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M
MikeArmstrong1-Visitor
1-Visitor
August 23, 2010
Question

Segment of a circle (Pipeline in radius of curvature)

  • August 23, 2010
  • 6 replies
  • 38016 views

I have a very unique problem. In the attached worksheet I am trying to calculate the height of an arc at any point along the curve, not just the center. The arc is actually replicating a pipeline which has been placed in a radius of curvature.

The length of the chord and arc radius are know from a AutoCAD drawing, the first graph is correct and has been checked against my AutoCAD drawing, the problem is the sag bend which should be the opposite to the hog bend shown on the graph.

More explanation with in the worksheet.

Mike

6 replies

J
jeanGiraud1-Visitor
1-Visitor
August 24, 2010

If you don't get replies, "saves as" 11 or lower versions,

or simply a diagram of the project.

Jean

M
MikeArmstrong1-VisitorAuthor
1-Visitor
August 24, 2010

Re-posted in M11 format.

Mike

J
jeanGiraud1-Visitor
1-Visitor
August 24, 2010

The very title "Segment of a circle" does not apply yet.

A very similar project was done in the former Mathcad collab [can't retrieve].

What is AutoCad doing in there ?

Here is my understanding, from total absence of analytical support in your work sheet.

MCADspline_0.gif

jmG

R
RichardJ19-Tanzanite
19-Tanzanite
August 24, 2010

Is this what you mean?

M
MikeArmstrong1-VisitorAuthor
1-Visitor
August 24, 2010

Richard Jackson wrote:

Is this what you mean?

Richard,

As usual you are very close to finding the solution. Have a look at the attached worksheet - easier to explain.

I would like to post an image of the problem, but due to the scale in Autocad it's hard to see when plotted.

Mike

M
MikeArmstrong1-VisitorAuthor
1-Visitor
August 24, 2010

Just got the "JIVE" error screen - made up now

J
jeanGiraud1-Visitor
1-Visitor
August 27, 2010

Mike,

My finger is itching to delete from my first reply.

We are message 25 and nothing to work with !.

Jean

J
jeanGiraud1-Visitor
1-Visitor
August 29, 2010

Mike

Back to your parabolic equation: it does not represent the shape of the constrained pipeline. The measurements from AutoCad is non sense if the shape is not extracted from an equation. If the shape is splined in Autocad, it does not makes more sense.unless proven so. The conclusion is that your project is from scratch, like reinventing the wheel because there are "codes", and otherwise formulations. It seems somebody in your outfit is either hiding information to you or simply pushing a pencil in a sheet of paper. Get the drawing and design approved by the "regulatory body".

MCADspline_3.gif

Jean

W
wayne12-Amethyst
12-Amethyst
August 29, 2010

From reading all the posts, I think the problem might be the problem statement itself.

I think this is what Mike's problem is??

There is a long pipe with multiple supports at intervals. Each support is placed such that the elevation of the
supports vary, by happenstance or desing, and that the elevation is known. In this case I think Mike is saying that
the supports are assumed to fit on a circular arc. Mike wants to know the equation for the location of the elastic
curve along the length between the supports. Mike indicated the he wants to check for buckling. (if his company makes
a conveyer with these support elevations, will the forced shape cause crimps in the pipe?)

Since there is no axial load, I assume that Mike is talking about local buckling, or crimping, of the pipe. The check for
this is based on stress, which is a function of curvature, or moment (M/EI)

Aside: A cable which supports its own weight forms a Canteary.
A cable, which supports a uniform load, and it's weigth can be ignored, forms a parabola.

The above problem is neither of these. The above is a problem of a continuous beam with support settlement.

If you do not have an analysis program, the problem can be solved by using the fromulas of a simple beam with a point load.

Take beam of length L with a point load representing the reaction at a roller, say 3 points along the length. That represents 3 unknowns.

Take each load one at a time and calculate the deflection at each uknown support (d11, d12, d13, then d21, d22, d23, d33, and d31, d32, d33)
This gives three equations in three unknowns.

Solve d1=d11+d21+d31, d2=d12+d22+d23, d3=d13+d23+d33, where d1, d2 and d3 are the support displacements from a chord connecting
the ends. The answers are the support reactions.

Then put the support reactions back in the standard beam solution, superimposing the results for each and you can find the equation of the line; but
is sounds like all you need is the moment, which a simple statics problem.

