ΘΕΩΡΗΤΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΓΕΩΤΕΧΝΙΚΗΣ ΣΕΙΣΜΙΚΗΣ ΜΟΝΩΣΗΣ ΒΑΘΡΩΝ ΓΕΦΥΡΩΝ ΣΕ ΑΒΑΘΗ ΘΕΜΕΛΙΑ ΚΑΙ ΠΑΣΣΑΛΟΥΣ

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1 ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ DEPARTMENT OF CIVIL ENGINEERING SCHOOL OF ENGINEERING UNIVERSITY OF PATRAS ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ ΘΕΩΡΗΤΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΓΕΩΤΕΧΝΙΚΗΣ ΣΕΙΣΜΙΚΗΣ ΜΟΝΩΣΗΣ ΒΑΘΡΩΝ ΓΕΦΥΡΩΝ ΣΕ ΑΒΑΘΗ ΘΕΜΕΛΙΑ ΚΑΙ ΠΑΣΣΑΛΟΥΣ DOCTOR OF PHILOSOPHY DISSERTATION THEORETICAL INVESTIGATION OF GEOTECHNICAL SEISMIC ISOLATION OF BRIDGE PIERS ON FOOTINGS AND PILES ΠΟΛΥΞΕΝΗ ΚΑΡΑΤΖΙΑ ΠΟΛΙΤΙΚΟΣ ΜΗΧΑΝΙΚΟΣ, M.Sc. POLYXENI KARATZIA CIVIL ENGINEER, M.Sc. ΟΚΤΩΒΡΙΟΣ, 16 OCTOBER, 16

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3 At a higher altitude with flag unfurled We reached the dizzy heights of that dreamed of world Encumbered forever by desire and ambition There's a hunger still unsatisfied Our weary eyes still stray to the horizon Though down this road we've been so many times The grass was greener The light was brighter The taste was sweeter The nights of wonder With friends surrounded The dawn mist glowing The water flowing The endless river Forever and ever High Hopes, 1994, Pink Floyd

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5 In Loving Memory of my Father

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7 Acknowledgements I would like to express my deepest gratitude to my Advisor Professor George Mylonakis for his excellent guidance, vast enthusiasm, countless hours and continuous support throughout my studies, since the time he served as my undergraduate Diploma Thesis Advisor, helping me to immerse deeply into real scientific research. Thanks for motivation, inspiration, mentorship and friendship. I would like to thank the members of my Advisory Committee Professor George Athanasopoulos and Professor Dimitris Karampalis for their academic interest and useful suggestions during the course of this work. I am greatly indebted to Professor George Bouckovalas for the ideas contributed, especially on Chapters & 3, the thorough remarks and the valuable discussions on the present work. Thanks for the opportunity you gave to me to work with you and for your triennial financial support. Special thanks go to Professor Costas Papantonopoulos for having always an answer to my questions and being one of the kindest and wisest person I have ever met, a genuine knowledge guru. I would like to thank Professor Demosthenes Polyzos for agreeing to serve on my Examination Committee. His help in giving me access to ISoBEM software is gratefully acknowledged, as well as the support of Professor Stephanos Tsinopoulos in using the software. Thank you both. Thanks are also due to Professor Colin Taylor, for agreeing to serve on my Examination Committee and for sponsoring my travel to Bristol. Special thanks go to Professor Dimitris Atmatzidis for his valuable advice and support during his Tenure as Director of the Geotechnical Laboratory at University of Patras.

8 I am thankful to Professor Takis Pelekis for his advice throughout my doctoral studies and the valuable discussions on scientific and philosophical matters. Thanks for your friendship. Special thanks to my uncle Dr. Nikos Labropoulos, I am grateful to have had such an awesome teacher like you. Thank you for setting the example. I thank all my friends and colleagues in the Geotechnical Lab for their advice, help, good cooperation and for providing a friendly working environment. Thanks to Drs-to-be Foteini Lyrantzaki, Vasilis Kitsis, Olga Theofilopoulou and Vasilis Vlachakis and to Drs. Constantine Thomas, Giannis Pantazopoulos, Panos Kloukinas, George Anoyatis and Tasos Batilas. Special thanks to Eirini P. for helping me prepare the list of symbols. I would like to thank my best friends Vivi S. and Georgia G. for their love, support and being always there for me for more than a decade. I am proud of being friends with you. Many thanks go to my dear friends Nikoleta I. and Nikos S. for the great leisure moments we had in Patras. Thanks to my friends Fares F., Christos R. and Elia M. for the most joyful moments we spent together, especially, during our undergraduate studies in Patras. I would like to thank Andrikos for his love, kindness, generosity, mystery, poetry and fathomless patience during the years of my doctorate. You are a unique human being with exceptional emotional intelligence. You are just different from any other I know and you will always be one of the most significant people in my life. Last, but no means least, I am deeply grateful to my mother Elena for her unconditional love, emotional and financial support during the years of my studies. I also thank my siblings Dimitra and Vasilis for their affectionate love, the great understanding and for always supporting my dreams. A million thanks to my dearest

9 sweet grandma. It would be impossible to accomplish this doctoral thesis without the endless love, help and support of my family. Part of the herein reported research was supported by the Research Funding Program THALES-NTUA (MIS 3843): Innovative Design of Bridge Piers on Liquefiable Soils with the use of Natural Seismic Isolation. This financial support is gratefully acknowledged.

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11 Εκτενής Περίληψη Στόχος διατριβής Σκοπός της παρούσας διδακτορικής διατριβής είναι η θεωρητική διερεύνηση μιας νέας τεχνικής αντισεισμικής προστασίας, μέσω συστημάτων Γεωτεχνικής Σεισμικής Μόνωσης (ΓΣΜ) σε βάθρα γεφυρών εδραζόμενων σε αβαθή θεμέλια και πασσάλους. Στόχος αποτελεί η εξέταση της εφαρμοσιμότητας των συγκεκριμένων συστημάτων στην πράξη («proof of concept»), καθώς και η ανάπτυξη απλής μεθοδολογίας για τον σχεδιασμό τους. Για θεμελίωση βάθρου σε μεμονωμένο πάσσαλο προτείνεται η χρήση προηγμένων βιομηχανικών προιόντων, όπως ο γεωαφρός EPS (διογκωμένη πολυστερίνη), ή μίγματα τεμαχισμένων ελαστικών και εδάφους (RSM) που μπορούν να τοποθετηθούν γύρω από το άνω τμήμα του πασσάλου, με στόχο τη μείωση των αναπτυσσόμενων σεισμικών δράσεων. Για βάθρο γέφυρας εδραζόμενο σε θεμέλιο επανεξετάζεται η ιδέα της εκμετάλλευσης υποκείμενου φυσικού ρευστοποιήσιμου εδαφικού στρώματος, που δρα ως φυσικός αποσβεστήρας / μονωτήρας της σεισμικής κίνησης, με επακόλουθο τη μείωση της κίνησης του ελεύθερου πεδίου και επομένως της αναπτυσσόμενης σεισμικής δράσης στην ανωδομή και θεμελίωση. Η έρευνα για το σύστημα γεωτεχνικής σεισμικής μόνωσης μέσω ρευστοποίησης διεξήχθη στο πλαίσιο του ερευνητικού προγράμματος με τίτλο «ΘΑΛΗΣ-ΕΜΠ (MIS 3843): Πρωτότυπος Σχεδιασμός Βάθρων Γεφυρών σε Ρευστοποιήσιμο Έδαφος με Φυσική Σεισμική Μόνωση». Στο Σχήμα 1, παρουσιάζονται τα δύο εξεταζόμενα συστήματα γεωτεχνικής σεισμικής μόνωσης του συστήματος θεμελίωσης-βάθρου γέφυρας. Για το πρώτο Π-1

12 Εκτενής Περίληψη σύστημα (Σχήμα 1α), ο κύριος στόχος της έρευνας συνίσταται στην αναλυτική και αριθμητική διερεύνηση, μέσω κατάλληλων προσομοιωμάτων, της αλληλεπίδρασης μεταξύ εδάφους-γεωαφρού-πασσάλου-βάθρου γέφυρας μέσω της ελαστοδυναμικής θεωρίας. Για το δεύτερο σύστημα (Σχήμα 1β), κύριος στόχος της παρούσας μελέτης αποτελεί η διερεύνηση της αλληλεπίδρασης μεταξύ ρευστοποιήσιμου εδάφουςθεμελίου-βάθρου μέσω ισοδύναμων γραμμικών προσεγγίσεων, οι οποίες διευκολύνουν τη μελέτη της δυναμικής συμπεριφοράς των επιμέρους μηχανικών συνιστωσών. Κατάστρωμα γέφυρας Κατάστρωμα γέφυρας Βάθρο Βάθρο Επιφανειακό θεμέλιο Γεωαφρός Ενισχυμένη επιφανειακή κρούστα Πάσσαλος Ρευστοποιήσιμο έδαφος Μη ρευστοποιήσιμο έδαφος (α) Προτεινόμενη μέθοδος σεισμικής μόνωσης για βάθρο στηριζόμενο σε πάσσαλο βάσει μαλακού παρεμβλήματος γεωαφρού στο άνω τμήμα του πασσάλου (β) Προτεινόμενη μέθοδος σεισμικής μόνωσης για βάθρο στηριζόμενο σε επιφανειακή θεμελίωση επί ρευστοποιήσιμου εδάφους παρουσία ενισχυμένης «κρούστας» Σχήμα 1: Συστήματα γεωτεχνικής σεισμικής μόνωσης που εξετάζονται στην παρούσα διατριβή. Σύντομη βιβλιογραφική ανασκόπηση Tα τελευταία χρόνια οι εφαρμογές της γεωτεχνικής σεισμικής μόνωσης προσελκύουν όλο και περισσότερο το ενδιαφέρον της επιστημονικής κοινότητας, ως εναλλακτικές Π-

13 Εκτενής Περίληψη λύσεις έναντι των συμβατικών μεθόδων σεισμικής μόνωσης της ανωδομής. Με τη γεωτεχνική σεισμική μόνωση επιτυγχάνεται, με σχετικά οικονομικό τρόπο, η μόνωση της θεμελίωσης έναντι της συμβατικής και δαπανηρής σεισμικής μόνωσης της ανωδομής. Ο όρος «Γεωτεχνική Σεισμική Μόνωση» αποδίδεται στον Tsang (9), ο οποίος τον εισήγαγε για να περιγράψει συστήματα μόνωσης κατασκευών που σχετίζονται με το έδαφος θεμελίωσης. Επίσης, ο ίδιος πρότεινε για πρώτη φορά ταξινόμηση των μεθόδων σεισμικής μόνωσης με βάση τη λειτουργία τους (Σχήμα ). Σχήμα : Παραδείγματα συστημάτων σεισμικής μόνωσης (από Tsang 9) Η ιδέα της ΓΣΜ φαίνεται ότι προέκυψε αρκετά χρόνια πριν, όταν οι Kavazanjian et al. (1991) και οι Yegian & Lahlaf (199), εργαζόμενοι ανεξάρτητα, πρότειναν την επένδυση της βάσης μιας κατασκευής με γεωσυνθετικά υλικά ως Π-3

14 Εκτενής Περίληψη οικονομικό και ταυτόχρονα αποτελεσματικό τρόπο σεισμικής μόνωσης. Τα φύλλα του γεωσυνθετικού υλικού (γεωυφάσματα / γεωμεμβράνες) που καλύπτουν ολόκληρη τη βάση της κατασκευής μπορούν να ολισθήσουν, λόγω χαμηλής τριβής, το ένα σε σχέση με το άλλο κατά τη διάρκεια της σεισμικής διέγερσης και με αυτό τον τρόπο να προστατέψουν την κατασκευή. Στη συνέχεια οι Yegian & Kadakal (4) πρότειναν την τοποθέτηση ενός λείου συνθετικού υφάσματος κάτω από την κατασκευή για τη μόνωση της θεμελίωσης. Στο ίδιο πνεύμα, οι Yegian & Catan (4) εξέτασαν τη δυνατότητα χρήσης φύλλων γεωσυνθετικού υλικού ανάμεσα σε εδαφικά στρώματα, τα οποία επιτρέπουν την ολίσθηση μεταξύ εδάφους και γεωσυνθετικού υλικού, για τη μείωση των οριζόντιων σεισμικών δράσεων. Καμιά από τις συγκεκριμένες προτάσεις δεν μπορεί να εφαρμοστεί σε υφιστάμενες κατασκευές λόγω της αδυναμίας τοποθέτησης των γεωσυνθετικών υλικών κάτω από θεμέλια. Ο Tsang (8) πρότεινε τη βελτίωση του εδάφους θεμελίωσης της κατασκευής μέσω μιγμάτων τεμαχισμένων ελαστικών-εδάφους για την απορρόφηση της σεισμικής ενέργειας. Η συγκεκριμένη πρόταση είχε δύο στόχους: πρώτον την αξιοποίηση των μεταχειρισμένων και άχρηστων ελαστικών, των οποίων η διαχείρηση αποτελεί περιβαλλοντικό πρόβλημα (και η συσσώρευση εστία μόλυνσης), και δεύτερον τη φθηνή σεισμική μόνωση, πρακτική που μπορεί να ευνοήσει ιδιαιτέρως τις αναπτυσσόμενες χώρες. Οι Xiong et al. (14) εξέτασαν πειραματικά σε δονητική τράπεζα τη δυναμική απόκριση κατασκευών που θεμελιώνονται σε έδαφος ενισχυμένο με τεμαχισμένα ελαστικά και σε καθαρή άμμο, και κατέληξαν στο συμπέρασμα ότι η κατασκευή με την εδαφική μόνωση υπόκειται σε μικρότερη σεισμική δράση σε σχέση με αυτή που εδράζεται σε καθαρή άμμο. Π-4

15 Εκτενής Περίληψη Ένα ενδιαφέρον σύστημα σεισμικής μόνωσης μέσω ρευστοποιημένου εδαφικού υλικού προτάθηκε απο τους Tang et al (1991). Οι ερευνητές πρότειναν τη χρήση μονωτήρων που αποτελούνται από τεχνητό εδαφικό μίγμα με χαμηλή σχετική πυκνότητα και γύρω τους υπάρχει αδιαπέρατος ελαστικός τοίχος που κρατά τον όγκο του εδάφους σταθερό. Το έδαφος στους μονωτήρες παραμένει κορεσμένο μέσω ενός συστήματος υδραυλικών αγωγών. Στην περίπτωση σεισμού, το έδαφος ρευστοποιείται, χάνει σημαντικό ποσοστό της στιφρότητάς του με αποτέλεσμα την αύξηση της ιδιοπεριόδου της κατασκευής πέρα από το «ενεργό» εύρος συχνοτήτων του φάσματος του σεισμού. Οι συγκεκριμένες μελέτες/εφαρμογές, παρότι ενδιαφέρουσες από ακαδημαικής πλευράς, δεν συμπεριλαμβάνουν τον έλεγχο των καθιζήσεων/στροφών της θεμελίωσης και επομένως δεν μπορούν να γίνουν αποδεκτές στην πράξη χωρίς περαιτέρω έλεγχο ειδικά για σημαντικές κατασκευές. Δομή διατριβής Στο Κεφάλαιο, εξετάζεται το πρόβλημα προσδιορισμού της δυναμικής δυσκαμψίας άκαμπτων επιφανειακών θεμελίων εδραζόμενων σε ρευστοποιήσιμο έδαφος, με χρήση ισοδύναμης ιξωδοελαστικής ανάλυσης. Το πρόβλημα είναι ιδιαίτερα πολύπλοκο, πρώτον λόγω του ίδιου του φαινομένου της ρευστοποίησης (το οποίο είναι εξόχως μη-γραμμικό), και κατά δεύτερο λόγο η παρουσία του ρευστοποιήσιμου εδαφικού στρώματος απαιτεί την εξέταση ενός πολύστρωτου εδαφικού σχηματισμού του οποίου τα στρώματα εμφανίζουν έντονη αντίθεση κυματικής εμπέδησης, πρόβλημα το οποίο δεν έχει μελετηθεί διεξοδικά στη βιβλιογραφία. Πλήθος δημοσιευμένων πειραματικών και αριθμητικών αποτελεσμάτων (Miwa & Ikeda 6, Theocharis 11) καταδεικνύουν ότι κατά τη διάρκεια Π-5

16 Εκτενής Περίληψη εκδήλωσης της ρευστοποίησης η ταχύτητα διάδοσης των διατμητικών κυμάτων μειώνεται δραστικά (στο 1 με 3% της αρχικής τιμής), ενώ ταυτόχρονα η απόσβεση υλικού αυξάνεται θεαματικά (στο % και παραπάνω). Βάσει των παραπάνω αποτελεσμάτων, το πρόβλημα της μη-γραμμικότητας μπορεί να ξεπεραστεί λαμβάνοντας υπόψη την επίδραση της ρευστοποίησης μέσω ισοδύναμης γραμμικώς ιξωδοελαστικής ανάλυσης, με την υπόθεση κατάλληλων τιμών για τη ταχύτητα διάδοσης διατμητικών κυμάτων και τον λόγο απόσβεσης στο ρευστοποιημένο στρώμα. Σε αυτό το πλαίσιο, διερευνάται η σύνθετη δυναμική δυσκαμψία σε κατακόρυφη, οριζόντια και λικνιστική κίνηση ενός επιφανειακού θεμελίου επί ρευστοποιήσιμου εδάφους, όπως περιγράφηκε προηγουμένως, μέσω του κώδικα CONAN (Wolf & Deeks 4), ο οποίος βασίζεται στη διάδοση κυμάτων σε κώνους. Η συγκεκριμένη προκαταρκτική έρευνα ακολουθείται από ακριβείς τριδιάστατες ελαστοδυναμικές αναλύσεις με χρήση συνοριακών στοιχείων κώδικας ISoBEM (Polyzos et al 1998, Tsinopoulos et al 1999), προκειμένου να ελεγχθεί η ακρίβεια των προηγούμενων αναλύσεων αλλά και να προκύψουν σχέσεις παλινδρόμησης, οι οποίες μπορούν να χρησιμοποιηθούν στην πράξη. Μέσω παραμετρικών αναλύσεων εξετάστηκε η επίδραση παραγόντων όπως το πάχος και η στιφρότητα του επιφανειακού μη ρευστοποιήσιμου εδαφικού στρώματος, το πάχος του ρευστοποιήσιμου στρώματος, και ο σχετικός λόγος των ταχυτήτων διάδοσης διατμητικών κυμάτων στο επιφανειακό και στο ρευστοποιήσιμο στρώμα. Τα αποτελέσματα παρουσιάζονται σε μορφή αδιάστατων διαγραμμάτων, πινάκων και απλών σχέσεων παλινδρόμησης που παρέχουν τη δυνατότητα κατανόησης της περίπλοκης φυσικής του προβλήματος και μπορούν να χρησιμοποιηθούν σε εφαρμογές Πολιτικού Μηχανικού για μια πρώτη διερεύνηση του προβλήματος. Επιπρόσθετα, το συγκεκριμένο κεφάλαιο περιλαμβάνει αναλυτική λύση, βάσει των Π-6

17 Εκτενής Περίληψη προσομοιωμάτων κώνου, για τον προσδιορισμό της κατακόρυφης, οριζόντιας και λικνιστικής στατικής στιφρότητας κυκλικού θεμελίου σε πολύστρωτο έδαφος. Η σύγκριση των προβλέψεων της προτεινόμενης λύσης βρίσκονται σε ικανοποιητική συμφωνία με πιο αυστηρές λύσεις τύπου συνοριακών στοιχείων, ωστόσο η συγκεκριμένη κλειστή λύση δεν μπορεί να εφαρμοστεί σε ρευστοποιήσιμα εδάφη, όπου χρησιμοποιείται μια εξαιρετικά χαμηλή τιμή για την ταχύτητα διάδοσης διατμητικών κυμάτων του ρευστοποιημένου στρώματος. Τα κύρια συμπεράσματα αυτού του κεφαλαίου συνοψίζονται στα εξής: Η ρευστοποίηση οδηγεί σε σημαντική μείωση της στατικής στιφρότητας του επιφανειακού θεμελίου. Η μείωση είναι δυνατό να φθάσει μέχρι 85% για την κατακόρυφη συνιστώσα, 55% για την οριζόντια και 6% για τον λικνισμό. Τα αποτελέσματα που προέκυψαν από τις δύο ανεξάρτητες λύσεις, μέσω των απλοποιημένων μοντέλων κώνου (κώδικας CONAN) και των συνοριακών στοιχείων (κώδικας ISoBEM), καταδεικνύουν τη σημαντική μείωση της στατικής στιφρότητας. Σχετικά με την επίδραση της ρευστοποίησης στις συναρτήσεις δυναμικής εμπέδησης, παρατηρείται σημαντική μείωση στη δυναμική δυσκαμψία και αύξηση στην απόσβεση για το χαμηλό εύρος των αδιάστατων συχνοτήτων ωh 1 /V s1 < και για συνήθη μεγέθη θεμελίων με πλάτος Β < 4m. Για το υψηλό εύρος αδιάστατων συχνοτήτων ωh 1 /V s1 > η δυναμική δυσκαμψία παρουσιάζει έντονες κυματώσεις, ενώ ο λόγος των αποσβεστήρων C i / C i (απόσβεση με ρευστοποίηση προς απόσβεση χωρίς ρευστοποίηση) τείνει στη μονάδα. Πρέπει να τονιστεί οτι οι παρατηρήσεις σχετικά με τη δυναμική δυσκαμψία βασίζονται στις αναλύσεις με το CONAN μόνο ένας περιορισμένος αριθμός Π-7

18 Εκτενής Περίληψη αναλύσεων πραγματοποιήθηκε με το ISoBEM προκειμένου να επιβεβαιωθεί η εγκυρότητα των προσεγγιστικών αποτελεσμάτων. Ο λόγος του πάχους του επιφανειακού μη ρευστοποιήσιμου στρώματος προς το πλάτος του θεμελίου (h 1 /B) αποδεικνύεται ως ο πιο σημαντικός παράγοντας για την απόκριση του θεμελίου τόσο σε στατικές όσο και δυναμικές συνθήκες. Επίσης, τα κανονικοποιημένα πάχη του επιφανειακού μη ρευστοποιήσιμου εδαφικού στρώματος (h 1 /B) και του ρευστοποιήσιμου (h /B) φαίνεται να ελέγχουν το ποσοστό μείωσης της στατικής στιφρότητας κατά τη διάρκεια της ρευστοποίησης. Αντίθετα, ο λόγος V s1 /V s φαίνεται να μην παίζει σημαντικό ρόλο. Με βάση τα αποτελέσματα της ανάλυσης με συνοριακά στοιχεία (Σχήματα.7.9 στο Κεφάλαιο ) προέκυψαν οι ακόλουθες σχέσεις παλινδρόμησης για την κατακόρυφη (K v ), οριζόντια (K h ) και λικνιστική (K r ) στατική στιφρότητα, αντίστοιχα: Για h /B =.5, Kv GB 1 V Exp.33 V s1 sliq h1 V s1 1Exp Exp.54 B V sliq (1α) Για 1 h /B, Π-8

19 Εκτενής Περίληψη Vs1.54 Kv h V sliq e GB 1 B h h1 h h1 h B B h B B e e B B (1β) K h h h GB 1 B B.45 V s V h s h V B B sliq Vsliq.3.46e.1 1e 1 e () 1 1 h h1 Kr Vs1 Vsliq Vs1 Vsliq B B.1 1 e 3 GB Vs 1 Vsliq (3) όπου Β το πλάτος του θεμελίου, h 1, G 1 και V s1 το πάχος, το μέτρο διάτμησης και η ταχύτητα διάδοσης διατμητικών κυμάτων της επιφανειακής στρώσης, αντίστοιχα, h και V sliq το πάχος και ταχύτητα διάδοσης διατμητικών κυμάτων του ρευστοποιήσιμου εδαφικού στρώματος. Τέλος, παρουσιάζεται μια απλοποιημένη αναλυτική λύση για τη στατική στιφρότητα κυκλικής θεμελίωσης σε πολύστρωτο έδαφος με χρήση των προσομοιωμάτων κώνου. Τα αποτελέσματα της προτεινόμενης λύσης βρίσκονται σε πολύ καλή συμφωνία με τα αποτελέσματα της πιο αυστηρής λύσης των συνοριακών στοιχείων για τον εδαφικό σχηματισμό χωρίς ρευστοποίηση. Παρόλα αυτά, η λύση αυτή αποκλίνει αρκετά στην περίπτωση του ρευστοποιήσιμου εδαφικού σχηματισμού. Στο Κεφάλαιο 3, εξετάζεται η χρήση του ρευστοποιήσιμου εδαφικού στρώματος ως φυσικού σεισμικού μονωτήρα βάθρου γέφυρας εδραζόμενου σε Π-9

