“f0” FOR PLASTIC BENDING CALCULATION METHOD

(FOR COZZONE METHOD)

TOSHIHIRO TANABE

"Cozzone method" is very useful method for plastic bending analysis. But

material properties “f0” is not shown in MMPDS or other material properties

data. Some material “f0” stress is shown in “ANALYSIS & DESIGN of FLIGHT

VEHICLE STRUCTURES” by Bruhn. But new material has not in there. Therefore,

we have to know how to get “f0” stress for new materials.

This report is shown “f0” stress calculate equation and how to create that

equation.

“f0” calculation equation:

Hence

fm: Maximum stress. (Ultimate: fm = Ftu, Limit: fm = Fty or Fcy)

: Maximum strain at maximum stress.

(Ultimate: elongation from MMPDS, Limit: )

’: Inelastic or plastic strain response.

(Ultimate: , Limit: 0.002)

E: Modulus of elasticity (from MMPDS)

n: Ramberg-Osgood number. (from MMPDS)

Reference

1) ANALYSIS & DESIGN of FLIGHT VEHICLE STRUCTURES; Bruhn

2) MMPDS (Metallic Materials Properties Development and Standardization)

3) Bending Strength in Plastic Range. By F.P.Cozzone, J.Aeronautical

Sci., May, 1943

4) NACA-TN-902; DESCRIPTION OF STRESS-STRAIN CURVES BY THREE

PARAMETERS BY Walter Ramberg and William R. Osgood

How to create “f0” equation

The Cozzone method is replaces the true bending variation curve by a

trapezoidal stress variation. The stress “f0” is fictions stress which is assumed

exist at the neutral axis or at zero strain. The value of “f0” is determined by

making the requirement that the internal moment of the true stress variation

must equal the moment of the assumed trapezoidal stress variation.

Equation “f0” strain

EQ. (1)

(Rectangle area moment) (Triangle area moment)

EQ. (2)

From equation (1)

Therefore

EQ. (3)

From equation (1) & (3)

FMAX = fm

FMAX = fm

N.A.

fm

fm

N.A.f0

f0

fm

f00

32

2

EQ. (4)

From equation (3)

EQ. (5)

From equations (2) & (5)

Therefore

Equation check result

Comparison of Reference(1) “f0” values and equation results.

Ref.(1)

Fig NoMaterial

Thickness

in

Ftu

ksiFty

E ×103

ksi

e

%

f0u

ksi

f0y

ksi

Calc. value

f0u

ksi

f0y

ksi

C3.9 2024-T3C SHT 0.01~0.062 59 39 9.5 15 50 18 50 16

C3.11 2024-T4C PLT 0.25~0.50 62 40 10.0 15 55 20 52 17

C3.13 2024-T3 SHT ~0.25 64 42 10.5 15 53 23 54 18

C3.15 7075-T6C SHT ~0.39 70 60 10.0 5 64 24 62 22

C3.17 7075-T6 EXT ~0.25 75 65 10.4 7 70 29 68 24

C3.16 7075-T6 SHT ~0.039 76 65 10.3 7 70 24 69 24

Calculated values are almost same as Reference 1) “f0” values.