Hybrid method for the computation of dynamic soil-structure ...

IntroductionFramework: dynamic soil-structure interactionIntroductionHybrid methodTime Domain BEMFE-BEConclusionsLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 2/44

IntroductionLinear and non-linear solution domainsIntroductionLinearHybrid methodTime Domain BEMLinear■ Traffic induced vibrations■ Machine vibrationsFE-BEConclusionsΣ bs⇒ Frequency domain, interface modes.Non-linearLinear■ Vibration related structural damage■ Earthquake engineeringΣ bs⇒ Time domain, interface modes.Non-linearNon-linearΣ bs■ Vibration related settlements■ Pile driving■ Earthquake engineering⇒ Time domain, no interface modes.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 3/44

IntroductionAlternatives for time domain dynamic SSIIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ Linear calculations: frequency domain boundary element method .■ Non-linear calculations:Fundamental solution Soil stiffness ResponseTime domain BEM ω → t t tHybrid domain methodusing soil flexibilityin the time domainHybrid frequency-timedomain analysisω ω → t tω ω ω ↔ tObjectiveInvestigate the applicability of time domain soil-structure interaction methods.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 4/44

IntroductionPresentation outlineIntroductionHybrid methodTime Domain BEM■ Dynamic soil-structure interaction methodology◆ Flexibility in the time domainyzxP ztFE-BEConclusions◆ Time domain boundary elements■ Implementation of a FE-BE couplingΩ bΩ sΣ rsΣ bszyxΩ rLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 5/44

IntroductionHybrid methodTime Domain BEMFE-BEConclusionsPart I: Dynamic soil-structure interactionin the time domainLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 6/44

Hybrid domain methodBenchmark problem: square foundationIntroductionHybrid method■ Dimensionless massTime Domain BEMFE-BEConclusionsyzx¯m = mρd 3■ Dimensionless frequencyd¯md■ Dimensionless timea 0 = ωdC sµ, ρ, ν = 1 3τ = tC sd■ Dimensionless responseµdu i , µd 3 ϕ iLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 7/44

Hybrid domain methodFrequency domain stiffnessIntroductionHybrid methodTime Domain BEMFE-BEˆf s (ω) = Ŝ(ω)û(ω) (1))Ŝ mn (a 0 ) = K mn(ˆkmn (a 0 ) + ia 0 ĉ mn (a 0 )(2)ConclusionsImpedance coefficient [−]10.80.60.40.2Impedance coefficient [−]10.80.60.40.2Impedance coefficient [−]10.80.60.40.200 2 4 6 8Frequency [−]00 2 4 6 8Frequency [−]00 2 4 6 8Frequency [−]ˆk HH (a 0 ), ĉ HH (a 0 ) ˆkV V (a 0 ), ĉ V V (a 0 ) ˆkRR (a 0 ), ĉ RR (a 0 )LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 8/44

Hybrid domain methodFrequency domain flexibilityIntroductionHybrid methodTime Domain BEMû(ω) = ˆF(ω)ˆf s (ω) (3)ˆF(ω) = (Ŝ)−1 (ω) (4)FE-BEConclusions1.51.51.5Flexibility [−]10.5Flexibility [−]10.5Flexibility [−]10.500 2 4 6 8Frequency [−]00 2 4 6 8Frequency [−]00 2 4 6 8Frequency [−]ˆF HH (a 0 ) ˆFV V (a 0 ) ˆFRR (a 0 )LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 9/44

Hybrid domain methodTime domain stiffness and flexibilityIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ Stiffness relation in the time domain.f s (t) =∫ τ0S(t − τ)u(τ) dτ◆ Ŝ(ω) Linear asymptotic behaviour for ω → ∞.◆ Therefore, S(t) contains singular terms for t = 0.■ Flexibility relation in the time domain.u(t) =∫ τ0F(t − τ)f s (τ)dτ◆ ˆF(ω) Converges to zero for ω → ∞.◆ However, large frequency content for large frequencies.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 10/44

