3D Frames, axes and stiffness matrices

Posted on 09/12/2016 by dougaj4

In the course of adding provision for spring end releases to the 3DFrame spreadsheet, I have also tidied up the code for generating the structure stiffness matrix, and added some documentation to the spreadsheet.  The revised spreadsheet (version 2.02) can be downloaded from:

3DFrame.zip

The original version of the spreadsheet, using code from Programming the Finite Element Method, used a beam local axis system with the local x axis aligned with the beam longitudinal axis, the local z axis parallel to the Global ZX plane, and the local y axis completing the right-handed axis system.  The beam stiffness matrix consistent with that system is shown below (click the image for a full sized view, or see the “Matrix examples” sheet of the download spreadsheet for a text version):

3dframe4-3

The Strand7 finite element program uses an alternative system (shown below), and because this program is used to check the results of the spreadsheet I have modified the code to use the same system:

Beam principal axes are defined as i1 to i3, where:

  • i3 – is the unit vector directed from Node 1 to Node 2.
  • i2 – is the unit vector arising from i2 = Z × i3 where Z is the unit vector in the global Z direction
  • i1 – completes the right-handed system such that i1 × i2 = i3

This procedure in effect creates the i2 Axis parallel to the XY plane, and the i1 Axis in the plane parallel to the Z axis; i.e. the Z axis is in effect defined as the vertical axis in the model, and the XY plane is horizontal.

  • If the i3 axis is parallel to the Z axis then the i2 axis is parallel to the Y axis in the positive direction.
  • Beam principal axes may be rotated about the i3 axis by a specified angle, Gamma.
  • Positive rotation is clockwise when looking in the positive i3 direction.

This system is shown in the screen shot below:

3dframe4-1

In Strand7 the web of I girders is by default aligned with 2 axis, so that the beam flexural stiffness values, I11 and I22, relate to the beam strong and weak axis respectively, as shown below:

3dframe4-2

The beam local stiffness matrix for this system is shown below:

3dframe4-4

To combine the beam stiffness matrices into a single global matrix they must all be rotated to a common set of axes, that is the Global XYZ system, using:

3dframe4-12

where T is the 12×12 rotation matrix, made up of 4 copies of the 3×3 matrix below:

3dframe4-5

where:

  • L is the beam length
  • XL, YL, ZL are the components of the local axes in the global system
  • Cg, Sg are the Cosine and Sine of Gamma, the angle of the 2 axis to the XY plane
  • Den = L * (XL ^ 2 + YL ^ 2) ^ 0.5

The rotation matrix transpose, TT is:
3dframe4-6

Application of this method to a single beam is shown in the screen shots below (from the Matrix Examples sheet of the download file).

The local stiffness matrix, KM1:

3dframe4-7

The rotation matrix, T:

3dframe4-8

The transpose of the rotation matrix, TT

3dframe4-9

The matrix CC = KM1.T

3dframe4-10

The global stiffness matrix KM2 = TT.CC

3dframe4-11

Alternative beam examples can be generated on the Matrix Examples sheet, by entering a different beam number in Cell C18:

3dframe4-13

Edit 19 Dec 2016: Formula for Den corrected

Posted in Arrays, Beam Bending, Excel, Finite Element Analysis, Frame Analysis, Newton, Strand7, UDFs, VBA | Tagged , , , , , , , , | 10 Comments

3DFrame with spring releases

Posted on 02/12/2016 by dougaj4

Following the previous post I have added similar functionality to the 3DFrame spreadsheet.  The new file can be downloaded from:

3DFrame.zip

As before, see Installing C# dll files, reminder for details of installing the supplied dll files, which provide much better performance for large frames. If these are not installed the spreadsheet will default to the built in VBA solver.

For the 3D analysis I have modified the original beam properties, rather than inserting new hinge members.  This required more extensive changes to the code than for the 2D frame (which will be described in more detail in later posts), but allows the releases to be added without introducing additional freedoms.  For the 3DFrame spreadsheet provision has been added for translational spring releases, as well as rotation about any axis.  The data input for the spring releases is shown below:

3dframe3-1

Note that end releases are now specified relative to the member principal axes, rather than the global axes.

There are also minor changes to the load input screen:

3dframe3-2

As in the previous version, there are three alternative solvers. The VBA version will work without installing any additional files, but is very slow for the large frames.

