Getting concrete shrinkage right

Posted on 21/03/2015 by dougaj4

At the end of last year, very quietly, the commentary to the Australian Concrete Structures Code (AS 3600) was published.  The commentary provides a wide range of explanatory and background information to the requirements in the main code, including some aspects that are neither obvious nor widely known.

Download Reinforced Concrete Design Functions for free open source Excel functions, including the shrinkage calculation described below.

An example is the calculation of design shrinkage strain from the standard 56 day test value.  The total design shrinkage for any concrete mix consists of two components:

  • The chemical (or autogenous) shrinkage, which is a property of the mix, and is specified in the code dependent on the concrete compressive strength.
  • The drying shrinkage, which is affected by the mix and also the environment and concrete section dimensions.

To find the design drying shrinkage the basic drying shrinkage is factored depending on environmental conditions, section thickness, and time since completion of curing.  Standard values are provided for the basic drying shrinkage (depending on location), but this value may also be derived from test data, using the standard 56 day shrinkage test to AS 1012.13.  Calculation of the basic drying shrinkage from the test value requires the following steps:

  1.  Find the Test Drying Shrinkage by deducting the part of the chemical shrinkage that occurred between completion of wet curing (7 days) and completion of the test (56 days).
  2. Find the code K1 and K4 factors.  For the standard test K1 = 1.446 and K4 = 0.7.
  3. The Basic Drying Shrinkage is then given by:
    (Test drying shrinkage) / (K1 * K4 * (1 – 0.008 * fc))

VBA code performing this calculation is shown below:

Function BDS(fc As Double, Testshrink As Double, Optional CureDays As Double = 7, Optional TestDays As Double = 56, _
            Optional HypeThick As Double = 37.5, Optional K_4 As Double = 0.7, Optional Out1 As Long) As Variant
    Dim Alpha As Double, ResA(1 To 6, 1 To 1) As Double, TestDShrink As Double
    Dim AFShrink As Double, AShrinkS As Double, AShrinkE As Double, BDShrink As Double, Alpha1 As Double, K_1 As Double, DShrink As Double, TShrink As Double
    Dim BetaRH As Double, Alphads1 As Double, Alphads2 As Double, Kh As Double, Betads As Double, TotalDays As Double

    AFShrink = (0.06 * fc - 1) * 50
    AShrinkS = AFShrink * (1 - Exp(-CureDays / 10))
    AShrinkE = AFShrink * (1 - Exp(-TestDays / 10))
    TestDShrink = Testshrink - (AShrinkE - AShrinkS)
    Alpha1 = 0.8 + 1.2 * Exp(-HypeThick * 0.005)
    K_1 = Alpha1 * TestDays ^ 0.8 / (TestDays ^ 0.8 + 0.15 * HypeThick)
    BDShrink = TestDShrink / (K_1 * K_4 * (1 - 0.008 * fc))

    ResA(1, 1) = BDShrink
    ResA(2, 1) = AShrinkS
    ResA(3, 1) = AShrinkE
    ResA(4, 1) = TestDShrink
    ResA(5, 1) = Alpha1
    ResA(6, 1) = K_1

    If Out1 = 0 Then
        BDS = ResA
    Else
        BDS = ResA(Out1, 1)
    End If

End Function

An example of the use of the function is shown in the screen shot below. Note that:

  •  For the standard 56 day test the only input required is the concrete grade and the test shrinkage value (in microstrain).
  • The function returns a column array of six values (as shown).  To display all six values enter as an array function.
  • The first value returned is the Basic Drying Shrinkage.  This must be factored as detailed in the code to find the design drying shrinkage, then add the chemical (autogenous) shrinkage to find the final Design Shrinkage.
  • The spreadsheet Shrink() function will return the Design Shrinkage after any given period, for specified Basic Drying Shrinkage, concrete grade, hypothetical thickness, and K4 values.

