DOTTORATO DI RICERCA IN INGEGNERIA GEOTECNICA XXII CICLO Consorzio tra: Università di Napoli Federico II, Università di Napoli Parthenope, Seconda Università di Napoli, Università di Salerno, Università del Sannio SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS Dottorando: ing. Raffaele Di Laora Tutore: prof. ing. Alessandro Mandolini SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS AIM OF THE THESIS TO BETTER UNDERSTAND THE MATHEMATICAL AND PHYSICAL MEANING OF KINEMATIC INTERACTION TO PROVIDE SIMPLIFIED FORMULAS FOR EVALUATING ITS EFFECTS IN TERMS OF BENDING MOMENTS (INTERFACE AND PILE HEAD) TO FIND SIMPLY RULES TO ESTIMATE THE IMPORTANCE OF “FILTERING EFFECT” EXERTED BY PILES ON FOUNDATION INPUT MOTION TO INVESTIGATE THE RELATIVE IMPORTANCE OF KINEMATIC VS. INERTIAL INTERACTION TO PROVIDE A CRITERION TO COMBINE INERTIAL AND KINEMATIC MAXIMUM EFFECTS SEISMIC SOIL-STRUCTURE INTERACTION FR PILE SUPPORTED SYSTEMS SOIL-STRUCTURE INTERACTION M u(t) Cu(t) K u(t) M I s(t) M u(t) K *u(t) M I s(t) Msoil uKI K *uKI Msoil ub (t ) M uII K *uII Mstructure uKI (t ) ub (t ) ONLY IN ELASTICITY SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SOIL-STRUCTURE INTERACTION M u(t) Cu(t) K u(t) M I s(t) M u(t) K *u(t) M I s(t) Msoil uKI K *uKI Msoil ub (t ) M uII K *uII Mstructure uKI (t ) ub (t ) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SOIL-STRUCTURE INTERACTION ONE COULD ALSO DEFINE KINEMATIC INTERACTION IN A DIFFERENT WAY. FOR EXAMPLE, KINEMATIC INTERACTION COULD BE THE PHENOMENON GENERATED, UNDER SEISMIC MOTION, WHEN ONLY SOIL AND PILES ARE PRESENT, WITHOUT ANY RESTRAINT AND STRUCTURE. IN THIS WAY, KINEMATIC INTERACTION HAVE SENSE ALSO, FOR EXAMPLE, IN PLASTICITY. UNFORTUNATELY, SUMMING THIS KINEMATIC INTERACTION TO THE INERTIAL ONE, ONE DOESN’T OBTAIN THE COMPLETE INTERACTION. THEN: KINEMATIC INTERACTION HAVE SENSE ONLY WITH REFERENCE TO ELASTICITY HYPOTHESIS AND RIGID RESTRAINT AT PILE HEAD SIMPLIFIED FORMULAS SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS DOBRY & O’ROURKE (1983) M 1,86E p I p 3/ 4 G1 1/ 4 1 F 1 c 1 c F 1 c c 1 c c 4 1 3 2 1/ 4 G c 2 G1 MYLONAKIS (2001) p 1 2 c c 1 3 Ek1 p h1 1 c c 1 1 d h 2c 4 1 d 1/ 4 p p / 1 / 1 0 NIKOLAOU ET AL. (2001) Ep 3 L M max ( ) 0.042c d d E1 0.3 0.65 V2 V1 0.5 M max ( t ) M max ( ) KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS NUMERICAL ANALYSES HYPOTESES: • LINEAR ELASTIC BEHAVIOUR OF PILES AND SOIL • BOUNDARY CONDITIONS: VERTICAL DISPLACEMENT RESTRAINED • MODAL DAMPING (10% FOR ALL MODES) • SOLUTION METHOD: MODE SUPERPOSITION • FREQUENCY DOMAIN ANALYSIS • FINITE ELEMENTS: 8-NODE BRICKS (ISOPARAMETRIC) KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS FREQUENCY DOMAIN ANALYSYS ADVANTAGES OF FREQUENCY DOMAIN ANALYSES: • MORE ACCURATE • VERY QUICK (ABOUT 100-200 TIMES FASTER) A FREQUENCY DOMAIN ANALYSIS IS PERFORMED: • PERFORMING A MODAL ANALYSIS EXTRACTING A CERTAIN NUMBER OF MODES; • APPLYING A UNIT ACCELERATION AT BEDROCK VARIABLE WITH FREQUENCY; • READING THE STEADY-STATE RESPONSE IN TERMS OF THE VARIOUS PARAMETERS OF INTEREST (DISPLACEMENT, STRESS AND SO ON); • MULTIPLYING THE STEADY-STATE RESPONSE BY THE FOURIER TRANSFORM OF THE BEDROCK SIGNAL • PERFORM A INVERSE FOURIER TRANSFORM TO OBTAIN THE RESULTS IN THE TIME DOMAIN KINEMATIC INTERACTION 2 Acceleration [m/s ] 2.5 Maximum acceleration [m/s 2] 1.3 0 0.0 -2.5 4 6 Time [s] 8 10 ACCELERATION TIME HISTORY AT SURFACE 2 15.0 5.0 2 3 Structural period [s] MAXIMUM ACCELERATION AGAINST DEPTH 30 0.0 