20 Εκτενής Περίληψη επιφανειακό θεμέλιο. Σε συνέχεια προηγούμενης έρευνας (Bouckovalas et al. 14a,b, Mylonakis et al. 14) και βάσει των αποτελεσμάτων που παρουσιάστηκαν στο Κεφάλαιο, εξετάζεται η ιδέα της εκμετάλλευσης του φυσικού, εν δυνάμει ρευστοποιήσιμου εδαφικού στρώματος, μετά από μερική εξυγίανση της επιφανειακής στρώσης («κρούστας») του εδαφικού σχηματισμού, για σεισμική προστασία των κατασκευών μέσω αναλύσεων στο πεδίο της συχνότητας και του χρόνου, με τη χρήση του τροποποιημένου λογισμικού SFIAB (Mylonakis et al. ). Πιο συγκεκριμένα εξετάζεται η επίδραση της ρευστοποίησης στην ιδιοπερίοδο ταλάντωσης και στη συνολική απόσβεση του συστήματος βάθρου γέφυρας-θεμελίου-εδάφους, καθώς επίσης και στη χρονοιστορία επιτάχυνσης της γέφυρας. Μέσω παραμετρικής ανάλυσης διερευνάται ο ρόλος του πάχους και της στιφρότητας της βελτιωμένης επιφανειακής στρώσης, καθώς και του ύψους του βάθρου στη δυναμική απόκριση της γέφυρας. Επιπρόσθετα παρουσιάζεται κλειστή αναλυτική λύση για τον υπολογισμό της θεμελιώδους ιδιοπεριόδου τρίστρωτου εδαφικού σχηματισμού. Η προτεινόμενη λύση βασίζεται στην υπόθεση μιας κατάλληλης συνάρτησης σχήματος για την πρώτη ιδιομορφή του εδαφικού σχηματισμού και στη συνέχεια στον υπολογισμό του πηλίκου του Rayleigh. Στο πλαίσιο της ισοδύναμης γραμμικής ελαστοδυναμικής ανάλυσης, η συγκεκριμένη λύση επιτρέπει τη διερεύνηση της επίδρασης της ρευστοποίησης στην ιδιοπερίοδο του εδαφικού σχηματισμού και επιβεβαιώνει τα αποτελέσματα που προκύπτουν από τις αναλύσεις μέσω του SFIAB. Με χρήση της γνωστής αναλυτικής λύσης των Veletsos & Meek (1974) και τα αποτελέσματα που παρουσιάστηκαν στο Κεφάλαιο, υπολογίζεται η ιδιοπερίοδος του συστήματος βάθρου-θεμελίου-ρευστοποιήσιμου εδάφους. Για τα αποτελέσματα που παρουσιάζονται σε μορφή γραφημάτων και πινάκων, παρατίθεται λεπτομερής σχολιασμός. Π-1

21 Εκτενής Περίληψη Τα κύρια συμπεράσματα αυτού του κεφαλαίου είναι: Για να μελετηθεί η επίδραση της ρευστοποίησης στη δυναμική απόκριση της γέφυρας, απαιτούνται δύο ξεχωριστές αναλύσεις, πριν και κατά τη διάρκεια εκδήλωσης της ρευστοποίησης. Τα αποτελέσματα στο πεδίο της συχνότητας δεν καταδεικνύουν κάποια θεαματική αύξηση στην ιδιοπερίοδο του συστήματος, όπως ίσως αναμενόταν, αλλά αντίθετα παρατηρείται μια αξιοσημείωτη μείωση στο συντελεστή ενίσχυσης. Τα αποτελέσματα στο πεδίο του χρόνου, καταδεικνύουν σημαντική μείωση τόσο στην κίνηση του ελεύθερου πεδίου όσο και στην κίνηση της γέφυρας. Ενδεικτικά αποτελέσματα παρατίθενται στο Σχήμα 3. Στην περίπτωση των υψίκορμων βάθρων γέφυρας (H c = 1m), η εκδήλωση ρευστοποίησης επιδρά οριακά στην ιδιοπερίοδο και την απόσβεση του συστήματος, ενώ η μέγιστη αναπτυσσόμενη επιτάχυνση στη γέφυρα φαίνεται να μην μεταβάλλεται. Η δυσκαμψία του επιφανειακού μη-ρευστοποιήσιμου στρώματος παίζει σημαντικό ρόλο στη δυναμική απόκριση του συστήματος βάθρου-θεμελίουεδάφους. Ένα στιφρό επιφανειακό στρώμα φαίνεται να δρα ως πάκτωση για το βάθρο της γέφυρας και, επομένως, η θεμελιώδης ιδιοπερίοδος του συστήματος είναι πολύ κοντά στη θεμελιώδη περίοδο για πακτωμένη βάση. Παρόλα αυτά, η επιτάχυνση της γέφυρας μειώνεται εξαιτίας του «μαλακού» εδαφικού σχηματισμού και της αυξημένης συνολικής απόσβεσης του συστήματος. Μια σημαντική παράμετρος που εντοπίστηκε είναι το πάχος της επιφανειακής εδαφικής στρώσης. Παρατηρείται ότι καθώς αυξάνει το πάχος, η αύξηση στη θεμελιώδη ιδιοπερίοδο του συστήματος είναι μικρή, ενώ η μείωση στο Π-11

22 Εκτενής Περίληψη συντελεστή ενίσχυσης είναι σημαντική. Η αύξηση του πάχους του μαλακού επιφανειακού στρώματος οδηγεί σε αύξηση της θεμελιώδους ιδιοπεριόδου του εδάφους, το οποίο συνεισφέρει επιπρόσθετα στην απομείωση της σεισμικής απόκρισης της γέφυρας. Δεδομένου ότι στη δυναμική ανάλυση και σχεδιασμό των γεφυρών χρησιμοποιείται συχνά η στατική δυσκαμψία της θεμελίωσης (αντί της πραγματικής δυναμικής δυσκαμψίας που μεταβάλλεται με τη συχνότητα) κρίθηκε σκόπιμη η μελέτη της επίδρασης αυτής της απλοποίησης στην απόκριση του συστήματος. Για τον σκοπό αυτό, το σύστημα αναλύθηκε με θεώρηση μόνο της στατικής στιφρότητας του θεμελίου, ενώ επίσης αμελήθηκε η απόσβεση ακτινοβολίας και η απόσβεση υλικού θεωρήθηκε η ίδια τόσο πριν όσο και κατά τη διάρκεια της ρευστοποίησης. Προφανώς, η θεμελιώδης ιδιοπερίοδος του συστήματος δεν μεταβάλλεται, ωστόσο δεν λαμβάνεται υπόψη η ευεργετική επίδραση της αύξησης της απόσβεσης υλικού στο ρευστοποιημένο στρώμα, το οποίο οδηγεί σε πιο συντηρητικά αποτελέσματα. Παρόλα αυτά, κατά τη διάρκεια της ρευστοποίησης η κίνηση της γέφυρας μειώνεται εξαιτίας της μείωσης της κίνησης στο ελεύθερο πεδίο. Από την άλλη πλευρά, εάν το επιφανειακό στρώμα είναι πολύ στιφρό, η ρευστοποίηση φαίνεται να έχει μικρή επίδραση στην επιτάχυνση της γέφυρας. Από τις παραπάνω παρατηρήσεις προκύπτει ότι η ρευστοποίηση επιδρά στο σύστημα βάθρου-θεμελίου-εδάφους με τρεις μηχανισμούς. Ο πρώτος αφορά στην αύξηση της θεμελιώδους ιδιοπεριόδου του εδαφικού σχηματισμού, λόγω της σημαντικής αύξησης της ενδοσιμότητας του εδάφους με την εκδήλωση της ρευστοποίησης, η οποία ενδέχεται να συνοδεύεται από μείωση της σεισμικής κίνησης. Ο δεύτερος μηχανισμός αφορά στην αύξηση της Π-1

23 Εκτενής Περίληψη απόσβεσης υλικού στο ρευστοποιήμενο εδαφικό στρώμα, το οποίο επίσης συμβάλλει στην απόσβεση της σεισμικής ενέργειας και οδηγεί σε μείωση της κίνησης του ελεύθερου πεδίου. Τον τρίτο μηχανισμό αποτελεί η μεταβολή στην ιδοπερίοδο του συστήματος βάθρου-θεμελίου-εδάφους, που ενδέχεται να μεταφέρει την ιδιοπερίοδο του συστήματος μακριά από το επικίνδυνο εύρος συχνοτήτων της σεισμικής κίνησης. Επειδή, από τα αποτελέσματα, δεν παρατηρήθηκε κάποια αξιόλογη αύξηση στην ιδιοπερίοδο του συστήματος, προκύπτει ότι οι πιο σημαντικοί εμπλεκόμενοι μηχανισμοί είναι η αύξηση της θεμελιώδους ιδιοπεριόδου του εδάφους και η αύξηση στην απόσβεση υλικού της ρευστοποιημένης στρώσης. Π-13

24 Εκτενής Περίληψη Συνάρτηση μεταφοράς ως προς την κίνηση στο ελεύθερο πεδίο πριν τη ρευστοποίηση Περίοδος (sec) μετά τη ρευστοποίηση Θεμέλιο Γέφυρα Επιτάχυνση (m/sec ) 4 - Γέφυρα πριν τη ρευστοποίηση μετά τη ρευστοποίηση Χρόνος (sec) Σχήμα 3: Σύγκριση των αρμονικών συναρτήσεων μεταφοράς και της σεισμικής επιτάχυνσης της γέφυρας, στην περίπτωση πριν και μετά τη ρευστοποίηση για κοντά βάθρα γέφυρας; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3. Στο Κεφάλαιο 4, παρουσιάζονται νέες θεωρητικές λύσεις σχετικά με τη δυναμική συμπεριφορά των πασσάλων. Αρχικά, εξετάζεται το πρόβλημα της εδαφικής ανομοιογένειας στην ακτινική διεύθυνση, με θεώρηση μιας ενδόσιμης ζώνης δακτυλιοειδούς μορφής περιμετρικά του πασσάλου υπό συνθήκες επίπεδης παραμόρφωσης. Παραδοσιακά, η ενδόσιμη ζώνη εισάγεται για να προσομοιώσει την επίδραση της μη γραμμικότητας υλικού του εδάφους εξαιτίας των υψηλών τάσεων Π-14

25 Εκτενής Περίληψη που αναπτύσσονται κοντά στην περίμετρο του πασσάλου, ή τη συντελούμενη πλαστικοποίηση της περιοχής που εκδηλώνεται με ολίσθηση ή και πλήρη διαχωρισμό μεταξύ των δυο υλικών. Εναλλακτικά και στο πνεύμα της παρούσας έρευνας, η ενδόσιμη ζώνη χρησιμοποιείται για να περιγράψει την επίδραση γεωσυνθετικού υλικού για τη γεωτεχνική σεισμική μόνωση του συστήματος πασσάλου-ανωδομής. Οι Novak & Sheta (198) ήταν αυτοί που πρότειναν την εισαγωγή αυτής της ενδόσιμης περιοχής στην ανάλυση των πασσάλων και ανέπτυξαν ένα διδιάστατο προσομοίωμα επίπεδης παραμόρφωσης για τον υπολογισμό της στιφρότητάς της. Με βάση το προσομοίωμα των Novak & Sheta, αναπτύχθηκαν κλειστές αναλυτικές λύσεις για: α) τη στιφρότητα του συστήματος πασσάλου-ενδόσιμης ζώνης, λαμβάνοντας υπόψη διαφορετικές συνοριακές συνθήκες στις διεπιφάνειες που διαχωρίζουν την ενδόσιμη ζώνη από τον πάσσαλο και το εξωτερικό σύνορο και β) τα πεδία τάσεων και παραμορφώσεων. Με βάση τη θεωρία της τεχνικής μηχανικής των υλικών, αναπτύχθηκαν δύο απλοποιημένα προσομοιώματα για τον έλεγχο των παραπάνω αποτελεσμάτων. Δεύτερον, μελετώνται οι συντελεστές δυναμικής εμπέδησης (δυναμική δυσκαμψία και απόσβεση) στην κεφαλή του πασσάλου για διαφορετικούς τύπους κατακόρυφης εδαφικής ανομοιογένειας. Για την αναλυτική επίλυση του προβλήματος, υιοθετείται το ελατηριωτό προσομοίωμα δοκού τύπου Euler-Bernoulli- Winkler. Το προσομοίωμα χρησιμοποιείται σε συνδυασμό με ενεργειακές μεθόδους και συναρτήσεις σχήματος που περιγράφουν ρεαλιστικά την ελαστική γραμμή του πασσάλου. Εξάγονται πρωτότυπες κλειστές λύσεις για: α) τη δυναμική δυσκαμψία και β) την αντίστοιχη απόσβεση στην κεφαλή του πασσάλου, 3) τους συντελεστές των κατανεμημένων ελατηρίων και αποσβεστήρων Winkler, 4) το ενεργό μήκος του πασσάλου, πέρα από το οποίο ο πάσσαλος συμπεριφέρεται ως απειρομήκης. Π-15

26 Εκτενής Περίληψη Προσδιορίζονται και οι τρεις συντελεστές δυσκαμψίας σε πλευρική μετάθεση, λικνισμό και σύζευξη πλευρικής μετάθεσης-λικνισμού. Τα αποτελέσματα της προτεινόμενης αναλυτικής λύσης βρίσκονται σε εξαιρετική συμφωνία με τα διαθέσιμα αποτελέσματα της βιβλιογραφίας. Οι λύσεις παρουσιάζονται με τη μορφή αδιάστατων εξισώσεων, διαγραμμάτων και πινάκων για την καλύτερη εποπτεία του προβλήματος και την εφαρμογή στο σχεδιασμό. Επίσης, για την καλύτερη κατανόηση της προτεινόμενης μεθοδολογίας παρουσιάζεται ένα εκτενές παράδειγμα εφαρμογής. Τρίτον, εξετάζεται αναλυτικά το σύνθετο πρόβλημα της κινηματικής αλληλεπίδρασης εδάφους-πασσάλου-κατασκευής. Συγκεκριμένα, προτείνεται μια απλή εύχρηστη λύση για την τροποποίηση της κίνησης του ελεύθερου πεδίου, η οποία αναπαριστάται μέσω φασμάτων απόκρισης, προκειμένου να ληφθεί υπόψη η κινηματική αλληλεπίδραση. Όλες οι προτεινόμενες λύσεις παρουσιάζονται με όσο το δυνατό απλούστερο τρόπο, με γνώμονα τις πρακτικές εφαρμογές, και μπορούν να εφαρμοστούν σε ρεαλιστικές συνθήκες. Τα κυριότερα συμπεράσματα αυτού του κεφαλαίου συνοψίζονται ως εξής: Εξήχθησαν κλειστές αναλυτικές λύσεις για το πρόβλημα της αλληλεπίδρασης πασσάλου-ενδόσιμης δακτυλιοειδούς ζώνης υπό πλευρική φόρτιση, με θεώρηση επίπεδης παραμόρφωσης και διαφορετικών συνοριακών συνθηκών αναφορικά με τη τραχύτητα στις διεπιφάνειες. Η γραφική απεικόνιση των αποτελεσμάτων δίνεται στο Σχήμα 4 της παρούσας περίληψης. Η δυσκαμψία του σύστηματος μειώνεται με την αύξηση του πάχους της χαλαρής ζώνης t/d, ενώ στις ακραίες περιπτώσεις t και t, η επίδραση των συνοριακών συνθηκών διάτμησης μηδενίζεται, και όλες οι λύσεις τείνουν στις ακόλουθες απλές εκφράσεις: Π-16

27 Εκτενής Περίληψη k 8 G1 v 3 4 vt/ r, k 8 G1 v 3 4vln b/ r (4α,β) Η λογαριθμική μεταβολή της στιφρότητας με το πάχος παρεμβλήματος στη σχέση (4β) δεν πρέπει να εκπλήσσει καθώς είναι εγγενής σε προβλήματα διδιάστατης ελαστικότητας. Σημειώνεται επίσης οτι η δεύτερη εξίσωση προκύπτει με ολοκλήρωση της πρώτης. Η δυσκαμψία αυξάνεται με την αύξηση του λόγου Poisson και μεγιστοποιείται για ασυμπίεστο υλικό (v=1/), ενώ ελαχιστοποιείται για v=. Η συγκεκριμένη συμπεριφορά είναι αναμενόμενη δεδομένων των ογκομετρικών περιορισμών που επιβάλλουν τα δύο κυκλικά σύνορα. Τα αποτελέσματα του προγράμματος πεπερασμένων στοιχείων ANSYS βρίσκονται σε εξαιρετική συμφωνία με αυτά της αναλυτικής λύσης, ακόμα και στην περίπτωση σχεδόν ασυμπίεστου υλικού (ν =.49). Τα πεδία τάσεων και παραμορφώσεων είναι εύκολο να προσδιοριστούν σε κάθε περίπτωση, με αντικατάσταση των συντελεστών A i στις Εξισώσεις (4.1)-(4.15) του Κεφαλαίου 4 και αποτελούν συναρτήσεις της επιβαλλόμενης οριζόντιας μετακίνησης, u. Αναπτύσσονται κλειστές αναλυτικές λύσεις για τη δυσκαμψία και απόσβεση στην κεφαλή του πασσάλου για φραγμένη και μη φραγμένη συνεχώς μεταβαλλόμενη ανομοιογένεια με το βάθος, έτσι όπως περιγράφεται από τις Εξισώσεις (4.41) και (4.46), καθώς και για πολύστρωτο εδαφικό προφίλ. Τα αποτελέσματα δίνονται σε αδιάστατη μορφή και περιλαμβάνουν τη σχετική συνεισφορά του πασσάλου και του εδάφους στη συνολική στιφρότητα του Π-17

28 Εκτενής Περίληψη συστήματος, κάτι το οποίο δεν είναι εφικτό με τη χρήση αριθμητικών λύσεων όπως τα πεπερασμένα στοιχεία. Σε αντίθεση με προηγούμενες προσεγγιστικές λύσεις στη βιβλιογραφία, όπου προσδιορίζεται ένας περιορισμένος αριθμός συντελεστών δυναμικής δυσκαμψίας, η παρούσα έρευνα καλύπτει και τους έξι συντελεστές (K hh, K rr, K hr, C hh, C rr, C hr ) για όλα τα εξεταζόμενα εδαφικά προφίλ. Βάσει του νέου διδιάστατου προσομοιώματος των απειροστά μικρών τομέων αναπτύχθηκε λύση για την κατανεμημένη απόσβεση ακτινοβολίας Winkler. Η προτεινόμενη προσεγγιστική εξίσωση δίνεται από τη σχέση: 1 c r V s.5.8 a dsvs V c.4 (5) η οποία ισχύει στην περιοχή.1 < a 1. Αναπτύχθηκε αναλυτική λύση για τις τιμές των κατανεμημένων ελατηρίων Winkler, με χρήση και επέκταση του τρισδιάστατου μοντέλου του Mylonakis (1). Σε ομοιογενές έδαφος, ο αδιάστατος συντελεστής Winkler λαμβάνει τιμές μεταξύ 1 E s (όπου E s το μέτρο ελαστικότητας του εδάφους) για πασσάλους ελεύθερους και πακτωμένους στην κεφαλή, ενώ σε ανομοιογενές έδαφος λαμβάνει τιμές μεταξύ 1 E sd (όπου E sd το μέτρο ελαστικότητας σε βάθος ίσο με μία διάμετρο πασσάλου) για πασσάλους πακτωμένους στην κεφαλή και 1.53 E sd για πασσάλους ελεύθερους να στραφούν στην κεφαλή (Σχήμα 5). Επίσης, αναπτύχθηκαν αντίστοιχες λύσεις για πασσάλους μερικώς περιορισμένους στην κεφαλή καθώς και για κινηματική φόρτιση. Προτάθηκε θεωρητική λύση για τον προσδιορισμό του ενεργού μήκους πασσάλου, σύμφωνα με την Εξίσωση (6) και τον Πίνακα 1, για διαφορετικά εδαφικά προφίλ. Η λύση εμπεριέχει μία παράμετρο σφάλματος (ε tol ) βάσει της Π-18

29 Εκτενής Περίληψη οποίας ερμηνεύονται οι διαφορές που παρατηρούνται μεταξύ των λύσεων της βιβλιογραφίας. Για τιμές της σχετικής στιφρότητας πασσάλου-εδάφους E p /E sd που κυμαίνονται μεταξύ 1 και 1 3, το ενεργό μήκος πασσάλου L a λαμβάνει τιμές μεταξύ 5 και 13d. Η μεγαλύτερη τιμή του L a προκύπτει για ομοιογενές έδαφος και η μικρότερη για έδαφος με γραμμικά μεταβαλλόμενη στιφρότητα με το βάθος. a L p sd n L L d E E (6) Πίνακας 1: Παράμετροι για το ενεργό μήκος πασσάλου Εδαφικό προφίλ χ 1 χ χ L n L ε = 1 ε = 1 3 Ομοιογενές Εκθετικό Παραβολικό Γραμμικό Τα αποτελέσματα για τη δυσκαμψία και απόσβεση στην κεφαλή του πασσάλου βρίσκονται σε εξαιρετική συμφωνία με αποτελέσματα αριθμητικών αναλύσεων ή εμπειρικών σχέσεων της βιβλιογραφίας. Σχετικά με τους συντελεστές δυσκαμψίας προτείνεται ο τρόπος γραφής τους σύμφωνα με την Εξίσωση (7). Οι αδιάστατοι συντελεστές S ij υποδηλώνουν ότι στην πλευρική μετάθεση το έδαφος συνεισφέρει στη συνολική στιφρότητα μεταξύ 1 με 3 φορές την αντίστοιχη συνεισφορά του πασσάλου. Για τις άλλες δύο μορφές ταλάντωσης, η συνεισφορά του εδάφους είναι μικρότερη, και κυμαίνεται Π-19

30 Εκτενής Περίληψη μεταξύ.4 και 1 στον όρο σύζευξης πλευρικής μετάθεσης-λικνισμού και.17 με.34 στον όρο του λικνισμού. 3 K 3 hh EpI p 1 Shh, Krr EpI p 1 Srr, Khr EpI p 1 Shr (7) Η προτεινόμενη μεθοδολογία μπορεί να εφαρμοστεί με τη βοήθεια απλών υπολογισμών και χρήση υπολογιστή τσέπης, ή φύλλα υπολογιστή/αριθμητικές πλατφόρμες και, ως εκ τούτου, μπορεί να χρησιμοποιηθεί σε υπολογισμούς ρουτίνας για το σχεδιασμό πασσάλων σε πλευρική δυναμική φόρτιση. Προτείνεται η Εξίσωση (8) για την τροποποίηση της φασματικής απόκρισης του ελεύθερου πεδίου, προκειμένου να ληφθεί υπόψη η κινηματική αλληλεπίδραση. 4 1 H 1 4 a d 4 eff c d a eff (8) Αποδεικνύεται ότι η στροφική κινηματική συνιστώσα είναι σημαντική για την δυναμική απόκριση συστημάτων που θεμελιώνονται σε μεμονωμένο πάσσαλο, και η παράλειψή της ενδέχεται να οδηγήσει σε εσφαλμένα αποτελέσματα. Π-

31 Εκτενής Περίληψη FEM τραχιά - τραχιά λεία - λεία τραχιά - λεία λεία - τραχιά τραχιά - τραχιά λεία - λεία τραχιά - λεία λεία - τραχιά Αδιάστατη δυσκαμψία, k/g ν =.1 ν =. ν =.3 ν =.4 ν =.45,,5 1,,,5 1, Αδιάστατο πάχος, t/d ν =.49,,5 1, Σχήμα 4: Επίδραση συνοριακών συνθηκών στη δυσκαμψία του συστήματος τεμάχους πασσάλου ελαστικής ζώνης. Σύγκριση με αποτελέσματα πεπερασμένων στοιχείων (ANSYS). Συντελεστής Winkler, δ Προτεινόμενη λύση Syngros (4) Γραμμικό Παραβολικό Πάκτωση στην κεφαλή 1. Ομοιογενές Ελευθερία στην κεφαλή Γραμμικό Παραβολικό Ομοιογενές E p /E sd E p /E sd Σχήμα 5: Αδιάστατος συντελεστής Winkler ως συνάρτηση της σχετικής στιφρότητας πασσάλου-εδάφους για εύκαμπτους πασσάλους με πάκτωση ή ελευθερία στροφής στην κεφαλή, για διαφορετικά εδαφικά προφίλ; v =.4. Π-1

32 Εκτενής Περίληψη 3 5 L a /d 15 1 L a /d προτεινόμενη λύση Gazetas (1991) Syngros (4) Fleming et al (199) Budhu & Davies (1987) Di Laora & Rovithis (15) Ομοιογενές προφίλ Γραμμικό ( =, n = ) ε tol =1 3 1 ε tol =1 5 Παραβολικό (n = ) Εκθετικό (b =, q = ) E p /E sd E p /E sd Σχήμα 6: Σύγκριση της προτεινόμενης λύσης για το ενεργό μήκος πασσάλου L a με αποτελέσματα από τη βιβλιογραφία, για διαφορετικά εδαφικά προφίλ. Στο Κεφάλαιο 5 προτείνεται μια πρωτοποριακή μέθοδος για τη γεωτεχνική σεισμική μόνωση βάθρου γέφυρας, εδραζόμενου σε μεμονωμένο πάσσαλο, με χρήση ελαστικού παρεμβλήματος, όπως π.χ γεωαφρού EPS, γύρω από το άνω τμήμα του πασσάλου. Βάσει προηγούμενης έρευνας (Papastylianou 1) και των όσων αναπτύχθηκαν στο Κεφάλαιο 4, παρουσιάζεται μια ολοκληρωμένη αναλυτική μεθοδολογία για τη μελέτη των δυναμικών χαρακτηριστικών του συστήματος βάθρου γέφυρας-πασσάλου-παρεμβλήματος-εδάφους. Αποδεικνύεται ότι το ελαστικό παρέμβλημα γύρω από το άνω τμήμα του πασσάλου λειτουργεί ως μηχανισμός μόνωσης, αυξάνοντας την ιδιοπερίοδο και μεταβάλλοντας την συνολική απόσβεση του συστήματος. Επιπλέον, πραγματοποιήθηκαν αναλύσεις στο πεδίο της συχνότητας και του χρόνου, μέσω κατάλληλου λογισμικού, για τη διερεύνηση της επίδρασης του παρεμβλήματος στη δυναμική απόκριση του συστήματος βάθρου γέφυρας-πασσάλου- Π-

