Παραμετρική διερεύνηση της σεισμικής απόκρισης αλλουβιακών κοιλάδων

Abstract

Η εργασία αφορά στην παραμετρική διερεύνηση του προβλήματος της (γεωμορφικής) επιδείνωσης της μέγιστης σεισμικής επιτάχυνσης στην επιφάνεια του εδάφους αλλουβιακών κοιλάδων. Για το σκοπό αυτό εκτελέσθηκαν 2Δ αριθμητικές αναλύσεις σεισμικής απόκρισης για ομοιόμορφες τραπεζοειδείς λεκάνες επί ιξωδο-ελαστικού υποβάθρου, με διεγέρσεις που αφορούν κατακορύφως προσπίπτοντα κύματα SV με χρονοϊστορίες που βασίζονται σε σεισμικές καταγραφές (Αίγιο, Κοζάνη 1995).

Ειδικότερα, στόχος της εργασίας είναι η αναίρεση ορισμένων περιορισμών που έχουν ληφθεί υπόψη στην αναλυτική και αριθμητική μελέτη του προβλήματος στη βιβλιογραφία. Πιο συγκεκριμένα, χάριν απλότητας, η βιβλιογραφία συνήθως επικεντρώνεται σε ομοιόμορφες συμμετρικές λεκάνες (διαφόρων σχημάτων), που έχουν οριζόντια επιφάνεια αναγλύφου και στις οποίες το έδαφος και ο βράχος θεωρούνται ομοιόμορφα (ιξωδο-) ελαστικά υλικά. Για την ποσοτικοποίηση της γεωμορφικής επιδείνωσης στην οριζόντια διεύθυνση ορίζεται ο λόγος Αh της μέγιστης οριζόντιας επιτάχυνσης σε κάθε θέση, προς την αντίστοιχη τιμή υπό 1Δ συνθήκες ταλάντωσης στην ίδια θέση. Αντίστοιχα, ορίζεται ο λόγος Av σε κάθε θέση, με τον αριθμητή να έχει ως τιμή τη μέγιστη (παρασιτική) κατακόρυφη επιτάχυνση σε κάθε θέση, και τον παρονομαστή να είναι εκείνος του Ah, δεδομένου ότι δεν υπάρχει κατακόρυφη ταλάντωση υπό 1Δ συνθήκες.

Έτσι, αρχικώς, μελετήθηκε η επίδραση της μη-συμμετρίας της εδαφικής λεκάνης (με τα δύο θαμμένα πρανή της να έχουν διαφορετική γωνία κλίσης i) στη γεωμορφική επιδείνωση της μέγιστης οριζόντιας Ah και της παρασιτικής κατακόρυφης Av επιτάχυνσης στην επιφάνεια του εδάφους. Για περιπτώσεις «στενών» εδαφικών λεκανών παρατηρείται ταύτιση της απόκρισης της μη-συμμετρικής κοιλάδας με εκείνες των αντιστοίχων συμμετρικών στην περιοχή των άκρων (αλλά και εκτός των ορίων της κοιλάδας, στους αναδυόμενους βράχους εκατέρωθεν), τόσο για τις οριζόντιες (Αh) όσο και για τις κατακόρυφες (Αv) επιδεινώσεις. Ταυτόχρονα όμως, παρατηρείται απόκλιση της σεισμικής απόκρισης στην περιοχή του κέντρου, ως αποτέλεσμα της έντονης αλληλεπίδρασης των ανόμοιων πρανών που βρίσκονται σε σχετικώς κοντινή απόσταση. Αντιθέτως, για περιπτώσεις «ευρειών» ή «πολύ ευρειών» (οιωνεί «μονοκλινών») εδαφικών λεκανών προκύπτει τέλεια ταύτιση της σεισμικής απόκρισης της μη-συμμετρικής λεκάνης με τις εκείνες των αντιστοίχων συμμετρικών, και στις δύο συνιστώσες της ταλάντωσης, και σε όλο το μήκος της λεκάνης, καθώς δεν υπάρχει αλληλεπίδραση των πρανών (πέραν του μη-μηδενισμού της Av στο κέντρο της μη-συμμετρικής κοιλάδας).

