Seismic Analysis of Buildings - MASES...

Seismic Analysis of BuildingsStorey-related functionality & IRS dynamic analysis

Chapter 1

Introduction 4

Reduced Analysis Model 5

Introduction: themodified IRSmethod 5

IRSanalysis vsdiaphragms 5

Benchmarks 6

Example 1: simple platform 6

Example 2: 7-storeybuilding 8

Using the IRSmodel in Scia Engineer 10

Enabling the reducedmodel 10

Definition of theR-nodes 11

Running the analysis&Resultsoutput 14

Storey Results 15

Introduction 15

Summarystorey results 15

Output Settings 15

Result typeStoreyData 16

Result typesDisplacements&Accelerations 17

Detailed storey results 21

Output Settings 21

Result type Internal forces 22

Result typeResulting forces– Location per member 27

Result typeResulting forces– Location per storey 29

Someexamples 30

Output of total vertical forces in all storeys (load descending) 30

Internal forces in all supportingmembersof a storey 31

Average forces in all supportingmembersof a storey 32

Resulting forces in all supportingmembersof a storey 33

Modal Superposition 34

Introduction – theoretical background 34

Modal superposition 34

Predominantmode&signed results 35

Modal superposition of resultants 36

Modal superposition in Scia Engineer 37

Signed results 37

4.2.1.2Modal superposition of resultants 38

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Accidental Eccentricity 40

Introduction 40

Definition of the accidental eccentricity 40

Value of the accidental eccentricity 41

Calculation of the effectsof eccentricity 42

Linear distribution of accelerations 43

Distribution of accelerations fromeigenshape 43

Accelerations frommodal superposition 44

Analysis& resultsof accidental eccentricity 45

Accidental eccentricity load cases 45

Load groups 46

Combinations 46

Generated combinations 47

References 48

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Chapter 1

IntroductionAsa general 3D structural analysis software, Scia Engineer is able to provide a detailed analysis of most civil engineeringstructures. However, the particular field of seismic design of buildings requires specific tools in order to improve the effi-ciencyof that process. The toolspresented in thisdocument are primarily intended for an effectivemodelization and analysisbuildingsunder seismicaction.

The overall seismicbehaviour of a building can veryoften be analyzed accuratelybyconsidering each floor asamass linkedto neighbouring floorsbywallsand columns. Therefore the concept of storeys is widelyused in this context. Several aspectsof data input, aswell asanalysismodel, result output and design functionalitywill be directly linked to the storeysof a building.

The IRS (ImprovedReduced System) analysis will be used for the computation of eigenmodes of the structure. That tech-nique allows reducing drastically the number of degreesof freedom of a large finite elementmesh. The computation time isreduced accordingly. Moreover, most of the local vibrationmodes, which are irrelevant for the overall seismic response ofthe structure, will not appear in the results– unlike in standard FEanalysis.

Storey results give a synthetic view of results in each storey. Displacements, accelerations, internal forces and resultantforcesmay be displayed per storey, either as bulkvalues for each storey, or as detailed results for each wall and column.That featuremayalso be used for fast and comprehensive load descending output.

Most seismicdesign codes require that torsional effectsdue to accidentalmasseccentricityare taken into account. Based onstoreydata, the effectsof accidental eccentricitymaybe included in themodel according to variousmethods.

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ReducedAnalysisModel

Reduced Analysis Model

Introduction: the modified IRS methodThe actual tendency in FE structural analysis isusing full 3Dmodelization of the considered structure. Scia Engineer obeysthat rule asstructuresare usuallymodelized in 3D using beamand shell elements, including buildings.

Once a detailed 3D modelization is ready for statical analysis of a structure, it is natural to use it also for dynamic analysisand, more specifically, for seismic design. A typical issue of full 3D modelization is, that seismic design regards mostly theoverall behaviour of the structurewhence the fullmesh of the structure provides a lot of information about local behaviours.More specifically for modal analysis, the fullmesh letsappear local vibration modes, which are irrelevant for the overall seis-mic response of the structure. It appearshence logical to use a different, reducedmesh for the dynamicanalysis.

Well knownmatrix condensation techniques (GuyanReduction, also known as static condensation [1]) allow obtaining veryefficiently a reduced system, but those methods are not well suited for dynamic analysis. An Improved Reduced System(IRS) method hasbeen developed [2] which takes into account not only the stiffnessmatrixof the system, but also themassmatrixduring the condensation process. Thatmethod proved to give excellent results for dynamic analysis, with bothmodalanalysisand direct time integrationmethods.

The algorithm implemented in Scia Engineer used the IRSmethod and consistsof 3 steps:

1. The IRSmethod isused for condensing themesh of the analysismodel.2. Themodal analysis isperformed using the reducedmesh, which has typically1’000 times lessdegreesof freedom

than the original fullmesh. Thismakes the calculation of eigenvaluesmassively faster on large structuresand alsoavoidsunwanted localmodes. The latter isparticularly interesting for seismicanalysis.

3. The resultsof the reduced systemare expanded to the original fullmesh, allowing for output of detailed results inthe entire structure.