BDShrink

Posted in Concrete, Excel, Newton, UDFs, VBA | Tagged , , , , , , | Leave a comment

Pi day of the Century

Posted on 14/03/2015 by dougaj4

In North America, where today is 3-14-15, it is Pi day, and since there are not 31 days in April, the rest of us have to go along with that as well.

Not only is it Pi day, it’s the best approximation to Pi we will get for the next 100 years, because the year ends in 15.

To mark the occasion I have set up a spreadsheet to calculate pi by drawing random lines, based on blog posts by Excel Hero, Hui at Chandoo.org, and also by Hui, another one from Excel Hero.

My spreadsheet is similar to Hui’s “hot-dog” throwing method, but since computers can throw lines much more accurately than people, I have adjusted it to produce a different looking chart.

In the spreadsheet I have set up a table to generate 1000 straight lines of length 1, with centre point exactly at X = 0, and Y between -0.5 and +0.5.  The lines have a random angle to the X axis so some will cross the X axis and some won’t.  The approximate value of Pi is given by: Pi = 2*N / X
where N is the total number of lines (1000 in this case) and X is the number of lines that cross the X axis.

The top of the table looks like:

HD-Pi1

Since 1000 lines does not give a very good approximation to Pi I have also written a UDF that will do the same calculation, but much more quickly because it doesn’t write all the line coordinates to the spreadsheet. With 10 million lines it gives a not too bad approximation:

HD-Pi2

If we plot an XY chart showing just the lines that cross the x-axis it forms a reasonable approximation to an ellipse:

HD-Pi3

Whereas the lines that do not cross form a square with a semi-circle at the top and bottom:

HD-Pi4

The spreadsheet may be downloaded from RandPi2.xlsb.

Have a great Pi Day.

Edit, Still Pi Day: I have just discovered that the term Pi was invented by a Welsh Man named William Jones.

Edit 21:40 15 Mar 2015; still Pi Day somewhere*:  I happened to click on an old comment and found a link to: Estimating pi using the MC technique

* The somewhere where it is still Pi day is:

The following countries use GMT-12 as their Time Zone.  NB No Summer Time / Daylight Saving Time is used.

Baker Island: an uninhabited atoll located just north of the equator in the central Pacific Ocean about 3,100 kilometres (1,900 miles) southwest of Honolulu.

Posted in Charts, Excel, Maths, Newton | Tagged , , , , , | 6 Comments

Continuous Beams with Specified Deflections

Posted on 26/02/2015 by dougaj4

A few weeks ago there was some discussion in a previous continuous beam post about how to analyse a continuous beam with specified deflections at the supports.  In this post I will look at three different ways of doing this with the current version of the ConbeamU spreadsheet, and in the next I will publish a new version that includes functions to do the job automatically.  The examples included in this post can be downloaded (including full open source code) from: Conbeam-Support Disp.zip

Probably the simplest approach is to use the Excel Solver to adjust the beam model to achieve the desired displacements at the supports.  The procedure in outline is:

  1. Set up a single cell with a formula that will act as a single target value for solver.
  2. Select a range (or separate cells) containing values that Solver will adjust to reach its goal.
  3. Enter starting values in this range.
  4. Open the Solver dialog, specify the required inputs (including any constraints) and run the Solver.

The first screenshot below shows this process with a three span beam, with cantilevers at each end, and specified displacements at each support.  In this example I have adjusted the support stiffness to get the required displacements (click on any image for a full size view):

DDeflect1

This example uses the ConBeamU User Defined Function (UDF) to find the beam deflections (also shear forces and moments) for the beam defined under “Beam Segments” and “Supports”, when subject to the specified Distributed Loads and Point Loads.  Tne main function output is in columns K to N (hidden by the Solver Dialog box), but the deflections at the four supports are extracted with simple formulas in cells K4:K7.  The required deflections are entered in cells J4:J7, and the difference of these two values squared is in L4:L7.  The difference is squared because this will always give a positive value.  In this case the ConBeamU function has been set up to return deflections in mm, but care must be taken to ensure consistent units.  The Solver target value (or “Objective”)  in Cell L8 is the sum of the 4 values in L4:L7.

in the Solver Dialog:

  1. The Objective is specified as the value  in L8
  2. This value is to be set to a value of 0
  3. The values to be adjusted to achieve this target are in the range F6:F9
  4. Each of these values must be greater than zero, because zero or negative values will be treated as a rigid support by the ConBeamU function.