Στη συνέχεια, μελετήθηκε η αλληλεπίδραση της τοπογραφίας αναγλύφου και μορφολογίας υποβάθρου στη σεισμική απόκριση. Προκύπτει πως η εδαφική λεκάνη δεν επηρεάζει τη σεισμική απόκριση στα αναδυόμενα πρανή υποβάθρου. Με άλλα λόγια, οι τιμές των οριζόντιων και κατακόρυφων ενισχύσεων της σεισμικής κίνησης, στην περιοχή εκτός των ορίων μιας εδαφικής λεκάνης που έχει υπερυψωμένα πρανή, εξαρτώνται μόνο από το πόσο έντονη είναι η τοπογραφία του αναγλύφου (δηλαδή πόσο είναι το ύψος του πρανούς που εξέχει από την επιφάνεια, ή και η κλίση του), κι όχι από το αν το υλικό στη βάση του πρανούς είναι βραχώδες ή εδαφικό. Αντιθέτως, η ύπαρξη υπερυψωμένων πρανών οδηγεί στην αύξηση των σημαντικών τιμών του συντελεστή Αh και την εμφάνιση των μεγίστων τιμών αυτού σε μεγαλύτερη απόσταση από τα όρια της λεκάνης. Η επίδραση αυτή είναι μικρή σε περιπτώσεις «μονοκλινών» ή «ευρειώ ν» κοιλάδων, αλλά σημαντική σε περιπτώσεις «στενών» κοιλάδων, όπου οι μέγιστες τιμές επιδείνωσης εμφανίζονται στο κέντρο τους. Σχετικά με το συντελεστή Av παρατηρείται πολύ μικρή μείωση των σημαντικών ενισχύσεων και εμφάνιση των μεγίστων τιμών του σε μεγαλύτερη απόσταση από τα όρια της λεκάνης κυρίως σε «στενές» και λιγότερο στις «ευρείες» ή «μονοκλινείς» κοιλάδες.

Τέλος, μελετήθηκε η επίδραση της χρήσης ενός μη-γραμμικού υστερητικού προσομοιώματος (τύπου Ramberg-Osgood) για το έδαφος, αντί για τη συνήθη θεώρηση ιξωδο-ελαστικής συμπεριφοράς. Η πιο ρεαλιστική αυτή προσομοίωση επιβεβαίωσε ότι η ιξωδο-ελαστική θεώρηση δίνει ποιοτικώς ορθά αποτελέσματα για τη γεωμορφική επιδείνωση, αλλά η ποσοτική της ακρίβεια περιορίζεται σε μικρής και μεγάλης έντασης διεγέρσεις. Πιο συγκεκριμένα, η αύξηση της έντασης της διέγερσης οδηγεί σε αύξηση των οριζόντιων (Αh) και κατακόρυφων (Αv) γεωμορφικών ενισχύσεων. Η αύξηση αυτή οφείλεται κυρίως στη μείωση των ενισχύσεων υπό 1Δ συνθήκες, δηλαδή στη μείωση του κοινού παρονομαστή των λόγων Αh και Av με την αύξηση της έντασης ως αποτέλεσμα της πιο έντονης μη-γραμμικής συμπεριφοράς του εδάφους (γίνεται πιο εύτμητο με μεγαλύτερη απόσβεση). Επιπλέον προκύπτει ότι η ανωτέρω αύξηση των Ah και Av γίνεται σημαντικότερη, όσο αυξάνουν οι σημαντικοί κύκλοι της διέγερσης, και όσο πιο μη-γραμμικό είναι το έδαφος (π.χ. για συνεκτικό έδαφος η ιξωδο-ελαστική θεώρηση είναι ποσοτικά ακριβής για μικρής και μέτριας έντασης διεγέρσεις, αλλά για μη-συνεκτικό έδαφος μόνο για μικρής έντασης).

