Manual
Starting from program version 1.19 Copyright Cubus AG, Zürich www.cubus.chTable of Contents
Part ABase Module
. . .
A−1
A 1 Introduction
. . .
A−1
A 1.1 Changes since CEDRUS4. . . A−1
A 2 Basic Theory
. . .
A−3
A 2.1 Element Model and Solution Method . . . A−3 A 2.2 Modelling . . . A−4 A 2.2.1 Geometry of the Plan Outline . . . A−4 A 2.2.2 Slab Thickness and Material . . . A−5 Isotropic Material . . . A−5 Drilling%Soft Material . . . A−5 Orthotropic Material . . . A−5 Downstanding Beams . . . A−6 A 2.2.3 Area%Supported Elements . . . A−7 A 2.2.4 Columns / Point Supports . . . A−7 Point Supports . . . A−7 Area Supports . . . A−7 A 2.2.5 Walls / Line Supports . . . A−8 A 2.2.6 Lines of Symmetry . . . A−9 A 2.2.7 Hinges. . . A−10 A 2.2.8 Loads . . . A−10 A 2.2.9 The FE Mesh . . . A−11 A 2.3 Actions and Limit State Specifications . . . A−12 A 2.3.1 Basic Considerations . . . A−12 A 2.3.2 Overview of the Limit State Specifications. . . A−12 A 2.3.3 Actions . . . A−13 Definition of the term ’Action’ . . . A−13 How actions are formed . . . A−13 The Dialog with the List of Actions . . . A−14 The Properties of an Action. . . A−14 The Action Specification . . . A−14 Automatically−generated Action Specifications . . . A−16 A 2.3.4 Limit Values of nonlinearly−determined Results. . . A−16 A 2.3.5 Limit State Specifications with Action Sets . . . A−17 A 2.3.6 Automatic Generation of Action Combinations . . . A−17 A 2.4 Load Transfer from Floor to Floor . . . A−20 A 2.4.1 Overview . . . A−20 Problem. . . A−20 Realization in CEDRUS%5A−20 . . . A 2.4.2 Load Export. . . A−21 Automatically generated Export Combinations . . . A−21 Manually created Export Combinations . . . A−22 Calculating the Export Combinations . . . A−22 A 2.4.3 Load Import. . . A−24 A 2.4.4 Checklist for the Load Transfer . . . A−25 A 2.5 Results . . . A−26 A 2.5.1 Raw Results . . . A−26
Required reinforcement . . . A−28 Forms of Presentation. . . A−29 A 2.6 Punching Shear Verification . . . A−31 A 2.6.1 The Punching Shear Resistance . . . A−31 Strengthened slabs . . . A−32 Required bending resistances . . . A−32 Notation for the tabular summary . . . A−33 SIA 162 − Specifics . . . A−34 SIA262 (Swisscodes) − Specifics . . . A−35 EC2 − Specifics . . . A−36 E%DIN 1045%1 − Specifics . . . A−36 DIN 1045 − Specifics . . . A−37 OeNorm B4700 − Specifics. . . A−37 A 2.6.2 The Punching Shear Load . . . A−38 A 2.6.3 The Punching Shear Verification . . . A−38
A 3 Working with CEDRUS5
. . .
A−40
A 3.1 Presentation Conventions for the Examples . . . A−40 A 3.2 Starting the Program . . . A−41 A 3.3 Opening a Calculation . . . A−41 A 3.4 The Control Tab Sheet. . . A−43 A 3.4.1 The Tab Sheet /Structure/ . . . A−44 Plan Outline, Openings . . . A−44 On the Input of Geometrical Objects . . . A−45 Material Properties . . . A−45 Undo / Redo . . . A−48 Supports . . . A−48 The Selection of Objects and the Right Mouse Button . . . A−50 Printed Documentation of the Structure Input . . . A−52 The CubusViewer . . . A−52 A 3.4.2 The Tab Sheet /Loads/ . . . A−53 The Dialog ’Actions’. . . A−55 The Dialog ’Load Import’ . . . A−55 A 3.4.3 The Tab Sheet /FE Mesh/ . . . A−56 A 3.4.4 The Tab Sheet /Calculation/ . . . A−57 A 3.4.5 The Tab Sheet /Results/ . . . A−58 Properties of Output Zones and Downstanding Beams . . . A−59 3D Presentation . . . A−59 A 3.5 The Layer Switches . . . A−60 The Layer Group DUser" . . . A−60 The Layer Group DResults" . . . A−60 A 3.6 The Documentation of a Calculation. . . A−61
Part BThe Graphics Editor
. . .
B−1
B 1 Overview
. . .
B−1
B 2 Interaction of Application with Graphics Editor
. . .
B−2
Table of Contents
B 2.2 The application window. . . B−2 B 2.2.1 The control of the application . . . B−2 B 2.2.2 The application status line. . . B−2 B 2.2.3 The window of the Graphics Editor. . . B−2 B 2.2.4 Toolbar of the Graphics Editor . . . B−3 B 2.2.5 The layer buttons. . . B−3
B 3 Screen Elements
. . .
B−4
B 3.1 Introduction. . . B−4 B 3.2 Toolbar . . . B−4 B 3.2.1 Graphics objects . . . B−4 B 3.2.2 Selection mode. . . B−5 B 3.2.3 Zoom functions . . . B−6 B 3.2.4 Undo / Redo functions . . . B−6 B 3.2.5 Settings . . . B−6 B 3.2.6 Working planes . . . B−6 B 3.2.7 Projection control . . . B−9 Parallel perspective. . . B−9 Point perspective. . . B−10 Eccentric perspective . . . B−10 The buttons for projection control . . . B−11 B 3.3 Layer buttons. . . B−13 B 3.3.1 Introduction . . . B−13 B 3.3.2 Layer groups . . . B−13 Close . . . B−13 Size adjustment . . . B−13 All layers visible . . . B−14 All layers invisible . . . B−14 New layer. . . B−14 Labels visible/ invisible . . . B−14 B 3.3.3 The layer group ’User’ . . . B−14 Active . . . B−14 Delete contents . . . B−15 Delete . . . B−15 B 3.3.4 Layer buttons. . . B−15 Visible . . . B−15 Grabbable . . . B−15 Selectability . . . B−15 Labels visible . . . B−15 Exclusively visible. . . B−15 Exclusively selectable . . . B−16 Select . . . B−16 Deselect. . . B−16 Sublayer visibility . . . B−16 Sublayer selectability . . . B−16 B 3.4 Coordinate fields . . . B−17 See also: . . . B−17 B 3.5 Context menus of the graphics area . . . B−18B 3.6.3 Paste . . . B−19 Introduction . . . B−19 Changing graphics object types . . . B−19 B 3.6.4 Export. . . B−20 B 3.6.5 Select all . . . B−20 B 3.6.6 Grabbing. . . B−20 B 3.6.7 Input options . . . B−21 B 3.6.8 Coordinates, distances . . . B−21 B 3.7 The Modify context menu . . . B−22 B 3.7.1 Attributes . . . B−22 B 3.7.2 Move . . . B−22 B 3.7.3 Moving labels. . . B−23 B 3.7.4 Rotating . . . B−23 B 3.7.5 Mirroring . . . B−24 B 3.7.6 Stretching . . . B−24 B 3.7.7 Duplicate . . . B−25 Duplicating in the Move tabsheet: . . . B−25 Duplicate in the Rotate tabsheet . . . B−25 Duplicate in the Fill tabsheet: . . . B−26 B 3.7.8 To the front / to the back . . . B−26 B 3.7.9 Cut. . . B−27 B 3.7.10 Delete . . . B−27 B 3.7.11 Copy . . . B−27 B 3.7.12 Deselect. . . B−27 B 3.7.13 Modify selection . . . B−27 B 3.8 The Point Input context menu. . . B−28
B 4 Input of Graphics Objects
. . .
