10.1:
Moments of Inertia for Areas
The moment of inertia for areas is a geometrical property that measures a cross-section's resistance to bending due to its shape about an axis.
Suppose the cross-section is subdivided into smaller area elements. An element's area moment of inertia is the square of its distance from a reference axis times the elemental area.
Integrating over the total area gives the second moment of area about the axis, where the first moment is the integral of area times distance.
The moments of inertia of areas about the x and y axes, can be used to obtain the polar moment of inertia by defining a perpendicular distance from the pole to the element.
In a beam subjected to equal and opposite couples, the internal resisting forces are linearly distributed with distance, and particles far from the center exhibit more resisting force, resulting in a higher second moment.
The larger the second moment, the greater the beam's resistance to bending.
In an I beam, most particles are located as far as possible from the bending axis, making it an efficient cross-section.
10.1:
Moments of Inertia for Areas
The second moment of area, also known as the moment of inertia of an area, is a geometric property of a shape that reflects its resistance to change. The moment of inertia of an area is expressed in terms of a single number and can be calculated for both two-dimensional and three-dimensional shapes. The moment of inertia of an area is calculated by taking the sum of the product of the area and the square of its distance from a chosen axis of rotation. The moment of inertia is expressed in units of length raised to the fourth power (m4, mm4, ft4, in4). For two-dimensional shapes, the moment of inertia can be expressed as a single equation in terms of the x and y axes of the shape's coordinates.
The equation for the moment of inertia of an area with respect to the x-axis is as follows:
where A is the shape's area, and y is the distance from the x-axis. A similar equation can be written for the y-axis:
For three-dimensional shapes, the moment of inertia can be expressed in a polar form, which takes into account the distance from the origin of the shape, as well as the angle of rotation around the origin. The equation for the moment of inertia of an area in polar form is as follows:
where r is the distance from the origin. The moment of inertia of an area has many practical applications in engineering, such as in the design of structures and machines. For example, when designing a bridge, engineers must consider the moment of inertia of the bridge's cross-section to ensure that it is strong enough to withstand the forces of an earthquake. Similarly, when designing a car engine, engineers must consider the moment of inertia of the pistons and other moving parts to ensure that they can handle the forces produced by the engine's combustion. In conclusion, the moment of inertia of an area is an important concept in mechanics and engineering, as it is used to calculate the stresses and strains on an object when the force of an applied torque is known.
Suggested Reading
- Meriam, J. L., Kraige, L. G., and Bolton, J.N. (2016). Engineering mechanics Volume 1 Statics Eighth edition, Wiley pp 434 – 436.
- Beer, F. P., Johnston Jr, E. R., Mazurek, D. F., Cornwell, and P. J., Sself, B. P. (2016) Vector mechanics for engineers Statics and dynamics Eleventh edition, McGraw Hill pp 487 – 488.
- Hibbeler, R. C. (2016). Engineering mechanics Statics Fourteenth edition, Pearson pp 529 – 530.