This thesis concerns the parametric investigation of the (geomorphic) aggravation of the peak seismic acceleration at the ground surface of alluvial valleys. For this purpose, 2D numerical seismic response analyses were performed for uniform trapezoidal valleys on visco-elastic bedrock with the finite difference method (FLAC). The excitations used were vertically impinging SV waves with time-histories based on seismic recordings (Aigion and Kozani, Greece, earthquakes of 1995) that were appropriately scaled to attain the desired predominant period.

To disallow artificial reflections at the boundaries, the finite difference mesh included the bedrock and extended to great lengths horizontally and vertically from the alluvial valley. For the same purpose, the excitations were imposed as time-histories of shear stress at the bottom (horizontal) boundary of the mesh, along with quiet boundaries (dashpots), while free-field boundaries were applied at its vertical (lateral) sides. Two (2) types of analyses were performed, depending on how the soil response was modeled: a) visco-elastic, where the soil was modeled elastic medium with Rayleigh damping, and b) non-linear, where the soil was modeled as a non-linear material obeying a Ramberg-Osgood hysteretic response.

In the former type of analyses, Rayleigh damping was calibrated to provide the desired damping ratio ξ at the frequencies of importance, namely between the predominant frequency of the excitation ωe and the fundamental period of the soil column (in 1D) ωs, since the final frequency of vibration is expected to lie in between these two values. For simplicity, this intermediate frequency ωmin for each analysis assumed equal to the average value of the 2 foregoing frequencies. In the non-linear analyses, the Ramberg-Osgood model used is known to accurately model the degradation of shear modulus G and the increase of hysteretic damping ξ with cyclic shear strain γ.

Performing an analysis of the 2D basin provides a full picture of the variability of seismic motion at the horizontal ground surface of the soil and the outcropping rock. Nevertheless, of importance here is the depiction of “basin” effects, i.e. of the geomorphic aggravation of the peak ground acceleration at each location of the horizontal ground for each exemplary case. This cannot be accurately depicted by a single analysis, because it is unclear whether the observed variability is an effect of the “basin” or of the soil or rock conditions at each location. Hence, for each 2D basin analysis, a couple of 1D analyses were also performed with FLAC for the same excitation and damping configuration, one where the soil layer is of infinite width (depicting free-field soil response under 1D conditions; denoted as 1D_soil) and one where the whole profile is rock (depicting the free field rock response under 1D conditions; denoted as 1D_rock). As a first step, from each 2D basin analysis the values the peak horizontal acceleration PHA and the peak parasitic vertical acceleration PVA at each location of the ground surface were kept in memory. Note that the term “parasitic” is introduced for the PVA, since the incident motion is purely horizontal vibration (vertically incident SV waves) and therefore any vertical vibration at the ground surface is purely a result of wave refraction at the included boundaries of the basin. Then, from the corresponding 1D_soil and 1D_rock analyses, the values of PHAs and PHAr were retrieved, i.e. the respective values of the peak horizontal acceleration. Obviously, the respective values of PVAs and PVAr are equal to zero, since they correspond to purely 1D condition.

Given the foregoing values, the horizontal geomorphic aggravation factor Ah¬ is defined at each location of the ground surface, as the ratio of the PHA over the value of PHA from the appropriate 1D analysis (i.e. PHAs or PHAr). In particular, if the location at the ground surface sits on soil, then the horizontal geomorphic aggravation factor Ah¬ = PHA/PHAs, while if the location at the ground surface sits on outcropping rock then Αh = PHA/PHAr. Similarly, the parasitic vertical geomorphic aggravation factor Αv is defined at each location of the ground surface, as the ratio of the PVA over the value of PHA from the appropriate 1D analysis (i.e. PHAs or PHAr), as for Ah. Obviously, this selection of denominator is based on the fact that PVAs = PVAr = 0, as explained above. Moreover, this selection for the denominator of Av provides a measure of the relative importance of the parasitic vertical motion, as compared to its horizontal counterpart.

