69003
THE BUCKLING OF FLAT PLATES UNDER NON-UNIFORM COMPRESSION SYMMETRIC DISTRIBUTIONS DUE TO INITIAL OR THERMAL STRESS
1.
NOTATION
a
plate length
m
in
b
plate width
m
in
E
Young’s modulus
N/m2
lbf/in2
f b
elasti stic buc buckli kling str stress ess of pla plate unde nder unif niform compr mpressi ssion
N/m2
lbf/in2
f c
value of f x at plate centre-line
N/ m 2
lbf/in2
f c b
value of f c at which plate first buckles
N/ m 2
lbf/in2
f e
value of f x at a t plate edge
N/m 2
lbf/in2
f x
axial stress on plate at distance y from plate edge, positive in compression
N/m2
lbf/in2
K
elastic buckling stress coefficient for uniform loading defined
m
in
2 by f b = K E -t - b
t
plate thickness
x , y
reference axes of plate
ν
Poisson’s ratio
Both SI and British units are quoted but any coherent system of units may be used.
2.
NOTES
This Data Item provides a method of checking the elastic stability of rectangular plates subject to stresses that are compressive at the centre-line and distributed parabolically across the width. The data are valid only for distributions, such as those arising from initial or thermal strain, that do not vary along the length of the plate. This Item is not therefore applicable to external load distributions that diffuse rapidly into the plate. Figure 1 gives values of K plotted against a / b for various edge conditions, assuming values values of ν that are appreci appreciably ably differ different ent,, K should should be multipli multiplied ed by 0.91/(1 0.91/(1– – ν2 ).
ν
= 0.3 . For
Figure 2 shows the variation of f c b / f b with f e / f c for the same range of edge conditions. Figure 3 presents the family of parabolas f x / f c corresponding to various values of f e / f c . The general expression for these distributions is f x f e f e 4 y ---- = ---- + ------ 1 – y -- 1 – ---- . f c f c b f c b
Issued January 1969 With Amendment A 1
69003 When calculating f b allowance should be made for possible variations with temperature in the modulus of the plate material. Where temperatures are non-uniform, the value of E selected should be appropriate to the mean temperature of the middle two thirds of the plate. The value of c b can be obtained for problems involving non-parabolic distributions by dividing x by c and then selecting an appropriate curve from Figure 3. The curve chosen should be a close fit over the middle two thirds of the plate width. The value of f e / f c for this curve is used in place of the actual value of f e / f c in determining c b . A more accurate solution for distributions that are non-parabolic can be obtained from the equations in the Derivation.
3.
DERIVATION
Using the Rayleigh-Ritz energy method, and assuming the buckling mode is the same as that of a plate under uniform compression, it can be shown that a
b
∫ ∫ ( ∂ w / ∂ x ) d x d y 2
f c b 0 0 ------- = ------------------------------------------------------------------------ , a b f b ( f x / f c ) ( ∂ w / ∂ x ) 2 d x d y
∫ ∫ 0
0
where w is the normal displacement of the plate. The accuracy of this expression has been confirmed by finite element analysis. It can be simplified further by assuming wave forms appropriate to the conditions of panel edge restraint, so that: where the panel sides are simply-supported (Cases (i) and (ii) of Figure 1) f c b b ------- = ------------------------------------------------------------------------------------------------------------- , b f b π y 2 ( f x / f c ) sin2 ------ d y b 0
∫
and where the panel sides are clamped (Cases (iii) and (iv) of Figure 1) f c b 3b ------- = ------------------------------------------------------------------------------------------------------------- . b f b π y 8 ( f x / f c ) sin4 ------ d y b 0
∫
This method of analysis is discussed in the following reference:
BENO BENOY Y, M.B. M.B.
An ene enerrgy solu soluti tion on to to the buc buckl klin ing g of rect rectan angu gula larr plat plates es und under er non non-u -uni nifo form rm in-plane loading. Jl R. aeronaut. Soc Soc ., Vol. 73, pp. 974-977, November 1969.
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4.