Hybrid domain methodAlternative 1: Gauss windowing techniqueIntroductionHybrid methodTime Domain BEMFE-BEConclusionsGauss window [−]10.80.60.40.200 0.1 0.2 0.3 0.4 0.5Frequency [−]Gauss window [−]0.250.20.150.10.050−10 −5 0 5 10Time [−]g(ω) = exp −σ 2 ω 2g(t) = 12σ √ π exp(− t4σ ) 2■ The flexibility is multiplied with the Gauss window.■ Problem: entire frequency domain influenced → large frequency range required.Alternative 2:Properly account for the asymptotic behaviour of the soil flexibility in the frequencydomain.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 11/44

Asymptotic expansion for ω → ∞Hybrid domain methodIntroductionHybrid methodTime Domain BEMyxyxFE-BEConclusionszz■ A direct stiffness approach is considered (Luco and Apsel 1983)■ Out-of-plane motion:√˜K SH (k x , ω) = iµω 2C 2 s− k 2 x ∼ iωρC s for ω → ∞ (5)■ In-plane motion:˜K P−SV (k x , ω) =[µkx 2 + k zs k zp∼ iωρ[ ]C s 00 C p]iks 2k zp k x (2kx 2 − k2 s + 2k zsk zp )k x (2kx 2 − ks 2 + 2k zs k zp ) iksk 2 zsfor ω → ∞ (6)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 12/44

Asymptotic expansion for ω → ∞Hybrid domain methodIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ Force-displacement relationship for ω → ∞:ũ x (k x , ω) = iωρC s˜t x (k x , ω) (7)ũ y (k x , ω) = iωρC s˜t y (k x , ω) (8)ũ z (k x , ω) = iωρC p˜t z (k x , ω) (9)■ The stiffnesses are independent of the wavenumber:û x (x, ω) = iωρC sˆt x (x, ω) (10)û y (x, ω) = iωρC sˆt y (x, ω) (11)û z (x, ω) = iωρC pˆt z (x, ω) (12)■ The result is extended to the 3D-case, using a Fourier series in the circumferentialdirection. The results remain valid.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 13/44

Asymptotic expansion for ω → ∞Hybrid domain methodIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ For the foundation under consideration, this is integrated:F HH ∼F V V ∼F RR ∼ 112F T T ∼ 8 3d2iωρC s(13)d2iωρC p(14)d 4iωρC p(15)d 4iωρC p(16)■ Similar results apply for a layered halfspace, if material damping is present. In thiscase, the wave velocities are rendered complex.■ For an embedded foundation, the 1D wave propagation is in the direction of thenormal on the interface.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 14/44

Hybrid domain methodAsymptotic expansionIntroductionHybrid method1Time Domain BEM0.83FE-BEConclusionsFlexibility [−]0.60.40.2Flexibility [m/N]4 x 10−6210Time [−]00 10 20 30 40 50Frequency [−]−1−0.5 0 0.5 1 1.5 2ˆF(ω)F(t)Different alternatives for the backtransformation:■ FFT (combined with analytical expression)■ Generalised Filon integration with unevenly distubted frequency domain sampling.(advantage: non-uniform sampling in time)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 15/44

Hybrid domain methodDiscretisation in timeIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ Linear temporal variation of the soil-structure interaction forces f s (t):■ Instantaneous soil flexibility:u N =N−1∑i=1F N−i f i s + F0 f N s (17)F 0 =■ Remaining influence matrices:∫ ∆t0τ ′∆t F(∆t − τ ′ ) dτ (18)F N−i =∫ ∆tτ ′+0∫ ∆t0∆t F ((N − i + 1)∆t − τ ′ ) dτ ′(1 − τ ′ )F ((N − i)∆t − τ ′ ) dτ ′ (19)∆tLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 16/44

