(Solution) - The bending of an elastic rod is described by the elastic -(2025 Original AI-Free Solution)

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The bending of an elastic rod is described by the elastic curve passing through centres of gravity of rod's cross-sections. At small bendings the equation of this curve takes the form N (x) = EI d2y/dx2, where N (x) is the bending moment of the elastic forces in the cross. section corresponding to the x coordinate, E is Young's modulus, I is the moment of inertia of the cross-section relative to the axis passing through the neutral layer (I = ?z2dS, Fig. 1.75). Suppose one end of a steel rod of a square cross-section with side a is embedded into a wall, the protruding section being of length l (Fig. 1.76). Assuming the mass of the rod to be negligible, find the shape of the elastic curve and the deflection of the rod ?, if its end A experiences
(a) The bending moment of the couple No;
(b) A force F oriented along the y axis.