# Optimizing support board placement on work table wing

I'm currently building a work table that will have "wings" on each side that are fixed to the mid section of the table via a hinge. These wings, in turn, have hinges that fix a support board to them which will rest against the mid section of the table. I'm currently in the process of deciding where to place the support and am wondering if anyone has any engineering input as to an optimal placement.

Because of how the rest of the table in constructed, I am limited as follows:

• $14'' \leq A \leq 19''$
• $B + C \leq 38''$

Then there's also obviously the fact that:

• $A^2 + B^2 = C^2$

I'm not sure what the optimal setup is here. The only scenario I've solved for is trying to make $A = B$ so that the angle $\theta$ between $A$ and $C$ is 45 degrees. This implies $\sin(\theta) = B / C = 1 / \sqrt{2}$. Plugging this into the equations I have above, I found that:

• $A = B$ implies that $14'' \leq A = B <= \frac{38}{1 + \sqrt{2}} \approxeq 15.3$

Are there any other more sophisticated approaches to this problem than the "make $\theta=$ 45 degrees" approach?

p.s. - I studied Physics in college, so have a math / physics background, but know very little about applied engineering like this. Also, let me know if you require / would like any additional information.  • There are hinges made for this type of table. Did you check folding brackets for tables? Here is a google image search google.com.tr/… . – Gokce Mehmet AY Sep 10 '15 at 8:50
• What are you trying to optimize for? As long as none of the pieces will break, there is little engineering that will inform the placement of the support. – hazzey Sep 10 '15 at 14:36
• @Gokce - thanks for the recommendation. I never did run across these in the limited planning I've done. However, these look like they'd work better if my "wing" was shorter than it was wide, which is not the case for me. – mknutso2 Sep 10 '15 at 23:33
• What materials would be used here? Wooden board on steel frame? – Wasabi Sep 10 '15 at 23:37
• @hazzey I guess it may be a case of "I don't know what I don't know". I assumed there would be some magical formula or principle to follow in choosing how to position the support, but perhaps that is not the case? The wing and the support board will be 1/2" plywood that could break under significant force, but the mid support could be assumed to be rigid in comparison (it's made of 2x4s). – mknutso2 Sep 10 '15 at 23:39

As @hazzey mentioned in a comment under the OP, if nothing is going to break, then the choice is mostly arbitrary. So I'm going to assume that things do break.

A structural element can basically collapse in three ways: excessive axial force (tension or compression), shear force or bending moment. Each of these is calculated differently.

For simple things like a table, one can assume a simple calculation for the maximum supported tensile force $$T_R$$: $$T_R = f_y\cdot A$$ where $$f_y$$ is the allowable tensile stress and $$A$$ is the total cross-section area.

Shear force can also be calculated in a similar fashion: $$Q_R = f_q\cdot A$$ where $$f_q$$ is the allowable shear stress. For steel one often uses $$f_q = 0.6f_y$$.

Bending moment has a series of complications due to lateral buckling. However, since that probably won't be the limiting factor in the design of your table, I'll skip over them as well. Therefore, we can take simply that the maximum stress at any point in the beam must be less than $$f_y$$. The applied stress can be obtained via $$\sigma = \dfrac{My}{12EI}$$ where $$M$$ is the bending moment, $$y$$ is the height from the centroid of the desired point, $$E$$ is the material's elastic modulus, and $$I$$, the cross-section's second moment of area (aka, moment of inertia).

I have left compression for last because it's the one with complications we can't avoid, since they will probably be the controlling factors. For short segments, the maximum allowable compressive force is equal to the maximum tensile force $$T_R$$ presented above. That being said, due to buckling, this usually isn't the case. Buckling is a pain and has a bunch of complications in and of themselves, but for a ballpark value, we can use Euler's equation: $$P_E = \dfrac{\pi^2EI}{(KL)^2}$$ where $$K$$ is a factor depending on the element's support conditions. $$P_E$$ is non-conservative, meaning the actual allowable compressive force will certainly be lower than this.

There is also another way the table can fail and that is if it doesn't suit your needs. It's possible for the table to support the desired load but to do so by deforming completely, making it unappealing and unusable. Deflections, however, don't have simple equations and depend on the current conditions.

So, now we need to find the stresses that occur in your table. This is its structural model (the partial dimensions are placeholders and change according to the selected angle and value of length $$A$$. The load is also a placeholder): The balancing act that is required is due to the fact that:

• if you increase the table span by moving the bracing closer to the base, you're going to increase the bending moment and shear force on the table. The deflections are also going to increase.
• if you lengthen the bracing (up to its maximum 19", for instance), you can reduce the stresses described above on the table, but lower the buckling load for the bracing.
• if you open up the bracing (increase its angle to the vertical), you can reduce the span of the table but you reduce the bracing's effectiveness.

The optimal solution needs to balance all of these considerations and it goes beyond the scope of this answer to actually obtain it (especially given that it depends on the materials and cross-sections). To perform the structural analyses, a 2D frame analysis tool like Ftool (free) can be used. That being said, 45 degrees is usually a good bet.

• So in summary: 45 degrees and don't worry about it. :) – hazzey Sep 18 '15 at 1:36

When looking at the piece from a force balancing analysis, the answer comes down to three cases:

If the angle is very small, then the board is primarily horizontal and most of the vertical load will be multiplied by a factor of the secant of the small angle. A skinny member like that does not take compressive loads like this well (buckling). The shelf will not take much in the way of load. However, since the end of the support will likely be near the end of the board, the plate functions like a simply supported beam and has minimal deflection.

If the angle is very large, then the board is primarily vertical and takes most of the compression without the secant multiplier adding to much. However, due to geometry, the board will struggle to reach the end of the plate, possibly turning it into cantilever, leading to poor deflection control of the plate, but it can take large amounts of loading.

Finally the middle case you described. This can handle both cases well, deflection of the plate minimized and holding a reasonable amount of load.

Typically in these cases a designer comes up with an angle and evaluates the loading the system can handle. Then, by comparing the performance of all members with regards to specified parameters (factor of safety on buckling, allowed deflection), the excess factors (high fos or low deflection) can sometimes be adjusted by changing the angle until the excess on each factor is the same.

Of course, this really depends on your specified parameters - what you want the system to do - high load carrying capability or minimal deflection.