Shear stress distribution for a tapered beam

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gmreit
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TL;DR
Shear stress distribution tapered beam
Free-body diagram of a differential slice of a tapered beam: a shaded region with dx across the top, shear stress tau at the cut face shown by arrows, normal stress sigma_x at the left face shown by arrows, and a height labelled h'dx at the right.

I am reading a journal article from Balduzzi about shear stress recover on tapered beams. I can not get the exact constants he gets and I am stumped. Above is solving for differential piece of the beam. Equations 16-18 are the ones I am trying to calculate. I attached my work and the article.
Thanks!
 
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  • Free-body diagram of a tapered beam element of length dx: a shaded hatched region above a blue-filled wedge with height h'dx, with arrows labelled tau at the interface f pointing right and a stack of left-pointing arrows labelled sigma x at f.
    1603990610823.png
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  • BALD - Tapered Beam.pdf
    BALD - Tapered Beam.pdf
    1,007.1 KB · Views: 328
  • Handwritten derivation on lined notebook paper for shear stress in a tapered beam, with equations for the axial force F, the stress sigma, and the constant C, and the substitution at y = -h(x)/2 - h_f.
    IMG_4303.jpg
    64.2 KB · Views: 280
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