SUPPLEMENT TO CHAP. XIII. f ABT. 2. 



For an unloaded span the weight P disappears, and 



f. _ M m 







For the shear just to the left of the right support of loaded span, 



For unloaded span, the weight P disappears, and 



_, M m M m _i 

 m = - - - . 

 *m 1 



S' m is then the shear to the left of any support m, and Sm that to the 

 right. The reaction at any support is therefore 



Rm = S'm + Sm 



These are the formulae already given in Art. 148. 



2. Equation of the Elastic Line. We can now easily make 

 out the equation of the elastic line for the continuous girder of constant 

 cross-section, or constant moment of inertia. 



The differential equation of the elastic line is,* 



where E is the coefficient of elasticity, and I the moment of inertia, 

 If now we insert in (3) the value of m, as given in (2), we have 



<Py_M r S f x + P T (x a) 

 cla?~ El 



Integrating f this between the limits * = and x, and upon the condition 



/7 At 



that x cannot be less than a, the constant of integration *= = t, = the tan- 



CL CC 



gent of the angle, which the tangent to the deflected curve makes with the 

 horizontal at r ; and we have, since we must take the / P, (x a) simul- 

 taneously between the limits x a and x for x = and x ; 

 dy_ 2M r g-8 r a' + P r fo-a)' 



2EI 



, , 



If we take the origin at a distance A, (see Fig.) above the support r, 

 then integrating again, the constant is A,, and we have 



8M r g > -B,te > + P r (g-g) m 



y = A, + t, x + - Q~EI - (*) 



which is the general equation of the elastic curve. If in this we make 



* See Supplement to Chapter VII., Art. 11. 



f Notice that when x = 0, a - 0, and hence (x a) = also. 



