198 JOHN C. KOCH 



1. The sum of the tensile forces = the sum of the compres- 

 sive forces. 



2. Resisting shear = vertical shear. 



3. Resisting moment = bending moment. 



37. Neutral surface and neutral axis. The above three theo- 

 retical laws do not furnish sufficient data for the full study of 

 beams. Experiment and experience show that when a hori- 

 zontal beam deflects one side becomes convex and the other 

 concave. It is shown that the tensile stresses are on the con- 

 vex side where the fibers have been elongated and compressive 

 stresses are on the concave side where the fibers have been short- 

 ened. By experiment it is found that any two parallel vertical 

 lines drawn on the beam before bending, remain straight after 

 bending of the beam; but are nearer together than before on the 

 compressive side and farther apart on the tensile side. These 

 experimental laws may be stated: 



4. The horizontal fibers on the convex side are elongated, 

 and those on the concave side are shortened, while near the 

 center there is a neutral surface which is unchanged in length. 



5. The elongation or shortening of any fiber is directly pro- 

 portional to its distance from the neutral surface. When the 

 elastic limit of the material is not exceeded, the stresses are 

 proportional to their changes in length; hence 



6. The horizontal stresses are directly proportional to their 

 distances from the neutral surface, provided all unit-stresses are 

 less than the elastic limit of the material. 



7. The neutral sm-face passes through the centers of gravity 

 of the cross sections. 



In order to make clear the application of these principles in 

 the analysis of beams, the following simple demonstration is 

 given. 



88. Moment of inertia. In figure 8d let s = unit-stress on 

 the horizontal fiber most remote from the neutral surface, at the 

 distance c from the neutral axis. 



From the sixth law above - ^ unit-stress at the distance 



c 



unity. 



