General Static Analysis of Finite Element
Structures
ASE Manual, Version 2014-9 Software Version SOFiSTiK 2014
Copyright c 2015 by SOFiSTiK AG, Oberschleissheim, Germany.
SOFiSTiK AG
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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.
Front Cover
Project: Yas Hotel, Abu Dhabi| Client: ALDAR Properties PJSC, Abu Dhabi | Structural Design and Engineering Gridshell: schlaich bergermann und partner| Architect: Asymptote Architecture | Photo: Björn Moermann
Contents
Contents i
1 Task Description 1-1
2 Theoretical Principles 2-1
2.1 General . . . 2-1 2.2 Implemented Elements - licence level . . . 2-2 2.3 Beam Elements . . . 2-3
2.3.1 Geometric nonlinear Theory 2nd
and 3r d
Order . . 2-3 2.3.2 Coordinate System of Forces, Center of Gravity . . 2-4 2.3.3 Warping torsion . . . 2-4 2.3.4 SOFiSTiK - T-Beam Philosophy . . . 2-6 2.4 Pile Elements . . . 2-11 2.4.1 Winkler Coefficient . . . 2-12 2.4.2 Numerical Solution and Accuracy . . . 2-12 2.5 Truss and Cable Elements . . . 2-14 2.6 Spring Elements . . . 2-14 2.7 Boundary Elements BOUN and FLEX . . . 2-15 2.8 Shell Elements . . . 2-16 2.8.1 Plate Structural Behaviour . . . 2-18 2.8.2 Membrane Structural Behaviour . . . 2-21 2.8.3 Elastic Foundation . . . 2-22 2.8.4 Rotations around the Shell Normal . . . 2-24 2.8.5 Twisted Shell Elements . . . 2-24 2.8.6 Eccentrically Connected Shell Elements . . . 2-24 2.8.7 Tendons in QUAD Elements . . . 2-25 2.8.8 Non-conforming Formulation . . . 2-25 2.9 Volume Elements . . . 2-26 2.10 Primary Load Cases . . . 2-27 2.11 Non-linear Analyses . . . 2-28 2.12 Non-linear Analysis of Plates and Shells . . . 2-29 2.12.1 Overview . . . 2-29 2.12.2 Input of the Materials . . . 2-30 2.12.3 Analysis Basics . . . 2-37 2.12.4 Rounding off over Punching Points . . . 2-50
2.12.5 Output of the Results . . . 2-51 2.12.6 Miscellaneous Information . . . 2-52 2.13 Membrane Structures: Formfinding and Static Analysis . . 2-55 2.13.1 Overview . . . 2-55 2.13.2 The Membrane Element . . . 2-55 2.13.3 Formfinding . . . 2-63 2.13.4 Static Analysis . . . 2-76 2.13.5 Unstable Membrane Forms . . . 2-80 2.13.6 Calculations of Cable Meshes . . . 2-82 2.13.7 Check List - Notes - Problem Solutions . . . 2-84 2.13.8 Overview about the Used Examples . . . 2-86 2.13.9 Necessary Program Versions . . . 2-86 2.14 Dynamic Modal Analysis . . . 2-87 2.15 Buckling Eigenvalues . . . 2-87 2.16 Masses . . . 2-88 2.17 Damping Elements . . . 2-88 2.18 Modal Damping and Modal Loads . . . 2-88
Literature 2-89
3 Input Description 3-1
3.1 Input Language . . . 3-1 3.2 Input Records . . . 3-2 3.3 CTRL – Control of the Calculation . . . 3-5 3.3.1 SOLV Equation solver . . . 3-18 3.3.2 CORE Parallel computation control . . . 3-22 3.4 SYST – Global Control Parameters . . . 3-24 3.5 STEP – Time Step Method Dynamics . . . 3-33 3.6 HIST – Storage STEP-LCST . . . 3-37 3.7 ULTI – Limit Load Iteration . . . 3-38 3.8 PLOT – Plot of a Limit Load Iteration . . . 3-41 3.9 CREP – Creep and Shrinkage . . . 3-43 3.10 GRP – Group Selection Elements . . . 3-46 3.11 GRP2 – Expanded Group Selection . . . 3-52 3.12 ELEM – Single Element Settings . . . 3-55 3.13 LEN0 – Unstressed Length . . . 3-56 3.14 HIGH – Membrane High Points . . . 3-57 3.15 PSEL – Selection of Piles . . . 3-59 3.16 TBEA – Reduction of the Width for T-Beams . . . 3-60 3.17 REIQ – Reinforcement in QUAD Elements . . . 3-61 3.18 STEX – External Stiffness . . . 3-63 3.19 OBLI – Inclination . . . 3-65 3.20 SLIP – SLIP Cable . . . 3-67
3.21 VOLU – Air Volume Element . . . 3-68 3.22 MOVS – Moving Spring . . . 3-70 3.23 LAUN – Incremental Launching . . . 3-72 3.24 SFIX – Fixing Stiffness . . . 3-74 3.25 LC – Load Case and Masses . . . 3-75 3.26 TEMP – Temperature from HYDRA . . . 3-78 3.27 LAG – Loads from Support Reactions . . . 3-80 3.28 PEXT – Prestress of External Cables . . . 3-82 3.29 LCC – Copy of Loads . . . 3-84 3.30 EIGE – Eigenvalues and -vectors . . . 3-86 3.31 MASS – Lumped Masses . . . 3-89 3.32 V0 – Initial Velocity . . . 3-91 3.33 REIN – Specification for Determining Reinforcement . . . . 3-92 3.34 DESI – Reinforced Concrete Design, Bending, Axial Force 3-97 3.35 NSTR – Non-linear Stress and Strain . . . 3-103 3.36 Non-linear Material Analysis in ASE . . . 3-105 3.37 ECHO – Output Control . . . 3-109
4 Output Description 4-1
4.1 Check List of the Generated Structure . . . 4-1 4.2 Check List of the Non-linear Parameters . . . 4-1 4.3 Check List of the Analysis Control Parameters . . . 4-1 4.4 Check Lists of the Loads . . . 4-2 4.5 Process of the Analysis . . . 4-2 4.6 Eigenvalues . . . 4-3 4.7 Element Results . . . 4-4 4.8 Non-linear Results . . . 4-7 4.9 Nodal Results and Support Reactions . . . 4-8 4.10 Internal Forces and Moments at Nodes . . . 4-10 4.11 Error Estimates . . . 4-11 4.12 Distributed Support Reactions . . . 4-12 4.13 Strain Energy of Groups . . . 4-13 4.14 Wind Load Generation . . . 4-13
1
Task Description
ASE calculates the static and dynamic effects of general loading on any type of structure. To start the calculations the user divides the structure to be analyzed into an assembly of individual elements interconnected at nodes (Finite Ele-ment Method). Possible types of eleEle-ments are : haunched beams, springs, ca-bles, truss elements, plane triangular or quadrilateral shell elements and three-dimensional continuum elements.
The program handles structures with rigid or elastic types of support. An elastic support can be applied to an area, a line or at nodal points. Rigid elements or skew supports can be taken into account.
ASE calculates the effects of nodal, line and block loads. The loads can be defined independently from the selected element mesh. The generation of loads from stresses of a primary load case allows the consideration of construction stages, redistribution and creep effects.
