Buckling of orthotropic plates with the two free edges loaded for the pure in-plane bending moment Текст научной статьи по специальности «Физика»

A. V. Lopatin, R. V. Avakumov Siberian State Aerospace University named after academician M. F. Reshetnev, Russia, Krasnoyarsk

BUCKLING OF ORTHOTROPIC PLATES WITH THE TWO FREE EDGES LOADED FOR THE PURE IN-PLANE BENDING MOMENT

In this paper we have solved the buckling problem of orthotropic plates with two free and two simply-supported edges loaded for the pure in-plane bending moment. We have used the finite difference method to solve the problem.

Keywords: orthotropic plates, finite difference method.

The buckling problem of rectangular plates involving two opposite edges loaded with distributed stress has been studied for a long time. Bubnov [1] and Timoshenko [2] were the first who solved this problem for isotropic plates. The same buckling problem of orthotropic plates was solved by Lekhnitskii [3]. These classical solutions in the form of trigonometric series are obtained for plates with simply-supported edges. The Ritz energy method is used to define critical loads because the partial differential equation of stability involving the floating factor makes integration difficult. Reddy [4] and Whitney [5] used the Ritz method for solving this buckling problem of composite plates with simply-supported edges loaded even with compressed stress.

Therefore, the buckling of plates was studied mostly in widespreadboundary conditions i.e. simply-supported edges. There is only one reference to Nolke’s paper [6] in Bloom and Coffin’s manual [7],in which the same problem is solved for the plate with two clamped edges.

So we have decided to solve the buckling problem of the orthotropic plate with two free and two simply-supported edges loaded for the pure in-plane bending moment. We have not been solving the buckling problem of orthotropic plate loaded for line-distributed forces due to the limits. The solutions for buckling of orthotropic plate loaded for even compressed forces are given in Leissa’s manual [8].

In our research the solution of an original equation for stability analysis isbased onthe Levy-type solutionprocedure. It allows the reduction of partial differential equation to ordinary differential equation. The latteris solvedby the finite difference method. Linearhomogeneous algebraic combined equations had been used. The determination of the critical load is reduced to the calculation of buckling coefficient correspondingto minimal eigenvalue of combined equations. The solutions for isotropic plate buckling and symmetrically reinforced orthotropic plate had been obtained as well.

The buckling solution. We have considered an orthotropic plate the middle plane of which is corresponds to the Cartesian coordinates xy. The dimensions of the plate are a and b as shown inthe figure below. Edges along the y= 0 andy=b are free, and edges loaded with the line-distributed stress along the x = 0 and x = a are simply-supported. The load distribution corresponds to two bending in-plane moments.

The equationforthe stability analysis of orthotropic plate is given as follows

R4 w / s. R4 w ^ R4 w

D11 %2(D12 %2D33)- -

Rx4

2^ 2 % D22

Rx Ry

Ry4

R2 w , _ R2 w _ R2 w n

-N—- - 2N----N —- = 0,

x Rx2 xy RxRy y Ry2

where w = w(x,y) is the transverse displacement, D11, D12, D22, D33 are the bending stiffnesses of the plate given in [9]. N°x, N°y, №xy are membrane stresses corresponding to the subcritical state of the plate.

We assume that the origin stressed state corresponds to it’s pure in-plane bend. Then membrane subcritical stresses could be define as

N°x — - N ^1 - 2 b j , — 0, Ny — 0, (2)

where N is the maximal stress value at edges y = 0, y = b.

-W

/ 1 1 1 1

t> 1 1 1 1

y

Plate stress

Equation(l) with (2) taken in account is depicted as

R4w _ R4w

D Rx4 % 2(Dl2 % 2D,,) '&■ Ry'

%D22 0 % N Vl - 2b 1^ = 0.

b 1 Rx

(3)

Letus considerthe boundary conditions. Displacements and bending moments are equal to null when the edges are simply-supported (x = 0 h x = a). Generalized forces and bending moments are equal to null when the edges are free (y = 0 h y = b) [9]. These boundary conditions could be derived from plate displacement w can be presented as w — 0, D11 %D12— 0 atx =0andx = a, (4)

and

(1)

Rr2 dy1

R2 w R2 w

d2 —~—% D99 —~ — 0

12 Rx 22 Ry2 ,

R3 w R3 w

D22 -TT%(D12 %4D33)= 0 Ry Rx dy

at y =0and y = b. (5)

Thereby, the buckling problem of plate is reduced to the definition of parameter N corresponding to nonzero solution of the boundary-value problem (3,4) and (5).

