# Fatigue Assessment of Prestressed Concrete Slab-Between-Girder Bridges

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## Abstract

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## Featured Application

**The results of this work can be used in the evaluation of existing prestressed concrete slab-between-girder bridges for fatigue.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Case Study Bridge

_{ck,cube}= 35 MPa) and B45 (f

_{ck,cube}= 45 MPa) for the girders. Testing of the cores taken from the deck slab resulted in an average f

_{cm,cube}= 98.8 MPa (f

_{ck,cube}= 84.6 MPa), as a result of the continued cement hydration. For the assessment calculations, we conservatively assumed that the mean compressive cylinder strength f

_{cm}= 65 MPa on the deck. The associated characteristic concrete compressive strength was f

_{ck}= 53 MPa.

#### 2.2. Live Load Models

_{Q}

_{1}× 300 kN in the first lane, α

_{Q}

_{2}× 200 kN in the second lane, and α

_{Q}

_{3}× 300 kN in the third lane [12]. For the Netherlands, the values of all the α

_{Qi}= 1, with i = 1 … 3. The uniformly distributed load acts over the full width of the notional lane of 3 m width, and it equals α

_{q}

_{1}× 9 kN/m

^{2}for the first lane, and α

_{qi}× 2.5 kN/m

^{2}for all the other lanes. In the Netherlands, for bridges with three or more notional lanes, the value of α

_{q}

_{1}= 1.15 and α

_{qi}= 1.4, with i > 1. Figure 2 shows a sketch of the live load model 1.

_{ik}for the axle loads, and 0.3q

_{ik}for the distributed lane loads. In other words, the axle load becomes 0.7 × 300 kN = 210 kN, and the load per wheel print becomes 105 kN. The distributed lane load is 0.3 × 1.15 × 9 kN/m

^{2}= 3.105 kN/m

^{2}. The fatigue load model has as a reference load of 2 million trucks per year. In the Netherlands, the guidelines for the assessment of bridges (RBK [32]) uses a higher number of passages: 2.5 million trucks per year. Over a lifespan of 100 years, the result is 250 million truck passages.

#### 2.3. Description of Experiments

_{cm,cube}= 75 MPa for the original slab in setup 1, f

_{cm,cube}= 68 MPa for the newly cast slab in setup 1, f

_{cm,cube}= 81 MPa for the first cast of setup 2, and f

_{cm,cube}= 79 MPa for the second cast of setup 2.

_{min}being 10% of the upper limit. A sine function was used with a frequency of 1 Hz. In the fatigue tests, the load was applied in a force-controlled way. If fatigue failure did not occur after a large number of cycles, the upper load level was increased (and the associated lower limit of 10% of the upper limit was adjusted as well).

## 3. Results

#### 3.1. Results of the Experiments

_{max}, the age of the concrete of the slab at the moment of testing, and the concrete cube compressive strength f

_{cm,cube}determined at the day of testing the slab.

_{max}(with P

_{max}from the static test) was given, as well as N, the number of cycles. For the variable amplitude fatigue tests, N was the number of cycles for the associated load level F/P

_{max}. After N cycles at load level F/P

_{max}, given in one row of Table 3, the test was continued with N cycles at another load level F/P

_{max}, given in the next row. The column “age” gives the age of the slab at the age of testing, and f

_{cm,cube}gives the associated cube concrete compressive strength. For fatigue tests that lasted several days, a range of ages was given in the column “age”, indicating the age of the concrete in the slab at the beginning of testing and at the end of testing. Similarly, a range of compressive strengths was given for f

_{cm,cube}, representing the strength determined at the beginning and the end of testing.

#### 3.2. Resulting Wöhler Curve

_{max}was plotted, see Figure 6. For this curve, we interpreted the variable amplitude loading tests as follows: if N

_{1}cycles at load level F

_{1}are applied, followed by N

_{2}cycles at load level F

_{2}, and then N

_{3}cycles to failure at F

_{3}, with increasing load levels F

_{1}< F

_{2}< F

_{3}, it is conservative to assume that the slab can withstand N

_{1}+ N

_{2}+ N

_{3}cycles at the load level F

_{1}, N

_{2}+ N

_{3}cycles at load level F

_{2}, and N

_{3}cycles at load level F

_{3}. This approach led to three datapoints for one variable amplitude fatigue test. As a result of this approach, we obtained 16 datapoints on the first setup and 28 datapoints on the second setup, resulting in 44 datapoints in Figure 6. The average value of the Wöhler curve is shown as “mean” in Figure 6, and it is described with the following expression, using S for the load ratio and N for the number of cycles to failure:

