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GENF Generation of Finite Elements and Beam Structures Version 10.20

E SOFiSTiK AG, Oberschleissheim, 2001

GENF

Definition of Finite Ele
ments

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.

Definition of Finite Elements

GENF

1

Task Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1−1

2 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Theoretical Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systems of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of the Element Types . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2−1 2−1 2−2 2−5 2−7 2−8 2−8 2−12 2−13 2−13

3 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11. 3.12. 3−27 3.13. 3.14. 3.15. 3.16. 3.17. 3.18. 3.19. 3.20. 3.21. 3.22. 3.23. 3.24. 3.25.

Input Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ECHO − Control of the Output . . . . . . . . . . . . . . . . . . . . . . . . . . . SYST − Global System Parameters . . . . . . . . . . . . . . . . . . . . . . . . NODE − Nodal Coordinates and Constraints . . . . . . . . . . . . . . . INTE − Intermediate Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KINE − Kinematic Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . MESH − Generation of Nodes and Quadrilateral Elements . . . IMES − Generation of Irregular Nodes, Quadrilateral Elements

3−1 3−1 3−1 3−2 3−2 3−3 3−7 3−8 3−12 3−18 3−22 3−23

CUBE − Nodes and Cubic Elements . . . . . . . . . . . . . . . . . . . . . . TRAN − Transformation of Nodes . . . . . . . . . . . . . . . . . . . . . . . . MIRR − Mirroring of Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ALIN − Node upon a Line (Projection to the Line) . . . . . . . . . . SECT − Node at Intersection of two Straight Lines . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NORM − Default Design Code . . . . . . . . . . . . . . . . . . . . . . . . . . . MAT − General Material Properties . . . . . . . . . . . . . . . . . . . . . . . MATE − Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MLAY − Layered Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BMAT − Elastic Support / Interface . . . . . . . . . . . . . . . . . . . . . . . NMAT − Nonlinear Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MEXT − Extra Materialconstants . . . . . . . . . . . . . . . . . . . . . . . . . i

. 3−29 3−31 3−33 3−36 3−39 3−41 3−43 3−44 3−45 3−48 3−49 3−52 3−63

GENF

Definition of Finite Elements

3.26. 3.27. 3.28. 3.29. 3.30. 3.31. 3.32. 3.33. 3.34. 3.35. 3.36. 3.37. 3.38. 3.39. 3.40. 3.41. 3.42. 3.43. 3.44. 3.45. 3.46. 3.47. 3.48. 3.49. 3.50. 3.51. 3.52. 3.53.

CONC − Properties of Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . STEE − Properties of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TIMB − Properties of Timber . . . . . . . . . . . . . . . . . . . . . . . . . . . . MASO − Masonry / Brickwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . SSLA − Stress−Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . SVAL − Cross−section values . . . . . . . . . . . . . . . . . . . . . . . . . . . . SREC − Rectangle, T−beam, Plate . . . . . . . . . . . . . . . . . . . . . . . SCIR − Circular and Annular Sections . . . . . . . . . . . . . . . . . . . . . BORE − Bore Profile of a Sondation . . . . . . . . . . . . . . . . . . . . . . BLAY − Layer of the Soil Strata . . . . . . . . . . . . . . . . . . . . . . . . . . BBAX − Input of Axial Subgrade Parameters . . . . . . . . . . . . . . . BBLA − Input of Lateral Subgrade Parameters . . . . . . . . . . . . . HING − Hinged Connection Combinations for Beams . . . . . . . GRP − Group Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TRUS − Truss−bar Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . CABL − Cable Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BEAM − Beam Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ADEF − Beginning of Beam Segment Definition . . . . . . . . . . . . BDIV − Input of Beam Segments . . . . . . . . . . . . . . . . . . . . . . . . . BSEC − Beam Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUPP − Definition of Support Sections . . . . . . . . . . . . . . . . . . . . QUAD − Plane Elements (Disks / Plates / Shells) . . . . . . . . . . . . BRIC − Three−dimensional Solid Elements . . . . . . . . . . . . . . . . SPRI − Spring Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BOUN − Distributed Elastic Support . . . . . . . . . . . . . . . . . . . . . . FLEX − General Elastic Element . . . . . . . . . . . . . . . . . . . . . . . . . DAMP − Damping Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MASS − Concentrated Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3−64 3−71 3−80 3−82 3−84 3−86 3−91 3−94 3−95 3−96 3−97 3−98 3−100 3−101 3−105 3−106 3−108 3−116 3−117 3−119 3−120 3−122 3−126 3−127 3−134 3−139 3−141 3−142

4 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.

Output Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nodal Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross−sectional Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Group Qualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Elements (2−D, QUAD) . . . . . . . . . . . . . . . . . . . . . . . . . . . Three−dimensional Solid Elements (3−D, BRIC) . . . . . . . . . . . Boundary Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric Definitions (Bedding Profiles) . . . . . . . . . . . . . . . . . . .

4−1 4−1 4−2 4−4 4−5 4−5 4−6 4−6 4−7 4−8

ii

Definition of Finite Elements

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4.10. 4.11. 4.12. 4.13.

Bending Beams and Piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truss−bar Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cable Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4−8 4−9 4−9 4−10

5 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angle Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pointwise Supported Ceiling Plate . . . . . . . . . . . . . . . . . . . . . . . . . Gridwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Frame, Restrained in Space . . . . . . . . . . . . . . . . . . . . . . . . . Shell Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reinforced Concrete Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calotte Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples in the Internet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5−1 5−1 5−3 5−7 5−9 5−12 5−14 5−18 5−24

iii

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Definition of Finite Elements

iv

Definition of Finite Elements 1

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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 Version 10.20

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Definition of Finite Elements

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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Definition of Finite Elements 2

Theoretical Principles

2.1.

Systems of Coordinates

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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. Version 10.20

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Definition of Finite Elements

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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Definition of Finite Elements

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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. Version 10.20

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Definition of Finite Elements

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 z−direction and/or tangential

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Definition of Finite Elements 2.3.

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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. Version 10.20

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Definition of Finite Elements

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.

2.4.

2−6

Plane Elements

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Definition of Finite Elements

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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. Version 10.20

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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: d a
+
a 1d 1
)
a 2d 2
)
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 Version 10.20

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Definition of Finite Elements

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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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Definition of Finite Elements

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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. Version 10.20

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Definition of Finite Elements

GENF 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

(1) O.C.Zienkiewicz (1984) 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: Number of nodes : Largest node number : Largest element number: Bore hole profiles : Hinge combinations : Segment definitions :

999 999 999 999 999 999 999 999 10 999

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).

Version 10.20

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Definition of Finite Elements 3

GENF

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
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Definition of Finite Elements

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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:

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Definition of Finite Elements − − − − −

GENF

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 TYPE T11

VAL GDIV T21

OPTI T31

FIXS T12

NDEL T22

GDIR T32

XREF T13

YREF T23

ZREF T33

NODE

NO COOR NO ND ND5 N1 CHNG N1 T4 N1 N FROM DNO FROM SNO NO NO

X

Y

(Z)

FIX

NREF

DX

DY

(DZ)

N1 ND1 FD5 N2 T1 N2

N2 FD1 ND6 N3 T2 INC1

TYPE ND2 FD6 N4 T3 N4

FD2

ND3

FD3

ND4

FD4

M T4 INC2

N

MNO

MPRO

NPRO

MNO

CHNG

T1

T2

N2 L TO CHNG TO CHNG NO1 NO1

N3 MNO INC

N4 CHNG DX

N5

N6

N7

N8

M

DY

(DZ)

ALPH

BETA

THET

INC

A

B

C

D

SMO

VAL

NO2 NO2

F NO3

NO3 NO4

REF

INTE KINE MESH IMES CUBE TRAN MIRR ALIN SECT

Version 10.20

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Definition of Finite Elements

GENF Records

Items

NORM MAT

DC NO MXY NO M90 NO T4 NO8 NO REF NO P8 NO NO ALFA NO ALFA TITL NO ALFA OAL NO SCM EPS

NDC E OAL E OAL T0 NO4 T9 C MREF TYPE P9 TYPE TYPE SCM TYPE SCM

COUN MUE OAF MUE OAF NO0 T5 NO9 CT H P1 P10 VAL FCN TYPR CLAS EPSY

TYPE SCM OAF STYP FCN SIG

NO S S1 TAND S1 P3

X MNO S2 KSIG S2 PMA1

MATE MLAY

BMAT NMAT MEXT CONC STEE

TIMB

BRWO SSLA BORE BLAY BBAX BBLA

3−4

G SPM G SPM T1 NO5 TITL CRAC

K TITL K FY NO1 T6

GAM

GAMA

ALFA

EY

GAM FT T2 NO6

GAMA TITL NO2 T7

ALFA

E90

T3 NO7

NO3 T8

YIEL

MUE

COH

DIL

GAMB

P2

P3

P4

P5

P6

P7

VAL1 FC FCR FY EPST

VAL2 FCT GC FT REL1

VAL3 FCTK GF FP REL2

VAL4 EC MUEC ES R

VAL5 QC TITL QS K1

CLAS FM TITL SCLA FC TYPE

EP FT0

G FT90

E90 FC0

QH FC90

QH90 FV

GAM FVR

MCLA FT TEMP

E FHS

G FTB

MUE TITL

GAM

ALFA

Y ICEX K0 D0 K0 PMA2

Z MNOR K1 D2 K1

NX ICRE K2

NY HWMI K3

NZ HWMA M0

ALF

TITL

C0

TANR

K2

K3

P0

P1

P2

GAM GAM FDYN

Version 10.20

Definition of Finite Elements

GENF

Records

Items

SVAL

NO CM WVZ NO MNO NO ITF NO

MNO YSC NPL H MRF RA DAS G1

A ZSC VYPL B ITF RI TITL G2

AY YMIN VZPL HO SAY SA

AZ YMAX MTPL BO SAZ SI

IT ZMIN MYPL SO DASO ASA

IY ZMAX MZPL SU DASU ASI

G3

G4

G5

G6

GRP

NOG TXY

T TD

MNO

MRF

STI

NR

TRUS CABL BEAM

NO NO NO NP NO DS NO NO TYBT NO T T3 NO MNO NO PRE NNO FROM RZ NO PHIZ NO NO MYZ

NA NA NA NCSE

NE NE NE

NCS NCS (NR)

PRE PRE NCS

AHIN

NCS X XFBM XFET N1 C T4 N1

STYP NCS XSBM XSET N2 STI

PRIN STYP TYBM TYET N3 NR

DIRE PRIN XFEM TO N4 POSI

LOC DIRE XSEM INC MNO CT

N2

N3

N4

NA GAP MNO TO TITL NO1 PHIW NA MX REF

N2 CRAC AR INC

DX YIEL

SREC SCIR HING

ADEF BDIV BSEC SUPP QUAD

BRIC SPRI

BOUN FLEX DAMP MASS

Version 10.20

IZ WT BCYZ ASO REF MNO

IYZ WVY TITL ASU TITL MRF

POSI

TX

TY

EHIN

DIV

NBD

LOC TYEM

XFBT

XSBT

DNO MRF

ENO T1

NNO T2

N5

N6

N7

N8

DY MUE

DZ COH

CP DIL

CT ENO

CM DNO

TYPE

CA

CE

REF

RX

RY

NO2

P

VX

VY

VZ

PHIX

PHIY

NE MY

D MZ

DT MXX

DM MYY

MZZ

MXY

MXZ

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GENF

Definition of Finite Elements

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. A description of each particular record follows.

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Definition of Finite Elements

3.6.

ECHO − Control of the Output

Item

Description

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

VAL

FULL

All the above options

NO PRIN

Nothing printed Print despite any input errors

Output extent NO no output YES regular output

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ECHO ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

LIT

FULL

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.

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Definition of Finite Elements

GENF See also: NODE

3.7. Item

TYPE

SYST − Global System Parameters Description

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

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ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ SYST ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

LIT

FRAM

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

Version 10.20

Definition of Finite Elements Item

Description

GDIV

Group divisior

GENF Dimension

Default

−

*

No numbering optimization Coarse optimization Fine optimization

LIT

FULL

Default values of nodal degrees of free
dom (see NODE) Unused nodes will be erased. YES NO

LIT

FREE

LIT

NO

GDIR

Direction of gravity load Literal XX,YY,ZZ,NEGX,NEGY,NEGZ

LIT

*

XREF YREF ZREF

Origin of coordinate system WCS

m m m

0.0 0.0 0.0

T11 T21 T31 ... T33

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

−

1.0 0.0 0.0

OPTI

FIXS NDEL

NO YES FULL

1.0

The SOFiSTiK system are assigned to a specific system type.

Special cases of the global system of coordinates Version 10.20

3−9

GENF

Definition of Finite Elements

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.

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Definition of Finite Elements

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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. OPTI NO must be input for partial structures not connected to each other.

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Definition of Finite Elements

GENF

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

3.8.

NODE − Nodal Coordinates and Constraints

Item

Description

NO

Node number

X Y Z

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ NODE ÄÄÄÄÄÄÄÄÄ

Dimension

Default

−

!

X−coordinate Y−coordinate Z−coordinate (omitted by 2−D−systems)

m/* m/* m/*

0 0 0

FIX

Node constraints

LIT

*

NREF

Node number of reference node

−

−

DX DY DZ

Directions for couplings or polar bound
ary conditions (DZ not necessary for 2−D−system)

m/* m/* m/*

* * *

COOR

System of coordinates CA Cartesian coordinates CY Cylindrical coordinates SP Spherical coordinates

LIT

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 The user can choose among three coordinate systems for the input:

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Definition of Finite Elements

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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 NODE NODE NODE

1 1 1 −1

12 6 4.243 6

45 45 4.243 45

30 10.392 10.392 10.392

COOR COOR COOR COOR

SP CY CA 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

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Definition of Finite Elements

GENF

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 PY PZ PR PT

Constraint of displacement in x Constraint of displacement in y Constraint of displacement in z Constraint of radial displacement Constraint of tangential displacement

MX MY MZ MR MT MB

Contstraint of rotation about x Contstraint of rotation about y Contstraint of rotation about z Contstraint of rotation about radial direction Contstraint of rotation about tangential direction Constraint of warping

XP YP ZP PP

= = = =

PY PX PX PX

+ + + +

PZ PZ PY PY + PZ

XM YM ZM MM

= = = =

MY MX MX MX

+ + + +

MZ MZ MY MY + MZ + MB

FREE F

= Deletion of all constraints = PP + MM

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Definition of Finite Elements DEL

GENF

= 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 KPY KPZ KPR KPT

Coupling of x−displacement only (ux = uxo) Coupling of y−displacement only (uy = uyo) Coupling of z−displacement only (uz = uzo) Coupling of radial displacement Coupling of tangential displacements

KMX KMY KMZ KMR KMT

Coupling of rotation about the x−axis (ϕx = ϕxo) Coupling of rotation about the y−axis (ϕy = ϕyo) Coupling of rotation about the z−axis (ϕz = ϕzo) Coupling of rotations about the radial direction Coupling of rotations about the tangential directions

KP KPPX KPPY KPPZ KPEX KPEY KPEZ

Articulated connection to rigid body at the reference node Connection of x displacement only (flexible yz−plane) Connection of y displacement only (flexible xz−plane) Connection of z displacement only (flexible xy−plane) Rotation about x−axis only (flexible, rigid yz−disk) Rotation about y−axis only (flexible, rigid xz−disk) Rotation about z−axis only (flexible, rigid xy−disk)

KL KQ

= KP + KMT = KP + KMR

KF

Fixed connection to rigid body at the reference node

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GENF

Definition of Finite Elements

KFEX KFEY KFEZ

Rotation about x−axis only (flexible, rigid yz−disk) Rotation about y−axis only (flexible, rigid xz−disk) Rotation about z−axis only (flexible, rigid xy−disk)

SYM ANTI CYCL

Symmetry conditions about the mid−perpendicular Anti−symmetry conditions about the mid−perpendicular 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:

ux = uxo + ϕyo ⋅ (z − zo) − ϕzo ⋅ (y − yo)

(1)

KPPY:

uy = uyo + ϕzo ⋅ (x − xo) − ϕxo ⋅ (z − zo)

(2)

KPPZ:

uz = uzo + ϕxo ⋅ (y − yo) − ϕyo ⋅ (x − xo)

(3)

KP:

KPPX + KPPY + KPPZ

KF additionally:

ϕx = ϕxo ϕy = ϕyo ϕz = ϕzo

(4) (5) (6)

KPEX:

uy = uyo − ϕxo ⋅ (z − zo) uz = uzo + ϕxo ⋅ (y − yo)

(7)

(8)

KFEX additionally:

ϕx = ϕxo

(9)

KPEY:

ux = uxo + ϕyo ⋅ (z − zo) uz = uzo − ϕyo ⋅ (x − xo)

(11)

KFEY additionally:

ϕy = ϕyo

(12)

KPEZ:

ux = uxo − ϕzo ⋅ (y − yo) uy = uyo + ϕzo ⋅ (x − xo)

(13)

(14)

ϕz = ϕzo

(15)

KFEZ additionally:

(10)

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.

3−16

Version 10.20

Definition of Finite Elements PR:

PT:

ut ⋅ n = 0 ux ⋅ dx + uy ⋅ dy + uz ⋅ dz = 0

KPT:

(16)

u⋅n=0 ux
+
u y
+
uz dx dy dz

KPR:

GENF

(u−uo)t ⋅ n = 0 (ux−uxo)⋅dx + (uy−uyo)⋅dy + (uz−uzo)⋅dz = 0

(17)

(18)

(u−uo) ⋅ n = 0 (u x * u xo)

dx


+


ǒuy * uyoǓ ( )
+
uz * uzo dy dz

(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:

ut ⋅ n = − uto ⋅ n

ANTI:

ut ⋅ n =

uto ⋅ 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

3−17

Definition of Finite Elements

GENF See also: KINE, NODE

3.9.

INTE − Intermediate Nodes

Item

Description

NO N1 N2

Number of intermediate node Number of a corner node Number of a corner node

TYPE

Type of Interpolation P Linear displacements F Linear displacements + constant rotations Q Quadratic displacements + linear rotations

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ INTE ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

− − −

! ! !

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.

3−18

Version 10.20

Definition of Finite Elements

GENF

u0 = u1 · DD + u2 · (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: Rotations: Application: TYPE F Displacements: Rotations:

Application:

linearly interpolated not defined mesh refinements TALPA

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

ϕx = D uz / d ϕy = Version 10.20

ϕy−m 3−19

Definition of Finite Elements

GENF

ϕz = − D ux / d TYPE Q Displacements: Rotations: Application:

quadratically interpolated linearly interpolated 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.

3−20

Version 10.20

Definition of Finite Elements

GENF

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

3−21

Definition of Finite Elements

GENF See also: INTE, NODE

3.10.

KINE − Kinematic Dependencies

Item

Description

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

ÄÄÄÄÄÄÄÄÄ KINE ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Dependent degree of freedom

LIT

!

Reference degree of freedom 1 Factor for reference degree of freedom 1

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 = ux 2 = uy 3 = uz 4 = ϕx 5 = ϕy 6 = ϕz e.g the record KINE

1003

13

1.0

25

0.5

means that the displacement uz 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.

3−22

Version 10.20

Definition of Finite Elements See also: IMES, CUBE, NODE, GRP

3.11.

MESH − Generation of Nodes and Quadrilateral Elements

Item

Description

N1 N2 N3 N4

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ MESH ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

Corner node Corner node Corner node Corner node

− − − −

! ! − −

M N

Partitions of N1−N2 or N3−N4 Partitions of N2−N3 or N4−N1

− −

1 M

MNO MPRO NPRO

Material number Progression for the subdivision M Progression for the subdivision N

− − −

* − −

CHNG

Change of previously defined nodes YES/NO/OFF

LIT

YES

T1 T2 T3 T4

Thickness at the four corner nodes (only when QUAD elements are gener
ated)

m/* m/* m/* m/*

* 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

3−23

GENF

Definition of Finite Elements

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.

3−24

Version 10.20

Definition of Finite Elements

GENF

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.

Circular mesh Version 10.20

3−25

GENF

Definition of Finite Elements

$ CIRCLE CENTER NODE 100 −4.00 0.00 FIX F $ POLAR COORDINATES N1−N2 NODE (−1 −9 −1) 4.00*SQR(2) (−45 11.25) NREF 100 $ CORNER NODES NODE 51 5.00 −4.00 4.00 NODE 59 5.00 MESH 1 9 59 51 M 8 5 MNO 1 CHNG OFF

3−26

Version 10.20

Definition of Finite Elements See also: MESH, CUBE, NODE, GRP

3.12.

IMES − Generation of Irregular Nodes, Quadrilateral Elements

Item

Description

N1 N2 INC1 N4 INC2

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ IMES ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

Corner node 1 Corner node 2 Increment for edge N1−N2 Corner node K4 Increment for edge N1−N4

− − − − −

! ! 1 ! 1

MNO

Group and material number (see MESH)

−

*

CHNG

Change of previously defined nodes YES/NO

LIT

YES

T1 T2 T4

Thickness at the three corner nodes (only when QUAD elements are gener
ated)

m/* m/* m/*

* 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

3−27

GENF

Definition of Finite Elements

IMES−generation

3−28

Version 10.20

Definition of Finite Elements See also: MESH, IMES, NODE

3.13.

CUBE − Nodes and Cubic Elements

Item

Description

N1 N2 N3 N4 N5 N6 N7 N8 M

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ CUBE ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

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

−

1

−

M

−

M

MNO

Group and material number (see MESH)

−

*

CHNG

Change of previously defined nodes YES/NO

LIT

YES

N L

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

3−29

GENF

Definition of Finite Elements

CUBE−generation

3−30

Version 10.20

Definition of Finite Elements See also: MIRR, ALIN, SECT, NODE

3.14.

TRAN − Transformation of Nodes

Item

Description

FROM TO INC

First node Last node Increment The nodes from FROM to TO by in
crements of INC are transformed.

DX DY DZ

Translation in X−direction Translation in Y−direction Translation in Z−direction (omitted in 2−D−systems)

ALPH BETA THET

Coning angle Rotation angle Nutation angle

DNO

Inkrement der Knotennummer

CHNG

Change of previously defined nodes YES/NO

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ TRAN ÄÄÄÄÄÄÄÄ

Dimension

Default

− − −

1 FROM 1

m/* m/* m/*

0. 0. 0.

Degrees Degrees Degrees

0. 0. 0.

−

100

LIT

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:

Version 10.20

3−31

Definition of Finite Elements

GENF ALPH: 0 90 0

BETA: 0 −90 phi

THET: phi phi 0

: Rotation about the X−axis : Rotation about the Y−axis : Rotation about the Z−axis

Only the angle BETA and the displacements DX, DY are used in plane cases.

3−32

Version 10.20

Definition of Finite Elements See also: TRAN, ALIN, SECT, NODE

3.15.

MIRR − Mirroring of Nodes

Item

Description

FROM TO INC

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ MIRR ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

First node Last node Increment The nodes from FROM to TO by incre− ments of INC are mirrored.

− − −

1 FROM 1

A B C D

Constants defining the plane of mirror
ing by A⋅x + B⋅y + C⋅z + D = 0.

−

0.

SMO

Partition point of node number

−

2

VAL

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

LIT

SV

SNO

Number for mirroring or addition

−

100

CHNG

Change of previously defined nodes YES/NO

LIT

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 Version 10.20

3−33

Definition of Finite Elements

GENF

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: by mirroring of the secondary number to: by addition to primary number to: by addition to secondary number to: by interchange to:

9323 777 5723 773 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

3−34

B = 1.0 D = −yo

Version 10.20

Definition of Finite Elements

GENF

Mirror plane

Version 10.20

3−35

Definition of Finite Elements

GENF

See also: SECT, MIRR, TRAN, NODE

3.16.

ALIN − Node upon a Line (Projection to the Line)

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ ALIN ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Node number

−

!

NO1 NO2 F NO3

Node at the beginning of the line Node at the end of the line Distance of node from node 1 Node away from the line

− − − −

! ! − NO

REF

Reference system for F

LIT

SS

Using ALIN one can define a node on the straight line connecting two other already defined nodes. With the same record an already existing node can be projected onto that line.

ALIN−intermediate nodes Generally there are two possibilities for the generation of nodes on a line: 1. Specification of a distance on a a line from a point (item F) 2. Definition of the position with a auxiliary point (item NO3) In the first case the item F which describes the distance of the new node to the node NO1 has to be defined. The auxiliary point number is input with the

3−36

Version 10.20

Definition of Finite Elements

GENF

item NO3 in the second case. Thus either a numerical value for F or a node number for NO3 can be specified. The new point has to be defined with an input of S,SS for REF directly on the line or on the auxiliary line. The input of a literal consisting of two same al
phabetic characters for REF ( e.g. XX, YY, ZZ) describes the definition of a ref
erence axis. However, the input of a literal consisting of two different alpha
betic characters (e.g. XY, YZ, XZ) defines a reference plane. 1. Input of F REF

Meaning

S XX YY ZZ

F F F F

XY XZ YZ

F in m on the projection of the line onto global xy−plane F in m on the projection of the line onto global xz−plane F in m on the projection of the line onto global yz−plane

SS

dimensionless 0 − 1 The node lies at NO1 + F ⋅ [NO2 − NO1]

in in in in

m m m m

on on on on

the the the the

straight line projection of projection of projection of

NO1 the the the

− NO2 line onto global x−axis line onto global y−axis line onto global z−axis

In case of S, the distance along the true length of the straight line from NO to NO1 is input. In case of XX, YY or ZZ, only the components along the respective axes are input. F 2.0 REF YY means, for example, that the Y−coordinate of NO is larger than the one of NO1 by 2.0. The missing coordinates result from the condition that NO lies upon the connecting straight line. In case of XY, XZ or YZ, the two coordinates are used together. In case of XY, for example, the distance in top view is input. The ratio as well as the missing Z−coordinate are again deduced from the connecting line. Lastly, SS defines a dimensionless input. 0.5 e.g. stands for a point exactly at the middle between NO1 and NO2. 2. Input of NO3

Version 10.20

3−37

Definition of Finite Elements

GENF

Node NO3 together with REF defines an auxiliary line, the intersection of which with NO1 − NO2 defines node NO. REF

Auxiliary line

S,SS

Perpendicular from NO3 to the line

XX

Parallel to yz−plane, i.e. NO and NO3 have the same x− coordinate similarly for xz−plane with the same y−coordinate similarly for xy−plane with the same z coordinate

YY ZZ XY YZ XZ

Perpendicular from the projection of NO3 to the projection of the line onto the xy−plane defines the x− and y−coordi− nates for NO. similarly for yz−plane similarly for xz−plane

In case of S and SS the perpendicular is uniquely defined in space, so long as NO3 does not lie onto the line connecting NO1 with NO2. In case of REF XX, YY or ZZ, NO and NO3 have these coordinates in common, respectively. The missing coordinates result from the condition that NO lies on the connecting straight line. In case of XY, XZ or YZ, the point NO3 as well as the straight line NO1−NO2 are projected onto the respective plane. In this plane then is drawn the per
pendicular from the projection of NO3 onto the projected line. The point NO lies exactly at the footpoint of the perpendicular to the connecting line.

3−38

Version 10.20

Definition of Finite Elements See also: ALIN, MIRR, TRAN, NODE

3.17.

SECT − Node at Intersection of two Straight Lines

Item

Description

NO

Node number

NO1 NO2 NO3 NO4

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ SECT ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

−

!

− − − −

! ! ! !

Node NO lies at the intersection of lines NO1−NO2 and NO3−NO4.

SECT−intersection points

Version 10.20

3−39

Definition of Finite Elements 3.18.

GENF

Materials

SOFiSTiK supports a large number of different material descriptions. All will be addressed by a unique material number and should be usable everywhere in general. The default for the material type is dependant on the selected de
sign code. The basic properties are input via the records: NORM MAT MATE CONC STEE TIMB MASO MLAY

Selection of a design code family General Materialdefinition (obsolete) General Materialdefinition including strength Concrete Material Steel and other metallic materials Timber/lumber Masonry / Brickwork Layered composite material for QUAD−Elements

These records are mutually exclusive but may be enhanced by other records: BMAT

Elastic support

NMAT

Nonlinear material properties for MAT/MATE (to be used in ASE/TALPA for QUAD and BRIC el
ements)

SSLA

uniaxial strain−stress law for materials CONC/STEE/ TIMB/BRWO

MEXT

Special material properties

Input of material is possible in all parts of the program system. However it is self−evident that not all parameters are used for all types of analysis or sys
tem. Each material has a standard name given by its classification, which might be extended by the user. If the user wants to replace the standard com
pletely, he has to start his own text with an equal sign (e.g. ’=my own Text’). Properties of materials must be distinguished according to whether they are to be kept as close as possible to real values (e.g. for dynamic calculations) or to be used with a safety coefficient for calculating an ultimate load−bearing capacity. Whereas the safety factors were formerly assigned more−or−less at random, sometimes to the load and sometimes to the material, more recent regulations (Eurocode) provide a clearer separation between safety factors for the loads and factors for the material. However, since the material safety Version 10.20

3−41

GENF

Definition of Finite Elements

factors still depend on the nature of the load or the type of design, it will not be possible to define all safety factors with the material itself. SOFiSTiK distinguishes therefore: • Properties and safety factors for the standard Design • Mean values or calculatoric values and safety factors for serviceability and deformation analysis If some design codes (DIN 18800, DIN 1045−1) apply additional safety−fac
tors to the mean values, this may be defined with the stress−strain relation via SSLA. The safety factor defined with the material will thus be used only for the full plastic forces in AQUA.

3−42

Version 10.20

GENF

Definition of Finite Elements 3.19.

NORM − Default Design Code

Item

Description

DC

Design code family EC Eurocodes DIN Deutsche Normen OEN Österreichische Normen SIA Schweizer Normen BS British Standard US US Standards (ACI etc.) JS Japanese Standard GBJ Chinese Building Codes IS Indian Standards

NDC

Number of a specific design code

COUN

Countrycode for boxed values within EC 30 = Greece 31 = Netherlands 32 = Belgium 33 = France 34 = Spain 39 = Italy 41 = Switzerland 43 = Austria 44 = Great Britain 45 = Danmark 46 = Sweden 47 = Norway 49 = Germany 351 = Portugal 352 = Luxembourg 353 = Ireland 358 = Suomi/Finland

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ NORM ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

LIT

EC

Lit16

−

−

*

Some properties of Eurocode are dependant on national variants (boxed va
lues). The country code may be used to select those values.

Version 10.20

3−43

GENF

3.20.

Definition of Finite Elements MAT − General Material Properties

Item

Description

NO

Material number

E MUE G K GAM GAMA ALFA EY MXY OAL

ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ MAT ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ

Dimension

Default

−

1

Elastic modulus Poisson’s ratio (between 0 and 0.49) Shear modulus Bulk modulus Specific weight Specific weight under buoyancy Thermal expansion coefficient

kN/m2 − kN/m2 kN/m2 kN/m3 kN/m3 −

* 0.2 * * 25 * E−5

kN/m2 − deg

E MUE 0

deg

0

SPM

Anisotropic elastic modulus Ey Anisotropic poisson’s ratio m−xy Meridian angle of anisotropy about the local x−axis Descent angle of anisotropy about the local x−axis Material safety factor

−

1.0

TITL

Material name

Lit32

−

OAF

The record MAT can define general materials that can be used for sections or QUAD and BRIC−elements. The material number must be unique for every material. This record has been superseeded by MATE, supplying input of strength and elasticity constants in MPa. Further comments are available there.

3−44

Version 10.20

GENF

Definition of Finite Elements 3.21.

MATE − Material Properties

Item

Description

NO

Material number

E MUE G K GAM GAMA ALFA E90 M90 OAL

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ MATE ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

−

1

Elastic modulus Poisson’s ratio (between 0 and 0.49) Shear modulus Bulk modulus Specific weight Specific weight under buoyancy Thermal expansion doefficient

MPa − MPa MPa kN/m3 kN/m3 −

* * * * 25 * E−5

kN/m3 − deg

E MUE 0

deg

0

SPM FY FT

Anisotropic elastic modulus Anisotropic poisson’s ratio Meridian angle of anisotropy about the local x−axis Descent angle of anisotropy about the local x−axis Material safety factor Design strength of material ultimate strength of material

− MPa MPa

1.0 − −

TITL

Material name

Lit32

−

OAF

Sometimes it is more convenient to define the elastic constants by other va
lues than the Elasticity modulus and the Poisson ratio. You may transform your values by the following formulas: E Elastic modulus Es subgrade modulus (horizontally cosntrained) K Bulk modulus G Shear modulus µ Poisson’s ratio

K+

E 3(1 * 2m)

E + 9·K·G (3K ) G) Version 10.20

G+

E 2(1 ) m)

m + 3K * 2G 6K ) 2G 3−45

GENF

Definition of Finite Elements

Es + E @

ǒ1 * mǓ (1 ) m)(1 * 2m)

G + 3·K·E 9·K * E

G+

3·K·(1 * 2m) 2·(1 ) m)

If not specified, missing values will be calculated according to these formulas. It is however possibel to define non consistent constants. If no values are given, E will default to 30000 MPa and MUE to 0.2. Orthotropy may be defined via material and thickness of QUAD−Elements. (confer record GRP in GENF and remarks in manuals to ASE, SEPP and TALPA). The description of a transversal orthotropie according to Lechnitzky has one direction that has different properties, while the description in the plane per
pendicular to this remains isotropic. This covers most practical problems like timber and rock. If this special direction is z it holds:

s å x + sx * m· y * m90· s z E E E 90 åy +

sy * m· sx * m90· s z E E E 90

(s ) s y) å z + sz * m90· x E90 E 90 Please mark, that the poisson ratio M90 is no longer bound to 0.5 and is strongly connected to the Elasticity modulus. For beams the main value of the fibres is the x−axis, perpendicular values y and z will be E90 and M90. For planar systems (TALPA) the basic values are in the x−z plane. The value OAF is the angle between the local x−direction and the global x−direction. You have to exchange the indices y and z in the above formula. For shells and plates (ex. plywood) we assume that the fibres are in both x and y direction. The anisotropy effects reduce to different shear moduli for in plane shear and the transverse directions. For three dimensional continua, the orientation is given by the meridina and decent angle, known from geology. They describe the deviation of the constant

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GENF

height lines to the north direction and the inclination of the layers. They are equivalent to first and third of the Euler angles. The transformation is de
fined by two rotations to be selected by the gravity direction. North is the cyclic permutation of the gravity. (ie. x−axis for GDIR ZZ or NEGZ, y−axis for GDIR XX or NEGX and z−axis for GDIR YY or NEGY). First the north− axis will be rotated about the vertical axis by the amount of OAF, then the rotated x’−y’ plane will be rotated about the x’−axis by the amount of OAL against the vertical.

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3.22.

Definition of Finite Elements

MLAY − Layered Material

Item

Description

NO T0 NO0 T1 NO1 ... T9 NO9 TITL

Number of composite material Thickness of first layer Material number of first layer Thickness of second layer Material number of second layer Thickness of 9th layer Material number of 9th layer Material Designation

ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ MLAY ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ

Dimension

Default

− * − * −

1 ! ! ! !

* − Lit32

− − −

With MLAY you may define for QUAD elements a composite layered material of up to 10 layers. Each layer may be defined with an positive absolute thick
ness or a negative relative one. The total thickness of the element will be cali
brated to the sum of the thicknesses of the material definition. If some layers have negative thickness only these layers will be adopted. Otherwise a uni
form scaling will take place. If you have a sandwich element with two outer laminates with a given thick
ness: MLAY 1

0.02 1 $$ −1.00 2 $$ 0.02 1 $$

upper laminate interior laminate lower laminate

then this data will be applied to match two QUAD elements with a total thick
ness of 0.10 or 0.15 as follows: MLAY 1

MLAY 1

0.02 0.06 0.02 0.02 0.11 0.02

1 2 1 1 2 1

$$ $$ $$ $$ $$ $$

upper laminate interior laminate if 0.10 total thickness untere Deckschicht upper laminate interior laminate if 0.15 total thickness lower laminate

A standard material definition will also be generated with mean values.

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Definition of Finite Elements 3.23.

BMAT − Elastic Support / Interface

Item

Description

NO

Material number

C CT CRAC YIEL MUE COH DIL GAMB

Elastic constant normal to surface Elastic constant tangential to surface Maximum tensile stress of interface Maximum stress of interface Friction coefficient of interface Cohesion of interface Dilatancy coefficient Equivalent mass distribution

REF

Reference PESS Plain stress condition PAIN Plain strain condition HALF circular disk on halfspace CIRC circular hole in infinite disk SPHE sperical hole in infinite space Number of a reference material Reference dimension (thickness/radius)

MREF H

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ BMAT ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

−

1

kN/m3 kN/m3 kN/m2 kN/m2 − kN/m2 t/m2

0. 0. 0. − − − 0. 0

LIT

−

− m

NO !

−

The bedding approach works according to the subgrade modulus theory (Winkler, Zimmermann/Pasternak). It facilitates the definition of elastic sup
ports by an engineering trick which, among others, ignores the shear de
formations of the supporting medium. The bedding effect may be attached to beam or plate elements, but in general it will be used as an independant single or distributed element. (see SPRI, BOUN, BEAM or QUAD element and the more general description of BORE−Profiles) The determination of a reasonable value for the foundation modulus often presents considerable difficulty, since this value depends not only on the ma
terial parameters but also on the geometry and the loading. One must always keep this dependance in mind, when assessing the accuracy of the results of an analysis using this theory. The subgrade parameters C and CT will be used for bedding of QUAD−El
ements or or the description of support or interface conditions. A QUAD el
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Definition of Finite Elements

ement of a slab foundation will thus have a concrete material and via BMAT the soil properties attached to the same material number. If subgrade parameters are assigned to the material of a geometric edge (GLN), spring elements will be generated along that edge based on the width and the distance of the support nodes. Instead of a direct value you may select a material and a reference dimension for some cases with constant pressure [1]: • Flat layer with horizontal constraints e.g. for elastic support by col
umns and supporting walls (plane stress condition):

1 Cs + E @ H (1 ) m)(1 * m)

1 Ct + E @ H 2(1 ) m)

• Flat layer with horizontal constraints for sttlements of soil strata (plane strain condition):

(1 * m) Cs + E @ H (1 ) m)(1 * 2m)

Ct + E @ 1 H (1 ) m)

• Equivalent circular disk with radius R on unlimited halfspace:

2 Cs + E @ R p(1 ) m)(1 * m) • Circular hole in unlimited disk with plane strain conditions

1 Cs + E @ R (1 ) m)(1 * 2m)

C t + Cs

• Spherical hole in infinite 3D elsatic continua

Cs + E @ 2 R (1 ) m)

C t + Cs

Including a dilatancy factor describing the normal strain induced by shear de
formations, we have for the stresses:

s + C s @ (u n ) DIL @ ut) t + Ct @ ut Nonlinear effects are controlled by CRAC, YIEL, MUE and COH:

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Cracking:

Upon reaching the failure load the interface fails in both the axial and the lateral direction. The failure load is always a tensile stress.

Yield load:

Upon reaching the yield stress, the deformation com
ponent of the interface in its direction increases with
out a corresponding increase of the stress.

Friction coefficient:If a friction and/or a cohesion coefficient are input, the lateral shear can not sustain forces greater than: Friction coefficient * normal stress + Cohesion If the axial interface has failed (CRAC), the lateral shear acts only if 0.0 has been input for both friction− coefficient and cohesion. The nonlinear effects can be taken into account only by a nonlinear analysis. The friction is an effect of the lateral interface, while all other effects act upon the normal stress. [1]

Katz, C., Werner, H. (1982) Implementation of nonlinear boundary conditions in Finite Element Analysis Computers & Structures Vol. 15 No. 3 pp. 299−304

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3.24.

Definition of Finite Elements

ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ NMAT ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ

NMAT − Non−linear Material

Item

Description

NO TYPE

Material number Kind of material law LINE Linear material MISE Mise / Drucker Prager law MOHR Mohr Coulomb law GUDE Gudehus law ROCK Rock material FAUL Faults in rock material LADE Lade law DUNC Duncan−Chang law HYPO Schad law SWEL Swelling MEMB Textile membrane

P1 P2 P3 P4 ... P10

1st parameter of material law 2nd parameter of material law 3rd parameter of material law 4th parameter of material law ... th 10 parameter of material law

Dimension

Default

− LIT

1 !

* * * *

− − − −

*

−

The types of the implemented material laws and the meaning of their para
meters can be found in the following pages. In a linear analysis the yield function for the nonlinear material is merely evaluated and output. This enables an estimation of the nonlinear regions for a subsequent nonlinear analysis. If TYPE LINE is given, the material remains linear.

3.24.1. Invariants of the stress tensor For the present chapter, as long as not specified differently, the following con
ventions hold: I1
+
s x ) s y ) s z Deviatoric stress tensor:

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Definition of Finite Elements sx
+
s x *

I1 3

sy
+
s y *

I1 3

sz
+
s z *

I1 3

GENF

J2
+
1 (s x2 ) s y 2 ) s z 2) ) t xy 2 ) t yz 2 ) t xz 2 2 J3
+
s xsysz ) 2t xyt yzt xz * sxt yz 2 * syt xz 2 * s zt xy 2

ȱ 3 Ǹ3 J 3ȳ q
+
1 sin *1ȧ* 3 ȧ 3 2 2J Ȳ 2 ȴ

*p vq vp 6 6

;

3.24.2. Material Law MISE Elastoplastic material after MISE or DRUCKER−PRAGER with associated flow rule. f
+
p2 @ I1 ) ǸJ2 *

p1 v0 Ǹ3

Application range: Metals and other materials without friction Parameters: P1 P2 P3 P4 P5

= = = = =

Comparison stress Friction parameter Hardening module Tensile strength β−z Compressive strength (cap) β−c

[kN/m2] [−] [kN/m2] [kN/m2] [kN/m2]

Several substitutes for P1 and P2 can be used for the calculation of common parameters in soil mechanics. Commonly used e.g. is the compression cone:

P1 +

6c cos ö 3 * sin ö

P2 +

2 sin ö Ǹ3 (3 * sin ö)

The values for the internal cone are better suited for plane strain conditions:

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GENF P1 +

Definition of Finite Elements 6c cos ö 3 ) sin ö

P2 +

2 sin ö Ǹ3 (3 ) sin ö)

By specification of parameter P5 the model can optionally be extended by a spherical cap (in principal stress space) that limits the volumetric compres
sive stress to a maximum value. This can be meaningful in particular for mainly hydrostatic straining. The cap is defined by: f
+
Ǹs 1 2 ) s 2 2 ) s 3 2 * ǸP 5 ) P 5 ) P 5 v 0 2

2

2

Reference: M.A.Chrisfield Non−linear Finite Element Analysis of Solids and Structures. Vol. I. Essentials. Chapter 14. Wiley & Sons (1991) M.A.Chrisfield Non−linear Finite Element Analysis of Solids and Structures. Ad
vanced Topics. Vol. II. Chapter 6. Wiley & Sons (1997)

3.24.3. Material Law MOHR Elastoplastic material with a prismatic yield surface and a non associated flow rule after MOHR−COULOMB. The model is extended by means of a spherical compression cap and plane tension limits. Formulation of yield condition and plastic potential using stress invariants: f
+
1 I1 sin ö ) ǸJ2 (cosq * 3

sin q sin ö ) * c cos ö v 0 Ǹ3

g
+
1 I1 sin n ) ǸJ2 (cosq * sin q sin n) Ǹ3 3 with:

Application range: soils with friction and cohesion Parameters: P1 = P2 = P3 =

3−54

Friction angle Cohesion Tensile strength

Default values: ϕ [degrees] c [kN/m2] β−z [kN/m2]

(0.) (0.) (0.)

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Definition of Finite Elements P4 P5 P6 P7 P8

= = = = =

ν [degrees] β−c [kN/m2] εu [o/oo] ϕu [grad] cu [kN/m2]

Dilatation angle Compressive strength (cap) plastic ultimate strain ultimate friction angle ultimate cohesion

(0.) (−) (0.) (P1) (P2)

Special comments: The following expressions are better suited for checking the yield criterion:

f = σI  m ⋅ σIII  b ≤ 0 m+

1 * sin ö 1 ) sin ö

b+

2c cos ö 1 ) sin ö

By specification of parameter P5 the model can optionally be extended by a spherical cap (in principal stress space) that limits the volumetric compres
sive stress to a maximum value. This can be meaningful in particular for mainly hydrostatic straining. The cap is defined by: f
+
Ǹs 1 2 ) s 2 2 ) s 3 2 * ǸP 5 ) P 5 ) P 5 v 0 2

2

2

Reference: M.A.Chrisfield Non−linear Finite Element Analysis of Solids and Structures. Ad
vanced Topics. Vol. II. Chapter 14. Wiley & Sons (1997) O.C.Zienkiewicz,G.N.Pande Some Useful Forms of Isotropic Yield Surfaces for Soil and Rock Mechanics. Chapter 5 in Finite Elements in Geomechanics (G.Gudehus ed.) Wiley & Sons (1977)

3.24.4. Material Law GUDE Elastoplastic material in its extended form after Gudehus with non asso
ciated flow rule.

f = q2 − c7 p2 + c6 p − c5 < 0 g = q2 − c9 p2 + c8 p with:

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Definition of Finite Elements p = (σx + σy + σz)/3 γ = (3−sinϕ)/(3+sinϕ)

ȱ 3 Ǹ3 @ J3ȳ q
+
1 ȧǒ g ) 1 Ǔ @ ǸJ 2 * ǒ g * 1 Ǔ @ 2 @ J2 ȧ 2gȲ ȴ c5 = (12c2cos2ϕ)/A ; A = (3−sin ϕ)2 c6 = (24c cosϕ sinϕ)/A c7 = (12 sin2ϕ)/A c8 = (24c cosϕ sinν)/B ; B = (3−sin ϕ)(3−sinν) c9 = (12 sinνsinϕ)/B Application range: soils with friction and cohesion Parameters: P1 P2 P3 P4 P5 P6 P7 P8

= = = = = = = =

Default values: ϕ [degrees] c [kN/m2] β−z [kN/m2] ν [degrees] β−c [kN/m2] εu [o/oo] ϕu [grad] cu [kN/m2]

Friction angle Cohesion Tensile strength Dilatation angle Compressive strength (cap) plastic ultimate strain ultimate friction angle ultimate cohesion

(0.) (0.) (0.) (0.) (−) (0.) (P1) (P2)

Special comments: This law is capable of describing a multitude of plane or curved yield surfaces. For g=1 a circle in the deviatoric plane is obtained. The dilatation angle is usually set to zero or equal to the friction angle. By specification of parameter P5 the model can optionally be extended by a spherical cap (in principal stress space) that limits the volumetric compres
sive stress to a maximum value. This can be meaningful in particular for mainly hydrostatic straining. The cap is defined by: f
+
Ǹs 1 2 ) s 2 2 ) s 3 2 * ǸP 5 ) P 5 ) P 5 v 0 2

2

2

Reference:

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GENF

W.Wunderlich, H.Cramer, H.K.Kutter, W.Rahn Finite Element Modelle für die Beschreibung von Fels Mitteilung 81−10 des Instituts für konstr.Ingenieurbau der Ruhr Universität Bochum, 1981

3.24.5. Material Law ROCK Elastoplastic material with marked shear surfaces

f1 = tan (p1) ⋅ σ − p2 + τ < 0 g1 = tan (p4) ⋅ σ + τ f2 = σ − p3 < 0 g2 = f2

(Kluftfläche/Fault)

f3 = tan (p6) ⋅ σ − p7 + τ < 0 g3 = tan (p9) ⋅ σ + τ f4 = σI − p8 < 0 g4 = f4

(Felsmaterial/Rock)

Application range: Plane strain conditions and anisotropic material Parameters: P1 P2 P3 P4 P5

= = = = =

P6 P7 P8 P9

= = = =

Crevasse friction angle Crevasse cohesion Crevasse tensile strength Crevasse dilatation angle Angle of crevasse direction with respect to x−axis (0−180) Rock friction angle Rock cohesion Rock tensile strength Rock dilatation angle

Default values: ϕ [degrees] c [kN/m2] β−z [kN/m2] ν [degrees] [degrees]

(0.) (0.) (0.) (0.) (*)

ϕ [degrees] c [kN/m2] β−z [kN/m2] ν [degrees]

(0.) (0.) (0.) (0.)

Special comments: This law ignores the effect of the third principal stress acting perpendicularly to the model. One can, however, specify the strength of the rock as well as the

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Definition of Finite Elements

strength of the slide surfaces, which are defined by the angle P5 (default value is that of an anisotropic material). The flow rule of the shear failure is non associated if P4 is different from P1. Any of the two limits can be deactivated in special occasions by specifying ϕ = c = 0.0. Reference: W.Wunderlich,H.Cramer,H.K.Kutter,W.Rahn Finite Element Modelle für die Beschreibung von Fels Mitteilung Nr. 81−10 des Instituts für konstruktiven Ingenieurbau der Ruhr Universität Bochum, 1981.

3.24.6. Material Law FAUL Discrete faults in materials

f1 = tan ϕ ⋅ σ − c + τ < 0 g1 = tan ν ⋅ σ + τ f2 = σ − βz < 0 g2 = f2 Application range: Additional discrete faults to a given rock material Parameters: P1 P2 P3 P4 P5 P6 P7 P8 P9

= = = = = = = = =

Crevasse friction angle Crevasse cohesion Crevasse tensile strength Crevasse dilatation angle Meridian angle of crevasse plane Descent angle of crevasse plane plastic ultimate strain ultimate friction angle ultimate cohesion

Default values: ϕ [degrees] c [kN/m2] β−z [kN/m2] ν [degrees] [degrees] [degrees] εu [o/oo] ϕu [grad] cu [kN/m2]

(0.) (0.) (0.) (0.) (*) (*) (0.) (P1) (P2)

Special comments: This material law may be specified up to three times in addition to any other nonlinear material to allow for the description of multiple faults.

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3.24.7. Material Law LADE Elastoplastic material after LADE with non associated flow rule.

ȡ *ȧ27 ) p1 @ Ȣ

ǒ

ȡ 3 g
+
I1 *ȧ27 ) p4 @ Ȣ

ǒ

f
+
I1

3

Ǔ ȣȧ@ I v 0 m

p
a
I1

Ȥ

3

Ǔ ȣȧ@ I

p
a
I1

m

Ȥ

3

with pa = 103.32 kN/m2 = atmospheric air pressure I1
+
* ǒs 1 * P 3Ǔ * ǒs 2 * P 3Ǔ * ǒs 3 * P 3Ǔ I3
+
* ǒs 1 * P 3Ǔ @ ǒs 2 * P 3Ǔ @ ǒs 3 * P 3Ǔ Application range: all materials with friction including rock and concrete Parameters: P1 P2 P3 P4 P5 P6 P7 P8

= = = = = = = =

Parameter "η" Exponent "m" Uniaxial tensile strength Parameter "η" for flow rule compressive strength (cap) plastic ultimate strain ultimate Parameter "η" ultimate Exponent "m"

Default values:

[kN/m2] βc [kN/m2] εu [o/oo]

(−) (−) (0.) (−) (−) (0.) (P1) (P2)

Special comments: Material LADE has shown very good accordance between analytical and ex
perimental results. In practice therefore, the parameters can be taken from experiments on the material’s strength. The law at hand can also describe concrete or ceramics. A simple comparison with the material parameters of the Mohr−Coulomb law can be made only if the invariant I1 is known. Calibration of the LADE yield function might −due to the non−physical para
meters − not seem straight forward at first sight. For this reason, the basic procedure for a material with known uniaxial tensile and compressive

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Definition of Finite Elements

strength (e.g. concrete) is described in the following. For this purpose, the curve resulting from intersection of the spatial yield surface with one of the principal planes (−> Kupfer Curve") is of particular relevance.
Parameter P2 (exponent) effects the curvature (convexity) of the yield sur
face towards the hydrostatic axis − the larger P2 the stronger the curvature. In this manner P2 determines the shape of the intersection curve. For most types of concrete a value of P2 between 1.0 and 2.0 is reasonable.
Using the known quantities uniaxial tensile and compressive strength and the chosen parameter P2, P1 can be determined from the condition: For the stress state corresponding to the uniaxial compressive stress limit the yield condition must be fulfilled. We rewrite the yield function as

ȡI3 ȣ P 2
+
ȧ 1 * 27ȧ@ ȢI3 Ȥ

ǒǓ I1 pa

m

The considered stress state is defined by (translated system): s I
+ s II
+
* ft s III
+
* ǒft ) f cǓ ft
ǒ+ P 3Ǔ and fc are the magnitudes of the uniaxial tensile and compressive strength, respectively. Computing the invariants I1 and I3 for this stress state according to the formulae above and inserting into the rewritten yield function yields the yet unknown parameter P2. The following table contains exemplary parameters for selected concrete types, derived from the procedure described above (classification according to EC2, Ultimate Limit State). Strength class C20/25 C30/37

3−60

fcd [kN/m2] 13333 20000

P3 (fctk;0.05) [kN/m2] 1500 2000

P2 [−] 1.0

P1 [−] 24669.11

1.5

324095.87

1.0

43466.02

1.5

689515.99

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Definition of Finite Elements C40/50 C50/60

26667

2500

33333

2900

1.0

63426.77

1.5

1153410.57

1.0

88162.15

1.5

1778218.62

By specification of parameter P5 the model can optionally be extended by a spherical cap (in principal stress space) that limits the volumetric compres
sive stress to a maximum value. This can be meaningful in particular for mainly hydrostatic straining. The cap is defined by: f
+
Ǹs 1 2 ) s 2 2 ) s 3 2 * ǸP 5 ) P 5 ) P 5 v 0 2

2

2

Reference: P.V.Lade Failure Criterion for Frictional Materials in Mechanics of Engineering Materials, Chap 20 (C.s.Desai,R.H.Gallagher ed.) Wiley & Sons (1984)

3.24.8. Material Law DUNC Hypoelastic material based on Duncan−Chang. Loading: 2

p6

ȱ ȱmaxǒǒp3 * s IǓ, 0Ǔȳ p7 @ ǒ1 * sinǒp1ǓǓ @ ǒs I * s IIIǓ ȳ Et
+
ȧ1 * ȧ @
p 4 @ȧ ȧ pa ǒ Ǔ ǒ Ǔ 2 @ p @ cos p * 2 @ s @ sin p I 2 1 1ȴ Ȳ Ȳ ȴ Unloading and reloading: p6

ȱmaxǒǒp3 * s IǓ, 0Ǔȳ Et
+
p 5 @ȧ ȧ pa Ȳ ȴ

pa = 103.32 kN/m2 = atmospheric air pressure Application range: Deformation analyses with little plastification and with stress paths not very different from a triaxial test.

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Definition of Finite Elements

Parameters: P1 P2 P3 P4

= = = =

P5 = P6 = P7 =

Friction angle Cohesion Tensile strength Reference elastic modulus during loading Reference elastic modulus during unloading Exponent (w 0) Calibration factor (w 0)

Default values: ϕ [degrees] c [kN/m2] β−z [kN/m2] [kN/m2]

(0.) (0.) (0.) (−)

[kN/m2]

(−)

[−] [−]

(−) (−)

Special comments: The model distinguishes between primary loading, unloading and reloading − different moduli for loading and un−/reloading can be specified. Loading is defined as an increase of the stress level S:

ǒ1 * sinǒp1ǓǓ @ ǒs I * s IIIǓ ȧ ȧ ȧ S
+
ȧ
ȧ 2 @ p @ cosǒp Ǔ * 2 @ s @ sinǒp Ǔ
ȧ ȧ I 2 1 1 ȧ The initial state should be calculated linearly − doing so, parameters defining the loading history are initialized and the resulting stress state is interpreted as loading". In case of unloading, after having passed a deviatoric stress minimum, a pri
mary loading branch is traced again −> simulation of cyclic loading behavior is possible. The original law according to DUNCAN/CHANG has been modified in order to allow for a better simulation of the plastic flow in soil materials. Poisson’s ratio is not kept constant but is defined as a function of the tangential modu
lus of elasticity and the bulk modulus. The bulk modulus is kept constant in this case. By P6=P7=0 one can define a law with each a constant elastic modulus for loading and unloading. In order to avoid numerical difficulties, the elastic modulus in the MAT re
cord should not be chosen smaller than the initial elastic modulus. Anisotropic materials are not possible with this model.

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Reference: J.M.Duncan, C.Y.Chang Nonlinear Analysis of Stress and Strains in Soils J.Soil.Mech.Found.Div. ASCE Vol 96 SM 5 (1970) ,1629−1653 C.S.Desai, J.T.Christian Numerical Methods in Geotechnical Engineering, 81−88 McGraw−Hill Book Company

3.24.9. Material Law HYPO Hypoelastic material after Schad. Bulk and shear moduli during loading:

K = p1 − p7 ⋅ p − p8 ⋅ qmax G = p2 − p5 ⋅ (σI+σIII) − p6 ⋅ q Bulk and shear moduli during unloading:

K = p3 G = p4 where:

p = (σx + σy + σz)/3 q = σI − σIII Application range: isotropic materials Parameters: P1 P2 P3 P4 P5 P6 P7 P8 P9

= = = = = = = = =

Initial bulk modulus Initial shear modulus Bulk modulus for unloading Shear modulus for unloading Parameter Parameter Parameter Parameter Tensile strength

Version 10.20

Default values: [kN/m2] [kN/m2] [kN/m2] [kN/m2] [−] [−] [−] [−] [kN/m2]

(−) (−) (−) (−) (−) (−) (−) (−) (0)

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Definition of Finite Elements

Special comments: This law must have a vanishing shear modulus at failure by Mohr−Coulomb, thus the following expressions are obtained:

p2 = p6 ⋅ 2 ⋅ c ⋅ cos ϕ p5 = p6 ⋅ sin ϕ Anisotropic Material constants are not possible with this model. Reference: H.Schad Nichtlineare Stoffgleichungen für Böden und ihre Verwendung bei der numerischen Analyse von Grundbauaufgaben. Mitteilungen Heft 10 des Baugrund−Instituts Stuttgart (1979)

3.24.10. Material law SWEL Additional Parameters for swelling of materials Application range: Selling of soils in case of unloading Relationship between stress and final state swelling strains:

ȡ 0






s i t s 0i ȧ logǒ s i Ǔ



s v s v * p q i 0i 2 s 0i åiR
+
* p1 @ ȥ
ȧ logǒ s c Ǔ


* p2 t s i s 0i Ȣ i
+
1..3

si = s 0i =

principal normal stresses equilibrium state of stress wrt swelling (initial condition), transformed to the direction of principal normal stresses s i

Parameters: P1 = P2 =

3−64

Default values:

modulus of swelling [−] (0.0033) magnitude of the smallest compressive stress below which no more increase of swelling occurs [kN/m2] (10.0)

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GENF

Special comments: Swelling of soils is a complex phemomenon that is influenced by various fac
tors. There are two swelling mechanisms of practical importance that can be distingushed − for which the presence of (pore−) water is a common prerequi
site. The first mechanism is termed as the osmotic swelling" of clay minerals, which basically is initiated by unloading of clayey sedimentary rock. The sec
ond mechanism takes place in sulfate−laden rock with anhydrite content. In this case the swelling effects are due to the chemical transformation of anhy
drite to gypsum− which goes along with a large increase in volume (61%). For both described mechanisms a principal dependency between the swell
ing−caused increase in volume and the state of stress was observed both in laboratory and in in−situ experiments. The formula employed here, repre
sents a generalization of the 1−dimensional stress−strain relationship, which HUDER and AMBERG derived from oedometer tests. The equilibrium state wrt swelling s 0 is defined by means of the GRP−record. Doing so, we use the option PLQ in order to reference a (previously calculated) load case as primary state for swelling". This state is in equilibrium wrt swelling (normally in−situ soil prior to construction work). In the course of construction work occuring unloading related to this primary state causes swelling strains according to the formula above. The SWEL record is specified in addition to a linear elastic or elastoplastic basic material. Anisotropy is not possible with this model. Reference: P.Wittke−Gattermann Verfahren zur Berechnung von Tunnels in quellfähigem Gebirge und Kalibrierung an einem Versuchsbauwerk. Dissertation RWTH−Aachen, Verlag Glückauf 1998 W.Wittke Grundlagen für die Bemessung und Ausführung von Tunnels in quellendem Gebirge und ihre Anwendung beim Bau der Wendeschleife der S−Bahn Stuttgart. Veröffentlichungen des Institutes für Grundbau, Bodenmechanik, Felsmechanik und Verkehrswasserbau der RWTH−Aachen 1978

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Definition of Finite Elements

W.Wittke, P.Rissler Bemessung der Auskleidung von Hohlräumen in quellendem Gebirge nach der Finite Element Methode. Veröffentlichungen des Institutes für Grundbau, Bodenmechanik, Felsmechanik und Verkehrswasserbau der RWTH−Aachen 1976, Heft 2, 7−46 Nichtlineare Stoffgleichungen für Böden und ihre Verwendung bei der numerischen Analyse von Grundbauaufgaben. Mitteilungen Heft 10 des Baugrund−Instituts Stuttgart (1979)

3.24.11. Material law MEMB Parameters for textile membranes P1

Factor for Stress change (only in special cases, cnf. ASE GRP FACS)

P2

Factor for compression stiffness 0.0 no compressive stress possible 1.0 full compressive stress possible 0.1 intermediate values for scaling the elasticity modulus

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Definition of Finite Elements 3.25.

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ MEXT ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

MEXT − Extra Materialconstants

Item

Description

NO TYPE VAL VAL1 VAL2 VAL3 VAL4 VAL5

Number of material Type of constant Value of material constant First additional material value Second additional material value Third additional material value 4th additional material value 5th additional material value

Dimension

Default

− LIT * * * * * *

1 ! − − − − − −

With MEXT you may define special material values for any type of material. The definition of TYPE selects one of the following possibilities: With KR VAL defines the equivalent roughness according to Table 10.8.1 of EC 1 part 2−4, needed especially for wind loads on circular sections:

Surface glas

Roughness k [mm]

0.0015

Surface

Roughness k [mm]

galvanised steel

0.2

polished metall

0.002

spinning concrete

0.2

smooth painting

0.006

cast in situ con
crete

1.0

spray painting

0.02

rust

2.0

blasted steel

0.05

masonry

3.0

cast iron

0.2

Hint: In table 4 of DIN 1055 part 4 slightly larger values are defined .

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3.26.

