a fibre beam-based approach for the evaluation of the seismic capacity of masonry arches

A fibre beam-based approach for the evaluation of the seismiccapacity of masonry arches

Stefano De Santis*,† and Gianmarco de Felice

Department of Engineering, Section of Civil Engineering, Roma Tre University, Via Vito Volterra 62, 00146, Rome, Italy

SUMMARY

The paper proposes the use of a fibre beam-based methodology to study the dynamic behaviour of masonryarches and arch bridges. The approach, previously used for load-carrying capacity assessment, allows for adetailed description of both the nonlinear material properties and the seismic action, with reasonablecomputational effort. As a first step, in order to validate the proposed method, the failure condition underacceleration pulse is analysed, and the results are compared with the solution provided by the mechanismmethod for different geometric configurations. The influence of material behaviour is also investigated.The seismic capacity of masonry arches is then assessed through pushover analyses under different loaddistributions and IDAs under natural accelerograms to identify a suitable representation of inertial forcesarising under earthquake ground motion. Finally, the in-plane seismic capacity of an existing seven-spanmasonry arch bridge is evaluated. Copyright © 2014 John Wiley & Sons, Ltd.

Received 11 August 2013; Revised 3 February 2014; Accepted 5 February 2014

KEY WORDS: masonry; arches; mechanism method; pushover analysis; IDA; fibre beams

1. INTRODUCTION

Damage and collapses caused by earthquakes have clearly shown that arches and vaults are one of themost vulnerable elements in historic masonry structures. Nevertheless, a deep knowledge of theirdynamic behaviour is still lacking, and the available assessment procedures based on elastic analysishave proved to be unreliable. As a consequence, radical substitutions or invasive strengtheningtechniques have been adopted in the past for the seismic retrofitting of arched structures. Reliablemethods for the seismic assessment of existing masonry arches are therefore needed for thepreservation of the built heritage exposed to earthquake risk.

The static analysis of masonry arches is traditionally based on Heyman’s assumptions of infinitecompressive strength and no tensile resistance [1]. According to this approach, the seismic action issimply described by horizontal body forces and the arch fails as soon as four hinges develop,turning the structure into a mechanism. Equivalent static methods predict the onset of motionthrough a limit analysis approach, providing a lower bound estimate of the actual capacity underearthquake ground motion, because of the variable nature of seismic signals. In order to take intoaccount such dynamic overstrength, a behaviour factor equal to 2 is provided by the Italian code forforce-based assessment [2]. As an alternative, a displacement-based method is also proposed, inwhich the displacement capacity of the mechanism is compared with the seismic demand, evaluatedby recurring to an equivalent elastic single DOF system [3]. In the recent past, static methods have

*Correspondence to: Stefano De Santis, Research Assistant, Department of Engineering, Section of Civil Engineering,Roma Tre University, Via Vito Volterra 62, 00146, Rome, Italy.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2014; 43:1661–1681Published online 28 February 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2416

been applied to predict the collapse mechanism and the seismic capacity of arched structures belongingto churches [4], and for damage limitation state assessment of monumental buildings [5].

Oppenheim proposed a first properly dynamic approach, overcoming the limits of static procedures,describing the arch as a four-bar linkage mechanism and deriving its failure domain under pulse baseacceleration [6]. Oppenheim’s model has then been extended to include free vibrations and sine baseaccelerations [7], and to take into account energy dissipation at impacts during cyclic response [8].It has also been used to predict the rocking stability of masonry arches under the so-called primaryimpulse of the base excitation, taken as a representative of earthquake time history [9].

As an alternative to analytical models, the distinct element method has been used to study thedynamic response of masonry arches [8], also including the presence of buttresses [10]. Distinctelement method describes the structure as an assemblage of rigid blocks and nonlinear interfacesand derives its response by means of the explicit integration of the equations of motion in the timedomain. The method shows great potentialities for studying the dynamics of discontinuousstructures, thus being particularly suitable for masonry, and makes it possible to account for largedisplacements [11, 12]. However, a detailed discretization at the block level results unfeasible forthree dimensional (3D) large structures and the explicit integration scheme may be very sensitive tocontact stiffness and damping values in time history analyses [13]. For these reasons, finite elements(FEs) and macroelements are often preferred for the seismic assessment of real case studies in thecurrent practice. On the one hand, 2D and 3D FE models allow for a faithful representation of thestructural geometry thus taking into account its influence under both in-plane [14] and out-of-planeloads [15, 16]. However, such accuracy requires extremely high computational efforts. On the otherhand, 1D elements and macroelements allow nonlinear dynamic analyses to be carried out withlower computational costs [17], thus achieving a detailed representation of earthquake input.

This latter approach is followed in the present work, in which the seismic capacity of masonryarches is evaluated through a fibre beam-based methodology, previously used for load-carryingcapacity assessment [18, 19]. The proposed strategy offers a good compromise between accuracyand simplicity, thus being suitable for both nonlinear static and dynamic analyses under earthquakeground motion. Differently from analytical methods, the fibre beam approach is able to predict thecollapse mechanism and to account for the effective material behaviour with acceptablecomputational effort, thanks to the intrinsic simplicity of the beam elements and to the discretizationof the cross-sections into fibres.

The paper is organized as follows. First, the capacity of masonry arches under impulse baseaccelerations is investigated. The theoretical formulation developed by Oppenheim [6] is retraced toderive a collapse condition for the arch, similar to the one proposed by Housner for the overturningof the rigid block [20]. Different arch geometries are considered to show the effects of size,slenderness and rise-to-span ratio. Then, the arch is modelled with fibre beams, and nonlineardynamic analyses are carried out under rectangular pulse accelerations. The results are comparedwith those provided by the procedure based on rigid block dynamics, to prove the reliability of theproposed approach. Numerical simulations are also repeated considering more refined constitutiverelationships accounting for finite crushing strength and limited ductility, to investigate the influenceof material properties on the failure domain of the arch. Aiming at studying the capacity underearthquake ground motion, pushover analyses are performed and compared with IDAs, to identifythe adequate representation of inertial forces arising during earthquake ground motion. Geometricnonlinearities and effective material properties are included, and their influence on the seismiccapacity estimate is discussed. Finally, the fibre beam methodology is used to assess the seismiccapacity of an existing multispan railway masonry arch bridge.

