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### 30120140505003

3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 28-44 © IAEME 30 Consider an elemental length PQ = ݀‫ݏ‬ of a curve. Let the tangents at P and Q make anglesΨ and dΨ + Ψ with the axis. Let the normal at P and Q meet at C. Then C is called the center of curvature of the curve at any point between P and Q on the curve. The distance CP = CQ = Ris called the radius of curvature at any point between P and Q on the curve. Obviously, s R d∂ = Ψ or , ܴ ൌ ݀‫ݏ‬ ݀߰ൗ but we know that if (‫,ݔ‬ ‫)ݕ‬ be the coordinate of P. ݀‫ݕ‬ ݀‫ݔ‬ ൌ tan Ψ ௗ௬ ௗ௦ ൌ ௗ௦ ௗ௫⁄ dΨ ௗ௫ൗ = ௦௘௖ట ೏ഗ ೏ೣ …………………………………..……………………………………………(1) Differentiating with respect to, we have ‫ܿ݁ݏ‬ଶ ‫.ݔ‬ ݀߰ ݀‫ݔ‬ ൌ ݀ଶ ‫ݔ݀/ݕ‬ଶ ௗట ௗ௫ ൌ ௗమ௬ ௗ௫మ / ‫ܿ݁ݏ‬ଶ ߰…………………………………………………………………………………………….. (2) Substituting in the equation (1) we have, ܴ ൌ ೞ೐೎ഗ ೏మ೤ ௗ௫మ ‫ܿ݁ݏ‬ଶ‫ݔ‬ ൌ ‫ܿ݁ݏ‬ଷ ߰ ݀ଶ‫ݕ‬ ݀‫ݔ‬ଶ⁄ Therefore, 1 ܴ ൌ ݀ଶ ‫ݕ‬ ݀‫ݔ‬ଶ /‫ܿ݁ݏ‬ଷ ߰ ଵ ோ ൌ ௗమ௬ ௗ௫మ /ሺ‫ܿ݁ݏ‬ଶ ߰ሻ య మor ଵ ோ ൌ ௗమ௬ ௗ௫మ /ሺ1 ൅ ‫݊ܽݐ‬ଶ ߰ሻ య మ For practical member bend due to bending moment the slope ‫߰݊ܽݐ‬ at any point is a small quantity, hence ‫݊ܽݐ‬ଶ ψ can be ignored. Therefore 1 ܴ ൌ ݀ଶ ‫ݕ‬ ݀‫ݔ‬ଶ If M be the bending moment which has produced the radius of curvature R, we have ெ ூ ൌ ா ோ , ଵ ோ ൌ ெ ாூ , ௗమ௬ ௗ௫మ ൌ ெ ாூ ‫ܯ‬ ൌ ‫ܫܧ‬ ௗమ௬ ௗ௫మ………………………………………………………………………………………..(3)