Investigation of Retrofit Strategies to Extend the Service Life of Bridge Structures through Active Control

Introduction

Active Control System Design

Actuation Load

Two active control strategies are formulated: quasi-static deflection control through EAT and vibration control through linear actuators integrated into the main load path (e.g., hangers for the tied-arch bridge and stays for the cable-stayed bridge). A conceptual illustration is given in Fig. 1. In both strategies, the structure is discretized in n^el beam elements. Depending on the retrofit strategy, n^act elements are active; i.e., they house a linear actuator installed in series (Böhm et al. 2020). This means that the actuator and the housing element are subjected to the same force, i.e., force-serial. An actuator can change length, thereby, depending on the structural topology, modifying the stress and displacement response under loading. In the absence of external loads, assuming small strains and displacements, when an actuator changes its length, stress develops due to the resistance offered by the stiffness of the other elements to which it is connected. A way to model the effect of force-serial actuators is to map the length changes to equivalent actuation loads ${\mathbf{p}}^{\mathrm{act}}\in {\mathbb{R}}^{n^d}$, where n^d is the number of degrees of freedom. ${\mathbf{p}}^{\mathrm{act}}$ contains the force couples caused by the length change $\mathbf{\Delta }{\mathbf{l}}^{\mathrm{act}}\in {\mathbb{R}}^{n^{\mathrm{act}}}$ of the actuators

$${\mathbf{p}}^{\mathrm{act}}=\mathbf{D}\mathbf{\Delta }{\mathbf{l}}^{\mathrm{act}}$$

(1)

The input matrix $\mathbf{D}\in {\mathbb{R}}^{n^d\times n^{\mathrm{act}}}$ contains the force couples produced by a unitary length change of the actuators

$$\mathbf{D}=\left[\begin{array}{ccccc}{\delta }_1& \dots & {\delta }_i& \dots & {\delta }_{n^{\mathrm{act}}}\end{array}\right]$$

$${\delta }_i={\mathrm{T}}_i^T{\left[\begin{array}{cccccc}-{\displaystyle \frac{E_i{A}_i}{l_i}}& 0& 0& {\displaystyle \frac{E_i{A}_i}{l_i}}& 0& 0\end{array}\right]}^T$$

(2)

where $\mathbf{T}\in {\mathbb{R}}^{6\times 6}$ = local-to-global transformation matrix for 2D beam elements. Assuming small displacements, linear-elastic equilibrium conditions can be expressed as follows:

$$\mathbf{K}{\mathbf{d}}^{\mathrm{act}}={\mathbf{p}}^{\mathrm{act}}$$

(3)

where $\mathbf{K}\in {\mathbb{R}}^{n^d\times n^d}$ = stiffness matrix; and ${\mathbf{d}}^{\mathrm{act}}\in {\mathbb{R}}^{n^d}$ = displacement compensation achieved through actuation

$${\mathbf{d}}^{\mathrm{act}}={\mathbf{K}}^{-1}{\mathbf{p}}^{\mathrm{act}}$$

(4)

Note that the actuator length change is modeled as a fictitious elastic strain in the housing element. However, in practice, it is inelastic because it is performed by the linear actuator. Therefore, to compute the forces ${\mathbf{f}}^{\mathrm{act}}\in {\mathbb{R}}^{n^{el}}$ that develop because of the actuator length changes, the inelastic strain must be eliminated from the total strain for the elements housing the actuators, which is expressed as follows:

$${\overline{\mathbf{f}}}_i^{\mathrm{act}}=\{\begin{array}{cc}{\overline{\mathbf{K}}}_i{\mathbf{T}}_i{\overline{\mathbf{d}}}_i^{\mathrm{act}}-{\left[\begin{array}{cccccc}-{\displaystyle \frac{E_i{A}_i}{l_i}}& 0& 0& {\displaystyle \frac{E_i{A}_i}{l_i}}& 0& 0\end{array}\right]}^T\mathrm{\Delta }{l}_i^{},& \mathrm{if}i\in S^{\mathrm{act}}\\ \overline{\mathbf{K}}{\mathbf{T}}_i{\overline{\mathbf{d}}}_i^{\mathrm{act}},& \mathrm{otherwise}\end{array}$$

(5)

where ${\overline{\mathbf{f}}}_i\in {\mathbb{R}}^6$, ${\overline{\mathbf{K}}}_i\in {\mathbb{R}}^{6\times 6}$, and ${\overline{\mathbf{d}}}_i^{\mathrm{act}}\in {\mathbb{R}}^6$ = force, stiffness, and displacement vector extracted from the corresponding full vectors according to the degrees of freedom of the ith element, respectively.

Case Studies

Mitigation of Corrosion-Induced Damage in Prestressed Beam Bridges

Petrulla Bridge is a three-lane multispan bridge built in the mid-1980s in Sicily (Italy). The cross-sectional dimensions and tendon profiles are given in Fig. 2 and further specifications are given in Table 3. According to Anania et al. (2018) and Bazzucchi et al. (2018), the primary cause for the formation of a plastic hinge in midspan was corrosion in the prestress tendons [Fig. 3(a)]. The investigation shows that the concrete in the I-girder cross section retained adequate mechanical properties and failure occurred due to the advanced state of corrosion in the prestress tendons caused by a noncorrect execution of the grout injection [Fig. 3(b)]. The spacing between adjacent prestress tendons did not meet code requirements (ACI-318-19, ACI 2022), causing an unwanted lack of concrete interface and resulting in a uniform rust distribution throughout the tendon area. Numerical analysis was carried out through nonlinear FEM (Anania et al. 2018). To simulate the effect of corrosion on the bridge response, the tendon cross-sectional area was progressively reduced. It was not possible to achieve convergence when the tendon cross-sectional area was reduced by 61%, indicating a drastic loss of stiffness and potential collapse (Anania et al. 2018). The residual cross-sectional area of 660 mm², corresponding to an equivalent tendon diameter of 29 mm and an estimated adherence value of 16.5 t/mm per meter, agrees with the evidence gathered through inspections.

