###### Abstract  
Stress is one of the most important physical indexes reflecting the mechanical behavior of concrete structures. In general, stress in structures cannot be directly monitored and can only be estimated through an established model of stress and strain. The accuracy of the estimated stress depends on the rationality of the established model for stress and strain. As the strain measured by sensors contains creep, shrinkage, and elastic strain, it is difficult to establish an analytical model for strain and stress. In this paper, a statistical inverse method was utilized to estimate the stress in continuously pre-stressed concrete beams based on the monitored strain. Stress in the beams and the model uncertainty factors were treated as model parameters. A linear-simplified method was adopted to determine the prior distribution of the stresses. The posterior distribution of the stresses at different locations during bridge construction can be obtained by the proposed method. A continuously pre-stressed concrete beam bridge was taken as the case study to verify the effectiveness of the proposed method. Additionally, the constitution of the total strain in the different construction stages was calculated. It was concluded that the creep strain is the dominant part of the total strain.  
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Footnote †: journal: Applied Sciences  
15 June 202111 October 20211 October 202129 October 202111 October 2021  
influencing factors are uncertain. The estimated results based on the relationship rarely match the monitored results for three reasons: (1) the uncertainty of the predicative model; (2) the randomness of the influencing factors; and (3) the randomness of the monitoring technology. Reason 3 can be improved by technical progress. Inverse analysis methods can be adopted to control the derivation from reasons 1 and 2.  
A number of researchers have worked on quantitative estimations based on inverse analysis [10]. Inverse analysis methods have been used in many fields of engineering such as geological process simulation [11], structural damage detection [12; 13; 14], drainage system design [15], excavation supporting system simulations [16], structural long-term deformation estimations [17], and scour risk analysis of bridge foundation [18]. The advantage of the methodology is that it is able to evaluate the hard-to-obtain variables according to the results of easily measured variables. Inverse methods can combine the measurement results and prior information of all the influencing factors, and thus increase the accuracy of the evaluation, especially for cases where the measurement variables have many uncertain influencing factors.  
This study proposed a statistical inverse method to evaluate stress in concrete beams based on the monitored strain. The methodology includes five steps: (1) establishing a forward model to describe the relationship between structural stress and monitored strain; (2) distinguishing the strain components and establishing the model space by introducing the uncertainty factors of the forward model; (3) establishing the data space according to the monitored strain of a pre-stressed concrete (PC) beam; (4) determining the prior distribution of the stress by means of the linear simplified method; (5) calculating the posterior distribution of the parameters based on the statistical inverse theory and comparing the posterior and prior distributions; and (6) deriving the value and weight of each component of the stress by means of the Monte-Carlo sampling method.