Optimum Design of RC Shells and Slab Question

[pyseng]

I have recently been interested in using FEA to model and design complex concrete shapes. Through my research, I've found that the Sandwich model, used by SAP2000 and ?Eurocode?, is very logically consistent to me and I would be comfortable using it under the so-called "engineering judgement/rational analysis" provisions of ACI.

However, I just want to make sure I fully understand the methodology before I move forward. I am struggling with one of the concepts presented in the derivation of this method. The figure below is taken from "Optimum Design of Reinforced Concrete Shells and Slabs" by Brondum-Nielsen (which is a primary source reference in the SAP2000 technical note).

Based on equilibrium, the author derives the equations:

Nxa = Nx + abs(Mxy)*cot(v)​

Nya = Ny + abs(Mxy)*tan(v)​

The author then says "If both Nxa and Nya are positive, the necessary reinforcement is proportional to Nxa + Nya. Minimum of reinforcement thus corresponds to v = 45 deg."

Therefore, cot(v) = tan(v) = 1 and the final equations are given as:

Nxa = Nx + abs(Nxy)​

Nya = Ny + abs(Nxy)​

Even though the author provides an explanation for taking v = 45 deg, I still just don't understand it. I was hoping that someone with some experience here might be able to offer an alternate explanation. Thank you!

 

[hardbutmild]

In plastic design you primarily need to satisfy the equilibrium. You can choose how much of the reinforcement you'd like in each direction as long as the combined reinforcement is able to resist the loads.
That being said, if you need reinforcement in both directions, then the minimum is achieved if the angle is 45°. This can be proved purely mathematically. Imagine a triangle with a right angle. If you vary the other two angles, you'll get different lengths of the hypotenuse. The longest hypotenuse will be in the case of equal sides (cathetus). It's purely geometrical consideration. If hypotenuse is c, and sides a and b, then
c[sup]2[/sup] = a[sup]2[/sup] + b[sup]2[/sup]
To get the maximum c, a needs to be equal to b. In that case c = 1,41*a.
So to achieve the same resultant c, you need the least a+b (in other words, the least combined reinforcement).

Someone correct me if I'm wrong.

 

[HTURKAK]

If we consider the membrane element with in biaxial tension, it is expected to provide steel reinforcement ( at lesat ) in two orthogonal directions and could be assumed the shear stresses will be resisted by concrete . In case of orthogonal reinforcement, 45 degr. will provide the lowest concrete strength requirement.

The yield line theory gives us to choose how much of the reinforcement we prefer in each direction .
Just for curious, is this approach always true ?. What are the limitations thta this approach true for R.C. Elements ?

Consider a RC concrete panel. The tension stresses before cracking in the concrete will remain elastic and the stresses in the reinforcement will be negligible. But after cracking , the tension stresses in the concrete will approach to zero and steel stresses will increase. We may assume that the concrete will not fail in compression . That is , the developed crack path will be stable until yield of the steel in one-direction then , the forces will be redistributed to provide balance.

However, what will be the acceptable limit for the amount of redistribution that is capable by the RC panel ?
IMO, the designer should question the ductility demand and , concrete elements should not be pushed more that can not be tolerable.

Imagine a typical tension- compression element. Taking v = 45 deg will lead to tension reinforcement in one direction and compression reinforcement in the other direction.. That is, choosing v = 45 deg . is not always true..

I would like to remind ;

(The author then says "If both Nxa and Nya are positive, the necessary reinforcement is proportional to Nxa + Nya. Minimum of reinforcement thus corresponds to v = 45 deg.")


Did you look EC 2 Part 2: Concrete bridges Annex LL ?