Hybrid domain methodCalculation of the structural responseIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ Soil-structure interaction force:■ At time N∆t:f N s = ( F 0) −1 u N − ( F 0) −1N−1∑i=1F N−i f i s (20)◆ The term −(F 0 ) −1 ∑ N−1i=1 FN−i f i s is a function of interaction forces f i s ofprevious time steps.◆ The term (F 0 ) −1 u N represents a constant stiffness relation between thedisplacements and the interaction force.■ For the soil-structure system:Mü N = f N ext + f N s (21)Mü N − ( F 0) −1 u N = f N ext− ( F 0) −1N−1∑i=1F N−i f i s (22)is solved by means of a Newmark time stepping scheme.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 17/44

Hybrid domain methodCalculation of the structural responseIntroduction■ Horizontal loadHybrid methodTime Domain BEMFE-BEConclusionsyP xxztP xDisplacement [−]0.50.40.30.20.1Rotation [−]0.20.150.10.0500 2 4 6 8 10Time [−]u x (t)00 2 4 6 8 10Time [−]ϕ y (t)■ Vertical loadyzxP z P ztDisplacement [−]0.50.40.30.20.100 2 4 6 8 10Time [−]u z (t)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 18/44

Time domain BEMTime domain fundamental solutionsIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• Discretisation■ Full-space solutions (Karabalis)■ Half-space solutions◆ Undamped analytical expressions (Triafantyllidis 1991, Bode 2000)◆ Numerical inverse of frequency domain solutions (Wolf and Obernhuber)FE-BEConclusionsP re θe rP re θe rP θe re θe ze ze zu G rru G zru G θθe θe zP zu G zze re re θe zP zu G rzLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 19/44

Time domain BEMFrequency domain Green’s functionsIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• DiscretisationFE-BEu G rzDisplacement [−]0.20.10−0.1−0.2Conclusions0 20 40 60 80 100 120 140 160 180 200Frequency [−]u G θθDisplacement [−]0.20.10−0.1−0.20 20 40 60 80 100 120 140 160 180 200Frequency [−]1µr uG ij (a 0) with a 0 = rωC sare independent of r.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 20/44

Time domain unit step response: ∆tC s /r = 2Time domain BEMIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• DiscretisationFE-BEConclusionsDisplacement [−]0.20.10−0.1−0.20 1 2 3 4Time [−]0.2µru rr (τ)Displacement [−]0.20.10−0.1−0.20 1 2 3 4Time [−]0.2µru rz (τ)Displacement [−]0.10−0.1Displacement [−]0.10−0.1−0.20 1 2 3 4Time [−]µru tt (τ)−0.20 1 2 3 4Time [−]µru zz (τ)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 21/44

Time domain unit step response: ∆tC s /r = 0.2Time domain BEMIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• DiscretisationFE-BEConclusionsDisplacement [−]0.20.10−0.1−0.20 1 2 3 4Time [−]0.2µru rr (τ)Displacement [−]0.20.10−0.1−0.20 1 2 3 4Time [−]0.2µru rz (τ)Displacement [−]0.10−0.1Displacement [−]0.10−0.1−0.20 1 2 3 4Time [−]µru tt (τ)−0.20 1 2 3 4Time [−]µru zz (τ)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 22/44

Time domain unit step response: ∆tC s /r = 0.02Time domain BEMIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• DiscretisationFE-BEConclusionsDisplacement [−]0.20.10−0.1−0.20 1 2 3 4Time [−]0.2µru rr (τ)Displacement [−]0.20.10−0.1−0.20 1 2 3 4Time [−]0.2µru rz (τ)Displacement [−]0.10−0.1Displacement [−]0.10−0.1−0.20 1 2 3 4Time [−]µru tt (τ)−0.20 1 2 3 4Time [−]µru zz (τ)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 23/44

