Neutral Axis of a Beam in Reinforced Concrete: Australian Standards AS3600

The neutral axis is the imaginary line within a beam section or cross section where no longitudinal stress occurs during bending. The neutral axis is defined as the intersection of the neutral surface with the cross-sectional plane. On one side of the neutral axis, the material is in compression, and on the other side, it is in tension. The exact distribution depends on the direction of bending, geometry of the cross section, and material properties. Different cross sections may have different neutral axes depending on their geometry and loading. This axis separates these zones and shifts in response to changes in structural behaviour.

A beam can be thought of as composed of many thin fibers or layers. When the beam bends, some fibers are in tension while others are in compression. The neutral axis is related to the neutral surface, which is the surface where fibers are neither stretched nor compressed.

When analysing reinforced concrete behaviour, three key stages can be defined depending on the condition of the material: (1) uncracked and linear elastic, (2) cracked but still linear elastic, and (3) cracked and inelastic. Each stage corresponds to a different structural response and requires its own method to locate the neutral axis. Understanding these stages is crucial for applying the correct calculations during serviceability and strength limit states.

In this article, we explore these stages in detail, highlight strain distributions, and compare code provisions across ACI 318, Eurocode 2, and AS3600.

Understanding the Neutral Axis of a Beam

The location of the neutral axis depends on the geometry of the cross section, the material properties, and whether the section is cracked or uncracked. It is a critical concept for determining how internal forces such as bending moment and shear force are resisted.

To find its position, we apply the equilibrium of internal forces: where the sum of compression forces is set equal to the sum of tension forces. This analysis allows us to evaluate the strain distribution and resulting stresses.

The position of the neutral axis can also be defined by the coordinates of a specific point \(zNA, yNA\) within the cross section, which relates to the geometry and stress distribution.

Transformed Section and the Neutral Axis in Reinforced Concrete

Concrete and steel are the primary materials in reinforced concrete, each with different stress-strain behaviours. To simplify calculations, we convert the steel areas into an equivalent area of concrete using the modular ratio. This yields a transformed section, making it easier to apply the inertia equation and determine bending stresses. Complex cross sections are often divided into simpler rectangular shapes to facilitate calculation of the neutral axis, which is a key part of structural analysis.

This method allows us to use a single material assumption when calculating the neutral axis and moment capacity. For each stage of section behaviour, a different neutral axis location is calculated.

The location of the neutral axis depends on the geometry and cracking of the RC section. The neutral axis can be found using equilibrium of internal forces, to which the section is subjected: \(\sum{C} - \sum{T} = 0\) where \(\sum{C}\) and \(\sum{T}\) are the sums of compression and tension forces, respectively.

Three different stages can be defined for the determination of the neutral axis, as provided below.

To simplify the calculation, we “transform” our section into an equivalent homogeneous section, that is, a section with one material (concrete) rather than two materials (concrete and steel). The \((n - 1)A_{st}\)​ and \((n - 1)A_{sc}\)​ terms in the below equations, where \(n=\frac{E_s}{E_c}\)​​ is the modular ratio, transforms the steel areas into equivalent concrete areas.

Stage 1: Uncracked and Linear Elastic

Basically, the depth of the neutral axis can be approximated using a simplified formula that depends on the main dimensions of the section, such as height and width.

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$$\begin{align}&\text{Compression in concrete, }&C_c &= \frac{1}{2} E_c \varepsilon_{comp} d_nb - E_c\varepsilon_{sc}A_{sc}  \\[1em]&\text{Compression in steel, } &C_s &= E_s\varepsilon_{sc}A_{sc}  \\[1em]&\text{Tension in concrete, }&T_c &= \frac{1}{2}E_c\varepsilon_{tens}(h-d_n)b - E_c\varepsilon_{st}A_{st} \\[1em] &\text{Tension in steel, } &T_s&=E_s\varepsilon_{st}A_st \\[1em] &\therefore &C_c + C_s&=T_c+T_s\end{align}$$

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Rearranging the above, the depth of the neutral axis, \(d_n\) is given by:

$$d_n= \dfrac{\frac{h}{2}bh + d(n-1)A_{st}+d_{sc}(n-1)A_{sc}}{bh+(n-1)A_{st}+(n-1)A_{sc}}$$

Here, the width (b) is a key parameter, and the calculation is based on the section's dimensions, including height (h) and width (b).

In the uncracked stage, the concrete section behaves in a linear elastic manner. The neutral surface aligns with the centroidal axis of the transformed cross section. Here, concrete resists both compressive and tensile stresses, and the entire cross section is active in resisting moment.

This stage applies to the serviceability limit state, particularly when evaluating deflection and crack control before any concrete cracking has occurred.

Neutral Axis Calculation in an Uncracked Beam Section

We calculate the neutral axis depth by using the first moment of area of the transformed section:

$$\bar{y} = \frac{\sum A_i y_i}{\sum A_i}$$

Note: This calculation assumes the section is fully uncracked and all materials behave elastically, with areas adjusted by the modular ratio as required.