B−29
B 4.1 Introduction. . . B−29 B 4.2 Point input methods . . . B−30 B 4.2.1 Introduction . . . B−30 B 4.2.2 Point input method ’Free’ . . . B−30 See also: . . . B−31 B 4.2.3 Point input method ’Absolute’ . . . B−31 See also: . . . B−31 Example of point input ’Absolute’ . . . B−31 B 4.2.4 Point input method ’Relative’ . . . B−32 See also: . . . B−32 Example of point input ’Relative’. . . B−32 B 4.2.5 Point input method ’Polar’ . . . B−33 1. Reference point . . . B−33 2. Direction . . . B−33 3. Distance . . . B−33 See also: . . . B−33 Example of point input ’Polar’ . . . B−33 B 4.2.6 Point input method ’Middle’. . . B−34 See also: . . . B−34 B 4.2.7 Point input method ’Intersection’. . . B−35 See also: . . . B−35 B 4.2.8 Point input method ’Normal’ . . . B−35 See also: . . . B−35 B 4.2.9 Point input method X−direction . . . B−35
Table of Contents
See also: . . . B−36 B 4.2.10 Point input method Y−Direction. . . B−36 See also: . . . B−36 B 4.2.11 Point input on a Help Line. . . B−36 B 4.3 Graphics objects. . . B−36 B 4.3.1 Introduction . . . B−37 B 4.3.2 Construction points . . . B−37 B 4.3.3 Handles. . . B−37 When moving . . . B−37 When inputting . . . B−38 B 4.4 Polygons. . . B−38 B 4.4.1 Start input. . . B−38 B 4.4.2 Correction. . . B−38 B 4.4.3 Completion. . . B−38 See also: . . . B−38 B 4.5 Circles and circular arcs . . . B−38 B 4.5.1 Introduction . . . B−38 B 4.5.2 Circular arc defined by 3 points . . . B−39 B 4.5.3 Circular arc defined by 2 points and centre of circle . . . B−39 B 4.5.4 Circular arcs defined by centre of circle and a point on the circumference . . . B−40 B 4.6 Dimension lines . . . B−41 B 4.7 Labels . . . B−41
B 5 Modifying Graphics Objects
. . .
B−42
B 5.1 Introduction . . . B−42 B 5.2 Selection . . . B−42 B 5.2.1 Introduction . . . B−42 Selection mode . . . B−42 Exclusive selection . . . B−43 Additive selection . . . B−43 Preselection . . . B−43 B 5.2.2 Select individual graphics objects. . . B−43 B 5.2.3 Select using a Window . . . B−43 Window from left to right . . . B−43 Window from right to left . . . B−43 B 5.2.4 Select with a polygon . . . B−44 B 5.2.5 Select with the context menus. . . B−44 B 5.2.6 Select with the keyboard . . . B−44 B 5.2.7 Seleting by means of search criteria . . . B−44 B 5.2.8 Modify selection . . . B−44 B 5.2.9 Cancel selection. . . B−44 B 5.3 Working with attributes dialogues . . . B−45 B 5.3.1 Introduction . . . B−45 B 5.3.2 Contents and position . . . B−45 B 5.3.3 Opening the attributes dialogues . . . B−45 B 5.3.4 Closing the attributes dialogues . . . B−45 B 5.3.5 Inspect attributes. . . B−46 B 5.3.6 Different attribute values . . . B−46
B 5.3.10 Range of values of input fields . . . B−48 B 5.3.11 Invalid input values . . . B−48 B 5.3.12 Selection using search criteria . . . B−49
B 6 The Help System
. . .
B−50
B 6.1 Introduction. . . B−50 B 6.2 Viewing help documents . . . B−50 B 6.3 Navigating in document collections. . . B−50 B 6.3.1 Hypertext links . . . B−50 Words . . . B−50
B 6.3.2 Accessing documents . . . B−51 B 6.3.3 Full text search in document collections . . . B−51 Accessing Documents . . . B−51
B 6.4 Help on WorldView . . . B−51
B 7 Options
. . .
B−52
B 7.1 Input options. . . B−52 B 7.1.1 Introduction . . . B−52 B 7.1.2 The tabsheet ’Coordinates’ . . . B−52 Introduction . . . B−52 Origin . . . B−53 Directions . . . B−53 Save . . . B−53 Coordinate axes and rulers . . . B−53 B 7.1.3 The tabsheet ’Grabbing’ . . . B−54 Introduction . . . B−54 Grab modes . . . B−54 Grid . . . B−55 Grab radius . . . B−55 B 7.1.4 The tabsheet ’Preselect’ . . . B−56 Preselected graphics objects . . . B−56 Preselect hint . . . B−56 B 7.2 Colours and line types. . . B−57 B 7.2.1 Introduction . . . B−57 B 7.2.2 Tabsheet . . . B−57 B 7.2.3 Style attributes. . . B−57 B 7.2.4 Copying a style table . . . B−57 B 7.2.5 Importing default values . . . B−58 B 7.3 Automatic save. . . B−58
B 8 Importing DXFData
. . .
B−59
B 8.1 Introduction. . . B−59 B 8.2 Use of imported graphics objects . . . B−59 B 8.2.1 As input help . . . B−59 B 8.2.2 Converting DXF elements. . . B−59 B 8.3 Large DXF files . . . B−59 B 8.4 The import dialogue . . . B−60 B 8.4.1 Button bar. . . B−60 B 8.4.2 List of DXF layers . . . B−61
Table of Contents
B 8.4.3 Dimensions of the visible layers . . . B−61 B 8.4.4 Circular arc subdivision . . . B−61 B 8.4.5 Tolerance value for short lines . . . B−61 B 8.4.6 Coordinate transformation . . . B−61
B 9 Key Combinations
. . .
B−63
Part CThe CubusViewer
. . .
C−1
C 1 Introduction
. . .
C−1
C 2 Creating a report
. . .
C−1
C 3 Preview of the report
. . .
C−1
C 4 Document styles
. . .
C−2
C 4.1 Introduction. . . C−2 C 4.2 Choosing a document style . . . C−3 C 4.3 Modify or create a document style . . . C−3 C 4.3.1 Paper format . . . C−3 C 4.3.2 Page borders . . . C−4 C 4.3.3 Header and footer. . . C−4
C 5 Modify the report
. . .
C−6
C 6 Printing a report
. . .
C−7
Part DReinforcement and
Ultimate Load Analysis
. . .
D−1
D 1 Introduction
. . .
D−1
D 1.1 Reinforcement Module. . . D−1 D 1.2 Ultimate Load Module . . . D−2
D 2 Basic Theory
. . .
D−3
D 2.1 Material Model. . . D−3 D 2.1.1 Moment%Curvature Diagram . . . D−3
D 2.1.4 Plastic Moment of Resistance . . . D−6 D 2.2 Reinforcement Model . . . D−7 D 2.2.1 Reinforcement Fields . . . D−7 D 2.2.2 Slab and Downstanding Beam Reinforcement . . . D−7 Downstanding beam reinforcement. . . D−8 D 2.2.3 Sloping Reinforcement. . . D−8 D 2.3 Solution Method . . . D−9 D 2.3.1 Nonlinear Analysis. . . D−9 Incremental, iterative calculation . . . D−10 Influence of the initial state and load history . . . D−10
D 3 Reinforcement Module
. . .
