Pushover analysis with ZSOIL taking soil into account. Stéphane Commend GeoMod Ing. SA, Lausanne

Pushover analysis with ZSOIL taking soil into account Stéphane Commend GeoMod Ing. SA, Lausanne scommend@geomod.ch

Contents Motivation Why pushover? Why take soil into account? Brief recall of pushover theory Application: 2-storey RC frame Structural-only analysis Nonlinear time history analysis (reference solution) Nonlinear pushover analysis Taking soil into account Nonlinear time history analysis with DRM (reference solution) Nonlinear pushover analysis Comparison of solutions Conclusions and perspectives

Motivation: why pushover? To design (new structures) or assess (existing structures) wrt seismic loading: - Replacement forces (linear) - Modal analysis (linear) - Nonlinear pushover analysis - Nonlinear time-history analysis -Nonlinear pushover analysis represents a good compromise between - replacement forces, where nonlinearity is taken into account by a single behavior coefficient q (too simple) - nonlinear time-history, very time consuming

Motivation: why pushover? Nonlinear pushover analysis (in ZSOIL, N2 approach [Fajfar]) - Until now structural only => applies mainly to buildings and bridges - Future: include soil => tunnels, retaining walls,...? - Returns a target displacement = maximal displacement during a certain earthquake, used in displacement-based seismic assessment (in Switzerland, since 2004, documented in CT SIA 2018) a eff = w Rd /w d (SIA CT 2018) a eff w Rd w d compliance factor allowable displacement (capacity of deformation) deformation during earthquake a eff < a min a min a eff a adm a adm a eff a min, a adm = f(structure type, lifetime) intervention necessary intervention if proportionate no intervention

Motivation: why take soil into account? Seismic action => inertia forces

Motivation: why take soil into account [Gazetas et al]? Taking soil into account in calculation (in Case 2) => Rocking allowed => Less damage in structure => Design with soil is more favorable than with structure only

Theory STEP 1: Define Seismic demand, elastic acceleration spectrum S ae Elastic ADRS demand spectrum f(structure type, soil conditions, zone) S ae S ae S de S de S ae

Theory STEP 2: Build structural model and apply gravity loads Vertical and horizontal members have to be modelled with nonlinear model (typically: reinforced concrete with f c and f t in concrete and steel)

Theory STEP 3: Choose and apply lateral load distribution and increase - Load pattern is applied in one direction at a time, meaning several calculations have to be conducted to fully assess the structure: - in x and z direction, in plus and minus directions - with different load patterns (uniform, linear or modal) F F represents the inertial forces which would be experienced by the structure during the earthquake

Theory STEP 4: Plot capacity curve F d V b Base shear, V b Capacity curve Top displacement, d

Theory STEP 5: build equivalent single degree of freedom (SDOF) model F d d* = d / Γ m* V b F* Equivalent SDOF

Theory Equation of motion for MDOF system (no damping assumed, influence will be adressed in design spectrum) M u + R = M 1 a Diagonal mass matrix Relative displacement Internal forces = f(u) N2 assumptions + some math. => Equation of motion for the equivalent SDOF system m d + F = m a Ground acceleration = f(t) F* = V b / G = force of SDOF system d* = D t / G = displacement of SDOF system

Theory From Capacity curve to Capacity spectrum Base shear, V b Capacity curve F* F y * d m *: assumed target displacement => Iterative procedure!! Assumption: post-yield stiffness = 0 E m * Top displacement, d Acceleration spectrum d y * = 2 (d m * - E m */F y *) T* = 2p SQRT(m* d y * / F y *) S a = F* m* F y * m* d y * d m * d* SDOF Capacity curve (bi-lin.) Capacity spectrum Elastic period of idealized bilinear SDOF system d y * d*

Theory STEP 6: compare demand and capacity spectra (*) and retrieve d t ADRS demand spectrum f(structure type, soil conditions, zone) F* m* T C Target displacement T* capacity spectrum d t = Γ d t * d t * d* Modal participation factor (*) not straightforward, because capacity spectrum is nonlinear

Application: 2-storey RC frame [Gelagoti et al, 2012]

Application: Model Pushover control node Dead and live loads: 3.3 kn/m 2

Application: Material definition for RC members

PUSHOVER TIME HISTORY Application: Seismic demand Accelerogram generated Synthetically (Sabetta) COMPATIBLE

Application: Time history analysis (reference solution)

Application: TH displacement time history ux max = 4 cm

Application: TH bending moment envelope during TH M max = +106 / -103 knm

Application: Pushover analysis ALSO TRIED

Application: Pushover analysis

Application: Pushover analysis 5.4 cm

Application: Pushover analysis Pushover analysis report Item Unit PSH 1/Default MDOF Free vibr. period...t [s] 0.445066 SDOF Free vibr. period...t* [s] 0.837848363 SDOF equivalent mass...m* [kg] 12369.6 Mass participation factor Gamma - 1.20401 Bilinear yield force value..fy* [kn] 48.9047294 Bilinear displ. at yield...dy* [m] 0.07030178 Target displacement...dm* [m] 0.166111577 SDOF displacement demand...dt* [m] 0.053940741 Energy...Em* [kn*m] 6.404596979 Reduction factor...qu - 1 Demand ductility factor...mi - 3.079519728 Capacity ductility factor...mic - 2.362836024 MDOF displacement demand...dt [m] 0.064945192 d t = 6.5 cm

