Hysteretic Uncertainty and Anomaly Quantification of Reinforced Concrete Beams Strengthened with Carbon Fiber Reinforced Polymer and Ultra-High-Performance Concrete in Thermocyclic Distress

Ju-Hyung Kim; Yail J. Kim

doi:10.1016/j.eng.2024.11.018

Engineering ›› 2026, Vol. 61 ›› Issue (6) :41 -54. DOI: 10.1016/j.eng.2024.11.018

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Hysteretic Uncertainty and Anomaly Quantification of Reinforced Concrete Beams Strengthened with Carbon Fiber Reinforced Polymer and Ultra-High-Performance Concrete in Thermocyclic Distress

Ju-Hyung Kim ^a^,^*
, Yail J. Kim ^b

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Abstract

Seismic strengthening with carbon fiber reinforced polymer (CFRP) sheets is a proven technique for improving the capacity and ductility of concrete members and has been used worldwide. In this study, the effects of multi-hazard loading on the behavior of reinforced concrete beams strengthened with CFRP and ultra-high-performance concrete (UHPC) jackets are investigated. This work is an unprecedented initiative in the rehabilitation sector. Based on a previously conducted experimental study, where load reversals are performed at elevated temperatures varying from 25 to 175 °C, analytical responses are investigated focusing on the performance degradation, uncertainty quantification, hysteresis, and pinching mechanisms of the retrofitted beams. The uncertainty index, which measures the extent of the anomaly caused by the multi-hazard loading, clarified that the pinching of hysteresis loops during loading processes is the predominant factor causing a loss of energy dissipation capacity in the thermocyclic distress. The development of uncertainty correlated with the degree of drift ratios, demonstrated by the increased uncertainty index of 0.37 at 175 °C, and the beam pinching is controlled by the retrofit schemes. The adjusted stiffness of the loops represents the accumulated damage and deformation resistance; meanwhile, the evolution of irreversible pinching alters hysteretic configurations in the subsequent unloading phases. The Eigen hysteretic properties of the beams are extracted to understand the contribution of individual modes to the progression of uncertainties. The first mode dominates the fourth mode by a factor of up to 127.9. Design recommendations are suggested to estimate thermocyclic damage in the strengthened beams with performance degradation factors ranging from 1.00 to 0.45, contingent upon temperature.

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Keywords

Anomaly detection / Carbon fiber reinforced polymer (CFRP) / Design recommendations / Hysteresis / Multi-hazard / Performance degradation / Pinching / Strengthening / Thermocyclic loading / Ultra-high-performance concrete (UHPC) / Uncertainty

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Ju-Hyung Kim, Yail J. Kim. Hysteretic Uncertainty and Anomaly Quantification of Reinforced Concrete Beams Strengthened with Carbon Fiber Reinforced Polymer and Ultra-High-Performance Concrete in Thermocyclic Distress. Engineering, 2026, 61 (6) : 41-54 DOI:10.1016/j.eng.2024.11.018

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1. Introduction

Multi-hazards negatively affect the functionality of built environments. From a structural engineering perspective, compared to circumstances under discrete loadings multi-hazards can increase the vulnerability of load-bearing members by accelerating the deterioration process [1]. The concept of multi-hazard design encompasses a wide spectrum that considers interactions between different risks and cumulative damage ensuring the safety of structures in a holistic manner [2]. However, conventional practices are inadequate to address the engineering challenges posed by multi-hazard loadings because these are intrinsically centered on single loadings at a time, and linear load combinations may not reflect the actual situation [3]. By dealing with concurrent threats in an integrated experimental or simulation framework, the performance of structural elements can be adequately evaluated in such mutually inclusive conditions. As demonstrated by the San Francisco Bay Area disasters that occurred on April 18, 1906, and October 17, 1989, earthquakes and subsequent fires, among others, are considered major causes of building damage [4], [5]. The Federal Emergency Management Agency (FEMA) also reports that seismic events associated with fires during the 1995 Kobe earthquake caused over 80 billion USD in damage, including 82 091 collapsed buildings and 7456 buildings damaged by fire [6]. Nevertheless, the consequences of seismic-fire-combined loadings have not yet been sufficiently researched and little information is available [7], [8].