D−11
D 3.1 Basic Concepts . . . D−11 D 3.1.1 Dimensioning for Ultimate Load. . . D−11 D 3.1.2 Elastic Design as an Optimization Task . . . D−12 D 3.1.3 Optimum Plastic Dimensioning. . . D−13 D 3.1.4 Dimensioning for Serviceability. . . D−14 D 3.2 Dimensioning Process . . . D−16 Introductory example. . . D−16 D 3.2.1 Input of the Reinforcement and Elastic Dimensioning. . . D−16 D 3.2.2 Preparation of the Output Zones . . . D−17 D 3.2.3 Creating a Layout . . . D−17 D 3.2.4 Input of Reinforcement Fields . . . D−17 Bottom reinforcement. . . D−17 Duplicating in another direction . . . D−19 Construction based on elastic reinforcement requirement. . . D−19 Additional reinforcement . . . D−20 Settings for the display of results . . . D−21 Top reinforcement . . . D−21 Additional reinforcement . . . D−21 Positioning of the field label . . . D−22 D 3.2.5 Elastic Dimensioning . . . D−22 D 3.2.6 Optimum Dimensioning . . . D−23 Design Monitor . . . D−23 D 3.2.7 Interpretation of the Results . . . D−24 D 3.2.8 Production of the Reinforcement Drawings. . . D−25
D 4 Ultimate Load Module
. . .
D−29
D 4.1 Basic Concepts . . . D−29 Nonlinear Calculation Method . . . D−29 D 4.1.1 Calculation Model . . . D−29 Nonlinear Moment%Curvature Relationship. . . D−30 Hardening . . . D−30 Load History . . . D−30 Partial Safety Factors . . . D−30 D 4.1.2 Termination Criteria for Calculation . . . D−31 D 4.1.3 Calculation Control . . . D−33
Part EPrestressing Module
. . .
E−1
E 1 Introduction
. . .
E−1
Table of Contents
E 1.2 Tendons and Supports. . . E−1 E 1.3 Input Procedure . . . E−1
E 2 Basics
. . .
E−3
E 2.1 Tendon Geometry . . . E−3 E 2.1.1 Geometrical description in the plan view . . . E−3 E 2.1.2 Geometrical description in the side view . . . E−3 Support and point attributes. . . E−4 Examples of generated tendon profiles . . . E−4 E 2.2 Force Variation along Tendon and Friction Losses . . . E−8 E 2.3 Deviation Forces . . . E−8 E 2.4 Dimensioning of the Non−Prestressed Reinforcement . . . E−9 E 2.4.1 Prestressing as an external action . . . E−9 E 2.4.2 Prestressing as self%equilibrating stress state . . . E−9 The dimensioning conditions . . . E−9 Dimensioning as beams . . . E−10 Cross sectional resistance . . . E−10 Constraint moments . . . E−12 Direction of the tendons and of the associated beam sections . . . E−12 E 2.4.3 Use of program . . . E−13 Input in the tabsheet ’Calculation’ . . . E−13 Input in the Tabsheet ’Results’ . . . E−14 Remarks on the Results Tables . . . E−15
E 3 Examples
. . .
E−17
E 3.1 Flat Slabs . . . E−17 E 3.1.1 Description of problem . . . E−17 E 3.1.2 Tabsheet ’Structure’ . . . E−17 E 3.1.3 Tabsheet ’Loads’ . . . E−17 E 3.1.4 Tabsheet ’Prestressing’ . . . E−18 Input of tendons and supports . . . E−19 Checks. . . E−21 E 3.1.5 Tabsheet ’Calculation’. . . E−24 E 3.1.6 Tabsheet ’Results’ . . . E−25 E 3.1.7 Checklist . . . E−25 E 3.2 Two Span Beams. . . E−27 E 3.2.1 Problem description . . . E−27 E 3.2.2 Results. . . E−28 E 3.3 Tips and Tricks . . . E−31 E 3.3.1 Duplication of groups. . . E−31 E 3.3.2 Generating possibilities . . . E−31 E 3.3.3 Checks. . . E−31 E 3.3.4 Detecting tendons lying on top of each other. . . E−31 E 3.3.5 Reference height of supports with different slab thicknesses . . . E−31
F 1.2 Mass Distribution . . . F−1 F 1.3 Input parameters. . . F−2 F 1.4 Output of Results . . . F−2
Part GWalls
. . .
G−1
G 1 Introduction
. . .
G−1
G 2 Basic Theory
. . .
G−1
G 2.1 Element Model and Solution Method. . . G−1 G 2.2 Modelling . . . G−2 G 2.2.1 Geometry. . . G−2 G 2.2.2 Thickness and Material . . . G−2 Isotropic Material . . . G−3 Orthotropiv Material . . . G−3 G 2.2.3 Point Supports. . . G−4 G 2.2.4 Line Supports . . . G−4 G 2.2.5 Lines of Symmetry . . . G−4 G 2.2.6 Loads . . . G−5 G 2.3 Results . . . G−5 G 2.3.1 Raw Results. . . G−5 G 2.3.2 The Structuring of the Output of Results . . . G−6 The quantities for load cases and load case combinations. . . G−6 Quantities for limit state specifications (envelope values) . . . G−7 Required reinforcement . . . G−7 Forms of Presentation. . . G−8
G 3 Slab with Normal Forces
. . .
G−9
G 3.1 Model . . . G−9 G 3.2 Changing the Structural Type . . . G−10 G 3.3 Results . . . G−10
Part ABase Module
A 1 Introduction
CEDRUS5 is a Finite Element program for the linear elastic, static and dynamic analysis of plate structures (both for bending and in−plane actions). It is especially conceived, however, for reinforced concrete slabs and provides the corresponding reinforcement for limit states as well as the reinforcement contents.
Besides the basic module for plates the following modules are available as options: S Plates subjected to in−plane actions (plane stress states)
S Ultimate load capacity for slabs
S Optimum design (with possible plastic redistribution) of the slab reinforcement S Dynamics
S Prestressing
The handbook available in printed form comprises Chapter 2 .Basic Theory" with a de scription of the general principles and scope of CEDRUS5, without treating matters of program handling, and Chapter 3 .Working with CEDRUS5", which with the help of an example gives an introduction to the use of the program.
. It is highly recommended that each user for getting started with CEDRUS-5 works through these two short chapters completely, before venturing on a serious calculation.
Besides these two printed chapters there is also the Help System of CEDRUS5, which during program execution provides contextsensitive help on all aspects of program handling. It is also equipped with search functions for any technical terms and thanks to the many hyperlinks (navigation aids to further information) one can obtain the re quired information quickly. Some information is also given in Chapter 3 on the use of the Help System.
CEDRUS5 is suitable for the solution of complex problems. But this means that there are various sources of error, from the static modelling to data input, numerical problems, interpretation of results and finally possible programming errors, which for such exten sive software cannot unfortunately be eliminated despite all care taken in the develop ment work. Thus the main requirements for the successful application of CEDRUS5 are an adequate theoretical background and checking the results by means of rough cal culations and plausibility considerations.
CEDRUS5 is continually being improved by the Cubus corporation. Therefore, criti cisms, suggestions and special wishes from the side of engineering practice are always welcome. Our clients will, of course, be informed of any major changes or develop ments.
We reserve the right of small deviations of the program from the printed description in the sense of selfevident changes in the dialogue.
tion generation. Besides these aspects a lot of productivity tools and functions have been implemented. The most important changes are listed here:
User Interface
S Functionality was added to the graphics editor: Objects can now also be stretched, rotated, searched for properties, selected with polygons, renumbered, labelled with the point coordinates (and the labels easily displaced via context menu). Dialogs do automatically shrink when objects are constructed. The structure can be rendered. Shortcuts <F12>/<F11> for ’Calculate’ and ’Create print entry’ and help documenta tion in pdf−format are now available. A ’direct conversion’ (from clipboard) function was introduced (e.g. making DXF−import even easier than before).
