Question

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.

Answer 1

We use the equation given above and use the result that when y is small

where we have used y = 0 for x = 0 to set the constant of integration at zero. This is the equation of a parabola. The sag of the free end is

(b) In this case N (x) =F (l- x) because the load F at the extremity is balanced by a similar force at F directed upward and they constitute a couple. Then