The method of calculating slab and shell reinforcement is based on the conception presented in the A.Capra and J-F. Maury's article titled "Calcul automatique du ferrailage optimal des plaques et coques en beton arme", Annales de l'Institut Technique du Batiment et des Travaux Publics, No.367, Decembre 1978.
Calculations Procedure
If the reinforcement values Ax and Ay (corresponding to two perpendicular directions x and y) are given, an equivalent reinforcement in any other direction (n) according to the following formula,
where α is an angle included between the direction x and the direction n.
The values of sectional forces (moments and membrane forces) M n , N n may be obtained from the following transformational formulas,
.
Thus, the below-presented inequality formulates the condition of correct reinforcement. The reinforcement that is able to carry the internal forces in an arbitrary section,
,
where
Φ (Mn, Nn) refers to the value of reinforcement required to carry the forces calculated for the direction 'n' - Mn, Nn.
Inequality
This determines on the plane (Ax, Ay) the area of 'admissible' values of reinforcement Ax, Ay (half-plane). If such area is determined for a sufficiently "dense" set of directions n (control is performed every 10), one obtains the area of admissible values Ax, Ay.
The adopted reinforcement is the minimal reinforcement which yields the minimal sum of surfaces Ax+Ay.
If a structure type or selection of calculation options causes reduction of internal forces, the reinforcement is calculated based on the following:
- Mn moments. Slab structure or simple bending option in a shell structure
- Nn membrane forces. Plane stress structure or compression/ tension option in a shell structure
- Complete set of Mn, Nn forces. Bending + compression/ tension option in a shell structure.
Note that in calculations of unidirectional reinforcement, the analytical method is limited to calculating the reinforcement only for the main reinforcement direction, without dividing it into n directions. A slab is designed only for the Mxx and Nxx set of forces.