**2. Failure mechanisms during hurricanes**

According to the Hurricane Hazard Mitigation Handbook for Public Facilities [34], winds, rains, and flood (or storm surge in coastal areas) are the three natural hazards or forces of hurricanes responsible for damages, including Na-techs. In line with [34], in the present study, hurricanes are considered as multi-hazard natural disasters consisting of strong winds, floods, and heavy rainfalls.

### **2.1 Flood**

Compared to seismic or wind related Na-techs, the one caused by floods have received relatively less attention; this has mainly been due to the scarcity of experimental or high-resolution field observations. Only for a limited number of floods were the flood inundation depths registered and even for fewer floods were the flood speeds recorded.

According to [14], 272 flood related Na-techs were reported in Europe and the U.S. from 1960 to 2007, with the aboveground storage tanks as the most frequently damaged equipment (74% of cases), including atmospheric storage tanks, floating roof tanks, and pressurized tanks. Displacement of storage tanks (due to rigid sliding or floatation) and subsequent disconnection or rupture of attached pipelines, shell rupture due to lateral forces or debris impact, and the collapse of equipment were reportedly the main failure modes during floods. Godoy [13] reported similar failure modes while investigating the process plants affected by the Hurricanes Katrina in Louisiana and Texas, U.S. These failure modes are depicted in **Figure 2**.

**Figure 2.** *Flood related failure modes for storage tanks.*

*Vulnerability Assessment of Process Vessels in the Event of Hurricanes DOI: http://dx.doi.org/10.5772/intechopen.109430*

Due to a lack of sufficiently accurate and reliable historical data, a majority of previous studies has employed analytical or numerical techniques to model the abovementioned failure modes and estimate the respective failure probabilities [12, 20, 22–25].

In these studies, limit state equations (LSEs) that were developed based on "physics of failure" models were used to calculate the failure probabilities. Such physics-offailure models consider forces (loads) exerted by floods on target vessels (e.g., buoyancy force, which tends to float the tank) and the resistance of the vessel (e.g., total weight of the tank, which tend to resist the floatation). The developed LSEs can then be coupled with Monte Carlo simulation to generate artificial databases that in turn can be used to develop fragility curves [23] or be combined with other QRA techniques, such as BN [12, 24] for probabilistic reasoning.

#### **2.2 Wind**

Shell buckling is the main failure mode caused by wind [35] especially if the storage tank is empty or less than half-filled. According to Godoy [13], several storage tanks were reported to suffer shell buckling during Hurricane Katrina, mostly due to strong winds rather than flood. High winds can enforce critical pressures on tank shells. The magnitude of this pressure depends on several parameters, the most important of which being the wind speed and the shape and size of the storage tank.

Considering cylindrical storage tanks, however, the height of the tank does not play a key role [36, 37] as does the tank's circumference. As such, the wind pressure is usually considered as constant along the tank's height [38]. Similar to the flood related failure modes, it has been a common practice to develop LSEs for wind related failure mode – i.e., shell buckling and shell damage due to windborne debris – based on the load-resistance relationships. Considering the shell buckling, for instance, external pressure exerted by wind (i.e., load) and the internal pressure of the tank (i.e., resistance) were taken into account to develop the related LSE [12].

#### **2.3 Rainfall**

Heavy rainfalls before, during and after passage of hurricanes were rarely accounted for as a separate natural hazard directly capable of causing damage to process vessels. In that sense, contribution of heavy rainfalls to Na-techs has been indirectly taken into account via the runover of water bodies and rivers due to heavy rainfall and consequent flooding of process plants.

Hurricane Harvey, however, demonstrated the potential of heavy rainfall as a natural hazard, which can cause significant damage to process vessels standalone or in combination with other forces of the hurricane. Large amount of rainfall that fell during Hurricane Harvey caused damage to 400 "floating roof storage tanks", including sinking of 14 roofs. A floating-roof storage tank is a tank the roof of which is not fixed and rather floats on the surface of chemicals stored inside the tank.

### **3. Bayesian network**

Bayesian network (BN) [26] is a directed acyclic graph consisting of random variable nodes and directed edges connecting the nodes, with the edges directed from

**Figure 3.**

*BN consisting of 5 random variables (nodes). X1 and X2 are the root nodes and parents of X3. X3 is an intermediate node, the child of X1 and X2 and the parent of X4 and X5. X4 and X5 are the leaf nodes and the children of X3.*

parents to children. A BN can mathematically be defined as BN = (G, θ), where G is the network structure – nodes and edges – and θ is the network parameters – conditional probabilities of child nodes given their parents. **Figure 3** shows an illustrative BN consisting of 5 nodes. X1 and X2 are the root nodes (equivalent to the basic events of a fault tree), X3 is an intermediate event (equivalent to an intermediate event of a fault tree), and X4 and X5 are the leaf nodes (equivalent to the top event of a fault tree). The root nodes are assigned marginal probability distributions while the other nodes are assigned conditional probability distributions. Such conditional probabilities reflect the type (usually causal) and strength of the impact parent nodes have on their child nodes.