Definition of Finite Elements

CONC − Properties of Concrete

Item

Description

NO TYPE

ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ CONC ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ

Dimension

Default

Material number (1−999) Type of concrete: C regular concrete LC light−weight concrete B,LB,SB concrete DIN 1045/4227 SIA,LSIA concrete SIA 162 BS concrete (BS 8110) ACI American Standard CBC Chinese Building Codes IS,IRC Indian Standards CE with constant E modulus

− LIT

1 *

FCN

Strength class fck/fcwk (nominal strength)

N/mm2

*

FC FCT FCTK EC QC GAM ALFA SCM TYPR

Design value of concrete strength Tensile strength of concrete Lower fractile strength value Elastic modulus Poisson’s ratio or shear modulus Unit weight Thermal expansion coefficient Typical material safety factor Type of service state line LINE = constant elastic modulus A,B = shorttime lines (Eurocode2) R = calc. mean values (DIN) RS = R with k=1.3 (SLWAC) Strength for nonlinear analysis Energy at break for compressive failure Energy at break for tensile failure Friction in cracks Material name

N/mm2 N/mm2 N/mm2 N/mm2 * kN/m3 − − LIT

* * * * 0.2 25 1E−5 * *

FCR GC GF MUEC TITL

N/mm2 N/mm N/mm N/mm Lit32

* * * * * *

3.26.1. Eurocode 2 According to Eurocode 2 the following types are available:

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GENF

= regular concrete = light−weight concrete

The cylindrical strength is to be input for FCN. The default value is 20. Some properties are dependant on national variants. The definition of NORM COUN is used to switch between those values. As EC2 and DIN 1045−1 differ considerably, you should use NORM to select the proper design code family, but you may also append the characters :DIN" or .EC" to the given class to set the given code explicitly for that material. The default values for design strength and elastic are derived as follows:

FC

= 0.85 ⋅ fck

FCT = 0.3 ⋅ fck 2/3

(fck < 55)

FCT = 2.12 ln((fck+8)/10+1)

(fck > 50)

EC

= 9500 ( fck + 8 ) 1/3

By light−weight concrete (LC) according to EC2−4, value EC must be defined explicitly or by means of GAM. The raw unit weight class can be input for GAM too, GAM and EC will then be defined appropriately. For the raw weight ρ in kg/m3 we have ρ = (γ−1.5)⋅100

EC = 9500 ( fck + 8 ) 1/3 ⋅ ( ρ/2200 ) 2 For detailed analysis of concrete according to appendix 1 you need the kind of cement. You may specify this by appending a Literal to the class of concrete N,R S RS

normal or rapid hardening cement (α = 0.0) slowly hardening cement (α = −1.0) with high strength cement (α = +1.0)

The usual stress−strain curve of the C types is the parabolic− rectangular stress−strain diagram of Eurocode 2 / DIN 1045 / OeNORM B 4200 / SIA 162. For nonlinear analysis or deformation analysis, there are other types A/B/R following the expression:

s + k·n * n 2 fc 1 ) (k * 2)·n with

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Definition of Finite Elements

n = ε / εc1 k = (1.1⋅EC) ⋅ εc1/fc For fc we have for the curves A and B the value fck+8, for R or RS the value 0.85αfck according to DIN 1045−1. The maximum strain is limited according to the strength. The B line does not possess a falling branch, and it is thus eventually more stable numerically. The safety factors SCM are preset to 1.5. In AQB, however, they must be se
lected explicitly, because they are dependent on the loading combinations. For concrete with high strength the factor will be increased by γ’, which will also be incorporated in the strain−stress laws, to allow a global safety factor to be used for the design. For nonlinear analysis with a constant safety factor according to DIN 1045−1 the strength of the concrete will be reduced, while those of the steel will be raised. This servicability work law is selected with the literal R at position TYPR. As DIN 1045−1 distinguishes between normal and light sand, the latter may be adressed with the literal RS.

3.26.2. DIN 1045 old / DIN 4227 / DIN 18806: The new DIN 1045 will be addressed by the national variant of Eurocode EC 2. The old DIN can be addressed with the old literals. B LB SB

= regular concrete (DIN) = light−weight concrete (DIN) = pre−stressed concrete (DIN)

The default FCN is 25 for B and LB, and 45 for SB. FCT is defined by:

FCT = 0.25 ⋅ FCN 2/3 Defaults in accordance with old DIN 1045 / DIN 4227: FCN

10

15

25

35

45

55

FC DIN 1045 (B) DIN 4227 (SB)

7 –

10.5 –

17.5 15.0

23 21

27 27

30 33

22000

26000

30000

34000

37000

39000

EC

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Definition of Finite Elements as well as the following high−strength concretes: FCN

65

75

85

95

105

115

FC EC

40.0 40500

45.0 42000

50.0 43000

55.0 44000

60.0 44500

64.0 45000

The elastic modulus or the weight has to be specified in case of light−weight concrete. However, the raw unit weight class according to DIN 4219 (1.0 − 2.0) may be input for item GAM. The default for GAM and EC then complies with DIN 1055. A bilinear stress−strain curve is usually employed for light− weight concrete. For standard concrete a parabola−rectangular diagram will be selected ac
cording to Eurocode EC2 / DIN 1045 / ÖNORM B4700 / SIA 162. SCM will de
fault to 1.00. If you analyse composite sections you might want to change the value. High strength concrete will have lesser ultimate strains.

3.26.3. ÖNORM B 4700 / B 4750 Although the OENORM B 4700 calls itself close to Eurocode, it deviates just with the classification of concrete based on the cubic stregth instead of the cy
lindrical strength. As the designation is C resp. LC the user has to select the option NORM OEN or append those literals to the class value. C LC

= regular concrete (ÖNORM 4700) = light−weight concrete (ÖNORM 4700)

The default FCN is 25. Defaults in accordance with OeNORM B 4700: FCN

20.0

25.0

30.0

40.0

50.0

60.0

FC FCT EC

15.0 1.9 27500

18.8 2.2 29000

22.5 2.6 30500

30.0 3.0 32500

27.5 3.5 35000

45.0 4.1 37000

SCM is preset to 1.5, FCTK to 0.7⋅FCT. The standard choice for regular concrete is the parabolic−rectangular stress− strain diagram in accordance with Eurocode 2 / DIN 1045 / OeNORM B 4700 / SIA 162. The value of SCM is preset to 1.5.

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Definition of Finite Elements

3.26.4. Swiss Standard SIA 162 (1989) As type we have SIA LSIA

= regular concrete (SIA 162) = light−weight concrete (SIA 162)

The nominal strength FCN is the mean cubical strength. The first value of the concrete class must thus be used (e.g. B 35/25 should be input as SIA 35). The elastic moduli are the mean values from Figure 31 in Section 5.18 der SIA. Half of the EC values are assigned to light−weight concrete. FCN FCN–min FC FCT EC

20.0

25.0

30.0

35.0

40.0

45.0

10.0 6.5 2.0 29000

15.0 10.0 2.0 31000

20.0 13.0 2.0 33500

25.0 16.0 2.5 35000

30.0 19.5 2.5 36000

30.0 23.0 2.5 37000

The default stress−strain diagram is the parabolic−rectangular one in ac
cordance with Eurocode 2 / DIN 1045 / OeNORM B 4200 / SIA 162. SCM will be preset with 1.2.

3.26.5. British Standard BS 8110 As type we have: BS

= normal weight concrete BS 8110

The nominal strength FCN is the cubical strength. The design strength is ob
tained by

FC = 0.67 FCN British Standards employ a parabolic rectangle curve, starting from a design cube strength β = FC/0.67 with 0.24 √β strain at full plasticity and an initial stiffness of 5.5 vβ according to Figure 2.1. The safety factor SCM is preset to 1.5.

3.26.6. American concrete institute ACI 318M−99 As type we have the specified compressive strength in MPa: ACI

= normal weight concrete ACI 318M

As the value of ǸfcȀ should not exceed the value of 25/3 MPa in general and different reductions have to be applied for lightweight concrete, we use the

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tensile stress to define the value of ǸfcȀ . The modulus of rupture fr is the upper fractile value of the tension strength. ACI 9.5.2.3 defines: fr + 0.75 * ǸfcȀ t 0.75 * 25ń3 or for lightweight concrete: fr + 0.70 * min(ǸfcȀ , 1.8 * f ct*m)

fr + 0.70 * 0.75 * Ǹf cȀ

The ratio of the fractiles is thus 1.26. The mean value fct−m will be preset to 0.5 * Ǹf cȀ. All other values will be derived from this value by a factor. If needed the lower fractile may be given, which will then set the upper value. But this value is only used for those cases where explicitly the value fr is used within a formula.

3.26.7. Chinese Standards As type we have: SGBJ

= Standard TB 10002.3−99 (Railway Bridge)

The nominal strength FCN (15 to 60) and the the design strength are taken from table 3.1.3. Youngs modulus is derived from 3.1.4.

3.26.8. Indian Standards IS 456 / IRC 21 As type we have: IS IRC

= Indian Standards IS 456 (10 bis 80) = Indian Roads Congress IRC 21 (15 bis 60)

The nominal strength FCN is the cubical strength. The design strength is ob
tained by

FC = 0.67 FCN Indish Standards employ either allowed stresses (IRC resp. Annex B of IS 456) or a parabolic rectangular curve with 2 and 2.5 o/oo strain. The allowed stresses will be converted to a serviceability stress strain law. The elasticity modulus is preset according to IS to 5000 * Ǹf ck, for IRC according to table 9.

The tensile strength is preset to 0.7 * Ǹf ck. The safety factor SCM is preset to 1.5.

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Definition of Finite Elements

3.26.9. Linear Elastic Concete A linear elastic material without tensile stresses is specified for CE. This can be used for servicability analysis, older design codes or stresses of founda
tions.

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Definition of Finite Elements 3.27.

STEE − Properties of Metals

Item

Description

NR TYPE

Material number (1−999) Type of the material S / PS reinforc./prestress. steel BST/PST Reinf./prestr. steel DIN FE / S / ST Structural steel EC/DIN GU Cast iron AL,AC,AW Aluminium alloy more types see comments Steel class or quality

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ STEE ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

− LIT

1 *

*

*

N/mm2 N/mm2 N/mm2 N/mm2 * kN/m3 −

SCM

Yield strength (f0.02) Tensile strength Elastic limit (f0.01) Elastic modulus Poisson’s ratio or shear modulus Unit weight Thermal expansion coefficient Default for AL: Typical material safety factor

* * * * 0.3 * 1.2E−5 2.38E−5 *

EPSY EPST REL1 REL2 R K1 FDYN TITL

Permanent strain at yield strength Ultimate strain Coefficient of relaxation (0.70 βΖ) Coefficient of relaxation (0.55 βΖ) Bond coefficient by DIN 4227 Table 8.1 Bond coefficient per EC 2 / Vol. 400 Allowed stress range Material name

o/oo o/oo % % − − N/mm2 Lit32

* * * 0 150/200 2.0/0.8 * *

CLAS FY FT FP ES QS GAM ALFA

The steel types S (partial), FE, ST, GU, BS, A and AL, AC, AW can be used for cross−sections. All other designations can be used only as reinforcement and prestressing tendons. The safety factors are considered by AQB first, be
cause they depend on the loading combination.

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Definition of Finite Elements

Defaults for structural steel: FY

FT

EPST

FP

EPSY

ES

Eurocode: * FE 360 FE 430 FE 510 FE 275 FE 355

235 275 355 275 355

360 430 510 430 510

– – – – –

– – – – –

– – – – –

210000 210000 210000 210000 210000

78.5 78.5 78.5 78.5 78.5

DIN: ST ST ST S S S

33 37 52 235 275 355

190 240 360 240 275 360

330 370 520 360 430 510

– – – – – –

– – – – – –

– – – – – –

210000 210000 210000 210000 210000 210000

78.5 78.5 78.5 78.5 78.5 78.5

* GU GU GU GU GU GU

52 17 20 200 240 400

260 240 300 200 240 250

520 370 500 380 450 390

– – – – – –

– – – – – –

– – – – – –

100000 210000 210000 210000 210000 169000

72.5 72.5 72.5 72.5 72.5 72.5

OENORM: ST 44 ST 55

285 355

430 540

– –

230 285

–.2 –.2

206000 206000

78.5 78.5

British Standard: BS 43 BS 50 * BS 55

275 180 140

430 500 550

– –

– –

– – –

205000 205000 205000

78.5 78.5 78.5

AISC; A 42 A 50 A 588

290 345 345

414 448 483

– – –

– – –

– – –

200000 200000 200000

78.5 78.5 78.5

IS/IRC IS 250 IRC ???

250 ???

250 – ???? –

– –

– –

211000 211000

77.0 77.0

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Definition of Finite Elements Defaults for aluminium alloy: FY

FT EPST

FP EPSY

ES

GAM REL1

Eurocode: AC 42100 AC 42200 AC 43200 AC 44100 AC 51300

190. 210. 80. 70. 90.

230 250 160 150 160

20. 10. 10. 40. 30

– – – – –

– – – – –

70000 70000 70000 70000 70000

28.0 28.0 28.0 28.0 28.0

AW AW AW AW AW AW AW AW AW AW AW

3103 5083 5052 5454 5754 6060 6061 6063 6005 6082 7020

120. 110. 160. 85. 80. 120. 240. 110. 200. 260. 280.

140 270 210 215 190 160 260 160 150 130 350

20 120 40 20 20 80 80 70 80 100 100

– – – – – – – – – – –

– – – – – – – – – – –

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0

18 20 22 25 27 28 31 35

80 100 160 180 140 200 200 290

180 200 215 250 270 275 310 350

– –

– –

– – – – –

– – – – –

– – – – – – – –

70000 70000 70000 70000 70000 70000 70000 70000

28.0 28.0 28.0 28.0 28.0 28.0 28.0 28.0

DIN: AL AL AL AL AL AL AL AL

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Definition of Finite Elements

Defaults for reinforcing and prestressing steel FY

FT EPST

FP EPSY

ES

GAM REL1

Eurocode: S 220 S 450 S 500 PS 800 PS 1050 PS 1350 PS 1500 PS 1600

220 450 500 800 1050 1350 1500 1600

220 486 540 1000 1250 1650 1800 1900

10 10 10 7 6 6 6 6

– – – 900 1125 1485 1620 1710

50. 50. 50. 3.5 3.5 3.5 3.5 3.5

200000 200000 200000 200000 200000 200000 200000 195000

78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5

2.5 2.5 8.0 8.0 8.0

DIN: BST BST BST PST PST PST PST PST PST

220 420 500 835 1080 1375 1420 1470 1570

220 420 500 835 1080 1375 1420 1470 1570

340 500 550 1030 1230 1570 1570 1670 1770

– – – 7 6 6 6 6 6

– – – 735 950 1150 1220 1250 1300

–.2 –.2 –.2 –.2 –.2 –.2 –.2 –.2 –.2

210000 210000 210000 205000 205000 205000 205000 205000 195000

78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5

3.3 3.3 7.5 2.0 7.5 7.5

OENORM: BSOE 240 BSOE 420 BSOE 500 BSOE 550 BSOE 600 PSOE 835 PSOE 1080 PSOE 1375 PSOE 1420 PSOE 1470 PSOE 1570

240 420 500 550 600 835 1080 1375 1420 1470 1570

360 500 550 620 670 1030 1230 1570 1570 1670 1770

17 10 10 10 10 7 6 6 6 6 6

– – – – – – – – – – –

.4 .4 .4 .4 .4 –.2 –.2 –.2 –.2 –.2 –.2

210000 210000 210000 210000 210000 205000 205000 205000 205000 205000 195000

78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5

3.3 3.3 7.5 2.0 7.5 7.5

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Definition of Finite Elements FY SIA: BSIA BSIA BSIA PSIA PSIA PSIA PSIA PSIA PSIA PSIA

FT EPST

235 500 550 830 1000 1410 1500 1590 1640 1670

220 460 520 830 1080 1410 1500 1590 1640 1670

360 550 580 1030 1230 1570 1670 1770 1820 1860

British Standard: SBS 250 SBS 460 SBS 500 PSBS 800 PSBS 1050 PSBS 1350 PSBS 1500 PSBS 1600

250 460 500 800 1050 1350 1500 1600

250 460 500 1000 1250 1650 1800 1900

ACI/ AASHTO: SACI 40 SACI 50 SACI 60 SACI 70 SACI 75

280 350 420 490 520

490 560 630 560 600

Version 10.20

FP EPSY

ES

GAM REL1

25 14 8 6.5 6.5 6 5 5 5 5

– – – – – – – – – –

–.2 –.2 –.2 –.2 –.2 –.2 –.2 –.2 –.2 –.2

210000 210000 210000 210000 210000 210000 210000 195000 195000 195000

78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5

3.3 3.3 1.8 7.5 7.5 7.5 7.5

7 6 6 6 6

– – – 800 1000 1320 1440 1520

– – – –.5 –.5 –.5 –.5 –.5

200000 200000 200000 200000 200000 200000 200000 195000

78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5

2.5 2.5 8.0 8.0 8.0

– – – – –

– – – – –

200000 200000 200000 200000 200000

78.5 78.5 78.5 78.5 78.5

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Definition of Finite Elements FY

GBJ: SCS SCS SCS PSCS

FT EPST

FP EPSY

ES

GAM REL1

I II IV IV

235 335 380 751

235 335 835 835

– – – –

– – – –

210000 210000 190000 190000

78.5 78.5 78.5 78.5

8.0 2.5 5.0 5.0

IS/IRC: SIS 240 SIS 415 SIS 500 PSIS 800 PSIS 1050 PSIS 1350 PSIS 1500 PSIS 1600

240 415 500 800 1050 1350 1500 1600

240 415 500 1000 1250 1650 1800 1900

– – – 800 1000 1320 1440 1520

– – – –.5 –.5 –.5 –.5 –.5

200000 200000 200000 200000 200000 200000 200000 195000

78.5 78.5 78.5 78.5 78.5 78.5 78.5 78.5

2.5 2.5 8.0 8.0 8.0

7 6 6 6 6

For the type BST you may attach to the class two extra characters switching to new DIN 1045−1: SA SB MA MB

Reinforcing bars with standard ductility Reinforcing bars with high ductility Reinforcing meshes with standard ductility Reinforcing meshes with high ductility

For nonlinear analysis with a constant safety factor according to DIN 1045−1 the strength of the concrete will be reduced, while those of the steel will be raised. If you attach an additional R to the steel type this special serviceabi
lity work law is selected and the corresponding safety factor is preset to 1.3. For prestressing steel we proceed in a similar way. As the old design code 4227 does not allow hardening, and the strength itself is not a sure identifier, the user has to append to the class specifier the letters M (mean values) or R (cal
culatoric values) for all steels according to new DIN 1045−1 to get a trilinear strain−stress law. The Stress−Strain−Law may have up to 4 segments: • Up to the proportional limit (FP/ES,FP) • Up to the yield limit (EPSY,FY) EPSY may be defined absolute (positiv) or relative to the strain limit (negativ)

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• Up to the tensile strength (EPST,FT) • constant to the infinite (1000 o/oo) Depending on the steel class the values EPSY and EPST and FP will be preset. With explicit definitions you may suppress: • If FP is not lesser than FY the first part will be omitted. • If EPST is not greater than EPSY the third part will be omitted. More general work laws are specified via record SSLA. In general the stress−strain laws are identical for servicability and ultimate limit design. However for EC2 and DIN 1045−1 there are numerous explicit changes. To select a prestressing steel according to DIN 1045−1, you have to select the literal PS and append the characters DIN to the class value. As the safety−factor concept will divide all stresses with the same safety fac
tor, the ultimate limit stress−strain−law will have an augmented initial modulus if only the strength should be reduced. Although the tensile strength for reinforcement steel with standard ductility will be reached at 25 o/oo, it is not allowed to use this in the design according to DIN 1045−1. The stress strain laws for design and nonlinear analysis differ therefore for those materials.

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Definition of Finite Elements

The bond coefficients R of reinforcements and prestressing steel are input as reference values for environment condition 1. The values for other environ
ment conditions are controlled through factors during design in AQB. The de
faults are:

Reinforcing steel Prestressing stee

R

K1

200 150

0.8 2.0

The safety factor SCM is preset to 1.1 for steel materials. For reinforcing and prestressing steel it is preset to 1.15 resp. 1.05. The safety factor becomes ef
fective for the calculation of the full plastic internal forces of steel and com
posite sections.

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Definition of Finite Elements 3.28.

TIMB − Properties of Timber

Item

Description

NO TYPE

Material number (1−999) Type of material see following table Quality class (1−3 for NA, 1−2 for BS) Strength by C and GL

CLAS EP G E90 QH QH90 GAM ALFA SCM FM FT0 FT90 FC0 FC90 FV FVR OAL OAF

Elastic modulus parallel to fibre Shear modulus Elastic modulus normal to fibre Poisson’s ratio (polywood panels) Poisson’s ratio yz (solid wood) Unit weight Temperature elongation coefficient Material safety factor Bending strength Tensile strength parallel to the fibre Tensile strength normal to the fibre Compressive strength parallel to fibres Compressive strength normal to fibres Shear strength at center (shear force) Shear strength at the edge (torsion) Meridian angle of anisotropy Descent angle of anisotropy

TITL

Material designation

Version 10.20

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ TIMB ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

− LIT

1 NA

−

2

N/mm2 N/mm2 N/mm2 − − kN/m3  − N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 degree degree

* * * * * * 0.0 1.3 * * * * * * * 0.0 0.0

Lit32

*

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Definition of Finite Elements

Types and defaults: TYPE (CLAS)

EP

C GL NA

see EC5 see EC5 10000

* (27) * (27) 1 2 3 BS 1 2 11 14 16 18 S 7 10 13 MS 10 13 17 LA – LB – LC – FTK – L – BE –

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G

GAM

Explanation

500

6

EC5 solid wood EC5 glued laminated wood DIN 1052 coniferous softwood

8000 11000

500

5

DIN 1052 glued laminated wood

11000 11000 12000 13000 8000 10000 10500 10000 11500 12500 12500 13000 17000 10000 11000 12000

550 600 650 700 500 500 500 500 550 600 1000 1000 1000 500 500 1000

5 5 5 5 6 6 6 6 6 6 8 ! ! 6 6 8

DIN 1052 A–1 laminated wood timber classes acc. DIN 4076

sorted timber classes DIN 4076 DIN 1052 deciduous hardwood DIN 1052 deciduous hardwood DIN 1052 deciduous hardwood OeNORM B3001 spruce, fir, pine OeNORM B3001 larch OeNORM B3001 beech, oak

Version 10.20

GENF

Definition of Finite Elements 3.29.

