length of unsupported bookshelves
Randall Kesselring
fireplace. After studying the situation, both my wife and I feel that the
shelves would look better without an additional support in the middle.
This would make the shelves 4"6" long, unsupported. Can anyone give me an
idea of how to accomplish this length without sag? The shelves will be
about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
middle of the underside of the shelf and attach a 1" or wider strip of
plywood to the dado. Are there any better ideas out there. Thanks in
advance.
Bud Noren
: I am starting a new project to replace the shelves on either side of my
I built a bookcase once from plans in a book. The shelves were also 3/4"
plywood. They are stiffened by a strip of hardwood that matches the
carcass. This strip is attached to the front of the shelves and is maybe
2" wide. The strip is flush with the top of the shelf and therefore hangs
below it about 1 1/4". The plans called for the hardwood strip to be
rounded over, though you don't really have to do that (I did and it looks
pretty nice). The ends of the shelves are hidden by the face frame of the
bookcase but if your shelves are all exposed you might have to figure out
a way to dress up the edges. The strip makes the shelves look much more
impressive and add a lot of strength. They span about 3' and are loaded with
textbooks and show no sign of sag.
-Bud
Ike Bottema
I built a 3' shelf out of 5/8" aspinite (sp?) about a year ago. I used a
one of those shelf rails used to allow adjustable shelf height rather than
the strip of plywood but same idea. I placed the dato about four inches from
the front of each shelf.
So far I have no observable sag. One of these shelves has my MS Office manuals
smack in the middle, surrounded by other equally weighty material.
Ike Bottema
Bryan Sutula
: I am starting a new project to replace the shelves on either side of my
As encouragement to do the job right, we also have some bookshelves
someone else made for us. They are 3/4" plywood, are 3' long, and have
no support other than the ends. There is noticeable sag, especially at
eye level. I would guess that we're seeing 3/8" of sag.
At this point, I'm considering a strip of hardwood at the front of each
shelf, perhaps 1.5 to 2" wide. If necessary, I'll add another strip at
the back of each shelf. Certainly, this would have been easier to do
before the shelves were finished, so do it right from the start.
The idea of gluing a single strip in the middle is also appealing. If
Ike Bottema could supply details on how thick his brace is, it would be
helpful.
Bryan Sutula
sut...@lvld.hp.com
Dave VanEss
Bryan Sutula <sut...@lvld.hp.com> wrote:
>Randall Kesselring (ran...@quapaw.astate.edu) wrote:
>: This would make the shelves 4"6" long, unsupported. Can anyone give me an
>: idea of how to accomplish this length without sag? The shelves will be
>: about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
>: middle of the underside of the shelf and attach a 1" or wider strip of
>: plywood to the dado. Are there any better ideas out there. Thanks in
>: advance.
>
>As encouragement to do the job right, we also have some bookshelves
>someone else made for us. They are 3/4" plywood, are 3' long, and have
>no support other than the ends. There is noticeable sag, especially at
>eye level. I would guess that we're seeing 3/8" of sag.
>
much matches this this definition) the sap is proportional to
the unsupported length to the 4th power
the inverse the the thickness to the 3rd power
the inverse of the width
the Uniform Force applied.
The stiffness of the material used.
Example:
If the 3ft shelf you desribed was instead 1" the sag would
be 3/8" *(.75 ^3) = 5/32
kay so if you want a shelf made of plywood 4'6" with the same sag as
a 3' the thinckness was be 1 & 9/32" thick
For 1/8 sag the thickness would have to be 1.86"
That is fairly thick. I would reccommend building a torsion box shelf
make a frame out of 1 by 2 material.
4.5'
|---------------------------------------------------|
| | | | | | | |
| | | | | | | |
----------------------------------------------------|
| | | | | | | | 11" or 12"
| | | | | | | |
|---------------------------------------------------|
Cover the Frame with 1/4" veneer plywood. Either put hardwood
around to cover frame or make the outer part of the frame out of
hard wood. Below is cross secton of shelf.
12"
HPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPH
HHH FFF HHH
HHH FFF HHH
HHH FFF HHH
2" HHH FFF HHH
HHH FFF HHH
HHH FFF HHH
HPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPH
Each letter 1/4" square
H = hardwood
F = framewood
P = 1/4" plywood
These shelves will be thick but the design can be worked out to
make it look nice. These shelves are also amazingly light.