Timely maintenance and rehabilitation measures play a crucial role in repairing physical damage and preventing catastrophic failure of existing structures [9]. Similarly, mitigation strategies should be actively sought as part of hazard mitigation planning to minimize potential problems that lead to operational disruptions and therefore unwanted costs to society. Numerous structures do not meet modern seismic standards and are at risk during large-scale earthquakes [10]. Seismic rehabilitation with high-performance materials is a recommended solution to upgrade these structures. Jacketing with cementitious materials is often performed to increase the cross-section of deficient structures [11], [12]. In most cases, ordinary concrete is used [13], whereas ultra-high-performance concrete (UHPC) can be a promising alternative [14]. The application of UHPC offers several advantages [15], [16], [17]: homogeneous composition, high strength and elastic modulus, durability, fracture resistance, recyclability, and dimensional stability. Seismic strengthening with carbon fiber reinforced polymer (CFRP) sheets, which consist of unidirectional carbon fibers and a resin matrix, is a proven technique for improving the capacity and ductility of concrete members and has been used worldwide [18]. Notwithstanding the favorable engineering properties of CFRP, such as its high strength-to-weight ratio, noncorrosiveness, reduced maintenance, and ease of installation [19], the performance of the polymeric composite is significantly degraded when the temperature exceeds the glass transition temperature of the material [20]. The behavior of concrete members strengthened with either CFRP or UHPC has been reported previously [21], [22], [23]. In contrast, the behavior of members retrofitted with both materials has rarely been documented [24], [25], and there is currently no citable bibliography on the response of CFRP/UHPC-retrofitted members subjected to seismic oscillations at elevated temperatures.

This study presents a comprehensive analytical research program to understand the effects of thermocyclic distress on the behavior of reinforced concrete beams strengthened with CFRP and UHPC. The program focuses on three main areas: damage quantification, hysteresis modeling, and energy dissipation mechanisms. The study is the first attempt of its kind to advance the state of the art of structural rehabilitation. Various metrics are used to demonstrate the adverse consequences of multi-hazard loading. Of particular interest is content related to uncertainties and anomalies, which are important to ensure reliable operation during the intended service life of structural members. The boundary of the anomalies in the present work is restricted to disparities between the predicted and experimental observations, without considering inconsequential outliers.

2. Research significance

There is a lack of design guidelines for multi-hazards because the effects of combined load on reinforced concrete are not explicitly accounted for in most cases [26], [27]. Moreover, most previous studies on multi-hazard loadings have focused on the behavior of structural members and not on conceivable discrepancies between experimental and predicted responses [28]. Consequently, probable irregularities in the model calculations remain to be known and hinder factual practices against real-life actions. Specifications and standards stipulate minimum requirements [29], which may be inadequate when interrelated stressors coexist. The complexity of multi-hazards is especially noteworthy in a thermocyclic environment [30]. Therefore, a variety of methodologies should be used to comprehensively interpret the hysteretic response of rehabilitated members at elevated temperatures, which will lead to the development of a new design paradigm.

3. Summary of the laboratory tests

An experimental program [31] was conducted to investigate the thermocyclic behavior of reinforced concrete beams upgraded by two retrofit schemes: CFRP and CFRP + UHPC jackets (Fig. 1(a)). The use of CFRP that enclosed the UHPC-strengthened region was intended to provide confinement and thus improve the performance of the brittle cementitious material under cyclic loading. The beams, 100 mm width × 165 mm depth × 1200 mm length, were longitudinally reinforced with four No. 3 steel bars (A_s = 71 mm² and f_y = 414 MPa, where A_s is the cross-sectional area of each bar and f_y is the yield strength) and transversely reinforced with No. 2 steel bars spaced at 75 mm intervals (A_s = 32 mm² and f_y = 250 MPa). The 28-day compressive strength of the concrete was f’_c = 24.9 MPa. After calculating the length of the plastic hinge according to Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures (ACI 440.2R-17) [19], the cantilever beams were confined using one layer of a CFRP sheet (CF beams) and a UHPC jacket plus a one-layer CFRP sheet (UC beams) to completely cover the hinge region of 350 mm from the support (Figs. 1(a) and (b)). The tensile strength and modulus of the unidirectional CFRP sheets were f_fu = 3800 MPa and E_f = 227 GPa, respectively, based on an equivalent fiber thickness of t_f = 0.165 mm and a glass transition temperature of T_g = 71 °C. The compressive strength and elastic modulus of the UHPC mixture, containing particle-optimized ingredients together with carbon nanofibers, were f’_c-UHPC = 120 MPa and E_UHPC = 30 GPa, respectively. To generate thermal loading from 25 to 175 °C at intervals of 50 °C, an electrical pad consisting of fiber glass-reinforced silicon and perfluoroalkoxy lead wires was used with a digital controller (Fig. 1(c), inset). Thermal equilibrium was achieved by a 10-minute preheating period to ensure that the beams reached the same temperatures as those applied, which is verified by thermocouple-wire readings. It was assumed that the strengthened members were insulated, which would slow the progression of heat conduction [32]. Cyclic load reversals were then applied according to the FEMA 461 protocol [33] (Fig. 1(c)) until the beams failed (Fig. 1(d)). Fig. 2 describes the hysteresis curves of the experimental beams, including the identification codes that convey the retrofit schemes at elevated temperatures. For example, CF125 and UC175 mean beams strengthened with CFRP and CFRP/UHPC tested at 125 and 175 °C, respectively. The load-carrying capacity of the beams declined with temperature by up to 53.3% and the change in hysteresis slope was indicative of the progressive damage of the strengthening system.

