Modeling Reinforced Concrete Stiffness
Part 3: FEA Modeling - Deflections

By Adam Stordahl, PE • Published August 19, 2025

In Part 1 of this series, we looked at the stiffness of concrete and reinforcing steel at the material level. Part 2 calculated the bending stiffness of a reinforced column section four different ways. In this installment we'll use those different cross-section stiffnesses to model columns in VisualAnalysis and look at the deflections produced. We've moved from materials to section, and now to structural elements.

We'll use the same cross-section as the previous post: 12" x 16" (f’c = 4 ksi) with (6) #7 bars (Grade 60).

Figure 1: Reinforced Concrete Column Section (click to expand)

We'll consider a cantilever column 16 feet tall with a 6.5 kip horizontal point load at the top of the column (bending the column about its strong axis). This configuration produces a unity value just under 1.0 when checked with the ACI spec—in other words, we are loading the column to its capacity.

Figure 2: Cantilever Column Model (click to expand)

Section Stiffness to FEA Input

Let's examine how we can use each of the cross-section stiffness approximations from Part 2 when modeling a column element in VisualAnalysis. The simplest way to model a concrete column in VA is to define the cross-section as a parametric rectangle and then pick a concrete material from the material database. This approach gives the gross concrete elastic stiffness from Part 2 of this series.

VisualAnalysis also lets you define a “Stiffness Factor” for each member, allowing you to easily scale stiffness values to match alternate section assumptions. This allows us to model all of our column elements as parametric rectangles and then assign a stiffness factor to get the desired EI. The table below summarizes this approach for the first three section stiffness approximations from Part 2.

Tangent Stiffness

A theme of this series has been that reinforced concrete stiffness depends heavily on the applied load—reinforced concrete softens as you load it. Of the approaches discussed, only the tangent stiffness method truly captures this behavior. Instead of a single stiffness value based on the layout of concrete and steel, the tangent stiffness is a function that varies with applied moment. Our cantilevered column has a linearly varying moment, so the tangent stiffness method cannot give us one value of EI to plug into VisualAnalysis. VisualAnalysis (and similar structural engineering FEA programs) use linear-elastic material models—they want a value for EI, not a curve.

The finite element method is fundamentally built on the concept of splitting complex problems into smaller parts and then approximating the solution over these finite regions. We'll follow that same principle here. By splitting the column into several segments we can find an average moment in each segment. We can then use our tangent stiffness plot from last time to find the stiffness that corresponds to each moment. Each stiffness can be divided by the gross elastic stiffness to produce a stiffness factor for each segment. The table below has the details for our example. A spreadsheet with the derivation of these values is available here.

Since the moment in the first segment is less than the cracking moment, I opted for the transformed elastic stiffness for that segment. There is also nothing magical about using seven segments. A little convergence testing showed this to be a reasonable number for our purposes.

Modeling in VisualAnalysis

We now have enough information to build our four column models in VisualAnalysis. You can download the VA project file here.

Figure 3: VisualAnalysis Column Model (click to expand)

Figure 4: VisualAnalysis Results (click to expand)

The table below summarizes the displacement of each column.

Conclusion

The main goal of this post was to demonstrate how the tangent stiffness curve developed in the previous post can be used in a program like VisualAnalysis. We were pretty easily able to handle the load-dependent stiffness of our reinforced concrete section by sub-dividing a column into several sub-elements. Hopefully this simple example gives you a sense of how you might use ConcreteSection and VisualAnalysis together.

Our more refined column model displaced nearly 8 times as much as the baseline model that did not attempt to consider the softening effects of yielding and cracking. The column is loaded close to capacity (see the VA unity values), so we'd expect yielding and cracking to play a big role—and we certainly captured that in the results.

Next Time

I think we should take this series one post further. Column deflections are nice, but how could we use these techniques to determine design forces in an indeterminate structure? Stay tuned.