# Finite Element Analysis of Reinforced Concrete Bridge Piers Including a Flexure-Shear Interaction Model

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## Abstract

**:**

## 1. Introduction

## 2. Objectives and Methods

## 3. Numerical Model

#### 3.1. Model Outline

#### 3.2. Flexural Behavior

_{y}and ε

_{y}are, respectively, the yield strength and strain of the reinforcement whilst E

_{0}is the elastic modulus and b

_{o}an adimensional parameter accounting for post-yield stiffening.

_{co}and f

_{cc}are, respectively, the strength of unconfined and confined concrete, whilst ε

_{co}and ε

_{cc}are the corresponding strains.

#### 3.3. Slipage Behavior

_{long}is the diameter of longitudinal rebar, f

_{y}is the yield strength of longitudinal rebar, u is the average tension on the surface between the longitudinal reinforcement and the concrete that can be calculated as $0.5\sqrt{{f}_{c}^{\prime}}$ where f’

_{c}is the concrete compressive strength and EI

_{eff}is the effective stiffness that can be evaluated by [27]:

_{g}is gross area of section, E is the Young’s module of the concrete and I is the section inertia moment (bh

^{3}/12).

#### 3.4. Shear Behavior

_{S,uncracked}can be calculated through the elasticity theory:

_{c,}and ν are respectively the shear, Young’s, and Poisson’s moduli of concrete, and A

_{v}is the shear effective area of the column.

_{S}

_{,cracked}can be calculated considering the deformation of transversal steel through the diagonal cracks. Park and Paulay [27] proposed an equivalent strut-model:

_{w}is the transversal steel reinforcement ratio, θ is the angle between the diagonal cracks and the member axis and Es is the Young’s modulus of steel.

_{c}and the transverse reinforcement V

_{s}:

_{c}is the compressive strength of concrete, A

_{g}is gross area of section, P is the axial load, a is the shear span (distance between the maximum moment section to point of inflection), d is effective depth of the section, A

_{w}is the transversal reinforcement area, f

_{y}is the yield strength of the transversal reinforcement. The factor k is the parameter which considers the variation of shear strength with the increase of displacement ductility and is defined to be equal to 1.0 for displacement ductility less than 2, to be equal to 0.7 for displacement ductility μ

_{Δ}exceeding 6, and to vary linearly for intermediate displacement ductility, as shown in Figure 6.

_{c}, transversal reinforcement V

_{s}and axial load V

_{P}:

_{tot}is the total longitudinal reinforcement ratio, h is the depth of the section, and c is the neutral axis deepth.

_{y}and θ

_{y}are respectively the yielding displacement and yielding rotation.

_{l}, and the γ factor is function of the ductility curvature μ

_{χ}, as shown in Figure 7, Figure 8 and Figure 9.

_{s}/L is the drift ratio at shear failure, Δ

_{s}is the displacement where the shear degradation begin, L is the height of the column, ρ

_{w}is the transverse reinforcement ratio and ν is the nominal shear stress (V

_{max}/A

_{g}).

#### 3.5. Interaction Model

_{n}of the pier is lower than its flexural strength V

_{f}, the pier will have a response controlled by the shear behavior as shown schematically in Figure 10b. Before reaching the shear capacity V

_{n}, shear response and flexural response will develop simultaneously in accordance with the solid line in the figure and once the shear demand reaches the shear strength, V

_{n}, a shear failure occurs and the shear response enters into descending range where significant deterioration behavior occurs, conditioning the overall response, that cannot develop forces higher than the shear strength.

_{n}of the column is higher than flexural strength V

_{f}, the pier will have a response controlled by the flexural behavior as shown schematically in Figure 10c. Before reaching the flexural capacity V

_{f}, shear response and flexural response will develop simultaneously in accordance with the solid line in the figure and once the shear demand reaches the flexural strength V

_{f}, a flexural failure occurs and the flexural response enters into descending range, conditioning the overall response, that cannot develop forces higher than the shear strength.

_{deg}, and the residual strength F

_{res}both indicated in Figure 11.

