# 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 axial forces as well as 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.

Such a design axial force and design moment are determined for each direction of reinforcement on each side of the surface. One or both internal forces can become zero – if searching for the optimal direction of the concrete compression strut in the determination of the design internal forces results in the fact that the reinforcement will not be activated in this direction.

When the design internal forces are determined for the respective direction of reinforcement, the focus is on that direction of reinforcement for which design **moments** are available.
Now, the program carries out a common one-dimensional design of a beam with the 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 those design directions where a design moment occurs have been determined in this preliminary design, the program determines the smallest lever arm for each plate surface.
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}.

Now, if we add half the axial force from the linear plate analysis that is perpendicular to the moment vector of the moment, which is divided by the lever arm of the internal forces, we obtain 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.

Once 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.

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 through Equation 2.7 as described in chapter 2.3.
The design membrane forces n_{α}, n_{β}, and n_{γ} are then assigned to the reinforcement directions φ_{1}, φ_{2}, and φ_{3}.
In this way, we obtain the design membrane forces n_{1}, n_{2}, and n_{3} in the reinforcement directions.

From the design membrane forces, we can determine the required amount of steel. To do this, the design membrane forces are divided by the steel stresses σ_{s} that are determined during the determination of 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 depth of the neutral axis x of concrete, which resulted from the determination of the lever arm.

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

If the resisting axial force n_{c} of 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 in 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 concerning the determination of the lever arm.