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 $E$. $I$ appears in the equation that gives the critical load at which a column will buckle. The area moment of inertia is an important parameter for any application that which involves bending of a structural member, which means it appears constantly in the analysis of beams and columns. Applications of the Area Moment of Inertia $$I_x = b\left[ \frac$ in the example above) correspond to the correct bending axis, or else you will first need to apply the parallel axis theorem to get $I$ values for the correct axis. To calculate $I_x$ all we have to do is integrate from the bottom of the rectangle at $y = -h/2$ to the top of the rectangle at $y = h/2$. Because the $y$ term is squared, the strips further away from the bending axis (the $x$ axis) contribute much more to $I$ than those close to the axis. This is why we are integrating – to calculate the effect of all of these really small strips. $I_x$ is given by the following equation:Įach strip contributes to the area moment of inertia. 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. 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.
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