Problem # p032

Shear Stress Distribution due to Bending, Q(y)/b(y)

The cross-section of beam shown in the figure is a symmetric triangle with "a" as the baseline and "h" as its height. The shearstress distribution in this beam due to bending is described by the quantity Q(y)/b(y) with the coordinate "y" as indicated in the figure.

  1. Derive the equation for Q(y)/b(y). For b(y) as indicated in the above figure you may use the following equation :

    b(y) = a*(2/3-y/h)

    Also, for a symmetric triangle the centroid is located at 1/3 of its height away from its baseline.

  2. Using the equation you developed under 1) find the expression for Q(y)/b(y) at y=-h/3 (top of cross-section).

  3. Using the equation you developed under 1) find the expression for Q(y)/b(y) at y=0 (neutral axis of cross-section).

  4. Using the equation you developed under 1) find the expression for Q(y)/b(y) at y=h/6.

  5. Using the equation you developed under 1) find the expression for Q(y)/b(y) at y=2h/3 (bottom of cross-section).

  6. At which of the investigated values of y is Q(y)/b(y) the largest ? ( You need to use differentiation to prove your assertion ).