Arborless Bender

Analytical Model

Arborless Bender Geometry

α

α

e

a

R1

u

L0

a

e

R1

u

α

a

R1

u

L0

R1

u

Clamp

Clamp

Swingarm

Pivot

Bend Radius

CoilInitial Coil Centerline

Offset

Bend Angle αa

Fixed ArmAdjust

Adjust

Diagram - top view Corresponding geometry

Arborless Bender Geometry

α

α

e

a

R1

u

L0

a

e

R1

u

α

a

R1

u

L0

R1

u

Clamp

Clamp

Swingarm

Pivot

Bend Radius

CoilInitial Coil Centerline

Offset

Bend Angle αa

Fixed ArmAdjust

Adjust

Diagram - top view Corresponding geometry

Constraint: Keep the arc length of the coil constant throughout the bend

Rf = Final Inside Bend Radiusf = Final Bend AngleR1= Intermed. Bend Radius = Intermed. Bend Angleu = Neutral Axis Offset

Neutral Axis LengthInitial: L0 = Rf f

Intermediate: L0 = (R1 +u)Bend Radius

Neutral Axis: (R1 + u) = (Rf f)/

Inside: R1 = (Rf f)/ – u

α

α

e

a

R1

u

L0

a

e

R1

u

Bend Geometry

Clamp Adjustment

Bend Geometrye = (R1 +u) tan(/2)a = R1 tan(/2)

a = [(Rf f)/ – u] tan(/2)

Clamp AdjustmentInitial: a0 = e0 = (Rf f)/2a = a – a0

a = [(Rf f)/– u] tan(/2) – (Rf f)/2

α

α

e

a

R1

u

L0

a

e

R1

u

Clamp Position Adjustment vs. Bend Angle InputArborless Bending Setup Outputby Dr Ken Chase, BYU ME Dept HeadingsJuly 2008

Operator Input Calculation of Adjusted Clamp Position vs Angle for a 90 Deg BendConduit Data Inputs Setup Angle-deg Angle-rad Bend Radius Clamp Position Clamp Adjust

OD 0.5 in in Alpha Alpha R1 a1 a1-a0Wall thick 0.042 in Length betw clamps 0 0 Infinity 3.14159 0

E 3.00E+07 psi 6.28319 10 0.17453 35.37500 3.09491 -0.04668Nu 0.282 in 20 0.34907 17.37500 3.06368 -0.07791Sy 45 kpsi Final Length of NA 30 0.52360 11.37500 3.04792 -0.09367Su 48 kpsi 6.28319 40 0.69813 8.37500 3.04825 -0.09334

45 0.78540 7.37500 3.05483 -0.08677Bend Specs Offset=dist from hinge 50 0.87266 6.57500 3.06597 -0.07562

Bend Angle 60 in pin to neutral axis 60 1.04720 5.37500 3.10326 -0.03833Inside Rad 5.375 in 70 1.22173 4.51786 3.16344 0.02184

Offset 0.625 Deg Mean radius= inside 80 1.39626 3.87500 3.25151 0.10992radius + offset 90 1.57080 3.37500 3.37500 0.23341

Mean Rad 6.000 in

Clamp Adjustment Spreadsheet

in Tables - Calculation of Adjusted Clamp Position vs Offset, Bend Radius and Bend Angledeg

Mean Radius=4.0 Thetaf=90 Adjustment vs OffsetAlpha Angle-rad u=0 u=0.5 u=0.625 u=1.0 u=1.25 u=1.5

