Seismic Design of Tall and Slender Structures Including Rotational Components of the Ground Motion: EN 1998-6 6 Approach 1
Chimneys Masts Towers
EN 1998-6: 005 TOWERS, CHIMNEYS and MASTS NUMERICAL MODELS of the STRUCTURE Prepared on the basis of finite elements 3 3D FRAME ELEMENTS SHELL ELEMENTS considering spatial modes of deformation
RESPONSE SPECTRUM METHOD + LINEAR ANALYSIS SEISMIC ACTION MODEL - 3 TRANSLATIONAL COMPONENTS of the GROUND MOTION 4 Elastic spectrum, type 1 Elastic spectrum, type FIXED BASE BOUNDARY CONDITIONS
TRANSLATIONAL COMPONENT of GROUND ACCELERATION m 0 { x} T ν = { 1 1... 1} VECTOR OF TRANSFERRED MOTION m k ( ) E = m Φ Γ S T x x ik k ik i e i DESIGN SEISMIC LOAD, MODE i 5 m 1 Tot T { x} [ ]{ x ν ν } M m m = = Γ = x i T { } [ ]{ x Φ } i m ν M i k k TOTAL MASS PARTICIPATION FACTOR, MODE i
TRANSLATIONAL COMPONENTS of GROUND ACCELERATION PEAK MODAL RESPONSES FOR SHEAR AND BENDING MOMENT AT THE E BASE x x m BASE SHEAR, MODE i 0 Vi = Γi Mi Se Ti ( ) ( ) K. A. GUPTA 6 m k m 1 M =Γ Γ MhS T V ρ nm = ( ) x x θ i i i i e i = ρ V V ( ) E = m Φ Γ S T x x ik k ik i e i x x x ij i j i j M BASE MOMENT, MODE i K. A. GUPTA NEED FOR MODAL COMBINATION - CQC RULE = ρ M M x x x ij i j i j 8ξ 1 ( + r ) 3/ nm nm ( ) 1 rnm + 4ξ rnm ( 1+ rnm ) r
SEISMIC INPUT for TRADITIONAL ANALYSIS: THREE TRANSLATIONAL COMPONENTS w&& g for GROUND ACCELERATIONS v&& g 7 u&& g
SURFACE SEISMIC WAVES (Rayleigh( and Love) 8
ROTATIONAL COMPONENTS DEFINITION θ&& gz θ&& gy 9 θ&& gx
SEISMIC ACTION MODEL TAKING INTO ACCOUNT SPATIAL VARIABILITY (SURFACE WAVES) ROTATIONAL COMPONENTS of GROUND ACCELERATION 10 Rayleigh waves Love waves
ROTATIONAL COMPONENTS of GROUND ACCELERATION ϕ&& w&& g θ&& z,g v&& g θ&& y,g 11 u&& g θ&& x,g Translational and rotational ground accelerations of the ground surface k θ c θ θ&& g Rotational SDOF for rotational spectrum definition
NEWMARK HALL SPECTRUM of the ROTATIONAL COMPONENT 1
ROTATIONAL COMPONENTS DEFINITION 13
S θ x ( T) ROTATIONAL COMPONENTS of GROUND ACCELERATION FIXED BASE BOUNDARY CONDITIONS Se ( T) θ = 1.7π S ( T) VT. s y ( T) Se 1.7π VT. θ = S ( T) s z ( T) Se =.0π VT. s 14 Response spectra S θ and scaled to DGA = 0.7g x S θ y Response spectrum S θ scaled to DGA = 0.7g z STRONG INFLUENCE on STIFF STRUCTURES (LOW PERIODS)
m 0 PEAK ACTION EFFECTS INDUCED by the ROTATIONAL COMPONENTS of GROUND ACCELERATION T { θ ν } = { 0... 1} VECTOR OF TRANSFERRED MOTION m k ( ) E = m Φ Γ hs T θ θ θ ik k ik i i DESIGN SEISMIC LOAD, MODE i 15 m 1 { θ} T [ ]{ θ} ( θ ν ν ν ) M m m = = Tot k k k Γ = θ i T { } [ ]{ θ Φ } i m ν M i TOTAL MASS PARTICIPATION FACTOR, MODE i
