Chapter 4 BENDING

Main contents: Deformation process of bending ; The location of neutral curvature of strain and the

minimum bending radius; The location of neutral curvature of stress; The

calculation of bending moment and bending force; Spring back of bending; The calculation of blank length of bends; The design of punch and die in bending. Structure and design of critical parts;

Key points &Difficult point

Key points: The location of neutral curvature of strain and

stress The minimum bending radius; Spring back of bending;

Difficult parts: The calculation of principal stress in bending

of wide plate

New words

bend radius/ 弯曲半径 , bend allowance/ 弯曲余量 ,

bend angle/ 弯曲中心角 , profile angle/ 弯曲角 ,

length of bend/ 弯曲宽度 , V-profile/V 形件 ,

U-profile/U 形件 , spring back/ 回弹bend allowance radius, axis of bend/ 弯曲轴线 ,

the pre-bend length/ 弯曲件展开长度 ,

curling/ 卷圆

Typical examples of sheet-metal bend parts

1. Deformation feature

2. Stress and strain state in deformation zone

3. Deformation degree

4. Defects

§4.1 Deformation process of bending

1. Deformation feature

1.1 Definition

uniformly straining flat sheets or strips of metal around a linear axis.

1.2 Types free bending/ 自由弯曲 bending with reinforcement/ 校正弯曲

1.3 Terminologies bend radius ri — measured on the inner surface of the bend.

bend angle — angle of the bend piece.

bend allowance—arc of the neutral bend line.

length of bend—the width of the sheet.

1. Deformation feature

1.4 deformation feature

1. Deformation feature

Deformation zone: arc parts change from rectangle to sector (inner contraction and outer extension)—neutral curvature surfaceThickness----r / t ↓→ t ↓

tt /1

1. Deformation feature

1.4 deformation feature

cross section: b/t >3 constant b/t <3 change(from rectangle to sector)

According to r / t and b / t, and using dV=0 ( )

wide plate (b / t >3) cubic stress and plane strain

narrow plate (b / t<3) plane stress and cubic strain

2. Stress and strain state in deformation zone

0321

3. Deformation degree

4. Defects

r/t,

)1/2(1

max tr

spring back: elastic deformation fracture: outer fiber thickness decreasing: length increasing : dV=0

tt /1

§4.2 The position of and , the calculation of the blank length of bends and the minimum bending radius

1. The position of neutral curvature surface

of strain

2. The minimum bend radius

3. The calculation of principal stress in

bending of wide plate

4. The position of neutral curvature surface

of stress

1. The position of neutral curvature surface

of strain

1.1 Definition and significance Fiber length is constant. 1.2Calculation From dV=0 obtain , r/t↓→ ↓(moving toward inside) 1.3 Usage calculation of blanking length of bends calculation of minimum bending radius rmin/ t

1.4 calculation of blanking length of bends Different profiles use different experimental formula

(see Chinese text book p58)

)2/( tr

2.1 Definition no fracture occurring limit radius

2.2 Theoretical way

2.3 Experimental way

rmin=c.t

2. Minimum bend radius

)1(2

22/min

tr

2. Minimum bend radius

2.4 Affecting factors (1) Material’s plasticity

(2) Relation between bending

axis and rolling/grain direction

(3) Edge state of plate

(4) Bend angle

3.1 3-D pure plastic bending (no hardening)r/t <3~5 and b/t>3three unknowns need three equationsmethod: principal stress methodassumption: plane cross-section, plane strain and similar

relation between stress and strain with uniaxial tension or compression

three equations: equation of equilibrium yield condition plane strain

3. The calculation of principal stress in bending of wide plate

sk

2/)( w

0 F

3. The calculation of principal stress in bending of wide plate

))/ln(21(

))/ln(1(

)/ln(

rk

rk

rk

sw

s

s

Inner

))/ln(21(

))/ln(1(

)/ln(

Rk

Rk

Rk

sw

s

s

Outer

3. The calculation of principal stress in bending of wide plate

3.2 3-D pure plastic bending (hardening)

