Structural members are usually classified according to the types of loads that they support. For instance, an axially loaded bar supports forces having their vectors directed along the axis of the bar, and a bar in torsion supports torques (or couples) having their moment vectors directed along the axis.
Beams are structural members subjected to lateral loads, that is, forces or moments having their vectors perpendicular to the axis of the bar.The beams shown in Fig. 1 are classified as planar structures because they lie in a single plane. If all loads act in that same plane, and if all deflections (shown by the dashed lines) occur in that plane, then we refer to that plane as the plane of bending.
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| FIG. 1 Examples of beams subjected to lateral loads |
Beams are usually described by the manner in which they are supported. For instance, a beam with a pin support at one end and a roller support at the other (Fig. 2a) is called a simply supported beam or a simple beam. The essential feature of a pin support is that it prevents translation at the end of a beam but does not prevent rotation. Thus, end A of the beam of Fig.2a cannot move horizontally or vertically but the axis of the beam can rotate in the plane of the figure. Consequently, a pin support is capable of developing a force reaction with both horizontal and vertical components (HA and RA), but it cannot develop a moment reaction.
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| FIG. 2 Types of beams: (a) simple beam, (b) cantilever beam, and (c) beam with an overhang |
At end B of the beam (Fig.2a) the roller support prevents translation in the vertical direction but not in the horizontal direction; hence this support can resist a vertical force (RB) but not a horizontal force. Of course, the axis of the beam is free to rotate at B just as it is at A. The vertical reactions at roller supports and pin supports may act either upward or downward, and the horizontal reaction at a pin support may act either to the left or to the right.
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| FIG.3 Beam supported on a wall: (a) actual construction, and (b) representation as a roller support. Beam-to-column connection:(c) actual construction, and (d) representation as a pin support. |
The beam shown in Fig.2b, which is fixed at one end and free at the other, is called a cantilever beam. At the fixed support (or clamped support) the beam can neither translate nor rotate, whereas at the free end it may do both. Consequently, both force and moment reactions may exist at the fixed support.
The third example in the figure is a beam with an overhang (Fig.2c). This beam is simply supported at points A and B (that is, it has a pin support at A and a roller support at B) but it also projects beyond the support at B. The overhanging segment BC is similar to a cantilever beam except that the beam axis may rotate at point B.
When drawing sketches of beams, we identify the supports by conventional symbols, such as those shown in Fig.2. These symbols indicate the manner in which the beam is restrained, and therefore they also indicate the nature of the reactive forces and moments. However, the symbols do not represent the actual physical construction. For instance, consider the examples shown in Fig.3. Part (a) of the figure shows a wide-flange beam supported on a concrete wall and held down by anchor bolts that pass through slotted holes in the lower flange of the beam. This connection restrains the beam against vertical movement (either upward or downward) but does not prevent horizontal movement.
Also, any restraint against rotation of the longitudinal axis of the beam is small and ordinarily may be disregarded. Consequently, this type of support is usually represented by a roller, as shown in part (b) of the figure.
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| Beam-to-column connection with one beam attached to column flange and other attached to column web (Joe Gough/Shutterstock) |
The second example (Fig.3c) is a beam-to-column connection in which the beam is attached to the column flange by bolted angles. (See photo.) This type of support is usually assumed to restrain the beam against horizontal and vertical movement but not against rotation (restraint against rotation is slight because both the angles and the column can bend). Thus, this connection is usually represented as a pin support for the beam (Fig.3d).
The last example (Fig.3e) is a metal pole welded to a base plate that is anchored to a concrete pier embedded deep in the ground. Since the base of the pole is fully restrained against both translation and rotation, it is represented as a fixed support (Fig.3f ).
The task of representing a real structure by an idealized model, as illustrated by the beams shown in Fig.2, is an important aspect of engineering work. The model should be simple enough to facilitate mathematical analysis and yet complex enough to represent the actual behavior of the structure with reasonable accuracy. Of course, every model is an approximation to nature. For instance, the actual supports of a beam are never perfectly rigid, and so there will always be a small amount of translation at a pin support and a small amount of rotation at a fixed support. Also, supports are never entirely free of friction, and so there will always be a small amount of restraint against translation at a roller support. In most circumstances, especially for statically determinate beams, these deviations from the idealized conditions have little effect on the action of the beam and can safely be disregarded.
Types of Loads
Several types of loads that act on beams are illustrated in Fig.2. When a load is applied over a very small area it may be idealized as a concentrated load, which is a single force. Examples are the loads P1, P2, P3, and P4 in the figure. When a load is spread along the axis of a beam, it is represented as a distributed load, such as the load q in part (a) of the figure. Distributed loads are measured by their intensity, which is expressed in units of force per unit distance (for example, newtons per meter or pounds per foot). A uniformly distributed load, or uniform load, has constant intensity q per unit distance (Fig.2a). A varying load has an intensity that changes with distance along the axis; for instance, the linearly varying load of Fig.2b has an intensity that varies linearly from q1 to q2. Another kind of load is a couple, illustrated by the couple of moment M1 acting on the overhanging beam (Fig.2c).
We assume in this discussion that the loads act in the plane of the figure, which means that all forces must have their vectors in the plane of the figure and all couples must have their moment vectors perpendicular to the plane of the figure. Furthermore, the beam itself must be symmetric about that plane, which means that every cross section of the beam must have a vertical axis of symmetry. Under these conditions, the beam will deflect only in the plane of bending (the plane of the figure).
Reactions
Finding the reactions is usually the first step in the analysis of a beam. Once the reactions are known, the shear forces and bending moments can be found, as described later in this chapter. If a beam is supported in a statically determinate manner, all reactions can be found from free-body diagrams and equations of equilibrium.
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| Internal releases and end supports in model of bridge beam (Courtesy of the National Information Service for Earthquake Engineering EERC, University of California, Berkeley.) |
In some instances, it may be necessary to add internal releases into the beam or frame model to better represent actual conditions of construction that may have an important effect on overall structure behavior. For example, the interior span of the bridge girder shown in Fig.4 is supported on roller supports at either end, which in turn rest on reinforced concrete bents (or frames), but construction details have been inserted into the girder at either end to ensure that the axial force and moment at these two locations are zero. This detail also allows the bridge deck to expand or contract under temperature changes to avoid inducing large thermal stresses into the structure.
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| FIG.4 Types of internal member releases for two-dimensional beam and frame members |
To represent these releases in the beam model, a hinge (or internal moment release, shown as a solid circle at each end) and an axial force release (shown as a C-shaped bracket) have been included in the beam model to show that both axial force (N) and bending moment (M), but not shear (V), are zero at these two points along the beam. (Representations of the possible types of releases for two-dimensional beam and torsion members are shown below the photo). As examples below show, if axial, shear, or moment releases are present in the structure model, the structure should be broken into separate free-body diagrams (FBD) by cutting through the release; an additional equation of equilibrium is then available for use in solving for the unknown support reactions included in that FBD.