Recovered_PDF_21.pdf

Generation of

Finite Elements

and Beam Structures

Version 10.20

This manual is protected by copyright laws. No part of it may be translated, copied or reproduced, in any form or by any means, without written permission from SOFiSTiK AG. SOFiSTiK reserves the right to modify or to release new editions of this manual. The manual and the program have been thoroughly checked for errors. However, SOFiSTiK does not claim that either one is completely error free. Errors and omissions are corrected as soon as they are detected.

The user of the program is solely responsible for the applications. We strongly encourage the user to test the correctness of all calculations at least by random sampling.

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1 Task Description . . . 1−1 2 Theoretical Principles . . . 2−1 2.1. Systems of Coordinates . . . 2−1 2.2. Overview of the Element Types . . . 2−2 2.3. Mesh Partitioning . . . 2−5 2.4. Plane Elements . . . 2−7 2.5. Solid Elements . . . 2−8 2.6. Boundary Conditions . . . 2−8 2.7. Girders . . . 2−12 2.8. Literature. . . 2−13 2.9. Limitations . . . 2−13 3 Input Description . . . 3−1 3.1. Nodes . . . 3−1 3.2. Elements . . . 3−1 3.3. Results . . . 3−2 3.4. Restart . . . 3−2 3.5. Input Records . . . 3−3 3.6. ECHO − Control of the Output . . . 3−7 3.7. SYST − Global System Parameters . . . 3−8 3.8. NODE − Nodal Coordinates and Constraints . . . 3−12 3.9. INTE − Intermediate Nodes . . . 3−18 3.10. KINE − Kinematic Dependencies . . . 3−22 3.11. MESH − Generation of Nodes and Quadrilateral Elements . . . 3−23 3.12. IMES − Generation of Irregular Nodes, Quadrilateral Elements . 3−27

3.13. CUBE − Nodes and Cubic Elements . . . 3−29 3.14. TRAN − Transformation of Nodes . . . 3−31 3.15. MIRR − Mirroring of Nodes . . . 3−33 3.16. ALIN − Node upon a Line (Projection to the Line) . . . 3−36 3.17. SECT − Node at Intersection of two Straight Lines . . . 3−39 3.18. Materials . . . 3−41 3.19. NORM − Default Design Code . . . 3−43 3.20. MAT − General Material Properties . . . 3−44 3.21. MATE − Material Properties . . . 3−45 3.22. MLAY − Layered Material . . . 3−48 3.23. BMAT − Elastic Support / Interface . . . 3−49 3.24. NMAT − Nonlinear Material . . . 3−52 3.25. MEXT − Extra Materialconstants . . . 3−63

3.26. CONC − Properties of Concrete . . . 3−64 3.27. STEE − Properties of Metals . . . 3−71 3.28. TIMB − Properties of Timber . . . 3−80 3.29. MASO − Masonry / Brickwork . . . 3−82 3.30. SSLA − Stress−Strain Curves . . . 3−84 3.31. SVAL − Cross−section values . . . 3−86 3.32. SREC − Rectangle, T−beam, Plate . . . 3−91 3.33. SCIR − Circular and Annular Sections . . . 3−94 3.34. BORE − Bore Profile of a Sondation . . . 3−95 3.35. BLAY − Layer of the Soil Strata . . . 3−96 3.36. BBAX − Input of Axial Subgrade Parameters . . . 3−97 3.37. BBLA − Input of Lateral Subgrade Parameters . . . 3−98 3.38. HING − Hinged Connection Combinations for Beams . . . 3−100 3.39. GRP − Group Control . . . 3−101 3.40. TRUS − Truss−bar Elements . . . 3−105 3.41. CABL − Cable Elements . . . 3−106 3.42. BEAM − Beam Elements . . . 3−108 3.43. ADEF − Beginning of Beam Segment Definition . . . 3−116 3.44. BDIV − Input of Beam Segments . . . 3−117 3.45. BSEC − Beam Sections . . . 3−119 3.46. SUPP − Definition of Support Sections . . . 3−120 3.47. QUAD − Plane Elements (Disks / Plates / Shells) . . . 3−122 3.48. BRIC − Three−dimensional Solid Elements . . . 3−126 3.49. SPRI − Spring Elements . . . 3−127 3.50. BOUN − Distributed Elastic Support . . . 3−134 3.51. FLEX − General Elastic Element . . . 3−139 3.52. DAMP − Damping Elements . . . 3−141 3.53. MASS − Concentrated Masses . . . 3−142 4 Output Description . . . 4−1 4.1. Nodal Values . . . 4−1 4.2. Material Values . . . 4−2 4.3. System Statistics . . . 4−4 4.4. Cross−sectional Overview . . . 4−5 4.5. Group Qualities . . . 4−5 4.6. Plane Elements (2−D, QUAD) . . . 4−6 4.7. Three−dimensional Solid Elements (3−D, BRIC) . . . 4−6 4.8. Boundary Elements . . . 4−7 4.9. Geometric Definitions (Bedding Profiles) . . . 4−8

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4.10. Bending Beams and Piles. . . 4−8 4.11. Truss−bar Elements . . . 4−9 4.12. Cable Elements . . . 4−9 4.13. Springs . . . 4−10 5 Examples . . . 5−1 5.1. Angle Plate . . . 5−1 5.2. Pointwise Supported Ceiling Plate . . . 5−3 5.3. Gridwork . . . 5−7 5.4. Plane Frame, Restrained in Space . . . 5−9 5.5. Shell Structure . . . 5−12 5.6. Reinforced Concrete Box . . . 5−14 5.7. Calotte Shell . . . 5−18 5.8. Examples in the Internet . . . 5−24

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1

Task Description

Any structure like e.g. a plane structure must in general be interpreted as a geometrically infinitely indeterminate structure. The Finite Element method consists in converting this infinite system into a finite one, in other words discretizing it.

A discrete solution consisting of n unknowns is computed in place of the con
tinuous solution. In case of static analysis these unknowns are for instance the displacements of particular points, the so−called nodes. These nodes are connected to each other by means of mechanically simplified members, the so−called elements. One can obtain the displacements of the entire region through interpolation of the nodal values inside the elements. The continuous plane structure is thus represented by a large − yet finite − number of el
ements.

The power of Finite Elements lies in their universal applicability to any geo
metrical shape and almost any loading. This is achieved by the following for
mulation principle. Individual elements, which describe parts of the struc
ture in a computer oriented manner, are assembled into a complete structure. Regular frame structures must be understood as a special case of this prin
ciple, in which a finite number of nodes leads to an exact solution.

The task of GENF is to carry out the first step of a FE−analysis, the mesh partitioning. The input data is supplied by means of a text file using the powerful generator language CADINP as well as additional geometrical func
tions. This input method presents certain advantages compared to graphical input by MONET or SOFiPLUS when it concerns the construction of vari
ations with parametric input or complicated special cases. Graphical and text input do not constitute either/or" methods, instead they complement one another.

The computation of the mechanical behaviour is generally based on an energy principle (minimisation of the deformation work). The result is a so−called stiffness matrix. This matrix specifies the reaction forces at the nodes of an element when these nodes are subjected to known displacements.

The global force equilibrium is then stated for each node in order to compute the unknowns. To each displacement corresponds a force in the same direc
tion, which is a function of this as well as other displacements. This leads to

a system of equations with n unknowns, where n can become very large. The local character, however, of the elementwise interpolation results in numeri
cally beneficial banded matrices.

The complete method is divided into five main parts:

1. Decomposition of the structure into individual parts (elements) 2. Computation of the element stiffness matrices.

3. Assembly of the global stiffness matrix and solution of the resulting system of equations.

4. Application of loads and solution for the displacements.

5. Computation of the element stresses and reaction forces based on the computed displacements.

Exactly one database exists for each system, and each module has unlimited access to its accumulated data. By system is understood the entirety of the parts forming a structure or a substructure, and co− operating statically dur
ing its lifespan. Sometimes a partial system can be analysed separately dur
ing the design.

Boundary conditions or material parameters as well as cross sections can be modified during a Restart. Elements and nodal coordinates though remain unchanged.

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2

Theoretical Principles

2.1.

Systems of Coordinates

The systems of coordinates and the notation conform to DIN 1080.

The nodal coordinates, displacements and rotations as well as loads and reac
tion forces are described in a global Cartesian right−handed system X−Y−Z. The input can be also given in polar, cylindrical or spherical coordinates which, however, are transformed automatically by the program to Cartesian ones. Local coordinate systems, which are described in the next section, exist for the elements as well.

The displacements and rotations are vectors with three components along the coordinate directions. These components are positive when they act in the positive coordinate axis direction. Rotational components are positive if they rotate clockwise about the given axis when observing along the positive direc
tion of the axis. The same holds for forces and moments.

System of coordinates

It is advised to define the global system of coordinates such that the Z−axis coincides with the direction of gravity. In case of plane structures only the X−Y plane is considered. In such case the Z−axis points towards the back or downwards. The resulting coordinate system has, in general, the X−axis pointing to the right and the Y−axis downward. In axisymmetric analysis the X−axis is the axis of rotation.

2.2.

Overview of the Element Types

2.2.1.

Truss and Cable Elements

The truss or cable element can only carry a constant axial force. In case of non
linear analysis, the cable element can not sustain any compression. The x− axis of the element direction is the only local coordinate axis.

2.2.2.

Beam and Pile Elements

These elements are defined through two nodes and their straight connection, which is also the centrobaric axis and the x−axis of the local coordinate sys
tem. The element in between can be prismatic or arbitrarily haunched via cross section jumps.

In case of plane structures the direction of the y−axis is defined such that bending occurs only about that axis. In case of three−dimensional structures, however, the orientation of the coordinate system must be specified explicitly.

Local coordinate system of a beam

The cross section is defined by the program AQUA in any parallely shifted y’− z’ system of coordinates. Internal forces and moments of the beams are posi
tive when they act in the positive direction upon a positive cross section.

2.2.3.

Plane Elements

The plane element of SOFiSTiK is a general quadrilateral element with four nodes (QUAD), which can degenerate to a triangle. As a rule, a significantly

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improved accuracy is achieved through non−conforming formulations, so that the introduction of the problematic six− to nine− noded elements is not necessary.

In the plane case the QUAD−element does not possess a coordinate system of its own, and the results are always output in the global coordinate system. Notice that in the notation used for moments and shear forces, their indices describe only their position and not their direction. Thus, to a plate moment m−xx corresponds a global moment MY.

Stress resultants of plates and shells

In the case of spatial structures there is a local coordinate system for internal forces and loads which is defined as follows:

The local z−axis is perpendicular to the midplane of the element and it is de
fined by the outer vector product of the node diagonals (3−1) x (2− 4). If one numbers the element nodes counterclockwise, then one is looking in the posi
tive z−direction from "above". Positive moments cause tension to the opposite bottom side of the plate.

The local x− and y−axes both lie within the element’s plane. The sign of the x− and y−axes is only useful for the results of shear forces.

The local x−axis can be oriented, upon request, inside the surface of an el
ement with a slight deviation with respect to the positive or negative direc
tion of any of the three global axes of coordinates.

If no such request is made, the local x−axis will lie in the element’s plane par
allel to the global X−Y plane, such that the angle of the projection on the X−Y plane and the global X−axis will not be larger than 90 degrees.

Local system of coordinates for plane elements

2.2.4.

Solid Elements

The solid element of SOFiSTiK is a general six−sided element with eight nodes (BRIC), which can degenerate, if necessary, to a tetrahedron. As a rule, a significantly improved accuracy is achieved through non− conforming for
mulations, so that the introduction of the 21−noded elements is not necessary. The element does not possess a local coordinate system of its own, and stresses are always output by their global components. The surfaces can be described through special QUAD−elements, which can be also employed for the display of stresses in the BRIC−elements.

2.2.5.

Spring Elements

Elastic elements with general properties are available in several variants: − Anisotropic spring element with nonlinear effects between two

nodes or as support condition (SPRI)

− Generalised stiffness with up to six nodes (FLEX)

− Elastic foundation along a line with a boundary element (BOUN) − Plane foundation for quadrilateral element (QUAD) in the local

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2.3.

Mesh Partitioning

The partitioning of a mesh is specified based on two requirements. On one hand the mesh should be as fine as possible, so as to obtain the most accurate results. The factors opposing that are:

− The computing times increase as n2, when the number of elements n is increased.

− In case of very fine partitioning, roundoff errors are amplified so much that the solution becomes unusable. As a rule of thumb, a logical partitioning of a free span consists of 5 up to 20 elements. − It is not logical in construction practice to attempt to model and

proportion all types of singularities. One should strive for a parti− tioning that is not too fine.

2.3.1.

Loads

The Finite−Element system is a discrete system and it can thus handle dis
crete loads only. Every loading must therefore be converted to so− called nodal loads. A nodal load should not be confused with a point load. The difference is illustrated in the following figure

Nodal loads

A mesh refinement leads to new nodal loads in case of a uniform load (a and d), but not in case of point loads.

On one hand, this means that a given mesh has a limited resolution for load
ings. The coarser mesh (a, b, c) can not make the differentiation between two point loads and a uniform load upon the element grid or a point load at the middle of the element.

It also means on the other hand, that a loading can be applied as a point load on a node only when its load induction area is smaller than the size of the ad
jacent elements. When inducing , for example, a point load upon a plate, each new mesh refinement in the area of the load will compute larger shear forces each time, due to the better modelling of the singularity. Therefore, one should either select an element size that will not be smaller than the plate thickness, or define the loads in the form of distributed loading with their actual contact surfaces.

2.3.2.

Beam Elements

An exact description of the geometry is possible to a very large degree in the case of beam elements. A single beam element may be used from one support to the other. A typical FE partitioning of the geometry is necessary, however, in the following cases:

− Coupling with elastic foundation (Boundary element) − Dynamic computations (nodal masses)

− Broken centrobaric axis (e.g. haunches)

− Large deformations according to 3rd order theory

The partitioning may become so fine that the length of the individual beams will approximately be the same as their cross section dimensions. When their length becomes smaller than that, it is required that the correct shear de
formation areas of the cross sections be input. Artificially large stiffnesses must be avoided too, as well as the direct combination of elements with very different lengths.

Results for this element can be obtained at all of its sections. Superposition
ing and proportioning can take place at these sections only.

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Conversion of triangular mesh to quadrilateral

This way for instance even a circular plate can be partitioned into quadrilat
eral elements easily:

Partitioning of a quarter−plate

Results for this element are obtained at the following points: − At the centre of the element

− At the so−called Gauss−points inside the element − As extrapolated average values of the nodes

The values at the element’s centre must be used for the proportioning of the element. The so computed value of the required reinforcement must then be applied to the entire area of the element. Through proper selection of the el
ement size and location, one can carry out direct calculations conforming to the diverse dimensioning rules. It is meaningful e.g. in case of wide supports or restraints, to place the centre of the element on top of the edge of the sup
port.

