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Alert on Grape FEA

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proppastie

proppastie

Well-Known Member
Joined
Feb 19, 2012
Messages
6,421
Location
NJ
I was recently made aware of possible problems using Grape FEA. The T recovery and J inertia must be calculated for non circular sections and input manually. As has been always mentioned here using automated methods without an understanding of the basic principals can get you in trouble. From Grape instructions manual below.

J Inertia


The polar moment of inertia as specified in the geometry property.

For a circular section, J is the polar moment of inertia given by the formula

( P i x R^4 ) / 2.

Radius R is the T Recovery distance

For a non circular section, GBW32 calculates the torsional shear stress using the formula

SHEAR STRESS = ( Torque x radius ) / J

and thus J is the torsional resistance of the non circular section referred to as K in several references such as Roark & Young ( see also Bibliography, Gieck, K and Gieck, R.). For a non circular section T-Recovery must be calculated.

T Recovery


The distance or radius to the outer most fibre for a circular section as specified in the geometry property. For a non circular section, use the following steps to determine the T-Recovery value.

a) There is an infinite number of possible non circular cross-sections and thus an infinite number of shear stress formulas that are appropriate for each section.

b) GBW32 uses only the formula

SHEAR STRESS = ( Torque x radius ) / J

to calculate the torsional shear stress. Units of stress for example are psi or Pascals.

c) Find an appropriate formula from a reference such as Roark & Young for the maximum shear stress desired. See Bibliography.

d) Set the equation in b) equal to the equation from the reference. The torque values cancel and you can solve for T-Recovery. This T-Recovery used in the standard torsional shear stress formula given in b) will give the required shear stresses due to torsion.

e) Note that the K value, the torsional resistance, should coincide with the standard formula for angle of twist given by

THETA = ( torque x length ) / ( G x K )

where G = the Modulus of Rigidity and THETA is in radians.


For example, let's calculate the t-recovery distance for an equilateral triangular section with each side 2 inch long. The torsional constant K is given by the formula

a^4 / 46.19 ( see Gieck in bibliography ).

Thus the torsional resistance is 2x2x2x2 / 46.19 = 0.3464 inch^4. The maximum torsional shear stress at the midpoint of any of the 3 sides of the section is given by

20 x T / a^3 where T = torque.

Since GBW32 uses the standard formula, T x t-recovery / K to calculate all shear stresses due to torque, we equate the two formulas as

T x t-recovery / ( a^4 / 46.19 ) = 20 x T / a^3.

Note that the torques T cancel and we solve for t-recovery

t-recovery = 20 x a / 46.19 ( the a^3 cancel ).

Thus t-recovery = 20 x 2 / 46.19 = 0.866 inch. This is the value entered for t-recovery of a 2 inch equilateral triangle cross section in the geometry database.



 
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