Considering the local dependencies and chain rule, the joint probability distribution of the random variables in a BN can be presented as the product of marginal and conditional probabilities considering only the immediate parents of a child node:

$$P(\mathbf{X}\_1, \mathbf{X}\_2, \dots, \mathbf{X}\_n) = \prod\_{i=1}^n P(\mathbf{X}\_i | pa(\mathbf{X}\_i)) \tag{1}$$

For example, for the BN in **Figure 3**, P (X1, X2, X3, X4, X5) = P (X1) P(X2) P(X3| X1, X2) P(X4|X3) P(X5|X3). BN can be used for either forward or backward reasoning. In forward reasoning, for example, the knowledge about X1 can be used to deduce about X5, whereas in backward reasoning, the knowledge about X5 can be used to infer about X1. The main advantages of BN over conventional techniques such as fault tree analysis are its capability of considering conditional dependencies, handling multistate variables, and applying Bayes' rule for belief updating (backward reasoning) [28]. Using the Bayes' theorem, BN can update the probabilities assigned a priori to the random variables in the presence of new information about the states of its nodes:

$$P(X|E) = \frac{P(X,E)}{P(E)}\tag{2}$$

The parameters of the BN—the marginal and conditional probabilities needed to quantify the model—can be assigned by subject matter experts or be estimated from a dataset by applying machine learning techniques such as the maximum likelihood estimation. **Table 2** summarizes some of the advantages of BN against fault tree.


#### **Table 2.**

*Modeling advantages of BN as opposed to fault tree.*

### **4. Vulnerability assessment of storage tanks**

As previously discussed, one way to analyze the failure mechanism and to generate failure data required for probabilistic vulnerability assessment of process vessels is application of "physics of failure" to develop LSEs. The benefit of LSEs is twofold: From one side, they help better understand random variables that contribute to the failure mechanisms and should thus be considered in the BN, and from the other side, they can be coupled with, for instance, Monte Carlo simulation to generate datasets required for estimating the parameters—especially, the conditional probabilities—of the BN [12, 23, 24].

To demonstrate the abovementioned methodology, we consider floatation as one of the frequently reported failure modes (failure mechanism) for above-ground atmospheric storage tanks. Since such storage tanks are usually unanchored (not bolted or cemented to their foundations), the buoyancy force of flood in Eq. (3) can make the storage tank float should it overcome the total weight of the tank—i.e., the weight of the tank in Eq. (4) plus the weight of chemical in the tank in Eq. (5)—as the only resisting force. Considering the bouncy as the only moving force and the total weight of the tank as the only resistance, the LSE of the tank due to the floatation can be modeled as in Eq. (6):

$$F\_B = \rho\_w \,\mathrm{g} \,\mathrm{.}\pi \,\frac{D^2}{4} \,\mathrm{S} \tag{3}$$

$$\mathcal{W}\_T = \rho\_T \lg \left( \pi DH + 2\pi \frac{D^2}{4} \right) . \tag{4}$$

$$\mathcal{W}\_L = \rho\_L \lg \pi \frac{D^2}{4} h \tag{5}$$

$$\text{LSE} = F\_B - \mathcal{W}\_L - \mathcal{W}\_T \tag{6}$$

where *FB* is the buoyancy force; *WT* and *WL* are the weight of the tank and the contained chemical, respectively; *ρw*, *ρT*, and *ρ<sup>L</sup>* are, respectively, the density of flood water, of the tank structure's material, and of the chemical; *g* is the gravitational acceleration; *D* and *H* are, respectively, the diameter and height of the tank; *S* is the depth of flood; *h* is the depth of chemical in the tank; t is the shell thickness of the tank. Given the LSE in Eq. (6), if LSE > 0, or simply *FB* >*WL* þ *WT*, the tank floats. The parameters in Eqs. (3)–(6) are depicted in **Figure 4**. Considering the depth of flood (S) and the depth of chemical (h) as the only random variables (the other parameters can reasonably be considered as constants), *FB* >*WL* þ *WT* can be further simplified as:

#### **Figure 4.**

*Parameters used to develop the LSE required for floatation failure mechanism of an unanchored atmospheric storage tank [23].*

$$S > ah + b \tag{7}$$

where *a* and *b* are constants:

$$a = \frac{\rho\_L}{\rho\_w} \tag{8}$$

$$b = \frac{W\_T}{\rho\_w \lg \pi. \frac{D^2}{4}}\tag{9}$$

According to Eq. (7), the probability of floatation *Pfloat* can be calculated as:

$$P\_{float} = P(\mathbf{S} > ah + b) = \mathbf{1} - P(\mathbf{S} < ah + b) \tag{10}$$

