Download - Seismic analysis with SOFiSTiK
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Seismic analysis with SOFiSTiK
Oliver Bruckermann, 28 February 2008
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Overview
1. Basics
2. Eurocode 8
3. Modal analysis in SOFiSTiK- definition of seismic action- dynamic analysis- superposition of results
4. Linear time history analysis in SOFiSTiK
5. Specials- SIR module- P-∆ effects
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Characterisation of seismic action - Time history plots
acceleration
velocity
displacement
The three graphs are equivalent.
Peak groundacceleration
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Peak ground accelerations
More detailed maps are available on the GSHAP website.http://www.seismo.ethz.ch/GSHAP/
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Characterisation of seismic action – Response spectra
Undamped natural period
mk
T π2=
Damping is always given in relation to critical damping.
Damping is viscous (proportional to the velocity).
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Smoothed elastic response spectrum
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Eurocode 8 elastic response spectrum
Pseudo-acceleration Se / peak ground accelerationParameters:
• Soil: A (hard) – E (soft)
• damping:(normal 5 % damping)
( ) 55.05/10 ≥+= ξη
Spring force:
F = M x Se
The peak ground acceleration is to be multiplied by an “importance factor”which is between 0.8 and 1.4, depending on the type of building.
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Eurocode 8 design spectrum
Behaviour factor q accounts for:
• inelastic behaviour
• energy dissipation (hysteretic damping)
• viscous damping different from 5 %
• q depends on type and material of the structure (sections 5 to 9 EC 8)
In essence, the design spectrum is obtained by dividing the elastic spectrum by q (slightly different formulas, see clause 3.2.2.5 EC 8).
There is also a threshold of 0.2 for the design spectrum.
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Modal analysis
For in-depth information see:
Earthquake design practice for buildings, E. Booth D. Key
Dynamics of structures,Anil K. Chopra
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Modal analysis with Sofistik
1. Create load case(s) containing loads additional to the self-weight (LC_add)
2. Run linear analysis of LC_add
3. Determine mode shapes with ASE or DYNA including mass of LC_add
4. Define the seismic action (horizontal / vertical design spectra)
5. Run dynamic analysis with appropriate superposition of modes (CQC) and subsequent superposition of directions (SRSS), result is the seismic load case LC_seismic
6. Superimpose LC_seismic with other actions (dead load, live load); thereby consider behaviour factor for seismic displacements
7. Design for ULS and check displacements
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Modal analysis, steps 2-3
+PROG ASEHEAD Computation of the loadcasesLC (1 5 1) $ LC 1 contains the self-weight only END $ LC 1 is needed later for superposition ,
$ but not for mass-conversion (is included automatically)
+PROG ASEHEAD Calculation of the mode shapesmass lc 2,3,4 PRZ 100 $ 100% of additional dead weightmass lc 5 PRZ 30 $ 30% of additional live load, see next slideeige 500 $ number of sought mode shapes (eigenmodes)end
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Masses
ikiEjk QG ,,, ∑∑ ⋅+ ψ
iiE 2,
Clause 3.2.4(2), EC8
ψϕψ ⋅=Clause 4.2.4(2), EC8
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Modal analysis, step 4
+PROG SOFILOAD headlc 11 type none titl 'Response Spectrum horizontal x'acce ax 3.2*1.20 ay 0 $a_g = 3.2 m/s2, importance factor = 1.2resp ec-1 clas C mod 1.5 $Type 1 spectrum, Soil class C, behaviour factor 1.5
lc 12 type none titl 'Response Spectrum horizontal y'acce ax 0 ay 3.2*1.20resp ec-1 clas C mod 1.5
lc 13 type none titl 'Response Spectrum vertical'acce ax 0 ay 0 az 3.2*1.2*0.9 $ a_vg / a_g = 0.90 (Type 1, Table 3.4, EC 8)resp ec-1 clas C mod 1.5 AH 0 $ AH 0 switches to the vertical spectrumend
The MOD parameter is used as a switch:All values < 1.0 are interpreted as modal damping and the elastic spectrum is generated
All values >1.0 are interpreted as a behaviour factor and the design spectrum is generatedIn the example we have a behaviour factor of 1.5 and the design spectrum is generated.
