Shortening
The shortening of members is caused by prestressing, shrinkage, and from flexure. The shortening at the centroid of the member is calculated using
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Flexure also effects the shortening at the ends of the member. For a member cambering up, this will make the top corners move further apart and the bottom corners moving closing together. This can be computed using some simple trigonometry as follows:
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Hand Calculation
Below is a hand calculation and the corresponding Eriksson Beam file showing the procedure Eriksson Beam using to calculation deflections, cambers, and shortening. Note that in the hand calculation, it assumes that when the member cracks, a percentage of the total load is carried by the cracked section. This is similar to the procedure shown in the PCI Design Handbook and results in conservative results. Following this approach you can see the total moment being applied to the cracked section, shown below as shaded in red, is much larger than doing it on a point by point basis as shown above.
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Because of this, in the hand calculation the stages that crack result in larger deflections than Eriksson Beam.
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Problem
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Hand Calculation
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Eriksson Beam File
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Prestressed Double T - Class C
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References
ACI Committee 209, ACI 209.2R-08, Guide for Modeling and Calculating Shrinkage and Creep in Hardened Concrete, American Concrete Institute, Detroit, 2008.