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The impacts of flood events that occurred in autumn 2011 in the Italian regions of Liguria and Tuscany revived the engagement of the public decisionmaker to enhance the synergy of flood control and land use planning. In this context, the design of efficient flood risk mitigation strategies and their subsequent implementation critically relies on a careful vulnerability analysis of the fixed and mobile elements exposed to flood hazard. In this paper we develop computation schemes enabling dynamic vulnerability and risk analyses for a broad typological variety of elements at risk. To show their applicability, a series of prime examples are discussed in detail, e.g. a bridge deck impacted by the flood and a car, first displaced and subsequently exposed to collision with fixed objects. We hold the view that it is essential that the derivation of the computational schemes to assess the vulnerability of endangered objects should be based on classical and fluid mechanics. In such a way, we aim to complement from a methodological perspective the existing, mainly empirical, vulnerability and risk assessment approaches and to support the design of effective flood risk mitigation strategies by defusing the main criticalities within the systems prone to flood risk.
Natural hazards, vulnerability and risk in mountain regions have increasingly become a focus of political attention in recent years [
Acknowledging the fact that flood risk governance is a multifaceted field of interdisciplinary activities [
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 [
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 subevents (e.g., bridge clogging and levee failure) [
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 timedependent 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, elastostatic 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 costbenefit analysis for the appraisal of flood risk mitigation projects [
It has repeatedly been claimed that there exists a lack of studies related to the spatiotemporal development of risk and the underlying vulnerability of the elements at risk [
Space is—along with time—a key factor when information on vulnerability and risk is assessed, since risk materializes only as a consequence of the physical overlap between hazardous events and elements at risk [
As a point of departure, we assume that the results of flood hazard analysis are given in the form of
The assessment of the timevarying vulnerability of the generic object
(1) Analysis of hydrodynamics, which entails the determination of the process intensities at the object location
(2) Mechanical analysis of the considered object, which entails:
Assessment of the geometrical and physical properties of the object in question, the transporting fluid and the environment,
Identification of the physical damage variables
Drawing the free body diagram of the object indicating the loading conditions (e.g., acting forces),
Choice of a proper coordinate system and check whether a 3D, 2D or a 1D analysis is suitable. (e.g., plane rigid body kinetics versus 3D motion).
Iterative analysis of the statics, elastostatics and kinetics of the considered object
(3) Valuation and economic assessment of vulnerability
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
Due to the uniform flow conditions, the following process intensities can be assumed:
(2) Mechanical analysis:
a. Assessment of the geometrical and physical properties
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
System sketch (
b. In this simplified setting the deformation depth
c. The free body diagram with the loading conditions and the reactive forces is shown in
d. The Cartesian x, y coordinate system is chosen in such a way that the xaxis coincides with the direction of potential translational motion of the rigid body: (compare
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 elastostatic analyses. Throughout we indicate the velocity of the object and of the fluid as
i. Floating condition check:
yields:
If this condition (Equation 2) is not satisfied then the sliding condition check is performed.
ii. Sliding condition check:
and
yields:
iii. Kinetic analysis for the floating case:
Expressing the Newton’s second law,
with
And integrating with respect to time the solution of Equation 5, which is the velocity of the floating object
iv. Kinetic analysis for the sliding case:
Analogously to the floating case, Newton’s second law,
or more concisely, as a differential equation of the form:
Equation 6 can be employed again to calculate the displacements.
The differential Equations 5 and 7 can be solved numerically by applying the RungeKutta Method. The solution [
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 flowdepths or velocities held constant both in space and time.
v. Computation of the deformation depth
Making the conservative assumption that a central impact takes place, we employ an empirical quadratic equation that links the impact energy (per unit width)
where
The kinetic energy per unit width of the object just before the obstacles collide:
Empirical crash tests [
yielding:
(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:
where
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
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 crosssectional velocities are known for four reference crosssections: proceeding in the downstream in upstream direction the first crosssection is located at a certain distance from the bridge where the flow is to be considered as fully expanded. The second control crosssection is placed immediately downstream of the bridge and represents the section where constriction flow switches to expansion flow. The third crosssection is placed immediately upstream of the bridge. The fourth crosssection is placed further upstream of the bridge where the backwater is fully developed. In
(2) Mechanical Analysis
a. Assessment of the geometrical and physical properties
The reader is referred to
Prospects of the bridge: hydrodynamic and geometrical parameters, necessary idealizations and definition of the control volume. (
b. Identification of the physical damage variables: The relevant physical damage variable is the displacement of the center of mass
if
Once
c, d. Free body diagram (compare
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,
The net hydrodynamic force is given by the difference,
The friction force can be expressed as,