Non-linear calculations enables the user to take the failure of particular elements into account, such as: cables in compression, uplifting of supported plates, yield-ing, friction or crack effects for spring and foundation elements. Non-linear ma-terials are available for three-dimensional and shell elements. Geometrical non-linear computations allow the investigation of 2nd
and 3r d
order theory effects by cable, beam, shell and volume structures.
In case of beam structures, the program can calculate warping torsion with up to 7 degrees of freedom per node.
The user of ASE should therefore gather experience from simple examples be-fore tackling more complicated structures. A check of the results through ap-proximate engineering calculations is imperative.
The basic version of ASE performs the linear analyses of beams, cables, truss-es, plane and volume structures. Plain strain and rotational systems can be analyzed with TALPA.
Extended versions of ASE offer calculations of: • Influence surfaces
• Non-linear analyses
• Pile elements with linear/parabolic soil coefficient distribution • Creep and shrinkage
• Forces from construction stages • Modal analysis, Time step method • Material non-linearities
• Geometrical non-linearities • Membrane elements
• Evaluation of collapse load • Non-linear dynamics
2
Theoretical Principles
2.1
General
A continuum or a plane structure can be interpreted as a statically or geometri-cally infinitely indeterminate structure. If an analytical solution is unknown, every numerically approximate method is based on converting this infinite system into a finite one, in other words to discretizing it.
The advantage of the finite elements lies in their universal applicability to any geometrical shape and almost to any loading. This is achieved by a modular principle. Single elements which describe parts of the structure in a computer oriented manner are assembled into a complete structure.
The continuous structure is represented thus by a large but finite number of el-ements. A discrete solution consisting of n unknowns is calculated instead of the continuous solution. In general, the approximate solution may represent the exact solution better with the use of more elements. The single elements of an area can be of arbitrarily small dimensions in comparison to the dimensions of the overall structure without giving rise to any incompatibilities with the present-ed theory. The refinement of the subdivision is, however, subjectpresent-ed to certain limitations due to numerical reasons.
The Finite Element Method (FEM) employed in ASE is a displacement method, meaning that the unknowns are deformation values at several selected points, the so-called nodes. Displacements can be obtained with an element-wise in-terpolation of the nodal values. The calculation of the mechanical behaviour is based generally 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 displace-ments.
The global force equilibrium is generated then for each node in order to deter-mine the unknowns. A force in the same direction which is a function of this or another displacement corresponds to each displacement. This leads to a system of equations with n unknowns, where n can become very large. Numer-ically beneficial banded matrices result, however, due to the local character of the element-wise interpolation.
The complete method is divided into four main parts: 1. Determination of the element stiffness matrices.
2. Assembly of the global stiffness matrix and solution of the resulting equation system
3. Application of loads and determination of the corresponding displacements. 4. Determination of the element stresses and support reactions due to the
computed displacements.
The second step is that with most CPU time. It may use up to 90 percent of the total CPU time. However, it has to occur only once for a static system.
The stresses jumps from element to element. The size of the jump is thus a direct measure of the quality of the FE analysis.
2.2
Implemented Elements - licence level
The elements shown in the following table are available in ASE. A non-linear analysis can occur also for some types of elements. A detailed list of the imple-mented nonlinear effects is written in section 3.36.
Program ASE runs with licences to ASE, SEPP and PFAHL. Depending on the licence not all elements can be used - see following table. Nonlinear analsis also require a higher licence level.
Non-linear Geometrical Element Material Non-linearity
SPRI yes yes
TRUS yes+tension failure yes
CABL yes+compression failure yes + cable sag
BEAM yes yes
PILE1 elastic bedding only yes
QUAD2 yes yes
BRIC12 yes yes
BOUN -
-FLEX12 -
-Halfspace2 yes
2 not available on licence PFAHL
2.3
Beam Elements
The beam element in ASE is an arbitrary haunched beam element including shear deformations and hinge effects. It can be defined also eccentrically to the node connecting line. For haunched cross sections in bridge analysis all nodes can be defined on the top face of the superstructure.
2.3.1 Geometric nonlinear Theory 2nd
and 3r d
Order
The following figures shall clarify the essential characteristics between SOFiSTiK theory 2nd and 3r d order (TH2 and TH3).
P
Pz
d− TH2
− TH3
Figure 2.1: Column geometric nonlinear theory2nd and3r d order
In the column example in figure 2.1 the effect of theory TH2 causes a stiffness reduction in the column due to the compression normal force (geometric stiff-ness). This creates an additional deflection dux in x direction (no duz!). The beam can get longer than in the original shape. The bottom bending moment increases due to the displacement of the vertical load Pz. This type of analysis is also known as pi-delta method.
In the complete geometric nonlinear analysis TH3 the column head follows the physically correct path. Equilibrium is iterated on the real deformed shape. In figure 2.2 a horizontally fixed bending girder is loaded vertically. In TH2 the girder just deflects vertical without a normal force N.
In the TH3 analysis the vertical displacement causes a lengthing of the beam. The created normal force N carries a part of the load and reduces the vertical deformation.
T H2 : N = 0 T H3 : N
Figure 2.2: girder geometric nonlinear theory2nd and3r d order
Examples see SYST
2.3.2 Coordinate System of Forces, Center of Gravity
In TH2 the forces are related to the original coordinate system. So the shear forces are transversal forces. In the column example in figure 2.1 N stays vertical and VZ stays horizontal.
In TH3 the forces are always related to the deformed beam coordinate system. The beam forces N, MY and MZ are related to the center of gravity of the actual active partial section (not to node connecting line). MT, VY and VZ are related to the center of shear.
Dead load is applied in the stiffness center (and not in the center of mass that may differ in case of composite sections).
2.3.3 Warping torsion
Warping torsion can be used for straight beam structures with CTRL WARP 1. Warping effects can also appear without warping support. In the following pic-ture to example ase11_girder_overturning.dat in loadcase 11 a single mo-ment MT=1 kNm (2*0.50 kNm) is applied in midspan of a single girder
90 % of it work via warping torsion as force pairs in the flanges - see MTs = 0.45 kNm. 10 % go directly into the section via Saint Venant shear - see MTp = 0.05 kNm. Warping parts (ASE output):
The total torsional moment Mt has 3 parts (MT= MTp +MTs +MTn): MTp - primary torsional moment from Saint Venant shear stresses
MTs - secondary tors. moment (flange shear from warping longitudinal stresses) MTn - theory 2. order torsional moment from twisted normal stresses
Mb warping moment (from warping longitudinal stresses - creates MTs at other beam sections)
the flanges opposite and transport a part of the loading via longitudinal stresses to the support.
The longitudinal warping stress due to Mb in midspan create a flange shear force at the supports. The corresponding torsional momennt MTs at the support is about 0.05 kNm. At the support itself, the longitudinal warping stresses are zero (free end) - see Mb = 0 kNm. The effects are as follows:
MT
MTp
MTs Mb
The warping effects are also explained in warping_mtp_mts_mtn.dat. There the interaction of MTs and Mb for a MT load on a cantilever is interpretet as follows:
• Please notice that at the cantlever end (beam 10 x= 0.4) already a part of the load MT= 10 kNm is carried by warping although there are no longitudinal
warping stresses at that location (free end)! • You can imagine this in the following way:
The top flange longitudinal warping stresses at beam 1 x=0 want to pull pack the top flange at the cantilever. The bottom flange longitudinal warping stresses want to pull pack the bottom flange as well.
