298 On the Calculation of the Equilibrium and Stiffness of Frames. 



the direction of the displacement to be determined. Call this 

 the value of q for each piece. 



4th. Find the tension on each piece due to unit of tension 

 along R, S, T, &c, the additional pieces of the frame. Call these 

 the values of r, s, t, &c. for each piece. 



5th. Find the extensibility of each piece and call it e } those of 

 the additional pieces being p, <t, t, &e. 



6th. R, S, T, &c. are to be determined from the equations 



R/> + R2(er 2 ) + S(m) + T2(«r/) + Fg(epr) = 0, 



So- + R2 (ers) + S {es 2 ) + TS [est) + FS (eps) = 0, 



Tt +RS(er/) + 8(est) + TS(«fj +¥S,(ept) = 0, 



as many equations as there are quantities to be found. 



7th. x, the extension required, is then found from the equa- 

 tion 



x=-YX(epq)-l&Z(erq)-S?,{eqs)-T2<{eqt). 



In structures acted on by weights in which we wish to deter- 

 mine the deflection at any point, we may regard the points of sup- 

 port as the extremities of pieces connecting the structure with 

 the centre of the earth ; and if the supports are capable of resist- 

 ing a horizontal thrust, we must suppose them connected by a 

 piece of equivalent elasticity. The deflection is then the shorten- 

 ing of a piece extending from the given point to the centre of 

 the earth. 



Example. — Thus in a triangular or Warren girder of length 

 /, depth d } with a load W placed at distance a from one end, ; to 

 find the deflection at a point distant h from the same end, due 

 to the yielding of a piece of the boom whose extensibility is e, 

 distant x from the same end. 



The pressure of the support at = W— j~; and if x is less 



W 



than a, the force at x will be -jjxij—a), or 



x(l— a) 



a(!—x) 



Similarly, if a? is less than b, 



x(l-b) 

 q dl > 



but if a? is greater than b, 



b(J-m) 

 q ~ dl ' 



If x is greater than a, 



