## 4 Seismic Damage Assessment based on Cloud Model  
### Theory of Cloud Model  
The assessment of seismic damage to pipelines has randomness and fuzziness. Since the concept of the damage degree of pipelines is fuzzy and the stress values obtained by dynamic response are uncertain, we introduce the concept of cloud to express the randomness and fuzziness through graphical features. The concept of cloud realizes the mutual transformation of qualitative and quantitative, and solves the evaluation result of boundary ambiguity and overlap. Suppose that \(U\) is a quantitative domain of exact numerical representation, and \(C\) is a qualitative concept on \(U\). For a quantitative value \(x\in U\), the membership degree \(u_{c}(x)\) is a description of the qualitative concept represented by \(x\) for \(C\). Thus, the distribution of membership degrees on the domain \(U\) is called the cloud. The value range of \(u_{c}(x)\) is \([0,1]\) and the cloud is a mapping from the domain \(U\) to the interval \([0,1]\), which can be expressed as [35]:  
\[u:U\rightarrow[0,1],\forall x\in U,x\to u(x) \tag{26}\]  
Each \(x\) is defined as a cloud drop, which is a quantitative description of a qualitative concept, and all the cloud drops are gathered into a cloud. The deterministic levels of different clouds reflect different ambiguities. Additionally, the cloud drop itself represents a random value. Therefore, the accuracy of the qualitative description depends on the number of clouds. For example, each cloud drop, as shown in Figure 5, represents the degree of certainty of the concept, indicating the quantitative position of the qualitative concept in the numerical space. Thus, the assessment value denotes that the value corresponding to the different evaluation of the object, and the certainty grade is the degree of membership. The cloud model uses the \"cloud\" to define the fuzziness and randomness of the data, constituting a transition between qualitative and quantitative. The three numerical characteristics of the cloud model are expectation \(E_{x}\), entropy \(E_{n}\), and super-entropy \(H_{e}\), which are quantitative concepts to describe the concept of qualitative. The evaluation results of the cloud model expression give the position of evaluated central value through the expectation, and the entropy and super-entropy express the randomness and dispersion of the evaluation results. Therefore, the evaluation results are more objective and accurate. The description of numerical characteristics for expectation, entropy, and super-entropy are shown in Figures 6-8, respectively. Figure 5: Cloud model chart. This paper establishes a cloud model-based approach to solve the uncertainty and fuzziness of structural damage assessment, and the damage evaluations of the pipeline with non-random earthquake are explained by the numerical characteristics of the model. Expectation is a qualitative description of a representative sample. In this paper, expectation represents the value of pipeline damage, which reflects the extent of damage. The entropy is used to measure the degree of uncertainty, and represents the measurability of qualitative concepts. Moreover, the entropy represents the range of values that can be accepted by the qualitative concept, which is a measure of the qualitative concept, and it is the fuzzy measure and the degree of dispersion of qualitative concept, reflecting the degree of uncertainty. Figure 8: Comparison of cloud model with different super-entropies. Figure 6: Comparison of cloud model with different expectations. Figure 7: Comparison of cloud model with different entropies. damage and the degree of dispersion of the stress values at each point of each pipeline relative to the evaluation stress value in this paper. The super-entropy is a measure of uncertainty state, reflecting the degree of aggregation of stress value. Therefore, the expectation gives the location of the evaluation center value, and the randomness and discreteness of the evaluation results are represented by entropy and super-entropy.