128 REINFORCED CONCRETE CONSTRUCTION 



including its own weight, if the beam is considered as simply supported? 

 A value of 8 will be taken for n. 



1 + 2np = 1 + (2) (8) (0.008) 



2 + 2np 2 +(2) (8) (0.008) 

 a =0.098 



D== _* _ . (24,000) (480) 3. _J__ 



(384) (30, 000,000) (18) (27.5) 3 0.098 



ll 



52. Economical Proportions. In the designing of a reinforced 

 concrete beam, the expression bd 2 appears. If one of the dimen- 

 sions represented in this expression is assumed, then the other 

 is also determined. Now the cost of a beam to resist a given 

 bending moment will vary with the proportions adopted for 

 breadth and depth, and it is useful to investigate this variation 

 in cost for different conditions. 



Without taking the matter of shear or web reinforcement into- 

 consideration, it is found from a mathematical study along this 

 line that the cost of a rectangular reinforced concrete beam 

 to support a given moment varies inversely with the depth, 

 directly with the square root of the breadth, and also directly 

 with the cube root of the ratio of breadth to depth. The 

 depth will, however, be limited in many ways. It may be 

 limited by the shearing stress fixing the value of bd, or it 

 may be limited by the head room required, or it may need 

 to be so chosen as to give a beam of satisfactory pro- 

 portions. In the case of floor slabs, the breadth (and like- 

 wise the depth) becomes fixed, as the breadth of beam to carry 

 the load coming upon a strip one foot wide is also one foot. 



In the case where the cross-section of beam is determined by 

 shear, the maximum depth theoretically permissible is that for 

 which bd is just large enough to carry the shear. With a beam 

 designed for moment alone, the cost decreases as the depth in- 

 creases, but the area of the cross-section becomes less. A point 

 must be reached when the beam will be of just the required 

 strength in moment and shear. (Art. 48.) The question which 

 now arises is whether or not a still greater depth will result in 

 greater economy. 



The quantity bd must now remain constant for the greater 

 depths. But bd 2 , on the other hand, is increased and the con- 

 crete stress (/ c ) decrease^!. A smaller value for f c permits the 