• This pair of forces (flange shear from warping longitudinal stresses) carries a part of the MT loading.
• So the longitudinal warping stresses at the beginning of the beam create a MTs at the end - and oposite.
• An MTs at a beam section x1 creates longitudinal warping stresses at an-other beam section x2.
2.3.4 SOFiSTiK - T-Beam Philosophy
Automatic addition of the T-beam parts for FE plates with beams Example see ase3_t_beam_test.dat
Usage in bridge construction see also tbeam_philosophy_e.pdf
Figure 2.3: T-Beam Philosophy
Attention: This model can not be used for influence line evaluation with ELLA because ELLA does not add the slab parts to the beam!
A 2D slab analysis is usually sufficient and desirable for beams and continuous beams with effective cross section widths in a slab. Only in a 2D slab analysis normal forces are not determined in the slab or in the beam! The advantage is that the slab can be simply designed (without normal forces) particularly for the shear checks. In addition the determined beam moments can be designed directly with the right T-beam cross section.
Procedure: The user or the graphical input program positions a centric defined beam in the node plane (with the T-beam cross section see picture on the right). The QUAD elements are also defined centric. As the beam is positioned in
the centre of gravity (a little bit below the slab center), the upper edge of the T-beam looks a little bit out of the slab - this is also visible in WinGRAF. The ANIMATOR displaces the cross section a little bit downwards, so that the upper edges beam+slab appear at the same position for a better visualization. So in the standard case the beam section is defined with the corresponding effective slab width. Looking from the side (see picture left below) you see that cross section parts and slab overlap and concrete areas are defined twice. These double parts are now corrected in the T-Beam philosophy:
Therefore in the stiffness analysis the slab part (I-slab = bVh3/12 with b=effective width = width of the cross section) is substracted automatically from the stiffness of the beam I-Tbeam. An equivalent (reduced) beam is used:
I–equivalent beam = I–Tbeam – I–slab
In the same way the deadload of the equivalent beam is modified to avoid double dead load.
Then the program at first determines a bending moment of this equivalent beam in a FE analysis. The internal forces parts of the slab (M-slab = m-slab V b) are added automatically immediately. Thus the complete T-beam internal forces are available for the following beam design:
M–Tbeam = M–equivalent beam + M–slab
The bending moments My and the shear forces Vz are added as default, for shells also the normal forces N. The torsional moment Mt is not added as default. Output:
• The parts of the slab are already included in the printout of the beam forces. • A statistic of the slab parts follows. The maximum slab parts are compared
with the maximum beam internal forces:
Statistic Beam - Additional Forces from a Slab Loadcase 2
The printed beam-forces include max. additional forces of a slab: max. beam-force without slab-addition max. slab-addition
cno bm Vz My Vz My
[m] [kN] [kNm] [kN] [kNm]
1 2.20 max 48.60 243.78 43.63 5.95 min -48.60 0.00 -43.63 0.00
For safety the internal forces are not reduced in the FE plate elements, although it would be possible about the amount of the increase of the beam internal
forces. So this method can be uneconomical for smaller beam heights.
Beams which are connected with kinematic constraints at the slab are also pro-cessed, if the beams are positioned in the slab plane.
Defaults for the addition of the plate internal forces to the beam internal forces: For slab structures:
• The single beam must have a cross section with a defined width at the start and the end. A defined width can be generated from a T-beam (e.g. record SREC in AQUA) and from general cross sections (e.g. AQUA record SECT and following). The maximum width of the cross section is used in each case (independent of the position of the plate, above or below). A cross section which is input without dimensions however with stiffnesses (e.g. with record SVAL) does not known any defined width. A plate part can therefore not be added for these beams!
• The single beam is connected generally directly with the nodes of the plate. • After an automatic mesh generation or a free mesh definition the straight
beam which is positioned in the plate plane can be combined also with the FE mesh via kinematic constraints.
• The beam reads the plate thickness and the modulus of elasticity from these plate nodes. Different plate thicknesses are possible at the beam start and end.
Additionally for three-dimensional slab structures (ASE):
The feature can be used also for three-dimensional slabs however with following restrictions:
• The beams must be positioned in the same plane as the plate. The plate parts are not added for beams which are connected eccentrically.
Special features with the input:
• The beam cross section must represent the effective cross section, therefore the web and the effective plate. If a concrete slab on a steel girder should be considered as a composite construction, the steel girder must be defined with the effective concrete plate as cross section! The determined internal forces and moments refer then to this composite cross section.
• The effective width (cross section width) should be chosen a little bit smaller than to large especially over the columns, because for the plate moment to be added only the moment near the node at which the beam and the plate
are combined is used (see CTRL PLAB V2). This plate moment is processed then unchangeable acting about the whole width. The internal forces and moments are therefore not exactly integrated about the effective width! • The plate stiffness I-plate (without the part of Steiner) is diverted from the
total cross section stiffness I-cross. If the subtrahend I-plate is bigger than 0.8·I-cross, a warning is printed and the minimum stiffness of 0.2·I-cross is used.
• For three-dimensional systems the subtrahend is maximal 0.9·A-cross for the area A-plate. At least 0.1·A-cross are available then for the fictitious beam in the FE system.
Special features with the output:
• The attenuated stiffnesses are printed with ECHO PLAB FULL. If a cross section is available at beams with different plate thicknesses (e.g. haunch-es), the attenuated stiffness is printed for the minimal and maximal plate thickness.
• The plate parts are already available in the printed beam internal forces and moments and can be designed directly.
beam at FE node
CTRL PLAB 0 beam which is connected with kinematic constr aints
added plate part s
Figure 2.4: Beam internal forces
For comparison a load case can be calculated once without input ofCTRLPLAB and the second time with CTRL PLAB 0 and another load case number. The beam internal forces and moments of both calculations can be represented then with the same scale in a picture.
(More precise) calculation possibilities:
Also with the above describes method, the normal forces occur in the compres-sion zone (plate) first during the design of the T-beam. Normal forces are not
considered during the calculation of the FE system. The effective width has to be estimated manually and defined. In reality the normal forces act from the supports into the plate. For a more precise calculation three possibilities are de-scribed here. For all three variants the effective width is realized automatically via the normal force calculation and has not to be input:
1. The web part which is positioned below the plate can be defined as a beam which lies eccentrically below the plate. Then two nodes lying upon each other are however necessary for the system input. This complicates the input. Problems occur also for the design, because the sum of the internal forces from web+plate including the parts of Steiner are necessary for a design of the total T-beam. The method is therefore only reasonable for composite slabs with eccentrically defined steel beams (see ASE example 5.3).
kinemat ic constraint
Figure 2.5: Eccentrical defined steel beam
2. The web can be also generated with shell elements. The same problems for the design result as for the eccentrical beam. In addition it should be noted that the area in the intersection point plate-web is not defined twice:
Figure 2.6: Shell elements
3. The SOFiSTiK offers the eccentrical plate elements as a real alternative. The system is generated here with different thick plate elements. The plate elements get a larger thickness in the area of the beams. A simply de-fined node plane which lies at the upper edge of the plate is here neces-sary in the input. All elements can be defined eccentrically below the node
plane. Thereby all elements have the same upper edge, the thicker beam elements stand only below out. Normal forces which are considered for the design are produced due to the eccentrical position of the elements. Thereby the usual plate design is done simultaneously the beam design -a speci-al be-am design is therefore not necess-ary. The FE -an-alysis uses here automatically the real effective width via the simultaneous analysis of the normal force distribution. This method is therefore applicable not only for the analysis of building slabs but also for analysis of concrete bridges. Each elements is processed for themselves alone during design and not the total T-beam cross section! This method is however only correct for beams with moderate thickness. The design can be uneconomical for larg-er beams (web height larglarg-er than 2.5·plate thickness), but it is in each case at the sure side. The simple method with fictitious beams lying in the plate is more practical for larger web heights.
