effects of shell-stiffening on the stability of circular ... · stiffening shell is attached to the...

Procedia Engineering 48 ( 2012 ) 46 – 55

1877-7058 © 2012 Published by Elsevier Ltd.Selection and/or peer-review under responsibility of the Branch Offi ce of Slovak Metallurgical Society at Faculty of Metallurgy and Faculty of Mechanical Engineering, Technical University of Košicedoi: 10.1016/j.proeng.2012.09.484

MMaMS 2012

Effects of Shell-stiffening on the Stability of Circular Plates

Dániel Burmeistera* aUniversity of Miskolc, Miskolc-Egyetemváros 3515, Hungary

Abstract

The present paper deals with the stability problem of a thin circular thin plate which is stiffened by a thin cylindrical shell on the external boundary or inside the plate. Assuming axisymmetric deformations and an axisymmetric in-plane load we determine the critical value of the load in order to clarify what is the effect of the stiffening shell. Using Kirchhoff's theory of thin shells and plates the paper presents the governing equations both for the circular plate and for the cylindrical shell, the boundary- and continuity conditions and last but not least numerical results as well. © 2012 The Authors. Published by Elsevier Ltd.

Selection and/or peer-review under responsibility of the Branch Office of Slovak Metallurgical Society at Faculty of Metallurgy and Faculty of Mechanical Engineering, Technical University of Košice.

Keywords: annular plates, shell, stiffened, stability, critical load

1. Introduction

In engineering practice we often meet thin plates loaded in their own plane. Stability problems of such structures are important as they determine the load capacity. Within this topic buckling of circular plates are especially interesting. A number of papers have been devoted to this question since the first paper was published: [1-9]. Here we have cited only a few of the papers devoted to the stability problem of circular plats. These contain further references.

In what follows we lay an emphasis only on those papers which deal with the influence of structural stiffening. The effect of elastic restrains on buckling has been investigated by Thevendran [10].

The stability of a structure can be increased in various ways. We can apply, for example, a corrugation on the structure. As regards the circular plates we can attach a stiffening to the plate along its diameter [11]. However the stability issues are left out of consideration in the paper cited. A further paper by Turvey and his co-authors deals with a ring stiffened circular plate [12]. This paper is concerned among others with experimental results, however the stability issues are again left out of consideration.

The influence of a stiffening ring on the stability on annular plates is investigated in two papers written by Frostig and his co-authors [13,14]. The ring model is based on the theory of curved beams. The stiffening ring is attached to the annular plate between the two boundaries. It turns out from the references that the authors do not know about the corresponding results of Szilassy [15,16].

Szilassy dealt with the stability of annular plates stiffened by a cylindrical shell on the outer boundary in his PhD. thesis [15] and in a further article [16]. It was assumed that (i) the load is an in-plane axisymmetric dead one and (ii) the deformations of the annular plate and the cylindrical shell are also axisymmetric. As regards the boundary value problem for

* Corresponding author. Tel.: +36-30-348-0250. E-mail address: [email protected].

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47 Dániel Burmeister / Procedia Engineering 48 ( 2012 ) 46 – 55

the shell he used the solution of a differential equation set up for the rotation field. For the cylindrical shell the solution was based on the theory of thin shells.

The main objective of the present paper is to clarify the effects of a stiffening shell on the stability of circular plates. The stiffening shell is attached to the plate either at the external boundary or at an arbitrary inner radius on the middle surface. We use the differential equation set up for the displacement field perpendicular to the middle plane of the plate.

The paper is organized in five sections. Section 2 shortly outlines the physical problem to be solved. Section 3 presents the governing equations both for the circular plate and for the cylindrical shell. The stability problem of a solid circular plate is then solved in Section 4. There we set up the boundary and continuity conditions which results in the nonlinear equation providing the critical load. Numerical solutions are also presented. Section 5 is a summary of the results.

2. Problem formulation

In Fig. 1. the cross section of a shell-stiffened circular plate is shown. The radii of the circular plate and the middle surface of the cylindrical shell are denoted by Re, and Rs, respectively. If the shell is attached on the outer boundary of the plate, then Rs=Re. The thicknesses of the plate and the shell are bp and bs. The shell is symmetric with respect to the middle plane of the plate. Its height is h. The load is a constant radial dead load distributed along the external boundary with the intensity fo. It is acting in the middle plane of the plate.

