# RF-CONCRETE Surfaces Version 5

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## RF-CONCRETE Surfaces Version 5

# 2.5.2 Lever Arm of Internal Forces

### Lever Arm of Internal Forces

A rectangular cross-section with a width of one meter is always designed. The design is carried out directly with the rectangular stress distribution (see EN 1992-1-1, Figure 3.5). An iterative procedure would take too much time because of the high number of necessary designs.

The lever arm z is determined for the figure above as follows.

$z=d-\frac{k\xb7x}{2}$

Figure 2.49 shows a state of strain that may arise when the moment and the axial force act simultaneously. Five states of strain are possible (see Figure 2.50).

This is a cross-section subjected to great bending.
The depth of the neutral axis has reached its maximum value (x = ξ _{lim} ⋅ d).
Another increase of the section modulus is only possible by using a compression reinforcement.

Compression predominantly occurs in this range.
The depth of the neutral axis is between the limits ξ _{lim} ⋅ d and h/k.

The applied moment is so small that the concrete compression zone (neutral axis) without compression reinforcement is able to provide a sufficient section modulus.
Depending on the applied moment, the limits for the neutral axis depth are between 0 and ξ _{lim} ⋅ d.

This range shows a fully compressed cross-section. The depth of the neutral axis is greater than h/k. This range also includes cross-sections that are subjected to compression forces only.

This state of strain is present if the tension force cracks a cross-section completely. This range also includes cross-sections that are subjected to tension forces only.

The lever arm is determined for each strain range. With this, the moments of the linear plate analysis can be divided into membrane forces.

For this range, the depth of the neutral axis is known: The concrete is fully utilized before a compression reinforcement is applied.

For the maximum depth of the neutral axis x, the resisting concrete compressive force F_{cd} is obtained according to the following equation:

${F}_{cd}=\kappa \xb7{f}_{cd}\xb7k\xb7{x}_{\text{lim}}\xb7b$

The limit section modulus m_{sd,lim}, which can be resisted by the cross-section without compression reinforcement, is determined as follows:

${m}_{sd,\text{lim}}={F}_{cd}\xb7\left(d-\frac{k\xb7{x}_{\text{lim}}}{2}\right)$

With the limit section modulus m_{sd,lim}, it is possible to determine the differential moment ∆m_{sd} that has to come from the compression reinforcement in order to reach an equilibrium with the applied moment m_{sd(1)}.

$\u2206{m}_{sd}={m}_{sd\left(1\right)}\xb7{m}_{sd,\text{lim}}$

The applied moment m_{sd(1)} relates to the centroid of the tension reinforcement.
It results from the applied moment m_{sd}, the acting axial force n_{sd}, and the distance z_{s(1)} between the centroidal axis of the cross-section and the centroidal axis of the tension reinforcement.

${m}_{sd\left(1\right)}={m}_{sd}-{n}_{sd}\xb7{z}_{s{\left(1\right)}_{}}$

With the differential moment Δm_{sd}, it is now possible to determine the required compression force F_{sd(2)} in a compression reinforcement.

${F}_{sd\left(2\right)}=\frac{\u2206{m}_{sd}}{d-{d}_{2}}$

where d is the effective depth of the tension reinforcement and d_{2} is the centroidal distance of the compression reinforcement from the edge of the concrete compression zone.

If we divide the applied moment m_{sd(1)}, which is related to the centroid of the tension reinforcement, by the concrete compression force F_{cd} and the force in the compression reinforcement F_{sd(2)}, we obtain the lever arm z.

$z=\frac{{m}_{sd}}{\left|{F}_{cd}+{F}_{sd\left(2\right)}\right|}$

In order to be able to determine the concrete neutral axis depth x, we first determine the design moment m_{sd(2)} about the centroid of the compression reinforcement.

${m}_{sd\left(2\right)}={m}_{sd}+{n}_{sd}+{z}_{s\left(2\right)}$

Now the sum of the moments about the centroid of the compression reinforcement is calculated.
These moments must be equal to zero.
On the side of the resistance, the moment is created only from the resulting force F_{cd} of the concrete compression zones multiplied by its distance.
In range II, there is no reinforcement in tension.