Dave VanEss
Vince Miller
>>I am starting a new project to replace the shelves on either side of my
>>fireplace. After studying the situation, both my wife and I feel that the
>>shelves would look better without an additional support in the middle.
>>This would make the shelves 4"6" long, unsupported. Can anyone give me an
>>idea of how to accomplish this length without sag? The shelves will be
>>about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
>>middle of the underside of the shelf and attach a 1" or wider strip of
>>plywood to the dado. Are there any better ideas out there. Thanks in
>>advance.
I think 36" is the best you should attempt with 3/4 ply, that assumes using
some kind of face support. I think ply is not the best material for
shelves. You only get halve of the wood fighting sag. I would go with some
thickish real wood, like 5/4 something, planed only as much as nesc to "
hit", that is to get a smooth face. To get this, you will need to find a
real wood dealer, like a lumber mill, or a cooperative store.
I have built 36" ply shelves, they don't sag, but it's clear they are just
about at that point.
Vince
Edwin Susbilla
I recall reading some where that a shelf should not be longer
then 30 inches, any longer then that will cause the shelf to
sag without center supports.
-Edwin-
Steven Bellovin
sag. As I recall, they even gave some figures. Does anyone have the
chart handy?
Ike Bottema
>Randall Kesselring (ran...@quapaw.astate.edu) wrote:
>: I am starting a new project to replace the shelves on either side of my
>: fireplace. After studying the situation, both my wife and I feel that the
>: shelves would look better without an additional support in the middle.
>: This would make the shelves 4"6" long, unsupported. Can anyone give me an
>: idea of how to accomplish this length without sag? The shelves will be
>: about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
>: middle of the underside of the shelf and attach a 1" or wider strip of
>: plywood to the dado. Are there any better ideas out there. Thanks in
>: advance.
>As encouragement to do the job right, we also have some bookshelves
>someone else made for us. They are 3/4" plywood, are 3' long, and have
>no support other than the ends. There is noticeable sag, especially at
>eye level. I would guess that we're seeing 3/8" of sag.
>The idea of gluing a single strip in the middle is also appealing. If
>Ike Bottema could supply details on how thick his brace is, it would be
>helpful.
The bracing I used is actually intended to be used as adjustable shelf
spacers. These are made of steel, with slots stamped every 1/2" (to
receive the shelf bracket), and bent to form a channel in the
following form:
|______|
That's the way I bought it (at Beaver Lumber as I recall). The depth of
the "notch" is 3/16" while the width is 5/8". Actually, I didn't dado the
notch as I call it. I left as much material under the channel as possible to
provide lateral support for the screws used to fix the channel inside the
notch. I made two parallel cuts spaced at 5/8" to a depth of 3/16" using
my table saw and a thin kerf saw blade. If I owned a dado :-( I would have
used it to remove the wood between the saw cuts. Since I don't, I used my
router to mill out the wood to a depth equal to the thickness of the channel.
Since I don't want to leave the impression that this method could be used to
correct the hubble mirror, I measured the sag in the most heavily loaded
shelf with a straight edge to be 3/32" over 2'. Remember, I used particle
board to construct my shelves. Plywood may not have the same compression
strength (to match the tensile force of the steel brace) as that of the
particle board.
Ike Bottema
kerf saw blade
Bob Moran
edge-grain stair stepping available from lumber yards.
Bob Moran, Rodale Woodworking Books
Charles A. Patrick
>Reply-To: Woodworking Discussions <WOOD...@ipfwvm.bitnet>
>Sender: Woodworking Discussions <WOOD...@ipfwvm.bitnet>
>Comments: Warning -- original Sender: tag was NET...@AUVM.AMERICAN.EDU
>From: Dave VanEss <da...@u.washington.edu>
>Organization: University of Washington
>Subject: Re: length of unsupported bookshelves
>To: Multiple recipients of list WOODWORK <WOOD...@ipfwvm.bitnet>
>
>In article <36s9lq$6...@hplvec.lvld.hp.com>,
>Bryan Sutula <sut...@lvld.hp.com> wrote:
>>Randall Kesselring (ran...@quapaw.astate.edu) wrote:
>>: idea of how to accomplish this length without sag? The shelves will be
>>: about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
>>: middle of the underside of the shelf and attach a 1" or wider strip of
>>: plywood to the dado. Are there any better ideas out there. Thanks in
>>: advance.
