Segment of a circle (Pipeline in radius of curvature)

[MikeArmstrong]

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

[jeanGiraud]

MIke Armstrong wrote:

jean Giraud wrote:

If that piece of maths is not entered in AutoCad, then the graph is erroneous and the project stops there, with the missing analytical justification.

Jean

Jean,

The pipeline radius is represented in Autocad as an arc with can be drawn at any desired radius. No maths is entered into Autocad.

I think Mike you are pedalling in butter.

Pedaling in butter? - Care to elaborate?

Mike

Yes pedaling in butter because Autocad can't draw the shape of a beam, so long and so supported at some spacing distance. The shape of the curve can't be any spline or else imagination ]. The shape comes from the appropriate codes or a formula. The shape resembles my blue curve drawing and it has nothing to do with an arc of circle. It it has the "catenary" shape, then not free ends parabolic as your equation plots. A very same project was done in the former Akiva collab, but can't retrieve.

[jeanGiraud]

Mike,

My finger is itching to delete from my first reply.

We are message 25 and nothing to work with !.

Jean

[jeanGiraud]

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".

Jean

[wayne]

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

[MikeArmstrong]

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

[wayne]

Mike,

Sorry for misunderstanding. The posts roam around a little.

You really are just looking at the geometry of two circular segments which intersect, top of one and bottom of other.

The problem with your Autocad sketch is that the tangents at the point of intersection are not the same, the curve has
a discontunity. If the radi are really the same, and you want the tangents to match then the chord must the the same for both,

the sectors are just mirror immages.

If the Radi are not equal, and you don't require a coninuous first derivative, then the problem is easily solved by applying the

equations you already have.

If the Radi are not equal and you want a continuous first derivative, then you need to assume an unknown, ex. one radius, or height of chord length,
and write the equations for the first derivative and solve for the unknown.

[jeanGiraud]

The shape of the beam is more closely related to the cubic spline than any nearest invented model . Which invention of model does not exists because it is not needed. It is not needed because the spacing of the supports goes by the "code", which code limits the sag vs all the variables. So, on the AutoCad drawing, the support are spaced by the codes and, if done correctly, in between there are the intermediate points for the AutoCad spline joins. Now, whether the AutoCad does spline Bézier or cspline, for the shape it has virtually no influence . All that monkey business of circular segment does not exist. For the arc length in between two support points, just integrate the Mathcad cspline fit. From recollection, Mike was after the sag along the drop between supports, then just calculate the spline Y ordinate(s) at whichever point or series of points. This project is like leveling the cannon to shoot a fly 3 ft distance. If the pipeline runs uphill/downhill, same applies. The pipeline is a rigid body, the electric line is a flexible body. Amazing that computers and graphic packages have "retarded" to many users in comparison to the years 1930 [per say].

[wayne]

Jean,

I agree with most of what you say, but not all.

After re-reading the posts and Mike's responses this is what I conclude:

The pipe is flexible (relative to the proposed shape) and is continuoulsy supported and will be in contact with the rollers over the full lenght.
Therfore, the pipe properties, with the exception of local buckling, have nothing to do with the problem, nor do codes.

He is trying to define a shape that begins at one elevation and ends at another an starts and ends with 0 slope; with
the constraint that the radus of curvature (actually the instantenious radius of curvature) does not exceed a set radius. There is no sag

between supports, because there is no "between supports". Apparently the only Code criteria here is with regard to the maximum radius of curvature.

You could do as you suggest, which is easy and I don't know why Mike doesn't do it that way; especially if only need to do only one time.

If this is needed many times, it may be advantegous to define with a curve, sine/cosine or polyonimal. But, what I was trying to say, is that you

can't really meet the stated criteria with circular segments with the same radi without having a discontinuity where the curves meet, which creates,

mathematically, an infinite radius of curvature.

I (and most other posts) missinterpreted his actual question, read much more into it than was there.

Sorry for that.

In WW2, General McCarther asked his Engineers to build a bridge across a river. He came back to them later and asked "where are the plans for
the bridge", The eingineers replied, "the bridge is built, but you will have to wait a couple of weeks for the plans". I think this applies here.

[MikeArmstrong]

Wayne Reid wrote:

The pipe is flexible (relative to the proposed shape) and is continuoulsy supported and will be in contact with the rollers over the full lenght.
Therfore, the pipe properties, with the exception of local buckling, have nothing to do with the problem, nor do codes.

