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Mohamed Obeidallah

Ultimate compression strength of human femur

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According to many sources, the ultimate compression strength of human femurs along its long axis (vertically) is 205 mpa or 29700 psi. I tried to convert 205 mpa to kg/cm² and the result is 205 mpa is equal to 2090.41824 kg/cm². So, I think the maximum weight that a human femur can support vertically before breaking is 2 tonnes. Isn't it?

But with different method of calculation, I got a drastically different answer. The maximum diameter of human femurs is 3 cm and some exceeded 3 cm by 5 mm or more especially in weightlifters. So I followed this calculation from https://openoregon.pressbooks.pub/bodyphysics/chapter/stress-and-strain-on-the-body/. First we divide the 3.00 cm femur diameter by two to find the femur radius, then we convert to standard units of meters. So 3.00 cm/2 (1 m/100 cm) equals to 0.0150 m.

Then, using the equation for the area of a circle we calculate the total area of the femur to be: (0.0150 m)2 equals to 7.0 x 10–4.

Finally we have to subtract off the area of the hollow middle part to get the net bone area. We used a ruler on the above picture of the femur cross-sections (available on the link I just shared) to see that the inner radius is roughly half of the outer radius, or 5.85 x 10^{-3}m so we calculate the missing inner area: (5.85 x 10^-3)^2 equals to 1.1 x 10^-4 m^2.

And subtract off the inner area from the total:

7.0 x 10^-4 m^2 - 1.1 x 10^-4 m^2 equals to 5.9 x 10^-4 m^2

Next, lets convert 12.3 tonnes to Newton. 12.3 tonnes is equal to 121000 N. An approximate minimum cross-sectional area of the femur is 5.9 x 10^-4 m^2. (*See the previous calculation example if you are interested in learning how I approximated this value). We divide the compressive force by the cross-sectional area to find the compressive stress on the bone.

So...

Stress = force/area = 121000 N/5.9 x 106-4 m^2 equals to 205084745 pa which is exactly 205 mpa, the ultimate compression strength of human femur. So, a human femur can vertically support 12 tonnes! Which is the weight of some mammoths and large carnivorous dinosaurs! Is this correct? Does this actually applies to human femurs in reality? That's actually quite unimaginable to be honest. 205 mpa isn't the only maximum result of compression testing on human femurs vertically if you keep looking on the Net, some put it at 215 mpa! Do you think this is realistic?

In this video https://www.youtube.com/watch?v=os98s9kBlOI, it is demonstrated that a moose femur bone (which is only the midshaft) can support nearly 10 tonnes vertically. Maybe it make sense because animals tend to have stronger bones than humans. But some people said the real strength could be much weaker because it is not a whole femur, it is just a midshaft so it is stable. Thoughts?

But according to my first statement, if the ultimate compression strength of human femurs is 205 mpa and it is equal to 2090.41824 kg/cm², the maximum weight that human femurs can support vertically before breaking is 2 tonnes. So, which one is correct? Is it the first statement or the second statement which includes calculations? Please correct any mistakes if there are any. Thank you.

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Please tell us what you actually want to achieve.

Then proper advice can be given.

There is a whole lot more to stress analysis than perhaps you realise.

In particular the femur is technically  a 'slender strut' that is also subject to significant torsion and bending.
This can 'magnify' the effects of direct compression dramatically.

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15 minutes ago, studiot said:

Please tell us what you actually want to achieve.

Then proper advice can be given.

There is a whole lot more to stress analysis than perhaps you realise.

In particular the femur is technically  a 'slender strut' that is also subject to significant torsion and bending.
This can 'magnify' the effects of direct compression dramatically.

I want to see answers whether human femurs with maximum diameter at 3 cm or a little bit more can really vertically support 12 tonnes or not. According to my calculations based on the link I've shared, human femurs with maximum diameter at 3 cm can hypothetically support 12 tonnes which is the weight of large carnivorous dinosaurs. I don't think this is actually realistic. But then I convert 205 mpa to kg/cm^2 using online converter and I got 2090.41824 kg/cm² which is 2 tonnes.

Which one is more realistic, is it human femurs with 3 cm diameter can vertically support 12 tonnes or 2 tonnes? My main question is I want to find what is the maximum weight that human femurs with 3 cm diameter can vertically support before crushing. Sorry if my English is that bad. 

Edited by Mohamed Obeidallah

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7 minutes ago, Mohamed Obeidallah said:

I want to see answers whether human femurs with maximum diameter at 3 cm or a little bit more can really vertically support 12 tonnes or not. According to my calculations based on the link I've shared, human femurs with maximum diameter at 3 cm can hypothetically support 12 tonnes which is the weight of large carnivorous dinosaurs. I don't think this is actually realistic. But then I convert 205 mpa to kg/cm^2 using online converter and I got 2090.41824 kg/cm² which is 2 tonnes.

Which one is more realistic, is it human femurs with 3 cm diameter can vertically support 12 tonnes or 2 tonnes? My main question is I want to find what is the maximum weight that human femurs with 3 cm diameter can vertically support before crushing. Sorry if my English is that bad. 

Thank you.

I will have to look some facts and figures up.

Did you understand my comment about the strut?

The femur is most likely to fail in buckling or twisting, rather than straight compression.

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26 minutes ago, studiot said:

Thank you.

I will have to look some facts and figures up.

Did you understand my comment about the strut?

The femur is most likely to fail in buckling or twisting, rather than straight compression.

Yes I understand. Bone is strongest in compression compared to shear and when under tension. Looking forward to your further responses. Thanks!

Edited by Mohamed Obeidallah

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On 10/3/2019 at 4:38 PM, studiot said:

Thank you.

I will have to look some facts and figures up.

Did you understand my comment about the strut?

The femur is most likely to fail in buckling or twisting, rather than straight compression.

Sorry I forgot to add, the diameter of the femur is 3 centimeters with the cortical thickness being 1.3 - 1.5 centimeters.

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