2. COLLAPSE UNDER ACCELERATION PULSE: MECHANISM METHOD

In this section, the failure of a circular arch under impulse base motion is evaluated by means of themechanism method. According to this approach, first developed by Oppenheim [6] and thenrecovered and extended by other authors [7, 8], the behaviour of the arch is described throughprinciples of rigid block dynamics. In the present work, it is used to derive the collapse condition

1662 S. DE SANTIS AND G. DE FELICE

Copyright © 2014 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2014; 43:1661–1681DOI: 10.1002/eqe

for the masonry arch, which is similar to the one proposed by Housner [20] for the overturning of therigid block. In the following section, the failure domains provided by the mechanism method willsupport the validation of the fibre beam modelling methodology. As damping is expressed through acoefficient of restitution at the impacts in the mechanism method and by means of an equivalentviscous term in the fibre beam method, the problem is solved only for the first half cycle of motionin order to compare the two approaches.

On the basis of Heyman’s assumptions of infinite crushing strength and no tensile resistance [1], thestructure is turned into a mechanism as soon as four plastic hinges (named A, B, C and D) form. InOppenheim’s model, after hinging occurs, the arch is represented by the four-bar linkage ABCDA(Figure 1), and its dynamic response is studied. In the initial configuration, the rotations of the linksare denoted as θAB,0, θBC,0 and θCD,0, while, in the deformed configuration, they assume genericvalues θAB, θBC and θCD. Diversely, because link DA is fixed, θDA does not change during motion.Moreover, rAB, rBC and rCD are the distances of the link centres of mass (GAB, GBC and GCD)measured from hinges A, B and D, respectively; ψAB, ψBC and ψCD are their angular offsetsmeasured from the rotations θAB, θBC and θCD, respectively. Finally, the masses of the links AB,BC and CD are named mAB, mBC and mCD, respectively.

Because the four-bar linkage has only one DOF, the deformed configuration, described by the linkrotations, depends on only one of them, arbitrarily chosen as the Lagrange parameter of the system; inthis case θAB = θ, that is the rotation of link AB (named driver link), is chosen. The rotations θBC andθCD and the rotational velocities _θBC and _θCD of links BC (coupler) and CD (follower) are written interms of θ and _θ through displacement and velocity analyses [21]:

θBC θð Þ ¼ a tanCD sinκ θð Þ

BC� CD cosκ θð Þ

� �� a tan

AB sin θ� θDAð ÞDA� AB cos θ� θDAð Þ

� �þ θDA (1)

θCD θð Þ ¼ π� a tanAB sin θ� θDAð Þ

DA� AB cos θ� θDAð Þ

� �� a cos

BC sinκ θð ÞCD� BCcosκ θð Þ

� �þ θDA (2)

where κ(θ), named transmission angle, is given by the following expression:

κ θð Þ ¼ a cosCD2 þ BC2 � AB2 � DA2

2BC� CDþ AB� DA

BC� CDcos θ� θDAð Þ

� �(3)

In the analytical procedure, the positions of the plastic hinges (identified by the angles βB and βC)are the ones yielding the lowest static multiplier (the horizontal load, normalized by the self-weight,

Figure 1. The circular arch as a four-bar linkage mechanism in its initial and deformed configurations.

EVALUATION OF THE SEISMIC CAPACITY OF MASONRY ARCHES 1663


turning the arch into a mechanism) and are found through a minimization procedure. Once themechanism is defined, the potential energy V θð Þ is derived; it depends on θ by means ofEquation (4) and is plotted in Figure 2(a) for an arch having radius R = 10m, thickness s = 0.15 Rand angle of embrace β = 157.5°.

V θð Þ ¼ g mABrAB sin θAB þ ψABð Þ þ g mBCAB sinθABþþ g mBCrBC sin θBC þ ψBCð Þ þ g mCDrCD sin θCD þ ψCDð Þ

(4)

Starting from the initial geometry, in which θ = θ(t = 0) = θ0, during motion, θ decreases and V θð Þincreases until it reaches a peak value when θ = θ1; until θ⩽ θ1, the self-weight does a negativework because it tends to bring the structure back to its undeformed configuration. On the contrary,when θ> θ1, it contributes to make the deformation diverge, and its work is positive. Thereby, thepeak rotation can be defined as a nonrecovery condition (point of instable equilibrium).

The considered forcing function is an acceleration a, constant within a time interval τ and null for t> τ,and the limit condition is represented by the maximum sustainable acceleration for a given duration. Thearch fails as soon as the total work carried out by the inertial forces in the duration τ (which is a positivework) is equal to the difference in potential energy between θ1 and θ0 (that is the maximum negative workthe self-weight can spend). A similar procedure is developed by Housner to identify the overturning of arigid block under constant base acceleration [20]. In the present case, the failure condition is expressed byEquation (5), in which vAB, vBC and vCD are the horizontal components of the velocities of the centres ofmass of the three links, stated in Equations (6)�(8):

∫τ

0a mAB vAB tð Þ þmBC vBC tð Þ þmCD vCD tð Þð Þdt ¼ V θ1ð Þ � V θ0ð Þ (5)

vAB tð Þ ¼ ∂∂t

rAB cos θ tð Þ þ ψABð Þ½ � ¼ �rAB sin θ tð Þ þ ψABð Þ½ � ∂θ∂t

(6)

vBC tð Þ ¼ ∂∂t

rBC cos θBC tð Þ þ ψBCð Þ þ ABcos θ tð Þð Þ½ � ¼

¼ �∂θ∂t

AB sin θ tð Þð Þ þ rBC sin θBC tð Þ þ ψBCð Þ∂θBC∂θ

� � (7)

vCD tð Þ ¼ ∂∂t

rCD cos θCD tð Þ þ ψCDð Þ½ � ¼ �rCD sin θCD tð Þ þ ψCDð Þ ∂θCD∂θ

∂θ∂t

(8)

Figure 2. Variation in (a) potential energy and (b) collapse domain of a circular arch with R = 10m,β = 157.5° and s = 0.15 R.



