Fatigue equation of structural materials and members under hot-wet environment and cyclic bending loads

Abstract

The main structural members of bridges are served under vehicle loads and environment coupling action. Therefore, there is a need for an effective method which can be used to analysis fatigue behavior of bridge structural materials and members under coupling action of hot-wet environment and cyclic loads. In this paper, a hot-wet environment fatigue equation was proposed for describing the fatigue behavior of steel fiber polymer structural concrete (SFPSC) and reinforced concrete (RC) beams strengthened with carbon fiber reinforced polymer (CFRP), and the fatigue experiments under the coupling and uncoupling action of different hot-wet conditions and cyclic bending loads were carried out. The research results showed that the proposed hot-wet environmental fatigue equation was effective and feasible for predicting the fatigue lives and the fatigue limits of the structural material and members of the bridges.

Keywords

Fatigue equation hot-wet environment steel fiber polymer structural concrete (SFPSC)carbon fiber reinforced polymer (CFRP)fatigue life

1. Introduction

Durability of bridge structures in service is a key issue in civil engineering [6,11]. As an important part of durability problems, study of environmental fatigue behavior of structural material and members has a great significance to ensure the durability and safety of bridge structures [1,2].

At present, durability studies of bridge structures are mainly based on experimental research referred to aging test procedures of related materials, which was the traditional experimental method with environment and loads uncoupling action [3,7]. Some studies [9,10] have been carried out to investigate the coupling effect of environment and sustained load. This research team has studied environmental fatigue behavior of bridge structures under coupling effects of temperature and fatigue loads [4,5,8]. Furthermore, this paper proposes a hot-wet environmental fatigue equation of steel fiber polymer structural concrete (SFPSC) and reinforced concrete (RC) members strengthened with fiber reinforced polymer (FRP) and provides a theoretical basis of fatigue design and fatigue life prediction for flexural members of the bridges.

2. Hot-wet environment fatigue equation

According to physical and mechanical behavior of materials used in SFPSC and RC beams strengthened with CFRP, hot-wet environment had a major effect on fatigue behavior of the materials and members. When hot-wet environment was varied, external load, temperature and humidity coupling was considered in calculation formula of the fatigue lives. Temperature and humidity affect the fatigue behavior of material and component were considered due to stress changes.

Therefore, for SFPSC specimens and RC beams strengthened with CFRP, the fatigue equation of under hot-wet environment and cyclic loads coupling action could be assumed as: $\begin{eqnarray}\displaystyle S^{m(T,H)}N=C, & & \displaystyle\end{eqnarray}$ (1) where, S is cyclic stress/load, m (T, H) is a function of material performance related to temperature T and humidity H, N is load cyclic count (fatigue life), C is constant.

Take logarithm on both sides of Eq. (1) and convert into series form: $\begin{eqnarray}\displaystyle S=\mathop{\sum }_{n=0}^{\infty }\frac{\left[\frac{A}{m(T,H)}(\log _{10}C-\log _{10}N)\right]^{n}}{n!}. & & \displaystyle\end{eqnarray}$ (2)

Constant and linear terms of the series in expansion of Eq. (2) were reserved and the equation was written as: $\begin{eqnarray}\displaystyle S\approx 1+\frac{A}{m(T,H)}(\log _{10}C-\log _{10}N). & & \displaystyle\end{eqnarray}$ (3) For three-point bending rectangular beam under concentrated load P, Eq. (3) was modified as: $\begin{eqnarray}\displaystyle P=\frac{2bh^{2}}{3L}\left[1+\frac{A}{m(T,H)}(\log _{10}C-\log _{10}N)\right], & & \displaystyle\end{eqnarray}$ (4) where, b, h, L was respectively width, height and span (distance between two supports) of the three-point bending beam, A is constant.

In hot-wet environment, m (T, H) was assumed as: $\begin{eqnarray}\displaystyle m(T,H)=\frac{A}{A_{1}+A_{2}e^{A_{4}T}+A_{3}e^{A_{5}H}}, & & \displaystyle\end{eqnarray}$ (5) where, A₁ ∼ A₅ are constants.

Take Eq. (5) into Eq. (4): $\begin{eqnarray}\displaystyle P=C_{1}+C_{2}f(T,H)+\left[C_{3}+C_{4}f(T,H)\right]f(N), & & \displaystyle\end{eqnarray}$ (6) where, hot-wet function f (T, H) and fatigue life function f (N) were respectively assumed as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle f(T,H)=C_{5}e^{C_{7}T}+C_{6}e^{C_{8}H}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle f(N)=\log _{10}N.\end{eqnarray}$ (8)

In Eq. (6) and (7), C₁ ∼ C₈ are constant coefficients to be determined by experiments. The unit of load P is kN, the unit of temperature T is °C, the unit of humidity H is % R ⋅ H.

