Beam

As the engineering books explain, a beam is a bar or structural member subjected to transverse loads that tend to bend it. Any structural members act as a beam if external transverse forces induce bending. A simple beam is a horizontal member that rests on two supports at the ends of the beam. All parts between the supports have free movement in a vertical plane under the influence of vertical loads.

There are fixed beams, constrained beams, or restrained beams rigidly fixed at both ends or rigidly fixed at one end and simply supported at the other. A continuous beam is a member resting on more than two supports. A cantilever beam is a member with one end projecting beyond the point of support, free to move in a vertical plane under the influence of vertical loads placed between the free end and the support.

When a simple beam bends under its own weight, the plastic or fibers in a plastic on the upper or concave side is shortened, with the stress acting on them is compression. The fibers on the under or convex side are lengthened, and the stress acting on them is tension. In addition, shear exists along each cross section, the intensity of which is greatest along the sections at the two supports and zero at the middle section. When a cantilever beam bends under its own weight the fibers on the upper or convex side are lengthened under tensile stresses. The fibers on the under or concave side are shortened under compressive stresses, the shear is greatest along the section at the support, and zero at the free end.

The neutral surface is that horizontal section between the concave and convex surfaces of a loaded beam, where there is no change in the length of the fibers and no tensile or compressive stresses acting upon them. The neutral axis is the trace of the neutral surface on any cross section of a beam. The elastic curve of a beam is the curve formed by the intersection of the neutral surface with the side of the beam, it being assumed that the longitudinal stresses on the fibers are within the elastic limit.

The reactions, or upward pressures at the points of support, are computed by applying certain conditions necessary for equilibrium of a system of vertical forces in the same plane. They are the algebraic sum of all vertical forces that must equal zero; that is, the sum of the reactions equals the sum of the downward loads. There is also the algebraic sum of the moments of all the vertical forces that equals zero.

The first condition applies to cantilever beams and to simple beams uniformly loaded, or with equal concentrated loads placed at equal distances from the center of the beam. In the candlever beam, the reaction is the sum of all the vertical forces acting downward, comprising the weight of the beam and the superposed loads. In the simple beam each reaction is equal to one-half the total load, consisting of the weight of the beam and the superposed loads. The second condition applies to a simple beam not uniformly loaded. The reactions are computed separately, by determining the moment of the several loads about each support. The sum of the moments of the load around one support is equal to the moment of the reaction of the other support around the first support.

The fundamental laws for the stresses at any cross-section of a beam in equilibrium are: (1) sum of the horizontal tensile stresses equal sum of horizontal compressive stresses, (2) resisting shear equal vertical shear, and (3) resisting moment equal bending moment. Bending moment at any cross-section of a beam is the algebraic sum of the moments of the external forces acting on either side of the section. It is positive when it causes the beam to bend convex downward, thus causing compression in upper fibers and tension in lower fibers of the beam. When the bending moment is determined from the forces that lie to the left of the section, it is positive if they act in a clockwise direction; if determined from forces on the right side, it is positive if they act in a counterclockwise direction. If the moments of upward forces are given positive signs, and the moments of downward forces are given negative signs, the bending moment will always have the correct sign, whether determined from the right or left side. The bending moment should be determined for the side for which the calculation will be the simplest.

The deflection of a beam as computed by the ordinary formulas is that due to flexural stresses only. The deflection in honeycomb and short beams due to vertical shear can be high, and should always be checked. Because of the nonuniform distribution of the shear over the cross section of the beam, computing the deflection due to shear by exact methods is difficult. It may be approximated by:

where ys = deflection, inch, due to shear; M = bending moment, lb-in, at the section where the deflection is calculated; A = area of cross section of beam, square inches; and Es = modulus of elasticity in shear, psi; For a rectangular section, the ratio of deflection due to shear to the deflection due to bending, will be less than 5% if the depth of the beam is less than one-eighth of the length.

In designing a beam an approach is: (1) compute reactions; (2) determine position of the dangerous section and the bending moment at that section; (3) divide the maximum bending moment (lb-in) by the allowable unit stress (psi) to obtain the minimum value of the section modulus; and (4) select a beam secdon with a secdon modulus equal to or slightly greater than the secdon modulus required.

Assumptions are made in simple beam-bending theory that involve (1) all deflections are small, so that planar cross-sections remain planar before and after bending; (2) the beam is initially straight, unstressed, and symmetrical; (3) its proportional limit is not exceeded; and (4) Young's modulus for the material is the same in both tension and compression. In the analysis maximum stress occurs at the surface of the beam farthest from the neutral surface (Fig. 2.32), as given by the following equation:

where M = the bending moment in in./lbs., c = the distance from the neutral axis to the outer surface where the maximum stress occurs in in., I = the moment of inertia in in.4, and Z = 1/c-, the section modulus in in.3. Observe that this is a geometric property, not to be confused with the modulus of the material, which is a material property.

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