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RE: Rigid plywood diaphragms

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Thanks for stating a limited and clear technical question within a topic
that has been of keen interest to me for more than 15 years. 

At 12:04 AM 11/20/98 -0800, you wrote:

In a one word answer, No.

In a two word answer, Somewhat, depending....

What follows is a failed attempt to explain as clearly and succinctly as
Judge Kenneth Starr did last Thursday in his opening statement to House
Judiciary. (Not that anyone had been expected to change his or her views as
to an ultimate outcome anyway.)

First, get hold of UBC Standards or its APA source material on plywood shear
wall deflections as a starting point. Some engineers now say this is no
longer "proper data", but we still need a starting point. And get an APA
Plywood Design Specification ("PDS") for its listings of sheathing
structural properties if not in UBC. Second, let's agree that the model
situation is a familiar 94 UBC Rw = 6 or 8, Zone 4 Seismic situation where
design loads are less than actual earthquake loads and everything lurches
back and forth vigorously. (Not Wind where nothing much is expected to
happen inelastically.) But also, let's agree that the building is intended
to perform with none of its shear walls reaching failure or breaking up, and
that it indeed achieves this customary goal. (To simplify further
discussion, let's assume all walls are the same height and on rigid
foundations.) And last, let's agree that the wall's "rated capacity" is the
familiar code-listed allowable shear load in lbs per foot, but "rigidity"
for the purpose at hand is stiffness against deflection to a total shear
force acting on a wall's length, not stiffness against a lbs per foot load. 

(To digress, your question reduces algebraically to, "Is relative rigidity
approximately proportional to a wall's total capacity in lbs." But let's not
go there.) 

Components of deflection:

Let's examine each separate component of shear wall deflection in turn to
see if it is consistent with your postulated rigidity relationship.
Following along with free-body diagrams would help in verifying each
separate condition.
I doubt that merely reading along will work.

There will be seen in the deflection formula at least four components that
add up to the total deflection in a given shear wall. Of these, only one is
BOTH linear elastic and independent of height to length aspect ratio. It is
the shear deformation internal to the sheets of sheathing. For a given type
and thickness of sheathing, the wall stiffness (against a total force) due
to this component is proportional to the length alone, and would not vary
according to the fastener- controlled rated capacity. Upgrading the
sheathing as per listed wall rated shear capacities would change this only a
little, so your proposition isn't met for this component of the total. 

Another component is leveraged axial stretch and compression of the vertical
boundary studs or posts. For a given actual wall shear stress (or rated
capacity) the axial force generated in the vertical boundary is a constant,
regardless of wall length, and the vertical PL/AE figure therefore would
vary only if the boundary member cross section changes, which it probably
wouldn't be compelled to do with an increased rated capacity, now that 4x
boundary members are routine. But happily, the wall deflection due to this
component does vary inversely with wall length, so the resulting rigidity
would once again be proportional to length, and again not proportional to
increases in rated lbs per foot capacity. Note also that this component of
deflection and rigidity is typically a minor one.

A third component is the leveraged result of vertical hold down device
stretch and of vertical downcrunch of the compression boundary member at its
sole plate bearing, etc. The gross vertical forces at these locations are
constant for a given actual shear stress or rated capacity, regardless of
wall length. Presuming that higher design shears or rated capacities cause
use of more robust hold downs (but not necessarily boundary members)
vertical deformations probably hold about constant as rated capacity
increases. But shear wall deflection for constant vertical hold down and
downcrunch deformation decreases linearly with wall length, hence this
component's rigidity to a total shear force increases proportional to wall
length,  with again little effect from changes in lbs per foot rated capacity. 

And now comes the "nail slip" component. Staying for the moment with older,
non-cyclic data, wall length doesn't matter to wall deflection at a
particular per foot shear stress. In other words, aspect ratio is not
relevant to this component. But no way is nail slip and resulting wall
deflection linear with variation in load for a given nail schedule. It is
non-linear for the entire range of load, according to figures in UBC
Standards I looked up and plotted years ago. But comfortingly, the "e-sub-n"
nail slip value listed for every nail size was right on 0.030 inches at the
nail shear corresponding to code-listed allowable wall shear stress for
every nail size and spacing and plywood thickness and grade appropriate to
that schedule. (Nail slip however was substantially less if seasoned framing
was used.) Nail loads in excess of what gave 0.030 inches of slip gave
rapidly increasing deformation, but not necessarily failure at any
particular multiple of "allowable", nor at any particular value of nail
slip. It appears that rated capacity for each combination was set at a
figure giving about the same nail slip value.  

Where all of the shear walls are actually to be loaded by design to their
rated capacity, even though that capacity varies from one wall to another,
then deflection of all of them due to nail slip will be essentially the
same, and not varying according to rated capacity. Each wall's rigidity to
total shear load for this idealized but usually desired condition will
therefore be proportional to wall length alone. And the proposed rigidity
relationship again does not hold.

Hence the one word answer, No.