If needed, you can wirte the moment equation, which is discontinuous at the supports, and use Mathcad to numerically integrate to get
curvature and then deflection at any point. You could also just use numerical integration instead of the simple beam formula, where the only

constant of integration you can't determine by inspeciton is the initial slope

Am I understanding the problem, or am I on a different boat.

Wayne

M
MikeArmstrong1-VisitorAuthor
1-Visitor
September 4, 2010

Wayne Reid wrote:

There is a long pipe with multiple supports at intervals. Each support is placed such that the elevation of the
supports vary, by happenstance or desing, and that the elevation is known. In this case I think Mike is saying that
the supports are assumed to fit on a circular arc. Mike wants to know the equation for the location of the elastic
curve along the length between the supports. Mike indicated the he wants to check for buckling. (if his company makes
a conveyer with these support elevations, will the forced shape cause crimps in the pipe?)

Since there is no axial load, I assume that Mike is talking about local buckling, or crimping, of the pipe. The check for
this is based on stress, which is a function of curvature, or moment (M/EI)

Wayne,

I don't want to check for buckling, that has already been done before the radius was chosen.

You are correct with your comments regarding local bucking, but I was after the conveyor heights along the curve.

Basically, because I have both a hog and sag curve throughout the pipeline launch line I was trying to solve for each of the conveyor heights. I can extract this information from Autocad, where the curve has initially be drawn, but I wanted to calculate it in Mathcad.

Cheers for taking the time to look at it. Richard and Jean where on the right path, but we never quite got there.

Mike

M
MikeArmstrong1-VisitorAuthor
1-Visitor
September 7, 2010

All who have contributed and posted, please find attached my latest effort to crack this irritating problem.

The Sag bend graph starts at the left hand side on the Hog bend graph, which is where the tangent/inflexion point is. I have checked all the numbers against the drawing from AutoCAD and they seem to tie-in. It should be one continuous spline / polyline, but with the help received so far I thought my latest work might be worth posting.

Mike

J
jeanGiraud1-Visitor
1-Visitor
September 8, 2010
Sorry Mike,
Your project is at stage minus 0
@ stage minus 0 because you don't have a data set to draw the shape. You will not get out of the right way of doing, i.e: space the support points as per the applicable code then interlace the sag figure given by the applicable code and interlace at the mid interval between the support points. From there, ask AutoCad to "spline" but here you have to specify the spline. You might need to provide a set of control points. Autocad is too far back in my skill. Whatever, that would be just to get the pencil of the printer not to forget that there is a line called pipeline between the LHS <==> RHS on the printed sheet. That stage of drawing a line is just to get a drawing but for the birds because the maths and the physics is the Mathcad cspline as the default starting design.
You must understand that AutoCad is dummy [it was ], it will not do what what it has no maths behind. So, you must draw the shape from a data set, which data set is to be supplied from some superior intelligence like Mathcad and in this case just supply the cspline data set as an educated design guess. My recollection fails about AutoCad but it would be wise checking if their "Polyline" is what is supposed to mean, i.e: convex . Why convex ? to insure that the minimal technical design for a path to robots will not scratch the object it will contour. SmartSketch as the complete Bézier spline from a supplementary set of control points. You insist in calculating something that you seem to have no reference for . What's the point to calculate "sag" anywhere in the course if you don't have some reference ? I didn't see such gyzmatic fancy from the code, do you have ? Note that the length of the shape comes out directly from applying the formula from the spline fit.
Jean
J
jeanGiraud1-Visitor
1-Visitor
September 8, 2010

I have added the Bézier spline. In AutoCad, when you specify to join points, you should have some options and you should have Bézier as a minimal option. This one is the Microsoft Bézier used to spline graph. This version is the original or very original version from Tom for Art [years ago]. A gorgeous piece of tool.

M
MikeArmstrong1-VisitorAuthor
1-Visitor
September 8, 2010

I think this is the answer. As Wayne stated "If the Radi are equal, and you don't require a continuous first derivative, then the problem is easily solved by applying the equations you already have." seems to have worked.

I have now combined both of the graph. Even though the project was a little pedantic - you know what's its like when something get you hooked.

I have checked all values against the Autocad drawing and they seem ok.

Cheers for all the input.

Mike

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