Structural Modeling - OdonataResearchLLC/NASTRAN-95 GitHub Wiki
Introduction
NASTRAN embodies a lumped element approach, wherein the distributed physical properties of a structure are represented by a model consisting of a finite number of idealized substructures or elements that are interconnected at a finite number of grid points, to which loads are applied. All input and output data pertain to the idealized structural model. The major components in the definition and loading of a structural model are indicated in Figure 1.1-1.
As indicated in Figure 1.1-1, the grid point definition forms the basic framework for the structural model. All other parts of the structural model are referenced either directly or indirectly to the grid points.
Two general types of grid points are used in defining the structural model. They are:
- Geometric grid point - a point in three-dimensional space at which three components of translation and three components of rotation are defined. The coordinates of each grid point are specified by you.
- Scalar point - a point in vector space at which one degree of freedom is defined. Scalar points can be coupled to geometric grid points by means of scalar elements and by constraint relationships.
The structural element is a convenient means for specifying many of the properties of the structure, including material properties, mass distribution, and some types of applied loads. In static analysis by the displacement method, stiffness properties are input exclusively by means of structural elements. Mass properties (used in the generation of gravity and inertia loads) are input either as properties of structural elements or as properties of grid points. In dynamic analysis, mass, damping, and stiffness properties may be input either as the properties of structural elements or as the properties of grid points (direct input matrices).
Structural elements are defined on connection cards by referencing grid points, as indicated on Figure 1.1-1. In a few cases, all of the information required to generate the structural matrices for the element is given on the connection card. In most cases the connection card refers to a property card, on which the cross-sectional properties of the element are given. The property card in turn refers to a material card which gives the material properties. If some of the material properties are stress dependent or temperature dependent, a further reference is made to tables for this information.
Various kinds of constraints can be applied to the grid points. Single- point constraints are used to specify boundary conditions, including enforced displacements of grid points. Multipoint constraints and rigid elements are used to specify linear relationships among selected degrees of freedom. Omitted points are used as a tool in matrix partitioning and for reducing the number of degrees of freedom used in dynamic analysis. Free-body supports are used to remove stress-free motions in static analysis and to evaluate the free-body inertia properties of the structural model.
Static loads may be applied to the structural model by concentrated loads at grid points, pressure loads on surfaces, or indirectly, by means of the mass and thermal expansion properties of structural elements or enforced deformations of one-dimensional structural elements. Due to the great variety of possible sources for dynamic loading, only general forms of loads are provided for use in dynamic analysis.
The following sections describe the general procedures for defining structural models. Detailed instructions for each of the bulk data cards and case control cards are given in Section 2. Additional information on the case control cards and use of parameters is given for each rigid format in Section 3.
Figure 1.1-1. Structural model
Grid Points
Grid Point Definition
Geometric grid points are defined on GRID bulk data cards by specifying their coordinates in either the basic or a local coordinate system. The implicitly defined basic coordinate system is rectangular, except when using axisymmetric elements. Local coordinate systems may be rectangular, cylindrical, or spherical. Each local system must be related directly or indirectly to the basic coordinate system. The CORD1C, CORD1R, and CORD1S cards are used to define cylindrical, rectangular, and spherical local coordinate systems, respectively, in terms of three geometric grid points which have been previously defined. The CORD2C, CORD2R, and CORD2S cards are used to define cylindrical, rectangular, and spherical local coordinate systems, respectively, in terms of the coordinates of three points in a previously defined coordinate system.
