2.3.1 Design Internal Forces

Design Internal Forces

Determining the design internal forces for walls and diaphragms is carried out according to Baumann's [1] method of transformation. In this method, the equations for determining the 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 analyzes the equilibrium conditions with the following wall element.

Figure 2.8 shows a rectangular segment of a wall. It 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.

$N_2=k·N_{1 }$

Three reinforcement directions are applied in the wall. The reinforcement directions are labelled x, y, and z. The angle included in clockwise direction by the first principal axial force N₁ and the reinforcement direction x is labelled 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 are 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), which form the angles α, β, γ with the larger main tensile force N₁ and are labelled Z_x, Z_y, Z_z (positive as tensile forces).

To determine these forces Z_x, Z_y (and Z_z in case of a third reinforcement direction), we first have to define a section parallel to the third reinforcement direction.

The value of the section length is applied as 1. With this section length, we can 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 b_x (perpendicular to the tension force Z_x) and b_y (perpendicular to the tension force Z_y).

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

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

$Z_{x }· b_x=\frac{1}{\sin (\beta -\alpha )} · (N_1 · b_{1 }· \sin \beta - N_{2 }· b_2 · \cos \beta )$

$Z_y · b_y=\frac{1}{\sin (\beta -\alpha )} · (-N_1 · b_1 · \sin \alpha - N_2 · b_2 · \cos \alpha )$

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

Graphically, we can determine the following equilibrium.

The equilibrium between the external forces (N₁, N₂) and the internal forces Z_z can then be expressed as follows.

$Z_z · b_z=\frac{1}{\sin (\beta - \gamma )} · (N_1 · b_1 · \sin \beta - N_2 · b_2 · \cos \beta )$

If you replace the projected section lengths b₁, b₂, b_x, b_y, b_z with the values shown in the figure and use k as the quotient of the principal axial force N₂ divided by N₁, you get the following equations.

$\frac{Z_x}{N_1}=\frac{\sin {\displaystyle }{\displaystyle \beta }{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle \gamma }{\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle k}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \cos }{\displaystyle }{\displaystyle \beta }{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \cos }{\displaystyle }{\displaystyle \gamma }}{\sin {\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle (}{\displaystyle \gamma }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}}$

$\frac{Z_y}{N_1}= \frac{\sin {\displaystyle }{\displaystyle \alpha }{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle \gamma }{\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle k}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \cos }{\displaystyle }{\displaystyle \alpha }{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \cos }{\displaystyle }{\displaystyle \gamma }}{\sin {\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \gamma }{\displaystyle )}}$

$\frac{Z_z}{N_1}=\frac{- \sin \alpha · \sin \beta - k · \cos \alpha · \cos \beta }{\sin {\displaystyle }{\displaystyle (}{\displaystyle \beta }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \gamma }{\displaystyle )}{\displaystyle }{\displaystyle ·}{\displaystyle }{\displaystyle \sin }{\displaystyle }{\displaystyle (}{\displaystyle \gamma }{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle \alpha }{\displaystyle )}}$

These equations are the core of the design algorithm for RF-CONCRETE Surfaces. You can thus determine the design internal forces Z_x, Z_y, and Z_z for the respective reinforcement directions from the acting internal forces N₁ and N₂.

By adding up Equation 2.5, Equation 2.6, and Equation 2.7, you get:

$\frac{Z_x}{N_1} + \frac{Z_y}{N_1} + \frac{Z_z}{N_1} = 1 + k$

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

$Z_x + Z_y + Z_z = N_1 + N_{2 }$