Cracking moment larger than moment of resistance? 1

[Woody1515]

Hello,

Is it possible for the cracking moment of a reinforced concrete beam to be larger than the ultimate moment of resistance? If I run the numbers with an 8” thick by 48” deep 30 MPa concrete beam with 2-15M bars top and bottom (very simple example) - I get a larger cracking moment than moment of resistance. This may be because the formula for the cracking moment has no resistance factors applied to it? Should I apply the 0.65 resistance factor to the cracking moment, since it is applied to the moment of resistance? I am trying to determine if the beam will crack under the factored ultimate loads.

This beam would be for a simple grade beam on piles supporting a new residential addition. The contractor wants to use a 48” deep beam as that is the only form he has available to him. The grade beam will be way stronger than what is required, but I want to know if the beam will even crack under the full design factored loads. Or in more general terms, can I compare the cracking moment (without a factor applied to it) to the factored loads?

 

[Tomfh]

Most design codes will provide a method for assessing this. The reinforced capacity is generally supposed to exceed the cracking moment, so as to ensure a ductile failure mode occurs. You don't want the beam falling down as soon as it cracks. If the beam can't fall down then the codes often allow a more lenient approach, e.g. assess it as a plain concrete member.

I would think you'd want more than 2 16mm bars in a 1200 deep beam regardless.

 

[Settingsun]

For whether it will crack, you need to account for stresses due to restraint of shrinkage, but the calculation of that is imprecise. The restraint will be external (eg the piles) or internal (the reinforcement). Some codes give methods for the reduction of cracking stress due to internal restraint eg Australian and Eurocode.

 

[IDS]

The cracking moment should not be used as the moment capacity of the section because it can be greatly reduced by shrinkage and differential stresses, and it is a brittle failure.

As Tomfh said, you need to provide sufficient tensile steel so that the design cracked ULS capacity exceeds the cracking moment, and for that purpose the cracking moment should not be factored down. Most codes require it to be factored up.

I have been involved in two jobs where the cracking moment exceeding the bending capacity caused major problems, and several others where it caused significant problems, or contributed to other major problems, so it is not something to be overlooked.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Tomfh]

IDS, what sort of problems were caused?

 

[IDS]

In one case, deflections about 5x greater than expected.

In the other, sudden total failure of a precast element during erection.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[Guest262653]

What reinforcement grade are you using? 2M16 in S500 steel should be fine to ensure minimum reinforcement and that the ULS resistance moment is over the cracking moment.

My back of the envelope calculations (EN1992):
- As,min = 0.26 fctm.b.d/fyk = 0.26 x 2900 x 0.20 x 1.15 / 500 = 3.5 cm2 (2M16 = 4.02cm2)
- MRd = 50/1.15 x As x 0.9 x d = 43.5 x 4.02 x 0.9 x 1.15 = 180 kNm
- Mcr = fctm x b x h^2 /6 = 2900 x 0.20 x 1.2^2 / 6 = 140 kNm

 

[Tomfh]

IDS,

Yeah I’ve seen precast snap and fall. Pretty scary.

How come the deflections were magnified?

 

[IDS]

How come the deflections were magnified?

Sudden reduction in stiffness after cracking resulting in dynamic forces which caused some plastic strain in the steel, but in this case not enough for total failure.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BridgeSmith]

Under the AASHTO bridge design spec, if the cracking moment is larger than the factored moment applied, the beam must be designed for the smaller of either the cracking moment or 1.33 times the factored applied moment. No resistance factors are applied to the calculated cracking moment.

Rod Smith, P.E., The artist formerly known as HotRod10

 

[r13]

Woody1515,

When the factored load is relatively small, and the concrete section is very large due to reasons other than strength requirement, then the cracking moment (Mcr) may exceed the ultimate moment of resistance (Mu, ultimate moment demand) due to size effect. If you have calculated both and find Mcr > Mu, the intuitive way is to design for the larger moment (Mcr in this case), or go through the code to determine a permissible lower design moment; however, this is not always done (calculate cracking moment), then the code took care of this matter by requiring the section must be reinforced with the specified minimum steel (As[sub]min[/sub]), or, in the US, 4/3 times of the steel required by analysis (in another term, 1.33 Mu), whichever is smaller. Please review avscorreia's and BridgeSmith's comment above, which covered this topic well.