Using loading level (stress level) S_R = P_max∕P_u to express load P in Eq. (6), which was rewritten as: $\begin{eqnarray}\displaystyle S_{R}=C_{1}^{\prime }+C_{2}^{\prime }f(T,H)+[C_{3}^{\prime }+C_{4}^{\prime }f(T,H)]f(N), & & \displaystyle\end{eqnarray}$ (9) where, $C_{i}^{\prime }=\frac{C_{i}}{P_{u}},(i=1,2,3,4)$ ; P_max is the maximum value of cyclic loads, P_u is the ultimate capacity of the specimens.

Using Eqs (6) ∼ (9), it is convenient to estimate the fatigue lives and fatigue limits of SFPSC specimens and RC beams strengthened with CFRP under hot-wet environment and cyclic bending loads.

3. Environment fatigue equation for SFPSC

Fatigue tests were carried out to study the fatigue performance and determine the constant coefficients of the environment fatigue equation for SFPSC.

The specimens were three-point bending beams as shown in Fig. 1. The mix design of SFPSC of test were finally determined as w (cement):w (sand):w (gravel):w (water):w (super-plasticizer) = 1:1.346:1.86:0.287:0.014. On the basis of this, 0.64 wt% of steel fiber and 0.015 wt% of polymer were mixed. In total 56 specimens were tested, which were divided into 7 groups.

Fig. 1.

Three-point bending SFPSC beam for fatigue test.

Fatigue tests for Group A were carried out under room temperature atmospheric environment as controlling group. Specimens in Groups B ∼ D were pretreated under 3 different hot-wet conditions for 144 hours and air-cooled for 48 hours before uncoupling fatigue experiments. Groups E ∼ G were tested under coupling action of hot-wet environments and cyclic bending loads, by using the intelligent environment simulation and controlling system developed by our research group [8].

Experimental results and S_R ∼ N curves under uncoupling action (Groups A ∼ D) are shown in Fig. 2, and those under coupling action (Groups E ∼ G) are shown in Fig. 3.

Fig. 2.

S_R ∼ N experimental curves of SFPSC under uncoupling action.

Fig. 3.

S_R ∼ N experimental curves of SFPSC under coupling action.

Using Eqs (7) ∼ (9), test data of 20 specimens (Groups B ∼ D) were simulated by Least Squares Method to determine coefficients $C_{1}^{\prime }\sim C_{8}^{\prime }$ . Then, fatigue equation of SFPSC under hot-wet environment and cyclic loads uncoupling action was established: $\begin{eqnarray}\displaystyle S_{R} & = & \displaystyle 0.343+0.238e^{-0.0174T}+0.851e^{-0.619H}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼(0.0137-0.00219e^{-0.0174T}+0.0657e^{-0.619H})\log _{10}N.\end{eqnarray}$ (10)

Similarly, fatigue equation of SFPSC under hot-wet environment and cyclic loads coupling action was established: $\begin{eqnarray}\displaystyle S_{R} & = & \displaystyle 0.543+1.12e^{-0.0998T}+0.412e^{-0.114H}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼(0.0294+0.0101e^{-0.0998T}+0.0221e^{-0.114H})\log _{10}N.\end{eqnarray}$ (11)

For general concrete structures, the corresponding loading value of N = 2 ×10⁶ cycles could be defined as the fatigue limit. Substitute N = 2 ×10⁶ cycles into Eqs (9) and (11), the relations of fatigue limits S_f of SFPSC and hot-wet conditions T, H could be obtained: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle S_{f}=0.257+0.252e^{-0.0174T}+0.437e^{-0.619H}\quad (\text{Uncoupling})\end{eqnarray}$ (12) $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle S_{f}=0.358+1.06e^{-0.0998T}+0.273e^{-0.114H}\quad (\text{Coupling}). \end{eqnarray}$ (13)

The experimental conditions of the hot-wet environments for another 8 groups of specimens were substituted into Eqs (12) and (13) to verify the effectiveness. As shown in Table 1, the experimental estimation of S_f well matched with the calculated values, and the average relative errors were presented as 2.28% (uncoupling action) and 1.55% (coupling action) respectively.

