The graphic method of obtaining the strains, as shown in Arts. 205 to 211, is, for its conciseness and simplicity, to be preferred to any other method; yet, on some accounts, the method of obtaining the strains by the parallelogram of forces and by arithmetical computations will be found useful, and will now be referred to.
By the parallelogram of forces, the weight of the roof is in proportion to the oblique thrust or pressure in the axis of the rafter as twice the height of the roof is to the length of the rafter; or -
R:Y:2h:l; or, transposing -
2h:l::R: Y=Rl/2 h; (97.)
where Y equals the pressure in the axis of the rafter, and R the weight of one truss and its load. Again, the weight of the roof is in proportion to the horizontal thrust in the tie-beam as twice the height of the roof is to half the span; or -
R: H: 2 h:s/2;
or, transposing -
2 h::s/2:: R: H = Rs/4 h; (98.)
where H equals the horizontal thrust in the tie-beam. To obtain R, the weight of the roof, multiply M, the load per foot, as in equation (96.), by s, the span, and by c, the distance from centres at which the trusses are placed; or -
R = M c s. With this value of R substituted for it, we have -
Y= Mcsl/2 h; (99.)
H =Mcs2/4 h; (100.)
in which Y equals the strain in the axis of the rafter, and H the strain in the tie-beam. These are the greatest strains in the rafter and tie-beam. At certain parts of these pieces the strains are less, as will be shown in the next article.