Six rectangular displacement components (3 translations and 3 rotations) are defined at each grid point. The local coordinate system used to define the directions of motion may be different from the local coordinate system used to locate the grid point. Both the location coordinate system and the displacement coordinate system are specified on the GRID card for each geometric grid point. The orientation of displacement components depends on the type of local coordinate system used to define the displacement components. If the defining local system is rectangular, the displacement system is parallel to the local system and is independent of the grid point location as indicated in Figure 1.2-1a. If the local system is cylindrical, the displacement components are in the radial, tangential, and axial directions as indicated in Figure 1.2-1b. If the local system is spherical, the displacement components are in the radial, meridional, and azimuthal directions as indicated in Figure 1.2-1c. Each geometric grid point may have a unique displacement coordinate system associated with it. The collection of all displacement coordinate systems is known as the global coordinate system. All matrices are formed and all displacements are output in the global coordinate system. The symbols T1, T2, and T3 on the printed output indicate translations in the 1, 2, and 3-directions, respectively, for each grid point. The symbols R1, R2, and R3 indicate rotations (in radians) about the three axes.
Provision is also made on the GRID card to apply single-point constraints to any of the displacement components. Any constraints specified on the GRID card will be automatically used for all solutions. Constraints specified on the GRID card are usually restricted to those degrees of freedom that will not be elastically constrained and hence must be removed from the model in order to avoid singularities in the stiffness matrix.
The GRDSET card is provided to avoid the necessity of repeating the specification of location coordinate systems, displacement coordinate systems, and single-point constraints, when all, or many, of the GRID cards have the same entries for these items. When any of the three items are specified on the GRDSET card, the entries are used to replace blank fields on the GRID card for these items. This feature is useful in the case of such problems as space trusses where one wishes to remove all of the rotational degrees of freedom or in the case of plane structures where one wishes to remove all of the out-of-plane or all of the in-plane motions.
Scalar points are defined either on an SPOINT card or by reference on a connection card for a scalar element. SPOINT cards are used primarily to define scalar points appearing in constraint equations, but to which no structural elements are connected. A scalar point is implicitly defined if it is used as a connection point for any scalar element. Special scalar points, called "extra points", may be introduced for dynamic analyses. Extra points are used in connection with transfer functions and other forms of direct matrix input used in dynamic analyses and are defined on EPOINT cards.
GRIDB is a variation of the GRID card that is used to define a point on a fluid-structure interface (see Section 1.7).
(a) Rectangular
(b) Cylindrical
(c) Spherical
Figure 1.2-1. Displacement coordinate systems
Grid Point Sequencing
The external identification numbers used for grid points may be selected in any manner you desire. However, in order to reduce the number of active columns, and, hence, to substantially reduce computing times when using the displacement method, the internal sequencing of the grid points must not be arbitrary. The best decomposition and equation solution times are obtained if the grid points are sequenced in such a manner as to create matrices having small numbers of active columns (see Section 2.2 of the Theoretical Manual for a discussion of active columns and the decomposition algorithm). The decomposition time is proportional to the sum of the squares of the number of active columns in each row of the triangular factor. The equation solution time (forward/backward substitution) is proportional to the number of nonzero terms in the triangular factor.
Manual Grid Point Resequencing
In order to allow arbitrary grid point numbers and still preserve sparsity in the triangular decomposition factor to the greatest extent possible, provision is made for you to resequence the grid point numbers for internal operations. This feature also makes it possible to easily change the sequence if a poor initial choice is made. All output associated with grid points is identified with the external grid point numbers. The SEQGP card is used to resequence geometric grid points and scalar points. The SEQEP card is used to sequence the extra points in with the previously sequenced grid points and scalar points.
In selecting the grid point sequencing, it is not important to find the best sequence; rather it is usually quite satisfactory to find a good sequence, and to avoid bad sequences that create unreasonably large numbers of active columns. For many problems a sequence which will result in a band matrix is a reasonably good choice, but not necessarily the best. Also, sequences which result in small numbers of columns with nonzero terms are usually good but not necessarily the best. A sequence with a larger number of nonzero columns will frequently have a smaller number of nonzero operations in the decomposition when significant passive regions exist within the active columns (see Section 2.2 of the Theoretical Manual).
Examples of proper grid point sequencing for one-dimensional systems are shown in Figure 1.2-2. For open loops, a consecutive numbering system should be used as shown in Figure 1.2-2a. This sequencing will result in a narrow band matrix with no new nonzero terms created during the triangular decomposition. Generally, there is an improvement in the accumulated round off error if the grid points are sequenced from the flexible end to the stiff end.
