A hybrid approach for prediction of long-term behavior of concrete structures

Abstract

Concrete displays long-term time-dependent behavior due to its rheological properties. The prediction of long-term behavior of concrete is difficult, even under laboratory conditions, due to the stochastic nature of its rheological phenomena. In concrete structures, long-term prediction is even more challenging due to the presence of uncontrolled conditions, such as variations in temperature, humidity, and loading. Current approaches for prediction of long-term time-dependent behavior at structural scale involve computationally intensive stochastic finite-element methods in which multiple creep and shrinkage models are implemented. However, these models are often calibrated using a database of experiments that are not the most informative for a specific structure. Structural health monitoring can improve prediction accuracy by providing structure-specific in-situ measurements of strain and temperature. Strain sensors, however, measure a multitude of effects simultaneously present in the structure, making it difficult to decouple effects of interest. In this work, a hybrid method employing probabilistic neural networks and engineering code models is proposed for the prediction of long-term behavior in concrete structures. A modular architecture is employed to decouple temperature-dependent environmental strain from long-term time-dependent strain. Generalized creep and shrinkage code models are fitted to the resulting time-dependent strain component data and used for prediction. The method is applied to a concrete pedestrian bridge instrumented with several embedded strain and temperature sensors. Excellent accuracy is achieved in the prediction of structural behavior multiple years beyond the training range. This, in turn, enables the detection of unusual structural behaviors with both gradual and sudden manifestation.

Appendix

At certain positions in Streicker bridge, a long-term time-dependent strain relaxation is observed (e.g., see Fig. 13a). This relaxation implies that at some time $t$, the rate of the long-term time-dependent effective strain ${\varepsilon }_{\rm EF}^{t}$ becomes positive. We show, under the simplified model outlined in the Methods section and using the CEB MC90-99 code, that the rate of ${\varepsilon }_{\rm EF}^{t}$ should remain negative, characterizing strain relaxation as anomalous structural behavior.

Taking the time derivative of Eq. (23) gives the effective strain rate

$$\frac{{\rm d}{\varepsilon }_{\rm EF}^{t}}{{\rm d}t}=-\frac{{\rm d}{\chi }^{t}}{{\rm d}t}{\varepsilon }_{R}^{t}+\left(1-{\chi }^{t}\right)\frac{{\rm d}{\varepsilon }_{R}^{t}}{{\rm d}t}+\frac{{\varepsilon }_{e,P}^{{\prime}}}{{\chi }^{{\prime}}}\frac{{\rm d}{\chi }^{t}}{{\rm d}t}.$$

(67)

The goal is to determine if the strain rate shown in Eq. (67) can be positive for some time $t$. That is nontrivial because of concrete aging and the recovery of creep strain due to prestress loss.

The time derivative of Eq. (17) is

$$\frac{{\rm d}{\chi }^{t}}{{\rm d}t}=-{\left(1+{\beta }^{t}\right)}^{-2}\frac{{\rm d}{\beta }^{t}}{{\rm d}t}.$$

(68)

Using the CEB MC90-99 model for the aging of concrete,

$${E}_{C}^{t}={E}_{C}^{28}{e}^{\frac{s}{2}\left(1-\sqrt{\frac{28}{t}} \right)},$$

(69)

in Eq. (14), gives

$${\beta }^{t}={\beta }^{28}{e}^{\frac{s}{2}\left(1-\sqrt{\frac{28}{t}} \right)},$$

(70)

where $s>0$ is a constant that depends on the type of concrete [15].

In view of Eq. (70), Eq. (68) can be expanded as

$$\frac{{\rm d}{\chi }^{t}}{{\rm d}t}=-\frac{s{\beta }^{28}}{4}{\left(1+{\beta }^{t}\right)}^{-2}{e}^{\frac{s}{2}\left(1-\sqrt{\frac{28}{t}} \right)}{t}^{-\frac{3}{2}}.$$

(71)

For Streicker bridge, $s=0.2,$ ${\beta }^{28}\approx 36$, and the expression above can be numerically evaluated for any $t>0.$ Notice that in general $\frac{{\rm d}{\chi }^{t}}{{\rm d}t}<0$ and, furthermore, because ${\varepsilon }_{R}^{t}<0$, the first term on the RHS of Eq. (67) is negative. Hence, by neglecting the negative contribution of the first term, the following inequality holds:

$$\frac{{\rm d}{\varepsilon }_{\rm EF}^{t}}{{\rm d}t}<\left(1-{\chi }^{t}\right)\frac{{\rm d}{\varepsilon }_{R}^{t}}{{\rm d}t}+\frac{{\varepsilon }_{e,P}^{{\prime}}}{{\chi }^{{\prime}}}\frac{{\rm d}{\chi }^{t}}{{\rm d}t}.$$