Property	Petrulla	Toutle River	Fleher
Span	38 m	96 m	368 m
Bridge width	11.3 m	9 m*	41 m
Girder spacing	2.825 m	—	—
Number of lanes	3	3	6
Year of construction	mid-1980s	1969	1978
Year of inspection	Collapse occurred before inspection	1998	2003 and 2019
Year of collapse	2014	—	—
Traffic load	Traffic loads taken from AASHTO (ASSHTO 2020)	33,000 crossings per day, 22% heavy trucks	75,000 crossings per day
Damage	Uniform rust distribution throughout the prestress tendon area	Fatigue cracks and perceivable vibration (Leland et al. 2002) under heavy trucks with speeds greater than 89 km/h	Stays were replaced due to corrosion (Marzahn et al. 2009); 1,158 fatigue cracks in the orthotropic steel deck (Straßen 2019)

*Assumed value since no data was found in the literature.

Serviceability Limit State

The SLS for prestressed concrete beam bridges must be satisfied to ensure user comfort and prevent damage from cracks and deflections. Vibrational motion is not considered because typical concrete bridges have much higher inertia and damping than steel bridges. To determine whether the SLS is met in the original design (Anania et al. 2018), stress limits to avoid crack formation are taken, assuming full prestressing as recommended in ACI-318-19; (ACI 2022). Compression and tension stress limits are

$$\mathrm{Compression}\begin{array}{c}{\sigma }^{\mathrm{SLS},u1}\ge -0.45f^c,\mathrm{under}\mathrm{permanent}\mathrm{load}{p}^p\\ {\sigma }^{\mathrm{SLS},u2}\ge -0.60f^c,\mathrm{under}\mathrm{total}\mathrm{load}{p}^{\mathrm{tot}}\end{array}$$

(21)

$$\mathrm{Tension}{\sigma }^{\mathrm{SLS},l}\le 0.62\sqrt{f^c},\mathrm{under}{p}^{\mathrm{tot}}$$

(22)

where f^c = concrete compressive strength after 28 days of curing. The stress caused by the SLS load combination in the upper and lower outermost fibers of the cross section are denoted σ^SLS,u and σ^SLS,l, respectively. All terms are defined in the Appendix. This is a more conservative constraint than partial prestressing, which typically does not require the adoption of stress limits. The deformation is calculated based on Nawy (2009) using the response factors that are recommended by the Prestressed/Precast Concrete Institute (PCI) (PCI 2014). These factors account for long-term effects in prestressed concrete members. The deformation limit for simply supported beams is set to a fraction of the span length L/800, as recommended in AASHTO (2020). For clarity, since this analysis concerns an existing bridge, the erection stage is not considered. This means that the stress and deflection after the prestress transfer from the tendon to the concrete are not evaluated.

Ultimate Limit State

According to Anania et al. (2018), the bridge failed after 30 years in service upon fracture of the prestressed tendon in the midspan. Consequently, the structure fails in a nonelastic state, and thus failure is strength-related (ULS). Bending moment and shear under ULS load are evaluated as follows:

$$\begin{array}{rl}& {\varphi }^M{M}^R\ge M^{\mathrm{ULS}}\\ & {\varphi }^V{V}^R\ge V^{\mathrm{ULS}}\end{array}$$

(23)

where M^R and V^R = moment and shear resistance, respectively; M^ULS and V^ULS = moment and shear caused by the ULS load combination, respectively (Table 2); and ϕ^M and ϕ^V = resistance reduction factor for bending moment and shear, respectively (ACI-318-19, ACI 2022). The resistance factor ϕ^M for prestressed concrete subjected to bending moment is a function of the strain in the outermost reinforcement, which in this case is assumed to be the rebar as recommended in ACI-318-19 (ACI 2022) or equivalently by AASHTO (2020). The resistance factor ϕ^V for shear verification is 0.75, as stated in ACI-318-19 (ACI 2022). Considering the high corrosion developed in the prestressed tendons, the moment and shear capacity based on ACI-318-19 (ACI 2022) are reformulated as functions of the prestressed tendon area percentage reduction β. The moment capacity is based on strain compatibility between the cross section and the prestressed tendon by considering both elements fully bounded and having an equal strain. Since it is assumed that there is full compatibility between the tendon and the surrounding concrete, the strain is expressed by referring to the tendon. The total strain ${\epsilon }^{t,t}$ in each tendon is the summation of the initial strain ${\epsilon }^{t,i}$, decompression strain ${\epsilon }^{t,d}$, and final strain ${\epsilon }^{t,f}$

$${\epsilon }_k^{t,t}(\beta )={\epsilon }_k^{t,i}(\beta )+{\epsilon }_k^{t,d}(\beta )+{\epsilon }_k^{t,f}(\beta )$$

(24)