4. Uncertainty quantification

As observed in the experimental program [31], CFRP/UHPC-strengthened beams are damaged by thermocyclic distress: capacity reductions alongside stiffness degradation during loading and unloading cycles in conjunction with pinching that manifests a localized nonlinear stiffness variation due to the accumulated damage. An analytical approach is developed to understand the hysteretic uncertainty of the beams and to determine the individual contributions of each segment phase.

4.1. Formulation

To quantify the uncertainty of the strengthened beams under thermocyclic loading, a reference model is created [34], and its prediction is compared with the measured behavior. In other words, differences between the modeled and measured responses are defined as uncertainty. The bilinear envelope of the model with a hysteresis rule (Fig. 3) can approximate the backbone curve of the experimental beams incorporating the effective stiffness (k_eff) (Fig. 3(a)) for the initial loading:

(1)$ k_{\mathrm{eff}}=P_{\mathrm{r}} / u_{\mathrm{r}}$

where P_r is the load corresponding to 75% of the ultimate load (P_u) in the envelope of the cyclically loaded specimen [35] and u_r is the displacement at P_r. The characteristic area under the bilinear curve corresponds to the dissipated energy of the test beam until the maximum displacement (u_u) is reached:

(2)$ \int_{0}^{u_{\mathrm{u}}} P_{\text {test }}(u) \mathrm{d} u=0.5 P_{\mathrm{y}} u_{\mathrm{y}}+P_{\mathrm{y}}\left(u_{\mathrm{u}}-u_{\mathrm{y}}\right)$

where u is the displacement, P_test is the measured load during the test, P_y is the load at yielding, and u_y is the displacement at P_y.

Following the approach of the energy-equivalent bilinear envelope, a hysteresis loop can be developed at the ith cycle (Fig. 3(b)):

(3)$ k_{\mathrm{un}}=k_{\mathrm{eff}}$

(4)$ k_{\mathrm{re}}=\frac{P_{\mathrm{y}}}{u_{i, \max }-u_{y}-u_{i, \min }}$

where k_un is the unloading stifness, k_re is the average stiffness of the beam during the thermocyclic loading, and u_i_,max and u_i_,min are the maximum and minimum displacements of the loading protocol at the ith cycle, respectively (Fig. 1(c)). The loading branch of the ith cycle (P_i) is parallel to the previous loading portion in the (i - 1)th cycle, and the sign convention of the hysteresis loop is determined by the directional slopes of the loading path: positive (dP_i/du_i ≥ 0) and negative (dP_i/du_i < 0), as listed in Table 1. To refine the level of uncertainty in the thermocyclic distress, the hysteresis loop of the beams in the ith cycle is divided into six phases (Fig. 4(a) and Table 1): Phases 1 and 4 represent loadings in the positive and negative directions, respectively, followed by Phases 2 and 5, in which transitions to Phases 3 and 6 occur for unloadings in the negative and positive directions, respectively. Upon acquiring the dissipated energy of the test and reference curves (χ_test and χ_ref, respectively) associated with each phase at the ith cycle, the fraction of the uncertainty coupled with Phase j at the ith cycle, in which j is the phase number varying from 1 to 6 (χ_ij = χ_test − χ_ref; Eq. (5)) in the hysteretic behavior is calculated (Fig. 4(b)) and the sum of the χ_ij terms shows the total anomaly of the hysteretic energy at the ith cycle (χ_i; Eq. (6)):

(5)$ \chi_{i j}=\int\left(P_{i j}^{\text {test }} u-P_{i j}^{\text {ref }} u\right) \mathrm{d} u$

(6)$ \chi_{i}=\sum_{j=1}^{6} \chi_{i j}$

where $ P_{i j}^{\text {test }}$ and $ P_{i j}^{\text {ref }}$ are the load for Phase j in the test and reference model at the ith cycle, respectively. The physical interpretation of Eqs. (5), (6) is that an increase in the hysteretic anomaly (χ_i) implies increased uncertainties for the thermocyclically loaded beams; scilicet, the actual behavior of the beams deviates from the predicted behavior and thus unacceptable risks may arise. Specifically, χ_i₁ and χ_i₄ are related to the extent of pinching, χ_i₂ and χ_i₅ pertain to unintentional damage when load reversals take place, and χ_i₃ and χ_i₆ refer to residual displacements when the applied load is zero. For a convenient assessment of the relative anomaly of the hysteretic energy dissipation, an uncertainty index (d_ij) is proposed, which is a ratio of the anomaly fraction of each phase (Eq. (5)) to the hysteretic energy of the reference model in a complete loop at the ith cycle:

