Radius of gyration

In structural engineering the Radius of Gyration is used to describe the distribution of cross sectional area in a column around its centroidal axis. • The structural engineering radius of gyration can be expressed as r = (I / A) 1/2 (1) where r = radius of gyration (m, mm, ft, in...) I = 4, mm 4, ft 4, in 4 ..) A = cross sectional area (m 2, mm 2, ft 2, in 2...) Some typical Sections and their Radius of Gyration Rectangle - with axis in center Radius of Gyration for a rectangle with axis in center can be calculated as r max = 0.289 h (1) where r max =max radius of gyration (strong axis moment of inertia) Rectangle - with excentric axis Radius of Gyration for a rectangle with excentric axis can be calculated as r = 0.577 h (2) Rectangle - with tilted axis Radius of Gyration for a rectangle with tilted axis can be calculated as r = b h / (6 (b 2 + h 2)) 1/2 (3) Rectangle - with tilted axis II Radius of Gyration for a rectangle with tilted axis can be calculated as r = (((h 2 + cos 2a) + (b 2 sin 2 a)) / 12) 1/2 (4) Hollow Square Radius of Gyration for a hollow square can be calculated as r = ((H 2 + h 2) / 12) 1/2 (5) Hollow Square - with tilted axis Radius of Gyration for a hollow square with tilted axis can be calculated as r = ((H 2 + h 2) / 12) 1/2 (6) Equilateral Triangle with excentric axis Radius of Gyration for a equilateral triangle can be calculated as r = h / (18) 1/2 (7) Triangle Radius of Gyration for a equilateral triangle can be calculated as r = h / (6) 1/2 ...

The radius of gyration or gyroradius of anybody about the axis of rotation is defined as that radial distance from the axis of rotation, at which the entire mass of that body is concentrated. Thus, the point will have a moment of inertia at this particular point also. If we describe the radius of gyration in mathematics, then we can say that the radius of gyration is the root -mean square distance of the part of the object from either its centre of mass or given axis. This depends upon the relevant application. This is actually called the perpendicular distance from the point mass to the axis of rotation. The radius of gyration can be used to characterize the particular distance travelled by a point. This is also referred to as the measure of the way in which the mass of a rotating rigid body can be distributed about its axis of rotation. The radius of Gyration of Thin Rod The moment of inertia (MOI) about any given axis will be the same as the actual body mass distribution. When seen in terms of mathematics, the “radius of gyration” is denoted as the square root of the mean square distance of parts of the object from the middle region of body mass or a specified axis that depends on the appropriate application. In other words, the radius of gyration is calculated as the perpendicular distance noted from the rotational axis to the point mass. The actual radial distance between the rotational axis and the point where the body mass is joined to it keeps the inertia of a rota...

The probability distribution of the radius of gyration of unbranched polymer chains is discussed. The characteristic function of the distribution is presented in closed form, and calculations are the made of the semi‐invariants, moments, and behavior of the probability at very large and very small values of the radius of gyration.

\(\require \) The radius of gyration can be thought of as the radial distance to a thin strip which has the same area and the same moment of inertia around a specific axis as the original shape. Compared to the moment of inertia, the radius of gyration is easier to visualize since it’s a distance, rather than a distance to the fourth power.

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When viewing the radius of gyration of a sphere, and trying to verify my own calculations $R_g^2=\frac25R^2$ against the $R_g^2=\frac35R^2$ and does not use the squared moment arm $r\sin\theta$ if rotating about $z-$axis. EDIT The derivation should br trivial in spherical coordinate system with the volume element $\mathrm + c$. Requested from Comments that I should present my own calculations. I checked $\frac25R^2$. I was puzzled and wondered to write to the author, but no contact information can be found. What tricks are hidden inside his derivation or some common definition rule I am not aware of?

This is a property of a section. It is also a function of the second moment of area. The radius of gyration gives the stiffness of a section. It is based on the shape of the cross-section. Normally, we use this for compression members such as a column. As shown in the diagram, the member bends in the thinnest plane. Using the radius of gyration, we can compare the behavior of various structural shapes under compression along an axis. It can be used to predict buckling in a compression member such as a column. The Formula for the Radius of Gyration – r Where I = the second moment of area A = cross-sectional area of the member The unit of measurement is mm. The smallest value of the radius of gyration is considered for the calculations of the structural stiffness of the member. That is the plane in which the member is most likely to fail or buckle. Square or circular shapes have the same radius of gyration about any plane. There is no smallest value. Therefore, these sections are ideal selections for columns. Calculating the radius of gyration The plan view of a column is shown below. First, we have to calculate the I value about x-x and y-y axes. I xx = 33.3 x 10 6 mm 4 I yy = 2.08 x 10 6 mm 4 A = cross sectional area = 50 mm x 200 mm = 10,000 mm 2 Substituting the value of I xx and cross-sectional area A in the above formula we can get r xx. This is the value of the r about the x-x axis. This is the value of the r about the y-y axis. Since r yy is smaller, probable failure...