Here is another "optimal shape / elasticity" problem of considerable interest. Consider a horizontal beam supported at N points: ========================== ^ ^ ^ Question: for each fixed N=1,2,3,..., what is the best choice of the N support points? (Assume thin uniform 1D beam with small-deflection approximation.) I presume the optimal points have even-symmetry and we can assume wlog the beam goes from x=-1 to x=+1. The beam curve shape y(x) will minimize gravitational+elastic energy: integral ( E*y''(x)^2 + g*y ) dx = minimum subject to constraint that y(x) = 0 whenever x is one of the N support points x=x1, x=x2, x=x3,..., x=xN. Here g is a gravitational-related constant and E an elastic-related constant. The Euler-Lagrange equation shows y(x) will satisfy the differential equation g + E*y''''(x) = 0 with solution y(x) = -x^4 * g/(E*24) + AnyCubicPolynomial(x). Thus y(x) + x^4 * g/(E*24) will simply be a "cubic spline" fit at the N support points (y, y', and y'' all continuous). The question then is where to place the points to minimize the maximum value of |y''(x)|, i.e. minimize the max-stress. It would be good to tabulate the points numerically for N=1,2,...,10. After all, civil engineers have been building these things for centuries, might as well figure out for them how to do it right. :) ---------- When N=1, obviously the answer is 1 midpoint at x=0. When N=2, the answer is not so obvious...