(7)$ d_{i j}=\frac{\left|\chi_{i j}\right|}{2 P_{y}\left(u_{u}-u_{y}\right)} \leq 1.0$

4.2. Implementation

Fig. 5 shows the uncertainty index of a representative experimental beam (UC175 was selected, which showed a remarkable pinching). For all phases, the indexes were initially zero and evolved with loading cycles. The increasing trend of d_i₁ and d_i₄ in the positive and negative loadings (Figs. 5(a) and (d), respectively) illustrates the decreasing energy capacity of the beam in conjunction with the pronounced pinching behavior. Given the insignificant increase in the uncertainty indexes in the transition phases (d_i₂ and d_i₅ in Figs. 5(b) and (e), respectively), the probability of anomalous responses within the load reversal region of the strengthened beam appears to be low. The amplified indexes of d_i₃ and d_i₆ in Figs. 5(c) and (f), respectively, clarify that the uncertainties in the residual displacement of the beam originated from the unloading component and not from other components such as pinching and transitional loadings.

The uncertainty indexes of all test beams, graphically illustrated in Fig. 6, help examine the influence of the strengthening schemes under the thermocyclic loading. For clarity, the paired indexes were averaged: d_i₁ and d_i₄ (loadings in the positive and negative directions), d_i₂ and d_i₅ (transitional loadings near the peak loads), and d_i₃ and d_i₆ (unloadings in the positive and negative directions). Regardless of the response phases, the indexes of the CF beams increased with increasing drift ratios and temperatures (Figs. 6(a)-(c)). The early deviation of the loading-phase indexes at a drift ratio of 1.54% (Fig. 6(a)) substantiates that the progression of pinching in the CFRP-strengthening system was more susceptible to the increased temperatures compared to the cyclic loading, reaffirming that thermal distress noticeably degrades the performance of externally bonded CFRP sheets [19]. The uncertainties related to the post-yield stiffness of the beams within the transition phases (Fig. 6(b)) gradually increased with the drift ratio, except for the CF25 beam where premature debonding failure occurred, whereas the effects of the temperatures were not as severe as those of the loading phases (Fig. 6(a)). The average indexes for the unloading phases (Fig. 6(c)) were positioned approximately in the middle of the previous cases (Figs. 6(a) and (b)) can be concluded that pinching was the most critical feature that led to uncertainties in the hysteresis loop of the CF beams when they were subjected to the thermocyclic loading. As shown in Figs. 6(d)-(f), analogous trends were observed for the UC beams; however, the magnitudes of the average indexes were generally lower than those of the CF beams. This observation suggests that the installation of UHPC reduced the uncertainty levels in the strengthened beams, especially in the transition and unloading phases.

5. Hysteretic system

Several topics related to the hysteresis of strengthened beams are elaborated focusing on deformation uncertainties, state-dependent responses, and factorized attributes. Mathematical expressions are used to describe the effects of mechanical stress involving elevated temperatures.

5.1. Deformation resistance

5.1.1. Formulation

The residual load (ΔP_ij) between the test and reference responses under the same displacement (Fig. 4(b)) can be linked by differentiating the uncertainty index (χ_ij) with respect to the displacement:

(8)$ P_{i j}^{\text {test }}=P_{i j}^{\text {ref }}+\frac{\mathrm{d} \chi_{i j}}{\mathrm{~d} u}=P_{i j}^{\text {ref }}+\Delta P_{i j}$

The incremental load at the (n₀ + 1)th data point (ΔP_ij,n0+1) is then expressed by:

(9)$ \begin{array}{l} \Delta P_{i j, n_{0}+1}=P_{i j, n_{0}+1}^{\mathrm{test}}-P_{i j, n_{0}+1}^{\mathrm{ref}} \\ =\left\{P_{i j, n_{0}}^{\mathrm{test}}+k_{i j, n_{0}}^{\mathrm{test}}\left(u_{i j, n_{0}+1}-u_{i j, n_{0}}\right)\right\}-\left\{P_{i j, n_{0}}^{\mathrm{ref}}+k_{i j, n_{0}}^{\mathrm{ref}}\left(u_{i j, n_{0}+1}-u_{i j, n_{0}}\right)\right\} \end{array}$

where $ k_{i j, n_{0}}^{\text {test }}$ is the tangent stiffness of the test model at the n₀th data point,$ k_{i j, n_{0}}^{\text {ref }}$ is the tangent stiffness of the reference model at the n₀th data point, and u_ij,n0+1 and u_ij,n0 are the displacement of the (n₀ + 1)th and n₀th data points in Phase j of the ith hysteresis loop.

