Neutral Axis with Axial Force and Bending Moment in Cracked Section

[bdbd]

Do you think that axial force and bending moment changes the neutral axis of a cracked section? If so, can you refer me to the relevant reference?
All the reinforced concrete books I have checked uses geometric equivalency and the neutral axis depth is not depending on the axial force and moments while concrete books in Denmark says otherwise.

 

[centondollar]

Yes, it changes the neutral axis. Also, I doubt that the books you read used "geometric equivalency"; rather, they probably showed examples of beams (which are typically not loaded by axial force) and derived the equations by strain compatibility and assumptions of regarding stress block shape, which is also at the core of the more general approach. There is not a closed-form solution to that problem AFAIK, but it can be computed by strain compatibility, i.e., the method used to derive RC column M-N diagrams.

The more general approach for uniaxial M+N is this: choose an ultimate concrete compression strain "ec_ult" and neutral axis depth "x". Calculate forces (concrete compression, steel tension/compression) in the section, sum them up and check for equilibrium. Iterate x until equilibrium is reached. Using the equilibrium values of force, calculate bending moment about some axis (e.g., zero-line or ultimate fiber) and divide with curvature (curvature=ec_ult / x) to get bending stiffness. Repeat for several values of concrete compression strain to get the moment-curvature and moment-bending stiffness curves.

For biaxial bending also the inclination angle of the N.A. must be found, and the calculation of cracked bending stiffness is not so straightforward anymore.

 

[bdbd]

Hi,
Thanks for the answer. I do that for ULS part without any problem. But I need to do similar thing for SLS to get steel stress. Thinking now, I cannot see any difference, but there is some that I cannot remember now I think we don't use rectangular block approach for SLS.

 

[Denial]

Depending on how deep you want to dive, my website (

https://rmniall.com)

has a free spreadsheet that analyses reinforced concrete beam-columns.

It performs a "plane sections remain plane" analysis under the combined actions of an axial force and biaxial bending moments.[ ] The outline of the concrete cross-section can be completely arbitrary (defined as any polygon).[ ] The stress-strain relationships for both the concrete behaviour and the steel behaviour can also be completely arbitrary (defined by a piecewise-linear relationship).

 

[bdbd]

Hi Denial,

Thanks for that! I will download and go through the steps you have used. Inhope you also have an SLS check.

 

[centondollar]

Thanks for the answer. I do that for ULS part without any problem. But I need to do similar thing for SLS to get steel stress. Thinking now, I cannot see any difference, but there is some that I cannot remember now I think we don't use rectangular block approach for SLS.

The only thing that changes is the assumed shape of the concrete compression block. The principle remains the same. If you use the triangular block, the concrete compression strain maximum value should be the SLS value (not 0.35%). If you use some other block shape, numerical integration might come in handy.

 

[bdbd]

Thanks for that. I am not using anything fancy. Just simple Eurocode. So, instead of yielding the concrete to 0.35%, we should try to reach an equilibrium with the forces, right? I think that's the problem I had.

 

[centondollar]

Well, the choice of ultimate concrete strain does not affect the computational steps. The calculations [iterate "x" until force balance --> calculate moment --> calculate EI = M/k = M/(e_c_ultimate/x)] are done for many values of compression strain (e.g., e_c_ultimate = 0.02%, 0.04%, 0.06%, 0.08%, 0.1%, 0.12%, ..., 0.2%), so that many values of EI are given. With 10 values of "e_c_ultimate", you get 10 values of "EI = M/k", and so on. Depending on the cross-section shape, rebar amount and strength, concrete strength and magnitude of axial force, more or less points may be needed.

 

[IDS]

The only thing that changes is the assumed shape of the concrete compression block. The principle remains the same. If you use the triangular block, the concrete compression strain maximum value should be the SLS value (not 0.35%).

I disagree, finding the neutral axis position and concrete strain in an elastic analysis is fundamentally different to finding the neutral axis for the ultimate load condition, where the concrete strain is specified.

