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Elastic-plastic analysis of tube bending
International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Elastic-plastic analysis of tube bending H.A. Al-Qureshi* Instituto Tecnologico de Aeronautica, Sa˜o Jose´ Dos Campos, 122280-900, Brazil Received 26 January 1998
Abstract The aim of the paper is to present a theoretical analysis of the elastic-plastic bending of tube. Analytical methods are given whereby approximate equations are derived to provide a quantitative method for predicting the spring back behaviour and residual stress distributions. Comparisons between experimental and theoretical results of spring back have shown remarkable agreement. Also, a special technique for the bending of thin-walled tubes using a flexible mandrel of elastomer rod, within the ‘Doubtful Bend Region 0/2R ⬍ 1.7’ is presented. The elastomer is employed as an internal pressure transmitting medium in the bending process. Theoretical analysis based on the upper-bound approach for predicting the required bending load is also presented. Generally, it is found that the theoretical results for the bending loads are in good agreement with experimental values. Both experiments were performed on materials with different work hardening characteristics. Various parameters associated with these techniques are also examined. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Tube bending; Elastomer forming; Spring-back factor; Residual stresses
1. Introduction The bending of tube has many applications, particularly in the aircraft industry. Tube bending was previously viewed as a craft, the work being done largely by hand by skilled labourers, whose art was based on long experience. However, the transition from handcraft to technical control has necessitated a careful study of the basic mechanics involved in bending tubes. In addition, it is frequently desired to bend a tube to a certain geometry profile. The well known phenomenon of spring back always makes this a certain job for which a good deal of ‘development time’ of the
* Tel: + 55-012-347-5900; Fax: + 55 012-347-5801; E-mail: [email protected] 0890-6955/99/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 1 2 - 1
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tooling is required. Needless to say, the principles of tube bending are similar to bar bending, except that internal support is normally needed. The internal support is provided by a mandrel inserted inside the tube prior to the bending operation [1]. In practice there exist several types of mandrels such as, rigid, flexible and articulated. The main purpose of the mandrel is to avoid wrinkling and collapse of the tube during the bending process. Generally, tube geometry and radius of bend determine whether a mandrel is needed and if so the type necessary. In fact, as a general guideline, when the ratio of the centre line radius to the outside diameter of the tube is less than 1.7, then collapse will take place for any ratio of outside diameter to wall thickness [1]. Bending of tubes using hydraulic pressure has been achieved successfully [2,3]. It was shown that thin-walled tubes can be bent at right angles with virtually zero bend radii. This is accomplished by special tooling employing internal pressure and external forces. In this technique a balance must be maintained between the internal pressure and the axial load to prevent tearing and/or buckling. This technique also presents other drawbacks, such as, high tooling costs, difficult design and operational problems in the form of high pressure seals and fatigue failure of the tooling. Recently however, a new process for bending of thin-walled tube has been developed, using a flexible mandrel of elastomer rod [4–6]. The elastomer is employed as an internal pressure transmitting medium in the bending process. Such internal support provided by the elastomer produced successful bends within the ‘Doubtful Bend Region, i.e. 0/2R ⬍ 1.7’. Needless to say, this outstanding feature offers a simple alternative to some conventional and non-conventional processes. However, due to lack of knowledge and ‘know-how’ elastomer has had limited use on the shop floor as a forming die, but the technique of forming with elastomer has been exploited and has found an increasing number of applications in the field of metal forming [4–8]. This is particularly true in cases where the cost of conventional tooling is prohibitive, or the product is required in small quantities.
2. Theoretical considerations 2.1. Spring back analysis The theory of bending of beams in plastic range has been developed and presented in many papers [8–13]. However, the present theory deals with the bending of tube beam having circular cross section. The bending moment and spring back can be solved as a function of other variables, by the use of the following general assumptions: 1. 2. 3. 4.
The cross section has an axis of symmetry perpendicular to the plane of external forces. Bauchinger effects, buckling, and tearing are absent. The material has elastic-perfectly plastic behaviour. Diameter of tube remains constant, i.e. plane strain condition.
Assuming that B(Y) denotes the width of the cross section at the distance Y from the neutral axis, and Ye denotes the distance of elastic-plastic boundary from the neutral axis, then the elasticplastic moment of the section may be expressed as [9],
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
M = 2KEIe + 2EKYeSp
89
(1)
where
冕
冕
Ye
Ie = Y2B(Y)dY
R
and
Sp = YB(Y)dY
0
(2)
Ye
From close examination of the geometry of the cross section shown in Fig. 1, the width B() at an angle , can be expressed as B() = 2Rsin − 2√(R − t)2 − R2cos2, for 0 ⱕ Y ⱕ R − t
(3)
B() = 2Rsin, for R − t ⱕ Y ⱕ R and dY = − Rsin where Y = Rcos The analytical expression of the elastic-plastic bending moment, Eq. (1), as a function of Ye, can be obtained by performing the integrations on the expressions of Ie and Sp [Eq. (2)]. Then, mathematical operations will be carried out separately. 2.1.1. Calculation of Ie Considering the situation when Y = 0, then Eq. (3) becomes 0 = Rcos⇒cos = 0⇒ =
2
(4)
Fig. 1. Geometry of tube.