 

[Agent666]

Part of the reason for code minimums is that the true concrete strength could be 1.5 to 2.0 times the minimum design value under longer term conditions.

As it happens I wrote a post on my blog on this a few days ago that might be of interest.

 

[Tomfh]

Agent666, do you think the code minimums adequately cover the true flexural strength of concrete?

 

[Agent666]

Well in NZ we seem to have it covered. Based on the post I linked to, given a modulus of rupture of 0.6f'c^0.5 it appears even at 2 x f'c that the minimum reinforcement provisions gave a reinforced strength that was at least 30% higher than the enhanced cracking moment capacity. This was based on a beam type member with d=0.95h.

As noted also, we recently doubled our minimum reinforcement requirements in the end regions of walls because it was observed in our 2011 Christchurch earthquakes that lightly reinforced walls suffered some progressive fracturing of the reinforcement under successive cycling due to the very effect being discussed in this thread. They now link the reinforcement required in the middle regions of walls to be a proportion of the end regions. End region is defined at 15% of the wall length.

In thinner members like slabs its certainty possible to get this margin over and above the cracking moment capacity closer to or below 1.0 due to lower effective depth to member depth ratios. With lower effective depth you also reduce the minimum reinforcement content required as minimum reinforcement provisions are linked to this.

 

[r13]

End region is defined at 15% of the wall length

A good rule. Very helpful in reinforcing the walls.

 

[Tomfh]

given a modulus of rupture of 0.6f'c^0.5

That’s the assumption I’m really wondering about. Concrete tends to rupture at far higher loads than that.

With say 32MPa conc that formula predicts 3.4MPa but in reality it might break at say 5MPa.

 

[IDS]

Tomfh - It's true that the short term tensile strength of concrete is often well above 0.6f'c^0.5, but taking into account the effects of shrinkage, creep, differential temperature, and stress concentrations the actual stress at first cracking is usually much lower than indicated by short term testing. That number works reasonably well in predicting deflections, which are highly sensitive to the cracking moment, so that suggests it is a reasonable approximation.

That said, the consequences of the cracking moment being higher under short term loading with early age concrete should also be considered, and a more conservative number used if the possible consequences warrant it.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BridgeSmith]

The value for the modulus of rupture in AASHTO is much lower than 0.6f'c[sup]0.5[/sup]. For calculation of Mcr it's .24f'c[sup]0.5[/sup].

Rod Smith, P.E., The artist formerly known as HotRod10

 

[Agent666]

Well NZS3101 does specifically state like IDS mentioned that this value for the modulus of rupture is appropriate for checking deflections. ACI318-19 has the same value, but no specific mention of it's importance in checking deflections.

The direct tensile strength of concrete is lower in NZS3101, this comes in at 0.38f'c^0.5 vs 0.6f'c^0.5. This is more appropriate for unreinforced concrete though.

 

[rapt]

AS3600's latest minimum reinforcing for walls under earthquake action are similar to what Agent666 is talking about for NZS.

They are no longer based on the lower bound concrete tensile strength of .6 f'c[sup]0.5[/sup], and there is an assumption that the real concrete strength might be up to about 30% higher (and this is limited in the code). For 50Mpa concrete this works out to a steel ratio just under .01, so about 4 - 5 times the normal flexural minimum based on .6f'c[sup]0.5[/sup]. And it is required in the end .15Lw of a free end of a wall as is required now in NZS. This requirement is in the plastic hinge zone for a tall wall and gradually reduces with height up the wall from that zone to the normal minimum of .0025.

I think ACI/American codes are now the only major design code that allows the 1.33Mu rule. AS3600 dropped it in 1988. It now requires a minimum of 1.2Mcr no matter what. Eurocode rules are based on 1.15Mcr, though simplified to a formula based on concrete strength. ACI has improved theirs to be based on concrete strength and cracking moment in the last few years.

The outlier was always BS8110 which persisted with .0013, independent of concrete strength, for slabs and walls and similar stupid numbers for beams until it was replaced by Eurocode. It was probably ok for concrete strengths up to about 20MPa cube strength.

BridgeSmith,
is the AASHTO rule imperial or metric units for concrete strength? There is a factor of 12 difference!