1866.] 



Mr. Tarn on the Stability of Domes. 



267 



supposed to be built of material weighing 125 lbs. to the cubic foot. The 

 thrust is calculated for the 180th part of the whole dome, being the por- 

 tion cut out by two planes which make an angle of 2° at the axis. 



Table showing the position of the weakest joint in domes of various forms, 

 and the horizontal thrust at that joint. 



Form of Dome. 



Span. 



Position of the weakest 

 joint, or joint where 

 thrust is greatest. 



Greatest hori- 

 zontal thrust, 

 or thrust at 

 weakest joint. 



Hemisphere ... 

 Gothic, a = 10° 

 Gothic, a = 22|< 

 Gothic, a =30° 



Parabolic ; height above 

 springing equals the half 

 span. 



Elliptical, major axis ver- 

 tical; ratio of major to 

 minor axis as 6 : 5. 



Ogival ; contour, the "curve 

 of sines." 



feet. 

 20 



20 



20 



20 



20 



20 

 I 20 



{Makes with springing line 

 an angle of 20'^. 

 r Makes with springing line 

 \ an angle of 17°. 

 r Makes with springing line 

 \ an angle of \o\°. 

 f Makes with springing line 

 I an angle of 11°. 



At springing line 



{One-third of semimajor 

 axis above springing 

 line. 



r One-sixteenth of tlie span 

 \ above springing line. 



lbs. 



92-01 

 88-7 

 80-418 

 77-27 



79-6 

 90-8G7 



In the preceding cases, except the last, the domes are assumed to be of 

 uniform thickness. The author finally applies his formulae to the case of 

 the spherical dome in which the thickness at the crown is one-half that at 

 the springing, the inner surface being generated by the revolution of a 

 circular quadrant whose centre is raised above the centre of that which 

 generates the outer surface by half the difference of the radii. Assuming 

 the outer and inner radii R, r to be respectively 11 feet and 10 feet, he 

 finds that the weakest point on a dome of this form appears to be at a 

 height equal to R. above the springing line ; and with the other nume- 

 rical values, assumed the same as before, he finds for the horizontal thrust 

 at the weakest joint 55*78 lbs. 



For each kind of dome the author forms what he calls the equation of 

 stability, giving, for an assumed value of the height of the pier, the least 

 thickness t which will permit of stability. The following are the results 

 for each kind of dome, the height of the pier being taken at 50 feet : — 



Spherical dome (from former paper) .... ^ = 2-45 



Gothic dome (a = 22|'') ^ = 2'259 



Parabaloidal dome ^=2*244 



Elhptic dome ^ = 2-44 



Ogival dome t = 2 



Spherical dome (thinner at crown) ^=1*9 



feet. 



