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The moments of inertia of a mass have units of dimension ML 2 ([mass] × [length] 2). It should not be confused with the second moment of area, which has units of dimension L 4 ([length] 4) and is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia or sometimes as the angular mass.
The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area. Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused.
Flexural rigidity of a plate has units of Pa·m 3, i.e. one dimension of length less than the same property for the rod, as it refers to the moment per unit length per unit of curvature, and not the total moment. I is termed as moment of inertia. J is denoted as 2nd moment of inertia/polar moment of inertia.
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory .
The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis. The unit of dimension of the second moment of area is length to fourth power, L 4, and should not be confused with the mass moment of inertia.
Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue) The defining feature of beams is that one of the dimensions is much larger than the other two. A structure is called a plate when it is flat and one of its dimensions is much smaller than the other two. There are several ...
The governing equations for the dynamics of a Kirchhoff-Love plate are , = ¨, + (,) = ¨ ¨, where are the in-plane displacements of the mid-surface of the plate, is the transverse (out-of-plane) displacement of the mid-surface of the plate, is an applied transverse load pointing to (upwards), and the resultant forces and moments are defined as
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draw on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions. [1] The typical thickness to width ratio of a plate structure is less than 0.1.