COLUMNS FIXED, ALGEBRAIC SOLUTION 115 



(see Chapter XIII), provided there is a joint at that point (point b, 

 Fig. 60). 



In Fig. 60 the reactions were calculated graphically and the stresses 

 in the portal were calculated by graphic resolution. Full lines in the 

 stress diagram represent required stresses in the members. Stresses 

 3-2 and 11-12 were determined by dropping verticals from points 3 and 

 ii to the load line 4-10. 



CASE II. STRESSES IN SIMPLE PORTALS: Columns 

 Fixed. The calculation of the stresses in a portal with columns fixed 

 at. the base is similar to the calculation of stresses in a transverse bent 

 with columns fixed at the base. The point of contra-flexure is at the 

 point 



measured up from the base of the column. The point of contra-flexure 

 is usually taken at a point a distance above the bases of the columns. 



The stresses in a portal with columns fixed may be calculated by 

 considering the columns hinged at the point of contra-flexure and solv- 

 ing as in Case I. 



Algebraic Solution. In Fig. 61 we have 



H=H*~*. 



R (h ~) 



and V= F 1 = -- - - 



Having found the reactions H and H*, V and F 1 , the stresses in 

 the members are found by taking moments as in (a) Fig. 58, consider- 

 ing the columns as hinged at the point of contra-flexure. 



The shear diagram for the columns is as shown in (a) and the mo- 

 ment diagram as in (c) Fig. 61. 



Anchorage of Columns. In order that the columns be fixed, the 

 anchorage of each column must be capable of developing a resisting 

 moment greater than the overturning moment M = -- 75 -~ , shown in 