MASO − Masonry / Brickwork

Item

Description

NO STYP

Material number (1−999) Type of brick stone SB Standard solid Brick LS Limestone Brick LC Lightweight concrete C concrete CC cellular concrete BS British Standard 5628−1 BS−2 Britisch Standard 5628−2 Strength of brick stone Group or strength of mortar i,ii,iia,iii,iiia,iv Standard mortar DM Thin bed mortar LM21,LM36 Light mortar numerical Qualified mortar

SCLA MCLA

E G MUE GAM ALFA SCM FCN FC FT FV FHS FTB

Elastic modulus Shear modulus Poisson’s ratio Unit weight Temperature elongation coefficient Material safety factor Nominal strength σo Compressive strength Tensile strength Shear strength Adhesional shear strength βHS=2σoHS Brick tensile strength βRZ

TITL

Material name

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ MASO ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ Dimension

Default

− LIT

1 MZ

N/mm2 LIT

* *

N/mm2 N/mm2 N/mm2 − kN/m3  − N/mm2 N/mm2 N/mm2 N/mm2 N/mm2 N/mm2

* * * * * 2.5 * σo/0.35 * * * *

Lit32

*

As there are not yet any specific design routines, the parameters follow DIN 1996−1−1 (EC6). According to DIN 1053−1 you should use a value of 2.5 for SCM and 2.67 so to FC. For masonry according to BS you have to attach the group A to D to the strength of the brick stone. Version 10.20

3−85

GENF

Definition of Finite Elements

For masonry according to BS 5628−1 the group A to D has to be a prefix to the stone class. You may then select the mortar designations I to IV. FT is the ten
sile strength for bending according Table 3 parallel to bed joints", FV is the vertical shear strength according pict. 2 resp clause (25, part 2), FHS is the basic shear value according clause 25 part 1, FTB is the bending tensile strength according Table 3 perpendicular to bed joints". FT and FTB vary considerably and should therefore be specified.

3−86

Version 10.20

GENF

Definition of Finite Elements 3.30.

SSLA − Stress−Strain Curves

Item

Description

EPS

Strain value or type of state in a headder record SERV Serviceability ULTI Ulimate Limit Stress value or safety factor in Headder record Type of vertex POL ultimate state & discontinuous slope SPL ultimate stat & continuous slope

SIG TYPE

TEMP

Temperature level for POL/SPL

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ SSLA ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

0/00 LIT

−

N/mm2

−

LIT

POL

grad

0

If the default stress−strain curves are not desirable, stress−strain curves must be defined immediately after the input of the material. Stress−strain diagrams can be specified for the checks in the ultimate state and the service state, each set may have multiple temperature levels, to be defined in ascend
ing order. A stress strain law starts with one of the two possible headers SSLA SERV safety_factor [TEMP tempval] SSLA ULTI safety_factor [TEMP tempval] The stress−strain curve follows. Each consists of several vertices in ordered sequence. The user must assure that a sufficiently large strain range gets cov
ered and that the zero point constitutes a vertex of its own. For each data point is specified whether it should behave as a vertex (polygo
nal line series) or it should be part of a smooth curve (quadratic or cubic parab
ola).

Version 10.20

3−87

GENF

Definition of Finite Elements

The stress−strain curves are only used for proportioning or for nonlinear strain calculations in AQB/STAR2. The TEMP−levels are only used in special applications for the time being.

3−88

Version 10.20

GENF

Definition of Finite Elements 3.31.

SVAL − Cross−section values

Item

Description

NO MNO

Cross−section number Material number or prefered beamtype CENT centric beam BEAM excentr. beam (Reference axis) TRUS only truss (no bending) CABL only cables

A AY AZ IT IY IZ IYZ CM YSC ZSC

Cross−sectional area Shear area for y Shear area for z Torsional moment of inertia Moment of inertia y Moment of inertia z Moment of inertia yz Warping modulus Coordinates of shear centre Relative to the gravity center

YMIN YMAX ZMIN ZMAX WT WVY WVZ

Ordinate of the left edge fibre Ordiante of the right edge fibre Ordinate of the top edge fibre Ordiante of the bottom edge fibre Shear stress due to Mt = 1.0 Shear stress due tp Qy = 1.0 Shear stress due to Qz = 1.0

NPL VYPL VZPL MTPL MYPL MZPL BCYZ TITL

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ SVAL ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

− −/LIT

1 1

m2 m2 m2 m4 m4 m4 m4 * * *

1.0 − − * 3 A /12 IY 0 0 0 0

* * * * N/mm2 N/mm2 N/mm2

* * * * * * *

Fully plastic axial force Fully plastic shear force Fully plastuc shear force Fully plastic torsional moment Fully plastic bending moment Fully plastic bending moment Buckling strain curve main+lateral

kN kN kN kNm kNm kNm LIT

− − − − − − C

Cross−section designation

Lit24

−

Version 10.20

3−89

GENF

Definition of Finite Elements

This command allows the input of cross−sections without the corresponding geometric data, which are necessary of course in detailed stress analysis, yield zone theory or reinforced concrete dimensioning. They can be used in the static analysis or simplified checks with full plastic internal forces. With NO and a Literal for MNO you may also specify which element type should be selected for automatic elements whith that section. This definition can be redone at any time for any section. All other input values will be igno
red in that case. Plastic internal forces may be needed for cross−sections with trial values. It is explicitly stated, however, that the use of this input command for dimen
sioning is by no means in accordance with the intentions of the program’s author for a consistent data input, and the user bears the sole responsibility in this case. If IT is defined zero, special attention should be paid so that the torsional de
gree of freedom during the assembly of the total static system does not lead to undefined rotation capability (Error message: Parts of the system can move freely.). The default for IY is equivalent to a rectangular section with a width of 1 m and the given area A. In accordance with Saint Venant’s estimate, the default value for the tor
sional moment of inertia is

A4 IT = 4 ⋅ Π 2 ⋅ (I y + I z ) This value is exact for circular and elliptical sections. Deviations for a rectangular section: a/b

1/1

2/1

10/1

exact

0.140

0.458

3.13

approx.

0.152

0.486

3.01

The defaults for ymin down to zmax start from a rectangular cross−section and they are derived by appropriate corrections from the radius of gyration. The default values for the full plastic internal forces come out of the cross−sec
tional area. WT enters the default for MTPL, while MYPL and MZPL make use of the extreme coordinates ymin through zmax.

3−90

Version 10.20

Definition of Finite Elements

GENF

All full plastic internal forces are without safety factor. In the case of the buckling strain lines the literals 0,A,B,C,D are used for the same curves in the main and the lateral direction and AB, BC and CD for dif
ferent curves. SVAL can make an identical copy of an already defined cross−section by in
putting SVAL NEWNO−OLDNO. This serves in accelarating the method, when different cross−section values must be applied later on. SVAL can also be used for defining a reduced cross section. This can be done either by using a negative NO to modify an already defined cross−section, or by making a copy of an existing cross−section by means of a negative MNO. The values A through ZSC are then viewed as factors for the corresponding values, and are thus preset to 1.0. The new cross−section has no geometric properties any more. Example: PROF PROF SVAL SVAL SECT

1 HEB 300 2 HEB 300 −1 IT 0.5 3 −2 IT 0.5 4; SV IT −0.5 ; PROF 1 HEB 300

Cross−section 1 receives 50% of the torsional moment of inertia. The ge
ometry of the cross−section gets erased. Cross section 3 has 50% of the tor
sional moment of inertia of cross−section 2, has no geometry, and is hence identical to cross−section 1. Cross−section 2 was not modified. Cross−section 4 is a cross−section with IT reduced by half and with complete geometry. (Only possible with AQUA)

Version 10.20

3−91

GENF 3.32.

Description

NO H B HO BO SO SU ASO ASU MNO MRF

IT SAY SAZ DASO DASU REF

TITL

ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ SREC ÖÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖÖ

SREC − Rectangle, T−beam, Plate

Item

RTYP

Definition of Finite Elements

Dimension

Default

Cross section number Total height Width for rectangular, T−beam Thickness of the plate (upper part) Thickness of the plate (lower part) Inset of top reinforcement Inset of bottom reinforcement Minimum top reinforcement− layer 2 Minimum bottom reinforcement − layer 1

− * * * * cm cm cm2 cm2

1 − 1m 0 0 H/10 SO 0 0

Material number Material number of reinforcement +1000 ⋅ material stirrup reinforcement Reinforcement subtype ASYM = assymmetric two sided ASYT = assymetric three sided SYM = symmetrical SYMT = symm.+ along the sides CU = perimetric reinforcement Factor for torsional moment of inertia Shear area for VY Shear area for VZ Diameter of top reinfrocement Diameter of bottom reinforcement Location of zero point C = gravity center R/L/M = right / left / middle UR/UL/UM = upper right/left/middle LR/LL/LM = lower right/left/middle PR/PL/PM = plate right/left/middle Cross−section designation

− −

* *

LIT

*

−/m4 −/m2 −/m2 mm mm LIT

−1. 0. 0. 28 DASO C

Lit32

*

Depending on the definition of values one of the following types of section is generated:

3−92

Version 10.20

Definition of Finite Elements H H,B H...BO

GENF

Plate with implied width of 1 m or width BO Rectangular cross section T−Beam cross section

Following this classification different design codes will be applied to the sec
tions. When nothing is input for REF, the zero point of the coordinate system of the cross−section is assumed to be at the gravity center.

The required dimensions of the cross−section can be calculated by AQB. For this task, B or H can be input negative when only that dimension should be changed. The full height of the web and the entire plate are used in determining the torsional moment of inertia and the torsional shear stresses; for the equival
ent hollow cross−section used in computing the torsion reinforcement only the web or only the plate is used, depending on which part is larger. The check of the shear stress due to shear force takes place at the most unfavourable location (at the height of the gravity center for the web or at the intersection of web and plate). For the interests of massive constructions the effective tor
sional moment of inertia can be reduced by IT. The input of a positive value specifies a value in m4, while a negative value is interpreted as a factor.

Version 10.20

3−93

GENF

Definition of Finite Elements

Shear areas are typically not used. They can be defined, however, by specify
ing SAY or SAZ. The input of a positive value specifies a value in m2, while a negative value is interpreted as a factor for the default value of the rec
tangular cross−section and the web or the plate. The lower reinforcement is layer 1, the upper one is layer 2. If the distance of the reinforcements is greater than 30 cm, addititonal reinforcement at the side of the web will be introduced with layer number 3. As it is only needed for torsion, you might suppress this by entering a value for RTYP or a zero value for IT. If not suppressed it might be activated for ultimate design in biaxial bending. Further we introduce a reinforcement at the lower side of the plate with layer number 4 if the upper layer is within the topmost quarter of the plate height. At RTYP there are various items available: ASYT three layers (1,2,3) at lower, upper, midsides ASYM two layers (1,2) at lower and upper side SYMT layer 0 at upper and lower side, layer 3 at midside SYM only layer 0 at upper and lower side CU layer 0 at all 4 sides (circumperipheral) CU0 as CU but without minimum shear reinforcement ASY0,SYM0 ASYM,SYM without minim. shear reinforcement AST0,SYT0 ASYT,SYMT without minim. shear reinforcement For symmetric reinforcement we have ASO applied for upper and lower side with layer 0, while layer 3 will be the optional torsional reinforcement at the sides. If CU is input, perimetric reinforcement gets generated. STB uses only the option SYM for ASO/ASU. MRF = 0 must be specified for unreinforced cross−sections. The input of MRF is not allowed for steel or wooden cross−sections. When MRF is specified to be smaller than 1000, the same material type will be assumed for the stirrup reinforcement as for the longitudinal one. The cover of the reinforcement from the side edge is equal to the cover from the upper or lower edge, but not larger than one−fourth of the width.

3−94

Version 10.20

GENF

Definition of Finite Elements 3.33.

SCIR − Circular and Annular Sections

Item

Description

NO RA RI SA SI ASA

ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ SCIR ÖÖÖÖÖÖÖÖ ÖÖÖÖÖÖÖÖ

Dimension

Default

Cross–section number Outer radius Inner radius Outer reinforcement inset (default: (ra–ri)/10) Inner reinforcement inset Outer reinforcement (positive values in cm2/m) (negative values in cm2) Inner reinforcement (omitted if nothing is input)

− m m cm cm *

1  0 * SA −

*

−

− −

* *

ITF DAS

Material number of cross–section Material number of reinforcement + 1000 ⋅ material of stirrup reinforcement Factor for torsional moment of inertia Diameter of reinforcement

− mm

1 28

TITL

Cross–section designation

Lit32

*

ASI MNO MRF

Shear deformations are not considered.

Circular cross section

Version 10.20

3−95

GENF

3.34.

Definition of Finite Elements BORE − Bore Profile of a Sondation

Item

Description

NR

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ BORE ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Number of bore profile

−

1

X Y Z

Coordinates of start point

m m m

0.0 0.0 0.0

NX NY NZ

Direction of bore profile Default: in gravity direction

− − −

* * *

ALF TITL

Rotation angle of local axis Title of bore profile

degree LIT32

0.0 *

With BORE we define a bore profil to be used for HASE and PFAHL to de
scribe the strata data of the soil.

3−96

Version 10.20

GENF

Definition of Finite Elements 3.35.

BLAY − Layer of the Soil Strata

Item

Description

S MNO ICEX

Ordinate along the profile axis (depth) Material number from this ordinate on Construction stage for excavation

MNOR ICRE

Materialnumber of refill Construction stage for refill

HWMI HWMA

Minimum ground water height Maximum griund water height

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ BLAY ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

m

*

m m

0.0 0.0

With BLAY you may specify the geological strata data including the construc
tion phases "excavation" and "backfill". All data is not yet used by any SOFiSTiK−Program.

Version 10.20

3−97

GENF

3.36.

Definition of Finite Elements BBAX − Input of Axial Subgrade Parameters

Item

Description

S1 S2

Starting parameter (depth) Ending parameter (depth)

K0 K1 K2 K3

Constants of subgrade reaction Parabula variation of subgrade reaction Linear variation of subgrade reaction Quadratic variation of subgrade reaction

M0 C0 TANR TAND KSIG D0 D2

Shaft resistance Cohesion coefficient Soil/pile friction angle coefficient Soil/pile dilatancy angle coefficient Pressure coefficient Constant rotational stiffness Linear rotational stiffness

3−98

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ BBAX ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

m m

* 999.99

kN/m2 kN/m2 kN/m2 kN/m2

0 0 0 0

kN/m kN/m − − − kNm kNm

0 0

0.0 0 0

Version 10.20

GENF

Definition of Finite Elements 3.37.

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ BBLA ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

BBLA − Input of Lateral Subgrade Parameters

Item

Description

Dimension

Default

S1 S2

Starting parameter (depth) Ending parameter (depth)

m m

* 999.99

K0 K1 K2 K3

Constants of subgrade reaction Parabula variation of subgrade reaction Linear variation of subgrade reaction Quadratic variation of subgrade reaction

kN/m2 kN/m2 kN/m2 kN/m2

0 0 0 0

P0 P1 P2 P3

Factors for variation along the periphery

− − − −

1 1 1 1

PMA1 PMA2

maximum foundation value at S1 maximum foundation value at S2

kN/m2 kN/m2

− −

Elastic support has many related parameters. That is why those values are combined to special property elements for a geometric line. All the corresponding GLBA and GLBL records follow the GLN record in the order defined by the s ordinate. All data for the s ordinate refer to the para
metric system of coordinates which defaults to the global z− axis. Within a section the subgrade modulus is interpolated by:

ǒ

z*z K + K0 ) K1 @ z * z1 2 1

Ǔ

1ń2

ǒ

Ǔ

ǒ

z*z z*z ) K2 @ z * z1 ) K3 @ z * z1 2 1 2 1

Ǔ

2

The subgrade modulus at the beginning of the section is K0, and the one at its end is K0+K1+K2+K3. The discrete values correspond to constant, para
bolic, linear and quadratic distributions. The default value for S1 is the latest S2 value. The initial default is −999.99. The factors for the variation along the periphery are effective at the angle (0, 90, 180 and 270 degrees). The angle is measured against the local z−axis. In linear analysis the factor (P0+P2)/2 is used for the primary bending (MY,VZ), while (P1+P3)/2 is used for the secondary bending (MZ,VY). Version 10.20

3−99

GENF

Definition of Finite Elements

For the axial reaction a rather sophisticated approach is available. The pres
sure allowing frictional support has many sources. s
+
KSIG @ s v ) K( x ) @ ƪv(x ) ) TAND @ u( x )ƫ t
+
K(x ) @ u( x )
t
TANR @ s ) C0 The first part of the pressure is given by the vertical earth pressure and the horizontal pressure coefficient. The second part is given by the elastic con
stants of the lateral stiff− ness and a combination of the displacements.

3−100

Version 10.20

Definition of Finite Elements See also: BEAM

3.38.

HING − Hinged Connection Combinations for Beams

Item

Description

NO

Number of definition (max. 10)

G1 G2 G3 G4 G5 G6

Up to 6 from the following literals can be entered making the corresponding inter
nal forces equal to zero. N VY VZ MT MY MZ MB

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ HING ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

−

1

LIT LIT LIT LIT LIT LIT

− − − − − −

Simple hinge actions can be entered directly for beams too.

Version 10.20

3−101

Definition of Finite Elements

GENF

See also: SYST, MESH, MAT, BEAM, QUAD, BRIC, SPRI, TRUS, CABL

3.39.

GRP − Group Control

Item

Description

NOG T

Group number Thickness of QUAD−elements

MNO

MRF

Material number The default is the lowest materia number that does not represent any reinforce
ment. Material number of the reinforcement The default is the lowest material number of any reinforcement in the data
base.

ÄÄÄÄÄÄÄÄÄ GRP ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

−

!

m/*

0.2

−

*

−

*

−

*

STI

Element formulation (refer to QUAD) 0 = geometry only 1 = plate action 2 = membrane action 3 = in−plane bending

NR

Direction of the local a−axis (refer to QUAD) Element position w.r.t. the nodes (refer to QUAD) CENT Element lies centred w.r.t. the nodes ABOV Element lies above the nodes BELO Element lies below the nodes

LIT

−

LIT

CENT

Thickness in local x−direction (QUAD) Thickness in local y−direction Thickness for lateral bending (refer to SEPP) Thickness for torsion (refer to SEPP) MIN−T = minimum of TX and TY

m/* m/* m/*

T T MIN−T

m/*

MIN−T

POSI

TX TY TXY TD

3−102

Version 10.20

Definition of Finite Elements

GENF

This is an attribute record. All elements input after it receive the current group provided that they are not categorized explicitly differently. The group number of an element is calculated by dividing the element number by the group divisor. The maximum group number amounts to 999. Beispiele: Gruppendivisor Elementnummer Gruppennummer 1000 1 0 1000 3569 3 2000 3569 1 5 3569 713 Examples: Group divisor Element number 1000 1 1000 3569 2000 3569 5 3569

Group number 0 3 1 713

With the specification of a group classification in the sentence SYST group 0 is preset. The input of GRP causes all following elements of the group 0 to be moved to the specified group by an appropriate change of their element number. The explicit input of elements with a group number deviating from the above is not reminded in order to maintain compatibility with old input data. If nothing different is input by the QUAD record, the subsequent QUAD− el
ements are assigned the properties defined by this record. The following restrictions hold for the input to TX through TD: TX and TY can have different values compared to T. The thickness T is used for the calculation of the gravity load always. The thicknesses TX through TD are used for orthotropic analysis. De
tailed analysis principles can be found in the particular program de
scriptions (SEPP, TALPA). The orthotropic thicknesses are rotated along through the input of and orthotropy angle OAL by the MAT record! Version 10.20

3−103

GENF

Definition of Finite Elements

If a different value is input for the thickness T by the QUAD record, the ortho
tropic thicknesses of the GRP record for this element get multiplied by the factor T−QUAD / T−GRP. The material number is provided as general option. An explicit input by el
ements as well as by MESH or IMES takes precedence.

3−104

Version 10.20

Definition of Finite Elements See also: CABL, SVAL, BEAM, GRP

3.40.

TRUS − Truss−bar Elements

Item

Description

NO

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ TRUS ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

Number of truss−bar

−

!

NA NE

Start node End node

− −

! !

NCS

Cross section number

−

1.

PRE

Prestress force

kN

0.

Truss−bar Truss−bar elements can not be processed by the program SEPP! A truss−bar can only sustain tensile or compressive forces. The input of a negative element number causes the deletion of an already de
fined element. If a cross section number is additionally input, the element is not deleted, and the cross section is changed instead. The prestress is active in all loadcases, thus it can not be used, as a rule, for load superpositioning by the program MAXIMA. It is better in that case to define it as a load. Version 10.20

3−105

Definition of Finite Elements

GENF

See also: TRUS, SVAL, BEAM, GRP

3.41.

CABL − Cable Elements

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ CABL ÄÄÄÄÄÄÄÄÄ Dimension

Default

Number of cable element

−

!

NA NE

Start node End node

− −

! !

NCS

Cross section number

−

1.

PRE

Prestress force

kN

0.

Cable element Cable elements can not be processed by the program SEPP! A cable element can only sustain tension. By linear analysis a cable element works like a truss−bar element. The internal cable slack of the element can be only taken into account, for the time being, in geometrically nonlinear analyses by ASE. In particular, it can enter an eigenvalue analysis only as a linear element with tensile and compressive forces. The geometric stiffness due to the prestress is, however, taken into consideration. The input of a negative element number causes the deletion of an already de
fined element. If a cross section number is additionally input, the element is not deleted, and the cross section is changed instead.

3−106

Version 10.20

Definition of Finite Elements

GENF

The prestress is active in all loadcases, thus it can not be used, as a rule, for load superpositioning by the program MAXIMA. It is better in that case to define it as a load.

Version 10.20

3−107

Definition of Finite Elements

GENF

See also: ADEF, HING, BSEC, SVAL, GRP

3.42.

BEAM − Beam Elements

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ BEAM ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Element number

−

!

NA NE

Number of start beam node Number of end beam node

− −

! !

