processing Technology, Volume 152, Issue 3 (2004), pp. 363-369

[43] N. C. Tang, Plastic-deformation analysis in tube bending, International Journal of

Pressure Vessels and Piping, Volume 77, Issue 12 (2000), pp.751-759.

II-1

CHAPTER 2

FUNDAMENTALS OF THE SHEAR BENDING PROCESS

2.1 Introduction

Various bending methods have been developed and are used depending on operation

efficiency and other requirements. However, conventional bending methods cannot meet

demands on precision in some cases, due to the tube roundness and the relatively flat

cross-section of bent tubes. Increasing global competition is resulting in requirements for

industries to decrease component cost by increasing production rates and reliability and to

improve component performance by offering improved mechanical characteristics. On the

other hands, various new demands in the manufacturing fields, such as in automobile

industries, necessitates increasing the number of components, which causes problems of

installation space [1,2]. Moreover, the weight reduction of automobiles is considered to be

an effective way of decreasing both fuel consumption and car emissions, which are the

main causes of the exhaustion of natural resources and global warming, respectively. It has

been reported that a weight reduction by 10% can save fuel consumption by 5–7%, while

10% decrease of aerodynamic resistance saves fuel consumption by only 2% [3].

Recently, in addition to various design demands in the manufacturing fields, the

occupied space of products has to be as small as possible. Also, manufacturers are

searching for ways to perform tighter-radii bending of thinner-wall tubes and wrinkle-free

bends with minimal wall thinning.

The problem that is facing tubing production industries is that with the customer's

II-2

demand on complex tubing parts and tight tolerances, there often exist defects and failures

of tubing parts, such as undesired deformation, inaccuracies of bend angles and geometry,

wall-thinning, flattening, wrinkling, cracks, etc. All of these are in close relationship with

the selection of bending methods, tool/die design, die set conditions, machine setup,

material effects, a number of bending process parameters such as minimum bending radius,

springback, wall factor, empty-bending factor, etc [4].

In order to reduce installation space and investment cost, research into pipe-bending

processes involving a small bending radius by theory and computer simulation is greatly

required [5]. There are several methods for cold tube bending production, such as rotary

draw bending, roll bending, push bending, stretch bending, etc. Applying these bending

methods, the minimum bending radius, is almost more than 1~2 times of the tube diameter

even using mandrels [6]. Bending on very small radii, undesirable forming defects such as

weakening or rupture of tube wall due to extreme thinning of bending outside, wrinkling in

bending inside resulted from high compression condition, cross section deformation and so

on may outbreak.

Tube shear bending is a beneficial technique to realize production of unified and

compact bent tubular parts through cold metal forming [6-9]. It is an appropriate

technology to realize considerable small bending radii, which is very difficult to be

achieved through common cold-bending methods. It can be applied as an effective means

when space limitation is a main design factor. Utilizing the shear bending process,

production of compact and unified elbow tubular parts having very small corner radii is

feasible (Figure 2.1). Therefore, this method can be used instead of ordinary methods

including casting or welding of mitered joints (Figure 2.2). Application of these

conventional methods carries disadvantages such as high production cost and time, low

quality due to creation of structural defects in the tube material and so on.

II-3

ro=15

R=60

Do=

30

rc=3

R/D0=2.0

Figure 2.1 A Simple Comparison between bending radius of the shear bending and

conventional bending methods.

(a) (b)

Figure 2.2 Tubular elbows with small corner radius made by (a) welding a mitered

joint; (b) casting

Until now a few research studies have been carried out in the field of shear bending.

Tanaka et al. investigated on the shear bending process applying hydraulic pressure inside

the tube and axial pushing force on it. They also analyzed the process using the upper

II-4

bound and the finite element methods [6.7]. SANGO Co. Ltd. has developed the process

using mandrels inside the tube instead of liquid pressure and employing the displacement

control technique. Changing the die corner radius and bending angle, the shear bending of

tubes with different dimension and material properties has been investigated. However the

mechanism of deformation has not been analyzed and illustrated clearly [8,9].

A shear bending apparatus was built and the process was studied both by experiments

and numerical simulations. In this work, mandrels were used inside the circular tube

instead of liquid pressure. Moreover, the displacement control technique was replaced by

application of a controllable axial pushing pressure on the tube. A series of experiments as

well as simulation were carried out. The deformation mechanism of the tube was clarified.

The theoretical, analytical and experimental values of strains were determined. The effects

of important factors on the deformation behavior and forming limit of the tube were

clarified.

2.2 Experiments

2.2.1 Experimental set-up

Figure 2.3 shows a schematic illustration of the shear bending equipment. Moreover,

photos of the actual equipment are shown in Figure 2.4. The main equipment consists of a

fixed die (#5), a moving die (#7), mandrel of fixed die (#6), mandrel of moving die (#8),

two load cells, two hydraulic cylinders and the circular tube.

Figure 2.5 shows the procedure of the shear bending schematically. The tube is inserted

into the dies (Figure 2.5 (1)) and the mandrels are positioned inside it (Figure 2.5 (2)). The

mandrel of moving die is fixed to the moving die. Then the hydraulic cylinder (#11) slides

the moving die and applies a shearing force on the tube. At the same time, another

II-5

hydraulic cylinder (#1) pushes the end of the tube and the material is supplied into the dies

during the process (Figure 2.5 (3)). Therefore, the tube will undergo a shearing

deformation continuously (Figure 2.5 (4)).

Two load cells measure the values of the pushing and shearing forces. The hydraulic

cylinders are connected to a hydraulic power pack and controlled independently.

#1: Hydraulic cylinder (pushing)#2: Load cell (pushing)#3: Tube pusher#4: Tube#5: Fixed die#6: Mandrel of fixed die#7: Moving die#8: Mandrel of moving die#9: Die pusher#10: Load cell (shearing)#11: Hydraulic cylinder (shearing)

#11

#9

#10

#1 #2 #3 #7

#8

#4 #5 #6

Figure 2.3 Schematic sketch of shear bending equipments

II-6

#11

#7

#5

#10

#2#1

#11 #7 #5#10

#2

#1

Figure 2.4 Photographs of shear bending apparatus

II-7

(1) (2)

(3) (4)

Figure 2.5 Schematic sketch of shear bending procedure

During the process the tube will be formed into a crank-shaped product as shown in

Figure 2.6. By cutting the web of the deformed tube, two 90º elbow parts can be obtained

as shown in Figure 2.7.

Fixed Die Moving Die

Tube

Pushing force

Shearing force

Fixed Die Moving Die

Tube

Mandrels

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PPushingpressure

Shea

ring

Stro

kerc

Shearingforce

rm

Pushingstroke t 0D

0

Fixed dieMoving die

Figure 2.6 Schematic illustration of shear bending process

Figure 2.7 90º elbow tubular part obtained by shear bending process

2.2.2 Experimental conditions

2.2.2.1 Material properties

In this research, the experiments were carried out using two kinds of A1050 circular

extruded aluminum tubes. Table 2.1 indicates the experimental conditions. The material

II-9

properties were determined from tensile test according to JIS Z 2201. Several tensile

specimens were prepared from the tubes’ wall in the longitudinal direction. The mean

values of the material properties obtained from the tensile tests were employed.

Table 2.1 Material properties

Material A1050 (1) A1050 (2)

Yield stress, Y/ MPa 30 95

Ultimate tensile strength, TS /MPa 103 120

Tube’s initial diameter, D0/ mm 30 30

Tube’s initial thickness, t0/ mm 1, 1.5, 2, 2.5, 3 1.5

Tube’s initial length, L0/ mm 270 270

2.2.2.2 Dies

In order to find out the effect of the die corner radius on the process, five sets of dies

with different corner radii rc=2, 3, 4, 5 &6mm were prepared and utilized in the

experiments. Keeping constant the outside diameter, the blank tubes with different original

thickness were utilized. The radial clearance between mandrel-tube and tube-dies c was

0.1mm.

2.3 Finite element simulation

To be able to control a bending process, say bending with respect to repeatability and

bendability, represents a great challenge in developing competitive design concepts.

Traditionally, experimental trial and error is required to design a process route, profile

geometry and tolling, although this may be time-consuming and expensive. It is, therefore,

II-10

of great interest to utilize theoretical investigations to gain insight into bending processes.

Hence, improving the properties of the product and reducing the time- to-market.

Simulation means a virtual imaging and optimization of actual business processes in the

computer with the aim of reaching a decision on a product, a process or a manufacturing

cycle before, for instance, investment determination takes place. Such optimization of

processes with the necessary tooling corrections by trial and error were responsible in

earlier times for lengthy development times in metal part manufacture for automobile

bodies. Today, simulation can shorten this development process considerably and thus

reduce costs [10].

The introduction of numerical methods, particularly finite element analysis, represents a

significant advance in metal forming operation. Numerical methods are used increasingly

to optimize product design and deal with problems in metal forming processes.

More and more sectors of the metal forming industry are beginning to recognize the

benefits that finite element analysis of metal deformation processes can bring in reducing

the lead time and development costs associated with the manufacture of new components.

Finite element analysis of non linear problems, such as metal deformation, require

powerful computing facilities and large amounts of computer running time but advances in

computer technology and the falling price of hardware are bringing these techniques within

the reach of even the most modest R&D departments [11-13].

The most important aims of process simulation are: (i) proving existing manufacturing

concepts concerning their feasibility; (ii) judgment of the product properties; and (iii)

improvement of the process knowledge for optimising purposes.

The application of process simulation to reach these aims is only useful if it is more

economical than carrying out experiments. This condition is given in the case of forming

technology, because the costs of forming machines and tools are much higher than the

expenses for high performance computers and modern software [14].

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In the present research, the numerical simulation is employed aiming to meet the above

items, especially improvement of the process knowledge for optimising purposes. In other

words, the numerical simulation is used to clarify the deformation behavior and the effect

of working conditions. One can uses a perfect model as a beneficial means to predict the

effects of working conditions on the deformation behavior.

Basically, the FEM implementations are either implicit or explicit. In an explicit code,

the accuracy of the solution is reported to be questionable and sensitive to scaling effects.

Implicit codes have proven to give reasonably springback results, although at unrealistic

high computational cost, but local deformations are generally underestimated. An implicit

algorithm solves the non-linear problem directly by use of a predictor-corrector method.

The stiffness matrix is updated throughout the entire analysis. In an explicit code, the

corrector step is omitted, i.e. no equilibrium check is performed. Stability requirements

provide that the drift from the correct solution is limited, as the time steps have to be

sufficiently small. The large benefit of explicit algorithms is their computational efficiency

and ability of handling contact problems. However, implicit codes are more effective in the

unloading phase [11].

During the shear bending, inside and outside of the tube is well constrained. Thus,

implementation of an implicit analysis is difficult. Therefore, in the present work, an

explicit analysis is employed. Introducing a reasonable time step and scaling factor into

simulation model results in adequate accuracy.

2.3.1 Simulation model

A 3D explicit analysis for the shear bending process of circular tubes was conducted

using a commercial finite element code ELFEN, which was developed by Rockfield

software Limited, Swansea. This code is widely used for the analysis of metal forming via

FEM [15-19]. Figure 2.8 shows the simulation model. Due to the symmetry, one half

II-12

model was considered. The model consists of the moving die, the fixed die, the mandrel of

moving die, the mandrel of fixed die and the circular tube.

Fixedside

Fixed die

Movingdie

Mandrel ofmoving die

Mandrel offixed die

P: Axialpushingpressure

Rigid bodymotion

Tube

Y

X

Figure 2.8 Simulation model for analysis of the tube shear bending process

The simulation parameters are detailed in Table 2.2. The material properties were

obtained from the conventional tensile test.

Table 2.2 Simulation parameters

Material A1050 (1) A1050 (2)

Young’s modulus, E/ GPa 70 70

Poisson’s ratio, v 0.34 0.34

Work hardening exponent, n 0.29 0.07

Strength coefficient, K / MPa 148 165

II-13

As a general rule, if the ratio of the tube diameter to its wall thickness is smaller than 20,

the assumption of a thin wall tube or shell cannot be employed [20]. Accordingly, the

circular tube was modeled using 8-node hexahedral elements. The number of elements in

the thickness direction was n=60t/D in which t and D denote the initial thickness and

diameter of the tube respectively. In this manner, the number of elements in the thickness

direction for t=2, 2.5 & 3mm were 4, 5 & 6 respectively. The number of elements in the

circumferential direction was 40.

