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More about bending

Galileo's (not quite right) theory of bending stress

Galileo developed a hypothesis concerning bending stress that was sensible but not correct. A better theory was not widely understood until more than 60 years later.

General Form of the Flexure Formula

• For non-rectangular sections, there is a more general derivation of the flexure formula.

• Moment of Inertia measures the size and "spread-outness" of a section with respect to an axis.

  
  
  

Determining Section Modulus and Moment of Inertia

• Rectangular sections

  • Direct calculation

  • Table lookup for standard sawn-lumber and glu-lam sections. (see handout)

    • Note that a table lists two values for I and S: Axis X-X and Axis Y-Y.

    • Axis X-X means that the X-X axis on the diagram is the neutral axis, and corresponds to the beam working in its stronger orientation. This is typcically called the Strong Axis orientation.

    • Axis Y-Y means that the Y-Y axis on the diagram is the neutral axis, and corresponds to the beam working in its weaker orientation. This is typcically called the Weak Axis orientation.

      

• Simple shapes (circles, circular tubes, etc.)

  • Appendix 5 of Schodek lists formulas for simple shapes.

  • For box sections where the center of the hole aligns with the center of the overall shape, the moment of inertia may be calculated by subtracting the moment of inertia of the hole from the moment of inertia of the overall.

    • The section modulus can then be calculated as I/c.

• Complex shapes

  • Appendix 5 of Schodek outlines the procedure for the "Parallel Axis Theorem".

  • Table lookup for standard steel sections. (see handout)

Beams and shear

Recall Shear in a column:

Shear in a beam:

Shear in bone:

Shear in a building: