Concrete shell design with CEB\fib 2010 (three-layer model) 2

[LonelyDeer]

Hello. I have a question about design of concrete shells with three-layer method described in p.7.3.2.2 fib Model Code for concrete structures.
The main idea of method is that we divide shell on three layers. The outer layers provide resistance to the in-plane effects of both the bending and the in-plane axial loading, while core provides a shear transfer between the outer layers.

The action effects of applied loads are expressed as eight components, three moments per unit width(m[sub]x[/sub],m[sub]y[/sub],m[sub]xy[/sub]), three axial forces per unit width (n[sub]x[/sub],n[sub]y[/sub],n[sub]xy[/sub]) and two shear forces per unit width ((v[sub]x[/sub],v[sub]y[/sub]) in directions parallel to the orthogonal reinforcement.
(there is positive directions of forces on picture)

Then we use some formula to find forces in the outer layers.

where:
The subscript notation inf and sup refer to the interior and superior faces on the element. The interior face is the tensile face for an element in positive bending.

v[sub]0[/sub]=sqrt(v[sub]x[/sub][sup]2[/sup]+v[sub]y[/sub][sup]2[/sup])

\theta is the inclination of compression stresses in the core layer

Here we have the picture for the equations:

And here is my question(s):
1. Why is plus sign in equations (7.3-1 - 7.3-3) in front of
/(v[sub]x[/sub])^2/(2*v[sub]0[/sub])
/ (v[sub]y[/sub])^2/(2*v[sub]0[/sub])
/ (v[sub]x[/sub]*v[sub]y[/sub])/(2*v[sub]0[/sub])
2. Why v[sub]0[/sub]*cot(\theta)/2 direction in sup part of shell is the same as inf part?

3.I don't understand how we got this part in equation.
Does anybody have any information about the three-layer model? Can you explain me this part?
Do you know any articles explained this?

 

[Guest262653]

Here are some additional links that may be of help:

https://repositorium.sdum.uminho.pt...einforcement design using linear analysis.pdf

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.918.9730&rep=rep1&type=pdf

 

[LonelyDeer]

Thank you, avscorreia!
Your link [URL unfurl="true"]http://citeseerx.ist.psu.edu/viewdoc/download?doi=...[/url] is very helpful. I've found some interesting information, but still I don't have an answer.
Now I'm trying to find original articles, writen by prof. Peter Marti:
1. MARTI, P. (1990) – Design of Concrete Slabs for Transverse Shear, ACI Structural
Journal, March-April, p. 180-190.
2. MARTI, P. (1991) – Dimensioning and Detail
ing, Structural Engineering, IABSE, 62,
Stuttgart, Germany, p. 411-443.

I've found only part of article #2, but without main idea of method.

Does anybody have any of these articles?

P.S.
I've wrote a letter to IABSE. Maybe they can help.

 

[LonelyDeer]

Hello.

I've got an answer from IABSE. Now I have article #2 (2. MARTI, P. (1991) – Dimensioning and Detailing, Structural Engineering, IABSE, 62, Stuttgart, Germany, p. 411-443.). After a week I've managed to find correct way to explain this. Here is a small pic with an answer.

The main idea, that picture 7.3-3 is really confusing. Shear component acts in opposite way, but the formula is correct.
I don't know, maybe this will help anybody.
Thank you for help, avscorreia.