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Modal analysis, step 4 (Ursula output)
Response spectra EC 8 Type 1, Soil Class CD[-] SA[-] SB[-] MIN[-] TB[sec] TC[sec] TD[sec] TE[sec] K1[-] K2[-] A[m/sec2]
1.5000 0.771 1.917 0.200 0.200 0.600 2.000 0.000 1.000 2.000 0.00
Loads acting on NodesNode A-X A-Y A-Z A-RX A-RY A-RZ
[m/sec2] [m/sec2] [m/sec2] [1/sec2] [1/sec2] [1/sec2]0 3.84
EC 8 Type 1, Soil Class C
[sec]
3.00
2.00
1.000.
00.
0
2.20
2.00
1.80
1.60
1.40
1.20
1.00
0.800
0.600
0.400
0.200
0.0
Note that the displayed response spectrum is normalised to the peak ground acceleration, ie in this example the maximum pseudo-acceleration is 1.917 x 3.84 = 7.36 m/s2
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Modal analysis, step 5
+PROG DYNA urs:45HEAD Calculation of all 3 directions and superposition of theseCTRL STYP 3 $ SRSS of the responses in 3 directionseige 500 rest $ Use 500 mode shapes (already calculated in step 3)lc 11lc 12lc 13$ modal superpositions with CQC (as per default)extr type u 53 STYP CQC $ deflections, not yet scaled with behaviour factorextr type N 301 STYP CQC $ actions in stick elementsextr type VY 302 STYP CQC $ ie. beam-elements and truss-elementsextr type VZ 303 STYP CQC extr type MY 304 STYP CQCextr type MZ 305 STYP CQCextr type NXX 321 STYP CQC $ actions in shell-elementsextr type NYY 322 STYP CQCextr type NXY 323 STYP CQCextr type MXX 324 STYP CQCextr type MYY 325 STYP CQCextr type P 326 STYP CQC $ spring forces (needed to get support reactions!)END
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SRSS / CQC – method for superposition of modes
∑=
=N
nntot RR
1
2SRSS (square root of the sum of the squares):
∑ ∑∑=
≠= =
××+=N
nni
N
i
N
nnintot RRRR
1)(
1 1
2 ρCQC (complete quadratic combination):
ρ is a factor between 0 and 1. The closer the period / frequencies of two modes the bigger ρ.
If the modes are widely spaced, SRSS and CQC give the same results.
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Axial (Normal) force N_x Bending moment M_y
LC 301(max N)
LC 304(max M_y)
The values for the leading action in one seismic result loadcase are always positive.
The other (non-leading actions) can take positive and negative values.
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”Special SRSS” in Sofistik for superposition of directions
1. For the leading action, e.g. N, “normal SRSS” is applied :
222zyxtot NNNN ++=
2. Multiplication factors are determined:
totzz
totyy
totxx
NNf
NNfNNf
/
//
=
==
3. Totals of non-leading actions are calculated, e.g.:
zzyyxxtot MfMfMfM ×+×+×=
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Mass activationCheck Ursula-output:Modal load contributions per functionfunct. mode R*V-factor [o/o] V*R*V-factor mode R*V-factor [o/o] V*R*V-factor
