collapse propagation in bridge structures. a semi-analytical model

Collapse propaga+on in bridge structures. A semi-‐analy+cal model

Michele Brun

Department of Mechanical Chemical and Material Engineering University of Cagliari

Brun – Collapse propaga+on in bridge structures.

2

Brun et al. Transition wave in a collapsing bridge

FIGURE 1 | Examples of gravitationally driven failure waves occurring in nature and in structural systems are shown. (A) A slab snow avalanche.(B) A falling domino. (C) The failure of the San Saba Bridge (Texas, May 2013). (D) The Tay Bridge disaster. Dundee, Scotland, 1879 (fromhttp://en.wikipedia.org/wiki/The_Tay_Bridge_Disaster).

row (Figure 1B), and the steady-state regime is independent onthe initial conditions (Maddox, 1987; Stronge, 1987).

The theoretical background developed by Brun et al. (Brunet al., 2012, 2013b,c) refers to long bridge structures as waveguidesrather than finite size elastic bodies. That approach enables one tobring the notion of so-called Floquet–Bloch waves from Physics(Brillouin, 1953; Kittel, 1996) in the areas such as Metamateri-als pioneered by Veselago (1968) and Pendry (2000). Importantrelated areas in Applied Mathematics involve averaging and high-frequency homogenization (Movchan and Slepyan, 2007; Crasteret al., 2010). Dispersion of waves and the pass band structureare of paramount importance in understanding of fundamentalmechanism of vibration of long bridges. For a model example, arecent analytical work (Brun et al., 2013a) presented the analysisof a class of functional equations of the Wiener–Hopf type thatdescribes transition waves in a periodic flexural system. The notionof configurational forces enables us to develop the model furtherto take into account the non-linear features of the physical prob-lem. The new mathematical approach has delivered an accurateestimate of the failure wave speed in the collapse of the HarmonyRidge Bridge as compared with the rare footage, which was takenduring propagation of the fault.

In this paper, we show an unusual, and unexpected to a cer-tain degree, phenomenon of a transition wave in a long collapsingbridge. We also demonstrate the link with a certain class of solu-tions, known as Bloch waves, in infinite periodic systems, whichprovide the best description of the influence of individual con-stituents and their interaction within the macro-system of thebridge. Transition waves and failure phenomena are also consid-ered in relation to damage and impact of structured solids, suchas honeycombs and bistable lattices. The approach to failure asa transition wave was advocated through lattice models (Slepyanand Troyankina, 1984; Fineberg and Marder, 1999; Balk et al., 2001;Cherkaev et al., 2005; Slepyan et al., 2005) including the advancedmolecular dynamics simulation (Abraham and Gao, 2000).

The importance of such problems is also apparent to eluci-date ways to prevent such destructions in earthquake protectionsystems.

MATERIALS AND METHODSUNWANTED VIBRATIONS OF LONG BRIDGESAn unusual example of a failure wave occurred in the unfortunatecollapse of the San Saba railway bridge. One would assume thatthe structure was optimally designed and capable of withstand-ing both quasi-static and dynamic loads. Nevertheless, a 300-yardbridge fell apart after catching fire in a dramatic collapse.

Even advanced engineering analysis and optimal design werenot sufficient to prevent a collapse of this relatively modern system.This was a hard and dramatic lesson to learn and a mathematicalmodel offered here shows an unexpected link to a notion of tran-sition waves, which would not be commonly used by structuralengineers and architects. An accurate finite element model (FEM)of San Saba Bridge has been developed for the eigenvalue analysispresented here. The bridge in its actual dimensions is displayed inFigure 2A. The standard procedures of engineering analysis wouldrequire identification of low-frequency resonance vibrations. As inFigure 2, the “dangerous” vibrations would normally be associatedwith horizontal motion of the main deck of the bridge. One wouldnot expect the vertical vibrations of the main deck to be of any con-cern, as the frequencies involved are relatively high. The detaileddiscussion of this data is given below, and the surprising outcomeis that the vertical vibrations play a significant role in formationof the transition wave.

ANALYSIS OF THE SAN SABA BRIDGE FAILUREIn elastic waveguides, the notion of Floquet–Bloch waves is com-monly used (Mead, 1970; Graff, 1991; Brun et al., 2013b) todescribe the rate of transmission of energy and to visualize thevibrating structure. Such waves are also proved to be essential inunderstanding the failure of systems with embedded structuralelements like lattices (Slepyan and Troyankina, 1984; Marder andGross, 1995; Fineberg and Marder, 1999; Slepyan, 2010) or sup-porting pillars (Brun et al., 2013a) as common for long bridges.Namely, the dispersion relations for Floquet–Bloch waves areembedded into the structure of the Wiener–Hopf equation thatdescribes propagation of a failure wave, which may occur in theform of fracture or a transition wave. In the particular case of

Frontiers in Materials | Mechanics of Materials September 2014 | Volume 1 | Article 12 | 2

Gravity driven Transi.on Wave


3

Gravity driven Transi.on Wave

In Fig. 2 we show an illustration of a bridge failed under a wave generated by an earthquake. A periodic pattern is clearlyvisible in the collapsed structures and it corresponds to damaged foundations of the bridges.

The paper is organized as follows: in Section 2 the uniform continuous model is presented, energy balance and thecomplete analytical solution for the displacement are given, three velocity regimes are defined; in Section 3 the solution forthe two discrete-continuous structures is obtained by means of the Wiener–Hopf technique. The critical displacement at thedamaged point is analyzed in Section 4 and the results for the different structures are compared. Finally, concluding remarksare included in Section 5. The symbol “tilde” is used for quantities normalized by the unit length and time defined in Eq. (11)and in Section 3, “bar” indicates increments with respect to the static values of the undamaged structure and “hat” is usedfor mass and stiffness ratios.

2. Uniform continuous model: a beam on an elastic foundation

We start with the simplest formulation of the structure as an infinite uniform elastic beam placed horizontally on auniform elastic foundation as given in Fig. 1a. The elastic foundation approximates the action of the supporting pillars on thebeam and represents the limiting case where the distance between the supports tends to zero. Among the different modelsof foundations described in Salvadurai (1979), we adopt the Winkler foundation, which is more adequate for the case ofdisjoint support pillars. The foundation is considered as massless; however, the corresponding ‘added mass’ is assumedto be included in the beam per-unit-length mass, m. The foundation is initially stressed by the beam under gravity forcesmg (g is the acceleration of gravity).

We consider the failure wave as the propagation of partial damage of the foundation. Namely, we assume that thefoundation stiffness drops at the wave front, η≡x−vt ¼ 0, from its initial value, ϰ1 (ahead of the front) to ϰ2oϰ1 (behind thefront), while the transverse displacementw reaches a critical value wc at the front, η¼ 0. The critical displacement is reachedas a result of dynamic excitation superimposed on the static load due to gravitational forces and, based on simple physicalconsiderations, wc4mg=ϰ1. In this case, tensionless elastic foundations for beams, like the ones analyzed in Nobili (2012)using a variational approach in statics, are not necessary. Also, the problem is considered assuming that a part of the addedmass disappears simultaneously with the jump discontinuity of the foundation stiffness. In this case, m¼m1 ahead of the

Fig. 3. Load q versus displacement w diagram for the beam on elastic foundation; q1 and q2 are the gravitational loads for the undamaged and damagedstructures of stiffnesses ϰ1 and ϰ2, respectively, and w1 ¼ q1=ϰ1, w2 ¼ q2=ϰ2 are the corresponding static displacements. The critical load and displacementat damage are qc and wc, respectively, whereas qn is the load corresponding to the displacement wc of the damaged structure. A is the work of thegravitational loads, E2 is the static strain energy in the damaged structure at η-−∞ and En is the energy loss at failure. The energy excess E0 is thedifference between the area of the two triangles P4P5P6 and P1P2P3 in the (w,q) space.

Fig. 2. View of the destroyed “Puente Viejo” over the Biobio river that links Concepciòn and San Pedro de la Paz, 500 km south of Santiago, Chile, after the8.8 magnitude earthquake on February 27, 2010. & Nicolás Piwonka, National Geographic.

M. Brun et al. / J. Mech. Phys. Solids 61 (2013) 2067–2085 2069

Energy Balance

Troyankina, 1984, 1988) were used. Waves in discrete bistable chains were then studied in Puglisi and Truskinovsky (2000),Slepyan (2000, 2001), Balk et al. (2001a, 2001b), Cherkaev et al. (2005), Slepyan et al. (2005), and Vainchtein and Kevrekidis(2012). Localized transition wave in a two-dimensional lattice model was considered by Slepyan and Ayzenberg-Stepanenko(2004).

The problem also has a strong physical connection with dynamic crack propagation where the energy lost under thesupport damage plays the role of the surface energy in fracture. Super-critical regimes for cracks are considered in manyworks, mainly for mode II shear fracture, see, e.g., Freund (1990), Huang et al. (1998), Broberg (1999), Needleman (1999),Gao et al. (2001), Geubelle and Kubair (2001), Samudrala et al. (2002), and Slepyan (2002). Mode I and II intersonic speedproblems were considered by Radi and Loret (2007, 2008) for a porous, liquid-saturated material, where ‘intersonic’ doesnot mean the shear-longitudinal range in the uniform solid material.

With regard to a large-scale long-length construction, the failure wave may be supported by the gravity forces. In thisconnection we refer to papers by Bažant and Zhou (2002), Bažant and Verdure (2007), and Bažant et al. (2008), which contain acomprehensive analysis of the collapse wave progress in the nine-eleven disaster. A bridge on pillars or a suspended bridge, anoverpass, long conveyers are examples of the constructions where the failure wavemay propagate taking energy from the gravityforces.

In this paper, we examine some simplified models of such a construction considering the latter as a beam on a discretesupport and on a continuous elastic foundation, where the failure wave is that of a partial damage of the supports.Mechanically, these models differ from the above-mentioned ones by the existence of the subsonic and intersonic regimesof the failure wave propagation: no elastic wave is excited in the steady-state regime, as far as it propagates in the subsonicspeed range, and there is such a wave only behind the failure wavefront in the intersonic regime. Note that the elastic waves,if exist, create wave resistance to the failure wave. Possibilities of the wave radiation, in dependence of the failure wavespeed, can be seen in the corresponding dispersion relations found for related structure in Brun et al. (2012). Dispersiondiagrams to compare are plotted in Fig. 10.

The supported beam model also differs from previous phase transition models by the fact that the failure wave speed canvary in a wide range depending on the structure and damage parameters, and it can be very small compared with that in theabove-mentioned bistable models. It is remarkable that the speed limit appears as low as the jump of the support stiffness islarge. Three related models are examined: (a) a dynamic beam on a continuous elastic foundation, (b) a discrete set ofmasses rested on elastic supports and connected by massless beams and (c) a dynamic beam on the set of discrete elasticsupports, Fig. 1. Stiffness of the support is assumed to drop when the stress in the pillars (or the beam transversedisplacement) reaches a critical value. The failure wave is also considered under the condition that, at the moment of thesupport damage, the value of the ‘added mass’, which reflects the dynamic response of the support, is dropped. We havefound the conditions for the failure wave to exist, to propagate uniformly or to accelerate. All speed regimes are considered,such where the failure wave is or is not accompanied by elastic waves excited by the jump in the stiffness and added mass.Limiting speeds are determined. We have shown that the effect of the change of the stiffness can essentially result in thefailure wave speed limitations.

Fig. 1. Supported beam models. Undamaged and damaged parts are indicated with subscripts 1 and 2, respectively. (a) Beam on an elastic foundation. Thebeam has mass density m1;2 ¼m0

1;2 þM1;2=a and bending stiffness D; the foundation has stiffness per unit length ϰ1;2 ¼ ϰ01;2=a. (b) Discrete set of massesM0

1;2 ¼m01;2aþM1;2 connected horizontally by massless beams of bending stiffness D and rested on elastic supports of stiffness ϰ0

1;2 placed at distance a. (c)Dynamic beam of density m0

1;2, added mass M1;2 and bending stiffness D, on the set of discrete elastic supports of stiffness ϰ01;2 placed at distance a. Thetransverse displacement is denoted as w.