To solve Equation (5), the equation of motion of the system is needed; it is expressed in terms of theLagrange parameter θ(t) and is derived starting from Hamilton’s principle:

∂∂t

∂T∂ _θ

� �� ∂T

∂θþ ∂V

∂θ¼ Q (9)

where T (θ, _θ) is the kinetic energy and Q(θ) is the forcing function. T is stated by Equation (10), inwhich IAB, IBC and ICD are the centroidal moments of inertia of the links, while Q, which is a linearfunction of the constant acceleration a, is given by Equation (11):

T ¼ 12mAB rAB _θ

� �2 þ IAB _θ2

h iþ

þ 12mBC AB _θ

� �2 þ 2ABrBC cos θ� θBC � ψBCð Þ _θ _θBC þ rBC _θBC� �2h i

þ

þ 12IBC _θBC

2 þ 12mCD rCD _θCD

� �2 þ ICD _θCD2

h i (10)

Q ¼ � a�mAB∂∂θrAB cos θ þ ψABð Þ � a�mBC

∂∂θ

rBC cos θBC þ ψBCð Þ þ AB cosθ½ ��

� a�mCD∂∂θrCD cos θCD þ ψCDð Þ

(11)

All the terms of Equations (4), (10) and (11) can be expressed in terms of θ(t) by using Equations (1)and (2), and by doing so, the equation of motion can be derived as in Equation (12). It expresses theequality between the works associated to kinetic and potential energies (on the left hand side) and tothe external load (on the right hand side) and contains four coefficients M(θ), L(θ), F(θ) and P(θ),depending on the geometry and nonlinear in θ, whose expressions are reported in Equations (13–16).

M θð Þ€θ þ L θð Þ _θ2 þ F θð Þg ¼ P θð Þa (12)

M θð Þ ¼ IAB þ AB2mBC þmABrAB2 þ 2AB mBC rBC cos ψBC þ θBC � θð Þ∂θBC∂θ

þ

þ IBC þmBCrBC2ð Þ ∂θBC∂θ

� �2

þ ICD þmCD rCD2ð Þ ∂θCD∂θ

� �2 (13)

L θð Þ ¼ �AB mBC rBC sin ψBC þ θBC � θð Þ ∂θBC∂θ

� 1

� �∂θBC∂θ

þ

þ AB mBC rBC cos ψBC þ θBC � θð Þ þ IBC þmBC rBC2ð Þ∂θBC∂θ

� �∂2θBC∂θ2

þ

þ ICD þmCD rCD2ð Þ∂θCD∂θ

∂2θCD∂θ2

(14)

F θð Þ ¼ mABrAB cos ψAB þ θð Þ þmBC AB cosθþ rBC cos ψBC þ θBCð Þ ∂θBC∂θ

� �þ

þ mCDrCD cos ψCD þ θCDð Þ ∂θCD∂θ

(15)

P θð Þ ¼ B mBC sinθþmABrAB sin ψAB þ θð Þþ

þ mBCrBC sin ψBC þ θBCð Þ∂θBC∂θ

þmCDrCD sin ψCD þ θCDð Þ∂θCD∂θ

(16)



Equations (13–16) can be simplified by writing the geometrical dimensions in terms of the radius Rand of the mass density γ and then by dividing all the terms by γR2. The equation of motion results tobe independent of the mass density; moreover, because M and L are proportional to R3 while F and Pare proportional to R2, in the simplified equation R multiplies the terms in €θ and _θ and not the otherones, thus indicating that a size effect has to be expected in the dynamic response of the arch butnot in the static one [6, 8].

To determine the expression of θ(t), an initial state of rest ( _θ=€θ = 0) is imposed, and instantaneous(constant) values of the coefficients are taken: M0 =M(θ0)/γR2, L0 = L(θ0)/γR2, F0 = F(θ0)/γR2 andP0 = P(θ0)/γR2. The latter assumption leads to a tangent approximation of the response, which isallowed in the small rotation field. The equation of motion is rewritten as in Equation (17) andsolved to find the function θ(t) as reported in Equation (18):

M0€θ þ L0

_θ2 þ F0 g ¼ P0 a (17)

θ tð Þ ¼ M0

L0ln cos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL0 F0 g� P0 að Þ

pM0

t

! !(18)

The failure condition can now be determined by substituting in Equation (4) the expression of θ(t)given by Equation (18). The solution for an arch having β = 157.5°, R = 10m and s/R = 0.15 is found bynumerical integration and plotted in Figure 2(b). It is seen that, for short impulse durations, highaccelerations are needed to induce the structural collapse; whereas for τ→∞, the curve tendsasymptotically to a limit value coinciding with the static multiplier aST/g, which can be easilyobtained as the ratio F0/P0 between the virtual works of horizontal and vertical loads, that is, bysolving the equation of motion (17) neglecting dynamic effects. Three domains can be identified aspointed out in [7]: If the ground motion amplitude is lower than the limit value (aST), the horizontalacceleration is not even sufficient to turn the arch into a mechanism as no hinging occurs; on thecontrary, when it is higher than aST, the onset of motion takes place. If the point (τ, a) is below thefailure curve, it represents an impulse that does not cause the structural collapse, that is, there ishinging, but the arch returns to its initial geometry; finally, if (τ, a) identifies a point above thecurve, the corresponding pulse acceleration makes the arch fail.

3. COLLAPSE UNDER ACCELERATION PULSE: FIBRE BEAM APPROACH

3.1. The fibre beam model

Starting from experimental results showing that plane sections remain plane after deformation [22],masonry elements under axial load and bending moment can be represented through a 1D model.This approach has already been used to assess the load-carrying capacity of masonry arch bridges[18, 19, 23]. The arch is described as a segmental beam, and a flexibility-based beam model withfibre cross-section is used (Figure 3). The fibre beam approach makes it possible to performstructural analyses under either static pushover loads or earthquake base accelerations, ensuring atthe same time low computational efforts thanks to the intrinsic simplicity of the beam elements andto the discretization of the cross-section into fibres. The constitutive characterization is madethrough uniaxial stress-strain (σ� ε) relationships assigned to the fibres. By doing so, the effectivebehaviour of the material can be efficiently taken into account, as the section force-deformation lawis derived by integration of fibre stress and stiffness values.

