Concrete Interaction Diagram

A P-M Diagram (Axial Force-Moment Interaction Diagram) is a critical tool in structural engineering that graphically represents the relationship between axial force \(P\) and bending moment \(M\) in reinforced concrete columns. It illustrates how a column behaves under combined axial and bending loads, helping engineers determine its load-carrying capacity and ensure safe and efficient designs. This diagram is particularly valuable for understanding the limits of structural performance, aiding in the assessment and optimization of columns under various loading conditions.

Concrete interaction in the context of the P-M Diagram (Axial Force-Moment Interaction Diagram) refers to how concrete and its reinforcement collectively respond to combined axial and bending loads. In reinforced concrete columns, the axial force \(P\) generates compressive stresses, while the bending moment \(M\) introduces tensile and compressive stresses on opposite sides of the section. The P-M Diagram captures this interplay, illustrating the structural capacity limits based on material behavior, such as the compressive strength of concrete and the tensile strength provided by reinforcement. Understanding this interaction is essential for optimizing the design and ensuring safety under complex loading scenarios.

\( \textbf{1.1}\space\)Axial Compression Strength

The relationship between axial load strength and the P-M Diagram arises from how a reinforced concrete column resists combined axial forces and bending moments. In pure compression, the column's maximum axial load capacity, often denoted as \(P_{max}\)​, occurs when no bending moment is applied. This represents the theoretical upper limit of the column's strength and is calculated as the sum of the compressive capacities of the concrete and reinforcement, reduced by safety factors.

As a bending moment is introduced, the load-carrying capacity of the column decreases because a portion of the column's cross-section must now resist the additional stresses caused by bending. The stress distribution in the cross-section shifts from being uniform under pure compression to having a linear gradient, with tension on one side and compression on the other. This interaction between axial load and bending moment creates a curved boundary in the P-M Diagram, which represents the maximum combination of these two forces that the column can safely resist. At a low load application rate, the accepted compressive strength for the concrete is 0.85 times the specified compressive strength 0.85 \(f'_c\​)

The ultimate axial strength of the element is attained when the concrete undergoes crushing failure, and the reinforcement steel yields:

$$ P_n=0.85f'_c(A_g-A_s)+f_yA_s $$

Where :

\( f'_c\hspace{2.5em}\) = (Compressive Strength of Concrete) : Compressive strength of concrete. It measures the maximum compressive stress that concrete can withstand under axial loads.
\( A_g\hspace{2.5em}\) = (Gross Sectional Area) : Total cross-sectional area of a structural element, such as a column or a beam. It includes the area of both the concrete and any reinforcing steel.
\( f_y\hspace{2.5em}\) = (Steel Yield Stress) : Yield strength of the reinforcing steel. It indicates the stress at which the steel exhibits significant plastic deformation.
\( A_s\hspace{2.5em}\) = (Total Longitudinal Reinforcement Area) : The total cross-sectional area of a concrete element's longitudinal (primary) reinforcement. It is the sum of the cross-sectional areas of all the individual longitudinal bars.

\( \textbf{1.2}\space\)Axial Tensile Strength

The concrete is fully fissured when the ultimate axial tensile strength is reached, rendering its contribution null. Therefore, the ultimate strength is attained when the reinforcement reaches its yield stress.

$$ P_{nt}=f_yA_s $$

In the tension-controlled region of the diagram, axial tensile loads are limited to the capacity of the steel reinforcement, as concrete has negligible tensile strength. As a result, the column's capacity in this region is much lower compared to the compression-controlled side. The P-M Diagram thus serves as a visual and analytical tool to ensure that the combination of axial load and bending moment in a column remains within safe limits, accounting for both material properties and section geometry.

This relationship is further influenced by factors such as material strength, the ratio of reinforcement, and the slenderness of the column. Higher concrete and steel strengths, for example, increase both axial and bending capacities, while higher reinforcement ratios can enhance strength but may reduce ductility. Slender columns, on the other hand, are more susceptible to buckling, which reduces the effective axial load capacity and shifts the interaction curve downward. By understanding and using the P-M Diagram, engineers can design columns to safely carry loads while accounting for complex interactions between axial and bending forces.

\( \textbf{1.3}\space\) Internal Force Equilibrium

Internal force equilibrium is the foundational principle for determining how internal axial forces and bending moments are distributed and resisted in a structural column. The P-M Diagram visually represents this equilibrium by showing the combinations of axial load and bending moment that a column can resist without failure. For every point on the diagram, there is a corresponding internal force equilibrium condition, which ensures the column remains stable and safe under combined loading. As the axial load and bending moment increase, the internal distribution of stress changes, and the column's ability to resist these forces decreases, which is captured by the curve in the P-M Diagram.

