ANALYSE NON LINEAIRE ELASTO-PLASTIQUE A COLLAPSE DES PONTES EN MURAGE A ARCADE MULTIPLE.

Transcription

1 ANALYSE NON LINEAIRE ELASTO-PLASTIQUE A COLLAPSE DES PONTES EN MURAGE A ARCADE MULTIPLE. NON LINEAR ELASTO-PLASTIC COLLAPSE ANALYSIS OF MULTI-SPAN MASONRY ARCH BRIDGES A. BRENCICH 1, U. DE FRANCESCO and L. GAMBAROTTA 3 1 Ass.nt Prof. Strutural Engineering DISEG University of Genoa ITALY Post-Graduate Student DISEG University of Genoa ITALY 3 Prof. Strutural Mehanis DISEG University of Genoa ITALY RESUME: Dans et travail vient présenté une proédure inrémentielle intérative pour l analyse non linéaire des ponts à ar en murage à arade multiple. Si on suppose pour le murage un modèle onstitutif sans résistane à tration (NTR) ave une réponse en ompression parfaitement élasto-plastique, l algorithme vient implémenté par un planning etérieur de un programme ommerial à éléments finis et vient, depuis, appliqué à ars de différente géométrie. Si on onsidér des modèles à arade double ou triple ave piles soit ourtes soit longues, vient étudié l effet sur le méanisme de ollapsé des paramètres géométriques de l ar et des propriétés méaniques du matériel. L approhe traditionnel rapporté à une unique arade vient omparé au résultats effetifs des ponts à arade multiple, on metten en évidene que la réponse elasto-plastique du matériel, l interation entre les arades adjaentes et la déformabilité des piles ont beauoup de relief sur la apaité de harge soit pour les ars surbaissés soit pour les ars en plein intre. En outre, l effet de es paramètres sur le méanisme du ollapse et sur la apaité de harge est différent pour les ars surbaissés ou sur les ars en plein intre, pareque dépendant aussi de la géométrie de le meme ar. ABSTRACT: An inremental-iterative proedure for the non-linear analysis of multi-span masonry arh bridges is disussed. Assuming masonry as a no tensile resistant (NTR) material with perfet elasto-plasti response in ompression, the algorithm is implemented by standard programming of a ommerial F.E. ode and applied to different arh geometries. The effet of the arh geometrial parameters and of the material mehanial properties on the ollapse mehanism is studied onsidering single- twin- and three-span models with both short and long piers. The lassial single-span assessment approah is ompared to the atual results for multispan bridges, showing that material elasto-plasti response, span interation and pier fleibility greatly affet the limit load of both shallow and deep arhes. Besides, the effet of these parameters on the ollapse mehanism and on the limit load differs for shallow and deep arhes, depending on the arh geometry as well. 1. INTRODUCTION The engineering methods of assessment for multi-span arh bridges refer to a single arh analysis with perfetly built-in ends at the skewbaks or three pinned elasti arhes with simplified load distributions (Hughes and Blakler, 1997). These proedures guarantee that an equilibrium onfiguration eists but give only rough estimates of the limit load and almost no information on the ollapse mehanism. The parameters relevant to the strutural response (masonry strength, mehanial harateristis of the fill, interation between adjaent spans, et.) are taken into aount by means of foggy orretive fators or are negleted at all. The mehanism approah, relying on the kinemati theorem, assumes the limit load as the one whih leads to the formation of a number of plasti hinges large enough to transform the arh in a mehanism (Heyman, 198, Harvey, 1988, Hughes, 1995, and Como, 1998). Suh an approah is reliable provided that the ollapse mehanism is ativated when the stresses in the arh are still below the material ompressive strength, whih is the ase of most deep arhes but not of