IRS analysis vs diaphragmsModelizing deckslabsasdiaphragms in a building for seismicanalysis isa common technique. It hasseveral purposes:

1. elimination of irrelevant, local bending vibrationmodes in the slabs2. elimination of unwanted frame effects from the structural behaviour

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Chapter 2

3. reduction of computation time4. easyhandling ofmasseccentricities for each deck

Items1, 3 and 4 are addressed directlyby the IRSanalysis:

l localmodes in all structural elementsare implicitly removed, due to the elimination of unwanted degreesof free-dom.Of course, addingmore reduction nodeswould allow for more detailed analysisof localmodes, but it is par-ticularly interesting for seismicanalysis to keep in the reducedmodel only the nodes that are strictlynecessary toreproduce the typical seismicbehaviour of a building. Ultimately, it is up to the user to choose the reduction points insuch away that thewanted eigenmodesare obtained.

l the computation time is reduced, due to the drastic reduction of the number of degreesof freedom; actually, thereduction isevenmore important thanwith diaphragms, assupportingmembersare also condensed. The reduc-tion of computation time affectsonly the computation of the eigenmodes itself. The pre- and post-processing timesremain practicallyunchanged, which include overall data preparation,mesh generation andmodal superposition oflocal results. These operationsare, obviously, still performed on the fullmesh of themodel and can be quite timeconsuming.

l the IRSanalysisusesa fullmassmatrix, which allows for exact implementation ofmasseccentricity in each node ofthe reduced system

Item2 – removal of frame effects– is not addressed by the IRSanalysis in itself, as it doesnot modify themechanical beha-viour of the structure. However, as unwanted local bending modes are implicitly removed from the reduced system, so-called flexible diaphragmsmaybe easily simulated bysignificantly reducing the bending stiffnessof deckslabs. Not onlydoesthat allow to obtains classical diaphragm behaviour by means of a very low bending stiffness, but also intermediate beha-viour where the bending stiffness is lessdrastically reduced and frame effectsare therefore reduced, but not completelyremoved.

Benchmarks

Example 1: simple platformThe structure is a simple platform, with 9 steel columnsand a thin plate on the top. This academicexample only intends toillustrate some aspectsof themethod.

The reducedmodel consistsof 2R-nodes, located at the top and at the bottomof themiddle column.

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The table below shows the natural frequenciesand eigenshapes for both fullmesh and IRSanalysis. The 11 first modesaredisplayed for the fullmesh analysis, alongwith the correspondingmodes from the IRSanalysis.

Asexpected, there are gaps in the IRS list, as the DoF of the reducedmodel cannot represent all the bending modesof theplate. Thosemodesare anyway irrelevant for the seismicbehaviour of the structure. Apart from the change of sign for mosteigenmodes,it is clear that the correspondence isexcellent forboth frequencyandmode shape for eigenmodes that are sig-nificant for seismicbehaviour.

FullMesh analysis IRS analysis Full Mesh analysis IRS analysis

Mode 1 – 1.15 Hz Mode 1 – 1.17 Hz Mode 7 – 2.10 Hz

N/A

Mode 2 – 1.15 Hz Mode 2 – 1.17 Hz Mode 8 – 2.15 Hz

N/A

Mode 3 – 1.21 Hz

N/A

Mode 9 – 2.27 Hz

N/A

Mode 4 – 1.23 Hz Mode 3 – 1.23 Hz Mode 10 – 2.30 Hz N/A

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Chapter 2

FullMesh analysis IRS analysis Full Mesh analysis IRS analysis

Mode 5 – 1.42 Hz Mode 4 – 1.42 Hz Mode 11 – 2.32 Hz Mode 5 – 2.32 Hz

Mode 6 – 2.10 Hz

N/A

The next table shows themodalmasses for bothmodels. Significantmodes for seismicbehaviour are spread up tomode 11for full mesh analysis. For IRS analysis, theyare all grouped are the beginning of the list. Note that the first two modesarepure bendingmodesandwould definitelynot occur in a building.

Another important point is, that the obtained totalmodalmass ishigher with the IRSanalysis. The 12 listedmodes representall the eigenmodes of the reduced model, as it has 12 DoF. That is the reasonwhy the IRS analysis obtains 100% of themass. For the fullmesh analysis, it is normal that the obtained modalmass is lower, as there aremanymoremodes that arenot listed here.

FullMesh analysis IRS analysis

Mode Freq. Wxi / Wyi / Wzi / Mode Freq. Wxi / Wyi / Wzi /

[Hz] Wxtot Wytot Wztot [Hz] Wxtot Wytot Wztot

1 1.15 0 0.0001 0 1 1.17 0 0.0003 0

2 1.15 0 0 0 2 1.17 0 0 0

3 1.21 0 0 0 3 1.23 0 0 0.8934

4 1.23 0 0 0.887 4 1.42 0 0.9987 0

5 1.42 0 0.9981 0 5 2.32 0.999 0 0

6 2.10 0 0.0001 0 6 2.71 0 0 0

7 2.10 0.0004 0 0 7 130.72 0 0.001 0

8 2.15 0 0 0 8 205.43 0.001 0 0

9 2.27 0 0 0 9 248.14 0 0 0

10 2.30 0 0 0.0001 10 413.53 0 0 0.1066

11 2.32 0.9644 0 0 11 711.66 0 0 0

12 2.47 0 0.0011 0 12 737.23 0 0 0

0.9648 0.9994 0.8871 1.00 1.00 1.00

Example 2: 7-storey buildingThisexample isa real, 7-storeybuilding. No particular simplification hasbeen done to create themodel for the seismic ana-lysis from the staticmodelization.