The reader is referred to the Appendix for the definition of each strain component and to Nawy (2009) for further details. Once the total strain is obtained, the constitutive relation between stress and strain is employed to compute the tendon stress f^t. However, since no information was available from the tendon supplier, the relationship was derived from the PCI (Precast Prestressed Concrete: Bridge Design Manual 2014):

$$f_k^t(\beta )=E^t{\epsilon }_k^{t,t}(\beta )\times \left(C^{t,A}+{\displaystyle \frac{(1-C^{t,A})}{({[1+{(C^{t,B}\cdot {\epsilon }_k^{t,t}(\beta ))}^{C^{t,C}}]}^{1/C^{{}^{t,C}}}}}\right)$$

(25)

where C^t,A, C^t,B, and C^t,B = constants defined in the PCI (Precast Prestressed Concrete: Bridge Design Manual 2014). Once the tendon stress is obtained, the concrete compressive height c^c(β) can be derived using force equilibrium, as illustrated in Fig. 4, and subsequently, the equivalent (simplified) height a^c(β) (ACI-318-19, ACI 2022) is computed. The concrete force F^c is then obtained via integration over the compression area. The moment capacity M^R(β) is obtained through equilibrium at the concrete neutral axis (plastic)

$$M^R(\beta )=F^c(\beta )z^c(\beta )+F^s{z}^s(\beta )+F^t(\beta )z^t(\beta )$$

(26)

The shear capacity is calculated based on ACI-318-19 (ACI 2022). In reinforced concrete, the total shear capacity comprises that of the concrete V^R,c(β) and transversal reinforcement V^R,s (i.e., stirrups)

$$V^R=V^{R,c}(\beta )+V^{R,s}$$

(27)

Two capacity types are considered: pure shear (web-shear) and flexure–shear interaction. The concrete shear contribution V^R,c(β) is the minimum value of the governing shear capacity between pure shear V^cw(β) and flexural–shear interaction V^ci(β). For simplicity, it is assumed that the shear capacity of the stirrup is not affected by corrosion since the effect on the prestressed tendon is dominant. Pure shear resistance is expressed as follows:

$$V^{cw}(\beta )=0.29\left(\sqrt{f^c}+0.3{\overline{f}}^c(\beta )\right)b^w{d}^t+V^t(\beta )$$

(28)

where b^w = web width. The compressive stress ${\overline{f}}^c$ in the concrete cross section in the centroid caused by the prestress tendon force and the vertical component V^t(β) of the effective prestress force P^t,e(β) are

$${\overline{f}}^c(\beta )={\displaystyle \frac{{\sum }_{k=1}^K{P}_k^{t,e}(\beta )}{A^c}}$$

(29)

$$V^t(\beta )=\sum _{k=1}^K{P}_k^{t,e}(\beta ){\theta }_k^t$$

(30)

where θ^t = prestress tendon slope, as shown in Fig. 5. Concerning flexure–shear interaction V^ci(β), the capacity is taken as the minimum between Eqs. (31) and (32)

$$V^{ci}(\beta )=0.05\sqrt{f^c}{b}^w{d}^t+V^d+{\displaystyle \frac{V^i{M}^{\mathrm{cre}}(\beta )}{M^{max,\mathrm{ULS}}}}$$

(31)

$$V^{ci}=0.14\sqrt{f^c}{b}^w{d}^t$$

(32)

where V^d = shear force caused by the unfactored dead load (p^SW + p^SIDL); M^max,ULS = maximum moment caused by the ULS load; Vⁱ = shear force occurring with M^max,ULS; and M^cre is the bending moment causing flexural cracks, which is given as follows:

$$M^{\mathrm{cre}}(\beta )={\displaystyle \frac{I^c}{y^u}\left[0.5\sqrt{f^c}+{\sigma }^{t,e}(\beta )-{\sigma }^d\right]}$$

(33)

where σ^t,e = concrete compressive stress caused by the effective prestress force P^t,e(β); and σ^d = concrete tensile stress caused by the unfactored dead load, both evaluated at the outermost lower fiber.

Corrosion Propagation over Time

When corrosion propagates in the prestressed tendon, it might cause a significant loss of cross-sectional area and therefore of structural capacity. A conceptual relationship between the deterioration level caused by corrosion and the structure service life is shown in Fig. 6. The initiation phase is the required time for carbon and/or chloride to go through the concrete cover and reach the reinforcement. An empirical expression of the corrosion rate for concrete reinforcement is taken from (Al-Mosawe et al. 2022; Val and Melchers 1997; Stewart and Rosowsky 1998)

$$i_0^{\mathrm{cor}}={\displaystyle \frac{37.8{(1-w/c)}^{-1.64}}{\mathrm{cover}}[\mu \mathrm{A}/\mathrm{c}{\mathrm{m}}^2]}$$

(34)

where $i_0^{\mathrm{cor}}$ = corrosion current density at the start of propagation t^p, which is a function of the concrete water per cement ratio w/c. The concrete cover is 50 mm for the Petrula Bridge case (Anania et al. 2018). The variation in time of the corrosion current density i^cor(t) during propagation is

$$i^{\mathrm{cor}}(t)=i_0^{\mathrm{cor}}0.85{(t)}^{-0.29}[\mu \mathrm{A}/\mathrm{c}{\mathrm{m}}^2]$$

(35)

where t = time that has passed since the start of corrosion initiation. Based on Anania et al. (2018), the corrosion in the tendon is characterized by a uniform rust distribution with a rate λ

$$\lambda =0.1163i^{\mathrm{cor}}$$

(36)