For closed loops, the grid points may be sequenced either as shown in Figure 1.2-2b or as shown in Figure 1.2-2c. If the sequencing is as shown in Figure 1.2-2b, the semiband will be twice that of the model shown in Figure 1.2-2a. The matrix will initially contain a number of zeroes within the band which will become nonzero as the decomposition proceeds. If the sequencing is as shown in Figure 1.2-2c, the band portion of the matrix will be the same as that for Figure 1.2-2a. However, the connection between grid points 1 and 8 will create a number of active columns on the right hand side of the matrix. The solution times will be the same for the sequence shown in Figure 1.2-2b or 1.2-2c, because the number of active columns in each sequence is the same.
Examples of grid point sequencing for surfaces are shown in Figure 1.2-3. For plain or curved surfaces with a pattern of grid points that tends to be rectangular, the sequencing shown in Figure 1.2-3a will result in a band matrix having good solution times. The semiband will be proportional to the number of grid points along the short direction of the pattern. If the pattern of grid points shown in Figure 1.2-3a is made into a closed surface by connecting grid points 1 and 17, 2 and 18, etc., a number of active columns equal to the semiband will be created. If the number of grid points in the circumferential direction is greater than twice the number in the axial direction, the sequencing indicated in Figure 1.2-3a is a good one. However, if the number of grid points in the circumferential direction is less than twice the number in the axial direction, the use of consecutive numbering in the circumferential direction is more efficient. An alternate sequencing for a closed loop is shown in Figure 1.2-3b, where the semiband is proportional to twice the number of grid points in a row. For cylindrical or similar closed surfaces, the sequencing shown in Figure 1.2-3b has no advantage over that shown in Figure 1.2-3a, as the total number of active columns will be the same in either case.
With the exception of the central point, sequencing considerations for the radial pattern shown in Figure 1.2-3c are similar to those for the rectangular patterns shown in Figures 3a and 3b. The central point must be sequenced last in order to limit the number of active columns associated with this point to the number of degrees of freedom at the central point. If the central point is sequenced first, the number of active columns associated with the central point will be proportional to the number of radial lines. If there are more grid points on a radial line than on a circumferential line, the consecutive numbering should extend in the circumferential direction beginning with the outermost circumferential ring. In this case, the semiband is proportional to the number of grid points on a circumferential line and there will be no active columns on the right hand side of the matrix. If the grid points form a full circular pattern, the closure will create a number of active columns proportional to the number of grid points on a radial line if the grid points are numbered as shown in Figure 1.2-3c. Proper sequencing for a full circular pattern is similar to that discussed for the rectangular arrays shown in Figures 3a and 3b for closed surfaces.
Sequencing problems for actual structural models can frequently be handled by considering the model as consisting of several substructures. Each substructure is first numbered in the most efficient manner. The substructures are then connected so as to create the minimum number of active columns. The grid points at the interface between two substructures are usually given numbers near the end of the sequence for the first substructure and as near the beginning of the sequence for the second substructure as is convenient.
Figure 1.2-4 shows a good sequence for the substructure approach. Grid points 1 through 9 are associated with the first substructure, and grid points 10 through 30 are associated with the second substructure. In the example, each of the substructures was sequenced for band matrices. However, other schemes could also be considered for sequencing the individual substructures. Figure 1.2-5 shows the nonzero terms in the triangular factor. The X's indicate terms which are nonzero in the original matrix. The zeros indicate nonzero terms created during the decomposition. The maximum number of active columns for any pivotal row is only five, and this occurs in only three rows near the middle of the matrix for the second substructure. All other pivotal rows have four or less active columns.
