A revisit to the bending problem of a thin elliptic aelotropic plate with simply supported edge and uniform lateral load

0044-2275/07/020318-12 DOI 10.1007/s00033-006-6040-4

c

° 2006 Birkh¨ auser Verlag, Basel

Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

A revisit to the bending problem of a thin elliptic aelotropic plate with simply supported edge and uniform lateral load

Kjell Eriksson

Abstract. The complex solution method of Okubo for the deflection of a thin circular aelotropic plate with simply supported edge and uniform lateral load was extended to an elliptic plate by Ohasi. In his work however several inconsistencies appear, of which at least one disqualifies a central part. From a revisit to the works of Okubo and Ohasi a new solution for the deflection of a thin elliptic aelotropic plate with simply supported edge and uniform lateral load emerged.

The solution is a generalisation of Okubo’s solution and is valid for any angle between material and geometric principal axes. Previously known solutions, including those for circular plates, are reproduced as special cases of the new solution and results of numerical calculations in new situations appear reasonable.

Mathematics Subject Classification (2000). 00L0

Keywords. Plate, anisotropic, simply supported, elliptic, circular.

1. Introduction

The bending theory of a thin circular anisotropic plate clamped on the boundary and carrying a uniform lateral load is surprisingly simple and straightforward, see e.g. Timoshenko and Woinowsky–Krieger [1]. This is also true for an elliptic plate in which the material and geometric principle axes coincide. The analysis of a simply supported circular or elliptic anisotropic plate, on the other hand, is much more complicated. Okubo [2] used the complex variable method of Morkovin [3]

to derive a solution for the circular plate and his method was later extended to

an elliptic plate by Ohasi [4]. However, several errors were found in [4] of which

at least one disqualifies a central result. The parts concerned were revisited and a

new solution of the bending problem of a thin elliptic aelotropic plate with simply

supported edge and uniform lateral load, which is valid for any angle between

the material and geometric axes, was derived. Numerical evaluations agree with

previously known results when available and yield reasonable results when applied

to new situations.

2. The plate equation and Okubo’s solution

For a material with three mutually perpendicular planes of elastic symmetry the plate equation takes the form [1]

D 1

∂ ⁴ w

∂x ⁴ + 2(D 2 + D 4 ) ∂ ⁴ w

∂x ² ∂y ² + D 3

∂ ⁴ w

∂y ⁴ = q (1)

in Okubo’s notation. Here w is the deflection of the plate, q the intensity of the lateral load and D i i = 1, · · 4 bending rigidities. The notation of Okubo is retained for simplicity and to facilitate comparison with previous works. ¹

A solution to Eq. (1) can be found with the complex variable method [3]. A simplified form of the homogeneous part w H of w, used by Okubo, is

w H = Re [f 1 (x + ik 1 y) + f 2 (x + ik 2 y)] (2) where f 1 and f 2 are arbitrary functions and k 1 and k 2 roots of the characteristic equation

D 1 − 2 (D 2 + D 4 ) k ² + D 3 k ⁴ = 0, (3) obtained through substitution of Eq. (2) in (1), i.e.:

k ² 1,2 = D 2 + D 4 ± p(D 2 + D 4 ) ² − D 1 D 3

D 3

. (4)

The general solution to Eq. (1) is then

w = w P + w H (5)

where w P is a particular solution.

For a circular plate with unit radius, the Cartesian coordinates are:

x + iy = e ^α+iβ (6a)

where e ^α = 1 and 0 ≤ β ≤ 2π.

The Cartesian coordinates are transformed into the following curvilinear coor- dinates:

x + ik 1 y = c ^′ cosh(α ^′ + iβ ^′ ) (6b)

x + ik 2 y = c ^′′ cosh(α ^′′ + iβ ^′′ ) (6c) with the geometric conditions

c ^′ cosh a ^′ = a c ^′ sinh a ^′ = ak 1 (7a)

1

In current notation D

= D

, D

= D

, D

= D

/2 and D

= D

.

c ^′′ cosh a ^′′ = a c ^′′ sinh a ^′′ = ak 2 . (7b) The boundary of the plate is here expressed by α ^′ = a ^′ and α ^′′ = a ^′′ , respec- tively. By allowing c ^′ , a ^′ etc. to be complex themselves and taking β real we can stipulate that β ^′′ = β ^′ = β on the boundary.

