Alexander and Jack.son — Polygons to Generate Diagrams, 4t. 2l 



mass, oriented north-south, mutually press normally on each other with 2^> 

 or q = p, in those directions, or on the barricades al, ccl, or, finally, on the 

 back of the rib everywhere normally with r = jJ = 2- 'i'hese are passive 

 stresses, the active stress being the vertical column of water h = p = q, or 

 h = op = oq for the column of earth, where the friction among the grains is 

 usually in this ratio of 3 to 1 in favour of the passive resistance. The state 

 of strain in the mass of earth itself is prolate spheroidal, since the rib, being 

 circular if balanced, compels q to be equal to p. 



For the earth (geostatic) load the cylindrical shaft might be elliptical to 

 a slight degree, that is, till q = h = op (oblate spheroidal stress). We may 

 suppose the circular rib to become rigid, and the four parallelograms 

 concentrated into four forces acting on it. And now let the figure be pulled 

 out east-west till the circle becomes an ellipse, with diameters in the ratio 

 v/o : 1. The east-west force for equilibrium will be increased ; and when 

 spread on the same barricade cd it gives the new value of q greater in the 

 ratio \/o : 1. The north-south force is unaltered by the pull out ; but when 

 spread out on the elongated barricade ab, the new value of p is reduced in 

 the ratio 1 : %/3. The ellipse of stress (trace of the spheroid) for the earth 

 itself has its axis in the duplicate ratio 3 : 1, the greatest safe ratio that 

 the earth may not '• run " and gradually allow the shaft to flatten. A 

 fine example is the ventilating shaft of the Hoosac Tunnel, Massachusetts 

 (Simm's " Tunnelling "), elliptical in form, the axes being 27 by 15 feet, which 

 are sensibly in the ratio v^3 ; 1. 



This is Kaukine's linear transformation of a structure. If one of the 

 conjugate stresses be a datum that cannot be supposed to decrease in 

 intensity, then the other must increase by the duplicate ratio, just as if we 

 multiplied loth our resultant new stresses by -v/o. 



For a formal proof of the rib of radius B, fig. la, being balanced, we 

 consider the north-west quadrant AS as rigid and hinged only at A and B ; 

 then Ta = qc, the whole of the Q area, and is tangential, and Ti = pc in the 

 same way, and each of these equals re, the normal resultant stress at A and 

 S, and, indeed, at all points. But for all (complete) ribs, with any conjugate 

 loads, the hoop thrust at A equals p^e, where p)^ is the normal component 

 stress at A. and c the radius of curvature of the rib at A, for the reason 

 that at A, as on fig. la, there is complete exposure to the stress ^^o ^t a, while 

 the exposure to q at d is nil. Heuce, if we know p^, at the a end of the barricade, 

 and the radius of curvature at A, their product gives us the algebraic sum 

 of the whole conjugate load Q on the other base eel, and also, if we know q 

 at c and the radius at S, their product is the algebraic total area of the 

 load P. 