2 shallow arhes or of weak mortar joints; in these ases the atual ompressive response of masonry needs to be taken into aount (Criesfield and Pakham, 1988). The elasto-plasti response of ompressed masonry an be dealt with on the basis of simplifying assumptions for the plasti stress distribution in the mortar joints (Clemente et al., 1995) or eperimental tests (Taylor and Mallinder, 1993, Boothby, 1997); in both ases yield surfaes are derived so that ollapse is met when the aial thrust and the bending moment, derived from the thrust line theory, lie on the limit surfae. Also F.E. proedures allow the ompressive response of masonry to be introdued into onedimensional (Criesfield, 1985, Bridle and Hughes, 1990, Choo et al., 1991, Molins and Roa, 1998.a, b) two-dimensional (Loo and Yang, 1991, Owen et al., 1998, among the latest results) or three-dimensional models (Rosson et al., 1998). The latter ones may give detailed information on loal phenomena provided that a relevant ompleity of the model and long omputing times are aepted; otherwise they need to be limited to very simple models (Faloner, 1994), so that only one-dimensional models are effetive for multi-span bridges (Molins and Roa, 1998.b). The effet of the adjaent spans on the loaded arh had also been studied eperimentally on 1:5 multi-span models (Royles and Hendry, 1991, Melbourne et al., 1995, Ponniah and Prentie, 1998). The results, negleting the ompres-sive response of masonry, show a redution of the limit load due to the presene of the adjaent arhes somewhere in-between 0% to 50% of the limit load for a single arh. The three-span model tests (Melbourne and Wagstaff, 1995) showed that the ollapse mehanism involves all the piers and spans so that dead weight added to the spans not diretly loaded resulted in a relevant inrease of the limit load. Twin-arh bridge ollapse has also been studied by means of a mehanism approah (Hughes, 1995). In this paper Castigliano s elasti method is implemented in a one-dimensional F.E. proedure and it is etended in order to take into aount the ompressive plasti response of the masonry (Brenih et al., 001.a, b). The proedure is applied to single-, two- and three-span models assuming piers of various height and different values for masonry ompressive strength. Comparisons with the standard methods of assessment are arried out.. ELASTO-PLASTIC EXTENSION OF THE CLASSICAL NTR APPROACH The lassial non linear model for masonry is due to Castigliano (1879) and relies on two basi assumptions: 1) the material has No Tensile Resistane (NTR model); ) the strains are linearly distributed on the ross setion (Navier plane ross setion hypothesis). Aording to these assumptions no limit is set to the ompressive stresses, figure 1. Suh a model is intended to represent dry assemblages of voussoirs or weak mortar joints but, in the first ase, it does not take into aount the rushing of the ompressed edges (Taylor and Mallinder, 1993), in the seond ase the global ompressive response depends on the whole assemblage of briks and joints and to the three-aial stress states that arise in the mortar joints and in the brik units due to the ompatibility onditions at the mortar/joint interfae. σ' σ' σ h ompression G N M h y G N M no-tensile resitant area Figure 1: NTR model, Castigliano (1879). Figure : NTR perfetly elasto-plasti model. The aial thrust N E and bending moment M E transmitted by the deformed ross setion are due to the ompressed part of the setion only: N E b E b h E h = σ', M = σ' = N, (1.a, b) 3 3