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Building of the ACPC, Fribourg, Switzerland

Courtesy of GIBES Engineering Group, Lausanne, Switzerland


Degrees of freedom 152’988 48

Requested modes 320 48

Total time: data generation,modal analysis, seismic analysis 31’27” 5’17”

Obtained total modal mass (X / Y / Z) 90% / 94% / 78% 95% / 96% / 98%

Nr ofmost significantmodes to obtain 90% modal mass 187 / 103 / >320 14 / 17 / 8

Note: 320 is the number of eigenmodes that is required in this case to obtain 90% of cumu-lated modal mass in both X and Y directions. For the vertical direction, over 400 modeswould be necessary (not calculated).


1stmost significantmode in X direction 3.12 Hz (3) 32.2% 3.12 Hz (3) 33.0%

2ndmost significantmode in X direction 7.38 Hz (18) 8.4% 14.56 Hz (30) 11.5%

1stmost significantmode in Y direction 1.71 Hz (1) 20.7% 1.71 Hz (1) 20.7%

2ndmost significantmode in Y direction 2.44 Hz (2) 13.2% 2.44 Hz (2) 13.2%

1stmost significantmode in Z direction 5.16 Hz (7) 9.6% 115.57 Hz (46) 36.4%

2ndmost significantmode in Z direction 5.14 Hz (6) 7.0% 5.60 Hz (7) 19.1%

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Chapter 2

About the Z direction: eigenmode 46@115Hz isnot represented in the fullmesh analysisbecause the last computedmode(320) reachesonly29Hz. That particular mode isanyway irrelevant for the seismicanalysisas it is completelyout of the fre-quency range of an earthquake. Themodes nr 7 in bothmodels, although their frequency slightlydiffers, show practicallythe same deformed shape.

The final internal forces after modal superposition for the seismic analysis show very close results. In the main wall of thestructure (picture below), the peakvaluesof the verticalmembrane force nyare only2% apart. The entire result distributionis close to identical.


nymax = 2’362 kN/m nymax = 2’416 kN/m

Using the IRS model in Scia EngineerUnlike the classical modal analysis, which typically uses a lumped mass matrix (only diagonal terms are non-zero), thereduced system usesa fullmass matrix , with non-zero values out of the diagonal. Thismeans thatmass eccentricitiescanbetaken into account easilyby the reduced system. The verysmallsize of the reduced systemallowsusing the fullmassmat-rix.

Therefore the reduction points– or so-calledR-nodes– that will constitute the reducedmodel do not need to be located in aparticularposition, such as themasscenter of each storey. Asthe structuremayhave tobe calculated several timeswith vari-ous distributions of the masses, the mass center of each storey is likely to be slightly different depending on the selectedmasscombination. Thanks to the use of a fullmassmatrix, the sameR-nodesmaybe used in all cases.

During the analysis, the reducedmodel is computed automatically from the fullmesh. Each node of the full mesh ismappedto the closest R-node of the reducedmodel.

It is up to the user to define howmany R-nodeswill be used for the analysis and, therefore, howmanydegrees of freedomthe reducedmodelwill have. For seismicanalysisof buildings, the typical reducedmodel consistsof 1R-node per storey, i.e.per deckslab.

Enabling the reduced modelFirst of all, the reduced model analysis must be enabled in the project. For this, tick the option “use reduced model” in thesolver settings.

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Definition of the R-nodesIn Scia Engineer 2013, the reducedmodel isdefined directly from the storeydata.

Storeysmust be defined in the modelization. The program will create the reduced analysis model bycreating one R-nodeper storey. Thismeans that the reduced analysis modelwill be valid for buildings that have one deckslab per storey. Eachdeckslabmaybemade of several 2Dmembers.

The programwill place aR-node at each storey, in themiddle of the bounding boxof the structure:

Note: this isa principle drawing. R-nodesare not displayed in Scia Engineer.

Further development isplanned, allowingmore advanced layout of R-nodes, i.e. more than one per storey. Scia Engineer2013.0 allows for oneR-node per storey.

Optionally, R-nodesmaybe placed at any level in each storey. The storey property “level of reduction point” allows to selectthe exact height of the reduction point for each storey separately. 0 corresponds to the bottom of the storey, 1 to the top ofthe storey.

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Chapter 2

With the default settings (see above), the deck slab of each storey is located at the bottom of the storey, and so is the cor-respondingR-node. It is recommended to keep it that way.

Asdescribed before, R-nodesare placed in each storey, at the specified level, in themiddle of the structure (allR-nodesarelocated on the same vertical axis).

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During the analysis, the reduced model is automaticallygenerated from the fullmesh of the structure. Each node of the fullmesh ismapped to the closest R-node. In a typical building configuration, this means that each R-nodewill receive the stiff-ness, loadsandmasses from the corresponding deckslab, from the top half of the supportingmembersbelow the slab andfrom the bottomhalf of the supportingmembersabove the slab.

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Chapter 2

Running the analysis & Results outputOnce the data hasbeen prepared, the analysis is run exactly in the samewayas for a standard analysis.

In Scia Engineer 2013, the IRS analysiswill be used only for the calculation of the eigenmodes of the structure. On the basisof those eigenmodes, the actual calculation of seismic load caseswill be performed on the original, fullmesh.

The use of the IRSanalysisappears in the calculation protocol:

There are fundamentally two typesof resultsavailable after an IRSanalysis:

l The resultsof the reducedmodel are automaticallyexpanded to the originalmesh and are accessible throughstandard output. Thiswill not be detailed here.

l Somededicated results, coming directly from the reducedmodel, are available in the new result service “SummaryStoreyResults” (see next chapter). This typicallygives information about themasses, displacementsand accel-erationsat each storey in the reducedmodel.

