Abstract
A novel method for obtaining complete interaction diagrams of arbitrarily shaped reinforced concrete and composite sections is presented. A mathematical optimization is used to apply the force criterion that maximizes the biaxial bending capacity of a section. A strain criterion for limiting the ultimate strain can be selectively applied through the bound constraint of the optimization problem. Three of the strain parameters for defining the deformation of the section are not fixed, to determine the true ultimate strength at the specified locations in the interaction diagram. A sequential quadratic programming algorithm is used to find the solution. A rotated atan2 function is introduced to overcome the discontinuity in the objective and constraint functions. A tilted axis is used for unsymmetrical sections. Interaction diagrams are obtained for various examples and demonstrate the robustness and versatility of the proposed method. The force criterion provides accurate interaction diagrams for the section, for both normal and high-strength materials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ACI (2014) Building code requirements for structural concrete. ACI 318-14, American Concrete Institute, Farmington Hills, MI, USA
Bonet JL, Barros MHFM, Romero ML (2006) Comparative study of analytical and numerical algorithms for designing reinforced concrete sections under biaxial bending. Computers & Structures 84:2184–2193, DOI: https://doi.org/10.1016/j.compstruc.2006.08.065
Caldas RB, Sousa JBM, Fakury RH (2010) Interaction diagrams for reinforced concrete sections subjected to fire. Engineering Structures 32:2832–2838, DOI: https://doi.org/10.1016/j.engstruct.2010.05.002
Carreira DJ, Chu KH (1985) Stress-strain relationship for plain concrete in compression. ACI Journal 82(6):797–804
CEN (2004) Design of concrete structures — Part 1–1: General rules and rules for buildings. Eurocode 2, European Committee for Standardization, Brussels, Belgium
Charalampakis AE, Koumousis VK (2008) Ultimate strength analysis of composite sections under biaxial bending and axial load. Advances in Engineering Software 39:923–936, DOI: https://doi.org/10.1016/j.advengsoft.2008.01.007
Chen WF, Shoraka MT (1972) Tangent stiffness method for biaxial bending of reinforced concrete columns. Report No 389.1, Fritz Engineering Laboratory, Lehigh University, Bethlehem, PA, USA
Chen SF, Teng JG, Chan SL (2001) Design of biaxially loaded short composite columns of arbitrary section. Journal of Structural Engineering 127(6):678–685, DOI: https://doi.org/10.1061/(ASCE)0733-9445(2001)127:6(678)
Chiorean CG (2010) Computerised interaction diagrams and moment capacity contours for composite steel-concrete cross sections. Engineering Structures 32:3734–3757, DOI: https://doi.org/10.1016/j.engstruct.2010.08.019
Fafitis A (2001) Interaction surfaces of reinforced-concrete sections in biaxial bending. Journal of Structural Engineering 127(7):840–846, DOI: https://doi.org/10.1061/(ASCE)0733-9445(2001)127:7(840)
Furlong RW, Hsu CTT, Mirza SA (2004) Analysis and design of concrete columns for biaxial bending-overview. ACI Structural Journal 101(3):413–423, DOI: https://doi.org/10.14359/13101
Hong HP (2000) Short reinforced concrete column capacity under biaxial and axial load. Canadian Journal of Civil Engineering 27(6): 1173–1182, DOI: https://doi.org/10.1139/l00-054
Jones E, Oliphant T, Peterson P (2001) SciPy: Open source scientific tools for Python. Retrieved August 15, 2020, http://www.scipy.org/
Kim CS, Park HG, Chung KS, Choi IR (2012) Eccentric axial load testing for concrete-encased steel columns using 800 MPa steel and 100 MPa concrete. Journal of Structural Engineering 138(8):1019–1031, DOI: https://doi.org/10.1061/(ASCE)ST.1943-541X.0000879
Kraft DA (1988) A software package for sequential quadratic programming. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center, Köln, Germany