From Eq. (7) the parameters c ^′ , a ^′ etc. are obtained a ^′ = atanh(k 1 ) c ^′ = a

q

1 − k 1 ² (8a)

a ^′′ = atanh(k 2 ) c ^′′ = a q

1 − k ² ₂ (8b)

where a = 1 for a plate of unit radius.

Further, from Okubo’s [2] assumption that

f 1 ^′′ (x + ik 1 y) =

∞

X

n=2

A n cosh 2n (α ^′ + iβ) (9a)

f 2 ^′′ (x + ik 2 y) =

∞

X

n=2

B n cosh 2n (α ^′′ + iβ) (9b) is obtained the deflection function w for a plate of unit radius, thickness and load intensity:

w = w H + w P

= c ^′2 4

∞

X

n=2

A n

½ cosh(2n + 2)α ^′

(2n + 2)(2n + 1) cos(2n + 2)β

−

· 1

(2n + 1)2n + 1 2n(2n − 1)

¸

cosh 2nα ^′ cos 2nβ + cosh(2n − 2)α ^′

(2n − 1)(2n − 2) cos(2n − 2)β

¾

+ c ^′′2 4

∞

X

n=2

B n

½ cosh(2n + 2)α ^′′

(2n + 2)(2n + 1) cos(2n + 2)β

−

· 1

(2n + 1)2n + 1 2n(2n − 1)

¸

cosh 2nα ^′′ cos 2nβ + cosh(2n − 2)α ^′′

(2n − 1)(2n − 2) cos(2n − 2)β

¾

+ C 1 x ⁴ + C 2 x ² y ² + C 3 y ⁴ + C 4 x ² + C 5 y ² + C 6 . (10)

Okubo’s solution can be applied to the special case in which material and

geometrical axes coincide. This condition is always fulfilled for a circular plate.

3. Arbitrarily oriented elliptic orthotropic plate (Ohasi)

In Okubu’s solution for a circular disk the material and geometric axes trivially coincide. Ohasi [4] considered a more general case in which material and geometric axes are different. In the first case, the functions f 1 and f 2 in Eq. (2) are complex conjugates, but in the second case this symmetry relation does not hold and thus a more general solution is required.

The most general form of solution of the thin plate differential equation, on a form involving odd-order derivatives can, according to Lechnitzky (see e.g. [5]), be expressed as

w H = f 1 (ζ 1 ) + ¯ f 1 (¯ ζ 1 ) + f 2 (ζ 2 ) + ¯ f 2 (¯ ζ 2 ) (11) where f 1 and f 2 are arbitrary functions and

ζ i = x + ik i y i = 1, 2, 3, 4 alternatively 1 ^′ , 1 ^′′ , 2 ^′ , 2 ^′′ (12) in which k i are solutions to Eq. (3)

k i = ± s

D 2 + D 4 ± p(D 2 + D 4 ) ² − D 1 D 3

D 3

(13) or k i = k 1 , k 2 , −k 2 , −k 1 where k 1 is chosen such that both its real and imaginary parts are positive. In the general case both k 1 and k 2 are complex and thus complex conjugates of each other. From the ordering of the roots it follows that

ζ 1 = x + ik 1 y (14a)

ζ 2 = x + ik 2 y (14b)

ζ 3 = x − ik 2 y (14c)

ζ 4 = x − ik 1 y (14d)

and further that

ζ 3 = ¯ ζ 1 (15a)

ζ 4 = ¯ ζ 2 . (16a)

We now consider Ohasi’s elliptic plate with the ratio of 3:1 of the major to the

minor axis, with lengths 2a and 2b, respectively. The bending stiffness values in

this case are [4]

D 1 = 10.19 D 2 = 6.687 D 3 = 22.62 D 4 = 6.667 which inserted in Eq. (13) yield

k 1,2 = 0.7942 ± i 0.2010.