Given the probability distribution functions of *S* and *h* as *f <sup>S</sup>*ð Þ*S* and *f <sup>h</sup>*ð Þ *h* , and considering that the level of chemical in the tank cannot exceed H (in practice, 15-20% of the tank's height is usually left for safety purposes), Eq. (10) can be expanded as:

$$P\_{float} = \mathbf{1} - P(\mathbf{S} < ah + b) = \mathbf{1} - \int\_{0}^{H} \left( \int\_{0}^{ah+b} f\_{\mathbf{S}}(\mathbf{S}) \, d\mathbf{S} \right) f\_{h}(h) \, dh \tag{11}$$

#### **4.1 Analytical approach**

For some types of probability distributions, P(S < ah +b) can easily be calculated with no need for calculating the double integral in Eq. (11). One type of such probability distributions is Normal distribution. Given two normally distributed random variables as X � Normal (*μx*, *σx*) and Y � Normal (*μy*, *σy*), any linear function of X and Y is a normally distributed random variable as Z � Normal (*μz*, *σz*). For instance, if Z = mX + nY, the mean value *μ<sup>z</sup>* and standard deviation *σ<sup>z</sup>* of Z can be calculated as:

*Vulnerability Assessment of Process Vessels in the Event of Hurricanes DOI: http://dx.doi.org/10.5772/intechopen.109430*

$$
\mu\_x = m\,\mu\_x + n\,\mu\_y \tag{12}
$$

$$
\sigma\_{\mathfrak{x}} = \sqrt{m^2.\sigma\_{\mathfrak{x}}^2 + n^2.\sigma\_{\mathfrak{y}}^2} \tag{13}
$$

Therefore, if S � Normal (*μs*, *σs*) and h � Normal (*μh*, *σh*), then Q = S – ah is a linear function of S and h and thus normally distributed as:

$$
\mu\_Q = \mu\_s - \mathfrak{a}\,\,\mu\_h \tag{14}
$$

$$
\sigma\_Q = \sqrt{\sigma\_s^2 + a^2 \sigma\_h^2} \tag{15}
$$

As a result:

$$\mathbf{P(S < \mathbf{a}h + \mathbf{b}) = P(S - \mathbf{a}h < \mathbf{b}) = P(Q < \mathbf{b}) = \Phi\_Q(b)}\tag{16}$$

where Φ*<sup>Q</sup>* ð Þ*:* is the cumulative distribution function for normal distribution. Consequently:

$$P\_{float} = P(\mathbb{S} > ah + b) = \mathbb{1} - P(\mathbb{S} < ah + b) = \mathbb{1} - \Phi\_{\mathbb{Q}}(b) \tag{17}$$

To demonstrate the application of Eq. (17), consider an unanchored crude oil storage tank with diameter of D = 91m, height of H = 6m, and shell thickness of t = 15mm. The characteristics of the tank, crude, and flood are presented in **Table 3**.

Given the values in **Table 3**, the values of a and b are calculated via Eqs. (8) and (9) as a = 0.88 and b = 0.26. Subsequently, using Eqs. (14) and (15):

$$
\mu\_Q = \mu\_s - a \,\,\mu\_h = \mathbf{1} - \mathbf{0.88} \times \mathbf{1} = \mathbf{0.62}
$$

$$
\sigma\_Q = \sqrt{\mathbf{0.1}^2 + \mathbf{0.88}^2 \times \mathbf{1}^2} = \mathbf{0.89}
$$

Knowing that Q is a normal random variable Q � Normal (0.62, 0.89), using Eq. (17), *Pfloat* ¼ 1 � Φ*<sup>Q</sup>* ð Þ¼ *b* 1 � Φ*<sup>Q</sup>* ð Þ¼ 0*:*26 0*:*66.

#### **4.2 Numerical approach**

Given the probability distribution of S and h, the probability of flotation in Eq. (10) can be approximated numerically using, among others, Monte Carlo


#### **Table 3.**

*Parameters used for the illustrative storage tank and hitting flood.*


#### **Table 4.**

*Sample of dataset generated by Monte Carlo simulation given the probability distributions of S and h.*

simulation. **Table 4** show part of the dataset generated for S and h given their probability distributions. Obviously, for each pair of S-h dataset, if Q > 0.26 the tank floats. The last column of **Table 4** assign 1 to each pair if Q > 0.26 (the tank floats) and 0 if Q < 0.26 (the tank does not float). Having a sufficiently large dataset, the mean value of the last column denotes the probability of floatation. Given 1000 samples, *Pfloat* = 0.676, which is close enough to the probability calculated using the analytical approach.
