Re: Will bone break? Given length of bone and cross area and force applied?

If one is to be really technical about this question, there is a whole lot more information you would need. Bone is too complex to be treated like typical engineering materials. To start, it's a composite material, made of hydroxyapatite crystals (strong and hard) embedded in a collagen (flexible) matrix. And further, a bone isn't a simple beam of material; it is a complex structure with different properties along its length. For example, most bones are hollow for much of the length, but solid at the ends - and are made of different materials at the ends than in the middle. Further, the properties of a bone change significantly whether it's in a living animal or it is a dry, prepared specimen.

But, if this is the extent of the information you've been given, you can take a good stab at answering the question by making some simple assumptions and simplifications.

Since a direction of the force was not specified, I'd assume that the force is purely axial (along the length of the bone). Further, since bone is fairly stiff (we'll get quantitative about its stiffness in a moment), we can probably assume it isn't going to buckle (bend in the middle, like a plastic straw would if you pressed on both ends) significantly.

This puts the bone in pure compression - no torque on the material, no bending or twisting. In pure compression, we can ignore the cross-sectional shape of the bone; the cross-sectional area is all that matters.

Your force is 3x10^4 newtons over an area of 3.6 cm^2, or 3.6 x 10^-4 m^2. So the pressure (aka stress) on the end is (3E4 / 3.6E-4) N per m^2 which equals 8.3E7 N/m^2, or 83 MPa. (MPa=megapascal. 1 Pascal = 1 N/m^ 2).

(Note: I'll write 8.3 X 10^7 as 8.3E7 sometimes, just to save space.)

The compression strength given for your bone is 1.7 x 10^6 N/m^2, or 1.7 MPa. The force applied is 55 times greater than the strength of the bone, so yes, it will break.

However - my textbooks give the compression strength of a typical bone (a human femur) at 170 MPa, not 1.7 Mpa. Other bones (various species, etc.) tend to vary from 100 MPa to 170 MPa. [1]

In that case, the compression strength of 170 Mpa is safely above the applied stress of 83 MPa, so the bone wouldn't break.

Which leads us to the second part of the question: If it doesn't break, how much will it shorten? To answer this, we need a quantity called the modulus of elasticity of the material. I don't have a modulus of elasticity for a human femur, but I'll use one for a horse femur; horse bones typically have similar material properties to human bones.

(Aside: the reason we have few data on human bone strengths is probably that bones change properties significantly when they dry out or when the organism dies. Which means, sadly, that to take accurate bone strength data one must slaughter an animal, remove the bone, and test it while it is still warm; obviously we cannot do this in humans. Fortunately, we only need to do this once and we have reasonable data that can be used in perpetuity. The data I am using date from the 1960's.)

Modulus of elasticity is defined as stress, or pressure, divided by strain, where strain is a fractional elongation under pressure: (change in length/original length before force was applied). The modulus of elasticity in compression of a horse femur is 9.4 x 10^9 N/m^2, 9.4 GPa. [1] (GPa = gigapascals) Let's use the measure to see what that means.

We take the force applied to your bone, 83 MPa, and divide it by the modulus of elasticity: 83E6 Pa / 9.4E9 Pa = 8.8E-3. The latter value is our strain. Strain has no units, since it's defined as (change in length)/(original length), or m/m. But multiplying our strain times the original length gives us the change in length: 8.8E-3 * 20cm = .18 cm, or about two millimeters. So, under the force described, the bone won't break, but will shorten by almost two millimeters.

I mentioned above that we'd get quantitative about the stiffness of bone; that's what the modulus of elasticity tells us. 9.4 GPa is a fairly stiff material: about the stiffness of most softwoods. Metals have elastic moduli into the hundreds of GPa, plastics vary widely but 1 GPa is a typical value. [2]

I hope this has been informative and of assistance. Please write if I can clear up anything else, and keep the questions coming!

Evan Dorn madsci@lrdesign.com

1. Fung, Y.C., 1993 Biomechanics: Mechanical Properties of Living Tissues. p 511.
2. Askeland, D. 1994 The Scienc and Engineering of Materials p. 141, 505.