Time domain BEMSquare load on a half spaceIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• DiscretisationFE-BEConclusionsyzx2dd■ Unit step loading t d C s /d = 4■ Response at r = 2d■ Regular Gauss integration:◆ 4 × 4 Gaussian points◆ 20 × 20 Gaussian points0.20.20.150.15Displacement [−]0.10.05Displacement [−]0.10.0500−0.050 1 2 3 4Time [−]µrd 2 u xx (τ)−0.050 1 2 3 4Time [−]µrd 2 u zz (τ)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 24/44

Time domain BEMBoundary integral equationIntroductionHybrid methodTime Domain BEM• Fundamental Sol.• Discretisationu j (x ′ , t) −■ Discretisation of tractions:∫Σu G ij (x′ , x, t) ⋆ t j (x, t)dx = 0 (23)FE-BEConclusionst(x, t) =N∑ Tn=1N I (e)∑k=1ψ n (t)M e k(ξ)t n k (24)■ Constant interpolation functions Mk e (ξ) are used■ Constant time interpolation functions ψ n (t)u j (x ′ , t) =∑N T∑N En=1 n=1N I (e)∑k=1∫e(uGij (x, ξ, t) ⋆ ψ n (t) ) M e k (ξ)J(ξ) dξ tn jk (25)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 25/44

Time domain BEMCalculation of the structural responseIntroduction■ The relationshipHybrid methodTime Domain BEM• Fundamental Sol.• Discretisationu N =N−1∑i=1F N−i t i + F 0 t N (26)FE-BEConclusionsis similar to the one resulting from the hybrid method.■ The computational time required for the evaluation of this equation is considerablylarger in the example considered.■ However, choices made during implementation seriously affect the computationaltime.■ In the case of a large soil-stucture interface, this advantage might get lost.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 26/44

Time domain BEMCalculation of the structural responseIntroduction■ Horizontal loadHybrid methodTime Domain BEM• Fundamental Sol.• DiscretisationFE-BEConclusionsyP xxztP xDisplacement [−]0.50.40.30.20.100 2 4 6 8 10Time [−]u x (t)Rotation [−]0.20.150.10.0500 2 4 6 8 10Time [−]ϕ y (t)■ Vertical loadyzxP z P ztDisplacement [−]0.50.40.30.20.100 2 4 6 8 10Time [−]u z (t)LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 27/44

IntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationPart II: FE-BE couplingConclusionsLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 28/44

FE-BE couplingProblem formulationIntroductionHybrid methodΩ bTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationyzΩ sΣ rsΣ bsxΩ rConclusionsΩ bΩ syzxΣ bsyzΩ sΣ rsΣ bsxΩ rΣ rsLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 29/44

FE-BE couplingIntroductionRoad propertiesHybrid methodTime Domain BEMWidth 2B = 4 mΩ sFE-BE• FE modelzΣ bs• Substructuring• FE equation• ImplementationTraffic plateauyxΩ r• Response• VerificationConclusionsTop length L = 10mHeight H = 0.12mSine shaped ramps with a lengthl = 1.2 m.Σ rsSoil propertiesC s = 200.00m/s, C p = 400.00m/s, C r = 186.40m/s, ρ s = 1750kg/m 3β s = β p = 0.025LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 30/44

FE-BE couplingTraffic induced incident wave fieldIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsYZXLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 31/44

FE-BE couplingFinite element model of the structureIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusions■ Dimensions: 12m × 6 m × 6 m.■ Concrete slabs with hinged joints atthe edges.■ Masonry walls.■ Concrete strip foundation.■ Concrete beams above windows anddoor opening.■ 6 degrees of freedom per node.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 32/44

FE-BE couplingSubstructuring techniqueIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsa. Wall A. b. Remaining part of the structure.A non-linear Drucker-Prager behaviour is considered for wall A, while the remainingpart of the structure is assumed to behave linear.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 33/44

