Basic Theory of Measuring Stress Concentration
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
RUAG Aerospace Services GmbH, Germany
Mechanical and Physical Journal
of the “Collegium” All World Academy of Sciences
Munich (Germany)
5 (2005), 1
Directly measuring stresses at zones of their concentration [1] by means of real physical devices with nonzero sizes such as strain gages leads to dangerously underestimating maximum stresses because of averaging the experimental data within measuring elements of the devices.
The purpose of the present theory is to reliably define true maximum stresses at their concentration zones. This purpose is achieved by designing and applying adequately modeling stress and strain states in typical elasticity problems on stress concentration. The distinctive feature of the proposed theory to measure stress concentrations is using the following new calculation formulas to determine true maximum stresses at concentration. The obtained experimental stress should be multiplied by the corresponding factor determined by solving the corresponding elasticity problem.
The Kirsch problem [2, 3] is determining stress concentration on the boundary of a round hole in a theoretically infinite plate extended in one direction x at infinity. The maximum stress holds on two generatrixes of the hole with radius r. But a strain gage best placed on the surface of the hole has a measuring mesh of a nonzero length 2rl (via introducing the corresponding number l) on which averaging takes place. To determine the true maximum stress, the measured stress has to be multiplied by the factor [4, 5]
K = 3/(1 + l-1sin2l).
Let two stresses σx , σy extend the plate in orthogonal directions x and y, and a uniform pressure p hold in the hole. By considering the Poisson ratio μof the plate material and the transverse sensitivity μt of the measuring mesh of the device at the same place, the factor is [4]
K = [(1 + 2μ)p + 3σy - σx)]/
{(1 + 2μ- μt)p + (1 - μμt)×
[σx + σy - l-1sin2l (σx - σy)]}.
If the measuring mesh with width βr is placed on a side of a plate with distance δr (δ a number) from the hole by x as a mesh symmetry axis, the factor is [4]
K = [(1 + 2μ)p + 3σy - σx)]/{μ(1 + μt)p +
(μt - μ)σx + (1 - μμt)σy + [(1 + μ)(1 - μt)p +
0.5(-1 + 3μ- 3μt + μμt)σx + 0.5(3 - μ +
3μt - 3μμt)σy]/(βl)[arctan l(1 + δ)-1 -
arctan l(1 + δ + β)-1)] + (1 + μ)(1 - μt)(σx - σy)/β ×
[(1 + δ)((1 + δ)2 + l2)-1 - (1 + δ + β)((1 + δ + β)2 + l2)-1 -
1.5(1 + δ)((1 + δ)2 + l2)-2 + 0.75(1 + (1 + δ + β)2(1 + δ)-2)×
(1 + δ + β)((1 + δ + β)2 + l2)-2 + (1 + δ)((1 + δ)2 + l2)-2 -
(1 + δ)2 + l2)(1 + δ)-2(1 + δ + β)3((1 + δ + β)2 + l2)-3 -
0.5β(1 + δ + 0.5β)(1 + δ + β)(3(1 + δ + β)2 - l2)(1 + δ)-2 ×
((1 + δ + β)2 + l2)-3]}.
For a round plate of radius r supported on the boundary and loaded by a one-side uniform pressure, for the true greatest stress at the center of the opposite side of the plate, the factor is [4]
K = 1/(1 - l2).
If the same plate is fixed on the boundary, the factor is [4]
K = 1/[1 - (1 + μ)/(3 + μ)l2].
For an infinitely long cylindrical shell with radius a and thickness h fixed on one edge, by measuring the greatest axial stress on the internal surface at the edge, the factor is [4]
K = kl exp kl /sin kl
where
k = [3(1 - μ2)]1/4/(ah)1/2.
For other types of stress concentration, the corresponding elasticity problems should be preliminarily solved. The proposed means of measuring maximum stresses at stress concentration zones apply to many problems in measurement technology and are especially useful and even urgent in modern engineering with extreme loading conditions.
[1] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998
[2] Kirsch G. Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. VDI Z., 42 (1898), 797-807
[3] Peterson's Stress Concentration Factors (2nd Edition). By: Pilkey, Walter D. John Wiley & Sons, 199
[4] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich (Germany), 2004
[5] Lev Gelimson. Equivalent Stress Concentration Factor. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to March 2005, Dr. Claudio Dalle Donne, EADS Corporate Research Center Germany, SC/IRT/LG-MT–2005-039 Technical Report, 30-32