Rearranging Eq. (9) yields the relationship of the tangent stiffness between the test and the reference models at the nth data point:

(10)$ k_{i j, n}^{\text {test }}=k_{i j, n}^{\text {ref }}+\frac{P_{i j, n}^{\text {ref }}-P_{i j, n}^{\text {test }}+\Delta P_{i j, n+1}}{u_{i j, n+1}-u_{i j, n}}=k_{i j, n}^{\text {ref }}+\frac{\Delta P_{i j, n+1}-\Delta P_{i j, n}}{u_{i j, n+1}-u_{i j, n}}$

where P_ij_,_n^ref and P_ij_,_n^test are the nth load data from a reference model and measured from test in Phase j of the ith hysteresis loop, k_ij_,_n^ref and k_ij_,_n^test are the nth stiffness data from a reference model and measured from test in Phase j of the ith hysteresis loop, u_ij_,_n and u_ij_,_n₊₁ are the nth and (n + 1)th displacement data, and ΔP_ij_,_n and ΔP_ij_,_n₊₁ are the difference between P_ij^test and P_ij^ref at nth or (n + 1)th data, respectively.

Eq. (10) is manipulated to obtain the adjusted stiffness of the beam ($\Delta k_{i j, n}=k_{i j, n}^{\mathrm{test}}-k_{i j, n}^{\mathrm{ref}}$), which characterizes the uncertainty of the deformation resistance at the data point during the loading and unloading phases:

(11)$ \Delta k_{i j, n}=\frac{\Delta P_{i j, n+1}-\Delta P_{i j, n}}{u_{i j, n+1}-u_{i j, n}}$

Mathematically, the adjusted stiffness is a forward finite-difference derivative of the residual load and the second-order derivative of the uncertainty index (χ_ij). The adjusted stiffness in each phase of the ith cycle can be computed from the hysteresis loops of the test and the reference model.

5.1.2. Implementation

The variation of the adjusted stiffness normalized by the effective stiffness (Δk_ij/k_eff) of the UC175 beam is plotted in Fig. 7. In the loading phases (Figs. 7(a) and (d)), the normalized stiffness declined with the increased cycles because of accrued damage: The Δk_ij/k_eff value was unity at the first cycle (drift ratio = 0.1%), while it approached zero at the 25th cycle (drift ratio = 3.5%). The flat-like behavior of the stiffness ratio indicates that the deformational resistance of the beam became virtually null during the loading phases: A plastic hinge was already formed through which a considerable amount of energy was dissipated. During the transitional loading phases (Figs. 7(b) and (e)), the stiffness values fluctuated with the loading cycles owing to the combined stiffening responses (Δk_ij/k_eff > 0) and softening (Δk_ij/k_eff < 0); stiffening corroborates that the reference model underestimated the initial stiffness of the beams; softening, in contrast, implies that the post-yield stiffness of the beams was in a downward direction. The asymmetric behavior of the beam in the first and third quadrants of the load-drift ratio curve (Fig. 2(h)) is responsible for the differences observed in Figs. 7(b) and (e). The decreasing stiffness ratios in Figs. 7(c) and (f) denote that the pinched hysteresis of the loading phases continues to affect the unloading phases.

5.2. Hysteresis model

5.2.1. Formulation

Two hysteresis models (mean and regression, Fig. 8(a)) are theorized by iteratively updating the reference model with the adjusted stiffness ($\Delta k_{i j, n}^{\text {update }}$):

(12)$ k_{i j, n}^{\mathrm{mod}}=k_{i j, n}^{\mathrm{ref}}+\Delta k_{i j, n}^{\text {update }}$

where $ k_{i j, n}^{\bmod }$ is the updated stiffness of the model. To simplify the updated stiffness in each cycle, Eqs. (13), (14) are used for the mean and regression models, respectively:

(13)$\Delta k_{i j, n}^{\text {update }}=\frac{1}{N} \sum_{n=1}^{N} \Delta k_{i j, n}=c$

(14)$\Delta k_{i j, n}^{\text {update }}=\alpha u_{i j, n}+\beta$

(15)$\mathrm{SSE}=\sum_{n=1}^{N}\left(\Delta k_{i j, n}-\Delta k_{i j, n}^{\mathrm{update}}\right)^{2}$

where N is the total number of data points in Phase j at the ith cycle; c is a constant; and α and β are regression coefficients determined by minimizing the sum of the squared error (SSE) (Eq. (15)) against the experimental stiffness in Fig. 7. Fig. 8(b) shows the proposed hysteresis rules for updating the stiffness of the reference model (Eq. (12)).