Having set up the Solver, click the Solve button, and the values in F6:F9 will be adjusted until L8 is very close to zero, as shown above.

The second example also uses the Solver, but in this case I have set the support stiffness values to a fixed (arbitrary) value, and created an artificial point load at each support:
DDeflect2

The set-up is the same as for the previous example except that:

  1. The “By Changing Variable Cells” range is now the values of the four additional point loads in G78:G81
  2. The constraints to keep all variable values greater than zero are no longer required.
  3. This procedure generates the correct shear forces, moments and deflections, because the artificial loads are applied directly to the supports, but to find the correct reactions the applied support loads must be added to the calculated reactions.

It can be seen (column N) that this method finds exactly the same Nett Reaction forces as the first example.

A similar approach, that does not require the use of the Solver, is to make the stiffness of the supports very much greater than the stiffness of the beam, and apply very large vertical point loads at the support positions:
DDeflect3

In the example shown above I have used the ConBeam UDF, which is not unit aware, so deflections are now in metres.  The support stiffness has been set to 1E10 kN/m, and the artificial point loads are 1E10 x required deflection in m (Range G24:G27).  The deflections are calculated immediately, and are a very close approximation to the required values (the maximum error is of the order of 1E-7 m).

To achieve this order of accuracy the stiffness of the supports must be chosen carefully.  If it is too low the relative beam stiffness will affect the results, but if it is too high round-off error has a significant effect.  The best value for any given beam can be found easily by trial and error.

The final method takes longer to set up, but is easily automated and is the one that has been incorporated in the new version of the spreadsheet.  The procedure used in this approach is:

  1. The SSSpanU UDF is used to calculate the beam deflection under the specified (real)loading when supported by the end supports only.
  2. Upward unit loads are applied separately at each internal support, and a flexibility matrix is set up showing the deflection at each node position due to each unit load.
  3. The flexibility matrix is inverted to yield a stiffness matrix.
  4. The stiffness matrix is multiplied by the deflection required at each internal node to return the single span deflections to the specified support deflection.
  5. The resulting values are the internal reaction loads, that may be applied in conjunction with the real applied loads to determine the resultant loads and deflections of the beam.
    DDeflect4

The screenshot above shows the input and results for the three initial calculations:

  1. The end supports are defined in range F5:G7, and the real applied loading is in ranges B20:E23 (distributed loads) and F20:H23 (point loads).
  2. The resulting beam actions and deflections for the single span are in range K24:N55
  3. The applied unit loads at the internal supports are in ranges F26:H27 and F30:H31, and the resulting output at the internal nodes is in ranges K11:N12 and K17:N18.

The final stage of the calculations is shown in the screenshot below:
DDeflect5

  1.  The support flexibility matrix is in P12:Q13, being the node deflections from cells, N12, N13 and N17 and N18.
  2. This matrix is inverted in range P16:Q17, using the MInverse function, to generate the support stiffness matrix.
  3. This matrix is then multiplied (using the MMult function) by the required beam deflections to find the support reaction at each internal support.  Note that the required settlements are measured from a datum line between the end supports after settlement.  The settlement of the Datum is shown in L5:L8, and the resulting required deflections at the internal supports at R16:R17.
  4. The internal support reactions are then applied together with the original loading in a single span analysis, with output in range Q25: T55.
  5. Finally the deflections must be adjusted for the deflection of the end nodes, as shown in Column U.  The final reactions and deflections at the supports are also transferred to M5:N8.

The results of the three different analysis methods, applied to a beam with the same loading and specified deflections, are shown in the graphs below, showing virtually identical results from each analysis.

