8th International Masonry Conference 2010 in Dresden
Control node identification in nonlinear seismic analysis of masonry arch bridges LEPROTTI, L.1; PELA’, L.2; APRILE, A.3; BENEDETTI, A.4 ABSTRACT: The structural analysis of masonry arch bridges involves several crucial issues related to the seismic safety assessment, since the application of both static and dynamic nonlinear analyses is not trivial. The reliability of the pushover analysis results is essential since the seismic structural safety is predicted by means of a simplified approach. In particular, the selection of the most significant vibration mode is not straightforward for massive structures. This work delves into several factors related to such analysis techniques, with reference to the case-study of an existing three-arched masonry bridge. First of all, the effectiveness of the nonlinear static analysis is evaluated by means of a comparison with the nonlinear dynamic analysis. Then, the choice of the control node on the FE structural model is investigated, in order to understand its influence on the variability of the results in terms of seismic capacity. Although the top node is usually considered as the most significant point of framed structures global behaviour, in case of arch bridges such a choice may not to be suitable. The critical discussion of the performed numerical analyses gives an important contribution to the understanding of the seismic assessment of the considered structural typology. Keywords:
Masonry Arch Bridges, Control Node, Pushover Analysis, Nonlinear Dynamic Analysis, Seismic Assessment.
NOTATION f c masonry compressive strength; ft masonry tensile strength; E Young’s modulus; v Poisson’s ratio;
φ c
friction angle; cohesion.
1 INTRODUCTION The seismic assessment of existing constructions is a fundamental issue nowadays. Engineers are often called to determine the structure vulnerability to earthquakes, then to design the necessary interventions in order to reach the safety standards according to current codes. This work concerns not only buildings but also infrastructure, due to the strategic importance. In many countries, masonry
1)
Research Assistant, University of Ferrara, ENDIF Department,
[email protected] Research Fellow, University of Bologna, DICAM Department,
[email protected] 3) Professor, University of Ferrara, ENDIF Department,
[email protected] 4) Professor, University of Bologna, DICAM Department,
[email protected] 2)
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arch bridges still play a central role in rail and road networks even if they were not designed according to seismic criteria. Appropriate analysis tools are necessary to obtain a reliable prediction of the complex structural behaviour during an earthquake. For this reason, several recent codes [1-4] include nonlinear procedures, which permit one to follow the structural behaviour in the inelastic range. Accordingly, much more information can be achieved than in the case of conventional linear approaches, e.g. forces redistribution, ductility, damage, collapse mechanisms, etc. However, such methodologies present some weak points, which have not been fully clarified yet. For instance, the nonlinear dynamic analyses (NLDAs) require a careful selection of the input earthquake data, since the results are strictly affected by their choice. Moreover, they are very complex during the pre- and postprocessing and considerably expensive from the computational point of view. On the other hand, the nonlinear static (pushover) analyses (NLSAs) often resort to conventional simplified rules, for instance concerning the definition of the equivalent capacity curve and the consideration of the higher modes influence [5]. Although seismic demands are best estimated using NLDA, approximate NLSA is becoming commonplace in performance assessment of existing structures, in order to avoid the intrinsic complexity and additional computational effort required by the former. While several comparisons between NLSA and NLDA of building frame systems are available in the existing scientific literature [6-8], the case of masonry arch bridges has not been fully investigated yet. In this paper, the ability of NLSAs to simulate the seismic performance of masonry arch bridges is explored through comparisons with results obtained from a comprehensive set of NLDAs, considering ground motions having different characteristics. In particular, this study focuses on the choice of the most significant control node, as already addressed in [9] for the same structural typology.
2 BRIDGE GEOMETRY AND STRUCTURAL IDENTIFICATION The bridge (Figure 1) is situated about 30 km northwest of Pistoia (Italy), in a village called S. Marcello Pistoiese. It is triple-arched stone bridges, built after the Second World War to cross the Lima River in the Tuscany region. The vaults are made of bricks, while sandstone blocks from the rocks of the surroundings were adopted for the others structural elements. Lime mortar was used for the deeper parts of the bridges; concrete mortar was used for all visible surfaces and for the bricks of the first bridge vaults.
