HOME

TheInfoList



OR:

The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area is typically denoted with either an I (for an axis that lies in the plane of the area) or with a J (for an axis perpendicular to the plane). In both cases, it is calculated with a
multiple integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
over the object in question. Its dimension is L (length) to the fourth power. Its
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
of dimension, when working with the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
, is meters to the fourth power, m4, or inches to the fourth power, in4, when working in the Imperial System of Units. In
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
, the second moment of area of a beam is an important property used in the calculation of the beam's
deflection Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflection ...
and the calculation of
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
caused by a
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
applied to the beam. In order to maximize the second moment of area, a large fraction of the
cross-sectional area In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The ...
of an
I-beam An I-beam, also known as H-beam (for universal column, UC), w-beam (for "wide flange"), universal beam (UB), rolled steel joist (RSJ), or double-T (especially in Polish, Bulgarian, Spanish, Italian and German), is a beam with an or -shap ...
is located at the maximum possible distance from the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
of the I-beam's cross-section. The planar second moment of area provides insight into a beam's resistance to bending due to an applied moment,
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
, or distributed load perpendicular to its
neutral axis The neutral axis is an axis in the cross section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the ne ...
, as a function of its shape. The polar second moment of area provides insight into a beam's resistance to torsional deflection, due to an applied moment parallel to its cross-section, as a function of its shape. Different disciplines use the term ''
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
'' (MOI) to refer to different moments. It may refer to either of the planar second moments of area (often I_x = \iint_ y^2\, dA or I_y = \iint_ x^2\, dA, with respect to some reference plane), or the polar second moment of area ( I = \iint_ r^2\, dA , where r is the distance to some reference axis). In each case the integral is over all the infinitesimal elements of ''area'', ''dA'', in some two-dimensional cross-section. In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, ''moment of inertia'' is strictly the second moment of mass with respect to distance from an axis: I = \int_ r^2 dm , where ''r'' is the distance to some potential rotation axis, and the integral is over all the infinitesimal elements of ''mass'', ''dm'', in a three-dimensional space occupied by an object . The MOI, in this sense, is the analog of mass for rotational problems. In engineering (especially mechanical and civil), ''moment of inertia'' commonly refers to the second moment of the area.


Definition

The second moment of area for an arbitrary shape  with respect to an arbitrary axis BB' is defined as J_ = \iint_ ^2 \, dA where * dA is the infinitesimal area element, and * \rho is the perpendicular distance from the axis BB'. For example, when the desired reference axis is the x-axis, the second moment of area I_ (often denoted as I_x) can be computed in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
as I_ = \iint_ y^2\, dx\, dy The second moment of the area is crucial in Euler–Bernoulli theory of slender beams.


Product moment of area

More generally, the product moment of area is defined as I_ = \iint_ yx\, dx\, dy


Parallel axis theorem

It is sometimes necessary to calculate the second moment of area of a shape with respect to an x' axis different to the centroidal axis of the shape. However, it is often easier to derive the second moment of area with respect to its centroidal axis, x, and use the parallel axis theorem to derive the second moment of area with respect to the x' axis. The parallel axis theorem states I_ = I_x + A d^2 where * A is the area of the shape, and * d is the perpendicular distance between the x and x' axes. A similar statement can be made about a y' axis and the parallel centroidal y axis. Or, in general, any centroidal B axis and a parallel B' axis.


Perpendicular axis theorem

For the simplicity of calculation, it is often desired to define the polar moment of area (with respect to a perpendicular axis) in terms of two area moments of inertia (both with respect to in-plane axes). The simplest case relates J_z to I_x and I_y. J_z = \iint_ \rho^2\, dA = \iint_ \left(x^2 + y^2\right) dA = \iint_ x^2 \, dA + \iint_ y^2 \, dA = I_x + I_y This relationship relies on the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
which relates x and y to \rho and on the
linearity of integration In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
.


Composite shapes

For more complex areas, it is often easier to divide the area into a series of "simpler" shapes. The second moment of area for the entire shape is the sum of the second moment of areas of all of its parts about a common axis. This can include shapes that are "missing" (i.e. holes, hollow shapes, etc.), in which case the second moment of area of the "missing" areas are subtracted, rather than added. In other words, the second moment of area of "missing" parts are considered negative for the method of composite shapes.


Examples

See
list of second moments of area The following is a list of second moments of area of some shapes. 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 ...
for other shapes.


