General Bearing Strength Theory
Strength Monograph
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
RUAG Aerospace Services GmbH, Germany
The “Collegium” All World Academy of Sciences Publishers
Munich (Germany)
Third Edition (2010)
Second Edition (2006)
First Edition (2004)
Part 1. General Bearing Strength Theory without Replacing Plate Parts with Washers
Bearing strength is commonly [1-3] investigated with two obvious and very important lacks:
regarding the only bearing stress which is one component of a complex three-dimensional stress state at a point under consideration;
regarding 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 by the design scheme in the following figure:
Here we see a plate with 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.
The present work is dedicated to creating general bearing strength theory with correcting the above lacks due to using fundamental mechanical and strength sciences [4-8].
By elasticity, elastic contact theory [9-12] could be used.
By plasticity, the following holds at the point (0, a):
The modulus (absolute value) of the average radial stress
σr = -F/(2at)
plays the bearing role and should be compared with the classical bearing strength, see Ref. [1-3].
The average shear stress
τ = F/(2et).
But the real distribution of τ is nonhomogeneous - theoretically vanishing at the ends and maximum in the middle [14]. By the parabolic distribution of τ [14], 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 symmetricity by the y axis and no friction.
The maximum tangential stress by elasticity
σt = F / (2at) × (e2 + a2) / (e2 - a2)
holds at the point (0, a) by the Lame formulae [14]. By plasticity, the maximum tangential stress holds at the point (0, e) with vanishing the radial stress σr [15, 16]. 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
σt = F / (2at) × a / (e - a) = F / [2(e - a)t]
by the equilibrium condition like that by a cylindrical shell [17, 18].
The remaining normal (axial) stress σz vanishes:
σz = 0.
The corresponding pure (dimensionless) stresses at the point (0, a) in fundamental mechanical and strength sciences [4-8] are
σ°r = σr / σLB = -F/(2atσLB),
τ° = τ / Rm = 0,
σ°t = σt / Rm = F / [2(e - a)tRm],
σ°z = σz / Rm = 0.
Here Rm is the ultimate strength by tension, σLB the ultimate bearing strength [2],
σ°r = σr / σLB
τ° = τ / Rm
σ°t = σt / Rm
σ°z = σz / Rm
with a little circle on the right-hand side above are the corresponding pure (dimensionless) stresses in fundamental mechanical and strength sciences [4-8] obtained from the usual ones by dividing them by their uniaxial ultimate values in the same direction with the same sign, τ being divided by the ultimate normal stress to conserve the form of a critical strength criterion generalizing the Huber-von Mises-Henky criterion [15, 16, 19] where the factor by each τ2 is the factor by each σ2 multiplied by 3. Dividing τ by τu (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 at the point (0, a) in fundamental mechanical and strength sciences [4-8] correcting and generalizing many known criteria [15, 16, 19, 20], too, is [21-28]
σ°e = (σ°t2 - σ°t σ°r + σ°r2 + 3 τ° 2)1/2 =
F/(2atRm) × [a2/(e - a)2 + a/(e - a) × Rm/σLB + (Rm/σLB)2]1/2.
The reserve factor [4-8]
n = 1/σ°e =
2atRm / F / [a2/(e - a)2 + a/(e - a) × Rm/σLB + (Rm/σLB)2]1/2.
Part 2. General Bearing Strength Theory by Replacing Plate Parts with Washers
The present work is dedicated to further extending general bearing strength theory by introducing washers between sheets and fasteners with correcting the above lacks due to using fundamental mechanical and strength sciences [4-8].
The essence of such an introduction, 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 tmax:
In this case, the stress states at the both points (0, a) and (0, b) should be considered.
By elasticity, elastic contact theory [9-12] could be used.
By plasticity, the following holds at the point (0, a):
The modulus (absolute value) of the average radial stress
σra = -F/(2at)
plays the bearing role and should be compared with the classical bearing strength [1-3].
The average shear stress
τa = [F / (2at) × 2at - F / (2bt) × 2at] /[2(b2 – a2)1/2 t] = F/(2bt) × [(b - a)/(b + a)]1/2.
