of the Thin-plate Beam. 355 



And d may be taken — without appreciable error — as the 

 deflexion at the point x = — — ; therefore, by considering 

 the deflexion over the portion cd of the beam we have : — 



r m(a+c) -«i(a+c) mM _ mM -. 



d = J£j±_ \ \ 2 + \ 3 ~( g o +*o ) I 



??l 3 E . I i m.a - m.a \ 



J 



+ 



l r '0 



w{3a + c)(a-c) _Q(a-c) 3 (2q + 3c) 

 8wi 2 E . I 48aE . I 



It is noteworthy that the magnitude of d, being relatively 

 very small, is only of theoretical significance, and, in the 

 application of the above principles to practical problems, 

 d may be neglected in comparison with I). 



Summarising, the equations to be used in the calculation 

 for strength of the beam are : — 



(I.) To find Q from :— 



_ /■ m.a —m.a , m.c — m.c 



Q(q--c) 8 (fl + 3c) = w.aj (% + *„ )-(e Q +e Q ) , w{a 2 -c 2 ) 

 12a m 3 L m. a -m.a j + 2 m 2 



(II.) To find t from :— 



nrt 2 a . c -,-. „ -p. A /a.c 



= JJH or D = mt\/ — — , 



o V o 



where 



/ii.a — m.a _ 



1 f tt?.q/ g +g -2 \ ^. a 2 Q(q-c) 2 (q + 2c) ') 

 E.ll m 3 \ »**-»■■ / + 2»i 2 12 J* 



(III.) To find the bending-moment from : — 

 Bending-moment at c 



, m.c —m.c 



= _w p.a(g +e Q ) | Qfq-c) 2 



m 2 j m M _-™.a j ~ ta ' 



In any specific case we have in (I.) and (II.) three 

 equations from which to determine four unknown quantities 

 Q, m, D, t. 



If we assume a value of m then Q, D, and t may be 

 determined, thence the bending-moment from (III.), and, 

 finally, the stresses due to bending and stretching. 



As an example take the case where : — 



q = ll inches, c= 10 inches. 



w= *007 ton per inch run (equal to 3G ft. head of fresh water). 



3B2 