S Automatic generation of a net box (default). Model
S Introduction of ’project materials’ and the corresponding material manager (Menu Settings>Materials). Project materials are containers for all material specific values (material class according to the code, mass, Poisson ratio, temperature deviation co efficient etc.).
S Stiffness factor for zones (material boxes, downstanding beams). S Analysis parameter sets (Menu Settings>Analysis parameters). S Support for Swisscode.
Loads, Actions, Limit State Specifications
S Actions and Limit State Specifications were reorganized, automatic generation ex tended to Swisscode/Eurocode.
S Load cases have now text Ids.
S Load case ’Dead load’ is autom. generated. The self weight is defined via acceleration loads (and the mass from the project material).
S Loads for differential temperature.
S Detailed legend of load objects with netto sum.
S Improved load transfer from floor to floor. (dialog ’Load export’ in the tabsheet ’Loading’).
Results, Output
S Envelop values for area supports. S Report generator.
S Result combinations of nonlinear calculations. S Dimensioning according to Swisscode. S Punching according to Swisscode. S Improved numerical output. S Output of edge stresses.
A 2 Basic Theory Part A Base Module
A 2 Basic Theory
A 2.1 Element Model and Solution Method
CEDRUS5 is a Finite Element (FE) program primarily for the calculation and design of reinforced concrete slabs. The linearelastic FE calculation also permits supports in capable to resisting tensile forces. The element models used are hybrid triangular and quadrilateral elements of arbitrary shape with the three displacement degrees of free dom vz (bending), rx, ry (rotations about the axes x,y) in the corner nodes.
ÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎ ÎÎÎÎÎÎÎ ÎÎÎÎÎ ÎÎÎÎÎ ÎÎÎÎÎ ÎÎÎÎÎ ÎÎÎÎÎ ÎÎÎÎÎ Z Y X vz ry rx
Bending actions are considered, but not membrane forces. The most important element properties are:
S Quadratic functions for moments within elements. S Cubic functions for displacements at element boundaries.
S Section forces in corner nodes as well as at the centre of an element. S Possibility of area supports (if desired without tensile fixture).
The elements are among the best available today for this area of application. They were first proposed by Pian and are to be found in numerous programs. For a detailed study of this element model refer to the following:
S J.P. Wolf: .Generalized Stress Models for Finite Element Analysis", Institute of Struc tural Engineering, ETH Zurich, Report Nr. 52, 1974 Birkhäuser, Basle.
S U. Walder: .Beitrag zur Berechnung von Flächentragwerken nach der Methode der Finiten Elemente", Institute of Structural Engineering, ETH Zurich, Report Nr. 77, 1977 Birkhäuser, Basle.
The FE method in CEDRUS5 essentially involves the following steps: 1. Determination of the element matrices for the hybrid method.
2. Determination of the load vectors (right hand sides of the system of equations). 3. Summation of the element stiffness matrices to form the global stiffness matrix. 4. Solution of the resulting system of equations for the unknown nodal displacement
parameters (possibly iteratively in the case of supports that do not take tension). 5. Calculation of the section forces in the elements for the now known nodal displace
ment parameters.
One should observe that the FE method is an approximate numerical method. The numerical solution, however, converges for an ever finer element mesh, within the limits of numerical accuracy, to the exact theoretical solution of Kirchhoff’s plate bending theory.
A 2.2 Modelling
A 2.2.1 Geometry of the Plan Outline
The geometry of the plan outline is basically fixed by the following conditions: S Outline: An arbitrary closed polygon.
S Recesses and openings: Arbitrary closed polygons.
S Downstanding beams: Wall−like lines or polygons of specified width and arbi trarily directed closure lines. Downstanding beams may intersect, but at most two at any one place.
S Walls: Wall−like lines or polygons of specified width and arbitrarily directed closure lines. Walls are modelled as line supports. The position of the support axis can be chosen anywhere within the wall (centrical, eccentrical). The wall outline has a vis ual function and is irrelevant to the calculation model.
S Columns: Rectangles or parallelograms to model columns. The user has the choice between point or area support.
S Material separators: Lines or polygons which divide up the slab into several zones with different material attributes.
S Hinges: Lines, along which and normal to the direction of the line only shear forces and no moments are transmitted.
S Lines of symmetry: Along these lines the slab may bend but normal to them it may not rotate. Thus one has a special type of linear support, which may only lie on the slab boundary.
ÏÏ ÏÏ ÏÏ ÏÏ ÏÏ ÏÏ Ï Ï ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏÏÏÏÏ ÏÏ ÏÏ ÏÏ ÏÏ Outline Opening Material separators Columns Hinge Wall Down standing beams
The distance between the corner or end points of the outline elements as well as the distance of these points to lines of the outline elements may not be less than a minimum permissible length. In the case of walls only the wall axis is relevant in this respect. This minimum length depends on the FE method and is preselected by the program. It can, if necessary, be changed by the user, provided appropriate attention is given to numeri cal effects.
Material properties and the slab thickness are constant within each zone. For later pro gram versions a linear variation within a zone is planned. Note, on this question one can get help within the program itself.
A 2 Basic Theory Part A Base Module
A 2.2.2 Slab Thickness and Material
A slab can be subdivided into different zones by material separators. Downstanding beams automatically create their own material zones and also act as material separators. In the case of columns one has the choice of giving them their own material properties or using the material of the zone in which they lie. In the first case the boundary of the column also acts as a material separator.
CEDRUS5 recognizes the following material models: isotropic, orthotropic, drilling% soft and downstanding beam which are described in detail in the following sections. Downstanding beams have implicitly a downstanding beams material, while columns with their own material are isotropic. Any of the material models listed above can be assigned to the other zones.
In the description of the material models the following notation is used: mx, my ,mxy: slab moments
kx, ky, kxy: slab curvatures
E,n : elastic modulus and Poisson’s ratio d thickness of the slab
Isotropic Material
Isotropic material is directionally independent and is completely described by two elas tic constants, i.e. the modulus of elasticity (or Young’s modulus) E and Poisson’s ratio n as in Kirchhoff’s plate theory. The relationship between plate curvatures and plate mo ments is governed by the following elasticity matrix (i.e. a matrix containing the elastic constants):
NJ
Ëx Ëy ËxyNj
+ 12f EEd3ȧȱȲ
1 * n 0 * n 1 0 0 0 2(1) n)ȧȳȴ
NJ
mx my mxyNj
whereby d = thickness of the element, fE = factor of the stiffness (deafult=1.0)
Drilling-Soft Material
If the third diagonal coefficient of the elasticity matrix for isotropic material is chosen to be very large, then the slab is very soft in drilling action with respect to selectable directions (x,y) and therefore no drilling moment mxy can be resisted. In special cases
this can be useful, e.g. if one wants to have no drilling reinforcement thereby accepting larger values of mx and my. It should be noted, however, that in certain cases without
drilling moments equilibrium is not possible.
Orthotropic Material
Orthotropic material exhibits different properties in the two directions x and y normal to one another. It is described by the following elasticity matrix:
NJ
Ëx Ëy ËxyNj
+ȧȱȲ
d11 d21 0 d12 d22 0 0 0 d33ȧȳȴ
NJ
mx my mxyNj
(d12+ d21x 0) (Ëxy+ 2 ē 2w ēxēy , w+ bendingdeflection)Downstanding Beams
Downstanding beams (just called beams in the following) are treated in CEDRUS5 as orthotropic elements using the following elasticity matrix:
NJ
Ëx Ëy ËxyNj
+ 1f Eȧȧȧ
ȧ
ȱ
Ȳ
12 Eh3 0 0 0 12 Ed3 0 0 0 24 Ed3ȧȧȧ
ȧ
ȳ
ȴ
NJ
mx my mxyNj
whereby h = fictitious height of underbeam
d = thickness of neighbouring element (=input value) fE = factor of the stiffness (deafult=1.0)
Thus the beam elements have the full stiffness in the longitudinal direction (on h see below), whereas for the transverse and drilling stiffness the thickness of the neighbour ing slab is used.