See also tbeam_philosophy_e.pdf
eccenticit y
plane of the node points
underside of the QUAD elements
centroid line of QUAD lying below the node plane
Figure 2.7: Eccentrical plate element
For all analysis methods the resultant internal forces and moments can be deter-mined with the program SIR (Sectional Results). Afterwards a design as beam cross section is possible, also for system 2 from folded structure elements. This is especially necessary in bridge design for checks of the ultimate limit state and for checks for safety against cracking.
Attention: This model can not be used for influence line evaluation with ELLA because ELLA does not add the slab parts to the beam!
Literature: KATZ AND STIEDA [11], WUNDERLICH ET AL. [18], BELLMANN [2],
KATZ [10]
2.4
Pile Elements
A single pile is idealized through a straight, elastically supported beam with s-hear deformations and 2nd
complete system of 12 differential equations. Pile elements get a minimal con-straint of the rotational spring in order to prevent instabilities.
Example see single_pile.dat
These equations are integrated numerically with the Runge-Kutta method. PLC analysis for piles:
In this way also system with pile elements can be used including creep in CSM. Shrinkage will never be taken into account for piles, creep acts for the pile and the bedding! If GRP PHIF is input this value is taken for both the pile and the bedding.
The pile element is not contained in the basic licence of ASE. 2.4.1 Winkler Coefficient
The definition of the bedding constants requires a good engineering understand-ing of the problem. For their definition it is most important to take into account that the Winkler coefficient is not a simple material property but depends on the system dimensions and the loading. The Winkler coefficient defines the stress caused by a given deformation whereas the influence of adjacent points (shear deformations) is not taken into account. The dimension of a bedding is therefore given as kN/ m3. A displacement causes a stress (kN/ m2).
However, the pile bedding is defined in kN/ m2 since the pile ”width” has to be
integrated into this value. In this case, a displacement causes a load (kN/ m).
Since the pile width influences also the Winkler coefficient, the pile dimensions are dropped possibly from the equation and the pile Winkler coefficient can be estimated also from the elastic modulus of the soil and a form factor.
For circular pile cross sections and a Poisson’s ratio of 0.4 a form factor of 1.12 can be derived. For a Poisson’s ratio of 0.0 the form factor would be 1.57. In EBK 82 of the Road Traffic Department in Rheinland-Pfalz the extreme values of the form factor are scheduled to be 0.5 and 2.0.
Simplifying to DIN 4014 a foundation modulus ks = Es/ D (atD > 1.0m D = 1.0m
may be used) can be determined and from that a beam foundation k= D · ks as
line-shaped foundation per m pile length. 2.4.2 Numerical Solution and Accuracy
In general, the set of differential equations can not be solved directly. Therefore, for each pile, these equations are integrated numerically by the Runge-Kutta method. The step width is controlled automatically to maintain a relative accura-cy of 0.01 percent. The setting of higher error limits results in a reduction of the
CPU time. The setting of smaller error limits is reasonable up to a certain value, which depends on the machine accuracy of the computer used. The reduction of the error limit below a certain value is not reasonable because the computa-tional error increases again due to the rounding errors in the high number of the necessary additions.
In the case of an unbedded beam the step width can be set very large. By contrast, for large Winkler coefficients the numerical calculation becomes more difficult. As a criterion the characteristic length is used, defined as:
L= p4 4 · E/ K (2.1)
This value is an estimation of the distance between the zero points of the solu-tion funcsolu-tion. For reasons with reference to the numerical integrasolu-tion, the char-acteristic length should not become less than 1/5 up to 1/10 of the element length. If this condition is not satisfied, great accuracy problems may occur resulting in differential forces in the nodes which are pointed out in an error message.
These problems can be overcome by subdividing a pile into more subelements. In the case that a pile is subdivided into more elements, the placing of the nodes at points of changing soil parameters (layers) is to be preferred over an uniform subdivision.
Pile elements get a minimal constraint of the rotational spring in order to prevent instabilities.
If otherwise nothing was specified, a linear analysis is performed. Non-linear effects are:
• Different bedding in various transverse directions (F1 - term of the series sequence)
• Limitation of the maximum bedding stresses • Second order theory
For these cases an iterative calculation has to be carried out.
The program uses the ”Quasi-Newton” method with constant stiffness matrix. To obtain a better convergence the single increments are modified according to the Crisfield method.
2.5
Truss and Cable Elements
Truss and cable elements can transfer only axial forces. In the case of non-linear analysis the cable elements can not sustain compressive forces.
Example see ase5_cable_trestle.dat
An internal cable sag is considered for geometrically non-linear analysis. In this case the transverse loading of the cable is calculated for the cable geometry (extensible plane prestressed cable). Without using internal cable sag a cable can be subdivided into shorter individual cables. The resulting cable chains can be analysed in a stable way with a prestress. For the control of the internal cable sag please look atCTRLCABL too.
2.6
Spring Elements
Spring elements idealize structural parts by means of a simplified force-displacement relationship. This is usually a linear equation which is based on the spring constant:
P= C · (2.2)
A spring is defined with a direction (dX, dY, dZ) and three spring constants. The here implemented element allows the following non-linear effects which are of course only usefully during a non-linear analysis:
• prestress (linear effect) • failure
• yield
• friction with cohesion • slip
• spring nonlinear work laws, please refer to section 3.36
Figure 2.8: Spring force-displacement diagrams
A prestress displaces the corresponding effects and produces always a loading which acts on the structure. A prestress should not be defined in the system generation because it acts in all loadcases. It is better to use the prestress in ASE...PREX. A prestressed spring gets a relaxation in the absence of external loading or constraints. The non-linear effects are considered both for rotational and displacement springs. Friction can be defined with a lateral spring. The force component perpendicular to the spring effect direction results from the product of the displacement component in the lateral direction multiplied by the lateral spring constant. The maximum value of this force, however, is equal to the force in the primary direction multiplied by the friction coefficient plus the cohesion. If the primary spring fails, the lateral spring gets eliminated too.
Spring loads are not included. A bearing lifting can be modelled in ASE with the group prestress GRP PREX also for coupling springs.