Figure 1.2-6 indicates the grid point sequencing using substructuring techniques for a square model, and Figure 1.2-7 shows the nonzero terms in the triangular factor. If the square model were sequenced for a band matrix, the number of nonzero terms in the triangular factor would be 129, whereas Figure 1.2-7 contains only 102 nonzero terms. The time for the forward/backward substitution operation is directly proportional to the number of nonzero terms in the triangular factor. Consequently, the time for the forward/backward substitution operation when the square array is ordered as shown in Figure 1.2-7 is only about 80% of that when the array is ordered for a band matrix. The number of multiplications for a decomposition when ordered for a band is 294, whereas the number indicated in Figure 1.2-7 is only 177. This indicates that the time for the decomposition when ordered as shown in Figure 1.2-6 is only 60% of that when ordered for a band.
Although scalar points are defined only in vector space, the pattern of the connections is used in a manner similar to that of geometric grid points for sequencing scalar points among themselves or with geometric grid points. Since scalar points introduced for dynamic analysis (extra points) are defined in connection with direct input matrices, the sequencing of these points is determined by direct reference to the positions of the added terms in the dynamic matrices.
(a) Consecutive numbering system for open loops.
(b) Sequencing of grid points for a closed loop (method 1).
(c) Sequencing of grid points for a closed loop (method 2).
Figure 1.2-2. Grid point sequencing for one-dimensional systems
(a) Grid-point sequencing for a rectangular surface (method 1).
(b) Grid-point sequencing for a rectangular surface (method 2).
(c) Grid-point sequencing for a radial pattern.
Figure 1.2-3. Grid point sequencing for surfaces
Figure 1.2-4. Grid point sequencing for substructures
X X X X X 0 X X 0 0 X X X 0 X X X 0 X X 0 0 X X X 0 X X X 0 X X 0 0 X X X X X X 0 X X 0 0 X X X 0 X X X 0 X X 0 0 X X X 0 X 0 X X 0 X 0 X 0 0 X 0 X X 0 X X X 0 X (Symmetric) X 0 0 X X X 0 X X X 0 X X 0 0 X X X 0 X X X 0 X X 0 0 X X X 0 X X X
Figure 1.2-5. Matrix for substructure example
Figure 1.2-6. Grid point sequencing for square model
X X X X 0 X X X X X 0 X 0 X 0 X X X X X 0 X X X X 0 X X 0 X 0 X X X X X 0 X X X X X 0 X 0 X 0 X X X X X 0 X X X X X 0 X 0 X 0 X (Symmetric) X X 0 0 X 0 0 0 0 X X X 0 0 0 0 0 X 0 0 0 0 X X X 0 0 0 X 0 0 X X X 0 X X X
Figure 1.2-7. Matrix for square model example
Automatic Grid Point Resequencing Using the BANDIT Procedure
If you want reduced matrix reduction and equation solution times, you can manually resequence your grid points by the use of SEQGP cards as per the guidelines outlined in the previous section. However, in order to relieve you of the burden of having to do so, an automatic resequencing capability has been provided in NASTRAN. This capability involves the use of the BANDIT procedure in NASTRAN. (See Reference 1 for details of the BANDIT procedure and Reference 2 for details of the manner in which it has been implemented in NASTRAN.)
The BANDIT procedure in automatically invoked in NASTRAN for all runs (except those indicated in Sections 1.2.2.2.2 and 1.2.2.2.3), unless specifically suppressed by you. (See the description of the BANDIT options in the next section.) The result of the BANDIT operations is a set of SEQGP cards that are automatically generated by the program. These SEQGP cards are added to your input data (replacing any SEQGP cards already input, if so specified) for subsequent processing by the program.
BANDIT Options
The execution of the BANDIT operations in NASTRAN is controlled by several parameters. These parameters can be specified by means of the NASTRAN card and are fully described in Section 2.1. All of these parameters have default values selected so that you normally do not have to explicitly specify any of them.
NASTRAN provides two methods to skip over the BANDIT operations. First, the NASTRAN BANDIT = -1 option can be used. The second method is to include one or more SEQGP cards in the Bulk Data Deck. In this second method, BANDIT would terminate since you have already stated your choice of SEQGP resequencing cards. However, the NASTRAN BANDTRUN = 1 option can be used to force BANDIT to generate new SEQGP cards to replace the old SEQGP set already in the input Bulk Data Deck. In all instances when BANDIT is executed, NASTRAN will issue a page of summary to keep you informed of the basic resequencing computations. You may refer to Reference 1 for the definition of the technical terms used.
