04 - Local Coordinate System

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Coordinate Systems

Introduction

This article discusses coordinate systems, and how they relate to piping systems and pipe stressanalysis. Additional information on this subject can be found in two issues of COADE's Mechanical

Engineering News- December 1992 and November 1994. These issues can be found on the COADEweb site at http://www.coade.com.

Many analytical models in engineering are based upon being able to define a real physical objectmathematically. This is accomplished by mapping the dimensions of the physical object into a similarmathematical space. Mathematical space is usually assumed to be either two-dimensional or three-dimensional. For piping analysis, the three dimensional space is necessary, since almost all pipingsystems are three dimensional in nature.

Two typical three-dimensional mathematical systems are shown below in Figure 1. Both of thesesystems are Cartesian Coordinate Systems. Each axis in these systems is perpendicular to all otheraxes.

Figure 1 Typical Cartesian Coordinate Systems

In addition, for these Cartesian coordinate systems, the right hand rule is used to define positive rotationabout each axis, and the relationship, or ordering, between the axes. Before illustrating the right handrule, there are several traits of the systems in Figure 1 that should be noted.

Each axis can be thought of as a number line, where the zero point is the point where all of theaxes intersect. While only the positive side of each axis is shown in Figure 1, each axis has anegative side as well.

The direction of the arrow heads indicates the positive direction of each axis.

In Figure 1, the X axis has one arrowhead, the Y axis has two arrowheads, and the Z axishas three arrowheads. The circular arcs labeled RX, RY, and RZ define the direction ofpositive rotation about each axis. (This point will be discussed later.)

Any point in space can be mapped to these coordinate systems by using its position along thenumber lines. For example, a point 5 units down the X axis would have a coordinate of (5.0,0.0, 0.0). A point 5 units down the X axis and 6 units down the Y axis would have a coordinateof (5.0, 6.0, 0.0).


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Notice that if the system on the right side of Figure 1 is rotated a positive 90 degrees about theX axis, the result is the system on the left side of Figure 1.

The coordinate system on the left side of Figure 1 is the default CAESAR II global coordinate system. Inthis system, the X and Z axes define the horizontal plane, and the Y axis is vertical. All furtherdiscussion in this article will target this default coordinate system, unless otherwise noted.

Other Global Coordinate Systems

There are other types of coordinate systems that can be used to mathematically map a physical object.

A "Polar" coordinate system maps points (in a two dimensional space) using a radius and arotation angle, (r, theta).

A Cylindrical coordinate system maps points using a radius, a rotation angle, and an elevation,(r, theta, z). The origin in this system could be considered the center of the bottom of a cylinder.Cylindrical coordinates are convenient to use when there is an "axis" of symmetry in the model.

A Spherical coordinate system maps points using a radius and two rotation angles, (r, theta,

phi). The origin in this system could be considered the center of a sphere. Spherical coordinatesare convenient to use when there is a "point" which is the center of symmetry in the model.

Typically, none of these coordinate systems are easily used to map piping systems. Most piping softwaredeals exclusively with the Cartesian coordinate system.

The Right Hand Rule

In the Cartesian coordinate system, each axis has a positive and a negative side, as previouslymentioned. Translations, straight-line movement, can be defined as movement along these axes.Rotation can also occur around these axes, as illustrated by the arcs in Figure 1.

A standard rule must be applied in order to define the direction of positive rotation about these axes. Thisstandard rule (known as the right hand rule) is: Put the thumb of your right hand along the axis, in the

positive direction of the axis. The direction your fingers curl is positive rotation about that axis. This isbest illustrated in Figure 2.

Figure 2 The Right Hand Rule


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The right hand rule can also be used to describe the relationship between the three axes.Mathematically, the relationship between the axes can be defined as:

X cross Y = Z eq 1Y cross Z = X eq 2Z cross X = Y eq3Where cross indicates the vector cross product.

Physically, using your right hand, what do the above equations mean? This question is best answered byFigure 3.

