Beams are subject to load that acts usually perpendicular to the long axis but is carried in bending along the long axis to vertical supports. Under gravity load beams are subject to bending moments that shorten the top in compression and elongate the bottom in tension. Most beams are also subject to shear, a sliding force, that acts both horizontally and vertically. Because beams and other bending elements are very common, the beam theory is important in structural design and analysis.

As for other structural elements, beam investigation may involve analysis or design; analysis, if a given beam is defined by architectural or other factors; design, if beam dimensions must be determined to support applied loads within allowable stress and deflection. Both, analysis and design, require to find the tributary load, reactions, shear, and bending moment. In addition, analysis requires to find deflections, shear- and bending stress, and verify if they meet allowable limits; by contrast design requires sizing the beam, usually starting with an estimated size.

The following notations are commonly used for bending and shear stress:

Allowable stresses are given in building codes for various materials.

Allowable stresses assumed in this chapter are:

The more complex design and analysis of concrete and masonry will be introduces-later.

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