Hit the nail on the head, all stresses have been checked before the radius of curvature was chosen, so the pipe properties have nothing to do with this specific problem.

He is trying to define a shape that begins at one elevation and ends at another an starts and ends with 0 slope; with
the constraint that the radus of curvature (actually the instantenious radius of curvature) does not exceed a set radius. There is no sag

between supports, because there is no "between supports". Apparently the only Code criteria here is with regard to the maximum radius of curvature.

Is the minimum radius, not the maximum. Correct the pipe is fully supported.

If this is needed many times, it may be advantegous to define with a curve, sine/cosine or polyonimal. But, what I was trying to say, is that you

can't really meet the stated criteria with circular segments with the same radi without having a discontinuity where the curves meet, which creates,

mathematically, an infinite radius of curvature.

I have managed to plot the first curve in my attached worksheets, but not managed to plot the second. Maybe its time to give up????

Mike

[MikeArmstrong]

Wayne Reid wrote:

You really are just looking at the geometry of two circular segments which intersect, top of one and bottom of other.

The problem with your Autocad sketch is that the tangents at the point of intersection are not the same, the curve has
a discontunity. If the radi are really the same, and you want the tangents to match then the chord must the the same for both,

the sectors are just mirror immages.

Wayne,

Correct the tangents at the intersection points are not the same. The radii are assumed to be the same, but I'm not sure if the tangents have to match.

Mike

[RichardJ]

I'm not sure if the tangents have to match.

If they don't match the pipe will have a kink in it.

[MikeArmstrong]

Richard,

The tangents are the same sorry, as is the radius. The only thing that is different is the chord length.

Mike

[jeanGiraud]

Mike,

My replies are the A ...Z of your project. It does not have 101 versions like the "Little Red Ridding hood". That was a good way to make the kids understand numbers at very early age. Is the pipe rubber for slurry transport ? Concrete pipe ...etc. Your zoomed drawing showed a sag value, where does this value comes from ? ... most probably from the code as applied by the designer. Again in case you missed it, the spline is a constrained route through points. So, get the data points the designer has plugged in, spline and all the impossible maths you are trying to do are in hand, short and hardly be contradicted. In your concept, you can join tangentially two arcs of circles but that does give a smooth curve and the derivatives will be discontinuous.

[MikeArmstrong]

Your zoomed drawing showed a sag value, where does this value comes from ?

The sag values have been extracted from Autocad.

Mike

[jeanGiraud]

MIke Armstrong wrote:

Your zoomed drawing showed a sag value, where does this value comes from ?

The sag values have been extracted from Autocad.

Mike

You mean from the AutoCad data base as entries ?

Next, I expect to see the XY data set on the ws.

Jean

[MikeArmstrong]

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

[jeanGiraud]

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

[jeanGiraud]

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.

[MikeArmstrong]

jean Giraud wrote:

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

Jean,

While I appreciate all your help I do not agree that this project is at minus 0.

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.

There is no application code. The supports points are governed by the Installation Stress Analysis performed priory to installation, obviously the larger the gap, the greater the bending moment.

Your first graph looks right for a pipe with supports every value of X, but if I do not calculate the "h" value, where would I get the correct Y values from. You have used 0 and 0.25.

Mike

[jeanGiraud]

>There is no application code. The supports points are governed by the Installation Stress Analysis performed priory to installation, obviously the larger the gap, the greater the bending moment.here is no application code. The supports points are governed by the Installation Stress Analysis performed priory to installation, obviously the larger the gap, the greater the bending moment.<

________________________________

YES there are codes ASME, EN 13480 ... about the stress analysis, which stress analysis results in the support types and distance between support points, which again creates a data set for the AutoCad draftsman. This is what I have simulated in the work sheet. How do you figure my colleagues piping designers did space the support points for a 48" steam pipe over 5 km long ... in years there was no Autocad, no Mathcad, no nothing to push buttons ? Only charts . You stick with the wrong concept that Engineering life just started yesterday afternoon. Your design or your team design and your Autocad drawing would be zapped to the recycle bin in few seconds if it not traceable to "codes" ... what I mean is that by inspection by QA, it does not meet the codes. Codes are for public safety, and an obligation from any Engineering firm to respect. You have proved nothing in that sense in 48 visits. With over 120,000 members, why not consult ASME or else reputable organisation to this end issue of "Pipeline supports" vs the application ?