M. Brun et al. / J. Mech. Phys. Solids 61 (2013) 2067–20852068

Work of gravita.onal load

Energy loss

Remote sta.c strein energy

front and m¼m2om1 behind the front. Under the moving discontinuity, elastic waves can be excited propagating behindor/and ahead of the failure wave front; however, the static values of the beam vertical displacement are equal to m1g=ϰ1(far ahead of the front) and to m2g=ϰ2 (far behind the front). This allows us to draw some conclusions based on energyconsiderations.

2.1. Energy considerations

Consider the loading diagram shown in Fig. 3, where q1;2, and qc are the initial, the final and the critical load at damageacting on the support, respectively. The other values pointed in the figure are

qn ¼ϰ2ϰ1

qc; w1 ¼q1ϰ1

; w2 ¼q2ϰ2

; wc ¼qcϰ1

; ð1Þ

where qn is the load at critical point η¼−0 in the damaged structure and w1;2 is the beam static displacement at η¼ 7∞. Itcan be seen that the work of the gravity forces, the static strain energy density at η¼−∞ (where the kinetic energy ofdecaying waves is zero) and the energy loss due to the failure are

A¼ 12 q1w1 þ q1ðwc−w1Þ þ q2ðw2−wcÞ; E2 ¼ 1

2 q2w2; En ¼ 12ðqc−qnÞwc: ð2Þ

Since the energy of the beam itself in the steady-state regime is invariable in η, the energy excess per unit length is (Slepyan,2002)

E0 ¼ A−E2−En ¼12

q22ϰ2

−q21ϰ1

! "þ

ðq1−q2Þqcϰ1

−q2c2ϰ1

1−ϰ2ϰ1

! ": ð3Þ

The energies are also shown in the load–displacement plane in Fig. 3. From this static value the energy radiated by elastic waves, ifexists, must be deducted. In this general case, for the uniform propagating failure wave the energy balance is (Hayes, 1977)

E0−U1ðc1=v−1Þ−U2ð1−c2=vÞ ¼ 0; ð4Þ

where U1;2 and c1;2 are the energy density and the group velocities of the waves at 7η40 respectively. Note that U1 ¼ 0 in thesub- and intersonic regimes and U2 ¼ 0 in the subsonic regime.

In this problem, there is a range of the failure wave speed, 0≤vov2, where no elastic wave can be radiated in steady-statemotion. The critical velocity v2, indicated in Fig. 4, is the resonant wave velocity corresponding to the lower dispersioncurve. At this regime the phase and group velocities are equal. The value of v2 is given in the next section. For this subsonicregime, we can conclude that the failure wave cannot propagate if E0o0. In the opposite case, where there is a positiveexcess of the energy, the steady-state regime is impossible, and the failure wavefront move unsteadily spending the energyexcess on the radiation of elastic waves, which arises under nonuniform motion. In the neutral case, E0 ¼ 0, any value of thespeed in the subsonic range satisfies the energy balance. However, in the latter case, the system parameters are connectedby a relation following from the equation E0 ¼ 0. In particular, if no loss of the mass is assumed, this relation is

wc ¼wnc≔w1

ffiffiffiffiffiffiffiffiffiffiffiffiffiϰ1=ϰ2

p: ð5Þ

In the subsonic regime, the wave propagates nonuniformly if the real critical displacement wc is less than the value wnc ,

otherwise, it cannot propagate. When wcownc , which means that E040, steady-state motion is possible at v4v2, solution

of (4) as shown in the example at the end of Section 2.

Fig. 4. Dispersion diagrams for the waves in uniform beams on an elastic foundation of stiffness ϰ. The dispersion curves are shown for the undamagedfoundation (ω1ðkÞ for stiffness ϰ¼ ϰ1), the damaged one (ω2ðkÞ for ϰ¼ ϰ2, with ϰ̂ ¼ ϰ2=ϰ1 ¼ 0:36) and the limiting case with ϰ¼ 0 (gray curve); no lossof mass is considered. Straight lines correspond to the different regimes of the failure wave speeds, v¼ω=k¼const. The group velocities c1;2 ¼ dω1;2=dkcoincide with the speed v at the resonant regimes, where v¼ v2 ¼

ffiffiffi2

pðϰ̂ Þ1=4 and v¼ v1 ¼

ffiffiffi2

p. The ðω; kÞ-points corresponding to the radiated waves are

marked by small bullets; such resonant points are marked by larger bullets. The non-dimensional variables are used corresponding to the length and timeunits introduced in (11).


Energy excess


Dispersion proper.es












front and m¼m2om1 behind the front. Under the moving discontinuity, elastic waves can be excited propagating behindor/and ahead of the failure wave front; however, the static values of the beam vertical displacement are equal to m1g=ϰ1(far ahead of the front) and to m2g=ϰ2 (far behind the front). This allows us to draw some conclusions based on energyconsiderations.

2.1. Energy considerations

Consider the loading diagram shown in Fig. 3, where q1;2, and qc are the initial, the final and the critical load at damageacting on the support, respectively. The other values pointed in the figure are

qn ¼ϰ2ϰ1

qc; w1 ¼q1ϰ1

; w2 ¼q2ϰ2

; wc ¼qcϰ1

; ð1Þ

where qn is the load at critical point η¼−0 in the damaged structure and w1;2 is the beam static displacement at η¼ 7∞. Itcan be seen that the work of the gravity forces, the static strain energy density at η¼−∞ (where the kinetic energy ofdecaying waves is zero) and the energy loss due to the failure are

A¼ 12 q1w1 þ q1ðwc−w1Þ þ q2ðw2−wcÞ; E2 ¼ 1

2 q2w2; En ¼ 12ðqc−qnÞwc: ð2Þ

Since the energy of the beam itself in the steady-state regime is invariable in η, the energy excess per unit length is (Slepyan,2002)

E0 ¼ A−E2−En ¼12

q22ϰ2

−q21ϰ1

! "þ

ðq1−q2Þqcϰ1

−q2c2ϰ1

1−ϰ2ϰ1

! ": ð3Þ

The energies are also shown in the load–displacement plane in Fig. 3. From this static value the energy radiated by elastic waves, ifexists, must be deducted. In this general case, for the uniform propagating failure wave the energy balance is (Hayes, 1977)

E0−U1ðc1=v−1Þ−U2ð1−c2=vÞ ¼ 0; ð4Þ

where U1;2 and c1;2 are the energy density and the group velocities of the waves at 7η40 respectively. Note that U1 ¼ 0 in thesub- and intersonic regimes and U2 ¼ 0 in the subsonic regime.

