Beer, Johnston, and DeWolf's Mechanics of Materials includes the following image of the distribution of shear stresses in a bending beam:Beer Jonston and Dewolf 6.12 (Axes and load vector carried forward from Fig 6.1 in the text by me for clarity in this question) This image represents the bottom part of a beam below a slice in an offset xy plane so the analysis of the shear stresses is being conducted in the top shown surface. The beam is being bent by the load V.

The text mentions that the shear stress along $D_1'-D_2'$ are a parabolic function of x. However, they never mention why. So my question is, why? I don't understand where this would pop out of the math or why this should be the case intuitively.

Consider this example. I have a book in front of me so that I'm looking at the cover. I fix the bound edge and push the opposite edge with a force perpendicular to the cover. The book will bend in a uniform profile at any slice parallel to the ground. This would seem to be to demonstrate that the stresses are independent of the height of the slice. Why is this wrong?


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