. The action of materials under stress; . stresses, shows that, forstresses within the elastic limit, equal increments of length-ening and shortening are occasioned by equal increments ofstress. If this beam has not been loaded so heavily as toproduce a unit stress on any particle in excess of the elasticlimit (and no working beam, one expected to last permanently,should be loaded to excess), the longitudinal unit stresses be-tween the particles will vary as the lengthening and shorten-ing of these fibres, that is, as the distance from the point ofno stress. Hence, at any section, the direct s
. The action of materials under stress; . stresses, shows that, forstresses within the elastic limit, equal increments of length-ening and shortening are occasioned by equal increments ofstress. If this beam has not been loaded so heavily as toproduce a unit stress on any particle in excess of the elasticlimit (and no working beam, one expected to last permanently,should be loaded to excess), the longitudinal unit stresses be-tween the particles will vary as the lengthening and shorten-ing of these fibres, that is, as the distance from the point ofno stress. Hence, at any section, the direct stress is uni-formly varying, with a max-imum tension on the convexside and a maximum com-pression on the concaveside. The stresses on differentforms of cross-section A C are shown in Fig. 25. Thetotal tension on the section is always equal to the totalcompression. 76. Neutral Axis.—The arrows in Figs. 24, 25 maybe taken to represent the unit stress at each point of the cross-section, varying as the distance from the plane of no Fig. 25. 66 STRUCTURAL MECHANICS. and constant in the direction z. To locate the point or planeof no stress or netctral axis for successive sections:— Let f^ and f^ be the unit stresses of compression and ten-sion between the particles at the extreme edge of any section,distant y^ and y^ from the point of no stress. It is plain thatfc • ft -- Jc • yt from similar triangles, and that the unit stress/ at any point distant jj/ from the point of no stress will be P = -ry, or 7- y, or, in general -~ y, from a similar proportion. If zdy is the area of the strip on which the unit stress p is exerted, z being the co-ordinate at right angles to x and y, fthe total force on zdy will be pzdy — — yzdy — cyzdy, where c is a constant, the unit stress at a unit distance. As the total normal tension on the section is to equal thetotal compression, or their sum is to be zero, § 74, the may be written \ pzdy = c \ yzdy O. Therefore the
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