In this problem, there is a range of the failure wave speed, 0≤vov2, where no elastic wave can be radiated in steady-statemotion. The critical velocity v2, indicated in Fig. 4, is the resonant wave velocity corresponding to the lower dispersioncurve. At this regime the phase and group velocities are equal. The value of v2 is given in the next section. For this subsonicregime, we can conclude that the failure wave cannot propagate if E0o0. In the opposite case, where there is a positiveexcess of the energy, the steady-state regime is impossible, and the failure wavefront move unsteadily spending the energyexcess on the radiation of elastic waves, which arises under nonuniform motion. In the neutral case, E0 ¼ 0, any value of thespeed in the subsonic range satisfies the energy balance. However, in the latter case, the system parameters are connectedby a relation following from the equation E0 ¼ 0. In particular, if no loss of the mass is assumed, this relation is

wc ¼wnc≔w1

ffiffiffiffiffiffiffiffiffiffiffiffiffiϰ1=ϰ2

p: ð5Þ

In the subsonic regime, the wave propagates nonuniformly if the real critical displacement wc is less than the value wnc ,

otherwise, it cannot propagate. When wcownc , which means that E040, steady-state motion is possible at v4v2, solution

of (4) as shown in the example at the end of Section 2.

Fig. 4. Dispersion diagrams for the waves in uniform beams on an elastic foundation of stiffness ϰ. The dispersion curves are shown for the undamagedfoundation (ω1ðkÞ for stiffness ϰ¼ ϰ1), the damaged one (ω2ðkÞ for ϰ¼ ϰ2, with ϰ̂ ¼ ϰ2=ϰ1 ¼ 0:36) and the limiting case with ϰ¼ 0 (gray curve); no lossof mass is considered. Straight lines correspond to the different regimes of the failure wave speeds, v¼ω=k¼const. The group velocities c1;2 ¼ dω1;2=dkcoincide with the speed v at the resonant regimes, where v¼ v2 ¼

ffiffiffi2

pðϰ̂ Þ1=4 and v¼ v1 ¼

ffiffiffi2

p. The ðω; kÞ-points corresponding to the radiated waves are

marked by small bullets; such resonant points are marked by larger bullets. The non-dimensional variables are used corresponding to the length and timeunits introduced in (11).


Three regimes

In particular

wð0Þ ¼v2−2ϰ̂−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiv4−4ϰ̂

p

2ϰ̂g: ð21Þ

Since wð0Þ depends on v, this allows us to satisfy the condition (10). The displacement profileswðηÞ for different values of thespeed v are presented in Fig. 6a.

In Fig. 6b we also plot the value of wð0Þ as a function of velocity v for ϰ̂ ¼ 0:5. The steady-state propagation speed vcorresponds to equality wð0Þ ¼wc40 when ϰ2=ϰ1o ðwc=g þ 1Þ−2. We note that such a speed v depends on the safety factorwc=g þ 1 (expressed in terms of non-dimensional values).

As can be seen in Fig. 6b the displacement, wð0Þ, monotonically decreases as the speed, v, increases from the criticalvalue, v¼ v2 ¼

ffiffiffi2

pðϰ̂Þ1=4, and becomes negative when the speed exceeds a value. These negative values correspond to a

hypothetic case where the energy lost at the moment of the damage is less than its real minimal value, that is the criticalload is less than the initial static load, qcomg.

The critical displacement wð0Þ versus v is shown in Fig. 7a for different values of ϰ̂ , while the total speed bound wð0Þ ¼ 0is given in the ðϰ̂ ; vÞ space in Fig. 7b. The steady propagation of the fault in the intersonic regime is not forbidden whenwð0Þ40, which implies that the velocity v satisfies the following constraint:

ffiffiffi2

pϰ̂1=4ovo

ffiffiffiffiffiffiffiffiffiffiffiffi1þ ϰ̂

p: ð22Þ

It is also noted that for the intersonic velocities within the interval ðffiffiffiffiffiffiffiffiffiffiffiffi1þ ϰ̂

p;

ffiffiffi2

pÞ the failure wave propagation is not possible.

2.2.3. Supersonic regimeFinally, we address the case when the velocity of the transition front v4

ffiffiffi2

p. In this case, elastic waves are generated in

both regions η40 and ηo0. The displacement takes the form

wðηÞ ¼ A1 cos β1ηþ B1 sin β1η ðη40Þ;wðηÞ ¼ A2 cos β2ηþ B2 sin β2ηþQ=ϰ̂ ðηo0Þ;

β1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=4−1

qr; β2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=2−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=4−ϰ̂

qr; ð23Þ

where the group velocity of the elastic wave ahead of the transition front is greater than the speed of the latter (see Fig. 4),and the constants defined by the continuity conditions are

A1 ¼− v2−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiv4−4ϰ̂

pffiffiffiffiffiffiffiffiffiffiffiv4−4

pþ


pQϰ̂; B1 ¼ 0;

A2 ¼−v2 þ

ffiffiffiffiffiffiffiffiffiffiffiv4−4


pþ


pQϰ̂; B2 ¼ 0: ð24Þ

Thus

wð0Þ ¼ −v2−



pþ


p1−ϰ̂ϰ̂

go0: ð25Þ

Since the transition can occur when wð0Þ40 this evidences that the wave cannot propagate at supersonic velocities. Thus,the absolute maximum of the failure wave speed is


p.

Fig. 8. The beam on an elastic foundation. Displacement at the transition point is given in black continuous lines as a function of velocities in the subsonic,intersonic and supersonic regimes for ϰ1=ϰ2 ¼ 0:35;0:45;0:75. Gray line indicates the critical displacement wc=g¼ 0:3 and black circles correspond to thesteady-state propagation of the fault at vn ¼ 1:106;1:164.