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On the other hand, when Y = Ye, hence Ye = Rcos⇒cos =
冉冊
Ye Ye ⇒ = arccos R R
(5)
It is clear from Fig. 1, that the width B() depends solely on the location of the elastic-plastic boundary (Ye). Therefore, either Ye ⱕ R − t or R − t ⱕ Ye ⱕ R, and each of these conditions will be examined independently. 2.1.2. Case (I)-when Ye ⱕ R − t The moment of inertia of the elastic region of the hollow circular section, 2Ie given in Eq. (3), with respect to the neutral axis is given by arccos
冉 冊
冕
Ie =
Ye R
R2cos2{2Rsin − 2√(R − t)2 − R2cos2}{ − Rsin.d}
(6)
2
By separating the terms of the above expression and integrating by parts, it was possible to derive an expression for Ie as a function of geometry and Ye, and is given by
再
冉冊
冉 冊冎
Ye Ye ] − 4arccos R R 16
R4 2 + sin[4arccos Ie =
+
−
冦
再
Y(3/2) √(R − t)2 − Y2e e 2
冧冦 −
(R − t)2Ye√(R − t)2 − Y2e 4
冉 冊冎
冧
(R − t)4 Ye arcsin 4 R−t
(7)
2.1.3. Calculation of Sp This term, [Eq. (2)], can be divided into two terms as
冕
R−t
Sp = S ip + S iip =
Ye
冕 R
B(Y)YdY +
B*(Y)YdY
R−t
Once again, the geometry of the section provides the following relations: for Ye ⱕ Y ⱕ R − t, gives
(8)
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
91
B() = 2Rsin − 2√(R − t)2 − R2cos2 R − t ⱕ Y ⱕ R, gives B*() = 2Rsin when
冉冊 冉 冊
Y = Ye⇒⇒arccos
Ye R
Y = R − t⇒ = arccos
R−t R
(9)
Y = R⇒ = 0 By using Y = Rcos then Eq. (8) can be written as
冉 冊 R−t R
arccos
冕
S ip =
再
冉 冊
arccos
冎
Rcos 2Rsin − 2√(R − t)2 − R2cos2 { − Rsind} Ye
(10)
R
and
冕 0
S iip =
arccos
2R2sincos(− Rsin)d =
冉 冊
冋 冉 冊册
2 3 3 R−t R sin arccos 3 R
R−t R
(11)
Finally, using Eqs. (8), (10) and (11) Sp can be expressed as Sp =
冋 冉 冊册
2 3 3 Ye 兵R sin arccos 3 R
− [(r
t)2 − Y2e ]3/2其
(12)
Hence, the resulting relationship for the elastic-plastic bending moment for the case where Ye ⱕ R − t, can be obtained by substituting Eqs. (7) and (12) into Eq. (1), then M = 2EK{Eq(7) + YeEq(12)}
(13)
2.1.4. Case (II)-R − t ⱕ Ye ⱕ R In this case the elastic-plastic boundary is situated at the upper part of the section. By repeating the same procedure as for Case (I), it is possible to calculate Ie in this region and is given by arccos
Ie =
冉 冊 R−t R
冕 2
R2cos2{2Rsin − 2√(R − t)2 − R2cos2}{ − Rsind}
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arccos
冉 冊
冕
+
Ye R
冉 冊
arccos
{R2cos2}{2Rsin}{− Rsind}
(14)
R−t R
The above relationship can be divided into two terms, and each term can be integrated by parts. The final expression for Ie in this region can be written as R4 (R − t)4 − − Ie = 8 8
冉冊
R4arccos
Ye R
4
冋
冉 冊册
R4sin 4arccos +
Ye R
(15)
16
On the other hand, the static moment of the plastic region taken with respect to neutral axis Sp is given by
冕 0
Sp =
arccos
2Rsin.Rcos(− Rsind) =
冉 冊 Ye
再 冉 冊冎
2 3 3 Ye R sin arccos 3 R
(16)
R
Therefore, the total elastic-plastic bending moment for the region (R − t ⱕ Ye ⱕ R), can be derived by substituting Eqs. (15) and (16) into Eq. (1) giving the following M = 2EK{Eq(15) + YeEq(16)}
(17)
It is evident from the present theoretical analysis that the expressions for Ie, Sp, and M must be continuous functions across the plastic boundary. Then the two proceeding sets of solutions [Cases (I) and (II)] will be matched at the junction of elastic-plastic boundary, i.e., IIe = IIIe =
再
冉 冊冎
R4 (R − t)4 R4 R−t − + sin 4arccos 8 8 16 R
−
冉 冊
R4 R−t arccos 16 R
(18)
However, by substituting Eq. (19) into Eqs. (7), (12), (13), (15), (16) and (17), the continuity equations become S Ip = S pII =
再 冉 冊冎
2 3 3 R−t R sin arccos 3 R
(19)
also MI = 2EK[IIe + (R − t)S Ip] MII = 2EK[IpII + (R − t)S pII]
(20)
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93
MI = MII where superscripts refer to the expression obtained either from Case (I) or Case (II). Needless to say, upon unloading the tube beam in which yielding has already taken place, the beam springs back or recovers elastically. The elastic bending moment is given as
=
MY I0 E⑀I0 = = or M = I0 Y Y
EI0
冉冊 Y
Y
=
EI0
(21)
When unloading occurs, there is a change in the radius of curvature due to elastic spring back (1/) expressed as [8] 1 1 1 = − 0 f
(22)
where 0 and f are the initial and final radii of curvature of the beam. By substituting Eq. (22) into Eq. (21) the final expression for the spring back is given by the following relationship, M = EI0
冉 冊
M0 1 1 0 − or =1− 0 f f EI0
(23)
where I0 is the moment of inertia of the hollow circular section and given by I0 =
4 [R − (R − t)4] 4
(24)
As mentioned previously, the position of the elastic-plastic boundary (Ye) controls primarily the determination of the bending moments. Consequently, it has direct influence on the elastic recovery of the section. Therefore, for the case where 0 ⱕ Ye ⱕ R − t, then Ye =
00 E
(25)
Hence, using Eqs. (13), (23) and (25) then the expression for the spring back in this region is written as 2A 0 =1− f I0 where
A=
冦
冋
冓
冉 冊冔
R4 2 + sin 4arccos
00 ER 16
− 4arccos
冉 冊册 00 ER
冧
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
03/2 0 + −
冪
冉 冊 00 E
(R − t)2 −
2
2E
−
冋
册 冉 冊册 冎
冉 冊 冪 再 冋 冉 冊册
(R − t)2 00 4 E
(R − t)2 −
00 E
2
(R − t)4 200 3 3 00 00 + R sin arccos arcsin 4 E(R − t) 3E ER
冋
00 − (R − t) − E 2
3 2
2
(26)
Whereas, for the case where R − t ⱕ Ye ⱕ R, and using Eqs. (17), (23) and (25), the expression for the spring back can be written as 2B 0 =1− f I0
(27)
where
B=
冤
+
R (R − t) − − 8 8 4
冤
4
再
冉 冊冎
R4sin 4arccos
冉 冊
R4arccos
00 ER
16
4
00 ER
冥
再 冉 冊冎
2R300sin3 arccos +
3E
00 ER
冥
(28)
On examination of Eqs. (26) and (28), it is evident that when 0/f = 0, then bending is wholly elastic. On the other hand, if 0/f = 1, then the section is totally plastic and there is no spring back. It is a known fact, that upon unloading after yielding has occurred, residual stresses will be present in the section. These stresses are totally elastic, and the net residual stress distribution can be expressed as
xr = x ± YE
冉 冊 1 1 − 0 f
(29)
It is worth mentioning that the correct sign for x and Y in Eq. (29) is essential. 2.2. Bending load requirement As mentioned previously, it is essential to support the diameter otherwise collapse and wrinkling will take place for any ratio of outside diameter to wall thickness of the tube. Therefore, the need for an internal mandrel with this operation becomes evident, and the most commonly used mandrels are flexible, such as elastomer rod.