The specific goal of this thesis was to investigate certain limitations of published pertinent literature, and on their effects on the spatial variability of Ah and Av factors for vertically impinging SV waves. In particular, for reasons of simplicity alone, published literature usually focuses on symmetric alluvial valleys (of various shapes), that have horizontal topography and in which both the soil and the rock are assumed uniform (visco-)elastic materials. Hence, the effect of non-symmetry of the alluvial valley is first investigated, by considering trapezoidal valleys with horizontal topography, having buried lateral boundaries with different slope angles. Then, the effect of non-horizontal outcropping bedrock topography was investigated in otherwise symmetric alluvial valleys, in an attempt to quantify the seismic interaction between surface topography and bedrock geomorphology. Finally, the effects of employing a non-linear hysteretic soil model (instead of a visco-elastic model) were investigated, in order to ascertain the limitations of the latter model.

In order to investigate the effects of non-symmetry of a trapezoidal alluvial valley, comparisons were made between the seismic responses of non-symmetric trapezoidal valleys and the corresponding symmetric valleys, i.e. the 2 valleys that have the same width B and buried lateral boundaries with the same slope angles as the non-symmetric one, under the same seismic excitation conditions. Such comparisons lead to the following conclusions, for “narrow” valleys: • Near the edges, the seismic response of the non-symmetric case is identical to that of the corresponding symmetric one, in terms of both the horizontal (Αh) and the parasitic vertical (Αv) geomorphic aggravations. In addition, identical seismic response (to its corresponding symmetric one) is observed and outside the valley, all along the horizontally outcropping neighboring bedrock. • At the central part of the valley, the seismic response of the non-symmetric case deviates from those of the corresponding symmetric ones, in terms of both the horizontal (Αh) and the parasitic vertical (Αv) geomorphic aggravations. This is considered an effect of the intense interaction of the non-symmetric buried slopes in cases of relatively “narrow” valleys, which is diminished at its edges. • The foregoing deviation in seismic response at the central part of the valley is quite important for Αh, mainly when one of the 2 bedrock slopes is small (e.g. 30o), and this because a small angle in one of the two slopes leads to significant Ah values, while the opposite occurs for low slope angles. If both slopes of a “narrow” valley are characterized by small angles, then high values of Ah are expected in both symmetric and non-symmetric valleys, and hence there is no practical usage of a “average” response. The deviation in seismic response at the central part of the valley is less important for Av and cannot be approximated by an “average” response. In any case, what Av variability at the central part is of little practical importance since the significant Av values are observed near the edges of the valley.

In particular, comparisons for “wide” valleys show that the seismic response of the non-symmetric case is identical to that of the corresponding symmetric one, in terms of both the horizontal (Αh) and the parasitic vertical (Αv) geomorphic aggravations, not only near the edges, but all along the valley width. This because there is no interaction of the two buried slopes, given the wide width of the valley (other than the local non-zeroing of Av at the very center of the non-symmetric valley).

Overall, with respect to the interaction of the two buried bedrock slopes of trapezoidal valleys, the following may be concluded: • The seismic response of a single-sloped (open ended) valley is identical to that of a “wide” symmetric (or not) valley, both in terms of Ah and Av values, all along the half-width of the valley (from normalized distance x/B=-0.5 to x/B=0) but also along the horizontally outcropping neighboring bedrock, while • The seismic response of a single-sloped (open ended) valley converges to that of a “narrow” valley (symmetric or not) only near its edges (and the horizontally outcropping neighboring bedrock). On the contrary, the interaction of the two buried bedrock slopes leads to deviations in the response, as the distance from its edges increases. • The interaction is less intense as the buried slope angles increases, while for i=90° slopes even a “narrow” valley with normalized width Β/λ=4 may show no interaction of its buried slopes.

