^{1}

^{2}

The pure shear strength for the all-simply supported plate has not yet been found ; what is described as pure shear in that plate, is, in fact, a pure-shear solution for another plate clamped on the “Y-Y” and simply supported on the long side, X-X. A new solution for the simply supported case is presented here and is found to be only 60-percent of the currently believed results. Comparative results are presented for the all-clamped plate which exhibits great accuracy. The von Misses yield relation is adopted and through incremental deflection-rating the effective shear curvature is targeted in aspect-ratios. For a set of boundary conditions the Kirchhoff’s plate capacity is finite and invariant for bending, buckling in axial and pure-shear and in vibration.

Plate-bending is expertly covered by Timoshenko and Krieger [

Timoshenko’s results for

N x y = D π 2 [ 5.35 + 4 / ( b / a ) 2 ]

this, as the all-simply-supported plate, appears to make no adjustment for the absence of the stiffeners. For a square plate “ N x y = 9.35 D π 2 ”, this formula persists to date.

Piscopo [

D ∇ 4 w = N x y ( β ∂ 2 w / ∂ x 2 + 2 ∂ 2 w / ∂ x ∂ y ) (1)

but a direct equilibrium solution for the latter has never been found. Incidentally, there appear to be some conflicts between Equation (1) and von Misses yield relation of Equation (2)

σ v m 2 = ( 1 / 2 ) [ ( σ 1 − σ 2 ) 2 + ( σ 2 − σ 3 ) 2 + ( σ 3 − σ 1 ) 2 ] (2)

σ v m = ( σ 1 2 + σ 2 2 + σ 1 σ 2 ) 1 / 2 , in the “ σ 1 − σ 2 ” plane

A degenerate form of this equation for “ σ compression ≫ σ tension ”, is

σ v m = β σ 1

or in terms of curvature,

X v m = β X 1 (3)

Find “ X 1 = X effective ” and the pure shear problem is solved in the “von Misses/Kirchhoff’s” framework. So Equation (1) is consumed by Equation (3), leading to,

D ∇ 4 w = ( N x y ) { β ( ∂ 2 w / ∂ x 2 ) effective } = ( N x y ) ( X V M ) (4)

Equation (4) is now the new standard pure-shear equation.

The present study starts with the buckling problem but assumes a great familiarity in the bending-deflection cases. Deflection factors are related to the desired curvatures. The deflection-factors employed emanate from the capacity of the Kirchhoff’s plate differentials which form the basis of all analyses, analytical or numerical finite-elements; these factors, when found, are easily recognizable, confirming that solutions are on track. Also, a fast approximate spot-buckling-solution is necessary for additional checks; Mohr’s loading curvature-circles (

The Kirchhoff’s plate capacity is constant whether in shear or axial compression, so there is no need to engage in a new extensive independent analysis in shear where von Misses shear solution can be invoked. The pure-shear plate buckling is hugely significant on account of the heavy demands on heavier and heavier ships and their platting.

Equation (5), is the existing uniaxial buckling equation. The biaxial case, Equation (6) ensues if “N_{xy}” = 0

D ( ∂ 4 w / ∂ x 4 + 2 ∂ 4 w / ∂ x 2 ∂ y 2 + ∂ 4 w / ∂ y 4 ) = H = ( N x ) ( ∂ 2 w / ∂ x 2 ) (5)

D { ∂ 4 w / ∂ x 4 + 2 ∂ 4 w / ∂ x 2 ∂ y 2 + ∂ 4 w / ∂ y 4 } = H = N x ∂ 2 w / ∂ x 2 + N x y ( 2 ∂ 2 w / ∂ x ∂ y ) + N y ∂ 2 w / ∂ y 2 (6)

The shear loading—Equation (7) is balanced in the same way as the biaxial case.