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
95
Generally, the power is supplied by pressurizing and bending the tube, which is caused by the axial compression of the elastomer rod. The upper bound technique is used here for estimating the necessary applied load to cause bending of the tube. The present analysis treats the case where surface discontinuity is absent; thus the power needed to bend the tube is mainly composed of the internal pressurization of the elastomer (Wa), plastic bending power (Wb), and the power losses to overcome friction between workpiece/die interface (Wf). Also, friction is assumed to be constant or sticking is ruled out, and the analysis is limited to plane strain problem, i.e. the diameter is constant. The power supplied can be estimated from the upper-bound theory by equating it to the power consumed which consists of the following powers: 2.2.1. Internal pressurization power (Wa) This is the power needed to be stored in the flexible mandrel or elastomer rod to cause bending of the tube inside the die, and is defined as
冕 x2
Wa = F(x)dx
(30)
x1
where F(x) =
ErArx C
(31)
Er is the compression modulus of elasticity, Ar is the cross sectional area and C is the total height of the elastomer rod when totally confined, and finally, x is the displacement of the elastomer during the bending operation. 2.2.2. Plastic bending power (Wb) To evaluate this power, the work hardening condition of the metal tube is assumed to be
¯ = K(⑀¯ )n
(32)
Therefore, the power can be expressed simply as Wb =
冕
K (⑀¯ )n + 1dV (n + 1)
(33)
v
where n is the strain hardening exponent, and dV is the incremental volume of the element. However, the volume of the element can be determined from the geometry of the bent section, Fig. 2, and can be expressed as dV = L(2Rt)d
(34)
where L is the circumferential length at the centre of the bend, i.e. the neutral axis of the bent tube, t is the initial thickness of the tube having external radius R. Hence, the power required for bending the tube is given by
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 2. Geometry of tube bending.
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
冢 冣
n+1
Wb =
2 √3
冉 冊 冕冉 再 2
K (2tRL) n+1
ln 1 +
Rsin R0
0
冎冊
97
n+1
d
(35)
where R0 is the radius of the neutral axis of the tube. Evidently, the above equation is solved numerically. 2.2.3. Friction power losses (Wf) The friction is assumed to be Coulomb type, and the power dissipated to overcome friction along the contact surface is given by Wf = 兰F¯d¯
(36)
where F¯ represents the friction force and d¯ is the infinitesimal displacement (¯ is the displacement vector where F¯ is applied). Both of the above parameters can be expressed as a function of geometry and constant parameters of the elastomer. The bending operation with elastomer is performed by compressing the elastomer/tube assembly simultaneously into the die block. Therefore, part of the tube descends into the radiused part of the bending die, whilst the rest of the tube is still moving against the guiding sleeve. Hence, the tube is in contact with two regions, i.e. the curved (bent) part and the straight part. As a result the friction force can be separated into: I. Friction in the cylindrical part which has not been bent, given by, Ffc = p2R(Lf − R0)
(37)
where p is the internal pressure created by pressurizing the elastomer, is the coefficient of friction, and Lf is the total length of the tube. Therefore, power dissipated against friction in this region (Fig. 2) is given by
冕
Wfc = Ffcd
(38)
0
or
冋
Wfc = 2pRR0 Lf(f − 0) − R0
冉
2f − 20 2
冊册
II. Friction force in the toroidal region, or the bent region is given by the following expression
冉
冊冉
R2 2f − 20 Wft = 2pRR0 R0 + 2R0 2
冊
(39)
Therefore, the total power dissipated to overcome friction during the bending operation is simply the sum of Eqs. (38) and (39).
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2.2.4. The external power (Wt) This power can be expressed as Wt = F.y
(40)
where F is the bending load and y( = 0.4 L) is the displacement. It is worth mentioning that the process here is considered quasi-static. It is evident from the definition of the upper bound theory, that the power balance can be written as Wt = Wf + Wb + Wa
(41)
Therefore, the solution for the compressive load required to bend the tube section is obtained from Eqs. (30), (35), (38), (39) and (41). 3. Experimental procedure and results In the present work two independent experiments were carried out on tube bending operations. The first, involves ‘simple bending’ of thin-walled tube to measure the effect of spring back and residual stress. The second experiment is performed on thin-walled tube using flexible mandrel such as elastomer to support the inside of the tube. Each of these experiments will be discussed thoroughly in this section. Firstly, the simple bending tests were performed using a CNC bending machine having 50 HP, coupled with a Laservision and XL Vector Measuring Machine. This equipment permits the geometrical comparison before and after the bending operation; thus, providing the elastic spring back values for each test. The experiments were carried out on materials with different work hardening characteristics and geometry and are listed in Table 1. The theoretical results of spring back ratio were calculated by the expressions given in the previous section, utilizing the properties listed in Table 1, for various bend radii. These values are compared with the experimental results and are given in Table 2. It can be observed that the greatest difference between theory and experiment is about 8.4% for the as received copper tubes. Needless to say, generally very close agreement between theoretical and experimental results is exhibited. Normally, the bending of tubes, while at first glance of a simple nature, involves several probTable 1 Mechanical properties of tubes Tube material (as received cond.) Aluminium Aluminium Stainless steel Stainless steel Copper Titanium
External radius, R (mm) 6.350 3.175 3.175 8.575 8.000 4.760
Thickness, t (mm) 1.02 0.89 0.71 2.31 1.40 0.51
Yield stress, 0 (Kgf/mm2)
Mod. of elast., E, (Kgf/mm2)
33 33 32.7 32.7 120 50
7200 7200 21 094 21 094 12 900 18 200
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
99
Table 2 Comparison between theoretical and experimental results for different tube materials Material
0 (mm)
0/f (experiment)
0/f (theory)
% Error
Aluminium Aluminium Stainless steel Stainless steel Copper Titanium
40 25 25 79.37 79.37 25.40
0.989 0.985 0.983 0.940 0.870 0.970
0.979 0.966 0.959 0.928 0.950 0.990
1.02 1.97 2.50 1.29 8.40 2.00
lems, the chief of which is the tendency of the material to wrinkle and buckle, this being in addition to the elastic recovery or the spring back. For this reason, the bending operations were performed by inserting the articulated-linked ball mandrel inside the tube. Evidently, the mandrel was used mainly to prevent collapse of the tube or uncontrolled flattening in the bend (Fig. 3). A generalized and normalized theoretical curve for the spring back factor (k = 0/f) as a function of tube factor (q = .0/E.R) is plotted in Fig. 4. It can be seen that the section, Fig. 4, is totally elastic when 0/f = 0, and completely plastic when 0/f = 1. However, for most materials having a spring back ratio between these limits, this value depends solely on the work hardening characteristics of the material and tool geometry. In practice spring back complicates tool design in that the die must be designed to compensate for it. Consequently, it is desirable to have a method of quantitatively predicting the magnitude of spring back as a function of the
Fig. 3. Typical bent tubes.