The tendon diameter D^t,t at time t is a function of the initial diameter D^t,0 and the corrosion rate λ

$$D^{t,t}(t)=\{\begin{array}{cc}{D}^{t,0},& t\le t^p\\ D^{t,0}-2\lambda (t-t^p),& t^p<t\le t^{\mathrm{end}}\\ 0,& t>t^{\mathrm{end}}\end{array}$$

(37)

where t^p = time at the start of corrosion propagation. The time of the expected end of service t^end is when there is no remaining tendon cross-sectional area

$$t^{\mathrm{end}}={\displaystyle \frac{D^{t,0}}{2\lambda }+t^p}$$

(38)

The prestress tendon area percentage reduction β(t) is expressed as follows:

$$\beta (t)=1-{\displaystyle \frac{A^{t,\mathrm{final}}}{A^{t,\mathrm{initial}}}=1-{\displaystyle \frac{(\pi /4){(D^{t,t}(t))}^2}{(\pi /4){(D^{t,0})}^2}}}$$

(39)

where A^t,final and A^t,initial = final and initial cross-sectional tendon areas, respectively.

Response Control through EAT in the Prestressed Concrete Bridge (Petrulla)

An illustration of the Petrulla bridge retrofitted with two EAT systems, each comprising n^act = 14 actuators for two adjacent I-girders, is given in Fig. 7. Following the investigation in Anania et al. (2018), key design parameters are identified. The minimum concrete cube compressive strength is set to 57.90 MPa ( $f_c^{\mathrm{\prime }}$ is 49.21 MPa for concrete cylindrical strength), while the effective prestress force P^t,e to 1,150 MPa. The cross-sectional dimensions of the I-girder and prestressed tendon are given in Table 4. Concerning the EAT system, the yield stress for strut and cable elements f^y are set to 350 and 1,470 MPa, respectively. The cross-sectional dimensions of the EAT system members, struts, and cable elements are given in Table 4. Uncontrolled and controlled displacement responses under characteristic loading conditions are illustrated in Fig. 8. The displacement caused by the permanent loads is assumed to be compensated by tensioning the EAT system cable through an external actuation system (e.g., jacking). The tension force required in the cable to reduce completely the displacements caused by the permanent loads is 7,222 kN. Cambering causes an initial compression state in the struts, which must be considered when sizing the actuator housing elements [Eq. (12)]. Concerning the state under vehicle load, the maximum control effort is required to compensate for the effect of a uniformly distributed load plus tandem load p^LLa applied in the midspan, as shown in Fig. 8(c). In this case, the maximum required force for the actuator close to the midspan is 12,622 kN with a Δl of 38 mm. The control forces could be reduced by considering additional actuators. That being said, modern electrohydraulic actuators can apply forces up to 22,000 kN (VHT 2022).

Element	Cross section	Inertia (mm4)	Area (mm2)
Petrulla
Beam bridge	I-beam	4.62 × 1011	6 × 103
	Prestress tendon diameter 37.4 mm	—	1 × 103
EAT system	Strut member, circular hollow section 92 × 9.2 mm	3.51 × 106	6 × 103
	Cable member diameter 110 mm	7.12 × 106	9 × 103
Toutle River
Tie chords	I-beam	1.52 × 1011	2 × 105
Arch	Rectangular hollow section	5.08 × 1010	2 × 105
Hanger	Circular bar	7.12 × 106	9 × 103
Fleher
Steel orthotropic deck	See Fig. 25(c)	7.38 × 1012	1 × 106
Concrete deck	See Fig. 25(d)	8.46 × 1013	4 × 107
Cable stay, 6 per row	Diameter 110 mm, see Fig. 25(b)	7.12 × 106	9 × 103
Pylon	Rectangular hollow section 5 m × 5 m × 1 m	3.08 × 1013	5 × 106

Due to a lack of data concerning the load that caused the collapse, lower and upper bound estimates are evaluated based on SLS and ULS, respectively. Since full prestressing is considered, when the SLS limit exceeds, the concrete subjected to tensile stress during service will likely develop cracks. This might cause corrosion of the prestressed tendon, resulting in a reduction of the cross-sectional area, which could lead to collapse. The concrete lower fiber tensile stress σ^SLS,l is 4.7 MPa [Eq. (22)], which exceeds the limit recommended in ACI-318-19 (ACI 2022) (4.3 MPa). The concrete upper fiber compressive stress σ^SLS,u1 and σ^SLS,u2 are 15.3 and 25.2 MPa [Eq. (21)], which are lower than the limits recommended in ACI-318-19 (ACI 2022) (22.3 and 29.7 MPa, respectively). The exceedance in tensile stress is likely to cause and/or worsen corrosion propagation. The evaluation of the ULS for prestressed concrete for uncontrolled and controlled states is plotted against the corrosion rate in Fig. 9.

This evaluation is carried out by reducing the prestress tendon force iteratively, simulating the effect of corrosion. Results are compared with those of Anania et al. (2018) concerning the moment capacity. According to Anania et al. (2018), who adopted nonlinear FEM analysis, no convergence was possible when the tendon area is reduced by 61%. This is in good accordance with the simulation carried out in this work, which indicates failure when the prestress tendon area is reduced by 50% under ULS and by 71% under SLS. This variance is likely due to differences in model fidelity. The benchmark in (Anania et al. 2018) accounts for material nonlinearity allowing stress redistribution within the element as the concrete reaches a plastic state. In contrast, the assessment in this work keeps mechanical properties in the elastic state, resulting in a more conservative evaluation. When the EAT system is employed, the moment M^ULS and shear V^ULS under ULS are reduced by 61% and 42%, respectively. This response reduction enables a loss of up to 80% in prestress tendon cross-sectional area. Concerning SLS, a loss of up to 90% could be tolerated using the EAT system, as shown in Fig. 10. Assuming a tolerance of 80% loss in prestress tendon cross-sectional area, the service life could be extended by 12 years or 40% from the recorded time of failure. The service life extension quantification is carried out using the corrosion rate model stated in Eqs. (34)–(38). The start of corrosion propagation is obtained through backward calculations from Eqs. (34) to (38). Knowing that the bridge collapsed after 30 years in operation when the prestress cross-sectional area was reduced by 50%, it can be estimated that corrosion propagation t_p initiates at year 18.

