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

line.

__________________________________________________ __________________________________________________ ______________________________

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

>example:

>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

>section.

>

>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

>plane.

>

> 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

>centroid.

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

>http://www.cis.upenn.edu/~goswami/papers/paper.html#journals):

>

>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

>section

>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

>stress

>(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

>formula.

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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

line.

__________________________________________________ __________________________________________________ ______________________________

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

>example:

>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

>section.

>

>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

>plane.

>

> 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

>centroid.

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

>http://www.cis.upenn.edu/~goswami/papers/paper.html#journals):

>

>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

>section

>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

>stress

>(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

>formula.

---------------------------------------------------------------

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

For information and archives: http://isb.ri.ccf.org/biomch-l

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