5.2.2. Implementation

As shown in Fig. 9, the hysteresis loop of the proposed models was compared with their experimental counterparts (for simplicity only UC25 and UC175 beams are shown). When the applied temperature was lower than the glass transition temperature of CFRP (T_g = 71 °C), the path-dependent pinching of the test beam was marginal (Figs. 9(a) and (b)); for this reason, minor differences were observed between the mean and regression models during the positive and negative loadings (Phases 1 and 4). However, the regression model overestimated the rising portion of the curves from Phases 1 to 2 because the abruptly increased load at the 32nd cycle affected the regression coefficients of Eq. (14). When the beam was subjected to a temperature of T = 175 °C (> T_g), pinching was apparent (Figs. 9(c) and (d)), and the mean model did not properly reflect such localized responses (Fig. 9(c)) compared to the regression model (Fig. 9(d)). These facts indicate that the simpler mean model is suitable when the hysteresis of the strengthened beam is examined under the condition of T < T_g; if not, the regression model should be used.

5.3. Eigen hysteretic properties

5.3.1. Formulation

Eigen hysteretic properties are determined using the singular value decomposition (SVD) method in linear algebra [36] to determine how elevated temperatures alter the cyclic behavior of the strengthened beams. As described in Fig. 10, a load matrix P ∈ R ^l ^× ^m (where l represents all data points from the initial to the last cycle, and m is the number of the variables: m = 4 in this study for the four temperatures) is decomposed by Ref. [37]:

(16)$\boldsymbol{P}=\boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^{\mathrm{T}}$

Twhere U and V are the unitary matrices (U ∈ R ^l ^× ^m and V ∈ R ^m ^× ^m), and Σ is the diagonal singular value matrix (Σ ∈ R ^m ^× ^m). The load matrix P consists of four component vectors (p_m) and generates non-zero diagonal singular values (σ_m in Fig. 10, where all off-diagonal values are zeros). Eq. (16) can be written as a linear combination of the four columns of the U and V matrices with the component vectors of u_m and v_m alongside the singular values denoting the contribution of the mth component to the load matrix (Fig. 10).

(17)$\boldsymbol{P}=\sum_{m=1}^{4} \sigma_{m} \boldsymbol{u}_{m} \boldsymbol{v}_{m}^{\mathrm{T}}$

As the length of the u_m vector in Fig. 10 corresponds to load magnitudes that depend on the applied displacement (Fig. 1(c)), the load component (p_m) becomes a linear combination of the u_m vectors:

(18)$\boldsymbol{p}_{m}=\sigma_{1} v_{m, 1} \boldsymbol{u}_{1}+\sigma_{2} v_{m, 2} \boldsymbol{u}_{2}+\sigma_{3} v_{m, 3} \boldsymbol{u}_{3}+\sigma_{4} v_{m, 4} \boldsymbol{u}_{4}$

where v_m_,1 to v_m_,4 are the elements of the v_m vector in Eq. (17).

5.3.2. Implementation

Fig. 11(a) shows the singular vectors (u_m) of the UC specimens computed from the test, which are analogous to those of the regression model (Fig. 11(b)). The first mode singular vector (u₁) resembled archetypal hysteresis curves; however, the second to fourth modes (u₂ to u₄) showed complex patterns. The increased coefficients of the vectors u₂ to u₄ (Eq. (18)) led to uncertainties in the hysteretic behavior of the strengthened beams. Figs. 11(c) and (d) chart the contribution of each modal coefficient (σ₁v_m_,1-σ₄v_m_,4) for the test and the model, respectively, at elevated temperatures. The coefficient for the first mode (σ₁v_m_,1) contributes primarily to the load component (p_m in Eq. (18)), while those for the second to fourth modes make secondary contributions. In addition, the first-mode coefficient, which is sensitive to the temperatures justifies the hysteresis curves of the beams degraded in the presence of thermal loadings.

6. Pinching mechanism

A deterioration process resulting from pinching is outlined to measure the diminished energy under multi-hazard loading. By classifying the loading stiffness of CFRP/UHPC-strengthened beams, a relationship between irreversible hysteresis and thermocyclic distress is constructed.

