Standard Stress Methods Doc: AA-SM-XX Revision: A Material Properties Page: 1 C.1 Derivation of Plastic Buckling Allowables C.1.1 Approximation of Stress/Strain Curves Eqn: C.1 Eqn: C.2 Eqn: C.3 Eqn: C.4 Eqn: C.5 In order for material plasticity effects on buckling behaviour to be derived the stress strain curve must be created. For this analysis the Ramberg Osgood method as defined in the Mil-Hdbk- This expression is valid for tension and compression. For Shear stress strain this expression can be adapted to the material shear behaviour in the following way The effect of material plasticity of buckling behaviour will be modelled using the following expressions. For shear and compression buckling the elastic allowable was derived using the method in NACA technical Note 3781, Equation 1, general buckling stress equation: A modification will be made to the shape factor in the plastic region to more accurately model the stress/strain curve: This equation can be derived from the figure on Page 9.79 of Mil-Hdbk-5H. This equation can be adapted for shear stress in the same manner as the Ramberg Osgood Stress/Strain Approximation:
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Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 1
C.1 Derivation of Plastic Buckling Allowables
C.1.1 Approximation of Stress/Strain Curves
Eqn: C.1
Eqn: C.2
Eqn: C.3
Eqn: C.4
Eqn: C.5
In order for material plasticity effects on buckling behaviour to be derived the stress strain curve must be created. For this analysis the Ramberg Osgood method as defined in the Mil-Hdbk-5H equation 9.3.2.4(b) will be used
This expression is valid for tension and compression. For Shear stress strain this expression can be adapted to the material shear behaviour in the following way
The effect of material plasticity of buckling behaviour will be modelled using the following expressions.
For shear and compression buckling the elastic allowable was derived using the method in NACA technical Note 3781, Equation 1, general buckling stress equation:
A modification will be made to the shape factor in the plastic region to more accurately model the stress/strain curve:
This equation can be derived from the figure on Page 9.79 of Mil-Hdbk-5H.This equation can be adapted for shear stress in the same manner as the Ramberg Osgood Stress/Strain Approximation:
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 2
C.1.2 Shear Buckling Correction for Material Plasticity effects
Eqn: C.6
Eqn: C.7
C.1.3
Eqn: C.8
Where Gs, the Secant Shear Modulus, is:
Other derivations used for Shear Buckling Data:
Shear Shape Factor (n)
The shear stress/strain curve shape factor can be derived from the known tension and compression shape factor using the following expression:
Where:ns = Shear Shape FactorncL =Compression shape factor in the longitudinal grain directionncLT = Compression shape factor in the longitudinal transverse grain directionntL = Tension shape factor in the longitudinal grain directionntLT = Tension shape factor in the longitudinal transverse grain direction
)
Gs
G
NACA recommends to use a shear plasticity factor based on a plastically corrected poissons ratio of Es/E. Bruhn chapter C5 recommends to use a Gs/G uncorrected for poissons ratio. As the shear stiffness of the material is based on the shear modulus G (even though the allowable stress is related to E - the experimentally determined ks factor accounts for this), and the poissons ratio correction is used in the elastic allowable calculation, it was decided to use a plasticity correction factor as follows:
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 3
Shear Yield Stress
the shear yield stress was estimated using equation 9.3.2.6(c) from Ref 1.1
Eqn: C.9
C.1.4 Compression Panel Buckling Correction for Material Plasticity effects
Eqn: C.10
Eqn: C.11
Eqn: C.12
Where Et, the tangent modulus, is
Where Es, the Secant Modulus, is:
Where ve, the plastic poissions ratio, is: (Ref NACA TN 3781 Equation A1)
There is some difficulty selecting a compression allowable plasticity correction factor. Most Aircraft webs and skins do not experience pure compression. Different panels usually experience either pure bending or a combination of bending and compression or bending and tension. Choosing a single plastic correction factor is not possible.After consideration it was decided to use NACA TN 3781 Equation A3 as it fitted closest the physical situation for most of the spar and would be slightly conservative for situations that deviate from pure compression. (Bruhn Fig C5.8)
ν=ν pl−(ν pl−νe )( EsE )
Eqn: C.13
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 4
C.1.5 Compression Flange Buckling Correction for Material Plasticity effects
Eqn: C.14
C.1.6 Column Buckling Correction Factor for Material Plasticity effects
Eqn: C.15
The plasticity correction factor for a compression flange will be taken from Bruhn Fig C5.7,
This plasticity correction factor depends on the Secant Modulus, this is defined by Eqn: C.12
This plasticity correction factor uses the poissons ratio correction factor in the plastic region, this is defined by Eqn: C.13
Bruhn section C2.4 gives a method for producing a correction for material plastcity effects based on the tangent modulus. Therefore the following plasticity correction factor will be used:
Et
E
In addition, the plastically corrected inter-rivet buckling allowable curve can be produced. For a rectangular section (Which is implicitly assumed in the inter rivet buckling method) a contant relationship between L/Rho and s/t exists of the square root of 12 (3.464).However where the fixity coefficient C does not equal 4 a correction factor of (4^½)/(C^½) must be applied.
ν=ν pl−(ν pl−νe )( EsE )
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 5
C.2 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in(Area <20in^2)
LOCATION TITLE
Mil-Hndbk-5F, change notice 2, Table 3.7.6.0 (g1):All Strength Data is A-basis
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 8
C.2.2 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Shear Buckling (continued)LOCATION TITLE
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
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Material Properties Page: 9
C.2.3 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Compressive Plate BucklingLOCATION TITLE
Plastic ElasticCompression Ramberg Tangent Secant v Plasticity Compression
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 10
C.2.3 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Compressive Plate Buckling (continued)LOCATION TITLE
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 11
C.2.4 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Compressive Flange BucklingLOCATION TITLE
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 12
C.2.4 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Compressive Flange Buckling (continued)LOCATION TITLE
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 13
C.2.5 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Column and Inter-Rivet BucklingLOCATION TITLE
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 14
C.2.5 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Column and Inter-Rivet Buckling (continued)LOCATION TITLE
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
80000
Elastic Euler Column Allow-ableTangent Modulus Based Column Al-lowable
Johnson Column Allowable based on Fcy
L/Rho
Col
umn
failu
re S
tres
s (P
si)
A490
R Abbott: Fcy *.5
Figure: C.2.4
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________
Standard Stress Methods Doc: AA-SM-XX
Revision: A
Material Properties Page: 15
C.2.5 Material Properties for Extruded 7075-T6511 (QQ-A-250/11) - .750/1.499in (Continued)Plastic Correction for Column and Inter-Rivet Buckling (continued)LOCATION TITLE
Note that the higher column allowables are likely to be limited by section crippling, therefore the above graph is only applicable for columns with stable cross sections.
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
80000
Elastic Euler Column Allow-ableTangent Modulus Based Column Al-lowable
Johnson Column Allowable based on Fcy
L/RhoC
olum
n fa
ilure
Str
ess
(Psi
)
0 5 10 15 20 25 30 35 400
10000
20000
30000
40000
50000
60000
70000
80000
s/t
Inte
r-R
ivet
Bucklin
g S
tress (
Psi)
Figure: C.2.5
s = fastener pitcht = sheet thickness
Prepared by: R. Abbott Date: Jan 2008 Checked by:___________________Date:__________