Then the total deflection at the end of the bar due to the bending all along the bar instead of for length Dx is given by the equation:
(6) The integral from x equals 1 to x equals o of 12Fx2 dx/wt3Y which on integration gives: 4Fl3/wt3Y
If we want to know the total deflection of any other point on the bar which is at a distance of D from the clamp, we multiply equation (5) by (x - 1 + D)/y instead of x/y and integrate from x equals 1 to x equals 1 - D
(7) Or the integral of [ 12Fx (x - 1 + D) dx ]/wt3Y from x equals 1 to x equals 1 - D
Integrating we get: 12F (x3 /2 - lx2/2 + Dx2/2)/wt3Y for limits of x equals 1 and x equals 1 - D
Substituting these limits we get:
12F (1D2/2 - D3/6) wt3Y which is the deflection of any point on the bar which is a distance D from the clamp.
Fig. 6a shows the form of bending of a bow constructed with limbs corresponding to the bar just discussed. It is shown with an 8 inch rigid section at the middle. A bow of this type would do most of its bending near the handle and would be considered worthless. Except for the rigid section at the middle, any uniform stick or limb would bend in this form.
Using the same method as in the preceding case we may determine the bending form for a bar of length l, thickness t and width W at the clamp but having no width at the loaded end. Fig. 3a shows such a bar loaded with a force F.
It may be shown as before that the deflection at any point P for a length Dx equals 12FxyDx/wt3Y, See equation (5).
However in this case w is a variable and by inspection it will be seen that w is equal to Wx/1
Substituting this value in the above we get:
(8) 12FlyDx/Wt3Y for the deflection at P
This deflection does not contain x and is therefore a constant at all points. This means that the curvature is constant for all parts of the bar or that the bar is bending in the arc of a circle.