3 where the supersript E stands for the fores equilibrated by an NTR material, for the height of the ompressed part of the setion and σ for the maimum ompressive stress in the setion. A first non linear model for masonry ompressive response should assume a ompressive limit stress σ ; in the following, referene is made to a perfetly elasto-plasti onstitutive model in ompression (fig. ) that allows the algorithm to be stable but does not aount for the softening response of ompressed masonry. The equilibrium equations of the setion are given as: b b h 1 N = σ ( + y) M = σ ( ) ( ) + y + y+ y. (.a,b) 3 If we assume that both the NTR-elasti and the NTR-elasto-plasti models refer to the same deformed ross setion, the plasti ut-off of figure results in unbalaned fores on the setion that an be omputed as the differene between the elasti response, eq.s (1), and the elastoplasti response, eq.s (): N M = N E = M N E M = σ = σ b y y b h y y y = N y h σ 3 b y 6 y (3.a) (3.b) Eq.s () and (3) are suitable for setting up an iterative proedure for the analysis of generi masonry arh-type strutures following a preditor/orretor iterative sheme. 1. The (k-1) th approimation of the solution is known, i.e. for eah i th setion the height h k-1 (i), its ompressed part k-1 (i), the plasti plateau y k-1 (i) and the eternal loads f k (i) to be applied in the net step, are known.. Predition. The geometry is updated reduing the setion height to the estimated ompressed part: h k (i) = k-1 (i). The eternal loads f k (i) are applied to the updated geometry for whih the linear elasti response is omputed; strains and displaements are alulated in every i th setion. 3. Corretion. On the basis of the alulated strains and displaements, in every i th setion the ompressed part (i) k, the plasti plateau y (i) -(i) k and the unbalaned part of the internal fores N k -(i) and M k are omputed via eq.s (3); it is worthwhile noting that the unbalaned fores are omputed as a funtion of the urrent strain state only. N -(i) k and M -(i) k are the effet of the plasti response of the material representing the part of the NTR-elasti response that annot be ahieved beause of the elasto-plasti response, therefore they an be onsidered as unbalaned fores to be added, in addition to the applied loads, to an NTR-elasti model to represent the plasti response; sine these fores are not diretly applied loads they are alled fititious eternal fores. The loads are updated: f k+1 = f 0 + f k, where f k stands for a vetor ontaining the whole of the fititious fores. The (k+1) th step an then start from point The initial step is the elasti response to the applied loads of the struture in its full geometry. In the generi ase of a redundant struture, the internal fores to be equilibrated are not known a priori; the onvergene riterion annot be diretly referred to the unbalaned joint fores but is rather related to the variation of the fititious fores between two subsequent steps: N ma N M M 1 k k+ k,, i ε (6) N M k k i i k + 1 where the subsript i stands for the generi setion and ε is the aepted tolerane. The proedure an be implemented by eternal programming of ommerial F.E. odes sine it is based on a series of elasti analysis of strutures different only beause of an updating algorithm modifies the geometry and varies the eternal fititious fores. An arh-type struture, therefore, an be divided in finite beam-type elements of onstant ross setion; tapered beams may not be neessary beause the varying height of the raked area is represented assuming different beam heights for the adjaent elements. In this way: 1. the aial thrust and bending moment, in every element, an be omputed by averaging the nodal fores;

4 . the strain and displaement states an be dedued from the averaged fores by means of the lassial elasti beam theory; Sine the proposed proedure an be regarded as a onstitutive model for masonry rather than a mehanial model for mortar joints, the finite elements disretizing the arh struture do not need to preserve the voussoir s dimension. 3. THE RESPONSE OF A SINGLE ARCH The proedure is applied to a single arh using the ANSYS 5.5 F.E. ode; the length of eah beam element used to disretize the struture is approimately that of two brik units. The geometry of the two sample arhes, shown in figure 3, was hosen so as to represent a typial shallow arh etrapolating the geometry of the sale models tested by Melbourne and Wagstaff (1995) and a typial deep arh of the same span. (a) (b) 80 P R=937.5 m R=798 m Figure 3: geometry for the (a) shallow and (b) deep arh. The shaded area represents the fill. The strutural response of the arh has been studied for a travelling knife-type load assuming, first, the NTR model, than assuming a ompressive strength σ of 5 MPa; the Young s modulus is assumed to be 0 MPa. The fill has been onsidered as a dead load on the arh and distributing ( ) the applied load in a limited area of the barrel and is given a density of 4.1 kn/m 3 ; the density of masonry is taken as kn/m 3. a) ollapse mehanism with effetive height b) plasti hinges distribution Figure 4: a) ollapse mehanism; b) plasti hinges yellow areas are the open part of the joints. Figure 4 represents, on a shallow arh, a typial output of the outlined proedure for the two types of arhes showing (a) the deformed shape of the effetive resistant portions of the arh at ollapse for /l=0.3; (b) the raked parts of the mortar joints (yellow areas). It is worthwhile noting that the plasti hinges are not point-like hinges but are rather diffuse in the arh. Great differene emerges between the shallow and deep arh responses onsidering the load-displaement and limit load-load position urves, figure 5. Shallow arhes undergo plasti strains far below the ollapse load and the ollapse mehanism is attained when the plasti mehanisms spread through a large part of masonry; in this ase the elasto-plasti ontribution is of great importane. Deep arhes, on the other side, undergo plasti strains very lose to the final ollapse revealing a strutural response hardly dependant on the masonry ompressive strength, figure 5. The sudden drop in the arh stiffness aounts for the ativation of a ollapse mehanism; sine the model does not onsider any tensile resistane, the limit load is reahed asymptotially from below. For both the arhes the weakest position for the load is loated nearby 1/3 of the arh span, while entral loading may be, for shallow arhes, the far mode resistant position.