>>
>>As encouragement to do the job right, we also have some bookshelves
>>someone else made for us. They are 3/4" plywood, are 3' long, and have
>>no support other than the ends. There is noticeable sag, especially at
>>eye level. I would guess that we're seeing 3/8" of sag.
>>
If the visual thickness of the shelves are going to be upto 2 inches, why
not make a Torsion Box?
Cheers.
Steve Rossanese
|> Like everything else, the length of those shelves may or may not work without
|> sagging depending on what you're putting on them. I built a bookcase a couple
|> of years ago with 32" shelves. A couple of those shelves hold National
|> Geographics and FWW issues, both of which are very heavy. The 3/4" plywood
|> shelves have deflected 3/8". I intend to replace those with something thicker
[snip]
So what you are saying is that it is not the length that matters, it's
the thickness that counts.
Could not pass that one up.
Steve
--
***************************************************************
* Standard * *
* Disclaimer * Steve Rossanese Voice: (613) 763-5711 *
* * Bell Northern Research Email: ros...@bnr.ca *
* * Ottawa, Ontario *
* * *
***************************************************************
Bruce Haugen
seemed like the better choice is to cut down the length of the shelf. At some
point everything will bend.
Bruce Haugen
sagging depending on what you're putting on them. I built a bookcase a couple
of years ago with 32" shelves. A couple of those shelves hold National
Geographics and FWW issues, both of which are very heavy. The 3/4" plywood
shelves have deflected 3/8". I intend to replace those with something thicker
could handle the stress. IMHO 4'8" is too wide for any shelf without some kind
of support, perhaps screws from the backside or unobtrusive supports under the
back at about 24-30" intervals PLUS about 1-1/4" lip on the front edge.
Experience is a good teacher, but some of those lessons can be a bit expensive.
Enjoy! bh
Norman R. Ellis
better yet smoothed 5/4 and your shelves will not sag over the distance
you have. Plywood will sag unless it is braced as you say. Norman Ellis
On Wed, 5 Oct 1994, Ike Bottema wrote:
> sut...@lvld.hp.com (Bryan Sutula) writes:
>
> >Randall Kesselring (ran...@quapaw.astate.edu) wrote:
> >: I am starting a new project to replace the shelves on either side of my
> >: fireplace. After studying the situation, both my wife and I feel that the
> >: shelves would look better without an additional support in the middle.
> >: This would make the shelves 4"6" long, unsupported. Can anyone give me an
> >: idea of how to accomplish this length without sag? The shelves will be
> >: about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
> >: middle of the underside of the shelf and attach a 1" or wider strip of
> >: plywood to the dado. Are there any better ideas out there. Thanks in
> >: advance.
>
> >As encouragement to do the job right, we also have some bookshelves
> >someone else made for us. They are 3/4" plywood, are 3' long, and have
> >no support other than the ends. There is noticeable sag, especially at
> >eye level. I would guess that we're seeing 3/8" of sag.
>
Gerald Chapman
>A few years ago, Woodsmith suggested using solid wood edging to prevent
>sag. As I recall, they even gave some figures. Does anyone have the
>chart handy?
>
>
Page 21 has the following chart:
Maximum span recommendation for 10" wide shelf full of books :
3/4" Particle Board 24"
3/4" Plywood 30"
4/4(13/16") Solid Stock 36"
6/4 (1 5/16 ") Solid Stock 60"
--------------------------------------------------------------
3/4" Plywood Reinforced with
1 1/4 " Wide face strip on edge 36"
1 1/4" wide face on side 32"
Aluminum strip underneath 36"
Molding strips underneath 36"
--
Jerry Chapman
Tom Baltz
Since you'll be removing the middle vertical support, you'll very likely
find out you will need to replace or repair the bookcase back. It may now
be in 2 pieces (1 for each side) depending upon construction. Even if it is
1 piece, you're very likely to find it unfinished where the support now is,
and probaly full of nail holes, dowels, screws, maybe even biscuits - again
depending on how it was originally built. (all assuming the case has a
back :-)
It can be repaired...but I'd suggest replacing it with 3/4" ply with
horizontal dados that will hold the long shelves back edge. This will give
tremendous support along its entire back edge - it cannot sag, at least in
the back.
