1. Introduction
Natural hazards, vulnerability and risk in mountain regions have increasingly become a focus of political attention in recent years [
1]. On a European level, the Directive on the Assessment and Management of Flood Risks addressed to the Member States (Flood Directive) was issued in 2007 as one of the three components of the European Action Programme on Flood Risk Management. It was recognized that flood events have the potential to severely compromise economic development and to undermine the economic activities of the community [
2,
3]. Due to the increasing imbalance between values at risk and level of protection in flood prone areas, concentrated action is needed at all levels to avoid severe flood impacts on human life and property. The analysis of flood risk at different scales with a commensurate degree of detail is a valuable basis, not only for priority setting and further financial and political decisions regarding flood risk management, but also for the identification on a technical level of the most promising solutions to the risk mitigation problems at hand [
4,
5].
Acknowledging the fact that flood risk governance is a multi-faceted field of interdisciplinary activities [
6,
7], an essential condition for success in defusing the criticalities of risk generation mechanisms is the engineering design of possible alternatives indicating feasible and cost-effective ways of reducing risks.
The societal need to reduce flood risks is not the same as an unbounded willingness to invest public money for any solution pathway proposed. Envisaged solutions must be convincing, both from a technical and economic viewpoint and also be sustainable from an ecological perspective [
8,
9]. A reliable and tailored risk assessment entails the definition of scenarios composed of different levels, namely: (1) exposure scenarios, (2) vulnerability scenarios, (3) analyses of the values at risk, resulting in (4) risk scenarios [
10,
11].
These components have multiple functional dependencies among each other, resulting in compound intersections both in space and time.
To remove the root causes of risk generation mechanisms it is necessary to mirror the spatial and temporal evolution of flood risk, in particular sub-events (e.g., bridge clogging and levee failure) [
12] which may trigger significant raising of hazard levels, leading to flooding configurations which may induce the uncontrolled floating of mobile objects (e.g., vehicles).
Departing from these premises and from a dynamic conceptualization of vulnerability, we approach its assessment from an engineering science perspective (entailing analyses based on fluid and classical mechanics), according to the following methodological skeleton: (1) Hydrodynamic computation of time-dependent flood intensities resulting for each element at risk in a succession of loading configurations; (2) Modeling the mechanical response of objects impacting against one another through static, elasto-static and kinetic analyses; (3) Characterizing the mechanical response through proper structural damage variables and (4) economic valuation of the losses as a function of the quantified damage variables.
We will exemplify the calculations for potentially mobile objects exposed to flood load exceeding a given probability.
Unveiling the significant risk generation mechanisms, both methodologically and computationally, is of great value for the planning of functionally efficient mitigation measures that are able to provide a higher degree of risk reduction than conventional mitigation strategies. These computational schemes will be embedded in the general framework of cost-benefit analysis for the appraisal of flood risk mitigation projects [
11,
13]. By making explicit risk dynamics and cost generation mechanisms, the scope of application of cost-benefit analysis is expanded beyond its classical role as a decision-support tool and linked to the core of the planning process.
3. Worked out Example Problems
In this section we elaborate two prime examples to illustrate the procedure outlined in the previous section. We prefer starting with a mainly didactical example containing in a simplified fashion: the full set of conceptual elements of the presented procedure. Thereafter, in a second step, we will approach a more complex problem.
The first problem is a stylized version of a process chain which occurred frequently during the catastrophic flash flood events in the Italian regions Liguria and Tuscany in autumn 2011, namely the so called “vehicle risk problem”.
A large number of vehicles were parked in dedicated parking zones on inclined planes prone to flooding. With increasing flow depths and velocities, incipient motion of these objects began. The objects were displaced either by sliding due to the reduced friction or more rapidly by floating as soon as the lift forces exceeded gravity. Along the displacement pathways the objects collided with fixed obstacles and were consequently severely damaged as a result of these impacts.
Let us consider in our simplified setting a vehicle in an initially resting condition on an inclined plane which is flooded uniformly with constant flow depth and constant velocity. A fixed obstacle is placed at a known distance in the direction of motion. The task consists of formulating a simple vulnerability model, assuming that the extent of damage depends only on the deformation due to the impact energy. Following the previously outlined general procedure we obtain:
(1) Hydrodynamic analysis: Determination of the process intensities at the object’s location
![Water 04 00196 i030]()
:
Due to the uniform flow conditions, the following process intensities can be assumed:
![Water 04 00196 i031]()
,
(2) Mechanical analysis:
a. Assessment of the geometrical and physical properties
![Water 04 00196 i033]()
and
![Water 04 00196 i034]()
respectively:
We approximate, for simplicity only, the geometrical shape of the vehicle as a rectangular solid and the obstacle is assumed to be a vertical wall (compare
Figure 1a,b).
