Continuing from the previous post, today I will be looking at the Macaulay function, that performs the underlying analysis, and the application of this function to cantilevers and single span beams.

There have been some significant changes to the Python code and the spreadsheet, so please download the latest versions from:

py_ConBeamU.zip

Also, see Python and pyxll for details of the pyxll package required to link the Python code to Excel, including a 10% discount code for new users.

All of the functions in the spreadsheet call the py_Macaulay function, to calculate the cumulative effect of applied loads along the beam, by integrating to find shear forces, bending moments, curvatures, slopes and deflections. For background information on Macaulay see: Beam actions and deflections by Macaulay’s Method.

For practical applications it is better to call the function related to the type of beam to be analysed, but the Macaulay function can also be called directly from Excel:

The beam to be analysed may be divided into any number of segments, with the bending stiffness, and optionally the shear stiffness, given for each segment. Any number of trapezoidal distributed loads, and point loads and moments may be applied. The function returns shear force, bending moments, and transverse slope and deflections along the beam, starting from zero at the left hand end.

Input for the py_Cantilever function is similar, with the addition of the input of a single support, defining the fixed end, and the transverse and flexure stiffness for the support, with rigid conditions indicated with -1:

In the example above the loading and section properties are the same as for the Macaulay example, with the beam fixed at the right hand end. The shear forces and bending moments are the same as the Macaulay output, but the slopes and deflections have been rotated about the right hand end, to give fixed conditions as specified.

The py_SSSpan function calculates end support loads (and deflections, for spring supports) for a single span beam, then finds actions and deflections along the beam, based on the Macaulay analysis:

For this function the supports must be at the beam ends. A single span with one or two end cantilevers may be analysed with the py_Conbeam function (to be covered in the next post).

The function also has a unit aware version, py_SSSpanU, allowing use of inconsistent units:

At the moment the range of available units is limited, but this will be expanded in future versions.

The end supports may be simply supported, or have spring or fixed restraints for one or both end. Output with rigid moment connections at both ends is shown below.