The polar moment of inertia is used to calculate shear stress and angular deflection: t = Taking the polar moment of inertia about the centroid translates the formula into: The polar moment of inertia is used in torsional strength and deflection problems. When both reference axes are asymmetrical, the values are found using: If either the X or Y axis is symmetrical, the product of inertia is zero. In this case, the product of inertia is used to calculate the maximum and minimum moments of inertia and the skew angle of the principal axes. When a beam is asymmetrical, the maximum and minimum moments of inertia are not around the X-Y axes. Formulas for these properties originally contain integrals but simplify to algebraic expressions when they’re referenced to the centroid. When the Lisp program calculates these fundamental sectional properties, they are used to find the product of inertia, polar moment of inertia, and radius of gyration. However, the moment of inertia is very useful when transferred to the centroid using the parallel axis theorem. The moment of inertia about an arbitrary set of axes is not useful in most strength calculations. (e.g., simply supported beam midspan load).
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Once the moment of inertia is transferred to the centroid, the remaining properties are calculated using: The final output locates and draws the centroidal X and Y axes on the screen after transferring the moment of inertia from the X-Y axes to the centroid using the parallel axis theorem. The results from the numerical integration calculations are the CAD program’s global Cartesian coordinates for centroid and moment of inertia about the X and Y axes. Sectional property formulas for a rectangle are: The differential rectangular properties are then referenced to the Y axis, at X = 0, using the parallel axis theorem. Since each differential element forms a rectangle, and formulas exist for determining rectangular sectional properties, these properties are determined about the centroid of each differential element. While the same process can be applied to the Y axis, using the X-axis data to calculate Y-axis properties saves execution time. Both values are found relative to the CAD program’s X axis, at Y = zero, using the equation: The moment of inertia is a measure of a shape’s ability to resist bending.
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The centroid is a sectional property used in bending stress calculations to help locate the neutral axis where normal (axial) stresses due to bending are zero. Next, the program finds the centroid and moment of inertia. Once all differential elements have been defined using the hatch command, the CAD program sums them as follows: By making the height of the differential element small enough, numerical integration meets accuracy requirements. The approach uses rectangular differential elements, a shape of known sectional properties, where height is the spacing between hatch lines and width is the length of the hatch line. This minimizes the number of hatch lines and reduces the number of program accesses to the drawing database during calculation. To quicken the execution, hatching is done in one direction only. Instead, the approach makes use of each hatch line as a completely independent entity. Therefore, the approach does not use conventional square differential elements. This is because the differential areas are essentially impossible to identify from the hatch pattern and are not always square. An initial step is to let the CAD program crosshatch the area and then extract the hatch information in a form that’s useful for additional calculations.Īfter examining the random methods that AutoCAD uses to hatch areas, it becomes apparent that the classical method of summing square differential areas in an X and Y direction is impractical. The challenge is writing a program to calculate sectional properties using standard AutoCAD hatching commands.īecause all other sectional properties can be determined once you find the area, centroid, and first and second moments of area, the task is to calculate these values first. One way to write such a program is with AutoCAD’s AutoLisp language, which directly accesses a drawing’s database. Programs can be written within CAD packages to find sectional properties for any cross section.
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But when shapes are complex, more advanced methods are required to determine the necessary values. In fact, formulas for common cross sections like I-beams and other shapes are found in most mechanical design reference books. In the mechanical design of equipment and systems, sectional properties, such as centroid and moment of inertia, are necessary for calculating stresses, deflections, and buckling.