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RE: Accidental Torsion

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Bill, some of the questions you've raised are quite penetrating.  In the
development of KeyLat(TM), Keymark struggled long and hard to find
solutions to these issues that made sense.  My response here is intended to
share with the community at large what direction our interpretation took.
This is also an attempt to verify the basis of some of the functions that
exist within KeyLat's rigid diaphragm analysis.  We welcome all feedback.


REGARDING A-SUB-X:
Ax is based on drift calculations that do not include the redundancy factor
"rho".  KeyLat's rigid analysis consists of four steps:
STEP 1:  KeyLat performs one analysis that distributes forces to shear
walls based on relative wall rigidity.**  
STEP 2:  Once the wall materials have been selected, KeyLat performs a
second analysis, using rho=1.0, to calculate drift.
STEP 3:  If torsional irregularity exists, KeyLat performs a third
analysis, augmenting accidental torsion by Ax.
STEP 4:  Finally, KeyLat performs a fourth analysis to recalculate the
deflection of the walls (which could have changed during step 3).

**Note that each step 1-4 occurs on every level of the structure, with
upper-level shear and torsion being transferred to lower levels.  Holddown
forces are also transmitted as point loads to lower levels.  KeyLat can
analyze up to 5 levels.


REGARDING 5% +/- EFFECTS OF ACCIDENTAL TORSION:
We feel we account for all possibilities by running the seismic loads in
four directions (N,S,E,W).

Picture a multi-story building, each level for which a center of rigidity
and a center of mass has been computed.  These physical locations are
fixed.  The third story might see a cancellation of "nominal" torsional
shear effects from the fourth and fifth stories if they in fact were
algebraically opposite to each other because of the centers of mass and
rigidity for each level.  The "accidental" torsional shears for levels
three to five, however, are accumulated and held in reserve by KeyLat on an
absolute value basis, and then given a sign to match the sign of the design
cumulative torsional shears for the level in question.  In this way, the
accidental torsion does its best to hurt the structure at every level down
through the building.  Since some users might feel this to be overly
conservative,  KeyLat includes the option of combining the accidental
torsion from each level not as an algebraic sum, but using SRSS (square
root of the sum of the squares) much as is frequently done for response
spectrum analysis.

In a typical rigid diaphragm analysis performed by hand, one "pass" is made
for a given direction (North/South or East/West) and the worst case
solution for a particular wall is computed directly.  The engineer may
intuitively understand which way to move the accidental eccentricity to get
the highest load on a particular wall.  In contrast, KeyLat performs this
calculation in four passes: load to the North, load to the South, load to
the East, and load to the West.  This may be performed for wind,
earthquake, or both, implying a possible eight passes in total.  For each
pass, the accidental eccentricity is given a sign that will tend to
increase the computed torsion for that level.  In this way, the program
brackets the solution of worst case for each wall without having to use
'judgment.'


REGARDING HOW TO ITERATE ON A SOLUTION
During an analysis, up to eight passes may be performed (N,S,E,W for wind
and seismic).  Each pass may consist of up to 5 levels.  It is possible
that a particular design from the first pass may survive through all the
other passes and control.  However, if on the eighth pass one wall on the
last level requires "bumping" to a higher capacity material, this negates
the validity of the first seven passes, since the distribution of relative
stiffness has changed.  Therefore, if on any pass a wall material is
bumped, the entire process starts over.  This cycle of passes and "bumping"
occurs during STEP 1 and STEP 3 (above). 


REGARDING HOW KEYLAT DEFINES SUCCESSFUL CONCLUSION TO AN ITERATION
The stiffness of the wall depends on the amount of force on the wall
(e-sub-n in the 4-part wall deflection equation).  But the amount of force
on the wall depends on the relative stiffness of the wall.  KeyLat assumes
that this iterative-type solution has "converged" if (1) no wall required
"bumping" of materials since the last iteration and (2) the force on any
wall in the current iteration is less than X plf compared the amount of
force on the wall during the previous iteration.  The value of "X" is
called a "convergence tolerance" and is under the control of the user.
Generally, we have found that a tolerance of 50-80 plf works well and
generates reasonable results (i.e. equilibrium is satisfied, all forces are
accounted for, etc).  Higher tolerances tend to yield results that are
suspect.  We have observed most of our solutions converge within 20-100
iterations.  We allow the user to specify an upper limit on number of
iterations (to avoid an endless loop).  However, systems that require more
than 100 iterations can generally be improved by
adding/subtracting/relocating shear walls.


I hope this gives some insight about how KeyLat performs the rigid
diaphragm calculations.  The obvious conclusion is that this process is
very cumbersome, and brain-straining.  We believe we have thought through
the issues and automated a solution that makes sense.  Verification is
tricky but possible with a sharp pencil (and lots of extra paper).

There are other issues involved, for example KeyLat currently implements Ax
on a level-by-level basis.  Is this really appropriate?  Are the effects of
torsional irregularity level-specific or should Ax somehow be applied to
the structure as a whole?

As always, your comments are welcome.

Sincerely,

Johnny Drozdek, E.I.T.
Keymark Enterprises, Inc.


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