The Gauss−points are necessary only for an optimally accurate capture of the elements stress state and they are not usually employed by the user.

The values at the element nodes can be extrapolated from the Gauss− points. Due to the approximate formulation of the FEM−solution these values are not identical at a node, therefore the average value is computed. These values are of prime importance for graphical representations. In case of coarse element partitioning, however, as well as in case of fixed edges or point supports the nodal values should be also taken into consideration in proportioning, be
cause otherwise the maximum values are not captured.

Special care should be taken for three−dimensional structures or load ap
plication regions in order to avoid the averaging of all the stress resultants at the nodes. In case of sudden changes in the element thickness as well, pro
portioning should take place, as a rule, separately on each side.

The nodal values can be also used in calculating an error indicator for the assessment of the accuracy of the solution, through the integration of the deviations between extrapolated and average values for each element.

2.5.

Solid Elements

Whatever was said for the QUAD−elements essentially holds for the solid el
ements as well.

2.6.

Boundary Conditions

The boundary conditions at the nodes are specified in the simplest case by suppressing the corresponding degrees of freedom. An elastic support is ob
tained by means of appropriate elements.

There is, however, a frequent need for special support conditions, which the engineer would like to model using infinitely large stiffnesses. Due to numeri
cal reasons the modelling should not be done with elements possessing very large stiffnesses, but with dependent degrees of freedom (kinematic con
straints) instead. The need for such constraints arises e.g. by oblique sup
ports or rigidly connected nodes. In general, every dependent degree of free
dom can be expressed as a linear combination of other displacements or rotations:

da
+
a₁d₁
)
a₂d₂
)
AAA

These conditions are taken into consideration explicitly in the assembly of the global stiffness matrix and, therefore, they are numerically more stable than artificially rigid elements.

These combinations can be directly formulated by the record KINE and they can become quite complex. However, the memory requirements for solving a problem increase with the number of constraints and especially with the number of recursive associations.

Coupling conditions can be defined recursive up to 99 levels. Cyclic references or duplicate references are not possible.

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Standard conditions are available for the most frequent cases of constraints in the form of the INTE record and the node coupling conditions.

Dependent degrees of freedom are designated by a * or a negative equation number in the node output. All displacements are always output, and they comply to the specified dependencies. Reaction forces can be calculated via

ECHO REAC for each node separately or in pairs for coupled nodes; in the latter case they represent the force transmitted through the coupling.

Attention: Inappropriate use of couplings of the KINE type or the slave coup
lings (KPX through KPZ) may lead to mechanically absurd results (forces moved by couplings may violate the moment equilibrium).

2.6.1.

Radial and Tangential Supports

A node can be supported in reference to some direction. By PR or MR, the dis
placement along or the rotation about this direction are, respectively, fixed; by PT or MT the respective displacement or rotation becomes the sole unre
stricted degree of freedom.

2.6.2.

Rigid Body Couplings

The couplings KP, KL, KQ and KF describe rigid bodies to which the depend
ent nodes are connected through a hinge (KP), or through a connection with fixed rotation about one (KQ), two (KL) or all three (KF) directions. One single plane may be activated in special cases (KPEX, KPEY, KPEZ and KFEX, KFEY, KFEZ). This is, for instance, the case when defining a plane of the structure which allows lateral bending but not in−plane distortion.

2.6.3.

Symmetry Conditions

Symmetry conditions are a rarely needed special case of coupling. Conditions of symmetry or anti−symmetry hold about the mid−perpendicular of the line connecting two nodes. In most cases the definition of a symmetry condition is easier through the use of a lateral support. The direction of the support must then be perpendicular to the symmetry plane. PRMT defines a sym
metry and PTMR an anti−symmetry.

2.6.4.

Eccentric Connections

Eccentric connections, e.g. between a beam and a plate, can be specified by KF.

2.6.5.

Slave Systems

A special class of couplings imposes the same displacements or rotations to several nodes (KPX to KMT). Their application is useful e.g. in the description

of rigid foundation plates, which are not allowed to rotate. These couplings act upon particular degrees of freedom and are thus more flexible. The danger on the other hand is that their inappropriate use can produce undesired offset moments.

2.6.6.

Mindlin Plate Boundary Conditions

The formulation of the boundary conditions of plate elements is not uncriti
cal. The Mindlin−element especially has some peculiarities which should be given attention.

According to Kirchhoff ’s theory two stress resultants exist on an edge, namely the bending moment and the equivalent shear force. The latter con
sists of the shear force and the torsional moment, and that is why both can have values along a free edge different from zero. By contrast, Mindlin’s theory recognises three support conditions for the three stress−resultants i.e. bending moment, torsional moment and shear force. A support for the tor
sional moment, for example, suppresses the rotations perpendicular to the edge.

Free edges

Free edges do not have any constraints of any type. The reaction forces along such edges are, within the bounds of computing accuracy, zero. The stress re
sultants inside the elements though are not always exactly zero, due to the numerical method.

Fixed edges

Perfectly fixed edges can be input without any problems. For the interpreta
tion of the results, however, it is important to know, that the torsional reac
tion moments must be taken up. This takes place automatically in the output of the BOUN−elements, where these are converted into corresponding sup
port loadings.

Simply supported edges

Here, one has a choice between the so−called soft support (only PZ) and the hard support (PZ+MT). In case of the soft support, shear deformations are still allowed along the edge, and thus a shear force too; this can lead in some cases to considerable deviations from Kirchhoff ’s plate theory. On the other hand, the soft support is more suitable for the manipulation of uplifting

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corners as well as of re−entrant corners. Particularly in the case of obtuse corners, the hard support leads to undesired fixing.

Simulation of support on masonry and concrete

There are generally four ways to describe such supports: • Point− or line−support

This type of support is mainly used for thin supports (width < plate thickness). The size of the adjacent elements should be selected in such a way that their gravity centre lies on the round section which is criti
cal for the punch−through check. The proportioning for the shear force takes place inside the element, whilst for the moment of the supported side at the nodes of the support.

• Rotatable column head support

The column is described through a node with fixed support and poss
ible rotational spring stiffness, which otherwise is not an element node. The column area is described by means of a single element as well as coupling conditions between the four element nodes and the col
umn node, which specify that the cross section will remain plane with
out a restraint for the moment (KP for columns, KQ for walls). The size of the element can be between 2/3 of the column area (e.g. for circular columns) and the actual column area (e.g. by rectangular column cross section). It goes with where one likes to arrange the resultant of the re
action pressure.

The central element has a zero shear force and thus a uniform moment corresponding to the moment of the section along the face of the col
umn. One should arrange additional elements for the shear force check with their gravity centre lying on the round section used for that check , or make a direct punch−through check.

• Elastic foundation

This variant is meaningful for elastic supports of large areas, for which a rounded moment above the support is desired. The use of large foundation coefficients (subgrade moduli), however, results into unde
sired restraints. The selection of the subgrade modulus is thus critical, and this variant should be applied to moderate foundations only. • Special conditions

In principle, any arbitrary conditions can be formulated through coup
lings. The effect though must be checked in every case.

2.7.