The tube blank was assumed to be an isotropic elastoplastic material following the

Von-Mises yield criterion and obeying the n-power law nKεσ = , where σ is the effective

stress, ε is the effective plastic strain, n is the strain hardening exponent and K is the

strength coefficient.

The dies and mandrels were defined by surfaces and created as rigid bodies. The fixed

die and its mandrel were completely constrained whereas a rigid body motion was applied

on the moving die and its mandrel.

The axial feed of the tube during the process was modeled using a force- based approach.

A constant axial pressure was applied on the end face of the tube whereas the displacement

of the other end of the tube was constrained.

A surface-to-surface contact model, which allows for sliding between these surfaces

with a coulomb friction model, was set in the simulation with a penalty algorithm, which

imposes a force constraint upon the nodes in order to prevent/minimize penetration.

A constant friction coefficient for all contact surfaces was set in the simulation.

Moreover, a uniform clearance between the tube and tooling was assumed.

Time scaling was used to secure a reasonable calculation time. For this purpose, the

scaling factor was taken as 500. The bending angle was 90º and the outside bending radius

ro was half of the tube diameter.

II-14

2.3.2 Failure criteria

Wrinkling and rupture are the major failure modes or defects encountered in the tube

shear bending process. Generally, rupture can be predicted by: (1) strain based criteria, e.g.

forming limit diagrams (FLDs) and maximum part thinning; (2) stress based criteria, e.g.

forming limit stress diagrams (FLSDs); and (3) ductile damage criteria [20]. In this study,

wall thinning is chosen as the rupture criterion. Therefore, a critical value of the tube

thinning means occurrence of rupture in the numerical simulation

h

Pushedside

Formingdirection

Z

XY

Figure 2.9 Measuring the wrinkle height in the FE simulation

Wrinkling can be predicted using various criteria [21-25]. In this study, the geometrical

method for prediction of wrinkling is adopted, which directly measures the wrinkle

dimensions of the deformed meshes. As shown in Figure 2.9, wrinkle amplitude is

measured by calculating the distance between the tops and bottoms of wrinkles in planes

perpendicular to the tube face and parallel to the forming direction.

Using experiment and FE simulation as the examination methods, the formability and

accuracy of process are interrogated in the following sections.

II-15

References

[1] S. Baudin, P. Ray, B.J. Mac Donald and M.S.J. Hashmi, Development of a novel

method of tube bending using finite element simulation, Journal of Materials Processing

Technology, Volumes153-154 (2004), pp.128-133.

[2] P. Gantner, H. Bauer, D. K. Harrison and A.K.M. De Silva, Free bending -A new

bending technique in the hydroforming process chain, Journal of Materials Processing

Technology, Volume 167 (2005), 302-308.

[3] H. C. Kwon, Y. T. Im, D. C. Ji and M. H. Rhee, The bending of an aluminum structural

frame with a rubber pad, Journal of Materials Processing Technology, Volume 113, Issues

1-3 (2001), pp. 786-791

[4] Z. Jin, S. Luo and X. Daniel Fang, KBS-aided design of tube bending processes,

Engineering Applications of Artificial Intelligence, Volume 14, Issue 5 (2001), pp.

599-606

[5] Z. Hu and J.Q. Li, Computer simulation of pipe-bending processes with small bending

radius using local induction heating, Journal of Materials Processing Technology, Volume

91 (1999), pp.75-79.

[6] M. Tanaka, M. Michino, K. Sano and M. Narita, Pipe bending technology with zero

bending radius, Journal of the Japan Society for Technology of Plasticity, Volume 35

(1994), p. 232 (In Japanese).

[7] M. Tanaka, M. Michino and K. Fujihira, Pipe forming with shearing deformation, in:

Proc. The Japanese Conference for the Technology of Plasticity, (1996), pp. 170-171, (In

Japanese).

[8] Japan patent, 210722A, (2000).

[9] Y. Hamanishi and K. Kato, Development of shear bending and its applications, in: Proc.

The 99th Japanese Meeting on Tube Forming Technology, (2002) (In Japanese).

[10] M. Geiger, Manufacturing science- driving force for innovations, Proceedings of the

II-16

7th ICTP (2002), Volume1, pp.17-30.

[11] F. Paulsen, T. Welo and O. P. Søvik, A design method for rectangular hollow sections

in bending, Journal of Materials Processing Technology, Volume 113, Issues 1-3 (2001),

pp.699-704.

[12] G.W. Rowe, Finite element plasticity and metalforming analysis, Cambridge

University press (1991).

[13] R. H. Wagnor and J. L. Chenot, Metal forming Analysis, Cambridge University press

(2001),

[14] F. Vollertsen, A. Sprenger, J. Kraus and H. Arnet, Extrusion, channel, and profile

bending: a review, Journal of Materials Processing Technology, Volume 87, Issues 1-3

(1999), pp. 1-27

[15] ELFEN 2D/3D numerical modeling package, Version 3.7.0, Rockfield Software Ltd.,

Technium, Swansea, UK.

[16] T. Okui, K. Kuroda and M. Akiyama, Die design for reducing tube outside diameter

by cold pressing and mechanism of thinning and bending phenomenon, Ironmaking &

Steelmaking, Volume 33, Issue 3 (2006), pp. 223-228.

[17] P. Fuschi, M. Dutko, D. Peri and D. R. J. Owen, On numerical integration of the

five-parameter model for concrete, Computers & Structures, Volume 53, Issue 4 (1994), pp.

773-1045.

[18] E. A. de Souza Neto, D. Peri , M. Dutko and D. R. J. Owen, Design of simple low

order finite elements for large strain analysis of nearly incompressible solids, International

Journal of Solids and Structures, Volume 33, Issues 20-22 (1996), pp. 3277-3296.

[19] M.S. Nielsen, N. Bay, M. Eriksen, J.I. Bech and M.H. Hancock, An alternative to the

conventional triaxial compression test, Powder Technology, Volume 161, Issue 3 (2006),

pp. 220-226.

[20] F.P. Beer and E. R. Johnston, Mechanics of materials (second ed.), McGraw-Hill Inc.,

II-17

New York (1992).

[21] Z Q Sheng, S. Jirathearanat and T. Altan, Adaptive FEM simulation for prediction of

variable blank holder force in conical cup drawing, International Journal of Machine Tools

and Manufactures., Volume 4 (2004), pp.487-494.

[22] P. Nordlund and B. Haggblad, Prediction of wrinkle tendencies in explicit sheet

metal-forming simulations, International Journal for Numerical Methods in Engineering.,

Volume 40 (1997), pp. 4079-4095.

[23] X. Wang and J. Cao, Wrinkling Limit in Tube Bending, Transactions of ASME,

Volume 123 (2001), pp. 430-435.

[24] R. Peek, Wrinkling of tubes in bending from finite strain three-dimensional continuum

theory, International Journal of Solids Structures. , Volume 39 (2002), pp. 709-723.

[25] H. Yang and Y. Lin, Wrinkling analysis for forming limit of tube bending processes, J

Mat. Process. Tech., Volume152, Issue 3 (2004), pp. 363-369.

III-1

CHAPTER 3

DEFORMATION BEHAVIOR OF A CIRCULAR TUBE SUBJECTED TO THE SHEAR BENDING PROCESS

3.1 Introduction

In this chapter, the deformation behavior of a tube during the shear bending process is

clarified. For this purpose, the Finite element Method is employed. Illustrating the

mechanism of deformation helps us to find out the effects of different working conditions

on the forming process.

3.2 Deformation behavior and strains distributions

In this part, the deformation behavior of a tube subjected to the shear bending process is

analyzed. Studying and comparing the states of deformations during basic forming

processes may help us to understand the deformation mechanism. For this purpose, at first,

the deformation mechanism during two simple processes: (1) pure bending and (2) pure

shearing is investigated. Comparing these two deformation models, the behavior of the

tube during the shear bending will be clarified.

3.2.1 pure bending process

Figure 3.1 presents a schematic illustration of deformation during a pure bending

III-2

process say push bending.

We suppose a square element of the blank (ABCD) and trace its deformation. The

element is bent on the bending zone (section b) and again straightened on the unbending

zone (section u). It is seen that a straight line (AB) remain straight and perpendicular to

forming route during the deformation (B'C' and B"C"). Moreover, between sections b and u

the deformation is in a steady state and the value of bending strain is

e=q/R .

A B

CDA

B'

C'

D

A

B''C''

D

R

q

b

u

Figure 3.1 Schematic representation of a pure bending process

3.2.2 Pure shearing

Figure 3.2 represents schematically a 2D shearing deformation through a die set. The

corners of passageway are sharp. The deformation behaviors of a square element (ABCD)

and a straight line, which has right angle respect to forming route before deformation (BC),

are analyzed. The deformation is occurred on MN which is called shearing plane. It is seen

III-3

that in the pure shearing deformation, the straight line remains straight and are sheared to

angle α. It is because during the shearing process, the linear velocity throughout the

deformation zone is constant.

A B

D C B'

A'

C'

D'

l

l

Shearing force

Y

Xl Tφ

M

N

Figure 3.2 Schematic representation of a 2D pure shearing process

As it has been illustrated in Figure 3.2, a small element, which initially has a square

shape is considered. Under the effects of shearing force, the element moves through the

passageway and deformed on the deformation zone.

It is assumed that each point of element travels with the same velocity. Therefore

AB=BC=A’B’=CD’=D’C’=l.

And the engineering shear strain

=2.

Therefore, the value of shear strain in the pure shearing process

TCTBBTC '/')''tan( =∠=γ

III-4

εsh=γ /2 = 1.

Consequently, the effective strain becomes

=1.15.

3.2.3 Actual shear bending

Now, we study the state of deformation in the actual shear bending. The deformed

meshes of the tube after the process have been shown in Figure 3.3. Route T, route L and

route B represent the top, lateral and bottom sides of tube respectively.

From the traces of the deformed grids, it can be seen that the deformation occurs through

a deformation zone. Passing through this zone, the meshes have been sheared across the

tube width. The grid distortion indicates the magnitude of deformation. In the vertical part

of the tube, the meshes around the Route L have been deformed intensively. Whilst a

steady state of deformation can be observed in the longitudinal direction of the tube’s

sheared part, the deformation in the peripheral direction is not homogenous. The elements

in the vicinity of the tube lateral side (Route L) have sustained more shearing deformation

compare to the top and bottom sides. Approaching to the Routes T and B, the degree of

deformation reduces and the elements in these regions have been subjected to significantly

less shear deformation.

Same as sections 3.2.1 and 3.2.2, we suppose a straight line and trace its deformation

during and after passing the deformation zone. As shown in Figure 3.4, the tube can be

divided into three regions in the peripheral direction from the inside of bending inside to

the outside:

-Region 1 (Bending inside): in this region, the mesh lines are horizontal and almost

perpendicular to the forming direction.

21

222 )

2(

32

⎥⎥⎦

⎤

⎢⎢⎣

⎡++= xy

yx

γεεε

III-5

Tube pushedside

Deformation zone　

D

C

F

E

Route T

Route L

Route B

View H

Less sheared zone

View H

Verticalpart

Figure 3.3 Finite element meshes after the deformation, (A1050 (2), D0=30mm,

t0=1.5mm, rc=3.0mm)

-Region 2 (Lateral side): mesh lines are tilted.

-Region 3 (Bending outside): mesh lines are horizontal again similar to the Region 1.

Remembering the deformation models during the pure bending (Figure 3.4(b)) and pure

shearing (Figure 3.4(c)) processes and comparing with the deformation behavior in the

III-6

Pushedside

Bendinginside

Bendingoutside

Forming

direction

e

f

e'

f '

e'' f ''

e

f

e'

f '

α

(a)

(b) (c)

e

f

e'

f '

Region 1 Region 3Region 2

Figure 3.4 Deformation behavior in (a)the actual shear bending; (b) pure bending; (c)

pure shearing

III-7

actual shear bending (Figure 3.4(a)), it is found that the deformation behavior in the

Regions 1 and 3 is similar to pure bending and in the Region 2 is close to pure shearing.

Therefore, the process of tube shear bending is governed by a combination of shearing and

bending deformations. Lateral side of the tube undergoes the maximum shearing

deformation whereas the deformation mode around the top and bottom sides of the tube is

bending.