Lai B, Liew JYR, Hoang AL, Xiong M (2019) A unified approach to evaluate axial force-moment interaction curves of concrete encased steel composite columns. Engineering Structures 201:109841, DOI: https://doi.org/10.1016/j.engstruct.2019.109841
Law A, Gillie M (2010) Interaction diagrams for ambient and heated concrete section. Engineering Structures 32:1641–1649, DOI: https://doi.org/10.1016/j.engstruct.2010.02.012
Legeron F, Paultre P (2003) Uniaxial confinement model for normaland high-strength concrete columns. Journal of Structural Engineering 129(2):241–252, DOI: https://doi.org/10.1061/(ASCE)0733-9445(2003)129:2(241)
Liu SW, Liu YP, Chan SL (2012) Advanced analysis of hybrid steel and concrete frames. Part 1: Cross-section analysis technique and second-order analysis. Journal of Constructional Steel Research 70:326–336, DOI: https://doi.org/10.1016/j.jcsr.2011.09.003
Meda A, Gambarova PG, Bonomi M (2002) High-performance concrete in fire-exposed reinforced concrete sections. ACI Structural Journal 99(3):277–287, DOI: https://doi.org/10.14359/11911
Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edition. Springer, New York, NY, USA
Pallares L, Miguel PF, Fernandez-Prada MA (2009) A numerical method to design reinforced concrete sections subjected to axial forces and biaxial bending on ultimate strain limits. Engineering Structures 31:3065–3071, DOI: https://doi.org/10.1016/j.engstruct.2009.08.006
Papanikolaou VK (2012) Analysis of arbitrary composite sections in biaxial bending and axial load. Computers & Structures 98–99:33–54, DOI: https://doi.org/10.1016/j.compstruc.2012.02.004
Parkinson AR, Balling RJ, Hedengren JD (2013) Optimization methods for engineering design. Brigham Young University, Provo, UT, USA
Rodrigues RV (2015) A new technique for ultimate limit state design of arbitrary shape RC sections under biaxial bending. Engineering Structures 104:1–17, DOI: https://doi.org/10.1016/j.engstruct.2015.09.016
Rodriguez JA, Ochoa JDA (1999) Biaxial interaction diagrams for short RC columns of any cross section. Journal of Structural Engineering 125(6):672–683, DOI: https://doi.org/10.1061/(ASCE)0733-9445(1999)125:6(672)
Rosati L, Marmo F, Serpieri R (2008) Enhanced solution strategies for the ultimate strength analysis of composite steel-concrete sections subject to axial force and biaxial bending. Computer Methods in Applied Mechanics and Engineering 197:1033–1055, DOI: https://doi.org/10.1016/j.cma.2007.10.001
Rotter JM (1985) Rapid exact inelastic biaxial bending analysis. Journal of Structural Engineering 111(12):2659–2674, DOI: https://doi.org/10.1061/(ASCE)0733-9445(1985)111:12(2659)
Sfakianakis MG (2002) Biaxial bending with axial force of reinforced, composite and repaired concrete sections of arbitrary shape by fiber model and computer graphics. Advances in Engineering Software 33:227–242, DOI: https://doi.org/10.1016/S0965-9978(02)00002-9
Stefan R Sura J, Prochazka J, Kohoutkova A, Wald F (2019) Numerical investigation of slender reinforced concrete and steel-concrete composite columns at normal and high temperatures using sectional analysis and moment-curvature approach. Engineering Structures 190:285–305, DOI: https://doi.org/10.1016/j.engstruct.2019.03.071
Yau CY, Chan SL, So AKW (1993) Biaxial bending design of arbitrarily shaped reinforced concrete column. ACI Structural Journal 90(3): 269–278, DOI: https://doi.org/10.14359/4235
Acknowledgments
This study was supported by Konkuk University in 2018 (2018-A019-0157).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, HS. Interaction Diagram of Arbitrarily Shaped Concrete Sections Determined by Constrained Nonlinear Optimization. KSCE J Civ Eng 25, 3823–3834 (2021). https://doi.org/10.1007/s12205-021-2008-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-021-2008-3