Girders

The modelling of girders in plate structures presents a special problem. Be
sides the option of modelling them with folded structure elements or solid el
ements, which is ruled out for practical processing, one has a choice between two other options:

• The girder is modelled as a beam eccentrically connected to a plane shell (plate− and disk action). The area of the girder and its moment of inertia are determined from the protruding part of the girder. For proportioning, the results of the shell and the beam should be com
bined into total stress resultants for a T−beam.

This method is general and always correct. It captures the co−operat
ing widths and their distribution in the structure.

• The girder is modelled as add−on element to a plate by defining all its cross sectional values (area, moment of inertia) as follows:

Add−on value = Total value of T−beam

− contribution of co−operating part of plate

The total stiffness is correctly modelled in this manner. For propor
tioning girders with small heights one should always make construc
tive observations, as for instance assembling the individual values and applying them to a T−beam cross section.

2.8.

Literature

Methode der finiten Elemente

2. Auflage , Hanser Verlag München (2) E.Ramm, J.Müller, K.Wassermann

Problemfälle bei FE−Modellierungen

Baustatik, Baupraxis Tagung Hannover 1990 (3) C.Katz, J.Stieda

Praktische FE−Berechnungen mit Plattenbalken Bauinformatik 3 (1992) Heft 1 S 30−34

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(4) M. Gupta

Error in Eccentric Beam Formulation

Int.Journ.Num.Meth. in Engineering 11 (1977) 1473 (5) O.C.Zienkiewicz, Zhu

A simple error estimate and adaptive procedure for practical engineering analysis. Int.Journ.Num.Meth. in Engineering 24 (1987) 337−357 (6) C. Katz Fehlerabschätzungen 1. FEM−Tagung, Kaiserslautern 1989

2.9.

Limitations

The following limits can not be exceeded in principle: Number of cross sections: 999

Number of nodes : 999 999 Largest node number : 999 999 Largest element number: 999 999

Bore hole profiles : 999

Hinge combinations : 10

Segment definitions : 999

Each computer has a finite computing precision. This is normally 7 digits in case of 32 bits per word, and 15 digits in case of double precision. It is nat
urally meaningless to want to discuss about the 7. decimal digit of a final re
sult. The danger, however, is that in FE−analyses, as in most cases in real life, it is not the absolute size of a displacement that is of interest, but the differ
ences.

Because of that , all numerical calculations are sensitive to large variations in stiffnesses or element dimensions, as well as to large numbers of elements between two boundary conditions (supports).

3

Input Description

The program GENF generates the basic structural system for plane or three− dimensional structures. On one hand the system consists of the nodes, de
fined by a number, their coordinates and geometric support conditions. On the other hand there are the elements, which are connected to each other at these nodes.

The number and the type of the elements can not be changed subsequently during a Restart of the program, whereas support conditions and material parameters can be arbitrarily modified. Any input data that includes el
ements always defines a new system.

Cross sections are usually defined by the program AQUA. For purely static analysis though (without proportioning or state II stiffness), the cross sec
tions can be defined with GENF as well. Each cross section must have been defined before an element can refer to it. Cross sectional data can be changed as often as the user likes, the latest input being valid at any time.

3.1.

Nodes

Nodes are provided with a number for identification. Node numbers need not be in a consecutive order. The maximum value of these numbers is limited to 99999 due to the output format. In addition, since some of the programs work with direct indices for quicker access, the highest possible node number is eventually limited by the available computer memory. The node numbering has normally no influence on the bandwidth of the stiffness matrix because the system’s generation is directly combined with an optimisation of the pro
file and the bandwidth of the stiffness matrix. If this operation is suppressed, the bandwidth is directly determined by the node numbers as they were de
fined by the user. Nodes which are not used by any elements, do not have any influence. Nodes can be defined as often as one likes, the last definition being valid at any time. Couplings, however, can not be defined more than once, when this would lead to a multiple dependency of the same degree of freedom.

3.2.

Elements

Elements are also identified by an arbitrary number within the selected el
ement group. An element number though can be used only once for each el

Version 10.20

3−2

ement type. Elements can be defined only once; if an element gets deleted, the same element number can not be used any more.

The element number contains the group number. The latter is the integer part of the element number divided by a freely defined divisor. The default value of this divisor GDIV by the record SYST is 99999, i.e. all elements are assigned to group 0. If the elements are subdivided into groups with a differ
ent value for GDIV, any elements of the group 0 that follow a group initiation by the record GRP are assigned to that group by their element number, i.e. the program changes the element numbers so as to adapt them to the active group.

Groups can be used in selecting a particular structural system or defining partial regions for post−processing or graphical representation. A sensible partitioning of a structure into such groups can be very helpful in studying stress resultants at nodes. In case of fold structures, one should arrange the elements of each disk into separate groups.

It is advantageous to number the elements in such a way that use can be made of generation options during the system selection (groups) and the loading input (refer to STAR2, beam groups).

The theoretical background of the elements is described in the calculation programs.

3.3.

Results

The created structure is stored in the database (project file) and it can be represented graphically by the program GRAF; this can be done even for er
ratic systems, so long as the program GENF has not terminated prematurely after the input. Further processing with other programs for analysis is poss
ible only when the structure is free of errors.

When no errors are detected, the structure’s data is output after being sorted, and a profile optimisation is performed on the stiffness matrix, in order to mi
nimise the cost of solving the system of equations for the structure at hand.

3.4.

Restart

After a static or dynamic analysis, boundary conditions, material parameters and cross sections can be modified with Restart. Elements and nodal coordi
nates, however, remain unchanged. A restart takes place with the explicit input SYST REST. The following can be included in a Restart input:

− Nodes, yet only constraints without coordinates − Couplings

− Material parameters and cross sections − Foundation profiles

− Flexibility of particular node supports

It is stressed here that all couplings must be redefined, in case of coupling input.

3.5.

Input Records

Input is made in free format by the CADINP input language (see General Manual).

Records Items

ECHO SYST

OPT VAL

TYPE GDIV OPTI FIXS NDEL GDIR XREF YREF ZREF

T11 T21 T31 T12 T22 T32 T13 T23 T33 NODE INTE KINE MESH IMES CUBE TRAN MIRR ALIN SECT NO X Y (Z) FIX NREF DX DY (DZ) COOR NO N1 N2 TYPE ND ND1 FD1 ND2 FD2 ND3 FD3 ND4 FD4 ND5 FD5 ND6 FD6

