The Physics of Hanging Pictures

Asked and Answered 3.3

This is the third and final article in my series about hanging picture frames.  The first post, Why Frames Tilt Forward, discusses why frames tilt at the top and what you should and should not do about it.   My next post, The “Hang It with Two Hooks” Calculator, presents an online calculator to help you hang pictures with less forward tilt, using two wall hooks and 45° wire angles.  This post completes the picture, so to speak.  Here I try to illustrate, with intuitive examples, the role of physics in picture hanging.  Most of all, I want to help you understand why it is a bad idea to string a wire tightly across a frame to keep it from tilting forward.

2hookicon2I will also discuss the picture-hanging hardware I like and why.  Finally, I will provide another online calculator — this one evaluates your frame’s margin of safety by estimating the tension in the wire and the tendency of your frame to bend.  Click the icon at right to go directly to my safety-factor calculator.

Physics-minded readers, relax.  This article is for general readership.  So I am not going to distinguish between mass and weight, I am ignoring the gravitational constant, and I will use pounds, not newtons, as the unit of force, because that is how the people around here hang pictures.

The Graphical Physical Tour

FIGURE 1

FIGURE 1

Everyone else can relax too.  I’m going to walk us through some basic physics that my wife and I learned in high school, before we started dating.  I’m sure she remembers all of this.

Let’s start with something simple.  In Figure 1 (click to zoom) we see a 1o-pound weight hanging from the ceiling via a wire.  The weight is at rest, neither rising or falling — this means that the upward force (or tension) in the wire must be exactly equal to the downward force w of the weight.  Hanging a 10-pound weight on a single wire produces 10 pounds of tension in the wire.

FIGURE 2

Onto our next example.  In Figure 2 (at right), the weight is the same as before (1o pounds) but it is now hanging by two identical wires instead of one.  Once again, the downward force w of the weight is balanced by the upward pull of the wires.  Because there are two wires, each individual wire carries just half the load.  So the tension in each wire is now w/2 (or 5 pounds in this case).

Okay, time to use your intuition.  If we were to weld together the two wires in Figure 2 at the top, would this change the tension?  No — the tension in each section of the wire would still be w/2.  Ponder this until you’re comfortable with the idea.

Now that (in our minds) the two wires are connected at the top, let us take one more step:  slice the weight down the middle, so that each end of the wire supports half the original weight.  This action should also have no effect on the tension in the wire.  Agree?

FIGURE 3

FIGURE 3

We are now prepared to consider this model of a frame suspended from one hook (Figure 3).  The total weight w is the same, but it is divided into two equal weights supported on either end of a single continuous wire.

The wire passes over pulleys at the top and sides. The top pulley represents the wall hook; the side pulleys are the D-rings attached to the frame.

The weights create forces that pull downward at the top and inward at the sides.  We will take a closer look at this in the diagram below (Figure 4) which focuses on the left side of the setup (the right side is a mirror image).  It may be helpful if you click on the figure to view it full size.

FIGURE 4

Once again, we have a system at rest: the weights are not rising or falling and the pulleys are not moving.  This means that the downward forces are balanced by equal and opposite upward forces — and the same is true for the horizontal forces.

Let’s zoom in on the force on the left side.  The weight w/2 exerts a downward force at the pulley, which must be offset by an equal force upward.  But the wire does not extend upward — instead it heads away from the pulley on a diagonal.  How can a diagonal wire produce an upward force?

It helps to imagine that the pull of the wire is composed of vertical and horizontal parts that add up, so to speak, to a total force (tension) in the diagonal direction.  In the figure, I denote the vertical and horizontal parts as Ty and Tx, respectively.  Because there is no net motion in the vertical direction, we know that Ty (the upward force) must equal w/2 (the downward force).

How do we find Tx, the force in the horizontal direction, and T, the tension in the wire?  Here, we have to use some trigonometry.  The wire tension T equals Ty times the cosecant (csc) of the wire angle α, and the horizontal force Tx equals Ty times the cotangent (cot) of the wire angle α.  If you did not take trigonometry in school, please accept this on trust.

