# How to calculate the influence on load bearing on beams cut/notched for connections?

I have trouble understanding what's the point in cutting beam endings in order to make wood construction connections. Assume a connection of a 50mm x 50 mm beam (green) to a 100mm x 100mm beam (pink)

with a leaf connection (my translation for - I guess - the simplest wood construction connection consisting of removing half of the diameter of the beam and putting that part into the counterpart connection cut out). I imagine that the forces where the beams connect are the same as on the whole green beam, but the connection must obviously hold because those connections are used in construction, so why not make the beam only half the diameter large/do for the whole beam what works at the location of the connection?

I further assume that there's an impact of the connection cut-out/notch, but it'll be smaller than making only 50% of the bearing load usable (otherwise every wood construction would generally waste up to 50% of its material). How can I estimate this impact.

The leaf connection is used in this example in order to reduce the material which has to be removed from the pink beam.

This might all be not ideal and not true; it's a question to improve my understanding - I'm not the contractor building the schools for your kids.

Beams are sized based on 1) Deflection, (bending) and 2) Shear. Notches can affect a beam differently in those two situations differently.

1) A beam deflects most at the center of the span. This puts the fibers along the top and bottom of the beam in the most stress.

Basically, those fibers along the extreme edges are stressed the most at the center of the span and the least (or not at all) at the ends of the beam. Therefore, a notch at the center of the span (like in the pink beam) could cause the beam to fail, but have no affect if notched at the ends (like the green beam).

2) A beam has the most horizontal shear at its ends and very little at the center of the span. That is to say, when the beam bends, the fibers in the wood want to tear apart mostly at the ends and has no stress in the exact center of the beam.

You can see this by stacking some boards flat between two supports and then standing on them. You’ll notice that the boards want to slide past one another at the ends, but they hardly move at the center of the span.

If you notch the center of the beam, you’ll have no effect on the horizontal stress, but if you notch the ends you’ll greatly affect the horizontal shear.

Summary:

Notching a beam does not affect its performance, unless it’s stressed to its maximum strength.

Beams with greater fiber strength can hold greater loads and therefore take a larger notch without failing. These fiber strengths are known for each species and grade of lumber and therefore can be calculated to determine maximum allowable notch sizes.

You're overlooking a couple of important factors, starting with deflection. Smaller structural members will flex more than thicker ones, especially in the middle of the span. Sag also adds lateral loads to the connections at the end since it they must resist the pull of the shorter curved member.

Those connections are the second thing you've overlooked. You do need to fasten things together eventually and butt joinery puts all of the load on the fasteners. Generally two thirds of a beam is quite a bit stronger than two or three nails. It's worth noting that it's fairly common to use angle iron and bolts when the work will be covered up.

I suspect that notching wooden members in new construction is pretty much limited to timber frame buildings. It's getting harder to find solid wood beams than it used to be, and modern structural joinery tends to incorporate steel fasteners to protect against extreme weather events. For hurricanes or tornadoes you want the joint tied down; for earthquakes you want it to deform rather than snap.

Getting back to your original question, structural engineers have access to lots of data to work on the estimate. They can look up the strength of a given material, average weight of a building per story, strength of a given joint compared, etc. From there it's fairly straightforward physics.

Simulation is also a common tool. Once the plans are drawn in CAD you can specify the building and wind loads and get a detailed report on the expected performance. If you're really, really interested Autodesk Fusion 360 is free for hobby use and will let you run some simulations. You'd have to deal with the learning curve and it's not the right tool to design a building, but it'll do for mucking about with simple joints. (Alternately you might find just the tutorial video on simulation entertaining.)

Non-engineers generally rely on experience and rules of thumb (or the building code in these more enlightened times) and gratuitous over-building. Nobody would notch anything as small as you've got in your sketch. In the US they wouldn't be called "beams" either as they're too small. Joists would be at least 150mm thick and wooden beams 200mm or more.

The lap joint is to position the top surface of both beams to the same level, and to provide a common surface for fasteners.

The ends of most beams can be reduced some without diminishing the maximum load capacity - usually found at the middle of a beam.