6.1. Formulation

As illustrated in Fig. 12(a), the experimental and predicted hysteresis curves of the strengthened beams can be simplified to evaluate the degree of pinching

(19)$k_{\mathrm{re}}=\frac{P_{u=0}}{u_{P=0}}$

where P_u₌₀ and uP=0 are the residual load and displacement, respectively. The characteristic stiffness associated with crack sliding, crack closure, and bond slip [38] may include a series of k_p and k_np (Fig. 12(b)):

(20)$\left\{\begin{array}{c} \frac{1}{k_{\mathrm{re}}}=\frac{1}{k_{\mathrm{p}}}+\frac{1}{k_{\mathrm{np}}} \\ k_{\mathrm{re}}=\frac{k_{\mathrm{np}} k_{\mathrm{p}}}{k_{\mathrm{np}}+k_{\mathrm{p}}} \end{array}\right.$

where k_p and k_np are the stiffness of the plastic hinge with and without pinching, respectively. Solving Eq. (20) for k_p gives

(21)$k_{\mathrm{p}}=\frac{k_{\mathrm{np}} k_{\mathrm{re}}}{k_{\mathrm{mp}}-k_{\mathrm{re}}}$

The k_np and k_re values are obtained from the test and the analytical model (Fig. 12(a)). The compliance of pinching (C_p = 1/k_p) is then determined as follows:

(22)$C_{\mathrm{p}}=C_{\mathrm{re}}-C_{\mathrm{np}}$

where C_re and C_np are the compliance of the system (C_re = 1/k_re) and the compliance without pinching (C_np = 1/k_np), respectively.

6.2. Implementation

The characteristic stiffness of selected beams is shown in Figs. 13(a) and (b). Although reasonable agreement was obtained between the test and predicted responses, local fluctuations in the model were observed owing to the simplified linear nature of the test response (Fig. 8(a)) and the adoption of the regression coefficients (Eq. (14)). The UC beam retained stiffness better than the CF beam at 25 °C; however, negligible differences were noted at 175 °C. Thus, the UHPC jacket was found to be beneficial for precluding the occurrence of pinching until thermal damage in the retrofit system is advanced. The literature shows that the interfacial bond between ordinary concrete and UHPC deteriorates when exposed to heat [39]. As for the development of the stiffness with a drift ratio, a similar propensity was recorded in both beams (Figs. 13(c) and (d)), implying that the progression of pinching was influenced more by the load-induced displacement of the beams than by the strengthening materials. Fig. 14 shows the compliance of the beams. For the CF beams, the compliance values with and without pinching (C_p and C_np, respectively) steadily increased with temperature (Fig. 14(a)), and the percentage of C_p was escalated from 39.3% to 52.7% at 25 and 175 °C, respectively, due to the evolution of pinching (Fig. 14(b)). For the UC beams, the magnitude of C_p was substantially lower than that of C_np up to 75 °C. Thereafter the trend reversed (Fig. 14(c)), which can be attributed to the aforementioned thermal damage between the concrete substrate and the UHPC. Consequently, the percentage of pinching compliance was susceptible to temperature (C_p = 28.8% at 25 °C and 55.6% at 175 °C, Fig. 14(d)).

7. Performance-based design

In accordance with the provisions of Minimum Design Loads and Associated Criteria for Buildings and Other Structures (ASCE/SEI 7-16) [29], design recommendations are developed for reinforced concrete beams retrofitted with CFRP and UHPC jackets subjected to repeated stress reversals at elevated temperatures. A performance metric is suggested and its application in practice is described.

7.1. Derivation

A performance degradation factor (ϕ) is proposed to evaluate the energy dissipation capacity of the strengthened beams under thermocyclic loading:

(23)$\phi=\frac{W_{T}}{W_{25}}=\frac{\int_{u} P_{T} \mathrm{~d} u}{\int_{u} P_{25} \mathrm{~d} u}$

where W₂₅ and W_T are the energy capacity of the beams at 25 °C and high temperatures, respectively; and P₂₅ and P_T are their load segments, repectively. Based on the SVD method described in Section 5.3, the dissipated energy can be decomposed into the primary (h_T_,p) and secondary (h_T_,s) components:

(24)$\phi=\frac{h_{T, p}+h_{T, s}}{h_{25, p}+h_{25, s}}$

(25)$h_{T, p}=\int_{u} \sigma_{1} v_{T, 1} \boldsymbol{u}_{1} \mathrm{~d} u$

(26)$h_{T, s}=\int_{u} \sigma_{2} v_{T, 2} \boldsymbol{u}_{2} \mathrm{~d} u+\int_{u} \sigma_{3} v_{T, 3} \boldsymbol{u}_{3} \mathrm{~d} u+\int_{u} \sigma_{4} v_{T, 4} \boldsymbol{u}_{4} \mathrm{~d} u$

where h_25,p and h_25,s indicate the primary and secondary dissipated energy components at T = 25 °C, and v_T_,1 to v_T_,4 are the first to fourth element of singular vector corresponding to the specimen at temperature T.

