ACI 3-18, Flexure

[bkal]

I am trying to assess an existing beam in flexure using ACI 3-18, Section 10.2. What surprised me is that there seems to be no limit on how small a distance from the compression fiber to the neutral axis is (parameter c). I am aware that the beam was not designed to modern standards.

Am I missing something, or is it prescribed somwhere else?

 

[JAE]

I'm not sure I totally understand your question. Distance c is simply the distance from the compression fiber to the neutral axis. It is simply a "fact" of the section - a property that is just there...not something that you need to place a minimum limit on.

 

[UcfSE]

JAE is right. It wouldn't make sense to put a limit on c. Keep in mind that c is not constant for a given section, it changes with load and concrete strength. The typical c we use corresponds to the nominal moment capacity. If you are looking at evaluating deflections, rotation, or post-yielding strength you probably won't have the same value for the distance from the extreme compression fiber to the neutral axis. Remember that as the section is loaded, more concrete cracks and "pushes" the neutral axis toward the compression side.

Are you getting an abnormally small c value or is this just a question about the code?

 

[bkal]

JAE, UcfSE,

thanks for your replies. Please bear in mind that I am talking about assessing an existing beam, not designing a new one.

The issue is that I end up with the depth of compression zone being small (10-20mm, theoretically), and would this not lead to spalling of the concrete. BS8110 indirectly limits compression zone depth by limiting lever arm to 0.95*d (d being a distance from the extreme compression fibre to the centoriod of tensile steel).

I am not saying that the beam I am assessing is well deigned, but I have to deal with is there.

 

[haynewp]

I've never heard of spalling of concrete because of a small compression zone depth. After tensile cracking, for a lot of beams the compression zone may be small at times because of a small amount of load that is on the beam compared to the design load.

Does the reinforcing meet the minimum ratio required?

 

[rapt]

bkal,

You are correct that BS8110 does limit x to .95*d. This rule is not in many other codes such as ACI318.

There are good reasons to limit x in this way. If you actually do the design properly from first principles rather than using rectangular stress blocks and black-box formulae as many designers do, you will find that for very small values of x, the strain in the reinforcement is very high (much higher than the breaking strain of the reinforcement). One way to limit this strain is to place a minimum limit on x.

This is then a defacto minimum reinforcement rule ensuring that the reinforcement strain is limited.

This limit should actually depend on the peak strain for the reinforcement being used. For lower ductility reinforcement the .95 factor should actually be less still.

Unfortunately too many designers have never done the real calculations to see the effect of the assumptions made in the approximations built into the codes to make design easy. Unfortunately all of these assumptions are not always conservative and can have adverse consequences when a design is outside the bounds of the approximation.

 

[VTPE]

I think what you are asking is actually "what haapens if my beam crushes?" When you are performing the calculations, you are seeing a beam that is over reinforced and therefore is subject to a brittle conpression failure rather than a ductile response to the plasticity of the reinforcing steel. Try running your calcs with the maximum steel ratio based on 0.75* the balanced steel ratio and see what that does to the compression zone. My guess is you are taking into account more steel than is allowed for a ductile failure.

 

[arnel123]

what i understand if u got a small value of c= 0.85fc'ab under a stress block diagram and equating it to T=AsFy, u will notice that the maximum moment is lesser amount since your beam maybe of big section..but you got a an underreinforced beam. is your calculation of steel ratio is lower than the minimum? then the tendency of your beam will crack. or maybe your shear reinforcement is not enough and maybe you might not considered some torsion that affect the beam..

 

[ajose]

bkal,

The limit is there but not in straight words and normally used, so used by any of us

For maximun stresses on concrete and steel

the codes gives

- concrete maximum compresion strain (side compresion)
- steel maximum traction strain (side in traction)
- the strain in concrete and reinforcement shall be directly proportional (except for deep beam behavior)

That is meaning there is a straight line defining a fixed "c"

so there is a limit.

When there is only traction rebar up to the yield strain the "c" is bigger than the minimun from the code.

Adding traction rebar has to be done fullfiting the straight line strain, that's meant when the traction steel has reached the yield strain and the moment it is not balanced. The steel is added based on the strain of the compresión and traction rebar.

