How much force would it take to break a 4 ft 2x4 supported at both ends?

Answer 1

Since wood is anisotropic (it has a different strength along different directions), it depends on how the grain of the wood is oriented. I'm going to assume the grain is oriented down the length of the board (most boards are cut this way).

If we break the board by pressing on the middle with increasing force until it breaks, there are two possibilities. Either we're pressing on the 4" side (requires less force to break), or we're pressing on the 2" side (requires more force to break).

The total force needed to break the beam by pushing in the middle is whatever force generates a tensile stress (on the underside of the beam) that is greater than the strength of the material. The underside will crack first, then a chain reaction will take place and the entire beam will break.

The moment of inertia of the beam is
I = 1/12*b*h^3
where b is the beam thickness from side-to-side, and h is the beam thickness in the direction you're applying the force. If we use the easier method, then
I = 1/12*(3.5 inches)*(1.5 inches)^3
I = 0.9844 in^4
since a 2x4 piece of lumber actually has 1/4-inch cut off of every side, making it a 1.5"x3.5" beam.
If we orient the beam the other way (trying to break it in the more difficult direction, the moment of inertia is
I = 1/12*(1.5 inches)*(3.5 inches)^3
I = 5.359 in^4
which is more than 5 times as large!

The stress in the beam depends on the bending moment we can apply, which is a function of the length of the beam. It takes less force to apply a large bending moment on a long beam, and more force to apply the same bending moment on a short beam.

The bending moment in a beam supported at both ends with a force F applied in the middle is
M = 1/2*F*L
M = 1/2*F*(4 feet)
M = F*(24 in)

The beam will fail when the maximum stress in the beam equals the ultimate strength for the material:
max stress = ultimate strength
M*h/(2*I) = 5800 lbs/in^2 (for typical Southern Pine wood)

If we break it in the easier direction:
F*(24 in)*(1.5 in)/(2*(0.9844 in^4)) = 5800 lbs/in^2
F = 317.2 lbs

If we break it in the harder direction:
F*(24 in)*(3.5 in)/(2*(5.359 in^4)) = 5800 lbs/in^2
F = 740.1 lbs

So, it takes 317 pounds to break the beam in the easier direction, and 740 pounds to break it in the hard direction. The ratio of these two numbers is the same as the ratio of the two cross-sectional dimensions: 3.5" / 1.5" = 2.3333

Answer 2

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RE:
How much force would it take to break a 4 ft 2x4 supported at both ends?

Answer 3

2x4 Strength

Answer 4

There is no good answer to this because the composition of wood changes even within a species. Furthermore, you have to specify whether the board is flat or on edge.