Fig. 1. The structure and its load

For our investigations we shall assume that the plate and the shell are thin. Consequently we can apply the Kirchhoff theory of plates and shells. We shall also assume that the kinematic equations and the material law are linear. Heat effects are not taken into account. The material of the plate and the shell is homogenous and isotropic for which Ep, Es and p, s are the Young modulus and the Poisson ratio.

Under the assumption of small, axisymmetric and linearly elastic deformations we shall determine (a) the critical load of the structure and (b) the effect of the stiffening shell on the critical load.

Equations for the cylindrical shell are presented in the coordinate system ( , , ). The coordinate surface =z, =0 coincides with the middle surface of the shell with radius Re. The polar angle is the same in the two coordinate systems (due to the axisymmetry it plays, however, no role in the investigations). The coordinate curves and the corresponding displacements u , u , and u on the middle surface of the shell are shown in Fig. 3.b.

We use the cylindrical coordinate system (R, ,z) for the plate. On the middle plane z=0. The displacements on the middle surface in the directions R, and z are denoted by u, v and w respectively - see Fig. 3.a. which shows the corresponding coordinate curves on a circle with radius R - =R/Re is a dimensionless coordinate.

We separate the structural elements mentally in order to solve the problem. Fig. 2. shows the outer part of the plate, the cylindrical shell and the inner part of the plate together with the inner forces fo, f1, f2 and the bending moments M1 and M2, which are acting between these elements.

If the deformations are axisymmetric and the load on the shell is exerted in direction , the only displacement component on the middle surface which is different from zero is u =u ( ). It is also obvious that u=u(R), w=w(R).


Fig. 2. Plate and shell in separation

Fig. 3. Coordinate curves in the coordinate systems

3. Governing equations

3.1. Governing equations for the cylindrical shell

If the deformations are axisymmetric, the displacement u of the middle surface of a cylindrical shell should fulfill the following differential equation [17, Chapter 15, p. 468]

44

41 1

d 14 ,

d s s s

u Nu p

I E Rζ ξ

ζβ νξ

+ = − − (1)

where p is the constant radial load exerted on the middle surface of the shell (its value is zero in the present case), N is the inner force in direction (its value is zero as well). We also introduce the following notations

( )1

1 22 3 24

1 1

13(1 ) , , /12 , / (1 .)e

o s o s s s s ss e

RI b E E

b Rν ν β ν ν= − = = = − (2)

The distributed loads f1, f2 and the bending moments M1 and M2 constitute the load of the shell, i.e. there is no load on the middle surface (p=0) and the load in the direction is also zero (N =0). The solution of equation (1) in the interval � [0,h] assumes the form


44

1 11

( ) ( ) ; / 0 ,i i p p s si

u aV u u p I Eζ ζ ζξ βξ β=

= + = − = (3)

where Vi( ) (i=1,…,4) denote the Krylov-functions - see Eqns. (42) in the appendix for their definitions. The shear force and bending moment in the shell can be given in terms of the solution for the displacement u :

3 2

1 1 1 13 2

d dand .

d ds s s s

u uQ I E M I Eζ ζ

ζ ξξ ξ= = − (4)

We can obtain the solution for u as a superposition of the following two partial loads: By assumption the distributed forces f1 and f2 are exerted on the intersection line of the two middle surfaces - see Fig. 4.a.

for details. The corresponding boundary (discontinuity) conditions are as follows:

1 2

0,

2

f fQζ ξ =

−= − (5)

0

d0 ,

d

uζ

ξξ =

= (6)

0 ,h

Qζ ξ == (7)

0 .h

Mξ ξ == (8)

The rotation about the axis is zero, because u ( )=u ( ) due to the load, - cf. equation (6). The other boundary conditions are obvious.

Fig. 4. The first partial load of the shell

The shell is subjected to the bending moments M1 and M2 as shown in Fig. 4.b. Now we have the following boundary (discontinuity) conditions:

0( ) 0 ,u ξζ

== (9)

2 1

0,

2

M MMξ ξ =

−= − (10)

0 ,h

Qζ ξ == (11)

0 .h

Mξ ξ == (12)

For this partial load it holds that u ( )=-u ( ). Consequently the displacement in the direction should be zero at =0. The other boundary conditions are again obvious.