But the real distribution of τ is nonhomogeneous - theoretically vanishing at the ends and maximum in the middle [12]. By the parabolic distribution of τ [12], 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
τa = [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 symmetricity by the y axis and no friction.
The maximum tangential stress by elasticity
σta = F / (2at) × (b2 + a2) / (b2 - a2) - F / (2bt) × 2b2 / (b2 - a2) =
F / (2at) × (b2 - ba + a2) / (b2 - a2)
holds at the point (0, a) by the Lame formulae [14]. By plasticity, the maximum tangential stress holds at the point (0, e) with vanishing the radial stress σr [15, 16]. 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
σta = [F / (2at) - F / (2bt)] × a / (b - a) =
F / (2at) × (1 - a/b) × a / (b - a) = F / (2bt)
by the equilibrium condition like that by a cylindrical shell [17, 18].
The remaining normal (axial) stress σz vanishes:
σz = 0.
The corresponding pure (dimensionless) stresses at the point (0, a) in fundamental mechanical and strength sciences [4-8] are
σ°r = σr / σLB = -F/(2atσLB),
τ° = τ / Rm = 0,
σ°t = σt / Rm = F / [2btRm],
σ°z = σz / Rm = 0.
Here Rm is the ultimate strength by tension, σLB the ultimate bearing strength [2],
σ°r = σr / σLB
τ° = τ / Rm
σ°t = σt / Rm
σ°z = σz / Rm
with a little circle on the right-hand side above are the corresponding pure (dimensionless) stresses in fundamental mechanical and strength sciences [4-8] obtained from the usual ones by dividing them by their uniaxial ultimate values in the same direction with the same sign, τ being divided by the ultimate normal stress to conserve the form of a critical strength criterion generalizing the Huber-von Mises-Henky criterion [15, 16,19] where the factor by each τ2 is the factor by each σ2 multiplied by 3. Dividing τ by τu (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 at the point (0, a) in fundamental mechanical and strength sciences [4-8] correcting and generalizing many known criteria [15, 16, 19, 20], too, is
σ°e = (σ°t2 - σ°t σ°r + σ°r2 + 3 τ° 2)1/2 =
F/(2atRm) × [(a/b)2 + a/b × Rm/σLB + (Rm/σLB)2]1/2.
The reserve factor [4-8]
n = R.F. = 1/σ°e =
2atRm / F / [(a/b)2 + a/b × Rm/σLB + (Rm/σLB)2]1/2.
By r ≥ b, the minimum thickness
tmin b ≥ tmin
and, to be conservative, namely tmin b should be used instead of t.
By elasticity, elastic contact theory [9-12] could be used.
By plasticity, the following holds at the point (0, b):
The modulus (absolute value) of the average radial stress
σrb = -F/(2btmin b)
plays the bearing role and should be compared with the classical bearing strength [1-3].
The average shear stress
τ = F/(2et)
But the real distribution of τ is nonhomogeneous - theoretically vanishing at the ends and maximum in the middle [14]. By the parabolic distribution of τ [14], 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 = 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 symmetricity by the y axis and no friction.
The maximum tangential stress by elasticity
σt = F / (2bt) × (e2 + b2) / (e2 - b2)
holds at the point (0, a) by the Lame formulae [14]. By plasticity, the maximum tangential stress holds at the point (0, e) with vanishing the radial stress σr [15, 16]. 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
σt = F / (2bt) × (e + b) / 2 / (e - b) = F × (e + b) / [4b(e - b)t]
by the equilibrium condition like that by a cylindrical shell [17, 18].
The remaining normal (axial) stress σz vanishes:
σz = 0.
The corresponding pure (dimensionless) stresses at the point (0, b) in fundamental mechanical and strength sciences [4-8] are
σ°r = σr / σLB = -F/(2btσLB),
τ° = τ / Rm = 0,
σ°t = σt / Rm = F × (e + b) / [4b(e - b)tRm],
σ°z = σz / Rm = 0.