Mitigation of Fatigue-Induced Damage in Steel Tied-Arch and Cable-Stayed Bridges

Mitigation of fatigue-induced damage through active control is tested on two types of bridges: tied-arch and cable-stayed. Toutle River is a tied-arch highway bridge located in Washington, DC, with further specifications given in Table 3. Fig. 11 shows a perspective view of the bridge and cracks in the tie chords located at the fillet welds. Figs. 12(a and b) show a clear correspondence between the crack location and the truck position. Most cracks developed due to the truck crossing at one-quarter and three-quarters of the span, which are the positions of the antinodes (i.e., points of maximum amplitude) for the first vibration mode.

Fleher is a cable-stayed highway bridge located in Düsseldorf, Germany. Fig. 13 shows a photograph of the bridge, with further specifications given in Table 3. Details of fatigue cracks that developed in the orthotropic steel deck can be found in Straßen (2019). The causes of such extensive damage have been attributed to vehicle-induced vibrations, inadequate material quality, and complex connection details in the built-up orthotropic deck.

Serviceability Limit State

The SLS for cable-stayed and tied-arch bridges, as well as generally stiffness-governed structures, is based on deflections and vibration limits. Response acceleration limits for vehicular bridges are categorized as perceptible and unpleasant for the user. The perceptible vibration limit is a function of the natural frequency f^v and is set to $0.5\sqrt{f^v}\mathrm{m}/{\mathrm{s}}^2$ (Nassif et al. 2011). The unpleasant vibration limit is taken as 1.3 m/s² (Nassif et al. 2011). The deformation limit is L/800, as in AASHTO (2020).

Ultimate Limit State

The ULS is not a critical limit state for stiffness-governed structures such as the cable-stayed and tied-arch bridges considered in this study. However, it was not possible to retrieve the structural element cross-sectional dimensions for both bridges from Roeder et al. (1998) and Straßen (2019). For this reason, the ULS criteria for bridge steel elements, as in AISC 360-22, are employed for element sizing and fatigue analysis. The ULS assessment includes evaluating the response in tension, compression, bending, and shear and the interaction between bending and axial forces. Further details are given in the Appendix.

Fatigue Limit State

Based on the damage assessment in Leland et al. (2002) and Straßen (2019), for both Fleher and Toutle bridges, the governing limit state is fatigue. Fatigue causes deterioration of the material mechanical properties induced by fluctuating stress. The repeated application of elastic stress below the yield value can cause significant deterioration through cracks, pitting, and deformations. The fatigue limit state is evaluated based on Stephens and Fuchs (2001), Russo et al. (2016), and Cui and Wang (2013). The dynamic response under loading is computed using time-history analysis. The fatigue assessment is based on the stress time-history domain. Rain-flow counting is employed to determine the governing stress range by identifying the closed cycle in the stress–strain hysteresis curve (ASTM 2017). The stress–life curve (S–N curve) is employed to determine the number of cycles for fatigue failure under cycling loading (AASHTO 2020). The fatigue life depends on several factors such as connection types and geometrical discontinuities of the structural members. In this work, the built-up members are considered in Category E, as described in Leland et al. (2002), for connections with fillet and groove welds that lack sufficiently smooth geometric transitions (ANSI/AISC 360-22, ANSI/AISC 2022).

Two fatigue limit states are considered: fatigue infinite service life (FLSI) and fatigue finite service life (FLSII). FLSI evaluates the bridge performance under the influence of a single truck per lane with a requirement to contain the stress below the constant amplitude fatigue limit (CAFL). FLSI is evaluated as follows:

$${\sigma }^{\mathrm{FLSI},rf}\le {\sigma }^{\mathrm{CAFL}}$$

(40)

where σ^FLSI,rf = stress range obtained from rain-flow counting under the FLSI load combination and σ^CAFL is 31 MPa (Category E). FLSII involves quantifying the fatigue-induced damage accumulation, which is obtained using the Palmgren–Miner model (Cui and Wang 2013)

$$D^{\mathrm{FLS}}=\sum _{i=1}^n{\displaystyle \frac{n_i^{\mathrm{FLSII},rf}}{n_i^{sn}}}$$

(41)

where $n_i^{\mathrm{FLSII},rf}$ = number of cycles for the ith stress level obtained from rain-flow counting under the FLSII load combination; and $n_i^{sn}$ = corresponding number of cycles to failure. If damage state D = 1, failure is considered to be caused by fatigue. The number of cycles $n_i^{sn}$ to failure for the ith stress level is obtained as follows:

$$\begin{array}{rl}& n_i^{sn}={\displaystyle \frac{C^{FL}}{{({\sigma }_i^{\mathrm{FLSII},rf})}^3},\mathrm{if}{\sigma }_i^{\mathrm{FLSII},rf}\ge {\sigma }^{\mathrm{CAFL}}}\\ & n_i^{sn}=\mathrm{\infty },\mathrm{if}{\sigma }_i^{\mathrm{FLSII},rf}<{\sigma }^{\mathrm{CAFL}}\end{array}$$