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 4. Theoretical curve for spring back factor.
properties of the material and geometry of tubes. Therefore, a generalized and simplified mathematical model of this effect is given in the present work. It is generally accepted knowledge that residual stresses are purely elastic even if they result from plastic deformation. They may be sufficiently large to cause fracture of the material in a brittle manner. Therefore, it is extremely important to determine the residual stress distributions throughout the section of the material. The typical theoretical residual stress distributions calculated by Eq. (29) for aluminium tubes are illustrated in Fig. 5, for 0 ⱕ Ye ⱕ ⱕ R − t and Fig. 6 for R − t ⱕ Ye ⱕ R. On close examination of these figures it can be observed that, as the plastic zone increases, it is accompanied by noticeable reduction in the residual stress distributions. To summarize, residual stresses may, under certain conditions, cause failure of the structure at a lower applied load than that predicted. The tooling used in conjunction with the second experiment is shown schematically in Fig. 7(A). The bend die was made of split configuration to facilitate removal after the bending operation, and was housed in a container. A guide was fixed at the top of the die to support and guide the tube/elastomer rod assembly into the bending die during the axial compression. All test tube specimens, as listed in Table 3, were closed at one end by a spherical segment steel cap [initial tube length = 145 mm and average diameter (2R) = 50.8 mm]. This permitted the elastomer to drag the rest of the tube through the die curvature [see Fig. 7(B)]. The elastomer rod had a 60 ‘A’ Shore hardness [modulus of elasticity (Er) = 1750 MPa], and was machined to fit tightly inside the tube. The outside of the tube and the die surfaces were lubricated with lanolin. The bending operation using this technique is simple to perform, and was carried out by compressing the tube/elastomer assembly into the die, thus descending into the curved (radiused) part of the bending die. A fully 90° bent tube inside the die and the steel cap
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 5.
Distribution of residual stresses for 0 ⬍ Ye ⬍ R − t.
Fig. 6.
Distribution of residual stresses for R − t ⬍ Ye ⬍ R.
101
attachment is shown in Fig. 8. Needless to say, the steel cap is provided with an air vent to permit the removal of the elastomer at the end of the operation. On examination of the loading procedure, it was observed that in all tests a steady state bending took place at maximum applied load (Fe), which is considered to be the experimental bending load. Table 3 lists all tested tubes. Also, a variation of about 1% in the diameter and the ovality
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Fig. 7.
Experimental set-up for tube bending using elastomer rod.
of the central radius was observed. In addition, an increase in thickness of about 11% in the compression region and a reduction of about 17% in the tension region was seen. For calculating the theoretical bending load, it was assumed that = 0.1 for graphite and = 0.6 for the other lubricants [5]. Together with the elastomer/tube mechanical properties and operational condition, it was possible to predict the bending load using the upper-bound technique. The experimental (Fe) and the theoretical (Ft) loads are numerated in Table 3, and close examination of these values reveals that there are slight differences between them. This evidently, may
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103
Table 3 Comparison between theoretical and experimental results of bending loads Tube materials and parameters
70/30 Brass ¯ = 839 (⑀¯ )0.46 (MPa)
Thickness, t 1.6 mm (mm) Length, L (mm) Wb, (102 J) Wf, (102 J) Wt, (102 J) y = 0.4 L (mm) Ft, 104 N Fe, 104 N
100
Stainless steel-AISI-304 ¯ = 1545 (⑀¯ )0.5 (MPa)
0.8 mm
60
110
1.00 mm
80
60
50
20
2.00 mm
30
18.1 25.4 43.5 40
10.9 13.7 24.6 24
19.9 28.6 48.5 44
7.37 2.23 9.60 32
5.53 1.54 7.07 24
4.61 1.22 5.83 20
3.7 5.1 8.8 8
10.9 8.06 19.0 12
10.9 9.8
10.2 10.0
11.0 8.8
3.0 2.6
2.94 2.8
2.92 2.1
11.0 12.3
15.8 16.9
Fig. 8.
Fully 90° bent tube and die assembly.
be attributed to the several assumptions which were made in setting out the present theory, particularly the pronounced effect of friction. It is clear that in this type of operation a great deal of power is lost against friction (Wf), as demonstrated in Table 3, and is more pronounced with stainless steel. 4. Conclusions In the present paper, theoretical expressions are derived in an attempt to provide a quantitative method with practical utility for predicting the spring back behaviour and residual stress distributions in a bent tube section as a function of die radius, wall thickness, and stress-strain character-
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istics of the materials. It was found that the agreement is very good for the majority of the tested tubes. The knowledge of these parameters associated with the bending is essential when designing the tools so that the contour of the product can be formed accurately. From the present theoretical analysis it can be concluded that the spring back factor and residual stresses are greatly influenced by the location of the elastic-plastic boundary. On the other hand, the use of elastomer as a flexible mandrel reflects the growing demand for low cost tooling and product quality. However, the main disadvantages of this process are limited production rate, and the difficulty in bending of large tube diameters. Over the range of tube materials tested, there is a good agreement between theoretical and experimental results of bending loads. In addition, the present theory can be used to choose the adequate elastomer, tools and press brake for a given tube material. Also, the use of suitable lubricant and elastomer is very essential, otherwise surface damage may take place. This clearly would reduce the operational life of the elastomer. Finally, it can be concluded that the use of elastomer in tube bending operation can be carried out successfully, specially in the ‘Doubtful Bend Region’. Evidently, this is an outstanding feature of this process, which offers a simple alternative to a number of conventional bending techniques. Acknowledgements The author wishes to thank CNPq for the financial support. Thanks are also extended to ITA/CTA for providing the facilities. References [1] American Society of Metals (ASTM) Metal Handbook, 8th ed., vol. 4, 1964. [2] G. Powell, B. Avitzur, Proceedings of the North American Metalworking Research Conference, USA, 1972, pp. 63–83. [3] F. Dohmann, C. Hartl, Proceedings of the 5th International Conference on Metal Forming 94, Birmingham, UK, 1994, pp. 377–382. [4] E.M. Bello, S.B. Sorensen, H.A. Al-Qureshi, Proceedings of the 27th International Machine Tool Design and Research Conference, MTDR, Manchester, UK, 1988, pp. 327–331. [5] L.A. Moreira Filho, J. Menezes, H.A. Al-Qureshi, Proceedings of the 5th International Conference on Metal Forming, 94, Birmingham, UK, 1994, pp. 383–388. [6] H.A. Al-Qureshi, Journal of Mechanical Working Technology 1 (3) (1978) 261–275. [7] H.A. Al-Qureshi, Machinery and Production Engineering 119 (1972) 189. [8] P.B. Mellor, W. Johnson, Engineering Plasticity, Van Nostrand, New York, 1973. [9] Aris Phillips, Introduction to Plasticity, The Ronald Press Co., 1956. [10] C.A. Queener, R.J. De Angelis, Transactions of the ASM 61 (1968) 757–768. [11] F.J. Gardiner, Transactions of the ASME January (1957) 1–9. [12] D.M. Woo, J. Marshall, The Engineer 28 August (1959) 135–136. [13] W. Schroeder, Transactions of the ASME November (1943) 817–827.