#### 3.3. Assessment of the Case Study Bridge for Punching

_{1}= 0.1, for C

_{Rd,c}= 0.18/γ

_{c}with γ

_{c}= 1.5, and for v

_{min}:

^{2}= 10.35 kN/m

^{2}. The contributions of the self-weight and asphalt were 25 kN/m

^{3}× 200 mm = 5 kN/m

^{2}and 23 kN/m

^{3}× 120 mm = 2.8 kN/m

^{2}, respectively. The area over which these loads were considered was the area within the punching perimeter, A

_{u}= (400 mm)

^{2}+ 4 × 162 mm × 400 mm + π(162 mm/2)

^{2}= 439,812 mm

^{2}= 0.4398 m

^{2}. The corresponding loads for the distributed lane load, self-weight, and asphalt then became 4.55 kN, 2.2 kN, and 1.23 kN, respectively, when the Eurocode wheel print was considered. For the smaller wheel print, the area within the punching perimeter became A

_{u}= 0.2613 m

^{2}, resulting in loads of 2.7 kN, 1.3 kN, and 0.7 kN, respectively, for the distributed lane load, the self-weight, and the asphalt.

_{Rd,c}, using the capacity obtained in the tests. To translate the capacity obtained in the test to a representative design capacity of the case study bridge, we had to consider the following (see Annex D of NEN-EN 1990:2002 [43]):

- The laboratory setup was a 1:2 scale of the case study bridge, resulting in a factor 2
^{2}; - Considering scaling laws, a scale factor of 1.2 [13] had to be included in the capacity;
- The partial factor derived from the experiments γ
_{T}had to be included.

_{T}. To calculate this factor, we compared the punching capacity obtained in the static experiments with the average punching stress capacity v

_{R,c}according to NEN-EN 1992-1-1:2005 [3]. The expression for v

_{R,c}was given in the background report of Eurocode 2 [44] as follows:

_{R,c}, the stress v

_{R,c}was then multiplied with u × d, where u was determined as shown in Figure 8, for the considered wheel print. Table 5 combines the experimental results V

_{exp}and the predicted capacities V

_{R,c}, as well as the ratio of the tested to predicted capacity V

_{exp}/V

_{R,c}. The average value of V

_{exp}/V

_{R,c}was 2.61, with a standard deviation of 0.296 and a coefficient of variation of 11%. This information led to the derivation of γ

_{T}as determined in Annex C of NEN-EN 1990:2002 [43]:

_{T}then becomes:

_{exp}could be scaled to the capacity of the bridge V

_{BB}as follows:

^{2}corrects for the 1:2 scale, and 1.2 is the scaling factor. The design capacity based on the test results is then:

_{BB}according to Equation (20) and V

_{BB,d}according to Equation (21), as well as the demand V

_{Ed}that corresponds to the wheel print in the experiment under consideration (see Table 4). The average value of V

_{BB,d}/V

_{Ed}= 3.06, which meant that the margin of safety was 3.23, or that the Unity Check was the inverse, UC = 0.33. When comparing this value based on the experiments to the values in Table 4, we observed the beneficial effect of compressive membrane action on the capacity of the thin, transversely prestressed concrete slabs.

#### 3.4. Assessment of Case Study Bridge for Fatigue

^{2}= 4, so that the concentrated load became 26.25 kN. The distributed lane load of the fatigue load model was 3.105 kN/m

^{2}. For the 1:2 scale model, the distributed lane load became 0.776 kN/m

^{2}.

_{eq}= 0.83 kN for a single wheel load, and F

_{eq}= 1.40 kN for a double wheel load. Then, the total load was F = 27.08 kN for a single wheel load and F = 27.65 kN for a double wheel load.

_{Rd}= 1.622. As such, the design capacity of the punching resistance with the punching perimeter around one wheel load, including the enhancing effect of compressive membrane action became 1.622 × 124.4 kN = 201.8 kN, for the most critical case (lowest capacity V

_{Rd,c}as a result of the lowest concrete compressive strength). To determine the capacity for punching with the case of a double wheel print, one could expect a double capacity. However, the results in Table 2 show that the capacity in the FAT8S2 was 1.64 times the capacity in FAT7S1. This ratio was used to determine the punching shear capacity. The capacity was now 1.64 × 201.8 kN = 331.0 kN.