Eq. (24) is reformulated into a product of hysteretic energy ratios

(27)$\phi=\psi_{1} \psi_{2}$

(28)$\psi_{1}=\frac{h_{T, p}}{h_{25, p}}$

(29)$\psi_{2}=\frac{1+h_{T, s} / h_{T, p}}{1+h_{25, s} / h_{25, \mathrm{p}}}$

where ψ₁ and ψ₂ are the primary and secondary energy ratios, respectively. As is evident in Figs. 11(c) and (d), the first mode of the singular vector can be the primary component and other modes are secondary (h_T,_p/h_T_,s ≈ 0) leading to

(30)$\psi_{1}=\frac{h_{T, p}}{h_{25, \mathrm{p}}}=\frac{\int_{u} \sigma_{1} v_{T, 1} \boldsymbol{u}_{1} \mathrm{~d} u}{\int_{u} \sigma_{1} v_{25,1} \boldsymbol{u}_{1} \mathrm{~d} u}=\frac{v_{T, 1}}{v_{25,1}}$

(31)$\psi_{2}=1$

where v_25,1 is the first element of the singular vector corresponding to the specimen at T = 25 °C.

Therefore, the performance degradation factor is obtained by the decomposed load magnitudes of the first mode at elevated temperatures

(32)$\phi=\frac{v_{T, 1}}{v_{25,1}}$

7.2. Application

The performance degradation factors of the CF and UC beams are shown in Fig. 15(a). Considering that the FEMA protocol (Fig. 1(c)) includes typical seismic loadings applied in practice, the factors ranging from 1.00 to 0.45 (rounded from the experimental and predicted results) may embody the performance degradation of the strengthened beams under thermocyclic distress (Table 2). These factors are conceptually different from the conventional strength reduction factors, which are used to determine the design strength of a structural member that is lower than its nominal strength for safety considerations. Fig. 15(b) confirms the adequacy of the degradation factors irrespective of displacement: Apart from the initial loading up to a displacement ratio of 0.1, which shows unstable experimental responses caused by the engagement of the test specimens in the support fixture, the φ factors are maintained with moderate irregularities. As instantiated in Fig. 15(c), the hysteresis curves of the strengthened beams are multiplied by the proposed degradation factors to estimate thermocyclically impaired performance. The design approach is also useful for quantifying the effectiveness of the retrofit methods. The energy dissipation capacity of the UC beams is higher than that of the CF beams (Fig. 15(d)).

8. Summary and conclusions

In this study, the performance of CFRP + UHPC-strengthening systems for reinforced concrete beams under multi-hazard loading is theoretically investigated, which is a pioneering achievement in the field of rehabilitation. As specified in FEMA 461 [33], excitations were applied at elevated temperatures varying from 25 to 175 °C. In accordance with previous laboratory tests, analytical modeling was performed to investigate the uncertainty, hysteresis, and pinching mechanisms of the strengthened beams. Design recommendations were derived and practical factors were proposed to estimate the degraded energy dissipation capacity of the beams under thermocyclic distress. The following conclusions are drawn:

(1)The increased uncertainty indexes of the strengthened beams correlate with a reduction in the energy capacity: The indexes of the CF and UC beams increase to as high as 0.35 and 0.37, respectively, at 175 °C. The anomalous responses of the beams are not remarkable during the transition phases between loading and unloading (the uncertainty indexes were less than 0.17 and 0.09 for the CF and UC beams at 175 °C, respectively), while the uncertainties related to the residual displacements are dominated by the unloading component.

(2)The degree of uncertainty is proportional to the drift ratio of the beams. The pinching of the CFRP-strengthened beams is accelerated at higher temperatures owing to the inclusion of the polymeric resin, as indicated by a compliance change from 39.3% to 52.7% at 25 and 175 °C, respectively, while the presence of UHPC reduces the uncertainties in the retrofit system.

(3)The adjusted stiffness of the hysteresis loop for the strengthened beams is indicative of damage accumulation and deformational resistance in the case of the formation of plastic hinges where the applied energy is dissipated. The pinched loops of the beams transform the configuration of the subsequent unloading phases.

(4)As claimed by the Eigen hysteretic properties of the beams, the first mode of the singular vector explains the temperature-sensitive hysteresis (with its contribution to the predicted Eigen properties of the UC beam being 127.9 and 5.3 times greater than that of the fourth mode at 25 and 175 °C, respectively), and other higher modes determine the uncertainty of the behavior.

(5)The magnitude of drift ratios is more responsible for the progression of pinching compared to the retrofit materials: The CF and UC beams show a reduced average pinching stiffness rate of up to 95.8% until the drift ratio attains 7.5%, at which the rates of these beams are almost identical. However, the installation of the UHPC jacket is still beneficial to maintain the stable pattern of hysteresis loops.

(6)The proposed performance degradation factors used to measure the effectiveness of the retrofit measures reproduce the thermocyclically damaged hysteresis curves of the strengthened beams. For real-world applications, the degradation factors can be chosen in the range of 1.00 to 0.45 in accordance with desired thermal exposure conditions.

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