Materials

www.elsevier.com/locate/matdes

Materials and Design 26 (2005) 549–554

&Design

Short Communication

Finite element analysis of bending occurring whilecutting with high speed steel lathe cutting tools

Abdullah Duran *, Muammer Nalbant

Gazi University, Mechanical Engineering Department, Technical Education Faculty, Bes�evler, Ankara 06500, TeknikokuIIar, Turkey

Received 4 February 2004; accepted 26 July 2004

Available online 6 October 2004

Abstract

The bending which occurs on a cutting tool during machining on a lathe affects tool life, surface roughness and dimension cor-

rectness. In this research, the bending which has been calculated by Castigliano theorem has been compared with the bending

obtained by finite element method. Under the constant cutting conditions, material C1060 has been machined with high speed steel

(HSS) lathe cutting tools having 60�, 75� and 90� of cutting edge angle. It was determined by using ANSYS finite elements program

that the bending of the cutting tool generated by the forces, which varied between 1360 N and 1325 N and occurred during cutting,

varied between 0.039958 and 0.04373 mm. According to the results, it has been observed that the bending that was calculated by

Castigliano theorem and that varied between 0.03542 and 0.034505 mm was almost the same with the bending determined by finite

elements method. In other words, it was seen that the calculated values approach to the analysed results up to 0.4% .

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Finite elements method; Bending; Cutting forces

1. Introduction

Various machine tools and cutting tools are used in

metal cutting. Cutting tools for turning can generally

be divided into two groups: high speed steels and cemen-

ted carbides. High speed steel (HSS) cutting tools can be

subdivided into three groups according to their manu-facture: single, brazed and indexable inserts [1,2].

Cutting tool geometry, chip geometry, cutting speed,

cutting tool and workpiece material, cutting speed and

cutting fluid are the main factors affecting the metal cut-

ting process [3,4].

Tool geometry, one of the most important factor

affecting metal cutting process, is determined by rake an-

gle, side clearance angle, side cutting edge angle andback rake angle. Tool geometry is an important factor

0261-3069/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.matdes.2004.07.028

* Corresponding author.

E-mail address: aduran@gazi.edu.tr (A. Duran).

having influence on cutting forces and tool life. For an

optimum turning operation, correct selection of cutting

parameters as well as the length of tool holder extending

from its post are essential [5]. That is because, incorrect

selection of cutting parameters leads to rapid tool wear,

breakage and plastic deformation. This increases ma-

chine tool idle time due to the changing of damaged cut-ting tools and causes some other problems such as poor

surface quality and wrong workpiece dimensions. This,

in turn, increases the overall cost [6].

Determination of cutting parameters through con-

ventional methods is mostly not possible. Therefore,

the use of high capacity computers is also gaining

importance as it is possible to find appropriate solutions

for metal cutting operation. Approximate solutionmethods giving very close results to those obtained by

the experimental work are appealing as they are easy

to use and there is no necessity to carry out costly exper-

imental work [7]. Parallel to the development in the

computer technology, finite element analysis (FEM)

550 A. Duran, M. Nalbant / Materials and Design 26 (2005) 549–554

method, one of the approximate solution methods, has

been increasingly used. Finite element method gives very

close results to the real values and, therefore, it is now a

well accepted numerical method [8–11].

In this study, deflection of HSS cutting tool during

turning was investigated using ANSYS version 5.4 pro-

gram based on the tool length extending through the

tool post. The deflection of the tool was also calculatedusing Castigliano theory and the results were compared

to those obtained using ANSYS program.

Fig. 2. Cutting forces acting on the cutting tool during turing.

2. Basics of the metal cutting

2.1. Cutting process

Metal cutting is a complex physical phenomenon

which involves elastic and plastic deformation, intense

friction and heat, chip formation and cutting tool wear.

In order to effect cutting action, the cutting tool (as

shown in Fig. 1(a)) is forced against the workpiece and

is moved relative to the workpiece. As the result, the

workpiece is initially deformed elastically (Fig. 1(b))

and when the workpiece�s yield strength is exceededplastic deformation occurs. Plastic deformation leads

to chip formation (Fig. 1(c)). The formed chip leaves

the workpiece through the cutting tool rake face. The

flow of the chip occurs in various ways depending on

the workpiece properties and cutting conditions [5].

2.2. Cutting forces

The necessary chip formation force required to over-

come the developed stresses during chip formation proc-

ess can be divided into three components: cutting force

(Fz), feed force (Fs) and radial force (Fp). Cutting forces

developed during turning are depicted in Fig. 2 [13].