Let x, y denote material axes and x ^′ , y ^′ geometrical axes in Cartesian coor- dinates. The angle between x and x ^′ is θ, taken positive clockcounterwise, and the major axis of the ellipse is oriented along the x ^′ -axis ² . The transformation between the two sets of axes is

x ^′ + iy ^′ = e ^{− iθ} (x + iy). (17) The equation of the ellipse takes the form

x ^′ + iy ^′ = c cosh (α o + iβ) (18) where α o and c are obtained from the relationships a = c cosh α o and b = c sinh α o . A pair (α, β) on and inside the ellipse is then mapped on (x, y) through the trans- formation

x + iy = e ^iθ c cosh(α + iβ). (19) The material coordinates are transformed into the following system of curvi- linear coordinates

x + ik i y = c i cosh (α i + iβ i ) . (20) As before, by allowing c i and α i to be complex, we can retain β real and stipulate that all β i = β with origin at x ^′ = a, y ^′ = 0 for the ellipse.

From the equation of the ellipse (18) and Eqs. (20) are obtained

c i cosh a i = c cosh α o (cos θ + ik i sin θ) (21a) c i sinh a i = c sinh α o (k i cos θ + i sin θ) (21b) and further the parameters

c i = c q

cosh ² α o (cos θ + ik i sin θ) ² − sinh ² α o (k i cos θ + i sin θ) ² (22a) a i = atanh µ k i cos θ + i sin θ

cos θ + ik i sin θ tanh α o

¶

(22b) where α i = a i on the boundary of (22).

When both k 1 and k 2 are complex, the solution scheme in Table 1 is obtained, that is, if k 1 yields a ^′ 1 and c ^′ 1 , then −k 2 yields a ^′′ 1 = −¯ a ^′ 1 and c ^′′ 1 = ¯ c ^′ 1 , etc.

It is easily seen that the functions f 1 and f 2 (as in [3]) allow a real-valued solution only in the cases when θ = 0 or π/2, that is, when the material and geometric axes coincide. Ohasi’s solution for θ = −π/4 is therefore in error or at

2

Note that Ohasi’s angle ϑ = −θ .

Table 1. Solution scheme.

i = 1

1

2

2

Root k

−k

k

−k

Parameter a

−¯ a

a

−¯ a

c

¯ c

c

¯ c

best approximate. Also, note that both root pairs (k 1 , −k 2 ) and (k 2 , −k 1 ) are required to allow a real-valued solution for an arbitrary angle between material and geometric axes. Further, it is not possible to determine four arbitrary constants (or sets of constants) of the solution with only two roots of Eq. (3), such as in Ohasi’s solution.

Inspired Okubo, we put

f 1 ^′′ (x + ik 1 y) =

∞

X

n=2

A n cosh 2n (α ^′ 1 + iβ) (23a) accompanied by

f ¯ ^′′ 1 (x − ik 2 y) =

∞

X

n=2

B n cosh 2n (α ^′′ 1 + iβ) (23b)

f 2 ^′′ (x + ik 2 y) = −i

∞

X

n=2

A ^′ n cosh 2n (α ^′ 2 + iβ) (23c)

f ¯ ^′′ 2 (x − ik 1 y) = −i

∞

X

n=2

B _n ^′ cosh 2n (α ^′′ 2 + iβ) (23d) from which are selected a symmetric part from the two first expressions and an anti-symmetric part from the latter to yield

w = w H + w P

= c ^{′ 2} 1

4

∞

X

n=2

A n

½ cosh(2n + 2)α ^′ 1

(2n + 2)(2n + 1) cos(2n + 2)β

−

· 1

(2n + 1)2n + 1 2n(2n − 1)

¸

cosh 2nα ^′ 1 cos 2nβ + cosh(2n − 2)α ^′ 1

(2n − 1) (2n − 2) cos(2n − 2)β

¾

+ c ^{′′ 2} 1

4

∞

X

n=2

B n

½ cosh(2n + 2)α ^′′ 1

(2n + 2)(2n + 1) cos(2n + 2)β

−

· 1

(2n + 1)2n + 1 2n(2n − 1)

¸

cosh 2nα ^′′ 1 cos 2nβ

+ cosh(2n − 2)α ^′′ 1

(2n − 1) (2n − 2) cos(2n − 2)β

¾

+ c ^{′ 2} 2

4

∞

X

n=2

A ^′ _n

½ sinh(2n + 2)α ^′ 2

(2n + 2)(2n + 1) sin(2n + 2)β

−

· 1

(2n + 1)2n + 1 2n(2n − 1)

