Interaction Diagram Concrete

[Auriville Cha]

Vertical members that are part of a building frame are subjected to combined axial loads and bending moments. These forces develop due to external loads, such as dead, live, and wind loads. Simply put, an interaction diagram (or curve) displays the combinations of the acceptable moment and axial capacities of a structural member.

To consider this curve SkyCiv considers the necessary number of intermediate points. Typically, there are three main points: maximum axial tension (point G), maximum axial compression (point A), and balanced condition (point D). Then intermediate points are considered from balanced condition to maximum tension (points D-G) and from balanced condition to maximum compression (points D-A). To calculate all that points as per design codes used the next assumptions:

The strength of a column cross-section can be determined from the geometry of the cross-section, the constitutive relationships of the concrete and steel and consideration of equilibrium and strain compatibility. For the calculation of intermediate M-N curve points that describe the strength of section the SkyCiv uses an iterative process. The next steps are involved in this process as per ACI code:

For the design of a column to be considered adequate (safe), the combination of action effects (M, P) must be less than the combination of design strengths (M, P) from the interaction curve. This means that if the position of the M,P point on the plot is outside of the curve it is deemed as not meeting this criterion and considered unsafe.

In SkyCiv RC Design Module, SkyCiv uses both the major and minor axis to calculate the balance point. The module defines the point of intersection with a 3D interaction diagram (green point in the below picture). The coordinates of this point provides the axial and flexure capacities for the section.

SkyCiv offers a fully featured Reinforced Concrete Design software that allows you to check concrete beam and concrete column designs as per ACI 318, AS 3600, and EN2 Design Standards. The software is easy-to-use and fully cloud-based; requiring no installation or downloading to get started!

Often a summary refresher helps keep us grounded in the fundamentals of elements that we commonly design. Owing to many requests from peers, this article is provided as a summary of the steps that may be taken for the development of a typical reinforced concrete column interaction diagram.

The methodology outlined below reflects the provisions of ACI 318, but it is not the only viable method. In fact, ACI 318 does not explicitly require an interaction diagram for column design. However, most structural engineers understand that such a tool is the most convenient form of expressing the nominal axial and flexural capacities, as well as the best tool for helping us know how axial loads and bending loads influence and affect one another.

Assume, just for the sake of argument, that the factored load effect for the design of a tied concrete column is a trivial matter, and that we have the results for Mu and Pu. The next step that we might follow would be to examine a generated interaction diagram and see whether our interactive load (represented by Mu and Pu) falls within the capacity boundary of our trial column.

The series of strains (tensile and compressive) are then assigned to the layer of reinforcement opposite the concrete compressive failure surface. For any one particular level of strain that we have arbitrarily assigned, we can follow ACI 318 criteria and connect the two opposing points of strain on a diagram with a straight line, thus assuming that the strain is distributed linearly across with column width. This makes for simple calculation of the strain in the remaining layers of reinforcement, using the simple formulas for similar triangles that we learned in high school math.

The result is a strain for each and every bar in the column as it correlates to the level of strain that was arbitrarily assigned to the layer of reinforcement opposite the compression surface. This also helps us know where the theoretical neutral axis for the column is for this strain condition, which occurs where the aforementioned line intersects the vertical axis of the strain diagram. Once the strain levels and the neutral axis are known, the design may proceed to the next step.

Once the stress in each layer of reinforcement is known, as well as the dimensions of the concrete compressive stress region, the resultant forces in each are calculated simply by multiplying the stresses by the respective areas. Summing the result yields a total force Pn, the nominal axial capacity of the column as it correlates to this level of strain. Multiplying these same forces by their relative distances from the centroid of the gross column section and summing the result yields the nominal moment capacity Mn.

The final step for this one iteration of design is to determine the strength reduction factor that is appropriate for the level of strain under consideration. This is a function of the net tensile strain arbitrarily assigned earlier; it has a value of 0.9 for net tensile strains of 0.005 or more and a value of 0.65 for net tensile strains of 0.002 or less. Intermediate values are linearly interpolated. The strength reduction factor is then multiplied by each of the Mn and Pn values calculated previously to determine the resulting φMn and φPn that define this one point on the interaction diagram.

The entire process is repeated several times, with varying levels of strain assigned to produce a series of points that define the interaction diagram boundary. Figure 1 depicts the superimposed strain conditions as recommended by prominent textbook authors. For each level of strain, the calculations described herein are repeated. For each level of strain, a corresponding point on an interaction diagram can be determined. Interconnecting the points results in the interaction diagram (potentially similar to Figure 3) on which we can plot Mu, Pu and assess whether the column is sufficient.

Jerod G. Johnson is a Principal at Reaveley Engineers in Salt Lake City. He was the engineer of record for recent updates to the base isolation system for the Salt Lake City & County Building. He was the principal investigator of the comprehensive isolator testing of May 2011. (jjoh...@reaveley.com)

The document provides information on constructing interaction diagrams for reinforced concrete columns. It defines an interaction diagram as a graph showing the relationship between axial load (Pu) and bending moment (Mu) for different failure modes of a column section. The document outlines the design procedure for constructing interaction diagrams, including considering pure axial load, axial load with uniaxial bending, and axial load with biaxial bending. An example is provided to demonstrate constructing the interaction diagram for a given reinforced concrete column cross-section.Read less

I would like to create an interaction diagram (axial force N & bending moment M) for a concrete beam that has a U shape steel girder inside. I have previously used Oasys Adsec and exported the results to MS Excel to plot the M-N curve. Is a simular function available in ROBOT or another Autodesk product?

The structural analysis software RFEM 6 is the basis of a modular software system. The main program RFEM 6 is used to define structures, materials, and loads of planar and spatial structural systems consisting of plates, walls, shells, and members. The program also allows you to create combined structures as well as to model solid and contact elements.

RSTAB 9 is a powerful analysis and design software for 3D beam, frame, or truss structure calculations, reflecting the current state of the art and helping structural engineers meet requirements in modern civil engineering.

Do you often spend too long calculating cross-sections? Dlubal Software and the RSECTION stand-alone program facilitate your work by determining section properties of various cross-sections and performing a subsequent stress analysis.

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When designing reinforced concrete members according to the ACI 318-19 [1], the moment interaction diagram is an essential tool. These diagrams represent the relationship between the bending moment and axial force at any given point along a reinforced concrete member. Valuable information is shown visually such as strength and how the concrete behaves under different loading conditions.

The moment interaction diagram is used to determine the maximum moment and axial force a member can resist, which is useful in calculating the ultimate strength. Generating a moment interaction diagram requires calculating the maximum axial force and moment. Then, theses points are plotted on an x-y graph. The y-axis represents the axial force, and the x-axis represents the bending moment. The interaction between these two forces is shown through a line/curve which represents the maximum resistance of the reinforced section. Any point on the curve represents a unique combination of bending moment and axial force that the reinforced section can resist. This curve is then further divided into regions based on failure points. For example, the upper region represents pure bending failure, and the lower region represents pure axial failure. This is shown in Image 01.
Moment Interaction Diagram

RFEM 6, with the Concrete Design Add-on, can analyze and design reinforced concrete structures. The add-on can create a moment interaction diagram automatically for any column or beam. The maximum moment and axial force a member can resist is calculated in the static analysis which is automatically considered in the Concreted Design add-on. Then, once the option is checked, a moment interaction diagram is generated based on properties such as section size and reinforcement layout.

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