The pure (dimensionless) equivalent stress at the point (0, b) in fundamental mechanical and strength sciences [4-8] correcting and generalizing many known criteria [15, 16, 19, 20], too, is
σ°e = (σ°t2 - σ°t σ°r + σ°r2 + 3 τ° 2)1/2 =
F/(2btRm) × {[(e + b)/2/(e - a)]2 + (e + b)/2/(e - a) × Rm/σLB + (Rm/σLB)2}1/2.
The reserve factor [4-8]
n = R.F. = 1/σ°e =
2btRm / F / {[(e + b)/2/(e - a)]2 + (e + b)/2/(e - a) × Rm/σLB + (Rm/σLB)2}1/2.
Calculation Results for General Bearing Strength
by Introducing Washers
The example calculation results (two for Hi-Loks and two for rivets with using [21, 22]) are represented in the following tables:
Hi-Lok | Washer of Steel 1.4544.9 | Sheet of Alu 3.1354T352 | ||||||||||||||||||
tmin | a | b | e | F | Rm | σLB | σra | σta | σza | τa | σ°ea | R.F. | Rm | σLB | σrb | σtb | σzb | τb | σ°eb | R.F. |
mm | mm | mm | mm | N | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | ||||
1.2 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -163 | 198 | 0 | 136 | 0.62 | 1.60 |
1.1 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -178 | 216 | 0 | 148 | 0.68 | 1.47 |
1.0 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -195 | 237 | 0 | 163 | 0.75 | 1.34 |
0.9 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -217 | 264 | 0 | 181 | 0.83 | 1.20 |
0.8 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -244 | 296 | 0 | 203 | 0.94 | 1.07 |
0.7 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -279 | 339 | 0 | 233 | 1.07 | 0.94 |
0.6 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -326 | 395 | 0 | 271 | 1.25 | 0.80 |
0.5 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -391 | 474 | 0 | 326 | 1.50 | 0.67 |
0.4 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -488 | 593 | 0 | 407 | 1.87 | 0.53 |
0.3 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -651 | 791 | 0 | 543 | 2.49 | 0.40 |
0.2 | 2.5 | 5 | 12 | 1953 | 500 | 430 | -326 | 163 | 0 | 188 | 0.96 | 1.04 | 430 | 650 | -977 | 1186 | 0 | 814 | 3.74 | 0.27 |
Hi-Lok | Washer of Steel 1.4544.9 | Sheet of Alu 3.1354T352 | ||||||||||||||||||
tmin | a | b | e | F | Rm | σLB | σra | σta | σza | τa | σ°ea | R.F. | Rm | σLB | σrb | σtb | σzb | τb | σ°eb | R.F. |
mm | mm | mm | mm | N | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | ||||
1.2 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -90 | 249 | 0 | 125 | 0.66 | 1.52 |
1.1 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -99 | 271 | 0 | 137 | 0.72 | 1.39 |
1.0 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -109 | 298 | 0 | 150 | 0.79 | 1.26 |
0.9 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -121 | 332 | 0 | 167 | 0.88 | 1.14 |
0.8 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -136 | 373 | 0 | 188 | 0.99 | 1.01 |
0.7 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -155 | 426 | 0 | 215 | 1.13 | 0.89 |
0.6 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -181 | 497 | 0 | 250 | 1.32 | 0.76 |
0.5 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -217 | 597 | 0 | 300 | 1.58 | 0.63 |
0.4 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -271 | 746 | 0 | 376 | 1.98 | 0.51 |
0.3 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -362 | 995 | 0 | 501 | 2.64 | 0.38 |
0.2 | 2.5 | 9 | 13 | 1953 | 500 | 430 | -326 | 90 | 0 | 136 | 0.86 | 1.16 | 430 | 650 | -543 | 1492 | 0 | 751 | 3.95 | 0.25 |
Rivet | Washer of Steel 1.4544.9 | Sheet of Alu 3.1354T352 | ||||||||||||||||||