The neutral axis depth is controlled by the eccentricity of the applied load, so it is affected by both the axial load and bending moment. If the axial load is zero then the NA depth can be found by equalising the compression and tension in the section, but if it is non-zero, and you want to find the stresses for a specified bending moment and axial load then the NA position must be found by adjusting the position so that the eccentricity of the reaction forces is equal to the eccentricity of the applied loads. This can be done by iteration (and usually is) but there is also a closed form solution, which is much quicker.

More details of the closed form solution at:

https://newtonexcelbach.com/2008/03/25/reinforced-concrete-section-analysis-1/

and the latest version of the associated spreadsheet can be found at:

https://newtonexcelbach.com/2022/11/11/rc-design-functions-9-03-compression-strut-angle-adjustment/

The spreadsheet includes functions for elastic and ULS analysis of rectangular sections. There are also other spreadsheets available for arbitrary sections, and biaxial bending.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[bdbd]

Hi Doug,
Your website was the first one I checked, but when I went through your calculation, I noticed that you have used quadratic equations, but I want to go through the steps one by one instead of the quadratic solution. Maybe I cannot understand that. But following the Ghali you referred in your note, I cannot understand this integral The integral for area, so it comes out of integral as y * (y-n) * Ac(above Neutral Axis), but y is not defined, it is coordinate system.

https://i.imgur.com/4I0Vciz.png

https://i.imgur.com/d8IOISZ.png

 

[HTURKAK]

I screened the responds rather than reading them in detail.. You did not mention if the axial force compression or tension and if the member is a column or beam..

But i feel free to guess .. Assuming the axial load is in compression, the first step you should check is if the concrete section is cracked or not..

In SLS loading the load comb. typically ( Characteristic value of dead loads + about 25% of Quasi-permanent imposed loads )
for office bldgs etc.

The assumptions for cracked section ;

- plane sections remain plane for calculation of strains , steel reinforcement is elastic with a modulus of elasticity of 200 GPa and concrete in compression is elastic with a secant modulus of elasticity Ecm.

The assumptions for uncracked section ;


- Concrete and steel are both considered to be elastic in tension and in compression. The analysis is similar to the cracked section analysis but the area of concrete below the neutral axis is uncracked.


For both cases the equilibrium shall be satisfied that is , the sum of the internal forces will be zero...







Tim was so learned that he could name a
horse in nine languages: so ignorant that he bought a cow to ride on.
(BENJAMIN FRANKLIN )

 

[IDS]

bdbd - Actually my solution uses cubic equations for a rectangular section, or quartic if the NA is in a trapezoidal section

But the equations are just closed form solutions to equation 7.10 in Ghali's book, which as far as I know everyone else solves by iteration.

I am not sure what you mean by y being a coordinate system. As shown in the diagram, the origin is placed at the centroid of the gross concrete cross-section, yn is the distance to the position of the NA (positive downwards) and yt is the distance to the compression face. For any value of yn you can evaluate the integrals between yt and yn, so you can adjust the assumed value of yn so that equation 7.10 evaluates to zero.

Having found the position of the NA you can find the axial force and the moment about the origin for any assumed stress at the compression face, and hence find the actual stress and strain for the actual applied loads. Note that the top and bottom lines of the equation are equal to the bending moment and axial force on the section respectively, assuming the stress at the compression face = yt, so it is easy to find the actual stress using:
stress = yt * (Applied Moment/Calculated Moment).



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[centondollar]

The neutral axis depth is controlled by the eccentricity of the applied load, so it is affected by both the axial load and bending moment. If the axial load is zero then the NA depth can be found by equalising the compression and tension in the section, but if it is non-zero, and you want to find the stresses for a specified bending moment and axial load then the NA position must be found by adjusting the position so that the eccentricity of the reaction forces is equal to the eccentricity of the applied loads. This can be done by iteration (and usually is) but there is also a closed form solution, which is much quicker.

This, with the exception of the last sentence, is what I explained in my first reply in this thread.

