10.6:
Moments of Inertia for Composite Areas
Composite areas comprise multiple basic shapes, such as rectangles, triangles, and circles, connected in some way.
Calculating the second moment of area for a composite shape involves subdividing it into basic shape components. The centroid for each component is calculated, from which the centroid for the composite can be obtained.
Next, the second moment of area for each component about a reference axis is calculated using the parallel axes theorem.
Finally, the summation of moments for each section gives the moment of inertia for the composite area.
For example, an L-shaped beam comprises two perpendicular rectangles. The centroid for each rectangular section is used to calculate the centroid for the beam.
The second moments of area for the rectangles about the respective centroidal axes are known.
The distance from each rectangle's centroid to the beam's centroid is substituted in the parallel axis theorem to estimate its second moment of area about the reference axis.
The summation of the moments for both rectangles gives the second moment of area for the beam about the reference axis.
10.6:
Moments of Inertia for Composite Areas
Composite areas are structures with multiple basic shapes connected in some way. These shapes usually include rectangles, triangles, circles, and other basic shapes that are connected in such a way as to form a single structure. Calculating the second moment of area for a composite area is essential when trying to understand the structure's overall stiffness.
The second moment of area, also known as the moment of inertia, measures a structure's resistance to bending. It is calculated by determining the distribution of the structure's area around an axis called the reference axis. For a composite area, finding the second moment of area involves splitting the composite shape into its fundamental components, finding their respective centroids, calculating the second moments of area along the reference axes for each component using the parallel axes theorem, and summing these moments to obtain the total.
To calculate the second moment of area for a composite area, such as an L beam, several steps must be taken. The first step involves identifying the centroids of the individual, perpendicular rectangular sections, followed by using these centroids to calculate the centroid for the entire beam. Next, the second moment of area for each rectangular section is determined by multiplying the width by the height cubed and dividing the result by twelve.
To obtain the total second moment of area for the beam, the distance from each rectangular section's centroid to the beam's centroid is substituted into the parallel axis theorem formula. This formula facilitates the calculation of each rectangular section's second moment of area along the reference axis by utilizing the distance between the section's centroid and the beam's centroid. Once the second moment of area for each rectangular section has been determined, it is necessary to add them together to obtain the beam's total second moment of area. This value is critical in assessing the beam's ability to withstand bending stress and must be accurately estimated for safe and effective design.
Suggested Reading
- R. C. , Hibbeler Engineering Mechanics Statics, Pearson. Pp. 540
- F.P. Beer, E.R. Johnston, D.F. Mazurek, P.J. Cornwell, B.P. Self, Vector Mechanics For Engineers Statics and Dynamics Engineering Mechanics Statics, Mc Graw-Hill Education. Pp. 499-501