RF-CONCRETE Surfaces Version 5

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RF-CONCRETE Surfaces Version 5

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2.3.1 Design Internal Forces

Design Internal Forces

The determination of the design internal forces for walls and diaphragms is carried out according to Baumann's method of transformation [1]. In this method, the equations for the determination of design internal forces are derived for the general case of a reinforcement with three arbitrary directions. Then, these forces can be applied to simpler cases such as orthogonal reinforcement meshes with two reinforcement directions.

Baumann analyzed the equilibrium conditions with the following wall element.

Figure 2.8 Equilibrium conditions according to Baumann

Figure 2.8 shows a rectangular segment of a wall. The wall is subjected to the principal axial forces N1 and N2 (tensile forces). The principal axial force N2 is expressed by means of the factor k as a multiple of the principal axial force N1.

N2=k·N1 

Three reinforcement directions are applied in the wall. The reinforcement directions are designated x, y, and z. The angle included clockwise by the first principal axial force N1 and the reinforcement direction x is designated α. The angle between the first principal axial force N1 and the reinforcement direction y is called β. The angle to the remaining reinforcement set is called γ.

Baumann writes in his thesis: If the shear and tension stresses in the concrete is neglected, the external loading (N1, N2 = k ⋅ N1) of a wall element can generally be resisted by three internal forces oriented in any direction. In a reinforcement mesh with three reinforcement directions, these forces correspond to the three reinforcement directions (x), (y), and (z). Those directions form the angles α, β, γ with the greater main tensile force N1, and are called Zx, Zy, Zz (positive as tensile forces).

To determine those forces Zx, Zy (and Zz in case of a third reinforcement direction), we first define a section parallel to the third reinforcement direction.

Figure 2.9 Section parallel to the third reinforcement direction z

The value of the section length is assumed as 1. With this section length, we determine the projected section lengths that run perpendicular to the respective force. In the case of the external forces, these are the projected section lengths b1 (perpendicular to force N1) and b2 (perpendicular to force N2). In the case of the tensile forces in the reinforcement, these are the projected section lengths bx (perpendicular to tension force Zx) and by (perpendicular to tension force Zy).

The product of the respective force and the corresponding projected section length yields the force that can be used to establish an equilibrium of forces.

Figure 2.10 Equilibrium of forces in a section parallel to the reinforcement in the direction z

The equilibrium between the external forces (N1, N2) and the internal forces (Zx, Zy) can thus be expressed as follows.

Zx · bx=1sin (β-α) · (N1 · b1 · sin β - N2 · b2 · cos β) 

Zy · by=1sin (β-α) · (-N1 · b1 · sin α - N2 · b2 · cos α) 

To determine the equilibrium between the external forces (N1, N2) and the internal force Zz in the reinforcement direction z, we define a section parallel to the reinforcement direction x.

Figure 2.11 Section parallel to the reinforcement direction x

Graphically, we can determine the following equilibrium.

Figure 2.12 Equilibrium of forces in a section parallel to the reinforcement in the direction x

Thus, the equilibrium between the external forces (N1, N2) and the internal forces Zz can be expressed as follows.

Zz · bz=1sin (β - γ) · (N1 · b1 · sin β - N2 · b2 · cos β) 

If we replace the projected section lengths b1, b2, bx, by, bz by the values shown in the figure and use k as the quotient of the principal axial force N2 divided by N1, we obtain the following equations.

ZxN1=sin β · sin γ + k · cos β · cos γsin (β - α) · sin (γ - α) 

ZyN1= sin α · sin γ + k · cos α · cos γsin (β - α) · sin (β - γ) 

ZzN1=- sin α · sin β + k · cos α · cos βsin (β - γ) · sin (γ - α) 

These equations are at the core of the design algorithm for RF-CONCRETE Surfaces. Thus, we can determine the design internal forces Zx, Zy, and Zz for the respective reinforcement directions from the acting internal forces N1 and N2.

By adding Equation 2.5, Equation 2.6, and Equation 2.7, we obtain:

ZxN1 + ZyN1 + ZzN1 = 1 + k 

By multiplying Equation 2.8 with N1 and substituting k for N2 / N1, we obtain the following equation that clarifies the equilibrium of the internal and external forces.

Zx + Zy + Zz = N1 + N2 

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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