33 Εκτενής Περίληψη εδάφους. Τα σημαντικότερα αποτελέσματα αυτής της έρευνας συνοψίζονται ακολούθως: Εξήχθησαν κλειστές αναλυτικές λύσεις για: α) τη συνολική δυσκαμψία του συστήματος βάθρου γέφυρας-πασσάλου-παρεμβλήματος-εδάφους μέσω του δυναμικού προσομοιώματος Winkler για την αλληλεπίδραση πασσάλουεδάφους, β) την ιδιοπερίοδο του συστήματος και γ) τη συνολική απόσβεση. Παρατηρείται αύξηση στην ιδιοπερίοδο του συστήματος όσο ο λόγος της σχετικής στιφρότητας παρεμβλήματος εδάφους E inc / E s μειώνεται (Σχήμα 7). Η αύξηση του αδιάστατου πάχους του παρεμβλήματος t / d οδηγεί σε αύξηση της ιδιοπεριόδου του συστήματος. Η αύξηση του αδιάστατου μήκους του παρεμβλήματος D e / L a δεν επιφέρει σημαντική αλλαγή στην ιδιοπερίοδο του συστήματος. Είναι αξιοσημείωτο όμως, ότι ακόμη και η ύπαρξη ενός μικρού τμήματος (D e / L a =.5) προκαλεί αλλαγή στην ιδιοπερίοδο. Αναφορικά με το ύψος του βάθρου, για κοντά βάθρα παρατηρείται μεγάλη αύξηση στην ιδιοπερίοδο, ενώ για υψίκορμες η μεταβολή είναι μικρότερη. Επίσης, για σταθερά γεωμετρικά χαρακτηριστικά παρεμβλήματος, η ιδιοπερίοδος του συστήματος αυξάνεται με την αύξηση της σχετικής στιφρότητας πασσάλου εδάφους (E p / E s ). Αναφορικά με τη συνολική απόσβεση του συστήματος, παρατηρήθηκε ότι αυτή εξαρτάται σημαντικά από την απόσβεση του παρεμβλήματος και αναλόγως μπορεί να αυξηθεί ή να μειωθεί. Η σύγκριση των αποτελεσμάτων της αναλυτικής λύσης με τα αποτελέσματα της αριθμητικής ανάλυσης μέσω του λογισμικού SPIAB είναι πολύ ικανοποιητική. Π-3

34 Εκτενής Περίληψη Μια σειρά από μονοβάθμιους ταλαντωτές υποβάλλονται σε οκτώ διαφορετικές σεισμικές διεγέρσεις για τη διερεύνηση της επίδρασης του παρεμβλήματος στη σεισμική τους απόκριση. Τα αποτελέσματα παρουσιάζονται σε μορφή αδιάστατων λόγων, S A (T ) / S A (T ) για τη τέμνουσα βάσης της κατασκευής, και M max (T ) / M max (T ) για τη μέγιστη καμπτική ροπή που ασκείται στο πάσσαλο, σαν συνάρτηση των λόγων συντονισμού T / T c και T / T c. Για κοντά βάθρα γέφυρας (H / d = 5) με T / T c.5, παρατηρείται σημαντική μείωση, περίπου 3 με 7%, στην τέμνουσα βάσης και τη μέγιστη καμπτική ροπή. Για T / T c <.5, η τέμνουσα βάσης αυξάνεται (Σχήμα 8). Για T / T c.5, η αύξηση του αδιάστατου πάχους του γεωαφρού t / d μειώνει περαιτέρω τις σεισμικές δράσεις ενώ, για T / T c <.5, οδηγεί σε αύξηση των σεισμικών δράσεων. Για υψίκορμα βάθρα (H / d = 1) με T / T c.6, παρατηρείται μείωση στη τέμνουσα βάσης και στη μέγιστη καμπτική ροπή. Η χρήση του παρεμβλήματος οδηγεί σε σημαντική μείωση της μέγιστης καμπτικής ροπής και αλλάζει την κατανομή των καμπτικών ροπών κατά μήκος του πασσάλου, ενώ το σημείο εκδήλωσης της μέγιστης καμπτικής ροπής μετατοπίζεται προς τα κάτω. Αποδεικνύεται ότι εάν το παρέμβλημα έχει μεγάλο λόγο απόσβεσης τότε η μείωση των σεισμικών δυνάμεων και των καμπτικών ροπών είναι οριακά μεγαλύτερη, ακόμη και για υψίκορμες κατασκευές. Διερευνάται επίσης η επίδραση της κινηματικής αλληλεπίδρασης, και αποδεικνύεται ότι η στροφική κινηματική συνιστώσα είναι σημανική στην περίπτωση του μεμονωμένου πασσάλου και η παράλειψή της ενδέχεται να Π-4

35 Εκτενής Περίληψη οδηγήσει σε υποεκτίμηση της τέμνουσας βάσης και των καμπτικών ροπών του πασσάλου, ειδικά για στιφρούς ταλαντωτές με χαμηλή τιμή του λόγου T / T c. Σχήμα 7: Ιδιοπερίοδος και απόσβεση συστήματος με αναφορά στο σύστημα με πακτωμένη βάση ως συνάρτηση της σχετικής στιφρότητας παρεμβλήματοςεδάφους E inc / E s ; β inc = 1%. Π-5

36 Εκτενής Περίληψη ~ S A ( T ) / S A ( T ) > ~ S A ( T ) / S A ( T ) > ~ S A ( T ) / S A ( T ) > t / d =.5 D e / L a =.5 t / d =.5 D e / L a = 1 t / d = 1 D e / L a = ~ M max ( T ) / M max ( T ) > ~ M max ( T ) / M max ( T ) > ~ M max ( T ) / M max ( T ) > H / d = 5, β inc =.1.5 E p / E s = 1 3, E inc / E s = T / T c T / T c Takatori 1995 Aegio 1995 Lefkada 3 El Centro 194 Northridge (Rinaldi) 1994 Kocaeli 1999 Northridge (Castaic) 1994 Loma Prieta 1989 Σχήμα 8: Τέμνουσα βάσης και μέγιστη καμπτική ροπή πασσάλου για κοντά βάθρα (H / d = 5) σαν συνάρτηση του λόγου συντονισμού T / T c ; β inc = 1%. Π-6

37 Table of Contents Chapter Introduction Overview of problem Objectives of research Background and literature review Thesis outline Chapter D Dynamic impedances of surface footings on liquefiable soil Introduction Problem definition Numerical results Results for static stiffness Results for dynamic impedance functions Comparison with BEM analyses Concluding remarks Parametric study by means of rigorous boundary element analysis Convergence and accuracy of BEM analysis Results for vertical static stiffness of rigid square footing on 3-layer liquefiable soil Results for horizontal static stiffness of rigid square footing on 3-layer liquefiable soil Results for rocking static stiffness of rigid square footing on 3-layer liquefiable soil Simplified analytical solution for static stiffness of circular foundation on multi-layer soil Conclusions i

38 Table of Contents Chapter Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil Introduction Bridge design concept proposed by Bouckovalas et al (14a,b) Parametric investigation of bridge pier seismic response on liquefiable soil Problem description Numerical results Application to squat bridge piers Application to tall bridge piers Influence of the stiffness of the surface non-liquefiable crust Influence of the thickness of surface non-liquefiable crust Bridge pier seismic response in view of SSI considering only static stiffness Verification of results via theoretical investigation Fundamental frequency of a three-layer soil deposit Fundamental period of the pier-foundation-soil system Conclusions Chapter Pile dynamics: some new theoretical solutions Introduction Horizontal soil reaction of a cylindrical pile segment with a soft zone Introductory remarks Problem definition Stiffness coefficient of the system Strength-of-materials modeling Results Importance of three-dimensional effects Concluding remarks Horizontal stiffness and damping of piles in inhomogeneous soil Problem definition ii

39 Table of Contents 4.3. Soil description Radiation damping model Pile stiffness model Winkler spring stiffness model Active pile length Stiffness and damping coefficients Application example Concluding remarks Kinematic modification factor Chapter Geotechnical isolation of bridges using elastic inclusions around piles Introduction Problem description Proposed solution Stiffness and damping coefficients of a pile enhanced with inclusion Vibrational properties of the pier-pile-inclusion-soil system Alternative derivation Effect of radiation damping and kinematic interaction Results of parametric study Effective period and damping Outline of method of analysis (computer code SPIAB) Steady-state response Time-domain response Simplified analytical procedure Outline of method Determination of inertial forces based on the response spectrum Determination of pile maximum bending moment Replacement oscillator and physical interpretation of wave parameter Conclusions iii

40 Table of Contents Chapter Conclusions, Limitations and Recommendations for future research Conclusions Limitations and Recommendations for future research References... R-1 APPENDIX A... A-1 APPENDIX B... B-1 APPENDIX C... C-1 iv

41 List of Figures FIGURE 1.1 Isolation systems investigated. 1-6 FIGURE 1. FIGURE 1.3 FIGURE 1.4 Shaking table apparatus testing geomembranes as base isolation (Yegian and Lahlaf, 199). 1-7 Smooth synthetic liner underneath the structure operates as foundation isolation (Yegian and Kadakal, 4). 1-8 Classification of seismic isolation systems after Tsang (9). 1-9 FIGURE 1.5 Isolation mechanism using liquefaction (Tang et al, 1991). 1-1 FIGURE.1 a) Physical interpretation of dynamic stiffness, b) Problem definition. -5 FIGURE. Post-liquefied vertical dynamic impedance coefficients of square footing normalized with the corresponding preliquefied impedance coefficients; Effect of a) thickness of surface crust, b) thickness of liquefiable soil layer, c) shear wave velocity ratio; h 1 /B =.5, h /B = 1, V s1 /V s = /3. -1 FIGURE.3 FIGURE.4 FIGURE.5 FIGURE.6 Post-liquefied horizontal dynamic impedance coefficients of square footing normalized with the corresponding preliquefied impedance coefficients; Effect of a) thickness of surface crust, b) thickness of liquefiable soil layer, c) shear wave velocity ratio; h 1 /B =.5, h /B = 1, V s1 /V s = / Post-liquefied rocking dynamic impedance coefficients of square footing normalized with the corresponding preliquefied impedance coefficients; Effect of a) thickness of surface crust, b) thickness of liquefiable soil layer, c) shear wave velocity ratio; h 1 /B =.5, h /B = 1, V s1 /V s = / Comparison of vertical and horizontal dynamic impedance functions (BEM vs Conan); h 1 /B = 1, h /B = 1, V s1 /V s = Comparison of vertical, horizontal and rocking dynamic stiffness coefficients of a square footing resting on a twolayer soil profile. -4 FIGURE.7 Effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized vertical static coefficient and results -7 v

42 List of Figures obtained from Eq. (.6) using regression analysis. FIGURE.8 FIGURE.9 FIGURE.1 FIGURE.11 FIGURE.1 FIGURE 3.1 FIGURE 3. FIGURE 3.3 FIGURE 3.4 FIGURE 3.5 FIGURE 3.6 Effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized horizontal static coefficient and results obtained from Eq. (.7) using regression analysis. -3 Effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized rocking static coefficient and results obtained from Eq. (.8) using regression analysis. -33 Cone model utilized for estimating vertical and horizontal static stiffness of a circular rigid foundation on multi-layer soil. -35 Cone models for various modes of vibration (Wolf & Deeks 4). -35 Cone model utilized for estimating rocking static stiffness of a circular rigid foundation on multi-layer soil. -38 Outcrop seismic excitation response spectra vs. the EC8 design spectrum for soil type A and PGA=.3g (left), and corresponding elastic response spectra at the liquefied ground surface (right) (Sextos et al. 14). 3-5 Relationship between the shear wave propagation velocity reduction ratio V sliq / V s and the factor of safety FS L (as proposed by Bouckovalas et al. 14b) 3-7 Comparison between cone model results and lumpedparameter approximation for the vertical oscillation mode (h 1 /B =, h /B =.5, V s1 /V s = /3): a) pre-liquefaction, b) post-liquefaction. 3-1 Bridge pier supported on spread footing on liquefiable soil with improved non-liquefiable surface crust Acceleration time history and five and ten percent damped spectra Harmonic steady-state transfer function for pre-liquefaction case and squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = / FIGURE 3.7 Harmonic steady-state transfer function for postliquefaction case and squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = FIGURE 3.8 Comparison of harmonic transfer functions in pre- and post- liquefaction case for squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4) vi

43 List of Figures FIGURE 3.9 FIGURE 3.1 FIGURE 3.11 FIGURE 3.1 FIGURE 3.13 FIGURE 3.14 FIGURE 3.15 FIGURE 3.16 FIGURE 3.17 FIGURE 3.18 FIGURE 3.19 Comparison of acceleration histories in pre- and postliquefaction case for squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). 3- Comparison of harmonic transfer functions in pre- and post- liquefaction case; H c = 1 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). 3- Comparison of bridge acceleration history in pre- and postliquefaction case; H c = 1 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). 3- Comparison of harmonic transfer functions in pre- and post- liquefaction case; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = 5/3 (1). 3-4 Comparison of bridge acceleration histories in pre- and post- liquefaction case; H = 5 m, B = 7 m, h1 / B =.5, h / B = 1, Vs1 / Vs = 5/3 (1). 3-4 Comparison of harmonic transfer functions in pre- and post- liquefaction case; H c = 5 m, B = 7 m, h 1 / B = 1, h / B = 1, V s1 / V s = /3 (4). 3-6 Comparison of bridge acceleration histories in pre- and post- liquefaction case; H c = 5 m, B = 7 m, h 1 / B = 1, h / B = 1, V s1 / V s = /3 (4). 3-6 Comparison of harmonic transfer functions in pre- and post- liquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = /3 (4). 3-8 Comparison of bridge acceleration history in pre- and postliquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = /3 (4). 3-9 Comparison of harmonic transfer functions in pre- and post- liquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = 5/3 (1). 3-3 Comparison of bridge acceleration history in pre- and postliquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = 5/3 (1). 3-3 FIGURE 3. Inhomogeneous three-layer soil deposit over a rigid base FIGURE 4.1 Problem considered and modeling approach 4-5 vii

44 List of Figures FIGURE 4. Problem geometry and parameter definition 4-6 FIGURE 4.3 FIGURE 4.4 FIGURE 4.5 FIGURE 4.6 FIGURE 4.7 FIGURE 4.8 FIGURE 4.9 FIGURE 4.1 FIGURE 4.11 FIGURE 4.1 FIGURE 4.13 Simple mechanistic analog models based on strength-ofmaterials theory 4-1 Magnitudes of stresses and strains as functions of radial distance from pile periphery and thickness of elastic material for rough inner and outer interfaces (case a); v =1/ Variation of stresses in elastic zone: comparison of predictions of the elasticity solution against those of two analog models based on strength-of-materials theory for the case of two rough interfaces (case a); v =1/ Magnitudes of stresses as functions of radial distance from pile periphery and Poisson s ratio, for rough inner and outer interfaces (case a); t/d= Effect of boundary conditions on variation of stresses and strains in the elastic medium for different boundary conditions at the inner and outer interfaces; t/d=.5, v =1/ Effect of boundary conditions on system stiffness as function of thickness and Poisson s ratio System stiffness: comparison of predictions of the elasticity solution against those of two analog models based on strength-of-materials theory for rough inner and outer interfaces (case a) a) Problem definition and active pile length, L a ; b) Unitary shape function for pile deflection due to unit head displacement under zero rotation, χ(z); c) Corresponding shape function due to unit head rotation under zero displacement, φ(z); d) Corresponding shape function due to unit head force under zero head moment, ψ(z). 4-8 Variation of soil stiffness with depth: (a) profile with unbounded parabolic increase in stiffness according to Eq. (4.41); (b) profile with bounded exponential increase in stiffness according to Eq.(4.46); (c) general multi-layer profile 4-31 Radiation damping models (modified from Gazetas & Dobry 1984) Comparison of Winkler radiation dashpot coefficient obtained with the proposed model versus results from the literature, for two values of Poisson s ratio viii

45 List of Figures FIGURE 4.14a FIGURE 4.14b FIGURE 4.15 FIGURE 4.16 FIGURE 4.17 FIGURE 4.18 FIGURE 4.19 FIGURE 4. FIGURE 4.1 FIGURE 4. FIGURE 4.3 FIGURE 4.4 FIGURE 4.5 FIGURE 4.6 Shape functions utilized in the solutions for the Winkler spring stiffness model for inertial problem. 4-4 Shape functions utilized in the solutions for the Winkler spring stiffness model for kinematic problem Winkler modulus versus pile-soil stiffness contrast for long free- and fixed-head piles in homogeneous soil; v = Normalized Winkler modulus versus pile-soil stiffness contrast, for long fixed- and free-head piles in different soil profiles; v = Normalized Winkler modulus versus pile-soil stiffness contrast, for long free-head piles subjected to an eccentric horizontal load; v = Normalized Winkler modulus versus pile-soil stiffness contrast, for long pile with its head elastically restrained against rotation, subjected to an eccentric horizontal load; v = Winkler modulus δ for homogeneous soil versus a) the dimensionless frequency a and b) the dimensionless frequency a eff - influence of E p /E s ratio; v = Winkler modulus δ versus the dimensionless frequency ratio ωh t /V sh, influence of inhomogeneity parameters n and a; v = Comparison of the proposed solution for active pile length La against results from available solutions in the literature, for various soil profiles Variation of coefficients S ij and w r ij with inhomogeneity parameters a, n and soil-pile stiffness contrast E p /E sd, for a generalized parabolic profile Variation of coefficients S ij and w r ij with inhomogeneity parameters q, b and soil-pile stiffness contrast E p /E sd, for a generalized exponential profile Comparison of predictions from the proposed analysis against available solutions Comparison of results from the proposed analysis against corresponding results from the literature for linear variation of Young s modulus with depth starting from zero value at the top (n = 1, a = ); v =.4, β s = Actual multi-layer soil profile, the assumed idealized linear ix

46 List of Figures and exponential soil profile (by Gazetas & Dobry 1984) 4-67 FIGURE 4.7 FIGURE 4.8 FIGURE 4.9 FIGURE 4.3 FIGURE 5.1 FIGURE 5. FIGURE 5.3 FIGURE 5.4 FIGURE 5.5 FIGURE 5.6 FIGURE 5.7 FIGURE 5.8 FIGURE 5.9 Predictions for the damping coefficients β ij using the actual soil profile, an equivalent idealized linear profile and a corresponding exponential profile; β p = Superstructure base displacement and rotation due to kinematic interaction Effect of slenderness ratio H c /d on the kinematic modifier for homogeneous soil conditions E p /E s = Effect of pile soil stiffness contrast on the kinematic modifier for homogeneous soil conditions; H c /d = a) The bridge-pier system founded on a single pile provided with inclusion; b) The associated beam-on- Winkler-foundation model. 5-5 a) The model for dynamic pile impedances; b) The resultant of the soil reaction on depth e from pile head; c) The reduced model with only two dynamic impedances (modified by Maravas et al 7) Period elongation of the system due to soil-structureinteraction and use of pile inclusion System period and damping with reference to the fixedbase system as function of inclusion-soil stiffness contrast E f / E s ; β inc = 5%. 5- System period and damping with reference to the fixedbase system as function of inclusion-soil stiffness contrast E f / E s ; β inc = 1%. 5-1 System period and damping with reference to the flexiblebase system as function of inclusion-soil stiffness contrast E f / E s ; β inc = 5%. 5- System period and damping with reference to the flexiblebase system as function of inclusion-soil stiffness contrast E f / E s ; β inc = 1%. 5-3 Parametric harmonic steady-state results for normalized shear forces as a function of dimensionless excitation circular frequency ω exc /ω: a) E p /E s = 1 3, H/d = 1, E inc /E s =.3, D e /L a =.5, b) E p /E s = 1 3, H/d = 1, E inc /E s =.3, t/d =.5, c) E p /E s = 1 3, E inc /E s =.3, D e /L a =.5, t/d =.5, d) H/d = 1, E inc /E s =.3, D e /L a =.5, t/d = Normalized shear forces as a function of pile-soil stiffness contrast E p /E s and the dimensionless height H/d; a) β inc = x

47 List of Figures.1, b) β inc = FIGURE 5.1 The eight motions used in the time-domain analyses. 5-3 FIGURE 5.11a FIGURE 5.11b The 5 per cent-damped response spectra of the acceleration histories (Takatori, Aegio, Lafkada, El Centro): β = 5% The 5 per cent damped response spectra of the acceleration histories (Northridge-Rinaldi, Kocaeli, Northridge-Castaic, Loma Prieta): β = 5%. 5-3 FIGURE 5.1 Definition of resonance ratio FIGURE 5.13 FIGURE 5.14 FIGURE 5.15 FIGURE 5.16 FIGURE 5.17 FIGURE 5.18 FIGURE 5.19 FIGURE 5. Results of the complete analysis: base shear forces and maximum pile bending moments for squatty structures (H / d = 5) as a function of the resonance ratio T / T c ; β inc = 1% Results of the complete analysis: Base shear forces and maximum pile bending moments for tall slender structures (H / d = 1) as a function of the resonance ratio T / T c ; βinc = 1% Results of the complete analysis: Base shear forces and maximum pile bending moments for squatty structures (H / d = 5) as a function of the resonance ratio T / Tc; β inc = 1% Results of the complete analysis: Base shear forces and maximum pile bending moments for tall slender structures (H / d = 1) as a function of the resonance ratio T / T c ; β inc = 1%. 5-4 Results of the complete analysis: Base shear forces and maximum pile bending moments for H / d =5 and 1 as a function of the resonance ratio T / T c ; β inc = % Results of the complete analysis: distribution with depth of pile bending strain for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 5, t / d =.5, β inc = 1% Results of the complete analysis: distribution with depth of pile bending strain for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 1, t / d =.5, β inc = 1% Results of the complete analysis: distribution with depth of normalized pile bending moment for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 5, t / d =.5, β inc = 1% FIGURE 5.1 Results of the complete analysis: distribution with depth of normalized pile bending moment for resonance ratios T / 5-47 xi

48 List of Figures T c =.,.5, 1., 1.5; H / d = 1, t / d =.5, β inc = 1%. FIGURE 5. FIGURE 5.3 FIGURE 5.4 FIGURE 5.5 FIGURE 5.6 FIGURE 5.7 FIGURE 5.8 Results of the complete analysis: distribution with depth of pile bending moment for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 5, t / d =.5, β inc = % Results of the complete analysis: distribution with depth of pile bending moment for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 1, t / d =.5, β inc = % Results of the complete analysis: distribution with depth of normalized pile bending moment for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 5, t / d =.5, β inc = %. 5-5 Results of the complete analysis: distribution with depth of normalized pile bending moment for resonance ratios T / T c =.,.5, 1., 1.5; H / d = 1, t / d =.5, β inc = % Superposition of kinematic and inertial bending strain developed along the pile, for a system (T / T c = 1) provided with and no EPS, in case of two extreme earthquake motions Effect of neglecting radiation damping and kinematic interaction on base shear forces - normalization with the corresponding force of a fixed-base system; H / d =5, Effect of neglecting radiation damping and kinematic interaction on bending strain; H / d =5, FIGURE 5.9 Parametric analysis results using Eqs. (5.38) and (5.39); E p /E s = 1 3, H/d = 1, γ =.5 and β inc = 5% FIGURE 5.3 Parametric analysis results using Eqs (5.38) and (5.39); E g /E s =.1, t/d =.5, D e /L a =.5 and β inc = 5%. 5-6 xii

49 List of Tables TABLE.1 Properties of soil layers (values during liquefaction are given in parenthesis) -7 TABLE. Normalized static stiffness of square rigid foundations on 3- layer liquefiable soil -1 TABLE.3 TABLE.4 TABLE.5 TABLE.6 TABLE.7 TABLE.8 TABLE.9 TABLE.1 TABLE.11 TABLE 3.1 TABLE 3. Comparison of vertical and horizontal static stiffness coefficients (BEM vs Cone) -18 Comparison of normalized static stiffness Kij/(GBm/) of square footing on halfspace -3 Comparison of normalized horizontal static stiffness Kh/(GB/) of rigid square footing on a two-layer soil profile -3 Normalized vertical static stiffness of rigid square footing on liquefiable soil before and after liquefaction percentage of decrease -6 Normalized horizontal static stiffness of rigid square footing on liquefiable soil before and after liquefaction percentage of decrease -9 Normalized rocking static stiffness of rigid square footing on liquefiable soil before and after liquefaction percentage of decrease -3 Comparison of the vertical static stiffness coefficient with BEM results -4 Comparison of the horizontal static stiffness coefficient with BEM results -4 Comparison of the rocking static stiffness coefficient with BEM results -43 Analytical computation of the fundamental natural period of the pier-foundation-soil system for pre-liquefaction 3-38 Analytical computation of the fundamental natural period of the pier-foundation-soil system for post-liquefaction 3-38 TABLE 4.1 Parameters for αc for various soil profiles 4-44 TABLE 4. Parameters for active pile length in various soil profiles 4-56 xiii

50 List of Tables TABLE 5.1 Soil, pile, inclusion and structural properties used in the analysis 5-4 TABLE 5. List of selected earthquake motions and their features 5-9 TABLE 5.3 Parameters investigated in the analyses 5-33 xiv