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StoreyResults

Storey Results

IntroductionScia Engineer 2013 introduces two new result services, dedicated for buildings.

The service “SummaryStoreyResults” provides resultsdirectlyproduced by the IRSanalysis (see previouschapter). At thistime, this service is dedicated to result output for the seismicanalysis of buildings. It providessingle resultsper storey, suchasmass, position ofmasscenter, displacement, acceleration…

The service “Detailed Storey Results” provides results from the full mesh analysis. At this time, it may be used for resultsfromany linear analysis, with or without dynamic analysis, with or without IRS analysis. It provides results in all supportingmembers, with easyselection ofmembersper storey.Wallsand columnsmaybe represented on the samedrawing. Typicalprovided resultsare: internal forces, resultantsper wall or per storey…

Tip: before using detailed storey results, make sure that all supporting members of thebuilding are properly allocated to storeys. That information is essential for proper handlingof storey results.

Summary storey resultsThisservice provides results directlyproduced by the IRS analysis (see previous chapter). At this time, this service is ded-icated to result output for the seismic analysis of buildings. It provides single results per storey, such as mass, position ofmasscenter, displacement, acceleration…

Pre-requisites for using SummaryStoreyResults:

l storeysmust be defined (pre-requisite for using the reduced analysismodel)l the reducedmodelmust be enabled in the solver settings (see chapter 2)

The servicemaybe found in theResults service. It isavailable onlyafter a successful dynamicanalysisusing the reducedmodel.

Output SettingsThe output settings are presented here according to the various types of results that can be obtained from this service.There are 3 typesof results:

l StoreyDatal Displacementsl Accelerations

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Chapter 3

Result type Storey DataStorey Data is available only for mass combinations. It displays for each storey the total mass and the coordinates of themasscenter.

Type of loadsselection of the type of loadMass combinationsselection of the combination ofmassgroupsor eigenmode for a given combination ofmassgroups.Selectiontype of selection; the possible choicesare

l All storeysl Named selectionl Single storey

Named selectionnamed selection that containsa set of storeys; only if Selection =Named selectionStoreydropdownmenu for selection of a single one; only if Selection =Single storeyDraw valuestick to draw valueson the drawingDraw unitstick to draw valueswith their unitson the drawingResult typetypesof results; possible choicesare

l Displacementsl Accelerationsl StoreyData

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StoreyResults

Result types Displacements & AccelerationsDisplacements& Accelerations are available for eigenmodesand seismic load cases. The valuesof displacement & accel-eration componentsare given at themasscenter of each storey.

Results for masscombinationsare raw, normalized results frommodal analysis, without effect of response spectrum.

Results for seismic load casesare valuesafter modal superposition.

Type of loadsselection of the type of load

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Chapter 3

l Masscombinationsl Load cases

Mass combinationlist of available eigenmodes for each computed combination of massgroups; only if Type of loads = Mass com-binationsLoad caselist of available seismic load cases; only if Type of loads=Load casesSelectiontype of selection; the possible choicesare


Named selectionnamed selection that containsa set of storeys; only if Selection =Named selectionStoreydropdownmenu for selection of a single one; only if Selection =Single storeyExtremefilter extreme results; possible choicesare

l Nol Section (relevant only for seismic load cases)l Member (relevant only for seismic load cases)l Global

Draw valuestick to draw valueson the drawingDraw unitstick to draw valueswith their unitson the drawingResult typetypesof results; possible choicesare

l Displacementsl Accelerationsl StoreyData

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StoreyResults

Valuesmain component to be displayed;for Result type =DisplacementsUx, Uy, Uz : displacement atmasscenter according toGCSaxesPhix, Phiy, Phiz : rotation atmasscenter aroundGCSaxesfor Result type =AccelerationsAx, Ay, Az : acceleration atmasscenter according toGCSaxesAlphaX, AlphaY, AlphaZ : rotational acceleration atmasscenter aroundGCSaxes

Additional valuesallow to showmore than one result component simultaneouslyon the drawing.When only the main component is selected (no additional value ticked), the selected component is drawn in thecorresponding direction.

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Chapter 3

When one or more additional value is ticked, all selected componentsare listed in the plane of the screen for bet-ter readability.

Limit valuesforeach result component, definition ofmin andmaxvalue for colour coding on the drawing. Coloursmaybe con-figured in Settings>Colours/Lines.Default colour coding isas follows:Values lower thanVmin are display in redValuesbetweenVmin andVmax are displayed in grayValuesgreater thanVmax are displayed in blue

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StoreyResults

Detailed storey resultsThisservice provides results from the fullmesh analysis. It may be used for results fromany linear analysis, with or withoutdynamicanalysis, with or without IRSanalysis. It provides results in all supportingmembers, with easyselection ofmembersper storey.Wallsand columnsmaybe represented on the samedrawing.

Typical provided resultsare: internal forces, resultantsper wall or per storey…

Pre-requisites for using SummaryStoreyResults:

l storeysmust be definedl supportingmembersmust be properlyallocated to storeys

The servicemaybe found in theResults service. It isavailable onlyafter a successful analysis if storeysare defined.