Sorry for the math, but I wanted to show how the wire angle has a multiplier effect on the tension T.  The smaller the wire angle (that is, the closer to horizontal the wire is strung), the greater the multiplier.

I have listed the multipliers for various wire angles in Figure 4.  The first column of the table is the wire angle, the second column is the tension multiplier, and the third column is the horizontal force multiplier.  These multipliers apply to Ty (which is w/2 in this case).

From the table, we see that if the wire is strung only 5° from horizontal, then the tension in the wire will be more than eleven times w/2.  For our 1o-pound frame, the wire tension would be 57.3 pounds and the inward pull on each side of the frame would be 57.1 pounds!

But if we were to string the wire 45° from horizontal, the wire tension would be 7.1 pounds and the force pulling in on the side would only be 5 pounds.  This shows why one should not string a wire tightly across a picture frame to reduce its forward tilt.

I like the idea of using two wall hooks and 45° wire angles, as discussed in my other posts, because it reduces both the wire tension and the inward pull on the frame.  But this does not mean that horizontal forces go away.  In any two-hook installation, there will be a net horizontal force on each hook, pulling them toward the center of the frame.

Figure 5: Forces on Wall Hook (Two-Hook Installation)

FIGURE 5

The diagram at left (Figure 5) depicts one wall hook in a two-hook setup.  The left end of the wire extends diagonally down to the frame, and the right end leads to the other hook.  The wire tension T is the same everywhere along the wire.

In the figure, the black arrows represent the forces that the wire exerts on the hook.  The net force on the hook Tz (red arrow) results from adding together the horizontal and vertical components of these forces.

Again, using some trigonometry, we find that the force Tz will always be somewhat higher than Ty (the exact formula is shown in Figure 5).  The direction of this force relative to vertical is one-half the wire angle.  In the case of our preferred 45° wire angle, the overall force on each hook would be 0.54 times the frame weight and the force would be directed 22.5° inward from vertical.   The horizontal component of this force would be 0.21 times the frame weight.  If one were to use a steeper wire angle — say, 60° from horizontal, as  some people suggest — it would increase the lateral force on each hook by almost 40%.

That’s it for the hard-core physics.  Let’s talk about what this means for picture framing.

Two-Hook Hardware

I do not intend to review all the various hardware available for hanging pictures.  Instead, I am going to focus on the parts and methods for a two-hook, low-forward-tilt installation. So the parts of interest here will be D-rings, wall hooks and wire.

FIGURE 6Let’s start with the hardware you use to attach the wire to the frame.  I much prefer D-rings (far left) because they lay flat and lead to less forward-tilt than eye-screws (right).  Also, D-rings are fastened to the frame with #6 or #8 screws which are larger and have deeper threads than the eye-screws that amateur framers often use.  This offers more resistance to sideways forces.

Next, the wall hook.  As I mentioned just a minute ago, each hook in a two-hook setup is subject to a lateral force.  When using 45° wire angles, the horizontal force on each hook will be about 20% of the weight of the frame.  But wall hooks are designed for vertical loads, not horizontal ones.  The wide base of the two-nail hook (Figure 5) offers extra stability in this situation.  I have not tested different brands but New England carpenter Doug Mahoney did, and Doug recommends the Floreat hangers sold by Ziabicki Imports.  I suggest you read his article on picture hangers – very thorough.

Finally, the wire.  I am always amazed by the types of wire I see on frames, old and new. Incredibly, I have seen framers re-use the wire from the customer’s old frame, even when the old wire was corroded and kinked.  I have also seen them use the thin consumer-grade wire that you find in drugstores and supermarkets.  Why do reputable people cut corners on a commodity like wire after so much money was put into the rest of the frame?

