Neutral Axis 6

[struggle66]

Hi
Recently I attended a two days course on RC design. The speaker who is a quite famous professor here used the centroidal axis(center of the mass) Y=(Summation AY/ Summation A)as neutral axis and find the stresses for both uncracked and cracked section. I am really really confused since I remember that during my university, I used C=T & strain compatibility to find neutral axis and stresses for cracked section. And for uncracked section I used MY/I to find stresses. Without any axial force, if y=0 in My/I, the stress will be zero @y=0. Does it mean centroidal axis = neutral axis for uncracked section?Neutral axis should vary according to the moment. Isn't it? I know this is really basis in engineering but can somebody help me?
Thanks

 

[rb1957]

RC = Reinforced Concrete ? if so different materials (steel and concrete) so rule of mixtures could come into play.

i'd've thought (very possibly wrongly) than you'd calc the NA based on the area of concrete in compression and the area of reinforcing rods in tension.

note, in "My/I" y is the distance from the NA, so when y = 0 (on the NA) then bending stress is zero.

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

 

[BAretired]

Unfortunately, the term "neutral axis" has a different definition for elastic design and ultimate strength design in concrete. In the case of elastic design, the neutral axis lies at the centroid of the transformed section. In the case of ultimate strength design, the neutral axis lies at the point of zero stress when the section reaches its ultimate capacity.

BA

 

[BAretired]

Sorry, I meant to say the point of zero strain in the last sentence.

BA

 

[IDS]

BA - I'd say the definition is the same for both elastic and ultimate (the NA is the line of zero strain), but this line only passes through the centroid if stress is proportional to strain.

But I don't think the question relates to an elastic vs ultimate stress issue.

For an uncracked section both steel and concrete are assumed to be linear elastic, and the NA will pass through the centroid of the section. If both the steel and concrete are symmetrical you can just find the centroid of the concrete section. If they are not the centroid of the concrete is usually used as a convenient approximation anyway.

For a cracked section the NA does not pass through the centroid of the uncracked section, but if there is no axial load you can take moments about anywhere and get the same result.

If there is an axial load then it does make a difference where you take moments about, and this needs to be consistent with the assumptions made in analysing the loads on the structure. Normally in a frame analysis the beams and columns are assumed to be located at the centroid of the uncracked concrete section even if the section is cracked, so the same axis is used in calculating the moment capacity of the section.





Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Yes Doug, I agree. The neutral axis is the point of zero strain, but it shifts when the concrete cracks and it shifts again if the stress becomes non-linear. So the definition is the same but the location of the N.A. varies.

BA

 

[struggle66]

Hi thanks all!
So that means professor was wrong about using the centroidal axis of the cracked section as neutral axis.

 

[BAretired]

No, he wasn't wrong; he was using WSD (Working Stress Design). In WSD, the cracked section is always assumed, i.e. concrete is assumed to take no tension. In WSD, the neutral axis is considered to be the centroid of the cracked transformed section.

In USD (Ultimate Strength Design) as it used to be called in the USA or LRFD (Load and Resistance Factor Design) as it is now called or LSD (Limit States Design) as we call it in Canada, the neutral axis moves toward the compression side of the member. The maximum compressive strain is set by code but the maximum tensile strain is not set. The tensile reinforcement is permitted to strain until the concrete compression block is just small enough to resist the factored moment.

The strain is considered to vary linearly from maximum compressive strain to maximum tensile strain. The neutral axis is located using geometry.

BA

 

[struggle66]

Thanks BA,
Sorry for taking long to get back. Normally when do you use WSD? Is it used when even the section is cracked but stress varies linearly and is still proportional to strain (Triangular concrete stress block)?

What I know is that when the section is cracked, I will assume the strain in concrete and assume the neutral axis depth until C=T. When C=T, I will get neutral axis depth. And from that I can find the moment & stress in reinforcement. And of course for USD, I already know maximum compressive strain from the code.

I still don't understand quite a lot of things. Consequences of not studying during university . Sorry to bother you with kind of questions. Anyway I will do simple manual calculations and study examples for each of every scenarios.

 

[BAretired]

WSD was used until about 1963. ACI 316-63 was the first code to permit both WSD and USD. Many engineers started using USD immediately while others preferred to stay with WSD. In Canada, USD or LSD as it is now called, became the only method specified for concrete design in CSA A23 in about 1980 (I don't remember the exact year). WSD is still used for masonry design in Canada. I don't know whether WSD is still used in the USA or not.

BA

 

[rapt]

Struggle,

You need to do working stress design for crack control to some codes as you need to know the stress in the reinforcement (eg Eurocode EN1992, Australian code, BS8110 part 2) to determine crack control limits.

You also need to do it if you want to do more accurate deflection calculations as we do in RAPT.

 

[struggle66]

Rapt,
Thanks
I understand that I need to find stress in reinforcement for crack control in EC2. My question is how can I find neutral axis. My understanding which I mentioned earlier is that to find the stress in reinforcement of a cracked section, I have two unknown(strain in concrete and neutral axis depth) so I have to assume strain in concrete and get the neutral axis depth from (C=T). And then find back the moment. Do I need to do that? OR just straight away consider neutral axis is at the centroid of the cracked section like BA mentioned earlier. Don't need to be troubled by repetitive calculations to find the neutral axis.