FE-BE couplingFinite element equationIntroductionHybrid methodTime Domain BEM[ ] {M b1 b 1M b1 b 2M b2 b 1M b2 b 2}ü b1 (t)ü b2 (t)+{ }f intb 1(t)f intb 2(t)={ }f extb 1(t)f extb 2(t)−{ }0Q b2(t)FE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsInterface modes are used:u b2 = Φα s (27)α s = (Φ T Φ) −1 Φ T u b2 = T u u b2 (28)Q b2= Φ(Φ T Φ) −1 f s = T q f s (29)Mode 6 Mode 9 Mode 15LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 34/44

FE-BE couplingThe dynamic soil-structure interaction forcesIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsFlexibility [1/NmHz]0.80.60.40.200 100 200 300 4001 x 10−5 Frequency [Hz]■ Flexibility relationship in the frequency domainˆα s (ω) = ˆF s (ω)ˆf s (ω)■ Flexibility relationship in time domainα s (t) =∫ t0F s (t − τ) f s (τ)dτ■ Asymptotic behaviour is accounted for[ˆF s (ω)] mm ∽ C mm1iωforω → ∞Flexibility [1/Nm]86420−20 0.05 0.1 0.1510 x Time [s]cfr. waves in 1D bar: ˆF s (ω) = (iωρC p ) −1 .■ Time domain flexibility F s (t)C r = 186.40m/s →6 m186.40m/s = 0.032sLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 35/44

FE-BE couplingThe dynamic soil-structure interaction forcesIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusions■ Linear temporal variation of the soil-structure interaction forces f s (t):■ Instantaneous soil flexibility:■ Influence matrices:α N s =N−1∑i=1F 0 s = ∫ ∆t0τ ′F N−is f i s + F 0 sf N s (30)∆t F s (∆t − τ ′ ) dτ (31)F N−is =∫ ∆t+τ ′0∫ ∆t0∆t F s ((N − i + 1)∆t − τ ′ ) dτ ′(1 − τ ′ )F s ((N − i)∆t − τ ′ ) dτ ′ (32)∆tLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 36/44

FE-BE couplingThe dynamic soil-structure interaction forcesIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusions■ Soil-structure interaction force:f N s = ( F 0 −1s)α N s − ( N−1F 0 ) −1∑s Fs N−i f i s (33)i=1■ The structural response is written in terms of the degrees of freedom u b2 :N−1Q N ( )b 2= T q F0 −1s T u u N ( )b 2− T q F0 −1∑si=1F N−is f i s (34)Q N b 2= T q(F0s) −1T u u N b 2− Q N b 2 hist (35)■ This is introduced into the finite element equation:[ ] } { }M b1 b 1M b1 b 2{üN b1f intNb+ 1+M b2 b 1M b2 b 2ü N b 2f intNb 2[] { }0 0 u N b 10 T q (F 0 s) −1 =T u u N b 2{ } {f extNb 1+f extNb 20Q N b 2 hist}LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 37/44

FE-BE couplingHybrid domain methodIntroductionHybrid methodANSYSMATLABMISSTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation1. PREPROCESSING■ The structure is modelledwith finite elements■ The mesh and mode shapesof the soil-structure interfaceΣ bs are exported to MatlabPass mesh and mode shapes ofthe soil-structure interface toMISS2. DYNAMIC SOIL-STRUCTURE INTERACTION■ Soil impedance■ Modal loads• Response• Verification4. CHANGE STIFFNESS MATRIX3. HYBRID DOMAIN METHODConclusions■ The instantaneous soil stiffness isadded to the structural stiffnessmatrix.5. SOLUTION■ Calculation of influence matricesusing a generalised Filonintegration■ Calculation of instantaneous soilstiffness■ The structural response iscalculated with a direct timeintegration procedure.■ The dynamic soil-structureinteraction forces are updated inevery time step.6. POSTPROCESSINGLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 38/44

