Have you ever had a cervical pain after staring at your phone for a long time, nodding off during classes, or working at your desk on your problem sets for several hours? Do you wonder why this would happen? Actually, this phenomenon can be explained by a simple structural engineering concept — bending moment.
Bending moment is the internal reaction happening in a structural element when an external force or moment, which can cause the element to bend, is applied. A very simple element in our buildings and infrastructures designed to resist bending moment is a beam.
To calculate the bending moment, we usually cut through the structure and solve the equilibrium equations. If the shear force (which is also an internal force) is known, we can also get the bending moment by integrating the shear force.
Since bending moment is an internal force, we need to cut the structure to find it. In the lower right structure below, the external load can only be balanced out by internal forces. And we can solve for the axial force N, shear force V, and bending moment M by applying the three equilibrium equations.
That's right! Internal forces have different sign conventions with external forces. For bending moments, moments that tend to make the beam "smile" are positive. It is okay to use other sign conventions. But the key point is to keep signs consistent.
Sounds confusing? Don't worry! We will go over an example soon.
Why can the bending moment in our neck cause cervical pain? This is because stress is generated when the structure is trying to resist bending. Think about when you are bending a plank: half of it tends to be extended while the other half tends to be contracted. These are caused by stress.
What happens to our neck when it is trying to resist bending? Right! Our muscles are in tension (since muscles are not good at resisting contraction), which is quite stressful for our neck if we keep the improper position for a long time. Structures experiencing bending moments also have stress, but the difference is that they don’t feel painful. (Or probably they do. Who knows?)
In order to understand why we sometimes have cervical pain, we can model our neck as a simple beam, and our head as the externally applied load. Considering the head as external to the neck might sound horrible, but at least you don’t need to tear them apart to view them as separate structural elements.
We have Stikky as our example! We know that Stikky's neck is 10cm long and 10cm in diameter, and Stikky's head weighs 50N.
Also we can assume Stikky's neck is uniform, which means his bones and muscles are the same. Not realistic for human beings.. But good enough for "stick beings"!
Now we can draw the free body diagrams. Assume the angle from Stikky's neck (the simple beam) to vertical is θ degree, and with the help of bones and muscles, it is fixed to Stikky's shoulder (meaning both rotation and displacement are restricted).
Then we will cut the beam (SORRY STIKKY!). Note when we look at the upper half, x=0 is the right-most point, so the local coordinate is rotated more than 180 degrees. Luckily the equations are easy:
We have 3 equations and 3 unknowns. So we can easily get:
The equation for stress caused by bending moment is -M*y/I, where M is the bending moment, y is the distance from the edge of bending to the center, and I is the moment of inertia. Since Stikky has a round neck, we can apply the I for circle here: I = (πD^4/64).
Thus, maximum Stress = - M(L)*(D/2) / (πD^4/64).
If Stikky nods off during class, and the angle θ is 60 degree, the maximum stress occurs at Stikky's neck will be:
- 50N * sin60 * 10cm * (-10cm/2) / (π * (10cm)^4 / 64) => ~ 44106 N/m^2
That's quite a lot, Stikky! Wake up and don't hurt yourself!
We will invite Stikky again! Drag the slider and see how the maximum bending moment and stress changes. Have fun!
Angle = 0 degree
Maximum Bending Moment = 0 N*m
Maximum Stress = 0 N
Note: All values are rounded to the closest integer
Stikky is a stick figure, which allows us to make a lot of idealizations. In reality, the human body is pretty complicated. Our muscles mainly resists tension while our bones mainly resists contraction. What's more, the vertebral structure makes our spine flexible and helps reduce bending moment.
Most likely, your neck won't undergo that much pressure as Stikky's does. However, sleeping in class can still put a lot of stress in your neck. Next time, try to stay awake and learn some structural analysis so you'll be able to calculate the stress yourself!