The results of the Section Analysis macro are compared against four kinds of source: closed-form relations that can be written by hand, independent implementations doing the same work, published worked examples, and the measured capacity of columns tested in the laboratory. Each comparison below states what it was made against and what it returned.
Closed-form solutions
In every row below, the quantity the calculation produces is worked out a second time from an independent relation that can be written by hand, and the two are compared. The reference section is 400×600 mm, C25 concrete, B420C steel, 8Ø20 bars; the mesh has 776 triangles.
| What was checked | Relation used | Agreement |
|---|---|---|
| Mesh area and first moments | and zero | — the floating-point limit |
| Mesh moment of inertia | ||
| Fiber integration, biaxial | Transformed-section relations | |
| Pure compression | 6155.6 / 6155.6 kN | |
| Pure tension | −1055.6 / −1055.6 kN | |
| Moment-curvature of a linear section | ||
| Resultant of the Hognestad compression block | Analytical integral of the curve | · |
| Cracking moment | 39.6 / 39.6 kN·m | |
| Bending at 45° | The triangle the block cuts from a square corner | 1.5% |
| Confined core with unconfined cover | Independent strip integration | 0.3% |
| The same section in pure bending | Equivalent rectangular stress block | 274.69 / 274.17 kN·m |
The figures in the agreement column vary with the character of what is being checked. Mesh area and moments agree to floating-point precision because the triangle centroid rule integrates constant and linear integrands without error. For the moment of inertia and the fiber integration the difference is at the limit decimal representation can carry.
The last three rows carry a measurable difference, and for different reasons. At 45° the compressed region is a triangle, so the triangles straddling the neutral axis are represented by a single centroid value. At the boundary between confined core and unconfined cover two different material curves meet, and the strip method resolves that boundary more finely. In the last row the two methods are entirely independent of each other: fiber integration and the code's equivalent rectangular block give 274.69 and 274.17 kN·m for the same section.
The same section, by a second implementation
The excel worksheets used in the reinforced concrete courses at universities and in engineering practice run the same material models as Section Analysis (Hognestad concrete, elastoplastic steel) but solve the section by an entirely different route: they divide it into slices (180 angular sectors for the circular section, 100 horizontal strips for the rectangular one) and close equilibrium with a goal-seek function.
| Section | Points compared | Largest difference | Pure compression |
|---|---|---|---|
| Circular Ø650, 16Ø28 | 15 | 0.5% | 0.08% |
| Beam 250×1200, 3Ø8 top / 6Ø8 bottom | 16 | 0.3% | exact |
On the circular section the comparison runs from pure tension to beyond the balance point. On the beam the top and bottom reinforcement differ, so the asymmetric bar layout enters the comparison as well. The region very close to pure compression is out of scope: there the worksheets' slicing loses accuracy and their axial force rises 4% above their own closed-form value.
Worked example
Example 2.1 in Ersoy & Özcebe's Betonarme solves the moment-curvature table of a 250×500 mm section with different top and bottom reinforcement, at two axial load levels.
| Axial load | Points printed | Result |
|---|---|---|
| Six | Within 0.4% | |
| kN | Five | Within 0.6%, one row at 2.8% |
The book stops its equilibrium iteration once the force imbalance drops below 2% of the total compression. The row deviating by 2.8% sits inside that allowance.
Published interaction diagram
Ersoy and Özcebe's 1997 paper publishes the interaction diagram of a 50×50 cm column. Both of the paper's curves are computed with models Section Analysis carries: the code's equivalent rectangular stress block at , and Modified Kent and Park for confined concrete.
| Curve | Points compared | Mean difference |
|---|---|---|
| Equivalent stress block | 19 | 1.1% |
| Confined concrete (Kent and Park) | 17 | 1.0% |
On the block curve the squash load comes out at 730 t against 729.5 and the largest moment at 72.04 against 71.99 t·m.
Columns tested in the laboratory
A measured moment-curvature curve
The same paper also publishes the measured behaviour of Sheikh and Üzümeri's specimen A-11: a 30.5 cm square section, 8Ø19 longitudinal bars, Ø6/10.8 cm ties, under 196 t of axial load.
| Curvature | Measured | Kent and Park |
|---|---|---|
| 0.0074 rad/m | 12.2 t·m | 12.1 |
| 0.0098 rad/m | 13.3 t·m | 14.0 |
| Peak | 13.9 t·m | 16.0 (+15%) |
| 0.0427 rad/m | 13.3 t·m | 13.5 |
| 0.0593 rad/m | 10.1 t·m | 10.3 |
| 0.0746 rad/m | 8.0 t·m | curve ended |
Up to yield the calculation tracks the measured behaviour within 10%. At the peak the confined models sit above the measurement; the paper's authors report the same direction for their own analytical models. Which curve to use is the user's choice, and the choice is written into the report.
The Kent and Park curve ends at 0.066 rad/m. Past that curvature the section can no longer carry the 196 t axial load, so no equilibrium exists.
Past the peak the computed curve is observed to stay closer to the measured behaviour than the paper's own analytical curve does.
PEER Database: measured capacity of thirty-nine specimens
The PEER Structural Performance Database publishes cyclic lateral load tests of reinforced concrete columns together with each specimen's measured concrete strength, steel strengths, tie layout, axial load and measured base moment. Specimens failing in shear, with non-rectangular tie layouts or with lap splices at the base were excluded, leaving 29 rectangular and 10 circular specimens. Concrete strength ranges from 21 to 102 MPa and the axial load ratio from 0 to 0.80.
Each specimen was solved with its own measured strengths, with behaviour curves and no material factors applied.
| Specimen set | Computed / measured | Scatter |
|---|---|---|
| 29 rectangular | 1.00 | 0.097 |
| 10 circular and spiral | 0.85 | 0.107 |
For scale: on the same specimens the ACI nominal capacity the database computes falls about 8% under the measured value.
Scope of this verification
The comparison against measured capacity is on peak moment. For each of the thirty-nine specimens a single value is compared; a measured curve shape is compared on one specimen (Sheikh & Üzümeri A-11).
The calculation departs from the data in the PEER database in two regions. On circular and spiral-confined sections it falls about 15% under the measured capacity, that is, on the conservative side. Above 70 MPa concrete strength at high axial load it rises 14% to 21% above it. A strength verdict must not be read from the behaviour curve in that region; the Column Design macro performs its strength check with the code's own diagram.
For circular sections both the Mander and the Kent & Park confined models were compared against the specimens' measured behaviour. Falling 15% under the measured capacity shows up in both models, so the difference does not come from choosing one of them. The worksheet of the section "The same section, by a second implementation" above meets this calculation within 0.5% on a circular section. The meshing and the integration of a circular section are therefore outside the 15% difference. The Mander and Kent & Park confinement models were developed on normal-strength concrete; for sections outside that range another confined curve can be selected.