Output SettingsMainly2 typesof resultsare available in this service:

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Chapter 3

l Internal forces in supportingmembersl Resulting forces

For resulting forces, the Locationmaybe selected:

l bymember: compute the resulting forces for each supportingmember separatelyl bystorey: compute the resulting forces for each entire storeyat once, combining 1D and 2Dmembers

Result type Internal forces

Type of loadsselection of the type of load

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StoreyResults

l Load casesl Combinationsl Class

Load caseslist of available load cases; only if Type of loads=Load casesCombinationslist available load case combinations; only if Type of loads=CombinationsClasslist of available result classes; only if Type of loads=ClassEnvelopeMaximum /Minimum; select themaximumor minimumvalue of an envelope; only if Type of loads=Combinationsor ClassSelectiontype of selection; the possible choicesare


Named selectionnamed selection that containsa set of storeys; only if Selection =Named selectionStoreydropdownmenu for selection of a single one; only if Selection =Single storeySection levellevelwhere the sectionmust bemade across the supportingmembers in each storey; possible choices:

l Top (section at the top of each storey)l Middle (section atmid-height of the each storey)l Bottom (section at the bottomof each storey)l User defined

User defined section levellevel of section in each storey; 0 =bottomof the storey, 1 = top of the storeyFilterfiltering of supportingmembers for the output; possible choicesare

l Nol Wildcardl Materiall Thickness / CSSl Layer

Wildcardwildcard for name filtering ofmembers; only if Filter =WildcardMateriallist of availablematerials for material filtering ofmembers; only if Filter =Material

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Chapter 3

Thicknessvalue of thickness for thickness filtering ofmembers; only if Filter =Thickness/CSS; 0 =all thicknessvaluesCSSlist of available cross-sections for filtering ofmembers; only if Filter =Thickness/CSSLayerlist of available layers for filtering ofmembers; only if Filter =LayerSystemselection of coordinate system for output of internal forces in 1Dmembers; possible choicesare

l principal (principal axesof the cross-section)l LCS (LCSof the 1Dmember)

for 2Dmembers, the LCS isalwaysusedExtremefilter extreme results; possible choicesare

l Nol Sectionl Memberl Global

Draw valuestick to draw valueson the drawingDraw unitstick to draw valueswith their unitson the drawing

Result typeselection of result type; possible choicesare

l internal forcesl resulting forces

Values on beamsmain component to be displayed on 1Dmembers (columns); possible choicesare

l N=axial forcel Vy=shear force according to Y-axisof selected systeml Vz=shear force according to Z-axisof selected systeml Mx= torsionalmomentl My=bendingmoment around theY-axisof selected systeml Mz=bendingmoment around the Z-axisof selected system

Additional valuesallow to show more than one result component simultaneously on the drawing for 1D members; possiblechoices: seeValueson beamsWhen only the main component is selected (no additional value ticked), the selected component is drawn in thecorresponding direction.When one or more additional value is ticked, all selected componentsare listed in the plane of the screen for bet-ter readability.

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StoreyResults

Limit valuesforeach result component on 1Dmembers, definition ofminandmaxvalue for colour coding on the drawing. Col-oursmaybe configured in Settings>Colours/Lines.Default colour coding isas follows:Values lower thanVmin are display in redValuesbetweenVmin andVmax are displayed in grayValuesgreater thanVmax are displayed in blueValues on slabsmain component to be displayed on 2Dmembers (walls); possible choicesare

l nx=membrane axial force in X-direction ofmember LCSl ny=membrane axial force in Y-direction ofmember LCSl nxy=membrane shear force inmember LCSl mx=bendingmoment aroundY-axisofmember LCSl my=bendingmoment aroundX-axisofmember LCSl mxy= torsionalmoment according tomember LCSl vx=shear force according to X-axisofmember LCSl vy=shear force according to Y-axisofmember LCS

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Chapter 3

Additional valuesallow to show more than one result component simultaneously on the drawing for 2D members; possiblechoices: seeValueson slabsLimit valuesfor each result component on 2Dmembers, definition ofmin andmaxvalue for colour coding on the drawing; seedetailed description aboveDiagramstyle of diagram for internal forces in 2Dmembers; possible choicesare

l Precise: raw computed results, without alterationl Trapezoidal: trapezoidal regression of diagram, for each 2Dmember separatelyl None: the diagram ishidden

Display total valuewhen ticked, the integral value of the displayed diagram iswritten, for each 2Dmember separatelyDisplay average valuewhen ticked, the average value of the displayed diagram is written and the corresponding uniform diagram isdrawn, for each 2Dmember separately

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StoreyResults

Result type Resulting forces – Location per member

Onlysettings that behave differently frompreviousparagraph are listed here. For information about settings that are not lis-ted here, please refer to previousparagraph “Internal forces”.



Location

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Chapter 3

selection of location type for resulting forces; possible choicesare

l bymember: the resulting forcesare computed for each supportingmember separatelyl bystorey: the resulting forcesare computed for each storey, considering all the supportingmembersat

once; 1D (columns) and 2Dmembers (walls) are taken into account together

Resulting forces in 1Dmembers (columns) are identical to internal forces in 1Dmembers.

Resulting forces in 2D members (walls) compute the resultant at the center of each wall, according to a dedicated localcoordinate system, regardlessof the Systemoutput setting. The coordinate system that isused is the same as the LCSof avertical rib placed in themiddle of thewall. It is also the same coordinate system that isused for integration strips.

l The localX-axis is vertical, upwardsl The localZ-axis is identical to the Z-LCSof thewalll Y=Z ^X

In thisway, resulting forces inwalls can be easilydisplayed together, consistentlywith internal forces in columnson a singledrawing.

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StoreyResults

Result type Resulting forces – Location per storey

Only settings that behave differently from paragraph “Internal forces” are listed here. For information about settings thatare not listed here, please refer to paragraph “Internal forces”.



