244 report — 1865. 



Section II. 

 Transverse Strains produced on a Ship under various conditions. 



6. The value of M, or the strain tending to rupture a beam, depends on 

 the magnitude and the relative position of the pressures applied to the beam. 



When the pressure "W is applied at the middle of the beam A B, supported 



at its extremities (fig. 1), the pressure p- -i 



W °' 



on the support at A is equal to — ; and > ■ j 



A Q ' C n 



the moment M, of this pressure tend- Q w ** 



ing to rupture the beam at any point, 



W 



Q, is equal to the product of the pressure -^ by its leverage A Q ; that is, 



— 



M 1= ^xAQ (4) 



Now this will become a maximum when AQ=AC; that is, the greatest 

 strain will take place at the centre of the beam, or at the point where the 

 weight is applied. 



7. A beam AB, fixed at the extremity B, is acted upon by a series of ver- 

 tical pressures, p v p 2 , p n , whose distances from B are respectively 



a 1} a 2 , a n . To find the value of M. 



Here the greatest strain must take place at B ; 



.-. TL=p x a x + . . . . .+p n a n =(p 1 + p n )QB), (5) 



where GB is the distance of the centre of gravity of all the pressures from 

 the point of support B. 



8. A beam AB, supported at its extremities, is acted upon by a series of 

 pressures on each side of the central pressures p 2 ,p 3 ,p v applied at the points 

 E, F, H. To find the great- 

 est value of M. Fl g- 2 - 



Let p 1 be the resultant of « |. « I D 1. 



the pressures applied be- a . * r ' * Ha *^ — jJ^X_gg_ b 



tween A and E, and p s the f^ D K Ci Q F H I J 



residtant of those applied jjj 4 



between B and H. p i l '* 



By the principle of the equality of moments, we get 



P lS =-L ( Pl x DB +p 2 X EB +p 3 x EB +p t x HB +p s x IB) 



A_b 



=«(*>■+ +A)GB=^? ( (0) 



which gives the pressure on the prop A, where "W is put for the sum of the 

 pressures, and G is their centre of gravity. 



Supposing Q, lying between the pressures p 2 and_p 3 , to be the point corre- 

 sponding to the greatest strain, or, in fact, the point where rupture would 

 take place in a beam of uniform dimensions. 



Now when the lever AQ, turns on Q as a centre, we get 



M=P 1 x AQ-fo X DQ+p 2 x EQ) 



=*( P i-JPi— Pa) A Q+Pi X AD+p 2 X AE 

 = A QJWxGB-ABCp 1 +^ 2 )} +p 1 AD+p 2 AE (7) 



by substituting the value of P x given in equation (6). 



