Hi All, I'm a few years out of school and out of practice on my Mechanics of Materials, so please pardon the fundamental question. Are there limitations on the use of the stress transformation equations in plane stress situations? For example I find the transformation equations don't yield the correct results when transforming the stress for a square plate, pulled in tension at it's corners (sectioned across it's diagonal) 45 degrees to get the stress when the plate is sectioned parallel to one pair of sides. Basically the transformation equations are yielding different results than if I were to just section the plate at 45 degrees and balance the internal forces. Do the stress transformation equations assume your original stresses are found at the smallest cross sectional area? I can't find that limitation in any textbooks. Thanks for the help.
Post the details of what you did, and somebody will probably explain where you went wrong. There shouldn't be any "limitations" with this.
I attached how I worked this out by hand. The top portion of the page just shows the square plate pulled in tension at the corners and sectioned along it's diagonal. On the left I found the stress on the plane parallel to the sides of the square using the method of sections. On the right I found the stress on the same plane (parallel to the sides) using stress transformation by force balancing on a small element. Page 2 is just the textbook stress transformation formula which yields the same result as the force balancing on the right side of page one. I included it as a check. I didn't include shear stresses in any of my calculations. Notice the left side results are different than the right side results. Can anyone explain where I'm going wrong with this? Thanks again.
That's where you went wrong. If you have a a stress field with Sx = some value and Sy = 0, and you make a cut at an angle, you will get a direct stress and a shear stress on the cut. The only situation where the shear stress is always 0, is when Sx and Sy both have the same value.
Thanks for the response Aleph! I agree with your statement regarding the shear stress that results when transforming from a plane with no shear to a plane at another angle. However, I don't believe this would affect my calculation of the normal stress. My issue is that the normal stress calculated from the stress transformation equations is not the same as the normal stress calculated from the method of sections. Any other insight would be greatly appreciated.