Using interactive methods for teaching principal axes and moments of inertia

Abstract

Engineering analysis and design often uses properties of plane sections in calculations. For example, in stress analysis of a beam under bending and torsional loads, you use the cross-sectional properties to determine the stress and displacement distributions in the beam cross section. In calculating the natural frequencies and shapes of a machine part, you also need to know the area, centroid, and various moments and products of inertia of a cross section or a composed cross section. The centroid, moments and products of inertia of an area made up of several common shapes (rectangles usually), may thus be obtained by adding the moments of inertia of the component areas. However the parallel-axes theorems should be used to transfer each moment of inertia to the desired axis. This procedure is used to calculate the second moments of structural shapes (U-shape, L-shape, C-shape,…) because the determination of their moments of inertia is necessary for design structural members. Many engineering students are introduced to the ideas and concepts of Mohr’s Circle when studying stress states due to various loading conditions on structures or machines. Mohr’s circle provides a visual representation of what is happening, and the positioning of the stress states on an element relative to a set of coordinate axes. In this paper we propose the use of interactive methods for teaching this part of Mechanics. A set of simulations designed with the Modellus 2.5 software are used for calculating centroids, moment and product of inertia and their properties. Mohr´s circles are also drawn with interactive simulations to obtain the principal axes and moments of inertia of structural shapes. For solving problems, students can interact in the simulation, by changing shapes, size, angles, etc.