410 PROFESSOR A. H. GIBSON ON 
2 [ 3 aul 
sa | M6, sin 6, — (R,7 — T,)(sin 6, — 0, cos 6,) + 5 \ 10 — 10 cos 6, — 2 sin? 6, — 39, sin 6, ; | 
Yo,= vo | (Ta —Rur)(sin 8, - 6, 00s 6,) + 2R,7(6, — sin 6) ~| | (25) 
1° San 38 2 sin? 
203 +M,(6; sin 8, +2 c0s 6, ~ 2) = | ae +16 cos 6, - 16 + % + 36, sin 6, 
At the centre, where 6, = a 
aa] Meg (Ror = Ree 365307" | 
Yeentre = 
2 
v2 
+ . 
[(, — Rr) + Ryr(w — 2) + M.( = 2) - 0350008 
_ wr) 1815 | 
Die ail CJ 
§ 7. GirDER witH UNsyMMeEtRicaL Loapine. 
Where the loading of a girder does not admit of being represented by a simple 
trigonometrical expression, or where the girder is not of uniform cross section through- 
out its length, a solution is most readily obtained by dividing the load, including the 
dead load due to the girder itself, into a series of comparatively short lengths, and by 
calculating the moments due to each of these portions of the load separately, by an 
application of the reasoning and results of § 4. In practice a first approximation to 
the moments would be obtained by assuming a likely value for the cross section and 
weight at each point, and by then applying these results. A girder then designed to 
withstand the moments as thus calculated, with equal stresses at each section, would, 
in the majority of cases, be sufficiently near for all practical purposes. If greater 
accuracy were required, a second approximation could be made, taking into account the 
weights of the girder calculated from the sections found necessary by the first 
approximation. 
§ 8. Bow-GirpDER Buitt In at THE ENps anp Restina on INTERMEDIATE SUPPORTS. 
Assuming all the supports to be at the same level, the reactions of the intermediate 
supports may be most readily obtained by expressing the fact that the upward 
deflections at these supports caused by their reactions, are equal to the downward 
deflections produced at the same points by the loading. 
(a) Gurder with Uniform Loading and Central Support. 
Let P be the reaction of this support. Let 180 —2¢, or 2a, be the angle subtended 
by the are of the girder. 
The upward deflection at the centre due to this reaction is given by equation (14), 
in which W =P, and in which M, and T, have the values given in the table on p. 12, 