3dframe3-3

The download zip file includes a file comparing output with results from the commercial program Strand7, which shows near exact agreement:

3dframe3-4

The check analysis used the small frame shown below, with a variety of different loads and end releases, and including inclined and rotated frame members:

3dframe3-8

Graphical output from the spreadsheet is shown below:

3dframe3-5

Beam actions or deflections can also be plotted along any selected beams:

3dframe3-7
The spreadsheet will also handle much larger frames, with the model shown below completing in about 15 seconds, using the sparse compiled solver (but note that the Excel drawing routines are painfully slow, with a re-draw of the display below taking several minutes).

3dframe3-6

Posted in Beam Bending, Excel, Finite Element Analysis, Frame Analysis, Newton, Strand7, VBA | Tagged , , , , | 8 Comments

Frame analysis with spring releases; 2D

Posted on 25/11/2016 by dougaj4

I have updated the Frame4 spreadsheet to allow a rotational spring stiffness to be specified for members with a rotational end release.  The new file can be downloaded from:

Frame4.zip

See Installing C# dll files, reminder for details of installing the supplied dll files, which provide much better performance for large frames. If these are not installed the spreadsheet will default to the built in VBA solver.

The previous version allowed insertion of beam end releases with zero rotational stiffness, which resulted in additional nodes being inserted in the frame model.  This allowed the addition of springs to the model, in effect inserting additional zero length spring members, connecting the rotational freedom at released ends to the rest of the model, by inserting the following values into the frame stiffness matrix:

frame4-8-1

using the following code:

'Add hinge stiffness factors
        k = BeamA(i, 5)
        If k > 0 Then
            For j = 1 To 2
                HingeK = BeamA(i, j + 2)
                If HingeK > 0 Then
                    gj = BeamER(k, (j - 1) * 2 + 8)
                    gk = BeamER(k, (j - 1) * 2 + 9)
                    KGCA(gj, gj) = KGCA(gj, gj) + HingeK
                    KGCA(gk, gk) = KGCA(gk, gk) + HingeK
                    KGCA(gj, gk) = KGCA(gj, gk) - HingeK
                    KGCA(gk, gj) = KGCA(gk, gj) - HingeK
                End If
            Next j
        End If

The screenshot below shows the input for the spring restraints:

frame4-8-2

I have compared results from the spreadsheet with those from the FEA program Strand7, using the model shown below:

frame4-8-6

The screenshot below shows near exact agreement for both beam end actions and deflections:

frame4-8-3

The spreadsheet also allows a lot of the frame deflected shape, and plots of arch actions and/or deflections along any slected beams.

Strand7 graphical output:

frame4-8-7

Spredsheet deflected shape plot:

frame4-8-4

Spreadsheet bending moment plots:

frame4-8-5

 

Posted in Beam Bending, Excel, Finite Element Analysis, Frame Analysis, Link to dll, Newton, Strand7, VBA | Tagged , , , , | 1 Comment

Can you spare a dime

Posted on 23/11/2016 by dougaj4

I enjoyed this rendition of a classic song by Martin Simpson:

I also enjoyed this comment:

daniel gorney1 year ago
Bravo. A very fine rendition. My daddy, Jay Gorney, who wrote the music, would have enjoyed it as much as I did. Keep on keeping on. A hell of a song. Daniel Gorney

They used to tell me I was building a dream
And so I followed the mob
When there was earth to plow or guns to bear
I was always there right on the job

They used to tell me I was building a dream
With peace and glory ahead
Why should I be standing in line
Just waiting for bread

Once I built a railroad, I made it run
Made it race against time
Once I built a railroad, now it’s done
Brother, can you spare a dime

Once I built a tower up to the sun
Brick and rivet and lime
Once I built a tower, now it’s done
Brother, can you spare a dime?

Once in khaki suits, gee we looked swell
Full of that Yankee-Doodly-dum
Half a million boots went slogging through Hell
And I was the kid with the drum

Say, don’t you remember, they called me “Al”
It was “Al” all the time
Why don’t you remember, I’m your pal
Say buddy, can you spare a dime

Once in khaki suits, ah gee we looked swell
Full of that Yankee-Doodly-dum
Half a million boots went slogging through Hell
And I was the kid with the drum

Oh, say, don’t you remember, they called me “Al”
It was “Al” all the time
Say, don’t you remember, I’m your pal
Buddy, can you spare a dime

Songwriters: E. Y. HARBURG, JAY GORNEY
Posted in Bach | Tagged , | Leave a comment

More on Lookups

Posted on 22/11/2016 by dougaj4

Following my recent post on nearest lookups, I came across a post from Charles Williams on the opposite problem; how to do an exact VLookup  on a large amount of data without taking all day, but still retaining the ability to return an error message if there was no exact  match.  The link is well worth a look:

VLookup tricks: why 2 VLOOKUPS are better than 1 VLOOKUP

Posted in Excel | Tagged , , | Leave a comment