D { ∂ 4 w / ∂ x 4 + 2 ∂ 4 w / ∂ x 2 ∂ y 2 + ∂ 4 w / ∂ y 4 } = H = N x y ( 2 ∂ 2 w / ∂ x ∂ y ) (7)

Under equivalent uniformly distributed transverse loading, q * , Equation (8) ensues

D { ∂ 4 w / ∂ x 4 + 2 ∂ 4 w / ∂ x 2 ∂ y 2 + ∂ 4 w / ∂ y 4 } = H = q * (8a)

Equations (5)-(8) are summarized as,

H x x + H x y + H y y = H = RHS ( loading ) (8b)

By giving finite values of the left-hand differentials, the capacity of a Kirchhoff’s plate ensues; this is achieved through valid deflection shape-functions, “w”

∂ 4 w / ( ∂ x 4 ) ( 1 ) = ∬ w ( ∂ 4 w / ∂ x 4 ) ∂ x ∂ y ∬ w ∂ x ∂ y = H x x (9)

( ∂ 4 w ) / ( ∂ y 4 ) ( 1 ) = ∬ w ( ∂ 4 w / ∂ y 4 ) ∂ x ∂ y ∬ w ∂ x ∂ y = H y y (10)

( 2 ∂ 4 w ) / ( ∂ x 2 ∂ y 2 ) ( 1 ) = ∬ 2 w ( ∂ 4 w / ∂ x 2 ∂ y 2 ) ∂ x ∂ y ∬ w ∂ x ∂ y = H x y (11)

( ∂ 2 w ) / ( ∂ x 2 ) ( 1 ) = ∬ w ( ∂ 2 w / ∂ x 2 ) ∂ x ∂ y ∬ w ∂ x ∂ y = X x (12)

( ∂ 2 w ) / ( ∂ y 2 ) ( 1 ) = ∬ w ( ∂ 2 w / ∂ y 2 ) ∂ x ∂ y ∬ w ∂ x ∂ y = X y (13)

X 1 , 2 = ( X x + X y ) / 2 ± [ { ( X x − X y ) 2 / 2 } 2 + { ( X x y ) / 2 } 2 ] ; principal curvatures;

(14a)

Or by reference to the Mohr’s circle,

X 1 , 2 = ( X x + X y ) / 2 ± R ; R = Mohr’s circle radius (14b)

The average curvature, ( X x + X y ) / 2 , is found significant as an intermediate loading curvature when “ X x < X y ” over the range of “ X y > X x ” in uni-axial X-loading. For the bi-axial case, X biaxial = ( X x + X y ) .

These integrals are the outcomes of criterion of buckling as relative-curvature/ deflection resonance. A typical buckling resistance integral is,

( ∂ 4 w ) / ( ∂ x 4 ) ( 1 ) = C x d 4 ( w x x - r / w ) = C x d 4 ( R x c d ) (15)

The ratio, “ ( w x x - r / w ) = R x c d ” must always be a scalar or else the function-w is inadmissible. The function-w is chosen as to make the ratio, ( w x x - r / w ), a scalar. The domain compliant factor at resonance, C_{xd}_{4}, is what is left to be found. Multiply both sides of Equation (16) and integrate to find it.

C x d 4 R x c d = [ ( ∂ 4 w / ∂ x 4 ) 1 ] = [ ∬ w ( ∂ 4 w / ∂ x 4 ) ∂ x ∂ y ∬ w ∂ x ∂ y ] (16)

Three possibilities are identified relative to X- and Y-axes in emulating the reactive potentials “ ∂ 4 w / ∂ x 4 ; 2 ∂ 4 w / ∂ x 2 ∂ y 2 ; ∂ 4 w / ∂ y 4 ”.

1) σ x X x

This is first in contention in uni-axial X-compression; this case easily solves Equation (5).

2) σ y X y

This is out of contention when no load is applied in the Y-axis, whatever the value of “ X y ”.

3) σ x X a v

This “average loading-curvature” situation will always happen and also in contention. (1) and (3) are identified in the Mohr’s diagram,

In effect, two curvature-loading circles ( X x , X a v ) are operative and the larger circle gives the required solution for “N_{x}”. This process softens the stiff constraint that the wave numbers “m, n”, must be whole

The solution of Equation (5) is easy when

( ∂ 2 w / ∂ x 2 ) ≥ ( ∂ 2 w / ∂ y 2 ) ; and X x = X c r = ( ∂ 2 w / ∂ x 2 )

When

( ∂ 2 w / ∂ x 2 ) < ( ∂ 2 w / ∂ y 2 )

“ X 1 ” may be interpreted as “effective principal loading curvature”.

So,

Relying on the deflection coefficients, Δ 1 , Δ 2 at two consecutive locations, “i” and “i + 1”,

( Δ 1 / A 1 ) ( X x , 1 ) = ( Δ 2 / A 2 ) ( X x , 2 ) (17)

A_{2}/A_{1} = C_{A}, stressed boundary lengths-ratio representing side areas:

Aspect ratio, “s*” gap of 25-percent can be tolerated.