Springs with a work law (see SOFIMSHA SARB) work with hysteresis by shift-ing the zeropoint of the work law curve after plastification. Examples see
a1_spring_overview.dat
e.g. spring_law_3_pkin_curve.dat
2.7
Boundary Elements BOUN and FLEX
The elastic boundary conditions do not represent actual elements. They de-scribe the additional stiffnesses of the structure. Results are not saved in the case of FLEX. The effect of the elements appears directly in the form of support reactions at the corresponding nodes. For BOUN additional distributed bound-ary forces are stored for graphical output.
Distributed support reactions are determined for boundary elements with num-ber (compare program SOFIMSHA/SOFIMSHC). If two boundaries are defined at an edge, the distributed support reactions are calculated once only and they are output for the boundary with the smaller boundary number. Single supports
can not be considered by boundary elements.
A boundary element interpolates linearly the displacements between two nodes. The resultant distribution of the stiffness matrix at the two nodes is
CR+ 3 · CL CR+ CL CR+ CL CL+ 3 · CR (2.3) with: CR= CA · L/12 CL= CB · L/12
CA,CB spring constants at beginning/end L distance of nodes.
2.8
Shell Elements
The shell element implemented in program ASE is a surface element. The in-dividual elements are plane and they lie in each case in a plane whose normal is generated through the vector product ((X3-X1)·(X2-X4)) of the diagonals. The deviation of the element’s plane from the nodes is taken into consideration by means of additional eccentricities. The local coordinate system is oriented in such a way that the z axis is given with the normal to the element’s plane and the local x axis can be selected freely. The default orientation is parallel to the global XY plane with an angle smaller-equal than 90 degrees to the global X ax-is. If the observer looks into the positive direction of the z axis (thus from ”above ”), then he watches the nodes numbered counter-clockwise. If the element’s plane coincides with the global XY plane, the local and the global coordinate systems are then identical.
Figure 2.9: Local coordinate system
The triangular element is considerably worse than the quadrilateral element and it should be used only, if no other choice of mesh partitioning can be found. Generally it should not to be used in the vicinity of supports.
Figure 2.10: Internal forces and moments
Because the normal element remains plane, the bending and the membrane structural behaviour of the individual element are decoupled. The element prop-erties can be defined thus separately for the both components. Additionally the consideration of the components of an elastic support and a numerically condi-tional stiffness for the rotations around the shell normal occurs still.
For a twisted element the membrane and plate parts are generated by decou-pling. Then they are coupled with each other via the twist of the element. Thus the element is able to represent curved shells very exactly. This was demon-strated with corresponding benchmarks.
The consideration of each structural behaviour can be specified in the program SOFIMSHA/SOFIMSHC for each particular element. The defaulted values are:
SYSTFRAM membrane structural behaviour only
SYSTGIRD plate structural behaviour only
SYSTSPAC additionally rotations around the normal
The elements defined in SOFIMSHA/SOFIMSHC without load bearing be-haviour are not considered for the structure. They can be referenced, however, in the case of load cases with free loads. In this way, a load area which consists of QUAD elements can be used for block loading of girders or three-dimensional elements.
The ASE element is defined as a general quadrilateral. The accuracy of the solution, however, depends on the geometry of the element, thus not all con-ceivable element shapes are permitted.
The optimum element is the square or the equilateral triangle. Rectangles and parallelograms are the second-best shape and the general quadrilateral the
third-best. General quadrilaterals with re-entrant corners are not allowed in the element formulation.
A rectangle with a large side ratio a/b has difficulties in the representation of the twisting moments and also for the bending near a corner. A ratio of 1:5 is still tolerated in the program SOFIMSHA/SOFIMSHC and it should be exceeded only in exceptions. The size ratio of two adjacent elements should not be smaller than approx. 1:5. However, this value is relatively uncritical.
The ratio thickness to element dimension is uncritical, because a shear correc-tion factor is applied. It should be clear to the user, however, that the shear deformations in the case of thick plates result in deviations from the Kirchhof-f’s theory. The ratio of the thicknesses of two adjacent elements should not be smaller than 1:10 due to its cubic effect.
2.8.1 Plate Structural Behaviour
The ASE element for the plate structural behaviour is based on Mindlin’s plate theory, as described in the implementations of HUGHES AND TEZDUYAR [8],
TESSLER AND HUGHES [16] and CRISFIELD [4], with an extension of a non-conforming formulation.
The cross sections remain plane also according to Mindlin’s theory, however, they are not perpendicular anymore to the neutral axis. The same shape func-tions as for the displacements are used for the additional shear rotafunc-tions. The total rotation is then the sum of the shear deformation and the bending rotation.
θ = δ δ + θS (2.4) with: deflection θ total rotation θS shear rotation
δ / δ derivative w.r.t. x (similarly for y)
For the curvature and the shear angle we then have:
k= δθ
ky= δθy δy (2.6) ky= δθ δ + δθy δy (2.7) θS = θ− δ δ (2.8) θSy = θy− δ δy (2.9)
A general orthotropic accretion which includes the thicknesses as well as the elastic moduli is formulated for the internal forces and moments:
m = −B· k− μ · By· ky (2.10) my = −By· ky− μ · By· k (2.11) my= −Bd· ky (2.12) and = S· θS (2.13) y= Sy· θSy (2.14)
with the stiffnesses
B = E· t3 12 · (1 − μ2) S = 5 6 G· t (2.15) By = Ey· t3 y 12 · (1 − μ2) Sy= 5 6 G· ty (2.16)
transverse bending stiffness By = E· t3 y 12 · (1 − μ2) (2.17) torsional stiffness Bd= G· t3 d 12 (2.18) with E, Ey elastic moduli G shear modulus μ Poisson’s ratio t, ty, ty, td plate thicknesses
In the isotropic case one must set t= ty= ty= td= t and E= Ey = E.
The orthotropic elastic moduli and thicknesses are rotated through the input of an orthotropy angle OAL in the record MAT!
a) For orthotropic material (e.g. mathematical cross section of prestressed con-crete or wood) it can be set:
By= Bd=ÆB· By (2.19)
To reach this the mathematical thickness for ty and td must be input in
ad-dition to the orthotropic input ofE and Ey.
ty= td= t· 3ÆBy/ B (2.20)
withB > By and ty= t.
b) For corrugated steel can be applied (Timoshenko)
z= ƒ · sin π· ; α= 1 + πƒ 2 2 (2.21) B = 1 α · E· t3 12 · (1 − μ2) (2.22)
By= 1 − 0.81 1 + 2.52ƒ 2 · E· t · ƒ2 2 (2.23) By≈ 0 (2.24) Bd= α 2 · E· t3 12 · (1 − μ2) (2.25)
c) For web plates (y-axis in longitudinal direction) one can set:
ty= t· 1 + b· t3 o · t3 1/ 3 (2.26) ty= t (2.27) Bd= Bd(t) + C/(2 · ) (2.28) where:
C torsional stiffness of the web, , b spacing and width of the web, t, to thickness of the plate and web.
Examples for orthotropic cases can be found e.g. in the book by TIMOSHENKO AND WOINOWSKY-KRIEGER [17].
2.8.2 Membrane Structural Behaviour
The element formulation of the membrane stress state occurs either via a classi-cal isoparametric formulation or probably via a similarly classiclassi-cal non-conforming formulation written by Wilson and Taylor.
The thicknesses as well as the elastic moduli in different directions are tak-en into consideration. The poisson ratio corresponding to Ex is used. For anisotropic poisson’s ratio see chapter membrane structures and example
membrane_poisson_ratio.dat
nyy= Sy· εy− μ · Sy· ε (2.30)
ny= G · ty· γy (2.31)
with the stiffnesses:
S= E· t 1 − μ2 (2.32) Sy= Ey· ty 1 − μ2 (2.33) Sy= E· ty 1 − μ2 (2.34) 2.8.3 Elastic Foundation
The QUAD element can be expanded with stiffness components in order to de-scribe an elastically supported area. Only appropriate inputs can activate this foundation component.
The foundation can be defined both perpendicularly and tangentially to the area. The non-linear effects like failure, yielding and friction may be specified.
An elastic foundation is an engineering trick used for the approximate modelling of subsiding structures. The method is known from foundation engineering, how-ever, it can be used also for the description of support conditions in structural engineering.
The foundation coefficient indicates the stress resulting at a point which is sub-jected to a certain displacement. It is not a material constant, it is calculated later with a settlement analysis. In principle, its value always consists of an elastic modulus together with a geometrical dimension. The displacements of adjacent points are independent of each other, since shear deformations are not taken into consideration with this method.
A more exact analysis of foundations according to the stiffness modulus method is possible with the program HASE.
calculation of the Winkler coefficient is achieved by applying a constant stress and by computing the resultant displacement. In the case of hindered lateral strain the result is
C= E h · (1 − μ) (1 + μ) · (1 − 2μ) = Es h (2.35)
In analog mode one can obtain Winkler coefficients for multi-layered systems. These coefficients are more acceptable as the layer becomes thinner in com-parison to its deformation. If, however, the layer is relatively thick in comcom-parison to the loaded area, or if it is infinitely thick, the Winkler coefficient has to be esti-mated in a settlement analysis at the point of interest. The horizontal foundation has usually the same order of magnitude.
Column heads are defined sometimes with elastic foundations, especially in the case of masonry. By defining the Winkler coefficient one must keep in mind, that a two-dimensional foundation develops a certain rotational spring effect which is more important to the loading of a plate than the perpendicular displacement spring.
A column of the height h which is supported articulated at its foot has a rotational stiffness equal to
Cϕ= 3 · E
h (2.36)
This stiffness should correspond to a rotational spring foundation with
Cϕ= C · (2.37)
From that follows
C= 3 · E
h (2.38)
The corresponding value for a column fixed at its foot is 4 · E/ h.
Therefore it is correct to define a foundation three till four times higher, instead of the Winkler coefficient E/h, in order to describe the rotational foundation prop-erly. If, however, the plate is supported articulated on the column, this type of foundation should not be used in any case because of its clamping effect a-gainst rotation. In this case it is recommended to use a single point support of a node and distribute the load by means of rigid or elastic elements (kinematic constraints).
nodes or as distributed foundation with a matrix. The use of single springs is advised in the case of very stiff foundations and severe load concentrations. The selection occurs with the input CTRLBTYP.
CTRLBTYP> 0 consistent foundation matrix (default)
CTRLBTYP< 0 single springs
Support reactions which result from a QUAD foundation are printed and stored as nodal support reactions. Thus a graphical check of the support reactions is facilitated.
2.8.4 Rotations around the Shell Normal
The rotational degree of freedom around the shell normal is not contained in both load bearing behaviours. In order to prevent numerical difficulties for three-dimensional structures, the Inplane-rotation of the nodes is coupled via a weak torsional spring at the displacements of the corner nodes in an intern way. 2.8.5 Twisted Shell Elements
If not all four nodes of an element lie in a plane (e.g. in the case of a hypershell), then the program defines an eccentric kinematic constraint of the corner nodes at a plane element in a median plane in an intern way. Three-dimensional curved structures may be analysed in this way with sufficient accuracy.
In the case of twisted shell elements as well as geometrically non-linear anal-yses (twisted elements are generated automatically with the latter), internal springs are used now instead of the rotational stiffnesses mentioned in the pre-vious paragraph. These springs convert the moment loading of a node around the shell normal to axial forces in the shell. The shear stiffness of the elements is modified slightly with this method, however, this is the only way to achieve moment equilibrium at the nodes of three-dimensional curved structures.
2.8.6 Eccentrically Connected Shell Elements
In the case of T-beams, it is an advantage to lay all nodes in the plane of the top surface of the plate and to connect the elements with different thicknesses eccentrically to this plane. Then the T-beam effect is realized correctly.
The position of the elements is input in the program SOFIMSHA/SOFIMSHC (e.g. QUAD ... POSI=BELO).
Skewed T-beam Bridge”.
2.8.7 Tendons in QUAD Elements
Prestressed cables defined with the program TENDON have the same element number as the QUAD element that contains them. They are characterised addi-tionally with a cable number and with construction stage numbers for installation, grouting and a possible removal. They possess their own stiffness and are pro-cessed independently from the QUAD elements. Thus not only the deflecting loads are applied to the structure, but also stress changes in the tendon are calculated. The input occurs by the means of GRP CS.
Prestressing cables in the QUAD elements can be used only in a geometrically linear analysis.
2.8.8 Non-conforming Formulation
The regular 4-node element is characterised through a bilinear accretion of the displacements and rotations. This accretion describes a uniform variation of the shear force and of the bending moment via a transformation. This element is called conforming, because the displacements and the rotations between ele-ments do not have any jumps. The results at the gravity centre of the element represent the actual internal force variation fairly well, whilst the results at the corners are relatively useless, especially the ones at the edges or at the corners of a region.
Taylor and Wilson came up with the idea to describe more stress states through additional functions that value is zero at all nodes. As a rule, these functions lead to a substantial improvement of the results, however, they violate the conti-nuity of displacements between elements. Thus they are called non-conforming elements.
Two element variations are available in the program ASE. The selection of the variations occurs via the CTRLoption QTYP.
QTYP 0 regular conforming element according to HUGHES AND TEZDUYAR
[8] or Zienkiewicz
QTYP 1 non-conforming element with six functions based on HUGHES AND
TEZDUYAR [8] or Wilson (default value)
Elements of type 0 can describe only uniform moments and membrane forces inside them. Elements of type 1 can describe a linear moment variation, if they are rectangular, whereas a general quadrilateral element can only do that ap-proximately. Membrane forces can vary linearly.
A corresponding non-conforming triangular element does not exist. Therefore the use of these elements in combination with triangles should be avoided, if possible.
More explanations of the element properties can be found in the manual of the program TALPA.
2.9
Volume Elements
The volume element (BRIC) represents an elastic body and it is defined by means of 8 nodes. Even uniform bending states of a structure can be realized exactly via non-conforming accretions using hexaeders.
Tetraeder should not be used as the used linear shape functions can not repre-sent a uniform bending states!
Please use WINTUBE for a graphical input or extrude Quad areas to a hexaedral mesh - see example SOFIMSHA/SOFIMSHC e.g. hex_handle.dat
But also with pure SOFIMSHA simple structures can be created using hexaeder, e.g. bric_bucl.dator more sofisticated water.dat
Orthotropic material properties can be defined with the help of a meridian and a descend angle.