The strength of wire depends mostly on its thickness, somewhat on its construction.  It is difficult to find technical data (as opposed to marketing claims) on the breaking strength of picture-hanging wire.  I wrote to Wire & Cable Specialties, the Pennsylvania-based manufacturer of the Super Softstrand vinyl-coated stainless steel wire that I like to use — they replied that the breaking strength for this wire was about 2.5 times the so-called “maximum picture weight” that is printed on the spool.  The following chart summarizes the breaking strength vs. Super Softstrand wire size:

Wire Size “Maximum Weight” Breaking Strength
#2 15 lb 37 lb
#3 20 lb 50 lb
#4 25 lb 62 lb
#5 43 lb 107 lb
#6 60 lb 150 lb

“Wire size” for picture-hanging wire is a vague term that has less to do with its diameter than its weight rating.  I have one spool each of the #4 and #5 Super Softstrand.  I almost always use the #5 wire unless I’m hanging something very small and light.  The #5 is a seven-stranded wire that measures about 0.040 inches diameter (equivalent to 18 gauge) without the vinyl coating, and about 0.060 inches including the coating.  In my opinion, the #5 wire is as easy to thread and knot as any other size.

Unless you frame thousands of pictures, you will not save much money using thinner wire: this online framing supply store will sell you 500 feet of #5 or 1125 feet of #3 wire for only $15.95.  If the average frame needs 3 feet of wire and you frame 300 pictures a year, you would spend $29 a year on #5 wire, compared to $13 on #3 wire.  This works out to 5 cents a frame.  Is it really worth 5 cents to use a lower-rated wire?

The Picture Frame Safety Factor Calculator

At last, the calculator.  This calculator lets you estimate the tension in the wire and the inward deflection of the sides of the frame, based on your dimensions and wiring setup.  This necessarily involves a number of assumptions, which I will discuss after presenting the calculator.

To evaluate the safety factors in your frame, you will need to enter the dimensions shown in the figure below:

Picture Frame Safety Factor Dimensions

First, indicate whether you have one wall hook or two.  (Before doing this, you might want to consult The “Hang It with Two Hooks” Calculator for my two-hook recommendation.)  Next, select whether you will enter the weight of your frame or let the calculator estimate the weight from its construction.

Now enter the frame dimensions, starting with the overall width and height (W and H) and the total length of wire (L).  If you are using D-rings, enter the length (D) from the hole to the tip.  But if your wire is attached directly to the frame, enter zero for that value.

Next, enter the distance (B) between the D-ring fastening screws (or wire fastening points if there are no D-rings).   If you indicated you are using two wall hooks, you will be asked to enter the distance (X) between the hooks.

Finally, enter the dimensions of the frame molding and the breaking strength of the wire.  It is possible you may not know these values, so here is some guidance:

For the cross-section of the molding, enter the face width of the molding (F) and the average thickness of the molding (T).  Frame moldings can have complicated profiles, so do your best to estimate average thickness.  The more curves in the molding profile, the greater uncertainty there will be in the estimated deflection.

For the breaking strength of the wire, enter the value if you know it; otherwise enter 2.5 times the rated weight.  If you don’t know that, make a conservative guess such as 50 lbs. or less.  Corroded or kinked wire is likely to have a lower breaking strength than new wire — any wire is only as strong as the weakest point along its length.

When you are finished, click CALCULATE to validate your entries and show the results. The calculator will estimate the tension in the wire and tell you what percentage of the breaking strength this represents.  (With wire and cable, it is common to use a 5x safety factor, which implies the tension should be no more than 20% of its breaking strength.)  The calculator will also estimate the inward deflection of the sides of your frame.  I suggest that if the deflection is more than one-third the typical clearance (1/16th-inch all around) between the frame and its contents, then you are in danger of damaging the artwork and/or glass.  Do not count on the glass to reinforce a frame: it is the job of the frame to support the art and the glass.

As promised, here is a list of my assumptions:

  1. The estimated weight (if selected) assumes 2.5 mm soda-lime glass (if selected) with 2.5 specific gravity, wood frame molding with 0.4 specific gravity, and other materials at 0.002 lb / in².
  2. Elongation of the wire due to tension in the wire is ignored.
  3. The force pulling inward on the side of the frame is assumed to be concentrated at a point one-third of the way down from the top of the frame.  The corners of the frame are assumed to be stationary.
  4. The calculator does not evaluate the integrity of the miter joints or the fasteners.
  5. The side of the frame is assumed to bend as if it had a rectangular cross-section.
  6. The amount of bend in the frame is inversely proportional to the elastic modulus of the wood.  The elastic modulus is assumed to be 1,500,000 lbf / in², which is a mid-range value for typical framing woods (see reference).

