RECTANGULAR BEAMS 97 



is the maximum allowable, the triangle abc represents the total 

 stress to be taken by the stirrups at each end of beam. This 

 may be proved as follows: Formula (1), Art. 41, shows the total 

 stress for a distance s along the beam (where V is the average 



2 Vs 

 shear) to be a s f s = -.-. in the case at hand s=Xi and, if v is con- 



O ya gyy 



sidered the unit shear at the end of beam = T~^, the total stress for 



9 V v 1 



the distance x is - ~ = ~. vbx lf which is the area of triangle abc. 

 o yd o 



The ordinate ab represents two-thirds of the shear at the support 

 per 1-in. length of beam. 



Some attention must be paid to the diameter of stirrup which 

 it will be possible to employ in any given case. Of course, the 

 diameter should not be so small that the stirrups will be placed 

 too close together for convenience in construction, nor yet so 

 far apart that the limiting value 3/4d is exceeded. But, in 

 addition to such consideration, the bond strength of the stirrup 

 must be carefully investigated since the danger of slipping 

 determines the maximum diameter which may be employed. 



Let us derive a formula for the maximum diameter to be used 

 in any given case. We shall consider straight stirrups only. 



Let i = diameter of stirrup. 

 a s = area of stirrup. 

 o = circumference of stirrup. 

 u = allowable bond stress per unit of surface of bar. 



The distribution of bond stresses developed on the surface of 

 the stirrups is indeterminate. Evidently it must not be expected 

 that tension will be transferred to the concrete until the com- 

 pression area of the beam is reached, or until a point but little 

 below is reached. Experiments show that it is safe to assume 

 the grip of a stirrup to be 0.6 the depth of beam. 



/ S a 8 =0.6 dou 

 or 







But, for round or square stirrups, 



o m 



