# Compressing the Femur

Opposite to tension forces, forces are provided by a material in response to being compressed rather than stretched. The resistance of materials to deformation is what causes the (support force) that we introduced in the unit on balance. For example, the femur is compressed while supporting the upper body weight of a person.



“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.”The average weight among adult males in the United States is 196 lbs (872 N).  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.

# Stress

The size of object affects how they deform in response to applied and forces. For example, 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 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 which is roughly circular and somewhat hollow (bone marrow fills the center space). 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



Larger bones and tendons can support more force, so in order to analyze the behavior of the bone material itself we would need to divide the force applied to by the ( ).  The resulting 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) # 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. Along with bone, concrete and chalk are other examples of materials with different compressive and tensile ultimate strengths.

### Everyday Example: Femur Ultimate Strength

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. An approximate minimum of the femur is . (*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. 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. Using the equation for the area of a circle we calculate the total area of the femur to be: 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 so we calculate the missing inner area: And subtract off the inner area from the total: # 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. Both tension and compressive stresses are applied to the Femur while standing. Image Credit: Blausen Medical via Wikimedia Commons 