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RE: EQ resistant Design

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A rational analysis just might show that the column is adequate with out
the 30% increase as required in section 1633.1. This code section is the
law and is the standard of care.

David Merrick, SE

The following is a draft and may have errors. The information is to
stimulate study and analysis. This text is not to be used for design.
Parts of the following may or may not be my professional opinion.

If I have a box like shear wall, and I push in the direction of a
principle axis there are two walls resisting. If I push on the box at a
45 degree angle I then have four walls resisting and four column
corners. The column force is (.707+.707)*(2walls/4walls)=.707. In other
words reduce the load by 30%. The code designed columns begin to look
over designed.

However there seems to be more benefit when looking any one particular
column that is shared by two walls. There is a lesser range of angles
that a force vector could result in added load on the column that is
near or above the equivalent column load for the principle axis. Rotate
the force vector in the other direction and the opposing wall has a
reversed load on the column. Lesser range of angles to add column load,
results in a less chance of exceeding the design load in the 50 year
period. In making that adjustment, the load from the perpendicular wall
could be reduced 20% from 41% to about 20% (values need to be checked)

But because a shared column means that there could have been two
columns. That would be a penalty due to the lack of redundancy. So lets
increase the demand by some arbitrary number such as 20%. (wild card)

There is the reduction assumed by the code to reduce by 10% the 41% to
30% based on the fact that there is probably not so many perfectly
square buildings.  It could be if some were to study all affects that
the column load from the principle axis earthquake could be reduced by
the accumulation of the mentioned affects and get: 100-30-20+20-10=60%
of column load. (not to be used) A rational analysis just might show
that the column is adequate with out the 30% increase as required in
section 1633.1.

The following is a start on the angles of influence and concluding a
reduction of force to reflect and equivalent chance of exceedance of 10%
in 50yrs for a column not shared. As a draft, I have skipped critical
steps that are needed if one were to find any conclusion possibly

A column at the end of one wall will have a force reduced when the shear
force vector is rotated. A column of shared walls has a higher load when
the earthquake vector is rotated in the quadrant toward a sharing wall.
If rotated away and into the opposing quadrant, the column load is
reduced! A greater reduction than that for the column that is not shared
by walls.  My first sense of this is that there is a benefit to the
column when it is confined by sharing walls. Clearly there is a force
direction that will increase the column load.

I estimated that the Chance of exceeding a 50 year occurrence as a
function of X a fraction of compliance with the code is 0.108*X^-2.37.
Assume the fraction of compliance as a function of Y chance of exceeding
the acceleration in 50 yrs is 0.303Ln(Y)+0.304

Column max load for either of two columns of walls not sharing columns.
A=Max (COS(a) and 1+COS(a+180))
Probability of exceeding in 50yrs is about...
PA=0.108(A) ^-2.37 for angles between ?90 and +90 degrees

Column load for a column sharing equal walls of a square plan.
Probability of exceeding in 50yrs is about...
PAA=0.108(COS+SIN)^-2.37 for angles between ?90 and +90 degrees

So if I integrate the chance of exceeding the design load over the range
of angles, I find that the percent increase of the percent chance of
exceeding the design load is about 50%. That is equivalent to an
increase of column load of about 20%, not the 41% found at the 45 degree
vector position.

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