tmin | a | b | e | F | Rm | σLB | σra | σta | σza | τa | σ°ea | R.F. | Rm | σLB | σrb | σtb | σzb | τb | σ°eb | R.F. |
mm | mm | mm | mm | N | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | ||||
1.2 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -109 | 122 | 0 | 83 | 0.39 | 2.53 |
1.1 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -118 | 133 | 0 | 91 | 0.43 | 2.32 |
1.0 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -130 | 146 | 0 | 100 | 0.47 | 2.11 |
0.9 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -145 | 163 | 0 | 111 | 0.53 | 1.90 |
0.8 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -163 | 183 | 0 | 125 | 0.59 | 1.69 |
0.7 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -186 | 209 | 0 | 143 | 0.68 | 1.48 |
0.6 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -217 | 244 | 0 | 167 | 0.79 | 1.27 |
0.5 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -260 | 293 | 0 | 200 | 0.95 | 1.06 |
0.4 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -326 | 366 | 0 | 250 | 1.18 | 0.84 |
0.3 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -434 | 488 | 0 | 334 | 1.58 | 0.63 |
0.2 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -651 | 732 | 0 | 501 | 2.37 | 0.42 |
Rivet | Washer of Steel 1.4544.9 | Sheet of Alu 3.1354T352 | ||||||||||||||||||
tmin | a | b | e | F | Rm | σLB | σra | σta | σza | τa | σ°ea | R.F. | Rm | σLB | σrb | σtb | σzb | τb | σ°eb | R.F. |
mm | mm | mm | mm | N | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | ||||
1.2 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -109 | 122 | 0 | 83 | 0.39 | 2.53 |
1.1 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -118 | 133 | 0 | 91 | 0.43 | 2.32 |
1.0 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -130 | 146 | 0 | 100 | 0.47 | 2.11 |
0.9 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -145 | 163 | 0 | 111 | 0.53 | 1.90 |
0.8 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -163 | 183 | 0 | 125 | 0.59 | 1.69 |
0.7 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -186 | 209 | 0 | 143 | 0.68 | 1.48 |
0.6 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -217 | 244 | 0 | 167 | 0.79 | 1.27 |
0.5 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -260 | 293 | 0 | 200 | 0.95 | 1.06 |
0.4 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -326 | 366 | 0 | 250 | 1.18 | 0.84 |
0.3 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -434 | 488 | 0 | 334 | 1.58 | 0.63 |
0.2 | 2 | 5 | 13 | 1302 | 500 | 430 | -271 | 109 | 0 | 142 | 0.76 | 1.31 | 430 | 650 | -651 | 732 | 0 | 501 | 2.37 | 0.42 |
Calculation Results for a Lug Model
by Introducing Washers
The limiting force for a lug Ref. [2]
FAL = min{FW; FS; FFL}
where:
FW = KW × (b - D) × s × Rm
is the limiting force by cheek failure,
KW factor by [2], 26101-01, P. 6 of 14, Fig. 2,
b width (an external diameter), in our case the distance between the centers of the next holes,
D internal diameter, in our case the external diameter of a washer,
s thickness earlier denoted by t;
FS = KS × D × s × Rm
is the limiting force by cutting in peak,
KS factor by [2], 26101-01, P. 6 of 14, Fig. 3,
D internal diameter, in our case the external diameter of a washer earlier denoted by 2b,
s thickness earlier denoted by t;
FFL = KF × Rp 0.2 / Rm × Fmin
is the limiting force by contact yielding,
KF factor by [2], 26101-01, P. 7 of 14, Fig. 4,
Fmin = min{FW; FS}.