(42)

where C^FL = constant; and σ^FLSII,rf = stress range obtained from rain-flow counting under the FLSII load combination. For Category E, C^FL is 3.61 × 10¹¹ MPa³. Following Tallin and Petreshock (1990), the probability density of the normalized gross vehicle weight (GVW) with respect to the reference truck HS-20, denoted PDF^GWV, is assumed to be a combined lognormal distribution (lognormal–lognormal) comprising heavy-weight and lightweight vehicles

$$\mathrm{PD}{\mathrm{F}}^{\mathrm{GWV}}=p^{\mathrm{ocr},L}\mathrm{PD}{\mathrm{F}}^L+(1-p^{\mathrm{ocr},L})\mathrm{PD}{\mathrm{F}}^H$$

(43)

where p^ocr,L = occurrence probability of lightweight vehicles; and PDF^L and PDF^H = probability density functions for lightweight and heavy-weight vehicles, respectively. Table 5 gives the estimated traffic lognormal distribution parameters (Tallin and Petreshock 1990).

Log normal	Median (μPDF)	Lognormal standard deviation (δPDF)	Probability (pocr)
Lightweight vehicle PDFL	0.39	0.255	0.598
Heavy-weight vehicle PDFH	0.84	0.172	0.402

The distribution of both normalized vehicle weight and velocity is given in Fig. 14. Since no data were reported in Leland et al. (2002) and Roeder et al. (1998), all lanes are assumed to have equal probability distribution in terms of vehicle weight and velocity. The vehicle velocity is assumed normally distributed with a mean and standard deviation of 83.7 km/h and 11.26, respectively (FHWA 2019). The correlation between vehicle weight and velocity is not considered in this study due to the lack of available additional traffic statistics.

The estimated increasing vehicle volume per day is taken from Leland et al. (2002) and Roeder et al. (1998). Based on Leland et al. (2002), in 1969, an average of 20,000 vehicles crossed the bridge daily. A linear trend, with an annual increment of 420 vehicles, was inferred from monitoring data from Roeder et al. (1998). Concerning the cable-stayed bridge, the average daily number of crossings is 75,000 (Baukunst-nrw 2024). Since no traffic volume trend is available, the yearly increase is assumed to be the same as that for the tied-arch bridge.

Vibration Control through Active Hangers in Toutle River Tied-Arch Bridge

It is assumed that linear actuators are retrofitted in the hangers. The bridge dimensions are extracted from Leland et al. (2002) and Roeder et al. (2000) and illustrated in Fig. 15, where the actuators are indicated with thick lines. The cross-sectional properties of the bridge elements are given in Table 4. Since most data regarding cross-sectional dimensions of the element are incomplete, they are obtained to satisfy ULS and are given in Table 8. The deck consists of the main chords (tie chords) and cross beams. Based on Leland et al. (2002) and AASHTO (2020), the detail categories for the tie chords are of Type “E.” To simulate a condition comparable to that determined in the damage assessment (Leland et al. 2002), it is assumed that fatigue and serviceability states are not met since the stress in the tie chords exceeds the CAFL prescribed for Category E (AASHTO 2020) (31 MPa). An illustration of the uncontrolled and controlled displacement responses under characteristic loading conditions is given in Fig. 16. The natural frequency is estimated as 1.6 Hz (period of 0.6s), and the first vibration mode is illustrated in Fig. 17. Three critical points are identified for the tie-chord members: (1) quarter span, (2) midspan, and (3) third-quarter span (Fig. 12). The critical locations for the arch where the stress range is the highest are also indicated in Fig. 17.

The load combinations for the fatigue limit states are defined in Table 2 and reported in Fig. 16. It is assumed that one truck is crossing per lane with a 10 s interval between consecutive crossings, allowing the bridge response to dissipate. Concerning FLSI, the stress response for the upper and lower fibers of the cross section of the tie-chord members at the critical points is shown in Figs. 18 and 19, respectively. Both uncontrolled and controlled responses are given. Similarly, the fatigue limit state is evaluated for the arch element. The peak stress response plot for the arch member at the critical locations is given in Fig. 20. Table 6 gives the stress range obtained from rain-flow counting for the uncontrolled and controlled states. While the response can be generally reduced at the critical points through active control, it could also cause a relative stress increase in other locations compared to the uncontrolled state. Since the structural configuration is internally statically indeterminate, the effect of the actuator actions propagates through all elements. An average 26% reduction in stress in the tie chords generates a 1.6× increase compared to the stress range in the controlled state for the arch members, from 17.86 to 30.21 MPa. However, the actuator-induced stress range is kept lower than the CAFL (31 MPa). To achieve so, the actuator force must be limited to 1,126 kN. Further information concerning the actuation system is given in Table 7.

Location	Maximum rain-flow cycle stress range (MPa)		Reduction (%)
	Uncontrolled	Controlled
Toutle River
Tie chords, lower fiber
¼ span	39.01a	28.34	26
½ span	24.74	18.74	24
¾ span	37.61a	28.10	25
Tie chords, upper fiber
¼ span	38.90a	26.97	31
½ span	21.44	16.33	24
¾ span	38.41a	28.21	27
Arch elements
Critical locations (Fig. 17)	20.21	30.87	−11
Fleher
Lower fiber	41.01a	30.25	26.23
Upper fiber	31.52a	22.76	27.79

a

Exceeding the stress limit for FLSI (31 MPa), AASHTO “Category E: welded built-up members.”