In order to evaluate the amount of strains during the shear bending process, we suppose

a 2D shearing process through a die set with round outside and sharp inside corners. As

seen in Figure 3.5, the shearing deformation spreads over a deformation zone rather than a

single shearing plane. A small element, which has a square shape initially, is considered.

Under the effects of pushing and shearing forces, the element moves through the

passageway and deformed on the deformation zone.

A B

D C

B'

A'

C'

D'

M

N

T

l

αβ

α

Pα

l

Deformationzone

Shearing force

Y

X

Figure 3.5 Schematic representation of a 2D shearing process

III-8

β⋅=− CMCNMA'

It is assumed that each point travels with the same velocity and passes on concentric

routes. Under these situations and from Figure 3.5 it can be seen that the distances traveled

by points B and C are

dC= CC’=CN+ND’+D’C’, (3-1)

dB=BB’=BM+MA’+A’B’ . (3-2)

in which AB=DC=D’C’=A’B’=l. As the traveled distances by all points are assumed to

be equal dC=dB. Therefore, the relations (3-1) and (3-2) yield

ND’=BM+MA’-CN. (3-3)

Also as we assumed that CN and MA’ are concentric curves:

,

, (3-4)

. (3-5)

Therefore, substituting the relations (3-4) and (3-5) into (3-3)

. (3-6)

Based on the deformation geometry, the amount of engineering shear strain is:

TCTBBTC '/')''tan( =∠=γ (3-7)

in which, B’T=B’P+PT ; B’P=ND’ ; PT=BM ; C’T=l.

Hence, introducing the relation above into (3-7)

γ = (ND’+BM) /l. (3-8)

Substituting the relations (3-5) and (3-6) into relation (3-8)

γ = 2tan(α)+β /cos(α).

Also from Figure 3.5 it is clear that β=π/2-2α.

Therefore, the amount of engineering shear strain is

γ = 2tan(α)+(π/2-2α) /cos(α). (3-9)

Consequently, the value of shear strain becomes

εsh =γ /2= tan(α)+(π/4-α) /cos(α). (3-10)

)cos(/ αlCM =

)tan(α⋅= lBM

)cos(/)tan(' αβα ⋅+⋅= llND

III-9

In the situation which the corner radius is half of the tube diameter

α=tan-1(1/2)=0.46 rad.

Therefore, from relation (3-10)

εsh ≈0.86.

In this case, the principal and effective strains take values of

ε1=0.86 ; ε 2=-0.86, (3-11)

ε =(2/3)( ε12+ε2

2- ε1ε2)1/2 ≈0.99. (3-12)

From the results above, it is found that during the shearing deformation through a die

set with round corner, deformation spreads through a deformation zone. Therefore,

comparing with the case of shearing through a sharp corners die, for which shearing occurs

on a single shear plane, the degree of shearing deformation decreases.

Figure 3.6 shows the distribution of effective strain obtained by simulation. In the

vertical part of the tube, which has undergone a shearing deformation, the steady state of

deformation is observed. In this region the effective strain along the tube longitudinal

direction is uniform whereas it varies in the tube’s circumferential direction.

The effective strain distribution across the tube’s vertical part obtained by simulation is

plotted in Figure 3.7. Also the principal strain components are shown in this figure. The

maximum effective strain is 1≈ε around θ=90º. Approaching to point B (bottom side of

the tube) or point T (top side of the tube), as the shearing deformation decreases, the

effective strain reduces.

III-10

Figure 3.6 Distribution of effective strain, (A1050 (2), D0=30mm, t0=1.5mm, rc=3.0mm)

0 45 90 135 180–1

0

1

Angular position,θ / deg

Stra

in

ε ε3 ε1 ε2

Figure 3.7 Strain distributions across the tube’s vertical part (BT in Figure 3.6)

(A1050 (2), D0=30mm, t0=1.5mm, rc=3.0mm)

III-11

C

D

F

E

Route T

Route B

shearedpart

Pushedside

A comparison between the simulation and experimental results of thickness strain along

the Route T and Route B of the deformed tube has been presented in Figure 3.8. As

mentioned before, the deformation mode around the bottom and top sides of the tube is

bending. Point C on the Route B and point F on the Route T are under compression in the

longitudinal direction and thickness strain in these regions is positive. On the contrary,

point D on the Route T and point E on the Route B are under tension and thickness strains

of these points are negative.

Almost good agreements between the simulation and experimental results are seen.

More agreement between these results can be achieved by accurate modeling of material

properties and the friction effects.

50 100 150 200

–0.2

0

0.2

0.4 Route T(FEM)A1050, t0=1.5mm, D0=30mm, rc= 5mm, c= 0.1mm

Distance from tube pushed side, ll /mm

Thic

knes

s stra

in, ε

t

C

D

F

E

Route T(Experiment)Route B(FEM) Route B(Experiment)

Figure 3.8 Distribution of thickness strain along Routes T and B (Figure 6)

III-12

3.3 Conclusion

In this section, the deformation behavior of a tube subjected to the shear bending process

was investigated. In was found that, during the process, a combination of shearing and

bending deformations occurs. In this manner, each part of the tube suffers a shearing

component as well as a bending moment. Lateral side of the tube undergoes the maximum

shearing deformation whereas the deformation mode around the top and bottom sides of

the tube is bending. Moreover, the values of strain were obtained by simulation. Good

agreements between the analytical and experimental results were found.

IV-1

CHAPTER 4

EFFECT OF AXIAL PUSHING FORCE ON THE SHEAR BENDING PROCESS

4.1 Introduction

As mentioned in chapter 2, there are few reports engaging the shear bending process.

These reports present the shear bending process applying hydraulic pressure inside the tube

and axial pushing force on it (Tanaka et al.) or a developed process using mandrels inside

the tube instead of liquid pressure and employing the displacement control technique

(SANGO Co. Ltd). In this research, the mandrels were used inside the circular tube instead

of liquid pressure. Moreover, the displacement control technique was replaced by

application of a controllable axial pushing pressure on the tube.

In this section, the effects the axial pushing pressure on the process are investigated both

by the experiment and the analytical methods.

4.2 Preliminary experiments

In the shear bending, material should be supplied into dies during the process. For this

purpose, a pushing force is exerted on the tube end side. Figure 4.1 shows the typical

products of shear bending obtained by applying various pushing forces on the blank tubes.

The parameter P indicates the amount of pushing pressure applied on the tube. The case (b)

IV-2

shows a sound product obtained by employing a suitable pushing force (○). In this case, no

failure is observed. When the pushing force is not enough, rupture occurs (Figure 4.1(a)).

Also, if the pushing force exceeds a critical amount, wrinkling happens (Figure 4.1(c)).

From these results it is found that a limit range of the appropriate pressures to perform a

successful forming process exists. If the value of applied pushing pressure is not selected

within the appropriate range, rupture or wrinkling occurs. Therefore, in order to perform

the forming process successfully and achieve a sound product without any failure, the

amount of the pushing force should be appropriate.

0 50mm(a) P=30MPa

(b) P=35MPa

(c) P=40MPa

Figure 4.1 Effect of the pushing force on the forming product, (a) rupture (×);

(b) success (○); (c) wrinkle (▲). (A1050 (2), D0=30mm, t0=1.5mm, rc=3.0mm)

Applying inappropriate pushing force on the tube, the following phenomena may occur:

(1) Rapture at low pushing force

(2) Wrinkling at high pushing force

In this section, the mechanism of deformation based on the analytical results is

examined to find out the causes of defects generation. In order to examine the deformation

IV-3

mechanism in the Region 2 (Figure 4.3), a schematic illustration of 2D shearing process in

a die set is presented in Figure 4.2. The outside and inside corners of dies are sharp to

eliminate the bending effects and provide the situation of a 2D shearing process.

τ

σ

τ

σ

E (σ E,τ E)

|σx | > |σy|σE < 0

|σx | < |σy|σE > 0

(a) (b)

E (σ E,τ E)

σx σyσx σy

A B

D C

B'

A'

C'

D'

M

N

Shearing force

YX

Pushing force

σy

σx

Figure 4.2 State of stresses for a typical element (E) subjected to shearing process

IV-4

A shearing force attendant upon an axial pushing force is applied on the workpiece.

Passing through the passageway, the unit element ABCD is sheared to A'B'C'D' abruptly on

the shear line (MN). The state of stresses for the typical element is shown in the figure,

which involves a tensile stress σy due to the shearing force and compressive stress σx

corresponding to the pushing force. The shearing deformation occurs on the shearing plane

(MN), which has 45˚ direction respect to the XY coordinate system.

When the absolute values of these stresses are equal, i.e. | σx| = | σy|, pure shearing

situation governs the deformation in which only a pure shearing stress acts on the shearing

planes and no thickness change happens. But when σx and σy have different absolute

values, the state of stresses on the shearing plane consists of a shearing component as well

as a normal stress introduced by (σE,τE). Hence, the situation of pure shearing is not

satisfied. Based on the value of σx, two different cases are possible. When | σx| < | σy|, the

normal component σE is positive. In this case, during the shearing deformation, thinning

occurs and the tendency of rupture increases. On the other hand, when | σx| > | σy|, the

normal component is compressive, which means the thickness increases and the material

tends to wrinkle.

As a result, it was understood that the pushing force plays an important role in the

forming process and affects the thickness distribution of the product. As mentioned in

chapter 3, the tube consists of three regions in the peripheral direction (Figure 4.3):

Region 1 (Bending inside); Region 2 (Lateral side) and Region 3 (Bending outside). The

deformation behavior in the Regions 1 and 3 is similar to the pure bending and in the

Region 2 is close to the pure shearing. When the pushing force is appropriate, it results in

an ideal deformation. However, inappropriate pushing pressures result in defect generation

as shown in Table 4.1.

IV-5

Pushedside

Bendinginside

Bendingoutside

Forming

direction

e

f

e'

f '

Region 1 Region 3Region 2

Figure 4.3 Deformation behavior during the actual shear bending

Table 4.1 Effects of the applied pushing force on thickness changes

Region Low pressure High pressure

1 ___ Thickening-wrinkling

2 Thinning Thickening

3 Thinning-rupture ___

4.3 The effect of the pushing force on working loads

The relation between the shearing force and shearing stroke (Figure 4.4) as a function of

the axial pushing pressure for t0=2mm has been plotted in Figure 4.5.

IV-6

Shea

ring

Stro

ke

Pushingforce

Shearingforce

Pushingstroke

Figure 4.4 Definition of working loads and strokes

It is seen that the shearing force decreases by increasing the applied pushing pressure. In

order explain the reason, the relation between the shearing and pushing strokes for the

same values of pushing pressures has been shown in Figure 4.6. From this figure, it can be

seen that increasing the pushing pressure, the slope of curves has been increasing. It means

that more material supplied into dies with the increase of the pushing pressure.

Remembering the deformation behavior and the state of stresses for a 2D shearing

process (Figure 4.2), the values of σx and σy can be calculated as:

σx=FP /A . (4-1)

σy=FS /A . (4-2)

where FP is the axial pushing force, FS is the shearing force and A corresponds to the tube’

section area. The shearing stress acting on the shearing plane is

τxy=(σx-σy)/2 . (4-3)

In order to perform a plastic deformation, employing the Tresca yield criteria, the shearing

stress must reach the flow stress of material. In other words,

τxy=Y/2 . (4-4) in which, Y is the flow stress of material.

IV-7

0 20 40 60 80

5

10

15

A1050, D0=30mm, t0=2mm, rc=5.0mm

Shea

ring

forc

e, F

S /k

N

Shearing stroke, SS /mm

P=25MPa, ruptureP=30MPaP=40MPaP=50MPaP=60MPa, wrinkleP=80MPa, wrinkle

Figure 4.5 Relation between shearing force and shearing stroke as a function of the

axial pushing pressure.

0 20 40 60 800

20

40

60

80

A1050, D0=30mm, t0=2mm, rc=5.0mm

Push

ing

stro

ke,

SP

/mm

Shearing stroke, SS /mm

P=40MPa

P=60MPa, wrinkleP=80MPa, wrinkle

P=50MPa

P=30MPaP=25MPa, rupture

Figure 4.6 Relation between shearing and pushing strokes as a function of the axial

pushing pressure.

IV-8

Introducing the relations (4-1) and (4-2) into (4-3) and using relation (4-4), finally the

following relation between the working loads can be derived

FS+FP=AY . (4-5)

This relation is a rough approximation. However, it indicates that by increasing the

pushing force, the shearing force decreases.