Elastic-plastic analysis of tube bending H.A. Al-Qureshi* Instituto Tecnologico de Aeronautica, Sa˜o Jose´ Dos Campos, 122280-900, Brazil Received 26 January 1998
Abstract The aim of the paper is to present a theoretical analysis of the elastic-plastic bending of tube. Analytical methods are given whereby approximate equations are derived to provide a quantitative method for predicting the spring back behaviour and residual stress distributions. Comparisons between experimental and theoretical results of spring back have shown remarkable agreement. Also, a special technique for the bending of thin-walled tubes using a flexible mandrel of elastomer rod, within the ‘Doubtful Bend Region 0/2R ⬍ 1.7’ is presented. The elastomer is employed as an internal pressure transmitting medium in the bending process. Theoretical analysis based on the upper-bound approach for predicting the required bending load is also presented. Generally, it is found that the theoretical results for the bending loads are in good agreement with experimental values. Both experiments were performed on materials with different work hardening characteristics. Various parameters associated with these techniques are also examined. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Tube bending; Elastomer forming; Spring-back factor; Residual stresses
1. Introduction The bending of tube has many applications, particularly in the aircraft industry. Tube bending was previously viewed as a craft, the work being done largely by hand by skilled labourers, whose art was based on long experience. However, the transition from handcraft to technical control has necessitated a careful study of the basic mechanics involved in bending tubes. In addition, it is frequently desired to bend a tube to a certain geometry profile. The well known phenomenon of spring back always makes this a certain job for which a good deal of ‘development time’ of the
* Tel: + 55-012-347-5900; Fax: + 55 012-347-5801; E-mail: [email protected] 0890-6955/99/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 8 ) 0 0 0 1 2 - 1
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tooling is required. Needless to say, the principles of tube bending are similar to bar bending, except that internal support is normally needed. The internal support is provided by a mandrel inserted inside the tube prior to the bending operation [1]. In practice there exist several types of mandrels such as, rigid, flexible and articulated. The main purpose of the mandrel is to avoid wrinkling and collapse of the tube during the bending process. Generally, tube geometry and radius of bend determine whether a mandrel is needed and if so the type necessary. In fact, as a general guideline, when the ratio of the centre line radius to the outside diameter of the tube is less than 1.7, then collapse will take place for any ratio of outside diameter to wall thickness [1]. Bending of tubes using hydraulic pressure has been achieved successfully [2,3]. It was shown that thin-walled tubes can be bent at right angles with virtually zero bend radii. This is accomplished by special tooling employing internal pressure and external forces. In this technique a balance must be maintained between the internal pressure and the axial load to prevent tearing and/or buckling. This technique also presents other drawbacks, such as, high tooling costs, difficult design and operational problems in the form of high pressure seals and fatigue failure of the tooling. Recently however, a new process for bending of thin-walled tube has been developed, using a flexible mandrel of elastomer rod [4–6]. The elastomer is employed as an internal pressure transmitting medium in the bending process. Such internal support provided by the elastomer produced successful bends within the ‘Doubtful Bend Region, i.e. 0/2R ⬍ 1.7’. Needless to say, this outstanding feature offers a simple alternative to some conventional and non-conventional processes. However, due to lack of knowledge and ‘know-how’ elastomer has had limited use on the shop floor as a forming die, but the technique of forming with elastomer has been exploited and has found an increasing number of applications in the field of metal forming [4–8]. This is particularly true in cases where the cost of conventional tooling is prohibitive, or the product is required in small quantities.
2. Theoretical considerations 2.1. Spring back analysis The theory of bending of beams in plastic range has been developed and presented in many papers [8–13]. However, the present theory deals with the bending of tube beam having circular cross section. The bending moment and spring back can be solved as a function of other variables, by the use of the following general assumptions: 1. 2. 3. 4.
The cross section has an axis of symmetry perpendicular to the plane of external forces. Bauchinger effects, buckling, and tearing are absent. The material has elastic-perfectly plastic behaviour. Diameter of tube remains constant, i.e. plane strain condition.
Assuming that B(Y) denotes the width of the cross section at the distance Y from the neutral axis, and Ye denotes the distance of elastic-plastic boundary from the neutral axis, then the elasticplastic moment of the section may be expressed as [9],
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
M = 2KEIe + 2EKYeSp
89
(1)
where
冕
冕
Ye
Ie = Y2B(Y)dY
R
and
Sp = YB(Y)dY
0
(2)
Ye
From close examination of the geometry of the cross section shown in Fig. 1, the width B() at an angle , can be expressed as B() = 2Rsin − 2√(R − t)2 − R2cos2, for 0 ⱕ Y ⱕ R − t
(3)
B() = 2Rsin, for R − t ⱕ Y ⱕ R and dY = − Rsin where Y = Rcos The analytical expression of the elastic-plastic bending moment, Eq. (1), as a function of Ye, can be obtained by performing the integrations on the expressions of Ie and Sp [Eq. (2)]. Then, mathematical operations will be carried out separately. 2.1.1. Calculation of Ie Considering the situation when Y = 0, then Eq. (3) becomes 0 = Rcos⇒cos = 0⇒ =
2
(4)
Fig. 1. Geometry of tube.