The calculation results (two for the same Hi-Loks and two for the same rivets) are represented in the following tables:
s =
hmin |
b | D | h | Rm | Rp 0.2 | KW | KS | FW | FS | Fmin | Fmin/
(DsRm) |
KF | FFL | FAL | F | R.F. |
mm | mm | mm | mm | MPa | MPa | N | N | N | N | N | N | |||||
1.2 | 25 | 10 | 12 | 430 | 290 | 0.86 | 1.1 | 6656 | 5676 | 5676 | 1.2 | 1.09 | 4173 | 4173 | 1953 | 2.14 |
1.1 | 25 | 10 | 12 | 430 | 290 | 0.86 | 1.0 | 6102 | 4730 | 4730 | 1.0 | 1.1 | 3509 | 3509 | 1953 | 1.80 |
1.0 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.9 | 5547 | 3870 | 3870 | 0.9 | 1.1 | 2871 | 2871 | 1953 | 1.47 |
0.9 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.8 | 4992 | 3096 | 3096 | 0.8 | 1.1 | 2297 | 2297 | 1953 | 1.18 |
0.8 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.7 | 4438 | 2408 | 2408 | 0.7 | 1.1 | 1786 | 1786 | 1953 | 0.91 |
0.7 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.7 | 3883 | 2107 | 2107 | 0.7 | 1.1 | 1563 | 1563 | 1953 | 0.80 |
0.6 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.6 | 3328 | 1548 | 1548 | 0.6 | 1.1 | 1148 | 1148 | 1953 | 0.59 |
0.5 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.5 | 2774 | 1075 | 1075 | 0.5 | 1.1 | 798 | 798 | 1953 | 0.41 |
0.4 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.4 | 2219 | 688 | 688 | 0.4 | 1.1 | 510 | 510 | 1953 | 0.26 |
0.3 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.3 | 1664 | 387 | 387 | 0.3 | 1.1 | 287 | 287 | 1953 | 0.15 |
0.2 | 25 | 10 | 12 | 430 | 290 | 0.86 | 0.2 | 1109 | 172 | 172 | 0.2 | 1.1 | 128 | 128 | 1953 | 0.07 |
s =
hmin |
b | D | h | Rm | Rp 0.2 | KW | KS | FW | FS | Fmin | Fmin/
(DsRm) |
KF | FFL | FAL | F | R.F. |
mm | mm | mm | mm | MPa | MPa | N | N | N | N | N | N | |||||
1.2 | 25 | 18 | 13 | 430 | 290 | 0.96 | 1.1 | 3468 | 10217 | 3468 | 0.373 | 1.1 | 2572 | 2572 | 1953 | 1.32 |
1.1 | 25 | 18 | 13 | 430 | 290 | 0.96 | 1.0 | 3179 | 8514 | 3179 | 0.373 | 1.1 | 2358 | 2358 | 1953 | 1.21 |
1.0 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.9 | 2890 | 6966 | 2890 | 0.373 | 1.1 | 2144 | 2144 | 1953 | 1.10 |
0.9 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.8 | 2601 | 5573 | 2601 | 0.373 | 1.1 | 1929 | 1929 | 1953 | 0.99 |
0.8 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.7 | 2312 | 4334 | 2312 | 0.373 | 1.1 | 1715 | 1715 | 1953 | 0.88 |
0.7 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.7 | 2023 | 3793 | 2023 | 0.373 | 1.1 | 1501 | 1501 | 1953 | 0.77 |
0.6 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.6 | 1734 | 2786 | 1734 | 0.373 | 1.1 | 1286 | 1286 | 1953 | 0.66 |
0.5 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.5 | 1445 | 1935 | 1445 | 0.373 | 1.1 | 1072 | 1072 | 1953 | 0.55 |
0.4 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.4 | 1156 | 1238 | 1156 | 0.373 | 1.1 | 857 | 857 | 1953 | 0.44 |
0.3 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.3 | 867 | 697 | 697 | 0.3 | 1.1 | 517 | 517 | 1953 | 0.26 |
0.2 | 25 | 18 | 13 | 430 | 290 | 0.96 | 0.2 | 578 | 310 | 310 | 0.2 | 1.1 | 230 | 230 | 1953 | 0.12 |
s =
hmin |
b | D | h | Rm | Rp 0.2 | KW | KS | FW | FS | Fmin | Fmin/
(DsRm) |
KF | FFL | FAL | F | R.F. |
mm | mm | mm | mm | MPa | MPa | N | N | N | N | N | N | |||||