For revision a similar aproach has to be taken.
That's meant
direct proportional strain
Force equilibrium (sure forces and moments)

Well known it is the method for trial and error, guess a "c", calculate the strain, stresses, forces, check the equilibrium. The output a Maximun Moment and "c"

Ajose

 

[rapt]

ajose,

Where does ACI 318 define a maximum design steel strain?

Also, the maximum compression strain is just that, a maximum. Sometimes it is necessary to limit the compression strain to less than this maximum if the strain in the tension steel is too high (even though ACI does not define the limit for this). In this case it is not possible to use the rectangular stress block for concrete or standard ultimate strength formulae. The real stress/strain diagram for the concrete must be used. This happens when the area of reinforcement is small and the value of c is small or when the reinforcement is not very ductile.

So a minimum value of c is not defined by the code. The only way the code limits this is by requiring a minimum area of reinforcement. Even then ACI has a 133% strength limit on this value. When this 133% limit is applied, the designer should investigate the real strength depending on the ductility of the reinforcement.

 

[ajose]

Dear rapt,

Strain is not just strain,

Strain*Elasticity Moduli= Stresses,

and allways will be.

The other question, well ....

It is hard answer the question without qoute the code, then reading on ACI 318-99:

Chapter 10 Flexure and axial loads
Section 10.2 Assumptions
Article 10.2.2 State the proporcionality
Article 10.2.3 State the maximun strain in concrete
Article 10.2.4 State that if steel is not yielding the steel stress is Strain*Elasticity Moduli
Article 10.2.6 State The stress distribution, let the designer choose between rectangular, trapezoidal parabolic or any other shape that fit

So i was not so wrong.

 

[rapt]

ajose,

ACI 318 requires that the reinforcement reach yield before the concrete reaches the maximum compression strain. Therefore the reinforcement is always yielded and the formula for stress = Strain * Es does not apply in this case (see your comment on 10.2.4). The only situation your stress formula will apply at ultimate strength for is a balanced design. If we are dealing with very low values of c then the design will not be balanced and the reinforcement stress/strain will be in the non-linear region so Es is meaningless.

If you do the calculations from first principles you will see that I am right. The code formulae make the assumption (unstated) that depth c is great enough so that the strain in the reinforcement is less that the peak strain and the concrete is at full compression strain. This is not always possible will low reinforcement ratios and medium to low ductility reinforcement.
For normal ductility reinforcement, the normal minimum reinfrocement rules will ensure that all is ok. For low ductility reinforcement, this is not the case and more accurate calculations are required even though ACI and most codes do not mention this.

 

[ajose]

rapt,


There is three phases on designing beam under flexion. The limits of these is when the traction rebar are yielding.

If the design moment is smaller than the moment that yield the traction steel, so the steel is not yielding and has a stress of strain*Es.

When the design moment is equal to the moment that yield the traction steel, well the the steel is yielding and reach fy.

When the design moment is biggerto the moment that yield the traction steel. The subtraction of both moment is used to to compensate. The area of aditional rebar is the aditional force in rebar divided by the stresses of the rebar. In compresion the stress is strain_on_compresion_rebar*Es but no bigger than fy and the total compresion rebar is Force_compresion/stress_compresion. The area aditional area of traction steel is the Force_traction/stress_traction.

Completly agree the revision of section under the code has not the same behavior, and have to be revised under the criteria the formule and trial and error (if the section give a formulation not easy to solve)

 

[rapt]

ajose,

You are missing the point. While the reinforcement stress in tension is limited to fy, the strain in the reinforcement is not limited to the yield strain.

If you look at the minimum limit discussed above for BS8110, with a lever arm of .95d, the strain in the concrete is limited to .0035 in BS8110, so with the compression at this level and a depth to neutral axis of .05d, the strain in the reinforcement would be
.0035*.95d/.05d = .0665. The yield strain for British steel is 460/205000 = .0224.
So this strain in the minimum case is 3 times the yield strain if the concrete compression strain is .0035.

If the peak strain for the reinforcement being used is less than .0665, then the reinfrocement will snap. In this case, the capacity must be calculated using a compression strain less than the maximum value to reduce the strain in the reinforcement. The code rules and general design equations available for design use do not allow a method of doing this or even indicate that they do not handle the situation.