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On the other hand, when Y = Ye, hence Ye = Rcos⇒cos =
冉冊
Ye Ye ⇒ = arccos R R
(5)
It is clear from Fig. 1, that the width B() depends solely on the location of the elastic-plastic boundary (Ye). Therefore, either Ye ⱕ R − t or R − t ⱕ Ye ⱕ R, and each of these conditions will be examined independently. 2.1.2. Case (I)-when Ye ⱕ R − t The moment of inertia of the elastic region of the hollow circular section, 2Ie given in Eq. (3), with respect to the neutral axis is given by arccos
冉 冊
冕
Ie =
Ye R
R2cos2{2Rsin − 2√(R − t)2 − R2cos2}{ − Rsin.d}
(6)
2
By separating the terms of the above expression and integrating by parts, it was possible to derive an expression for Ie as a function of geometry and Ye, and is given by
再
冉冊
冉 冊冎
Ye Ye ] − 4arccos R R 16
R4 2 + sin[4arccos Ie =
+
−
冦
再
Y(3/2) √(R − t)2 − Y2e e 2
冧冦 −
(R − t)2Ye√(R − t)2 − Y2e 4
冉 冊冎
冧
(R − t)4 Ye arcsin 4 R−t
(7)
2.1.3. Calculation of Sp This term, [Eq. (2)], can be divided into two terms as
冕
R−t
Sp = S ip + S iip =
Ye
冕 R
B(Y)YdY +
B*(Y)YdY
R−t
Once again, the geometry of the section provides the following relations: for Ye ⱕ Y ⱕ R − t, gives
(8)
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
91
B() = 2Rsin − 2√(R − t)2 − R2cos2 R − t ⱕ Y ⱕ R, gives B*() = 2Rsin when
冉冊 冉 冊
Y = Ye⇒⇒arccos
Ye R
Y = R − t⇒ = arccos
R−t R
(9)
Y = R⇒ = 0 By using Y = Rcos then Eq. (8) can be written as
冉 冊 R−t R
arccos
冕
S ip =
再
冉 冊
arccos
冎
Rcos 2Rsin − 2√(R − t)2 − R2cos2 { − Rsind} Ye
(10)
R
and
冕 0
S iip =
arccos
2R2sincos(− Rsin)d =
冉 冊
冋 冉 冊册
2 3 3 R−t R sin arccos 3 R
R−t R
(11)
Finally, using Eqs. (8), (10) and (11) Sp can be expressed as Sp =
冋 冉 冊册
2 3 3 Ye 兵R sin arccos 3 R
− [(r
t)2 − Y2e ]3/2其
(12)
Hence, the resulting relationship for the elastic-plastic bending moment for the case where Ye ⱕ R − t, can be obtained by substituting Eqs. (7) and (12) into Eq. (1), then M = 2EK{Eq(7) + YeEq(12)}
(13)
2.1.4. Case (II)-R − t ⱕ Ye ⱕ R In this case the elastic-plastic boundary is situated at the upper part of the section. By repeating the same procedure as for Case (I), it is possible to calculate Ie in this region and is given by arccos
Ie =
冉 冊 R−t R
冕 2
R2cos2{2Rsin − 2√(R − t)2 − R2cos2}{ − Rsind}
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arccos
冉 冊
冕
+
Ye R
冉 冊
arccos
{R2cos2}{2Rsin}{− Rsind}
(14)
R−t R
The above relationship can be divided into two terms, and each term can be integrated by parts. The final expression for Ie in this region can be written as R4 (R − t)4 − − Ie = 8 8
冉冊
R4arccos
Ye R
4
冋
冉 冊册
R4sin 4arccos +
Ye R
(15)
16
On the other hand, the static moment of the plastic region taken with respect to neutral axis Sp is given by
冕 0
Sp =
arccos
2Rsin.Rcos(− Rsind) =
冉 冊 Ye
再 冉 冊冎
2 3 3 Ye R sin arccos 3 R
(16)
R
Therefore, the total elastic-plastic bending moment for the region (R − t ⱕ Ye ⱕ R), can be derived by substituting Eqs. (15) and (16) into Eq. (1) giving the following M = 2EK{Eq(15) + YeEq(16)}
(17)
It is evident from the present theoretical analysis that the expressions for Ie, Sp, and M must be continuous functions across the plastic boundary. Then the two proceeding sets of solutions [Cases (I) and (II)] will be matched at the junction of elastic-plastic boundary, i.e., IIe = IIIe =
再
冉 冊冎
R4 (R − t)4 R4 R−t − + sin 4arccos 8 8 16 R
−
冉 冊
R4 R−t arccos 16 R
(18)
However, by substituting Eq. (19) into Eqs. (7), (12), (13), (15), (16) and (17), the continuity equations become S Ip = S pII =
再 冉 冊冎
2 3 3 R−t R sin arccos 3 R
(19)
also MI = 2EK[IIe + (R − t)S Ip] MII = 2EK[IpII + (R − t)S pII]
(20)
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93
MI = MII where superscripts refer to the expression obtained either from Case (I) or Case (II). Needless to say, upon unloading the tube beam in which yielding has already taken place, the beam springs back or recovers elastically. The elastic bending moment is given as
=
MY I0 E⑀I0 = = or M = I0 Y Y
EI0
冉冊 Y
Y
=
EI0
(21)
When unloading occurs, there is a change in the radius of curvature due to elastic spring back (1/) expressed as [8] 1 1 1 = − 0 f
(22)
where 0 and f are the initial and final radii of curvature of the beam. By substituting Eq. (22) into Eq. (21) the final expression for the spring back is given by the following relationship, M = EI0
冉 冊
M0 1 1 0 − or =1− 0 f f EI0
(23)
where I0 is the moment of inertia of the hollow circular section and given by I0 =
4 [R − (R − t)4] 4
(24)
As mentioned previously, the position of the elastic-plastic boundary (Ye) controls primarily the determination of the bending moments. Consequently, it has direct influence on the elastic recovery of the section. Therefore, for the case where 0 ⱕ Ye ⱕ R − t, then Ye =
00 E
(25)
Hence, using Eqs. (13), (23) and (25) then the expression for the spring back in this region is written as 2A 0 =1− f I0 where
A=
冦
冋
冓
冉 冊冔
R4 2 + sin 4arccos
00 ER 16
− 4arccos
冉 冊册 00 ER
冧
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
03/2 0 + −
冪
冉 冊 00 E
(R − t)2 −
2
2E
−
冋
册 冉 冊册 冎
冉 冊 冪 再 冋 冉 冊册
(R − t)2 00 4 E
(R − t)2 −
00 E
2
(R − t)4 200 3 3 00 00 + R sin arccos arcsin 4 E(R − t) 3E ER
冋
00 − (R − t) − E 2
3 2
2
(26)
Whereas, for the case where R − t ⱕ Ye ⱕ R, and using Eqs. (17), (23) and (25), the expression for the spring back can be written as 2B 0 =1− f I0
(27)
where
B=
冤
+
R (R − t) − − 8 8 4
冤
4
再
冉 冊冎
R4sin 4arccos
冉 冊
R4arccos
00 ER
16
4
00 ER
冥
再 冉 冊冎
2R300sin3 arccos +
3E
00 ER
冥
(28)
On examination of Eqs. (26) and (28), it is evident that when 0/f = 0, then bending is wholly elastic. On the other hand, if 0/f = 1, then the section is totally plastic and there is no spring back. It is a known fact, that upon unloading after yielding has occurred, residual stresses will be present in the section. These stresses are totally elastic, and the net residual stress distribution can be expressed as
xr = x ± YE
冉 冊 1 1 − 0 f
(29)
It is worth mentioning that the correct sign for x and Y in Eq. (29) is essential. 2.2. Bending load requirement As mentioned previously, it is essential to support the diameter otherwise collapse and wrinkling will take place for any ratio of outside diameter to wall thickness of the tube. Therefore, the need for an internal mandrel with this operation becomes evident, and the most commonly used mandrels are flexible, such as elastomer rod.
