114 SUPPLEMENT TO CHAP. VII. [CHAP. HI. 



Integrating once (Art. 5) we have 



, 

 dx 2 



where O is the constant of integration to be determined (Art. 6) by the 



given conditions. Now by the condition in this case, when x = I, -^ 



d x 



must be zero, because the end is fixed, and the tangent there must therefore 



P f 



be horizontal (Art. 8). Hence O = -- , and 



2 



1SI dy = Paf_PP. 



dx ~ 3 2 ' 



We have thus introduced the condition that x cannot be greater than L 

 Integrating again (Art. 5) 



Pa* Pl*x 



Here again we have a constant to be determined, and here again we have 

 the condition that for x = I, y must be zero, since at the fixed end there can 



P I* 



be no deflection. Therefore, O = - and 



8 



=-? (2 I + x) (I - z). 



The deflection will evidently be greatest at the free end, and here, therefore, 

 for x = 0, we have 



PZ 



If the cross-section is rectangular, I = 6 h 9 (Art. 10), and the maximum 



12 



deflection A = 



- 



(&) Breaking weight. 



T I 



We have also from equation (8) M = - , where T is the tensile strain 



t> 



in any fibre distant from the centre. For = , T is the tensile strain in 



I 



2 T I h 



the outer fibre, and M = - . For = we have the compressive 

 h 2 



2 O I 

 strain in the outer fibre upon the other side, or M = - . Theoretically 



the two should be equal. Practically they are not In fact, if we put for 



[P X = 



Tbh* 



2 T I 



M its value, we have P x = , or for a rectangular cross-section P x 



h . 



- - T I h*. This is greatest for x = I, hence the breaking weight P = y- 



(J O 6 



O "n 7 



From thi* we have T = y-^-. Now experimenting with beams of various 