Steady propaga.on












In particular

wð0Þ ¼v2−2ϰ̂−


p

2ϰ̂g: ð21Þ

Since wð0Þ depends on v, this allows us to satisfy the condition (10). The displacement profileswðηÞ for different values of thespeed v are presented in Fig. 6a.

In Fig. 6b we also plot the value of wð0Þ as a function of velocity v for ϰ̂ ¼ 0:5. The steady-state propagation speed vcorresponds to equality wð0Þ ¼wc40 when ϰ2=ϰ1o ðwc=g þ 1Þ−2. We note that such a speed v depends on the safety factorwc=g þ 1 (expressed in terms of non-dimensional values).

As can be seen in Fig. 6b the displacement, wð0Þ, monotonically decreases as the speed, v, increases from the criticalvalue, v¼ v2 ¼

ffiffiffi2

pðϰ̂Þ1=4, and becomes negative when the speed exceeds a value. These negative values correspond to a

hypothetic case where the energy lost at the moment of the damage is less than its real minimal value, that is the criticalload is less than the initial static load, qcomg.

The critical displacement wð0Þ versus v is shown in Fig. 7a for different values of ϰ̂ , while the total speed bound wð0Þ ¼ 0is given in the ðϰ̂ ; vÞ space in Fig. 7b. The steady propagation of the fault in the intersonic regime is not forbidden whenwð0Þ40, which implies that the velocity v satisfies the following constraint:

ffiffiffi2

pϰ̂1=4ovo


p: ð22Þ

It is also noted that for the intersonic velocities within the interval ðffiffiffiffiffiffiffiffiffiffiffiffi1þ ϰ̂

p;

ffiffiffi2

pÞ the failure wave propagation is not possible.

2.2.3. Supersonic regimeFinally, we address the case when the velocity of the transition front v4

ffiffiffi2

p. In this case, elastic waves are generated in

both regions η40 and ηo0. The displacement takes the form

wðηÞ ¼ A1 cos β1ηþ B1 sin β1η ðη40Þ;wðηÞ ¼ A2 cos β2ηþ B2 sin β2ηþQ=ϰ̂ ðηo0Þ;

β1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=4−1

qr; β2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=2−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2=4−ϰ̂

qr; ð23Þ

where the group velocity of the elastic wave ahead of the transition front is greater than the speed of the latter (see Fig. 4),and the constants defined by the continuity conditions are

A1 ¼− v2−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiv4−4ϰ̂


pþ


pQϰ̂; B1 ¼ 0;

A2 ¼−v2 þ

ffiffiffiffiffiffiffiffiffiffiffiv4−4


pþ


pQϰ̂; B2 ¼ 0: ð24Þ

Thus

wð0Þ ¼ −v2−



pþ


p1−ϰ̂ϰ̂

go0: ð25Þ

Since the transition can occur when wð0Þ40 this evidences that the wave cannot propagate at supersonic velocities. Thus,the absolute maximum of the failure wave speed is


p.

Fig. 8. The beam on an elastic foundation. Displacement at the transition point is given in black continuous lines as a function of velocities in the subsonic,intersonic and supersonic regimes for ϰ1=ϰ2 ¼ 0:35;0:45;0:75. Gray line indicates the critical displacement wc=g¼ 0:3 and black circles correspond to thesteady-state propagation of the fault at vn ¼ 1:106;1:164.



Discrete structures













3.3. Factorization

To facilitate this action we assume that a small dissipation exists in proportion to the strain rate, that is, we correct theexpression for the bending moment to be

M¼∂2wðx; ηÞ

∂x2þ α

∂3wðx; ηÞ∂t∂x2

; ð49Þ

where α is a small time parameter of viscosity. The non-dimensional beam equation after the Fourier transform on ηbecomes

ð1þ ikvαÞ∂4wF ðx; kÞ

∂x4−v2k2wF ðx; kÞ ¼ 0; ð50Þ

that results in the change of the parameter λ:

λ⟹λ

ð1þ ikvαÞ1=4¼

m0

m1

! "1=4 ffiffiffiffiffiffiffiffiffiffiffiffiffikv−i0

p

ð1þ ikvαÞ1=4¼Oð kj1=4Þ ðv40; k-∞Þ:

$$ ð51Þ

Now the product

LðkÞ ¼M2

M1L0ðkÞ-1 ðk-7∞Þ ð52Þ

has no zeroes on the real k-axis (if ϰ0240). It means that its index is zero, i.e.

Ind LðkÞ ¼12π

½Arg Lð∞Þ−Arg Lð−∞Þ& ¼ 0: ð53Þ

Besides, the integral of ln LðkÞ over the real k-axis converges. This allows us to use the Cauchy type integral for thefactorization of this function, that is to represent

LðkÞ ¼ limIk-0

LþðkÞL−ðkÞ; L7 ðkÞ ¼ exp 712πi

Z ∞

'∞

ln LðξÞξ−k

dξ% &

ð7Ik40Þ: ð54Þ

Eq. (48) can be represented in the form

LþðkÞwþðkÞ þM2w−ðkÞM1L−ðkÞ

¼ GðkÞM1

M2LþðkÞ−

1L−ðkÞ

% &;

GðkÞ ¼M2C

M1ð0þ ikÞ½1−ϰ2=ϰ1 þ ðM1−M2Þð0þ ikvÞ2&: ð55Þ

First consider the case M1 ¼M2 ¼M. In this case, we have

GðkÞ LþðkÞ−1

L−ðkÞ

% &¼ C1 þ C2; C1 ¼

gik½LþðkÞ−Lþð0Þ&;