Fig. 1. Chip formatio

Cutting force (Fs) acts against the workpiece turningmotion and forces the cutting tool downwards perpen-

dicular to the workpiece axis. Feed force (Fv) acts paral-

lel to the workpiece turning axis and is in the reverse

direction of the feed. Radial force (Fp) acts perpendicu-

larly to the machined surface and forces the cutting tool

backwards.

If a cutting tool fixed on the tool post is considered a

cantilever beam as shown in Fig. 3, the deflection of thetool (dB) due to Fs force should be ideally zero during

cutting. However, this is not possible in real cutting con-

ditions and a deflection value of <0.1 mm is permitted

for rough machining operations [14].

2.3. Castigliano theory

According to Castigliano theory, partial derivation ofthe total internal stress energy with respect to the one of

the applied external forces gives the displacement of the

point of applied force in its application direction [15].

The displacement of the equal cross-sectioned beam at

point B (Fig. 4(a)) under Fs force can be found using

Castigliano theory.

n process [3,12].

Fig. 3. Representation of the cutting tool as a cantilever beam.

A. Duran, M. Nalbant / Materials and Design 26 (2005) 549–554 551

The bending moment at any point of the beam in Fig.4(b) can be found using the following equation:

M ¼ xF s: ð1ÞIf theMmoment changes along the beam, internal stress

energy developed as the result of basic bending moment

in the beam then dx length element strain energy isfound using

du ¼ M2 dx2EI

: ð2Þ

The total strain energy is calculated in the following

way:

u ¼Z L

X¼0

M2

2EIdx: ð3Þ

The bending in the beam (dB) can be found by rear-

ranging Eq. (2) in the following form:

dB ¼ oy toF s

¼Z L

x¼0

2ðxF sÞx2EI

dx ¼ F s

EIx3

3

����L

0

¼ F sL3EI

: ð4Þ

Fig. 5. Octagonal ring dynamometer capable of measuring three

cutting force components [12].

2.4. Finite elements

In finite elements method, it is assumed that a part is

divided into so many small parts. This method is, there-

Fig. 4. Single side ca

fore, called finite element method [10]. The main idea in

finite element method is to solve a complex problem by

replacing it with a simple problem. By using the existing

mathematical methods, in practice real or approximate

solutions cannot be found for many problems. However,

finite element method can be used to find approximate

solutions for these problems. In finite element method,

the solution zone is composed of adjacent sub partswhich are called finite element. It can be assumed that

these sub parts are held together by nuts and screws.

It is also assumed that when the bonding is removed,

the sub parts are separated.

3. Materials and methods

3.1. The materials used for the tests and the cutting forces

A HSS cutting tool of 12 · 12 · 100 mm in dimen-

sions was used as shown in Fig. 5. This tool was fixed

on the tool post of a dynamometer capable of measuring

cutting forces between 5 and 300 N. The tool was

ground to give the following angles:

Rake angle ðcnÞ ¼ 12�

Side relief angle ðaÞ ¼ 6�

ntilever beam.

Fig. 6. Application of the forces to the cutting edge angle tool.

552 A. Duran, M. Nalbant / Materials and Design 26 (2005) 549–554

End relief angle ðaiÞ ¼ 6�

Included angle ðeÞ ¼ 87�

In this study, cutting tests were performed using three

different side cutting edge angles: 60�, 75� and 90�.C 1060 steel was selected as the workpiece material

with a tensile strength of 700 N/mm2. The length of

the cutting tool extending from the tool post was 30 mm.

The analysis of deflection occuring during turning

using a HSS cutting tool (B 12 · 100 TS 95/1 HSS-Co)12 · 12 · 100 mm in dimensions was analysed using fi-

nite element method. The cutting force values required

for the analysis were recorded during turning of C

1030 steel at a cutting speed of 30 m/min, depth of cut

of 2 mm and feed rate of 0.2 mm/rev by using three dif-

ferent side cutting edge angles (60�, 75� and 90�). Theturning tests were carried out without coolant and the

cutting forces were measured using the dynamometershown in Fig. 5.

3.2. Division of the solution zone into the finite elements

and the stress analysis

Modulus of elasticity = 2 · 105 N/mm2;

Poisson ratio (l) = 0.3;

Tensile strength = 850 N/mm2;Density = 78.33 N/dm3.