Figure 3 The Right Hand Rule - Continued

The left pane of Figure 3, corresponds to vector equation 3 above. Similarly, the center pane in Figure 3also corresponds to vector equation 3 above. The right hand pane in Figure 3 corresponds to vectorequation 2 above. All panes of Figure 3 refer to the left hand image of Figure 1.

Straight-line movement along any axis can be therefore described as positive or negative, depending onthe direction of motion. This straight-line movement accounts for three of the six degrees of freedomassociated with a given node point in a model. (Analysis of a model requires the discretization of the

model into a set of nodes and elements. Depending on the analysis and the element used, theassociated nodes have certain degrees of freedom. For pipe stress analysis, using 3D Beam Elements,each node in the model has six degrees of freedom.) The other three degrees of freedom are therotations about each of the axes. In accordance with the "right hand rule", positive rotation about eachaxis is defined as shown in Figures 1 and 2.


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When modeling a system mathematically, there are two coordinate systems to deal with, a global (ormodel) coordinate system and a local (or elemental) coordinate system. The global or model coordinatesystem is fixed, and can be considered a constant characteristic of the analysis at hand. The localcoordinate system is defined on an elemental basis. Each element defines its own local coordinatesystem. The orientation of these local systems varies with the orientation of the elements. An importantconcept here (which will be reiterated later) is the fact that local coordinate systems are definedby, and therefore associated with, elements. Local coordinate systems are not defined for, orassociated with, nodes.

Pipe Stress Analysis Coordinate Systems

As noted previously, most pipe stress analysis computer programs utilize the 3D Beam Element. Thiselement can be described as an infinitely thin stick, spanning between two nodes. Each of these nodeshas six degrees of freedom - three translations and three rotations. Piping systems (models) areconstructed by defining a series of elements, connected by nodes. These pipe elements are typicallydefined as vectors, in terms of "delta dimensions" referenced to a global coordinate system. Severalexample pipe elements are shown below in Figure 4.

Figure 4 - Example Pipe Elements

For most pipe stress applications, there are two dominant global coordinate systems to choose from,either "Y" axis or "Z" axis up. These two systems are depicted in Figure 1. As previously noted, theglobal coordinate system is fixed. All nodal coordinates and element delta dimensions are referenced tothis global coordinate system. For example, in Figure 4 above, the pipe element spanning from node 10to node 20 is defined with a DX (delta X) dimension of 5 ft. Additionally, node 20 has a global "X"coordinate 5 ft greater that the global "X" coordinate of node 10. Similar statements could be made aboutthe other two elements in Figure 4, only these elements are aligned with the global "Y" and global "Z"axes.

In CAESAR II, the user can choose between the two global coordinate systems shown in Figure 1. Bydefault, the CAESAR II global coordinate system puts the global "Y" axis vertical, as shown in the left halfof Figure 1, and in Figure 4. There are two ways to change the CAESAR II global coordinate system sothat the global "Z" axis is vertical.

The first method is to modify the configuration file in the current data directory. This can be accomplishedfrom the Main Menu, by selecting "Tools\Configure Setup". Once the configuration dialog appears, select


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the "Geometry" tab, as shown in Figure 5. On this tab, check the "Z Axis Up" check box, as shown in theFigure.

Figure 5 - Geometry Configuration

Once the "Z Axis Vertical" switch is activated, the CAESAR II global coordinate system will be inaccordance with the right half of Figure 1. This configuration affects all new jobs created in this datadirectory. Existing jobs with the "Y" axis vertical are not affected by this configuration change.

The second method to obtain a global coordinate system with the "Z" axis vertical is to switch coordinatesystems from within the input for the specific job at hand. This can be accomplished from the "SpecialExecution Parameters" dialog of the piping input processor. This dialog is shown below in Figure 6.


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Figure 6 - Special Execution Parameters Dialog

Checking the "Z Axis Vertical" checkbox will immediately change the orientation of the global coordinatesystem axis, with corresponding updates to the element delta dimensions. However, the relativepositions and lengths of the elements are not affected by this switch.