Regarding closed form solutions, they are not typically available for arbitrary shapes (e.g., non-symmetrical), which is why I did not mention them. The iterative procedure also avoids tedious algebra and gives a precise enough solution for all practical purposes. A simple loop ("until Force_Balance < tol, increase N.A. depth in increments of "x"") can be programmed to satisfy the force balance in an iterative algorithm, so the iterative procedure (properly programmed) does not actually require any more work than a closed-form solution. With modern computers, the computing time will also remain reasonable for reasonable values of tolerance, N.A. increment and points on the M-k curve.

EDIT: the article written by you uses a trapezoidal-shaped stress block only (correct me if I am wrong), so it doesn't generalize to situations where a different stress block, concrete stress-strain diagram and steel stress-strain diagram is to be used.

 

[IDS]

This, with the exception of the last sentence, is what I explained in my first reply in this thread.

But it isn't. The important difference is that you said the neutral axis should be adjusted so that the total reaction force is equal to the applied axial load, and this should be done for multiple values of concrete compression strain, so you can interpolate to find the strain and neutral axis depth for the applied axial load and moment.

If you find the NA depth so that the eccentricity of the reaction is equal to the eccentricity of the applied load (either by iteration or using a closed form solution), then the concrete strain can be found by simple proportion so that either the axial force or moment reactions are equal to the applied loads, and no further calculation is necessary.

Yes, this only works assuming liner elastic behaviour for the steel and the concrete compression, but if you are checking reinforcement stress and/or section curvature this is normally a serviceability check where linear elastic properties are normally used.

Regarding the closed form solution:
- The method can be used for any shape divided into trapezoidal (or triangular) layers where the neutral axis is parallel to the x axis. For multi layered sections you have to step through the layers to find the one containing the neutral axis, you can then find its exact position within that layer.
- Both the steel and concrete are assumed to have linear elastic properties, with zero tension for the concrete.
-A similar closed form solution can be used for Ultimate Limit State analysis, but in this case it finds the neutral axis depth to equalise the axial load and reaction force, using the code specified strain at the compression face.
- For a rectangular section a simple formula can be used to find rectangular stress block parameters that will give exactly equal results to a parabolic or parabolic-linear stress block.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[centondollar]

But it isn't. The important difference is that you said the neutral axis should be adjusted so that the total reaction force is equal to the applied axial load, and this should be done for multiple values of concrete compression strain, so you can interpolate to find the strain and neutral axis depth for the applied axial load and moment.

No, I said that the force balance should be satisfied and that the process involves finding neutral axis depth and curvature for each chosen value of concrete compression fibre strain. From this, a moment-curvature and moment-bending stiffness diagram (secant or tangent EI) can be constructed. If the normal axis is added as a parameter to be changed (from e.g., 0 to a suitable value) for each case of ultimate compression strain moment-curvature, then a M-N interaction diagram can also be obtained. This process is simple, explained in all reinforced concrete textbooks and suitable to any geometry and stress-strain diagram. If hardening is included, there is an extra iterative check included, but for bi-linear elastic-perfectly plastic steel material law, the procedure contains only the one iteration: find the neutral axis so that force balance is satisfied.

- The method can be used for any shape divided into trapezoidal (or triangular) layers where the neutral axis is parallel to the x axis. For multi layered sections you have to step through the layers to find the one containing the neutral axis, you can then find its exact position within that layer.

This is quite restrictive and seems to involve some sort of iteration ("stepping through layers"), despite you claiming earlier that it is non-iterative.

- Both the steel and concrete are assumed to have linear elastic properties, with zero tension for the concrete.

This is also restrictive. A numerical iterative analysis, using e.g., constant rectangular strips, can easily be modified to accommmodate various concrete and steel stress-strain diagrams.

 

[IDS]

OK, I give up.

I didn't say the closed form solution was suitable for everything. It's only applicable to about 99% of the section analyses I do.

For others it may be even less.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/