5 Limit load [kn Loading at /s =0.3 Displaement under the applied load [m] NTR NTR Shallow Shallow Arh (5 MPa) (5 MPa) Shallow Arh NTR Deep Arh (5 MPa) Deep Arh (a) load-displaement response Loading position /s NTR Shallow Arh (5 MPa) Shallow Arh NTR Deep Arh (5 MPa) Deep Arh (b) limit load vs. load position Figure 5: load-displaement and limit load-load position urves for shallow and deep arhes. 4. THE RESPONSE OF MULTI-SPAN MODELS Figure 6 shows the twin- and three-span models with short (stiff) and long (fleible) piers whih resembles that of the sale models tested by Melbourne and Wagstaff (1995) but for the ring thikness, whih has been redued to a more typial value; figure 6 represents the deep arh (rise/span = 0.35) models only but also shallow arh models have been tested with same geometry (rise/span = 0.5). P P R=798 m r = P P (a) (b) 0 80 r = r = Figure 6: - and 3-span bridge models with short and long piers. The shaded area represents the fill. () (d) Figure 7: ollapse mehanisms for twin-span and three-span models for entral and lateral loading.

6 Figure 7 shows the ollapse mehanisms for the twin- and three-span model with shallow arhes and short piers for entral and lateral loading. It is worthwhile noting that in all the ases the last hinge to be ativated is loated at the base of the pier. The ollapse mehanisms for both deep and shallow arhes show that the ollapsing span is always the loaded span, while it is worthwhile noting that adjaent spans are lifted up in the final ollapse. This irumstane is in agreement with the eperimental observations made by Melbourne and Wagstaff and allows to assume that any other load added to the bridge (i.e. the remaining part of the train) is likely to stabilise the struture and rise the limit load. Figure 8 represents the limit load vs. load position diagrams for the four multi-span models of figure 6. If we ompare the diagrams (a) and () on one hand and (b) and (d) on the other hand (omparison among the diagrams in the horizontal position) we an learly identify the effet of the interation between adjaent spans, where the interation with a third span aounts for a relevant redution in the ollapse load. Comparison of plot (a) with plot (b) and of plot () with plot (d) shows the net effet of the pier height on the ollapse load, whih results in a muh more pronouned redution of the ollapse load than the one produed by span interations. The limit loads for models with different ompressive strength are markedly different, showing that the elasto-plasti ompressive response of masonry annot be negleted. Besides, it is found that the weakest position for multi-span models is the entral one, as opposed to the single span analysis where the lowest ollapse load is attained at 1/3 of the span. (a) Loading position /s () Limit load [kn Loading position / s (b) Loading position /s (d) Limit load [kn NTR (5 MPa) ( MPa) Loading position /s Figure 8: limit load vs. load position for deep arh models: a) span model with short pier; b) span model with long pier; ) 3 span model with short pier; d) 3 span model with long pier. The ompressive strength of MPa is dramatially low, but it has been used so as to represent the effet of this mehanial property of masonry. Similar plots are shown in figure 9 for the shallow arh models. With short piers the effet of the ompressive strength is muh more pronouned than for deep arhes; when the piers are long (fleible) the limit load dramatially drops down and beomes less sensitive to the material ompressive strength. For some loading positions, the elasto-plasti ompressive response aounts for 30% of the limit load but for other positions its effet is hardly notieable. The weakest position is found nearby the entral loading but the limit load for the entral position is not muh different from the minimum one with long piers. These differenes probably arise from the stiffness differene between the shallow and the deep arhes. The latter are muh