51 List of Symbols Chapter Latin symbols A, A(z) = footing area B C ij, C i, C i c ij = width of footing = dynamic damping corresponding to the foundation resistance to mode i for motion along the mode j = pre-liquefied and post-liquefied damping value, corresponding to the foundation resistance to mode i for motion along the same mode = dimensionless dashpot coefficient D, D i = modulus of the soil stiffness depending on the oscillation mode (for each layer i) E f ij f = Young s modulus = dimensionless factor dependent solely on Poisson s ratio = excitation frequency G, G i, G = shear modulus for layer i and for small strains h i i = thickness of soil layer i = imaginary unit I, I(z) = moment of inertia of a circular footing K ij K ij K i, K i k ij, k ĩj m M P R R S ij S r T exc T s1 u = static stiffness corresponding to the foundation resistance to mode i for motion along the mode j = dynamic stiffness = pre-liquefied and post-liquefied dynamic stiffness corresponding to the foundation resistance to mode i for motion along the same mode = pre-liquefied and post-liquefied dimensionless spring coefficient = coefficient depending on oscillation mode = moment acting on the footing = axial load acting on the footing = equivalent circular radius = radius of the footing = complex dynamic impedance of the footing = degree of saturation = excitation period = fundamental natural period of the surface soil layer = horizontal displacement xv

52 List of Symbols V p, V pi V s, V si V sliq w z z z p Greek symbols a = compressional wave propagation velocity = shear wave propagation velocity = shear wave propagation velocity of the liquefied soil layer = vertical displacement = depth measured from the cone top = apex height = depth measured from the ground surface = dimensionless frequency β, β i = material damping ratio (for each layer i) θ λ λ v, λ h, λ r (λ i ) = rotation = wavelength = opening angle pertaining to vertical, horizontal and rocking mode (for each layer i) v,v i = Poisson s ratio (for each layer i) ρ, ρ i = soil mass density (for each layer i) ω Chapter 3 Latin symbols A b C c d D r EI FS L = circular excitation frequency = the amplitude of the bridge pier = inhomogeneity parameter = dimensionless parameter = constant dashpot value = column diameter = soil density = flexural rigidity of the pier column = factor safety against liquefaction G, G B = shear modulus, shear modulus at the bottom of the layer H c H H 1, H H 1, H K st k 1,,6 k M = height of the pier column = total depth of soil deposit = depth of the first and second soil layer, respectively = dimensionless depth of the first and second soil layer, respectively = stiffness of the pier column = dimensionless parameters = finite value of dynamic stiffness coefficient at infinity = degree of polynomial xvi

53 List of Symbols m b = mass of spread footing m s = mass of superstructure P, Q = polynomials r u,design = excess pore pressure ratio in the improved zone S r (a ), S s (a ) = regular and singular part of the complex dynamic stiffness S al, S anl, S apred = response spectrum acceleration T = period T st = fixed-base fundamental period of the structure T 1, T = fundamental and second natural period of the soil deposit T = fundamental natural period of the pier-foundation-soil system V s, V sb = shear wave propagation velocity at the bottom of the layer V s,o = the initial shear wave propagation velocity in the liquefiable soil layer z, z = depth and dimensionless depth measured from ground surface Greek symbols a 1, a = dimensionless parameters a PGA = dimensionless coefficient = effective damping of the pier-foundation-soil system γ, γ(z) = shear strain γ' = Buoyant unit weight ζ 1, ζ = dimensionless coefficients λ = the predominant wave length of the seismic waves propagating through the liquefiable soil layer ψ, ψ(z) = dimensionless shape function = natural circular frequency of soil deposit ω n Chapter 4 Latin symbols A i, i=1,,4 A t* a, a d B 11, B 1 b C 1, C C hh, C rr, C hr, C ij c c r, c r (z) = integration constants = hypergeometric function = dimensionless frequency & frequency at depth z=d = real part of generalized Gamma functions Γ 11 and Γ 1, respectively = the outer boundary of soft zone, dimensionless inhomogeneity parameter for exponential soil profile = constants = damping coefficients in swaying, rocking and cross swaying-rocking at pile head = constant = Winkler radiation dashpot modulus xvii

54 List of Symbols c rd c ri d dk E E s, E s (z) E s, E sd, E s E p E p I p E c, E st Ẽ p F up, F um, F θp, F θm f G = Winkler radiation dashpot modulus at depth of one pile diameter = dashpot modulus of soil layer i = pile diameter = infinitesimal amount of stiffness = Young s modulus of soft zone = soil Young s modulus = soil Young s modulus at soil surface, at depth of one pile diameter, and at infinite depth, respectively = pile Young s modulus = pile flexural stiffness = Young s modulus of concrete and steel pile, respectively = Young s modulus of an equivalent solid non hollow pile = pile flexibility coefficients = excitation frequency = shear modulus of soft zone G s, G s (z),g sh = soil shear modulus and modulus at the base of the layer (z = H t ) H H t H (), H 1 () H p1, H p h i i ij J 1, J = eccentricity of load = total thickness of soil profile = zero- and first-order Hankel functions of the second kind = hypergeometric functions = thickness of soil layer i = imaginary unit, number of soil layer in multilayered soil (subscript) = refers to different vibrational modes = imaginary and real part of hypergeometric function A 1, respectively J 3 = imaginary part of hypergeometric function A 1 J v (), Y v () = Bessel functions of the first and second kind and order v k, k(z) = Winkler spring modulus k d, k k i = Winkler spring modulus at depth z = d and at infinite depth, respectively = Winkler spring modulus of soil layer i k, k, k = stiffness of soft zone, asymptotic value in limits t and t K (), K 1 () K r K hh, K rr, K hr, K ij K p ij l m = modified Bessel function of the nd kind and order zero and one, respectively = rotational spring stiffness = pile head stiffness coefficients in swaying, rocking and cross swayingrocking at pile head = contribution to overall head stiffness of pile flexural stiffness = exponent 1, or 3 depending on oscillation mode = pile mass per unit length xviii

55 List of Symbols M Ν n n 1, n, n* n δ, n L = moment applied at pile head = number of homogeneous layers = dimensionless inhomogeneity exponent for generalized parabolic soil profile = dimensionless parameters = dimensionless coefficients o, p = arguments of generalized Gamma function P q q c Q Q 11, Q 1 r r R θ s u S ij t t w = lateral force on pile segment = dimensionless inhomogeneity exponent for exponential soil profile = proportionality coefficient = horizontal force applied at pile head = imaginary part of generalized Gamma functions Γ 11 and Γ 1, respectively = polar coordinate = pile segment radius = rocking term = dimensionless coefficient = dimensionless stiffness coefficient expressing the contribution of the restraining action of soil to the overall head stiffness = thickness of soft zone = wall thickness of hollow pile t* = argument of hypergeometric function u(z) u k (z) u(r, θ) v(r,θ) V c V s, V sd, V sh V La w p ij,w r ij Y o Y, Y(z) = pile deflection z z t,i, z b,i Greek symbols a cor a = lateral displacement function of a free head pile = deflected shape of the pile due to kinematic loading = radial component of displacement = tangential component of displacement = soil compressional wave propagation velocity = soil shear wave propagation velocity and velocity at depth z=d and z=h t = Lysmer s analog wave propagation velocity = weight factors expressing the contribution of pile material and radiation damping, respectively, to the overall damping = amplitude of motion at pile head = depth = elevation of upper (t) and lower (b) face of soil layer i = correction factor = dimensionless inhomogeneity parameter for generalized parabolic soil profile xix

56 List of Symbols a eff a c β p, β s β rd β ij β ri γ γ k (z) Γ op δ ε tol ε z ε ij, ε ij, ε c η u (z) θ θ * = dimensionless frequency parameter = dimensionless stiffness parameter = pile and soil hysteretic damping = radiation damping coefficient at depth of one pile diameter = normalized damping coefficients at pile head = dimensionless damping coefficient of soil layer i = Euler s number = distribution of shear strain within the inhomogeneous soil = generalized Gamma function = Winkler spring coefficient = tolerance parameter = vertical normal strain = strains, normalized strains, average strain = compressibility coefficient = normalized deflected shape of a free-head pile due to unit head horizontal displacement produced by a moment applied at the pile head = polar coordinate = dimensionless coefficient κ, κ * = wave numbers λ, λ(z) = Winkler wavenumber parameter (1/Length) λ d λ i, λ i (z) Λ μ v ρ s σ ij, σ ij, σ c σ z τ rz φ(z) = Winkler wavenumber parameter at depth of one pile diameter = Winkler wavenumber parameter of soil layer i = complex-valued dimensionless function of frequency = shape parameter = Poisson s ratio = dimensionless parameter = soil mass density = stresses, normalized stresses, average stress = vertical normal stress = stress component = polar angle in global reference system = normalized deflected shape of a fixed-head pile due to unit head rotation under zero displacement χ, χ(z) = normalized deflected shape of a fixed-head pile due to unit head displacement under zero rotation χ i, χ j χ δ, χ L, χ 1, χ Χ 1 (z), Χ (z) = any of the shape functions χ(z) and φ(z) = dimensionless coefficients = shape functions xx

57 List of Symbols ψ(z) ψ 1, ψ ω = normalized deflected shape of a free-head pile due to a horizontal head force = potential functions = cyclic excitation frequency Chapter 5 Latin symbols D e D(β) d E p, E s, E inc, E str e f FF acc H = depth of inclusion = damping modifier = column and pile diameter = pile, soil, inclusion & structure Young s modulus = eccentricity = natural frequency of the fixed-base oscillator = free field acceleration = height of pier column I, I p = moment of inertia of the pier column and the pile K K f K f K K hh, K rr, K hr K * hh, K * rr, K * hr K hhe, K rre, K hre k inc, k s k s = stiffness of a fixed-base bridge pier = stiffness of a rigid bridge pier on a flexible foundation = stiffness of SSI system = stiffness of the SSI-inclusion system = static stiffness atop the pile pertaining to swaying, rocking and crossswaying-rocking = complex impedances atop the pile pertaining to swaying, rocking and cross-swaying-rocking = modified dynamic impedances = inner stiffness of inclusion, outer stiffness of soil = stiffness of inclusion-soil system L, L a = pile length and pile active length m M w M(z), M max () M kin (z), M in (z) M b s hh, s rr, s hr S A () t t s, t sf = mass of superstructure = moment magnitude = bending moment and maximum pile bending moment = kinematic and inertial pile bending moment, respectively = overturning moment = dimensionless static stiffness terms pertaining to swaying, rocking and cross-swaying-rocking = maximum base shear force = thickness of inclusion = wave travel times for the soil and the fictitious medium T, T, T = fixed-base fundamental period, SSI period, SSI-inclusion period xxi

58 List of Symbols T c = dominant excitation period V, Ṽ, V (V norm ) = base shear force pertaining to fixed-base, SSI and SSI-inclusion system (normalized shear force) V b V s, V sf w z Greek symbols = base shear force (computed based on response spectrum) = shear wave propagation velocity of the soil and the fictitious medium of the superstructure = parameter referring to fixity conditions at the deck = depth measured from the ground surface β, β inc, β s, β p = material damping ratio of structure, inclusion, soil and pile, respectively β f β, β hh, β rr, β hr β hhe, β rre, β hre β r γ δ λ 1, λ, λ μ = effective damping ratio of a rigid bridge pier on a flexible foundation = effective damping ratio of SSI system and SSI-inclusion system, respectively = damping coefficients atop the pile terms pertaining to swaying, rocking and cross-swaying-rocking = modified damping coefficients = radiation damping ratio = relative mass density of the structure and the soil = Winkler spring coefficient = Winkler parameters = shape parameter v, v inc = soil and inclusion Poisson s ratio ρ s, ρ p = soil and pile mass density 1/σ = wave parameter χ 1,...,7 = dimensionless quantities xxii

59 Chapter 1 Introduction 1.1 Overview of problem Determining seismic forces imposed on piles or spread footings supporting bridge piers through the prism of soil-structure interaction (SSI) involving soil liquefaction and/or seismic isolation principles, is a significant unsolved problem in earthquake engineering. Under linear or equivalent linear conditions, the response of the soil-foundation-structure system can be viewed as the superposition of two simultaneous, yet distinct phenomena: kinematic and inertial interaction. Kinematic interaction is associated with the assumption of zero superstructure mass and results in a modification of the foundation response relative to the free-field due to the scattering of the impinging seismic waves. Inertial interaction is defined as the response of the complete soil-foundation-structure system subjected to inertial (D Alembert) forces on the superstructure developing due to kinematic interaction. Examples of bridge collapses during extreme earthquake events in the past, often demonstrate the significant role of soil in the failure. For instance, during the San Fernando earthquake of 1971, Youd and Olsen (1971) report that Subsidence of 1-1

60 Chapter 1 the approaches to highway bridges was very common in the northern San Fernando Valley and probably resulted from seismic compaction or lateral spreading of the highway fills or underlying soils. During the Loma Prieta earthquake of 1989, three major bridges (Oakland Bay Bridge, Cypress Street Viaduct and Struve Slough) suffered severe damage due to soft soil at bridge sites, which caused amplification of the ground motion. Another example is the Northridge earthquake of 1994 which caused heavy damage to highway bridges. According to Yegian et al (1995), in some cases the motion at the bridge pier was larger than the motion of the free-field (San Bernardino s Hwy I-1/15 Interchange Bridge free-field PGA of.1g was amplified to.14g on the footing). They also report that the interaction between the abutment retaining walls, the bridge deck and the piers may have caused failure in some bridges. The spectacular failure of 18 piers of Hanshin Expressway (Fukae section) in the 1995 Kobe earthquake has been attributed to the detrimental role of soil-structure interaction (Mylonakis et al 6). During Kobe earthquake, anchorages and tower foundations of the suspended bridge Akashi Kaikyo, being under construction at that time, were subjected to permanent lateral movements and rotations (Elnashai and Di Sarno 15). Last but not least, several collapses of bridges were caused by soil liquefaction during the 1964 Niigata earthquake. A particular example was the collapse of the Showa Bridge due to failure of the pile foundation owing to lateral spreading. Soil liquefaction occurred extensively around the bridge and caused lateral spreading of the surface ground as far as 1m along the Shinano River (Elnashai and Di Sarno 15 ). More recently, significant soil-structureinteraction effects were also observed in Chile during the M8.8 earthquake of 1, and in Japan during the M9. earthquake of

61 Introduction The widespread occurrence of such failures provided the impetus for extensive research to understand the role of SSI in the seismic performance of structures. The major conclusion drawn from studies conducted during over the last two decades, is that the dynamic response of structures including bridge piers depends on certain site conditions and the frequency content of seismic excitation (Mylonakis 1995; Mylonakis et al 1997; Gazetas & Mylonakis 1998; Stewart and Fenves 1998; Stewart et al 1999a,b, Nikolaou et al 1; Syngros 4; Mylonakis et al 6; Ghalibafian 6; Gerolymos et al 8). Additionally, since numerous bridge failures have occurred as a result of soil liquefaction and lateral spreading during large earthquakes, research has been oriented towards exploring the types of liquefaction-induced failure mechanisms, the impact of them on seismic performance of bridges and ways to address the problem (Cooke ; Boulanger et al 3; Brandenberg et al 5; Cubrinovski et al 9). Owing to the great importance of bridges to economy and society, various innovative, technologically advanced and expensive isolated systems are utilized to prevent damage in strong earthquakes. Isolating systems in bridge engineering have been widely incorporated in seismic design. However, their performance in the context of SSI and liquefaction requires basic research since many bridge failures have been triggered by the failure of the isolating system. Approximate analytical methods have been developed in relation with the performance of seismically isolated bridges in line with SSI (Vlassis and Spyrakos 1; Usac and Tsopelas 8). It is has been demonstrated that ignoring SSI often leads to underestimation of the response of the isolated bridge (Makris and Zhang 4), therefore SSI should always be considered in soft soil conditions regardless of bridge type (Dicleli et al. 5). Also, SSI decreases the efficiency of the lead rubber bearings (LRBs), which means 1-3

62 Chapter 1 that this type of isolation is effective when the bridge is founded on stiff soil (Wang et al. 14). Regarding liquefaction, Wang et al (14) show that the displacement of LRBs increases and the failure probability of pile foundation rise in comparison to nonliquefiable soils. Notwithstanding the extensive research and the publications, outcomes of these studies have only recently been implemented in design methods or code-type provisions (NIST 1; CalTrans 13b) while SSI effects are commonly neglected in dynamic analysis and design of bridges. Commonly, for simple bridges, elastoplastic design is performed (EC8: Design of structures for earthquake resistance. Part: Bridges) and piers are allowed to develop plastic hinges at, as possible as, accessible points for inspection and repair. For conventional design, the response spectrum analysis is utilized providing the maximum response (acceleration, velocity, displacement) of a single degree of freedom (SDOF) system having the same oscillation period and damping with that of the actual structure. The ductile behavior of the bridge is considered by means of a factor q1, where q = 1 corresponds to elastic behavior. Yet another topic of consideration is the type of bridge foundation. The usage of deep foundations for supporting bridge piers is a common practice. FHWA (14) reports that Survey results from geotechnical engineers in 44 states indicated that the average distribution of bridge foundation types considered by State Departments of Transportation (DOTs) across the United States was approximately 4 percent (%) spread footings (11.5% on soils, 1.5% on rock) and 76% deep foundations (56.5% driven piles and 19.5% drilled shafts). Recently, there is an effort to promote the use of spread footings in highway bridges due to their many advantages, among them, low cost of construction and maintenance, quick procedure, simple and flexible design 1-4

63 Introduction (FHWA 14, 1b, 6a). However, these instructions recommend avoiding the use of shallow foundations on liquefiable soils. If the use of spread footings is deemed necessary, the standard practice involves mitigation of the liquefaction hazards by improving the strength, density, and/or drainage characteristics of the entire liquefiable soil layer below the foundation, which is an expensive process. 1. Objectives of research The objective of this dissertation is to investigate the feasibility of using alternative methods for seismic protection of pile- and spread footing-supported bridge piers, which actually aim at ensuring foundation isolation. The method involves use of modern industrial materials (such as geofoam) around the upper part of a pile, for reducing the seismic forces imposed on the system, and exploitation of the existing liquefiable soil layer as a natural base isolation system for structures founded on a shallow foundation (Figure 1.1). The proposed means of seismic isolation may be classified in the realm of Geotechnical Seismic Isolation (GSI). Moreover, the seismic performance of the geotechnical seismically isolated piers is explored considering soil-structure interaction. 1-5

64 Chapter 1 (a) Proposed isolation method for pilesupported bridge pier (b) Proposed isolation method for spread footing-supported bridge pier FIGURE 1.1: Isolation systems investigated 1.3 Background and literature review In recent years, the concept of GSI has gained momentum among researchers as an alternative to conventional Structural Seismic Isolation applications. Although the idea may have existed for a long time, the term GSI was introduced by Tsang (9) to describe isolating systems associated with the soil under foundation. The first innovative idea in this field was put forth by Kavazanjian et al. (1991) and Yegian and Lahlaf (199), who worked independently and suggested the use of geosynthetic materials to provide a cost-effective frictional base isolation for structures (Figure 1.). For instance, two sheets of geomembrane, having lowinterface friction angle, underneath the structure can protect the structure from earthquake ground motion by allowing sliding between the two sheets, as an inexpensive sliding isolator. 1-6

65 Introduction FIGURE 1.: Shaking table apparatus testing geomembranes as base isolation (Yegian and Lahlaf, 199) Later, Yegian and Kadakal (4) proposed placing a smooth synthetic liner underneath the foundation of the structure to dissipate earthquake energy, as shown in Figure 1.3. To this end, a single-story model with and without foundation isolation was experimentally tested, via shaking table tests, and it was proved that the dynamic response of the model based on the synthetic liner was substantially smaller than the corresponding response of the fixed-base model. In the same spirit, Yegian and Catan (4) examined the potential of using geotextiles between soil layers, thus allowing sliding between soil and geotextile, to reduce horizontal ground motions at the expense of some increased compliance. 1-7

66 Chapter 1 FIGURE 1.3: Smooth synthetic liner underneath the structure operates as foundation isolation (Yegian and Kadakal, 4) Tsang (8) suggested soil improvement around the foundation of building structures by means of rubber-soil mixtures (RSM) for absorbing seismic energy. The use of scrap tires as the rubber material can contribute to the effective management of the stockpiles of scrap tires worldwide, which is a significant threat to the environment. Additionally, this inexpensive seismic isolation can benefit especially the developing countries having limited resources and technology. Also, Tsang (9) made an attempt to classify the conventional structural and the new geotechnical isolation systems, as depicted in Figure

67 Introduction FIGURE 1.4: Classification of seismic isolation systems after Tsang (9) Xiong et al. (1, 14) investigated experimentally, via shaking table tests, the effectiveness of GSI with rubber-soil mixtures during earthquakes and concluded that the dynamic response of the superstructure may be attenuated due to GSI. An interesting method of base isolation using natural liquefiable soil material was proposed by Tang et al (1991). To the best of author s knowledge, this is the first work in which it is demonstrated that liquefied soil can operate as an isolation material under certain circumstances. They suggested the use of isolators made of an artificial soil mixture with low relative density, and around the soil an impermeable elastic wall which keeps soil volume constant. By means of a network of pipes, the soil in the isolators is constantly saturated. The isolation mechanism was tested via shaking table tests, and the configuration of the system is illustrated in Figure 1.5. It is shown that during a strong earthquake, the soil in the isolator has lost its stiffness due 1-9

68 Chapter 1 to liquefaction, the natural period of the isolated building is shifted towards to the high range and, therefore, out of the harmful frequency range of the earthquake. These applications, although interesting from an academic view point, do not provide control of foundation settlements and tilting, therefore, do not fulfill design requirements for important structures. FIGURE 1.5: Isolation mechanism using liquefaction (Tang et al, 1991) 1.4 Thesis outline In Chapter 1, an introduction to the problem and a brief literature review, as well as an outline of the current thesis, are presented. In Chapter, the problem of determining the dynamic impedance of surface rigid foundations on liquefiable soil by means of equivalent linear approach is investigated. The problem studied is particularly complex, firstly due to occurrence of liquefaction, which is a strongly non-linear phenomenon, and secondly since the presence of a liquefiable soil layer dictates the existence of a certain stratigraphy (non-liquefiable surface crust liquefiable zone non-liquefiable base soil stratum) i.e., a multi-layer soil profile, which has not been considered in the literature. Based on recently published experimental and numerical results which demonstrate that during liquefaction shear wave propagation velocity is significantly diminished and 1-1

69 Introduction the soil material damping ratio is amplified, one may attempt to overcome the problem of non-linearity by assessing the effect of liquefaction through an equivalent linear elastic analysis by employing a well selected low value of shear wave propagation velocity and, likewise, a corresponding high value of material damping ratio for the liquefied soil layer. The dynamic impedance (vertical, horizontal and rocking) of a surface rigid foundation on a liquefiable soil profile, as it was described before, is explored parametrically by means of the computer software CONAN (Wolf & Deeks 4), which is based on wave propagation in cones. This preliminary study is followed by more rigorous 3D numerical analyses using boundary elements, software ISoBEM (Polyzos et al 1998), to check the accuracy of the above and to provide regression relations regarding the static stiffness of a foundation on liquefiable soil. The influence of problem parameters such as thickness of the surface non-liquefiable soil, thickness of the liquefiable soil layer and shear wave propagation velocity ratio of the surface crust to the liquefiable soil layer, are examined. Results are presented in terms of dimensionless graphs, tables and simple equations that provide insight into the complex physics of the problem and can be used in practice as a seismic demand mitigation measure. As a final remark, this Chapter also includes an analytical solution based on cone models for deriving the vertical, horizontal, and rocking static stiffness of a foundation on a multi-layer soil profile. The predictions of the proposed solution compare satisfactorily with BEM results; however this solution does not seem applicable, without modifications, in liquefiable soil profiles, where an extremely low value for the shear wave propagation velocity of the liquefied soil stratum is utilized. In Chapter 3, the feasibility of a new geotechnical isolation method of bridge piers on shallow foundations founded on soils susceptible to liquefaction is examined. 1-11

70 Chapter 1 Following up on previous research (Bouckovalas et al. 14a,b, Mylonakis et al. 14) and the developments presented in Chapter, the idea of exploiting the natural potentially liquefiable soil after partial remediation of the surface layer for seismic protection of structures, is further studied by means of frequency and time domain analyses via a modified version of SFIAB computer code (Mylonakis et al. ). The effect of liquefaction on the vibrational characteristics (fundamental period and effective damping) of the pier-foundation-liquefiable soil system, as well as the acceleration history of the structure are investigated. A parametric study is carried out to explore the role of thickness and stiffness of the surface remediation zone, as well as the height of the superstructure on dynamic response of the bridge pier. Furthermore, an analytical closed-form solution using the Rayleigh quotient and an approximate shape function for the first oscillation mode of soil, is derived for predicting the fundamental natural period of a three-layer soil deposit. In the context of linear elastodynamic analysis, the pertinent solution allows for a thorough investigation of the effect of liquefied layer on the fundamental properties of the site, and validates results obtained through SFIAB. Furthermore, utilizing the footing impedances on liquefiable soil presented in Chapter, the fundamental natural period of the pier-foundation-soil system is compared against the well-known analytical formula of Veletsos and Meek (1974). Results from the above investigation are presented in the form of graphs and tables and are discussed in detail. In Chapter 4, some new theoretical developments on pile dynamics are presented. First, the problem of soil inhomogeneity in the radial direction is revisited by considering an annular zone of weak material around the pile under plane strain conditions. The weak zone accounts in an approximate way for soil nonlinearity due to high stresses in the vicinity of the pile, as well as, pile separation or slippage and 1-1

