# Paper stiffness

As piece of paper is folded and unfolded, the stiffness of the sheet may seem to be greatly increased.

To those of you who don't recognise it: take a sheet of A4 paper, grasp one of the edges and move your hand vertically. Now fold the paper in half, unfold it and repeat.

Is anyone aware of any formal or heuristic explanation of what changed?

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Without information on orientation of folding and lift Your question is meaningless. – Georg May 30 '11 at 8:44

You have stiffened the paper by greatly increasing its bending moment of intertia. This can occur in at least two ways:

1.) When you unfolded the paper, it still had some residual bend which made the paper form a very shallow 'V'. Even though shallow. it is much deeper than the thickness of the original paper.

2.) When you unfolded the paper, it still had some residual bend which made the paper form a very shallow 'V' which was very localized to the area next to the bend. Even if you refold the paper in the opposite way, to take out or minimize the original fold, there is still a shallow 'V'. Even though it might be much shallower than in case 1, it is much deeper than the thickness of the original paper.

In both cases, the increased resistance to bending comes from the new geometry of the paper, more specifially, the geometry of a cross-section of the paper which goes through the bend.

Think of the unbent paper as a beam. Its resistance to bending is proportional to b(d^3), where 'b' is the width of the beam and 'd' is the depth. If you take a piece of 8.5" x 11" piece of paper and lay it flat over a pencil on the table, the paper will flop so that both ends touch the table. The paper forms a beam: 'b' is 8.5" or 11" (depending on how you laid the paper) and 'd' is the thickness of the sheet (say, about one one-hundreth of an inch).

How to improve the stiffness and strength of this beam? Fold the paper, accordion-style, with sharp 1/2" folds. Then lay it across the pencil, so that the folds are perdendicular to the pencil.

Assume that you folded the paper into an accordion that was 1.1" wide (one tenth of the original unfolded 11"), had about 22 folds and was 8.5" long. The 'b' for this new beam is 0.5", which is 50 times more than the (1/100) of an inch thickness of the paper. The new beam is (50)(50)(50)/10 = 12500 times stiffer than the unfolded sheet.

Now, let's go back to your sheet with the one bend. When you folded that sheet, you increased 'd'. The amount is hard to quantify, but I would ballpark it at a factor of 5, even if you tried to straighten out the bend by refolding it the opposite way. Again, guesstimate that 'b' of the localized area affected by the bend is 1/10 of the original sheet width. So the new resistance to bending increased by a factor of (5)(5)(5)/10 = 12.5 times the unfolded sheet.

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