Torsional Analysis Calculations

The force F_i on shear line line i is given by

max (F_i+, F_i-)

where

F_i+ = F_di + F_ti+

F_i- = F_di + F_ti-

where

F_di is the direct component given by
F_di = F * K_i / S K_i,

where
- F is the total applied force in the direction parallel to the shear lines and
- K_iis the rigidity of shear line i
F_ti is the torsional component given by
F_ti+ = T+ * K_i * d_i / (J_x + J_y)

F_ti- = T- * K_i * d_i / (J_x + J_y)

where
- d_iis the distance from the shear line to the center of rigidity CR (see below)
- J_xand J_y are the torsional rigidities, in the x- and y- directions, given by
  J = S K_i * d_i^a
- T is the torsion, given by
  T+ = (CL - CR + ae) * F
  
  T- = (CL - CR - ae) * F
  
  where
  - CL = Center of load = centroid of applied loads in force direction relative to the edge of the building. For seismic loading, this is equivalent the center of the building mass at that level.
  - CR = Center of rigidity relative to the edge of the building given by
    CR = S K_i * l_i / S K_i,
    
    where
    - l_i is the distance to the shear line from the edge of the building
    - K_iis the rigidity of shear line i
  - ae = accidental eccentricity, which is a proportion of the building width perpendicular to force direction. See Diaphragm Flexibility Cases for the values for different design cases.

In the direction of loading, torsional forces add to the direct shear forces for some walls and subtract from the direct shear forces for other walls, depending on the direction of the torsional moment created by the difference in the center of load and rigidity.

Torsional analysis also creates torsional forces in shear lines orthogonal to the force direction. Unless we are designing for loads from both directions simultaneously, as we do for low-rise wind loads, the program discards the torsional forces in the orthogonal direction on the assumption they are smaller than the total force from torsional analysis of loads in that direction.