_{char}= 0.348. For two wheel loads, using Equation (5) with N = 250 million cycles gave S

_{char}= 0.447. The outcome of the assessment was that the margin of safety for one wheel print was 0.348/0.134 = 2.60, or that inversely, the UC = 0.39. For the case with two wheel prints, the margin of safety was 0.447/0.167 = 2.68 or inversely UC = 0.37. Thus, the results for one and two wheel prints were very similar. The conclusion of the assessment was that based on the experimental results, we found that the case study bridge met the code requirements for fatigue.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Notations

b | width |

c | concrete cover |

d | average effective depth |

d_{l} | effective depth to the longitudinal reinforcement |

d_{t} | effective depth to the transverse reinforcement |

f_{ck,cube} | characteristic cube concrete compressive strength |

f_{cm,cube} | average cube concrete compressive strength |

f_{ck} | characteristic cylinder concrete compressive strength |

f_{cm} | average cylinder concrete compressive strength |

h | height |

k | size effect factor |

k_{1} | factor on effect of axial stresses |

l | length |

l_{span} | span length |

q_{ik} | distributed lane load |

u | punching perimeter length |

v_{min} | lower bound of shear capacity |

v_{R,c} | mean capacity for punching shear |

v_{Rd,c} | design capacity for punching shear |

A_{s,l} | longitudinal reinforcement area |

A_{sp} | area of prestressing steel |

A_{s,t} | transverse reinforcement area |

A_{u} | area within punching perimeter |

B_{Rd} | design capacity derived from statistical results of experiments |

COV | coefficient of variation |

C_{Rd,c} | constant in punching capacity equation |

DL | dead load |

DW | superimposed dead load |

F | applied load |

F_{eq} | equivalent load |

F_{min} | lower limit of the load as used in the fatigue tests |

LL | live load |

M_{dist,1wheel} | bending moment caused by distributed lane load for influence area of one wheel load |

M_{dist,2wheel} | bending moment caused by distributed lane load for influence area of two wheel loads |

N | number of cycles |

P_{max} | load at failure |

Q_{ik} | axle load of design tandem |

S | load ratio |

S_{char} | characteristic value of load ratio (5% lower bound Wöhler curve) |

U | load combination |

UC | Unity Check |

V_{BB} | average capacity of deck of Van Brienenoord Bridge based on experiments |

V_{BB,d} | design capacity of deck of Van Brienenoord Bridge based on experiments |

V_{R,c} | mean value of the punching shear capacity |

V_{Rd,c} | design value of the punching shear capacity |

V_{Ed} | design value of punching shear demand |

V_{exp} | experimental punching capacity |

α | factor that considered effect of experiments |

α_{qi} | factor on distributed lane loads |

α_{Qi} | factor on design tandem |

β | reliability index |

γ_{T} | partial factor derived from experiments |

μ | mean value of experimental results |

ρ_{avg} | average reinforcement ratio |

ρ_{l} | longitudinal reinforcement ratio |

ρ_{t} | transverse reinforcement ratio |

σ_{cp} | average axial stress |

σ_{cx} | longitudinal axial stress |

σ_{cy} | transverse axial stress |

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**Figure 1.**Van Brienenoord Bridge: (

**a**) sketch of the elevation of the entire bridge structure, showing the approach slabs, as well as the steel arch; (

**b**) cross-section of the slab-between-girder approach bridge. Dimensions in cm.

**Figure 3.**Dimensions of first 1:2 scale model: (

**a**) top view; (

**b**) cross-section view. Figure adapted from Reference [34]. Reprinted with permission. This figure was originally published in Vol. 116 of the ACI Structural Journal.

**Figure 4.**Overview of the second 1:2 scale setup: (

**a**) top view; (

**b**) cross-section view. Figure adapted from Reference [37]. Reprinted with permission. This figure was originally published in Vol. 116 of the ACI Structural Journal.

**Figure 6.**Relation between the number of cycles N and the applied load ratio F/P

_{max}in all the fatigue experiments, adapted from Reference [37]. Reprinted with permission. This figure was originally published in Vol. 116 of the ACI Structural Journal.

