This post is a continuation from The Dome of Santa Maria del Fiore in Florence
Different sources give a wide variety of different dimensions for the dome, but there seems to be reasonable agreement that the “across flats” diameter of the base of the structure is 42 metres.  Using this as a basis I have determined the dimensions of both shells of the dome by scaling from the section shown below, using the DigitGraph spreadsheet:

Dome cross section

The method of use of the spreadsheet is:

• Insert the image to be scaled on the spreadsheet.
• Move and stretch the “axes” shape so the bottom left and top right corners are over points separated by known horizontal and vertical dimensions
• Draw freeform shapes with nodes over each point to be scaled.
• The User Defined Function (UDF) can then be used to find the coordinates of each node of the inserted shapes.

See the spreadsheet and How to digitise a scanned image for more details.

In this case I have found the coordinates of three points on each face of the inner shell, and the outer face of the outer shell, using the centre point of the base of the dome as the origin.  This yielded the data shown below:

Dome shell face coordinates, and derived dimensions and radii

Having found the coordinates of three points on each surface the UDF ArcCenP3, from the IP2 spreadsheet, was used to find the centre and radius of the arc passing through these points.  The thickness of each shell was also checked from the DigitGraph sheet, and the results were cross-checked against a similar analysis using a photograph of the exterior of the dome:

DigitGraph analysis of the outer shell

This data was used to determine the arc centre and radius data shown below, which was in turn used to generate the 3D coordinates of 21 points along the edge of a segment of the dome:

Generated coordinates along the edge of an arch segmet.

The final stages were:

• Generate the 3D coordinates of an additional 20 points at each level across the full width of each face of one segment of the dome, and assign node numbers to these points.
• Generate the node numbers defining “brick elements” forming the inner and outer shells, and also four ribs connecting the shells.
• Transfer the node and brick data into a finite element program (Strand 7), forming a model of one segment of the dome.  This was done using the Strand7 Applications Programming Interface (API), and the spreadsheet S7 API Tools, which allows Strand7 models to be generated automatically from spreadsheet data.
• Assign appropriate boundary restraints to the base of the segment, and assign materials properties.
• Make four contiguous copies of the segment, each rotated by 45 degrees.
• Trim the ends of the end segments, leaving a half dome with a vertical plane cut face, and assign symmetry restraint conditions to this face.
• The model is now ready for analysis.

This process is shown in the screenshots below:

Node List in S7 API Tools spreadsheet

Brick element list in S7 API Tools spreadsheet

Arch segment generated from spreadsheet data

Segment cut to show connecting ribs between inner and outer shell

Completed model of half dome, ready for analysis