RECTANGULAR BEAMS 53 



inary formula to be solved before the formula for resisting moment 

 may be employed. Solving this preliminary formula locates 

 the position of the neutral axis which is in the same position only 

 for beams of a given concrete and of a given percentage of steel 

 reinforcement. 



The two most general varieties of flexure formulas in practical 

 use will be taken up in Arts. 33 and 34. In each of these two 

 classes of formulas, tension in the concrete is neglected. For 

 purposes of discussion, the subject of beams will first be treated 

 with reference only to the horizontal reinforcement. The in- 

 clined tensile stresses will be considered separately. Analysis 

 "and results of experiments discussed in succeeding assignments 

 prove beyond any doubt that the rational formulas which we 

 will now develop may be used safely and economically in design- 

 ing. No empirical formula is needed. 



The student should realize that the following assumptions 

 must be made in order to derive working formulas: (1) the union 

 between the steel and the concrete is sufficient to cause the two 

 materials to act as one material; (2) no initial stresses are con- 

 sidered in either the concrete or the steel due to temperature or 

 shrinkage; (3) the applied forces are parallel to each other and 

 perpendicular to the neutral surface of the beam before bending; 

 (4) sectional planes before bending remain plane surfaces after 

 bending within the elastic limit of the steel. 



33. Flexure Formulas for Working Loads Based on Rectilinear 

 Variation of Stress in Concrete. The loads being working loads, 

 the unit stress in the steel is within the elastic limit, 'and the 

 unit stresses in the concrete may be considered without material 

 error to vary as the ordinates to a straight line. The following 

 notation will be employed referring to Fig. 28. 



Let f c = maximum intensity of compressive stress in the 

 concrete under a given load. It is represented by 

 the distance AA" '. 



f 8 = maximum intensity of tensile stress in the metal 

 under the same load (the area of reinforcement is 

 assumed to be so small with reference to the total 

 area of cross-section of the beam that the stress in 

 the metal is practically uniform) . 



AA f represent the deformation, or deformation per unit 

 length, in the concrete which is stressed to the 

 amount / c . 