C2 ¼g

0þ ikLþð0Þ−

1L−ðkÞ

% &; ð56Þ

where C1;2 are regular in the upper (C1) and lower (C2) half-planes of k. The solution follows as

wþðkÞ ¼gikLþðkÞ−Lþð0Þ

LþðkÞ;

w−ðkÞ ¼g

0þ ik½Lþð0ÞL−ðkÞ−1&: ð57Þ

In particular, using the limiting relations

wð70Þ ¼ limk-7 i∞

ð∓ikÞw7 ðkÞ; L7 ð7 i∞Þ ¼ 1 ð58Þ

we find that

wðþ0Þ ¼wð−0Þ ¼wð0Þ ¼ g½Lþð0Þ−1& ð59Þ

with

L7 ð0Þ ¼ffiffiffiffiffiϰ1ϰ2

rexp 7

1π

Z ∞

0

Arg LðkÞk

dk% &

Lð0Þ ¼ϰ1ϰ2

! "; ð60Þ

where Arg LðkÞ is equal to zero in a vicinity of k¼0 and exponentially decreasing as k-∞. So the above integral converges.In the case M2oM1, the function G(k) in (55) can be split by the corresponding terms as follows:

GðkÞ ¼ C1ðkÞ þ C2ðkÞ;

C1ðkÞ ¼ C0LþðkÞ−Lþð0Þ

0þ ik−12

LþðkÞ−LþðβÞ0þ iðk−βÞ

þLþðkÞ−Lþð−βÞ0þ iðkþ βÞ

! "% &;


Wiener-‐Hopf equa.on

Cri.cal Displacement













and hence, when anoxoaðn−1Þ, the displacement is cubic in x. For x¼an the equations are the same as in (36) and (37).The Wiener–Hopf equation for the one-sided Fourier transform of the displacement has the form (48) with M1 ¼M2.

Taking into account Eq. (44) we deduce

L1ðkÞ ¼ 1þ ð0þ ikvÞ2 þ12a4

ð1− cos kaÞ2

2þ cos ka;

L2ðkÞ ¼ ϰ2=ϰ1 þ ð0þ ikvÞ2 þ12a4

ð1− cos kaÞ2

2þ cos ka: ð65Þ

Consequently, the dispersion equation describing the waves for η40 and ηo0 are

ω1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ12ð1− cos kaÞ2

a4ð2þ cos kaÞ

s

;

ω2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϰ2ϰ1

þ12ð1− cos kaÞ2

a4ð2þ cos kaÞ

s

; ð66Þ

respectively. The corresponding dispersion curves are shown in Fig. 10.

Fig. 11. The piecewise constant (a) Arg½L1&, (b) Arg½L2& and (c) Arg½L& as a function of the normalized wavenumber ka. Results are given for a¼2.0 andv¼0.23. Jumps occur at the intersection point wavenumbers kð1Þ1 ;…; kð1Þ3 and kð2Þ1 ;…; kð2Þ5 highlighted in Fig. 10.

Fig. 12. Critical displacement wð0Þ as a function of the speed v. The uniform continuous model (Fig. 1a, Eqs. (18), (21) and (25)) is compared with thediscrete massless beam model (Fig. 1c, Eq. (67)). Results are given for ϰ2=ϰ1 ¼ 0:5 and in the discrete model a¼2. The velocities vI, vII, vIII and vIV arereported in the dispersion diagram in Fig. 10. The critical displacement wc=g¼ 0:3 is denoted by the gray line.


Discrete and discrete-‐con.nuous beam


and that for k-þ i∞

1LþðkÞ

∼1− 12πi

Z ∞

$∞

ln LðξÞξ−k

dξ; ð70Þ

we check the range of velocity where

_wð0Þ ¼vwð0Þ2π

Z ∞

$∞ln LðkÞ dk40: ð71Þ

By using the symmetries of L1ðkÞ and L2ðkÞ, the integral in (71) is given byZ ∞

$∞ln LðkÞ dk¼ 2

Z ∞

0lnðjLðkÞjÞ dk: ð72Þ

which is integrable but the integrand has logarithmic singularities. Then, the integral can be split asZ ∞

0¼

Z k1

0þZ k2

k1þ⋯þ

Z kM

k2ðn1þn2þ1Þ

þZ ∞

kMð73Þ

where ðk1; k2;…; k2ðn1þn2þ1ÞÞ are the 2n1 þ 2n2 þ 2 intersection points between the line ω¼ kv and the dispersion curves ω1

and ω2, such that kiokj for io j. For the parameters considered in Fig. 10, at v¼0.23, the wavenumbers are ðkð2Þ1 ; kð2Þ2 ;kð2Þ3 ; kð1Þ1 ; kð1Þ2 ; kð2Þ4 ; kð2Þ5 ; kð1Þ3 Þ. The first 2n1 þ 2n2 þ 2 integrals can be computed numerically using standard quadrature rules forintegrand having logarithmic singularities at the endpoints and kM has been chosen to be sufficiently large to guarantee thatjR∞kM

lnðjLðkÞjÞ dkj=jR kM0 lnðjLðkÞjÞ dkjo10−6.

These values are reported in Table 1. We see that for v≤0:55, _wð0Þo0 (except at v¼0.2), showing that in this range nosteady-state propagation, w¼wðx; ηÞ, can exist.

Following the behavior of the transversal displacement wð0Þ at higher velocities, it can be seen that there is a sort ofplateau in the range vIovovII with a drastic drop in the range vIIovovIII . This dependence shows strong similarities withthe behavior of the continuous model.

We can conclude that steady-state propagation, when possible, occurs in a range of velocity which is exceptionally wellapproximated by the narrow range of velocities (22), within the intersonic regime, provided by the continuous model of beam on

Fig. 13. Critical transversal displacement wð0Þ as a function of the speed v. The discrete massless beam model (continuous gray lines) and the uniformcontinuous model (dashed black line) are represented with ϰ2=ϰ1 ¼ 0:5. In the discrete model a¼ 0:25;1:0;2:0;4:0; and 8:0.

Fig. 14. Dispersion diagrams ω1;2ðkÞ for the discrete massless beam model. Results are given for a¼ 1:4;1:2;1:0;þ0 and ϰ2=ϰ1 ¼ 0:5.


elastic foundation. In such a velocity range, the steady-state solution for the discrete case is linearly stable and satisfies theconditions (71) and the critical one wð0Þ ¼wc.

In Fig. 13 the displacement wð0Þ is given as a function of velocity v for different values of the normalized span length a.We note the good agreement between the continuous and the discrete models for sufficiently small values of a, which isexpected on physical ground, an issue that will be better discussed in Section 4.2. Also, the continuous model is approachedfrom below at decreasing values of a.

