Center of Gravity or mass? 1

[Bubik]

Hi Everyone

Would anybody be able to explain if bending in a section of a member, for example RC Beam, occurs about center of gravity?? To my understanding if a section is of uniform material bending will occur about centroidal axis. What about if an RC beam with different reinforcement quantities in top and bottom? what axis will bending occur about, is it centroidal, gravitational etc.???

 

[racookpe1978]

Be realistic: Your design will be used on earth, in a constant gravitational field.

1. IF the material is of uniform density, the "center of strength" (center of resistance to force) will be proportional to the center of mass which will be proportional to the center of gravity will be proportional to the center of volume. (You're actually using the I and S shape factors from the structural tables)

2. IF the member is NOT of uniform density and uniform strength (such as a concrete beam with rebar more heavily positioned in the bottom or top of a deep beam), you have to separate the materials and calculate each based on its distance from a convenient point of calculation.

 

[MotorCity]

Don't confuse the center of gravity with the neutral axis. The center of gravity is based purely on the mass and position of mass about its centroid. The neutral axis in a flexural member could be anywhere within the cross section of a member, as it defines the point of zero stress, or the boundary between the compression side and tension side of a beam

 

[racookpe1978]

Good point: I was considering symmetry about the neutral axis.

 

[AnimusVox]

If your beam has more or less steel reinforcement in the top or bottom portion, you'll need to use the substitution method (using a ratio of their E values combined with the cross sectional area of the rebar) to find the neutral axis.

For a cross section of a uniform material, the only reason you'd need the center of gravity is if it were tall enough to extend into space, i.e. in a nonuniform gravity field.

 

[rb1957]

"Would anybody be able to explain if bending in a section of a member, for example RC Beam, occurs about center of gravity??" ... no it doesn't (because an RC beam doesn't have a uniform homogeneous density).

bending of an RC beam (as I understand it) depends on the neutral axis defined by the re-bar and the concrete in compression.

another day in paradise, or is paradise one day closer ?

 

[JoelTXCive]

I'm not sure how to do for other materials, but I do know for concrete.

It is my understanding that for a RC beam, bending occurs around the elastic neutral axis until the concrete cracks. The elastic neutral axis is at the same location as the geometric neutral axis, which is the geometric centroid of the cross section.

After the concrete cracks, the neutral axis shifts upwards and an elastic analysis is required to determine the new location. For a given loading, you have to balance the compression and tension stresses to find the new neutral axis. I think the easiest way is set up a spreadsheet in excel and then use solver to iterate the neutral axis until you reach force balance.

 

[JoelTXCive]

I forgot.... the reason the neutral axis moves is because once the concrete cracks, the moment of inertia for the cross section changes.

 

[rb1957]

ok, that says to me that concrete has an allowable tension stress (I thought of it as compression only); maybe due to pre-tension in the re-bar ? then the concrete fails in tension but the beam sustains tension in the re-bar.

but then if uniform re-bar, then using center of mass isn't incorrect (though the logic of why is important to remember).

another day in paradise, or is paradise one day closer ?

 

[IDS]

I don't think it is useful to bring centre of mass or centre of gravity into it. For a section with uniform density and thickness the centre of mass is the same location as the centre of area, and in a uniform gravitational field this is the same location as the centre of gravity, but since beam sections often have materials of different density it is better to think of the centre of area, and forget about mass and gravity.

The other important points are:
[ul]
[li]If the section has materials of different elastic modulus the areas must be multiplied by the modular ratio. The neutral axis then passes through the centroid of the adjusted section, assuming linear elastic behaviour.[/li]
[li]For cracked reinforced concrete the concrete in tension is taken as having zero stiffness, so calculating the position of the centroid becomes a little more complicated, since you need to know the position of the centroid to know what area should be excluded from the calculation. You can do it by iteration, but there are not very complicated closed form solutions that are easy to set up on a spreadsheet (or your preferred maths program).[/li]
[li]For non-linear behaviour in any part of the section the neutral axis no longer passes through the centroid of the section, since the elastic modulus is affected by the stress.[/li]
[/ul]

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Bubik]

If a section is subjected to an eccentric load, the neutral axis however is not in the same location as the line about which the bending occurs ( for example columns in bending and axial loading). I assume then neutral axis has nothing to do with where bending occurs, although I always thought it is the case. What do you think???

 

[IDS]

If a section is subjected to an eccentric load, the neutral axis however is not in the same location as the line about which the bending occurs ( for example columns in bending and axial loading). I assume then neutral axis has nothing to do with where bending occurs, although I always thought it is the case. What do you think???

What do you mean by "the line about which the bending occurs"? The neutral axis is the line of zero strain, so it separates the part of the section in compression from the part in tension. That could be considered "the line about which the bending occurs", but it will usually be in a different location to the centroid of the section, which is usually the line which load eccentricity is measured from.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Bubik]

Sorry for confusion ..regarding line about which bending occurs I meant axis. How to calculate the location of that axis (about which moment occurs) for example in case of RC Section with different reinforcement area in top and bottom??

 

[rb1957]

doesn't JoelTX's post (1st Sept) answer this ?

another day in paradise, or is paradise one day closer ?

 

[IDS]

doesn't JoelTX's post (1st Sept) answer this ?

I find that post rather confusing:

It is my understanding that for a RC beam, bending occurs around the elastic neutral axis until the concrete cracks. The elastic neutral axis is at the same location as the geometric neutral axis, which is the geometric centroid of the cross section.

I don't know what the "geometric neutral axis" is. The neutral axis is the line of zero strain, and that's all it is. For a section where all stresses are linear elastic with zero nett axial force the neutral axis passes through the centroid of the transformed section, but if there is any non-linearity usually it doesn't.

For a section with combined axial load and bending you can take moments about wherever you like, as long as you use the same axis for calculating applied loads and balancing internal forces and moments. For a reinforced concrete column we usually use the centroidal axis of the concrete section, regardless of whether the section is cracked, or if the reinforcement is asymmetric. To calculate the section stresses and forces we need to find the location or the neutral axis, but that is not normally the line used for calculating applied moments.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[JoelTXCive]

IDS-

I could have said better. Another way of saying is:

For prismatic RC beams (cross section does not change along length of beam); It does not matter what your cross sectional shape is, or how many different layers of steel you have; or even the locations of the steel. The bending will be about geometric centroid of the cross section UNTIL the concrete cracks.

The geometric centroid (aka elastic neutral axis or geometric neutral axis) is also the point of zero strain UNTIL concrete cracking occurs.

As soon as the concrete cracks (which doesn't take long); life gets more difficult and you have to an elastic analysis to find the new neutral axis.