Mechanics of Mozzarella: Section Properties of Pizza in SOLIDWORKS
My mother always told me to never play with my food, but little does she know, you can get a lesson in engineering from it! In this article, I will discuss this with one of my favorite foods: pizza! As delicious as it is, it can prove to be a little bit difficult to eat it. We’ve all been there, unless your pizza is super thin and crispy, the slice will droop.
Still, humans are resourceful and have found ways to tackle the challenge. First, there are some who try to duck their heads and ambush the slice from below. While this gets the job done, it can be messy if the maneuver is executed incorrectly. Not to mention, then you become that person that droops a pizza over his/her face to eat it.
Some have found that they were able to subdue their pizza by using both of their hands. While effective, this technique can be dangerous to wielder. Many recount near death experiences, coming from the brink of dehydration as they were unable to hold a drink with what would, otherwise, be a free hand.
Another controversial technique is to leave the pizza on the plate and bring tools called “forks” and “knives” to it! This is polarizing among activists, who note that cutting up a pizza with cutlery is inhumane to the pizza, which should be treated with more respect. Once again, you become THAT person that eats a pizza with a knife and fork.
Surely, there must be a better way! We have technology! We have SOLIDWORKS!!! Well, the most recent development from the world’s top pizza researchers gives us this technique: the pinch.
This is when you pick up the slice of pizza by the crust and pinch the piece of pizza in half. This magically coerces the pizza to stand at attention, ready to be enjoyed.
But why does this work? Let us step back and ask the question: Why do pizzas droop in the first place? When we hold a slice of pizza by the crust, the force of gravity is pulling down on the pizza, the average location of the force at the center of mass. This force is far forward from the fixed point (human hand) that is holding the crust. This situation is analogous to a cantilever beam and results in a net bending moment that makes the pizza droop (annoyingly) downwards!
This makes sense, except how does crimping the pizza on the end help with this at all?? Crimping does not change the mass (or gravity of the Earth) so the magnitude of the weight force remains the same. The location of the center of mass hasn’t changed either (assuming you haven’t taken a bite out of it while reading this, of course), and the location of the hand hasn’t changed either. So, the net bending moment is the same too! What gives?!
There is another property that determines the net deflection on a cantilever beam: section properties specifically. That’s right, the shape of the beam affects how a beam bends under load. It is this property that explains why civil engineers love the I-beam so much. The I beam is really good at resisting bending moment because of its I shape. Specifically, the I shape has a high second moment of area (or principal moments of inertia of the area) property. If you look on this page, it gives the formulas for many cross section shapes. I looked for hours and was unable to find the equation for pizza. That’s not a problem though, because we can calculate it with SOLIDWORKS!
Here, I’ve made a you guys a slice of pizza! But this pizza isn’t for eating, its for science…and then eating. So how can we find the second moment of area of this pizza at a particular location? It is a section property, so we need to section it first! Click the section tool at the top of the heads-up view toolbar. I’ll place a section plane about an inch from the crust (no I didn’t eat it already)!
Now, the cross section here is basically a rectangle and can be manually calculated if you’re a nerd and like math but it can also be computed with SOLIDWORKS. Go to the Evaluate tab and click on “Section Properties”. A window not unlike mass properties appears. Select the section face and click “Recalculate”. SOLIDWORKS will display the results in that window.
Many numbers are displayed in the window, because SOLIDWORKS calculates the moments of inertia with respect to many axes at once, but the value we are looking for is listed as Ix = 3759 mm^4. This is the principal moment of inertia with respect to the x axis (the neutral axis).
Now, let’s apply our secret technique and then recalculate Ix! Here is my masterful crimp of this slice as well as its section:
That cross section looks wildly different, but let’s see what the numbers say!
We can see that, even though the cross-sectional area remained virtually the same at 925mm², the Principal Moment of Inertia of the Area Ix ballooned to over 130982mm^4, an increase of 35 times!
So that’s good and all, but how exactly does a large Principal Moment of Inertia correspond to resistance in bending for a cantilever pizza?
Let’s consider the equation that describes the maximum deflection in a cantilever beam under uniform distributed load where:
- delta (δ) is maximum deflection
- q is the uniform distributed load
- L is the length of the beam
- E is the Modulus of Elasticity (a material property)
- And I is the Principal Moment of Inertia of the Area.
The real take away is this: the equation for deflection is inversely dependent on the Moment of Inertia. In layman’s terms, when the value of Ix gets large, the value of the deflection diminishes…and that’s exactly what we like in our pizza. Thanks for following along!
Author: Robert Maldonado, Application Engineer at DesignPoint
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