## Elastic Biaxial Bending

Spreadsheets performing Ultimate Limit State analysis with biaxial moments have been presented previously, most recently here: Biaxial bending update

I have now modified the Beam Design Functions spreadsheet to carry out a biaxial moment analysis with linear elastic material properties. The spreadsheet, including full open-source code, can be downloaded from:

Beam Design Functions Biax.zip

Input for a square section is shown below (the cross section is plotted with an xy graph, so x and y axes are not plotted to equal scale). The Biax function adjusts the position of the neutral axis so that the stress at one end is close to zero. The other stresses are then calculated based on the input neutral axis angle. Clicking the “Adjust NA Angle” button adjusts the angle so that the stress at both ends are near equal.

Maximum concrete compression and steel tension are plotted below for a resultant input moment angle to the X axis between 0 and 90 degrees.

Input and results below are for a rectangular section, 2000 mm wide by 1000 mm deep.

Non-rectangular sections may also be analysed. In some cases the “Adjust NA Angle” routine may fail to find a solution. Adjusting the initial input angle, so that the NA stresses are both close to zero should allow the routine to work:

Detailed output results are given on the “Elastic Out” sheet, including:

• Section properties for the concrete in compression, reinforcement, and the combined section. Note that the listed properties are calculated with the section rotated so that the neutral axis is parallel to the X axis.
• Concrete stresses at each node of the section in compression.
• Reinforcement stresses at each end of each layer.

The spreadsheet also allows for input of steel prestress forces:

The image below shows the same section input with coordinates rotated through 90 degrees, and with the same moment applied about the Y axis. The neutral axis angle is also rotated through 90 degrees, giving identical results for concrete and reinforcement stresses:

The results have also been checked against the “Elastic” single axis bending function, showing exact agreement:

A prestressed example with biaxial bending is shown below. Note that for complex shapes the Excel goal-seek function (used by the Adjust NA Angle routine) is more likely to fail to find a solution, and some initial manual adjustment of the NA angle may be required:

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