The fact that there are edges simply supported makes the representationof solutionpossible (3) inLevy-type form

w

(^ y) = ( wm (y)sin Zmx .

(6)

a)2wt - 2pn2At %----- Bt - "t, wt — 0 .

Here m is the amount ofhalf-waves along side a, wm(y) is an unknown function, Z = m)/a. However, there no whereti = 1-2(i-1)/n

m

necessity to approximate the solution with series (6) in the problem we overlook. It’s enough to keep one term of the series corresponding to m = 1. Indeed, the buckling of the plate doesn’t experience resistance along side a. Therefore, the plate’s bend takes place with formation of only one halfwave along side a. This half-wave has a maximal amplitude at the edge y = 0, which decrease along y to the edge y = b.

Now the solution of the equation (3) canbe presented as w (x, y) — w (y) sin Zx . (7)

Here w(y) = w1(y), l = )/a.

Substituting of (7) into equation (3) gives the ordinary differential equation

a— fD! b_ p= D12 % 2D33

V D22 a VD11D22

Tl =-

Nb2

(15)

(16) (17)

4D11D22

Here " isa non dimensional buckling coefficient.

Equation (15) writing for all points (i = 1, 2, ..., k) represents linear algebraic combined equations, however, including outside edge points. The substituting of i = 1 and i = k to equalities (14) gives

A1 — w0 - 2w1 % w2,

D„Z w - 2(D % 2D33) D22 d4 w , „ y

d w ~dy2~'

and

2 4 -N1 1 -2^- Iw — 0.

Z2 dy4 I b

(8)

B1 — w-1 - 4w0 % 6w1 - 4w2 % w3

Ak — wk-1 - 2wk % wk%l,

Bk — wk-2 - 4wk-1 % 6wk - 4wk%1 % wk%2.

(18)

Here w = w(y).

Substituting of (7) into boundary conditions (5) gives

D12 Z w - D22

d w ~dy2~

d w

— 0.

dw

-D22 —3 % (D12 % 4D33 )Z — — 0

dy

dy

(9)

(10)

(19)

Therefore, w-1, w0, wk+1, wM are outside edge points. It should be noted that unknown w0 and wk+1 are also including in B2h Bk-1, giving

B2 — w0 - 4w1 % 6w2 - 4w3 + w4,

Bk-1 — wk-3 - 4wk-2 % 6wk-1 - 4wk % wk%1.

(20)

Four equations need to define four outside edge unknowns. These equations can be obtained by finite-The difference method is used to solve equation (8). Let’s difference approximation ofboundary conditions (9,10). By divide side b into numerous equal parts. The points of substitutingof(12)into(9),andof(10)with i =1h i = k, we

where y = 0, y = b.

The difference divide side b int division have the following coordinates

obtain

(11)

yt — s(i -1), i — 1,2, ..., k .

Here s = b/n and is subinterval, k = n + 1.

The approximationof derivatives corresponding arbitrary point yi gives

dwj 1 , s

~T I — 2“ (-wi-1 % wi%1), dy )t 2s

P1) w1--w - 2 w1 % w2) — 0,

a

n2

M1)2w-(wk-1 - 2wk % w%1) — 0,

a

where

dy2

P1 —

D,.

(1 )

4D11D2:

Equations (21) give

w0 — (r1 % 2)w1 - w2,

— (r1 % 2)wk - wk-1 =

(21)

(22)

(23)

— ^T(wi-2 - 4wi-1 % 6wi - 4wi%1 % wi%2 ) s

Here w! — w(yt).

Substituting of (12) into (8) gives the following finite difference approximation of equation (8):

D„Z2 w, - 2 D^

where r1 = aM1)2/n2.