m 0 PEAK ACTION EFFECTS INDUCED by the ROTATIONAL COMPONENTS of GROUND ACCELERATION V =Γ Γ M hs T ( ) θ x θ θ i i i i i BASE SHEAR, MODE i m k ( ) ( ) M = Γ MhS T θ θ θ i i i i ( ) E = m Φ Γ hs T θ θ θ ik k ik i i BASE MOMENT, MODE i 16 m 1 V NEED FOR MODAL COMBINATION - CQC RULE = ρ V V M ijmi M j = ρ θ θ θ θ θ θ ij i j i j i j NEED FOR COMPONENT COMBINATION OF ACTION EFFECTS - SRSS x x = ( ) + ( ) max M = ( M ) + ( M θ ) maxv V V θ
h = 60 m ( < 80 m nationally determined parameter, National Annex) a g S = 0.7g > 0.5g Elastic design spectrum Type 1 ANALYSIS DATA 17 Elastic design response spectrum for horizontal accelerations, ground type C, a g = 0.7g, ξ = 5% Elastic design response spectrum for rotational accelerations around horizontal axis, ground type C, a g = 0.7g, ξ = 5%
i M i M ( Γ ) i 1 Tot NUMERICAL EXAMPLE CRITERION FOR EVALUATION THE SUFFICIENCY OF THE MODES INCLUDED 18 ROTATIONAL GROUND MOTION: UNI-MODAL ANALYSIS USING FUNDAMENTAL MODE IS SUFFICIENT
BASE MOMENT: MODAL CONTRIBUTION assuming only translational component (solid line) and assuming only rotational component (dashed line) 19
BASE MOMENT: assuming both translational and rotational components (solid line) and assuming only translational component (dashed line) 0 INCLUSION of ROTATIONAL COMPONENT OVERESTIMATES 11% the RESULTS for BASE MOMENT and BASE SHEAR 11%
LARGE PERIOD STRUCTURES - AFFECTED BY: 1) LARGE MAGNITUDE DISTANT EARTHQUAKES ) CONTRIBUTION of LONG PERIOD COMPONENTS Time histories of 1977 Vrancea Earthquake, record Incerc, Bucharest on epicentral distance 110 km, representing distant large magnitude earthquakes (T. Petrovski and B. Dimiskovska) 1
HUNGARY 46 CROATIA 44 Sarajevo 0 Belgrade SERBIA 8 Nish V VI VII 4 ROMANIA BULGARIA 6 Bucharest 8 46 44 Varna Black Sea LARGE MAGNITUDE LONG DISTANT EARTHQUAKE PEHCEVO KRESNA (T. Petrovski and B. Dimiskovska from IZIIS - Skopje) MONTENEGRO Sofia Burgas Podgorica VIII Plovdiv Skopje 4 V VI IX VII VIII Shtip X VIII Kresna VII VI V Pehcevo MACEDONIA Strumica TURKEY Adriatic Sea Tirana Ohrid Bitola 4 140 km from Sofia ALBANIA Thessaloniki VII 40 40 38 VIII 0 Ionian Sea GREECE VI Aegean Sea Athens 4 Seismic intensity isolines MSK Scale X IX VIII VII VI V 0 50 100 km 6 38 8 Isoseismal map of Pehcevo- Kresna catastrophic Earthquake of April 04, 1904 at 10:5 (after Hadzievski,, D., 1974), compiled by Petrovski,, T., and late Petrovski,, J.
