Poisson's expansion strength

[lejam]

Given a 3000 psi cylinder block with 4" diameter and 8" height in a compression test machine. When the test compression is equal to 1500 psi, what is the corresponding tranverse poisson's strength or pressure (in psi) as it expand outward in let's say the middle of the cylinder (transversely)? How do you calculate this?

 

[BAretired]

If the coefficient of friction between the test machine and the concrete sample is zero, the horizontal stress in the sample is zero. Otherwise, there is a lateral compression near the ends of the cylinder where it is restrained from expanding freely according to its Poisson ratio.

BA

 

[lejam]

It is empirically stated that "Until a unaxial stress of about 0.95fc, the Poisson ratio of concrete is approximately constant and close to 0.20. Above that stress level it increases fast, reaching a secant value of about 0.4 at ultimate strength. The underlying physical reason that, as failure approaches, pre-existing microcracks at the interface of aggregates with the hardened cement paste extent into the latter in a direction parallel to that of the applied uniaxial compression stress and tend to join up as macro-cracks in that direction. The opening up of these macro-cracks soon leads to ultimate strength, manifesting itself as a precipitous increase in the apparent lateral strain. The volumetric contraction, which so far had been continuously increasing, starts decreasiong. Right after ultimate strength it gives way to volumetric expansion (dilation). This mechanism has important implications for the enhancement of concrete strength through confinement.".

Let's go back to the concrete cylinder. If you do the test. You will notice that the concrete fails by crushing and shearing outwards along inclined planes at middle of cylinder (not a bottom). Can you explain why the hour glass remaining is at center and not at bottom?

How familiar with you on the different models of concrete confinement? Confinement says if the lateral expansion after ultimate strength is restrained, there is enhancement of ultimate and compressive strength. So if the compressive strength of 3000 psi is applied to the test cylinder, and the bottom (or middle) is restrained by retrofitting it with steel. What is the formula that relates the compressive stress applied on top of the cylinder and the stress that must be resisted laterally (to increase compressive strength capacity) at right angle in middle (or bottom?) of cylinder at ultimate strength to oppose the large Poisson expansion arising from the opening up of internal macro-cracks parallel to the predominant compression stress?

 

[BAretired]

Can you explain why the hour glass remaining is at center and not at bottom?

A cone failure results when friction at the platens of the testing machine restrains lateral expansion of the concrete as the vertical compressive force is applied. This restraint confines the concrete near the platens and results in two relatively undamaged cones when the cylinder is tested to fracture. If the friction were eliminated, the cylinder would expand more laterally and exhibit a splitting failure. Such vertical splitting has been observed in numerous tests on high-strength specimens made of mortar or neat cement paste, but the effect is less common in ordinary concrete when coarse aggregate is present (Neville, A., Properties of Concrete, 4th Ed., Prentice Hall, 1995).

If a cylinder is laterally confined throughout its length during the test, the strength will be higher than that of an unconfined test. The increase in strength is related to the confining pressure. A Mohr's circle for 3D stress would be needed to determine its magnitude.

BA

 

[Ron]

The reason the profile is different at the center than at the bottom is partially because of the lack of restraint at the center as compared to the ends. The high compressive stress at the end creates a contract friction condition that does not exist at the center.

As for the stress level....

Assuming your stress level in near the center of the "almost linear" elastic range of the concrete, then you can compute the stress from f'c x mu or 1500 x 0.15 or about 225 psi. The difficulty is determining the area over which this stress acts. Secondly, the 225 psi might exceed the localized shear strength of the concrete, thus precipating failure.

 

[Ron]

Sorry..didn't finish my answer.

The triaxial stress state of the cylinder under compression yields a "shear-cone" failure mode when the loading is consistent and inordinate stress concentrations resulting from contact stresses or anomalous internal concrete conditions do not exist. When such anomalies exist, the failure might be a "slicing shear" failure or some other configuration of failure, resulting from stress concentrations that occur within or at the contact surface of the cylinder.