4.1.1. Dispersion diagram transformation and the continuous limitLet a-þ 0. Expanding in ka we deduce from (66) that in any finite range of the wavenumber the dispersion relations

take the form

ω1∼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ k4

q; ω2∼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϰ2=ϰ1 þ k4

q; ð74Þ

while the remaining periodic part of the dispersion curves moves away to infinity (see Fig. 14). In this transformation, thetwo branches, ω1 and ω2 (66), coalesce, becoming closer and closer, and the separation ω1ðkÞ−ω2ðkÞ-0 as a-0. Hence in thelimit the cross points other than those corresponding to the asymptotic formulae (74), do not contribute to the product in(67) for the displacement wð0Þ. At the same time, these asymptotic expressions coincide with the dispersion relations (15)derived for the continuous model, which has the same average density and the same average support stiffness as the discretemodel. Thus, the dynamic response of the discrete flexural system is shown to converge to the continuum limit as a-0.

4.2. Inertial beam model

For the case of dynamic beam on the set of discrete elastic supports (Fig. 1c) the dispersion diagrams for the undamagedand damaged structures have infinite numbers of curves (see, for example, Brun et al., 2012) and Eq. (67) is still valid, butwith n1;n2-∞. To compute the displacement wð0Þ is then convenient to introduce some dissipation as in (49) and make useof Eqs. (59) and (60).

In Fig. 15 the displacement wð0Þ is given as a function of velocity v for different values of the normalized span length a.Computations have been done assuming α¼ 0:01 and the integral in (60) was truncated imposing a relative error smallerthan 10−6. The comparative analysis between the results of Figs. 13 and 15 shows larger differences at increasing values of a,where the difference between the coefficients in (43) and in (44) is more pronounced and the wave propagation is betterdescribed by the beam inertial model.

Despite the fact that the interval of velocity where oscillations are present increases with a, all the curves display adrastic drop in correspondence of the intersonic regime identified by the model of beam on elastic foundation, thus giving auniversal property for the steady-state failure propagation in such mono-dimensional structures.

Finally, we report some data deduced from real life bridges, showing the range of possible engineering applications of theproposed model. We first consider a bridge of average dimension, the S'Adde bridge, whose geometrical and materialproperties are given in the caption of Fig. 4 of Brun et al. (2012), with a span length of 90 m. In such a case the normalizedspan length is a¼2.02 for vertical flexural waves and a¼0.83 for horizontal flexural waves. In addition, we also pay attention to the Millau viaduct, one of the tallest vehicular bridges and the longest multiple cable-stayed bridge in the world(Magalhães et al., 2012). The steel structure has the Young modulus E¼210,000 MPa, moments of inertia Jy¼1137.7 m4 andJz¼8267.5 m4, vertical and transverse stiffnesses ϰz ¼ 20;000 MPa m and ϰy ¼ 41:84 MPa m, respectively, and span length of342 m. Then, the structure has normalized span length a¼7.61 for vertical flexural waves and a¼0.99 for the horizontalones, showing that, also for this extreme structure, we are in the range of a where the presented model can be used tomodel the steady-state failure propagation.

Fig. 15. Critical displacement wð0Þ as a function of the speed v. The discrete inertial beam model (continuous gray lines) and the uniform continuous model(dashed black line) are represented with ϰ2=ϰ1 ¼ 0:5. In the discrete-continuous model m0=m1 ¼ 0:67 and a¼ 0:25;1:0;2:0;4:0;8:0.

M. Brun et al. / J. Mech. Phys. Solids 61 (2013) 2067–2085 2083























Collapse of San Saba bridge


Transition wave in the collapse of the San Saba bridge Michele Brun 1,2, Gian Felice Giaccu3, Alexander B. Movchan2*, Leonid I. Slepyan 4,5

1Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari, Via Marengo 3, 09123 Cagliari, Italy. 2Department of Mathematical Sciences, University of Liverpool, L69 7ZL Liverpool, UK. 3Dipartimento di Architettura, Design e Urbanistica, Facoltà di Architettura, Università di Sassari, Palazzo del Pou Salit, Piazza Duomo 6, 07041 Alghero, Italy. 4School of Mechanical Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. 5Department of Mathematics and Physics, Aberystwyth University, Aberystwyth, Ceredigion, SY23 3BZ, UK.

*Correspondence to: Alexander B. Movchan, [email protected] Abstract: A domino wave is a well-known illustration of a transition wave, which appears to reach a stable regime of propagation. Nature also provides spectacular cases of gravity driven transition waves at large scale, observed in snow avalanches and landslides. On a different scale, the micro-structure level interaction between different constituents of the macro-system may influence critical regimes leading to instabilities in avalanche-like flow systems. Most transition waves observed in systems such as bulletproof vests, racing helmets under impact, shock-wave driven fracture in solids, are transient. For some structured waveguides a transition wave may stabilize to achieve a steady regime. Here we show that the failure of a long bridge is also driven by a transition wave that may allow for steady-state regimes. The recent observation of a failure of the San Saba Bridge in Texas provides experimental evidence supporting an elegant theory based on the notion of transition failure wave. No one would think of an analogy between a snow avalanche and a collapsing bridge. Despite an apparent controversy of such a comparison, these two phenomena can both be described in the framework of a model of the dynamic gravity driven transition fault. Introduction The long San Saba railway bridge (Fig. 1C) in the Central Texas (also known as Harmony Ridge Bridge, 31°14'07" N, 98°33'52" W) collapsed in May 2013 as a result of initial damage caused by fire. A 300-yard bridge fell apart after catching fire in a dramatic collapse captured on video (1). This dramatic event was the subject of attention worldwide, when it was featured on BBC News and other News programs across the globe. The video footage provides the data for measuring the speed of propagation of the failure, and it is apparent that this failure reaches a steady state regime. Although the phenomenon of collapse of a long bridge is extraordinarily complicated, we show that it can be analysed in the framework of an analytical model, which refers to gravity driven transition waves. Furthermore, an explicit simple formula has been derived for the speed ! of the steady state propagating fault:

! = !!!!!

