R

It can then be shown that the maximum stress set up is the tangential stress at r — R\ of value

7rt2

loge

If the outside edge of the plate is clamped instead of freely supported the maximum stress becomes

7.11. Summary of end conditions

Axes can be selected to move with the plate as shown in Fig. 7.7(a) or stay at the initial, undeflected position Fig. 7.7(b).

For the former case, i.e. axes origin at the centre of the deflected plate, the end conditions which should be used for solution of the constants of integration are:

plate where r = R (Load points move relative to centre of plate)

Fig. 7.7(a). Origin of reference axes taken to move with the plate.

Fig. 7.7(a). Origin of reference axes taken to move with the plate.

Fig. 7.7(b). Origin of reference axes remaining in the undeflected plate position.

Edges freely supported:

(i) Slope 6 and deflection y non-infinite at the centre. .'. C2 — 0.

The maximum deflection is then that given at x — R. Edges clamped:

(i) Slope 0 and deflection y non-infinite at the centre. :.C2 = 0.

Again the maximum deflection is that given at x = R.

7.12. Stress distributions in circular plates and diaphragms subjected to lateral pressures

It is now convenient to consider the stress distribution in plates subjected to lateral, uniformly distributed loads or pressures in more detail since this represents the loading condition encountered most often in practice.

Figures 7.8(a) and 7.8(b) show the radial and tangential stress distributions on the lower surface of a thin plate subjected to uniform pressure as given by the equations obtained in §§7.5 and 7.6.

Fig. 7.8(a). Radial and tangential stress distributions in circular plates with clamped edges.
Fig. 7.9. Comparison of bending stresses in circular plates for clamped and freely supported edge conditions.

The two figures may be combined on to common axes as in Fig. 7.9 to facilitate comparison of the stress distributions for freely supported and clamped-edge conditions. Then if ordinates are measured from the horizontal axis through origin Oc, the curves give the values of radial and tangential stress for clamped-edge conditions.

Alternatively, measuring the ordinates from the horizontal axis passing through origin Of in Fig. 7.9, i.e. adding to the clamped-edges stresses the constant value |qR2/t2, we obtain the stresses for a simply supported edge condition. The combined diagram clearly illustrates that a more favourable stress distribution is obtained when the edges of a plate are clamped.

The results of the preceding paragraphs are summarised in Table 7.1 at the start of the chapter. From this table the following approximate relationships are seen to apply:

(1) The maximum deflection of a uniformly loaded circular plate with freely supported edges is approximately four times that for the clamped-edge condition.

(2) Similarly, for a central concentrated load, the maximum deflection in the freely supported edge condition is 2.5 times that for clamped edges.

(3) With clamped edges the maximum deflection for a central concentrated load is four times that for the equivalent u.d.l. (i.e. F = q x ttR1) and the maximum stresses are doubled.

(4) With freely supported edges, the maximum deflection for a central concentrated load is 2.5 times that for the equivalent u.d.l.

It must be remembered that the theory developed in this chapter has been based upon the assumption that deflections are small in comparison with the thickness of the plate. If deflections exceed half the plate thickness, then stretching of the middle surface of the plate must be considered. Under these conditions deflections are no longer proportional to the loads applied, e.g. for circular plates with clamped edges deflections <5 can be determined from the equation

For very thin diaphragms or membranes subjected to uniform pressure, stresses due to stretching of the middle surface may far exceed those due to bending and under these conditions the central deflection is given by

In the design of circular plates subjected to central concentrated loading, the maximum tensile stress on the lower surface of the plate is of prime interest since the often higher compressive stresses in the upper surface are generally much more localised. Local yielding of ductile materials in these regions will not generally affect the overall deformation of the plate provided that the lower surface tensile stresses are kept within safe limits. The situation is similar for plates constructed from brittle materials since their compressive strengths far exceed their strength in tension so that a limit on the latter is normally a safe design procedure. The theory covered in this text has involved certain simplifying assumptions; a full treatment of the problem shows that the limiting tensile stress is more accurately given

7.13. Discussion of results - limitations of theory

by the equation

7.14. Other loading cases of practical importance

In addition to the standard cases covered in the previous sections there are a number of other loading cases which are often encountered in practice; these are illustrated in Fig. 7. iot. The method of solution for such cases is introduced briefly below ^

Fig. 7.10. Circular plates and diaphragms: various loading cases encountered in practice. In all the cases illustrated the maximum stress is obtained from the following standard

For loads concentrated around the edge of the central hole,

^ S. Timoshenko, Strength of Materials, Part II, Advanced Theory and Problems, Van Nostrand.

Similarly, the maximum deflections in each case are given by the following equations: For uniformly distributed loads,

For loads concentrated around the central hole,

The values of the factors k\ and k2 for the loading cases of Fig. 7.10 are given in Table 7.2, assuming a Poisson's ratio v of 0.3.

Table 7.2. Coefficients k\ and k2 for the eight cases shown in Fig. 7.10(").
0 0

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