N1 N2 N3 N4 M N MNO MPRO NPRO

CHNG T1 T2 T3 T4

N1 N2 INC1 N4 INC2 MNO CHNG T1 T2

T4

N1 N2 N3 N4 N5 N6 N7 N8 M

N L MNO CHNG

FROM TO INC DX DY (DZ) ALPH BETA THET

DNO CHNG

FROM TO INC A B C D SMO VAL

SNO CHNG

NO NO1 NO2 F NO3 REF

Version 10.20

3−4

NO E MUE G K GAM GAMA ALFA EY

MXY OAL OAF SPM TITL

NO E MUE G K GAM GAMA ALFA E90

M90 OAL OAF SPM FY FT TITL

NO T0 NO0 T1 NO1 T2 NO2 T3 NO3

T4 NO4 T5 NO5 T6 NO6 T7 NO7 T8

NO8 T9 NO9 TITL

NO C CT CRAC YIEL MUE COH DIL GAMB

REF MREF H

NO TYPE P1 P2 P3 P4 P5 P6 P7

P8 P9 P10

NO TYPE VAL VAL1 VAL2 VAL3 VAL4 VAL5

NO TYPE FCN FC FCT FCTK EC QC GAM

ALFA SCM TYPR FCR GC GF MUEC TITL

NO TYPE CLAS FY FT FP ES QS GAM

ALFA SCM EPSY EPST REL1 REL2 R K1 FDYN TITL

NO TYPE CLAS EP G E90 QH QH90 GAM

ALFA SCM FM FT0 FT90 FC0 FC90 FV FVR

OAL OAF TITL

NO STYP SCLA MCLA E G MUE GAM ALFA

SCM FCN FC FT FHS FTB TITL

EPS SIG TYPE TEMP

BORE BLAY BBAX

BBLA

NO X Y Z NX NY NZ ALF TITL

S MNO ICEX MNOR ICRE HWMI HWMA

S1 S2 K0 K1 K2 K3 M0 C0 TANR

TAND KSIG D0 D2

S1 S2 K0 K1 K2 K3 P0 P1 P2

Records Items SVAL SREC SCIR HING NO MNO A AY AZ IT IY IZ IYZ

CM YSC ZSC YMIN YMAX ZMIN ZMAX WT WVY WVZ NPL VYPL VZPL MTPL MYPL MZPL BCYZ TITL

NO H B HO BO SO SU ASO ASU

MNO MRF ITF SAY SAZ DASO DASU REF TITL

NO RA RI SA SI ASA ASI MNO MRF

ITF DAS TITL

NO G1 G2 G3 G4 G5 G6

GRP NOG T MNO MRF STI NR POSI TX TY TXY TD TRUS CABL BEAM ADEF BDIV BSEC SUPP QUAD BRIC SPRI BOUN FLEX DAMP MASS NO NA NE NCS PRE NO NA NE NCS PRE

NO NA NE (NR) NCS AHIN EHIN DIV NBD

NP NCSE NO

DS NCS STYP PRIN DIRE LOC

NO X NCS STYP PRIN DIRE LOC

NO XFBM XSBM TYBM XFEM XSEM TYEM XFBT XSBT TYBT XFET XSET TYET TO INC

NO N1 N2 N3 N4 MNO DNO ENO NNO

T C STI NR POSI CT MRF T1 T2

T3 T4

NO N1 N2 N3 N4 N5 N6 N7 N8

MNO

NO NA N2 DX DY DZ CP CT CM

PRE GAP CRAC YIEL MUE COH DIL ENO DNO NNO MNO AR

FROM TO INC TYPE CA CE REF RX RY

RZ TITL

NO NO1 NO2 P VX VY VZ PHIX PHIY

PHIZ PHIW

NO NA NE D DT DM

NO MX MY MZ MXX MYY MZZ MXY MXZ

Version 10.20

3−6

The records can be input in any order; however, certain data (e.g. nodes) must have been already introduced before any reference can be made to them (e.g.

MESH). As an exception, the records ADEF and BDIV as well as BORE,

BBAX and BBLA are meaningful only in a specific order.

The parameters between parentheses Z, DZ and NR are not applicable to two−dimensional structures, therefore they are omitted from the input. In case of the record NODE for a two−dimensional system, the parameter FIX must be specified in the fourth place.

3.6.

ECHO − Control of the Output

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

ECHO

Item Description Dimension Default

OPT A literal from the following list: GEOD Geometric definitions NODE Node parameters MAT Material properties GROU Group properties SECT Cross sections QUAD 2−D−elements BRIC 3−D−elements

BEAM Flexible beams and piles SPRI Spring elements

TRUS Truss−bar elements CABL Cable elements BOUN Boundary elements SYST System values

FULL All the above options NO Nothing printed

PRIN Print despite any input errors

LIT FULL

VAL Output extent

NO no output YES regular output

LIT YES

The command name ECHO must always be repeated, otherwise confusion may occur with other records with the same names (e.g. NODE).

The default value corresponds to regular output so long as the system has been generated error free.

Version 10.20

3−8

See also: NODE

3.7.

SYST − Global System

Parameters

SYST

Item Description Dimension Default

TYPE FRAM Plane frame or disk

(system lies in the XY−plane) PAIN Plane strain condition

PESS Plane stress condition

(system lies in the XY−plane) AXIA Axisymmetric stress condition

(system lies in the XY−plane, rotation around x)

GIRD Gridwork or plate

(system lies in the XY−plane) SPAC Spatial frames or shells and

folded structures

REST Restart of the system with new material and cross sec− tional properties or boundary conditions

Item Description Dimension Default GDIV OPTI FIXS NDEL Group divisior NO No numbering optimization YES Coarse optimization

FULL Fine optimization

Default values of nodal degrees of free
dom (see NODE)

Unused nodes will be erased. YES NO − LIT LIT LIT * FULL FREE NO GDIR XREF YREF ZREF T11 T21 T31 ... T33

Direction of gravity load

Literal XX,YY,ZZ,NEGX,NEGY,NEGZ Origin of coordinate system WCS

Transformation matrix WCS −> UCS Default: T11 T12 T13 1.0 0.0 0.0 T21 T22 T23 = 0.0 1.0 0.0 T31 T32 T33 0.0 0.0 1.0 LIT m m m − * 0.0 0.0 0.0 1.0 0.0 0.0 1.0 The SOFiSTiK system are assigned to a specific system type.

Version 10.20

3−10

It is advised to orientate the axes of coordinates so that the direction of grav
ity coincides with the z−axis for three−dimensional systems, with the y−axis for FRAM systems and with the z−axis for GIRD systems. For some of the pro
grams (PILE, TALPA, ASE, ELSE) this orientation of the coordinate system is mandatory.

XREF through T33 can be used in order to describe the position of the GENF− coordinate system relative to the world coordinate system WCS.

In the case of plane structures of the type FRAM/GIRD and/or PAIN/PESS/ AXIA the output of out−of−plane deformations and stress−resultants is sup
pressed. Therefore, plane frames or gridworks, the axes of which do not co
incide with the principal axes of their cross sections, can be analyzed correctly in three dimensions only.

Changes in an existing database (Restart) can be made by SYST REST. This is necessary for instance when changing the support conditions due to differ
ent construction stages. The type and number of the elements and their nodes can not be changed in such case.

The following can be defined in a Restart−input: − Node constraints (no coordinates!), couplings

− Material values and cross sections, foundation profiles − Flexibilities of individual nodes

Take notice that all couplings must be redefined, if any couplings are input. Groups can be moved for the selection of a static system or for the definition from subareas in case of evaluations or graphic representations here. In par
ticular can during the determination of internal forces and moments of nodes a reasonable group division be helpful. With folded structures itself is recom
mended to arrange the elements of the individual discs into separate groups. The element number includes the group number implicit through the integral part of the element number divided by a freely definable divisor. In the default this divisor GDIV from sentence SYST has the value 99999, that is all el
ements are assigned to the group 0. If the elements are defined without an explicit specificated value of the group (= group 0), than the elements follow
ing after a group inauguration with the sentence GRP are classified with their element number in this group. It means the element number is changed from the program in such a way that it is a part of the activ group.

That one is preset from historical grounds temporarily still for data records without every input to GRP formerly firm group divisor 1000. With that many data records can be employed as before more further, and/or an only input

GRP suffices.

