Max Reinforcing for CMU Shearwalls 1

[Boof19]

NCMA TEK 14-4b has this equation for the maximum reinforcement ratio in CMU shearwalls:

I'm hoping one of you can help me understand how this equation isn't blowing up and going to infinity. The way I'm understanding it, that equation is dividing by 0 once my steel reinforcement reaches it's yield limit. (Which in a special wall is pretty damn quick, e.g. the intent of the alpha factor...)

I've found the same equation in ACI 530 (TMS 402), so I'm faaaairly certain it's not a typo. But I haven't found any evidence online of people screaming about an error, so I'm erring on the side of myself just not understanding the correct intent.

 

[Agent666]

In ACI350, this eqn is preceeded with the provision that it applies when there is an equal area of tension and compression reinforcement. In this case I'd expect the other term in the Min criteria to possibly govern (not epsilon_y)? It would I assume be very unusual for say a masonry wall with evenly distributed vertical reinforcement to end up with equal areas of reinforcement in compression and tension.

If this equal criteria doesn't apply then it's just f_y on the bottom as I read it from ACI350.

 

[Deker]

It may be helpful here to plot the equation against some values of d'/d to get an idea of what is happening. The blue line in the charts below plot your equation without the ε_y cap in the denominator. The orange line keeps the cap. I have assumed the following values:

f'm = 2 ksi
fy = 60 ksi
Es = 29000 ksi
ε_mu = 0.0025
ε_y = 0.002
α = 4
P/bd = 0

Notice the discontinuity that occurs when ε_s = ε_y at around d'/d = 0.045. Now let's plot the uncapped equation with an assumed P/bd = 0.5 ksi. Since this produces a negative numerator, we should expect ρ_max to be negative, thereby alerting us that the wall needs to be reconfigured to avoid brittle failure prior to achieving the desired ductility. However, if we look at the uncapped equation where ε_s > ε_y at small values of d'/d, we see that ρ_max flips back to positive. This might lead a designer to the incorrect conclusion that the wall configuration would be permitted.

Edit:

The way I'm understanding it, that equation is dividing by 0 once my steel reinforcement reaches it's yield limit. (Which in a special wall is pretty damn quick, e.g. the intent of the alpha factor...)

I thought I should also clarify that ε_s is the strain in the compression reinforcement without application of the α factor...just in case that was the source of confusion.

 

[Boof19]

Deker,

Thank you so much, yes that was the source of my confusion. Thanks for the response everyone, much appreciated