Finite Element Mesh Size Effect on Deformation Predictions of Reinforced Concrete Bridge Girder


  • Viktor Gribniak Dept of Bridges and Spec Structures, Vilnius Gediminas Technical University, Saulėtekio al. 11, 10223 Vilnius, Lithuania
  • Gintaris Kaklauskas Dept of Bridges and Spec Structures, Vilnius Gediminas Technical University, Saulėtekio al. 11, 10223 Vilnius, Lithuania
  • Siim Idnurm Dept of Bridge Constructions, Tallinn University of Technology, Ehitajate 5, Tallinn, 19086 Estonia
  • Darius Bačinskas Dept of Bridges and Spec Structures, Vilnius Gediminas Technical University, Saulėtekio al. 11, 10223 Vilnius, Lithuania



reinforced concrete (RC), finite element size, deformations, tension-stiffening, crack band model


Present research was dedicated to investigation of finite element size effect on deformation predictions of reinforced concrete bending members. Experimental beams have been modelled by commercial finite element software ATENA, using two main approaches for simulating tension-stiffening: stress-crack width (fracture mechanics approach) and average stress-average strain relationships. The latter approach uses the ultimate strain adjusted according to the finite element size. It was shown that the modelled post-cracking behaviour of the beams is dependent on the finite element mesh size. To reduce this effect, a simple formula has been proposed for adjusting the length of the descending branch of the constitutive relationship. Post-cracking behaviour of a reinforced concrete bridge girder has been investigated assuming different finite element mesh sizes. The analysis has shown that the proposed technique allows reducing the dependence of calculation results on the finite element size.


Barenblatt, G. I. 1962. Mathematical Theory of equilibrium Cracks in Brittle Fracture, Advances in Applied Mechanics 7: 55–129. doi:10.1016/S0065-2156(08)70121-2

Barros, M.; Martins, R. A. F.; Ferreira, C. C. 2001. Tension Stiffening Model with Increasing Damage for Reinforced Concrete, Engineering Computations 18(5–6): 759–785. doi:10.1108/02644400110393616

Bažant, Z. P.; Oh, B. H. 1983. Crack Band Theory for Fracture of Concrete, Materials and Structures 16(3): 155–177.

Bažant, Z. P.; Planas, J. 1998. Fracture and Size Effect in Concrete and Other Quasibrittle Structures. Boca Raton: CRC Press. 640 p.

Bischoff, P. H. 2001. Effects of Shrinkage on Tension Stiffening and Cracking in Reinforced Concrete, Canadian Journal of Civil Engineering 28(3): 363–374. doi:10.1139/cjce-28-3-363

Broberg, K. B. 1999. Cracks and Fracture. San Diego: Academic Press. 752 p.

Cervenka, V. 1995. Mesh Sensitivity Effects in Smeared Finite Element analysis of concrete fracture, in Proc of the 2nd International Conference Fracture Mechanics of Concrete Structures. Freiburg: AEDIFICATIO Publishers, 1387–1396.

Cervenka, V.; Cervenka, J.; Pukl, R. 2002. ATENA – A tool for engineering Analysis of Fracture in Concrete, Sādhanā 27(4): 485–492. doi:10.1007/BF02706996

Cervenka, J.; Chandra Kishen, J. M.; Saouma, A. E. 1998. Mixed Mode Fracture of Cementitious Biomaterial Interfaces. Part II: Numerical Simulation, Engineering Fracture Mechanics 60(1): 95–107. doi:10.1016/S0013-7944(97)00094-5

Cervenka, V.; Jendele, L.; Cervenka, J. 2003. ATENA Program Documentation. Theory. Prague: Cervenka Consulting. 129 p.

Cervenka, V.; Pukl, R. 1994. SBETA Analysis of Size Effect in Concrete Structures, in Size Effect in Concrete Structure. London: E&FN Spon, 323–333.

Dugdale, D. S. 1960. Yielding of Steel Sheets Containing Slits, Journal of the Mechanics and Physics of Solids 8(2): 100–104. doi:10.1016/0022-5096(60)90013-2

Durbin, J. 1961. Some Methods of Constructing Exact Tests, Biometrika 48(1–2): 41–65. doi:10.1093/biomet/48.1-2.41

Ebead, U. A.; Marzouk, H. 2005. Tension Stiffening Model for FRP-Strengthened RC Concrete Two-Way Slabs, Material and Structures 38(2): 193–200. doi:10.1007/BF02479344

Elices, M.; Guinea, G. V.; Gómez, J.; Planas, J. 2002. The Cohesive Zone Model: Advantages, Limitations and Challenges, Engineering Fracture Mechanics 69(2): 137–163. doi:10.1016/S0013-7944(01)00083-2

Gribniak, V. 2009. Shrinkage Influence on Tension-Stiffening of Concrete Structures. PhD thesis, Vilnius Gediminas Technical University, Vilnius, Lithuania. 146 p.