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When a column is subjected to pure axial load \(P\) with no bending moment where (M=0), the compressive force within the column will be resisted entirely by concrete and reinforcements. The internal force equilibrium is met when the applied axial load is balanced by the internal forces and the steel reinforcements.

The summation of forces adheres to the internal axial force equilibrium, where \(P_n​\) represents the ultimate axial strength force.

$$ P_n\ =\ 0.85f'_cab\ +\ A'_sf'_s\ -\ A_sf_s $$

\(0.85f'_cab\) Stands for the concrete compression force given by the Whitney equivalent rectangular stress distribution and:
$$ a\ = \beta_1c $$

When a column is subjected to pure axial load \(P\) with no bending moment, the compressive force within the column is resisted entirely by concrete and reinforcement. where the column's maximum axial load capacity is determined by the material strengths \(f'_c\) and the steel yield strength \(f_y\) and the cross - sectional area of concrete and reinforcements. The internal force equilibrium is then met when applied axial load is finally balanced by the internal forces

The summation of internal flexural moments and forces originating from the center of the span must be equal and opposite to the internal force \(P_n\)

$$ M_n\ =\ P_ne\ =\ 0.85f'_cab(\frac{h}{2}-\frac{a}{2})\ +\ A'_sf'_s(\frac{h}{2}-d')\ +\ A_sf_s(d-\frac{h}{2})$$

\( \textbf{1.4}\space\) Compatibility of Strains

Compatibility of strains ensures that the deformations (strains) in different parts of a structural element, such as concrete and steel reinforcement, are consistent with the geometry of the structure and with each other. This concept is essential in ensuring that the internal stresses and deformations are well-distributed across the element and that the concrete and reinforcement work together effectively to resist applied loads. Strain compatibility is fundamental in the design and analysis of reinforced concrete structures, as it governs the interaction between concrete and steel, influencing factors such as bending strength, cracking behavior, and overall stability.

In reinforced concrete design, the structural element is typically made up of concrete and steel reinforcement. Concrete will experience compressive strain while steel reinforcement experiences both tensile and compressive strains depending on the load conditions. Moreover, when bending moment is applied on to the concrete section, the element will undergo a curvature while the strain distribution across the section is linear, the maximum compression on the side of the section opposite to the direction of bending and maximum tension on the side of the section perpendicular to the direction of bending.

Strain combability can be expressed mathematically by relating the strain in concrete and steel reinforcement based on the total deformation of the section and the use of Hooke's Law.

Geometrically:

\(\epsilon_s=\epsilon_u(\frac{d-c}{c})\)
\(\epsilon'_s=\epsilon_u(\frac{c-d'}{c})\)

Hooke's Law :

\(f_s=E_s\epsilon_s\)
\(f'_s=E_s\epsilon'_s\)

Where:

\(E_s\hspace{2.5em}\) =  Modulus of elasticity of the reinforcing steel.

\( \textbf{1.5}\space\)Balanced Failure of Strains

In reinforced concrete columns, the concept of "balanced failure in flexo-compression" refers to a condition where the concrete in compression and the tensile reinforcement reach their limit states simultaneously. In this balanced state, the column is designed to maximize the utilization of both materials, ensuring an efficient use of concrete and steel.

$$ c\ =\ c_b\ =\ d\frac{\epsilon_u}{\epsilon_u\ +\ \epsilon_y}$$

\( \textbf{2.1}\space\)Flexo-Compression Design

Flexo-compression design involves designing reinforced concrete columns to safely resist a combination of axial compression and bending moments. In this design, the column's capacity is determined using the P-M Diagram, which illustrates the interaction between the axial load  and the bending moment. The goal is to ensure that the applied combination of axial load and bending moment lies within the safe region of the interaction curve. As axial load and bending moment increase, the axial load capacity decreases, and this relationship must be considered to prevent failure. The design process typically includes calculating the column's strength under various load combinations, ensuring that it can safely carry both compressive forces and bending without exceeding the column's ultimate strength, while considering factors like material properties, reinforcement layout, and load eccentricities. A P-M diagram can be seen as explained below in Figure 1 :

• For the given section \(bxh\), axially loaded with \(P_n\) and an eccentricity \(e\):

• The distribution of strains in a section (a-a) for an immediate failure state assumes that plain sections remain plain and vary linearly with the distance 'c' from the neutral axis.

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By ensuring the compatibility of strains, the strains experienced by the reinforcement at any location are identical to those exhibited by the surrounding concrete.

Where:

\(\epsilon_u\hspace{2.5em}\) = (Ultimate Strain of Concrete) : The ultimate strain or deformation that concrete can undergo before failure. It represents the maximum deformation the concrete material can experience under load.
\(\epsilon'_s,\epsilon_s\hspace{2.5em}\) = (Reinforcement Bar Strain): denotes the strain experience by the reinforcement bars.