1.2 | 25 | 10 | 13 | 430 | 290 | 0.86 | 1.1 | 6656 | 5676 | 5676 | 1.2 | 1.09 | 4173 | 4173 | 1302 | 3.20 |
1.1 | 25 | 10 | 13 | 430 | 290 | 0.86 | 1.0 | 6102 | 4730 | 4730 | 1.0 | 1.1 | 3509 | 3509 | 1302 | 2.70 |
1.0 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.9 | 5547 | 3870 | 3870 | 0.9 | 1.1 | 2871 | 2871 | 1302 | 2.21 |
0.9 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.8 | 4992 | 3096 | 3096 | 0.8 | 1.1 | 2297 | 2297 | 1302 | 1.76 |
0.8 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.7 | 4438 | 2408 | 2408 | 0.7 | 1.1 | 1786 | 1786 | 1302 | 1.37 |
0.7 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.7 | 3883 | 2107 | 2107 | 0.7 | 1.1 | 1563 | 1563 | 1302 | 1.20 |
0.6 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.6 | 3328 | 1548 | 1548 | 0.6 | 1.1 | 1148 | 1148 | 1302 | 0.88 |
0.5 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.5 | 2774 | 1075 | 1075 | 0.5 | 1.1 | 798 | 798 | 1302 | 0.61 |
0.4 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.4 | 2219 | 688 | 688 | 0.4 | 1.1 | 510 | 510 | 1302 | 0.39 |
0.3 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.3 | 1664 | 387 | 387 | 0.3 | 1.1 | 287 | 287 | 1302 | 0.22 |
0.2 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.2 | 1109 | 172 | 172 | 0.2 | 1.1 | 128 | 128 | 1302 | 0.10 |
s =
hmin |
b | D | h | Rm | Rp 0.2 | KW | KS | FW | FS | Fmin | Fmin/
(DsRm) |
KF | FFL | FAL | F | R.F. |
mm | mm | mm | mm | MPa | MPa | N | N | N | N | N | N | |||||
1.2 | 25 | 10 | 13 | 430 | 290 | 0.86 | 1.1 | 6656 | 5676 | 5676 | 1.2 | 1.09 | 4173 | 4173 | 1302 | 3.20 |
1.1 | 25 | 10 | 13 | 430 | 290 | 0.86 | 1.0 | 6102 | 4730 | 4730 | 1.0 | 1.1 | 3509 | 3509 | 1302 | 2.70 |
1.0 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.9 | 5547 | 3870 | 3870 | 0.9 | 1.1 | 2871 | 2871 | 1302 | 2.21 |
0.9 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.8 | 4992 | 3096 | 3096 | 0.8 | 1.1 | 2297 | 2297 | 1302 | 1.76 |
0.8 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.7 | 4438 | 2408 | 2408 | 0.7 | 1.1 | 1786 | 1786 | 1302 | 1.37 |
0.7 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.7 | 3883 | 2107 | 2107 | 0.7 | 1.1 | 1563 | 1563 | 1302 | 1.20 |
0.6 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.6 | 3328 | 1548 | 1548 | 0.6 | 1.1 | 1148 | 1148 | 1302 | 0.88 |
0.5 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.5 | 2774 | 1075 | 1075 | 0.5 | 1.1 | 798 | 798 | 1302 | 0.61 |
0.4 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.4 | 2219 | 688 | 688 | 0.4 | 1.1 | 510 | 510 | 1302 | 0.39 |
0.3 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.3 | 1664 | 387 | 387 | 0.3 | 1.1 | 287 | 287 | 1302 | 0.22 |
0.2 | 25 | 10 | 13 | 430 | 290 | 0.86 | 0.2 | 1109 | 172 | 172 | 0.2 | 1.1 | 128 | 128 | 1302 | 0.10 |
1. General bearing strength theory [23-31] applies to replacing plate parts with washers, too.
2. General bearing strength theory gives results similar to those of the lug model.
3. Unlike the lug model, general bearing strength theory provides investigating washer strengths, too.
4. Unlike the lug model, general bearing strength theory provides optimizing the choice of washer geometry and material by considering different possibilities.