Locationselection of location type for resulting forces; possible choicesare

l bymember: the resulting forcesare computed for each supportingmember separatelyl bystorey: the resulting forcesare computed for each storey, considering all the supportingmembersat

once; 1D (columns) and 2Dmembers (walls) are taken into account together

Systemselection of coordinate system for output of resulting forcesbystorey; possible choicesare

l GCSl UCS

UCSselection of aUCS from theUCS library, to be used as reference system for output of resulting forcesbystoreyValuesmain component of resulting force; possible choicesareFx, Fy, Fz= resulting force according to X,Y,Z axisof selected coordinate system

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Chapter 3

Mx,My,Mz= resultingmoment aroundX,Y,Z axisof selected coordinate system

Some examples

Output of total vertical forces in all storeys (load descending)

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StoreyResults

Internal forces in all supporting members of a storey

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Chapter 3

Average forces in all supporting members of a storey

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StoreyResults

Resulting forces in all supporting members of a storey

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Chapter 4

Modal Superposition

Introduction – theoretical background

Modal superpositionThe response spectrummethod isone of themost widelyused methods for the seismicanalysis of structures. It hasmanyadvantageson other methods:

l unlike equivalent lateral forces (ELF) method, it takesseveral vibrationmodes into account and allowsanalyzingstructurewithout restriction on the geometry

l unlike explicit time historydynamicanalysis, it takes into account awhole range of earthquakes, thuscovering anenvelope of possible earthquakes for a given site; a time historyanalysis takesonlyone earthquake into account ata time (one input accelerogram)

l the computation cost of the response spectrummethod ismuch lower than for time historyanalysis, especiallywhen nonlinearitiesneed accounting for

The response spectrum method uses a modal superposition of the relevant eigenmodes of the structure. The methodallows to calculate the magnitude of each mode, but not their phase shift. The values of phase shift actually depend on thereal accelerogram that will be applied to the structure. Asa response spectrum represents the envelope of a familyof accel-erograms, it isnot possible to define unique valuesof phase shift for eachmode: these variablesare random.

This iswhere the concept ofmodal superposition comes in: variousstatistical techniquesallow to determine envelope valueswhich cover the real behaviour with a reasonable probability of occurrence.

Themost widely used techniques are the SRSS (Square Root of Sum of Squares) andCQC (Complete Quadratic Com-bination).

Thesemethods have the advantage of providing very easily design values of all results (displacements, internal forces…)without knowing the real phase shift values, but onlypart of the information isavailable:

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ModalSuperposition

l likelymin andmaxvaluesof any result can be determinedl the actual sign of a result cannot be definedl the concomitance of separate results cannot be defined

The last point above is actually not entirely accurate, as probabilistic correlations can be established between results, butthose techniquesare currentlynot commonlyused in practice and are beyond the scope of thisdocument.

The lossof concomitance and sign of results isan issue typicallywhen computing resulting forces in shear walls: it is not pos-sible to compute a resultant from internal forcesafter modal superposition, as typicallyall raw resultsare positive.

Computing resultant forces in one of those shear walls would typically give near-zero moments and extremely over-estimated axial forces.

Predominant mode & signed resultsTo obtain usable valuesof resulting forces, a possibility is the so-called “signed results” method.

It consistsof applying some signature scheme to raw results of themodal superposition. A classical approach uses the signof themost significant eigenmode.

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Chapter 4

Applying this to shear walls, it is possible to “sign” the internal forces,making themsuitable for computation of resultingforces:

Modal superposition of resultantsThemethod of signed resultsmight be seen asaworkaround to obtain usable resulting forces. Thismethodwill be referredto as “calculation of resultantswith modal pre-superposition”. It is convenient froma point of view of computation, becausemodal superposition needs to be done onlyonce on all local resultsand the resultant can be computed directly froma uniqueset of results.

However, the rigorousmethod for computation of resultants in the context of the response spectrum method can be sum-marized as follows:

l compute the local internal forces for each eigenmodel compute the resultant force for each eigenmode separatelyl apply themodal superposition to the obtainedmodal resultant values

Thismethodwill be referred to as “calculation of resultantswithmodepost-superposition”.

When proceeding so, no result signature is necessary to obtain correct valuesof resulting forces.Moreover there are caseswhere the method described in the previous paragraph gives overestimated results of most result components and cantherefore onlybe seen asan approximation. Themethod described here is clearlymore robust and accurate.

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ModalSuperposition

As shown in the example below, structures with predominant torsional behaviour are especially sensitive to that phe-nomenon.

Important discrepancies can be seen on most resultant components when using the calculation of resulting forces fromsuperposed signed results (up to 4 times the reference result).

On the other hand, all values obtained by modal superposition of resulting forcescomputed in each mode separately areveryclose to the referencemodel (max6% difference).

Forces at the bottom of the core Fx Fy Fz Mx My Mz

Resultant from results signed after superposition 518 855 15 2509 2732 1691

Modal superposition after resultant calculation 249 198 26 1900 2394 1614

Reference model (1Dmember) 264 209 25 1911 2429 1640

Modal superposition in Scia EngineerScia Engineer offersallmethodsdescribed above. All related settingsare located in the propertiesof the seismic load case.

Signed resultsThe signature of results can be enabled using the setting “Predominantmode”. The eigenmode to be used as reference forthe signature can be selected automaticallyor manually.

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Chapter 4

Note that the automatic selection of the mode shape will select the modewhich has the highestmodal mass, regardlessofthe actual direction of the earthquake. This is most of the time correct for 2D modelizations (2D frames) but oftenwrong for3Dmodelizations! For 3Dmodels, it is recommended to select themode shapemanually.