If the calculator warns you about tension or excessive bending of your frame, I suggest you buy some #5 vinyl-coated wire and consult The “Hang It with Two Hooks” Calculator to find a more frame-friendly wiring method.  Also, be aware that the taller the frame or the narrower the molding, the more that its sides will bend inward for a given tension.

And now I must add my usual disclaimer.  This calculator makes it easy for you to estimate the safety factors in your framing situation — but because of the assumptions involved, the results should only be treated as estimates.   The calculator may indicate a problem where there is none, or it may fail to warn you that a problem exists.  I offer this calculator as a convenience but I assume no liability for damage of any kind, even if my suggestions are followed exactly.  You bear full responsibility for choosing to use this information.

That concludes my three-part series on framing with wall hooks and wires.  I believe this is one of the most exhaustive (hopefully not exhausting) treatments of this topic that you will find on the internet.  I have tried to make it as accessible as possible.  Please let me know if you find the calculator useful, or if you have problems or discover bugs while using it.

Share your thoughts about this post (below).
More in Art and Photo, Asked & Answered | Subscribe.

10 Responses to The Physics of Hanging Pictures

  1. Sue Ellen Collins says:

    My sweet, lovable, caring husband is nothing short of a genius and a renaissance man!

  2. Enrique says:

    This is brilliant. I could see the three posts combined into a single monograph. There is a book by Faraday on the chemistry of candles, so why not one on the physics of hanging picture frames. (I wish I knew an art museum curator who could publish it.)

    • Craig says:

      Thank you for your comment, Enrique. I think there are some who would prefer to treat this endeavor as an art rather than science. I hope I have added some of the latter into the mix.

  3. Joyce says:

    Dear Craig

    Your wife’s right, this is genius, and I have a math degree so I don’t say this lightly. I wonder if I might expand the application of your genius, though. My walls can’t be screwed, drilled or nailed into. Tinted, high-polish plaster (marmorino) that couldn’t be patched later w/o great expense. But I do have exposed beams running all around the ceiling joint.

    On a smaller picture, my first, I screwed a D-ring (actually, the smaller triangle version) into the ceiling beam, so that the screwed portion was horizontal and the triangle hung down at a 90 degree angle, flush against the wall. I then bent a cable (actually, thin plastic, b/c the picture was small enough to use the transparent stuff), threaded the two ends through the triangle and through the bend in the wire and viola, the two cables hung flat against the wall from this one point, like an upside-down V. I attached them to the D-rings where the framer had set them. Nope! So then I moved them on the frame to about one inch from the horizontal edges. Better. Not flat, but flatter, and flat enough. Maybe 2 cables hanging parallel, from two ceiling attachment points, would have created an even flatter picture, but whatevs, small picture, not a big deal.

    In another place I got the idea to hang two pictures, one atop the other. I’m guessing, though I haven’t yet tried it, that hanging them railroad, two parallel cables, threading through a small eye on the top picture, might will increase the tension – the weight of the bottom one will pull the top one flatter against the wall. But setting that aside for a moment:

    Grandiose ambitions, maybe hubris, lead me to want to hang a far larger picture with glass – 21×27 – also, gallery style, from cables attached at the joint between the wall and the ceiling. Hmm. Thing’s much heavier, so vinyl coated wire for the cables, I get that. But how and where to attach the cables? (I’m calling them that to distinguish them from the wire on the picture, even though it might use the same stuff and indeed, in some setup, it all might be a single length of wire.)

    What thinks the genius?