7 more fleible than shallow ones and, therefore, already the single span model is prone to ollapse mehanisms in whih the stresses reah the ompressive threshold only in the very proimity of ollapse. For this reason the interation with adjaent spans and the elasto-plasti response are of minor importane. Shallow arhes ehibit, instead, a muh higher stiffness; ollapse ours a long deal after the ompressive strength has been reahed in many setions and, therefore, turns out to be greatly affeted by the masonry ompressive strength and by all those fators whih affet the fleibility of the skewbaks (pier height and interation with adjaent spans). (a) /s 0 1 () Limit load [k Loading position / s (b) /s 0 1 (d) Loading position /s NTR 15 MPa 10 MPa 5 MPa Figure 9: limit load vs. load position for shallow arh models: a) span model with short pier; b) span model with long pier; ) 3 span model with short pier; d) 3 span model with long pier Loading position /s (a) twin-span model NTR Shallow Arh NTR Deep Arh Loading position /s (5 MPa) Shallow Arh (5 MPa) Deep Arh (b) three-span model Figure 10: limit load of shallow and deep arh bridges vs. load position and masonry strength (short piers). For the short pier models, the geometri effet of the arh shape is shown in figure 10, where the limit load vs. load position diagrams are given for shallow and deep arhes in twin- and threespan models. Again it is found that the elasto-plasti ompressive response as well as the effet of interation between adjaent spans is relevant for shallow arhes muh more than for deep ones; the latter are less sensitive to the number of spans of the model and to the masonry ompressive strength.

8 5. DISCUSSION AND CONCLUSION Figures 11 and 1 represent the limit load vs. load position for deep and shallow arh bridges. For the shallow arh model with short piers, in the weakest positions somewhere in-between 0.5 /s 0., the Castigliano s model (NTR) aounts for a limit load that is 15% to 0% (-span) and 15% to 30% (3-span) higher than the values predited by the weakest, but realisti, elastoplasti model. In the entral loading position, the NTR model gives a largely unrealisti estimate of the ollapse load whih ould be unbounded for some span to ring thikness ratios (a) NTR /s Twin-arh model h=7,5m Three-arh model h=7,5m Twin-arh model h=15m Three-arh model h=15m Single-arh (b) σ =MPa /s Figure 11: limit loads for the single-, two- and three-span deep arh models with short and long piers vs. loading position for the NTR model and for a ompressive strength of σ = MPa /s (a) NTR /s Twin-arh model h=7,5m Three-arh model h=7,5m Twin-arh model h=15m Three-arh model h=15m Single-arh (b) σ =5MPa Figure 1: limit loads for the single-, two- and three-span shallow arh models with short and long piers vs. loading position for the NTR model and for a ompressive strength of σ = 5 MPa. The deep arh behaviour is markedly different, sine the differenes between the single-span and multi-span geometries are not only less pronouned, but also it is found that the multi-span struture sometimes presents a higher ollapse loads that the single-arh. Speifi values are given in tables 1 and. Table 1: Deep arh bridges with short piers: single-arh limit load / multi-span ollapse load ratio (in brakets the figures referring to the long pier models). NTR /s= arh /-arh -5 (+17)% -4(+3)% -9(+31)% +10(+71)% -15(+18)% 0(0)% +(0)% 1-arh /3-arh +4 (+19)% +3(+3)% -6(+31)% +(+)% -6(+6)% +3(+9)% +4(+15)% σ = MPa /s= arh /-arh -(+)% +3(+7)% +3(+53)% +18(+)% -9(+30)% 0(+14)% 0(+4)% 1-arh /3-arh +6(+9)% +10(+50)% +7(+57)% +8(+79)% +7(+57)% +10(+50)% +6(+9)%

