Distribution of Seismic Forces to Primary LFRS
- Now that we have the base shear force, what type of induced forces
act through the height of the building?
- How to model the inertial force that acts opposite to yank of paper on the cereal box?
- Recall for wind loads
- First, calculate loads/pressures over the height of building.
- Then developed base values.
- These values are at the allowable stress level.
- In contrast, with seismic -
- First, determine base force.
- Then determine and distribute forces over the height of the building, called story forces, Fx.
- There are two different sets of story forces distributed to the
primary LFRS:
- For vertical elements, use Fx.
- For horizontal elements, use Fpx.
- Recall that the primary LFRS for a box building = horizontal diaphragms and vertical shear walls.
- Then adjust these strength level forces by a redundancy/reliability factor, r, and an allowable stress factor of 1.4 discussed further in item d, below.
- Story forces for vertical elements.
- Used in design of shear walls and shear wall anchorage at the foundation.
- Determined before Fpx's.
- Applied simultaneously at all levels.
- Results in a triangular distribution of forces over a multi-story
building that has approximately equal floor masses.
and
Where:-
Ft = roof level force accounting for whiplash effect.
Ft { .07TV £ .25V or 0 if T £ .7 sec. wx, wi = tributary weights at levels x and i.
hx, hi = height above base to levels x and i.
- further detail can be found in '97 UBC 1630.5.
- Story forces for horizontal elements.
- At roof level, Fpx = Fx.
- At other levels, Fpx > Fx.
- Accounting for the possibility that larger instantaneous forces can occur on individual diaphragms.
- Applied individually to each level for the design of that diaphragm.
where wpx = weight of diaphragm and elements tributary to it at level x. - For masonry buildings (and concrete) supported by flexible diaphragms, the R factor used to determine V must be reduced to 4.0 from 4.5 ('97 UBC 1633.2.9.3).
- For more information see '97 UBC 1630.6.
- The single story building is a special case.
- In most cases, T £ .7 and Ft then is taken as zero.
- From equation 30-15:
- From equation 33-1:
- Consequently, F1 = Fp1 = V for the case of wood frame buildings.
- For masonry buildings, Fp, is based upon a slightly
larger V due to R changing from 4.5 to 4.0 according to '97 UBC
1633.2.9.3. In this case, then: F1 = V and Fp1
= 1.125 V.
- Redundancy/reliability factor and the 1.4 ASD adjustment:
- In the load combination
equations as discussed in the last sub-module in the load module
of this site, all earthquake forces are generically called E.
- Where:
Eh = load developed from V, (like Fx or Fpx) or Fp, (the design force on a part of a structure).
Ev = 0 for ASD
r = redundancy/reliability factor, discussed below.
- Where:
- E is at strength level and must be divided by 1.4 for use in allowable
stress design.
- The application of 1.4 and p are shown in example one of this sub-module.
- The redundancy/reliability factor penalizes structures in seismic
zones 3 and 4 that do not have a reasonable number and distribution
of lateral force resisting elements, such as shear walls. These
structures with a limited number of shearwalls are referred to as
non-redundant structures where the failure of one wall loads to
the total collapse of the structure.
Where:
-
AB = the ground floor area of the structure in ft2.
- r = 1 when in seismic zones 0, 1, or 2.
- r = 1 when calculating drift.
- Upon careful inspection of the r and ri equation
with application to a single story, regular building, we see:
- To maintain a r = 1.0, the
minimum length of the most heavily loaded shear wall is
fixed as:
- If a flexible diaphragm, a common controlling case will
be when Rwall/Rstory = .5.
In this case then
to keep r = 1.0.
- To maintain a r = 1.0, the
minimum length of the most heavily loaded shear wall is
fixed as:
rmax = maximum element-story shear ratio, ri, occurring at any story level in bottom 2/3 of the structure. rmax identifies the least redundant story.
ri = Rwall/Rstory(10/lw)
Where:-
Rwall = shear in most heavily loaded wall
Rstory = total story force, Fx
lw = length of most heavily loaded shear wall. - Although the Breyer, et al book uses the subscript "u"
to distinguish strength-level vs. allowable stress-level loads,
I have opted for a different convention that I believe is simpler.
- Upon modifying the various Eh values by r
and 1.4, Eh becomes E'h. For our single
story building, the shear wall forces and diaphragm forces at
ASD level would look like:
-
F'1 = rF1
(1/1.4)
F'1 = rFp1 (1/1.4)
- Upon modifying the various Eh values by r
and 1.4, Eh becomes E'h. For our single
story building, the shear wall forces and diaphragm forces at
ASD level would look like:
- In the load combination
equations as discussed in the last sub-module in the load module
of this site, all earthquake forces are generically called E.
