2.5.1 Design Concept

Design Concept

In terms of their internal forces, shells are a combination of walls (chapter 2.3) and plates (chapter 2.4) because they contain both axial forces and moments.

That way, a design axial force and design moment are determined for each reinforcement direction on each side of the surface. One or both of the internal forces can become zero – if searching for the optimal direction of the concrete compression strut when determining the design internal forces results in the reinforcement not being activated in this direction.

When the design internal forces for the respective reinforcement direction are determined, the focus is on the direction of reinforcement for which design moments are available. For it, the program now carries out a common one-dimensional design of a beam with a width of one meter. The goal of this design, however, is not to find a required reinforcement but to determine the lever arm of the internal forces.

As soon as all lever arms of the design directions where a design moment occurs have been determined in this preliminary design, the program determines the smallest lever arm for each plate side. With this eccentricity, the moments of the linear plate analysis can now be transformed into membrane forces. To this end, the moments of the linear plate analysis are simply divided by the smallest lever arm z_min.

If you now add half of the axial force from the linear plate analysis that is perpendicular to the moment vector of the moment that is divided by the lever arm of the internal forces, you get the final membrane force. This process can be expressed as follows:

$$\begin{gathered} n_{xs}=\frac{m_x}{z_{min}} + \frac{n_x}{2} \\ {n}_{ys }= \frac{m_y}{z_{min}} + \frac{n_y}{2} \\ {n}_{xys} = \frac{m_{xy}}{z_{min}} + \frac{n_{xy}}{2} \end{gathered}$$

The moments at the top and bottom surface of the plate are considered with different algebraic signs.

When the moments m_x, m_y, and m_xy, as well as the axial forces n_x, n_y, and n_xy of the linear plate analysis have been substituted by the membrane forces n_xs, n_ys, and n_xys by means of the lever arm z_min from the preliminary design, the principal membrane forces n_Is and n_IIs can be determined from these membrane forces for the bottom and top surface of the plate.

As described in chapter 2.3, the design membrane forces n_α, n_β, and n_γ are determined from the principal membrane forces n_Is and n_IIs according to Equation 2.5 to Equation 2.7. These design membrane forces n_α, n_β, and n_γ are then assigned to the reinforcement directions φ₁, φ₂, and φ₃. This way, the design membrane forces n₁, n₂, and n₃ are obtained in the reinforcement directions.

The required amount of steel can be determined from the design membrane forces by dividing them by the steel stresses σ_s that have resulted during the determination of the minimum lever arm z_min in the respective reinforcement direction.

$$\begin{gathered} a_{s1} = \frac{n_1}{{\sigma }_s} \\ {a}_{s2} = \frac{n_2}{{\sigma }_s} \\ {a}_{s3} = \frac{n_3}{{\sigma }_s} \end{gathered}$$

If the design membrane force is a compression force, the concrete's resisting axial force n_c is first determined with the concrete neutral axis depth x, which has resulted from determining the lever arm.

$n_c = f_{cd} · b · x$

If the resisting axial force n_c of the concrete is not sufficient, a compression reinforcement is determined for the differential force between the acting axial force and the resisting axial force. The design stress for this compression reinforcement results from the deformation of the compression reinforcement during the determination of the lever arm z.

If the lever arm was determined under the assumption of the strain range III, no compression reinforcement will be determined because it was not assumed. The strain ranges I through V are described in the following chapter, in the part regarding the determination of the lever arm.