[/No, he wasn't wrong; he was using WSD (Working Stress Design). In WSD, the cracked section is always assumed, i.e. concrete is assumed to take no tension. In WSD, the neutral axis is considered to be the centroid of the cracked transformed section.]

 

[IDS]

To find the centroid of the cracked section you need know where the NA is, because the concrete in tension is ignored.

Your understanding of locating the NA based on equilibrium and strain compatibility is correct, but there are closed form solutions to the problem, and possibility this is what the professor you mentioned in the OP was talking about.

Also note that for a given cross section and reinforcement the depth of the NA is constant from the cracking moment until the steel (or concrete) goes non-linear, so you can find the NA by equating the compression and tension forces for some arbitrary strain, calculate the moment for that strain, then calculate the actual strains and stresses without further adjustment of the NA.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Signious]

Struggle66:

Through LSD method to find the N.A. location is fairly easy. There are two general design cases (balanced reinforcement and properly-reinforced) which dictate how you will proceed. If you have compression reinforcement as well, the process becomes iterative.

If you have the time/ money/ inclination I suggest 'Reinforced Concrete Design: A Practical Approach', the book is specifically tailored to Canadian codes though. It is very useful and goes through very detailed examples that pretty much walk you through concrete element design.

 

[IDS]

Through LSD method to find the N.A. location is fairly easy. There are two general design cases (balanced reinforcement and properly-reinforced) which dictate how you will proceed. If you have compression reinforcement as well, the process becomes iterative.

For a rectangular section with a rectangular concrete stress block you can find the depth of the NA with a closed form solution requiring no more than a quadratic equation, including compressive reinforcement and non-zero axial load. With a triangular stress block it requires solving a cubic, but that's easy on a computer.

For a non-rectangular section that can be divided into trapezoidal layers you need an iterative method to find the layer containing the NA, but can then use a closed form solution.

Open source spreadsheets with examples can be found at:

https://newtonexcelbach.wordpress.com/2013/05/10/using-rc-design-functions-1/

https://newtonexcelbach.wordpress.c...2-sls-design-of-reinforced-concrete-sections/

https://newtonexcelbach.wordpress.c...3-uls-design-of-reinforced-concrete-sections/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rapt]

Struggle,

The NA of a cracked RC section is not the centroid of area, it is the point of zero strain. The iterative approach you have mentioned above is correct, but if you assume a triangular stress concrete compression you can solve it by formula, based on the formulae for C and T and M which are all dependant on NA depth and extreme compression strain.
Once complexities like multiple layers of reinforcement, non linear concrete stress block, PT or other materials are added, the iterative approach is required. For RC sections, the NA hardly moves after cracking (assuming no concrete tensile strength), so it is basically the same as the ultimate NA location.

 

[BAretired]

It has been a few years since I've done Working Stress Design, but if I remember correctly, the NA is not only the point of zero strain but it is also the centroid of the transformed area. The transformed area includes the area of concrete above the NA plus n.As where n is the modular ratio and As is the area of tensile steel. If there is also compression steel, add (n-1)As'.

BA

 

[BAretired]

I believe that the NA moves substantially after cracking as the ultimate capacity is approached, particularly in an under-reinforced member.

BA

 

[IDS]

The NA is the line of zero strain, and hence zero stress.

It therefore divides the section into compression and tensile zones and the algebraic sum of the forces on these two zones must be equal to the applied axial load.

If the axial load is zero the absolute value of the force on the two sections must be equal.

The force is the integral of stress x da, and if the materials are in the linear range, stress is proportional to the distance from the NA and the Elastic modulus, E, so if we transform the stress in proportion to E, the force on the section is proportional to the first moment of area.

It follows that for zero axial load and elastic materials the NA passes exactly through the centroid of the transformed section.

If the transformation includes multiplying the area of concrete in tension by zero then this remains true for the section after cracking, although obviously the centroid will be a different location to that when the concrete in tension is included.

Since the position of the centroid depends only on the areas and E values, it will remain in exactly the same position from cracking up to the moment that first causes non-linear behaviour, in either the steel or concrete.

When either or both materials go non-linear stress is no longer proportional to the distance from the NA, so the NA will no longer pass through the centroid of the transformed section.

So, in short, I agree with both of BA's points, with the addition that the NA does not start to move away from the centroid until the material properties become non-linear.

One other point: the "concrete takes no tension" assumption provides a good approximation for calculating stresses, forces and moments. For calculating deflections accurately we have to take account of the tension stiffening effect of the concrete, which is highly non-linear, so the fixed NA position no longer applies.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

Once complexities like multiple layers of reinforcement, non linear concrete stress block, PT or other materials are added, the iterative approach is required

Multiple layers of reinforcement, post-tensioning, and non-zero axial load can all be dealt with easily with a closed form solution. Non-linear concrete behaviour is a bit more difficult, but for typical code based non-rectangular stress blocks this can also be handled.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/