So much for looking at it with a hands-off policy. Now let's intervene and
deliberately design in some alterations and see if useful things happen.

Alterations from usual practice:

If a wall has chosen for it (or for other reasons has) a rated capacity well
in excess of the design shear it otherwise needs, big rigidity benefits in
the form of reduced nail slip result. These benefits are not merely linear,
so it would not be strictly true that rigidity would be directly
proportional to the ratio of understressing the nailing, but the general
tendency is there. So now one can reasonably say that rigidity is
proportional to wall length, times some measure of surplus rated capacity,
for this nailing component of deflection.

One also can specify hold down devices that have low-deformation properties,
and specify oversized examples of such devices to further reduce vertical
deformations in the device and its fasteners. These benefits appear to be
decidedly non-linear in a way similar to nail slip benefits. One can get
this better hold-down rigidity as a by-product of designing all of the wall
to a much higher rated capacity than the building design otherwise required,
or one can specify super hold downs without comparably enhancing shear
nailing. Improvements to specially limit downward deflection at shear wall
boundary corners is of course also appropriate when acting to limit upward
deflection at hold downs. This might consist of stouter end posts and thus
less cross-grain bearing stress below, enhanced bearing provisions in
framing below, etc.  By using these interventions, one has grounds to say
that for this component, rigidity is now somewhat proportional to a rated
capacity surplus in boundary details as well as to wall length. 

Similarly, one can beef up vertical boundary members as a discretionary
by-product of overdesigning the rated capacity of the whole wall, or as a
separate act. The effect would be relatively linear, and rigidity for this
component would improve according to the AE overdesign improvement obtained. 

Sheathing specification overdesign appears to offer minimal opportunities at
first, because the listed "effective thickness" figures for structural
plywood shear deflection purposes do not increase much until thicknesses
approaching 3/4 inch are obtained. (I have specified 3/4 and 1-1/8 plywood
for wall shear use on occasion, when this property and another one unrelated
to shear were of advantage.) For ordinary thicknesses, Struct I has a listed
shear modulus better than lesser grades. How OSB compares remains to be
looked up. Enhanced sheathing really can help with rigidity, but the
enhancement has to be well in excess of what mere overdesign in rated
capacity would automatically obtain. Yet your proposed rigidity relationship
is there in part with respect to the excess capacity.

A provisional conclusion:

With all four of the deflection components discussed so far, but still
avoiding really narrow aspect ratios and cyclic loading results, it appears
that rigidity is proportional to wall length alone for walls that fully
employ rated capacity, and that rigidity is somewhat proportional as well to
perhaps the ratio of rated capacity -to- actual design stress demand. But
not proportional to straight rated capacity without respect to demand.

Recent considerations:

OK, what about narrow panels? As others point out, the leveraged effects of
vertical deformations at boundary details really matter with a vengeance.
Reliance on an Rw factor of 8 instead of 6 or less means the design
overturning force couple falls that much further short of of the real thing,
so that adverse deflection effects are worsened more than linearly. The
obvious remedy is to depart from habit and design these features very
conservatively for realistic forces and with superior devices and details to
reduce shear wall deflection to a minimum. Pity, but code now prohibits
trying this for ratios narrower than 2:1.  Codewise, insult follows injury.

And what about cyclic testing's lessons? One is that again, hold down
performance and that of the other boundary sources of vertical deformation
really matters. Another lesson is that as a panel nears failure, it begins
to fail. Rigidity of a failing panel isn't very predictable, among other
disadvantages. A remedy for this might be to build these things strong
enough that they don't approach failure. That was one of the understandings
set out at the beginning of this discussion. 

Another result of cyclic testing is that nails may wiggle and slip more than
had been seen in one-directional testing years ago. This implies that the
nail slip component of deflection and rigidity is an even greater proportion
of the total, and even more non-linear. A compensation would be to not load
those nails as much as has been customary, by having a higher ratio of rated
capacity to design demand than formerly...yet another incentive for a more
conservative nailing-based shear wall rated capacity. 

The two word "somewhat, depending..." answer came from these design
intervention and overdesign possibilities.

One more component of shear wall deflection has been ignored till now:
slippage of the sole plate on its support due to looseness around anchor
bolts or due to other anchorage deformations. The magnitude of this
component can be large compared to the others and is subject to large
variation according to  workmanship and other imponderables. And rocking of
the foundation was intentionally excluded but may contribute  significantly.
So how seriously should we take the task of calculating relative rigidities
with precision?  Given also that a shear wall loaded by more than its
expected share incurs more nail slip and softens its relative rigidity in a
self protective way, rather than exploding in anger, how accurate a rigidity
analysis is worthwhile? 

The above addressed a single shear wall rigidity question, not horizontal
diaphragms or the whole interacting system. But a general principle is offered: 

The main purpose of seismic-resistant design is that the building performs.
It is a sideshow to nitpick how the designer himself performs, if the main
purpose gets fulfilled.

Charles O. Greenlaw  SE     Sacramento CA