The iterative solution process is based on the Newton–Raphson method and is organized into threenested phases corresponding to the three levels of structure, element and cross-section. The externalloop gives the total displacements and resisting forces corresponding to an applied external loadincrement. The middle loop returns the element resisting forces and stiffness starting from a given



nodal displacement increment. Finally, the internal loop yields the section deformations and stiffnesscorresponding to a given section force increment.

In the generic cross-section at abscissa x of the fibre beam, the vector comprising the strains of the nfibres e(x) = {ε1(x, y1, z1)… εn(x, yn, zn)}T is obtained as l(y,z)d(x), being d(x) = {χz(x) χy(x) ε(x)}T asthe section deformation vector, collecting the two curvatures and the axial strain, and l(y,z) as ageometric matrix containing the position of the fibres; l is linear because a plane section behaviouris assumed. At each step j of the element state determination process, the fibre deformationincrement is obtained from the element deformation increment, and the fibre strain is updated:

e j xð Þ ¼ e j�1 xð Þ þ Δe j xð Þ ¼ e j�1 xð Þ þ l y; zð ÞΔdj xð Þ (19)

The stress and the tangent modulus of each fibre is determined from its σ� ε law; then the fibrestresses are collected in vector Σj(x,y,z) and the elastic moduli in the diagonal matrix Ej(x,y,z).Finally, the section resisting force vector DR

j (x) = {Mz(x) My(x) N(x)}T, containing the two bending

moments and the axial force, and the section tangent stiffness matrix kj(x) are obtained by means ofthe numerical integrals (Equations (20) and (21)), in which A(x) is a diagonal matrix comprising theareas of the fibres:

DRj xð Þ≅

Xn xð Þ

k¼1

lT yk; zkð Þ Ak xð Þ Σj x; yk; zkð Þ (20)

kj xð Þ≅Xn xð Þ

k¼1

lT yk; zkð Þ Ej x; yk; zkð Þ Ak xð ÞlT yk; zkð Þ (21)

3.2. Fibre beam simulations for the masonry arch under impulse base motion

The failure of circular arches under an acceleration pulse is estimated by means of the modellingmethodology based on the use of fibre beam elements; the results are compared with those providedby the mechanism method, as shown in the previous section, to verify the reliability of the proposedapproach. The investigation of the failure condition is made by repeated nonlinear dynamicanalyses, carried out for several impulse durations so as to find as many collapse accelerationvalues. Aiming at reproducing the same assumptions of the mechanism method, no damping isincluded, and the material assigned to the fibres has no tensile resistance and infinite compressivestrength. The result is found to be independent of the number of beam elements and of fibres in the

Figure 3. The fibre beam element.



cross-section, provided that an adequate discretization is ensured that does not affect the geometry ofthe arch (insufficient number of elements) or the stress distribution within the cross-section(insufficient number of fibres). In the present case, the model is defined in a 2D domain, and thearch is discretized into 100 beam elements whose cross-section is divided into 150� 1 fibres(discretization is only in the bending direction).

The result of a fibre beam analysis is represented in Figure 4 for an arch having mass density equalto 2000 kg/m3, R = 10m, s = 0.15R and β = 157.5° (which is the same arch considered in the previoussection) under a pulse acceleration with amplitude ranging from 0.73 to 0.77 g and duration ofτ = 0.60s. The collapse is assumed to occur when the rotation of link AB (Figure 4(c)) or thehorizontal displacement of the crown (Figure 4(d)) diverges, that is, does not go back to zero afterthe peak, and so no equilibrium solution can be found any more.

The four-hinge mechanism characterizing the collapse configuration is shown in Figure 4(a) andcorresponds to that predicted by equivalent static analyses. The positions of the hinge sections areeasily identified by the peaks in the curvature diagram (Figure 4(b)), where a slight spreading is dueto the continuum nature of the modelling approach. Finally, the stress field in sections A, B, C andD is plotted in Figure 5, pointing out the high level of partialization of the hinges where the loadresultant is very close to the section edge at the arch intrados and extrados, alternatively.

3.3. Influence of arch geometry: results and comparisons

The comparison between rigid block dynamics and fibre beam approach is carried out for archeshaving different geometries; the sensitivity to the variation of size, slenderness and rise-to-span ratio

Figure 4. Results of the fibre beam model for an arch having R = 10m, s/R = 0.15, β = 157.5° under basepulse acceleration with τ = 0.60 s: (a) collapse configuration, (b) curvature of the beam elements, (c) timehistory of the driver link rotation and (d) time history of the horizontal displacement of the key node.



is investigated (Figures 6–8). The failure condition provided by the mechanism method is plotted withthe solid lines, whereas the results of FE simulations are represented by the marks. A good agreementis found for all the considered configurations; even if for very short impulses, it may be difficult to findstable numerical solutions.

Curves corresponding to different values of R (Figure 6) show an increase of resistance towardsshort impulses when the size increases: When τ = 0.6 s, for example, the collapse acceleration isequal to 0.58, 0.77 and 1.05 g for R = 5, 10 and 20m, respectively. Such a difference becomessmaller for long impulses and asymptotically tends to zero because all the arches have an identicalstatic multiplier aST = 0.348 g. The static solution does not depend on the size while the dynamic onedoes, as discussed before.

The arch capacity towards both dynamic actions and static loads results to be also related to theaspect ratio s/R, which is a measure of the slenderness (Figure 7): As the thickness increases, theresistance becomes higher for both short and long impulses. Given a duration τ = 0.6 s, the collapseacceleration amplitude is a = 0.53 g for s/R = 0.12 and a = 1.12 g for s/R = 0.20; likewise, the limitvalue aST grows passing from 0.255 to 0.507 g. The arch with s = 0.20 R displays a larger error thanthe others (Figure 7). This may depend on the larger difference (θ0� θ1) between the instable

Figure 5. Fibre beam model results for an arch having R= 10m, s/R = 0.15, β = 157.5° under impulse basemotion with a = 0.75 g and τ = 0.60 s: stress field in the hinge sections at the instant of maximum

displacement of the crown section.

