2.4.1 Design Internal Forces

Design Internal Forces

The most important formulas for determining the design axial forces from the principal axial forces are presented in Equation 2.5 through Equation 2.7 in chapter 2.3. According to Baumann [1], these formulas can also be used for moments, because they are nothing more than a force couple with the same absolute value, situated at a certain distance from each other and with a diametrical direction.

Among other things, plates differ from walls in that the actions result in stresses with different algebraic signs on two opposing sides of the plate. It would therefore make sense to provide plates with reinforcement meshes with different directions for both sides of the plates. Since the principal moments m₁ and m₂ are determined in the centroidal plane of the surface, they must be distributed to the plate sides in order to be able to determine the design moments for the reinforcement of the respective plate side.

We look at a plate element with its loading. The surface's local coordinate system is located in the centroidal plane of the plate.

The surface axes can be switched on in the Display navigator by selecting Model → Surfaces → Surface Axis Systems x,y,z or in the shortcut menu of surfaces (see Figure 3.28).

In RFEM, the principal internal forces m₁ and m₂ are determined for the centroidal plane of the plate.

The principal moments are displayed as simple arrows. They are oriented like the reinforcement that would be required to resist them.

To obtain design moments for the reinforcement mesh at the bottom surface of the plate from these principal moments, the principal moments are shifted to the plate's bottom surface without alteration. For the design, they are described with Roman indexes as m_I and m_II.

To obtain the principal moments for determining the design moments for the reinforcement mesh at the top side of the plate, the principal moments are shifted to the plate's top surface. At the same time, their direction is rotated by 180°.

Since the principal moment is usually referred to as m₁, which is larger with respect to the algebraic sign (see Figure 2.27), the designations of the principal moments at the top side of the plate have to be reversed.

Thus, the principal moments for determining the design moments at both plate sides are represented as follows:

If the principal moments for both plate sides are known, the design moments can be determined. To that end, the first step is to determine the differential angle of the reinforcement directions to the direction of the principal moment at each plate side.

The smallest differential angle specifies the positive (clockwise) direction. All other angles are determined in this positive direction and then sorted by size. In RF-CONCRETE Surfaces, they are designated as α_m,+z, β_m,+z, and γ_m,+z as shown in the following example. The index +z indicates the bottom surface.

Then, Equations 2.5 through 2.7 are used according to Baumann [1] in order to determine the design moments:

$$\begin{gathered} m_{\alpha } = m_l · \frac{\sin \beta · \sin \gamma + k · \cos \beta · \cos \gamma }{\sin (\beta - \alpha ) · \sin (\gamma - \alpha )} \\ {m}_{\beta } = m_l · \frac{\sin \alpha · \sin \gamma + k · \cos \alpha · \cos \gamma }{\sin (\beta - \alpha ) · \sin (\beta - \gamma )} \\ {m}_{\gamma } = m_l · \frac{-\sin \alpha · \sin \beta + k · \cos \alpha · \cos \beta }{\sin (\beta - \gamma ) · \sin (\gamma - \alpha )} \end{gathered}$$

In RF-CONCRETE Surfaces, the design moments m_α,+z, m_β,+z, and m_γ,+z for the bottom surface of the plate are output as follows:

In this example, one of the design moments is less than zero. The program now searches for a reinforcement mesh consisting of two reinforcement layers that is stiffened by a concrete compression strut.

The first assumed reinforcement mesh consists of the two reinforcement sets in the directions α_m and β_m. The direction γ of the stiffening concrete compression strut (the stiffening moment that produces compression at this side of the plate) is assumed to be exactly between these two reinforcement directions.

${\gamma }_{1,am} = \frac{{\alpha }_m + {\beta }_m}{2}$

With the adapted Equations 2.5 through 2.7, the program once more determines the design moments in the selected reinforcement directions of the mesh and the moment that stiffens them. In the example, the result for the plate's bottom side is the following.

The assumption of the reinforcement mesh results in a viable solution, because the direction of the compression strut is valid.

The analysis of additional compression strut directions must show whether it is the energetic minimum with the least required reinforcement. These analyses are carried out analogously.

Once all sensible possibilities for a reinforcement mesh consisting of two reinforcement directions and a stiffening concrete compression strut are analyzed, the sums of the absolute design moments are shown. For the example above, the overview looks as follows.

The Smallest Energy for all Valid Cases ∑_min,+z is given as the minimum absolute sum of the determined design moments. In the example, the reinforcement mesh from the reinforcement layouts for the differential angle β_m,+z,2a yields the most favorable solution for the bottom side of the plate.

The design details also show the direction of the governing compression strut. This direction is related to the definition of the differential angles according to Baumann. Hence, the program also gives the direction φ_strut in relation to the reinforcement direction. In the example, the following compression strut angle is determined for the plate's bottom side:

For an optimized direction of the design moment that stiffens the reinforcement mesh (see Figure 3.47), the design moments according to Baumann are obtained for the example above. These design moments are applied to the defined reinforcement directions as shown in the following figure.