## 4. Numerical Validation

_{w}= 0.0007) for both specimens. The axial force was equal to P = 503 kN (ν = P/(A

_{g}·f’

_{c}) = 0.09). The concrete compressive strength was equal to f’

_{c}= 26 MPa, whilst the longitudinal steel yielding strength was equal to f

_{yl}= 335 MPa and the transversal steel yielding strength was equal to f

_{yw}= 400 MPa.

_{S,cracked}has been set equal to the one of the uncracked one, K

_{S,uncracked}, following. Therefore, the black dash line of Figure 4, as given by Equation (3).

_{exp}= 0.01 and displacement of about Δ

_{s,expt}= 30.4 mm (square mark in Figure 13a).

_{calc}= 0.024 and Δ

_{s,calc}= 71.4 mm, respectively (circle mark in Figure 13a).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Nuti, C.; Rasulo, A.; Vanzi, I. Seismic assessment of utility systems: Application to water, electric power and transportation networks. In Proceedings of the Safety, Reliability and Risk Analysis: Theory, Methods and Applications Proceedings of the Joint ESREL and SRA-Europe Conference, Valencia, Spain, 22–25 September 2008; Volume 3, pp. 2519–2529. [Google Scholar]
- Nuti, C.; Rasulo, A.; Vanzi, I. Seismic safety of network structures and infrastructures. Struct. Infrastruct. Eng.
**2010**, 6, 95–110. [Google Scholar] [CrossRef] - Rasulo, A.; Testa, C.; Borzi, B. Seismic risk analysis at urban scale in Italy. Lect. Notes Comput. Sci.
**2015**, 9157, 403–414. [Google Scholar] [CrossRef] - Rasulo, A.; Fortuna, M.A.; Borzi, B. Seismic risk analysis at urban scale in Italy. Lect. Notes Comput. Sci.
**2016**, 9788, 198–213. [Google Scholar] [CrossRef] - Lavorato, D.; Pelle, A.; Fiorentino, G.; Nuti, C.; Rasulo, A. A nonlinear material model of corroded rebars for seismic response of bridges, COMPDYN 2019. In Proceedings of the 7th ECOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Crete Island, Grecee, 24–26 June 2019. [Google Scholar]
- Lavorato, D.; Fiorentino, G.; Pelle, A.; Rasulo, A.; Bergami, A.V.; Briseghella, B.; Nuti, C. A corrosion model for the interpretation of cyclic behavior of reinforced concrete sections. Struct. Concr.
**2019**. [Google Scholar] [CrossRef] - Lynn, A.C.; Moehle, J.P.; Mahin, S.A.; Holmes, W.T. Seismic evaluation of existing reinforced concrete building columns. Earthq. Spectra
**1996**, 12, 715–739. [Google Scholar] [CrossRef] - Priestley, M.J.N.; Seible, F.; Verma, R.; Xiao, Y. Seismic shear strength of reinforced concrete columns. In Structural Systems Research Project Report No. SSRP 93/06; University of California: San Diego, CA, USA, 1993. [Google Scholar]
- Sezen, H.; Moehle, J.P. Shear strength model for lightly reinforced concrete columns. J. Struct. Eng.
**2004**, 130, 1692–1703. [Google Scholar] [CrossRef] - Filippou, F.C.; D’Ambrisi, A.; Issa, A. Nonlinear static and dynamic analysis of reinforced concrete subassemblages. In Earthquake Engineering Research Center Report No. UCB/EERC-92/08; University of California: Berkeley, CA, USA, 1992. [Google Scholar]
- Cassese, P.; De Risi, M.T.; Verderame, G.M. A degrading shear strength model for RC columns with hol-low circular cross-section. Int. J. Civ. Eng.
**2019**, 17, 1241–1259. [Google Scholar] [CrossRef] - Cassese, P.; De Risi, M.T.; Verderame, G.M. A modelling approach for existing shear-critical RC bridge piers with hollow rectangular cross section under lateral loads. Bull. Earthq. Eng.
**2019**, 17, 237–270. [Google Scholar] [CrossRef] - Guedes, J.; Pinto, A.V. A numerical model for shear dominated bridge piers. In Proceedings of the Second Italy–Japan Workshop on Seismic Design and Retrofit of Bridges, Rome, Italy, 27–28 February 1997. [Google Scholar]
- Ranzo, G.; Petrangeli, M. A fibre finite beam element with section shear modelling for seismic analysis of RC structures. J. Earthq. Eng.