4.4 The effect of the axial pushing force on the distribution of thickness

strain

Figure 4.8 shows the effect of the axial pushing pressure on the distribution of thickness

strain for t0=3mm along the Route T, Route L and Route B (Figure 4.7). Routes T, B and L

represent the top, bottom and lateral sides of the tube respectively. The thickness strain is

calculated as )ln( 0ttt =ε , where t0 is the tube’s initial thickness.

Figures 4.8(a) and (b) indicate that increasing the pushing pressure the amount of

thickness reduction decreases. Moreover, the tube thinning along the Route L decreases by

increasing the pushing pressure as shown in Figure 4.8(c). The excessive material due to

high pushing pressures is redistributed in the tube wall. In other words, the thickness

reduction of the deformed tube decreases. From the results above it is found that, the

applied pushing pressure affects the thickness distribution of the deformed tube.

IV-9

ab

c d Top side:Route T

Lateral side:Route L

Bottom side:Route B

BB Tθ

T

shearedpart

Pushedside

Figure 4.7 Schematic representation of the deformed tube and thickness measurement

zones

–0.5

0

0.5

A1050, D0=30mm, t0=3mm, rc=5.0mm

Thic

knes

s stra

in,

ε t

P=60MPaP=80MPa

P=40MPa

b ca dMeasuring position

Figure 4.8 (a) Distribution of thickness strain along the Route T

IV-10

–0.5

0

0.5

A1050, D0=30mm, t0=3mm, rc=5.0mm

Thic

knes

s stra

in,

ε t

P=60MPaP=80MPa

P=40MPa

b ca dMeasuring position

Figure 4.8 (b) Distribution of thickness strain along the Route B

–0.5

0

0.5

A1050, D0=30mm, t0=3mm, rc=5.0mm

Thic

knes

s stra

in,

ε t

P=60MPaP=80MPa

P=40MPa

cba dMeasuring position

Figure 4.8(c) Distribution of thickness strain along the Route L

IV-11

4.5 The effect of the axial pushing force on the cross section

deformation of the deformed tube

Figure 4.9 presents the relation between the axial pushing pressure and the cross section

ovality of the deformed tube obtained by experiments as well as simulation. The ovality of

the deformed tube is evaluated as

η =|η1|+|η2| (4-6) in which, η1 and η2 are calculated according to Figure 4.10 and using the following

relations: η1=(D v-D 0 ) /D 0 η2=(D 0-D h ) /D 0 .

30 40 500

2

4

6A1050, D0=30mm, t0=2mm, c=0.1mm

Axial pushing pressure, P /MPa

Max

imum

ova

lity

/% Exp.

Sim., µ =0

Figure 4.9 Effect of the axial pushing pressure on the cross section ovality of the

deformed tube

The lower the pushing pressure, the lesser the supplied material and the higher the

thinning of the tube wall is. In other words, during shearing deformation under the effect of

IV-12

low pushing pressures, the tube undergoes the elongation and its cross section deforms.

As the result, in order to prevent the tube from high ovality, the value of the pushing

pressure must be as high as possible within a range in which wrinkling dose not occur.

Pushedside

Dh

Dv

Fixed die

Figure 4.10 Definition of the tube diameters to evaluate its cross section ovality

4.6 Conclusions

In this chapter, the effects of the axial pushing pressure on the shear bending process of

circular tube were investigated both by the experiments and numerical simulation. It is

concluded that

1. A limited range of appropriate pressures to perform a successful forming process

exists. If the value of the applied pushing pressure is not selected within the appropriate

range, rupture or wrinkling occurs. Therefore, in order to perform the forming process

successfully and achieve a sound product without any failure, the amount of the pushing

IV-13

force should be appropriate.

2. Applying a higher pushing pressure within the forming limit results in decrease of the

thickness reduction and the cross section deformation of the tube.

V-1

CHAPTER 5

EFFECT OF DIE CORNER RADIUS ON THE SHEAR BENDING PROCESS

5.1 Introduction

Te shear bending of tubes is an appropriate technology to realize unified bent tubular

parts with considerable small bending radii through cold metal forming.

It is important to examine the effects of the die radius in the shear bending process.

Because various requirements exist for bending radius, e.g. small radius is demanded for

reduction of occupied space. On the other hands, larger one would be preferable for liquids

or gases to flow through smoothly. Therefore, it is necessary to clarify the range of bending

radii, which can be obtained applying the shear bending process. This range makes clear

how effective is the shear bending process to meet the various demands in the tube forming

technologies.

In this chapter, the experiment and the finite element simulation are employed as

examination methods aiming to clarify the influence of the die corner radius on the

forming process and deformation accuracy. Moreover, analytical method is helpful to

illustrate the mechanism of defect generation.

In the next section, the methods of examination are introduced and the corresponding

results are presented.

V-2

5.2 Experimental conditions

The experiments were carried out using A1050 (2) extruded aluminum tubes. Figure 5.1

shows the die corner radius. Five sets of dies having different corner radii rc = 2, 3, 4, 5

and 6mm were prepared and utilized in the experiments.

Figure 5.1 The corner radius of a die

5.3 Simulation parameters

The Finite element is employed as an examination method to clarify the influence of the

die corner radius on the forming process. The circular tube was modeled using 8-node

hexahedral elements. The numbers of elements in the circumferential and thickness

directions of the tube were 40 and 3 respectively.

V-3

The experimental results show that for the aluminum material used in the experiments,

wall thinning almost more than 20%, i.e. 2.0)(

0

0 ⟩−

ttt , results in rupture (t0 is the tube’s

initial thickness). Therefore, the critical value of 20% for tube thinning means occurrence

of rupture in the simulation

The critical value of wrinkle amplitude (h/t0) for wrinkling recognition is set to be at 5%

of the tube’s initial thickness in the present simulation model.

Using the experiment and the FE simulation as examination methods, the formability

and accuracy of process are interrogated in the following sections.

5.4 Formability of the tube

5.4.1 Preliminary experiments

A series of preliminary experiments were carried out using a die set with rc=3mm and

applying various pushing pressures on the tube. The corresponding results were presented

in 4.2. From these results it was found that a limited range of appropriate pressures to

perform a successful forming process exists. If the value of the applied pushing pressure is

not selected within the appropriate range, rupture or wrinkle occurs.

In the conventional tube bending processes, bending with small radii may results in

various defects such as rupture in the bending outside and wrinkle or buckle in the bending

inside. Wrinkling and rupture might be suppressed by the increase of the bending radius [1].

In order to investigate the role of the die radius to prevent the tube from rupture and

wrinkling during the shear bending, the dies with different corner radii were utilized and

the pushing pressures of P=30 & 40 MPa were applied on the blank tubes. Figure 5.2 (a)

and (b) show the results. The results are similar to the case of rc=3mm. It means, wrinkling

V-4

and rupture occur regardless the value of the die radius. Against the expectation, wrinkling

and rupture cannot be avoided even by increasing the die radius.

rc =6mm

rc =4mm

rc =2mm

0 50mm

0 50mm

rc =6mm

rc =4mm

rc =2mm

(a)

(b)

Figure 5.2 Typical products using dies with various corner radii, (a) P=30MPa;

(b) P=40MPa

Hereby, we clarified the generation mechanism of rupture and wrinkling .In the

V-5

θ

Pushed side

following sections, we will discuss the effects of the die radius on the tube formability and

defect generation.

5.4.2 The effects of the die radius on the deformation

To investigate the effects of the die radius, the distribution of effective plastic strain

along the circumferential direction of the tube’s sheared part is plotted in Figure 5.3.

Moreover, the distribution of thickness strain obtained by simulation is presented in Figure

5.4. Thickness strain is calculated as )ln( 0ttt =ε , where t0 is the tube’s initial thickness.

90 1800

0.5

1

A1050, D0=30mm, t0=1.5mm, c=0.1mm

Effe

ctiv

e pl

astic

stra

in, ε

p

rc=2mm

rc=4mmrc=3mm

Angular position, θ/ deg

rc=5mm

Figure 5.3 Effect of the die radius on the distribution of effective plastic strain along the

circumferential direction of the tube’s sheared part

When the die radius is small, the tube material near the bending inside (b) undergoes a

severe bending deformation. Therefore, the thickness strain and the level of effective strain

in this region increase. However, the effect of the die radius on the strains distributions in

other regions of the deformed tube is insignificant.

V-6

0 90 180

–0.2

–0.1

0

0.1

0.2A1050, D0=30mm, t0=1.5mm, c=0.1mm

Thic

knes

s stra

in, ε

t rc=3mm rc=5mmrc=2mm

Angular position, θ/ deg

rc=4mm

Figure 5.4 Distribution of thickness strain across peripheral direction of deformed tube

obtained by simulation

The state of stresses for a typical element of tube is shown in the Figure 5.5. The state of

stress on the shearing plane consists of a shearing component as well as a normal stress. In

fact, the amount of the die radius has only a slight effect on the state of stresses. The

position where the tube becomes the thinnest is located far from the die corner. Therefore,

the die radius has a low effect on the occurrence of rupture. The tube rupture is mainly

affected by the value of the applied pushing force.

5.4.3 FEM results of the tube formability

Actually, It is difficult to examine the states of stress, strain and deformation by

experiment, especially those during the deformation. Therefore, using the Finite Element

Simulation and based on the failure criteria explained before, the effects of the die corner

radius and axial pushing pressure on the formability of the tube during the shear bending

V-7

τ

σ

τ

σ

E (σ E,τ E)

|σx | > |σy|σE < 0

|σx | < |σy|σE > 0

(a) (b)

E (σ E,τ E)

σx σyσx σy

A B

D C

B'

A'

C'

D'

M

N

Shearing force

YX

Pushing force

σy

σx

Fig. 5.5 State of stresses for a typical element (E) subjected to shearing process

are investigated.

Figure 5.6 shows the results for rc= 5mm. Various pushing pressures were applied on the

tubes and the maximum thinning and the wrinkle height were determined. From this graph

it is clearly seen that the applied pushing pressure strongly affects the process. Raising the

V-8

pushing pressure, the wrinkle height increases whereas the tube thinning decreases.

Therefore, the tube tends to wrinkle. On the contrary, lowering the applied pushing

pressure, the tendency to rupture increases. It confirms the results of the preliminary

experiments that there is a limited range of appropriate pushing pressures to perform the

forming process successfully.

30 40 50 600

5

10

15

20

25

30

35

40

45

Axial pushing pressure, P /MPa

Wrin

kle

heig

ht a

nd th

inin

g, /

%

A1050,D0=30mm,t0=1.5mm,c=0.1mm,µ=0.1

Wrinkle height, h/t0

WrinkleRupture

Maximum thinning, (t0–t)/t0

Success

Figure 5.6 Effect of the axial pushing pressure on the maximum values ot tube thinning

and wrinkle height obtained by simulation (rc=5mm)

For various die radii, the same graphs were prepared and the formability of the tubes

was interrogated. Figure 5.7 presents the results. This figure shows the effects of the die

radius and the pushing pressure on the results of the forming process using FE simulation

and employing the failure criteria. This figure indicates that the formability of the tubes is

mainly governed by the value of the applied pushing pressure and dose not depends on the

V-9

value of the die radius.

After carrying out the preliminary experiments and clarifying the tube formability by the

simulation, the final experiments are performed and the results will be presented in the

next section.

2 3 4 5 60

20

40

60

80

100

Die corner radius,rc /mm

Push

ing

pres

sure

, P

/MPa

A1050,D0=30mm,t0=1.5mm,c=0.1mm,µ=0.1

Rupture Success Wrinkle

Figure 5.7 Effect of the die corner radius and the pushing pressure on the results of the

shear bending process obtained by simulation

5.4.4 Results of the experiments

A series of experiments were performed using dies with different corner radii and

applying various pushing pressures on the tubes. Figure 5.8 shows the results of

experiments. A typical photo of the successfully deformed tubes is shown in Figure 5.9.

The salient role of the pushing pressure in generation of defects is observed well.

Moreover, for the die radii less than 6mm, the appropriate values of pushing pressures are

almost the same. That is to say, the effect of the die radius on the suitable values of pushing

pressure is insignificant.

