Surface-Loading on the Flexure of Beams. 493 



It has been proved that the state ot strain along the normal 

 at the point of contact due to the surface-loading may be 

 represented by an hyperbola whose asymptotes are re- 

 spectively the normal itself and a line parallel to the axis of 

 the beam at a distance 6 from the point of contact. Let 

 C, D in fig. 3 represent these asymptotes, E = ; let an 

 hyperbola be drawn whose ordinates parallel to D represent 

 the shear at any point along E C for a given load : since the 

 shear is proportional to the compressive stress, these ordinates 

 may be considered as proportional to the compressive stress at 

 any point along E C. 



By our (a) assumption we may represent the stresses at any 

 point along E C, due to bending, by a right line drawn through 

 C, the centre of the depth. 



Let C K be such a line, drawn on the same scale as the 

 hyperbola, so that E K represents the shear (vertical stretch) 

 at E due to bending *, while E M represents the shear (vertical 

 squeeze) due to loading. 



These two curves must intersect at some point N ; at the 

 corresponding point P on the normal the shear (vertical 

 squeeze) due to the loading is equal to the shear (vertical 

 stretch) due to the bending : an element of volume at P will 

 therefore be subject to voluminal compression only, and the 

 shear will be zero, there will therefore be no birefringent 

 action, and when viewed with crossed nicols there should be 

 a dark spot on a white field. 



If the load is kept constant and the span diminished, E K 

 will decrease until C K cuts the hyperbola at a second point ; 

 we should now get two points of darkness. As the span is 

 still diminished these dark points should rise and fall re- 

 spectively until they coincide, when C K is a tangent to the 

 hyperbola ; after this they should separate out at right 

 angles. 



Plate II. fig. 2 gives the results of an experiment (5) made 

 with constant load and varying spans. The beam was 128 

 millim. x 19 millim. deep x 5'5 millim. thick, supported on 

 two steel rollers 2 millim. in diameter and centrally loaded 

 over a similar roller : the nicols were crossed and at 45° to 

 the axis. The following table gives the spans : — 



* The compressive stress due to bending, at any point on CE, produces 

 a shear (vertical stretch) and a voluminal compression, and both are pro- 

 portional to the stress, similarly for the shear (vertical squeeze) and 

 voluminal compression produced by the stress due to the loading ; so for 

 this purpose it is indifferent whether the ordinates of the two curves 

 represent the compressive stresses or the shears produced. 



