Flexural strength calculations for composite beams
The design of composite beams for bending is based on a comparison of ultimate bending strength and the applied bending moment under factored loads.
In common with the design of many structural elements, the design rules for composite beams are specified in terms of limit state principles. The design bending moment is therefore calculated on the basis of factored loads. This is then compared with the ultimate moment of resistance of the composite section based on the design strength of both the steel and concrete. In the following sections a procedure for determining the ultimate moment of resistance is described.
The contribution of the slab to composite beam action is assumed to be limited to an "effective width", related to the span and spacing of beams.
The composite beam is similar to a reinforced concrete T- beam, with the steel and slab representing the stalk and flange respectively of a T- section. The flange thickness is determined by the details of the slab, but its width must be quantified. This is referred to as the "effective width", Be, and is taken as the lesser of:
one fifth of the span of the beam, and
the distance between centres of beams.
The stress distribution in the beam cross-section at collapse is idealised as a uniform tensile stress in the steel beam, and uniform compressive stress in the top of the concrete.
When the beam is loaded up to its collapse condition the stress distribution is rather complex, but to a reasonable degree of accuracy can be represented as a uniform stress level of tension in the steel and compression in the concrete, with tensile stresses in the concrete ignored.
The tensile stress in the steel at this limit state is taken as its design strength, py. This is dependent solely on the steel grade and appropriate values are given in BS 5950.
The compressive stress in the concrete is assumed to be 0.45fcu where fcu is the characteristic strength of the concrete.
The depth of concrete in compression can be found by equating the compression in the concrete with the tension in the steel cross-section.
From considerations of equilibrium the total forces in compression and tension must balance. The tensile force in the steel section, Fs, is given by
Fs = py . As
where As is the cross-sectional area of the section.
The compressive force in the concrete, Fc, is given by
Fc = 0.45 fcu . Be . x
where x is the depth of concrete in compression.
From equilibrium, the compressive and tensile forces must be equal. Thus
py. As = 0.4 fcu . Be . x
and hence x = (py . As) / (0.4 fcu . Be)
The bending strength corresponds to the couple produced by this system of forces.
The ultimate moment of resistance, Mu, is equal to the couple produced by this force system, given by
Mu = Fs . (D/2 + hc - x/2)
The composite beam section is satisfactory if the ultimate bending strength is greater than the applied bending moment under factored loads.
By comparing this with the design bending moment, the suitability of the composite beam can be determined. If Mu is greater than the design bending moment the section is adequate, but if this is not so a larger beam section must be tried.