In closure, it may be stated that the peak geomorphic aggravation factors Αhmax and Αvmax are always observed near the edge of a non-symmetric valley that is characterized by the largest buried slope angle. The only exception to this rule is the Ahmax for “narrow” valleys which is observed near the centerline, as in symmetric valleys of the same width.

Then, the emphasis was put on studying the effect of non-horizontal outcropping bedrock topography on the seismic ground response of otherwise symmetric alluvial valleys. This investigation targeted the quantification of the seismic interaction between surface topography and bedrock geomorphology.

It may be concluded that the alluvial valley does not affect the seismic response on the outcropping bedrock slopes. In other words, the values of the horizontal and vertical aggravations of seismic ground motion are affected only from the surface topography characteristics (e.g. the height, and possibly the inclination) and not from what type of ground material may be found at its toe (i.e. whether it is soil or rock).

In particular, the presented (as well as similar) comparisons show a distinct effect, namely the existence of outcropping bedrock slopes leads to: • An increase of the significant values of Ah and the dislocation of the location of where Ahmax is observed further from the edges of the valley. These effects are small in cases of single-sloped (open ended) or “wide” valleys, but significant in cases of “narrow” valleys where Ahmax is observed at the center. In “narrow” valleys where interaction of the sloped bedrock faces is significant, the existence of merely one outcropping bedrock slope suffices for increasing the peak values of Ah, while the existence of a second outcropping bedrock slope leads to even further enhancement of the peak values, due to even more intense interaction, • A very small increase of the significant values of Av (practically negligible) and the dislocation of the location of where Avmax is observed further from the edges of the valley, especially in “narrow” valleys and less so in “wide” or single-sloped (open ended) ones.

It should be underlined that lack of symmetry of the alluvial valley corresponds to non-symmetry of the buried bedrock, while the existence of one, two or none outcropping bedrock slopes corresponds to non-symmetry of the outcropping bedrock.

Hence, both cases comprise non-symmetrical geomorphology features. In this respect, this study shows that the seismic response of a non-symmetrical geomorphology feature is identical to that of a single-faced (open ended) similar feature, at least at area outside the alluvial valley, i.e. where the bedrock outcrops.

The response is identical in terms of horizontal aggravation factor Ah all along the alluvial valley if this is “wide”, but only near the edges if this is “narrow”. On the contrary, for the parasitic vertical aggravation factor Av the response is practically identical even within the alluvial valley, essentially irrespective of its width.

