New Formulas for Computing Incomplete Elliptic Integrals of the First and Second Kind

1959 ◽  
Vol 6 (4) ◽  
pp. 515-526 ◽  
Author(s):  
A. R. DiDonato ◽  
A. V. Hershey
Keyword(s):  
Elliptic Integrals ◽  
Incomplete Elliptic Integrals

The Incomplete Elliptic Integrals F(k,ϕ) AND E(k,ϕ)

2008 ◽  
pp. 653-669
Author(s):  
Keith B. Oldham ◽  
Jan C. Myland ◽  
Jerome Spanier
Keyword(s):  
Elliptic Integrals ◽  
Incomplete Elliptic Integrals

Calculated Deflected Contours and Load-Deflection Curves for Tires

1968 ◽  
Vol 41 (4) ◽  
pp. 977-987
Author(s):  
S. D. Gehman
Keyword(s):  
Approximation Method ◽  
Elliptic Integrals ◽  
Deflection Curve ◽  
Load Axis ◽  
Incomplete Elliptic Integrals ◽  
Load Deflection Curve

Abstract The tire contour equation is derived for a flexible tire body with radial cords and a complete circumferential deflection, corresponding to the case of a radial-ply tire or a tire inflated inside a constraining cylinder. Equations are derived to calculate the load on a single cord in the deflected region using parameters of the un-deflected contour. It is then shown how the reasoning and equations can be generalized for a tire body with any cord path having a complete circumferential deflection so that the load-deflection curve for a single cord can be calculated. A new approximation method is described so that the integrals involved in these calculations can be evaluated in terms of incomplete elliptic integrals of the first and second kind with any desired accuracy. Finally, a procedure is given for summing cord loads in the usual spot deflection of a tire so that the load-deflection curve for a flexible tire body can be calculated from contour parameters of the undeflected tire. An illustrative calculation is included. Although the load-deflection curve for a single cord is convex toward the load axis, that for the tire is concave because more cords are involved as deflection progresses. A calculated contour does not exist beyond a limiting deflection, at which, presumably, buckling starts above the bead.

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