69003 EXAMPLE
During deceleration from supersonic flight, transient thermal compressive stresses are set up in an aircraft panel. It is required to determine whether or not, at any time during the deceleration, buckling occurs. The panel is made from DTD 5070 material, it has ends clamped and sides simply-supported, and the principal dimensions are in, a = 12 in
b = 6.5 in,
t = 0.080 in.
It is necessary to check the stability of the panel under a number of temperature and stress distributions, corresponding to various points in time. One of these stress distributions is illustrated below. It has f c = 7000 lbf/in2, f e / f c = 0.2 , and the mean temperature in the middle of the plate is 100°C.
For the purpose of analysis this stress st ress distribution is compared with the th e curves of Figure 3 (as shown above), and an equivalent parabola is assumed having f e / f c = 0.6 . Reference to material properties data indicates that at 100°C the elastic modulus of DTD 5070 falls to 97 6 per cent of its room temperature value, given as 10.6 × 10 lbf/in 2. Hence,
E = 0.97
×
10.6
×
10
6
= 10.3
×
6
10 lbf/in2.
12 / 6 6.5 = 1.85 , From Figure 1, at a / b = 12 / K = 4.57 , 2
so that
f b = 4.57
------------- × 10.3 × 106 0.080 6.5 3
= 7130 lbf/in2
69003 From Figure 2, at f e / f c = 0.6 f cb ------- = 1.055 . f b
Thus
cb
= 1.055
×
7130 = 7520 lbf/in2.
The panel is therefore found to be stable since c = 7000 lbf/in 2, and the process is repeated for other potentially critical stress distributions.
4
69003
11 Side
End
10
b
a
9
8
(iv) Ends and sides clamped
7 Asymptotic to 6.31 (iii) Ends simply-supported, sides clamped
6
K
(ii) Ends clamped, sides simply-supported
5
4
Asymptotic to 3.62 (i) Ends and sides simply-supported
3
2
1
0 0.0
0.5
1.0
1.5 a b
FIGURE 1
5
2.0
2.5
3.0
69003
b
y
b
y
x x
f c f c
f x
f x
a a
f e
f e
1.4 f cb f b
1.3
(i) Ends and sides simply-supported (ii) Ends clamped, sides simply-supported simply-supporte d
1.2
(iii) Ends simply-supported, sides clamped (iv) Ends and sides clamped 1.1
1.0
0.9
0.8
− 1.0
− 0. 5
0. 0
0. 5 f e f c FIGURE 2
6
1. 0
1. 5
69003
1.4
1.2
1.0
0.8
f e
f e
f c
f c
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.6
0.4
0
−0.2 −0.4 −0.6 −1.0
f x f c
0.2
Compression
−0.2 −0.4 −0.6 −1.0
0.0 0.0
0 .1
0. 2
0.3
0.4
0. 5
−0.2
0 .6
0 .7
y b
0.8
0.9
Tension
−0.4
−0.6
−0.8
−1.0
Plate edge
Centre-line of plate
FIGURE 3
7
Plate edge
1. 0
69003 THE PREPARA P REPARATION OF O F THIS DATA DATA ITEM
The work on this particular Data Item was monitored and guided by the Aerospace Structures Committee which first met in 1940 and now has the following membership: Chairman Prof. W.S. Hemp
– University of Oxford
Vice-Chairman Mr F. Tyson
– Handley Page Ltd
Members Mr H.L. Cox Mr K.H. Griffin Mr N.F. Harpur
– National Physical Laboratory – College of Aeronautics – British Ai Aircraft Corporation (Filton) Ltd
Mr P.J. McKe cKenzie nzie Dr G.G. Pope Mr I.C. Taig Mr A.W. To Torry
– – – –
Hawker wker Sidd iddeley Av Aviation ion Ltd, td, King ingsto ston Royal Aircraft Establishment British Aircraft Corporation (Preston) Ltd Hawker Si Siddeley Av Aviation Lt Ltd, Ha Hatfield.
The members of staff of the Engineering Sciences Data Unit concerned were Mr A.G. A.G.R. R. Thom Thomsson Mr M.B. Benoy
– Head Head of Mech Mechan anic icss of of Soli Solids ds Grou Group p – Mechanics of Solids Group.
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