ln'0 Es

so yield condition

ln'( 0 Ek

.,;,

4. The position of neutral curvature of stress

Fibre where stress discrete ,

From ,

obtain

When r / t ↑ ,

when r / t ↓

EI

rR

§4.3 The calculation of bending moment and bending force

1. Moment of bending1.1 Moment of bending in elastic-plastic domain

1.2 Moment of bending in purely plastic domain

2. Bending forces2.1 Forces for U-die

2.2 Forces for V-die

3. Equipment choosing

1. Moment of bending

1. Moment of bending

Suppose: long, thin, straight beam

cross-section:bxT, length: L

bent into a curve by moments M

1.1 Moment of bending in elastic-plastic domain

1.2 Moment of bending in purely plastic domain4

2bTM s

4

2bTnM b

2. Bending forces

Aim: press choosing and die design

Influence factors: material strength, die opening, length and thickness of the piece, bend profile, bending way

2.1 Forces for U-dieFree bending:

Bending with reinforcement:

F=Ap

bp tr

BtF

2

7.0

2. Bending forces

2.3 Forces for V-die

Free bending:

Bending with reinforcement: F=Ap

bp tr

BtF

2

6.0

3. Equipment choosing

General rule:

free bend Fp ≧ Fa + Fe

coin bend Fp ≧ Fc

Load-punch travel curve for coin bending

§4.4 Bend allowance

3.1 Bend allowanceDefinition: length of the arc of the neutral bending line for a

given degree value.Formula:

3.2 Pre-bend length L=L1+Ln+L2

180

nL

§4.5 Springback

1. Definition

2. Calculation

2.1 10< r / t <100

2.2 r / t <3~5

2.3 bending with reinforcement

3. Affecting factors

4. Methods to decrease the spring back of bending

1. Definition

Every elastic deformation is followed by elastic recovery.

The final angle after springback is samller

The final bend radius is larger

The bend allowance of the

neutral line is same.

if

if RR

2. Calculation

.2.1 10< r / t <100

both Δα and ΔK must be took into consideration

2.2 r / t <3~5

ΔK↓↓so only Δα is took into consideration

2.3 bending with reinforcement

Spring back is very small, can be neglected.

, ,

EtK s3

f

f

rK

rr

1 1 K

3. Affecting factors

1) Material property: σs ↑E ↓→Δα and ΔK↑(σ~ε curve)

2) Deformation degree: r/t↓→Δα and ΔK↓(σ~ε curve)

3) Bending angle: α↑→Δα↑4) Bending way: free bending Δα and ΔK↑;bottoming (coining the bend area) Δα and ΔK↑5) Shape of bends: single bend V↑, double bend or multiple

bend U↓6) State of working (friction between die and sheet) μ↑Δα and

ΔK↑

3. Affecting factors

Loading and unloading curve for bending

4. Methods to decrease the spring back of bending

According to the affecting factors to determine the

way of decreasing the value of spring back.

1)  workpiece: material--σs↓,E↑; structure—rib

2)  bending way:(bottoming instead of free bending)

3) compensation: reverse of Δα and ΔK(negative springback)

4)  stretch bending: subjected to tension

§4.6 The design of bending punch and die

1. Die radius

2. Clearance: c=tmax-0.1

3. Die width

4. Bendability analysis (handbook, p124) Homework: P71 No.3 and 1 supplement

1. Die radius

rp=r>rmin

rd >3mm

3. Die width

dimension mark (outside or inside)

dLbd

)2

( dLbd

)4

3(

pLbp

)

2(

pLbp

)

4

3(

a)

d)c)

b)

Homework

P71 No.3 (Chinese text book) 1 supplement

Fig.1 Single bend

A Bending work piece is shown in Fig.1. Known: material 20 carbon steel, thickness

t=1.2mm, width of bend b=20mm.(1) Calculate the blank length before

bending and the springback value when it is not bottom sized.

(2) Design the radius and angle of bending punch.

(3) Calculate the bending force needed in two conditions: free bending and bottom sizing.

Homework