[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] Bruhn, E. F.: Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc., Indianapolis (IN), 1973
[4] Lev Gelimson: General Strength Theory. Drukar Publishers, Sumy, 1993
[5] Lev Gelimson: Basic New Mathematics. Drukar Publishers, Sumy, 1995
[6] Lev Gelimson: General strength theory. Abhandlungen der Wissenschaftlichen Gesellschaft zu Berlin, Publisher Prof. Dr. habil. V. Mairanowski, 3 (2003), 56-62
[7] Lev Gelimson: Elastic Mathematics. General Strength Theory. The ”Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2004
[8] Lev Gelimson: Correction and Unification Approaches and Methods for Strength Criteria. The ”Collegium” All World Academy of Sciences Publishers, Munich (Germany), 2005
[9] Hertz, H.: Über die Berührung fester elastischer Körper. J. reine und angewandte Mathematik, 92 (1882), 156-171
[10] Huber, M. T.: Zur Theorie der Berührung fester elastischer Körper. Annalen der Physik, 14/1 (1904), 153-163
[11] Huber, M. T., Fuchs, S: Spannungsverteilung bei der Berührung zweier elastischer Zylinder. Physikalische Zeitschrift, 15/6 (1914), 298-303
[12] Johnson, K. L.: Contact Mechanics. Cambridge University Press, N. Y. etc., 1985
[13] Love, A. E. H.: A Treatise on the Mathematical Theory of Elasticity. Vols. I, II. Cambridge University Press, Cambridge, 1892, 1893
[14] Lamé, G.: Lecons sur la theorie mathematique de l’élasticite des corps solides. Gauthier-Villars, Paris, 1852
[15] von Mises, R.: Mechanik der festen Körper im plastisch-deformablen Zustand. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttingen, 4 (1913), 582-592
[16] Henky, H.: Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Zeitschrift angewandter Mathematik und Mechanik, 4 (1924), 323-334
[17] Timoshenko, S. P., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York, 1969
[18] Pisarenko, G. S. y otros. Manual de Resistencia de Materiales. Moscú: Editorial MIR, 1989
[19] Huber, M. T.: Die spezifische Formänderungsarbeit als Maß der Anstrengung eines Materials. Czasopismo Techniczne, Lemberg (Lwow), 20 (1904), 81-83
[20] Tresca, H. E.: Memoire sur l'ecoulement des corps solides soumis a de fortes pressions. Comptes Rendus de l’Academie des Sciences, Paris, 59 (1864), 754-758
[21] AN, AND & MS Standards: June 96-20
[22] Standards Committee for Hi-Lok Products. 2002
[23] Lev Gelimson: Maximum Rivet Contact Pressure. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 32-33
[24] Lev Gelimson: General Reserve Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 55-56
[25] Lev Gelimson: Critical State Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period March 2003 to May 2005, Ed. Dr. Claudio Dalle Donne, SC/IRT/LG-MT-2005-039 Technical Report, Aeronautical fatigue, ICAF2007, EADS Corporate Research Center Germany, 2005, 67-68
[26] Lev Gelimson: Providing Helicopter Fatigue Strength: Flight Conditions [Overmathematics and Other Fundamental Mathematical Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416
[27] Lev Gelimson: Providing Helicopter Fatigue strength: Unit Loads [Fundamental Mechanical and Strength Sciences]. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 589-600
[28] Lev Gelimson: Regarding the Ratio of Tensile Strength to Shear Strength in General Strength Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 44-46
[29] Lev Gelimson: Correcting and Further Generalizing Critical State Criteria in General Strength Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 47-48
[30] Lev Gelimson: General Bearing Strength Theory. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 22-24
[31] Lev Gelimson: General Bearing Strength Theory by Replacing Plate Parts with Washers. In: Review of Aeronautical Fatigue Investigations in Germany During the Period May 2005 to April 2007, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2007-042 Technical Report, Aeronautical fatigue, ICAF2007, EADS Innovation Works Germany, 2007, 24-26