Interaction

• Bending stresses and shear stresses in beams can usually be considered separately, since the maximum stress levels occur in different parts of the cross section.

  • Maximum bending stresses occur at the extreme edge of the section.

  • Maximum shear stresses occur at the center of the section (on the neutral axis.

• Bending and axial stresses must be considered together.

• Axial and bending stresses add together when linear elastic. Under linear elastic conditions, axial compression stresses will add to compression stresses due to bending, and counteract tension stresses due to bending.

• Interaction formula. Using the working stress approach to member design, members with both axial stress and bending stress are designed using the interaction formula:

  

  Where:

  (bending stress under service loads)

  (axial stress under service loads)

  F_b = allowable bending stress.

  F_a = allowable axial stress.

  • In the case of compression, the allowable axial stress must account for buckling, and will be the smallest of the allowable crushing stress, the allowable weak axis critical stress, and the allowable strong axis critical stress.

Pre-stressing and Post-tensioning

• Compression prestressing can reduce or eliminate tension due to bending. Since axial compression can eliminate tension in a beam, it can be useful in concrete construction to remove tension and thus limit cracking.

• This strategy is generally called pre-stressing, although that term can also refer to the specific strategy where steel tendons are tensioned before the concrete hardens (also known as pre-tensioning).

  • When tendons are tightened after the concrete hardens, the technique is commonly called post-tensioning.

• When the tendons are located eccentrically with respect to the centroid of the cross section, they can be used to produce deflections and stress patterns opposite those resulting from the applied load.

  

• Variable eccentricity. When the eccentricity is varied to match the moment patterns due to applied load, the member can have near zero deflection and work in near uniform compression.

• The variation in eccentricity can be achieved by either varying the trajectory of the tendon, or the profile of the member.