*3.3. Modulus of Elasticity and Durability*

The secant modulus of elasticity in the range 0 to 40% of the ultimate strength (according to EN 1992-1-1 [49] standard) is assumed 22,890 MPa (mean value of the modulus of elasticity, see Ambroziak et al. [30]) with regard to structural analysis of old arch concrete bridge. The ASTM C469 M standard [50] guideline was used to determine the modulus of elasticity. Diamond-drilled concrete cores with a length to diameter ratio of 1.50 were used in a compressometer device to measure the static modulus of elasticity. The regulations [36], applicable in time of the bridge erection indicated that the concrete compressive strength greater or equal to 13.73 MPa corresponds to the modulus of elasticity 14,715 MPa (150,000 kg/cm<sup>2</sup> ). The modulus of elasticity assumed for investigated old concrete is 1.55 times greater than the value specified in regulations [36], possibly due to various aggregates types in concrete mixes.

Based on the early experimental investigation performed by Ambroziak et al. [30] it can be concluded that the 95-year-old concrete has good freezing resistance. The chloride content of the old concrete did not exceed 0.2% by mass of cement; thus, the old concrete arch bridge was not exposed to chloride attack. The pH values for the old concrete indicated that there is no corrosion of the steel rebar's. Nevertheless, the old concrete has large variations in depth of the carbonated zone ranging from 20 to 55 mm. Despite the large depth of the carbonated zone, the pH of the old concrete is still in the safety range.

### **4. Structural Analysis**

### *4.1. Description of FEM Model*

The three-dimensional finite element model of the 95-year-old concrete arch bridge is built (see Figure 7), followed by numerical calculations and structural analysis. The SOFiSTiK structural engineering system is applied in numerical calculations. This software is frequently used in the design and analysis of bridge structures [51–55]. In the present structural analysis, the traffic loads class C are assumed, according to standard PN-S-10030 [56]. This standard was applicable in Poland during the reconstruction design process of the old bridge. The traffic load is composed of q and K types of loads, see Figure 8. The load values of q and K type are equal to 2 kPa and 400 kN (8.P = 8.50 kN = 400 kN), respectively. The distribution of q load (arbitrary distribution) and location of K load have to produce the largest responses in analyzed structural elements. Besides this traffic load, permanent loads are also imposed (dead-weight and bridge equipment's loads). The shell (SH3 D) and 3 D beam (B3 D) finite elements (FE) are applied in the FEM model of the arch bridge. The finite elements adopted in this study are 4-node isoparametric shell finite elements of DKMQ type and 2-node 3 D beam elements of the Timoszenko type, C0 class with linear shape functions. The mesh independence study of the bridge finite element model is carried out to ensure that the results of an analysis are not affected by changing the size of the mesh. The FEM model of the old concrete arch bridge is meshed by 45,075 shell elements (modeled slab and concrete arch) and 776 3 D beam elements (girders), the model includes 42,223 nodes with and 940 support constraints (Figure 7). It should be noted, that the authors' experiences in the design and monitoring of bridge structures are also exploited during the construction of the FEM model [57,58]. It can be pointed, that more detailed numerical models are created to investigate the individual structural elements of bridges and comparisons them with laboratory tests [59,60].

**Figure 8.** Traffic loads class C (q and K types) according to standard PN-S-10030 [56].

### *4.2. Results of Structural Analysis*

The numerical analysis allows to evaluate the internal forces acting on the sections and specifically their largest values, which in turns make it possible to define the geometry of the members and later to proceed with the verifications at the Serviceability Limit State (SLS) under the service loads, and at the Ultimate Limit State (ULS) under the ultimate loads. The maps of bending moments in the deck slab and the concrete arch under the dead weight of the structure and loads of the bridge equipment are determined in Figures A1 and A2 respectively. On the other hand, for the moving loads q and K of class C (see Figure 8), the envelope maps of characteristic values of bending moments in the deck slab (see Figure A3) and the envelope maps of characteristic values of forces in the concrete arch (see Figure A4) are presented. In Figures A5 and A6 the characteristic values of bending moments for inner and outer girders are shown. The largest bending moments and axial forces at the chosen, critical cross-sections of a superstructure correspond to the most unfavorable combinations of permanent and live loads according to PN-S-10030 [56], these values are presented in Table 4. The lack of symmetry in the results of internal forces between the right and left span girders results from different spans (see Figure 1). The results for the outer girder and the inner girder are similar in the longer span (right span, see Table 4) where they have a similar height, while in the shorter span (left span) the results differ significantly between the outer girder and inner girder because the girders have different constructional heights.

The site inspection and inventory of old arch bridge make it possible to assess the amount of reinforcement in structural elements, see Table 5. Concrete detector for rebar localization, depth measurement, and size estimation are necessary for the proper estimation of steel reinforcements in the existing old concrete bridge. New technologies, methods, and systems are still developed for the detection of steel rebars in concrete structures [61–64].


**Table 4.** Characteristic and design value of internal forces under designed loads.

**Table 5.** Materials characteristic and calculated extreme stresses in concrete and steel rebar's.


The design yield strength of reinforcement (*f* yd) and design compressive strength of old concrete (*f* cd) are determined according to specified characteristic yield strength of reinforcement (*f* yk), characteristic compressive strength of concrete (*f* ck), partial safety factors for reinforcement (*γ*s) and concrete (*γ*c) material properties. The partial safety factors *γ*<sup>s</sup> and *γ*<sup>c</sup> are assumed according to Table NA.2 guidelines in National Annex to PN-EN 1992-1-1 [65]. The characteristic yield strength of steel rebars and the characteristic compressive strength of concrete are specified in laboratory tests.