**1998**, 2, 443–473. [Google Scholar] [CrossRef] - Petrangeli, M. Fiber element for cyclic bending and shear of RC structures. II: Verification. J. Eng. Mech.
**1999**, 125, 1002–1009. [Google Scholar] [CrossRef] - Marini, A.; Spacone, E. Analysis of reinforced concrete elements including shear effects. ACI Struct. J.
**2006**, 103, 645–655. [Google Scholar] - Ceresa, P.; Petrini, L.; Pinho, R.; Sousa, R. A fibre flexure-shear model for seismic analysis of RC-framed structures. Earthq. Eng. Struct. Dyn.
**2009**, 38, 565–586. [Google Scholar] [CrossRef] - McKenna, F.; Fenves, G.L.; Scott, M.H.; Jeremic, B. Open System for Earthquake Engineering Simulation (OpenSEES); University of California: Berkeley, CA, USA, 2000. [Google Scholar]
- Menegotto, M.; Pinto, P.E. Method of Analysis of Cyclically Loaded RC Plane Frames including Changes in Geometry and Non-Elastic Behavior of Elements under Normal Force and Bending; International Association for Bridge and Structural Engineering: Zurich, Switzerland, 1973; pp. 15–22. [Google Scholar]
- Popovics, S. A numerical approach to the complete stress-strain curve of concrete. Cem. Concr. Res.
**1973**, 3, 583–599. [Google Scholar] [CrossRef] - Mander, J.B.; Priestley, M.J.; Park, R. Theoretical stress-strain model for confined concrete. J. Struct. Eng.
**1988**, 114, 1804–1826. [Google Scholar] [CrossRef] [Green Version] - Eligehausen, R.; Bertero, V.V.; Popov, E.P. Local bond stress-slip relationships of deformed bars under generalized excitations: Tests and analytical model. In Report No EERC; Earthquake Engineering Research Center, University of California: Berkeley, CA, USA, 1983; pp. 1–169. [Google Scholar]
- Sezen, H.; Setzler, E.J. Reinforcement slip in reinforced concrete columns. ACI Struct. J.
**2008**, 105, 280. [Google Scholar] - CEB-FIB. Model Code 2010-Final Draft; The International Federation for Structural Concrete (FIB CEB-FIP): Lausanne, Switzerland, 2010. [Google Scholar]
- Elwood, K.J. Shake Table Tests and Analytical Studies on the Gravity Load Collapse of Reinforced Concrete Frames. Level of. Ph.D. Thesis, University of California, Berkeley, CA, USA, January 2002. [Google Scholar]
- Elwood, K.J.; Eberhard, M.O. Effective Stiffness of Reinforced Concrete Columns. In Research Digest No 2006-1. ACI Struct. J.
**2009**, 106, 476–484. [Google Scholar] - Park, R.; Paulay, T. Reinforced Concrete Structures; John Wiley & Sons: Hoboken, NJ, USA, 1975. [Google Scholar]
- Biskinis, D.E.; Roupakias, G.K.; Fardis, M.N. Degradation of Shear Strength of Reinforced Concrete Members with Inelastic Cyclic Displacements. ACI Struct. J.
**2004**, 101, 773–783. [Google Scholar] - Elwood, K.J.; Moehle, J.P. Drift Capacity of Reinforced Concrete Columns with Light Transverse Reinforcement. Earthq. Spectra
**2005**, 21, 71–89. [Google Scholar] [CrossRef] - Kowalsky, M.J.; Priestley, M.J.N. Improved Analytical Model for Shear Strength of Circular Reinforced Concrete Columns in Seismic Regions. ACI Struct. J.
**2000**, 97, 388–396. [Google Scholar] - Lynn, A. Seismic Evaluation of Existing Reinforced Concrete Building Column. Ph.D. Thesis, University of California at Berkeley, Berkeley, CA, USA, 2001. [Google Scholar]
- Saatcioglu, M.; Ozcebe, G. Response of reinforced concrete columns to simulated seismic loading. Struct. J.
**1989**, 86, 3–12. [Google Scholar] - Calvi, G.M.; Pavese, A.; Rasulo, A.; Bolognini, D. Experimental and numerical studies on the seismic response of R.C. hollow bridge piers. Bull. Earthq. Eng.
**2005**, 3, 267–297. [Google Scholar] [CrossRef]

