Learn how to calculate the flexural and shear design capacities for a concrete beam, according to AS3600-2018.
Introduction
Beams are an important structural element in reinforced concrete (RC) structures. They are designed to provide resistance to external loads that cause shear forces, bending moments and, in some cases, torsion across their length.
Concrete is strong in compression but weak in tension, so reinforcement is added to take the tensile stresses induced when beams are loaded. In a sagging beam the tensile stresses are along the bottom of the beam, so theoretically, no top reinforcement is required. However most beams will still have top reinforcement for constructability reasons, so that the ligs can hang from the top bars.
The latest Australian Standard for concrete structures, AS3600-2018, has undergone significant updates and changes since the 2009 edition, mainly concerning:
- Rectangular stress-block configuration (Section 8.1.3)
- Values of the capacity reduction factor Φ, for different action effects (Table 2.2.2)
- Shear and torsion design (Section 8.2)
Generally, the limit state checks for RC beams include the flexural, shear, deflection and crack control. This design guide covers the first two checks: flexural and shear capacity.
Ultimate Flexural Capacity
Rectangular Stress Block
The flexural (also referred to as 'bending' or 'moment') capacity of a beam cross-section is determined using the Rectangular Stress Block method (Cl. 8.1.2). The stress distribution in concrete under bending is curved in reality, however it can be converted to an equivalent rectangular stress block by the use of reduction factors
By taking moments about the concrete compression force
The design flexural capacity is given by
The
The
Where, as per Cl 8.1.3:
Minimum Strength,
The ultimate flexural strength at critical sections should not be less than the minimum required strength in bending
Where:
: section modulus of theuncracked cross-section : characteristic flexural tensilestrength of the concrete : total effective prestress force allowing forall losses of prestress : eccentricity of the prestressing force
Where there is no pre-stressing,
Also according to Cl 8.1.6, for reinforced concrete sections, the above requirement is deemed to be satisfied if the total area of the provided tensile reinforcement satisfies the following:
For rectangular sections,
Ductility
To finalise your ultimate bending capacity limit state, youmust check your section is ductile. A ductile section ensures the reinforcementwill yield before the concrete crushes, which is deemed a less dangerousfailure mechanism since the failure occurs relatively slow compared to a suddenbrittle failure. As per Cl 8.1.5, a section is ductile if:
Ultimate Shear Capacity
Section 8.2 outlines methods for calculating strength of beams in shear. We will leave out torsion from this design guide, but it must be checked if the beam sees in-plane rotation.
The total nominal shear capacity of a section
The design shear capacity is given by
In this design guide, we will not consider prestress i.e.
Truss Analogy
AS3600 (similar to other international codes) use a truss analogy of shear resistance, where the diagonal concrete strut is in compression and the vertical tie (provided by shear reinforcement) is in tension.
The strut angle
- a smaller
means the strut crosses more shear ligs and therefore a smaller area of shear ligs is needed - a larger
means more of the applied shear force is taken by the vertical tie (shear ligs) and therefore a larger area of shear ligs is needed
It is common practice to ensure
Shear Reinforcement Requirement
As per Cl 8.2.1.6, shear links are required if either of the following is true:
If shear links are deemed to be required, the minimum shear cross-sectional area is calculated as per Cl. 8.2.1.7:
Where:
: centre-to-centre spacing of shear reinforcement parallel to the longitudinal axis of the member : yield strength of shear reinforcement : total effective prestress force allowing for all losses of prestress
It is useful to work in values of
Contribution to Shear Strength from Concrete,
As per Cl 8.2.4, the concrete contribution to shear strength is given by:
The factor
- No prestress and no applied axial tension
- Strength of concrete, f'c, is less than 65 MPa
- Size of aggregates, kdg, is not less than 10mm
- Yield strength of longitudinal reinforcement does not exceed 500 MPa
For most design scenarios, the above conditions will be satisfied. If not, general method must be used as per Cl. 8.2.4.2.
Effective shear width
Contribution to Shear Strength from Reinforcement,
As per Cl 8.2.5, for shear reinforcement placed at an inclination, the shear reinforcement resistance is given by:
Where:
: angle between the inclined shear reinforcement and the longitudinal tensile reinforcement : angle between the axis of the concrete compression strut and the longitudinal axis of the member (Cl 8.2.4)
The strut angle,
For shear reinforcement placed perpendicular to the longitudinal axis of the member (
Shear Strength Limited by Web Crushing,
The sum of contribution to shear strength by concrete and steel (