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
95
Generally, the power is supplied by pressurizing and bending the tube, which is caused by the axial compression of the elastomer rod. The upper bound technique is used here for estimating the necessary applied load to cause bending of the tube. The present analysis treats the case where surface discontinuity is absent; thus the power needed to bend the tube is mainly composed of the internal pressurization of the elastomer (Wa), plastic bending power (Wb), and the power losses to overcome friction between workpiece/die interface (Wf). Also, friction is assumed to be constant or sticking is ruled out, and the analysis is limited to plane strain problem, i.e. the diameter is constant. The power supplied can be estimated from the upper-bound theory by equating it to the power consumed which consists of the following powers: 2.2.1. Internal pressurization power (Wa) This is the power needed to be stored in the flexible mandrel or elastomer rod to cause bending of the tube inside the die, and is defined as
冕 x2
Wa = F(x)dx
(30)
x1
where F(x) =
ErArx C
(31)
Er is the compression modulus of elasticity, Ar is the cross sectional area and C is the total height of the elastomer rod when totally confined, and finally, x is the displacement of the elastomer during the bending operation. 2.2.2. Plastic bending power (Wb) To evaluate this power, the work hardening condition of the metal tube is assumed to be
¯ = K(⑀¯ )n
(32)
Therefore, the power can be expressed simply as Wb =
冕
K (⑀¯ )n + 1dV (n + 1)
(33)
v
where n is the strain hardening exponent, and dV is the incremental volume of the element. However, the volume of the element can be determined from the geometry of the bent section, Fig. 2, and can be expressed as dV = L(2Rt)d
(34)
where L is the circumferential length at the centre of the bend, i.e. the neutral axis of the bent tube, t is the initial thickness of the tube having external radius R. Hence, the power required for bending the tube is given by
96
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 2. Geometry of tube bending.
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
冢 冣
n+1
Wb =
2 √3
冉 冊 冕冉 再 2
K (2tRL) n+1
ln 1 +
Rsin R0
0
冎冊
97
n+1
d
(35)
where R0 is the radius of the neutral axis of the tube. Evidently, the above equation is solved numerically. 2.2.3. Friction power losses (Wf) The friction is assumed to be Coulomb type, and the power dissipated to overcome friction along the contact surface is given by Wf = 兰F¯d¯
(36)
where F¯ represents the friction force and d¯ is the infinitesimal displacement (¯ is the displacement vector where F¯ is applied). Both of the above parameters can be expressed as a function of geometry and constant parameters of the elastomer. The bending operation with elastomer is performed by compressing the elastomer/tube assembly simultaneously into the die block. Therefore, part of the tube descends into the radiused part of the bending die, whilst the rest of the tube is still moving against the guiding sleeve. Hence, the tube is in contact with two regions, i.e. the curved (bent) part and the straight part. As a result the friction force can be separated into: I. Friction in the cylindrical part which has not been bent, given by, Ffc = p2R(Lf − R0)
(37)
where p is the internal pressure created by pressurizing the elastomer, is the coefficient of friction, and Lf is the total length of the tube. Therefore, power dissipated against friction in this region (Fig. 2) is given by
冕
Wfc = Ffcd
(38)
0
or
冋
Wfc = 2pRR0 Lf(f − 0) − R0
冉
2f − 20 2
冊册
II. Friction force in the toroidal region, or the bent region is given by the following expression
冉
冊冉
R2 2f − 20 Wft = 2pRR0 R0 + 2R0 2
冊
(39)
Therefore, the total power dissipated to overcome friction during the bending operation is simply the sum of Eqs. (38) and (39).
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2.2.4. The external power (Wt) This power can be expressed as Wt = F.y
(40)
where F is the bending load and y( = 0.4 L) is the displacement. It is worth mentioning that the process here is considered quasi-static. It is evident from the definition of the upper bound theory, that the power balance can be written as Wt = Wf + Wb + Wa
(41)
Therefore, the solution for the compressive load required to bend the tube section is obtained from Eqs. (30), (35), (38), (39) and (41). 3. Experimental procedure and results In the present work two independent experiments were carried out on tube bending operations. The first, involves ‘simple bending’ of thin-walled tube to measure the effect of spring back and residual stress. The second experiment is performed on thin-walled tube using flexible mandrel such as elastomer to support the inside of the tube. Each of these experiments will be discussed thoroughly in this section. Firstly, the simple bending tests were performed using a CNC bending machine having 50 HP, coupled with a Laservision and XL Vector Measuring Machine. This equipment permits the geometrical comparison before and after the bending operation; thus, providing the elastic spring back values for each test. The experiments were carried out on materials with different work hardening characteristics and geometry and are listed in Table 1. The theoretical results of spring back ratio were calculated by the expressions given in the previous section, utilizing the properties listed in Table 1, for various bend radii. These values are compared with the experimental results and are given in Table 2. It can be observed that the greatest difference between theory and experiment is about 8.4% for the as received copper tubes. Needless to say, generally very close agreement between theoretical and experimental results is exhibited. Normally, the bending of tubes, while at first glance of a simple nature, involves several probTable 1 Mechanical properties of tubes Tube material (as received cond.) Aluminium Aluminium Stainless steel Stainless steel Copper Titanium
External radius, R (mm) 6.350 3.175 3.175 8.575 8.000 4.760
Thickness, t (mm) 1.02 0.89 0.71 2.31 1.40 0.51
Yield stress, 0 (Kgf/mm2)
Mod. of elast., E, (Kgf/mm2)
33 33 32.7 32.7 120 50
7200 7200 21 094 21 094 12 900 18 200
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
99
Table 2 Comparison between theoretical and experimental results for different tube materials Material
0 (mm)
0/f (experiment)
0/f (theory)
% Error
Aluminium Aluminium Stainless steel Stainless steel Copper Titanium
40 25 25 79.37 79.37 25.40
0.989 0.985 0.983 0.940 0.870 0.970
0.979 0.966 0.959 0.928 0.950 0.990
1.02 1.97 2.50 1.29 8.40 2.00
lems, the chief of which is the tendency of the material to wrinkle and buckle, this being in addition to the elastic recovery or the spring back. For this reason, the bending operations were performed by inserting the articulated-linked ball mandrel inside the tube. Evidently, the mandrel was used mainly to prevent collapse of the tube or uncontrolled flattening in the bend (Fig. 3). A generalized and normalized theoretical curve for the spring back factor (k = 0/f) as a function of tube factor (q = .0/E.R) is plotted in Fig. 4. It can be seen that the section, Fig. 4, is totally elastic when 0/f = 0, and completely plastic when 0/f = 1. However, for most materials having a spring back ratio between these limits, this value depends solely on the work hardening characteristics of the material and tool geometry. In practice spring back complicates tool design in that the die must be designed to compensate for it. Consequently, it is desirable to have a method of quantitatively predicting the magnitude of spring back as a function of the
Fig. 3. Typical bent tubes.