Figure 1. View and geometry of the analyzed bridge [9].
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Control node identification in nonlinear seismic analysis of masonry arch bridges
The foundations of the piers are reinforced concrete footings on high load bearing capacity rocks. The waste material obtained from excavation of the foundations constitutes the fill above the vaults and between upstream and downstream spandrel walls. The material properties were assessed according to laboratory and field testing results. The assessment of the characteristic strength of the existing mortar was carried out by means of the penetrometric mechanical in situ test. Seven stone cores were drilled from the bridge and were subjected to compression or splitting tests, in order to evaluate the stone elastic modulus, the compressive and tensile strength. On the basis of the tests carried out on components, the masonry overall strengths f c and ft were assessed equal to 4.5 MPa and 0.3 MPa, respectively. Furthermore, the bridge was subjected to on-site non destructive tests, based on the dynamic response analysis to impelling force. Such an experimental approach allows for the identification of the principal dynamic characteristics of the structure, in order to set the mechanical parameters and the restraint conditions of the numerical model. An impelling force was transversally applied to the bridge next to the middle arch crown. In this position, an accelerometer PCB/393A03 (voltage sensitivity: 1 V/g; accuracy: 0.00001 g rms) was placed in order to record the consequent horizontal acceleration values. For sake of data identification, a three-dimensional finite element model was developed using eight-node or six-node brick solid elements. The FEM code STRAND7 Release 2.3.3 [10] was used for the analysis. The comparison between experimental and numerical results in terms of vibration frequencies and modal shapes allowed for the calibration of the mechanical parameters and the restraint conditions of the finite element model [11]. Figure 2 shows the acceleration signals recorded in the proximity of the middle arch crown (a), with the respective power spectral density (b) and modal shape numerically determined (c). The impelling force applied at the middle arch crown of the the S. Marcello Pistoiese Bridge excites only the mode with frequency equal to about 4 Hz (Figure 2b), which is the first mode in the transversal direction of the bridge (Figure 2c).
Figure 2. Dynamic on-site test of S. Marcello Pistoiese Bridge [9]: (a) registered horizontal acceleration; (b) power spectral density of the dynamic response; (c) Mode 1: Frequency=3.998 Hz.
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3 STRUCTURAL MODELLING The nonlinear FEM analyses of the bridge have been carried out by adopting a macro-modelling strategy, in which masonry is represented as a homogeneous continuum, without making any distinction between elements and mortar [12]. The parameters considered in calculations for the different materials are reported in Table 1; they must be understood as average response properties and have been determined on the basis of the experimental tests or following engineering judgement. In particular, the materials' secant Young's moduli were set equal to one half of the determined dynamic ones [13]. The constitutive model is elastic-plastic, with a Drucker-Prager isotropic criterion and associated flow rule. The parameters φ and c have been assessed for each different tested material and implemented in the numerical model. The soil reaction at the pier footing was modelled with particular elements, called cut-off bars, which properly account for the plastic behaviour of these critical sections. Cut-off bars present an elastic-plastic constitutive law, with different yielding strength in traction and compression. These elements are defined by fixing axial stiffness and cut-off values (maximum permissible tension and compression forces), so that they can simulate the soil reaction beneath the plinth. This expedient permits us to localize the damage at the pier end for a better numerical control when the limit state of overturning is about to be approached. Table 1.
Structural parameters adopted in the numerical model of the bridge.
