![]() ![]() ![]() The plank will be much less stiff when the load is placed on the longer edge of the cross-section. This is because resistance to bending depends on how the material of the cross-section is distributed relative to the bending axis. The plank on the left has more material located further from the bending axis, which makes it much stiffer. The same plank is much less stiff when the load is applied to the long edge of the cross-section. This resistance to bending can be quantified by calculating the area moment of inertia of the cross-section.Īs we will soon see, this is related to the area moment of inertia. It is denoted using the letter $I$, has units of length to the fourth power, which is typically $mm^4$ or $in^4$. It reflects how the area of the cross section is distributed relative to a particular axis. It’s not a unique property of a cross section – it varies depending on the bending axis that is being considered. 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. 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. $I_x$ is given by the following equation:Įach strip contributes to the area moment of inertia. This is why we are integrating – to calculate the effect of all of these really small strips. ![]() 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. 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 $y = h/2$. ![]()
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