Figure 6. Collapse domains of arches with s = 0.15R, β = 157.5° and radius ranging from R= 5m toR= 20m: comparison between mechanism method (MM) and fibre beam approach (FB).



equilibrium configuration and the initial configuration, in which the equation of motion has beenlinearized. Tangent approximations of the response in different positions (e.g. in the nonrecoveryrotation, as suggested by [24]), could lead to smaller errors. Finally, as expected, deep arches resultto be more resistant towards both vertical and horizontal static loads (note that the crushing strengthof masonry is assumed to be infinite). Deep arches are found to be stronger also under dynamicactions as it is shown in Figure 8 for arches having the same thickness s = 0.06 L and different rise-to-span ratios. With the decrease of r/L, not only aST but also the capacity towards pulses grows: Onthe one hand, the static multiplier passes from 0.248 to 0.516 g and 1.462 g for r/L equal to 0.4, 0.3and 0.2, respectively; on the other hand, the collapse acceleration for a given duration τ = 0.5 sresults to be 0.61, 1.09 and 2.48 g for the three considered configurations.

The positions of the plastic hinges and the nonrecovery rotations depend on the arch shape (s/R and r/L)and not on its size (R): A higher slenderness results in a decrease of angles βB and βC thus indicatingthat the plastic hinges move towards the first one (point A), as it is found also in [7]. A variation not

Figure 7. Collapse domains of arches with R = 10m, β = 157.5° and thickness ranging from s = 0.12 tos = 0.20R: comparison between mechanism method (MM) and fibre beam approach (FB).

Figure 8. Collapse domains of arches with s = 0.06 L and rise-to-span ratio ranging from r = 0.2 L to r = 0.4 L:comparison between mechanism method (MM) and fibre beam approach (FB).



only of capacity but also of mechanism shape is found when passing from deep arches (r/L = 0.4) toshallow arches (r/L = 0.2). Larger nonrecovery rotations are related to lower s/R ratios (because thepotential energy is proportional to the link masses, and therefore, low variations of V are related tosmall thickness values, given a rotation θ0� θ) and to higher r/L values (deep arches are moredeformable than shallow arches, in the sense that higher vertical components are associated to agiven rotation).

3.4. Effect of material properties

The effective mechanical response of masonry can be significantly far from the elastic-no tensileresistant (ENT) constitutive model, assumed in the mechanism method. Experimental tests onhistoric brickwork under compression show that the material displays not only a limited crushingstrength but also a strongly degrading postpeak behaviour [22]. Thanks to the agreement found withthe mechanism method, the fibre beam approach is used in this section to investigate the effect ofmaterial properties on the capacity of an arch under impulse base motion. A circular arch havingR= 10m, s/R = 0.15, β = 157.5° is considered, and its failure is analysed considering an elastic-plastic (EP) and a Kent&Park (K&P) [25] constitutive models in addition to the ENT relationship.The former is characterized by an initial elastic phase followed by an unlimited horizontal branchwith constant stress equal to the compressive strength (fc). The latter presents an initial parabolicascending branch, ending when the strain correspondent to the compressive strength εc0 = ε(fc) isreached; it has an initial stiffness E0 = 2fc/εc0 and a final horizontal slope. The following postpeakbehaviour is described by a linear descending branch, connecting the peak to the ultimate point(εu, 0). In this work, a crushing strength fc = 7.1N/mm2 and an initial elastic stiffness E0 = 747N/mm2

are assumed. In the K&P constitute law, the available ductility εcu/εc0 is set equal to 3.0. In all theconsidered constitutive relationships, the tensile strength is null.

When the assumptions of infinite strength and unlimited ductility are removed, passing from ENT toEP material and from EP to K&P, a decrease of the arch resistance towards both horizontal static loadsand impulse base accelerations is found (Figure 9). The limit value passes from 0.348 to 0.336 g and0.325 g; at the same time, such an asymptote is reached more rapidly, indicating lower resistance todynamic actions if the effective material behaviour is taken into account than if the EP assumptionis made or an infinitely resistant material is assumed. Thus, the capacity to pulse base acceleration

Figure 9. Collapse domains of an arch with R = 10m, s/R = 0.15, β = 157.5°, considering different materials:elastic-no tensile resistant (ENT), elastic perfectly plastic (EP) and Kent&Park (K&P).



results more sensitive to the material properties in the range of short-to-medium pulse durations. Forexample, for τ = 1 s, the base acceleration making the arch fail is equal to 0.531, 0.440 and 0.375 gfor the three constitutive laws: In this case, neglecting the effective material properties provides anoverestimate of the collapse pulse acceleration of about 155% and 375% for EP and ENTconstitutive laws, respectively. It appears from these results that, as for the assessment of load-carrying capacity [18, 19, 23], the material properties play a non-negligible role also in the case ofdynamic actions. Such influence could be lower for arches with a smaller span than that consideredin the present section, while it is expected to be particularly relevant under high loads or for weakmasonry, as a consequence of crushing strength and postpeak deterioration [26].

4. SEISMIC CAPACITY UNDER EARTHQUAKE MOTION

4.1. Cyclic constitutive relationship for masonry under compression

Aiming at studying the response of masonry arches to cyclic loads, a constitutive uniaxial relationshipis defined in the average stress-strain plane to describe the macroscopic response of the material,considered as an equivalent homogeneous medium, under cyclic compression. The skeleton curvecoincides with the K&P relationship [25], whereas the unloading and reloading branches are definedas a bilinear function (ABC) and a linear function (CA), respectively (Figure 10). The resultingcyclic constitutive law is defined by the seven parameters collected in Table I.

4.2. Pushover analyses

Pushover analysis under different load distributions and nonlinear dynamic (step integration) analysisis performed to evaluate the seismic capacity of masonry arches under earthquake ground motion.The structure is the same arch considered in the previous section, having radius R = 10m, thicknesss/R = 0.15 and angle of embrace β = 157.5°. The same model is also used consisting of 100 beamelements, whose section is divided into 150� 1 fibres.