The largest stresses in reinforced steel and old concrete in representative structural elements of the arch bridge are determined and collected in Table 5. The highest stress in steel rebar's equals 145.6 MPa, approximately equal to 57% of the design yield strength of reinforcement, *f* yd = 257 MPa, see Table 5. The specified maximal compressive stress in old concrete arch bridge is 9.3 MPa, approximately equal to 96% of the design compressive strength of concrete *f* cd = 9.7 MPa.

The largest displacement under the SLS combination of the traffic loadings (see Figure A7) is equal to about 1.3 mm. According to the assumptions of PN-S-10042 [66] standard, the permissible deflection of reinforced concrete beam elements in continuous systems (in this case the largest spans of the analysed bridge, see Figures 1a and A7) is *f* perm = *L*/1000 = (6380 + 2·250)/1000 = 6.88 mm. Determined largest deflections fulfils the SLS condition *f* max = 1.32 mm < *f* perm = 6.88 mm.

Structural analysis indicated that the 95-year old concrete arch bridge structure satisfies SLS and ULS requirements under the traffic loads class C according to PN-S-10030 [56] standard. Sometimes the car traffic on old bridges is not permitted and after reconstructions are able to meet the requirements for footbridges only [67]. The structural analysis confirmed the validity of the adopted design solutions. On top of the existing old deck structure, a new deck slab with pavement covers is designed and the entire surface of the old concrete is protected with repair mortars. The task of the new deck slab is not only to strengthen the structure but also to increase the resistance to present environmental influences and also the shear capacity of the deck slab. The higher strength class of concrete is designed and used to fulfill the EN 206 [47] standard requirement of the present exposure classes.

All newly erected bridges and old rebuilt bridges need a detailed assessment and dynamic analysis. In this paper, this analysis is not included but the dynamic analysis is performed to properly assess the properties of the 95-year-old arch concrete bridge. It should be noted, that improper design of bridge structure cause the risk of excessive structural vibrations throughout the operation [68–73].

### **5. Conclusions**

A 95 year-old concrete arch bridge in Jagodnik (northern Poland) is examined in this paper, from the properties of the materials—by testing samples in a lab—to the structural behavior—by finite-element modeling, with the following results:


This paper provides scientists, engineers, and designers with experimental and structural assessments of the 95-year-old concrete. The study confirms the old arch bridge ability to carry newly designed traffic loads and helped give him a new structural life and extend his working life. Lots of existed old and historic bridge structures require restoration and reconstruction, which require not only a sound financial plan and the expertise of civil engineers, but—more often than not—the expertise of the scientific community, for an

appropriate assessment of structural safety. Old bridge structures should return like the Phoenix from their ashes and recovery their past appearance. The protection of old bridges coincides with the preservation of their cultural heritage.

**Author Contributions:** Conceptualization, A.A.; methodology, A.A. and M.M.; validation, A.A. and M.M.; formal analysis, A.A. and M.M.; investigation, A.A. and M.M.; resources, A.A. and M.M.; data curation, A.A. and M.M.; writing—original draft preparation, A.A. and M.M.; writing—review and editing, A.A.; visualization, A.A.; supervision, A.A.; project administration, A.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All laboratory test results (data) are included in Table 1 in the present paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A FEM Calculation Results**

The chosen results of internal forces used in the verification and design process (see Table 4) are presented in Figures A1–A6. In Figure A7 extreme displacement under the SLS combination of traffic loadings is given.

**Figure A1.** Maps of characteristic values of bending moments *M*xx (kNm) in the deck slab: (**a**) Dead weight; (**b**) bridge equipment loads.

**Figure A2.** Maps of characteristic values of bending moments *M*xx (kNm) in the concrete arch: (**a**) Dead weight; (**b**) bridge equipment loads.

**Figure A3.** *Cont*.

**Figure A3.** The envelope maps of characteristic values of bending moments *M*xx (kNm) in the deck plate: (**a**) Max Mxx (kNm) under the moving load q of class C; (**b**) min Mxx (kNm) under the moving load q of class C; (**c**) max Mxx (kNm) under the moving load K of class C; (**d**) min Mxx (kNm) under the moving load K of class C.

**Figure A4.** *Cont*.

**Figure A4.** Envelope maps of characteristic values of forces *N*xx (kN) in the concrete arch: (**a**) Max Nxx (kN) under the moving load q of class C; (**b**) min Nxx (kN) under the moving load q of class C; (**c**) max Nxx (kN) under the moving load K of class C; (**d**) min Nxx (kN) under the moving load K of class C.

**Figure A5.** Characteristic values of bending moments *M* (kNm) under the moving load q of class C in the right span: (**a**) Outer girder; (**b**) inner girder.

**Figure A6.** Characteristic values of bending moments *M* (kNm) under the moving load K of class C in the right span: (**a**) Outer girder; (**b**) inner girder.

**Figure A7.** *Cont*.

**Figure A7.** Extreme displacement *f* (mm) under the SLS combination of traffic loadings class C according to PN-S-10030 [56] standard: (**a**) 3 D structure view; (**b**) displacement map on the deck slab.

### **References**

