Clive James finally succumbed to a long illness this week. He was mainly known for his humorous work on TV and in writing, but he was also a prolific poet, and in his early career teamed up with Pete Atkin to produce musical versions of his work. The sample below is from their first album, Beware the Beautiful Stranger:

]]>- Stress block and capacity reduction factors updated for AS 3600-2018
- ACI factors for US customary units used when stress is entered in ksi or psi units
- Example data updated

The revised spreadsheet is included in the file:

I have checked the spreadsheet results against a paper published by Enercalc:

Reinforced Concrete Columns in Biaxial Bending

ULS Design Functions-biax results:

After adjusting the Enercalc moments to the centroid of the uncracked concrete section, the results are in near exact agreement.

The results below compare the spreadsheet results with the simplified procedures given in AS 3600 and Eurocode 2, using a rectangular section and plotting moment capacity results under different axial loads:

The AS 3600 and Eurocode provisions for biaxial bending take exactly the same form:

The two codes give significantly different results however because:

- In AS 3600 the Phi factor is defined as 0.65 for all axial loads, whereas the Eurocode uses the design Mux and Muy values for the applied axial load.
- The definition of the exponent alpha is different, resulting in AS 3600 having a lower value for low axial loads, but a higher value for high axial loads.

For the purposes of comparison, the Eurocode Mx and My values used the design moment capacities under the applicable axial load to the AS 3600 rules.

For an axial load well below the balance load both codes have an alpha value of 1, resulting in conservative capacity values under biaxial loading. The AS 3600 requirement to use a Phi factor of 0.65 results in a further substantial conservatism:

Increasing the axial load to close to the balance load increases the alpha values above 1, and brings the AS 3600 Phi factor down close to 0.65, reducing the conservatism of the simplified approach:

A further increase in the axial load increases the alpha value for AS 3600 to close to 2, so that the AS 3600 results are no longer conservative. The Eurocode alpha value is lower, and it is still conservative:

A further increase in the axial load shows the same trend, with the AS 3600 results increasingly unconservative, and the Eurocode results closer to the detailed calculation, but still conservative:

For very high axial loads, where the base of the rectangular stress block is outside the concrete section, there is a large reduction in the moment capacity from the detailed calculation. This is not reflected in the simplified method from either code, which both have an alpha value of 2 at these loads:

Further complications arise with non-rectangular sections:

In this case the simplified method from both codes gives results that are highly unconservative for some moment combinations, even for axial loads well below the design limit:

It should be noted that all calculations assumed a rectangular stress block to the AS 3600 code. Use of a parabolic linear stress block would result in further significant reductions in the design moment capacity when the concrete compression zone was triangular.

]]>Have a look for all the keyboard shortcuts you didn’t know/ had forgotten about.

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Stefan Grossman responded in the only way possible, with the assassination of John Fahey:

Why John Fahey chose to assassinate Stefan Grossman in the first place, nobody knows, not even Stefan Grossman.

]]>In the first example each of the three cubes is evaluated, and the resulting strings are concatenated into a single string function that is evaluated:

The mp-Eval function also allows values to be assigned to symbols, which can then be evaluated. In this example the letters a to d are replaced with the values in the cells to the right, and substituted into the function in cell A41:

Using the same approach, a rather simpler solution to the problem can be found:

Further discussion of the background to this problem can be found at The Math Less Travelled, which has links to on-line calculators to solve the problem summing to 42, and it is also pointed out that that the formula can be evaluated directly in Python:

Also worth a read is the Wikipedia article on the sum of three cubes, which contains further examples and numerous links.

]]>The video is short, easy to follow, and doesn’t need the sound working to understand it!

The procedure on my machine was pretty much as shown in the video except:

- The Device Manager was not listed when I clicked on the Windows Icon. Just start typing “Device Manager” and it appears.
- The Sound driver listed for my machine was “Realtek(R) Audio”
- I had to restart the computer after re-installing the driver.

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I hadn’t encountered that problem before, but a quick check confirmed that it does raise an error, whatever the data type of str1 (including Integer).

There is a detailed answer to this question at:

https://stackoverflow.com/questi…

In short, the problem is that the result data type defaults to the largest type of the values being operated on, so if they are all integers the result is an integer, and since the largest value for an integer is 32767 you get an overflow.

Alternatives ways to avoid the problem include:

- Add a decimal point to at least one number, converting it to a double.
- Convert at least one value to a Long: str1 = CLng(24) * 60 * 30
- Declare at least one value as a Long constant:

`Const HrPerDay As Long = 24`

str1 = HrPerDay * 60 * 30

The variable used to receive the result of the calculation must of course also be at least a Long, since a value greater than 32767 cannot be assigned to an integer.

It might be expected that a similar problem would arise if one value was converted to a double, so that the result of the calculation was a double, and this was assigned to a Long, but in this case the data type is automatically converted by VBA to suit the variable receiving the value.

]]>The indexing process takes time, but once completed it is updated in the background with no noticeable effect on performance.

Once indexing is complete, the Windows File Explorer provides very fast searches of all indexed folders, and fast preview of selected files:

]]>The actual question, to which the answer is 42, is not known, but a lesser known long standing question involving this number has recently been solved:

The original problem, set in 1954 at the University of Cambridge, looked for Solutions of the Diophantine Equation x^3+y^3+z^3=k, with k being all the numbers from one to 100. 42 was the last remaining number (for which a solution is possible), but recently a solution was found by a team led by the University of Bristol and Massachusetts Institute of Technology. The answer is:

(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

If we try to check this result in Excel we find that the available 15 significant figure accuracy is not quite up to the job:

But calling the mpmath package, via pyxll, finds that the solution is indeed correct:

]]>The spreadsheet (including open source VBA code) may be downloaded from:

Note that to keep the download file a reasonable size the file has been saved with a low resolution image. 350 x 350 resolution gives a good compromise between image quality and speed of processing and plotting.

Required input is:

- Centre point coordinates
- Plot width (= height)
- Plot resolution
- Number of iterations

The results are plotted as a scatter chart with 7 data ranges, plotting markers only.

The plot below was generated with 28 iterations, and a resolution of 350×350. Calculation time was 0.33 seconds:

The second example was taken from:

How To Quickly Compute The Mandelbrot Set In Python

Higher resolution (1000×1000) with 2048 iterations resulted in much longer calculation time, but times are 2-3 times faster than the plain Python code given in the link:

For faster plotting compiled code is required. A good site providing fast high resolution plots for user selected location, scale and number of iterations is:

The site text says the plots will not work in Microsoft browsers, but they worked with no problem for me in Internet Explorer. The image below was generated in the spreadsheet using parameters from the link above.