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 crosssection 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 λ, if its end A experiences
(a) the bending moment of the couple N0;
(b) a force F oriented along the y axis.