NR

Direction data of Y−axis (parameter omitted by plane systems)

LIT/ degrees

0

NCS

Cross section number

−

1

AHIN EHIN

Input of hinge at start node Input of hinge at end node

−/LIT −/LIT

* *

DIV

Partitioning of beam into DIV ( 0. If NP is −1 a referenced beam element is generated. The axis of the beam is then the connection of the origin of the sectional coordinate system! Beam coordinate system Each beam or pile has a local system of coordinates x, y, z (refer to section 2.2). The longitudinal axis of the beam NA−NE defines the positive x−direction. The following cases can be distinguished regarding the orientation of the other two axes: 1. Plane frame The structure lies in the global XY−plane. The local y−axis of the beam is par
allel to the global Z−axis but in the opposite direction. The local z−axis is per
pendicular to the axis of the beam and to the right of the direction of the beam.

2. Gridwork The structure lies in the global XY−plane. The local z−axis is parallel to the global Z−axis. The local y−axis is perpendicular to the axis of the beam and to the right of the direction of the beam. Version 10.20

3−109

GENF

Definition of Finite Elements

3. Space frame In a three−dimensional system the orientation of the local y−axis must be de
fined by the user. The parameter NR is available for this purpose. The local z−axis is perpendicular to the local x− and y−axes. Its direction is determined by the three−finger right hand rule. The following possibilities exist: 3.1. NR=0 (Default) The local y−axis is parallel to the global XY−plane and perpendicular to the axis of the beam, thus to the right of the direction of the beam. This is inde
pendant of the gravity direction always clockwise with respect to the global Z−axis.

3−110

Version 10.20

Definition of Finite Elements

GENF

Standard orientation In case the axis of the beam is parallel to the global Z−axis, then the local z− axis is parallel to the global y−axis.

Special cases of orientation 3.2. NR negative (0) A positive value for NR is interpreted as a reference node. The local y−axis lies in a plane defined by the nodes NA−NE−NR. Therefore, NR can not lie on the straight line NA−NE.

Direction node If a non−integer number is input for NR, its decimal part, multiplied by 1000, is interpreted as additional negative rotation of the beam coordinate system

3−112

Version 10.20

Definition of Finite Elements

GENF

about the beam axis in degrees. Thus, 5.090 rotates the z−axis in the plane defined by node 5. 3.4. NR as a literal If one of the literals XX, YY, ZZ, NEGX, NEGY or NEGZ is input for NR the local y−axis will be placed on a plane defined by the axis of the beam and the coordinate axis corresponding to that literal. Haunches and sections Beams can have segments and variable cross sections. Not all of the programs though can take into consideration all the effects resulting from that. One must occasionally settle with an average value (e.g. rotation of the principal axes or shear center). Haunches can be defined in simple cases by a special input format for NCS. Namely, if a decimal number is entered for NCS (e.g. 1.02), the two decimal digits define the cross section at the end of the beam element (1.2 describes cross sections 1 and 20!). The parameter NCSE must be used in case of three digit cross section numbers. The user generally has a whole range of input options. Combinations are al
lowed, but duplicate section definitions are usually ignored. 1. Input of DIV The beam is partitioned into an integer number of parts. Each sec
tion acquires the cross section number of its predecessor. Beams with haunches therefore can not be partitioned this way. 2. ADEF and BDIV can define a pattern of segment lengths and cross section numbers, which can be suited upon several beams through a scaling of the individual segment lengths or the sum of them. Cross section jumps can be solely defined by means of this method. 3. Input of SUPP The use of SUPP generates the sections that are usually necessary for a check (edge of support and critical section for shear). 4. Input of BSEC It can be used for explicit section definitions. The output or the proportioning of internal section forces is usually possible only for cross sections specified by a segment definition or a partitioning. For nonlinear analysis it is necessary to define a large number of sections. Version 10.20

3−113

Definition of Finite Elements

GENF

All sections can include additional information for controlling the processing of the sections during the proportioning and the static analysis. Two para
meters are provided for this purpose: STYP

Typ des Schnitts ABSC ANSC AGEL AIND SCHU

STYP

Normaler Schnitt Anschnitt eines biegesteifen Anschlusses Anschnitt eines Auflagers mit gelenkigem Anschluß (Mauerwerk) Anschnitt eines indirekten Auflagers Für die Schubbemessung maßgebender Schnitt

Type of section SECT FACE HFAC IFAC SHEA

Regular section Section at the face of a clamped connection Section at the face of a support with articulated connection (masonry) Section at the face of an indirect support Critical section for shear proportioning

and PRIN

Ausgabekategorie des Schnittes JA NEIN

PRIN

Output category of the section YES NO

RICH

Section data will be output always Section data will be output only when a corresponding ECHO−option is set.

Richtung von STYP HAUP QUER VOLL

3−114

Schnitt wird immer ausgegeben Schnitt wird nur gedruckt, wenn eine entsprechende ECHO−Option gesetzt ist.

Hauptrichtung (Vz,My) Querrichtung (Vy,Mz) Beide Richtungen Version 10.20

Definition of Finite Elements DIRE

Direction of STYP MAIN TRAN BOTH

ORT

Principal direction (Vz,My) Transverse direction (Vy,Mz) Both directions

Lokalisierung des Schnitts ANFA ENDE

LOC

GENF

zum Stabanfang gehörend zum Stabende gehörend

Localization of the section BEG END

belongs to beginning of beam belongs to end of beam

An assignment of the section to the beginning or the end of the beam is necess
ary for support sections, when these lie on the wrong side with respect to the midpoint of the beam. This is, for example, a case when a beam should be subdivided into more elements. The ’beginning’ then of the third beam el
ement could lie short of the beams end.

Version 10.20

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Definition of Finite Elements

GENF

See also: BDIV, BEAM, BSEC

3.43.

ADEF − Beginning of Beam Segment Definition

Item

Description

NO

Number of definition

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ ADEF ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

−

1

Up to 99 segment definitions can be entered. A segment definition consists of one ADEF record and any number of BDIV records. It can describe one or more entire beams. The same segment definition can be used for beams with different lengths. The length adjustment is carried out according to one of two methods: 1. None of the segment lengths DS is negative. The segment definition is built by measuring the lengths from the centre of the beam. 2. At least one segment has a negative length. All other segment lengths are the same except for the negative lengths, which are adjusted so that the sum of the segment lengths will equal the total beam length. SUPP and BSEC are independent of ADEF and they can be defined in addi
tion to ADEF. The sections that are generated this way get sorted into ADEF in order to prevent duplicate definitions.

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Definition of Finite Elements See also: BSEC, ADEF, BEAM

3.44.

BDIV − Input of Beam Segments

Item

Description

DS

Segment length In case of cross section jumps DS = 0.

NCS

Cross section number at the end of seg
ment Default: cross section specidied for beam

STYP

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ BDIV ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

m/*

!

−

*

Type of section (refer to BEAM)

LIT

SECT

PRIN

Output category of section YES Output desire NO Output only upon request

LIT

YES

DIRE

Direction of STYP MAIN/TRAN/BOTH

LIT

BOTH

LOC

Localisation of section BEG/END

LIT

*

A cross section jump at the beginning or at the end of a beam is not permitted.

Version 10.20

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GENF

Definition of Finite Elements

Segment definition

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Definition of Finite Elements See also: BDIV, ADEF, BEAM

3.45.

BSEC − Beam Sections

Item

Description

NO X NCS

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ BSEC ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

Beam number Section location Cross section number Default: last cross section number

− m/* −

! ! *

STYP

Type of section (refer to BEAM)

LIT

SECT

PRIN

output category of section YES Output desired NO Output only upon request

LIT

YES

DIRE

Direction of STYP MAIN/TRAN/BOTH

LIT

BOTH

LOC

Localisation of section BEG/END

LIT

*

BSEC can be used, in addition to DIV and BDIV, to define sections in beams for output or dimensioning purposes.

Version 10.20

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Definition of Finite Elements

GENF

See also: BDIV, ADEF, BEAM

3.46.

SUPP − Definition of Support Sections

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ SUPP ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Beam number

−

!

XFBM

X−distance of the face of the support at the beginning of the beam in the prin− cipal direction (bending about the local y−axis) 0.0 none section defined > 0.0 distance from beginning of beam

m

0.

XSBM

X−distance of the shear section at the be
ginning of teh beam in the principal direction 0.0 none section defined > 0.0 distance from beginning of beam < 0.0 generation of shear section at distance −(0.9⋅d⋅XSBM)

m

−0.5

TYBM

Type of connection at the beginning of the beam in the principal direction FACE build in connection HFAC articulated connection (hinge) IFAC indirect support

LIT

FACE

XFEM

X−distance of the face of the support at the end of the beam in the principal direction similar to XFBM similar to XSBM similar to TYBM

m

XFBM

m m

XSBM TYBM

XSEM TYEM

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Version 10.20

Definition of Finite Elements Item

Description

XFBT XSBT TYBT

GENF Dimension

Default

Transverse direction; similar to XFBM similar to XSBM similar to TYBM

m m m

XFBM XSBM TYBM

XFET XSET TYET

similar to XFEM similar to XSEM similar to TYEM

m m m

XFEM XSEM TYEM

TO INC

End beam number Increment of beam number

− −

NO 1

In addition to the options offered by the commands ADEF/BDIV or BSEC, the support sections that are necessary for the dimensioning can be easily input by SUPP.

Version 10.20

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Definition of Finite Elements

GENF

See also: MESH, IMES, MAT, BMAT, GRP

3.47.

QUAD − Plane Elements (Disks / Plates / Shells)

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ QUAD ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Element number

−

!

N1 N2 N3 N4

Number of node 1 Number of node 2 Number of node 3 Number of node 4 N4 must not be input for triangular el
ements.

− − − −

! ! ! −

MNO

Element material number

−

*

DNO ENO NNO

The element number gets increased by DNO until ENO is reached. The node numbers are increased at each step by NNO.

− − −

1 NO 1

T

Thickness t

m/*

*

C

Elastic foundation 0 absolut foundation coefficient

kN/m3

−1

STI

Element formulation 0 = geometry only 1 = plate action 2 = membrane action 4 = in−plane bending

−

*

NR

Direction of local x−axis

LIT

*

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Version 10.20

Definition of Finite Elements Item

Description

POSI

GENF Dimension

Default

Element position w.r.t. the nodes CENT Element lies centred w.r.t. the nodes ABOV Element lies above the nodes BELO Element lies below the nodes

LIT

*

CT

Foundation constant tangentially to the surface 0 absolute foundation coefficient

kN/m3

−1

MRF

Reinforcement material number

−

*

T1 T2 T3 T4

Thickness at the four nodes

m/* m/* m/* m/*

T T T T

QUAD plane element In case of no specific input, the values for MNO, T, C, STI, NR, POSI, CT and MRF are copied from the records GRP, MAT and BMAT. The element number is arbitrary, duplicate numbers though are not allowed. The input of a record with a negative number results in the deletion of an al
ready defined element. Recesses can be subsequently added this way to an ex
tensive generated mesh. If the element number is negative and some value is input for MNO, T, C, CT or MRF the element is not deleted, but the material number and the thick
ness or the foundation get changed, respectively. Version 10.20

3−123

GENF

Definition of Finite Elements

The element’s geometry is checked by the program for node numbering order, re−entrant corners and side ratios smaller than 1 : 5. Entire rows of elements can be generated by a single record using DNO, ENO and NNO and, in combination to the generation capabilities of the input lan
guage, one can even generate multi−dimensional meshes. If the record QUAD specifies a thickness T other than the one input by GRP, the gravity load and the isotropic stiffness are computed with this new thick
ness. In case of orthotropic thickness input by the record GRP, the orthotropic thicknesses of the GRP record are multiplied for computing the stiffness by the factor T−QUAD / T−GRP. The input of a negative value for C or CT represents a factor for the foundation coefficient of the BMAT record, while a positive value determines an absolute foundation coefficient for this element. Several element actions can be selected through the value of STI. Addition of these values results in combined action. The preset values are: SYST SYST SYST

FRAM GIRD SPAC

2 1 7

Input of this value is meaningful, as a rule, only for three−dimensional sys
tems. The orientation of the coordinate system is described in section 2.2.3. By three−dimensional systems the local x−axis can be oriented toward a particu
lar direction with respect to the global axes of coordinates; this can be of im
portance in the assessment of the results for the nodes. The following can be defined as reference direction by NR: XX,YY,ZZ for the positive directions NEGX,NEGY,NEGZ for the negative directions nodenumber of a node within local xz−plane A negative value for NR defines an angle by which the default coordinate sys
tem is rotated aginst the local z−axis. By POSI=BELO the elements are eccentrically below the nodes. This can be useful in case of plates with girders for modelling the T−beam action by differ
ent element thicknesses. The input in such case is as easy as for a regular

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Definition of Finite Elements

GENF

plate due to the constant altitude of the nodes lying at the upper edge of the plate. Because of the necessary axial force effects the analysis of eccentric plates can be done only with SYST SPAC by the program ASE. The thicknesses T1 through T4 at the nodes are used for proportioning at the nodes only. Variable thicknesses at the nodes can not be input at the same time with an orthotropy from GRP.

Version 10.20

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Definition of Finite Elements

GENF

See also: CUBE, MAT, GRP

3.48.

BRIC − Three−dimensional Solid Elements

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ BRIC ÄÄÄÄÄÄÄÄÄ Dimension

Default

Element number

−

!

N1 N2 N3 N4 N5 N6 N7 N8

Number of node 1 Number of node 2 Number of node 3 Number of node 4 Number of node 5 Number of node 6 Number of node 7 Number of node 8

− − − − − − − −

! ! ! ! ! ! ! !

MNO

Material number of the element

−

1

BRIC solid element The element number is arbitrary, duplicate numbers though are not allowed. The input of a record with a negative element number causes the deletion of an already defined element. Recesses can be subsequently added this way to a generated large surface mesh. If a value for MNO is input, the element is not deleted, but the material number gets changed instead. The six sides of the element are checked geometrically by the program as QUAD−elements.

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Version 10.20

Definition of Finite Elements See also: BOUN, FLEX, DAMP, GRP

3.49.

SPRI − Spring Elements

Item

Description

NO NA N2

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ SPRI ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

Spring number Number of node where the spring acts Number of a second node

− − −

! ! −

DX DY DZ

X−component of dierction Y−component of direction Z−component of direction

m m m

0 0 0

CP CT CM

Axial spring constant Lateral spring constant Rotational spring constant

kN/m kN/m kNm/rad

0 0 0

PRE

Prestress force

0

GAP

Spring gap (slip)

CRAC

Spring failure load

YIEL

Spring yield load FLIE < 0 only in compression FLIE > 0 tension and compression

kN bzw. kNm m bzw. rad kN bzw. kNm kN bzw. kNm

MUE COH DIL

Friction coefficient for the lateral spring Cohesion value for the lateral spring Dilatation value for the lateral spring

− kN −

− 0 −

ENO DNO NNO

Highest number for element generator Increment of element numbers Increment of node numbers

− − −

NO 1 1

MNO AR

Material number of stress−strain curve Reference area

− m2

− −

Version 10.20

0 − −

3−127

GENF

Definition of Finite Elements

Spring element Springs can be defined as support conditions or as coupling springs between two nodes. The second node number must not be input in the first case. The spring is defined by means of an axial direction (DX,DY,DZ) and two spring constants. The affected forces or deformations are then analysed into a component in the axial direction and a remainder or lateral component per
pendicular to the axial one. The spring constants CP and CT are assigned to the axial and the lateral direction, respectively. Two lateral springs perpen
dicular to each other with the same spring constants are likely to be assigned in this manner. Since the directions of these springs can be freely selected, one can speak of an isotropic lateral spring stiffness. The total spring force has the following form :

ȧPXȧ ȧ ȧ ȧ
+
ȧ ȧDX ȧ@ PH )ȧ ȧPTX ȧ P
+
ȧ PY DY PTY ȧPZȧ ȧDZȧ ȧPTZȧ ȧ ȧ ȧ ȧ ȧ ȧ PT
+
ǒPTX 2 ) PTY 2 ) PTZ 2Ǔ

1ń2

The values of PH and PT affect the various combined nonlinear effects. In the general three−dimensional case the lateral force can be input only in compo
nents or as an amount PT. If one wants to prescribe different lateral spring stiffnesses, one should pre
scribe two or three separate axial springs. Since the input parameter CM usually does not agree neither in its direction of action nor regarding the non
linear effects with CP, it should not be input with a single element except for very few cases. As a rule therefore, separate rotational springs (CP=CT=0.0) should be defined.

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Version 10.20

Definition of Finite Elements

GENF

The direction of coupling springs is derived from the difference of the two nodes (N2−NA). In cases of coinciding nodes or in cases of support springs, the direction must be input explicitly. The method for choosing in this case the signs of DX, DY and DZ, which are important for nonlinear computations (compression or tension?), is to imagine that the second node is offset from the first by this amount. By defining a dilatation value, all lateral shear displacements will induce an axial displacement DIL⋅u−t. Nonlinear effects are controlled by PRE, CRAC, GAP, YIEL, MUE and COH: Prestress: The spring exerts a force or a moment upon the node even in its initial position. The failure and yield loads are appropriately modified by the amount of prestress. A prestress for the lateral spring is not defined. Gap: The spring transmits forces along its axis only after its deformation has exceeded the gap. Failure load: Upon reaching the failure load the spring fails in both the axial and the lateral direction. The failure load is always a tensile force or a posi
tive moment. Yield load: Upon reaching the yield load, the deformation component of the spring in its direction increases without a corresponding increase of the spring force. Friction coefficient: If a friction and/or a cohesion coefficient are input, the lateral spring can not sustain forces greater than: Friction_coeff. * Compressive_force + Cohesion If the axial spring has significant tension or has failed (CRAC), the lat
eral force acts only if 0.0 has been input for both friction and cohesion. Version 10.20

3−129

GENF

Definition of Finite Elements

Spring force−displacement curves The nonlinear effects can be taken into account only by a nonlinear analysis (STAR2/TALPA/ASE). The friction is an effect of the lateral spring, while all other effects act upon the axial spring (CP or CM). The program SEPP handles springs as linear only without prestress. Instead of these simple nonlinear effects you may assign a nonlinear material worklaw to a spring element by the definition of MNO. You will then also need a influence area AR. The force of the spring is the product of the stress and this area. For the stiffness or the strain you will need a length of the element L: L := CP := CT := P := s := e :=

(dx2+dy2+dz2)1/2 E−Modulus⋅AR/L G−Modulus⋅AR/L CP⋅u P/AR u/L

This value A/L will also be used indenpendent of MNO by HYDRA for the con
struction of equivalent thermal or seepage conductivities. For a torsional spring however AR will be equivalent to the torsional inertia in m4: CM := M := σ :=

G−Modulus⋅AR/L G−Modulus⋅AR⋅φ/L M/AR/L

The stress−strain curve is established for a fictive stress in dependance from the torsional strain φ/L.

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Definition of Finite Elements

GENF

The definition of a negative value for AR will select an inverse treatement of the nonlinear effects. The force in the axial direction will not become greater than COH+MUE⋅of the vectorlength of the combined transvers force, which will not become greater than YIEL. This is usefull for the description of an
chors and piles in soil mechanics. With an input of MNO CP and Ct are calculated automatically. If one or both values, e.g. at an input of CM, are not used, the value 0 is entered for CP and/or CT. ATTENTION: The above effects and the dilatation are new and will become implemented in the various modules step after step. Please read the relevant htm−files. Special instructions More complicated force−displacement curves can be generated by combining several nonlinear springs in parallel or in series. A change of the direction component signs can be very helpful in these combinations. These signs have an effect on the sign of the spring force. Negative spring constants can be used for the modelling of suspended pendu
lum towers according to 2nd order theory or for similar effects. Attention should be given in this case to the correct consideration of the load safety fac
tor. Additional springs with small spring constants can possibly make the conver
gence easier or even enable it in the first place during nonlinear computa
tions. The prestress is the same for all loadings, therefore it can not be used, in gen
eral, for load superpositioning by the program MAXIMA. The element generator (ENO, DNO and NNO) generates springs along the specified nodes, which are defined by the geometry of their distances from one another. All spring constants and material properties must be input in this case per unit length. The program computes automatically the resulting dis
cretised spring stiffnesses from the varying distances between nodes. The input of a record with a negative spring number results in the deletion of an already defined element. Support displacements or rotations can be described for the FE−programs SEPP, TALPA and ASE by a spring of stiffness 1E20 and a loading equal to Version 10.20

3−131

GENF

Definition of Finite Elements

displacement times 1E20. This technique avoids the rebuilding of the equa
tion system for each particular loadcase.

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Version 10.20

Definition of Finite Elements See also: SPRI, FLEX, GRP, NODE

3.50.

BOUN − Distributed Elastic Support

Item

Description

FROM TO INC

Starting node or element number End node number Increment All nodes from FROM to TO in in
crements of INC are elastically sup
ported.

TYPE

Direction of support CX Support in global X−dirction CY Support in global Y−direction CZ Support in global Z−direction CN Support in longitudinal direc− tion (axial force) CT Support in both transverse directions (shear force) DX Fixing about global X− direction DY Fixing about global Y− direction DZ Fixing about global Z− direction DN Fixing about longitudinal direction DT Fixing about both transverse directions

CA CE

Spring constants at the beginning Spring constants at the end

Version 10.20

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ BOUN ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

− − −

! FROM !

LIT

!

kN/m2 or kN

! CA

3−133

Definition of Finite Elements

GENF

Item

Description

REF

Axis for the increase XX global X−axis YY global Y−axis ZZ global Z−axis The input is required only for CE differ
ent from CA and for broken boundaries.

