. The strength of materials; a text-book for engineers and architects. Fig. 121. This is the complete relation between the stresses in beams,the bending moment, and the radius of curvature. In practicewe do not so much want to know the radius of curvature atvarious points of a beam, but we require the deflection, andso we will next find the relation between radius of curvatureand deflection, and then find the deflections for various kindsof loading. Our investigation now divides itself into two parts accordingas we consider it from the graphical or the mathematicalstandpoint, and we will deal
. The strength of materials; a text-book for engineers and architects. Fig. 121. This is the complete relation between the stresses in beams,the bending moment, and the radius of curvature. In practicewe do not so much want to know the radius of curvature atvarious points of a beam, but we require the deflection, andso we will next find the relation between radius of curvatureand deflection, and then find the deflections for various kindsof loading. Our investigation now divides itself into two parts accordingas we consider it from the graphical or the mathematicalstandpoint, and we will deal with it in this order. 250 THE STRENGTH OF IVIATERIALS INVESTIGATION FROM GRAPHICAL STANDPOINT* Preliminary Note on Curvature.—Let a b (Fig. 122)represent any curve, and let p p^ be points on it at a shortdistance s apart. Draw tangents p Q, p^ q^ to meet any baseline making angles 0 and 0-^ with it. and draw lines perpen-dicular to the tangents, then the point of intersection of theseperpendiculars is the centre of curvature of the short arc p Fig. 122. Then the angle subtended by p p^ at the centre will beequal to {0 — 6-^). •. if R is the radius of curvature R x (^ — ^j) = s. s R = e - 0^ or ^ 0 - 0^ _ 1 R 1 . Then - is called the curvature at the given point, or rather R A A the curvature is the value which ~ approaches as 5 gets smaller and smaller. Mohrs Theorem. — Now imagine a b to be a cableloaded vertically in any manner, and let the load between * The reader may take either the mathematical or the graphicalreasoning. Each is complete in itself. DEFLECTIONS OF BEAMS 251 / the points p, p^ be equal to w. Then it follows from Sthe laws of graphic statics that the cable takes up theshape of the link polygon, for the load system on it,(drawn with a polar distance equal to the horizontal pullin the let the tension in the cable at the points P, p^ be T, T^.Then the horizontal components of these tensions must beequal, since there is no horizontal fo
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