Defining a Model

Using the CAESAR II default coordinate system (Y axis vertical), and assuming the system shown belowin Figure 7, the corresponding element definitions given in Figure 8.

Figure 7 - Sample Piping Model


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Figure 8 - Sample Piping Model Element Definitions

For this sample model, most of the element definitions are very simple:

The first element, 10-20, is defined as 5 ft in the positive global "X" direction. This element startsat the model origin.

The second element, 20-30, is defined as 5 ft in the positive global "Y" direction. This elementbegins at the end of the first element, since both elements share node 20.

The third element, 30-40, is defined as 5 ft in the negative global "Z" direction. Note in Figure 8

that the delta dimension for this element is a negative number. This is necessary to define theelement in a negative direction.

The fourth element, 40-50, runs in both the positive global "X" and negative global "Y" directions,this element slopes to the right and down. This element is defined with delta dimensions in boththe DX and DY fields. Notice that these delta dimensions are equal in magnitude; therefore thiselement slopes at 45 degrees.

Continuing the model, from node 50, along the same 45 degree slope can be rather tedious,since most often only the overall element length is know, not its components in the globaldirections. In CAESAR II this can be best accomplished by activating the "Direction Cosine"dialog box, shown below in Figure 9. Using this dialog box, the element length can be entered,and CAESAR II will determine the appropriate components in the global directions, based on the

preceding element.

Figure 9 - Direction Cosine Dialog

CAESAR II provides an additional coding tool, for longer runs of pipe with uniform node spacing.An "element break" option is provided, which allows an element to be broken into equal lengthsegments, given a node number increment.


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In the preceding example, the model is defined solely using "delta dimensions". By constructing themodel in this fashion, it is assumed that the world coordinates of node 10 (the first node in the model) areat (0., 0., 0.). This assumption is acceptable in all but a one instance, when environmental loads areapplied to the model. In this instance, the elevation of the model is critical to the determination of theenvironmental loads, and therefore must be specified. In CAESAR II, the specification of the startingnode of the model can be accomplished using the [Alt+G] key combination. Regardless of whether ornot the global coordinates of the starting node are specified, the model geometry will plot the same.

Once a model has been defined, there are a number of operations that can be performed on the entiresystem, or on any section of the system. These operations include:

Translating the model: translation can be accomplished by specifying the global coordinates ofthe starting node of the model. If the model consists of disconnected segments, CAESAR IIrequests the coordinates of the starting node of each segment.

Rotating the model: rotation can be accomplished by using the [LIST] processor. The [LIST]processor presents the model in a spreadsheet, or grid, format, as shown in Figure 8. Options inthis processor allow the model (or any sub-section of the model) to be rotated about any of thethree global axes, a specified amount. For example, if the model shown in Figures 7 and 8 isrotated a (negative) -90 degrees about the global "Y" axis, the result is as shown in Figure 10.

Figure 10 - Example of Model Rotation

Duplicating the model: duplication can also be accomplished by using the [LIST] processor. Theentire model, or any sub-section of the model, can be duplicated.

Using Local Coordinates

When analyzing a piping system, there are a number of items that must be checked and verified. Theseitems include:

Operating loads on restraints and terminal points Maximum operating displacements

Hanger design results Codes stresses for code casesEquipment evaluation Vessel nozzle evaluationExpansion joint evaluation

Restraint loads and displacements are checked in the global coordinate system. This is necessarybecause restraint loads and displacements are nodalquantities. Element loads and stresses are mostoften evaluated in their local coordinatesystem. A good example illustrating the use of a local (element)coordinate system is the free body diagram, of forces and moments. The forces and moments in this free


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body diagram remain the same, regardless of the position of the element in the global coordinate system.Note however, that each element has its own local coordinate system. Furthermore, the local coordinatesystem of one element may be different from the local coordinate system of a different element.