And since you'll be using ply for the shelves, hide the ugly front edge
with solid wood strips at least 1 1/4" wide. The top will be flush with the
shelf, bottom hanging down 1/2" or so. It would be preferable to join this
to the shelve with a rabbet, allowing wood to support the shelve from
underneath. Route a roundover on each side, or ogee if you prefer for a
nice looking detail.
With the dado in the back, solid strip support in front, it won't sag.
This is exactly how I built my 5' wide garage cabinet out of flexible 5/8"
particle board. 3 years now and no sagging...loaded with many heavy
gallons of paint, anti-freeze, quarts of oil & a huge selection of
stain/varnish/finishes from other woodworking projects.
- Tom Baltz t...@cray.com
Dave Halliwell
>I am starting a new project to replace the shelves on either side of my
>fireplace. After studying the situation, both my wife and I feel that the
>shelves would look better without an additional support in the middle.
>This would make the shelves 4"6" long, unsupported.
_Completely_ unsupported? That would be a neat trick! I assume you
mean they will be supported only at the ends! :-)
> Can anyone give me an
>idea of how to accomplish this length without sag? The shelves will be
>about 11" deep. My best idea was to use 3/4" plywood, run a dado down the
>middle of the underside of the shelf and attach a 1" or wider strip of
>plywood to the dado. Are there any better ideas out there. Thanks in
>advance.
This is a possibility, but I think the presence of a "beam" in the
middle of the shelf would look a bit odd. I can think of three
alternatives, but whether they would fit your design depends on a number
of factors.
1) if you plan to have some sort of back to the project, make it solid
enough to provide some support, and attach the shelves to the back along
their entire length.
2) Do the "beam thing" you suggest for the middle of each board, but
make it two beams per shelf - one front and one back - and cut them all
curvy so they become a design element.
3) Go for a pseudo-cantilever design, where the supports are some
distance in from the ends. Part of the shelf sticks out at each end. This
makes the shelf stronger at the support, since you have a continuous
piece of wood instead of a joint to deal with. It also makes the span a
bit shorter, and puts the "sag" on either side of the supports instead of
all in the middle.
--
Dave Halliwell I don't speak for my employers, and you
Edmonton, Alberta shouldn't expect them to speak for me.
douglas rhodes
All bookshelves will deflect regardless of their thickness. The thicker ones
will deflect less. There are some simple calculations you can use to
determine how much a "beam" (shelf) will deflect based on its dimensions
and modulus of elasticity. Refer to "Understanding Wood" by R. Bruce Hoadley
by FWW for complete details. This is what I use to determine thickness,
wood species, etc.
Hope this helps,
Doug Rhodes
AT&T Bell Labs
(do...@aloft.att.com)
Steve Saxe
: I am starting a new project to replace the shelves on either side of my
I don't know if it's better, but when I built a bookcase I had a similar
problem. I used two 3/4" shelving boards, screwed together. I ran a
facing strip of lath along the front. The double thickness shelves don't
sag and look kinda elegant, if I do say so myself.
--Steve
Elliott Wolin
like that (a USDA publication advertised in lots of WW magazines). There's
a section on loading plywood boards (transverse and longitudinal, fixed or simply
supported ends), lots of formulas and recommendations, etc.
Indeed this is the best reference I know of concerning structural
aspects of wood, at least if you have an engineering turn of mind.
Elliott Wolin
wo...@wmheg.physics.wm.edu
Paul J. Georgeson
>The strip is flush with the top of the shelf and therefore hangs
>below it about 1 1/4".
Depending on what you're putting on the shelves, you may want to put a
strip on the back that's flush with the bottom, thereby providing a lip to
prevent 'stuff' from falling/rolling off the back of the shelf.
--
Paul Georgeson e-mail: pa...@network.com
Network Systems Corp. ph.: (612)-391-1088
7625 Boone Ave N, MS-613
Minneapolis, MN 55428-1099
Norm Slothouber
Some more suggestions and a method I've used successfully.
Take a piece of material and place it on supports in the same pattern
and span that the shelves will be used. Place the supports on your
shopfloor. (pattern: refers to your method of support - both ends, or
both ends and back). Now start loading it up with bookks and magazines
etc. and pile some more on top. Measure at what load the deflection
starts, that is when the center of the shelf begins to sag.