1 15 ANSYS EERA 25 ANSYS EERA 0 2 20 ACCELERATION SPECTRUM AT SURFACE 10.0 5 10 Depth [m] 2 1 0 ANSYS EERA -1.3 0 Spectral acceleration [m/s ] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS COMPARISON WITH EERA 4 HOMOGENEOUS SOIL Vs = 400 m/s RIGID BEDROCK DEPTH : 30 m INPUT SIGNAL: TOLMEZZO DAMPING b = 10% 3 KINEMATIC INTERACTION Bending moment [kNm] Bending moment [kNm] 0 10 20 30 0 40 1 2 3 4 5 0 0 2.7 m 3 4 1 2 0.6 m 10 HOMOGENEOUS SOIL 5 Depth [m] 1.8 m 5 Depth [m] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS GROUP EFFECTS HOMOGENEOUS SOIL Vs = 400 m/s 10 Vs = 100 m/s 15 15 pile n. 1 pile n. 2 pile n. 3 pile n. 4 20 20 pile n. 1 pile n. 2 pile n. 3 pile n. 4 NO DIFFERENCE AMONG PILES OF THE SAME GROUP KINEMATIC INTERACTION Bending moment [kNm] 0 20 40 60 80 100 0 AGAIN, LITTLE DIFFERENCE BETWEEN SINGLE PILE AND PILE IN A GROUP 5 Depth [m] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS 2-LAYERS SOIL Vs1 = 100 m/s BENDING MOMENTS MUCH LARGER WITH RESPECT TO HOMOGENEOUS SOIL 10 Vs2 = 400 m/s FAR FROM THE INTERFACE PILE DEFORMATIONS ARE EQUAL TO SOIL DEFORMATIONS 15 pile n. 1 single pile FF curvature 20 THE INTERFACE IS ONLY A SINGULARITY AND ITS EFFECTS VANISH WITH DISTANCE KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PARAMETRIC STUDY N. 1 FIXED PARAMETERS L= d= H= E1 = gp = gs1 = gs2 = βp = βs1 = βs2 = νp = νs1 = νs2 = 24 m 1m 30 m 50 MPa 25 kN/m3 16 kN/m3 18 or 20 kN/m3 0.1 0.1 0.1 0.2 0.3 0.3 VARIABLE PARAMETERS SIGNALS Ep/E1 Vs2/Vs1 h1/d BORGO CERRETO 300 1.5 2 STURNO 1000 2 4 NOCERA UMBRA 10000 3 8 6 16 TOLMEZZO SAN ROCCO TARCENTO 288 CASES KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PARAMETRIC STUDY N. 2 FIXED PARAMETERS H= 30 m d= 0.5 m Ep = gp = 30 GPa 25 kN/m3 gs1 = VARIABLE PARAMETERS Ep/E1 Vs2/Vs1 h1/d L/d BORGO CERRETO 150 1.5 4 24 16 kN/m3 STURNO 666 2 8 40 gs2 = 1.8 or 2.0 kN/m3 NOCERA UMBRA 1500 3 16 βp = 0.1 TOLMEZZO βs1 = 0.1 SAN ROCCO βs2 = 0.1 TARCENTO νp = 0.2 νs1 = 0.3 νs2 = 0.3 SIGNALS 324 CASES KINEMATIC INTERACTION soil layer 1 soil layer 1 soil layer 2 PILE CAP soil layer 2 30 Bending moment at pile cap [kNm] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS RESULTS – EFFECT OF DEPTH OF INTERFACE BORGO CERRETO STURNO H 1 < LA 25 H 1 > LA 20 15 NOCERA UMBRA TOLMEZZO 10 SAN ROCCO 4 5 8 TARCENTO 16 2 0 0 5 10 15 20 25 h1/d E1 = 50 MPa, Vs2/Vs1=1.5, Ep/E1 = 300, d = 1 m LA = 1.5 d (Ep/E1)0.25 WHEN H1 < LA BENDING MOMENT AT PILE CAP INCREASES WHEN INCREASING H1. THIS IS NOT DUE TO AN INCREASE IN SURFACE ACCELERATION, BUT THE CONSTRAINT APPLIED BY STIFFER LAYER LIMITS DEFORMATIONS THE MORE IT IS CLOSE TO THE PILE HEAD (RESTRAINED) WHEN H1 > LA eP DEPENDS ONLY ON THE ACCELERATION AT SURFACE, THE SHEAR WAVE VELOCITY OF THE FIRST LAYER, AND THE DIAMETER OF THE 30 PILE. THE SECOND LAYER AFFECTS THE ACCELERATION AT SURFACE. KINEMATIC INTERACTION INTERFACE 40 Bending moment at interface [kNm] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS RESULTS – EFFECT OF DEPTH OF INTERFACE 16 H 1 < LA 30 H 1 > LA STURNO 25 NOCERA UMBRA 8 20 WHEN H1 < LA BENDING MOMENT AT INTERFACE CAN DECREASE OR INCREASE BECAUSE OF THE INTERACTION WITH CAP RESTRAINT, BUT ALWAYS SMALLER THAN IN THE CASE IN WHICH H1 = LA BORGO CERRETO 35 TOLMEZZO 15 SAN ROCCO WHEN H1 > LA IT INCREASES UP TO A CERTAIN DISTANCE FROM PILE TOE, THEN IN DECREASES. 