While the global coordinate system is typically referred to using the capital letters 'X", "Y", and "Z", localcoordinate systems use a variety of nomenclature. In almost all cases, local coordinate systems uselower case letters. Typical local coordinate system axes are: "xyz", "abc", and "uvw". CAESAR II uses"xyz" to denote the local element coordinate system.

The local coordinate system for an element is related to the global coordinate system through a rule.There may be a number of such rules, depending on the type of element. In CAESAR II, the followingrules are used to define the local coordinate systems of the piping elements in a model.

CAESAR II Local Coordinate Definitions

Rule 1 - Straight Pipe: For straight pipe elements, the local "x" axis always points from the "From Node"to the "To Node". The local "y" axis can be found by the vector cross product of the local "x" axis with theglobal "Y" axis. Applying the "right hand rule", this local "y" axis can be found by:

1. Lay your right hand on the pipe, with the wrist at the "From Node", and the fingers pointing to the

"To Node".

2. Align or rotate your hand so that the global "Y" axis points perpendicularly out from the palm.

3. The thumb is now aligned with the local "y" axis for this element.

The local "z" axis can be found by the vector cross product of the local "x" and local "y" axes.

An exception to this ruleis the case of a vertical element. In this case, the local "x" axis is still alignedin the "From - To" direction. However, you can't "cross" a vertical element into global "Y", so the local "y"axis was arbitrarily assigned to align with the global "X" axis.

The straight elements of the model in Figure 7 are reproduced below in Figure 11, along with their local

coordinate systems. Notice that each of these straight elements has its own local coordinate system, andthat in this model, they are all aligned differently.

Figure 11 - Local Coordinate Systems for Straight Elements (1)


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In Figure 11, the positive direction of the local "x" axis for each element is defined according to the "From- To" definition of the element. For example, the local "x" axis of element 10-20 is aligned with thepositive global "X" axis, because that is the direction defined in moving from node 10 to node 20. Thelocal "x" axis of element 30-40 is aligned with the negative global "Z" axis, because that is the directiondefined in moving from node 30 to node 40. Figure 11 should be studied to ensure a good understandingof how the local element coordinate system can be defined based on the definition of the element,especially with regard to the skewed element 40-50.

As an additional example, the local element coordinate systems for the rotated system of Figure 10 areshown below in Figure 12.

Figure 12 - Local Coordinate Systems for Straight Elements (2)

Rule 2 - Bend Elements: For the near weld line of bend elements, the local "x" axis is directed alongthe incoming tangent, in the From To direction. The local z axis points to the center of the circledescribed by the bend. For the far weld line of bend elements, the local "x" axis is directed along theoutgoing tangent, in the From To direction. The local z axis points to the center of the circledescribed by the bend. In both cases, the local y axis can be found by applying the right hand rule.The local coordinate system for the bends in the example model of Figure 7 are shown below in

Figure 13.


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Figure 13 Local Coordinate Systems for Bend Elements

Rule 3 - Tee Elements: For tees, there is no element or "fitting" as there is in a CAD application. Ratherdesignating a node as a tee simply applies code defined SIFs at that point, for the three elements framinginto the tee node. In order to be able to determine the "in-plane" versus "out-of-plane" directions for thetee (necessary to apply the proper SIFs and moments), the following procedure is used to define the localcoordinate systems of the elements framing into the tee node.

The local "x" axis corresponds to the axis of the pipe, in the "From -To" direction. The local "y" axisdefines the "in-plane" plane of the tee. The local "z" axis is found by applying the "right hand rule", andpoints perpendicularly out of the plane of the fitting. Examples of local coordinates for elements framing

into tees are depicted below in Figure 14.

Figure 14 - Local Coordinate Systems for Tee Elements


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Applications - Utilizing Global and Local Coordinates

Global coordinates are used most often when dealing with piping models. Global coordinates are used todefine the model and review nodal results. Even though element stresses are defined in terms of axialand bending directions, which are "local coordinate system terms", local coordinates are rarely used. Atypical piping analysis scenario is as follows.

