100 SUPPLEMENT TO CITAP. TIT. [CHAP. I 



strains in these three pieces must hold in equilibrium the reaction at A and 

 all the forces between A and the section. 



Now the principle of statical moments is simply that, when any number 

 of forces in a plane are in equilibrium, the algebraic sum of their moments 

 with respect to any point in that plane must be zero. 



The application of this principle is simply so to choose this point of 

 moments as to get rid- of all the unknown strains in the pieces cut, except one 

 only ; and then the other forces being known in intensity, position, and 

 direction of action, we can easily find this one ; since, when multiplied by 

 its known lever arm, it must be equal and opposite to the sum of the 

 moments of the known forces. 



In a properly constructed frame it will, in general, always be possible to 

 pass a section cutting only three pieces. Then, by taking as a centre of 

 moments the intersection of any two, we can easily find the strain in the 

 third. 



Even if any number of pieces are thus cut, if all but one meet at a com- 

 mon point, the strain in this one can be determined. 



Thus, in Fig. IV., PL 1 of the Appendix, a section may be made cutting 

 2 3, d h, h c and c Y. But all these pieces, except the last, meet in 2, and 

 the strain in this last piece may, therefore, be easily determined. 



The above is all that is necessary to be said as to this method. The ex- 

 amples already referred to will make all points of application and detail 

 plain as we proceed. We see no reason why the reader who has mastered 

 Chapter I. and diligently followed out the examples as given in the Appen- 

 dix, should not now be able to both calculate and diagram the strains in 

 any framed structure all of whose outer forces are known. 



3. Idethod by Resolution of Forces. We have also yet another 

 method of calculation, based upon the principle that, if any number of 

 forces in a plane are in equilibrium, the sum of their vertical and hori- 

 zontal components are respectively zero. In structures all the forces acting 

 upon which are vertical, and such are all bridge and roof trusses, etc., of 

 single span, we have only to regard the vertical components. 



In this connection we have to call attention to the following terms and 

 considerations. The shear or shearing force at any point is the algebraic 

 sum of all the outer forces acting between that point and one end. These 

 outer forces are the weights and reactions at the ends. At any apex of a 

 framed structure, where several pieces meet, the horizontal components of 

 the strains in these pieces must balance, or the structure would move ; and 

 for the same reason, the algebraic sum of the vertical components must be 

 equal and opposite to the shear. The shear being known, if the strains in 

 all the pieces but one are also known, that one can be easily found. Thus 

 the algebraic sum of all the vertical components of the strains in the other 

 pieces being found, and added or subtracted from the shear, as the case 

 may be, the resultant shear, multiplied by the secant of the angle made by 

 the piece in question with the vertical, gives at once its strain. 



This method is also fully explained in the Appendix, Art. 16 (4), and a 

 practical rule is there given for properly adding the vertical components 

 and determining whether the result is to be added to or subtracted from 