 

[lejam]

Thanks Ron.

what is the "mu" (can't be moment) in your mentioned "f'c x mu or 1500 x 0.15 or about 225 psi"?

So this 225 psi is the tranverse stress when 1500 psi is applied on top of the test cylinder?

Now if you can restrain the 225 psi on all sides of the cylinder, then the compression strength capacity can be increased (up to the limit of the restraining element) such that if you surround the concrete cylinder with diamond, the cylinder can have compressive strength of 50,000 psi just like sand in a drum is being restrained by the wall of the drum when sand alone can't support any load?

 

[Ron]

Mu is Poisson's ratio. I use 0.15 for this value as I have done lots of rigid pavement analysis and this is the accepted value for Poisson's ratio.

You are correct in that if you restrain the cylinder's lateral expansion, it will withstand more lateral stress, depending on the material used for the restraint.

 

[lejam]

I have thought of that initially. But Poisson's ratio is ratio of the transverse strain to the axial strain. Not the stress ratio. What's your proof that the relationship modulus=stress/strain supports Poisson's ratio = transverse stress/axial stress? I don't see this being use. Only transverse strain/transverse stress.

Most of us use codes for minimum spacing of ties (such as ACI). Do you have empirical formula that get the tranverse stresses that spacing of ties, bars sizes and patterns of confinement can resist (for example.. ties of certain sizes and spacing can resist 150 psi tranverse stresses)?

 

[BAretired]

Suppose the axial stress in a cylinder is 1500 psi. Then the axial strain is 1500/E[sub]c[/sub] and the transverse strain is μ*1500/E[sub]c[/sub] for a laterally unconfined test.

The radial pressure required to completely prevent transverse strain while maintaining 1500 psi axial pressure is greater than 1500*μ because the cylinder is being compressed in three dimensions. Also, the axial strain is less than the unconfined value.

Ties act as confinement steel and may theoretically resist a small portion of transverse strain, but it is not normally considered in design because (a) transverse strain is very small and (b) the ties must stretch in order to carry any load so that the differential strain is negligible.

BA

 

[Ron]

Lejam...in the range where stress and strain are proportional the ratio applies to either stress or strain

 

[lejam]

Thanks BA,

But the purpose of concrete jacket retrofitting is to resist the transverse strain... which is very small amount... do you think they can effectively do that? All concrete jackets have longitudinal bars at the ends and stirrups just like that of a column but it is wrapped to the original column. I wonder how can it effectively resist the tiny transverse strain of the orignal column at the ultimate strength when concrete is crushing.

If concrete jackets can't resist that tiny transverse strain (can it?). Maybe it's more practical usage is in avoiding the crushed concrete from shearing outwards by its confinement?

 

[Ron]

Sorry BA...I should have read your post before posting mine. Didn't mean to repeat what you had already more clearly stated!

 

[BAretired]

No problem Ron. But maybe you'd like to field the latest question about concrete jacket retrofitting as it is a little outside my area of expertise.

BA

 

[Ron]

BA...I'm no expert in it, but have a few points to offer.

First, the stress levels in the columns should not be close to f'c, unless this is a renovation/change of use with higher load considerations. If the loading can be determined properly, then a solution can be effected.

For jacketing, let's consider a round column, concentrically loaded. If the lateral strain is computed properly, then the lateral strain from the column will impose a stress on the jacket. That stress is hoop stress. Tensile reinforcement can then be determined to resist the hoop stress. Using this approach, there is inherent strain compatibility in the repair.

If there is no concern about localized crushing at the top or bottom of the column and it is only a mid-height concern developed from Poisson's ratio, then the jacket should be isolated from the top and bottom to remove direct axial load from the loading at the top or bottom of the column. That simplifies the condition and the analysis. Note that some axial load will still exist in the jacket from contact and bonding friction of the column face. This must be considered.

If the columns are rectangular (including square!), the stress distribution is a little different but the same concepts apply.

For jacketing like this, I would consider a carbon fiber overlay, provided the differential stresses are reasonable.