V-10

2 3 4 5 60

20

40

60

80

100

Die corner radius,rc /mm

Push

ing

pres

sure

, P

/MPa

A1050, D0=30mm, t0=1.5mm, c=0.1mmRupture Success Wrinkle

Figure 5.8 Results of the experiments using dies with different corner radii and applying

various pushing pressures on the tubes

Figure 5.9 Typical photo of successful shear bending products using dies with different

corner radii

rc=2

rc=3

rc=4

rc=5

rc=6

V-11

This result was predicted by simulation. However, it is seen that for rc=6mm no

successful product could be obtained. In this case, variation of the applied pushing pressure

results in rupture or wrinkle only. It seems that increasing the die corner radius raises the

wrinkling tendency of the tube. The numerical simulation could not predict this result.

Actually, using hexahedral elements, the occurrence of wrinkling in FE simulation is

delayed and its prediction is difficult.

To explain the reason of wrinkling occurrence at large die radii, we refer to Figure 5.10.

Pushedside

Bendinginside

Bendingoutside

Forming

direction

e

f

e'

f '

Region 1 Region 3Region 2

Figure 5.10 Deformation behavior in the actual shear bending

As mentioned before, the deformation in the shear bending is a combination of bending

and shearing. The tube consists of three regions in the peripheral direction from the

bending inside to the outside (Figure 4.3): Region 1 (Bending inside); Region 2 (Lateral

V-12

side) and Region 3 (Bending outside). The deformation behavior in the Regions 1 and 3 is

similar to pure bending and in the Region 2 is close to pure shearing. It can be said that the

total deformation energy is dedicated suitably to the shearing as well the bending

components, in such a manner that the total energy becomes minimum. As the bending on

small radii needs a huge energy, when the die radius is too small the portion of bending in

the total deformation decreases. On the contrary, increasing the die radius, the portion of

the bending deformation increases and the bending region is expanded over the

deformation zone. It means that a larger volume of the tube sustains the bending

deformation during the process and the territory of the Region 1 develops. However, the

bending radius is still very small with respect to the tube diameter. These conditions

heighten the wrinkling potential of the tube.

According to this section, it was understood that a limit range of appropriate die radii

exists. That is to say, there is critical value of die radius on which carrying out the shear

bending results in only rupture or wrinkling.

In the following section, the effects of the die corner radius on the dimensional accuracy

of the deformed tube regarding cross section ovality and thickness strain is studied.

5.5 Dimensional accuracy

5.5.1 Cross section deformation

In order to investigate the effects of the die corner radius on the section deformation, the

flattening factors of the deformed tubes are calculated as

η =|η1|+|η2| in which, η1 and η2 are calculated according to Figure 5.11 and using the following relations:

η1=(D v-D 0 ) /D 0 , η2=(D 0-D h ) /D 0 .

V-13

Pushedside

Dh

Dv

Fixed die

Figure 5.11 Definition of the tube diameters to evaluate its cross section ovality

Figure 5.12 shows the relation between the die corner radius and the maximum values of

the tube’s flattening factors obtained by simulation and experiments. From this figure it is

found that increasing the die corner radius, the value of η2 and consequently the ovality of

the deformed tube decreases.

As mentioned before, the smaller the die radius, the greater the energy required for

bending and straightening is. Therefore, when the die radius is too small, the tube material

cannot follow the die profile during the straightening stage. In other words, the tube

undergoes diametral shrinkage.

As a result, utilizing a die with too small corner radius is not suitable to obtain products

with high section roundness.

V-14

2 3 4 5 60

1

2

3

4

5

6A1050, D0=30.0mm, t0=1.5mm, c=0.1mm

Die corner radius, rc /mm

Max

imum

ova

lity

/%η, Exp.η2, Exp.η2, Sim.η1, Exp.η1, Sim.

η, Sim.

Figure 5.12 Relation between the die radius and the maximum section flattening of the

deformed tube

5.5.2 Thickness change

The experimental results of thickness strain distribution ( )ln( 0ttt =ε ) along the Route T

and B of the deformed tube were plotted in Figure 5.13. It is seen that the effect of the die

radius of the thickness distribution is negligible.

According to section 5.5, it can be concluded that increasing the value of the die radius

promotes the accuracy of the deformed tube regarding the cross section roundness.

V-15

a b

c d Route T

Route B

shearedpart

Pushed side

–0.2

0

0.2

0.4

A1050, D0=30mm, t0=1.5mm, c=0.1mm

Thic

knes

s stra

in,

ε t rc=3mmrc=4mm

rc=2mm

b ca dMeasuring position

rc=5mm

(1)

–0.2

0

0.2

A1050, D0=30mm, t0=1.5mm, c=0.1mm

Thic

knes

s stra

in,

ε t rc=3mmrc=4mmrc=2mm

b ca dMeasuring position

rc=5mm

(2)

Figure 5.13 Experimental results of thickness strain distribution along (1) Route T;

(2) Route B of the deformed tubes

V-16

5.6 Conclusions

To investigate the effects of the die corner radius on the shear bending process, a series

of experiments were carried out using circular A1050 aluminum tubes and utilizing dies

with different corner radii. Aiming to clarify the deformation mechanism, a 3D FE

simulation was conducted. The appropriate working conditions depending on the die radius

were examined. Moreover, the influence of the die radius on the dimensional accuracy was

investigated. The principle method is proposed on the selection of the die corner radius and

the appropriate pushing forces for different die radii. As conclusions:

1. In order to perform the shear bending process successfully in which rupture and

wrinkle does not occur, an appropriate pushing force should be applied on the tube.

2. The appropriate pushing force is almost constant regardless the value of the die

radius.

3. There is a limit range of suitable die radii. Forming on dies with radii larger than a

critical value, results in only rupture or wrinkle in tube.

4. The effect of the die radius on the thickness changes of the deformed tube is low.

5. Whilst a small bending radius results in high cross section deformation, increasing the

die corner radius, the wrinkling tendency of the tube increases. Thus, if the die radius is

allowed to be selected the larger radius within the suitable range of radii should be

employed because it would improve the precision.

V-17

Reference

[1] L.Gao and M. Strano, FEM analysis of tube pre-bending and hydroforming, Journal of

Material Processing Technology, Volume 151, Issues1-3 (2004), pp.294-297

VI-1

CHAPTER 6

EFFECT OF INITIAL THICKNESS ON THE SHEAR BENDING PROCESS

6.1 Introduction

In this chapter, the effects of the initial thickness on the shear bending process are

investigated. For this purpose, the role of the initial thickness on the prevention of the

defect generation during the shear bending process is illustrated by the experimental and

analytical approaches. The influence of the initial thickness on the forming limit of A1050

aluminum tubes is clarified by the experiment. Moreover, the effects of initial thickness on

the deformation accuracy regarding thickness change and section deformation are clarified.

6.2 Experiments

Circular A1050 (1) aluminum tubes with diameter of D0=30mm were used in the

experiments. Also, the die corner radius was rc=5mm.

6.2.1 Preliminary experiment

As was mentioned in the previous chapters, in order to perform the shear bending

process successfully in which either rupture or wrinkle does not occur, an appropriate

pushing force should be applied on the tube. In the preliminary experiments, the

VI-2

appropriate values of pushing pressure for thin aluminum tubes with t0=1mm are to be

determined. Different pushing pressures were exerted on the blank tubes. However,

variation of applied pushing forces results in only rupture or wrinkle. In other words, an

appropriate value of pushing pressure for successful forming cannot be found for t0=1mm.

In conventional bending processes, undesirable defects such as rupture and wrinkle

might be avoided by employing thick tubes [1]. In order to check the validity of this

criterion in the shear bending, blank tubes with t0=1.5mm were used. Figure 6.1 shows the

typical results. It seems that occurrence of wrinkle can be restrained by employing a

thicker tube and applying a suitable pushing pressure. More examinations based on the

analytical and theoretical approaches are presented in the next sections.

(c)P=50MPa

(a)P=30MPa

(b)P=40MPa

Figure 6.1 Typical products by applying various pushing pressures on the tubes:

(a) Rupture (×); (b) success (○); (c) wrinkle (▲); (A1050 (1), D0=30mm, t0=1.5mm,

rc=5.0mm)

VI-3

6.3 Results of simulation

The FEM is used as an investigation method to analyze the effects of the tube thickness

on the deformation behavior. At first, the conditions of the preliminary experiments are

simulated. Various pushing pressures were applied on tubes with t0=1mm. The deformed

meshes are inspected to recognize the existence of wrinkles in the tubes. Moreover, the

thickness reduction around the outside of bending where the tube becomes thinnest is

determined. Figure 6.2 shows the effect of the axial pushing pressure on the wrinkle height

and the maximum thinning of the tube. Increasing the value of pushing pressure, the

thickness reduction decreases. However, severe wrinkle tendency is observed even

applying low pushing pressures.

10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

Axial pushing pressure, P / MPa

Wrin

kle

heig

ht a

nd th

inni

ng Wrinkle height, h/t0Maximum thinning, (t0–t)/t0

A1050,D0=30mm,rc=5mm,t0=1mm,µ=0

Figure 6.2 Prediction of the wrinkle height and maximum thinning of tubes under the

effect of various pushing pressures obtained by simulation, t0=1mm

VI-4

Now, the effect of the initial thickness on prevention of wrinkle is investigated. Under

the influence of the same pushing pressure, the relation between the tube thickness and the

wrinkle height obtained by simulation is shown in Figure 6.3. It can be seen that increasing

the initial thickness, the wrinkle height shortens. It means the tube resistance against

wrinkling is enhanced.

1 1.5 2 2.50

0.5

1

1.5

2

2.5

Initial thickness, t0 / mm

Wrin

kle

heig

ht,

h / m

m

A1050,D0=30mm,rc=5mm,P=40MPa,µ=0

Figure 6.3 Effect of the initial thickness on the wrinkle height obtained by simulation,

P=40MPa

Naturally, wrinkling is a phenomenon of compressive instability at the presence of

excessive in-plane compression and lack of boundary constraints. During the shear bending

process, material should be supplied into dies. Performing the process without sufficient

axial feed will cause the tube to rupture while excessive application of axial force leads to

tube wrinkle. That is to say, in order to prevent the tube from extreme thinning and rupture,

a compensating feed of material is necessary. However, it may results in excessive material

in other regions of tube. For prevention of wrinkling, the excessive material must be

VI-5

suppressed by appropriate constraint or absorbed by the tube wall through thickening

deformation.

The critical zone of tube during shear bending respect to wrinkling failure is shown in

Figure 6.4. This region is under compression and insufficient constraint. Both simulation

and experiment indicate that wrinkle will be initiated from this region. In order to find out

how increasing in thickness prevents the tube from wrinkling, firstly the deformation

behavior of the tube is illustrated. Then, the influences of the initial thickness on the

forming velocity and deformation energy of the critical zone are illustrated.

Fixed die

Critical zone

Pushingforce

Shearingforce

Figure 6.4 Schematic illustration of the critical zone of tube in the shear bending.

6.4 Deformation behavior

A schematic configuration of the tube during the shear bending is shown in Figure 6.5.

To analyze the behavior of the tube, the concept of equivalent cross-section is employed. It

is assumed that the tube section can be discretized into several straight segments.

With respect to the applied loads, the tube is divided into 3 regions. The segment 1 has a

low section modulus compared to other segments and tends to be bent. On the contrary, the

segment 3 shows high resistance against bending and sustains shearing deformation with

VI-6

respect to the applied loads. Therefore, the total deformation of the tube is a combination

of bending and shearing. Depending on the amount of supplied material by pushing force,

the thickness distribution is affected. In the case of excessive feeding, the segments 2 and 3

may undergo wrinkling rather than thickening.

TopBottomθ

Pushed side

Lateral side

1

23

Route T

Route L

Route B

ab

cd

X

Y

Z

Figure 6.5 Schematic representations of the deformed tube and its equivalent section

As mentioned before, the constraint in the critical zone is weak. Therefore, redistribution

of the excessive material in the tube wall is the only way to prevent wrinkling. The effect

VI-7

of the initial thickness on prevention of wrinkling appears in the following ways:

(1) Excessive material is redistributed in the critical zone.

(2) Redistribution of excessive material in other zones. In other words, the field of

deformation velocity changes.

In the following section, we will discuss these two possible ways.

6.5 Forming velocity

Figure 6.6 shows the contours of normalized forming velocity in the deformation zone

when t0=1.5mm. The normalized velocity is calculated as yV

V in which |V| is the velocity

of material and Vy is the velocity of the moving die.