The volume width and/or the profile of the stiffness matrix has decisive influ
ence on the CPU time and the storage requirement to the solution of the Fi
nite−Element−system of equations. In the volume width and/or profile op
timization are minimized these sizes in a heuristic procedure. There the volume width of the greatest difference in the FE−net occurring (intern, for the user not visible) node numbers of an element derives, it is attempted to form the numbering so that neighboring nodes have numbers resting with each other near. The quality of the volume width depends in this case also on the choice of the start node.

In the ’standard optimization’ a probably well suitable node is chosen for this purpose heuristically. In the expanded optimization is started (fundamental) from every node and preserved with that a i.a. better result, however, at the expense of a larger CPU time for the optimization. This larger expenditure rewards for i.a., when during the following FE−calculation the system of equations is very often to be solved (many loads, non−linear calculation) or boots onto the boundaries of the available CPU time or storeroom capacity, to itself then. Near the iterative equation solvers the volume width is needed only for an estimate of the memory requirement.

Version 10.20

3−12

See also: SYST, MESH, IMES, ALIN, SECT, TRAN, MIRR, INTE, KINE

3.8.

NODE − Nodal Coordinates and

Constraints

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

NODE

Item Description Dimension Default

NO X Y Z FIX NREF DX DY DZ COOR Node number X−coordinate Y−coordinate

Z−coordinate (omitted by 2−D−systems) Node constraints

Node number of reference node

Directions for couplings or polar bound
ary conditions

(DZ not necessary for 2−D−system) System of coordinates CA Cartesian coordinates CY Cylindrical coordinates SP Spherical coordinates − m/* m/* m/* LIT − m/* m/* m/* LIT ! 0 0 0 * − * * * CA Remarks

Coordinates or constraints for all the nodes can be defined as often as one likes with MESH, IMES, CUBE, TRAN, MIRR or NODE. The last input is valid at any time. Only support conditions can be modified by RESTART; couplings, however, can not be partially redefined, thus in RESTART either all couplings and dependent boundary conditions (PR, PT, MR and MT) must be input again or none at all. When only the constraints or certain coordinates are being modified, a − (default value) must be input for the rest of the coordi
nates.

The nodes need not be numbered in a consecutive order. Coordinates

Input coordinate systems

The input values for y or z by CY or SP are interpreted as angles in degrees. The default system of coordinates is CA. Any definition holds for all following nodes until a new explicit definition is given. While in CA mode, one can switch to cylindrical coordinates for certain nodes through the use of negative node numbers for them.

Regardless of the input mode, the coordinates of the nodes are immediately converted to Cartesian ones and they are the only ones used thereafter. As an example, the following definitions of coordinates are equivalent:

NODE 1 12 45 30 COOR SP NODE 1 6 45 10.392 COOR CY NODE 1 4.243 4.243 10.392 COOR CA NODE −1 6 45 10.392 COOR CA

If a reference node is defined, all coordinates are considered to be relative to those of the reference node.

An earlier defined node can be translated with respect to its old position, if a reference node with the same node number is input. The input of a coupling is not allowed in such case.

Beispiel: KNOT 15 1 2 KREF 15

Der Knoten 15 wird relativ zu seiner bisherigen Lage um 1 Meter in X− und um 2 Meter in Y−Richtung verschoben.

Version 10.20

3−14

Example: NODE 15 1 2 NREF 15

Node 15 is translated 1 m in the X− and 2 m in the Y−direction with respect to its previous position.

It is impossible to specify couplings to a reference node and absolute coordi
nates in the same record. It is best, in principle, first to define all the nodal coordinates and then all the couplings (without coordinates).

Nodal constraints

All the constraints of a node can be described by any combination of the follow
ing literals (limited up to 8 characters). Any degree of freedom not included in a 2−D system gets fixed. The default constraint is the value defined by FIXS in SYST.

PX Constraint of displacement in x PY Constraint of displacement in y PZ Constraint of displacement in z PR Constraint of radial displacement PT Constraint of tangential displacement MX Contstraint of rotation about x

MY Contstraint of rotation about y MZ Contstraint of rotation about z

MR Contstraint of rotation about radial direction MT Contstraint of rotation about tangential direction MB Constraint of warping XP = PY + PZ YP = PX + PZ ZP = PX + PY PP = PX + PY + PZ XM = MY + MZ YM = MX + MZ ZM = MX + MY MM = MX + MY + MZ + MB FREE = Deletion of all constraints F = PP + MM

DEL = Node will be deleted

usefull for auxiliary nodes, which should not appear in the graphs nor the results.

A boundary condition on a symmetry or an anti−symmetry axis can be defined by PRMT or PTMR, respectively, if the direction of the coupling is defined per
pendicular to the axis. A direction must be defined in case of PR, PT, MR, MT by means of DX, DY, DZ or the reference node.

Support conditions can be also defined in relation to another node (reference node). The following input is therefore allowed only in conjunction with the parameter NREF. Combinations with other literals are not allowed. Opposite to constraints, coupling conditions can not be subsequently overwritten; addi
tional couplings, however, can be defined so long as no multiple definition oc
curs.

KPX Coupling of x−displacement only (ux = uxo)

KPY Coupling of y−displacement only (u_y = u_yo) KPZ Coupling of z−displacement only (u_z = u_zo) KPR Coupling of radial displacement

KPT Coupling of tangential displacements

KMX Coupling of rotation about the x−axis (ϕ_x = ϕ_xo) KMY Coupling of rotation about the y−axis (ϕ_y = ϕ_yo) KMZ Coupling of rotation about the z−axis (ϕ_z = ϕ_zo) KMR Coupling of rotations about the radial direction

KMT Coupling of rotations about the tangential directions KP Articulated connection to rigid body at the reference node KPPX Connection of x displacement only (flexible yz−plane) KPPY Connection of y displacement only (flexible xz−plane) KPPZ Connection of z displacement only (flexible xy−plane) KPEX Rotation about x−axis only (flexible, rigid yz−disk) KPEY Rotation about y−axis only (flexible, rigid xz−disk) KPEZ Rotation about z−axis only (flexible, rigid xy−disk) KL = KP + KMT

KQ = KP + KMR

Version 10.20

3−16

KFEX Rotation about x−axis only (flexible, rigid yz−disk) KFEY Rotation about y−axis only (flexible, rigid xz−disk) KFEZ Rotation about z−axis only (flexible, rigid xy−disk) SYM Symmetry conditions about the mid−perpendicular

ANTI Anti−symmetry conditions about the mid−perpendicular CYCL Cyclic symmetry conditions

Coupling conditions describe infinitely stiff elements and special boundary conditions which are numerically stable. Their application area is the for
mulation of boundary conditions for plates and shells and the modelling of very stiff structural parts. General kinematic constraints can be defined using the records KINE and INTE. Kinematic constraints can not take care of any non−linear geometric analysis.

Kinematic conditions of couplings

KPPX: u_x = u_xo + ϕ_yo ⋅ (z − z_o) − ϕ_zo ⋅ (y − y_o) (1) KPPY:

u

= u

+

ϕ

(x − x

) −

ϕ

(z − z

)

u

= u

+

ϕ

⋅ (y − y

) −

ϕ

⋅ (x − x

)

ϕ

=

ϕ

ϕ

=

ϕ

ϕ

=

ϕ

u

= u

−

ϕ

(z − z

)

u

= u

+

ϕ

(y − y

)

(8)

ϕ

=

ϕ

u

= u

+

ϕ

(z − z

)

u

= u

−

ϕ

(x − x

)

(11)

ϕ

=

ϕ

u

= u

−

ϕ

⋅ (y − y

)

u

= u

+

ϕ

(x − x

)

(14)

ϕ

=

ϕ

The conditions PR and PT, KPR and KPT as well as their counterparts for mo
ments are not explicitly but implicitly defined. The programs themselves create an appropriate explicit form.