2 10 4 TARCENTO 5 0 0 5 10 15 20 25 h1/d E1 = 50 MPa, Vs2/Vs1=1.5, Ep/E1 = 300, d = 1 m LA = 1.5 d (Ep/E1)0.25 30 KINEMATIC INTERACTION 80 Bending moment at interface [kNm] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS RESULTS – EFFECT OF STIFFNESS CONTRAST 6 70 GENERALLY, INCREASING STIFFNESS CONTRAST PILE BENDING MOMENT INCREASES BORGO CERRETO 3 60 STURNO 2 50 NOCERA UMBRA 1.5 40 TOLMEZZO 30 SAN ROCCO ONLY WHEN H1 > LA IT IS ALWAYS TRUE. 20 TARCENTO 10 0 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 Vs2/Vs1 E1 = 50 MPa, h1/d = 16, Ep/E1 = 300, d = 1 m 12.0 KINEMATIC INTERACTION WHEN H1 < LA MOMENT AT INTERFACE CAN DECREASE WITH INCREASING STIFFNESS CONTRAST, BECAUSE EVEN IF SOIL SHEAR STRAINS GENERALLY INCREASES, TRANSMISSBILITY DROPS DUE TO INTERFERENCE BETWEEN CAP AND STIFFER LAYER RESTRAINTS 300 Bending moment at interface [kNm] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS RESULTS – EFFECT OF STIFFNESS CONTRAST 1.5 BORGO CERRETO STURNO 250 2 200 NOCERA UMBRA TOLMEZZO 150 3 100 SAN ROCCO 6 TARCENTO 50 0 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 Vs2/Vs1 E1 = 50 MPa, h1/d = 4, Ep/E1 = 10000, d = 1 m BENDING MOMENT AT PILE CAP DEPENDS ON SURFACE ACCELERATIONS KINEMATIC INTERACTION BENDING MOMENT ALWAYS INCREASES WHEN INCREASING STIFFNESS RATIO BOTH AT PILE CAP AND AT INTERFACE. 1.5 BORGO CERRETO 300 e p at interface (x 10 4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS RESULTS – EFFECT OF PILE-SOIL STIFFNESS RATIO STURNO 1000 1.0 NOCERA UMBRA 10000 0.5 TOLMEZZO SAN ROCCO TARCENTO 0.0 0 5000 10000 15000 20000 Ep /E1 Es = 50 MPa, h1/d = 16, Vs2/Vs1 = 2, d = 1 m IF Ep/E1 INCREASES BECAUSE OF AN INCREASE IN Ep WITH E1 CONSTANT, FREE FIELD SHEAR STRAINS REMAIN CONSTANT, RESULTING IN SMALLER STRAINS IN THE PILE DUE TO ITS GREATER STIFFNESS, BUT THEY DECREASE LESS THAN LINEARLY INCREASING STIFFNESS RATIO. KINEMATIC INTERACTION 3000 Bending moment at interface [kNm] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS RESULTS – EFFECT OF PILE-SOIL STIFFNESS RATIO 10000 NOCERA UMBRA 2500 TOLMEZZO 2000 SAN ROCCO 1500 TARCENTO 1000 1000 STURNO THIS IS ALSO EVIDENT IF EP IS CONSTANT AND E1 DECREASES, BECAUSE THE SOIL STRAINS INCREASE. 300 BORGO CERRETO 500 0 0 5000 10000 15000 AS BENDING MOMENT IS PROPORTIONAL TO PILE BENDING STRAIN AND ITS YOUNG MODULUS, IT INCREASES WITH INCREASING STIFFNESS RATIO. 20000 Ep /E1 Es = 50 MPa, h1/d = 16, Vs2/Vs1 = 2, d = 1 m KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF DIAMETER AND LENGHT IF THE INTERFACE IS LOCATED OUTSIDE THE ACTIVE LENGTH OF THE PILE, BENDING MOMENTS VARY CUBICALLY AGAINST PILE DIAMETER THIS IS DUE TO THE FACT THAT THE PILE BENDING STRAIN AT OUTER FIBER IS NOT DEPENDENT ON PILE DIAMETER. OF COURSE ITS INCREMENT INCREASES THE ACTIVE LENGTH AND THE INTERFACE CAN LOCATE INSIDE THAT, SO THE εp CAN REDUCE. THE SAME HAPPENS FOR THE PILE LENGTH: IF THE INTERFACE REMAINS OUTSIDE THE ACTIVE LENGTH, THE εp IN NOT SENSITIVE TO IT, BUT IF THE LENGTH DECREASES, THE PART OF PILE EMBEDDED IN THE SECOND LAYER CAN BECAME TOO SHORT AND RESULT TO A LOWER PILE BENDING STRAIN DUE TO A LOWER CONSTRAINT OF THE STIFFER LAYER. KINEMATIC INTERACTION 3.0 2.5 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF INTERFACE MOMENTS 2.0 Ep = 50 GPa ; h1/d = 16 1.5 Ep = 15 GPa ; h1/d = 8 y = 0.256x 1.0 2 R = 0.904 Ep = 15 GPa ; h1/d = 16 Ep = 30 GPa ; h1/d = 16 0.5 0.0 0 5 10 g 1(Vs2/Vs1)0.5(Ep /E1)-0.25 (x 10-4) H1 > L A LA = 1.5 d (Ep/E1)0.25 15 20 KINEMATIC INTERACTION 3.0 2.5 Ep = 500 GPa ; h1/d = 16 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF INTERFACE MOMENTS 2.0 Ep = 50 GPa ; h1/d = 8 Ep = 50 GPa ; h1/d = 16 1.5 Ep = 15 GPa ; h1/d = 8 1.0 Ep = 15 GPa ; h1/d = 16 Ep = 30 GPa ; h1/d = 8 0.5 Ep = 30 GPa ; h1/d = 16 0.0 0 5 10 15 g 1(Vs2/Vs1)0.5(Ep /E1)-0.25(h1/La)n (x 10-4) H1 > 0.8 LA 20 KINEMATIC INTERACTION ALTERNATIVELY, INSTEAD OF γ1, IT CAN BE REPLACED THE QUANTITY (γ1- γ2) 3.0 2.5 Ep = 500 GPa ; h1/d = 16 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF INTERFACE MOMENTS 2.0 Ep = 50 GPa ; h1/d = 8 y = 0.296x 1.5 R2 = 0.939 1.0 Ep = 50 GPa ; h1/d = 16 Ep = 15 GPa ; h1/d = 8 Ep = 15 GPa ; h1/d = 16 Ep = 30 GPa ; h1/d = 8 0.5 Ep = 30 GPa ; h1/d = 16 0.0 0 5 10 15 (g 1-g 2)(Vs2/Vs1)0.5(Ep /E1)-0.25(h1/La)n (x 10-4) 20 KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF INTERFACE MOMENTS APPARENTLY, THE LAST FORMULATION HAS A DRAWBACK WITH RESPECT TO THE PREVIOUS ONE: WITH REFERENCE TO AN HOMOGENEOUS SOIL, (γ1- γ2) TENDS TO 0, AND NO BENDING MOMENT IS PREDICTED ALONG THE PILE, IN CONTRAST WITH THE EVIDENCE. ACTUALLY, EVEN IF THE PREVIOUS FORMULATION PREDICTS A NON-ZERO MOMENT IN THIS CASE, ITS NUMERICAL VALUE IS EVIDENTLY WRONG, BECAUSE IT BECOMES: Ep p Es 0.25 THAT IS INCORRECT. THE INTERFACE ACTS AS A FURTHER “SOURCE OF BENDING MOMENT”, RULED BY OTHER PARAMETERS, AND THIS MOMENT ADDS TO THE MOMENT ACTING WHEN THE PILE IS EMBEDDED IN AN HOMOGENEOUS SOIL; THIS MECHANISM IS RULED BY DIFFERENT PARAMETERS (FOR EXAMPLE THE BENDING STRAIN IS PROPORTIONAL TO THE DIAMETER). IT IS EVIDENT THAT, WHEN THE STIFFNESS CONTRAST IS SHARP, THE INTERFACE PRACTICALLY BECOMES THE UNIQUE “SOURCE OF BENDING MOMENT” , AND (γ1- γ2) TENDS TO γ2. KINEMATIC INTERACTION ACCORDING TO THE SPEAKER, PILE BENDING STRAIN DEPENDS ALSO ON SOIL SHEAR STRAINS AROUND THE INTERFACE 3.0 2.5 Ep = 500 GPa ; h1/d = 16 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF INTERFACE MOMENTS 2.0 Ep = 50 GPa ; h1/d = 8 y = 0.324x R2 = 0.965 1.5 1.0 Ep = 50 GPa ; h1/d = 16 Ep = 15 GPa ; h1/d = 8 Ep = 15 GPa ; h1/d = 16 Ep = 30 GPa ; h1/d = 8 0.5 Ep = 30 GPa ; h1/d = 16 0.0 0 5 10 15 (g 1* -g 2* )(Vs2/Vs1)0.5(Ep /E1)-0.25(h1/La)n (x 10-4) γ1* = γ1(h1-d) γ2* = γ2(h1+d) 20 KINEMATIC INTERACTION WHEN H1 < LA, NO CORRELATION EXISTS BETWEEN PILE BENDING STRAIN AND FREE FIELD SHEAR STRAIN 0.5 Ep = 15 GPa ; h1/d = 2 0.4 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF INTERFACE MOMENTS Ep = 50 GPa ; h1/d = 2 Ep = 500 GPa ; h1/d = 2 0.3 0.2 0.1 0.0 0.0 1.0 2.0 3.0 g 1 (x 10-4) THE STRAINS (BOTH FREE-FIELD AND PILE) ARE VERY SMALL THE EFFECT OF THE PILE STIFFNESS IS LIMITED. THE PILE, IN FACT, HAS NO SPACE TO DEFORM (ONLY 2 DIAMETERS), BECAUSE CONSTRAINED, AND THEN IT IS ANYWAY RIGID, REGARDLESS OF ITS YOUNG MODULUS KINEMATIC INTERACTION WHEN H1 > LA, PILE FOLLOWS FREE-FIELD ALSO IN 2-LAYERS SOILS PILE CURVATURE AT PILE CAP IS EQUAL TO FREE-FIELD SOIL CURVATURE 3.0 2.5 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF PILE CAP MOMENTS 2.0 y = 0.994x Ep = 50 GPa ; h1/d = 16 2 1.5 R = 0.994 Ep = 15 GPa ; h1/d = 8 1.0 Ep = 15 GPa ; h1/d = 16 Ep = 30 GPa ; h1/d = 8 0.5 Ep = 30 GPa ; h1/d = 16 0.0 0.0 0.5 1.0 1.5 2.0 2.5 g ' r (x 10-4) ' a ff Vs12 NO SEISMIC LOCAL RESPONSE ANALYSIS REQUIRED! 