I

O1.151.101.051.00.950.90.850.80.750.70.650.6

Pushed side

Formingdirection

Deformation zone

Figure 6.6 Contours of the forming velocity

VI-8

In the pure shearing deformation, linear velocity is uniform and no thickness change

occurs. During actual shear bending, as the tube material is under a combination of

bending and shearing deformations, the forming velocity varies throughout the

deformation zone.

Figure 6.7 presents the effect of the tube thickness on the distribution of the forming

velocity in the deformation zone along I-O in Figure 6.6. One can see that this effect is

insignificant and the suppression of wrinkling does not arise due to change of the forming

velocity. In other words, prevention of wrinkling depends on redistribution of the excessive

material in the wall of the tube critical zone.

0.4

0.6

0.8

1

1.2

A1050,D0=30mm,rc=5mm,P=40MPa,µ=0

Nor

mal

ized

form

ing

velo

city

I OMeasuring position

t0=2mmt0=3mmt0=4mm

Figure 6.7 Effect of the initial thickness on the distribution of forming velocity in the

deformation zone along I-O (Figure 6.6)

6.6 Forming energy

To estimate the potential of the tube wall to absorb the surplus material, the necessary

VI-9

energy for thickening is evaluated. For this purpose, a strip of metal in the critical zone,

which is under a uniaxial compression, is considered. Figure 6.8 presents the states of

deformation.

δ lw t

2s

s

(a) thickening (b) wrinkle

R

l0w0

t0

Figure 6.8 Schematic illustrations of a strip metal under uniaxial compression;

(a) thickening deformation (b) wrinkling

The width of strip (w) is assumed to be constant during deformation. Under compression,

the strip with original length of l0 is initially thickened and shortened to l (Figure 6.8(a)).

Under these conditions, the principal strains becomes

εI = -εII = -δ /l .

The deformation energy is calculated as

dVdUV

εσε

∫ ∫=0

.

VI-10

Assuming an elastic-linear hardening behavior for the material ( εσ KY += ), the total

forming energy for thickening

tl

KYwU )2

(3

2 2

1δδ += . (6-1)

If the feed material (δ) exceeds a critical value, the strip will wrinkle. The wrinkle shape

is schematically shown in Figure 6.8(b). In order to estimate the wrinkling energy, it is

easily assumed that the strip is wrinkled through a bending deformation. Hence, the

bending energy can be simply calculated from Figure 6.8(b) as

322 )

242(

32 lt

RKYwU +

= . (6-2)

From the geometry of deformation

)4

sin(4 0

0

Rl

lR δ−= . (6-3)

The relations (6-1) and (6-2) indicate that the thickening and wrinkling energies depend

on the initial thickness.

For example, replacing Y=30MPa, K=150MPa, l0=20mm, w=1mm and δ=3mm in the

equations above, the calculated values of U1 and U2 are plotted in Figure 6.9. This graph

shows that the thicker the strip, the higher the energy for wrinkling is.

Using these simple assumptions, it was verified that aiming to prevent the tube from

wrinkling, the tube wall must be thick enough.

A series of experiments using tubes with different thickness were carried out. As the

result, forming limit of tubes is presented in the next section.

VI-11

1 2 3 4–1

0

1

2

Initial thickness, t0 / mm

Ener

gy,

U /

kJ

U1,Thickening U2,Bending

Buckling Thickening tendency tendency

U1–U2

Figure 6.9 Comparison between the energy values necessary for bending or thickening

of a strip metal under uniaxial compression

6.7 Forming limit

A series of experiments were carried out in which tubes with different wall thickness

were utilized and various pushing pressures were applied on the tubes. Figure 6.10 shows

the results of experiments. The shear bending process is strongly affected by the value of

the pushing pressure exerted on the tube. When the pushing pressure is not enough, rupture

(x) occurs. Also if the pushing force exceeds the critical amount, wrinkle (▲) outbreaks.

For the case of t0=1mm, no successful product could be obtained.

For t0=1.5mm, a narrow range of suitable pushing pressures exists. It is seen that

increasing the initial thickness, the forming limit of the process is extended and improved

because the buckling resistance of the tube against the applied pushing pressures is

VI-12

enhanced. In the case of t0=3mm, even when increasing the pushing pressure up to

120MPa no failure occurred.

1 1.5 2 2.5 30

40

80

120

160

Tube wall thickness, t0 /mm

Push

ing

pres

sure

, P

/MPa

A1050, D0=30mm, rc=5.0mm, c=0.1mmRupture Success Wrinkle

Figure 6.10 Results of experiments using tubes with different wall thickness and

applying various pushing pressures

6.8 Forming accuracy

6.8.1 Cross section deformation

In conventional bending, the ovality of cross section decreases with the increase of the

initial thickness. However, there is few research works on it regarding the shear bending.

Therefore, in this section the effects of the initial thickness on the distribution of thickness

strain and the cross section deformation of the deformed tube are clarified. For this purpose,

the same value of pushing pressure (P=40MPa) is applied on tubes with different initial

thickness.

VI-13

Figure 6.11 shows the experimental and simulated results of cross section configurations

of the deformed tubes with different initial thickness when pushing pressure is P=40MPa.

The thicker the initial tube, the greater its cross section deformation is, particularly

around the bending inside. It is completely against the tendency of conventional bending.

Increasing the initial thickness enhances the bending rigidity of the tube and the bending

becomes tighter accordingly. Therefore, the tube material cannot follow the die profile well

and the tube section sustains more deformation. As shown in Figure 6.11(c), a gap is

formed between the tube and the die wall due to diametral shrinkage. On the other hand,

increasing in the applied pushing pressure can prevent the cross section from high ovality.

t0=2.0mm

Mov

ing

die

Fixe

d di

et0=2.5mm

t0=3.0mm

(a) Simulation (b) Experiment (c)Experiment (t0=3.0mm)

Form

ing

dire

ctio

n

Figure 6.11 Cross section configurations of the deformed tubes with different initial

thickness obtained by simulation and experiment

6.8.2 Distribution of thickness strain

Figure 6.12 shows the effect of the initial thickness on the thickness strain (εt=ln(t/t0))

of a typical section of tube’s sheared part obtained by simulation and experiment. Both

results indicate that thickness strain decreases with the increase of the initial thickness.

VI-14

1.5 2 2.5 3

–0.4

–0.2

0

0.2

0.4

A1050,D0=30mm,P=40MPa,rc=5mm

Tube initial thickness,t0 / mm

Thic

knes

s stra

in, ε

t=ln

(t/t 0)

Inside,Sim.Lateral side,Sim.Outside,Sim.

Inside,Exp.Lateral side,Exp.Outside, Exp.

Figure 6.12 Effect of initial thickness on the thickness strain of tube sheared part

obtained by simulation and experiment

To explain the reason, the relation between the pushing and shearing strokes as a

function of the tube thickness obtained by simulation is plotted in Figure 6.13. Moreover,

Figure 6.14 shows the effect of the initial thickness on shearing load. The shearing load is

calculated as Fs/A0 in which A0 is the cross section area of the original tube. The slope of

curves in Figure 6.13 decreases by increasing the initial thickness. It means that lesser

amount of material is supplied into dies by increasing the initial thickness. Therefore,

under the effect of the same pushing pressure, the tube suffers more elongation and as

shown in Figure 6.14, the shearing load increases by increasing the tube’s initial thickness.

Thus, the overall tube undergoes thinning deformation as shown in Figure 6.12. In other

words, comparing to a thin tube, a thick wall tube has more tendency to thinning and

rupture.

VI-15

0 20 40 600

20

40

60

80A1050, D0=30mm, rc=5.0mm, P=40MPa

Push

ing

stro

ke,

S P /m

m

Shearing stroke,SS /mm

t0=2.0mmt0=2.5mmt0=3.0mm

Figure 6.13 Relation between the pushing and shearing strokes as a function of the

tube’s initial thickness obtained by simulation

0 20 40 60

40

80

A1050, D0=30mm,rc=5.0mm, P=40MPa

Shea

ring

load

, F S/

A 0 /M

Pa

Shearing stroke,SS /mm

t0=2.0mmt0=2.5mmt0=3.0mm

Figure 6.14 Effect of the tube’s initial thickness on the shearing load obtained by

simulation

VI-16

The possibility of prevention from thinning by applying a higher pushing force on the

tube is examined here. The pushing pressure of P=45MPa was applied on a tube with

t0=3mm. Figure 6.15 shows the thickness strain distribution across the hoop direction of

the tube’s sheared part. Moreover, a comparison has been shown between these results and

the results for P=40MPa and t0=2mm. These results confirm that applying a higher pushing

pressure on thick tubes can prevent the wall from thinning. As the result, in order to

prevent the tube from thinning, the applied pushing pressure must be as high as possible

within a range in which wrinkle does not occur.

0 45 90 135 180

–0.2

0

0.2

A1050, D0=30mm, rc=5.0mm, µ=0

Thic

knes

s stra

in, ε

t=ln

(t/t 0) t0=3.0mm,P=40MPa

Angular position,θ/ deg

t0=3.0mm,P=45MPat0=2.0mm,P=40MPa

Figure 6.15 Effect of the initial thickness and the pushing pressure on the distribution of

thickness strain along the hoop direction of the tube’s sheared part obtained by simulation

To explain the role of the pushing pressure, the distribution of effective plastic strain

along the hoop direction of the tube’s sheared part is plotted in Figure 6.16. Increasing the

tube initial thickness, the level of the tube deformation and consequently the plastic strain

increases. Therefore, the tube material becomes more work-hardened and the flow stress of

VI-17

material increases. As the result, comparing to a thin tube, higher value of pushing pressure

should be applied on a thick tube to perform a successful plastic deformation and obtain

the same thickness distribution.

0 45 90 135 1800

0.5

1

A1050, D0=30mm, rc=5.0mm, P=40MPaEf

fect

ive

plas

tic st

rain

, ε p

t0=2mm

Angular position, θ/ deg

t0=4mmt0=3mm

Figure 6.16 Effect of the initial thickness on the distribution of effective plastic strain of

a tube’s sheared part obtained by simulation.

6.9 Conclusions

The effects of the initial thickness on the shear bending process of circular tubes were

investigated employing different examination methods. According to the analysis and

experiments, it can be concluded that:

1. Thin tubes show a high wrinkle tendency during the shear bending.

2. Increasing the initial thickness, the energy necessary for wrinkling rises and the tube

resistance against high pushing pressures heightens. Thus, the forming limit of the

VI-18

tube expands.

3. As increasing the tube thickness enhances its bending rigidity, the tube cannot

follow the die profile and the cross section ovality increases accordingly.

4. From the point of prevention from thinning and section ovality to promote the

dimensional accuracy, applying higher pushing pressure on the tube is preferred

within a range in which wrinkle does not occur.

VI-19

Reference

[1] N. Utsumi and S. Sakaki, Countermeasures against undesirable phenomena in the

draw-bending process for extruded square tubes, Journal of Materials Processing

Technology, Volume 123, Issue 2 (2002), pp.264-269.

VII-1

CHAPTER 7

EFFECT OF MATERIAL PROPERTIES ON THE SHEAR BENDING PROCESS

7.1 Introduction

The behavior of the metal tube as it is bent depends on the mechanical properties of the

metal and the characteristics of the bend. If the metal is very brittle or the bend is

excessively sharp, the metal is likely to break. If the metal is too hard or a section too

heavy, bending forces required on a particular machine may be excessive.

In considering any material for its cold bending suitability, a general rule is to use the

following equation as a guide to determining the elongation necessary in the metal to make

a given bend

e=0.5D/R

Where e is necessary elongation, R is bending radius and D is outside diameter.

Simple forming operations also can be performed on magnesium, titanium and certain

copper/nickel alloys.

Generally, most common metals can undergo cold bending, providing they have

sufficient elongation to achieve the desired angle and radius before reaching their ultimate

strength. Metals commonly formed without difficulty include low carbon and stainless

steel, aluminum, brass and copper.

Since stainless usually has greeter elongation than mild steel, it is generally capable of

being formed to greater angles and on smaller radii than comparable carbon-steel material.

VII-2

Copper tubes extruded or drawn are bent by many fabricators. Pieces in the range

between fully annealed and half-hard are commonly used for small radius bending. Binary

alloys of copper and zinc are known as brasses. Brass is widely used in bending, especially

to manufacture plumbing waste traps and elbows. Annealed material is best for bending

light-wall brass tubing to centerline radii that are one-to-two times the diameter [1].

