Materials and Structures

, Volume 43, Issue 9, pp 1245–1256 | Cite as

Optimal reinforcement of RC columns for biaxial bending

  • Luisa María Gil-Martín
  • Enrique Hernández-Montes
  • Mark Aschheim
Original Article


The Reinforcement Sizing Diagram (RSD) approach to determining optimal reinforcement for reinforced concrete beam and column sections subjected to uniaxial bending is extended to the case of biaxial bending. Conventional constraints on the distribution of longitudinal reinforcement are relaxed, leading to an infinite number of reinforcement solutions, from which the optimal solution and a corresponding quasi-optimal pragmatic is determined. First, all possibilities of reinforcement arrangements are considered for a biaxial loading, including symmetric and non-symmetric configurations, subject to the constraint that the reinforcement is located in a single layer near the circumference of the section. This theoretical approach establishes the context for obtaining pragmatic distributions of reinforcement that are more suitable for construction, in which distributions having double symmetry are considered. This contrasts with conventional approaches for the design of column reinforcement, in which a predetermined distribution of longitudinal reinforcement is assumed, even though such a distribution generally is non-optimal in any given design. Column and wall sections that are subjected to uniaxial or biaxial loading may be designed using this method. The solutions are displayed using a biaxial RSD and can be obtained with relatively simple algorithms implemented in widely accessible software programs such as Mathematica® and Excel®. Several examples illustrate the method and the savings in reinforcement that can be obtained relative to conventional solutions.


Ultimate strength design Optimal reinforcement Biaxial bending 

List of symbols


Cross sectional area of concrete section


Area of bottom reinforcement

\( A^{\prime}_s\)

Area of top reinforcement


Area of prestressing tendon


Nominal axial strength


Design value of the applied axial force


Bending moment applied at the center of gravity of the gross section


Design value of the applied bending moment


Flexural moment strength about x-axis


Design value of the bending moment applied about the x-axis


Flexural moment strength about y-axis


Design value of the bending moment applied about the x-axis


Characteristic compressive strength of concrete


Characteristic yield strength of reinforcement


Distance between centroids of consecutive bars of the top and bottom reinforcement


Distance between centroids of consecutive bars of side reinforcement


Depth to neutral axis from top fiber of cross section


Vertical coordinate measures from the center of gravity of the gross section


Stress in concrete


Stress in prestressing tendon


Stress in bottom reinforcement

\( \sigma^{\prime}_s\)

Stress in top reinforcement


Intersection of the neutral axis with the y-axis


Angle of the neutral fiber


Bar diameter


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Copyright information

© RILEM 2009

Authors and Affiliations

  • Luisa María Gil-Martín
    • 1
  • Enrique Hernández-Montes
    • 1
  • Mark Aschheim
    • 2
  1. 1.Department of Structural MechanicsUniversity of GranadaGranadaSpain
  2. 2.Department of Civil EngineeringSanta Clara UniversitySanta ClaraUSA

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