To determine the fictitious beam height h model 4 described below is used. It lies be tween the extreme cases represented by models 2 and 3.
Model 1: Tsection with effective width B
This model corresponds best with reality. It cannot, however, be applied, since CE DRUS5 does not take membrane effects (i.e. membrane strains) into account.
Model 2: Beam eccentric
Since the neutral axis in slabs lies in the middle, the stiffness here is too high.
In* Inslab+ bh 3
12 ) bhe
2 (I
nslab+ IpartofslabofwidthB * b)
Model 3: Beam centric
The stiffness of this model is too small, since the eccentricity is neglected.
In* Inslab+ bh 3
12
Model 4: Lies between model 2 and model 3 = CEDRUS5 model
In* Inslab+ bh 3 12 + 12
ǒ
bh3 12 ) bhe 2) bh3 12Ǔ
; h+ h 3) 6e2h 3 ǸFor intersecting beams, in the zone of intersection an orthotropic material is used which exhibits in the direction of the higher underbeam, if present, the bending stiffness, while normal to it as well as for drilling action that of the other beam. More than two beams may not intersect at a point. If an beam connects laterally to another one, it shold inter sect this one (and not just touch) in order to permit a correct introduction of its shear force. x y d Illustration b B h d n h n e h n h n h
A 2 Basic Theory Part A Base Module
A 2.2.3 Area-Supported Elements
Areasupported slab zones exhibit a uniformly distributed reaction pressure for each finite element and the average settlement is determined on the basis of the given modu lus of subgrade reaction and the support pressure. If the modulus of subgrade reaction ks has a very high value, an element may theoretically be free to bend and rotate, but
on average it cannot exhibit settlement. With an areasupported element alone or even several acting on one line, therefore, a stable support of the slab is not guaranteed. A good visual model for the areasupported element is the liquid filled cushion:
ÏÏÏÏ
liquid filled cushion stiff plate elastic support Element Subgrade reaction ks: ks = [kN/m3] support pressure settlement
A use of areasupported elements is the supports of columns. The support elements are described in the next section.
Another use of areasupported elements is the Modulus of Subgrade Reaction Method. Areasupported elements can be employed with or without tension capabil ities. In the latter the program reaches the correct solution by means of iteration.
A 2.2.4 Columns / Point Supports
The column object of the CEDRUS5 Input Module offers the following modelling possi bilities. Special columns, which cannot be modelled in this way, must be modelled with other support types.
For each column the settlement sdz (or the modulus of subgrade reaction ks for areasup
ports) and rotational stiffnesses srx, sry can be specified. If one provides reasonable va
lues of elasticity modulus and column height, then the stiffnesses can be calculated automatically by the program using the following formulas:
sdz+ EAh ; srx+
4EIx
h ; sry+
4EIy
h E: elasticity modulus of the column
A: sectional area of column = area of input column object Ix,Iy: second moments of area of column section
h: column height
There are two support models for columns: area supports and point supports.
Point Supports
Since point supports, because of the moment singularities resulting from plate theory, involve rather problematic modelling, they should mainly be used only when an area support is difficult to include in the input. Whereas for an area support the sides of the column quadrilateral represent fixed lines for the FE mesh, with point supports it is only a question of a point. With point supports the column dimensions are used for punching shear verification and to determine the stiffnesses of the column.
ÏÏ ÏÏ column section supported area ÏÏ ÏÏ 1) 2) ÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏ Support area subdivided into several elements b a ks
with the effective column cross section, which can either be a rectangle or a circular section. In the following cases it is better if the supported zone is not identical with the column section:
1) Columns of circular section of rather small diameters cannot be modelled with a rea sonable FE mesh.
2) For small diameter columns one chooses with advantage the support zone to be somewhat larger than the column section, in order to obtain a more homogeneous FE mesh and also more realistic column moments.
The user has to decide whether an area support should be with a single finite element or with subdivision into several elements.
Area supports with one element are necessary if a column should not resist moments. If in spite of this a bending stiffness is specified, then this is distributed by the programm automatically in the form of bending stiffnesses to the four corner nodes.
Area supports distributed over several elements are unavoidable if their sides are divided up by other geometrical input objects like material separators. They may be de sirable if one wants a finer FE mesh over a column. Distribution over several elements, however, means that such a column always exhibits a certain bending stiffness, since each element acts like an independent support spring. With the value of ks for the modu
lus of subgrade reaction of the supported elements the bending stiffness for a rectangu lar column section resulting from the area support (see figure alongside) is given ap proximately by:
sr+ ab 3
12 ks
If one determines ks from the settlement of the column (see following figure), then as
suming that the supported zone is identical with the rectangular column section ks=E/h
and thus
sr+ Eab 3
12h .
If the column section deviates from the shape of the support zone, then these consider ations about stiffness can be adapted accordingly.
A column of the same dimensions fixed at its base (column extreme right in figure), on the other hand, exhibits a bending stiffness four times higher:
sr+ Eab 3 3h ÏÏÏ ÏÏÏ h a ÏÏÏ ÏÏÏ ÏÏÏ ÏÏÏ ÏÏÏ ÏÏÏ 1 1 1 b ks+ N ab+ Eh N+ Eab h M+ 1 @ sr M+ 1 @ sr
Since it is not allowed to fall below the stiffness sr , this value represents a minimum
for the input. Higher stiffnesses, as for a column with one element, are modelled by a rotational stiffness uniformly distributed over all column nodes.
A 2.2.5 Walls / Line Supports
Line supports are input in CEDRUS5 basically as wall objects. Wall objects are polygons of selectable width with information about modulus of elasticity and wall height.
A 2 Basic Theory Part A Base Module
The computational model, however, is a line support, whose axis can be feely chosen within the inputted wall. These support axes are fixed lines in the FE mesh and thus are subject to the conditions of minimum distance between each other and to other structure objects. The wall outline only serves a visual purpose and is not bound to any geometrical consistency condition.
The settlement stiffness sdz and the bending stiffnesses srx and sry of the walls are de
fined per unit length of wall and can be freely input or determined by the program on the basis of wall thickness and height and elasticity modulus. They are defined as fol lows:
sdz: force per unit settlement; sdz = Et/h
srx: moment for a unit rotation about support axis; srx = Et3/3h
sry: moment for a unit rotation normal to support axis
The program does not calculate any stiffness for sry. This stiffness component is difficult
to model, but is only of secondary importance and usually is either set to .blocked" or .free".
A 2.2.6 Lines of Symmetry
Lines of symmetry are special line supports, along which the slab is free to bend, about whose axis, however, it cannot rotate. Symmetry conditions are really only meaningful at slab boundaries, which is why lines of symmetry are only allowed to lie on the plan outline.
Lines of symmetry are used to demarcate parts of the slab in the model, be it a genuine line of symmetry or if by means of a symmetry condition one obtains the most favour able boundary condition. Note, that with symmetry conditions the loads too always act symmetrically.
An example of genuine symmetry is given by the circular slab − even if with the condi tion of a symmetrically acting load it is rather academic. Here the input of a sector with the corresponding symmetry conditions along the radial boundaries suffices.
Lines of symmetry on walls are not allowed. If a wall, however, has to act as a line of symmetry then the rotation about its support axis is blocked. But the user is respon sible for reducing the settlement stiffness by 50%.