9 When long piers are introdued into the model, representing a viadut rather than a bridge, the differenes between the NTR and elasto-plasti models, as well as between the twin- and threespan models almost disappear, showing the pier height to be a soure of major onern in omputing the ollapse load of a viadut. Being this the ase largely with the lowest limit loads, it is the long pier model that largely fails to be treated by means of a single arh analysis; the single arh approah may overestimate the ollapse load up to Table : Shallow arh bridges with short piers: single-arh limit load / multi-span ollapse load (in brakets the figures referring to the long pier models).. NTR /s= arh /-arh 0(+5)% +15(+60)% +6(+145)% +30(+18)% 0(+41)% 0(0)% 1-arh /3-arh +14(+8)% +41(+6)% +95(+147)% +95(+147)% +41(+6)% +14(+8)% σ = 5 MPa /s= arh /-arh +5(+6)% +9(+7)% +70(+137)% +40(+118)% 0(+47)% 0(+8)% 1-arh /3-arh +6(+35)% +50(+)% +100(+137)% +100(+137)% +50(+)% +6(+35)% The effet of abutment ompliane on the limit load had been studied by Faloner (1994) referring to a three-span bridge and Hughes (1995) analysing a twin-span model. Faloner estimated the redution fator to aount for abutment ompliane as a funtion of the pier height-to-thikness ratio only, while Hughes suggests a redutive fator that depends on all the relevant geometri fators of the bridge. For the speifi values of the studied geometry, the two approahes foresee an overestimation of the single-arh analysis vs. the real limit load of 35% for both shallow and deep arhes in Faloner s approah and 5% (knife type load) and 33% (for a 0.6m wide load footing) for the shallow arh and % (knife type load) and 31% (for a 0.6m wide load footing) for the deep arh in the seond ase (Hughes); the MEXE-MOT method would suggest an overestimation of the single-arh analysis from 0% to 5%. The omparison of the estimated ollapse loads for single, twin and three-arh multi-span bridges for shallow arhes showed that the lassial single-arh approah dramatially overestimates the atual limit load. This happens when 1) the load is nearby the entre of the span; ) the material is poor with low ompressive strength; 3) the pier height is not adequately taken into aount; 4) the ring thikness is high enough, in the single arh analysis, to make it possible for the thrust line to be straight and, nevertheless, entirely ontained inside the arh thikness. For shallow arh bridges, the available estimates of the abutment ompliane effet seem to underestimate the multi-span effet. The presented results point out that the limit loads and the ollapse mehanisms for two-span and three-span shallow bridges are quite different, pointing out that, in this ase, the number of spans involved in the ollapse mehanism seems to be of relevant importane for establishing the limit load. For deep arhes, on the other hand, the ratio single arh/multi-span ollapse load is not far from unity, also taking into aount the ompressive elasto-plasti response of the material. For this lass of bridges the single arh approah seems, therefore, to be reliable provided the pier height is taken into aount. The presented elasto-plasti models suffers of two limitations: 1) the perfetly elasto-plasti model does not adequately represent masonry whih ehibits an elasto-softening ompressive response; ) shear ollapse mehanisms are not onsidered being the ollapse mehanisms pure bending ones. 6. ACKNOWLEDGMENTS This researh had been arried out by partial finanial support of the National Earthquake Defene Group (GNDT) of the National Institute for Geophysis and Vulanology (INGV) inside the VIA researh projet Seismi vulnerability redution of infra-strutural systems and environment (Riduzione della vulnerabilità sismia di sistemi infrastrutturali e ambiente fisio), a part of the Biennial Researh Program.

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