PROCEEDINGS OF SECTION H. 
Se 
wt 
Lo 
3. That if a heavy load is applied at the end of the partial 
load, the effect is the same as if this heavy load is 
applied direct to the arch ring ; 
4. That there are no spandril walls— 
it is shown that, if we call u,v the radial and tangential dis- 
placements at any point (@) of the arch ring, the differential 
equation for w is 
au Gu au Tpa es 
763 ie 78 yr ere sin 0 7 aps (1) 
where yp is the weight of filling per cubic foot, 
«a ,, radius of the arch ring, 
is) PIIOE a 4 x i.¢., the moment of 
either of the balancing couples (per unit width of arch ring) 
required to produce unit change of curvature in the ring, 
= ;, Y# where Y is Young’s modulus: and ¢ is the thickness 
of ring (supposed uniform). 
The complete solution of (1) is 
BY 
o= A. 44(B +.C 6)-cos. 6. = (D4 E 0) sin 0 ~ 7 6° cos @ ... (2) 
If the arch ring is incompressible, as it practically is in all 
ordinary cases, 
dv Z 
u tae =) 19 ue at ie (3) 
so that v is known at all points, to an arbitrary constant, F. 
The bending moment (per unit width of arch ring) is, for the 
unloaded segment, 
2KE ae 
Me a * gpa’) cos a (2 = ype 0) sin 6 (4) 
a* ay a- 
while for the loaded segment, it 1s 
M = — Ss = (" K - * gpa’) cos0 + i a = -9 pate) sn 6 (5) 
a- ae a" 
From the conditions of continuity at the junction of the loaded 
and unloaded segments, it is shown that we have 
1 gpa(b—b')+W sin pt 
Oe ee S | 9 pa(h—b') sin B +w} 
3 
x a 
{= E - = 
2K 
where 6 is the angle to the point on the arch ring vertically 
below the end of the partial load; } and 0’ the heights from the 
crown of the arch to the free surfaces in the unloaded and loaded 
portions respectively. 
x gpa(b-b’)cos Bp (6) 
