53 Strength of Human Bones

The Femur

“In human anatomy, the femur (thigh bone) is the longest and largest bone. Along with the temporal bone of the skull, it is one of the two strongest bones in the body. The average adult male femur is 48 cm (18.9 in) in length and 2.34 cm (0.92 in) in diameter and can support up to 30 times the of an adult.”[1]

The Human Femur. Image Credit: Anatomography via Wikimedia Commons

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Compression

When you place an object on top of a structure, the object’s weight tends to compress the structure. Any push that tends to compress a structure is called a . The average weight among adult males in the United States is 196 lbs (872 N)[3].  According to the statement that the femur can support 30x body weight, the adult male femur can support roughly 6,000 lbs of compressive force!  Such high forces are rarely generated by the body under its own power, thus motor vehicle collisions are the number one cause of femur fractures[4].

Tension

When you hang an object from a structure the object’s will tend to stretch the structure. The structure responds by providing a force to hold up the object. Tension forces are produced in response to materials being stretched. Non-rigid objects like ropes, cables, chains, muscles, tendons, can effectively provide tension forces only, while rigid object can supply and tension forces. For example, the biceps muscle is providing a tension (T) force on the thumb-side forearm bone (radius bone).

Figure is a schematic drawing of a forearm rotated around the elbow. A 50 pound ball is held in the palm. The distance between the elbow and the ball is 13 inches. The distance between the elbow and the biceps muscle, which causes a torque around the elbow, is 1.5 inches. Forearm forms a 60 degree angle with the upper arm.
The elbow joint flexed to form a 60° angle between the upper arm and forearm while the hand holds a 50 lb ball . Image Credit: Openstax University Physics

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Stress

The maximum or forces that a bone can support depends on the size of the bone. More specifically, the more area available for the force to be spread out over, the more force the bone can support. That means the maximum forces bones, (and other objects) can handle are to the c of the bone that is (90°) to the direction of the force. For example, the force that the femur can support vertically along its length depends on the area of its horizontal cross-section which is roughly circular and somewhat hollow (bone marrow fills the center space).

 

The femur cross-section on right shows significant reduction in bone thickness and reduced density near the inner surface.
These cross sections show the midshaft of the femur of an 84-year-old female with advanced osteoporosis (right), compared to a healthy femur of a 17-year-old female (left). Image Credit: Smithsonian National Museum of Natural History

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Larger bones can support more force, so in order to analyze the behavior of the bone material itself we need to divide the force applied to the bone by the minimum (A_x). This quantity is known as the  (σ) on the material. Stress has units of force per area so the units are (N/m2) which are also known as . Units of pounds per square inch (PSIlbs/in2) are common in the U.S.

(1)   \begin{equation*} stress  = \frac{F}{A_x} \end{equation*}

Reinforcement Exercises

Estimate the stress within  a  1.0 cm x 2.0 cm Lego block when you step on it with full body in units of pb_glossary id=”3977″]Pascals[/pb_glossary]. [Hint: We want the result in units, so convert the length and width to meters before calculating the and use SI units for your weight.]

Ultimate Strength of the Femur

The maximum that bone, or any other material, can experience before the material begins or is called the . Notice that material strength is defined in terms of stress, not force, so that we are analyzing the material itself, without including the effect of how much material is present. For some materials the ultimate strength is different when the stress is acting to crush the material () versus when the forces are acting to stretch the material under , so we often refer to ultimate tensile strength or ultimate compressive strength. For example, the ultimate compressive strength for human femur bone is measured to be 205 MPa (205 Million Pascals) under compression along its length. The ultimate tensile strength of femur bone under tension along its length is 135 MPa.[7] Along with bone, concrete and chalk are other examples of materials with different compressive and tensile ultimate strengths.

Reinforcement activity

Try to crush a piece of chalk by using your fingers to push on the ends and compress it along the long axis, no bending allowed. Any luck?

Now use your fingers to break the chalk by pulling it apart, straight along the long axis, again no bending allowed. Any luck?

Record your results and explain what they tell you about the compressive and tensile of chalk.

Compare and contrast the behavior you observed for chalk with the known behavior of bone and concrete. Cite your sources.

Everyday Example

Let’s check to see if the measured values for compressive agree with the claim that the human femur can support 30x the adult body , or roughly 6,000 lbs 

First let’s to convert the claimed 6,000 lbs force to and work in SI units.

    \begin{equation*} 6,000\, \bold{lbs}  = 6,000\, \cancel{\bold{lbs}} \left(\frac{4.45\, \bold{N}}{1\, \cancel{\bold{lb}}}\right) = 26,166 \bold{N}. \end{equation*}

An approximate minimum of the femur is 3.2\times 10^{-4}\, \bold{m^2}. (*See the bottom of this example if you are interested in learning how we approximated this value). We divide the compressive force by the cross-sectional area to find the compressive stress on the bone.

    \begin{equation*} Stress = \frac{force}{area} = \frac{26,166\, \bold{N}}{3.2\times 10^{-4}\, \bold{m^2}} = 80,000,000\, \bold{Pa}  = 80\, \bold{MPa} \end{equation*}

Our approximate value for the of bone that would be required to support 30x body weight was 80 MPa, which is actually less than the measured value of 205 MPa, so the claim that the femur can support 30x body weight seems reasonable.


*This is how we approximated the femur cross-sectional area, skip this if you aren’t interested:

First we divide the 2.34 cm femur diameter quoted earlier by two to find the femur radius, then we convert to standard units of meters.

    \begin{equation*} r = \frac{2.34\, \cancel{\bold{cm}}}{2}\left(\frac{1\, \bold{m}}{100\, \cancel{\bold{cm}}}\right) = 0.0117\, \bold{m} = 1.17 \times 10^{-2} \end{equation*}

Using the equation for the area of a circle we calculate the total area of the femur to be:

    \begin{equation*} A_{out} = \pi r^2 =  \pi \left(0.0117\, \bold{m}\right)^2 = 0.00043\, \bold{m^2} = 4.3 \times 10^{-4}\, \bold{m^2} \end{equation*}

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 to see that the inner radius is roughly half of the outer radius, or 5.85\, \times 10^{-3} \bold{m} so we calculate the missing inner area:

    \begin{equation*} A_{in} = \pi r^2 =  \pi \left(5.85 \times 10^{-3}\, \bold{m}\right)^2 = 1.1 \times 10^{-4}\, \bold{m^2} \end{equation*}

And subtract off the inner area from the total:

    \begin{equation*} A_{x} = 4.3 \times 10^{-4}\, \bold{m^2} -1.1 \times 10^{-4}\, \bold{m^2} = 3.2\times 10^{-4}\, \bold{m^2} \end{equation*}

Transverse Ultimate Strength

So far we have discussed along the long axis of the femur, known as the direction. Some materials, such as bone and wood, have different ultimate strengths along different axes. The ultimate compressive strength for bone along the short axis ( direction) is 131 MPa, or about 36% less than the 205 MPa longitudinal value. Materials that have different properties along different axes are known as . Materials that behave the same in all directions are called .

An interesting fact to finish up this chapter: when a person stands the femur actually experiences compressive and tensile stresses on different sides of the bone. This occurs because the structure of the hip socket applies the load of the body off to the side rather than directly along the long axis of the bone.

A cross section of a human femur. The force of body weight acts downward on the top of the bone where it attaches to the hip. The side of the bone closer to the body is under compression, the farther side is under tension.
Both tension and compressive stresses are applied to the Femur while standing. Image Credit: Blausen Medical via Wikimedia Commons

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