elasticity is equal under the two kinds of stress. Fig. 1 7
shows the cross section usually adopted for cast-iron girders.
§18. Neutral Axis.—The line ZZ 1 , fig. 14, perpendicular to the plane in which a beam is bent, and passing through the centre of gravity of any given cross section, is called the neutral axis of the beam at that point. The surface containing all the neutral axes is the neutral surface. Prac tical engineers sometimes apply the term neutral axis to the longitudinal line showing the neutral surface on a sids elevation, but in this article, as in Eankine s works, the words will be used as denned above.
When the assumption is made that the modulus of elasticity is the same for any given material whether under tension or compression, notwithstanding any difference in the ultimate strength to resist tension or compression, then it follows, as has been shown, that the neutral surface of a bent beam will separate it into two parts, one of which is compressed while the other is extended. The neutral axis in any cross section then contains the only part of the material which is neither extended nor compressed.
If, however, the average value of E for stresses varying between zero and the maximum intensity of compression to which the beam is subjected be different from the average value between zero and the maximum intensity of tension, then the neutral axis as above denned will not be the un strained axis ; the neutral axis is determined as soon as the cross section of the beam is known, being independent of the material used; the unstrained axis may differ in beams of the same cross section but made of different ma terials ; for if the average of E be greater say for compres sion than for tension, this will raise the unstrained axis above the neutral axis. It is not improbable that the position of the unstrained axis may vary in the same beam with loads similarly distributed, but of different magnitude, and also with different distributions of load. Until ex periments shall have accurately determined the relation of E to the intensity of stress we have no means of deter mining accurately the position of the unstrained axis. Even when E is constant the neutral axis, as above denned, will not always in practice correspond with the unstrained axis ; for instance, in a beam which was not only bent across, but also compressed endways, the unstrained surface would no longer contain the neutral axis as above denned. The un strained surface might be near one edge of the beam, or, indeed, if the general compression were large and the bend ing "small, the whole beam might be under compression, so that no part was unstrained. By restricting the use of the words neutral axis to the above definition, and using the words unstrained axis, or unstrained surface, for the second idea all ambiguity will be avoided.
The actual position of the unstrained axis in any beam of any material subject to a bending moment depends on the relative values of the modulus of elasticity for the material under all stresses positive and negative, great and small ; but as the simple hypothesis of a constant modulus is borne out by experiment in the most important materials, it is unnecessary to pursue this subject further.
§19. Shearing Stress.—The theory hitherto given shows the relation between longitudinal stresses (such as are re sisted by the india-rubber in the model) and the load on the girder ; but in designing a girder we have also to provide for the shearing stress or transverse force tending at each imaginary cross section to make the more heavily loaded of the two parts into which it divides the bridge slide down past the other. This shearing stress was resisted by the tongues of the model. The total shearing stress at any section is the sum of all the vertical forces acting on the beam on one side of the section. The shearing stress at any section will be called positive if the sum of the external forces on the right hand part of the beam tends to lift that portion up. Diagrams may be conveniently used to show shearing stresses, and, as in the case of bending moments, the shearing stress at any section due to two or more loads is simply the algebraic sum of the shearing stresses due to each load.
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Fig. 18.
Example 1. Load W at centre between supports (fig. IS); weight of beam neglected. The shearing stress is equal to W all along the beam, being the reaction at one pier ; the stress is positive to the right of the load, nega tive to the left.
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Fig. 19.
Example 2. Uniformly distributed load ^v per foot run (fig. 18rt), The shearing stress is tUcL at the points of support, and zero at the centre of the span ; at any section distant c from the centre of the beam it is we.
Example 3. A single load rolling from right to left of a beam of span L (fig. 186). When the load is at the distance x from the right hand support the shearing stress to the right L a; Wx of the load is W-^ , and to the left it is - . The JU L maximum stress for each section occurs when the load reaches that section ; it is positive for the right half, and negative for the left half of the beam.