The following options are available in extensions:
• Yield criteria for plastic analyses including analytical primary stress states • Import of temperature fields from program HYDRA (they can be applied to
the structure as loading)
Material laws of AQUA-NMAT are implemented especially for tunnel analysis, e.g.:
• Mise-Drucker-Prager (also for steel) • Mohr Coulomb
• Lade (for concrete)
Materials CONC and STEE are only computed linear in volume elements! For a nonlinear analysis with concrete or steel see:
Concrete: bric_concrete.dat
2.10
Primary Load Cases
For the analysis of construction stages or for the definition of load steps in ge-ometrical non-linear analyses it is possible to use a previous load case. The parameters of the primary stress state are defined group-wise for this purpose. A detailed description of the method is also given in the TALPA manual.
Construction stages can be considered with different accuracies. The easiest way, of course, is to analyse the construction stages with the respective struc-tural system independently on each other and then proceed with the superpo-sition and the design of the structure. The different statical systems can be selected through the assignment of the elements in groups.
ASE has, however, also a very efficient possibility to use stresses and deforma-tions of a primary load case which allow the complete consideration of effects from creep or system change. See also module CSM Construction stage man-ager.
During application the user must keep in mind that each stress state in a single element corresponds to an external loading of the element and is in equilibrium with that loading.
ASE calculates now equivalent forces from the internal forces or stresses of the elements and can apply them as loading (GRP...FACL). These forces create a deformation state which counteracts the internal forces and makes them to zero when the statical system is not changed. If a system change has taken place in the meanwhile or if these loads have been applied with different factors, corresponding inherent stress states result.
Following principal cases have to be distinguished:
1. If the old loading is activated together with the primary state with a factor
GRP...FACL=1.0, new loads do not result. The stresses remain the same, the deformations are zero. According toSYST...PLC the total deformations or atSYST...PLC=0 only the addition deformations are output.
2. If only the primary state is applied as loading with a factorGRP... FACL=0, the resultant loading is the primary load case with inverted sign. This gives rise to unloading deformations that generate a stress state which becomes zero together with the primary stress state in the case of free deformability. This FACL=0 method should only be used in special cases. FACL=1 is the usual default.
A graphical explanation to this can also be found in figure 2.18 :taking over the primary load case
If some elements are removed from the system of the primary stress state along with their corresponding loads, the initial equilibrium is disturbed and forces arise at the boundary nodes of the removed parts. The remaining elements expand to the direction of the removed parts. If the primary state is generated analyti-cally, the removed parts do not have to be defined once, because all necessary information can be extracted from the remaining elements.
Using ECHO LOAD EXTR one can obtain an output of the internally generated loading at every node. This option should be used generally during analyses with primary states, because it is the best means for tracking down errors in the description of the states. The really applied nodal loads (nodal load vectors) can be represented with the program WinGRAF.
Further instructions can be found in the description of the record GRP in the TALPA manual or in the examples.
See also figure 2.18 :taking over the primary load case
2.11
Non-linear Analyses
Non-linear effects can be analysed only with iterations. This is done in ASE usually with a modified Newton method with constant stiffness matrix. The ad-vantages of the method are that the stiffness matrix does not need to be de-composed more than once and that the system matrix remains always positive definite. The speed of the method is increased through an accelerating algo-rithm written by Crisfield. This method notices the residual forces developing during the iterations and calculates the coefficients e and f for the displacement increments of the current and the previous step. A damping of the method can be specified in the case of critical systems (SYST...FMAX<1.0).
For a geometric nonlinear analsis a Line Search technique with an update of stiffness is used - see CTRL ITER.
Examples see -> Summary of example overviews
Following non-linear effects are implemented currently: please also refer to N-STR in section 3.36:
• Spring elements (failure, yield, slip, friction, work laws) • QUAD foundation elements (failure, yield, slip, friction) • Cable elements ( compression failure, material work laws) • Truss elements (tension failure, material work laws)
• Nonlinear beam elements
• Non-linear material laws for QUAD and BRIC elements
• Geometrically non-linear analyses for all elements, cable sag, membranes Tendons defined in the QUAD elements with the program TENDON can be used only in geometrically linear analysis.
For TRUS, SPRI, CABL, BEAM, QUAD and BRIC in a geometrically non-linear analysis the initial stress matrix is added to the stresses of the primary stress state (for TRUS, SPRI and CABL without reference to a primary stress state, the prestress from the program SOFIMSHA/SOFIMSHC is used for this purpose -see CTRL CABL). Thereby the iterations are markedly more stable when refer-ring to a primary load case and the ultimate load can be calculated more pre-cisely. A stability failure is recognized also in this way, even in the cases without unplanned initial deformation (an unstable system is reported, if the stresses of the primary state exceed the buckling load, i.e. the total stiffness matrix is neg-ative). Since it is reported here, that the PLC was actually unstable, this feature is only meaningful in the case of small load steps.
A module for the ultimate load calculation ULTIincreases or decreases the load step-by-step until it reaches a still sustained loading.
Initial deformations of the structure can be read as results of already anal-ysed load cases with the record SYST...PLC...FACV. With GRP...FACL=0 and FACP=0 the initial deformation is applied without stresses. This can also be done more clearly with OBLI (the OBLI oblique position or predeformation can also be mixed with a primary stress state SYST PLC). In the stored results, the initial deformation is added to the incremental displacements of the actual load-case. With CTRL DIFF the increment can be stored separately. Deformations from a modal analysis (bucling eigenvalue) can be used as initial deformation via scaling with FACV or OBLI see ase9_quad_euler_beam.dattask (ULTI iteration ... with predeformation)
Non-linear analyses are not possible with the basic version.
2.12
Non-linear Analysis of Plates and Shells
2.12.1 Overview
The Layer-Model allows the layering of the material properties in a QUAD-shell element. The model can be implemented for laminated glass, laminated wood plates or other composite plates. The layer technique can be also implemented for the non-linear calculation of elements consisting of a homogeneous
materi-al. In this case it is used to establish the positions of the individual layers. This method is especially suited for the non-linear calculation of plates and shells consisting of steel and reinforced concrete. Up to now the non-linear construc-tion material models, steel and concrete, have been implemented for the shell-elements.
Example Input file
Concrete a1_introduction_example.dat
Steel ase12_buckling_slab.dat
Ulimate load - verification nonlinear_quad_concrete_beam.dat
pdf for that nonlinear_quad_concrete_beam.pdf
Arch pure concrete arch_bridge.dat
Steel_fibre_concrete steel_fibre_concrete.dat
Sector tank concrete_tank_cracked.dat
Fire design quads_on_fire_1.dat
pdf for that quads_on_fire_1_english.pdf
-> Example overviews ->Summary of example overviews
The relaxation in individual layers, due to former plastification, is considered by consistently saving the results in all the layers of the elements (hysteresis effect for the bending of plates). This could create residual stresses over the cross-sectional height, even after total relaxation.
By means of the concrete law one can even consider creep and shrinkage ef-fects for a cracked shell-element (The redistribution of stress, from concrete to the reinforced steel, due to creep and shrinkage).