Figure 1.
System sketch (a and b) and free body diagram (c).
Figure 1.
System sketch (a and b) and free body diagram (c).
b. In this simplified setting the deformation depth
(after collision) is the only relevant physical damage variable of interest.
c. The free body diagram with the loading conditions and the reactive forces is shown in
Figure 1c.
![Water 04 00196 i037]()
is the drag force,
![Water 04 00196 i038]()
is gravity, whereas
![Water 04 00196 i039]()
is the force due to friction and
![Water 04 00196 i040]()
is the lift force.
d. The Cartesian x, y coordinate system is chosen in such a way that the x-axis coincides with the direction of potential translational motion of the rigid body: (compare
Figure 1). The rigid body kinetics are described according to the conceptual scheme outlined in
Scheme 1.
Scheme 1.
Conceptual scheme for the analysis of the rigid body kinetics.
Scheme 1.
Conceptual scheme for the analysis of the rigid body kinetics.
According to this scheme the kinetic problem is analyzed as a planar problem to check whether incipient uplift displacement or sliding in the x direction may take place.
If neither the incipient floating nor the sliding condition is given, equilibrium conditions are satisfied.
e. Static, kinetic and elasto-static analyses. Throughout we indicate the velocity of the object and of the fluid as
![Water 04 00196 i042]()
and
![Water 04 00196 i043]()
respectively. The drag force in flow direction is generally expressed as
![Water 04 00196 i044]()
, where
![Water 04 00196 i045]()
is the drag coefficient,
![Water 04 00196 i046]()
is the density of the fluid,
![Water 04 00196 i047]()
is the flow depth (or the submerged object depth,
![Water 04 00196 i048]()
, once floating occurs) and
![Water 04 00196 i049]()
is the width of the object. The force due to friction acting in the opposite direction is expressed as
![Water 04 00196 i050]()
, where
![Water 04 00196 i051]()
indicates the static or dynamic friction coefficient, depending on whether the object is in motion or not,
![Water 04 00196 i052]()
is the acceleration due to gravity,
![Water 04 00196 i053]()
are the geometrical parameters of the object and
![Water 04 00196 i054]()
is the inclination angle of the plane. The force due to gravity is given by
![Water 04 00196 i055]()
, and the lift force that the object is subjected to, is
![Water 04 00196 i056]()
(
![Water 04 00196 i057]()
in the case of floating).
i. Floating condition check:
![Water 04 00196 i058]()
or
![Water 04 00196 i059]()
yields:
![Water 04 00196 i060]()
(2)
If this condition (Equation 2) is not satisfied then the sliding condition check is performed.
ii. Sliding condition check:
![Water 04 00196 i061]()
(3)
and
![Water 04 00196 i062]()
or
![Water 04 00196 i063]()
yields:
![Water 04 00196 i064]()
(4)
iii. Kinetic analysis for the floating case:
Expressing the Newton’s second law,
![Water 04 00196 i065]()
, as
![Water 04 00196 i066]()
, yields the differential equation:
![Water 04 00196 i067]()
.
with
![Water 04 00196 i068]()
, we obtain the differential equation for the sliding case:
![Water 04 00196 i069]()
(5)
And integrating with respect to time the solution of Equation 5, which is the velocity of the floating object
![Water 04 00196 i070]()
, we obtain its displacement as follows:
![Water 04 00196 i071]()
(6)
iv. Kinetic analysis for the sliding case:
Analogously to the floating case, Newton’s second law,
![Water 04 00196 i072]()
, can be expressed as:
or more concisely, as a differential equation of the form:
![Water 04 00196 i074]()
(7)
Equation 6 can be employed again to calculate the displacements.
The differential Equations 5 and 7 can be solved numerically by applying the Runge-Kutta Method. The solution [
6] is straightforward.
In Annex 1, analytical solutions for velocities (compare Equations 5 and 7) and displacements (compare Equation 6) for both the floating and the sliding kinetics are provided for flow-depths or velocities held constant both in space and time.
v. Computation of the deformation depth
![Water 04 00196 i075]()
due to collision of object
![Water 04 00196 i012]()
with the fixed obstacle:
Making the conservative assumption that a central impact takes place, we employ an empirical quadratic equation that links the impact energy (per unit width)
![Water 04 00196 i076]()
with the deformation depth
![Water 04 00196 i075]()
:
![Water 04 00196 i077]()
(8)
where
![Water 04 00196 i078]()
and
![Water 04 00196 i079]()
are empirically defined stiffness coefficients, with units [N/m] and [N/m
2].