11 1 -2.604E-12 0.0 -3.173E-26 11 1.222E-14 0.0 -3.522E-262 2.974E+01 96.9 -3.826E+00 12 2.969E-02 0.0 -9.858E-023 -1.739E-14 0.0 -2.867E-30 13 4.339E-02 0.0 -1.416E+004 -8.909E-13 0.0 -3.072E-26 14 -2.490E-02 0.0 -5.420E-015 -5.178E+00 2.9 -3.148E+00 15 -4.943E-16 0.0 -4.006E-276 -1.055E+00 0.1 -3.387E-01 16 -4.694E-15 0.0 -7.172E-267 2.791E-13 0.0 -5.849E-26 17 5.424E-03 0.0 -1.671E-018 1.411E-01 0.0 -2.974E-01 18 1.632E-02 0.0 -1.336E+009 -2.267E-01 0.0 -2.203E+00 19 -3.748E-14 0.0 -2.914E-21
10 8.740E-16 0.0 -1.483E-28 20 8.561E-04 0.0 -4.222E-02Sq.Sum 9.122E+02 100.0 -1.341E+01
12 1 -2.597E+01 73.9 -3.191E+00 11 8.600E-02 0.0 -2.334E+002 -2.454E-12 0.0 -2.612E-26 12 -1.744E-14 0.0 -1.300E-253 -1.248E+01 17.1 -6.555E-01 13 -4.552E-15 0.0 -9.261E-274 -9.056E+00 9.0 -3.329E+00 14 -5.508E-15 0.0 -1.671E-255 4.589E-13 0.0 -2.271E-26 15 3.197E-02 0.0 -8.991E-016 1.596E-13 0.0 -1.082E-25 16 2.788E-02 0.0 -2.023E+007 4.804E-01 0.0 -2.606E+00 17 1.688E-14 0.0 -1.901E-248 -8.700E-15 0.0 -2.855E-27 18 2.764E-15 0.0 -1.225E-259 5.466E-15 0.0 -1.039E-26 19 -1.494E-02 0.0 -9.449E-01
10 1.185E-01 0.0 -9.125E-01 20 1.733E-12 0.0 -5.154E-19Sq.Sum 9.122E+02 100.0 -1.689E+01
The activated mass needs to be bigger than 90% (clause 4.3.3.3.1(3)).If it is less, then you need to consider more modes!
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Seismic design situation, Eurocode 0
iki
iEdj
jk QAPG ,1
,21
, ∑∑≥≥
+++ ψ
G : characteristic permanent loads (i.e. dead load)
P : Prestress
A: Earthquake
Q: variable loads (live load)
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Modal analysis, step 6
+PROG MAXIMAhead Ultimate limit state beamscomb 1 stanlc 21 G 1.0 $ dead load (LC 21 to be created in Sofiload)lc 5 Q 0.6 $ live load lc 301 A1 1.0lc 301 A1 -1.0lc 302 A1 1.0lc 302 A1 -1.0lc 303 A1 1.0lc 303 A1 -1.0lc 304 A1 1.0lc 304 A1 -1.0lc 305 A1 1.0lc 305 A1 -1.0
supp 1 extr mami etyp beam type n lc 1001 titl 'ULS_N_beam'supp 1 extr mami etyp beam type vy lc 1003 titl 'ULS_Vy_beam'supp 1 extr mami etyp beam type vz lc 1005 titl 'ULS_Vz_beam'supp 1 extr mami etyp beam type my lc 1007 titl 'ULS_My_beam'supp 1 extr mami etyp beam type mz lc 1009 titl 'ULS_Mz_beam'end
Combination rule
Each mami-superposition generates two result-loadcases!
Exclusive alternative loadcase “A1”
Superpositioncommands
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Modal analysis, step 6+PROG MAXIMA urs:71head ULS Displacementscomb 4 stanlet#q 1.5 $ behaviour factorlet#nu 0.4 $ reduction factor (EC 8, 4.4.3.1)lc 21 G 1.0lc 5 Q 0.6lc 53 A1 fact #q*#nu $ seismic displacements x q x nulc 53 A1 fact -#q*#nusupp 4 extr mami etyp node type ux lc 1101 titl 'displ ux'supp 4 extr mami etyp node type uy lc 1103 titl 'displ uy'supp 4 extr mami etyp node type uz lc 1105 titl 'displ uz'end
Combination rule
Superpositioncommands
The reduction fator ν takes into account the lower return period of the seismic action associated with the damage limitation requirement.
