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Chapter Six Shearing Stresses in Beams and Thin-Walled Members

Chapter Six Shearing Stresses in Beams and Thin-Walled Members. 6.1 Introduction. -- In a long beam, the dominating design factor: . -- Primary design factor. -- Minor design factor. [due to transverse loading]. -- In a short beam, the dominating design factor: .

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Chapter Six Shearing Stresses in Beams and Thin-Walled Members

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  1. Chapter Six Shearing Stresses in Beams and Thin-Walled Members

  2. 6.1 Introduction -- In a long beam, the dominating design factor: -- Primary design factor -- Minor design factor [due to transverse loading] -- In a short beam, the dominating design factor:

  3. Equation of equilibrium: (6.1) (6.2) -- Shear stress xy is induced by transverse loading. -- In pure bending -- no shear stress

  4. Materials weak in shear resistance shear failure could occur.

  5. 6.2 Shear on the Horizontal Face of a Beam Element Knowing and We have (6.3)

  6. Since Defining Q = max at y = 0 Therefore, (6.4) and = shear flow = horizontal shear/length (6.5) here Q = the first moment w.t.to the neutral axis

  7. 6.3 Determination of the Shearing Stresses in a Beam = ave. shear stress (6.6)

  8. xy = 0 at top and bottom fibers Variation of xy < 0.8% if b  h/4 -- for narrow rectangular beams

  9. 6.4 Shearing Stresses xy in Common Types of Beams -- for narrow rectangular beams t = b (6.7) (6.8) Also, Hence,

  10. Knowing A = 2bc, it follows (6.9) This is a parabolic equation with @ y = c @ y = 0 -- i.e. the neutral axis At y = 0, (6.10)

  11. Special cases: American Standard beam (S-beam) or a wide-flange beam (W-beam) For the web: (6.6) -- over section aa’ or bb’ -- Q = about cc’ For the flange: (6.11)

  12. 6.5 Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam (6.12) (6.13)

  13. 6.6 Longitudinal Shear on a Beam Element of Arbitrary Shape xz = ?

  14. Using similar procedures in Sec. 6.2, we have (6.4) = shear flow (6.5)

  15. 6.7 Shearing Stresses n Thin-Walled Members These two equations are valid for thin-walled members: (6.4) (6.5)

  16. From Sec. 6.2 （6.4) We have: （6.6)

  17. (6.6) -- This equation can be applied to a variety of cross sections.

  18. 6.8 Plastic Deformation (6.14)

  19. (6.15) (6.16)

  20. 6.9 Unsymmetric Loading Of Thin-Walled Members; Shear Center (4.16) (6.6)

  21. (6.17) (6.18) (6.19)

  22. (6.20) (6.21) (6.22) (6.23)

  23. (6.23) (6.25) (6.26) (6.27)

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