Substituting of (12) into (10) with i = 1 and i = k gives

-—(-w-1 %2w0 -2w2 %w3)%(M1 %4M2))2(-w0 %w2) — 0,(24) L n2

-----(-wk-2 % 2wk-1 - 2wk%1 % wk%2) %

a

Zv Bi- N I1 - 2 wi— 0.(13)

where

%(M1 % 4M2 ))2 (-wk-1 % wk%1 ) — 0,

Here A and B are defined as

P2 —-

D3

(25)

a, — w,-1 - 2w, % w, %l,

b, — w,-2 - 4w,-1 % 6w, - 4w,%1 % w,%2. (14)

The transformation of equation (8) into dimensionless form gives

VD11D22

Equations (24) in accordance with (23) give w-1 — (2 + r1 + 4r2 )(r1 % 2)w1 - 2(2 % r1 % 4r2)w2 % w3, (26)

wk%2 — wk-2 - 2(2 % r1 % 4r2 )wk-1 %(2 % r1 % 4r2 )(r1 % 2)wk,

where r = aM2)2/n2.

So, equalities (23) and (26) define four outside edge unknowns. Substituting of (23) and (26) into (18,19) and (20) gives

Ai = ri w^ Ak = ri wk (27)

and

Bi = gwi - 2fw2 + 2W3,

B2 = vw1 + 5w2 - 4w3 + w4, (28)

Bk-1 = Wk-3 - 4wk-2 + 5wk-i + vwt,

Bk = 2wk-2 - 2fwk-i + gwk .

Here g = 6-4u + uf,f =2 + r1 + 4r2, u = r 1 + 2,n = r2 — 2. It should be noted that A. (i = 2, ..., k — 1) and B. (i = 3, ..., k — 2) are still definedby equations (14).

Thereby, homogeneous algebraic linear combined equations

n4

a$2wt - 2%n2Ai +----- Bt -&ti.wi. = 0 ,

a$

(i =1,2, ..., k) (29)

the approximated differential equation (8) contains unknown w1, w 2,..., wk only.

Combined equation (29) could be presented as the following matrix equation

(D-&T) W = 0, (30)

where

D = a$2E - 2%n2 A + —- B .

Here

W =

w1 _t1 0 0 0 *

w2 -, T = 0 t2 0 0

M 0 0 0 0

wk 0 0 0 Ik /

A =

b =

r1 M

1 -2 1 M

1 -2 1 M

1 -2 1 M

<0 0 0 0 0 •M 0 0 0 0 0

M 1 -2 1

M 1 -2 1

M 1 -2 1++

) g 2 f 2 M M r1.

v 5 -4 1 M

1 -4 6 -4 1 M

1 -4 6 -4 1 M

A 0 0 0 0 0 •M 0 0 0 0 0 0

M 1 -4 6 -4 1

M 1 -4 6 -4 1+

M 1 -4 5 v++

M 2 -2 f g

So, the problem we have considered has been reduced to an eigenvalue problem (30). Minimal eigenvalue of linear combined equations determine the value of the non dimensional buckling coefficient hcr. The accuracy of calculations is estimated by the comparison of results obtaining different values k. Critical stress Ncr is defined by the following equation

N \/D11D22

Ncr =&cr—b------------------------------. (32)

The value of &cr depends on parameters a, %1, %2 which contain all the information about the size of the plate and its bending stiffnesses.

Examples. The first example we overlook is that of an isotropic plate. Bending stiffnesses at this case are defined by the following expressions.

-h3

D11 = D22 = E —,

11 22 12

D12 =2E—, D33 = 2E— .

12 12 33 2 12

(33)

(31)

where h — is the thickness of plate, E =E/(1 — 22). Here is is the modulus of elasticity, m is Poisson’s coefficient.

Expressions (16), (22) and (25) give us

a = 1/ c2, % = 1,

%1 =2, %2 = (1 -2)/2.

Here c = a/b is the ratio of the plate’s sides. In fact, the buckling coefficient depends on parameter c only. Dependence of &cr on c has been studied for m=0,3, n = 50. Parameter c has varied within 1.5. The data is shown in table 1.

The buckling coefficient could also be representedby an expressionobtainedby the least-squares method, fromwhich we get

18.9695

&cr =7-----------TF955T . (34)

(c - 0.2726)

In the second example we have overlooked the plate formed from unidirectional or orthogonal reinforced layers which reinforce axes form angles 3 4 with axis x.