    Yours in physics and admiration…

    • Craig says:

      Hi Joyce. From what you described, I think you would be happiest in the long run going with a ceiling-mount track rail mounted to the bottom face of the beam (from your description, I assume the beam has an exposed bottom face where it meets the wall). Take a look at the ceiling-mounted track offered by AS Hanging (no I don’t work for them) at this site:

      http://www.ashanging.com/en/ceiling-track/

      Two of the cable-hook designs for this track bend back toward the wall to keep the cable close to the wall. I would use two cables per frame, so that the cables hang vertically, and attach the D-rings on the side members of the frame just below the bottom edge of the top member. Hook the cables right onto the D-rings and remove any existing wire on the frame. If the picture is heavy, consider reinforcing the joints of the frame with flat corner braces.

      This will give you the flexibility to adjust the height of the picture, move it sideways, exchange it with other pictures, etc. without removing and reinstalling hardware on the beam.

      I have a wall-mount rail not a ceiling-mount rail. My biggest concern with a ceiling-mount rail is that the load pulls straight down on the threads of the screws holding up the rail. You really need to make sure that the beam is solid and that you are using screws that are long and large-diameter. I don’t know what size screws they supply with these rails but whatever they provide, I might consider using larger ones.

      Since this is a post on physics, I will mention that the withdrawal force of screws can be given by the formula (courtesy of Douglas Rammer, US Forest Service)

      p = 15,700 * G^2 * D * L

      where p is maximum withdrawal load (lb), G is the specific gravity of the wood at 12% moisture content, D is shank diameter of the screw (inches) and L is the length of penetration of the threaded part of the screw (inches). For instance, a 3/4-inch long #8 screw (0.164-inch shank) screwed into white pine (about 0.4 specific gravity) would have a maximum withdrawal load of 309 lbs according to this formula. If you used a 5x safety factor, that would translate to about 60 lbs. So, a lot depends on the beam and its condition. Hope this has been helpful.

  4. Bob says:

    I have heard that it is easier to keep the picture hanging straight when using the two-hook method. Do you have any comments on this claim?

    • Craig says:

      Left-to-right movement depends on factors like friction between the hook(s) and wire, friction between the frame and the wall, and outboard torque (caused by the frame hanging off-center). Using two hooks would increase the friction on the wire and reduce the outboard torque, both of which would help keep the frame in position if it were slightly off-center.

  5. Joseph Dee says:

    I do not find general web information on the use of one strand of wire loosely looped through a D mount on one side creating a parallel (two parallel strands) wire pattern with both end terminated on the opposite side of the frame. With two wall hangers close to the horizontal spaced about 1/4 each or so back from the frame edge verticals, put just one strand over one hanger hook, creating the normal “W” hanging pattern. It holds level an distributes the weight.

  6. Christine Hartmann says:

    I agree that this is genius, and I love the physics.

    My hanging conundrum doesn’t involve framed art. I’d like to hang a pot rack (coincidentally, it is the exact item hanging above the stove in the Big Bang kitchen of Leonard and Sheldon). The rack itself weighs in the range of about 4 lb. Each pan that I’d like to hang is in the range of 4 lb – 5 lb, so I’m looking at a total of about 25 lb or so. I was thinking (intuitively for me) that it would be best to distribute the weight across two (or more) hooks, but I don’t really know if that’s the way to go or not.

    Also looking to do as little damage and destruction as possible, so was considering the high and mighty hooks. Do you have any interest in commenting?

    Christine

    • Craig says:

      Hi Christine, thanks for reading and your comment. Although this post is about picture hanging, I will offer a physics opinion on your question. I took a look at the “Big Bang” pots-and-pans hanging unit you mentioned. I had thought that it might be a simple horizontal bar that attaches to the wall, but no, your unit is sort of clamshell-shaped and most of the weight on the unit (when the pans are on it) is far away from the wall. This will create a substantial force trying to pull the top attachment point out and away from the wall — that component of the force will be much greater than the 25 lbs that the pots and pans weigh. Think how much harder it is to hold a cast-iron pan out at arm’s length from your body vs holding it straight down — same principle.

      I think a push-in hanger like the High and Mighty hook, which is designed for vertical loads, might be the worst possible hook for this situation! I don’t know what kind of mounting holes or hooks your unit offers, or how wide your unit is, but I would want to hang such a unit onto two or more lag bolts screwed into studs. Good luck.

Leave a Reply