E
Material
v
φ
c
( MPa ) ( − ) ( deg ) ( MPa )
Masonry of stone and lime mortar (piers, spandrel walls, abutments, parapets) Masonry of stone and concrete mortar (arch cornice) Masonry of bricks and concrete mortar (vaults) Backfill
5000
0.2
61
0.58
6000 5000 500
0.2 0.2 0.2
61 55 20
0.58 0.05 0.05
4 NON LINEAR STATIC ANALYSES (NLSAs) A comprehensive study concerning the pushover analysis of S. Marcello Pistoiese Bridge pointed out the influence of the control node selection on the safety factor assessment [9]. The structural model shown in Figure 2c, with the material parameters reported in Table 1, was subjected to gravitational loads and then to a system of lateral transversal forces proportional to the mass distribution. Three different capacity curves were evaluated in terms of: i) the top of the structure displacement (TOP), ii) the centre of mass displacement (CM) and iii) the virtual energy equivalent displacement (EN) [14]. The performance displacements were calculated with the N2 procedure [15] comparing the capacity curves with the seismic demands resulting from the spectra defined by the EC8 [1]. The study showed that although the top displacement is normally considered in common practice, it always leads to the highest estimate of the safety factor. The choice of the centre of mass or the choice of the energy equivalent displacement, on the other hand, led to lower estimates of the safety factor, see Figure 3. Such points correspond to positions located around the piers top, whose displacements are relevant for this bridge typology because they dominate the out-of-plane collapse displacement shape. The bridge centre of mass was preferred as control point because it brings results that are very close to the energy equivalent ones, which provided conservative estimations in terms of safety factor, but it has a clearer geometrical interpretation and requires less computational effort. This work considers a single seismic scenario with the aim of making a comparison between NLSAs and NLDA results. The seismic demand for the NLSAs is defined by the EC8 spectrum corresponding to a peak ground acceleration value equal to 0.15g and for a ground type A (rock or other rock-like geological formations). This choice is quite in compliance with the hazard level 1116
Control node identification in nonlinear seismic analysis of masonry arch bridges
established by the recent Italian seismic mapping [2] for the S. Marcello Pistoiese bridge site. The performance points (PP) are evaluated making reference to the capacity curves drawn for the TOP and CM displacements (see Figures 5a-b afterwards) and result equal to 0.024 m and 0.016 m, respectively.
Figure 3. Comparison between the safety factors determined in [9] considering the top displacement (TOP), the centre of mass displacement (CM), the virtual energy equivalent displacement (EN) and the N2 displacement (N2).
5 NON LINEAR DYNAMIC ANALYSES (NLDAs) While seismic demands using NLSAs can be computed directly from a site-specific hazard spectrum, NLDAs require an ensemble of ground motions and an associated probabilistic assessment in order to account for the uncertainty of the earthquake recordings. The horizontal components of a set of 28 earthquakes (Table 2) are used as input for the NLDAs. Their selection have been carried out using the REXEL software [16], a tool for automatic selection of real recordings from the European Strong-motion Database [17]. REXEL allows to select records checking the compatibility of the average spectrum with a selected reference spectrum by means of a tolerance parameter, either for scaled or unscaled sets. In this study, the selected reference spectrum is the same that was considered for NLSAs, i.e. the EC8 one, and the tolerance parameter is assumed equal to 10%. Figure 4a shows the average spectrum of the 28 records considered (black line), the EC8 elastic design spectrum (gray line), the upper and lower bound curves whose ordinates are equal to ±10% of such spectrum ones (dashed light gray lines). Figure 4b shows the Fast Fourier Transforms (FFTs) of the 28 records used for NLDAs. According to the selected accelerometer signals, 28 NLDAs have been performed along the transversal direction of the bridge model (Figure 2c) in order to permit a comparison with NLSA results. For each analysis, the maximum pseudo-acceleration and displacement values at both TOP and CM nodes have been evaluated and then gathered in Figures 5a-b, respectively. As shown, the results are quite scattered, revealing a significant influence of the input earthquake record on the bridge response. Table 3 reports the comparison between mean and standard deviation values of the computed TOP and CM maximum displacements.
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Table 2.
Earthquakes used for non-linear dynamic analyses.