71 Introduction was first proposed by Novak & Sheta (198). Based on this model, closed form solutions are obtained for: a) the stiffness of soft zone, considering different boundary conditions at the interfaces separating the annular zone from the pile and the outer material, and b) the strain and stress fields. Based on strength-of-materials considerations, two simple models are developed to compare and develop insight into results. Second, the impedance coefficients (i.e., the dynamic stiffness and damping) at the head of a laterally-loaded pile in different types of vertically inhomogeneous soil, are studied. A simple analytical formulation based on the Beam-on-Dynamic- Winkler-Foundation (BDWF) model is employed to solve the problem analytically. The model is used in conjunction with a virtual work approximation implemented by means of pertinent shape functions for the deflected shape of the pile, which are analogous to those employed in energy solutions such as the Rayleigh-Ritz method and FEM. Explicit closed-form solutions are derived for: (1) the dynamic stiffness; () the corresponding damping coefficient, at the pile head; (3) the moduli of the distributed Winkler springs and dashpots and (4) the active length beyond which the pile behaves as an infinitely long beam. Both swaying and rocking oscillation modes are considered and all associated impedance coefficients in planar oscillations (i.e., swaying, rocking, cross swaying-rocking) are determined. Results compare favorably with numerical solutions and are presented in terms of simple formulae and dimensionless graphs that provide insight into the physics of the problem. The implementation of the proposed method is illustrated with the help of a worked example. Third, the complex problem of soil-pile-structure kinematic interaction is analytically tackled. A simple solution for modifying free-field ground motions to account for kinematic interaction effects by response spectrum analysis is proposed. 1-13

72 Chapter 1 All the above solutions are presented in a simple, design-oriented way and can be used in applications. In Chapter 5, an innovative geotechnical isolation method for reducing seismic forces imposed on pile-supported bridge piers using elastic inclusions, such as geofoam material around a pile foundation, is presented. Following up on previous research (Papastylianou 1) and developments presented in Chapter 4, a complete analytical formulation for studying the vibrational properties of the pier-pileinclusion-soil system, is developed. It is shown that the elastic inclusion around the upper part of a pile foundation operates effectively as an isolation mechanism by increasing the fundamental natural period of the system and altering its effective damping. Analytical closed-form solutions are derived for: a) the overall compliance of the pier-pile-geofoam-soil system by means of a dynamic Winkler model of pilesoil interaction, b) the fundamental period and c) the overall damping of the system. The influence of problem parameters such as stiffness, thickness and depth of inclusion are examined. To investigate the seismic response of the pier-pile-geofoamsoil system, a systematic parametric study was conducted on an idealized bridge model. From this study, a substantial decrease in base shear and maximum bending moment for the common case of squat structures is observed. Finally, based on a properly modified response spectrum method, the inertial forces acting at the pile head are determined. The increase in displacements due to the presence of geofoam is acknowledged, but not investigated here. In Chapter 6, conclusions, limitations and recommendations for future research are presented. 1-14

73 Chapter 3D Dynamic impedances of surface footings on liquefiable soil.1 Introduction For non-liquefiable soil profiles, soil-structure interaction in practical applications is traditionally considered by means of lumped springs and dashpots attached to the foundation. Contrary to approximate Winkler models employed for the analysis of piles and retaining walls, these formulations can be rigorous, as the stiffness and damping coefficients are obtained from exact numerical solutions of the corresponding boundary value problems. Considering the importance of dynamic soilstructure interaction in earthquake and foundation vibration problems, a vast number of analytical and numerical solutions for determining foundation impedances for footings of different shapes, embedment and soil stratification have been developed (Wong & Luco 1985, Pais & Kausel 1988, Gazetas 1991, Meek & Wolf 199, Vrettos 1999, Mylonakis et al 6). Published results refer to the common cases of a footing resting on a halfspace, on a soft or stiff layer overlying a halfspace (Ahmad & Rupani 1999), and a soil layer over rigid bedrock. Cases involving a smooth variation of soil -1

74 Chapter properties with depth have also been considered (Guzina and Pak 1998, Vrettos 1999, Mylonakis et al 6). Clearly, such solutions are not applicable to liquefiable soils for two main reasons: a) the existence of a multi-layer soil profile (e.g. a non liquefiable surface crust followed by a liquefiable soil layer and then a non-liquefiable base stratum) with sharp impedance contrast between the layers, leading to the entrapment of seismic waves within the liquefied soil layer, and b) the mostly unknown mechanisms of seismic wave propagation within liquefied soil layers, where shear-induced dilation under extremely low effective stresses leads to significant variation in excess pore pressure and seismic wave propagation velocity, even within the same loading cycle. The aim of this study is to investigate the dynamic impedance of a rigid surface square footing resting on a liquefiable sandy soil layer, sandwiched between two stiff cohesive layers, under external harmonic loading. Results are obtained by means of elastodynamic analyses using pertinent values for the material constants. Key to analyzing a strongly non-linear phenomenon such as liquefaction using elastodynamic tools lies in the proper selection of shear wave propagation velocity during the course of liquefaction. There is ample experimental and analytical evidence that shear wave propagation velocity can be reduced to 1 3% of its initial value (Miwa & Ikeda 6, Theocharis 11) while at the same time the soil material damping ratio may increase from 5% or so to over %, in agreement with a substantial increase in shear strains due to liquefaction. Based on the above observations, one may assess the effects of liquefaction through an equivalent linear elastic analysis using a low value of shear wave propagation velocity and a corresponding high value of hysteretic damping ratio for the liquefied soil stratum. Note that in the absence of sufficient soil permeability above and below the liquefied -

75 3D Dynamic impedances of surface footings on liquefiable soil layer, this change in soil stiffness and damping may be considered to be permanent for the purposes of an earthquake dynamic analysis. As a starting point, it is briefly recalled that the static stiffness of a rigid foundation in two dimensions may be expressed by means of appropriate stiffness coefficients, K ij, corresponding to the foundation resistance to mode i for motion along mode j (i.e., along the vertical, K vv, horizontal, K hh, rocking, K rr, and torsional, K tt, degree of freedom). In the common case of combined action of a horizontal force and an overturning moment, a coupling term between horizontal and rocking stiffness K hr is also present. Nevertheless, this coupling term is significant for embedded foundations and is omitted here. Furthermore, in the case of rigid foundations and linear or equivalent-linear soil, static coefficients are functions of the soil shear modulus G, foundation width B, and soil Poisson s ratio v. For static conditions, the stiffness terms are written in the following dimensionally consistent form K ij = G B m f ij (v) (.1) where K ij stands for the force or moment along the degree of freedom i for a unit displacement or rotation along the degree of freedom j, while m = 1 for the translational degrees of freedom and m = 3 for the rotational. f ij (v) is a dimensionless factor dependent on footing geometry and soil Poisson s ratio. In evaluating Eq.(.1) for use in practical applications, the degradation of shear modulus due to increasing shear strain amplitude should be taken into account (ASCE 1, FEMA 9, NIST 1). For small strains, the shear modulus, G, can be computed as G = V s ρ (.) -3

76 Chapter where V s is the shear wave propagation velocity obtained by geophysical measurements and ρ is the soil mass density. For non-homogeneous soil, the usual practice is to compute an average effective value of V s up to a certain depth, z p, and then to correct this value according to the amplitude of deformation, as described in NIST (1). When the foundation is subjected to a set of external harmonic loads with circular excitation frequency ω along each degree of freedom, the interacting soilfoundation system is represented by springs and dashpots, with the amplitude of the associated coefficients varying with frequency, as depicted in Figure.1a. The frequency-dependence of the spring and dashpot coefficients stems from the elimination (condensation) of the infinite dynamic degrees of freedom in the soil. The dynamic impedance of foundation for each oscillation mode can thus be written in the generic form S ij (ω) = K ij (ω) + i ωc ij (ω) (.3) where K ij and C ij are the dynamic stiffness and damping, respectively, and i is the imaginary unit ( 1), which indicates a phase lag of 9 o between maximum spring force and corresponding dashpot force. It is common to describe frequency by means of the dimensionless coefficient a = ωb/v s, where B refers to a typical width of the foundation (full or half-width). This coefficient is interpreted as the ratio of B to 1/6 of wavelength λ for excitation frequency ω. Parameter a is essentially unique for halfspace conditions (where B is the only parameter carrying units of length to normalize ω), but may not be optimal in the presence of bedrock at shallow depth (Anoyatis & Mylonakis 1), or in the -4

77 3D Dynamic impedances of surface footings on liquefiable soil presence of a significantly stiffer surface soil crust as will be discussed in later sections of this Chapter. Alternatively, the dynamic impedance can be decomposed as S ij (a ) = K ij [ k ij (a ) + i a c ij (a )] (.4) where K ij is the static stiffness and k ij, c ij are dimensionless stiffness and damping coefficients, respectively, as a function of the dimensionless frequency a. The dimensionless stiffness and damping coefficients are real-valued. It is noted that, whereas the k ij coefficient may also become negative at times (e.g. in case of phase lag between excitation and response greater than 9 o under dynamic conditions), c ij is always a positive number so as to comply with thermodynamic constraints. It is mentioned that for a given foundation shape, the above dynamic impedance coefficients are functions of the Poisson s ratio v of soil, the dimensionless frequency a, and the material damping ratio β. (a) Pe iωt Linear elastic soil, G,, ρ Rigid footing (b) Non liquefiable soil B h 1 Liquefiable soil layer h H Pe iωt Non liquefiable soil h 3 K(ω) C(ω) Bedrock FIGURE.1: a) Physical interpretation of dynamic stiffness, b) Problem definition -5

78 Chapter. Problem definition The problem considered is depicted in Figure.1b: a rigid surface footing resting on liquefiable soil, subjected to dynamic loading. A three-layer soil profile consisting of a surface clayey crust overlying a liquefiable sandy layer followed by a stiff base stratum is considered. Owing to the sharp impedance contrast between these layers, which results in strong wave reflections, kinetic and potential energy is trapped within the profile in the form of stress waves. A numerical study for determining the dynamic impedance of the footing is conducted for three oscillation modes (vertical, horizontal and rocking). Numerical analyses refer to square footings of various sizes used for the foundation of both ordinary structures and bridge piers. Excitation frequencies cover the frequency range of importance in earthquake engineering. To gain insight and provide comparisons, the elastodynamic analysis is performed in two stages: dynamic analysis without liquefaction and dynamic analysis with liquefaction, where the properties of the liquefiable soil stratum have been properly adjusted over those prior to liquefaction. As already mentioned, the principal parameter which needs to be degraded due to the occurrence of liquefaction and also dictates to a great extent the dynamic response of the footing is the shear wave propagation velocity of the liquefiable sandy layer, V sliq. In the current analyses and except specifically otherwise indicated, it is assumed that V sliq = 5m/s. With reference to material damping of the liquefied soil, it has been observed that the energy loss due to material damping during liquefaction increases dramatically, and the values of the associated damping ratio may increase from less than 5% to over %. In the present analyses, material damping ratio in the liquefiable soil stratum is taken at 3% without liquefaction and % in presence of liquefaction. Given the impermeable nature of the layers above and below the -6

79 3D Dynamic impedances of surface footings on liquefiable soil liquefied zone, no pore water pressure dissipation effects (and associated coupling of the stress and diffusion problem) are considered. Regarding Poisson s ratio, the soil is considered fully saturated (S r = 1%). Accordingly, the apparent compressional wave propagation velocity is V p = 15 m/s, i.e. equal to the corresponding wave propagation velocity in water. Considering the pore and solid phase as a single medium, from Hooke s law we obtain: v = [(V p /V s ) /1]/[(V p /V s ) 1] (.5) For instance, assuming V s = 15 m/s without liquefaction and V sliq = 5 m/s with liquefaction, Eq. (.5) yields v =.49 and.499, respectively. In the analyses reported herein, a uniform value ν =.49 is employed, corresponding to a nearly incompressible liquefied medium. TABLE.1: Properties of soil layers (values during liquefaction are given in parenthesis) Non liquefiable surface crust Liquefiable sand layer Non liquefiable base layer h 1 /B V s1 β 1 1 h /B V s β h 3 /B V s3 β (5).3 (.).49 H/B h 1 +h B shear wave propagation velocity V si in m/s, density ρ = Mg/m 3 for all soil layers With reference to Figure.1b, the characteristic parameters of each soil layer are the soil thickness, h i, the shear wave propagation velocity, V si, the mass density, ρ i, the Poisson s ratio, v i, and the material damping, β i. For numerical simulation purposes and for compatibility with previous research (Theocharis 11), it is -7

80 Chapter assumed that the dimensionless ratio of the total soil profile equals H/B = 15, thus, the presence of bedrock does not affect the dynamic response of the footing. Among the aforementioned parameters, the parametric investigation focused upon the effect of the thickness of the liquefied stratum, as well as the thickness and stiffness of the nonliquefiable surface crust (which should meet the bearing capacity requirements under gravity loading). The dimensional analysis that follows justifies the choice of the above critical problem properties. The values of the problem properties considered in the parametric analyses are summarized in Table.1. A reduction of the independent variables of the problem to be investigated is possible by means of dimensional analysis. The specific problem involves six major dimensional parameters (M = 6): the thickness of surface crust, h 1, the thickness of liquefiable layer, h, the shear wave propagation velocity of non-liquefiable surface crust, V s1, and the corresponding velocity of liquefiable layer, V s, the footing width, B, and the excitation frequency, f. The rest of parameters illustrated in Figure.1b, including the stiffness of the base layer and the total thickness of the soil profile have a second-order influence on the response and are not explored parametrically in the ensuing. In light of the two fundamental dimensions, length [L] and time [T] (N = ), application of Buckingham s theorem (1914) (M N = 4) leads to four dimensionless ratios controlling the response of the footing. The governing dimensionless groups were selected to be (h 1 /B), (h /B), (V s1 /V s ), (ωh 1 /V s1 ). These ratios suffice for fully describing the dynamic impedance of the footing. An extensive parametric study is conducted to elucidate the role of the aforementioned parameters on the impedance functions. -8

81 3D Dynamic impedances of surface footings on liquefiable soil.3 Numerical results The dynamic impedance of a rigid square footing on liquefiable soil is investigated numerically using simplified cone models based on Strength-of-Materials theory. This method, which is based on viscoelastic wave propagation in cones, was developed by Wolf and Deeks (4) (implemented in the computer code CONAN) and can be used to determine the dynamic impedance functions of surface or embedded rigid disks. However, various foundation shapes can be also analyzed by considering an equivalent circular radius to match footing area for translational oscillation modes (R = B/ π), or moment of inertia for rotational oscillation modes (R = B/ 4 (3π)) (Mylonakis et al 6). With cone models, the complex threedimensional elastodynamic problem is reduced to a simpler problem described, in a satisfactory manner, by one-dimensional wave propagation. A soil profile with linear elastic behavior and hysteretic material damping may consist of any number of horizontal layers overlying a halfspace or bedrock. Dynamic impedance is evaluated for any single frequency for vertical, horizontal, rocking and torsional degrees of freedom. The accuracy of the cone model predictions has been verified in a number of studies (Wolf 1994, Wolf & Deeks 4, Hiltunen et al 7)..3.1 Results for static stiffness Table. presents results for normalized static stiffness of square rigid footings for both pre-liquefaction (V s1 /V s =.67 and 1.67) and post-liquefaction (V s1 /V s = 4 and 1) conditions. All main oscillation modes (vertical, horizontal, rocking) are investigated. Static stiffness is normalized with the shear modulus of the surface non-liquefiable soil crust (G 1 ) and the width of the footing (B) according to Eq. (.1). For convenience, the results are presented in the form of tables and graphs and are expressed in terms of dimensionless quantities. -9

82 Chapter TABLE.: Normalized static stiffness of square rigid foundations on 3-layer liquefiable soil K ij / (G 1 B m ) Vertical (m=1) Horizontal (m=1) Rocking (m=3) V s1 / V s Preliquefaction Postliquefaction Preliquefaction Postliquefaction Preliquefaction Postliquefaction h 1 /B h /B It may be observed that: a) The stiffness degradation of the liquefiable soil stratum during liquefaction (i.e. the increase of shear wave velocity ratio V s1 /V s ) results in a decrease of the static stiffness coefficient, ranging from 8% to 78% for the vertical mode, from 14% to 55% for the horizontal mode and from % to 38% for the rocking mode. The highest decrease is observed for initial shear wave velocity ratio V s1 /V s =.67. This is anticipated, since a stiff top layer, in contrast with a soft one, can provide sufficient rigidity to the foundation even in presence of a soft underlying soil stratum. b) The values of post-liquefaction static stiffness coefficient increase as the crust thickness ratio h 1 /B increases. This is reasonable if one considers that for a thick top layer the pressure bulb beneath the loaded area (about 1.5 B in diameter) does not extend to the soft underlying liquefied soil. -1

83 3D Dynamic impedances of surface footings on liquefiable soil c) The increase in thickness of liquefiable soil (h /B) seems to further reduce the static stiffness. This is because the stiff non-liquefiable clayey base layer, providing extra rigidity to the foundation, is located deeper in the soil, thus reducing stiffening of foundation response..3. Results for dynamic impedance functions Results for dynamic impedance functions of square footings on liquefiable three-layer soil profile are depicted in normalized form in Figures..4. K i and C i refer to post-liquefied dynamic stiffness and damping while K i and C i refer to preliquefied stiffness and damping values, respectively. Hence, K i / K i and C i / C i ratios are selected to be graphically examined in order to isolate the influence of liquefaction on the dynamic impedance functions. In the graphs, the stiffness ratio K i / K i and damping ratio C i / C i are given as a function of the dimensionless frequency ωh 1 / V s1. -11

84 Chapter FIGURE.: Post-liquefied vertical dynamic impedance coefficients of square footing normalized with the corresponding pre-liquefied impedance coefficients; Effect of a) thickness of surface crust, b) thickness of liquefiable soil layer, c) shear wave velocity ratio; h 1 /B =.5, h /B = 1, V s1 /V s = /3. -1

85 3D Dynamic impedances of surface footings on liquefiable soil FIGURE.3: Post-liquefied horizontal dynamic impedance coefficients of square footing normalized with the corresponding pre-liquefied impedance coefficients; Effect of a) thickness of surface crust, b) thickness of liquefiable soil layer, c) shear wave velocity ratio; h 1 /B =.5, h /B = 1, V s1 /V s = /3. -13

86 Chapter FIGURE.4: Post-liquefied rocking dynamic impedance coefficients of square footing normalized with the corresponding pre-liquefied impedance coefficients; Effect of a) thickness of surface crust, b) thickness of liquefiable soil layer, c) shear wave velocity ratio; h 1 /B =.5, h /B = 1, V s1 /V s = /3. Inspection of Figure. for the vertical impedance functions reveals the existence of two regions, (I) and (II). Region (I), defined for ωh 1 / V s1 <, refers to footings having small to moderate width, B = 1 3 m, (h 1 is comparable to B), profiles with soft soil crust, V s1 = 1 15 m/s, and frequency range f = 1 Hz. On the other hand, region (II) with ωh 1 / V s1 > corresponds to footings of large width, B > 4 m, profiles with soft to moderate soil crust, V s1 = 1 5 m/s, and high -14

87 3D Dynamic impedances of surface footings on liquefiable soil frequency range f > 15 Hz. The same applies for the horizontal and rocking oscillation modes (Figures.3 and.4), with the only difference being that the boundary between these two regions for the horizontal mode is reduced to ωh 1 / V s1 = 1. It is noteworthy that common buildings and structures fall into region (I). It is further observed that the dynamic stiffness is considerably reduced in region (I), while the corresponding dynamic damping ratio C i / C i increases well above unity. For region (II), dynamic stiffness seems to amplify exhibiting sharp undulations while C i / C i ratio tends to unity. The following additional noteworthy trends become also evident from Figures..4: (a) The variation in thickness of the surface crust has a significant effect on the dynamic impedance functions. Specifically, as the h 1 /B ratio decreases dynamic stiffness decreases in region (I). For the second region, the h 1 /B ratio does control the undulations of impedance functions. (b) With reference to damping, a significant increase in post-liquefied damping is observed, which is attributed to the increase in material damping of the liquefied layer. Increase in the thickness of the surface non liquefiable crust results in amplification of damping for the vertical mode, while the opposite trend is noticed for the other two modes. Interestingly, vertical damping coefficient exhibits a peak at around ωh 1 / V s1.75, which suggests development of a kind of resonance at T exc T s1 (i.e. T s1 = 4h 1 /V s1 ). However, this is not observed in the K i / K i ratio. (c) The change in thickness of the liquefiable soil stratum does not affect the dynamic stiffness of the footing, and the K i / K i curves almost coincide. On the contrary, -15

88 Chapter h /B ratio seems to affect damping ratio C i / C i which increases as the h /B ratio decreases, especially in the horizontal mode. (d) In Figures c 4c it is noted that the initial shear wave velocity ratio V s1 /V s remains constant (= /3) while for the liquefied layer assumed V sliq = 1, 5 and 35 m/s being 7%, 17% and 3% of the initial shear wave velocity. Surprisingly, V s1 /V sliq ratio appears to affect only slightly the dynamic impedance functions. (e) Regarding the vertical dynamic stiffness, it is noted that for some frequencies K v / K v ratio admits negative values (with K v being negative), which suggests that the phase lag between response and excitation is greater than 9 o. Moreover, at the high frequency range and for h 1 /B =.5, the dynamic stiffness coefficients are extremely high. It seems that the analysis provides unstable solutions at the high frequency range and further investigation is required to ensure dependable results..3.3 Comparison with BEM analyses To validate the herein reported analyses, a rigorous Boundary Element Method in 3 dimensions implemented in software platform ISoBEM is employed. In this case, the layered medium is solved by considering each soil stratum as a separate homogeneous region, developing BEM equations independently and then assembling and solving the associated set of simultaneous equations by respecting equilibrium and compatibility across a common interface. Note that ISoBEM has been successfully used to study several related problems in applied mechanics (Polyzos et al. 1998, 3, 5, Tsinopoulos et al. 1999) and soil mechanics (Heidarzadeh et al. 15). Three-dimensional models simulating a rigid square footing on a three-layer liquefiable soil in ISoBEM were set up to provide comparisons. In the case of vertical and harmonic oscillations, a comparison is performed between static stiffness coefficients obtained by cone models and ISoBEM analysis. -16

89 3D Dynamic impedances of surface footings on liquefiable soil The comparison is summarized in Table.3 as the ratio of values obtained from cone models over the corresponding values obtained from BEM analysis. It is shown that BEM typically predicts higher values for static stiffness. The discrepancies observed are anticipated in light of the complexity of the problem and the extremely low value of shear wave velocity considered for the liquefied stratum (V sliq = 5 m/s). For the pre-liquefied case (V s1 /V s =.67), results from the two analyses exhibit differences of less than 1% or so. Regardless of the observed discrepancies, it is stressed that the percentage of reduction in static stiffness triggered by the occurrence of liquefaction, is comparable in the two approaches. Hence, rigorous elastodynamic boundary element results verify the significant loss of stiffness of the foundation during liquefaction. From an engineering viewpoint, results are generally comparable and the results from CONAN analyses can be used for a preliminary assessment of the effect of liquefaction on the stiffness of the footing in an equivalent linear sense. -17

90 Chapter TABLE.3: Comparison of vertical and horizontal static stiffness coefficients (BEM vs Cone) Vertical K ij,(bem) / K ij,(cone) Horizontal V s1 / V s h 1 /B h /B BEM / Cone Figure.5 depicts a comparison between dynamic stiffness and damping coefficients obtained by means of Cone and BEM solutions. For the purposes of this discussion, the case where h 1 /B = 1, h /B = 1 and V s1 /V s = 4 is depicted. Spring coefficient k ij and dashpot coefficient c ij are plotted, in accordance with Eq. (.4), against the dimensionless frequency a, i.e., ωr/v s1 with R = B/ π being the equivalent circular radius, stemming from the cone solution, and ω(b/)/v s1 for results obtained using the boundary element method. It is observed that agreement between Cone model predictions and BEM results is generally quite good. For vertical oscillations, CONAN predictions for spring and dashpot coefficients are in satisfactory agreement with the rigorous BEM results. Curves exhibit similar trends, except perhaps for k ṽ in the high frequency range. For horizontal oscillations, there is a consistent over-prediction of dynamic stiffness from the Cone method at all frequencies. However, the curves exhibit the same trend. For dashpot coefficient, accord is better. -18

91 3D Dynamic impedances of surface footings on liquefiable soil FIGURE.5: Comparison of vertical and horizontal dynamic impedance functions (BEM vs Conan); h 1 /B = 1, h /B = 1, V s1 /V s = Concluding remarks The following conclusions could be drawn from the preliminary investigation at hand: a) Results obtained from simplified cone models (CONAN software) compare well against more rigorous BEM results (ISoBEM software) for the pre-liquefied case. Nevertheless, for the post-liquefied case results deviate considerably, perhaps owning to the extremely low value considered for the shear wave velocity of the liquefied stratum. From a geotechnical engineering viewpoint, results are generally comparable and both methods demonstrate a significant decrease in stiffness of the footing accompanied by a considerable increase in damping due to occurrence of liquefaction. -19