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 4. Theoretical curve for spring back factor.
properties of the material and geometry of tubes. Therefore, a generalized and simplified mathematical model of this effect is given in the present work. It is generally accepted knowledge that residual stresses are purely elastic even if they result from plastic deformation. They may be sufficiently large to cause fracture of the material in a brittle manner. Therefore, it is extremely important to determine the residual stress distributions throughout the section of the material. The typical theoretical residual stress distributions calculated by Eq. (29) for aluminium tubes are illustrated in Fig. 5, for 0 ⱕ Ye ⱕ ⱕ R − t and Fig. 6 for R − t ⱕ Ye ⱕ R. On close examination of these figures it can be observed that, as the plastic zone increases, it is accompanied by noticeable reduction in the residual stress distributions. To summarize, residual stresses may, under certain conditions, cause failure of the structure at a lower applied load than that predicted. The tooling used in conjunction with the second experiment is shown schematically in Fig. 7(A). The bend die was made of split configuration to facilitate removal after the bending operation, and was housed in a container. A guide was fixed at the top of the die to support and guide the tube/elastomer rod assembly into the bending die during the axial compression. All test tube specimens, as listed in Table 3, were closed at one end by a spherical segment steel cap [initial tube length = 145 mm and average diameter (2R) = 50.8 mm]. This permitted the elastomer to drag the rest of the tube through the die curvature [see Fig. 7(B)]. The elastomer rod had a 60 ‘A’ Shore hardness [modulus of elasticity (Er) = 1750 MPa], and was machined to fit tightly inside the tube. The outside of the tube and the die surfaces were lubricated with lanolin. The bending operation using this technique is simple to perform, and was carried out by compressing the tube/elastomer assembly into the die, thus descending into the curved (radiused) part of the bending die. A fully 90° bent tube inside the die and the steel cap
H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 5.
Distribution of residual stresses for 0 ⬍ Ye ⬍ R − t.
Fig. 6.
Distribution of residual stresses for R − t ⬍ Ye ⬍ R.
101
attachment is shown in Fig. 8. Needless to say, the steel cap is provided with an air vent to permit the removal of the elastomer at the end of the operation. On examination of the loading procedure, it was observed that in all tests a steady state bending took place at maximum applied load (Fe), which is considered to be the experimental bending load. Table 3 lists all tested tubes. Also, a variation of about 1% in the diameter and the ovality
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
Fig. 7.
Experimental set-up for tube bending using elastomer rod.
of the central radius was observed. In addition, an increase in thickness of about 11% in the compression region and a reduction of about 17% in the tension region was seen. For calculating the theoretical bending load, it was assumed that = 0.1 for graphite and = 0.6 for the other lubricants [5]. Together with the elastomer/tube mechanical properties and operational condition, it was possible to predict the bending load using the upper-bound technique. The experimental (Fe) and the theoretical (Ft) loads are numerated in Table 3, and close examination of these values reveals that there are slight differences between them. This evidently, may
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103
Table 3 Comparison between theoretical and experimental results of bending loads Tube materials and parameters
70/30 Brass ¯ = 839 (⑀¯ )0.46 (MPa)
Thickness, t 1.6 mm (mm) Length, L (mm) Wb, (102 J) Wf, (102 J) Wt, (102 J) y = 0.4 L (mm) Ft, 104 N Fe, 104 N
100
Stainless steel-AISI-304 ¯ = 1545 (⑀¯ )0.5 (MPa)
0.8 mm
60
110
1.00 mm
80
60
50
20
2.00 mm
30
18.1 25.4 43.5 40
10.9 13.7 24.6 24
19.9 28.6 48.5 44
7.37 2.23 9.60 32
5.53 1.54 7.07 24
4.61 1.22 5.83 20
3.7 5.1 8.8 8
10.9 8.06 19.0 12
10.9 9.8
10.2 10.0
11.0 8.8
3.0 2.6
2.94 2.8
2.92 2.1
11.0 12.3
15.8 16.9
Fig. 8.
Fully 90° bent tube and die assembly.
be attributed to the several assumptions which were made in setting out the present theory, particularly the pronounced effect of friction. It is clear that in this type of operation a great deal of power is lost against friction (Wf), as demonstrated in Table 3, and is more pronounced with stainless steel. 4. Conclusions In the present paper, theoretical expressions are derived in an attempt to provide a quantitative method with practical utility for predicting the spring back behaviour and residual stress distributions in a bent tube section as a function of die radius, wall thickness, and stress-strain character-
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H.A. Al-Qureshi / International Journal of Machine Tools & Manufacture 39 (1999) 87–104
istics of the materials. It was found that the agreement is very good for the majority of the tested tubes. The knowledge of these parameters associated with the bending is essential when designing the tools so that the contour of the product can be formed accurately. From the present theoretical analysis it can be concluded that the spring back factor and residual stresses are greatly influenced by the location of the elastic-plastic boundary. On the other hand, the use of elastomer as a flexible mandrel reflects the growing demand for low cost tooling and product quality. However, the main disadvantages of this process are limited production rate, and the difficulty in bending of large tube diameters. Over the range of tube materials tested, there is a good agreement between theoretical and experimental results of bending loads. In addition, the present theory can be used to choose the adequate elastomer, tools and press brake for a given tube material. Also, the use of suitable lubricant and elastomer is very essential, otherwise surface damage may take place. This clearly would reduce the operational life of the elastomer. Finally, it can be concluded that the use of elastomer in tube bending operation can be carried out successfully, specially in the ‘Doubtful Bend Region’. Evidently, this is an outstanding feature of this process, which offers a simple alternative to a number of conventional bending techniques. Acknowledgements The author wishes to thank CNPq for the financial support. Thanks are also extended to ITA/CTA for providing the facilities. References [1] American Society of Metals (ASTM) Metal Handbook, 8th ed., vol. 4, 1964. [2] G. Powell, B. Avitzur, Proceedings of the North American Metalworking Research Conference, USA, 1972, pp. 63–83. [3] F. Dohmann, C. Hartl, Proceedings of the 5th International Conference on Metal Forming 94, Birmingham, UK, 1994, pp. 377–382. [4] E.M. Bello, S.B. Sorensen, H.A. Al-Qureshi, Proceedings of the 27th International Machine Tool Design and Research Conference, MTDR, Manchester, UK, 1988, pp. 327–331. [5] L.A. Moreira Filho, J. Menezes, H.A. Al-Qureshi, Proceedings of the 5th International Conference on Metal Forming, 94, Birmingham, UK, 1994, pp. 383–388. [6] H.A. Al-Qureshi, Journal of Mechanical Working Technology 1 (3) (1978) 261–275. [7] H.A. Al-Qureshi, Machinery and Production Engineering 119 (1972) 189. [8] P.B. Mellor, W. Johnson, Engineering Plasticity, Van Nostrand, New York, 1973. [9] Aris Phillips, Introduction to Plasticity, The Ronald Press Co., 1956. [10] C.A. Queener, R.J. De Angelis, Transactions of the ASM 61 (1968) 757–768. [11] F.J. Gardiner, Transactions of the ASME January (1957) 1–9. [12] D.M. Woo, J. Marshall, The Engineer 28 August (1959) 135–136. [13] W. Schroeder, Transactions of the ASME November (1943) 817–827.