The kinetic energy per unit width of the object just before the obstacles collide:
![Water 04 00196 i080]()
(9)
Empirical crash tests [
17] indicate that reasonable restitution coefficient values for approximately inelastic collisions are close to zero, hence we assume that the kinetic energy is entirely available for deformation, thus:
yielding:
![Water 04 00196 i082]()
(10)
(3) Valuation and economic assessment of vulnerability
A suitable functional relationship between the damaged state of the considered object and the corresponding vulnerability is of the type:
![Water 04 00196 i083]()
(11)
where
![Water 04 00196 i084]()
corresponds to a deformation depth, which, once reached, completely destroys the market value of the vehicle.
The second problem addressed in this paper has demonstrated its relevance in the recent flood events in Italy. A large number of bridges were clogged by driftwood. In addition to inundations triggered by backwater effects, severe direct structural damage also resulted.
In extreme cases, bridge decks (superstructures) were displaced from their supports, resulting in significant direct losses and restrictions to normal economic activity.
The procedure outlined in the previous section is now applied to the bridge deck problem. Again it is assumed that the damage extent is well captured by the displacement of the bridge deck. It is assumed that dynamic vulnerability equals unity as soon as the displacement of the superstructure reaches a critical value for which the equilibrium of the moments of the forces acting on the structure can no longer be satisfied.
(1) Hydrodynamic analysis: Determination of the Process Intensities at the Location of the Object
![Water 04 00196 i085]()
.
We assume throughout that steady water profiles corresponding to the design discharge are given. These can easily be computed applying the energy equation. It is essential that flow depths and the associated average cross-sectional velocities are known for four reference cross-sections: proceeding in the downstream in upstream direction the first cross-section is located at a certain distance from the bridge where the flow is to be considered as fully expanded. The second control cross-section is placed immediately downstream of the bridge and represents the section where constriction flow switches to expansion flow. The third cross-section is placed immediately upstream of the bridge. The fourth cross-section is placed further upstream of the bridge where the backwater is fully developed. In
Figure 2(a–d) the structural characteristics and all relevant process intensity values are indicated for a control volume with defined cross-sections.
(2) Mechanical Analysis
a. Assessment of the geometrical and physical properties
![Water 04 00196 i033]()
and
![Water 04 00196 i034]()
respectively:
The reader is referred to
Figure 2 were all geometrical and physical properties are indicated. The control volume for the mechanical analysis is confined by sections 2 and 3 respectively. It is assumed that the flow depths
![Water 04 00196 i086]()
and
![Water 04 00196 i087]()
remain unaltered during the possible displacement of the bridge deck. Expressed another way, the hydrodynamic loadings at the boundaries of the moving control volume are held constant (compare
Figure 2c).
Figure 2.
Prospects of the bridge: hydrodynamic and geometrical parameters, necessary idealizations and definition of the control volume. (a) side view; (b) top down view; (c) font view; (d) control volumes.
Figure 2.
Prospects of the bridge: hydrodynamic and geometrical parameters, necessary idealizations and definition of the control volume. (a) side view; (b) top down view; (c) font view; (d) control volumes.
b. Identification of the physical damage variables: The relevant physical damage variable is the displacement of the center of mass
![Water 04 00196 i089]()
, which ranges from 0 (stable bridge, no displacement) to a maximum value
![Water 04 00196 i090]()
where the equilibrium condition
![Water 04 00196 i091]()
(
![Water 04 00196 i092]()
= moments around P) is no longer satisfied:
![Water 04 00196 i093]()
(12)
if
![Water 04 00196 i094]()
and
![Water 04 00196 i095]()
, which corresponds to the severest loading condition before the bridge starts to be submerged,
![Water 04 00196 i096]()
simplifies to:
![Water 04 00196 i097]()
(13)
Once
![Water 04 00196 i098]()
, we assume that the damage corresponds to the reinstatement value of the bridge deck and hence
c, d. Free body diagram (compare
Figure 3) and coordinate system.
Figure 3.
Free body diagram of the bridge deck impacted by the flood.
Figure 3.
Free body diagram of the bridge deck impacted by the flood.
e. Iterative analysis of the statics, elastostatics and kinetics.
i. Sliding condition:
The sliding condition,
![Water 04 00196 i101]()
, entails a comparison between the net hydrodynamic force on the bridge structure
![Water 04 00196 i102]()
and the reactive friction force
![Water 04 00196 i103]()
.
