78 c. M. i;k()Omai,i. : 



quantities neglected by the equations are, as a rule, small, and 

 for practical purposes the fornuilas give near enough the value 

 and direction of the maximum stresses. If the beam were 

 absolutely rigid and carried no load but its own weight, so that 

 each element of it carried an equal share of the load, and if 

 the supporting forces were distributed over the ends of the 

 beam, then the equations wotild be absolutely correct. Let us 

 assume we are dealing with such an ideal beam. 



Referring now to the figure, the direct stress curves are 

 shown solid and the shear curves dotted. On any line of ten- 

 sion or compression the amount of stress increases as the 

 inclination from the vertical increases, being zero where the 

 line is vertical and a maximum where it is horizontal. The 

 tension curves are concave upward and those of compression 

 concave downward, any two lines cutting each other at right 

 angles. Likewise any two shear lines cut each other at right 

 angles. Any line of direct stress cuts any line of shear at an 

 angle of 45°. The arrows in the figure indicate the kind of 

 shear existing at various points and the direction in which 

 distortion begins to take place. No simple rule can be given 

 for the variation of shear along these curves. The best that 

 can be said is that the shear has its maximum value at the 

 middle of the ends of the beam, reduces to zero at the central 

 point and the four corners, and has various intermediate values 

 at other points. At any point the two shears are always equal. 

 Generally speaking, the tension and compression are greater 

 toward the middle of the beam, while the shears are greater 

 toward the ends. Of course, at any point of the beam, stresses- 

 are acting in all sorts of directions at the same time. The 

 important matter, however, is to know the amount and direc- 

 tion of the maximum stresses, and this is what the theory gives. 



Inspection of the figure will show that there are two types 

 of direct stress curves and also two types of shear curves. We 

 may speak of these as the bowed and pointed types. If the 

 mind pictures the space between the lines filled in with other 

 more or less similar lines, it will be seen that one type of curve 