If your test shows that the material will not support a full load of
paper plus a percentage safety factor without any sag, do one of
several things.
use a heavier material. (for your suggested project consider 1-1/4 in
dressed msmt. pine.)
when using plywood consider a lateral support - full length - at the
back (against the wall). If that still does not eliminate deflection,
make a frame of (dressed msmts) 3/4 X 1-3/4 in and repeat test.
Regards,
Norm Slothouber
Waterloo, Ontario, Canada
"Experientia docet stultos" ... but only the wise get wiser.
Robert Mingee
<gcha...@HEARTLAND.BRADLEY.EDU> writes:
> The Woodsmith in question is issue number 49 dated Feb. 1987.
> Page 21 has the following chart:
>
> Maximum span recommendation for 10" wide shelf full of books :
>
> 3/4" Particle Board 24"
> 3/4" Plywood 30"
> 6/4 (1 5/16 ") Solid Stock 60"
> Jerry Chapman
>
> gcha...@heartland.bradley.edu
With regards to the 4/4 solid stock, what kind of wood would be acceptable
to meet up to those sag specs? I am going to be building a bookcase for
our study in the next few weeks, and I was considering one of 2 options:
poplar, painted to match/coordinate with room
maple, stained and finished with some kind of oil varnish
My guess is that the first option is cheaper and easier, but the second
option will be much less suceptible (sp?) to sagging. But will the first
option sag unacceptably? I am considering a case that has 11" wide
shelves, 3' wide, 6' high, with a middle shelf fixed for extra case
strength and 2 adjustable shelves above and below the fixed middle. I will
also be putting a 1/4 plywood back rabetted in.
Any finishing recommendations? I remember seeing yesterday that oil paint
should be used instead of latex, but what about recs for the clear
protective coat for option 2? Does maple tend to blotch when staining?
Thanks in advance for any guidance,
-- robert
--
---------------------------------------------------------------------------
----
Robert Mingee | "The problem with the future is that it keeps
Alcatel Network Systems | turning into the present" -- Hobbes
rmi...@aud.alcatel.com |
Steven Bakalis
the shelf strength to resist deflection I installed a 3/4" wide by 1/2"
deep by 3/8" thick aluminum T-beams. This was installed into the front
and back of the shelf by first cuting a 3/8" rabit along the front
and back edge of the shelf and pushing the T-beam in. I then faced the
front with veneer to hide the aluminum.
__ ________________________________ __
| |__ __| |
T-Beam --> | __| Side view of shelf |__ |
|__|________________________________|__|
Steven Bakalis
Paul Houtz
----------
Why not just use steel for the whole shelf, and glue on veneer on
all sides to make it look like wood?
Charles A. Patrick
>Subject: no subject (file transmission)
>To: pat...@statcan.ca
>Date: Mon, 10 Oct 1994 16:58:06 -0500 (CDT)
>Mime-Version: 1.0
>
>To: pat...@STATCAN.CA (Charles A. Patrick)
>Subject: Re: length of unsupported bookshelves
>Organization: Ripco BBS, Free Trial account (312) 665-0065
>X-Newsreader: TIN [version 1.2 PL2]
>
>: >Subject: Re: length of unsupported bookshelves
>: >To: Multiple recipients of list WOODWORK <WOOD...@ipfwvm.bitnet>
>: >
>: >In article <36s9lq$6...@hplvec.lvld.hp.com>,
>
>: not make a Torsion Box?
>
>by 30 " wide by 4 inches thick. Made out of common pine (1x4) and 1/4" top
>and bottom. screwed and glued. It weighed about 35 pounds. We supported it
>at each end in her room. two football players could do chin-ups in the center
>with no discernable sag. She put everything on it you could imagine.
>
>If you assume the tensile strength of common pine as a starting point,
>Machinery's handbook or any college text on strength of materials can lead you
>to a design to meet your needs.
>
>The torsion box is amazingly strong for its weight. But it must be well made
>as each piece depends on all the others. One weak glue joint will spell
>disaster.
>--
>
>
>
>
John J. Desmond
What's a torsion box?