Finally, the focus was set on employing a non-linear hysteretic soil model (instead of a visco-elastic model) for the simulation of alluvial soil response. The target of this investigation was to test the accuracy and the limitations of employing (visco-)elastic soil models in the study of geomorphic aggravation of seismic ground motion, i.e. the most usually selected assumption in published literature so far. To do so, a number of viscoelastic seismic ground response analyses (for symmetric trapezoidal valleys on horizontally outcropping bedrock) were repeated for different levels of outcropping bedrock acceleration PHAr (0.02g, 0.10g, 0.25g), by employing the aforementioned non-linear hysteretic model for the alluvial soil but with the same elastic properties as in the visco-elastic analysis. Then, comparisons were made in terms of the spatial variation of geomorphic aggravation factors Ah and Av between the results of the viscoelastic analysis (with a set value of Rayleigh damping, irrespective of PHAr) and those of the non-linear analyses for the 3 different acceleration levels. Similar comparisons lead to the following conclusions: • The spatial variation of horizontal Ah and parasitic vertical Av geomorphic aggravation factors from non-linear analyses are qualitatively similar with the pertinent results from visco-elastic analyses. • Yet, the use of a non-linear model leads to quantitative differences in both Ah and Av values. More specifically, the intensity of shaking (i.e. PHAr) leads to:  slight or significant increase in the values of Ah from non-linear analyses, at least in the usual cases when the alluvial soil column is flexible enough to have an (elastic) fundamental period larger or equal to the predominant excitation period. The amount of increase in Ah values is governed mainly from the effect of PHAr level on seismic amplification under 1D shaking conditions. In particular, an increase in PHAr leads to slight decrease in 1D site amplification when the soil layer is far from resonance or even when the soil layer is near resonance but exhibits mild non-linear response. This slight decrease in 1D site amplification leads to a slight increase in the values of Ah. On the contrary, an increase in PHAr leads to significant decrease in 1D site amplification when the soil layer is near resonance and the soil is a highly non-linear material (e.g. non-cohesive with PI=0%). This results in a significant increase of Ah values with an increase of PHAr.  significant increase in the values of Av from non-linear analyses. These increases are due to both an increase in parasitic vertical accelerations, as well as in the slight or significant decrease in 1D site amplification described above. These larger parasitic vertical accelerations in non-linear analyses are not new in published literature, but still are an issue that requires further investigation in order to ascertain whether these are a product of high frequency harmonics (which are of little importance for civil engineering works). • The effect of dimensionless valley thickness ratio λ/Η in non-linear analyses is similar to that from visco-elastic analyses. In particular, a change in λ/Η affects the spatial variation of Ah and Av along the valley. More specifically, as the thickness of the valley decreases (i.e. λ/H increases), the locations of peak geomorphic aggravation are dislocated towards the edges of the valley, and this for all tested shaking intensities. Concurrently, the values of Ah and Av generally decrease, with this decrease being independent of PHAr level for the former and less intense as PHAr increases for the latter aggravation factor. • The effect of dimensionless valley width ratio Β/λ in non-linear analyses is qualitatively similar to that from visco-elastic analyses, but there are quantitative differences. In particular, a change in Β/λ affects the spatial variation of Ah and Av along the valley. More specifically, as the width of the valley increases (i.e. B/λ increases), the locations of peak geomorphic aggravation are dislocated towards the edges of the valley, and this for all tested shaking intensities. In parallel, for medium (PHAr = 0.10g) and high intensity (PHAr = 0.25g) shaking, “narrow” valleys leads to higher Ah and Av values as compared to “wide” valleys, a differentiation that is not observed in low intensity shaking (PHAr = 0.02g) or visco-elastic analyses. • The number of significant excitation cycles does not qualitatively affect the spatial variation of geomorphic aggravation. Quantitatively, the excitation with 4 significant cycles (Kozani) leads to systematically higher Ah and Av values, as compared to those for the excitation with 1 significant cycle (Aigio). This result stems from non-linear analyses and is more or less consistent irrespective of PHAr level. It should be mentioned that visco-elastic analyses depicted no effect of number of significant cycles, especially in the horizontal direction. • The increase of soil plasticity index PI(%) is generally related to a decrease in its non-linear nature, as depicted by the empirical curves of Vucetic & Dobry (1991) included in Fig. 1. By comparing results for intensely non-linear (e.g. PI=0%) and mildly non-linear (e.g. PI=50%) alluvial valleys, one does not observe qualitative differences in the spatial variation of Ah and Av factors. Yet, geomorphic aggravation factors increase with the intensity of non-linearity, i.e. Ah and Av factors increase as the PI(%) of the soil decreases, for the same level of shaking intensity PHAr. For the same reason, the differentiation in Ah and Av values for non-cohesive and cohesive soils increases, as the intensity of shaking PHAr increases. • As a general rule, results show that visco-elastic analyses satisfactorily approximate non-linear geomorphic aggravation of valleys with intensely non-linear soil (e.g. PI=0%) for low intensity excitations (e.g. PHAr = 0.02g), while they under-predict the values of Ah for higher shaking intensities. On the contrary, for the most usual case of valleys with mildly non-linear soil (e.g. PI > 0%), results show that visco-elastic analyses are more in tune with mildly intense shaking (e.g. PHAr = 0.10g), while they under-predict the values of Ah only for high intensity shaking. In accordance, visco-elastic analyses seem conservative for low intensity shaking of such valleys. For Av, visco-elastic analyses always seem to under-predict the non-linear results, with the possible exception of low intensity motions.