ν = 0.5 (Importance classes I and II) ν= 0.4 (Importance classes III and IV)
Check interstorey drift: displ < 0.005 h (brittle non-structural elements)
displ < 0.0075 h (ductile non-structural elements)
displ < 0.01 h (no non-struct. elements or special connections)
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Time-history analysis
1. Time history of ground accelerations (accelerogram) is required- record of past earthquakes in the region- artificial accelerogram
2. No need to determine mode shapes, but it is good to know naturalperiods/frequencies of the main modes
3. Determine Rayleigh damping factors such that main modes are damped at desired damping ratio
4. Run dynamic analysis (direct integration)
5. Checks
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Artificial accelerogram generator SIMQKE-1
Available from NISEE, MIT (http://nisee.berkeley.edu/)
- FORTRAN computer code
- Text based input
- American units
- Target : Velocity response spectrum in in./sec
- Several options regarding the shape of the accelerogram
SIMULATION OF EARTHQUAKE (OPTION 1).1 4.0 .1 4.0 .1 500.2 2.0 15. 20. 0. 0. 0. 0.01 0.391437308869 2585 1 20 1 200 40 0 00.050.1 3.228213648860.2 9.223467568160.3 13.83520135220.4 18.44693513630.5 23.05866892040.6 27.67040270450.7 27.6704027045...
Example input
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Artificial accelerogram generator SIMQKE-1, PYTHON GUI
FUNK 0.0 0.0FUNK 0.01 -0.008829FUNK 0.02 0.0FUNK 0.03 0.006867FUNK 0.04 0.002943FUNK 0.05 -0.00981FUNK 0.06 -0.01962FUNK 0.07 -0.017658FUNK 0.08 -0.011772FUNK 0.09 -0.016677...
20 seconds of ground motion are generated.Other shapes/durations possible, but require scripting.
Output file:
“accelerogram.dat”
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Rayleigh dampingFor time-history analysis, it is not possible to prescribe a constant damping ratio!
Mass-proportional damping
Stiffness-proportional damping
Raleigh Damping
00.010.020.030.040.050.060.070.080.090.1
0 2 4 6 8 10 12 14 16
Frequency [Hz]
Dam
ping
ratio
[-]
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Sofistik Teddy-script for time-history analysis
+PROG DYNAKOPF Time History horizontal x-dir
STEU ELF 3001 $ result load cases start from 3001STEP 800 0.025 A 1.0 B 0.00175 $ 800 steps of 0.025 s = 20 secondsLF 26 $ arbitrary number of load case#include accelerogram.datACCE no 0 ax 1.0 ay 0.0 az 0.0 $ accelerations in x-dirENDE
The time-increments should be smaller than . Recommended isto take 0.1 x Tmin. π
minT
If results are to be compared with modal analysis, the displacements need to be factored by the behaviour factor q (and the reduction factor ν)!
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P-∆-effects
Second order effects can be neglected if :
10.0≤⋅⋅
=hVdP
tot
rtotθ Clause 4.4.2.2 (2)
totP Total vertical load for seismic design situation
totV Total shear in the storey
d Interstorey drift
h Storey height
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PROG SIRECHO FORC FULLCTRL AQUA 0LC 21,22,3500SECT NO XS XM YM ZM NX NY NZ NR YMIN YMAX ZMIN ZMAX
1 - 0 0 18.0 0 0 1 YY -30 5. -5. 67END
Getting the forces
Ursula output for program SIRForces and moments section 1 XS = 0.000
LC Type No N[kN] Vy[kN] Vz[kN] Mt[kNm] My[kNm] Mz[kNm] Mb[kNm2]...3500 QUAD 60351 -2003.3 -44.65 285.17 -5921.79 -61444.21 -51116.55 0.00
QUAD 60353 -26.2 0.88 10.20 -283.23 -703.68 -668.89 0.00QUAD 60357 -539.9 -7.47 130.83 -3123.80 -15168.64 -13764.14 0.00QUAD 60358 -1601.5 -29.23 467.23 -11102.9 -43833.59 -40832.01 0.00QUAD 60359 -1824.5 123.27 595.16 -18404.3 -47942.48 -46499.21 0.00
3500 SUM 4159.1 -37650.5 302.70 741204.4 9235.73 -306374.9 0.00