Earthquake name Tabas Montenegro Montenegro Campano-Lucano Campano-Lucano Duzce 1 Off coast of Magion Oros peninsula Izmit South Iceland South Iceland South Iceland (aftershock) South Iceland (aftershock) Vrancea Bingol Campano-Lucano Kalamata Vrancea South Iceland Campano-Lucano Friuli (aftershock) Umbria-Marche (aftershock) Umbria-Marche (aftershock) Friuli (aftershock) Valnerina Valnerina South Iceland Friuli (aftershock) Calabria
Date 16/09/1978 15/04/1979 15/04/1979 23/11/1980 23/11/1980 12/11/1999 06/08/1983 17/08/1999 17/06/2000 17/06/2000 21/06/2000 21/06/2000 30/08/1986 01/05/2003 23/11/1980 13/10/1997 30/08/1986 17/06/2000 23/11/1980 11/09/1976 14/10/1997 03/10/1997 16/09/1977 19/09/1979 19/09/1979 17/06/2000 16/09/1977 11/03/1978
Mw 7.3 6.9 6.9 6.9 6.9 7.2 6.6 7.6 6.5 6.5 6.4 6.4 7.2 6.3 6.9 6.4 7.2 6.5 6.9 5.3 5.6 5.3 5.4 5.8 5.8 6.5 5.4 5.2
Epicentral distance 12 21 65 23 127 34 76 47 13 13 14 15 49 14 32 48 49 32 25 8 12 5 11 22 5 5 9 10
0.45 Average spectrum (28 accel.)
0.40
EC8 elastic spectrum (3-A)
0.35
Low er bound (-10%)
Sa [g]
0.30
Upper bound (+10%)
0.25 0.20 0.15 0.10 0.05 0.00 0.00
0.50
1.00
1.50
2.00
T (s)
Figure 4. a) Average spectrum of the records used for NLDAs and EC8 spectrum.
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Control node identification in nonlinear seismic analysis of masonry arch bridges
0.25
Calabria Valnerina C.Lucano Umbria-Marche Friuli Izmit S.Iceland S.Iceland S.Iceland Vrancea Bingol C.Lucano Duzce C.Lucano
Amplitude [g]
0.2
0.15
Valnerina C.Lucano Friuli Umbria-Marche Friuli S.Iceland S.Iceland Kalamata S.Iceland Vrancea Tabas Montenegro Montenegro M.Oros
0.1
0.05
0 0
5
10
15
20
25
30
35
Frequency (Hz)
Figure 4. b) FFTs of the records used for NLDAs. 6
6
5
5 Pushover curve Equivalent bilinear system Elastic spectrum 3-A NLSA Performance Point NLDAs Mean of NLDAs
3
Pushover curve Equivalent bilinear sys tem Elas tic Spectrum 3-A NLSA Performance Point NLDAs Mean of NLDAs
4
Acceleration (m/s 2)
2
Acceleration (m/s )
4
2 1 0
3 2 1 0
0.00
0.05
0.10
0.15
Displacement (m)
(a)
0.00
0.05
(b)
0.10
0.15
Displacement (m)
Figure 5. Comparison between NLSAs and NLDs results: a) TOP and b) CM control node.
Table 3.
TOP and CM maximum displacements in NLDAs.
Control node
Mean
Standard deviation
TOP CM
0.021 m 0.009 m
0.0119 0.0057
6 PROBABILISTIC COMPARATIVE STUDY A reasonable result emerges from the comparison between the NLSAs performance points and the NLDAs, i.e. the simplified NLSAs slightly overestimates in a conservative way the mean of the maximum estimations derived by NLDAs [18]. This feature is valid both for pseudo-acceleration and displacement values, as shown in Figures 5a-b. However, in this study the attention will be focused on the latter ones, according to the modern concept of Displacement-Based Design [19].
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Table 4 reports the comparison between the NLSA performance displacements and the mean of the maximum displacements recorded in NLDAs, both for TOP and CM control nodes. As shown, in the case of TOP control node, the NLSA displacement overestimates of the 14% the NLDA mean of the maximum displacements, whereas for CM the gap is more marked (+78%). This scatter is calculated with reference to NLDA values since the main objective is to assess the accuracy of the simplified pushover procedures. Table 4.