It follows from the symmetry of the problem that it is sufficient to determine the solution for the shell in the interval � [0,h]. The distributed forces f1 and f2, the bending moments M1 and M2 are involved in the solutions for the partial loads as

unknown parameters. We can calculate these quantities from the geometrical continuity conditions, which should be prescribed on the intersection line of the middle surfaces of the plate and the shell. These have the following forms:

0 00s sR R R Ru u uζ ξ= − = +=

= = (13)

and

0 00 00

dd d.

d d ds s

s s

R R R RR R R R

uw w

R Rζ

ξ

ϑ ϑξ= − = +

= − = +=

= − = = = − (14)

After some paper and pencil calculations, which are not presented here, we obtain from the continuity conditions (13) and (14) that

( ) ( )1

3 2

2 1 2 120

d cos 2 cosh 2 2 1

d sinh 2 sin 2o s

s s

u R h hM M M M

E b h h bζ

ξ

κ

ν β β κξ β β=

+ += − − = − −−

(15)

and

3

2

1 2 1 20

cos 2 cosh 2 2( ) ( ) ,

2 sin 2 sinh 2o s

s

R h hu f f f f

E b h hζ ξ

α

ν β β αβ β=

+ += − − = − −+

(16)

where and are defined by the above relations.

3.2. Deformation of the annular plate, governing equations for the in-plane load

If the in-plane load exerted on the outer boundary is axisymmetric, the inner forces NR, N , NR in the plate are axisymmetric as well. Here we use the cylindrical coordinate system (R, ,z). In terms of the dimensionless coordinate , the inner forces NR and N take the form

2 2, .R

B BN A N Aϕρ ρ

= − + = − − (17)

Since the problem is an axisymmetric one it holds that NR =0. The integration constants A and B depend on the boundary conditions. For the inner circular plate

2 and 0 .A f B= = (18)

For the outer annular plate, both loads appear in the constants as follows

( )2 2

112 2

and ,1 1o s s

os s

f fA B f f

ρ ρρ ρ

−= = −− −

(19)

where s=Rs/Re. The radial displacement can also be given in terms of A and B:

( ) ( )21 1 .

R Bu A

bE Rν ν= − − − + (20)


Substituting (19) into (20) and utilizing then the continuity condition (13) and equation (16) we obtain that the inner forces (that is the constants A and B) in the outer plate can also be given in terms of one load parameter. For example

( ) ( )( )2 2

22

111 1 ,

22 1s s s

p s

RA f

Eb

νρ ν νρνα ρ

−− + −= + − +−

(21)

( ) ( )2 2 2 2

22

1 11 1 1 .

2 2 1s s s s s

p s

RB f

Eb

ν ρ νρ ρ ρν να ρ

− + + − −= + − + −−

(22)

3.3. Deformation of the annular plate, equations for the displacement field after stability loss

Under the assumption of axisymmetric deformations, all quantities depend only on the radial coordinate R. The displacement w on the middle plane of the plate in the direction z (the deflection) should satisfy the differential equation

2

21 1 1 1

1 d 1 d

d,

1

dH H R z

w ww N N p

I E R R R I EϕΔ Δ − + = (23)

where pz=0 and

23 2

1 1 2

d 1 d 1 d d/12 , / (1 ) , .

d d d dp p p p p HI b E E RR R R R R R

ν= = − Δ = + = (24)

By introducing the notations

2 2

1 1 2 2

, , ,ep p

f A BR

I E f f= = =A B (25)

and using equation (17) together with the dimensionless variable , equation (23) takes the form

2 2

2 2 2 2

d 1 d d 1 d0 ,

d d d d

w ww

ρ ρ ρ ρ ρ ρ ρ ρΔΔ + − + + − − = Δ = +B B

A A (26)

For a circular plate the above equation has a closed form solution (this is not valid for an annular plate)

1 1 2 2 3 3 4 4( ) ,w c Z c Z c Z c Zρ = + + + (27)

1 2 3 41, ln , ( ), ( ) ,o oZ Z Z J Z Yρ ρ ρ= = = = (28)

where ci, (i=1,…,4)$ are undetermined constants of integration while Zi denote the linearly independent particular solutions in which Jn(

½ ) and Yn(½ ), n=0,1,2,3,… are the Bessel functions of integer order.

If the plate is an annular one, the solution of equation (26) is determined by using numerical methods. We transform it into four differential equations of order one. Then we compute four linearly independent particular solutions numerically - the numerical solution involves four integration constants as well - and use them in the solution algorithm. For the sake of brevity further details are, however, omitted here.