A decision is made as to how the global coordinate system for the piping model will align with theplant coordinate system. Usually, one of the two horizontal axes is selected to correspond to the"North" direction. However, if this results in a majority of the system being skewed with respect tothe global axes, one should consider realigning the model. It is best to have most of the systemaligned with one of the global coordinate axes.

The piping system is then assigned node points at locations where: there is a change in direction,a support, a terminal point, a point of cross section change, a point of load application, or anyother point of interest.

Once the nodes have been assigned the piping model can be defined using the "deltadimensions" as dictated by the orientation of the global coordinate system. Analysts should takeadvantage of the tools provided by CAESAR II in constructing the model - this includes theelement "break" option, the LIST rotate and duplicate options, and the "direction cosine" facility.

After verifying the input, confirming the load cases, and analyzing the model, output reviewcommences.

Output review involves checking various output reports to ensure the system responds within certainlimits. These checks include:

Checking that operating displacements make sense and are within any operational limits (to avoidponding etc.). Displacements being "nodal quantities", are reviewed in the global coordinatesystem. There is no local coordinate system associated with nodes. For the model definedin Figures 7 and 8, the operating displacements are shown in Figure 15 below.

Figure 15 - Operating Displacements


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This report shows the movements of all of the nodes in the model, in each of the six degrees offreedom, in the global coordinate system.

Checking that the restraint loads for the "structural load cases" are reasonable. This includesensuring that the restraints can be designed to carry the computed load. Restraints being "nodalquantities", are reviewed in the global coordinate system. There is no local coordinate system

associated with restraints. For the model defined in Figures 7 and 8, the operating / sustainedrestraint summary is shown in Figure 16 below.

Figure 16 - Operating / Sustained Restraint Summary

This report shows the loads on the anchor at 10 and the nozzle at 50, for all six degrees offreedom, for the two selected structural load cases, in the global coordinate system.

Checking the "Code cases" for codes stress compliance. Typically the "code stress" is comparedto the "allowable stress" for each node on each element. Occasionally, when there is anoverstress condition, a review of axial, bending, and torsion stresses are necessary. Thesestresses (axial, bending, and torsion) are "local coordinate system terms", and therefore relate tothe element's local coordinate system. For the model defined in Figures 7 and 8, a portion of the

sustained stress report is shown in Figure 17 below.


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Figure 17 - Sustained Stress Report

These reports provide sufficient information to evaluate the pipe elements in the model, to ensure properbehavior and code compliance. However, the analyst's job is not complete, loads and stress must still beevaluated at terminal points, where the piping system connects to equipment or vessel nozzles.Depending on the type of equipment or nozzle, various procedures and codes are applied. These includeAPI-610 for pumps and WRC-107 for vessel nozzles, as well as others. In the case of API-610 andWRC-107, a local coordinate system specific to these codes is employed. These local coordinatesystems are defined in terms of the pump or nozzle/vessel geometry.

When the equipment coordinate system aligns with the global coordinate system of the piping model, thenozzle loads from the restraint report (node 50 in Figure 14) can be used in the nozzle evaluation.However, when the equipment nozzle is skewed (as it is in the case of node 50 in Figure 14), theapplication of the loads is more difficult. In this case, it is best to use the loads from the element's

force/moment report, in local coordinates. The only thing to remember here is to flip the signs on all ofthe forces and moments, since the element force/moment report shows the loads on the pipe element,not on the nozzle. For the element from node 40 to node 50, the local element force/moment report isshown in Figure 18 below.


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Figure 18 - Local Element Force/Moment Report

Because the correlation between the pipe model's coordinate systems and those of equipment codes(API, WRC, etc) are often times tedious and error prone, CAESAR II provides an option in its equipmentmodules to acquire the loads on the nozzle directly from the static output. The user simply has to selectthe node and the load case; CAESAR II will acquire the loads and rotate them into the proper coordinatesystem as defined by the applicable equipment code. The user really does not have to be concerned with

the transformation from global to local coordinates, even for skewed components. This is illustratedbelow, in Figure 19. In this figure, the API-610 nozzle loads at node 50 have been acquired by clicking onthe [Get Loads from Output File] button.