92 Chapter b) The Cone method seems to be appropriate for a preliminary assessment of the problem at hand. c) Static stiffness of the footing is reduced dramatically under liquefied conditions. The decrease ranges from 8% to 78% for the vertical mode, from 14% to 55% for the horizontal mode and from % to 38% for the rocking mode. d) Key parameters of the problem emerging from this study are the dimensionless thickness of the surface non-liquefiable crust, h 1 /B, the dimensionless thickness of the liquefiable soil layer, h /B, the relative stiffness of the surface layer, V s1 /V s, and the dimensionless excitation frequency ωh 1 / V s1. e) The influence of liquefaction on the dynamic impedance functions is investigated through the dimensionless ratios K i / K i and C i / C i. Results demonstrate the existence of two distinct regions. For the low frequency range and footings used for the foundation of common structures (ωh 1 /V s1 < 1), significant reduction in dynamic stiffness is observed, accompanied by a considerable increase in damping. For ωh 1 / V s1 >, dynamic stiffness exhibits sharp undulations, while damping ratio C i / C i tends to unity. f) h 1 /B ratio does control the undulations of the impedance functions for all three oscillation modes. Naturally, static stiffness increases with increasing h 1 /B. g) An increase in h /B ratio results in a decrease in static stiffness but it does not affect considerably the dynamic stiffness coefficient. However, damping ratio C i / C i seems to be more sensitive to this ratio. h) Remarkably, V s1 /V sliq ratio seems to be of secondary importance since it only slightly affects the dynamic impedance functions of the footing. -

93 3D Dynamic impedances of surface footings on liquefiable soil.4 Parametric study by means of rigorous boundary-element analysis The objective of this section is to perform a more accurate and in-depth systematic analysis of the vertical, horizontal and rocking static stiffness of a rigid square footing resting on the layered soil deposit depicted in Figure.1b. An advanced algorithm of the boundary element method (ISoBEM) incorporating isoparametric four-noded linear quadrilateral elements (for surfaces) was utilized for this investigation. The dimensionless ratios h 1 /B =.5, 1,, h /B =.5, 1,, V s1 /V sliq = 4, 7, 1, are explored. It is noted that the investigation reported in the preceding sections, was performed for two values of the dimensionless ratio V s1 /V sliq (equal to 4 and 1). In the present analysis the intermediate value V s1 /V sliq = 7 is additionally investigated. All properties of the soil layers are the same as in Table.1, with the exception of Poisson s ratio v, being.499 for the second and third layer in the current analyses..4.1 Convergence and accuracy of BEM analysis As it has already been mentioned, the boundary element method has been proven an effective numerical technique for solving a wide variety of engineering problems. In this technique, the problem is represented by a set of boundary integral equations, being an exact formulation in the realm of a viscoelastic continuum analysis. Approximations arise only from the numerical implementation of the integral equations. For linear problems and in the absence of body forces, only the boundaries of the domain need to be discretized and, hence, the dimensionality of the problem is reduced by one (from 3D to D), by means of Gauss s theorem. Various 3D models (half and quarter) and mesh sizes, for each oscillation mode, were examined to obtain the optimum model for the square footing resting on -1

94 Chapter the given soil stratigraphy. Because of lack of solutions for stiffness of square footings on 3-layer soil profiles, the performance of the models was checked a) by applying to all soil layers the same properties to obtain the stiffness of a rigid square footing on a halfspace, and b) by applying to the second and third layer the same properties, forming a two-layer soil profile, and comparing the results for stiffness with corresponding results in literature. Taking advantage of the symmetry of geometry and loading, for the vertical (symmetric) oscillation mode, only one quarter of the system is analyzed. For the horizontal and rocking modes (antisymmetric), analyzing a half model is essential. To this end, footing surface and the region around the footing up to a distance 3B is discretized using isoparametric four-noded quadrilateral linear elements with element length about 1/1 of the shear wavelength. Moreover, the surface beyond this region up to a distance of 5B needs fine mesh (element length = 1/8 of shear wavelength). The same also applies for the first interface (between first and second layer). In addition, discretization of the ground surface and the interfaces up to a distance of at least 1B to 15B beyond the foundation is necessary for obtaining accurate results. Coarser elements with lengths of about 1/3 to 1/ of the shear wavelength are adequate for more distant locations (Ahmad & Rupani 1999). For the vertical and horizontal oscillation mode, a uniform unit vertical and horizontal displacement, respectively, is applied to all element nodes of the footing and the resulting load is computed from the tractions developed on the element nodes (Appendix A). For the rocking mode, a unit rotation is applied. To establish the accuracy of the present BEM analysis, comparative studies are conducted with available published results. Table.4 provides the comparison of ISoBEM results for normalized static stiffness of rigid square footing on halfspace, -

95 3D Dynamic impedances of surface footings on liquefiable soil for all three oscillation modes, with the empirical formulas from Pais & Kausel (1988). In Table.5, results for the horizontal static stiffness of a two-layer soil (v 1 = v =.4) are compared against those obtained from Ahmad & Rupani (1999). Figure -4 depicts the comparison of ISoBEM results for horizontal, vertical and rocking impedance of a square footing resting on a uniform soil layer over a halfspace (V s1 /V s =.8, H/B/ = 1, v 1 = v =.33, ρ /ρ 1 = 1.13, β 1 =.5, β =.3) with those reported by Wong and Luco (1985). Evidently, results from the present BEM analyses are in reasonable agreement with the published results. TABLE.4. Comparison of normalized static stiffness K ij /(G 1 B m /) of a rigid square footing on halfspace Oscillation mode Pais & Kausel (1988) ISoBEM Difference (%) Vertical (m=1) Horizontal (m=1) Rocking (m=3) TABLE.5: Comparison of normalized horizontal static stiffness K h /(G 1 B/) of a rigid square footing on a two-layer soil profile H/B/ V s1 /V s =.5 V s1 /V s = ISoBEM Ahmad & Rupani (1999) Diff. (%) ISoBEM Ahmad & Rupani (1999) Diff. (%)

96 Chapter Dynamic stiffness coefficients 14 1 Real part Imaginary part vertical Dynamic stiffness coefficients 14 Wong & Luco (1985) 1 ISoBEM horizontal a = ωb / V s a = ωb / V s Dynamic stiffness coefficients 16 rocking a = ωb / V s FIGURE.6: Comparison of vertical, horizontal and rocking dynamic stiffness coefficients of a square footing resting on a two-layer soil profile..4. Results for vertical static stiffness of rigid square footing on 3-layer liquefiable soil Results for the normalized vertical static stiffness obtained by means of ISoBEM using 3656 isoparametric four-noded linear quadrilateral elements meshing one-quarter of the domain, are presented in Table.6. The outcome of this rigorous elastostatic analysis is in accord with the preliminary results obtained in the previous section and confirms the significant decrease in the vertical static stiffness during liquefaction. -4

97 3D Dynamic impedances of surface footings on liquefiable soil According to Table.6, the decrease ranges from 11% to 84% for the vertical mode, being comparable with these predicted from the cone analyses. It is observed that the highest decrease occurs in case of a very thin surface clay crust (h 1 /B =.5) over a thick liquefiable sandy layer (h /B = ) regardless of the shear wave propagation velocity ratio. The minimum decrease is observed in the opposite case, where a thin liquefiable sandy layer (h /B =.5) underlies a thick surface clay zone (h 1 /B = ). Figure.7 depicts the variation of normalized vertical static stiffness, K v /G 1 B, with the thickness of liquefiable soil profile, h /B, for three values of the improved surface zone, h 1 /B, and three values for the shear wave propagation velocity V s1 /V sliq. Although results have been discussed in detail in the previous section, we reiterate that: a) increase in V s1 /V sliq ratio entails a decrease in normalized static stiffness, b) a decrease in the thickness of the surface zone leads to a decrease in normalized static stiffness and c) a increase in h /B ratio also leads to a decrease in K v /G 1 B ratio. -5

98 Chapter TABLE.6: Normalized vertical static stiffness of rigid square footing on liquefiable soil before and after liquefaction percentage of decrease K vv / (G 1 B) V s1 /V sliq = 4 V s1 /V sliq = 7 V s1 /V sliq = 1 (%) (%) h 1 /B h /B Preliquef. Postliquef. Preliquef. Postliquef. Preliquef. Postliquef. (%) Using non-linear regression analysis in the results of Fig..7, the following predictive equations are derived For h /B =.5, Kv GB 1 V Exp.33 V s1 sliq h1 V s1 1Exp Exp.54 B V sliq (.6a) For 1 h /B, -6

99 3D Dynamic impedances of surface footings on liquefiable soil Vs1.54 Kv h V sliq e GB 1 B h h1 h h1 h B B h B B e e B B (.6b) which can be used in applications as a preliminary estimate of the vertical footing stiffness in liquefiable soil h / B =.5 h / B = 1 4. K vv / (G 1 B) V s1 / V sliq = regression h 1 / B K vv / (G 1 B) h 1 / B h / B = K vv / (G 1 B) h 1 / B FIGURE.7: Effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized vertical static coefficient and results obtained from Eqs. (.6a) & (.6b) using regression analysis. -7

100 Chapter.4.3 Results for horizontal static stiffness of rigid square footing on 3-layer liquefiable soil Results for the normalized horizontal static stiffness obtained by means of ISoBEM, using 496 isoparametric four-noded linear quadrilateral elements meshing half-quarter of the domain, are presented in Table.7. It is observed that the decrease in stiffness due to liquefaction ranges from 11% to 56%. The highest decrease occurs in the case where the thickness of the surface crust is minimum, the corresponding thickness of the liquefiable soil layer is maximum and the V s1 /V sliq ratio is equal to 4. However, it is apparent that the percentage of decrease is about the same for all three V s1 /V sliq for a given set of h 1 /B and h /B ratios. The effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized horizontal static coefficient is depicted in Figure.8. Similar behavior with the normalized vertical static coefficient can be observed. Nevertheless, it is noted that the thickness of liquefiable soil layer appears to not affect substantially the horizontal stiffness. Moreover, for high values of h 1 /B ratio, V s1 /V sliq ratio seems to be of secondary importance. In the same vein, applying non-linear regression analysis in the results of Figure.8 yields K h h h GB 1 B B V s V h s h V B B sliq Vsliq.3.46e.1 1e 1 e (.7) -8

101 3D Dynamic impedances of surface footings on liquefiable soil TABLE.7: Normalized horizontal static stiffness of rigid square footing on liquefiable soil before and after liquefaction percentage of decrease K hh / (G 1 B) V s1 /V sliq = 4 V s1 /V sliq = 7 V s1 /V sliq = 1 (%) (%) h 1 /B h /B Preliquef. Postliquef. Preliquef. Postliquef. Preliquef. Postliquef. (%)

102 Chapter h / B =.5 h / B = 1 K hh / (G 1 B) h 1 / B 3. V s1 / V sliq = regression h / B = K hh / (G 1 B) h 1 / B K hh / (G 1 B) h 1 / B FIGURE.8: Effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized horizontal static coefficient and results obtained from Eq. (.7) using regression analysis..4.4 Results for rocking static stiffness of rigid square footing on 3-layer liquefiable soil With reference to the rocking static stiffness obtained by means of ISoBEM, using 3656 isoparametric four-noded linear quadrilateral elements meshing halfquarter of the domain, are presented in Table.8. Evidently, the decrease ranges from 1% to 59%. It is worth mentioning that the corresponding analysis with CONAN did not reveal such a reduction. -3

103 3D Dynamic impedances of surface footings on liquefiable soil Likewise, the maximum decrease occurs in the case where the thickness of the surface crust is minimum and the corresponding thickness of the liquefiable soil layer is maximum. V s1 /V sliq ratio does not influence the percentage of reduction in rocking stiffness. In case of a very thick surface crust, i.e. h /B =, the reduction is negligible. The effect of the dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized horizontal static coefficient is depicted in Figure.9. It is shown that the thickness of the liquefiable soil does not affect the rocking static stiffness, and as the thickness of the surface crust increases neither V s1 /V sliq ratio plays a significant role. In case of h 1 /B =, the normalized rocking static stiffness is approximately constant and equal to.75. In the same spirit as before, using non-linear regression analysis in the results of Figure.9 the following fitted formula was obtained 1 1 h h1 Kr Vs1 Vsliq Vs1 Vsliq B B.1 1 e 3 GB Vs 1 Vsliq (.8) The above equations can be used in applications as a preliminary assessment of the problem. It is noted that the regression formulae are valid for the postliquefaction case, i.e., V s1 /V sliq > 1, and for the parameter range examined in the analyses i.e.,.5 h /B.. -31

104 Chapter TABLE.8: Normalized rocking static stiffness of rigid square footing on liquefiable soil before and after liquefaction percentage of decrease K rr / (G 1 B 3 ) V s1 /V sliq = 4 V s1 /V sliq = 7 V s1 /V sliq = 1 (%) (%) h 1 /B h /B Preliquef. Postliquef. Preliquef. Postliquef. Preliquef. Postliquef. (%)

105 3D Dynamic impedances of surface footings on liquefiable soil K rr / (G 1 B3 ) h / B =.5 h / B = 1. K rr / (G 1 B3 ) h 1 / B K rr / (G 1 B3 ) h / B =. V s1 / V sliq = regression h 1 / B h 1 / B FIGURE.9: Effect of dimensionless ratios h 1 /B, h /B and V s1 /V sliq on the normalized rocking static coefficient and results obtained from Eq. (.8) using regression analysis..5 Simplified analytical solution for static stiffness of circular rigid footing on multi-layer soil The simplified method of cone model is utilized to obtain the static stiffness of a rigid circular footing resting on a multilayer soil. The concept of cone models was first introduced by Ehlers (194) to model a surface disk on a halfspace under horizontal loading and was later extended to rotational loading (Meek and Veletsos 1974, Veletsos and Nair 1974). Though a simplified concept, cone model is the -33

106 Chapter cornerstone of numerous studies associated primarily with dynamic stiffness of foundations (Meek and Wolf 199, 1994, Wolf and Meek 1993, 1994, Jaya and Prasad, Pradhan et al 3). A detailed description on the use of cone models on foundation vibration analysis is given by Wolf (1994), while a computer software platform, based on wave propagation in cones, was developed by Wolf and Deeks (4) for computing the dynamic impedance functions of footings. Figure.1 illustrates the concept of the developed solution. The soil is considered a linear elastic medium and consists of N horizontal layers. Soil Mechanics problems usually demand the use of soil properties such as the soil Young s modulus, E, and the Poisson s ratio, v. Nevertheless, utilizing the theory in the dynamic regime requires using more appropriate parameters such as propagation velocities of compression and shear waves. For each degree of freedom, soil is idealized as a semi-infinite elastic cone of different apex height z, as shown in Figure.11. The aspect ratio of the height to footing radius z /R defines the opening angle λ i. For homogeneous soil conditions, the aspect ratio is determined by equating the exact solution for the static stiffness of a footing resting on elastic halfspace with the corresponding value obtained from the cone model. Based on Strength-of-Materials theory, it is assumed that plane sections remain plane and the material being out of the cone is ignored. Depending on the type of deformation (horizontal or shear), shear wave velocity (for horizontal and torsional oscillation modes) or longitudinal wave velocity (vertical and rocking oscillation modes) is employed. -34

107 3D Dynamic impedances of surface footings on liquefiable soil R z = R / λi z D 1, v 1 λ i u w A Ground surface 1 st layer D 3, v 3 D, v h i nd layer 3 nd layer D N, v N N th layer FIGURE.1: Cone model utilized for estimating vertical and horizontal static stiffness of a circular rigid foundation on multi-layer soil. FIGURE.11: Cone models for various modes of vibration (after Wolf & Deeks 4). -35

108 Chapter The footing area is A = π R and the variation of footing area with depth is A z i z A1 R (.7) with z measured from cone top. Supposing that P denotes the axial load acting on the footing and u is the associated displacement, it is easy to show that the relationship between displacement and force is u P P R dz (.8) DAz DAi where D is the modulus of soil stiffness depending on the response mode. Accordingly, the stiffness is obtained P D Ai K (.9) u R To compute the opening angle λ i of the cone, Eq. (.11) is set equal to the exact solution for stiffness. For the vertical oscillation mode, Eq. (.11) takes the following form K Vp Ai (.1) R The exact value of the static stiffness of a footing on an elastic halfspace is given by 4GR K, exact 1 v (.11) By equating Eq. (.1) with Eq. (.13), the opening angle of the cone pertaining to the vertical degree of freedom is derived 8 V s 1 vv p (.1) -36

109 3D Dynamic impedances of surface footings on liquefiable soil Note that, it is considered V p =V s for v 1/3 and V p =V s for v > 1/3. given by In case of a multilayer soil, the relationship between displacement and force is w P 1 N zi dz A z (.13) i1 i1 Di 1 z R It is straightforward to show that the static stiffness for vertical loading is K P w A h N i i1 ivp, i 1 zi 1 R 1 zi R (.14) ρ i, V pi being soil density and compressional wave velocity of each layer with thickness h i. For computing the opening angle of the cone, an average value of the Poisson s ratios of the layers should be employed. In the same vein, for homogeneous soil conditions and horizontal loading the stiffness of the footing using the cone model is K h GAi (.15) R The corresponding exact value of the static stiffness of a circular footing on an elastic halfspace is given by K h, exact 8GR v (.16) By equating Eq. (.17) with Eq. (.18), the opening angle of the cone pertaining to the horizontal degree of freedom is obtained 16 h v (.17) -37

110 Chapter obtained For multilayer soil, the relationship between displacement and force is u P 1 N zi dz A z (.18) i1 i1 Gi 1 hz R Accordingly, K h P u N A hi G 1 z R 1 z R i1 i h i1 h i (.19) where G i is the shear modulus of each layer. R z = R / λ i z λ i θ Ground surface D 1, v 1 z 1 st layer D 3, v 3 D, v h i nd layer 3 nd layer D N, v N N th layer FIGURE.1: Cone model utilized for estimating rocking static stiffness of a circular rigid foundation on multi-layer soil. -38

111 3D Dynamic impedances of surface footings on liquefiable soil Figure.1 depicts the cone model for rotational loading. If I is the moment of inertia of the circular footing, the variation of I with depth is given by the following expression I z i z I 1 R 4 (.) with I being equal to πr 4 /4. For homogeneous soil, the relationship between rotation θ and moment M is M DI z dz (.1) M R 3 i DI Therefore, the static stiffness is K r 3i DI (.) R The corresponding exact value of stiffness for rocking is K r, exact 8GR 31v 3 (.3) By equating Eq. (.4) with Eq. (.5), the opening angle for the rocking mode is obtained in the form 64 V s r 9 1 vv p (.4) As in the case of vertical loading, it is V p =V s for v 1/3 and V p =V s for v > 1/3. For multilayer soil, the relationship between rotation and moment is written -39

112 Chapter M 1 N zi dz 4 I z (.5) i1 i1 Di 1 rz R Likewise, K r M B N 3r I V z R z R 3 3 i1 i p, i 1 r i1 1 r i (.6) TABLE.9: Comparison of the vertical static stiffness coefficient with BEM results K v / (G 1 B) V s1 /V s = /3 V s1 /V s = 7/6 V s1 /V s = 5/3 h 1 /B h /B BEM Eq. (.16) Diff. (%) BEM Eq. (.16) Diff. (%) BEM Eq. (.16) Diff. (%)

113 3D Dynamic impedances of surface footings on liquefiable soil Tables.9.11 provide comparisons between the proposed analytical solution and BEM results for a square footing on the three-layer soil profile studied earlier for the pre-liquefied case. It is reminded that the analytical solution refers to circular footings, thus, for analyzing square footings an equivalent circular radius is needed. However, for the normalization of the static stiffness, the width B of the actual square footing is used. It is noted that for the implementation of the analytical solution the Poisson s ratio of each soil stratum instead of a mean value is utilized. Overall, predictions of the cone model compare satisfactorily with rigorous numerical results. For the vertical mode, it is observed that the highest discrepancy is about 19%, which is acceptable from a geotechnical engineering view point. Note that, the analytical solution overpredicts the vertical static stiffness. In the horizontal mode, the difference between the results is smaller below 1% for the majority of the cases studied. It is noted that the proposed solution overpredicts the value of horizontal static stiffness coefficient except for the case of V s1 /V s = 5/3. With reference to the rocking static stiffness, discrepancies between results obtained from analytical solution and those obtained from BEM analyses are below 1%. The proposed solution seems to predict satisfactorily the static stiffness of footings on multi-layer soil profiles, with its layers having comparable soil properties, such as in the pre-liquefied case, i.e, V s1 /V s = /3, 7/6, 5/3 and V s /V s3 = 1/. If the impedance contrast between the soil layers is sharper, such as in a liquefied soil profile, with V s1 /V s = 4, 7, 1 and V s /V s3 = 1/1, the closed-form solution does not provide satisfactory results. -41

114 Chapter TABLE.1: Comparison of the horizontal static stiffness coefficient with BEM results K h / (G 1 B) V s1 /V s = /3 V s1 /V s = 7/6 V s1 /V s = 5/3 h 1 /B h /B BEM Eq. (.1) Diff. (%) BEM Eq. (.1) Diff. (%) BEM Eq. (.1) Diff. (%)

115 3D Dynamic impedances of surface footings on liquefiable soil TABLE.11: Comparison of the rocking static stiffness coefficient with BEM results K r / (G 1 B 3 ) V s1 /V s = /3 V s1 /V s = 7/6 V s1 /V s = 5/3 h 1 /B h /B BEM Eq. (.8) Diff. (%) BEM Eq. (.8) Diff. (%) BEM Eq. (.8) Diff. (%) Conclusions A parametric investigation was conducted using elastodynamic methods to explore the dynamic response of a surface rigid square footing on a three-layer liquefiable soil under harmonic excitation. Although liquefaction is a strongly nonlinear phenomenon and advanced methods are necessary for a rigorous treatment of the problem, elastodynamic analyses assuming proper values for the shear wave velocity and material damping of the liquefied layer, can shed light into the physics of dynamic impedance functions of footings under the conditions at hand. The main conclusions of the study may be summarized as follows: -43

116 Chapter Liquefaction occurrence leads to significant degradation in static stiffness of a surface footing. For the parameters examined, the reduction may reach 84% for the vertical mode, 56% for the horizontal and 58% for the rocking mode. Results obtained from two independent analyses by means of simplified cone models (CONAN software) and boundary elements (ISoBEM software), manifest this significant loss of stiffness. Regarding the influence of liquefaction on the dynamic impedance functions, a significant decrease in dynamic stiffness and considerable increase in damping are observed for the low frequency range and common sizes of footings. For the high frequency range, dynamic stiffness exhibits sharp undulations while damping ratio C i / C i tends to unity. It is mentioned that these observations are based on cone model analyses and only a limited number of rigorous 3D analyses with boundary elements was performed to check the validity of these. A further important task would be to perform more boundary element analyses for a number of frequencies, to obtain a more accurate behavior for the dynamic impedance functions in the frequency domain. h 1 /B ratio proved to be the most significant factor for the response of the foundation in both static and dynamic regime. Both h 1 /B and h /B ratio seem to control the percentage of reduction in static stiffness during liquefaction, while V s1 /V s ratio does not play any role. An attempt was made to obtain regression relations for the static stiffness (vertical, horizontal and rocking) in the post-liquefaction case, based on numerical boundary element results (Figures.7.9). Equations (.6), (.7) -44

117 3D Dynamic impedances of surface footings on liquefiable soil and (.8) were derived via non-linear regression analysis and can be applied to engineering practice. A simplified analytical solution for static stiffness of a circular foundation on multi-layer soil by means of cone models was developed. The proposed analytical solution compares satisfactorily with the more rigorous boundary results in case of a multi-layer soil profile with its layers having comparable soil properties i.e., in the pre-liquefied case. For multi-layer soil profiles with layers exhibiting sharp impedance contrast between them such as in liquefied soil, the proposed solution does not provide satisfactory results. -45

118 Chapter -46

119 Chapter 3 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil 3.1 Introduction Seismic codes specify that the use of shallow foundations on liquefiable soils may be considered only after appropriate ground improvement. Nevertheless, there is a lack of instructions regarding the depth and width of ground improvement, or what provisions should be taken in case of a non-liquefiable surface layer. Hence, the standard practice involves mitigation of the liquefaction hazards by improving the strength, density, and/or drainage characteristics of the entire liquefiable soil layer below the foundation. In this light, deep foundations remain the conventional, yet expensive, approach for such extreme soil conditions, as they bypass the liquefiable strata and transfer the loads of the superstructure into deeper and more stable soil strata. Contrary to current engineering practice, recent experimental and theoretical studies suggest that pile installation may be avoided, provided that the existence of a non-liquefiable layer on top of the liquefiable sand is appropriately considered. It is 3-1

120 Chapter 3 experimentally established that the presence of a soil layer of adequate thickness and shear strength on the ground surface, may restrain the accumulation of excessive seismic settlements and prevent post-shaking bearing capacity failure owing to the liquefied subsoil. This evidence has given rise to attempts for a complete, rational design approach for the performance-based design of shallow foundations resting on soil profiles susceptible to liquefaction (Karamitros et al 13a c). The idea proposed herein suggests the intentional soil liquefaction under an improved non-liquefiable soil stratum, so that the liquefied soil layer performs as a natural damper and reduces the induced seismic ground motions transmitted to the superstructure. An established bridge design concept exploiting the natural liquefiable soil by means of a response spectrum analysis and deformation-based performance criteria, has been developed by Bouckovalas et al. (14a,b). Based on this methodology, Sextos et al. (14) presented a preliminary demonstration study for three typical structural systems (a statically determinate, a statically indeterminate concrete bridge and a steel overpass) and obtained that the bridge systems studied tolerate the liquefaction-induced deformations without any significant structural damage or loss of serviceability. The aim of this chapter is to examine the feasibility of the proposed geotechnical isolation method of bridge piers in the context of frequency and time domain analyses using a modified version of the code SFIAB (Mylonakis et al. ). The code SFIAB has been modified to take into account the dynamic impedance of the foundation (frequency-dependent springs and dashpots) obtained by elastodynamic analyses by means of programs, such as CONAN and ISoBEM, and solve for the response of the superstructure. Results for the dynamic impedance functions of a footing on liquefiable soil obtained in Chapter can be used directly as 3-