The rotation, the shear force and the bending moment can all be given in terms of the solution for w as follows

d

d,

w

Rϑ = − (29)

2

1 1 2

d 1 d,

d dR p p

w wM I E

R R Rν= − + (30)

2

1 1 2

d d 1 d d.

d d d dR p p R

w w wQ I E N

R R R R R= + − (31)


4. Numerical results

4.1. Stability of a circular plate stiffened on the outer boundary

If the plate is stiffened on its external boundary, - see Fig. 5. - the solution for the deflection is the closed form solution.

Fig. 5. Circular plate stiffened on the outer boundary

The boundary conditions for the plate are as follows ( s=1)

0finite ,w ρ =

= (32)

00 ,ρϑ

== (33)

10 ,RQ ρ =

= (34)

1 1.RMρ ρϑ κ

= == − (35)

Boundary condition (34) follows from the global equilibrium of the structure while condition (35) reflects relation (15). Let us substitute solution (27) into relations (29), (30) and (31) valid for the physical quantities and then utilize the above

boundary conditions. After some calculations, in which the properties of the Bessel functions should be taken into account, we get a non-linear equation for the critical load

1 11 1( ) (1 ) ( ) ( ) 0 .p p

oe

I EJ J J

Rκ ν− − − = (36)

In accordance with the notations introduced let o = Re2fo/I1pE1p and 2 = Re

2fo/I1pE1p be dimensionless quantities which belong to fo and f2. We remark that the critical value of a quantity (load or an inner force) is denoted by the subscript cr, for instance o cr.

Fig. 6. Critival load o cr against the height of the shell

A program has been written in Fortran 90 to solve the non-linear equation (36) for the dimensionless critical load o cr

and compute 2 cr, fo cr, f2 cr. The computational results are presented in Fig. 6. if the thickness of the plate and shell are the same i.e. bp=bs. We remind the reader that the material of the plate and the shell are also the same Ep=Es, p= s. It is clear


from the graphs that the shell significantly increases the stiffness of the plate. The curves are asymptotic i.e. the height of the plate does not affect the critical load if the height is larger than a certain value.

If the plate and the shell are made of different materials, the effect of the stiffening on the critical load 2 depends on the ratio Es/Ep. The results are shown in Fig. 7. It is also clear from the graph that the critical load of plates stiffened by high shells tends to that of those plates clamped on the outer boundary - see the red horizontal line in Fig. 7. The greater the quotient Es/Ep the faster the convergence of 2 cr to this limit.

Fig. 7. Effect of the material of the shell

The effect of the thickness of the stiffening shell is presented in Fig. 8., where 2 cr is shown against the quotient h/Re

for various values of bs/bp. The effect of the changes in the thickness values is more significant than that of the different materials. High shells with greater or equal thickness than the thickness of the plate provide critical loads closer to the critical load of clamped plates.

Fig. 8. Effect of the thickness of the shell

4.2. Stability of a circular plate stiffened at an arbitrary radius

In order to determine the critical load of a plate which is stiffened by a cylindrical shell at a radius inside the plate, we shall clarify the boundary- and continuity conditions. The conditions in the center of the plate are the same as in equation


(32) and (33). The conditions prescribed on the intersection line of the middle surfaces of the shell and the plate include the continuity of the displacement and rotation and the influence of the shell on the bending moment -- see eq. (15).

0 0,

s sw wρ ρ ρ ρ= − = +

= (37)

0 0,

s sρ ρ ρ ρϑ ϑ= − = +

= (38)

( )0 0.

s s sR RM Mρ ρ ρ ρ ρ ρϑ κ

= = − = += − − (39)

The outer boundary of the plate is free, consequently the bending moment and the shear force vanish if =1

10 ,RM ρ =

= (40)

10 .RQ ρ =

= (41)

The solutions for w involve two constants (one for the inner part of the plate and one for the outer part of the plate). These represent the rigid body motions in the vertical direction. One of them can be set to zero, the other should meet a continuity condition if = s<1. Consequently there are 7 homogeneous boundary and continuity conditions for determining the 7 integration constants. These equations involve as a parameter. The critical value of can be determined from the condition that the system determinant should vanish.

The computational results for such a plate are presented in Fig.9. The graph shows the critical load o of the structure against the radius of the shell. It is clear from the graph that the largest value of the critical load belongs to a certain radius inside the plate.