First, the influence of second order effects is evaluated. Pushover analyses are carried out withENT constitutive relationship (infinite compressive strength and no tensile resistance) under auniform distribution of horizontal loads (forces proportional to masses) accordingly to the conditionsassumed in the mechanism method. Both a linear and corotational geometric transformations are used.Geometric transformation determines the conversion rule from the local reference system (in whichthe equilibrium of each beam element is evaluated) to the global reference system (in which the

Figure 10. Unloading–reloading branches in the uniaxial constitutive relationship for historic brickwork un-der cyclic compression.



entire equilibrium problem is solved at each step of the analysis). The linear transformation ruleprovides a unique criterion that is set at the beginning of the analysis and is based on theundeformed structural configuration and is suitable for solving problems in the field of smallrotations and small displacements. On the contrary, the corotational method accounts for thedisplacements and the rotations of the end nodes of the beam elements to update the global stiffnessmatrix at every analysis step. Thus, it allows second order effects to be taken into account. Thecapacity curves resulting from these two geometric transformations are shown in Figure 11, inwhich the horizontal displacement of the key node (dk) and the resultant base shear normalized bythe arch self-weight (Vb/W) are on the x-axis and on the y-axis, respectively. In the same graph, thecapacity curve provided by the mechanism method in nonlinear kinematics, that is, by solvingEquation (12) neglecting dynamic effects, is also represented. The mechanism method assumes thearch to be infinitely rigid up to the activation of the collapse mechanism, which occurs under aresultant of horizontal loads equal to the static multiplier (aST = 0.348 g). The latter defines the firstpoint of the curve, whereas the intersection with the x-axis corresponds to the horizontaldisplacement of the key node (dk = 681mm) in the instable equilibrium deformed configuration, thatis, for a rotation of the first link equal to the nonrecovery rotation (θAB= 0.086 rad, Figure 2(a)). Thecapacity provided by the fibre beam approach agrees with that of the mechanism method if a lineargeometric transformation is used. A non-null displacement of the control node is recorded as thematerial is not infinitely rigid, but the ultimate capacity is independent from the elastic stiffness.In reverse, if second order effects are taken into account, the same capacity is achieved only asan asymptotic result when the elastic stiffness of the material tends to infinite, whereas a

Table I. Parameters of the modified Kent&Park cyclic constitutive relationshipfor masonry.

Symbol Description Range

fc Crushing strength <0εc0 Strain at peak stress <0fcu Residual strength ⩽0εcu Strain at residual strength ⩽0λ Slope of the reloading brancha >0ξ1 Initial slope of the unloading brancha >λξ2 Final slope of the unloading brancha ⩽ξ1aWith respect to the initial tangent stiffness E0 = 2fc/εc0.

Figure 11. Capacity curves provided by the mechanism method (MM) in the field of finite rotations and bythe fibre beam approach (FB) with linear and corotational geometric transformation rules (ENT material), for

an arch having R = 10m, s/R = 0.15, β = 157.5°.



lower capacity results for more realistic stiffness values as it is shown in Figure 11, in whichE0 = 747N/mm2 is assumed. In this case, the reduction with respect to the first order solution is inthe order of 23% (0.27 vs 0.35 g).

Second, the capacity curves corresponding to different loading profiles are compared. Pushoveranalyses are carried out with a corotational transformation rule and the modified K&P constitutivelaw described in the previous section. The following three distributions are used:

(1) Horizontal loads proportional to masses;(2) Horizontal loads proportional to the product of masses and displacements of the first mode; and(3) Horizontal and vertical loads, proportional to the product of masses and displacements of the first

mode. This distribution accounts for the presence in the first eigenvector of non-negligibledisplacements in both vertical and horizontal directions, even if the participating mass in thevertical direction is zero (the algebraic sum of the vertical nodal displacements results to be null).

The capacity curves are represented in Figure 12, for the following set of parameters: fc =�7.1N/mm2,εc0 =�0.019, fcu = 0N/mm2, εcu =�0.0405, λ=4, ξ1 = 6 and ξ2 = 1.25. Because the structure issymmetric, the capacity curves are symmetric, too. The seismic capacity resulting under the firstdistribution is 0.23W, which is 15% lower than that of ENT material (corotational transformation),thus indicating the effect of considering the finite crushing strength and the limited ductility (postpeakdegradation). Clearly, this effect depends on the load rating, that is, on the stress level in the arch withrespect to the crushing strength, as failure in arches built with strong material or displaying low stresslevel under vertical loads is due to the activation of a collapse mechanism rather than to crushing.Distributions (2) and (3) provide a capacity in the order of 0.2 and 0.15W, respectively, which are 13%and 35% lower than that resulting from the simple application of horizontal body forces. Finally, thecyclic loading leads to narrow cycles in the response, indicating a low hysteretic dissipation, similarlyto the response of a block in rocking motion.

4.3. Incremental nonlinear dynamic analyses

In this section, the seismic capacity of a masonry arch is evaluated by means of nonlinear IDAs. Theconsidered arch is the same investigated in the previous section of the paper, as well as the numericalmodel with fibre beams. A set of 30 natural accelerograms are selected. To this purpose, anacceleration response spectrum is first defined according to Eurocode 8 [27] assuming a designground acceleration ag = 0.15 g and soil type B. The signals are then chosen from the EuropeanStrong Motion Database and scaled (only on the y-axis) such that their average accelerationspectrum matches the target spectrum, that is, has a maximum spread from the target spectrum of

Figure 12. Pushover curves under different load distributions for an arch having R = 10m, s/R = 0.15,β = 157.5° and effective material properties (K&P constitutive law).



±10% in the 0.15–2.0 s range. Accelerograms are selected using REXEL software [28], from events withmoment magnitude between 5 and 7, on recording stations at least 20 km far from the source. Weakevents and near field records are therefore avoided.