0 0 0 0 0 0 0 010 0.17453 0.00800 -0.03575 -0.04668 -0.07949 -0.10136 -0.1232320 0.34907 0.03229 -0.05587 -0.07791 -0.14403 -0.18812 -0.2322030 0.52360 0.07380 -0.06018 -0.09367 -0.19415 -0.26114 -0.3281340 0.69813 0.13414 -0.04785 -0.09334 -0.22983 -0.32082 -0.4118245 0.78540 0.17212 -0.03499 -0.08677 -0.24210 -0.34565 -0.4492050 0.87266 0.21582 -0.01733 -0.07562 -0.25049 -0.36706 -0.4836460 1.04720 0.32251 0.03383 -0.03833 -0.25484 -0.39918 -0.5435270 1.22173 0.45947 0.10937 0.02184 -0.24073 -0.41578 -0.5908480 1.39626 0.63436 0.21481 0.10992 -0.20474 -0.41452 -0.62429

Rf >> 90 1.57080 0.85841 0.35841 0.23341 -0.14159 -0.39159 -0.64159

Offset u=0.625 Thetaf=90 Adjustment vs Final Bend RadiusAlpha Angle-rad Rf=4 Rf=5 Rf=6 Rf=7 Rf=8 Rf=9

0 0 0 0 0 0 0 010 0.17453 -0.04668 -0.04468 -0.04268 -0.04068 -0.03868 -0.0366820 0.34907 -0.07791 -0.06984 -0.06176 -0.05369 -0.04562 -0.0375530 0.52360 -0.09367 -0.07522 -0.05677 -0.03832 -0.01987 -0.0014240 0.69813 -0.09334 -0.05981 -0.02627 0.00726 0.04080 0.0743345 0.78540 -0.08677 -0.04374 -0.00071 0.04232 0.08535 0.1283850 0.87266 -0.07562 -0.02166 0.03229 0.08625 0.14020 0.1941660 1.04720 -0.03833 0.04229 0.12292 0.20355 0.28417 0.3648070 1.22173 0.02184 0.13671 0.25158 0.36645 0.48132 0.5961980 1.39626 0.10992 0.26851 0.42710 0.58569 0.74427 0.90286

Rf >> 90 1.57080 0.23341 0.44801 0.66261 0.87721 1.09181 1.30642Setup: Length betw clamps 6.28319 7.85398 9.42478 10.99557 12.56637 14.13717

Length of conduit 12.28319 13.85398 15.42478 16.99557 18.56637 20.13717

Offset u=0.625 Rf=6 Adjustment vs Final Bend AngleAlpha Angle-rad Thetaf=30 Thetaf=45 Thetaf=60 Thetaf=90

0 0 0 0 0 010 0.17453 -0.05068 -0.04868 -0.04668 -0.0426820 0.34907 -0.09406 -0.08598 -0.07791 -0.0617630 0.52360 -0.13057 -0.11212 -0.09367 -0.0567740 0.69813 -0.12688 -0.09334 -0.0262745 0.78540 -0.12980 -0.08677 -0.0007150 0.87266 -0.07562 0.0322960 1.04720 -0.03833 0.1229270 1.22173 0.2515880 1.39626 0.4271090 1.57080 0.66261

Setup: Length betw clamps 3.14159 4.71239 6.28319 9.42478Length of conduit 9.14159 10.71239 12.28319 15.42478

Offset

Bend Radius

Bend Angle

Calculated Clamp Adjustment vs Bend Specifications

(1/2 Coil Thickness)

(Inside)

Moment-Curvature Relationship

Finding the moment at the onset of yielding

( )

Y

Y

Y

S

b h

h

b hS

M

62

1 223

==

c

IS

MY

Y

=

I

cM

SY

Y==

m a xσ

I

cM

SY

Y==

m a xσ

General Case For a Solid Rectangular Bar

Estimating Springback

Decrease in curvature due to springback is

Where R0 is the forming radius, R1radius at the first onset of yielding, and K is the ratio of limit bending to bending moment at the onset of yielding.

YS

RR⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=−

ρ

1

2

311

10

hE

S

c

YY

SY

21

==⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ε

ρ

YS

K

RR⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=−

ρ

111

10

General Case For a Solid Rectangular Bar

Springback Calculation SpreadsheetInput

Output

€

1R0

− 1R1

=KSY

Ec

€

R1 =R0Ec

Ec−KR0SY