H.Lincoln
plywood! They will sag and I think they will even with extra strip of
ply underneath. You might try channel iron underneath!! I don't like
to go over three feet for a book shelf. Books are heavy! Good luck.
Harry Lincoln
Jerry Davis
: Sir: Four feet six inches is very long for a book shelf using 3/4"
The easiest way out of this dilemma is not to use plywood. It is not
built for this. Just go to the store and get some hardwood of some kind.
--
----------------------------------------------------------------------
| Jerry & Ellen Davis | If work were like my hobbies, then I would |
| Wauwatosa, WI | want to WORK all the time! |
----------------------------------------------------------------------
Chris Wolf
"H.Lincoln" <BG0...@BINGVMB.BITNET> writes:
We had builtin bookshelves added during remodeling, and the contractor made
them 41" long out of 3/4" cherry plywood. They sagged about 1/2" in the
center as soon as we put books on them. He had to come back and add
vertical supports in the middle of each one, giving us 20" shelves that are
very rigid.
Paul Houtz
>H.Lincoln (BG0...@BINGVMB.BITNET) wrote:
>: Sir: Four feet six inches is very long for a book shelf using 3/4"
>: plywood! They will sag and I think they will even with extra strip of
>: ply underneath. You might try channel iron underneath!! I don't like
>: to go over three feet for a book shelf. Books are heavy! Good luck.
>
>The easiest way out of this dilemma is not to use plywood. It is not
>built for this. Just go to the store and get some hardwood of some kind.
It would be nice if it were that simple, but it's not.
I have an entertainment center with a 32" long shelf that
is supported by a 1X2 with the narrow side attached to
the shelf. That means a 2" wide structure supporting
it. On top of that is a lazy susan with a 27" TV monitor.
It now has over 1/2" sag across its length.
Simply switching to hardwood will not work.
If it is really just a book shelf, the only way to handle this problem
is to make the shelf shorter. Why not make the shelf 2' long instead
of over four? The books don't care. It doesn't make the shelf any
less usable.
Of course, you might argue that it won't look exactly the way you
want, but this is a bit silly, since the look you want isn't possible
the way you want to do it. You would have to use steel, and then
to make it look like wood, glue veneer to it, and the thing is going
to look artificial no matter what.
I have seen elegant libraries in upscale homes that have solid oak
hardwood shelves. They are usually supported by uprights that
are placed every 16" - 24". They look great.
Norm Slothouber
without any desire to advance counterargumentation, consider the following:
Each shelf design and each shelf application is unique, unless you are in
the mass production of vinyl-covered-pc-board-boxes.
Secondly, the number of sides of the shelf that are supported, or not
supported is as much a factor and maybe more of a factor than the length
of the shelf, the material used, the thickness etc., etc.
I have designed and built 36 inch shelves that one could dance on with nary
a deflection.
If you don't want your particular shelf to sag, do a test before you begin
cutting the stock. It maybe good to remember also that it makes a difference
whether or not the shelf is fastened to its supports, and whether or not an
eventual back-length-support is (or can be) fastened to the wall or back of
the case. Mimick as many of these factors in your test as you need to stop
sagging or deflection under the required load.
If you are going to be held liable for any disasters resulting from your
design, you're a fool if you don't take some such precautions.
Norm Slothouber, Waterloo, Ont.=====no...@normspc.ia.mks.com
=================================================================
Paul Houtz
>Secondly, the number of sides of the shelf that are supported, or not
>supported is as much a factor and maybe more of a factor than the length
>of the shelf, the material used, the thickness etc., etc.
This statement is a little troublesome for me. Assuming a material
of uniform composition, the actual sag is uniform across the material
no matter where it is supported.
The total sag in the material depends upon the distance between supports,
so it is fixed. By supporting the back of the shelf, you move the
supports closer, but you do not eliminate sag. You simply reduce the
total sag across the length of the shelf.
However, for a shelf that has very noticeable sag across its length,
merely supporting the back may lead to a shelf that sags only in
front. I have seen this.
Perhaps a good compromise if a very long shelf is desired, would be to
fit the back of the shelf into a dado in the cabinet back, and then
support the front of the shelf with an 2" wide piece of hardwood.
Of course, this would restrict the access to the space below the shelf.
Seems like it would be easier to simply add vertical supports every
16 inches, but to each his own, and variety is the spice of life.