RX RY RZ

Action direction N of the boundary el
ement in global coordinates. This input is required only when the direction N does not coincide with the line connecting the nodes.

TITL

Boundary element designation

Dimension

Default

LIT

*

−

−

LIT24

−

Boundary elements are used for the description of elastic or rigid supports, independently of other elements, along a line of nodes. All nodes are appropri
ately fixed in case of a rigid support, thus BOUN defines a node line with a designation and an element number only for the output. The input of the sup
port parameters can be omitted in such case. A boundary consists of an initial record that allows input only for FROM (= element number NO) and TITL, followed by any number of records that de
scribe the segments of the boundary. As a rule, the first node number of a new record must coincide with the last number of the previous record. Example: BOUN 1001 TITL ’WALL ALONG AXIS K−K’ BOUN 1 10 1 CZ 3.2E5 10 90 10 == 90 95 1 ==

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Version 10.20

Definition of Finite Elements

GENF

BOUN boundary element In three−dimensional systems the direction T of the boundary is not uniquely defined. The program therefore introduces supports in two directions of the boundary perpendicular to each other. This corresponds to a uniform support in all directions perpendicular to the line connecting the nodes. If one wishes to have only one of the directions elastically supported, one can select the direction N and input the desired direction by means of RX, RY and RZ. Elastic foundation The boundary element works according to the subgrade modulus theory. It facilitates the definition of elastic supports in any direction independently of the employed elements. The formulation of the subgrade modulus theory is an engineering trick which, among others, ignores the shear deformations of the supporting medium. The determination of a reasonable value for the foundation modulus often presents considerable difficulty, because this value depends not only on the material parameters but on the geometry and the loading as well. One must always keep this in mind, when assessing the accu
racy of the results of an analysis using this theory. The BOUN−element accounts for the continuous foundation through a spring matrix, which is the result of an energy contemplation, when the displace
ments between two nodes are linearly interpolated (infinitely rigid load dis
tribution beams with hinges at the nodes). This is a compromise between a support with single springs and an exact solution of the differential equation. A static connection thus is only possible at the nodes of the system. Therefore, a FE−typical partitioning of the system must be chosen. Version 10.20

3−135

GENF

Definition of Finite Elements

Foundation models This formulation is fully compatible with the QUAD and BRIC elements, meaning that there are no gaps between element and foundation. Only the single springs are available for nonlinear effects. The use of the el
ement generator for them along with a subgrade modulus results in the auto
matic computation of the width between nodes by GENF. A small gap results in the case of beam elements due to the cubic formulation of the displacements, but it is usually unimportant. More serious is that the load is applied only at the nodes. The following thus occur between two beam nodes: 1. The distribution of shear forces due to the foundation is uniform. The value is correct for the middle of the beam, but it lacks at the edges amounts equal to half the beam length times the foundation stress. 2. The distribution of moments due to the foundation is linear. 3. Beam loads generate a garland−shaped moment distribution. The appropriate loading is nodal loads. This is why it is suggested, at least for the boundary region, to make the boundary and the beam elements approximately double as long as the dis
tance of the shear section from the support. For numerical reasons, however, the beam elements should not be shorter than the height of the cross section. An important upper limit for the length of the boundary element results from the stiffness of the supported structure. For elastically supported beams, this is described by the characteristic length L.

3−136

Version 10.20

Definition of Finite Elements L
+


GENF

Ǹǒ4c@@EIb Ǔ 4

The solution for the displacement under concentrated loads is a wave, which always has a zero crossing at distance L. For the boundary element to be able to approximate that, its segments must be smaller than one quarter of that length. When this is not the case, the boundary element searches to represent that distribution through a wild oscillation. The results are severely disturbed in this manner. If the structure’s stiffness is zero, the characteristic length is zero as well, and the boundary element can not be used for concentrated loads at all. Comparative analyses have shown, that an elastically supported beam is sig
nificantly better approximated by boundary elements with continuous foundation than by single springs.

Version 10.20

3−137

Definition of Finite Elements

GENF

See also: BOUN, SPRI, GRP

3.51.

FLEX − General Elastic Element

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ FLEX ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Element number

−

!

NO1 NO2

Node number 1 Node number 2

− −

! !

P

Loading direction PX / PY / PZ MX / MY / MZ / MB

LIT

!

VX VY VZ

Displacement due to load 1

m m m

0 0 0

PHIX PHIY PHIZ

Rotation due to load 1

rad rad rad

0 0 0

PHIW

Warping

rad/m

0

FLEX can be used for the input of any arbitrary element matrices or supports (e.g. pile stand) with up to six nodes. An element consists of several FLEX re
cords with the same element number. Not all the components need to be de
fined. The following variants exist: 1. Flexibility matrix (NO2=0) The displacements in the affected directions are input for each unit load with direction ’P’. Components with diagonal terms 0 are not pro
cessed, and fixed supports must be defined by means of nodal con
straints. 2. Stiffness matrix (NO2>0) One row or one column of the submatrix of the nodes NO1−NO2 is input directly each time in the form of stiffness values in (kN,m). No check of whether the matrix is "positive definite" takes place.

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Definition of Finite Elements

GENF

3. Identifier for substructuring (NO2 = "SUBS") When this literal is input, node NO1 is identified as the main node of the substructure. The inverse values of the stiffness can then be pre
scribed by the displacement values (see HASE). Only one record per node is allowed in this case.

Version 10.20

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Definition of Finite Elements

GENF See also: SPRI, MASS

3.52.

DAMP − Damping Elements

Item

Description

NO

ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ DAMP ÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ

Dimension

Default

Element number

−

1

NA NE

Start node End node

− −

1 0

D DT DM

Damping in axial direction Damping in lateral direction Damping moment about the axis

kNsec/m kNsec/m kNsec

0. 0. 0.

The direction of the damping element is defined by the nodes NA and NE. The damping coefficient DT acts perpendicularly to it. The damping element is used in dynamic analysis only.

Damping element

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Version 10.20

Definition of Finite Elements

3.53.

MASS − Concentrated Masses

Item

Description

NO

GENF

ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ MASS ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ

Dimension

Default

Node number

−

1

MX MY MZ

Translational mass

t t t

0. MX MX

MXX MYY MZZ MXY MXZ MYZ

Rotational mass

tm2 tm2 tm2 tm2 tm2 tm2

0. 0. 0. 0. 0. 0.

REF

Reference of masses GLOB Global coordinate system LOCA Local coordinate system

LIT

GLOB

Masses are used as points without stiffness in the generation of dead loading and in dynamic analysis as additional load. A mass usually acts the same in all three coordinate directions, they should be usually input in all three of them (default). The gravity load of the entire structure is always defined in the form of translational masses. If needed, rotational masses must be defined separ
ately by MASS. In case the gravity load of a structure must be neglected, the gravity load in the appropriate material or cross section records must be set to zero as well. Off diagonal terms of the rotational masses are taken into account only for a fully consistent mass matrix, which might not be available in all cases.

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Definition of Finite Elements This page intentionally left blank

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Definition of Finite Elements 4

GENF

Output Description

In case of error−free input, all input records are output as well. This can be prevented by setting ECHO NO; by contrast, setting ECHO PRIN results in full output regardless of whether the input is correct or not. By use of the options of the ECHO record, one can obtain a selective output, e.g., ECHO NODE NO means that node data should not be printed. The results of the profile optimisation of the stiffness matrix are output on the screen as well as in the output file. The bandwidth and the profile are out
put here without consideration of the degree of freedom per node.

4.1.

Nodal Values

With ECHO NODE YES all nodal coordinates and the relevant supports (node celebration attributes) and the kinematic constraints are spent. Meaning nodal coordinates and supports in the list: Number X Y Z Support Conditions MIN MAX

Node number X−coordinate in m Y−coordinate in m Z−coordinate in m Support conditions of a node (Explanation of the short cut see record NODE) minimal nodal value maximal nodal value

According to the input the list of the kinematic constraints is made addition
ally. With input of couplings at NODE and/or of intermediate nodes with INTE the list results followingly: Node LV type

reference Version 10.20

Node number Level of coupling with recursive definition Coupling condition (Explanation of the short cut see record NODE) or with definition model intermediate node of the interpolation INTtyp (Explanation of typ see record INTE) Reference node

4−1

Definition of Finite Elements

GENF dx,dy,dz df

Direction of the couplings or effective dis− tances between the nodes general factor, indicates where the inter− mediate node (INTE) lies between the reference nodes

With input of the kinematic dependences with the record KINE following list is made: ND LV ND1

FD1 . . . ND6 FD6

4.2.

Dependent degree of freedom indicated with node number/local degree of freedom Level of coupling with recursive definition Reference degree of freedom 1 indicated with node number/local degree of freedom with X,Y,Z for the displacements an XX,YY,ZZ for the rotations Factor for reference degree of freedom 1

Material Values

In the normal case (ECHO MAT YES) a list from the tables of the material va
lues is following: General Material Values No Material number Young−module Elastic modulus Poisson−Ratio Poisson’s ratio Shear−module Shear modulus Compress.module Bulk modulus Weight Specific weight Weight buoyancy Specific weight under buoyancy, only for soil mechanics Temperat. coeff. Thermal expansion coefficient Y−Modulus E−y Anisotropic elastic modulus Poisson R. m−yz Anisotropic Poisson’s ratio Angle Rotation Meridian angle of anisotropy

4−2

Version 10.20

Definition of Finite Elements Angle Precission Safetyfactor calc strength fy ult. strength ft

GENF

Descent angle of anisotropy Material safety factor Calculation strength Final strength

Non−linear Material Values With input of a non−linear material (NMAT) law the parameters of the material law are spent to the general material values additionally (Parameter discription in the manual TALPA). Bedding Attributes No Cs Ct pr py tan c w CONCRETE Strength fc Nominal strength Tens.Str. fctm Tens.Str. fctk Compr.fail.ener. Tens.fail.energ. Friction crack

Material number Elastic constant normal to surface Elastic constant tangential to surface Maximum tensile stress of interface Maximum stress of interface Friction coefficient of interface Cohesion of the interface Equivalent mass distribution MATERIAL Design value of concrete strength Cube strength or cylinder compressive strength average tensile strength Fractile of the tensile strength Fracture energy for compression failure Fracture energy for tensile failure Friction in the crack

STEEL MATERIAL Yield stress fy Yield stregth Tensile str. ft Tensile strength Plastic strain Failure strain Relaxation .55ft Coefficient of relaxation (0.55 ßz) Relaxation .70ft Coefficient of relaxation (0.70 ßz) nat. bond coeff. Bond coefficient by DIN 4227 Table 8.1 EC2 bondcoeff K1 Bond coefficient per EC2 / Vol. 400 Hardening module Hardening module Version 10.20

4−3

Definition of Finite Elements

GENF

TIMBER MATERIAL Bendingstr fm Bending strength Tensionstr ft,0 Tensile strength parallel to the fibre Tensionstr ft,90 Tensile strength normal to the fibre Compress. fc,0 Compressive strength parallel to the fibre Compress. fc,90 Compressive sterength normal to the fibre Shearstr. fv Shear strength at center (shear force) Shearstr. fv,T Shear strength at edge (torsion) The entered stress−strain curves are not spent. An output of the stress− strain curves occurs only with input in the programm AQUA. Graphic repre
sentations of the stress−strain curves are possible with AQUP. A layerd material was entered with the record MLAY, so the general material values of this material are determined and spent. The layer are entered addi
tionally: Layer thickness Material No.

Layer thickness in m Material number and material specification of the layer

Up to ten shifts are spent according to the input.

4.3.

System Statistics

A list of several cross sectional types according to input of BEAM, TRUS, CABL or plane elements according to input QUAD is made with ECHO SYST YES. Meaning summary of the several cross sectional types in the list: No. Total Length Total Weight max. Length Title

Number of the cross section Total Length in m Total Weigth in t Maximum Length in m Title

Meaning summary of used plane elements in system in the list: Grp Total Area Total Weight max. area Total Volume Material No.

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Number of the group Total Area in m2 Total Weight in t Maximum area of an element in m2 Total Volume in m3 Material number Version 10.20

Definition of Finite Elements 4.4.

GENF

Cross−sectional Overview

With ECHO SECT YES a overview of the cross−sectional values is spent. In that means: CROSS SECTIONS − STATIC PROPERTIES No MNo MNs A It Ay,Az,Ayz Iy,Iz,Iyz ys,zs y,z−sc modules gam

Number of the cross section Material number of the cross section Material number of the reinforcement Cross−sectional area in m2 Torsional moment of inertia in m4 Shear areas in m2 Moments of inertia y,z,yz in m4 Coordinates of centre of gravity in m Coordinates of shear centre in m Elastic and shear modulus in kN/m2 Specific weight in kN/m3

Provided that a cross−section designation was defines, this is appended so be
hind every cross section. An output of the supplementary cross−sectional values (moments of resis
tance, fully plastic internal forces and moments, reinforcement etc.) is poss
ible only after input with the program AQUA.

4.5.

Group Qualities

An output of the group qualities occurs with ECHO GROU YES. In the list of the group qualities means: No MNo Mrf Ansatz Posi Direction x−axis Thick Orthotropic thickn. DX,DY,DXY,DD

Version 10.20

Number of the group Material number Material number of the reinforcement Element formulation Position of the elements to the nodes Direction of the local x−axis Thickness of the elements in m Orthotropic thicknesses of the elements in m (Output occurs only in case od differences to the thickness.)

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Definition of Finite Elements

GENF 4.6.

Plane Elements (2−D, QUAD)

An output of the group qualities occurs with ECHO QUAD YES. In the list of the group qualities means: Grp Number Nodeno Nodeno Nodeno Nodeno MNo t C Orthotropic thickn. TX,TY,TXY,TD Variable thickness t1 − t4

Number of the group Element number Node numbers of the corner nodes

Material number Thickness of the element in m Elastic foundation in kN/m3 Orthotropic thicknesses of the element in m (Output occurs only in case od differences to the thickness.) Variable thicknesses of the elements in m (An output occurs only with different thick− nesses in the nodes.)

Footnotes to the several output values are explained in the ERG−file.

4.7.

Three−dimensional Solid Elements (3−D, BRIC)

The three−dimensional solid elements are spent after input of ECHO BRIC YES in following list: Grp Number Nodeno Nodeno Nodeno Nodeno Nodeno Nodeno Nodeno Nodeno MNo

4−6

Number of the group Element number Number of the node 1 Number of the node 2 Number of the node 3 Number of the node 4 Number of the node 5 Number of the node 6 Number of the node 7 Number of the node 8 Material number of the element Version 10.20

Definition of Finite Elements 4.8.

GENF

Boundary Elements

With the record ECHO BOUN YES lists of the boundary elements (BOUN), the element matrixes and supports (FLEX), the damping elements (DAMP) and the single masses (MASS) are spent. After the input of the record BOUN − Distributed Elastic Support following list appears: from to inc type ref CA/CB Title/Direction Total Length

Starting node or element number End node number Increment Direction of support Axis for the increase Spring constants at the beginning/end Boundary element designation/direction Total Length in m

After input of the record FLEX − General Elastic Support an element matrix follows in the output listing: Number Nodeno

Number of the element Node number, negatively at Substructering

F−xx,F−yy,F−zz, FR−xx,FR−yy,FR−zz, FR−ww

Values of the flexibility matrix

S−xx,S−yy,S−zz, SR−xx,SR−yy,SR−zz

Values of the stiffness matrix

At the substructure−technique the shift−values of the stiffness are spent. If damping elements were entered with the record DAMP, a list of the damp
ing elements is spent: Grp Number Nodeno Nodeno dX,dY,dZ DP Version 10.20

Number of the group Element number Star node End node Indication of direction of the damping element Damping in axial direction in kNsec/m

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Definition of Finite Elements

GENF DT DM

Damping in lateral direction in kNsec/m Damping moment about the axis in kNsec

If an input of single masses with the record MASS occurs, these are spent so as follows: Nodeno Group M−X,M−Y,M−Z M−XX,M−YY,M−ZZ

4.9.

Node number Number of the group Translational masses in t Rotational masses in tm2

Geometric Definitions (Bedding Profiles)

Bedding profiles with the records BORE, BBAX, BBLA were entered in con
nection with the pile elements or for analysis of settlement, the output of the values with ECHO GEOD YES in a list of the structural axis / bedding axis occurs so. In that means: Line s x,y,z L K0−a,K1−a,K2−a, K3−a M0 C0 TANR TAND KSIG K0−t,K1−t,K2−t, K3−t P0,P1,P2,P3 Pmax

4.10.

Line number of the structural/bedding axis Starting and/or ending depth Coordinates of the well site in m Length in m Constants of the foundation profile Mantle friction in kN/m Cohesion coefficient in kN/m Soil/pile friction angle in degrees Dilatation angle in degrees (SOFIMESH only) Lateral pressure value (SOFIMESH only) Constants of foundation profile in tangential direction (lateral) in kN/m2 Factors for circumferential variation Maximum foundation value at starting and ending depth in kN/m

Bending Beams and Piles

An output of the bending beams and piles occurs with ECHO BEAM YES. For the entered bending beams and piles (BEAM) following list appears:

4−8

Version 10.20

Definition of Finite Elements Grp Number Nodeno x NoS Ref hinges direction local y−axis

GENF

Number of the group Beam number Number of the start and/or end beam node Distance of the beam beginning in m Cross section number Reference axis (at genrated pile elements geometry or bedding number) Hinge combination (Explanation of the short cut see records HING and BEAM) Direction data of local y−axis

After definition of beam sections with the records ADEF and BDIV or of beam sections with the record BSEC or of supporting sections with the record SUPP additional to the list of bending beams the type, the direction and the place of the defined beam sections are spent (Explanations see BEAM, ADEF, BDIV, BSEC and SUPP).

4.11.

Truss−bar Elements

A list of truss−bar elements is made with ECHO TRUS YES. In that means: Grp Number Nodeno Nodeno section L N−p

4.12.

Number of the group Number of the truss−bar Start node End node Cross section number Length in m Prestress force in kN

Cable Elements

With ECHO CABL YES a following list of cable elements is spent: Grp Number Nodeno Nodeno section L N−p Version 10.20

Number of the group Number of the cable element Start node End node Cross section number Length in m Prestress force in kN

4−9

Definition of Finite Elements

GENF 4.13.

Springs

Spring values are spent with ECHO SPRI YES in two lists. The general spring values appear in the list SPRING ELEMENTS: Grp Number Nodeno Nodeno dX,dY,dZ CP CT CM N−p

Number of the group Spring number Number of node upon which the spring acts Number of a second node Components of direction X,Y,Z Axial spring constant in kN/m Lateral spring constant in kN/m Rotational spring constant in kNm/rad Prestress force in kN and/or kNm

The list NONLINEAR EFFECTS is made only by input of one of these values: Grp Number Prestress Gap T−Cutoff Yielding Friction Cohesion

4−10

Number of the group Spring number Prestress force in kN and/or kNm Spring gap (slip) in mm Spring failure load in kN and/or kNm Sprinf yield load in kN and/or kNm Friction coefficient for the lateral spring Cohesion value for the lateral spring in kN

Version 10.20

Definition of Finite Elements 5

Examples

5.1.

Angle Plate

GENF

A completely regular element mesh can be used for this system. The input for GENF is therefore very simple:

Angle plate A completely regular element mesh can be used for this system. The input for GENF is therefore very simple: PROG GENF HEAD ANGLE PLATE SYST GIRD $ CORNER NODES OF THE NODE 1 9.00 0.00 ; 7 61 4.00 0.00 ; 67 115

MESH = 6.00 ; 12 = 11.00 = 6.00 ; 72 = 11.00 0 6.00 ; 120 = 11.00

$ MATERIAL AND PLATE−THICKNESS MAT 1 ; GRP 0 T 0.24

Version 10.20

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GENF

Definition of Finite Elements

$ GENERATION OF THE MESH MESH 61 67 7 1 6 5 1 ; 67 72 12 7 5 5 1 ; 115 120 72 67 5 4 1 $ SUPPLEMENT OF THE BOUNDARY CONDITIONS NODE (61 66 1) − − PZMX ; 67 − − F ; (79 115 12) − − PZMY ( 1 3 1) − − PZMX ; (9 11 1) == ; 12 − − F (24 36 12) − − PZMY ; 115 − − F ; (116 120 1) − − PZMX $ FIXING ACTION OF THE BOUNDARIES $ ROTATIONAL FOUNDATION CONSTANT D = 3EI/L= 3*3E7*0.24**3/12/3 BOUN 1 TITL ’INTERNAL WALL’; BOUN 1 3 1 DN 34560 9 12 1 == ; 12 36 12 == BOUN 2 TITL ’EXTERNAL WALL’; BOUN 61 67 1 == 67 115 12 == ; 115 120 1 == END

The input for the program GRAF is as follows: PROG HEAD VIEW COLO SCHH STRU END

GRAF GRAPHICS STAN 0 0 −1 NEGX C5 6000 H3 0.4 1 1 MARK 0

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Definition of Finite Elements

GENF

The following system results:

Mesh partitioning

5.2.