The BANDIT procedure automatically counts the number of grid points used in a NASTRAN job and sets up the exact array dimensions needed for its internal computations. However, if your structural model uses more grid points in the connecting elements than the total number of grid points as defined on the GRID cards, BANDIT will issue a fatal message and terminate the job. In the case where non-active grid points (that is, grid points defined on the GRID cards but nowhere used in the model) do exist, BANDIT will add them to the end of the SEQGP cards, and their presence will not cause termination of a job. (If necessary, the NASTRAN HICORE parameter can be used on the UNIVAC version to increase the amount of open core available for the BANDIT operations.)
Multipoint constraints (MPCs) and rigid elements are included in the BANDIT computations only when the BANDTMPC = 1 (or 2) option is selected. (The use of the dependent grid points of MPCs and/or rigid elements is controlled by the BANDTDEP option.) However, as noted in Reference 1, it should be emphasized here that only in rare cases would it make sense to let BANDIT process MPCs and rigid elements. The main reasons for this are that BANDIT does not consider individual degrees of freedom and, in addition, cannot distinguish one MPC set from another.
Cases for Which BANDIT Computations are Skipped
The BANDIT computations in NASTRAN are unconditionally skipped over if any of the following conditions exists:
- There are errors in input data.
- The Bulk Data Deck contains any of the following types of input:
- Axisymmetric (CONEAX, TRAPAX, or TRIAAX) elements
- Fluid (FLUID2, FLUID3, or FLUID4) elements
- DMI (Direct Matrix Input) data
- It is a substructure Phase 2 run.
BANDIT in Restarts
At the beginning of a NASTRAN job, the Preface (or Link 1) modules read and process the Executive, Case Control, and Bulk Data decks. The SEQGP cards generated by BANDIT are added directly to the NASTRAN data base (specifically, the GEOM1 file) at a later stage. Since these SEQGP cards are not part of the original Bulk Data Deck, they are not directly written on to the NPTP (New Problem Tape) in a checkpoint run and, therefore, are not available as such for use on the OPTP (Old Problem Tape) in a restart.
In the light of the above comments, the following points about the use of BANDIT in NASTRAN restarts should be noted:
- BANDIT is automatically skipped if the restart job has no input data changes with respect to the checkpoint job. However, the previously generated SEQGP cards, if any, are already absorbed into the NASTRAN data base (data blocks such as EQEXIN, SIL, etc.). A message is printed to inform you that the BANDIT computations are not performed. (BANDIT can be executed if the restart job contains one or more of the appropriate BANDIT options on the NASTRAN card, for example, NASTRAN BANDMTH = 2.)
- BANDIT is executed (except for the cases indicated in Section 1.2.2.2.2) if the restart job has input data changes with respect to the checkpoint job, unless specifically suppressed by you. (The BANDIT = -1 option on the NASTRAN card can be used to stop BANDIT execution unconditionally.)
Grid Point Properties
Some of the characteristics of the structural model are introduced as properties of grid points, rather than as properties of structural elements. Any of the various forms of direct matrix input are considered as describing the structural model in terms of properties of grid points.
Thermal fields are defined by specifying the temperatures at grid points. The TEMP card is used to specify the temperature at grid points for use in connection with thermal loading and temperature-dependent material properties. The TEMPD card is used to specify a default temperature, in order to avoid a large number of duplicate entries on a TEMP card when the temperature is uniform over a large portion of the structure. The TEMPAX card is used for conical shell problems.
Mass properties may be input as properties of grid points by using the concentrated mass element (see Section 5.5 of the Theoretical Manual). The CONM1 card is used to define a 6x6 matrix of mass coefficients at a geometric grid point in any selected coordinate system. The CONM2 card is used to define a concentrated mass at a geometric grid point in terms of its mass, the three coordinates of its center of gravity, the three moments of inertia about its center of gravity, and its three products of inertia, referred to any selected coordinate system.
