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:

GBearing.gif

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) × RmLB + (RmLB)2]1/2.

The reserve factor [4-8]

n = 1/σ°e =

2atRm / F / [a2/(e - a)2 + a/(e - a) × RmLB + (RmLB)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:

GBWasher.gif

In this case, the stress states at the both points (0, a) and (0, b) should be considered.

Washer Strength

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 × RmLB + (RmLB)2]1/2.

The reserve factor [4-8]

n = R.F. = 1/σ°e =

2atRm / F / [(a/b)2 + a/b × RmLB + (RmLB)2]1/2.

Plate Strength

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) × RmLB + (RmLB)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) × RmLB + (RmLB)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:

Hi-Lok
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

Hi-Lok
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

Rivet
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

Rivet
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

Conclusions

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.

References

[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

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[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

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[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