BEAMS AND BENDING 109 
than 2h, the tooth will break at the root, and then Q =o , but if b is 
_ greater than 2h, the tooth will break at an oblique section, and then 
116. Bending beyond the Elastic Limit—Modulus of Rupture.— 
The Ca for the moment of resistance of a beam to bending, viz. 
. fy or hoy was deduced on the assumptions that the stress varied 
Eetomiy from zero at the neutral axis, and that the material was not 
strained beyond the elastic limit. In the case of a ductile material, such 
as wrought-iron or mild steel, permanent set will first take place either at 
the top or bottom of the section, and as the strain increases the distribu- 
tion of stress will change, tending to become more uniform. 
The change in the distribution of the stress as the beam is strained 
beyond the elastic limit is:shown approximately in Fig. 148. At (qa) is 
shown the distribution of stress before permanent set takes place. At (b) 
the material has taken a permanent set in tension and compression, the 
ons AC and 
D being strained 
beyond the elastic 
limit, but the por- 
tions OC and OD 
still obey Hooke’s 
law. At (c) the 
whole of the ma- 
terial has been 
strained beyond the elastic limit. If the material is very plastic beyond 
_ the elastic limit the distribution of stress approximates to that shown 
_ at (d), the tensile and compressive stresses being both uniformly dis- 
_ tributed. If permanent set takes place at A before it takes place at B, 
the distribution of stress will be as shown at (e), the neutral axis moving 
nearer to B. 
s The value of f in the formula M=/Z, when M is the bending moment 
_ which fractures the beam, i is called the modulus of rupture of the beam. 
The modulus of rupture is generally greater than the value of f deter- 
mined by experiments on bars in direct tension or compression, and the 
difference depends on the form of the cross section of the beam, being 
small for a flanged section and greatest for a circular section. 
117. Reinforced Concrete Beams.—Good concrete, having the com- 
_ position, cement 1, sand 2, and broken stone 4, will carry safe working 
___ Stresses of 60 and 500 Ibs. per square inch in tension and compression 
respectively. Beams made of this material offer small resistance to 
bending on account of the low value of the allowable tensile stress. 
The resistance of a concrete beam to bending may, however, be greatly 
increased by embedding steel bars in the concrete near the face of maxi- 
mum tension. The cross section of such a beam is shown to the right in 
Fig. 149, the black squares representing the reinforcement. 
Tn calculating the moment of resistance of a reinforced concrete beam 
it is usual to assume that the steel reinforcement carries the whole of the 
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