Rectangle with centroid at the origin

Consider a rectangle with base b and height h whose
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
is located at the origin. I_x represents the second moment of area with respect to the x-axis; I_y represents the second moment of area with respect to the y-axis; J_z represents the polar moment of inertia with respect to the z-axis. \begin I_x &= \iint_ y^2\, dA = \int^\frac_ \int^\frac_ y^2 \,dy \,dx = \int^\frac_ \frac\frac\,dx = \frac \\ I_y &= \iint_ x^2\, dA = \int^\frac_ \int^\frac_ x^2 \,dy \,dx = \int^\frac_ h x^2\, dx = \frac \end Using the
perpendicular axis theorem The perpendicular axis theorem (or plane figure theorem) states that the moment of inertia of a planar lamina (i.e. 2-D body) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about t ...
we get the value of J_z. J_z = I_x + I_y = \frac + \frac = \frac\left(b^2 + h^2\right)


Annulus centered at origin

Consider an
annulus Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
whose center is at the origin, outside radius is r_2, and inside radius is r_1. Because of the symmetry of the annulus, the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
also lies at the origin. We can determine the polar moment of inertia, J_z, about the z axis by the method of composite shapes. This polar moment of inertia is equivalent to the polar moment of inertia of a circle with radius r_2 minus the polar moment of inertia of a circle with radius r_1, both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius r with respect to the origin. In this case, it is easier to directly calculate J_z as we already have r^2, which has both an x and y component. Instead of obtaining the second moment of area from
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
as done in the previous section, we shall calculate I_x and J_z directly using
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
. \begin I_ &= \iint_ y^2\,dA = \iint_ \left(r\sin\right)^2\, dA = \int_0^\int_0^r \left(r\sin\right)^2\left(r \, dr \, d\theta\right) \\ &= \int_0^\int_0^r r^3\sin^2\,dr \, d\theta = \int_0^ \frac\,d\theta = \fracr^4 \\ J_ &= \iint_ r^2\, dA = \int_0^\int_0^r r^2\left(r\,dr\,d\theta\right) = \int_0^\int_0^r r^3\,dr\,d\theta \\ &= \int_0^ \frac\,d\theta = \fracr^4 \end Now, the polar moment of inertia about the z axis for an annulus is simply, as stated above, the difference of the second moments of area of a circle with radius r_2 and a circle with radius r_1. J_z = J_ - J_ = \fracr_2^4 - \fracr_1^4 = \frac\left(r_2^4 - r_1^4\right) Alternatively, we could change the limits on the dr integral the first time around to reflect the fact that there is a hole. This would be done like this. \begin J_ &= \iint_ r^2 \, dA = \int_0^\int_^ r^2\left(r\, dr\, d\theta\right) = \int_0^\int_^ r^3\, dr\, d\theta \\ &= \int_0^\left frac - \frac\right, d\theta = \frac\left(r_2^4 - r_1^4\right) \end


Any polygon

The second moment of area about the origin for any simple polygon on the XY-plane can be computed in general by summing contributions from each segment of the polygon after dividing the area into a set of triangles. This formula is related to the shoelace formula and can be considered a special case of
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orie ...
. A polygon is assumed to have n vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct. \begin I_y &= \frac\sum_^ \left( x_i y_ - x_ y_i\right)\left( x_i^2 + x_i x_ + x_^2 \right) \\ I_x &= \frac\sum_^ \left( x_i y_ - x_ y_i\right)\left( y_i^2 + y_i y_ + y_^2 \right) \\ I_ &= \frac\sum_^ \left( x_i y_ - x_ y_i\right) \left( x_i y_ + 2 x_i y_i + 2 x_ y_ + x_ y_i \right) \end where x_i,y_i are the coordinates of the i-th polygon vertex, for 1 \le i \le n. Also, x_, y_ are assumed to be equal to the coordinates of the first vertex, i.e., x_ = x_1 and y_ = y_1.


See also

*
List of second moments of area The following is a list of second moments of area of some shapes. 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 ...
* List of moments of inertia *
Moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
*
Parallel axis theorem The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body abou ...
*
Perpendicular axis theorem The perpendicular axis theorem (or plane figure theorem) states that the moment of inertia of a planar lamina (i.e. 2-D body) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about t ...
*
Radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...


References

{{Commons category, Second moments of area Geometry Structural analysis Physical quantities Moment (physics)