This equation is similar to Equation (5) as “(Force/Area)(curvature) = Constant” = Kirchhoff’s plate-capacity.

From Equation (5),

D ( ∂ 4 w / ∂ x 4 + 2 ∂ 4 w / ∂ x 2 ∂ y 2 + ∂ 4 w / ∂ y 4 ) = H = ( N x ) ( ∂ 2 w / ∂ x 2 ) = q *

For a given plate-function the “LHS” is invariant and once computed can be used for bending, buckling and vibration; “ q * ” is equivalent uniform transverse pressure. That is, deflection,

Δ = ( 1 / H ) ( w shape )

the primitive value is sufficient. The familiarity of “ Δ -value” gives confidence the solution is on track.

Invoke the already known solution on the inner square, _{cr}.= 10.66𝜌

2N_{xy}cos45 = (10.66) (1/(0.707b)^{2}); in Ref [

N_{xy} = 15.08; cf. {14.81ρ, [

For automatic values at any aspect-ratio, simply multiply “ X e ” by the von Misses shear factor of “ 1 / 2 ” and complete

s*^{ } (1) | Δ s ∗ (2) | C_{A} (3) | X Δ (4) | H (5) | K_{cr} ([Ref. [ | ( δ X / δ Δ ) = monitor (7) | 2 X x y = 0.707 X Δ (8) | N_{xy} = (5)/(8) “Pure-shear” (9) |
---|---|---|---|---|---|---|---|---|

1 | 0.00128 | 29.61 | 3116.8 | 10.66ρ, [10.4] | - | 20.93 | 15.0; ( [ | |

1.25 | 0.00186 | 1.05 | 21.72 | 2146.2 | 10.01ρ, [9.9] | −13, 600.0 | ||

1.5 | 0.00229 | 1.071 | 19.00 | 1746.0 | 9.3ρ, [9.3] | − 6, 325.6 | ||

1.75 | 0.00258 | 1.055 | 18.06 | 1547.85 | 8.68ρ, [8.6] | − 3, 241.4 | ||

2.0 | 0.002784 | 1.053 | 17.88 | 1436.65 | 8.14ρ, [8.0] | −900.0 | 12.64 | 11.5 |

2.25 | 0.002923 | 1.051 | 17.88 | 1368.3 | 7.75ρ, [7.7] | 0 | ||

2.35 | 0.002967 | 1.019 | 17.88 | 1348.2 | 7.64𝜌 | 0 | ||

2.45 | 0.003005 | 1.019 | 17.88 | 1331.05 | 7.54𝜌 | 0 | ||

2.5 | 0.003022 | 1.009 | 17.88 | 1323.39 | 7.50ρ, [7.5] | 0 | ||

2.75 | 0.00309 (0.00302) | 1.045 | 18.24 (17.88) | 1292.5 | 7.32𝜌 | +493.2, δ X / δ Δ = min, so, δ Δ = 0 | 12.64 | 10.3; ( [ |

3 | 0.00302 | 1.0526 | 18.70 (17.88) | 1292.5 | 7.32ρ, [7.35] | “H, X ”, move together |

s* (1) | Δ (2) | C_{A} (3) | X Δ (4) | H (5) | K_{cr}; (Ref. [ | ( δ X / δ Δ ) = monitor (7) | 2 X x y = 0.7 X Δ (8) | N_{xy} (ref. [ | Mass = m ∗ = ∬ w w / ∬ w _{ } (10) | ω^{2} = fundamental m = n = 1 (5)/(10) (11) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 0.00416 | - | 6.0875 | 240.34 | 4.0ρ [4ρ] | - | 4.3 | 5.66ρ [9.3ρ] | 0.617 | 389 |