The cutting tool was modelled using Pro/ENGI-

NEER version 16.0 as a solid model and the deflection

of cutting tool under the cutting forces was analysed

using ANSYS version 5.4 finite element analysis pro-

gram. Having created the solid model of the cutting

tool, this model was imported into ANSYS and divided

Fig. 7. Deflection values of 60� s

into the finite elements. After this stage, boundary con-

ditions were applied. As was mentioned earlier on, the

cutting tool was like a cantilever beam with 30 mm

length. Cutting forces were applied through the cutting

edge as a uniform loads in x, y and z directions as shown

in Fig. 6. The cutting tool was regarded as having iso-tropic properties and some of them required for the

solution were entered to the program through isotropic

material properties icon. In order to calculate the total

stress of the defined component, global stiffness matrix

becomes 3n lines and 3n columns as the component

was 3 freedom level.

After this stage, Fig. 7, showing the amount of deflec-

tion of the each point of the cutting tool, was obtained.As can be seen from this figure, nine different colours

were obtained on the cutting tool having 60� side cuttingedge angle. These colours show the amounts of deflec-

tions or in other words they show clearly the deflections

in decreasing order from the cutting tool edge towards

the back. As can be seen from the figure, the maximum

ide cutting edge angle tool.

Fig. 8. Deflection values of 75� side cutting edge angle tool.

Fig. 9. Deflection values of 90� side cutting edge angle tool.

A. Duran, M. Nalbant / Materials and Design 26 (2005) 549–554 553

deflection was observed at the tool tip, 0.04373 mm.

Similarly, Fig. 8 shows the deflection of 75� side cuttingedge angle tool, while Fig. 9 shows that of 90� side cut-

ting edge angle tool.

4. Measurement results

C 1060 steel workpiece was held between the chuck

and tailstock without run-out. Cutting force measure-

ment system was set up and it was set to the conditions

(the same gain, feeding voltage and number of dataread) prevailing during the system calibration. Then,

the system was run idle for 30 min so that steady state

was reached before taking the readings. For each posi-

tion of the cutting tool given in Section 3.1, cutting tests

were performed at constant cutting speed, feed rate and

depth of for 5 s. The strain values corresponding to the

Fs, Fv and Fp values were recorded on a computer during

the cutting [5]. This practice was performed for each of

Table 1

The measured cutting forces [13]

Side cutting edge angle The mean forces measured by the

dynamometer (N)

(Fs) (Fv) (Fp)

60� 136 540 270

75� 1330 530 265

90� 1325 525 265

Table 2

Deflection of the cutting tool

Side cutting

edge angle

Deflection values

obtained by

ANSYS (mm)

Deflection values

obtained by

Castigliona

method (mm)

Difference

(% deviation)

60� 0.04373 0.03542 8

75� 0.04065 0.0344 6

90� 0.03995 0.0345 4

Fig. 10. Cutting tool side cutting edge angles.

554 A. Duran, M. Nalbant / Materials and Design 26 (2005) 549–554

the side cutting edge angle. The resulting cutting forces

are given in Table 1. The highest analysis values ofdeflections of the cutting tools having 60�, 75� and 90�side cutting edge angles found by finite element method

are compared to theoretically obtained ones through

Castigliano method explained in Section 2.2, Table 2.

As shown in Table 2, the least deviation was observed

for the 90� side cutting edge angle tool. That is because

this cutting tool had less changes in its cross-sectional

area than 60� and 75� side cutting edge angle toolsdepending on side cutting edge angle. This is shown in

Fig. 10 schematically. In addition, in calculations with

Castigliano theory Fs was applied as a point load at

the cutting edge of the cutting tool, while in ANSYS

program all the cutting force components Fs, Fp and

Fv forces occurring during real cutting operations were

applied as uniformly distributed load along the cutting

surface as stated in Fig. 6.

5. Conclusions

The differences between the cutting tool deflection

values obtained by ANSYS program and Castigliano

theory were approached by 0.4%, 0,6% and 0.8% for

90�, 75� and 60� side cutting edge angles, respectively.

Therefore, it can be said that ANSYS program and Cas-

tigliano theory produced almost the same values. The

least deviation between the results obtained by ANSYS

and Castigliano theory for the tool having 90� side cut-

ting edge angle can be explained by the cutting tool

cross-sections used in the both methods which were

the most similar for this cutting tool.The resulting deviations do not only stem from the

cutting tool cutting angles. That is because the applica-

tion of forces to the cutting tool are different. That is, in

the computer based method all the three force compo-

nents are applied uniformly along the cutting edge as

in real cutting operation, while in Castigliano theory

only Fs force is applied at the cutting edge as the point

load.However, all the results show that Castigliano theory

used for calculations of the deflections of square and

rectangular cross-sectioned cantilever beam can also be

used to determine the deflections of single part HSS

tools having 90�, 75� and 60� side cutting edge angles.

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