**Figure 1.**Idealized components of horizontal displacement: (

**a**) Original undeformed configuration as a cantilever column; (

**b**) bending deformation; (

**c**) shear deformation and (

**d**) bonding deformation.

**Figure 3.**Properties of a RC section: (

**a**) Fiber discretization of the section (y and z are the element local axis); (

**b**) steel reinforcement and (

**c**) cover (unconfined) and core (confined) concrete constitutive law (the meaning of the symbols is given in the text).

**Figure 4.**ATC model for shear-strength degradation (d

_{y}: yielding displacement). Three cases are considered: (

**A**) shear failure before flexural yielding (pure shear failure); (

**B**) shear failure after flexural yielding (shear-flexural failure); (

**C**) flexural failure.

**Figure 7.**The factor α after Kowalsky and Priestley [30] (in Equation (14)).

**Figure 8.**The factor β after Kowalsky and Priestley [30] (in Equation (14)).

**Figure 9.**The factor γ after Kowalsky and Priestley [30] (in Equation (14)).

**Figure 10.**Combination of the single response components in shear-flexure interaction model: (

**a**) Two-component model; (

**b**) Response controlled by shear failure and (

**c**) Response controlled by flexural failure.

**Figure 12.**Specimen 2CLH18: (

**a**) Experimental behaviour. (

**b**) Comparison of numerical cyclic response with experimental results.

**Figure 13.**Specimen 3CLH18: (

**a**) Experimental behaviour. The square and circle marks represent respectively the experimental (Table 2) and calculated (Equation (17)) initiation of shear degradation. (

**b**) Comparison of numerical cyclic response with experimental results.

Rectangular Section | ||||||||||||

Reference | Specimen | b (mm) | h (mm) | s (mm) | a/d (---) | ρ_{l}(---) | ρ_{w}(---) | f’_{c} (MPa) | f_{yl} (MPa) | f_{yw} (MPa) | ν(---) | Test Type |

Lynn A.C. [31] | 2CLH18 | 457.2 | 457.2 | 457.2 | 3.74 | 0.02 | 0.0007 | 26.8 | 330.7 | 399.6 | 0.09 | DC |

3CLH18 | 457.2 | 457.2 | 457.2 | 3.74 | 0.03 | 0.0007 | 26.8 | 330.7 | 399.6 | 0.09 | DC | |

3SLH18 | 457.2 | 457.2 | 457.2 | 3.74 | 0.03 | 0.0007 | 26.8 | 330.7 | 399.6 | 0.09 | DC | |

2SLH18 | 457.2 | 457.2 | 457.2 | 3.71 | 0.02 | 0.0007 | 27.5 | 330.7 | 399.6 | 0.07 | DC | |

3CMH18 | 457.2 | 457.2 | 457.2 | 3.74 | 0.03 | 0.0007 | 25.4 | 330.7 | 399.6 | 0.28 | DC | |

3CMD12 | 457.2 | 457.2 | 457.2 | 3.74 | 0.03 | 0.0017 | 25.4 | 330.7 | 399.6 | 0.28 | DC | |

Saatcioglu Ozcebe [32] | U1 | 350.0 | 350.0 | 150.0 | 3.28 | 0.03 | 0.0085 | 43.6 | 430.0 | 470.0 | 0.00 | SC |

