TABLE OF FACTORS AND TERMS FOR BENDING FORMULAS

B = degree of bend

E = feathered edge thickness

F_b = bend difficulty factor

F_d = "D" of bend

F_w = wall factor

K_r = constant for rigidity

K_s = constant for minimum clamp length

K_z = constant for feathered edge

L_c = clamp length

L_p = pressure die length

M_b = mandrel ball diameter

M_d = mandrel nose diameter

M_m = mandrel body diameter

M_r = mandrel nose radius

P_e = percentage of elongation at arc

P_t = percentage of wall-thinning

P_w = wall thickness after thinning

R = centerline radius

R_i = inside radius

R_o = outside radius

S = maximum set-up depth

T = tube outside diameter

T_i = tube inside diameter

W = wall thickness

W_i = thickness of inside lamination

W_o = thickness of outside lamination

Tube inside diameter:

T_i = T – ( W x 2 )

Inside radius:

R_i = R – ( T / 2 )

Outside radius:

R_o = R + ( T / 2 )

Wall factor:

F_w = T / W

“D” of bend:

F_d = R / T

Bend difficulty rating (the higher the value, the more difficult the bend is to make; rule of thumb only):

Where “K_r” = a constant for material rigidity (assign the same value to “K_r” as you would to calculate pressure die length; a value of 2 is suitable for most applications; click here for more information) and “n₁” through “n₄” are values to adjust the weight of each factor in the equation (see below for our recommended weighting):

General formula:  F_b = [ ( n₁ x K_r ) + ( n₂ x F_w ) + ( ( n₃ x B ) / 180 ) ) ] / [ n₄ x F_d ]

Formula with recommended weighting:  F_b = [ 2K_r + .2F_w +  ( B / 180 ) ] / [ F_d ]

Note: A bend difficulty rating (calculated with our recommended weighting) of 7 or less indicates a bend that is relatively simple to produce with the rotary-draw method.  Factors in excess of 7 typically require either additional precision in set-up or close attention during production in order to hold the set-up parameters.

Wall-thinning of extrados at outside radius after bending (rule of thumb only):

Where “P_t” = percentage of wall-thinning and “P_w” = targeted thickness of wall after thinning out from bending:

P_t = ( R_o – R ) / R_o

P_w = W x ( 1 – P_t )

Percentage of elongation at arc of the bend (rule of thumb only):

P_e = ( R_o / R ) – 1

Mandrel nose diameter for single-wall tubing:

M_d = T – ( W x 2.21 )

Mandrel nose diameter for double-wall tubing:

Where “W_o” = wall thickness of outside lamination and “W_i” = wall thickness of inside lamination:

M_d = ( T – ( W_o x 2 ) ) – ( W_i x 2.21 )

Mandrel nose radius:

if F_w < 50 then M_r = M_d x .1 else M_r = M_d x .02

Mandrel body diameter:

M_m = M_d x .995

Mandrel ball diameter:

M_b = M_d x .998

Maximum set-up depth of mandrel nose relative to the line of tangency, as measured from nose end (including nose radius):

S = [ ( R + ( T / 2) – W )² – ( R + ( M_d / 2 ) )² ]^1/2 + M_r

Wiper feathered edge thickness (simple-sweep geometry only):

Where “K_z” = a constant approaching zero depending upon limitations of material and method of manufacturing (with current technology, a value of .0025 is reasonable for “K_z”):

if T x K_z > .006* then E = T x K_z else E = .006*

* Inches.  For metric applications, substitute .15 millimeters.

Clamp length:

Where “K_r” = a constant for material rigidity (assign a value of 2 to “K_r” for most applications; click here for more information) and “K_s” = a constant limiting the minimum clamp length depending upon the surface of the cavity (assign to “K_s” the value of 2 for smooth cavities and 1 for serrated cavities; click here for more information):

if ( T x ( K_r x 2.5) ) – R < T x K_s then L_c = T x K_s else L_c = ( T x ( K_r x 2.5) ) – R

Pressure die length:

Where “K_r” = a constant for material rigidity (assign a value of 2 to “K_r” for most applications; click here for more information):

L_p = ( R x 3.14 x ( B / 180 ) ) + ( T x K_r )

Springback and radial growth:

We are frequently asked for formulas to calculate springback and radial growth.  While there are rules of thumb -- e.g., a radius will increase 1% for every "D" of bend -- they are not effective, as a true formula would be, in reducing the prove-out needed to lock in the parameters of a machine set-up.

Unfortunately, effective formulas for springback and radial growth have not been developed, because the factors involved include not only tube and bend specifications but also machine settings -- especially the radial pressure and axial pressure applied by the pressure die to the tube and the placement of the mandrel nose relative to the line of tangency.  How an operator sets these things on a particular make and model of machine alters where the neutral axis of a tube bend lies in relationship to the centerline of the radius, and it is the location of the neutral axis that determines how much springback and radial growth there will be.  Moreover, springback and radial growth are the result of fundamentally non-linear processes, which would make any effective formula that does account for all these factors fairly complex.  Presently, finite element analysis (FEA) is the only tool up to this task, and it is not yet practical for everyday use in the bend shop.

Fortunately, the trial-and-error needed to adjust for springback and radial growth does not have to be repeated for every set-up of a tube bend.  By using the "Four-Step Set-Up Method" to employ the "forward mandrel, low pressure" set-up for rotary-draw tube-bending, the parameters of a successful set-up can be recorded and then duplicated with little or no trial-and-error to prove out future set-ups of the same or similar tube bends.