IDAs are performed by applying the input signals of the set with increasing scaling factor (SF)ranging from 0.1 to 2, with the Newmark time-step integration method with 0.005 s time-step.Damping is represented by means of a Rayleigh viscous term. Because hysteretic damping isalready included in the material constitutive law, Rayleigh damping represents the dissipative effectsinduced by other phenomena that are not explicitly described, such as local inelastic deformationsdue to inhomogeneous stress distribution at the microscale, which may occur in a range of averagemacroscopic stresses that are well below the elastic threshold. The equivalent viscous damping ratiofor dynamic analyses on masonry structures is a particularly challenging issue, also because theymay exhibit a rocking behaviour, in which energy dissipation mainly occurs at impacts [29]. Valuesof equivalent viscous damping ranging from 2% to 10% have been proposed on the basis ofexperimental tests (see, among others, [30]). In the current work, the Rayleigh parameters arechosen to achieve an equivalent damping ratio of 5% at the first and fourth modal frequencies(f1 = 2.25Hz and f4 = 8.40Hz), which are related to the highest participating masses in horizontaldirection. By doing so, damping is in the order of 5% within the range of the significant frequencies(in terms of spectral content) of both the input signal and the structural response, whereas highervalues are set for low frequencies (<0.2Hz), which are usually reset to zero in the filtering processof sampled recorded signals, as well as for high frequencies (>50Hz), which may result fromnumerical instabilities and, anyway, have negligible importance for the structural dynamic response.

The maximum horizontal displacement of the crown (dk) and the corresponding (same instant)normalized resultant base shear (Vb/W) are chosen as structure state variables and are recorded foreach analysis, in order to synthetically represent the overall response and to make comparisons withpushover analyses. The comparison between pushover and nonlinear dynamic analyses is shown inFigure 13. The grey area represents the set of all the individual simulations, whereas the black linewith round marks is the corresponding mean IDA curve. The mark at dk = 35mm and Vb = 0.15W,corresponding to the integral application of the signals (SF = 1), represents the predicted averageresponse of the considered arch to the selected set of accelerograms. Finally, the three lines (dottedblack, solid grey and solid black) correspond to the capacity curves of the three load distributions ofpushover analyses, as in Figure 12. A good agreement is found between static and dynamicapproaches provided that both horizontal and vertical loads proportional to the product of massesand modal displacements are applied in pushover analyses (load distribution (3)). Therefore, thisappears, among the considered loading profiles, the adequate representation of the inertial forcesarising in the dynamic response of the arch.

Figure 13. IDA curve and pushover (POA) curves under different load distributions for an arch havingR = 10m, s/R = 0.15, β = 157.5° and effective material properties (K&P constitutive law).



5. APPLICATION TO A MULTISPAN VIADUCT

5.1. Description of the bridge fibre beam modelling

The seismic capacity of a multispan masonry arch bridge is evaluated in this section through the fibrebeam approach. The structure under study, named Ronciglione viaduct, belongs to the Rome–Viterborailway and was built at the end of the 19th century [18, 19]. It has a rectilinear layout and is made outof seven circular barrel vaults and six piers (Figure 14). The arches, built with clay bricks and hydrauliclime and pozzolan mortar, have 18m span, 9m rise, 1.07m thickness and 4.60m depth. The masonryof the piers is in rough tuff stone with squared units on the external face; there are corners andhorizontal chains every 3m made of squared flint stones. The central pillars are about 35m high. Allthe piers have a vertical slope of 3.5% in longitudinal direction and varying from 5% to 6% intransversal one. The second and the fifth piers are provided with buttresses in transversal directionand are dimensioned 1.50m larger in the longitudinal one to be able to carry asymmetric archthrusts during the buildings phases; this device allowed the vaults to be made in three subsequentsteps making use of the same wood centrings. The spandrel walls are 75 cm thick and 11m highfrom the springing sections and are made of regular courses of tuff squared stones; finally, thebacking height is about 4.70m. The bridge is founded on flint and pozzolan concrete plinths basedon volcanic tuff soil.

Ronciglione viaduct is modelled using 100 fibre beams for each arch and 50 for each pier, whilebacking, spandrels and abutments are described by using horizontal and diagonal nonlinear trusselements, in order to represent their stiffening effect under in-plane loads. The cross-section ofvaults and piers is discretized into 100� 1 fibres, whereas the cross-section of the truss elements isdivided into 30� 1 fibres (Figure 15). Finally, shear flexibility is assumed for the cross-section ofpiers. The mesh is chosen after performing adequate sensitivity analyses and ensures good numericalstability. The mass of soil and spandrels is represented through point masses connected by rigid

Figure 14. Longitudinal view of Ronciglione viaduct.



links to the underlying vaults. The fibre beam model is defined in a 3D domain and is able to performanalyses under both longitudinal and transversal loads (note that analyses under transversal loadsrequire the discretization of the fibre section in both directions). The latter condition, however, is outof the scope of the present work, and only longitudinal analyses are therefore presented hereafter. Acorotational geometric transformation is adopted to take into account second order effects, and themodified K&P constitutive model described in the previous sections of the paper is used with thesame set of parameters, calibrated to fit experimental results (Figure 16) derived on brickworkspecimens representative of the same masonry type of that of the bridge under study [22].

5.2. Modal analysis

As a first step, the dynamic behaviour of Ronciglione viaduct is investigated in the elastic range by alinear 3D FE model; the results in terms of modal shapes and natural frequencies are compared withthose provided by the fibre beam model to investigate its capability in representing the dynamicresponse of a multispan arch bridge. The 3D model is made out of brick FEs with eight nodes. AYoung’s modulus E0 = 750N/mm2 and a self-weight γ = 1650 kg/m3 are assigned to vault brickwork,whereas E = 750N/mm2, γ= 1500 kg/m3 and E = 200N/mm2, γ= 1500 kg/m3 are chosen for pier tuff

Figure 15. Modelling of masonry arch bridges through fibre beam elements.

Figure 16. Comparison between experimental response curve and uniaxial constitutive relationship forhistoric brickwork under cyclic compression.



masonry and fill soil, respectively. The first modal shape in the longitudinal direction is represented inFigure 17 having period T = 0.68 s and 37% participating mass. It is characterized by deformation of allthe seven spans and by large displacements of the central piers, because of their height. Within thedynamic characterization phase, the same mechanical properties adopted for the 3D model areassigned to the fibres of the 1D model, and the stiffness of the diagonal trusses representing thespandrel walls is divided by 2, to balance the fictitious contribution provided by the elements intension. A satisfactory agreement is found in terms of both modal shape and period. As for thelatter, a certain overestimate results from the fibre beam model (T = 0.72 s), which may be related tothe simplified representation of the effective stiffening effect provided by spandrels and fill soil.

