Objectives:
Reference: §4.1 - 4.12 of Mechanics of Materials, 3rd Edition, Beer, Johnston, & DeWolf, McGraw-Hill
Definitions and Concepts:
Stress, Strain, & Section Modulus:
Pure bending occurs when a structural member is subjected to equal and
opposite couples which act in the same longitudinal plane. Please see Figure
4.6 a and b on page 210. When the bending moment M is such that the normal
stresses remain below the yield stress 
,
no permanent deformation occurs. If the stress anywhere in the material
exceeds the yield stress, elastoplastic behavior occurs and there is permanent
deformation and possibly residual stress when the moment is removed.
The details on page 216 show that, if the normal stresses remain less than the yield stress, those stresses vary linearly on either side of the neutral surface (cf Figure 4.13). "c" is the maximum distance from the neutral axis and may be measured on either side of the neutral axis. Note the directions of the arrows. The "beam" is in compression above the neutral axis and in tension below it. Since compressive stresses are negative, for values of positive y the normal stress is negative:
When y = c, the formula provides the maximum stress. While the figure shows a general case - the neutral axis is not shown passing through the centroid of the section - your text proves that if the following two conditions are satisfied, the neutral axis does pass through the centroid of the section:
The elastic section modulus S is defined to be I/c, so that the formula for the maximum stress can be written:
Section moduli are tabulated for various rolled steel shapes in Appendix C. Since the maximum stress is inversely proportional to S, beams with as large a value of S as possible (economically and physically justifiable) should be used. American standard beams (S-beams) and wide-flange (W-beams) have large portions of their cross sections located far from the neutral axis and consequently have large values of S.
Radius of Curvature: Your text also
shows that the radius of curvature 
of
a beam is given by
While a bending moment applied to a beam
causes compression above the neutral axis and tension below, as stated
earlier, the transverse cross section of the beam is also affected. Figure
4.21 on page 221 shows that the transverse section also has a radius of
curvature 
;
the reciprocal of this is called the anticlastic curvature:
Composite Beams: Beams are often composed of more than one material (composite beams). Figure 4.23 on page 230 shows a bending moment M applied to a beam composed of materials 1 and 2 bonded together. Figure 4.24 shows that, while the strain varies linearly (or else we get delamination), there is an abrupt change in the stress at the boundary. Now we cannot assume that the neutral axis passes through the centroid of the composite section!
As you look at Figure 4.24 and the figures below, notice that the strain plot is linear and continuous. But because
at the value of y for which the material changes, E also changes, and there is a discontinuity in the stress plot.
Notice that the neutral axis is defined by the surface where the strain
is zero, but it is not coincident with the centroid of area of
the cross section.
Looking at Figure 4.25, suppose that the lower (red) section is steel and the upper is brass.The sections have equal width b. Since the steel can withstand more stress before yielding, its equivalent area (as brass) is larger, as shown in the transformed section of the figure. Thus a differential area dA becomes n dA, as shown. Example 4.03 on page 232 involves brass and steel with specific areas. As the theory shows, n is used to compute the equivalent area:
Prestressed Concrete: A special case of a composite beam is concrete with embedded steel reinforcing rods (rerods). From doing the problems, you know that the portion of the concrete below the neutral axis is disregarded, since concrete in tension cracks and the steel carries all of the stress. A better solution is to place tension on the rerods until the concrete has reached a certain percentage of its full strength. When the rods are released, they place the concrete in compression. Since the rods are usually placed near the bottom of a beam where the concrete would normally be in tension, this has the effect of causing the entire cross section to be in compression. The diagram below shows a concrete/steel composite subject to a positive bending moment:
Assuming that this is not prestressed, much of the concrete would be in tension and would not be included in the calculations:
Just as in Figure 4.25, the transformation in Example 4.03 occurs in a direction parallel to the neutral axis. Once we have the transformed section, its centroid can be found since the material is essentially homogeneous (see Figure 4.26). Sample Problem 4.3 on page 236 shows that the centroid and hence the position of the neutral axis is computed from
which is the method you learned in your course in Statics. The next step is to compute the centroidal moment of inertia; note that the parallel-axis theorem has to be used! Finally, the maximum stress can be computed since the values of c and I have been computed.
Plastic Deformations: Since the
additional force required to continue to stretch a material after the yield
point has been reached is minimal, we usually draw the stress-strain curve
as in Figure 4.39 on page 246, where the stress is shown as constant after 
.
This is an approximation; compare it with Figure 2.65 on page 109 and the
figures on pages 52 and 53.
Please look at Figure 4.40 on page 247.
As the bending moment is increased, the stress increases (case a). Eventually
the stress at the points farthest from the neutral axis reaches the yield
stress
(case
b). If one continues to apply torque, the material begins to yield in areas
farthest from the neutral axis and the width of the plastic region increases
(case c). As additional torque (bending moment) is applied, the yield stress
is reached at points closer and closer to the neutral axis and the elastic
core is reduced. Finally (case d), both regions become fully plastic. You
can see this three-dimensionally on page 248, where R represents the resultant
of the forces that act on each cross section; these R-forces act through
the centroids for a total moment of
This applies to Figure 4.41a. Since the centroids (where R acts) are different for case (b), the formula is also different. Can you see that the area to be multiplied by the yield stress is ½ bc for case (a) and bc for case (b)?
Residual Stresses: Plastic zones will develop in an elastoplastic material if the bending moment is large enough. Please refer to Examples 4.05 and 4.06 on pages 249 and 251. In the first example, the bending moment is increased sufficiently (to 36.8 kN-m) so that plastic zones 20 mm wide develop on each end. (Your text uses special formulas developed for beams of rectangular cross section. I like to do it more generally; you may want to look at my solution.) In Example 4.06, the bending moment is reduced to zero; this occurs elastically as shown on page 109. The effect is as if we added the stress diagrams of Fig. 4.45a and 4.45b to get 4.45c. Note that, at y = 40 mm, the original stress is -240 MPa (the yield stress); the stress to be added at y = 40 mm is +204.5 MPa; the sum is -35.5 MPa.. At y = 60 mm, the original stress is still -240 MPa; adding 306.7 MPa (representing the removal of bending moment), we get a sum of +66.7 MPa. See Example 1 to see how to determine the plastic moment; for determining residual stresses and radius of curvature, see Example 2.
Eccentric Axial Loading: For these cases, an axial force P is applied "off-axis". When it is moved so that it acts through the centroid, a moment must be included. The axial stress P/A must be added to the stress arising from the bending moment. Here's another example (3), but those in the book are very good; be sure to study them!
And here are a few more examples...
Example 4 - Stresses that result from a bending moment - in the elastic range
Example 5 - Stresses resulting from a bending moment - in the elastic range
Example 6 - Bending of members made from two or more materials
Example 7 - Concrete beam with reinforcing
Example 8 - Stress Concentrations
Example 9 - Plastic Deformations