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Theoretical Relationships
Opening Stress

The greatest stressing of the ring occurs during assembly, as the ring must be stretched open sufficiently to enable the inner contour to slip over the outer diameter of the piston.

The standard formula for the opening stress σ found in the literature makes the assumption that the closure stress and opening stress are equal. This assumption is only true for a particular value of the total free gap m and a given d/a relationship. Use of the formula for any other value of m can lead to significant errors.
When determining the ring dimensions given in the DIN/ISO piston ring standards a new formula for calculating the opening stress was used which is valid for all values of m and also makes allowance for assembly by means of a sleeve. However, it is rather complicated to use.

Based on the mathematically exact relationship
 (11)

σ = the stress in the outermost element
ρ1 = radius of curvature of the unstressed ring
ρ2 = radius of curvature of the stressed ring, measured at the same point

a more convenient expression for the opening stress was derived, since the solution to the mathematically exact relationship would be too complicated.
The constants in equations (12) to (15) were determined by means of computed tables and assure a high degree of conformity between the equations in the stated form and the exact solution:

The following applies to rings of rectangular cross section:
1. Opening the piston ring solely under tangential force: (12)
 (12)

 (12a)

and k = piston ring parameter (equation 10)

2. Opening the piston ring by means of a sleeve:
 (13)

 (13a)

k = piston ring parameter (equation 10)
dH = diameter of sleeve

The following applies to slotted oil control rings:
1. Opening the piston ring solely under tangential force:
 (14)

 (12a)

2. Opening the piston ring by means of a sleeve:
 (15)

 (13a)

lu = moment of inertia of the unslotted cross section
ls = moment of inertia of the slotted cross section
lm =
x2 = twice the distance between centre of gravity and inside diameter
k = piston ring parameter (equation 10)
dH = diameter of sleeve

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