1.25 | 0.00619 | 1.06 | 4.337 | 161.60 | 3.78 [>4ρ] | −862 | ||||

1.50 | 0.00798 | 1.06 | 3.566 | 125.36 | 3.56 [>4ρ] | −431 | ||||

1.75 | 0.00948 | 1.055 | 3.174 | 105.73 | 3.375 [>4ρ] | −265 | ||||

2.0 | 0.0106 | 1.051 | 2.977 | 93.73 | 3.195 [4ρ] | −173 | 0.617 | 152 | ||

2.25 | 0.0116 | 1.051 | 2.859 | 86.11 | 3.05 [>4ρ] | −118 | ||||

2.50 | 0.01236 | 1.0048 | 2.812 | 80.85 | 2.91ρ | −61.8 | ||||

2.75 | 0.0130 | 1.046 | 2.797 | 77.02 | 2.79ρ | −23.4 | ||||

3.00 | 0.0135 | 1.043 | 2.797 | 74.18 | 2.69 [4ρ] | −0.0 | ||||

3.25 | 0.0139 (0.0135) | 1.04 | 2.83 (2.797) | 72.0 | 2.60ρ | +82.5, ( δ X / δ Δ = min, so, δ Δ = 0) | 2.0 | 3.65ρ [5.3ρ] | 0.617 | 117 |

3.50 | 0.0135 | 2.60 | (“H, X ” move together) |

The “SSSS” case for “s* = 1” in

1) W = ( sin G x / a + A x / a ) ( sin n π y / b ) ; n = 1 , 2 , 3 , ⋯ ; G = 4.5; A = 0.977.

2) ∬ w ∂ x ∂ y = 0.482 。

3) ∬ w ( ∂ 4 w / ∂ x 4 ) ∂ x ∂ y 1 = 0.225 G 4 / a 4 .

4) ∬ w ( ∂ 4 w / ∂ y 4 ) ∂ x ∂ y 1 = 0.384 n 4 π 4 / b 4 .

5) ∬ 2 w ( ∂ 4 w / ∂ x 2 ∂ y 2 ) ∂ x ∂ y 1 = 0.45 n 2 G 4 / a 2 b 2 .

6) ∬ w ( ∂ 2 w / ∂ x 2 ) ∂ x ∂ y 1 = 0.225 G 2 / a 2 .

7) ∬ w ( ∂ 2 w / ∂ y 2 ) ∂ x ∂ y 1 = 0.384 n 2 π 2 / b 2 .

H = H x x + H x y + H y y = 191.42 + 186.6 + 77.6 = 455.6 ; { Δ fund = w s h / H = 0.00278 q * b 4 / D } ; X x = 9.45 ; X y = 7.687 ; X x , c r = 9.45 ; so: N x , c r = 455.6 / 9.45 = 48.2 = ( 4.88 ρ ) ( ρ ) .

Confirm that “ Δ fund ” is correct, cf., (0.0028) [

Since, also, “ X x > X y ” the result found must be exact or near-exact. Ref. [

Check Pure Shear, N x y = H / ( 2 X x y )

Find “ 2 X x y ” by von Misses as

2 X x y = ( 1 / 2 ) ( X x -effective ) = 0.707 × 9.45 = 6.681

N x y = 455.6 / 6.681 = 6.91 ρ

for “SSCS at a/b = 1”.

W = ( 1 − cos m π x / a ) ( 1 − cos n π y / b ) ; m , n = 2 , 4 , 6 , ⋯

a / b = 1 ;

1) Uniaxial compression; H_{xx} = 1168.8; H_{xy} = 779.2; X x = X y = X c r = 29.61 ; H_{yy} = 1168.8; H = 3116.8; N s * = 1 = 31168.8 / 29.61 = 10.66 ρ ; Δ c = 0.00128 ; cf, (0.00126) [_{x}” depends. Further, the exactness of the w-function is verified.

2) In pure-shear

N x y = 3116.8 / ( 0.707 × 29.61 ) = 15.08 ρ ; cf (14.81ρ), [

Note that the “CCCC” plate is summarized in

The pure-shear strength values varied from “15.0ρ” at s* = 1 to “10.3ρ” at s* = 2.5 (near infinity); these compare well with those of Ref. [

The trend is similar to the “CCCC” in

This is elaborated in Columns 10 and 11 in

W = ( 1 − cos m π x / a ) ( sin m π y / b ) ; m = 2 , 4 , 6 , ⋯ , n = 1 , 2 , 3 , ⋯

a/b = 1: H_{xx} = 612; H_{xy} = 306; H_{yy} = 114.75; H = 1032.75; Δ c = 0.001936 , cf, 0.00192 [

In Tables 1-3, the factor, δ X / δ Δ , exhibits a critical stationary-point,

s* (1) | Δ s ∗ _{ } ([Ref. [ | C_{A} (3) | X Δ (4) | H (5) | K_{cr}; (Ref. [ | ( δ X / δ Δ ) = monitor (7) | 2 X x y = 0.707 X Δ (8) | N_{xy} = (5)/(8) “Pure-shear” (9) |
---|---|---|---|---|---|---|---|---|

