To simplify the equations of equilibrium of an elastic body under bending, the stress function, which may be viewed as a potential for the stress components, is introduced. This shows a circle with a unit radius and a number of options for the Mathematica function Plot. Using the cross-section plotting function CrossSectionPlot, introduced in the torsional analysis package TorsionAnalysis, you can visualize a number of cross sections included in the package BeamAnalysis. For example, you can draw pictures of a cross section and a deflected beam by using the function CrossSectionPlot from the torsional analysis package TorsionAnalysis and the function BendingPlot from the BeamAnalysis package. A number of two- and three-dimensional graphical functions are also available to generate illustrative representations of deflected beams under bending loads. Structural Mechanics discusses closed-form solutions for the following set of beam cross sections:įor these cross sections, you can calculate the bending stress function, bending stresses, and the deflection of the center line of a beam. This stress function approach is adapted to obtain the stress and displacement distribution for various beam cross sections. The analytical solutions are obtained by using the Saint-Venant's semi-inverse method. The type of beam under consideration is also known as the Timoshenko beam due to the assumptions made in generating the equations of equilibrium for a beam. This chapter considers the bending of a static cantilever beam of a constant cross section by a force at the end of the beam. In designing engineering structures, such as buildings and bridges, cantilever beams are a main structural element receiving bending forces. Finance, Statistics & Business Analysis.Wolfram Knowledgebase Curated computable knowledge powering Wolfram|Alpha. Wolfram Universal Deployment System Instant deployment across cloud, desktop, mobile, and more. There are two different methods for finding the moment of inertia of any object, that is, the parallel axis theorem and the other one is the perpendicular axis theorem.Wolfram Data Framework Semantic framework for real-world data. However, when we change the location of the axis of rotation the formula as well as the value of the moment of inertia of a rectangle changes with it. To sum up, the formula for finding the moment of inertia of a rectangle is given by I=bd³ ⁄ 3, when the axis of rotation is at the base of the rectangle. H is the depth and b is the base of the rectangle. In this case, the formula for the moment of inertia is given as, The variables are the same as above, b is the width of the rectangle and d is the depth of it.įormula when the axis is passing through the centroid perpendicular to the base of the rectangle When the axis is passing through the base of the rectangle the formula for finding the MOI is, The formula for finding the MOI of the rectangle isĭ = depth or length of the rectangle Formula when the axis is passing through the base of the rectangle When the axis of rotation of a rectangle is passing through its centroid. Formula when the axis is passing through the centroid Let us see when we change the axis of rotation, and then how the calculation for the formula changes for it. Therefore, the equation or moment of inertia of a rectangular section having a cross-section at its lower edge as in the figure above will be, Similar to mathematical derivations, as we found the MOI for the small rectangular strip ‘dy’ we’ll now integrate it to find the same for the whole rectangular section about the axis of rotation CD. If we see the area of a small rectangular strip having width ‘dy’ will beĪnd the moment of inertia of this small area dA about the axis of rotation CD according to a simple moment of inertial formula which is And after finding the moment of inertia of the small strip of the rectangle we’ll find the moment of inertia by integrating the MOI of the small rectangle section having boundaries from D to A. Involvement of this ‘dy’ will make the assumptions and calculations easier. Now, let us find the MOI about this line or the axis of rotation CD.Īlso, consider a small strip of width ‘dy’ in the rectangular section which is at a distance of value y from the axis of rotation. Consider the line or the edge CD as the axis of rotation for this section. Where b is the width of the section and d is the depth of the section. Consider a rectangular cross-section having ABCD as its vertices.
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