Elastic buckling of plates subjected to distributed tangential loads

Cqm&rs & Sfrwtvns Vol. 41, No. 1, pp. 151-W Printed in Great Britain.

004s1949/91 $3.00 + 0.00 Pergsmon Press plc

1991

ELASTIC BUCKLING DISTRIBUTED

OF PLATES SUBJECTED TANGENTIAL LOADS

TO

C. J. BROWN Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K. (Recetved 14 August 1990)

Abstract-This paper examines the elastic buckling behaviour of plates with a variety of different boundary conditions under the actions of loads applied tangentially to the surface within the boundaries, effectively distributed loads. Conclusions are drawn about the effect of distributed as opposed to end-concentrated loads. but the imwrtance of dete~ining the appropriate bending boundary conditions is emphasized. ’

NOTATION X,YJ t

z

&Ol PC,

k 2 K

C %lbx

P

Cartesian co-ordinate axes plate dimension in x-direction plate dimension in y-direction plate aspect ratio = a/b applied load total applied load load to cause elastic instability plate thickness modulus of elasticity Poisson’s ratio plate flexural rigidity buckling coefficient constant for load type Maximum value of stress in the plate pressure (load) applied to the plate 1. INTRODUCFION

Thin plates are often used in situations where the loads they carry are not only normal to the plane of the plate but also tangential to the plane of the plate itself. Many authors have addressed the problems of determining the elastic stability of plates under ‘end’ loading and results are widely available [l-4]. Some authors have investigated the problems of point or concentrated end loads applied to plates[5-73, but little work has been carried out to address the problem of the elastic stability of plates subject to uniform or non-uniform distributed in-plane loadings. Such situations might occur in the flat walls of bins or silos. This paper looks at such problems, and provides results for two specific cases of loading and in-plane support condition along with a range of possible plate bending boundary conditions. The conjugate load~ispla~ment method [8,9] is used throughout to provide the results. 2. PLATRS WITH UNDWRMLY DISTRIRUTED LOAD

Figure 1 shows a typical configuration of a plate subjected to uniformly distributed in-plane load. The plate has uniform in-plane supports along its bottom

edge (edge 4 in Fig. 1) and the distributed load is applied by means of discrete loads at the nodal points. The applied load is proportional to the rectangular area surrounding the node, and critical values for elastic instability are given in terms of the total applied load, rather than the applied pressure or maximum internal stress value. Applied pressure (load) is independent of x-coordinate, and so L1 pTOT

pb dx =pab

=

s0 a,,,

=

PTOT . bt

K is defined as

j,P,,b =:

n2D’

Values of X for various plate geometries and boundary conditions are given in Table 1 where P, is the .. crtttcal value of PTOT.S represents a simply supported edge, F a free edge and B an edge restrained against rotation as well as out-of-plane displacement. The edges are referred to in turn as shown in Fig. 1. In implementing the conjugate load/displacement method of analysis, a square mesh has been used throughout, and the value of a has been retained as 1000 mm. E is taken as 210 kN/mm2 and p = 0.3. The plate has been taken to be 10 mm thick. C#J is defined as the ratio a/b. If the effect of the differing boundary conditions is first considered, it can be seen from Table 1 that the elastic critical load is vastly reduced when the ends normal to the direction of the applied load are simply supported rather than built-in. Where the plate is otherwise simply supported and for low aspect ratios particularly, the effect is most marked, but even in cases where the aspect ratio, 4, is high the effect can still be observed. Because it can be noted that results

152

C. 3. BROWN

0

sequence used in defining boundary conditions

Fig. I. Configuration of plate.

for BSBB, BFBB and BBBB rapidly converge for aspect ratios greater than 1 it can also be assumed that the significant differences shown in results for BSBS are due to the boundary condition on edge (4). Figure 2 shows two typical (SSSS, SSSB) curves for K against 4. The fact that the curves tend towards a limiting value is to be expected in view of the fact that elastic buckling is taking place within each square panel, or rather one critical square panel of width b. In computing results, the total number of elements has been kept of the same order (between 216 and 320), rather than retaining the same number of square elements in any one chosen direction. For panels with aspect ratios less than unity, the boundary condition on edge 2 can be seen to reduce significantly the elastic critical load, and for SFSS or BFBB this is quite noticeable, but the effect is not as marked as the effect of the other end boundary condition referred to above. Comparative results for different loading cases are available for many of the boundary conditions and aspect ratios given above, but it is perhaps pertinent