Parameter	Toutle River	Fleher
Actuator force	1,126 kN	Precamber: 5,579 kN
		Traffic load: 2,366 kN
		Total: 7,945 kN
Stroke	25 mm	Precamber:30 mm
		Traffic load: 24 mm
		Total: 54 mm
Velocity	18 mm/s	46 mm/s
Available electrohydraulic actuators	Max. 2,000 kN, velocity of 80 mm/s (VHT 2022)	Max. 8,000 kN, velocity of 40 mm/s (VHT 2022)

Concerning FLSII, the traffic probability distribution parameters in Table 5 were taken, considering 30,000 crossings. Fig. 21 shows the stress response for a 300-vehicle crossing sample. All trucks are assumed to cross the bridge with a velocity of 80 km/h. Fig. 22 shows (1) the uncontrolled and controlled stress response at the critical locations for the arch and (2) the variation of the maximum required actuation force, which reaches 593 kN in compression.

The fatigue life at the critical points in the tie chords is calculated using the Palmgren–Miner model (Mouritz 2012) and Monte Carlo simulation. The fatigue damage accumulation curve plot is given in Fig. 23(a). After 30 years in operation, damage accumulation is estimated to be up to 44% . This figure matches well with that provided in the reference damage assessment (Roeder et al. 1998), which estimates it at 52%. Damage accumulation for the controlled state indicates a significant service life extension up to and above 75 years because most of the stress range obtained from rain-flow counting is reduced below the CAFL threshold of 31 MPa (Category E). As observed for FLSI, a greater damage accumulation occurs for the arch members since the stress at the critical locations increases in the controlled state [Fig. 23(b)]. However, the increased damage accumulation due to active control is not critical for the arch elements and the service life is still expected to be greater than 75 years. The peak response plots in service (SLS) in terms of acceleration and displacement for the uncontrolled and controlled states, considering 30,000 vehicles crossing, are given in Figs. 24(a and b), respectively. Peak values are indicated in Table 8. A significant mitigation of the response can be obtained through the active hangers. Referring to Figs. 24(c and d), in the uncontrolled state, the acceleration response is above the unpleasant and perceptible thresholds in the 98th and 93rd percentiles, respectively. Through active control, the controlled acceleration response is reduced by 95% from the uncontrolled state so that is always below the perceptible threshold. The maximum displacement is also reduced by 39% through active control.

Response	Uncontrolled	Controlled	Reduction (%)
Toutle River
Displacement	44 mm	27 mm	39
Acceleration	2.4 m/s2a	0.12 m/s2	95
Fleher
Displacement	668 mm	457 mm	32
Acceleration	0.585 m/s2b	0.188 m/s2	68

a

Exceeding unpleasant vibration limit of 1.3 m/s².

b

Exceeding perceptible threshold of 0.37 m/s².

Vibration Control through Active Stays in the Fleher Cable-Stayed Bridge

It is assumed that linear actuators are retrofitted in the stays. Unlike the previous case study, this bridge features an orthotropic deck with complex connections, as shown in Fig. 25. The data concerning sizing and reinforcement of the deck elements and pylon cannot be sourced. Element sizing is carried out to comply with ULS as recommended in ANSI/AISC 360-22 (ANSI/AISC 2022). The cross-sectional dimensions are given in Table 4. An illustration of uncontrolled and controlled displacement responses under characteristic loading conditions is given in Fig. 26. The natural frequency is estimated as 2.7 Hz (period of 0.37 s). The first vibration mode is illustrated in Fig. 27. The critical point for analysis is indicated with a circle mark. This is the location for the highest stress range occurrence under all load combinations.

Following AASHTO (2020), an additional load factor of 1.5 was considered to estimate the fatigue limit state for the orthotropic deck [Fig. 26(f)]. It is assumed that the primary welding connection for the orthotopic deck elements is Category type E as in the previous case study. The combination applied to the truck moving load for the fatigue limit state is detailed in Fig. 26. A vehicle speed of 100–120 km/h is considered as per the records in Gericht stoppt Fleher-Brücke-Blitzer (2014). The peak stress response plot in the uncontrolled and controlled states for FLSI is given in Fig. 28. Table 6 gives the stress range obtained from rain-flow counting for the uncontrolled and controlled states. Through active control, a reduction of up to 27% in stress for the orthotropic deck can be achieved. This effectively reduces upper and lower fiber stress ranges below the CAFL threshold, thereby providing a significant fatigue life extension. Actuator specifications are given in Table 7. Concerning FLSII, traffic parameters are based on those in Table 5 and Fig. 14. A crossing of 30,000 trucks per lane is considered, with the axle load for each truck and velocity determined using the probability density given in Table 5. Fig. 29 shows the stress response for a sample sequence of 300 trucks per lane with a uniform velocity of 100 km/h. Through active control, the peak stress in the upper fiber can be reduced by 30% and that in the lower fiber by 60%. Fig. 30 shows the plot of the fatigue service life. The orthotropic deck fatigue life is calculated using the Palmgren–Miner model (Mouritz 2012). A service life of 50 years is estimated without active control. Conversely, for the controlled state, the fatigue life can be extended beyond 75 years since most of the stress range obtained from rain-flow counting is reduced below the CAFL threshold (Category E).