Several other advantages of the layer technique become apparent during the visualisation of the results. Besides the output of the numerical results in the different layers of the element one also has the option to graphically view the stresses over the element thickness in the program called ANIMATOR (choose a loadcase and double klick an element).
2.12.2 Input of the Materials
The calculation program ASE can evaluate an analysis for either the working-or the failure-stress level. It is advisable to use the option ECHO MAT YES in ASE, which checks the material values. The really used stress-strain curves of the material are plotted then and the significant values are printed.
Work law input Input file
Spring work law a1_spring_overview.dat
.z.B. kinematic hardening spring_law_3_pkin.dat
Concrete work law steel_fibre_concrete.dat
Steel work law ase12_buckling_slab.dat
Layer Sperrholzplatte ase.dat/english/special/timber_quad_layer.dat
Layer Hohlkörperbeton bubble_deck.dat
-> Example overviews ->Summary of example overviews
Preset Stress-Strain Curves in AQUA
Without any different defaults for the material parameters one gets from AQUA with following input:
ECHO MAT FULL $ for output of the stress-strain curves $
NORM DIN 1045-1 $ acc. to DIN 1045-1 $
$---Concrete:---$
CONC 1 TYPE C 25 $ standard C25/30 $
STEE 2 BST 500SA $ reinforcement $
the stress-strain curves which are represented below for the desired concrete. Here are according to chapter 9.1.5 of DIN 1045-1 (02.07):
sig-u (red) Stress-strain-curve for the cross section design (parabola-rectangle-diagram) according to equation (65) and (66) [ 4] .
sig-r (blue) Stress-strain-curve for non-linear methods of the determina-tion of internal forces and moments according to equadetermina-tion (62) with fc = fcR [ 4] .
sig-m (green) Stress-strain-curve for non-linear methods of deformation analysis according to equation (62) with fc = fcm [ 4] .
Figure 2.11: AQUA plot of the standard stress-strain curves for concrete C 25/30
according to DIN 1045-1 (07.02)
In analog mode one gets the stress-strain curves for the reinforcement according to chapter 9.2.3 and 9.2.4 of DIN 1045-1 (07.02):
Figure 2.12: AQUA plot of the standard stress-strain curves for reinforcement
500S(A) according to DIN 1045-1 (07.02)
ASE uses the strain curves from AQUA. In this way also arbitrary stress-strain curves which are defined manually can be considered.
Following requirements are to be considered for the input of the stress-strain curve type in order to select the correct curve during calculation in ASE with record NSTR. The stress-strain curve for concrete as well as for steel is defined with the item KSV in record NSTR and without the input for KSB. If a
stress-strain curve is defined for KSV and for KSB, KSV sets the curve for concrete and KSB for the reinforcement. In this way arbitrary combinations are possible. Types and designations of the stress-strain curves in AQUA and ASE
Designation of the stress-strain curve
Type in AQUA record SSLA
Selection in ASENSTR
without/with safety coefficient
sig-u (red): design ULTI UL / ULD
sig-r (blue): non-linear internal forces and
moments
CALC CAL / CALD
sig-m (green): non-linear deformations
SERV SL / SLD
Following AQUA input defines a new serviceability stress-strain curve for con-crete as well as for reinforcement with the safety 1.3:
$-- Input of an example stress-strain curve for serviceability limit state: $
SSLA SERV 1.3 $ first SSLA record defines the type of the stress-strain curve $
$ The value after type of the stress-strain curve sets the corresponding $
$ safety coefficient $
SSLA EPS SIG TYPE
0.30 0.0 $ tensile zone $ 0.09 2.1 0 0 $---$ -1.1 -17.8 spl -2.0 -24.0 spl -3.5 -23.0 $ compression zone $ -4.5 0 $--- reinforcement: ---$ STEE 2 BST 500SA
$-- Input of an example stress-strain curve for serviceability limit state: - $
SSLA SERV 1.3 $ first SSLA record defines the type of the stress-strain curve $
$ The value after type of the stress-strain curve sets the corresponding $
$ safety coefficient $
SSLA EPS SIG TYPE=POL
-50 -525 $ compression zone $
-2.3 -500
0 0 $---$
2.3 500
25 525 $ tensile zone $
50 525
The stress-strain curves which are input in this way can be seen and checked as modified serviceability stress-strain curve (sig-m / green) in the AQUA output of the material values and in the plot of the stress-strain curves:
Figure 2.13: AQUA plot with manually defined stress-strain curve sig-m (green) for
concrete
Temporary Material Control Parameters in ASE
In ASE record CTRL item CONC there are extended input possibilities for the material law for non-linear reinforced concrete. On the one hand the control parameters can be input here for consideration of the multiaxial stress state. On the other hand a temporary modification of the in AQUA defined material values FCT and FCTK, which is only valid in the current ASE calculation, can be done here also.
Selection of a Stress-Strain Curve for an ASE Calculation
The selection of a preset or manually defined stress-strain curve is done with an input in the ASE record NSTR (items KSV and /or KSB). Possible temporarily different inputs for the concrete tensile strengths and the consideration of the multiaxial stress state can be done with record CTRLCONC.
Check of the Material Values in ASE
In order to increase the transparency of the calculation the material values and further definitions for the non-linear material law which is in each case used in the calculation are also output in ASE. For this purpose it is necessary to set ECHO MAT YES. Then it follows here a definition of the analysis method for consideration of the crack widths and the tension stiffening as well as the output of all relevant parameters. In addition a presentation of the actually used stress-strain curves of the materials as well as a detailed plot of the concrete stress-strain curve in tensile zone are printed in the URSULA output.
Figure 2.14: Plot of the used concrete stress-strain curve in ASE
Figure 2.15: Detailed plot of the tensile zone of the concrete stress-strain curve in
ASE
For laminated timber or laminated glass calculations a QUAD element can be defined about the height also with variable material composition. The materials for the individual layers are saved at first in AQUA how usual in separate
material numbers. Then MLAY is used to define a composite material, which is input according to the layer arrangement. First the layer-thicknesses t0, t1, t2, t3 and t4 are defined, which are then followed by the respective material numbers:
Layer t0 6 mm thick out of material 11,
Layer t1 3 mm thick out of material 12 etc... : PROG AQUA
MATE 11 E 60e3 MUE 0.2 $ glass $
MATE 12 E 0.8e3 MUE 0.3 $ plastic $ $ glass-plastic-glass $ MLAY NO 1 T0 0.006 11 $$ T1 0.003 12 $$ T2 0.003 12 $$ T3 0.003 12 $$ T4 0.006 11 END
Figure 2.16: Heterogeneous Layers
The intermediate layers t2+t3 were defined only for a more clear output! The layer material No. 1 can be used only for QUAD elements.
Example MLAY input see bubble_deck.dat
or for a timber slab: timber_quad_layer.dat
Note: The analysis is according to plate theory, i.e. assuming that the cross-section does not have planar deformation! The displacement of the plates be-tween each other is not taken into account. For this one would have to couple the plates with springs!
This model is not suited for the analysis of local failure at the coupling points of laminated glass plates, because for such an analysis the planar deformation of the cross-sections is very important. At these points one could evaluate a spatial stress-state, which can only be depicted by volume elements.
Any arbitrary material can be used basically also orthotropic as layer for non-linear analyses.
