Evaluate Function

Posted on 22/04/2008 by dougaj4

It would often be convenient to evaluate a function entered as text; for instance if we have the function for the deflection of a cantilever under point loading at the end:

F*L^3/(3*E*I)

then it would be convenient to be able to allocate different values to F, L, E and I, and calculate the value of the function, without having to re-enter it.

Unfortunately Excel provides no such function.  Fortunately it is quite easy to write one in VBA.

Function Eval(form As String, RepA As Variant, ParamA As Variant)
Dim Eform As String, i As Long

GetArray RepA
GetArray ParamA
Eform = form
For i = 1 To UBound(RepA, 1) - LBound(RepA, 1) + 1
Eform = Replace(Eform, RepA(i, 1), ParamA(i, 1))
Next i
Eval = Evaluate(Eform)
End Function

Usage of this function is illustrated in this screen shot:

The UDF parameters are:
The cell containing the function to be evaluated
A range containing the function parameters
A range containing the values to be substituted into each corresponding parameter

Note that the GetArray function can be found here: GetArray Function

A working version of this function can also be found in the section properties spreadsheet here: Section Properties Spreadsheet

The idea for this function was borrowed from: Lamda Function

Posted in Excel, Maths, UDFs | Tagged , , | 9 Comments

The history of the theory of beam bending – Part 2

Posted on 20/04/2008 by dougaj4

Before moving on from Galileo, Mariotte and Parent, let’s examine a peculiarity arising from the testing of their formulas.

Jacques Heyman in “The Science of Structural Engineering” states:

“… then the new calculation of bending strength [by Parent] gave a coefficient or 1/6 instead of the 1/2 of Galileo or 1/3 of Mariotte. For the first time, a logical and correct mathematical description had been given of the way a beam might fracture, but in fact none of the three values of coefficient accorded with tests – 1/2 might be better for stone and 1/3 for wood, while the mathematically correct value of 1/6 seemed to be useless as a predictor of fracture.”

This statement seems to be based on the work of Coulomb, who in 1773 (60 years after Parent’s work) reinvented the theory that had already been discovered by others. Coulomb found that stone and wood behaved in different ways, and presented two theories, based on a common approach, that satisfied the requirements of mechanics. He concluded that Galileo’s coefficient of 1/2 seemed best for stone, but he could not verify his (and Parent’s) coefficient of 1/6 for timber. By the turn of the century the standard text of the Ecole Polytechnique still stuck to Galileo’s formula for stone, and Mariotte’s coefficient of 1/3 for wood (Heyman).

This is curious, because although timber might approximate a plastic material, and thus Mariotte’s coefficient of 1/3 would give a good approximation of the bending strength, stone is a brittle material that will fail in tension while the stress distribution is close to triangular, and thus the coefficient of 1/6 should be appropriate, rather than 1/2.

Unfortunately Heyman gives no details of Coulomb’s experimental work, but the only way I can explain this anomaly is to assume that either Coulomb calculated the tensile strength of stone by back-calculation from bending tests, using Galileo’s formula, or that he carried out tensile tests that gave results much lower than the correct value, perhaps due to stress concentrations in the testing apparatus.

Either way, the end result was that by the early 19th century, over 300 years after Leonardo Da Vinci published the basis of the correct theory of bending, engineering texts still recommended a formula for the bending strength of stone that was incorrect by a factor of 3.

Reference:
The Science of Structural Engineering – Jacques Heyman

Posted in Beam Bending, Newton | Tagged , , , , , , , | 3 Comments

Bach and Newton

Posted on 04/04/2008 by dougaj4

Bach’s Music and Newtonian Science:

A Composer in Search of the Foundations of

His Art

CHRISTOPH WOLFFhttp://www.bachnetwork.co.uk/ub2/wolff.pdf

Posted in Bach, Newton | Leave a comment

Difference of Opinion

Posted on 28/03/2008 by dougaj4

Found whilst browsing:

http://diff-op.blogspot.com/

about Difference of Opinion

The following articles, reprinted by agreement with IOP Publishing Ltd, formed a regular diary column in Scientific Computing World magazine betweeen September 1997 and February 2000.Difference of Opinion was co-written by Ray Girvan and Felix Grant under the pen names Babbage and Lovelace. The two personae do not represent the two authors; every diary entry was jointly written by us both, as a team. Rather, they represent different aspects of the computing world: loosely, Babbage is interested in hardware, and Lovelace in software and philosophical aspects.

Posted in Computing - general, Newton | Leave a comment

Reinforced Concrete Section Analysis – 1

Posted on 25/03/2008 by dougaj4

Increased yield strengths of reinforcing steels have increased the importance of the Serviceability Limit State in reinforced concrete design, with SLS reinforcement stresses, crack widths, or deflections often controlling the design.

Elastic analysis of a rectangular section under pure bending can be solved easily with a quadratic equation, but for more complex shapes or combined bending and axial load an iterative process is normally used to determine the position of the neutral axis and section strains and stresses.

Click on: RC NA Depth for a paper presenting closed form solutions for any symmetrical reinforced concrete section, with any number of layers of reinforcement, under combined bending and axial load.

An extract from the paper is shown below:
NA Depth

Future posts will describe how these equations can be conveniently solved in an Excel VBA UDF, including calculation of concrete and reinforcement strains and stresses.

Posted in Beam Bending, Newton | 4 Comments