4.2.1.2 Modal superposition of resultantsIn order to enable the calculation of resultantsusing modal post-superposition, the option “enable advanced modal super-position for seismic load cases” needsenabling in the solver settings.

Bydefault, that option is

l enabled for new projectsl disabled for projects created in previousversionsof Scia Engineer

The table below showswhichmethod isused for calculation of resultants.

Solver setting “Enableadvancedmodalsuperposition for seis-mic load cases”

Predominant mode setting in seismicload case

Method used for calculation ofresultants

OFF AllPre-superposition

ONON

OFF Post-superposition

The flowcharts below summarize the process for both methods. Note that although the post-superposition method mightlookmuch simpler than the pre-superposition method, actuallyall the part related to local results must be done anyway inorder to provide local results for seismic load cases. It is just not represented on the flowchart because it is irrelevant in thiscontext.

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ModalSuperposition

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Chapter 5

Accidental Eccentricity

IntroductionMost of the seismic codes require that structures are checked for torsion due to mass eccentricity including an additionaleccentricity– so-called accidental eccentricity. Thisis required to cover inaccuraciesbetweenthe real structure and themod-elization, aswell as the fact thatmasses that are linked to service loadsmayvaryduring the life of the structure.

Two typesof eccentricitymust be distinguished for the analysis: the structural eccentricityand the accidental eccentricity.

The structural eccentricity is the offset between the center ofmassand the center of stiffness of the structure. It ispart of thestructure. In a simplified seismic analysis via 2D models, where typically the X andY directionsare analyzed separately, theimpact of the structural eccentricity is taken into account by manually distributing the torsional effects on the structure. Anadditional safety factor isusuallyapplied to the structural eccentricity to cover inaccuraciesdue to that simplifiedmethod.

When using a 3D modelization of the structure, the structural eccentricity is automatically taken into account due to the factthat the XandYand linked and analyzed together, allowing torsional effects to appear directly in the analysiswithout havingto add themmanuallyafterwards.

The accidental eccentricityaccounts for inaccuracies in the distribution ofmasses in the structure. Design codesusually takeit into account asan additionalmasseccentricity that isdefined asa fraction of the size of the structure.

In the Eurocode 8, the accidental eccentricity for a given floor is defined as5% of thewidth of the floor perpendicularly to thedirection of the acting seismicaction.

In simplified modelizationswhere the structural eccentricity appearsexplicitly, it is very simple to add the accidental eccent-ricity in the calculation. In general 3D modelizations, the structural eccentricity does not appear as such and it is thereforemore difficult to account for itseffects in such a case.

In Scia Engineer, using the IRS condensed model allows introducing accidental eccentricity easily, since the condensedmodel uses onlyone R-node per storey. The accidental eccentricitymay be taken into account either as real mass eccent-ricityor asadditional torsional actions (simplifiedmethod according to the design codes).

However, themethod using realmasseccentricity in themodal analysishasnot been implemented yet in Scia Engineer.

For now, only the simplifiedmethod using additional torsionalmoment isavailable.

Definition of the accidental eccentricityFirst of all, please note, that accidental eccentricity maybe used only together with the reduced model analysis. See how toenable it in Enabling the reducedmodel.

The accidental eccentricity settings are defined in the propertiesof the considered seismic load case, in the sub-group .”Ac-cidental eccentricity”. Accidental eccentricity isdisabled bydefault.

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AccidentalEccentricity

Themethod for the calculation of the accidental eccentricity can be selected in the combobox “Accidental eccentricity”:

Depending on the selectedmethod, varioussettingsmust be defined.

Eccentricityvalue of the accidental eccentricity, defined asa fraction of thewidth of the considered storey in the direction per-pendicular to the seismicaction;most seismicdesign codesspecifya value of 0.05 for this ratio (EN1998-1 § 4.3.2(1)Pand formula (4.3) ).Mode shapein case the distribution of accelerations is calculated according to a mode shape, the user must specify whichmode should be used for that purposeLoad casein case the distribution of accelerations is calculated according to the deformed shape of the structure under theaction of static loads, a static load casemust be specified here

These settingswill explainedmore in detail hereafter.

Value of the accidental eccentricityRegardlessof the selectedmethod, the value of the accidental eccentricitymust be specified. That value isdefined asa frac-tion of thewidth of the considered storey in the direction perpendicular to the seismicaction (EN1998-1 § 4.3.2(1)P).

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Chapter 5

The actual eccentricity is then calculated as follows:

e e b=A i Ar i i, , ⋅

where eAR,i is the user defined value of the relative eccentricity, e.g. 0.05, and bi is thewidth of the considered storey. Thisvalue is computed separately for each storey.

Calculation of the effects of eccentricityThe accidental eccentricity is taken into account as follows:

a dynamicanalysisof the structure isperformedwithout accidental eccentricity, using the response spectrummethod

the effectsof accidental eccentricityare added byapplying static torsionalmoments to the structure about the vertical axisofeach storey. Thismethod isdescribed in detail in the Eurocode 8 (EN1998-1 § 4.3.3.3.3)

The general principle for the calculation of torsionalmoments isas follows:

F F=j base

α m

α m∑

j j

k k k

⋅

⋅

⋅

M F e=z j j A j, ,⋅

where

Fj is the horizontal force acting on storey j

Fbase is the total horizontal force acting on the structure (aka base shear) in the considered earthquake direction obtainedfrom the response spectrumanalysisof the structure

mj is themassof storey j

αj is the distribution key of the accelerations; this depends on the selected method; at this time, 3 methods are offered todefine the distribution of accelerations (see below)

eA,j is the accidental eccentricityof storey j, asdefined in the previousparagraph

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Mz,j is the applied torsionalmoment about the Z-axis for the storey j

Linear distribution of accelerations

Inthis case, the distribution of acceleration isassumed to be linear, proportional to theheight. The reference level is the over-turning level defined in the seismic load case properties.

