ERRORS OF THE GECKELER APPROXIMATION FOR CONICAL SHELLS

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ERRORS OF THE GECKELER APPROXIMATION FOR CONICAL SHELLS

By

M. ORVOS, L. LEGEZA., and P. REUSS

Department of Chemical Machines and Agricultural Industries, Technical University, Budapest (Received May 15, 1975)

Presented by Prof. Dr. S. SZENTGYORGYI

Introduction

Stress analysis ofaxisymmetric conical shells is rather tedious by exact methods. Approximations of different accuracies may mean a simplification.

For the analysis of stresses and deformations in axisymmetric conical shells, one of the most 'widely extended approximations is that by Geckeler.

Outputs from a Geckeler approximation will be compared to those from a numerical method likely to be exact. Varying the wall thickness of the conical shell, and the half cone apex angle as parameters, the relative error of the approximation is determined, and geometry limits of its applicability are sug- gested. A case of an isotropic, homogeneous conical shell under axisymmetric edge force and edge moment is analysed.

Edge influence coefficients

Relationship based on the Geckeler approximation for edge forces and edge moments lending themselves for examining the variation of stresses and deformations along the generatrix have been published in [1].

Stress and deformation maxima are at the edge, therefore, often it is sufficient to restrict analyses there. The edge influence coefficients are [2]:

cos²(X;

W H = - - - .H, 2B k³ cos(X; M

WM = 2Bk₂ ' ,

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178

where

M. ORVOS ef al.

Ea

³

B = - - - - - 12(1 p..2)

4

k = V::-;3(-=-1 -_-fl,=2)

liRa

Assuming f-l

=

0,3, edge influence coefficients can be written in dimen- sionless form:

-.!:L ^w = 2,57 -

(R ^{)312 -}

1!coso:· - -

H

=

wH·H, -

R

a

ER

The calculation method for shells of revolution, serving as basis of com- parison, starts from the numerical solution of the couple of differential equa- tions deduced by REISSNER [3], based on the method of finite differences that will not be described here. For further applications see [4].

Discussion of results

Deformations obtained by the numerical method and by the G-eckeler approximation are compared in diagrams plotting the course of deformations along the cone generatrix (Figs 1, 2, 3, 4). Results from both methods are displayed on the same diagram.

Deformations are linearily dependent on edge loads, and subject to the principle of superposition. For the sake of simplicity, effect of unit edge force or unit edge moment has been examined. Courses are seen to be of identical character, and the approximation is irrelevant to the damping length.

Figs 5 and 6 show the percentage relative error of the G-eckeler approximation. The numerical method can practically be considered as exact, therefore, its results can be taken as references. Diagrams show edge deformation errors, with curve family parameters of wall thickness ratio

Rio

and half cone apex angle 0:, respectively, where 0: ranges from 10° to 85° and the wall thick-

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30

[if

[xlrYl

20

10

G

- 10

-20

ERRORS OF THE GECKELER APPROXThfATION

,

I

I i

~ I

i

A

ⁱ

lA

I ⁱ

~ ^~

I I

i

^"I-

I I b ,~

I ' . 'v

I P;,: =100 : / ^I

I

I ^o ^. I

I Ma=1 ^0{⁼^40' _I

I

I I I I I

I

,...--..

lUO,?5 0,5 0,7

^x

! ^[

/ !

1\ / i

I ... " :

!f

I

if

Ho= 10 - - Geckeler's approximation - - - numerical method

i

Fig. 1

.:0

r---~---~l---l

(xl0"J - - Geckeler's approximation

- - - numerical method I

O~~~-=~---~----~ _0,25 0,5

I

15. L

- 50 H - - - t _ _

Fig. 2

179

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180 101. ORVOS ., al.

w

!x1cf3]

30h---+---+---~

3;=100

0(= 80° Ra

Mr-::C:;,

^{: 1} ^{H;? Ho}

Ho=!

~=TI

20 H~---_;_- ~f i~ -,

- - GeCke:er's approximation ^1,1 - - - numerical method

10 r-~.----;---

O~---\~~~~==~~~~--~O'7~t~ ^I

-10~---1---+---__,

- 50

Iy.:

I ^Ho

^,

^Ho

- 100 ^~I

I

I !ft '"

¹⁰⁰

Mo'" 1 I ₀₍₌_80°

- 150 I _I

- - Geckeler's approximation I

I

i I

! - - - numjeriCdl method \ - 200

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-10

Li%

30

20

10

o

- 10

ERRORS OF THE GECKELER APPROXIMATION

I

^I

I I

20 40 60

Fig. 5

! ^I

i I

c£=t\

⁸⁰

₁ !\

i

'l

I

~

i

60~ I

i ^I

"---

^jⁱ

304- I ⁱ_I

30

4

ⁱ

80-DJ"'~ ^I

6~ _, _~ 8

(of

•

! 20 50

-

!

I

I i

, I I i

100 Fig. 6

80

i

!

i

I

i

150

0(0

I

: I i I

}

Ll~1 H

I

i

: i I

11

200 &.

d

181

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182 M. 6RV6s et al.

Error functions show the error of the displacement due to edge force to be the greatest. In the tested range, the error of angular rotations is below 5%, hence limits of applicability of the Geckeler approximation are governed by the error of the radial displacement due to the edge force. Accordingly, it can be stated that admitting a deviation of 6%, the half apex angle can be about rx 40°, and the wall thickness ratio RID;;:;;;: 50. The same error is com- mitted for r;.

<

^{55° and}^RID

>

100. The error only exceeds 10%, if con- ditions RID

<

150 and rx

>

80° coincide.

Legend:

B bending stiffness E modulus of elasticity H radial edge force k shell constant

L distance of cone edge from apex 1"\1[ meridional edge moment

R radius of the cone at the edge w radial displacement

x arc length along the generatrix, from the edge

r;. half cone apex angle

f3

angular rotation

o

wall thickness

Ll~ error of radial displacement due to an edge moment Llr error of angular rotation due to an edge moment Ll{!, error of radial displacement due to an edge force Ll~ error of angular rotation due to an edge force

Summary

Results from the Geckeler approximation are compared to those from a numerical method considered as exact. Varying the wall thickness ratio and the half cone apex angle as parameters, the relative err01:" of the approximation is numerically determined, and a sugges- tion is made on the geometrical limitations of the approximation applicability.

References

1. VARGA, L.: Naherungsverfahren zur Bestimmung der an drehsymmetrischen randbelaste- ten Kegelschalen angreifenden Schnittkriifte und Schnittmomente. Per. Po!. M. E.

Vol. 6, 4 (1962).

2. HAlIIPE: Statik rotationssymmetrischer Flachentragwerke. Berlin, 1967.

3. REISSNER, E.: On the Theory of Thin Elastic Shells, Reissner Anniversary Volume, J. W.

Edwards, Ann Arbor, Mich., 1949.

4. REUSS, P.: Elastic Stresses in Toriconical Pressure Vessel Heads. Per. Pol. M. E. Vol. 19, 1 (1975).

Maria ORVOS

l

Laszl6 LEGEZA

Dr. Pal

H-1521 Budapest

ERRORS OF THE GECKELER APPROXIMATION FOR CONICAL SHELLS