Point supports on lines of symmetry are permitted. Their xdirection, however, has to coincide with the direction of the lines of symmetry. The program then blocks the rotation about the xaxis, but does not reduce any stiffnesses due to symmetry. The user has to supply the corresponding values. A simple possibility is to halve the elasticity modulus of the column.
Point supports at the intersection of two lines of symmetry are also allowed. If these form an angle of 180 degrees, then one has the same case as if the line of symmetry would run right through (previous section). Otherwise the input direction of the column is unimportant and the program blocks both rotations. The user is again responsible for the correct input of the settlement stiffness.
Lines of symmetry
A 2.2.7 Hinges
Normal to hinges only shear forces and no moments are transmitted.
Hinges are preset (i.e. fixed) lines in the FE mesh and thus are subject to the conditions of minimum distance between each other and to other structure objects. Several hinges in a chain are allowed but no branching.
The hinge lines are implemented in the program as double nodes with the correspon ding nodal connections.
A 2.2.8 Loads
CEDRUS5 permits the following types of load: 1) Area loads (i.e. loads per unit area)
Rectangular or arbitrary polygons for: − body force (e.g. dead or selfweight) − uniformly distributed force
− differential temperature loading 2) Line loads
Constant or trapezium distributed − forces
− moments (about loading line) 3) Point loads
− forces
− moments about x or ydirection
4) Displacements of point supports (the corresponding displacement parameters of the support nodes have to be blocked)
− settlements
− rotations in x or ydirection 5) Influence fields for
− moments mx, my, mxy in the global coordinate directions
− bearing pressure of areasupported elements
By introducing a unit point load or a unit support displacement one can also obtain influence fields for point displacements and point reaction forces, respectively.
The load types 1) to 3) are independent of the FE mesh, so that they can be arbitrarily arranged geometrically.
Loads are combined to individual load cases, which can be combined or superimposed in any way for the calculation of the results. The load cases can be assigned to particular action types, like dead weight loads, surcharge loads, imposed loads etc., whereby in standard cases a load superposition automatically carried out by the program to deter mine the design section quantities is possible (see Chapter 2.3).
The influence field load cases are a special case. Each influence field requires a load case, which cannot be combined with other load cases. It should be noted that in the region of the influence point a very fine FE mesh is needed to obtain sufficiently accu rate results.
A 2 Basic Theory Part A Base Module
A 2.2.9 The FE Mesh
An important part of modelling in an FE calculation is to have a suitable mesh subdivi sion, so that accurate results can be obtained with acceptable input, calculation and data storage expenditure. There is no simple recipe. One factor is element quality (see Ch. 2.1). Further, in an FE calculation it is always important to know which results are of primary interest. To want to obtain all results exact in every detail according to plate theory is uneconomic and also not necessarily desirable (e.g. infinitely large moments at point supports and in recessed corners).
In CEDRUS5 the FE mesh is automatically created within zones if certain parameters are provided. The zone boundaries are automatically defined by polygon line chains (mesh separators), which themselves represent mesh lines and thus fulfil the conditions regarding the distance of separation between structure objects. Automatic mesh gener ation can also be suppressed in individual zones. In these zones the mesh is input man ually.
The following parameters influence automatic mesh generation within a mesh zone: S Maximum element side length: This important parameter defines the fineness of the
mesh.
S Minimum element side length: The minimum element side length is usually gov erned by the structure’s geometry. If in regions without geometrical restrictions of the structure one does not want, if possible, to go below a certain element side length, then with this optional parameter one can specify a corresponding value. But only values in the range 0.1 to at most 0.5 of the given maximum element side length are meaningful.
S Direction of the mesh lines. The mesh is created by two families of lines in the given directions, whereby all input structure objects have to be taken into account and ad justed to. It is recommended, if possible, to choose these directions orthogonal to one another.
The program considers area supports to be separate mesh zones, which are automati cally treated taking into account the input column attributes.
A 2.3 Actions and Limit State Specifications
A 2.3.1 Basic Considerations
The aim of all structural analysis is ultimately the dimensioning of a structure. This is based on limit states, which requires among other things the selection of design situ% ations with the associated load cases.
Each load case is characterised by a leading action and a simultaneously acting accom panying action and thus consists of a weighted combination of actions.
An experienced engineer can − at least for preliminary dimensioning − often limit the consideration to a few points and also without much effort can recognize the critical load cases for the investigated design situations.
The strength of a program however lies in the systematic treatment of numerous sections or points. For many dimensioning tasks it is best to work with limit values of section forces, reactions or displacements. These are determined by the Cubus programs on the basis of limit state specifications, which uniquely describe the combination rules for the individual loading. How these limit state specifications are arrived at is described below.
A 2.3.2 Overview of the Limit State Specifications
A simple limit state specification at the highest level looks as follows in the programs:
The considered actions are dealt with in the left half of the dialogue, and in the right half the investigated combinations of these actions with the corresponding combination factors. How these combinations were obtained is clearly seen here: the permanent ac tions ’Dead Load’ and ’Surcharge/Live Loads’ are investigated with the factors γsup (here
1.35) und γinf (0.8). In addition there are the variable actions, of which on the one hand
the snow loads as leading action (γQ=1.5) and the wind load as accompanying action
(ψ0=0.6) and on the other hand the wind loads as leading (γQ=1.5) and the snow loads
as accompanying (ψ0=0.88) actions are considered. The load and accompanying action
factors depend on S the code S the actions
S the design situation
A 2 Basic Theory Part A Base Module
The design situation and the limit state are specified on creating a new limit state specifi cation by the user .
The list of actions in the left part of the dialogue is created automatically on the basis of the input loads, each of which is assigned to an action. The right part of the action combinations can be automatically generated, but also be arbitrarily defined by adding to, deleting or modifying columns. Regarding automatic generation see Chapter A 2.3.6. The programs CEDRUS5 and STATIK5 automatically create a limit state specification for the limit state (Type 2) of the ultimate limit state (structural safety) for the standard design situation.
The actions, which have not yet been discussed in detail, will be treated in the next chapter.
A 2.3.3 Actions
Each individual load case is strictly speaking an action. But as is evident from the previ ous chapter, the term ’Action’ is defined more narrowly here. Before it is defined pre cisely, the following terms are once again clearly defined:
Loads: As loads all elementary load elements are meant, which are available in a pro gram as actions on a structure (see also above in A 2.3.1).
Examples: concentrated loads, line loads, etc.
Loads are always summed up together in loadings (see below).
Load Cases are a type of container for individual loads. On the load side they represent the basic unit for which results can be calculated, and also from which actions are formed.
Definition of the term ’Action’
Actions are load cases grouped to form individual categories like dead loads, live loads, wind loads, snow loads, etc., which then finally are combined to form design situations in the limit state specification.
How actions are formed
During input new load cases are always assigned to an action. Thereby the most used actions available in the codes for the chosen structural type can be selected, whereby the user can also define his/her own actions.
For the creation of limit state specifications all load cases assigned to the same action are treaded as one. A user is still free to choose how the loadings interact to form an action, i.e. whether e.g. they can all act together or are mutually exclusive (e.g. action truck load: each position of the truck is a load case that is mutually exclusive). The corre sponding specification is called an action specification and is explained in detail later.
The Dialog with the List of Actions
In order that on the one hand one has an overview of the actions that can arise in an analysis with their properties and on the other hand to be able to define ones own ac tions, the programs have been provided with a special dialog:
Code prescribed action categories
(cannot be changed)
Actions created by user
Load factors and combination factors
The Properties of an Action
Action type
For an action type there are the following selection possibilities: ’permanent’, ’variable’, ’prestressing’, ’accidental’ or ’undefined’. The type influences the way in which action combinations are formed in the limit state specification.