Notice that the loads shown in Figure 19 are in the CAESAR II global coordinate system. This can beeasily verified by comparing these values to those in the restraint summary (for the Operating load case)as shown previously in Figure 16


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Figure 19 - API-610 Nozzle Load Acquisition

In the corresponding output report for this API-610 analysis, both the global and API local loads arereported. This is shown below in Figure 20.

Figure 20 - API-610 Nozzle Output Report Segments

Notice in Figure 20, that each report segment indicates which values are related to the global coordinatesystem and which are related to the local API coordinate system.


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Transforming from Global to Local

Converting (or transforming) values from the CAESAR II global coordinate system to a local coordinatesystem involves applying a number of rotation matrices to the global values. Matrix mathematics is not atrivial task, and one must exercise the utmost care to arrive at the correct result. For those that want toundertake this task themselves, a small utility (discussed in the July 2001 issue of COADE's MechanicalEngineering News) can be downloaded from the COADE web site to perform this transformation. Theuse of this utility (GlbtoLocal) is illustrated here, using the nozzle at node 50 as an example.

The element 40-50 is defined with the delta coordinates of:

DX = 3 ft. 6.426 inDY = -3 ft. 6.426 inDZ = 0.0

The global restraint forces at node 50, in global coordinates, for the operating case are:

FX = 323. MX = -953.FY = 4. MY = -9.FZ = -271. MZ = -548.

Using this data as input to GlbtoLocal, the utility yields the forces on the restraint in the element'slocalcoordinate system. This is shown in Figure 21 below.

Figure 21 - Example Global to Local Transformation

The set of values labeled "Rotated Displacements / Load Vector" can be compared with the "LocalElement Force / Moment" report, as shown in Figure 18. Note however, that a change in sign isnecessary, since the restraint report shows loads acting on the restraint, while the element report showsloads acting on the element.

Frequently Asked Questions

What are global coordinates? Global coordinates define the mapping of a physical system into amathematical system. For any given model, the global coordinate system is fixed for the entire model. In


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CAESAR II, there are two alternative global coordinate systems that can be applied to a model. Bothcoordinate systems follow the "right hand rule" and use "X", "Y", and "Z" as mutually perpendicular axes.The first alternative uses the "Y" axis vertical, while the second uses the "Z" axis as vertical.

What are local coordinates? Local coordinates represent the mapping for a single element. Localcoordinate systems are used to define positive and negative directions and loads on elements. Localcoordinate systems are aligned with the elements, and therefore vary throughout the model.

What coordinates are used to plot and view the model? The model's global coordinate system isused to generate plots of the model. This is necessary since each element has its own local coordinatesystem, and these local systems can vary from element to element. Local coordinate systems are anelement property, not a system property.

How do you obtain restraint loads in local coordinates? In general, you don't - this doesn't make anysense. Restraint loads are a nodalproperty. Nodes don't have local coordinate systems, elements do.While an argument can be made that the local coordinate system of the connecting element should beused, this is only valid if one single element frames into the restraint. As soon as multiple elements frameinto the restraint, there are multiple local coordinate systems to deal with. The lone exception is when asingle element frames into a nozzle. In this instance, the restraint loads in this single element'scoordinate system can be obtained from the element's local force / moment report, with a change in sign.

How do you obtain nodal displacements in local coordinates? In general, you don't - this doesn'tmake any sense. Displacements are a nodalproperty. Nodes don't have local coordinate systems,elements do. Refer to the preceding discussion on restraint loads for additional details.

What do you do with local coordinates? In most instances nothing. The only time local coordinatesare useful in CAESAR II is when dealing with a skewed nozzle. The CAESAR II software interfacemakes the use of local coordinates unnecessary except in this one instance.

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04 - Local Coordinate System