Second Moment of Inertia

IntroductionHave you ever wondered why many structural elements like beams aren't commonly manufactured as fully-filled simple shapes like circles or squares?  Well, it's all about how efficiently a cross-section resists bending...and that's got a lot to do with a shapes Second Moment of Inertia (I).  
A column or a beam must typically contend with biaxial moments and flexural and shear forces. A member requires a minimum stiffness or, analogously, a minimum cross-sectional moment of inertia to resist these forces and avoid buckling or deflecting.  
So what exactly is the Second Moment of Inertia (I), what determines it, and how can engineers use this to their advantage? Let's take a look at an example to understand it fully.
Suppose you have a rectangular plank of wood and are using it to cross a void. You can orientate it flat, as shown on the right below, or rotate it vertically onto its side, as shown on the left.
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Intuitively, you know that the arrangement on the left will perform better,  but the plank cross-sections and the area are the same, and so are its material properties. So what's changed? Well, by rotating the plank onto its side, its geometry and resultant moment of inertia have changed completely.
Some cross-sections are better at resisting bending than others, and the orientation of a shape has a major influence on a cross-section's ability to resist bending. This is precisely what the moment of inertia or the second moment of the area describes. When quantified, it measures how much resistance the cross-section has to bend. 
Let's unpack this example further by looking at the cross-sectional geometry in 2D.
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The further the area of a material is concentrated away from its bending axis, also known as the neutral axis (N.A.), the stiffer the section is and the higher its moment of inertia will be, relative to that axis. Just by orientating an identical wooden plank on its side (left), the cross-section has more material located away from the (N.A.), thereby improving its resistance to bending. 
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Take any small area in that plank of wood, it will either experience compressive or tension stresses, which increase linearly as you move up or down from the N.A. It will try to resist bending when subjected to a load by creating a moment around the neutral axis. This moment exerted increases linearly as you move away from the N.A. This means that the resistance to bending provided by any point in the cross-section is directly proportional to the distance from the neutral axis squared. Once you integrate the force times the distance from the N.A. on the entire surface, you get the total resistance to bending specific to the axis specified.
This is a critical phenomenon that we can use to our advantage. Let's take it further and add an I beam into the mix:
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An I-beam maximizes the amount of material located far away from the NA. This gives it a bigger moment of inertia and greater stiffness for the same mass of material. As a result, it performs most efficiently in bending out of all three cross-sections shown. This is a reason why I-sections are so commonly seen in construction.
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While the moment of Inertia (I) is a cross-sectional property that partially determines a section's resistance to bending and deflection, it also has many uses in other engineering applications.
The moment of inertia is used in beam and column analysis, calculating bending stresses and flexural rigidity. It is also important to note that it is not a fixed value that is unique to each cross-section; it changes depending on the location of the bending (or reference) axis. In general, most engineers will take the neutral axis x-x and y-y, located at the centroid of a cross-section profile, as the reference axis used to calculate the second moment of inertia.
How do you calculate the Second Moment of Inertia (I)?Now that you understand what factors influence a cross-section's Second Moment of Inertia, we will outline the general steps required to calculate the moment of inertia.
1.  Understand the fundamental mathematical expressionsThe Second Moment of Inertia is synonymous with the second moment of the area because it is calculated by taking the moment about the area twice: 
'I' is expressed mathematically as:
Ix = ∫ Ay2dA
Iy = ∫ Ax2dA
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Where,
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dA=area consideredy=perpendicular distance taken from the X−axis to area dAx=perpendicular distance taken from the Y−axis to area dAUnits=Units4 i.e,mm4,m4,in4dA=\text{area considered}\\ y=\text{perpendicular distance taken from the X−axis to area dA}\\ x=\text{perpendicular distance taken from the Y−axis to area dA}\\ Units=Units^4\ i.e, mm^4, m^4, in^4dA=area consideredy=perpendicular distance taken from the X−axis to area dAx=perpendicular distance taken from the Y−axis to area dAUnits=Units4 i.e,mm4,m4,in4
Now, you could explore the next steps in calculating the second moment of inertia by hand (which we recommend you do). Still, in case you find yourself pressed for time, we have compiled a summary of second moment of inertia formulas for common shapes and beam sections:
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Table 1: Second moment of Area formulas for common shapes
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2.  Break the cross-section up into segmentsWhen calculating the Moment of Inertia, we must calculate the Moment of Inertia of smaller segments. Try to break them into simple rectangular sections or other sections for which the formula already exists. Here we can break this I-beam into three components:
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Or this hollow rectangle into an equivalent outer rectangle minus the inner:
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3. Find the Neutral Axis (NA)    The Neutral Axis (N.A.) or the horizontal x-x axis is located at the centroid or center of mass. Most engineering cross-sectional shapes are symmetrical, so you will only need to find the x-x neutral axis.
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Cx=A1x1+A2x2+...AnxnA1+A2+...An=1A∫Ax.dAwhereA=area of a segment x=vertical distance to the centroid of the segment\large{Cx=\frac{A_1x_1+A_2x_2+...A_nx_n}{A_1+A_2+...A_n} =\frac{1}{A}\int_{A}{x.dA}} \\ \small{\text{where}} \\ A = \text{area of a segment } \\ x = \text{vertical distance to the centroid of the segment}Cx=A1​+A2​+...An​A1​x1​+A2​x2​+...An​xn​​=A1​∫A​x.dAwhereA=area of a segment x=vertical distance to the centroid of the segment
4.  Calculate the Moment of Inertia (I)Now let's combine the segments using the parallel axis theorem with the formula:
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Itotal=∑(Ii+Aidi2)whereIi=The moment of inertia of the individual segment about its own centroid axisAi=The area of the individual segmentdi=The vertical distance from the centroid of the segment to the Neutral Axis (N.A)\large{I_{total}=∑(I_i+A_id_i^2)} \\ \small{\text{where}} \\ \small{I_i=\text{The moment of inertia of the individual segment about its own centroid axis}} \\ A_i= \text{The area of the individual segment}\\ d_i=\text{The vertical distance from the centroid of the segment to the Neutral Axis (N.A)}\\Itotal​=∑(Ii​+Ai​di2​)whereIi​=The moment of inertia of the individual segment about its own centroid axisAi​=The area of the individual segmentdi​=The vertical distance from the centroid of the segment to the Neutral Axis (N.A)
Looking AheadThe second moment of inertia is one of the most important material properties of structural elements because it determines an engineer's section selections. Since this value is used to predict the resistance of beams to bending, torsion, and deflection, you'll see it pop up in quite a few engineering concepts and formulas, namely:
Bending stress: the stress that an object encounters when it is subjected to a large load at a particular point, causing it to bend and become fatigued.
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σb=MyIσb-bending stressM- Calculated bending momenty−Vertical distance away from the neutral axisI−Moment of inertia around the neutral axisσ_b=  {\frac{M_y}{I}} \\\\σ_b \text {-bending stress} \\ M \text {- Calculated bending moment} \\y-\text{Vertical distance away from the neutral axis}\\ I-\text{Moment of inertia around the neutral axis}σb​=IMy​​σb​-bending stressM- Calculated bending momenty−Vertical distance away from the neutral axisI−Moment of inertia around the neutral axis
Deflection: Deflection is also known as displacement and can occur from externally applied loads or the structure's self-weight. If the deflection is large enough, you can visualize the degree to which an element/structure changes shape or is displaced from its original position- this is something we want to avoid at all costs!
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