RF-CONCRETE Surfaces Version 5

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RF-CONCRETE Surfaces Version 5

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2.4.1 Design Internal Forces

Design Internal Forces

The most important formulas used for the determination of 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 couple of diametrically opposed forces with the same absolute value situated at a certain distance from each other.

Plates differ from walls in that, among other things, the actions result in stresses with different algebraic signs on the opposing surfaces of the plate. Therefore, it would make sense to provide plates with reinforcement meshes with different directions for both surfaces of the plates. The principal moments m1 and m2 are determined in the centroidal plane of the surface. Hence, they must be distributed on the surfaces of the plate in order to determine the design moments for the reinforcement of the respective plate surface.

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

Figure 2.24 Plate element with local surface coordinate system in the centroidal plane of the plate

In RFEM, the bottom surface is always in the direction of the positive local surface axis z. Accordingly, the top surface is defined in the direction of the negative local z-axis.

Top and Bottom surface

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

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

Figure 2.25 Principal moments m1 and m2 in the centroidal plane of the plate

The principal moments are indicated by 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 mI and mII.

Figure 2.26 Principal moments shifted to the bottom surface of the plate

To obtain the principal moments needed to determine the design moments for the reinforcement mesh at the top surface of the plate, the principal moments are shifted to the plate's top surface. In addition, their direction is rotated 180°.

Figure 2.27 Principal moments shifted to the top surface of the plate

The principal moment is usually denoted m1, which, considering the algebraic sign, is the greater one (see Figure 2.27). Hence, the denotations of the principal moments at the top surface of the plate must be reversed.

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

Figure 2.28 Final principal moments at bottom and top surface of the plate

If the principal moments for both plate surfaces 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 surface.

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

Figure 2.29 Differential angle according to [1] for bottom surface of plate (in this case, for three directions of reinforcement)

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

mα = ml · sin β · sin γ + k · cos β · cos γsin (β - α) · sin (γ - α)mβ = ml · sin α · sin γ + k · cos α · cos γsin (β - α) · sin (β - γ)mγ = ml · -sin α · sin β + k · cos α · cos βsin (β - γ) · sin (γ - α) 

RF-CONCRETE Surfaces obtains the following design moments mα,+z, mβ,+z, and mγ,+z for the bottom surface of the plate:

Figure 2.30 Design moments according to [1] for the bottom surface of the plate

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

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

γ1,am = αm + βm2 

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

Figure 2.31 First assumption for the direction γ of the concrete compression strut

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

The analysis of further compression strut directions must show whether this is the energetic minimum with the least required reinforcement. These analyses are carried out in a similar way.

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

Figure 2.32 Sum of the absolute design moments

The Smallest Energy for all Valid Cases ∑min,+z is given as minimum absolute sum of the determined design moments. In the example, the reinforcement mesh from the reinforcement directions for the differential angle βm,+z,2a yields the most favorable solution for the bottom surface 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 φstrut related to the reinforcement direction. In the example, the following compression strut angle is determined for the plate's bottom surface:

Figure 2.33 Governing compression strut

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

Figure 2.34 Final design moments for bottom surface of plate
Literatur
[1] Deutscher Ausschuss für Stahlbeton, Heft 217: Tragwirkung orthogonaler Bewehrungsnetze beliebiger Richtung in Flächentragwerken aus Stahlbeton (von Theodor Baumann). Verlag Ernst & Sohn, Berlin, 1972.

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