(890 3) 
XX.—The Equilibrium of the Circular-Arc Bow-Girder. By Prof. A. H. Gibson, 
D.Se., A.M.I.C.E., University College, Dundee. (With Eleven Diagrams.) 
(MS. received March 2, 1912. Read June 3,1912. Issued separately August 17, 1912.) 
CONTENTS. 
PAGE PAGE 
1. Introduction . ‘ 391 6. Cireular-are girder built in at two ends and 
2. Circular-are cantilever with Toad ai free oe 392 carrying a uniformly loaded platform . 407 
3. 5 » uniform loading. 394 7. Circular-are girder built in at two ends with 
4 a bieden built in at two ends avd unsymmetrical loading . 3 410 
with single load ~ . 396 8. Circular-are girder built in at two onl and 
5. Circular-are girder built in at ae ends “ts resting on intermediate supports. . 410 
uniform Teadige ; ; E ; . 403 9. Applicability to sections other than circular. 415 
10. Conclusions . : ‘ ; ; ; . 416 
§ 1. InrRopuction. 
A girder built in to supports at one or at both ends and forming an arc of a circle 
in plan, is subjected, at each section, to both bending and twisting moments. At the 
supports, fixing moments of both kinds are called into play, and until these are known 
the resultant moment tending to cause rupture at any section is indeterminate. The 
following investigation is devoted to a consideration of the general state of elastic 
equilibrium of such a girder under various systems of loading. 
The investigation is based on the theorem that in a straight beam, fixed horizontally 
at some point, the slope at any other point distant S is given by the area of the 
= diagram between the two points. Where a girder is circular in plan and is sub- 
jected to both bending and twisting moments this theorem requires modification. 
Thus (fig. 1) if M, and T, be the bending and twisting moments at a point distant 
6 (in angular measure) from the support, since a given slope at @ in the direction of 
the tangent at this point only gives rise to a slope of cos(@,—96) times its magnitude 
at 0, in the direction of the tangent at 0,, while an angular displacement y at 0, due 
to a torque between the support and this point, produces a slope y sin (@,—4) at @,, 
in the direction of the tangent at 0,, the resultant slope at the latter point, assuming 
the slope at the support to be zero, is given by 
Me Te 
GaN cas (eee Uy 
(F).- | EI, °° (0, — 6)ds + Gy, sin (0, — 0)ds 
Here EK and C are respectively the moduli of elasticity and of shear for the material 
of the beam, while I, and J, are the moments of inertia of its section at @ about the 
axes of bending and of twisting. 
TRANS. ROY. SOC. EDIN., VOL. XLVIII. PART II. (NO. 20). 59 
