OP SCARPING, &,C. 269 



part of a scarfed beam, slnxned in the direction of its length, and 

 put togetiuT without i'olfi*. Now, it is plain that the strengih of 

 the part c 6, must bo exactly equal to the force that would ^ ause 

 the fibres to slide at the dotted line c d ; for if the part c d, were 

 shorter, the ioint would be less strong than it is possible to make it. 

 Also, if the depth of the indent a c be too small, it would be crush- 

 ed by the strain ; consequently, the parts must have a certain pro- 

 portion, so that the joint may be equally strong in each part. 



In the first degrees of exlcnsion and compression the resistance 

 is equal, therefore the depth of tlie indent a c must be equal to 

 the part c 6, in order that the strain may be equal ; and it is evi- 

 dent, that when there is only one indent, as in this example, the 

 depth a c should be one-third the whole depth. Also, let d be 



d 

 the depth of the beam, and m the number of indents ; the — = the 



3in 

 depth of each indent. Or the sum of the depth of the indents must 

 be equal to one third of the depth of the beam. 



To determine the length of the part c d, we must know the ratio 

 between the force to resist sliding, and the direct cohesion of the 

 material. Let the ratio be as 1 : n •, then c d must be equal to n 

 times c b ^ that is, in oak, ash, or elm, c d must be equal to from 

 8 to 1 times c h. 



In fir, and other straight-grained woods, c d must be equal to 

 from 16 to 20 times c 6. 



Hence may be derived some maxims that will be sufficiently ac- 

 curate for practical purposes : 



i. In oak, ash, or elm, the whole length of the scarf should be 

 six times the depth or thickness of the beam when there are no 

 bolts. 



ii. In fir the whole length of the scarf should be about twelve 

 times the thickness of the beam, when there are no bolts. 



iii. In oak, ash, or elm, the whole length of a scarf depending 

 on bolts only, should be about three times the breadth of the beam ; 

 and for fir beams it should be six times the breadth. 



iv. When both bolts and indents are combined, the whole length 

 of the scarf for oak and hard woods may be twice the depth ; and 

 that for fir, or soft woods, four times the depth. 



Of lengthening Beams that are intended to resist cross strains. 



Beams to resist cross strains require to be lengthened more fre- 

 quently than any others, and from the nature of the strain, a diffe- 

 rent form must be adopted for the scarf from that which is best for 

 a strain in the direction of the length. There are cases where 

 beams are exposed to both strains at the same time, but the cross 

 strain is generally that of the most importance. Of this we have 

 an example in the tie-beam of a roof, where the strain in the direc- 

 tion of the length is very small compared with the cross strain. 



