Pedestal Design using Interaction Diagrams

Pedestals are transition elements between the superstructure columns and the foundations. When the superstructure is being designed, the foundation bearing level or the foundation height might not yet be defined; therefore, the pedestal is responsible for transferring loads between both elements. This is not a trivial task, as this load transfer can occur over a considerable dimension, which amplifies the bending moments generated by shear forces.

Pedestals essentially behave like columns and must be designed as such. In particular, the interaction between bending and axial stresses is of great relevance, as both generate significant compressive stresses in the concrete. When combined, these can lead to a concrete crushing failure.

Understanding the Interaction Diagram

An interaction diagram is a complex chart used to easily determine if a column (or pedestal) is correctly designed. In principle, ensuring that all load combinations fall within the design curve is sufficient to deem the design adequate, but we can extract much more information if we truly understand the diagram.

Nominal Curve vs. Design Curve

The nominal curve of the interaction diagram is constructed through a series of points that define different possible failure modes of the column. Different combinations of axial load ($P$) and bending moment ($M$) will trigger specific failure mechanisms. Column failure is defined by two primary scenarios:

• Concrete Crushing: Occurs in the concrete of the compression face when it reaches an ultimate strain of $\varepsilon_{cu} = -0.003$.

• Steel Yielding: Occurs in the reinforcement near the tension face when it reaches a yield strain of $\varepsilon_y = 0.002$.

On the other hand, the design curve corresponds to a reduction of the nominal curve using a variable strength reduction factor $\phi(\varepsilon_s)$. This factor aims to ensure that the column's performance under ultimate load states remains safely away from actual failure. This factor varies based on the net tensile strain of the steel ($\varepsilon_s$), as design codes seek to be even more conservative regarding brittle failures.

Thus, when the failure is governed by concrete crushing, the strength reduction factor is $\phi = 0.65$. Conversely, when failure is governed by steel yielding, the factor is $\phi = 0.9$. This is because steel yielding is much more ductile, and visible cracks appear long before failure, allowing time for structural repairs.

Key Points on the Interaction Diagram

While the entire interaction curve is a sequence of points determining column failure, specific points have very tangible physical meanings:

• Pure Compression ($P_0$): This is the highest point on the diagram (vertical axis). It represents the pedestal's maximum capacity if it were subjected only to axial load without eccentricity. In practice, codes limit this load to a lower value ($P_{n,max} = 0.8 P_0$) to account for accidental eccentricities.

• Balanced Point ($P_b, M_b$): This is the "knee" of the diagram. At this point, the concrete reaches its ultimate strain ($\varepsilon_{cu} = 0.003$) exactly at the same time the tension steel begins to yield ($\varepsilon_y$).

• Pure Bending ($M_0$): The point where the curve crosses the horizontal axis. It represents the pedestal's capacity to resist moment without any axial load.

• Pure Tension: The lowest point on the diagram. Although pedestals rarely work in tension, this value defines the lower limit of the steel's capacity, as the concrete's contribution to tensile capacity is neglected.

Which Load Combinations to Consider?

Design is not performed with a single force but with a "cloud" of points. Each point represents a factored load combination ($P_u, M_u$) obtained from structural analysis (e.g., using LRFD combinations like $1.2D + 1.6L$ or $1.2D + 1.0E + L$). However, it is vital to ensure that the evaluated pairs ($P_u, M_u$) are compatible with each other.

In structural design, it is common to perform a Modal Response Spectrum Analysis (MRSA). This analysis estimates the peak modal response of the structure for each vibration mode, and the total response is obtained by combining these modes. Consequently, the maximum estimate for axial force might not occur simultaneously with the maximum bending moment. Evaluating all four possible combinations ($\pm P_u, \pm M_u$) is conservative but may include physically incompatible combinations, which could over-design the pedestal.

Symmetry in the Interaction Diagram

It is important to note that the interaction diagrams we are reviewing are 2D diagrams. This means positive and negative moments are represented within the same direction. Therefore, it is common for column interaction diagrams to be symmetric, as columns usually have identical reinforcement on opposite faces. If the compression steel is identical to the tension steel, the interaction diagram will be symmetric.

How to Construct an Interaction Diagram

The process of obtaining an interaction diagram is fundamentally simple: impose a failure state and solve for equilibrium. However, understanding the internal mechanics of the column is more complex.

One way to cover the full spectrum of possible failures is to impose various strain profiles across the section. For instance, we can set a constant strain at the compression edge of $\varepsilon_{cu} = -0.003$ and take discrete values for $\varepsilon_s \in (-0.003, 0.05)$. This allows us to trace all points on one half of the interaction curve.

The imposed strain profile across the section is typically represented as a linear gradient. From this, using direct proportionality or similar triangles, we can determine the neutral axis depth ($c$) and the strain in the compression steel. Then, assuming a bilinear behavior for the steel, the force exerted by each bar is defined as $f_s = \min(|\varepsilon_s|E_s, f_y)$.

Note: This model is a simplification using a single layer of steel on each face. In reality, multiple layers may exist, requiring the calculation of an $\varepsilon_{si}$ for each layer.

Concrete behavior is more complex. Instead of a bilinear constitutive relationship, it follows a piecewise function (parabolic and linear parts). However, since we start by imposing that the concrete is at its ultimate compression state, we can simplify its forces using the Equivalent Rectangular Stress Block (Whitney stress block). Concrete is assumed to have uniform compression in this block of length $\beta_1 c$, where $\beta_1$ is determined by the concrete strength:

With the stresses of all participating elements defined, the pair $(P_n, M_n)$ for a point on the nominal curve is obtained from the section's equilibrium (sum of forces and moments). This defines the nominal interaction curve. To obtain the design interaction curve, the variable reduction factor $\phi(\varepsilon_s)$ must be applied.

This explains why the nominal and design curves are quite distant at high axial loads (compression-controlled failure) and much closer at low axial loads (tension-controlled failure).

Note: For circular columns with spiral reinforcement, the section has better confinement, resulting in a minimum $\phi$ value of 0.75 instead of 0.65.

How to Use the Interaction Diagram for Design?

Obtaining an interaction diagram requires modeling the behavior of each element in the section through multiple steps. When designing a column for flexo-compression, multiple diagrams must be generated as the reinforcement ratio changes during design iterations. Essentially, the procedure involves increasing the reinforcement ratio until all load combination points fall within the design curve.

How Does Foundaxis Do It?

In Foundaxis, the pedestal design process using interaction diagrams is fully automated. Pedestal dimensions are automatically determined based on the column dimensions in your structural model.

The process begins by reinforcing the pedestal symmetrically in both directions, considering both corner and interior bars. If any pair $(P_u, M_u)$ falls outside the design curve, the reinforcement ratio is increased in that direction until the design curve expands sufficiently. Notably, Foundaxis designs the pedestal independently in both directions, calculating two interaction curves for each.

The interaction diagrams generated by Foundaxis are included in the automatically generated calculation report and can also be reviewed directly from the reinforcement design menu.

 

Discover how pedestal design is integrated into the complete Foundaxis workflow:

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