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Overview

1 Introduction

2 Theoretical Background

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.

Image 2.49 Calculation parameters of the design

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

$z = d - \frac{k \cdot x}{2}$

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

Image 2.50 Ranges of strain distribution

Range I

This range shows a cross-section strongly subjected to 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.

Range II

In this range, compression predominantly occurs. The depth of the neutral axis ranges between the limits ξ _lim ⋅ d and h/k.

Range III

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

Range IV

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 only subjected to compression forces.

Range V

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

The lever arm is determined for each strain range. This makes it possible to divide the moments of the linear plate analysis into membrane forces.

Lever arm for range I

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

Image 2.51 Lever arm z for maximum depth of neutral axis of concrete

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

$F_{c d} = κ \cdot f_{c d} \cdot k \cdot x_{lim} \cdot b$

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

$m_{s d, lim} = F_{c d} \cdot (d - \frac{k \cdot x_{lim}}{2})$

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).

$∆ m_{s d} = m_{s d (1)} \cdot m_{s d, lim}$

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.

$m_{s d (1)} = m_{s d} - n_{s d} \cdot z_{s {(1)}_{}}$

With the differential moment Δm_sd, you can now determine the required compression force F_sd(2) in a compression reinforcement.

$F_{s d (2)} = \frac{∆ m_{s d}}{d - d_{2}}$

Here, d is the effective depth of the tension reinforcement and d₂ is the centroidal distance of the compression reinforcement from the edge of the concrete compression zone.

If you 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), the desired lever arm z is obtained.

$z = \frac{m_{s d}}{|F_{c d} + F_{s d (2)}|}$

Lever arm for range II

In order to be able to determine the concrete's neutral axis depth x, we first determine the design moment m_sd(2) about the centroid of the compression reinforcement.

$m_{s d (2)} = m_{s d} + n_{s d} + z_{s (2)}$

The sum of the moments about the centroid of the compression reinforcement is now calculated. These moments must amount 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.

$\underset{}{\sum m = F_{c d} \cdot (\frac{k \cdot x}{2} - d_{2}) + m_{s d (2)} = 0}$

The depth x of the concrete neutral axis is also contained in the resulting concrete compression force F_cd.

$F_{c d} = κ \cdot f_{c d} \cdot k \cdot x \cdot b$

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

$κ \cdot f_{c d} \cdot k \cdot x \cdot b \cdot (\frac{k \cdot x}{2} - d_{2}) + m_{s d (2)} = \frac{κ \cdot f_{c d} \cdot k^{2} \cdot x^{2}}{2} - κ \cdot f_{c d} \cdot k \cdot x \cdot b \cdot d_{2} + {m_{s d (2)}}_{} =_{} 0 x^{2} = \frac{2 \cdot d_{2} \cdot x}{k} + \frac{2 \cdot m_{s d (2)}}{κ \cdot f_{c d} \cdot b \cdot k^{2}} = 0 \Rightarrow x = \frac{d_{2}}{k} + \sqrt{{(\frac{d_{2}}{k})}^{2} - \frac{2 \cdot m_{s d (2)}}{κ \cdot f_{c d} \cdot b \cdot k^{2}}}$

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

$z = d - \frac{k \cdot x}{2}$

Lever arm for range III

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.

$m_{s d (1)} = m_{s d} + n_{s d} + z_{s (1)}$

The sum of the moments about the tension reinforcement's centroid is now calculated. These moments must amount to zero. On the side of the resistance, the moment is calculated only from the resulting force F_cd of the concrete compression zone times its distance. Then the equilibrium of the moments about the position of the tension reinforcement is calculated.

$\underset{}{\sum m = F_{c d} \cdot (d - \frac{k \cdot x}{2}) - m_{s d (1)} = 0}$

The depth x of the concrete's neutral axis is also contained in the resulting concrete compression force F_cd (see Equation 2.52).

$κ \cdot f_{c d} \cdot k \cdot b \cdot d \cdot x - (\frac{κ \cdot f_{c d} \cdot k^{2} \cdot b}{2}) - {m_{s d (1)}}_{} =_{} x^{2} - \frac{2 d}{k} \cdot x + {\frac{2 m_{s d (1)}}{κ \cdot f_{c d} \cdot k^{2} \cdot b}}_{} =_{} 0$

This quadratic equation can be solved as follows.

$x = \frac{d}{k} + \sqrt{\frac{d^{2}}{k^{2}} - \frac{2 \cdot m_{s d (1)}}{κ \cdot f_{c d} \cdot k^{2} \cdot b}} = 0$

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

$z = d - \frac{k \cdot x}{2}$

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.

Lever arm for range IV

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 that ε_s = ε_cu.

When 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).

Lever arm for range V

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

Parent section

2.5 Shells