Load Factors and Combination Factors
To each action category there belong load factors and, depending on the type, combina tion factors. In the case of the actions prescribed in the codes these values cannot be modified in the dialog. If this is necessary a user−defined action must be defined.
Action Sets and Action Groups
Several actions can together form an action set (e.g. several load models for a bridge). Such actions are not considered separately in forming the action combination, but the set appears as a whole in an action combination (e.g. as leading or accompanying ac tion).
The codes speak in this connection of action groups. This term is also supported in a general sense in the Cubus programs. By it is understood the combinations, in which the actions of a set have to be differentiated.
Example: The vertical and the horizontal live loads of a road bridge should be able to occur as follows:
1) vertical loads full, without horizontal loads, 2) vertical loads * 0.75, horizontal loads full.
Solution: The loads are defined by two actions, which together form a set. The set ap pears in two so called action groups with the following combination factors,: (1,0) and (0.75,1).
Which combinations should together form a set is specified in the above action dia logue. All actions which have the same names in the Set column form together an action set.
The Action Specification
To the full definition of an action there belongs the combination specification for the participating loadings. Whereas permanent actions often consist of just one loading, for
A 2 Basic Theory Part A Base Module
variable actions a complicated superposition may be necessaray. Take for example as an action the live loads acting on a multispan beam, for which loads have to be con sidered in an unfavourable position in the individual spans and in addition with a ve hicle load in different positions.
One can imagine as a simple combination scheme − denoted here by E1 − the compari son of all possible loading combinations. This may be represented in a loading scheme sequence as follows: Action = ( E1 ) or Loading combination where ( E1 ) Loading combination = Loading ( * factor )
Example: Spans with unfavourably applied live load for a three span beam
A B C
3 loadings A, B and C are introduced for the individual spans.
The specification of the action Live Load according to the scheme E1is: A or B or C or AB or AC or BC or ABC
In the case of a five span beam one would already have 31 load case combinations to compare one with another. If two spans and perhaps a vehicle load in n possible posi tions were added, then the user would find it rather challenging in terms of combina torial analysis and soon lose track of the number of loading combinations.
According on the other hand to the extended superposition scheme − denoted here by E2 − a compact and clear definition of all possible loadings is possible. This is best illustrated in a loading scheme sequence:
Action = = Loading combination wobei: ( E2 ) or Loading step permanent optional plus Loading step = Loading combination Loading ( * factor )
The scheme is based on an unconditional (permanent) or optional superposition of loading steps. In contrast to the scheme E1, no complete loading combinations are de scribed, but instead there is an instruction on how the results have to be superimposed in forming the limit values and how the limit values are formed.
From optional loading steps the value of the result for an extreme value in a point is only considered if it is decisive, that is the extreme value is increasedby the correspon ding amount. Thus a positive value increases a maximum value and a negative one de creases a minimum value.
A loading step consists in the simple case of a single loading or of a loading combina tion. It can however also consist of a series of loadings or loading combinations, of
A plus B plus C
In the case of a five span beam with loading in the spans A,B,C,D,E and an additional vehicle in 9 positions (a,b,c,..,i) the specification is:
A plus B plus C plus D plus E plus a or b or c or d or e or f or g or h or i
The user can specify an arbitrary number of actions following the superposition scheme E2.
. The scheme E1 is contained in E2 as a special case (without ’PERMANENT’ and ’PLUS’).
Automatically–generated Action Specifications
The programs automatically create for each action (exception: prestressing actions) an action specification according to the folowing rules:
Permanent actions: all associated loadings are added up. Such an action consists therefore of a fixed loading combination.
Variable actions: Each loading that is assigned to a variable action is also given during input the superposition attribute, ’additive’ or ’exclusive’. Additive loadings (a1,a2,...) can occur simultaneously, exclusive (e1,e2,...) are mutually exclusive (e.g. vehicle in different positions). These actions are formed corresponding to the superposition scheme E2 described above as follows:
a1 plus a2 plus a3 plus . . . plus e1 or e2 or e3 or ...
In cases, which are not covered by this simple rule, the specification must be done man ually.
In the programs these specifications are performed in a dialogue, which looks as follows and is self−explanatory:
A 2.3.4 Limit Values of nonlinearly–determined Results
When using loading combinations or the specification ’plus’ in the above schemes the programs superimpose results that were obtained for the participating loadings. Such
A 2 Basic Theory Part A Base Module
superpositions however are not permitted in nonlinear analyses as well as those ob tained by second order theory.
In order nevertheless to obtain useful limit values for such cases, one has to consider the following points:
S One has to limit oneself to a single action, so that in the limit state specification no action combinations result,
S In the specification of the single action one may only use the superposition scheme E1, that is only B1 OR B2 OR B3 ... , whereby the B1 may only be single loadings. In order nevertheless to be able to work with the input loadings, the programs pro vide the combination loadings. Here the loadings are combined before the analysis with the necessary factors to form a new loading, which is then equivalent to a nor mal loading.
A 2.3.5 Limit State Specifications with Action Sets
Working with action sets is necessary or recommendable when all loading configur ations of an action cannot be obtained or only with a lot of effort using the superposition scheme described above for action specifications. This case is certainly necessary for the following example:
The automatically generated action combina tions each consider the action set with the two defined groups (lower dialogue)
Excerpt from the dia logue ’Actions’
In the case of a multi−span bridge, the action Road Traffic Loading with the condition that it must be considered in the two above groups, could scarcely be specified without dividing it into two separate sub−actions. This way of handling the problem also gives a better overview, as one can follow more easily what has actually been done.
They are formed according to the following scheme (code−dependent), whereby it is assumed that an action is always defined in the same way as a leading action, as also an accompanying action. The correctness of the hazard scenarios formed in this way has always to be checked in each case by the engineer and if necessary adapted to the actual requirements.
Eurocode/E%DIN:
Basic combination (without accidental loading):
Sd+
ȍ
gG,j@ Gk,j) gQ,1ƪ
Qk,1)ȍ
iu1y0,i@ Qk,i
ƫ
() gp@ Pk)Accidental action combination:
Sd,A+
ȍ
Gk,j) Ad) y1,1@ Qk,1)ȍ
iu1y2,i@ Qk,i() gp@ Pk)
Gk : char. values of the permanent loads (actions dead load, surcharges)
Qk,1 ,Qk,i: char. value of the first or further varaiable actions
Pk : prestressing
Ad: design value of the accidental action
gG : partial factors for permanent actions (1.35 and 1.0)
gQ : partial factors for varaible actions
gP : partial factors for actions due to prestressing
A 2 Basic Theory Part A Base Module
SIA 260:
Standard design situation
Ed+ E(gGGk,gPPk ,gQ1 Qkl ,y0i Qki) (4.4.3.4)
gG : load factor for permanent actions
Gk : characteristic value of the permanent action
gP : load factor for prestressing actions
Pk : characteristic value of the prestressing action
gQ : load factors for variable actions
Qk : characteristic value of a variable action
y0 : reduction factors for variable actions
Note: For a variable accompanying action y0iQki (the decisive one) is considered.
Accidental design situation
Ed+ E(Gk, Pk , Ad,y2i Qki) (4.4.3.5) Ad : design value of the accidental action
Reaktionen: floor i with wall ans column supports:
Loads on floor i−1:
A 2.4 Load Transfer from Floor to Floor
A 2.4.1 Overview
Problem
The transfer of the vertical loads is a central problem in the dimensioning of buildings. It deals with the dimensioning of vertical elements, foundations and floor slabs. The problem is very complex and it is not easy to find a good model with a reasonable effort. Here rae a fews points that should be noted:
S Why not just build a finite element model of the hole building, introduce loads everywhere, press a button and print out the results? This approach is not unrealistic and has been implemented a long time ago (with little success in practice however). It has advantages and disadvantages:
+ You have a simple model: Equilibrum is always garanteed and you could even con sider soil−structure interaction and the influence of horizontal forces.
