. Journal. —^—- +-^— tor a square,o or ! 8— V~t+^) or t20-53 (r - t) + 0-26 — for an octagon. We may in general neglect the last term containing t2, T — t T t making K for a circle = —-—. The values of —~— for dif- £ A ferent proportions of r and t are shown in the diagram Fig. 5, But a = -j- nn<i <* = ^ When the limit of no tension in tlio windward edge is reached, S0 = 0 or I x — • rfl Calling this value of x K, we have K--L. r a where K = the semidiameter in the direction X X of the kernel of the section. W Hence ?-«( + £) It is easily found that for a hollow figure of seroidiameter r
. Journal. —^—- +-^— tor a square,o or ! 8— V~t+^) or t20-53 (r - t) + 0-26 — for an octagon. We may in general neglect the last term containing t2, T — t T t making K for a circle = —-—. The values of —~— for dif- £ A ferent proportions of r and t are shown in the diagram Fig. 5, But a = -j- nn<i <* = ^ When the limit of no tension in tlio windward edge is reached, S0 = 0 or I x — • rfl Calling this value of x K, we have K--L. r a where K = the semidiameter in the direction X X of the kernel of the section. W Hence ?-«( + £) It is easily found that for a hollow figure of seroidiameter r and thickness t,of circular section, r2 + (r - 02 ! = £(»•*- (•O4), Q = *(r*- (r-02), K = *—tiof octagonal section,I= 4(^2-1) (3-^2)^ _(r_tyyQ = 8(^_1)^_(r_t)^ K = 3 - V2Or (r2+(r -<)2); of square section, I = |(r4-(r-04) fl = 4(r2 -(r - 02) K = A ( r* + (r - <)2) 160 STABILITY OF CIIIMNEY SnAFTS. Fig. 5. Diagram of values of K fop Circle jipprccamatzcTv -Hcm/cinc. ?%¥$? STABILITY OF CHIMNEY SHAFTS. 161 together with the exact values of K. It is evident that r — t is a very close approximation to K in the cases which mostfrequently occur in practice, viz. when t does not exceed one-third of K, and that in all cases the difference is on the safeside. Rankine gives the same formula as above, but adds that asI is generally very small as compared with the semidiameterit may be assumed = 0, and hence he gets for a circle K = -T 2 The values of — are shown in the diagram .Fig. 5. They are not as close an approximation as the r- t , even when t is very small, and in all cases they are on the unsafe side. The following table gives some notable instances of therelative proportions of t and r, and shows that t is far from being negligible as regards r, St. ..TownsendsMechernichWest Cumberland Height. r t 435 6 454 0397 3248 0 t tt 20 016 012 412 9 2 7h 5 l\ 6 7 1 ioi In the case of the Mechernich shaft the values of K are :— Ranki
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