**Figure 7.**Relation between the number of cycles N and applied load level F/P

_{max}for (

**a**) a single wheel load; and (

**b**) a double wheel load, from Reference [37]. Reprinted with permission. This figure was originally published in Vol. 116 of the ACI Structural Journal.

Dimension | Value |
---|---|

Thickness h | 200 mm |

Concrete cover c | 30 mm |

Longitudinal reinforcement | ϕ8 mm–250 mm |

Effective depth longitudinal d_{l} | 166 mm |

Area of longitudinal reinforcement A_{s,l} | 201.1 mm^{2}/m |

Longitudinal reinforcement ratio ρ_{l} | 0.12% |

Transverse reinforcement | ϕ8 mm–200 mm |

Effective depth transverse d_{t} | 158 mm |

Area of transverse reinforcement A_{s,t} | 251.3 mm^{2}/m |

Transverse reinforcement ratio ρ_{t} | 0.16% |

Average effective depth d | 162 mm |

Average reinforcement ratio ρ_{avg} | 0.14% |

Prestressing reinforcement | 462 mm^{2}–800 mm |

Area of prestressing steel A_{sp} | 0.5775 mm^{2}/mm |

Test Number | Size Load (mm × mm) | P_{max}(kN) | Age (days) | f_{cm,cube}(MPa) |
---|---|---|---|---|

BB1 | 200 × 200 | 348.7 | 96 | 80.0 |

BB2 | 200 × 200 | 321.4 | 99 | 79.7 |

BB7 | 200 × 200 | 345.9 | 127 | 80.8 |

BB19 | 200 × 200 | 317.8 | 223 | 79.9 |

FAT1S1 | 150 × 115 | 347.8 | 94 | 82.2 |

FAT7S1 | 150 × 115 | 393.7 | 240 | 88.8 |

FAT8S2 | 2 of 150 × 115 | 646.1 | 245 | 88.6 |

Test Number | Setup | Size Load (mm × mm) | Wheel | F/P_{max} | N | Age (days) | f_{cm,cube}(MPa) |
---|---|---|---|---|---|---|---|