In dynamic analysis, mass, damping and stiffness properties may be provided, in part or entirely, as properties of grid points through the use of direct input matrices. The DMIG card is used to define direct input matrices for use in dynamic analysis. These matrices may be associated with components of geometric grid points, scalar points, or extra points introduced for dynamic analysis. The TF card is used to define transfer functions that are internally converted to direct matrix input. The DMIAX card is an alternate form of direct matrix input that is used for hydroelastic problems (see Section 1.7).
References
- Everstine, G. C., "BANDIT User's Guide", COSMIC Program No. DOD-00033, May 1978.
- Chan, G. C., "BANDIT in NASTRAN," Eleventh NASTRAN Users' Colloquium, NASA Conference Publication, May 1983, San Francisco, California, pp. 1-5.
1.3 STRUCTURAL ELEMENTS
1.3.1 Element Definition
Structural elements are defined on connection cards that identify the grid points to which the elements are connected. The mnemonics for all such cards have a prefix of the letter "C", followed by an indication of the type of element, such as CBAR and CROD. The order of the grid point identification defines the positive direction of the axis of a one-dimensional element and the positive surface of a plate element. The connection cards include additional orientation information when required. Except for the simplest elements, each connection card references a property definition card. If many elements have the same properties, this system of referencing eliminates a large number of duplicate entries.
The property definition cards define geometric properties such as thicknesses, cross-sectional areas, and moments of inertia. The mnemonics for all such cards have a prefix of the letter "P", followed by some, or all, of the characters used on the associated connection card, such as PBAR and PROD. Other included items are the nonstructural mass and the location of points where stresses will be calculated. Except for the simplest elements, each property definition card will reference a material property card.
In some cases, the same finite element can be defined by using different bulk data cards. These alternate cards have been provided for your convenience. In the case of a rod element, the normal definition is accomplished with a connection card (CROD) which references a property card (PROD). However, an alternate definition uses a CONROD card which combines connection and property information on a single card. This is more convenient if a large number of rod elements all have different properties.
In the case of plate elements, a different property card is provided for each type of element, such as membrane or sandwich plates. Thus, each property card contains only the information required for a single type of plate element, and in most cases, a single card has sufficient space for all of the property information. In order to maintain uniformity in the relationship between connection cards and property cards, a number of connection card types contain the same information, such as the connection cards for the various types of triangular elements. Also, the property cards for triangular and quadrilateral elements of the same type contain the same information.
The material property definition cards are used to define the properties for each of the materials used in the structural model. The MAT1 card is used to define the properties for isotropic materials. The MAT1 card may be referenced by any of the structural elements. The MATS1 card specifies table references for isotropic material properties that are stress dependent. The TABLES1 card defines a tabular stress-strain function for use in piecewise linear analysis. The MATT1 card specifies table references for isotropic material properties that are temperature dependent. The TABLEM1, TABLEM2, TABLEM3, and TABLEM4 cards define four different types of tabular functions for use in generating temperature-dependent material properties.
The MAT2 card is used to define the properties for anisotropic materials. The MAT2 card may only be referenced by triangular or quadrilateral membrane and bending elements. The MAT2 card specifies the relationship between the inplane stresses and strains. The material is assumed to be infinitely rigid in transverse shear. The angle between the material coordinate system and the element coordinate system is specified on the connection cards. The MATT2 card specifies table references for anisotropic material properties that are temperature dependent. This card may reference any of the TABLEM1, TABLEM2, TABLEM3, or TABLEM4 cards.
The MAT3 card is used to define the properties for orthotropic materials used in the modeling of axisymmetric shells. This card may only be referenced by CTRIARG, CTRIAAX, CTRAPRG, CTRAPAX, and PTORDRG cards. The MATT3 card specifies table references for use in generating temperature-dependent properties for this type of material.