UBC, Vol. (Tall buildings) TRESHOLD ACCELERATIONS Spectral acc. Spectral disp. 3 T, [s]
Ground Type C determination of threshold β Spectral acc., [m/s/s] Spectral disp., [m] 4 EN 1998-6 PARAMETER β = 0.10 Nationally determined T, [s]
Ground Type D determination of threshold β Spectral acc., [m/s/s] Spectral disp., [m] 5 EN 1998-6 PARAMETER β = 0.10 Nationally determined T, [s]
Ground Type E determination of threshold β Spectral acc., [m/s/s] Spectral disp., [m] 6 EN 1998-6 PARAMETER β = 0.10 Nationally determined T, [s]
UPPER BOUNDS for DRIFTS at DL LIMIT STATE d r 0,00 ΔH For precast members/towers d r 0,01 ΔH For continuous towers 7
RESPONSE AMPLIFICATION ZONES Amplification area (effect of rotational components of the ground motion) Amplification area due to increased large period contribution (large magnitude distant earthquakes) 0 1 3.5 T, [s] 8 Medium period structures (moderate amplification) What is the good solution? The use of passive energy dissipation devices
SOIL FOUNDATION STRUCTURE INTERACTION : IMPEDANCE FUNCTIONS + SUBSTRUCTURE APPROACH e j. Ω. t e j. Ω. t ρ, G, ν, VS ρ, G, ν, VS 9 Solutions provided by Luco and discussed by J. Stewart k c ( Ω= ) α ( Ω). S X X X X S. r V X 1 ( Ω ) = β ( Ω). X S c R k ( Ω ) = α ( Ω). S R R R SR.. r V ( Ω ) = β ( Ω) R S
30 Ω - Input frequency unit force - compliance functions unit displacement impedance functions Horizontal motion Rocking motion (translation along X) (rotational motion) S X r 1 8. G. r ν 8. G 3. 1 ν = 1 S =. r R ( ) A = f 4 r = π 4. I f π Complex stiffness k k j.. c k Ω = k Ω + j. Ω. c Ω ( Ω ) = ( Ω ) + Ω ( Ω ) ( ) ( ) ( ) X X X Hysteretic damping Ω. c X ( Ω ) horisontal direction Ω ( Ω) R R R. rotational direction c R
Dimensionless frequency Shear waves velocity VS Frequency dependent impedances (real and imaginary) X direction: real part (stiffness impedance) a 0 = Ω V S r 31
X direction: damping impedance 3
Ω. c X 100 90 80 70 60 50 40 30 0 ( Ω) X direction: imaginary part (hysteretic damping impedance) 33 10 0 0 50 100 150 00 Ω
R direction: real part (stiffness impedance) 34
R direction: damping impedance 35
Ω. c X 100 ( Ω) R direction: imaginary part (hysteretic damping impedance) 1000 800 600 400 00 36 0 0 0 40 60 80 100 10 140 160 180 00 Ω
Vertical stress settlement theory Applied in steady-state analysis - John Lysmer static analysis F steady-state analysis σ = F A Soil layer Soil layer Bedrock Bedrock F σ = F A σ 37 Bedrock Kelvin-Voigt element k 1 s
FOUNATION SOIL SYSTEM IMPEDANCES rotational flexibility conditions horizontal flexibility conditions 38 foundation impedances spring-dashpot system
KINEMATIC INTERACTION CONDITIONS: Veletsos and co-authors (steady-state analysis) 1. h δ R ( Ω ) = δ X ( Ω) K ( Ω) δ R F = j t e Ω F = j t e Ω F = j t e Ω h EI θ = 1. h K R ( Ω) EI 39
δ ~ KINEMATIC INTERACTION CONDITIONS ( Ω) = δ + δ X ( Ω) + δ ( Ω) R superposition rule ~ k k ( Ω) = k h k 1+ + k ~ 1 k 1 ~ ω = 1 + + ( Ω) k k ( Ω) k ( Ω) = 1 X ( Ω) ( Ω) X k R 1 ( Ω) ω ( Ω) ( Ω) h R 1 1 + + ω X ω R stiffness of the flexible foundation stiffness equation frequency equation 40 k ( ) X ( Ω) ω Ω = ω ( Ω) X m h ( ) k R R = Ω m ω = k m
