# RF-CONCRETE Surfaces Version 5

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## RF-CONCRETE Surfaces Version 5

# 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.

All 3D model types (see Figure 2.1) are designed as shells. RF-CONCRETE Surfaces proceeds as follows: First, as shown in chapter 2.3 and chapter 2.4, the design axial forces and design bending moments are determined separately. They are once again based on the principal axial forces and principal bending moments of the linear RFEM plate analysis.

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 that direction of reinforcement for which design **moments** are available,
for which 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 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:

${n}_{xs}=\frac{{m}_{x}}{{z}_{min}}+\frac{{n}_{x}}{2}\phantom{\rule{0ex}{0ex}}{n}_{ys}=\frac{{m}_{y}}{{z}_{min}}+\frac{{n}_{y}}{2}\phantom{\rule{0ex}{0ex}}{n}_{xys}=\frac{{m}_{xy}}{{z}_{min}}+\frac{{n}_{xy}}{2}$

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 side of the plate.

From the principal membrane forces n_{Is} and n_{IIs}, the *design membrane forces* n_{α}, n_{β}, and n_{γ} are determined according to Equation 2.5 to Equation 2.7 as described in chapter 2.3.
These design membrane forces n_{α}, n_{β}, and n_{γ} are then assigned to the reinforcement directions φ_{1}, φ_{2}, and φ_{3}.
This way, the design membrane forces n_{1}, n_{2}, and n_{3} are obtained in the reinforcement directions.

You can determine the required amount of steel from the design membrane forces by dividing them by the steel stresses σ_{s} that result when determining the minimum lever arm z_{min} in the respective reinforcement direction.

${a}_{s1}=\frac{{n}_{1}}{{\sigma}_{s}}\phantom{\rule{0ex}{0ex}}{a}_{s2}=\frac{{n}_{2}}{{\sigma}_{s}}\phantom{\rule{0ex}{0ex}}{a}_{s3}=\frac{{n}_{3}}{{\sigma}_{s}}$

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.

${n}_{c}={f}_{cd}\xb7b\xb7x$

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 when determining 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.