PR:

u

n = 0

u

dx + u

dy + u

dz = 0

u

n = 0

(17)

(u_x−u_xo)⋅dx + (u_y−u_yo)⋅dy + (u_z−u_zo)⋅dz = 0 (18)

KPT:

(u−u

)

n = 0

(19)

The symmetry and anti−symmetry conditions are given in the following equations in vectorial form. A presentation by their components is not in
cluded here:

SYM:

u

n = − u

n

u

n = u

n

The directional or differential vector n = (dx,dy,dz) is built from the differ
ences of the node coordinates. These coordinate differences can also be speci
fied explicitly by means of DX, DY and DZ.

Certain degrees of freedom that have been coupled can be constrained again with input of a constraint after the coupling condition.

Version 10.20

3−18

See also: KINE, NODE

3.9.

INTE − Intermediate Nodes

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

INTE

Item Description Dimension Default

NO N1 N2 TYPE

Number of intermediate node Number of a corner node Number of a corner node Type of Interpolation P Linear displacements F Linear displacements + constant rotations Q Quadratic displacements + linear rotations − − − LIT ! ! ! F

In case of mesh refinement or in cases of stiff cross−girders there may arise a need for nodes that lie between two others and depend on them. This kind of dependency can be described by INTE.

INTE−couplings

The INTE−coupling is a constraint with special attributes. Herein, opposite to node couplings, one node (the middle node) becomes dependent on two other nodes. The displacements and rotations of the middle node are interpo
lated from the corresponding ones of the adjacent nodes.

u

= u

· DD + u

· (1−DD)

When the deflections of the outer nodes are somehow prescribed, e.g. fixed or provided with a certain stiffness, the deflection of the middle node is pre
scribed in the same way too. The coupling is rigid only when both nodes can not displace relatively to each other. A rigid body with three nodes must be described by means of two KP/KF couplings; the INTE−coupling can not be used in that case.

There are several variants of interpolation used by INTE−couplings, which are described in the following.

TYPE P

Displacements: linearly interpolated Rotations: not defined

Application: mesh refinements TALPA TYPE F

Displacements: linearly interpolated as in TYPE P

Rotations: torsion" linearly interpolated, other rotations com− puted from displacement differences divided by the respective node distances

Application: connection of beam elements onto disks stiff cross−girders between two supports

In the general three−dimensional case, if one draws the lines connecting the two nodes in the initial undeformed as well as in their deformed state, two rotational components are defined exactly by the secant angles of those. The third yet undetermined rotational component has the direction of the con
necting line (torsion), and it is normally interpolated. The general expression is very complicated; however, INTE−couplings parallel to the axes of coordi
nates can be expressed by much simpler expressions, e.g.,

DX = 0. DY = d DZ = 0. results in:

ϕ

=

D u

/ d

ϕ

=

ϕ

Version 10.20

3−20

ϕ

= −

D u

/ d

TYPE Q

Displacements: quadratically interpolated Rotations: linearly interpolated

Application: mesh refinements of plates and shells

In mesh refinements of plates and shells there is a problem in coupling the translational and rotational degrees of freedom. Very poor elements function with a plain interpolation. Due to the peculiarities exhibited by the formula
tion of the SEPP/ASE−elements, even in its simplest form, the INTE−condi
tions must be accordingly complicated. In case of regular elements by Kirch
hoff’s theory for example, a cubic interpolation of the displacements and two of the rotations must be employed. Mindlin elements also work with the so− called Kirchhoff constraints. In principle of course, translations and rotations are interpolated independently of one another, yet proper additional condi
tions are used to make sure that the shear force corresponds to the derivative of the moment.

A quadratic distribution of the bending deflection along with a linear dis
tribution of the rotations can be accomplished through the introduction of an additional translational degree of freedom at the middle of an element’s side. This additional degree of freedom can be later eliminated. This method is also employed by V−couplings. Although the formulation is consistent and leads to considerably better results than the older methods, it is not recommended unlimitedly. In particular, it should not be used with non−conforming el
ements.

The application of INTE in the direct vicinity of singularities is generally not recommended.

Finally, here is an example of modelling a rigid cross−beam in a bridge struc
ture with oblique axes of supports. The cross−beam is 5 m long and it is posi
tioned at an angle of 45 degrees with respect to the X−axis; one of its supports allows movement in all directions, while the other can only translate at an angle of 105 degrees with respect to the X− axis.

NODE 1 0.0 0.0 FIX PTMM DX COS(105) DY SIN(105)

NODE −3 5.0 45 FIX KPR 1 ; 3 FIX PZMT DX 1 DY 1

NODE −2 2.5 45 ; INTE 2 1 3 TYPE F

The constraint PT determines the translational freedom. The two perpen
dicular directions as well as Z are fixed. MM is important, so that no movable system results. Node 3 is defined in polar mode with respect to 1. KPR defines a fixed distance. The constraint PZ overwrites here part of the coupling, thus it must certainly come after that. MT on the other hand does not conflict with KPR, therefore it could have been input earlier as well. Node 3 can now move only in a circle about node 1 in the X−Y plane. The constraint MM of node 1 has no influence on that. Node 2 which is defined by INTE has now all its de
grees of freedom defined. It is still free to rotate about the 1−3 axis through the MT constraint of node 3. A fixed support condition could have been defined by MM on node 3.

Version 10.20

3−22

See also: INTE, NODE

3.10.

KINE − Kinematic

Dependencies

KINE

Item Description Dimension Default

ND ND1 FD1 ND2 FD2 ... ... ND6 FD6

Dependent degree of freedom Reference degree of freedom 1

Factor for reference degree of freedom 1

LIT LIT

! − −

In special cases kinematic dependencies can be described explicitly too: (ND) = (ND1) · FD1 + ... + (ND6) · FD6

The degrees of freedom are defined by:

nodenumber · 10 + local degree of freedom

1 = u

2 = u

3 = u

4 =

ϕ

5 =

ϕ

6 =

ϕ

KINE 1003 13 1.0 25 0.5

means that the displacement u_z of node 100 is prescribed to be the sum of the displacement uz of node 1 and one−half of the rotation ϕy of node 2.

If a positive number is entered for ND, the same coupling holds for the reac
tion forces too. Therefore, no reaction forces arise at coupled nodes. If ND is negative, however, the coupling holds for the displacements only. Rigid bodies are typical cases of the first variant, oblique supports are typical of the second one.

See also: IMES, CUBE, NODE, GRP

3.11.

MESH − Generation of Nodes

and Quadrilateral Elements

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

MESH

Item Description Dimension Default

N1 N2 N3 N4 M N MNO MPRO NPRO CHNG T1 T2 T3 T4 Corner node Corner node Corner node Corner node Partitions of N1−N2 or N3−N4 Partitions of N2−N3 or N4−N1 Material number

Progression for the subdivision M Progression for the subdivision N Change of previously defined nodes

YES/NO/OFF

Thickness at the four corner nodes (only when QUAD elements are gener
ated) − − − − − − − − − LIT m/* m/* m/* m/* ! ! − − 1 M * − − YES * T1 T1 T1 A region is described by three or four already defined nodes, which are the corner points of a quadrilateral. This region is partitioned with MESH into m by n elements. The nodenumber differences must be perfectly divisible by m and n. If this is not the case, the last interval will be increased.

