by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

RUAG Aerospace Services GmbH, Germany

Strength and Engineering Journal

of the “Collegium” All World Academy of Sciences

Munich (Germany)

6 (2006), 2

The present work is dedicated to further extending general bearing strength theory [1] by introducing washers between sheets and fasteners, e.g., by corrosion in the plate about the hole, is removing some ring a ≤ r ≤ b (see the following figure) with most corrosion, removing the remaining corrosion by r ≥ b, and then placing a new, specially manufactured ring a ≤ r ≤b with the initial thickness t_max :

In this case, the stress states at the both points (0, a) of the washer and (0, b) of the hole should be considered. Analogously to [1], in the washer, the maximum shear stress

τ_a = [F / (2at) * 2at - F / (2bt) * 2at] / [(b² – a²)^1/2 t] = F/(bt) * [(b - a)/(b + a)]^1/2.

However, such values of τ can hold by x = a and x = -a but not by x = 0. Therefore, by determining the equivalent stress at the point (0, a), we take τ = 0 assuming symmetry by the y-axis and no friction. The normal (axial) stress σ_z vanishes: σ_z= 0. The average tangential stress

σ_ta = [F / (2at) -F / (2bt)] * a / (b - a) = F / (2at) * (1 - a/b) * a / (b - a) = F / (2bt).

The corresponding pure (dimensionless) stresses in general strength theory [4] are

σ°_r = σ_r / σ_LB = -F/(2atσ_LB), τ° = τ / R_m = 0,

σ°_t = σ_t / R_m = F / [2btR_m], σ°_z = σ_z / R_m = 0.

The pure (dimensionless) equivalent stress at the point (0, a) in general strength theory [4] and the reserve factor are

σ°_e = (σ°_t² - σ°_tσ°_r + σ°_r² + 3 τ° ²)^1/2 = F/(2atR_m) * [(a/b)² + a/b * R_m/σ_LB + (R_m/σ_LB )²]^1/2,

n = R.F. = 1/σ°_e  =2atR_m / F / [(a/b)² + a/b * R_m/σ_LB + (R_m/σ_LB )²]^1/2.

By r ≥ b, the minimum thickness t_{min b} ≥ t_min and, to be conservative, namely t_{min b} should be used instead of t. For the plate strength at the point (0, b), the average radial stress σ_rb = -F/(2bt_{min b}), and the maximum shear stress τ = F/(et). However, such values of τ can hold by x = b and x = -b but not by x = 0. Therefore, by determining the equivalent stress at the point (0, b), we take τ = 0 assuming symmetry by the y-axis and no friction. The normal (axial) stress σ_z vanishes: σ_z = 0. The average tangential stress

σ_t = F / (2bt) * (e + b) / 2 / (e - b) = F * (e + b) / [4b(e - b)t].

The pure (dimensionless) stresses at the point (0, b) in general strength theory [4] are

σ°_r = σ_r / σ_LB= -F/(2btσ_LB), τ° = τ / R_m = 0,

σ°_t = σ_t / R_m = F * (e + b) / [4b(e - b)tR_m], σ°_z = σ_z / R_m = 0.

The pure (dimensionless) equivalent stress at the point (0, b) in general strength theory [4] and the reserve factor are

σ°_e = (σ°_t² - σ°_tσ°_r + σ°_r² + 3 τ°²)^1/2 =

F/(2btR_m) * {[(e + b)/2/(e - a)]² + (e + b)/2/(e - a) * R_m/σ_LB+ (R_m/σ_LB)²}^1/2,

n = R.F. = 1/σ°_e = 2btR_m / F / {[(e + b)/2/(e - a)]² + (e + b)/2/(e - a) * R_m/σ_LB + (R_m/σ_LB)²}^1/2.

Many comparisons have shown that general bearing strength theory gives results similar to those of the lug model and unlike it, provides investigating washer strengths and optimizing the choice of washer geometry and material by considering different possibilities.

[1] Lev Gelimson. General Bearing Strength Theory.In: Review of Aeronautical Fatigue Investigations in Germany During the Period 2005 to 2007, Dr. Claudio Dalle Donne, EADS Corporate Research Center Germany

[2] Military Handbook. Metallic Materials and Elements for Aerospace Vehicle Structures. MIL-HDBK-5H,1998

[3] Handbuch Struktur-Berechnung. Prof. Dr.-Ing. L. Schwarmann. Industrie-Ausschuss-Struktur-Berechnungsunterlagen, Bremen, 1998

[4] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich (Germany), 2004