$\underset{}{\sum m={F}_{cd}\xb7\left(\frac{k\xb7x}{2}-{d}_{2}\right)+{m}_{sd\left(2\right)}=0}$

The depth x of the concrete neutral axis is also contained in the resulting concrete compression force F_{cd}.

${F}_{cd}=\kappa \xb7{f}_{cd}\xb7k\xb7x\xb7b$

Thus, the equation for the determination of x is obtained as:

$\kappa \xb7{f}_{cd}\xb7k\xb7x\xb7b\xb7\left(\frac{k\xb7x}{2}-{d}_{2}\right)+{m}_{sd\left(2\right)}=\frac{\kappa \xb7{f}_{cd}\xb7{k}^{2}\xb7{x}^{2}}{2}-\kappa \xb7{f}_{cd}\xb7k\xb7x\xb7b\xb7{d}_{2}+{{m}_{sd\left(2\right)}}_{}{=}_{}0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{x}^{2}=\frac{2\xb7{d}_{2}\xb7x}{k}+\frac{2\xb7{m}_{sd\left(2\right)}}{\kappa \xb7{f}_{cd}\xb7b\xb7{k}^{2}}=0\Rightarrow x=\frac{{d}_{2}}{k}+\sqrt{{\left(\frac{{d}_{2}}{k}\right)}^{2}-\frac{2\xb7{m}_{sd\left(2\right)}}{\kappa \xb7{f}_{cd}\xb7b\xb7{k}^{2}}}$

With the depth x of the neutral axis, it is possible to determine the lever arm z by subtracting half the neutral axis depth x from the effective height d, which is reduced by the factor k:

$z=d\xb7\frac{k\xb7x}{2}$

To determine the depth x of the neutral axis, we first determine the design moment m_{sd(1)} about the centroid of the tension reinforcement.

${m}_{sd\left(1\right)}={m}_{sd}+{n}_{sd}+{z}_{s\left(1\right)}$

Now the sum of the moments about the tension reinforcement's centroid is calculated.
These moments must be equal to zero.
On the resistance side, the moment is calculated only from the resulting force F_{cd} of the concrete neutral axis times its distance.
Then, the equilibrium of the moments about the position of the tension reinforcement is calculated.

$\underset{}{\sum m={F}_{cd}\xb7\left(d-\frac{k\xb7x}{2}\right)-{m}_{sd\left(1\right)}=0}$

The depth x of the concrete neutral axis is also contained in the resulting concrete compression force F_{cd} (see Equation 2.52).

$\kappa \xb7{f}_{cd}\xb7k\xb7b\xb7d\xb7x-\left(\frac{\kappa \xb7{f}_{cd}\xb7{k}^{2}\xb7b}{2}\right)-{{m}_{sd\left(1\right)}}_{}{=}_{}{x}^{2}-\frac{2d}{k}\xb7x+{\frac{2{m}_{sd\left(1\right)}}{\kappa \xb7{f}_{cd}\xb7{k}^{2}\xb7b}}_{}{=}_{}0$

This quadratic equation can be solved as follows.

$x=\frac{d}{k}+\sqrt{\frac{{d}^{2}}{{k}^{2}}-\frac{2\xb7{m}_{sd\left(1\right)}}{\kappa \xb7{f}_{cd}\xb7{k}^{2}\xb7b}}=0$

With the depth x of the concrete neutral axis, the lever arm z can be determined by subtracting half the depth of the neutral axis x, which is reduced by the factor k, from the effective height d:

$z=d\xb7\frac{k\xb7x}{2}$

If the steel strain ε_{s} is greater than the maximum allowable steel strain ε_{ud}, x is calculated iteratively from the equilibrium conditions.
The conversion factors κ and k for the concrete neutral axis are directly derived from the concrete's parabola-rectangle diagram.

In a fully compressed cross-section, the lever arm is assumed as the distance between both reinforcements.

$z=d-{d}_{2}$

For this range, a maximum utilization of the reinforcement is specified, meaning ε_{s} = ε_{cu}.

In the case that the compression is approximately concentric (e_{d}/h ≤ 0.1), the mean compressive strain should be limited to ε_{c2} according to EN 1992-1-1, clause 6.1 (5).

In a fully cracked cross-section, the lever arm is also assumed as the distance between the two reinforcements (see Equation 2.60).