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This is why transmission shafts are hollow – the central portion of the cross section doesn’t contribute much to the torsional resistance, so it’s more efficient to use hollow shafts.Īn interesting observation we can make is that the polar moment of inertia about an axis passing through a specific point on the cross-section is equal to the sum of the area moments of inertia for two perpendicular axes that pass through the same point. Like with $I_x$ and $I_y$, the $\rho$ term is squared in the equation, which means that areas of the cross-section that are located far from the axis of rotation contribute most to the value of $J$. The polar moment of inertia is calculated based on the distribution of the area of the cross-section relative to a twisting axis ($z$ in the image above) The $\rho$ term is the distance from the $z$ axis (which is pointing out of the screen in the image below) to an element dA. $J$ accounts for how the area of the cross-section is distributed radially relative to the rotation or twisting axis $z$. It is often used in problems involving torsional deformation, which is the twisting of a beam or shaft. This is called the polar moment of inertia, and it is usually denoted either using the letter $J$ or as $I_z$. In addition to calculating the area moment of inertia for $x$ and $y$ axes that are in the same plane as the cross-section, we can also calculate the area moment of inertia for an axis this is perperdicular to the cross-section. The flexural rigidity $EI$ represents the total stiffness of the cross-section. $I$ represents the stiffness of a beam cross-section due to its geometry, and $E$ represents the stiffness of the cross-section due to its material. The term $EI$ is given the name flexural rigidity. If you’re interested in learning more about these applications, check out the beam deflection and buckling pages.Īs demonstrated by the two examples shown above, the area moment of inertia often appears in equations alongside Young’s modulus $E$. $I_x$ is given by the following equation: $$I_x = \int$$ The area of one strip is: $$dA = b \cdot dy$$ This is one of the reasons the I-beam is such a commonly used cross-section for structural applications – most of the material is located far from the bending axis, which makes it very efficient at resisting bending whilst using a minimal amount of material. Let’s compare $I$ values calculated for a few different cross-sections, for the bending axis shown below: Area moment of inertia values (in mm 4) for three shapesĬross-sections that locate the majority of the material far from the bending axis have larger moments of inertia – it is more difficult to bend them. It’s not a unique property of a cross section – it varies depending on the bending axis that is being considered. It reflects how the area of the cross section is distributed relative to a particular axis. It is denoted using the letter $I$, has units of length to the fourth power, which is typically $mm^4$ or $in^4$. This resistance to bending can be quantified by calculating the area moment of inertia of the cross-section. As we will soon see, this is related to the area moment of inertia. The same plank is much less stiff when the load is applied to the long edge of the cross-section. The plank on the left has more material located further from the bending axis, which makes it much stiffer. This is because resistance to bending depends on how the material of the cross-section is distributed relative to the bending axis. The plank will be much less stiff when the load is placed on the longer edge of the cross-section.
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Video can’t be loaded because JavaScript is disabled: Understanding the Area Moment of Inertia ()Ĭonsider a thin plank that supports a 100 kg load.