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 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

§4.1 Deformation process of bending Typical examples of sheet-metal bend parts 1. Deformation feature 2. Stress and strain state in deformation zone 3. Deformation degree 4. Defects

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

1. Deformation feature 1.4 deformation feature cross section: b/t >3 constant b/t <3 change(from rectangle to sector)

2. Stress and strain state in deformation zone 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

3. Deformation degree 4. Defects r/t, spring back: elastic deformation fracture: outer fiber thickness decreasing: length increasing : dV=0

§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)

no fracture occurring limit radius 2.2 Theoretical way 2. Minimum bend radius 2.1 Definition no fracture occurring limit radius 2.2 Theoretical way 2.3 Experimental way rmin=c.t

(1) Material’s plasticity 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. The calculation of principal stress in bending of wide plate 3.1 3-D pure plastic bending (no hardening) r/t <3~5 and b/t>3 three unknowns need three equations method: principal stress method assumption: 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 Inner Outer

3. The calculation of principal stress in bending of wide plate 3.2 3-D pure plastic bending (hardening) so yield condition

4. The position of neutral curvature of stress Fibre where stress discrete , From , obtain When r / t ↑ , when r / t ↓

§4.3 The calculation of bending moment and bending force Moment of bending 1.1 Moment of bending in elastic-plastic domain 1.2 Moment of bending in purely plastic domain 2. Bending forces 2.1 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 domain

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-die Free bending: Bending with reinforcement: F=Ap

2. Bending forces 2.3 Forces for V-die Free bending: Bending with reinforcement: F=Ap

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 allowance 3.2 Pre-bend length Definition: length of the arc of the neutral bending line for a given degree value. Formula: 3.2 Pre-bend length L=L1+Ln+L2

§4.5 Springback 1. Definition 2. Calculation 2.1 10< r / t <100 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.

2. Calculation 2.1 10< r / t <100 2.2 r / t <3~5 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. , ,

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)

Homework P71 No.3 (Chinese text book) 1 supplement Fig.1 Single bend

Homework 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.