1 | 0.001936 [0.00192] | - | 15.5 | 1032.18 | 6.75ρ, [6.75] | - | 10.96 | 9.5ρ |

1.15 | 0.0029 | - | 11.63 | 688.72 | 6.0ρ, [ | 8.22 | 8.5ρ | |

1.4 | 0.00465 | 1.059 | 7.68 | 430.35 | 5.68ρ, [ | −2257 (−2144) | 5.43 | 8.0ρ |

1.65 | 0.00646 | 1.057 | 5.84 | 309.7 | 5.37 | −1016 (−903) | 4.13 | 7.6ρ |

1.90 | 0.0081 | 1.054 | 4.91 | 246.5 | 5.09ρ, [ | −581 (−468) | 3.47 | 7.2ρ |

2.15 | 0.0095 | 1.051 | 4.54 | 209.6 | 4.68ρ, [ | −257 (−144) | 3.21 | 6.6ρ |

2.4 | 0.0107 | 1.049 | 4.23 | 186.3 | 4.46ρ | −258 (−145) | 2.99 | 6.3ρ |

2.65 | 0.0117 | 1.046 | 4.05 | 170.7 | 4.27ρ | −180 (−77) | 2.863 | 6.04ρ |

2.9 | 0.0125 | 1.043 | 3.96 | 159.8 | 4.09ρ; stop!! | −113 (0)** | 2.80 | 5.78ρ |

3.15 | 0.0132 | 1.043 | 3.92 | 151.79 | 3.92 | −243 (−130) | 2.77 | 5.55ρ |

is indicated at that point, whatever the values of “m, n or s*”. The relative weakness of a plate is indicated in bending-buckling-vibration-analyses [

For example: s* = 2.5, try two waves, placed between the actions, 1) _{2.5,1}; and 2) m = 3,

1) C_{2.5,1}: H_{xx} = 1.54; H_{xy} = 19.2; H_{yy} = 60.1; H = 80.8;

2) C_{2.5,3}: H_{xx} = 374; H_{xy} = 519; H_{yy} = 180.2; H = 1073;

Combining by Dunkerley’s:

So, it can be said that the waves “m = 1 and m = 3” combine for the aspect ratio, s* = 2.5; N_{2.5,1,3} = 2.77ρ. This result is very different from the reference value of “4ρ” [

1) The finite “Capacity” of the Kirchhoff’s plate differentials is constant in shear, in plate-buckling, in pure-shear plate-buckling, among others; compliant deflection functions supply domain relations.

2) The von Misses shear condition was shown to correlate exactly with the behavior of all-clamped rectangular plate in pure shear.

3) Using the same method new results are found for the “SSSS” plate; they are about 60-percent of currently held values. The imposition of stiffeners introduces boundary conditions different from the “SSSS” and so the present results, without stiffeners, appear more realistic.

4) The presently held “SSSS” shear values are, here, found corresponding to those of a plate clamped on Y-Y and simply-supported on the long side, X-X, with very good accuracy.

5) It is, therefore, concluded that the pure-shear results for the “SSSS” plate had not been found until the new results presented here: for “a/b = 1”, N_{xy} = 5.66ρ and not 9.3ρ. The difference is huge with respect to safety and frequency of maintenance of vessels.

The study reported here is original and there is no conflict of interest whatsoever.

Johnarry, T.N. and Ebitei, F.W. (2020) Von Misses Pure Shear in Kirchhoff’s Plate Buckling. Open Journal of Civil Engineering, 10, 105-116. https://doi.org/10.4236/ojce.2020.102010

a, b: rectangular plate dimensions in X, Y

s*: aspect ratio, a/b

E: Young’s modulus of elasticity

t: thickness of plate

D: flexural rigidity of plate, (isotropic);

w: deflection symbol; w_{sh} = shape-function value

w_{xx}_{-r}; w_{yy}_{-r}: relative curvature in X-direction; Y-direction

w_{xx}_{-r}/w: relative-curvature/deflection ratio; must be a scalar for any solution

XX-SC, YY-CC: plate simply and clamped on X-X; and clamped-clamped on Y-Y

r_{cap}: capacity ratio of axes as,

m, n: wave numbers

_{cr}: critical stress symbol; critical buckling load symbol