Application: Pushover analysis M max = +94 knm

Contents Motivation Why pushover? Why take soil into account? Brief recall of pushover theory Application: 2-storey RC frame Structural-only analysis Nonlinear time history analysis (reference solution) Nonlinear pushover analysis Taking soil into account Nonlinear TH analysis with DRM (reference solution) Nonlinear pushover analysis Comparison of solutions Conclusions and perspectives

Taking soil into account: soil hypotheses Soil (HSS model) E 50 = 80 MPa, E ur = 320 MPa, E 0 = 800 MPa s h,ref = 100 kpa g = 20 kn/m 3, c = 0 kpa, f = 30, y = 10 Interface elements (optional) h = 0.5 m b = 1.4 m

Contents Motivation Why pushover? Why take soil into account? Brief recall of pushover theory Application: 2-storey RC frame Structural-only analysis Nonlinear time history analysis (reference solution) Nonlinear pushover analysis Taking soil into account Nonlinear TH analysis with DRM (reference solution) Nonlinear pushover analysis Comparison of solutions Conclusions and perspectives

Taking soil into account: Time history analysis with DRM DRM => two models: a reduced model and a background model Seismic input on reduced model = free-field motion of background model

Taking soil into account: Time history analysis with DRM Background model (elastic) and free-field motion Horizontal acceleration read at top of soil column 45 m 15 m 10% Rayleigh damping on 2 Hz and 6 Hz => a = 1.88 and b = 0.004 Seismic input: linear deconvolution of Fig. 12.3 accelerogram Periodic BCs: u x left = u x right

Taking soil into account: Time history analysis with DRM Reduced model Interior domain (HSS model) Boundary layer (elastic) Exterior domain (elastic) Viscous dampers

Taking soil into account: Time history analysis with DRM ux max = -8.0 cm

Taking soil into account: Time history analysis with DRM M max = +47 / -43 knm

Taking soil into account: pushover analysis V b max << V b max (struct. Only) Structural only

Taking soil into account: pushover analysis

Taking soil into account: pushover analysis Pushover analysis report Item Unit PSH 1/Default MDOF Free vibr. period...t [s] 0.529679 SDOF Free vibr. period...t* [s] 1.1692452 SDOF equivalent mass...m* [kg] 14036 Mass participation factor Gamma - 1.28476 Bilinear yield force value..fy* [kn] 19.58028002 Bilinear displ. at yield...dy* [m] 0.048308883 Target displacement...dm* [m] 0.075135711 SDOF displacement demand...dt* [m] 0.075181918 Energy...Em* [kn*m] 0.998227533 Reduction factor...qu - 1.556275235 Demand ductility factor...mi - 1.556275235 Capacity ductility factor...mic - 1.555318745 MDOF displacement demand...dt [m] 0.096590721 d t = 9.7 cm > d t (struct. only) = 6.5 cm

Taking soil into account: pushover analysis M max = +31 / -37 knm

STRUCTURAL ONLY TAKING SOIL INTO ACCOUNT Comparison PLAY MOVIES

mean stress level under foundation [-] absolute foundation displacement [m] bending moment in column just above foundation [knm] Comparison: «case 2» vs. «case 1» found 1.4 m M struct 1.20E+02 1.00E+02 M struct Mean SL disp 8.00E+01 6.00E+01 Strip (BIG) found 4.00E+01 M struct 2.00E+01 Mean SL disp d(target) = 8 cm d(target) = 9.7 cm 0.00E+00 0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 Top floor displacement [m] found 1.4 m (right) strip (BIG) found struct only 1.2 1 Mean Stress Level 0.007 0.006 disp 0.005 0.8 0.004 0.6 0.003 0.4 0.002 0.2 0.001 0 0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 Top floor displacement [m] 0 0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01 top floor displacement [m] found 1.4 m (right) strip (BIG) found found 1.4 m (right) strip (BIG) found

Comparison Time history (with DRM) Pushover analysis Difference PO vs. TH d top min/max drift min/max M min/max d t drift(d t ) M max (d t ) on d max on M max [cm] [cm] [knm] [cm] [cm] [knm] [%] [%] structural only -2.3/+4.0-1.1/+2.0-103/+106 6.5 3.1 94 63-11 with soil -8.0/+4.0-4.8/+2.3-43/+47 9.7 5.9 37 21-21 with soil, BIG foundation -4.8/+8.0-1.4/+3.3-81/+99 8 4.1 95 0-4

Conclusions and perspectives Taking soil into account (if in case 2!) leads to: d >> design M(struct) << => gain! Needs more benchmarking Thank you Thomas for ideas