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Cross-section Shape (section A-A)

Bending Stress (linear distribution) (side view)

Free-Body Showing Loads (side view)

Fig. 11-31. Bending and Shear In a Cantllevered Beam. Bending stresses vary linearly, peaking in the parts of the section that are farthest from the neutral axis (centroldal bending axis). Shear stresses vary nonlinearly and are maximum at the neutral axis. Magnitudes of both bending and shear stresses vary for different cross-sections.

The shear force also relates to the change in bending moment along the beam.

dM dx

When the applied force is continuous so that Vcan be differentiated, the following is also true:

The force w(x) in Fig. 11-31 is neither tensile nor compressive as it is not applied along the beam's axis. However, from the shape of the deflected beam, we can see that the upper surface of the beam is stretched; this material is in tension. Likewise, the bottom material is shortened and is in compression. The tensile and compressive stresses are necessary to react the applied bending load. The bending moment increases for sections of the beam closer to the fixed end. For any individual cross-section of the beam, the tensile and compressive stresses are maximum at the upper and lower surfaces. Provided the maximum stress remains below the proportional limit, these stresses vary linearly for parallel surfaces inward from the extremities, finally reaching zero at a line called the neutral axis (Fig. 11-31C). Shearing stresses vary nonlinearly along the cross-section and, unlike bending stress, reach a maximum at the neutral axis. Extensive beam equations for stress, deflection, and reactions to applied loads are in Roark and Young [1975].

We can quantify a beam's ability to resist bending loads using the second moment of area of a cross-section. It is usually referred to as area moment of inertia, /, (or moment of inertia for the section) and should not be confused with mass moment of inertia used in control system analysis. The area moment of inertia about an arbitrary axis is where y is the distance from the centroid to the infinitesimal area, dA. Figure 11-32 presents values of / for several commonly used sections. For boxes and tubes, we find the I for a section by subtracting the / of the "hole" from the total.

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