3.0 KINEMATIC INTERACTION 3.0 2.5 soil layer 1 L/d = 24 e p (x 10 -4) SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS SIMPLIFIED EVALUATION OF PILE CAP MOMENTS 2.0 L/d = 40 L/d = 40 ; V s2 = variable 1.5 1.0 0.5 y = 0.994x 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 a ff r / V s12 (x 10-4) WHEN H1 > LA, PILE BENDING STRAIN AT PILE CAP IS NOT DEPENDENT ON PILE SLENDERNESS THE PILE FOLLOWS THE SOIL ALSO IF THERE ARE MORE THAN TWO LAYERS BELOW THE INTERFACE KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION KINEMATIC INTERACTION BETWEEN SOIL AND PILES RESULTS IN TWO “KINEMATIC” EFFECTS: MODIFY OF THE SEISMIC SIGNAL AT THE BASE OF THE SUPERSTRUCTURE A ROTATIONAL COMPONENT OF MOTION RISES (IMPORTANT OLY FOR VERY TALL STRUCTURES) THE IMPORTANCE OF THE “FILTERING EFFECT” EXERTED BY THE PILES DEPENDS ON VARIOUS PARAMETERS; WITH REFERENCE TO TWO-LAYERS SOIL DEPOSITS, THE PRIMARY VARIABLES INFLUENCING THE PHENOMENON ARE, IN ANALOGY WITH BENDING MOMENTS: THE STIFFNESS CONTRAST THE DEPTH OF THE INTERFACE THE STIFFNESS RATIO BETWEEN PILE AND SOIL THE FREQUENCY OF EXCITATION KINEMATIC INTERACTION 12 10 pile 8 Ampli SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION free field 6 4 2 0 0 1 2 3 w /w 1 d = 0.5 m, h1/d = 16, Vs2/Vs1 = 3, Ep/Es1 = 1500 GENERALLY PILES EXERT A FILTERING EFFECT, CUTTING ONLY HIGH FREQUENCIES 4 KINEMATIC INTERACTION 12 10 pile 8 Ampli SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION free field 6 4 2 0 0 1 2 3 4 w /w 1 d = 1 m, h1/d = 2, Vs2/Vs1 = 6, Ep/Es1 = 10000 IN EXTREME CASES, FILTERING EFFECT IS VERY IMPORTANT ALSO FOR LOW FREQUENCIES (IN CONTRAST WITH GAZETAS, 1984) KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION THE FOUNDATION AND FREE-FIELD MOTION TEND TO BE MORE AND MORE DIFFERENT: INCREASING THE STIFFNESS RATIO INCREASING THE STIFFNESS CONTRAST INCREASING THE IMPORTANCE OF HIGH FREQUENCIES OF THE BEDROCK SIGNAL DECREASING THE VALUE OF h1/LA KINEMATIC INTERACTION ANALYSING THE RESULTS OF THE TWO PARAMETRIC STUDIES, LOOKING AT FREE FIELD AND FOUNDATION SIGNALS (612 x 2 = 1224 SPECTRA), IT FOLLOWS THAT IN VERY RARE CASES IN WHICH THE FOLLOWING CONDITIONS: h1 < L A Vs2/Vs1 > 3 Ep/Es1 ≥ 10000 ARE CONTEMPORARY PRESENT, IT IS POSSIBLE TO NEGLECT THE FILTERING EFFECT (WITHOUT A TOO LARGE CONSERVATISM) ONLY FOR STRUCTURAL PERIODS T1 > 0.5 s 25 25 20 20 15 15 free field BORGO CERRETO 10 5 0 0 1 2 T [s] 3 free field 10 5 0 pile Sa [m/s2] pile Sa [m/s2] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION 4 STURNO 0 1 2 T [s] d = 1 m, h1/d = 2, Vs2/Vs1 = 6, Ep/Es1 = 10000 3 4 KINEMATIC INTERACTION IN OTHER (FREQUENT) CASES, IT IS POSSIBLE TO NEGLECT THE FILTERING EFFECT EXERTED BY THE PILES ONLY WHEN THE STRUCTURAL PERIOD IS GREATER THAN 0.3 s 10 10 8 8 pile 6 Sa [m/s2] Sa [m/s2] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION free field BORGO CERRETO 4 6 2 0 0 1 2 T [s] 3 4 free field 4 2 0 pile STURNO 0 1 2 T [s] d = 1 m, h1/d = 16, Vs2/Vs1 = 6, Ep/Es1 = 1000 3 4 KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS MODIFY OF THE FOUNDATION INPUT MOTION IT’S IMPORTANT TO CLARIFY THAT THE ANALYSES HAVE BEEN UNDERTAKEN CONSIDERING THE SUPERSTRUCTURE AS A SDOF SYSTEM. THESE CONCLUSIONS CAN BE CONSIDERED STILL VALID IF THE STRUCTURE (THE REAL MDOF SYSTEM) IS REGULAR, SO THE CONTRIBUTION OF THE OTHER MODES (CHARACTERISED BY HIGHER FREQUENCIES) IS NOT SIGNIFICANT MOREOVER, THE ANALYSES ARE ELASTIC; IT IS EVIDENT THAT NON LINEARITY CAN INCREASE THIS DIFFERENCE, FOR EXAMPLE, INCREASING THE STIFFNESS RATIO. IT’S ANYWAY IMPORTANT TO KEEP IN MIND THAT REFERRING TO FREEFIELD MOTION FOR INERTIAL INTERACTION ANALYSES IS ALWAYS CONSERVATIVE. SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS KINEMATIC INTERACTION IT HAS BEEN HIGHLIGHTED THE ROLE PLAYED BY THE DIFFERENT PARAMETERS THAT HAVE A SIGNIFICANT INFLUENCE ON KINEMATIC INTERACTION