The GENEL card is used to define general elements whose properties are defined in terms of deflection influence coefficients or stiffness matrices, and which can be connected between any number of grid points. One of the important uses of the general element is the representation of part of a structure by means of experimentally measured data. No output data is prepared for the general element. Detail information on the general element is given in Section 5.7 of the Theoretical Manual.
Dummy elements are provided in order to allow you to investigate new structural elements with a minimum expenditure of time and money. A dummy element is defined with a CDUMi (i = index of element type, 1 <= i <= 9) card and its properties are defined with the PDUMi card. The ADUMi card is used to define the items on the connection and property cards. Detailed instructions for coding dummy element routines are given in Section 6.8.5 of the Programmer's Manual.
1.3.2 Beam Elements
1.3.2.1 Simple Beam or Bar Element
The simple beam or bar element is defined with a CBAR card and its properties (constant over the length) are defined with a PBAR card. The bar element includes extension, torsion, bending in two perpendicular planes, and the associated shears. The shear center is assumed to coincide with the elastic axis. Any five of the six forces at either end of the element may be set equal to zero by using the pin flags on the CBAR card. The integers 1 to 6 represent the axial force, shearing force in Plane 1, shearing force in Plane 2, axial torque, moment in Plane 2, and moment in Plane 1, respectively. The structural and nonstructural mass of the bar are lumped at the ends of the element, unless coupled mass is requested with a PARAM COUPMASS card (see PARAM bulk data card). Theoretical aspects of the bar element are treated in Section 5.2 of the Theoretical Manual.
The element coordinate system is shown in Figure 1.3-1a. End a is offset from grid point a an amount measured by vector wa and end b is offset from grid point b an amount measured by vector wb. The vectors wa and wb are measured in the global coordinates of the connected grid point. The x-axis of the element coordinate system is defined by a line connecting end a to end b of the bar element. The orientation of the bar element is described in terms of two reference planes. The reference planes are defined with the aid of vector v. This vector may be defined directly with three components in the global system at end a of the bar or by a line drawn from end a to a third referenced grid point. The first reference plane (Plane 1) is defined by the x-axis and the vector v. The second reference plane (Plane 2) is defined by the vector cross product (x x v) and the x-axis. The subscripts 1 and 2 refer to forces and geometric properties associated with bending in planes 1 and 2, respectively. The reference planes are not necessarily principal planes. The coincidence of the reference planes and the principal planes is indicated by a zero product of inertia (I12) on the PBAR card. If shearing deformations are included, the reference axes and the principal axes must coincide. When pin flags and offsets are used, the effect of the pin is to free the force at the end of the element x-axis of the beam, not at the grid point. The positive directions for element forces are shown in Figure 1.3-1b. The following element forces, either real or complex (depending on the rigid format), are output on request:
- Bending moments at both ends in the two reference planes.
- Shears in the two reference planes.
- Average axial force.
- Torque about the bar axis.
The following real element stresses are output on request:
- Average axial stress.
- Extensional stress due to bending at four points on the cross-section at both ends. (Optional; calculated only if you enter stress recovery points on PBAR card.)
- Maximum and minimum extensional stresses at both ends.
- Margins of safety in tension and compression for the whole element. (Optional; calculated only if you enter stress limits on MAT1 card.)
Tensile stresses are given a positive sign and compressive stresses a negative sign. Only the average axial stress and the extensional stresses due to bending are available as complex stresses. The stress recovery coefficients on the PBAR card are used to locate points on the cross-section for stress recovery. The subscript 1 is associated with the distance of a stress recovery point from plane 2. The subscript 2 is associated with the distance from plane 1.
The use of the BAROR card avoids unnecessary repetition of input when a large number of bar elements either have the same property identification number or have their reference axes oriented in the same manner. This card is used to define default values on the CBAR card for the property identification number and the orientation vector for the reference axes. The default values are used only when the corresponding fields on the CBAR card are blank.