Aluminum is another commonly formed metal. Unalloyed aluminum has many desirable

characteristics, including its lightweight, pleasing appearance, malleability, formability and

resistance to corrosive attack by industrial and marine atmospheres, many chemicals and

food products, but has relatively low strength and hardness levels.

Characteristics of aluminum space frames such as low specific weight, high specific

strength, good corrosion resistance, good recyclability and excellent formability [2,3].

In this section, the effects of material properties on the characteristics of the shear

bending are investigated both by the experiments and the numerical simulation. In the

experiments, circular aluminum and copper tubes were utilized. The forming limit and the

forming loads were clarified.

7.2 Experiments

Aiming to study the effects of the material properties, the shear bending process using

circular aluminum and copper tubes was carried out. Table 7.1 indicated the material

properties obtained by simple tension tests.

The stress-strain curves of materials are shown in Figure 7.1. Moreover, the dimensions

of the blank tubes are detailed in Table 7.2.

The die corner radius rc of 5mm was selected.

VII-3

Table 7.1 Material properties

Yield Stress, Y/ MPa Tensile stress, TS/ MPa Elongation / (%)

A1050 (1) 30 103 34

A6063-O 38 115 32

C1100-O 52 350 52

Table 7.2 Dimensions of the blank tube

Diameter, D0/ mm 30

Thickness, t0/mm 2

Length, L0/ mm 270

0.1 0.2 0.3 0.4 0.50

100

200

300

400

True

stre

ss,

σ/ M

Pa

A6063–OC1100–O

A1050

True strain, ε

Figure 7.1 Stress-Strain curves of material used in the experiments

VII-4

7.2.1 Forming limit

A series of experiments were performed aiming to find the forming limit of the shear

bending using different materials. Various pushing pressures were applied on the aluminum

and copper tubes. Figure 7.2 presents the forming limit. The typical shear bending products

are show in Figure 7.3.

20 40 60 80 100 120 140

C1100-O

A6063-O

A1050

Rupture Success Wrinkling

Axial pushing pressure, P/MPa

Figure 7.2 Forming limit of the shear bending using tubes with different materials

C1100-O

A6063-O

A1050

Figure 7.3 Typical products of the shear bending using different tube materials

VII-5

According to Table 7.1, the order of material regarding the yield stress is

A1050 < A6063-O < C1100-O. Thus, the lower limit of the appropriate pushing pressures

Pl is in the order of

Pl (A1050) < Pl (A6063-O) < Pl (1100-O).

Furthermore, the relation between the yield stress and the working load for different

materials subjected to the shear bending is shown in Figure 7.4. The working loads include

the appropriate pushing pressure and the shearing force during a successful shear bending

process. As shown in the figure, the working loads increase with increase in the yield

stress.

20 40 60 80 1000

100

200

D0=30mm, t0=2mm, c=0.1mm

Wor

king

load

s

Pushing pressure, PShearing load, Fs /A0

Yield stress, Y

Figure 7.4 Relation between the yield stress and the working load for different materials

subjected to the shear bending

The order of materials regarding the tensile stress is A1050 < A6063-O < C1100-O.

Correspondingly, the upper limit of the appropriate pushing pressures Pu is in the order of

Pu (A1050) < Pu (A6063-O) < Pu (C1100-O).

The relation between the tensile strength and the working loads for different materials

VII-6

subjected to the shear bending is shown in Figure 7.5. Furthermore, the relation between

σ (ε =1) and the working loads for different materials is shown in Figure 7.6. σ (ε =1) is

the value of stress when ε=1 and obtained by extrapolating the stress-strain curves over the

tensile strength.

100 200 300 400 500 6000

100

200

D0=30mm, t0=2mm, c=0.1mm

Wor

king

load

s

Pushing pressure, PShearing load, Fs /A0

Tensile strength, TS

Figure 7.5 Relation between tensile strength and working load for different materials

subjected to shear bending

7.2.2 Thickness strain

Figures 7.7 (a)~(d) show the distributions of thickness strain of the deformed tubes with

different material properties. The values of the axial pushing pressure applied on A1050,

A6063 and C1100 tubes were P=50MPa, P=60MPa and P=120MPa respectively.

From these figures it is observed that the distribution of thickness strain along the

Routes L and B are almost the same. However, along the Route T, where the tube

undergoes high thickness reduction, the amount of thickness reduction for the copper tube

is minimum.

VII-7

100 200 300 400 500 6000

100

200

D0=30mm, t0=2mm, c=0.1mm

Wor

king

load

s

Pushing pressure, PShearing load, Fs /A0

Extrapolated stress at ε =1

Figure 7.6 Relation between σ(ε=1) and the working loads for different materials

subjected to the shear bending

ab

c

Rou

te T

Rou

te B

sheared part

Pushedside

90

Rou

te L

Figure 7.7 (a) Thickness measurement zones

VII-8

–0.4

–0.2

0

0.2 D0=30mm, t0=2mm, c=0.1mm

Thic

knes

s stra

in,

ε t A6063–OC1100–O

A1050

b caMeasuring position

Figures 7.7(b) Distribution of thickness strain using tubes with different material

properties along theRoute T

–0.2

0

0.2

0.4

D0=30mm, t0=2mm, c=0.1mm

Thic

knes

s stra

in,

ε t A6063–OC1100–O

A1050

b caMeasuring position

Figures 7.7(c) Distribution of thickness strain using tubes with different material

properties along the Route L.

VII-9

–0.2

0

0.2

0.4

D0=30mm, t0=2mm, c=0.1mm

Thic

knes

s stra

in,

ε t A6063–OC1100–O

A1050

b caMeasuring position

Figures 7.7(d) Distribution of thickness strain using tubes with different material

properties along the Route B

7.3 Effect of the work hardening exponent

The behavior of tube bending depends on the properties of the material and the nature of

the bend. The strain-hardening exponent is an indicator of the formability of the material. A

higher value of this exponent is desirable in conventional bending since it results in a better

distribution of strain and delays onset of strain localization [4].

Among material properties, in this section, the role of the strain-hardening exponent on

the deformation characteristics of the tube during the shear bending process is investigated

using finite element simulation.

7.3.1 Simulation Parameters

The aluminum tubes with different hardening exponents are considered. Figure 7.8

VII-10

shows the stress-strain curves of the aluminum tubes used in the simulation.

The axial pushing pressure of 40MPa was applied on the blank tubes. Table 7.3 indicates

the simulation parameters.

0 0.5 10

50

100

150

200

250Ef

fect

ive

stre

ss,

σ/ M

Pa

Effective plastic strain,εp

n=0.01n=0.15n=0.30

Figure 7.8 Stress-Strain curves of the aluminum tubes used in the simulation

Table 7.3 Simulation parameters

Work hardening exponent, n 0.01, 0.15, 0.30

Strength coefficient, K/ MPa 200

Young’s Modulus, E/ GPa 70

Poisson’s ratio, v 0.34

Tube’s initial diameter, D0 /mm 30.0

Tube’s initial thickness, t0 /mm 2.0

Die corner radius, rc /mm 5.0

Friction coefficient, µ 0.0

VII-11

The contours of the current flow stress of the tube are shown in Figure 7.9. As shown,

for n=0.01, the flow stress overall the sheared part of the tube is almost constant.

n = 0.01 n = 0.30

20019017015013011095806040

Figure 7.9 Effect of the hardening exponent on the distribution of current flow stress of

material after the deformation

0 90 1800

0.5

1

Effe

ctiv

e St

rain

, ε

Angular position,θ/ deg

n=0.01 n=0.3

1.15

Inside Lateral side Outside

Figure 7.10 Effect of the hardening exponent on the distribution of effective strain across

peripheral direction of the deformed tube

VII-12

Figure 7.10 shows the distribution of effective strain in the peripheral direction of the

tube for n=0.01 and n=0.30. Higher n-value results in lower strain around the lateral side of

the deformed tube. The value of effective strain in this region is closer to the theoretical

value ( ε =1.15 which was obtained in chapter 3). However, the value of effective strain

near the bending inside and bending outside for higher n-value material is greater.

To make clear the effect of the hardening exponent on the deformation behavior, the

distributions of effective plastic strain for n=0.01 and n=0.3 are shown in Figure 7.11.

The effective plastic strain pε is a quantity, which takes into account the history of

deformation including the effects of bending and unbending. It can be seen that the higher

the n-value, the larger the difference between the effective strain ε and the effective

plastic strain pε is. In other words, for higher work-hardening exponent the tube

material sustains more bending and unbending deformations, which heightens the level of

effective plastic strain. However, the amount of shearing in the total deformation decreases.

Therefore, increasing the n-value, the effective plastic strain in the vicinity of the bending

inside and outside increase whereas the level of shear strain and consequently the effective

strain around the lateral side of the tube becomes smaller than the corresponding values for

the pure shearing deformation ( ε =1.15).

The contours of the effective plastic strain after the deformation for n=0.01 and n=0.3

are shown in Figure 7.12. It is seen that for n=0.01 the deformation zone is a narrow region

near (A) similar to the shearing plane in the pure shearing process.

For n=0.3, the deformation zone (B) is larger and the deformation occurs on a spread

zone rather than a narrow plane. This result confirms that the smaller the n-value, the

closer the deformation mode to a shearing process is. For the material with small n-value,

localization of strain, especially at the early stages of the deformation (C) can be observed.

VII-13

0 90 1800

0.5

1

Stra

in

Angular position,θ/ deg

εεp

1.15

n=0.01

0 90 1800

0.5

1

Stra

in

Angular position,θ/ deg

ε εp

1.15

n=0.3

Figure 7.11 Effect of the hardening exponent on the distribution of effective strain and

effective plastic strain across the peripheral direction of the deformed tube

VII-14

A B

1.21.080.960.840.720.60.480.360.240.120.0

n=0.01 n=0.30

C

n=0.01 n=0.30

Figure 7.12 Effect of the hardening exponent on the distribution of effective plastic strain

This also can be seen in Figure 7.13, which is a plot of the effective strain history during

the process.. As shown in the figure, for small n-value, the strain value before the

deformation zone (around point a) is almost equal zero. Advancing the process, the tube

material is strained abruptly and reaches a high value close to the theoretical one. For

higher n-value, the material is strained smoothly and the value of strain after leaving the

deformation zone in the steady state region is less than the value for non-hardening

material.

VII-15

0

0.5

1

1.5

Effe

ctiv

e pl

astic

stra

in,

ε p

Position /mm

n=0.01 n=0.30

a b

Figure 7.13 Effect of the hardening exponent on the history of effective plastic strain

Another evidences, which prove that the bending deformation is magnified increasing

the hardening exponent, are shown in Figures 7.14 and 7.15. Figure 7.14 presents the

distributions of thickness strain across the peripheral direction of the tube’s sheared part.

Thickness strain is calculated as εt=ln(t/t0) where t0 is the initial thickness. The figure

shows that the higher the work hardening, the greater the variation of thickness strain in the

b

a

VII-16

peripheral direction is. In other words, the tube material sustains more bending

deformation. The thickness change for material with lower work hardening is less. It is due

to higher shear deformation rather than bending deformation. In other words, lower

n-values prevent the deformed tube from large thickness changes.

0 90 180

–0.2

0

0.2

0.4

Thic

knes

s stra

in,

ε t

n=0.01

Angular position,θ/ deg

n=0.15n=0.30

Figure 7.14 Effect of the hardening exponent on the distribution of thickness strain

across the peripheral direction of the deformed tube

Figure 7.15 shows the configurations of the deformed tubes with different n-values

respect to the fixed die obtained by simulation. This figure indicates that the deformation

accuracy regarding the section deformation is higher for larger n-value material. As it was

mentioned, the smaller the n-value, the smaller the bending deformation is. Consequently,

the tube material cannot follow the die profile well and undergoes diametral shrinkage,

which results in the ovality of cross section. In other words, the cross section deformation

can be reduced using materials with higher n-values.

VII-17

n=0.3n=0.01

Fixed die

Fixed die

Figure 7.15 Configurations of the deformed tubes with different n-values

7.4 Conclusions

The effects of the material properties on the shear bending process were investigated in

this chapter. The experiments were performed using copper and two kinds of aluminum

tubes. The forming limits of the tubes with different materials were obtained. Moreover,

the FEM was employed to find out the role of the work hardening exponent on the forming

process. As conclusion:

1. Experimental results show that by providing a material with sufficient elongation

implementation of a successful shear bending process is feasible.