− Such a model does only seem to deliver high accuracy and the results are exremly sensible to stiffness assumptions (especially with high rising structures, see notes below). It does in no way take into account the nonlinear behaviour of the structure (cracking of concrete, long term behaviour) and construction stages. Therefor it can not give more accurate results than the much simpler model, which analyses isolated parts (i.e. floors) of the structure and then transfers the loads from one floor to the next below.
S Both models, the hole structure model and the from−by−floor model, have to deal with the problem of stiffness assumptions. If the vertical elements, supporting a slab, are irregularly distributed, it can lead to stress concentrations in the slab. These stresses and the corresponding reactions do not match the reality, because due to the nonlinear behaviour of the slab (i.e. cracking, creep etc.) stress redistribution (and with it also reaction redistribution) takes place. In order to overcome this prob lem one must choose a reasonably high stiffness for the vertical elements.
S The distribution of the reactions of a slab are known to be rather ’wild’ on the supporting walls. For the floor below these reactions should be intoduced as loads with a much simpler distribution, ’flatened’trough the walls.
S The selfweight of a wall is usually modeled as a constant line load. The engineer has to decide, if this model is realistic for his concrete problem.
S In order to model the load transfer for most buildings it is sufficient to have, besides the dead load, a single load distribution for all the variable loads. It’s up to the engineer to decide, if this simplification is accurate enough or if a series of loading patterns have to be investigated and propagated to the underlying floors.
S According to the national codes the loads effective for the dimensioning of the structure usually consist of several different actions with different partial safety factors (e.g. snow load, live load cathegories like office space, storage space etc.). For the dimensioning of the individual floors it can be usefull, to model all of them accordingly. For a hole structure (i.e. by load transfer from floor to floor) however one could end up with hunderets of load combinations to investigate at the bottom floor, leading to ’strange’ results. A single variable load transferred from floor to floor is a better approach.
S National codes allow for reduced partial safety factors when transferring live load from floor to floor (e.g. if more than two floors share the same live load cathegory).
Realization in CEDRUS-5
For a slab calculation CEDRUS5 models pure bending action only, i.e. the vertical load transfer can only be realized with importing the reactions from the floor(s) above as
A 2 Basic Theory Part A Base Module
loads and exporting the reactions of the walls and columns as loads to the floor below. In previous versions of CEDRUS this scheme was also supported, but it was not as tightly integrated into the application, since the user had to do all the steps by hand. This is no longer the case. In CEDRUS5 the calculation of a hole building, i.e. the load transfer from the top to the bottom floor, is very straightforward:
How the load transfer is implemented and how easy it is to actually do it, is explained in the following two sections.
Permanent loads marked for export (incl. coresp. import loads)
actual slab
(floor) Solver (result generation) Reactions of
Export comb. G
Export load G Action ’Dead load’ Export combination G
Live loads marked for export (incl. corresp. import loads)
Export load Q Action ’Live load − general’
Export combination Q Import load cases =
The floor below can import the export loads G,Q and P as import load cases
Prestress loads marked for export (incl. corresp. import loads)
Export combination P
Dead load of walls and columns
Reactions of Export comb. Q Reactions of Export comb. P Export load P Action ’Prestressing’ Export combinations of the selected floors directely standing on top of the actual
A 2.4.2 Load Export
The load export, i.e. the transfer of the reaction forces at wall and column supports as well as the dead load of these elements to the underlying floor, is realized over so−called export combinations. These are special load case combinations, that are automatically solved and make the ’loads to be exported’ ready for import in the underlying floor. Which load cases are combined to what export combination can be controlled by the user. Since load cases usually have different action cathegories and you may want to analyse them individually, one export combination would be needed for each of the actions. Although this is possible to do (and can be done by hand), this leads to a system too complex. Therefore CEDRUS does automatically generate three export combinations only: one for all the permanent loads, one for all the variable loads and one for prestressing.
. For the treatement of the dead load of the vertical elements (i.e. walls and columns) see the section ’Calculating the Export Calculations’.
Automatically generated Export Combinations
For each load case the user can specify with a check box in the dialog, if it should be included in the corresponding export combination (e.g. a variable load case goes into ’!Exp−Q’) and what factor should be used for it (see next section).
Newly created load cases and import load cases are automatically activated for export. All the export combinations are inculded in the list of load cases.
Reduction factor for load export
Some national codes allow for a reduced load transfer to the underlying floor, if a number of floors share the same type of action (e.g. ’Live load − office space’). By specifying a value < 1.0 this reduction can be taken into account. The reduction factor is used for the generation of the automatic export combinations only.
. Automatically generated export combinations are only supported for the structure type ’Building’.
Manually created Export Combinations
The user can define his own export combination by creating a load case of type ’Export combination’. The specification does not differ from a normal load combination. Manually created export combinations are treated just like the automatically generated.
Calculating the Export Combinations
For the current floor export combinations are normal load combinations. However, besides the load elements that are actually exported and shown in the tabsheet ’loads’, the user cannot get any results for these combinations.
The self weight of the walls and columns is automatically added to the export combination ’!Exp−G’ if
1. a load case ’dead load’ with an acceleration load for the hole slab is specified and
2. the self weight is activated for the walls and columns.
For a newly created calculation the first condition is always fullfilled, since CEDRUS automatically generates the load case ’dead load’. For the second condition however, the user must make sure that the shown interface elements in the dialogs of walls and columns are set accordingly:
The calculated reactions along a wall can vary a lot form node to node (see figure below). The direct transfer of these reactions to the underlying floor is not a realistic
A 2 Basic Theory Part A Base Module
model, because the force distribution is ’flattened’ in some way by the wall. CEDRUS does automatically equalize the reactions by calculating an uniformely distributed force per section.
finite element nodes Reactions (raw):
Reactions (flattened for export)
Length of section = ca. height of wall (at least 1 section per input wall)
Nonlinear Export Combinations
In the load case dialog you could activate an export combination, like any ther load case, to be solved nonlinearly, in order to avoid tension in the suporting walls and columns. Although this is possible it seldom makes sense, since the tension part of the reactions is usually eliminated by the procedure described above.
Transfer of the Exported Reactions
After changes on the structure or loading the load export is automatically started whenever the system is solved, i.e. when the user requests an result output in the tabsheet ’Result’ or he presses the ’flash’ button in the tabsheet ’Calculation’. The calculated reactions from the export combinations are then ready to be imported by the underlying floor.
. Changes in a floor effect all floors below that do import loads. Therefore you must recalculate all the dependent floors in the right order.
A 2.4.3 Load Import
As discussed in the last section, by default no steps must be taken for export of the reac tions of a slab. However, in order to import these reactions in the floor below, there you must define the source, i.e. the floor above it. This is done in the dialog ’Load Im port’ that opens when you click on the corresponding button in the tabsheet /Loads/. Since more than one source is possible (e.g. departement complex with several houses as sources for the underlying slab of the parking garage), here you can specify a number of slabs as sources. Every export combination of the specified slabs will become an (read only) import load case listed in the dialog. The actual import is performed by pressing on the ’Update’ button, what you should do everytime changes where made in the upper floors.
The imported load cases have take their action types form the export combination and will be treaded accordingly.
. The exported loads are imported in their original coordinate system. If the origin of the upper floor does not match the actual, you can specify the offset dX and dY, which are added to the (imported) coordinates X and Y upon import.
