## 3 Stress Estimation Based on the Monitored Strain
### Establishment of the Forward Model
#### 3.1.1 Stress-Independent Strain  
When concrete structures are not constrained, the stress-independent strain is expressed as  
\[\varepsilon_{ST}=\varepsilon_{s}(t)+\varepsilon_{T}. \tag{7}\]  
However, in actual continuous beam bridges, strains caused by temperature and shrinkage effect are inhibited, leading to smaller actual structural deformation than free structural deformation. In the cross-section of the concrete beam, the development of the concrete temperature strain and shrinkage strain is constrained by reinforcements, while the structural system is constrained by the supports. The two constraints are referred to as the internal constraint and external constraint [23], respectively. The intensity of a constraint can be identified by using the restraint coefficient \(\xi\). The total restraint coefficient \(\xi\) is related to the internal restraint coefficient \(\xi_{i}\) and external restraint coefficient \(\xi_{e}\):  
\[\frac{1}{\xi}=\frac{1}{\xi_{i}}+\frac{1}{\xi_{e}}. \tag{8}\]  
where \(\xi\) is within the range of 0-1. The concrete structure is strongly constrained when \(\xi\) approaches 1. Otherwise, the concrete structure is slightly constrained.  
The external constraint is caused by the longitudinal pushing-resistance stiffness of the support and flexural stiffness of the pier top. Correspondingly, \(\xi_{e}\) is expressed as  
\[\xi_{e}=\frac{1}{1+\frac{kl_{0}}{[E_{c}(t)A_{c}+E_{s}A_{e}]}}, \tag{9}\]  
where \(E_{c}(t)\) is the elastic modulus of concrete at the age of \(t\); \(A_{c}\) is the cross-sectional area of a concrete beam; \(E_{s}\) is the elastic modulus of the reinforcement in the concrete beam; \(A_{s}\) is the cross-sectional area of the reinforcement; \(l_{0}\) is the whole length of the concrete beam; and \(k\) is the external constraint stiffness, which is composed of the longitudinal pushing-resistance stiffness of support \(k_{s}\) and the flexural stiffness of pier \(k_{p}\):  
\[\frac{1}{k}=\frac{1}{k_{s}}+\frac{1}{k_{p}} \tag{10}\]  
Assume that the arrangement of the reinforcements in the concrete beams is uniform and constant along the longitudinal direction. The internal constraint coefficient is expressed as  
\[\xi_{i}=\frac{1}{1+\frac{E_{s}A_{s}}{E_{c}(t)A_{c}}}. \tag{11}\]  
In actual concrete bridges with expansion joints, since the axial stiffness of a concrete beam, \(E_{c}(t)A_{c}+E_{s}A_{s}\), is much larger than the longitudinal pushing-resistance stiffness of support, \(k_{s}l_{0}\), or longitudinal flexural stiffness of pier, \(k_{p}l_{0}\), \(\frac{kl_{0}}{[E_{c}(t)A_{c}+E_{s}A_{s}]}\) is much smaller than 0.1 so \(\xi_{e}\) approaches 1.0. This shows that the external constraint is often not significant. For the internal constraint, the reinforcement ratio of the concrete beam is often 0.5-2.5% and \(E_{s}/E_{c}\) is 6-9, so \(\xi_{i}\) can be calculated as 0.82-0.97. This shows that the internal constraint cannot be neglected for actual concrete bridges. As a result, the stress-independent strain can be modified as shown in Equation (12) by introducing the constraint coefficient:  
\[\varepsilon_{sT}(t)=\xi[\varepsilon_{s}(t)+\varepsilon_{T}]. \tag{12}\]