U2 | 350.0 | 350.0 | 150.0 | 3.28 | 0.03 | 0.0085 | 30.2 | 453.0 | 470.0 | 0.16 | SC | |

U3 | 350.0 | 350.0 | 75.0 | 3.28 | 0.03 | 0.0169 | 34.8 | 430.0 | 470.0 | 0.16 | SC | |

Hollow Rectangular Section | ||||||||||||

Reference | Specimen | b (mm) | h (mm) | s (mm) | a/d (---) | ρ_{l}(---) | ρ_{w}(---) | f’_{c} (MPa) | f_{yl} (MPa) | f_{yw} (MPa) | ν(---) | Test Type |

Calvi et al. [33] | T250 | 450.0 | 450.0 | 75.0 | 3.00 | 0.0177 | 0.0025 | 30.3 | 550.0 | 550.0 | 0.07 | SC |

T500A | 450.0 | 450.0 | 75.0 | 3.00 | 0.0177 | 0.0025 | 29.7 | 550.0 | 550.0 | 0.15 | SC | |

T500B | 450.0 | 450.0 | 75.0 | 3.00 | 0.0177 | 0.0025 | 32.7 | 550.0 | 550.0 | 0.15 | SC | |

T750 | 450.0 | 450.0 | 75.0 | 3.00 | 0.0177 | 0.0025 | 30.8 | 550.0 | 550.0 | 0.21 | SC |

_{l}: longitudinal reinforcement ratio; ρ

_{w}: transverse reinforcement ratio; f’

_{c}: compressive strength of concrete; f

_{yl}: yield strength of longitudinal steel; f

_{yw}: yield strength of transverse steel; ν: axial load ratio; Test type: SC (single cantilever), DC (double cantilever).

**Table 2.**Experimental values needed for the calibration of the shear spring (Figure 4).

Rectangular Section | |||||

Reference | Specimen | Δ_{s}/L(---) | K_{s,deg} (kN/mm) | V_{s}(kN) | V_{res}(kN) |

Lynn A.C. (2001) [31] | 3CLH18 | 0.0100 | −7.5 | 277.0 | 50.0 |

3SLH18 | 0.0077 | −4.4 | 270.0 | 48.7 | |

2SLH18 | 0.0088 | −6.2 | 231.6 | − | |

3CMH18 | 0.0100 | −12.4 | 324.4 | 94.3 | |

3CMD12 | 0.0085 | −4.1 | 355.8 | − | |

Saatcioglu, Ozcebe [32] | U1 | 0.044 | −2.8 | 273.0 | 70.0 |

U2 | 0.021 | −6.0 | 280.0 | 90.0 | |

U3 | 0.044 | −3.7 | 255.0 | − | |

Hollow Rectangular Section | |||||

Reference | Specimen | Δ_{s}/L(---) | K^{t}_{deg} (kN/mm) | V_{s}(kN) | V_{res}(kN) |

Calvi et al. [33] | T250 | 0.022 | −8.9 | 217.2 | 47.9 |

T500A | 0.010 | −6.3 | 209.0 | − | |

T500B | 0.022 | −7.8 | 228.0 | − | |

T750 | 0.020 | − | 257.9 | − |

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**MDPI and ACS Style**

Rasulo, A.; Pelle, A.; Lavorato, D.; Fiorentino, G.; Nuti, C.; Briseghella, B.
Finite Element Analysis of Reinforced Concrete Bridge Piers Including a Flexure-Shear Interaction Model. *Appl. Sci.* **2020**, *10*, 2209.
https://doi.org/10.3390/app10072209

**AMA Style**

Rasulo A, Pelle A, Lavorato D, Fiorentino G, Nuti C, Briseghella B.
Finite Element Analysis of Reinforced Concrete Bridge Piers Including a Flexure-Shear Interaction Model. *Applied Sciences*. 2020; 10(7):2209.
https://doi.org/10.3390/app10072209

**Chicago/Turabian Style**

Rasulo, Alessandro, Angelo Pelle, Davide Lavorato, Gabriele Fiorentino, Camillo Nuti, and Bruno Briseghella.
2020. "Finite Element Analysis of Reinforced Concrete Bridge Piers Including a Flexure-Shear Interaction Model" *Applied Sciences* 10, no. 7: 2209.
https://doi.org/10.3390/app10072209