( )α z z= max − ; 0j j overturn

where zj is the level of themasscenter of storey j. zoverturn isdefined by the user in the propertiesof the seismic load case.

Thismethod corresponds to the simplified approach defined in EN1998-1 § 4.3.3.2.3(3) and formula (4.11).

Distribution of accelerations from eigenshape

In this case, the distribution of accelerations is assumed to be proportional to the displacements of the structure in the rel-evantmode shape. The user must specify the referencemode (fundamentalmode).

When themode shape selection is set as “automatic”, the programselects the mode that has the highest modalmass in thedirection of the seismicaction.

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Chapter 5

α U=j G j,

where UG,j is the modal displacement of the mass center of storey j in the direction of the seismic action, obtained frommodal analysisof the reducedmodel.

This is the preferred approach in Eurocode 8, defined in EN1998-1 § 4.3.3.2.3(2) and formula (4.10).

Accelerations from modal superposition

Inthis case, no distribution key isused. The accelerationsare obtained directly from the seismic load case after modal super-position. The acting forceson storeysare obtained as follows:

F a m=j Gj j⋅

where aGj is the acceleration at the mass center of storey j in the direction of the seismic action obtained from the modalsuperposition in the reducedmodel.

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Thisapproach isnot described in theEurocode 8. It is more conservative than the other approaches, as it usesan envelopeof accelerations instead of a distribution of the resultant base shear. However, it has the advantage of covering caseswherehigher order modescannot be neglected for accidental eccentricity.

Analysis & results of accidental eccentricity

Accidental eccentricity load casesWhen enabling accidental eccentricity in a seismic load case, the program will automatically create an accidental eccentricityload case.

Accidental eccentricity load cases are read-only and cannot be deleted. To remove an AE load case, disable accidentaleccentricity in the corresponding seismic load case.

All properties of AE load cases are read-only, except their name and description which may be edited by the user. Thedefault valuesof the propertiesare:

Namenameof the source seismic load casewith added suffix “_AE”Description“Accidental eccentricity for EQ*” where EQ* is the nameof the source seismic load caseAction typeVariableLoad groupidentical to Name; see next paragraph for detailsLoad typestatic; the torsional effect is indeed computed asa set of static loads (moments) applied to the structureSpecificationSeismicaccidental eccentricityDurationShortMaster load casethe source seismic load case; thanks to this, the accidental eccentricitywill be applied in envelopes onlywhen thecorresponding seismicaction isalso present in the envelope

The content of anAE load case cannot be viewed nor edited. No loads can be added to it. The appliedmoments due to theaccidental eccentricitywill be automatically computed during the analysis.

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Chapter 5

ThegeneratedAE load casesmaybe inserted in seismicenvelopecombinations in order to accountfor AEeffects. Their res-ultsmayalso be viewed separately for validation.

Load groupsFor each seismic load casewith enabled accidental effect, the programwill create automaticallya read-only load groupwiththe following properties:

Namesame as the corresponding AE load caseRelationsame as the relation of the load group of the source seismic load caseLoadSeismicaccidental eccentricity

This load group isautomaticallyassigned to the corresponding AE load case.

CombinationsWhen using an AE load case in an envelope combination together with its source seismic load case, it will automatically becombined with the seismic action, assigning + and – sign to it. The AE load case will not be taken into account without itssource seismic load case.

TheAE load casesmust be addedmanually to combinations.

A typical use case:

Possible combinations

LC Description 1 2 3 4 5 6

SW Self weight (static) 1 1 1 1 1 1

EQX Seismic load case (dynamic) 1 -1 1 -1 1 -1

EQX_AE Accidental eccentricity load case for EQX (static) 1 1 -1 -1

Reminder: this table shows the principle of combination, but it is not applied strictlyassuch.When using a seismicenvelope combination in checks, the programalso uses other caseswhere the sign of internal forces components are switched independently to account forthe non-concomitance of extremesafter modal superposition in the response spectrummethod.

Reminder: when dealing with seismic load cases, for the reason above, use only “envel-ope” or “code” combinations. Do not use “linear” combinations, nor explode envelope orcode combinations to linear.

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Generated combinationsAsmentioned previously, AE load casesmust be addedmanually to load case combinations.

However, for convenience, the program generatesautomaticallyan envelope combination for each seismic load case, con-taining the source seismic load case and itsAE load case and named after the source seismic load case. Thisallows to checkeasily the results for the full seismicaction, including the effectsof accidental eccentricity.

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Chapter 6

References[1]Guyan, R.J., Reduction of StiffnessandMassMatrices, AIAAJournal, Vol. 3, No. 2, February, 1965[2]O’Callahan, j., AProcedure for an ImprovedReducedSystem (IRS) Model, Proceedingsof the 7th InternationalModalAnalysisConference, LasVegas, Nevada, February, 1989[3]EC-EN 1998-1, Eurocode 8 - Design of structures for earthquake resistance - Part 1: General rules, seismicactionsand rules for buildings

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