RECTANGULAR BEAMS 43 



2vbh 

 tension of T~ and on AB a total compression of the same 



amount. Since the length of the diagonals is h V2, the in- 

 tensities of these inclined stresses are equal and with a magni- 



2vbh 

 tude of f/~\ ,, ,~r . = v. It, therefore, follows that at the neu- 



tral plane there exists a tension and compression at angles of 

 45 degrees to the horizontal, and that the intensity of these 

 forces is equal to that of the shear. 



Above and below the neutral plane the direction and mag- 

 nitude of the inclined stresses are not as above found, due to 

 the fact that the final value of the tension or compression at any 

 point would have to be obtained by combining the horizontal 

 fiber stresses due to bending with the inclined stresses due to 

 shear. At the end of a beam, however, where the shear is a 

 maximum and the bending moment a minimum, these stresses lie 

 practically at 45 degrees to the horizontal throughout the entire 

 depth of beam. Also, at the section of maximum moment the 

 shear is zero and the stresses become horizontal. 



It is proved in treatises on mechanics that if / represents the 

 intensity of horizontal fiber stress and v the intensity of vertical 

 or horizontal shearing stress at any point in a beam, the intensity 

 of the inclined stress will be given by the formula 



and the direction of this stress by the formula 



tan 2K=^ 



where K is the angle of the stress with the horizontal. 



The two formulas given above are general formulas and apply 

 when / is either tension or 'compression. By their use the lines of 

 maximum stress may be traced throughout the beam. With the 



I P ! US 1 sign before the radical in the first formula, the result- 

 minus 



(K i 1 

 ing value of t is the maximum tension aboye r the maximum 



compression j * bove j the neutral plane. The formula for K 



shows that two values of K, differing by 90 degrees, will 

 satisfy the equation. At any given point, then, maximum 

 compressive stress and maximum tensile stress make an angle 

 of 90 degrees with each other. Fig. 16 shows approx- 