Pointwise Supported Ceiling Plate

A real system from the design practice is presented in the following example. The statical system and the relevant assumptions are shown in the figure (refer also to SEPP−Manual Example 5.6.) Version 10.20

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GENF

Definition of Finite Elements

Ceiling plate The input for the program GENF is printed in the following. PROG GENF HEAD CEILING PLATE SYST GIRD

$ EXAMPLE 2 SEE MANUAL GENF 5−2

MAT 1 3.0E7 GRP 0,1 D .3

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Version 10.20

Definition of Finite Elements

GENF

LET#1 5.28+6.56+.2 $ Y−COORDINATE OF RIGHT BOUNDARY $ Y−COORDINATES OF THE FIXED NODES NODE NO 1 0 0; 2 = .55; 6 = 5.28−.35; 7 = 5.25+.35; 12 0 #1 $ GENERATION OF THE NODES LYING IN BETWEEN MESH 2 6 M 4 ; 7 12 M 5 $ X−COORDINATES OF THE FIXED NODES NODE 141 6.25−.35 0 ; 161 6.25+.35 = ; 261 6.25+5.5−.35 = 281 6.25+5.5+.35 = ; 401 6.25+5.5+3.8 $ GENERATION OF THE NODES LYING IN BETWEEN MESH 1 141 M 7 ; 161 261 M 5 ; 281 401 M 6 $ NODE 101 EXACTLY 1.65 M UNDER NODE 141 BECAUSE OF ADDITIONAL LOAD NODE 101 −1.65 0 NREF 141 $ MESH IMES 1 12 1 401 20 1 $ CORRECTIONS $ NODE AXIS 4/D NODE 412 6.25+5.5+6.44 #1−3.36−.4 $ EDGE AXIS D MESH 401 412 M 11 $ TRANSLATION OF NODES 292 AND 312 WALL EDGE SECT 292 281 282 192 412 ; 312 301 302 192 412 $ REBUILT MESH OF THE OBLIQUE REGION MESH 301 307 407 401 6 5 ; 307 312 412 407 5 5 188 192 292 288 4 5 $ SUPPORT CONDITIONS AT EXTERNAL WALL NODE (1 12 1) FIX PZ ; (401 412 1) == ; (12 412 20) == $ ELASTIC RESTRAINT AT EXTERNAL WALL BOUN 1 TITL ’EXTERNAL WALL’ BOUN 1 12 1 DN 58000; 12 412 20 DN 60000; 412 401 −1 DN 62000 $ SUPPORTING COLUMNS NODE 13 6.25 .2 PZ ; 14 = 5.28 = 15 6.25+5.5 .2 PZ ; 16 = 5.28 = $ LINKING OF COLUMNS TO THE SUPPORT NODES NODE 141,142,161,162 FIX KP 13 146,147,166,167 FIX KP 14 261,262,281,282 FIX KP 15 266,267,287,286 FIX KP 16 $ ELASTIC CLAMPING OF THE PERIMETER COLUMNS SPRI 113,115 13,15 DX 1 CM 28000 ; 213,215 13,15 DY 1 CM 28000 END

Version 10.20

5−5

GENF

Definition of Finite Elements

The system along with its node− and element partitioning, is presented in the following figures. The necessary input for the program GRAF is as follows: PROG HEAD VIEW COLO SCHH STRU END

GRAF GRAPHICS STAN X 0 0 −1 AXIS NEGX C5 6000 H3 0.20 H4 0.175 1 0 MARK 0

Element mesh for ceiling plate

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Version 10.20

Definition of Finite Elements 5.3.

GENF

Gridwork

Gridworks lie in the X−Y plane and they are stressed perpendicularly to their plane. The Z−axis in this case points downwards. The resulting basic forces are Vz, Mt and My (corresponding to the cross sectional parameters Az, It and Iy).

Gridwork A 2−span gridwork with 3 main girders serves as example. The input to GENF reads: (refer also to STAR2−Manual Example 5.3.) PROG GENF HEAD GRID FRAMEWORK SYST GIRD NODE 1 0 7 ; 7 24 7 ; 21 0 0 ; 27 24 0 MESH 1 7 27 21 6 2 $ SUPPORTINGS NODE (1 21 10) − − PZ ; (4 24 10) − − PZ ; (7 27 10) − − PZ CONC 1 C 25 SVAL 1 1 1 − − .024279 .030375 SVAL 2 1 1 − − .002794 .003125 BEAM ( 1 6 1) ( 1 1) ( 2 1) 1 − − 4 BEAM (11 16 1) (11 1) (12 1) 1 − − 4 BEAM (21 26 1) (21 1) (22 1) 1 − − 4 BEAM (31 37 1) ( 1 1) (11 1) 2 − − 4 BEAM (41 47 1) (11 1) (21 1) 2 − − 4 END

The GENF−output is as follows: Version 10.20

5−7

GENF

Definition of Finite Elements

M A X I M U M N O D A L P A R A M E T E R S max−No Xmin/Xmax Ymin/Ymax Zmin/Zmax 27 .000 .000 .000 24.000 7.000 .000 NODAL COORDINATES AND SUPPORTS Number X (m) Y (m) Z (m) 1 .000 7.000 .000 2 4.000 7.000 .000 3 8.000 7.000 .000 4 12.000 7.000 .000 5 16.000 7.000 .000 6 20.000 7.000 .000 7 24.000 7.000 .000 11 .000 3.500 .000 12 4.000 3.500 .000 ... ...

Support Conditions PX PY PZ PX PY PX PY PX PY PZ PX PY PX PY PX PY PZ PX PY PZ PX PY

MZ MZ MZ MZ MZ MZ MZ MZ MZ

MB MB MB MB MB MB MB MB MB

M A T E R I A L S No. 1 C 25 (Eurocode EC2) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Young−module 30472 [MPa] Safetyfactor 1.50 [−] Poisson−Ratio .20 [−] Strength fc 21.25 [MPa] Shear−module 12696 [MPa] Nominal Strength 25.00 [MPa] Compress.module 16929 [MPa] Tens.Str. fctm 2.56 [MPa] Weight 25.0 [kN/m3] Tens.Str. fctk 1.80 [MPa] Weight buoyancy .0 [kN/m3] Tens.Str. fctm 3.33 [MPa] Temperat. coeff. 1.00E−05 [−] Compr.fail.ener. 20.00 [kN/m] Tens.fail.energ. .05 [kN/m] Friction crack .20 [−] C R O S S − S E C T I O N S

S T A T I C

No MNo A[m2] Ay/Az/Ayz Iy/Iz/Iyz MNs It[m4] [m2] [m4] 1 1 1.0000E+00 3.038E−02 2.428E−02 3.038E−02 2 1 1.0000E+00 3.125E−03 2.794E−03 3.125E−03 B E A M E L E M E N T S beam− node− x NP NCO No. No. (M) 1 1 .000 1 1.000 1 2.000 1

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ys/zs [m] .000 .000 .000 .000

P R O P E R T I E S

hinges

y/z−sc [m] .000 .000 .000 .000

modules gam [MPa] [kN/m3] 30472 25.0 12696 30472 25.0 12696

direction of local y−axis .00 1.00

.00

Version 10.20

Definition of Finite Elements 2 2

2

3

5.4.

3.000 4.000 .000 1.000 2.000 3.000 4.000 . . .

1 1 1 1 1 1 1

.00

GENF

1.00

.00

Plane Frame, Restrained in Space

Plane frame A two−aisle gable frame serves as example of a beam structure. The exterior columns are horizontally restrained at the top through wall connections, whilst the middle column can deform perpendicularly to the plane of the frame for the lack of support. The support conditions of the frame are defined in space because a stability check will be carried out subsequently. The cross sections should be input, as a rule, by the program AQUA, since the cross sec
tions defined in GENF can not be proportioned. A different method was chosen here in order to obtain a complete example. Especially interesting in the input is the record SUPP, by which 10 proportion
ing sections are generated with very few input data. Notice that no constraint Version 10.20

5−9

GENF

Definition of Finite Elements

in the transverse direction exists above the middle support (bending about the local z−axis)! Input: PROG HEAD SYST NODE

GENF TWO AISLE PLANE FRAME; RESTRAINED IN SPACE SPAC (1 3 1) (0 5) 0 5 F (11 13 1) (0 5) FIX PY 12 FIX FREE

MAT 1 SREC no mno h b 1 1 .4 .4 2 1 .5 .4 SUPP 11 .2 −1 XFET 0 0 ; 12 .2 −1 XFBT 0 0 (1 1) (11 1) XX 1 DIV 5 BEAM (1 3 1) (11 12 1) (11 1) (12 1) − 2 END

Output: M A X I M U M N O D A L P A R A M E T E R S max−No Xmin/Xmax Ymin/Ymax Zmin/Zmax 13 .000 .000 .000 10.000 .000 5.000 NODAL COORDINATES AND SUPPORTS Number X (m) Y (m) Z (m) 1 .000 .000 5.000 2 5.000 .000 5.000 3 10.000 .000 5.000 11 .000 .000 .000 12 5.000 .000 .000 13 10.000 .000 .000

Support Conditions PX PY PZ MX MY PX PY PZ MX MY PX PY PZ MX MY PY

MZ MZ MZ

MB MB MB

PY

M A T E R I A L S No. 1 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Young−module 30000 [MPa] Safetyfactor 1.00 [−] Poisson−Ratio .20 [−] Shear−module 12500 [MPa] Compress.module 16667 [MPa] Weight 25.0 [kN/m3] Weight buoyancy 15.0 [kN/m3] Temperat. coeff. 1.00E−05 [−]

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Version 10.20

Definition of Finite Elements C R O S S − S E C T I O N S

S T A T I C

No MNo A[m2] Ay/Az/Ayz MNs It[m4] [m2] 1 1 1.6000E−01 3.584E−03 = 40.0/ 40.0 [cm] 2 1 2.0000E−01 5.472E−03 = 40.0/ 50.0 [cm] B E A M E L E M E N T S beam− node− x NP NCO No. No. (M) 1 1 .000 1 1.000 1 2.000 1 3.000 1 4.000 1 11 5.000 1 2 2 .000 1 1.000 1 2.000 1 3.000 1 4.000 1 12 5.000 1 3 3 .000 1 1.000 1 2.000 1 3.000 1 4.000 1 13 5.000 1 11 11 .000 2 .200 2 .560 2

12

12 12

13

.650 4.350 4.800 5.000 .000 .200 .650 4.350 4.440

2 2 2 2 2 2 2 2 2

4.800 5.000

2 2

Version 10.20

GENF

P R O P E R T I E S

Iy/Iz/Iyz [m4] 2.133E−03 2.133E−03

ys/zs [m] .000 .000

y/z−sc [m] .000 .000

4.167E−03 2.667E−03

.000 .000

.000 .000

30000 12500

direction of local y−axis 1.00 .00

.00

hinges

modules gam [MPa] [kN/m3] 30000 25.0 12500

1.00

.00

.00

1.00

.00

.00

25.0

.00 1.00 .00 face of fixed support Princ.+Transv. direction critical section for shear Transversal direction critical section for shear Principal direction critical section for shear Principal direction face of fixed support Principal direction .00 1.00 .00 face of fixed support Principal direction critical section for shear Principal direction critical section for shear Principal direction critical section for shear Transversal direction face of fixed support Princ.+Transv. direction

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GENF 5.5.

Definition of Finite Elements Shell Structure

A cylindrical masonry dam was generated as example of a Finite−Element generation; its analysis is documented in the ASE−manual. The input for GENF reads: PROG HEAD SYST NODE

GENF E CYLINDRICAL DAM SPAC −1 43.25 0 0.0 PYYM ; ( −2 −5 −1) = −6 = 53 = F −7 43.25 0 7.5 PYYM ; ( −8 −11 −1) = −12 = 53 = F −13 43.25 0 15.0 PYYM ; ( −14 −17 −1) = −18 43.25 0 22.5 PYYM ; ( −19 −21 −1) = −22 43.25 0 30.0 F ; ( −23 −24 −1) = NODE 17 − − − F ; 21 == MAT 1 MUE 0.15 QUAD (1 5 1) (7 1) (8 1) (2 1) (1 1) (6 9 1) (13 1) (14 1) (8 1) (7 1) ; 10 (11 13 1) (18 1) (19 1) (14 1) (13 1) ; (16 17 1) (22 1) (23 1) (19 1) (18 1) ; END

(6.625 13.25) = (6.625 13.25) = (6.625 13.25) = (6.625 13.25) = (6.625 13.25) = =

17 12 11 14 21 16 17 18 24 21 20

The following picture was created by this GRAF−input: PROG SIZE VIEW STRU END

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GRAF −LP 0 $ LASER−PRINTER STAN 40 20 15 POSZ 1 1

Version 10.20

Definition of Finite Elements

GENF

Structure Version 10.20

5−13

GENF 5.6.

Definition of Finite Elements Reinforced Concrete Box

Box This example deals with a high cube out of reinforced concrete with dimen
sions 10 * 10 * 10 meters. It is based flat on the ground, and the foundation stiffness is different in the vertical and the horizontal directions. A finer partitioning should be selected in practice, here however a partitioning of each side into 2 * 2 elements suffices to illustrate the principle. The input: PROG GENF HEAD ELASTIC FOUNDED HOLLOW CONCRETE CUBE SYST SPAC GDIV 1000 OPTI NO NODE 1 0 10 ; 3 10 10 ; 21 0 0 ; 23 10 0 TRAN 1 23 1 DZ −10 DNO 200 MAT 1,2 ; BMAT 2 C 50000 MAT 3 GRP 1 MESH 1 3 23 21 2 2 2 GRP 6 MESH 201 203 223 221 2 2 1 GRP 2 NR XX MESH 1 3 203 201 2 2 1 GRP 3 NR NEGY MESH 3 23 223 203 2 2 1

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GRP 4 NR NEGX MESH 23 21 221 223 2 2 1 GRP 5 NR YY MESH 21 1 201 221 2 2 1 END

In order to subsequently interpret the stress resultants and the proportion
ing results correctly, it is important to select an appropriate coordinate sys
tem for the shell elements. The direction of the local X−axis is determined by the input to the parameter NR of the MESH record, while the direction of the Z−axis is determined by the direction of rotation around the elements. This on the other hand is determined by the sequence of the corner nodes in the input of the MESH record. With the exception of the base plate, the local Z− axis always point to the interior of the box. This means that the lower rein
forcement layer lies at the inner side of the walls. Since the stress resultants of shell structures are always local, they must be calculated at the nodes separately for each side of the cube. This is achieved by subdividing the structure into 6 groups. The group divisor (GDIV) in the record SYST is set to 1000. During the element generation by the record MESH the elements are assigned each time to the current group (record GRP) − the element numbers are increased by the corresponding amount of thousands (see page after next). In case of larger structures with more than 1000 nodes, duplicate element numbers may arise inside the groups. In such cases a negative MNO− value should be used in order to select a numbering that will be independent of the node numbers. M A X I M U M N O D A L P A R A M E T E R S max−No Xmin/Xmax Ymin/Ymax Zmin/Zmax 223 .000 .000 −10.000 10.000 10.000 .000 NODAL COORDINATES Number X (m) 1 .000 2 5.000 3 10.000 11 .000 12 5.000 13 10.000 21 .000 22 5.000 23 10.000

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AND SUPPORTS Y (m) Z (m) 10.000 .000 10.000 .000 10.000 .000 5.000 .000 5.000 .000 5.000 .000 .000 .000 .000 .000 .000 .000

Support Conditions PY PZ

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GENF 101 102 103 111 113 121 122 123 201 202 203 211 212 213 221 222 223

.000 5.000 10.000 .000 10.000 .000 5.000 10.000 .000 5.000 10.000 .000 5.000 10.000 .000 5.000 10.000

10.000 10.000 10.000 5.000 5.000 .000 .000 .000 10.000 10.000 10.000 5.000 5.000 5.000 .000 .000 .000

−5.000 −5.000 −5.000 −5.000 −5.000 −5.000 −5.000 −5.000 −10.000 −10.000 −10.000 −10.000 −10.000 −10.000 −10.000 −10.000 −10.000

M A T E R I A L S No. 1 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Young−module 30000 [MPa] Safetyfactor 1.00 [−] Poisson−Ratio .20 [−] Shear−module 12500 [MPa] Compress.module 16667 [MPa] Weight 25.0 [kN/m3] Weight buoyancy 15.0 [kN/m3] Temperat. coeff. 1.00E−05 [−] No. 2 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Young−module 30000 [MPa] Safetyfactor 1.00 [−] Poisson−Ratio .20 [−] Shear−module 12500 [MPa] Compress.module 16667 [MPa] Weight 25.0 [kN/m3] Weight buoyancy 15.0 [kN/m3] Temperat. coeff. 1.00E−05 [−] No. 3 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Young−module 30000 [MPa] Safetyfactor 1.00 [−] Poisson−Ratio .20 [−] Shear−module 12500 [MPa] Compress.module 16667 [MPa] Weight 25.0 [kN/m3]

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Definition of Finite Elements Weight buoyancy Temperat. coeff.

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15.0 [kN/m3] 1.00E−05 [−]

E L A S T I C B E D D I N G No. Cs[kN/m3] Ct[kN/m3] pr[kN/m2] py[kN/m2] 2 5.0000E+04 0.0000E+00 .00 .00

F L A T E L E M E N T S EL−No Nodes 1001 1 2 12 11 1002 2 3 13 12 1011 11 12 22 21 1012 12 13 23 22 2001 1 2 102 101 2002 2 3 103 102 2101 101 102 202 201 2102 102 103 203 202 3003 3 13 113 103 3013 13 23 123 113 3103 103 113 213 203 3113 113 123 223 213 4022 22 21 121 122 4023 23 22 122 123 4122 122 121 221 222 4123 123 122 222 223 5011 11 1 101 111 5021 21 11 111 121 5111 111 101 201 211 5121 121 111 211 221 6201 201 202 212 211 6202 202 203 213 212 6211 211 212 222 221 6212 212 213 223 222 H = Bedding normal (local z)

MNo 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T(m) .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200 .200

tan[−] .00

C(kN/m3) 5.000E+04H 5.000E+04H 5.000E+04H 5.000E+04H

c[kN/m2] w[kN/m3] .00 .0

local−x (xyz) 1.000 .000 1.000 .000 1.000 .000 1.000 .000 1.000 .000 1.000 .000 1.000 .000 1.000 .000 .000 −1.000 .000 −1.000 .000 −1.000 .000 −1.000 −1.00 .000 −1.00 .000 −1.00 .000 −1.00 .000 .000 1.000 .000 1.000 .000 1.000 .000 1.000 1.000 .000 1.000 .000 1.000 .000 1.000 .000

.000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000

Lastly, the following figure shows a graph of the structure by GRAF with all nodes and element numbers, along with a graph of the structure with hidden edges and the local element coordinate systems on the right.

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Box − Structure graph

5.7.

Calotte Shell

The analysis of a calotte shell with quadrilateral elements is a sophisticated FE−problem. It is obvious that one should arrange element edges along the length and the width directions. This way, however, triangular elements re
sult at the North and the South Pole.

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Standard mesh This partitioning is not optimum for several reasons. On one hand, the ob
tained results are naturally not fully symmetric. More important on the other hand is the different stiffness of the triangles in case of very thin shells. A mesh partitioning though with quadrilaterals only is perfectly possible. The input of such a mesh "on foot" can be done in a very elegant way, even if it does not look that easy. For this purpose let one think as a first step of a cube inscribed in the sphere. Its numbering is the rational result of distinct in
crements in the three edge directions, which would likewise be selected in a full three−dimensional discretization with BRIC elements.

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Definition of Finite Elements

Cube in octants As a next step all nodes are stretched radially onto the sphere surface. Un
fortunately, this is not a practical method because neither GENF nor MONET or AutoCAD offer such a function. An alternative construction subdivides the sphere surface into 8 segments. Great circles are drawn from the piercing points of the eight space diagonals to the corners of the spherical triangles. Each of these arcs can then be uni
formly subdivided, thus the total of the 24 quadrilaterals can be subdivided through great circles into n x n quadrilaterals.

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Definition of Finite Elements

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Great circles for mesh partitioning A relatively easy method by means of CADINP has been presented here. The transformation by means of spherical coordinates is a basic idea. The se
quence of the two rotation angles should not be overlooked, because it is criti
cal during a generation. A special transformation matrix of the incremental rotation is employed therefore, instead of the classical spherical coordinates. If one subdivides all rotation angles into small increments and then rotates in turn about one of the coordinate axes each time, one obtains a transform
ation which no longer depends on the sequence of rotations. With the rotations φi = tanϕi the matrix of transformation is: 2 2 ȧ ȧ 1 * ǒfy ) fz Ǔ @ CC
fz @ SS ) f xfy @ CC
−fy @ SS ) f xfz @ CC
ȧ ȧ ȧ ȧ ȧ
−fz @ SS ) f xfy @ CC
1 * ǒf2x ) f2z Ǔ @ CC
fx @ SS ) f yfz @ CC
ȧ ȧ ȧ ȧ ȧ 2 2 ȧ fy @ SS ) f xfz @ CC
−fx @ SS ) f yfz @ CC
1 * ǒfx ) fy Ǔ @ CC
ȧ ȧ ȧ

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ψ2 = φx2 + φy2 + φz2 CC = (1−cosψ)/ψ2 SS = sinψ/ψ if φx = 0, and the original vector is (1,0,0), then the resulting coordinates of the rotation are: | cosψ

−φz·SS

φy·SS |

A Macro is defined for the input, which defines the nodes of a quadrilateral field each time. The starting point for the Macro is the X− axis, and rotations of 45 degrees about the Y− and Z−axis take place each time. The other three faces of an octant are described via cyclic interchange of the node labels. The variable #90 stands for the chord angle ψ. During calculation of SS the special case ψ=0 must be controlled by a special LOOP, which should execute only once or not at all. PROG GENF HEAD QUAD−PARTITIONING OF A SPHERICAL SURFACE syst spac MAT 1 e 2.1E8 GRP 0 T 0.005 LET#4 4 LET#9 10.

$ Mesh partitioning (