BB17 | 1 | 200 × 200 | S | 0.80 | 13 | 147 | 82.6 |

BB18 | 1 | 200 × 200 | S | 0.85 | 16 | 56 | 82.6 |

BB23 | 1 | 200 × 200 | S | 0.60 | 24,800 | 301 | 79.9 |

BB24 | 1 | 200 × 200 | S | 0.45 | 1,500,000 | 307–326 | 79.9 |

BB26 | 1, new | 150 × 115 | S | 0.48 | 1,405,337 | 35–59 | 70.5–76.7 |

BB28 | 1, new | 150 × 115 | S | 0.48 | 1,500,000 | 68–97 | 76.8–77.1 |

0.58 | 1,000,000 | 97–113 | 77.1–77.3 | ||||

0.70 | 7144 | 113 | 77.3 | ||||

BB29 | 1, new | 150 × 115 | S | 0.58 | 1,500,000 | 117–136 | 77.3–77.5 |

0.64 | 264,840 | 136–139 | 77.5–77.6 | ||||

BB30 | 1, new | 150 × 115 | D | 0.58 | 100,000 | 143–144 | 77.6 |

0.50 | 1,400,000 | 144–162 | 77.6–77.8 | ||||

0.58 | 750,000 | 162–171 | 77.8–77.9 | ||||

0.67 | 500,000 | 171–177 | 77.9–78.0 | ||||

0.75 | 32,643 | 177 | 78.0 | ||||

BB32 | 1, new | 150 × 115 | S | 0.70 | 10,000 | 184 | 78.1 |

0.58 | 272,548 | 185–187 | 78.1 | ||||

FAT2D1 | 2 | 150 × 115 | S | 0.69 | 100,000 | 102–144 | 82.6–84.6 |

0.58 | 2,915,123 | ||||||

0.69 | 100,000 | ||||||

0.75 | 150,000 | ||||||

0.81 | 20,094 | ||||||

FAT3D1 | 2 | 150 × 115 | S | 0.69 | 200,000 | 149–168 | 84.9–85.8 |

0.58 | 1,000,000 | ||||||

0.69 | 100,000 | ||||||

0.75 | 300,000 | ||||||

0.81 | 6114 | ||||||

FAT4D1 | 2 | 150 × 115 | S | 0.58 | 1,000,000 | 169–190 | 85.8–86.8 |

0.69 | 200,000 | ||||||

0.75 | 100,000 | ||||||

0.81 | 63,473 | ||||||

FAT5D1 | 2 | 150 × 115 | S | 0.71 | 10,000 | 192–217 | 91.6–89.6 |

0.51 | 1,000,000 | ||||||

0.61 | 100,000 | ||||||

0.66 | 1,000,000 | ||||||

0.71 | 1424 | ||||||

FAT6D1 | 2 | 150 × 115 | S | 0.71 | 10,000 | 219–239 | 89.6–88.8 |

0.51 | 1,000,000 | ||||||

0.61 | 100,000 | ||||||

0.71 | 160,000 | ||||||

0.51 | 410,000 | ||||||

0.71 | 26,865 | ||||||

FAT9D2 | 2 | 150 × 115 | D | 0.59 | 500,000 | 246–255 | 88.5–88.2 |

0.65 | 209,800 | ||||||

FAT10D2 | 2 | 150 × 115 | D | 0.63 | 100,000 | 260–284 | 90.2–91.3 |

0.56 | 1,000,000 | ||||||

0.63 | 950,928 | ||||||

FAT11D2 | 2 | 150 × 115 | D | 0.67 | 100,000 | 288–315 | 91.5–92.8 |

0.60 | 1,000,000 | ||||||

0.67 | 1,100,000 | ||||||

0.75 | 1720 | ||||||

FAT12D1 | 2 | 150 × 115 | S | 0.89 | 30 | 318 | 85.9 |

FAT13D1 | 2 | 150 × 115 | S | 0.86 | 38 | 319 | 85.8 |

Wheel Print | V_{Ed} (kN) | V_{Rd,c} (kN) | Unity Check |
---|---|---|---|

400 mm × 400 mm | 236 | 337 | 0.70 |

230 mm × 300 mm | 232 | 287 | 0.81 |

**Table 5.**Comparison between the mean predicted punching capacity and punching capacity in experiment.

Test Number | Wheel Print (mm × mm) | V_{exp}(kN) | V_{R,c}(kN) | V_{exp}/V_{R,c} |
---|---|---|---|---|

BB1 | 200 × 200 | 348.7 | 141.9 | 2.458 |

BB2 | 200 × 200 | 321.4 | 141.9 | 2.266 |

BB7 | 200 × 200 | 345.9 | 141.9 | 2.438 |

BB19 | 115 × 150 | 317.8 | 121.6 | 2.613 |

FAT1S1 | 115 × 150 | 347.8 | 124.4 | 2.795 |

FAT7S1 | 115 × 150 | 393.7 | 127.4 | 3.091 |

Test Number | V_{exp}(kN) | V_{BB}(kN) | V_{BB,d}(kN) | V_{Ed}(kN) | V_{BB,d}/V_{Ed} |
---|---|---|---|---|---|

BB1 | 348.7 | 1162.3 | 721.9 | 236.0 | 3.06 |

BB2 | 321.4 | 1071.3 | 665.4 | 236.0 | 2.82 |

BB7 | 345.9 | 1153.0 | 716.1 | 236.0 | 3.03 |

BB19 | 317.8 | 1059.3 | 658.0 | 232.0 | 2.84 |

FAT1S1 | 347.8 | 1159.3 | 720.1 | 232.0 | 3.10 |

FAT7S1 | 393.7 | 1312.3 | 815.1 | 232.0 | 3.51 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lantsoght, E.O.L.; Koekkoek, R.; van der Veen, C.; Sliedrecht, H.
Fatigue Assessment of Prestressed Concrete Slab-Between-Girder Bridges. *Appl. Sci.* **2019**, *9*, 2312.
https://doi.org/10.3390/app9112312

**AMA Style**

Lantsoght EOL, Koekkoek R, van der Veen C, Sliedrecht H.
Fatigue Assessment of Prestressed Concrete Slab-Between-Girder Bridges. *Applied Sciences*. 2019; 9(11):2312.
https://doi.org/10.3390/app9112312

**Chicago/Turabian Style**

Lantsoght, Eva O.L., Rutger Koekkoek, Cor van der Veen, and Henk Sliedrecht.
2019. "Fatigue Assessment of Prestressed Concrete Slab-Between-Girder Bridges" *Applied Sciences* 9, no. 11: 2312.
https://doi.org/10.3390/app9112312