¸

sinh 2nα ^′ 2 sin 2nβ + sinh(2n − 2)α ^′ 2

(2n − 1)(2n − 2) sin(2n − 2)β

¾

− c ^{′′ 2} 2

4

∞

X

n=2

B ^′ n

½ sinh(2n + 2)α ^′′ 2

(2n + 2)(2n + 1) sin(2n + 2)β

−

· 1

(2n + 1)2n + 1 2n(2n − 1)

¸

sinh 2nα ^′′ 2 sin 2nβ + sinh(2n − 2)α ^′′ 2

(2n − 1)(2n − 2) sin(2n − 2)β

¾

+ C 1 x ^{′ 4} + C 2 x ^′3 y ^′ + C 3 x ^{′ 2} y ^′2 + C 4 x ^′ y ^′3

+ C 5 y ^′4 + C 6 x ^′2 + C 7 x ^′ y ^′ + C 8 y ^′2 + C 9 (24) where the particular solution w P is taken from [4].

The general solution, Eq. (24), inserted into the differential equation (1) yields

{3D 1 C 1 + (D 2 + D 4 ) C 2 + 3D 3 C 5 } cos ⁴ θ

+ 3 {(D 2 + D 4 − D 1 ) C 2 − (D 2 + D 4 − D 3 ) C 4 } cos ³ θ sin θ

+ {3 (D 1 + D 3 ) C 3 + 2 (D 2 + D 4 ) (3D 1 − 2D 3 + 3D 5 )} cos ² θ sin ² θ + 3 {(D 2 + D 4 − D 1 ) C 4 − (D 2 + D 4 − D 3 ) C 2 } cos θ sin ³ θ

{3D 3 C 1 + (D 2 + D 4 ) C 2 + 3D 1 C 5 } sin ⁴ θ = q/8. (25) Equation (24) must also satisfy the boundary conditions vanishing deflection and vanishing bending moment M α on the plate edge. The solution constants A n etc. are here called boundary condition constants and as such denoted b.c.

constants. The condition that w vanishes at the plate boundary has the structure

£3C 1 c ⁴ cosh ⁴ α o + · · · ¤ +

·

C 1 c ⁴ cosh ⁴ α o + · · · + A 2

12 c ^{′ 2} 1 cosh 2a ^′ 1 + B 2

12 c ^{′′ 2} 1 cosh 2a ^′′ 1

¸ cos 2β +

·

C 2 c ⁴ sinh α o cosh ³ α o + · · · − 4A 2

15 c ^{′ 2} 1 cosh 4a ^′ 1 − 4B 2

15 c ^{′′ 2} 1 cosh 4a ^′′ 1

¸ cos 4β + 1

2

·

C 1 c ⁴ cosh ⁴ α o + · · · − A ^′ 2

6 c ^{′ 2} 2 sinh 2a ^′ 2 + B 2 ^′

6 c ^{′′ 2} 2 sinh 2a ^′′ 2

¸

sin 2β

+

·

C 2 c ⁴ sinh α o cosh ³ α o + · · · + 4A ^′ 2

15 c ^{′ 2} 2 sinh 4a ^′ 1 − 4B ^′ 2

15 c ^{′′ 2} 2 sinh 4a ^′′ 1

¸

sin 4β = 0 (26a) and further

½ A n−1

2n − 1 −

µ 1

2n − 1 + 1 2n + 1

¶

A n + A n+1

2n + 1

¾

c; ² 1 cosh 2na ^′ 1

+ ½ B n−1

2n − 1 −

µ 1

2n − 1 + 1 2n + 1

¶

B n + B n+1

2n + 1

¾

c ^{′′ 2} 1 cosh 2na ^′′ 1 = 0 (26b)

½ A ^′ n−1

2n − 1 −

µ 1

2n − 1 + 1 2n + 1

¶

A ^′ n + A ^′ n+1

2n + 1

¾

c ^{′ 2} 2 cosh 2na ^′ 2

−

½ B ^′ n−1

2n − 1 −

µ 1

2n − 1 + 1 2n + 1

¶

B n ^′ + B n+1 ^′

2n + 1

¾

c ^{′′ 2} 2 cosh 2na ^′′ 2 = 0. (26c) In order to fulfil the boundary condition for any β the sum of all terms inside a bracket in Eq. (26a) must be zero for all brackets. All terms pertaining to the particular solution are identical to those of Ohasi and therefore not reproduced here. For brevity, only the terms of the homogeneous part of the present solution and their scaling are given. Equations (26) applies exclusively to the present solution.

