I have replaced elastic in several sheets, I just did the same thing Alice did, worked fine for me... I had some fitted sheets, that I never used, and the elastic just lost all the stretch, so I figured the regular 1/4 inch could not be any worse. I just sewed iit the same as Alice did...and it holds the sheets fine..
I have also, bought flat cream or white sheets and added extension to the deepness of the pockets , because after adding a feather bed and thick mattress cover , they were not deep enough , So I just took off old elastic and added new to the sewn on extension. It worked fine, the strip you add too, is under your mattress anyway.