Question by MG (LinkedIn)
"I had a question in regards to your Bearing Theory for Column Transitions.
In regards to A1 and A2, How do you calculate A2 as AS3600 notes that A1 and A2 should be geometrically similar yet from your analogy it doesn't appear they are?

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Answers

Answers

The A1/A2 analogy originates from ACI . This analogy is essentially used to disperse stress from a smaller area (A1) to a larger area (A2), as the surrounding concrete confines. The stress limit on A1 can be increased by the factor (A2/A1)^0.5, but not beyond a factor of 2, or ultimately the upper bound of factor*fc, as this is the maximum crushing or squashing stress concrete can withstand. ACI does not specify a depth for the dispersion prism or frustum, except in cases where the underlying area is limited, such as a column near an edge. In such cases, the dispersion depth (or A2) should be limited by, or stops at if you wish, at the discontinuous edge. Consider an exaggerated case where the drop panel’s depth is as tall as a full storey. In such a scenario, we could imagine a “geometrically similar” area, A2, created by constructing 45-degree lines from the four corners of the columns to the planar limits of the drop panel. This setup could theoretically achieve an enhancement factor of 2 if the drop panel in plan is only about 75 mm larger all around (for a 200x100 blade). Although we might argue this fits within the code’s lingo for any drop panel depth, it doesn’t align with the first principles or the fundamental intent of this clause—namely, stress dispersion or spread assisted by confinement. Stress dispersion within the drop panel limits can be easily verified using finite element models. We would like to think of this problem as two columns pushing against the drop panel, shooting beams of light or stress, and these intersect at a new area within the middle depth of the drop panel, making a diamond shape, this make a lot more sense. On a related note, the AS3600 standard suggests a concession for stepped or sloped surfaces, allowing A2 to be defined as “the area of the base of the largest frustum of a right pyramid or cone.” How should we interpret this? Could this be viewed as removing the requirement for geometric similarity? Interpretation is left to the reader. More generally, the code itself is nothing but guide, and responsibility of the design lies at the structural engineer’s judgement. In all cases, we clearly mentioned that this tool is a complementary bearing check, and we recognise that this theory may be open to challenge. However, it is not a replacement for the strut-and-tie analogy; both theories, introduced by RKALC, aim to assist the engineering community. One could adopt an extremely conservative approach by considering the small square area between two columns and applying the stress directly to it. However, this approach would only be practical for very large columns and would effectively eliminate the need for a drop panel altogether, provided the column is confined enough to take the stress concentration. Technically, there is nothing wrong with this approach.

Answered by: RKALC Admin | Date: 11/4/2024 6:13:14 PM

The A1/A2 analogy originates from ACI . This analogy is essentially used to disperse stress from a smaller area (A1) to a larger area (A2), as the surrounding concrete confines. The stress limit on A1 can be increased by the factor (A2/A1)^0.5, but not beyond a factor of 2, or ultimately the upper bound of factor*fc, as this is the maximum crushing or squashing stress concrete can withstand.
ACI does not specify a depth for the dispersion prism or frustum, except in cases where the underlying area is limited, such as a column near an edge. In such cases, the dispersion depth (or A2) should be limited by, or stops at if you wish, at the discontinuous edge.
Consider an exaggerated case where the drop panel’s depth is as tall as a full storey. In such a scenario, we could imagine a “geometrically similar” area, A2, created by constructing 45-degree lines from the four corners of the columns to the planar limits of the drop panel. This setup could theoretically achieve an enhancement factor of 2 if the drop panel in plan is only about 75 mm larger all around (for a 200x100 blade).
Although we might argue this fits within the code’s lingo for any drop panel depth, it doesn’t align with the first principles or the fundamental intent of this clause—namely, stress dispersion or spread assisted by confinement. Stress dispersion within the drop panel limits can be easily verified using finite element models.
We would like to think of this problem as two columns pushing against the drop panel, shooting beams of light or stress, and these intersect at a new area within the middle depth of the drop panel, making a diamond shape, this make a lot more sense.
On a related note, the AS3600 standard suggests a concession for stepped or sloped surfaces, allowing A2 to be defined as “the area of the base of the largest frustum of a right pyramid or cone.” How should we interpret this? Could this be viewed as removing the requirement for geometric similarity? Interpretation is left to the reader.
More generally, the code itself is nothing but guide, and responsibility of the design lies at the structural engineer’s judgement. In all cases, we clearly mentioned that this tool is a complementary bearing check, and we recognise that this theory may be open to challenge. However, it is not a replacement for the strut-and-tie analogy; both theories, introduced by RKALC, aim to assist the engineering community.
One could adopt an extremely conservative approach by considering the small square area between two columns and applying the stress directly to it. However, this approach would only be practical for very large columns and would effectively eliminate the need for a drop panel altogether, provided the column is confined enough to take the stress concentration. Technically, there is nothing wrong with this approach.