The condition that M α vanishes at the disk boundary yields

[D 1 + D 2 + (D 1 − D 2 ) cos 2 (ϕ + θ)] ∂ ² w

∂x ² + [D 2 + D 3 + (D 2 − D 3 ) cos 2 (ϕ + θ)] ∂ ² w

∂y ² + 2D 4 sin 2 (ϕ + θ) ∂ ² w

∂x∂y = 0 (27) where ϕ is the angle between the normal to the curve α = α o and the x ^′ -axis and

∂ ² w

∂x ² =

∞

X

n=2

(A n cosh 2na ^′ 1 + B n cosh 2na ^′′ 1 ) cos 2nβ

+

∞

X

n=2

(A ^′ _n sinh 2na ^′ 2 − B _n ^′ sinh 2na ^′′ 2 ) sin 2nβ + [ · · · ] (28a)

∂ ² w

∂y ² = −

∞

X

n=2

¡A n k 1 ² cosh 2na ^′ 1 + B n k 2 ² cosh 2na ^′′ 1 ¢ cos 2nβ

−

∞

X

n=2

¡A ^′ _n k 2 ² sinh 2na ^′ 2 − B _n ^′ k ² 1 sinh 2na ^′′ 2 ¢ sin 2nβ + [ · · · ] (28b)

∂ ² w

∂x∂y = −

∞

X

n=2

(A n k 1 sinh 2na ^′ 1 − B n k 2 sinh 2na ^′′ 1 ) sin 2nβ

+

∞

X

n=2

(A ^′ _n k 2 cosh 2na ^′ 2 + B _n ^′ k 1 cosh 2na ^′′ 2 ) cos 2nβ + [ · · · ] . (28c) The terms inside the brackets pertain to the particular solution; they are iden- tical to those of Ohasi [4] and therefore not repeated here.

The first approximation, or the particular solution only, yields C 1 = 0.0000874 q C 4 = 0.000244 q C 7 = −0.000244 qb ²

C 2 = 0.0000271 q C 5 = 0.00235 q C 8 = −0.01341 qb ²

C 3 = 0.001048 q C 6 = −0.002016 qb ² C 9 = 0.01106 qb ⁴ .

At the boundary, w vanishes but the moment boundary condition is not fully satisfied, leaving a residual bending moment

M α = − 3.00 cos 4β + 2.60 sin 4β

1.250 − cos 2β 10 ⁻³ qb ² .

The second approximation, involving the particular solution and the first term of the series expansion of the homogeneous solution, yields

C 1 = 0.0000882 q C 4 = 0.000242 q C 7 = 0.000220 qb ²

C 2 = 0.0000235 q C 5 = 0.00235 q C 8 = −0.01340 qb ²

C 3 = 0.001034 q C 6 = −0.002026 qb ² C 9 = 0.0111 qb ⁴

A 2 = ¡1.010 · 10 ⁻⁴ + i 5.872 · 10 ⁻⁵ ¢

qb ² B 2 = ¡1.010 · 10 ⁻⁴ − i 5.872 · 10 ⁻⁵ ¢ qb ²

A ^′ 2 = ¡1.948 · 10 ⁻⁵ + i 1.2408 · 10 ⁻⁵ ¢

qb ² B 2 ^′ = ¡1.948 · 10 ⁻⁵ − i 1.2408 · 10 ⁻⁵ ¢ qb ² and the residual moment is

M α = − 3.67 cos 6β + 3.54 sin 6β

1.250 − cos 2β 10 ⁻⁷ qb ² .

A comparison with the residual moment for the first approximation indicates

that the solutions converge quickly.

In the cases of θ = 0 and π/2 we have

M α = − 3.67 cos 6β

1.250 − cos 2β 10 ⁻⁷ qb ² and M α = − 5.40 sin 6β

1.250 − cos 2β 10 ⁻⁷ qb ² respectively. The bending moments at the centre of the plate Eq. (4:21) ³ , with the replacement,

∂ ² w

∂x∂y = C 6 sin 2θ + C 7 cos 2θ − C 8 sin 2θ + A ^′ 2 k 2 + B 2 ^′ k 1 (29) are shown in Table 2.