(72)

We would like to show that a conservative upper bound on $\frac{{\rm d}{\varepsilon }_{\rm EF}^{t}}{{\rm d}t}$ is still negative, such that the effective strain rate must necessarily be negative. Since $\left(1-{\chi }^{t}\right)>0$, towards more conservative upper bounds, further considerations should seek to make the rheological strain rate $\frac{{\rm d}{\varepsilon }_{R}^{t}}{{\rm d}t}$ a larger number.

The rheological strain is given by

$${\varepsilon }_{R}^{t}={\varepsilon }_{\rm Cr}^{t}+{\varepsilon }_{Sh}^{t},$$

(73)

where ${\varepsilon }_{\rm Cr}^{t}$ is the creep strain, as given in Eq. (12), and ${\varepsilon }_{Sh}^{t}$ is the shrinkage strain. Introducing Eq. (12) in Eq. (73) gives

$${\varepsilon }_{R}^{t}={\varepsilon }_{e,P}^{{\prime}}\varphi \left(t,{t}^{{\prime}}\right)+{\varepsilon }_{Sh}^{t}+{\int }_{{t}^{{\prime}}}^{t}\varphi \left(t,u\right){\rm d}{\varepsilon }_{e,P}^{u}.$$

(74)

The first two terms contribute with negative values, while the integral term contributes with positive values, corresponding to the creep recovery associated with prestress loss.

From Eq. (24)

$${\rm d}{\varepsilon }_{e,P}^{u}=\left(\frac{{\varepsilon }_{e,P}^{{\prime}}}{{\chi }^{{\prime}}}-{\varepsilon }_{R}^{u}\right){\rm d}{\chi }^{u}-{\chi }^{u}{\rm d}{\varepsilon }_{R}^{u}.$$

(75)

Introducing Eq. (75) into Eq. (74)

$${\varepsilon }_{R}^{t}={\varepsilon }_{e,P}^{{\prime}}\varphi \left(t,{t}^{{\prime}}\right)+{\varepsilon }_{Sh}^{t}-{\int }_{{t}^{{\prime}}}^{t}\varphi \left(t,u\right){\chi }^{u}{\rm d}{\varepsilon }_{R}^{u}+{\int }_{{t}^{{\prime}}}^{t}\varphi \left(t,u\right)\left(\frac{{\varepsilon }_{e,P}^{{\prime}}}{{\chi }^{{\prime}}}-{\varepsilon }_{R}^{u}\right){\rm d}{\chi }^{u}.$$

(76)

In general, the integral terms cannot be solved analytically, so it is typical to consider a discretization of the integral terms [1], yielding

$$\begin{aligned} \varepsilon _{R}^{t} & = \varepsilon _{{e,P}}^{'} \varphi \left( {t,t^{\prime}} \right) + \varepsilon _{{Sh}}^{t} ~ \\ & \quad + ~\Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \varphi \left( {t,u_{i} } \right)\left( {\frac{{\varepsilon _{{e,P}}^{'} }}{{\chi ^{\prime}}} - \varepsilon _{R}^{{u_{i} }} } \right)\frac{{d\chi ^{{u_{i} }} }}{{{\rm d}t}} \\ & \quad - \Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \varphi \left( {t,u_{i} } \right)\chi ^{{u_{i} }} \frac{{d\varepsilon _{R}^{{u_{i} }} }}{{{\rm d}t}}, \\ \end{aligned}$$

(77)

where $N$ is the number of time steps $\Delta t$, which can be taken arbitrarily small. Taking the time derivative of Eq. (77)

$$\begin{aligned} \frac{{d\varepsilon _{R}^{t} }}{{{\rm d}t}} & = \varepsilon _{{e,P}}^{'} \frac{{{\rm d}\varphi \left( {t,t^{\prime}} \right)}}{{{\rm d}t}} + \frac{{d\varepsilon _{{Sh}}^{t} }}{{{\rm d}t}} \\ & \quad + ~\Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{{\rm d}t}}\left( {\frac{{\varepsilon _{{e,P}}^{'} }}{{\chi ^{\prime}}} - \varepsilon _{R}^{{u_{i} }} } \right)\frac{{d\chi ^{{u_{i} }} }}{{{\rm d}t}} \\ &\quad - \Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{{\rm d}t}}\chi ^{{u_{i} }} \frac{{d\varepsilon _{R}^{{u_{i} }} }}{{{\rm d}t}}. \\ \end{aligned}$$