The net hydrodynamic force is given by the difference,
![Water 04 00196 i104]()
, whereas
![Water 04 00196 i105]()
and
![Water 04 00196 i106]()
are the forces acting on the control volume in section 3 and 2 respectively, thus:
![Water 04 00196 i107]()
and
![Water 04 00196 i108]()
, therefore:
The friction force can be expressed as,
![Water 04 00196 i110]()
, where
![Water 04 00196 i111]()
is the force due to gravity,
![Water 04 00196 i112]()
and
![Water 04 00196 i113]()
are the lift force components (compare
Figure 3):
![Water 04 00196 i114]()
and
![Water 04 00196 i115]()
.
Thus the sliding condition can be written as:
![Water 04 00196 i116]()
(14)
ii. Kinetics of sliding:
According to Newton’s second law the differential equation for the sliding case is:
![Water 04 00196 i117]()
with
![Water 04 00196 i118]()
and
![Water 04 00196 i119]()
.
Inserting and simplifying, one obtains:
which can compactly be written as:
![Water 04 00196 i121]()
(15)
with the coefficients
![Water 04 00196 i122]()
,
![Water 04 00196 i123]()
,
In Annex 1 analytical solutions for velocities (compare Equation 15) and displacements (compare Equation 6) are provided for the sliding kinetics for flow-depths or velocities held constant at the cross sections delimiting the control volume.
(3) Valuation and economic assessment of vulnerability
A suitable functional relationship between the damaged state of the considered object and the corresponding vulnerability is of the type:
![Water 04 00196 i125]()
(16)
5. Conclusions
The evolution in Europe in recent centuries, mainly consisting in a shift from a strong agricultural orientation to a service industry and leisure-centered society, has carried ever-increasing pressures in terms of usage of alpine areas and nearby regions for settlement, industry and recreation. These dynamics resulted into conflicts between human needs and their satisfaction on the one hand and naturally-determined conditions on the other [
22]. The economic bottlenecks emerging on a global level further restricted the margins of public investments to reduce flood hazard by realizing costly technical protection measures. Partly as a reaction and partly in anticipation, the European Flood Directive was issued, introducing the concept of risk as a basis for the management of natural hazards. Moreover, it was reaffirmed that an informed decision-making process regarding public investments to mitigate natural hazards could take advantage of sound cost-benefit analyses. Such an in-depth analysis entails the quantification of net benefits of risk mitigation strategies, which mainly derive from the associated reductions in terms of flood risk in relation to the prior level of risk and the expenditures needed for implementation. Besides the probabilities and intensities of the flood hazards, the determination of the physical vulnerability of impacted objects is a crucial element in risk analysis.
Therefore, in this paper, we have discussed in-depth the fundamental notion of physical vulnerability from a dynamic perspective, introducing a methodological and analytical apparatus to derive computational schemes for different categories of elements at risk.
We worked out two prime examples demonstrating the full applicability of the suggested procedural workflow. In the “vehicle risk” problem we illustrated how to subsume under the vulnerability concept the damage generation mechanism given by the interplay of static, kinetic and elasto-static effects. Concerning the “bridge deck displacement problem”, we could neglect elasto-static effects in analyzing the damage generation mechanism.
In a dedicated annex we provided analytic solutions for special cases of the two example problems.
In our opinion, linking the vulnerability assessment to engineering mechanics furthers the idea that the utility of cost-benefit analysis goes far beyond pure selection of the optimal management option out of an available bundle, if employed in earlier phases of the risk management process as an additional planning tool. Analyzing the time-varying vulnerability of elements at risk having a crucial impact on the expected consequences of flood impacts is increasingly becoming essential for a broad spectrum of activities within the risk governance process [
23,
24].
Intervention planning for example, which is recognized to be an effective tool to mitigate flood risk, is strongly based on the quality of the analysis of both the spatial and the temporal dynamics either of the flood hazard process or of the corresponding damaging impacts. Hazard and risk studies are valuable tools, especially if they contain an accurate time-varying representation of vulnerability for land use planning.
As mentioned earlier, we embedded the dynamic notion of vulnerability and risk into the formal framework of cost-benefit analysis. By making explicit risk dynamics and cost generation mechanisms, we have contributed to an expansion of the classical scope of application of cost-benefit analysis, promoting its use earlier in the planning process to enhance the search for both technically-feasible and economically-efficient solutions. Strengthening the link between physics and the economics of risk and expressing in monetary terms the annual risk reduction achievable by the envisaged investment projects may support a rational prioritization of public investment flows for the mitigation of flood risk (i.e., risk-based decision making).
In order to improve the risk-based selection of optimal mitigation strategies, an economic valuation of the elements at risk is necessary. In a dedicated section we have reviewed suitable existing valuation techniques.