Composite concrete beams are widely used in both building and bridge structures; typically precast pretensioned beams are combined with an in-situ reinforced concrete slab to provide structures that are both efficient in use of materials and fast and easy to construct.
If pre-stressed beams are used for the first stage, and if the section remains uncracked under all stages of loading, then analysis is straightforward since each stage may be analysed separately and linearly combined. On the other hand if the section is cracked at any stage the bending behaviour becomes non-linear, and the closed form solutions presented previously in the Beam Design Functions spreadsheet, are no longer applicable.
In this post I will describe an iterative procedure for this analysis, and in the following post I will cover a spreadsheet to carry out the analysis using a VBA User Defined Function (UDF).
The section below shows a simple reinforced concrete composite beam consisting of a first stage rectangular section of 800 deep x 300 wide, with a second stage slab of 200 deep x 1000 wide.
Analysis of the first stage is straightforward, allowing for the self weight of the beam, the weight of the in-situ top slab wet concrete, plus any permanent formwork and any other loads applied at the time of pouring the second stage concrete. This can be done with the Beam Design Functions spreadsheet (link above), or for a rectangular section the RC Design Functions spreadsheet can be used.
The graph below shows the strain distribution at each stage:
- Self-weight plus wet in-situ concrete load on Stage 1 (blue line).
- The incremental strain due to the additional Stage 2 load on the composite section (green line).
- The combined strain due to the total load (red line)
The procedure used to generate these strain diagrams was:
- Find the Neutral Axis depth and section curvature (top face strain/depth of NA) using one of the closed form solutions linked above.
- For a first estimate of the strain increment due to the Stage 2 loads on the composite section apply the incremental load to the composite section using one of the closed form solutions. Note that this will not give the correct final result unless the section is uncracked at both stages of loading.
- Adjust the Neutral Axis depth and section curvature for the Stage 2 loading so that the total stress distribution in the composite section is in equilibrium with the total applied loading.
The next post will provide an Excel UDF that will perform this analysis for a Stage 1 beam of any cross section, with or without prestress, and for any combination of applied moment and axial load.