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A statically determinate beam, bending under an evenly distributed load.
A statically determinate beam, bending under an evenly distributed load.

A beam is a structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight and external reactions to these loads is called a bending moment.

Beams generally carry vertical gravitational forces but can also be used to carry horizontal loads (i.e., loads due to an earthquake or wind). The loads carried by a beam are transferred to columns, walls, or girders, which then transfer the force to adjacent structural compression members. In Light frame construction the joists rest on the beam.

Beams are characterized by their profile (the shape of their cross-section), their length, and their material. In contemporary construction, beams are typically made of steel, reinforced concrete, or wood. One of the most common types of steel beam is the I-beam or wide-flange beam (also known as a "universal beam" or, for stouter sections, a "universal column"). This is commonly used in steel-frame buildings and bridges. Other common beam profiles are the C-channel, the hollow structural section beam, the pipe, and the angle.

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Determination of the bending moment

The magnitude of the bending moment along the length of a beam varies depending on the loading on the beam and the location and type of supports on which the beam is located. Typically the bending moment is calculated by making an imaginary cut through the beam where the bending moment and shear force in the beam are required to be calculated. The application and mathematical manipulation of Newton's Laws to tiny elements of material at the cut reveal the existence of the bending moment in the beam. A positive bending moment induces compressive forces above the so called neutral axis, while tensile forces are induced in the material below the neutral axis. The compressive and tensile forces result in shortening and lengthening of the material respectively above and below the neutral axis resulting in the characteristic bending found in beams.

Structural characteristics

Diagram of stiffness of a simple square beam (A) and universal beam (B). The universal beam flange sections are three times further apart than the solid beam's upper and lower halves. The second moment of inertia of the universal beam is nine times that of the square beam of equal cross section (universal beam web ignored for simplification)
Diagram of stiffness of a simple square beam (A) and universal beam (B). The universal beam flange sections are three times further apart than the solid beam's upper and lower halves. The second moment of inertia of the universal beam is nine times that of the square beam of equal cross section (universal beam web ignored for simplification)

Internally, beams experience compressive, tensile and shear stresses as a result of the loads applied to them. Typically, under gravity loads, the original length of the beam is slightly reduced to enclose a smaller radius arc at the top of the beam, resulting in compression, while the same original beam length at the bottom of the beam is slightly stretched to enclose a larger radius arc, and so is under tension. The same original length of the middle of the beam, generally halfway between the top and bottom, is the same as the radial arc of bending, and so it is under neither compression nor tension, and defines the neutral axis (dotted line in the beam figure). Above the supports, the beam is exposed to shear stress. There are some reinforced concrete beams that are entirely in compression. These beams are known as prestressed concrete beams, and are fabricated to produce a compression more than the expected tension under loading conditions. High strength steel tendons are stretched while the beam is cast over them. Then, when the concrete has begun to cure, the tendons are released and the beam is immediately under eccentric axial loads. This eccentric loading creates an internal moment, and, in turn, increases the moment carrying capacity of the beam. They are commonly used on highway bridges.

The primary tool for structural analysis of beams is the Euler-Bernoulli beam equation. Other mathematical methods for determining the deflection of beams include "method of virtual work" and the "slope deflection method". Engineers are interested in determining deflections because the beam may be in direct contact with a brittle material such as glass. Beam deflections are also minimized for aesthetic reasons. A visibly sagging beam, though structurally safe, is unsightly and to be avoided. A stiffer beam (high modulus of elasticity and high second moment of area) produces less deflection. Mathematical methods for determining the beam forces (internal forces of the beam and the forces that are imposed on the beam support) include the "moment distribution method", the force or flexibility method and the matrix stiffness method.

General shapes

Mostly the beams have rectangular cross sections in reinforced concrete buildings, but the most efficient cross-section is a universal beam. The fact that most of the material is placed away from the neutral axis (axis of symmetry in case of universal beam) increases the second moment of area of the beam which in turn increases the stiffness.

A universal beam is only the most efficient shape in one direction of bending: up and down looking at the profile as an I. If the beam is bent side to side , it functions as an H where it is less efficient. The most efficient shape for both directions in 2D is a box (a square shell) however the most efficient shape for bending in any direction is a cylindrical shell or tube. But, for unidirectional bending, the universal (I or wide flange) beam is king.

Efficiency means that for the same cross sectional area (Volume of beam per length) subjected to the same loading conditions, the beam deflects less.

Other shapes, like L (angles), C (Channels) or tubes, are also used in construction when there are special requirements.

See also

External links

References
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