2. The smaller the hardening exponent, the larger the shearing deformation is.

Consequently, more uniform thickness distribution can be obtained. However, the

cross section deformation of the tube increases.

VII-18

References

[1] K. Lange, Handbook of metal forming, McGraw-Hill (1985), USA

[2] H. C. Kwon, Y. T. Im, D. C. Ji and M. H. Rhee, The bending of an aluminum structural

frame with a rubber pad, Journal of Materials Processing Technology, Volume 113, Issues

1-3 (2001), pp. 786-791

[3] A. H. Clausen, O. S. Hopperstad and M. Langseth, Stretch bending of aluminium

extrusions for car bumpers, Journal of Materials Processing Technology, Volume 102,

Issues 1-3 (2000), pp. 241-248

[4] G. T. Kridli, L. Bao, P. K. Mallick and Y. Tian, Journal of Materials Processing

Technology, Volume 133, Issue 3 (2003), pp. 287-296

VIII-1

CHAPTER 8

THE SHEAR BENDING OF A CIRCULAR TUBE SUBJECTED TO AN ECCENTRIC AXIAL PUSHING

FORCE

8.1 Introduction

As mentioned in Chapter 1, during bending deformation, the walls along the outside

radius of the bend tend to thin, while the walls along the inside radius thicken. An extra

axial compressive stress will make the neutral layer have an excursion in the outside

direction, which is beneficial for decreasing the wall thickness reduction [1]. However,

applying an extra pushing force increases the wrinkling tendency of the inside bending of

the tube during bending deformation. A suitable value of extra pushing force is highly

required in which prevent the tube from extreme thinning without any increase in the

wrinkling potential of it. One way to realize this concept is the idea of applying an

eccentric pushing force on the tube. It means the axial pushing force is applied on a part of

the tube end rather than on the whole section.

Employing Finite Element Simulation, the effects of an eccentric axial pushing force on

the tube shear bending process of circular tubes are investigated in this chapter.

8.2 Simulation parameters

The FEM is used as an investigation method to analyze the influence of an eccentric

VIII-2

axial pushing force on the deformation behavior of the tube during the shear bending.

Circular aluminum A1050 (1) tubes with t0=1.5mm and D0=30.0mm are considered.

It is assumed that the eccentric value is 50%. It means the axial pushing force is applied

on half of the tube section area (Figure 8.1).

The tubes with initial length of 100, 150 and 300mm were considered.

P

L0

Figure 8.1 Applying an eccentric axial pushing force on the tube

8.3 Stress distribution

The distributions of effective stress for tubes with different initial length are shown in

Figure 8.2. The case (a) shows the stress distribution applying a “centric” pushing pressure

of P=40MPa when L0=300mm. The same value of pushing force, which equals to

P=80MPa, is applied on the tube with L0=300mm eccentrically. Case (b) shows the result.

Comparing to the case (a), it can be seen that the contours are different in the vicinity of

the tube’s pushed side. However, the distributions of stress in other regions, especially in

the deformation zone, are almost the same.

The same value of pushing force (P=80MPa) is applied on a tube with L0=150mm

eccentrically. The case (c) presents the result. Comparing to case (b), more changes in the

stress counters can be observed. However, in the deformation zone and the tube’s sheared

VIII-3

part these changes are not considerable. In fact, subjected to an end loading, the stress

distribution in regions far from the loaded zone is independent of the end effects (the

Saint-Venant’s principle). Therefore, it is found that applying an eccentric stress may have

an insignificant effect on the shear bending of long tubes.

1601451301151008570554025100

(a)

(b)

(c)

Figure 8.2 Distributions of effective stress (in MPa) for a tube subjected to (a) centric

P=40MPa, L0=300mm; (b) eccentric P=80MPa, L0=300mm; (c) eccentric P=80MPa,

L0=150mm

VIII-4

θ

Pushed side

The analysis of the thickness change is the other way to investigate the effects of

applying an eccentric load on the tube.

8.4 Thickness distribution

In order to analyze the effect of an eccentric axial pressure on the forming process, the

thickness distributions of tubes with different initial length are investigated in this section.

Figure 8.3 shows a comparison between the thickness strain distribution (εt=ln(t/t0)) of a

tube under centric pushing force and tubes with different initial length loaded eccentrically.

The thickness is measured in the peripheral direction of the tube’s sheared part.

0 90 180–0.2

0

0.2

A1050, D0=30mm, t0=1.5mm, µ=0

Thic

knes

s stra

in, ε t P=80MPa, eccentric, L0=150mm

P=40MPa, centric, L0=300mm

P=80MPa, eccentric, L0=300mm

Angular position,θ/ deg

P=80MPa, eccentric, L0=100mm

Figure 8.2 Distributions of thickness strain across the peripheral direction of the sheared

part of the deformed tubes

VIII-5

Only a small effect on the thickness strain distribution of the tube with L0=100mm can

be observed. This means that applying an eccentric load may affect the thickness of tubes,

which are short enough.

8.5 Conclusions

Conducting the Finite Element Simulation as an investigation method, the effects of

applying an eccentric load on the tube during the shear bending process were studied. The

results show that loading a tube eccentrically may affect the thickness distribution if the

tube is short enough. In this case, a suitable value of extra pushing force may prevent the

tube from extreme thinning.

VIII-6

References

[1] Y. Zeng and Z. Li, Experimental research on the tube push-bending process,

Journal of Materials Processing Technology, Volume 122, Issues 2-3 (2002), pp.237-240.

IX-1

CHAPTER 9

SUMMARY

The tube shear bending is a beneficial technique for the production of unified and

compact bent tubular parts through cold metal forming technology. It is an appropriate

technology to realize considerable small bending radii.

In this research, the shear bending of circular tubes applying an axial pushing force on

the tube was studied theoretically and experimentally.

In chapter 1, an introduction to the metal forming, especially bending deformation, was

presented. Moreover, the main characteristics of the conventional tube bending methods

were given and summarized.

In chapter 2, the tube shear bending method was introduced and the examination

methods including the experimental procedure and the simulation model were introduced.

In chapter 3, the main characteristics of the shear bending process were clarified. The

differences between the pure bending, pure shearing and the actual shear bending were

highlighted

The effects of the axial pushing force, the die corner radius, the initial thickness, the

material properties and the eccentric axial pushing force, as the main forming parameters,

on the forming limit and the dimensional accuracy of the shear bending process were

clarified within chapters 4, 5, 6, 7& 8.

Table 9.1 summarizes the results. Moreover, a simple comparison between the shear

bending and conventional bending methods is presented in this table.

In chapter 4, the effects of the axial pushing pressure were investigated. It was found

that a limited range of appropriate pressures to perform a successful forming process exists.

IX-2

Applying a pushing force beyond this appropriate range results in rupture (low pushing

force) or wrinkle (high pushing force). Furthermore, employing a higher pushing pressure

within the appropriate range of pressures results in decrease of thickness reduction and

cross section deformation of the tube.

In chapter 5, the effects of the die corner radius on the process were illustrated. It was

understood that the appropriate pushing force is almost constant regardless the value of the

die corner radius. However, there is a limit range of the die radii suitable for performing

the process. Forming on dies with radii larger than a critical value results in either rupture

or wrinkle in the tube. Moreover, whilst a small bending radius results in high cross section

deformation, increasing the die corner radius, the wrinkling tendency of the tube increases.

If the die radius is allowed to be selected the larger radius within the suitable range of radii

should be employed because it would improve the precision.

In chapter 6, the influences of the initial thickness on the shear bending process were

clarified. The results demonstrate that increasing the initial thickness, the forming limit of

the tube expands as the tube resistance against high pushing pressures heightens. It was

proven that in order to prevent the tube from thinning and section ovality to promote the

dimensional accuracy, applying higher pushing pressure is preferred within the appropriate

range in which wrinkle does not occur.

In chapter 7, the effects of the tube material on the process were analyzed utilizing tubes

with different material properties. It was concluded that implementation of a successful

shear bending process is feasible by providing a material with sufficient elongation.

Moreover, simulation results show that the smaller the hardening exponent, the larger the

shearing deformation is.

In chapter 8, the effects of applying an eccentric axial pushing force on the tube were

investigated. Using finite element simulation, it was proven that exerting an eccentric load

might affect the tubes, which are short enough.

IX-3

Table 9.1 Effects of forming parameters on the shear bending and conventional bending

Increased

parameter

Effect on conventional bending Effect on shear bending

Initial

diameter (D0)

Minimum bending radius:

Rmin >1D0

Minimum bending radius:

Rmin <6mm

Pushing Force

(P)

Thinning (Rupture): decreases

Thickening (Wrinkle): increases

Thinning (Rupture): decreases

Thickening (Wrinkle): increases

Ovality: decreases

Die Corner

Radius

(rc)

Rupture: decreases

Wrinke: decreases

Ovality: decreases

Forming limit: expands

Rupture: low effect

Wrinkle: increases

Ovality: decreases

Forming limit: low effect

Initial

Thickness

(t0)

Wrinkle: decreases

Ovality: decreases

Forming limit: expands

Rupture: slightly decreases

Wrinkle: decreases

Ovality: decreases

Forming limit: expands

Eccentric

stress

Short tubes: prevention from

extreme thinning & thickening

Long tubes: low effect

Short tubes: prevention from

extreme thinning & thickening

Long tubes: low effect

IX-4

PUBLICATIONS

Based on this dissertation, the following publications have been derived:

- Journals

1. M. Goodarzi, T. Kuboki and M. Murata: Deformation Analysis for the Shear Bending

Process of Circular Tubes, Journal of Materials Processing Technology, 162-163

(2005), 492-497.

2. M. Goodarzi, T. Kuboki & M. Murata: Effect of Die Corner Radius on Formability

and Dimensional Accuracy of Tube Shear Bending, International Journal of Advanced

Manufacturing Technology, (In press).

3. M. Goodarzi, T. Kuboki & M. Murata: Effect of Tube Thickness on Shear Bending

Process of Circular Tubes, Journal of Materials Processing Technology, (In press).

4. M. Goodarzi, T. Kuboki and M. Murata: Formability of A1050 Aluminum Tubes with

Different Thickness in Shear Bending Process, (In progress).

- International l conferences:

1. M. Goodarzi, T. Kuboki and M. Murata: The Effect of Pushing Force on the Shear

Bending Process of Circular Tubes, Advanced Technology of Plasticity, (2005) Italy,

(In CD).

2. M. Goodarzi, T. Kuboki and M. Murata: Deformation Analysis for the Shear Bending

Process of Circular Tubes, Advanced in Materials Processing Technology,

(AMPT2005) Poland, (In CD).

3. M. Goodarzi, T. Kuboki & M. Murata: The Effect of Tube Thickness on the Shear

Bending of Circular Tubes, Advanced in Materials Processing Technology,

(AMPT2006) USA.

PUBLICATIONS

Based on this dissertation, the following publications have been derived:

- Journals

1. M. Goodarzi, T. Kuboki and M. Murata: Deformation Analysis for the Shear Bending

Process of Circular Tubes, Journal of Materials Processing Technology, 162-163

(2005), 492-497.

2. M. Goodarzi, T. Kuboki & M. Murata: Effect of Die Corner Radius on Formability

and Dimensional Accuracy of Tube Shear Bending, International Journal of Advanced

Manufacturing Technology, (In press).

3. M. Goodarzi, T. Kuboki & M. Murata: Effect of Tube Thickness on Shear Bending

Process of Circular Tubes, Journal of Materials Processing Technology, (In press).

- International conferences

1. M. Goodarzi, T. Kuboki and M. Murata: The Effect of Pushing Force on the Shear

Bending Process of Circular Tubes, Advanced Technology of Plasticity, (2005) Italy,

(In CD).

2. M. Goodarzi, T. Kuboki and M. Murata: Deformation Analysis for the Shear Bending

Process of Circular Tubes, Advanced in Materials Processing Technology,

(AMPT2005) Poland, (In CD).

3. M. Goodarzi, T. Kuboki & M. Murata: The Effect of Tube Thickness on the Shear

Bending of Circular Tubes, Advanced in Materials Processing Technology,

(AMPT2006) USA.