(78)

The first two terms are negative rates associated with the creep due to the initial prestressing and shrinkage, respectively, while remaining terms are associated with creep recovery due to prestress loss. However, notice that in the summation terms

$$\frac{{\rm d}\varphi \left(t,{u}_{i}\right)}{{\rm d}t}\left(\frac{{\varepsilon }_{e,P}^{{\prime}}}{{\chi }^{{\prime}}}-{\varepsilon }_{R}^{{u}_{i}}\right)\frac{{\rm d}{\chi }^{{u}_{i}}}{{\rm d}t},$$

(79)

the rheological strain ${\varepsilon }_{R}^{{u}_{i}}<0$ contributes towards a more negative strain rate $\frac{{\rm d}{\varepsilon }_{R}^{t}}{{\rm d}t}$, since $\frac{{\rm d}\varphi \left(t,{u}_{i}\right)}{{\rm d}t}>0$ and $\frac{{\rm d}{\chi }^{{u}_{i}}}{{\rm d}t}<0$. Thus, the following inequality, obtained by disregarding this negative contribution, holds:

$$\begin{gathered} \frac{{d\varepsilon _{R}^{t} }}{{{\rm d}t}} < \varepsilon _{{e,P}}^{'} \frac{{{\rm d}\varphi \left( {t,t^{\prime}} \right)}}{{{\rm d}t}} + \frac{{d\varepsilon _{{Sh}}^{t} }}{{{\rm d}t}} \hfill \\ + ~\Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{{\rm d}t}}\frac{{\varepsilon _{{e,P}}^{'} }}{{\chi ^{\prime}}}\frac{{d\chi ^{{u_{i} }} }}{{{\rm d}t}} \hfill \\ - \Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{{\rm d}t}}\chi ^{{u_{i} }} \frac{{d\varepsilon _{R}^{{u_{i} }} }}{{{\rm d}t}}. \hfill \\ \end{gathered}$$

(80)

Expanding the rheological strain rate $\frac{{d\varepsilon_{R}^{{u_{i} }} }}{{\rm d}t}$ in terms of its creep and shrinkage components gives

$$\begin{gathered} \frac{{d\varepsilon _{R}^{t} }}{{{\rm d}t}} < \varepsilon _{{e,P}}^{'} \frac{{{\rm d}\varphi \left( {t,t^{\prime}} \right)}}{{{\rm d}t}} + \frac{{d\varepsilon _{{Sh}}^{t} }}{{{\rm d}t}} \hfill \\ + ~\Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{{\rm d}t}}\frac{{\varepsilon _{{e,P}}^{'} }}{{\chi ^{\prime}}}\frac{{d\chi ^{{u_{i} }} }}{{{\rm d}t}} \hfill \\ - \Delta t\mathop \sum \limits_{{i = 1}}^{{N - 1}} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{{\rm d}t}}\chi ^{{u_{i} }} \left( {\frac{{d\varepsilon _{{\rm Cr}}^{{u_{i} }} }}{{{\rm d}t}} + \frac{{d\varepsilon _{{Sh}}^{{u_{i} }} }}{{{\rm d}t}}} \right). \hfill \\ \end{gathered}$$

(81)

Taking the time derivative of Eq. (12) gives the creep strain rate

$$\frac{{\rm d}{\varepsilon }_{\rm Cr}^{t}}{{\rm d}t}={\varepsilon }_{e,P}^{{\prime}}\frac{{\rm d}\varphi \left(t,{t}^{{\prime}}\right)}{{\rm d}t}+\frac{d\left({\int }_{{t}^{{\prime}}}^{t}\varphi \left(t,u\right){\rm d}{\varepsilon }_{e,P}^{u}\right)}{{\rm d}t},$$