Fig 9. Influence of the radius of the shell

5. Concluding remarks

In accordance with the objectives in the introduction we have summed up the differential equations for the stability problem of shell stiffened circular plates and shell stiffened annular plates under the assumption of axisymmetric deformations. In addition we have clarified what the continuity conditions are between the plate and the cylindrical shell. We have developed appropriate algorithms which make possible to determine numerical solutions for the critical load.

Numerical solutions are provided for a solid circular plate for various types of shell-stiffening. According to the results obtained the stiffening shell increases the value of the critical load significantly.

The numerical solutions for a plate stiffened at an inner radius shows the influence of the radius of the shell. The value of the critical load depends on s and has a maximum if s 0.72.


Acknowledgement

This work has been carried out as part of the TÁMOP-4.2.2/B-10/1-2010-0008 project in the framework of the New Hungarian Development Plan. The realization of this project is supported by the European Union, co-financed by the European Social Fund.

References

[1] Bryan, G. H., 1890. On the stability of a plate under thrust in its own plane with applications to the "buckling" of the sides of a ship, Proceedings of the London Mathematical Society, p. 54-67.

[2] Nádai, A., 1915. Über das Ausbeulen von Kreisförmingen Platten, Zeitschrift des Vereines deutscher Ingenieure, 11, p. 221-224. [3] Meissner, E., 1933. Über das Knicken kreis-ringförrmiger Scheiben, Schweizerische Bauzeitung, 101, p. 87-89. [4] Kawamoto, M., 1936. The stability of a thin circular plate with a concentric circular hole, Journal of the Japan Society of Mechanical Engineers, 39,

p. 449-450. [5] Iwato, S., 1939. Stability of circular plate with a centre hole, Journal of the Japan Society of Mechanical Engineers, 42, p. 663-664. [6] Yamaki, N., 1959. Buckling of a thin annular plate under uniform compression, Journal of Applied Mechanics, ASME, p. 267-273. [7] Mansfield, E.H., 1960. On the buckling of an annular plate, The Quarterly Journal of Mechanics and Applied Mathematics, 13, p. 16-23. [8] Majumdar, S., 1971. Buckling of a thin annular plate under uniform compression, AIAA Journal, 9, p. 1701-1707. [9] Ramaiah, G.K., Vijayakumar, K., 1975. Elastic stability of annular plates under uniform compressive forces along the outer edge, AIAA Journal, 13,

p. 832-834. [10] Thevendran, V., Wang, M. C., 1996. Buckling of Annular Plates Elastically Restrained against Rotation along Edges, Thin-Walled Structures, 25(3),

p. 231-246. [11] Turvey, G. J., Salehi, M., 2008. Elasto-plastic large deflection response of pressure loaded circular plates stiffened by a single diameter stiffener,

Thin-Walled Structures, 46, p. 996-1002. [12] Turvey, G. J., Der Avanessien, N. G. V., 1989. Axisymmetric elasto-plastic large deflection response of ring siffened circular plates, International

Journal of Mechanical Sciences, 31(11-12), p. 905-924. [13] Frostig, Y., Simitses, G. J., 1987. Effect of boundary conditions and rigidities on the buckling of annular plates, Thin-Walled Structures, 5(4),

p. 229-246. [14] Frostig, Y., Simitses, G. J., 1988. Buckling of ring-stiffened multi-annular plates, Computers & Structures, 29(3), p. 519-526. [15] Szilassy, I., 1971. Stability of circular plates with a hole loded on their outer boundary, PhD thesis, University of Miskolc (in Hungarian) [16] Szilassy, I., 1976. Stability of an annular disc loaded on its external flange by an arbitrary force system, Publ. Techn. Univ. Heavy Industry. Ser. D.

Natural Sciences, 33, p. 31-55. [17] Timoshenko, S., Woinowsky-Krieger, S., 1987. Theory of Plates and Shells, McGraw-Hill, 2nd edition, New-York

Appendix A. Krylov functions

For completeness we present the definition of the Krylov functions here:

( )

( )

1 2

3 4

1cosh cos , cosh sin sinh cos ,

21 1

sinh sin , cosh sin sinh cos .2 4

V V z

V V z

βξ βξ βξ βξ β βξ

βξ βξ βξ βξ β βξ

= = +

= = −(42)

effects of shell-stiffening on the stability of circular ... · stiffening shell is attached to the...

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