Table 2. Bending moments M

qb

at plate centre.

Angle θ M

M

M

0 0.4140 0

−π/4 0.3771 0.2299 -0.0952

π/2 0.6601 0

Figure 1. Deflections along the geometrical axes of a 1:3 plate of Ohasis’s material.

The expressions for the deflection along the x ^′ - and y ^′ -axes do not contain any primed b.c. constants and are therefore identical to Eqs. (4:22, 23). The deflections along the two axes are shown graphically in Fig. 1. Also for convenience of comparison and in addition to provide an improved impression of the geometrical

3

Here (x : y) means Eq. (y) in [x].

effect of both types of boundary conditions the deflections for a plate with a clamped edge are also included. The values of the maximum deflection for all cases are shown in Table 3.

Table 3. Maximum deflection w 10

qb

.

Angle Boundary condition

Simply supported Clamped [4]

0 0.7983 0.1755

1.103 0.2542

π/2 1.532 0.3636

Discussion

This work is to some extent drawn on [4] where several results could not be repro- duced. First, the roots k _1,2 ² of the characteristic equation (3) do not correspond to the bending stiffness values given. Incidentally, they are identical to the roots given by Okubo in [2] (where, in turn, no bending stiffness values are stated).

Second, Eq. (4:13), which is proposed to yield the coordinate transformation con- stant β 1 , is valid only for real but not complex roots k of Eq. (3) and invalidates therefore the subsequent solution. It is easily seen that Eq. (4:13), as it stands, yields a complex solution.

Third, Ohasi’s solution is incompatible in the general case in that it contains four arbitrary constants (or sets of constants) to be determined but only two roots of Eq. (3). Fourth, there are terms missing or erroneous in Eq. (4:18), which is obtained by substituting the plate deflection expression (24) in Eq. (1).

Ohasi does not provide any numerical values of the deflections. Although not very accurate, values measured in Fig. 1 in [4] are close to but not identical to those in Table 3 for the simply supported plate. Because of symmetry of the solution functions, Ohasi’s solution should be valid in the two special cases θ = 0 and π/2 in which cases all primed b.c. constants are zero. However, the erroneous Eq.

(4:18) prevents an accurate solution. Further, a calculation for θ = −π/4 using Ohasi’s solution with Eq. (4:18) replaced by Eq. (25) still yields, for obvious reasons, complex material coefficients and accompanying complex b.c. constants that do not conjugate pair-wise. In a valid solution the b.c. complex constants must conjugate pair-wise in order to yield real material coefficients. As some of the omitted imaginary parts are of the same order as the corresponding real parts, such real parts, taken as an approximate solution, cannot be very accurate. Ohasi’s deflection curve is however a close approximation to the present solution but the reason for this relation remains unknown.

In the course of the analysis of Ohasi’s work a new solution to the simply

supported aeolotropic thin plate problem emerged, which is fundamentally differ-

ent from the previous attempt. The present solution is based on the observation that the solution of a fourth order differential equation, with four independent integration constants to be determined, requires an equal number of roots of the characteristic equation (1).

References

[1] Timoshenko, S.P., Woinowsky-Krieger, S., Theory of Plates and Shells, Int. Ed., McGraw- Hill, Singapore 1959.

[2] Okubo, H., Bending of a thin plate of an aeolotropic material under uniform lateral load (Supported edge), Journal of Applied Physics 20 (1949), 1151–1154.

[3] Morkovin, V., On the deflection of anisotropic thin plates, Q. Appl. Math. 1 (1943), 116–129.

[4] Ohasi, Y., Bending of a thin elliptic plate of an orthotropic material under uniform lateral load, Z. angew. Math. u. Phys. 3 (1953), 212–224.

[5] Huber, M. T., Probleme der Statik technisch wichtiger orthotroper Platten, Gebethner &

Wolff, Warszawa 1929.

Kjell Eriksson

Department of Solid Mechanics Lule˚ a University of Technology SE 971 87 Lule˚ a

Sweden

e-mail: kjell.eriksson@ltu.se

(Received: March 16, 2006)

Published Online First: September 8, 2006