Gribniak, V.; Bačinskas, D.; Kaklauskas, G. 2006. Numerical Simulation Strategy of Bearing Reinforced Concrete Tunnel Members in Fire, The Baltic Journal of Road and Bridge Engineering 1(1): 5–9.

Gribniak, V.; Kaklauskas, G.; Bačinskas, D. 2008. Shrinkage in Reinforced Concrete Structures: a Computational Aspect, Journal of Civil Engineering and Management 14(1): 49–60. doi:10.3846/1392-3730.2008.14.49-60

Gribniak, V.; Kaklauskas, G.; Sokolov, A.; Logunov, A. 2007. Finite Element Size Effect on Post-Cracking Behaviour of Reinforced Concrete Members, in Proc of the 9th International Conference Modern Building Materials, Structures and Techniques. Vilnius: Technika, 2: 563–570.

Hillerborg, A.; Modéer, M.; Petersson, P.-E. 1976. Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements, Cement and Concrete Research 6(6): 773–782. doi:10.1016/0008-8846(76)90007-7

Kaklauskas, G. 2001. Integral Flexural Constitutive Model for Deformational Analysis of Concrete Structures. Vilnius: Technika. 140 p.

Kaklauskas, G. 2004. Flexural Layered Deformational Model of Reinforced Concrete Members, Magazine of Concrete Research 56(10): 575–584.

Kaklauskas, G.; Bačinskas, D.; Gribniak, V.; Geda, E. 2007. Mechanical Simulation of Reinforced Concrete Slabs Subjected to Fire, Technological and Economic Development of Economy 13(4): 295–302.

Kaklauskas, G.; Girdžius, R.; Bačinskas, D.; Sokolov, A. 2008. Numerical Deformation Analysis of Bridge Concrete Girders, The Baltic Journal of Road and Bridge Engineering 3(2): 51–56. doi:10.3846/1822-427X.2008.3.51-56

Kaplan, M. F. 1961. Crack Propagation and The Fracture of Concrete, ACI Journal Proceedings 58(11): 591–610.

Kesler, C. E.; Naus, D. J.; Lott, J. L. 1972. Fracture Mechanics – its Applicability to Concrete, in Proc of the International Conference on the Mechanical Behavior of Materials, Kyoto, 1971, IV: 113–124.

Lin, C. S.; Scordelis, A. C. 1975. Nonlinear Analysis of RC Shells of General Form, Journal of Structural Engineering 101(3): 523–538.

Prakhya, G. K. V.; Morley, C. T. 1990. Tension-Stiffening and Moment-Curvature Relations of Reinforced Concrete Elements, ACI Structural Journal 87(5): 597–605.

Rashid, Y. R. 1968. Ultimate Strength Analysis of Prestressed Concrete Pressure Vessels, Nuclear Engineering and Design 7(4): 334–344. doi:10.1016/0029-5493(68)90066-6

Reinhardt, H.-W. 1996. Uniaxial Tension, in Fracture Mechanics of Concrete Structures. Freiburg: Balkema, III: 1871–1881.

Rots, J. G. 1988. Computational Modeling of Concrete Structures. PhD thesis. Delft University of Technology, Delft, The Netherlands. 127 p.

Suidan, M.; Schnobrich, W. C. 1973. Finite Element Analysis of Reinforced Concrete, ASCE Journal of the Structural Division 99(10): 2109–2122.

Mang, H. A.; Jia, X. ; Hoefinger, G. 2009. Hilltop Buckling as the Alfa and Omega in Sensitivity Analysis of the Initial Postbuckling Behavior of Elastic Structures, Journal of Civil Engineering and Management 15(1): 35–46. doi:10.3846/1392-3730.2009.15.35-46

Vořechovský, M. 2007. Interplay of Size Effects in Concrete Specimens under Tension Studied via Computational Stochastic Fracture Mechanics, International Journal of Solids and Structures 44(9): 2715–2731. doi:10.1016/j.ijsolstr.2006.08.019




How to Cite

Gribniak, V., Kaklauskas, G., Idnurm, S., & Bačinskas, D. (2010). Finite Element Mesh Size Effect on Deformation Predictions of Reinforced Concrete Bridge Girder. The Baltic Journal of Road and Bridge Engineering, 5(1), 19-27.