. A treatise on the mathematical theory of elasticity . expression for w may be written Ww=T — jrf * {iz — \z^) — ^a; + SoX — EI X- das + xy^ ....(26) The term t(j> corresponds with the twisting of the beam by the load, andwe know that it represents a distortion of the cross-sections into curvedsurfaces. The terms — x{W{Iz — ^z)/EI + /3] represent a displacement bywhich the cross-sections become at right angles to the strained term s„x represents a displacement by which each cross-section is turnedback, towards the central-line, through an angle So. as explained
. A treatise on the mathematical theory of elasticity . expression for w may be written Ww=T — jrf * {iz — \z^) — ^a; + SoX — EI X- das + xy^ ....(26) The term t(j> corresponds with the twisting of the beam by the load, andwe know that it represents a distortion of the cross-sections into curvedsurfaces. The terms — x{W{Iz — ^z)/EI + /3] represent a displacement bywhich the cross-sections become at right angles to the strained term s„x represents a displacement by which each cross-section is turnedback, towards the central-line, through an angle So. as explained in (c) remaining terms in W/EI represent a distortion of the cross-sectionsinto curved surfaces, independent of that which depends upon t^. If weconstruct the surface which is given by the equation = -m{^-(Mh^]^t (2^) and suppose it to be placed so that its tangent plane at the origin coincideswith the tangent plane of a strained cross-section at its centroid, the strainedcross-section will coincide with this 232, 233] IN A BENT BEAM 347 In the case of a circular boundary the value of the right-hand member of (27) is and the contour lines of the strained cross-section are found by equating this expressionto a constant. Some of these lines are traced in Fig. 29. 233. Distribution of shearing stress. The importance of the transverse component Y^ of the tangential tractionon the cross-sections may be seen in the case of the elliptic boundary. Whena is large compared with h, the maximum value of Y^ is small compared withthat of Xg; as the^ratio of 6 to a increases, the ratio of the maximum of Y^to that of Xj increases; and, when h is large compared with a, the maximumof Fj is large compared with that of X;,. Thus the importance of Fj increasesas the shape of the beam approaches to that of a plank. We may illustrate graphically the distribution of tangential traction on the cross-sections by tracing curves, which are such that the tangent to any on
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