Problem 1.301

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 \[ N(x)=E I \frac{d^{2} y}{d x^{2}} \] 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 \(\left(I=\int z^{2} d S,\right.\) Fig. \(\left.1.75\right)\)

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 \(\lambda,\) if its end \(A\) experiences (a) the bending moment of the couple \(N_{0}\); (b) a force \(F\) oriented along the \(y\) axis.

1.301. (a) Here \(N\) is independent of \(x\) and equal to \(N_{0} .\) Integrating twice the initial equation with regard to the boundary conditions \(d y / d x(0)=0\) and \(y(0)=0,\) we obtain \(y=\left(N_{0} / 2 E I\right) x^{2} .\) This is the equation of a parabola. The bending deflection is \(\lambda=\) \(=N_{0} l^{2} / 2 E I,\) where \(I=a^{4} / 12\) (b) In this case \(N(x)=F(l-x)\) and \(y=(F / 2 E I)(l-x / 3) x^{2}\) \(\lambda=F l^{3} / 3 E I,\) where \(I\) is of the same magnitude as in (a).

Elastic deformations