Seismic analysis with SOFiSTiK

Oliver Bruckermann, 28 February 2008

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

Characterisation of seismic action - Time history plots

acceleration

velocity

displacement

The three graphs are equivalent.

Peak groundacceleration

Peak ground accelerations

More detailed maps are available on the GSHAP website.http://www.seismo.ethz.ch/GSHAP/

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).

Smoothed elastic response spectrum

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.

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.

Modal analysis

For in-depth information see:

Earthquake design practice for buildings, E. Booth D. Key

Dynamics of structures,Anil K. Chopra

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

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

Masses

ikiEjk QG ,,, ∑∑ ⋅+ ψ

iiE 2,

Clause 3.2.4(2), EC8

ψϕψ ⋅=Clause 4.2.4(2), EC8

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.

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

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

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.

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.

”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 ×+×+×=

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!

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)

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

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)

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

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

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”

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

[-]

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)!

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

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