Basic Bearing Strength Theory

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), 1

Common bearing strength theory [1, 2] regards the only bearing stress in a complex three-dimensional stress state at a point under consideration) and only selected values of the e/a ratio, namely 4 and 3, with linearly interpolating between them, declaring the necessity of special investigations between 3 and 2, and forbidding any values less than 2, see the following figure:

A plate has thickness t and a round hole of radius a with minimum distance e from the hole center to the border (edge) of the plate. Force F acts on the point (0, a) normally to the border (edge) of the plate. By elasticity, elastic contact theory could be used. By plasticity, at the point (0, a), the modulus (absolute value) of the average radial stress σ_r = -F/(2at) plays the bearing role and should be compared with the classical bearing strength [1, 2]. The average shear stress τ = F/(2et). But the real distribution of t is nonhomogeneous – theoretically vanishing at the ends and maximum in the middle [3]. By the parabolic distribution [3] of τ, its maximum is the above average value multiplied by 1.5, by stepwise linear distribution – multiplied by 2. To be conservative, take factor 2 the greatest among the both. Then  τ = F/(et). 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 maximum tangential stress by elasticity

σ_t = F / (2at) * (e² + a²) / (e² - a²)

holds at the point (0, a) by the Lame formulae. By plasticity, the maximum tangential stress holds at the point (0, e) with vanishing the radial stress σ_r . Therefore, it is conservative to consider the point (0, a) with the real radial stress whose modulus (absolute value) achieves its maximum and with the average tangential stress

σ_t = F / (2at) * a / (e - a) = F / [2(e - a)t]

by the equilibrium condition like that by a cylindrical shell. The remaining normal (axial) stress σ_z vanishes: s_z = 0. The corresponding pure (dimensionless) stresses at the point (0, a) in general strength theory [3] are

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

σ°_t = σ_t / R_m = F / [2(e - a)tR_m], σ°_z = σ_z / R_m = 0.

Here R_m is the ultimate strength by tension, σ_LB the ultimate bearing strength [2],

σ°_r = σ_r / σ_LB , τ° = τ / R_m , σ°_t = σ_t / R_m , σ°_z = σ_z / R_m

with a little circle on the right-hand side above are the corresponding pure (dimensionless) stresses in general strength theory [3] obtained from the usual ones by dividing them by their uniaxial ultimate values in the same direction with the same sign, τbeing divided by the ultimate normal stress to conserve the form of a critical strength criterion generalizing the Huber-von-Mises-Henky criterion where the factor by each τ² is the factor by each σ² multiplied by 3. Dividing τby τ_u(the ultimate shear stress) with replacing this factor 3 by 1 would give the same result but the criterion formula for the pure (dimensionless) stresses would be not similar to the criterion formula for the usual stresses. The pure (dimensionless) equivalent stress σ°_e at the point (0, a) in general strength theory [3] correcting and generalizing many known criteria, too, and the reserve factor [3] n are

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

F/(2atR_m) * [a²/(e - a)² + a/(e - a) * R_m/σ_LB + (R_m/σ_LB)²]^1/2,

n = 1/σ°_e  = 2atR_m / F / [a²/(e - a)² + a/(e - a) * R_m/σ_LB + (R_m/σ_LB)²]^1/2.

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

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

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