(82)

where the first term contributes negatively to the creep strain rate and is associated with prestressing at time $t^{\prime}$, while the second term contributes positively to the creep strain rate, as it corresponds to the rate of creep recovery associated with prestress loss. Thus, the following inequality holds:

$$-\frac{{\rm d}{\varepsilon }_{\rm Cr}^{t}}{{\rm d}t}<-{\varepsilon }_{e,P}^{{\prime}}\frac{{\rm d}\varphi \left(t,{t}^{{\prime}}\right)}{{\rm d}t}.$$

(83)

Then, in view of inequalities (81) and (83), the following inequality obtained by introducing a more positive contribution on the RHS holds:

$$\begin{gathered} \frac{{d\varepsilon_{R}^{t} }}{{\rm d}t} < \varepsilon_{e,P}^{{\prime}} \frac{{{\rm d}\varphi \left( {t,t^{\prime}} \right)}}{{\rm d}t} + \frac{{d\varepsilon_{Sh}^{t} }}{{\rm d}t} \hfill \\ + {{ \Delta }}t\mathop \sum \limits_{i = 1}^{N - 1} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{\rm d}t}\frac{{\varepsilon_{e,P}^{{\prime}} }}{{\chi^{\prime}}}\frac{{d\chi^{{u_{i} }} }}{{\rm d}t} \hfill \\ - \Delta t\mathop \sum \limits_{i = 1}^{N - 1} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{\rm d}t}\chi^{{u_{i} }} \varepsilon_{e,P}^{{\prime}} \frac{{{\rm d}\varphi \left( {u_{i} ,t^{\prime}} \right)}}{{\rm d}t} \hfill \\ - {{ \Delta }}t\mathop \sum \limits_{i = 1}^{N - 1} \frac{{{\rm d}\varphi \left( {t,u_{i} } \right)}}{{\rm d}t}\chi^{{u_{i} }} \frac{{d\varepsilon_{Sh}^{{u_{i} }} }}{{\rm d}t}. \hfill \\ \end{gathered}$$

(84)

The CEB MC90-99 creep coefficient is of the form

$$\varphi \left(t,{t}^{{\prime}}\right)={\varphi }_{0}{\beta }_{\rm Cr}\left(t,{t}^{{\prime}}\right),$$

(85)

where ${\varphi }_{0}>0$ is the notional creep coefficient, and the shrinkage strain is of the form

$${\varepsilon }_{Sh}^{t}={\varepsilon }_{cs0}{\beta }_{Sh}\left(t,{t}^{{\prime}}\right),$$

(86)

where ${\varepsilon }_{cs0}<0$ is the notional shrinkage coefficient. The time-dependent coefficients ${\beta }_{\mathrm{\rm Cr}}\left(t,{t}^{{\prime}}\right)$ and ${\beta }_{\mathrm{Sh}}\left(t,{t}^{{\prime}}\right)$ are monotonically increasing functions.

Introducing Eq. (85) and (86) in inequality (84) gives

$$\frac{{\rm d}{\varepsilon }_{R}^{t}}{{\rm d}t}<{\varepsilon }_{e,P}^{{\prime}}{\varphi }_{0}\cdot {c}_{\rm Cr}\left(t\right)+{\varepsilon }_{cs0}\cdot {c}_{Sh}\left(t\right),$$

(87)

where

$${c}_{\rm Cr}\left(t\right)=\frac{{\rm d}{\beta }_{\rm Cr}\left(t,{t}^{{\prime}}\right)}{{\rm d}t}+\Delta t{\sum }_{i=1}^{N-1}\frac{{\rm d}{\beta }_{\rm Cr}\left(t,{u}_{i}\right)}{{\rm d}t}\frac{1}{{\chi }^{{\prime}}}\frac{{\rm d}{\chi }^{{u}_{i}}}{{\rm d}t}-{\varphi }_{0}\Delta t{\sum }_{i=1}^{N-1}\frac{{\rm d}{\beta }_{\rm Cr}\left(t,{u}_{i}\right)}{{\rm d}t}{\chi }^{{u}_{i}}\frac{{\rm d}{\beta }_{\rm Cr}\left({u}_{i},{t}^{{\prime}}\right)}{{\rm d}t},$$

(88)

and

$${c}_{Sh}\left(t\right)=\frac{{\rm d}{\beta }_{Sh}\left(t,{t}^{{\prime}}\right)}{{\rm d}t}$$

$$-{\varphi }_{0}\Delta t{\sum }_{i=1}^{N-1}\frac{{\rm d}{\beta }_{\rm Cr}\left(t,{u}_{i}\right)}{{\rm d}t}{\chi }^{{u}_{i}}\frac{{\rm d}{\beta }_{Sh}\left({u}_{i},{t}^{{\prime}}\right)}{{\rm d}t},$$

