144 COMBINED AND ECCENTRIC STRESSES 



The following nomenclature will be used: 



Let P = total direct loading on member in pounds ; 



/ = length of member in inches ; 



L length of member in feet; 



/ =. moment of inertia of member; 



y 1 = distance from neutral axis to remote fibre on side for 

 which stress is desired ; 



= modulus of elasticity of the material; 



e = eccentricity of P, i.e. distance from line of action of 

 P to neutral axis of member in inches ; 



v = deflection of member in inches ; 



A = area of member in square inches ; 



f 1 = fibre stres's due to cross bending; 



P 

 / 2 = - = direct fibre stress ; 



A 



M = total bending moment; 

 M 2 = bending moment due to deflection, = P v; 

 MI = bending moment due to external forces and is equal to 

 y*Wlm (a) and (b) ; % w I- in (c) and (d) ; and P e 

 in (e), (f), (g) and (h) Fig. 77. 



Now M=M 2 M 1 =P^M l = (72) 



But 



in which c is a constant depending upon the condition of the ends, and 

 the manner in which the beam is loaded. 



Substituting this value of v in (72) we have 