SIMPLIFIED FORMULAS HAVE BEEN PROPOSED TO SIMPLY EVALUATE KINEMATIC BENDING MOMENT AT INTERFACE AND PILE HEAD WHEN INTERFACE IS LOCATED BEYOND THE ACTIVE LENGTH OF THE PILE IN THE CASES IN WHICH H1 < LA ONE CAN ESTIMATE (CONSERVATIVELY) THESE MOMENTS PLACING THE INTERFACE AT A DEPTH EQUAL TO THE ACTIVE LENGHT IT HAS BEEN FOUND A CRITERION TO ESTABLISH IN WHICH CASES IT IS POSSIBLE TO NEGLECT THE FILTERING EFFECT EXERTED BY PILES SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS KINEMATIC VS. INERTIAL INTERACTION RECENT ITALIAN CODES IMPOSE TO EVALUATE THE EFFECTS OF KINEMATIC INTERACTION ONLY UNDER CERTAIN CONDITIONS, BUT REGARDINGLESS OF THE AMOUNT AND THE DISTRIBUTION OF MASSES OF THE STRUCTURE. IT IS EVIDENT THAT IT SHOULD BE ALLOWABLE TO NEGLECT KINEMATIC INTERACTION WHEN ITS EFFECTS ARE NEGLIGIBLE WITH RESPECT TO INERTIAL ONES; FOR THIS REASON THE CODES INDICATIONS APPEARS TO BE INCOMPLETE, AS THE INERTIAL EFFECTS ARE PROPORTIONAL TO THE STRUCTURE MASS, WHILE KINEMATIC ONES ARE EVIDENTLY NOT DEPENDENT ON IT. TO INVESTIGATE THE RELATIVE IMPORTANCE OF INERTIAL VS. KINEMATIC INTERACTION (AND THEN TO VERIFY THE VERACITY OF THE INDICATIONS OF THE CODES), ONCE IT HAS BEEN ESTIMATED THE EFFECTS OF THE LATTER THROUGH THE FORMULAS PROPOSED, IT HAS BEEN BUILT A SIMPLE TOOL FOR EVALUATING THE FOUNDATION IMPEDANCE AND THE RESULTING INERTIAL EFFECTS AT PILES TOP. SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS KINEMATIC VS. INERTIAL INTERACTION Cuu Cu Cu C 0 2 ,1 uu Cuu u2,1Cu 0 Cu C 0 2 ,1 u 0 2 ,1 ....... ....... nn ,1 uu Cuu unn ,1Cu 0 1 0 nn ,1 0 0 1 0 0 nn ,1 vv nn ,2 uu nn ,2 u nn ,2 u nn ,2 0 0 1 d1,G 0 0 0 1 0 d 2 ,G nn ,1 u Cu C Cvv 0 0 Cuu 0 Cu Cvv 0 ....... ....... C 0 Cu C 0 ....... 0 0 Cvv 0 0 Cvv ....... 0 0 ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... 1 0 0 1 0 0 ....... 1 0 0 0 0 0 1 d1,G 0 1 d 2 ,G ....... 0 1 d nn ,G 0 0 0 Cuu 0 Cu Cu 1,2 uu 1,2 u 1,2 u 1,2 1,2 vv 2 ,1 vv Cuu Cu Cu C Cvv nn ,2 vv Cvv T1 0 M 1 0 N1 0 . . . . . . . . . . . . . . uG H PG G M PG RANDOLPH’S EXPRESSIONS FOR BOTH PILE-SOIL COMPLIANCES AND INTERACTION COEFFICIENTS HOMOGENEOUS SOIL r= L= s= n= Ep = 0.3 m 20 m 1.8 m 9 30 GPa Es = γ= aff = m= T= Hstr = 30 MPa 16 kN/m3 1.1 m/s2 500 t 0.5 s 0m INERTIAL VS. KINEMATIC INTERACTION 80 A B 25 D C RATIO FOR PILE A RATIO FOR PILE B RATIO FOR PILE C RATIO FOR PILE D 70 KINEMATIC 60 20 50 15 40 30 10 20 5 10 0 0 0 10 20 30 Ep [GPa] 40 50 60 Kinematic moment at pile top [kNm] 30 Ratio inertial/kinematic moment [-] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF PILE YOUNG MODULUS INERTIAL VS. KINEMATIC INTERACTION 150 A C B D RATIO FOR PILE A RATIO FOR PILE B RATIO FOR PILE C RATIO FOR PILE D KINEMATIC 100 100 50 50 0 0 200 400 600 800 Es [MPa] D C B 1000 0 1200 Kinematic moment at pile top [kNm] 150 Ratio inertial/kinematic moment [-] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF SOIL YOUNG MODULUS INERTIAL VS. KINEMATIC INTERACTION 200 KINEMATIC (INTERFACE) VS2/VS1 = 3 Bending moment [kNm] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF SOIL YOUNG MODULUS KINEMATIC (PILE HEAD) INERTIAL PILE A (PILE HEAD) 150 100 A B C D 50 0 0 200 400 600 800 1000 Es1 [MPa] C B A 1200 INERTIAL VS. KINEMATIC INTERACTION 2500 A B 35 C 30 D RATIO FOR PILE A RATIO FOR PILE B RATIO FOR PILE C RATIO FOR PILE D 2000 KINEMATIC 25 1500 20 1000 15 10 500 5 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Pile radius [m] 0.7 0.8 0.9 1.0 Kinematic moment at pile top [kNm] 40 Ratio