to consider the cases of square plates and ‘long’ plates both simply supported and fully fixed against bending, subjected to end loads only. The ‘standard’ case for a simply supported square plate subjected to uniform end load is K = 4.00, where K takes the meaning described above. This gives a factor of increase on tota critical load of 1.62 for uniformly dist~but~ loading compared to end loading. For built-in plates the ratio is 1.74. For plates with longer aspect ratios, the value of 4.91 for simply supported plates compares with 4.00, a ratio of 1.23, while for built-in plates the computed value is 10.29 to compare with 6.97 according to Roark and Young [3] a ratio of 1.48. A sketch showing the displaced form for the first mode for a square plate (SSSS) with 400 elements is shown in Fig. 3(a). It can be concluded from the results given above that, as expected, the effect of distributing an in-plane load over the surface of the plate as opposed to appl~ng the load on the ends only, is to increase the elastic critical load of a plate, but perhaps of more importance is that the effect of varying the boundary

Table 1 4 0.50 0.67 1.00 1.50 2.00 3.00 4.00 5.00

SSSS

SFSS

Boundary conditions SSSB BSBB BSBS BFBB BBBB

11.48 8.37 6.50 5.83 5.47 5.13 4.91 4.78

4.46 5.37 6.35 5.81 5.45 5.13 4.91 4.78

27.56 18.07 11.72 9.26 7.97 6.85 6.23 5.86

30.38 13.70 22.25 11.51 17.53 10.46 14.17 9.57 12.51 9.07 11.15 8.70 10.29 8.40 9.77 8.21

17.80 21.72 16.69 14.16 12.51 11.15 10.29 9.77

37.00 24.69 17.54 14.23 12.52 11.15 10.29 9.77

Elastic h~kling of plates subjected to ~s~but~

25 i

---

1

Hence

i

l5-

IO-

p = cx.

i i i

f

&

SSSB

i

t 20-

$!

ssss

-*-

i

1.53

loaded edge of the plate (x = 0), i.e. the tangential pressure increases linearly from zero at edge (2) to maximum at edge (4) and is constant with y. Thus,

3or 1 i i

tangential loads

\

\ I\ \ ‘\ \, i... Again the values given in Table 2 are K, where \---

5-

---------

----

oAspect

ratio

Fig. 2. K vs t# for plate with uniformly distributed load. conditions, pa~icularly on the edges normal to the direction of the applied load, can have the same or even greater significance.

3. PLATES

WITH LINEARLY INCREASING D~~~~D LOAD

The loading is applied in the same way as that described above, except that the value of load (pressure) is dependent on the distance, x from the un-

and P, is the critical value of PmT. The same range of boundary conditions has been used. It can be seen that the comments made about the importance of boundary conditions in respect of unifo~ly distributed loads can be made for plates subjected to linearly increasing load. In all cases, the K values given in Table 2 are slightly greater than those given in Table 1. The factors for the ‘standard cases given above are, for square simply supported plate I .79, for the built-in edges 2.20 and for the long pfates the corresponding factors are 1.3 1 and 1.71. A sketch showing the displaced form for the first mode for a square plate (SSSS) with 400 elements is shown in Fig. 3(b), and the slight modification in mode shape between the two types of loading can be

Fig. 3. Mode shapes for plate with uniformly distributed load (a) and with linearly increasing distributed load (b).