121 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil input. The above analyses are essential because they reveal the actual impact of liquefaction on the response of a structure and can provide quantitative information on the effective fundamental period and damping of the pier-foundation-liquefied soil system. 3. Bridge design concept proposed by Bouckovalas et al (14a,b) In this section, the bridge design concept of the natural seismic isolation, based on a response spectrum analysis and deformation-based performance criteria is briefly described. For more details the reader can consult recent work by Bouckovalas et al. (14a,b) and Sextos et al. (14). Following up on previous research by Karamitros et al. (13a c), Bouckovalas et al (14a, b) developed a rational methodology for performancebased design of shallow foundations in a liquefaction regime. As it has already been mentioned, the design is based on the idea of a permeable (natural or artificial) crust which needs not extend over the whole depth of the liquefiable sand in order to take advantage of the observed benefits of settlement reduction (Acacio et al 1, Sitar and Hausler 1) but at a reduced thickness and width. The key feature of this method is the design of the foundation and superstructure to respect, mutatis mutandis, the pertinent seismic code provisions to fully account for the effect of liquefaction on the design inertial loads and displacements. According to the proposed procedure, the first step in determining whether a shallow or a deep foundation will be designed for supporting a bridge pier is to ensure that the thickness of the potentially liquefiable soil stratum suffices for providing seismic protection. To this end, researchers suggest a site specific seismic response analysis of the liquefiable soil profile. Alternatively, Bouckovalas et al. (13) 3-3

122 Chapter 3 suggest that the thickness of the liquefiable soil stratum beneath the non-liquefiable surface zone should be extended at depth at least equal to (1/5 1/15)λ, λ (= V s,o T exc ) being the predominant wave length of the seismic waves propagating through the liquefiable soil layer and V s,o the initial (prior to liquefaction) average shear wave propagation velocity. Since the liquefiable soil layer is sufficiently thick, the shallow foundation below each bridge pier has to be carefully designed. For each bridge footing, the soil properties such as soil density D r, excess pore pressure ratio in the improved zone, r u,design, buoyant unit weight γ', should be taken into account. Moreover, the geometric limits of the surface improved zone, the thickness of the liquefiable soil layer, the earthquake ground motion (maximum acceleration, predominant period, number of cycles), the footing size and shape, as well as the maximum tolerable displacements and rotations should be specified. Given that the maximum tolerable deformations are not known a priori, the recurrence period of the earthquake should be decided. This will define the limits that the above deformations should not exceed based on the current code provisions regarding the life prevention and safety. The most crucial step is to determine the design inertial forces and displacements. Apparently, the design spectra available in modern seismic codes, such as EC8, are not appropriate for design in a liquefiable regime. It is understood that, upon liquefaction, the natural fundamental period of the soil deposit will be elongated due to the increase of excess porewater pressures and the ensuing degradation of soil stiffness. This shift in site period reduces the potential amplification of excitation motion, leading to a decrease in peak ground acceleration. Therefore, since the liquefied soil layer attenuates the seismic excitation, the available design spectrum should be modified accordingly, to accommodate the nonlinear 3-4

123 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil hysteretic response of the liquefiable soil. This can be attained by considering the earthquake hazard at the given site, the geological characteristics of the region and the return period of the earthquake, depending on the importance of the structure. Having identified the above features, a sequence of input rock outcrop excitation motions have to be cautiously selected (possibly also scaled), so that their average response spectrum is consistent with the design spectrum for soil type that approximately resembles bedrock conditions, for the desired peak ground acceleration (PGA) level. To estimate the peak ground acceleration and the corresponding mean 5% damped elastic spectrum, one-dimensional nonlinear site response and liquefaction analyses need to be conducted. Figure 3.1 illustrates a typical example from these analyses for a specific site and earthquake excitations with 1 years recurrence period (Bouckovalas et al., 14b). For the present study, it is evident that the peak spectral acceleration suggested by EC8 is quite the double of the one predicted by analyses. FIGURE 3.1: Outcrop seismic excitation response spectra vs. the EC8 design spectrum for soil type A and PGA=.3g (left), and corresponding elastic response spectra at the liquefied ground surface (right) (Sextos et al. 14). 3-5

124 Chapter 3 Alternatively, Bouckovalas et al. (14b) proposed an analytical methodology for obtaining the spectral acceleration of the liquefied ground. The associated steps are: 1. Estimation of the factor of safety against liquefaction FS L from CPT (Cone Penetration Test) or SPT (Standard Penetration Test) results.. Performance of equivalent linear analysis for non-liquefiable soil profile and computation of the response spectrum, Sa NL. 3. Based on FS L and the Figure 3., estimation of the proper shear wave propagation velocity of the liquefied soil profile. 4. Performance of equivalent linear analysis for the liquefied soil profile employing the shear wave propagation velocity obtained from Step 3 and G/G max = 1, and computation of the response spectrum, Sa L. 5. Computation of coefficient a PGA based on FS L : a PGA.7 1 FS 1 cos L.65 (3.1) 6. Computation of coefficient a for each period value, T: 1a 1a PGA PGA tanh 1T.8 at (3.) 7. Computation of the predicted spectral acceleration of the liquefied soil deposit, for each period value: (3.3) Sa T Sa T a T Sa T Sa T PRED NL NL L 3-6

125 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil FIGURE 3.: Relationship between the shear wave propagation velocity reduction ratio V sliq / V s and the factor of safety FS L (as proposed by Bouckovalas et al. 14b) Having defined the seismic scenario, the design of the foundation and the associated imposed forces, the bridge-foundation-soil system should be analyzed in view of soil-structure interaction taking into account the stiffness degradation and nonlinear hysteretic response of the liquefiable soil. As it has been discussed in Chapter, determining the dynamic response of surface or embedded foundations often involves complicated three-dimensional elastodynamic analyses using rigorous methods, such as the finite element, the finite difference, and the boundary element method. These models typically involve hundreds or even thousands degrees of freedom, and the frequency-dependent dynamic stiffness of the foundation is evaluated in the frequency domain. In order to reproduce in the time domain the frequency-dependent impedance functions and, additionally, to consider non-linear behavior of the superstructure, lumped-parameter models are often considered. Lumped-parameter models represent the soil-foundation interaction via a suitable 3-7

126 Chapter 3 combination of few springs, dashpots and masses, all with real-valued, frequencyindependent coefficients. In other words, these assemblies of springs, dashpots and masses imitate the complex-valued dynamic response of the foundation. The incorporation of these models to the structural model is relatively simple. Each degree of freedom at the foundation node of the superstructure is joined to a lumpedparameter model that may consist of additional internal degrees of freedom. The idea was first introduced in the pioneering work of de Barros and Luco (199), and was subsequently extended by Wolf (1991a,b), Wolf & Paronesso (1991), Wolf (1994), Wu & Lee (, 4), Andersen (), Ibsen & Liingaard (6), Saitoh (1) and others. The procedure to calculate the springs and dashpots is described in the Technical Report by Mylonakis et al (14) and involves the following steps: 1. Determine the frequency-dependent impedance or the dynamic stiffness S(a ) (Eq..4) by means of the finite element, finite difference or the boundary element method, a standing for the familiar dimensionless frequency (ω R/ V s ). To this end, dynamic stiffness should be decomposed into a singular part, S s (a ), and a regular part, S r (a ). The singular part S s (a ) represents the asymptotic value of dynamic stiffness at a, and it is usually considered to involve a vanishing real part (k = ) and a finite imaginary part (c = finite), c being a constant dashpot value determined using alternative procedures such as asymptotic analysis of wave propagation, least squares or conservation of momentum. The difference between S(a ) and S s (a ) defines the regular part S r (a ).. Approximate the regular part S r (a ) by the ratio of two polynomials in dimensionless variable (i a ), P and Q. The degree of the polynomial in the 3-8

127 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil denominator is M and the degree in the numerator is (M 1) to ensure vanishing values at infinity. The approximation of the regular part S r (a ) contains (M 1) unknown real-valued coefficients, which are determined by a curve-fitting technique based on the least-squares method. 3. Establish the lumped-parameter model from the (M 1) real coefficients. The lumped-parameter model may contain several zero-order, first-order and second-order discrete-element models. Definitions for these coefficients are provided in Appendix B. In this summary, it is sufficient to state that the zero-order model contains no internal degrees of freedom (simple springs and dashpots), the first-order model contains one internal degree of freedom (e.g., a spring and a dashpot attached in parallel) and the second-order model contains two internal degrees of freedom. 4. Finally, the lumped-parameter model is assembled by using springs, dashpots and masses with real-valued constants, which can be incorporated directly into dynamic programs such as SAP, ETABS, ANSYS, OPENSEES etc. The four steps in the procedure are explained in detail in the main text of the Technical Report by Mylonakis et al. (14). The report also includes a series of lumped-parameter models that have been developed accounting for frequencydependence of the springs and dashpots of a foundation on liquefiable soil. Typical results of the lumped-parameter approximation is shown in Figure 3.3. It should be pointed out that lumped-parameter models do not provide any information of the stresses or strains in the embedded foundations or the surrounding subsoil, yet they faithfully describe the global compliance of the foundation to dynamic loading, within a pre-specified frequency range. 3-9

128 Chapter 3 (a) (b) FIGURE 3.3: Comparison between cone model results and lumped-parameter approximation for the vertical oscillation mode (h 1 /B =, h /B =.5, V s1 /V s = /3): a) pre-liquefaction, b) post-liquefaction. The next step involves an elastic analysis for the static loads imposed on the bridge and a response spectrum analysis for the seismic loads. It is noted that the dynamic analysis is performed for both pre-liquefaction and post-liquefaction, 3-1

129 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil considering the appropriate, for each case, design spectra and dynamic impedance coefficients. In a second step, the liquefaction-induced ground displacements and rotations should be considered as a separate loading case and, hence, they should be applied to the footing. Finally, the structural response under the liquefaction-induced imposed ground deformations should be checked according to the provisions of the applied code. A back-verification of the bridge design for the design earthquake is deemed necessary by Sextos et al (14) who propose a nonlinear static pushover analysis to check that the bridge can accommodate the imposed deformations without damage. The aforementioned research demonstrated the applicability of the proposed isolation method and concluded that all three bridges studied accepted with safety the soil deformations without any structural damage or loss of serviceability. As a final remark, techno-economic analyses proved that the proposed geotechnical isolation method, depending on the bridge type (concrete, steel) may lead to a reduction of construction cost up to % in comparison with conventional design (Psycharis 15, Kappos 15, Gantes 15). 3.3 Parametric investigation of bridge pier seismic response on liquefiable soil Problem description The structural system examined is illustrated in Figure 3.4 and may be considered as an elementary model of an actual bridge. The pier is single-column type with diameter d = 1.3 m, founded on a spread footing resting on soil prone to liquefaction under seismic action. The axial gravity load accommodated by the system 3-11

130 Chapter 3 is 35 kn, typical of a two lane highway bridge with spans of about 35 m. The bridge deck is free to rotate. Two values for the height of the column bent are considered, H c = 5 and 1 m, to explore the influence of the proposed geotechnical isolation method to squat-like and slender-like structural elements, respectively. It is also supposed that the shape of the spread footing is square with width B = 7 m while its mass is equal to 1 Mg, which is a reasonable value for a large foundation. To be compatible with the results for the dynamic impedance functions obtained in Chapter, it is further assumed that the foundation has no embedment. The liquefiable soil profile has been described in detail in Chapter. It is stressed that two thickness values, h 1 /B =.5 and 1, and shear wave propagation velocities, V s1 = 1 and 5 m/s, of the improved non-liquefiable surface soil layer are examined. Note that the depth setting the rigid bedrock is assumed invariable and equal to 18.5 m. The soil-foundation-pier system is excited by vertically propagating S-waves corresponding to a horizontal rock outcrop motion. Both frequency and time domain analyses are conducted to investigate the impact of liquefaction on: a) the vibrational characteristics (ie., fundamental period, overall damping) of the foundation-bridge pier system, and b) the response of the system to real earthquake motion. In time domain analyses, the Pacoima (Northridge 1994 earthquake) excitation time history having a peak horizontal acceleration (PGA) of.4g, is used. The acceleration time history and the five and ten percent damped spectra are shown in Figure 3.5. Owing to the common practice by structural engineers to utilize the static stiffness of the foundation for design analyses, it is deemed necessary to explore the effect of employing only the static stiffness of the foundation, instead of the actual 3-1

131 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil frequency-dependent stiffness, on bridge response. To this study, kinematic interaction is ignored, which leads to slight conservative results. FIGURE 3.4: Bridge pier supported on spread footing on liquefiable soil with improved non-liquefiable surface crust. 3-13

132 Chapter 3 Ground Acceleration (g) Spectral Acceleration (g) Pacoima, Northridge (1994) Time (sec) Period (sec) FIGURE 3.5: Acceleration time history and five and ten percent damped spectra Numerical results Application to squat bridge piers The harmonic steady-state and transient seismic response of a squat pier, H c = 5 m, with a soft improved surface zone, V s1 = 1 m/s, prior to the onset and during the course of liquefaction, are depicted in Figures The following noteworthy observations can be made with the help of these graphs: 1. For pre-liquefaction case, the fundamental natural period, T 1, of the soil deposit appears to be 1.46 s, which is quite close to the natural period of the thick and stiff clayey base stratum (i.e, 4h 3 / V s3 = 498/ s). The second natural period, T, of the soil deposit is.49 s, which is exactly 1/3 of the fundamental period T 1. The fundamental period of the pier-foundation-soil 3-14

133 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil system is equal to T =.54 s, much close to the second resonance soil, that seems to be crucial to the response of the pier. The amplitude of pier response is about A = 11.3, providing a value of about 4.4 % (A 1/) for the overall damping of the pier-foundation-soil system, which seems reasonable if one considers that the hysteretic material damping alone has been set at 3%.. For the post-liquefaction case, the harmonic steady-state transfer functions display two peaks at 1.78 s and 1. s. The first resonance in the soil takes place at 1.78 s and the second one at 1. s exhibiting a marginally greater amplification. The latter period is due to resonance of the liquefiable sandy layer (i.e, 4h / V sliq = 47/5 1.1 s) which controls the response at this frequency range. The fundamental period of the pier-foundation-soil system is T =.61 s very close to the third-mode resonance occurring at.59 s. The amplification factor for the bridge pier being about A = 4.74 implies that the effective damping of the overall system reaches up to 1.5%. Finally, it is mentioned that the maximum amplification in the soil is about two times smaller compared to that without liquefaction. 3. Comparison of amplification functions given in Figure 3.8 highlights two important features of the effect of liquefaction on seismic bridge response: a) the fundamental oscillation period marginally increases from T =.54 s to T =.61 s (about 11.5%) and b) the amplitude of response decreases considerably, by 58%. 4. The fundamental period of pier-foundation-soil system T for both preliquefaction and post-liquefaction is quite close to the fixed-base fundamental period of the superstructure 3-15

134 Chapter 3 T st ms.41s (3.4) 3 EI / H 3 c This may be attributed to the fact that the deck is free to rotate and, thus, the response is controlled to a greater extent by the characteristics of the superstructure. Moreover, the dominant periods of the rock outcrop motion are within the range.15 to.5 s, which also plays an important role on response. 5. In the time domain (Figure 3.9), comparison of acceleration histories shows a remarkable de-amplification of the seismic motion during liquefaction. As a result, liquefaction has a beneficial effect on seismic response of the pier. 3-16

135 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil Transfer function wrt rock outcrop motion Free Field Footing Bridge Transfer function wrt free-field surface motion Footing Bridge Period (sec) FIGURE 3.6: Harmonic steady-state transfer function for pre-liquefaction case and squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /

136 Chapter 3 Transfer function wrt rock outcrop motion Free Field Footing Bridge Transfer function wrt free-field surface motion Footing Bridge Period (sec) FIGURE 3.7: Harmonic steady-state transfer function for post-liquefaction case and squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s =

137 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil 1 Transfer function wrt free-field surface motion pre-liquefaction post-liquefaction Footing Bridge Period (sec) FIGURE 3.8: Comparison of harmonic transfer functions in pre- and postliquefaction case for squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). 3-19

138 Chapter 3 4 Bridge pre-liquefaction post-liquefaction - Acceleration (m/sec ) Footing Free Field Time (sec) FIGURE 3.9: Comparison of acceleration histories in pre- and post- liquefaction case for squatty bridge pier; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). 3-

139 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil Application to tall bridge piers Results presented in this section refer to a bridge pier with the double height of the previous studied, i.e. H c = 1 m. The following observations can be made: 1. As anticipated, the fundamental vibrational features of the soil do not change, and only the characteristics of the superstructure are modified.. For pre-liquefaction, the fundamental period of the pier-foundation-soil system is equal to T = 1.3 s, closer to the first resonance of the soil; the amplification factor is equal to A 1.6, as shown in Figure 3.1. For postliquefaction, the fundamental period shifts to T = 1.41 s and the amplification factor attains the value A 8.9. Evidently, the period shift is even smaller compared to that for the squat pier, an increase of about 6.4%, and the amplitude decrease is about 16%. 3. The minor effect of liquefaction on seismic response of the pier is also apparent in Figure This is also anticipated if we recall the response spectrum of rock outcrop motion (Figure 3.5). Both the resulting fundamental periods, 1.3 and 1.41s, are far beyond the harmful range of the excitation where the response spectrum tends to be almost flat, which implies that any increase in period has little impact. 3-1

140 Chapter 3 Transfer function wrt free-field surface motion pre-liquefaction post-liquefaction Footing Bridge Period (sec) FIGURE 3.1: Comparison of harmonic transfer functions in pre- and postliquefaction case; H c = 1 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). Acceleration (m/sec ) 4 - Bridge pre-liquefaction post-liquefaction Time (sec) FIGURE 3.11: Comparison of bridge acceleration history in pre- and postliquefaction case; H c = 1 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4). 3-

141 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil Influence of stiffness of the non-liquefiable surface crust The purpose of this section is to elucidate the role of the non-liquefiable improved surface zone on bridge response. To this end, a stiffer surface zone with shear wave propagation velocity V s1 = 5 m/s is examined. The analysis refers to the squat pier. Important findings from Figures 3.1 and 3.13 are summarized below: 1. Evidently, the stiffness of the thin surface soil layer does not alter substantially the free field response and, hence, the fundamental period of the whole soil deposit seems to be invariable. This is observed for both the pre- and postliquefaction case. However, the increase in stiffness of the surface layer does modify the fundamental period of the pier, which seems to be equal to T =.43 s and.45 s for pre- and post-liquefaction conditions, respectively, bringing it closer to the fixed-base fundamental period, i.e, T st =.41 s. This indicates that the stiff surface soil layer operates essentially as a fixity condition for the superstructure, minimizing the influence of the underlying liquefiable soil layer.. With reference to amplitude, in case of no liquefaction the amplification factor appears to be A = 9.9, leading to a damping ratio which is almost the same as that of the superstructure. In case of liquefaction, amplitude diminishes by approximately %, highlighting the important role of the amplified damping. 3. With reference to Figure 3.13, bridge response is de-amplified, mainly due to de-amplification of free field ground response. 3-3

142 Chapter 3 1 Transfer function wrt free-field surface motion pre-liquefaction post-liquefaction Footing Bridge Period (sec) FIGURE 3.1: Comparison of harmonic transfer functions in pre- and postliquefaction case; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = 5/3 (1). Acceleration (m/sec ) 4 - Bridge pre-liquefaction post-liquefaction Time (sec) FIGURE 3.13: Comparison of bridge acceleration histories in pre- and postliquefaction case; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, V s1 / V s = 5/3 (1) Influence of thickness of non-liquefiable surface crust In the following, the influence of the thickness of the improved surface zone on the vibrational characteristics of the bridge pier is studied. Results refer to a squat 3-4

143 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil bridge pier with the properties of the surface zone being h 1 = 7m and V s1 = 1 m/s. Based on Figures 3.14 and 3.15, the following trends are observed: 1. It is notable that the elongation of fundamental period of the pier-foundationsoil system upon liquefaction is trivial.. The decrease in bridge amplification factor is significant and equal to about 4%. However, the discrepancy is lower compared to the predicted for h 1 = 3.5 m. This is anticipated, as the surface zone increases in thickness and, hence, liquefaction has a smaller influence on free-field response. 3. Bridge acceleration histories, in contrast to those in case of h 1 = 3.5 m, show a substantial increase for pre-liquefaction and decrease upon liquefaction. This seems to be reasonable because, when the thickness of the soft soil overlying the stiff stratum doubles, the free-field motion is aggravated. On the contrary, for post-liquefaction, the increase in thickness of the surface soft soil leads to an elongation in fundamental site period, T 1 =.16 s (see Appendix B), resulting in significant de-amplification of seismic motion. 4. The bridge seismic motion attenuates significantly upon liquefaction. 3-5

144 Chapter 3 1 Transfer function wrt free-field surface motion pre-liquefaction post-liquefaction Footing Bridge Period (sec) FIGURE 3.14: Comparison of harmonic transfer functions in pre- and postliquefaction case; H c = 5 m, B = 7 m, h 1 / B = 1, h / B = 1, V s1 / V s = /3 (4). Acceleration (m/sec ) 4 - Bridge pre-liquefaction post-liquefaction Time (sec) FIGURE 3.15: Comparison of bridge acceleration histories in pre- and postliquefaction case; H c = 5 m, B = 7 m, h 1 / B = 1, h / B = 1, V s1 / V s = /3 (4). 3-6

145 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil 3.4 Bridge pier seismic response in view of SSI considering only static stiffness Common practice in the seismic analysis and design of bridges is to neglect the frequency dependence of stiffness and to consider only the static value, i.e., stiffness for ω =. This section examines how the use of static stiffness influences bridge response. Based on the more rigorous BEM results in Chapter, the static stiffness values for the case h 1 / B =.5, h / B = 1, V s1 / V s = /3 (4) and H c = 5 m, considering only the horizontal and rocking mode (vertical stiffness is ignored) are utilized. It is noted that for the present analysis, for both pre- and post-liquefaction, soil damping ratio is assumed equal to 1%, which seems to be a more reasonable value for a soft soil. Moreover, radiation damping is ignored. From Figures 3.16 and 3.17, it is observed that: 1. Evidently, the fundamental period of the pier-foundation-soil system is expected not to change. However, it should be pointed out that, given the deviation in static stiffness between CONAN and BEM results which was discussed in the previous Chapter, the current fundamental period of the system, considering no liquefaction, is computed as T =.5 s instead of T =.54 s. The discrepancy is insignificant and it is interesting, that even though the difference in static stiffness is appreciable (in this case 18% for the horizontal and 5% for the rocking mode), there is little impact on pier response.. The same also applies for post-liquefaction conditions. The fundamental period of the system is estimated as T =.59 s instead of T =.61 s. Using BEM results, the shift in period appears to be slightly greater, equal to about 15%. 3-7

146 Chapter 3 3. With reference to the amplification factor, it is observed that for both cases the amplification factor is relatively low, due to the assumed value of soil material damping (1%). Considering only static stiffness and no radiation damping, the decrease in amplitude is less than about 11%. 4. The considerable high peaks in the footing harmonic transfer functions in the part with the very short periods, is probably a result of secondary resonance between the strong-short period part of the excitation motion and the resonance of the first soil layer (i.e, 4h 1 / V s1 = 43.5/1.14 s). However, the impact of this part of footing motion on bridge response is insignificant. 5. Comparison of bridge acceleration history in pre- and post- liquefaction shows that, under the effect of liquefaction, bridge motion decreases. 1 Transfer function wrt free-field surface motion pre-liquefaction post-liquefaction Footing Bridge Period (sec) FIGURE 3.16: Comparison of harmonic transfer functions in pre- and postliquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = /3 (4). 3-8

147 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil Acceleration (m/sec ) 4 - Bridge pre-liquefaction post-liquefaction Time (sec) FIGURE 3.17: Comparison of bridge acceleration history in pre- and postliquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = /3 (4). In the same vein, the case h 1 / B =.5, h / B = 1, V s1 / V s = 5/3 (1) and H c = 5 m is studied. Results are depicted in Figures 3.18 and Evidently, employing static stiffness has no effect on the fundamental period of the system. Further, assuming no radiation damping leads to a small reduction in amplitude, equal approximately to 6%. In the time domain, it is apparent that liquefaction has little impact on the bridge acceleration history. It is noteworthy that the reduction in static stiffness due to occurrence of liquefaction assuming either a soft or a stiff improved surface zone is about the same. However, the reduction in stiffness appears to be of secondary importance, with the stiffness of the surface layer controlling the seismic response of the pier. Overall, using static stiffness in seismic analysis and design of bridges may lead to conservative results. 3-9

148 Chapter 3 1 Transfer function wrt free-field surface motion pre-liquefaction post-liquefaction Footing Bridge Period (sec) FIGURE 3.18: Comparison of harmonic transfer functions in pre- and postliquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = 5/3 (1). Acceleration (m/sec ) 4 - Bridge pre-liquefaction post-liquefaction Time (sec) FIGURE 3.19: Comparison of bridge acceleration history in pre- and postliquefaction case considering only static stiffness; H c = 5 m, B = 7 m, h 1 / B =.5, h / B = 1, β = 1%, V s1 / V s = 5/3 (1). 3-3

