View Full Version : Summary of discussion on the first moment of an area

Anwar Upal
05-20-2002, 10:16 AM
This posting is a summary of the replies which I received in response to
the following question:

>The first moment of an area is defined as the integral of the (distance
>from axis about which the moment is being calculated), where the
>integration is carried out over the area. The second moment of an area,
>also known as the moment of inertia, is defined as the integral of the
>(distance from axis)^2 where the integration is carried out over the area.
>I am familiar with the second moment of an area and its uses in mechanics,
>however I am not familiar with the first moment.
>Can someone refer me to a resource which explains the physical
>significance of the first moment of an area and also for what calculations
>is this property of an area used?

Much thanks to everyone that responded.
Anwar Upal

The responses are attached below this
__________________________________________________ __________________________________________________ ______________________________

Glen Niebur wrote:
>Two applications:
>1. The first moment is zero when calculated about the centroid. Thus it
>can be used to find the centroid of an area.
>2. In a beam, the shear stress varies from 0 at the outer surfaces to a
>maximum in the center. The shear stress at any point in the beam is
>dependent on the first moment of the portion of the beam above that
>location: Tau = VQ/Ib where Tau is shear stress, V is the shear force, Q
>is the first moment, I the second moment, and b, the width of the
>cross-section (only works for symmetric beams). See, e.g., Gere,
>"Mechanics of Materials"

Eduardo Borges Pires wrote:
>The 1st moment of an area or of a volume is required to evaluate the center of
>mass of such domain.
>Any book on Statics or Vector Mechanics for Engineers such as Beer & Johnston
>(McGraw-Hill) will do.

Antonio Perez wrote:
>It is used to calculate the center of mass (com) of a planar body, for
>distance of com to a line= (1st moment wrt this line )/ (area of the body)

> Bensaci wrote:
>You can find some things about this subject and its definition in the
>famous book of Berkley Physics series (Part-I; Mechanics)

>Jim Funk wrote:
>The first moment of area is better known as the centroid, which is also the
>center of gravity if you have uniform density.

>Necip Berme wrote:
>The first moment of an area about any axis passing through the center of
>gravity of that area is zero. So it can be used to calculate the centroid
>(i.e. center of gravity) of the area. I cannot think of any other
>significance. Strength of material, and machine design text book should
>cover this topic.

Anders Eriksson wrote:
>as a few examples, there are three aspects where the first moment of area
>is used:
>1) As a definition of the center of gravity;
>2) In common engineering expressions for evaluating beam shear stresses;
>3) In the calculation of a bending moment for a fully plasticised beam
>There are surely more applications, but I think that these three, and in
>particular the first, cover most applications. I don't think you will find
>any particular references on the topic, but
>the expression 'pops up' here and there.

Paolo de LEVA wrote:
> As far as I know, the second moment of an area is not at all the moment
>of inertia. The latter is defined as the second moment of >>>a massdm is the mass of a particle of a body.
> The first moment of mass (with respect to carthesian planes, in this
>case) is useful for locating the center of mass (CM) of a body (Varignon
>Theoreme). If you divide a first moment of mass by the total mass of the
>body, you obtain the distance of the CM from the considered carthesian
> Consider, also, that the particles of a body fill a volume, not just an
>area. Second, in a human body they are not uniformly distributed, i.e.
>density is not constant.

Dr. Robert Wm. Soutas-Little wrote:
>The first moment of an area is used to compute the centroid or
>centroidal axis of an area. If the first moment of an area is divided
>by the are you will obtain the distance from the reference axis to the

Ambarish Goswami wrote:
>The second moment is just one in an infinite series of moments
>which can be used to fully describe a shape. The lower moments
>(1,2,3) easily identifiable as physical quantities, the higher
>moments less so because they are less used. They are in a way
>analogous to a Fourier Frequency series. For a discussion, take
>a look at the following paper of mine (downloadable from
>A new gait parameterization technique by means of cyclogram
>moments: Application to human slope walking
>A. Goswami
>Gait & Posture, Vol. 8, No. 1, 1998.

Mark Gillies wrote:
>try Miriam and Craig, or popoff engineering mechanics volume 1 statics

Danny Levine wrote:
>I'm not sure that I can offer any grand definition of the physical
>significance of the first moment of area, but I can tell you
>about a couple of uses.
>In the calculation of the centroid of a compound area (such as the cross
>of an I-beam) the
>formula uses the first moment of area in the numerator. You'll find this
>in any
>good book on statics.
>In the calculation of shear stress on a beam cross section the formula is
>(tau) = VQ/It, where
>V = shear force, Q = first moment of area, I = second moment of area and t =
>thickness at the location
>where the stress is to be computed. This can be found in a first book on
>strength of materials.

>Paul Bourassa wrote:
> you may use the first moment to find out where the center of mass is
>located on a 2 or 3 dim body. Also you may use the first moment to find out
>where the force due to a pressure distribution is acting against a surface.
> Ex ^ressure on a dam, on an airplane wing etc. Standard textbook on
>statics for engineers give numerous examples.

>Nilay Mukherjee wrote:
>When calculating shear stress in a beam due to a transverse shear load,
>the formula = Shear stress = VQ/ It
>V is the shear force, Q is the first moment......I is the moment of
>inertia and t is the thickness of the beam
>Any "Mechanics of Materials" book... e.g. by Beer and Johnston (McGraw
>Hill) will have this formula.
>Also the first moment is used to calculate the position of the centroid of
>the body.
>A * y(bar) = Q
>Where A is area of the body in question, y(bar) is the coordinate of the
>centroid and Q is the first moment wrt a defined coordinate system.
>Check any book on Statics (Beer and Johnston have a book too) for this

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