C. J. BROWN

154

Table 2 4

SSSS

SFSS

Boundary conditions SSSB BSBB BSBS

BFBB

BBBB

0.50 0.67 1.00 1.50 2.00 3.00 4.00 5.00

12.70 9.19 7.16 6.31 5.93 5.47 5.23 5.07

5.36 6.01 6.72 6.31 5.93 5.47 5.23 5.07

35.80 23.40 15.28 11.60 9.95 8.24 7.36 6.82

15.06 12.46 11.30 10.29 9.74 9.16 8.82 8.60

27.04 26.79 21.98 17.59 15.28 13.09 11.88 11.12

47.22 31.39 22.16 17.62 15.28 13.09 11.88 11.12

39.32 28.49 22.10 17.59 15.28 13.09 11.88 11.12

Table 3 4

SSSS

SFSS

Boundary conditions SSSB BSBB BSBS

BFBB

BBBB

0.50 0.67 1.oo 1.50 2.00 3.00 4.00 5.00

14.98 10.28 7.60 6.53 6.07 5.54 5.27 5.09

7.21 7.57 7.31 6.53 6.07 5.54 5.27 5.09

41.30 25.49 15.79 11.77 10.00 8.24 7.36 6.81

39.68 31.15 22.61 17.73 15.34 13.10 11.88 11.12

55.84 34.51 22.84 17.77 15.34 13.10 11.88 11.12

compared. These results reinforce the conclusion reached above, namely that the distribution of the load away from concentrated has the effect of increasing the elastic critical load as one would expect, but more pertinently that the use of the appropriate of the boundary conditions is essential for the correct evaluation of elastic critical load.

4. PLATESWITH LOAD VARYINGSINUSOIDALLY ACROSSTHE WIDTH In order to investigate the effect of variation of the load distribution across the plate (in the y-direction), the load was applied such that is was distributed sinusoidally across the plate with a mean value of that used in the cases given in Sec. 3 above, but such that the maximum occurred at y = 0 and y = b, and was 50% above the mean value, or

p =cx(l+O.5(sin(~-F)).

The total load applied is the same as that used for the non-varying load given above. From the results presented in Table 3 it can again be seen that all the K-values are greater than or equal to those given in Tables 1 and 2, and again that the boundary conditions and particularly the end boundary conditions are all-important. The factors on the ‘standard’ cases are 1.90 for square simply supported, 2.25 for square built-in and for long plates the results are effectively the same as those given above. This highlights the fact that any small variations which do occur in the value of K are most noticed when the

46.98 31.58 22.80 17.74 15.34 13.10 11.88 11.12

18.42 14.25 12.13 10.71 10.01 9.31 8.92 8.67

aspect ratio is small. As the aspect ratio increases, results approach those obtained for non-variation transverse distribution.

the in

5. CONCLUSIONS

This paper has investigated the elastic critical load for plates with a variety of boundary conditions pertaining to the bending characteristics of the plate, but also with three different distributions of the in-plane loading-uniform, linearly increasing in the direction of the load, and linearly increasing in the direction of the load while varying sinusoidally across the plate. It concludes that the variation in loading will lead to variations in the elastic critical load. The more the load tends away from concentrated end point or line-loads to distributed systems, the greater will be the K-value and hence total load to cause elastic instability. Furthermore this will be the case for all boundary conditions and aspect ratios investigated. It has been highlighted that the factor which most affects the elastic critical load is the determination of the pertinent boundary conditions, particularly those which have effect on the edges normal to the direction of the applied load.

REPERRNCRR 1. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability. McGraw-Hill, New York (1977). 2. G. Gerard and H. Becker, Handbook of Structural Stability-Part 1. Buckling of Flat Plates. N.A.C.A. Technical Note 3781 (1957). 3. R. J. Roark and W. C. Young, Formulas for Stress and Strain. McGraw-Hill, Kogakusha, Tokyo (1975). 4. P. S. Bulson, The Stability of Flat Plates. Chatto and Windus, London (1970).

Elastic buckling of plates subjected to distributed tangential loads 5. D. M. A. Leggett, The effect of two isolated forces on the elastic stability of a flat rectangular plate. Proc. Camb. Phil. Sot. 33, 325-339 (1937). 6. N. Yamaki, Buckling of a rectangular plate under locally distributed forces applied on the two opposite edges. First Report (Report No. 26 in European Languages). Institute of High Speed Mechanics, Tohoku University (1953).

155

7. C. J. Brown, Elastic stability of plates subjected to concentrated loads. Compuf. Struct. 33, 1325-1327 (1989). 8. A. L. Yettram and C. J. Brown, The elastic stability of square perforated plates. Comput. Struct. 21,1267-1272 (1985). 9. C. J. Brown and A. L. Yettram, The elastic stability of stiffened plates using the conjugate load/displacement method. Comput. Struct. 23, 385-391 (1986).