1.3.2.2 Curved Beam or Elbow Element
The curved beam or elbow element is a three-dimensional element with extension, torsion, and bending capabilities and the associated shears. No offset of the elastic axis is allowed nor are pin releases permitted to eliminate the connection between motions at the ends of the element and the adjacent grid points.
The elbow element was initially developed to facilitate the analysis of pipe networks by using it as a curved pipe element. However, the input format is general enough to allow application to beams of general cross section. An important assumption in the development of the element is that the radius of curvature is much larger than the cross section depth.
The element is defined with a CELBOW card and its properties (constant over the length) are defined with a PELBOW card. There are six degrees of freedom at each end of the element: translations in the local x, y, z directions and rotations about the local x, y, z axes. The structural and nonstructural mass of the element are lumped at the ends of the element.
The specified properties of the elbow element are its area; its moments of inertia, I1 and I2 (the product of inertia is assumed to be zero); its torsional constant, J; the radius of curvature; the angle between end-a and end-b; the factors K1 and K2 for computing transverse shear stiffness; the nonstructural mass per unit length, NSM; the stress intensification factor, C; and the flexibility correction factors, Kx, Ky, and Kz. The stress intensification factor C is applied to the bending stress only. The flexibility correction factors Kx, Ky, and Kz are generally greater than 1.0 and are used as divisors to reduce the respective moments of inertia. These are discussed further towards the end of this section.
The material properties, obtained by reference to a materials properties table, include the elastic moduli, E and G, density, rho, and the thermal expansion coefficient, \( \alpha \), determined at the average temperature of the element.
The plane of the element is defined by two grid points, A and B, and a vector v from grid point A directed toward the center of curvature. Plane 1 of the element cross section lies in this plane. Plane 2 is normal to Plane 1 and contains the vector v. The area moments of inertia, I1 and I2, are defined as for the BAR element. The cross product of inertia, I12, is neglected. This assumption requires that at least one axis of the element cross section be an axis of symmetry.
The following element forces are output on request:
- Bending moments at both ends in the two reference planes
- Transverse shear force at both ends in the two reference planes
- Axial force at both ends
- Torque at both ends
The following element stresses are output on request:
- Average axial stress at both ends
- Bending stresses at four points on the cross section at both ends. The points are specified by you.
- Maximum and minimum extensional stresses at both ends.
- Margins of safety in tension and compression (Optional, output only if you enter stress limits on MAT1 card)
Stress Intensification Factor and Flexibility Correction Factors
When a plane pipe network, consisting of both straight and curved sections, is analyzed by simple beam theory as an indeterminate system, the computed support reactions are greater than actually would be measured in an experiment. The apparent decrease in stiffness in such a case is due to an ovalization of the pipe in the curved sections. The ovalization also yields a stress distribution different from that computed by simple beam theory.
When a curved beam or elbow element is used as a curved pipe element, there are two factors available that can be specified to account for the differences in its behavior compared to curved beams. These are the stress intensification factor and the flexibility correction factors.
The maximum stress, \( \sigma_{max} \), in a curved pipe element is given by
\begin{equation} \sigma_{max} = C \frac{Mc}{I}\end{equation}
where C is a stress intensification factor,
M = bending moment,
c = fiber distance, and
I = plane (area) moment of inertia of the cross section.
In general, the factor C mentioned above may be regarded as a stress correction factor in curved beam analysis.
The effect of the ovalization of the pipe in curved sections is to reduce the stiffness parameter EI (E: modulus of elasticity) of the curved pipe to a fictitious value. Thus, for the elbow element,
\begin{equation} (EI1)^{'} = \frac{EI1}{K_y} \mbox{ , } (1.0 < K_y) \mbox{, and} \end{equation}`
\begin{equation} (EI2)^{'} = \frac{EI2}{K_z} \mbox{ , } (1.0 < K_z) \end{equation}
where Ky and Kz are the stiffness correction factors corresponding to planes 1 and 2, respectively. The stiffness correction factor, Kz, corresponds to the torsional behavior and is generally taken to be 1.0.