More on Beam Bending

• The general form of the flexure formula:

  f_b = Mc / I

  Where I is defined as Moment of Inertia, a section that measures the size and "spread-outness" of a section with respect to an axis.

• Tables for standard steel and timber sections list two values for moment of inertia

  • A strong axis value called I_xx, for the section bending in its strongest orientation.

  • A weak axis value called I_yy, for the section bending in its weakest orientation.

  • The general definition of section modulus:

    S = I/c

    Where c is the distance from the neutral axis to the extreme fiber of the section.

  • Section modulus is also defined in terms of strong axis and weak axis properties:

    S_xx = I_xx / c_xx

    S_yy = I_yy / c_yy

Shear in Beams

• Shear causes a racking deformation, inducing diagonal tension and compression on mutually perpendicular axes.

  

• Shear failure in beams may manifest itself in several forms

  • Diagonal cracking (concrete).

  • Diagonal buckling (thin plates in steel beams).

  • Horizontal cracking (timber).

• In beams, the shear stresses are maximum at the neutral axis because this is where the tension and compression resultants of the unbalanced moment create the greatest horizontal sliding action.

  • Since maximum bending stresses occur at the extreme edge of a beam section while maximum shear stresses occur at the neutral axis, shear and bending stresses can be considered separately in design. They are uncoupled.

Example

• Contours of bending stress:

  Overall:
  

  At mid-span:
  

• Contours of shear stress:

  Overall:
  

  At the end support:
  

• Question:

  Bending
  

  Shear
  

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