Jean

[MikeArmstrong]

YES there are codes ASME, EN 13480 ... about the stress analysis, which stress analysis results in the support types and distance between support points, which again creates a data set for the AutoCad draftsman. This is what I have simulated in the work sheet. How do you figure my colleagues piping designers did space the support points for a 48" steam pipe over 5 km long ... in years there was no Autocad, no Mathcad, no nothing to push buttons ? Only charts . You stick with the wrong concept that Engineering life just started yesterday afternoon. Your design or your team design and your Autocad drawing would be zapped to the recycle bin in few seconds if it not traceable to "codes" ... what I mean is that by inspection by QA, it does not meet the codes. Codes are for public safety, and an obligation from any Engineering firm to respect. You have proved nothing in that sense in 48 visits. With over 120,000 members, why not consult ASME or else reputable organisation to this end issue of "Pipeline supports" vs the application ?

Jean

Jean,

I have already stated that the pipeline was checked against the corresponding British standards for the temporary stresses induced throughout the installation, as the supports are continuous, a check for bending was not required. I know exactly how your pipe designers work because I work for a International Pipeline Installation company, who have recently just completed a 48" gas pipeline over 25km long.

I am well aware that such Engineering has been going long before I was even born, half of my colleagues have more years experience that I am old. The design and corresponding drawings have been approved by the client.

I wasn't after a design check, I was just trying to plot the launch line and obtain every conveyor height.

Mike

[RichardJ]

Here. I think you need to borrow one of my handy animated gifs:

[MikeArmstrong]

If your sure you don't mind???

[RichardJ]

Why would I mind? I nicked it off someone else's website anyway

[MikeArmstrong]

I would have used it many times in this thread if I knew how too.

Mike

[RichardJ]

Right click on the image and open it in a new tab or window. Then you can right click and save it as an animated gif (this two step process is the only way I can see of doing this, because of the crappy way this forum handles animated gifs). When you post you can then insert it as an image from the file. If you do that it will not be animated unless someone clicks on it.

Strangely though, if you have a URL for the image on the web, for example the one that shows in the new tab or window above, and you insert the image from the web (third tab on the insert image dialogue) then it is animated!

[MikeArmstrong]

Test

Message was edited by: MIke Armstrong

[RichardJ]

Don't attach it, insert it as an image using the toolbar in the edit dialogue:

[jeanGiraud]

MIke Armstrong wrote:

YES there are codes ASME, EN 13480 ... about the stress analysis, which stress analysis results in the support types and distance between support points, which again creates a data set for the AutoCad draftsman. This is what I have simulated in the work sheet. How do you figure my colleagues piping designers did space the support points for a 48" steam pipe over 5 km long ... in years there was no Autocad, no Mathcad, no nothing to push buttons ? Only charts . You stick with the wrong concept that Engineering life just started yesterday afternoon. Your design or your team design and your Autocad drawing would be zapped to the recycle bin in few seconds if it not traceable to "codes" ... what I mean is that by inspection by QA, it does not meet the codes. Codes are for public safety, and an obligation from any Engineering firm to respect. You have proved nothing in that sense in 48 visits. With over 120,000 members, why not consult ASME or else reputable organisation to this end issue of "Pipeline supports" vs the application ?

Jean

Jean,

I have already stated that the pipeline was checked against the corresponding British standards for the temporary stresses induced throughout the installation, as the supports are continuous, a check for bending was not required. I know exactly how your pipe designers work because I work for a International Pipeline Installation company, who have recently just completed a 48" gas pipeline over 25km long.

I am well aware that such Engineering has been going long before I was even born, half of my colleagues have more years experience that I am old. The design and corresponding drawings have been approved by the client.

I wasn't after a design check, I was just trying to plot the launch line and obtain every conveyor height.

Mike

Then, better than better if the supports constitute a solid continuous channel. So, your AutoCad drawing is incorrect showing bending and again the curve is not of circular segment. So, in the mass of information there is lie. If there is bending as you pretend, then apply the formulation for the radius of curvature [see osculating circle].

[MikeArmstrong]

jean Giraud wrote:

Then, better than better if the supports constitute a solid continuous channel. So, your AutoCad drawing is incorrect showing bending and again the curve is not of circular segment. So, in the mass of information there is lie. If there is bending as you pretend, then apply the formulation for the radius of curvature [see osculating circle].

Sorry Jean,

Bending stresses are generated when the pipeline is subject to a radius of curvature as are longitudinal stresses, but the continuous support prevents additional bending moments being generated between conveyors.

Mike