!/!= 24.3 m/sec (1)

where D is the flexural rigidity and ! the linear mass density of the bridge and ! the stiffness of

Brun et al. Transition wave in a collapsing bridge

is a monotonically decreasing function of the crack speed. Inparticular, when w(0) = 0 we have

V = V =✓

⇠

⌧

◆ r1 + 2

1, (3)

which becomes V = (D 1/⇢2)

1/4 = ⇠/⌧ in the limit of a com-plete failure of the damaged foundation (i.e.,2 ! 0) as in Figure 4

and in Eq. (1). The analytical model of Brun et al. (2013a) predictsthe speed of steady propagation to be in the left neighborhood ofthe upper limit V . In the following, we will show that the pre-dicted speed matches amazingly with the observation recordedduring the failure of the bridge.

RESONANCE MODES AND “INSIGNIFICANT” FLEXURAL VIBRATIONSA direct transient analysis for a failing bridge would involve alarge-scale computational model and is not considered to be fea-sible in the engineering practice. Of course, the choice of initialconditions and evolution of the structure becomes an importantand challenging part of the computational procedure. A con-ventional engineering approach would allow an extensive anddetailed analysis of eigenfrequencies and resonant modes. Howuseful would this information be in the circumstances related toSan Saba Bridge?

To answer this question, we have done a complete eigenfre-quency analysis in the framework of an FEM based on the indus-trial grade tool Strand7. The computational model has not beensimplified in any way, and every technical detail has been embed-ded in a full three-dimensional FEM computational domain asshown in Figure 2A. The details of computational parameters aresupplied in the supplementary material. The computed eigenfre-quencies accurately represent the resonant vibrations of the actualSan Saba Bridge.

For an undamaged structure, the vertical flexural modes wouldnot attract much significant attention of a Structural Engineer,since they correspond to relatively high-frequency range (f = 10.4–11.1 Hz) compared to the modes involving a horizontal motionof the upper deck of the bridge (f = 1.6–8.9 Hz). In Figure 2,the first types of eigenmodes are shown: an example of a trans-verse flexural mode is represented in Figure 2D, a mode wherevibrations are localized within the supporting beam is shown inFigure 2B, a typical torsional mode is reported in Figure 2C and,finally, a vertical flexural mode is given in Figure 2E. The overalldiagram with eigenfrequencies (Figure 2F) suggests that the ver-tical flexural vibrations would be in the highest frequency rangeamong the identified vibrations. The three-dimensional computa-tion has revealed that the low-frequency vibrations of the San SabaBridge correspond to horizontal modes, transverse (Figure 2D)and longitudinal. On the contrary, the vertical flexural vibrations(Figure 2E) occur in a narrow band at much higher frequenciesand also take into account the effect of the longitudinal stiffness ofthe supporting pillars. The corresponding resonance frequenciesare in the same range of the frequencies associated with local-ized vibration of the pillars (f = 8.5–10.6 Hz) and torsional vibra-tions (f = 10.9–14.4 Hz), making difficult to distinguish differenteigenmodes.

FIGURE 4 | Vertical displacement w (0) at the failure point as a functionof the velocity v . The curves are shown for stiffness ratios 2/1 ! 0(similar to the real bridge) and 2/1 = 0.25. The velocityv = V /(⇠/⌧ ) =

p(1 + 2/1), corresponding to w (0) = 0 is an upper limit for

the steady-state velocity of propagation. Steady-state propagation ispossible only in the intersonic velocity regime, the velocity interval wherethe curves are monotonically decreasing. The steady-state failureconfiguration (Figure 3B) shows that the vertical displacement w (0) is ofsmall magnitude and the critical velocity is in the left neighborhood ofv = V /(⇠/⌧ ) = 1.

DISCUSSIONPREDICTION OF THE SPEED OF THE TRANSITION WAVEThe first impression gained from the computational model is thatthe vertical flexural motion is less relevant to the identification ofdangerous vibrations within the dynamic design process, and themain attention should be given to the low-frequency transversemodes. As follows from the physical evidence, the vertical flexuralmode that is driven by gravitational forces, is the one, which leadsto a failure wave. This also suggests that the standard, althoughadvanced, engineering techniques would not lead to the right con-clusion in the explanation of the failure wave in the San SabaBridge. However, the information provided by the finite elementcomputations, combined with the knowledge of flexural Blochwaves, leads to the correct answer, and prediction of the steadyregime also includes an accurate estimate of the speed of the fail-ure wave. The comparison between the FEM and the simplifiedwaveguide model for an elastic beam structure, supported by theelastic foundation (Figures 3D,E), shows that the parameters ofthe system are chosen so that the frequencies generated by FEM(Figure 2F) match well with the pass band interval identified forFloquet–Bloch waves in the periodic waveguide model (Brillouin,1953; Brun et al., 2013a).

The movie taken for the wooden section of the San Saba Bridge,together with the measurements, show that the length of the failedsection is around 209 m, and the speed of the steady-state prop-agation approaches V = 22.4 m/s (see Figure 3C). To comparewith the analytical model, we require the evaluation of the inter-nal unit length ⇠ = 0.382 m and unit time ⌧ = 0.157 ⇥ 10�1 s,which has been estimated form the FEM implementation asdetailed in the supplementary material (in Figures 3D,E theparameters are: a = 4.26 m, D = 0.8 ⇥ 106 Nm2, 1 = 37.5 MPa,

www.frontiersin.org September 2014 | Volume 1 | Article 12 | 5

Thank you for your aBen.on


Movchan, A.B., Brun, M., Slepyan, L.I., Giaccu, G.F. 2015 “Dynamic mul.-‐structure in modelling a transi.on flexural wave” Mathema'ka, 61(2), 444-‐456.

Brun, M., Giaccu, G.F., Movchan, A.B., Slepyan, L.I. 2014 “Transi.on wave in the collapse of the San Saba bridge” Fron'ers Materials, 1:12.

Brun, M., Movchan, A.B., Slepyan, L.I. 2013 “Transi.on wave in a supported heavy beam” Journal of the Mechanics and Physics of Solids, 61, 10, 2067–2085.

Projects

H2020-‐MSCA-‐IF-‐2016: CAT-‐FLAPP Catastrophic Failure in Fexural La\ce Problems