Table 1

Relative fatigue limits S_f of SFPSC under hot-wet environments

Testing method	Temperature/°C	Humidity/% R ⋅ H	Relative fatigue limit S_f		Relative error/%
			Calculated values	Experimental estimation
Coupling action	23	78	0.715	0.709	0.846
	50	80	0.614	0.616	0.325
	50	95	0.611	0.594	2.86
	35	95	0.635	0.649	2.16
Uncouping action	23	78	0.696	0.709	1.83
	50	80	0.629	0.649	3.08
	50	90	0.613	0.597	2.68
	50	95	0.605	0.596	1.51

4. Environment fatigue equation for RC beams strengthened with CFRP

Fatigue tests were carried out to study the fatigue performance and determine the constant coefficients of environment fatigue equation for RC beams strengthened with CFRP, as shown in Fig. 4. In total 30 beams were divided into two parts: 10 beams for the uncoupling action of hot-wet environment and cyclic loads, 20 beams for coupling action. Specimens in Group H were pretreated under 50 °C, 95% R ⋅ H for 144 hours and air-cooled for 48 hours before uncoupling fatigue experiments. On the other hand, the coupling experimental specimens (Groups I and J) were tested under coupling action of hot-wet environment and cyclic bending loads, by using the intelligent environment simulation and the controlling system [8].

Experimental results and S_R ∼ N curves of the strengthened beams were shown in Fig. 5. The test data at indoor atmosphere were set as that of other group which the average values of environment parameters were 23 °C and 78% R ⋅ H from reference [4].

Then, using Eqs (7) ∼ (9), test data of 20 specimens of group I and J and 20 specimens in reference [4] were simulated by Least Squares Method to determine coefficients $C_{1}^{\prime }\sim C_{8}^{\prime }$ . Fatigue equation of RC beams strengthened with CFL under hot-wet environment and cyclic loads coupling action was established: $\begin{eqnarray}\displaystyle S_{R} & = & \displaystyle 0.469+0.948e^{-0.100T}+3.09e^{-1.61H}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼(0.0725+0.0973e^{-0.100T}+0.134e^{-1.61H})\log _{10}N.\end{eqnarray}$ (14)

Using Eq. (14), it is convenient to calculate fatigue lives and fatigue limit of RC beams strengthened with CFRP under hot-wet environment and load coupling action. Substitute N = 2 ×10⁶ cycles into Eq. (14), the relations of fatigue limits S_f of the strengthened beams and hot-wet conditions T, H could be obtained: $\begin{eqnarray}\displaystyle S_{f}=0.0121+0.335e^{-0.0100T}+2.25e^{-1.61H}. & & \displaystyle\end{eqnarray}$ (15)

The experimental conditions of the hot-wet environments for another 3 groups of specimens were substituted into Eq. (15) to verify the effectiveness. As shown in Table 2, the experimental estimation of S_f well matched with the calculated values, and the average relative error was 3.37%.

Fig. 4.

RC beam strengthened with CFRP.

Fig. 5.

S_R ∼ N experimental curves of RC beam strengthened with CFRP.

Table 2

Relative fatigue limits S_f of CFRP-RC beam under hot-wet environment

Testing method	Temperature/°C	Humidity/% R ⋅ H	Relative fatigue limit S_f		Relative error/%
			Calculated values	Experimental estimation
Coupling action	23	78	0.687	0.644	6.68
	50	85	0.587	0.573	2.44
	50	95	0.501	0.506	0.99

5. Conclusions

This paper proposed a hot-wet environmental fatigue equation to describe the fatigue behavior of SFPSC and RC beams strengthened with CFRP under different hot-wet conditions and loading levels. The following conclusions were obtained.

(1) Hot-wet environment had a significant effect on fatigue/durability behavior of bridge structural materials and members.

(2) Environmental equations were proposed both of SFPSC and RC beams strengthened CFRP. Fatigue lives and fatigue limits under hot-wet environment and load coupling/uncoupling action were conveniently calculated with better accuracy by the environmental fatigue equations.

(3) Using test data of current fatigue/durability experimental method of environment and load uncoupling action, negative influence of environment was underestimated.

For further study, the environmental fatigue equations proposed in this paper should be verified on a greater environmental scale and include more varied kinds of fatigue loads.

Footnotes

Acknowledgements

The project is supported by the National Key R&D Program of China (No. 2017YFC0806000) and the National Natural Science Foundation of China (Nos 11627802, 51678249, 11132004).

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