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Strength of Beams 56

beam, forces, neutral, called, convex and applied

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56. Kinds of Loads Considered. The loads that are applied to a horizontal beam are usually vertical, but sometimes forces are applied otherwise than at right angles to beams. Forces acting on beams at right angles are called transverse forces; those applied parallel to a beam ure called longitudinal forces ; and others are called inclined forces. For the present we deal only with beams subjected to transverse forces (loads and reactions).

57. Neutral Surface, Neutral Line, and Neutral Axis. When a beam is loaded it may be wholly convex up (concave down), as a cantilever; wholly convex down (concave up), as a simple beam on end supports; or partly convex up and partly convex down, as a simple beam with overhanging ends, a restrained beam, or a con tinuous beam. Two vertical parallel lines drawn close together on the side of a beam before it is loaded will not be parallel after it is loaded and bent. If they are on a convex-down portion of a beam, they will be closer at the top and farther apart below than when drawn (Fig. 32a), and if they are on a convex:up portion, they will be closer below and farther apart above than when drawn (Fig. 32b).

The "fibres " on the convex side of a beam are stretched and therefore under tension, while those on the concave side are short ened and therefore under compression. Obviously there must be some intermediate fibres which are neither stretched nor shortened, i. e., under neither tension nor compression. These make up a sheet of fibres and define a surface in the beam, which surface is called the neutral surface of the beam. The intersection of the neutral surface with either side of the beam is called the neutral line, and its intersection with any cross-section of the beam is called the neutral axis of that section. Thus, if ab is a fibre that has been neither lengthened nor shortened with the bending of the beam, then nn is a portion of the neutral line of the beam; and, if Fig. 32c be taken to represent a cross-section of the beam, NN is the neutral axis of the section.

It can be proved that the neutral axis of any cross-section of a loaded beam passes through the center of gravity of that section, provided that all the forces applied to the beam are transverse, and that the tensile and compressive stresses at the cross-section are all within the elastic limit of the material of the beam.

58. Kinds of Stress at a of a Beam. It has already been explained in the preceding article that there are ten sile and compressive stresses in a beam, and that the tensions are on the convex side of the beam and the compressions on the con cave ( see Fig. 33). The forces T and C are exerted upon the portion of the beam represented by the adjoining portion to the right (not shown). These, the student is reminded, are often called fibre stresses.

Besides the fibre stresses there is, in general, a shearing stress at every cross-section of a beam. This may be proved as follows: Fig. 34 represents a simple beam on end supports which has actually been cut into two parts as shown. The two parts can maintain loads when in a horizontal position, if forces are applied at the cut ends equivalent to the forces that would act there if the beam were not cut. Evidently in the solid beam there are at the section a compression above and a tension below, and such forces can be applied in the cut beam by means of a short block C and a chain or cord T, as shown. The block furnishes the compressive forces and the chain the tensile forces. At first sight it appears as if the beam would stand up under its load after the block and chain have been put into place. Except in certain cases*, how ever, it would not remain in a horizontal position, as would the solid beam. This shows that the forces exerted by the block and chain (horizontal compression and tension ) are not equivalent to the actual stresses in the solid beam. What is needed is a vertical force at each cut eLid.

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