5.3. Pushover analyses and IDA

Pushover analyses are carried out on Ronciglione viaduct under in-plane loads with the three loaddistributions described in the previous section: (1) is proportional to nodal masses and (2) isproportional to nodal masses times horizontal displacements of longitudinal principal mode. Finally,(3) also includes vertical loads proportional to modal displacements. Forces are applied to the nodesof arches and piers, as well as to those representing fill soil and spandrels, within an incrementalquasistatic analysis, after the application of the entire self-weight. Analyses are carried out underdisplacement control and adopting an energy increment-based convergence criterion. The resultingcapacity curves are plotted in Figure 18, in which the horizontal displacement of the springingsection of the central span (dk) is on the horizontal axis, and the normalized base shear (Vb/W) is onthe vertical one. Similar to the single arch considered in the previous section, the capacity curveunder load distribution (1) is higher and stiffer than that of distribution (2). The ultimate loads are inthe order of 0.33 and 0.14W, respectively, with a reduction of about 57%. Conversely, theestimated seismic capacity of the bridge under the loading profile (3) is about 0.13W. Therefore, inthis specific case, a smaller mismatch between (2) and (3) is found, because of the height of thepiers, whose bending deflection mainly governs the in-plane response, whereas the deformation ofthe arches is partially constrained by the stiffening effect of spandrels and infill.

Incremental nonlinear dynamic analyses were carried out with the same set of natural signals used inthe previous section, applied with increasing SF, ranging from 0.2 to 2. The mark corresponding to theintegral application of the earthquake input (SF = 1) is at dk = 41mm and Vb = 0.045W and representsthe estimated average response of Ronciglione viaduct under the selected group of signals. The averageIDA curve is in good agreement with those corresponding to load distributions (2) and (3),confirming that a loading profile proportional to modal displacements is the best representation ofthe inertial forces among those considered in the present work. The runtime required for300 nonlinear analyses (30 signals with 48 s mean duration, 10 SFs) on the bridge, at 0.005 stime-step, is about 14 h (corresponding to 2′48″ average runtime per analysis) in a standard PCequipped with a 3.00GHz processor.

Figure 17. First longitudinal modal shape of Ronciglione viaduct: comparison between 3D and fibre beammodels.



Clearly, the fibre beam model provides a simplified description of the bridge. Apart from therepresentation of piers and barrel vaults through 1D elements, the abutments and the foundations areconsidered as perfectly fixed, which may yield to a strength overestimate. Moreover, the effect ofspandrels and fill soil in terms of stiffness and strength is taken into account in a very simplifiedway, which may instead lead to an underestimate of the actual capacity. Nevertheless, thanks to thelow computational effort required, the proposed method turns out to be able of providing anestimate of the bridge’s response under earthquake ground motion, with an accurate description ofboth seismic input and effective mechanical properties of historic masonry.

6. CONCLUSIONS

A modelling approach based on fibre beam elements is proposed to study the dynamic behaviour ofmasonry arches and arch bridges and to evaluate their seismic capacity under in-plane actions.Starting from Oppenheim’s dynamic model [6], in which the arch is described as a four-bar linkagemechanism, a failure condition under impulse base motion is derived, and the three domainsidentifying onset of motion, hinging and collapse are derived for different geometric configurations.Higher capacity is found for larger arches (given the shape), thicker arches (given the size) and forarches with lower rise-to-span ratios (given the span), which agrees with the results of previousresearch works [7, 8]. The masonry arch is then modelled as a segmental fibre beam and its failureunder pulse base accelerations is investigated through nonlinear dynamic analyses. A goodagreement is found with the solution provided by rigid block dynamics, proving the reliability of thefibre beam approach, which, differently from analytical procedures, does not need the collapsemechanism to be defined a priori.

The fibre beam methodology takes into account the effective material properties in terms of stress-strain relationship, which are found to play a relevant role in the capacity of the arch under both staticand dynamic loads. Indeed, not only the static multiplier but also the maximum sustainable impulseamplitude decreases if finite crushing strength and limited ductility are accounted for, thusovercoming the simplifying constitutive assumptions of analytical procedures. As a further step, asuitable constitutive law is defined to represent the response of masonry under cyclic compression,and the seismic capacity of an arch under seismic input is assessed including geometricnonlinearities. IDAs are performed under a set of 30 natural accelerograms, and the average IDAcurve is in good agreement with pushover results, provided that in this latter a load distribution isused that includes both horizontal and vertical loads proportional to masses and modal displacements.

Finally, the seismic capacity of an existing seven-span railway masonry viaduct is evaluated. Thefibre beam approach is validated by comparison to a 3D FE model in terms of modal shapes and

Figure 18. IDA curve and pushover (POA) curves under different load distributions for Ronciglione viaduct.



natural frequencies. Afterward, pushover analyses under different loading profiles and IDAsunder natural accelerograms are carried out in the longitudinal direction. Because the bendingdeflection of the slender piers mainly governs the dynamic behaviour of the viaduct, the verticaldisplacements of the barrel vaults do not result to be as significant as for the arch alone. Thecomparison between pushover and IDA curves, however, confirms the necessity of considering aload distribution proportional to masses and fundamental modal shape in order to derive a reliableseismic capacity estimate.

The results of the present work indicate that neglecting the inertial forces related to modaldisplacements in a pushover based seismic assessment of a masonry arched structure may be unsafe,as it leads to an overestimate of the actual capacity. Despite the simplifications in the representationof structural geometry, the proposed methodology allows for a detailed description of both materialproperties and earthquake input with reasonable computational effort, thus appearing a promisingapproach for a preliminary assessment of the seismic capacity of arched masonry structures.

ACKNOWLEDGEMENTS

The financial support of the research projects ‘MIUR PRIN-2009, Analysis and Modelling Strategies for theConservation of multi-leaf Historical Masonry’ and ‘ReLUIS-DPC 2010–2013, Thematic Area 1 – Seismicassessment of built heritage, Task 1.1 – Masonry structures and cultural heritage’ are kindly acknowledged.

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a fibre beam-based approach for the evaluation of the seismic capacity of masonry arches

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