Where:

• \(A_i\): Area of each transformed component (adjusted for modular ratio)
• \(y_i\): Distance from reference axis to centroid of \(A_i\)

Code Reference: AS3600 Clause 8.1.3, ACI 318-19 Section 6.4

Stage 2: Cracked and Linear Elastic

$$\begin{align}\text{Compression in concrete, }C_c &= \frac{1}{2} E_c\varepsilon_{comp}d_nb - E_c\varepsilon_{sc}A_{sc}  \\[1em] \text{Compression in steel, } C_s &= E_s\varepsilon_{sc}A_{sc}\\[1em] \text{Tension in concrete, }T_c &=0 \\[1em] \text{Tension in steel, }T_s&=E_s\varepsilon_{st}A_{st} \\[1em] \therefore C_c + C_s&=T_s\end{align}$$

The force equilibrium can be presented as a quadratic equation with respect to the depth of the neutral axis \(d_n\) and can be calculated as:

$$\frac{1}{2}bd_n^2+[(n-1)A_{sc}+nA_{st}]d_n-[(n-1)A_{sc}d_{sc}+nA_{st}d_{st}]=0$$

The position where the neutral axis lies determines the balance between compression and tension forces in the cracked section, indicating whether the section is balanced, under-reinforced, or over-reinforced.

Once the concrete cracks in tension, it no longer contributes below the neutral axis. Steel carries all the tensile force. The stress distribution remains linear, forming a triangular compression block.

Neutral Axis of a Cracked Beam in the Linear Stage

We assume strain compatibility and elastic behaviour of steel:

$$f_s = E_s \cdot \epsilon_s = E_s \cdot \epsilon_c \cdot \frac{d - c}{c}$$

Compression in concrete (triangular distribution):

$$C_c = \frac{1}{2} f’_c \cdot b \cdot c$$

Equilibrium:

$$\frac{1}{2} f’_c \cdot b \cdot c = A_s \cdot f_s$$

The position of the neutral axis may shift depending on the type and magnitude of loading applied to the beam, as different loading conditions influence the stress distribution and crack development.

Code Reference: AS3600 Clause 8.1.6.1, ACI 318-19 Clause 22.2.2.2

Stage 3: Cracked and Inelastic

$$\begin{align}\text{Compression in concrete, } C_c &= \alpha_2 f'_c \gamma \ b \ d_n \\[1em] \text{Compression in steel, }C_s &= \varepsilon_{cu} E_s A_{sc} \frac{(d_n - d_{sc})}{d_n} \\[1em] \text{Tension in concrete, }T_c &=0 \\[1em] \text{Tension in steel, } T_s &= f_{sy}A_{st}\\[1em] \therefore C_c + C_s&=T_s\end{align}$$

The force equilibrium can be presented as a quadratic equation with respect to the depth of the neutral axis (d_n) and can be calculated as:

$$\alpha_2 f'_c \gamma b {d_n}^2 + \varepsilon_{cu} E_s A_{sc} (d_n - d_{sc}) - f_{sy} A_{st} = 0$$

When analyzing the orientation or inclination of the neutral axis, it is important to apply positive and negative sign conventions to correctly represent the direction of stresses and positions in the coordinate system.

At ultimate conditions, we assume a non-linear stress distribution and use a rectangular stress block as prescribed by design codes.

Calculating Neutral Axis at Ultimate Strength in a Beam

Force equilibrium:

$$\begin{align} A_s f_y &= 0.85 f’_c b a \\[1em] a &= \beta_1 c\end{align}$$

Ultimate moment capacity:

$$M_u = A_s f_y (d - a/2)$$

Accurate calculation of the neutral axis is essential in structural engineering, as it directly impacts the design, safety, and performance of load-bearing elements.

Code Reference: AS3600 Clause 8.1.6.4, ACI 318-19 Clause 22.2.2.1

Code Comparisons: ACI vs Eurocode vs AS3600 for Neutral Axis Design

• ACI 318-19: Rectangular block, varies with concrete strength. For sections with symmetry, code provisions may be simplified since the neutral axis passes through the centroid.
• Eurocode 2: Parabolic-rectangular block, partial safety factors.
• AS3600: Similar to ACI, with different equations for and strength limits.

Cross Sections and Axis Behaviour in Beams

Different cross sections—rectangular, symmetric, curved, or circular—affect the neutral axis location. Factors like the second moment of area, centroid, and reinforcement layout all play a role.

Fibre Stresses and Strain Layers Across the Neutral Axis

Each layer within the section responds based on its position relative to the neutral axis. The strain varies linearly, with tension and compression occurring on opposite sides.

Shear Stress and the Role of the Neutral Axis

Though shear stress peaks near the neutral axis, it's not uniform across depth. Proper detailing with stirrups is crucial to handle peak shear forces.

Practical Applications and Structural Mechanics of the Neutral Axis

Correct identification of the neutral axis is essential in all stages of design:

• Serviceability (crack control, deflection)
• Ultimate strength (moment and shear capacity)

In practical applications, the position of the neutral axis is influenced by the magnitude and type of load applied to the structure, as different loads affect stress distribution and deformation.

Codes like ACI 318-19 and AS3600 also govern spacing, anchorage, and development length based on the position of the neutral axis.

FAQs About the Neutral Axis in a Beam

What is the neutral axis of a beam?
The neutral axis is the line in a structural member under bending where the longitudinal stress is zero.

How do you calculate the neutral axis of a beam?
Use equilibrium of internal forces; the method depends on whether the section is uncracked, cracked-linear, or cracked-inelastic.

Why is the neutral axis important?
It determines internal force distribution, affects stiffness, crack control, and flexural strength.

Which codes use neutral axis analysis?
All major design codes—ACI 318, Eurocode 2, and AS3600—require it for bending calculations.

Summary: Why the Neutral Axis of a Beam Matters

The neutral axis is central to understanding and designing reinforced concrete under flexural loads. Accurately locating it ensures safe, efficient, and compliant structures.

By mastering the concept across different design stages and codes, engineers can confidently navigate the transition from elastic analysis to ultimate strength design using AS3600, ACI 318-19, and Eurocode 2.