The peak response plot in service (SLS) in terms of displacement and acceleration for the uncontrolled and controlled states considering 30,000 vehicles crossing is shown in Figs. 31(a and b), respectively. Peak values are indicated in Table 8. The response is significantly mitigated through active control via the stays. The displacement is reduced by 32% from the uncontrolled state. The acceleration in the uncontrolled state is above the perceptible threshold in the 99th percentile [Figs. 31(c and d)]. Through active control, the acceleration is reduced by 68% so that it is always below the perceptible threshold. The analysis done in this study employed a relatively low-fidelity model (2D beam elements 3dofs/node). Results showed that the upper fiber of the orthotropic deck is not as important as the lower fiber. However, crack propagation has also occurred at the upper fiber, according to the inspection documented in Baukunst-nrw (2024). This suggests that a more detailed analysis using shell and 3D beam elements to model more accurately connections and a transversal analysis within the orthotropic deck could be carried out to validate the findings of this study.

Discussion

Conclusion

Appendix. SLS Stress Derivation

Data Availability Statement

Acknowledgments

Notation

$\mathbf{A}$

system matrix (state-space);

A^c

concrete cross-sectional area;

A^r

longitudinal rebar cross-sectional area;

A^t

prestress tendon cross-sectional area;

A^v

steel cross-sectional shear area;

a^c

concrete equivalent compressive height;

$\mathbf{B}$

control matrix;

b^w

concrete cross-sectional web width;

$\mathbf{C}$

damping matrix;

C^FL

fatigue constant;

C^t,A, C^t,B, C^t,C

constants to compute the prestress tendon strain;

c^c

concrete compressive height;

$\mathbf{D}$

input matrix;

D^FLS

fatigue damage accumulation;

D^t

prestress tendon diameter;

D^t,0

initial prestress tendon diameter;

D^t,t

remaining prestress tendon reduced diameter due to corrosion;

${\mathbf{d}}^{\mathrm{act}}$

displacement compensation achieved through actuation;

${\mathbf{d}}^{\mathrm{load}}$

displacement response caused by the external load;

d^t

distance of the prestress tendon centroid to the outermost compressive fiber;

E^c

concrete Young modulus;

e^t

prestress tendon eccentricity with respect to the girder centroid;

F^c

concrete equivalent force (cross section);

F^s

longitudinal rebar force (cross section);

F^t

prestress tendon force (cross section);

f^c

concrete cylindrical compressive strength after 28 days of curing;

f^v

structural fundamental frequency;

f^y

steel yield strength;

$\mathbf{I}$

identity matrix;

I^c

concrete second moment of area;

i^cor

corrosion current density;

$\mathbf{K}$

stiffness matrix;

k

prestress tendon index;

$\mathbf{M}$

mass matrix;

M^cre

bending moment causing flexural cracks;

M^R

bending moment resistance;

M^ULS

bending moment caused by the ULS load combination;

M^ULS,max

maximum moment caused by external loads from ULS combination;

$\mathbf{N}$

shape function matrix;

N^R

axial resistance;

N^ULS

axial force caused by the ULS load combination;

n^sn

fatigue life expressed in the number of cycles obtained from the S–N curve;

P^t,e

prestress tendon effective force;

PDF^GVW

probability density function for a gross weight vehicle;

PDF^H

probability density function for a heavy-weight vehicle;

PDF^L

probability density function for a lightweight vehicle;

p^act

actuator load;

p^ca

camber load;

p^LLa

live load UDL and tandem load;

p^LLb

fatigue load, HS-20 (20-t three-axle highway semitrailer);

p^p

permanent load;

p^SIDL

superimposed-dead load;

p^SW

self-weight load;

p^t

tendon prestress load;

p^tot

total load;

$\mathbf{Q}$

response weighting matrix (LQR);

$\mathbf{R}$

actuation weighting matrix (LQR);

$\mathbf{T}$

transformation matrix;

t

time;

V^cw

pure shear resistance;

Vⁱ

ULS factored shear caused by external loads corresponding to M^ULS,max_;

V^R

shear resistance;

V^R,c

shear resistance from concrete cross-sectional contribution;

V^R,s

shear resistance from transversal rebar contribution;

V^ULS

shear caused by the ULS load combination;

w/c

concrete water per cement ratio;

y^low

distance of the cross-sectional centroid to the lower outermost fiber;

y^up

distance of the cross-sectional centroid to the upper outermost fiber;

$\mathbf{z}$

state vectors;

β

prestress tendon cross-sectional area percentage loss;

δ^PDF

lognormal standard deviation;

${\epsilon }^{t,d}$

prestress tendon decompression strain;

${\epsilon }^{t,f}$

prestress tendon final strain;

${\epsilon }^{t,i}$

prestress tendon initial strain;

${\epsilon }^{t,t}$

prestress tendon total strain;

λ

corrosion rate;

μ^PDF

lognormal median;

σ^d

concrete tensile stress caused by the unfactored dead load (p^SW and p^SIDL) at the outermost fiber;

σ^FLS,rf

cyclic stress range obtained from rain-flow counting under FLS load combination;

σ^FLS,R

fatigue stress amplitude limit;

σ^SLS,l

stress under SLS load combination in the lower outmost fiber;

σ^SLS,u

stress under SLS load combination in the upper outmost fiber;

σ^t,e

concrete compressive stress caused by the effective prestress force P^t,e at the outermost compressive fiber;

ϕ^M

bending moment resistance factor;

ϕ^N

axial resistance factor;

ϕ^V

shear resistance factor; and

$\mathbf{\Delta }{\mathbf{l}}^{\mathrm{act}}$

length changes of actuators.

References