41 INPUT FREQUENCY DEPENDENT QUANTITIES ( Ω) ( Ω) ( ) ( ) mu.&& % + cu %.%& + ku %.% = ma..sin Ω. t mu.&& + cu.& + ku. = ma..sin Ω. t 1 1 % = δ ( Ω) 1 1 + + k k k x 1 k% ( Ω ) = k. k k. h 1+ + kx( Ω) kr( Ω) % δ ( Ω ) k k. h = 1 + + δ kx( Ω) kr( Ω) % ω ( Ω) 1 ψ ( Ω ) = = ω k 1+ + k g g R x h k. h ( Ω) k ( Ω) inertial interaction (flexible base system) displacement of the flexible base system R inertial interaction (fixed base system) stiffness of the flexible base system displacement ratio: flexible / fixed base system frequency ratio: flexible / fixed base system
4 INPUT FREQUENCY DEPENDENT QUANTITIES ~ T T ( Ω) T = = π 1+ k m k ~ m T ( Ω) = π ~ k k + h ( Ω) ( Ω) X k R ( Ω) ~ ξ ξ ( Ω) = ξ + T Ω T ~ 0.05 ξ ( Ω) = T ~ ( Ω) 3 T 0 ~ ( ) 3 k period ratio: flexible base / fixed base system natural period, fixed base system natural period, flexible base system damping ratio, flexible base system according to Veletsos damping ratio, flexible base system, 5 % damping (Veletsos)
INPUT FREQUENCY DEPENDENT QUANTITIES % ( ) dynamic amplification ratio Ω Ω 1 + ξ β Ω ω ω = ψ ( Ω ) β Ω Ω ψ ( Ω) + % ξψ ( Ω) ω dynamic displacement ratio ω 43 Ω Ω % 1 + ξ β( Ω) % δ ( Ω) ω ω β δ ω k h k = ψ ( Ω ) 1 + + k ( ) ( ) Ω Ω X Ω kr Ω ψ ( Ω) + % ξψ ( Ω) ω
ANALYSIS DATA cantilever beam 44 = 3. EI. k = = 157.9 kn / m 3 h m 1 kn. s / m m T =. π. = 0.5 k [ ] [ s] [ ] c=. ξω.. m= 1.55 kn. s/ m
ANALYSIS DATA 45 d = 0.5 h= 3 r 1 r A = = 0. π 4. I π 4 0 = = Foundation - Soil System [ m ] [ m] 0.5 [ m] [ m] π. d I0 E m 64 4 3 4 = = 3.06796 G kn m = 40 000 /
ANALYSIS DATA 46 ρ 15 9.81 3 = = = 1.59 t/ m ( ν ) [ ] ν = 0. V s γ g Foundation - Soil System G = = 161.74 / ρ [ m s] 8. G Sx =. r1 = 44 444.45 ν [ kn / m] 8. G SR =. r = 083.33 3. 1 kn. m/ rad [ ]
Frequency ratio (flexible base / fixed base) ψ (Ω) ratio 47 ω% ω 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 0 0 0.5 1 1.5.5 3 3.5 4 4.5 5 5.5 6 a0
Displacement ratio (flexible base / fixed base) Displacement Ratio 48 % δ δ 0 18 16 14 1 10 8 6 4 0 0 0.5 1 1.5.5 3 3.5 4 4.5 5 5.5 6 a0
Stiffness ratio (cantilever / foundation, horizontal) Kx Ratio 0.006 0.005 K K X 0.004 0.003 0.00 0.001 49 0 0 1 3 4 5 6 7 8 9 10 a0
Stiffness ratio (cantilever / foundation, rocking) Kr Ratio 50 h. K K R 5 4 3 1 0-1 - -3-4 -5 0 1 3 4 5 6 7 8 9 10 a0
Period ratio in terms of stiffness ratios T% T Re Im 51 K K X ht. K R
Damping ratio (flexible base system) Damping Coefficient 0.03 ξ% 0.05 0.0 0.015 0.01 0.005 5 0 0 0.5 1 1.5.5 3 3.5 4 4.5 5 5.5 a0
Damping ratio in terms of period ratio (flexible base / fixed base) % ξ ξ 1 0.9 0.8 0.7 0.6 0.5 0.4 53 0.3 0. 0.1 0 1 1. 1.4 1.6 1.8 T% T
Dynamic coefficient ratio (flexible base / fixed base) versus input frequency 54
Dynamic displacement magnification ratio (flexible base / fixed base) versus input frequency 55
Dynamic Amplification Factor (flexible base / fixed base) 56
Displacement Histories (flexible base / fixed base) 57
Simple Soil-Foundation-Structure Interaction Model Taking into account structure foundation dynamic coupling m 58
Implementation of Flexible Base Boundary Conditions 59 m 0 0 u&& 0 0 0 u&& f 0 0 0 && θ f c c hc u& k k hk u + c c+ c ( Ω) hc u& + k k+ k ( Ω) hk u X f X f hc hc h c cr( ) θ f hk hk hk kr( ) θ + Ω & + Ω f m 0 0 1 0 0 0 = 1 a 0 0 0 0 ( ), ( ), ( ), ( ) X X R R g sin Ωt k 3EI 3 h = - Cantilever stiffness c - Cantilever damping c Ω k Ω c Ω k Ω - Foundation Impedances - Small added mass / moment of inertia
Application to MDOF systems through FEM analysis 60
Application to MDOF systems through FEM analysis [ m]{ u&& } + ( ) { u& } + ( ) { u} = [ m]{ ν } a sin c Ω k Ω X g Ωt 61
ANY QUESTIONS? 6