Following a recent comment I have updated the documentation for finding the section properties of groups of shapes in the Section-Properties-Group spreadsheet. The revised file can be downloaded from:

__Section Properties07-Group.zip__

The screenshots below illustrate use of the Group function to find the composite section properties for a circular concrete section with steel reinforcement, taking account of the relative stiffness of the two materials.

First define the concrete shape on the DefShapes sheet:

Note that X and Y axes are tangential to the circle. To simplify the generation of the reinforcement we want the circle centre at the origin, so in the groups table on the same sheet enter an offset of -R (450mm) for X and Y. Enter 1 for the elastic modulus, then click the “Create new group” button:

The area should now be the same as in the section property results, but the centroid is moved to the origin, and the associated properties about the X and Y axes will be different.

The reinforcement diameter should now be defined in the section properties input:

The details of the reinforcing bars are now added in the group table. Note that:

- The bar data overwrites the data for the concrete.
- The X and Y offsets are defined so that the centre of all the bars is on the Y axis at the required level (in this case 450 mm – 50 mm cover -24 mm bar diameter).
- The individual bars are defined by a rotation angle about the origin.
- The elastic modulus is defined as the modulus for steel/concrete (200/30) -1, because the steel is displacing concrete.

After entering the required details click the “Add shapes to group” button. The Group Properties table will update:

The combined shape and group properties can now be seen on the “Coords_Group” sheet; click the “Recalc Group Properties” button, then the “Plot Group” button:

Details of each shape can also be viewed on the “Group data” sheet. This table should always be checked to ensure that the correct number of elements are included. Note that if shapes are accidentally added twice at the same location, this will not be obvious on the plot, but they will appear twice in the group data table.

The shape can also be modified by adding additional groups, or subtracting shapes using a negative elastic modulus. In the example below the section properties are adjusted for the bottom region of the concrete being in tension, and therefore deducted from the concrete area. The tension region is defined as a circular segment with a chord angle of +- 60degrees from the X axis:

This is then entered in the group table, rotated through -90 degrees, so that the chord is parallel to the X axis:

Finally the three reinforcement bars now in the tension zone are no longer displacing active concrete, so the area of these bars should be added back, with an elastic modulus of 1:

The revised cross-section and section properties may now be plotted on the Coords_Group sheet: