The mechanics of bending a strip of paper

Here we study the physical problem in which a thin bar, such as a strip of paper, is bent under several types of loads and boundary conditions. Initially, we consider lengthwise compression of the strip due to concentrated forces applied on its ends. We present this problem for the three boundary conditions: doubly-clamped, hinged-clamped, and doubly-hinged ends constrained on a flat surface. We also consider both analytically and experimentally the problems of distributed loading such as the heavy cantilever and the post-buckling heavy column. We find the equations that govern the shapes of the bent strips via a minimization flow of the bending energy. In all cases, we find that photographically-captured profiles agree very well with the theoretical predictions. We also validate the theory with measurements of the deflection of the middle or the tip of the strip as its length is varied.