THEORY OF FLEXURE OF CONCRETE BEAMS. The theory of flexure for a homogeneous material, like steel or Wood, is based upon two fundamental principles, viz.: (1) a plane cross section of an unloaded beam remains a plane after bending, and hence the unit deformations of the fibers at any section of a beam are proportional to their dis tances from the neutral surface; and (2) the stress is proportional to the strain or deformation, and hence the unit stresses in the fibers at any section of a beam are proportional to their distances from the neutral surface.
Fig. 22, page 222, represents these two laws as applied to a beam when subject to a vertical load acting down upon it. According to the first law, if ab represents the unit shortening at the top face of the beam, el will be the deformation at a distance ae below the upper surface; and similarly, if cd represents the unit extension of the lower fiber, gh will be the extension of a fiber at a distance dh above the bottom. According to the second law, if ab represents the com pressive stress in a fiber at the top of the beam, el will be the com pressive stress on a fiber at a distance ae below the top.
In concrete the deformations are not proportional to the stresses producing them, and consequently the second law as above is not applicable to concrete beams. Fig. 23 shows the character istic relation between the deformations and the stresses for a material in which the deformations are not proportional to the stresses pro ducing them. The deformations are obtained by testing specimens
in direct compression and also in direct tension. Fig. 23 is drawn by plotting unit stresses as abscissas and unit deformations as ordinates.
Fig. 24 shows the stress diagram corresponding to the stress deformation curve of Fig. 23. Fig. 24 is constructed as follows: Since, according to principle 1 above, the deformations of the fibers are proportional to the distances from the neutral axis, the distances 01, 02, 08, and OA will represent to some scale the deforma tions; and if the unit deformation at the point 1 in Fig. 24 is repre sented by 01, the corresponding stress can be determined from the stress-deformation diagram in Fig.
23 by using the proper scale.
The distance la in Fig. 24 rep resents the stress at the point 1, and similarly for points 2, 8, and A.
The lower branch of the curve is determined in a similar manner.
Connecting the points A'cbaOB' gives the stress-deformation diagram for the section AB. Notice that the stress-deformation diagram of Fig. 24 is really only the stress-deformation curve of Fig. 23 drawn to a new scale.