Thus the sliding condition can be written as:
ii. Kinetics of sliding:
According to Newton’s second law the differential equation for the sliding case is:
Inserting and simplifying, one obtains:
which can compactly be written as:
with the coefficients
In Annex 1 analytical solutions for velocities (compare Equation 15) and displacements (compare Equation 6) are provided for the sliding kinetics for flowdepths 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:
Major interrelated requirements have to be met by the risk management process [
Based on the fully probabilized flood scenarios,
Assuming that different location matrices
where
The overall risk, quantified on an annual basis, resulting from the expected monetary losses for all elements at risk, considering the entire set of exposure scenarios and flood impacts, can be written as:
where
For practical purposes it can be convenient to track the dynamic risk in time as follows:
In the adopted conceptualization of flood hazard risk (compare Equations 17 to 19), the expected losses are expressed monetarily, which entails an economic valuation of the elements at risk. We restrict our analysis to tangible loss, namely damage to capital stocks or resource flows which can be specified in monetary terms, neglecting damages to assets which are not traded and are therefore difficult to transfer into monetary values [
With reference to object categories valued through economic approaches using market values (e.g., reinstatement value for structures) we report a general scheme to structurally dissect complex objects and make them accessible to economic valuation in risk assessment.
Hence, in dissecting a complex object (e.g., a production plant) we distinguish between:
a. vertically extending fixed structures (e.g., walls of the buildings) impacted directly by the flood process;
b. particular superstructures impacted directly (e.g., bridge decks) or indirectly (e.g., roofs) by the flood process.
c. installations and/or mobile objects (e.g., machines and cars) impacted directly by the flood process.
For completeness two supplementary categories have to be considered that are also affected by flooding: sediment and wood deposition;
d. surfaces (areas) for different land use purposes (e.g., agricultural land, but also parking areas and roads); and
e. biotic systems (e.g., wood, but also orchards).
The direct economic reference for a valuation of object parts belonging to the categories (a,b) is the determination of the reinstatement value,
where NV is the reinstatement value of the considered object:
For the category (c), the estimation of the market value,
where MV is the most probable market value of the equipment component under consideration.
For category (d) it is relevant to determinate the costs of clearingup operations and the necessary reinstatements to reestablish the original functionally.
In case of object category (e), the economic valuation is carried out by determining the capital value of the biotic system under consideration through suitable capitalization formulas.
The next methodological step consists of embedding the dynamic risk notion into the mathematical apparatus of costbenefit analysis.
We assume throughout that the public sector seeks to create system modifications maximizing the difference between annual savings in terms of risk reductions and the annual costs of the investment project through flood risk mitigation investments, entailing construction costs at the beginning of the investment lifecycle and maintenance expenditures throughout the lifecycle.
The flood risk mitigation performance,
Graphical representation of the risk mitigation performance
As shown in
The annual costs
Mathematically, a cost plan is a vector containing the foreseen expenditures as elements for each year, namely
The net present value,
Normally, as shown in
Graphical representation of the risk mitigation performance
In the case illustrated by
Graphical representation of the risk mitigation performance
From a mathematical perspective we need (i) a cost plan (vector containing as elements the foreseen expenditures for each year
The net present value,
The evolution in Europe in recent centuries, mainly consisting in a shift from a strong agricultural orientation to a service industry and leisurecentered society, has carried everincreasing 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 naturallydetermined conditions on the other [
Therefore, in this paper, we have discussed indepth 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 elastostatic effects. Concerning the “bridge deck displacement problem”, we could neglect elastostatic 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 costbenefit 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 timevarying 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 [
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 timevarying representation of vulnerability for land use planning.
As mentioned earlier, we embedded the dynamic notion of vulnerability and risk into the formal framework of costbenefit analysis. By making explicit risk dynamics and cost generation mechanisms, we have contributed to an expansion of the classical scope of application of costbenefit analysis, promoting its use earlier in the planning process to enhance the search for both technicallyfeasible and economicallyefficient 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 (
In order to improve the riskbased 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.
Let us recall Equations 5 and 7 for the case of a floating or sliding free rigid body
and the Equation of sliding for the bridge case (Equation 15):
In each of the above cases, the evolution of the system is described by the following Cauchy problem, for a suitable choice of the constant parameters
The choice for the variable and the parameters in the various cases is reported in the following table:
Case 





Floating rigid body 




Sliding rigid body 




Bridge 




The analytical solution can be found by separation of variables and gives rise to different forms of the solution, depending on the sign of the constant
Writing explicitly the integral and inverting we have
The equation for the position can then be found by integration. For the free rigid body we have
while for the bridge we just have
Let us now specialize the above results for the three different cases.
We have
and
In this case
and
If
and
for
And
In this case
and since
and