DDeflect6 DDeflect7 DDeflect8

Posted in Excel | Tagged , , , , , | 1 Comment

7 Year Report

Posted on 24/02/2015 by dougaj4

As in previous years, I have downloaded the statistics for this blog for the previous year, and pasted them into a worksheet. The link to each post is preserved in the spreadsheet, so it makes a convenient index to what has been posted over the year. The spreadsheet has been uploaded to OneDrive, so you should be able to access the links in the window below, or open the file in your browser or Excel, or download it.

Link not displaying? If you get a blank space above, rather than a Onedrive spreadsheet, please let me know, together with your browser type and version, and Flash version. Thanks

Of the 2014 posts, the most popular overall was The angle between two vectors, Python version

The most popular in the Newton category was Frame Analysis with Excel,

and the most popular in the Bach category was The Deserter

From the “deserving but sadly neglected category” I have chosen (so go and have a look/listen):

Newton: Numerical Integration; Tanh-Sinh Quadrature v. 4.3

Excel: Binary functions, combining text, and finding unique columns

Bach: More from Martin Simpson

Most frequent referrers to this site came from:

Referrers2014

Posted in Bach, Excel, Newton | Tagged | 2 Comments

Reducing Risk in Structural Concrete Design

Posted on 23/02/2015 by dougaj4

The Concrete Institute of Australia is running a series of full day seminars on Structural Concrete Design for Extreme Events in each of the mainland capitals:

See:  Structural Concrete Design for Extreme Events for details of dates and locations in each state.

The aim of these seminars is not just to focus on design for earthquake and fire, but to encourage a focus on a design approach that will reduce the risks arising from all sorts of unexpected events.

In recent years the legal focus on risk reduction has moved from consideration of the possible risks towards a focus on the available precautions, regardless of the probability of the associated risks. The latter approach requires that risks should be eliminated or minimised “so far as is reasonably practicable”, whereas the risk focussed approach requires that the overall risk should be “as low as reasonably practicable”.

The precaution based approach is presented as providing a clear definition of the risk reduction measures that must be taken, but the question remains, how is it to be decided if a given precautionary measure is “reasonably practicable” or not? A precaution based approach has much to commend it, but from the engineer’s perspective it moves the risk minimisation process from one where correct procedures are reasonably well defined in specifications and codes of practice, to one where decisions must be made on the basis of judgement and experience.

The engineer may see it as unfortunate that two radically different approaches towards the minimisation of risks are described by phrases that at face value have identical meanings, but as engineers we must recognise the world as it is, and find ways to deal with it. A precaution based approach to risk minimisation requires not just a thorough knowledge of structural theory and code requirements, but also a knowledge of techniques that will improve the robustness of structures at low cost, and a knowledge of what is considered current best practice.

Regardless of an increased focus on precautionary measures, risk minimisation also continues to require an awareness of the possible risks, a best estimate of their probability, and a knowledge of the likely consequences. This also requires knowledge outside that found in codes of practice, which are necessarily restricted in the scope of their data, and have a significant time lag in incorporating new research and procedures.

The first of the Concrete Institute’s National Seminars this year is entitled “Structural Concrete Design for Extreme Events” and will deliver detailed guidance on designing robust structures that will provide increased resistance to not just earthquakes and fire, but also other infrequent but possibly calamitous events such as impact or blast loading, extreme storm loading, or gross foundation movements.

The seminars will be presented by three of Australia’s leading experts in their respective fields.

  • Professor Stephen Foster is Head of School at the University of New South Wales, and has developed an international reputation for his work on high strength concrete and structural robustness.
  • Professor John Wilson is Executive Dean at Swinburne University and a past president and continuing active member of the Australian Earthquake Engineering Society. He has a long-standing interest in earthquake engineering, structural dynamics and blast engineering.
  • Professor José Torero is Professor of Civil Engineering and Head of School at the University of Queensland, and also has a truly international reputation as a leader in research and education in his area of fire engineering.

These seminars present a rare opportunity to hear from three such distinguished presenters on a subject of essential importance to all practicing structural engineers, and I encourage all to attend.

Posted in Concrete, Newton | Tagged , , , | Leave a comment