If there is a large amount oflayers, the plate’s structure couldbe considered as homogeneous and orthotropic. Then the bending stiffnesses are defined as

D11 = A11 —, 11 11 12 '

D12 = A12 —, 12 12 12

D = A hi

-^22 ^22 12 ’

D33 = A33 —, 33 33 12

(35)

where

+ 2E 12 c42 s92 ,

A11 = E 1c44 + E 2

A12 = E1 212 + (E 1 + E 2 - 2E 12)c^ V ,

A22 = E1 s94 + E 2 c; + 2E12 c92 s92,

A„ =

= (E 1 + E 2 - 2E 1212)c4 s4 + G12(c4 - s4 )

E 12 E 1 M"12 + 2G12

E1

E1

The unspecified array of the matrix’s cells A and B are equal to null. E is the identity matrix.

E2

E

1-212 221

c4 =cos 4, s4 =sin 4 .

value of the coefficient depends on plate’s geometric and elastic parameters. The finite difference method had been used to solve the aforementioned problem. The buckling coefficient had been defined as a minimal eigenvalue of homogeneous linear combined equations approximating the boundary-value problem; it has been solved forthe isotropic plate and the orthotropic symmetrically reinforced plate. Different elongation angles of optimal reinforcement for the plates had been defined. We have completed studying the influence of the side ratio and reinforcement angles on wave generation.

Bibliography

1. Bubnov, I. G. Theory of Structures of Ships. Vols. I and II /1. G. Bubnov. St. Petersburg, 1912,1914.

2. Timoshenko, S. P. Theory of Elastic Stability / S.P.Timoshenko, J.M.Gere. 2nded,N.Y:McGrawHill, 1961.

3. Lekhnitskii, S. G. Anisotropic Plates / S. G. Lekhnitskii. N.Y.: GordonandBreachPub. Co., 1968.

4. Reddy, J. N. Theory and analysis of elastic plates / J. N. Reddy. Philadelphia: Taylor&Francis, 1998.

5. Whitney, J. M. Structural Analysis of Laminated Anisotropic Plates / J. M. Whitney. Lancaster, Pa: Technomic PublishingCo.,Inc., 1987.

6. Bloom, F. Handbook of Thin Plate Buckling and Postbuckling / F. Bloom, D. Coffin. N. Y.: Chapman & Hall/ CRC,2001.

7. Nolke, K. Biegungsbeulung der Rechteckplatte mit eingepannten Langsrandern /K.Nolke//Der Bauingenieur. 1936.17. S.111.

8. Leissa, A. W. Buckling oflaminated composite plates and shell panels : technical report AFWAL-TR-85-3069 / A. W. Leissa. 1985.

9. Vasiliev, V V. Mechanics of Composite Structures / V V Vasiliev. Bristol: Taylor&Francis, 1993.

Table 1

The buckling coefficient Tici,(c)

c 1 2 3 4 5

'cr 25.71 11.25 7.27 5.39 4.29

Table 2

The dependence of parameters c, (p, ijcr

c 1.0 2.0 3.0 4.0 5.0

+ 14.0 22.0 22.0 22.0 22.0

'cr 16.17 6.29 3.94 2.88 2.27

© LopatinA. V., AvakumovR. V., 2009

Here Ev E2 are modulus of elasticity along the reinforce-direction and along the transverse direction respectively, G12 is the rigidity modulus, [12, [21 are Poison’s coefficients. The substituting of (35) into (16), (22) and (25) gives

M A12 % 2A33 a =— , p = - 12 33

c

(37)

Thereby, the buckling coefficient depends on the parameter c and the angle + for the orthotropic plate. A transformation of the expression

=vdd: (38)

to

'cr = (39)

makes the parametrical analysis more convenient. Here 'cr = 'crV f11 f22 , fu = An / E , /22 = A22 / E ,

h3

D = E1 —. (40)

We have finished studying the dependence of 'cr on c and +. The calculation has been done for carbon-filled plastic with elastic characteristics Ex = 142.8 GPa, E2 = 9.13 GPa, G12 = 5.49 GPa, Ex = 142.8 GPa, [12 = 0.02, [21 = 0.32. The data is shownintable 2.

The maximal buckling coefficient for the square plate realizes with angle + = 14E. The optimal angle tends to 22E with an elongation increase.

The buckling problem of orthotropic plates with two free unloaded edges and two simply-supported edges loaded for in-plane line-distributed stress had been solved. The definition of the critical load has been reduced to the calculation of a non dimensional buckling coefficient. The