Scatter in NLSAs performance displacements toward the NLDAs means of the maximum displacements. Analysis type
TOP
CM
NLSA (performance displacement) NLDA (mean of the maximum displacements) Scatter in NLSA
0.024 m 0.021 m +14%
0.016 m 0.009 m +78%
Although the previous result would point in favour of TOP node choice in NLSAs, actually the most significant result is the one reported in Table 3, i.e. the standard deviation of the NLDAs displacements is lower in the case of CM node choice. This feature means that the probabilistic distribution of the maximum displacements for an assigned set of earthquake records results less scattered if the CM control node is selected. Moreover, the more the NLSA performance point exceeds the distribution main value, the estimate is conservative. In order to understand this very important feature, it is necessary to draw the probabilistic distribution of the NLDAs maximum displacements. The Gaussian curves, for TOP and CM nodes, are depicted in Figure 6 and compared with the NLSAs performance points. It can be noticed that the probability that any NLDA maximum displacement results are greater than the NLSA performance point is lower if the CM displacement is considered in calculations. Therefore, the CM node is the best choice for NLSAs because it leads to a more reliable estimation. Since the set of NLDAs maximum displacements does not lie around the Gaussian distribution mean value (see Figure 6), the Gamma distribution has been also considered (Figure 7). Such probabilistic function is much more representative: the better agreement with data has been verified through the Chi-square test. The cumulative probability of having points that exceed the PP value decreases moving from the Gaussian to the Gamma distributions, as presented in Table 5. Therefore, the assumption of this more suitable distribution stresses even more that CM performance displacements derived by NLSAs are more reliable than TOP ones. 80 70 G(x) TOP NLSA PP TOP NLDAs TOP G(x) CM NLSA PP CM NLDAs CM
60
G(x)
50 40 30 20 10 0 -0.020
0.000
0.020
0.040
0.060
Displacement (m)
Figure 6. Gaussian distributions of TOP and CM maximum displacements in NLDAs. Comparison with NLSAs Performance Points (PP).
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Control node identification in nonlinear seismic analysis of masonry arch bridges
-0.02
(a)
0.00
Gamma distrib. Gaussian distrib.
0.02 0.04 Displacement (m)
100 90 80 70 60 50 40 30 20 10 0
G(x) - f(x, a, b)
G(x) - f(x, a, b)
100 90 80 70 60 50 40 30 20 10 0
-0.02
0.06
0.00
Gamma distrib. Gaussian distrib.
0.02
0.04
0.06
Displacement (m)
(b)
Figure 7. Gaussian and Gamma distributions of the maximum displacements in NLDAs: a) TOP and b) CM.
Table 5.
Probability of exceeding PP value for Gaussian and Gamma distributions.
Probability of exceeding PP value
TOP
CM
Gaussian distribution
41.2%
9.1%
Gamma distribution
35.5%
10.0%
7 CONCLUSIONS This paper investigates the effectiveness of NLSAs in predicting the seismic performance of a triplearched masonry bridge through comparison with benchmark responses obtained from a comprehensive set of NLDAs. The input adopted for NLDAs is selected in order to satisfy the EC8 provisions; this allows a rigorous comparison between the demand computed by NLDA and the one associated to NLSA for the same design spectrum. A suite of 28 natural earthquakes is selected. The following significant results can be pointed out: - The performance displacement obtained via simplified NLSAs slightly overestimates in a conservative way the mean of the maximum displacements obtained by 28 NLDAs. This is valid both for TOP and CM control nodes. - The choice of the TOP node provides a performance displacement close to the mean value of the probabilistic distribution of the maximum NLDAs displacements. On the other hand, the probability that any NLDA maximum displacement results greater than the NLSA performance point is great. - The CM control node is a reliable choice to deal with the simplified seismic assessment of the considered masonry arch bridge. In this case, the probability that any NLDA maximum displacement results greater than the NLSA performance point is low. - Although the considered NLSAs do not account for the effect of higher vibration modes to the seismic response, the obtained results are acceptable and conservative if compared to the distribution of the NLDAs maximum displacements.
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