inertial/kinematic moment [-] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF PILE RADIUS INERTIAL VS. KINEMATIC INTERACTION 100 A B D C RATIO FOR PILE A RATIO FOR PILE B RATIO FOR PILE C RATIO FOR PILE D 80 KINEMATIC 10 60 40 5 20 0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Pile spacing s/d [-] 8.0 0 9.0 10.0 11.0 12.0 Kinematic moment at pile top [kNm] 15 Ratio inertial/kinematic moment [-] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF PILE SPACING INERTIAL VS. KINEMATIC INTERACTION 50 9 RATIO FOR CENTRAL PILE 8 RATIO FOR CORNER PILE 7 KINEMATIC 40 6 30 5 4 20 3 9 25 2 10 49 1 0 0 0 10 20 30 Number of piles [-] 40 50 60 Bending moment at pile top [kNm] 10 Ratio inertial/kinematic moment [-] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF NUMBER OF PILES INERTIAL VS. KINEMATIC INTERACTION 100 10 RATIO FOR PILE A RATIO FOR PILE B RATIO FOR PILE C RATIO FOR PILE D 80 KINEMATIC A 5 B 60 C 0 D 40 -5 20 -10 -15 0 0 5 10 M / H [m] 15 20 Kinematic moment at pile top [kNm] 15 Ratio inertial/kinematic moment [-] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS EFFECT OF M / H INERTIAL VS. KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS IN GENERAL, TIME HISTORIES OF KINEMATIC AND INERTIAL STRESSES ARE OUTOF-PHASE. IF KINEMATIC AND/OR INERTIAL EFFECTS ARE ESTIMATED ONLY WITH REFERENCE TO THEIR MAXIMA, IT RISES A PROBLEM: HOW TO COMBINE THEM? SUMMING THEIR MAXIMA, IN FACT, CAN BE TOO OVERCONSERVATIVE. IS IT POSSIBLE TO EVALUATE THE PHASE LAG A PRIORI? INERTIAL VS. KINEMATIC INTERACTION KS H I KS I -6.28 -9.42 H AP IS H I A I ACCELERATION STRESS -12.56 0 Phase H I A Phase I 2 3 4 5 6 7 w /w 1 [-] PhaseH I A PhaseH I KS Phase IS Phase H AP IS 1 3.14 PhaseKS PhaseH I KS PhaseI 0 STRESS ACCELERATION 1.57 -1.57 INERTIAL 0.00 PhaseIS PhaseKS PhaseH APIS PhaseIS PhaseKS PhaseH I AS -3.14 0.0 0.5 1.0 1.5 2.0 w /w st [-] (IF THE MOMENT DUE TO RAFT ROTATION PREVAILS) Acceleration phase [rad] IS H AP IS AP H AP IS A KINEMATIC -3.14 Phase [rad] A H I A I 0.00 Stress phase [rad] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS INERTIAL VS. KINEMATIC INTERACTION 3.14 IS IMPORTANT TO NOTICE THAT THE PHASE LAG OF EACH HARMONIC IS INDEPENDENT OF KINEMATIC INTERACTION, DEPENDS ONLY ON THE INERTIAL ONE, THROUGH THE STRUCTURAL PERIOD Phase [rad] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS 1.57 x = 5% 0.00 0.0 0.5 1.0 1.5 2.0 w /w st [-] NEVERTHLESS, THE PHASE LAG DEPENDS ALSO ON INTERACTION, BECAUSE IT DEPEND ON WHERE ARE PREVALENT HARMONICS OF THE SURFACE FREE-FIELD SIGNAL (RIGOROUSLY FOUNDATION MOTION). ONE CAN JUDGE A PRIORI THE PHASE LAG BETWEEN KINEMATIC AND INERTIAL INTERACTION PERFORMING A FREE-FIELD ANALYSIS LOOKING AT THE FOURIER SPECTRA OF SURFACE SIGNAL COMPARING IT WITH THE STRUCTURAL PERIOD INERTIAL VS. KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS NUMERICAL ANALYSES Esoil[Mpa] wsoil [Hz] wstr (with SSI) [Hz] A 30 0.71 1.45 B 120 1.42 1.45 C 120 1.42 1.02 D 120 1.42 0.72 HYPOTESES: • LINEAR ELASTIC BEHAVIOUR OF PILES AND SOIL • BOUNDARY CONDITIONS: VERTICAL DISPLACEMENT RESTRAINED • MODAL DAMPING (10% FOR ALL MODES) • SOLUTION METHOD: MODE SUPERPOSITION • FREQUENCY DOMAIN ANALYSIS • FINITE ELEMENTS: 8-NODE BRICKS (ISOPARAMETRIC) INERTIAL VS. KINEMATIC INTERACTION IN THE CASES SHOWN IN THE FIGURES THE MOMENT DUE TO RAFT ROTATION PREVAILS Fourier Amplitude [m/s] 2.5 2.0 1.5 E1 = 120 MPa STURNO SIGNAL 1.0 E1 = 30 MPa 0.5 0.0 0 2 4 6 8 10 w [Hz] 2.0 Fourier Amplitude [m/s] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS 1.5 E1 = 120 MPa 1.0 TOLMEZZO SIGNAL E1 = 30 MPa 0.5 0.0 0 2 4 6 w [Hz] 8 10 LOOKING AT THE SURFACE FOURIER SPECTRUM WITH REFERENCE, FOR ISTANCE, TO CASE A – STURNO SIGNAL, IT CAN BE NOTICED THAT PREDOMINANT HARMONICS HAVE LOWER FREQUENCIES (WITH RESPECT TO THE STRUCTURE). FOLLOWING THIS METHOD, THEN, KINEMATIC AND INERTIAL EFFECTS HAVE TO BE OUTOF-PHASE. IN THE CASE B – TOLMEZZO SIGNAL, PREDOMINANT HARMONIC OF THE SURFACE FREE FIELD SIGNAL ARE CLOSE TO THE STRUCTURE FREQUENCY (THAT IS EQUAL TO THE GROUND ONE→ RESONANCE). THERE SHOULD BE A PHASE LAG EQUAL TO 90°. INERTIAL VS. KINEMATIC INTERACTION Pile stress [kPa] 1500 CASE A – STURNO 1000 500 0 TOTAL -500 KINEMATIC -1000 INERTIAL -1500 0 1 2 3 4 5 6 1500 Pile stress [kPa] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS 7 8 9 10 11 12 13 14 15 Time [s] CASE B – TOLMEZZO 1000 500 0 TOTAL -500 KINEMATIC -1000 INERTIAL -1500 0 1 2 3 4 5 6 7 8 Time [s] 9 10 11 12 13 14 15 INERTIAL VS. KINEMATIC INTERACTION Fourier Amplitude [m/s] 2.0 1.5 E1 = 120 MPa 1.0 TOLMEZZO SIGNAL E1 = 30 MPa 0.5 0.0 0 2 4 6 8 10 w [Hz] 2.0 Fourier Amplitude [m/s] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS 1.5 E1 = 120 MPa 1.0 TOLMEZZO SIGNAL E1 = 30 MPa 0.5 0.0 0 2 4 6 w [Hz] 8 10 IN THE CASES, FOR ISTANCE, OF TOLMEZZO SIGNAL, MOVING THE STRUCTURE FREQUENCY ON THE LEFT OF THE PREDOMINANT HARMONICS OF THE SURFACE FOURIER SPECTRUM (CASES C AND D), IT IS EXPECTED THAT KINEMATIC AND INERTIAL INTERACTION ARE IN-PHASE INERTIAL VS. KINEMATIC INTERACTION Pile stress [kPa] 1500 1000 500 0 -500 TOTAL KINEMATIC -1000 INERTIAL -1500 0 1 2 3 4 5 6 1500 Pile stress [kPa] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS 7 8 9 10 11 12 13 14 15 9 10 11 12 13 14 15 Time [s] 1000 500 0 -500 TOTAL KINEMATIC -1000 INERTIAL -1500 0 1 2 3 4 5 6 7 8 Time [s] INERTIAL VS. KINEMATIC INTERACTION Fourier Amplitude [m/s] 2.5 2.0 1.5 STARTING FROM CASE A, DECREASING THE STRUCTURE HEIGHT UNTIL 5 METERS, THE BENDING MOMENT DUE TO RAFT TRANSLATION PREVAILS, AND THEN KINEMATIC AND INERTIAL INTERACTION HAVE TO BE IN-PHASE. E1 = 120 MPa STURNO SIGNAL 1.0 E1 = 30 MPa 0.5 0.0 0 2 4 6 8 10 w [Hz] 2000 Pile stress [kPa] SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS 1500 TOTAL 1000 KINEMATIC INERTIAL 500 0 -500 -1000 -1500 -2000 0 1 2 3 4 5 6 7 8 Time [s] 9 10 11 12 13 14 15 INERTIAL VS. KINEMATIC INTERACTION SEISMIC SOIL-STRUCTURE INTERACTION FOR PILE SUPPORTED SYSTEMS PHASE LAG BETWEEN KINEMATIC AND INERTIAL EFFECTS ACCORDING TO WHAT IS WRITTEN ABOVE, THE PHASE LAG DECREASES (FIXING OTHER CONDITIONS): - DECREASING THE FREQUENCY OF THE STRUCTURE; - INCREASING THE FREQUENCY OF THE SOIL; - MOVING THE FREQUENCY CONTENT OF INPUT SIGNAL TO HIGHER FREQUENCIES. IF THE BENDING MOMENT DUE TO RAFT TRANSLATION PREVAILS, OPPOSITE CONCLUSIONS HAVE TO BE DERIVED; IN THIS CASE, THE THREE CONDITIONS ABOVE HAVE TO INCREASE THE PHASE LAG. BASING ON THE DESCRIBED METHOD, IN DESIGN IT IS POSSIBLE TO COMBINE MAXIMUM KINEMATIC AND INERTIAL EFFECTS IN THE FOLLOWING WAY: - IN THE CASE THEY ARE IN-PHASE, SUM THE MAXIMA; - IF THE PHASE LAG IS APPROXIMATELY 180°, TAKE THE MAXIMUM BETWEEN THE KINEMATIC AND THE INERTIAL CONTRIBUTION; - IN THE RARE CASE IN WHICH THE PHASE LAG IS ABOUT 90°, IT IS STILL LICIT TO SUM MAXIMUM EFFECTS WITHOUT COMMITTING LARGE ERRORS; IT IS WORTHY NOTING THAT IN THIS CASE INERTIAL INTERACTION IS PARTICULARLY IMPORTANT BECAUSE OF THE RESONANCE BETWEEN FOUNDATION INPUT MOTION AND STRUCTURE.