149 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil 3.5 Verification of results via theoretical investigation Fundamental frequency of a three-layer soil deposit Using the Rayleigh quotient procedure, a generalized closed-form solution is developed for the prediction of the fundamental natural period of a three-layer soil deposit. A different inhomogeneity coefficient for each layer and a zero shear modulus at the ground surface are assumed, as shown in Figure 3.. Following previous research (Durante et al. 15), a shape function obtained from the lateral equilibrium of a soil column having the same layering and inhomogeneity properties, is utilized. G 1,B G,B G 3,B G H 1 z G1z G1, B H b1 H H z Gz G, B H b z b3 z G3z G3, B H FIGURE 3.: Inhomogeneous three-layer soil deposit over a rigid base. 3-31

150 Chapter 3 The current derivation pertains to a three-layer deposit over a rigid base. The mass density is assumed constant within each layer, while the shear modulus increases with depth according to the equation z G z GB H b (3.5) where G B refers to the shear modulus at the bottom of the layer, b is an inhomogeneity parameter and z is depth measured from ground surface. Assuming an inhomogeneous soil column and one-dimensional shear wave propagation under harmonic oscillations leads to the governing equation d du Gz u dz dz (3.6) Based on Mylonakis et al (13), the natural frequencies of the system may be derived using the Rayleigh quotient H H G z u z dz u z (3.7) By substituting u(z) = u ψ(z), Eq. (3.7) yields H H G z z dz z (3.8) where ψ(z) is a dimensionless unitary shape function representing the mode shape corresponding to a particular natural frequency of the inhomogeneous soil. For the fundamental mode, the shape function is obtained by means of a simple method in which the soil column is modeled as a shear beam and the distributed horizontal load 3-3

151 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil acting on the soil column is equal to the self-weight of the soil. The shape mode is determined as the lateral movement of the soil column under the distributed load, considering the origin of displacement axis at the top of the soil column. This formulation allows considering the variations of both unit weight and shear modulus with depth. Therefore, the soil lateral displacement is Gz z z z gz u z z dz dz (3.9) where γ(z) is shear strain. The shape function should be unitary at ground surface and zero at the base, to satisfy the boundary conditions of the problem. Therefore, z 1 z H Gz Gz z gz dz z gz dz (3.1) Specific forms of the above equation, for the single layer and the two-layer soil profile are provided in Durante et al. (15) and are provided in Appendix B. Shape function for a three-layer soil deposit For a three-layer soil deposit, Eq. (3.1) becomes H z H H 1H 1 H z 1H z a a H zh D b b a b3 1 z, H z1 b3 b1 b1 b b b3 1 1 a1 a, z H 1 b1 b b3 b b b3 1, 1 3 (3.11) 3-33

152 Chapter 3 with H 1 = H 1 /H and H = H /H being the dimensionless depth of the first and second layer of the soil deposit, z = z/h is the dimensionless depth measured from the ground surface. On the above equations, H H H 1H D a a b1 b b b b1 b b3 (3.1) V s1, B V s1, B a1, a V s, B V s3, B (3.13) with V s,b = (G B /ρ) 1/ being the shear wave propagation velocity at the bottom of the layer. Computation of Rayleigh quotient Using the shape function in Eq. (3.11), the solution to Eq. (3.8) becomes b1 b b3 H1 z H z H z 1, B ', B ' 3, B ' H H1 H H H n H1 H H 1 z dz z dz3 z dz H 1 H G z dz G z dz G z dz (3.14) Evaluating the integrals, Eq. (3.14) takes the following form 1 1 k1 a1 1k a V k3 s1, B n k4 1k5 k6 H (3.15) where 3 b1 H 1 k1 3 b 1 (3.16) k H H 3b 3b 1 a1 3 b (3.17) 3-34

153 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil k 1 H 3b3 3 a 3 b3 (3.18) H H k C H C 5b1 3b (3.19) 3b15b1 3b1 H H 1H C a a b b b3 1 1 b b3 (3.) k b 3 1 a H H H 1 5 b3 5 b 5 b b 3b 3b a 1 4 b H H H1 H 1 H H H H 1 b 5b 3b b3 aa H 1 H H H H 3 b H H b 3b b b b b 3 (3.1) k a 1H 1H (3.) 5b3 3b3 6 1H b3 5b3 3b3 3 1, 1 (3.3) 1 In case of a three-layer soil deposit with each layer being homogeneous i.e., having constant properties with depth (b 1 = b = b 3 = ), the above parameters simplify to 1 k H k a H H k a H , 1, 3 (3.4a,b,c) 5 3 H 1 H 1 b H 3 1 b3 k4 a1h 1 H a1h a1h 1 H a1h (3.5) 3-35

154 Chapter 3 1 k5 H 1 H 1aa 1 H 1 H H 1 H 1H 15a 1H 6 a1 H 1 H 3H 1 9H 1H 8H (3.6) a 3 k 6 1H 89H 3H (3.7) 6 The above methodology is applied to the case of the liquefiable soil deposit which is examined here. For the pre-liquefied case and soft improved surface zone (V s1 = 1 m/s), referring to applications and 3.3.., the computed fundamental natural period of the soil deposit is T 1 = 1.44 s, almost the same value as that obtained using SFIAB (1.46 s). With reference to the post-liquefied case, the fundamental natural period of the soil deposit is estimated as T 1 = 1.64 s which, as expected, is slightly lower than the obtained numerical value (1.78 s). Referring to application , where the stiffness of the improved zone is enhanced (V s1 = 5 m/s), the fundamental natural period of the soil deposit for both pre-liquefaction and post-liquefaction does not change, indicating that the thick and stiff base stratum controls the response of the free field. With reference to application , the thickness of the surface zone is doubled and in that case Eq. (3.15) yields T 1 = 1.45 s for pre-liquefaction and T 1 = 1.9 s for post-liquefaction, values very close to those obtained by means of SFIAB, 1.46 s and.16 s, respectively. The predictions of the Rayleigh method naturally underestimate the actual values provided by the numerical analysis. From the above results, it is evident that liquefaction leads to an elongation of the fundamental natural period of the soil deposit, which is attributed to the softening of the middle soil layer. The extent of the elongation depends on the thickness of the liquefied soil and the improved surface zone. As the thickness increases the period 3-36

155 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil shift is greater. Evidently, this mechanism may attenuate free-field response and therefore, the seismic forces induced to the superstructure Fundamental period of the pier-foundation-soil system The effect of liquefaction on the fundamental period of the bridge pierfooting-soil system can be investigated via the analytical solution of Veletsos and Meek (1974). The corresponding analytical formula is Kst KhHc T Tst 1 1 (3.8) Kh Kr where H c, K st, and T st are the height, stiffness and fixed-base fundamental period of the superstructure, and K h, K r are the horizontal and rocking stiffness of the foundation, respectively. Results obtained from the analytical formula in Eq. (3.8) are presented in Tables 3.1 and 3. for each case studied in the previous sections. It is shown that the agreement between the predicted values and the corresponding numerical values obtained via SFIAB are in excellent agreement. The analytical solution verifies that liquefaction has little impact on the fundamental period of the system and the stiffness of the improved zone seems to control the vibrational features. 3-37

156 Chapter 3 TABLE 3.1: Analytical computation of the fundamental natural period of the pier-foundation-soil system for pre-liquefaction conditions. Pre-liquefaction Application K h (kn/m) K r (knm) T (s) (3.3..1) BEM (3.3..3) BEM TABLE 3.: Analytical computation of the fundamental natural period of the pier-foundation-soil system for post-liquefaction conditions Post-liquefaction Application K h (kn/m) K r (knm) T (s) (3.3..1) BEM (3.3..3) BEM Conclusions A novel seismic bridge design concept using geotechnical isolation via a natural liquefiable soil, was revisited following the studies by Bouckovalas et al 3-38

157 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil (14a,b) and Sextos et al. (14). The feasibility of the proposed geotechnical isolation of bridge piers on shallow foundations was further studied by means of frequency and time domain analyses via a modified version of code SFIAB. The above analyses are significant because they can reveal the actual effect of liquefaction on the response of a structure and can shed light to the vibrational characteristics of the pier-foundation-liquefiable soil system. To verify the SFIAB results, a theoretical solution based on the Rayleigh quotient and considering an appropriate shape function, is developed for estimating the fundamental natural period of a three-layer soil deposit. Furthermore, the fundamental natural period of the system studied is checked through a well-known analytical formula. A parametric study was conducted to elucidate the role of thickness and stiffness of the surface improved zone, as well as the height of the superstructure on the dynamic response of the bridge pier. The main conclusions of the study are: To assess the impact of liquefaction on dynamic response of bridge piers, two separate analyses, prior and post liquefaction, are necessary. In the frequency domain, results did not show a spectacular elongation in fundamental period of the structural system, as might have been expected, but on the other hand a significant decrease in amplification factor was observed. Time-history analyses revealed a significant decrease in free field motion and an analogous decrease in bridge response. In case of tall piers, the occurrence of liquefaction does not notably modify the vibrational characteristics of the pier-foundation-soil system. A small shift in fundamental period of the system and a marginal decrease in amplitude are observed, while the peak acceleration developing in the bridge seems to be unaffected. 3-39

158 Chapter 3 The stiffness of the non-liquefiable surface soil layer plays a significant role in the dynamic response of the pier-foundation-soil system. A stiff improved zone appears to act as base fixity for the pier and, hence, the fundamental period of the system is very close to the fixed-base value. Nevertheless, the bridge response is de-amplified, due to the enhanced effective damping of the system and the softening of the soil deposit. Another important factor is the thickness of the surface soil layer. The increase in thickness has a minor effect on the elongation of the fundamental period of the system. Since static foundation stiffness, instead of the actual frequency-dependent impedance functions, are commonly used in dynamic structural analysis and design, an investigation of the consequences of this practice was deemed necessary. For this purpose, zero radiation damping and equal soil material damping was assumed for both pre- and post-liquefaction cases. It is shown that using the static value instead of the frequency-dependent stiffness, has apparently little effect on the fundamental period of the system. However, the beneficial effect of the enhanced damping of the liquefied soil was not considered, leading to conservative results. Nevertheless, upon liquefaction, the bridge motion decreases due to the softening of the soil that affects the free-field response. On the other hand, if the improved zone is very stiff, liquefaction has little impact on the bridge motion. The main conclusion drawn from the above investigation is that the impact of liquefaction on the pier response is threefold. The first mechanism is the elongation of the site period due to the softening of soil which indicates that the triggering of liquefaction may attenuate the seismic motion. The second 3-4

159 Geotechnical isolation of bridges on surface footings exploiting the natural liquefiable soil mechanism refers to the increase of damping during liquefaction, which also dissipates seismic energy leading to a reduced free-field motion. The third mechanism is the shift in fundamental period of the pier-foundation-soil system, which may transfer the period of the system out of the harmful frequency range of the earthquake excitation. Given that no significant elongation in the fundamental period of the system is observed, it is concluded that the most important mechanisms are the elongation of the site period and the corresponding increase in damping. As a final remark, the above analysis shows some important trends and the interpretation of the results helps in understanding the role of liquefaction in bridge response. However, these results should not be generalized to any pier-foundationliquefiable soil system. It is pointed out that the effect of liquefaction on a structural system depends on the geometry of structure, the soil properties and site features, and the characteristics of the seismic excitation. Therefore, each case needs a separate investigation. 3-41

160 Chapter 3 3-4

161 Chapter 4 Pile dynamics: some new theoretical solutions 4.1 Introduction The objective of the present chapter is to summarize a series of novel analytical solutions which can be used for the design of pile foundations against dynamic loads. The theoretical solutions developed herein lead to simple closed-form expressions which can be implemented by means of a pocket calculator or simple computer spreadsheets and, thereby, can be used in routine engineering design. Emphasis is placed on the problem of a laterally-loaded cylindrical pile segment surrounded by a circular zone of soft elastic material, accounting for an inclusion or material nonlinearity in the vicinity of the pile, under plane strain conditions. To this end, the seminal model of Novak & Sheta (198) is utilized to derive the stiffness of the system with reference to different boundary conditions at the interfaces separating the annular zone from the pile and the outer material. The predictions of the model are compared against those of two simple models based on strength-of-materials theory while three-dimensional effects and numerical simulation aspects are discussed. The ability of the model to predict the stiffness of the soft 4-1

162 Chapter 4 (inner) zone is of paramount importance, as it gives thrust to a new idea of geotechnical seismic isolation. Employing a soft material with known properties in this zone renders the pile-structure system more flexible, resulting in an increase in the fundamental natural period of the system and an associated change in damping. This issue is investigated in detail in Chapter 5. In this chapter, an approximate, practically-oriented analysis procedure for estimating the dynamic stiffness and damping of a laterally-loaded pile in different types of vertically inhomogeneous soil is presented (Karatzia and Mylonakis 16a). To this end, a dynamic Winkler model is adopted in conjunction with a virtual-work scheme associated with approximate shape functions for the pile deflection under imposed head displacements and rotations. The expressions of pile stiffness are given in the form of simple formulae and compare favorably with numerical solutions. Formulae for determining the moduli of the distributed Winkler springs and dashpots are also presented. The implementation of the equations is illustrated with the help of a worked example. An additional topic regarding kinematic pile-soil interaction is also specifically addressed. Kinematic interaction stems from the presence of a stiff foundation resting on or embedded in the soil, which leads to a considerable deviation between foundation and free-field motion. For shallow or embedded footings, there is an established methodology for modifying the free-field motion to account for kinematic interaction (NIST 1). For pile-supported foundations, the interaction between piles and wave passage and the potential settlement of the soil beyond the base of the structure render the problem highly complex. For this purpose, an analytical solution is presented to incorporate the effect of kinematic interaction on 4-

163 Pile dynamics: some new theoretical solutions the response of the superstructure. The kinematic modification factors developed can be readily applied in design. 4. Horizontal soil reaction of a cylindrical pile segment with a soft zone 4..1 Introductory remarks In their seminal work, Novak and Sheta (198) model contact effects around an embedded cylindrical pile by considering an elastic or viscoelastic annular zone of finite thickness surrounding the pile. The body of the pile is considered sufficiently long and the gradient of its response with depth sufficiently small, so that the assumption of plane strain conditions is justified. As a rigorous approach to account for material and geometric nonlinearities in the vicinity of the pile is highly complex, the approximate theory of Novak and Sheta and its variants have gained wide acceptance by engineers and researchers for application in both static and dynamic problems, leading to a number of closed-form solutions for horizontal, vertical, rocking and torsional response modes (Lakshmanan and Minai 1981, Veletsos and Dotson 1986, 1988, Poulos 1988, Novak & Han 199, Han and Sabin 1995, Michaelides et al 1998, El-Naggar, Karatzia et al. 14). These developments have been incorporated into commercial and academic computer codes such as DYNA and SPIAB (Novak et al 1993, Mylonakis 1995). With reference to boundary conditions, the assumption of perfect bonding at the interface separating the pile from the annular zone (inner interface, Figure 4.1a) as well as that separating the annular zone from the outer domain (outer interface, Figure 4.1a) provides practical means for solving the problem under vertical, torsional and 4-3

164 Chapter 4 rocking loading. For horizontal loading, however, four pairs of shear boundary conditions are possible: a. Perfectly bonded inner interface and perfectly bonded outer interface b. Perfectly smooth inner interface and perfectly smooth outer interface c. Perfectly bonded inner interface and perfectly smooth outer interface d. Perfectly smooth inner interface and perfectly bonded outer interface Of these conditions, the first (a) has been explored by Novak and Sheta and it is the one adopted in most of the literature. Some results pertaining to a smooth inner interface have been reported by Klar and Osman (8), whose work, however, is limited to an unbounded incompressible medium. To the best of the author knowledge, conditions (b), (c) and (d) have not been investigated in the past. The lack of knowledge as to the physics of the solution for these cases and the associated implications in piling engineering and soil-structure interaction provided the initial motivation for the herein reported work. It should be noted that in the realm of the specific model, the stiffness of the overall system is determined by combining the compliance of the soft annular zone with that of the outer medium in the form of a pair of springs attached in a series (Figure 4.1b). This treatment is approximate for the distortion of the interfaces is not accounted for, thus leading to a somewhat stiffer system. However, as the material in the outer domain is typically over an order of magnitude stiffer than that in the inner one (Novak & Sheta 198, Poulos 1988, Novak et al 1993), the overall compliance practically coincides with that of the inner zone. Accordingly, the present work focuses exclusively on the behavior of this region. Discussions on the stiffness of the outer domain can be found, among other studies, in Baguelin et al (1977), Novak et al 4-4

165 Pile dynamics: some new theoretical solutions (1978), Roesset (198), Novak and Sheta (198) and Mylonakis (1). Threedimensional effects, notably the stress component τ rz acting at the upper and lower faces of the soil slice (Figure 4.1b) which couples the response of horizontal layers at different elevations, are discussed in the ensuing. FIGURE 4.1: Problem considered and modeling approach 4..1 Problem definition The problem under consideration is depicted in Figure 4.: a rigid circular pile segment of radius r, surrounded by an annular zone of homogeneous, isotropic elastic material of shear modulus G, Poisson s ratio v and width t. The radial component of displacement at an arbitrary point in the elastic domain is denoted by u and the tangential by v. The problem is characterized by point symmetry, thereby polar coordinates (r, θ) are employed, varying from r to (b r ) and from to π, respectively. The pile is subjected to a static load P acting in the radial direction θ =, resulting in a displacement vector u(r, ) = u, v(r, ) =, at the inner interface. The 4-5

166 Chapter 4 outer boundary is located at a distance b = r + t from the pile center and is considered rigid in the radial sense, thereby u(b, θ) =. Different assumptions are employed as to the tangential (shear) tractions and the corresponding displacements along the inner and outer interfaces, as discussed in the following. FIGURE 4.: Problem geometry and parameter definition Following the approach of Novak and Sheta, the planar displacement field (u, v) is decomposed in terms of a pair of potential functions (ψ 1, ψ ) as: ur 1 (, ) [4(1 ) 1(, ) ( 1)] G v r r r (4.1) 1 1 v( r, ) [4(1 ) (, ) ( 1)] G v r r r (4.) For the conditions at hand, these functions have to satisfy the harmonic equation (Timoshenko & Goodier, 197): ( 1, ) (4.3) where 1 1 r r r r (4.4) 4-6

167 Pile dynamics: some new theoretical solutions is the Laplacian operator. Employing the separation of variables: 1(, r ) 1()cos r, (, r ) ()sin r (4.5a,b) corresponding to an even (ψ 1 ) and an odd (ψ ) function of θ, it is straightforward to show that the above potentials admit the solutions: ( r, ) [ ln( r/ r ) ( r/ r ) ( r/ r ) ]cos (4.6) ( r, ) [ ln( r/ r ) ( r/ r ) ( r/ r ) ]sin (4.7) in which A 1, A, A 3, A 4 are integration constants to be determined from the boundary conditions. It should be noticed that due to the two-dimensional nature of the problem, the logarithmic term ln(r/r ) is inherent in the solution. Also, the presence of characteristic lengths r and b makes the problem not self-similar and, thereby, does not allow establishing the exponents of the terms (r/r ) by dimensional means. From Eqs. (4.1), (4.), (4.6) and (4.7), the displacement components u and v are determined as: ur (, ) G [(34 v)ln( r/ r) 1] (3 4 v) (1 4 v)( r/ r) (5 4 v)( r/ r) cos (4.8) 1 v( r, ) 1(34 v)ln( r/ r) (3 4 v) 3(5 4 v)( r/ r) 4(54 v)( r/ r) sin G (4.9) which coincide to those in the original paper by Novak and Sheta. On the basis of the standard relations in polar coordinates ε rr = u / r, ε θθ = ( v / θ + u) / r and are: r v/ r v/ r1/ r( u/ ), the strains in the medium rr(34) v r(14) v r(54) v cos (4.1) rr Gr r 4-7

168 Chapter 4 r rr(1) v r r(54) v sin (4.11) Gr r rr r(34 v) r(54 v) cos (4.1) Gr r From Hooke s law, it is straightforward to show that the corresponding stresses are: rr rr(3 v) r r(5 4 v) cos (4.13) rr r rr(1) v r r(54) v sin (4.14) rr rr(1) v 6r r(54) v cos (4.15) rr Note that due to symmetry, normal stresses σ rr and σ θθ are even functions of θ, whereas the shear stress τ rθ is an odd function. Under the assumption of plane strain conditions, ε zz = ε zr = ε zθ = and σ zz = v (σ rr + σ θθ ). Naturally, all stresses and strains are independent of the rigid body displacement component Α (3 4v) Stiffness coefficient of the system Case a: Rough inner and outer interfaces This is the problem considered by Novak & Sheta (198) and it is revisited here as a reference case. The boundary conditions pertaining to the specific assumptions are: ur ( r, ) u, v( r r, / ) u (4.16a,b) ur ( b, ), v( r b, ) (4.16c,d) The first condition (Eq. 4.16a) describes a radial displacement u of the inner boundary in the direction θ =. The second condition (Eq. 4.16b) corresponds to a tangential displacement (u ) of the inner boundary in the direction θ=π/, as required 4-8

169 Pile dynamics: some new theoretical solutions by the assumption of a perfectly bonded inner interface. Equations (4.16c) and (4.16d) express the fixity conditions at the outer interface. Enforcing these constraints in Eqs. (4.8) and (4.9) yields a set of simultaneous equations, the solution of which is given in Appendix C. Assuming a unit displacement at the inner boundary (u = 1), the stiffness of the system can be determined from the traction resultant (Baguelin et al, 1977, Veletsos & Younan, 1994): ( rr )cos r ( )sin (4.17) k r r r d which, naturally, is measured in units of stress (as the above equation is implicitly normalized by u ). Upon integration over θ, Eq. (4.17) simplifies to: rr r k r ( r,) ( r, /) (4.18) Using this result in conjunction with Eqs (4.13), (4.14) and the computed integration constants A i (Appendix C) yields the stiffness coefficient of the system: 3 4 ln / 8 G 34v 1v b r k (4.19) b r v b r b r which coincides with the solution of Novak and Sheta. The stress and strain fields (not reported by Novak and Sheta) are determined from Eqs (4.1) (4.15) as functions of applied displacement and given in Appendix C. Case b: Smooth inner and outer interfaces The boundary conditions pertaining to this case are: ur ( r, ) u, r r r (, ) (4.a,b) ur ( b, ), r ( r b, ) (4.c,d) 4-9

170 Chapter 4 of which Eqs. (4.a) and (4.c) have already been discussed. The conditions in Eqs. (4.b) and (4.d) define zero shear stresses at the inner and outer interfaces, respectively. The corresponding stiffness coefficient is: k 8 G b r 1v 1v b r 34vb r ln b/ r (4.1) Case c: Rough inner interface, smooth outer interface This configuration constitutes an intermediate case between those in a and b. The corresponding boundary conditions are: ur ( r, ) u, v( r r, / ) u (4.a,b) ur ( b, ), r ( r b, ) (4.c,d) the meaning of which has already been described. Likewise, k G 1 v b 3 4v r 4 4 b r b v34vr 7v4v 34vb r 34vln b/ r (4.3) Case d: Smooth inner interface, rough outer interface In this last case to be examined as part of the particular model, the pertinent boundary conditions are: ur ( r, ) u, r r r (, ) (4.4a,b) ur ( b, ), v( r b, ) (4.4c,d) Accordingly, k G 1v b 34v r 4 4 b r b 7v4v rv34v34vb 34vr ln b/ r (4.5) 4-1

171 Pile dynamics: some new theoretical solutions For all cases studied, the solutions for stress and strain fields are presented in the Appendix C, however the graphical representation for stresses is thoroughly commented below Strength-of-materials modeling It is instructive to compare the above solutions against simpler models based on strength-of-materials theory, as shown in Figure 4.3. The first model (Figure 4.3a) considers a square pile segment surrounded by four trapezoidal zones having total area equal to that in the original domain [i.e., ( x.89b) π b ]. In this approach, the imposed load is resisted upon compression and tension of two zones of elastic material located in the front and the back of the pile segment, respectively, and shearing of the two zones on the sides. This configuration is analogous to the one employed by Gazetas & Dobry (1984) for investigating pile radiation damping in an unbounded medium, an approach which was later extended to other related problems by Wolf and co-workers (Wolf 1994). Under the familiar assumption that plane sections remain plane, it is straightforward to show that the stiffness of the system in Figure 4.3a is given by 3 v k 4G ln 1 t/ d (4.6) The variation of stresses in the two trapezoidal zones is 1 E P 4 E Gr, 1 4 G P E Gr (4.7a,b) In the second model (Figure 4.3b) an annular zone identical to the one in the original problem is utilized. Unlike the exact solution, however, the domain is divided into an infinite number of thin independent sectors, with each sector possessing an infinitesimal amount of stiffness according to the strength-of-materials solution 4-11

172 Chapter 4 1 dk E G d ln 1 / cos sin t d (4.8) where θ is the polar angle defined in Fig. 4.. Upon integration it is simple to show that the overall stiffness is 3 v k G ln 1 t/ d (4.9) which, remarkably, is identical to that in Eq. (4.6) except for a different multiplier (π instead of 4). The corresponding expressions for normal and shear stresses in the infinitesimal sectors are 1 E P E G r cos, 1 G P E G r sin (4.3a,b) The absence of radii r and b from the expressions for stresses in last two models is noteworthy and allows establishing the (1/r) attenuation of stresses in the elastic medium by pure dimensional arguments. FIGURE 4.3: Simple mechanistic analog models based on strength-of-materials theory 4-1