Version 10.20

3−24

MESH−generation Remark

The number assigned to the elements is the node number of the corner node oriented towards N1. In case a record of the GRP type has been previously input (or GDIV in record SYST), the numbers get changed appropriately. The default value for MNO can be set by a preceding GRP record. If a negative MNO is input, the elements are not assigned the number of the corresponding N1 corner node, instead they are numbered consecutively in the active group NOG of the GRP record. The first element of the mesh is assigned the number GDIV * NOG + 1. The group divisor is defined in the record SYST.

Regions with partitions varying like geometric progressions can be defined by MPRO or NPRO. Beginning from side N4−N1, each segment is MPRO times the previous one. If MPRO is negative, a symmetric partitioning takes place (length of first segment equal to that of the last one).

Recesses can be defined afterwards with QUAD. Node constraints are not af
fected by MESH.

If only N1, N2, M and possibly MPRO are given, then only nodes on the line connecting N1 and N2 will be generated.

MESH−One−dimensional generation

In cases several MESH regions are defined adjacent to each other, the numbering of the nodes on the common edges must be identical in order to ensure the mechanical connection of the various parts.

Normally, all the nodes acquire the computed coordinates. By CHNG NO though, the coordinates of all previously defined nodes remain unchanged. By CHNG OFF, in addition, the previously defined nodes of the edges N1− N2 and N3−N4 are being used for generating the intermediate nodes. This can be very useful in generating systems with circular boundaries.

Example:

A region with corner nodes 1, 9, 51 and 59 is partitioned into 8 by 5 elements. Nodes 1 through 9 lie on a circular arc, the rest of the edges are straight lines, and the centre of the circular arc is at node 100.

Version 10.20

3−26

See also: MESH, CUBE, NODE, GRP

3.12.

IMES − Generation of Irregular

Nodes, Quadrilateral Elements

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

IMES

Item Description Dimension Default

N1 N2 INC1 N4 INC2 MNO CHNG T1 T2 T4 Corner node 1 Corner node 2

Increment for edge N1−N2 Corner node K4

Increment for edge N1−N4

Group and material number (see MESH) Change of previously defined nodes

YES/NO

Thickness at the three corner nodes (only when QUAD elements are gener
ated) − − − − − − LIT m/* m/* m/* ! ! 1 ! 1 * YES * T1 T1 By IMES, opposite to MESH, all the nodes on the edges (N1−N2) and (N1− N4) are defined instead of the corner nodes. An irregularly partitioned region is generated through a parallel translation of the edges with the above nodes towards the corresponding nodes of the opposite edges. An edge can consist of any number of nodes, and it can be broken as well. Recesses can be later introduced by QUAD. Node constraints are not affected by IMES.

Version 10.20

3−28

See also: MESH, IMES, NODE

3.13.

CUBE − Nodes and Cubic

Elements

CUBE

Item Description Dimension Default

N1 N2 N3 N4 N5 N6 N7 N8 M N L MNO CHNG Corner node Corner node Corner node Corner node Corner node Corner node Corner node Corner node Partitions of N1−N2, N3−N4, N5−N6, N7−N8 Partitions of N2−N3, N4−N1, N6−N7, N8−N5 Partitions of N1−N5, N2−N6, N3−N7, N4−N8

Group and material number (see MESH) Change of previously defined nodes

YES/NO − − − − − − − − − − − − LIT ! ! ! ! ! ! ! ! 1 M M * YES

Nodes N1 through N8 are the corner nodes of an 8−cornered solid region. This region is subdivided by CUBE into L by M by N elements. The differences (N1−N2), (N3−N4), (N5−N6) and (N7−N8) must be divisible by M; similarly for N and L. Recesses can be defined later on by the BRIC record. Node con
straints are not affected by CUBE.

Version 10.20

3−30

See also: MIRR, ALIN, SECT, NODE

3.14.

TRAN − Transformation of

Nodes

TRAN

Item Description Dimension Default

FROM TO INC DX DY DZ ALPH BETA THET DNO CHNG First node Last node Increment

The nodes from FROM to TO by in
crements of INC are transformed. Translation in X−direction Translation in Y−direction Translation in Z−direction (omitted in 2−D−systems) Coning angle Rotation angle Nutation angle

Inkrement der Knotennummer Change of previously defined nodes

YES/NO − − − m/* m/* m/* Degrees Degrees Degrees − LIT 1 FROM 1 0. 0. 0. 0. 0. 0. 100 YES

Using TRAN it is possible to generate new nodes from the rotation and translation of old ones. The number of a transformed node is the initial node
number plus DNO. By entering DNO 0, nodes that were defined in any system of coordinates convenient for their input, can be now displaced and rotated to any desired location in the global system of coordinates.

TRAN does not define or modify constraints. ALPH, BETA and THET are Eulerian angles. Any rotation in the three−dimensional space consists of three individual components:

1. Rotation ALPH about the Z−axis, 2. Rotation THET about the new X−axis, 3. Rotation BETA about the new Z−axis. The most usual cases are given by:

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ALPH: BETA: THET:

0 0 phi : Rotation about the X−axis

90 −90 phi : Rotation about the Y−axis

0 phi 0 : Rotation about the Z−axis

See also: TRAN, ALIN, SECT, NODE

3.15.

MIRR − Mirroring of Nodes

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MIRR

Item Description Dimension Default

FROM TO INC A B C D SMO VAL SNO CHNG First node Last node Increment

The nodes from FROM to TO by incre− ments of INC are mirrored.

Constants defining the plane of mirror
ing by

A⋅x + B⋅y + C⋅z + D = 0.

Partition point of node number

Transformation for new node number SV Mirroring of primary number SN Mirroring of secondary

number

AV Addition of primary number AN Addition of secondary number T Interchange of primary and

secondary numbers Number for mirroring or addition Change of previously defined nodes

YES/NO − − − − − LIT − LIT 1 FROM 1 0. 2 SV 100 YES

Using MIRR one can generate new nodes from the mirroring of other already existing nodes. Constraints can neither be set nor changed by MIRR.

The procedure for calculating the new node number is relatively complicated in order to account for all possible cases. To begin with, the node number is partitioned into the so−called primary and secondary number. The point of partition is specified by SMO. The secondary number is defined by as many

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of the last digits as SMO, while the primary number is built by the rest of the digits at the beginning of the node number.

The mirror of a number is defined as:

NO NEW = SNO + (SNO − NO OLD) SNO can also differ from a whole number by 1/2.

The user now has a choice among several transformation options:

By SV or SN the primary or secondary part of the old node number, respect
ively, will be mirrored with respect to SNO, whilst by AV or AN, SNO will be added to the primary or secondary part of the old node number, respectively. Example : Nodenumber 723, with SMO=2 and SNO=50

Primary number 07, Secondary number 23

is transformed by mirroring of the primary number to: 9323 by mirroring of the secondary number to: 777 by addition to primary number to: 5723 by addition to secondary number to: 773

by interchange to: 2307

The range FROM TO must define as exactly as possible the range of the mir
rored nodes, so that the generated nodes lie in the permissible range for node numbers. As a rule, an input with TO = 9999 does not satisfy this require
ment.

S: y = yo

B = 1.0

D = −yo