(89)

are, respectively, the creep and shrinkage strain rate upper bound coefficients. Since ${\varepsilon }_{e,P}^{{\prime}}{\varphi }_{0}<0$ and ${\varepsilon }_{cs0}<0,$ if ${c}_{\rm Cr}\left(t\right)>0$ and ${c}_{Sh}\left(t\right)>0$, the rheological strain rate must remain negative per inequality (87). Using the CEB MC90-99 creep and shrinkage models with Streicker Bridge properties gives a notional creep coefficient of ${\varphi }_{0}=2.08$, and the time-dependent coefficient for creep

$${\beta }_{\rm Cr}\left(t,u\right)={\left(\left(t-u\right){\left(1126+\left(t-u\right)\right)}^{-1}\right)}^{0.3},$$

(90)

and shrinkage

$${\beta }_{Sh}\left(t,u\right)={\left(\left(t-u\right){\left(10976+\left(t-u\right)\right)}^{-1}\right)}^{0.5}.$$

(91)

The derivatives of the time-dependent coefficients of creep and shrinkage are, respectively

$$\frac{{\rm d}{\beta }_{\rm Cr}\left(t,u\right)}{{\rm d}t}=0.3\frac{{\left(t-u\right)}^{-0.7}}{{\left(1126+\left(t-u\right)\right)}^{0.3}}-0.3\frac{{\left(t-u\right)}^{0.3}}{{\left(1126+\left(t-u\right)\right)}^{1.3}},$$

(92)

$$\frac{{\rm d}{\beta }_{Sh}\left(t,u\right)}{{\rm d}t}=0.5\frac{{\left(t-u\right)}^{-0.5}}{{\left(10976+\left(t-u\right)\right)}^{0.5}}-0.5\frac{{\left(t-u\right)}^{0.5}}{{\left(10976+\left(t-u\right)\right)}^{1.5}}.$$

(93)

Thus, Eqs. (88) and (89) can be numerically evaluated for any time $t$. Figures 

Fig. 16

Creep strain rate upper bound coefficient

16 and

Fig. 17

Shrinkage strain rate upper bound coefficient

17 show, respectively, ${c}_{\rm Cr}(t)$ and ${c}_{Sh}\left(t\right)$ for ${t}^{{\prime}}<t\le 3\times 365$ days, with $\Delta t=1 min.$ The figures show that both ${c}_{\rm Cr}\left(t\right)$ and ${c}_{Sh}(t)$ remain positive over this period, and, therefore, the rheological strain rate remains negative.

Then, considering inequality (72), and that the shrinkage component ${\varepsilon }_{cs0}{c}_{Sh}\left(t\right)<0,$ the following inequality, obtained by disregarding the negative shrinkage contribution, holds,

$$\frac{{\rm d}{\varepsilon }_{\rm EF}^{t}}{{\rm d}t}<{\varepsilon }_{e,P}^{{\prime}}{c}_{\rm EF}\left(t\right),$$

(94)

where

$${c}_{\rm EF}\left(t\right)=\left(1-{\chi }^{t}\right){\varphi }_{0}{c}_{\rm Cr}\left(t\right)+\frac{1}{{\chi }^{{\prime}}}\frac{{\rm d}{\chi }^{t}}{{\rm d}t},$$

(95)

is the effective strain rate upper bound coefficient. Since ${\varepsilon }_{e,P}^{{\prime}}<0$, if ${c}_{\rm EF}\left(t\right)>0$, the time-dependent effective strain rate must remain negative in view of inequality (94). Figure 

Fig. 18

Effective strain rate upper bound coefficient

18 shows that the resulting values of ${c}_{\rm EF}(t)$ computed numerically for ${t}^{{\prime}}<t\le 3\times 365$ days, with $\Delta t=1 min,$ remain positive, implying that

$$\frac{{\rm d}{\varepsilon }_{\rm EF}^{t}}{{\rm d}t}<0,$$

(96)

over the period of interest here and, thus, characterizing the positive rates, observed between 730 and 1000 days (e.g., see Fig. 13a), at some locations in Streicker Bridge as anomalous structural behavior.