FE-BE couplingDirect domain methodIntroductionHybrid methodTime Domain BEMANSYSMATLABFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusions1. PREPROCESSING■ The structure is modelledwith finite elements.■ The mesh and mode shapesof the soil-structure interfaceΣ bs are exported to Matlab.4. CHANGE STIFFNESS MATRIX■ The instantaneous soil stiffness isadded to the structural stiffnessmatrix.2. DYNAMIC SOIL-STRUCTURE INTERACTION■ Influence matrices in thetime domain■ Equivalent load on theinterface5. SOLUTION■ The structural response iscalculated with a direct timeintegration procedure.■ The dynamic soil-structureinteraction forces are updated inevery time step.6. POSTPROCESSINGLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 39/44

FE-BE couplingThe structural responseIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsThe response in point B:Displacement [m]−9−9.05x 10 −4−9.14 5 6 7 8Time [s]Velocity [m/s]10−1−24 5 6 7 82 x 10−4 Time [s]−1.06−1.07−1.08−1.09−1.14 5 6 7 8−1.05 x 105 Time [s]a. uz(t). b. vz(t). c. σ 2 (t).Stress [Pa]BLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 40/44

FE-BE couplingThe structural responseIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsLMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 41/44

FE-BE couplingRetrospective frequency domain analysisIntroductionHybrid methodTime Domain BEMFE-BE• FE model• Substructuring• FE equation• Implementation• Response• VerificationConclusionsThe proposed methodology is based on an approximation. The error induced by thisapproximation is assumed to be limited:■ The frequency content of the incident wave field is limited to frequencies up to30 Hz.■ A retrospective frequency domain analysis can be performed: the relationshipˆα s (ω) = ˆF s (ω)ˆf s (ω)in the frequency domain is applied to verify the displacements on the modalcoordinates ˆα s (ω) based on calculated interaction forces ˆf s (ω).■ The method should be applied cautiously. A visual inspection of the least-squaresapproximations of the dynamic soil flexibility is advised.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 42/44

FE-BE couplingRetrospective frequency domain analysisIntroductionHybrid methodTime Domain BEMThe relationshipˆα s (ω) = ˆF s (ω)ˆf s (ω)FE-BE• FE modelis verified.• Substructuring• FE equation• Implementation• Response• VerificationConclusionsForce [Nm]−1.6 x 105 Time [s]−1.7−1.8−1.9FFTForce [Nm/Hz]10 x 107 Frequency [Hz]8642Flexibility [1/N/m/Hz]5432100 10 20 306 x Frequency [Hz]−24 5 6 7 800 10 20 30x 10 −50.050.05Displacement [−]−8.5−9−9.5−10−10.5FFTDisplacement [1/Hz]0.040.030.020.01Displacement [1/Hz]0.040.030.020.01−114 5 6 7 8Time [s]00 10 20 30Frequency [Hz]00 10 20 30Frequency [Hz]LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 43/44

ConclusionsDynamic soil-structure interaction in the time domainIntroductionHybrid methodTime Domain BEMFE-BEConclusions■ A solution technique for dynamic soil-structure interaction forces in the timedomain has been presented and applied to the calculation of traffic inducedvibrations in buildings.■ A hybrid domain approach has been used, i.e. the relationship between thedisplacements and dynamic interaction forces at the soil-stucture interface havebeen derived from the frequency domain dynamic soil flexibility. The method is agood alternative for the time domain boundary element method, certainly if afrequency domain formulation is available.■ The method allows to account for non-linear structural behaviour in the structure.Standard finite element techniques, such as the Irons-Guyan method and aNewmark direct time integration procedure are applicable.■ Present research:◆ The usefullness of this method in modelling vibration induced structuraldamage is limited, as long-term foundation settlements are involved.◆ The method is being employed in a numerical model for the calculation ofvibrations induced by pile driving. In this case, short-term non-linear soilbehaviour is important.LMSSMat Hybrid method for the computation of dynamic soil-structure interaction in the time domain - p. 44/44